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\section{Introduction} \label{submission} Few-shot learning, the problem of learning under data scarcity, is an important challenge in deep learning as large number of training instances may not be available in many real-world settings. While the recent advances in meta-learning made it possible to obtain impressive performance on few-shot learning tasks~\cite{hou2019cross, li2019learning,lifchitz2019dense}, it still remains challenging in cases where we are given very little information (e.g. one-shot learning). Some of the metric-based meta-learning approaches tackle this problem using \emph{transductive learning} or \emph{semi-supervised learning}, by leveraging the structure of the unlabeled instances at the inference time~\cite{hou2019cross,kim2019edge,l2ST,liu2018learning,ren2018meta}. Popular approach for these problem includes leveraging nearest neighbor graph for propagating labels~\cite{kim2019edge,liu2018learning,DPGN}, or using predicted soft or hard labels on unlabeled samples to update the class prototype~\cite{hou2019cross,ren2018meta}. However, all these transductive or semi-supervised inference approaches are fundamentally limited by the intrinsic \emph{unreliability} of the labels predicted on the unseen samples. In this work, we aim to tackle this problem by proposing a novel confidence-based transductive inference scheme for metric-based meta-learning models. Specifically, we first propose to meta-learn the distance metric (or metric) to assign different confidence scores to each query (or test) instance for each class, such that the updated prototypes obtained by confidence-weighted averaging of the queries improve classification of the query samples. This is done by learning a metric length-scale term for each individual instance or a pair of instances. However, the confidence prediction on the test instances for \emph{unseen} task should be inevitably unreliable, since the samples come from an unknown distribution. To account for such uncertainties of prediction on an unseen task, we further propose to generate various model and data perturbations, such as random dropping of residual blocks and random augmentations. This randomness helps the model better learn the confidence measure by considering various uncertainties for an unseen task (see Figure~\ref{fig:concept_3way}), and also allows us to take an ensemble over the confidence measures under random perturbations at test time. We refer to this transductive inference using meta-learned input-adaptive confidence under various perturbations as \emph{Meta-Confidence Transduction} (MCT). To further enhance the reliability of the output confidence, we introduce additional regularizations to enforce consistency among the transformed samples in the embedding space. Specifically, we compose episodes with differently augmented support and query set and train the model to enforce the distribution of these two sets to be close to each other. Moreover, we also enforce consistency among the dimension-wise classification of the high-dimensional embedding vectors, such that their predictions are coherent. We validate our transductive inference scheme for metric-based meta-learning models on four benchmark datasets against existing transductive approaches, which shows that the models using meta-learned confidence significantly outperform existing transductive inference methods, and obtain new state-of-the-art results. We further verify the generality of our MCT on semi-supervised learning tasks, where we assign confidence scores to unlabeled data. The results show that MCT outperforms relevant baselines by large margins, which shows the efficacy of our method. Further ablation studies show that both meta-learning of the input-adaptive distance metric and various perturbations are crucial in the success of our method in assigning correct confidence to each test sample. Our main contributions are as follows: \vspace{-0.05in} \begin{itemize} \item We propose to \textbf{meta-learn} an \textbf{input-adaptive distance metric}, which allows to output an accurate and reliable confidence for an unseen test samples that can directly improve upon the transductive inference performance. \vspace{-0.03in} \item To further enhance the reliability of the learned confidence, we introduce various types of \textbf{model} and \textbf{data perturbations} during meta-learning, such that the meta-learned confidence can better account for uncertainties at unseen tasks. \vspace{-0.03in} \item We suggest \textbf{consistency regularizations} across different perturbations and predictions for each embedding dimension, which improves the consistency of the embeddings. \item We validate our model on four benchmark datasets for few-shot classification and achieve \textbf{new state-of-the-art} results, largely outperforming all baselines. Further experimental validation of our model on semi-supervised few-shot learning also verifies its efficacy. \end{itemize} \begin{figure}[t!] \vspace{-0.1in} \centering \hfill \subfigure[ProtoNets (+4.44\%)]{\includegraphics[width=4.3cm]{pdfs/embedding_PN_444.pdf}\label{fig:fig1_PN}} \hfill \subfigure[Instance-wise metric (+8.89\%) ]{\includegraphics[width=4.3cm]{pdfs/embedding_IMS_889.pdf}\label{fig:fig1_IMS}} \hfill \subfigure[MCT (+15.56\%)]{\includegraphics[width=4.3cm]{pdfs/embedding_MCT_1556.pdf}\label{fig:fig1_MCT}} \hfill \vspace{-0.13in} \caption{ \small Transductive inference with confidence scores. We visualize t-SNE embeddings on a 3-way 1-shot task, where each color stands for different class. The numbers show the accuracy increase after transduction for this task. The transparency shows the confidence scores for \emph{red} class.} \vspace{-0.20in} \label{fig:concept_3way} \end{figure} \section{Related Work} \vspace{-0.05in} \paragraph{Distance-based meta-learning for few-shot classification} The goal of few-shot classification is to correctly classify query set examples given only a handful of support set examples. Due to its limited amount of data, each task-specific classifier should resort to the meta-knowledge accumulated from the previous tasks, which is referred to as meta-learning~\cite{thrun98}. Meta-learning of few-shot classification can roughly be divided into several categories such as optimization-based method~\cite{finn2017model,grant2018recasting,lee2018gradient,opt_as_model,leo,cavia}, distance-based approaches~\cite{snell2017prototypical,sung2018learning,vinyals2016matching}, class or task-wise network modulation with amortization~\cite{gordon2018meta,cnaps}, or some combination of those approaches~\cite{das2019two,l2b,oreshkin2018tadam,triantafillou2019meta}. We use a distance-based approach in this work, which allows us to directly compare distance between examples on a metric space. For example, Matching Networks~\cite{vinyals2016matching} use cosine distance, whereas Prototypical Networks~\cite{snell2017prototypical} use euclidean distance with each class prototype set to the mean of support embeddings. \vspace{-0.15in} \paragraph{Transductive learning} Since few-shot classification is intrinsically challenging, we may assume that we can access other unlabeled query examples, which is called transductive learning~\cite{vapnik98}. Here we name a few recent works. TPN~\cite{liu2018learning} constructs a nearest-neighbor graph and propagate labels to pseudo-label the unlabeled query examples. EGNN~\cite{kim2019edge} similarly constructs a nearest-neighbor graph, but utilizes both edge and node features in the update steps. On the other hand, Hou \textit{et al.}~\cite{hou2019cross} tries to update class prototypes by picking top-$k$ confident queries with their own criteria. Our approach also updates class prototypes for each transduction step, but makes use of all the query examples instead of a small subset of $k$ examples. \vspace{-0.15in} \paragraph{Semi-supervised learning} In the few-shot classification, semi-supervised learning can access additional large amount of unlabeled data. Ren \textit{et al.}~\cite{ren2018meta} proposed several variants of soft $k$-means method in prototypical networks~\cite{snell2017prototypical}, where soft label is predicted confidence of unlabeled sample. Li \textit{et al.}~\cite{l2ST} proposed the self-training method with pseudo labeling module based on gradient descent approaches~\cite{finn2017model, MTL}. Basically, if an unlabeled query set is used for few-shot classification instead of an additional unlabeled set, it becomes transductive learning, and vice versa. Our approach has connection to soft $k$-means method of Ren \textit{et al.}~\cite{ren2018meta}, but we predict the confidence with input-adaptive distance metric and use meta-learned confidence under various perturbations. \section{Preliminaries} \vspace{-0.05in} \subsection{Few-shot Classification} \vspace{-0.05in} We start by introducing notations. In the conventional $C$-way $N$-shot classification, we first sample $C$ classes randomly from the entire set of classes, and then sample $N$ and $M$ examples from each class for the support set and query set, respectively. We define this sampling distribution as $p(\tau)$. As a result, we have a support set $\mathcal{S} = \{(\mathbf{x}_i,y_i)\}_{i=1}^{C \times N}$ and query set $\mathcal{Q}=\{(\tilde{\mathbf{x}}_i,\tilde{y}_i)\}_{i=1}^{C \times M}$, where $y,\tilde{y} \in \{1,\dots,C\}$ are the class labels. If some portion of the support set is unlabeled, then the problem becomes semi-supervised learning. The convention for the evaluation of few-shot classification models is to use $N\in\{1,5\}$ (i.e. $1$- or $5$-shot) and $M=15$. The goal of few-shot classification is to correctly classify query examples in $\mathcal{Q}$ given the support set $\mathcal{S}$. Since $\mathcal{S}$ includes only a few examples for each class, conventional learning algorithms will mostly fail due to overfitting (e.g. consider 1-shot classification). Thus, most existing approaches tackle this problem by meta-learning over a task distribution $p(\tau)$, such that the later tasks can benefit from the knowledge obtained over the previous training episodes. One of the most popular and successful approaches for few-shot classification is the metric-based approach, in which we aim to learn an embedding function ${f}_{\theta}(\mathbf{x}) \in \mathbb{R}^{l}$ that maps an input $\mathbf{x}$ to a latent embedding $\mathbf{z}$ in an $l$-dimensional metric space (which is usually the penultimate layer of a convolutional network). Support set and query examples are then mapped into this space, such that we can measure the distance between class prototypes and query embeddings. \vspace{-0.05in} \subsection{Transductive Inference with Soft $k$-means} \vspace{-0.05in} We now describe and discuss transductive inference using the confidence scores of query examples computed by soft $k$-means algorithm~\cite{ren2018meta}. Suppose that we are given an episode consisting of support set $\mathcal{S}$ and query set $\mathcal{Q}$. We also define $\mathcal{S}_c$ as the set of support examples in class $c$ and $\mathcal{Q}_x = \{\tilde{\mathbf{x}}_1,\dots,\tilde{\mathbf{x}}_{C \times M}\}$ as the set of all query instances. Starting from prototypical networks~\cite{snell2017prototypical}, we first compute the initial prototype $P_c^{(0)} = \frac{1}{|\mathcal{S}_c|} \sum_{x \in \mathcal{S}_c} f_\theta(\mathbf{x})$ for each class $c = 1,\dots,C$. Then, for each step $t=1,\dots,T$, and for each query example $\tilde{\mathbf{x}} \in Q_x$, we compute its confidence score, which denote the probability of it belonging to each class $c$, as follows: \begin{align} q_{c}^{(t-1)}(\tilde{\mathbf{x}}) = \frac{\exp(-d(f_\theta(\tilde{\mathbf{x}}), P_c^{(t-1)} ))} {\sum_{c'=1}^C \exp(-d(f_\theta(\tilde{\mathbf{x}}), P_{c'}^{(t-1)}))} \label{eq:conf} \end{align} where $d(\cdot,\cdot)$ is Euclidean distance and $P^{(t-1)}$ denotes $t-1$ steps updated prototype. We then update the prototypes of class $c$ based on the confidence scores (or soft labels) $q_c^{(t-1)}(\tilde{\mathbf{x}})$ \emph{for all} $\tilde{\mathbf{x}} \in \mathcal{Q}_x$: \begin{align} P_c^{(t)}=\frac{\sum_{\mathbf{x} \in \mathcal{S}_c}1 \cdot f_\theta(\mathbf{x})+\sum_{\tilde{\mathbf{x}} \in \mathcal{Q}_x}q_c^{(t-1)}(\tilde{\mathbf{x}}) \cdot f_\theta(\tilde{\mathbf{x}})}{\sum_{\mathbf{x} \in \mathcal{S}_c}1+\sum_{\tilde{\mathbf{x}} \in \mathcal{Q}_x}q_c^{(t-1)}(\tilde{\mathbf{x}})} \label{eq:update} \end{align} which is the weighted average that we previously mentioned. Note that the confidence of the support examples is always 1, since their class labels are observed. We repeat the process until $t=1,\dots,T$. \vspace{-0.05in} \paragraph{Questions} However, confidence-based transduction, such as soft $k$-means, leads to a couple of new questions, which is the focus of this work: \emph{1) Is using the confidence of the model indeed helpful in transductive inference?} \emph{2) Can we trust the model confidence that is output from the few-shot task?} \vspace{-0.05in} \section{Approach} \subsection{Meta-Confidence Transduction} \vspace{-0.05in} In order to address the first question, we propose to \emph{Meta-Confidence Transduction} (MCT). As shown in the method overview in Figure~\ref{fig:overview}, we \emph{meta-learn} the distance metric by learning an \emph{input-dependent} temperature scaling for confidence, using the various perturbations on confidence in training. \vspace{-0.05in} \paragraph{Meta-learning confidence with input-adaptive distance metric} We first propose to meta-learn the input-adaptive metric by performing transductive inference during training with query instances, to obtain a metric that yield performance improvements when performing transductive inference using it. Specifically, we meta-learn the distance metric $d_\phi$ in Eq.~\eqref{eq:metric}, which we define as Euclidean distance with normalization and instance-wise metric scaling $g_\phi^I$, or pair-wise metric scaling $g_\phi^P$: \begin{align} d^I_\phi(\mathbf{a}_1, \mathbf{a}_2) = \left\|\frac{\mathbf{a}_1/\|\mathbf{a}_1\|_2}{g^I_\phi(\mathbf{a}_1)} - \frac{\mathbf{a}_2/\|\mathbf{a}_2\|_2}{g^I_\phi(\mathbf{a}_2)}\right\|_2^2,\quad d^P_\phi(\mathbf{a}_1, \mathbf{a}_2) = \left\|\frac{\mathbf{a}_1/\|\mathbf{a}_1\|_2}{g^P_\phi(\mathbf{a}_1,\mathbf{a}_2)} - \frac{\mathbf{a}_2/\|\mathbf{a}_2\|_2}{g^P_\phi(\mathbf{a}_1,\mathbf{a}_2)}\right\|_2^2 \label{eq:metric} \end{align} for all $\mathbf{a}_1,\mathbf{a}_2 \in \mathbb{R}^l$. Note that the normalization allows the confidence to be mainly determined by metric scaling. In order to obtain the optimal scaling function $g_\phi \in \{g_\phi^I,g_\phi^P\}$ for transduction, we first compute the query likelihoods after $T$ transduction steps, and then optimize $\phi$, the parameter of the scaling function $g_\phi$ by minimizing the following instance-wise loss for $d_\phi \in \{d_\phi^I, d_\phi^P\}$: \begin{align} L_I^\tau(\theta,\phi) &= \frac{1}{|\mathcal{Q}|}\sum_{(\tilde{\mathbf{x}},\tilde{y}) \in \mathcal{Q}} -\log p(\tilde{y}|\tilde{\mathbf{x}}, \mathcal{S};\theta,\phi) \\ &= \frac{1}{|\mathcal{Q}|}\sum_{(\tilde{\mathbf{x}},\tilde{y}) \in \mathcal{Q}} \left\{d_\phi(f_\theta(\tilde{\mathbf{x}}), P_c^{(T)} ) + \sum_{c'=1}^C \exp(-d_\phi(f_\theta(\tilde{\mathbf{x}}), P_{c'}^{(T)}))\right\}. \label{eq:inst_loss} \end{align} As for $g_{\phi}$, we simply use a CNN with fully-connected layers which takes either the feature map of an instance or the concatenated feature map of a pair of instances as an input. We set the number of transduction steps to $T=1$ for training to minimize the computational cost, but use $T=10$ for test. \begin{figure}[t] \centering \includegraphics[width=1\linewidth]{pdfs/concept_figure4.pdf} \vspace{-0.20in} \caption{\small \textbf{Overview.} (a) To capture data uncertainty, we randomly apply horizontal flip augmentation to the whole data in episode. (b) Along with data uncertainty, we randomly drop the last residual block to capture the model uncertainty. (c) In order to efficiently train the confidence under these perturbations, we \emph{meta-learn} the input-adaptive distance metric.} \vspace{-0.2in} \label{fig:overview} \end{figure} \input{self_ensemble} \vspace{-0.1in} \paragraph{Model and data perturbations} The model confidence from few-shot tasks is intrinsically unreliable due to the data scarcity problems, even if the model has been meta-learned over similar tasks. One way to output more reliable and consistent confidence scores is to enforce the model to output consistent predictions while perturbing either the model or the data. In this work, we consider the following two sources of perturbations: \vspace{-0.05in} \begin{itemize} \item \textbf{Model perturbation:} We consider two confidence scores, one from the full network (full-path) and the other from a sub-network generated by dropping a block (drop-path) ~\cite{dropout_as_bayesian,veit2016residual,blockdrop} from the full network. \item \textbf{Data perturbation:} We also consider two confidence scores, one from the original image and the other from horizontally flipped image. \end{itemize} \vspace{-0.05in} By jointly considering these two sources of perturbations, we can have a total of four ($2 \times 2$) scenarios (or sources) of possible transductive inferences. As shown in Algorithm \ref{algo:train_MCT}, at training time, we randomly select a source of confidence and simulate a single transduction step. However at test time, we perform transductive inference for all scenarios using the ensemble confidence obtained from all perturbed sources. This process is done $T$ times to get the final confidence scores. By doing so, we can enforce the model to consistently perform well under various transduction scenarios with different perturbations, leading to better performance due to the ensemble effect of meta-learned confidences (see the Section~\ref{app:C} of the appendix for more details). \vspace{-0.05in} \subsection{Consistency Regularization}\label{sec:consist} \vspace{-0.05in} In order to address the second question, we suggest consistency regularization for data and embedding. The quality of the confidence scores can be improved with consistency regularization. In semi-supervised learning, consistency regularization is one of the most popular techniques as it allows for unlabeled examples to output consistent predictions under various perturbations, thereby improving the quality of the confidence scores~\cite{temporal, fixmatch,mean_teacher}. \paragraph{Consistency over data perturbation} We also propose to enforce the model to output consistent predictions under various perturbations of the query examples. The idea is that, even though we perturb the query examples by large amount, a good and discriminative embedding space for transductive inference should be able to correctly classify them. Specifically, we apply only the horizontal flipping and shifting to the examples in the support set (\emph{weak} augmentation), whereas we apply horizontal flipping, shifting, RandAugment~\cite{randaugment} and CutOut~\cite{cutout} to the examples in the query set (\emph{strong} augmentation), and perform classification with those augmentations. This approach is related to FixMatch~\cite{fixmatch} algorithm for semi-supervised learning, but we apply various augmentations to disjoint sets rather than to the same instance, which allows to achieve the same effect as a regularization without an explicit consistency loss. \vspace{-0.05in} \paragraph{Consistency over dimensions of embedding space} Dense classification (DC) \cite{lifchitz2019dense} achieves successful performance improvement in few-shot classification. However, they apply spatial pooling to feature maps, in order to make embeddings at testing. This causes unnecessary bottlenecks, making it difficult to completely use the learned spatial information. To alleviate this problem, we reinterpret DC as a regularizer on the high dimensional embedding being learned. In other words, we do not apply spatial pooling at both training and testing, and then use flattened feature map as the embedding for each instance. We found that computing the distance with densely matching the spatial embeddings improves performance, without any additional parameters. When training with DC, we additionally compute dimension-wise loss $L_D^{\tau}$, the average classification loss for each dimension of embedding (e.g. 64-way classification for miniImageNet). Hence, final learning objective is $L=E_{p(\tau)}[\lambda L_I^{\tau}+L_D^{\tau}]$, where $L_I^{\tau}$ is the instance-wise loss in Eq.~\ref{eq:inst_loss} and $\lambda$ is the balancing factor. \section{Experiments} \paragraph{Dataset} We validate our method on four popular benchmark datasets for few-shot classification. \textbf{1) miniImageNet}, \textbf{2) tieredImageNet}, \textbf{3) CIFAR-FS}, and \textbf{4) FC100}. Please see the Section~\ref{app:A.1} of the appendix regarding the detailed information for each of the datasets. \input{table_tex/main_result1} \input{table_tex/main_result2} \vspace{-0.05in} \paragraph{Experimental setting} Here we mention a few important experimental settings of our model. During training, we apply the weight decay of $0.0005$, and unless otherwise indicated, apply the augmentations proposed in Section~\ref{sec:consist} by default. When the image size is $32\times 32$, we apply max-pooling only to the second and the fourth layer to increase the dimensionality of the final embedding space. For our full models, we evaluate the expectation over task distribution $p(\tau)$ via Monte-Carlo (MC) approximation with a single sample during training to obtain the learning objective, where we set $\lambda=0.5$ which we found with a validation set. More details (e.g. learning rate scheduling, detailed network architectures and settings for semi-supervised experiment) can be found in the Section~\ref{app:A} and~\ref{app:B} of the appendix. \subsection{Main Results} \paragraph{Inductive inference} We first examine the results of inductive inference. We define \emph{Meta-Confidence Induction (MCI)} as an our proposed metric with consistency regularizations only. The top rows of Table~\ref{tbl:mini_tiered} and Table~\ref{tbl:cifar_fc100} show the accuracy of MCI and the existing inductive inference methods for few-shot classification. Our model achieves \textbf{new state-of-the-art results} for inductive inferecne models on all four benchmark datasets with significant margins. This performance gain is coming from both the consistency regularization over the data perturbation and on the dimensions of the embeddings. We analyze each component in detail in the following sections. \input{table_tex/ablation_transduction} \paragraph{Transductive inference} The bottom rows of Table~\ref{tbl:mini_tiered} and Table~\ref{tbl:cifar_fc100} show the results of transductive inference with the baselines and our full model, Meta-Confidence Transduction (MCT), which performs transductive inference with the meta-learned confidence. We again achieve \textbf{new state-of-the-art results} on all the datasets, with particularly good performance on one-shot classification. For fair comparsion against TPN~\cite{liu2018learning} and EGNN~\cite{kim2019edge} that use different backbone networks, we further perform an additional experiments using shallow backbone networks in Table~\ref{tbl:ablation_transduction}. Again, our model largely outperforms all baselines. Note that we use MCT without model perturbation (block drop) since ConvNet-64 and ConvNet-256 do not have skip connections. \input{table_tex/semi_result} \paragraph{Semi-supervised inference} We also perform experiments on semi-supervised classification in Table~\ref{tbl:semi_result} to further validate the effectiveness and generality of our MCT. We follow the same experimental setting described in Li \textit{et al.}~\cite{l2ST}. In the semi-supervised setting, instead of computing the confidence scores of query examples, we compute the confidence scores of unlabeled support examples in order to update the class prototype. Again, our MCT largely outperforms all the baselines including the recent LST model. The results demonstrate the effectiveness of our consistency regularizations and the distance metric scaling for correctly assigning confidence scores to unlabeled examples. \subsection{Ablation Studies} We next perform ablation studies of our model on miniImageNet dataset to identify from where the performance improvements come from. We use prototypical networks (PN) with ResNet-12 backbone networks for these experiments, including only a single component of MCT at a time. \input{table_tex/ablation_metric} \begin{wrapfigure}{t}{0.36\textwidth} \vspace{-0.25in} \centering \subfigure{\includegraphics[width=0.49\linewidth]{pdfs/histogram_IMS.pdf}\label{fig:scale_IMS}} \subfigure{\includegraphics[width=0.49\linewidth]{pdfs/histogram_PMS.pdf}\label{fig:scale_PMS}} \vspace{-0.25in} \caption{\small Histogram of metric scale, on a miniImageNet 5-way 5-shot task. $\sigma$ corresponds to $g_{\phi}$.}\label{fig:scale_histogram} \vspace{-0.10in} \end{wrapfigure} \paragraph{Effect of the distance metrics} We first study the effect of the distance metric in Table~\ref{tbl:ablation_metric}. The performance in the transductive inference columns correspond to each of the models with the transductive inference with naive soft $k$-means algorithm~\cite{ren2018meta} without model and data perturbations. We see that the PN with metric scaling underperforms the plain PN with Euclidean distance. On the other hand, the proposed instance-wise and pair-wise metric significatly outperform both distances in both inductive and transductive inference settings, demonstrating the effectiveness of our input-dependent metric scaling methods over globally shared metric scaling. In Figure~\ref{fig:scale_histogram}, we observe that instance-wise metric scaling assigns various scales to different inputs, whereas the pair-wise metric scaling assigns low values between the samples from the same class and high values between samples from different classes. \input{table_tex/ablation_ensemble} \paragraph{Effect of the model / data perturbation} In Table~\ref{tbl:ablation_ensemble}, We analyze the contribution of each type of uncertainty to the reliability of confidence. We observe that the performance of transductive inference improves as we add in each type of uncertainties. We use negative log-likelihood (NLL) as the quality measure for the confidence scores: the lower the NLL, the closer the confidence scores to the target label. We observe that both types of uncertainties are helpful in improving the reliability of the output confidence. \begin{wrapfigure}{t}{0.5\textwidth} \vspace{-0.23in} \centering \subfigure{\includegraphics[width=0.51\linewidth]{pdfs/aug_1shot.pdf}\label{aug_in}} \hspace{-0.1in} \subfigure{\includegraphics[width=0.49\linewidth]{pdfs/aug_5shot.pdf}\label{aug_trans}} \vspace{-0.15in} \caption{\small Test accuracy with various augmentation pair. "w" and "s" denote "weak" and "strong", respectively. Detailed results with confidence interval can be found in Section~\ref{app:D} of the appendix.}\label{fig:aug} \vspace{-0.20in} \end{wrapfigure} \paragraph{Effect of the perturbation of query examples} We next analyze the effect of our augmentation strategy. We see from Table~\ref{fig:aug} that applying weak augmentations (horizontal flipping and shifting) to the examples in the support set while applying strong augmentations (horizontal flipping, shifting, RandAugment~\cite{randaugment} and CutOut~\cite{cutout}) to the examples in the query set (weak-strong pair, i.e. w/s) outperforms other possible combinations of weak and strong augmentations. This result is reasonable since the class prototypes should remain as a stable target for stable training, while for query examples it may be beneficial to make them as diverse as possible, for the meta-learned confidence to account for various uncertainties with transductive inference. \input{table_tex/ablation_DC} \paragraph{Effect of the across-dimension consistency} Lastly, we compare the effect of our consistency regularization across embedding dimensions with existing dense classification (DC) methods. In Table~\ref{tbl:ablation_DC}, we see that embedding without global average pooling (GAP) outperforms the model with GAP, demonstrating the effectiveness of dense matching of spatial features that gets rid of unnecessary bottlenecks. Also, unlike the existing DC which train pixel-wise classifiers during training and instance-wisely predict at test time, our method has a consistent framework as it has an additional instance-wise loss term (Eq.~\eqref{eq:inst_loss}) that is used both at training and test time. \section{Conclusion} \vspace{-0.05in} Using unlabeled data for few-shot learning, either test instances themselves (transductive) or others (semi-supervised) could help with predictions. Yet, they should be assigned correct confidence scores for optimal performance gains. In this work, we proposed to tackle them by meta-learning confidence scores, such that the prototypes updated with meta-learend scores optimize for the transductive inference performance. Specifically, we proposed to \emph{meta-learn} the parameter of the length-scaling function, such that the proper \emph{distance metric} for the confidence scores can be automatically determined. We also consider model and data-level uncertainties for unseen examples, for more robust confidence estimation. Moreover, to enhance the quality of confidence scores, we suggest a consistency regularization for data and embedding, which allows for consistent prediction under various perturbations. We experimentally validate our transductive inference model on four benchmark datasets and obtain state-of-the-art performances on both transductive and semi-supervised few-shot classification tasks. Further ablation studies confirm the effectiveness of each component. \section*{Broader Impact} In real world scenarios, we may not have large amount of labeled data to train an accurate and reliable model on the target task, but should nevertheless obtain desired level of accuracy. To learn an accurate prediction model under such data-scarce scenarios, we may further exploit the unlabeled data given at test time (transductive inference), or extra unlabeled data during training (semi-supervised learning). Our model is especially helpful when using such unlabeled data to aid the learning with scarce data, as it is able to output accurate confidence scores for the unlabeled examples such that they help with the transductive inference or semi-supervised learning. Such low-resource learning can greatly reduce either the training time (for transductive inference) or human labeling cost (for semi-supervised learning) since we only need a few training data points to train a classifier that obtains the desirable level of accuracy. \section{Detailed Explanation of Meta-Confidence Transduction}\label{app:C} \subsection{Design choices} \paragraph{Model perturbation} To generate the model uncertainty, we drop the last residual block in residual network (ResNet). As discussed in Veit \textit{et al.}~\cite{veit2016residual}, dropping single upper block in ResNet doesn't significantly affect model performance. Furthermore, we empirically found that block drop allows us to obtain the model with less dependency rather than dropout. \paragraph{Data perturbation} There can be various choices of augmentation method to perturb the data. However, we found that large transformation from the raw image can degrade classification accuracy at inference, causing large information loss on few data regime. Thus, we choose horizontal flip augmentation, which can perturb the data without losing information, to obtain perturbed confidences in training and testing consistently. \paragraph{Optimization} The reason we optimize only a single full-path is as follows. First, since we randomly apply horizontal flipping to the whole data in each episode, perturbed spaces with flipped images are optimized through the sequence of episodes. Secondly, as drop-path is one of the ensemble path of full-path, it is jointly optimized with full-path~\cite{veit2016residual}. \input{inference} \subsection{Transductive inference} As shown in Algorithm~\ref{algo:test_MCT}, we update the class prototypes by considering various types of uncertainties. Given an episode consisting of raw images, we generate another episode by flipping the original images. First, prototypes of full-path and drop-path are obtained by averaging embedding of support set. By using these prototypes, we compute the confidence scores for each space and class, respectively. With the ensemble confidence score obtained from various spaces and queries, we update prototypes of each space. Then, we repeatedly update the prototype $T$ times by using an averaged confidence. Finally, $q^{(T)}(\tilde{\mathbf{x}})$ is used for inference. \input{table_perturbation} \vspace{-0.2in} \subsection{Importance of model and data perturbations} To further investigate the effect of the model and data perturbations on confidence, we report more results of MCI and MCT in Table~\ref{tbl:MCI_MCT}. MCI is the model that is trained with inductive manner, whereas MCT is transductively trained with model and data perturbations. We see that the MCT outperforms MCI in both inductive and transductive settings. It means that meta-learned confidence with perturbations allows us to obtain more reliable confidence, which is further helpful for transduction. \vspace{+0.1in} \section{Detailed Result for Augmentation Strategy}\label{app:D} \begin{table*}[h!] \centering \footnotesize \caption{Result of augmentation strategy on pair-wise distance metric.} \vspace*{0.03in} \begin{tabular}{c c | c c c c } \hline Support & Query & \multicolumn{2}{c}{Inductive} & \multicolumn{2}{c}{Transductive} \\ Aug & Aug & 1-shot & 5-shot & 1-shot & 5-shot \\ \hline \hline weak & weak & 58.21\tiny$\pm$0.61 & 75.99\tiny$\pm$0.52 & 64.43\tiny$\pm$0.62 & 78.49\tiny$\pm$0.50 \\ strong & strong & 60.93\tiny$\pm$0.65 & 76.59\tiny$\pm$0.45 & 69.55\tiny$\pm$0.71 & 79.15\tiny$\pm$0.52 \\ strong & weak & 56.75\tiny$\pm$0.64 & 74.32\tiny$\pm$0.50 & 66.12\tiny$\pm$0.60 & 75.71\tiny$\pm$0.52 \\ weak & strong & \textbf{61.81\tiny$\pm$0.58} & \textbf{77.67\tiny$\pm$0.50} & \textbf{71.95\tiny$\pm$0.81} & \textbf{81.06\tiny$\pm$0.51} \\ \hline \end{tabular} \label{tbl:ablation_aug_PMS} \end{table*} \begin{table*}[h!] \centering \footnotesize \caption{Result of augmentation strategy on instance-wise distance metric.} \vspace*{0.03in} \begin{tabular}{c c | c c c c } \hline Support & Query & \multicolumn{2}{c}{Inductive} & \multicolumn{2}{c}{Transductive} \\ Aug & Aug & 1-shot & 5-shot & 1-shot & 5-shot \\ \hline \hline weak & weak & 58.90\tiny$\pm$0.61 & 74.76\tiny$\pm$0.47 & 67.25\tiny$\pm$0.89 & 76.68\tiny$\pm$0.55 \\ strong & strong & 59.74\tiny$\pm$0.62 & 75.59\tiny$\pm$0.49 & 69.01\tiny$\pm$0.93 & 78.99\tiny$\pm$0.53 \\ strong & weak & 58.09\tiny$\pm$0.64 & 74.03\tiny$\pm$0.48 & 67.94\tiny$\pm$0.88 & 77.24\tiny$\pm$0.55 \\ weak & strong & \textbf{61.08\tiny$\pm$0.66} & \textbf{77.26\tiny$\pm$0.46} & \textbf{69.45\tiny$\pm$0.86} & \textbf{79.54\tiny$\pm$0.54} \\ \hline \end{tabular} \label{tbl:ablation_aug_IMS} \end{table*} \section{Experimental Setup}\label{app:A} \subsection{Datasets}\label{app:A.1} We validate our method on four benckmark datasets for few-shot classification. \textbf{1) miniImageNet.} This dataset~\cite{vinyals2016matching} consists of a subset of $100$ classes sampled from the ImageNet dataset~\cite{russakovsky2015imagenet}. Each class has $600$ images, resized to $84\times84$ pixels. We use the split of $64/16/20$ for training/validation/test. \textbf{2) tieredImageNet.} This dataset~\cite{ren2018meta} is another subset of ImageNet, that consists of $779,165$ images of $84\times{84}$ pixels collected from $608$ classes. The task is to generalize the few-shot classifier over $34$ different superclasses. Thus the entire dataset is split into $20/6/8$ superclasses for training/validation/test, where each superclass contains $351$, $97$, and $160$ low-level classes, respectively. \textbf{3) CIFAR-FS.} This dataset~\cite{bertinetto2018metalearning} is a variant of CIFAR-100 dataset used for few-shot classification, which contains $100$ classes that describe general object categories. For each class, there are $600$ images of $32\times{32}$ pixels. The dataset is split into $64/16/20$ classes for training/validation/test. \textbf{4) FC100.} This is another few-shot classification dataset~\cite{oreshkin2018tadam} compiled by reorganizing the CIFAR-100 dataset. The task for this dataset is to generalize across $20$ superclasses, as done with the tieredImageNet dataset. The superclasses are divided into $12/4/4$ classes for training/validation/test, each of which contains $60/20/20$ low-level classes, respectively. \subsection{Network architectures}\label{app:A.2} We consider ResNet-12 backbone and conventional 4-block convolutional networks with 64-64-64-64 (ConvNet-64) or 64-96-128-256 (ConvNet-256) channels for each layer. We implement the metric scaling function as a single convolutional block followed by two fully-connected layers (FC-layers). The convolutional block consists of 3x3 convolution, batch normalization, ReLU activation and 2x2 max pooling. The first FC-layer is followed by batch normalization and ReLU activation, whereas the last FC-layer followed by sigmoid function to ensure non-negativity. Finally, in order to balance the effect of the scaling and normalized distance on confidence, we apply scaling ($exp(\alpha)$) and shifting ($exp(\beta)$) to the output of the sigmoid function, where $\alpha$ and $\beta$ are initialized to 0. \subsection{Hyperparameters}\label{app:A.3} We apply dropout to each layer with the ratio of $0.1$. We use SGD optimizer with the Nesterov momentum of $0.9$ and set the weight decay to $0.0005$. Following Snell \textit{et al.}~\cite{snell2017prototypical}, we use higher way ($15$-way) classification for training and $5$-way for test. The number of query examples for each class is set to $8$ for training and $15$ for test. For \textbf{miniImageNet}, \textbf{CIFAR-FS} and \textbf{FC100}, we set the initial learning rate to $0.1$ and cut it to $0.006$ and $0.0012$ at $25,000$ and $35,000$ episodes, respectively. For \textbf{tieredImageNet}, we set the initial learning rate to $0.1$ and decay it by a factor of 10 at every $20,000$ episode until convergence. \section{Settings for Semi-Supervised Few-shot Classification}\label{app:B} We split both miniImageNet and tieredImageNet into labeled and unlabeled sets, following previous works~\cite{l2ST, ren2018meta}. Before we train the model with semi-supervised learning, we pre-train the model with conventional supervised manner (e.g. 64-way classification for miniImageNet). At the training phase, we additionally use 15 instances for each class. At test phase, we use 30 and 50 unlabeled instances for each class on 1-shot and 5-shot task, respectively, following Li \textit{et al.}~\cite{l2ST}. For fair comparison with masked soft $k$-means of Ren \textit{et al.}~\cite{ren2018meta}, we use single update step with unlabeled set for both training and testing. \section{Qualitative Analysis} \vspace{-0.2in} \begin{figure}[h!] \small \centering \subfigure[\small Raw image, Full-path, Local conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_OriginFull_local.pdf}} \hfil \subfigure[\small Raw image, Full-path, Ensemble conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_OriginFull_global.pdf}} \vspace{-0.1in} \subfigure[\small Raw image, Drop-path, Local conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_OriginDrop_local.pdf}} \hfil \subfigure[\small Raw image, Drop-path, Ensemble conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_OriginDrop_global.pdf}} \vspace{-0.1in} \subfigure[\small Flipped image, Full-path, Local conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_FlipFull_local.pdf}} \hfil \subfigure[\small Flipped image, Full-path, Ensemble conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_FlipFull_global.pdf}} \vspace{-0.1in} \subfigure[\small Flipped image, Drop-path, Local conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_FlipDrop_local.pdf}} \hfil \subfigure[\small Flipped image, Drop-path, Ensemble conf]{\includegraphics[width=0.42\linewidth]{pdfs_sup/embedding_FlipDrop_global.pdf}} \caption{ \small Visualization of incorrectly classified query examples, on a miniImageNet 3-way 1-shot task. The size of circles shows the confidence score for the red class. Every figure is visualized by same task. conf denotes confidence. In each row, we show the transduction with local confidence and the transduction with ensemble confidence, where local confidence is derived from each space. Best viewed in color.} \label{fig:my figure} \end{figure}

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\section{Introduction} \label{sec:intro} The explosive ubiquity of digital photography brings a phenomenal surge in videos captured every day. Confronted with the enormous amount of video data, apparently, it is impractical for end-user to browse the videos completely to understand the content. To address this issue, video summarization has gained increasing attention to figure out how to browse, manage and retrieve videos efficiently \cite{mahasseni2017unsupervised,yuan2019cycle,zhang2016summary}. The objective of video summarization is to create a short synopsis that preserves the most important and relevant content of the original video with minimal redundancy \cite{zhu2020dsnet,rochan2018video,zhang2018retrospective}. \begin{figure}[t!] \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{ours_simple.pdf} \caption{MHSCNet (ours)} \label{ours_simple} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.4\textwidth]{other_simple.pdf} \caption{Previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip}} \label{other_simple} \end{subfigure} \caption{(a) The proposed MHSCNet builds adaptive shot-aware frame-level representation by modeling the frame-to-frame interaction and incorporating the hierarchical shot-level representation into the frame-level representation. (b) In contrast, previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip} utilize different temporal modeling layers (LSTM, Attention, GCN, etc.) to produce the frame-level representation containing the long-range temporal dependency.} \label{simple_compare} \end{figure} Typically, the most recent video summarizers follow the three steps: 1) video segmentation, 2) importance score prediction, and 3) key shot selection. Regarding video segmentation, KTS (Kernel Temporal Segmentation) is widely used to segment the coherent frame sequence into several shots \cite{potapov2014category}. As for key shot selection, it is tackled by 0-1 knapsack algorithm to generate a key-shot based summary. The most challenging step is importance score prediction, which is designed to highlight the most important parts according to the predicted scores for the video content. Previous works have devised different approaches to cope with this key challenge. They can be roughly categorized into unsupervised \cite{jung2019discriminative,zhou2018deep}, weakly-supervised \cite{panda2017weakly,kim2014reconstructing} and supervised methods \cite{gygli2015video,zhang2016summary}. Recent unsupervised methods attempt to reconstruct the video based on the key-shot summary generated by the frames' importance score. \cite{mahasseni2017unsupervised,he2019unsupervised}. Also, the diversity and sparsity of output summary \cite{panda2017collaborative,panda2017diversity,rochan2019video,yuan2019cycle,panda2017multi} are introduced as repelling regularizers in the training loss. For the weakly-supervised methods, the video-level metadata, such as video titles \cite{panda2017weakly,zeng2016generation}, web priors \cite{cai2018weakly,chu2015video} and video categories \cite{potapov2014category}, work as the auxiliary information to put additional constraints for video summarizers. Although unsupervised and weakly-supervised methods have reached promising performance, the supervised methods always perform better benefiting from the manually annotated dataset. To manipulate video frames for supervised video summarizers, it is desirable to handle temporal dependency among video frames effectively. To achieve this, recent efforts exploited LSTM units \cite{zhang2018retrospective,zhang2016video,zhao2018hsa} to capture long-range temporal dependency among video frames. Besides, some methods \cite{he2019unsupervised,fajtl2018summarizing,liu2020transforming} resorted to the attention mechanism to propagate the temporal dependency information in the frame level. To better perform relationship reasoning among the frames, Park et al. \cite{park2020sumgraph} introduced a recursive graph structure and achieved the state-of-the-art performance. Apart from the unimodal approaches, some bimodal methods \cite{palaskar2019multimodal,libovicky2018multimodal} extract important frame by fusing the textual as well as video metadata and remarkable performance could be achieved. Recently, the work \cite{narasimhan2021clip} proposed a language-guided bimodal transformer via fusing information across the video and language modalities. Despite the promising success, existing video summarization methods still encounter two major challenges. First, \textit{how to construct a more powerful and robust frame-wise representation for video summarization?} Essentially, most of the above methods \cite{narasimhan2021clip,zhu2020dsnet,park2020sumgraph} only adopted visual cues or textual video metadata to construct the frame-wise representation. However, almost all videos include an audio track, which could provide vital complementary information for the frame-wise representation. For instance, in a scene when a man has a conversation with a woman, we only see the two people sit aside but the audio helps us to understand the dialogue content. Meanwhile, a caption usually highlights the subject of a video, which can help us to select the relevant shots. Therefore, it is critical for a model to consider the multimodal information to construct a more powerful and robust frame-wise representation for video summarization. Second, \textit{how to predict the frame-level importance score in a fair and comprehensive manner?} The previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip} mainly focused on extracting the long-range temporal dependencies to predict the importance score of the frames from the global view, which inevitably overlooks the important frames within the local shot. As we know, video temporal structures are intrinsically layered, i.e., a video is composed of shots, and a shot is composed of several frames. Suppose a shot of blowing candles consists of three sample frames which represent taking out the lighter, lighting candles, and blowing out candles, respectively. Considering the visual coherence of the key-shot summary, the importance scores of the three consecutive frames will influence each other internally. Thus, lacking in modeling the frame-to-frame interaction within the shot the previous methods unfairly evaluate the frame importance score and generate inaccurate summary results. To address the above challenges, we present a multimodal hierarchical shot-aware convolutional network, referred to \textbf{MHSCNet}, to exhaustively exploit the available modality information of the video to enhance the frame representation, and model the frame-to-frame interactions explicitly within the shot to pay attention to the important frames both in the local and global view. The shot-aware network means that the inner operation among the network is defined at the shot level. As illustrated in Fig. \ref{Fig_framework}, it first encodes the multimodal inputs individually with pre-trained models to generate separate representations and then aggregates them hierarchically to explore the semantic relations. Subsequently, to answer the first challenge, MHSCNet fully exploits the modality information including the audio and the caption of the video to construct a robust and informative frame-wise representation. Moreover, to answer the second challenge, we design a hierarchical shot-aware convolutional network with the progressive cross-shot padding mechanism to generate an adaptive shot-aware representation, which contains both short-range and long-range temporal dependency. As illustrated in Fig. \ref{ours_simple}, MHSCNet builds the adaptive shot-aware frame-level representation which perceives shot-level information by modeling the frame-to-frame interaction in the hierarchical shot-level and incorporating the shot-level representation into the frame-wise representation. In contrast, previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip} generally investigated the long-range temporal dependency to construct the frame-wise representation by different temporal modeling layers (LSTM, Attention, GCN, etc.) leading to ignoring the important frames in the local view. Benefiting from the adaptive shot-aware representation, our MHSCNet can evaluate the frame-level importance score in a fair and comprehensive view. The contributions of this work are summarized as follows: - We propose a multimodal framework to fully explore the semantic relations among the modalities and construct a powerful and robust frame-wise representation for video summarization. - We design a hierarchical ShotConv network with the cross-shot padding mechanism to produce the adaptive shot-aware representation, which offers a fair and comprehensive view to predict the importance scores of the frames. Due to the hierarchical design, the shot-aware representation containing both the short-range and long-range temporal dependency. - The experiment results indicate that our proposed method outperforms existing state-of-the-art methods on two benchmarks. Besides, we also conduct extensive ablation studies and detailed analyses to verify the effectiveness of our method. Our code will be available for further comparison. \section{Related Work} \label{sec:rw} Given an input video, video summarization intends to generate a shortened version that contains the most important information, resulting in outputs such as video synopses \cite{pritch2007webcam}, time-lapses \cite{joshi2015real,poleg2015egosampling,kopf2014first}, montages \cite{kang2006space,sun2014salient}, or storyboards \cite{liu2010hierarchical,lee2012discovering}. Early works in video summarization primarily relied on hand-crafted heuristics \cite{kim2014reconstructing,lee2012discovering,lu2013story,ngo2003automatic,panda2017collaborative,khosla2013large}, including importance, representativeness and diversity to select representative frames to produce a summary video. Nowadays, deep neural networks for video summarization have achieved great success. Focusing on the utilized modality information, we can briefly outline two categories of video summarization methods: 1) unimodal methods, 2) bimodal methods. \subsection{Unimodal Video Summarization} In the past, most researchers \cite{zhang2018retrospective,zhang2016video,zhao2018hsa} utilized the unimodal visual information to construct the frame representation and resorted to the recurrent neural network like LSTM to exploit the long-range temporal dependency. Although LSTM-based approaches have been applied to video data, recurrent operations are sequential, limiting the processing all the frames simultaneously. To tackle this problem, Fajtl et al. \cite{fajtl2018summarizing} proposed a self-attention mechanism based network for video summarization which executes the entire sequence to sequence transformation. He et al. \cite{he2019unsupervised} built their framework upon a generative adversarial network which is composed of the self-attention module. To capture multiple concepts, Liu et al. \cite{liu2020transforming} introduced the multi-head attention mechanism. Rochan et al. \cite{rochan2018video} constructed a fully convolutional sequence model and formulated video summarization as a sequence labeling problem. Different from these methods, Zhu et al. \cite{zhu2020dsnet} proposed a Detect-to-Summarize network which predicted the corresponding importance scores and temporal locations of segments via defining video summarization problem as a temporal interest detection process. Park et al. \cite{park2020sumgraph} utilized a recursive graph to perform relationship reasoning among frames. However, these unimodal approaches only adopt the visual modality information, leading to the insufficient frame representation. \subsection{Bimodal Video Summarization} Unlike unimodal video summarization methods, existing bimodal video summarization methods fused additional audio or caption information to enhance the frame representation. Zhou et al. \cite{zhou2018video} combined the category-related information with the frame feature. Huang et al. \cite{huang2021gpt2mvs} designed an bimodal framework consisting of a contextualized video summary controller and bimodal attention mechanisms to explore the power of text modality. Yuan et al. \cite{yuan2017video} constructed a latent subspace to find the meaningful frame-level feature with side semantic information. Zhao et al. \cite{zhao2021audiovisual} developed an AudioVisual Recurrent Network to jointly exploit the audio and visual information for the video summarization. Recently, Narasimhan et al. \cite{narasimhan2021clip} proposed a language-guided bimodal transformer for both generic and query-focused video summarization. While these methods resort to the bimodal information and the long-range temporal dependency, ignoring the potential of fully incorporating available multimodal information and the short-range temporal dependency they miss the key-shot summary. To construct a powerful and robust representation and fully consider the locally-global side information, we build our MHSCNet with both the audio and the caption modal information. \section{Methodology} \label{sec:method} \begin{figure*}[tb!] \centering{ \includegraphics[width=0.80\textwidth]{framework.pdf}} \caption{Overview of our multimodal hierarchical shot-aware convolutional network (MHSCNet), consisting of the multimodal feature fusion and a hierarchical ShotConv network. In inference, the summary of the original video is generated by transforming frame scores to shot scores and a 0-1 knapsack algorithm is used to select high-scoring shots in the limited summary length.} \label{Fig_framework} \end{figure*} \subsection{Motivation and Approach Overview} Manual video summarization generally extracts key information by means of video segmentation and subsequent ordinal or unordered shot-shot pairwise significance ordering \cite{gygli2014creating,song2015tvsum}, which indicates that the frames within the same shot will interact with each other and all determined to be chosen or not for the key-frame summary. Meanwhile, the human annotator will consider the subject of the video via the audio and the textual information to judge the relevance between the frame and the video. The aforementioned methods mainly consider the long-range temporal dependency to enhance the frame-level representation via little additional modality information, ignoring the mutual relations between the contiguous frames within the same shot. Motivated by the manual annotation procedure, we propose a hierarchical ShotConv network that takes the aggregated multimodal feature as input and outputs the adaptive shot-aware representation preserving the short-range temporal dependency within the shot and the long-range temporal dependency across the shot via the proposed cross-shot padding mechanism. During the inference stage, it is hard to set the fixed length of the shot due to the variability of the shot segmentation. To make our network adaptive to the variable video length, we fix the different shot lengths for our sub-ShotConv modules. Overall, our framework mainly has two modules as shown in Fig. \ref{Fig_framework}, where the first module is multimodal feature fusion, and the second module is a hierarchical ShotConv network. During the inference, the video is firstly segmented to the several shots by the Kernel Temporal Segmentation algorithm \cite{potapov2014category}. Then, the importance scores of the frames are predicted by our MHSCNet. Finally, the 0-1 knapsack algorithm is utilized to generate a key-shot based summary. \subsection{Feature Extraction} Given a video $V$ as a sequence of $T$ frames, we first employ CNN (e.g. GoogLeNet) without the last three layers to extract feature $F_{V} =[f_{v_{1}}, ..., f_{v_{i}}, ..., f_{v_{T}}] \in \mathbb{R}^{T\times N}$. For audio information $F_{A} =[f_{a_{1}}, ..., f_{a_{i}},$ $..., f_{a_{T}}] \in \mathbb{R}^{T\times M}$, the VGGish \cite{hershey2017cnn} pre-trained on Audioset is adopted for audio feature extraction and the sampling frequency is set for keeping the same sequence length with the frame feature. To capture generic caption information, we use the Bi-Modal Transformer \cite{iashin2020better} to generate dense video captions for the whole video. We use the CLIP \cite{radford2021learning} model to encode caption vector $F_{C} \in \mathbb{R}^{1 \times N}$ and average pooling is used to balance multiple sentences. \subsection{Multimodal Feature Fusion} \subsubsection{Audio Feature Fusion} Firstly we fuse the image feature with the audio feature to explore the semantic relations. For the i-th frame, we project the audio vector $f_{a_{i}}$ to the same embedding space with the image feature $F_{V}$ by a trainable matrix $W_{audio}$ like Eq. \ref{audio_fusion}. \begin{equation} f_{va_{i}} = f_{v_{i}} + f_{a_{i}}W_{audio} +b_{audio} \label{audio_fusion} \end{equation} where $W_{audio} \in \mathbb{R}^{M \times N}$, $b_{audio}\in \mathbb{R}^{N}$ are the trainable matrices and the trainable bias, respectively. \subsubsection{Caption Feature Fusion} After fusing the audio information, we also incorporate prior knowledge into the frame feature. To this end, we employ the multi-head attention mechanism to fuse the caption feature $F_{C}$. We regard the caption feature $F_{C}$ as both the Key and the Value and the enhanced local feature $F_{VA}$ as the Query. Therefore, the aggregated multimodal frame-level feature $F_{AM}$ can be calculated as follows: \begin{equation} F_{AM} = Concat(head_{1},...,head_{h})W^{O} \label{multi_head} \end{equation} \begin{equation} head_{j} = Attention(QW_{j}^{Q},KW_{j}^{K},VW_{j}^{V}) \label{head_cal} \end{equation} \begin{equation} Attention(Q,K,V) = softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V \label{att_cal} \end{equation} where $W^{O} \in \mathbb{R}^{N \times N}$, $W_{j}^{Q} \in \mathbb{R}^{N \times d_{k}}$, $W_{j}^{K} \in \mathbb{R}^{N \times d_{k}}$ and $W_{j}^{V} \in \mathbb{R}^{N \times d_{k}}$ are trainable matrices, $d_{k}=N/h$ and $h$ is the number of attention heads. \subsection{Hierarchical ShotConv Network} \label{3.4} \subsubsection{Cross-shot Padding \& Split} \begin{figure}[tb!] \centering \includegraphics[width=1.0\linewidth]{crossshot.pdf} \caption{The cross-shot padding \& split module.} \label{cross_shot} \end{figure} As shown in Fig. \ref{cross_shot}, we firstly pad the specified number of frames in the boundary of the each shot to maintain the consistent temporal information. Given a padding ratio $\eta$ and the shot number $S$, $\eta \times \lfloor \frac{T}{S} \rfloor$ frames from the end of the $(S$-1)-th shot will be padded at the start of the $S$-th shot. For the consistency, we pad the parts of frames from the last shot to the start of the first shot. Then a 1$\times$1 convolution is utilized to increase the channel of the frame from $N$ to $\lambda N$ and the frame-level feature $F_{CP}^{l} =[f_{cp_{1}}^{l},...,f_{cp_{i}}^{l},...,f_{cp_{(1+\eta) T}}^{l}] \in \mathbb{R}^{(1+\eta) T\times \lambda N}$ in the $l$-th layer could be obtained via the cross-shot padding mechanism. Then, we reshape the dimension from $(1+\eta) T\times \lambda N$ to $S \times (D_{S} \cdot \lambda N)$, where $D_{S} = \lfloor \frac{(1+\eta) T}{S} \rfloor$ and $\lfloor . \rfloor$ is the round down function to transform the frame-level representation to the inflated shot-level representation $F_{ISR}^{l}$. Specially, the $s$-th shot contains the features of inside frames as follows: \begin{equation} F_{ISR_{s}}^{l} = \{f_{cp_{k}}^{l}: k=(s-1)D_{S} + 1 ,...,sD_{S} ,sD_{S} +1 \} \label{shot_split} \end{equation} Lastly, an average pooling operator will be used on the dimension $D_{S}$ of the frame number to obtain the concentrated shot-level representation $F_{SR}^{l} \in \mathbb{R}^{S \times \lambda N}$. To make our network adaptive to the video length, we propose three scales cross-shot padding, and the shot number $S$ will be set to $S_{l}$, $S_{m}$, $S_{s}$ for the long scale, middle scale, and short scale to generate the corresponding representation $F_{SR_{l}}^{l}$, $F_{SR_{m}}^{l}$ and $F_{SR_{s}}^{l}$. \subsubsection{Inner ShotConv Module} As shown in Fig. \ref{shotconv}, we propose an inner shot convolution module to model the frame interaction within the shot. The inner shot convolution module is composed of four convolution operators with different kernel sizes and other postprocessing modules. The strides of the four convolution operators are set to 4. The 5$\times$5 and 7$\times$7 convolution operators are implemented by the dilated convolution \cite{yu2015multi}. We firstly perform these four convolution operators on the shot-level feature $F_{SR}^{l}$. Then we will concatenate them in the channel dimension and feed it to a 1$\times$1 convolution layer to reduce the channel dimension from $\lambda N$ to $N$. Each shot-level representation will be assigned to the frames in the same shot to get the enhanced frame-level representation $F_{EF}^{l}$ via the expand operation. For the three scales ShotConv modules, the enhanced frame-level representation $F_{EF_{l}}^{l}$, $F_{EF_{m}}^{l}$ and $F_{EF_{s}}^{l}$ are generated for the long scale, middle scale and short scale, respectively. \begin{figure}[tb!] \centering \includegraphics[width=1.0\linewidth]{shotconv.pdf} \caption{The inner shot convolution module.} \label{shotconv} \end{figure} As shown in Fig. \ref{shotconv}, the adaptive shot-aware representation $F_{ASF}^{l} \in \mathbb{R}^{T \times N}$ is obtained after the multi-scale ShotConv modules by Eq. \ref{asf_add}. \begin{equation} F_{ASF}^{l} = F_{EF_{l}}^{l} + F_{EF_{m}}^{l} + F_{EF_{s}}^{l} + F_{AM}^{l} \label{asf_add} \end{equation} where $F_{AM}^{l}$ will be replaced by $F_{ASF}^{l-1}$ except the first layer. We stack $L$ layers of the hierarchical ShotConv network and the output from each layer will be the input of the next layer. \subsubsection{Implicit Global Information Propagation} \label{3.4.3} In each layer of the ShotConv network, cross-shot padding will propagate information between two adjacent shots. Benefitting from this cross-shot padding mechanism, the frames in the local shot will learn the information from the other adjacent shot. After several stacked ShotConv networks, the frames in the local shot can learn the information from the whole video since the global information will be passed in the deeper layers. Formally, the global information will be passed in a local shot when the number of stacked layers and the shot number meet the criteria as illustrated in Eq. \ref{global_condition}. \begin{equation} L+1 >= S_{l} \label{global_condition} \end{equation} where $L$ is the number of the stacked hierarchical ShotConv network and $S_{l}$ is the shot number of the long scale ShotConv module. \subsection{Loss Function} We apply the focal loss \cite{lin2017focal} for keyframe classification, which deals with the imbalance between the number of keyframes and the number of background frames. Moreover, the difference of the hard/easy sample will be highlighted. The loss is defined by \begin{equation} \begin{split} \mathcal{L} = -\frac{1}{T}\sum_{t=1}^{T}[\alpha y_{t}(1-p_{t})^{\gamma}\log(p_{t})+ (1-\alpha)(1-y_{t})p_{t}^{\gamma}\log(1-p_{t}))] \end{split} \label{loss} \end{equation} where $y_{t}$ is the groundtruth label of the $t$-th frame. $\alpha$ and $\gamma$ are the re-weighting factor. $p_{t}$ is the prediction of our proposed model. \section{Experiments} \subsection{Experiment Setup} \noindent \textbf{Implementation Details.} \ To ensure fair comparisons with prior works \cite{zhu2020dsnet,park2020sumgraph,rochan2018video}, we use the ImageNet-pretrained GoogleNet \cite{szegedy2015going} for frame feature extraction, where the 1024-d feature are extracted after the 'pool5' layer. The dimension of the audio feature extracted by the pre-trained VGGish \cite{hershey2017cnn} is 128. For the generated multiple captions by the BMT \cite{iashin2020better}, we use CLIP \cite{radford2021learning} to encode them and concatenate the embedding in the channel dimension. The dimension of the first fully-connected layer is 128. The number of multi-head in the caption feature fusion module is set to 32. We set $ S_{l} = 5$, $S_{m}=10$, $S_{s}=15$, $\eta=0.05$, $L = 4$ and $\lambda = 8$ in Sec. \ref{3.4}. $\alpha$ and $\gamma$ in the focal loss are set to 0.25 and 2. All the experiment results are reported in 5-fold cross validation. Adam works as the optimizer with an initial learning rate of $1e^{-4}$, and a weight decay of $1e^{-5}$. All the comparative models are trained for 100 epochs in total and tested on the same machine, which has a single NVIDIA GeForce Titan V GPU and Intel Core i7-8700K CPU. \noindent \textbf{Datasets.} \ We evaluate the performance of our approach on two standard video summarization datasets: SumMe \cite{gygli2014creating} and TVSum \cite{song2015tvsum}. SumMe dataset contains 25 personal videos obtained from YouTube which covers multiple categories such as cooking and sports. Each video ranges from 1 to 6 minutes and each reference summary is generated by 15 to 18 persons. TVSum dataset consists of 50 YouTube videos which contain 10 categories including changing vehicle tire, parade, and dog show. The video lengths of TVSum which are a little longer than SumMe vary from 2 to 10 minutes. Beside, two additional datasets including the YouTube \cite{de2011vsumm} and the OVP \cite{de2011vsumm} are used to augment the training data. The OVP dataset contains 50 video sequences and the YouTube dataset consists of 40 video sequences. In detail, we follow \cite{zhang2016video} to sample the frame at 2 fps to handle temporal redundancy and reduce computation. The pre-processed datasets including the data splits can be found in the URLs\footnote{https://github.com/li-plus/DSNet;https://github.com/KaiyangZhou/pytorch-vsumm-reinforce} provided by the previous researchers \cite{zhu2020dsnet,zhou2017reinforcevsumm}. \noindent \textbf{Data configuration.} \ We apply three data configurations adopted in \cite{zhang2016summary,zhou2018deep,rochan2018video} within our experiments including standard data setting, augmented data setting, and transfer data setting. \noindent \textbf{Standard}: In standard data setting, the training and testing data come from the same benchmark. Typically, we randomly select 80\% of videos for training and validation while 20\% of videos for testing. The results of the selected dataset split stay the same with previous works \cite{zhang2016video,zhu2020dsnet}. \noindent \textbf{Augment}: In the augment data setting, the videos of YouTube and OVP datasets are utilized to augment the training data. Thanks to more abundant training data, the model performance will be improved in this data setting \cite{rochan2018video,zhang2018retrospective}. \noindent \textbf{Transfer}: In the transfer data setting, the videos of three datasets will be used as the training data and the videos of one another dataset (e.g. SumMe or TVSum) will be used as the testing data. For reliable comparisons, all the experiments are conducted five times and the average performance is reported in our paper. We report the results at F-score in all of the settings. Experiments at other metrics and the ablation study are all conducted in a standard supervision manner. \noindent \textbf{Evaluation Criteria.} \ Following the previous works \cite{song2015tvsum,gygli2015video,otani2016video}, F-score is used to evaluate the agreement between the generated summary and the user summary(GT). Let $Y$ and $Y^{*}$ be the ground-truth summary and the generated keyshot summary, respectively. The precision and the recall are calculated as follows: \begin{equation} P = \frac{length(Y \cap Y^{*})}{length(Y^{*})} , R = \frac{length(Y \cap Y^{*})}{length(Y)} \label{recall} \end{equation} We compute the F-score to evaluate the quality of the generated keyshot summary based on the precision and the recall like Eq. \ref{fscore}. \begin{equation} F = \frac{2\times P \times R}{P+R} \label{fscore} \end{equation} For datasets with multiple human annotations, we follow standard approaches described in \cite{mahasseni2017unsupervised,gygli2015video,song2015tvsum,zhang2016video} to calculate the metrics for the videos. Similar to \cite{zhou2018deep,zhu2020dsnet}, we evaluate the diversity score of the generated summary on the SumMe and TVSum datasets. In recent work, Otani et al. \cite{otani2019rethinking} showed that randomly generated summaries achieve a similar F-score with the state-of-the-art methods. Therefore, two rank correlation coefficients, precisely Kendall's $\tau$ \cite{kendall1945treatment} and Spearman's $\rho$ \cite{zwillinger1999crc} are also used to evaluate the effectiveness of our model. \subsection{Comparisons to the State-of-the-Arts} \begin{table*}[htbp!] \footnotesize \caption{Comparisons of $F$-Score (\%) and parameters (in million) with peer supervised techniques on SumMe \cite{gygli2014creating} and TVSum \cite{song2015tvsum}, with various data configurations including standard data, augmented data, and transfer data settings. The scores in bold indicate the best values. "-" denotes that the results are not available. Bimodal approaches are marked with $\diamond$. For some unsupervised learning methods, we report their results in a supervised manner. As CLIP-It, we report their results with two different backbone networks. For fair comparisons with previous methods, we mainly adopt GoogLeNet features.} \centering \setlength\tabcolsep{11pt}{ \begin{tabular}{lcccccccc} \toprule[0.75pt \multicolumn{1}{l}{\multirow{2}{*}{Method}} &\multicolumn{1}{c}{\multirow{2}{*}{Backbone}} & \multicolumn{3}{c}{SumMe} & \multicolumn{3}{c}{TVSum} & \multicolumn{1}{c}{\multirow{2}{*}{Params}} \\ \cline{3-8} \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{Standard} & \multicolumn{1}{l}{Augment} & \multicolumn{1}{l}{Transfer} & \multicolumn{1}{l}{Standard} & \multicolumn{1}{l}{Augment} & \multicolumn{1}{l}{Transfer} \\ \hline\hline Random summary \cite{otani2019rethinking} & - & 41.0 & - & - & 57.0 & - & - & 0.0 \\ SUM-GAN \cite{mahasseni2017unsupervised} & GoogLeNet \cite{szegedy2015going} & 41.7 & 43.6 & - & 56.3 & 61.2 & - &295.9 \\ SUM-FCN \cite{rochan2018video} &GoogLeNet \cite{szegedy2015going} & 47.5 & 51.1 & 44.1 & 56.8 & 59.2 & 58.2 &116.5 \\ SUM-DeepLab \cite{rochan2018video} &GoogLeNet \cite{szegedy2015going} & 48.8 & 50.2 & 45.0 & 58.4 & 59.8 & 58.9 & - \\ DR-DSN \cite{zhou2018deep} &GoogLeNet \cite{szegedy2015going} & 42.1 & 43.9 & 42.6 & 58.1 & 59.8 & 58.9 &2.6 \\ ACGAN \cite{he2019unsupervised} &GoogLeNet \cite{szegedy2015going} & 47.2 & - & - & 59.4 & - & - & - \\ $\diamond$DQSN \cite{zhou2018video} &GoogLeNet \cite{szegedy2015going} & - & - & - & 58.6 & - & - & - \\ $\diamond$DSSE \cite{yuan2017video} &AlexNet \cite{krizhevsky2012imagenet} & - & - & - & 57.0 & - & - & - \\ $\diamond$AVRN \cite{zhao2021audiovisual} &GoogLeNet \cite{szegedy2015going} & 44.1 & 44.9 & 43.2 & 59.7 & 60.5 & 58.7 & - \\ VASNet \cite{fajtl2018summarizing} &GoogLeNet \cite{szegedy2015going} & 49.7 & 51.1 & - & 61.4 & 62.4 & - &7.4 \\ DSNet-AB \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 50.2 & 50.7 & 46.5 & 62.1 & 63.9 & 59.4 &8.5 \\ DSNet-AF \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 51.2 & 53.3 & 47.6 & 61.9 & 62.2 & 58.0 & 4.3 \\ MC-VSA \cite{liu2020transforming} &GoogLeNet \cite{szegedy2015going} & 51.6 & 53.0 & 48.1 & 63.7 & 64.0 & 59.5 &- \\ SumGraph \cite{park2020sumgraph} &GoogLeNet \cite{szegedy2015going} & 51.4 & 52.9 & 48.7 & 63.9 & 65.8 & 60.5 & 5.5 \\ $\diamond {\rm CLIP-It}^{*}$ \cite{narasimhan2021clip} &GoogLeNet \cite{szegedy2015going} & 51.6 & 53.5 & 49.4 & 64.2 & 66.3 & 61.3 & - \\ $\diamond {\rm CLIP-It}$ \cite{narasimhan2021clip} &CLIP-ViT-B/32 \cite{radford2021learning} & 54.2 & 56.4 & \textbf{51.9} & 66.3 & 69.0 & \textbf{65.5} & - \\\hline MHSCNet (ours) &GoogLeNet \cite{szegedy2015going} & \textbf{55.3} &\textbf{56.9} & 49.8 & \textbf{69.3} & \textbf{69.4} & 61.0 &134.6 \\ \toprule[0.750pt \end{tabular}} \label{table1} \end{table*} \noindent \textbf{Quantitative comparison.} \ To validate the effectiveness of our proposed method on the video summarization problem, we quantitatively compare our proposed model against other state-of-the-art methods including unimodal methods and bimodal methods on the SumMe and TVSum datasets. As reported in Table \ref{table1}, our MHSCNet achieves a significant improvement ranging from 3.7\% to 13.6\% on F-score in the standard data setting compared to the state-of-the-art methods equipped with the same backbone network. The unimodal methods can be roughly classified into four categories: 1) LSTM-based methods(dppLSTM \cite{zhang2016video}, SUM-GAN \cite{mahasseni2017unsupervised}, DR-DSN \cite{zhou2018deep}); 2) Attention-based methods(ACGAN \cite{he2019unsupervised}, VASNet \cite{fajtl2018summarizing}, MC-VSA \cite{liu2020transforming}, DASN \cite{fu2021video}); 3) Convolution-based methods(SUM-FCN \cite{rochan2018video}, SUM-DeepLab \cite{rochan2018video}); 4) Other methods(DSNet-AB \cite{zhu2020dsnet}, DSNet-AF \cite{zhu2020dsnet}, SumGraph \cite{park2020sumgraph}). Random summary \cite{otani2019rethinking} is generated by means of random importance scores and KTS video segmentation algorithm \cite{potapov2014category}. Specifically, existing LSTM-based methods achieve better performance on the TVSum dataset and relatively worse performance on the SumMe dataset against other methods. Such poorly-balanced performance indicates that the long-range dependence feature extracted by LSTM-based networks cannot work on the SumMe dataset and the relation across the shots is weak on the SumMe dataset. Different from the LSTM-based methods, attention-based methods perform slightly better on the SumMe dataset. Benefitting from the designed structure, the attention-based methods can implicitly capture the local information within the shot. Similarly, convolution-based methods extract the short-range temporal dependency within the shot implicitly. Moreover, DSNet formulate video summarization as an interest detection problem. The proposed center-ness score loss leads the model to learn more information within the shot. Besides, SumGraph perform better than DSNet by explicitly modeling relationships among the frames using a recursive graph. Benefitting from the multimodal informantion and adaptive shot-aware representation, our MHSCNet outperforms SumGraph by a wide margin. Besides the unimodal methods, the bimodal methods equipped with the same backbone (GoogleNet) drop the performance ranging from 3.7\% to 12.3\% in the standard data setting compared to our proposed MHSCNet because they overlook the inner-shot information and involve extra modality information excessively. We also analyze the diversity to verify the generated summary by MHSCNet contains more diverse key-frames. As shown in Table \ref{table2}, our method MHSCNet outperforms LSTM-based methods (dppLSTM \cite{zhang2016video} and DR-DSN \cite{zhou2018deep}) and other methods (DSNet-AF \cite{zhu2020dsnet} and DSNet-AB \cite{zhu2020dsnet}) with improvement ranging from 5.3\% to 14.1\%. \begin{table}[tb!] \footnotesize \caption{The diversity scores of generated summaries on the SumMe\cite{gygli2014creating} and TVSum \cite{song2015tvsum} datasets.} \centering \setlength\tabcolsep{12pt}{ \begin{tabular}{lccc} \toprule[0.75pt] Method &Backbone & SumMe & TVSum\\ \hline\hline dppLSTM \cite{zhang2016video} &GoogLeNet \cite{szegedy2015going} & 0.591 & 0.463 \\ DR-DSN \cite{zhou2018deep} &GoogLeNet \cite{szegedy2015going} & 0.594 & 0.464 \\ DSNet-AF \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 0.642 & 0.476 \\ DSNet-AB \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 0.664 & 0.477 \\ \hline MHSCNet (ours) &GoogLeNet \cite{szegedy2015going} & \textbf{0.732} & \textbf{0.530} \\ \toprule[0.75pt] \end{tabular}} \label{table2} \end{table} \begin{table}[tb!] \footnotesize \caption{Kendall's $\tau$ \cite{kendall1945treatment} and Spearman's $\rho$ \cite{zwillinger1999crc} correlation coefficients computed on the TVSum benchmark \cite{song2015tvsum}.} \centering \setlength\tabcolsep{6pt}{ \begin{tabular}{lccc} \toprule[0.75pt] Method &Backbone & Kendall's $\tau$ & Spearman's $\rho$\\ \hline\hline Random summary \cite{otani2019rethinking} &GoogLeNet \cite{szegedy2015going} & 0.000 & 0.000 \\ dppLSTM \cite{zhang2016video} &GoogLeNet \cite{szegedy2015going} & 0.042 & 0.055 \\ DR-DSN \cite{zhou2018deep} &GoogLeNet \cite{szegedy2015going} & 0.020 & 0.026 \\ HSA-RNN \cite{zhao2018hsa} &VGG16 \cite{simonyan2014very} & 0.082 & 0.088 \\ SumGraph \cite{park2020sumgraph} &GoogLeNet \cite{szegedy2015going} & 0.094 & 0.138 \\ MC-VSA \cite{liu2020transforming} &GoogLeNet \cite{szegedy2015going} & 0.116 & 0.142 \\ AVRN \cite{zhao2021audiovisual} &GoogLeNet \cite{szegedy2015going} & 0.096 & 0.104 \\ DASN \cite{fu2021video} &GoogLeNet \cite{szegedy2015going} & 0.058 & 0.065 \\ ${\rm CLIP-It}$ \cite{narasimhan2021clip} &CLIP-ViT-B/32 \cite{radford2021learning} &0.108 &0.147\\ Human \cite{otani2019rethinking} & - & \textbf{0.177} & 0.204 \\\hline MHSCNet (ours) &GoogLeNet \cite{szegedy2015going} & 0.154 & \textbf{0.225} \\ \toprule[0.75pt] \end{tabular}} \label{table3} \end{table} As Otani et al.\cite{otani2019rethinking} has indicated that random summary could achieve similar results compared to the state-of-the-art methods due to the limitation of the video segmentation methods and the 0-1 knapsack algorithm. Therefore, we directly measure importance scores using two rank correlation coefficients including Kendall's $\tau$ and Spearman's $\rho$ between the predicted ordering and the human annotators' ordering. The results of quantitative comparisons are displayed in Table \ref{table3}. We observe that MHSCNet surpasses the state-of-the-art methods including the unimodal methods and the bimodal methods with improvements of 3.8\% and 7.8\% at least on Kendall's $\tau$ and Spearman's $\rho$, respectively. From the result of human annotators, we infer that our model possesses comparable discrimination ability with the single experienced annotator. In summary, as MHSCNet jointly exploits the relations among the frames within the shot and across the shots explicitly, more discriminated and robust information could be captured. Various experiment results verify the superiority of our method. \begin{figure}[tb!] \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{a_2.pdf} \caption{DSNet-AB \cite{zhu2020dsnet}} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{b_2.pdf} \caption{MHSCNet} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{c.pdf} \caption{DSNet-AF \cite{zhu2020dsnet}} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{d.pdf} \caption{MHSCNet} \end{subfigure} \caption{Qualitative results on the SumMe benchmark \cite{gygli2014creating} and the TVSum benchmark \cite{song2015tvsum}: (a) DSNet-AF \cite{zhu2020dsnet}, (b) MHSCNet for video number 25 on the SumMe benchmark, (c) DSNet-AF \cite{zhu2020dsnet}, (d) MHSCNet for video number 34 on the TVSum benchmark. The ground-truth importance scores are shown as gray background and the selected subset shots are shown as red background. Five selected frames are randomly shown for the summary results.} \label{vis} \end{figure} \noindent \textbf{Qualitative Analysis.} \ We present the groundtruth importance scores and the selected frames generated by DSNet \cite{zhu2020dsnet} and MHSCNet in Fig. \ref{vis}. To visualize the summaries, we randomly sample five frames from the selected key shot summaries. The gray bars represent the groundtruth importance scores and the red bars represent the selected summary. Compared to DSNet-AF \cite{zhu2020dsnet}, the summaries produced by MHSCNet are more diverse and informative. Moreover, the generated summary captures most of the peak regions of the groundtruth scores. This phenomenon indicates that our MHSCNet captures the meaningful short-range and long-range temporal dependency via the hierarchical ShotConv network. \subsection{Ablation Study} \noindent We conduct a series of experiments on the SumMe \cite{gygli2014creating} dataset and the TVSum \cite{song2015tvsum} datasets to better understand the proposed model and verify the contribution of each component. \noindent \textbf{Multi Modality.} \ We first study the influence from the additional modality information of our MHSCNet. Without the audio modality and the caption modality, our model also achieve the best result on the SumMe benchmark and the TVSum benchmark compared to the state-of-the-art methods. As observed in Table \ref{table1}, the second-best method equipped with the same backbone network(GoogLeNet) achieve 51.6 \% on the SumMe dataset and 64.2\% on the TVSum dataset. The results are shown in Table \ref{abl_modal}. With the help of the local information via the audio modality, the model achieves 2.5\% and 0.2\% gain on two benchmarks, respectively. Similarly, the caption modality provides complementary global information which enhances the performance by 1.8\% and 0.3\% for our model. Furthermore, our model combining the audio modality and the caption modality can achieve a higher F-score. \begin{table}[ht!] \footnotesize \caption{Ablation study for various information of the modality in MHSCNet with F-score evaluation metric on the SumMe \cite{gygli2014creating} benchmark and the TVSum \cite{song2015tvsum} benchmark.} \centering \setlength\tabcolsep{16pt}{ \begin{tabular}{lcc} \toprule[0.75pt] Modality & SumMe & TVSum\\ \hline\hline Image & 52.0 & 68.0 \\ Image+Audio & 54.5 & 68.2 \\ Image+Caption & 53.8 & 68.3 \\ \hline Ours (Image+Audio+Caption) & \textbf{55.3} & \textbf{69.3} \\ \toprule[0.75pt] \end{tabular}} \label{abl_modal} \end{table} \noindent \textbf{Hierarchical ShotConv Network.} \ Then, we study the influence at different scales of the shots in the hierarchical ShotConv network and show the results in Table \ref{abl_scale}. Compared to the double-scale ShotConv networks, the single scale ShotConv networks perform worse. However, the network with the single middle-scale shot convolution still achieves the competitive performance on the SumMe dataset and the TVSum dataset due to the split shot length being matched with the initial shot length of the video. Similarly, the network with a single long scale shot convolution obtains a 67.6\% F-score which is slightly lower than the best results of the double scale networks. Moreover, we observe that a double scale network such as M+S drops the performance compared to a single scale network such as M. The negative transfer occasionally happen in the double scale network. However, our MHSCNet equipped with three scales can achieve the best results due to the strong adaptivity to the video length. \begin{table}[h!] \footnotesize \caption{Ablation study on our hierarchical ShotConv network with F-score metric on SumMe\cite{gygli2014creating} and TVSum \cite{song2015tvsum} benchmarks. L, M and S means long-scale, middle-scale and short-scale ShotConv module, respectively.} \centering \setlength\tabcolsep{25pt}{ \begin{tabular}{lcc} \toprule[0.75pt] Scale & SumMe & TVSum\\ \hline\hline L & 52.2 & 67.6 \\ M & 54.1 & 67.2 \\ S & 52.1 & 65.4 \\ L+M &54.4 & 67.5 \\ L+S &53.0 & 66.0 \\ M+S & 53.6 & 67.7 \\ \hline Ours (L+M+S) & \textbf{55.3} & \textbf{69.3} \\ \toprule[0.75pt] \end{tabular}} \label{abl_scale} \end{table} \noindent \textbf{The Layers Number.} \ Besides, we provide a ablation study for the layers number of the hierarchical ShotConv network. Benefitting from the cross-shot padding and the stacked hierarchical ShotConv network, the global information will propagate within the shot implicitly as mentioned in main text Sec. \ref{3.4.3}. The quantitative comparison can be found in Table \ref{abl_layer}. Note that the results with respect to number of layer 1 represent the results without any global information propagation inner the shot. We observe that the deeper hierarchical ShotConv networks (1 $\rightarrow$ 4) achieve the better performance. When the number of layers is 4, the global information from all shots can be passed within one shot. Due to the overfitting problem, the deeper hierarchical ShotConv networks (4 $\rightarrow$ 6) lead to the worse results. In addition, we also show the qualitative results corresponding to the number of hierarchical ShotConv networks in Fig. \ref{abl_vis_2}. In each illustrated summary, we visualize 5 sampled frames from the highlighted predicted frames. \begin{figure}[tb!] \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b1.pdf} \caption{One Layer, $F$-score = 66.1} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b2.pdf} \caption{Two Layers, $F$-score = 70.8} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b3.pdf} \caption{Three Layers, $F$-score = 71.2} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b4.pdf} \caption{Four Layers, $F$-score = 74.0} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b5.pdf} \caption{Five Layers, $F$-score = 73.1} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b6.pdf} \caption{Six Layers, $F$-score = 67.4} \end{subfigure} \caption{Qualitative results on the the TVSum benchmark \cite{song2015tvsum}: (a) - (f) : one - six layers which represent the stacked hierarchical ShotConv network for video number 1 on the TVSum benchmark. The ground-truth importance scores are shown as gray background and the selected subset shots are shown as red background. Five selected frames are randomly shown for the summary results.} \label{abl_vis_2} \end{figure} \begin{table}[tb!] \footnotesize \caption{Ablation study for the layers number $L$ of the hierarchical ShotConv network with F-score (\%) evaluation metric, parameters (M) and runtime (ms).} \centering \setlength\tabcolsep{13pt}{ \begin{tabular}{lcccc} \toprule[0.75pt] $L$ & SumMe & TVSum &Params &Runtime \\ \hline\hline 1 & 50.8 & 66.1 &\textbf{33.8} &\textbf{6.6} \\ 2 & 51.9 & 67.1 &67.4 &12.1 \\ 3 & 52.3 & 68.2 &101.0 &17.7 \\ 5 & 53.5 & 68.5 &168.2 &32.8 \\ 6 & 51.8 & 67.9 &201.7 &36.3 \\\hline 4 & \textbf{55.3} & \textbf{69.3} &134.6 &23.7 \\ \toprule[0.75pt] \end{tabular}} \label{abl_layer} \end{table} \noindent \textbf{Cross-shot Padding Ratio.} \ Last, we study the cross-shot padding ratio for the hierarchical ShotConv network. With a larger padding ratio (0 $\rightarrow$ 0.05), the performance of the F-score on the two benchmarks raises accordingly. When the larger padding ratio is considered (0.05 $\rightarrow$ 0.09), the performance drop accordingly. We think the reason is that the larger padding ratio reduces the weight of the initial frame within the shot. We also observe that the network without the cross-shot padding achieve the worst results because of the lack of the global information propagation. \begin{figure}[h!] \centering \includegraphics[width=0.7\linewidth]{abl.pdf} \caption{Ablation study of the cross-shot padding ratio on the SumMe\cite{gygli2014creating} benchmark and the TVSum \cite{song2015tvsum} benchmark. } \label{abl_ratio} \end{figure} \section{Conclusion} In this paper, we propose a novel multimodal framework with a hierarchical ShotConv network for video summarization. Unlike existing unimodal or bimodal methods which only adopt visual cues with one additional modal information to construct the frame representation, our MHSCNet exploits the caption and the audio modality to enhance the frame-wise representation and explore the semantic relations. To evaluate the frame-level importance score in a fair and comprehensive manner, we further propose a hierarchical ShotConv network with the cross-shot padding mechanism to model the frame-to-frame interaction both within and across the shot. The hierarchical ShotConv network generates an adaptive shot-aware representation that captures the short-range temporal dependency explicitly and long-range temporal dependency implicitly. Extensive experimental results on multiple datasets with comprehensive evaluation criteria demonstrate the superiority of our approach over most existing state-of-the-art methods. \bibliographystyle{ACM-Reference-Format} \section{Introduction} \label{sec:intro} The explosive ubiquity of digital photography brings a phenomenal surge in videos captured every day. Confronted with the enormous amount of video data, apparently, it is impractical for end-user to browse the videos completely to understand the content. To address this issue, video summarization has gained increasing attention to figure out how to browse, manage and retrieve videos efficiently \cite{mahasseni2017unsupervised,yuan2019cycle,zhang2016summary}. The objective of video summarization is to create a short synopsis that preserves the most important and relevant content of the original video with minimal redundancy \cite{zhu2020dsnet,rochan2018video,zhang2018retrospective}. \begin{figure}[t!] \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{ours_simple.pdf} \caption{MHSCNet (ours)} \label{ours_simple} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.4\textwidth]{other_simple.pdf} \caption{Previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip}} \label{other_simple} \end{subfigure} \caption{(a) The proposed MHSCNet builds adaptive shot-aware frame-level representation by modeling the frame-to-frame interaction and incorporating the hierarchical shot-level representation into the frame-level representation. (b) In contrast, previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip} utilize different temporal modeling layers (LSTM, Attention, GCN, etc.) to produce the frame-level representation containing the long-range temporal dependency.} \label{simple_compare} \end{figure} Typically, the most recent video summarizers follow the three steps: 1) video segmentation, 2) importance score prediction, and 3) key shot selection. Regarding video segmentation, KTS (Kernel Temporal Segmentation) is widely used to segment the coherent frame sequence into several shots \cite{potapov2014category}. As for key shot selection, it is tackled by 0-1 knapsack algorithm to generate a key-shot based summary. The most challenging step is importance score prediction, which is designed to highlight the most important parts according to the predicted scores for the video content. Previous works have devised different approaches to cope with this key challenge. They can be roughly categorized into unsupervised \cite{jung2019discriminative,zhou2018deep}, weakly-supervised \cite{panda2017weakly,kim2014reconstructing} and supervised methods \cite{gygli2015video,zhang2016summary}. Recent unsupervised methods attempt to reconstruct the video based on the key-shot summary generated by the frames' importance score. \cite{mahasseni2017unsupervised,he2019unsupervised}. Also, the diversity and sparsity of output summary \cite{panda2017collaborative,panda2017diversity,rochan2019video,yuan2019cycle,panda2017multi} are introduced as repelling regularizers in the training loss. For the weakly-supervised methods, the video-level metadata, such as video titles \cite{panda2017weakly,zeng2016generation}, web priors \cite{cai2018weakly,chu2015video} and video categories \cite{potapov2014category}, work as the auxiliary information to put additional constraints for video summarizers. Although unsupervised and weakly-supervised methods have reached promising performance, the supervised methods always perform better benefiting from the manually annotated dataset. To manipulate video frames for supervised video summarizers, it is desirable to handle temporal dependency among video frames effectively. To achieve this, recent efforts exploited LSTM units \cite{zhang2018retrospective,zhang2016video,zhao2018hsa} to capture long-range temporal dependency among video frames. Besides, some methods \cite{he2019unsupervised,fajtl2018summarizing,liu2020transforming} resorted to the attention mechanism to propagate the temporal dependency information in the frame level. To better perform relationship reasoning among the frames, Park et al. \cite{park2020sumgraph} introduced a recursive graph structure and achieved the state-of-the-art performance. Apart from the unimodal approaches, some bimodal methods \cite{palaskar2019multimodal,libovicky2018multimodal} extract important frame by fusing the textual as well as video metadata and remarkable performance could be achieved. Recently, the work \cite{narasimhan2021clip} proposed a language-guided bimodal transformer via fusing information across the video and language modalities. Despite the promising success, existing video summarization methods still encounter two major challenges. First, \textit{how to construct a more powerful and robust frame-wise representation for video summarization?} Essentially, most of the above methods \cite{narasimhan2021clip,zhu2020dsnet,park2020sumgraph} only adopted visual cues or textual video metadata to construct the frame-wise representation. However, almost all videos include an audio track, which could provide vital complementary information for the frame-wise representation. For instance, in a scene when a man has a conversation with a woman, we only see the two people sit aside but the audio helps us to understand the dialogue content. Meanwhile, a caption usually highlights the subject of a video, which can help us to select the relevant shots. Therefore, it is critical for a model to consider the multimodal information to construct a more powerful and robust frame-wise representation for video summarization. Second, \textit{how to predict the frame-level importance score in a fair and comprehensive manner?} The previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip} mainly focused on extracting the long-range temporal dependencies to predict the importance score of the frames from the global view, which inevitably overlooks the important frames within the local shot. As we know, video temporal structures are intrinsically layered, i.e., a video is composed of shots, and a shot is composed of several frames. Suppose a shot of blowing candles consists of three sample frames which represent taking out the lighter, lighting candles, and blowing out candles, respectively. Considering the visual coherence of the key-shot summary, the importance scores of the three consecutive frames will influence each other internally. Thus, lacking in modeling the frame-to-frame interaction within the shot the previous methods unfairly evaluate the frame importance score and generate inaccurate summary results. To address the above challenges, we present a multimodal hierarchical shot-aware convolutional network, referred to \textbf{MHSCNet}, to exhaustively exploit the available modality information of the video to enhance the frame representation, and model the frame-to-frame interactions explicitly within the shot to pay attention to the important frames both in the local and global view. The shot-aware network means that the inner operation among the network is defined at the shot level. As illustrated in Fig. \ref{Fig_framework}, it first encodes the multimodal inputs individually with pre-trained models to generate separate representations and then aggregates them hierarchically to explore the semantic relations. Subsequently, to answer the first challenge, MHSCNet fully exploits the modality information including the audio and the caption of the video to construct a robust and informative frame-wise representation. Moreover, to answer the second challenge, we design a hierarchical shot-aware convolutional network with the progressive cross-shot padding mechanism to generate an adaptive shot-aware representation, which contains both short-range and long-range temporal dependency. As illustrated in Fig. \ref{ours_simple}, MHSCNet builds the adaptive shot-aware frame-level representation which perceives shot-level information by modeling the frame-to-frame interaction in the hierarchical shot-level and incorporating the shot-level representation into the frame-wise representation. In contrast, previous approaches \cite{fajtl2018summarizing,park2020sumgraph,narasimhan2021clip} generally investigated the long-range temporal dependency to construct the frame-wise representation by different temporal modeling layers (LSTM, Attention, GCN, etc.) leading to ignoring the important frames in the local view. Benefiting from the adaptive shot-aware representation, our MHSCNet can evaluate the frame-level importance score in a fair and comprehensive view. The contributions of this work are summarized as follows: - We propose a multimodal framework to fully explore the semantic relations among the modalities and construct a powerful and robust frame-wise representation for video summarization. - We design a hierarchical ShotConv network with the cross-shot padding mechanism to produce the adaptive shot-aware representation, which offers a fair and comprehensive view to predict the importance scores of the frames. Due to the hierarchical design, the shot-aware representation containing both the short-range and long-range temporal dependency. - The experiment results indicate that our proposed method outperforms existing state-of-the-art methods on two benchmarks. Besides, we also conduct extensive ablation studies and detailed analyses to verify the effectiveness of our method. Our code will be available for further comparison. \section{Related Work} \label{sec:rw} Given an input video, video summarization intends to generate a shortened version that contains the most important information, resulting in outputs such as video synopses \cite{pritch2007webcam}, time-lapses \cite{joshi2015real,poleg2015egosampling,kopf2014first}, montages \cite{kang2006space,sun2014salient}, or storyboards \cite{liu2010hierarchical,lee2012discovering}. Early works in video summarization primarily relied on hand-crafted heuristics \cite{kim2014reconstructing,lee2012discovering,lu2013story,ngo2003automatic,panda2017collaborative,khosla2013large}, including importance, representativeness and diversity to select representative frames to produce a summary video. Nowadays, deep neural networks for video summarization have achieved great success. Focusing on the utilized modality information, we can briefly outline two categories of video summarization methods: 1) unimodal methods, 2) bimodal methods. \subsection{Unimodal Video Summarization} In the past, most researchers \cite{zhang2018retrospective,zhang2016video,zhao2018hsa} utilized the unimodal visual information to construct the frame representation and resorted to the recurrent neural network like LSTM to exploit the long-range temporal dependency. Although LSTM-based approaches have been applied to video data, recurrent operations are sequential, limiting the processing all the frames simultaneously. To tackle this problem, Fajtl et al. \cite{fajtl2018summarizing} proposed a self-attention mechanism based network for video summarization which executes the entire sequence to sequence transformation. He et al. \cite{he2019unsupervised} built their framework upon a generative adversarial network which is composed of the self-attention module. To capture multiple concepts, Liu et al. \cite{liu2020transforming} introduced the multi-head attention mechanism. Rochan et al. \cite{rochan2018video} constructed a fully convolutional sequence model and formulated video summarization as a sequence labeling problem. Different from these methods, Zhu et al. \cite{zhu2020dsnet} proposed a Detect-to-Summarize network which predicted the corresponding importance scores and temporal locations of segments via defining video summarization problem as a temporal interest detection process. Park et al. \cite{park2020sumgraph} utilized a recursive graph to perform relationship reasoning among frames. However, these unimodal approaches only adopt the visual modality information, leading to the insufficient frame representation. \subsection{Bimodal Video Summarization} Unlike unimodal video summarization methods, existing bimodal video summarization methods fused additional audio or caption information to enhance the frame representation. Zhou et al. \cite{zhou2018video} combined the category-related information with the frame feature. Huang et al. \cite{huang2021gpt2mvs} designed an bimodal framework consisting of a contextualized video summary controller and bimodal attention mechanisms to explore the power of text modality. Yuan et al. \cite{yuan2017video} constructed a latent subspace to find the meaningful frame-level feature with side semantic information. Zhao et al. \cite{zhao2021audiovisual} developed an AudioVisual Recurrent Network to jointly exploit the audio and visual information for the video summarization. Recently, Narasimhan et al. \cite{narasimhan2021clip} proposed a language-guided bimodal transformer for both generic and query-focused video summarization. While these methods resort to the bimodal information and the long-range temporal dependency, ignoring the potential of fully incorporating available multimodal information and the short-range temporal dependency they miss the key-shot summary. To construct a powerful and robust representation and fully consider the locally-global side information, we build our MHSCNet with both the audio and the caption modal information. \section{Methodology} \label{sec:method} \begin{figure*}[tb!] \centering{ \includegraphics[width=0.80\textwidth]{framework.pdf}} \caption{Overview of our multimodal hierarchical shot-aware convolutional network (MHSCNet), consisting of the multimodal feature fusion and a hierarchical ShotConv network. In inference, the summary of the original video is generated by transforming frame scores to shot scores and a 0-1 knapsack algorithm is used to select high-scoring shots in the limited summary length.} \label{Fig_framework} \end{figure*} \subsection{Motivation and Approach Overview} Manual video summarization generally extracts key information by means of video segmentation and subsequent ordinal or unordered shot-shot pairwise significance ordering \cite{gygli2014creating,song2015tvsum}, which indicates that the frames within the same shot will interact with each other and all determined to be chosen or not for the key-frame summary. Meanwhile, the human annotator will consider the subject of the video via the audio and the textual information to judge the relevance between the frame and the video. The aforementioned methods mainly consider the long-range temporal dependency to enhance the frame-level representation via little additional modality information, ignoring the mutual relations between the contiguous frames within the same shot. Motivated by the manual annotation procedure, we propose a hierarchical ShotConv network that takes the aggregated multimodal feature as input and outputs the adaptive shot-aware representation preserving the short-range temporal dependency within the shot and the long-range temporal dependency across the shot via the proposed cross-shot padding mechanism. During the inference stage, it is hard to set the fixed length of the shot due to the variability of the shot segmentation. To make our network adaptive to the variable video length, we fix the different shot lengths for our sub-ShotConv modules. Overall, our framework mainly has two modules as shown in Fig. \ref{Fig_framework}, where the first module is multimodal feature fusion, and the second module is a hierarchical ShotConv network. During the inference, the video is firstly segmented to the several shots by the Kernel Temporal Segmentation algorithm \cite{potapov2014category}. Then, the importance scores of the frames are predicted by our MHSCNet. Finally, the 0-1 knapsack algorithm is utilized to generate a key-shot based summary. \subsection{Feature Extraction} Given a video $V$ as a sequence of $T$ frames, we first employ CNN (e.g. GoogLeNet) without the last three layers to extract feature $F_{V} =[f_{v_{1}}, ..., f_{v_{i}}, ..., f_{v_{T}}] \in \mathbb{R}^{T\times N}$. For audio information $F_{A} =[f_{a_{1}}, ..., f_{a_{i}},$ $..., f_{a_{T}}] \in \mathbb{R}^{T\times M}$, the VGGish \cite{hershey2017cnn} pre-trained on Audioset is adopted for audio feature extraction and the sampling frequency is set for keeping the same sequence length with the frame feature. To capture generic caption information, we use the Bi-Modal Transformer \cite{iashin2020better} to generate dense video captions for the whole video. We use the CLIP \cite{radford2021learning} model to encode caption vector $F_{C} \in \mathbb{R}^{1 \times N}$ and average pooling is used to balance multiple sentences. \subsection{Multimodal Feature Fusion} \subsubsection{Audio Feature Fusion} Firstly we fuse the image feature with the audio feature to explore the semantic relations. For the i-th frame, we project the audio vector $f_{a_{i}}$ to the same embedding space with the image feature $F_{V}$ by a trainable matrix $W_{audio}$ like Eq. \ref{audio_fusion}. \begin{equation} f_{va_{i}} = f_{v_{i}} + f_{a_{i}}W_{audio} +b_{audio} \label{audio_fusion} \end{equation} where $W_{audio} \in \mathbb{R}^{M \times N}$, $b_{audio}\in \mathbb{R}^{N}$ are the trainable matrices and the trainable bias, respectively. \subsubsection{Caption Feature Fusion} After fusing the audio information, we also incorporate prior knowledge into the frame feature. To this end, we employ the multi-head attention mechanism to fuse the caption feature $F_{C}$. We regard the caption feature $F_{C}$ as both the Key and the Value and the enhanced local feature $F_{VA}$ as the Query. Therefore, the aggregated multimodal frame-level feature $F_{AM}$ can be calculated as follows: \begin{equation} F_{AM} = Concat(head_{1},...,head_{h})W^{O} \label{multi_head} \end{equation} \begin{equation} head_{j} = Attention(QW_{j}^{Q},KW_{j}^{K},VW_{j}^{V}) \label{head_cal} \end{equation} \begin{equation} Attention(Q,K,V) = softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V \label{att_cal} \end{equation} where $W^{O} \in \mathbb{R}^{N \times N}$, $W_{j}^{Q} \in \mathbb{R}^{N \times d_{k}}$, $W_{j}^{K} \in \mathbb{R}^{N \times d_{k}}$ and $W_{j}^{V} \in \mathbb{R}^{N \times d_{k}}$ are trainable matrices, $d_{k}=N/h$ and $h$ is the number of attention heads. \subsection{Hierarchical ShotConv Network} \label{3.4} \subsubsection{Cross-shot Padding \& Split} \begin{figure}[tb!] \centering \includegraphics[width=1.0\linewidth]{crossshot.pdf} \caption{The cross-shot padding \& split module.} \label{cross_shot} \end{figure} As shown in Fig. \ref{cross_shot}, we firstly pad the specified number of frames in the boundary of the each shot to maintain the consistent temporal information. Given a padding ratio $\eta$ and the shot number $S$, $\eta \times \lfloor \frac{T}{S} \rfloor$ frames from the end of the $(S$-1)-th shot will be padded at the start of the $S$-th shot. For the consistency, we pad the parts of frames from the last shot to the start of the first shot. Then a 1$\times$1 convolution is utilized to increase the channel of the frame from $N$ to $\lambda N$ and the frame-level feature $F_{CP}^{l} =[f_{cp_{1}}^{l},...,f_{cp_{i}}^{l},...,f_{cp_{(1+\eta) T}}^{l}] \in \mathbb{R}^{(1+\eta) T\times \lambda N}$ in the $l$-th layer could be obtained via the cross-shot padding mechanism. Then, we reshape the dimension from $(1+\eta) T\times \lambda N$ to $S \times (D_{S} \cdot \lambda N)$, where $D_{S} = \lfloor \frac{(1+\eta) T}{S} \rfloor$ and $\lfloor . \rfloor$ is the round down function to transform the frame-level representation to the inflated shot-level representation $F_{ISR}^{l}$. Specially, the $s$-th shot contains the features of inside frames as follows: \begin{equation} F_{ISR_{s}}^{l} = \{f_{cp_{k}}^{l}: k=(s-1)D_{S} + 1 ,...,sD_{S} ,sD_{S} +1 \} \label{shot_split} \end{equation} Lastly, an average pooling operator will be used on the dimension $D_{S}$ of the frame number to obtain the concentrated shot-level representation $F_{SR}^{l} \in \mathbb{R}^{S \times \lambda N}$. To make our network adaptive to the video length, we propose three scales cross-shot padding, and the shot number $S$ will be set to $S_{l}$, $S_{m}$, $S_{s}$ for the long scale, middle scale, and short scale to generate the corresponding representation $F_{SR_{l}}^{l}$, $F_{SR_{m}}^{l}$ and $F_{SR_{s}}^{l}$. \subsubsection{Inner ShotConv Module} As shown in Fig. \ref{shotconv}, we propose an inner shot convolution module to model the frame interaction within the shot. The inner shot convolution module is composed of four convolution operators with different kernel sizes and other postprocessing modules. The strides of the four convolution operators are set to 4. The 5$\times$5 and 7$\times$7 convolution operators are implemented by the dilated convolution \cite{yu2015multi}. We firstly perform these four convolution operators on the shot-level feature $F_{SR}^{l}$. Then we will concatenate them in the channel dimension and feed it to a 1$\times$1 convolution layer to reduce the channel dimension from $\lambda N$ to $N$. Each shot-level representation will be assigned to the frames in the same shot to get the enhanced frame-level representation $F_{EF}^{l}$ via the expand operation. For the three scales ShotConv modules, the enhanced frame-level representation $F_{EF_{l}}^{l}$, $F_{EF_{m}}^{l}$ and $F_{EF_{s}}^{l}$ are generated for the long scale, middle scale and short scale, respectively. \begin{figure}[tb!] \centering \includegraphics[width=1.0\linewidth]{shotconv.pdf} \caption{The inner shot convolution module.} \label{shotconv} \end{figure} As shown in Fig. \ref{shotconv}, the adaptive shot-aware representation $F_{ASF}^{l} \in \mathbb{R}^{T \times N}$ is obtained after the multi-scale ShotConv modules by Eq. \ref{asf_add}. \begin{equation} F_{ASF}^{l} = F_{EF_{l}}^{l} + F_{EF_{m}}^{l} + F_{EF_{s}}^{l} + F_{AM}^{l} \label{asf_add} \end{equation} where $F_{AM}^{l}$ will be replaced by $F_{ASF}^{l-1}$ except the first layer. We stack $L$ layers of the hierarchical ShotConv network and the output from each layer will be the input of the next layer. \subsubsection{Implicit Global Information Propagation} \label{3.4.3} In each layer of the ShotConv network, cross-shot padding will propagate information between two adjacent shots. Benefitting from this cross-shot padding mechanism, the frames in the local shot will learn the information from the other adjacent shot. After several stacked ShotConv networks, the frames in the local shot can learn the information from the whole video since the global information will be passed in the deeper layers. Formally, the global information will be passed in a local shot when the number of stacked layers and the shot number meet the criteria as illustrated in Eq. \ref{global_condition}. \begin{equation} L+1 >= S_{l} \label{global_condition} \end{equation} where $L$ is the number of the stacked hierarchical ShotConv network and $S_{l}$ is the shot number of the long scale ShotConv module. \subsection{Loss Function} We apply the focal loss \cite{lin2017focal} for keyframe classification, which deals with the imbalance between the number of keyframes and the number of background frames. Moreover, the difference of the hard/easy sample will be highlighted. The loss is defined by \begin{equation} \begin{split} \mathcal{L} = -\frac{1}{T}\sum_{t=1}^{T}[\alpha y_{t}(1-p_{t})^{\gamma}\log(p_{t})+ (1-\alpha)(1-y_{t})p_{t}^{\gamma}\log(1-p_{t}))] \end{split} \label{loss} \end{equation} where $y_{t}$ is the groundtruth label of the $t$-th frame. $\alpha$ and $\gamma$ are the re-weighting factor. $p_{t}$ is the prediction of our proposed model. \section{Experiments} \subsection{Experiment Setup} \noindent \textbf{Implementation Details.} \ To ensure fair comparisons with prior works \cite{zhu2020dsnet,park2020sumgraph,rochan2018video}, we use the ImageNet-pretrained GoogleNet \cite{szegedy2015going} for frame feature extraction, where the 1024-d feature are extracted after the 'pool5' layer. The dimension of the audio feature extracted by the pre-trained VGGish \cite{hershey2017cnn} is 128. For the generated multiple captions by the BMT \cite{iashin2020better}, we use CLIP \cite{radford2021learning} to encode them and concatenate the embedding in the channel dimension. The dimension of the first fully-connected layer is 128. The number of multi-head in the caption feature fusion module is set to 32. We set $ S_{l} = 5$, $S_{m}=10$, $S_{s}=15$, $\eta=0.05$, $L = 4$ and $\lambda = 8$ in Sec. \ref{3.4}. $\alpha$ and $\gamma$ in the focal loss are set to 0.25 and 2. All the experiment results are reported in 5-fold cross validation. Adam works as the optimizer with an initial learning rate of $1e^{-4}$, and a weight decay of $1e^{-5}$. All the comparative models are trained for 100 epochs in total and tested on the same machine, which has a single NVIDIA GeForce Titan V GPU and Intel Core i7-8700K CPU. \noindent \textbf{Datasets.} \ We evaluate the performance of our approach on two standard video summarization datasets: SumMe \cite{gygli2014creating} and TVSum \cite{song2015tvsum}. SumMe dataset contains 25 personal videos obtained from YouTube which covers multiple categories such as cooking and sports. Each video ranges from 1 to 6 minutes and each reference summary is generated by 15 to 18 persons. TVSum dataset consists of 50 YouTube videos which contain 10 categories including changing vehicle tire, parade, and dog show. The video lengths of TVSum which are a little longer than SumMe vary from 2 to 10 minutes. Beside, two additional datasets including the YouTube \cite{de2011vsumm} and the OVP \cite{de2011vsumm} are used to augment the training data. The OVP dataset contains 50 video sequences and the YouTube dataset consists of 40 video sequences. In detail, we follow \cite{zhang2016video} to sample the frame at 2 fps to handle temporal redundancy and reduce computation. The pre-processed datasets including the data splits can be found in the URLs\footnote{https://github.com/li-plus/DSNet;https://github.com/KaiyangZhou/pytorch-vsumm-reinforce} provided by the previous researchers \cite{zhu2020dsnet,zhou2017reinforcevsumm}. \noindent \textbf{Data configuration.} \ We apply three data configurations adopted in \cite{zhang2016summary,zhou2018deep,rochan2018video} within our experiments including standard data setting, augmented data setting, and transfer data setting. \noindent \textbf{Standard}: In standard data setting, the training and testing data come from the same benchmark. Typically, we randomly select 80\% of videos for training and validation while 20\% of videos for testing. The results of the selected dataset split stay the same with previous works \cite{zhang2016video,zhu2020dsnet}. \noindent \textbf{Augment}: In the augment data setting, the videos of YouTube and OVP datasets are utilized to augment the training data. Thanks to more abundant training data, the model performance will be improved in this data setting \cite{rochan2018video,zhang2018retrospective}. \noindent \textbf{Transfer}: In the transfer data setting, the videos of three datasets will be used as the training data and the videos of one another dataset (e.g. SumMe or TVSum) will be used as the testing data. For reliable comparisons, all the experiments are conducted five times and the average performance is reported in our paper. We report the results at F-score in all of the settings. Experiments at other metrics and the ablation study are all conducted in a standard supervision manner. \noindent \textbf{Evaluation Criteria.} \ Following the previous works \cite{song2015tvsum,gygli2015video,otani2016video}, F-score is used to evaluate the agreement between the generated summary and the user summary(GT). Let $Y$ and $Y^{*}$ be the ground-truth summary and the generated keyshot summary, respectively. The precision and the recall are calculated as follows: \begin{equation} P = \frac{length(Y \cap Y^{*})}{length(Y^{*})} , R = \frac{length(Y \cap Y^{*})}{length(Y)} \label{recall} \end{equation} We compute the F-score to evaluate the quality of the generated keyshot summary based on the precision and the recall like Eq. \ref{fscore}. \begin{equation} F = \frac{2\times P \times R}{P+R} \label{fscore} \end{equation} For datasets with multiple human annotations, we follow standard approaches described in \cite{mahasseni2017unsupervised,gygli2015video,song2015tvsum,zhang2016video} to calculate the metrics for the videos. Similar to \cite{zhou2018deep,zhu2020dsnet}, we evaluate the diversity score of the generated summary on the SumMe and TVSum datasets. In recent work, Otani et al. \cite{otani2019rethinking} showed that randomly generated summaries achieve a similar F-score with the state-of-the-art methods. Therefore, two rank correlation coefficients, precisely Kendall's $\tau$ \cite{kendall1945treatment} and Spearman's $\rho$ \cite{zwillinger1999crc} are also used to evaluate the effectiveness of our model. \subsection{Comparisons to the State-of-the-Arts} \begin{table*}[htbp!] \footnotesize \caption{Comparisons of $F$-Score (\%) and parameters (in million) with peer supervised techniques on SumMe \cite{gygli2014creating} and TVSum \cite{song2015tvsum}, with various data configurations including standard data, augmented data, and transfer data settings. The scores in bold indicate the best values. "-" denotes that the results are not available. Bimodal approaches are marked with $\diamond$. For some unsupervised learning methods, we report their results in a supervised manner. As CLIP-It, we report their results with two different backbone networks. For fair comparisons with previous methods, we mainly adopt GoogLeNet features.} \centering \setlength\tabcolsep{11pt}{ \begin{tabular}{lcccccccc} \toprule[0.75pt \multicolumn{1}{l}{\multirow{2}{*}{Method}} &\multicolumn{1}{c}{\multirow{2}{*}{Backbone}} & \multicolumn{3}{c}{SumMe} & \multicolumn{3}{c}{TVSum} & \multicolumn{1}{c}{\multirow{2}{*}{Params}} \\ \cline{3-8} \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{Standard} & \multicolumn{1}{l}{Augment} & \multicolumn{1}{l}{Transfer} & \multicolumn{1}{l}{Standard} & \multicolumn{1}{l}{Augment} & \multicolumn{1}{l}{Transfer} \\ \hline\hline Random summary \cite{otani2019rethinking} & - & 41.0 & - & - & 57.0 & - & - & 0.0 \\ SUM-GAN \cite{mahasseni2017unsupervised} & GoogLeNet \cite{szegedy2015going} & 41.7 & 43.6 & - & 56.3 & 61.2 & - &295.9 \\ SUM-FCN \cite{rochan2018video} &GoogLeNet \cite{szegedy2015going} & 47.5 & 51.1 & 44.1 & 56.8 & 59.2 & 58.2 &116.5 \\ SUM-DeepLab \cite{rochan2018video} &GoogLeNet \cite{szegedy2015going} & 48.8 & 50.2 & 45.0 & 58.4 & 59.8 & 58.9 & - \\ DR-DSN \cite{zhou2018deep} &GoogLeNet \cite{szegedy2015going} & 42.1 & 43.9 & 42.6 & 58.1 & 59.8 & 58.9 &2.6 \\ ACGAN \cite{he2019unsupervised} &GoogLeNet \cite{szegedy2015going} & 47.2 & - & - & 59.4 & - & - & - \\ $\diamond$DQSN \cite{zhou2018video} &GoogLeNet \cite{szegedy2015going} & - & - & - & 58.6 & - & - & - \\ $\diamond$DSSE \cite{yuan2017video} &AlexNet \cite{krizhevsky2012imagenet} & - & - & - & 57.0 & - & - & - \\ $\diamond$AVRN \cite{zhao2021audiovisual} &GoogLeNet \cite{szegedy2015going} & 44.1 & 44.9 & 43.2 & 59.7 & 60.5 & 58.7 & - \\ VASNet \cite{fajtl2018summarizing} &GoogLeNet \cite{szegedy2015going} & 49.7 & 51.1 & - & 61.4 & 62.4 & - &7.4 \\ DSNet-AB \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 50.2 & 50.7 & 46.5 & 62.1 & 63.9 & 59.4 &8.5 \\ DSNet-AF \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 51.2 & 53.3 & 47.6 & 61.9 & 62.2 & 58.0 & 4.3 \\ MC-VSA \cite{liu2020transforming} &GoogLeNet \cite{szegedy2015going} & 51.6 & 53.0 & 48.1 & 63.7 & 64.0 & 59.5 &- \\ SumGraph \cite{park2020sumgraph} &GoogLeNet \cite{szegedy2015going} & 51.4 & 52.9 & 48.7 & 63.9 & 65.8 & 60.5 & 5.5 \\ $\diamond {\rm CLIP-It}^{*}$ \cite{narasimhan2021clip} &GoogLeNet \cite{szegedy2015going} & 51.6 & 53.5 & 49.4 & 64.2 & 66.3 & 61.3 & - \\ $\diamond {\rm CLIP-It}$ \cite{narasimhan2021clip} &CLIP-ViT-B/32 \cite{radford2021learning} & 54.2 & 56.4 & \textbf{51.9} & 66.3 & 69.0 & \textbf{65.5} & - \\\hline MHSCNet (ours) &GoogLeNet \cite{szegedy2015going} & \textbf{55.3} &\textbf{56.9} & 49.8 & \textbf{69.3} & \textbf{69.4} & 61.0 &134.6 \\ \toprule[0.750pt \end{tabular}} \label{table1} \end{table*} \noindent \textbf{Quantitative comparison.} \ To validate the effectiveness of our proposed method on the video summarization problem, we quantitatively compare our proposed model against other state-of-the-art methods including unimodal methods and bimodal methods on the SumMe and TVSum datasets. As reported in Table \ref{table1}, our MHSCNet achieves a significant improvement ranging from 3.7\% to 13.6\% on F-score in the standard data setting compared to the state-of-the-art methods equipped with the same backbone network. The unimodal methods can be roughly classified into four categories: 1) LSTM-based methods(dppLSTM \cite{zhang2016video}, SUM-GAN \cite{mahasseni2017unsupervised}, DR-DSN \cite{zhou2018deep}); 2) Attention-based methods(ACGAN \cite{he2019unsupervised}, VASNet \cite{fajtl2018summarizing}, MC-VSA \cite{liu2020transforming}, DASN \cite{fu2021video}); 3) Convolution-based methods(SUM-FCN \cite{rochan2018video}, SUM-DeepLab \cite{rochan2018video}); 4) Other methods(DSNet-AB \cite{zhu2020dsnet}, DSNet-AF \cite{zhu2020dsnet}, SumGraph \cite{park2020sumgraph}). Random summary \cite{otani2019rethinking} is generated by means of random importance scores and KTS video segmentation algorithm \cite{potapov2014category}. Specifically, existing LSTM-based methods achieve better performance on the TVSum dataset and relatively worse performance on the SumMe dataset against other methods. Such poorly-balanced performance indicates that the long-range dependence feature extracted by LSTM-based networks cannot work on the SumMe dataset and the relation across the shots is weak on the SumMe dataset. Different from the LSTM-based methods, attention-based methods perform slightly better on the SumMe dataset. Benefitting from the designed structure, the attention-based methods can implicitly capture the local information within the shot. Similarly, convolution-based methods extract the short-range temporal dependency within the shot implicitly. Moreover, DSNet formulate video summarization as an interest detection problem. The proposed center-ness score loss leads the model to learn more information within the shot. Besides, SumGraph perform better than DSNet by explicitly modeling relationships among the frames using a recursive graph. Benefitting from the multimodal informantion and adaptive shot-aware representation, our MHSCNet outperforms SumGraph by a wide margin. Besides the unimodal methods, the bimodal methods equipped with the same backbone (GoogleNet) drop the performance ranging from 3.7\% to 12.3\% in the standard data setting compared to our proposed MHSCNet because they overlook the inner-shot information and involve extra modality information excessively. We also analyze the diversity to verify the generated summary by MHSCNet contains more diverse key-frames. As shown in Table \ref{table2}, our method MHSCNet outperforms LSTM-based methods (dppLSTM \cite{zhang2016video} and DR-DSN \cite{zhou2018deep}) and other methods (DSNet-AF \cite{zhu2020dsnet} and DSNet-AB \cite{zhu2020dsnet}) with improvement ranging from 5.3\% to 14.1\%. \begin{table}[tb!] \footnotesize \caption{The diversity scores of generated summaries on the SumMe\cite{gygli2014creating} and TVSum \cite{song2015tvsum} datasets.} \centering \setlength\tabcolsep{12pt}{ \begin{tabular}{lccc} \toprule[0.75pt] Method &Backbone & SumMe & TVSum\\ \hline\hline dppLSTM \cite{zhang2016video} &GoogLeNet \cite{szegedy2015going} & 0.591 & 0.463 \\ DR-DSN \cite{zhou2018deep} &GoogLeNet \cite{szegedy2015going} & 0.594 & 0.464 \\ DSNet-AF \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 0.642 & 0.476 \\ DSNet-AB \cite{zhu2020dsnet} &GoogLeNet \cite{szegedy2015going} & 0.664 & 0.477 \\ \hline MHSCNet (ours) &GoogLeNet \cite{szegedy2015going} & \textbf{0.732} & \textbf{0.530} \\ \toprule[0.75pt] \end{tabular}} \label{table2} \end{table} \begin{table}[tb!] \footnotesize \caption{Kendall's $\tau$ \cite{kendall1945treatment} and Spearman's $\rho$ \cite{zwillinger1999crc} correlation coefficients computed on the TVSum benchmark \cite{song2015tvsum}.} \centering \setlength\tabcolsep{6pt}{ \begin{tabular}{lccc} \toprule[0.75pt] Method &Backbone & Kendall's $\tau$ & Spearman's $\rho$\\ \hline\hline Random summary \cite{otani2019rethinking} &GoogLeNet \cite{szegedy2015going} & 0.000 & 0.000 \\ dppLSTM \cite{zhang2016video} &GoogLeNet \cite{szegedy2015going} & 0.042 & 0.055 \\ DR-DSN \cite{zhou2018deep} &GoogLeNet \cite{szegedy2015going} & 0.020 & 0.026 \\ HSA-RNN \cite{zhao2018hsa} &VGG16 \cite{simonyan2014very} & 0.082 & 0.088 \\ SumGraph \cite{park2020sumgraph} &GoogLeNet \cite{szegedy2015going} & 0.094 & 0.138 \\ MC-VSA \cite{liu2020transforming} &GoogLeNet \cite{szegedy2015going} & 0.116 & 0.142 \\ AVRN \cite{zhao2021audiovisual} &GoogLeNet \cite{szegedy2015going} & 0.096 & 0.104 \\ DASN \cite{fu2021video} &GoogLeNet \cite{szegedy2015going} & 0.058 & 0.065 \\ ${\rm CLIP-It}$ \cite{narasimhan2021clip} &CLIP-ViT-B/32 \cite{radford2021learning} &0.108 &0.147\\ Human \cite{otani2019rethinking} & - & \textbf{0.177} & 0.204 \\\hline MHSCNet (ours) &GoogLeNet \cite{szegedy2015going} & 0.154 & \textbf{0.225} \\ \toprule[0.75pt] \end{tabular}} \label{table3} \end{table} As Otani et al.\cite{otani2019rethinking} has indicated that random summary could achieve similar results compared to the state-of-the-art methods due to the limitation of the video segmentation methods and the 0-1 knapsack algorithm. Therefore, we directly measure importance scores using two rank correlation coefficients including Kendall's $\tau$ and Spearman's $\rho$ between the predicted ordering and the human annotators' ordering. The results of quantitative comparisons are displayed in Table \ref{table3}. We observe that MHSCNet surpasses the state-of-the-art methods including the unimodal methods and the bimodal methods with improvements of 3.8\% and 7.8\% at least on Kendall's $\tau$ and Spearman's $\rho$, respectively. From the result of human annotators, we infer that our model possesses comparable discrimination ability with the single experienced annotator. In summary, as MHSCNet jointly exploits the relations among the frames within the shot and across the shots explicitly, more discriminated and robust information could be captured. Various experiment results verify the superiority of our method. \begin{figure}[tb!] \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{a_2.pdf} \caption{DSNet-AB \cite{zhu2020dsnet}} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{b_2.pdf} \caption{MHSCNet} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{c.pdf} \caption{DSNet-AF \cite{zhu2020dsnet}} \end{subfigure} \begin{subfigure}{0.5\textwidth} \centering \includegraphics[width=0.8\textwidth]{d.pdf} \caption{MHSCNet} \end{subfigure} \caption{Qualitative results on the SumMe benchmark \cite{gygli2014creating} and the TVSum benchmark \cite{song2015tvsum}: (a) DSNet-AF \cite{zhu2020dsnet}, (b) MHSCNet for video number 25 on the SumMe benchmark, (c) DSNet-AF \cite{zhu2020dsnet}, (d) MHSCNet for video number 34 on the TVSum benchmark. The ground-truth importance scores are shown as gray background and the selected subset shots are shown as red background. Five selected frames are randomly shown for the summary results.} \label{vis} \end{figure} \noindent \textbf{Qualitative Analysis.} \ We present the groundtruth importance scores and the selected frames generated by DSNet \cite{zhu2020dsnet} and MHSCNet in Fig. \ref{vis}. To visualize the summaries, we randomly sample five frames from the selected key shot summaries. The gray bars represent the groundtruth importance scores and the red bars represent the selected summary. Compared to DSNet-AF \cite{zhu2020dsnet}, the summaries produced by MHSCNet are more diverse and informative. Moreover, the generated summary captures most of the peak regions of the groundtruth scores. This phenomenon indicates that our MHSCNet captures the meaningful short-range and long-range temporal dependency via the hierarchical ShotConv network. \subsection{Ablation Study} \noindent We conduct a series of experiments on the SumMe \cite{gygli2014creating} dataset and the TVSum \cite{song2015tvsum} datasets to better understand the proposed model and verify the contribution of each component. \noindent \textbf{Multi Modality.} \ We first study the influence from the additional modality information of our MHSCNet. Without the audio modality and the caption modality, our model also achieve the best result on the SumMe benchmark and the TVSum benchmark compared to the state-of-the-art methods. As observed in Table \ref{table1}, the second-best method equipped with the same backbone network(GoogLeNet) achieve 51.6 \% on the SumMe dataset and 64.2\% on the TVSum dataset. The results are shown in Table \ref{abl_modal}. With the help of the local information via the audio modality, the model achieves 2.5\% and 0.2\% gain on two benchmarks, respectively. Similarly, the caption modality provides complementary global information which enhances the performance by 1.8\% and 0.3\% for our model. Furthermore, our model combining the audio modality and the caption modality can achieve a higher F-score. \begin{table}[ht!] \footnotesize \caption{Ablation study for various information of the modality in MHSCNet with F-score evaluation metric on the SumMe \cite{gygli2014creating} benchmark and the TVSum \cite{song2015tvsum} benchmark.} \centering \setlength\tabcolsep{16pt}{ \begin{tabular}{lcc} \toprule[0.75pt] Modality & SumMe & TVSum\\ \hline\hline Image & 52.0 & 68.0 \\ Image+Audio & 54.5 & 68.2 \\ Image+Caption & 53.8 & 68.3 \\ \hline Ours (Image+Audio+Caption) & \textbf{55.3} & \textbf{69.3} \\ \toprule[0.75pt] \end{tabular}} \label{abl_modal} \end{table} \noindent \textbf{Hierarchical ShotConv Network.} \ Then, we study the influence at different scales of the shots in the hierarchical ShotConv network and show the results in Table \ref{abl_scale}. Compared to the double-scale ShotConv networks, the single scale ShotConv networks perform worse. However, the network with the single middle-scale shot convolution still achieves the competitive performance on the SumMe dataset and the TVSum dataset due to the split shot length being matched with the initial shot length of the video. Similarly, the network with a single long scale shot convolution obtains a 67.6\% F-score which is slightly lower than the best results of the double scale networks. Moreover, we observe that a double scale network such as M+S drops the performance compared to a single scale network such as M. The negative transfer occasionally happen in the double scale network. However, our MHSCNet equipped with three scales can achieve the best results due to the strong adaptivity to the video length. \begin{table}[h!] \footnotesize \caption{Ablation study on our hierarchical ShotConv network with F-score metric on SumMe\cite{gygli2014creating} and TVSum \cite{song2015tvsum} benchmarks. L, M and S means long-scale, middle-scale and short-scale ShotConv module, respectively.} \centering \setlength\tabcolsep{25pt}{ \begin{tabular}{lcc} \toprule[0.75pt] Scale & SumMe & TVSum\\ \hline\hline L & 52.2 & 67.6 \\ M & 54.1 & 67.2 \\ S & 52.1 & 65.4 \\ L+M &54.4 & 67.5 \\ L+S &53.0 & 66.0 \\ M+S & 53.6 & 67.7 \\ \hline Ours (L+M+S) & \textbf{55.3} & \textbf{69.3} \\ \toprule[0.75pt] \end{tabular}} \label{abl_scale} \end{table} \noindent \textbf{The Layers Number.} \ Besides, we provide a ablation study for the layers number of the hierarchical ShotConv network. Benefitting from the cross-shot padding and the stacked hierarchical ShotConv network, the global information will propagate within the shot implicitly as mentioned in main text Sec. \ref{3.4.3}. The quantitative comparison can be found in Table \ref{abl_layer}. Note that the results with respect to number of layer 1 represent the results without any global information propagation inner the shot. We observe that the deeper hierarchical ShotConv networks (1 $\rightarrow$ 4) achieve the better performance. When the number of layers is 4, the global information from all shots can be passed within one shot. Due to the overfitting problem, the deeper hierarchical ShotConv networks (4 $\rightarrow$ 6) lead to the worse results. In addition, we also show the qualitative results corresponding to the number of hierarchical ShotConv networks in Fig. \ref{abl_vis_2}. In each illustrated summary, we visualize 5 sampled frames from the highlighted predicted frames. \begin{figure}[tb!] \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b1.pdf} \caption{One Layer, $F$-score = 66.1} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b2.pdf} \caption{Two Layers, $F$-score = 70.8} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b3.pdf} \caption{Three Layers, $F$-score = 71.2} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b4.pdf} \caption{Four Layers, $F$-score = 74.0} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b5.pdf} \caption{Five Layers, $F$-score = 73.1} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.8\textwidth]{abl_b6.pdf} \caption{Six Layers, $F$-score = 67.4} \end{subfigure} \caption{Qualitative results on the the TVSum benchmark \cite{song2015tvsum}: (a) - (f) : one - six layers which represent the stacked hierarchical ShotConv network for video number 1 on the TVSum benchmark. The ground-truth importance scores are shown as gray background and the selected subset shots are shown as red background. Five selected frames are randomly shown for the summary results.} \label{abl_vis_2} \end{figure} \begin{table}[tb!] \footnotesize \caption{Ablation study for the layers number $L$ of the hierarchical ShotConv network with F-score (\%) evaluation metric, parameters (M) and runtime (ms).} \centering \setlength\tabcolsep{13pt}{ \begin{tabular}{lcccc} \toprule[0.75pt] $L$ & SumMe & TVSum &Params &Runtime \\ \hline\hline 1 & 50.8 & 66.1 &\textbf{33.8} &\textbf{6.6} \\ 2 & 51.9 & 67.1 &67.4 &12.1 \\ 3 & 52.3 & 68.2 &101.0 &17.7 \\ 5 & 53.5 & 68.5 &168.2 &32.8 \\ 6 & 51.8 & 67.9 &201.7 &36.3 \\\hline 4 & \textbf{55.3} & \textbf{69.3} &134.6 &23.7 \\ \toprule[0.75pt] \end{tabular}} \label{abl_layer} \end{table} \noindent \textbf{Cross-shot Padding Ratio.} \ Last, we study the cross-shot padding ratio for the hierarchical ShotConv network. With a larger padding ratio (0 $\rightarrow$ 0.05), the performance of the F-score on the two benchmarks raises accordingly. When the larger padding ratio is considered (0.05 $\rightarrow$ 0.09), the performance drop accordingly. We think the reason is that the larger padding ratio reduces the weight of the initial frame within the shot. We also observe that the network without the cross-shot padding achieve the worst results because of the lack of the global information propagation. \begin{figure}[h!] \centering \includegraphics[width=0.7\linewidth]{abl.pdf} \caption{Ablation study of the cross-shot padding ratio on the SumMe\cite{gygli2014creating} benchmark and the TVSum \cite{song2015tvsum} benchmark. } \label{abl_ratio} \end{figure} \section{Conclusion} In this paper, we propose a novel multimodal framework with a hierarchical ShotConv network for video summarization. Unlike existing unimodal or bimodal methods which only adopt visual cues with one additional modal information to construct the frame representation, our MHSCNet exploits the caption and the audio modality to enhance the frame-wise representation and explore the semantic relations. To evaluate the frame-level importance score in a fair and comprehensive manner, we further propose a hierarchical ShotConv network with the cross-shot padding mechanism to model the frame-to-frame interaction both within and across the shot. The hierarchical ShotConv network generates an adaptive shot-aware representation that captures the short-range temporal dependency explicitly and long-range temporal dependency implicitly. Extensive experimental results on multiple datasets with comprehensive evaluation criteria demonstrate the superiority of our approach over most existing state-of-the-art methods. \bibliographystyle{ACM-Reference-Format}

BkiUdxg5qdmDMTwvPmRD

\section{Introduction} Let $(S,\mathcal B)$ be a measurable space and let $(X, Y)\in S\times\left\{-1,1\right\}$ be a random couple with unknown distribution $P$. The marginal distribution of the design variable $X$ will be denoted by $\Pi$. Let $\eta(x):=\mathbb E(Y|X=x)$ be the regression function. The goal of {\it binary classification} is to predict label $Y$ based on the observation $X$. Prediction is based on a {\it classifier} - a measurable function $f:S\mapsto\left\{-1,1\right\}$. The quality of a classifier is measured in terms of its generalization error, $R(f)=\Pr\left(Y\ne f(X)\right)$. In practice, the distribution $P$ remains unknown but the learning algorithm has access to the {\it training data} - the i.i.d. sample $(X_i,Y_i), \ i=1\ldots n$ from $P$. It often happens that the cost of obtaining the training data is associated with labeling the observations $X_i$ while the pool of observations itself is almost unlimited. This suggests to measure the performance of a learning algorithm in terms of its {\it label complexity}, the number of labels $Y_i$ required to obtain a classifier with the desired accuracy. {\it Active learning} theory is mainly devoted to design and analysis of the algorithms that can take advantage of this modified framework. Most of these procedures can be characterized by the following property: at each step $k$, observation $X_k$ is sampled from a distribution $\hat\Pi_k$ that depends on previously obtained $(X_i,Y_i), \ i\leq k-1$(while passive learners obtain all available training data at the same time). $\hat\Pi_k$ is designed to be supported on a set where classification is difficult and requires more labeled data to be collected. The situation when active learners outperform passive algorithms might occur when the so-called {\it Tsybakov's low noise assumption} is satisfied: there exist constants $B,\gamma>0$ such that \begin{equation}\label{noise1} \forall \ t>0, \ \Pi(x: |\eta(x)|\leq t)\leq Bt^{\gamma} \end{equation} This assumption provides a convenient way to characterize the noise level of the problem and will play a crucial role in our investigation.\\ The topic of active learning is widely present in the literature; see \citet{balcan1}, \citet{hanneke2}, \citet{castro1} for review. It was discovered that in some cases the generalization error of a resulting classifier can converge to zero exponentially fast with respect to its label complexity(while the best rate for passive learning is usually polynomial with respect to the cardinality of the training data set). However, available algorithms that adapt to the unknown parameters of the problem($\gamma$ in Tsybakov's low noise assumption, regularity of the decision boundary) involve empirical risk minimization with binary loss, along with other computationally hard problems, see \citet{hanneke1}, \citet{hanneke2}. On the other hand, the algorithms that can be effectively implemented, as in \citet{castro1}, are not adaptive. \\ The majority of the previous work in the field was done under standard complexity assumptions on the set of possible classifiers(such as polynomial growth of the covering numbers). \citet{castro1} derived their results under the regularity conditions on the decision boundary and the noise assumption which is slightly more restrictive then (\ref{noise1}). Essentially, they proved that if the decision boundary is a graph of the H\"{o}lder smooth function $g\in \Sigma(\beta,K,[0,1]^{d-1})$ (see section \ref{preliminaries} for definitions) and the noise assumption is satisfied with $\gamma>0$, then the minimax lower bound for the expected excess risk of the active classifier is of order $ C\cdot N^{-\frac{\beta(1+\gamma)}{2\beta+\gamma(d-1)}} $ and the upper bound is $ C (N/\log N)^{-\frac{\beta(1+\gamma)}{2\beta+\gamma(d-1)}}, $ where $N$ is the label budget. However, the construction of the classifier that achieves an upper bound assumes $\beta$ and $\gamma$ to be known. \\ In this paper, we consider the problem of active learning under classical nonparametric assumptions on the regression function - namely, we assume that it belongs to a certain H\"{o}lder class $\Sigma(\beta,K,[0,1]^d)$ and satisfies to the low noise condition (\ref{noise1}) with some positive $\gamma$. In this case, the work of \citet{tsyb2} showed that plug-in classifiers can attain optimal rates in the {\it passive} learning framework, namely, that the expected excess risk of a classifier $\hat g={\rm sign~} \hat\eta$ is bounded above by $CN^{-\frac{\beta(1+\gamma)}{2\beta+d}}$ (which is the optimal rate), where $\hat \eta$ is the local polynomial estimator of the regression function and $N$ is the size of the training data set. We were able to partially extend this claim to the case of active learning: first, we obtain minimax lower bounds for the excess risk of an active classifier in terms of its label complexity. Second, we propose a new algorithm that is based on plug-in classifiers, attains almost optimal rates over a broad class of distributions and possesses adaptivity with respect to $\beta,\gamma$(within the certain range of these parameters). \\ The paper is organized as follows: the next section introduces remaining notations and specifies the main assumptions made throughout the paper. This is followed by a qualitative description of our learning algorithm. The second part of the work contains the statements and proofs of our main results - minimax upper and lower bounds for the excess risk. \section{Preliminaries}\label{preliminaries} Our {\it active learning} framework is governed by the following rules: \begin{enumerate} \item Observations are sampled sequentially: $X_k$ is sampled from the modified distribution $\hat \Pi_k$ that depends on $(X_1,Y_1),\ldots,(X_{k-1},Y_{k-1})$. \item $Y_k$ is sampled from the conditional distribution $P_{Y|X}(\cdot|X=x)$. Labels are conditionally independent given the feature vectors $X_i, \ i\leq n$. \end{enumerate} Usually, the distribution $\hat \Pi_k$ is supported on a set where classification is difficult.\\ Given the probability measure $\mathbb Q$ on $S\times\left\{-1,1\right\}$, we denote the integral with respect to this measure by $\mathbb Q g:=\int g d\,\mathbb Q$. Let $\mathcal F$ be a class of bounded, measurable functions. The risk and the excess risk of $f\in \mathcal F$ with respect to the measure $\mathbb Q$ are defined by \begin{align*} & R_{\mathbb Q}(f):=\mathbb Q \mathcal I_{y\ne {\rm sign~} f(x)} \\ & \mathcal{E}_{\mathbb Q}(f):=R_{\mathbb Q}(f)-\inf\limits_{g\in \mathcal F}R_{\mathbb Q}(g), \end{align*} where $\mathcal I_{\mathcal A}$ is the indicator of event $\mathcal A$. We will omit the subindex $\mathbb Q$ when the underlying measure is clear from the context. Recall that we denoted the distribution of $(X,Y)$ by $P$. The minimal possible risk with respect to $P$ is $$ \ R^*=\inf\limits_{g:S\mapsto[-1,1]}\Pr\left(Y\ne {\rm sign~} g(X)\right), $$ where the infimum is taken over all measurable functions. It is well known that it is attained for any $g$ such that ${\rm sign~} g(x)={\rm sign~} \eta(x)$ $\Pi$ - a.s. Given $g\in \mathcal F, \ A\in \mathcal B, \ \delta>0$, define $$ \mathcal F_{\infty,A}(g;\delta):=\left\{f\in \mathcal F: \ \|f-g\|_{\infty,A}\leq \delta\right\}, $$ where $\|f-g\|_{\infty,A}=\sup\limits_{x\in A}|f(x)-g(x)|$. For $A\in \mathcal B$, define the function class $$ \mathcal F|_{A}:=\left\{f|_{A}, \ f\in \mathcal F\right\} $$ where $f|_{A}(x):=f(x)I_A(x)$. From now on, we restrict our attention to the case $S=[0,1]^d$. Let $K>0$. \begin{definition} \label{holder3} We say that $g:\mathbb R^d\mapsto\mathbb R$ belongs to $\Sigma(\beta,K,[0,1]^d)$, the $(\beta,K,[0,1]^d)$ - H\"{o}lder class of functions, if $g$ is $\lfloor\beta\rfloor$ times continuously differentiable and for all $x, x_1\in [0,1]^d$ satisfies $$ |g(x_1)-T_x(x_1)|\leq K\|x-x_1\|_{\infty}^{\beta}, $$ where $T_x$ is the Taylor polynomial of degree $\lfloor\beta\rfloor$ of $g$ at the point $x$. \end{definition} \begin{definition} $\mathcal P(\beta,\gamma)$ is the class of probability distributions on \\ $\mathbb [0,1]^d\times\left\{-1,+1\right\}$ with the following properties: \begin{enumerate} \item $\forall \ t>0, \ \Pi(x: |\eta(x)|\leq t)\leq Bt^{\gamma}$; \item $\eta(x)\in \Sigma(\beta,K,\mathbb [0,1]^d)$. \end{enumerate} \end{definition} We do not mention the dependence of $\mathcal P(\beta,\gamma)$ on the fixed constants $B,K$ explicitly, but this should not cause any uncertainty.\\ Finally, let us define $\mathcal P_U^*(\beta,\gamma)$ and $\mathcal P_U(\beta,\gamma)$, the subclasses of $\mathcal P(\beta,\gamma)$, by imposing two additional assumptions. Along with the formal descriptions of these assumptions, we shall try to provide some motivation behind them. The first deals with the marginal $\Pi$. For an integer $M\geq 1$, let $$ \mathcal G_{M}:=\left\{\left(\frac{k_1}{M},\ldots,\frac{k_d}{M}\right), \ k_i=1\ldots M, \ i=1\ldots d\right\} $$ be the regular grid on the unit cube $[0,1]^d$ with mesh size $M^{-1}$. It naturally defines a partition into a set of $M^d$ open cubes $R_i, \ i=1\ldots M^d$ with edges of length $M^{-1}$ and vertices in $\mathcal G_{M}$. Below, we consider the nested sequence of grids $\left\{\mathcal G_{2^{m}}, \ m\geq 1\right\}$ and corresponding dyadic partitions of the unit cube. \begin{definition} We will say that $\Pi$ is $(u_1,u_2)$-regular with respect to $\left\{\mathcal G_{2^{m}}\right\}$ if for any $m\geq 1$, any element of the partition $R_i, \ i\leq 2^{dm}$ such that $R_i\cap {\rm supp}(\Pi)\ne \emptyset$, we have \begin{equation}\label{t-reg} u_1\cdot 2^{-dm}\leq \Pi\left(R_i\right)\leq u_2\cdot 2^{-dm}. \end{equation} where $0<u_1\leq u_2<\infty$. \end{definition} \begin{assumption}\label{regular} $\Pi$ is $(u_1,u_2)$ - regular. \end{assumption} In particular, $(u_1,u_2)$-regularity holds for the distribution with a density $p$ on $[0,1]^d$ such that $0<u_1\leq p(x)\leq u_2<\infty$.\\ Let us mention that our definition of regularity is of rather technical nature; for most of the paper, the reader might think of $\Pi$ as being uniform on $[0,1]^d$( however, we need slightly more complicated marginal to construct the minimax lower bounds for the excess risk). It is know that estimation of regression function in sup-norm is sensitive to the geometry of design distribution, mainly because the quality of estimation depends on the {\it local} amount of data at every point; conditions similar to our {\it assumption} \ref{regular} were used in the previous works where this problem appeared, e.g., {\it strong density assumption} in \citet{tsyb2} and {\it assumption D} in \citet{gaiffas1}.\\ Another useful characteristic of $(u_1,u_2)$ - regular distribution $\Pi$ is that this property is stable with respect to restrictions of $\Pi$ to certain subsets of its support. This fact fits the active learning framework particularly well. \begin{definition} We say that $\mathbb Q$ belongs to $\mathcal P_U(\beta,\gamma)$ if $\mathbb Q\in \mathcal P(\beta,\gamma)$ and {\it assumption} \ref{regular} is satisfied for some $u_1, u_2$. \end{definition} The second assumption is crucial in derivation of the upper bounds. The space of piecewise-constant functions which is used to construct the estimators of $\eta(x)$ is defined via $$ \mathcal F_m=\left\{\sum\limits_{i=1}^{2^{dm}} \lambda_i I_{R_i}(\cdot): \ |\lambda_i|\leq 1, \ i=1\ldots 2^{dm}\right\}, $$ where $\left\{R_i\right\}_{i=1}^{2^{dm}}$ forms the dyadic partition of the unit cube. Note that $\mathcal F_m$ can be viewed as a $\|\cdot\|_{\infty}$-unit ball in the linear span of first $2^{dm}$ Haar basis functions in $[0,1]^d$. Moreover, $\left\{\mathcal F_m, \ m\geq 1\right\}$ is a nested family, which is a desirable property for the model selection procedures. By $\bar\eta_m(x)$ we denote the $L_2(\Pi)$ - projection of the regression function onto $\mathcal F_m$.\\ We will say that the set $A\subset [0,1]^d$ {\it approximates the decision boundary} $\left\{x:\eta(x)=0\right\}$ if there exists $t>0$ such that \begin{equation}\label{approx} \left\{x:|\eta(x)|\leq t\right\}_{\Pi}\subseteq A_{\Pi}\subseteq \left\{x:|\eta(x)|\leq3t\right\}_{\Pi}, \end{equation} where for any set $A$ we define $A_{\Pi}:=A\cap {\rm supp}(\Pi)$. The most important example we have in mind is the following: let $\hat \eta$ be some estimator of $\eta$ with $ \|\hat \eta-\eta\|_{\infty,{\rm supp}(\Pi)}\leq t, $ and define the $2t$ - band around $\eta$ by $$ \hat F=\left\{f: \ \hat \eta(x)-2t\leq f(x)\leq \hat \eta(x)+2t \ \forall x\in[0,1]^d\right\} $$ Take $ A=\left\{x: \ \exists f_1,f_2\in \hat F \text{ s.t. }{\rm sign~} f_1(x)\ne{\rm sign~} f_2(x)\right\} $, then it is easy to see that $A$ satisfies (\ref{approx}). Modified design distributions used by our algorithm are supported on the sets with similar structure.\\ Let $\sigma(\mathcal F_m)$ be the sigma-algebra generated by $\mathcal F_m$ and $A\in \sigma(\mathcal F_m)$. \begin{assumption} \label{holder2} There exists $B_2>0$ such that for all $m\geq 1$, $A\in \sigma(\mathcal F_m)$ satisfying (\ref{approx}) and such that $A_{\Pi}\ne\emptyset$ the following holds true: \begin{align*} & \int\limits_{[0,1]^d}\left(\eta-\bar\eta_m\right)^2 \Pi(dx|x\in A_{\Pi}) \geq B_2 \|\eta-\bar\eta_m\|^2_{\infty,A_{\Pi}} \end{align*} \end{assumption} Appearance of {\it assumption \ref{holder2}} is motivated by the structure of our learning algorithm - namely, it is based on adaptive confidence bands for the regression function. Nonparametric confidence bands is a big topic in statistical literature, and the review of this subject is not our goal. We just mention that it is impossible to construct adaptive confidence bands of optimal size over the whole $\bigcup\limits_{\beta\leq 1}\Sigma\left(\beta, K,[0,1]^d\right)$. \citet{low1,nickl2} discuss the subject in details. However, it is possible to construct adaptive $L_2$ - confidence balls(see an example following Theorem 6.1 in \citet{kolt6}). For functions satisfying {\it assumption \ref{holder2}}, this fact allows to obtain confidence bands of desired size. In particular, \begin{enumerate}[(a)] \item functions that are differentiable, with gradient being bounded away from 0 in the vicinity of decision boundary; \item Lipschitz continuous functions that are convex in the vicinity of decision boundary \end{enumerate} satisfy {\it assumption} \ref{holder2}. For precise statements, see Propositions \ref{holder2:example1}, \ref{holder2:example2} in Appendix \ref{examples}. A different approach to adaptive confidence bands in case of one-dimensional density estimation is presented in \citet{nickl1}. Finally, we define $\mathcal P_U^*(\beta,\gamma)$: \begin{definition} We say that $\mathbb Q$ belongs to $\mathcal P_U^*(\beta,\gamma)$ if $\mathbb Q\in \mathcal P_U(\beta,\gamma)$ and {\it assumption} \ref{holder2} is satisfied for some $B_2>0$. \end{definition} \subsection{Learning algorithm} Now we give a brief description of the algorithm, since several definitions appear naturally in this context. First, let us emphasize that {\it the marginal distribution $\Pi$ is assumed to be known to the learner.} This is not a restriction, since we are not limited in the use of unlabeled data and $\Pi$ can be estimated to any desired accuracy. Our construction is based on so-called {\it plug-in} classifiers of the form $\hat f(\cdot)={\rm sign~} \hat \eta(\cdot)$, where $\hat\eta$ is a piecewise-constant estimator of the regression function. As we have already mentioned above, it was shown in \citet{tsyb2} that in the passive learning framework plug-in classifiers attain optimal rate for the excess risk of order $N^{-\frac{\beta(1+\gamma)}{2\beta+d}}$, with $\hat \eta$ being the local polynomial estimator. \par Our active learning algorithm iteratively improves the classifier by constructing shrinking confidence bands for the regression function. On every step $k$, the piecewise-constant estimator $\hat\eta_k$ is obtained via the model selection procedure which allows adaptation to the unknown smoothness(for H\"{o}lder exponent $\leq1$). The estimator is further used to construct a confidence band $\hat {\mathcal F}_k$ for $\eta(x)$. The {\it active set} assosiated with $\hat{\mathcal F}_k$ is defined as $$ \hat A_k=A(\hat {\mathcal F}_k):=\left\{x\in {{\rm supp}(\Pi)}: \ \exists f_1,f_2\in \hat {\mathcal F}_k, {\rm sign~} f_1(x)\ne{\rm sign~} f_2(x)\right\} $$ Clearly, this is the set where the confidence band crosses zero level and where classification is potentially difficult. $\hat A_k$ serves as a support of the modified distribution $\hat \Pi_{k+1}$: on step $k+1$, label $Y$ is requested only for observations $X\in \hat{A}_k$, forcing the labeled data to concentrate in the domain where higher precision is needed. This allows one to obtain a tighter confidence band for the regression function restricted to the active set. Since $\hat A_k$ approaches the decision boundary, its size is controlled by the low noise assumption. The algorithm does not require a priori knowledge of the noise and regularity parameters, being adaptive for $\gamma>0, \beta\leq 1$. \begin{figure}[ht] \centering \label{fig} \subfigure{\label{fig:1}\includegraphics[scale=0.31]{fig1.pdf}} \subfigure{\label{fig:2}\includegraphics[scale=0.30]{fig2.pdf}} \caption{Active Learning Algorithm} \end{figure} Further details are given in section \ref{upper}. \subsection{Comparison inequalities} Before proceeding to the main results, let us recall the well-known connections between the binary risk and the $\|~\cdot~\|_{\infty}$, $\|\cdot~\|_{L_2(\Pi)}$ - norm risks: \begin{proposition}\label{risk_bound} Under the low noise assumption, \begin{align} &\label{sup} R_{P}(f)-R^*\leq D_1\|(f-\eta)\mathcal I\left\{{\rm sign~} f\ne {\rm sign~} \eta\right\}\|_{\infty}^{1+\gamma}; \\ &\label{square} R_{P}(f)-R^*\leq D_2\|(f-\eta)\mathcal I\left\{{\rm sign~} f\ne {\rm sign~} \eta\right\}\|_{L_2(\Pi)}^{\frac{2(1+\gamma)}{2+\gamma}}; \\ &\label{noise} R_P(f)-R^*\geq D_3\Pi({\rm sign~} f\ne {\rm sign~} \eta)^{\frac{1+\gamma}{\gamma}} \end{align} \end{proposition} \begin{proof} For (\ref{sup}) and (\ref{square}), see \citet{tsyb2}, lemmas 5.1, 5.2 respectively, and for (\ref{noise})---\citet{kolt6}, lemma 5.2. \end{proof} \section{Main results} The question we address below is: what are the best possible rates that can be achieved by active algorithms in our framework and how these rates can be attained. \subsection{Minimax lower bounds for the excess risk} The goal of this section is to prove that for $P\in \mathcal P(\beta,\gamma)$ no active learner can output a classifier with expected excess risk converging to zero faster than $N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}}$. Our result builds upon the minimax bounds of \citet{tsyb2}, \citet{castro1}. \\ {\bf Remark} The theorem below is proved for a smaller class $\mathcal P_U^*(\beta,\gamma)$, which implies the result for $\mathcal P(\beta,\gamma)$. \begin{theorem}\label{lower_bound} Let $\beta,\gamma,d$ be such that $\beta\gamma\leq d$. Then there exists $C>0$ such that for all $n$ large enough and for any active classifier $\hat f_n(x)$ we have $$ \sup_{P\in \mathcal{P}_U^*(\beta,\gamma)}\mathbb E R_P(\hat f_n)-R^*\geq C N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}} $$ \end{theorem} \begin{proof} We proceed by constructing the appropriate family of classifiers $f_{\sigma}(x)={\rm sign~} \eta_{\sigma}(x)$, in a way similar to Theorem 3.5 in \citet{tsyb2}, and then apply Theorem 2.5 from \citet{tsyb3}. We present it below for reader's convenience. \begin{theorem} \label{tsyb2.5} Let $\Sigma$ be a class of models, ${\rm d}: \Sigma\times \Sigma\mapsto \mathbb R$ - the pseudometric and $\left\{P_{f}, \ f\in \Sigma\right\}$ - a collection of probability measures associated with $\Sigma$. Assume there exists a subset $\left\{f_{0},\ldots,f_{M}\right\}$ of $\Sigma$ such that \begin{enumerate} \item $d(f_{i},f_{j})\geq 2s>0 \ \forall 0\leq i<j\leq M$ \item $P_{f_{j}}\ll P_{f_{0}}$ for every $1\leq j\leq M$ \item $\frac 1 M \sum_{j=1}^{M}{\rm KL}(P_{f_{j}},P_{f_{0}})\leq \alpha \log M, \quad 0<\alpha<\frac 18$ \end{enumerate} Then $$ \inf_{\hat f}\sup_{f\in \Sigma}P_{f}\left({\rm d}(\hat f,f)\geq s\right)\geq \frac{\sqrt{M}}{1+\sqrt{M}}\left(1-2\alpha-\sqrt{\frac{2\alpha}{\log M}}\right) $$ where the infimum is taken over all possible estimators of $f$ based on a sample from $P_{f}$ and ${\rm KL}(\cdot,\cdot)$ is the Kullback-Leibler divergence. \end{theorem} Going back to the proof, let $q=2^{l}, \ l\geq 1$ and $$ G_q:=\left\{\left(\frac{2k_1-1}{2q},\ldots,\frac{2k_d-1}{2q}\right), \ k_i=1\ldots q, \ i=1\ldots d\right\} $$ be the grid on $[0,1]^d$. For $x\in [0,1]^d$, let \begin{align*} n_q(x)&={\rm argmin~}\left\{\|x-x_k\|_2: \ x_k\in G_q\right\} \end{align*} If $n_q(x)$ is not unique, we choose the one with smallest $\|\cdot\|_2$ norm. The unit cube is partitioned with respect to $G_q$ as follows: $x_1,x_2$ belong to the same subset if $n_q(x_1)=n_q(x_2)$. Let $'\succ'$ be some order on the elements of $G_q$ such that $x\succ y$ implies $\|x\|_{2}\geq \|y\|_{2}$. Assume that the elements of the partition are enumerated with respect to the order of their centers induced by $'\succ'$: $[0,1]^d=\bigcup\limits_{i=1}^{q^d}R_i$. Fix $1\leq m\leq q^d$ and let \begin{align*} & S:=\bigcup_{i=1}^m R_i \end{align*} Note that the partition is ordered in such a way that there always exists $1\leq k\leq q\sqrt{d}$ with \begin{equation} \label{cube} B_+\left(0,\frac{k}{q}\right)\subseteq S\subseteq B_+\left(0,\frac{k+3\sqrt{d}}{q}\right), \end{equation} where $B_+(0,R):=\left\{x\in \mathbb R^d_+: \ \|x\|_2\leq R \right\}$. In other words, (\ref{cube}) means that that the difference between the radii of inscribed and circumscribed spherical sectors of $S$ is of order $C(d) q^{-1}$.\\ Let $v>r_1>r_2$ be three integers satisfying \begin{equation} \label{constants} 2^{-v}<2^{-r_1}<2^{-r_1}\sqrt{d}<2^{-r_2}\sqrt{d}<2^{-1} \end{equation} Define $u(x):\mathbb R\mapsto \mathbb R_+$ by \begin{equation} \label{smooth} u(x):=\frac{\int_x^{\infty}U(t)dt}{\int\limits_{2^{-v}}^{1/2} U(t)dt} \end{equation} where $$ U(t):=\left\{\begin{array}{c l} \exp\left(-\frac{1}{(1/2-x)(x-2^{-v})}\right), & x\in(2^{-v},\frac12)\\ 0 & \text{else.} \\ \end{array}\right. $$ Note that $u(x)$ is an infinitely diffferentiable function such that $u(x)=1, \ x\in [0,2^{-v}]$ and $u(x)=0, \ x\geq \frac12$. Finally, for $x\in \mathbb R^d$ let $$ \Phi(x):=C u(\|x\|_{2}) $$ where $C:=C_{L,\beta}$ is chosen such that $\Phi\in \Sigma(\beta, L, \mathbb R^d)$. \\ Let $ r_S:=\inf\left\{r>0: \ B_+(0,r)\supseteq S\right\} $ and $$ A_0:=\left\{\bigcup\limits_{i}R_i: \ R_i\cap B_+\left(0,r_S+q^{-\frac{\beta\gamma}{d}}\right)=\emptyset\right\} $$ \begin{figure}[t] \label{minmax} \begin{center} \includegraphics[width=0.33\textwidth]{minimax4.jpg} \end{center} \caption{Geometry of the support} \end{figure} Note that \begin{equation} \label{radius} r_S\leq c\frac{m^{1/d}}{q}, \end{equation} since ${\rm Vol}\, (S)=mq^{-d}$. \\ Define $ \mathcal H_m=\left\{P_{\sigma}: \sigma\in\left\{-1,1\right\}^m\right\} $ to be the hypercube of probability distributions on $[0,1]^d\times \left\{-1,+1\right\}$. The marginal distribution $\Pi$ of $X$ is independent of $\sigma$: define its density $p$ by $$ p(x)=\left\{\begin{array}{c l} \frac{2^{d(r_1-1)}}{2^{d(r_1-r_2)}-1}, & \ x\in B_{\infty}\left(z,\frac{2^{-r_2}}{q}\right)\setminus B_{\infty}\left(z,\frac{2^{-r_1}}{q}\right), \ z\in G_q\cap S, \\ c_0, & x\in A_0, \\ 0 & \text{else}. \end{array}\right. $$ where $B_{\infty}(z,r):=\left\{x: \ \|x-z\|_{\infty}\leq r\right\}$, $c_0:=\frac{1-mq^{-d}}{{\rm Vol}(A_0)}$(note that $\Pi(R_i)=q^{-d}\quad\forall i\leq m$) and $r_1, r_2$ are defined in (\ref{constants}). In particular, $\Pi$ satisfies {\it assumption} {\ref{regular}} since it is supported on the union of dyadic cubes and has bounded above and below on ${\rm supp}(\Pi)$ density. Let $$ \Psi(x):=u\left(1/2-q^{\frac{\beta\gamma}{d}}{\rm dist}_2(x,B_+(0,r_S))\right), $$ where $u(\cdot)$ is defined in (\ref{smooth}) and ${\rm dist}_2(x,A):=\inf\left\{\|x-y\|_2, \ y\in A\right\}$.\\ Finally, the regression function $\eta_{\sigma}(x)=\mathbb E_{P_{\sigma}}(Y|X=x)$ is defined via $$ \eta_{\sigma}(x):= \left\{\begin{array}{l l} \sigma_i q^{-\beta}\Phi(q[x-n_q(x)]), & x\in R_i, \ 1\leq i\leq m \\ \frac{1}{C_{L,\beta}\sqrt{d}}\, {\rm dist}_2(x,B_+(0,r_S))^{\frac{d}{\gamma}}\cdot \Psi(x), & x\in [0,1]^d\setminus S. \end{array} \right. $$ The graph of $\eta_{\sigma}$ is a surface consisting of small ''bumps" spread around $S$ and tending away from 0 monotonically with respect to ${\rm dist}_{2}(\cdot,B_+(0,r_S))$ on $[0,1]^d\setminus S$. Clearly, $\eta_{\sigma}(x)$ satisfies smoothness requirement, since for $x\in[0,1]^d$ $$ {\rm dist}_2(x,B_+(0,r_S))=\|x\|_2-r_S $$ and $\frac{d}{\gamma}\geq \beta$ by assumption. \footnote{$\Psi(x)$ can be replaced by 1 unless $\beta\gamma=d$ and $\beta$ is an integer, in which case extra smoothness at the boundary of $B_+(0,r_S)$, provided by $\Psi$, is necessary.} Let's check that it also satisfies the low noise condition. Since $|\eta_{\sigma}|\geq Cq^{-\beta}$ on support of $\Pi$, it is enough to consider $t=Czq^{-\beta}$ for $z>1$: \begin{align*} \Pi(|\eta_{\sigma}(x)|\leq Czq^{-\beta})&\leq mq^{-d}+\Pi\left({\rm dist}_{2}(x,B_+(0,r_S))\leq Cz^{\gamma/d}q^{-\frac{\beta\gamma}{d}}\right)\leq \\ & \leq mq^{-d}+C_2\left(r_S+Cz^{\gamma/d}q^{-\frac{\beta\gamma}{d}}\right)^d\leq \\ & \leq mq^{-d}+C_3 mq^{-d}+C_4 z^{\gamma}q^{-\beta\gamma}\leq \\ & \leq \widehat C t^{\gamma}, \end{align*} if $mq^{-d}=O(q^{-\beta\gamma})$. Here, the first inequality follows from considering $\eta_{\sigma}$ on $S$ and $A_0$ separately, and second inequality follows from (\ref{radius}) and direct computation of the sphere volume. \\ Finally, $\eta_{\sigma}$ satisfies {\it assumption} \ref{holder2} with some $B_2:=B_2(q)$ since on ${\rm supp}(\Pi)$ $$ 0<c_1(q)\leq \|\nabla \eta_{\sigma}(x)\|_2\leq c_2(q)<\infty $$ The next step in the proof is to choose the subset of $\mathcal H$ which is ``well-separated'': this can be done due to the following fact(see \citet{tsyb3}, Lemma 2.9): \begin{proposition}[Gilbert-Varshamov] For $m\geq 8$, there exists $$ \left\{\sigma_0,\ldots,\sigma_M\right\}\subset \left\{-1,1\right\}^m $$ such that $\sigma_0=\left\{1,1,\ldots,1\right\}$, $\rho(\sigma_i,\sigma_j)\geq \frac{m}{8} \ \forall \ 0\leq i<k\leq M$ and $M\geq 2^{m/8}$ where $\rho$ stands for the Hamming distance. \end{proposition} Let $\mathcal H':=\left\{P_{\sigma_0},\ldots, P_{\sigma_M}\right\}$ be chosen such that $\left\{\sigma_0,\ldots,\sigma_M\right\}$ satisfies the proposition above. Next, following the proof of Theorems 1 and 3 in \citet{castro1}, we note that $\forall \sigma\in \mathcal H', \ \sigma\ne\sigma_0$ \begin{align} \label{KL} & {\rm KL}(P_{\sigma,N}\|P_{\sigma_0,N})\leq 8N\max_{x\in[0,1]}(\eta_{\sigma}(x)-\eta_{\sigma_0}(x))^2\leq 32 C_{L,\beta}^2 N q^{-2\beta}, \end{align} where $P_{\sigma,N}$ is the joint distribution of $\left(X_i,Y_i\right)_{i=1}^N$ under hypothesis that the distribution of couple $(X,Y)$ is $P_{\sigma}$. Let us briefly sketch the derivation of (\ref{KL}); see also the proof of Theorem 1 in \citet{castro1}. Denote \begin{align*} & \bar X_k:=(X_1,\ldots,X_k), \\ & \bar Y_k\,:=(Y_1,\ldots,Y_k) \end{align*} Then $d P_{\sigma,N}$ admits the following factorization: \begin{align*} & d P_{\sigma,N}(\bar X_N,\bar Y_N)=\prod_{i=1}^N P_{\sigma}(Y_i|X_i)dP(X_i|\bar X_{i-1},\bar Y_{i-1}), \end{align*} where $dP(X_i|\bar X_{i-1},\bar Y_{i-1})$ does not depend on $\sigma$ but only on the active learning algorithm. As a consequence, \begin{align*} {\rm KL}(P_{\sigma,N}\|P_{\sigma_0,N})&= \mathbb E_{P_{\sigma,N}}\log\frac{dP_{\sigma,N}(\bar X_N,\bar Y_N)}{dP_{\sigma_0,N}(\bar X_n,\bar Y_N)}= \mathbb E_{P_{\sigma,N}}\log\frac{\prod_{i=1}^N P_{\sigma}(Y_i|X_i)}{\prod_{i=1}^N P_{\sigma_0}(Y_i|X_i)}=\\ & =\sum_{i=1}^N \mathbb E_{P_{\sigma,N}} \left[\mathbb E_{P_{\sigma}}\left(\log\frac{P_{\sigma}(Y_i|X_i)}{P_{\sigma_0}(Y_i|X_i)}\left\lvert X_i\right.\right)\right]\leq \\ & \leq N\max_{x\in[0,1]^d}\mathbb E_{P_{\sigma}}\left(\log\frac{P_{\sigma}(Y_1|X_1)}{P_{\sigma_0}(Y_1|X_1)}\left\lvert X_1=x\right.\right)\leq \\ & \leq 8 N \max_{x\in[0,1]^d}(\eta_{\sigma}(x)-\eta_{\sigma_0}(x))^2, \end{align*} where the last inequality follows from Lemma 1, \citet{castro1}. Also, note that we have $\max_{x\in[0,1]^d}$ in our bounds rather than the average over $x$ that would appear in the passive learning framework.\\ It remains to choose $q, m$ in appropriate way: set $q\simeq\lfloor C_1 N^{\frac{1}{2\beta+d-\beta\gamma}}\rfloor$ and $m=\lfloor C_2 q^{d-\beta\gamma}\rfloor$ where $C_1, \ C_2$ are such that $q^d\geq m\geq 1$ and $32C_{L,\beta}^2 N q^{-2\beta}<\frac{m}{64}$ which is possible for $N$ big enough. In particular, $mq^{-d}=O(q^{-\beta\gamma})$. Together with the bound (\ref{KL}), this gives $$ \frac{1}{M}\sum_{\sigma\in \mathcal H'}{\rm KL}(P_{\sigma}\|P_{\sigma^0})\leq 32 C_u^2 N q^{-2\beta} < \frac{m}{8^2}=\frac18\log|\mathcal H'|, $$ so that conditions of Theorem \ref{tsyb2.5} are satisfied. Setting $$ f_{\sigma}(x):={\rm sign~} \eta_{\sigma}(x), $$ we finally have $\forall \sigma_1\ne\sigma_2\in \mathcal H'$ $$ d(f_{\sigma_1},f_{\sigma_2}):=\Pi({\rm sign~} \eta_{\sigma_1}(x)\ne{\rm sign~} \eta_{\sigma_2}(x))\geq \frac{m}{8q^d}\geq C_4 N^{-\frac{\beta\gamma}{2\beta+d-\beta\gamma}}, $$ where the lower bound just follows by construction of our hypotheses. Since under the low noise assumption $R_P(\hat f_n)-R^*\geq c\Pi(\hat f_n\ne {\rm sign~}\eta)^{\frac{1+\gamma}{\gamma}}$(see (\ref{noise})), we conclude that \begin{align*} &\inf_{\hat f_N}\sup_{P\in \mathcal P_U^*(\beta,\gamma)} \Pr\left(R_{P}(\hat f_n)-R^*\geq C_4 N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}}\right)\geq \\ &\geq \inf_{\hat f_N}\sup_{P\in \mathcal P_U^*(\beta,\gamma)} \Pr\left(\Pi(\hat f_n(x)\ne{\rm sign~} \eta_{P}(x))\geq \frac{C_4}{2}N^{-\frac{\beta\gamma}{2\beta+d-\beta\gamma}}\right)\geq \tau>0. \end{align*} \end{proof} \subsection{Upper bounds for the excess risk} \label{upper} Below, we present a new active learning algorithm which is computationally tractable, adaptive with respect to $\beta,\gamma$(in a certain range of these parameters) and can be applied in the nonparametric setting. We show that the classifier constructed by the algorithm attains the rates of Theorem \ref{lower_bound}, up to polylogarithmic factor, if $0<\beta\leq 1$ and $ \beta\gamma\leq d$ (the last condition covers the most interesting case when the regression function hits or crosses the decision boundary in the interior of the support of $\Pi$; for detailed statement about the connection between the behavior of the regression function near the decision boundary with parameters $\beta, \ \gamma$, see Proposition 3.4 in \citet{tsyb2}). The problem of adaptation to higher order of smoothness ($\beta>1$) is still awaiting its complete solution; we address these questions below in our final remarks. \\ For the purpose of this section, the regularity assumption reads as follows: there exists $0<\beta\leq 1$ such that $\forall x_1,x_2\in[0,1]^d$ \begin{align}\label{holder1} & |\eta(x_1)-\eta(x_2)|\leq B_1\|x_1-x_2\|_{\infty}^{\beta} \end{align} Since we want to be able to construct non-asymptotic confidence bands, some estimates on the size of constants in (\ref{holder1}) and {\it assumption \ref{holder2}} are needed. Below, we will additionally assume that \begin{align*} & B_1\leq \log N\\ & B_2\geq \log^{-1}N, \end{align*} where $N$ is the label budget. This can be replaced by any known bounds on $B_1,B_2$.\\ Let $A\in \sigma(\mathcal F_m)$ with $A_{\Pi}:=A\cap {\rm supp}(\Pi)\ne \emptyset$. Define $$ \hat \Pi_A(dx):=\Pi(dx|x\in A_{\Pi}) $$ and $d_m:=\dim \mathcal F_m|_{A_{\Pi}}$. Next, we introduce a simple estimator of the regression function on the set $A_{\Pi}$. Given the resolution level $m$ and an iid sample $(X_i,Y_i), \ i\leq N$ with $X_i\sim \hat \Pi_A$, let \begin{equation} \label{est} \hat \eta_{m,A}(x):= \sum_{i:R_i\cap A_{\Pi}\ne \emptyset} \frac{\sum_{j=1}^N Y_j \mathcal I_{R_i}(X_j)}{N\cdot\hat\Pi_A(R_i)}\mathcal I_{R_i}(x) \end{equation} Since we assumed that the marginal $\Pi$ is known, the estimator is well-defined. The following proposition provides the information about concentration of $\hat \eta_m$ around its mean: \begin{proposition}\label{supnorm} For all $t>0$, \begin{align*} \Pr\Bigg(\max_{x\in A_{\Pi}}\lvert\hat\eta_{m,A}(x)-& \bar\eta_m(x)\rvert\ \geq t\sqrt{\frac{2^{dm}\Pi(A)}{u_1 N}}\Bigg)\leq \\ & \leq 2d_m\exp\left(\frac{-t^2}{2(1+\frac t3\sqrt{2^{dm}\Pi(A)/u_1 N})}\right), \end{align*} \end{proposition} \begin{proof} This is a straightforward application of the Bernstein's inequality to the random variables $$ S_N^i:=\sum_{j=1}^N Y_j\mathcal I_{R_i}(X_j), \ i\in\left\{i:R_i\cap A_{\Pi}\ne \emptyset\right\}, $$ and the union bound: indeed, note that $\mathbb E(Y \mathcal I_{R_i}(X_j))^2=\hat\Pi_A(R_i)$, so that $$ \Pr\left(\left|S_N^i-N\int_{R_i}\eta d\hat\Pi_A\right|\geq tN\hat\Pi_A(R_i)\right) \leq 2\exp\left(-\frac{N\hat\Pi_A(R_i)t^2}{2+2t/3}\right), $$ and the rest follows by simple algebra using that $\hat\Pi_A(R_i)\geq \frac{u_1}{2^{dm}\Pi(A)}$ by the $(u_1,u_2)$-regularity of $\Pi$. \end{proof} Given a sequence of hypotheses classes $\mathcal G_m, \ m\geq 1$, define the index set \begin{align} \label{index_set} & \mathcal J(N):=\left\{m\in \mathbb N: \ 1\leq \dim \mathcal G_m\leq \frac{N}{\log^2 N}\right\} \end{align} - the set of possible ``resolution levels'' of an estimator based on $N$ classified observations(an upper bound corresponds to the fact that we want the estimator to be consistent). When talking about model selection procedures below, we will implicitly assume that the model index is chosen from the corresponding set $\mathcal J$. The role of $\mathcal G_m$ will be played by $\mathcal F_m|_A$ for appropriately chosen set $A$. We are now ready to present the active learning algorithm followed by its detailed analysis(see Table 1). \\ \begin{table}[t] \centering \label{alg1} {\footnotesize \begin{tabular}{|l|} \hline $\mbox{\bf Algorithm 1a}$ \\ \hline $\mbox{{\bf input } label budget } N; \ \text{confidence }\alpha$;\\ $\hat m_0=0, \ \mathcal{\hat F}_0:=\mathcal{F}_{\hat m_0}, \ \hat\eta_0\equiv 0$; \\ $LB:=N$; \hspace{2cm} // {\it label budget} \\ $N_0:=2^{\lfloor\log_2 \sqrt{N}\rfloor}$;\\ $s^{(k)}(m,N,\alpha):=s(m,N,\alpha):=m(\log N+\log\frac{1}{\alpha})$;\\ $k:=0$;\\ \bf{while} $LB\geq 0$ {\bf do} \\ $k:=k+1$; \\ $N_k:=2N_{k-1}$;\\ $\hat A_k:=\left\{x\in [0,1]^d: \ \exists f_1,f_2\in \hat{\mathcal{F}}_{k-1}, {\rm sign~}(f_1(x))\ne{\rm sign~}(f_2(x))\right\};$ \\ $\mbox{\bf if } \hat A_k\cap {\rm supp}(\Pi)=\emptyset \mbox{ \bf or } LB<\lfloor N_k\cdot \Pi(\hat A_k) \rfloor\mbox{ \bf then } $ \\ $\qquad\qquad\mbox{ {\bf break; output }} \hat g:={\rm sign~} \hat \eta_{k-1}$ \\ $\qquad\qquad \mbox{ \bf else}$ \\ $\mbox{{\bf for }} i=1\ldots \lfloor N_{k}\cdot \Pi(\hat A_k)\rfloor$\\ $\mbox{{\bf sample i.i.d} } \left(X_{i}^{(k)},Y_i^{(k)}\right) \mbox{{\bf with }} X_i^{(k)} \sim\hat\Pi_k:=\Pi(dx|x\in\hat A_k);$ \\ $\mbox{{\bf end for}}$;\\ $LB:=LB-\lfloor N_{k}\cdot \Pi(\hat A_k)\rfloor$; \\ $\hat P_k:=\frac{1}{\lfloor N_{k}\cdot \Pi(\hat A_k)\rfloor} \sum\limits_{i}\delta_{X^{(k)}_{i},Y^{(k)}_{i}}$ \quad // {\it ''active'' empirical measure} \\ $\hat m_k:={\rm argmin~}_{m\geq \hat m_{k-1}}\left[\inf_{f\in\mathcal F_m} \hat P_k(Y-f(X))^2+K_1\frac{2^{dm}\Pi(\hat A_k)+s(m-\hat m_{k-1},N,\alpha)}{\lfloor N_{k}\cdot \Pi(\hat A_k)\rfloor}\right]$ \\ $\hat \eta_k:=\hat \eta_{\hat m_k,\hat A_{k}}$ \qquad // {\it see (\ref{est})} \\ $\delta_{k}:=\tilde D\cdot \log^2\frac{N}{\alpha}\sqrt{\frac{2^{d\hat m_k}}{N_k}}$; \\ $\hat{\mathcal F}_k:=\left\{f\in \mathcal F_{\hat m_k}: \ f|_{\hat A_k}\in\mathcal{F}_{\infty,\hat A_k}(\hat \eta_k;\delta_{k}), \ \ f|_{[0,1]^d\setminus {\hat A}_k}\equiv\hat\eta_{k-1}|_{[0,1]^d\setminus {\hat A}_k}\right\}$;\\ $\mbox{\bf{end; }}$ \\ \hline \end{tabular} } \caption{Active Learning Algorithm} \end{table} \par {\bf Remark} Note that on every iteration, {\bf Algorithm 1a} uses the whole sample to select the resolution level $\hat m_k$ and to build the estimator $\hat \eta_k$. While being suitable for practical implementation, this is not convenient for theoretical analysis. We will prove the upper bounds for a slighly modified version: namely, on every iteration $k$ labeled data is divided into two subsamples $S_{k,1}$ and $S_{k,2}$ of approximately equal size, $|S_{k,1}|\simeq |S_{k,2}|\simeq\left\lfloor \frac 12 N_{k}\cdot \Pi(\hat A_k)\right\rfloor$. Then $S_{1,k}$ is used to select the resolution level $\hat m_k$ and $S_{k,2}$ - to construct $\hat \eta_k$. We will call this modified version {\bf Algorithm 1b}. \par As a first step towards the analysis of {\bf Algorithm 1b}, let us prove the useful fact about the general model selection scheme. Given an iid sample $(X_i,Y_i), \ i\leq N$, set $s_m=m(s+\log\log_2 N), \ m\geq1$ and \begin{align}\label{res1} & \hat m:=\hat m(s)={\rm argmin~}_{\substack{m \in \mathcal J(N)}} \left[ \inf_{f\in\mathcal F_m} P_N(Y-f(X))^2+K_1\frac{ 2^{dm}+ s_m }{N} \right] \\ & \bar m:=\min\left\{ m\geq 1: \ \inf_{f\in \mathcal F_m}\mathbb E(f(X)-\eta(X))^2\leq K_2 \frac{ 2^{dm}}{N} \right\} \end{align} \begin{theorem}\label{opt_selection} There exist an absolute constant $K_1$ big enough such that, with probability $\geq 1-e^{-s}$, $$ \hat m\leq \bar m $$ \end{theorem} \begin{proof} See Appendix \ref{res_lvl}. \end{proof} Straightforward application of this result immediately yields the following: \begin{corollary}\label{resol_bound} Suppose $\eta(x)\in \Sigma(\beta,L,[0,1]^d)$. Then, with probability $\geq 1-e^{-s}$, $$ 2^{\hat m}\leq C_1\cdot N^{\frac{1}{2\beta+d}} $$ \end{corollary} \begin{proof} By definition of $\bar m$, we have \begin{align*} \bar m& \leq 1+\max\left\{ m : \ \inf_{f\in \mathcal F_m} \mathbb E(f(X)-\eta(X))^2 > K_2\frac{2^{dm}}{N} \right\}\leq \\ & \leq 1+\max\left\{ m:\ L^2 2^{-2\beta m}> K_2\frac{2^{dm}}{N} \right\}, \end{align*} and the claim follows. \end{proof} With this bound in hand, we are ready to formulate and prove the main result of this section: \begin{theorem}\label{main} Suppose that $P\in \mathcal P_U^*(\beta,\gamma)$ with $B_1\leq \log N, \ B_2\geq \log^{-1}N$ and $\beta\gamma\leq d$. Then, with probability $\geq 1-3\alpha$, the classifier $\hat g$ returned by {\bf Algorithm 1b} with label budget $N$ satisfies $$ R_{P}(\hat g)-R^*\leq {\rm Const}\cdot N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}}\log^p\frac{N}{\alpha}, $$ where $p\leq\frac{2\beta\gamma(1+\gamma)}{2\beta+d-\beta\gamma}$ and $B_1, \ B_2$ are the constants from (\ref{holder1}) and {\it assumption} \ref{holder2}. \end{theorem} {\bf Remarks } \begin{enumerate} \item Note that when $\beta\gamma>\frac d 3$, $N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}}$ is a {\it fast rate}, i.e., faster than $N^{-\frac{1}{2}}$; at the same time, the passive learning rate $N^{-\frac{\beta(1+\gamma)}{2\beta+d}}$ is guaranteed to be fast only when $\beta\gamma>\frac d 2$, see \citet{tsyb2}. \\ \item For $\hat\alpha\simeq N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}}$ {\bf Algorithm 1b} returns a classifier $\hat g_{\hat \alpha}$ that satisfies $$ \mathbb E R_{P}(\hat g_{\hat \alpha})-R^*\leq {\rm Const}\cdot N^{-\frac{\beta(1+\gamma)}{2\beta+d-\beta\gamma}}\log^p N. $$ This is a direct corollary of Theorem \ref{main} and the inequality \\ $$ \mathbb E|Z|\leq t+\|Z\|_{\infty}\Pr(|Z|\geq t) $$ \end{enumerate} \begin{proof} Our main goal is to construct high probability bounds for the size of the active sets defined by {\bf Algorithm 1b}. In turn, these bounds depend on the size of the confidence bands for $\eta(x)$, and the previous result(Theorem \ref{opt_selection}) is used to obtain the required estimates. Suppose $L$ is the number of steps performed by the algorithm before termination; clearly, $L\leq N$.\\ Let $N_k^{{\rm act}}:=\lfloor N_k\cdot \Pi(\hat A_k)\rfloor$ be the number of labels requested on $k$-th step of the algorithm: this choice guarantees that the ''density'' of labeled examples doubles on every step. \\ Claim: the following bound for the size of the active set holds uniformly for all $2\leq k\leq L$ with probability at least $ 1-2\alpha $: \begin{align} \label{Z3} \Pi(\hat A_{k})&\leq \, C N_k^{-\frac{\beta\gamma}{2\beta+d}}\left(\log \frac{N}{\alpha}\right)^{2\gamma} \end{align} It is not hard to finish the proof assuming (\ref{Z3}) is true: indeed, it implies that the number of labels requested on step $k$ satisfies $$ N_k^{{\rm act}}=\lfloor N_k\Pi(\hat A_k)\rfloor\leq C\cdot N_k^{\frac{2\beta+d-\beta\gamma}{2\beta+d}} \left(\log \frac{N}{\alpha}\right)^{2\gamma} $$ with probability $\geq 1-2\alpha$. Since $\sum\limits_k N_k^{{\rm act}}\leq N $, one easily deduces that on the last iteration $L$ we have \begin{equation}\label{Z7} N_L\geq c\left(\frac{N}{\log^{2\gamma} (N/\alpha)}\right)^{\frac{2\beta+d}{2\beta+d-\beta\gamma}} \end{equation} To obtain the risk bound of the theorem from here, we apply inequality (\ref{sup}) \footnote{alternatively, inequality (\ref{square}) can be used but results in a slightly inferior logarithmic factor.} from proposition \ref{risk_bound}: \begin{equation}\label{final} R_{P}(\hat g)-R^*\leq D_1\|(\hat\eta_L-\eta)\cdot\mathcal I\left\{{\rm sign~} \hat\eta_L\ne {\rm sign~} \eta\right\}\|_{\infty}^{1+\gamma} \end{equation} It remains to estimate $\|\hat \eta_L-\eta\|_{\infty,\hat A_L}$: we will show below while proving (\ref{Z3}) that $$ \|\hat \eta_L-\eta\|_{\infty,\hat A_L}\leq C\cdot N_{L}^{-\frac{\beta}{2\beta+d}}\log^2 \frac{N}{\alpha} $$ Together with (\ref{Z7}) and (\ref{final}), it implies the final result. \par To finish the proof, it remains to establish (\ref{Z3}). Recall that $\bar \eta_k$ stands for the $L_2(\Pi)$ - projection of $\eta$ onto $\mathcal F_{\hat m_k}$. An important role in the argument is played by the bound on the $L_2(\hat\Pi_k)$ - norm of the ``bias'' $(\bar \eta_{k}-\eta)$: together with {\it assumption} \ref{holder2}, it allows to estimate $\|\bar \eta_{k}-\eta\|_{\infty,\hat A_k}$. The required bound follows from the following oracle inequality: there exists an event $\mathcal B$ of probability $\geq 1-\alpha$ such that on this event for every $1\leq k\leq L$ \begin{align} \label{oracle} \|\bar{\eta}_{k}-\eta\|_{L_2(\hat\Pi_k)}^2\leq \inf_{m\geq \hat m_{k-1}} \Bigg[ \inf_{f\in \mathcal F_m}&\|f-\eta\|_{L_2(\hat\Pi_k)}^2+ \\ & \nonumber +K_1\frac{2^{dm}\Pi(\hat A_k) +(m-\hat m_{k-1})\log(N/\alpha)}{N_k \Pi(\hat A_k)} \Bigg] \end{align} It general form, this inequality is given by Theorem 6.1 in \citet{kolt6} and provides the estimate for $\|\hat\eta_k-\eta\|_{L_2(\hat\Pi_k)}$, so it automatically implies the weaker bound for the bias term only. To deduce (\ref{oracle}), we use the mentioned general inequality $L$ times(once for every iteration) and the union bound. The quantity $2^{dm}\Pi(\hat A_k)$ in (\ref{oracle}) plays the role of the dimension, which is justified below. Let $k\geq 1$ be fixed. For $m\geq \hat m_{k-1}$, consider hypothesis classes $$ \mathcal F_m|_{\hat A_k}:=\left\{f\mathcal I_{\hat A_k}, \ f\in \mathcal F_m\right\} $$ An obvious but important fact is that for $P\in \mathcal P_U(\beta,\gamma)$, the dimension of $\mathcal F_m|_{\hat A_k}$ is bounded by $u_1^{-1}\cdot 2^m \Pi(\hat A_k)$: indeed, $$ \Pi(\hat A_k)=\sum_{j:R_j\cap \hat A_k\ne\emptyset} \Pi(R_j)\geq u_1 2^{-dm}\cdot \#\left\{j:R_j\cap \hat A_k\ne\emptyset\right\}, $$ hence \begin{equation}\label{dimension} {\rm dim }\, \mathcal F_m|_{\hat A_k}=\#\left\{j:R_j\cap \hat A_k\ne\emptyset\right\}\leq u_1^{-1}\cdot 2^m \Pi(\hat A_k). \end{equation} Theorem \ref{opt_selection} applies conditionally on $\left\{X_i^{(j)}\right\}_{i=1}^{N_j}, \ j\leq k-1$ with sample of size $N_k^{{\rm act}}$ and $s=\log(N/\alpha)$: to apply the theorem, note that, by definition of $\hat A_k$, it is independent of $X_i^{(k)}, \ i=1\ldots N_k^{{\rm act}}$. Arguing as in Corollary \ref{resol_bound} and using (\ref{dimension}), we conclude that the following inequality holds with probability $\geq 1-\frac{\alpha}{N}$ for every fixed $k$: \begin{align} & \label{Z1} 2^{\hat m_k}\leq C\cdot N_k^{\frac{1}{2\beta+d}}. \end{align} Let $\mathcal E_{1}$ be an event of probability $\geq 1-\alpha$ such that on this event bound (\ref{Z1}) holds for every step $k$, $k\leq L$ and let $\mathcal E_2$ be an event of probability $\geq 1-\alpha$ on which inequalities (\ref{oracle}) are satisfied. Suppose that event $\mathcal E_1\cap \mathcal E_2$ occurs and let $k_0$ be a fixed arbitrary integer $2\leq k_0\leq L+1$. It is enough to assume that $\hat A_{k_0-1}$ is nonempty(otherwise, the bound trivially holds), so that it contains at least one cube with sidelength $2^{-\hat m_{k_0-2}}$ and \begin{equation}\label{Z5} \Pi(\hat A_{k_0-1})\geq u_1 2^{-d\hat m_{k_0-1}}\geq c N_{k_0}^{-\frac{d}{2\beta+d}} \end{equation} Consider inequality (\ref{oracle}) with $k=k_0-1$ and $2^m\simeq N_{k_0-1}^{\frac{1}{2\beta+d}}$. By (\ref{Z5}), we have \begin{equation} \label{Z8} \|\bar\eta_{k_0-1}-\eta\|^2_{L_2(\hat\Pi_{k_0-1})}\leq C N_{k_0-1}^{-\frac{2\beta}{2\beta+d}} \log^2 \frac{N}{\alpha} \end{equation} For convenience and brevity, denote $\Omega:={\rm supp}(\Pi)$. Now {\it assumption \ref{holder2}} comes into play: it implies, together with (\ref{Z8}) that \begin{align} \label{supnorm1} C N_{k_0-1}^{-\frac{\beta}{2\beta+d}}\log\frac{N}{\alpha} &\geq \|\bar \eta_{k_0-1}-\eta\|_{L_2(\hat\Pi_{k_0-1})}\geq B_2\|\bar \eta_{k_0-1}-\eta\|_{\infty,\Omega\cap\hat A_{k_0-1}} \end{align} To bound $$ \|\hat\eta_{k_0-1}(x)-\bar\eta_{k_0-1}(x)\|_{\infty,\Omega\cap\hat A_{k_0-1}} $$ we apply Proposition \ref{supnorm}. Recall that $\hat m_{k_0-1}$ depends only on the subsample $S_{k_0-1,1}$ but not on $S_{k_0-1,2}$. Let $$ \mathcal T_{k}:=\left\{\left\{X_i^{(j)},Y_i^{(j)}\right\}_{i=1}^{N_j^{{\rm act}}}, \ j\leq k-1; \ S_{k,1}\right\} $$ be the random vector that defines $\hat A_{k}$ and resolution level $\hat m_{k}$. Note that $ \mathbb E(\hat\eta_{k_0-1}(x)|\mathcal T_{k_0-1})=\bar\eta_{\hat m_{k_0-1}}(x) \quad \forall x \ \text{a.s.} $ \\ Proposition \ref{supnorm} thus implies \begin{align*} \nonumber \Pr\Bigg(\max_{x\in\Omega\cap\hat A_{k_0-1}}\lvert\hat\eta_{k_0-1}(x)-\bar\eta_{\hat m_{k_0-1}}(x)\rvert\ &\geq K t\sqrt{\frac{2^{d\hat m_{k_0-1}}}{N_{k_0-1}}} \, \Bigg|\, \mathcal T_{k_0-1}\Bigg)\leq \\ & \leq N \exp\left(\frac{-t^2}{2(1+\frac t3 C_3)}\right). \end{align*} Choosing $t=c\log(N/\alpha)$ and taking expectation, the inequality(now unconditional) becomes \begin{equation}\label{supnorm2} \Pr\left(\max_{x\in\Omega\cap\hat A_{k_0-1}}\lvert\hat\eta_{\hat m_{k_0-1}}(x)-\bar\eta_{\hat m_{k_0-1}}(x)\rvert\ \leq K \sqrt{\frac{2^{d\hat m_{k_0-1}}\log^2(N/\alpha)}{N_{k_0-1}}}\right) \geq 1-\alpha \end{equation} Let $\mathcal E_3$ be the event on which (\ref{supnorm2}) holds true. Combined, the estimates (\ref{Z1}),(\ref{supnorm1}) and (\ref{supnorm2}) imply that on $\mathcal E_{1}\cap \mathcal E_{2}\cap \mathcal E_3$ \begin{align}\label{Z4} \nonumber \|\eta-\hat \eta_{k_0-1}\|_{\infty,\Omega\cap\hat A_{k_0-1}}&\leq \|\eta-\bar\eta_{k_0-1}\|_{\infty,\Omega\cap\hat A_{k_0-1}}+ \|\bar\eta_{k_0-1}-\hat \eta_{k_0-1}\|_{\infty, \Omega\cap\hat A_{k_0-1}}\\ & \leq \frac{C}{B_2} N_{k_0-1}^{-\frac{\beta}{2\beta+d}}\log \frac{N}{\alpha} +K\sqrt{\frac{2^{d\hat m_{k_0-1}}\log^2 (N/\alpha)}{N_{k_0-1}}} \leq \\ & \nonumber \leq (K+C)\cdot N_{k_0-1}^{-\frac{\beta}{2\beta+d}}\log^2 \frac{N}{\alpha} \end{align} where we used the assumption $B_2\geq \log^{-1}N$. Now the width of the confidence band is defined via \begin{equation}\label{width} \delta_k:= 2(K+C)\cdot N_{k_0-1}^{-\frac{\beta}{2\beta+d}}\log^2 \frac{N}{\alpha} \end{equation} (in particular, $\tilde D$ from {\bf Algorithm 1a} is equal to $2(K+C)$). With the bound (\ref{Z4}) available, it is straightforward to finish the proof of the claim. Indeed, by (\ref{width}) and the definition of the active set, the necessary condition for $x\in \Omega\cap\hat A_{k_0}$ is $$ |\eta(x)|\leq 3(K+C)\cdot N_{k_0-1}^{-\frac{\beta}{2\beta+d}}\log^2 \frac{N}{\alpha}, $$ so that \begin{align*} \Pi(\hat A_{k_0})=\Pi(\Omega\cap\hat A_{k_0})& \leq \Pi\left(|\eta(x)|\leq 3(K+C)\cdot N_{k_0-1}^{-\frac{\beta}{2\beta+d}}\log^2 \frac{N}{\alpha}\right)\leq \\ & \leq \tilde B N_{k_0-1}^{-\frac{\beta\gamma}{2\beta+d}}\log^{2\gamma} \frac{N}{\alpha} \end{align*} by the low noise assumption. This completes the proof of the claim since $\Pr\left(\mathcal E_1\cap \mathcal E_2\cap \mathcal E_3\right)\geq 1-3\alpha$. \end{proof} We conclude this section by discussing running time of the active learning algorithm. Assume that the algorithm has access to the sampling subroutine that, given $A\subset [0,1]^d$ with $\Pi(A)>0$, generates i.i.d. $(X_i,Y_i)$ with $X_i\sim \Pi(dx|x\in A)$. \begin{proposition} \label{runningtime} The running time of {\bf Algorithm 1a(1b)} with label budget $N$ is $$ \mathcal O(dN\log^2 N). $$ \end{proposition} {\bf Remark } In view of Theorem \ref{main}, the running time required to output a classifier $\hat g$ such that $ R_{P}(\hat g)-R^*\leq \varepsilon $ with probability $\geq 1-\alpha$ is $$ \mathcal O\left(\left(\frac{1}{\varepsilon}\right)^{\frac{2\beta+d-\beta\gamma}{\beta(1+\gamma)}}{\rm poly}\left(\log\frac{1}{\varepsilon\alpha}\right)\right). $$ \begin{proof} We will use the notations of Theorem \ref{main}. Let $N_k^{{\rm act}}$ be the number of labels requested by the algorithm on step $k$. The resolution level $\hat m_k$ is always chosen such that $\hat A_k$ is partitioned into at most $N_k^{{\rm act}}$ dyadic cubes, see (\ref{index_set}). This means that the estimator $\hat \eta_k$ takes at most $N_k^{{\rm act}}$ distinct values. The key observation is that for any $k$, the active set $\hat A_{k+1}$ is always represented as the union of a finite number(at most $N_{k}^{{\rm act}}$) of dyadic cubes: to determine if a cube $R_j\subset \hat A_{k+1}$, it is enough to take a point $x\in R_j$ and compare ${\rm sign}(\hat\eta_{k}(x)-\delta_{k})$ with ${\rm sign}(\hat\eta_{k}(x)+\delta_{k})$: $R_j\in \hat A_{k+1}$ only if the signs are different(so that the confidence band crosses zero level). This can be done in $\mathcal O(N_k^{{\rm act}})$ steps.\\ Next, resolution level $\hat m_k$ can be found in $\mathcal O(N_k^{{\rm act}}\log^2 N)$ steps: there are at most $\log_2 N_k^{{\rm act}}$ models to consider; for each $m$, $\inf_{f\in \mathcal F_m}\hat P_k(Y-f(X))^2$ is found explicitly and is achieved for the piecewise-constant $$ \hat f(x)=\frac{\sum_i Y_i^{(k)}\mathcal I_{R_j}(X_i^{(k)})}{\sum_i \mathcal I_{R_j}(X_i^{(k)})}, \ x \in R_j. $$ Sorting of the data required for this computation is done in $\mathcal O(dN_k^{{\rm act}}\log N)$ steps for each $m$, so the whole $k$-th iteration running time is $\mathcal O(dN_k^{{\rm act}}\log^2 N)$. Since $\sum\limits_k N_k^{{\rm act}}\leq N$, the result follows. \end{proof} \section{Conclusion and open problems} \label{conclusion} We have shown that active learning can significantly improve the quality of a classifier over the passive algorithm for a large class of underlying distributions. Presented method achieves fast rates of convergence for the excess risk, moreover, it is adaptive(in the certain range of smoothness and noise parameters) and involves minimization only with respect to quadratic loss(rather than the $0-1$ loss). \\ The natural question related to our results is: \begin{itemize} \item Can we implement adaptive smooth estimators in the learning algorithm to extend our results beyond the case $\beta\leq 1$? \end{itemize} The answer to this second question is so far an open problem. Our conjecture is that the correct rate of convergence for the excess risk is $ N^{-\frac{\beta(1+\gamma)}{2\beta+d-\gamma(\beta\wedge 1)}}, $ up to logarithmic factors, which coincides with presented results for $\beta\leq 1$. This rate can be derived from an argument similar to the proof of Theorem \ref{main} under the assumption that on every step $k$ one could construct an estimator $\hat\eta_k$ with $$ \|\eta-\hat\eta_k\|_{\infty, \hat A_k}\lesssim N_k^{-\frac{\beta}{2\beta+d}}. $$ At the same time, the active set associated to $\hat\eta_k$ should maintain some structure which is suitable for the iterative nature of the algorithm. Transforming these ideas into a rigorous proof is a goal of our future work. \begin{comment} This conjecture comes from the following heuristic argument. Suppose $\beta>1$ and suppose that $\hat \eta_k$ is an estimator of $\eta$ based on a sample of size $N_k$ such that $\|\eta-\hat\eta_k\|_{\infty}\lesssim N_k^{-\frac{\beta}{2\beta+d}}\log^r N$. The natural resolution level for the active set $\hat A_k$ associated with this estimator is of order $N_k^{-\frac{\beta}{2\beta+d}}$, ignoring log-terms (compare to $N_k^{-\frac{1}{2\beta+d}}$ for $\beta\leq 1$). The size of $\hat A_k$ is controlled by the low noise assumption: $\Pi(\hat A_k)\lesssim N_k^{-\frac{\beta\gamma}{2\beta+d}}$. In order to improve the estimator on the active set, let $N_{k+1}:=N_k^{\beta}$, generate a new sample of size $\simeq N_{k+1}\Pi(\hat A_k)\lesssim N_k^{\beta}N_k^{-\frac{\beta\gamma}{2\beta+d}}$ from $\Pi(dx|\hat A_k)$ and use it to construct $\hat \eta_{k+1}$. Applying the bound $\sum_k N_k\leq N$ and arguing as in the proof of Theorem \ref{main}, one easily gets the conjectured rate as a result. Transforming this argument into a rigorous proof is a goal of our future work. \end{comment} \section*{Acknowledgements} I want to express my deepest gratitude to my Ph.D. advisor, Dr. Vladimir Koltchinskii, for his support and numerous helpful discussions.\\ I am grateful to the anonymous reviewers for carefully reading the manuscript. Their insightful and wise suggestions helped to improve the quality of presentation and results.\\ I would like to acknowledge support for this project from the National Science Foundation (NSF Grants DMS-0906880 and CCF-0808863) and by the Algorithms and Randomness Center, Georgia Institute of Technology, through the ARC Fellowship.

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\section{Introduction} In the analysis of the long term behavior of geometric heat flows, it is often useful to consider initial geometries with symmetry, since those symmetries are preserved and so the nonlinear flow equations are greatly simplified \cite{CIJ, IJ, LS, MCF}. If, given a topology and appropriate notion of stability, the flow solution starting at some metric $g_0$ does converge to a stable fixed-point geometry, does it follow that solutions starting at geometries nearby converge to a stable fixed-point geometry as well? Since geometric heat flows are generally weakly parabolic PDE systems, and since one expects such systems to have a well-posed initial value problem with solutions depending continuously on initial data, it is plausible that flow solutions starting near $g_0$ \emph{do} always converge. In this work, we make these ideas precise, and apply them to the Ricci flow on compact manifolds. Given a Riemannian manifold $(\mathcal{M},g),$ the Ricci flow is the weakly parabolic system \begin{equation} \begin{cases} \label{ricciflow} \frac{\partial}{\partial t} g(t) &= -2 \Rc(t) \\ g(0) &= g_0, \\ \end{cases} \end{equation} where $\Rc$ is the Ricci curvature tensor. We work in the topology of little H\"older spaces of metrics as in \cite{GIK}. We define a geometric heat flow to be \emph{convergence stable} if for every initial geometry $g_0$ whose resulting solution converges to a stable fixed-point geometry, there is a neighbourhood of $g_0$ such that every solution starting in that neighbourhood also converges to a stable fixed-point geometry. The key to proving convergence stability for a given flow is the verification that the flow is a well-posed initial value problem, with well-posedness encompassing continuous dependence of the solutions on initial data in addition to local existence and uniqueness. The continuous dependence of the Ricci flow for metrics on compact manifolds has been broadly assumed, but we have been unable to find a complete proof of it in the literature (e.g. see \cite{MOF}). Since our results here crucially involve continuous dependence in a certain topology, in Section \ref{Sec:ContDepRF} we provide a careful proof of this property. A complete understanding in the compact case is necessary to a subsequent project in which we study the flow on non-compact manifolds, and we therefore include an amount of detail that additionally provides an accessible entry into this theory for geometric analysts. In the course of this exposition, we in particular prove a crucial resolvent bound whose proof is either omitted or incomplete in various treatments of stability for geometric flows, see Proposition \ref{resolvent-estimate}. We now describe our results. We endow the bundle of symmetric two tensors, $\sym$, with a topology given by H\"older norms. In what follows, the spaces $h^{k,\alpha}(\sym)$, $k \in \mathbb{N}_0, \alpha \in (0,1)$, denote the completion of the space of smooth symmetric two tensors with respect to the $(k,\alpha)$-H\"older norm (see the Appendix for a detailed definition of these spaces). Given a metric $g_0$, we let $\tau(g_0) \in (0,\infty]$ denote the maximal time of existence of the Ricci flow starting at $g_0$. We prove the following theorem: \ \noindent{\bf Theorem A.} (Continuous Dependence of the Ricci Flow). \label{thm:ctsdep} {\it Let $(\mfld,g_0)$ be a smooth compact Riemannian manifold. Let $g_0 (t)$ be the maximal solution of the Ricci flow \eqref{ricciflow} for $t \in [0,\tau(g_0))$, \ $\tau(g_0) \le \infty$. Choose $0 < \tau < \tau(g_0)$. Let $k \geq 2$. There exist positive constants $r$ and $C$ depending only on $g_0$ and $\tau$ such that if $$ ||g_1 - g_0||_{h^{k+2,\alpha}} \le r,$$ then for the unique solution $g_1(t)$ of the Ricci flow starting at $g_1$, the maximal existence time satisfies $$\tau(g_1) \ge \tau,$$ and \begin{equation*} ||g_1(t) - g_0(t) ||_{h^{k,\alpha}} \le C || g_1 - g_0 ||_{h^{k+2,\alpha}} \end{equation*} for all $t \in [0,\tau].$} \ The proof of this theorem involves the Ricci-DeTurck flow, to which parabolic PDE theory may be applied; our results give a semigroup proof of well-posedness. Since we work in finite regularity spaces, the passage from the Ricci-DeTurck flow to the Ricci flow accounts for the drop in regularity in the theorem above. Note that we are not asserting a drop in regularity of the solution $g_1(t)$ which is smooth after $t>0$, only the differentiability of the dependence estimate. After giving some analytic background in Section \ref{Sec:Analytic}, in Section \ref{Sec:RDF} we present a detailed exposition of the continuous dependence of the Ricci-DeTurck flow using analytic semigroup theory. Working within the setting of little H\"older spaces, we can simplify various parts of the semigroup-based treatment of the well-posedness of parabolic equations; see, e.g., \cite{Lun95}. In Section \ref{Sec:ContDepRF}, we return to the Ricci flow and prove Theorem A. Having established the continuous dependence of the Ricci flow for metrics on compact manifolds, we prove convergence stability of the Ricci flow on closed $\mathbb{T}^n$ and $\mathbb{H}^n$ in Section \ref{ConvStab}. Denote the Ricci flow solution starting at $g_0$ by $g_0(t)$. We say {$g_0(t)$ {\it converges to $g_{\infty}$ in $h^{2,\alpha}$} if $\tau(g_0)=\infty$ and \[ \lim_{t \to \infty} \| g_0(t) - g_{\infty} \|_{h^{2,\alpha}} = 0. \] Note that by results of Hamilton, this is sufficient to obtain convergence in $C^{\infty}(\mfld)$, \cite[Section 17]{Hamilton}. We say $g_{\infty}$ is a {\it stable fixed point} if there is some $\varepsilon > 0$ so that for every $g_0$ within the $\varepsilon$-ball about $g_{\infty}$ in $h^{2,\alpha}$, the Ricci flow starting at $g_0$ converges to a fixed point. D. Knopf and two of the authors proved that any flat metric on the torus $\mathbb{T}^n$ is a stable fixed point for the Ricci flow in this sense, see \cite[Theorem 3.7]{GIK}. In \cite{KnopfYoung}, the D. Knopf and A. Young prove that a compact hyperbolic metric is also a stable fixed point for a normalized Ricci flow in this sense. The statement of the $\mathbb{T}^n$ result is as follows: \ \noindent{\bf Theorem B.} (Convergence Stability (Flat)). \label{thm:cstab-flat} {\it Let $(\mfld, g_0)$ be a smooth closed torus, and let $g_0(t)$ be a solution of the Ricci flow \eqref{ricciflow} starting at $g_0$ that converges to a flat metric $g_\infty$. There exists $r>0$ depending only on $g_0$ such that if $g_1 \in B_r(g_0) \subset h^{2,\alpha}(\sym)$, then the solution $g_1(t)$ with $g(0) = g_1$ of the Ricci flow exists for all time and converges to a flat metric.} \ This theorem shows that the set of metrics which converge to a flat fixed point is \emph{open} in a certain topology of metrics on $\mathbb{T}^n$. We emphasize that the metrics contained in these sets need not have symmetries, nor are they generally small perturbations from a fixed flat metric. We apply this to the results of J. Lott and N. Sesum, who showed that for two families of geometries on $\mathbb{T}^3$ defined by isometries, every Ricci flow solution which starts at a metric contained in one of these families must converge exponentially quickly to a flat metric on $\mathbb{T}^3$ \cite{LS}. By Theorem B, we obtain open neighbourhoods of the set of these metrics such that Ricci flow solutions which start with initial geometries in either of these neighbourhoods must converge to flat metrics. We discuss this and other applications in Section \ref{ConvStab}. We strongly expect that the notion of convergence stability holds for a wide range of other geometric heat flows. We hope to return to these questions in a subsequent work. In the Appendix we provide some of the background definitions and results needed in this paper including the basic function spaces which arise and a few results from elliptic PDE theory. \textbf{Acknowledgments}{ The authors would like to thank Dan Knopf, Jack Lee, Rafe Mazzeo, and Haotian Wu for helpful conversations related to this work. This work was supported by grants from the Simons Foundation (\#426628, E. Bahuaud and \#283083, C. Guenther). J. Isenberg was partially supported by the NSF grant DMS-1263431. This work was initiated at the 2015 BIRS workshop {\it Geometric Flows: Recent Developments and Applications} (15w5148).} \section{Analytic background} \label{Sec:Analytic} In this section, we introduce some of the ideas and results from analytic semigroup theory that we need in order to prove our main results on continuous dependence and convergence stability for Ricci flow. An overview of stability techniques and results for geometric flows, and in particular maximal regularity theory, can be found in Chapter 35 of \cite{RFV4}. An excellent reference for this theory in a general setting is provided by \cite{Lun95}. A careful exposition of the geometric ideas that we use, including how to define the relevant function spaces of tensors, as well as certain key facts from elliptic PDE theory, can be found in Appendix A. As motivation of the semigroup approach, we recall that in the analysis of systems of ordinary differential equations, for a constant $n$ by $n$ matrix $A$ over $\mathbb C$, and for a vector-valued function of time $x = x(t) \in \mathbb{C}^n$, the solution to the initial value problem \begin{equation*} \begin{cases} x'(t) &= A x(t), \\ x(0) &= x_0,\\ \end{cases} \end{equation*} exists for all time and can be written in terms of the matrix exponential in the following form: $x(t) = e^{t A} x_0$. One readily verifies that the long-time behavior of solutions near equilibrium points depends on the eigenvalues of $A$. We discuss an extension of this setting to general unbounded operators acting on Banach spaces, as well as nonlinear generalizations. Let $\mathcal{X}$ be a Banach space, and let \[ A: \mathcal{D}(A) \subset \mathcal{X} \longrightarrow \mathcal{X}, \] be a closed linear operator. A linear operator $A$ is {\it closed} if its graph is a closed subset of $\mathcal{X} \times \mathcal{X}$. In that case $\mathcal{D}(A)$ with the graph norm, $\|x\|_{\mathcal{D}(A)} := \|x\| + \|Ax\|$, is a Banach space. The \emph{spectrum} of $A$ with respect to the Banach space $\mathcal{X}$ is defined to be the set \[ \sigma_\mathcal{X}(A) := \{ \lambda \in \mathbb{C}: \lambda I - A \; \; \mbox{does not have a bounded inverse} \}, \] while the corresponding \emph{resolvent set} is defined as \[\rho_\mathcal{X}(A) := \mathbb{C} \smallsetminus \sigma_\mathcal{X}(A).\] For a chosen value $\lambda$ in the resolvent set $\rho_\mathcal{X}(A)$, the corresponding resolvent operator is the map \begin{equation*} R_{\lambda} := (\lambda I - A)^{-1}: \mathcal{X} \longrightarrow \mathcal{D}(A) \subset \mathcal{X}. \end{equation*} We denote by $\mathcal{L}(\mathcal{X})$ the set of bounded linear operators on a normed space $\mathcal{X}$, and we denote by $\mathcal{L}(\mathcal{X},\mathcal{Y})$ the set of bounded operators from $\mathcal{X}$ to $\mathcal{Y}$. A \emph{sector} is defined to be the set \[ S_{\theta,\omega} = \{ \lambda \in \mathbb{C}: \lambda \neq \omega, |\arg(\lambda -\omega)| < \theta\}.\] The linear operator $A$ is \emph{sectorial} if there exist constants $\omega \in \mathbb{R}$ and $\theta \in (\pi/2, \pi)$ and $M > 0$ such that \ \begin{enumerate} \item[R1.] (The resolvent set contains a sector): the set $S_{\theta,\omega}$ is a subset of $\rho_\mathcal{X}(A).$ \item [R2.] (Resolvent estimate): the inequality \begin{equation} \label{ResEst} \| R_{\lambda} \|_{\mathcal{L}(\mathcal{X})} \leq \frac{M}{|\lambda - \omega|} \end{equation} holds for all $\lambda \in S_{\lambda,\omega}$. \end{enumerate} \ Now consider the Banach space valued initial value problem \begin{equation} \label{IVP} u'(t) = A u(t), \; \; u(t_0) = u_0, \end{equation} where $u_0 \in \mathcal{D}(A).$ If $A$ is sectorial, then for an appropriate contour $\gamma$ in the complex plane and for $t>0$, one may define the exponential of $tA$ by \[ e^{tA} := \frac{1}{2\pi i} \int_{\gamma} e^{t \lambda} R_\lambda \; d\lambda, \] and analyzing the mapping properties of $e^{tA} $, one obtains well-posedness for the linear IVP \eqref{IVP}. Sectoriality is important for nonlinear initial value problems as well. In particular, beginning with a continuous embedding of Banach spaces $\mathcal{D} \subset \mathcal{X}$, we let $\mathcal{O} \subset \mathcal{D}$ be an open subset, and we consider the evolution equation \[ u'(t) = F(u(t)), \; \; u(t_0) = u_0, \] where \[ F: \mathcal{O} \longrightarrow \mathcal{X}. \] \ We require three conditions on the flow: \begin{itemize} \label{hypothesis} \item[H0.] $F$ is continuous and Fr\'echet differentiable. \item[H1.] For each $u \in \mathcal{O}$, the Fr\'echet derivative $F_u$ of $F$ is sectorial in $\mathcal{X}$, and its graph norm is equivalent to the norm of $\mathcal{D}$. \item[H2.] For each $u \in \mathcal{O}$, there exist positive constants $R$ and $L$ depending on $u$ such that the inequality \[ \| F_u(v) - F_u(w)\|_{\mathcal{L}(\mathcal{D},\mathcal{X})} \leq L \| v - w\|_\mathcal{D}, \] holds for all $v, w \in B_R(u) \subset \mathcal{D}$. \end{itemize} \ To state a well-posedness theorem for a nonlinear map $F$ which satisfies conditions H1 and H2, one needs to address a technicality which arises in the study of parabolic evolution systems: A key feature of parabolic systems is \emph{parabolic regularity}, which is the propensity of solutions of parabolic evolution solutions to become smooth even if the initial data is significantly less regular. Hence, in the formulation of a well-posedness theorem, one must account for diminished regularity as $t$ approaches zero. Various strategies are used for dealing with this issue in the semigroup setting: one approach is to work with certain weighted function spaces involving powers of $t$ which precisely capture regularity of the solutions down to $t = 0$. A second approach is to use continuous interpolation spaces between the Banach spaces $\mathcal{D}$ and $\mathcal{X}$. For our work the first approach is adequate. We thus introduce a series of function spaces that are used only in this section. For a time interval $(a,b)$, a H\"older exponent $\alpha \in (0,1)$, and a Banach space $\mathcal{X}$, we introduce a semi-norm on curves from $(a,b)$ into $\mathcal{X}$ by \[ [ f ]_{C^{\alpha}( (a,b); \mathcal{X} )} = \sup_{ s,t \in (a,b), s \neq t} \frac{\| f(t) - f(s) \|_\mathcal{X}}{{|t-s|^{\alpha}}}, \] and a corresponding norm by \[ \|f\|_{C^{\alpha}( (a,b); \mathcal{X} )} = \sup_{t \in (a,b)} \|f(t)\|_\mathcal{X} + [ f ]_{C^{\alpha}( (a,b); \mathcal{X} )}. \] Denote the space of continuous curves for which this norm is finite by $C^{\alpha}( (a,b); \mathcal{X} )$. The (time) weighted space $C^{\alpha}_{\alpha}( (a,b); \mathcal{X})$ is given by curves for which the norm \[ \|f\|_{C^{\alpha}_{\alpha}( (a,b); \mathcal{X} )} = \sup_{t \in (a,b)} \|f(t)\|_\mathcal{X} + [ (t-a)^{\alpha} f ]_{C^{\alpha}( (a,b); \mathcal{X} )} \] is finite. We state an existence theorem in a form suitable for our work here. The estimate \eqref{well-posedness local estimate} below is the key to the well-posedness result in Theorem \ref{well-posedness}. We denote an initial condition by $u_i$, and the solution that starts at $u_i$ by $u_i(t)$. \begin{theorem}[Amalgamation of Theorems 8.1.1 and Corollary 8.1.2 in \cite{Lun95}] \label{existence} For $\mathcal{D} \subset \mathcal{X}$ a continuous embedding of Banach spaces, and for $\mathcal{O}$ an open subset of $\mathcal{D}$, let \[ F: \mathcal{O} \longrightarrow \mathcal{X} \] satisfy hypotheses $H0$, $H1$, and $H2$ above. For every $u_0 \in \mathcal{O}$ such that $F(u_0) \in \overline{\mathcal{D}}$, there exists $\delta = \delta(u_0)>0$ and $r = r(u_0) > 0$ such that for any $t_0 \in [0, r)$ there is a unique solution $u$ to the (nonlinear) initial value problem \begin{equation} \label{F-IVP} u'(t) = F(u(t)), \; \; u(t_0) = u_0. \end{equation} The solution $u$ satisfies the regularity condition \[ u \in C( [t_0,t_0+\delta]; \mathcal{D} ) \cap C^{1}( [t_0,t_0+\delta]; \mathcal{X} ), \] and moreover $u \in C^{\alpha}_{\alpha}( (t_0, t_0+\delta]; \mathcal{D})$ and $u$ is the unique solution to the IVP in $$\bigcup_{\beta \in (0,1)}~C^{\beta}_{\beta}( (t_0, t_0+\delta]; \mathcal{D}) \cap C( [t_0,t_0+\delta]; \mathcal{D} ).$$ Additionally there is a constant $k = k(u_0) > 0$ such that if $u_1\in \mathcal{O}$ satisfies the inequality \[ \| u_1 -u_0\| \leq r, \; \; F(u_1) \in \overline{\mathcal{D}}, \] then the solution $u_1(t)$ satisfies the inequality \begin{equation} \label{well-posedness local pre-estimate} \|u_1(t) - u_0(t)\|_{C^{\alpha}_{\alpha}((t_0,t_0+\delta],\mathcal{D})} \leq k \| u_1- u_0\|_\mathcal{D}. \end{equation} \end{theorem} This theorem guarantees the short-time existence, uniqueness, and continuous dependence on initial conditions - in short {\it well-posedness} - of the nonlinear initial value problem \eqref{F-IVP}. In view of the definition of the norm for the space $C^{\alpha}_{\alpha}((t_0,t_0+\delta],\mathcal{D})$, we may translate the estimate from equation \eqref{well-posedness local pre-estimate} to the more useful form \begin{equation} \label{well-posedness local estimate} \|u_1(t) - u_0(t)\|_{\mathcal{D}} \leq k \| u_1 - u_0\|_\mathcal{D}, \; \; \mbox{for all} \; t \in (t_0,t_0+\delta). \end{equation} \section{The Ricci-DeTurck flow} \label{Sec:RDF} As is well-known, the Ricci flow PDE system is not itself parabolic; to prove local existence (and well-posedness) for Ricci flow, one uses the Ricci-DeTurck flow as an intermediary. In this section, we review the setup of the Ricci-DeTurck flow system, with an emphasis on the analysis which leads to a proof that the Ricci-DeTurck initial value problem is well-posed (including continuous dependence on initial data) for data in the Banach spaces of interest for this work. Let $\mfld$ be the manifold on which the evolving metrics $g(t)$ are defined, and let $\widetilde{g}$ be a fixed reference metric on $\mfld$. Denoting the connection coefficients, the covariant derivative operators, and the curvatures corresponding to the reference metric $\widetilde{g}$ with a tilde, and those corresponding to the evolving metric $g$ without a tilde, we define the DeTurck vector field \[ W^k = W^k(g,\widetilde{g}) := g^{pq} \left( \Gamma_{pq}^k - \widetilde{\Gamma}_{pq}^k \right), \] and then write the initial value problem for the Ricci-DeTurck flow as follows: \[ \partial_t g = F^{\widetilde{g}}(g), \; \; g(0) = g_0; \] here $F^{\widetilde{g}}(g) := - 2 \Rc (g) - 2 \mathcal{L}_W g$, and the superscript on $F$ emphasizes the choice of background metric appearing through the vector field $W$. If we study the Ricci-DeTurck flow for metrics $g$ close to the reference metric $\widetilde{g}$, then it is useful to consider the linearization of this flow relative to $\widetilde{g}$; we obtain (for $h := g-\widetilde{g}$) the linear evolution system \[\partial_t h= A^{\widetilde{g}}_{\widetilde{g}} h,\] where the operator $A^{\widetilde{g}}_{\widetilde{g}}$ is the Lichnerowicz Laplacian (relative to $\widetilde{g}$): \[ A^{\widetilde{g}}_{\widetilde{g}} h := D_h F^{\widetilde{g}}(\widetilde{g}) = \left. \frac{d}{ds} \right|_{s=0} F^{\widetilde{g}}( \widetilde{g} + sh ) = \tilde \Delta_L h. \] One readily verifies the following explicit formula for $A^{\widetilde{g}}_{\widetilde{g}} = \tilde \Delta_L$: \[\tilde \Delta_L h_{ij} := {\widetilde{g}}^{ab} \widetilde{\nabla}_a \widetilde{\nabla}_b h_{ij} + 2 \widetilde{\Rm}_{iabj} h^{ab} - \widetilde{\Rc}_{ia} h^{a}_j - \widetilde{\Rc}_{aj} h^{a}_i. \] We also consider solutions that start near a fixed metric $\mathfrak{g}$ which is distinct from the reference metric $\widetilde{g}$. For this, we linearize the operator $F^{\widetilde{g}}$ about the metric $\mathfrak{g}$. Labeling this linearization operator $A^{\widetilde{g}}_{\mathfrak{g}}$, we calculate (following Proposition 3.2 of \cite{GIK}) \begin{equation*} A^{\widetilde{g}}_{\mathfrak{g}} h:= D_h F^{\widetilde{g}}(\mathfrak{g}) = \left. \frac{d}{ds} \right|_{s=0} F^{\widetilde{g}}( \mathfrak{g} + sh ) = \Delta_L^{\mathfrak{g}} h - \Psi_{\widetilde{g}} h, \end{equation*} where $\Psi_{\widetilde{g}} h$ is a first-order linear operator in $h$ whose coefficients involve $\mathfrak{g}$-contractions of at most two $\mathfrak{g}$-covariant derivatives of $\widetilde{g}$. This can be written in the following form \begin{equation} \label{form-of-A} A^{\widetilde{g}}_{\mathfrak{g}} h = a(\mathfrak{g}) \partial^2 h + b(\mathfrak{g}, \partial \mathfrak{g}, \widetilde{g}, \partial \widetilde{g}) \partial h + c(\mathfrak{g}, \partial \mathfrak{g}, \partial^2 \mathfrak{g}, \widetilde{g}, \partial \widetilde{g}, \partial^2 \widetilde{g}) h \end{equation} for suitable components $a, b, c$. A calculation shows that $A^{\widetilde{g}}_{\mathfrak{g}}$ is elliptic. Having identified the operators of interest, we now discuss the spaces of tensors on which we apply Theorem \ref{existence}. Let $\sym$ denote the bundle of symmetric $2$-tensor fields on $\mfld$. Since $A^{\widetilde{g}}_{\mathfrak{g}}$ is a second-order elliptic operator on $\sym$, the H\"older spaces $C^{2,\alpha}(\sym)$ are a natural choice. Unfortunately, $C^{2,\alpha}(\sym)$ is not a closed subset of $C^{0,\alpha}(\sym)$ in the $\alpha$-norm, and this makes the hypothesis of Theorem \ref{existence} involving the closure of the domain $\mathcal{D}$ difficult to check for generic initial data. Instead we use ``little H\"older spaces'', since the inclusion $h^{2,\alpha} \hookrightarrow h^{0,\alpha}$ is continuous and dense (see Appendix \ref{Geometric background} for a detailed introduction of little H\"older spaces). In particular let $$P := A_{\mathfrak{g}}^{\widetilde{g}},$$ and set \[ \mathcal{D}(P) = h^{2,\alpha}(\sym), \] with \[ \mathcal{X} = h^{0,\alpha}(\sym). \] Observe that the closure of $\mathcal{D}(P)$ in $\mathcal{X}$ is $\mathcal{X}$: \[ \overline{\mathcal{D}(P)}= \mathcal{X}. \] In particular for any $g \in \mathcal{D}(P)$, $F(g) \in \mathcal{X} = \overline{\mathcal{D}(P)}$. We could also take the pair $(\mathcal{D},\mathcal{X}) = (h^{k+2,\alpha}(\sym), h^{k,\alpha}(\sym))$, but to keep the notation simple we restrict to $k=0$. We now build toward verifying the hypotheses of Theorem \ref{existence} for a certain open subset of $\mathcal{D}(P)$. \subsection{Sectoriality of the DeTurck operator} We must show that $P$ is sectorial between these function spaces. Since this point has been treated tersely (and at times incorrectly) in the existing literature, we start at the beginning. Note that the arguments we give in this section use the compactness of $\mfld$ in an essential way. Assume that the chosen reference metric $\widetilde{g}$ and the metric $\mathfrak{g}$ at which we are linearizing, are smooth. Then $P = A_{\mathfrak{g}}^{\widetilde{g}}$ is an unbounded self-adjoint elliptic operator with smooth coefficients on $L^2(\sym)$, with maximal domain $\mathcal{D}_{L^2}(P) := \{ w \in L^2(\sym): P w \in L^2(\sym) \}$. We know that there exists a finite $K \in \mathbb{R}$ for which $\sigma_{L^2}(P) \subset (-\infty, K]$ (see e.g. Appendix Theorem \ref{thm:L2-spectrum}). Thus the resolvent set $\rho_{L^2}(P)$ contains a sector in the complex plane. We now analyze the spectrum of $P$ acting as an operator on H\"older spaces $C^{k,\alpha}(\sym)$ for $k \in \mathbb{N}$ and $\alpha \in (0,1)$. By analogy with the $L^2$ theory, we consider the maximal domain for $P$: \[ \mathcal{D}_{C^{0,\alpha}}(P) := \{ u \in C^{0,\alpha}(\sym) | Pu \in C^{0,\alpha}(\sym)\}. \] By definition, the $C^{0,\alpha}$-resolvent set for $P$ is given by \[ \rho_{C^{0,\alpha}}(P) = \left\{ \lambda \in \mathbb{C}| \lambda I - P: \mathcal{D}(P) \to C^{0,\alpha}(\sym) \; \mbox{has bounded inverse} \right\}, \] and the corresponding spectrum is \[\sigma_{C^{0,\alpha}}(P) = \mathbb{C} \smallsetminus \rho_{C^{0,\alpha}}(P).\] For $\lambda \in \rho_{C^{0,\alpha}}(P)$ we have the resolvent operator, \begin{equation} \label{ResOp} R_{\lambda} := (\lambda I - P)^{-1}: C^{0,\alpha}(\sym) \to \mathcal{D}(P) \subset C^{0,\alpha}(\sym). \end{equation} We next give a proof of the following Proposition, which guarantees that the spectrum of $P$ on $C^{0,\alpha}(\sym)$ coincides with its $L^2$ spectrum. This result relies on elliptic regularity and on the Sobolev embedding theorem, as well as the compactness of $\mfld$. \begin{prop} \label{spectra-coincide} Let ($\mfld,g)$ be a closed Riemannian manifold, and $P = A_{\mathfrak{g}}^{\widetilde{g}}$. For any $\alpha \in (0,1)$, the $L^2$ and $C^{0,\alpha}$ spectra coincide: \[ \sigma_{L^2(\sym)}( P ) = \sigma_{C^{0,\alpha}(\sym)}( P ). \] \end{prop} \begin{proof} We prove that these subsets of the complex plane coincide. We first choose any $\lambda \in \sigma_{L^2}(P)$. Since the $L^2$ spectrum of $P$ on a compact manifold is purely point spectrum, there is an $L^2$-eigentensor field $u$ corresponding to $\lambda$ for which \[ (\lambda I - P) u = 0. \] Elliptic regularity implies that $u$ is smooth, and thus that $u$ is an element of $\mathcal{D}_{C^{0,\alpha}}(P)$. It follows that $\lambda I - P$ as an operator on $C^{0,\alpha}$ has a non-trivial kernel; consequently $\lambda \in \sigma_{C^{0,\alpha}}(P)$. This tells us that $\lambda$ is an eigenvalue for $P$ acting on $C^{0,\alpha}$ as well as on $L^2$. We now choose $\lambda \in \rho_{L^2}( P )$. We know that $(\lambda I - P): L^2(\sym) \to L^2(\sym)$ is invertible with bounded inverse on its domain; i.e., there exists a constant $C_1 > 0$ such that for all $v \in L^2(\sym)$, one has \begin{equation} \label{l2-est} \| (\lambda I - P)^{-1} v \|_{L^2} \leq C_1 \|v\|_{L^2}. \end{equation} Since $C^{0,\alpha}(\sym) \subset L^2(\sym)$ on a compact manifold, the maximal H\"older domain $\mathcal{D}_{C^{\alpha}}(P)$ is defined and is contained in the maximal $L^2$ domain, $\mathcal{D}_{L^2}(P)$. To show that if $\lambda \in \rho_{L^2}( P )$ then $\lambda \in \rho_{C^{0,\alpha}(\sym)}(P)$ (thereby completing the proof of this proposition), we need to verify that the estimate of equation \eqref{l2-est} holds in the H\"older norms, as well as in the $L^2$ norms. Indeed, we first note that for every $ v \in C^{0,\alpha}(\sym)$, $u := (\lambda I - P)^{-1} v$ is a well-defined element of $L^2(\sym)$, for which the following elliptic PDE holds \begin{equation} \label{vu} v = (\lambda I - P) u. \end{equation} We now begin a bootstrap procedure. Since $C^{0,\alpha}(\sym) \subset L^p(\sym)$ for all $p$, we may regard $v \in L^p(\sym)$. Hence, elliptic regularity guarantees that $u \in L^{2,p}(\sym)$. Since $L^{2,p}(\sym) \subset L^p(\sym)$ is a compact embedding, and since $L^p(\sym) \subset L^2(\sym)$ is a continuous inclusion for $p \geq 2$, the elliptic estimate corresponding to \eqref{vu} yields \[ \|u \|_{L^{2,p}} \leq C ( \|v\|_{L^p} + \|u\|_{L^2} ); \] here we note that this constant $C$ depends on $\lambda$, since the operator appearing in \eqref{vu} involves $\lambda$. Now, for $p$ sufficiently large (i.e., for $p$ large enough so that $2- \frac{n}{p} > \alpha)$), Sobolev embedding tells us that $L^{2,p}(\sym) \subset C^{0,\alpha}(\sym)$. Consequently, we find that \[ \|u\|_{C^{0,\alpha}} \leq \|u \|_{L^{2,p}} \leq C ( \|v\|_{L^p} + \|u\|_{L^2} )\leq C' \|v\|_{C^{0,\alpha}}. \] We thus verify that $\| (\lambda I - P)^{-1} v \|_{C^{0,\alpha}} \leq C \|v\|_{C^{0,\alpha}}.$ This shows that $\lambda \in \rho_{C^{0,\alpha}(\sym)}(P)$, which completes the proof. \end{proof} Proposition \ref{spectra-coincide} combined with the fact that the $L^2$-spectrum of $P$ is contained in the ray $(-\infty, K]$ show that $\rho_{C^{0,\alpha}}(P)$ contains a sector. The second part of the proof of the sectoriality of a given operator is the verification of the resolvent estimate \eqref{ResEst}. This requires that we obtain uniform control of the constant $M$, and thus requires some care. In the literature it has been stated that the resolvent estimate follows from standard Schauder theory; however, since the operator $\lambda I - P$ depends on $\lambda$, use of the Schauder estimates results in constants which depend on $\lambda$. Instead, we adapt a scaling argument from \cite{BM} to obtain the resolvent estimate. The reader may compare our discussion with that in Chapter 3 of \cite{Lun95}; there, resolvent estimates (on $\mathbb{R}^n$) are obtained from the classical Agmon-Douglis-Nirenberg estimates for operators with less regular coefficients. \begin{prop} \label{resolvent-estimate} Let $(\mfld,g)$ be a compact smooth Riemannian manifold, and let $ \Delta_L$ denote the Lichnerowicz Laplacian with respect to $g$ acting on symmetric $2$-tensors. For any positive constant $K$ greater than the maximum eigenvalue of $ \Delta_L$ and for a fixed $\alpha \in (0,1)$, there exists $M > 0$ such that for all $\lambda$ with $Re(\lambda) > K$, and for all $f \in C^{0,\alpha}(\sym)$, the resolvent operator $R_\lambda$ (see \eqref{ResOp}) satisfies the inequality \[ \| R_{\lambda} f \|_{\mathcal{L}(C^{0,\alpha})} \leq \frac{M}{|\lambda|} . \] \end{prop} \begin{proof} As a preliminary step, by rescaling the metric we may assume that the injectivity radius is bounded below by $1$. We proceed by contradiction: If the estimate is false, then for every $n \in \mathbb{N}$ it must be possible to find a number $\lambda_n$ with $Re(\lambda_n) > K$ and a tensor field $f_n \in C^{0,\alpha}(\sym)$ for which the following holds: \begin{equation} \label{eqn:bwoc-holder} \| R_{\lambda_n} f_n \|_{C^{0,\alpha}} > \frac{n}{|\lambda_n|} \| f_n \|_{C^{0,\alpha}}. \end{equation} If we further assume that the sequence of numbers $\lambda_n$ remains in a compact set, then some subsequence must converge to some $\lambda_*$ with $ Re(\lambda_*) \geq K$. But then along this subsequence, the estimate of equation \eqref{eqn:bwoc-holder} above shows that $\|R_{\lambda_*}\|_{\mathcal{L}(C^{0,\alpha})}$ must diverge, which is impossible since $\lambda_*$ is not contained in the H\"older spectrum of $\Delta_L$. Consequently, we may now assume that $\lambda_n \to \infty$ (note that $\lambda_n \in \mathbb{C}$, so this means that $|\lambda_n| \to +\infty$). We consider the sequence of tensor fields \[ w_n(x) := \frac{R_{\lambda_n} f_n(x)}{\| R_{\lambda_n} f_n \|_{C^{0,\alpha}}}. \] Applying the operator $I - \lambda_n^{-1} \Delta_L$ to each $w_n$, and recalling the definition of the resolvent operator $R_{\lambda_n}$ (see \eqref{ResOp}), we obtain \begin{equation} \label{eqn:limit} (I - \lambda_n^{-1} \Delta_L) w_n = \lambda_n^{-1} ( \lambda_n I - \Delta_L )w_n = \lambda_n^{-1} \frac{f_n(x)}{\|R_{\lambda_n} f_n\|_{C^{0,\alpha}}}. \end{equation} It is an immediate consequence of inequality \eqref{eqn:bwoc-holder} that the $C^{0,\alpha}$ norm of the right hand side of equation \eqref{eqn:limit} converges to zero as $n \to \infty$. Pulling the analysis back to normal coordinates centred at certain points, we now show that this limiting behavior leads to a contradiction. It follows from the compactness of the manifold $\mfld$ and from the continuity of $R_{\lambda_n}f_n$ that for each $n \in \mathbb{N}$, there is a point $x_n$ at which the supremum of $R_{\lambda_n} f_n$ is achieved; i.e., we have \[ \| R_{\lambda_n} f_n \|_{L^{\infty}(\mfld)} = \sup_{x\in \mfld} |R_{\lambda_n} f_n(x)| = |R_{\lambda_n} f_n(x_n)| \] (of course the norm of a $2$-tensor $\zeta_{ij}$ is given by $| \zeta |^2 = g^{ij}g^{kl} \zeta_{ik} \zeta_{jl}$). Also as a consequence of the compactness of $\mfld$, we know that there exists a subsequence of the sequence $\{x_n\}$ which converges to some point $x_\infty \in \mfld$. Relabeling this subsequence as $\{x_n\}$, and letting $B_n$ denote the unit ball with centre $x_n$, we now introduce geodesic normal coordinates $z$ in each ball, centred at $x_n$ (to simplify the notation, we suppress any labelling of these geodesic coordinates related to the index $``n"$). In terms of these coordinates (in each ball $B_n$), the metric components takes the special form $g_{ij} = \delta_{ij} + O(|z|^2)$, and moreover since compact smooth Riemannian manifolds are of bounded geometry, any number of derivatives of the metric are uniformly bounded in normal coordinates \cite[Proposition 1.2]{Eichhorn}. Finally the tensor bundle $\sym$ is trivialized in terms of these coordinates over $B_n$. Now return to equation \eqref{eqn:limit} and for each $n$ express in these normal coordinates. The components then read \begin{equation} \label{eqn:components1} \lambda_n (w_n)_{ij} - (\Delta_L w_n)_{ij} = \frac{(f_n(z))_{ij}}{\|R_{\lambda_n} f_n\|_{C^{0,\alpha}}}. \end{equation} In terms of geodesic coordinates (in each ball), the Lichnerowicz Laplacian---acting on a symmetric tensor field $u$--- takes the following form \begin{align*} \label{DeltaLz} (\Delta_L u)_{ij} &= {g}^{kl} \nabla_k \nabla_l u_{ij} + 2 \Rm_{iklj} u^{kl} - \Rc_{ik} u^{k}_j - \Rc_{kj} u^{k}_i\\ &= \delta^{kl} \frac{\partial^2 u_{ij}}{\partial z^k \partial z^l} + (S(u) + 2\Rm*u - 2\Rc*u)_{ij}, \end{align*} where $S$ is a second-order operator in $u$ with bounded smooth coefficients that vanishes to order $|z|^2$ and $*$ indicates contractions with respect to the metric $g$. Using this we find equation \eqref{eqn:components1} now reads \begin{equation} \label{eqn:components2} \lambda_n (w_n)_{ij} - \delta^{kl} \frac{\partial^2 (w_n)_{ij}}{\partial z^k \partial z^l} - (S(w_n) + 2\Rm*w_n - 2\Rc*w_n)_{ij} = \frac{(f_n(z))_{ij}}{\|R_{\lambda_n} f_n\|_{C^{0,\alpha}}}, \end{equation} which we regard as a sequence of a system of equations on $B_1(0)$. Introduce the operator $Z$ whose action on $w_n$ is given by \[ Z w_n := S(w_n) + 2\Rm*w_n - 2\Rc*w_n. \] Now rescale each system by introducing coordinates $y := \sqrt{|\lambda_n|} z$. In doing so, we observe that since the coordinates $z$ are valid on $B_1(0)$, it follows that the coordinates $y$ are valid on $B_{|\lambda_n|}(0)$. Computing $S$ from \eqref{eqn:components2} in these rescaled coordinates using the chain rule $\partial_z = \sqrt{|\lambda_n|} \partial_y$ and $\partial^2_z = {|\lambda_n|} \partial^2_y$ and the asserted facts about the metric in normal coordinates allows us to write \[ Z w_n = |\lambda_n|\widetilde{Z} w_n, \] where $\widetilde{Z}$ has coefficients that converge to zero on compact subsets. Thus we rewrite equation \eqref{eqn:components2} in the rescaled $y$ coordinates as \begin{equation} \label{eqn:components3} \lambda_n (w_n)_{ij} - |\lambda_n| \delta^{kl} \frac{\partial^2 (w_n)_{ij}}{\partial y^k \partial y^l} - |\lambda_n| \widetilde{Z}(w_n)_{ij} = \frac{\left(f_n\left( \frac{1}{\sqrt{|\lambda_n|}} y \right)\right)_{ij}}{\|R_{\lambda_n} f_n\|_{C^{0,\alpha}}}. \end{equation} Set \[ \mathcal{E}_1 = \frac{f_n\left( \frac{1}{\sqrt{|\lambda_n|}} y \right)}{ |\lambda_n| \cdot \|R_{\lambda_n} f_n\|_{C^{0,\alpha}}}, \] from here we can write equation \eqref{eqn:components3} as \begin{equation} \label{eqn:components4} \frac{\lambda_n}{|\lambda_n|} (w_n)_{ij} - \delta^{kl} \frac{\partial^2 (w_n)_{ij}}{\partial y^k \partial y^l} + \widetilde{Z}(w_n)_{ij} = [\mathcal{E}_1]_{ij}. \end{equation} Now consider the behaviour of the right hand side as $n \to \infty$. By the remarks following equation \eqref{eqn:limit}, we have $\| \mathcal{E}_1 \|_{L^{\infty}(B_{|\lambda_n|}(0))} \leq \| \mathcal{E}_1 \|_{C^{0,\alpha}(B_{|\lambda_n|}(0))} \to 0$. To summarize the argument so far, from equation \eqref{eqn:components4} we have a sequence of equations on $\mathbb{R}^n$ that are eventually defined on a ball of any radius, and whose right hand side converges to zero uniformly on compact sets. Thus on $B_1(0)$ we may apply local elliptic regularity to write \begin{equation} \label{eqn:loc-elliptic-reg} \| w_n \|_{C^{2,\alpha}(B_1(0))} \leq c \left( \|\mathcal{E}_1 \|_{ {C^{0,\alpha}(B_2(0)) }} + \|w_n\|_{C^{0,\alpha}(B_2(0))} \right), \end{equation} where $c$ is uniformly bounded in $n$ since the coefficient $\lambda_n /|\lambda_n|$ is of unit modulus. By compactness of the unit sphere and the Arzela-Ascoli theorem we may obtain a subsequence $\lambda_{n_1}$ of $\lambda_n$ and a subsequence $w_{n_1}$ of $w_n$ so that both $\lambda_{{n_1}_k} / |\lambda_{{n_1}_k}|$ converges to a unit modulus element $e \in \mathbb{C}$, and $w_{n_1}$ converges in $C^{2,0}(B_1(0))$. Now return to equation \eqref{eqn:loc-elliptic-reg}, modify the radii involved and repeat this procedure to obtain a subsequence $n_2$ of $n_1$ so that $w_{n_1}$ converges in $C^{2,0}(B_2(0))$. Proceeding inductively we obtain a subsequences $n_k$ of $n_{k-1}$ so that $w_{n_k}$ converges in $C^{2,0}(B_k(0))$. Finally, take the diagonal subsequence $w_{{k_k}}$ and set $w_* = \lim_k w_{{k_k}}$, which we may regards as a $C^2$ vector-valued function $w_*$ defined on flat $\mathbb{R}^n$ that satisfies \[ (e-\Delta) w_* = 0, \] where here $\Delta$ acts as the flat Laplacian on each component. Moreover, $ \|w_*\|_{L^{\infty}} \leq 1$ and $|w_*(0)| = 1$, so $w_* \neq 0$. However note that $e\neq -1$ since $Re(\lambda_n) > K$. A short computation using Fourier analysis shows that no solution can exist, since the total symbol of $e - \Delta$ is nonvanishing. This contradiction concludes the proof. \end{proof} We are now ready to prove that $\Delta_L$ is a sectorial operator in H\"older spaces. Recall that $P = A_{\mathfrak{g}}^{\widetilde{g}}$ is in fact $\Delta_L$ when $\mathfrak{g}=\widetilde{g}.$ \begin{theorem} Let $(\mfld,g)$ be a closed Riemannian manifold. Then for any $\alpha \in (0,1)$, \[ \Delta_L: \mathcal{D} \subset C^{0,\alpha}(\sym) \to C^{0,\alpha}(\sym) \] is a sectorial operator. \end{theorem} \begin{proof} Since $\Delta_L$ is an unbounded operator on $L^2$, then Theorem \ref{thm:L2-spectrum} implies there exists $K \in \mathbb{R}$ so that $\sigma_{L^2}(\Delta_L) \subset (-\infty, K]$. Thus the $L^2$ resolvent set contains a sector, and Proposition \ref{spectra-coincide} allows us to conclude that the $C^{0,\alpha}$ resolvent set also contains a sector. Proposition \ref{resolvent-estimate} gives the resolvent estimate in a half-plane. However knowing the resolvent set contains a half-plane and a uniform estimate in this half-plane is a sufficient condition for sectoriality by Proposition 2.1.11 of \cite{Lun95}. \end{proof} \begin{corollary} If $(\mfld,g)$ is a closed Riemannian manifold. Then for any $\alpha \in (0,1)$, \[ \Delta_L: \mathcal{D} \subset h^{0,\alpha}(\sym) \to h^{0,\alpha}(\sym) \] is a sectorial operator. \end{corollary} \begin{proof} This simply follows by continuity of the resolvent and the fact that smooth tensor fields are dense in $h^{0,\alpha}$. \end{proof} \subsection{Continuous Dependence of the Ricci-DeTurck flow} We are in a position to prove continuous dependence of the Ricci-DeTurck flow. Choose any smooth initial metric $g_0$ and set the reference metric $\widetilde{g} = g_0$. Recall that in this exposition we take $\mathcal{D} = h^{2,\alpha}(\sym)$ and $\mathcal{X} = h^{0,\alpha}(\sym)$, but we may also use $\mathcal{D} = h^{k+2,\alpha}(\sym)$ and $\mathcal{X} = h^{k,\alpha}(\sym)$. The Ricci-DeTurck operator is a nonlinear map $F^{g_0}: \mathcal{D} \longrightarrow \mathcal{X}$, and is Fr\'echet differentiable at any point since it is a polynomial contraction of up to two covariant derivatives of its argument. Consider the open ball in $\mathcal{D}$ defined by \[ \mathcal{O} := B_r( g_0 ) = \{ g \in \mathcal{D}: \|g - g_0\|_{h^{2,\alpha}(\sym)} < r \}, \] where we choose $r > 0$ sufficiently small that the following three conditions are satisfied (see also \cite{KnopfYoung}): \begin{enumerate} \item Every $g$ in $B_r(g_0)$ is a Riemannian metric. \item For each $g$ in $B_r(g_0)$, the linearization $A^{g_0}_g$ is uniformly elliptic. \item For all $g \in \mathcal{O}$, $h \in h^{2,\alpha}(\sym)$, \[ \| (A^{g_0}_g - A^{g_0}_{g_0}) h \|_{h^{0,\alpha}(\sym)} < \frac{1}{M+1} \|h\|_{h^{2,\alpha}(\sym)}, \] \end{enumerate} where $M$ is the constant from Proposition \ref{resolvent-estimate}. We remark that for item (2), the considerations that lead to equation \eqref{form-of-A} show that the principal symbol of $A^{g_0}_g$ involves only the inverse of the metric, $g^{-1}$. See \cite{KnopfYoung, GIK} for further details. For item (3), standard estimation (see Section 4 of \cite{KnopfYoung}) allows one to conclude that such a bound exists. We next verify hypothesis H1 of Theorem \ref{existence}. Our argument so far only proves the sectoriality of $A^{g_0}_{g_0}$, but we need sectoriality of $A^{g_0}_g$ in an open set. In view of the third item above, for any $g \in \mathcal{O}$, $A^{g_0}_g = (A^{g_0}_g - A^{g_0}_{g_0}) + A^{g_0}_{g_0}$ is a small perturbation of a sectorial operator on $\mathcal{X}$ and hence is sectorial on $\mathcal{X}$ by \cite[Proposition 2.4.2]{Lun95}. Moreover, by elliptic regularity the graph norm of $A^{g_0}_{g}$ is equivalent to the $h^{2,\alpha}$-norm. This verifies hypothesis H1. The hypothesis H2 follows from the structure of $A$ as a polynomial contraction of the components of equation \eqref{form-of-A} and may be verified in a manner similar to Section 4 of \cite{KnopfYoung}. We have satisfied the hypotheses of Theorem \ref{existence}, and thus have existence, uniqueness, and continuous dependence on the initial conditions for a short time. We next study continuous dependence on initial conditions for the entire interval of existence. In what follows, $g_i$ means the fixed metric at $t=0$, and $g_i(t)$ is the flow that starts at $g_i$. Let $\tau(g_0)$ be the \textit{maximal time of existence} of a solution that starts at $g_0$. We conclude this section with the proof of continuous dependence of the Ricci-DeTurck flow. We use the pair $(\mathcal{D},\mathcal{X}) = (h^{k+2,\alpha}(\sym), h^{k,\alpha}(\sym))$ in this theorem since our eventual application next section requires $k=2$. \begin{theorem}[Continuous Dependence of the Ricci-DeTurck flow] \label{well-posedness} Let $(\mfld,g_0)$ be a compact Riemannian manifold, $\mathcal{D} = h^{k+2,\alpha}(\sym)$, $k \ge 0$, and let ${g_0}(t), \ {g_0}(0) = g_0$ be a solution of the Ricci-DeTurck flow with background metric $g_0$, so the flow exists on a maximal time interval $[0,\tau(g_0)). $ Let $\tau < \tau(g_0).$ Then there exist constants $r > 0$ and $C>0$ depending on $g_0$ and $\tau$, such that if $$ ||g_1 - g_0||_{h^{k+2,\alpha}(\sym)} \le r,$$ then \[\tau(g_1) \ge \tau,\] and \begin{equation*} ||{g}_1(t) - g_0(t) ||_{h^{k+2,\alpha}(\sym)} \le C || g_1 - g_0 ||_{h^{k+2,\alpha}(\sym)} \end{equation*} for all $t \in [0,\tau].$ \end{theorem} \begin{proof} The argument we give is an adaptation of the covering argument of \cite[Proposition 8.2.3]{Lun95}. Given $g_0(s)$ for $s \in [0,\tau]$ as in the statement, observe from Theorem \ref{existence} that for each $s\in[0,\tau]$ there exists positive constants $r_s, \delta_s$ and $c_s$ so that if $|t_0 - s| \leq r_s$ and $g_1 \in D$ with $\| g_1 - g_0(s) \|_{\mathcal{D}} \leq r_s$ and $F(g_1) \in \overline{\mathcal{D}}$, then the flow has a unique solution on $[s,s+\delta_s]$, with \begin{equation*} ||{g}_1(s) - g_0(s) ||_{h^{k+2,\alpha}(\sym)} \le c_s || g_1 - g_0 ||_{h^{k+2,\alpha}(\sym)}, \end{equation*} on that short interval. As the set \begin{equation*} \label{cpt set} [0,\tau] \times \{g_0(t): t \in [0,\tau]\} \end{equation*} is compact in $\mathbb R \times \mathcal{D}$, it is covered by finitely many open subsets $\{ U_i: i=1, \cdots, N\}$ where \[ U_i = (s_i - r_{s_i}, s_i + r_{s_i}) \times B_{r_{s_i}}^{\mathcal{D}}(g_0(s_i)). \] Obtain positive constants $\delta = \min_{1 \leq i \leq N} \delta_{s_i}$, $c = \max_{1 \leq i \leq N} c_{s_i}$, $r = \min_{1 \leq i \leq N} r_{s_i}$. Now $(0,g_0) \in U_{i_0}$ for some $i_0$. Thus for $\|g_1 - g_0\|_{\mathcal{D}} \leq r$, \[ ||g_1(t) - g_0(t)||_\mathcal{D} \le c ||g_1 - g_0||_\mathcal{D}, \; \; \mbox{for} \; t \in [0,\delta]. \] If $\delta > \tau$ the proof is complete, otherwise observe $(\frac{1}{2}\delta,g_0(\frac{1}{2}\delta)) \in U_{i_1}$ for some $i_1$, and we may solve and obtain estimates on a further interval of time of length $\delta$ to obtain $\tau(g_0) > \frac{3}{2} \delta$ and \[ ||g_1(t) - g_0(t)||_\mathcal{D} \le c_1 ||g_1 - g_0||_\mathcal{D}, \; \; \mbox{for} \; t \in \left[0,\frac{3}{2}\delta\right], \] for a new constant $c_1$. This process terminates after finitely many steps. \end{proof} \section{Continuous Dependence of the Ricci flow} \label{Sec:ContDepRF} In this section we prove continuous dependence of the Ricci flow on initial conditions. We assume closeness of initial conditions in a slightly more regular space, due to loss of regularity when moving from the Ricci-DeTurck flow to the Ricci flow. The next theorem quantifies the idea that a sufficiently small perturbation (in $h^{4,\alpha}$) of a smooth metric yields a Ricci flow that remains nearby, as quantified by a $h^{2,\alpha}$ dependence estimate. This perturbed Ricci flow is smooth after $t = 0$. \ \noindent{\bf Theorem A} (Continuous Dependence of the Ricci Flow). \label{Continuous dependence of the Ricci flow} \emph{Let $(\mfld,g_0)$ be a compact Riemannian manifold, with $g_0$ a smooth metric, and let $g_0(t), \ \ g(0) = g_0$ be the maximal solution of the Ricci flow that exists for time $t \in [0,\tau(g_0))$, with $\tau(g_0) \le \infty$. Choose $\tau < \tau(g_0)$, $k \geq 2$. Then there exist positive constants $r$ and $C$ depending only on $g_0$ and $\tau$ such that if \[ ||g_1 - g_0||_{h^{k+2,\alpha}(\sym)} \le r,\] then for $g_1(t)$ the unique solution of the Ricci flow starting at $g_1$, \[ \tau(g_1) \ge \tau, \] and \begin{equation*} ||g_1(t) - g_0(t) ||_{h^{k,\alpha}(\sym)} \le C || g_1 - g_0 ||_{h^{k+2,\alpha}(\sym)} \end{equation*} for all $t \in [0,\tau].$} \ \begin{proof} We prove this in the case $k = 2$, but the proof is the same in the general case. We use both the Ricci flow and the Ricci-DeTurck flow. Unadorned metrics depending on time, for example $g(t)$, refer to solutions of the Ricci flow, whereas time-dependent metrics with hats, for example $\hat{g}(t)$ are solutions to the associated Ricci-DeTurck flow. Let $g_0(t), \ g(0) = g_0$ be the solution of the Ricci flow defined on $[0,\tau(g_0)),$ where $ \ \tau(g_0) \le \infty$. This generates a maximal solution to the Ricci-DeTurck flow $\hat{g}_0(t)$ with background metric equal to $g_0$, that starts at $g_0$ and exists at least until time $\tau(g_0)$. To see this, note that if $g(t)$ is a solution of the Ricci flow and $\phi(t)$ satisfies the harmonic map heat flow \begin{equation*} \frac{\partial}{\partial t} \phi_t = \Delta_{g(t),g_0} \phi_t, \end{equation*} then $\hat{g}(t) = (\phi_t)_* g(t)$ satisfies the Ricci-DeTurck flow \cite[pg. 121]{HRF}. Existence of the Ricci-DeTurck flow then follows from existence for the harmonic map heat flow. Let $\tau < \tau(g_0)$. By Theorem \ref{well-posedness}, there exists $\delta > 0$ so that if $g_1$ is a metric that satisfies \[ \| g_1 - g_0 \|_{h^{4,\alpha}} < \delta, \] then there is a unique Ricci-DeTurck flow $\hat{g}_1(t)$ starting at $g_1$ that exists at least until time $\tau$ and there exists a constant $L > 0$ such that \begin{equation} \label{RDF-dependence} \| \hat{g}_1(t) - \hat{g}_0(t) \|_{h^{4,\alpha}} \leq L \| g_1 - g_0 \|_{h^{4,\alpha}}. \end{equation} Note that $\hat{g}_1(t)$ is smooth for any fixed time $t > 0$ by parabolic regularity. Given $\delta$ above, fix $g_1$ with $\| g_1 - g_0 \|_{h^{4,\alpha}} < \delta$. The DeTurck vector field with respect to the background metric $g_0$ is \begin{equation} \label{WDef} W^k_i = W^k_i(t, \hat{g}_i(t), g_0) =\hat{g_i}(t)^{pq} ( \Gamma(\hat{g_i}(t))_{pq}^k - \Gamma(g_0)_{pq}^k ) \end{equation} for $i = 0, 1$. If we solve the following ODE for the (time-dependent) diffeomorphisms $F_i(t)$ generated by $W_i$, \begin{equation} \label{diffeo} \partial_t F_i(t) = -W_i \circ F(t), \; \; \; F_i(0) = Id, \end{equation} where $Id$ is the identity map, we find that the metric $g_i(t) = F^*_i \hat{g}_i(t)$ is a solution of the Ricci flow starting at $g_i$, $i = 0, 1$. Note that at this point, we have lost control of one spatial derivative of our estimates due to \eqref{WDef}, as the $W^k_i$ are controlled in $h^{3,\alpha}$. \subsubsection{Estimates for the DeTurck vector field and diffeomorphism} This subsection is rather technical, involving estimates for the $W$ and $F$. Let us begin with the easier $W_0$. We have supposed that the Ricci-DeTurck flow $\hat{g}_0(t)$ exists on $[0,\tau]$, so the curvature tensor is bounded on $[0,\tau]$, i.e. that there exists $K > 0$ where $|\Rm^{\hat{g}_0(t)}(x,t)|\leq K$ for all $x \in \mfld$ and $t \in [0,\tau]$. Thus by Proposition 6.48 of \cite{ChowKnopf}, choosing $g_0$ as the reference metric, for any $m \in \mathbb{N}$ there is a constant $C_m = C_m( \tau, K ) > 0$ where $| \nabla^m_{g_0} \hat{g}_0(x,t)|_{g_0} \leq C_m$ for all $x \in \mfld$ and $t \in [0,\tau]$. In short: any number of derivatives of $\hat{g}_0$ are bounded by a constant depending on $\tau$ and $K$, i.e. there exists a constant $C(\tau, K, k, \alpha) > 0$ where \begin{equation} \label{bounds-on-g0} \| \hat{g}_0(t) \|_{C^{k,\alpha}(\mfld)} \leq C(\tau, K, k, \alpha), \end{equation} for all $t \in [0,\tau]$. This then implies that any norm of $W_0$ satisfies similar bounds on $[0,\tau]$. Regarding $W_1$, we must estimate in terms of the difference $\hat{g}_1(t) - \hat{g}_0(t)$ and $\hat{g}_0(t)$. Omitting indices and time-dependence we must estimate \begin{align*} W_1 = \hat{g_1}^{-1} ( \Gamma(\hat{g_1}) - \Gamma(g_0) ) \end{align*} by interpolating differences. Since $W_1$ is a polynomial contraction in $\hat{g}_1^{-1}$ and $\partial \hat{g}_1$, one obtains abstractly \[ W_1 = \mathcal{P}( \hat{g}_1 - \hat{g}_0, \partial(\hat{g}_1 - \hat{g}_0), \hat{g}_0, \partial \hat{g}_0 ), \] where $\mathcal{P}$ denotes polynomial contractions in the tensors indicated. In view of equation \eqref{RDF-dependence} and equation \eqref{bounds-on-g0}, we then see there is a constant depending on $\hat{g}_0$, $\tau$, $K$, $L$ and the algebraic structure of $\mathcal{P}$ so that \[ \| W_1(t) \|_{h^{3,\alpha}} \leq C(\tau, K, L) \| {g}_1 - {g}_0 \|_{h^{4,\alpha}}, \] for all $t \in [0,\tau]$, and similarly for the difference $W_1 - W_0$. Taking a spatial derivative of $W_1$ denoted here by $\partial$, we obtain estimates of the form \[ \| \partial W_1(t) \|_{h^{2,\alpha}} \leq C(\tau, K, L) \| {g}_1 - {g}_0 \|_{h^{4,\alpha}}, \] and similarly for $W_1 - W_0$. To obtain estimates for $F$, we may write equation \eqref{diffeo} abstractly as \[ F_i(t) = -\int_0^t W_i( F(s) ) ds + Id. \] From here we can argue that any number of derivatives of $F_0(t)$ remain bounded by a constant that grows at worst linearly in $\tau$. Up to three spatial derivatives of $F_1(t)$ remain bounded by a constant that grows linearly in $\tau$, and the same for the difference $F_1 - F_0$. In summary, we have an estimate of the form \begin{equation} \label{loc-est-f1} \| \partial F_1(t) \|_{h^{2,\alpha}} \leq C'(\tau, K, L) \| {g}_1 - {g}_0 \|_{h^{4,\alpha}}, \end{equation} and \begin{equation} \label{loc-est-f1-f0} \| \partial F_1(t) - \partial F_0(t) \|_{h^{2,\alpha}} \leq C''(\tau, K, L) \| {g}_1 - {g}_0 \|_{h^{4,\alpha}}. \end{equation} The precise dependence on $\tau$ does not matter for finite time (which is all that is required). \subsubsection{Returning to the Ricci flow} We are now ready to compare $g_1(t)$ and $g_0(t)$. We work in coordinates, and we begin with a generic computation. Suppose $\hat{g}$ is a metric expressed in local coordinates $\{(V,y^i)\}$ by \[ \hat{g} = \hat{g}_{ij}(y) dy^i dy^j. \] Let $F$ be a diffeomorphism $F: U \to V$, $y = F(x)$. The pullback metric is \[ g = F^* \hat{g} = \hat{g}_{ij}(F(x)) d ( y^i \circ F ) d ( y^j \circ F ) = \hat{g}_{ij}(F(x)) \pd{F^i}{x^a} \pd{F^j}{x^b} dx^a dx^b, \] which leads to the classical transformation law \[ g_{ab}(x) = \hat{g}_{ij}(F(x)) \pd{F^i}{x^a}(x) \pd{F^j}{x^b}(x), \] where $F^i = y^i \circ F$ is the $i$th component of $F$. Now return to the Ricci flow argument and let \[ F_i: (\mfld,\{x^{a}\},g_i(t)) \longrightarrow (\mfld,\{y^j\},\hat{g}_i(t)).\] We have, in a reference coordinate patch after suppressing time-dependence (note the inclusion of indices below is for reference) \[ \| (g_1)_{ab} - (g_0)_{ab} \|_{h^{2,\alpha}} = \left\| (\hat{g}_1)_{ij}(F_1(x)) \pd{F_1^i}{x^a}(x) \pd{F_1^j}{x^b}(x) - (\hat{g}_0)_{ij}(F_0(x)) \pd{F_0^i}{x^a}(x) \pd{F_0^j}{x^b}(x) \right\|_{h^{2,\alpha}}, \] so that the standard interpolation trick lets us write \begin{align*} \| (g_1)_{ab} - (g_0)_{ab} \|_{h^{2,\alpha}} &\leq \left\| \left[(\hat{g}_1)_{ij}(F_1(x)) - (\hat{g}_0)_{ij}(F_0(x))\right] \pd{F_1^i}{x^a}(x) \pd{F_1^j}{x^b}(x) \right\|_{h^{2,\alpha}} \\ &+\left\| (\hat{g}_0)_{ij}(F_0(x)) \left(\pd{F_0^i}{x^a}(x) - \pd{F_1^i}{x^a}(x) \right) \pd{F_1^j}{x^b}(x) \right\|_{h^{2,\alpha}} \\ &+\left\| (\hat{g}_0)_{ij}(F_0(x)) \pd{F_0^i}{x^a}(x) \left(\pd{F_0^j}{x^b}(x) - \pd{F_1^i}{x^b}(x) \right) \right\|_{h^{2,\alpha}}. \end{align*} Taking this inequality, reinstating time-dependence, setting $r = \delta$ and then combining estimates \eqref{RDF-dependence}, \eqref{bounds-on-g0}, \eqref{loc-est-f1} and \eqref{loc-est-f1-f0} above we obtain a new constant, $C > 0$ independent of $g_1$, where for all $t \in [0,\tau]$, \begin{align*} \| (g_1)_{ab}(t) - (g_0)_{ab}(t) \|_{h^{2,\alpha}} & \leq C \| g_1 - g_0 \|_{h^{4,\alpha}}. \end{align*} Finally, we remark that by chasing the regularity of the metric throughout the proof above, one obtains a smooth dependence result with an estimate in terms of $h^{k,\alpha}$ norm of the initial data. \end{proof} \section{Convergence Stability of the Ricci Flow and Applications} \label{ConvStab} In this section, we apply the continuous dependence result to prove convergence stability of the Ricci flow in several cases. We begin with the case of a flat metric, $g_0$, on a torus $\mathbb{T}^n$. Since the operator has eigenvalues with zero real part, one expects a centre manifold. The stability of the Ricci flow on a torus is given in the following theorem (see Theorem 3.7 in \cite{GIK}): \ \noindent{\bf Theorem} (Stability (Flat)\cite{GIK}). \label{thm-GIK} {\it Given a flat fixed point $g_0$ of the Ricci flow on a closed torus $\mathbb{T}^n$, there exists a neighbourhood $\mathcal{U} \subset h^{2,\alpha}(\sym)$ of $g_0$ such that for any initial metric $g_1 \in \mathcal{U}$, the Ricci flow $g_1(t)$ converges in $h^{2,\alpha}(\sym)$ to a metric in the centre manifold of flat metrics at $g_0$.} \ With this theorem and continuous dependence in hand, we obtain convergence stability (this is essentially Corollary 3.8 of \cite{GIK}, with the continuous dependence of the Ricci flow now proved). \ \noindent{\bf Theorem B.} (Convergence Stability (Flat)). \label{Convergence to Flat} {\it Let $(\mathbb{T}^n, g_0)$ be a smooth closed torus, where $g_0(t), \ g(0) = g_0$ is a solution of the Ricci flow that converges to a flat metric $g_\infty$. Then there exists $r$ such that if $g_1 \in B_r(g_0) \subset h^{2,\alpha}(\sym)$, then the solution $g_1(t), \ g(0) = g_1$ of the Ricci flow exists for all time and converges to a flat metric.} \ \begin{proof} This follows from the continuous dependence of the Ricci flow, together with stability of the fixed point $g_\infty$. Let $g_0(t), \ g(0) = g_0$ be a solution of the Ricci flow that converges to a flat metric $g_\infty$. Since $g_\infty$ is a flat fixed point of the Ricci flow, there exists a stability neighbourhood $\mathcal{U} \subset h^{2,\alpha}$ around $g_\infty$ by the stability theorem above. That is, the Ricci flow of any initial metric within this set converges to a flat metric in the centre manifold at $g_\infty$. Since $g_0(t)$ converges to $g_\infty$, we know that there exists finite $\tau$ such that $g_0(t) \in \mathcal{U}$ for all $\tau \le t \le \tau(g_0)$. Since $\mathcal{U}$ is open, there exists $\rho > 0$ such that $B_\rho(g_0(\tau)) \subset \mathcal{U}$. By the continuous dependence of the Ricci flow, we can choose $r > 0$ so that if $g_1$ is a metric that satisfies $||g_0 - g_1||_{h^{4,\alpha}} < r,$ then $||g_0(\tau) - g_1(\tau)||_{h^{2,\alpha}} \le C ||g_0 - g_1||_{h^{4,\alpha}}$. This bound persists by shrinking the distance between $g_0$ and $g_1$; thus, if $||g_0 - g_1||_{h^{4,\alpha}} < \min\{\rho/C, r\}$, we have $g_1(\tau) \in B_\rho(g_0(\tau)) \subset \mathcal{U}$. Then the Ricci flow starting at $g_1(\tau)$ is an extension of the flow at $g_1$ that converges to a flat metric. \end{proof} As an application, consider two classes of metrics on $\mathbb{T}^3$ with symmetry that were studied by Lott-Sesum \cite{LS}. For the first class, suppose that $(\mfld^3,g)$ is obtained as a warped product $\mfld = \mathbb{S}^1 \times N$, with $N = \mathbb{T}^2$ with metric \[ g = e^{2u} d\theta^2 + h, \] where $h$ is a metric on $\mathbb{T}^2$, $\theta$ is the standard coordinate on $\mathbb{S}^1$ and $u \in C^{\infty}(\mathbb{T}^2)$. The Ricci flow $g(t)$ starting at such a warped product metric preserves this symmetry and exists for all time. Moreover, by \cite[Theorem 1.1 case (iii)]{LS}, $\lim_{t\to\infty} g(t)$ exists and converges exponentially fast to a flat metric. Theorem B thus implies there is a small neighbourhood of the initial metric on which the flow exists for all time and still converges. We emphasize that metrics in a $h^{2,\alpha}$-neighbourhood of a warped product metric need not be warped product metrics. Moreover, a calculation (see \cite{LS} for details) shows that the scalar curvature of $g$ is given by \[ \scal(g) = \scal(h) - 2 \Delta u - 2 |\nabla u|^2. \] Choose a flat metric $(N,h)$ and highly oscillatory warping function on the torus factor. It is possible to create a sequence of warped product metrics $g_n$ with arbitrarily large scalar curvature that leaves any (fixed) neighbourhood of a flat metric. Yet each of these metrics possesses a neighbourhood on which every metric converges exponentially quickly to a flat metric. For the second class of metrics, we assume that $\mfld$ fibres over $\mathbb{S}^1$ with $\mathbb{T}^2$ fibres. Choose an orientation of $\mathbb{S}^1$ and let $H \in SL(2,\mathbb{Z})$ be the holonomy of the torus bundle. We further assume that $H$ has finite order. Now regard $\mfld$ as the total space of a twisted principal $U(1)\times U(1)$ bundle with twist determined by $H$. The Ricci flow $g(t)$ starting at a metric that is invariant under the $U(1) \times U(1)$ action exists for all time, and moreover $\lim_{t\to\infty} g(t)$ exists and converges exponentially fast to a flat metric (see \cite[Theorem 1.6, case (i)]{LS}). Once more, convergence stability thus implies that there is a small neighbourhood of the initial metric on which the flow exists for all time, and still converges. Calculations in \cite{CIJ} show that the scalar curvature of a class of these metrics can be made arbitrarily large. Finally, we conclude with an example on $\mathbb{H}^3$. Consider a compact Riemannian manifold $(\mfld^3,g)$ possessing a metric of constant negative curvature, which we rescale to be $-1$. In \cite{KnopfYoung}, the authors study a suitably normalized Ricci flow \[ \partial_t g = - 2\Rc(g) - 4 g, \] on a compact manifold. An integration by parts argument using Koiso's Bochner formula allows us to bound the $L^2$ spectrum of the linearized DeTurck operator from above by a negative constant. The results of Section 3 then show that $g$ is a stable fixed point of this flow: \ \noindent{\bf Theorem} (Stability (Hyperbolic) \cite{KnopfYoung}). {\it Let $(\mfld,g)$ be a closed Riemannian manifold, where $g$ is a metric of constant negative curvature. Then there exists $\delta > 0$ such that for all $g_0 \in B^{2+\rho}_{\delta}(g)$ the solution to the normalized flow exists for all time and converges exponentially quickly to a constant curvature hyperbolic metric.} \ Similarly to the proof the flat case we then obtain \begin{theorem} [Convergence Stability (Hyperbolic)] \label{Convergence to Hyperbolic} Let $(\mfld,g_0)$ be a closed Riemannian manifold with $g_0 \in h^{4,\alpha}(\sym)$, and let $g_0(t), \ g(0) = g_0$ be a solution of the Ricci flow that converges to a constant curvature hyperbolic metric $g_\infty$. Then there exists $\epsilon$ such that if $g_1 \in B_\epsilon(g_0) \subset h^{4,\alpha}(\sym)$, then the solution $g_1(t), g(0) = g_1$ of the Ricci flow exists for all time and converges to $g_\infty$. \end{theorem} We will discuss the non-compact case for hyperbolic metrics in a subsequent work.

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\section{Introduction} Recent progress in the studies of single atoms and molecules on ultrathin-insulating films supported by a metal substrate have opened up a new frontier in atomic scale science. Not only are these films able to decouple the electronic states of the adsorbates from the metal substrate states, but they also provide sufficient tunnelling current to be able to characterise, manipulate and image the adsorbed species by a scanning tunnelling microscope. Some important examples include charge state control of adsorbed species \cite{RepMeyOlsPer04,OlsPaaPerRepMey07}, imaging of frontier orbitals \cite{RepMeyStoGouJoa05,RepMeyPaaOlsPer06}, coherent electron-nuclear coupling in molecular wires \cite{RepLilMey10} and tunnelling-induced switching of adsorbed molecules \cite{liljeroth,mohn}. In particular, polar films such as NaCl bilayers are able to support multiple charge states of adsorbed species, which can be switched in a controlled manner by attachment and detachment of tunnelling electrons. Furthermore, these processes have been suggested to be responsible for the observed reversible bond formation in molecular switches \cite{mohn}. To fully exploit the new potential opportunities provided by these systems, there is a need of theory to unravel the electronic and geometric structure and the excited state potentials of anionic and cationic states of the adsorbed species. Some important advances in this respect have been made by using Density Functional Theory (DFT) \cite{HK,KS} calculations~\cite{RepMeyOlsPer04,RepMeyPaaOlsPero5,RepMeyPaaOlsPer06,OlsPaaPerRepMey07,mohn}. However, the delocalisation error in the current exchange-correlation functionals and the system size make the calculations very challenging. In fact, this error in current density functionals can result in fractional occupation of the adsorbed species and makes it difficult to identify various multiply charged states \cite{CohenScienceDFTfailures08}. Even with these limitations, important progress has been made by using DFT+U \cite{anisimov,cococcioni} to correct for the self-interaction behind the delocalisation error for a Ag atom \cite{OlsPaaPerRepMey07}, despite it is generally a bit dubious for delocalised states such as for $s$ states of atoms \cite{sit}. There has also been some developments in constraining the occupancy of adsorbate orbitals, but this requires that one has to identify the appropriate Kohn-Sham orbital, which can still be challenging due to the mixing of adsorbate and substrate states albeit being very small \cite{RepMeyPaaOlsPero5}. However, when considering, for example, organic molecules adsorbed on an ultrathin-insulating film supported by a metal surface, the number of metal atoms and the associated number of metal electrons makes the calculations to become prohibitively large. As a first step to try to overcome these limitations, we present a new simplified DFT method for the calculation of the total energy, ionic (Hellman-Feynman) forces and electronic structure of a charged system placed in front of a metal surface. In this method, the charged system is fully treated using DFT, the interaction between the two systems is assumed to be electrostatic and the density response of the metal surface to the electrostatic interaction is treated to linear order. The proposed method has two main advantages. Firstly, the computational effort is significantly decreased, since the metal electron states are not treated explicitly but only implicitly through their density response to an external electrostatic field, which can be captured using a simple model. Here, we have explicitly considered the classical perfect conductor (PC) model for the metal response in which the screening length by the conduction electrons is assumed to be zero and the screening charge only resides on the perfect conductor plane. Secondly, different charge states of the charged system can be handled directly using this method. We have implemented this new methodology in the VASP code~\cite{vasp} using the simple PC model for the metal response. A critical test of this implementation is provided by the example of the Na$^{+}$ ion outside a perfect conductor. Here, the six electrons in the $p$ semi-core of the Na$^{+}$ ion were included in the calculations. The results show an excellent agreement between the calculated Hellman-Feynman force on the ion and the negative gradient of the interaction energy even in the region close to the perfect conductor plane, where the electron density of the ion overlaps with the induced charge density at perfect conductor plane. Furthermore, the small polarisability of the ion makes the calculated interaction energy to be in close agreement with the analytical result for a point charge model outside the perfect conductor. In applying this method to study charged adsorbates deposited over ultrathin-insulating films supported by a metal substrate, we need to extend the PC model by including the interactions between the film and the metal substrate arising from overlapping electron densities and van der Waals interactions. We plan to capture these interactions through a simple parametrised force field between the metal substrate and the ions in the film, where parameters will be obtained by fitting the force field to DFT calculations of the forces between the film (without adsorbates) and the metal substrate. A successful implementation of this force field should enable us to describe, for example, the observed reversible bond formation of a molecular switch induced by a tunnelling electron and hole attachment to form ionic adsorbate states~\cite{mohn}. The presentation of this scheme and associated results are deferred to a future publication. This paper is organised as follows. In the next Section \ref{sec:theory}, we present the theoretical background of the proposed method. First, the effective energy density functional and the associated Hellman-Feynman forces for the charged system are derived in Subsection \ref{sub:Approx-energy-DF} under the assumption that the electrons of the charged system and the metal surface are distinguishable. The total energy of the metal surface is expanded up to second order in the resulting electrostatic interaction between the two systems. In the next Subsection \ref{sub:PBC-dipole}, the non-trivial effects on the electrostatics when imposing periodic boundary conditions are analysed and appropriate dipole corrections for the total energy and the Hellman-Feynman forces are derived. The classical perfect conductor model for the metal surface response is then introduced in the next Subsection \ref{sub:PC-model}. Here, we also present analytical expressions for the interaction energy and force in the case of a point charge (Subsection \ref{sub:IntPointCharge}) in the presence of periodic boundary conditions, and discuss possible refinements of the perfect conductor model (Subsection \ref{sub:beyondPC}). In Section \ref{sec:compimpl}, the keys steps in implementing the PC model in the plane wave code VASP are presented. As a first test of this methodology, we show the results of the computational implementation of the PC model for the case of Na$^{+}$ ion outside a PC (Section \ref{sec:Results}) and finally give some concluding remarks in Section \ref{sec:conc}. \section{Theoretical framework} \label{sec:theory} In this Section we develop a density functional theoretical description of a charged system placed in front of a metal surface based on the assumption that the electrons of both fragments are distinguishable, so that the interaction between the two systems is only electrostatic. The corresponding approximation becomes valid when the electron densities of the two systems are well separated. We analyse how the system can be treated in a supercell geometry and derive expressions for the dipole correction of the Kohn-Sham potential and the Hellmann-Feynman forces. As an explicit model for the metal surface, we focus on the classical perfect conductor (PC) model of the metal surface response, but also discuss possible extensions of this approximation. For the particular case of a point charge, we present explicit expressions for the interaction energy with a perfect conductor. Throughout the paper, we will make use of electrostatic units. \subsection{Approximate energy density functional\label{sub:Approx-energy-DF}} We consider a closed, charged system $S$ of electrons and ions outside a metal surface $M$. In contrast to the system $S$, the metal surface will be treated as an open system, kept at a constant chemical potential $\mu$. The electron densities of $S$ and $M$ are assumed to be non-overlapping and are given by $n_{s}({\bf r})$ and $n_{m}({\bf r})$, respectively. In this case the kinetic energy functional of the non-interacting electrons is additive in the electron densities of $S$ and $M$. Furthermore, we will neglect any non-local contributions between $S$ and $M$ such as the van der Waals interaction so that the exchange-correlation energy functional is also additive in these electron densities. The total energy density functional for the combined system is then given by the following expression: \begin{eqnarray} E[n_{m},n_{s}]=E_{s}[n_{s}]+E_{m}[n_{m}]- \mu N_{m}+\int\rho_{m}({\bf r})\phi_{s}({\bf r})d{\bf r}\ .\label{eq:EtotApp} \end{eqnarray} Here $E_{s}[n_{s}]$ and $E_{m}[n_{m}]$ are the energy density functionals for the isolated $S$ and $M$, respectively, whereas $\rho_{m}({\bf r})$ is the charge density of $M$ and $N_{m}$ the total number of electrons of $M$. The potential $\phi_{s}({\bf r})$ is defined as the electrostatic potential generated from the total charge density $\rho_{s}({\bf r})=-en_{s}({\bf r})+\rho_{i}({\bf r})$ of electrons and ions in $S$ and is given by, \begin{equation} \phi_{s}({\bf r})=\int\frac{\rho_{s}({\bf r^{\prime}})d{\bf r^{\prime}}}{|{\bf r}-{\bf r^{\prime}}|}\ .\label{eq:phiDef} \end{equation} Note that the approximation in Eqn.(\ref{eq:EtotApp}) is still meaningful for overlapping electron densities. Furthermore, any neglected non-additive contributions to the kinetic energy and the exchange-correlation potential including any non-local contribution between $M$ and $S$, such as the van der Waals interaction, will be accounted by introducing a force field between $M$ and $S$. An effective total energy density functional for $S$ outside $M$ in term of the electron density $n_{s}({\bf r})$ is obtained by minimising the energy density functional in Eqn.(\ref{eq:EtotApp}) with respect to $n_{m}$ for fixed $n_{s}$ and $\mu$, \begin{equation} \frac{\delta E_{m}}{\delta n_{m}({\bf r})}=\mu+e\phi_{s}({\bf r})\label{eq:muDef} \end{equation} and, since $\mu$ is constant, we conclude that the ground state electron density of $M$ in the presence of $S$ is a functional of $n_{s}$, i. e. $n_{m}=n_{m}[n_{s}]$. The change in energy of $M$ induced by the presence of $S$ is defined as, \begin{eqnarray} \Delta E_{m}[n_{s}]&=&(E_{m}[n_{m}]-E_{m}[n_{m0}])- \mu(N_{m}-N_{m0}) +\int\rho_{m}({\bf r})\phi_{s}({\bf r})d{\bf r}\label{eq:DeltaEDef} \end{eqnarray} where $n_{m0}$ and $N_{m0}$ are the unperturbed ground state electronic density and total number of electrons of $M$, respectively. Expanding $\Delta E_{m}[n_{s}]$ up to second order in $\phi_{s}({\bf r})$, one obtains \begin{eqnarray} \Delta E_{m}[n_{s}]&=&\int\frac{\delta\Delta E_{m}}{\delta\phi_{s}({\bf r})}\phi_{s}({\bf r})d{\bf r}+ \frac{1}{2}\int\int\frac{\delta^{2}\Delta E_{m}}{\delta\phi_{s}({\bf r})\delta\phi_{s}({\bf r}^{\prime})}\phi_{s}({\bf r})\phi_{s}({\bf r}^{\prime})d{\bf r}d{\bf r}^{\prime}. \label{eq:DeltaEApp} \end{eqnarray} The two terms in Eqn.(\ref{eq:DeltaEApp}) can now be cast in a more familiar form using the result \begin{equation} \rho_{m}({\bf r})=\frac{\delta\Delta E_{m}}{\delta\phi_{s}({\bf r})}\ ,\label{eq:RhomRes} \end{equation} which follows from the fact that $\Delta E_{m}[n_{s}]$, defined in Eqn.(\ref{eq:DeltaEDef}), is stationary with respect to variations in $n_{m}({\bf r})$ for fixed $\phi_{s}$ . Thus, the first term of the r.h.s in Eqn.(\ref{eq:DeltaEApp}) is determined by the unperturbed charge density $\rho_{m0}$ of the metal $M$, whereas the second term is given by the induced charge density $\rho_{ind}({\bf r})$ in $M$, which, to linear order in $\phi_{s}$, is given by \begin{equation} \rho_{ind}({\bf r})=\int\frac{\delta\rho_{m}({\bf r})}{\delta\phi_{s}({\bf r^{\prime}})}\phi_{s}({\bf r^{\prime}})d{\bf r^{\prime}}\ .\label{eq:rhoindDef} \end{equation} Note that the symmetry of the kernel in Eqn.(\ref{eq:DeltaEApp}) and Eqn.(\ref{eq:RhomRes}) give rise to the following important symmetry condition for the density response kernel in Eqn.(\ref{eq:rhoindDef}), \begin{equation} \frac{\delta\rho_{m}({\bf r})}{\delta\phi_{s}({\bf r^{\prime}})}=\frac{\delta\rho_{m}({\bf r}^{\prime})}{\delta\phi_{s}({\bf r})}\ .\label{eq:rhoindSymm} \end{equation} Finally, we obtain the following effective total energy functional for $S$ outside $M$ from Eqns.(\ref{eq:DeltaEApp}) and (\ref{eq:rhoindDef}) as, \begin{eqnarray} E_{eff}[n_{s}]\equiv E_{s}[n_{s}]+\Delta E_{m}[n_{s}]= \nonumber \\ E_{s}[n_{s}] + \int\rho_{m0}({\bf r})\phi_{s}({\bf r})d{\bf r}+\frac{1}{2}\int\rho_{ind}({\bf r})\phi_{s}({\bf r})d{\bf r}\ .\label{eq:EeffDef} \end{eqnarray} The interaction energy between $S$ and $M$ is then simply defined as \begin{eqnarray} E_{int} & \equiv & E_{eff}[n_{s}]-E_{s}[n_{s0}]=\label{eq:EintDef}\\ & & E_{s}[n_{s}]-E_{s}[n_{s0}]+\int\rho_{m0}({\bf r})\phi_{s}({\bf r})d{\bf r} + \frac{1}{2}\int\rho_{ind}({\bf r})\phi_{s}({\bf r})d{\bf r}\ ,\label{eq:EintExpr} \end{eqnarray} where $n_{s0}$ is the unperturbed ground state electron density of $S$. The contribution from the first two terms in Eqn.(\ref{eq:EintExpr}) gives the interaction energy from the polarisation of the system $S$, whereas the contributions from the third and fourth terms give the interaction energies of $S$ with the unperturbed surface density and the polarisation of the metal surface, respectively. The condition that the chemical potential of $M$ should be constant imposes an important constraint on the electrostatic potential from the induced charge densities in $M$ and $S$. From the functional derivative of Eqn.(\ref{eq:muDef}) it follows \begin{equation} \mu=\frac{\delta T_{0}}{\delta n_{m}({\bf r})}+\frac{\delta E_{xc}}{\delta n_{m}({\bf r})}-e\phi_{m}-e\phi_{s}({\bf r})\ ,\label{eq:muRes} \end{equation} where $\phi_{m}({\bf r})$ is the electrostatic potential \begin{equation} \phi_{m}({\bf r})=\phi_{m0}({\bf r})+\phi_{ind}({\bf r})\ \label{eq:phimDef} \end{equation} obtained from $\rho_{m}({\bf r})=\rho_{m0}({\bf r})+\rho_{ind}({\bf r})$. Inserting Eqn.(\ref{eq:phimDef}) in Eqn.(\ref{eq:muRes}), one finally has \begin{equation} \mu=\frac{\delta T_{0}}{\delta n_{m}({\bf r})}+\frac{\delta E_{xc}}{\delta n_{m}({\bf r})}-e\phi_{m0}-e[\phi_{ind}({\bf r})+\phi_{s}({\bf r})]\ .\label{eq:phimDef2} \end{equation} We now assume that the metal surface $M$ is perpendicular to the z-axis and it is a semi-infinite system that extends to $-\infty$. The contribution from the first three terms of the r.h.s. of Eqn.(\ref{eq:phimDef2}) gives the chemical potential in the absence of the perturbation by $S$ and, since the electron density is unperturbed far inside the metal surface ($z\rightarrow-\infty$), this contribution is constant. Therefore, the contribution from the last two terms has to be zero in this region, \begin{equation} \phi_{s}({\bf r})+\phi_{ind}({\bf r})\rightarrow0\ ,\ z\rightarrow-\infty\ .\label{eq:PhiZero} \end{equation} The next step is to determine how the Kohn-Sham (K-S) potential of $S$ changes in the presence of $M$. Since the K-S potential is determined by the functional derivative of the electrostatic energy and exchange correlation energy terms with respect to the electron density, we need to investigate how these terms are modified. The exchange correlation energy term is a universal functional of the density and does not change. On the other hand, according to Eqn.(\ref{eq:EeffDef}), the electrostatic energy term in $E_{s}[n_{s}]$, is defined here as, \begin{equation} E_{el}[n_{s}]=\frac{1}{2}\int\rho_{s}({\bf r})\phi_{s}({\bf r})d{\bf r}\ ,\label{eq:EHDef} \end{equation} and changes in the presence of $M$ to the effective electrostatic energy \begin{eqnarray} E_{el}^{eff}[n_{s}]&=&E_{el}[n_{s}]+\int\rho_{m0}({\bf r})\phi_{s}({\bf r})d{\bf r}+ \frac{1}{2}\int\rho_{ind}({\bf r})\phi_{s}({\bf r})d{\bf r}\ .\label{eq:EeleffDef} \end{eqnarray} Thus, only the electrostatic potential energy term $-e\phi_{s}({\bf r})$ in the K-S potential is modified and is now given by, \begin{eqnarray} -e\phi_{el}^{eff}[n_{s}]&\equiv&\frac{\delta E_{el}^{eff}[n_{s}]}{\delta n_{s}({\bf r})}=-e[\phi_{s}({\bf r})+\phi_{m0}({\bf r})+\phi_{ind}({\bf r})]\ .\label{eq:phieffRes} \end{eqnarray} In deriving this result from Eqns.(\ref{eq:DeltaEApp}), (\ref{eq:RhomRes}) and (\ref{eq:rhoindDef}) , we have made use of the symmetry condition in Eqn.(\ref{eq:rhoindSymm}). Therefore, the only change of the K-S potential in the presence of $M$ is simply the inclusion of the electrostatic potential from the metal surface. The presence of the metal surface will now change the forces ${\bf F}_{I}$ on the nuclei $I$ at positions $R_{I}$ and charges $eZ_{I}$ in $S$, and this modification will be only through the electrostatic potential from the metal surface. Since the effective total energy functional is stationary around the ground state electron density and only the electrostatic term has an explicit dependence on the nuclei positions, the forces ${\bf F}_{I}$ are given by \begin{equation} {\bf F}_{I}=-\nabla_{I}E_{el}^{eff}=-\int\frac{\delta E_{el}^{eff}}{\delta\rho_{i}({\bf r})}\nabla_{{\bf {R}_{I}}}\rho_{i}({\bf r})\ d{\bf r}.\label{eq:Fi} \end{equation} In a similar manner as for the functional derivative with respect to the electron density in Eqn.(\ref{eq:phieffRes}), one obtains that \begin{equation} \frac{\delta E_{el}^{eff}}{\delta\rho_{i}({\bf r})}=\phi_{s}({\bf r})+\phi_{m}({\bf r}). \label{eq:FuncDerEeff} \end{equation} The expression for the forces in Eqn.(\ref{eq:Fi}) can now be cast in a more familiar Hellman-Feynman form using Eqn.(\ref{eq:FuncDerEeff}) and, since \begin{equation} \nabla_{{\bf R}_{I}}\rho_{i}({\bf r})=-eZ_{I}\nabla\delta({\bf r}-{\bf R}_{I}), \end{equation} one obtains, after partial integration, the total force acting on the ion $I$ \begin{equation} {\bf F}_{I}=-eZ_{I}\nabla[\phi_{s}+\phi_{m}]({\bf R}_{I}).\label{eq:FiRes} \end{equation} Therefore, the force on a nuclei is still given by the total electrostatic field at the nuclei, but now includes the electrostatic contribution from the metal surface. \subsection{Periodic boundary conditions and dipole corrections\label{sub:PBC-dipole}} Since the combined system will be represented in a finite supercell, we need to discuss the effects of imposing periodic boundary conditions (PBCs) \cite{ashcroft,kittel}. Firstly, we will introduce PBCs in the lateral directions along the planar surface of the semi-infinite metal and state some general key properties of the behaviour of the electrostatic potentials. In addition, the system will have a net dipole moment component along the direction perpendicular to the metal surface and, when imposing PBCs in this direction, one has to introduce appropriate dipole corrections for the total energy, K-S potential and the forces on the nuclei as detailed in the following.\\ In discussing the behaviour of the electrostatic fields and charge densities in the presence of lateral PBCs, it is convenient to introduce a plane wave representation over two-dimensional (2D) reciprocal lattice vectors ${\bf G}$ of the two dimensional surface unit cell $\mathcal{A}$ with area $A$. The 2D plane wave expansion of a charge density $\rho({\bf r})$ is here defined as, \begin{equation} \rho({\bf r})=\sum_{{\bf G}}\rho(z,{\bf G})\exp[i{\bf G}.{\bf R}]\label{eq:2DPlaneWave} \end{equation} where the plane wave coefficients $\rho(z,{\bf G})$ are given by \begin{equation} \rho(z,{\bf G})=\frac{1}{A}\int_{\mathcal{A}}\rho({\bf R},z)\exp[-i{\bf G}.{\bf R}]d{\bf R}\label{eq:2DCoeff} \end{equation} and $({\bf R},z)\equiv{\bf r}$. The electrostatic potential from the charge density can now be obtained using the associated plane wave representation of the Coulomb kernel, \begin{eqnarray} \frac{1}{|{\bf r}-{\bf r}^{\prime}|}&=&-\frac{2\pi}{A}|z-z^{\prime}|+ \sum_{{\bf G}\neq0}\frac{2\pi}{AG}\exp[-G|z-z^{\prime}|]\exp[i{\bf G}.({\bf R}-{\bf R}^{\prime})]\ \label{eq:2DFourierCoulomb} \end{eqnarray} where $|{\bf G}|=G$. In particular, the laterally averaged potential, $\bar{\phi}(z)=\phi(z,{\bf G}=0)$ is determined by the laterally averaged density, $\bar{\rho}(z)=\rho(z,{\bf G}=0)$, as, \begin{equation}\label{eq:AvPhi} \bar{\phi}(z)=-2\pi\int|z-z^{\prime}|\bar{\rho}(z^{\prime})dz^{\prime}+\phi_{0}\ . \end{equation} Note that $\bar{\phi}(z)$ is only determined up to a constant $\phi_{0}$, which has to be fixed by the boundary conditions as discussed later. Finally, the remaining laterally varying part of the potential, which has contributions only from the non-zero ${\bf G}$ plane wave coefficients, is given by \begin{eqnarray} \phi^{\prime}({\bf r})\equiv\phi({\bf r})-\bar{\phi}(z)= \sum_{{\bf G}\neq0}\frac{2\pi}{G}\int\exp[-G|z-z^{\prime}|]\exp[i{\bf G}.{\bf R}]\rho(z^{\prime},{\bf G})dz^{\prime}, \label{eq:latvarPhi} \end{eqnarray} and decays exponentially away from a localised charge distribution. In the case of a semi-infinite metal surface, the external electric field from the charged system sets up an electron current towards the surface and builds up a localised surface charge distribution and an electric field that opposes the external field. In a stationary situation, the perpendicular components of the electric field and current far inside the surface have to be zero. This electric field is determined by the laterally averaged electrostatic potential $\bar{\phi}(z\rightarrow-\infty)$ and, according to Eqn.(\ref{eq:AvPhi}), it is given by \begin{equation} \bar{E}_{z}(z)=-\frac{\partial\bar{\phi}}{\partial z}(z)\rightarrow2\pi\int\bar{\rho}(z^{\prime})dz^{\prime}\ ,\ z\rightarrow-\infty\label{eq:Ez} \end{equation} Therefore, in order for the electric field to be zero far inside in the metal, the total charge of $S$ and $M$ has to be zero. Since the surface charge of the unperturbed metal is zero \begin{equation} \int_{\mathcal{A}}\int\rho_{m0}({\bf r})d{\bf R}dz=0\ ,\label{eq:rhm0Charg} \end{equation} the total induced surface charge has to be equal in magnitude to the total charge $Q_{s}$ of $S$ but with opposite sign, \begin{equation} \int_{\mathcal{A}}\int\rho_{ind}({\bf r})d{\bf R}dz=-Q_{s}\ .\label{eq:SurfCharg} \end{equation} The energy of the vacuum level for the unperturbed metal is defined to be equal to zero, $(\bar{\phi}_{m0}(z)\rightarrow0$, $z\rightarrow\infty$), then \begin{equation} \bar{\phi}_{m0}(z)=-2\pi\int[|z-z^{\prime}|+z']\bar{\rho}_{m0}(z^{\prime})dz^{\prime}\ .\label{eq:AvPhim0} \end{equation} The undetermined constant $\phi_{0}$ in $\bar{\phi}_{s}(z)+\bar{\phi}_{ind}(z)$ can now be determined from the condition in Eqn.(\ref{eq:PhiZero}) that the chemical potential of the metal is constant: \begin{equation} \bar{\phi}_{s}(z)+\bar{\phi}_{ind}(z)\rightarrow0\ ,\ z\rightarrow-\infty\ .\label{eq:phizerores} \end{equation} Using Eqns.(\ref{eq:AvPhi}) and (\ref{eq:phizerores}), this condition gives, \begin{eqnarray} \bar{\phi}_{s}(z)+\bar{\phi}_{ind}(z)= -2\pi\int[|z-z^{\prime}|-z^{\prime}][\bar{\rho}_{s}(z^{\prime})+\bar{\rho}_{ind}(z^{\prime})]dz^{\prime}\label{eq:phizeroRes} \end{eqnarray} Since the screening length in a metal is typically short on the order of 1 \AA{}, the surface charge distribution of the semi-infinite metal surface $M$ is highly localised and the combined system is neutral, so that the total system can be confined in a supercell $\mathcal{V}$ of volume $V$ by introducing a PBC in the perpendicular direction. However, the separation of charges between $M$ and $S$ gives rise to a dipole moment and a long-ranged potential $\bar{\phi}(z)$ that has to be treated carefully. In addition, one also has to ensure that the length $L$ of the supercell along the z-direction is large enough so that the short-ranged part of the potential $\phi^{\prime}({\bf r})$, Eqn.( \ref{eq:latvarPhi}), is confined within $\mathcal{V}$ . In order to proceed, we first need to discuss the electrostatics in a supercell. The electrostatic potential $\phi({\bf r})$ from a charge distribution $\rho({\bf r})$ is most efficiently obtained by introducing a three dimensional (3D) plane wave representation of the densities and electrostatic potential over the 3D reciprocal lattice vectors ${\bf g}$ of the supercell. The plane wave expansion and coefficients $\rho({\bf g})$ of a density $\rho({\bf r})$ are defined here as, \begin{equation} \left\{ \begin{array}{lcl} \rho({\bf r}) & = & \sum_{{\bf g}}\rho({\bf g})\exp[i{\bf g}.{\bf r}]\\ \rho({\bf g}) & = & \frac{1}{V}\int_{\mathcal{V}}\rho({\bf r})\exp[-i{\bf g}.{\bf r}]d{\bf r\ .} \end{array}\right.\label{eq:3DPlaneWave} \end{equation} The plane wave coefficients of the electrostatic field $\phi({\bf g})$ from the density $\rho({\bf r})$ are obtained from the solution of the Poisson equation in reciprocal space and are given by, \begin{eqnarray} \phi({\bf g})=\left\{ \begin{array}{ll} 0 & ,\ {\bf g}=0\\ \frac{4\pi}{g^{2}}\rho({\bf g}) & ,\ {\bf g}\neq0 \end{array}\right.\label{eq:3DCoulomb} \end{eqnarray} Note that this solution for $\phi({\bf r})$ from a $\rho({\bf r})$ with a net charge corresponds to the electrostatic potential from $\rho({\bf r})$ compensated with a uniform neutralising background, $-\rho({\bf g=0})$ inside the supercell. Furthermore, the undetermined constant of $\phi({\bf r})$ is set so that the average value $\phi({\bf g}=0)$ of $\phi({\bf r})$ is zero. In the case when the system has a net dipole moment, an artificial uniform electrical field has to be generated in the supercell in order to fulfil the PBCs. An efficient solution to this problem was first proposed by Neugebauer and Scheffler \cite{scheffler} and later corrected by Bengtsson \cite{bengtsson}. In this approach, a net dipole moment of a charge distribution $\rho({\bf r})$ (perpendicular to the z-axis) within the supercell is compensated by introducing a dipole layer at $z=z_{dip}$, \begin{equation} \rho_{dip}({\bf r})=m\delta^{\prime}(z-z_{dip}) \end{equation} where the surface dipole moment density is defined as, \begin{equation} m=\frac{1}{A}\int_{\mathcal{V}}\rho({\bf r})zd{\bf r}\ \label{eq:dipA} \end{equation} and the resulting dipole potential in the supercell is given by, \begin{equation} \phi_{dip}(z)=4\pi m\left[\frac{z}{L}-\frac{1}{2}\right]\ ,\ 0<z<L\ .\label{eq:dippot} \end{equation} when $z_{dip}=L$. By correcting the electrostatic potential in the supercell using this dipole potential, one obtains the dipole-corrected potential \begin{equation} \phi_{dipcor}({\bf r})=\phi({\bf r})+\phi_{dip}(z)\ .\label{eq:dipcorrpot} \end{equation} This corrected potential in the supercell is now equal, up to a constant, to the electrostatic potential from $S$ and $M$ in the absence of the corresponding PBC in the direction perpendicular to the surface. The dipole-corrected effective electrostatic energy is obtained by correcting the electrostatic potentials $\phi_{s}({\bf r})+\phi_{ind}({\bf r})$ and $\phi_{m0}({\bf r})$ in $E_{el}^{eff}$ (Eqn. \ref{eq:EeleffDef}) as \begin{eqnarray} E_{el}^{eff,dip} & = & \int_{\mathcal{V}}\rho_{s}({\bf r})[\phi_{m0}({\bf r})+\phi_{dip0}(z)+\phi_{0}]d{\bf r}+ \\ && \frac{1}{2}\int_{\mathcal{V}}\rho_{s}({\bf r})[\phi_{s}({\bf r})+\phi_{ind}({\bf r})+\phi_{dip1}(z)+\phi_{1}]d{\bf r}, \label{eq:Eldipeff} \end{eqnarray} where $\phi_{dip1}(z)$ and $\phi_{dip0}(z)$ are the dipole potentials from the surface dipole moment densities of $\rho_{s}({\bf r})+\rho_{ind}({\bf r})$ and $\rho_{m0}({\bf r})$, respectively. The constants $\phi_{1}$ and $\phi_{0}$ have to be determined such that the conditions in Eqns. (\ref{eq:AvPhim0}) and (\ref{eq:phizeroRes}) are fulfilled, corresponding to \begin{eqnarray} \phi_{1} & = & -\phi_{s}({\bf r}_{in})-\phi_{ind}({\bf r}_{in})-\phi_{dip1}(z_{in})\label{eq:phi1}\\ \phi_{0} & = & -\phi_{m0}({\bf r}_{out})\label{eq:phi0} \end{eqnarray} where ${\bf r}_{in}$ and ${\bf r}_{out}$ refer to well inside and outside of the metal surface, respectively. The dipole corrections of the K-S potential and the Hellman-Feynman forces in Eqns.(\ref{eq:phieffRes}) and (\ref{eq:FiRes}) are now simply obtained by replacing the electrostatic potential $\phi_{m}({\bf r})+\phi_{s}({\bf r})$ by the dipole corrected potential $\phi_{m}({\bf r})+\phi_{s}({\bf r})+\phi_{dip0}(z)+\phi_{dip1}(z)+\phi_{0}+\phi_{1}$. Before closing this section, we present the expressions for the double counting terms used to evaluate the total energy. In general, the kinetic energy is obtained from the one-electron sum and the double counting term as, \begin{equation} T_{0}[n_{s}]=\sum_{i:occ}\epsilon_{i}-\int_{\mathcal{V}}n_{s}({\bf r})v({\bf r})d{\bf r}.\label{eq:T0Res} \end{equation} Note that adding a constant to the K-S potential, $v({\bf r})$, does not change the kinetic energy and $\phi_{0}+\phi_{1}$ does not give a contribution in this case. The electrostatic part of the double counting term is given by, \begin{eqnarray} E_{el}^{DC} = -\int_{\mathcal{V}}\rho_{es}({\bf r})[\phi_{m}({\bf r})+\phi_{s}({\bf r})+\phi_{dip0}(z)+\phi_{dip1}(z)]d{\bf r} \label{eq:EDCDef} \end{eqnarray} where $\rho_{es}({\bf r})=-en_{s}({\bf r})$ is the charge density from the electrons. Adding this term to the dipole-corrected, electrostatic potential energy in Eqn.(\ref{eq:Eldipeff}), one obtains, \begin{eqnarray} E_{el}^{eff,dip}+E_{el}^{DC} = \nonumber \\ \frac{1}{2}\int_{\mathcal{V}}\rho_{i}({\bf r})\phi_{i}({\bf r})d{\bf r}-\frac{1}{2}\int_{\mathcal{V}}\rho_{es}({\bf r})\phi_{es}({\bf r})d{\bf r}+\label{eq:Ewaldionelectron}\\ \frac{1}{2}\int_{\mathcal{V}}[\rho_{i}({\bf r})-\rho_{es}({\bf r})-\rho_{ind}({\bf r})]\phi_{dip1}(z)d{\bf r}+\label{eq:ECDCdip1}\\ \frac{1}{2}\int_{\mathcal{V}}\rho_{ind}({\bf r})[\phi_{i}({\bf r})-\phi_{es}({\bf r})+\phi_{dip1}(z)]d{\bf r}+\label{eq:ECDCind}\\ \int_{\mathcal{V}}\rho_{i}({\bf r})[\phi_{m0}({\bf r})+\phi_{dip0}(z)]d{\bf r}+\label{eq:ECDCdip0}\\ (\phi_{0}+\frac{1}{2}\phi_{1}) Q_{s}. \label{eq:ECDCconstant} \end{eqnarray} where $\phi_{i}({\bf r})$ is the ionic potential. Note that in the frozen core approximation $\rho_{es}({\bf r})$ is replaced by the valence charge density and $\rho_{i}({\bf r})$ includes the frozen core charge density. \subsection{Classical perfect conductor model for the metal surface response\label{sub:PC-model}} Now we turn to a specific model for the unperturbed charge density $\rho_{m0}({\bf r})$ of the metal surface and its density response $\rho_{ind}({\bf r})$ to an external electrostatic field. In the first application of the proposed scheme, we will use the classical perfect conductor (PC) model. At the end of this Section, we will discuss how to go beyond this simplistic model for the metal surface by using knowledge obtained from previous DFT calculations of the semi-infinite jellium model of a metal surface, in which the charge of the nuclei is smeared out to a positive homogeneous background. The classical PC model is based on a semi-infinite jellium model, here assumed to occupy the half-space $z<z_{p}$, but makes the bold assumption that the screening length is zero, that is, the metallic electrons are able to screen out perfectly any spatial variation of the electric field inside the metal. This assumption implies that \begin{equation} \rho_{m0}({\bf r})=0 \end{equation} and also that $\rho_{ind}({\bf r})$ will be localised at the jellium edge at $z=z_{p}$, \begin{equation} \rho_{ind}({\bf r})=\sigma_{ind}({\bf R})\delta(z-z_{p}).\label{eq:rhomsurf} \end{equation} The surface charge distribution $\sigma_{ind}({\bf R})$ can now be determined by imposing the condition of Eqn.(\ref{eq:phizerores}) that the electrostatic potential inside the metal should be zero, \begin{equation} \phi({\bf r})=0\ ,\ z<z_{p},\label{eq:phimzero} \end{equation} and the laterally averaged part of the induced charge density $\bar{\sigma}(z)$ has to be equal to \begin{equation} \bar{\sigma}(z)=\sigma_{ind}({\bf G}=0)=-\frac{Q_{s}}{A}\label{eq:sigavRes} \end{equation} where $Q_{s}$ is the total charge of $S$. The laterally varying part of the surface charge distribution $\sigma^{\prime}({\bf r})$, as obtained from the 2D plane wave coefficients with non-zero reciprocal lattice vectors (${\bf G}\neq0$), can be determined from the condition in Eqn.(\ref{eq:phimzero}). Since $\phi_{s}({\bf r})$ satisfies the Laplace equation in the metal, the $z$ dependencies of their plane wave coefficients for ${\bf G}\neq0$ are given by, \begin{equation} \phi_{s}(z,{\bf G})=\phi_{s}(z_{p},{\bf G})\exp[G(z-z_{p})]\ ,\ z<z_{p}\ .\label{eq:phisG} \end{equation} Using the plane wave representation of the Coulomb kernel in Eqn.(\ref{eq:2DFourierCoulomb}), the corresponding coefficients of the induced electrostatic potential from the PC are given in this region by, \begin{equation} \phi_{ind}(z,{\bf G})=\frac{2\pi}{G}\exp[G(z-z_{p})]\sigma_{ind}({\bf G})\ ,\ z<z_{p}\ .\label{eq:phiindG} \end{equation} Using (\ref{eq:phisG}) the condition in Eqn.(\ref{eq:phimzero}) will now be obeyed if and only if, \begin{equation} \sigma_{ind}({\bf G})=-\frac{G}{2\pi}\phi_{s}(z_{p},{\bf G})\ ,\label{eq:sigG} \end{equation} which in a real space corresponds to the following condition on the laterally varying parts of $\phi_{s}({\bf r})$ and $\sigma({\bf R})$, \begin{equation} \sigma_{ind}^{\prime}({\bf R})=-\frac{1}{2\pi}\frac{\partial\phi_{s}^{\prime}}{\partial z}({\bf R},z_{p})\ .\label{eq:latvarsigma} \end{equation} Note that the derivations of Eqns.(\ref{eq:sigG}) and (\ref{eq:latvarsigma}) are based on the assumption that the electron densities of $S$ is not overlapping. In the case of such an overlap, the result in Eqn.(\ref{eq:latvarsigma}) in contrast to the result in Eqn.(\ref{eq:sigG}) does not obey the symmetry condition for the response kernel in Eqn.(\ref{eq:rhoindSymm}). Since in practise a small overlap cannot be avoided we will use the result in Eqn.(\ref{eq:sigG}) rather than the result in Eqn.(\ref{eq:latvarsigma}). The induced electrostatic potential $\phi_{ind}({\bf r})$ can now be obtained outside the PC from the induced surface charge distribution using Eqns.(\ref{eq:sigavRes}) and (\ref{eq:sigG}) but some care is needed to handle the boundary condition for the laterally averaged part of the induced potential. The laterally averaged potential $\bar{\phi}_{s}(z)$ from a charge distribution of $S$ that is well-separated from the PC is given by, \begin{equation} \bar{\phi}_{s}(z)=\frac{2\pi Q_{s}}{A}(z-\bar{z}_{s})\ ,\ \ z\leq z_{p}\ , \end{equation} where $\bar{z}_{s}$ is the centroid of the charge distribution of $S$ defined as, \begin{equation} \bar{z}_{s}=\frac{1}{Q_{s}}\int\bar{\rho}_{s}(z)zdz. \end{equation} Since according to the condition in Eqn.(\ref{eq:phimzero}), $\phi_{ind}(z_{p})=-\phi_{s}(z_{p})$, the resulting laterally averaged induced electrostatic potential from Eqn.(\ref{eq:sigavRes}) in the region outside the PC is given by, \begin{equation} \bar{\phi}_{ind}(z)=\frac{2\pi Q_{s}}{A}(z+\bar{z}_{s}-2z_{p})\ ,\ z>z_{p}\ .\label{eq:latavphimRes} \end{equation} The plane wave coefficients from the laterally varying part of the induced electrostatic potential in the region outside the perfect conductor is now obtained directly from Eqns.(\ref{eq:latvarPhi}) and (\ref{eq:sigG}) as \begin{equation} \phi_{ind}(z,{\bf G})=-\phi_{s}(z_{p},{\bf G})\exp[-G(z-z_{p})]\ ,z>z_{p}\ .\label{eq:latvarphimRes} \end{equation} Note that according to Eqns.(\ref{eq:latavphimRes}) and (\ref{eq:latvarphimRes}), the induced electrostatic potential in real space is given outside the PC by the classical image potential corresponding to the electrostatic potential from the mirror image of the charge distribution of $S$ in the plane $z=z_{p}$ as given by, \begin{equation} \phi_{ind}({\bf R},z)=-\phi_{s}({\bf R},2z_{p}-z)\ ,\ \ z>z_{p}\ . \end{equation} \subsubsection{Interaction with a point charge\label{sub:IntPointCharge}} In evaluating the proposed scheme based on a perfect conductor (PC) model, it is useful to have the result for the interaction energy $E_{int}(z_{0})$ between the PC and an external point charge $\rho_{s}({\bf r})=Q_{s}\delta({\bf r}-{\bf r}_{0})$, located at a position ${\bf r}_{0}$ in the supercell. The interaction energy is obtained from Eqns.(\ref{eq:EintExpr}) and (\ref{eq:Eldipeff}) and can be decomposed into one contribution arising from the lateral averaged potential, $\bar{E}_{int}(z_{0})$, and one contribution arising from the laterally varying part of the potential, $E_{int}^{\prime}(z_{0})$, \begin{eqnarray} \bar{E}_{int} & = & \frac{Q_{s}}{2}[\bar{\phi}_{ind}(z_{0})+\phi_{dip1}(z_{0})+\phi_{1}]\label{eq:EintAvDef}\\ E_{int}^{\prime} & = & \frac{Q_{s}}{2}\phi_{ind}^{\prime}(z_{0})\label{eq:EintVarDef} \end{eqnarray} where, according to Eqns.(\ref{eq:dippot}) and (\ref{eq:phi1}), the dipole corrections terms are determined by, \begin{eqnarray} & &\phi_{dip1}(z) =\frac{4\pi Q_{s}}{A}(z_{0}-z_{p})[\frac{z}{L}-\frac{1}{2}]\label{eq:PhiDip1Res}\\ & & \phi_{1} = -[\bar{\phi}_{ind}(z_{p})+\bar{\phi}_{s}(z_{p})+\phi_{dip1}(z_{p})]. \label{eq:PhiDip1ConRes} \end{eqnarray} Note that in the absence of perpendicular periodic boundary conditions $\bar{\phi}_{ind}(z_{0})$ is given by Eqn.(\ref{eq:latavphimRes}) and $\phi_{dip1}(z_{0})+\phi_{1}=0$. The laterally averaged electrostatic potentials are given in the supercell by, \begin{eqnarray} \bar{\phi_{s}}(z) = \frac{2\pi Q_{s}}{A}[-|z-z_{0}|+\frac{(z-z_{0})^{2}}{L}+\frac{L}{6}] & & \label{eq:latavphisPBC}\\ \bar{\phi}_{ind}(z) = -\frac{2\pi Q_{s}}{A}[-|z-z_{p}|+\frac{(z-z_{p})^{2}}{L}+\frac{L}{6}]. & & \label{eq:latphiindPBC} \end{eqnarray} Inserting these results into Eqns.(\ref{eq:PhiDip1ConRes}), one obtains that \begin{eqnarray} \bar{\phi}_{ind}(z_{0})+\phi_{dip1}(z_{0})+\phi_{1} = \frac{4\pi Q_{s}}{A}[(z_{0}-z_{p})-\frac{L}{12}]\ ,\label{eq:PhiIndSuper} \end{eqnarray} and the laterally averaged interaction energy is finally given by, \begin{equation} \bar{E}_{int}(z_{0})=\frac{2\pi Q_{s}^{2}}{A}[(z_{0}-z_{p})-\frac{L}{12}]\ .\label{eq:EintAvRes} \end{equation} Note that this result differ from the result obtained in the absence of perpendicular periodic boundary conditions, \begin{equation} \bar{E}_{int}(z_{0})=\frac{2\pi Q_{s}^{2}}{A}(z_{0}-z_{p})\ .\label{eq:EintAvInftyRes} \end{equation} by the extra term \begin{equation} \frac{Q_{s}}{2}\bar{\phi}_{s}(z_{0})=-\frac{\pi Q_{s}^{2}L}{6A}\ .\label{eq:PBCselfenergy} \end{equation} Furthermore, it is noteworthy that this term diverges when $L\rightarrow\infty$ for fixed $A$. The interaction energy $\bar{E}_{int}$ in Eqn.(\ref{eq:EintAvInftyRes}) is nothing else than the electrostatic energy of a parallel plate capacitor where the two plates, each with an area $A$, are separated by the distance $z_{0}-z_{p}$ and having charges $-Q_{s}$ and $Q_{s}$. This repulsive interaction energy vanishes in the limit $A\rightarrow\infty$. In the case of the laterally varying part, $\phi_{ind}^{\prime}({\bf r})$ is obtained from $\rho_{s}(z,{\bf G})=\frac{Q_{s}}{A}\delta(z-z_{0})$ and Eqns.(\ref{eq:latvarPhi}) and (\ref{eq:latvarphimRes}) as, \begin{eqnarray} \phi_{ind}^{\prime}({\bf r})= -\frac{2\pi Q_{s}}{A}\sum_{{\bf G}\neq0}\frac{\exp[-G(z+z_{0}-2z_{p})]}{G}\exp\left[i{\bf G.({\bf R}-{\bf R}_{0})}\right] \nonumber \end{eqnarray} when the induced potential is well-localised within the supercell corresponding to $2\pi(L-z_{0})/L_{\|}\gg1$ and $2\pi z_{0}/L_{\|}\gg1$, with $L_{\|}=L_{x,y}$ the length of the supercell in the $x$ and $y$ directions, parallel to the PC plane. The resulting laterally varying part of the interaction energy, $E_{int}^{\prime}$, in Eqn.(\ref{eq:EintVarDef}) is then given directly by, \begin{equation} E_{int}^{\prime}(z_{0})=-\frac{2\pi Q_{s}^{2}}{A}\sum_{{\bf G}\neq0}\frac{\exp[-2G(z_{0}-z_{p})]}{2G}.\label{eq:EintvarRes} \end{equation} This part of the interaction energy is always attractive and crosses over into the classical image potential, \begin{equation} E_{int}(z_{0})=-\frac{Q_{s}^{2}}{4(z-z_{0})}\\ ,\label{eq:ClassImage} \end{equation} of the point charge in the limit $A\rightarrow\infty$. The Hellman-Feynman force $F_{int}(z_{0})$ on the point charge is determined by the dipole corrected electric field at the point charge as, \begin{eqnarray} {\bf F}_{int}(z_{0})=-Q_{s}\nabla[\bar{\phi}_{ind}(z_{0})+ \phi_{ind}^{\prime}({\bf r}_{0})+\phi_{dip1}(z_{0})+\phi_{1}]\label{eq:FpointChargeDef} \end{eqnarray} Using the expressions for $\bar{\phi}_{ind}(z_{0})$ and the plane wave coefficients $\phi_{ind}^{\prime}({\bf r})$ in Eqns.(\ref{eq:latavphimRes}), (\ref{eq:latvarphimRes}) and (\ref{eq:PhiDip1Res}), one obtains directly, \begin{eqnarray} {\bf F}^{z}_{int}(z_{0})=-\frac{2\pi Q_{s}^{2}}{A}\left(1 + \sum_{{\bf G}\neq0}\exp[-2G(z_{0}-z_{p})]\right) \label{eq:FpointChargeRes} \end{eqnarray} Note that this result for the force is consistent with the result ${\bf F}^{z}_{int}(z_{0})=-\nabla E_{int}(z_{0})\hat{z}$ obtained from the interaction energy $E_{int}(z_{0})=\bar{E}_{int}(z_{0})+E_{int}^{\prime}(z_{0})$ in Eqns.(\ref{eq:EintAvRes}) and (\ref{eq:EintvarRes}). \subsubsection{Beyond the perfect conductor model\label{sub:beyondPC}} The first step in going beyond the prefect conductor approximation for the metal surface is to account for the non-zero screening length of the conduction electrons. Important information and concepts about the behaviour of this response have been drawn in the pioneering DFT studies of the semi-infinite jellium model based on the LDA by Lang and Kohn\cite{kohn_lang}. They showed that there is a spill-out of electrons into the vacuum region that creates an extended dipole layer with a surface charge density $\rho_{m0}(z)$. The dipole moment of this distribution was shown to determine the surface contribution to the work function. This charge contribution is not accounted for in the PC model. From their calculations of the response of the semi-infinite jellium to an external homogeneous electric field, they showed that the classical image plane is located at the centroid $z_{im}$ of the induced density and not at the jellium edge as in the perfect conductor model. However, this effect is easily accounted for in the PC model by choosing $z_{p}=z_{im}$. The static response of the semi-infinite jellium to a laterally varying external potential can be characterised by a wave-vector dependent reflection coefficient $g(G)$\cite{LiebRefl}. The 2D plane wave coefficient of the induced electrostatic potential outside the induced metal density is given by, \begin{eqnarray} \phi_{ind}(z,{\bf G}) & = & -g(G)\phi_{ext}({\bf G})\exp[-Gz] \label{eq:reflDef} \end{eqnarray} where $\phi_{ext}({\bf G})$ is the 2D plane wave coefficient of the external electrostatic potential. For example, the interaction energy of a single point charge $Q_{s}$ at $z=z_{0}$ with the metal surface becomes, \begin{equation} E_{int}(z_{0})=-\frac{Q_{s}^{2}}{2}\intop_{0}^{\infty}dG\exp(-2Gz_{0})g(G)\ .\label{eq:EintRefl} \end{equation} Since the PC model assumes perfect screening for all parallel wave vectors, $g(G)=\exp(2gz_{p})$, according to Eqn.(\ref{eq:latvarphimRes}) where $z_{p}$ gives the position of the PC plane and Eqn.(\ref{eq:EintRefl}) reduces to the classical image potential in Eqn.(\ref{eq:ClassImage}). Calculations by the stabilised jellium model have shown that PC result for $g(G)$ is an excellent approximation for $G$ up to $G_{c}=$0.8 \AA{}$^{-1}$ in the range of 2-4 a$_{0}$ for the electron gas density parameter $r_{s}$\cite{LiebSat}. According to Eqn.(\ref{eq:EintRefl}), this suggests the PC model is still a good approximation for the image potential down to distances of about $1/2G_{C}=0.6$ \AA{}. \section{Computational details and implementation \label{sec:compimpl}} The proposed scheme that is based on a perfect conductor model has been implemented in the plane wave code VASP\cite{vasp}. The key quantity to compute is the induced electrostatic potential $\phi_{ind}({\bf r})$, which determines how the total energy, K-S potential and Hellmann-Feynman forces change in the presence of the PC. The first step is to generate $\bar{\sigma}_{ind}$ from the total charge of $S$, Eqn.(\ref{eq:sigavRes}), and the Fourier components of $\sigma_{ind}^{\prime}({\bf R})$ from Eqn.(\ref{eq:sigG}). The induced density of charge $\rho_{ind}({\bf r})=\sigma_{ind}({\bf R})\delta(z-z_{p})$ was then represented at a plane of grid points corresponding to the position of the PC plane, from which $\rho_{ind}({\bf g})$ was generated by the standard routine in VASP based on Eqn.(\ref{eq:3DCoulomb}). The surface dipole moment $m$ that determines $\phi_{dip}$ through Eqn.(\ref{eq:dippot}) was obtained from Eqn.(\ref{eq:dipA}). The dipole correction term in Eqn.(\ref{eq:ECDCdip1}) is computed from the standard dipole correction subroutine by $\rho_{ind}$ to $\rho_{es}$. Finally, the computation of energy term of Eqn.(\ref{eq:ECDCind}) is carried out in reciprocal space. As a simple illustration and test of the proposed scheme, we have considered the Na$^{+}$ ion outside the PC. The electron-ion interaction was described by the projector augmented wave method \cite{paw}. The six electron in the $p$ semi-core states were treated as valence electrons. The electronic exchange and correlation effects were treated within the PBE version \cite{pbe} of the generalised gradient approximation. The plane wave cut-off energy was set to the standard value of 400 eV and the Brillouin zone was sampled by a 3x3x1 $k$-points. \section{Results\label{sec:Results}} \begin{figure} \centering \includegraphics[scale=0.75]{fig1_upper_panel.eps} \includegraphics[scale=0.75]{fig1_lower_panel.eps} \caption{Induced surface charge density at the perfect conductor plane for Na$^{+}$ ion placed at 1.5 \AA{} (Upper panel) and 2.0 \AA{} (Lower panel) away from the perfect conductor. The size of the surface unit cell is 10 \AA{} x 10 \AA{} and the height of the supercell is 20 \AA{}.} \label{fig:ind_charge} \end{figure} The behaviour of the induced surface electron density, $\sigma_{ind}({\bf R})$, at two different distances of the Na$^{+}$ ion from the perfect conductor plane is shown in Fig. \ref{fig:ind_charge}, for a supercell with transversal area $A=100$ \AA{}$^{2}$ and a height L of 20 \AA{}. It is evident that the induced electron density becomes laterally more extended as the ion separates from the surface but the net total charge remains constant and is equal to $-e$. In other words, the contributions from plane-wave coefficients $\sigma({\bf G}\neq0)$ decrease with the distance and $\sigma_{ind}({\bf R})$ becomes practically uniform when this distance is sufficiently large. \begin{figure} \centering \includegraphics[scale=0.115]{fig2.eps} \caption{Laterally averaged electrostatic potential, $\bar{\phi}(z)=\bar{\phi}_{s}(z)+\bar{\phi}_{m}(z)+\phi_{dip}(z)+\phi_{1}$, as a function of the $z$ coordinate for a Na$^{+}$ ion placed 2 \AA{} away from the perfect conductor plane located at $z=7$ \AA{}. The dipole layer is located at $z$ = 20 \AA{}. Same supercell as in Fig. \ref{fig:ind_charge}.} \label{fig:pot} \end{figure} The characteristic behaviour of the laterally averaged, dipole-corrected electrostatic potential, $\bar{\phi}(z)=\bar{\phi}_{s}(z)+\bar{\phi}_{ind}(z)+\phi_{dip}(z)+\phi_{1}$, is illustrated in Fig. \ref{fig:pot} as a function of the $z$ coordinate when the Na$^{+}$ ion is placed 2 \AA{} outside the perfect conductor plane located at $z=7$ \AA{}. Again, the supercell is the same as in Fig. \ref{fig:ind_charge}. The potential is constant and equal to zero inside the perfect conductor as dictated by the condition in Eqn.(\ref{eq:phimzero}). The discontinuous change in slope at the perfect conductor plane (located at $z=7$\AA) is induced by the laterally averaged surface charge density $\bar{\sigma}$. In the region between the perfect conductor plane and the ion (placed at $z=9$\AA), the slope is constant corresponding to a constant electric field. The spatial extension of the charge distribution of the ion is reflected by the deviation from this linear behaviour. The calculated value of the slope in the potential in this region is 1.81 eV/\AA{}, which is precisely the value of $4\pi\bar{\sigma}/\epsilon_{0}$ = 1.81 eV/\AA{} for the z-component of the electrical field inside a capacitor with plates having opposite surface charge densities of $\bar{\sigma}=e/A=0.01$ e\AA{}$^{-2}$, since $A=100$ \AA{}$^{2}$. At the dipole layer located at $z=$ 20 \AA{} the potential makes a jump, so that the potential is periodic across the supercell boundaries at $z=0$ and 20 \AA{}. Note that the representation of the surface charge on a single plane of grid points give rise to a well-behaved electrostatic potential in contrast to the dipole layer, which has to be smeared out in the $z$ direction in order to damp out Gibbs oscillations. To facilitate the discussion of the behaviour of the calculated interaction energy, $E_{int}$, between the Na$^{+}$ ion and the PC as defined in Eqn.(\ref{eq:EintExpr}), we have decomposed this energy into the contributions $\bar{E}_{int}$ and $E_{int}^{\prime}$ from the laterally averaged, $\bar{\sigma}_{ind}$, and the laterally varying surface charge density, $\sigma_{ind}^{\prime}({\bf R})$, respectively. Note that $E_{s}[n_{s0}${]} in Eqn.(\ref{eq:EintExpr}) has been obtained for the ion in the supercell with a uniform neutralising background. The two contributions $\bar{E}_{int}$ and $E_{int}^{\prime}$ are shown in Figs. \ref{fig:EG0} and \ref{fig:EneG0} for different surface areas of the supercell. Since we shall compare DFT results for the Na$^{+}$ with the analytical results for a point charge in Eqns.(\ref{eq:EintAvRes}) and (\ref{eq:EintvarRes}), we need to avoid a significant overlap between the electron density of the ion and the induced electron density at the perfect conductor plane. With this, we shall only consider distances of the Na$^{+}$ from the perfect conductor plane larger than 0.8 \AA{}. As shown in Fig. \ref{fig:EG0} (Upper panel), the contribution from the uniform surface charge distribution results in a linear variation of the interaction energy with the ion distance to the PC surface. This linear behaviour and the decrease in magnitude of the associated force (given by the negative gradient) with increasing transversal area is in agreement with the corresponding result of Eqn.(\ref{eq:EintAvRes}) in the point charge model. The relative difference between the latter result and the computed interaction energy is smaller than the 2\%, as shown in Fig. \ref{fig:EG0} (Lower panel). These small differences can in part be attributed to the polarisation of the semi-core of the Na$^{+}$and at shorter distances in part the spatial extension of the charge distribution of the ion. Note that $\bar{E}_{int}$ for the different $A$ do not cross at the perfect conductor plane but at a distance of about 1.75 \AA{} outside the perfect conductor plane due to the extra energy term of Eqn.(\ref{eq:PBCselfenergy}). In fact this term gives that the crossing is located at $L/12$, which is in excellent agreement with the result in Fig. \ref{fig:EG0} (Upper panel). \begin{figure} \centering \includegraphics[scale=0.11]{fig3_upper_panel.eps} \includegraphics[scale=0.11]{fig3_lower_panel.eps} \caption{(Upper panel) Calculated contribution, $\bar{E}_{int}$, to the interaction energy from the laterally averaged part of the induced surface charge density, $\bar{\sigma}$, as a function of the distance to the perfect conductor plane for the different sizes of the surface unit cell. (Lower panel) Calculated energy difference with respect to the point charge case result of Eqn.(\ref{eq:EintAvRes}). The height of the supercell is 20 \AA{}.} \label{fig:EG0} \end{figure} In contrast to the laterally average interaction energy $\bar{E}_{int}$, the contribution to the interaction energy from the laterally varying part $E_{int}^{\prime}$ is always attractive, becomes more dominant with increasing surface area and decay rapidly with the distances from the perfect conductor, as shown in Fig. \ref{fig:EneG0} (Upper panel). This result is in close agreement with the result in Eqn.(\ref{eq:EintvarRes}), obtained from the corresponding contribution in the point charge model, as shown by the difference between these two results in Fig. \ref{fig:EneG0} (Lower panel). In fact, relative energy differences between DFT and the point charge case are smaller than 2\%. In the limit of infinite surface area, the interaction tends to the classical image interaction. \begin{figure} \centering \includegraphics[scale=0.11]{fig4_upper_panel.eps} \includegraphics[scale=0.11]{fig4_lower_panel.eps} \caption{(Upper panel) Calculated contribution, $E_{int}^{\prime}$, to the interaction energy from the laterally varying part of the induced surface charge density, $\sigma^{\prime}({\bf R})$, as a function of the distance of the ion to the perfect conductor plane for the different sizes of the surface unit cell.(Lower panel) Calculated energy difference with respect to the point charge model of Eqn.(\ref{eq:EintvarRes}). The height of the supercell is 20 \AA{}.} \label{fig:EneG0} \end{figure} We now present the calculated Hellmann-Feynman force along the $z$ direction in Fig. \ref{force_DFT}. The force is always attractive and becomes constant for large distances of the ion from the perfect conductor plane. As the ion approaches to perfect conductor, the effect of the laterally varying components of the induced charge density become increasingly more important, thus leading to an increased attraction. Clearly, the magnitude of this attraction increases with the transversal area. In the limit of infinite surface area, the induced force will tend to the classical force exerted by the image charge. We observe that the differences with respect to the point charge case are smaller than 2.5\%. \begin{figure} \centering \includegraphics[scale=0.115]{fig5.eps} \caption{Calculated Hellman-Feynman force on the Na$^{+}$ ion as a function the distance, $z_{0}$, from the perfect conductor plane for the different sizes of the surface unit cell. The height of the super cell is 20 \AA{}. Differences with respect to the point charge case are lower than 2.5\%.} \label{force_DFT} \end{figure} \begin{figure} \centering \includegraphics[scale=0.115]{fig6.eps} \caption{Test of the consistency between the Hellman-Feynman force on the ion and the force from the gradient of the total energy. $\Delta F$ is the difference between these two forces and values are smaller than the 0.06\% of the computed forces. Same supercell as in Fig. \ref{fig:ind_charge}.} \label{fig:SCforces} \end{figure} Finally, as a critical test of the implementation of the perfect conductor model, we analyse the consistency between the computed Hellman-Feynman force on the ion and the gradient of the total energy. In Fig. \ref{fig:SCforces}, we show the difference between the force, $F_{z}$, computed along the $z$ direction using Eqn.(\ref{eq:FiRes}) and the numerical derivative of the interaction energy, $-\partial E_{int}/\partial z_{0}$, as a function of the Na$^{+}$ distance to the perfect conductor. Ideally, this difference should be zero but, due to numerical errors, small deviations are always expected. In fact, errors are essentially smaller than the 0.06\% of the computed forces, showing that sufficient numerical consistency has been achieved. This level of self-consistency is also obtained for smaller distances where we have a more substantial overlap of the electron densities. Note that this high degree of consistency even in the case of overlapping electron densities is guaranteed by using the density response in Eqn.(\ref{eq:sigG}) that obeys the symmetry condition in Eqn.(\ref{eq:rhoindSymm}). The above results suggest that we have derived and implemented a DFT method for a charged system in front of a perfect conductor. Small differences with respect to the the point charge case could be attributed to the differences in the spatial extension of the densities of charge and that the point charge model does not include any polarisation, which will be inevitably present when placing atoms or molecules under the electrical field generated by the induced potential. We believe that the use of this new DFT methodology becomes particularly convenient since it allows to the possibility of computing DFT problems of different charge states in a controllable way, that is, by defining (at will) the amount of charge transfer between the system and the perfect conducting plane. However, we have shown that the use of the perfect conductor approximation only induces attractive interactions, which would make the system to move closer and closer to the perfect conductor plane. This represent a limitation if one aims to make use of this DFT methodology to simulate realistic problems involving metallic surfaces. In fact, the metallic surface will exert repulsive forces for sufficiently close distances, thus avoiding the system to collapse with the surface \cite{pauli}. This lack of repulsion is a direct consequence of having considered only the electrostatic interactions in the approximate energy functional of Eqn.(\ref{eq:EtotApp}). In a following publication \cite{IS_MP}, we propose a new procedure to surmount this limitation. \section{Concluding Remarks\label{sec:conc}} To the final purpose of developing a simplified density functional theory (DFT) method for treating charged atoms and molecules on an ultrathin-insulating film supported by a metal substrate, we have presented a new approximate DFT methodology for the calculation of the total energy, ionic (Hellman-Feynman) forces and electronic structure of a charged system placed in front of a metal surface. In this new methodology, the electron densities of the metal surface and the charged system are assumed to be non-overlapping and there is only an electrostatic interaction between these two fragments. Whereas the charged system is treated fully using DFT, the metal surface is approximated within linear response, corresponding to an expansion of the energy of the metal surface to second order in the electrostatic potential from the charged system. In particular, we have carried out a careful analysis of the effect of periodic boundary conditions and derived appropriate dipole corrections for the total energy and the ionic forces. The proposed method have two main advantages. First, the metal electron states are not treated explicitly but only implicitly through their density response to an external electrostatic field, which can be captured in a simple model. Here, we have explicitly considered the perfect conductor approximation for the metal response although the method is not strictly limited to this approximation. In the perfect conductor model the screening length by the conduction electron is assumed to be zero and the screening charge resides on a plane. Second, different charge states of the charged system can be handled directly. Based on the simple perfect conductor approximation for the metal response, we have implemented this method in the density functional theory VASP code. A simple illustration and test of this method is provided by the case of the Na$^{+}$ ion outside a perfect conductor. The six electrons in the $p$ semi-core of the ion were included in the calculations. The success of our implementation is demonstrated by the excellent agreement between the calculated Hellman-Feynman force and the gradient of the interaction energy, even in the region close to the perfect conductor plane where the electron density of the ion overlaps with the induced charge density at the perfect conductor plane. Furthermore, the small polarisability of the ion makes the calculated interaction energy to be in close agreement to the analytical result for a point charge outside the perfect conductor within a supercell of finite size. Finally, to fulfil our overall aim, we need to include the missing interactions in our simplified scheme arising from over-lapping densities and van der Waals interactions between the ultrathin-insulating film with charged adsorbates and the metal substrate. We plan to do this by developing a simple parametrised force field between the metal substrate and the atoms in the adjacent layer of the film, where the parameters are obtained by fitting the resulting interactions to DFT calculations of the film (without adsorbates) over the metal substrate. The presentation of this scheme and associated results of this model are deferred to a separate publication. \ack This work is supported by the grant (F/00 025/AG) from the Leverhulme Trust. Mats Persson is grateful for the support from the Swedish Research Council (VR) and the EU project ARTIST. \section*{References}

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\section{Introduction} Let $X_1$ and $X_2$ be two random variables, such that $X_1$ is \emph{stochastically larger} than $X_2$. This means that $F_1(x) \le F_2(x)$, $x \in \mathbb{R}$, for their cumulative distribution functions (cdfs) $F_1$ and $F_2$, respectively. It is well known that this is the case if and only if $X_1$ and $X_2$ can be realized on the same probability space, so that $X_1\geq X_2$ almost surely. The objective of this paper is to characterize all random vectors $(X_1, X_2)$ such that $X_1$ and $X_2$ have given cdfs and $\mathbb{P}(X_1\geq X_2)=1$. As a first observation, we establish a representation of joint distributions of ordered random variables as distributions of the order statistics sampled from an exchangeable bivariate law. \begin{theorem} \label{thr:order} A random vector $(X_1,X_2)$ with marginal cdfs $F_1$ and $F_2$ satisfies $\mathbb{P}(X_1\geq X_2)=1$ if and only if $X_1=\max(V_1,V_2)$ and $X_2=\min(V_1,V_2)$ for a random vector $(V_1,V_2)$ with exchangeable components and such that $\mathbb{P}(V_1\leq x,V_2\leq x)=F_1(x)$ and $\mathbb{P}(V_i\leq x)=G(x)$ for all $x$, $i\in\{1,2\}$, where \begin{equation} \label{eq:2} G(x):=\frac{1}{2}(F_1(x)+F_2(x)), \quad x\in\mathbb{R}. \end{equation} \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}}] The vector $(V_1,V_2)$ obtained as the random permutation of $(X_1,X_2)$ is exchangeable and its marginal distributions are $G$. Furthermore, \begin{displaymath} \mathbb{P}(V_1\leq x,V_2\leq x) =\mathbb{P}(X_1\leq x,X_2\leq x)=\mathbb{P}(X_1\leq x). \end{displaymath} Conversely, if $X_1$ and $X_2$ are order statistics from $(V_1,V_2)$, then $X_1\geq X_2$ a.s., and \begin{align*} \mathbb{P}(X_1\leq x)&=\mathbb{P}(V_1\leq x,V_2\leq x)=F_1(x),\\ \mathbb{P}(X_2\leq x)&=\mathbb{P}(V_1\leq x,V_2\leq x) +\mathbb{P}(V_1\leq x,V_2>x)+\mathbb{P}(V_1>x,V_2\leq x)\\ &=2 G(x)-F_1(x)=F_2(x). \qedhere \end{align*} \end{proof} This theorem complements already known results deriving the distribution of order statistics from general multivariate laws, see \cite{nav:spiz10,leb:dut14,diet:fuc:sch16}. If the supports of $X_1$ and $X_2$ are disjoint intervals, then any dependency structure between them is possible. Otherwise, restrictions are necessary, e.g.\ $X_1$ and $X_2$ cannot be independent. In Section~\ref{sec:char-stoch-order}, we give a complete description of the joint distribution of $X_1$ and $X_2$. This description is given in terms of copulas and their diagonal sections. In Section~\ref{sec:maxim-indep-pairs}, we identify bounds on the joint distribution of $(X_1,X_2)$ in terms of concordance ordering. Then in Section~\ref{sec:nonp-corr-coeff}, we determine the smallest possible nonparametric correlation coefficients. The joint distribution of $(X_1,X_2)$ with the maximal entropy is found in Section~\ref{sec:maxim-entr-solut}, followed by examples in Section~\ref{sec:examples}. A multivariate extension and a generalization to random elements in partially ordered spaces are presented in Section~\ref{sec:generalizations}. \section{Characterization of stochastically ordered copulas} \label{sec:char-stoch-order} A (bivariate) copula $C:[0,1]^2 \to [0,1]$ is the cdf of a random vector $(U_1,U_2)$ with standard uniformly distributed marginals. The joint cdf of each random vector $(X_1,X_2)$ can be written as \[F(x_1,x_2)=C(F_1(x_1),F_2(x_2))\] for a copula $C$ with $F_1$ and $F_2$ being the marginal cdfs. A copula $C$ is called \emph{symmetric} on a set $A \subset [0,1]$ if $C(u,v) = C(v,u)$ for all $u,v \in A$. For $A = [0,1]$, symmetry of the copula is equivalent to the pair $(U_1,U_2)$ being exchangeable. The following theorem provides a characterization of all dependence structures that are compatible with the stochastic ordering of the marginals. \begin{theorem} \label{thm:one} Let $(X_1,X_2)$ be a random vector with marginals $X_1$ and $X_2$ having cdfs $F_1$ and $F_2$, respectively. Then $\mathbb{P}(X_1\geq X_2)=1$ if and only if $F_1(x)\leq F_2(x)$ for all $x\in\mathbb{R}$ and the joint cdf of $(X_1,X_2)$ is given by \begin{equation} \label{eq:1} F(x_1,x_2)= \begin{cases} F_1(x_1), & x_1\leq x_2,\\ 2\tilde{C}(G(x_1),G(x_2))-F_1(x_2),& x_1 > x_2, \end{cases} \end{equation} for all $x_1,x_2\in\mathbb{R}$, where $G$ is given by \eqref{eq:2} and $\tilde{C}$ is a symmetric copula on the range of $G$ such that \begin{equation} \label{eq:8} \tilde{C}(G(x),G(x))=F_1(x),\quad x\in\mathbb{R}. \end{equation} \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}} Sufficiency. Let $(V_1,V_2)$ be distributed according to the symmetric bivariate cdf $\tilde{C}(G(x_1),G(x_2))$. By construction, $V_1$ and $V_2$ are identically distributed with cdf $G$. Furthermore, $\mathbb{P}(V_1 \le x,V_2 \le x) = \tilde{C}(G(x),G(x)) = F_1(x)$. The distribution of $(\tilde{X}_1,\tilde{X}_2)=(\max\{V_1,V_2\},\min\{V_1,V_2\})$ is given by \eqref{eq:1}, so sufficiency follows from Theorem~\ref{thr:order} because $\mathbb{P}(\tilde{X}_1 \ge \tilde{X_2}) = \mathbb{P}(X_1 \ge X_2)$. \noindent Necessity. Let $(V_1,V_2)$ be as in Theorem~\ref{thr:order}. Then, any copula $\tilde{C}$ of $(V_1,V_2)$ satisfies \begin{align*} \tilde{C}(G(x),G(y)) &= \mathbb{P}(V_1 \le x, V_2 \le y) = \mathbb{P}(V_1 \le y, V_2 \le x) = \tilde{C}(G(y),G(x)),\\ F_1(x) &= \mathbb{P}(V_1 \le x, V_2 \le x) = \tilde{C}(G(x),G(x)) \end{align*} for all $x,y \in \mathbb{R}$. \end{proof} Diagonal sections of copulas, i.e. the functions that arise as $\delta(t)=C(t,t)$, $t\in[0,1]$, for some copula $C$ are characterized by the following properties, see \cite{DuranteMesiarETAL2005}. \begin{defn} A function $\delta:[0,1] \to [0,1]$ is a \emph{diagonal section} if \begin{description} \item[\textnormal{(D1)}] $\delta(0)=0$, $\delta(1)=1$; \item[\textnormal{(D2)}] it is increasing; \item[\textnormal{(D3)}] $|\delta(t)-\delta(s)|\le 2|t-s|$ for all $t,s \in [0,1]$; \item[\textnormal{(D4)}] $\delta(t)\le t$ for all $t \in [0,1]$. \end{description} \end{defn} For an increasing function $F:\mathbb{R} \to [0,1]$, denote by \begin{displaymath} F^-(t):=\inf\{x:\; F(x)\geq t\} \in [-\infty,+\infty],\quad t\in[0,1], \end{displaymath} the generalized inverse of $F$, where $\inf \emptyset = +\infty$. Note that \begin{equation}\label{eq:15} F\circ F^-(t)=t \end{equation} for all $t$ from the range of $F$, see \cite[Proposition 2.3(4)]{EmbrechtsHofert2013}. For notational convenience, we set $F(-\infty) = 0$, $F(\infty) = 1$. The following result follows from the representation \eqref{eq:8} of the diagonal section of the copula $\tilde{C}$ and identity \eqref{eq:15}. \begin{corollary} \label{cor:one} Let $G$ be given by \eqref{eq:2}. The function \begin{equation} \label{eq:3} D := F_1\circ G^-, \end{equation} given by the composition of $F_1$ and the generalized inverse of $G$, is the restriction of a diagonal section to the range of $G$. \end{corollary} We can also provide a converse to Corollary~\ref{cor:one}. \begin{proposition} \label{prop:two} Let $\delta$ be a diagonal section. Then there are cdfs $F_1$ and $F_2$ such that $F_1 \le F_2$ and $\delta = F_1 \circ G^-$ with $G$ defined at \eqref{eq:2}. \end{proposition} \begin{proof}[\textnormal{\textbf{Proof}}] We can extend $\delta$ to an increasing function on $\mathbb{R}$ with range $[0,1]$. Its generalized inverse $\delta^-$ is left-continuous. For $x \in [0,1)$, we define \begin{displaymath} \delta^-(x+) = \lim_{y \downarrow x} \delta^-(y), \qquad \delta^-(1+) = 1. \end{displaymath} The function $x\mapsto \delta^-(x+)$ is increasing and right-continuous, $\delta(\delta^-(x+))=x$, and $\delta^-(x+) \ge \delta^-(x)$ for all $x \in [0,1]$. Let $x \le y$, $x, y \in [0,1]$. Then, by (D3), \[ 0 \le y - x = \delta(\delta^-(y+)) - \delta(\delta^-(x+)) \le 2(\delta^-(y+) - \delta^-(x+)). \] Set $F_1(x) = x$ for $x \in [0,1]$ and $F_2(x) = 2\delta^-(x+) - x$ for $x \in [0,1]$. The function $F_2$ is a cdf by the above arguments. By (D4), we obtain $F_1 \le F_2$. We have $G(x) = \delta^-(x+)$, $x \in [0,1]$. It remains to be checked that $G^- = \delta$. The function $\delta$ is constant on $[\delta^-(x),\delta^-(x+)]$ for any $x \in [0,1]$. This implies, for any $t \in (0,1]$, \begin{align*} G^-(t) &= \inf\{x:\; G(x) \ge t\} = \inf\{x:\; \delta^-(x+) \ge t\} = \inf\{\delta(\delta^-(x+)):\; \delta^-(x+) \ge t\} \\ &= \delta(\inf\{\delta^-(x+):\; \delta^-(x+) \ge t\}) = \delta(t).\qedhere \end{align*} \end{proof} Equation~\eqref{eq:8} specifies the diagonal section $\tilde{C}(t,t)$ of the copula $\tilde{C}$ for all $t$ from the range of $G$. If both $X_1$ and $X_2$ are non-atomic, then this range is $[0,1]$ and so the diagonal section of $\tilde{C}$ is uniquely specified. \begin{example} \label{ex:archi} An Archimedean generator is a decreasing convex function $\psi:[0,\infty) \to [0,1]$ with $\psi(0)=1$ and $\lim_{x \to \infty}\psi(x) = 0$, see \cite[Theorem 6.3.2]{SchweizerSklar1983}. Note that we define an Archimedean generator following \cite{McNeilNeslehova2009}. For the Archimedean copula $C(u,v) = \psi(\psi^-(v) + \psi^-(u))$, $u,v \in [0,1]$, with generator $\psi$, the diagonal section is $\delta(t) = \psi(2\psi^-(t))$. There are many parametric families of Archimedean generators. For example, the Gumbel family of copulas is generated by $\psi_\theta(t) = \exp(-t^{1/\theta})$, $\theta \in [1,\infty)$. Then, the cdf constructed in the proof of Proposition~\ref{prop:two} is $F_2(t) = 2t^a - t$, $t \in [0,1]$, with $a = 2^{-1/\theta}$. \end{example} \begin{example}[Identical distributions] \label{ex:equal} If $X_1$ and $X_2$ are identically distributed, then $F_1=F_2=G$. In this case, the diagonal section of $\tilde{C}$ in Theorem~\ref{thm:one} is given by $D(t)=t$ for all $t$ from the range of $G$. For $x_1>x_2$, by \eqref{eq:8}, \begin{displaymath} F_1(x_2)=\tilde{C}(G(x_2),G(x_2))\leq \tilde{C}(G(x_1),G(x_2))\leq \tilde{C}(1,G(x_2)) =G(x_2), \end{displaymath} hence, $\tilde{C}(G(x_1),G(x_2)) = F_1(x_2)$. Therefore, by \eqref{eq:1}, the joint law of $(X_1,X_2)$ satisfies $F(x_1,x_2)=F_1(\min\{x_1,x_2\})$, meaning that $X_1=X_2$ almost surely. \end{example} \begin{example}[Discrete distributions] \label{ex:discrete} Assume that $X_1$ and $X_2$ have discrete distributions, say supported on $\{0,1\}$ with masses $p,1-p$ and $q,1-q$, respectively, and such that $p\leq q$. The range of $G$ is $\{0,(p+q)/2,1\}$. The condition \eqref{eq:8} on $\tilde{C}$ in Theorem~ \ref{thm:one} is \[\tilde{C}((p+q)/2,(p+q)/2)= p.\] While there are clearly many copulas that satisfy this constraint, the condition is sufficient to uniquely determine the joint law of $(X_1,X_2)$. By \eqref{eq:1}, $\mathbb{P}(X_1=1,X_2=0)=2\tilde{C}(1,(p+q)/2)-2p=q-p$. \end{example} \begin{example}[Disjoint supports] \label{ex:disjoint} Assume that $X_1$ is uniformly distributed on $[1,2]$ and $X_2$ on $[0,1]$. In this case, all kinds of dependency structures between $X_1$ and $X_2$ are allowed. For $x_1\in[1,2]$ and $x_2\in[0,1]$, \eqref{eq:1} yields that \begin{displaymath} F(x_1,x_2)=2\tilde{C}(x_1/2,x_2/2). \end{displaymath} As prescribed by \eqref{eq:8}, $\tilde{C}(t,t)=0$ for $t\in[0,1/2]$ and $\tilde{C}(t,t)=2t-1$ for $t\in(1/2,1]$. This is the diagonal section of the Fr\'echet-Hoeffding lower bound. It is not a contradiction that any copula $C$ yields a possible bivariate law $F(x_1,x_2) = C(F_1(x_1),F_2(x_2))$ of $(X_1,X_2)$ such that $X_1 \ge X_2$ almost surely, but in the representation of Theorem~\ref{thm:one}, there are restrictions on the diagonal of the symmetric copula $\tilde{C}$. The copula $\tilde{C}$ is the copula of the random permutation $(V_1,V_2)$ of $(X_1,X_2)$, and as such it cannot put any mass on the squares $[0,1/2]^2$ or $[1/2,1]^2$. \end{example} \section{Pointwise bounds on the joint cdf} \label{sec:maxim-indep-pairs} By Theorem~\ref{thm:one}, the range of all possible bivariate cdfs of random vectors $(X_1,X_2)$ with given marginals $F_1$ and $F_2$ and such that $X_1\geq X_2$ a.s.\ depends on the choice of a symmetric copula $\tilde{C}$ satisfying \eqref{eq:8}, equivalently, having the diagonal section \eqref{eq:3} on the range of $G$. For a general diagonal section $\delta$, the following result holds. \begin{theorem}[\cite{nel:ques:rod:04,KlementKolesarova2005}] \label{thm:KK05} Each copula $\tilde{C}$ with diagonal section $\delta$ satisfies \begin{displaymath} B_\delta(u,v)\leq \tilde{C}(u,v),\quad u,v\in[0,1], \end{displaymath} where \begin{equation} \label{eq:lowerB} B_\delta(u,v) := \min\{u,v\} - \inf\{t - \delta(t) : t \in [\min\{u,v\},\max\{u,v\}]\}, \quad u,v\in[0,1], \end{equation} is the \emph{Bertino copula}. The copula $B_\delta$ has diagonal section $\delta$. \end{theorem} Denote \begin{equation} \label{eq:11} H(x):=F_2(x)-F_1(x), \quad x\in\mathbb{R}. \end{equation} \begin{theorem} \label{cor:KK05} Each random vector $(X_1,X_2)$ with marginal cdfs $F_1$ and $F_2$, and such that $X_1\geq X_2$ a.s., has a joint cdf $F$ satisfying \begin{equation} \label{eq:4} L(x_1,x_2)\leq F(x_1,x_2)\leq \min\{F_1(x_1),F_2(x_2)\},\quad x_1,x_2\in\mathbb{R}, \end{equation} where both bounds are attained, and \begin{equation} \label{eq:5} L(x_1,x_2) := \begin{cases} F_1(x_1), & x_1\leq x_2,\\ F_2(x_2)-\inf_{x_2\leq s\leq x_1} H(s), & x_1 > x_2, \end{cases} \end{equation} \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}}] The upper bound in \eqref{eq:4} is the Fr\'echet--Hoeffding one; it corresponds to complete dependence between $X_1$ and $X_2$, so that $X_1=F_1^-(U)$ and $X_2=F_2^-(U)$ for a standard uniformly distributed random variable $U$. For the lower bound, let $\delta$ be a diagonal section which is equal to $D$ at \eqref{eq:3} on the range $R_G$ of $G$. We continuously extend $D$ to the closure $\operatorname{cl}(R_G)$ of $R_G$. Continuity of $\delta$ implies that $\delta$ is equal $D$ on $\operatorname{cl}(R_G)$. The function $\delta$ is bounded below by $\delta_G$ defined as \begin{displaymath} \delta_G(x)=\max\{D(x^-),D(x^+)-2(x^+-x)\}, \end{displaymath} where $x^-=\sup R_G\cap[0,x]$ and $x^+=\inf R_G\cap[x,1]$. This function is itself a diagonal section which is equal to $D$ on $\operatorname{cl}(R_G)$. For all $u,v \in [0,1]$ it holds that $B_{\delta}(u,v) \ge B_{\delta_G}(u,v)$. Therefore, Theorem~\ref{thm:KK05} and \eqref{eq:1} imply for $x_1>x_2$, \begin{equation} \label{eq:43} F(x_1,x_2)\geq 2B_{\delta_G}(G(x_1),G(x_2))-F_1(x_2) =2(G(x_2)-\inf\{t-\delta_G(t): t\in[G(x_2),G(x_1)]\})-F_1(x_2). \end{equation} Since \begin{displaymath} t - \delta_G(t) = \min\{t^- - D(t^-) + t-t^-,t^+ - D(t^+)+t^+ - t\}, \end{displaymath} we can restrict the infimum in \eqref{eq:43} to $R_G$. Hence, \begin{align*} F(x_1,x_2)&\geq F_2(x_2)-2\inf\{G(x)-F_1\circ G^-\circ G(x): x\in[x_2,x_1]\} =L(x_1,x_2). \end{align*} The last equality holds because $G^-\circ G(x) \le x$ always holds (see \cite{EmbrechtsHofert2013}) and $G^-\circ G(x) < x$ only happens if there is an $\varepsilon > 0$ such that $G$ is constant on $(x-\varepsilon,x]$. But if $G$ is constant on some interval, then $F_1$ is necessarily also constant on this interval. \end{proof} The lower bound in \eqref{eq:4} corresponds to the Bertino copula and so yields the least possible dependence between $X_1$ and $X_2$. If the function $H$ at \eqref{eq:11} is unimodal, this corresponds to the assumption that the Bertino copula \eqref{eq:lowerB} is simple, compare \cite{FredricksNelsen2002}. The unimodality condition (which also appears in Theorem~\ref{thm:SpearmanW}) applies in many examples, and simplifies the structure of the distribution $L$ considerably. \begin{corollary} \label{cor:unimodal} Assume that the function $H$ is unimodal, that is, $H$ increases on $(-\infty,r]$ and decreases on $[r,\infty)$ for some $r$. Then \begin{equation} \label{eq:10} L(x_1,x_2) = \begin{cases} F_1(x_1), & x_1\leq x_2,\\ F_1(x_1)-\min\{F_1(x_1)-F_1(x_2),F_2(x_1)-F_2(x_2)\}, & x_1 > x_2. \end{cases} \end{equation} \end{corollary} \begin{proof}[\textnormal{\textbf{Proof}}] Let $x_1 > x_2$. By the unimodality, the infimum of $H$ over $[x_2,x_1]$ is attained at one of the end-points $x_2$ or $x_1$. Therefore, \begin{align*} F_2(x_2)-\inf_{x_2\leq s\leq x_1} H(s) &= F_2(x_2) - \min\{F_2(x_1)-F_1(x_1),F_2(x_2) - F_1(x_2)\} \\&= F_1(x_1) - \min\{F_2(x_1)-F_1(x_1)-F_2(x_2) + F_1(x_1),F_2(x_2) - F_1(x_2)-F_2(x_2) + F_1(x_1)\}, \end{align*} which yields \eqref{eq:10}. \end{proof} \begin{example}[Disjoint supports -- Example \ref{ex:disjoint} continued]\label{ex:disjoint2} We assume that $X_1$ is uniformly distributed on $[1,2]$ and $X_2$ on $[0,1]$. Then, \begin{equation}\label{eq:disjoint21} H(x) = F_2(x) - F_1(x) = \begin{cases} x, & x \in [0,1],\\ 2 - x, &x \in [1,2], \\ 0, & \text{otherwise},\end{cases} \end{equation} which is clearly unimodal. Therefore, by Corollary \ref{cor:unimodal}, and for $x_1, x_2 \in [0,2]$ \begin{equation}\label{eq:disjoint22} L(x_1,x_2) = \begin{cases} 0, & x_1 \in [0,1],\\ x_1 - 1, & x_1,x_2 \in [1,2],\\ \max\{x_1+x_2-2,0\}, & x_1 \in [1,2], x_2 \in [0,1].\\ \end{cases} = \max\{F_1(x_1) + F_2(x_2) - 1,0\}, \end{equation} which corresponds to choosing the Fr\'echet-Hoeffding lower bound as the dependence structure for $(X_1,X_2)$. \end{example} As shown by Rogers in \cite{rog99}, if $H$ is unimodal, the distribution given by \eqref{eq:10} maximizes the payoff (or transportation cost) $\mathbb{E}\phi(|X_1-X_2|)$ over all strictly convex decreasing functions $\phi:\mathbb{R}_+\mapsto\mathbb{R}_+$. Without unimodality assumption, the joint distribution maximizing the payoff is given by \begin{displaymath} P(x_1,x_2):= \begin{cases} F_1(x_1), & x_1\leq x_2,\\ \sup_{v\leq x_2}\Big[F_2(v)-\inf_{v\leq s\leq x_1} H(s)\Big], & x_1 > x_2. \end{cases} \end{displaymath} This joint distribution satisfies $L(x_1,x_2)\leq P(x_1,x_2)\leq \min\{F_1(x_1),F_2(x_2)\}$; it provides the joint distribution with the largest mass concentrated on the diagonal, see \cite[Th.~7.2.6]{rac:rus98}. The following result concerns the support of the random vector with distribution $L$ in the case when the cdfs $F_1$ and $F_2$ are continuous. In the general case, the support of $L$ is more intricate to describe. \begin{lemma} \label{lemma:support} Suppose that $F_1$ and $F_2$ are continuous. The support of the distribution $L$ given at \eqref{eq:5} is the set \begin{align} \label{eq:9} A&=\Big\{(x_1,x_2)\in\mathbb{R}^2:\; x_2 < x_1,\; H(x_2)=H(x_1)< H(s) \;\text{for all $s \in (x_2,x_1)$} \Big\}\notag \\ & \quad \cup \Big\{(x_1,x_2)\in\mathbb{R}^2:\; x_2 < x_1,\; H(x_2)=H(x_1)\le H(s) \;\text{for all $s \in (x_2,x_1)$}\notag \\& \qquad \quad\text{and $H(s) < H(x_1)=H(x_2)$ for $s \in (x_2-\varepsilon,x_2) \cup (x_1,x_1+\varepsilon)$ for some $\varepsilon>0$} \Big\}\notag \\& \quad \cup \Big\{(x,x) \in \mathbb{R}^2:\; x \in (S_1 \cap S_2)\backslash T\Big\}, \end{align} where $S_1$ and $S_2$ are the supports of the distributions $F_1$ and $F_2$, respectively, and \begin{displaymath} T= \Big\{ x \in \partial S_1 \cap \partial S_2:\; \text{$(x,x+\varepsilon)\cap S_1 = \emptyset$ and $(x-\varepsilon,x) \cap S_2 = \emptyset$ for some $\varepsilon>0$}\Big\}. \end{displaymath} Here, $\partial S_i$, denotes the topological boundary of $S_i$, $i=1,2$. \end{lemma} \begin{proof}[\textnormal{\textbf{Proof}}] Let $(X_1,X_2)$ have distribution $L$. Let $x \in \mathbb{R}$ and $\varepsilon > 0$. Then, \begin{align}\label{eq:765} \mathbb{P}((X_1,X_2)\in (x-\varepsilon,x+\varepsilon]^2)&=F_1(x+\varepsilon)-F_2(x-\varepsilon)+\inf_{x-\varepsilon\leq s\leq x+\varepsilon} H(s)\\ & =\inf_{x-\varepsilon\leq s\leq x+\varepsilon} \big(F_1(x+\varepsilon) - F_1(s) + F_2(s) - F_2(x-\varepsilon)\big)\nonumber \end{align} The right hand side can only be strictly positive if \begin{displaymath} \min\{F_1(x+\varepsilon)-F_1(x-\varepsilon),F_2(x+\varepsilon)-F_2(x-\varepsilon)\}>0, \end{displaymath} which is the case whenever $\mathbb{P}(X_1\in(x-\varepsilon,x+\varepsilon])>0$ and $\mathbb{P}(X_2\in(x-\varepsilon,x+\varepsilon])>0$. Thus, only points $(x,x)$ with $x$ belonging to $S_1$ and $S_2$ may be in the diagonal parts of the support of $L$. If $x \in T$ and $\varepsilon > 0$ is small enough, then \[ \inf_{x-\varepsilon\leq s\leq x+\varepsilon} \big(F_1(x+\varepsilon) - F_1(s) + F_2(s) - F_2(x-\varepsilon)\big) = \big(F_1(x+\varepsilon) - F_1(x) + F_2(x) - F_2(x-\varepsilon) = 0, \] so $(x,x)$ cannot belong to the support of $L$. Conversely, assume that $x \in S_1 \cap S_2$. Since $H$ is continuous, the infimum in \eqref{eq:765} is attained at some $s_0(\varepsilon) \in [x-\varepsilon,x+\varepsilon]$, if $s_0(\varepsilon) < x$, then $x \in (s_0(\varepsilon),x+\varepsilon)$, hence $F_1(x+\varepsilon) - F_1(s_0(\varepsilon)) > 0$ because $x \in S_1$. One can argue analogously if $s_0(\varepsilon) > x$. If $s_0(\varepsilon) = x$, then we distinguish two cases. If $x \in \operatorname{int} S_1$ or $x \in \operatorname{int} S_2$, then one can argue as previously. Here $\operatorname{int}(S_i)$ denotes the interior of $S_i$, $i\in\{1,2\}$. If $x \in \partial S_1 \cap \partial S_2$ and \[ \inf_{x-\varepsilon\leq s\leq x+\varepsilon} \big(F_1(x+\varepsilon) - F_1(s) + F_2(s) - F_2(x-\varepsilon)\big) = \big(F_1(x+\varepsilon) - F_1(x) + F_2(x) - F_2(x-\varepsilon) = 0, \] then $x \in T$ which yields the claim concerning the diagonal part of the support of $L$. Now assume that $x_2<x_1$ and $0 < \varepsilon < (x_1-x_2)/2$. Then \begin{equation} \label{eq:6} \mathbb{P}((X_1,X_2)\in (x_1-\varepsilon,x_1+\varepsilon]\times(x_2-\varepsilon,x_2+\varepsilon]) =\min\{a,b,c\}+b-\min\{a,b\}-\min\{b,c\}, \end{equation} where \[ a:=\inf_{x_2-\varepsilon\leq s\leq x_2+\varepsilon} H(s),\quad b:=\inf_{x_2+\varepsilon\leq s\leq x_1-\varepsilon} H(s),\quad c:=\inf_{x_1-\varepsilon\leq s\leq x_1+\varepsilon} H(s). \] The probability in \eqref{eq:6} is strictly positive if and only if $a\leq c< b$ or $c\leq a< b$. The point $(x_1,x_2)$ belongs to the support of $L$ if and only if $\max\{a,c\} < b$ for all $\varepsilon>0$ small enough. Letting $\varepsilon$ converge to zero, we find that a necessary condition is that \[ \max\{H(x_1),H(x_2)\} \le \min\{H(x_2),H(x_1)\}, \] hence $H(x_1) = H(x_2)$. It is not hard to check that the conditions on $x_2$ and $x_1$ in $A$ are necessary and sufficient to ensure that $\max\{a,c\} < b$ is fulfilled for all $\varepsilon > 0$ small enough. \end{proof} The set $A$ from \eqref{eq:9} is illustrated in the top-left panel of Fig.~\ref{fig:ex} using points sampled from $L$. \begin{example}[Disjoint supports -- Example \ref{ex:disjoint} continued] We assume that $X_1$ is uniformly distributed on $[1,2]$ and $X_2$ on $[0,1]$. The function $H$ and the distribution $L$ are given at \eqref{eq:disjoint21} and \eqref{eq:disjoint22}, respectively. The support of $L$ is given by \[ \Big\{(2-x,x) \in \mathbb{R}^2:\; x \in [0,1]\Big\}. \] This follows from Example~\ref{ex:disjoint2} or Lemma~\ref{lemma:support}. The set $A$ at \eqref{eq:9} consists of three parts. The first set is $\{(2-x_2,x_2)\in \mathbb{R}^2:\; x_2 \in [0,1)\}$, the second set is empty, and the third set is $\{(x,x) \in \mathbb{R}^2:\; x = 1\}$ because $T$ is the empty set. \end{example} \section{Nonparametric correlation coefficients} \label{sec:nonp-corr-coeff} Dependence measures quantitatively summarize the degree of dependence between two random variables $X_1$ and $X_2$. Kendall's tau and Spearman's rho are arguably the two most well-known measures of association whose sample versions are purely based on ranks. If the marginal distributions of $X_1$ and $X_2$ are continuous then the population versions of Kendall's tau and Spearman's rho only depend on the copula of $(X_1,X_2)$. In this section, we assume that both, $F_1$ and $F_2$ are continuous, and hence the copula of $(X_1,X_2)$ is uniquely defined. We refer the reader to \cite{Neslehova2007} for details concerning problems that arise in the case of arbitrary marginal distributions. Kendall's tau of a copula $C$ is given by \begin{equation} \label{eq:Ktau} \tau := 4\int_{[0,1]^2} C(u,v) \,\mathrm{d} C(u,v) - 1, \end{equation} and, Spearman's rho is given by \begin{equation} \label{eq:Srho} \rho := 12 \int_{[0,1]^2} C(u,v)\,\mathrm{d} u \,\mathrm{d} v - 3 = 12 \int_{[0,1]^2} uv \,\mathrm{d} C(u,v) - 3. \end{equation} These two correlation coefficients are monotonic with respect to pointwise, or, concordance ordering of copulas \cite{Neslehova2007}. They take the value one for the joint cdf given by the upper bound in \eqref{eq:4}. For the copula $C(u,v):=L(F_1^-(u),F_2^-(v))$ with $L$ given by \eqref{eq:5}, these dependence measures attain their lowest values calculated as follows. \begin{theorem} \label{thm:KendallW} Suppose that $F_1$ and $F_2$ are continuous. The smallest possible Kendall's tau of $(X_1,X_2)$ that satisfies the conditions of Theorem~\ref{thm:one} is \begin{displaymath} \tau=4\;\mathbb{E} F_1(X_2)-1. \end{displaymath} \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}}] Writing $C(u,v)=L(F_1^-(u),F_2^-(v))$ yields that \begin{displaymath} \tau=4\;\mathbb{E} L(X_1,X_2)-1, \end{displaymath} where $(X_1,X_2)$ has cdf $L$ given by \eqref{eq:5}. By Lemma~\ref{lemma:support}, the support of $(X_1,X_2)$ is given by the set $A$ at \eqref{eq:9}. On the set $A$, \begin{displaymath} L(X_1,X_2)=F_2(X_2)-(F_2(X_2)-F_1(X_2)), \end{displaymath} hence the result. \end{proof} \begin{theorem} \label{thm:SpearmanW} Suppose that $X_1$ and $X_2$ have continuous cdfs $F_1$ and $F_2$ with the same support $[x^L,x^U]$ and that the function $H(s)=F_2(s) - F_1(s)$ from \eqref{eq:11} is unimodal, strictly increases on $(x^L,r]$ and strictly decreases on $[r,x^U)$ for some $r$. Then the smallest possible Spearman's rho of $(X_1,X_2)$ satisfying the conditions of Theorem~\ref{thm:one} is \begin{align} \label{eq:sp-rho} \rho= 12\; \Bigg[\int_{x^L}^r F_1(s)F_2(s)\,\mathrm{d} F_1(s) &+ \int_r^{x^U} F_1(s)F_2(t(s))\,\mathrm{d} F_1(s) + \int_r^{x^U} F_1(s)(F_2(s)-F_2(t(s)))\,\mathrm{d} F_2(s)\Bigg] -3, \end{align} where $F_2(t(s))-F_1(t(s))=F_2(s)-F_1(s)$ and $t(s)<s$ for $s\in(r,x^U]$ and $t(r)=r$. \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}}] Spearman's rho is given by \begin{displaymath} \rho=12\;\mathbb{E} \big(F_1(X_1)F_2(X_2)\big)-3. \end{displaymath} The set $A$ from \eqref{eq:9}, consists of 3 pieces: $\{(s,s):\; s\in(x^L,r]\}$ with the push-forward of $F_1(s)$, $s\in(x^L,r]$; $\{(s,s):\; s\in(r,x^U)\}$ with the distribution being the image of the measure on $(r,x^U)$ with push-forward of $F_2(s)-F_2(r)$; and $\{(s,t(s)):\; s\in(r,x^U)\}$ with the distribution being the push-forward of $H(r)-H(s)$. The push-forward is the image of the measure on the line by the specified map, e.g., the third part if the image of the measure $\mu$ on $(r,x^U)$ with $\mu((r,s])=H(r)-H(s)$ under the map $s\mapsto(s,t(s))$. The result is obtained by splitting the above expectation into these 3 parts. \end{proof} \section{Maximum entropy distributions} \label{sec:maxim-entr-solut} We assume that both $X_1$ and $X_2$ have full supports on a (possibly infinite) interval $[x^L,x^U]$ and that their cdfs $F_1$ and $F_2$ are absolutely continuous with densities $f_1$ and $f_2$. Amongst all joint absolutely continuous laws $F$ of $(X_1,X_2)$ with given marginals $F_1$ and $F_2$ and such that $X_1 \ge X_2$ a.s., we characterize those maximizing the differential entropy (see \cite[Ch.~8]{cov:thom06}) given by \begin{equation} \label{eq:Shannon} \ent(F) := -\int f(x_1,x_2)\log f(x_1,x_2)\,\mathrm{d} x_1\,\mathrm{d} x_2. \end{equation} These copulas correspond to the least informative (most random) joint distributions, equivalently, to the distributions minimizing the Kullback--Leibler divergence with respect to the uniform distribution. We use the common convention $0\log 0 = 0$. Independently of our work, maximum entropy distributions of order statistics in the multivariate case have been studied in \cite{ButuceaDelmasETAL2018}. Note that the function $G$ from \eqref{eq:2} is absolutely continuous with density $g = (f_1 + f_2)/2$. By Theorem~\ref{thm:one}, the joint law $F$ of $(X_1,X_2)$ is absolutely continuous if and only if the associated symmetric copula $\tilde{C}$ is absolutely continuous. We denote its density by $\tilde{c}$. By the symmetry of $\tilde{C}$, \begin{equation} \label{eq:HFHC} \ent(F) = -\int_{[0,1]^2} \tilde{c}(z_1,z_2) \log \tilde{c}(z_1,z_2)\,\mathrm{d} z_1\,\mathrm{d} z_2 - \log 2 - 2 \int_{x^L}^{x^U} g(z) \log g(z)\,\mathrm{d} z. \end{equation} Therefore, maximizing $\ent(F)$ over all $F$ is equivalent to maximizing $\ent(\tilde{C})$ over all symmetric copulas $\tilde{C}$ with diagonal section $D=F_1 \circ G^-$. Note that the smallest entropy $-\infty$ arises as the limit by considering absolutely continuous distributions approximating the distribution of $X_1=F_1^-(U)$ and $X_2=F_2^-(U)$ for a uniformly distributed $U$. Butucea et al.~\cite{ButuceaDelmasETAL2015} characterize the maximum entropy copula with a given diagonal section $\delta$. We recall some of their notation in order to be able to state our result. For a diagonal section $\delta$ with $\delta(t) < t$ for all $t \in (0,1)$, define for $u, v \in [0,1]$, $u \le v$, \begin{equation} \label{eq:cdelta_simple} \bar{c}_\delta(u,v) := \frac{\delta'(v)(2-\delta'(u))} {4\sqrt{(v-\delta(v))(u-\delta(u))}} \exp\left(-\frac{1}{2}\int_{u}^v \frac{1}{s-\delta(s)}\,\mathrm{d} s\right), \end{equation} and for $u \ge v$, set $\bar{c}_\delta(u,v) = \bar{c}_\delta(v,u)$. Butucea et al.~\cite[Proposition~2.2]{ButuceaDelmasETAL2015} show that $\bar{c}_\delta$ is the density of a symmetric copula with diagonal section $\delta$. Note that the derivative $\delta'$ of $\delta$ exists almost everywhere as $\delta$ is Lipschitz continuous. For a general diagonal section $\delta$, due to its continuity, the set $\{t \in [0,1]: \delta(t) < t\}$ is the union of disjoint open intervals $(\alpha_j,\beta_j)$, $j \in J$ for an at most countable index set $J$. Note that $\delta(\alpha_j)=\alpha_j$ and $\delta(\beta_j)=\beta_j$. For $u,v \in [0,1]$, define \begin{equation} \label{eq:cdelta_general} c_\delta(u,v) := \sum_{j \in J}\frac{1}{\beta_j - \alpha_j} \bar{c}_{\delta_j}\left(\frac{u-\alpha_j}{\beta_j - \alpha_j}, \frac{v-\alpha_j}{\beta_j - \alpha_j}\right)\mathbbm{1}_{(\alpha_j,\beta_j)^2}(u,v), \end{equation} where $\bar{c}$ is given at \eqref{eq:cdelta_simple}, and \begin{displaymath} \delta_j(t) := \frac{\delta(\alpha_j + t(\beta_j-\alpha_j))-\alpha_j}{\beta_j - \alpha_j}, \quad t \in [0,1]. \end{displaymath} Based on the results of \cite{ButuceaDelmasETAL2015}, we arrive at the following theorem. Recall that $H=F_2-F_1$. \begin{theorem} \label{thm:maxentro} Let $(X_1,X_2)$ be a random vector with marginals $X_1$ and $X_2$ satisfying $\mathbb{P}(X_1 \ge X_2) = 1$. Suppose that $X_1$ and $X_2$ have identical support being a (possibly unbounded) interval $[x^L,x^U]$, and that their cdfs $F_1$ and $F_2$ are absolutely continuous with densities $f_1$ and $f_2$. If \begin{equation} \label{eq:7} -\int_{x^L}^{x^U}\log H(z) \,\mathrm{d} G(z) < \infty, \end{equation} then \begin{displaymath} -\infty < \sup_F \ent(F) = \max_F \ent(F) < \infty, \end{displaymath} where the supremum is taken over all possible joint laws of $(X_1,X_2)$. The maximum is attained when the joint density of $(X_1,X_2)$ is given by \begin{displaymath} f(x_1,x_2) = 2 c_D(G(x_1),G(x_2))g(x_1)g(x_2), \quad x^L\leq x_2\leq x_1\leq x^U, \end{displaymath} and $c_D$ is defined at \eqref{eq:cdelta_general}. If \eqref{eq:7} does not hold, then $\sup_F \ent(F)=-\infty$. \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}}] By substitution, \begin{displaymath} -\int_0^1 \log(t-D(t))\,\mathrm{d} t = -\int_{x^L}^{x^U}\log \frac{H(z)}{2}\,\mathrm{d} G(z). \end{displaymath} The result now follows from \cite[Th.~2.5]{ButuceaDelmasETAL2015} in combination with \eqref{eq:HFHC} and Theorem~\ref{thm:one}. \end{proof} The condition $D(t) < t$ for all $t \in (0,1)$ is equivalent to $F_2(z) > F_1(z)$ for all $z \in (x^L,x^U)$. If this condition holds, then $c_D = \bar c_D$ and the formula for the entropy maximizing $f$ in Theorem~\ref{thm:maxentro} simplifies to \begin{displaymath} f(x_1,x_2) = \frac{f_1(x_1)f_2(x_2)}{\sqrt{H(x_1){H(x_2)}}} \exp\left(-\int_{x_2}^{x_1}\frac{1}{H(s)}\,\mathrm{d} G(s)\right) \end{displaymath} for $x_1,x_2 \in [x^L,x^U]$, $x_1 \ge x_2$. \section{Examples} \label{sec:examples} \begin{example}\label{ex:6n1} Let $X_2$ be uniformly distributed on $[0,1]$, and let $X_1$ be distributed as the maximum of $X_2$ and another independent uniformly distributed random variable, that is, $F_1(x)=x^2$. In this case $H(s)=s-s^2$ is unimodal with the maximum at $r=1/2$, and $t(s)=1-s$ for $s\in(r,1]$. The top-left panel of Fig.~\ref{fig:ex} shows a sample from the distribution $L$. It is easily seen that these values belong to the set $A$ given by \eqref{eq:9} which consists here of the diagonal of the square $[0,1]^2$ and the lower part of the off-diagonal. \begin{figure}[htbp] \centering \includegraphics[width=0.45\textwidth]{Example61L.eps} \includegraphics[width=0.45\textwidth]{Example61Lhist.eps} \includegraphics[width=0.45\textwidth]{Example61ent.eps} \caption{Top-left panel: Sample of size $n=300$ of points (dark and light gray) from the maximally independent joint distribution $L$ for Example~\ref{ex:6n1}; Top-right panel: Histograms of two subsamples of a sample of $L$ of size $n=50000$ depending on whether the points are on the diagonal (light gray) or the off-diagonal (dark gray); Bottom panel: Sample of size $n=1000$ of points from the entropy maximizing distribution in Example~\ref{ex:6n1}. \label{fig:ex}} \end{figure} The smallest values for Kendall's tau and Spearman's rho are $1/3$ and $1/4$, respectively. The joint density with the maximal entropy is given by \begin{equation}\label{eq:maxe61} f(x_1,x_2)=\frac{2(1-x_1)}{(1-x_2)^2},\quad 0\leq x_2\leq x_1\leq 1. \end{equation} A sample from this distribution is shown in the bottom panel of Fig.~\ref{fig:ex}. Note that in this example it is easy to simulate from the distribution $L$, and also from the distribution with density $f$ given at \eqref{eq:maxe61}. To simulate a random vector $(X_1,X_2)$ with distribution $L$, generate a random variable $U$ which is uniformly distributed on $[0,1]$ and set \[ (X_1,X_2) = \begin{cases} (\sqrt{U},\sqrt{U}), & U \le 1/4,\\ ((1 + \sqrt{4U - 1})/2,(1 - \sqrt{4U - 1})/2), & 1/4 < U < 1/2,\\ (U,U), & U \ge 1/2.\end{cases} \] A random vector $(X_1,X_2)$ with distribution given by the density $f$ at \eqref{eq:maxe61} is obtained by simulating independent random variables $U,V$ both uniformly distributed on $[0,1]$ and defining \[ (X_1,X_2) = (1 - \sqrt{V}(1-U),U). \] \end{example} \begin{example} Theorem~\ref{thr:order} establishes a relationship between the distribution of $(X_1,X_2)$ and the order statistics of a suitably chosen exchangeable pair $(V_1,V_2)$. Assume that $V_1$ and $V_2$ are independent. Then $\mathbb{P}(V_1\leq x,V_2\leq x)=G(x)^2$. By Theorem~\ref{thr:order}, $F_1(x)=G(x)^2$, whence \begin{displaymath} F_2(x)=2\sqrt{F_1(x)}-F_1(x), \quad x\in\mathbb{R}. \end{displaymath} Then \begin{displaymath} H(x)=2\sqrt{F_1(x)}(1-\sqrt{F_1(x)}) \end{displaymath} is always unimodal. If $F_1$ is continuous the maximum attained at any lower quartile of $X_1$. The smallest possible Kendall's tau equals $-1/3$; it does not depend on $F_1$. If we assume additionally that the support of $F_1$ is an interval, Theorem \ref{thm:SpearmanW} applies and the equation used to find $t(s)$ turns into $\sqrt{F_1(t(s))}=1-\sqrt{F_1(s)}$. Substituting this in \eqref{eq:sp-rho} yields that Spearman's rho equals $-1/2$ for all $F_1$. If $F_1$ is absolutely continuous with density $f_1$ the maximum entropy is attained on the density $f(x_1,x_2)=2g(x_1)g(x_2)$, $x_1\geq x_2$, where $g(x)=f_1(x)/(2\sqrt{F_1(x)})$. \end{example} \begin{example} \label{ex:6.1} Let $X_1$ be uniform on $[0,1]$, and let $X_2=X_1^{1/\alpha}$ with $\alpha\in(0,1]$. Then $F_1(x)=x$, $F_2(x)=x^\alpha$, and \begin{displaymath} F(x_1,x_2)=2\tilde{C}((x_1+x_1^\alpha)/2,(x_2+x_2^\alpha)/2)-x_2 \end{displaymath} for $x_1\geq x_2$, see \eqref{eq:1}. In this case, \eqref{eq:5} yields that \begin{displaymath} L(x_1,x_2) = \begin{cases} x_1, & x_1\leq x_2,\\ x_2^\alpha-\min\{x_2^\alpha-x_2,x_1^\alpha-x_1\}, & \text{otherwise}, \end{cases} \end{displaymath} The function $F_2-F_1$ is unimodal and attains its maximum at $r=\alpha^{(1-\alpha)^{-1}}$. The smallest Kendall's tau is \begin{displaymath} \tau=3-4 \;\mathbb{E} X_1^\alpha=\frac{3\alpha-1}{1+\alpha}. \end{displaymath} Note that $\tau=1$ if $\alpha=1$, $\tau=0$ if $\alpha=1/3$, and $\tau\to-1$ as $\alpha\downarrow0$. We are not able to provide an explicit formula for Spearman's rho in terms of $\alpha$ but Fig.~\ref{fig:1} shows $\tau$ and $\rho$ as a function of $\alpha$. \end{example} \begin{figure} \begin{center} \includegraphics[width=0.5\textwidth]{Uniform.eps} \caption{Kendall's tau $\tau$ and Spearman's rho $\rho$ as functions of $\alpha \in (0,1]$ for the distributions in Example~\ref{ex:6.1}.\label{fig:1}} \end{center} \end{figure} \begin{example}[Unimodal densities] \label{ex:6.2} Let $Z$ be a random variable with cdf $F$ and unimodal density $f$ whose support is $\mathbb{R}$. Let $X_i=Z+\mu_i$, $i=1,2$, with $\mu_1\geq \mu_2$, hence $F_i = F(\cdot-\mu_i)$, $i\in\{1,2\}$. Then Theorem~\ref{thm:KendallW} yields for Kendall's tau of the distribution $L$ at \eqref{eq:5} \begin{displaymath} \tau=3-4\;\mathbb{E} F(Z+\mu_1-\mu_2)=4\int \check{F}(\mu_2-\mu_1-z)\,\mathrm{d} F(z) -1, \end{displaymath} which is the convolution of $\check{F} (z):=1-F(-z)$ and $F$. Let us additionally assume that $f$ is symmetric about its mode at zero. Then $\check{F} = F$, and the unimodal function $F_2-F_1$ has its maximum at $r=(\mu_1 + \mu_2)/2$. Hence, the function $t:(r,\infty) \to (-\infty,r)$ in Theorem \ref{thm:SpearmanW} is given by $t(s) = \mu_2 + \mu_1 - s$. If $F = \Phi$ is the standard Gaussian cdf, then $\Phi * \Phi(x) = \Phi(x/\sqrt{2})$, and, therefore, \begin{displaymath} \tau = 4 \Phi((\mu_2-\mu_1)/\sqrt{2}) - 1. \end{displaymath} In particular, $\tau = 1$ if $\mu_1 = \mu_2$, $\tau = 0$ if $\mu_1 - \mu_2 = \sqrt{2}\Phi^{-1}(3/4) \approx 0.954$, and if $\mu_1 - \mu_2 \to \infty$, then $\tau \to -1$. For Spearman's rho, we numerically computed the integrals in \eqref{eq:sp-rho} for $F = \Phi$. The values of $\tau$ and $\rho$ as functions of $\mu_1 - \mu_2 \ge 0$ are displayed in Fig.~\ref{fig:2}. \end{example} \begin{figure} \begin{center} \includegraphics[width=0.5\textwidth]{Normal.eps} \caption{Kendall's tau $\tau$ and Spearman's rho $\rho$ as functions of $\mu_1 - \mu_2 \in [0,4]$ for $F = \Phi$ in Example~\ref{ex:6.2}.\label{fig:2}} \end{center} \end{figure} \begin{example}[Exponential marginal distributions] \label{ex:6.3} Let $Z$ be an exponential random variable with cdf $F(x)=1-e^{-x}$, and let $X_i=Z/\lambda_i$, $i\in\{1,2\}$, with $\lambda_1\le \lambda_2$. Considering $\log(X_i)$, $i\in\{1,2\}$ shows that we are in the same setting as in Example~\ref{ex:6.2} because Kendall's tau and Spearman's rho are invariant under monotone transformations of the marginals and the stochastic ordering is preserved if we transform both marginals with the same increasing function. However, we can also compute $\tau$ and $\rho$ directly. Kendall's tau is given by \begin{displaymath} \tau = 4\int_0^1 (1-u)^{\lambda_2/\lambda_1} \,\mathrm{d} u - 1 = \frac{3\lambda_1-\lambda_2}{\lambda_1 + \lambda_2}. \end{displaymath} Note that $\tau=1$ if $\lambda_1=\lambda_2$, $\tau = 0$ if $\lambda_1/\lambda_2 = 1/3$, and $\tau \to -1$ as $\lambda_1/\lambda_2 \downarrow 0$. The function \begin{displaymath} F_2(x)-F_1(x)=F(\lambda_2 x)-F(\lambda_1 x)=e^{-\lambda_1 x}-e^{-\lambda_2 x} \end{displaymath} is unimodal on $[0,\infty)$ with maximum at \begin{displaymath} r = (\log\lambda_2-\log\lambda_1)/(\lambda_2 - \lambda_1). \end{displaymath} Therefore, we can use \eqref{eq:sp-rho} to compute $\rho$. Considering the increasing transformation $\lambda_1 X_i$, $i\in\{1,2\}$, we see that $\tau$ and $\rho$ only depend on $\lambda_1/\lambda_2 \le 1$. Fig.~\ref{fig:3} provides plots of $\tau$ and $\rho$ as functions of $\lambda_1/\lambda_2 \in (0,1]$. \end{example} \begin{figure} \begin{center} \includegraphics[width=0.5\textwidth]{Exponential.eps} \caption{Kendall's tau $\tau$ and Spearman's rho $\rho$ in terms of $\lambda_1/\lambda_2$ for exponentially distributed marginal distributions as in Example \ref{ex:6.3}.\label{fig:3}} \end{center} \end{figure} \section{Generalizations} \label{sec:generalizations} A multivariate version of Theorem~\ref{thr:order} is the following. \begin{theorem} \label{thr:orderM} A random vector $(X_1,\dots,X_n)$ with marginal cdfs $F_1,\dots,F_n$ satisfies $\mathbb{P}(X_1\geq \dots \geq X_n)=1$ if and only if $X_{i} = V_{(i)}$ where $V_{(1)} \ge \dots \ge V_{(n)}$ are the order statistics of a random vector $(V_1,\dots,V_n)$ with exchangeable components and such that for $j =1,\dots,n$ \[ \mathbb{P}(V_1\leq x,\dots,V_j\leq x) = \frac{1}{\binom{n}{j}}\sum_{\ell=1}^{n-j+1}\binom{n-\ell}{j-1}F_\ell(x). \] \end{theorem} \begin{proof}[\textnormal{\textbf{Proof}}] Let $(V_1,\dots,V_n)$ be an exchangeable random vector that satisfies the above condition. By \cite[Proposition 4.4.1]{Lange2010}, we have that \begin{align*} \mathbb{P}(V_{(i)} \le x) &= \sum_{j=n-i+1}^n (-1)^{j-(n-i+1)}\binom{j-1}{n-i}\binom{n}{j} \mathbb{P}(V_1\leq x,\dots,V_{j}\leq x)\\ &= \sum_{j=n-i+1}^n (-1)^{j-(n-i+1)}\binom{j-1}{n-i} \sum_{\ell=1}^{n-j+1}\binom{n-\ell}{j-1}F_\ell(x)\\ &= \sum_{\ell=1}^i F_{\ell}(x) \binom{n-\ell}{n-i} \sum_{j=n-i+1}^{n-\ell+1}(-1)^{j-(n-i+1)}\binom{i-\ell}{n-\ell-j+1}\\ &= F_i(x). \end{align*} Conversely, if the vector $(V_1,\dots,V_n)$ is obtained as the random permutation of $(X_1,\dots,X_n)$, then it is exchangeable and the formula for $\mathbb{P}(V_1\leq x,\dots,V_j\leq x)$ is essentially the inversion of the first equality in the above equation. \end{proof} A variant of Theorem~\ref{thr:order} applies to random elements in a lattice $E$ with partial order $\preceq$, and with $\vee$ being the maximum and $\wedge$ being the minimum operation. Endow $E$ with the $\sigma$-algebra generated by $\{y:\; y\preceq x\}$ for all $x\in E$. Since these events form a $\pi$-system, the values $\mathbb{P}(X\preceq x)$, $x\in E$, uniquely determine the distribution of an $E$-valued random element $X$. In this case, Theorem~\ref{thr:order} admits a direct generalization. Namely $X_1\preceq X_2$ a.s.~if and only if $X_1=V_1\vee V_2$ and $X_2=V_1\wedge V_2$ for a pair $(V_1,V_2)$ of exchangeable random elements in $E$ such that \begin{displaymath} \mathbb{P}(V_i\preceq x)=\frac{1}{2}\Big(\mathbb{P}(X_1\preceq x)+\mathbb{P}(X_2\preceq x)\Big) \end{displaymath} and $\mathbb{P}(V_1\preceq x,V_2\preceq x)=\mathbb{P}(X_1\preceq x)$. \section*{Acknowledgement} The problem of characterizing bivariate copulas with stochastically ordered marginals was brought to the attention of the second author by Nicholas Kiefer from the Economics Department at Cornell University. The authors are grateful to the referees for spotting mistakes in the original version of this paper and suggesting numerous improvements. \bibliographystyle{elsarticle-num.bst}

BkiUe9HxK1ThhAcYmHx_

\section{INTRODUCTION} Nonequilibrium quantum physics is much less understood than equilibrium quantum physics. Advances for the first will impact fields as diverse as condensed matter physics, molecular dynamics, and cosmology. In this paper, we focus on a crucial aspect of this vast subject, namely the unitary dynamics of isolated many-body quantum systems initially far from equilibrium. This topic has gained enormous attention due to recent experiments with cold atoms in optical lattices~\cite{experimentRef,Trotzky2008,Trotzky2012,FukuharaPhys,Fukuhara}, where coherent evolution can be studied for long times. Knowing the maximum speed at which many-body systems can evolve~\cite{Bhattacharyya1983,Pfeifer1993,GiovannettiALL} is also central for the development of algorithms for quantum optimal control~\cite{Caneva2009}. In experiments with optical lattices, the evolution of the system initiates after changing instantaneously (quenching) a certain initial Hamiltonian $\hat{H}_I$ to a new final Hamiltonian $\hat{H}_F$. In this context there is evidence that the relaxation dynamics shows a power law behavior in some disordered systems~\cite{Khatami2012}, noninteracting integrable systems~\cite{Gramsch2012,HeSantos2013}, and close to critical points~\cite{Venuti2010,criticalRef}. Here we study deterministic (clean) quantum systems with interaction. For different final Hamiltonians, we identify features that lead to similar relaxation processes and can therefore contribute to a long aspired universal picture. In 1984 Peres proposed that the notion of irreversibility in the quantum domain could be explained in terms of sensitivity of the quantum states to perturbations added to the Hamiltonian~\cite{Peres1984}. Such sensitivity is quantified via fidelity (Loschmidt echo) between states evolved under an unperturbed Hamiltonian and states evolved by a perturbed Hamiltonian~\cite{Gorin2006,LoschRef}. Several studies confirmed Peres's expectations and showed that the fidelity decayed exponentially when the perturbation induced chaos~\cite{expRef,Cerruti2002,Flambaum2001ab,Weinstein2003}. However, an exponential behavior was observed also in integrable systems when the initial state was sufficiently delocalized in the energy representation~\cite{Emerson2002,Santos2012PRL,Santos2012PRE}. This suggests that, rather than a chaotic Hamiltonian, the main cause for a fast statistical relaxation must be the chaotic structure of the initial state with respect to the Hamiltonian that dictates its evolution. To establish a relationship between the level of delocalization of the initial state and the characteristics of the relaxation dynamics after a quench, we employ concepts from many-body quantum chaos. We associate the unperturbed Hamiltonian with $\hat{H}_I$ and the perturbed Hamiltonian with $\hat{H}_F$. Our initial state $|\text{ini} \rangle$ is one of the eigenstates $ |n \rangle$ of $\hat{H}_I$. Its projection on the eigenstates $|\psi_{\alpha} \rangle$ of $\hat{H}_F$ leads to $ | \text{ini} \rangle= \sum_{\alpha} C_{\alpha}^{\text{ini}} |\psi_{\alpha} \rangle$, and its energy is given by ${E_{\text{ini}} = \langle \text{ini} |\hat{H}_F | \text{ini} \rangle = \sum_{\alpha} |C_{\alpha}^{\text{ini}}|^2 E_{\alpha} }$. The distribution $P^{\text{ini}}_{\alpha}$ of the components $|C_{\alpha}^{\text{ini}}|^2$ in the eigenvalues $E_{\alpha}$ is known as local density of states (LDOS)~\cite{Flambaum2000}. For $E_{\text{ini}}$ close to the middle of the spectrum, as the perturbation increases from zero, this distribution broadens from a delta function to a Lorentzian form and eventually approaches a Gaussian of width $\sigma_{\text{ini}}$, identified as the energy shell~\cite{noteShell,ZelevinskyRep1996,Flambaum1997,Flambaum2000,Flambaum2001ab,Santos2012PRL,Santos2012PRE,Torres2013,noteGauss}. Contrary to full random matrices, where the states are fully delocalized, in realistic systems with few-body interactions, the energy shell gives the maximum possible spreading of the LDOS. LDOS is a key concept in nuclear physics~\cite{ZelevinskyRep1996}, where it is measured experimentally. It is likely to gain a especial role also in quantum thermodynamics, since it is related with the probability distribution of work~\cite{work} In this paper, we investigate LDOS, the fidelity decay, and the short-time dynamics of few-body observables in the limit of strong perturbation. The fidelity corresponds to the Fourier transform of the LDOS. It therefore decays exponentially when $P^{\text{ini}}_{\alpha}$ is Lorentzian~\cite{expRef,Cerruti2002,Flambaum2001ab,Weinstein2003,Emerson2002}, although at very short-times it behaves as $1 - \sigma_{\text{ini}}^2 t^2$, as expected from perturbation theory~\cite{Flambaum2001ab,Cerruti2002}. Interestingly, consensus has been reached that even when $P^{\text{ini}}_{\alpha}$ was Gaussian, a Gaussian decay would occur for a certain time and then necessarily switch to exponential before saturation~\cite{Flambaum2001ab,Santos2012PRL,Santos2012PRE,Castaneda}. Here, we show that when the initial state fills the energy shell substantially, the Gaussian expression for the fidelity, $\exp (-\sigma_{\text{ini}}^2 t^2)$, {\em can persist until saturation}, independent of the regime (integrable or chaotic) of $\hat{H}_F$. This is illustrated numerically for initial states that can be prepared in experiments with cold atoms, but can hold for any $|\text{ini} \rangle$ where $P^{\text{ini}}_{\alpha}$ has a Gaussian shape~\cite{preparation,preparationTorres}. The absence of an exponential decay also has been noticed in recent works~\cite{Genway} for models and initial states different from the ones we consider. The Gaussian expression sets the lower bound for the fastest fidelity decay in realistic systems with two-body interactions that are strongly perturbed. We also discuss the scenario where $\hat{H}_F$ is a full random matrix~\cite{Brody1981}. Although less realistic, the latter sets the ultimate bound for the fidelity decay in many-body quantum systems after a quench. The analysis of the dynamics of few-body observables is more challenging and at the same time indispensable to establishing a connection with current experiments. We provide examples of observables that although evolving under entirely different Hamiltonians show almost identical short-time dynamics. We identify the conditions for such {\em general behavior}. \section{MODEL} We consider a one-dimensional lattice of interacting spins 1/2 with open boundaries and an even number $L$ of sites. The Hamiltonian has nearest-neighbor (NN) and possibly also next-nearest-neighbor (NNN) couplings: \begin{align} \hat{H} = &\hat{H}_{NN} + \lambda \hat{H}_{NNN} , \label{ham} \nonumber\\ \hat{H}_{NN} = &\sum_{i=1}^{L-1} J \left(\hat{S}_i^x \hat{S}_{i+1}^x + \hat{S}_i^y \hat{S}_{i+1}^y +\Delta \hat{S}_i^z \hat{S}_{i+1}^z \right) , \\ \hat{H}_{NNN} =& \sum_{i=1}^{L-2} J \left(\hat{S}_i^x \hat{S}_{i+2}^x + \hat{S}_i^y \hat{S}_{i+2}^y +\Delta \hat{S}_i^z \hat{S}_{i+2}^z \right) . \nonumber \end{align} This is a prototype many-body quantum model, simulated in optical lattices~\cite{Trotzky2008,Trotzky2012,FukuharaPhys,Fukuhara} and also mappable onto systems of spinless fermions or hardcore bosons~\cite{Jordan1928}. $\hat{S}^{x,y,z}_i$ are the spin operators on site $i$; $\hbar=1$. The coupling strength $J$, the anisotropy $\Delta$, and the ratio $\lambda$ between NNN and NN exchanges are positive; $\hat{S}_i^x \hat{S}_{i+1 (i+2)}^x + \hat{S}_i^y \hat{S}_{i+1 (i+2)}^y$ is the flip-flop term and $\hat{S}_i^z \hat{S}_{i+1 (i+2)}^z $ is the Ising interaction. The Hamiltonian conserves total spin in the $z$ direction, $[\hat{H},{\cal \hat{S}}^z]=0$, where ${\cal \hat{S}}^z = \sum_{i=1}^L \hat{S}_i^z$. The noninteracting XX model ($\Delta = \lambda = 0$) is trivially solvable. The interacting XXZ case ($\Delta \neq 0$, $\lambda =0$) is solved with the Bethe ansatz ~\cite{Bethe1931}. The system undergoes a crossover to the chaotic regime~\cite{Brody1981} as $\lambda $ increases~\cite{Gubin2012}. We investigate the dynamics of the system for the following choices of parameters for $\hat{H}_F$: \begin{enumerate} \itemsep-0.4em \item[(1)] Integrable isotropic Hamiltonian, $\hat{H}_{\Delta=1,\lambda=0}$. \item[(2)] Integrable anisotropic Hamiltonian, $\hat{H}_{\Delta=0.5,\lambda=0}$. \item[(3)] Weakly chaotic isotropic Hamiltonian, $\hat{H}_{\Delta=1,\lambda=0.4}$. \item[(4)] Strongly chaotic isotropic Hamiltonian, $\hat{H}_{\Delta=1,\lambda=1}$. \item[(5)] Strongly chaotic anisotropic Hamiltonian, $\hat{H}_{\Delta=0.5,\lambda=1}$. \end{enumerate} Independent of the regime, the density of states of the five Hamiltonians above has a Gaussian shape, as typical of systems with two-body interactions~\cite{ZelevinskyRep1996,Izrailev1990,Kota2001}. This implies that the majority of the states concentrate in the middle of the spectrum. This is the region where the eigenstates are expected to reach their highest level of delocalization. \vskip -0.5 cm \begin{table}[h] \caption{Energy of $|\text{ini}\rangle$ and width of its energy distribution.} \begin{center} \begin{tabular}{ccc} \hline \hline & $E_{\text{ini}}$ & $\sigma_{\text{ini}}$ \\ [0.1 cm] \hline $ |\rm{NS}\rangle$ & \hspace{0.2 cm} $ \frac{\displaystyle J\Delta}{\displaystyle 4} [ -(L-1) + (L-2)\lambda ]$ & $\frac{\displaystyle J}{\displaystyle 2} \sqrt{L-1}$ \\ [0.2 cm] $ |\rm{PS}\rangle$ & $ - \frac{\displaystyle J\Delta}{\displaystyle 4} [ 1 + (L-2)\lambda ]$ & $\frac{\displaystyle J}{\displaystyle 2} \sqrt{ \frac{\displaystyle L}{\displaystyle 2} + (L-2)\lambda^2}$ \\ [0.2 cm] $|\rm{DW}\rangle$ & \hspace{0.2 cm} $\frac{\displaystyle J\Delta}{\displaystyle 4} [(L-3) + (L-6)\lambda]$ & $ \frac{\displaystyle J}{\displaystyle 2} \sqrt{1+2\lambda^2}$ \\ [0.2 cm] \hline \hline \end{tabular} \end{center} \label{table:initial} \end{table} \vskip -0.5 cm \section{INITIAL STATE}. Our analysis focuses on $|\text{ini} \rangle$'s where each site has a spin either pointing up or down in the $z$ direction~\cite{Zangara2013,Santos2011,Pozsgay}. The chosen states enhance the effects of $\Delta$ or $\lambda$: \begin{enumerate} \itemsep-0.1em \item[(1)] N\'eel state, $ |\rm{NS}\rangle= | \downarrow \uparrow \downarrow \uparrow \ldots \downarrow \uparrow \downarrow \uparrow \rangle$, \item[(2)] Pairs of parallel spins, $ |\rm{PS}\rangle=| \downarrow \uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow \ldots \rangle$. \item[(3)] Sharp domain wall, $|\rm{DW}\rangle = | \uparrow \uparrow \uparrow \ldots \downarrow \downarrow \downarrow \rangle$, \end{enumerate} They belong to the same subspace with ${\cal S}^z =0$ and dimension ${\cal D} = L!/(L/2)!^2$. These states can be prepared in optical lattices: $|\rm{DW}\rangle$ requires a magnetic field gradient~\cite{Weld2009} and $|\rm{NS}\rangle$ was used in~\cite{Trotzky2008,Koetsier2008,Trotzky2012,Mathy2012}. Our $\hat{H}_I$ therefore coincides with the Ising part of (\ref{ham}). Its eigenstates are referred to as site-basis vectors. The final Hamiltonian is written in this basis. $\hat{H}_F$ is in the nonperturbative regime, since the off-diagonal elements are much larger than the average level spacing. \section{ENERGY SHELL} The energy shell is a Gaussian centered at $E_{\text{ini}}$ of width \begin{eqnarray} \sigma_{\text{ini}} &=& \sqrt{\sum_{\alpha} |C_{\alpha}^{\text{ini}} |^2 (E_{\alpha} - E_{\text{ini}})^2} =\sqrt{\sum_{n \neq \text{ini}} |\langle n |\hat{H}_F | \text{ini}\rangle |^2 } \nonumber \\ &=& \frac{J}{2} \sqrt{M_1 + \lambda^2 M_2} . \label{deltaE} \end{eqnarray} The second equality above shows that $\sigma_{\text{ini}}$ can be obtained from $\hat{H}_F$ before diagonalization. \begin{figure}[htb] \centering \includegraphics*[width=0.45\textwidth]{Fig01new_Shell.eps} \caption{(Color online) Distribution of $|C_{\alpha}^{\text{ini}} |^2$ in $E_{\alpha}$ ({\em i.e.} LDOS) for $ |\rm{NS}\rangle$ (top) and $ |\rm{DW}\rangle$(bottom); $L=16$. The Hamiltonians are: $\hat{H}_{\Delta=1,\lambda=0}$ (a, d); $\hat{H}_{\Delta=1,\lambda=0.4}$ (b, e); $\hat{H}_{\Delta=0.5,\lambda=1}$ (c, f); and $\hat{H}_{\Delta=0.5,\lambda=0}$ [inset of (d)]. The solid line is the energy shell.} \label{fig:shell} \end{figure} The last equality holds for the models and initial states studied here. Our $\sigma_{\text{ini}}$ does not depend on the anisotropy parameter. The connectivity $M_1$ ($M_2$) corresponds to the number of states directly coupled to $|\text{ini}\rangle$ via the NN (NNN) flip-flop term. The total connectivity $M$ of any site-basis vector is low, $M\!=\!M_1 + M_2 \propto L \ll {\cal D}$. For $ |\rm{NS}\rangle$, the five Hamiltonians considered lead to the same $\sigma_{\text{ini}}$ (see Table~\ref{table:initial}), since $M_2=0$. For $ |\rm{PS}\rangle$ and $|\rm{DW}\rangle$ only Hamiltonians with the same $\lambda$ give the same $\sigma_{\text{ini}}$. The domain wall is directly coupled to only one (three) state(s) when $\hat{H}_F$ is integrable (chaotic), independent of $L$. It has the smallest $\sigma_{\text{ini}}$ among the states investigated. Figure~\ref{fig:shell} displays the energy distribution of $ |\rm{NS}\rangle$ and $|\rm{DW}\rangle$ (see $ |\rm{PS}\rangle$ in Refs.~\cite{Zangara2013,preparation}). As visible, $P^{\text{ini}}_{\alpha}$ depends on both $|\text{ini} \rangle $ and the $\hat{H}_F$ that evolves it. This dependence is reflected also in the relaxation dynamics of the system. For the N\'eel state, $\sigma_{\text{ini}}$ is always the same, but the shell gets better filled as $\lambda$ increases from zero and $\Delta$ decreases [Fig.~\ref{fig:shell} from (a) to (c)], since its energy is brought closer to the middle of the spectrum (cf. Table~\ref{table:initial}), where the density of states is larger. For $|\rm{DW}\rangle$, $\Delta$ plays a major role [cf. main panel and inset of Fig.~\ref{fig:shell} (d)]. At the critical point ($\Delta=1$) or above it, this state approaches the right edge of the spectrum. In this region, $|\rm{DW}\rangle$ and the few states directly coupled to it are more localized. As a result, in addition to the narrow energy shell, the latter is also poorly filled. $|\rm{DW}\rangle$ should therefore decay very slowly when $\Delta \geq 1$ \cite{Zangara2013,Santos2011,DWstudies}. For $|\rm{PS}\rangle$ the worst filling occurs for $\Delta=1$ and $\lambda=1$, since this combination pushes the distribution to the edge of the spectrum (cf. Table~\ref{table:initial}). \section{FIDELITY} The probability of finding $|\text{ini} \rangle$ at time $t$ defines the fidelity. The latter corresponds to the Fourier transform of $|C_{\alpha}^{\text{ini}} |^2$. When the energy distribution of $|\text{ini} \rangle$ fills the energy shell, $|C_{\alpha}^{\text{ini}} |^2$ is replaced by a Gaussian of $\sigma_{\text{ini}}$ (\ref{deltaE}) \cite{Flambaum2000,Flambaum2001ab}, so \begin{eqnarray} &&F(t) \equiv |\langle \text{ini} | e^{-i \hat{H}_F t} |\text{ini} \rangle |^2 \! = \! \left|\sum_{\alpha} |C_{\alpha}^{\text{ini}} |^2 e^{-i E_{\alpha} t} \right|^2 \nonumber \\ && \approx \left| \frac{1}{ \sqrt{ 2 \pi \sigma^2_{\text{ini}} } } \int_{-\infty}^{\infty} \! e^{-\frac{(E-E_{\text{ini}})^2}{ 2 \sigma^2_{\text{ini}} } } \!e^{-i E t} dE \right|^2 \!=\! e^{-\sigma_{\text{ini}}^2 t^2} . \label{eq:fidelity} \end{eqnarray} Above, the sum becomes an integral because ${\cal D}$ is large \cite{noteVenuti}. \begin{figure}[htb!] \centering \includegraphics[width=0.45\textwidth]{Fig02_fidelityC.eps} \caption{(Color online) Fidelity decay for the final Hamiltonians: $\hat{H}_{\Delta=1,\lambda=0}$ (up triangle), $\hat{H}_{\Delta=0.5,\lambda=0}$ (down triangle), $\hat{H}_{\Delta=1,\lambda=0.4}$ (cross), $\hat{H}_{\Delta=1,\lambda=1}$ (circle), and $\hat{H}_{\Delta=0.5,\lambda=1}$ (square); $L=16$. [Inset: $\hat{H}_{\Delta=0.5,\lambda=1}$, $L\!=\!24$]. Solid lines are for Eq.~(\ref{eq:fidelity}). Dashed lines give $\overline{F}=\text{IPR}_{|\rm{NS}\rangle}^{-1}$: $\hat{H}_{\Delta=1,\lambda=0}$ (top) and $\hat{H}_{\Delta=0.5,\lambda=1}$ (bottom).} \label{fig:fidelity} \end{figure} As seen in Fig.\ref{fig:fidelity}, $F(t)$ for $|\rm{NS}\rangle$ is similar for the five Hamiltonians, since $\sigma_{\text{ini}}$ is equal for all of them. The agreement between the Gaussian expression (\ref{eq:fidelity}) and the numerical results is excellent and holds until saturation (shown with dashed lines). This is not an artifact of small system sizes, as confirmed by the inset for larger $L$. This behavior contrasts previous studies, where $F(t)$ was Gaussian for some time and then exponential before saturation~\cite{expRef,Cerruti2002,Flambaum2001ab,Weinstein2003,Emerson2002,Santos2012PRL,Santos2012PRE,Castaneda}. For $|\rm{DW}\rangle$ (Fig.~\ref{fig:fidelity}) and $|\rm{PS}\rangle$ (not shown), $F(t)$ coincides only for $\hat{H}_F$ with equal NNN flip-flop strength, since $\sigma_{\text{ini}}$ now depends on $\lambda$. The fidelity decay for $|\rm{DW}\rangle$ is slow due to its narrow energy distribution. For both, $|\rm{DW}\rangle$ and $|\rm{PS}\rangle$, the filling of the energy shell is poorer when $\Delta=1$, so the Gaussian expression holds for longer in the anisotropic case. The filling is also overall worse than for $ |\rm{NS}\rangle$, which explains the transition of $F(t)$ to an exponential decay as $t$ increases. Identifying the critical time for this transition is far from trivial and strongly dependent on the initial state. It was discussed in~\cite{Flambaum2001ab} and estimates were made in Ref.~\cite{Castaneda}. The fidelity eventually saturates to its infinite time average $\overline{F}\!=\!\sum_{\alpha} |C_{\alpha}^{\text{ini}} |^4\!=\!\text{IPR}_{\text{ini}}^{-1} < 3/{\cal D}$. After reaching this point, $F(t)$ fluctuates around $\overline{F}$, the fluctuations decreasing exponentially with $L$ \cite{Zangara2013}. The inverse participation ratio~\cite{Izrailev1990,ZelevinskyRep1996}, $\text{IPR}_{\text{ini}} $, measures the level of delocalization of $|\text{ini}\rangle $ in the energy eigenbasis. Better filling of the energy shell necessarily implies larger $\text{IPR}_{\text{ini}} $. For $ |\rm{NS}\rangle$, as expected from Fig.~\ref{fig:shell}, $\overline{F}$ decreases significantly from integrable to chaotic Hamiltonians (see Fig.~\ref{fig:fidelity}), whereas for $|\rm{DW}\rangle$ and $|\rm{PS}\rangle$, $\overline{F}$ is actually largest for the chaotic $\hat{H}_{\Delta=1,\lambda=1}$, where $E_{\text{ini}}$ is closest to the border of the spectrum. Thus, the width of the energy shell and the extent of its filling are more important in the characterization of the relaxation process than the actual regime of $\hat{H}_F$. \begin{figure}[h!] \centering \includegraphics[width=0.45\textwidth]{Fig03_GOE2.eps} \caption{(Color online) Energy distribution of a generic initial state (a) and fidelity decay (b) for full random matrices from a Gaussian Orthogonal Ensemble~\cite{Brody1981}; ${\cal D} \!=\! 12870$. The distribution is normalized so that $\sigma_{\rm{ini}} \!=\!{\cal E}/2\!=\!J$. Solid lines: $P^{\text{ini}}_{\text{FRM}}(E)$ (a) and $F_{\text{FRM}}(t)$ (b); dot-dashed: Eq.~(\ref{eq:fidelity}); dashed: saturation; circles: numerical data.} \label{fig:RMT} \end{figure} We verified that the Gaussian decay of $F(t)$ until saturation is valid for initial states from different $\hat{H}_I$, such as XX and XXZ models~\cite{preparation}, and for different $\hat{H}_F$, such as disordered models~\cite{preparationTorres}, provided $P^{\text{ini}}_{\alpha}$ be Gaussian. Here $t_R=\sqrt{\ln(\text{IPR}_{\text{ini}}) }/\sigma_{\text{ini}}$ sets the minimum time for an initial state with a single-peaked energy distribution and evolving under a two-body-interaction Hamiltonian to relax to equilibrium. The fidelity decay can however be faster when more-body interactions are included. To identify the ultimate bound for the fidelity decay in the general scenario of strong perturbations, we resort to full random matrices. They do not describe realistic systems, because they imply the simultaneous interactions of many particles, but they provide the extreme case. The maximum LDOS in two-body-interaction Hamiltonians is Gaussian, reflecting these systems' density of states. The energy distribution of initial states evolving under full random matrices is broader, showing a semicircular shape, $P^{\text{ini}}_{\text{FRM}}(E)=\frac{2}{\pi {\cal E}} \sqrt{1 - \left(\frac{E}{{\cal E}}\right)^2}$, where $2{\cal E}$ is the length of the spectrum [Fig.\ref{fig:RMT} (a)]. This reflects the semicircular form of the density of states of full random matrices~\cite{Brody1981}. The Fourier transform of the semicircular LDOS leads to the following fidelity decay: \begin{equation} F_{\text{FRM}}(t) = [{\cal J}_1( 2 \sigma_{\text{ini}} t)]^2/(\sigma_{\text{ini}}^2 t^2), \end{equation} where ${\cal J}_1$ is the Bessel function of the first kind. In this case, the fidelity behavior is clearly faster than Gaussian, as made explicit by the comparison between the two shown in Fig.~\ref{fig:RMT} (b) \cite{noteCos}. \section{FEW-BODY OBSERVABLES} Many factors come into play when describing the dynamics of few-body observables $\hat{A}$. Nevertheless, a simple general picture becomes viable when $[\hat{A}, \hat{H}_I]=0$ (we recall that $\hat{H}_I$ defines the basis and $|\text{ini}\rangle$ is one of the basis vectors). In this case, the evolution depends on the fidelity: $A(t) = F(t) A(0) + \sum_{n\neq |\text{ini} \rangle} | \langle n| e^{-i \hat{H}_F t}|\text{ini}\rangle|^2 A_{nn} $, where $A_{nn} = \langle n |\hat{A} |n \rangle$. For the $\hat{H}_I$ considered here, this implies observables in the $z$ direction, whereas for systems quenched from the XX model, this would mean $\hat{A}$ in the $xy$ plane, such as the kinetic energy. One can then infer from the expansion, \begin{equation} A(t) \! \simeq \! \left( 1 \!-\! \sigma_{\text{ini}}^2 t^2 \right) \! A(0) + \sum_{n\neq \text{ini}} |\langle n |\hat{H}_F |\text{ini} \rangle|^2 t^2 A_{nn} , \label{eq:Oexpansion} \end{equation} that any observable will show a very similar short-time dynamics for $\hat{H}_F$ inducing comparable squared off-diagonal elements $|\langle n |\hat{H}_F |\text{ini} \rangle|^2$, and thus similar $\sigma_{\text{ini}}$, even when the Hamiltonians have very different properties. When $|\text{ini}\rangle$ is a site-basis vector and $\hat{A}$ is in a direction perpendicular to $z$, $A(0)\!=\!0$, so $F(t)$ does not play a part in the evolution. In this case, if $A_{nn'}$ is imaginary, the dominant terms in $A(t)$ are ${\cal O}(t)$ and similar dynamics emerges for distinct $\hat{H}_F$ with comparable $\langle n |\hat{H}_F |\text{ini} \rangle$ \cite{preparation}. This happens {\em e.g.} for the spin-current~\cite{Santos2011,Zotos1997}. If $A_{nn'}$ is real, the dominant terms are ${\cal O}(t^2)$ and contributions from the diagonal elements of $\hat{H}_F$ (thus $\Delta$) complicate the picture. However, even for observables where $[\hat{A}, \hat{H}_I]\neq 0$, one can always construct particular $|\text{ini}\rangle$'s that lead to expressions equivalent to Eq.~(\ref{eq:Oexpansion}). As an example, we consider the spin-spin correlation in $x$ between sites $\frac{L}{2},\frac{L}{2}+1$ for a Bell-type state $|\text{BS}\rangle\!=\! (|\ldots\uparrow_{L/2} \downarrow_{L/2+1} \ldots \rangle \!+\! |\ldots\downarrow_{L/2} \uparrow_{L/2+1} \ldots \rangle )/\sqrt{2}$, where apart from $\frac{L}{2},\frac{L}{2}+1$ the two bases are equal. \begin{figure}[htb] \centering \includegraphics[width=0.38\textwidth]{Fig04new_Obs.eps} \caption{(Color online) $C^{z,x}_{L/2, L/2+1}$ and $s^z_f(\pi)$ for $ |\rm{NS}\rangle$ (top), $ |\rm{PS}\rangle$ (c), and $ |\rm{BS}\rangle$ (d); $\hat{H}_F$ as in Fig.~\ref{fig:fidelity}; $L=16$. Solid lines give Eq.~(\ref{eq:Oexpansion}).} \label{fig:ONeel} \end{figure} Figure~\ref{fig:ONeel} shows the spin-spin correlation in $z(x)$, $\hat{C}^{z(x)}_{L/2,L/2+1} \!\!=\!\! \hat{S}_{L/2 }^{z(x)} \hat{S}_{L/2+1}^{z(x)}$, and the structure factor in $z$, $\hat{s}_f^{z}(k) \!=\! \sum_{l,j=1}^L e^{i k (l-j) } \hat{S}_l^{z} \hat{S}_{j}^{z}/L$, with momentum $k$. The first observable is local and the second is nonlocal in position. Despite their differences, for the initial states considered in the figure, their short-time dynamics is dictated by Eq.~(\ref{eq:Oexpansion}). The top panels of Fig.~\ref{fig:ONeel} depict results for $|\rm{NS}\rangle$. For short-times, $\hat{C}^{z}$ and $\hat{s}_f^z$ are insensitive to the differences between the five $\hat{H}_F$, the curves almost coinciding. Strikingly, this behavior persists for $s_f^{z}(\pi)$ until the steady state. For $\hat{C}^{z}$, the saturation points follow the filling of the energy shell: correlations remain for the integrable Hamiltonians, being larger for $\hat{H}_{\Delta=1,\lambda=0}$, and get close to zero for strongly chaotic cases. The bottom panels of Fig.~\ref{fig:ONeel} give results for $\hat{C}^{z}$ [$ |\rm{PS}\rangle$ in (c)] and $\hat{C}^{x}$ [$ |\rm{BS}\rangle$ in (d)]. For both observables the dynamics is very similar for Hamiltonians leading to the same $ \sigma_{\text{ini}}$. Other observables and initial states reiterate these findings~\cite{preparation}. \section{CONCLUSIONS} We studied the fidelity decay and the evolution of few-body observables in isolated interacting quantum systems initially far from equilibrium. The Hamiltonians dictating the evolution, the initial states $|{\text{ini}} \rangle $, and the observables analyzed are accessible to experiments with optical lattices and with nuclear magnetic resonance~\cite{Franzoni,Ramanathan2011}. We revised the commonly accepted picture that in these systems the fidelity decay had to transition to an exponential behavior before saturation. We demonstrated that when the energy distribution of $|{\text{ini}} \rangle $ is Gaussian, substantially filling the energy shell, the numerical results for the {\em fidelity decay can be Gaussian until saturation} and show excellent agreement with the analytical expression. This behavior sets the lower bound for the fidelity decay of isolated systems with two-body interactions and $|{\text{ini}} \rangle $ consisting of a single peaked energy distribution. We also derived an expression for the fidelity decay of full random matrices, which establishes the ultimate bound for quenched systems with many-body interactions. The width $\sigma_{\text{ini}}$ plays a major role in the short-time dynamics of few-body observables that commute with $\hat{H}_I$. We thus find {\em observables showing a remarkable similar behavior} despite evolving according to very different final Hamiltonians (distinct anisotropies and regimes). The general features unveiled here for fidelity and observables constitute crucial steps towards a complete description of the relaxation process in realistic quantum systems of interacting particles. Our results are also key for quantum control methods~\cite{Bhattacharyya1983,Pfeifer1993,GiovannettiALL,Caneva2009} and quantum thermodynamics due to the link between LDOS and work distribution function~\cite{work}. \section{ACKNOWLEDGMENTS} This work was supported by the NSF grant No.~DMR-1147430. E.J.T.H. acknowledges partial support from CONACyT, Mexico. We thank Felix Izrailev, Eduardo Mascarenhas, and Simone Montangero for discussions.

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\section{Introduction} Aiming at high-level and structured description of quantum computation/information, many \emph{quantum programming languages} have been proposed and their semantics studied~\cite{Gay2006,Valiron2013}. As one of pioneering works, in 2004 Selinger~\cite{Selinger2004} proposed a first-order functional quantum programming language QPL (or QFC), and gave its denotational semantics rigorously in terms of categories. He (jointly with Valiron) successively started to study a higher-order quantum programming language, or the \emph{quantum lambda calculus}~\cite{SelingerV2006,SelingerV2008,SelingerV2009}. It turned out to be challenging to give a denotational semantics for the quantum lambda calculus (with the full features, such as the $\oc$ modality and recursion). The first denotational semantics was given via Geometry of Interaction~\cite{HasuoH2011}; but there are several different approaches~\cite{MalherbeSS2013,PaganiSV2014}. As is stated in \cite[\S1]{PaganiSV2014}, the problem lies in that quantum computation is typically modelled by using finite dimensional Hilbert spaces, and hence it is difficult to model \emph{infinite} structures in computation. The present paper proposes a novel denotational semantics for a quantum programming language by \emph{operator algebras}. Operator algebras, specifically $C^*$-algebras and $W^*$-algebras (the latter are also known as von Neumann algebras), give an alternative formulation for quantum theory (sometimes called the \emph{algebraic} formulation~\cite{Landsman2009}). It is worth mentioning that von Neumann himself, who formulated quantum theory by Hilbert spaces~\cite{Neumann1955}, developed the theory of operator algebras~\cite{MurrayN1936,MurrayN1937, Neumann1940,MurrayN1943,Neumann1949} (some of them jointly with Murray), and later preferred the algebraic approach for quantum theory~\cite{Redei1996}. Operator algebras have been successfully used in areas such as quantum statistical mechanics~\cite{BratteliR1987all2dois} and quantum field theory~\cite{HaagK1964,Haag1996,Araki1999}. They have also been of growing importance in the area of quantum information~\cite{Keyl2002}; for example, in~\cite{DArianoKSW2007}, the impossibility of quantum bit commitment is (re)examined in the algebraic formalism. \subsection*{Contributions and related work} In this paper it is shown that the category $\mathbf{Wstar}_\mathrm{CPSU}$ of $W^*$-algebras and normal completely positive subunital maps is a $\Dcppo_\bot$-enriched symmetric monoidal category with $\Dcppo_\bot$-enriched finite products. It follows that its opposite $(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$ is an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched \emph{elementary quantum flow chart category}. As a consequence, it gives rise to a denotational semantics for a first-order functional quantum programming language QPL designed by Selinger~\cite{Selinger2004}. Selinger himself gave a denotational semantics for QPL by the category $\mathcal{Q}$~\cite{Selinger2004}. In comparison to his original semantics, our semantics by operator algebras has the following two advantages. First, our semantics accommodates infinite structures such as a type $\mathbf{nat}$ of natural numbers. This is because we discuss general $W^*$-algebras and do not restrict them to finite dimensional ones. In \S\ref{sect:embedding} we will see that our model can be considered as an infinite dimensional extension of Selinger's model. Second, it is known that \emph{commutative} $C^*$-algebras (resp.\ $W^*$-algebras) correspond to and certain type of topological (resp.\ measure) spaces, and therefore we can interpret classical computation as a map between commutative algebras. This correspondence (or \emph{duality}) is studied well for $C^*$-algebras e.g.\ in~\cite{FurberJ2013}, but not yet very well for $W^*$-algebras. In this paper, hence, we restrict it to deterministic computation (in the category $\mathbf{Set}$), and give rudimentary results in \S\ref{sect:infinite-type} as a first step. It will enable us to handle classical and quantum computations in a unified way. Although the present paper has not yet been able to give a semantics for a \emph{higher-order} quantum programming language, compared to such works~\cite{HasuoH2011,MalherbeSS2013,PaganiSV2014}, our semantics is quite simple: a type is interpreted just as a $W^*$-algebra, and a program as a map of them (in opposite direction). However, the theory of operator algebras itself could be complicated enough. As is the case in previous works~\cite{Selinger2004,HasuoH2011,MalherbeSS2013,PaganiSV2014}, quantum computation is usually modelled by using finite dimensional Hilbert spaces $\mathbb{C}^n$ (or matrix algebras $\mathcal{M}_n\cong\mathcal{B}(\mathbb{C}^n)$). It seems that the (explicit) use of operator algebras is not so common in the area of quantum computation. Recently there are several works using $C^*$-algebras~\cite{FurberJ2013,Jacobs2013}, which led the author to the present work. The use of $W^*$-algebras in this context appeared independently and coincidentally in Rennela's thesis~\cite{Rennela2013Master} and the present work (or the author's thesis~\cite{Cho2014Master}). In~\cite[Theorem~3.8]{Rennela2013Master}, he also observed that the category $\mathbf{Wstar}_\mathrm{PSU}$ of $W^*$-algebras and normal positive subunital maps are $\mathbf{Dcppo}$-enriched, which is almost the same result as Theorem~\ref{thm:wstarcppu-dccpobot-enriched} in the present paper. In his latest paper~\cite{Rennela2014}, he further showed that $\mathbf{Wstar}_\mathrm{PSU}$ is algebraically compact for a certain class of functors. This result enables us to have inductively defined types. Some related technical results have appeared in~\cite{ChiribellaTU2013}, which studies spaces of maps (`quantum operations') between $W^*$-algebras, and maps (`quantum supermaps') between them. For instance, \cite[Proposition~7]{ChiribellaTU2013} states (in our terms) that $\mathbf{Wstar}_\mathrm{CP}(M,N)$ is bounded directed complete. \subsection*{Organisation of the paper} First we review complete partial orders in \S\ref{sec:cpo}, and then review Selinger's work on QPL in \S\ref{sec:QPL}. In \S\ref{sec:opalg} we also review the basics of operator algebras, and then look at the order-theoretic perspective of operator algebras. We show in \S\ref{sec:dcpo-enrichment} that $\mathbf{Wstar}_\mathrm{CPSU}$ is a $\Dcppo_\bot$-enriched symmetric monoidal category with $\Dcppo_\bot$-enriched finite products and in \S\ref{sec:sem-by-wstar} that its opposite $(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$ is an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched elementary quantum flow chart category, hence gives a denotational semantics for QPL. Section~\ref{sec:Schr-vs-Heis} discuss a duality between Selinger's and our semantics, and in \S\ref{sect:embedding} it is shown that Selinger's model is contravariantly embedded into our model. In \S\ref{sect:infinite-type} we see that our model can accommodate infinite types, and also classical computation as commutative structures. Section~\ref{sec:concl-future} concludes the paper with future work. This paper is based on the author's master thesis~\cite{Cho2014Master}, in which one may find further details. \section{Complete partial orders} \label{sec:cpo} In this section, we will briefly review the notion of complete partial orders, which is essential in domain theory~\cite{AbramskyJ1994}, and is fundamental for the denotational semantics of programming languages. \begin{mydefinition} A poset is \begin{enumerate} \item \emph{directed complete} if every directed subset has a supremum; \item \emph{bounded directed complete} if every directed subset that is bounded from above has a supremum; \item \emph{$\omega$-complete} if every $\omega$-chain ($(x_n)_{n\in\omega}$ with $x_n\le x_{n+1}$) has a supremum; \item \emph{pointed} if it has a least element (denoted by $\bot$). \end{enumerate} A (bounded) directed complete poset is abbreviated as a \emph{(b)dcpo}, and an $\omega$-complete poset as a \emph{$\omega$cpo}. \end{mydefinition} \begin{mydefinition} A map between posets (in \ref{def-enum:strict}, pointed posets) is \begin{enumerate} \item \emph{Scott-continuous} if it preserves suprema of directed subsets; \item \emph{$\omega$-(Scott-)continuous} if it preserves suprema of $\omega$-chains; \item\label{def-enum:strict} \emph{strict} if it preserves the least element. \end{enumerate} \end{mydefinition} Note that every dcpo is an $\omega$cpo, and every Scott-continuous map is $\omega$-continuous. The next theorem is a fundamental tool to give an interpretation of a program with recursion and loop. \begin{mytheorem} Every $\omega$-continuous endomap $f$ on a pointed $\omega$cpo has a least fixed point, which is given by $\bigvee_n f^n(\bot)$. \qed \end{mytheorem} We here fix the notations of categories we use in this paper. \begin{mydefinition} We denote by $\Dcppo_\bot$ the category of pointed dcpos and strict Scott-continuous maps, and by $\boldsymbol{\omega}\mathbf{Cppo}$ the category of pointed $\omega$cpos and $\omega$-continuous maps. \end{mydefinition} Both categories $\Dcppo_\bot$ and $\boldsymbol{\omega}\mathbf{Cppo}$ have products, which are given by cartesian products of underlying sets with coordinatewise order. When we speak of $\Dcppo_\bot$- (or $\boldsymbol{\omega}\mathbf{Cppo}$)-enrichment of categories, these cartesian structures are taken as monoidal structures. \section{Selinger's QPL and its semantics} \label{sec:QPL} In \cite{Selinger2004}, Selinger proposed a quantum programming language QPL and its denotational semantics. It is a first-order functional language with loop and recursion, and is described by flow chart syntax, besides by textual syntax.\footnote{% Strictly speaking, the name QPL is reserved for the textual syntax. In the present paper we do not distinguish the two syntaxes because semantics are given in the same manner.} He showed a denotational semantics for QPL is given by the following category. \begin{mydefinition}[{\cite[\S6.6]{Selinger2004}}] \label{def:elem-quantum-flow-chart-cat} An \emph{elementary quantum flow chart category} is a symmetric monoidal category $(\cat{C}, \otimes, I)$ with traced finite coproducts $(\oplus, 0, \Tr)$ such that: \begin{itemize} \item For each $A\in\cat{C}$, $A\otimes (-)$ is a traced monoidal functor. \item $\cat{C}$ has a distinguished object $\mathbf{qbit}$ with arrows $\iota\colon I\oplus I\to\mathbf{qbit}$ and $p\colon \mathbf{qbit}\to I\oplus I$ such that $p\circ \iota = \mathrm{id}$. \end{itemize} \end{mydefinition} \begin{mytheorem}[{\cite[\S6.6]{Selinger2004}}] \label{thm:eqfcc-give-sem-qpl} Let $\cat{C}$ be an elementary quantum flow chart category. Suppose we have an assignment $\eta$ of built-in unitary operator symbols $S$ of arity $n$ to arrows $\eta_S\colon\mathbf{qbit}^{\otimes n}\to \mathbf{qbit}^{\otimes n}$ in $\cat{C}$. Then we have an interpretation of QPL programs without recursion. If $\cat{C}$ is additionally $\boldsymbol{\omega}\mathbf{Cppo}$-enriched, then we can also interpret QPL programs with recursion. \qed \end{mytheorem} Selinger also gave a concrete model for QPL, constructing an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched elementary quantum flow chart category $\mathcal{Q}$ as follows. \begin{mydefinition} \label{def:selingers-cat} We write $\mathcal{M}_n$ for the set of complex $n\times n$ matrices. \begin{enumerate} \item The category $\CPM_s$ is defined as follows. \begin{itemize} \item An object is a natural number. \item An arrow $f\colon n\to m$ is a completely positive map $f\colon \mathcal{M}_n\to \mathcal{M}_m$. \end{itemize} \item The category $\mathbf{CPM}$ is the finite biproduct completion of $\CPM_s$. Specifically: \begin{itemize} \item An object is a sequence $\vec{n}=(n_1,\dotsc,n_k)$ of natural numbers. \item An arrow $f\colon \vec{n}\to \vec{m}$ is a matrix $(f_{ij})$ of arrows $f_{ij} \colon n_j\to m_i$ in $\CPM_s$. \end{itemize} \item \label{def:cat-Q} The category $\mathcal{Q}$ is a subcategory of $\mathbf{CPM}$ such that \begin{itemize} \item Objects are the same as $\mathbf{CPM}$. \item An arrow is $f\colon \vec{n}\to \vec{m}$ in $\mathbf{CPM}$ which is trace-nonincreasing, i.e. \[ \sum\nolimits_i \sum\nolimits_j \tr(f_{ij}(A_j)) \le \sum\nolimits_j\tr(A_j) \] for all $(A_j)_j$ with positive $A_j\in\mathcal{M}_{n_j}$. \end{itemize} \end{enumerate} \end{mydefinition} The category $\mathcal{Q}$ turns out to have the monoidal structure $(\otimes, I)$ and finite coproducts $(\oplus, 0)$, and furthermore to be $\boldsymbol{\omega}\mathbf{Cppo}$-enriched. The category is also equipped with a monoidal trace w.r.t.\ $(\oplus, 0)$, which is obtained from its $\boldsymbol{\omega}\mathbf{Cppo}$-enriched structure. Then it forms an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched elementary quantum flow chart category with $\mathbf{qbit}\coloneqq 2$. As Selinger mentioned~\cite[\S6.4]{Selinger2004} (though he did not give a proof), the construction of monoidal trace from $\boldsymbol{\omega}\mathbf{Cppo}$-enriched structure works for every $\boldsymbol{\omega}\mathbf{Cppo}$-enriched category with finite coproducts (satisfying certain condition). Specifically, we have the following theorem. \begin{mytheorem} \label{thm:cppo-enriched-cart-cat} \mbox{} \begin{enumerate} \item\label{thm-enum:cppo-enriched-cart-cat} Every $\boldsymbol{\omega}\mathbf{Cppo}$-enriched cocartesian category with right-strict composition (i.e.\ $f\circ\bot=\bot$) is traced. \item Let $\cat{C}$ and $\cat{D}$ be $\boldsymbol{\omega}\mathbf{Cppo}$-enriched cocartesian categories with right-strict composition, which are traced by \ref{thm-enum:cppo-enriched-cart-cat}. Every $\boldsymbol{\omega}\mathbf{Cppo}$-enriched cocartesian functor between $\cat{C}$ and $\cat{D}$ satisfying $F\bot=\bot$ is traced. \end{enumerate} \qed \end{mytheorem} Here, a cocartesian category refers to a monoidal category whose monoidal structure is given by finite coproducts. The proof is found in Appendix~\ref{sec:trace-on-omcppo-enrieched-cat}. Combining Theorem~\ref{thm:cppo-enriched-cart-cat} with Definition~\ref{def:elem-quantum-flow-chart-cat}, we obtain a sufficient condition for a category to be an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched elementary quantum flow chart category, which gives a semantics for QPL programs with recursion. \begin{mytheorem} \label{thm:omcppo-eqfcc} A category is an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched elementary quantum flow chart category if it is an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched symmetric monoidal category $(\cat{C}, \otimes, I)$ satisfying the following conditions. \begin{itemize} \item $\cat{C}$ has $\boldsymbol{\omega}\mathbf{Cppo}$-enriched finite coproducts $(\oplus, 0)$; \item the composition is right-strict; \item for each $A\in\cat{C}$, a functor $A\otimes (-)$ preserves finite coproducts and bottom arrows; \item $\cat{C}$ has a distinguished object $\mathbf{qbit}$ with arrows $\iota\colon I\oplus I\to\mathbf{qbit}$ and $p\colon \mathbf{qbit}\to I\oplus I$ such that $p\circ \iota = \mathrm{id}$. \end{itemize} \qed \end{mytheorem} \section{Operator algebras: $C^*$-algebras and $W^*$-algebras} \label{sec:opalg} \subsection{$C^*$-algebras and $W^*$-algebras} The theory of operator algebras is usually concerned with $C^*$-algebras and $W^*$-algebras (the latter are often studied as von Neumann algebras). Due to limitations of space, here we just fix notations and terminology, and list the basic results. A reader who is not familiar with these topics may consult \cite{Takesaki2002,Sakai1998} for the standard theory and \cite{Meyer2008,Kornell2012} for the categorical perspective. In this paper, $C^*$-algebras are always assumed to be unital. Note also that a `map' usually refers to a \emph{linear} map. For a $C^*$-algebra $A$, we write $\sa{A}$ for the set of self-adjoint elements, and $[0,1]_A\coloneqq \{x\in A\mid 0\le x\le 1\}$ for the ``unit interval'', or the set of \emph{effects}. We write $\mathbf{Cstar}$ for the category of $C^*$-algebras and linear maps. We will denote subcategories of $\mathbf{Cstar}$, with the same objects but different maps, by adding subscripts to $\mathbf{Cstar}$ as follows: M for `multiplicative'; I for `involutive'; P for `positive'; CP for `completely positive'; U for `unital'; SU for `subunital'\footnote{% A map $f$ between $C^*$-algebras is said to be \emph{subunital} if $f(1)\le 1$. Note that in the author's thesis~\cite{Cho2014Master} (and the preliminary version of this paper), `pre-unital' is used instead of `subunital'.}. For example, $\mathbf{Cstar}_\mathrm{MIU}$ is the category of $C^*$-algebras and unital $*$-homomorphisms (i.e.\ multiplicative involutive maps), while $\mathbf{Cstar}_\mathrm{CPSU}$ is the category of $C^*$-algebras and completely positive subunital maps. Note that there are inclusions like $\mathbf{Cstar}_\mathrm{MI}\subseteq\mathbf{Cstar}_\mathrm{CP}\subseteq\mathbf{Cstar}_\mathrm{P}$. A \emph{$W^*$-algebra} is a $C^*$-algebra that has a (necessarily unique) predual. Every $W^*$-algebra is equipped with the weak* topology introduced by its predual, which is called the \emph{ultraweak} topology. A map between $W^*$-algebras is said to be \emph{normal} if it is ultraweakly continuous. We write $\mathbf{Wstar}$ for the category of $W^*$-algebras and normal maps. We denote subcategories of $\mathbf{Wstar}$ in the same manner as $\mathbf{Cstar}$. For example, $\mathbf{Wstar}_\mathrm{MIU}$ is the category of $W^*$-algebras and normal unital $*$-homomorphisms, which is a \emph{non-full} subcategory of $\mathbf{Cstar}_\mathrm{MIU}$ (since maps in $\mathbf{Wstar}$ are required to be normal). We denote \emph{direct sum}~\cite[Definition~1.1.5]{Sakai1998} of $C^*$- and $W^*$-algebras by $\oplus$. It forms categorical products in categories $\mathbf{Cstar}_\mathrm{MIU}$, $\mathbf{Cstar}_\mathrm{CP}$, $\mathbf{Cstar}_\mathrm{CPSU}$, $\mathbf{Wstar}_\mathrm{MIU}$, $\mathbf{Wstar}_\mathrm{CP}$, $\mathbf{Wstar}_\mathrm{CPSU}$ etc. The nullary direct sum is the zero space $0$. There are several kinds of tensor products for $C^*$- and $W^*$-algebras. In this paper we use the \emph{spatial $C^*$-tensor product}~\cite[Definition~1.22.8]{Sakai1998}, denoted by $\otimes$, for $C^*$-algebras, and the \emph{spatial $W^*$-tensor product}~\cite[Definition~1.22.10]{Sakai1998}, denoted by $\mathop{\overline{\otimes}}$, for $W^*$-algebras. The spatial $C^*$-tensor product makes categories $\mathbf{Cstar}_\mathrm{MIU}$, $\mathbf{Cstar}_\mathrm{CP}$, $\mathbf{Cstar}_\mathrm{CPSU}$ etc.\ symmetric monoidal categories with the unit object $\mathbb{C}$, but not $\mathbf{Cstar}_\mathrm{P}$. In fact, a tensor product of positive maps can be unbounded~\cite[Proposition~3.5.2]{BrownO2008}. This is why we need the notion of the \emph{complete} positivity. Similarly, the spatial $W^*$-tensor product makes categories $\mathbf{Wstar}_\mathrm{MIU}$, $\mathbf{Wstar}_\mathrm{CP}$, $\mathbf{Wstar}_\mathrm{CPSU}$ etc.\ symmetric monoidal categories. The spatial $C^*$- and $W^*$-tensor products distribute over finite direct sums, i.e.\ \[ A\otimes (B\oplus C)\cong (A\otimes B)\oplus (A\otimes C) \enspace,\quad A\otimes 0\cong 0 \enspace,\quad M\mathop{\overline{\otimes}} (N\oplus L)\cong (M\mathop{\overline{\otimes}} N)\oplus (M\mathop{\overline{\otimes}} L) \enspace,\quad M\mathop{\overline{\otimes}} 0\cong 0 \enspace, \] for $C^*$-algebras $A,B,C$ and $W^*$-algebras $M,N,L$. These properties seem to be known results, but to be missing in the standard literature. For the sake of completeness, the proofs are included in Appendix~\ref{sec:dist-tensorprod-directsum}. For our purpose, it is useful to restate them as follows. \begin{myproposition} \label{prop:tensor-distr} \mbox{} \begin{enumerate} \item For each $C^*$-algebra $A$, a functor $A\otimes(-)$ on $\mathbf{Cstar}_\mathrm{MIU}$ preserves finite products. \item\label{prop:enum:wtensor-distr} For each $W^*$-algebra $M$, a functor $M\mathop{\overline{\otimes}}(-)$ on $\mathbf{Wstar}_\mathrm{MIU}$ preserves finite products. \end{enumerate} \qed \end{myproposition} \subsection{Order theory in operator algebras} Recall each $C^*$-algebra is equipped with a partial order $\le$ defined by: $a\le b$ $\Longleftrightarrow$ `$b-a$ is positive'. Many concepts on operator algebras can be rephrased in terms of the orders. Observe, for instance, the following easy proposition. \begin{myproposition} A map between $C^*$-algebras is positive if and only if it is monotone. \qed \end{myproposition} In fact, the orders on $W^*$-algebras have a very nice property, called \emph{monotone completeness}, which distinguishes $W^*$-algebras from $C^*$-algebras. \begin{mydefinition} A $C^*$-algebra $A$ is \emph{monotone complete} (or \emph{monotone closed}) if every norm-bounded directed subset of $\sa{A}$ has the supremum in $\sa{A}$. \end{mydefinition} \begin{myproposition}[{\cite[Lemma~1.7.4]{Sakai1998}}] \label{prop:wstar-monotone-complete} Every $W^*$-algebra is monotone complete. Moreover, suprema are obtained as ultraweak limits. \qed \end{myproposition} \noindent We can also rephrase the notion of normality of positive (i.e.\ monotone) maps between $W^*$-algebras. \begin{myproposition}[{\cite[Corollary~46.5]{Conway2000}}] \label{prop:normal-sup-pres-prelim} Let $f\colon M\to N$ be a positive map between $W^*$-algebra. Then $f$ is normal (i.e.\ ultraweakly continuous) if and only if it preserves the supremum of every norm-bounded directed subset of $\sa{M}$. \qed \end{myproposition} \noindent Therefore, we shall say a positive map $f\colon A\to B$ between monotone complete $C^*$-algebras is \emph{normal} if it preserves the supremum of every norm-bounded directed subset of $\sa{A}$. Then $W^*$-algebras can be characterised as follows. \begin{mytheorem}[{\cite[Theorem~III.3.16]{Takesaki2002}}] A $C^*$-algebra is a $W^*$-algebra if and only if it is monotone complete and admits sufficiently many normal positive functionals (i.e.\ they separate the points). \qed \end{mytheorem} Now, we shall recapture the order structures in operator algebras from a more order-theoretic (or \emph{domain-theoretic}~\cite{AbramskyJ1994}) point of view. \begin{myproposition} \label{prop:equiv-monotone-closed} Let $A$ be a $C^*$-algebra. The following are equivalent. \begin{enumerate} \item\label{prop-enum:norm-bounded} $A$ is monotone complete. \item\label{prop-enum:order-bounded-above} $\sa{A}$ is bounded directed complete. \item\label{prop-enum:unit-interval} $[0,1]_A$ is directed complete. \end{enumerate} \end{myproposition} \begin{myproof} Without loss of generality, we may assume directed subsets are bounded from below. Then, \ref{prop-enum:norm-bounded} $\Longleftrightarrow$ \ref{prop-enum:order-bounded-above} follows from the fact that norm-boundedness and order-boundedness coincide (see Proposition~\ref{prop:norm-bdd-vs-order-bdd} in Appendix~\ref{sec-apx:other-omitted-proofs}). \ref{prop-enum:order-bounded-above} $\Longrightarrow$ \ref{prop-enum:unit-interval} is trivial. For the converse, note that $\sa{A}$ is an ordered vector space over $\mathbb{R}$. Hence we can obtain the supremum of a bounded directed subset of $\sa{A}$ by transforming it into a directed subset of $[0,1]_A$ by shifting and scaling. \end{myproof} \noindent Consequently, for every $W^*$-algebras $M$, $\sa{M}$ is a bdcpo, and $[0,1]_M$ is a pointed dcpo. We have a corresponding result for normal maps, which is proved in a similar way, using Proposition~\ref{prop:normal-sup-pres-prelim}. \begin{myproposition} \label{prop:normal-sup-pres} Let $f\colon M\to N$ be a positive map between $W^*$-algebra. The first two of the following are equivalent. They are also equivalent to the third when $f$ is subunital. \begin{enumerate} \item $f$ is normal. \item The restriction $f|_{\sa{M}}\colon\sa{M}\to\sa{N}$ is Scott-continuous. \item The restriction $f|_{[0,1]_M}\colon [0,1]_M\to [0,1]_N$ is Scott-continuous. \end{enumerate} \qed \end{myproposition} \section{$\Dcppo_\bot$-enrichment of the category of $W^*$-algebras} \label{sec:dcpo-enrichment} In this section we will show that the category $\mathbf{Wstar}_\mathrm{CPSU}$ is $\Dcppo_\bot$-enriched. We also see that the monoidal product $(\mathop{\overline{\otimes}}, \mathbb{C})$ and finite products $(\oplus, 0)$ on $\mathbf{Wstar}_\mathrm{CPSU}$ are $\Dcppo_\bot$-enriched. \begin{mydefinition} \label{def:order-hom} Let $M,N$ be $W^*$-algebras. We define a partial order $\sqsubseteq$ on $\mathbf{Wstar}_\mathrm{CPSU}(M,N)$ by \[ f\sqsubseteq g \stackrel{\mathrm{def}}{\iff} g-f \text{ is completely positive} \enspace. \] \end{mydefinition} \begin{myproposition} \label{prop:hom-wstarcppu-is-dcppo} For any $W^*$-algebras $M$ and $N$, $\mathbf{Wstar}_\mathrm{CPSU}(M,N)$ with the order $\sqsubseteq$ is a pointed dcpo. \end{myproposition} \begin{myproof} First of all, it is easy to see that the zero map is the least element of $\mathbf{Wstar}_\mathrm{CPSU}(M,N)$, hence it is pointed. Note that a positive subunital map $f\colon A\to B$ between $C^*$-algebras restricts to $[0,1]_A\to [0,1]_B$, while a ``linear''\footnote{% $g(0)=0$, $g(x+y)=g(x)+g(y)$ for $x+y\le 1$, $g(rx)=r g(x)$ for $r\in[0,1]$.} map $g\colon [0,1]_A\to [0,1]_B$ extends to a positive subunital map $A\to B$. Hence such maps correspond bijectively (cf.\ \cite[Lemma~2]{FurberJ2013}). Let $(f_i)$ be a monotone net in $\mathbf{Wstar}_\mathrm{CPSU}(M,N)$. For each $x\in[0,1]_M$, $(f_i(x))$ is a monotone net in $[0,1]_N$, which is a dcpo. Hence define $f(x)\coloneqq\sup f_i(x)$. It is easy to see $f$ is a ``linear'' map $[0,1]_M\to[0,1]_N$, so that we obtain a positive subunital map $f\colon M\to N$. Normality of $f$ is proved as follows. For a monotone net $(x_j)$ in $[0,1]_M$, \[ f\Bigl(\sup x_j\Bigr) = \sup_i f_i\Bigl(\sup_j x_j\Bigr) = \sup_i \Bigl(\sup_j f_i(x_j)\Bigr) = \sup_j \Bigl(\sup_i f_i(x_j)\Bigr) = \sup_j f(x_j) \enspace. \] Note that we can exchange of the order of $\sup$ (\cite[Proposition~2.1.12]{AbramskyJ1994}). Therefore $f\colon[0,1]_M\to[0,1]_N$ is Scott-continuous, and $f\colon M\to N$ is normal by Proposition~\ref{prop:normal-sup-pres}. To show that $f$ is completely positive (hence $f\in\mathbf{Wstar}_\mathrm{CPSU}(M,N)$) and that $f$ is indeed the supremum of $(f_i)_i$, we need some more results on $W^*$-algebras. The remaining proof is found at Proposition~\ref{prop:hom-sup-cp} in Appendix~\ref{sec-apx:other-omitted-proofs}. \end{myproof} This shows that every hom-set of $\mathbf{Wstar}_\mathrm{CPSU}$ is an object in $\Dcppo_\bot$. We can also show that the composition in $\mathbf{Wstar}_\mathrm{CPSU}$ is bi-strict Scott-continuous (see Proposition~\ref{prop:composition-scott-conti} in Appendix~\ref{sec-apx:other-omitted-proofs}). Therefore, now we prove: \begin{mytheorem} \label{thm:wstarcppu-dccpobot-enriched} The category $\mathbf{Wstar}_\mathrm{CPSU}$ is $\Dcppo_\bot$-enriched. Moreover, the composition is bi-strict, i.e.\ $\bot\circ f=\bot$ and $f\circ\bot=\bot$ for each arrow $f$. \qed \end{mytheorem} \noindent Moreover, finite products $(\oplus,0)$ and the monoidal product $(\otimes,0)$ are also suitably enriched: \begin{mytheorem} \label{thm:finprod-dcppobot-enriched} Finite products in $\mathbf{Wstar}_\mathrm{CPSU}$ are $\Dcppo_\bot$-enriched. \qed \end{mytheorem} \begin{mytheorem} \label{thm:monoprod-dcppobot-enriched} The symmetric monoidal product $(\mathop{\overline{\otimes}},\mathbb{C})$ on $\mathbf{Wstar}_\mathrm{CPSU}$ is $\Dcppo_\bot$-enriched. Moreover, the tensor product of maps is bi-strict, i.e.\ $\bot\mathop{\overline{\otimes}} f=\bot$ and $f\mathop{\overline{\otimes}}\bot=\bot$. \qed \end{mytheorem} \noindent The details are found in Propositions~\ref{prop:tupling-scott-conti} and~\ref{prop:wtensor-scott-conti} in Appendix~\ref{sec-apx:other-omitted-proofs}, respectively. \begin{myremark} It is worth noting that the category $\mathbf{Cstar}_\mathrm{CPSU}$ is never $\mathbf{Dcppo}_\bot$-enriched, nor $\boldsymbol{\omega}\mathbf{Cppo}$-enriched. This is because we have an order-isomorphism $\mathbf{Cstar}_\mathrm{CPSU}(\mathbb{C},A)\cong [0,1]_A$, whereas there exists a $C^*$-algebra such that $[0,1]_A$ is not $\omega$-complete (take $A=\mathit{C}([0,1])$ for example). \end{myremark} \section{Semantics for QPL by $W^*$-algebras} \label{sec:sem-by-wstar} We have proved that $\mathbf{Wstar}_\mathrm{CPSU}$ is an $\Dcppo_\bot$-enriched symmetric monoidal category with $\Dcppo_\bot$-enriched finite products. Now we can show the following theorem. \begin{mytheorem} The opposite category of $\mathbf{Wstar}_\mathrm{CPSU}$ is an $\boldsymbol{\omega}\mathbf{Cppo}$-enriched elementary quantum flow chart category. \end{mytheorem} \begin{myproof} We apply Theorem~\ref{thm:omcppo-eqfcc}. All requirements follow from Theorems~\ref{thm:wstarcppu-dccpobot-enriched}, \ref{thm:finprod-dcppobot-enriched}, \ref{thm:monoprod-dcppobot-enriched} ($\Dcppo_\bot$-enrichment implies $\boldsymbol{\omega}\mathbf{Cppo}$-enrichment), and Proposition~\ref{prop:tensor-distr}.\ref{prop:enum:wtensor-distr}, except a distinguished object $\mathbf{qbit}$ with arrows $\iota$, $p$. We take $\mathbf{qbit}\coloneqq \mathcal{M}_2$, the algebra of complex $2\times 2$-matrices. We define two maps $\iota$, $p$ by \[ \iota \left( \begin{bmatrix} x & y \\ z & w \end{bmatrix} \right) = (x,w) \enspace,\qquad\qquad p(x,y) = \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix} \enspace. \] It is straightforward to see the two maps are positive, hence completely positive by \cite[Corollary~IV.3.5 and Proposition~IV.3.9]{Takesaki2002} (notice that $\mathbb{C}\oplus\mathbb{C}$ is commutative). They are normal because they are maps between finite dimensional $W^*$-algebras. Moreover they are clearly unital. Therefore $\iota$ and $p$ are arrows in $\mathbf{Wstar}_\mathrm{CPSU}$. It is clear that $\iota\circ p = \mathrm{id}$, hence $p\circ\iota = \mathrm{id}$ in $(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$. \end{myproof} Every unitary operator $S\colon (\mathbb{C}^2)^{\otimes n}\to(\mathbb{C}^2)^{\otimes n}$ of arity $n$ determines an arrow $\eta_S\colon (\mathcal{M}_2)^{\mathop{\overline{\otimes}} n}\to (\mathcal{M}_2)^{\mathop{\overline{\otimes}} n}$ in $\mathbf{Wstar}_\mathrm{CPSU}$ by \[ (\mathcal{M}_2)^{\mathop{\overline{\otimes}} n}\cong \mathcal{M}_{2^n} \longrightarrow \mathcal{M}_{2^n}\cong (\mathcal{M}_2)^{\mathop{\overline{\otimes}} n} \enspace,\qquad x\longmapsto S^\dagger x S \enspace, \] where $S$ is seen as a complex $2^n\times 2^n$ matrix. By Theorem~\ref{thm:eqfcc-give-sem-qpl}, finally, we show: \begin{mytheorem} The opposite category of $\mathbf{Wstar}_\mathrm{CPSU}$ gives a denotational semantics for QPL (with recursion). \qed \end{mytheorem} \section{Schr\"odinger vs.\ Heisenberg picture} \label{sec:Schr-vs-Heis} We have shown the category $(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$ gives a semantics for QPL. From now on we will discuss a comparison between two semantics by Selinger's $\mathcal{Q}$ and our $(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$. Recall that for a Hilbert space $\mathcal{H}$, $\mathcal{B}(\mathcal{H})$, i.e.\ the set of bounded operators on $\mathcal{H}$, is a $W^*$-algebra with the predual $\mathcal{T}(\mathcal{H})$, i.e.\ the set of trace class operators on $\mathcal{H}$. For every normal map $\mathcal{E}\colon\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$, there exists a corresponding bounded map $\mathcal{E}_*\colon\mathcal{T}(\mathcal{K})\to\mathcal{T}(\mathcal{H})$ between preduals. They are related in the following way: \begin{equation} \label{eq:duality} \tr(\mathcal{E}(S)\cdot T) = \tr(S\cdot \mathcal{E}_*(T)) \end{equation} for all $S\in\mathcal{B}(\mathcal{H})$ and $T\in\mathcal{T}(\mathcal{K})$. Furthermore the following holds. \begin{myproposition}[{\cite[\S4.1.2]{HeinosaariZ2012}}] Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces. Suppose a normal map $\mathcal{E}\colon\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$ and a bounded map $\mathcal{E}_*\colon\mathcal{T}(\mathcal{K})\to\mathcal{T}(\mathcal{H})$ related as above. Then \begin{enumerate} \item $\mathcal{E}$ is completely positive if and only if $\mathcal{E}_*$ is completely positive. \item $\mathcal{E}$ is unital if and only if $\mathcal{E}_*$ is trace-preserving. \item $\mathcal{E}$ is subunital if and only if $\mathcal{E}_*$ is trace-nonincreasing. \end{enumerate} \qed \end{myproposition} Hence, a normal completely positive subunital map $\mathcal{E}\colon\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$, which is an arrow in $\mathbf{Wstar}_\mathrm{CPSU}$, corresponds to a completely positive trace-nonincreasing map $\mathcal{E}_*\colon\mathcal{T}(\mathcal{K})\to\mathcal{T}(\mathcal{H})$, which is known as a \emph{quantum operation} (see e.g.~\cite[\S8.2]{NielsenC2000}, \cite[Chap.~4]{HeinosaariZ2012}). This is the well-known duality between the Heisenberg and Schr\"odinger pictures: one transforms observables (i.e.\ self-adjoint operators), while another transforms states (i.e.\ density operators). Hence, it is understood that our semantics for QPL by $\mathbf{Wstar}_\mathrm{CPSU}$ is given in the Heisenberg picture, while Selinger's semantics by $\mathcal{Q}$ is given in the Schr\"odinger picture. In the words of \cite{DHondtP2006}, our semantics can also be thought of as the \emph{weakest precondition} semantics. This is because a positive subunital map $\mathcal{E}\colon\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$ can be restricted to a map $\mathcal{E}\colon\mathcal{E}\mspace{-4mu}f(\mathcal{H})\to\mathcal{E}\mspace{-4mu}f(\mathcal{K})$ between their effects, where $\mathcal{E}\mspace{-4mu}f(\mathcal{H})\coloneqq[0,1]_{\mathcal{B}(\mathcal{H})}$ is the set of effects on $\mathcal{H}$, and coincides with the set of \emph{predicates} in \cite{DHondtP2006}. \section{Embedding $\mathcal{Q}$ into $(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$} \label{sect:embedding} As seen in the previous section, the two semantics by $\mathbf{Wstar}_\mathrm{CPSU}$ and $\mathcal{Q}$ can be considered as different viewpoints (Schr\"odinger vs.\ Heisenberg) for the same phenomena. We can state it categorically: the category $\mathcal{Q}$ can be contravariantly embedded into $\mathbf{Wstar}_\mathrm{CPSU}$. First we show the following embedding. \begin{mytheorem} There is a full embedding $I\colon\mathbf{CPM}\to(\mathbf{Wstar}_\mathrm{CP})^\mathrm{op}$. \qed \end{mytheorem} \begin{myproof} Observe the following bijective correspondences. \[ \renewcommand{\arraystretch}{1.2} \begin{array}{c} f\colon (n_1,\dotsc,n_k)\longrightarrow (m_1,\dotsc,m_l) \quad\text{in $\mathbf{CPM}$} \\ \hline\hline f_{ij}\colon n_j\longrightarrow m_i \quad\text{in $\CPM_s$, for each $i,j$} \\ \hline\hline f_{ij}\colon \mathcal{M}_{n_j}\longrightarrow \mathcal{M}_{m_i} \quad\text{completely positive, for each $i,j$} \\ \hline\hline (f_{ij})^*\colon \mathcal{M}_{m_i}\longrightarrow \mathcal{M}_{n_j} \quad\text{completely positive, hence an arrow in $\mathbf{Wstar}_\mathrm{CP}$, for each $i,j$} \\ \hline\hline I (f) \colon \bigoplus_{i=1}^l\mathcal{M}_{m_i} \longrightarrow \bigoplus_{j=1}^k\mathcal{M}_{n_j} \quad\text{in $\mathbf{Wstar}_\mathrm{CP}$} \end{array} \] For the third correspondence, note that the self-duality of finite dimensional spaces: \[ \mathcal{M}_{n_j} \cong \mathcal{B}(\mathbb{C}^{n_j}) \cong \mathcal{T}(\mathbb{C}^{n_j})^* \cong (\mathcal{M}_{n_j})^* \enspace. \] The last correspondence comes from the fact finite direct sums are biproducts in $\mathbf{Wstar}_\mathrm{CP}$. Hence the mapping $I(n_1,\dotsc,n_k)=\bigoplus_{j=1}^k\mathcal{M}_{n_j}$ defines a contravariant functor $I\colon\mathbf{CPM}\to(\mathbf{Wstar}_\mathrm{CP})^\mathrm{op}$, which is full and faithful by definition, and clearly injective on objects. \end{myproof} We use the following lemma. \begin{mylemma} \label{lem:pos-trace} Let $\mathcal{H}$ be a Hilbert space. A bounded operator $T\in\mathcal{B}(\mathcal{H})$ is positive if and only if $\tr(TS)\in\mathbb{R}^+$ ($=[0,\infty)$) for all positive $S\in\mathcal{T}(\mathcal{H})$. \qed \end{mylemma} \begin{mytheorem} There is a full embedding $I'\colon\mathcal{Q}\to(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$. \end{mytheorem} \begin{myproof} The functor $I\colon\mathbf{CPM}\to(\mathbf{Wstar}_\mathrm{CP})^\mathrm{op}$ restricts to a full and faithful functor $I'\colon\mathcal{Q}\to(\mathbf{Wstar}_\mathrm{CPSU})^\mathrm{op}$ as follows. \begin{align*} & \text{$f\colon \vec{n}\to \vec{m}$ is trace-nonincreasing (Definition~\ref{def:selingers-cat}.\ref{def:cat-Q})} \\ &\iff \sum\nolimits_i \sum\nolimits_j \tr(f_{ij}(A_j)) \le \sum\nolimits_j\tr(A_j) \text{ for all $(A_j)_j$ with positive $A_j\in \mathcal{M}_{n_j}$} \\ &\iff \sum\nolimits_i \tr(f_{ij}(A)) \le \tr(A) \text{ for each $A\in \mathcal{M}_{n_j}$, for each $j$} \\ &\stackrel{\star}{\iff} \sum\nolimits_i \tr\Bigl(\bigl((f_{ij})^*(1)\bigr)A\Bigr) \le \tr(A) \text{ for each $A\in \mathcal{M}_{n_j}$, for each $j$} \\ &\iff \tr\Bigl( \bigl(1 - \sum\nolimits_i (f_{ij})^*(1)\bigr)A\Bigr) \ge 0 \text{ for each $A\in \mathcal{M}_{n_j}$, for each $j$} \\ &\stackrel{\star\star}{\iff} 1 - \sum\nolimits_i (f_{ij})^*(1) \ge 0 \text{ for each $j$} \\ &\iff \sum\nolimits_i (f_{ij})^*(1) \le 1 \text{ for each $j$} \\ &\iff I(f)\bigl((1)_i\bigr) \le (1)_j \\ &\iff I(f)\colon \bigoplus\nolimits_i\mathcal{M}_{m_i} \longrightarrow \bigoplus\nolimits_j\mathcal{M}_{n_j} \text{ is subunital} \enspace, \end{align*} where $\stackrel{\star}{\Longleftrightarrow}$ is by the equation \eqref{eq:duality} and $\stackrel{\star\star}{\Longleftrightarrow}$ is by Lemma~\ref{lem:pos-trace}. \end{myproof} In fact, we can say more about the embedding. Notice that $I(\vec{n})$ is a finite dimensional $W^*$-algebra for each $\vec{n}\in \mathbf{CPM}$. Hence the embedding is restricted to $I_{\mathrm{fd}}\colon\mathbf{CPM}\to(\mathbf{FdWstar}_\mathrm{CP})^\mathrm{op}$, where $\mathbf{FdWstar}_\mathrm{CP}$ denotes the category of finite dimensional $W^*$-algebras and normal completely positive maps. In the same way we have an embedding $I'_{\mathrm{fd}}\colon\mathcal{Q}\to(\mathbf{FdWstar}_\mathrm{CPSU})^\mathrm{op}$. \begin{mytheorem} The embeddings \[ I_{\mathrm{fd}}\colon\mathbf{CPM}\to(\mathbf{FdWstar}_\mathrm{CP})^\mathrm{op} \enspace,\qquad I'_{\mathrm{fd}}\colon\mathcal{Q}\to(\mathbf{FdWstar}_\mathrm{CPSU})^\mathrm{op} \] give equivalences of categories: \[ \mathbf{CPM}\simeq(\mathbf{FdWstar}_\mathrm{CP})^\mathrm{op} \enspace,\qquad \mathcal{Q}\simeq(\mathbf{FdWstar}_\mathrm{CPSU})^\mathrm{op} \enspace. \] \end{mytheorem} \begin{myproof} \cite[Theorem~I.11.2]{Takesaki2002} implies that $I_{\mathrm{fd}}$ and $I'_{\mathrm{fd}}$ are essentially surjective. A full, faithful and essentially surjective functor is a part of equivalence~\cite[Theorem~IV.4.1]{MacLane1998}. \end{myproof} \section{Infinite types and classical computation} \label{sect:infinite-type} Because Selinger's category $\mathcal{Q}$ is (contravariantly) embedded into $\mathbf{Wstar}_\mathrm{CPSU}$, the category $\mathbf{Wstar}_\mathrm{CPSU}$ can be thought of as an infinite dimensional extension of $\mathcal{Q}$. Therefore, working in the category $\mathbf{Wstar}_\mathrm{CPSU}$ rather than $\mathcal{Q}$ enables us to handle infinite types. Recall that a type $\mathbf{bit}$ is interpreted by $\sem{\mathbf{bit}}=\mathbb{C}\oplus\mathbb{C}$. It is easily extended to an interpretation of $\mathbf{trit}$ by $\sem{\mathbf{trit}}=\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}$, and in general, an interpretations of a type of $n$-level classical system by $\bigoplus_{i=1}^n\mathbb{C}$. Then, it is natural to interpret a type $\mathbf{nat}$ of natural numbers by $\sem{\mathbf{nat}}=\bigoplus_{i\in\mathbb{N}}\mathbb{C}$, as Selinger also suggested in~\cite[\S7.3]{Selinger2004}. Actually we have infinite direct sums in $\mathbf{Wstar}_\mathrm{CPSU}$, but not in $\mathcal{Q}$. We can also consider quantum analogue. Interpretations $\sem{\mathbf{qbit}}=\mathcal{M}_2\cong\mathcal{B}(\mathbb{C}^2)$, $\sem{\mathbf{qtrit}}=\mathcal{M}_3\cong\mathcal{B}(\mathbb{C}^3)$, $\dotsc$, could be extended to an interpretation of a type of countable level quantum system (or ``quantum natural numbers''?) by $\mathcal{B}(\mathcal{H}_\mathbb{N})$, where $\mathcal{H}_\mathbb{N}$ is the Hilbert space of countable dimension. This type cannot be interpreted in $\mathcal{Q}$. In what follows, we will generalise the former observation on classical types. \begin{mydefinition} For a set $S$ and for a real number $p\ge 1$, we define \[ \ell^p(S) \coloneqq \Bigl\{\varphi\colon S\to \mathbb{C} \Bigm| \sum_{s\in S}\abs{\varphi(s)}^p <\infty \Bigr\} \enspace, \qquad \ell^\infty(S) \coloneqq \Bigl\{\varphi\colon S\to \mathbb{C} \Bigm| \sup_{s\in S}\abs{\varphi(s)} <\infty \Bigr\} \enspace. \] It is a standard fact that these are Banach spaces with coordinatewise operations, and norms $\norm{\varphi}_p=(\sum_{s\in S}\abs{\varphi(s)}^p)^{1/p}$ and $\norm{\varphi}_\infty=\sup_{s\in S}\abs{\varphi(s)}$ respectively. \end{mydefinition} Then, notice that $\sem{\mathbf{bit}}=\mathbb{C}\oplus\mathbb{C}\cong\ell^\infty(2), \sem{\mathbf{trit}}=\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\cong\ell^\infty(3), \dotsc,\sem{\mathbf{nat}}=\bigoplus_{i\in\mathbb{N}}\mathbb{C}\cong\ell^\infty(\mathbb{N})$. We have the following general result. \begin{myproposition} \label{prop:countable-comm-wstar} Let $S$ and $T$ be sets. \begin{enumerate} \item\label{lem-enum:ell-infi-wstar} $\ell^\infty(S)$ is a $W^*$-algebra with the predual $\ell^1(S)$. \item\label{lem-enum:ell-infi-starhom} Any function $f\colon S\to T$ induces a normal unital $*$-homomorphism $\ell^\infty(f)\colon \ell^\infty(T)\to \ell^\infty(S)$ by $\ell^\infty(f)(\varphi)=\varphi\circ f$. \item\label{lem-enum:wtensor-and-prod} There is a (normal unital) $*$-isomorphism: $\ell^\infty(S)\mathop{\overline{\otimes}}\ell^\infty(T)\cong \ell^\infty(S\times T)$. \end{enumerate} \end{myproposition} \begin{myproof} \ref{lem-enum:ell-infi-wstar}: It is straightforward to see $\ell^\infty(S)$ is a $C^*$-algebra. The duality $\ell^\infty(S)\cong\ell^1(S)^*$ is standard.\footnote{% A really standard result is the case where $S=\mathbb{N}$, but its proof is easily generalised. Alternatively, one can think of $\ell^\infty(S)\cong\ell^1(S)^*$ as a special case of $L^\infty(X,\Sigma,\mu)\cong L^1(X,\Sigma,\mu)^*$ for a localisable measure space $(X,\Sigma,\mu)$ (see e.g.~\cite[\S243]{Fremlin2010vol2}), using the counting measure on $S$.} \auxproof{% To see $\ell^1(S)$ is the predual of $\ell^\infty(S)$, define a map $\iota\colon\ell^\infty(S)\to\ell^1(S)^*$ by $\iota(\varphi)(\psi)=\sum_{s\in S}\varphi(s)\psi(s)$. Then it is straightforward to check this map is an isometry. It is also surjective as follows. For any $\Psi\in\ell^1(S)^*$, $\Psi\circ \delta$ is in $\ell^\infty(S)$, where $\delta\colon S\to \ell^1(S)$ is defined by $\delta(s)(s)=1$ and $\delta(s)(s')=0$ ($s\ne s'$). To prove $\iota(\Psi\circ\delta)=\Psi$, it suffices to show $\iota(\Psi\circ\delta)(\psi)=\Psi(\psi)$ for all $\psi\colon S\to\mathbb{C}$ with finite support, because they form a dense subset of $\ell^1(S)$. For $\psi\colon S\to\mathbb{C}$ with finite support $F\subseteq S$, \begin{align*} \iota(\Psi\circ\delta)(\psi) &= \sum_{s\in S}(\Psi\circ\delta)(s)\psi(s) \\ &= \sum_{s\in S}\Psi(\delta(s))\psi(s) \\ &= \sum_{s\in F}\Psi(\delta(s))\psi(s) \\ &= \sum_{s\in F}\Psi(\psi(s)\delta(s)) \\ &= \Psi\Bigl(\sum_{s\in F}\psi(s)\delta(s)\Bigr) \\ &= \Psi(\psi) \enspace. \end{align*} } \ref{lem-enum:ell-infi-starhom}: It is easy to see $\ell^\infty(f)$ is a unital $*$-homomorphism. To see it is normal, we can give explicitly the predual map $\ell^1(f)\colon \ell^1(S)\to\ell^1(T)$ by $\ell^1(f)(\varphi)(t)=\sum_{s\in f^{-1}(t)}\varphi(s)$, which is bounded linear, and makes the following diagram commute. \[ \xymatrix@C=3pc{ \ell^\infty(T) \ar[d]_\cong \ar[r]^{\ell^\infty(f)} & \ell^\infty(S) \ar[d]^\cong \\ \ell^1(T)^*\ar[r]^{\ell^1(f)^*} & \ell^1(S)^* } \] \ref{lem-enum:wtensor-and-prod}: First we will show $\ell^1(S)\otimes\ell^1(S)\cong \ell^1(S\times T)$, i.e.\ the preduals are isometrically isomorphic. It is known that, when at least one of $W^*$-algebras $M,N$ is commutative, the dual spatial $C^*$-norm on $M_*\odot N_*$ coincides with the projective (i.e.\ greatest) cross norm~\cite[\S1.22]{Sakai1998}. Hence $\ell^1(S)\otimes\ell^1(T)$ is the projective tensor product of $\ell^1(S)$ and $\ell^1(T)$, which possesses the universality to represent bounded bilinear maps $\ell^1(S)\times\ell^1(T)\to X$ (see \cite[Theorem~2.9]{Ryan2002} for detail). Therefore, to prove $\ell^1(S)\otimes\ell^1(S)\cong \ell^1(S\times T)$, it suffices to show the following: there is a short bilinear map $\eta\colon\ell^1(S)\times\ell^1(T)\to\ell^1(S\times T)$ and for each short bilinear map $f\colon\ell^1(S)\times\ell^1(T)\to X$, there exists a unique short map $\bar{f}\colon\ell^1(S\times T)\to X$ such that $f=\bar{f}\circ\eta$.\footnote{ A bilinear map $f$ is short if $\norm{f(x,y)}\le\norm{x}\norm{y}$; and a linear map $g$ is short if $\norm{g(x)}\le\norm{x}$.} We define $\eta(\varphi,\psi)(s,t)=\varphi(s)\psi(t)$, then it is easy to check $\eta$ is short bilinear. Let $f\colon\ell^1(S)\times\ell^1(T)\to X$ be a short bilinear map. Let \[ F(S\times T)=\{\varphi\colon S\times T\to \mathbb{C}\mid \text{$\varphi$ has finite support}\} \] be a vector space, which is a dense subspace of $\ell^1(S\times T)$. Note that $\{\delta(s,t)\mid s\in S, t\in T\}$ is a basis of this space, where $\delta\colon S\times T\to F(S\times T)$ is Kronecker's delta. Then we define a linear map $f'\colon F(S\times T)\to X$ by $f'(\delta(s,t))=f(\delta(s),\delta(t))$. With respect to $\ell^1$-norm, it is short because \begin{align*} \norm[\Big]{f\Bigl(\sum_{i=1}^n a_i \delta(s_i,t_i)\Bigr)} &= \norm[\Big]{\sum_{i=1}^n a_i f(\delta(s_i),\delta(t_i))} \\ &\le \sum_{i=1}^n \abs{a_i} \norm{f(\delta(s_i),\delta(t_i))} \\ &\le \sum_{i=1}^n \abs{a_i} \norm{\delta(s_i)}\norm{\delta(t_i)} \\ &= \sum_{i=1}^n \abs{a_i} = \norm[\Big]{\sum_{i=1}^n a_i \delta(s_i,t_i)} \enspace. \end{align*} Hence by density $f'$ extends to a short map $\bar{f}\colon \ell^1(S\times T)\to X$. From $\bar{f}(\eta(\delta(s),\delta(t))) =\bar{f}(\delta(s,t))=f(\delta(s),\delta(t))$, (bi)linearity and continuity, it follows that $\bar{f}\circ \eta=f$, and that such $\bar{f}$ is unique. We conclude that $\ell^1(S)\otimes\ell^1(T)\cong\ell^1(S\times T)$. By dualising, we obtain an isomorphism \[ \ell^\infty(S)\mathop{\overline{\otimes}}\ell^\infty(T)\cong (\ell^1(S)\otimes\ell^1(T))^*\cong \ell^1(S\times T)^*\cong \ell^\infty(S\times T) \enspace. \] Finally we have to show this isomorphism is a $*$-isomorphism. Let $\Theta\colon \ell^\infty(S)\odot\ell^\infty(T)\to \ell^\infty(S\times T)$ be a map defined by $\Theta(\varphi\otimes\psi)(s,t)=\varphi(s)\varphi(t)$. Then it is easy to check $\Theta$ is a $*$-homomorphism. Now it suffices to show the following diagram commutes, \[ \xymatrix{ \ell^\infty(S)\odot\ell^\infty(T) \ar@{>->}[d] \ar[dr]^\Theta & \\ \ell^\infty(S)\mathop{\overline{\otimes}}\ell^\infty(T) \ar[r]_-\cong& \ell^\infty(S\times T) } \] because the canonical embedding $\ell^\infty(S)\odot\ell^\infty(T)\to \ell^\infty(S)\mathop{\overline{\otimes}}\ell^\infty(T)$ is ultraweakly dense. Checking this commutativity is straightforward. \auxproof{(This is an old proof.) Note that $\ell^2(S)$ is a Hilbert space, and $\ell^\infty(S)$ has a canonical normal unital faithful representation $\pi\colon\ell^\infty(S)\to\mathcal{B}(\ell^2(S))$ by $\pi(\varphi)(\psi) = \varphi\psi$ (pointwise multiplication). We can identify the spatial $W^*$-tensor product $\mathop{\overline{\otimes}}$ with the tensor product of von Neumann algebras. Note the isomorphism $\ell^2(S)\otimes\ell^2(T)\cong \ell^2(S\times T)$ of Hilbert spaces. By the identification $\mathcal{B}(\ell^2(S)\otimes\ell^2(T))\cong \mathcal{B}(\ell^2(S\times T))$, we have an inclusion $\ell^\infty(S)\odot\ell^\infty(T)\subseteq \ell^\infty(S\times T)$.\footnote{% Here the symbol $\odot$ denotes the algebraic tensor product.} The (ultra)weak denseness of the inclusion proves $\ell^\infty(S)\mathop{\overline{\otimes}}\ell^\infty(T)\cong \ell^\infty(S\times T)$.} \end{myproof} \begin{mycorollary} \label{cor:embed-set-wstar} There is an embedding $\ell^\infty\colon\mathbf{Set}\to(\mathbf{CWstar}_\mathrm{MIU})^\mathrm{op}$, from the category of sets and functions to the category of commutative $W^*$-algebras and unital $*$-homomorphisms. Moreover, it maps binary product of sets to the spatial $W^*$-tensor product. \qed \end{mycorollary} \begin{myproof} By Proposition~\ref{prop:countable-comm-wstar}, $\ell^\infty$ defines a functor which maps binary product of sets to the spatial $W^*$-tensor product. The injectivity on objects is obvious. We show it is faithful. Let $f,g\colon S\to T$ be functions with $\ell^\infty(f)=\ell^\infty(g)$. Let $\delta\colon T\to\ell^\infty(T)$ be Kronecker's delta. Then, for each $s\in S$, \[ \delta(f(s))(g(s)) =\ell^\infty(g)(\delta(f(s)))(s) =\ell^\infty(f)(\delta(f(s)))(s) =\delta(f(s))(f(s)) =1 \enspace, \] so that $f(s)=g(s)$. Hence $f=g$. \end{myproof} As a consequence, we can embed sets $S_1,\dotsc,S_n,T$ and a function $f\colon S_1\times\dotsb\times S_n\to T$ contravariantly into the category $\mathbf{CWstar}_\mathrm{MIU}$ (hence into $\mathbf{Wstar}_\mathrm{CPSU}$) as: \[ \ell^\infty(f)\colon \ell^\infty(T) \longrightarrow \ell^\infty(S_1\times\dotsb\times S_n) \cong \ell^\infty(S_1) \mathop{\overline{\otimes}}\dotsb\mathop{\overline{\otimes}} \ell^\infty(S_n) \enspace. \] Therefore, any classical data type and function between them interpreted in $\mathbf{Set}$ can also be interpreted in $\mathbf{Wstar}_\mathrm{CPSU}$. This enables us, for example, to build in a function symbol $f\colon \mathbf{nat}^{\otimes n}\to \mathbf{nat}$ with an (classical) interpretation $\sem{f}\colon\mathbb{N}^n\to \mathbb{N}$ into a quantum programming language. \section{Conclusions and future work} \label{sec:concl-future} We have given a novel denotational semantics for a first-order functional quantum programming language QPL by operator algebras. Technically, we showed that the category $\mathbf{Wstar}_\mathrm{CPSU}$ of $W^*$-algebras and normal completely positive subunital maps is a $\Dcppo_\bot$-enriched symmetric monoidal category with $\Dcppo_\bot$-enriched finite products, and therefore its opposite is an elementary quantum flow chart category. Our model can be considered as an infinite dimensional extension of Selinger's original model $\mathcal{Q}$, and is more flexible model to accommodate infinite structures and to unify classical computation as commutative structures. In parallel with the present work, Rennela~\cite{Rennela2014} recently showed that $\mathbf{Wstar}_\mathrm{PSU}$ is algebraically compact for a certain class of (``von Neumann'') functors. These results demonstrate that the use of operator algebras, especially $W^*$-algebras, for modelling quantum computation is quite promising. One of the most important future work is to give a denotational semantics by operator algebras for a \emph{higher-order} quantum programming language, or the quantum lambda calculus. In an (unpublished) paper~\cite{Kornell2012}, it is showed that the symmetric monoidal category $((\mathbf{Wstar}_\mathrm{MIU})^\mathrm{op},\mathop{\overline{\otimes}},\mathbb{C})$ is \emph{closed}. This result can be helpful. Another future work is, extending the results in \S\ref{sect:infinite-type}, to study commutative $W^*$-algebras for modelling classical computation. It is known that commutative $W^*$-algebras are characterised as $L^\infty$ spaces $L^\infty(X,\Sigma,\mu)$ of localisable measure spaces $(X,\Sigma,\mu)$ (see e.g.~\cite[\S1.18]{Sakai1998}). Hence, by analogy with the categorical (Gelfand) duality between commutative $C^*$-algebras and compact Hausdorff spaces, it is natural to expect the categorical duality between commutative $W^*$-algebras and certain measure (or measurable) spaces. In that case, Proposition~\ref{prop:countable-comm-wstar} and Corollary~\ref{cor:embed-set-wstar} could be obtained as a special case for counting measures. We further expect a similar result to \cite{FurberJ2013}, showing that $\mathbf{Wstar}_\mathrm{CPSU}$ can also accommodate probabilistic computation. \subsection*{Acknowledgements} This paper is based on my master thesis~\cite{Cho2014Master} at the University of Tokyo. I am deeply grateful to my supervisor Ichiro Hasuo and colleagues at the time for valuable discussions. I also wish to thank my current colleagues at Radboud University Nijmegen and the QPL reviewers for helpful comments. \input{qpl2014.bbl}

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\section*{Figure captions} \noindent Figure 1: An SU(1,1) interferometer. Two light modes $a_{1}$ and $a_{2}$ are mixed by four-wave mixer FWM1, accumulate phase shifts $\phi_{1}$ and $\phi_{2}$, respectively, and then they are again mixed by four-wave mixer FWM2. The photons in output modes are counted by detectors D1 and D2. \end{document}

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\section{\label{intro}Introduction} The $\psi(3770)$ resonance is a vector state that was first detected at SPEAR \cite{prl39p526} in 1977. The signal was fitted with a pure $P$-wave Breit-Wigner distribution. More recently, the interest on this state was revived, and the parameters are nowadays fitted by the Particle Data group (PDG) as $3773.13\pm 0.35$ MeV for the mass and $27.2\pm1.0$ MeV for the width \cite{pdg}. According to the `quark model review' of the PDG, the $\psi(3770)$ resonance is classified as a $1\ ^{3}D_{1}$ charmonium state \cite{prd32p189}, the first one above the $D\bar{D}$ threshold, which is the reason for its relatively large width. As a consequence, besides kinematic interference, additional nonperturbative effects are expected. Indeed, the line-shape of the $\psi(3770)$ resonance turned out to be quite anomalous. Although other experiments have been performed with observations in $D\bar{D}$ channel \cite{prl93p051803,prl97p121801,prd76p111105,plb668p263,plb711p292}, the deformation on the line-shape of the $\psi(3770)$ was made clear by the BES Collaboration data in the $e^{+}e^{-}$ annihilation to hadrons \cite{prl101p102004}. The existence of a second resonance was suggested, although possible dynamical effects, generated by the $D\bar{D}$ threshold were not discarded. A deformation of a line-shape due to the superposition of two resonances has been discussed before e.g., for the scalar kaon \cite{npb909p418}, where, besides the dominant $\bar{q}q$ state, an additional dynamically generated state arises from the continuum, the well known $K_{0}^{\ast}(800)$. Such effect has not been discussed before for the vector charmonium. Various analysis of the $\psi(3770)$ have been performed. In Refs.~\cite{prd86p114013,prd87p057502} fits were computed taking into account not only the $D\bar{D}$ interference but also the tail of the $\psi(2S)$. Such inclusion is natural since the $\psi(3770)$ should be a mixed state $\ket{ 1\ ^{3}D_{1}}$-$\ket{2\ ^{3}S_{1}}$. In Ref.~\cite{prd80p074001}, the deformation from the right side of the resonance, i.e. a dip structure, is explained by the interference with the $D\bar{D}$ kinematical background which is higher for larger relative momentum. The same dip is reproduced in \cite{prd86p114013}, using $D\bar{D}$ background only, in \cite{plb747p321}, where in addition, the continuum of light hadrons is removed, and in Refs.~\cite{1408.5600} and \cite{1410.1375}, using Fano resonances. In \cite{plb641p145} BES measured an unexpectedly large branching fraction $\psi(3770)\rightarrow$ non$-D\bar{D}$ of about $15\%$, a result that is not contradicted by CLEO in \cite{prl96p092002}, within errors.\textbf{ }Predictions for such non-$D\bar{D}$ continuum were made including the tails of $J/\psi$ and $\psi(2S)$, $\tau\tau$ and $uds$ decays \cite{plb769p187}, other excited $\psi$ states \cite{prd81p011501}, the $D\bar{D}^{\ast}$ channel \cite{prd88p014010}, other hadronic decays and radiative decays \cite{prd87p057502}, and final state interactions \cite{plb675p441}, which in any case do not sum up to the value of $15\%$. Yet, since the phase space to $D\bar{D}$ is not excessively large, it is likely that the missing decays are simply the sum of all the many Okubo-Zweig-Iizuka (OZI)-suppressed hadronic decays, that have not been studied systematically in the theory. Estimations for the $\psi(3770)$ production via $p\bar{p}$ annihilation are made in \cite{prd90p032007,epjc76p192,prd91p114022}, which may possibly be measured at the PANDA experiment \cite{0903.3905}. Mass estimations for the $\psi(3770)$ have also been made on the lattice \cite{jhep09p089,prd77p034501}. For a review of the charmonium states, see Ref.~\cite{ppnp54p615}. In this work, we study the properties of the $\psi(3770)$ by analyzing its production through electron-positron annihilation, and subsequent decay into $D\bar{D}$ pairs (for a preliminary study, see Ref.~\cite{1708.02041}). Our starting point is a vector charmonium {\it seed} state, which gets dressed by ``clouds'' of $D^+D^-$ and $D^0\bar{D}^0$ mesons. Our aim is to study the deformation seen on the left side of the resonance in Ref.~\cite{prl101p102004} with mesonic loops combined with the nearby thresholds. To this end, we use an effective relativistic Lagrangian approach in which a single vector state $\psi\equiv\psi(3770)$ is coupled to channels $D^+D^-$ and $D^0\bar{D}^0$, as well as to lepton pairs. The propagator of $\psi$ is calculated at the resummed one-loop level and fulfills unitarization requirements. Then, we perform a fit of the four parameters of our approach, i.e. the effective couplings of $\psi$ to $D\bar{D}$ and of $\psi$ to leptons, the mass of $\psi$, and a cutoff responsible for the finite dimension of the $\psi$ meson, to the experimental cross-section of the reaction $e^{+}e^{-}\rightarrowD\bar{D}$ in Refs.~\cite{plb668p263} and \cite{prl97p121801}, in the energy region up to 100 MeV above the $D^0\bar{D}^0$ threshold. We assume that the $\psi(3770)$ resonance dominates in this energy range. We obtain a very good description of the data, which in turn allows us to determine in a novel and independent way the mass, width, and branching ratios of the $\psi(3770)$. Moreover, we study in detail, to our knowledge for the first time, the poles of this state. In fact, a pole was found in Ref.~\cite{1410.1375}, using Fano resonances, at about $3778-i14$ MeV, though in lesser detail. Quite remarkably, we find \emph{two} poles for this resonance, one which roughly corresponds to the peak of the resonance, the seed pole, and one additional dynamically generated pole, responsible for the enhancement left from the peak, which emerges due to the strong coupling between the seed state and the mesonic loops. This is a companion pole, similar to the one found in the kaonic system in Ref. \cite{npb909p418}. As a consequence of the determined parameters, we also show that the cross sections $e^{+}e^{-}\rightarrow$ $D^+D^-$ and $e^{+}e^{-}\rightarrow D^0\bar{D}^0$ agree separately to data. Being the resonance not an ideal Breit-Wigner one, different definitions for the partial widths are compared to each other. As a verification of our theoretical approach, we artificially vary the intensity of the coupling in order to show its effect on the line-shape of the resonance. Smaller couplings lead to a narrower Breit-Wigner-like shape, while larger couplings to an even larger deformation, which eventually gives rise to two peaks, as in Ref.~\cite{prd93p014002}. In addition, the trajectory of each pole is analyzed, confirming the effects on the line-shape. The paper is organized as it follows. In Sec.~\ref{model} the model is introduced and various theoretical quantities, such as the resummed propagator, spectral function, cross-section, and computation of the poles, are presented. In Sec.~\ref{results} we show our results: first, we present a fit to data and examine its consequences, then we study the position and trajectory of the poles. In Sec.~\ref{conclusion} conclusions are drawn. Some important technical details and comparative results are discussed in the Appendices. \section{\label{model}An Effective Model} In this section, we present the Lagrangian of the model and evaluate the most relevant theoretical quantities, namely the resummed propagator and the spectral functions, needed to calculate the cross section for the processes $e^{+}e^{-}\rightarrow D^+D^-$ and $e^{+}e^{-}\rightarrow D^0\bar{D}^0$, via production of the $\psi(3770)$, and with $D^0\bar{D}^0+D^+D^-$ one-loops. \subsection{\label{model.1}The strong Lagrangian and dispersion relations} We aim to write down a Lagrangian for the strong interaction between a single vectorial resonance and two pseudoscalar mesons. The bare resonance $\psi\equiv\psi(3770)$ is identified with a bare quarkonium $c\bar{c}$ state, with predominant quantum numbers $N\ ^{2S+1}L_J=1\ ^{3}D_{1}$, but admixtures of other quantum numbers, most notably the $2\ ^{3}S_{1},$ are possible. In our model, however, neither the quarkonium structure nor the orbital angular momentum mixing are explicit, since we work with mesonic degrees of freedom in the basis of total angular momentum. The effective strong Lagrangian density, for the charmonium state $\psi$, reads \begin{equation} \begin{split} \mathcal{L}_{\psiD\bar{D}}& =ig_{\psi D^0\bar{D}^0}\psi_{\mu }\Big(\partial^{\mu}D^0\bar{D}^0-\partial^{\mu}\bar{D}^{0}D^{0}% \Big)\\ & +ig_{\psi D^+D^-}\psi_{\mu}\Big(\partial^{\mu}D^+D^--\partial^{\mu }D^{-}D^{+}\Big),\label{lagi}% \end{split} \end{equation} which corresponds to the simplest interaction among $\psi$ and its main decay products, the pseudoscalar pairs $D^0\bar{D}^0$ and $D^{-}D^{+}$, each vertex represented in Fig.~\ref{diag2}. In this work, we shall take into account the small mass difference between $D^{0}$ and $D^{+}$, i.e.~isospin breaking for the masses, but we shall keep the same coupling constant, i.e.~isospin symmetry for the decays, and we define $g_{\psi D^{0}\bar{D} ^{0}}=g_{\psi D^+D^-}\equiv g_{\psi D\bar{D}}$. The full Lagrangian can be found in \ref{A}. \begin{figure}[ptb] \centering \begin{tabular}[c]{cc} {\small\hspace*{2.3cm}$D^0$}&{\small \hspace*{2.3cm}$D^+$}\\[0mm] \hspace*{-1.cm}$\psi$&\hspace*{-1.cm}$\psi$\\[-3mm] \resizebox{!}{30pt}{\includegraphics{vertex1.pdf}}& \resizebox{!}{30pt}{\includegraphics{vertex1.pdf}}\\ {\small\hspace*{2.3cm}$\bar{D}^0$}&{\small \hspace*{2.3cm}$D^-$}\\[1mm] \end{tabular} \caption{\label{diag2}Interaction vertices $\psi\to D^0\bar{D}^0$ and $\psi\to D^+\bar D^-$, corresponding to the Lagrangian in Eq.~\eqref{lagi}.} \end{figure} \begin{figure}[ptb] \centering \begin{tabular}[c]{c}% {\small\hspace*{2.3cm}$D^0$\hspace*{1.7cm}$D^+$}\\[1mm] \resizebox{!}{18pt}{\includegraphics{diag2ch.pdf}}\\[-1mm] {\small\hspace*{2.3cm}$\bar{D}^0$\hspace*{1.7cm}$D^-$}\\[3mm] \vspace*{2mm} \resizebox{!}{18pt}{\includegraphics{diagloopDD.pdf}} \end{tabular} \caption{\label{diag3}Upper diagram: one-loop $D\bar{D}=D^0\bar{D}^0+D^+D^-$. Lower diagram: Full one-loop expansion (cf.~Fig.~\ref{diag1}).}% \end{figure} The free propagator of a vector field $\psi$, without any loops, with mass $m_{\psi}$ and momentum $p$, is given by \begin{equation} G_{\mu\nu}(p)=\frac{1}{p^{2}-m_{\psi}^{2}+i\varepsilon}\Big(-g_{\mu\nu }+\frac{p_{\mu}p_{\nu}}{m_{\psi}^{2}}\Big), \label{vprop}% \end{equation} where the term in parenthesis is the sum over the three polarization states of a vector. We include the resummed one-loop effect as shown in Fig.~\ref{diag3}, leading to the full propagator of $\psi$ \begin{equation} \Delta_{\mu\nu}(p)=G_{\mu\nu}(p)+G_{\mu\mu^{\prime}}% (p)\Pi_{\mu^{\prime}\nu^{\prime}}(p)G_{\nu^{\prime}\nu}(p)+\cdots, \label{1loop}% \end{equation} where $\Pi_{\mu\nu}(p)$ is the loop-function, consisting of two contributions, the $D^0\bar{D}^0$ loops and the $D^+D^-$ loops. In particular, we get% \begin{equation} \Pi_{\mu\nu}(p)=g_{\psi D\bar{D}}^2\{\Sigma_{\mu\nu}(p,m_{D^{0}% })+\Sigma_{\mu\nu}(p,m_{D^{+}})\}, \end{equation} where the function $\Sigma_{\mu\nu}(p,m)$, with $\int_q\equiv\int\frac{d^{4}q}{(2\pi)^{4}}$, reads \begin{equation}% \Sigma_{\mu\nu}(p,m)=i \int_q\frac{4\ q_{\mu}q_{\nu}f_{\Lambda}^{2}% (\xi)}{[(q+p/2)^{2}-m^{2}+i\varepsilon][(q-p/2)^{2}-m^{2}+i\varepsilon]}, \label{se}% \end{equation} where $m$ is the mass of the meson circulating in the loop (either $D^{0}$ or $D^{+}$), $q=(q^{0},\mathbf{q})$ is the four-momentum of the loop, and $\xi=4(\mathbf{q}^{2}+m^{2})$ (see \ref{B} for computation details). Note, in the reference frame of the decaying particle, it holds the relation% \begin{equation} \label{rfdp} s=p^{2}=E^{2},\ \ p=(E,\vec{0}).% \end{equation} An important element for our discussion is the vertex (or cutoff) function $f_{\Lambda}(\xi)$ entering in Eq.~\eqref{se}. This is a form-factor that is needed to account for the unknown black vertices in Fig.~\ref{diag2}, due to the fact that mesons are not elementary particles \cite{prc76p065204}. The cutoff function must ensure the convergence of the integral. We choose the following Gaussian form \begin{equation} f_{\Lambda}(\xi)=e^{-\xi/(4\Lambda^{2})}\times e^{\left( m_{D^{0}}^{2}+m_{D^{+}% }^{2}\right) /\left( 2\Lambda^{2}\right) },\label{cutoff}% \end{equation} where the second term is built up for convenience. In the isospin limit, $f_{\Lambda}$ reduces to $e^{-{\mathbf q}^{2}/\Lambda^{2}}$. The parameter $\Lambda$, introduced in Eq.~\eqref{cutoff}, is on the same level of all the other parameters of our model and will be evaluated through our fit to data. Moreover, even if our form factor depends on the three-momentum $\mathbf{q}$ only, hence strictly valid in the rest frame of the decaying particle, covariance can be recovered by properly generalizing the vertex function, see details in Ref.~\cite{apps9p467} \footnote{One can formally introduce the vertex function already at the Lagrangian level by rendering it nonlocal, see the discussion in Refs.~\cite{npa582p655,prd68p014011,prc71p025202,prc76p065204}, where it is also pointed out that the special form of the form factor is not important, as long as fast convergence is guaranteed.}. However, the exponential form for the cutoff is very typical and has been used in many different approaches in hadron physics, e.g. Ref.~\cite{prd68p014011}. For completeness, in \ref{C} we shall also present the results for a different vertex function. We now turn back to the study of the propagator. In \ref{B}, it is shown that the relevant quantity is the transverse part of the propagator that, in the rest frame of the decaying particle, and for a single loop, reads% \begin{align} \label{loopint} &\Sigma(s,m) =\frac{1}{3}\left( -g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^2% }\right) \Sigma_{\mu\nu}(p,m)\nonumber\\ & =-\frac{i}{3}\int_q\frac{4\ \mathbf{q}^{2}\ f_{\Lambda }^{2}(\xi)}{[(q+\frac{p}{2})^{2}-m^{2}+i\varepsilon][(q-\frac{p}{2})^{2}-m^{2}+i\varepsilon ]}\text{,}% \end{align} and similarly for the sum% \begin{equation} \Pi(s)=\frac{1}{3}\left( -g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^{2}}\right) \Pi_{\mu\nu}(p,m)\text{ } =g_{\psi D\bar{D}}^2\{\Sigma(s,m_{D^{0}})+\Sigma(s,m_{D^{+}})\}. \end{equation} The scalar part of the one-loop resummed dressed propagator reads% \begin{equation} \Delta(s)=\frac{1}{s-m_{\psi}^{2}+\Pi(s)}. \end{equation} Note, this expression of the propagator fulfills unitarity (cf.~Sec.~\ref{model.4} below) and it is accurate as long as further contributions to the self-energy are small \footnote{ The next contribution would be represented by a loop in which the unstable state $\psi$ would be exchanged by $D$ mesons circulating in the loop. Such contributions are typically very small in hadron physics, as shown for instance in Ref.~\cite{npb888p287}. In addition, one could include also other channels, such as many other possible but suppressed decays of the $\psi(3770)$ (e.g., $J/\psi\pi^{+}\pi^{-}$, etc...), as well as subthreshold channels such as $D^{\ast}\bar{D}+h.c.$ that can contribute to the real part of the propagator.}. The self-energy in Eq.~\eqref{se} can either be computed through the integration (as in \ref{B}), or using dispersion relations. We follow the latter. To this end, we first decompose $\Sigma(s,m)$ into its real and imaginary parts \begin{equation} \Sigma(s,m)=R(s,m)+iI(s,m),\ \ R,\ I\in\Re.\label{loop}% \end{equation} According to the optical theorem, the imaginary part $I(s,m)$ (dispersive term) is given by \begin{equation} I(s,m)=\frac{k(s,m)}{8\pi \sqrt{s}\ g_{\psiD\bar{D}}^2}|\mathcal{M}_{\psi\rightarrow D\bar{D}}|^{2},\label{I} \end{equation} with $k(s,m)=\sqrt{s/4-m^{2}}$ being the center-of-mass momentum of the final mesons. The partial decay widths of $\psi\rightarrow D^0\bar{D}^0$ and $\psi\rightarrow D^+D^-$ are then calculated as% \begin{equation} \Gamma_{\psi\rightarrow D^0\bar{D}^0}(s)=\frac{k(s,m_{D^{0}})}{8\pi s}|\mathcal{M}_{\psi\rightarrow D^0\bar{D}^0}|^{2}\\ =g_{\psiD\bar{D}}^{2}\frac{I(s,m_{D^{0}})}{\sqrt{s}},% \end{equation}% \begin{equation} \Gamma_{\psi\rightarrow D^+D^-}(s)=\frac{k(s,m_{D^{+}})}{8\pi s}|\mathcal{M}_{\psi\rightarrow D^+D^-}|^{2}\\ =g_{\psi D\bar{D}}^{2}\frac{I(s,m_{D^{+}})}{\sqrt{s}},% \end{equation} where the Lorentz invariant amplitudes squared, computed from $\mathcal{L}% _{I}$ in Eq.~\eqref{lagi}, are given by \begin{align} &|\mathcal{M}_{\psi\to D^0\bar{D}^0}|^{2}=g_{\psiD\bar{D}}^{2}\frac{4}{3}k^{2}(s,m_{D^{0}})f_{\Lambda}^{2}(s)\ ,\label{pamp1}\\ &|\mathcal{M}_{\psi\to D^+D^-}|^{2}=g_{\psi D\bar{D}}% ^{2}\frac{4}{3}k^{2}(s,m_{D^{+}})f_{\Lambda}^{2}(s)\text{ .}% \label{pamp2} \end{align} Note, by choosing the form factor as function of the energy [$f_{\Lambda }(\xi =4(\mathbf{q}^{2}+m^{2}))$], when the imaginary part of the loop in Eq.~\eqref{loopint} is taken, the replacement $\xi \rightarrow s$ is performed (see Eqs.~\eqref{pamp1} and \eqref{pamp2}). Then, the function $f_{\Lambda }(s)$ directly enters in various expressions. Finally, the on-shell partial and total decay widths are \begin{align} \label{osw1} &\Gamma_{\psi\rightarrow D^0\bar{D}^0}^{\text{on-shell}}=\Gamma _{\psi\rightarrow D^0\bar{D}^0}(m_{\psi}^{2})\ ,\\ \label{osw2} &\Gamma_{\psi\rightarrow D^+D^-}^{\text{on-shell}}=\Gamma_{\psi\rightarrow D^+D^-}(m_{\psi}^{2})\ , \\ \label{oswt} &\Gamma_{\psi\to D\bar{D}}^{\text{on-shell}}=\Gamma_{\psi\rightarrow D^0\bar{D}^0}^{\text{on-shell}}+\Gamma_{\psi\rightarrow D^+D^-}^{\text{on-shell}}\text{ }. \end{align} Once the imaginary part of the loop is known, the real part, the function $R(s,m)$, is computed from the dispersion relation \begin{equation} R(s,m)=\frac{PP}{\pi}\int_{s_{th}=4m^2}^{\infty}\frac{I(s^{\prime},m)}% {s^{\prime}-s}\ \mathrm{d}s^{\prime},\label{dr}% \end{equation} where $I(s^{\prime},m)$ is zero below threshold. Convergence is guaranteed by the cutoff function. As we shall see in Sec.~\ref{results}, the real part $R(s,m)$ causes a distortion in the line-shape due to the continuous shifting of the physical mass of the resonance with the energy. Note, when the energy $E=z$ is complex, the function $\Sigma(s=z^{2},m)$ reads (away from the real axis):% \begin{equation} \Sigma(z^{2},m)=\frac{1}{\pi}\int_{s_{th}=4m^2}^{\infty}\frac{I(s^{\prime}% ,m)}{s^{\prime}-z^{2}}\ \mathrm{d}s^{\prime}. \label{eq21} \end{equation} This formula clearly shows that in the first Riemann Sheet (RS) the complex function $\Sigma(z^{2},m)$ is regular everywhere on the $z^{2}$-complex plane, besides a cut from $4m^{2}$ to infinity. In particular, $\Sigma(z^{2}\rightarrow\infty,m)\to 0$ in all directions \footnote{In order to avoid misunderstanding, we recall that, in the 1st Riemann Sheet, the function $\Sigma (z^{2},m)$ is an utterly different complex function than $f_{\Lambda}^{2}(z^{2})\propto e^{-z^{2}/2\Lambda^{2}}$. Namely, Eq.~\eqref{eq21} implies solely that $\operatorname{Im}\Sigma(x^{2},m)$ contains $e^{-x^{2}/2\Lambda^{2}}$ for $z^{2}=x^{2}$ being real. This fact is also clear by noticing that, while $\Sigma(z^{2}\rightarrow\infty,m)\to 0$ in any direction, $f_{\Lambda}^{2}(z^{2}\rightarrow\infty)$ diverges for $\operatorname{Re}\ z^{2}<0$.}. Furthermore, although it is not strictly needed, since the integral in Eq.~\eqref{dr} is already convergent, we use the once-subtracted dispersion relation, with subtraction in point $m_{\psi}^{2}$, for our convenience. Hence, the total loop-function is given by \begin{equation} \Pi_1(s)=\Pi(s)-R(m_{\psi}^{2},m_{D^{0}})-R(m_{\psi}% ^{2},m_{D^+})\text{ ,}\label{1sub1}% \end{equation} and, the final dressed propagator of the $\psi$ meson is% \begin{equation} \Delta(s)=\frac{1}{s-m_{\psi}^{2}+\Pi_1(s)}\text{ }% .\label{propfinal}% \end{equation} In this way, the parameter $m_{\psi}$ in the propagator corresponds to the mass of the particle defined as% \begin{equation} \operatorname{Re}\ \Delta_{\psi}(s)^{-1}=0\rightarrow s=m_{\psi}^{2}% \text{.}\label{mass1}% \end{equation} Other definitions for the mass are possible, such as the position of the peak, or the real part of the pole, as we shall see below. \subsection{\label{model.2} Poles} In order to find poles, the energy $E$ is analytically continued to the complex plane ($E\rightarrow z$). In case of two decay channels, the Riemann surface is composed by four RSs. Poles are found in the unphysical sheet which results in the physical sheet when the energy is real and the resonance can be seen. Above both thresholds, this corresponds to the condition $\mathrm{Im}\ k_{D^0}<0$ and $\mathrm{Im}\ k_{D^+}<0$ when $\mathrm{Re}\ E>2m_{D^0}$ and $\mathrm{Re}\ E>2m_{D^+}$, i.e., the third RS. Poles are given when the denominator of the propagator in Eq.~\eqref{propfinal} is zero in the correct RS, i.e., \begin{equation} z^{2}-m_{\psi}^{2}+\Pi_{1, III}(z^2)=0,\ z\in\mathbb{C}, \end{equation} where \begin{equation} \Pi_{1, III}(z^2)=\Pi_{1,I}(z^{2}) -2ig_{\psi D\bar{D}}^2[I_{I}(z^{2},m_{D^{0}})+I_{I}(z^{2},m_{D^{+}})]=0\ , \end{equation} where the subscripts $I$ and $III$ stand for the first and third RS \footnote{ In other terms, $z_{pole}$ corresponds to $\sqrt{s_{pole}},$ see Eq.~\eqref{rfdp}.}. While in the first RS $\Pi_{1}(z^{2})$ is regular everywhere apart from the cut(s) on the real axis (see Eq.~\eqref{eq21} and subsequent discussion), in the other RSs the imaginary part continued to complex plane, $I(z^{2},m),$ appears. As a consequence, the vertex function $f_{\Lambda}% ^{2}(z^{2})\propto e^{-z^{2}/2\Lambda^{2}}$ generates a singular point at the complex infinity. When other choices for the vertex function are made, such as the one in \ref{C}, other singularity types are generated. In general, any nontrivial function $f_{\Lambda}^{2}(z^{2})$ will display singularities appearing in RSs different from the first one. \subsection{\label{model.3}Coupling to leptons} The available experimental data comes from the production process $e^{+}e^{-}\rightarrow\psi\rightarrow D\bar{D}$ (see Fig.~\ref{diag1}), therefore we need to couple the state $\psi(3770)$ to leptons. The corresponding interaction Lagrangian is defined by \begin{equation} \mathcal{L}_{\psi l^{+}l^{-}}=g_{\psi e^{+}e^{-}}\ \psi_{\mu}\sum_{l=e% ,\mu,\tau}\bar{\Psi}_{l}\gamma^{\mu}\Psi_{l},\label{lagii}% \end{equation} which is the simplest interaction among a massive vector field $\psi$ and a fermion ($\Psi$)-antifermion pair. The coupling $g_{\psi e^{+}e^{-}}$, between $\psi$ and the electron-positron pair, is here considered to be the same between $\psi$ and all leptonic pairs. It is the overall strength for the annihilation of the leptonic pair $l^+l^-$ into one photon, and further conversion into the $\psi$ vector. In Eq.~\eqref{lagii}, we describe the process $e^{+}e^{-}% \rightarrow\psi$ through a single effective vertex proportional to $g_{\psi e^{+}e^{-}}$, but, as shown in Fig.~\ref{diag4}, this coupling constant emerges via an intermediate virtual photon that converts into a charmonium state. \begin{figure}[ptb] \centering \begin{tabular} [c]{c}% {\small\hspace*{-4cm}$e^+$\hspace{3.2cm}$e^+$}\\[2mm] \hspace*{.9cm}$\psi$\hspace*{3.2cm}$\gamma$\hspace*{.9cm}$\psi$\\[-6mm] \hspace*{-4mm}\resizebox{!}{55pt}{\includegraphics{vertexll.pdf}}\\[-1mm {\small\hspace*{-4cm}$e^-$\hspace{3.2cm}$e^-$}\\[1mm] \end{tabular} \caption{\label{diag4} Interaction vertex $e^{-}e^{+}\to\psi(3770)$.}% \label{diag4}% \end{figure} \begin{figure}[ptb] \centering \begin{tabular}[c]{c}% {\small\hspace*{0cm}$e^-$\hspace*{3.5cm}$D$}\\ \resizebox{!}{40pt}{\includegraphics{diagDD.pdf}}\\[-1mm]% {\small\hspace*{0cm}$e^+$\hspace*{3.5cm}$\bar{D}$}\\[1mm] \end{tabular} \caption{\label{diag1}Reaction $e^{-}e^{+}\rightarrow\psi(3770)\rightarrow D\bar{D}$ with $D^0\bar{D}^0+D^+D^-$ loops (cf.~Figs.~\ref{diag2}, \ref{diag3} and \ref{diag4}).}% \end{figure} The corresponding decay into leptons reads \begin{equation} \Gamma_{\psi\rightarrow l^{+}l^{-}}(m_\psi^2)=\frac{k(m_\psi^2,m_l)}{8\pi m_\psi^2}\frac{4}% {3}\Big(m_\psi^2+2m_{l}^{2}\Big)g_{\psi e^{+}e^{-}}^{2}, \label{gamll}% \end{equation} where $m_{l}$ is the leptonic mass. In principle, the loops of leptons should be included in the total one-loop function of resonance $\psi(3770)$, obtaining $\Pi_{tot}% (s)=\Pi(s)+\sum_{l}\Pi_{l}(s)+\cdots$, where dots refer to other possible but suppressed hadronic decay channels. However, the loop contribution of $\Pi_{l}(s)$ is definitely negligible w.r.t. $\Pi(s).$ Here, we shall simply consider $\Pi_{tot}% (s)=\Pi(s).$ \subsection{\label{model.4}Spectral function and cross section} The unitarized spectral function as a function of energy $E$, equivalent to the running mass of the $\psi(3770)$ resonance, is given by \begin{equation} \begin{split} &d_{\psi}(E) =-\frac{2E}{\pi}\mathrm{Im}\ \Delta(E)\\ & =\frac{2E^{2}}{\pi}\frac{\Gamma_{\psi\rightarrow D^0\bar{D}^0}% (E^{2})+\Gamma_{\psi\rightarrow D^+D^-}(E^{2})}{[E^{2}-m_{\psi}% ^{2}+\operatorname{Re}\Pi_{1}(E^{2})]^{2}+[\operatorname{Im}\Pi _{1}(E^{2})]^2}\text{ }, \end{split} \end{equation} that has the general shape of a relativistic Breit-Wigner distribution, distorted by the loop-function $\Pi_1$ (cf.~Eq.~\eqref{1sub1}). When no poles below threshold emerge, the normalization above threshold is guaranteed (unitarity):% \begin{equation} \label{unit} \int_{2m_{D^{0}}}^{\infty}d_{\psi}(E)dE=1\text{ .}% \end{equation} The quantity $d_{\psi}(E)dE$ is interpreted as the probability that the state $\psi(3770)$ has a mass between $E$ and $E+dE.$ The cross section for $e^{+}e^{-}\rightarrow D\bar{D}=D^+D^-+D^0\bar{D}^0$ takes the form \begin{equation} \sigma_{e^{+}e^{-}\rightarrow D\bar{D}}=\frac{\pi}{2E}g_{\psi e^{+}e^{-}}% ^{2}d_{\psi}(E)=-g_{\psi e^{+}e^{-}}^{2}\mathrm{Im}\Delta(E)\text{ .}\label{cs}% \end{equation} Hence, the experimental data for this cross section give us direct access to the imaginary part of the propagator of the meson $\psi(3770)$. The corresponding amplitude, leading to $\sigma_{e^{+}e^{-}\rightarrow D\bar{D}}$, is depicted in Fig.~\ref{diag1}. One also defines the partial spectral functions as \cite{fp42p1262}:% \begin{align} &d_{\psi\rightarrow D^+D^-}(E)=\frac{2E^{2}}{\pi}\frac{\Gamma_{\psi\rightarrow D^+D^-}(E^{2})} {[E^{2}-m_{\psi}^{2}+\operatorname{Re}\Pi_{1}(E^{2})]^{2}+[\operatorname{Im}\Pi_{1}(E^{2}% )]^2}\ ,\\ &d_{\psi\rightarrow D^0\bar{D}^0}(E)=\frac{2E^{2}}{\pi} \frac{\Gamma_{\psi\rightarrow D^0\bar{D}^0}(E^{2})}{[E^{2}-m_{\psi}% ^{2}+\operatorname{Re}\Pi_{1}(E^{2})]^{2}+[\operatorname{Im}\Pi _{1}(E^{2})]^2}\ .% \end{align} Then, the partial cross sections are given by:% \begin{align} &\sigma_{e^{+}e^{-}\rightarrow D^+D^-} =\frac{\pi}{2E}g_{\psi e^{+}e^{-}}^{2}d_{\psi\rightarrow D^+D^-}(E)\text{ ,}\\ &\sigma_{e^{+}e^{-}\rightarrow D^0\bar{D}^0} =\frac{\pi}{2E}g_{\psi e^{+}e^{-}}^{2}d_{\psi\rightarrow D^0\bar{D}^0}(E)\text{ ,}\\ &\sigma_{e^{+}e^{-}\rightarrow D\bar{D}}=\sigma _{e^{+}e^{-}\rightarrow D^+D^-}+\sigma_{e^{+}e^{-}\rightarrow D^{0}\bar {D}^{0}}\ . \end{align} \section{\label{results} Line-shapes and poles} In this section, we present our fit of Eq.~\eqref{cs} to experimental data and its consequences, and the notable existence, position, and trajectory of two poles underlying the $\psi(3770)$. \subsection{Fit to data and consequences} As a first necessary step, we determine the four free parameters of our model% \begin{equation} \label{para} \{g_{\psiD\bar{D}},\text{ }m_{\psi},\text{ }\Lambda,\text{ }g_{\psi e^{+}e^{-}}\}\text{ ,}% \end{equation} defined in Sec.~\ref{model}, by performing a fit to the cross section data for the process $e^{+}e^{-}% \rightarrow\psi\rightarrow D\bar{D}=D^0\bar{D}^0+D^+D^-$. We use 14 experimental points published in Ref.~\cite{plb668p263}, and the theoretical expression in Eq.~\eqref{cs}. Note, we use only the data of Ref.~\cite{plb668p263} for the fit, since the data of Ref.~\cite{prl97p121801} are contained in the data samples of Ref.~\cite{plb668p263} (the data sets of March 2001 quoted in these two papers are the same \cite{pc}). The fit to data is shown in Fig.~\ref{fitsum}, and the values of the parameters in the set \eqref{para} are presented in Table \ref{fitpar}. We get the value $\chi^{2}/d.o.f.\simeq 1.03$, which shows that a very good description of the data is achieved. The errors of the parameters entering the fit are estimated by the square root of the corresponding Hessian matrix (this is a result of the standard procedure according to in which one defines a new set of parameters that diagonalize the Hessian matrix, see e.g.~Ref.~\cite{prd96p054033} for details). There are various consequences of the fit that we discuss in detail:\\ \begin{figure}[ptb] \begin{center}% \begin{tabular} [c]{c}% \resizebox{!}{200pt}{\includegraphics{fitsDD.pdf}} \end{tabular} \end{center} \caption{Data: $\bullet$ BES \cite{plb668p263}, $\ast$ BES \cite{prl97p121801}% , $\star$ BaBar \cite{prd76p111105}. Solid line: our fit to data in Ref.~\cite{plb668p263} (cf.~Table \ref{fitpar}).}% \label{fitsum}% \end{figure} \begin{table} \begin{center} \begin{tabular}{c c} \hline $m_\psi$ (MeV)&$3773.19\pm 0.46$\\[1mm] $\Lambda$ (MeV)&$272.3\pm 2.3$\\[1mm] $g_{\psiD\bar{D}}$&$30.73\pm 0.58$\\[1mm] $g_{\psi e^+e^-}$&$(1.055\pm 0.015)\times 10^{-3}$\\[1mm] $\chi^2$&$10.28$\\[1mm] $\chi^2/d.o.f$&$1.028$\\ \hline \end{tabular} \caption{\label{fitpar} Fitting parameters.} \end{center} \end{table} \begin{figure}[ptb] \begin{center}% \begin{tabular} [c]{c}% \resizebox{!}{200pt}{\includegraphics{fitsDpDm.pdf}}\\[-10mm]% \resizebox{!}{200pt}{\includegraphics{fitsD0D0.pdf}}\\[0mm]% \end{tabular} \end{center} \caption{Data: $\bullet$ BES \cite{plb668p263}, $\ast$ BES \cite{prl97p121801}% , $\star$ BaBar \cite{prd76p111105}. Solid line: our result, using the parameters in Table \ref{fitpar}.}% \label{fit}% \end{figure} \begin{table} \begin{center} \begin{tabular}{ccc} \hline &$\Gamma(m_\psi^2)$ &$\Gamma^{\mathrm{average}}$\\[1mm] \hline $\Gamma_{D^0\bar{D}^0}$ (MeV)&$11.5\pm 0.4$&$10.87$\\[1mm] $\Gamma_{D^+D^-}$ (MeV)&$8.0\pm 0.3$&$6.95$\\[1mm] $\Gamma_{D\bar{D}}$ (MeV)&$19.5\pm 0.7$&$17.83$\\[1mm] $\Gamma_{D^0\bar{D}^0}/\Gamma_{D^+D^-}$&$1.44$&$1.57$\\[1mm] $\Gamma_{D^0\bar{D}^0}/\Gamma_{D\bar{D}}$&$0.59$&$0.61$\\[1mm] $\Gamma_{D^+D^-}/\Gamma_{D\bar{D}}$&$0.41$&$0.39$\\[1mm] \hline \end{tabular} \caption{\label{widths} Decay widths and branching ratios using Eqs.~\eqref{osw1}-\eqref{oswt}, for the on-shell widths, and Eqs.~\eqref{avw1}-\eqref{avw3}, for the average widths.} \end{center} \end{table} a) The value of the mass $m_{\psi}=$ $3773.19\pm 0.46$ MeV (corresponding to the zero of the real part of the inverse's propagator, see Eq. (\ref{mass1})) is in well agreement with the current PDG fit of $3773.13\pm0.35$ MeV \cite{pdg}. \bigskip b) In Fig.~\ref{fit} we present a comparison between theory and the data for the cross sections $\sigma_{e^{+}e^{-}\rightarrow D^+D^-}$ and $\sigma_{e^{+}% e^{-}\rightarrow D^0\bar{D}^0}$, using the parameters in Table \ref{fitpar}. The good agreement shows that the theory is able to describe the two partial cross sections separately, without the need of some extra parameter, and in agreement with isospin symmetry. \bigskip c) The branching ratios and partial and total ($D\bar{D}$) widths are presented in Table \ref{widths}. The partial widths, evaluated on-shell, are defined in Eqs.~\eqref{osw1}-\eqref{oswt}. The branching ratios are in agreement with those quoted in PDG for the $D\bar{D}$. The ratio $\Gamma_{\psi\rightarrow D^0\bar{D}^0}^{\text{on-shell}}/\Gamma_{\psi\rightarrow D^+D^-}^{\text{on-shell}}$ of $1.44$ is compatible with the PDG values, which range from $1.04$ to $1.51$. However, as shown in Fig.~\ref{gam}, the functions $\Gamma_{\psi\rightarrow D^0\bar{D}^0% }(E^2)$ and $\Gamma_{\psi\rightarrow D^+D^-}(E^2)$ strongly vary in the region of interest. Alternatively, one may use a different definition, cf.~\cite{prc76p065204}, to obtain the partial decay widths and the full decay width to $D\bar{D}$, by integrating over the spectral function as \begin{align} &\Gamma_{\psi\rightarrow D^0\bar{D}^0}^{\text{average}}=\int _{2m_{D^{0}}}^{\infty}\Gamma_{\psi\rightarrow D^0\bar{D}^0}(E^{2})\ d_{\psi }(E)\ dE\ ,\label{avw1}\\ &\Gamma_{\psi\rightarrow D^+D^-}^{\text{average}}=\int_{2m_{D^{+}}% }^{\infty}\Gamma_{\psi\rightarrow D^+D^-}(E^{2})\ d_{\psi}(E)\ dE\ ,\label{avw2}\\% &\Gamma_{\psi\toD\bar{D}}^{\text{average}}=\Gamma_{\psi\rightarrow D^0\bar{D}^0% }^{\text{average}}+\Gamma_{\psi\rightarrow D^+D^-}^{\text{average}}\label{avw3}\ . \end{align} \begin{figure}[ptb] \begin{center} \resizebox{!}{200pt}{\includegraphics{gama.pdf}}\\[5mm] \end{center} \caption{Partial widths using the cutoff function in Eq.~\eqref{cutoff}, using parameters in Table \ref{fitpar}. The width $\Gamma_{D\bar{D}}$ is the sum of the width in each channel.}% \label{gam}% \end{figure} A somewhat naive but still useful alternative determination of the mass is the value corresponding to the peak and the half-height width:% \begin{equation} \begin{split} &m_{\psi}^{\text{peak}}\simeq 3774.3\pm 1.0\text{ MeV, }\\ &\Gamma_{\psi\to D\bar{D}}^{\text{half-height}}\simeq 21.5\pm 1.0\text{ MeV ,} \end{split} \end{equation} where we estimated the error of the peak mass as $\sim 1$ MeV, close to the determined error for the parameter $m_{\psi}$, and the error of the distribution width as $\sim 1$ MeV as well, coincident to the error for $\Gamma_{D\bar{D}}$ (see Table \ref{widths}). These evaluations show that the resonance $\psi(3770)$ is far from being an ideal Breit-Wigner. While all these different approaches coincide for states with a very small width, here the distortions are sizable, hence clear definitions of mass and width, as well as branching ratios, are difficult. A commonly used approach to get a uniform result, makes use of poles in the complex plane, as we show in Sec.~\ref{results.2}. This discussion shows that there is no unique definition of the total and the partial widths. This fact also renders a direct comparison with the fit of the PDG quite difficult. Namely, according to PDG, one has $\Gamma _{\psi (3770)\rightarrow D^{0}\bar{D}% ^{0}}^{PDG}=14.1_{-1.5}^{+1.2}$ MeV, $\Gamma _{\psi (3770)\rightarrow D^{+}D^{-}}^{PDG}=11.2_{-1.7}^{+1.7}$ MeV, and $\Gamma _{\psi (3770)\rightarrow DD}^{PDG}=25.2_{-2.6}^{+2.1}$ MeV \cite{pdg}, hence our results for the partial decay widths evaluated on shell and presented in Table \ref{widths}, first column, are somewhat smaller. However, as visible in Fig.~\ref{gam}, the corresponding function vary strongly in the energy region of interest. Moreover, instead of using theoretical partial widths which cannot be uniquely defined, one should stress that our approach correctly describes the data for channels $D^{0}\bar{D}^{0}$ and $% D^{+}D^{-}$ separately, as visible in Fig.~\ref{fit}, and as described in point b) above. Another commonly used approach to circumvent all these definition problems mentioned above, is to move away from the real axis and to study the pole(s) in the complex plane. As we shall see in detail in Sec.~\ref{results.2}, the seed pole of $\psi (3770)$ corresponds to a larger decay width of $24.6$ MeV which is in very well agreement with PDG. \bigskip d) Using the coupling $g_{\psi e^{+}e^{-}}$ that outcomes from the fit, the widths $\Gamma_{\psi\rightarrow l^{+}l^{-}}$ can be easily computed from Eq.~\eqref{gamll}. Results are shown in Table \ref{lw}. The value for $\Gamma_{\psi\rightarrow e^{+}e^{-}}$ is smaller than the one given by the PDG fit of $262\pm 18$ eV by a factor two, but it is compatible with the analysis in Ref.~\cite{plb711p292}, that gives $154^{+79+21}_{-58-27}$ eV. The mismatch of our result with the PDG estimate, that nevertheless lists a quite broad range of values from different experiments, shows that this decay rate should be further investigated in the future. Also the experimental identification of the decays into $\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$ could help to distinguish among different models. \begin{table} \centering \begin{tabular}{ccc} \hline $\Gamma_{e^+e^-}$ (eV)&$\Gamma_{\mu^+\mu^-}$ (eV)&$\Gamma_{\tau^+\tau^-}$ (eV)\\[1mm] \hline $111.4\pm 3.3$&$111.4\pm 3.3$&$54.0\pm $\\[1mm] \hline \end{tabular} \caption{\label{lw}Leptonic widths computed with Eq.~\eqref{gamll}.} \end{table} \bigskip e) The Gaussian vertex function together with the cutoff value $\Lambda =272.3\pm 2.3$ MeV takes into account in a simple way the composite nature of the resonance $\psi(3770)$ and its nonlocal interaction with the $D$ mesons. Microscopically the vertex function emerges from the nonlocal nature of both $\psi (3770)$ and of the decay products $D$ (for point-like part interaction, $\Lambda $ goes to infinity), see Refs.~\cite{prd68p014011,0903.3905} for an explicit treatment. (Indeed, the rather small value of $\Lambda $ emerging from our fit is intuitively understandable by the fact that the resonance $\psi (3770)$ is predominantly a $D$ wave. A detailed study of this issue by using a microscopic model such as the one in Ref.~\cite{prd94p014016} will be given elsewhere). The vertex function is a crucial ingredient that defines our model and, among other properties, guarantees the finiteness of the results. The exponential form used here emerges naturally from various microscopic approaches, see Refs.~\cite{prd94p014016,prd53p295} and refs.~therein. However, our results do not strongly depend on the precise choice of the vertex function as long as it is smooth and at the same time falls sufficiently fast. As we show in \ref{C}, a hard cutoff would imply that the spectral function would fall abruptly to zero above a certain threshold; this unphysical behavior does not lead to any satisfactory description of data. On the other hand, as discussed in \ref{D}, the avoidance of a form factor through a three-times subtracted dispersion relation does not lead to satisfactory results, thus confirming that the need of some vertex function for our treatment of mesonic loops is needed. Once the vertex function is fixed, the model is mathematically consistent at any energy: in fact, the important normalization condition reported in Eq.~\eqref{unit} is obtained by formally integrating up to infinity (numerically, the normalization is verified by integrating up to $10$ GeV). In turn, this means that, from a mathematical perspective, the value of the momentum of the emitted mesons $D$ can be larger than $\Lambda .$ A different issue is the maximal energy up to which we shall trust our model. In fact, even if mathematically consistent, the model is limited because it takes into account only one vector resonance and neglects the contributions of other resonances, most notably $\psi (2S),$ as well as the background (see the comments h) and i) about these topics) and the next opening thresholds, such as $D\bar{D}^{\ast }$. We therefore trust the model in the energy range starting from the lowest threshold, $2m_{D^{0}}$, up to at most $3.8$ GeV, when the line-shape goes down and, as described in Refs.~\cite{prd86p114013,prd87p057502}, the role of the background becomes important. \bigskip f) In our fit, we consider only the $\psi(3770)$ resonance as a virtual state of the process $e^{+}e^{-}\rightarrow D\bar{D}$. As discussed above, a possible mixing between bare $c\bar{c}$ states with quantum numbers $1\ ^{3}D_{1}$ and $2\ ^{3}S_{1}$ is automatically taken into account in our bare field $\psi$ entering into the Lagrangian defined in Eq.~\eqref{lagi}. Indeed, the bare field, i.e., the field prior to the dressing by mesonic loops, represents the diagonal quark-antiquark state, while the $\psi(3770)$ is a mixture of $1\ ^{3}D_{1}$ and $2\ ^{3}S_{1}$ configurations, with the $\psi(2S)$ as its orthogonal state. Yet, the role of the $\psi(2S)$ as an additional exchange in the reaction $e^{+}e^{-}\rightarrow\psi(2S)\rightarrow D\bar{D}$ was not taken into account. Being $\psi(2S)$ off-shell in the energy of interest, the propagator is expected to suppress this amplitude; moreover, also its coupling to channel $D\bar{D}$ is expected to be small, due to the form-factor. Indeed, in Refs.~\cite{plb718p1369,prd81p034011}, the $\psi(2S)$, $D\bar{D}$ loops, and $D\bar{D}$ rescattering have been considered, leading to fits that are in good agreement with the experiment in Ref.~\cite{plb668p263}. In both references the $\psi(2S)$ contribution is quite small, in agreement with our results. However in their case the $\psi(2S)$ was still necessary to obtain a good fit, whereas in our model it is not. Note, in Refs.~\cite{plb718p1369,prd81p034011} the pole structure was not examined. \bigskip {g) In this work we did not consider a four-leg interaction process of the type $D\bar{D}\rightarrow D\bar{D},$ that can be formally described by a four-vertex proportional to $(\partial_{\mu}D\bar{D}-\partial_{\mu}\bar {D}D)^{2}$. Such a rescattering describes elastic scattering, induced by a direct four-body interaction, and also as an effective description of various $t$-channel exchanges between $D\bar{D}$. Our model can describe properly the data without this contribution, hence the inclusion of a four-leg interaction is not needed to improve the agreement with the data. This does not necessarily mean that its magnitude is small, but simply that one can hardly disentangle its role from the $D\bar{D}$-loops that we have considered in our approach. In addition, there is a more formal point that should be taken into account: one can in principle redefine the field $\psi_{\mu}\rightarrow\psi_{\mu }+\alpha(\partial_{\mu}D\bar{D}-\partial_{\mu}\bar{D}D),$ where $\alpha$ is a parameter on which no physical quantity should depend on \cite{pr130p776,epja44p93}. The redefinition would however lead to some rescattering terms. Of course, the independence on the field redefinition can be fully accomplished only if one could solve the theory exactly, since in any approximation scheme differences would still persist at a given order. In conclusion, the study of a four-leg interaction term (as well as other effects such as the mixing with $\psi(2S)$) should be performed when more precise data will be available. \bigskip h) Here, we discuss more closely the differences between our work and the one in Ref.~\cite{plb718p1369}. The main difference is the treatment of the propagator of the $\psi(3770)$ meson, in particular for what concerns the self-energy. In Ref.~\cite{plb718p1369} the imaginary part of the self-energy (i.e., the energy-dependent decay width) grows indefinitely with increasing $s.$ In this way, the K\"{a}ll\'{e}n--Lehmann representation does not lead to a normalized spectral function. Moreover, the real part of the self-energy depends on an additional subtraction constant $\mu$ (linked to the regularization of the loop in Ref.~\cite{plb718p1369}, hence ultimately corresponding to our cutoff $\Lambda$).\ When $\mu$ is set to zero (their so-called minimal subtraction scheme) the real part of the self energy simply vanishes (hence, it is trivial). When $\mu$ is different from zero, the real part also grows indefinitely for increasing $s.$ Indeed, in the approach of Ref.~\cite{plb718p1369}, the propagator of $\psi(3770)$ is not enough to describe data, also when the rescattering is present; the inclusion of the $\psi(2S)$ state is necessary. Hence, our approach is different under many aspects: since it contains a fully consistent treatment of $D\bar{D}$ loops, the dispersion relations are fulfilled and the normalization of the spectral function is naturally obtained. In this way, nontrivial effects which modify the form of the spectral function automatically arise. Moreover, in our case neither the $\psi(2S)$ nor the rescattering are needed to describe the data, hence the physical picture of the two models is rather different. In conclusion, the fit in Ref.~\cite{plb718p1369} is performed with six parameters (repeated for different values of the subtraction constant $\mu$, for a total of seven), while we use only 4. Both approaches can describe data: in the future, better data are needed to understand which effects are more important. Our work suggests that the nearby $\bar{D}D$ threshold and $\bar{D}D$ loops are necessary to correctly understand the $\psi(3770)$. Furthermore, the presence of two poles is a stable prediction of our approach, see Secs.~\ref{results.2} and \ref{results.3} below. \bigskip i) As a last comment, we compare our result with the approach of Refs.~\cite{prd86p114013,prd87p057502}. Interestingly, in Ref.~\cite{prd87p057502} it is shown that a suppression of the cross-section at about $3.81$ GeV is obtained by a destructive interference of the resonance $\psi (3770)$ with the background. This outcome is useful, since it sets a physical limit for our approach, in which no background is present: our description of data cannot be expected above this upper limit (and indeed we perform the fit below it). However, for what concerns the treatment of $\psi (3770)$ our model is quite different: in Refs.~\cite{prd86p114013,prd87p057502} rescattering is taken into account. When their rescattering parameter is set to zero, the real part of the $D\bar{D}$ loop vanishes. In contrast, in our approach, even without an explicit rescattering term, the $D% \bar{D}$ loops are taken into account in a way that guarantees the unitarity expressed by Eq.~\eqref{unit} (its fulfillment is one of the most relevant technical aspects of our approach) and they are found to be important for a proper description of the $\psi (3770)$. \subsection{\label{results.2} Pole positions} As renowned, a theoretically (and lately experimentally as well, see e.g. the $f_0(500)$ in PDG) stable approach to describe unstable states is based on the determination of the position of the corresponding poles. In the present case, one obtains, quite remarkably, two poles on the III Riemann Sheet. As seen in Sec.~\ref{model.2}, this is the sheet relevant for a system with two decay channels at energy above both thresholds. In the isospin limit, the III RS reduces to the usual II RS. In the following, the indeterminacy of the pole masses is estimated to be $\sim 1.0$ MeV, close to the error obtained for the parameter $m_{\psi}$, and the indeterminacy of the pole width(s) as $\sim 1.0$ MeV, coincident with the error for $\Gamma_{D\bar{D}}$ (see Table \ref{widths}). In fact, the errors of these strictly correlated quantities must have a similar magnitude. The closest pole to the Riemann axis reads:% \begin{equation} \label{pole1} \text{First pole: }\ E=3777.0-i12.3\text{ MeV, }% \end{equation} hence% \begin{equation} \begin{split} &m_{\psi}^{\text{pole}}\simeq 3777.0\pm 1.0\text{ MeV and }\\ &\Gamma_{\psi}^{\text{pole} }\simeq 24.6\pm 1.0\text{ MeV,} \end{split} \end{equation} which agrees quite well with the PDG values \cite{pdg}. This pole is closely related to the {\it seed} charmonium state and to the position and width of the peak. Furthermore, a second broader pole appears at lower energy:% \begin{equation} \label{pole2} \text{Second pole: }\ E=3741.1-i18.4\text{ MeV.}% \end{equation} This pole is \emph{dynamically generated}, and it is related to the the deformation of the signal on the left side. It is also referred to as a companion pole emergent due to the strong dynamics. The situation is very much reminiscent of the light scalar $\kappa$ state at lower energy, which was interpreted as a dynamically generated companion pole of $K^*_0$ in Refs.~\cite{npb909p418,apps9p467}. For additional recent works on this state, see Refs.~\cite{epjc77p431,prd93p074025}, and references therein. This similarity is interesting also in view of an important difference between the two systems: the decay of a scalar kaon into two pseudoscalars is a $s$-wave ($\Gamma\sim k$) while that of a vector charmonium into two pseudoscalars is a $p$-wave ($\Gamma\sim k^{3}$). We mention again \ref{C}, where a different form factor is used. The results turn out to be very similar and also in that case two poles, very close to the ones discussed here, are found. No other poles could be numerically found in the complex plane. As discussed in comment e), the energy scale $\Lambda $ does not represent a limit of validly of our approach. Yet, since the numerical value of $272$ MeV preferred by the fit is rather small (the value of the emitted momentum $k(m_{\psi }^{2},m_{D^{0}})$ is comparable to it), it is interesting to investigate if two poles exist also when artificially increasing its value. Upon fixing $\Lambda =350$ MeV, one gets $\chi ^{2}=35.05,$ hence the description of data is worsened ($\chi ^{2}/d.o.f=3.5$% ) but still qualitatively acceptable; nevertheless, there are still two poles: the seed one corresponds to $3.770-i0.018$ GeV (somewhat too large) and the broad dynamically generated one to $3.755-i0.040$ GeV. Upon increasing $\Lambda $ even further to $400$ MeV, the $\chi ^{2}$ worsens, i.e.~$\chi ^{2}/d.o.f=4.98$ (which is already beyond the border of a satisfactory description of data), but there are still two poles; the seed pole reads $3.769-i0.016$ GeV and the companion pole $3.767-i0.062$ GeV. This exercise shows that the presence of two poles is a stable result also when changing the value of $\Lambda $, even to values which are safely larger than the value of the modulus of the outgoing momentum of the $D$ meson (even if the data description gets worse for increasing $\Lambda $). \subsection{\label{results.3} Pole Trajectories} Here we study the effect of varying the coupling constant over the line-shape and over the pole positions. We define $\tilde{g}=\lambda g_{\psi D\bar{D}}$ as the variable parameter (upon varying the scaling factor $\lambda$) and let $g\equiv g_{\psi D\bar{D}}$ be fixed to the value in Table \ref{fitpar}. In Quantum Chromodynamics (QCD), a change in such coupling is equivalent to vary the number of colors according to the scaling $\lambda\propto N_{c}^{-1/2}$. Results are shown in Fig.~\ref{gvar} for the cross-section, and in Table \ref{t4} for pole positions. The pole trajectories, as a function of $\tilde{g}$, are depicted in Fig.~\ref{traj}. As expected, as $\tilde{g}$ decreases, the cross section approaches a Breit-Wigner-like shape, see case where $\tilde{g}=0.7g$ in Fig.~\ref{gvar}. Figure \ref{traj} shows that, as $\tilde{g}$ gets smaller, the first pole, from the seed, gets closer to the real axis, while the second pole, dynamically generated, moves deeper in the complex plane until it disappears at the $D^0\bar{D}^0$ threshold for a certain critical value, in this case $\tilde{g}<26.9$ (corresponding to $N_c=3.9$). On the contrary, when $\tilde{g}$ increases, the first pole runs away from the real axis, while the second pole approaches it. Eventually, the dynamically generated pole gets even closer to the real axis than the seed pole, as it is shown for the case where $\tilde{g}=1.3g$. For such a value of $\tilde{g}$ (or larger), the second pole originates a second peak in the line-shape (see Fig.~\ref{gvar}). \begin{figure}[ptb] \begin{center} \resizebox{!}{200pt}{\includegraphics{gvarls31.pdf}} \end{center} \caption{Variation of the cross section in channel $D\bar{D}$ with the coupling $\tilde{g}$. Here, $g\equiv g_{\psiD\bar{D}}=30.73$, and the other parameters are found in Table \ref{fitpar}. See corresponding poles in Table \ref{t4}.}% \label{gvar}% \end{figure} \begin{table}[ptb] \begin{center}% \begin{tabular} [c]{cccc}\hline $\tilde{g}$ & $0.7{g}$ & $g\equiv g_{\psiD\bar{D}}$ & $1.3{g}$\\\hline Pole 1 & $-$ & $3741.1-i18.4$ & $3741.0-i9.5$\\ Pole 2 & $3773.7-i5.5$ & $3777.0-i12.3$ & $3785.0-i17.1$\\\hline \end{tabular} \end{center} \caption{\label{t4}Variation in the poles position with the coupling $\tilde{g}$, for parameters in Table \ref{fitpar} (cf.~Fig.~\ref{gvar}).}% \end{table} \begin{figure}[ptb] \begin{center} \resizebox{!}{200pt}{\includegraphics{poles.pdf}} \end{center} \caption{Pole trajectories. Here, $g\equiv g_{\psiD\bar{D}}=30.7$, and the other parameters can be found in Table \ref{fitpar}. For the numerical values of the poles at positions $\bullet$, $\diamond$, and $\circ$, see Table \ref{t4}. The vertical lines correspond to the $D^0\bar{D}^0$ and $D^+D^-$ thresholds, from left to right. The arrows show the direction of increasing the coupling (cf.~text).}% \label{traj}% \end{figure} \section{\label{conclusion} Summary, Conclusions, and Perspectives} Vector mesons can be produced by a single virtual photon, hence they play a crucial role in the experimental study of QCD. The renowned charmonium vector state $J/\psi$ opened a new era in hadron physics and it is one of the best studied resonances (cf.~\cite{pdg}). Here, we have focused on an orbital excitation of the $J/\psi$, the meson $\psi(3770)$, whose mass lies just above the $D\bar{D}$ threshold. For this reason, this resonance is extremely interesting. The cross-section $e^{+}e^{-}\rightarrow\psi\rightarrow D\bar{D}$ encloses nonperturbative phenomena nearby the OZI-allowed open decay channel $D\bar{D}$. The production and decay to all hadrons shows evidence of a deformation on the line-shape of the resonance \cite{prl101p102004}. In this work, we presented an unitarized effective Lagrangian model, with one bare $\bar{c}c$ orbital seed state $\psi$ dressed by $D\bar{D}$ mesonic loops, that has never been employed before to this system. In particular, our model accounts for one-loop $D^+D^-+D^0\bar{D}^0$ contributions by fulfilling unitarity of the spectral function. Using four parameters only, we obtain a satisfactory fit to the $D\bar{D}$ cross-section data ($\chi^{2}/$ ${d.o.f.}\sim 1.03$). Moreover, the theoretical cross section is in good agreement with data in channels $D^+D^-$ and $D^0\bar{D}^0$ separately. Partial widths are also in agreement with the PDG values. The partial decay to leptons is smaller than the average in PDG, but still reasonable (and in agreement with Ref.~\cite{plb711p292}). The cutoff parameter $\Lambda$, inversely proportional to the size of the wave-function, turns out to be rather small, possibly due to the fact that the $\psi(3770)$ is a $D$-wave. Other effects, such as final state rescattering modelled by a four-leg interaction term, or the contribution of the tail of the $\psi(2S)$ to the amplitude, are not taken into account, since our fit is already very good without them (this represents a difference with Ref.~\cite{plb718p1369}, where these terms were necessary to describe data but where the normalization of the spectral function of $\psi(3770)$ was not guaranteed). The detailed study of the role of these additional effects within our framework needs more precise data and is left for the future. An important result of our work is the evaluation of two poles at $3777-i12$ and $3741-i19$, the first is a standard seed pole, very close to the one determined by BES in Ref.~\cite{prl101p102004}, and the second broader pole is dynamically generated, being an additional companion pole. The presence of a second pole explains the deviation from a pure Breit-Wigner line-shape. By varying the coupling constant $g_{\psi D\bar{D}}$, we studied the pole trajectories and its influence over the line-shape. For small couplings the dynamical pole disappears and the seed pole approaches the real axis, leading to a Breit-Wigner-like line-shape, while for larger couplings the dynamical pole approaches the real axis while the seed pole moves down and to the right, and a second peak becomes evident (see Figs.~\ref{gvar} and \ref{traj} for details).} The width of about 24 MeV to channel $D\bar{D}$, that we obtain from the first pole, is consistent with the experimental measurements to this channel, and with the branching fraction estimated to be about $85\%$ in Ref.~\cite{plb659p74}, considering that measurements of the width in decays to all hadrons are a few MeV higher (cf.~Ref.~\cite{pdg}). A natural outlook is the inclusion of the many OZI-suppressed hadronic decay channels. A related topic is the evaluation of mixing with the vector glueball, whose mass is predicted to be at about 3.8 GeV by lattice QCD \cite{prd73p014516}. Finally, the same theoretical approach used in this work can be applied to other resonances whose nature is not yet clarified, the so-called $X,$ $Y,$ and $Z$ states \cite{xyz,pr668p1,epja36p189}. It is well possible that some of these resonances arise as companion poles due to a strong coupling of a standard charmonium seed state to lighter mesonic resonances. The strong dressing of the original state with meson-meson clouds generates new poles and new peaks. An intriguing possibility is that the additional peak arises at a higher energy than the seed state, see Refs.~\cite{zpc30p615,prd65p114010,zpc68p647,prd95p034015} for a phenomenological description of this phenomenon in the light sector. Hence, detailed studies of the dynamics underlying each line-shape, that consider poles and interference with the main hadronic degrees of freedom, may shed light on some newly discovered resonances in the charmonium region. An interesting outlook should be the description of the whole sector $1^{--}\ $above the $D\bar{D}$ threshold. To this end, one should develop a unique treatment of the four standard charm-anticharm resonances above the $D\bar{D}$ threshold $\psi (3770)$, $% \psi (4040)$, $\psi (4160)$, and $\psi (4415)$ (and also the resonance $\psi (2S) $ below). In this way, following and extending the discussion in Refs.~\cite{prd94p014016,plb718p1369}, all quantum mixing terms should be properly taken into account. The aim of such an ambitious project would be the simultaneous description of the whole sector as well as the description of (some of the) newly observed $% Y$ resonances in this energy region, which might emerge as dynamically generated states. \section*{Acknowledgements} We thank G.~Rupp and E.~van Beveren for useful discussions. This work was supported by the \textit{Polish National Science Center} through the project OPUS no.~2015/17/B/ST2/01625.

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\section{Introduction} Yang--Baxter (YB) sigma model was originally introduced by Klim\v{c}\'ik \cite{Klimcik:2002zj} as a class of Poisson--Lie symmetric sigma models. It is characterized by a classical $r$-matrix that satisfies the modified classical YB equation (mCYBE). It was later shown to be integrable by constructing the Lax pair \cite{Klimcik:2008eq}. The original YB sigma model can be applied only to sigma models on group manifolds, but it was later generalized to coset sigma models in \cite{Delduc:2013fga} and to the case of the homogeneous classical YB equation (CYBE) in \cite{Matsumoto:2015jja}. An interesting application of YB deformations is an integrable deformation of type IIB superstring theory on the $\AdS5\times \rmS^5$ background \cite{Delduc:2013qra,Delduc:2014kha,Kawaguchi:2014qwa}, that has been studied in the context of the AdS/CFT correspondence. Through various examples \cite{Matsumoto:2014nra,Matsumoto:2014gwa,Matsumoto:2015uja,vanTongeren:2015soa,vanTongeren:2015uha,Kyono:2016jqy}, it turned out that, when we employ an Abelian classical $r$-matrix, the YB-deformed $\AdS5\times \rmS^5$ superstring can be described as type IIB superstring on a TsT-transformed\footnote{A TsT transformation is a sequence of two Abelian $T$-dualities with a coordinate shift in between.} $\AdS5\times \rmS^5$ background \cite{Lunin:2005jy,Frolov:2005dj,Hashimoto:1999ut,Maldacena:1999mh,Herzog:2008wg,Maldacena:2008wh,Adams:2008wt} (see \cite{Osten:2016dvf} for a clear explanation and generalizations). Namely, Abelian YB deformation was found to be equivalent to a TsT-transformation. For non-Abelian classical $r$-matrices, the deformations of the $\AdS5\times \rmS^5$ background have not been understood clearly; some deformed backgrounds were obtained through non-commuting TsT-transformations (see for example \cite{Borsato:2016ose}) and some were obtained through a combination of diffeomorphisms and $T$-dualities \cite{Orlando:2016qqu}, but it is not clear whether an arbitrary YB deformation can be realized as a combination of Abelian $T$-dualities and gauge symmetries of the supergravity (it was recently shown in \cite{Hoare:2016wsk,Borsato:2016pas,Hoare:2016wca,Borsato:2017qsx,Lust:2018jsx} that YB deformations can be also reproduced from non-Abelian $T$-dualities \cite{Fridling:1983ha,Fradkin:1984ai,delaOssa:1992vci,Gasperini:1993nz,Giveon:1993ai,Alvarez:1994np,Elitzur:1994ri,Sfetsos:2010uq,Lozano:2011kb,Itsios:2013wd}). As shown in a seminal paper \cite{Borsato:2016ose}, at least when an $r$-matrix satisfies a certain criterion called unimodularity, the deformed $\AdS5\times \rmS^5$ background are solutions of type IIB supergravity. Moreover, for a non-unimodular $r$-matrix, the deformed $\AdS5\times \rmS^5$ background was shown to satisfy the generalized supergravity equations of motion (GSE) \cite{Arutyunov:2015mqj,TW}, and a Killing vector $I^m$ appearing in the GSE was determined for a general $r$-matrix. In a recent paper \cite{Wulff:2018aku} (which appeared a few days after this manuscript was posted to the arXiv), a more tractable expression for $I^m$ has been given for $r$-matrices consisting only of bosonic generators (see Appendix \ref{app:conventions} for more details). On the same day, the second version of \cite{Hong:2018tlp} appeared on the arXiv, which has derived the GSE from the dual sigma model of \cite{Elitzur:1994ri} in a general NS-NS background, and has determined the Killing vector $I^m$ in the non-Abelian $T$-dualized backgrounds. Recently, in \cite{Sakatani:2016fvh,Baguet:2016prz,Sakamoto:2017wor,Sakamoto:2017cpu,Fernandez-Melgarejo:2017oyu}, the GSE and YB deformations were studied from a viewpoint of a manifestly $T$-duality covariant formulation of supergravity, called the double field theory (DFT) \cite{Siegel:1993xq,Siegel:1993th,Siegel:1993bj,Hull:2009mi,Hull:2009zb,Hohm:2010pp} and its extensions. Through various examples of non-Abelian YB deformations, it was noticed that YB deformations are equivalent to (local) $\beta$-transformations of the $\AdS5\times \rmS^5$ background \cite{Sakamoto:2017cpu}. The local $\beta$-transformations may be realized as gauge transformations in DFT, known as the generalized diffeomorphism, and for many examples of non-Abelian $r$-matrices, YB-deformed backgrounds were reproduced by acting generalized diffeomorphisms to the $\AdS5\times \rmS^5$ background. However, until now the equivalence between YB deformations and local $\beta$-transformations has not yet been proven. In this paper, we show the equivalence for YB deformations of the $\AdS5\times \rmS^5$ superstring. To be more precise, we show that, for a classical $r$-matrix consisting of the bosonic generators and satisfying the homogeneous CYBE, the YB deformed $\AdS5\times \rmS^5$ superstring action can be regarded as the Green--Schwarz (GS) type IIB superstring action \cite{Green:1983wt} defined in a $\beta$-transformed $\AdS5\times \rmS^5$ background. During the proof, we perform a suitable identification of the deformed vielbein and make a redefinition of the fermionic variable. These procedures can be clearly explained by using the double-vielbein formalism of DFT \cite{Siegel:1993xq,Jeon:2011cn,Jeon:2011vx,Jeon:2011sq,Jeon:2012kd,Jeon:2012hp}. We also find a manifestly $\OO(10,10)$-invariant string action that reproduces the conventional GS superstring action up to quadratic order in fermions. In the previous works, $T$-duality covariant string theories with the worldsheet supersymmetry were studied in \cite{Hull:2006va,Blair:2013noa,Driezen:2016tnz}. The GS-type string actions were also constructed in \cite{Bandos:2015cha,Park:2016sbw,Bandos:2016jez} but the target space was assumed to be flat and have no the R-R fluxes. Our GS-type string action can apply to arbitrary curved backgrounds with the R--R fields and is a generalization of the previous ones (another $T$-duality manifest GS superstring action in a general background was proposed in \cite{Hatsuda:2014aza,Hatsuda:2015cia} although the relation to our action is unclear so far). We expect that the equivalence between YB deformations and $\beta$-deformations will hold in more general backgrounds beyond the $\AdS5\times \rmS^5$ background. As a non-trivial example, we study local $\beta$-deformations of the $\AdS3\times \rmS^3\times \TT^4$ background that contains a non-vanishing $H$-flux. In this case, due to the presence of $H$-flux, it is not straightforward to perform YB deformations.\footnote{There are several works \cite{Kawaguchi:2011mz,Kawaguchi:2013gma,Delduc:2014uaa,Delduc:2017fib,Demulder:2017zhz} where YB deformations of the WZ(N)W model based on the mCYBE have been studied.} Therefore, we do not show the equivalence in this paper. However, thanks to the homogeneous CYBE for the local $\beta$-deformations, all examples of the $\beta$-deformed backgrounds are shown to satisfy the equations of motion of DFT, or the (generalized) supergravity. This paper is organized as follows. In section \ref{sec:DFT}, we review the double-vielbein formalism of DFT and find a simple $\beta$-transformation rule for the Ramond--Ramond (R--R) fields. We also find the action of the double sigma model for type II superstring that reproduces the conventional GS superstring action. In section \ref{sec:YB-AdS5xS5}, we concisely review YB deformations of $\AdS{5} \times \rmS^5$ superstring and show the equivalence of homogeneous YB deformations and local $\beta$-transformations. In section \ref{sec:AdS3}, we perform $\beta$-transformations of the $\AdS3\times \rmS^3\times \TT^4$ background and obtain various solutions. We also discuss a more general class of local $\OO(10,10)$ transformations that are based on the homogeneous CYBE. Section \ref{sec:discussion} is devoted to conclusions and discussions. Various technical computations are explained in the Appendices. \section{Local \texorpdfstring{$\beta$}{\textbeta}-deformations in DFT} \label{sec:DFT} In this section, we review the basics of the type II DFT and find a simple transformation rule for bosonic fields under local $\beta$-deformations. We also find a manifestly $\OO(10,10)$-invariant superstring action that reproduces the conventional GS type II superstring action. \vspace{-\Pskip} \subsection{DFT fields and their parameterizations} Bosonic fields in DFT are the generalized metric $\cH_{MN}$, the $T$-duality-invariant dilaton $d$, and the R--R potential $\bisC$, which is an $\OO(1,D-1)\times \OO(D-1,1)$ bispinor. In this subsection, we review their definitions and basic properties by turning off fermions (such as gravitino). Here, we employ the double-vielbein formalism developed in \cite{Siegel:1993xq,Jeon:2011cn,Jeon:2011vx,Jeon:2011sq,Jeon:2012kd,Jeon:2012hp} (see also \cite{Hohm:2010xe}), which is quite suitable for discussing YB deformations. \paragraph{NS--NS fields:} The generalized metric $\cH_{MN}$ $(M,N=0,\dotsc,2D-1)$ is defined as \begin{align} \begin{split} \cH &\equiv (\cH_{MN}) \equiv E\,\mathsf{S}\, E^\rmT \,,\qquad E = (E_M{}^N) \in \OO(D,D)\,, \\ \mathsf{S}&\equiv (\mathsf{S}_{MN}) \equiv \diag(\underbrace{-1,\,+1,\cdots,+1}_{D},\,\underbrace{-1,\,+1,\dotsc,+1}_{D}) \,, \end{split} \end{align} where the $\OO(D,D)$ property of the generalized vielbein $E$ is defined as \begin{align} E \, \ODDeta\, E^\rmT = \ODDeta = E^\rmT\,\ODDeta\, E \,,\qquad \ODDeta\equiv (\ODDeta_{MN})\equiv \begin{pmatrix} 0 & \delta_m^n \\ \delta^m_n & 0 \end{pmatrix} \qquad (m,n=0,\dotsc,D-1). \end{align} The familiar properties of the generalized metric \begin{align} \cH^\rmT = \cH \,,\qquad \cH^\rmT\,\ODDeta\,\cH = \ODDeta\,, \label{eq:H-properties} \end{align} follow from the above definitions. For an $\OO(D,D)$ matrix $h$ satisfying $h^\rmT\,\mathsf{S}\,h = \mathsf{S}$, namely an $\OO(1,D-1)\times \OO(D-1,1)$ matrix $h$, both $E$ and $E\,h$ give the same generalized metric $\cH$. Thus, the generalized metric $\cH$ can be regarded as a representative of a coset \begin{align} \frac{\OO(D,D)}{\OO(1,D-1)\times \OO(D-1,1)} \,, \end{align} where $\OO(1,D-1)\times \OO(D-1,1)$ is known as the double local Lorentz group \cite{Jeon:2011sq}. We raise or lower the $\OO(D,D)$ indices by using the $\OO(D,D)$-invariant metric $\ODDeta$ like $\cH_M{}^N \equiv \cH_{MP}\,\ODDeta^{PN}$\,, and then \eqref{eq:H-properties} indicates that the matrix $\cH_M{}^N$ has eigenvalues $\pm1$\,. Then, we introduce the double (inverse) vielbeins $\GV_{\Loa}{}^{M}$ and $\brGV_{\Lobra}{}^M$ ($\Loa,\,\Lobra=0,\dotsc,D-1$) as the eigenvectors \begin{align} \cH^{M}{}_{N}\,\GV_{\Loa}{}^N =+\GV_{\Loa}{}^M\,,\qquad \cH^{M}{}_{N}\,\brGV_{\Lobra}{}^N=-\brGV_{\Lobra}{}^M\,. \end{align} Since the eigenvalues are different, they are orthogonal to each other \begin{align} \cH_{MN}\,\GV_{\Loa}{}^M\,\brGV_{\Lobrb}{}^{N}=0 \,,\qquad \ODDeta_{MN}\,\GV_{\Loa}{}^M\,\brGV_{\Lobrb}{}^{N}=0\,. \label{eq:H-eta-orthogonal} \end{align} Following \cite{Jeon:2011cn,Jeon:2011vx,Jeon:2011sq,Jeon:2012kd,Jeon:2012hp}, we normalize the double vielbeins as \begin{align} \begin{split} \eta_{\Loa\Lob}&=\ODDeta_{MN}\,\GV_{\Loa}{}^M \,\GV_{\Lob}{}^{N} = \cH_{MN}\,\GV_{\Loa}{}^M \,\GV_{\Lob}{}^{N} = \diag(-1,\,+1,\dotsc,+1) \,, \\ \breta_{\Lobra\Lobrb}&=\ODDeta_{MN}\,\brGV_{\Lobra}{}^M \,\brGV_{\Lobrb}{}^{N} =-\cH_{MN}\,\brGV_{\Lobra}{}^M \,\brGV_{\Lobrb}{}^{N} = \diag(+1,\,-1,\dotsc,-1) \,. \end{split} \end{align} By introducing $2D\times 2D$ matrices, \begin{align} (\GV_A{}^M) \equiv \begin{pmatrix} \GV_{\Loa}{}^{M}\\ \brGV_{\Lobra}{}^{M} \end{pmatrix}\,,\qquad (\eta_{AB}) \equiv \begin{pmatrix} \eta_{\Loa\Lob}& 0\\ 0& \breta_{\Lobra\Lobrb} \end{pmatrix} \,,\qquad (\cH_{AB}) \equiv \begin{pmatrix} \eta_{\Loa\Lob}& 0\\ 0& -\breta_{\Lobra\Lobrb} \end{pmatrix} \,, \end{align} where $\{A\}\equiv\{\Loa,\,\Lobra\}$, the above orthonormal conditions are summarized as \begin{align} \eta_{AB} = \GV_A{}^M\, \ODDeta_{MN}\, (\GV^\rmT)^N{}_B \,, \qquad \cH_{AB} = \GV_A{}^M\, \cH_{MN}\, (\GV^\rmT)^N{}_B \,. \label{eq:DV-ortho-normal} \end{align} The matrix $\GV_A{}^M$ is always invertible and the inverse matrix is given by \begin{align} (\GV^{-1})_M{}^A = \ODDeta_{MN}\,(\GV^\rmT)^N{}_B\,\eta^{BA} \,, \end{align} which indeed satisfies \begin{align} \GV_A{}^M\,(\GV^{-1})_M{}^B = \ODDeta_{MN}\, \GV_A{}^M\,\GV_C{}^N \,\eta^{CB} = \delta_A^B \,. \end{align} As long as we raise or lower the indices $M,\,N$ with $\ODDeta_{MN}$ and $A,\,B$ with $\eta_{AB}$ (namely, $\Loa,\,\Lob$ and $\Lobra,\,\Lobrb$ with $\eta_{\Loa\Lob}$ and $\eta_{\Lobra\Lobrb}$, respectively), there is no difference between $\GV_A{}^M$ and $(\GV^{-\rmT})_A{}^M\equiv \eta_{AB}\,\ODDeta^{MN}\,(\GV^{-\rmT})^B{}_N$\,. Thus, in the following, we may not show the inverse or the transpose explicitly. When $D\times D$ matrices, $\GV^m{}_{\Loa}$ and $\brGV^m{}_{\Lobra}$, are invertible, we can parameterize the double vielbeins as \begin{align} (\GV^M{}_{\Loa}) =\frac{1}{\sqrt{2}} \begin{pmatrix} (e^{-\rmT})^m{}_{\Loa} \\ E_{mn}\,(e^{-\rmT})^n{}_{\Loa} \end{pmatrix} \,, \qquad (\brGV^M{}_{\Lobra}) =\frac{1}{\sqrt{2}} \begin{pmatrix} (\bre^{-\rmT})^m{}_{\Lobra} \\ \bar{E}_{mn}\,(\bre^{-\rmT})^n{}_{\Lobra} \end{pmatrix} \,, \end{align} where we introduced matrix notations, $e\equiv (e_m{}^{\Loa})$ and $\bre\equiv (\bre_m{}^{\Lobra})$. From \eqref{eq:DV-ortho-normal}, we find \begin{align} \bar{E}_{mn} = - E^\rmT_{mn}\,,\qquad \CG_{mn} \equiv E_{(mn)} = (e \,\eta \, e^\rmT)_{mn} = - (\bre\,\breta \, \bre^\rmT)_{mn}\,. \end{align} By denoting $B_{mn}\equiv E_{[mn]}$, the parameterizations of the double vielbeins and the generalized metric become \begin{align} &(\GV^M{}_{\Loa}) =\frac{1}{\sqrt{2}} \begin{pmatrix} (e^{-\rmT})^m{}_{\Loa} \\ (\CG+B)_{mn}\,(e^{-\rmT})^n{}_{\Loa} \end{pmatrix} \,, \qquad (\brGV^M{}_{\Lobra}) =\frac{1}{\sqrt{2}} \begin{pmatrix} (\bre^{-\rmT})^m{}_{\Lobra} \\ -(\CG-B)_{mn}\,(\bre^{-\rmT})^n{}_{\Lobra} \end{pmatrix} \,, \label{eq:dV-geometric} \\ &\cH = \begin{pmatrix} (\CG-B\,\CG^{-1}\,B)_{mn} & (B\,\CG^{-1})_m{}^n \\ -(\CG^{-1}\,B)^m{}_n & \CG^{mn} \end{pmatrix} = \begin{pmatrix} \delta_m^p & B_{mp} \\ 0 & \delta^m_p \end{pmatrix} \begin{pmatrix} \CG_{pq} & 0 \\ 0 & \CG^{pq} \end{pmatrix} \begin{pmatrix} \delta^q_n & 0 \\ -B_{qn} & \delta^q_n \end{pmatrix} \,. \label{eq:H-geometric} \end{align} The dual parameterization, that can be prescribed when $\GV_m{}^{\Loa}$ and $\brGV_m{}^{\Lobra}$ are invertible, is \begin{align} \begin{split} \GV_M{}^{\Loa} &=\frac{1}{\sqrt{2}} \begin{pmatrix} \tilde{e}_m{}^{\Loa} \\ (\OG^{-1}-\beta)^{mn}\, \tilde{e}_n{}^{\Loa} \end{pmatrix} \,, \qquad \brGV_N{}^{\Lobra} =\frac{1}{\sqrt{2}} \begin{pmatrix} \tilde{\bre}_m{}^{\Lobra} \\ -(\OG^{-1}+\beta)^{mn}\, \tilde{\bre}_n{}^{\Lobra} \end{pmatrix} \,, \\ \OG_{mn}&\equiv (\tilde{e}\,\eta\, \tilde{e}^\rmT)_{mn} = - (\tilde{\bre} \,\breta \, \tilde{\bre}^\rmT)_{mn}\,,\qquad \beta^{mn}=-\beta^{nm} \,, \end{split} \end{align} and it provides the dual parameterization of the generalized metric \begin{align} \cH = \begin{pmatrix} \OG_{mn} & (\OG\, \beta)_m{}^{n} \\ -(\beta\,\OG)^m{}_{n} & (\OG^{-1}-\beta\,\OG\,\beta)^{mn} \end{pmatrix} = \begin{pmatrix} \delta_m^p & 0 \\ - \beta^{mp} & \delta^m_p \end{pmatrix} \begin{pmatrix} \OG_{pq} & 0 \\ 0 & \OG^{pq} \end{pmatrix} \begin{pmatrix} \delta^p_n & \beta^{pn} \\ 0 & \delta_p^n \end{pmatrix} \,. \label{eq:H-non-geometric} \end{align} When both parameterizations are possible, comparing \eqref{eq:H-geometric} and \eqref{eq:H-non-geometric}, we obtain \begin{align} \begin{split} E^{mn} &\equiv (E^{-1})^{mn} = \OG^{mn} - \beta^{mn} \qquad \bigl(E_{mn} \equiv \CG_{mn} + B_{mn}\bigr)\,, \\ \CG_{mn}&= E_{mp}\,E_{nq}\,\OG^{pq}\,,\qquad B_{mn}= E_{mp}\,E_{nq}\,\beta^{pq}\,. \end{split} \label{eq:relation-open-closed} \end{align} In the following, we raise or lower the indices of $\{e_m{}^{\Loa},\,\bre_m{}^{\Lobra},\,\tilde{e}_m{}^{\Loa},\,\tilde{\bre}_m{}^{\Lobra}\}$ as \begin{align} \begin{alignedat}{2} e^m{}_{\Loa}&=\CG^{mn}\,e_n{}^{\Lob}\,\eta_{\Lob\Loa}\,,&\qquad \bre^m{}_{\Lobra}&=\CG^{mn}\,\bre_n{}^{\Lobrb}\,\breta_{\Lobrb\Lobra}\,, \\ \tilde{e}^m{}_{\Loa}&=\OG^{mn}\,\tilde{e}_n{}^{\Lob}\,\eta_{\Lob\Loa}\,,&\qquad \tilde{\bre}^m{}_{\Lobra}&=\OG^{mn}\,\tilde{\bre}_n{}^{\Lobrb}\,\breta_{\Lobrb\Lobra}\,, \end{alignedat} \end{align} and then we obtain relations like $(e^{-\rmT})^m{}_{\Loa}=e^m{}_{\Loa}$\,. We can then omit the inverse or the transpose without any confusions as long as the indices are shown explicitly. By using the two metrics, $\CG_{mn}$ and $\OG_{mn}$, we also introduce two parameterizations of the dilaton $d$, \begin{align} \sqrt{\abs{\OG}}\,\Exp{-2\tilde{\phi}} = \Exp{-2d} = \sqrt{\abs{\CG}}\,\Exp{-2\Phi} \,. \label{eq:DFT-dilaton} \end{align} \paragraph{Ramond--Ramond fields:} In order to study the ten-dimensional type II supergravity, let us consider the case $D=10$\,. Associated with the double local Lorentz group $\OO(1,9)\times\OO(9,1)$, we introduce two sets of gamma matrices, $(\Gamma^{\Loa})^{\SPa}{}_{\SPb}$ and $(\brGamma^{\Lobra})^{\SPbra}{}_{\SPbrb}$, satisfying \begin{align} \begin{split} &\{\Gamma^{\Loa},\,\Gamma^{\Lob}\} =2\,\eta^{\Loa\Lob}\,,\qquad \{\brGamma^{\Lobra},\,\brGamma^{\Lobrb}\} = 2\,\breta^{\Lobra\Lobrb} \,, \\ &(\Gamma^{\Loa})^\dagger = - \Gamma^{0}\,\Gamma^{\Loa}\,(\Gamma^{0})^{-1} = \mp \Gamma^{\Loa}\,, \qquad \Loa=\biggl\{\begin{array}{l} 0 \\[-1mm] 1,\dotsc,9 \end{array} , \\ &(\brGamma^{\Lobra})^\dagger = + \brGamma^{0}\,\brGamma^{\Lobra}\,(\brGamma^{0})^{-1} = \pm \brGamma^{\Lobra}\,, \qquad \Lobra=\biggl\{\begin{array}{l} 0 \\[-1mm] 1,\dotsc,9 \end{array} . \end{split} \end{align} We also introduce the chirality operators \begin{align} \begin{split} &\Gamma^{11}\equiv \Gamma^{012\cdots 9}\,,\qquad \brGamma^{11}\equiv \brGamma^{012\cdots 9}\,,\qquad (\Gamma^{11})^\dagger=\Gamma^{11}\,,\qquad (\brGamma^{11})^\dagger=\brGamma^{11}\,, \\ &\{\Gamma^{\Loa},\,\Gamma^{11}\} =0\,,\qquad \{\brGamma^{\Lobra},\,\brGamma^{11}\} = 0\,,\qquad (\Gamma^{11})^2 =1\,,\qquad (\brGamma^{11})^2=1\,, \end{split} \end{align} and the charge conjugation matrices $C_{\SPa\SPb}$ and $\brC_{\SPbra\SPbrb}$ satisfyin \footnote{In order to follow the convention of \cite{Arutyunov:2015qva}, we employ the charge conjugation matrices $C_-$ and $\bar{C}_-$ of \cite{Jeon:2012kd} rather than $C_+$ and $\bar{C}_+$. They are related as $C_-=C_+\,\Gamma^{11}$ and $\bar{C}_- = \bar{C}_+\,\brGamma^{11}$.} \begin{align} \begin{alignedat}{2} &C\,\Gamma^{\Loa}\,C^{-1} = -(\Gamma^{\Loa})^\rmT \,, &\qquad C &= - C^\rmT = - C^{-1}\,, \qquad C^*=C \,, \\ &\brC\,\brGamma^{\Lobra}\,\brC^{-1} = -(\brGamma^{\Lobra})^\rmT \,,&\qquad \brC &= -\brC^\rmT = - \brC^{-1}\,, \qquad \brC^*=\brC \,. \end{alignedat} \label{eq:CC-properties} \end{align} We can show $C\,\Gamma^{11}\,C^{-1}=-\Gamma^{11}$ and $\brC\,\brGamma^{11}\,\brC^{-1}=-\brGamma^{11}$ by using \begin{align} C\,\Gamma^{\Loa_1\cdots\Loa_n}\,C^{-1} = (-1)^{\frac{n(n+1)}{2}} \,(\Gamma^{\Loa_1\cdots\Loa_n})^\rmT \,, \qquad \brC\,\brGamma^{\Lobra_1\cdots\Lobra_n}\,\brC^{-1} = (-1)^{\frac{n(n+1)}{2}} \,(\brGamma^{\Lobra_1\cdots\Lobra_n})^\rmT \,. \end{align} We raise or lower the spinor indices by using the charge conjugation matrices like \begin{align} \begin{alignedat}{2} (\Gamma^{\Loa})_{\SPa\SPb} &\equiv (\Gamma^{\Loa})^{\SPc}{}_{\SPb}\, C_{\SPc\SPa}\,,&\qquad (\Gamma^{\Loa})^{\SPa\SPb} &\equiv C^{\SPb\SPc}\,(\Gamma^{\Loa})^{\SPa}{}_{\SPc} \,, \\ (\brGamma^{\Lobra})_{\SPbra\SPbrb} &\equiv (\brGamma^{\Lobra})^{\SPbrc}{}_{\SPbrb}\,\brC_{\SPbrc\SPbra}\,,&\qquad (\brGamma^{\Lobra})^{\SPbra\SPbrb} &\equiv \brC^{\SPbrb\SPbrc}\,(\brGamma^{\Lobra})^{\SPbra}{}_{\SPbrc} \,, \end{alignedat} \end{align} and then from \eqref{eq:CC-properties} we have \begin{align} (\Gamma^{\Loa})_{\SPa\SPb} = (\Gamma^{\Loa})_{\SPb\SPa}\,,\qquad (\brGamma^{\Lobra})^{\SPbra\SPbrb} = (\brGamma^{\Lobra})^{\SPbrb\SPbra} \,. \end{align} We define the R--R potential as a bispinor $\bisC^{\SPa}{}_{\SPbrb}$ with a definite chirality \begin{align} \Gamma^{11}\,\bisC\,\brGamma^{11} = \pm\, \bisC \,, \end{align} where the sign is for type IIA/IIB supergravity. The R--R field strength is defined as \begin{align} \begin{split} &\bisF{}^{\SPa}{}_{\SPbrb} \equiv \cD_+ \bisC{}^{\SPa}{}_{\SPbrb} \equiv \frac{1}{\sqrt{2}}\,\bigl(\Gamma^M\,\cD_M \bisC + \Gamma^{11}\,\cD_M \bisC\,\brGamma^M\bigr){}^{\SPa}{}_{\SPbrb}\,, \\ &\Gamma^M\equiv \GV^M{}_{\Loa}\,\Gamma^{\Loa}\,,\qquad \brGamma^M\equiv \brGV^M{}_{\Lobra}\,\brGamma^{\Lobra}\,, \end{split} \label{eq:F-C-relation} \end{align} where $\cD_+$ is a nilpotent operator introduced in \cite{Jeon:2012kd}, and the covariant derivative $\cD_M$ for a bispinor $\bm{\cT}^{\SPa}{}_{\SPbrb}$ and the spin connections are defined as \cite{Jeon:2011cn,Jeon:2011vx,Jeon:2012kd} \begin{align} \begin{split} &\cD_M \bm{\cT}^{\SPa}{}_{\SPbrb} \equiv \partial_M \bm{\cT}^{\SPa}{}_{\SPbrb} + \Phi_{M}{}^{\SPa}{}_{\SPc}\,\bm{\cT}^{\SPc}{}_{\SPbrb} - \bm{\cT}^{\SPa}{}_{\SPbrc}\,\brPhi_{M}{}^{\SPbrc}{}_{\SPbrb}\,, \\ &\Phi_{M}{}^{\SPa}{}_{\SPb} \equiv \frac{1}{4}\,\Phi_{M\Loc\Lod}\,(\Gamma^{\Loc\Lod})^{\SPa}{}_{\SPb}\,,\qquad \brPhi_{M}{}^{\SPbra}{}_{\SPbrb}\equiv \frac{1}{4}\,\brPhi_{M\Lobrc\Lobrd}\,(\Gamma^{\Lobrc\Lobrd}){}^{\SPbra}{}_{\SPbrb} \\ &\Phi_{M\Loc\Lod} \equiv \GV^N{}_{\Loc}\,\nabla_M \GV_{N\Lod} = \GV^N{}_{\Loc}\,\bigl(\partial_M \GV_{N\Lod} - \Gamma_{M}{}^P{}_N\,\GV_{P\Lod} \bigr) \,, \\ &\brPhi_{M\Lobrc\Lobrd} \equiv \brGV^N{}_{\Lobrc}\,\nabla_M \brGV_{N\Lobrd} = \brGV^N{}_{\Lobrc}\,\bigl(\partial_M \brGV_{N\Lobrd} - \Gamma_{M}{}^P{}_N\,\brGV_{P\Lobrd} \bigr) \,, \end{split} \end{align} where $\nabla_M$ is the (semi-)covariant derivative in DFT \cite{Jeon:2010rw,Jeon:2011cn,Hohm:2011si} (see also \cite{Sakatani:2016fvh} which employs the same convention as this paper). Since $\cD_+$ flips the chirality, $\bisF$ has the opposite chirality to $\bisC$ \cite{Jeon:2012kd} \begin{align} \Gamma^{11}\,\bisF\,\brGamma^{11} = \mp\, \bisF \,. \end{align} As it has been shown in \cite{Jeon:2012kd}, $\bisF$ transforms covariantly under the $\OO(1,9)\times\OO(9,1)$ double Lorentz transformations, and transforms as a scalar under generalized diffeomorphisms. Further, from the nilpotency of $\cD_+$, $\bisF$ is invariant under gauge transformations of R--R potential \begin{align} \delta \bisC = \cD_+ \bm{\lambda} \,,\qquad \Gamma^{11}\,\bm{\lambda}\,\brGamma^{11} = \mp \bm{\lambda} \,, \end{align} and the Bianchi identity is given by \begin{align} \cD_+ \bisF = 0 \,. \end{align} As in the case of the democratic formulation \cite{Fukuma:1999jt,Bergshoeff:2001pv}, the self-duality relation \begin{align} \bisF = -\Gamma^{11}\,\bisF \ \bigl(= \pm\, \bisF \,\brGamma^{11}\bigr)\,, \label{eq:F-self-dual} \end{align} for type IIA/IIB supergravity is imposed by hand at the level of the equations of motion. \paragraph{Section condition and gauge symmetry:} In DFT, fields are defined on the doubled spacetime with the generalized coordinates $(x^M)=(x^m,\,\tilde{x}_m)$, where $x^m$ are the standard ``physical'' $D$-dimensional coordinates and $\tilde{x}_m$ are the dual coordinates. For the consistency of DFT, we require that arbitrary fields or gauge parameters $A(x)$ and $B(x)$ satisfy the so-called section condition \cite{Siegel:1993th,Hull:2009mi,Hull:2009zb}, \begin{align} \ODDeta^{MN}\,\partial_M A(x)\,\partial_N B(x) = 0\,,\qquad \ODDeta^{MN}\,\partial_M \partial_N A(x) = 0 \,. \end{align} In general, under this condition, fields and gauge parameters can depend on at most $D$ coordinates out of the $2D$ coordinates $x^M$. We frequently choose the ``canonical solution'' where all fields and gauge parameters are independent of the dual coordinates; $\tilde{\partial}^m \equiv \frac{\partial}{\partial\tilde{x}_m}=0$\,. In this case, DFT reduces to the conventional supergravity. Instead, if all fields depend on $(D-1)$ coordinates $x^i$ and only the dilaton $d(x)$ has an additional linear dependence on a dual coordinates $\tilde{z}$, DFT reduces to the generalized supergravity as discussed in \cite{Sakatani:2016fvh,Sakamoto:2017wor}. When the section condition is satisfied, the gauge symmetry of DFT is generated by the generalized Lie derivative \cite{Siegel:1993th,Hull:2009zb} \begin{align} \gLie_V W^M \equiv V^N\,\partial_N W^M - \bigl(\partial_N V^M -\partial^M V_N \bigr)\,W^N\,. \end{align} This symmetry is interpreted as diffeomorphisms in the doubled spacetime, $x^M\to x^M + V^M(x)$\,. Indeed, under the canonical section $\tilde{\partial}^m=0$\,, this symmetry consists of the conventional diffeomorphisms and $B$-field gauge transformations. If we parameterize the diffeomorphism parameter as $(V^M)=(v^m,\,\tilde{v}_m)$, the vector $v^m$ corresponds to the $D$-dimensional diffeomorphism parameter while the 1-form $\tilde{v}_m$ corresponds to the gauge parameter of the $B$-field gauge transformation, $B_2\to B_2 + \rmd \tilde{v}_1$\,. Under the canonical section, this is the whole gauge symmetry, but if we choose a different section, the generalized diffeomorphism may generate other local $\OO(D,D)$ transformations, such as $\beta$-transformations. For more details, the reader may consult a concise review \cite{Hohm:2013bwa}. \vspace{-\Pskip} \subsection{Diagonal gauge fixing} \label{sec:diagonal-gauge} In this subsection, we review the diagonal gauge fixing introduced in \cite{Jeon:2011cn,Jeon:2012kd}. \subsubsection{NS--NS fields} In order to constrain the redundantly introduced two vielbeins $e_{m}{}^{\Loa}$ and $\bre_{m}{}^{\Lobra}$\,, we implement the diagonal gauge fixing \begin{align} e_{m}{}^{\Loa} = \bre_{m}{}^{\Lobra} \,, \label{eq:diagonal-gauge} \end{align} which is important to reproduce the conventional supergravity. Before the diagonal gauge fixing, the double vielbeins transform as \begin{align} \GV_{\Loa}{}^M \to h^M{}_N\,\,\GV_{\Loa}{}^N\,,\qquad \brGV_{\Lobra}{}^M \to h^M{}_N\,\,\brGV_{\Lobra}{}^N \,, \end{align} under a global $\OO(10,10)$ rotation or a finite generalized diffeomorphism. We parameterize the $\OO(10,10)$ matrix $h^M{}_N$ as \begin{align} \begin{split} &h_M{}^N = \begin{pmatrix} \bmp_{m}{}^{n} & \bmq_{mn} \\ \bmr^{mn} & \bms^{m}{}_{n} \end{pmatrix} \,,\qquad h^M{}_N = \begin{pmatrix} \bms^{m}{}_{n} & \bmr^{mn} \\ \bmq_{mn} & \bmp_{m}{}^{n} \end{pmatrix} \\ &\bigl(\bmp\,\bms^\rmT + \bmq\,\bmr^\rmT =\bm{1}\,,\quad \bmr\,\bms^\rmT + \bms\,\bmr^\rmT =0 \,,\quad \bmp\,\bmq^\rmT + \bmq\,\bmp^\rmT =0 \bigr)\,, \end{split} \end{align} and then obtain the following transformation rule: \begin{align} \begin{alignedat}{2} e_m{}^{\Loa} &\to \bigl[\bigl(\bms^\rmT + E^\rmT\,\bmr^\rmT\bigr)^{-1}\bigr]_m{}^n\, e_n{}^{\Loa}\,,&\qquad \bre_m{}^{\Lobra} &\to \bigl[\bigl(\bms^\rmT - E\,\bmr^\rmT\bigr)^{-1}\bigr]_m{}^n\, \bre_n{}^{\Lobra}\,, \\ \tilde{e}_m{}^{\Loa} &\to \bigl(\bmp + \bmq\, E^{-1}\bigr)_m{}^n\,\tilde{e}_n{}^{\Loa}\,,&\qquad \tilde{\bre}_m{}^{\Lobra} &\to \bigl(\bmp - \bmq\, E^{-\rmT}\bigr)_m{}^n\,\tilde{\bre}_n{}^{\Lobra}\,, \\ E_{mn} &\to [(\bmq+\bmp\,E)\,(\bms+\bmr\,E)^{-1}]_{mn} \,,&\qquad E^{mn} &\to [(\bmr + \bms\,E^{-1})\,(\bmp + \bmq\,E^{-1})^{-1}]^{mn} \,. \label{eq:double-vielbein-transf} \end{alignedat} \end{align} At the same time, the dilaton transforms as \begin{align} \Exp{-2d} \to \abs{\det(\bmp_{m}{}^{n})}\,\Exp{-2d} \,, \end{align} and the bispinors of R--R fields, $\bisC$ and $\bisF$, are invariant. As we can see from \eqref{eq:double-vielbein-transf}, under a (geometric) subgroup (where $\bmr^{mn}=0$), \begin{align} h_M{}^N = \begin{pmatrix} \bmp_{m}{}^{n} & \bmq_{mn} \\ 0 & (\bmp^{-\rmT})^{m}{}_{n} \end{pmatrix} \,,\qquad h^M{}_N = \begin{pmatrix} (\bmp^{-\rmT})^{m}{}_{n} & 0 \\ \bmq_{mn} & \bmp_{m}{}^{n} \end{pmatrix} \qquad \bigl(\bmp\,\bmq^\rmT = - \bmq\,\bmp^\rmT \bigr)\,, \end{align} $e_m{}^{\Loa}$ and $\bre_m{}^{\Lobra}$ transform in the same manner. However, if we perform a general $\OO(10,10)$ transformation with $\bmr^{mn}\neq 0$, even if we choose the diagonal gauge in the original duality frame $(e_m{}^{\Loa}=\bre_m{}^{\Lobra})$, after the transformation, $e_m{}^{\Loa}\to e'_m{}^{\Loa}$ and $\bre_m{}^{\Lobra}\to \bre'_m{}^{\Lobra}$, we obtain \begin{align} \bre'_m{}^{\Lobra} = (\Lambda^{-1})^{\Lobra}{}_{\Lob}\,e'_m{}^{\Lob} \,,\qquad \Lambda^{\Loa}{}_{\Lobrb} \equiv \bigl[e^{\rmT}\,(\bms + \bmr\,E)^{-1}\,(\bms - \bmr\,E^\rmT)\,e^{-\rmT} \bigr]^{\Loa}{}_{\Lobrb} \in \OO(9,1) \,. \label{eq:Lambda-def} \end{align} In order to maintain the diagonal gauge \eqref{eq:diagonal-gauge}, we shall simultaneously perform an $\OO(9,1)$ local Lorentz transformation for barred tensors that compensates the deviation of $\bre_m{}^{\Lobra}$ from $e_m{}^{\Loa}$. Namely, we modify the $\OO(10,10)$ transformation as \cite{Jeon:2012kd} \begin{align} \GV_M{}^{\Loa} \to h_M{}^N\, \GV_{N}{}^{\Loa}\,,\qquad \brGV_M{}^{\Lobra} \to h_M{}^N\, \Lambda^{\Lobra}{}_{\Lobrb}\,\brGV_{N}{}^{\Lobrb} \,. \label{eq:modified-O(10-10)} \end{align} After the diagonal gauge fixing, since there is no more distinction between $\{\Loa,\,\SPa\}$ and $\{\Lobra,\,\SPbra\}$, we may simply replace $\{\Lobra,\,\SPbra\}$ by $\{\Loa,\,\SPa\}$. In this replacement, we should be careful about the signature \begin{align} \breta_{\Loa\Lob} = -\eta_{\Loa\Lob}\,,\qquad \brC_{\SPa\SPb} = C_{\SPa\SPb} \,. \end{align} In addition, we relate the two sets of gamma matrices as \begin{align} \brGamma^{\Loa}=\Gamma^{11}\,\Gamma^{\Loa}\qquad \bigl(\,\{\brGamma^{\Loa},\,\brGamma^{\Lob}\} = - \{\Gamma^{\Loa},\,\Gamma^{\Lob}\} = 2\,\breta^{\Loa\Lob}\, \bigr)\,,\qquad \brGamma^{11}=-\Gamma^{11}\,. \end{align} \subsubsection{R--R fields} According to the diagonal gauge fixing, there is no distinction between the two spinor indices $\SPa$ and $\SPbra$, and we can convert the bispinors into polyforms: \begin{align} \bisC^{\SPa}{}_{\SPb} = \sum_{n} \frac{1}{n!}\,\hat{\cC}_{\Loa_1\cdots \Loa_n}\,(\Gamma^{\Loa_1\cdots \Loa_n})^{\SPa}{}_{\SPb} \,,\qquad \bisF^{\SPa}{}_{\SPb} = \sum_{n} \frac{1}{n!}\,\hat{\cF}_{\Loa_1\cdots \Loa_n}\,(\Gamma^{\Loa_1\cdots \Loa_n})^{\SPa}{}_{\SPb} \,. \end{align} From the identity, \begin{align} \Gamma^{11}\,\Gamma^{\Loa_1\cdots \Loa_p} = \frac{(-1)^{\frac{p(p+1)}{2}}}{(10-p)!}\,\epsilon^{\Loa_1\cdots \Loa_p \Lob_1\cdots \Lob_{10-p}}\,\Gamma_{\Lob_1\cdots \Lob_{10-p}} \,, \end{align} where $\epsilon_{0\cdots 9}=-\epsilon^{0\cdots 9}=1$\,, the self-duality relation \eqref{eq:F-self-dual} can be expressed as \begin{align} \hat{\cF}_p = (-1)^{\frac{p(p-1)}{2}}\, * \hat{\cF}_{10-p} \,. \end{align} Here, we have defined \begin{align} \begin{alignedat}{2} \hat{\cF} &\equiv \sum_p \hat{\cF}_p\,,&\qquad \hat{\cF}_p &\equiv \frac{1}{p!}\,\hat{\cF}_{m_1\cdots m_p}\,\rmd x^{m_1}\wedge \cdots \wedge\rmd x^{m_p}\,, \\ \hat{\cC} &\equiv \sum_p \hat{\cC}_p\,,&\qquad \hat{\cC}_p &\equiv \frac{1}{p!}\,\hat{\cC}_{m_1\cdots m_p}\,\rmd x^{m_1}\wedge \cdots \wedge\rmd x^{m_p}\,, \end{alignedat} \end{align} where the R--R fields with the curved indices are defined as \begin{align} \hat{\cF}_{m_1\cdots m_p} \equiv e_{m_1}{}^{\Loa_1}\cdots e_{m_p}{}^{\Loa_p}\,\hat{\cF}_{\Loa_1\cdots \Loa_p}\,,\qquad \hat{\cC}_{m_1\cdots m_p} \equiv e_{m_1}{}^{\Loa_1}\cdots e_{m_p}{}^{\Loa_p}\,\hat{\cC}_{\Loa_1\cdots \Loa_p}\,. \label{eq:B-curved-flat} \end{align} In addition, if we define the components of the spin connections as \begin{align} \begin{split} \Phi_{\Loa\Loc\Lod} &\equiv \GV^M{}_{\Loa}\,\Phi_{M\Loc\Lod} \,,\qquad \brPhi_{\Lobra\Lobrc\Lobrd} \equiv \brGV^M{}_{\Lobra}\,\brPhi_{M\Lobrc\Lobrd} \,, \\ \Phi_{\Lobra\Loc\Lod} &\equiv \brGV^M{}_{\Lobra}\,\Phi_{M\Loc\Lod} \,,\qquad \brPhi_{\Loa\Lobrc\Lobrd} \equiv \GV^M{}_{\Loa}\,\brPhi_{M\Lobrc\Lobrd} \,, \end{split} \end{align} and compute their explicit forms under the canonical section $\tilde{\partial}^m=0$ as \begin{align} \begin{split} &\!\begin{alignedat}{2} \Phi_{\Lobra\Lob\Loc} &= \frac{1}{\sqrt{2}}\,\Bigl(\omega_{\Loa\Lob\Loc} + \frac{1}{2}\, H_{\Loa\Lob\Loc}\Bigr) \,,&\qquad \brPhi_{\Loa\Lobrb\Lobrc} &= \frac{1}{\sqrt{2}}\,\Bigl(-\omega_{\Loa\Lob\Loc} + \frac{1}{2}\, H_{\Loa\Lob\Loc}\Bigr)\,, \\ \Phi_{[\Loa\Lob\Loc]} &= \frac{1}{\sqrt{2}}\,\Bigl(\omega_{[\Loa\Lob\Loc]} + \frac{1}{6}\,H_{\Loa\Lob\Loc}\Bigr)\,,&\qquad \brPhi_{[\Lobra\Lobrb\Lobrc]} &= \frac{1}{\sqrt{2}}\,\Bigl(-\omega_{[\Loa\Lob\Loc]} + \frac{1}{6}\,H_{\Loa\Lob\Loc}\Bigr)\,, \end{alignedat} \\ &\eta^{\Loa\Lob}\,\Phi_{\Loa\Lob\Loc} = \frac{1}{\sqrt{2}}\,\bigl(\eta^{\Loa\Lob}\,\omega_{\Loa\Lob\Loc} -2\,e_{\Loc}{}^m\,\partial_m\Phi\bigr) = \breta^{\Lobra\Lobrb}\,\brPhi_{\Lobra\Lobrb\Lobrc} \,, \\ &\omega_{\Loa\Lob\Loc}\equiv e_{\Loa}{}^{m}\,\omega_{m\Lob\Loc} \,,\qquad \omega_m{}^{\Loa\Lob} \equiv 2\,e^{n[\Loa}\,\partial_{[m} e_{n]}{}^{\Lob]} - e^{\Loa p}\,e^{\Lob q}\,\partial_{[p} e_{q]}{}^{\Loc}\,e_{m\Loc} \,, \\ &H_{\Loa\Lob\Loc}\equiv e_{\Loa}{}^{m}\,e_{\Lob}{}^{n}\,e_{\Loc}{}^{p}\,H_{mnp} \,, \qquad H_{mnp}\equiv 3\,\partial_{[m}B_{np]}\,, \end{split} \label{eq:spin-connections} \end{align} we can show that the relation \eqref{eq:F-C-relation} between $\bisF$ and $\bisC$ can be expressed as \cite{Jeon:2012kd}\footnote{Here, we have used the following identities for type IIA/IIB theory: \begin{align*} \begin{split} &\frac{1}{2}\, \bigl(\Gamma^{m}\,\partial_{m}\bisC \mp \partial_m\bisC\,\Gamma^{m}\bigr) = \sum_n \frac{1}{n!}\,(\rmd \cC)_{\Loa_1\cdots\Loa_n}\,\Gamma^{\Loa_1\cdots\Loa_n} \,, \\ &\frac{1}{2}\, \partial_m\Phi\,\bigl(\Gamma^{m}\,\bisC \mp \bisC\,\Gamma^{m}\bigr) = \sum_n \frac{1}{n!}\,(\rmd\Phi \wedge \cC)_{\Loa_1\cdots\Loa_n}\,\Gamma^{\Loa_1\cdots\Loa_n} \,, \\ &\frac{1}{8}\, \omega_{m\Loa\Lob}\,\bigl[\Gamma^{\Loa}\,(\Gamma^{\Lob\Loc}\,\bisC-\bisC\,\Gamma^{\Lob\Loc})\mp (\Gamma^{\Lob\Loc}\,\bisC-\bisC\,\Gamma^{\Lob\Loc})\,\Gamma^{\Loa}\bigr] = -\sum_n \frac{\omega_{[\Loa_1}{}^{\Lob}{}_{\Loa_2}\,\cC_{|\Lob|\Loa_3\cdots\Loa_n]}}{2!\,(n-2)!} \,\Gamma^{\Loa_1\cdots\Loa_n} \,, \\ &\frac{1}{16}\,H_{\Loa\Lob\Loc}\,\Bigl[\frac{1}{3}\, \bigl(\Gamma^{\Loa\Lob\Loc}\,\bisC \mp \bisC\,\Gamma^{\Loa\Lob\Loc}\bigr) + \bigl(\Gamma^{\Loa}\,\bisC\,\Gamma^{\Lob\Loc} \mp \Gamma^{\Lob\Loc}\,\bisC\,\Gamma^{\Loa}\bigr) \Bigr] = \sum_n \frac{1}{n!}\,(H_3\wedge \cC)_{\Loa_1\cdots\Loa_n}\,\Gamma^{\Loa_1\cdots\Loa_n} \,. \end{split} \end{align*} } \begin{align} \hat{\cF} = \rmd \hat{\cC} - \rmd \Phi\wedge \hat{\cC} + H_3\wedge \hat{\cC} \,. \label{eq:cF-cC-relation} \end{align} Originally, the R--R fields were invariant under global $\OO(10,10)$ transformations or generalized diffeomorphisms, but after the diagonal gauge fixing, according to the modified transformation rule \eqref{eq:modified-O(10-10)}, they transform as \begin{align} \bisC \to \bisC \, \Omega^{-1} \,,\qquad \bisF \to \bisF \, \Omega^{-1} \,, \label{eq:R-R-beta-transf} \end{align} where $\Omega$ is the spinor representation of the local Lorentz transformation \eqref{eq:Lambda-def}, \begin{align} \Omega^{-1}\,\brGamma^{\Loa}\,\Omega = \Lambda^{\Loa}{}_{\Lob}\, \brGamma^{\Lob} \qquad \bigl(\Lambda^{\Loa}{}_{\Lob} = \bigl[e^{\rmT}\,(\bms + \bmr\,E)^{-1}\,(\bms - \bmr\,E^\rmT)\,e^{-\rmT} \bigr]^{\Loa}{}_{\Lob} \bigr)\,. \label{eq:Omega-def} \end{align} For later convenience, we here introduce several definitions of R--R fields that can be summarized as follows: \begin{align} \vcenter{ \xymatrix@R=35pt{ (\check{C},\,\check{F};\,\check{\bm{C}},\,\check{\bm{F}}) \ar@/_/[rr]_{\Atop{\beta\text{-twist}}{(\text{for polyform} \Exp{-\beta\vee})}} \ar@/_/[d]_{\Atop{\tilde{\phi}\text{-untwist}}{\Exp{\tilde{\phi}}}} & \qquad\quad & \ar@/_/[ll]_{\Atop{\beta\text{-untwist}}{(\text{for polyform} \Exp{\beta\vee})}} (A,\,F;\,\sla{A},\,\sla{F}) \ar@/^/[rr]^{\Atop{B\text{-untwist}}{(\text{for polyform} \Exp{-B_2\wedge})}} &\qquad & (\hat{C},\,\hat{F};\,\hat{\bm{C}},\,\hat{\bm{F}}) \ar@/^/[ll]^{\Atop{B\text{-twist}}{(\text{for polyform} \Exp{B_2\wedge})}} \ar@/_/[d]_{\Atop{\Phi\text{-untwist}}{\Exp{\Phi}}} \\ (\check{\cC},\,\check{\cF};\,\check{\bm{\cC}},\,\check{\bm{\cF}}) \ar@/_/[u]_{\Atop{\tilde{\phi}\text{-twist}}{\Exp{-\tilde{\phi}}}} \ar@{<->}[rrrr]^{\text{Eqs.~\eqref{eq:R-R-relation1} and \eqref{eq:R-R-relation2}}}_{\text{(local Lorentz transformation)}} & \qquad & &\qquad\quad & (\hat{\cC},\,\hat{\cF};\,\bisC,\,\bisF) \ar@/_/[u]_{\Atop{\Phi\text{-twist}}{\Exp{-\Phi}}} } } \label{eq:R-R-diagram} \end{align} The quantities at the lower right, polyforms $(\hat{\cC},\,\hat{\cF})$ and bispinors $(\bisC,\,\bisF)$, are already defined, which we call $(B,\,\Phi)$-untwisted fields. There, the curved indices and flat indices are interchanged by using the usual vielbein $e_m{}^{\Loa}$ like \eqref{eq:B-curved-flat}. The quantities at the upper right, which we call the $B$-untwisted fields, are defined as \begin{align} \hat{\bm{C}} \equiv \Exp{-\Phi}\bisC\,,\qquad \hat{\bm{F}} \equiv \Exp{-\Phi}\bisF\,,\qquad \hat{C}\equiv \Exp{-\Phi}\hat{\cC}\,,\qquad \hat{F}\equiv \Exp{-\Phi}\hat{\cF}\,. \end{align} The curved and flat indices are again related as \begin{align} \hat{C}_{m_1\cdots m_n}\equiv e_{m_1}{}^{\Loa_1}\cdots e_{m_n}{}^{\Loa_n}\, \hat{C}_{\Loa_1\cdots \Loa_n}\,,\qquad \hat{F}_{m_1\cdots m_n}\equiv e_{m_1}{}^{\Loa_1}\cdots e_{m_n}{}^{\Loa_n}\, \hat{F}_{\Loa_1\cdots \Loa_n}\,. \end{align} The $B$-untwisted fields are rather familiar R--R fields satisfying \begin{align} \hat{F} = \rmd \hat{C} + H_3\wedge \hat{C} \,, \end{align} which can be shown from \eqref{eq:cF-cC-relation}. We also define a polyform $A$ and its field strength $F$ as \begin{align} A= \Exp{-\Phi} \Exp{B_2\wedge} \hat{\cC} = \Exp{B_2\wedge} \hat{C} \,,\qquad F= \Exp{-\Phi} \Exp{B_2\wedge} \hat{\cF} = \Exp{B_2\wedge} \hat{F} \,. \end{align} These are utilized in \cite{Fukuma:1999jt,Hassan:1999mm,Hohm:2011dv} to define R--R fields as $\OO(D,D)$ spinors (see also \cite{Sakamoto:2017wor}) \begin{align} \sla{A} \equiv \sum_n \frac{1}{n!}\, A_{m_1\cdots m_n}\,\gamma^{m_1\cdots m_n}\vert0\rangle\,,\qquad \sla{F} \equiv \sum_n \frac{1}{n!}\, F_{m_1\cdots m_n}\,\gamma^{m_1\cdots m_n}\vert0\rangle\,. \end{align} By using the dual fields $(\tilde{e}_m{}^{\Loa},\,\beta^{mn},\,\tilde{\phi})$\,, we can also introduce the dual R--R fields, \begin{center} \begin{tabular}{ll} \underline{$\bullet$ $\beta$-untwisted fields:} & polyforms $(\check{C},\,\check{F})$ and bispinors $(\check{\bm{C}},\,\check{\bm{F}})$\,, \\[2mm] \underline{$\bullet$ $(\beta,\,\tilde{\phi})$-untwisted fields:} \qquad\quad& polyforms $(\check{\cC},\,\check{\cF})$ and bispinors $(\check{\bm{\cC}},\,\check{\bm{\cF}})$\,. \end{tabular} \end{center} By introducing an operator $\beta\vee F\equiv \frac{1}{2}\,\beta^{mn}\,\iota_m\,\iota_n F$, we define these polyforms as \begin{align} \check{C} \equiv \Exp{\beta\vee} A \,,\qquad \check{F} \equiv \Exp{\beta\vee} F \,. \qquad \check{\cC} \equiv \Exp{\tilde{\phi}} \Exp{\beta\vee} A \,,\qquad \check{\cF} \equiv \Exp{\tilde{\phi}} \Exp{\beta\vee} F \,, \end{align} and their flat components as \begin{align} \begin{split} \check{C}_{\Loa_1\cdots \Loa_p} &\equiv \tilde{e}_{\Loa_1}{}^{m_1}\cdots \tilde{e}_{\Loa_p}{}^{m_p}\,\check{C}_{m_1\cdots m_p}\,,\qquad \check{F}_{\Loa_1\cdots \Loa_p} \equiv \tilde{e}_{\Loa_1}{}^{m_1}\cdots \tilde{e}_{\Loa_p}{}^{m_p}\,\check{F}_{m_1\cdots m_p}\,, \\ \check{\cC}_{\Loa_1\cdots \Loa_p} &\equiv \tilde{e}_{\Loa_1}{}^{m_1}\cdots \tilde{e}_{\Loa_p}{}^{m_p}\,\check{\cC}_{m_1\cdots m_p}\,,\qquad \check{\cF}_{\Loa_1\cdots \Loa_p} \equiv \tilde{e}_{\Loa_1}{}^{m_1}\cdots \tilde{e}_{\Loa_p}{}^{m_p}\,\check{\cF}_{m_1\cdots m_p}\,, \end{split} \label{eq:dual-RR-flat-components} \end{align} by using the dual vielbein $\tilde{e}_m{}^{\Loa}$\,. Their corresponding bispinors are defined as \begin{align} \begin{split} \check{\bm{C}} &\equiv \sum_n \frac{1}{n!}\,\check{C}_{\Loa_1\cdots \Loa_n}\,\Gamma^{\Loa_1\cdots \Loa_n} \,,\qquad \check{\bm{F}} \equiv \sum_n \frac{1}{n!}\,\check{F}_{\Loa_1\cdots \Loa_n}\,\Gamma^{\Loa_1\cdots \Loa_n} \,, \\ \check{\bm{\cC}} &\equiv \sum_n \frac{1}{n!}\,\check{\cC}_{\Loa_1\cdots \Loa_n}\,\Gamma^{\Loa_1\cdots \Loa_n} \,,\qquad \check{\bm{\cF}} \equiv \sum_n \frac{1}{n!}\,\check{\cF}_{\Loa_1\cdots \Loa_n}\,\Gamma^{\Loa_1\cdots \Loa_n} \,. \end{split} \end{align} \subsubsection{Single \texorpdfstring{$T$}{T}-duality} As a simple application of the formula \eqref{eq:R-R-beta-transf}, let us explain how the R--R fields transform under a single $T$-duality along the $x^z$-direction, \begin{align} (h^M{}_N) = \begin{pmatrix} \bm{{1_{10}}} - \bm{e}_z & \bm{e}_z \\ \bm{e}_z & \bm{{1_{10}}} - \bm{e}_z \end{pmatrix}\,, \qquad \bm{e}_z \equiv \diag (0,\dotsc,0,\, \overset{z\text{-th}}{1},\,0,\dotsc,0)\,. \end{align} In this case, the vielbein and the dilaton transform as \begin{align} e' = \bigl[\bm{{1_{10}}} - (\bm{{1_{10}}}-E^\rmT)\, \bm{e}_z \bigr]^{-1} \, e = \bigl[\bm{{1_{10}}} + \CG_{zz}^{-1}\, (\bm{{1_{10}}}-E^\rmT)\, \bm{e}_z \bigr] \, e \,, \qquad \Exp{\Phi'} = \frac{1}{\sqrt{\CG_{zz}}} \Exp{\Phi} \,, \label{eq:abelian-NS-NS} \end{align} and the Lorentz transformation matrix is \begin{align} \Lambda \equiv (\Lambda^{\Loa}{}_{\Lob}) = e^{\rmT}\,\bigl[\bm{{1_{10}}} - \bm{e}_z \,(\bm{{1_{10}}} - E)\bigr]^{-1}\,\bigl[\bm{{1_{10}}} - \bm{e}_z \,(\bm{{1_{10}}}+ E^\rmT)\bigr]\,e^{-\rmT} \,. \end{align} This can be simplified as \begin{align} \Lambda^{\Loa}{}_{\Lob} = \delta^{\Loa}_{\Lob} - 2\,\frac{e_z{}^{\Loa} \,e_{z\Lob}}{\CG_{zz}} \,, \end{align} and we can easily see that the R--R field transforms under the $T$-duality as \cite{Hassan:1999mm} \begin{align} \bisC' = \bisC\, \Omega_z^{-1} \,,\qquad \Omega_z \equiv \frac{e_{z \Loa}}{\sqrt{\CG_{zz}}}\,\brGamma^{\Loa}\,\brGamma^{11} = \frac{1}{\sqrt{\CG_{zz}}}\,\Gamma_z = \Omega_z^{-1} \quad (\Gamma_m\equiv e_m{}^{\Loa}\,\Gamma_{\Loa})\,, \end{align} where we have supposed $\CG_{zz}\geq 0$. From the identity \eqref{eq:Gamma-n-1}, we obtain \begin{align} \bisC' = \bisC\, \Omega_z^{-1} = \frac{1}{\sqrt{\CG_{zz}}}\sum_n \frac{1}{n!}\biggl( n\,\hat{\cC}_{[\Loa_1\cdots\Loa_{n-1}}\,e_{\Loa_n]z} + \hat{\cC}_{\Loa_1\cdots\Loa_n\Lob}\,e_z{}^{\Lob} \biggr)\,\Gamma^{\Loa_1\cdots\Loa_n}\,. \label{eq:C-transformation-single} \end{align} By using the $B$-untwisted R--R potentials, $\hat{\bm{C}} = \Exp{-\Phi}\bisC$ and $\hat{C}=\Exp{-\Phi}\hat{\cC}$, \eqref{eq:C-transformation-single} is expressed as \begin{align} \hat{\bm{C}}' = \sum_n \frac{1}{n!}\biggl( n\,\hat{C}_{[\Loa_1\cdots\Loa_{n-1}}\,e_{\Loa_n]z} + \hat{C}_{\Loa_1\cdots\Loa_n\Lob}\,e_z{}^{\Lob} \biggr)\,\Gamma^{\Loa_1\cdots\Loa_n}\,, \end{align} where we have used \eqref{eq:abelian-NS-NS}. For the curved components, using the transformation rule of the vielbein \eqref{eq:abelian-NS-NS}, we obtain \begin{align} \hat{C}'_{m_1\cdots m_n} &= e'_{m_1}{}^{\Loa_1}\cdots e'_{m_n}{}^{\Loa_n}\, \bigl(n\,\hat{C}_{[\Loa_1\cdots\Loa_{n-1}}\,e_{\Loa_n]z} + \hat{C}_{\Loa_1\cdots\Loa_n\Lob}\,e_z{}^{\Lob} \bigr) \nn\\ &=n\,\hat{C}_{[m_1\cdots m_{n-1}}\,\CG_{m_n]z} + \hat{C}_{m_1\cdots m_{n}z} \nn\\ &\quad +n\,\CG_{zz}^{-1}\, \bigl[\hat{C}_{[m_1\cdots m_{n-1}}\,\CG_{zz}-(n-1)\,\hat{C}_{[m_1\cdots m_{n-2}|z|}\,\CG_{m_{n-1}|z|}\bigr]\,\bigl(\delta^z_{m_n]}-E^{\rmT}_{m_n]z}\bigr) \nn\\ &= \hat{C}_{m_1\cdots m_{n}z} +n\, \biggl[\hat{C}_{[m_1\cdots m_{n-1}} -(n-1)\,\frac{\hat{C}_{[m_1\cdots m_{n-2}|z|}\,\CG_{m_{n-1}|z|}}{\CG_{zz}}\biggr]\,\bigl(\delta^z_{m_n]}+B_{m_n]z}\bigr)\,. \end{align} This reproduces the famous transformation rule, \begin{align} \begin{split} &\hat{C}'_{i_1\cdots i_n} = \hat{C}_{i_1\cdots i_{n}z} +n\,\hat{C}_{[i_1\cdots i_{n-1}}\,B_{i_n]z} + n\,(n-1)\,\frac{\hat{C}_{[i_1\cdots i_{n-2}|z|}\,B_{i_{n-1}|z|}\,\CG_{i_n]z}}{\CG_{zz}} \,, \\ &\hat{C}'_{i_1\cdots i_{n-1}z} = \hat{C}_{i_1\cdots i_{n-1}} - (n-1)\,\frac{\hat{C}_{[i_1\cdots i_{n-2}}\,\CG_{i_{n-1}]z}}{\CG_{zz}} \,, \end{split} \end{align} where we have decomposed the coordinates as $\{x^m\}=\{x^i,\,x^z\}$. It is also noted that, under the single $T$-duality after taking the diagonal gauge, an arbitrary $\OO(1,9)$ spinor $\Psi_1^{\SPa}$ and an $\OO(9,1)$ spinor $\Psi_2^{\SPbra}$ transform as \begin{align} \Psi_1 \ \to \ \Psi'_1=\Psi_1 \,,\qquad \Psi_2 \ \to \ \Psi'_2=\Omega \,\Psi_2 = \frac{e_{z \Loa}}{\sqrt{\CG_{zz}}}\, \brGamma^{\Loa}\,\brGamma^{11}\,\Psi_2 \,. \label{eq:Psi-Tdual} \end{align} When we consider a single $T$-duality connecting type IIA and type IIB superstring, these transformations are applied to the spacetime fermions $\Theta_1$ and $\Theta_2$ introduced later. \vspace{-\Pskip} \subsection{\texorpdfstring{$\beta$}{\textbeta}-transformation of R--R fields} In this subsection, we consider local $\beta$-transformations \begin{align} h_M{}^N = \begin{pmatrix} \bm{{1_{10}}} & \bm{{0_{10}}} \\ \bmr^{mn}(x) & \bm{{1_{10}}} \end{pmatrix} \,,\qquad h^M{}_N = \begin{pmatrix} \bm{{1_{10}}} & \bmr^{mn}(x) \\ \bm{{0_{10}}} & \bm{{1_{10}}} \end{pmatrix} \qquad \bigl(\bmr^{mn}=-\bmr^{nm}\bigr) \,. \end{align} From the general transformation rule \eqref{eq:R-R-beta-transf}, the R--R fields should transform as $\bisC \to \bisC'=\bisC\, \Omega^{-1}$ and $\bisF \to \bisF' = \bisF\, \Omega^{-1}$\,. We here find an explicit form of $\Omega$ associated with $\beta$-transformations [the final result is obtained in \eqref{eq:Omega-general}]. \subsubsection{Gauge fixing for dual fields} Let us first specify the dual vielbein $\tilde{e}_m{}^{\Loa}$ explicitly. As we can see from \eqref{eq:double-vielbein-transf}, under $\beta$-transformations, we have the following transformation rules:\footnote{The transformation rule of $e_m{}^{\Loa}$ given \eqref{eq:beta-rule-NS} make sense only when $(E^{-\rmT})^{mn}$ is not singular. When $(E^{-\rmT})^{mn}$ is singular, we should express it as $e_m{}^{\Loa} \to e'_m{}^{\Loa}=\bigl[\bigl(\bm{1} - E^\rmT\, \bmr\bigr)^{-1}\bigr]_m{}^n\, e_n{}^{\Loa}$\,. When both $E_{mn}$ and $E^{mn}$ are singular, we should choose another parameterization of the double vielbein, although we do not consider such cases in this paper.} \begin{align} \begin{split} e_m{}^{\Loa} &\ \to\ e'_m{}^{\Loa}=\bigl[\bigl(E^{-\rmT} - \bmr\bigr)^{-1}\,E^{-\rmT}\bigr]_m{}^n\, e_n{}^{\Loa}\,, \\ \tilde{e}_m{}^{\Loa} &\ \to\ \tilde{e}'_m{}^{\Loa}=\tilde{e}_m{}^{\Loa} \,, \qquad E^{mn}\ \to\ E'^{mn} = E^{mn} + \bmr^{mn} \,. \end{split} \label{eq:beta-rule-NS} \end{align} Then, we can consistently relate $e_m{}^{\Loa}$ and $\tilde{e}_m{}^{\Loa}$ as \begin{align} \tilde{e}_{m}{}^{\Loa} = E_{mn}\,e^n{}_{\Lob}\,\eta^{\Lob\Loa} \,. \label{eq:tilde-e-gauge} \end{align} This is equivalent to a direct identification of two parameterizations, \begin{align} \frac{1}{\sqrt{2}} \begin{pmatrix} e^m{}_{\Lob}\,\eta^{\Lob\Loa} \\ (\CG+B)_{mn}\,e^n{}_{\Lob}\,\eta^{\Lob\Loa} \end{pmatrix} = \GV^{M\Loa}= \frac{1}{\sqrt{2}} \begin{pmatrix} (\OG^{-1}-\beta)^{mn}\, \tilde{e}_n{}^{\Loa} \\ \tilde{e}_m{}^{\Loa} \end{pmatrix} \,, \end{align} and consistent with the relation \eqref{eq:relation-open-closed}. If we introduce the flat components of $E^{mn}$ as \begin{align} \Einv^{\Loa\Lob} \equiv \tilde{e}_m{}^{\Loa}\, \tilde{e}_n{}^{\Lob}\,E^{mn} \equiv \eta^{\Loa\Lob} - \beta^{\Loa\Lob} \,, \label{eq:E-cEinv-cE} \end{align} we obtain \begin{align} E_{mn} = \tilde{e}_m{}^{\Loa}\,\tilde{e}_n{}^{\Lob}\,(\Einv^{-1})_{\Loa\Lob} = e_m{}^{\Loa}\,e_n{}^{\Lob}\, (\Einv^{\rmT})_{\Loa\Lob}\,. \end{align} Namely, we have simple expressions, \begin{align} \begin{alignedat}{2} \CG_{mn} &= e_{m}{}^{\Loa}\,e_{n}{}^{\Lob}\,\eta_{\Loa\Lob} \,,&\qquad B_{mn} &= e_{m}{}^{\Loa}\,e_{n}{}^{\Lob}\,\beta_{\Loa\Lob}\,, \\ \OG_{mn} &= \tilde{e}_{m}{}^{\Loa}\,\tilde{e}_{n}{}^{\Lob}\,\eta_{\Loa\Lob} \,,&\qquad \beta^{mn} &= \tilde{e}^m{}_{\Loa}\,\tilde{e}^n{}_{\Lob}\,\beta^{\Loa\Lob}\,. \end{alignedat} \label{eq:g-G-B-beta} \end{align} In terms of $\Einv^{\Loa\Lob}$, the relation \eqref{eq:tilde-e-gauge} can also be expressed as \begin{align} e_m{}^{\Loa} = \tilde{e}_{m}{}^{\Lob}\,(\Einv^{-\rmT})_{\Lob}{}^{\Loa}\,. \label{e-etilde-Einv} \end{align} From \eqref{e-etilde-Einv}, the relation \eqref{eq:DFT-dilaton} between the two dilatons, $\Phi$ and $\tilde{\phi}$, can be expressed as \begin{align} \Exp{\Phi} = (\det\Einv_{\Loa}{}^{\Lob})^{-\frac{1}{2}} \Exp{\tilde{\phi}}\,. \label{eq:two-dilatons} \end{align} \subsubsection{Relation between untwisted R--R fields} From \eqref{eq:R-R-diagram}, the relation between $(B,\,\Phi)$-untwisted R--R polyforms and the $(\beta,\,\tilde{\phi})$-untwisted R--R polyforms can be expressed as \begin{align} \begin{alignedat}{2} \check{\cF} &= \Exp{\tilde{\phi}-\Phi}\Exp{\beta\vee} \Exp{B_2\wedge} \hat{\cF} \,, \qquad& \hat{\cF} &= \Exp{\Phi-\tilde{\phi}}\Exp{-B_2\wedge} \Exp{-\beta\vee} \check{\cF} \,, \\ \check{\cC} &= \Exp{\tilde{\phi}-\Phi}\Exp{\beta\vee} \Exp{B_2\wedge} \hat{\cC} \,, \qquad& \hat{\cC} &= \Exp{\Phi-\tilde{\phi}}\Exp{-B_2\wedge} \Exp{-\beta\vee} \check{\cC} \,. \end{alignedat} \label{eq:R-R-relation1} \end{align} As we show in Appendix \ref{app:formula-proof} by a brute force calculation, if rephrased in terms of bispinors, these relations have quite simple forms \begin{align} \begin{alignedat}{2} &\bisF = \check{\bm{\cF}}\,\Omega_0^{-1}\,,\qquad \bisC = \check{\bm{\cC}}\,\Omega_0^{-1}\,,\qquad & &\check{\bm{\cF}} = \bisF \,\Omega_0\,,\qquad \check{\bm{\cC}} = \bisC \,\Omega_0\,, \label{eq:R-R-relation2} \\ &\Omega_0^{-1} =(\det \Einv_{\Loc}{}^{\Lod})^{-\frac{1}{2}} \text{\AE}\bigl(-\tfrac{1}{2}\,\beta^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr)\,,\qquad& &\Omega_0 =(\det \Einv_{\Loc}{}^{\Lod})^{-\frac{1}{2}} \text{\AE}\bigl(\tfrac{1}{2}\,\beta^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr)\,, \end{alignedat} \end{align} where $\text{\AE}$ is an exponential-like function with the gamma matrices totally antisymmetrized \cite{Hassan:1999mm} \begin{align} \text{\AE}\bigl(\tfrac{1}{2}\,\beta^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr) \equiv \sum^5_{p=0}\frac{1}{2^{p}\,p!}\, \beta_{\Loa_1\Loa_2}\cdots\beta_{\Loa_{2p-1}\Loa_{2p}}\,\Gamma^{\Loa_1\cdots \Loa_{2p}}\,. \end{align} In fact, this $\Omega_0$ is a spinor representation of a local Lorentz transformation,\footnote{Note that $\brGamma^{\Loa}=\Gamma^{11}\,\Gamma^{\Loa}$ also satisfies the same relation, $\Omega_0^{-1} \,\brGamma^{\Loa}\,\Omega_0 = \bigl(\Einv^{-1}\, \Einv^\rmT\bigr)^{\Loa}{}_{\Lob}\, \brGamma^{\Lob}$\,.} \begin{align} \Omega_0^{-1} \,\Gamma^{\Loa}\,\Omega_0 = \bigl(\Einv^{-1}\, \Einv^\rmT\bigr)^{\Loa}{}_{\Lob}\, \Gamma^{\Lob} \,, \end{align} as we can show by employing the formula provided below \cite{Hassan:1999mm} (see Appendix \ref{app:omega} for a proof). In this sense, the $(B,\,\Phi)$-untwisted fields and the $(\beta,\,\tilde{\phi})$-untwisted fields are related by a local Lorentz transformation. \noindent\textbf{\underline{Formula:}} For an arbitrary antisymmetric matrix $a_{\Loa\Lob}$, the spinor representation of a local Lorentz transformation \begin{align} \Lambda^{\Loa}{}_{\Lob} \equiv \bigl[(\eta+a)^{-1}\,(\eta-a)\bigr]^{\Loa}{}_{\Lob} = \bigl[(\eta-a)\,(\eta+a)^{-1}\bigr]^{\Loa}{}_{\Lob}\ \in \OO(1,D-1)\,, \end{align} is given by \begin{align} \begin{split} \Omega_{(a)} &=\bigl[\det (\delta_{\Loc}^{\Lod}\pm a_{\Loc}{}^{\Lod})\bigr]^{-\frac{1}{2}} \text{\AE}\bigl(-\tfrac{1}{2}\,a_{\Loa\Lob}\,\Gamma^{\Loa\Lob}\bigr)\,, \\ \Omega_{(a)}^{-1} &=\bigl[\det (\delta_{\Loc}^{\Lod}\pm a_{\Loc}{}^{\Lod})\bigr]^{-\frac{1}{2}} \text{\AE}\bigl(\tfrac{1}{2}\,a_{\Loa\Lob}\,\Gamma^{\Loa\Lob}\bigr) \,, \qquad \Omega_{(a)}^{-1} \,\Gamma^{\Loa}\,\Omega_{(a)} = \Lambda^{\Loa}{}_{\Lob}\, \Gamma^{\Lob} \,. \end{split} \label{eq:Hassan-formula} \end{align} \subsubsection{General formula for \texorpdfstring{$\Omega$}{\textOmega}} Now, let us find the explicit form of $\Omega$ for $\beta$-transformations [recall \eqref{eq:Omega-def}], satisfying \begin{align} \Omega^{-1}\,\brGamma^{\Loa}\,\Omega = \Lambda^{\Loa}{}_{\Lob}\, \brGamma^{\Lob} \qquad \bigl[\Lambda=e^{\rmT}\,(1+ \bmr\,E)^{-1}\,(1- \bmr\,E^\rmT)\,e^{-\rmT}\bigr]\,. \end{align} A key observation is that by using \eqref{eq:tilde-e-gauge}, \eqref{eq:E-cEinv-cE}, and \eqref{e-etilde-Einv}, $\Lambda^{\Loa}{}_{\Lob}$ can be decomposed into a product of two Lorentz transformations, \begin{align} \Lambda = \Lambda'\, \Lambda^{-1}_{0} \,,\qquad \Lambda' \equiv \eta^{-1}\,\Einv'^{-1}\, \Einv'^\rmT\,\eta \,,\qquad \Lambda_{0}\equiv \eta^{-1}\,\Einv^{-1}\, \Einv^\rmT\,\eta\,, \label{eq:Lambda-decomp} \end{align} where $\Einv'$ is defined by \begin{align} \Einv'^{\Loa\Lob} \equiv \bigl[\tilde{e}^\rmT\,(E^{-1}+\bmr)\,\tilde{e}\bigr]^{\Loa\Lob} \equiv \eta^{\Loa\Lob} - \beta'^{\Loa\Lob} \qquad \bigl(\beta'^{\Loa\Lob}\equiv \beta^{\Loa\Lob}- \bmr^{mn}\,\tilde{e}_m{}^{\Loa}\,\tilde{e}_n{}^{\Lob} \bigr)\,. \end{align} Then, we can check the following relations associated with $\Einv'^{\Loa\Lob}$: \begin{align} \begin{alignedat}{2} \Einv'^{\Loa\Lob}&= \tilde{e}_m{}^{\Loa}\,\tilde{e}_n{}^{\Lob}\, E'^{mn} = e'^{\Loa m}\,e'^{\Lob n}\, E'^\rmT_{mn} \,,&\qquad E'^{mn}&=E^{mn}+\bmr^{mn} \equiv \OG'^{mn} - \beta'^{mn}\,, \\ e'_m{}^{\Loa} &\equiv E'^\rmT_{mn}\,\tilde{e}^{n\Loa} = (\Einv'^{-\rmT})_{\Lob}{}^{\Loa}\,\tilde{e}_{m}{}^{\Lob}\,,&\qquad E'_{mn} &= \bigl[(E^{-1}+\bmr)^{-1}\bigr]_{mn} \equiv \CG'_{mn} + B'_{mn} \,, \end{alignedat} \end{align} where $\CG'_{mn}$ and $B'_{mn}$ are the $\beta$-transformed metric and $B$-field, respectively. From the invariance of $d$, $\tilde{e}_m{}^{\Loa}$, and $\tilde{\phi}$ under $\beta$-transformations, the dilaton $\Phi$ in the $\beta$-transformed background becomes \begin{align} \Exp{\Phi'} = (\det\Einv'_{\Loa}{}^{\Lob})^{-\frac{1}{2}} \Exp{\tilde{\phi}} = \frac{(\det\Einv'_{\Loa}{}^{\Lob})^{-\frac{1}{2}}}{(\det\Einv_{\Loc}{}^{\Lod})^{-\frac{1}{2}}} \Exp{\Phi} \,. \label{eq:Phi-beta} \end{align} Corresponding to the decomposition \eqref{eq:Lambda-decomp}, we can also decompose $\Omega$ as \begin{align} \begin{split} \Omega &= \Omega' \,\Omega_{0}^{-1} = \bigl[\det (\Einv'\,\Einv)_{\Loe}{}^{\Lof}\bigr]^{-\frac{1}{2}} \, \text{\AE}\bigl(\tfrac{1}{2}\,\beta'^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr)\,\text{\AE}\bigl(-\tfrac{1}{2}\,\beta^{\Loc\Lod}\,\Gamma_{\Loc\Lod}\bigr)\,, \\ \Omega^{-1} &= \Omega_0 \,\Omega'^{-1} = \bigl[\det (\Einv'\,\Einv)_{\Loe}{}^{\Lof}\bigr]^{-\frac{1}{2}} \,\text{\AE}\bigl(\tfrac{1}{2}\,\beta^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr)\, \text{\AE}\bigl(-\tfrac{1}{2}\,\beta'^{\Loc\Lod}\,\Gamma_{\Loc\Lod}\bigr)\,. \end{split} \label{eq:Omega-general} \end{align} where we have defined \begin{align} \Omega' \equiv (\det \Einv'_{\Loc}{}^{\Lod})^{-\frac{1}{2}} \text{\AE}\bigl(\tfrac{1}{2}\,\beta'^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr)\,,\qquad \Omega'^{-1} = (\det \Einv'_{\Loc}{}^{\Lod})^{-\frac{1}{2}} \text{\AE}\bigl(-\tfrac{1}{2}\,\beta'^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr)\,. \end{align} This gives the desired local Lorentz transformation, \begin{align} \Omega^{-1} \,\brGamma^{\Loa}\,\Omega = \Omega_{0}\,\Omega'^{-1} \,\brGamma^{\Loa}\,\Omega'\,\Omega_{0}^{-1} = (\Lambda'\,\Lambda^{-1}_{0})^{\Loa}{}_{\Lob}\, \brGamma^{\Lob} = \Lambda^{\Loa}{}_{\Lob}\, \brGamma^{\Lob} \,. \end{align} The $\beta$-transformed R--R field is then expressed as \begin{align} \bisF' = \bisF\, \Omega^{-1} \,. \label{eq:beta-F-transf} \end{align} In terms of the differential form, we can express the same transformation rule as \begin{align} \hat{\cF}' = \Exp{\Phi'-\Phi}\Exp{-B'_2\wedge} \Exp{\bmr\vee} \Exp{B_2\wedge} \hat{\cF} \qquad \bigl(\hat{\cF}'_{m_1\cdots m_n} \equiv e'_{m_1}{}^{\Loa_1}\cdots e'_{m_n}{}^{\Loa_n}\,\hat{\cF}'_{\Loa_1\cdots \Loa_n}\bigr)\,. \end{align} In terms of the $B$-untwisted field $\hat{F}$, the $\beta$-untwisted field $\check{F}$, and the $(\beta,\,\tilde{\phi})$-untwisted field $\check{\cF}$, we can express the above formula as \begin{align} \hat{F}' = \Exp{-B'_2\wedge} \Exp{\bmr\vee} \Exp{B_2\wedge} \hat{F} \,,\qquad \check{F}' = \check{F}\,,\qquad \check{\cF}' = \check{\cF} \,. \end{align} Namely, the $\beta$- or $(\beta,\,\tilde{\phi})$-untwisted field is invariant under $\beta$-transformations, which has been shown in \cite{Sakamoto:2017cpu} (see also \cite{Sakamoto:2017wor}) by treating the R--R fields, $A$ and $F$, as $\OO(D,D)$ spinors. Specifically, if the $B$-field and the dilaton $\Phi$ are absent before the $\beta$-transformation, we have $\beta^{\Loa\Lob}=0$\,, $\Einv_{\Loa}{}^{\Lob}=\delta_{\Loa}^{\Lob}$\,, and $\beta'^{\Loa\Lob}= - \bmr^{mn}\,\tilde{e}_m{}^{\Loa}\,\tilde{e}_n{}^{\Lob}$. Then, \eqref{eq:Omega-general} becomes \begin{align} \Omega = (\det\Einv'_{\Loc}{}^{\Lod})^{-\frac{1}{2}}\,\text{\AE}\bigl(-\tfrac{1}{2}\,\bmr^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr) \qquad (\bmr^{\Loa\Lob}\equiv \bmr^{mn}\,\tilde{e}_m{}^{\Loa}\,\tilde{e}_n{}^{\Lob}) \,. \end{align} In section \ref{sec:YB-AdS5xS5}, we see that this $\Omega$ plays an important role in YB deformations of $\AdS{5}\times\rmS^5$ superstring [see Eq.~\eqref{eq:omega} where $2\,\eta\,\lambda^{\Loa\Lob}$ plays the same role as $\bmr^{\Loa\Lob}$ here]. \vspace{-\Pskip} \subsection{\texorpdfstring{$T$}{T}-duality-invariant Green--Schwarz action} In section \ref{sec:YB-AdS5xS5}, we study homogeneous YB deformations of the GS type IIB superstring action and show that YB deformations are equivalent to $\beta$-deformations of the target space. In order to show the equivalence, it will be useful to manifest the covariance of the GS superstring theory under $\beta$-transformations. In this section, we provide a manifestly $\OO(10,10)$ $T$-duality-covariant formulation of the GS type II superstring theory. A manifestly $T$-duality covariant formulations of string theory, the so-called double sigma model (DSM), has been developed in \cite{Duff:1989tf,Tseytlin:1990nb,Tseytlin:1990va,Hull:2004in,Hull:2006va,Copland:2011wx,Lee:2013hma} for the bosonic string. More recently, the DSM for the GS type II superstring theory was formulated in \cite{Park:2016sbw} (see also \cite{Hull:2006va,Blair:2013noa,Bandos:2015cha,Driezen:2016tnz,Bandos:2016jez} for other approaches to supersymmetric DSMs). The action by Park, in our convention, is given by \begin{align} S &= \frac{1}{4\pi\alpha'}\int \Bigl[\,\frac{1}{2}\, \cH_{MN}\,\Pi^M\wedge *_{\gga} \Pi^N - D X^M\wedge \bigl(\cA_{M} + \Sigma_{M} \bigr)\Bigr] \nn\\ &= -\frac{1}{4\pi\alpha'}\int \sqrt{-\gga}\,\rmd^2\sigma\, \Bigl[\,\frac{1}{2}\,\gga^{\WSa\WSb}\,\cH_{MN}\,\Pi_{\WSa}^M\,\Pi_{\WSb}^N + \varepsilon^{\WSa\WSb}\, D_{\WSa}X^M\,\bigl(\cA_{\WSb M} + \Sigma_{\WSb M} \bigr)\Bigr] \,, \label{eq:GS-DSM-Park} \end{align} where $\gga_{\WSa\WSb}$ is the intrinsic metric on the string worldsheet and \begin{align} \begin{split} &\Pi^M \equiv DX^M + \Sigma^M \,,\qquad DX^M \equiv \rmd X^M-\cA^M \,,\qquad \varepsilon^{01}\equiv \frac{1}{\sqrt{-\gga}}\,, \\ &(X^M)\equiv \begin{pmatrix} X^m \\ \tilde{X}_m \end{pmatrix} \,,\qquad \Sigma^M \equiv \begin{pmatrix} \Sigma^m \\ \tilde{\Sigma}_m \end{pmatrix} \equiv \frac{\ii}{\sqrt{2}}\,\bigl(\brTheta_1\,\Gamma^M\,\rmd \Theta_1 + \brbrTheta_2\,\brGamma^M\,\rmd \Theta_2\bigr) \,, \end{split} \end{align} and a worldsheet 1-form $\cA^M(\sigma)$ is defined to satisfy, \begin{align} \cA^M \, \partial_M T = 0\,,\qquad \cA^M\,\cA_M = 0\,, \label{eq:cA-condition} \end{align} for arbitrary supergravity fields or gauge parameters $T(x)$\,. Here, the Dirac conjugates for the spacetime fermions $\Theta_1^{\SPa}$ and $\Theta_2^{\SPbra}$ are defined respectively as \begin{align} \brTheta_1 \equiv \Theta_1^\dagger\,\Gamma^0\,,\qquad \brbrTheta_2 \equiv -\Theta_2^\dagger\,\brGamma^0\,, \end{align} which indeed transform as \begin{align} \brTheta_1 \ \to \ \brTheta_1 \, \Exp{-\frac{1}{4}\,\omega_{\Loa\Lob}\,\Gamma^{\Loa\Lob}}\,,\qquad \brbrTheta_2 \ \to \ \brbrTheta_2 \, \Exp{-\frac{1}{4}\,\bromega_{\Lobra\Lobrb}\,\brGamma^{\Lobra\Lobrb}} \,, \end{align} under a double Lorentz transformation $\Theta_1\to \Exp{\frac{1}{4}\,\omega_{\Loa\Lob}\,\Gamma^{\Loa\Lob}} \Theta_1$ and $\Theta_2\to \Exp{-\frac{1}{4}\,\bromega_{\Lobra\Lobrb}\,\brGamma^{\Lobra\Lobrb}} \Theta_2$\,. The Majorana--Weyl conditions are defined as\footnote{The non-standard factor $-\brGamma^{11}$ is introduced in the Majorana condition for $\Theta_2$ such that the condition becomes the standard Majorana condition after the diagonal gauge fixing; $\Theta_2 = C\, (\Gamma^0)^\rmT\,\Theta_2^*$\,.} \begin{align} \begin{alignedat}{2} &\Theta_1 = C\,(\Gamma^0)^\rmT\,\Theta_1^*\,,\qquad& &\Theta_2 = -\brGamma^{11}\,\brC\,(\brGamma^0)^\rmT\,\Theta_2^* \,, \\ &\Gamma^{11}\,\Theta_1 = \Theta_1\,,\qquad& &\brGamma^{11}\,\Theta_2 = \pm \Theta_2\qquad (\text{IIA/IIB})\,, \end{alignedat} \end{align} and then we obtain \begin{align} \brTheta_1 = \Theta_1^\dagger\,\Gamma^0 = \Theta_1^\rmT\,C \,,\qquad \brbrTheta_2 = -\Theta_2^\dagger\,\brGamma^0 = -\Theta_2^\rmT\,\brC\, \brGamma^{11} \,. \end{align} In \cite{Park:2016sbw}, the target space was assumed to be flat, but here we generalize the action to arbitrary curved backgrounds. In order to consider the superstring action in the presence of fluxes, such as the $H$-flux and the R--R fluxes, we introduce generalized tensors, \begin{align} \begin{split} \cK_{MN}^{(1)}&\equiv -\frac{\ii}{\sqrt{2}}\,\GV_{(M}{}^{\Loa}\,\brGV_{N)}{}^{\Lobrb}\, \brTheta_1\,\Gamma_{\Loa}\,\Gamma^{\Loc\Lod}\,\Theta_1 \, \Phi_{\Lobrb\Loc\Lod} \,, \\ \cK_{MN}^{(2)}&\equiv -\frac{\ii}{\sqrt{2}}\,\brGV_{(M}{}^{\Lobra}\,\GV_{N)}{}^{\Lob}\, \brbrTheta_2\,\brGamma_{\Lobra}\,\brGamma^{\Lobrc\Lobrd}\,\Theta_2 \, \brPhi_{\Lob\Lobrc\Lobrd} \,, \\ \cK^{\text{\tiny(RR)}}_{MN} &\equiv \frac{\ii}{4}\,\GV_{(M}{}^{\Loa}\,\brGV_{N)}{}^{\Lobrb}\,\brTheta_{1}\, \Gamma_{\Loa}\, \bisF \, \brGamma_{\Lobrb} \Theta_{2} \,. \end{split} \end{align} Then, we add the following term to the DSM action \eqref{eq:GS-DSM-Park}: \begin{align} \Delta S \equiv \frac{1}{8\pi\alpha'}\int \cK_{MN}\,\Pi^M\wedge *_{\gga} \Pi^N\,, \qquad \cK_{MN}\equiv \cK_{MN}^{(1)}+\cK_{MN}^{(2)}+\cK_{MN}^{\text{\tiny(RR)}}\,. \end{align} By choosing the diagonal gauge, the explicit form of $\cK_{MN}$ becomes \begin{align} \cK_{MN} &= \begin{pmatrix} -(\CG\,\kappa^{\text{s}}\,\CG+B\,\kappa^{\text{s}}\,B+B\,\kappa^{\text{a}}\,\CG+\CG\,\kappa^{\text{a}}\,B)_{mn} & (B\,\kappa^{\text{s}} +\CG\,\kappa^{\text{a}})_m{}^n \\ - (\kappa^{\text{s}}\,B+\kappa^{\text{a}}\,\CG)^m{}_n & (\kappa^{\text{s}})^{mn} \end{pmatrix} \,, \\ \kappa_{mn} &\equiv -\frac{\ii}{4}\,\Bigl(\sqrt{2}\,\brTheta_1\,\Gamma_{m}\,\Gamma^{\Loa\Lob}\,\Theta_1\,\Phi_{n\Loa\Lob} + \sqrt{2}\,\brbrTheta_2\,\brGamma_{n}\,\brGamma^{\Lobra\Lobrb}\,\Theta_2 \, \brPhi_{m\Lobra\Lobrb} - \frac{1}{2}\,\brTheta_{1}\, \Gamma_m\, \bisF \, \brGamma_n \Theta_{2}\Bigr)\,, \end{align} where we defined $\kappa^{\text{s}}_{mn}\equiv \kappa_{(mn)}$ and $\kappa^{\text{a}}_{mn}\equiv \kappa_{[mn]}$ and their indices are raised or lowered with the metric $\CG_{mn}$\,. Note that $\cK_{MN}$ is an $\OO(10,10)$ matrix up to quadratic order in $\Theta_I$ ($I=1,2$). The modification of the DSM action, $S\to S+\Delta S$, is equivalent to the replacement of the generalized metric \begin{align} \cH_{MN} \ \to\ \cM_{MN}\equiv \cH_{MN} + \cK_{MN} \,. \label{eq:replacement} \end{align} The explicit form of $\cM_{MN}$ is given by \begin{align} (\cM_{MN}) &= \begin{pmatrix} \delta_m^p & B_{mp} \\ 0 & \delta^m_p \end{pmatrix} \begin{pmatrix} \CG_{pq}-\kappa^{\text{s}}_{pq} & (\kappa^{\text{a}})_{p}{}^q \\ -(\kappa^{\text{a}})^p{}_q & \CG^{pq}+(\kappa^{\text{s}})^{pq} \end{pmatrix} \begin{pmatrix} \delta^q_n & 0 \\ -B_{qn} & \delta_q^n \end{pmatrix} \nn\\ &= \begin{pmatrix} \delta_m^p & \hat{B}_{mp} \\ 0 & \delta^m_p \end{pmatrix} \begin{pmatrix} \hat{\CG}_{pq} & 0 \\ 0 & (\hat{\CG}^{-1})^{pq} \end{pmatrix} \begin{pmatrix} \delta^q_n & 0 \\ -\hat{B}_{qn} & \delta_q^n \end{pmatrix} + \cO(\Theta^4)\,, \end{align} where we defined \begin{align} \hat{\CG}_{mn}\equiv \CG_{mn}-\kappa^{\text{s}}_{mn}\,,\qquad \hat{B}_{mn}\equiv B_{mn}+\kappa^{\text{a}}_{mn} \,. \end{align} Then, we consider an action \begin{align} S = \frac{1}{4\pi\alpha'}\int \Bigl[\,\frac{1}{2}\, \cM_{MN}\,\Pi^M\wedge *_{\gga} \Pi^N - D X^M\wedge \bigl(\cA_{M} + \Sigma_{M} \bigr)\Bigr]\,. \label{eq:GS-DSM-ours} \end{align} Before we choose the diagonal gauge, spinors $\Theta_I$, R--R fields $\cF^{\SPa}{}_{\SPbrb}$, and the spin connections $\Phi_{\Lobra\Lob\Loc}$, $\brPhi_{\Loa\Lobrb\Lobrc}$ are invariant under global $\OO(10,10)$ transformations or (finite) generalized diffeomorphisms, while $\cH_{MN}$, $\cK_{MN}$, $\Pi^M$, and $\Sigma^M$ transform covariantly and the action is invariant \footnote{More precisely, as discussed in \cite{Lee:2013hma,Park:2016sbw}, $\cA^M$ does not transform covariantly because $\rmd X^M$ does not transform covariantly, and $D X^M\wedge \cA_{M}$ is not invariant under generalized diffeomorphisms. However, the variation is only the total-derivative term and the action is invariant under generalized diffeomorphisms.} The global double Lorentz symmetry is manifest but the local one is not manifest because $\Sigma^M$ contains non-covariant quantity $\rmd\Theta_I$ (or $\cK_{MN}$ contains the spin connection). The local symmetry becomes manifest only after eliminating the auxiliary fields. On the other hand, if we choose the diagonal gauge fixing, although the global $\OO(10,10)$ transformations are manifest, the covariance under generalized diffeomorphisms are lost, because the barred indices are rotated under the compensating local Lorentz transformation. Indeed, the transformation rule of fermionic fields after the diagonal gauge fixing is \begin{align} \Theta_1 \ \to \ \Theta_1 \,,\qquad \Theta_2 \ \to \ \Omega\, \Theta_2 \,, \label{eq:Theta-transf} \end{align} where $\Omega$ is the one given in \eqref{eq:Omega-def}, and in general, it is non-constant. Accordingly, $\rmd\Theta_2$ (and thus $\Sigma^M$ also) does not transform covariantly. It is interesting to note that all information on the curved background is contained in the generalized metric $\cM_{MN}$\,. The usual generalized metric $\cH_{MN}$ contains only the $P$-$P$ or $\brP$-$\brP$ components [see \eqref{eq:H-eta-orthogonal}] while other quantities such as the R--R fluxes are contained in the $P$-$\brP$ or $\brP$-$P$ components $\cK_{MN}$\,. In the following, we show that the action \eqref{eq:GS-DSM-ours} reproduces the conventional GS superstring action \cite{Cvetic:1999zs} up to quadratic order in fermions $\Theta_I$. \subsubsection{Classical equivalence to the type II GS action} In order to reproduce the conventional action, we choose the canonical section $\tilde{\partial}^m =0$. Then, the condition \eqref{eq:cA-condition} for $\cA^M$ indicates that $\cA^M$ takes the form $(\cA^M)=(0,\,A_m)$ and $DX^M$ becomes \begin{align} DX^M = \begin{pmatrix} \rmd X^m \\ \rmd \tilde{X}_m -A_m \end{pmatrix} \equiv \begin{pmatrix} \rmd X^m \\ P_m \end{pmatrix} \,, \end{align} where, for simplicity, we defined $P_m$ and treated it as a fundamental variable rather than $A_m$\,. The action then becomes \begin{align} S &= \frac{1}{4\pi\alpha'}\int \Bigl[\,\frac{1}{2}\, \cM_{MN}\,\Pi^M\wedge *_{\gga} \Pi^N - P_{m}\wedge \bigl(\rmd X^m+\Sigma^{m}\bigr)\Bigr] \nn\\ &\quad - \frac{1}{4\pi\alpha'}\int \bigl(\rmd X^m\wedge \tilde{\Sigma}_m + \rmd X^m\wedge \rmd \tilde{X}_m \bigr) \,. \end{align} We can expand the first line as \begin{align} &\frac{1}{2}\, \cM_{MN}\,\Pi^M\wedge *_{\gga} \Pi^N - P_{m}\wedge \bigl(\rmd X^m+\Sigma^{m}\bigr) \nn\\ &=\frac{1}{2}\, \hat{\CG}_{mn}\,\bigl(\rmd X^m+\Sigma^{m}\bigr)\wedge *_{\gga} \bigl(\rmd X^n+\Sigma^n\bigr) \nn\\ &\quad +\frac{1}{2}\, \hat{\CG}^{mn}\,\bigl[P_m+\tilde{\Sigma}_{m} -\hat{B}_{mp}\,(\rmd X^p+\Sigma^{p})\bigr]\wedge *_{\gga} \bigl[P_n+\tilde{\Sigma}_{n} -\hat{B}_{nq}\,(\rmd X^q+\Sigma^{q})\bigr] \nn\\ &\quad - P_{m}\wedge \bigl(\rmd X^m+\Sigma^{m}\bigr) \nn\\ &= \hat{\CG}_{mn}\,\bigl(\rmd X^m+\Sigma^{m}\bigr)\wedge *_{\gga} \bigl(\rmd X^n+\Sigma^n\bigr) + \bigl[\tilde{\Sigma}_n + \hat{B}_{mn}\, (\rmd X^m+\Sigma^{m})\bigr]\wedge \bigl(\rmd X^n+\Sigma^n\bigr) \nn\\ &\quad +\frac{1}{2}\, \hat{\CG}^{mn}\,\bigl[P_m+\tilde{\Sigma}_{m} -\hat{B}_{mp}\,(\rmd X^p+\Sigma^{p})-\hat{\CG}_{mp}\,*_{\gga} (\rmd X^p+\Sigma^{p})\bigr] \nn\\ &\quad\qquad\qquad \wedge *_{\gga} \bigl[P_n+\tilde{\Sigma}_{n} -\hat{B}_{nq}\,(\rmd X^q+\Sigma^{q}) -\hat{\CG}_{nq}\,*_{\gga} (\rmd X^q+\Sigma^{q})\bigr]\,, \end{align} and eliminating the auxiliary fields $P_m$, we obtain \begin{align} S &= \frac{1}{4\pi\alpha'}\int \bigl[\,\hat{\CG}_{mn}\,\bigl(\rmd X^m+\Sigma^{m}\bigr)\wedge *_{\gga} \bigl(\rmd X^n+\Sigma^n\bigr) + \hat{B}_{mn}\,\bigl(\rmd X^m+\Sigma^{m}\bigr)\wedge \bigl(\rmd X^n+\Sigma^n\bigr) \nn\\ &\qquad\qquad\quad - 2\, \rmd X^m\wedge \tilde{\Sigma}_{m}- \Sigma^{m} \wedge \tilde{\Sigma}_{m} - \rmd X^m\wedge \rmd \tilde{X}_m \bigr] \,. \end{align} By using the explicit expression for $\Sigma^M$, \begin{align} \Sigma^M = \begin{pmatrix} \Sigma^m \\ \tilde{\Sigma}_m \end{pmatrix} \equiv \begin{pmatrix} \Sigma^m \\ \hat{\Sigma}_m + B_{mn}\, \Sigma^n \end{pmatrix}\,,\quad \begin{pmatrix} \Sigma^m \\ \hat{\Sigma}_m \end{pmatrix} \equiv \frac{\ii}{2} \begin{pmatrix} \brTheta_1\,\Gamma^m\,\rmd \Theta_1 + \brbrTheta_2\,\brGamma^m\,\rmd \Theta_2 \\ \brTheta_1\,\Gamma_m\,\rmd \Theta_1 - \brbrTheta_2\,\brGamma_m\,\rmd \Theta_2 \end{pmatrix} \,, \end{align} and neglecting quartic terms in $\Theta$ and the topological term, the action becomes \begin{align} \begin{split} S &= \frac{1}{2\pi\alpha'}\int \rmd^2\sigma \sqrt{-\gga}\,\cL \,, \\ \cL&=-\frac{1}{2}\,\bigl({\gga}^{\WSa\WSb}- \varepsilon^{\WSa\WSb}\bigr)\,(\hat{\CG}_{mn}+ \hat{B}_{mn})\, \partial_{\WSa}X^m \, \partial_{\WSb}X^n - \CG_{mn}\, \partial_{\WSa}X^m \, \bigl(\gga^{\WSa\WSb}\,\Sigma_{\WSb}^n + \varepsilon^{\WSa\WSb}\, \hat{\Sigma}_{\WSb}^n \bigr) \,. \end{split} \end{align} In order to compare the obtained action with the conventional GS superstring action, let us further expand the Lagrangian as \begin{align} \begin{split} \cL &=- \Pg_{-}^{\WSa\WSb}\,(\CG_{mn}+ B_{mn})\, \partial_{\WSa}X^m \, \partial_{\WSb}X^n \\ &\quad - \ii\,\Pg_{+}^{\WSa\WSb}\, \partial_{\WSa}X^m \, \brTheta_1\,\Gamma_m\,\Bigl(\partial_{\WSb} \Theta_1 +\frac{1}{4}\, \partial_{\WSb}X^n\,\omega_{+n\Loa\Lob} \, \Gamma^{\Loa\Lob} \,\Theta_1\Bigr) \\ &\quad - \ii\,\Pg_{-}^{\WSa\WSb}\, \partial_{\WSa}X^m \, \brTheta_2\,\Gamma_m\,\Bigl(\partial_{\WSb} \Theta_2 - \frac{1}{4} \, \partial_{\WSb}X^n\,\omega_{-n\Loa\Lob} \, \brGamma^{\Loa\Lob}\,\Theta_2\Bigr) \\ &\quad + \frac{\ii}{8} \,\Pg_{+}^{\WSa\WSb}\, \brTheta_{1}\, \Gamma_m\, \bisF \, \brGamma_n \Theta_{2} \, \partial_{\WSa}X^m\,\partial_{\WSb}X^n \,. \end{split} \end{align} where we have defined \begin{align} \Pg_{\pm\WSa\WSb}\equiv \frac{\gga_{\WSa\WSb}\pm \varepsilon_{\WSa\WSb}}{2}\,,\qquad \brTheta_2\equiv \Theta_2^\dagger\,\Gamma^0 \qquad \bigl(\brbrTheta_2\,\brGamma_m = \brTheta_2\,\Gamma_m\bigr)\,, \end{align} and used the explicit form of the spin connection \eqref{eq:spin-connections}, \begin{align} \Phi_{m\Loa\Lob} = \frac{1}{\sqrt{2}}\,\omega_{+m\Loa\Lob}\,,\qquad \brPhi_{m\Lobra\Lobrb}= -\frac{1}{\sqrt{2}}\,\omega_{-m\Loa\Lob}\,, \qquad \omega_{\pm m\Loa\Lob}\equiv \omega_{m\Loa\Lob} \pm \frac{1}{2}\, e_m{}^{\Loc}\,H_{\Loc\Loa\Lob}\,. \label{eq:Phi-omega-pm} \end{align} Further using \begin{align} \brGamma^{\Loa\Lob}=-\Gamma^{\Loa\Lob}\,,\qquad \bisF \, \brGamma_m = \mp\,\bisF \, \Gamma_m \quad (\text{IIA/IIB})\,, \end{align} we obtain the type II superstring action \begin{align} \begin{split} \cL_{\text{IIA/IIB}} &= - \Pg_{-}^{\WSa\WSb}\, (\CG_{mn}+B_{mn})\, \partial_{\WSa}X^m\,\partial_{\WSb}X^n \\ &\quad - \ii\, \bigl(\Pg_{+}^{\WSa\WSb} \,\partial_{\WSa} X^m\, \brTheta_{1}\, \Gamma_m\, D_{+\WSb}\Theta_{1} + \Pg_{-}^{\WSa\WSb} \,\partial_{\WSa} X^m\, \brTheta_{2}\, \Gamma_m\, D_{-\WSb}\Theta_{2} \bigr) \\ &\quad \mp \frac{\ii}{8}\, \Pg_{+}^{\WSa\WSb}\, \brTheta_{1}\, \Gamma_m\, \bisF \, \Gamma_n\, \Theta_{2} \, \partial_{\WSa} X^m\, \partial_{\WSb} X^n \,, \end{split} \label{eq:GS-action-rewriting} \end{align} where we defined \begin{align} D_{\pm \WSa} \equiv \partial_{\WSa} + \frac{1}{4}\, \partial_{\WSa} X^m\, \omega_{\pm m}{}^{\Loa\Lob}\, \Gamma_{\Loa\Lob} \,. \end{align} For type IIA superstring, defining $\Theta\equiv \Theta_1+\Theta_2$\,, we obtain a simple action \begin{align} \begin{split} \cL_{\text{IIA}} &=- \Pg_{-}^{\WSa\WSb}\,(\CG_{mn}+ B_{mn})\, \partial_{\WSa}X^m \, \partial_{\WSb}X^n \\ &\quad - \frac{\ii}{2}\, \gga^{\WSa\WSb} \,\partial_{\WSa} X^m\,\brTheta\,\Gamma_m\, \bm{D}_{\WSb}\Theta - \frac{\ii}{2}\, \varepsilon^{\WSa\WSb}\,\partial_{\WSa} X^m\,\brTheta\, \Gamma^{11} \, \Gamma_m\,\bm{D}_{\WSb}\Theta \,, \end{split} \end{align} where we defined \begin{align} \bm{D}_{\WSa} \equiv \partial_{\WSa} + \frac{1}{4}\, \partial_{\WSa} X^m\, \omega_m{}^{\Loa\Lob}\, \Gamma_{\Loa\Lob} - \frac{1}{8}\,\partial_{\WSa} X^m\, H_m{}^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\, \Gamma^{11} + \frac{1}{16} \, \partial_{\WSa}X^m\, \bisF \, \Gamma_m \,. \end{align} On the other hand, for type IIB superstring, using the Pauli matrices $\sigma_i^{IJ}$ ($i=1,2,3$), we can rewrite the action in a familiar form \begin{align} \begin{split} \cL_{\text{IIB}} &= - \Pg_{-}^{\WSa\WSb}\, (\CG_{mn}+B_{mn})\, \partial_{\WSa}X^m\,\partial_{\WSb}X^n \\ &\quad - \frac{\ii}{2}\, \bigl(\gga^{\WSa\WSb}\,\delta^{IK}+\varepsilon^{\WSa\WSb}\,\sigma_3^{IK}\bigr) \,\brTheta_{I}\, \partial_{\WSa} X^m\, \Gamma_m\, D^{KJ}_{\WSb}\Theta_{J} \,, \end{split} \label{eq:GS-action-conventional} \end{align} where we used \eqref{eq:bTGGGT} and defined \begin{align} \begin{split} D^{IJ}_{\WSa} &\equiv \delta^{IJ}\, \Bigl(\partial_{\WSa} + \frac{1}{4}\, \partial_{\WSa} X^m\, \omega_m{}^{\Loa\Lob}\, \Gamma_{\Loa\Lob} \Bigr) + \frac{1}{8}\,\sigma_3^{IJ}\, \partial_{\WSa} X^m\, H_{m\Loa\Lob}\,\Gamma^{\Loa\Lob} \\ &\quad - \frac{1}{8}\, \Big(\epsilon^{IJ}\, \bisF_1 + \sigma_1^{IJ}\, \bisF_3 +\frac{1}{2}\,\epsilon^{IJ}\, \bisF_5 \Big)\, \partial_{\WSa} X^n\, \Gamma_n \,, \\ \bisF_p&\equiv \frac{1}{p!}\,\hat{\cF}_{\Loa_1\cdots\Loa_p}\,\Gamma^{\Loa_1\cdots\Loa_p}\,,\qquad (\epsilon^{IJ}) \equiv \footnotesize{\begin{pmatrix} 0 & \,1\, \\ -1 & 0 \end{pmatrix}}\,. \end{split} \end{align} As discussed around \eqref{eq:Psi-Tdual}, under a single $T$-duality along the $x^z$-direction, the fermionic variables transform as \begin{align} \Theta_1 \ \to \ \Theta_1 \,,\qquad \Theta_2 \ \to \ \frac{1}{\sqrt{\CG_{zz}}}\,\Gamma_z\, \Theta_2 \,. \end{align} Since it flips the chirality of $\Theta_2$, it maps type IIA and IIB superstring to each other. \section{YB deformations of \texorpdfstring{$\AdS{5} \times \rmS^5$}{AdS\textfiveinferior{x}S\textfivesuperior} superstring} \label{sec:YB-AdS5xS5} In this section, we revisit the homogeneous YB deformations of the $\AdS{5} \times \rmS^5$ superstring. After a concise review of the supercoset construction of $\AdS{5}\times\rmS^5$ superstring, we show that the action of the YB sigma model can be expressed as the GS superstring action in the $\beta$-deformed $\AdS{5} \times \rmS^5$ background. \vspace{-\Pskip} \subsection{A supercoset construction of \texorpdfstring{$\AdS{5} \times \rmS^5$}{AdS\textfiveinferior{x}S\textfivesuperior} superstring} \subsubsection{Metsaev--Tseytlin action} Before considering YB deformations, we review a supercoset construction of the $\AdS{5} \times \rmS^5$ superstring action based on the supercoset \begin{align} \frac{\PSU(2,2|4)}{\SO(1,4)\times \SO(5)}\,. \end{align} In order to perform the supercoset construction, we introduce a coset representative $g\in\SU(2,2|4)$\footnote{In order to obtain a matrix representation of the $\PSU(2,2|4)$ group (which is necessary in Appendix \ref{app:kappa}), we consider the $\SU(2,2|4)$ group and project out the central generator $\gZ$ as explained in Appendix \ref{app:psu-algebra}.} and define the left-invariant current $A$ as \begin{align} A=g^{-1}\,\rmd g \,, \end{align} which satisfies the Maurer--Cartan equation \begin{align} \rmd A + A\wedge A = 0\,. \end{align} By using the projections $P^{(i)}$ $(i=0,1,2,3)$ to the $\mathbb{Z}_4$-graded components of $\alg{g}\equiv\alg{su}(2,2|4)$ [see \eqref{eq:P-i-projector}], we decompose the left-invariant current as \begin{align} A = A^{(0)} +A^{(1)} +A^{(2)} + A^{(3)} \,. \end{align} We also define projection operators $d_\pm$ as \begin{align} d_{\pm} \equiv \mp P^{(1)}+2\,P^{(2)}\pm P^{(3)} \,, \label{eq:dpm} \end{align} which satisfy \begin{align} \str \bigl[ X\,d_\pm(Y) \bigr] = \str \bigl[ d_\mp (X)\, Y \bigr] \,. \label{eq:dpm-transpose} \end{align} Now, we consider the sigma model action \cite{Metsaev:1998it} \begin{align} S&=\frac{\dlT}{2}\int \str\bigl(A^{(2)}\wedge *_{\gga} A^{(2)} - A^{(1)}\wedge A^{(3)} \bigr) \nn\\ &=-\frac{\dlT}{2}\int \rmd^2\sigma\,\sqrt{-\gga}\,\Pg_{-}^{\WSa\WSb} \, \str \bigl[\,A_{\WSa} \, d_-(A_{\WSb})\,\bigr]\,, \label{AdS5S5-action} \end{align} where $\dlT \equiv R^2/2\pi\alpha'$ ($R$\,: the radius of $\AdS{5}$ and $\rmS^5$) is the dimensionless string tension. In order to relate the supercoset sigma model action to the $\AdS{5} \times \rmS^5$ superstring action, let us prescribe a concrete parameterization of $g$ and expand the action up the second order in fermions. We first decompose the group element into the bosonic and the fermionic parts, \begin{align} g =g_{\bos}\cdot g_{\fer}\,, \end{align} and parameterize the bosonic part $g_{\bos}$ as (see Appendix \ref{app:psu-algebra} for the details) \begin{align} \begin{split} &g_{\bos} = g_{\AdS5}\cdot g_{\rmS^5}\,,\qquad g_{\AdS5} \equiv \exp(x^\mu\,P_\mu)\cdot \exp(\ln z \,D)\,, \\ &g_{\rmS^5} \equiv \exp(\phi_1\, h_1+\phi_2\,h_2+\phi_3\,h_3)\cdot \exp(\xi\,\gJ_{56})\cdot \exp(r\,\gP_5)\,. \end{split} \label{eq:group-parameterization} \end{align} Here, $P_{\mu}$ ($\mu,\nu=0,\dotsc,3$) and $D$ are the translation and dilatation generators in the conformal algebra $\alg{so}(2,4)$, and $h_{i}\, (i=1,2,3)$ are Cartan generators of the $\alg{so}(6)$ algebra, given by \begin{align} h_1 \equiv \gJ_{57}\,,\qquad h_2 \equiv \gJ_{68}\,,\qquad h_3 \equiv \gP_9\,. \end{align} On the other hand, we parameterize the fermionic part $g_{\fer}$ as \begin{align} g_{\fer} = \exp(\gQ^I\, \theta_I) \,, \qquad \gQ^I\,\theta_I=(\gQ^I)^{\check{\SPa}\hat{\SPa}}\,\theta_{I\check{\SPa}\hat{\SPa}} \,, \end{align} where the supercharges $(\gQ^I)^{\check{\SPa}\hat{\SPa}}$ $(I=1,2)$ are labeled by two indices $(\check{\SPa}\,,\hat{\SPa}=1,\dotsc, 4)$ and $\theta_{I\check{\SPa}\hat{\SPa}}\,(I=1,2)$ are $16$-components Majorana--Weyl fermions. Then, we can expand the left-invariant current $A$ as \begin{align} A &= g_{\fer}^{-1}\,A_{(0)}\,g_{\fer} + \gQ^I\,\rmd\theta_I \nn\\ &= A_{(0)} + [A_{(0)},\,\gQ^I\,\theta_I] + \frac{1}{2}\,\bigl[[A_{(0)},\,\gQ^I\,\theta_I],\,\gQ^J\,\theta_J\bigr] + \gQ^I\,\rmd\theta_I + \cO(\theta^3) \label{eq:A-AdS5xS5} \\ &= \Bigl(e^{\Loa}+\frac{\ii}{2}\,\brtheta_I\,\hat{\gamma}^{\Loa}\,D^{IJ}\theta_J\Bigr)\,\gP_{\Loa} -\frac{1}{2}\,\Bigl(\omega^{\Loa\Lob}-\frac{1}{4}\,\epsilon^{IK}\,\brtheta_I\,\gamma^{\Loc\Lod}\,R_{\Loc\Lod}{}^{\Loa\Lob}\,D^{KJ}\theta_J\Bigr)\,\gJ_{\Loa\Lob} +\gQ^I\,D^{IJ}\theta_J + \cO(\theta^3)\,, \nn \end{align} where we have defined \begin{align} \begin{split} A_{(0)}&\equiv g_{\bos}^{-1}\,\rmd g_{\bos} = \Bigl(e_m{}^{\Loa}\,\gP_{\Loa} - \frac{1}{2}\, \omega_m{}^{\Loa\Lob}\,\gJ_{\Loa\Lob}\Bigr)\,\rmd X^m \,, \\ D^{IJ} &\equiv \delta^{IJ}\,\Bigl(\rmd + \frac{1}{4}\,\omega^{\Loa\Lob}\,\gamma_{\Loa\Lob}\Bigr)+\frac{\ii}{2}\,\epsilon^{IJ}\,e^{\Loa}\,\hat{\gamma}_{\Loa}\,, \qquad (X^m)=(x^\mu,\,z,\,r,\,\xi,\,\phi_1,\,\phi_2,\,\phi_3)\,, \end{split} \end{align} and used $\delta^{IJ}\,\brtheta_I\,\hat{\gamma}^{\Loa}\,\rmd \theta_J = 0$ and $\epsilon^{IJ}\,\brtheta_I\,\gamma^{\Loa\Lob}\,\rmd \theta_J = 0$.\footnote We have also used \begin{align*} &[A_{(0)},\,\gQ^I\,\theta_I]= \gQ^I\,\Bigl(\frac{1}{4}\,\delta^{IJ}\,\omega^{\Loa\Lob} \,\gamma_{\Loa\Lob} + \frac{\ii}{2}\,\epsilon^{IJ}\, e^{\Loa}\, \hat{\gamma}_{\Loa} \Bigr)\,\theta_J \,, \\ &\bigl[[A_{(0)},\,\gQ^I\,\theta_I],\,\gQ^J\,\theta_J\bigr] = \ii\,\brtheta_I\,\hat{\gamma}^{\Loa}\,\Bigl(\frac{1}{4}\,\delta^{IJ}\,\omega^{\Loc\Lod}\, \gamma_{\Loc\Lod} + \frac{\ii}{2}\,\epsilon^{IJ}\, e^{\Lob}\, \hat{\gamma}_{\Lob} \Bigr)\,\theta_J\,\gP_{\Loa} \\ &\qquad\qquad\qquad\qquad\qquad + \frac{1}{4}\,\epsilon^{IK}\, \brtheta_I\, \gamma^{\Loc\Lod}\,\Bigl(\frac{1}{4}\,\delta^{KJ}\,\omega^{\Loa\Lob}\, \gamma_{\Loa\Lob} + \frac{\ii}{2}\,\epsilon^{KJ}\, e^{\Loa}\, \hat{\gamma}_{\Loa} \Bigr)\,\theta_J\,\rmd X^m\, R_{\Loc\Lod}{}^{\Loe\Lof}\,\gJ_{\Loe\Lof} \\ &\qquad\qquad\qquad\qquad\qquad + \text{(irrelevant terms proportional to the central charge $\gZ$)}\,. \end{align* } The vielbein $e^{\Loa}=e_{m}{}^{\Loa}\,\rmd X^m$ takes the form \begin{align} e^{\Loa} = \biggl(\frac{\rmd x^0}{z},\frac{\rmd x^1}{z} ,\frac{\rmd x^2}{z},\frac{\rmd x^3}{z}, \frac{\rmd z}{z}, \rmd r, \sin r\,\rmd \xi, \sin r\,\cos\xi\,\rmd\phi_1,\sin r\,\sin\xi\,\rmd\phi_2,\cos r\,\rmd\phi_3\biggr) \,, \end{align} and $\omega^{\Loa\Lob}=\omega_m{}^{\Loa\Lob}\,\rmd X^m$ and $R_{\Loa\Lob\Loc\Lod}$ are the corresponding spin connection and the Riemann curvature tensor (see Appendix \ref{app:conventions} for our conventions). Using the expansion \eqref{eq:A-AdS5xS5}, we can straightforwardly obtain \begin{align} \frac{1}{2}\,\str\bigl[\,A_{\WSa}\,d_-(A_{\WSb})\,\bigr] &= \eta_{\Loa\Lob}\,e_{\WSa}{}^{\Loa}\,e_{\WSb}{}^{\Lob} + \ii\, \bigl[\,e_{\WSb}{}^{\Loa}\, (\brtheta_1\,\hat{\gamma}_{\Loa}\, \partial_{\WSa} \theta_1) + e_{\WSa}{}^{\Loa}\, (\brtheta_2\,\hat{\gamma}_{\Loa}\, \partial_{\WSb} \theta_2) \,\bigr] \nn\\ &\quad + \frac{\ii}{4}\,\Bigl[e_{\WSb}{}^{\Lob}\, e_{\WSa}{}^{\Loa}\,\omega_{\Loa}{}^{\Loc\Lod} \,(\brtheta_1\,\hat{\gamma}_{\Lob}\,\gamma_{\Loc\Lod} \,\theta_1) + e_{\WSa}{}^{\Loa}\, e_{\WSb}{}^{\Lob}\, \omega_{\Lob}{}^{\Loc\Lod} \, (\brtheta_2\,\hat{\gamma}_{\Loa}\,\gamma_{\Loc\Lod}\,\theta_2) \Bigr] \nn\\ &\quad - e_{\WSb}{}^{\Loa} \,e_{\WSa}{}^{\Lob}\, \brtheta_1\,\hat{\gamma}_{\Loa}\, \hat{\gamma}_{\Lob}\,\theta_2 \,, \end{align} where $e_{\WSa}{}^{\Loa}\equiv e_m{}^{\Loa}\,\partial_{\WSa}X^m$\,. Further using Eqs.~\eqref{eq:lift-32-AdS5-1}, \eqref{eq:lift-32-AdS5-2}, and \eqref{eq:lift-32-AdS5-3}, we obtain \begin{align} \frac{1}{2}\,\str\bigl[\,A_{\WSa}\,d_-(A_{\WSb})\,\bigr] &= \CG_{mn}\,\partial_{\WSa} X^m\,\partial_{\WSb} X^n + \ii\, \bigl[\,e_{\WSb}{}^{\Loa}\, \brTheta_1\,\Gamma_{\Loa}\, \partial_{\WSa} \Theta_1 + e_{\WSa}{}^{\Loa}\, \brTheta_2\,\Gamma_{\Loa}\, \partial_{\WSb} \Theta_2 \,\bigr] \nn\\ &\quad + \frac{\ii}{4}\,\Bigl[e_{\WSb}{}^{\Lob}\, e_{\WSa}{}^{\Loa}\,\omega_{\Loa}{}^{\Loc\Lod} \, \brTheta_1\,\Gamma_{\Lob}\,\Gamma_{\Loc\Lod} \,\Theta_1 + e_{\WSa}{}^{\Loa}\, e_{\WSb}{}^{\Lob}\, \omega_{\Lob}{}^{\Loc\Lod} \, \brTheta_2\,\Gamma_{\Loa}\,\Gamma_{\Loc\Lod}\,\Theta_2 \Bigr] \nn\\ &\quad - \frac{\ii}{8}\,e_{\WSb}{}^{\Loa} \,e_{\WSa}{}^{\Lob}\, \brTheta_I\,\Gamma_{\Loa}\,\bisF_5\,\Gamma_{\Lob}\,\Theta_J\,, \end{align} where $\bisF_5$ is a bispinor \begin{align} \bisF_5 \equiv 4\,(\Gamma^{01234}+\Gamma^{56789})\,. \end{align} Then, the action \eqref{AdS5S5-action} becomes the GS type IIB superstring action \eqref{eq:GS-action-conventional} with the target space given by the familiar $\AdS{5} \times \rmS^5$ background \footnote{Here, we have defined $(\eta_{\mu\nu})=\diag(-1,1,1,1)$ and \begin{align*} \begin{split} &\rmd s_{\rmS^5}^2\equiv \rmd r^2 + \sin^2 r\, \rmd\xi^2 + \cos^2\xi\,\sin^2 r\, \rmd\phi_1^2 + \sin^2r\,\sin^2\xi\, \rmd\phi_2^2 + \cos^2r\, \rmd\phi_3^2\,, \\ &\omega_{\AdS5} \equiv - \frac{\rmd x^0\wedge\rmd x^1\wedge \rmd x^2\wedge\rmd x^3\wedge\rmd z}{z^5}\,, \\ &\omega_{\rmS^5} \equiv \sin^3r \cos r \sin\xi \cos\xi\,\rmd r\wedge \rmd\xi\wedge\rmd \phi_1\wedge \rmd\phi_2\wedge\rmd \phi_3\qquad (\omega_{\AdS{5}} = *_{10}\omega_{\rmS^5}) \,. \end{split} \end{align*} \begin{align} \rmd s^2 = \frac{\eta_{\mu\nu}\,\rmd x^\mu\,\rmd x^\nu + \rmd z^2}{z^2}+\rmd s_{\rmS^5}^2\,, \qquad B_{mn} =0\,,\qquad \Exp{\Phi}\hat{F}_5 = 4\,\bigl(\omega_{\AdS5}+\omega_{\rmS^5}\bigr) \,. \end{align} From the supergravity equations of motion (or the Weyl invariance of string theory), the dilaton is determined as $\Phi=0$\,. \subsubsection{Killing vectors} For later convenience, let us calculate the Killing vectors $\hat{T}_i\equiv\hat{T}_i^m\,\partial_m$ associated with the bosonic symmetries $T_i$ of the $\AdS{5}$ background. From the general formula \eqref{eq:Killing-formula} explained in Appendix \ref{app:homogeneous-space}, the Killing vectors can be expressed as \begin{align} \hat{T}_{i}=\hat{T}_{i}{}^{m}\,\partial_{m} =\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_{i}{}^{\Loa}\, e_{\Loa}{}^{m}\, \partial_{m} = \str\bigl(g_{\bos}^{-1}\,T_{i}\,g_{\bos}\,\gP_{\Loa}\bigr)\,e^{\Loa m}\, \partial_{m}\,, \label{eq:Killing-Formula} \end{align} where we introduced a notation $g\,T_i\,g^{-1}\equiv [\Ad_{g}]_{i}{}^{j}\,T_j$\,. By using our parameterization \eqref{eq:group-parameterization}, the Killing vectors on the $\AdS{5}$ background are given by \begin{align} \begin{split} \hat{P}_\mu &\equiv \str\bigl(g_{\bos}^{-1}\,P_{\mu}\,g_{\bos}\, \gP_{\Loa} \bigr)\,e^{\Loa m}\,\partial_m = \partial_\mu \,, \\ \hat{K}_\mu &\equiv \str\bigl(g_{\bos}^{-1}\,K_{\mu}\,g_{\bos}\, \gP_{\Loa} \bigr)\,e^{\Loa m}\,\partial_m = \bigl(x^\nu\,x_\nu +z^2\bigr)\,\partial_\mu - 2\,x_\mu\,(x^\nu\,\partial_\nu+z\,\partial_z)\,, \\ \hat{M}_{\mu\nu} &\equiv \str\bigl(g_{\bos}^{-1}\,M_{\mu\nu}\,g_{\bos}\, \gP_{\Loa} \bigr)\,e^{\Loa m}\,\partial_m = x_\mu\,\partial_\nu -x_\nu\,\partial_\mu \,, \\ \hat{D} &\equiv \str\bigl(g_{\bos}^{-1}\,D\,g_{\bos}\, \gP_{\Loa} \bigr)\,e^{\Loa m}\,\partial_m = x^\mu\,\partial_\mu + z\,\partial_z \,. \end{split} \end{align} The Lie brackets of these vector fields satisfy the same commutation relations \eqref{eq:so(2-4)-algebra} as the conformal algebra $\alg{so}(2,4)$ (with negative sign, $[\hat{T}_i,\,\hat{T}_j]=-f_{ij}{}^k\,\hat{T}_k$): \begin{align} \begin{split} &[\hat{P}_\mu,\, \hat{K}_\nu]= -2\,\bigl(\eta_{\mu\nu}\, \hat{D} - \hat{M}_{\mu\nu}\bigr)\,,\quad [\hat{D},\, \hat{P}_{\mu}]= -\hat{P}_\mu\,,\quad [\hat{D},\,\hat{K}_\mu]= \hat{K}_\mu\,, \\ &[\hat{M}_{\mu\nu},\, \hat{P}_\rho] = -\eta_{\mu\rho}\, \hat{P}_\nu +\eta_{\nu\rho}\, \hat{P}_\mu \,,\quad [\hat{M}_{\mu\nu},\, \hat{K}_\rho] = -\eta_{\mu\rho}\,\hat{K}_\nu + \eta_{\nu\rho}\,\hat{K}_\mu\,, \\ &[\hat{M}_{\mu\nu},\,\hat{M}_{\rho\sigma}] = -\eta_{\mu\rho}\,\hat{M}_{\nu\sigma}+\eta_{\mu\sigma}\,\hat{M}_{\nu\rho} + \eta_{\nu\rho}\,\hat{M}_{\mu\sigma}-\eta_{\nu\sigma}\,\hat{M}_{\mu\rho}\,. \end{split} \end{align} \vspace{-\Pskip} \subsection{YB deformed \texorpdfstring{$\AdS{5} \times \rmS^5$}{AdS\textfiveinferior{x}S\textfivesuperior} backgrounds} \label{sec:YB-sigma} Let us now consider (homogeneous) YB deformations of the $\AdS{5} \times \rmS^5$ superstring action. A key ingredient that characterizes the YB deformation is an $R$-operator. It is a linear operator $R:\alg{g} \to \alg{g}$, that solves the homogeneous CYBE \begin{align} \CYBE(X,Y) \equiv [R(X),\,R(Y)] - R([R(X),\,Y]+[X,\,R(Y)])=0\,, \label{eq:CYBE} \end{align} where $X,\,Y \in\alg{g}$\,. We also define the dressed $R$-operator $R_g$ as \begin{align} R_g(X)\equiv g^{-1}\, R(g\,X\,g^{-1})\,g \,, \end{align} which also satisfies the homogeneous CYBE \eqref{eq:CYBE}, \begin{align} \CYBE_g(X,Y) \equiv [R_g(X),\,R_g(Y)] - R_g([R_g(X),\,Y]+[X,\,R_g(Y)])=0\,, \label{eq:CYBE-g} \end{align} as long as $R$ satisfies the homogeneous CYBE. Then, the action of YB-deformed $\AdS{5} \times \rmS^5$ superstring is given by \begin{align} S_{\YB}=-\frac{\dlT}{2}\int \rmd^2\sigma\,\sqrt{-\gga}\,\Pg_{-}^{\WSa\WSb}\, \str\bigl[A_{\WSa}\, d_-\circ\cO_-^{-1}(A_{\WSb})\bigr]\,, \label{eq:YBsM} \end{align} where we defined linear operators $\cO_{\pm}$ as \begin{align} \cO_{\pm}=1\pm\eta\, R_g\circ d_\pm\,, \end{align} and $\eta\in\mathbb{R}$ is a deformation parameter. This action reduces to the undeformed $\AdS{5}\times \rmS^5$ action \eqref{AdS5S5-action} by taking $\eta=0$\,. In this paper, we consider a class of $R$-operators that can be specified by using a skew-symmetric classical $r$-matrix. By introducing an $r$-matrix $r\in \alg{g}\otimes \alg{g}$ of the form \begin{align} r=\frac{1}{2}\,r^{ij}\,T_{i}\wedge T_{j}\,, \qquad r^{ij}=-r^{ji}\,, \qquad T_{i}\in \alg{g}\,, \end{align} the $R$-operator is defined as \begin{align} R(X) = r^{ij}\, T_{i}\,\str(T_{j}\,X) \,, \qquad X\in\alg{g} \,. \label{eq:R-operator} \end{align} Then, the YB deformations are characterized only by the $r$-matrix. In terms of the $r$-matrix, the homogeneous CYBE \eqref{eq:CYBE} can be expressed as \begin{align} f_{l_1l_2}{}^i\,r^{jl_1}\,r^{kl_2} + f_{l_1l_2}{}^j\,r^{kl_1}\,r^{il_2} + f_{l_1l_2}{}^k\,r^{il_1}\,r^{jl_2} =0\,, \label{eq:CYBE-r} \end{align} where $f_{ij}{}^k$ is the structure constant, $[T_i,\,T_j]=f_{ij}{}^k\,T_k$\,. \subsubsection{Preparations} In the following, we rewrite the YB-deformed action in the form of the conventional GS action, and show that the target space is a $\beta$-deformed $\AdS{5} \times \rmS^5$ background. In order to determine the deformed background, it is sufficient to expand the action up to quadratic order in fermions, \begin{align} S_{\YB}=S_{(0)}+S_{(2)}+\cO(\theta^4)\,. \end{align} This kind of analysis has been performed in \cite{Arutyunov:2015qva} for the $q$-deformation of $\AdS{5}\times \rmS^5$ and in \cite{Kyono:2016jqy} for homogeneous YB deformations. However, the obtained deformed actions in the previous works are very complicated and it is not easy to read off the deformed background explicitly. In this paper, we provide a general formula for the deformed background for an arbitrary $r$-matrix satisfying homogeneous CYBE, though our analysis is limited to the cases where the $r$-matrices are composed only of the bosonic generators of the superalgebra $\alg{g}$\,.\footnote{Rewriting of the YB sigma model action to the standard GS form based on the $\kappa$-symmetry was done in \cite{Borsato:2016ose} to full order in fermionic variables, and there, the deformed background associated with a general $r$-matrix was determined.} In order to expand the YB sigma model action \eqref{eq:YBsM}, let us introduce some notations. Since we are supposing that the $r$-matrices are composed of the bosonic generators, the dressed $R$-operator $R_{g_{\bos}}$ acts as \begin{align} R_{g_{\bos}}(\gP_{\Loa}) =\lambda_{\Loa}{}^{\Lob}\,\gP_{\Lob}+\frac{1}{2}\,\lambda_{\Loa}{}^{\Lob\Loc}\,\gJ_{\Lob\Loc}\,, \quad R_{g_{\bos}}(\gJ_{\Loa\Lob}) =\lambda_{\Loa\Lob}{}^{\Loc}\,\gP_{\Loc}+\frac{1}{2}\,\lambda_{\Loa\Lob}{}^{\Loc\Lod}\,\gJ_{\Loc\Lod}\,, \quad R_{g_{\bos}}(\gQ^I) =0\,. \label{eq:Rg-operation} \end{align} According to the definition \eqref{eq:R-operator}, the (dressed) $R$-operator is skew-symmetric \begin{align} \str\bigl[R_{g_{\bos}}(X)\,Y\bigr] = r^{ij}\,\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_i{}^k\,\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_j{}^l\, \str(T_{k}\,Y)\,\str(T_{l}\, X) = - \str\bigl[X\,R_{g_{\bos}}(Y)\bigr] \,, \end{align} and by choosing $X$ and $Y$ as $\gP_{\Loa}$ or $\gJ_{\Loa\Lob}$, we obtain the following relations: \begin{align} \lambda_{\Loa\Lob} \equiv \lambda_{\Loa}{}^{\Loc}\,\eta_{\Loc\Lob} = -\lambda_{\Lob\Loa}\,,\quad \lambda_{\Loa\Lob}{}^{\Loc}=-\frac{1}{2}\,\eta^{\Loc\Lod}\,R_{\Loa\Lob\Loe\Lof}\,\lambda_{\Lod}{}^{\Loe\Lof}\,,\quad \lambda_{\Loa\Lob}{}^{\Loe\Lof}\,R_{\Loe\Lof\Loc\Lod} = - \lambda_{\Loc\Lod}{}^{\Loe\Lof}\,R_{\Loe\Lof\Loa\Lob}\,. \label{eq:lambda-properties} \end{align} For later convenience, we introduce the deformed currents as \begin{align} J_{\pm} \equiv \cO_{\pm}^{-1}\,A_{\pm}\,. \label{eq:J-O-inv-A} \end{align} By using the results of Appendix \ref{app:expansion-O}, it can be expanded as \begin{align} J_\pm&=\cO^{-1}_{\pm(0)}(A_{(0)})+\cO^{-1}_{\pm(0)}(A_{(1)})+\cO^{-1}_{\pm(1)}(A_{(0)})+\cO(\theta^2) \nn\\ &=e_{\pm}^{\Loa}\,\gP_{\Loa}-\frac{1}{2}\,W_{\pm}^{\Loa\Lob}\,\gJ_{\Loa\Lob} + \gQ^I\,D^{IJ}_{\pm}\theta_J +\cO(\theta^2)\,, \label{eq:Jpm-expansion} \end{align} where we have defined \begin{align} &e_{\pm}^{\Loa}\equiv e^{\Lob}\,k_{\pm \Lob}{}^{\Loa}\,,\qquad k_{\pm \Loa}{}^{\Lob} \equiv \bigl[(1\pm 2\,\eta\,\lambda)^{-1}\bigr]{}_{\Loa}{}^{\Lob}\,, \qquad W_{\pm}^{\Loa\Lob} \equiv \omega^{\Loa\Lob}\pm 2\,\eta\,e_{\pm}^{\Loc}\,\lambda_{\Loc}{}^{\Loa\Lob}\,, \label{eq:e-torsionful-spin-pm} \\ &D^{IJ}_{\pm}\equiv \delta^{IJ}\,D_{\pm} +\frac{\ii}{2}\,\epsilon^{IJ}\,e_{\pm}^{\Loa}\,\hat{\gamma}_{\Loa}\,, \qquad D_{\pm}\equiv \rmd+\frac{1}{4}\,W_{\pm}^{\Loa\Lob}\, \gamma_{\Loa\Lob}\,. \end{align} As it turns out, $e_{\pm}^{\Loa}$ and $W_{\pm}^{\Loa\Lob}$\,, respectively, play the roles of the two vielbeins and the torsionful spin connections $\omega_{\pm}$ \eqref{eq:Phi-omega-pm} in the deformed background.\footnote{More precisely, we have $W_{\pm \Loa\Lob}\wedge e_{\mp}^{\Loa}\wedge e_{\mp}^{\Lob}=\bigl(\omega_{[\mp] \Loa\Lob} \pm\frac{1}{2}\,e_{\mp}^{\Loc}\,H'_{\Loc\Loa\Lob}\bigr)\wedge e_{\mp}^{\Loa}\wedge e_{\mp}^{\Lob}$\,, where $\omega_{[\pm]}$ represent the spin connections associated with the vielbeins $e_{\pm}$ and $H'_3$ represents the $H$-flux in the deformed background.} \subsubsection{NS--NS sector} \label{sec:YB-NS-NS} Let us first consider the NS--NS part of the YB sigma model action, \begin{align} S_{(0)}=-\frac{\dlT}{2}\int \rmd^2\sigma\,\sqrt{-\gga}\,\Pg_{-}^{\WSa\WSb}\, \str\bigl[A_{\WSa (0)}\, d_-\circ\cO_{-(0)}^{-1}(A_{\WSb(0)})\bigr]\,. \end{align} From \eqref{eq:Oinv-0}, we can easily see that it takes the form \begin{align} S_{(0)} =-\dlT\int \rmd^2\sigma\,\sqrt{-\gga}\,\Pg_{-}^{\WSa\WSb}\, \eta_{\Loa\Lob}\,e_{\WSa}{}^{\Loa}\,e_{\WSb}{}^{\Loc}\,k_{-\Loc}{}^{\Lob}\,. \label{eq:action-order0} \end{align} By comparing this with the NS--NS part of the GS action [i.e.~the first line of \eqref{eq:GS-action-conventional}], we can regard \eqref{eq:action-order0} as the NS--NS part of the string sigma model on a deformed background \begin{align} \CG'_{mn} = e_{(m}{}^{\Loa}\,e_{n)}{}^{\Lob}\, k_{+\Loa\Lob} \,,\qquad B'_{mn} = e_{[m}{}^{\Loa}\,e_{n]}{}^{\Lob}\, k_{+\Loa\Lob} \,. \label{eq:G-B-prime} \end{align} Then, we obtain \begin{align} E'^{mn} \equiv \bigl[(\CG' + B')^{-1}\bigr]^{mn} = (k_+^{-1})^{\Loa\Lob}\, e_{\Loa}{}^{m}\,e_{\Lob}{}^{n} = (\eta + 2\,\eta\,\lambda)^{\Loa\Lob}\, e_{\Loa}{}^{m}\,e_{\Lob}{}^{n} \,. \end{align} In the original $\AdS{5}\times\rmS^5$ background, the $B$-field is absent and we have \begin{align} E_{mn}= \CG_{mn} = e_{m}{}^{\Loa}\,e_{n}{}^{\Lob}\,\eta_{\Loa\Lob}\,, \qquad E^{mn}= \eta^{\Loa\Lob}\, e_{\Loa}{}^{m}\,e_{\Lob}{}^{n} \,. \end{align} Therefore, the deformation can be summarized as \begin{align} E^{mn} \ \to \ E'^{mn} = E^{mn} + 2\,\eta\,\lambda^{\Loa\Lob}\, e_{\Loa}{}^{m}\,e_{\Lob}{}^{n} \,. \end{align} By comparing this with the $\beta$-transformation rule \eqref{eq:beta-rule-NS}, we can regard the YB deformation as $\beta$-deformation with the parameter \begin{align} \bmr^{mn} = 2\,\eta\,\lambda^{\Loa\Lob}\, e_{\Loa}{}^{m}\,e_{\Lob}{}^{n} \,. \label{eq:bmr-YB} \end{align} If we compute the dual field in the deformed background, we obtain \begin{align} \OG'_{mn}&=\eta_{\Loa\Lob}\,e_m{}^{\Loa}\,e_n{}^{\Lob}\,, \qquad \beta'^{mn}=-2\,\eta\,\lambda^{\Loa\Lob}\,e_{\Loa}{}^m\,e_{\Lob}{}^n\,. \label{eq:beta} \end{align} The dual metric is invariant under the deformation $\OG_{mn}\to \OG'_{mn} = \OG_{mn}$\,, while the $\beta$-field, which is absent in the undeformed background, is shifted as $\beta^{mn}=0\ \to\ \beta'^{mn}= -\bmr^{mn}$\,. In addition, the YB-deformed dilaton $\Phi'$ that is consistent with the kappa invariance (or supergravity equations of motion) has been proposed in \cite{Kyono:2016jqy,Borsato:2016ose} as \begin{align} \Exp{\Phi'} = (\det k_{+})^{\frac{1}{2}}=(\det k_{-})^{\frac{1}{2}} \,. \label{eq:dilaton-YB} \end{align} In order to compare this with the $\beta$-transformation law of the dilation, we consider the two vielbeins $e_{\pm m}{}^{\Loa}=e^{\Lob}\,k_{\pm \Lob}{}^{\Loa}$ introduced in \eqref{eq:e-torsionful-spin-pm}. Here, we can rewrite $k_{\pm \Loa}{}^{\Lob}$ as \begin{align} k_{\pm \Loa}{}^{\Lob} \equiv \bigl[(1\pm 2\,\eta\,\lambda)^{-1}\bigr]{}_{\Loa}{}^{\Lob} = e_{\Loa}{}^m\,\bigl[(1\pm \CG\,\bmr)^{-1}\bigr]_m{}^n\,e_n{}^{\Lob} = e_{\Loa}{}^m\,\bigl[(1\pm E\,\bmr)^{-1}\bigr]_m{}^n\,e_n{}^{\Lob} \,, \end{align} by using $\bmr^{mn}$ of \eqref{eq:bmr-YB} and $B_{mn}=0$ in the undeformed background. Then, $e_{\pm m}{}^{\Loa}$ becomes \begin{align} e_{\pm m}{}^{\Loa} = \bigl[(E^{-\rmT} \pm \bmr)^{-1}\,E^{\rmT}\bigr]{}_{m}{}^{n}\,e_n{}^{\Loa} \,. \end{align} Comparing this with the $\beta$-transformation rule \eqref{eq:beta-rule-NS}, we can identify $e_{-m}{}^{\Loa}$ as the $\beta$-deformed vielbein $e'_m{}^{\Loa}$\,. Similarly, $e_{+m}{}^{\Loa}$ can be identified as the $\beta$-deformed barred vielbein $\bre'_m{}^{\Loa}$, \begin{align} e_{-m}{}^{\Loa}\ \leftrightarrow\ e'_m{}^{\Loa}\,,\qquad e_{+m}{}^{\Loa}\ \leftrightarrow\ \bre'_m{}^{\Loa}\,. \label{eq:e+-eebar} \end{align} Namely, we can express the deformed metric as \begin{align} \CG'_{mn} = e_{+m}{}^{\Loa}\,e_{+n}{}^{\Lob}\,\eta_{\Loa\Lob} = e_{-m}{}^{\Loa}\,e_{-n}{}^{\Lob}\,\eta_{\Loa\Lob}\,, \end{align} and the invariance of $\Exp{-2d}=\Exp{-2\Phi}\sqrt{-\CG}$ under $\beta$-deformations shows \begin{align} \Exp{-2\Phi'} = \frac{\sqrt{-\CG}}{\sqrt{-\CG'}} \Exp{-2\Phi} = \frac{\det (e_m{}^{\Loa})}{\det (e_{\pm m}{}^{\Loa})} \Exp{-2\Phi} = (\det k_{\pm})^{-1} \Exp{-2\Phi} \,. \end{align} Recalling $\Phi=0$ in the undeformed background, the transformation rule \eqref{eq:dilaton-YB} can be understood as the $\beta$-transformation. Therefore, NS--NS fields are precisely $\beta$-deformed under the homogeneous YB deformation. For later convenience, let us rewrite $\bmr^{mn}$ of \eqref{eq:bmr-YB} by using the $r$-matrix instead of $\lambda^{\Loa\Lob}$\,. From the definition, $\lambda^{\Loa\Lob}$ can be expressed as \begin{align} \lambda^{\Loa\Lob} = \str\bigl[ R_{g_{\bos}}(\gP^{\Loa})\,\gP^{\Lob} \bigr] \,. \end{align} By using the $r$-matrix $r=\frac{1}{2}\,r^{ij}\,T_i\wedge T_j$\,, this can be expressed as \begin{align} \str\bigl[ R_{g_{\bos}}(\gP^{\Loa})\,\gP^{\Lob} \bigr] &= r^{ij}\, \str\bigl(g_{\bos}^{-1}\,T_{i}\,g_{\bos}\,\gP_{\Lob})\, \str(g_{\bos}^{-1}\,T_{j}\,g_{\bos}\,\gP_{\Loa}) = -r^{ij}\,\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_i{}^{\Loa}\,\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_i{}^{\Lob}\,, \end{align} and we obtain \begin{align} \bmr^{mn} = -2\,\eta\,r^{ij}\,\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_i{}^{\Loa}\,\bigl[\Ad_{g_{\bos}^{-1}}\bigr]_j{}^{\Lob}\, e_{\Loa}{}^{m}\,e_{\Lob}{}^{n} \,. \end{align} By using the Killing vectors \eqref{eq:Killing-Formula} we obtain a very simple expression \begin{align} \bmr^{mn} = -2\,\eta\,r^{ij}\,\hat{T}_i^m\, \hat{T}_j^n \,. \label{eq:bmr-YB2} \end{align} The $\beta$-field after the deformation takes the form \begin{align} \beta'^{mn} = - \bmr^{mn} = 2\,\eta\,r^{ij}\,\hat{T}_i^m\, \hat{T}_j^n \,,\qquad \beta' \equiv \frac{1}{2}\,\beta'^{mn}\,\partial_m\wedge\partial_n = 2\,\eta\,\biggl(\frac{1}{2}\,r^{ij}\,\hat{T}_i \wedge \hat{T}_j\biggr) \,, \end{align} and we can calculate the associated non-geometric $R$-flux \begin{align} R^{mnp} \equiv 3\,\beta^{[m|q}\,\partial_q \beta^{|np]} \,. \end{align} By using the Lie bracket for the Killing vector fields $[\hat{T}_i,\,\hat{T}_j]=-f_{ij}{}^k\,\hat{T}_k$\,, we obtain \begin{align} R^{mnp} = -8\,\eta^2\,\bigl(f_{l_1l_2}{}^i\,r^{jl_1}\,r^{kl_2} + f_{l_1l_2}{}^j\,r^{kl_1}\,r^{il_2} + f_{l_1l_2}{}^k\,r^{il_1}\,r^{jl_2}\bigr)\,\hat{T}_i^m\,\hat{T}_j^n\,\hat{T}_k^p = 0\,, \end{align} upon using the homogeneous CYBE \eqref{eq:CYBE-r}. This shows the absence of the $R$-flux in homogeneous YB-deformed backgrounds as noted in \cite{Sakamoto:2017cpu}. \subsubsection{R--R sector} \label{sec:YB-RR} Next, we determine the R--R fields from the quadratic part of the YB sigma model action $S_{(2)}$\,, and show that the R--R fields are also $\beta$-deformed with the $\bmr^{mn}$ given in \eqref{eq:bmr-YB2}. As noticed in \cite{Arutyunov:2015qva,Kyono:2016jqy}, the deformed action naively does not have the canonical form of the GS action \eqref{eq:GS-action-conventional}, and we need to choose the diagonal gauge and perform a suitable redefinition of the bosonic fields $X^m$\,. Since the analysis is considerably complicated, we relegate the details to Appendix \ref{app:deformed-Lagrangian}, and here we explain only the outline. The quadratic part of the deformed action $S_{(2)}$ can be decomposed into two parts \begin{align} S_{(2)} = S_{(2)}^{\rmc} + \delta S_{(2)}\,. \end{align} For a while, we focus only on the first part $S_{(2)}^{\rmc}$ since the second part $\delta S_{(2)}$ is completely cancelled after some field redefinitions. The explicit expression of $S_{(2)}^{\rmc}$ is given by \begin{align} S_{(2)}^{\rmc} &=-\ii\,\dlT\int \rmd^2\sigma\,\sqrt{-\gga}\, \biggl[ \Pg_+^{\WSa\WSb}\,e_{-\WSa}{}^{\Loa}\,\brtheta_1\,\hat{\gamma}_{\Loa}\,D_{+\WSb}\theta_1 +\Pg_-^{\WSa\WSb}\,e_{+\WSa}{}^{\Loa}\,\brtheta_2\,\hat{\gamma}_{\Loa}\,D_{-\WSb}\theta_2 \nn\\ &\qquad\qquad\qquad\qquad\quad +\ii\, \Pg_+^{\WSa\WSb}\,\epsilon^{IJ}\,\brtheta_1\,e_{-\WSa}{}^{\Loa}\,\hat{\gamma}_{\Loa}\,e_{+\WSb}{}^{\Lob}\,\hat{\gamma}_{\Lob}\,\theta_2\biggr]\,. \label{eq:L-order2} \end{align} This action contains the two deformed vielbeins $e_{\pm m}{}^{\Loa}$ similar to the DSM action \eqref{eq:GS-DSM-ours} prior to taking the diagonal gauge. As we observed in \eqref{eq:e+-eebar}, these vielbeins $e_{-m}{}^{\Loa}$ and $e_{+m}{}^{\Loa}$ correspond to the two vielbeins $e'_m{}^{\Loa}$ and $\bre'_m{}^{\Lobra}$ introduced in \eqref{eq:dV-geometric}, respectively. In order to rewrite the action into the canonical form of the GS action, we need to choose the diagonal gauge $e_m{}^{\Loa}=\bre_m{}^{\Lobra}$\,. For this purpose, we first rewrite the action \eqref{eq:L-order2} in terms of the $32\times 32$ gamma matrices. By using relations \eqref{eq:lift-32-AdS5-1}, \eqref{eq:lift-32-AdS5-2}, and \eqref{eq:lift-32-AdS5-3}, we obtain \begin{align} \begin{split} S^{\rmc}_{(2)}&=-\ii\,\dlT\int \rmd^2\sigma\,\sqrt{-\gga}\,\biggl[ \Pg_+^{\WSa\WSb}\,\brTheta_1\,e_{-\WSa}{}^{\Loa}\,\Gamma_{\Loa}\,D_{+\WSb}\Theta_1 +\Pg_-^{\WSa\WSb}\,\brTheta_2\,e_{+\WSa}{}^{\Loa}\,\Gamma_{\Loa}D_{-\WSb}\Theta_2 \\ &\qquad\qquad\qquad\qquad\quad -\frac{1}{8}\,\Pg_+^{\WSa\WSb}\,\brTheta_1\,e_{-\WSa}{}^{\Loa}\,\Gamma_{\Loa} \,\bisF_5\, e_{+\WSb}{}^{\Lob}\,\Gamma_{\Lob}\,\Theta_2 \biggr]\,, \end{split} \end{align} where $D_{\pm\WSa}\Theta_I\equiv \bigl(\partial_{\WSa}+\frac{1}{4}\,W_{\pm\WSa}{}^{\Loa\Lob}\, \Gamma_{\Loa\Lob}\bigr)\,\Theta_I$ and $\bisF_5$ is the undeformed R--R $5$-form field strength \begin{align} \bisF_5 =\frac{1}{5!}\,\bisF_{\Loa_1\cdots \Loa_5}\,\Gamma^{\Loa_1\cdots \Loa_5} =4\,\bigl(\Gamma^{01234}+\Gamma^{56789}\bigr)\,. \end{align} Next, we eliminate the barred vielbein $e_{+m}{}^{\Loa}$ by using \begin{align} e_{+m}{}^{\Loa}=(\Lambda^{-1})^{\Loa}{}_{\Lob}\,e_{-m}{}^{\Lob} = \Lambda_{\Lob}{}^{\Loa}\,e_{-m}{}^{\Lob}\,,\qquad \Lambda_{\Loa}{}^{\Lob} \equiv (k_-^{-1})_{\Loa}{}^{\Loc}\, k_{+\Loc}{}^{\Lob}\in \SO(1,9)\,, \label{eq:e--e+-relation} \end{align} which follows from \eqref{eq:e-torsionful-spin-pm}. By further using the identity [recall the formula \eqref{eq:Hassan-formula}] \begin{align} \Omega^{-1} \,\Gamma_{\Loa}\,\Omega = \Lambda_{\Loa}{}^{\Lob} \, \Gamma_{\Lob}\,, \qquad \Omega =(\det k_-)^{\frac{1}{2}}\,\text{\AE}\bigl(-\eta\,\lambda^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\bigr) \,, \label{eq:omega} \end{align} the action becomes \begin{align} \begin{split} S^{\rmc}_{(2)}&=-\ii\,\dlT\int \rmd^2\sigma\,\sqrt{-\gga}\,\biggl[ \Pg_+^{\WSa\WSb}\,\brTheta_1\,e'_{\WSa}{}^{\Loa}\,\Gamma_{\Loa}\,D_{+\WSb}\Theta_1 +\Pg_-^{\WSa\WSb}\,\brTheta_2\,\Omega^{-1}\,e'_{\WSa}{}^{\Loa}\,\Gamma_{\Loa}\,\Omega\,D_{-\WSb}\Theta_2 \\ &\qquad\qquad\qquad\qquad\quad -\frac{1}{8}\,\Pg_+^{\WSa\WSb}\,\brTheta_1\,e'_{\WSa}{}^{\Loa}\,\Gamma_{\Loa} \,\bisF_5\,\Omega^{-1}\, e'_{\WSb}{}^{\Loc}\,\Gamma_{\Loc}\,\Omega\,\Theta_2 \biggr]\,. \end{split} \end{align} We then perform a redefinition of the fermionic variables $\Theta_I$\,, \begin{align} \Theta'_1 \equiv \Theta_1\,,\qquad \Theta'_2 \equiv \Omega \,\Theta_2 \,, \label{eq:fermi-redef} \end{align} which corresponds to the transformation rule of fermions \eqref{eq:Theta-transf} under the diagonal gauge fixing. As the result of the redefinition, we obtain \begin{align} S^{\rmc}_{(2)}&=-\dlT\int \rmd^2\sigma\,\sqrt{-\gga}\,\biggl[ \Pg_+^{\WSa\WSb}\,\ii\,\brTheta'_1 \,e'_{\WSa}{}^{\Loa}\,\Gamma_{\Loa}\,D'_{+\WSb} \Theta'_1 +\Pg_-^{\WSa\WSb}\,\ii\,\brTheta_2\,e'_{\WSa}{}^{\Loa}\,\Gamma_{\Loa}\,D'_{-\WSb} \Theta'_2 \nn\\ &\qquad\qquad\qquad\qquad\quad -\frac{1}{8}\,\Pg_+^{\WSa\WSb}\,\ii\,\brTheta_1\,e'_{\WSa}{}^{\Loa}\,\Gamma_{\Loa}\,\bisF_5\,\Omega^{-1}\, e'_{\WSb}{}^{\Lob}\,\Gamma_{\Lob}\,\Theta'_2 \biggr]\,, \label{eq:ScYB2-2} \end{align} where the derivatives $D'_{\pm}$ are defined as \begin{align} \begin{split} D'_+ &\equiv D_+ = \rmd + \frac{1}{4}\,W_+^{\Loa\Lob}\,\Gamma_{\Loa\Lob}\,, \\ D'_- &\equiv \Omega\circ D_-\circ \Omega^{-1} = \rmd + \frac{1}{4}\,W_-^{\Loa\Lob}\,\Omega\,\Gamma_{\Loa\Lob}\,\Omega^{-1} + \Omega\,\rmd \Omega^{-1} \\ &= \rmd + \frac{1}{4}\,\bigl[\Lambda^{\Loa}{}_{\Loc}\,\Lambda^{\Lob}{}_{\Lod}\,W_-^{\Loc\Lod} +(\Lambda\,\rmd\Lambda^{-1})^{\Loa\Lob}\bigr] \,\Gamma_{\Loa\Lob} \,. \end{split} \end{align} As we show in Appendix \ref{app:torsionful-spin-connections}, the spin connection $\omega'^{\Loa\Lob}$ associated with the deformed vielbein $e'^{\Loa}$ and the deformed $H$-flux $H'_{\Loa\Lob\Loc}$ satisfy \begin{align} \begin{split} \omega'^{\Loa\Lob}+\frac{1}{2}\,e'_{\Loc}\,H'^{\Loc\Loa\Lob} &=W_{+}^{\Loa\Lob}\,, \\ \omega'^{\Loa\Lob}-\frac{1}{2}\,e'_{\Loc}\,H'^{\Loc\Loa\Lob} &= \Lambda^{\Loa}{}_{\Loc}\,\Lambda^{\Lob}{}_{\Lod}\,W_-^{\Loc\Lod} +(\Lambda\,\rmd\Lambda^{-1})^{\Loa\Lob} \,, \end{split} \label{eq:torsionful-spin} \end{align} and $D'_{\pm}$ can be expressed as \begin{align} D'_{\pm} = \rmd + \frac{1}{4}\,\Bigl(\omega'^{\Loa\Lob}\pm\frac{1}{2}\,e'_{\Loc}\,H'^{\Loc\Loa\Lob}\Bigr)\,\Gamma_{\Loa\Lob}\,. \end{align} Then, the deformed action \eqref{eq:ScYB2-2} becomes the conventional GS action at order $\cO(\theta^2)$ by identifying the deformed R--R field strengths as \begin{align} \bisF' =\bisF_5\,\Omega^{-1} \,. \label{eq:YBRR0} \end{align} The transformation rule \eqref{eq:YBRR0} has originally been given in \cite{Borsato:2016ose} but our expression of $\Omega$ \eqref{eq:omega} may be more useful to recognize the homogeneous YB deformations as $\beta$-twists. Indeed, \eqref{eq:YBRR0} is precisely the $\beta$-transformation rule of the R--R field strengths \eqref{eq:beta-F-transf}. Another evidence for the equivalence between YB deformations and local $\beta$-deformations, based on the $\kappa$-symmetry variations, is given in Appendix \ref{app:kappa}. Finally, let us consider the remaining part $\delta S_{(2)}$\,. This is completely canceled by redefining the bosonic fields $X^m$ \cite{Arutyunov:2015qva,Kyono:2016jqy}, \begin{align} X^m\ \to\ X^m + \frac{\eta}{4}\,\sigma_1^{IJ}\,e^{\Loc m}\,\lambda_{\Loc}{}^{\Loa\Lob}\,\brtheta_I\,\gamma_{\Loa\Lob}\,\theta_J + \cO(\theta^4)\,, \label{eq:bosonic-shift} \end{align} as long as the $r$-matrix satisfies the homogeneous CYBE. Indeed, this redefinition gives a shift $S_{(0)} \to S_{(0)} + \delta S_{(0)}$\,, and as explained in Appendix \ref{app:bosonic-shift}, the deviation $\delta S_{(0)}$ satisfies a quite simple expression (the shift of $S_{(2)}$ is higher order in $\theta$) \begin{align} &\delta S_{(0)}+\delta S_{(2)} \nn\\ &=\frac{\eta^2\,\dlT}{2}\int \rmd^2\sigma\,\sqrt{-\gga}\, \Pg_-^{\WSa\WSb}\,\sigma_1^{IJ}\, \bigl[\CYBE^{(0)}_g\bigl(J_{+m}^{(2)},J_{-n}^{(2)}\bigr)\bigr]^{\Loa\Lob}\,\brtheta_I\, \gamma_{\Loa\Lob}\,\theta_J\,\partial_{\WSa}X^{m}\,\partial_{\WSb}X^n \,, \end{align} where $\CYBE^{(0)}_g (X,Y)$ represents the grade-$0$ component of $\CYBE_g (X,Y)$ defined in \eqref{eq:CYBE-g}. This shows that $\delta S_{(2)}$ is completely cancelled out by $\delta S_{(0)}$ when the $r$-matrix satisfies the homogeneous CYBE. \section{\texorpdfstring{$\beta$}{\textbeta}-deformations with \texorpdfstring{$H$}{H}-flux: \texorpdfstring{$\AdS3\times \rmS^3\times \TT^4$}{AdS\textthreeinferior{x}S\textthreesuperior{x}T\textfoursuperior}} \label{sec:AdS3} In the previous section, we have shown that the YB sigma model on the $\AdS{5} \times \rmS^5$ background associated with an $r$-matrix $r=\frac{1}{2}\,r^{ij}\,T_i\wedge T_j$ can be regarded as the GS superstring theory defined on a $\beta$-deformed $\AdS{5} \times \rmS^5$ background with the $\beta$-deformation parameter $\bmr^{mn}=-2\,\eta\,r^{ij}\,\hat{T}_i^m\,\hat{T}_j^n$\,. The same conclusion will hold also for other backgrounds in string theory. In this section, we study deformations of an AdS background with $H$-flux. In the presence of $H$-flux, it is not straightforward to define the YB sigma model, and we shall concentrate only on $\beta$-deformations. As an example, we here consider the $\AdS3\times \rmS^3\times \TT^4$ solution \begin{align} \begin{split} &\rmd s^2 = \frac{-(\rmd x^0)^2+(\rmd x^1)^2+\rmd z^2}{z^2} + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,, \\ &B_2= \frac{\rmd x^0\wedge \rmd x^1}{z^2} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,,\qquad \Phi=0\,, \\ &\rmd s^2_{\rmS^3} \equiv \frac{1}{4}\,\bigl[\rmd \theta^2 + \sin^2\theta\, \rmd \phi^2 + \bigl(\rmd \psi + \cos\theta\,\rmd \phi)^2\bigr] \,, \end{split} \label{eq:AdS3-S3-T4} \end{align} which contains the non-vanishing $H$-flux \begin{align} H_3 =-2\,\frac{\rmd x^0\wedge \rmd x^1 \wedge \rmd z}{z^3}-\frac{1}{4}\,\sin\theta\rmd\phi \wedge \rmd \psi \wedge \rmd \theta\,. \end{align} Using the Killing vectors $\hat{T}_i$ of the $\AdS3\times \rmS^3\times \TT^4$ background, we consider local $\beta$-deformations with deformation parameters of the form, $\bmr^{mn}=-2\,\eta\,r^{ij}\,\hat{T}_i^m\,\hat{T}_j^n$\,. We consider several $r$-matrices $r^{ij}$ satisfying the homogeneous CYBE, and show that all of the $\beta$-deformed backgrounds satisfy the equations of motion of (generalized) supergravity. In order to find the Killing vectors explicitly, we introduce a group parameterization for $\AdS3\times \rmS^3$ (for simplicity, we do not consider the trivial $\TT^4$ directions) \begin{align} \begin{split} &g =g_{\AdS{3}}\cdot g_{\rmS^3}\cdot g_{\TT^4}\,, \qquad g_{\AdS{3}} =\exp(x^{\mu} P_{\mu})\cdot\exp(\ln z\,D)\qquad (\mu=0,1)\,, \\ &g_{\rmS^3} =\exp(\phi\,T^L_4)\cdot\exp(\theta\,T^L_3)\cdot\exp(\psi\,T^R_4) \,. \end{split} \end{align} Here, similar to the $\AdS{5}\times\rmS^5$ case (see Appendix \ref{app:psu-algebra}), we have introduced the $\alg{so}(2,2)\times \alg{so}(4)$ generators $(\gP_{\check{\Loa}}\,, \gP_{\hat{\Loa}}\,,\gJ_{\check{\Loa}\check{\Lob}}\,, \gJ_{\hat{\Loa}\hat{\Lob}})$ $(\check{\Loa},\,\check{\Lob}=0,1,2;\,\hat{\Loa},\,\hat{\Lob}=3,4,5)$ as the following $8\times 8$ supermatrices: \begin{align} \begin{alignedat}{3} \gP_{\check{\Loa}} &= \begin{pmatrix} \frac{1}{2}\,\bm{\gamma}_{\check{\Loa}} & \bm{0_4} \\ \bm{0_4} & \bm{0_4} \end{pmatrix},& \qquad \gJ_{\check{\Loa}\check{\Lob}} &= \begin{pmatrix} -\frac{1}{2}\,\bm{\gamma}_{\check{\Loa}\check{\Lob}} & \bm{0_4} \\ \bm{0_4} & \bm{0_4} \end{pmatrix},& \qquad \bm{\gamma}_{\check{\Loa}} &\equiv \begin{pmatrix} +\gamma_{\check{\Loa}}& \bm{0_2} \\ \bm{0_2} &-\gamma_{\check{\Loa}} \end{pmatrix}, \\ \gP_{\hat{\Loa}} &= \begin{pmatrix} \bm{0_4} & \bm{0_4} \\ \bm{0_4} & -\frac{\ii}{2}\,\bm{\gamma}_{\hat{\Loa}} \end{pmatrix}& , \qquad \gJ_{\hat{\Loa}\hat{\Lob}} &= \begin{pmatrix} \bm{0_4} & \bm{0_4} \\ \bm{0_4} & -\frac{1}{2}\,\bm{\gamma}_{\hat{\Loa}\hat{\Lob}} \end{pmatrix}& , \qquad \bm{\gamma}_{\hat{\Loa}}&\equiv \begin{pmatrix} -\gamma_{\hat{\Loa}} & \bm{0_2} \\ \bm{0_2} &+\gamma_{\hat{\Loa}} \end{pmatrix}, \end{alignedat} \end{align} where $2\times 2$ gamma matrices $\gamma_{\check{\Loa}}$ and $\gamma_{\hat{\Loa}}$ are defined as \begin{align} \{\gamma_0,\,\gamma_1,\,\gamma_2\} = \{\ii\,\sigma_3\,, \sigma_1\,, \sigma_2\}\,,\qquad \{\gamma_3,\,\gamma_4,\,\gamma_5\} = \{\sigma_1\,, \sigma_2\,, \sigma_3\} \,. \end{align} We have also defined the conformal basis $\{P_{\mu}\,, M_{\mu\nu}\,, D\,, K_{\mu}\}$ as \begin{align} P_{\mu}\equiv \gP_{\mu}+\gJ_{\mu2}\,,\qquad K_{\mu}\equiv \gP_{\mu}-\gJ_{\mu2}\,,\qquad M_{\mu\nu}\equiv \gJ_{\mu\nu}\,,\qquad D\equiv \gP_{2}\,. \end{align} The generators of $\alg{su}(2)_L\times \alg{su}(2)_R\simeq \alg{so}(4)$ are defined as \begin{align} \begin{split} T^L_3&=\frac{1}{2}\,(\gP_3-\gJ_{4,5})\,,\qquad T^R_3=\frac{1}{2}\,(\gP_3+\gJ_{4,5})\,, \\ T^L_4&=\frac{1}{2}\,(\gP_4-\gJ_{5,3})\,,\qquad T^R_4=\frac{1}{2}\,(\gP_4+\gJ_{5,3})\,, \\ T^L_5&=\frac{1}{2}\,(\gP_5-\gJ_{3,4})\,,\qquad T^R_5=\frac{1}{2}\,(\gP_5+\gJ_{3,4})\,, \end{split} \end{align} which satisfy the commutation relations, \begin{align} [T^{L}_{i},\,T^{L}_{j}]=-\epsilon_{ijk}\,T^{L}_{k}\,,\qquad [T^{R}_{i},\,T^{R}_{j}]=\epsilon_{ijk}\,T^{R}_{k}\,,\qquad [T^{L}_{i},\,T^{R}_{j}]=0\quad (\epsilon_{345}=1)\,. \end{align} By computing the Maurer--Cartan 1-form $A=g^{-1}\,\rmd g$, and using the supertrace formula \begin{align} \begin{alignedat}{2} &\str(\gP_{\Loa}\,\gP_{\Lob}) =\eta_{\Loa\Lob}\,,&\qquad &\str(\gJ_{\Loa\Lob}\,\gJ_{\Loc\Lod})=R_{\Loa\Lob\Loc\Lod} \,, \\ &R_{\check{\Loa}\check{\Lob}}{}^{\check{\Loc}\check{\Lod}} \equiv -2\, \delta_{[\check{\Loa}}^{[\check{\Loc}}\,\delta_{\check{\Lob}]}^{\check{\Lod}]}\,,& \qquad &R_{\hat{\Loa}\hat{\Lob}}{}^{\hat{\Loc}\hat{\Lod}} \equiv 2\, \delta_{[\hat{\Loa}}^{[\hat{\Loc}}\,\delta_{\hat{\Lob}]}^{\hat{\Lod}]} \,, \end{alignedat} \end{align} we can reproduce the above metric \eqref{eq:AdS3-S3-T4}. Then, we can find the Killing vectors $\hat{T}_i$ of this background associated with the generator $T_i$ by using the formula \eqref{eq:Killing-Formula}, or $\hat{T}_i^m = \str\bigl[ g^{-1}\,T^i\,g\,\gP_{\Loa} \bigr]\,e^{\Loa m}$\,. The result is summarized as \begin{align} \begin{split} \hat{P}_{\mu}&=\partial_{\mu}\,,\qquad \hat{M}_{\mu\nu}=x_{\mu}\,\partial_{\nu}-x_{\nu}\,\partial_{\mu}\,,\qquad \hat{D}=x^{\mu}\,\partial_{\mu}+z\,\partial_z\,, \\ \hat{K}_{\mu}& =(x^{\nu}\,x_{\nu}+z^2)\,\partial_{\mu}-2\,x_{\mu}\,(x^{\nu}\,\partial_{\nu}+z\,\partial_z)\,, \\ \hat{T}^L_3& =\cos\phi\,\partial_\theta+\sin\phi\,\Bigl(-\frac{1}{\tan\theta}\,\partial_\phi+\frac{1}{\sin\theta}\,\partial_{\psi}\Bigr) \,, \\ \hat{T}^R_3& =\cos\psi\,\partial_\theta+\sin\psi\,\Bigl(\frac{1}{\sin\theta}\,\partial_\phi-\frac{1}{\tan\theta}\,\partial_{\psi}\Bigr)\,, \\ \hat{T}^L_4&=\partial_{\phi}\,,\qquad \hat{T}^R_4=\partial_{\psi}\,, \\ \hat{T}^L_5& =\sin\phi\,\partial_\theta+\cos\phi\,\Bigl(\frac{1}{\tan\theta}\,\partial_\phi-\frac{1}{\sin\theta}\,\partial_{\psi}\Bigr) \,, \\ \hat{T}^R_5&=-\sin\psi\,\partial_\theta+\cos\psi\,\Bigl(\frac{1}{\sin\theta}\,\partial_\phi-\frac{1}{\tan\theta}\,\partial_{\psi}\Bigr) \,. \end{split} \end{align} We note that among the AdS isometries, $\hat{P}_{\mu}$, $\hat{M}_{01}$, and $\hat{D}$ are symmetry of the $B$-field, \begin{align} \Lie_{\hat{P}_{\mu}}B_2 =\Lie_{\hat{M}_{01}}B_2 =\Lie_{\hat{D}}B_2=0\,, \end{align} while the special conformal generators $\hat{K}_{\mu}$ change the $B$-field by closed forms, \begin{align} \Lie_{\hat{K}_{0}}B_2 = -\frac{2\,\rmd x^1\wedge \rmd z}{z}\,,\qquad \Lie_{\hat{K}_{1}}B_2 = \frac{2\,\rmd x^0\wedge \rmd z}{z}\,. \label{eq:K-B2-closed} \end{align} In the following, we first study $\beta$-deformations by using Killing vectors $\hat{P}_{\mu}$, $\hat{M}_{01}$, $\hat{D}$, and $T^R_4$\,. Then, non-trivial cases using the Killing vectors $\hat{K}_{\mu}$ are studied in section \ref{sec:generalized-isometries}. \vspace{-\Pskip} \subsection{Abelian deformations} Let us begin by studying simple examples associated with Abelian $r$-matrices. As it has been known well \cite{Matsumoto:2014nra,Matsumoto:2014gwa,Matsumoto:2015uja,vanTongeren:2015soa,vanTongeren:2015uha,Kyono:2016jqy,Osten:2016dvf}, YB deformations associated with Abelian $r$-matrices can be also realized as TsT-transformations. \subsubsection{\texorpdfstring{$r=\frac{1}{2}\,P_0\wedge P_1$}{r=P\textzeroinferior{\textwedge}P\textoneinferior}} Let us first consider an Abelian $r$-matrix \begin{align} r=\frac{1}{2}\,P_0\wedge P_1 \,. \end{align} From $\hat{P}_0=\partial_0$ and $\hat{P}_1=\partial_1$, the $\beta$-transformation parameter is $\bmr^{mn}= -\eta\,(\delta_0^{m}\,\delta_1^{n}-\delta_1^{m}\,\delta_0^{n})$\,. After the $\beta$-transformation, we obtain the background \begin{align} \begin{split} \rmd s^2&=\frac{-(\rmd x^0)^2+(\rmd x^1)^2}{z^2+2\,\eta}+\frac{\rmd z^2}{z^2} +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,, \\ B_2&=\frac{\rmd x^0\wedge \rmd x^1}{z^2+2\,\eta}+\frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi\,,\qquad \Exp{-2\Phi} = \frac{z^2+2\,\eta}{z^2} \,. \end{split} \end{align} As we have mentioned above, we can also obtain the background by a TsT transformation from the background \eqref{eq:AdS3-S3-T4}; (1) T-dualize along the $x^1$-direction, (2) (active) shift $x^0\to x^0+\eta\,x^1$\,, (3) T-dualize along the $x^1$-direction. This background is of course a solution of supergravity. As a side remark, noted that this background interpolates a linear dilaton background in the UV region ($z \sim 0$) and the undeformed $\AdS{3} \times \rmS^3\times \TT^4$ background in the IR region ($z \to \infty$). Indeed, by performing a coordinate transformation \begin{align} x^{\pm}=\frac{x^0\pm x^1}{\sqrt{2}} \,,\qquad z=\Exp{\rho}\,, \end{align} the deformed background becomes \begin{align} \begin{split} \rmd s^2&=-\frac{2\Exp{-2\rho}}{1+2\,\eta\Exp{-2\rho}}\,\rmd x^+\,\rmd x^- + \rmd \rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,,\qquad \Exp{-2\Phi} = 1+2\,\eta\,\Exp{-2\rho}\,, \\ B_2&=-\frac{\Exp{-2\rho}}{1+2\,\eta\Exp{-2\rho}}\,\rmd x^+\wedge\rmd x^- + \frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi \,. \label{eq:TT-AdS} \end{split} \end{align} In the asymptotic region $\Exp{-2\rho} \gg \eta^{-1}$ (i.e.~$z\sim 0$), the background approaches to a solution that is independent of the deformation parameter $\eta$ \begin{align} \begin{split} \rmd s^2&=-2\,\rmd x^+\rmd x^-+\rmd \rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,,\qquad \Phi =\rho\,, \\ B_2&=-\rmd x^+\wedge\rmd x^-+\frac{1}{4}\cos \theta\,\rmd \phi\wedge \rmd \psi \,, \end{split} \label{eq:TT-AdS-limit} \end{align} where we ignored the constant part of the dilaton and rescaled light-cone coordinates $x^{\pm}$ as \begin{align} x^{\pm}\to \sqrt{2\,\eta}\,x^{\pm}\,. \end{align} The $\AdS{3}$ part of the background \eqref{eq:TT-AdS} is precisely the geometry obtained via a null deformation of $\SL(2,\mathbb{R})$ WZW model \cite{Forste:1994wp} (see also \cite{Israel:2003ry}), which is an exactly marginal deformation of the WZW model (see \cite{Giveon:2017nie,Giveon:2017myj,Asrat:2017tzd,Giribet:2017imm} for recent studies). Note also that, under a formal $T$-duality along the $\rho$-direction, the solution \eqref{eq:TT-AdS-limit} becomes the following solution in DFT: \begin{align} \begin{split} \rmd s^2&=-2\,\rmd x^+\rmd x^-+\rmd \rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,,\qquad \Phi =\tilde{\rho}\,, \\ B_2&=-\rmd x^+\wedge\rmd x^-+\frac{1}{4}\cos \theta\,\rmd \phi\wedge \rmd \psi \,, \end{split} \end{align} where the dilaton depends linearly on the dual coordinate $\tilde{\rho}$\,. This background can be also interpreted as the following solution of GSE: \begin{align} \begin{split} \rmd s^2&=-2\,\rmd x^+\rmd x^-+\rmd \rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,,\qquad \Phi=0\,, \\ B_2&=-\rmd x^+\wedge\rmd x^-+\frac{1}{4}\cos \theta\,\rmd \phi\wedge \rmd \psi \,,\qquad I =\partial_\rho \,. \end{split} \end{align} \subsubsection{\texorpdfstring{$r=\frac{1}{2}\,P_+\wedge T^R_4$}{r=P\textplusinferior{\textwedge}T\textfourinferior R}} As the second example, let us consider an Abelian $r$-matrix \begin{align} r = \frac{1}{2}\,P_+\wedge T^R_4\qquad \Bigl(P_+ \equiv \frac{P_0 + P_1}{\sqrt{2}}\Bigr)\,. \end{align} For convenience, let us change the coordinates such that the background \eqref{eq:AdS3-S3-T4} becomes \begin{align} \begin{split} \rmd s^2&=-2\Exp{2\rho} \rmd x^+\,\rmd x^- + \rmd \rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,, \qquad \Phi =0\,, \\ B_2&=-\Exp{2\rho}\rmd x^+\wedge\rmd x^- + \frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi \,. \end{split} \end{align} In this coordinate system, the Killing vectors take the form, $\hat{P}_+=\partial_+$ and $\hat{T}^R_4=\partial_\psi$. Then, the associated $\beta$-deformed (or TsT-transformed) background is given by \begin{align} \begin{split} \rmd s^2&=-2\Exp{2\rho} \rmd x^+\,\rmd x^- + \rmd \rho^2 +\frac{\eta}{2}\Exp{2\rho} \rmd x^-\,(\rmd\psi+2\,\cos\theta\,\rmd\phi) +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,, \quad \Phi =0\,, \\ B_2&=-\Exp{2\rho}\rmd x^+\wedge\rmd x^- + \frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi -\frac{\eta}{4}\,\Exp{2\rho}\rmd x^-\wedge (\rmd \psi+2\,\cos\theta\,\rmd\phi) \,. \end{split} \end{align} This background has been studied in \cite{Azeyanagi:2012zd}, where the twist was interpreted as a spectral flow transformation of the original model in the context of the NS--R formalism. \subsubsection{\texorpdfstring{$r = \frac{1}{2}\,D\wedge M_{01}$}{r=D{\textwedge}M\textzeroinferior\textoneinferior}} \label{sec:AdS3-DwM} Let us also consider a slightly non-trivial example $r = \frac{1}{2}\,D\wedge M_{01}$, which is also an Abelian $r$-matrix. The associated $\beta$-deformed background is given by \begin{align} \begin{split} \rmd s^2&= \frac{\eta_{\mu\nu}\,\rmd x^\mu\,\rmd x^\nu - 2\,\eta\,z^{-1}\,x_\mu\,\rmd x^\mu\, \rmd z +(1+\frac{2\,\eta\,x_\mu\,x^\mu}{z^2}) \,\rmd z^2}{z^2-\eta\,(\eta-2)\,x_\mu\,x^\mu} + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,, \\ \Exp{-2\Phi} &= \frac{z^2-\eta\,(\eta-2)\,x_\mu\,x^\mu}{z^2}\,, \\ B_2 &= \frac{\rmd x^0\wedge\rmd x^1-\eta\,z^{-1}\, (x^1\,\rmd x^0-x^0\,\rmd x^1)\wedge \rmd z}{z^2-\eta\,(\eta-2)\,x_\mu\,x^\mu} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,. \end{split} \end{align} We can easily check that this is a solution of the supergravity. In order to obtain the same background by performing a TsT transformation, we should first change the coordinates such that the Killing vectors $\hat{D}$ and $\hat{M}_{01}$ become constant, and perform a TsT transformation, and then go back to the original coordinates. The $\beta$-transformation is much easier in this case. In order to describe the same $\beta$-deformation in the global coordinates \begin{align} \begin{split} \rmd s^2&=-\cosh^2\rho\,\rmd \tau^2+\sinh^2\rho\,\rmd \chi^2+\rmd\rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,,\qquad \Phi=0\,, \\ B_2&=\cosh^2\rho\,\rmd\tau\wedge \rmd \chi +\frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi\,, \end{split} \end{align} we change the group parameterization as \begin{align} g_{\AdS{3}}= \exp(\ii\,\tau\, D+\ii\,\chi\, M_{01})\cdot\exp(\rho\, \gP_1)\,. \end{align} In this case, we can compute the Killing vectors as \begin{align} \hat{D}=-\ii\,\partial_{\tau}\,,\qquad \hat{M}_{01}=-\ii\,\partial_{\chi}\,. \end{align} Then, the $\beta$-deformed background becomes \begin{align} \begin{split} \rmd s^2 &=\frac{-\cosh^2\rho\,\rmd \tau^2+\sinh^2\rho\,\rmd \chi^2}{1+\eta\,(\eta-2)\,\cosh^2\rho}+\rmd\rho^2 +\rmd s_{\rmS^3}^2+\rmd s_{\TT^4}^2\,, \\ \Phi&=\frac{1}{2}\,\ln\Bigl[\frac{1}{1+\eta(\eta-2)\cosh^2\rho}\Bigr] \,, \\ B_2&=(1-\eta)\,\frac{\cosh^2\rho\,\rmd\tau\wedge \rmd \chi}{1+\eta\,(\eta-2)\,\cosh^2\rho} +\frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi \,. \end{split} \end{align} If the deformation parameter $\eta$ and the angular coordinate $\chi$ are replaced as \begin{align} \eta\to 1-\sqrt{\alpha}\,,\qquad \chi\to \sqrt{\alpha}\, \chi\,, \end{align} the AdS part of this background reproduce the background obtained in \cite{Giveon:1993ph,Israel:2003ry} through a current-current deformation of the $\SL(2,\mathbb{R})$ WZW model (see Eqs.~(5.1)--(5.3) in \cite{Israel:2003ry}). \vspace{-\Pskip} \subsection{Non-unimodular deformations} \label{sec:non-unimodular} Let us next consider $\beta$-deformations associated with non-Abelian $r$-matrices. In particular, we consider non-unimodular $r$-matrices, namely non-Abelian $r$-matrices satisfying \begin{align} \cI \equiv \eta\,r^{ij}\,[T_i,\,T_j] = \eta\,r^{ij}\,f_{ij}{}^k\,T_k \neq 0 \,. \label{eq:cI-formula} \end{align} In general, as was shown in \cite{Borsato:2016ose}, YB deformations associated with non-unimodular $r$-matrices give backgrounds that do not satisfy the usual supergravity equations but rather the GSE \cite{Arutyunov:2015mqj,TW,Sakatani:2016fvh,Baguet:2016prz,Sakamoto:2017wor}, which include non-dynamical Killing vector $I^m$ (see Appendix \ref{app:conventions}). As it was observed experimentally \cite{Araujo:2017jkb,Araujo:2017jap,Araujo:2017enj,Fernandez-Melgarejo:2017oyu}, the extra vector $I^m$ typically takes the form (see Appendix \ref{app:conventions} for a derivation in the case of the $\AdS{5}\times \rmS^5$ superstring) \begin{align} \begin{split} &I^m = \hat{\cI}^m \equiv - \eta\,r^{ij}\,[\hat{T}_i,\,\hat{T}_j]^m = \eta\,r^{ij}\,f_{ij}{}^k\,\hat{T}_k^m \\ &\quad\bigl(\,[\hat{T}_i,\,\hat{T}_j]^m = \Lie_{\hat{T}_i}\hat{T}_j^m= - f_{ij}{}^k\,\hat{T}_k^m \,\bigr)\,. \end{split} \label{eq:experimental} \end{align} Using $\bmr^{mn} = -2\,\eta\,r^{ij}\,\hat{T}_i^m\, \hat{T}_j^n$ obtained in \eqref{eq:bmr-YB2} and the Killing property of $\hat{T}_i$, we can also express the experimental formula as \cite{Araujo:2017jkb,Araujo:2017jap,Sakamoto:2017cpu,Araujo:2017enj,Fernandez-Melgarejo:2017oyu} \begin{align} I^m = \sfD_n \bmr^{nm} \,, \end{align} where $\sfD_n$ is the usual covariant derivative associated with the undeformed $\AdS3\times \rmS^3\times \TT^4$ background. Interestingly, as we explain in section \ref{sec:AdS3-S3-T4MwedgeP}, in some examples, even for non-unimodular $r$-matrices, the $\beta$-deformed backgrounds satisfy the usual supergravity equations of motion. Such example has not been observed in the case of the $\AdS5\times \rmS^5$ background \footnote{See a recent paper \cite{Wulff:2018aku} for a general analysis of such backgrounds, called the ``trivial solutions'' of GSE.} and this is due to a particular property of the $\AdS3\times \rmS^3\times \TT^4$ background as explained below. \subsubsection{\texorpdfstring{$r = \frac{1}{2}\,\bar{c}^\mu\,D\wedge P_\mu$}{r=D{\textwedge}P}} Let us consider the simplest non-unimodular $r$-matrix $r = \frac{1}{2}\,\bar{c}^\mu\,D\wedge P_\mu$\,, satisfying \begin{align} \cI = \eta\,\bar{c}^\mu\,[D,\,P_\mu] = c^\mu P_\mu \neq 0 \qquad \bigl(c^\mu\equiv\eta\,\bar{c}^\mu\bigr)\,. \end{align} The $\beta$-deformed background becomes \begin{align} \begin{split} \rmd s^2&= \frac{\eta_{\mu\nu}\,\rmd x^\mu\,\rmd x^\nu + 2\,z^{-1}\,(c^0\,\rmd x^1 -c^1\,\rmd x^0)\,\rmd z + \bigl[1+2\,z^{-2}\,(c^1\,x^0-c^0\,x^1)\bigr]\,\rmd z^2}{z^2 + c_\mu\,c^\mu + 2\,(c^1\,x^0 - c^0 \,x^1)} \\ &\quad + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,,\qquad \Exp{-2\Phi} = \frac{z^2 + c_\mu\,c^\mu + 2\,(c^1\,x^0 - c^0 \,x^1)}{z^2}\,, \\ B_2 &= \frac{\rmd x^0\wedge\rmd x^1 - z^{-1}\, c_\mu\,\rmd x^\mu\wedge \rmd z}{z^2 + c_\mu\,c^\mu + 2\,(c^1\,x^0-c^0\,x^1)} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,, \end{split} \end{align} where $c_\mu \equiv \eta_{\mu\nu}\,c^\nu$\,. Although this is not a solution of the usual supergravity, by introducing a Killing vector, \begin{align} I = c^\mu \hat{P}_\mu = c^\mu\,\partial_\mu \,, \end{align} it becomes a solution of the GSE. \subsubsection{\texorpdfstring{$r = \frac{1}{2}\,\bar{c}^\mu\,M_{01}\wedge P_\mu$}{r=M\textzeroinferior\textoneinferior{\textwedge}P}} \label{sec:AdS3-S3-T4MwedgeP} The next example is a non-unimodular $r$-matrix $r = \frac{1}{2}\,\bar{c}^\mu\,M_{01}\wedge P_\mu$\,, satisfying \begin{align} \cI = - c^\mu P_\mu \neq 0 \qquad \bigl(c^\mu\equiv\eta\,\bar{c}^\mu\bigr)\,. \end{align} The $\beta$-deformed background becomes \begin{align} \begin{split} \rmd s^2&= \frac{\eta_{\mu\nu}\,\rmd x^\mu\,\rmd x^\nu}{z^2 - 2\, c_\mu\,x^\mu} + \frac{\rmd z^2}{z^2} + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,,\qquad \Exp{-2\Phi} = \frac{z^2 - 2\, c_\mu\,x^\mu}{z^2}\,, \\ B_2 &= \frac{\rmd x^0\wedge \rmd x^1}{z^2 - 2\, c_\mu\,x^\mu} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,. \label{eq:AdS3_M^P} \end{split} \end{align} where $c_\mu \equiv \eta_{\mu\nu}\,c^\nu$\,. As usual, by introducing \begin{align} I = - c^\mu \hat{P}_\mu = -c^\mu\,\partial_\mu \,, \end{align} this background satisfies the GSE. Here, note that the defining properties of $I^m$, \begin{align} \Lie_{I} \CG_{mn} = 0 \,,\qquad \Lie_{I} B_{mn} = 0 \,, \end{align} require that the parameters should satisfy $c^0=\pm c^1$\,. In terms of DFT, the above deformed background can be expressed as \begin{align} (\cH_{MN})= \begin{pmatrix} (\CG-B\,\CG^{-1}\,B)_{mn} & (B\,\CG^{-1})_{m}{}^n \\ -(\CG^{-1}\,B)^m{}_n & \CG^{mn} \end{pmatrix},\qquad d = \Phi -\frac{1}{2}\,\ln \sqrt{-\CG} + I^\mu\,\tilde{x}_\mu \,. \end{align} This solves the equations of motion of DFT for arbitrary parameters $c^\mu$\,, but they satisfy the strong constraint \begin{align} \partial_P \cH_{MN}\,\partial^Pd=0\,, \end{align} only when $c^0=\pm c^1$\,. Therefore, we have to choose $c^0=\pm c^1$\,. In fact, this background has a distinctive feature that has not been observed before. According to the classification of \cite{Borsato:2016ose}, the condition for a YB-deformed background to be a standard supergravity background is the unimodularity condition. However, in this example, the background \eqref{eq:AdS3_M^P} satisfies the GSE even if we perform a rescaling $I^m \to \lambda\,I^m$ with arbitrary $\lambda\in\mathbb{R}$. In particular, by choosing $\lambda=0$, the background \eqref{eq:AdS3_M^P} without $I^m$ satisfies the usual supergravity equations of motion. As we explain below, the reason for the unusual behavior is closely related to the degeneracy of $(\CG \pm B)_{mn}$. According to \cite{Sakatani:2016fvh}, the condition for a solution of the GSE to be a standard supergravity background is given by \begin{align} \gLie_{\bm{Y}} \cH_{MN} = 0\,, \qquad \cH_{MN}\,\bm{Y}^M\,\bm{Y}^N= \nabla_M \bm{Y}^M \,, \label{eq:condition-GSE-to-SUGRA} \end{align} where $\nabla_M$ is the (semi-)covariant derivative in DFT and \begin{align} \bm{X}^M \equiv \begin{pmatrix} I^m \\ 0 \end{pmatrix},\qquad \bm{Y}^M \equiv \cH^M{}_N\,\bm{X}^N = \begin{pmatrix} -(\CG^{-1}\,B)^m{}_n\,I^n \\ (\CG-B\,\CG^{-1}\,B)_{mn}\,I^n \end{pmatrix} \,. \end{align} In our example with $c^0=\pm c^1$, $(\CG\pm B)_{mn}\,I^n=0$ is satisfied, and this leads to $\bm{Y}^M = \pm \bm{X}^M$\,. Then, from the null and generalized Killing properties of $\bm{X}^M$ \begin{align} \cH_{MN}\,\bm{X}^M\,\bm{X}^N=0\,,\qquad \gLie_{\bm{X}} \cH_{MN} = 0\,, \qquad \nabla_M \bm{X}^M =0 \,, \end{align} the condition \eqref{eq:condition-GSE-to-SUGRA} is automatically satisfied, and our GSE solution is also a solution of the standard supergravity. If we regard the background \eqref{eq:AdS3_M^P} as a solution of supergravity, the strong constraint is satisfied for an arbitrary $c_\mu$ and it is not necessary to require $c^0=\pm c^1$. \subsubsection{\texorpdfstring{$r = \frac{1}{2}\,\bigl(\bar{a}^\mu D\wedge P_\mu + \bar{b}^\mu M_{01}\wedge P_\mu\bigr)$}{r=a D{\textwedge}P+b M\textzeroinferior\textoneinferior{\textwedge}P}} As a more general class of $r$-matrices, let us consider \begin{align} r = \frac{1}{2}\,\bigl(\bar{a}^\mu D\wedge P_\mu + \bar{b}^\mu M_{01}\wedge P_\mu\bigr) \,. \end{align} The homogeneous CYBE requires \begin{align} \bar{a}^0\,\bar{b}^1 - \bar{a}^1\,\bar{b}^0 = 0 \,,\qquad -\bar{a}^0\,\bar{b}^0 + a^1\,\bar{b}^1 = 0 \,, \end{align} and we consider a non-trivial solution \begin{align} r = \frac{1}{2}\,\bigl(\bar{c}\,D + \bar{d}\,M_{01}\bigr)\wedge (P_0\pm P_1) \,. \end{align} The non-unimodularity becomes \begin{align} \cI = (c-d)\,(P_0\pm P_1) \qquad \bigl(c \equiv\eta\, \bar{c}\,,\quad d\equiv \eta\,\bar{d}\,\bigr)\,. \end{align} The corresponding $\beta$-deformed background is given by \begin{align} \begin{split} \rmd s^2&= \frac{\eta_{\mu\nu}\,\rmd x^\mu\,\rmd x^\nu \mp 2\,c \,z^{-1}\,(\rmd x^0 \mp \rmd x^1)\,\rmd z + \bigl[1\pm 2\,z^{-2}\,(c\pm d)(x^0\mp x^1)\bigr]\,\rmd z^2}{z^2 \pm 2\,(c\pm d)(x^0\mp x^1)} \\ &\quad + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,,\qquad \Exp{-2\Phi} = \frac{z^2 \pm 2\,(c\pm d)(x^0\mp x^1)}{z^2}\,, \\ B_2 &= \frac{\rmd x^0\wedge\rmd x^1 + c \, z^{-1}\,(\rmd x^0\mp \rmd x^1) \wedge \rmd z}{z^2\pm 2\,(c\pm d)(x^0\mp x^1)} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,. \label{eq:D+M^P-pm} \end{split} \end{align} By introducing a Killing vector, \begin{align} I = (c-d)\,(\hat{P}_0\pm \hat{P}_1) = (c-d)\,(\partial_0\pm \partial_1) \,, \end{align} this background becomes a solution of the GSE. In particular, when $c=d$, this becomes a supergravity background. Similar to the previous example, the Killing vector again satisfies $(\CG\pm B)_{mn}\,I^n=0$, and even if we rescale the Killing vector as $I^m\to \lambda\,I^m$, this is still a solution of the GSE. As a particular case $\lambda=0$, the background \eqref{eq:D+M^P-pm} becomes a solution of the usual supergravity. Let us also consider the case, \begin{align} r=\frac{1}{2}\,\bar{a}^{\mu} D\wedge P_{\mu}\qquad \bigl(\text{i.e., }\bar{b}^\mu=0\bigr)\,. \end{align} In this case, the $\beta$-deformed background becomes \begin{align} \begin{split} \rmd s^2&=\frac{\eta_{\mu\nu}\,\rmd x^\mu\,\rmd x^\nu +2\,z^{-1}\,(a^0\,\rmd x^1-a^1\,\rmd x^0)\,\rmd z +[1+2\,z^{-2}\,(a^1 x^0-a^0 x^1)]\,\rmd z^2}{z^2+a^{\mu} a_{\mu}+2(a^1 x^0-a^0 x^1)}\\ &\quad + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,,\qquad \Exp{-2\Phi} = \frac{z^2+a^{\mu}\,a_{\mu}+2\,(a^1\,x^0-a^0\,x^1)}{z^2} \,, \\ B_2&=\frac{\rmd x^0\wedge \rmd x^1+z^{-1}\rmd z\wedge (-a^0\,\rmd x^0+a^1\,\rmd x^1)}{z^2+a^{\mu}\,a_{\mu}+2\,(a^1\,x^0-a^0\,x^1)}+\frac{1}{4}\,\cos \theta\,\rmd \phi\wedge \rmd \psi \,,\quad I =a^{\mu}\,\partial_{\mu}\,, \end{split} \end{align} and this is a solution of the GSE for arbitrary $a^\mu \equiv \eta\,\bar{a}^{\mu}$\,. In this case, we can freely rescale the Killing vector $I^m$ only when $a^0=\pm a^1$\,. \vspace{-\Pskip} \subsection{``\texorpdfstring{$\beta$}{\textbeta}-deformations'' with generalized isometries} \label{sec:generalized-isometries} In the previous subsections, we have not considered the special conformal generators $\hat{K}_{\mu}$\,. As in the case of $\AdS5\times\rmS^5$ background, if there is no $B$-field, we can obtain various solutions from $\beta$-deformations using $\hat{K}_{\mu}$. However, in the $\AdS3\times \rmS^3\times \TT^4$ background, we cannot naively use $\hat{K}_\mu$ according to $\Lie_{\hat{K}_{\mu}}B_2\neq 0$\,. Indeed, even for a simple Abelian $r$-matrix, such as $r=\frac{1}{2}\,K_0\wedge K_1$ or $r=\frac{1}{2}\,K_+\wedge P_+$, the $\beta$-deformed background does not satisfy the supergravity equations of motion. In this subsection, we explain how to utilize the special conformal generators, and obtain several solutions from (generalization of) $\beta$-deformations. In the canonical section $\tilde{\partial}^m=0$, if there exists a pair $(v^m,\,\tilde{v}_m)$ satisfying \begin{align} \Lie_v \CG_{mn} = 0 \,,\qquad \Lie_v B_2 + \rmd \tilde{v}_1 = 0\,,\qquad \Lie_v \Phi =0 \,, \end{align} it means that the background admits a generalized Killing vector $(V^M)=(v^m,\,\tilde{v}_m)$ satisfying \begin{align} \Lie_V \cH_{MN} = 0\,,\qquad \Lie_V d = 0\,. \end{align} Then, the equation \eqref{eq:K-B2-closed} shows that there exist generalized Killing vectors $\hat{\mathsf{K}}_\mu^M$ associated with the Killing vectors $\hat{K}_\mu^m$\,. Since a generalized vector of the form $V^M=\partial^M f(x)$, which we call a trivial Killing vector, is always a generalized Killing vector, there is ambiguity in the definition of the generalized Killing vector. Using the ambiguity, we can find a set of generalized Killing vectors $\hat{\mathsf{T}}_i=(\hat{\mathsf{T}}_i^M)$ that satisfy \begin{align} \gLie_{\hat{\mathsf{T}}_i}\cH_{MN} = \gLie_{\hat{\mathsf{T}}_i}d =0\,,\qquad \gLie_{\hat{\mathsf{T}}_i}\hat{\mathsf{T}}_j^M + \gLie_{\hat{\mathsf{T}}_j}\hat{\mathsf{T}}_i^M =0 \,, \end{align} as well as the conformal algebra $\alg{so}(2,2)$ by means of the C-bracket \begin{align} [V,\,W]_{\rmC}^M \equiv \frac{1}{2}\, (\gLie_V W - \gLie_W V)^M \,. \end{align} Note that, according the requirement $\gLie_{\hat{\mathsf{T}}_i}\hat{\mathsf{T}}_j^M + \gLie_{\hat{\mathsf{T}}_j}\hat{\mathsf{T}}_i^M =0$, the C-bracket coincides with the D-bracket, $[V,\,W]_{\rmD}^M\equiv \gLie_V W^M$. We can find the following set of generalized Killing vectors: \begin{align} \begin{split} &\hat{\mathsf{D}} \equiv x^+\,\partial_+ + x^-\,\partial_- + z\,\partial_z \,, \qquad \hat{\mathsf{P}}_+ \equiv \partial_+\,, \qquad \hat{\mathsf{P}}_- \equiv \partial_-\,, \\ &\hat{\mathsf{M}}_{+-} = x^+\,\partial_+ - x^-\,\partial_- + z^{-1}\,\tilde{\partial}^z \,, \\ &\hat{\mathsf{K}}_+ = z^2\,\partial_+ + 2\,(x^-)^2\,\partial_- + 2\,x^-\,z\,\partial_z + 2\,\tilde{\partial}^- -\frac{2\, x^-}{z}\,\tilde{\partial}^z \,, \\ &\hat{\mathsf{K}}_- = 2\,(x^+)^2\,\partial_+ + z^2\,\partial_- + 2\, x^+\, z\,\partial_z -2\,\tilde{\partial}^+ + \frac{2\,x^+}{z}\,\tilde{\partial}^z \,, \end{split} \end{align} which satisfy \begin{align} \eta_{MN}\,\hat{\mathsf{K}}_\pm^M\, \hat{\mathsf{P}}_\mp^N = \pm 2 \,, \qquad \eta_{MN}\,\hat{\mathsf{D}}^M\, \hat{\mathsf{M}}_{+-}^N = 1\,, \qquad \eta_{MN}\,\hat{\mathsf{T}}_i^M\,\hat{\mathsf{T}}_j^N = 0 \quad (\text{others}) \,. \end{align} If we could find generators $\hat{\mathsf{T'}}_i$ which satisfy \begin{align} \eta_{MN}\,\hat{\mathsf{T'}}_i^M\,\hat{\mathsf{T'}}_j^N = 0\,, \end{align} they are on a common $D$-dimensional section, and we can find a duality frame where the generalized Killing vectors take the form $(\mathsf{T'}_i^M) =(\mathsf{T'}_i^m,\,0)$. If it is possible, the generalized Killing vectors reduces to the usual Killing vector and we can consider the usual $\beta$-deformations in such duality frame. However, it seems unlikely to be the case in the $\AdS3\times \rmS^3\times \TT^4$ background, and in the following, we employ the above set of generalized Killing vectors. \subsubsection{\texorpdfstring{$r=\frac{1}{8}\,\mathsf{K}_+\wedge \mathsf{P}_+$}{r=K\textplusinferior{\textwedge}P\textplusinferior}} Let us first consider an Abelian $r$-matrix $r=\frac{1}{8}\,\mathsf{K}_+\wedge \mathsf{P}_+$ associated with the Abelian generalized isometries; $[\hat{\mathsf{K}}_+,\, \hat{\mathsf{P}}_+]_{\rmC}=0$. Since $\hat{\mathsf{K}}_+$ has the dual components, it is not clear how to perform a ``$\beta$-deformation.'' We thus change the generalized coordinates such that the dual components disappear. We here employ the simple coordinate transformation law by Hohm and Zwiebach \cite{Hohm:2012gk}. Namely, under a generalized coordinate transformation $x^M\to x'^M$, the generalized tensors are transformed as \begin{align} \begin{split} &\cH'_{MN}(x') = \cF_M{}^K(x',x)\,\cF_N{}^L(x',x)\,\cH_{KL}(x) \,,\qquad \Exp{-2d'(x')} = \Bigl\lvert\det\frac{\partial x^M}{\partial x'^N}\Bigr\rvert\,\Exp{-2d'(x')}\,, \\ &\cF_M{}^N(x',x)\equiv \frac{1}{2}\,\biggl(\frac{\partial x'_M}{\partial x_P}\,\frac{\partial x^N}{\partial x'^P}+\frac{\partial x^P}{\partial x'^M}\,\frac{\partial x'_P}{\partial x_N}\biggr)\,. \end{split} \end{align} We can easily check that a generalized coordinate transformation \begin{align} \tilde{z}' = \tilde{z} + \frac{\ln x^-}{z}\,, \qquad x'^M = x^M \quad (\text{others})\,, \label{eq:gen-coord-trsf} \end{align} indeed removes the dual components; $(\mathsf{K'}_+^M)=(\hat{K'}_+^m,\,0)$ and $(\mathsf{P'}_+^M)=(\hat{P'}_+^m,\,0)$. In fact, this transformation $\cH'_{MN} = \cF_M{}^K\,\cF_N{}^L\,\cH_{KL}$ with \begin{align} (\cF_M{}^N) = \begin{pmatrix} \bm{1_{10}} & \bmq_{mn} \\ 0 & \bm{1_{10}} \end{pmatrix}\,,\qquad \bmq_{-z}=-\bmq_{z-} = -\frac{1}{x^-\,z}\,, \end{align} is precisely a $B$-field gauge transformation, \begin{align} B_2 \ \to \ B_2 - \frac{\rmd x^- \wedge \rmd z}{x^-\,z} \,. \end{align} In the transformed background, the $B$-field is shifted \begin{align} \begin{split} \rmd s^2&= \frac{-2\,\rmd x^+\,\rmd x^- + \rmd z^2}{z^2} + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,, \\ B_2&= \frac{\rmd x^- \wedge (x^-\,\rmd x^+ - z\,\rmd z)}{x^-\,z^2} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,, \qquad \Exp{-2\Phi}=1\,, \end{split} \end{align} we can check the isometries \begin{align} \Lie_{\hat{K}_+}\CG_{mn}=\Lie_{\hat{K}_+}B_{mn}=\Lie_{\hat{K}_+}\Phi=0\,,\qquad \Lie_{\hat{P}_+}\CG_{mn}=\Lie_{\hat{P}_+}B_{mn}=\Lie_{\hat{P}_+}\Phi=0\,. \end{align} Then, we can perform the usual $\beta$-deformation associated with $r=\frac{1}{8}\,K_+\wedge P_+$, \begin{align} \begin{split} \rmd s^2&= \frac{-2\,\rmd x^+\,\rmd x^- + \eta\,\rmd x^-\,(\rmd x^- - 2\,x^-\,z^{-1}\,\rmd z)}{z^2 + \eta\,(x^-)^2} + \frac{\rmd z^2}{z^2} + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,, \\ B_2&= \frac{\rmd x^- \wedge (x^-\,\rmd x^+ - z\,\rmd z)}{x^-\,[z^2 + \eta\,(x^-)^2]} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,, \qquad \Exp{-2\Phi}=\frac{z^2 + \eta\,(x^-)^2}{z^2}\,. \end{split} \end{align} Finally, we go back to the original coordinates, $\cH_{MN} = (\cF^{-1})_M{}^K\,(\cF^{-1})_N{}^L\,\cH_{KL}$, and obtain \begin{align} \begin{split} \rmd s^2&= \frac{-2\,\rmd x^+\,\rmd x^- + \eta\,\rmd x^-\,(\rmd x^- - 2\,x^-\,z^{-1}\,\rmd z)}{z^2 + \eta\,(x^-)^2} + \frac{\rmd z^2}{z^2} + \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2 \,, \\ B_2&= \frac{\rmd x^- \wedge \rmd x^+ + \eta\,x^-\,z^{-1}\,\rmd x^- \wedge \rmd z}{z^2 + \eta\,(x^-)^2} + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,, \qquad \Exp{-2\Phi}=\frac{z^2 + \eta\,(x^-)^2}{z^2}\,. \end{split} \label{eq:K_+wP_+} \end{align} This is a new solution of the usual supergravity. \subsubsection{General procedure} In general, it is not easy to find a generalized coordinate transformation like \eqref{eq:gen-coord-trsf}, which removes the dual components of the generalized Killing vectors. However, in fact, it is not necessary to find such a coordinate transformation. As it is clear from the above procedure, for an $r$-matrix, $r=\frac{1}{2}\,r^{ij}\,\mathsf{T}_i\wedge \mathsf{T}_j$, associated with the generalized Killing vectors, the previous deformation is simply a transformation \begin{align} \cH_{MN} \ \to \ \cH'_{MN} = h_M{}^K\,h_N{}^L\,\cH_{KL}\,,\qquad h_{M}{}^N \equiv \delta_M^N - 2\,\eta\,r^{ij}\,\hat{\mathsf{T}}_{iM}\,\hat{\mathsf{T}}_j^N\,. \label{eq:general-Odd} \end{align} Requiring the generalized Killing vectors $\hat{\mathsf{T}}_i$ contained in the $r$-matrix to be mutually orthogonal (i.e.~$\eta_{MN}\,\hat{\mathsf{T}}_i^M\,\hat{\mathsf{T}}_j^N = 0$), we can easily see that the transformation matrix $h_{M}{}^N$ is an $\OO(D,D)$ matrix. In general, this $\OO(D,D)$ transformation is a combination of a $\beta$-transformation and diffeomorphisms, but in particular, when all of $\hat{\mathsf{T}}_{i}$ do not have the dual components, this $h_{M}{}^N$ reduces to the usual $\beta$-transformation matrix. We can easily check that the above solution \eqref{eq:K_+wP_+} can be obtained from the original background in the single step \eqref{eq:general-Odd}. When we consider a non-unimodular $r$-matrix, we suppose that the formula \eqref{eq:cI-formula} will be correct in a duality frame where $\hat{\mathsf{T}}_{i}$ take the form $(\hat{\mathsf{T}}_{i}^M)=(\hat{T}_{i}^m,\,0)$. Then, the deformed background will be a solution of modified DFT (mDFT) \cite{Sakatani:2016fvh} with \begin{align} \bm{X}^M=\hat{\cI}^M \equiv \begin{pmatrix} \hat{\cI}^m\\ \hat{\cI}_m \end{pmatrix} \equiv \eta\,r^{ij}\,[\hat{\mathsf{T}}_i,\,\hat{\mathsf{T}}_j]_{\rmC}^M \,. \end{align} In terms of the GSE, it is a solution with $I^m=\hat{\cI}^m$ and $Z_m=\partial_m\Phi +I^n\,B_{nm} + \hat{\cI}_m$\,. \subsubsection{\texorpdfstring{$r=\frac{1}{2}\,\mathsf{K}_+\wedge \mathsf{K}_-$}{r=K\textplusinferior{\textwedge}K\textminusinferior}} For an Abelian $r$-matrix $r=\frac{1}{2}\,\mathsf{K}_{+}\wedge \mathsf{K}_-$, we do not find a generalized coordinate system where dual components of both $\hat{\mathsf{K}}_+^M$ and $\hat{\mathsf{K}}_-^M$ vanish. However, from the general procedure \eqref{eq:general-Odd}, we can easily obtain the deformed background \begin{align} \begin{split} \rmd s^2 &= \frac{-2\,\rmd x^+\,\rmd x^- + \rmd z^2 + 2\,\eta\,[2\,(x^-\,\rmd x^+ + x^+ \,\rmd x^-) - (2\,x^+\,x^- + z^2)\,\frac{\rmd z}{z}]^2}{z^2+2\,\eta\,(z^2- 2\,x^- \, x^+)^2} \\ &\quad+ \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2\,, \qquad \Exp{-2\Phi} =\frac{z^2+2\,\eta\,(2\,x^+\,x^- -z^2)^2}{z^2}\,, \\ B_2&= \frac{\rmd x^-\wedge \rmd x^+ -4\,\eta\,(2\,x^+\,x^- - z^2) (\rmd x^- \wedge\rmd x^+ + (x^-\,\rmd x^+ - x^+\,\rmd x^-)\wedge\frac{\rmd z}{z}}{z^2+2\,\eta\,(z^2- 2\,x^- \, x^+)^2} \\ &\quad + \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \,. \end{split} \end{align} We can easily see that this is a solution of the usual supergravity. \subsubsection{\texorpdfstring{$r=\frac{1}{2}\,\mathsf{M}_{+-}\wedge \mathsf{K}_+$ or $r=\frac{1}{2}\,\mathsf{D}\wedge \mathsf{K}_+$}{r=M\textplusinferior\textminusinferior{\textwedge}{K\textplusinferior} or r=D{\textwedge}K\textplusinferior}} Let us next consider a non-unimodular $r$-matrix $r=\frac{1}{2}\,\mathsf{M}_{+-}\wedge \mathsf{K}_+$, satisfying \begin{align} \cI = \eta\, [\mathsf{M}_{+-},\,\mathsf{K}_+] = \eta\,\mathsf{K}_+ \,. \end{align} In this case, the deformed background \begin{align} \rmd s^2 &= \frac{-2\,\rmd x^-\,\rmd x^+ + \rmd z^2 - 2\,\eta\, (\rmd x^- - x^-\,\frac{\rmd z}{z})\,[2\,x^+\,\rmd x^- + 2\,x^-\, \rmd x^+ - (2\,x^+\, x^- + z^2)\,\frac{\rmd z}{z}]}{z^2 - 2\,\eta\,x^-\, (2 x^- x^+ - z^2)} \nn\\ &\quad+ \rmd s^2_{\rmS^3} + \rmd s_{\TT^4}^2\,, \qquad \Exp{-2\Phi} = \frac{z^2-2\,\eta\,x^-\,(2\,x^+\,x^- -z^2)}{z^2} \,, \nn\\ B_2&= \frac{1}{4}\,\cos\theta\,\rmd \phi \wedge \rmd \psi \nn\\ &\quad + \frac{(1+2\,\eta\,x^-)\,\rmd x^-\wedge \rmd x^+ + \eta\, [\,2\,(x^-)^2\,\rmd x^+ - (4\,x^+\,x^- - z^2)\,\rmd x^-\,]\wedge \frac{\rmd z}{z}}{z^2 - 2\,\eta\,x^-\, (2\,x^+\,x^- -z^2)} \,, \label{eq:MwedgeK+} \end{align} satisfies the equations of motion of mDFT with $\bm{X}^M =\eta\,\hat{\mathsf{K}}_+^M$\,. Similar to the example studied in section \ref{sec:AdS3-S3-T4MwedgeP}, we can freely rescale $\bm{X}^M$ as $\bm{X}^M\to \lambda\,\bm{X}^M$ $(\lambda\in\mathbb{R})$, and in a particular case $\lambda=0$, \eqref{eq:MwedgeK+} can be regarded as a solution of the usual supergravity. Interestingly, we can obtain the same background also by considering an $r$-matrix $r=\frac{1}{2}\,\mathsf{D}\wedge \mathsf{K}_+$, satisfying $\cI = \eta\, [\mathsf{D},\,\mathsf{K}_+] = -\eta\,\mathsf{K}_+$\,. This also may be related to the degeneracy of $(\CG \pm B)_{mn}$ in the $\AdS3\times \rmS^3\times \TT^4$ background. \subsubsection{Non-orthogonal case: \texorpdfstring{$r=\frac{1}{2}\,\mathsf{D} \wedge \mathsf{M}_{+-}$}{r=D{\textwedge}M\textplusinferior\textminusinferior}} Let us finally comment on an Abelian $r$-matrix $r=\frac{1}{2}\,\mathsf{D} \wedge \mathsf{M}_{+-}$, which is a generalized version of the example considered in section \ref{sec:AdS3-DwM}. In this case, the associated generalized Killing vectors $\hat{\mathsf{D}}^M$ and $\hat{\mathsf{M}}_{+-}^N$ are not orthogonal to each other, $\eta_{MN}\,\hat{\mathsf{D}}^M\,\hat{\mathsf{M}}_{+-}^N\neq 0$, and $h_M{}^N$ in \eqref{eq:general-Odd} is not an $\OO(10,10)$ matrix. Accordingly, the deformed background is not a solution of the usual supergravity. In order to reproduce the example of section \ref{sec:AdS3-DwM} from the general procedure \eqref{eq:general-Odd}, we need to utilize trivial Killing vectors. Indeed, by introducing a trivial Killing vector, $\hat{\mathsf{N}}^M \equiv \partial^M \ln z$, $\hat{\mathsf{M}}'^M_{+-}\equiv \hat{\mathsf{M}}_{+-}^M -\hat{\mathsf{N}}^M$ takes the form $(\hat{\mathsf{M}}'^M_{+-})=(M_{+-}^m,\,0)$\,, and the transformation matrix $h_M{}^N$ associated with an $r$-matrix $r=\frac{1}{2}\,\mathsf{D} \wedge \hat{\mathsf{M}}'_{+-}$ is an $\OO(10,10)$ matrix. The deformed background reproduces the same background obtained in section \ref{sec:AdS3-DwM}. As this example indicates, the null condition $c_{ij}\equiv \eta_{MN}\,\hat{\mathsf{T}}_i^M\,\hat{\mathsf{T}}_j^N = 0$, is very important. Since a gauge transformation in DFT (i.e.~generalized diffeomorphism) is a local $\OO(D,D)$ rotation, in order to realize the deformation $h_M{}^N$ as a gauge symmetry in DFT, $h_M{}^N$ should be an $\OO(D,D)$ matrix. This requirement is equivalent to a condition \begin{align} r^{ik}\,c_{kl}\,r^{lj}\,\hat{\mathsf{T}}_{i}^M\,\hat{\mathsf{T}}_{j}^N = 0\,. \end{align} It is interesting to find a set of generalized Killing vectors $\hat{\mathsf{T}}_{i}^M$ and an $r$-matrix (satisfying CYBE) that satisfy the above condition with $c_{ij}\neq 0$, but here we simply require $c_{ij}=0$\,. Note that the same the null condition $c_{ij}=0$ is known in the context of the non-Abelian $T$-duality. As it has been studied in \cite{Hull:1989jk,Hull:1990ms}, the null condition played an important role in gauging non-Abelian isometries. \subsubsection{Short summary} Let us summarize this subsection. Usually, we prepare a bi-vector $\hat{r}=\frac{1}{2}\,r^{ij}\,\hat{T}_i\wedge \hat{T}_j$ satisfying the homogeneous CYBE (or the Poisson condition) \begin{align} [\hat{r},\,\hat{r}]_{\rmS} \equiv r^{ij}\,r^{kl}\, [\hat{T}_i,\,\hat{T}_k]\wedge T_j\wedge T_l = - r^{ij}\,r^{kl}\,f_{ik}{}^m\,\hat{T}_m\wedge \hat{T}_j\wedge \hat{T}_l = 0 \,, \end{align} where $[\cdot,\cdot]_{\rmS}$ is the Schouten bracket, and perform a local $\beta$-transformation \begin{align} h_M{}^N = \begin{pmatrix} \bm{{1_{10}}} & \bm{{0_{10}}} \\ \bmr^{mn}(x) & \bm{{1_{10}}} \end{pmatrix},\qquad \bmr^{mn}(x) = -2\,\eta\,r^{ij}\,\hat{T}_i^m\, \hat{T}_j^n \,. \end{align} In this subsection, in order to allow for the non-standard Killing vectors $\hat{K}_{\mu}$ satisfying \eqref{eq:K-B2-closed}, we have generalized the Killing vectors $\hat{T}_i$ into the generalized Killing vectors $\hat{\mathsf{T}}_i$\,. The generalized Killing vectors $\hat{\mathsf{T}}_i$ are defined such that their C-bracket satisfy the same commutation relations as those of $\hat{T}_i$\,. The homogeneous CYBE is generalized by replacing the usual Lie bracket with the C-bracket, and performing $\OO(D,D)$ transformations \begin{align} h_{M}{}^N = \delta_M^N - 2\,\eta\,r^{ij}\,\hat{\mathsf{T}}_{iM}\,\hat{\mathsf{T}}_j^N\,, \end{align} we have obtained several new solutions of DFT. In particular, when all of the generalized Killing vectors $\hat{\mathsf{T}}_{i}^M$ do not have the dual components, this generalized transformation reduces to the usual local $\beta$-transformations. \section{Conclusions} \label{sec:discussion} In this paper, we have shown that, after suitable field redefinitions, a homogeneous YB-deformed $\AdS{5}\times \rmS^5$ superstring action associated with bosonic isometries can be always express as the usual GS superstring action up to quadratic order in fermions. The deformations were made only in the supergravity backgrounds and they were identified as local $\beta$-deformations. We have also found a DSM action that reproduces the GS type II superstring action up to quadratic order in fermions. After taking the diagonal gauge, the spacetime fermion is transformed as $\Theta_2\to \Omega\,\Theta_2$ under $\beta$-transformations. We found an explicit form of $\Omega$ in terms of the $\beta$-transformation parameter and the NS--NS fields. Moreover, $\beta$-deformations of $H$-fluxed AdS backgrounds were also studied and various solutions are obtained. In several examples of non-unimodular deformations, we unexpectedly obtained solutions of the usual supergravity. In this paper, we have mainly focused on the YB deformations of the $\AdS{5}\times \rmS^5$ superstring, but the equivalence between homogeneous YB deformations and local $\beta$-deformations will be shown also for other backgrounds.\footnote{See a recent paper \cite{Bakhmatov:2018apn} for a generalization of the ``YB deformations'' (or the open-closed string map) beyond the coset spaces.} Indeed, if we observe the NS--NS part of the deformed action discussed in section \ref{sec:YB-NS-NS}, it is clear that we have not used specific properties of the $\alg{psu}(2,2|4)$ algebra. At least when the $B$-field is absent, and the algebra $\alg{g}$ admits a projection $P$ to the bosonic coset generators $\sfP_{\sfa}$ and $\kappa_{\sfa\sfb}\equiv \str(\sfP_{\sfa}\,\sfP_{\sfb})$ is non-degenerate, the bosonic part of a coset YB sigma model associated with a skew-symmetric $R$-operator becomes \begin{align} S_{(0)} &\sim \int \rmd^2\sigma\,\sqrt{-\gga}\,\Pg_{-}^{\WSa\WSb}\, \str\bigl[A_{\WSa}\, P\circ\cO_{-}^{-1}(A_{\WSb})\bigr]\bigr\rvert_{\theta=0} = \int \rmd^2\sigma\,\sqrt{-\gga}\,\Pg_{-}^{\WSa\WSb}\,e_{\WSa}{}^{\sfa}\,e_{\WSb}{}^{\sfb}\,k_{+\sfa\sfb} \,, \end{align} where $P(A_{\WSa})\equiv e_{\WSa}{}^{\sfa}\,\sfP_{\sfa}$\,, $R_g(\sfP_{\sfa})\equiv \lambda_{\sfa}{}^{\sfb}\,\sfP_{\sfb}$\,, and $k_{+\sfa\sfb} \equiv [(\kappa^{-1}+\eta\,\kappa^{-1}\,\lambda)^{-1}]_{\sfa\sfb}$\,. According to the skew-symmetry $\lambda^{\sfa\sfb}=-\lambda^{\sfb\sfa}$ [$\lambda^{\sfa\sfb}\equiv (\kappa^{-1}\,\lambda)^{\sfa\sfb}$]\,, the YB deformation \begin{align} E^{mn}=\OG^{mn} \ \to \ E'^{mn} = \OG^{mn} + \eta\, \lambda^{\sfa\sfb}\, e_{\sfa}{}^{m}\,e_{\sfb}{}^{n}\qquad \bigl(\OG_{mn} \equiv e_{m}{}^{\sfa}\,e_{n}{}^{\sfb}\,\kappa_{\sfa\sfb}\bigr)\,, \end{align} can be regarded as a (local) $\beta$-deformation. The YB deformations of R--R fields are rather non-trivial, but we expect that the equivalence between YB deformations and local $\beta$-deformation will be shown in more general cases, as long as the $r$-matrix consists of bosonic generators. In a general analysis performed in \cite{Borsato:2016ose}, the YB sigma model action associated with a general $r$-matrix satisfying the homogeneous (or modified) CYBE has been expressed in the standard form of the GS superstring to all order in fermions. When the $r$-matrix contains fermionic generators, we expect the deformation of the target space can be no longer regarded as a local $\beta$-deformation. It will be an interesting future direction to clarify what kind of deformations are made in such cases. As we showed in section \ref{sec:AdS3}, all of the $\beta$-deformed $\AdS3\times \rmS^3\times \TT^4$ backgrounds are solutions of (generalized) supergravity and string theory is well-defined in such backgrounds. In this sense, the local $\beta$-deformations are certain duality transformations in string theory. It will be interesting future work to check the integrability of string sigma model on the $\beta$-deformed $\AdS3\times \rmS^3\times \TT^4$ backgrounds obtained in this paper. It is also important to formulate the homogeneous YB deformations of type II superstring in $H$-fluxed background and show the equivalence to $\beta$-transformations. In this paper, we obtained the $\OO(10,10)$-invariant DSM action for type II superstring. Although we considered only up to the quadratic order in fermionic variables, it is important to obtain the complete action. In fact, a $T$-duality manifest GS superstring action has been proposed also in \cite{Hatsuda:2014aza,Hatsuda:2015cia}. In our approach, the R--R field strengths are contained in the $P$-$\brP$ or $\brP$-$P$ components of the generalized metric $\cM_{MN}$\,, but also in the approach of \cite{Hatsuda:2014aza,Hatsuda:2015cia}, they will appear in the ``left-right'' mixing terms. It is interesting future work to make the connection between the double-vielbein formalism and the approach of \cite{Hatsuda:2014aza,Hatsuda:2015cia} clearer, and obtain a $T$-duality manifest GS superstring action in arbitrary curved backgrounds. In the conventional GS superstring, from the requirement of the kappa invariance, the (generalized) supergravity equations of motion has been obtained \cite{TW}. By generalizing this analysis, it will be important to derive the type II DFT equations of motion from the $T$-duality manifest GS superstring action. It is also important to formulate the DSM that manifests the symmetry of non-Abelian $T$-dualities. When the target space is a group manifold, such DSM has been formulated in \cite{Klimcik:1995dy,Klimcik:1995ux} and its symmetry is clearly discussed in \cite{ReidEdwards:2010vp} (see also \cite{Hassler:2017yza,Lust:2018jsx} for relevant recent works). Its generalization to coset space will be important for a clearer understanding of YB deformations. In the usual DSM, the generalized vector $DX^M$ has the form \begin{align} DX^M \equiv \begin{pmatrix} \rmd X^m \\ \rmd \tilde{X}_m -A_m \end{pmatrix} = \begin{pmatrix} \rmd X^m \\ B_{mn}\, \rmd X^n + \CG_{mn}\,*_{\gga} \rmd X^n \end{pmatrix}, \end{align} upon using the equations of motion. In order to keep the integrability of $\rmd X'^m$ after an $\OO(D,D)$ transformation \begin{align} 0= \rmd^2 X'^m = \rmd \bigl(\bms^{m}{}_{n}\,\rmd X^n + \bmr^{mn}\,B_{np}\, \rmd X^p + \bmr^{mn}\,\CG_{np}\,*_{\gga} \rmd X^p\bigr)\,, \label{eq:integrability} \end{align} we have to require the existence of a set of generalized Killing vectors \cite{Rennecke:2014sca} \begin{align} V^{(m)N} \equiv \bigl(\bmr^{(m)n},\,\bms^{(m)}_n\bigr)\,,\qquad \gLie_{V^{(m)}}\cH_{MN} = 0 \,. \label{eq:integrability-requirement} \end{align} As discussed in \cite{Rennecke:2014sca}, in the case of $\beta$-transformations ($\bms^{m}{}_{n}=\delta^m_n$ and $\bmr^{mn}=-\bmr^{nm}$), by requiring the set of generalized Killing vectors $V^{(m)}$ to form a closed algebra by means of the C-bracket, the homogeneous CYBE for the bi-vector $[\hat{r},\,\hat{r}]_{\rmS}=0$ ($\hat{r}\equiv \frac{1}{2}\,\bmr^{mn}\,\partial_m\wedge\partial_n$) is required. In the case of an Abelian $r$-matrix, we can find a coordinate system where all of the relevant Killing vectors $\hat{T}_i$ are constant vector, and the bi-vector $\bmr^{mn} = -2\,\eta\,r^{ij}\,\hat{T}_i^m\, \hat{T}_j^n$ automatically satisfies the requirement \eqref{eq:integrability-requirement} and we can perform the $\beta$-deformation without breaking the integrability. This is the usual constant $\beta$-shift in the presence of Abelian isometries, that can also realized as a TsT-transformation. In the case of non-Abelian $r$-matrices, since we cannot find a coordinate system where all of the Killing vectors $\hat{T}_i$ are constant vectors, the requirement \eqref{eq:integrability-requirement} is too restroctive. It will be interesting future work to relax the requirement \eqref{eq:integrability} by reformulating the DSM that manifests the symmetry of non-Abelian isometries. \section*{Acknowledgment} We would like to thank Machiko Hatsuda, Jeong-Hyuck Park, Shozo Uehara, and Kentaroh Yoshida for valuable discussions. We also thank Eoin \'O.~Colg\'ain and Linus Wulff for useful comments on the first version of the manuscript. The work of J.S.\ was supported by the Japan Society for the Promotion of Science (JSPS).

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\section{Appendices} \end{document} \section{Introduction} Unsupervised pretraining models, such as GPT and GPT-2 \citep{Radford2018ImprovingLU,radford2019language}, ELMo \citep{peters-etal-2018-deep}, and BERT \citep{devlin2018bert} yield state-of-the-art performance on a wide range of natural language processing tasks. All these models rely on language modeling (LM) objectives that exploit the knowledge encoded in large text corpora. BERT \cite{devlin2018bert}, as one of the current state-of-the-art models, is pretrained on a joint objective consisting of two parts: (1) masked language modeling (MLM), and (2) next sentence prediction (NSP). Through both of these objectives, BERT still consumes only the distributional knowledge encoded by word co-occurrences. While several concurrent research threads are focused on making BERT optimization more robust~\citep{liu2019roberta} or on imprinting external world knowledge on its representations~\citep[\textit{inter alia}]{sun2019ernie,zhang2019ernie,sun2019ernie2,liu2019k,peters2019knowledge}, no study yet has been dedicated to mitigating a severe limitation that contextualized representations and unsupervised pretraining inherited from static embeddings: every model that relies on distributional patterns has a tendency to conflate together pure lexical semantic similarity with broad topic relatedness \cite{Schwartz:2015conll,Mrksic:2017tacl}. In the past, a plethora of models have been proposed for injecting linguistic constraints (i.e., lexical knowledge) from external resources to static word embeddings \cite[\textit{inter alia}]{Faruqui:2015naacl,Wieting:2015tacl,Mrksic:2017tacl, ponti2018adversarial} in order to emphasize a particular lexical relation in a \textit{specialized} embedding space. For instance, lexically informed word vectors specialized for pure semantic similarity result in substantial gains in a number of downstream tasks where such similarity plays an important role, e.g., in dialog state tracking \cite{Mrksic:2017tacl,Ren:2018emnlp} or for lexical simplification \cite{glavavs2018explicit,ponti2019cross}. Existing specialization methods are, however, not directly applicable to unsupervised pretraining models because they are either (1) tied to a particular training objective of a static word embedding model, or (2) predicated on the existence of an embedding space in which pairwise distances can be modified. In this work, we hypothesize that supplementing unsupervised LM-based pretraining with clean lexical information from structured external resources may also lead to improved performance in language understanding tasks. We propose a novel method to inject linguistic constraints, available from lexico-semantic resources like WordNet~\cite{Miller:1995:WLD:219717.219748} and BabelNet~\cite{navigli2012babelnet}, into unsupervised pretraining models, and steer them towards capturing word-level semantic similarity. To train Lexically Informed BERT (LIBERT), we (1) feed semantic similarity constraints to BERT as additional training instances and (2) predict lexico-semantic relations from the constraint embeddings produced by BERT's encoder \cite{vaswani2017attention}. In other words, LIBERT adds lexical relation classification (LRC) as the third pretraining task to BERT's multi-task learning framework. We compare LIBERT to a lexically blind ``vanilla'' BERT on the GLUE benchmark \citep{wang-etal-2018-glue} and report their performance on corresponding development and test portions. LIBERT yields performance gains over BERT on 9/10 GLUE tasks (and is on a par with BERT on the remaining one), with especially wide margins on tasks involving complex or rare linguistic structures such as Diagnostic Natural Language Inference and Linguistic Acceptability. Moreover, we assess the robustness and effectiveness of LIBERT on 3 different datasets for lexical simplification (LS), a task proven to benefit from word-level similarity specialization \cite{ponti2019cross}. We report LS improvements of up to 8.2\% when using LIBERT in lieu of BERT. For direct comparability, we train both LIBERT and BERT from scratch, and monitor the gains from specialization across iterations. Interestingly, these do not vanish over time, which seems to suggest that our specialization approach is suitable also for models trained on massive amounts of raw text data. \section{Related Work} \subsection{Specialization for Semantic Similarity} \label{ss:rwspec} The conflation of disparate lexico-semantic relations in \textit{static} word representations is an extensively researched problem. For instance, clearly discerning between true semantic similarity and broader conceptual relatedness in static embeddings benefits a range of natural language understanding tasks such as dialog state tracking \cite{Mrksic:2017tacl}, text simplification \cite{glavavs2018explicit}, and spoken language understanding \cite{Kim:2016slt}. The most widespread solution relies on the use of specialization algorithms to enrich word embeddings with external lexical knowledge and steer them towards a desired lexical relation. \textit{Joint specialization} models \cite[\textit{inter alia}]{Yu:2014,Kiela:2015emnlp,Liu:EtAl:15,Osborne:16,Nguyen:2017emnlp} jointly train word embedding models from scratch and enforce the external constraints with an auxiliary objective. On the other hand, \textit{retrofitting} models are post-processors that fine-tune pretrained word embeddings by gauging pairwise distances according to the external constraints \cite{Faruqui:2015naacl,Wieting:2015tacl,Mrksic:2016naacl,Mrksic:2017tacl,Jo:2018extro}. More recently, retrofitting models have been extended to specialize not only words found in the external constraints, but rather the entire embedding space. In \textit{explicit retrofitting} models \citep{glavavs2018explicit}, a (deep, non-linear) specialization function is directly learned from external constraints. \textit{Post-specialization} models \cite{Vulic:2018naaclpost,ponti2018adversarial,kamath2019specializing}, instead, propagate lexico-semantic information to unseen words by imitating the transformation undergone by seen words during the initial specialization. This family of models can also transfer specialization across languages \cite{glavavs2018explicit,ponti2019cross}. The goal of this work is to move beyond similarity-based specialization of static word embeddings only. We present a novel methodology for enriching unsupervised pretraining models such as BERT \cite{devlin2018bert} with readily available discrete lexico-semantic knowledge, and measure the benefits of such semantic specialization on similarity-oriented downstream applications. \subsection{Injecting Knowledge into Unsupervised Pretraining Models} Unsupervised pretraining models do retain some of the limitations of static word embeddings. First, they still conflate separate lexico-semantic relations, as they learn from distributional patterns. Second, they fail to fully capture the world knowledge necessary for human reasoning: masked language models struggle to recover knowledge base triples from raw texts \citep{petroni2019language}. Recent work has, for the most part, focused on mitigating the latter limitation by injecting structured world knowledge into unsupervised pretraining and contextualized representations. In particular, these techniques fall into the following broad categories: i) \textit{masking} higher linguistic units of meanings, such as phrases or named entities, rather than individual WordPieces or BPE tokens \citep{sun2019ernie}; ii) including an \textit{auxiliary task} in the objective, such as denoising auto-encoding of entities aligned with text \citep{zhang2019ernie}, or continuous learning frameworks over a series of unsupervised or weakly supervised tasks (e.g.,\ capitalization prediction or sentence reordering) \citep{sun2019ernie2}; iii) \textit{hybridizing} texts and graphs. \citet{liu2019k} proposed a special attention mask and soft position embeddings to preserve their graph structure while preventing unwanted entity-word interactions. \citet{peters2019knowledge} fuse language modeling with an end-to-end entity linker, updating contextual word representations with word-to-entity attention. As the main contributions of our work, we incorporate external lexico-semantic knowledge, rather than world knowledge, in order to rectify the first limitation, namely the distortions originating from the distributional signal. In fact, \citet{liu2019k} hybridized texts also with linguistic triples relating words to sememes (minimal semantic components); however, this incurs into the opposite effect of reinforcing the distributional signal based on co-occurrence. On the contrary, we propose a new technique to enable the model to distinguish between purely similar and broadly related words. \section{Specializing for Word-Level Similarity} LIBERT, illustrated in Figure~\ref{fig:bert}, is a \textit{joint} specialization model. It augments BERT's two pretraining tasks -- masked language modeling (1. MLM) and next sentence prediction (2. NSP) -- with an additional task of identifying (i.e., classifying) valid lexico-semantic relations from an external resource (3. LRC). LIBERT is first pretrained jointly on all three tasks. Similarly to BERT, after pretraining, LIBERT is fine-tuned on training datasets of downstream tasks. For completeness, we first briefly outline the base BERT model and then provide the details of our lexically informed augmentation. \begin{figure}[t!] \centering \includegraphics[width=1.0\linewidth,trim=0.2cm 0cm 0.2cm 0cm]{img/libert.pdf} \vspace{-8mm} \caption{Architecture of LIBERT -- lexically-informed BERT specialized with semantic similarity constraints.} \label{fig:bert} \vspace{-3mm} \end{figure} \subsection{BERT: Transformer-Based Encoder} The core of the BERT model is a multi-layer bidirectional Transformer \citep{vaswani2017attention}, pretrained using two objectives: (1) masked language modeling (MLM) and (2) next sentence prediction (NSP). MLM is a token-level prediction task, also referred to as \emph{Cloze} task \citep{taylor:cloze}: among the input data, a certain percentage of tokens is masked out and needs to be recovered. NSP operates on the sentence-level and can, therefore, be seen as a higher-level sequence modeling task that captures information across sentences. NSP predicts if two given sentences are adjacent in text (negative examples are created by randomly pairing sentences). \subsection{LIBERT: Lexically-Informed (Specialized) Pretraining} The base BERT model consumes only the distributional information. We aim to steer the model towards capturing true semantic similarity (as opposed to conceptual relatedness) by exposing it to clean external knowledge presented as the set of \textit{linguistic constraints} $C = \{(w_1, w_2)_i\}^N_{i = 1}$, i.e., pairs of words that stand in the desired relation (i.e., true semantic similarity) in some external lexico-semantic resource. Following the successful work on semantic specialization of static word embeddings (see \S\ref{ss:rwspec}), in this work we select pairs of synonyms (e.g., \textit{car} and \textit{automobile}) and direct hyponym-hypernym pairs (e.g., \textit{car} and \textit{vehicle}) as our semantic similarity constraints.\footnote{As the goal is to inform the BERT model on the relation of {true semantic similarity} between words \cite{Hill:2015cl}, according to prior work on static word embeddings \cite{Vulic:2018injecting}, the sets of both synonym pairs and direct hyponym-hypernym pairs are useful to boost the model's ability to capture true semantic similarity, which in turn has a positive effect on downstream language understanding applications.} We transform the constraints from $C$ into a BERT-compatible input format and feed them as additional training examples for the model. The encoding of a constraint is then forwarded to the relation classifier, which predicts whether the input word pair represents a valid lexical relation. \vspace{1.6mm} \noindent \textbf{From Linguistic Constraints to Training Instances.} We start from a set of linguistic constraints $C = \{(w_1, w_2)_i\}^N_{i = 1}$ and an auxiliary static word embedding space $\mathbf{X_{aux}} \in \mathbb{R}^d$. The space $\mathbf{X_{aux}}$ can be obtained via any standard static word embedding model such as Skip-Gram \cite{Mikolov2013distributed} or fastText \cite{Bojanowski:2017tacl} (used in this work). Each constraint $c = (w_1, w_2)$ corresponds to a true/positive relation of semantic similarity, and thus represents a \textit{positive} training example for the model. For each positive example $c$, we create corresponding negative examples following prior work on specialization of static embeddings \cite{Wieting:2015tacl,glavavs2018explicit,ponti2019cross}. We first group positive constraints from $C$ into mini-batches $B_p$ of size $k$. For each positive example $c = (w_1, w_2)$, we create two negatives $\hat{c}_1 = (\hat{w}_1, w_2)$ and $\hat{c}_2 = (w_1, \hat{w}_2)$ such that $\hat{w}_1$ is the word from batch $B_p$ (other than $w_1$) closest to $w_2$ and $\hat{w}_2$ the word (other than $w_2$) closest to $w_1$, respectively, in terms of the cosine similarity of their vectors in $\mathbf{X_{aux}}$. This way we create a batch $B_n$ of $2k$ negative training instances from a batch $B_p$ of $k$ positive training instances. Next, we transform each instance (i.e., a pair of words) into a ``BERT-compatible'' format, i.e., into a sequence of WordPiece \cite{wu2016google} tokens.\footnote{We use the same 30K WordPiece vocabulary as \newcite{devlin2018bert}. Sharing WordPieces helps our word-level task as lexico-semantic relationships are similar for words composed of the same morphemes.} We split both $w_1$ and $w_2$ into WordPiece tokens, insert the special separator token (with a randomly initialized embedding) before and after the tokens of $w_2$ and prepend the whole sequence with BERT's sequence start token, as shown in this example for the constraint (\textit{mended}, \textit{regenerated}):\footnote{The sign \# denotes split WordPiece tokens.} \vspace{1.5mm} \begin{center} \setlength{\tabcolsep}{2pt} \footnotesize{ \begin{tabular}{c c c c c c c c} \texttt{[CLS]} & \texttt{men} & \texttt{\#ded} & \texttt{[SEP]} & \texttt{reg} & \texttt{\#ener} & \texttt{\#ated} & \texttt{[SEP]} \\ \texttt{0} & \texttt{0} & \texttt{0} & \texttt{0} & \texttt{1} & \texttt{1} & \texttt{1} & \texttt{1} \end{tabular} } \end{center} \vspace{1.5mm} \noindent As in the original work \cite{devlin2018bert}, we sum the WordPiece embedding of each token with the embeddings of the segment and position of the token. We assign the segment ID of \texttt{0} to the \texttt{[CLS]} token, all $w_1$ tokens, and the first \texttt{[SEP]} token; segment ID \texttt{1} is assigned to all tokens of $w_2$ and the final \texttt{[SEP]} token. \normalsize \vspace{1.8mm} \noindent \textbf{Lexical Relation Classifier.} Original BERT feeds Transformer-encoded token representations to two classifiers: MLM classifier (predicting the masked tokens), and the NSP classifier (predicting whether two sentences are adjacent). LIBERT introduces the third pretraining classifier: it predicts whether an encoded word pair represents a desired lexico-semantic relation (i.e., a positive example where two words stand in the relation of true semantic similarity -- synonyms or hypernym-hyponym pairs) or not. Let $\textbf{x}_\mathit{CLS} \in \mathbb{R}^H$ be the transformed vector representation of the sequence start token [CLS] that encodes the whole constraint ($w_1, w_2$). Our lexical relation predictor (LRC) is a simple softmax classifier formulated as follows: \begin{equation} \hat{\mathbf{y}} = \mathrm{softmax}(\mathbf{x}_{\mathit{CLS}}\mathbf{W}_{\mathit{LRC}}^\top + \mathbf{b}_{\mathit{LRC}})\,, \end{equation} with $\mathbf{W}_{\mathit{LRC}} \in \mathbb{R}^{H \times 2}$ and $\mathbf{b}_{\mathit{LRC}} \in \mathbb{R}^{2}$ as the classifier's trainable parameters. Relation classification loss $L_{\mathit{LRC}}$ is then simply the negative log-likelihood over $k$ instances in the training batch: \begin{equation} L_{\mathit{LRC}} = -\sum_{k}{\ln \mathbf{\hat{y}}_k \cdot \mathbf{y}_k}. \label{eq:lrc} \end{equation} \noindent where $\mathbf{y} \in \{[0, 1], [1, 0]\}$ is the true relation label for a word-pair training instance. \section{Language Understanding Evaluation} To isolate the effects of injecting linguistic knowledge into BERT, we train base BERT and LIBERT in the same setting: the only difference is that we additionally update the parameters of LIBERT's Transformer encoder based on the gradients of the LRC loss $L_{\mathit{LRC}}$ from Eq.~\eqref{eq:lrc}. In the first set of experiments, we probe the usefulness of injecting semantic similarity knowledge on the well-known suite of GLUE tasks \cite{wang-etal-2018-glue}, while we also present the results on lexical simplification, another task that has been shown to benefit from semantic similarity specialization \cite{glavavs2018explicit}, later in \S\ref{s:lexsimp}. \subsection{Experimental Setup} \noindent \textbf{Pretraining Data.} We minimize BERT's original objective $L_\mathit{MLM} + L_\mathit{NSP}$ on training examples coming from English Wikipedia.\footnote{We acknowledge that training the models on larger corpora would likely lead to better absolute downstream scores; however, the main goal of this work is not to achieve state-of-the-art downstream performance, but to compare the base model against its lexically informed counterpart.} We obtain the set of constraints $C$ for the $L_{\mathit{LRC}}$ term from the body of previous work on semantic specialization of static word embeddings \cite{Zhang:2014emnlp,Vulic:2018naaclpost,ponti2018adversarial}. In particular, we collect 1,023,082 synonymy pairs from WordNet \cite{Miller:1995:WLD:219717.219748} and Roget's Thesaurus \cite{Kipfer:2009book} and 326,187 direct hyponym-hypernym pairs \cite{vulic2018specialising} from WordNet.\footnote{Note again that similar to work of \citet{Vulic:2018injecting}, both WordNet synonyms and direct hyponym-hypernym pairs are treated exactly the same: as positive examples for the relation of true semantic similarity.} \vspace{1.8mm} \noindent \textbf{Fine-Tuning (Downstream) Tasks.} We evaluate BERT and LIBERT on the the following tasks from the GLUE benchmark \citep{wang-etal-2018-glue}, where sizes of training, development, and test datasets for each task are provided in Table~\ref{tbl:glue}: \vspace{1.4mm} \noindent \textbf{CoLA} \cite{warstadt2018neural}: Binary sentence classification, predicting if sentences from linguistic publications are grammatically acceptable; \vspace{1.3mm} \noindent \textbf{SST-2} \cite{socher2013recursive}: Binary sentence classification, predicting sentiment (positive or negative) for movie review sentences; \vspace{1.3mm} \noindent \textbf{MRPC} \cite{dolan2005automatically}: Binary sentence-pair classification, predicting whether two sentences are mutual paraphrases; \vspace{1.3mm} \noindent \textbf{STS-B} \cite{cer-etal-2017-semeval}: Sentence-pair regression task, predicting the degree of semantic similarity for a pair of sentences; \vspace{1.3mm} \noindent \textbf{QQP} \cite{chen2018quora}: Binary classification task, recognizing question paraphrases; \vspace{1.3mm} \noindent \textbf{MNLI} \cite{williams2018broad}: Ternary natural language inference (NLI) classification of sentence pairs. Two test sets are given: a matched version (MNLI-m) in which the test domains match with training data domains, and a mismatched version (MNLI-mm) with different test domains; \vspace{1.3mm} \noindent \textbf{QNLI}: A binary classification version of the Stanford Q\&A dataset \citep{rajpurkar-etal-2016-squad}; \vspace{1.3mm} \noindent \textbf{RTE} \cite{bentivogli2009fifth}: Another NLI dataset, ternary entailment classification for sentence pairs; \vspace{1.3mm} \noindent \textbf{AX} \cite{wang-etal-2018-glue}: A small, manually curated NLI dataset (i.e., a ternary classification task), with examples encompassing different linguistic phenomena relevant for entailment.\footnote{Following \newcite{devlin2018bert}, we do not evaluate on the Winograd NLI (WNLI), given its well-documented issues.} \vspace{1.8mm} \noindent \textbf{Training and Evaluation.} We train both BERT and LIBERT from scratch, with the configuration of the BERT$_{BASE}$ model \cite{devlin2018bert}: $L=12$ transformer layers with the hidden state size of $H=768$, and $A=12$ self-attention heads. We train in batches of $k = 16$ instances;\footnote{Due to hardware restrictions, we train in smaller batches than in the the original work \cite{devlin2018bert} ($k = 256$). This means that for the same number of update steps, our models will have observed less training data than the original BERT model of \newcite{devlin2018bert}.} the input sequence length is $128$. The learning rate for both models is $2 \cdot 10^{-5}$ with a warm-up over the first $1,000$ training steps. Other hyperparameters are set to the values reported by \newcite{devlin2018bert}. LIBERT combines BERT's MLM and NSP objectives with our LRC objective in a multi-task learning setup. We update its parameters in a balanced alternating regime: (1) we first minimize BERT's $L_\mathit{MLM} + L_\mathit{NSP}$ objective on one batch of masked sentence pairs and then (2) minimize the LRC objective $L_\mathit{LRC}$ on one batch of training instances created from linguistic constraints. \begin{table*}[!t] \centering \def0.97{0.95} \small{ \begin{tabularx}{\linewidth}{l X X X X X X X X X X} \toprule & CoLA & SST-2 & MRPC & STS-B & QQP & MNLI-m & MNLI-mm & QNLI & RTE & AX\\ \midrule \# Train & 8,551 & 67,349 & 3,668 & 5,749 & 363,870 & 392,702 & 392,702 & 104,743 & 2,490 & -- \\ \# Dev & 1,042 & 872 & 408 & 1,501 & 40,431 & 9,815 & 9,832 & 5,463 & 278 & --\\ \# Test & 1,063 & 1,821 & 1,725 & 1,379 & 390,964 & 9,796 & 9,847 & 5,463 & 3,000 & 1,104\\ \bottomrule \end{tabularx} }} \vspace{-0.5mm} \caption{dataset sizes for tasks in the GLUE benchmark \citep{wang-etal-2018-glue}.} \label{tbl:glue} \vspace{-0.5mm} \end{table*} \newcolumntype{g}{>{\columncolor{white}}l} \setlength{\tabcolsep}{2pt} \begin{table*}[!t] \def0.97{0.97} \centering {\fontsize{8pt}{8pt}\selectfont \begin{tabularx}{\linewidth}{l l l X X X X X X X X X X} \toprule & & & CoLA & SST-2 & MRPC & STS-B & QQP & MNLI-m & MNLI-mm & QNLI & RTE & AX\\ & & & MCC & Acc & F1/Acc & Pears & F1/Acc & Acc & Acc & Acc & Acc & MCC\\ \midrule & \multirow{3}{3em}{Dev} & BERT & 29.4 & 88.7 & 87.1/81.6 & 86.4 & 85.9/89.5 & 78.2 & \textbf{78.8} & 86.2 & 63.9 & -- \\ & & LIBERT & \textbf{35.3} & \textbf{89.9} & \textbf{87.9/82.6} & \textbf{87.2} & \textbf{86.3/89.8} & \textbf{78.5} & 78.7 & \textbf{86.5} & \textbf{65.3} & --\\ \rowcolor{Gray} & & $\Delta$ & +5.9 & +1.2 & +0.8/+1.0 & +0.8 & +0.4/+0.3 & +0.3 & -0.1 & +0.3 & +1.4 & -- \\ \multirow{-2}{3em}{1M} & \multirow{3}{3em}{Test} & BERT & 21.5 & 87.9 & 84.8/78.8 & \textbf{80.8} & 68.6/87.9 & 78.2 & \textbf{77.6} & 85.8 & 61.3 & 26.8 \\ & & LIBERT & \textbf{31.4} & \textbf{89.6} & \textbf{86.1/80.4} & 80.5 & \textbf{69.0/88.1} & \textbf{78.4} & 77.4 & \textbf{86.2} & \textbf{62.6} & \textbf{32.8} \\ \rowcolor{Gray} & & $\Delta$ & +9.9 & +1.7 & +1.3/+1.6 & -0.3 & +0.4/+0.2& +0.2 & -0.2& +0.4 & +1.3 & +6.0\\ \midrule & \multirow{3}{3em}{Dev} & BERT & 30.0 & 88.5 & 86.4/81.1 & 87.0 & 86.3/89.8 & 78.8 & 79.3 & 86.6 & 64.3 & -- \\ & & LIBERT & \textbf{37.2} & \textbf{89.3} & \textbf{88.7}/\textbf{84.1} & \textbf{88.3} & \textbf{86.5/90.0} & \textbf{79.6} & \textbf{80.0} & \textbf{87.7} & \textbf{66.4} & --\\ \rowcolor{Gray} & & $\Delta$ & +7.2 & +0.8 & +2.3/+3.0 & +1.3 & +0.2/+0.2 & +0.8 & +0.7 & +1.1 & +2.1 & --\\ \multirow{-2}{3em}{2M} &\multirow{3}{3em}{Test} & BERT & 28.8 & 89.7 & 84.9/79.1 & 81.1 & 69.0/88.0 & 78.6 & 78.1 & \textbf{87.2} & 63.4 & 30.8\\ & & LIBERT & \textbf{35.3} & \textbf{90.8} & \textbf{86.6/81.7} & \textbf{82.6} & \textbf{69.3/88.2} & \textbf{79.8} & \textbf{78.8} & \textbf{87.2} & \textbf{63.6} & \textbf{33.3}\\ \rowcolor{Gray} & & $\Delta$ & +6.5 & +1.1 & +1.7/+2.6 & +1.5 &+0.3/+0.2 & +1.2 & +0.7 & +0.0 & +0.2 & +2.5\\ \bottomrule \end{tabularx} \vspace{-0.5mm} \caption{Results on 10 GLUE tasks after 1M and 2M MLM+NSP steps with BERT and LIBERT.} \label{tbl:results} \vspace{-1.5mm} \end{table*} \begin{figure*}[!t] \centering \begin{subfigure}[t]{0.48\linewidth} \centering \includegraphics[width=0.99\linewidth]{img/SST2_new.pdf} \caption{SST-2} \label{fig:sst2} \end{subfigure} \begin{subfigure}[t]{0.48\textwidth} \centering \includegraphics[width=0.99\linewidth]{img/MRPC_new.pdf} \caption{MRPC} \label{fig:mrpc} \end{subfigure} \vspace{-0.5mm} \caption{Accuracy over time for BERT and LIBERT on (a) SST-2 and (b) MRPC on the corresponding dev sets.} \vspace{-0.5mm} \label{fig:similar} \end{figure*} \begin{table*}[th] \centering \small{ \begin{tabularx}{\linewidth}{XX|X|XXXX|XXXXXX} \toprule & & & \multicolumn{4}{c|}{Coarse-grained} & \multicolumn{6}{c}{Fine-grained} \\ & Model & All & LS & PAS & Lo & KCS & LE & MN & Fa & Re & NE & Qu \\ \hline \multirow{3}{*}{1M} & BERT & 26.8 & 24.5 & 38.8 & 19.6 & 12.8 & 17.5 & 29.3 & 04.9 & 22.5 & 15.6 & \textbf{57.2} \\ & LIBERT & \textbf{32.8} & \textbf{35.2} & \textbf{39.7} & \textbf{25.3} & \textbf{19.4} & \textbf{28.5} & \textbf{51.4} & \textbf{18.7} & \textbf{59.2} & \textbf{18.0} & {56.9} \\ \rowcolor{Gray} & $\Delta$ & 6.0 & 10.7 & 0.9 & 5.7 & 6.6 & 11.0 & 22.2 & 13.8 & 36.7 & 2.4 & -0.3\\ \hline \multirow{3}{*}{2M} & BERT & 30.8 & 31.3 & 40.0 & 21.7 & \textbf{19.7} & 21.2 & 51.3 & 09.1 & 59.2 & \textbf{21.0} & 60.5 \\ & LIBERT & \textbf{33.3} & \textbf{40.6} & 39.9 & \textbf{24.5} & 18.3 & \textbf{33.2} & \textbf{72.0} & \textbf{21.0} & 59.2 & 18.3 & \textbf{68.4} \\ \rowcolor{Gray} & $\Delta$ & 2.5 & 9.3 & -0.1 & 2.8 & -1.4 & 12.0 & 20.7 & 11.9 & 0.0 & -2.7 & 7.9 \\ \bottomrule \end{tabularx} }% \caption{Linguistic analysis on the Diagnostic dataset. The scores are $R_3$ coefficients between gold and predicted labels, scaled by 100, for sentences containing linguistic phenomena of interest. We report all the coarse-grained categories: \textit{Lexical Semantics} (\textbf{LS}), \textit{Predicate-Argument Structure} (\textbf{PAS}), \textit{Logic} (\textbf{Lo}), and \textit{Knowledge and Common Sense} (\textbf{KCS}). Moreover, we report fine-grained categories for Lexical Semantics: \textit{Lexical Entailment} (\textbf{LE}), \textit{Morphological Negation} (\textbf{MN}), \textit{Factivity} (\textbf{Fa}), \textit{Redundancy} (\textbf{Re}), \textit{Named Entities} (\textbf{NE}), and \textit{Quantifiers} (\textbf{Qu}).} \label{tab:diagnostic} \end{table*} During fine-tuning, for each task, we independently find the optimal hyperparameter configurations of the downstream classifiers for the pretrained BERT and LIBERT: this implies that it is valid to compare their performances on the downstream development sets. Finally, we evaluate fine-tuned BERT and LIBERT on all 10 test sets. \subsection{Results and Discussion} \iffalse We report the results on the development and test datasets after pretraining LIBERT and BERT for 1M and 2M MLM+NSP steps in Table~\ref{tbl:results}.\footnote{As we search for both models' best hyperparameter configurations on the development sets, the respective results serve as a fair approximation of the differences, but they do not provide an absolute estimate of the models' performances.} As it can be seen from the scores, after 1M MLM+NSP steps, the lexically informed BERT model outperforms the vanilla BERT on 8/10 GLUE benchmark tasks on the test set, and on 8/9 tasks on the development set. After 2M MLM+NSP steps, LIBERT is superior in 9/9 tasks on the development sets and 9/10 tasks on the test set. For the tenth task, LIBERT is en par with BERT. The highest gains are achieved for CoLA and AX To further analyze the learning process, we compare the two models on two of the benchmark tasks (SST-2 and MRPC) over time on the development sets. We chose the two tasks as two divers representatives of the GLUE benchmark, with SST-2 being a single-sentence and MRPC a sentence-pair classification task. The results are depicted in Figures~\ref{fig:sst2} and \ref{fig:mrpc}, respectively. For both tasks, we see that the additional pretraining objective leads to performance gains over the vast majority of evaluation points. These gains do not seem to vanish over time, which suggests that the ingestion of complementary knowledge from linguistic knowledge bases will also lead to performance increases over a longer training period. \fi \textbf{Main Results.} The main results are summarized in Table~\ref{tbl:results}: we report both dev set and test set performance. After 1M MLM+NSP steps, LIBERT outperforms BERT on 8/9 tasks (dev) and 8/10 tasks (test). After 2M MLM+NSP steps, LIBERT is superior in 9/9 tasks (dev) and 9/10 tasks (test). For the test set of the tenth task (QNLI), LIBERT is on a par with BERT. While large gains are reported on CoLA, AX, and visible gains appear on SST-2 and MRPC, it is encouraging to see that slight and consistent gains are observed on almost all other tasks. These results suggest that available external lexical knowledge can be used to supplement unsupervised pretraining models with useful information which cannot be fully captured solely through large text data and their distributional signal. The results indicate that LIBERT, our lexically informed multi-task method, successfully blends such curated linguistic knowledge with distributional learning signals. It also further validates intuitions from relevant work on specializing static word embeddings \cite{Wieting:2015tacl,Mrksic:2017tacl} that steering distributional models towards capturing true semantic similarity (as also done here) has a positive impact on language understanding applications in general. \vspace{1.8mm} \noindent \textbf{Fine-grained Analysis.} To better understand how lexical information corroborates the model predictions, we perform a fine-grained analysis on the Diagnostic dataset \cite{wang-etal-2018-glue}, measuring the performance of LIBERT on specific subsets of sentences annotated for the linguistic phenomena they contain. We report the results in Table~\ref{tab:diagnostic}. As expected, \textit{Lexical Semantics} is the category of phenomena that benefits the most (+43.7\% for 1M iterations, +29.7\% for 2M), but with significant gains also in phenomena related to \textit{Logic} (+29.1\% for 1M and +29.1\% for 2M) and \textit{Knowledge \& Common Sense} (+51.7\% for 1M). Interestingly, these results seem to suggest that knowledge about semantic similarity and lexical relations also partially encompasses factual knowledge about the world. By inspecting even finer-grained phenomena related to \textit{Lexical Semantics}, LIBERT outdistances its baseline by a large margin in: i) \textit{Lexical Entailment} (+62.9\% for 1M, +56.6\% for 2M), as expected from the guidance of hypernym-hyponym pairs; ii) \textit{Morphological Negation} (+75.8\% for 1M, +40.4\% for 2M). Crucially, the lower performance of BERT cannot be explained by the low frequency of morphologically derived words (prevented by the WordPiece tokenization), but exactly because of the distributional bias. iii) \textit{Factivity} (+281.7\% for 1M, +130.8\% for 2M), which is a lexical entailment between a clause and the entire sentence it is embedded in. Since it depends on specific lexical triggers (usually verbs or adverbs), it is clear that lexico-semantic knowledge better characterizes the trigger meanings. The improvement margin for \textit{Redundancy} and \textit{Quantifiers} fluctuate across different amounts of iterations, hence no conclusions can be drawn from the current evidence. \vspace{1.8mm} \noindent \textbf{Performance over Time.} Further, an analysis of performance over time (in terms of MLM+NSP training steps for BERT and LIBERT) for one single-sentence task (SST-2) and one sentence-pair classification task (MRPC) is reported in Figures~\ref{fig:sst2}-\ref{fig:mrpc}. The scores clearly suggest that the impact of external knowledge does not vanish over time: the gains with the lexically-informed LIBERT persist at different time steps. This finding again indicates the complementarity of useful signals coded in large text data versus lexical resources \cite{Faruqui:2016thesis,Mrksic:2017tacl}, which should be investigated more in future work. \section{Similarity-Oriented Downstream Evaluation: Lexical Simplification} \label{s:lexsimp} \setlength{\tabcolsep}{3.7pt} \begin{table*}[!t] \centering \small{ \begin{tabular}{c l | c c c c c c c c c | c c c \toprule & & \multicolumn{9}{c}{Candidate Generation} & \multicolumn{3}{c}{Full Simplification Pipeline}\\ & & \multicolumn{3}{c}{BenchLS} & \multicolumn{3}{c}{LexMTurk} & \multicolumn{3}{c}{NNSeval} & BenchLS & LexMTurk & NNSeval \\ \# Steps & & P & R & F1 & P & R & F1 & P & R & F1 & A & A & A\\ \midrule \multirow{3}{3em}{1M} & BERT & .2167 & .1765 & .1945 & .3043 & .1420 & .1937 & .1499 & .1200 & .1333 & .3854 & .5260 & .2469\\ & LIBERT & \textbf{.2348} & \textbf{.1912} & \textbf{.2108} & \textbf{.3253} & \textbf{.1518} & \textbf{.2072} & \textbf{.1646} & \textbf{.1318} & \textbf{.1464} & \textbf{.4338} & \textbf{.6080} & \textbf{.2678}\\ \rowcolor{Gray} & $\Delta$ & .0181 & .0147 & .0163 & .0210 & .0098 & .0135 & .0147 & .0118 & .0131 & .0484 & .0820 & .0209\\ \midrule \multirow{3}{3em}{2M} & BERT & .2408 & .1960 & .2161 & .3267 & .1524 & .2079 & .1583 & .1267 & .1408 & .4241 & .5920 & .2594\\ & LIBERT & \textbf{.2766} &\textbf{.2252} & \textbf{.2483} & \textbf{.3700} & \textbf{.1727} & \textbf{.2354} & \textbf{.1925} & \textbf{.1541} & \textbf{.1712} & \textbf{.4887} & \textbf{.6540} & \textbf{.2803} \\ \rowcolor{Gray} & $\Delta$ & .0358 & .0292 & .0322 & .0433 & .0203 & .0275 & .0342 & .0274 & .0304 & .0646 & .0620 & .0209\\ \bottomrule \end{tabular} \caption{Results on the lexical simplification candidate generation task and for the full pipeline on three datasets: BenchLS, LexMTurk, and NNSeval. For each dataset we report the performance after 1M and 2M MLM+NSP steps (\# Steps) with BERT and LIBERT in terms of Precision (P), Recall (R) and F1-measure (F1) for candidate generation and accuracy (A) for the full pipeline.} \label{tbl:results_simpl} \end{table*} \textbf{Task Description.} The goal of lexical simplification is to replace a target word $w$ in a context sentence $S$ with simpler alternatives of equivalent meaning. Generally, the task can be divided into two main parts: (1) generation of substitute candidates, and (2) candidate ranking, in which the simplest candidate is selected \citep{paetzold2017survey}. Unsupervised approaches to candidate generation seem to be predominant lately \citep[e.g.,][]{glavas-stajner-2015-simplifying,ponti2019cross}. In this task, discerning between pure semantic similarity and broad topical relatedness (as well as from other lexical relations such as antonymy) is crucial. Consider the example: ``\textit{Einstein unlocked the door to the atomic age,}'' where \textit{unlocked} is the target word. In this context, the model should avoid confusion both with related words (e.g.\ \textit{repaired}) and opposite words (e.g.\ \textit{closed}) that fit in context but alter the original meaning. \vspace{1.8mm} \noindent \textbf{Experimental Setup.} In order to evaluate the simplification capabilities of LIBERT versus BERT, we adopt a standard BERT-based approach to lexical simplification \citep{qiang2019BERTLS}, dubbed BERT-LS. It exploits the BERT MLM pretraining task objective for candidate generation. Given the complex word $w$ and a context sentence $S$, we mask $w$ in a new sequence $S'$. Next, we concatenate $S$ and $S'$ as a sentence pair and create the BERT-style input by running WordPiece tokenization on the sentences, adding the \texttt{[CLS]} and \texttt{[SEP]} tokens before, in-between, and after the sequence, and setting segment IDs accordingly. We then feed the input either to BERT or LIBERT, and obtain the probability distribution over the vocabulary outputted by the MLM predictor based on the masked token $p(\cdot|S,S'\backslash\{w\} )$. Based on this, we select the candidates as the top $k$ words according to their probabilities, excluding morphological variations of the masked word. For the substitution ranking component, we also follow \citet{qiang2019BERTLS}. Given the set of candidate tokens $C$, we compute for each $c_i$ in $C$ a set of features: (1) BERT prediction probability, (2) loss of the likelihood of the whole sequence according to the MLM when choosing $c_i$ instead of $w$, (3) semantic similarity between the fastText vectors~\citep{Bojanowski:2017tacl} of the original word $w$ and the candidate $c_i$, and (4) word frequency of $c_i$ in the top $12$ million texts of Wikipedia and in the Children's Book Test corpus.\footnote{A detailed description of these features can be found in the original work.} Based on the individual features, we next rank the candidates in $C$ and consequently, obtain a set of ranks for each $c_i$. The best candidate is chosen according to its average rank across all features. In our experiments, we fix the number of candidates $k$ to $6$. \vspace{1.8mm} \noindent \textbf{Evaluation Data.} We run the evaluation on three standard datasets for lexical simplification: \vspace{1.3mm} \noindent (1) LexMTurk \citep{horn-etal-2014-learning}. The dataset consists of 500 English instances, which are collected from Wikipedia. The complex word and the simpler substitutions were annotated by $50$ crowd workers on Amazon Mechanical Turk. \vspace{1.3mm} \noindent (2) BenchLS \citep{paetzold-specia-2016-benchmarking} is a merge of LexMTurk and LSeval~ \citep{de2010text} containing 929 sentences. The latter dataset focuses on text simplification for children. The authors of BenchLS applied additional corrections over the instances of the two datasets. \vspace{1.3mm} \noindent (3) NNSeval \citep{paetzold2017survey} is an English dataset focused on text simplification for non-native speakers and consists in total of 239 instances. Similar to BenchLS, the dataset is based on LexMTurk, but filtered for a) instances that contain a complex target word for non-native speakers, and b) simplification candidates that were found to be non-complex by non-native speakers. \vspace{1.3mm} We report the scores on all three datasets in terms of Precision, Recall and F1 for the candidate generation sub-task, and in terms of the standard lexical simplification metric of \textit{accurracy} (A) \cite{horn-etal-2014-learning,glavas-stajner-2015-simplifying} for the full simplification pipeline. This metric computes the number of correct simplifications (i.e., when the replacement made by the system is found in the list of gold standard replacements) divided by the total number of target complex words. \vspace{1.3mm} \noindent \textbf{Results and Discussion.} The results for BERT and LIBERT for the simplification candidate generation task and for the full pipeline evaluation are provided in Table~\ref{tbl:results_simpl}. We report the performance of both models after 1M and 2M MLM+NSP pretraining steps. We observe that LIBERT consistently outperforms BERT by at least 0.9 percentage points across all evaluation setups, measures, and for all three evaluation sets. Same as in GLUE evaluation, the gains do not vanish as we train both models for a longer period of time (i.e., compare the differences between the two models after 1M vs.\ 2M training steps). On the contrary, for the candidate generation task, the gains of LIBERT over BERT are even higher after 2M steps. The gains achieved by LIBERT are also visible in the full simplification pipeline: e.g., on LexMTurk, replacing BERT with LIBERT yields a gain of 8.2 percentage points. In sum, these results confirm the importance of similarity specialization for a similarity-oriented downstream task such as lexical simplification. \iffalse \subsection{Dialog State Tracking} \setlength{\tabcolsep}{10pt} \begin{table}[!t] \centering \small{ \begin{tabular}{c l | r r \toprule \# Steps & & \multicolumn{1}{c}{DSTC2} & \multicolumn{1}{c}{WOZ 2.0} \\ \midrule \multirow{3}{3em}{1M} & BERT & \\ & LIBERT & \\ \rowcolor{Gray} & $\Delta$ & \\ \midrule \multirow{3}{3em}{2M} & BERT & 58.83 \% & 73.63 \% \\ & LIBERT & \textbf{60.81} \% & \textbf{73.69} \% \\ \rowcolor{Gray} & $\Delta$ & 1.98 \% & 0.06 \% \\ \bottomrule \end{tabular} \caption{Results on the dialog state tracking task for two datasets: DSTC2 and WOZ 2.0. For each dataset we report the performance after 1M and 2M MLM+NSP steps (\# Steps) with BERT and LIBERT in terms of joint goal accuracy.} \label{tbl:results_dst} \end{table} \setlength{\tabcolsep}{8pt} \begin{table}[!t] \centering \small{ \begin{tabular}{c c c c c c \toprule \multicolumn{3}{c}{DSTC2} & \multicolumn{3}{c}{WOZ 2.0} \\ Train & Dev & Test & Train & Dev & Test \\ \midrule 1,612 & 506 & 1,117 & 600 & 200 & 400 \\ \bottomrule \end{tabular} \caption{Number of dialogs for the dialog state tracking datasets DSTC2 and WOZ 2.0.} \label{tbl:dst_stats} \end{table} \paragraph{Experimental Setup.} In dialog state tracking, the idea is to maintain the user's intentions throughout the course of a task-oriented dialog. A user's intention -- a state -- is represented by a collection of slot-value pairs. We consider the specific task setting, in which no domain ontology defining the set of possible slots is given. Instead, a slot value can be any word segment in the dialog context and consequently, the task is to extract the specific slot value from the text (therefore predicting the specific span) or to indicate that the slot is not going to be filled (returning \emph{none} or \emph{dontcare} as prediction). We implement the BERT-DST approach~\citep{chao2019bert} and replace the encoder with our pretrained versions of BERT and LIBERT accordingly: Description of the model ... We evaluate our models on two datasets: \vspace{0.5em} \noindent (1) DSTC2~\citep{henderson-etal-2014-second}: The task-oriented dialog state tracking dataset consisting of in total 3,235 dialogs in the restaurant domain. \vspace{0.5em} \noindent (2) WOZ 2.0~\citep{wen2017network}: Another dataset in the restaurant domain, which consists of 1,200 dialogs. They where collected by applying the Wizard-of-Oz paradigm~\citep{Kelley:1984:IDM:357417.357420} on Amazon Mechanical Turk. \vspace{0.5em} \noindent Both datasets ask for values for the same slots: \emph{area}, \emph{price range}, and \emph{food}. Table~\ref{tbl:dst_stats} list number of dialogs of the training, development and test splits for each of the two datasets. We train the models using the Adam optimizer with an initial learning rate of $2e^{-5}$ and apply early stopping on the validation set. \paragraph{Results.} The joint goal accuracy of BERT vs. LIBERT after 1M and 2M MLM+NSP steps is given in Table~\ref{tbl:results_dst}. On both datasets, LIBERT outperforms BERT. While on WOZ 2.0 the performance gain is only marginal, on DSTC2, LIBERT outperforms BERT by 1.98 \%. \fi \section{Conclusion} We have presented LIBERT, a lexically informed extension of the state-of-the-art unsupervised pretraining model BERT. Our model is based on a multi-task framework that allows us to steer (i.e., specialize) the purely distributional BERT model to accentuate a lexico-semantic relation of true semantic similarity (as opposed to broader semantic relatedness). The framework combines standard BERT objectives with a third objective formulated as a relation classification task. The gains stemming from such explicit injection of lexical knowledge into pretraining were observed for 9 out of 10 language understanding tasks from the GLUE benchmark, as well as for 3 lexical simplification benchmarks. These results suggest that complementing distributional information with lexical knowledge is beneficial for unsupervised pretraining models. In the future, we will work on more sophisticated specialization methods, and we will investigate methods to encode the knowledge on asymmetric relations such as meronymy and lexical entailment. Finally, we will port this new framework to other languages and to resource-poor scenarios. We will release the code at: \url{[URL]}

BkiUeDTxK1yAgWt7_FqW

\section{Introduction} Although field theories quantized on the light-front (LF) have been studied for many years (see \cite{pb85,osu:all} and also \cite{bpp98,brazil} for a review), recent developments in non-perturbative string theory have generated additional interest. The first surprise was M(atrix) theory \cite{bfss97}, which was conjectured to be a non-perturbative description of M-theory formulated in the infinite momentum frame. Motl and Susskind provided additional insight by suggesting that the finite $N$ version of matrix theory was in fact the discrete light-cone quantization (DLCQ) of M-theory \cite{suss97}. Soon afterwards, the validity of the matrix theory conjecture was seemingly strengthened by the works of Seiberg and Sen \cite{nati97,sen97}, but it was pointed out by Hellerman and Polchinski \cite{hell97} that a correct interpretation of their results required a detailed understanding of the (typically complicated) dynamics of zero longitudinal momentum modes in the light-like compactification limit. In general, it was observed, ``DLCQ is not a free lunch''. The question we wish to address in this paper is the following: ``When is the light-like limit a free lunch?'' Under a reasonable class of assumptions, we argue that the zero-mode degrees of freedom in some supersymmetric field theories decouple, and so omitting them in a DLCQ calculation leads to no inconsistency if the decompactification limit is taken prior to the light-like limit. This observation is intriguing, since it suggests that the complicated zero-mode degrees of freedom studied in \cite{hell97} might become totally irrelevant in the continuum limit if enough supersymmetry is present. Moreover, the ``correctness of matrix theory'' argument provided by Seiberg may depend on this special property of supersymmetric theories. Another issue that we address is Lorentz invariance. We show that in the light-front formulation, Lorentz invariance is maintained after a careful treatment of zero modes. However, for the special case of supersymmetric theories, the boson and fermion zero modes that ensure Lorentz symmetry cancel at least perturbatively! Thus, we are free to exclude them from the outset. All of these observations suggest that the implementation of DLCQ in the absence of zero modes yields no inconsistency for supersymmetric theories. In general, however, one needs to integrate out the zero-mode degrees of freedom to derive an effective Hamiltonian. We discuss these issues next. \section{Tadpole improved light-front quantization} It has been known for a long time that field theories quantized on a light-front $x^+ \equiv \left(x^0 + x^3\right)/\sqrt{2} =0$ leads to a subtle treatment of the zero modes (modes which are independent of $x^-\equiv \left(x^0 - x^3\right)/\sqrt{2}$ \cite{my}). This result holds both in the continuum, when zero-modes are discarded but also in DLCQ when the theory is formulated in a finite ``box'' in the $x^-$ direction with periodic boundary conditions. Various schemes have been invented to define LF quantization through a limiting procedure in order to investigate these issues. For example, one can study LF perturbation theory by starting from covariant Feynman diagram expressions and then ``derive'' the LF Hamiltonian perturbation theory by carefully integrating over all energies $k^-$ in loop integrations first (see for example Refs. \cite{mb:rot,mb:sg} and references therein). An alternative prescription starts by quantizing the fields on a near light-like surface (using so-called $\varepsilon$-coordinates) and then studying the evolution of the states as one takes the LF-limit in an infinite volume \cite{eps}. The basic upshot of these investigations is that, at least for theories without massless degrees of freedom, zero-modes become high-energy degrees of freedom and ``freeze out''. However, this does not mean that zero-modes disappear completely, since there is still a strong interaction present among the zero-modes, giving rise to non-trivial vacuum structure even in the LF limit. Nevertheless, because of the high energy scale for excitations within the zero-mode sector, one has been able to derive {\it effective LF Hamiltonians}, where the zero-modes have been integrated out, which act only on non-zero-mode degrees of freedom. Thus even though these effective LF Hamiltonians contain only non-zero-mode degrees of freedom, they yield the same Green's functions as a covariant calculation provided one considers only Green's functions where all external momenta have $k^+\neq 0$. \subsection{Self-interacting scalar fields} As an example, let us consider a scalar field theory with cubic (plus higher order) self-interactions. The presence of cubic self interactions gives rise to ``tennis racket'' Feynman diagrams (Fig. \ref{fig:tennis}a). \begin{figure} \unitlength1.cm \begin{picture}(5,5.3)(0,-7.5) \special{psfile=tennis.ps angle=-90 hscale=80 vscale=80} \end{picture} \caption{Typical self-energy diagrams for scalar fields which vanish if zero-modes are not included. a) ``tennis racket'' shaped tadpole in $\phi^3$ theory, b) generalized tadpole diagram in $\phi^4$ theory. The grey blob represents an arbitrary self-energy insertion.} \label{fig:tennis} \end{figure} If zero-modes are excluded then obviously all tennis racket diagrams (which do contribute to Feynman perturbation theory) have no analog in LF Hamiltonian perturbation theory. However, the crucial observation is that tennis racket diagrams are momentum independent and only lead to a mass renormalization proportional to $\langle 0 |\phi |0\rangle$. Similarly, all tennis racket insertions into $n$-point interactions only lead to a renormalization of the $(n-1)$-point interaction term, i.e. all these diagrams can be easily integrated out. More generally, one can show that for self-interacting scalar fields, \footnote{Here and in the following we will implicitly restrict ourselves to Green's functions where all external momenta have a non-vanishing plus-component.} zero-modes contribute only to diagrams with generalized tadpole topology. As a result, zero-modes can be integrated out easily. For a polynomial self-interaction \begin{equation} {\cal L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - V(\phi) \label{eq:lager} \end{equation} where \begin{equation} V(\phi) = \sum_{k\leq n} c_k \frac{\phi^k}{k!} \end{equation} one thus finds for the effective LF Hamiltonian \cite{mb:sg} \begin{equation} P^- = V_{eff}(\phi), \label{eq:Peff} \end{equation} where the effective potential is also a polynomial of the same degree \begin{equation} V_{eff}(\phi)=\sum_{k\leq n} c_k^{eff} \frac{\phi^k}{k!} \label{eq:Veff} \end{equation} but with coefficients that are renormalized due to integrating out zero-mode degrees of freedom \begin{equation} c_k^{eff} = \sum_{l=k}^n c_l \langle 0|\frac{\phi^{l-k}}{(l-k)!}|0\rangle . \label{eq:ceff} \end{equation} As an illustration of how Eq. (\ref{eq:ceff}) arises, let us consider a theory with quartic self-interactions, i.e. $V(\phi )= \frac{1}{2} \mu^2 \phi^2 + \frac{1}{4!}\lambda^2 \phi^4$. \footnote{For the general case, see Ref. \cite{mb:sg}.} In this case, the only Feynman diagrams which are improperly handled (they are set to zero!) when the zero-mode region ($k^+=0$) is cut out are the generalized tadpole diagrams (Fig. \ref{fig:tennis}b). In order to see why these diagrams give only a zero-mode contribution, let us consider the sum of all generalized tadpole diagrams, which can be easily done by using the full propagator for the scalar fields for which we write down a spectral representation \cite{bd} \begin{equation} \Delta_F(p) = \int_0^\infty \! dM^2\, \frac{i \rho(M^2)} {p^2-M^2+i\varepsilon} \label{eq:bspectr} \end{equation} with spectral density $\rho (M^2)$. As a side remark, for later use, we would like to point out that the spectral density has a very simple representation in terms of the LF Fock states. Upon inserting a complete set of eigenstates of the LF Hamiltonian into the scalar two-point function \cite{bd}, one finds (Appendix A) \begin{eqnarray} \rho (M^2) &=& 2\pi \sum_n \delta\left(\frac{M^2}{2P^+} - P_n^-\right) \left|\langle 0|\phi (0)|n,P^+\rangle \right|^2 \nonumber\\ &=&2\pi \sum_n \delta\left(M^2 - M_n^2\right) 2P^+ \left|\langle 0|\phi (0)|n,P^+\rangle \right|^2 \nonumber\\ &=& \sum_n \delta\left(M^2 - M_n^2\right) b_n \label{eq:bspectr2} \end{eqnarray} where $|n,P^+ \rangle$ is a complete set of eigenstates of $P^-$ (with eigenvalues $P_n^-=\frac{M_n^2}{2P^+_n}$) which we take to be normalized to $1$ and where $b_n$ is the probability that the state $n$ is in its one boson Fock component (one boson which carries the whole momentum $P^+$). The sum can be evaluated at arbitrary but fixed total momentum $P^+$ (assuming we work in the continuum limit). Using Eq. (\ref{eq:bspectr}), one finds for the sum of all generalized tadpole diagrams \footnote{Note that this result holds regardless whether or not fermions pairs contribute to the spectral density of the bosons!} \begin{equation} -i\Sigma^{tadpole} = \frac{\lambda^2}{2} \int_0^\infty \! dM^2 \rho(M^2)\int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2-M^2+i\varepsilon}. \label{eq:tadpole} \end{equation} The crucial point is that for $k^+ \neq 0$, all poles lie only on one side of the real $k^-$ axis and the result is thus zero (up to a contribution from the semi-circle at infinity, which disappears if one subtracts the one loop result). In order to compensate for the omission of all generalized tadpole diagrams in naive LF quantization, we thus add a counter-term equal to the sum of all these omitted diagrams, i.e. a calculation that omits all explicit zero-mode degrees of freedom, but adds a mass counter-term $\delta \Sigma = \Sigma^{tadpole}$ will give the same results as a calculation that includes all zero modes explicitly. The connection with Eq. (\ref{eq:ceff}) can now be seen by noting that the vacuum expectation value of $\phi^2$ is (up to a combinatoric factor) identical to the r.h.s. of Eq. (\ref{eq:tadpole}). In summary, one finds that (for self-interacting scalar fields) \cite{mb:sg} \begin{itemize} \item zero-modes contribute to n-point functions involving only $k^+ \neq 0$ modes only through generalized tadpole (sub-)diagrams. By generalized tadpole diagrams we mean diagrams where a sub-diagram is connected to the rest of the diagram only at one single point and hence there is no momentum transfer through that point. \item n-point functions calculated with the ``tadpole improved'' effective LF-Hamiltonian (\ref{eq:Peff},\ref{eq:Veff},\ref{eq:ceff}) and without explicit zero-mode degrees of freedom is equivalent to covariant perturbation theory generated by ${\cal L}$ (\ref{eq:lager}) to all orders in perturbation theory. \end{itemize} \subsection{Yukawa interactions} As a generic example for a theory with fermions, let us now consider a Yukawa theory with scalar couplings \begin{equation} {\cal L} = \bar{\psi} \left( i\gamma^\mu \partial_\mu - m_F - g\phi\right) \psi - \frac{1}{2}\phi \left( \Box + m_B^2\right)\phi . \end{equation} If zero modes are excluded then two classes of Feynman diagrams (to be discussed below) are treated improperly in the LF Hamiltonian perturbation series. Obviously, LF theory without zero-modes cannot generate any tadpole (i.e. tennis racket) self energies for the fermions. Since the above Lagrangian contains a scalar Yukawa coupling, such diagrams are in general non-zero. Their omission in naive LF quantization can be easily compensated by replacing \begin{equation} m_F\longrightarrow m_F^{eff} \equiv m_F + g \langle 0|\phi|0\rangle . \end{equation} The second class of diagrams which cannot be generated by a zero-mode free LF field theory is more subtle. As an example, let us consider the one loop fermion self energy \footnote{For simplicity, we will write down the expressions only in 1+1 dimension, but it should be emphasized that the conclusions are also valid in 3+1 dimension \cite{mb:rot}.} \begin{eqnarray} -i\Sigma (p) &=& g^2 \!\! \int \frac{d^2k}{(2\pi)^2} \frac{\gamma^\mu k_\mu+m_F}{k^2 - m_F^2 + i\varepsilon} \frac{1}{(p-k)^2 - m_B^2+ i \varepsilon} \nonumber\\ &=& -i\Sigma_{LF} + g^2 \!\! \int \frac{d^2k}{(2\pi)^2} \frac{\gamma^+}{2k^+} \frac{1}{(p-k)^2 - m_B^2+ i \varepsilon} \label{eq:self} \end{eqnarray} where \begin{equation} -i\Sigma_{LF}= g^2\!\! \int \frac{d^2k}{(2\pi)^2} \frac{\gamma^\mu \tilde{k}_\nu+m_F}{k^2 - m_F^2 + i\varepsilon} \frac{1}{(p-k)^2 - m_B^2+ i \varepsilon} \label{eq:sigmaLF} \end{equation} and $\tilde{k}^+=k^+$ while $\tilde{k}^- = \frac{m_F^2}{2k^+}$ is the on mass shell energy for the fermion. Obviously, Eq. (\ref{eq:self}) is a mere algebraic rewriting of the original Feynman self-energy. The important point is that the second term on the r.h.s. of Eq. (\ref{eq:self}) has the same pole structure as a tadpole diagram and thus cannot be generated by a LF Hamiltonian. Indeed, as one can easily verify, second order perturbation theory with the canonical LF Hamiltonian yields only (the matrix elements of) $\Sigma_{LF}$ and a disagreement between self-energies calculated in covariant perturbation theory and those calculated in LF-Hamiltonian perturbation theory (without zero-modes) emerges. Before we proceed to analyze more general diagrams which suffer from a similar problem, let us understand intuitively how this second term arises: In the LF formulation, not all components of the fermion field are independent degrees of freedom. Multiplying the Dirac equation \begin{equation} \left( i\gamma^\mu \partial_\mu - m_F - g\phi\right) \psi =0 \end{equation} by $\gamma^+$ one finds that \begin{equation} 2 i \partial_- \psi_{(-)} = \left(m_F+g\phi\right) \gamma^+ \psi_{(+)}, \label{eq:constr} \end{equation} where $\psi_{(\pm)} \equiv \frac{1}{2} \gamma^\mp \gamma^\pm \psi$ . Eq. (\ref{eq:constr}) is a constraint equation and it is often used to eliminate the dependent component $\psi_{(-)}$ prior to quantization. This gives rise to ``induced'' four point interactions \begin{equation} {\cal L}^{(4)} = -g^2 \psi_{(+)}^\dagger \phi \frac{1}{i\sqrt{2} \partial_-} \phi \psi_{(+)} \label{eq:L4} \end{equation} in the Lagrangian after eliminating the constrained field $\psi_{(-)}$ and hence it is possible to generate ``induced tadpoles'' diagrams by contracting for example the two scalar fields in Eq. (\ref{eq:L4}). Before discussing the general case, it is very instructive to investigate the one loop fermion self-energy in more detail. First one notes that the $2^{nd}$ order perturbation theory result (\ref{eq:sigmaLF}) is divergent at $k^+ \rightarrow 0$ \begin{equation} \Sigma_{LF} = \frac{g^2}{8\pi} \int_0^{p^+} \frac{dk^+}{k^+(p^+-k^+)} \frac{ k^+\gamma^- + \frac{m_F^2}{2k^+}\gamma^+ + m_F} {p^--\frac{m_F^2}{2k^+} - \frac{m_B^2}{2(p^+-k^+)} }. \label{eq:sigmaLF2} \end{equation} This divergence is cancelled by the self-induced inertia term, which arises from normal ordering Eq. (\ref{eq:L4}) \begin{equation} \Sigma_{n.o.} = \frac{g^2}{8\pi}\int_0^{p^+}\frac{dk^+}{k^+} , \end{equation} yielding \begin{eqnarray} \Sigma_{LF}+\Sigma_{n.o.} &=& \frac{g^2}{4\pi}\int_0^1 dx \frac{xp^\mu \gamma_\mu + m_F}{x(1-x)p^2 - m_F^2(1-x) -m_B^2 x} \nonumber\\ &+&\frac{g^2}{4\pi} \frac{\gamma^+}{p^+} \ln \frac{m_B^2}{m_F^2}. \label{eq:LFno} \end{eqnarray} Several important observations can be made from Eq. (\ref{eq:LFno}). First of all, even though including the normal ordering term renders the self-energy finite, the final result disagrees in general with the covariantly calculated result [the first term on the r.h.s. in Eq. (\ref{eq:LFno})]. Furthermore, the additional term breaks covariance (parity invariance). \footnote{This fact has been used in Ref. \cite{mb:parity} to determine the necessary counterterm non-perturbatively by demanding covariance for physical amplitudes.} However, most importantly, the unwanted term vanishes for $m_F=m_B$, which indicates already a crucial cancellation between bosonic zero-modes and fermionic zero-modes. In the rest of this paper, we will demonstrate for the case of certain supersymmetric theories, that this cancellation goes beyond the one loop result. After this more intuitive discussion of zero-mode effects for fermions, let us now formally derive the counter-terms that arise for a theory with Yukawa interactions. For this purpose, it is useful to identify those Feynman diagrams (external momenta nonzero) where zero-modes in internal lines give a nonzero contribution to the total amplitude. Diagrams which suffer from the same problem as the one-loop fermion self-energy are all diagrams where the internal lines in the fermion self-energy are dressed by arbitrary self-interactions (Fig. \ref{fig:fself}). \begin{figure} \unitlength1.cm \begin{picture}(5,6)(-0.5,1) \special{psfile=fself.ps angle=0 hscale=80 vscale=80} \end{picture} \caption{Fermion self-energy diagram, which is treated improperly when zero-modes are excluded. The shaded blobs represent arbitrary self-energy insertions.} \label{fig:fself} \end{figure} Let us assume that all counter-terms that are necessary to achieve agreement between LF perturbation theory (no zero modes) and covariant perturbation theory have been added to all sub-loops in Fig. \ref{fig:fself}, i.e. we assume that there exists a covariant spectral representation for fermion propagators within the loop \begin{eqnarray} S_F(p) &=& i\int_{0}^\infty \!\!dM^2 \,\frac{\gamma^\mu p_\mu \rho_1 (M^2) + M \rho_2(M^2)}{p^2-M^2 + i\varepsilon} \label{eq:lehmann} \end{eqnarray} Similar to the scalar case, the fermion spectral density has a very simple representation in terms of the eigenstates of the LF Hamiltonian as well (Appendix A) \begin{eqnarray} \rho_1 (M^2) &=& \frac{2\pi}{2P^+} \sum_n \delta\left(\frac{M^2}{2P^+} - P_n^-\right) \left|\langle 0|\Psi_-(0)|n,P^+\rangle \right|^2 \nonumber\\ &=& \sum_n \delta\left(M^2 - M_n^2\right) f_n. \label{eq:fspectr} \end{eqnarray} The spectral representation for bosons (\ref{eq:bspectr}) from the previous section is also still valid (of course with a different spectral function since we now deal with a different theory). For later use, we also note that completeness of the LF eigenstates implies the normalization condition \begin{equation} \int_0^\infty dM^2 \rho_1(M^2) = \int_0^\infty d\mu^2 \rho (\mu^2) =1 \label{eq:norm} \end{equation} for the spectral densities. Using the above spectral representations [Eqs. (\ref{eq:bspectr}) and (\ref{eq:fspectr})] for the internal propagators, we now calculate the necessary counter-term self consistently. The covariant self-energy for the diagram in Fig. \ref{fig:fself} thus reads \begin{eqnarray} -i\Sigma_F &=& g^2 \!\!\int_0^\infty \!\!dM^2 \int_0^\infty \!\!d\mu^2 \int \frac{d^2k}{(2\pi)^2} \frac{\gamma^\nu k_\nu \rho_1(M^2)+M\rho_2(M^2)}{k^2 - M^2 + i\varepsilon} \nonumber\\ & &\quad \times \frac{\rho(\mu^2)}{(p-k)^2 - \mu^2+ i \varepsilon} . \label{eq:sigmacov} \end{eqnarray} We will now calculate the piece which is missed when the vicinity of both $k^+=0$ and $p^+-k^+=0$ is omitted in the integration in Eq. (\ref{eq:sigmacov}) (naive LF quantization with omission of fermion and boson zero-modes respectively). Using the one-loop analysis as a guide, it is clear that the only problems arise in the $\gamma^+$ component of the self-energy. In order to further isolate the troublemaker, we use the algebraic identity \begin{eqnarray} & &\!\!\!\!\frac{k^-}{k^2-M^2+i\varepsilon} \frac{1}{(p-k)^2 - \mu^2 +i\varepsilon} = \nonumber\\ & &\quad \quad \quad \quad \frac{1}{2p^+} \frac{ 2(p^+-k^+)p^- + M^2-\mu^2}{ \left(k^2-M^2+i\varepsilon \right) \left((p-k)^2-\mu^2+i\varepsilon \right)} \nonumber\\ & & \quad \quad \quad \quad + \frac{1}{2p^+}\left[ \frac{1}{k^2-M^2+i\varepsilon} - \frac{1}{(p-k)^2 - \mu^2 +i\varepsilon} \right] . \label{eq:alg} \end{eqnarray} Obviously, the first term on th r.h.s. of Eq. (\ref{eq:alg}) can be straightforwardly integrated over $k^-$ and, for this term, the ``zero-mode regions'' ($k^+=0$ and $p^+-k^+=0$) can be omitted without altering the result of the integration. However, the two last terms on the r.h.s. of Eq. (\ref{eq:alg}) have the pole structure of simple tadpoles and hence their only contribution to the $k$ integration is from zero-modes of the fermions $k^+ =0$ as well as the bosons $p^+-k^+=0$. This simple observation implies that the zero-mode counter-term from the class of diagrams in Fig. \ref{fig:fself} reads \cite{mb:adv} \begin{eqnarray} -i\delta \Sigma_F &=& \frac{g^2 \gamma^+}{2p^+} \int \frac{d^2k}{(2\pi)^2} \int_0^\infty d \mu^2 \frac{\rho (\mu^2)} {k^2 - \mu^2 +i\varepsilon} \nonumber\\ &-& \frac{g^2 \gamma^+}{2p^+} \int \frac{d^2k}{(2\pi)^2} \int_0^\infty d M^2 \frac{\rho_1 (M^2)} {k^2 - M^2 +i\varepsilon} \label{eq:fself0} \end{eqnarray} where we made use of the normalization of the spectral functions (\ref{eq:norm}). A similar zero-mode counter-term arises from ``vacuum-polarization'' type self-energies for the bosons where the fermion and anti-fermion lines may be dressed but where there is no interaction among the fermion and anti-fermion (Fig. \ref{fig:bself}). \begin{figure} \unitlength1.cm \begin{picture}(5,5)(-.8,1) \special{psfile=bself.ps angle=0 hscale=80 vscale=80} \end{picture} \caption{Boson self-energy diagram, which is treated improperly when zero-modes are excluded. The full and dashed lines are fermion and boson propagators respectively. The shaded blobs represent arbitrary self-energy insertions.} \label{fig:bself} \end{figure} Using again the above spectral representation (\ref{eq:lehmann}) one finds for this class of Feynman diagrams \begin{eqnarray} -i\Sigma &=& -2g^2 \int_0^\infty \!\!\!dM_A^2 \int_0^\infty \!\!\!dM_B^2 \int \!\frac{d^2k}{(2\pi^2)} \label{eq:bself}\\ &\times& \frac{ k \cdot (k-p) \rho_1(M_A^2) \rho_1 (M_B^2) + M_A M_B \rho_2 (M_A^2) \rho_2 (M_B^2)} {\left( k^2 -M_A^2 + i\varepsilon \right) \left( (p-k)^2 -M_B^2 + i\varepsilon \right)} .\nonumber \end{eqnarray} The part of Eq. (\ref{eq:bself}) where zero-mode contributions are crucial arise from the $k\cdot (p-k)$ term in the numerator. In order to see this, let us write \begin{eqnarray} 2k\cdot (k-p) &=& \left( M_A^2 + M_B^2 - p^2 \right) \nonumber\\ & &+ \left( k^2-M_A^2 \right) +\left( (p-k)^2-M_B^2 \right) \label{eq:alg2} \end{eqnarray} and we note that the first term on the r.h.s. yields no zero-mode contribution when inserted in Eq. (\ref{eq:bself}). However, the other two terms on the r.h.s. of Eq. (\ref{eq:alg2}) cancel one of the energy denominators and thus again yield a tadpole like pole structure. One thus finds for the contribution from the zero-modes \begin{equation} -i\delta \Sigma = -g^2 \int_0^\infty dM^2 \int \frac{d^2k}{(2\pi)^2} \frac{\rho_1 (M^2) }{k^2-M^2 + i\varepsilon}. \label{eq:floop0} \end{equation} Empirical studies of Feynman diagrams up to three loops \cite{mb:rot} in Yukawa theories have shown that zero-modes play a role for $k^+\neq 0$ modes only in 2-point functions (except of course through sub-diagrams). Furthermore, of all the diagrams contributing to the two point functions, only the very simple sub-class of diagrams discussed above seems to be affected when zero-modes are cut out. Diagrams with a more complicated topology, such as crossed diagrams (except of course through sub-diagrams), require no zero-mode counter-terms when the region $k^+=0$ is cut out. Although no rigorous analytical proof exists at this stage, it is reasonable to assume that these are the only diagrams yielding contributions from zero-modes. In the following, we discuss the consequences of this assumption for supersymmetric field theories. \section{Zero-Modes in Supersymmetric Theories} In order to study the implications of supersymmetry on zero-mode renormalization, let us consider a concrete example, namely a matrix model in 1+1 dimensions with action \cite{igor} \begin{equation} S = \int d^2x \mbox{Tr}\left[ \frac{1}{2} \left(\partial_\mu \phi \right)^2 +\frac{1}{2} \bar{\Psi} i \partial\!\!\!\!\!\!\not\;\, \Psi -\frac{1}{2}V^2(\phi) -\frac{1}{2}V^\prime(\phi) \bar{\Psi}\Psi\right] \end{equation} where $V(\phi) = \mu \phi - \frac{\lambda}{\sqrt{N}} \phi^2$. The canonical LF-Hamiltonian for this model has been discussed in Ref. \cite{igor} and we refer the reader to this paper for details. Obviously this model contains both three-point and four-point interactions for the scalar field as well as a Yukawa coupling between the scalar field and the (Majorana) fermion field, i.e. we can now directly apply above zero-mode analysis to this model. First we note that ``tennis racket'' tadpole diagrams (Fig. \ref{fig:tennis}a) must all vanish in a covariant calculation, since $\langle \phi \rangle \neq 0$ would break the global matrix symmetry of the model. On the LF, without zero-modes, these diagrams are automatically zero for simple kinematic reasons and therefore, there is no need to add any zero-mode counter-terms for tennis racket diagrams to the LF Hamiltonian. Since we have already seen that all tennis racket diagrams vanish in this model, the only diagrams that could still give rise to zero-mode counter-terms are the classes of self-energy diagrams depicted in Figs. \ref{fig:tennis}b, \ref{fig:fself} and \ref{fig:bself}. For the zero-mode contributions to the fermion self-energy, matrix symmetry is not sufficient to prove that the zero-mode counter-term vanishes and we have to invoke supersymmetry. Using the explicit expression for the supercharge $Q_-$ in terms of the LF fields \begin{equation} Q_- \equiv \int dx^- :\mbox{Tr} \left[ \sqrt{2} (\partial_-\phi )\Psi_-\right]: \label{eq:Q-}, \end{equation} operators we obtain the supersymmetry transformation \begin{eqnarray} && [Q_-, \phi]=-\frac{i}{\sqrt 2}\Psi_-, \\ && [Q_-, \Psi_-]= \sqrt 2 \partial_{-}\phi, \end{eqnarray} which gives rise to $Q_-^2= P^+$. Let us show that the spectral densities $\rho$ and $\rho_1$ defined in the previous section is equal owing to the supersymmetry. First note that the states of non-zero energy are paired by the action of supercharge. Namely, \begin{eqnarray} &&Q_-|n,P^+\rangle_{B}=\sqrt{P^+} |n,P^+\rangle_{F}, \\ &&Q_-|n,P^+\rangle_{F}=\sqrt{P^+} |n,P^+\rangle_{B}, \end{eqnarray} where the $B$ and $F$ denote bosonic and fermionic state, respectively. The fermionic state is normalized if the bosonic state is, i.e. $_B\langle n,P^+|n,P^+\rangle_{B}=1$ since $Q_-^2= P^+$. Now we can easily find \begin{eqnarray} \rho_1(M^2) &=& 2\pi \sum_n \delta\left(M^2 - M_n^2 \right) \left|\langle 0|\Psi_-(0)|n,P^+\rangle \right|^2 \nonumber\\ &=& 2\pi \sum_n \delta\left(M^2 - M_n^2 \right) \left|\langle 0|\sqrt{2} [Q_-, \phi(0)]|n,P^+\rangle \right|^2 \nonumber\\ &=& 2\pi \sum_n \delta\left(M^2 - M_n^2 \right) 2P^+ \left|\langle 0|\phi(0)|n,P^+\rangle \right|^2, \nonumber\\ &=& \rho (M^2). \end{eqnarray} Therefore the spectral densities $\rho$ and $\rho_1$ must be equal. Since fermions and bosons contribute with opposite signs (but equal strength) to the zero-mode part of the fermion self-energy [Eq. (\ref{eq:fself0})], the zero-mode contributions from bosons and fermions to the fermion self-energy cancel exactly! \footnote{Note that there is a flaw in the discussion of the two loop fermion self-energy for the SUSY Wess-Zumino model in Ref. \cite{mb:rot} which arises because subtraction procedure employed in Ref. \cite{mb:rot} breaks the supersymmetry. The unsubtracted result in Ref. \cite{mb:rot} is consistent with the above findings of cancellation between bosonic and fermionic zero-mode contributions.} The boson self-energy is more complicated, since we have to consider two different classes of diagrams where zero-modes contribute: tadpoles from $\phi^4$ interactions (Fig. \ref{fig:tennis}b) as well as the vacuum polarization type graphs (Fig. \ref{fig:bself}). Using the results from the previous two sections, we find that the zero-mode contribution from tadpoles to the mass reads (\ref{eq:tadpole}) \begin{equation} \delta \mu^2 _{boson\, ZM}= 4 \lambda^2 \int_0^\infty \!\!\!dM^2 \rho(M^2) \int \frac{d^2k} {(2\pi)^2} \frac{1}{k^2-M^2 +i\varepsilon} . \label{eq:bsuper} \end{equation} For the contribution from zero-modes in fermion loops to the boson self one finds instead \begin{equation} \delta \mu^2 _{fermion\, ZM}= -4 \lambda^2 \int_0^\infty \!\!\!dM^2 \rho_1(M^2) \int \frac{d^2k} {(2\pi)^2} \frac{1}{k^2-M^2 +i\varepsilon} \label{eq:fsuper} \end{equation} and invoking again supersymmetry, we find that the contributions from boson and fermion zero-modes again cancel. Note that supersymmetry has played a dual role in obtaining this fundamental result. First of all, it relates the Yukawa coupling and the scalar four-point coupling and thus the coefficients of Eqs. (\ref{eq:bsuper}) and (\ref{eq:fsuper}) are the same. But the cancellation between Eqs. (\ref{eq:bsuper}) and (\ref{eq:fsuper}) happens only because the spectral densities are the same. \section{summary} Even for theories with massive particles, where zero-modes are high energy degrees of freedom, they cannot be completely discarded. However, they can be integrated out, which gives rise to an effective (tadpole improved) LF Hamiltonian. In supersymmetric theories, there is scope for a complete cancellation between effective interactions induced by bosonic zero-modes and those induced by fermionic zero-modes. There is of course the possibility of spontaneous symmetry breaking, in which fields acquire a non-zero expectation value. In such a scenario, the fermion-boson cancellation may not occur, and we are left with the (difficult) task of deriving an effective Hamiltonian. However, our observations suggest that for theories with enough supersymmetry, the zero-mode degrees of freedom may be ignored. As a result, as long as one is interested only in the dynamics of $k^+\neq 0$ modes in such massive supersymmetric theories, zero modes can be discarded. This implies that for such theories DLCQ (in the continuum $K \rightarrow \infty$ limit, and with the zero-modes discarded) leads to the same Green's functions for $k^+\neq 0$ modes as a covariant formulation. Clearly, it would be interesting to understand the precise connection between the decoupling of zero-modes in supersymmetric theories, and various non-renormalizations theorems that are known to exist. We leave this for future work. \acknowledgements One of us (M.B.) was supported by the D.O.E. under contract DE-FG03-96ER40965 and in part by TJNAF.

BkiUeEHxK1ThhBMLhG21

\section{introduction} Using the unfolded $\bar B \to D^* \ell\, \bar\nu$ spectra from Belle~\cite{Abdesselam:2017kjf}, several theory papers~\cite{Bernlochner:2017jka, Bigi:2017njr, Grinstein:2017nlq} could perform fits to the data for the first time, using different theoretical approaches. Using the BGL parametrization~\cite{Boyd:1995sq, Boyd:1997kz} for the $\bar B \to D^* \ell \, \bar\nu$ form factors, a substantial shift in the extracted value of $|V_{cb}|$ was found~\cite{Bigi:2017njr, Grinstein:2017nlq}, compared to the Belle~\cite{Abdesselam:2017kjf} analysis using the CLN~\cite{Caprini:1997mu} parametrization, \begin{subequations} \begin{eqnarray} |V_{cb}|_{\rm CLN} &=& (38.2\pm 1.5)\times 10^{-3}\,, \quad\,\text{\cite{Abdesselam:2017kjf}}\,, \label{belleVcb}\\ |V_{cb}|_{\rm BGL} &=& (41.7^{+2.0}_{-2.1})\times 10^{-3}\,, \qquad \text{\cite{Bigi:2017njr}}\,, \label{gambinoVcb}\\ |V_{cb}|_{\rm BGL} &=& (41.9^{+2.0}_{-1.9})\times 10^{-3}\,, \qquad \text{\cite{Grinstein:2017nlq}}\,. \label{benVcb} \end{eqnarray} \end{subequations} The main result in Ref.~\cite{Abdesselam:2017kjf} was $|V_{cb}|_{\rm CLN} = (37.4\pm 1.3)\times 10^{-3}$, obtained from a fit inside the Belle framework, before unfolding. Only Eq.~(\ref{belleVcb}) quoted in the Appendix of~\cite{Abdesselam:2017kjf} can be directly compared with Eqs.~(\ref{gambinoVcb}) and (\ref{benVcb}). These papers, as well as this work, use the same fixed value of ${\cal F}(1)$~\cite{Bailey:2014tva} (see Eq.~(\ref{dGdw}) below), so the differences in the extracted values of $|V_{cb}|$ are due to the extrapolations to zero recoil, where heavy quark symmetry gives the strongest constraint on the rate~\cite{Isgur:1989vq, Isgur:1989ed, Shifman:1987rj, Nussinov:1986hw, Luke:1990eg}. Intriguingly, the BGL fit results for $|V_{cb}|$ are compatible with those from inclusive $B\to X_c\ell\bar\nu$ measurements~\cite{HFAG}. If one assumed, naively, a 100\% correlation between the fits yielding Eqs.~(\ref{belleVcb}), (\ref{gambinoVcb}), and (\ref{benVcb}), then the tension between Eqs.~(\ref{belleVcb}) and (\ref{gambinoVcb}) or between Eqs.~(\ref{belleVcb}) and (\ref{benVcb}) would be above $5\sigma$. The BGL~\cite{Boyd:1995sq, Boyd:1997kz} fit implements constraints on the $B\to D^*\ell\bar\nu$ form factors based on analyticity and unitarity~\cite{Bourrely:1980gp, Boyd:1994tt, Boyd:1995cf}. The CLN~\cite{Caprini:1997mu} fit imposes, in addition, constraints on the form factors from heavy quark symmetry, and relies on QCD sum rule calculations~\cite{Neubert:1992wq, Neubert:1992pn, Ligeti:1993hw} of the subleading Isgur-Wise functions~\cite{Luke:1990eg, Falk:1990pz}, without accounting for their uncertainties. Ref.~\cite{Bernlochner:2017jka} performed combined fits to $\bar B\to D^* \ell\, \bar \nu$ and $\bar B\to D \ell\, \bar \nu$, using predictions of the heavy quark effective theory (HQET)~\cite{Georgi:1990um, Eichten:1989zv}, including all ${\cal O}(\ensuremath{\Lambda_{\rm QCD}}\xspace/m_{c,b})$ uncertainties and their correlations for the first time. The effect of relaxing the QCD sum rule inputs in the CLN fit was found to be small compared to the difference of the CLN and BGL results. The recent papers using the BGL parametrization~\cite{Bigi:2017njr, Grinstein:2017nlq} assert that the higher values obtained for $|V_{cb}|$ are due to the too restrictive functional forms used in the CLN fits. It was previously also noticed that the CLN gives a poorer fit to the $B\to D\ell\bar\nu$ data than BGL~\cite{Bigi:2016mdz}. The effects on $|V_{cb}|$ due to additional theoretical inputs were also explored in Refs.~\cite{Bigi:2017jbd, Jaiswal:2017rve}. Based on our work in Ref.~\cite{Bernlochner:2017jka}, we explore which differences between the BGL and CLN fits are responsible for the different extracted $|V_{cb}|$ values, study the consistency and compatibility of the fits, and the significance of the shift in the extracted value of $|V_{cb}|$. \begin{table*}[bt] \begin{tabular}{c|cccc} \hline\hline form factors & BGL & CLN & CLNnoR\xspace & noHQS\xspace \\ \hline axial $\propto \epsilon^*_\mu$ & $b_0,\, b_1$ & $h_{A_1}(1),\ \rho_{D^*}^2$ & $h_{A_1}(1),\ \rho_{D^*}^2$ & $h_{A_1}(1),\ \rho_{D^*}^2,\ c_{D^*}$ \\ vector & $a_0,\, a_1$ & \multirow{2}{*}{$\bigg\{ R_1(1),\, R_2(1)$} & \multirow{2}{*}{$\bigg\{$ \hspace*{-10pt} \begin{tabular}{l} $R_1(1),\ R'_1(1)$ \\ $R_2(1),\ R'_2(1)$\end{tabular}} & \multirow{2}{*}{$\bigg\{$ \hspace*{-10pt} \begin{tabular}{l} $R_1(1),\ R'_1(1)$ \\ $R_2(1),\ R'_2(1)$\end{tabular}} \\ ${\cal F}$ & $c_1,\, c_2$ \\ \hline\hline \end{tabular} \caption{The fit parameters in the BGL, CLN, CLNnoR\xspace, and noHQS\xspace fits, and their relationships with the form factors.} \label{fitsummary} \end{table*} \section{definitions} The $B\to D^{*}\ell\bar\nu$ form factors which occur in the standard model are defined as \begin{align}\label{formfactors} \ampBb{D^*}{\bar{c} \gamma^\mu b} & = i\sqrt{m_B m_{D^*}}\, h_V\, \varepsilon^{\mu\nu\alpha\beta}\, \epsilon^*_{\nu}v'_\alpha v_\beta \,, \nonumber\\* \ampBb{D^*}{\bar{c} \gamma^\mu \gamma^5 b} & = \sqrt{m_B m_{D^*}}\, \big[h_{A_1} (w+1)\epsilon^{*\mu} \\* & \quad - h_{A_2}(\epsilon^* \cdot v)v^\mu - h_{A_3}(\epsilon^* \cdot v)v'^\mu \big] , \nonumber \end{align} where $v$ is the four-velocity of the $B$ and $v'$ is that of the $D^{*}$. The form factors $h_{V,A_{1,2,3}}$ depend on $w = v\cdot v' = (m_B^2+m_{D^{*}}^2-q^2) / (2m_B m_{D^{*}})$. Neglecting lepton masses, only one linear combination of $h_{A_2}$ and $h_{A_3}$ is measurable. In the heavy quark limit, $h_{A_1} = h_{A_3} = h_V = \xi$ and $h_{A_2} = 0$, where $\xi$ is the Isgur-Wise function~\cite{Isgur:1989vq, Isgur:1989ed}. Each of these form factors can be expanded in powers of $\ensuremath{\Lambda_{\rm QCD}}\xspace / m_{c,b}$ and $\alpha_s$. It is convenient to parametrize deviations from the heavy quark limit via the form factor ratios \begin{equation}\label{eqn:R1R2Def} R_1(w) = \frac{h_V}{h_{A_1}}\,, \qquad R_2(w) = \frac{h_{A_3} + r_{D^*} h_{A_2}}{h_{A_1}}\,, \end{equation} which satisfy $R_{1,2}(w) = 1 + {\cal O}(\ensuremath{\Lambda_{\rm QCD}}\xspace/m_{c,b},\, \alpha_s)$ in the $m_{c,b} \gg \ensuremath{\Lambda_{\rm QCD}}\xspace$ limit, and $r_{D^*} = m_{D^*}/m_B$. The $B\to D^*\ell\bar\nu$ decay rate is given by \begin{align}\label{dGdw} \frac{{\rm d} \Gamma}{{\rm d} w} & = \frac{G_F^2|V_{cb}|^2\, m_B^5}{48 \pi^3}\, (w^2-1)^{1/2}\, (w + 1)^2\, r_{D^*}^3 (1- r_{D^*})^2 \nonumber \\* & \quad \times \bigg[1 + \frac{4w}{w+1} \frac{1- 2 wr_{D^*} + r_{D^*}^2}{(1 - r_{D^*})^2} \bigg] \mathcal{F}(w)^2\,, \end{align} and the expression of ${\cal F}(w)$ in terms of the form factors defined in Eq.~(\ref{formfactors}) is standard in the literature~\cite{Manohar:2000dt}. In the heavy quark limit, ${\cal F}(w) = \xi(w)$. We further denote \begin{equation} \rho_{D^*}^2 = -\frac1{h_{A_1}(1)} \frac{{\rm d} h_{A_1}(w)}{{\rm d} w}\bigg|_{w=1} \,, \end{equation} which is a physical fit parameter in the CLN approach, and is a derived quantity in the other fits. \section{New fits, lattice QCD, and their tensions} The constraints built into the CLN fit can be relaxed by ignoring the QCD sum rule inputs and the condition $R_{1,2}(w) = 1 + {\cal O}(\ensuremath{\Lambda_{\rm QCD}}\xspace/m_{c,b},\, \alpha_s)$ following from heavy quark symmetry. (Ref.~\cite{Bernlochner:2017jka} showed that only ignoring the QCD sum rule inputs, and using only $w=1$ lattice QCD data, leaves $|V_{cb}| = (38.8 \pm 1.2) \times 10^{-3}$.) Thus, we write \begin{align} R_1(w) & = R_1(1) + (w-1) R_1'(1) \,, \nonumber\\* R_2(w) & = R_2(1) + (w-1) R_2'(1) \,, \end{align} and treat $R_{1,2}(1)$ and $R_{1,2}'(1)$ as fit parameters. We refer to this fit as ``CLNnoR\xspace''. It has the same number of fit parameters as BGL, and allows ${\cal O}(1)$ heavy quark symmetry violation, but the constraints on the form factors are nevertheless somewhat different than in BGL. While this CLNnoR\xspace fit is a simple modification of the CLN fit widely used by BaBar and Belle, it still relies on heavy quark symmetry and model-dependent input on subleading Isgur-Wise functions. The reason is that both CLN and CLNnoR\xspace use a cubic polynomial in $z = (\sqrt{w+1} - \sqrt2) / (\sqrt{w+1} + \sqrt2)$ to parametrize the form factor $h_{A_1}$, with its four coefficients determined by two parameters, $h_{A_1}(1)$ and $\rho_{D^*}^2$, derived from unitarity constraints on the $B\to D$ form factor. Therefore, we also consider a ``noHQS\xspace'' scenario, parametrizing $h_{A_1}$ by a quadratic polynomial in $z$, with unconstrained coefficients, \begin{equation} h_{A_1}(w) = h_{A_1}(1) \big[ 1 - 8 \rho_{D^*}^2 z + (53.\, c_{D^*} - 15.) z^2 \big] , \end{equation} keeping the same prefactors as in CLN, to permit comparison between $\rho_{D^*}^2$ and $c_{D^*}$ (in the CLN fit $c_{D^*} = \rho_{D^*}^2$). The fit parameters in the BGL, CLN, CLNnoR\xspace, and \mbox{noHQS\xspace} fits are summarized in Table~\ref{fitsummary}. The results of these fits for $|V_{cb}|$, $\rho^2_{D^*}$, $c_{D^*}$, $R_{1,2}(1)$, and $R_{1,2}'(1)$ are shown in Table~\ref{tab:resVcbDs}. The BGL, CLNnoR\xspace, and noHQS\xspace results are consistent with each other, including the uncertainties, and the fit quality. The correlations of these four fit results for $|V_{cb}|$ are shown in Table~\ref{tab:corrVcbDs} and have been derived by creating a bootstrapped~\cite{Hayes:1988xc} ensemble of the unfolded distributions of Ref.~\cite{Abdesselam:2017kjf}, using the published covariance. Each set of generated decay distributions in the ensemble is fitted with the BGL, CLN, CLNnoR\xspace, and \mbox{noHQS\xspace} parametrizations, and the produced ensemble of $|V_{cb}|$ values is used to estimate the covariance between them. The correlation of the CLN fit with either BGL, CLNnoR\xspace, or noHQS\xspace is substantially below 100\%. This reduces the tension between these fits to below $3\sigma$. \begin{table}[t]\tabcolsep 2pt \resizebox{\linewidth}{!}{ \begin{tabular}{c|cccc} \hline\hline & CLN & CLNnoR\xspace & noHQS\xspace & BGL \\ \hline $|V_{cb}| \!\times\! 10^{3}$ & $38.2 \pm 1.5$ & $41.5 \pm 1.9$ & $41.8 \pm 1.9$ & $41.5 \pm 1.8$ \\ $\rho^2_{D^*}$ & $1.17 \pm 0.15$ & $1.6 \pm 0.2$ & $1.8 \pm 0.4$ & $1.54 \pm 0.06$ \\ $c_{D^*}$ & $\rho^2_{D^*}$ & $\rho^2_{D^*}$ & $2.4 \pm 1.6$ & fixed: 15./53. \\ $R_1(1)$ & $1.39 \pm 0.09$ & $0.36 \pm 0.35$ & $0.48 \pm 0.48$ & $0.45 \pm 0.28$ \\ $R_2(1)$ & $0.91 \pm 0.08$ & $1.10 \pm 0.19$ & $0.79 \pm 0.36$ & $1.00 \pm 0.18$ \\ $R_1'(1)$ & fixed: $-0.12$ & $5.1 \pm 1.8$ & $4.3 \pm 2.6$ & $4.2 \pm 1.2$ \\ $R_2'(1)$ & fixed: 0.11 & $-0.89 \pm 0.61$ & $0.25 \pm 1.3$ & $-0.53 \pm 0.42$ \\ \hline $\chi^2$\,/ ndf & 35.2 / 36 & 27.9 / 34 & 27.6 / 33 & 27.7 / 34 \\ \hline\hline \end{tabular} } \caption{Summary of CLN, CLNnoR\xspace, noHQS\xspace, and BGL fit results.} \label{tab:resVcbDs} \end{table} \begin{table}[t] \begin{tabular}{l|cccc} \hline\hline & $|V_{cb}|_{\rm CLN}$ & $|V_{cb}|_{\rm CLNnoR\xspace}$ & $|V_{cb}|_{\rm noHQS\xspace}$ & $|V_{cb}|_{\rm BGL}$ \\ \hline $|V_{cb}|_{\rm CLN}$ & 1. & 0.75 & 0.69 & 0.76 \\ $|V_{cb}|_{\rm CLNnoR\xspace}$ & & 1. & 0.95 & 0.97 \\ $|V_{cb}|_{\rm noHQS\xspace}$ & & & 1. & 0.97 \\ $|V_{cb}|_{\rm BGL}$ & & & & 1. \\ \hline\hline \end{tabular} \caption{Correlation matrix of the four extracted $|V_{cb}|$ values. For BGL the outer functions of Ref.~\cite{Grinstein:2017nlq} were used. All results are derived by bootstrapping~\cite{Hayes:1988xc} the unfolded distributions of Ref.~\cite{Abdesselam:2017kjf} using the published covariance.} \label{tab:corrVcbDs} \end{table} \begin{figure*}[th] \centerline{ \includegraphics[width=0.49\textwidth]{figures/R1_filled_latt} \hfil \includegraphics[width=0.51\textwidth]{figures/R2_filled_latt}} \caption{The form factor ratios $R_1(w)$ (left) and $R_2(w)$ (right) for the BGL (red long dashed), CLN (gray dashed), CLNnoR\xspace (orange dotted) fits, and noHQS\xspace (purple dot-dot-dashed). The BGL, CLNnoR\xspace, and noHQS\xspace fits for $R_1$ suggest a possibly large violation of heavy quark symmetry, in conflict with lattice QCD predictions. The blue lines show our estimated bounds, based on preliminary FNAL/MILC lattice results~\cite{BDsLatticeAllw}. The black data point for $R_1(1)$ follows from the FNAL/MILC $B\to D\, \ell\, \bar \nu$ result and heavy quark symmetry (see details in the text).} \label{fig:R1_R2} \end{figure*} As soon as $R'_{1,2}(1)$ are not constrained to their values imposed in the CLN framework, large deviations from those constraints are observed. The BGL, CLNnoR\xspace, and noHQS\xspace results favor a large value for $R_1'(1)$, in tension with the heavy quark symmetry prediction, $R_1'(1) = {\cal O}(\ensuremath{\Lambda_{\rm QCD}}\xspace/m_{c,b},\, \alpha_s)$. These aspects of the BGL, CLNnoR\xspace, and noHQS\xspace fits are also in tension with lattice QCD results. Recently the first preliminary lattice results were made public on the $B\to D^*\ell\bar\nu$ form factors away from zero recoil, at finite lattice spacing~\cite{BDsLatticeAllw}. The results are fairly stable over a range of lattice spacings. Assuming that the continuum extrapolation will not introduce a sizable shift (the chiral logs are not large~\cite{Chow:1993hr, Randall:1993qg}) we can estimate the projections for the $R_{1,2}(w)$ form factor ratios. We approximate the predicted form factors in a narrow range of $w$ using a linear form, with a normalization and slope chosen such that they encompass all reported lattice points and uncertainties in Ref.~\cite{BDsLatticeAllw}. At zero recoil we obtain the estimates $R_1(1) \simeq 1.5 \pm 0.2$ and $R_2(1) \simeq 0.95 \pm 0.45$, which should be viewed as bounds on these values, as the actual lattice QCD results will likely have smaller uncertainties. Figure~\ref{fig:R1_R2} shows $R_{1,2}(w)$ derived from the results of our fit scenarios, as well as these lattice QCD constraints. We can obtain another independent prediction for $R_1(1)$ based on lattice QCD and heavy quark symmetry, using the result for the $B\to D\, \ell \bar\nu$ form factor~\cite{Lattice:2015rga}. Using the $\mathcal{O}(\ensuremath{\Lambda_{\rm QCD}}\xspace/m_{c,b},\, \alpha_s)$ expressions~\cite{Bernlochner:2017jka}, the $f_+$ form factor (see Eq.~(2.1) in Ref.~\cite{Lattice:2015rga}) and the subleading Isgur-Wise function $\eta$ are related at zero recoil via \begin{align} \frac{ 2 \sqrt{r_D} }{1+r_D} f_+(1) & = 1 + \hat \alpha_s \bigg( C_{V_1} + C_{V_2}\, \frac{2r_D}{1+r_D} + C_{V_3}\, \frac{2}{1+r_D} \bigg) \nonumber\\ & \quad - (\varepsilon_c - \varepsilon_b)\, \frac{1-r_D}{1+r_D}\, [2\, \eta(1)-1] + \ldots ,\!\!\!\! \end{align} since other subleading Isgur-Wise functions enter suppressed by $w-1$. Here $r_D = m_D/m_B$, $\varepsilon_{c,b} = \bar\Lambda/m_{c,b}$ is treated as in Ref.~\cite{Bernlochner:2017jka}, and hereafter the ellipsis denotes $\mathcal{O}(\varepsilon_{c,b}^2,\, \alpha_s\, \varepsilon_{c,b},\, \alpha_s^2)$ higher order corrections. Using $f_+(w = 1) = 1.199 \pm 0.010$~\cite{Lattice:2015rga} one finds $\eta(1) = 0.35 \pm 0.10$. The uncertainty in this relation and the extracted value of $\eta(1)$ is dominated by $\mathcal{O}(\ensuremath{\Lambda_{\rm QCD}}\xspace^2/m_c^2)$ corrections parametrized by several unknown matrix elements~\cite{Falk:1992wt}, which we estimate with $\varepsilon_c^2 \sim0.05$. Thus, \begin{equation}\label{R11eq} R_1(1) = 1.34 - 0.12 \, \eta(1) + \ldots = 1.30 \pm 0.05 \,. \end{equation} (Recall that both the $\alpha_s$ terms and a $\bar\Lambda/(2m_c)$ correction enhance $R_1(1)$.) This estimate is shown with the black dot and error bar in the left plot in Fig.~\ref{fig:R1_R2}. It shows good consistency with our estimate from the preliminary direct calculation of the $B\to D^*\ell\bar\nu$ form factors, as shown in the region bounded by the blue curves. Another clear way to see that the central values of the BGL, CLNnoR\xspace, and noHQS\xspace fit results cannot be accommodated in HQET, without a breakdown of the expansion, is by recalling~\cite{Bernlochner:2017jka} that besides Eq.~(\ref{R11eq}), also \begin{align}\label{R121} R_2(1) &= 0.98 - 0.42\, \eta(1) - 0.54\, \hat\chi_2(1) + \ldots \,,\nonumber\\ R'_1(1) &= -0.15 + 0.06\, \eta(1) - 0.12\, \eta'(1) + \ldots\,, \\ R'_2(1) &= 0.01 - 0.54\, \hat\chi'_2(1) + 0.21\,\eta(1) - 0.42\, \eta'(1) + \ldots\,.\nonumber \end{align} Here $\eta$ and $\hat\chi_2$ are subleading Isgur-Wise functions. Eqs.~(\ref{R11eq}) and (\ref{R121}) have no solutions close to the BGL, CLNnoR\xspace, or noHQS\xspace fit results in Table~\ref{tab:resVcbDs} with ${\cal O}(1)$ values for $\eta(1)$, $\eta'(1)$, $\hat\chi_2(1)$, and $\hat\chi'_2(1)$. Figure~\ref{fig:dGdw} shows ${\rm d}\Gamma/{\rm d} w$ in the four fit scenarios, as well as the Belle data~\cite{Abdesselam:2017kjf}. The shaded bands show the uncertainties of the CLN and noHQS\xspace fits, which are comparable to the uncertainties of the other two fits. The BGL, CLNnoR\xspace, and noHQS\xspace fits show larger rates near zero and maximal recoil, in comparison to CLN. The CLN fit shows a larger rate at intermediate values of $w$. \begin{figure}[t] \centerline{\includegraphics[width=\columnwidth]{figures/w_spectra_BGL_CLNp}} \caption{${\rm d}\Gamma/{\rm d} w$ for the fit scenarios shown in Fig.~\ref{fig:R1_R2}.} \label{fig:dGdw} \end{figure} \section{Conclusions} Our results show that the tensions concerning the exclusive and inclusive determinations of $|V_{cb}|$ cannot be considered resolved. The central values of the BGL, CLNnoR\xspace, and noHQS\xspace fits, which all give good descriptions of the data, suggest possibly large deviations from heavy quark symmetry. These results are also in tension with preliminary lattice QCD predictions for the form factor ratio $R_1$, which use the same techniques as for the determination of ${\cal F}(1)$ used to extract $|V_{cb}|$ from $B\to D^*\ell\bar\nu$. If the resolution of the tension between lattice QCD and the fits for $R_1$ is a fluctuation in the data, then we would expect the extracted value of $|V_{cb}|$ to change in the future. If the resolution of the tension is on the lattice QCD side, then it may also affect the calculation of ${\cal F}(1)$ used to extract $|V_{cb}|$. We look forward to higher statistics measurements in the future, and a better understanding of the composition of the inclusive semileptonic rate as a sum of exclusive channels~\cite{Bernlochner:2012bc, Bernlochner:2014dca}, which should ultimately allow unambiguous resolution of these questions. \acknowledgments FB and ZL thank Prof.\ Toru Iijima for organizing the ``Mini-workshop on $D^{(*)}\tau\nu$ and related topics", and the kind hospitality in Nagoya, where this work started. We also thank the Aspen Center of Physics, supported by the NSF grant PHY-1066293, where this paper was completed. We thank Ben Grinstein and Bob Kowalewski for helpful conversations, not only over sushi and sake. FB was supported by the DFG Emmy-Noether Grant No.\ BE~6075/1-1. ZL and MP were supported in part by the U.S.\ Department of Energy under contract DE-AC02-05CH11231. DR acknowledges support from the University of Cincinnati. \bibliographystyle{apsrev4-1}

BkiUeHA4ubnhC4Ri7g1M

\section{Introduction} The best actuator design can improve performance and significantly reduce the cost of the control in distributed parameter systems; see for example \cite{morris2015comparison}. The optimal actuator design problem of linear systems has been reviewed in various contexts, see \cite{frecker2003recent,van2001review}. For linear partial differential equations (PDEs), the existence of an optimal actuator location has been proven in the literature. In \cite{morris2011linear}, it is proven that an optimal actuator location exists for a linear system with quadratic cost function if the input operator is compact and continuously depends on actuator locations. Further conditions on operators and cost functions are needed to guarantee the convergence in numerical schemes \cite{morris2011linear}. Similar results have been obtained for $H_2$ and $H_\infty$ controller design objectives \cite{DM2013,kasinathan2013h}. Nonlinearities can have a significant effect on dynamics, and such systems cannot be accurately modelled by linear differential equations. Control of systems modelled by nonlinear partial differential equations (PDE's) has been studied for a number of applications, including wastewater treatment systems \cite{martinez2000}, steel cooling plants \cite{unger2001}, oil extraction through a reservoir \cite{li2003}, solidification models in metallic alloys \cite{boldrini2009}, thermistors \cite{homberg2010optimal}, Schl\"ogl model \cite{buchholz2013,casas2013}, FitzHugh–Nagumo system \cite{casas2013}, micro-beam model \cite{edalatzadeh2016boundary}, static elastoplasticity \cite{reyes2016}, type-II superconductivity \cite{yousept2017optimal}, Fokker-Planck equation \cite{fleig2017}, Schr\"odinger equation with bilinear control \cite{ciaramella2016}, Cahn-Hilliard-Navier-Stokes system \cite{hintermuller2017optimal}, wine fermentation process \cite{merger2017optimal}, time-dependent Kohn-Sham model \cite{sprengel2018investigation}, elastic crane-trolley-load system \cite{kimmerle2018optimal}, and railway track model \cite{edalatzadeh2019stability}. A review of PDE-constrained optimization theory can be found in the books \cite{hinze2008optimization,leugering2012constrained,troltzsch2010optimal}. State-constrained optimal control of PDEs has also been studied. In \cite{bergounioux2003structure}, the authors investigated the structure of Lagrange multipliers for state constrained optimal control problem of linear elliptic PDEs. Research on optimal control of PDEs, such as \cite{casas1997pontryagin,raymond1999hamiltonian}, has focused on partial differential equations with certain structures. Optimal control of differential equations in abstract spaces has rarely been discussed \cite{meyer2017optimal}. This paper extends previous results to abstract differential equations without an assumption of stability. Few studies have discussed optimal control for general classes of nonlinear distributed parameter systems; and even less have looked into actuator design problem of such systems. Using a finite dimensional approximation of the original partial differential equation model, optimal actuator location has been addressed for some applications. Antoniades and Christofides \cite{antoniades2001integrating} investigated the optimal actuator and sensor location problem for a transport-reaction process using a finite-dimensional model. Similarly, Lou and Christofides \cite{lou2003optimal} studied the optimal actuator and sensor location of Kuramoto-Sivashinsky equation using a finite-dimensional approximation. Other research concerned with optimal actuator location for nonlinear distributed parameter systems can be found in \cite{armaou2008robust,moon2006finite,saviz2015optimal}. To our knowledge, there are no theoretical results on optimal actuator design of nonlinear distributed parameter systems. The results of this paper apply to the Kuramoto-Sivashinsky (KS) equation. This equation was derived by Kuramoto to model angular phase turbulence in reaction–diffusion systems \cite{kuramoto1975formation}, and by Sivashinsky for modeling plane flame propagation \cite{sivashinsky1977nonlinear}. It also models film layer flow on an inclined plane \cite{craster2009dynamics}, directional solidification of dilute binary alloys \cite{novick1987interfacial}, growth and saturation of the potential of dissipative trapped-ion \cite{laquey1975}, and terrace edge evolution during step-flow growth \cite{bena1993}. From system theoretic perspective, Christofides and Armaou studied the global stabilization of KS equation using distributed output feedback control \cite{christofides2000}. Lou and Christofides investigated the optimal actuator/sensor placement for control of KS equation by approximating the model with a finite dimensional system \cite{lou2003optimal}. Gomes et al. also studied the actuator placement problem for KS equation using numerical algorithms \cite{gomes2017}. The feedback control as well as optimal actuator arrangement of multidimensional KS equation has been studied in \cite{tomlin2019point}. Controllability of KS equation has also been studied \cite{cerpa2010,cerpa2011}. Optimal control of KS equation using maximum principle was studied in \cite{sun2010}. Optimal control of KS equation with point-wise state and mixed control-state constraints was studied in \cite{gao2016}. Liu and Krstic studied boundary control of KS equation in \cite{liu2001stability}. Al Jamal and Morris studied the relationship between stability and stabilization of linearized and nonlinear KS equation \cite{al2018linearized}. The paper is organized as follows. Section 2 is a short section containing notation and definitions. Section 3 discusses the existence of an optimal input together with an optimal actuator design to nonlinear parabolic systems. In section 4, the worst initial condition is discussed. In section 5 and 6, Kuramoto-Sivashinsky equation and nonlinear heat equation are discussed, respectively. \section{Notation and Definitions} Let $\ss$ be a reflexive Banach space. The notation $\ss_1\hookrightarrow \ss_2$ means that the space $\ss_1$ is densely and continuously embedded in $\ss_2$. Also, letting $I\subset \mathbb{R}$ to be a possibly unbounded interval, the Banach space $C^s(I;\ss)$ consists of all H\"older continuous $\ss$-valued functions with exponent $s$ equipped with norm \begin{equation} \norm{{\bm{x}}}{C^s(I;\ss)}=\norm{{\bm{x}}}{C(I;\ss)}+\sup_{t,s\in I}\frac{\normm{{\bm{x}}(t)-{\bm{x}}(s)}}{|t-s|^s}. \end{equation} The Banach space $c^{s}(I,\ss)$ is the space of little-H\"older continuous functions with exponent $s$ defined as all ${\bm{x}}\in C^s(I;\ss)$ such that \begin{equation} \lim_{\delta\to 0} \sup_{t,s\in I, |t-s|\le \delta}\frac{\normm{{\bm{x}}(t)-{\bm{x}}(s)}}{|t-s|^s}=0. \end{equation} Also, $W^{m,p}(I;\ss)$ is the space of all strongly measurable functions ${\bm{x}}:I\to \ss$ for which $\norm{{\bm{x}}(t)}{\ss}$ is in $W^{m,p}(I,\mathbb{R})$. For simplicity of notation, when $I$ is an interval, the corresponding space will be indicated without the braces; for example $C([0,\tau];\ss)$ will be indicated by $C(0,\tau;\ss) . $ Let ${\mathcal{A}}$ be the generator of an analytic semigroup $e^{{\mathcal{A}} t}$ on $\ss$. For every $p\in [1,\infty]$ and $\alpha\in (0,1)$, the interpolation space $D_{{\mathcal{A}}}(\alpha,p)$ is defined as the set of all ${\bm{x}}_0 \in \ss$ such that the function \begin{equation} t \mapsto v(t)\coloneqq\normm{t^{1-\alpha-1/p}{\mathcal{A}} e^{t{\mathcal{A}}}{\bm{x}}_0} \end{equation} belongs to $L^p(0,1)$ \cite[Section 2.2.1]{lunardi2012analytic}. The norm on this space is $$\norm{{\bm{x}}_0 }{D_{{\mathcal{A}}}(\alpha,p)}=\normm{{\bm{x}}_0 }+\norm{v}{L^p(0,1)}.$$ The Banach space ${\mathbb{W}}(0,\tau)$ is the set of all $ {\bm{x}} (\cdot ) \in W^{1,p}(0,\tau;\ss)\cap L^p(0,\tau;D({\mathcal{A}}))$ with norm \cite[Section II.2]{bensoussan2015book} \begin{equation}\notag \norm{{\bm{x}}}{{\mathbb{W}}(0,\tau)}=\norm{\dot{{\bm{x}}}}{L^p(0,\tau;\ss)}+\norm{{\mathcal{A}} {\bm{x}}}{L^p(0,\tau;\ss)}. \end{equation} \begin{definition}\label{def-maximal} The operator $\mc{A}:D({\mathcal{A}})\to \ss$ is said to have maximal $L^p$ regularity if for every $\bm{f}\in L^p(0,\tau;\ss)$, $1<p<\infty$, the equation \begin{equation}\label{eq-linear} \left\{\begin{array}{l} \dot{{\bm{x}}}(t)={\mathcal{A}}{\bm{x}}(t)+\bm{f}(t),\quad t>0,\\ {\bm{x}}(0)={\bm{x}}_0, \end{array}\right. \end{equation} admits a unique solution in ${\mathbb{W}}(0,\tau)$ that satisfies (\ref{eq-linear}) almost everywhere on $[0,\tau]$. \end{definition} Every generator of an analytic semigroup on a Hilbert space has maximal $L^p$ regularity \cite[Theorem 4.1]{dore1993p}. \section{Nonlinear Parabolic Systems}\label{sec-main results} Let ${\bm{x}}(t)$ and ${\bm{u}}(t)$ be the state and input taking values in reflexive Banach spaces $\ss$ and $\cs$, respectively. Also, let ${\bm{r}}$ denote the actuator design parameter that takes value in a compact set $K_{ad}$ of a topological space $\as$. Consider the following initial value problem (IVP): \begin{equation}\label{eq-IVP} \begin{cases} \dot{{\bm{x}}}(t)=\mc{A}{\bm{x}}(t)+\mc{F}({\bm{x}}(t))+\mc{B}({\bm{r}}){\bm{u}}(t),\quad t>0,\\ {\bm{x}}(0)={\bm{x}}_0. \end{cases} \end{equation} The linear operator ${\mathcal{A}}:D({\mathcal{A}})\to \ss$ is assumed to have maximal $L^p$ regularity. In particular, if ${\mathcal{A}}$ is associated with a sesquilinear form that is bounded and coercive with respect to ${\mathbb{V}} \hookrightarrow \ss, $ it generates an analytic semigroup on $\ss$ \cite[Lemma 36.5 and Theorem 36.6]{sell2013dynamics}. The nonlinear operator ${\mathcal{F}}(\cdot)$ maps a reflexive Banach space ${\mathbb{V}}$ to $\ss$ where $D_{{\mathcal{A}}}(1/p,p)\hookrightarrow {{\mathbb{V}}}\hookrightarrow \ss.$ The operator $\mc{F}(\cdot)$ is locally Lipschitz continuous; that is, for every bounded set $D$ in ${\mathbb{V}}$, there is a positive number $L_{{\mathcal{F}}}$ such that \begin{equation} \normm{{\mathcal{F}}({\bm{x}}_2)-{\mathcal{F}}({\bm{x}}_1)}_{\ss} \le L_{{\mathcal{F}}} \norm{{\bm{x}}_2-{\bm{x}}_1}{{\mathbb{V}}}, \; \forall{\bm{x}}_1, {\bm{x}}_2\in D. \label{eq-nonlinear-assn} \end{equation} When there is no ambiguity, the norm on $\ss$ will not be explicitly indicated. For each ${\bm{r}}\in \as$, the input operator $\mc{B}({\bm{r}})$ is a linear bounded operator that maps the input space $\cs$ into the state space $\ss$ and it is continuous with respect to ${\bm{r}} :$ \begin{equation}\label{B-cont} \lim_{{\bm{r}}_n \to {\bm{r}}_0} \| \mc B ({\bm{r}}_n ) - \mc B({\bm{r}}_0) \| = 0 , \end{equation} where the convergence ${\bm{r}}_n\to {\bm{r}}_0$ is with respect to the topology on $\as .$ \begin{comment} {For every ${\bm{u}}\in L^p(0,\tau;\cs)$, the operator $\mc{B}(\cdot){\bm{u}}:K\to L^p(0,\tau;\ss)$ is weakly continuous with respect to ${\bm{r}} .$ In other words, if $ \lim_{n\to \infty} {\bm{r}}_n = {\bm{r}}_0$ in the topology on $\as ,$ then, letting $q$ be such that $\frac{1}{q}+\frac{1}{p} =1 ,$ for all $\phi (\cdot ) \in L^q (0,\tau; \ss ) , $ \begin{equation}\label{as-B(r)} \lim_{n \to \infty} \int_0^\tau \inn{ \mc{B}({\bm{r}}_n){\bm{u}} (s ) - \mc{B}({\bm{r}}_0){\bm{u}} (s) }{\phi (s)} ds = 0 . \end{equation} If $\cs$ is finite-dimensional, as is usual in applications, then this assumption is equivalent to ???? } \end{comment} For any positive numbers $R_1$ and $R_2$, define the sets \begin{flalign}\label{ad sets} \Bu&=\left\{{\bm{u}}\in L^p(0,\tau;\cs): \norm{{\bm{u}}}{p}\le R_1 \right\},\\ B_{{\mathbb{V}}}(R_2)&=\left\{{\bm{x}}_0 \in {\mathbb{V}}: \norm{{\bm{x}}_0}{{\mathbb{V}}}\le R_2 \right\}. \end{flalign} \begin{definition}\cite[Definition 3.1.i]{bensoussan2015book}(strict solution) The function ${\bm{x}}(\cdot)$ is said to be a {\em strict solution} of (\ref{eq-IVP}) if ${\bm{x}}(0)={\bm{x}}_0$, ${\bm{x}}\in {\mathbb{W}}(0,\tau)$, and ${\bm{x}}(t)$ satisfies (\ref{eq-IVP}) for almost every $t\in[0,\tau]$. \end{definition} \begin{lemma}\cite[Proposition 2.2 and Corollary 2.3]{clement1993abstract}\label{lem-clement} Let $\tau_0>\tau$ and $p\in (1,\infty)$ be given. If ${\mathcal{A}}$ has maximal $L^p$ regularity, then there exists a constant $c_{\tau_0}$ independent of $\tau$ such that for all $\tau\in (0,\tau_0]$ and ${\bm{v}}\in W^{1,p}(0,\tau;\ss)\cap L^p(0,\tau;D({\mathcal{A}})) ,$ \begin{flalign*} \norm{\dot{{\bm{v}}}}{L^2(0,\tau;\ss)}&+\norm{{\mathcal{A}}{\bm{v}}}{L^2(0,\tau;\ss)}\\ &\le M_{\tau_0}\left(\norm{\dot{{\bm{v}}}+{\mathcal{A}}{\bm{v}}}{L^2(0,\tau;\ss)}+\norm{{\bm{v}}(0)}{D_{{\mathcal{A}}}(1/p,p)}\right) \, . \end{flalign*} Furthermore, if ${\bm{v}}(0)=0$, \begin{equation*} \norm{{\bm{v}}}{C(0,\tau;D_{{\mathcal{A}}}(1/p,p))}\le M_{\tau_0}\left(\norm{\dot{{\bm{v}}}}{L^2(0,\tau;\ss)}+\norm{{\mathcal{A}}{\bm{v}}}{L^2(0,\tau;\ss)}\right). \end{equation*} \end{lemma} \begin{theorem}\label{thm-existence} For every pair $R_1>0$ , $R_2 >0 ,$ there is $\tau>0$ and $\delta>0$ such that the IVP (\ref{eq-IVP}) admits a unique strict solution ${\bm{x}}\in{\mathbb{W}}(0,\tau)$, $\norm{{\bm{x}}}{{\mathbb{W}}(0,\tau)}\le \delta$ for all $({\bm{u}},{\bm{r}},{\bm{x}}_0)\in \Bu\times K_{ad}\times B_{{\mathbb{V}}}(R_2)$. \end{theorem} \begin{proof} The proof of this theorem follows the same line as that of \cite[Theorem 2.1]{clement1993abstract}\label{thm-strict} with some modifications. Let ${\bm{w}}$ solve the linear equation \begin{equation} \begin{cases} \dot{{\bm{w}}}(t)={\mathcal{A}} {\bm{w}}(t)+{\mathcal{F}}({\bm{x}}_0)+{\mathcal{B}}({\bm{r}}){\bm{u}}(t), \quad t\in(0,\tau],\\ {\bm{w}}(0)={\bm{x}}_0. \end{cases} \end{equation} Define for an arbitrary number $\rho>0$ the set \begin{equation} \Sigma_{\rho,\tau}=\left\{{\bm{v}}\in {\mathbb{W}}(0,\tau): {\bm{v}}(0) = {\bm{x}}_0, \; \norm{{\bm{v}}-{\bm{w}}}{{\mathbb{W}}(0,\tau)}\le \rho \right\}. \label{eq-sigma-rho} \end{equation} Because ${\bm{w}} (\cdot ) \in {\mathbb{W}} (0, \tau ) , $ ${\bm{w}} (\cdot ) \in C ( 0, \tau ; {\mathbb{V}} ) . $ Define $\phi(\tau;R_1,R_2)=\norm{{\bm{w}} -{\bm{x}}_0}{C(0,\tau;{\mathbb{V}})}$ where here ${\bm{x}}_0$ indicates the constant function in $C(0,\tau ;{\mathbb{V}}) $ that equals ${\bm{x}}_0 .$ Note that \begin{equation} \lim_{\tau \to 0 } \phi(\tau;R_1,R_2) = 0 . \end{equation} According to \Cref{lem-clement}, there is a constant $M$ independent of $\tau$ such that \begin{equation} \norm{{\bm{v}}-{\bm{x}}_0}{C(0,\tau;{\mathbb{V}})}\le M\rho+\phi(\tau;R_1,R_2),\quad \forall {\bm{v}}\in \Sigma_{\rho,\tau}. \end{equation} Consider the mapping $\gamma:{\mathbb{W}}(0,\tau) \to {\mathbb{W}}(0,\tau)$, ${\bm{x}} (\cdot ) \mapsto {\bm{v}} (\cdot ) $ defined by \begin{equation} \begin{cases} \dot{{\bm{v}}}(t)={\mathcal{A}} {\bm{v}}(t)+{\mathcal{F}}({\bm{x}}(t))+{\mathcal{B}}({\bm{r}}){\bm{u}}(t), \; t\in(0,\tau],\\ {\bm{v}}(0)={\bm{x}}_0. \end{cases} \end{equation} It will now be shown that for some numbers $\rho$ and $\tau$ the mapping $\gamma$ defines a contraction on $\Sigma_{\rho,\tau}$ and hence has a unique fixed point. Consider the linear equation \begin{equation*} \begin{cases} \dot{{\bm{v}}}(t)-\dot{{\bm{w}}}(t)={\mathcal{A}} ({\bm{v}}(t) -{\bm{w}}(t) ) +{\mathcal{F}}({\bm{x}}(t)), \; t\in (0,\tau],\\ ({\bm{v}}-{\bm{w}})(0)=0, \end{cases} \end{equation*} Use \Cref{lem-clement} together with Lipschitz continuity of ${\mathcal{F}}$, let $L_{\mathcal{F}}$ be the Lipschitz constant of ${\mathcal{F}}$ over the ball $B({\bm{x}}_0,M\rho+\phi(\tau;R_1,R_2))$. It follows that \begin{flalign} \norm{{\bm{v}}-{\bm{w}}}{{\mathbb{W}}(0,\tau)}&\le M\norm{{\mathcal{F}}({\bm{x}}(t))-{\mathcal{F}}({\bm{x}}_0)}{p}\notag \\ &\le M L_{{\mathcal{F}}} \tau^{\frac{1}{p}} \norm{{\bm{x}}-{\bm{x}}_0}{C(0,\tau;{\mathbb{V}})}\notag\\ &\le M^2 L_{{\mathcal{F}}} \tau^{\frac{1}{p}} (M\rho+\phi(\tau;R_1,R_2)) \label{ineq1}. \end{flalign} Furthermore, for any ${\bm{x}}_1 , {\bm{x}}_2 \in \Sigma_{\rho,\tau}$, define ${\bm{v}}_1=\gamma({\bm{x}}_1)$ and ${\bm{v}}_2=\gamma({\bm{x}}_2)$, then \Cref{lem-clement} yields \begin{flalign} \norm{{\bm{v}}_2-{\bm{v}}_1}{{\mathbb{W}}(0,\tau)}&\le M \norm{{\mathcal{F}}({\bm{x}}_2)-{\mathcal{F}}({\bm{x}}_1)}{p}\notag\\ &\le M L_{{\mathcal{F}}} \tau^{\frac{1}{p}} \norm{{\bm{x}}_2-{\bm{x}}_1}{C(0,\tau;{\mathbb{V}})}\notag \\ &\le M^2 L_{{\mathcal{F}}} \tau^{\frac{1}{p}} \norm{{\bm{x}}_2-{\bm{x}}_1}{{\mathbb{W}}(0,\tau)}\label{ineq2}. \end{flalign} Choose $\rho$ and $\tau$ so that \begin{gather*} M^2 L_{{\mathcal{F}}} \tau^{\frac{1}{p}}< 1,\\ M^2 L_{{\mathcal{F}}} \tau^{\frac{1}{p}} (M\rho+\phi(\tau;R_1,R_2))\le \rho. \end{gather*} The Contraction Mapping Theorem ensures that the mapping $\gamma$ has a unique fixed point in $\Sigma_{\rho,\tau}$. This fixed point is the unique solution ${\bm{x}}$ to (\ref{eq-IVP}). Also, from the definition (\ref{eq-sigma-rho}), every ${\bm{x}}$ in $\Sigma_{\rho,\tau}$ satisfies \begin{equation} \norm{{\bm{x}}}{{\mathbb{W}}(0,\tau)}\le \norm{{\bm{w}}}{{\mathbb{W}}(0,\tau)}+\rho. \end{equation} Let $L_{\mathcal{F}}$ be the Lipschitz constant of ${\mathcal{F}}$ over the ball $B(0, \| {\bm{x}}_0 \| )$. Proposition 2.2 in \cite{clement1993abstract} yields \begin{flalign*} &\norm{{\bm{w}}}{{\mathbb{W}}(0,\tau)}\le M(\norm{{\bm{x}}_0}{{\mathbb{V}}}+\norm{{\mathcal{F}}({\bm{x}}_0)+{\mathcal{B}}({\bm{r}}){\bm{u}}(t)}{p})\notag \\ &\quad \le M(\norm{{\bm{x}}_0}{{\mathbb{V}}}+\tau^{\frac{1}{p}}L_{{\mathcal{F}}}\norm{{\bm{x}}_0}{{\mathbb{V}}}+\norm{{\mathcal{B}}({\bm{r}})}{\mc L (\ss,\cs)}\norm{{\bm{u}}(t)}{p})\notag\\ &\quad \le \underbrace{M(R_2+\tau^{\frac{1}{p}}L_{{\mathcal{F}}}R_2+R_1\max_{{\bm{r}}\in K_{ad}}\norm{{\mathcal{B}}({\bm{r}})}{\mc L (\ss,\cs)}) }_{\delta} . \notag \end{flalign*} Defining $$\delta = M(R_2+\tau^{\frac{1}{p}}L_{{\mathcal{F}}}R_2+R_1\max_{{\bm{r}}\in K_{ad}}\norm{{\mathcal{B}}({\bm{r}})}{\mc L (\ss,\cs)}) ,$$ yields the required upper-bound on $\norm{{\bm{x}}}{{\mathbb{W}}(0,\tau)}$. \end{proof} \begin{definition} Let ${\bm{x}}(t)$ be the strict solution to (\ref{eq-IVP}). The mapping $\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0):\Bu\times K_{ad} \times B_{{\mathbb{V}}}(R_2) \to{\mathbb{W}}(0,\tau)$, $({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)\mapsto {\bm{x}}(t)$, is called the solution map. \end{definition} An embedding $D({\mathcal{A}})\hookrightarrow\ss$ where $D({\mathcal{A}})$ is compact in $\ss$ ensures that the space $W^{1,p}(0,\tau;\ss)\cap L^p(0,\tau,D({\mathcal{A}}))$ is compactly embedded in $c^{s}(0,\tau;{{\mathbb{V}}})$, $0\le s<1$ \cite[Theorem 5.2]{amann2000compact}. Since $c^{s}(0,\tau;{{\mathbb{V}}})\hookrightarrow C(0,\tau;{\mathbb{V}})$, it follows that the space $W^{1,p}(0,\tau;\ss)\cap L^p(0,\tau,D({\mathcal{A}}))$ is compactly embedded in $C(0,\tau;{{\mathbb{V}}})$. \begin{theorem}\label{thm-weak} If the embedding $D({\mathcal{A}})\hookrightarrow\ss$ is compact then the solution map is weakly continuous with respect to $({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)\in L^p(0,\tau;\cs)\times \as \times {\mathbb{V}}$. \end{theorem} \begin{proof} The weak continuity of the solution map with respect to ${\bm{u}}(t)$ is shown in \cite[Lemma 2.12]{meyer2017optimal}. Weak continuity with respect to $({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)$ follows from a similar proof. Choose any weakly convergent sequences $\{{\bm{u}}_n(t)\}\subset L^p(0,\tau;\cs)$, $\{ {\bm{x}}_0^n\} \subset {\mathbb{V}} $, and $\{{\bm{r}}_n \}\subset \as .$ Since sets $\Bu$ and $B_{{\mathbb{V}}}(R_2)$ are bounded, closed, convex subsets of Banach spaces $L^p(0,\tau;\cs)$ and ${\mathbb{V}}$, respectively; these sets are weakly closed \cite[Theorem 2.11]{troltzsch2010optimal}. This implies that there are ${\bm{u}}^o\in \Bu$ and ${\bm{x}}_0\in B_{{\mathbb{V}}}(R_2)$ such that \begin{flalign}\label{eq-3} {\bm{u}}_n & \rightharpoonup {\bm{u}}^o \text{ in } \Bu,\\ {\bm{x}}_0^n &\rightharpoonup{\bm{x}}_0 \text{ in } B_{{\mathbb{V}}}(R_2). \end{flalign} Since the set $K_{ad}$ is a compact subset of $\as$ \begin{equation} {\bm{r}}_n\to {\bm{r}}^o \text{ in } K_{ad}. \end{equation} It will be shown that ${\mathcal{B}}_{{\bm{r}}_n}{\bm{u}}_n(t)$ converges weakly to ${\mathcal{B}}_{{\bm{r}}^o}{\bm{u}}^o(t)$ in $L^p(0,\tau;\ss)$. For every ${\bm{z}} \in L^q(0,\tau;\ss)$, $1/q=1-1/p$, \begin{flalign}\notag I:=&\inn{{\bm{z}}}{{\mathcal{B}}_{{\bm{r}}_n}{\bm{u}}_n-{\mathcal{B}}_{{\bm{r}}^o}{\bm{u}}^o}_{L^q(0,\tau;\ss^{^*}),L^p(0,\tau;\ss)}\\ =&\inn{{\bm{z}}}{{\mathcal{B}}_{{\bm{r}}_n}{\bm{u}}_n-{\mathcal{B}}_{{\bm{r}}^o}{\bm{u}}_n}_{L^q(0,\tau;\ss^{^*}),L^p(0,\tau;\ss)}\\ &+\inn{{\bm{z}}}{{\mathcal{B}}_{{\bm{r}}^o}{\bm{u}}_n-{\mathcal{B}}_{{\bm{r}}^o}{\bm{u}}^o}_{L^q(0,\tau;\ss^{^*}),L^p(0,\tau;\ss)}.\notag \end{flalign} Taking the adjoint and norm yield \begin{flalign*} I \le \norm{{\mathcal{B}}_{{\bm{r}}_n}-{\mathcal{B}}_{{\bm{r}}^o}}{\mc L(\cs,\ss)}\int_0^\tau \norm{{\bm{u}}_n(t)}{\cs}\normm{{\bm{z}}(t)}dt\\ +{\Big |}\int_0^\tau\inn{{\mathcal{B}}^*_{{\bm{r}}^o}{\bm{z}}(t)}{{\bm{u}}_n(t)-{\bm{u}}^o(t)}_{\cs^{^*},\cs}dt{\Big |}. \end{flalign*} Use H\"older inequality and let ${\bm{v}}(t)={\mathcal{B}}^*_{{\bm{r}}^o}{\bm{z}}(t)$, it follows that \begin{flalign*} I \le \norm{{\mathcal{B}}_{{\bm{r}}_n}-{\mathcal{B}}_{{\bm{r}}^o}}{\mc L(\cs,\ss)}\norm{{\bm{u}}_n}{L^p(0,\tau;\cs)}\norm{{\bm{z}}}{L^q(0,\tau;\ss)}\\ +|\inn{{\bm{u}}_n-{\bm{u}}^o}{{\bm{v}}}_{L^p(0,\tau;\cs),L^q(0,\tau;\cs^{^*})}|. \end{flalign*} The convergence of the first term follows from \eqref{B-cont}. The second term converges to zero because ${\bm{u}}_n\rightharpoonup {\bm{u}}^o$ in $L^p(0,\tau;\cs)$. Combining these yields \begin{equation}\label{weak-B} {\mathcal{B}}_{{\bm{r}}_n}{\bm{u}}_n\rightharpoonup {\mathcal{B}}_{{\bm{r}}^o}{\bm{u}}^o \text{ in }L^p(0,\tau;\ss). \end{equation} Using \Cref{thm-existence}, the corresponding solution ${\bm{x}}_n(t)$ is a bounded sequence in the reflexive Banach space $L^p(0,\tau;D({\mathcal{A}}))\cap W^{1,p}(0,\tau;\ss)$. Thus, there is a subsequence of ${\bm{x}}_n(t)$ such that \begin{equation}\label{eq-1} {\bm{x}}_{n_k}\rightharpoonup{\bm{x}} \text{ in }{\mathbb{W}}(0,\tau). \end{equation} This in turn implies that the sequence ${\bm{x}}_n(t)$ strongly converges to ${\bm{x}}(t)$ in $C(0,\tau;{{\mathbb{V}}})$. This together with Lipschitz continuity of ${\mathcal{F}}(\cdot)$ yields \begin{equation}\label{eq-d1} {\mathcal{F}}({\bm{x}}_{n_k}(t)) \to {\mathcal{F}}({\bm{x}}(t)) \quad \text{in } L^p(0,\tau;\ss). \end{equation} This strong convergence also yields weak convergence in the same space, that is \begin{equation}\label{eq-d2} {\mathcal{F}}({\bm{x}}_{n_k}(t)) \rightharpoonup {\mathcal{F}}({\bm{x}}(t)) \quad \text{in } L^p(0,\tau;\ss). \end{equation} Now apply (\ref{eq-3}), \eqref{weak-B}, (\ref{eq-1}), and (\ref{eq-d2}) to the IVP (\ref{eq-IVP}); take the limit; notice that a solution to the IVP is unique; it follows that ${\bm{x}}=\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0)$. Deleting elements $\{{\bm{x}}_{n_k}(t)\}$ from $\{{\bm{x}}_n(t)\}$ and repeating the previous processing, knowing that a weak limit is unique, it follow that ${\bm{x}}_n(t)\rightharpoonup {\bm{x}}(t)$ in ${\mathbb{W}}(0,\tau)$. \end{proof} \section{Optimal Actuator Design} Consider a cost function $J({\bm{x}},{\bm{u}},{\bm{r}}):{\mathbb{W}}(0,\tau)\times L^p(0,\tau;\cs )\times \as \to \mathbb{R}$ that is bounded below and weakly lower-semicontinuous with respect to ${\bm{x}}$, ${\bm{u}}$, and ${\bm{r}}$. For a fixed initial condition ${\bm{x}}_0\in B_{{\mathbb{V}}}(R_2)$, consider the following optimization problem over the admissible input set $U_{ad}$ and actuator design set $K_{ad}$ \begin{equation} \left\{ \begin{array}{ll} \min&J({\bm{x}},{\bm{u}},{\bm{r}})\\ \text{s.t.}& {\bm{x}}=\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0),\\ &({\bm{u}},{\bm{r}}) \in U_{ad}\times K_{ad}. \end{array} \right. \tag{P} \label{eq-optimal problem} \end{equation The set $U_{ad}$ will be assumed a convex and closed set contained in the interior of $\Bu$. \begin{theorem} \label{thm-existence optimizer} For every ${\bm{x}}_0\in B_{{\mathbb{V}}}(R_2)$, there exists a control input ${\bm{u}}^o\in U_{ad}$ together with an actuator design ${\bm{r}}^o\in K_{ad}$ that solve the optimization problem \eqref{eq-optimal problem}. \end{theorem} \begin{proof} The proof of this theorem follows from standard analysis; see for example, \cite[Theorem 1.45]{hinze2008optimization} and \cite[Theorem 4.1]{edalatzadehSICON} for a similar argument. Define \begin{equation} j({\bm{x}}_0):=\inf_{({\bm{u}},{\bm{r}}) \in U_{ad}\times K_{ad}}J(\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0),{\bm{u}},{\bm{r}}). \end{equation} and let $({\bm{u}}_n,{\bm{r}}_n)$ be the minimizing sequence: \begin{equation} \lim_{n\to \infty}J(\mc S({\bm{u}}_n,{\bm{r}}_n,{\bm{x}}_0),{\bm{u}}_n,{\bm{r}}_n)=j({\bm{x}}_0). \end{equation} The set $U_{ad}$ is closed and convex in the reflexive Banach space $L^p(0,\tau;\cs)$, so it is weakly closed. This implies that there is a subsequence of ${\bm{u}}_n$, denote it by the same symbol, that converges weakly to some elements ${\bm{u}}^o$ in $U_{ad}$. Because of compactness of $K_{ad}$, there is also a subsequence of ${\bm{r}}_n$, denote it by the same symbol, that strongly converges to ${\bm{r}}^o$. \Cref{thm-existence} and \Cref{thm-weak} state that the solution map is bounded and weakly continuous in each variable. Thus, the corresponding state ${\bm{x}}_n=\mc S({\bm{u}}_n,{\bm{r}}_n;{\bm{x}}_0)$ also weakly converges to ${\bm{x}}^o= \mc S({\bm{u}}^o,{\bm{r}}^o;{\bm{x}}_0)$ in ${\mathbb{W}}(0,\tau)$. The cost function is weakly lower semi-continuous with respect to each ${\bm{x}}$, ${\bm{u}}$, and ${\bm{r}}$, this ensures that $({\bm{x}}^o,{\bm{u}}^o,{\bm{r}}^o)$ minimizes the cost function. Therefore, $({\bm{u}}^o,{\bm{r}}^o)$ is a solution to the optimization problem \eqref{eq-optimal problem}. \end{proof} \begin{definition}\cite[Definition 1.29]{hinze2008optimization} The operator ${\mathcal{G}}:\ss\to \Yb$ is said to be G\^ateaux differentiable at ${\bm{x}}\in \ss$ in the direction ${\bm{p}}\in \ss$, if there is a linear bounded operator ${\mathcal{G}}'_{\bm{x}}$ such that for all real $\epsilon$ \begin{equation} \lim_{\epsilon\to 0}{\|{\mathcal{G}}({\bm{x}}+\epsilon{\bm{p}})-{\mathcal{G}}({\bm{x}})-\epsilon{\mathcal{G}}'_{\bm{x}}{\bm{p}}\|_{\Yb}}=0. \end{equation} \end{definition} The optimality conditions are derived next after assuming that the problem has certain properties. Consider the assumptions: \begin{enumerate} \item[A1.] The spaces $\ss$ and $\cs$ are Hilbert spaces and $p=2$. The space $\as $ is a Banach space. \item[A2.] Let $a:{\mathbb{V}}\times {\mathbb{V}}\to \mathbb{C}$ be a sesquilinear form (see \cite[Chapter 4]{lang2012real}), where ${\mathbb{V}}\hookrightarrow \ss$, and let there be positive numbers $\alpha$ and $\beta$ such that \begin{flalign*} |a({\bm{x}}_1,{\bm{x}}_2)|&\le \alpha \norm{{\bm{x}}_1}{{\mathbb{V}}}\norm{{\bm{x}}_2}{{\mathbb{V}}}, &\forall&{\bm{x}}_1,{\bm{x}}_2\in {\mathbb{V}},\\ \text{Re} \; a({\bm{x}},{\bm{x}})&\ge \beta \norm{{\bm{x}}}{{\mathbb{V}}}^2, &\forall&{\bm{x}}\in {\mathbb{V}}. \end{flalign*} The operator ${\mathcal{A}}$ has an extension to $\bar{{\mathcal{A}}}\in\mc L({\mathbb{V}},{\mathbb{V}}^{^*})$ described by \begin{equation} \inn{\bar{{\mathcal{A}}} {\bm{v}}}{{\bm{w}}}_{{\mathbb{V}}^{^*},{\mathbb{V}}}=a({\bm{v}},{\bm{w}}), \quad \forall {\bm{v}}, {\bm{w}} \in {\mathbb{V}}, \end{equation} where ${\mathbb{V}}^{^*}$ denotes the dual of ${\mathbb{V}}$ with respect to pivot space $\ss$. \item[A3.] \label{as-diff J}The cost function $J({\bm{x}},{\bm{u}},{\bm{r}})$ is continuously Fr\'echet differentiable with respect to each variable. \item[A4.] \label{as-diff F} The nonlinear operator ${\mathcal{F}}(\cdot)$ is G\^ateaux differentiable. Indicate the G\^ateaux derivative of ${\mathcal{F}}(\cdot)$ at ${\bm{x}}$ in the direction ${\bm{p}}$ by ${{\mathcal{F}}}_{{\bm{x}}}^\prime{\bm{p}}$. Furthermore, the mapping ${\bm{x}}\mapsto {{\mathcal{F}}}_{{\bm{x}}}^\prime$ is bounded; that is, bounded sets in ${{\mathbb{V}}}$ are mapped to bounded sets in $\mc L({{\mathbb{V}}},\ss)$. \item[A5.] \label{as:diff B} The control operator $\mc{B}({\bm{r}})$ is G\^ateaux differentiable with respect to ${\bm{r}}$ from $K_{ad}$ to $\mc{L}(\cs,\ss)$. Indicate the G\^ateaux derivative of $\mc{B}({\bm{r}})$ at ${\bm{r}}^o$ in the direction ${\bm{r}}$ by $\mc{B}'_{{\bm{r}}^o}{\bm{r}}$. Furthermore, the mapping ${\bm{r}}^o\mapsto {\mathcal{B}}'_{{\bm{r}}^o}$ is bounded; that is, bounded sets in $\as$ are mapped to bounded sets in $\mc L(\as,\mc L(\cs,\ss))$. \end{enumerate} Using these assumptions, the G\^ateaux derivative of the solution map with respect to a trajectory ${\bm{x}}(t)=\mc S({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)$ is calculated. The resulting map is a time-varying linear IVP. Let $\bm g\in L^p(0,\tau;\ss)$, consider the time-varying system \begin{equation}\label{eq-time var} \begin{cases} \dot{{\bm{h}}}(t)=({\mathcal{A}}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime) {\bm{h}}(t)+\bm g(t),\\ {\bm{h}}(0)=0. \end{cases} \end{equation} \begin{lemma}\cite[Corollary 5.2]{dier2015}\label{lem-estimate} Let assumptions A1 and A2 hold. For any $\tau>0$, let $\mc P(\cdot):[0,\tau]\to \mc L ({\mathbb{V}},\ss)$ be such that $\mc P(\cdot){\bm{x}}$ is weakly measurable for all ${\bm{x}}\in {\mathbb{V}}$, and there exists an integrable function $h:[0,\tau]\to [0,\infty)$ such that $\norm{\mc P(t)}{\mc L({\mathbb{V}},\ss)}\le h(t)$ for all $t\in [0,\tau]$. Then for every ${\bm{x}}_0\in {\mathbb{V}}$ and $\bm g\in L^2(0,\tau;\ss)$, there exists a unique ${\bm{x}}$ in ${\mathbb{W}}(0,\tau)$ such that \begin{equation} \begin{cases} \dot{{\bm{x}}}(t)=({\mathcal{A}} +\mc P(t)) {\bm{x}}(t)+\bm g(t),\\ {\bm{x}}(0)={\bm{x}}_0. \end{cases} \end{equation} Moreover, there exists a constant $c>0$ independent of ${\bm{x}}_0$ and $\bm g(t)$ such that \begin{equation} \norm{{\bm{x}}}{{\mathbb{W}}(0,\tau)}^2\le c \left(\norm{\bm g}{L^2(0,\tau;\ss)}^2+\norm{{\bm{x}}_0}{{\mathbb{V}}}^2\right). \end{equation} \end{lemma} Since ${\mathbb{W}}(0,\tau)$ is embedded in $C(0,\tau;{\mathbb{V}})$, the state ${\bm{x}}(t)$ is bounded in ${\mathbb{V}}$ for all $t\in [0,\tau]$. This together with G\^ateaux differentiablity of ${\mathcal{F}}(\cdot)$ ensures that there is a positive number $M_{{\mathcal{F}}}$ such that \begin{equation} \sup_{t\in [0,\tau]}\norm{{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime}{\mc L({\mathbb{V}},\ss)}\le M_{{\mathcal{F}}}. \end{equation} Thus, replacing the operator $\mc P(t)$ with ${{\mathcal{F}}}_{{\bm{x}}(t)}^\prime$ and noting that \begin{equation}\label{d1} \norm{\mc P(t)}{\mc L({\mathbb{V}},\ss)}\le M_{{\mathcal{F}}}, \end{equation} shows that the conditions of \Cref{lem-estimate} hold. Thus, there is a positive number $c$ independent of $\bm g$ such that \begin{equation}\label{eq-estimate} \|{\bm{h}}\|_{{\mathbb{W}}(0,\tau)}\le c \norm{\bm g}{L^2(0,\tau;\ss)}. \end{equation} \begin{proposition}\label{prop-diff} Under assumptions A1-A5, the solution map $\mc S({\bm{u}}(t),{\bm{r}};{\bm{x}}_0)$ is G\^ateaux differentiable with respect to each ${\bm{u}}(t)$ and ${\bm{r}}$ in $U_{ad}\times K_{ad}$. Let ${\bm{x}}(t)=\mc S({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)$. \begin{enumerate} \item[a.] The G\^ateaux derivative of $\mc S({\bm{u}}(t),{\bm{r}};{\bm{x}}_0)$ at ${\bm{r}}$ in the direction $\tilde{{\bm{r}}}$ is the mapping $\mc S'_{\bm{r}} :\as\to L^2(0,\tau;D({\mathcal{A}}))\cap W^{1,2}(0,\tau;\ss)$, $\tilde{{\bm{r}}}\mapsto {\bm{z}}(t)$, where ${\bm{z}}(t)$ is the strict solution to \begin{equation} \begin{cases} \dot{{\bm{z}}}(t)=({\mathcal{A}}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime) {\bm{z}}(t)+({\mathcal{B}}'_{\bm{r}}\tilde{{\bm{r}}}){\bm{u}}(t),\\ {\bm{z}}(0)=0. \end{cases} \end{equation} \item[b.] The G\^ateaux derivative of $\mc S({\bm{u}}(t),{\bm{r}};{\bm{x}}_0)$ at ${\bm{u}}(t)$ in the direction $\tilde{{\bm{u}}}(t)$ is the mapping $\mc S'_{\bm{u}} :L^2(0,\tau;\cs )\to L^2(0,\tau;D({\mathcal{A}}))\cap W^{1,2}(0,\tau;\ss)$, $\tilde{{\bm{u}}}(t)\mapsto {\bm{h}}(t)$, where ${\bm{h}}(t)$ is the strict solution to \begin{equation} \begin{cases} \dot{{\bm{h}}}(t)=({\mathcal{A}}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime) {\bm{h}}(t)+{\mathcal{B}}({\bm{r}})\tilde{{\bm{u}}}(t),\\ {\bm{h}}(0)=0. \end{cases} \end{equation} \end{enumerate} \end{proposition} \begin{proof} a) Let $\epsilon$ be sufficiently small such that ${\bm{r}}+\epsilon\tilde{{\bm{r}}}\in K_{ad}$. Define ${\bm{x}}_\epsilon(t)=\mc S({\bm{u}}(t),{\bm{r}}+\epsilon\tilde{{\bm{r}}},{\bm{x}}_0)$, this state solves \begin{equation} \begin{cases} \dot{{\bm{x}}}_\epsilon(t)=\mc{A}{\bm{x}}_\epsilon(t)+\mc{F}({\bm{x}}_\epsilon(t))+\mc{B}({\bm{r}}+\epsilon\tilde{{\bm{r}}}){\bm{u}}(t),\quad t>0,\\ {\bm{x}}_\epsilon(0)={\bm{x}}_0. \end{cases} \label{eq-xe} \end{equation} Similarly, ${\bm{x}} (t)= \mc S({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)$ solves \eqref{eq-xe} with $\epsilon =0 .$ Define $\bm{{\bm{e}}}_{{\mathcal{F}}}(t)$ and $\bm{{\bm{e}}}_{{\mathcal{B}}}$ as \begin{subequations} \begin{flalign} \bm{{\bm{e}}}_{{\mathcal{F}}}(t)&\coloneqq \frac{1}{\epsilon}\left({\mathcal{F}}({\bm{x}}(t))-{\mathcal{F}}({\bm{x}}_\epsilon(t))-{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime({\bm{x}}(t)-{\bm{x}}_\epsilon(t))\right),\label{eF}\\ \bm{{\bm{e}}}_{{\mathcal{B}}} &\coloneqq \frac{1}{\epsilon}\left({\mathcal{B}}({\bm{r}}+\epsilon\tilde{{\bm{r}}})-{\mathcal{B}}({\bm{r}})\right)-{\mathcal{B}}'_{{\bm{r}}}\tilde{{\bm{r}}}. \end{flalign} \end{subequations} The state ${\bm{e}}(t)=({\bm{x}}(t)-{\bm{x}}_\epsilon(t))/\epsilon-{\bm{z}}(t)$ satisfies \begin{equation}\label{edot} \begin{cases} \dot{{\bm{e}}}(t)=(\mc{A}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime){\bm{e}}(t)+\bm{{\bm{e}}}_{{\mathcal{F}}}(t)+\bm{{\bm{e}}}_{{\mathcal{B}}}{\bm{u}}(t),\; t>0,\\ {\bm{e}}(0)=0. \end{cases} \end{equation} Assumption A4 and A5 ensure that as $\epsilon\to 0$ \begin{subequations} \begin{gather} \normm{\bm{{\bm{e}}}_{{\mathcal{F}}}(t)}\to 0, \quad \forall t\in [0,\tau],\label{e(t)2}\\ \norm{\bm{{\bm{e}}}_{{\mathcal{B}}}}{\mc{L}(\cs,\ss)} \to 0.\label{eB} \end{gather} \end{subequations}% It will be shown that $\lim_{\epsilon\to 0} \norm{{\bm{e}}}{{\mathbb{W}}(0,\tau)}=0.$ First, consider ${\bm{x}}(t)-{\bm{x}}_\epsilon(t)$, which satisfies \begin{flalign}\label{eq-7} \begin{cases} \dot{{\bm{x}}}(t)-\dot{{\bm{x}}}_\epsilon(t)=&{\mathcal{A}}({\bm{x}}(t)-{\bm{x}}_\epsilon(t))+{\mathcal{F}}({\bm{x}}(t))-{\mathcal{F}}({\bm{x}}_\epsilon(t))\notag \\ &+\left({\mathcal{B}}({\bm{r}})-{\mathcal{B}}({\bm{r}}+\epsilon\tilde{{\bm{r}}})\right){\bm{u}}(t),\\ {\bm{x}}(0)-{\bm{x}}_\epsilon(0)=&0.\notag \end{cases} \end{flalign} Lemma \ref{lem-clement} implies that there is a number $c_\tau$ depending only on $\tau$ such that for all $t\in [0,\tau]$ \begin{flalign} &\norm{{\bm{x}}(t)-{\bm{x}}_\epsilon(t)}{{\mathbb{V}}} \\ &\quad \le c_\tau \left(\norm{\dot{{\bm{x}}}-\dot{{\bm{x}}}_\epsilon}{L^2(0,t;\ss)}+\norm{{\mathcal{A}}({\bm{x}}-{\bm{x}}_\epsilon)}{L^2(0,t;\ss)}\right).\label{eq11} \end{flalign} Also, use \cite[Proposition 2.2]{clement1993abstract}, there is a number $d_\tau$ depending only on $\tau$ such that for all $t\in [0,\tau]$ \begin{flalign}\label{eq12} \norm{\dot{{\bm{x}}}-\dot{{\bm{x}}}_\epsilon}{L^2(0,t;\ss)}&+\norm{{\mathcal{A}}({\bm{x}}-{\bm{x}}_\epsilon)}{L^2(0,t;\ss)} \\ &\quad \le d_\tau \norm{\dot{{\bm{x}}}-\dot{{\bm{x}}}_\epsilon-{\mathcal{A}}({\bm{x}}-{\bm{x}}_\epsilon)}{L^2(0,t;\ss)}\notag. \end{flalign} Combine (\ref{eq11}) and (\ref{eq12}) to obtain \begin{flalign} \norm{{\bm{x}}(t)-{\bm{x}}_\epsilon(t)}{{\mathbb{V}}}&\le c_\tau d_\tau \norm{{\mathcal{F}}({\bm{x}})-{\mathcal{F}}({\bm{x}}_\epsilon)}{L^2(0,t;\ss)} \\ &\quad +c_\tau d_\tau \norm{\left({\mathcal{B}}({\bm{r}})-{\mathcal{B}}({\bm{r}}+\epsilon\tilde{{\bm{r}}})\right){\bm{u}}}{L^2(0,t;\ss)}.\notag \end{flalign} \Cref{thm-existence} implies that the states ${\bm{x}}(t)$ and ${\bm{x}}_\epsilon(t)$ belong to some bounded set in ${\mathbb{W}}(0,\tau)$ and so in $C(0,\tau,{\mathbb{V}})$. Let $D\subset {\mathbb{V}}$ be a bounded set that contains the trajectories ${\bm{x}}(t)$ and ${\bm{x}}_\epsilon(t)$. Let $L_{\mathcal{F}}$ be the Lipschitz constant of ${\mathcal{F}}(\cdot)$ on $D$. Since the set $K_{ad}$ is compact and ${\mathcal{B}}({\bm{r}})$ satisfies assumption A5, the number $L_{{\mathcal{B}}}$ defined as \begin{equation} L_{{\mathcal{B}}}=\sup_{{\bm{r}}\in K_{ad}}\norm{{\mathcal{B}}'_{{\bm{r}}}}{\mc L(\as,\mc L(\cs,\ss))}. \end{equation} is finite. This together with \cite[Theorem 12.1.1 and Corollary 3]{wouk1979course} yields \begin{equation} \norm{{\mathcal{B}}({\bm{r}})-{\mathcal{B}}({\bm{r}}_\epsilon)}{\mc L(\cs,\ss)}\le L_{{\mathcal{B}}}\norm{{\bm{r}}-{\bm{r}}_\epsilon}{\as}\le L_{{\mathcal{B}}}\epsilon. \end{equation} Use these to obtain the inequality \begin{flalign}\label{eq-8} \norm{{\bm{x}}(t)-{\bm{x}}_\epsilon(t)}{{\mathbb{V}}}^2&\le 2c^2_\tau d^2_\tau L^2_{\mathcal{F}} \int_0^t \norm{{\bm{x}}(s)-{\bm{x}}_\epsilon(s)}{{\mathbb{V}}}^2ds \\ &\quad +2c^2_\tau d^2_\tau L^2_{{\mathcal{B}}}\epsilon^2 \norm{{\bm{u}}}{L^2(0,\tau;\cs)}^2\notag \end{flalign} Applying Gronwall's lemma yields \begin{equation}\label{eq-9} \norm{{\bm{x}}(t)-{\bm{x}}_\epsilon(t)}{{\mathbb{V}}}\le \sqrt{2}e^{c^2_\tau d^2_\tau L^2_{\mathcal{F}}}c_\tau d_\tau L_{{\mathcal{B}}}\epsilon\norm{{\bm{u}}}{L^2(0,\tau;\cs)}. \end{equation} Define $$M_{{\mathcal{F}}}:=\sup_{t\in [0,\tau]} \|{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime\|_{\mc L({\mathbb{V}},\ss)}.$$ Assumption A4 ensures that $M_{{\mathcal{F}}}$ is finite. Take the norm of the right side of \eqref{eF} in $\ss$. It follows that \begin{equation}\label{eq13} \normm{{\bm{e}}_{{\mathcal{F}}}(t)} \le (L_{{\mathcal{F}}}+M_{{\mathcal{F}}}c_e)\sqrt{2}e^{c^2_\tau d^2_\tau L^2_{\mathcal{F}}}c_\tau d_\tau L_{{\mathcal{B}}}\norm{{\bm{u}}}{L^2(0,\tau;\cs)}. \end{equation} This and (\ref{e(t)2}) together with the Bounded Convergence Theorem ensure that \begin{equation}\label{eq-10} \lim_{\epsilon\to 0}\int_0^\tau \normm{{\bm{e}}_{{\mathcal{F}}}(t)}^2dt= 0. \end{equation} Statements (\ref{eq-10}) and (\ref{eB}), and \Cref{lem-estimate} can be applied to conclude \begin{equation} \lim_{\epsilon\to 0} \norm{{\bm{e}}}{{\mathbb{W}}(0,\tau)}=0. \end{equation} This shows that $\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0)$ is G\^ateaux differentiable at ${\bm{r}}$ in the direction $\tilde{{\bm{r}}}$ with derivative ${\bm{z}}(t)=\mc S'_{\bm{r}} \tilde{{\bm{r}}}$. b) This part is proven in \cite[Theorem 3.4]{meyer2017optimal} assuming that $\partial_t+{\mathcal{A}}$ is invertible. However, the result is still true without assuming the invertibility of $\partial_t+{\mathcal{A}}$. Let $\epsilon$ be sufficiently small such that ${\bm{u}}+\epsilon\tilde{{\bm{u}}}\in U_{ad}$. Define ${\bm{x}}_\epsilon(t)=\mc S({\bm{u}}(t)+\epsilon\tilde{{\bm{u}}}(t),{\bm{r}},{\bm{x}}_0)$, this state solves \begin{equation} \begin{cases} \dot{{\bm{x}}}_\epsilon(t)=\mc{A}{\bm{x}}_\epsilon(t)+\mc{F}({\bm{x}}_\epsilon(t))+\mc{B}({\bm{r}})({\bm{u}}(t)+\epsilon\tilde{{\bm{u}}}(t)),\quad t>0,\\ {\bm{x}}_\epsilon(0)={\bm{x}}_0. \end{cases} \label{eq-xe} \end{equation} Let ${\bm{e}}(t)=({\bm{x}}(t)-{\bm{x}}_\epsilon(t))/\epsilon-\bm{h}(t)$. Following the same steps as in part (a) yields \begin{equation} \lim_{\epsilon\to 0} \norm{{\bm{e}}}{{\mathbb{W}}(0,\tau)}=0. \end{equation} This means that $\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0)$ is G\^ateaux differentiable at ${\bm{u}}$ in the direction $\tilde{{\bm{u}}}$ with derivative ${\bm{h}}(t)=(\mc S'_{\bm{u}} \tilde{{\bm{u}}})(t)$. \end{proof} Assumption A1 implies that the dual of each of $\ss$ and $\cs$ will be identified with the space itself. For each ${\bm{u}}$, the operator $(\mc{B}'_{{\bm{r}}^o}{\bm{u}})^*:\ss\to \mathbb{K}^*$ is defined by \begin{equation}\label{B-adj} \inn{(\mc{B}'_{{\bm{r}}^o}{\bm{u}})^*{\bm{p}}}{{\bm{r}}}_{\as^*,\as}=\inn{{\bm{p}}}{(\mc{B}'_{{\bm{r}}^o}{\bm{r}}){\bm{u}}}, \; \forall ({\bm{u}},{\bm{p}},{\bm{r}})\in \cs\times\ss\times \mathbb{K}.\notag \end{equation} \begin{theorem}\label{thm-optimality} Suppose assumptions A1-A5 hold, and writing the derivatives $J'_{{\bm{x}}}$, $J'_{{\bm{u}}}$, and $J'_{{\bm{r}}}$ by elements $\bm j_{\bm{x}}\in {\mathbb{W}}(0,\tau)^*$, $\bm j_{\bm{u}}\in L^2(0,\tau;\cs)$ and $\bm j_{\bm{r}} \in \as^*$, respectively. For any initial condition ${\bm{x}}_0\in\ss$, let the pair $({\bm{u}}^o,{\bm{r}}^o)\in U_{ad}\times K_{ad}$ be a local minimizer of the optimization problem \eqref{eq-optimal problem} with the optimal trajectory ${\bm{x}}^o=\mc{S}({\bm{u}}^o;{\bm{r}}^o,{\bm{x}}_0)$ and let ${\bm{p}}^o(t)$ indicate the strict solution in ${\mathbb{W}}(0,\tau)^*$ of the final value problem \begin{equation}\label{adj} \dot{{\bm{p}}}^o(t)=-(\mc{A}^*+{{\mathcal{F}}_{{\bm{x}}^o(t)}^\prime}^*){\bm{p}}^o(t)-\bm j_{{\bm{x}}^o}(t), \quad {\bm{p}}^o(\tau)=0. \end{equation} Then $({\bm{u}}^o,{\bm{r}}^o)$ satisfy \begin{subequations} \begin{equation}\notag \begin{aligned} &\inn{\bm j_{{\bm{u}}^o}+\mc{B}^*({\bm{r}}^o){\bm{p}}^o}{{\bm{u}}-{\bm{u}}^o}_{L^2(0,\tau;\cs )}\ge 0,\\ &\inn{\bm j_{{\bm{r}}^o}+\int_0^{\tau} (\mc{B}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*{\bm{p}}^o(t)\, dt}{{\bm{r}}-{\bm{r}}^o}_{\as^*,\as}\ge 0. \end{aligned} \end{equation} \end{subequations} for all ${\bm{u}} \in U_{ad}$ and ${\bm{r}}\in K_{ad}$. \end{theorem} \begin{proof} Let $$\mc G({\bm{u}},{\bm{r}})=J(\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0),{\bm{u}},{\bm{r}}).$$ The G\^ateaux derivative of $\mc G({\bm{u}},{\bm{r}})$ with respect to ${\bm{u}}$ has been obtained in the proof of \cite[Proposition 4.13]{meyer2017optimal}. Using the chain rule to take the G\^ateaux derivative of $\mc G({\bm{u}},{\bm{r}})$ at ${\bm{u}}^o$ in the direction $\tilde{{\bm{u}}}$ yields \begin{equation} \mc G'_{{\bm{u}}^o}\tilde{{\bm{u}}}= J'_{{\bm{u}}^o}\tilde{{\bm{u}}} + J'_{{\bm{x}}^o}\mc S'_{{\bm{u}}^o}\tilde{{\bm{u}}} .\label{opt1} \end{equation} Identify the functionals $\mc G'_{{\bm{u}}^o}:L^2(0,\tau;\cs)\to \mathbb{R}$ and $J'_{{\bm{u}}^o}:L^2(0,\tau;\cs)\to \mathbb{R}$ with elements of $L^2(0,\tau,\cs)$. That is \begin{flalign} \mc G'_{{\bm{u}}^o}\tilde{{\bm{u}}}&=\inn{\bm g_{{\bm{u}}^o}}{\tilde{{\bm{u}}}}_{L^2(0,\tau;\cs)},\label{Ju}\\ J'_{{\bm{u}}^o}\tilde{{\bm{u}}}&=\inn{\bm j_{{\bm{u}}^o}}{\tilde{{\bm{u}}}}_{L^2(0,\tau;\cs)}.\label{ju} \end{flalign} Also, identifying the functional $J'_{{\bm{x}}^o}:L^2(0,\tau;\ss)\to \mathbb{R}$ with an element of ${\mathbb{W}}(0,\tau)^*=L^2(0,\tau;D({\mathcal{A}}^*))\cap W^{1,2}(0,\tau;\ss)$ yields \begin{equation} J'_{{\bm{x}}^o}\mc S'_{{\bm{u}}^o}\tilde{{\bm{u}}}=\inn{\bm j_{{\bm{x}}^o}}{\mc S'_{{\bm{u}}^o}\tilde{{\bm{u}}}}_{L^2(0,\tau;\ss)}.\label{jx} \end{equation} The adjoint operator $\mc S'^*_{{\bm{u}}^o}$ can be obtained as follows. Use \eqref{adj} in the following inner product and let $\bm{h}(t)=\mc S'_{{\bm{u}}^o}\tilde{{\bm{u}}}$ \begin{flalign*} &\inn{\bm j_{{\bm{x}}^o}}{\mc S'_{{\bm{u}}^o}\tilde{{\bm{u}}}}_{L^2(0,\tau;\ss)}\\ &\quad =\int_0^\tau \inn{-\dot{{\bm{p}}}^o(t)-(\mc{A}^*+{{\mathcal{F}}_{{\bm{x}}^o(t)}^\prime}^*){\bm{p}}^o(t)}{\bm{h}(t)}dt. \end{flalign*} Taking the adjoint and integration by parts yield \begin{flalign*} &\inn{\bm j_{{\bm{x}}^o}}{\mc S'_{{\bm{u}}^o}\tilde{{\bm{u}}}}_{L^2(0,\tau;\ss)}\\ &\quad=\int_0^\tau \inn{{\bm{p}}^o(t)}{\dot{\bm{h}}(t)-({\mathcal{A}}+{{\mathcal{F}}_{{\bm{x}}^o(t)}^\prime}) \bm{h}(t)}dt\\ &\quad=\int_0^\tau \inn{{\bm{p}}^o(t)}{{\mathcal{B}}({\bm{r}})\tilde{{\bm{u}}}(t)}dt\\ &\quad=\int_0^\tau \inn{{\mathcal{B}}^*({\bm{r}}){\bm{p}}^o(t)}{\tilde{{\bm{u}}}(t)}_{\cs}dt. \end{flalign*} This implies \begin{equation} \mc S'^*_{{\bm{u}}^o} \bm j_{{\bm{x}}^o}= {\mathcal{B}}^*({\bm{r}}){\bm{p}}^o(t).\label{adj3} \end{equation} Combine \eqref{Ju}, \eqref{ju}, \eqref{jx} and use \eqref{adj3}, equation \eqref{opt1} is written using the functionals as \begin{equation} \inn{\bm g_{{\bm{u}}}}{\tilde{{\bm{u}}}}_{L^2(0,\tau;\cs)}=\inn{\bm j_{{\bm{u}}^o}+\mc{B}^*({\bm{r}}^o){\bm{p}}^o}{\tilde{{\bm{u}}}}_{L^2(0,\tau;\cs)}. \end{equation} Applying \cite[Theorem 1.46]{hinze2008optimization} and letting $\tilde{{\bm{u}}}={\bm{u}}-{\bm{u}}^o$ for all ${\bm{u}} \in U_{ad}$ yields \begin{equation} \inn{\bm j_{{\bm{u}}^o}+\mc{B}^*({\bm{r}}^o){\bm{p}}^o}{{\bm{u}}-{\bm{u}}^o}_{L^2(0,\tau;\cs )}\ge 0. \end{equation} Using the chain rule to take the G\^ateaux derivative of $\mc G({\bm{u}},{\bm{r}})$ at ${\bm{r}}^o$ in the direction $\tilde{{\bm{r}}}$ yields \begin{equation} \mc G'_{{\bm{r}}^o}\tilde{{\bm{r}}}= J'_{{\bm{r}}^o}\tilde{{\bm{r}}} + J'_{{\bm{x}}^o}\mc S'_{{\bm{r}}^o}\tilde{{\bm{r}}} . \end{equation} Write the functionals $\mc G'_{{\bm{r}}^o}:\as\to \mathbb{R}$ and $J'_{{\bm{r}}^o}:\as \to \mathbb{R}$ as elements of $\bm g_{{\bm{r}}^o}$ and $\bm j_{{\bm{r}}^o}$ in $\as^*$, respectively, and take the adjoint of $\mc S'_{{\bm{r}}^o}$. It follows that \begin{equation} \bm g_{{\bm{r}}^o}=\mc S'^*_{{\bm{r}}^o}\bm j_{{\bm{x}}^o}(t)+\bm j_{{\bm{r}}^o}. \end{equation} An explicit representation of the adjoint operator $\mc S'^*_{{\bm{r}}^o}$ will be derived. Consider the inner product \begin{equation*} \inn{\bm j_{{\bm{x}}^o}}{\mc{S}'_{{\bm{r}}^o}\tilde{{\bm{r}}}}_{L^2(0,\tau;\ss )}=\int_0^\tau \inn{\bm j_{{\bm{x}}^o}(t)}{\mc{S}'_{{\bm{r}}^o}\tilde{{\bm{r}}}}dt. \end{equation*} Write ${\bm{z}} (t)=\mc{S}'_{{\bm{r}}^o}\tilde{{\bm{r}}}$. Substitute for $\bm j_{{\bm{x}}^o}(t)$ from \eqref{adj} into this integral. Perform integration by parts to obtain \begin{flalign} &\int_0^\tau\inn{-\dot{{\bm{p}}}^o(t)-(\mc{A}^*+{{\mathcal{F}}_{{\bm{x}}^o(t)}^\prime}^*){\bm{p}}^o(t)}{{\bm{z}}(t)}dt\notag \\ &\quad =\int_0^\tau \inn{{\bm{p}}^o(t)}{\dot{{\bm{z}}}(t)-(\mc{A}+{{\mathcal{F}}}_{{\bm{x}}^o(t)}^\prime){\bm{z}}(t)}dt \notag \\ &\quad =\int_0^\tau \inn{{\bm{p}}^o(t)}{({\mathcal{B}}'_{{\bm{r}}^o}\tilde{{\bm{r}}}){\bm{u}}^o(t)}dt\notag \\ &\quad =\inn{\int_0^\tau ({\mathcal{B}}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*{\bm{p}}^o(t)dt}{\tilde{{\bm{r}}}}_{\as^{*},\as}. \end{flalign} Thus, \begin{equation} \mc S'^*_{{\bm{r}}^o}\bm j_{{\bm{x}}^o}(t)=\int_0^\tau ({\mathcal{B}}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*{\bm{p}}^o(t)dt. \end{equation} As a result, the G\^ateaux derivative of $\mc G({\bm{u}},{\bm{r}})$ at ${\bm{r}}^o$ in the direction $\tilde{{\bm{r}}}$ is \begin{equation} \bm g'_{{\bm{r}}^o}=\int_0^\tau ({\mathcal{B}}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*{\bm{p}}^o(t)dt+\bm j_{{\bm{r}}^o}. \end{equation} The optimality conditions now follow by substituting the G\^ateaux derivatives $\bm g'_{{\bm{r}}^o}$ in \cite[Theorem 1.46]{hinze2008optimization}. \end{proof} \begin{corollary}\label{cor} Let the cost $J({\bm{x}},{\bm{u}},{\bm{r}})$ be \begin{equation} J({\bm{x}},{\bm{u}},{\bm{r}})=\int_0^\tau\inn{\mc Q{\bm{x}}(t)}{{\bm{x}}(t)}+\inn{\mc R {\bm{u}}(t)}{{\bm{u}}(t)}_{\cs}dt, \end{equation} where $\mc Q$ is a positive semi-definite, self-adjoint bounded linear operator on $\ss$, and $\mc{R}$ is a coercive, self-adjoint linear bounded operator on $\cs$. If the minimizer $({\bm{u}}^o,{\bm{r}}^o)$ is in the interior of $U_{ad}\times K_{ad}$, then the following set of equations characterizes $({\bm{x}}^o,{\bm{p}}^o,{\bm{u}}^o,{\bm{r}}^o)$: \begin{equation}\notag \left\lbrace\begin{array}{ll} \dot{{\bm{x}}}^o(t)=\mc{A}{\bm{x}}^o(t)+{\mathcal{F}}({\bm{x}}^o(t))+\mc{B}({\bm{r}}^o){\bm{u}}^o(t),& {\bm{x}}^o(0)={\bm{x}}_0,\\[2mm] \dot{{\bm{p}}}^o(t)=-(\mc{A}^*+{{\mathcal{F}}_{{\bm{x}}^o(t)}^\prime}^*){\bm{p}}^o(t)-\mc{Q}{\bm{x}}^o(t),& {\bm{p}}^o(\tau)=0,\\[2mm] {\bm{u}}^o(t)=-\mc{R}^{-1}\mc{B}^*({\bm{r}}^o){\bm{p}}^o(t),\\[2mm] \int_0^{\tau} (\mc{B}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*{\bm{p}}^o(t)\, dt=0. \end{array}\right. \end{equation} \end{corollary} \begin{proof} If the optimizer $({\bm{u}}^o,{\bm{r}}^o)$ is in the interior of $U_{ad}\times K_{ad}$, then the optimality conditions of \Cref{thm-optimality} hold if and only if \begin{flalign} \bm j_{{\bm{u}}^o}+\mc{B}^*({\bm{r}}^o){\bm{p}}^o&=0, \label{op1}\\ \bm j_{{\bm{r}}^o}+\int_0^{\tau} (\mc{B}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*{\bm{p}}^o(t)\, dt&=0.\label{op2} \end{flalign} The derivatives $J'_{{\bm{x}}^o}(t):L^2(0,\tau;\ss)\to \mathbb{R}$ and $J'_{{\bm{u}}^o}(t):L^2(0,\tau;\cs)\to \mathbb{R}$ are \begin{flalign} J'_{{\bm{x}}^o}\tilde{{\bm{x}}}&=\inn{\mc Q {\bm{x}}^o}{\tilde{{\bm{x}}}}_{L^2(0,\tau;\ss)},\\ J'_{{\bm{u}}^o}\tilde{{\bm{u}}}&=\inn{\mc R {\bm{u}}^o}{\tilde{{\bm{u}}}}_{L^2(0,\tau;\cs)}. \end{flalign} Identify these functionals with elements $\bm j_{{\bm{x}}^o}=\mc Q {\bm{x}}^o(t)$ and $\bm j_{{\bm{u}}^o}= \mc R {\bm{u}}^o(t)$, and notice that $\bm j_{{\bm{r}}^o}=0$. Substituting the derivatives in \eqref{op1} and \eqref{op2} yields the optimality conditions. \end{proof} \chgs{For all ${\bm{x}}_1$ and ${\bm{x}}_2$ in $D({\mathcal{A}})$ and $t\in (0,\tau)$, let $\Pi(t)$ be the solution to the differential Riccati equation \begin{equation}\label{ric} \begin{cases} \frac{d}{dt}\inn{{\bm{x}}_2}{\Pi(t){\bm{x}}_1}=-\inn{{\bm{x}}_2}{\Pi(t){\mathcal{A}}{\bm{x}}_1}-\inn{{\mathcal{A}} {\bm{x}}_2}{\Pi(t){\bm{x}}_1}\\ \qquad -\inn{\mc Q{\bm{x}}_2}{{\bm{x}}_1}+\inn{\Pi(t){\mathcal{B}}({\bm{r}})\mc R^{-1}{\mathcal{B}}^*({\bm{r}})\Pi(t){\bm{x}}_2}{{\bm{x}}_1}, \\ \Pi(\tau)=0. \end{cases} \end{equation} It is well-known, \cite[Chapter 6]{curtain2012introduction} and \cite[Chapter 1]{lasiecka2000control}, that if the system is linear then the adjoint trajectory state ${\bm{p}}^o(t)$ satisfies \begin{equation} {\bm{p}}^o(t)=\Pi(t){\bm{x}}^o(t). \end{equation} As a result, the optimal input and actuator design satisfy in this case \begin{equation}\notag \left\lbrace\begin{array}{ll} {\bm{u}}^o(t)=-\mc{R}^{-1}\mc{B}^*({\bm{r}}^o)\Pi(t){\bm{x}}^o(t),\\[2mm] \int_0^{\tau} (\mc{B}'_{{\bm{r}}^o}{\bm{u}}^o(t))^*\Pi(t){\bm{x}}^o(t)\, dt=0. \end{array}\right. \end{equation}} \section{Worst Initial Condition} \chgs{In this section, sets $U_{ad}$ and $K_{ad}$ and numbers $\tau$ and $R_2$ are the same sets and numbers as in the previous section.} The worst initial condition maximizes $J({\bm{x}},{\bm{u}},{\bm{r}})$ over all choices of initial conditions in $B_{{\mathbb{V}}}(R_2)$ subject to IVP (\ref{eq-IVP}) for a fixed input ${\bm{u}}\in U_{ad}$ and fixed actuator design ${\bm{r}} \in K_{ad}$. Formally, define $\mc G(\cdot):{\mathbb{V}}\to \mathbb{R}$ as $$\mc G({\bm{x}}_0)=J(\mc S({\bm{u}},{\bm{r}};{\bm{x}}_0),{\bm{u}},{\bm{r}}),$$ the worst initial condition over $B_{{\mathbb{V}}}(R_2)$ is the solution to \begin{equation} \begin{cases} \displaystyle\max&\mc G({\bm{x}}_0)\\ \text{s.t.}& {\bm{x}}_0\in B_{{\mathbb{V}}}(R_2). \end{cases} \tag{P1} \label{P1} \end{equation} \begin{lemma}\label{lem-x0} For every ${\bm{u}}\in U_{ad}$ and ${\bm{r}}\in K_{ad}$, the optimization problem (P1) admits a maximizer. \end{lemma} \begin{proof} As in the proof of \Cref{thm-existence optimizer}, define \begin{equation} j:=\sup_{{\bm{x}}_0\in B_{{\mathbb{V}}}(R_2)}\mc G({\bm{x}}_0). \end{equation} Extract a maximizing sequence ${\bm{x}}^n_0$ in $B_{{\mathbb{V}}}(R_2)$. The set $B_{{\mathbb{V}}}(R_2)$ is closed and convex in the reflexive Banach space ${\mathbb{V}}$, it is therefore weakly closed. This implies that ${\bm{x}}^n_0$ has a subsequence that converges weakly to some element $\bar{{\bm{x}}}_0$ in $B_{{\mathbb{V}}}(R_2)$. Also, according to \Cref{thm-existence} and \Cref{thm-weak}, the solution map is bounded and weakly continuous in ${\bm{x}}_0$. The cost function is also convex and continuous in ${\bm{x}}_0$, so it is weakly lower semi-continuous in ${\bm{x}}_0$. These imply that $\bar{{\bm{x}}}_0$ solves \eqref{P1}. \end{proof} \begin{proposition}\label{prop-x0} Under assumptions A1-A4, the solution map $\mc S({\bm{u}}(t),{\bm{r}};{\bm{x}}_0)$ is G\^ateaux differentiable with respect to ${\bm{x}}_0\in B_{{\mathbb{V}}}(R_2)$. Let ${\bm{x}}(t)=\mc S({\bm{u}}(t),{\bm{r}},{\bm{x}}_0)$, the G\^ateaux derivative of $\mc S({\bm{u}}(t),{\bm{r}};{\bm{x}}_0)$ at ${\bm{x}}_0$ in the interior of $B_{{\mathbb{V}}}(R_2)$ in the direction $\tilde{{\bm{x}}}_0$ is the mapping $\mc S'_{\bm{x}} ({\bm{u}}(t),{\bm{r}};\cdot):{\mathbb{V}}\to{\mathbb{W}}(0,\tau)$, $\tilde{{\bm{x}}}_0\mapsto {\bm{q}}(t)$, where ${\bm{q}}(t)$ is the strict solution to \begin{equation} \begin{cases} \dot{{\bm{q}}}(t)=({\mathcal{A}}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime) {\bm{q}}(t),\\ {\bm{q}}(0)=\tilde{{\bm{x}}}_0. \end{cases} \end{equation} \end{proposition} \begin{proof} Let the number $\epsilon>0$ be small enough such that ${\bm{x}}_0+\epsilon\tilde{{\bm{x}}}_0\in B_{{\mathbb{V}}}(R_2)$. Define ${\bm{x}}_\epsilon(t):=\mc S({\bm{u}}(t),{\bm{r}},{\bm{x}}_0+\epsilon\tilde{{\bm{x}}}_0)$, it solves \begin{equation} \begin{cases} \dot{{\bm{x}}}_\epsilon(t)=\mc{A}{\bm{x}}_\epsilon(t)+\mc{F}({\bm{x}}_\epsilon(t))+\mc{B}({\bm{r}}){\bm{u}}(t),\quad t>0,\\ {\bm{x}}_\epsilon(0)={\bm{x}}_0+\epsilon\tilde{{\bm{x}}}_0. \end{cases} \end{equation} Define $\bm{{\bm{e}}}_{{\mathcal{F}}}(t)$ as \begin{equation}\notag \bm{{\bm{e}}}_{{\mathcal{F}}}(t)\coloneqq \frac{1}{\epsilon}\left({\mathcal{F}}({\bm{x}}(t))-{\mathcal{F}}({\bm{x}}_\epsilon(t))-{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime({\bm{x}}(t)-{\bm{x}}_\epsilon(t))\right). \end{equation} Let ${\bm{e}}(t)=({\bm{x}}(t)-{\bm{x}}_\epsilon(t))/\epsilon-{\bm{q}}(t)$, it satisfies \begin{equation}\label{edot2} \begin{cases} \dot{{\bm{e}}}(t)=(\mc{A}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime){\bm{e}}(t)+\bm{{\bm{e}}}_{{\mathcal{F}}}(t),\\ {\bm{e}}(0)=0. \end{cases} \end{equation} Assumption A4 ensures that as $\epsilon\to 0$ \begin{equation} \normm{\bm{{\bm{e}}}_{{\mathcal{F}}}(t)}\to 0, \quad \forall t\in [0,\tau].\label{e(t)3}\\ \end{equation} The convergence in (\ref{e(t)3}) is uniform; to show this, note that ${\bm{x}}(t)-{\bm{x}}_\epsilon(t)$ satisfies \begin{equation}\notag \begin{cases} \dot{{\bm{x}}}(t)-\dot{{\bm{x}}}_\epsilon(t)={\mathcal{A}}({\bm{x}}(t)-{\bm{x}}_\epsilon(t))+{\mathcal{F}}({\bm{x}}(t))-{\mathcal{F}}({\bm{x}}_\epsilon(t)), \\ {\bm{x}}(0)-{\bm{x}}_\epsilon(0)=\epsilon \tilde{{\bm{x}}}_0. \end{cases} \end{equation} According to \cite[Proposition 2.2]{clement1993abstract}, there is $d_\tau$ depending only on $\tau$ such that for all $t\in [0,\tau]$ \begin{flalign}\label{eq14} \norm{\dot{{\bm{x}}}-\dot{{\bm{x}}}_\epsilon}{L^2(0,t;\ss)}&+\norm{{\mathcal{A}}({\bm{x}}-{\bm{x}}_\epsilon)}{L^2(0,t;\ss)}\\ &\le d_\tau \left(\norm{{\mathcal{F}}({\bm{x}})-{\mathcal{F}}({\bm{x}}_\epsilon)}{L^2(0,t;\ss)} + \epsilon\norm{\tilde{{\bm{x}}}_0}{{\mathbb{V}}}\right)\notag \end{flalign} Also, letting $c_\tau$ be the embedding constant of ${\mathbb{W}}(0,\tau) \hookrightarrow C(0,\tau;{\mathbb{V}})$, ${\bm{x}} -{\bm{x}}_\epsilon$ satisfies \begin{flalign}\notag \norm{{\bm{x}} -{\bm{x}}_\epsilon}{C(0,t;{\mathbb{V}})}\le c_\tau &\left(\norm{\dot{{\bm{x}}}-\dot{{\bm{x}}}_\epsilon}{L^2(0,t;\ss)}\right. \notag \\ &\quad \left. +\norm{{\mathcal{A}}({\bm{x}}-{\bm{x}}_\epsilon)}{L^2(0,t;\ss)}\right).\label{eq15} \end{flalign} \Cref{thm-existence} implies that the states ${\bm{x}}(t)$ and ${\bm{x}}_\epsilon(t)$ belong to some bounded set $D$; so let $L_{\mathcal{F}}$ be the Lipschitz constant ${\mathcal{F}}(\cdot)$ on $D$. Combining this with inequalities (\ref{eq14}) and (\ref{eq15}) yield \begin{flalign} \norm{{\bm{x}}(t)-{\bm{x}}_\epsilon(t)}{{\mathbb{V}}}^2\le& 2 c_\tau^2 d_\tau^2 L_{\mathcal{F}}^2 \int_0^t \norm{{\bm{x}}(s)-{\bm{x}}_\epsilon(s)}{{\mathbb{V}}}^2ds \notag \\ &+ 2 \epsilon^2 \norm{\tilde{{\bm{x}}}_0}{{\mathbb{V}}}^2. \end{flalign} Applying Gronwall's lemma to this inequality yields \begin{equation}\label{eq-14} \norm{{\bm{x}}(t)-{\bm{x}}_\epsilon(t)}{{\mathbb{V}}}\le \sqrt{2} e^{c_\tau^2 d_\tau^2 L_{\mathcal{F}}^2} \epsilon \norm{\tilde{{\bm{x}}}_0}{{\mathbb{V}}}. \end{equation} Take the norm of ${\bm{e}}_{{\mathcal{F}}}(t)$ in $\ss$, use (\ref{eq-14}), define $$M_{{\mathcal{F}}}:=\sup\{\|{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime\|_{\mc L({\mathbb{V}},\ss)}:t\in [0,\tau]\}.$$ It follows that \begin{equation}\notag \normm{{\bm{e}}_{{\mathcal{F}}}(t)}\le (L_{{\mathcal{F}}}+M_{{\mathcal{F}}})\sqrt{2} e^{c_\tau^2 d_\tau^2 L_{\mathcal{F}}^2} \norm{\tilde{{\bm{x}}}_0}{{\mathbb{V}}} <\infty. \end{equation} The Bounded Convergence Theorem now ensures that \begin{equation}\label{eq-15} \lim_{\epsilon\to 0}\int_0^\tau \norm{{\bm{e}}_{{\mathcal{F}}}(t)}{{\mathbb{V}}}^2dt = 0. \end{equation} \Cref{lem-estimate} together with (\ref{eq-15}) gives \begin{equation} \lim_{\epsilon\to 0} \norm{{\bm{e}}}{{\mathbb{W}}(0,\tau)}=0. \end{equation} This shows that $\mc S({\bm{u}},{\bm{r}},{\bm{x}}_0)$ is G\^ateaux differentiable at ${\bm{x}}_0$ in the direction $\tilde{{\bm{x}}}_0$. \end{proof} \begin{theorem}\label{thm-worst ini} Suppose assumptions A1-A4 hold, and identify the derivative $J'_{{\bm{x}}}$ by element $\bm j_{\bm{x}}\in {\mathbb{W}}(0,\tau)^*$. Let ${\bm{u}}\in U_{ad}$, ${\bm{r}} \in K_{ad}$, and ${\bm{x}}=\mc S({\bm{u}},{\bm{r}};{\bm{x}}_0)$. Also, let ${\bm{p}}(t)$, the adjoint trajectory state, satisfy \begin{flalign}\label{adj2} \begin{cases} \dot{{\bm{p}}}(t)=-({\mathcal{A}}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime){\bm{p}}(t)-\bm j_{\bm{x}}(t), \quad t\in [0,\tau),\\ {\bm{p}}(\tau)=0. \end{cases} \end{flalign} If ${\bm{x}}_0$ is a worst initial condition over $B_{{\mathbb{V}}}(R_2)$, then, there is a non-negative number $\mu$ such that \begin{flalign}\label{opt x0} \begin{cases} \mu \left(\normm{{\bm{x}}_0}-R_2 \right)=0,\\ {\bm{p}}(0)+\mu {\bm{x}}_0=0.\\ \end{cases} \end{flalign} \end{theorem} \begin{proof} Define $f({\bm{x}}_0):=\frac{1}{2}(\normm{{\bm{x}}_0}^2_{{\mathbb{V}}}-R_2^2).$ Rewrite \eqref{P1} as \begin{equation} \begin{cases} \displaystyle\max&\mc G({\bm{x}}_0)\\ \text{s.t.} &f({\bm{x}}_0)\le 0. \end{cases} \label{P2} \end{equation} The constraint $f({\bm{x}}_0)\le 0$ satisfies Robinson's regularity condition \cite[Section 1.7.3.2]{hinze2008optimization}. This allows one to apply \cite[Theorem 1.56]{hinze2008optimization}. Let $\lambda$ be a non-negative number, and define the Lagrangian \begin{equation} \mf L({\bm{x}}_0,\lambda):=\mc G({\bm{x}}_0,\lambda) + \mu f({\bm{x}}_0). \end{equation} Let $\mf L'_{{\bm{x}}_0}:{\mathbb{V}}\to \mathbb{R}$ be the G\^ateaux derivative of $\mf L({\bm{x}}_0,\lambda)$ at ${\bm{x}}_0$. Identify $\mf L'_{{\bm{x}}_0}$ with an element $l_{{\bm{x}}_0}\in {\mathbb{V}}$. Theorem 1.56 of \cite{hinze2008optimization} ensures that the worst initial condition satisfies for all $\tilde{{\bm{x}}}_0\in {\mathbb{V}}$ the conditions \begin{subequations}\label{optim} \begin{flalign} f({\bm{x}}_0)&\le 0,\\ \mu &\ge 0,\\ \mu f({\bm{x}}_0)&=0,\label{slackness}\\ \inn{l_{{\bm{x}}_0}}{\tilde{{\bm{x}}}_0-{\bm{x}}_0}_{\mathbb{V}}&\ge 0.\label{ineq optim} \end{flalign} \end{subequations} In the following, an explicit expression for $l_{{\bm{x}}_0}$ will be derived. First, the G\^ateaux derivative of $f({\bm{x}}_0)$ at ${\bm{x}}_0$ along $\tilde{{\bm{x}}}_0$ is \begin{equation} f'_{{\bm{x}}_0}\tilde{{\bm{x}}}_0=\inn{\tilde{{\bm{x}}}_0}{{\bm{x}}_0}_{{\mathbb{V}}}.\label{der1} \end{equation} This implies that the functional $f'_{{\bm{x}}_0}:{\mathbb{V}} \to \mathbb{R}$ can be identified with the element ${\bm{x}}_0$. The G\^ateaux derivative of $\mc G({\bm{x}}_0)$ at ${\bm{x}}_0$ along $\tilde{{\bm{x}}}_0$ is derived using the chain rule, \begin{equation}\label{opt2} \mc G'_{{\bm{x}}_0}\tilde{{\bm{x}}}_0=J'_{\bm{x}} \mc{S}'_{{\bm{x}}_0}\tilde{{\bm{x}}}_0. \end{equation} The functionals $\mc G'_{{\bm{x}}_0}:{\mathbb{V}}\to \mathbb{R}$ and $J'_{\bm{x}}:L^2(0,\tau;\ss)\to \mathbb{R}$ can be identified with some elements $\bm g_{{\bm{x}}_0}\in {\mathbb{V}}$ and $\bm j_{{\bm{x}}}\in L^2(0,\tau;\ss)$, respectively. Then, equality \eqref{opt2} implies that \begin{equation} \bm g_{{\bm{x}}_0}=\mc{S}'^*_{{\bm{x}}_0}\bm j_{\bm{x}}(t). \end{equation} The adjoint operator $\mc{S}'^*_{{\bm{x}}_0}$ will be derived. Let $\mc{S}'_{{\bm{x}}_0}\tilde{{\bm{x}}}_0={\bm{q}}(t)$. Consider the inner-product \begin{flalign} &\inn{\bm j_{\bm{x}}}{\mc{S}'_{{\bm{x}}_0}\tilde{{\bm{x}}}_0}_{L^2(0,\tau;\ss)}=\int_0^\tau\inn{\bm j_{{\bm{x}}}(t)}{{\bm{q}}(t)}dt\notag \\ &=\int_0^\tau\inn{-\dot{{\bm{p}}}(t)-(\mc{A}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime){\bm{p}}(t)}{{\bm{q}}(t)}dt. \end{flalign} Using \Cref{prop-x0} and applying integration by parts yield \begin{flalign} &\inn{\bm j_{\bm{x}}}{\mc{S}'_{{\bm{x}}_0}\tilde{{\bm{x}}}_0}_{L^2(0,\tau;\ss)}= \inn{{\bm{p}}(0)}{{\bm{q}}(0)}_{{\mathbb{V}}}-\inn{{\bm{p}}(\tau)}{{\bm{q}}(\tau)}_{{\mathbb{V}}}\notag \\ &\quad +\int_0^\tau \inn{{\bm{p}}(t)}{\dot{{\bm{q}}}(t)-(\mc{A}+{{\mathcal{F}}}_{{\bm{x}}(t)}^\prime){\bm{q}}(t)}dt\notag\\ & =\inn{{\bm{p}}(0)}{\tilde{{\bm{x}}}_0}_{{\mathbb{V}}}. \label{eq-16} \end{flalign} It follows that $\mc{S}'^*_{{\bm{x}}_0}\bm j_{\bm{x}}={\bm{p}}(0)$, and so \begin{equation} \bm g_{{\bm{x}}_0}={\bm{p}}(0).\label{der2} \end{equation} Combining \eqref{der1} and \eqref{der2} yield \begin{equation} l_{{\bm{x}}_0}={\bm{p}}(0)+\mu {\bm{x}}_0. \end{equation} Substituting this in \eqref{optim} yields \begin{equation}\label{ineq optim 2} \inn{{\bm{p}}(0)+\mu {\bm{x}}_0}{\tilde{{\bm{x}}}_0-{\bm{x}}_0}_{\mathbb{V}}\ge 0, \quad \forall \tilde{{\bm{x}}}_0\in {\mathbb{V}}. \end{equation} Since $\tilde{{\bm{x}}}_0\in {\mathbb{V}}$ is arbitrary, the inequality condition \eqref{ineq optim 2} becomes an equality condition. This together with \eqref{slackness} yields \eqref{opt x0}. \end{proof} For linear systems with quadratic cost, the adjoint trajectory state satisfies ${\bm{p}}^o(t)=\Pi(t){\bm{x}}^o(t)$ where $\Pi(t)$ solves \eqref{ric}. Consequently, the optimality condition ${\bm{p}}^o(0)+\mu {\bm{x}}_0=0$ becomes \begin{equation} \Pi(0){\bm{x}}_0=-\mu {\bm{x}}_0. \end{equation} This implies that the worst initial condition is an eigenfunction of the operator $\Pi(0)$. \section{Kuramoto–Sivashinsky equation} For every actuator location $r\in (0,1)$, let the function $b(\cdot;r)$ be in $C^1[0,1]$. Consider the controlled Kuramoto–Sivashinsky equation with Dirichlet boundary conditions and initial condition $w_0(\xi)$ on $\xi\in [0,1]$ and some number $\lambda$ \begin{flalign}\notag \begin{cases} \begin{aligned} &\frac{\partial w}{\partial t} + \frac{\partial^4 w}{\partial \xi^4} +\lambda \frac{\partial^2 w}{\partial \xi^2} + w\frac{\partial w}{\partial \xi} =b(\xi;r)u(t),\ \ &t&> 0,\\[2mm] &w(0,t) = w(1,t)=0,\quad &t&\ge 0,\\[2mm] &\frac{\partial w}{\partial \xi}(0,t) = \frac{\partial w}{\partial \xi}(1,t)=0,\quad &t&\ge 0,\\[2mm] &w(\xi,0)= w_0(\xi), \quad &\xi&\in [0,1]. \end{aligned} \end{cases} \end{flalign} Define the state ${\bm{x}}(t)\coloneqq w(\cdot,t)$, the state space $\ss\coloneqq L^2(0,1)$. Let the state operator ${\mathcal{A}}:D({\mathcal{A}})(\subset \ss)\to \ss$ be \begin{flalign} &{\mathcal{A}} w\coloneqq -w_{\xi\xi\xi\xi}-\lambda w_{\xi\xi},\notag\\ &D({\mathcal{A}})=H^4(0,1)\cap H^2_0(0,1). \end{flalign} Also, the control space is $\cs\coloneqq\mathbb{R}$. The actuator design space is $\as\coloneqq\mathbb{R}$. Define ${{\mathbb{V}}}\coloneqq H^1_0(0,1)$; the nonlinear operator ${\mathcal{F}}(\cdot):{{\mathbb{V}}}\to \ss$ and the input operator ${\mathcal{B}}(\cdot):\as\to \mc{L}(\cs,\ss)$ are defined as \begin{flalign} {\mathcal{F}}(w)&\coloneqq -ww_\xi,\label{F-KS}\\ {\mathcal{B}}(r)u&\coloneqq b(\xi,r)u. \end{flalign} The state space representation of the model will then be (\ref{eq-IVP}). The operator ${\mathcal{A}}:D({\mathcal{A}})\to \ss$ is a self-adjoint operator, is bounded from below, and has compact resolvent. According to Theorem \cite[Theorem 32.1]{sell2013dynamics}, ${\mathcal{A}}$ generates an analytic semigroup on $\ss$. Since the operator ${\mathcal{A}}$ is analytic on a Hilbert space, Theorem 4.1 in \cite{dore1993p} ensures that this operator enjoys maximal parabolic regularity. Also, by Rellich-Kondrachov compact embedding theorem \cite[Chapter 6]{adams2003sobolev}, the space $D({\mathcal{A}})$ is compactly embedded in $\ss$. The operator ${\mathcal{A}}$ is also associated with a form described in A2. \begin{lemma} The nonlinear operator ${\mathcal{F}}(\cdot)$ is G\^ateaux differentiable from ${{\mathbb{V}}}$ to $\ss$. The G\^ateaux derivative of ${\mathcal{F}}(\cdot)$ at $w$ in the direction $f$ is ${{\mathcal{F}}}^\prime_wf=-wf_\xi-w_\xi f$. \end{lemma} \begin{proof} The operator ${{\mathcal{F}}}^\prime_w$, if exists, needs to satisfy \begin{equation} \lim_{\epsilon\to 0}\norm{\frac{1}{\epsilon}\left({\mathcal{F}}(w+\epsilon f)-{\mathcal{F}}(w)\right)-{{\mathcal{F}}}^\prime_wf}{L^2}=0. \end{equation} Substituting in (\ref{F-KS}), inside the limit becomes \begin{flalign} &\norm{\frac{1}{\epsilon}(ww_{\xi}-(w+\epsilon f)(w_{\xi}+\epsilon f_{\xi}))-wf_\xi-w_\xi f}{L^2}\notag \\ &\qquad=\norm{\epsilon f f_{\xi}}{L^2}. \end{flalign} Note that $f\in H^1_0(0,1)$. Embedding $H^1_0(0,1)\hookrightarrow C[0,1]$ means that $f$ is a continuous function over $[0,1]$. This implies that $ff_{\xi}$ is in $L^2(0,1)$, thus \begin{equation} \lim_{\epsilon\to 0}\epsilon\norm{f f_{\xi}}{L^2}=0. \end{equation} The lemma now follows from the uniqueness of G\^ateaux derivative. \end{proof} Note that $D_{{\mathcal{A}}}(1/2,2)=H_0^2(0,1)\hookrightarrow {{\mathbb{V}}}$ (see \cite[Corollary 4.10]{chandler2015interpolation}). The operator ${\mathcal{F}}(\cdot):{{\mathbb{V}}}\to \ss$ is not however weakly continuous, and does not satisfy assumption B1 of \cite{edalatzadehSICON}. For all functions $f$ and $w$ in $H^1_0(0,1)$ and $g$ in $H^1(0,1)$, the adjoint of ${{\mathcal{F}}}_w^\prime$ satisfies \begin{equation} \inn{f}{{{\mathcal{F}}_w^\prime}^*g}_{L^2}=\inn{{{\mathcal{F}}}_w^\prime f}{g}_{L^2}=\int_0^1 (-wf_{\xi}-w_{\xi}f)g d\xi . \end{equation} Performing integration by parts yields \begin{flalign} \int_0^1 (-wf_{\xi}-w_{\xi}f)g d\xi=-\int_0^1 wg_{\xi}fd\xi. \end{flalign} The operator ${{\mathcal{F}}_w^\prime}^*$ maps $D({{\mathcal{F}}_w^\prime}^*)=H^1(0,1)$ to $L^2(0,1)$ as follows \begin{equation} {{\mathcal{F}}_w^\prime}^*g=-wg_{\xi}. \end{equation} In addition, \begin{flalign} {\mathcal{B}}^*(r)w&=\int_{0}^{1}b(\xi,r)w(\xi)d\xi, &\forall& w\in {\mathbb{V}},\quad\\ ({\mathcal{B}}'_ru)^*f&=u\int_{0}^{1} b_r(\xi;r)f(\xi) d\xi, &\forall& f\in {{\mathbb{V}}}.\quad \end{flalign} Also, define \begin{equation} K_{ad}:=\left\{r\in[a,b]: 0<a<b<1\right\}. \end{equation} Global stability of an uncontrolled KS equation has been studied extensively, see e.g. \cite{al2018linearized,liu2001stability,fantuzzi2016,ahmadi2016}. Theorem 2.1 of \cite{liu2001stability} proves that for $\lambda< 4\pi^2$, the uncontrolled KS equation is globally exponentially stable. Proof of this theorem can be modified to ensure that there is solution to the controlled KS equation over $[0,\tau]$ for all initial conditions in ${\mathbb{V}}$. The following lemma ensures that for some parameters $\lambda$ there is a solution to the KS equation for all initial conditions and inputs over arbitrary time intervals. \begin{lemma}\label{lem-KS} Let $\lambda<4\pi^2$ and $\sigma(\lambda)$ be the smallest eigenvalue of $-{\mathcal{A}}$. For all initial conditions $w_0\in {\mathbb{V}}$ and inputs $u\in L^2(0,\tau)$, the strict solution to the KS system satisfies \begin{equation}\notag \normm{w(\tau)}^2\le \normm{w_0}^2+\frac{1}{\sigma(\lambda)} \norm{u}{L^2(0,\tau)}^2\max_{\xi\in[0,1]}b^2(\xi;r). \end{equation} \end{lemma} \begin{proof} \Cref{thm-existence} ensures that there is a solution $w\in {\mathbb{W}}(0,\tau)$ over $[0,\tau]$ to the KS system with initial condition $w_0\in {\mathbb{V}}$ and input $u\in L^2(0,\tau)$. Consider the Lyapunov function \begin{equation} E(t):=\int_0^1 w^2(\xi,t)\, d\xi. \end{equation} Since $w\in W^{1,2}(0,\tau;\ss)$, the function $E(t)$ is differentiable. Taking the derivative of $E(t)$ and applying \cite[Lemma 3.1]{liu2001stability} yield \begin{equation}\label{eq1} \dot{E}(t)\le -2\sigma(\lambda) E(t)+2\int_0^1w(\xi,t)b(\xi;r)u(t)d\xi. \end{equation} Apply Young's inequality to the integral term, for every $\epsilon>0$, \begin{equation} \dot{E}(t)\le (-2\sigma(\lambda)+\epsilon) E(t)+\frac{1}{\epsilon}\int_0^1b^2(\xi;r)u^2(t)d\xi. \end{equation} Let $\epsilon= \sigma(\lambda)$. Taking an integral over $[0,\tau]$ yields the desired inequality in the lemma. \end{proof} Since the KS system satisfies assumptions A1-A5, \Cref{cor} can be applied to obtain the optimality conditions. The cost function to be optimized is \begin{equation} J({\bm{x}},{\bm{u}},{\bm{r}})=\int_0^\tau\int_0^1w^2(\xi,t)d\xi dt+\int_0^\tau u^2(t)dt. \end{equation} Letting ${\bm{p}}(t)=f(\cdot,t)$, the optimizer $(u^o,r^o,w^o,f^o)$ with initial condition $w_0(\xi)\in H^1_0(0,1)$ satisfies {\small\begin{flalign}\label{optimizers KSE} &\begin{cases}\notag \begin{aligned} &\p{w^o}{t}{}+\p{w^o}{\xi}{4}+\lambda \p{w^o}{\xi}{2}+w^o\p{w^o}{\xi}{}=b(\xi;r^o)u^o(t), &t&>0\\[2mm] &w^o(0,t)=w^o(1,t)=0, &t&>0\\[2mm] &\p{w^o}{\xi}{}(0,t)=\p{w^o}{\xi}{}(1,t)=0, &t&>0\\[2mm] &w^o(\xi,0)=w_0(\xi),&& \end{aligned} \end{cases}\\ &\begin{cases}\notag \begin{aligned} &\p{f^o}{t}{}-\p{f^o}{\xi}{4}-\lambda \p{f^o}{\xi}{2}-w^o\p{f^o}{\xi}{}=-w^o(\xi,t), &t&>0\\[2mm] &f^o(0,t)=f^o(1,t)=0, &t&>0\\[2mm] &\p{f^o}{\xi}{}(0,t)=\p{f^o}{\xi}{}(1,t)=0, &t&>0\\[2mm] &f^o(\xi,\tau)=0,&& \end{aligned} \end{cases}\\ &\begin{cases}\notag \begin{aligned} &u^o(t)=-\int_0^{1}b(\xi;r^o)f^o(\xi,t)\, d\xi, \quad t>0,\\[2mm] &\int_0^{\tau}\int_{0}^{1}u^o(t)b_{r}(\xi;r^o)f^o(\xi,t)\,d\xi dt=0. \end{aligned} \end{cases} \end{flalign} The worst initial condition over a unit ball satisfies \begin{equation} \begin{cases} \mu\left( \norm{{w_0}}{H_0^1(0,1)}- 1 \right)=0,\\[2mm] f^o(\xi,0)+\mu{w}_0(\xi)= 0. \end{cases} \end{equation} \section{Nonlinear Diffusion} Consider the transfer of heat in a bounded, open, connected set $\Omega\subset \mathbb{R}^2$. It is assumed that $\Omega$ has a Lipschitz boundary separated into $\partial \Omega=\overline{\Gamma_0\cup\Gamma_1}$ where $\Gamma_0\cap\Gamma_1=\emptyset$ and $\Gamma_0\neq \emptyset$. Denote by $\nu$ the unit outward normal vector field on $\partial \Omega$. The class of nonlinear heat transfer models is, for actuator shape $r \in C^1 (\overline{\Omega} ) ,$ \begin{equation}\small\notag \begin{cases} \begin{aligned} &\p{w}{t}{}(\xi,t)=\\ &\qquad\Delta w(\xi,t)+F(w(\xi,t))+r(\xi)u(t), &(\xi,t)&\in \Omega\times (0,\tau],\\ &w(\xi,t)=0, &(\xi,t)&\in \Gamma_0\times [0,\tau],\\ &\p{w}{\nu}{}(\xi,t)=0, &(\xi,t)&\in \Gamma_1\times [0,\tau],\\ &w(\xi,0)=w_0(\xi), &\xi &\in \Omega. \end{aligned} \end{cases} \end{equation} Defining $\as=L^2(\Omega),$ a set of admissible actuator shapes is $$K_{ad} = \{ r \in C^1 (\overline{\Omega} ) : \norm{r}{C^1 (\overline{\Omega})}\leq 1 \} . $$ The set $K_{ad}$ is compact in $\as$ with respect to the norm topology \cite[Chapter 6]{adams2003sobolev}. Let $\ss:= L^2(\Omega)$, $\cs\coloneqq\mathbb{R}$, and the state ${\bm{x}}(t):=w(\cdot,t)$. The operator ${\mathcal{A}}:D({\mathcal{A}})\to \ss$ is defined as \begin{subequations} \begin{gather} {\mathcal{A}} w=\Delta w,\\ D({\mathcal{A}})=\left\{w\in H^2(\Omega)\cap H^1_{\Gamma_0}: \p{w}{\nu}{}=0 \text{ on }\Gamma_1\right\}. \end{gather} \end{subequations} The operator ${\mathcal{A}}$ self-adjoint, non-negative and has compact resolvent. Thus, it generates an analytic semi-group on the Hilbert space $L^2(\Omega)$ \cite[Theorem 32.1]{sell2013dynamics}, and has maximal $L^p$ regularity \begin{comment} Different nonlinear functions $F(\cdot)$ are used in different applications. If $F(\zeta)$ has sub-linear growth, i.e., there is a positive number $a$ such that \begin{equation} |F(\zeta)|\le a(1+|\zeta|), \end{equation} then, ${\mathbb{V}}=\ss$. However, in most applications, $F(\zeta)$ has super-linear growth, so $ {\mathbb{V}} \subsetneq \ss$. \end{comment} Define ${\mathbb{V}}=H^1_{\Gamma_0}(\Omega)$ and assume that the nonlinear operator ${\mathcal{F}}(\cdot) : {\mathbb{V}} \to \ss .$ The proof of the following lemma is the same as that of \cite[Lemma 7.1.1]{edalatzadehSICON}. \begin{lemma}\label{lem-heat-F} Let ${\mathbb{V}}=H^1_{\Gamma_0}(\Omega)$. Assume that \begin{enumerate} \item $F(\zeta)$ is twice continuously differentiable over $\mathbb{R}$; denote its derivatives by ${\mathcal{F}}^\prime(\zeta)$ and $F^{\prime\prime} (\zeta)$; \item \label{C2}there are numbers $a_0>0$ and $b>1/2$ such that $|F^{\prime \prime}(\zeta)|\le a_0(1+|\zeta|^b)$. \end{enumerate} Then ${\mathcal{F}}(\cdot)$ is G\^ateaux differentiable from ${\mathbb{V}}$ to $\ss$. The G\^ateaux derivative of ${\mathcal{F}}(\cdot)$ at $w(\xi)$ in the direction $f(\xi)$ is ${{\mathcal{F}}^\prime}_wf=F^\prime(w)f$. \end{lemma} It is straightforward to show that the operator ${\mathcal{F}}_w^\prime:{\mathbb{V}}(\subset \ss)\to \ss$ is self-adjoint, i.e., \begin{equation} \inn{{{\mathcal{F}}_w^\prime}^*g}{f}=\inn{g}{{{\mathcal{F}}_w^\prime}f}, \quad \forall f,g \in {\mathbb{V}}. \end{equation} Define $\cs =\mathbb{R}$ and the input operator ${\mathcal{B}}(r)\in \mc{L}(\cs ,\ss)$ maps $u$ to $r(\xi)u$. Also, for all $f$ in $\ss$ \begin{flalign} {\mathcal{B}}^*(r)f&=\int_{\Omega}r(\xi)f(\xi)d \xi,\\ ({\mathcal{B}}'_ru)^*f&=uf. \end{flalign} For every initial condition in ${\mathbb{V}}$, a strict solution over $[0,\tau]$ to the nonlinear heat equation is not guaranteed. The following lemma states a condition under which there is a solution to the diffusion equation for all initial conditions and inputs over arbitrary time intervals. \begin{lemma}\label{lem-heat} If the function $F(\zeta)$ satisfies $\zeta F(\zeta)\le 0$ for all $\zeta\in \mathbb{R}$, then there is $c_\Omega>0$ such that the strict solution to the nonlinear heat equation satisfies \begin{equation}\notag \normm{w(\tau)}^2\le \normm{w_0}^2+\frac{4}{c_\Omega} \norm{u}{L^2(0,\tau)}^2\norm{r}{\as}^2. \end{equation} \end{lemma} \begin{proof} Theorem 1 in \cite{mazenc2011strict} proves that the nonlinear equation in one spatial dimension is input-to-state stable. This lemma extends \cite[Theorem 1]{mazenc2011strict} to two-spatial dimension. Using the same idea of proof, consider the Lyapunov function \begin{equation} E(t):=\int_{\Omega}w^2(\xi,t) \, d\xi. \end{equation} The function $E(t)$ is differentiable since $w\in W^{1,2}(0,\tau;\ss)$. Take the derivative of this function, substitute for $\dot{w}(\xi,t)$ from the heat equation, and perform integration by parts as follows \begin{flalign} \dot{E}(t)=&2\int_{\Omega}w(\xi,t)\left( \Delta w(\xi,t)+F(w(\xi,t))+r(\xi)u(t) \right)\, d\xi\notag\\ =&2\int_{\Gamma}w(\xi,t)\p{w}{\nu}{}(\xi,t)d\xi-2\int_{\Omega}\left(\nabla w(\xi,t)\right)^2\, d\xi\notag\\ &+2\int_{\Omega}w(\xi,t)\left( F(w(\xi,t))+r(\xi)u(t) \right)\, d\xi. \end{flalign} Apply the boundary conditions. Use Poincar\'e inequality and let $c_\Omega$ be its constant. Also, use Young's inequality for all $\epsilon>0$ \begin{equation}\label{eq2} \dot{E}(t)\le -2\left(c_{\Omega}-\epsilon\right)E(t)+\frac{2}{\epsilon} u^2(t)\norm{r}{2}^2. \end{equation} Set $\epsilon=c_\Omega/2$. Taking the integral over $[0,\tau]$ of (\ref{eq2}) then yields the desired inequality. \end{proof} The nonlinear heat equation satisfies assumptions A1-A5, and thus, \Cref{cor} can be applied to obtain the optimality conditions. The cost function to be optimized is \begin{equation} J({\bm{x}},{\bm{u}},{\bm{r}})=\int_0^\tau \int_{\Omega} w^2(\xi,t)d\xi dt +\int_0^\tau u^2(t)dt. \end{equation} Letting ${\bm{p}}(t)=f(\cdot,t)$, The optimizer $(u^o,r^o,w^o,f^o)$ with initial condition $w_0\in H^1_{\Gamma_0}(\Omega)$ satisfies {\small\begin{flalign}\label{optimizers heat} &\begin{cases}\notag \begin{aligned} &\p{w^o}{t}{}(\xi,t)=\\ &\qquad\Delta w^o(\xi,t)+F(w^o(\xi,t))+r^o(\xi)u^o(t), &(\xi,t)&\in \Omega\times (0,\tau],\\[1mm] &w^o(\xi,t)=0, &(\xi,t)&\in \Gamma_0\times [0,\tau],\\[1mm] &\p{w^o}{\nu}{}(\xi,t)=0, &(\xi,t)&\in \Gamma_1\times [0,\tau],\\[1mm] &w^o(\xi,0)=w_0(\xi), &\xi &\in \Omega. \end{aligned} \end{cases}\\ &\begin{cases}\notag \begin{aligned} &\p{f^o}{t}{}(\xi,t)=-\Delta f^o(\xi,t)\\ &\quad -F^\prime(w(\xi,t))f^o(\xi,t)-w^o(\xi,t), &(\xi,t)&\in \Omega\times (0,\tau],\\[1mm] &f^o(\xi,t)=0, &(\xi,t)&\in \Gamma_0\times [0,\tau],\\[1mm] &\p{f^o}{\nu}{}(\xi,t)=0, &(\xi,t)&\in \Gamma_1\times [0,\tau],\\[1mm] &f^o(\xi,\tau)=0, &\xi &\in \Omega, \end{aligned} \end{cases}\\ &\begin{cases}\notag \begin{aligned} &u^o(t)=-\int_{\Omega}r^o(\xi)f^o(\xi,t)\, d\xi, &t&\in [0,\tau],\\[2mm] &\int_0^{\tau}u^o(t)f^o(\xi,t) dt=0, &\xi&\in \Omega. \end{aligned} \end{cases} \end{flalign} The worst initial condition over a unit ball satisfies \begin{equation} \begin{cases} \mu\left(\norm{{w}_0}{H^1_{\Gamma_0}(\Omega)}- 1\right)=0,\\[2mm] f^o(\xi,0)+\mu w_0(\xi)= 0. \end{cases} \end{equation} \section{Conclusion} Optimal actuator design for quasi-linear infinite-dimensional systems with a parabolic linear part was considered in this paper. It was shown that the existence of an optimal control together with an optimal actuator design is guaranteed under natural assumptions. With additional assumptions of differentiability, first-order necessary optimality conditions were obtained. The theory was illustrated by application to the Kuramoto-Sivashinsky (KS) equation and nonlinear heat equations. Current work is concerned with developing numerical methods for solution of the optimality equations. Extension of these problems to situations where the input operator is not bounded on the state space is also of interest.

BkiUeHXxK7FjYEB4WTK5

\section{Introduction} \label{sec:intro} When a computational problem cannot be solved exactly within the desired time budget, a frequent solution is to employ approximation algorithms~\cite{Gonzalez:2007:HAA:1199638}. With large data sets being the rule and not the exception today, approximation is frequently applied, even to polynomial-time problems~\cite{borassi2016kadabra}. We focus on a particular subclass of approximation algorithms: \emph{sampling algorithms}. They sample data according to some (usually algorithm-specific) probability distribution, perform some computation on the sample and induce a result for the full data set. More specifically, we consider \emph{adaptive} sampling (ADS\xspace) algorithms (also called \emph{progressive} sampling algorithms). Here, the number of samples that are required is not statically computed (e.\,g.,\xspace from the input instance) but also depends on the data that has been sampled so far. While non-adaptive sampling algorithms can often be parallelized trivially by drawing multiple samples in parallel, adaptive sampling constitutes a challenge for parallelization: checking the stopping condition of an ADS\xspace algorithm requires access to all the data generated so far and thus mandates some form of synchronization. \paragraph*{Motivation and Contribution.} Our initial motivation was a parallel implementation of the sequential state-of-the-art approximation algorithm \tool{KADABRA}\xspace~\cite{borassi2016kadabra} for betweenness centrality (BC\xspace) approximation. BC\xspace is a very popular centrality measure in network analysis, see Section~\ref{sub:bc-approx} for more details. To the best of our knowledge, parallel adaptive sampling has not received a generic treatment yet. Hence, we propose techniques to parallelize ADS\xspace algorithms in a generic way, while scaling to large numbers of threads. While we turn to \tool{KADABRA}\xspace to demonstrate the effec\-tive\-ness of the proposed algorithms, our techniques can be adjusted easily to other ADS\xspace algorithms. We introduce two new parallel ADS\xspace algorithms, which we call \emph{local-frame} and \emph{shared-frame}. Both algorithms try to avoid extensive synchronization when checking the stopping condition. This is done by maintaining multiple copies of the sampling state and ensuring that the stopping condition is never checked on a copy of the state that is currently being written to. \emph{Local-frame} is designed to use the least amount of synchronization possible -- at the cost of an additional memory footprint of $\Theta(n)$ per thread, where $n$ denotes the size of the sampling state. This algorithm performs only atomic \texttt{load-acquire} and \texttt{store-release} operations for synchronization, but no expensive read-modify-write operations (like \texttt{CAS} or \texttt{fetch-add}). \emph{Shared-frame}, in turn, aims instead at meeting a desired tradeoff between memory footprint and synchronization overhead. In contrast to \emph{local-frame}, it requires only $\Theta(1)$ additional memory per thread, but uses atomic read-modify-write operations (e.\,g.,\xspace \texttt{fetch-add}) to accumulate samples. We also propose the deterministic \emph{indexed-frame} algorithm; it guarantees that the results of two different executions is the same for a fixed random seed, regardless of the number of threads. \nprounddigits{1} Our experimental results show that local-frame, shared-frame and indexed-frame achieve parallel speedups of $\numprint{15.879639123178466}\times$, $\numprint{18.081267115073295}\times$, and $\numprint{10.818961592817336}\times$ on 32 cores, respectively. Using the same number of cores, our OpenMP-based parallelization (functioning as a baseline) only yields a speedup of $\numprint{6.27616521473421}\times$; thus our algorithms are up to $\numprint{2.8809418644086824}\times$ faster. Moreover, also due to implementation improvements and parameter tuning, our best algorithm performs adaptive sampling $\numprint{65.29488898389054}\times$ faster than the existing implementation of \tool{KADABRA}\xspace (when all implementations use 32 cores). \npnoround \section{Preliminaries and Baseline for Parallelization} \label{sec:prelim} \subsection{Basic Definitions} \label{sub:basic-defs} \paragraph{Memory model.} Throughout this paper, we target a multi-threaded shared-memory machine with $T$ threads. We work in the C11 memory model~\cite{ISO:2012:III} (more details in Appendix~\ref{app:c11_model}); in particular, we assume the existence of the usual atomic operations, as well as \texttt{load-acquire} and \texttt{store-release} barriers. \paragraph{Adaptive sampling.} \begin{algorithm}[bt] \caption{Generic Adaptive Sampling} \label{algo:adaptive} \begin{minipage}[t]{0.50\textwidth} Variable initialization: \begin{algorithmic} \State $d \gets$ new sampling state structure \State $d\mathtt{.data} \gets (0, \ldots, 0)$ \Comment{Sampled data.} \State $d\mathtt{.num} \gets 0$ \Comment{Number of samples.} \end{algorithmic} \end{minipage} \hspace{0.05\textwidth} \begin{minipage}[t]{0.45\textwidth} Main loop: \begin{algorithmic} \While{\textbf{not} \Call{checkForStop}{$d$}} \State $d\mathtt{.data} \gets d\mathtt{.data} \circ \Call{sample}{\null}$ \State $d\mathtt{.num} \gets d\mathtt{.num} + 1$ \EndWhile \end{algorithmic} \end{minipage} \end{algorithm} For our techniques to be applicable, we expect that an ADS\xspace algorithm behaves as depicted in Algorithm~\ref{algo:adaptive}: it iteratively samples data (in \Call{sample}{}) and aggregates it (using some operator $\circ$), until a stopping condition (\Call{checkForStop}{}) determines that the data sampled so far is sufficient to return an approximate solution within the required accuracy. This condition does not only consider the number of samples ($d\mathtt{.num}$), but also the sampled data ($d\mathtt{.data}$). Throughout this paper, we denote the size of that data (i.\,e.,\xspace the number of elements of $d.\mathtt{data}$) by $n$. We assume that the stopping condition needs to be checked on a \emph{consistent} state, i.\,e.,\xspace a state of $d$ that can occur in a sequential execution.\footnote{That is, $d\mathtt{.num}$ and all entries of $d\mathtt{.data}$ must result from an integral sequence of samples; otherwise, parallelization would be trivial.} Furthermore, to make parallelization feasible at all, we need to assume that $\circ$ is associative. \subsection{Betweenness Centrality and its Approximation} \label{sub:bc-approx} \emph{Betweenness Centrality} (BC\xspace) is one of the most popular vertex centrality measures in the field of network analysis. Such measures indicate the importance of a vertex based on its position in the network~\cite{boldi2014axioms} (we use the terms \emph{graph} and \emph{network} interchangeably). Being a centrality measure, BC\xspace constitutes a function $\mathbf{b}: V \to \mathbb{R}$ that maps each vertex of a graph $G = (V, E)$ to a real number -- higher numbers represent higher importance. To be precise, the BC\xspace of $u \in V$ is defined as $ \mathbf{b}(u) = \sum_{s \neq t \in V \setminus \{u\}} \frac{\sigma_{st}(u)}{\sigma_{st}}, $ where $\sigma_{st}$ is the number of shortest $s$-$t$-paths and $\sigma_{st}(u)$ is the number of shortest $s$-$t$-paths that contain $u$. Unfortunately, BC\xspace is rather expensive to compute: the standard exact algorithm~\cite{brandes2001faster} has time complexity $\Theta(|V||E|)$ for unweighted graphs. Moreover, unless the Strong Exponential Time Hypothesis fails, this asymptotic running time cannot be improved~\cite{borassi2016into}. Numerous approximation algorithms for BC\xspace have thus been developed (we refer to Section~\ref{sec:related_work} for an overview). The state of the art of these approximation algorithms is the $\tool{KADABRA}\xspace$ algorithm~\cite{borassi2016kadabra} of Borassi and Natale, which happens to be an ADS\xspace algorithm. With probability $(1 - \delta)$, \tool{KADABRA}\xspace approximates the BC\xspace values of the vertices within an additive error of $\epsilon$ in nearly-linear time complexity, where $\epsilon$ and $\delta$ are user-specified constants. While our techniques apply to any ADS\xspace algorithm, we recall that, as a case study, we focus on scaling the \tool{KADABRA}\xspace algorithm to a large number of threads. \subsection{The \tool{KADABRA}\xspace algorithm} \label{sub:kadabra_algo} \tool{KADABRA}\xspace samples vertex pairs $(s, t)$ of $G = (V, E)$ uniformly at random and then selects a shortest $s$-$t$-path uniformly at random (in \Call{sample}{} in Algorithm~\ref{algo:adaptive}). After $\tau$ iterations, this results in a sequence of randomly selected shortest paths $\pi_1, \pi_2, \dots, \pi_\tau$; from those paths, BC\xspace is estimated as: \begin{equation*} \widetilde{\mathbf{b}}(v) = \frac{1}{\tau}\sum_{i = 1}^{\tau}x_i(v), \quad x_i(v) = \begin{cases} 1 & \text{ if } v \in \pi_i\\ 0 & \text{otherwise.} \end{cases} \end{equation*} $\sum_{i = 1}^\tau x_i$ is exactly the sampled data ($d\mathtt{.data}$) that the algorithm has to store (i.\,e.,\xspace the accumulation $\circ$ in Algorithm~\ref{algo:adaptive} sums $x_i$ over $i$). To compute the stopping condition (\Call{checkForStop}{} in Algorithm~\ref{algo:adaptive}), \tool{KADABRA}\xspace maintains the invariants \begin{equation} \label{eq:kad-inv} \Pr(\mathbf{b}(v) \le \widetilde{\mathbf{b}}(v) - f) \le \delta_{L}(v) \text{ ~and~ } \Pr(\mathbf{b}(v) \ge \widetilde{\mathbf{b}}(v) + g) \le \delta_{U}(v) \end{equation} for two functions $f = f(\widetilde{\mathbf{b}}(v), \delta_{L}(v), \omega, \tau)$ and $g = g(\widetilde{\mathbf{b}}(v), \delta_{U}(v), \omega, \tau)$ depending on a maximal number $\omega$ of samples and per-vertex probability constants $\delta_L$ and $\delta_U$ (more details in the original paper~\cite{borassi2016kadabra}). The values of those constants are computed in a preprocessing phase (mostly consisting of computing an upper bound on the diameter of the graph). $\delta_L$ and $\delta_U$ satisfy $\sum_{v \in V} \delta_L(v) + \delta_U(v) \leq \delta$ for a user-specified parameter $\delta \in (0, 1)$. Thus, the algorithm terminates once $f, g < \epsilon$; the result is correct with an absolute error of $\pm \epsilon$ and probability $(1 - \delta)$. We note that checking the stopping condition of \tool{KADABRA}\xspace on an inconsistent state leads to incorrect results. For example, this can be seen from the fact that $g$ is increasing with $\widetilde{\mathbf{b}}$ and decreasing with $\tau$, see Appendix~\ref{app:kadabra}. \subsection{First Attempts at \tool{KADABRA}\xspace Parallelization} \label{sub:first-attempts} In the original \tool{KADABRA}\xspace implementation\footnote{ Available at: \url{https://github.com/natema/kadabra}}, a lock is used to synchronize concurrent access to the sampling state. As a first attempt to improve the scalability, we consider an algorithm that iteratively computes a fixed number of samples in parallel (e.\,g.,\xspace using an OpenMP \texttt{parallel for} loop), then issues a synchronization barrier (as implied by the \texttt{parallel for} loop) and checks the stopping condition afterwards. While sampling, atomic increments are used to update the global sampling data. This algorithm is arguably the \enquote{natural} OpenMP-based parallelization of an ADS\xspace algorithm and can be implemented in a few extra lines of code. Moreover, it already improves upon the original parallelization. However, as shown by the experiments in Section~\ref{sec:experiments}, further significant improvements in performance are possible by switching to more lightweight synchronization. \section{Scalable Parallelization Techniques} \label{sec:algorithms} To improve upon the OpenMP parallelization from Section~\ref{sub:first-attempts}, we have to avoid the synchronization barrier before the stopping condition can be checked. This is the objective of our \emph{epoch-based} algorithms that constitute the main contribution of this paper. In Section~\ref{sub:epoch_based}, we formulate the main idea of our algorithms as a general framework and prove its correctness. The subsequent subsections present specific algorithms based on this framework and discuss tradeoffs between them. \subsection{Epoch-based Framework} \label{sub:epoch_based} \begin{figure}[t]% \begin{subfigure}[t]{.4\textwidth} \quad \fbox{\begin{tabular}{r@{\quad}l@{\quad}l} \textsf{int} & \texttt{epoch} &$\gets e$ \\ \textsf{int} & \texttt{num} &$\gets 0$\\ \textsf{int} & \texttt{data}[$n$] &$\gets (0, \ldots, 0)$ \end{tabular}}% \caption{Structure of a state frame (SF) for epoch $e$. \texttt{num}: Number of samples, \texttt{data}: Sampled data}% \label{fig:sf} \end{subfigure}% \begin{subfigure}[t]{.6\textwidth} \qquad \qquad \begin{tabular}{r@{\quad}l@{\quad}l} \textsf{bool} & \texttt{stop} &$\gets$ \textsf{false}\\ \textsf{int} & \texttt{epochToRead} &$\gets 0$\\ SF $\ast$ & \texttt{sfFin[$T$]} &$\gets (\textsf{null}, \ldots, \textsf{null})$ \end{tabular}% \caption{Shared variables}% \label{fig:globalvars} \end{subfigure}% \caption{Data structures used in epoch-based algorithms, including initial values}% \label{fig:epoch-state} \end{figure} In our epoch-based algorithms, the execution of each thread is subdivided into a sequence of discrete \emph{epochs}. During an epoch, each thread iteratively collects samples; the stopping condition is only checked at the end of an epoch. The crucial advantage of this approach is that the end of an epoch \emph{does not} require global synchronization. Instead, our framework guarantees the consistency of the sampled data by maintaining multiple copies of the sampling state. As an invariant, it is guaranteed that that no thread writes to a copy of the state that is currently being read by another thread. This is achieved as follows: each copy of the sampling state is labeled by an epoch \emph{number} $e$, i.\,e.,\xspace a monotonically increasing integer that identifies the epoch in which the data was generated. When the stopping condition has to be checked, all threads advance to a new epoch $e + 1$ and start writing to a new copy of the sampling state. The stopping condition is only verified after all threads have finished this transition and it only takes the sampling state of epoch $e$ into account. More precisely, the main data structure that we use to store the sampling state is called a \emph{state frame} (SF). Each SF $f$ (depicted in Figure~\ref{fig:sf}) consists of (i) an epoch number ($f.\mathtt{epoch}$), (ii) a number of samples ($f.\mathtt{num}$) and (iii) the sampled data ($f.\mathtt{data}$). The latter two symbols directly correspond to $d.\texttt{num}$ and $d.\mathtt{data}$ in our generic formulation of an adaptive sampling algorithm (Algorithm~\ref{algo:adaptive}). Aside from the SF structures, our framework maintains three global variables that are shared among all threads (depicted in Figure~\ref{fig:globalvars}): (i) a simple Boolean flag \texttt{stop} to determine if the algorithm should terminate, (ii) a variable \texttt{epochToRead} that stores the number of the epoch that we want to check the stopping condition on and (iii) a pointer \texttt{sfFin}[$t$] for each thread $t$ that points to a SF finished by thread $t$. Incrementing $\mathtt{epochToRead}$ is our synchronization mechanism to notify all threads that they should advance to a new epoch. This transition is visualized in Figure~\ref{fig:advance} in Appendix~\ref{app:epoch-transition}. \begin{algorithm}[t] \caption{Epoch-based Approach} \label{algo:epoch-based} \begin{minipage}{.49\textwidth} Per-thread variable initialization: \begin{algorithmic} \State $e_\mathrm{sam} \gets 1$ \State $f_\mathrm{sam} \gets$ new SF for $e_\mathrm{sam} = 1$ \If{$t = 0$} \State $e_\mathrm{chk} \gets 0$ \State $inCheck \gets$ \textsf{false} \EndIf \end{algorithmic} Main loop for thread $t$: \begin{algorithmic}[1] \Loop \State $doStop \rlxmove \mathtt{stop}$ \label{line:main-loop} \If{$doStop$} \State \textbf{break} \EndIf \State $f_\mathrm{sam}\mathtt{.data} \gets f_\mathrm{sam}\mathtt{.data} \circ \Call{sample}{\null}$ \State $f_\mathrm{sam}\mathtt{.num} \gets f_\mathrm{sam}\mathtt{.num} + 1$ \State $r \rlxmove$ \texttt{epochToRead} \label{line:advance-epoch} \If{$r =$ $e_\mathrm{sam}$} \State reclaim SF of epoch $e_\mathrm{sam} - 1$ \label{line:reclaim} \State \texttt{sfFin}[t] $\storerel$ $f_\mathrm{sam}$ \label{line:publish} \State $e_\mathrm{sam}$ $\gets$ $e_\mathrm{sam}$ $+ 1$ \State $f_\mathrm{sam}$ $\gets$ new SF for $e_\mathrm{sam}$ \label{line:new-sf} \EndIf \If{$t = 0$} \label{line:thread-zero-call} \State \Call{checkFrames}{\null} \EndIf \EndLoop \algstore{epochbased} \end{algorithmic} \end{minipage}\hspace{.02\textwidth} \begin{minipage}{.49\textwidth} Check of stopping condition by thread $0$: \begin{algorithmic}[1] \algrestore{epochbased} \Procedure{checkFrames}{\null} \If{\textbf{not} $inCheck$} \label{line:check-cycle} \State $e_\mathrm{chk} \gets e_\mathrm{chk} + 1$ \State \texttt{epochToRead} $\rlxmove e_\mathrm{chk}$ \State $inCheck \gets$ \textsf{true} \EndIf \For{$i \in \{1, \ldots, T\}$} \label{line:check-frames} \State $f_\mathrm{fin} \loadacq$ \texttt{sfFin}[t] \label{line:subscribe} \If{$f_\mathrm{fin} = \mathsf{null}$} \State \Return \EndIf \If{$f_\mathrm{fin}$.\texttt{epoch} $\neq e_\mathrm{chk}$} \State \Return \EndIf \EndFor \State $d \gets$ new SF for accumulation \For{$i \in \{1, \ldots, T\}$} \label{line:accumulate} \State $f_\mathrm{fin} \rlxmove$ \texttt{sfFin}[t] \State $d\mathtt{.data} \gets d\mathtt{.data} \circ f_\mathrm{fin}\mathtt{.data}$ \State $d\mathtt{.num} \gets d\mathtt{.num} + f_\mathrm{fin}\mathtt{.num}$ \EndFor \If{\Call{checkForStop}{$d$}} \label{line:convergence} \State $\mathtt{stop} \rlxmove$ \textsf{true} \label{line:do-stop} \EndIf \State $inCheck \gets$ \textsf{false} \EndProcedure \end{algorithmic} \end{minipage} \end{algorithm} Algorithm~\ref{algo:epoch-based} states the pseudocode of our framework. By $\rlxmove$, $\loadacq$ and $\storerel$, we denote relaxed memory access, \texttt{load-acquire} and \texttt{store-release}, respectively (see Sections~\ref{sub:basic-defs} and Appendix~\ref{app:c11_model}). In the algorithm, each thread maintains an epoch number $e_\mathrm{sam}$. To be able to check the stopping condition, thread 0 maintains another epoch number $e_\mathrm{chk}$. Indeed, thread 0 is the only thread that evaluates the stopping condition (in \textsc{checkFrames}) after accumulating the SFs from all threads. \textsc{checkFrames} determines whether there is an ongoing check for the stopping condition ($inCheck$ is \textsf{true}; line~\ref{line:check-cycle}). If that is not the case, a check is initiated (by incrementing $e_\mathrm{chk}$) and all threads are signaled to advance to the next epoch (by updating $\texttt{epochToRead}$). Afterwards, \textsc{checkFrames} only continues if all threads $t$ have published their SFs for checking (i.\,e.,\xspace $\mathtt{sfFin}[t]$ points to a SF of epoch $e_\mathrm{chk}$; line~\ref{line:check-frames}). Once that happens, those SFs are accumulated (line~\ref{line:accumulate}) and the stopping condition is checked on the accumulated data (line~\ref{line:convergence}). Eventually, the termination flag (\texttt{stop}; line~\ref{line:do-stop}) signals to all threads that they should stop sampling. The main algorithm, on the other hand, performs a loops until this flag is set (line~\ref{line:main-loop}). Each iteration collects one sample and writes the results to the current SF ($f_\mathrm{sam}$). If a thread needs to advance to a new epoch (because an incremented \texttt{epochToRead} is read in line~\ref{line:advance-epoch}), it publishes its current SF to \texttt{sfFin} and starts writing to a new SF ($f_\mathrm{sam}$; line~\ref{line:new-sf}). Note that the memory used by old SFs can be reclaimed (line~\ref{line:reclaim}; however, note that there is no SF for epoch 0). How exactly that is done is left to the algorithms described in later subsections. In the remainder of this subsection, we prove the correctness of our approach. \begin{proposition} Algorithm~\ref{algo:epoch-based} always checks the stopping condition on a consistent state; in particular, the epoch-based approach is correct. \end{proposition} \begin{proof} The order of lines~\ref{line:publish} and~\ref{line:new-sf} implies that no thread $t$ issues a store to a SF $f$ which it already published to $\mathtt{sfFin}[t]$. Nevertheless, we need to prove that all stores by thread $t$ are visible to \Call{checkFrames}{} before the frames are accumulated. \Call{checkFrames}{} only accumulates $f.\mathtt{data}$ after $f$ has been published to $\mathtt{sfFin}[t]$ via the \texttt{store-relase} in line~\ref{line:publish}. Furthermore, in line~\ref{line:subscribe}, \Call{checkFrames}{} performs at least one \texttt{load-acquire} on $\mathtt{sfFin}[t]$ to read the pointer to $f$. Thus, all stores to $f$ are visible to \Call{checkFrames}{} before the accumulation in line~\ref{line:accumulate}. The proposition now follows from the fact that $\circ$ is associative, so that line~\ref{line:accumulate} indeed produces a SF that occurs in some sequential execution. \qed \end{proof} \subsection{Local-frame and Shared-frame Algorithm} We present two epoch-based algorithms relying on the general framework from the previous section: namely, the \emph{local-frame} and the \emph{shared-frame} algorithm. Furthermore, in Appendix~\ref{app:deterministic}, we present the deterministic indexed-frame algorithm (as both local-frame and shared-frame are non-deterministic). Local-frame and shared-frame are both based on the pseudocode in Algorithm~\ref{algo:epoch-based}. They differ, however, in their allocation and reuse (in line~\ref{line:reclaim} of the code) of SFs. The local frame algorithm allocates one pair of SFs per thread and cycles through both SFs of that pair (i.\,e.,\xspace epochs with even numbers are assigned the first SF while odd epochs use the second SF). This yields a per-thread memory requirement of $\mathcal{O}(n)$; as before, $n$ denotes the size of the sampling state. The shared-frame algorithm reduces this memory requirement to $\mathcal{O}(1)$ by only allocating $F$ pairs of SFs in total, for a constant number $F$. Thus, $T/F$ threads share a SF in each epoch and atomic \texttt{fetch-add} operations need to be used to write to the SF. The parameter $F$ can be used to balance the memory bandwidth and synchronization costs -- a smaller value of $F$ lowers the memory bandwidth required during aggregation but leads to more cache contention due to atomic operations. \subsection{Synchronization Costs} \label{sub:sync-costs} In Algorithm~\ref{algo:epoch-based}, all synchronization of threads $t > 0$ is done wait-free in the sense that the threads only have to stop sampling for $\Theta(1)$ instructions to communicate with other threads (i.\,e.,\xspace to check $\mathtt{epochToRead}$, update per-thread state and write to $\mathtt{sfFin}[t]$). At the same time, thread $t = 0$ generally needs to check all $\mathtt{sfFin}$ pointers. Taken together, this yields the following statement: \begin{proposition} In each iteration of the main loop, threads $t > 0$ of local-frame and shared-frame algorithms spend $\Theta(1)$ time to wait for other threads. Thread $t = 0$ spends up to $\mathcal{O}(T)$ time to wait for other threads. \end{proposition} In particular, the synchronization cost does not depend on the problem instance -- this is in contrast to the OpenMP parallelization in which threads can idle for $\mathcal{O}(\mathcal{S})$ time, where $\mathcal{S}$ denotes the time complexity of a sampling operation (e.\,g.,\xspace $\mathcal{S} = \mathcal{O}(|V| + |E|)$ in the case of \tool{KADABRA}\xspace). Nevertheless, this advantage in synchronization costs comes at a price: the accumulation of the sampling data requires additional evaluations of $\circ$. $\mathcal{O}(Tn)$ evaluations are required in the local-frame algorithm, whereas shared-frame requires $\mathcal{O}(Fn)$. No accumulation is necessary in the OpenMP baseline. As can be seen in Algorithm~\ref{algo:epoch-based}, we perform the accumulation in a single thread (i.\,e.,\xspace thread 0). Compared to a parallel implementation (e.\,g.,\xspace using parallel reductions), this strategy requires no additional synchronization and has a favorable memory access pattern (as the SFs are read linearly). A disadvantage, however, is that there is a higher latency (depending on $T$) until the algorithm detects that it is able to stop. Appendix~\ref{app:latency} discusses how a constant latency can be achieved heuristically. \section{Experiments} \label{sec:experiments} \nprounddigits{1} \begin{figure}[t]% \begin{subfigure}[t]{0.48\textwidth}% \includegraphics{plots/original-su} \caption{Average speedup (preprecessing + ADS, geom.\ mean) of OpenMP baseline over the original sequential implementation of \tool{KADABRA}\xspace} \label{fig:original_su} \end{subfigure}\hspace{0.04\textwidth}% \begin{subfigure}[t]{0.48\textwidth}% \includegraphics{plots/time_bar} \caption{Breakdown of sequential \tool{KADABRA}\xspace running times into preprocessing and ADS (in percent) on instances orkut-links~(O), wikipedia\_link\_de~(W), and dimacs9-COL~(D)} \label{fig:rt_split} \end{subfigure}% \caption{Performance of OpenMP baseline} \end{figure} \begin{figure}[t] \begin{subfigure}[t]{0.48\textwidth}% \includegraphics{plots/adaptive-su} \caption{Average ADS\xspace speedup (geom.\ mean) of epoch-based algorithms over sequential OpenMP baseline} \label{fig:adaptive_su} \end{subfigure}\hspace{0.04\textwidth}% \begin{subfigure}[t]{0.48\textwidth}% \includegraphics{plots/time_memory} \caption{Average ADS\xspace speedup (over 36-core local-frame, geom.\ mean) and memory consumption of shared-frame, depending on the number of SFs} \label{fig:mem_rt} \end{subfigure}% \caption{Performance of epoch-based algorithms} \end{figure} The platform we use for our experiments is a Linux server equipped with 1.5TB RAM and two Intel Xeon Gold 6154 CPUs with 18 cores (for a total of 36 cores) at 3.00GHz. Each thread of the algorithm is pinned to a unique core; hyperthreading is disabled. Our implementation\footnote{When this paper is accepted, we are keen to publish our code on the \tool{NetworKit}\xspace GitHub repository: \url{https://github.com/kit-parco/networkit}. In the meantime, our code is available at \url{https://gist.github.com/angriman/cfb729c1c369198b8a1a36aad1f52fcc}.} is written in C++ building upon the \tool{NetworKit}\xspace toolkit~\cite{staudt2016networkit}. We use 27\xspace undirected real-world graphs in the experiments (see Appendix~\ref{app:instances} for more details). The error probability for \tool{KADABRA}\xspace is set to $\delta = 0.1$ for all experiments. Absolute running times of our experiments are reported in Appendix~\ref{app:raw-data}. In a first experiment, we compare our OpenMP baseline against the original implementation of \tool{KADABRA}\xspace (see Section~\ref{sub:first-attempts} for these two approaches). We set the absolute approximation error to $\epsilon = 0.01$. The overall speedup (i.\,e.,\xspace both preprocessing and ADS) is reported in Figure~\ref{fig:original_su}. The results show that our OpenMP baseline outperforms the original implementation considerably (i.\,e.,\xspace by a factor of $\numprint{6.89846099713451}\times$), even in a single-core setting. This is mainly due to implementation tricks (see Appendix~\ref{app:impr_seq}) and parameter tuning (as discussed in Appendix~\ref{app:tuning}). Furthermore, for 32 cores, our OpenMP baseline performs $\numprint{13.544344311728693}\times$ better than the original impelementation of \tool{KADABRA}\xspace\ -- or $\numprint{22.66442436431893}\times$ if only the ADS phase is considered. Hence, for the remaining experiments, we discard the original implementation as a competitor and focus on the parallel speedup of our algorithms. To understand the relation between the preprocessing and ADS phases of \tool{KADABRA}\xspace, we break down the running times of the OpenMP baseline in Figure~\ref{fig:rt_split}. In this figure, we present the fraction of time that is spent in ADS on three exemplary instances and for different values of $\epsilon$. Especially if $\epsilon$ is small, the ADS running time dominates the overall performance of the algorithm. Thus, improving the scalability of the ADS phase is of critical importance. For this reason, we neglect the preprocessing phase and only consider ADS when comparing to our local-frame and shared-frame algorithms. In Figure~\ref{fig:adaptive_su}, we report the parallel speedup of the ADS phase of our epoch-based algorithms relative to the OpenMP baseline. All algorithms are configured to check the stopping condition after a fixed number of samples (see Appendix~\ref{app:latency} for details). The number $F$ of SF pairs of shared-frame has been configured to $2$, which we found to be a good setting for $T = 32$. On 32 cores, local-frame and shared-frame achieve parallel speedups of $\numprint{15.879639123178466}\times$ and $\numprint{18.081267115073295}$; they both significantly improve upon the OpenMP baseline, which can only achieve a parallel speedup of $\numprint{6.27616521473421}\times$ (i.\,e.,\xspace local-frame and shared-frame are $\numprint{2.5301499530156892}\times$ and $\numprint{2.8809418644086824}\times$ faster, respectively; they also outperform the original implementation by factors of $\numprint{57.34439224050918}$ and $\numprint{65.29488898389054}$, respectively). The difference between local-frame and shared-frame is insignificant for lower numbers of cores; this is explained by the fact that the reduced memory footprint of shared-frame only improves performance once memory bandwidth becomes a bottleneck. For the same reason, both algorithms scale very well until 16 cores; due to memory bandwidth limitations, this nearly ideal scalability does not extend to 32 cores. This bandwidth issue is known to affect graph traversal algorithms in general~\cite{bader2005architectural,lumsdaine2007challenges}. The indexed-frame algorithm is not as fast as local-frame and shared-frame on the instances depicted in Figure~\ref{fig:adaptive_su}: it achieves a parallel speedup of $\numprint{10.818961592817336}\times$ on 32 cores. However, it is still considerably faster than the OpenMP baseline (by a factor of $\numprint{1.7238172072682618}\times$). There are two reasons why the determinism of indexed-frame is costly: index-frame has similar bandwidth requirements as local-frame; however, it has to allocate more memory as SFs are buffered for longer periods of time. On the other hand, even when enough samples are collected, the stopping condition has to be checked on older samples first, while local-frame and shared-frame can just check the stopping condition on the most recent sampling state. In a final experiment, we evaluate the impact of the parameter $F$ of shared-frame on its performance. Figure~\ref{fig:mem_rt} depicts the results. The experiment is done with 36 cores; hence memory pressure is even higher than in the previous experiments. The figure demonstrates that in this situation, minimizing the memory bandwith requirements at the expense of synchronization overhead is a good strategy. Hence for larger numbers of cores, we can minimize memory footprint and maximize performance at the same time. \npnoround \section{Related Work} \label{sec:related_work} Our parallelization strategy can be applied to arbitrary ADS\xspace algorithms. ADS\xspace was first introduced by Lipton and Naughton to estimate the size of the transitive closure of a digraph~\cite{lipton1989estimating}. It is used in a variety of fields, e.\,g.,\xspace in statistical learning~\cite{provost1999efficient}. In the context of BC\xspace, ADS\xspace has been used to approximate distances between pairs of vertices of a graph~\cite{oktay2011distance}, to approximate the BC\xspace values of a graph~\cite{bader2007approximating,riondato2018abra,borassi2016kadabra} and to approximate the BC\xspace value of a single vertex~\cite{chehreghani2018novel}. An analogous strategy is exploited by Mumtaz and Wang~\cite{mumtaz2017identifying} to find approximate solutions to the group betweenness maximization problem. Regarding more general (i.\,e.,\xspace not necessarily ADS\xspace) algorithms for BC\xspace, a survey from Matta et al.\xspace~\cite{matta2019comparing} provides a detailed overview of the the state of the art. The \tool{RK}\xspace~\cite{riondato2016fast} algorithm represents the leading non-adaptive sampling algorithm for BC\xspace approximation; \tool{KADABRA}\xspace was shown to be 100 times faster than \tool{RK}\xspace in undirected real-world graphs, and 70 times faster than \tool{RK}\xspace in directed graphs\cite{borassi2016kadabra}. McLaughlin and Bader~\cite{mclaughlin2014scalable} introduced a work-efficient parallel algorithm for BC\xspace approximation, implemented for single- and multi-GPU machines. Madduri et al.\xspace~\cite{madduri2009faster} presented a lock-free parallel algorithm optimized for specific architectures to approximate or compute BC\xspace exactly in massive networks. The SFs used by our algorithms are concurrent data structures that enable us to minimize the synchronization latencies in multithread environments. Devising concurrent (lock-free) data structures that scale over multiple cores is not trivial and much effort has been devoted to this goal~\cite{boyd2010analysis,michael2004hazard}. A well-known solution is the Read-Copy-Update mechanism (RCU); it was introduced to achieve high multicore scalability on read-mostly data structures~\cite{mckenney1998read}, and was leveraged by several applications~\cite{arbel2014concurrent,clements2012scalable}. Concurrent hash tables~\cite{DBLP:conf/sosp/DavidGT13} are another popular example. \section{Conclusions and Future Work} \label{sec:concl} In this paper, we found that previous techniques to parallelize ADS algorithms are insufficient to scale to large numbers of threads. However, significant speedups can be achieved by employing adequate concurrent data structures. Using such data structures and our epoch mechanism, we were able to devise parallel ADS algorithms that consistently outperform the state of the art but also achieve different trade-offs between synchronization costs, memory footprint and determinism of the results. Regarding future work, a promising direction for our algorithms is parallel computing with distributed memory; here, the stopping condition could be checked via (asynchronous) reduction of the SFs. In the case of BC\xspace this, might yield a way to avoid bottlenecks for memory bandwidth on shared-memory systems. \bibliographystyle{splncs04}

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\section{Proof of the First Zonklar Equation} \ifCLASSOPTIONcaptionsoff \newpage \fi \bibliographystyle{IEEEtran} \section{Combining Frame-Synchronous and Label-Synchronous Systems} \label{sec:combine} \begin{figure*}[t] \centering \includegraphics[width=0.7\linewidth]{fig/pipeline.pdf} \caption{Pipeline for combining a frame-synchronous system and a label-synchronous system. Rounded box indicates trainable models and rectangular box indicates an algorithm or a procedure.} \label{fig:isca} \end{figure*} \subsection{Motivation} \label{ssec:motivation} Frame-synchronous and label-synchronous systems exhibit distinctive advantages due to their different modelling and decoding strategies. Frame-synchronous systems such as NN-HMMs and CTC assume the calculations of different $P(s_t|\mathbf{x}_t)$ are independent. In contrast, label-synchronous systems such as AED models calculates $P(\mathcal{C}|\mathcal{O})$ based on \eqn{seqprob} without making any label independence assumption. Consequently, frame-synchronous systems with phone-based acoustic models can easily incorporate structured phonetic, linguistic and contextual knowledge by using a lexicon. This is very helpful for the robustness of the ASR systems such as rare words. Because of the label independence assumption, NN-HMMs and CTC can produce a rich and compact graph-based representation of the hypothesis space for each utterance such as a lattice and a confusion network. Recent studies show that by limiting the history context, \emph{i.e.} imposing some degree of label independence assumption, RNN-Ts can also effectively generate lattices with comparable performance to the full-context RNN-Ts~\cite{Prabhavalkar2020LessIM}. It is more difficult for label-synchronous systems to generate lattices since the previous decoding output needs to be fed into the auto-regressive neural decoder to obtain the probability distribution over the next token. The hypothesis space explored by label-synchronous systems is normally limited to the top $N$ candidates, which is significantly less than lattices from frame-synchronous systems. Although it is more challenging to adapt AED models to process streaming data due to its label-synchronous nature~\cite{Miao2020OnlineHC,Moritz2020StreamingAS}, AED models can jointly model acoustic and textual information and the neural decoder can model long-range dependencies across labels without the independence assumption. Therefore, it is desirable to leverage the advantages of both types of systems and exploit their complementarity to improve the performance of the final ASR system. In this work, we propose a generic two-pass ASR combination method, which uses the frame-synchronous systems to perform the first-pass decoding and uses the label-synchronous system to rescore the first-pass results. In the first-pass decoding with NN-HMMs or CTC systems, the model can process streaming data, leverage structured knowledge in particular contextual knowledge, and recover from search errors made in early decoding steps. In the second-pass rescoring with AED systems, complete acoustic and language information from each entire utterance can be used jointly to refine the estimation of hypothesis probabilities. Furthermore, since NN-HMM or CTC systems prune most of the less possible hypotheses, rescoring with AED can be more robust and efficient than decoding with AED models. \subsection{Combination Framework} \label{ssec:framework} As shown in \fig{isca}, the acoustic features of an utterance are first passed through the acoustic model of the frame-synchronous system. Together with a language model and a lexicon, the frame-synchronous decoder generates the hypotheses in the form of an $N$-best list or a lattice with corresponding acoustic and language model scores. The same acoustic features can be forwarded through the neural encoder of the label-synchronous system. The neural decoder then uses the hidden representations of the acoustic signal and scores the hypotheses from the first-pass similar to the training mode. Finally, applying either $N$-best rescoring or lattice rescoring with tuned interpolation coefficients yields the best hypothesis of the combined system. Generically, if a frame-synchronous system $P_f$ and a label-synchronous system $P_l$ are available, the final score of a hypothesis $\mathcal{W}$ for an utterance $\mathcal{O}$ is \begin{align} \mathcal{S}(\mathcal{W}|\mathcal{O}) = \log P_f(\mathcal{W}|\mathcal{O}) + \alpha \log P_l(\mathcal{W}|\mathcal{O}) \end{align} where $\alpha$ is the interpolation coefficient. For HMM-based frame-synchronous systems, AM scores and $n$-gram LM scores are available as in \eqn{scmodel}. When combining with label-synchronous systems, the LM scaling factor can also be tuned together with other interpolation coefficients. For example, when rescoring with an RNNLM $P_{\text{RNNLM}}$, a label-synchronous system with an RNN decoder $P_{l-\text{RNN}}$, and a label-synchronous system with a Transformer decoder $P_{l-\text{TFM}}$, the final score becomes \begin{align} \mathcal{S}(\mathcal{W}|\mathcal{O}) = & \log p(\mathcal{O}|\mathcal{W}) \nonumber\\ & + \gamma \log P_{\text{$n$-gram}}(\mathcal{W}) + \lambda\log P_{\text{RNNLM}}(\mathcal{W}) \nonumber\\ & + \alpha\log P_{\text{RNN}}(\mathcal{W}|\mathcal{O}) + \beta\log P_{\text{TFM}}(\mathcal{W}|\mathcal{O})\nonumber \\ & + \kappa |\mathcal{W}| \label{eqn:score} \end{align} where $\gamma,\lambda$ are the coefficients for two LMs, $\alpha,\beta$ are the coefficients for two label-synchronous systems, $\kappa$ is the insertion penalty, and $|\mathcal{W}|$ is the number of words in the hypothesis. Based on \eqn{score}, label-synchronous systems can be viewed as audio-grounded language models. For other frame-synchronous systems, corresponding scores from various systems can be interpolated in a similar fashion. In contrast to the proposed framework, standard combination methods for ASR systems are not suitable for combining with label-synchronous systems. For example, confusion network combination~\cite{Evermann2000PosteriorPD} requires decoding lattices and ROVER~\cite{Fiscus1997APS} requires comparable confidence measures from both types of system. The benefits of using hypothesis-level combination may be limited, because the hypothesis space generated from the frame-synchronous system is generally much larger than the label-synchronous system. \begin{algorithm*}[t] \caption{Lattice Rescoring Using a Label-Synchronous Model\protect\footnotemark} \label{alg:latrescore} \begin{algorithmic}[1] \Procedure{LatticeRescore}{lattice, model, utt, ngram, collar} \State cache$\gets$ \textsc{CreateCache}(model, utt, collar) \Comment{Initialise the two-level cache} \For{node $n_i$ in lattice.nodes} \Comment{Initialise expanded nodes and arcs} \State $n_i$.expanded\_nodes $\gets$ []; $n_i$.expanded\_arcs $\gets$ [] \EndFor \State lattice.nodes[0].expanded\_nodes.Append(lattice.nodes[0].Duplicate()) \Comment{Initialise the starting node} \For{node $n_i$ in lattice.nodes} \Comment{Lattice traversal, expansion and rescoring} \For{node $\tilde{n}_j$ in $n_i$.expanded\_nodes} \For{arc $a_k$ in $n_i$.exits} \State $n_k$ $\gets$ $a_k$.dest \State hist $\gets$ \textsc{LastNItems}([$\tilde{n}_j$.hist, $n_k$.word], ngram-1) \If{$\exists$ node $\tilde{n}_l\in n_k$.expanded\_nodes such that $\tilde{n}_l$.hist = hist} \State $\tilde{a}_l$ $\gets$ $a_k$.Duplicate($\tilde{n}_j$, $\tilde{n}_l$) \Comment{Create arc between expanded nodes} \Else \State $\tilde{n}_l$ $\gets$ $n_k$.Duplicate(hist) \Comment{Create expanded node with new history} \State $n_k$.expanded\_nodes.Append($\tilde{n}_l$) \State $\tilde{a}_l$ $\gets$ $a_k$.Duplicate($\tilde{n}_j$, $\tilde{n}_l$) \Comment{Create arc between expanded nodes} \EndIf \State post $\gets$ \textsc{LastNItems}([cache.GetPost($\tilde{n}_j$.hist, $\tilde{n}_j$.time), $\tilde{a}_l$.post], ngram-1) \If{cache.Lookup(hist, $n_k$.time) fails} \State cache.Renew($\tilde{n}_j$.hist, $\tilde{n}_j$.time, $\tilde{n}_l$.hist, $\tilde{n}_l$.time, post) \Comment{Cache miss -- new ngram or timestamp} \ElsIf{\textsc{Sum}(post) $>$ \textsc{Sum}(cache.GetPost($\tilde{n}_l$.hist, $\tilde{n}_l$.time))} \State cache.Renew($\tilde{n}_j$.hist, $\tilde{n}_j$.time, $\tilde{n}_l$.hist, $\tilde{n}_l$.time, post) \Comment{Cache update -- use the more likely path} \EndIf \State $\tilde{a}_l$.model\_score $\gets$ cache.GetPred($\tilde{n}_j$.hist, $\tilde{n}_j$.time, $\tilde{n}_l$.word) \Comment{Assign score from label-sync model} \State $\tilde{n}_j$.exits.Append($\tilde{a}_l$); $\tilde{n}_l$.entries.Append($\tilde{a}_l$) \Comment{Connect new arc to expanded nodes} \EndFor \EndFor \EndFor \State lattice $\gets$ \textsc{BuildExpandedLattice}(lattice) \Comment{Construct new lattice from expanded nodes and arcs} \EndProcedure \end{algorithmic} \end{algorithm*} \subsection{Training} \label{ssec:training} The frame-synchronous system and the label synchronous system can be trained separately or jointly in a multi-task fashion by sharing the neural encoder with the acoustic model. For multi-task trained models~\cite{Watanabe2017HybridCA}, the total number of parameters in the entire system is smaller due to parameter sharing. Although multi-task training can be an effective way of regularisation, setting the interpolation weights between the two losses and configuring the learning rate to achieve good performance for both models may not be straightforward. Moreover, multi-task training also limits the model architectures or model-specific training techniques that can be adopted for individual systems. For example, the acoustic model can be a unidirectional architecture for streaming purposes but the neural encoder for the label-synchronous system can be bi-directional for second-pass rescoring. Acoustic models in frame-synchronous systems normally have a frame subsampling rate of 3 in a low frame-rate system~\cite{Pundak2016LowerFR,Povey2018SemiOrthogonalLM} but neural encoders normally have a frame rate reduction of 4, by using convolutional layers~\cite{Gulati2020ConformerCT} or pyramidal RNNs~\cite{Chan2016ListenAA}, for better performance. Frame-level shuffling~\cite{Su2013ErrorBP} is important for the optimisation of HMM-based acoustic models, whereas label-synchronous systems have to be trained on a per utterance basis. Triphone units are commonly used for HMM-based acoustic models whereas word-pieces are widely used for attention-based models~\cite{Chiu2018StateoftheArtSR}. Overall, sharing the acoustic model and the neural encoder in a multi-task training framework hampers both systems from reaching the best possible performance. Therefore, this work will focus on different systems that are trained separately. \subsection{N-best Rescoring} \label{ssec:nbest} The frame-synchronous system generates the top $N$ hypotheses for each utterance. The word sequence $\mathcal{W}$ can be tokenised into the set of word-pieces modelled by the label-synchronous system. The word-piece sequence $\mathcal{C}$ can be forwarded through the neural decoder to obtain the probability for each token $P(c_t|c_{1:t-1},\mathcal{O})$. By tuning the interpolation coefficients in \eqn{score} to have the lowest WER on the development set, the final hypothesis is the one with the highest score among the top $n$ candidates. \begin{equation} \mathcal{W}^{*} = \argmax_{\mathcal{W}} \mathcal{S}(\mathcal{W}|\mathcal{O}) \end{equation} When rescoring the $N$-best hypotheses with a label-synchronous system with an RNN-based decoder, the time complexity is $O(L)$ and the space complexity is $O(1)$ because of the sequential nature of RNNs. In contrast, a Transformer-based decoder has a time complexity of $O(1)$ and space complexity of $O(L^2)$ as during Transformer training. Since the entire hypothesis is available, self-attention can be directly computed across the whole sequence for each token. \subsection{Improved Lattice Rescoring} \label{ssec:lattice} \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{fig/lattice_exp.pdf} \caption{Example of a 2-gram lattice expanding to a 3-gram lattice. The hollow node from the left is expanded into two hollow nodes on the right, which corresponds to line 16-18 in \alg{latrescore}.} \label{fig:latexp} \end{figure} \footnotetext{Our code is available at \url{https://github.com/qiujiali/lattice-rescore}.} For a fixed number of candidate hypotheses, the number of alternatives per word in the sequence is smaller when the hypothesis becomes longer. This means the potential for improvement diminishes for longer utterances. Therefore, in order to have the same number of alternatives per word, the size of $N$-best lists needs to grow exponentially with respect to the utterance length. However, lattice rescoring can effectively mitigate this issue. Lattices are directed acyclic graphs where nodes represent words and edges represent associated acoustic and language model scores. A complete path from the start to the end of a lattice is a hypothesis. Because a various number of arcs can merge to or split from a node, a lattice generally contains a far greater number of hypotheses than a limited $N$-best list. The size of lattices is measured by the number of arcs per second of speech, also known as the \emph{lattice density}. One commonly used lattice rescoring approach for RNNLMs is on-the-fly lattice expansion with $n$-gram based history clustering~\cite{Liu2016TwoEL}. Similar to RNNLMs, attention-based models are auto-regressive models where the current prediction depend on all history tokens. This means approximations must be made when assigning scores on edges of lattices because each word in the lattice may have numerous history sequences. Although lattices can be expanded to allow each word to have a more unique history, a trade-off between the uniqueness of the history and computational efficiency need to be considered. $n$-gram based history clustering~\cite{Liu2016TwoEL} assumes that the history before the previous $n-1$ words has little impact on the probability of the current word. As illustrated in \fig{latexp}, a lattice can be expanded such that the $n-1$ history words of each word in the lattice are unique. During rescoring, a hash table-based cache is created, where the key is the $n-1$ history words and the value is the corresponding hidden state of RNNLM and the output distribution. When the same history appears again during rescoring, repetitive computation can be avoided, regardless of the more distant history. However, for label-synchronous systems based on attention mechanisms, $n$-gram based history clustering can lead to undesirable behaviour. For example, ``\texttt{i think those are are wonderful things to have but i think in a big company}'' (from SwitchBoard), the phrase ``\texttt{i think}'' appears twice in the utterance. If the original trigram based history clustering is used, at the second occurrence of the phrase, the algorithm will restore the cache from the first occurrence including the attention context and the decoder state from the first occurrence and then continue to score the rest of the utterance. Consequently, scores for the second half of the utterance will be wrong because of the incorrect attention context. To this end, a time-dependent two-level $n$-gram cache is proposed. When looking up the history phrase in the cache, a secondary level cache indexed by the corresponding frame number of the word is used. A collar of $\pm9$ frames is used when looking up the secondary cache to accommodate a small difference in alignment. More specifically, when looking up the cache, the timestamp of the current word is also used. The cache is only hit when the timestamp falls within the vicinity of one of the timestamps in the cache. Otherwise, the cache is missed and a new entry is created in the cache with the timestamp and the corresponding decoder states (line 22 in \alg{latrescore}). For all the functions related to the cache, a timestamp must be provided. Another key detail is that when there is a cache hit, the sum of arc posteriors of the current $(n-1)$-gram is compared with the one stored in the cache. If the current posterior is larger, indicating the current $(n-1)$-gram is on a better path, then the cache entry is updated to store the current hidden states (line 24 in \alg{latrescore}). For the example in \fig{latexp}, when the lower path is visited after the upper path, the cache entry ``\texttt{sat on}'' should already exist. If the lower path has a higher posterior probability, then the cache entry will be updated, so that future words in the lattice will adopt the history from the lower path. For label-synchronous systems with RNN decoders, lattice rescoring has $O(L)$ for time complexity and $O(1)$ for space complexity as the RNN hidden states can be stored and carried forward at each node in the lattice. However, since lattice rescoring operates on partial hypotheses, Transformer decoders have to run in the decoding mode as in \tbl{complexity}. Because self-attention need to be computed with all previous tokens, lattice rescoring with the Transformer-based decoder has $O(L^2)$ time and space complexities. \section{Conclusion} \label{sec:conclusion} In this paper, we have proposed to combine frame-synchronous with label-synchronous systems in a two-pass manner. Frame-synchronous systems are used as the first pass, which can process streaming data and integrate structured knowledge such as lexicon. Label-synchronous systems are viewed as audio-grounded language models to rescore hypotheses from the first pass. Since the two highly complementary systems are integrated at the word level, each one can be trained independently for optimal performance. An improved Lattice rescoring algorithm is proposed for label-synchronous systems, which generally outperform $N$-best rescoring. Label-synchronous systems with RNN decoders are better suited for lattice rescoring while Transformer decoders are more time-efficient for $N$-best rescoring. On AMI and Switchboard datasets, the WERs of the combined systems are around 30\% relatively lower than individual systems. \section{Experimental Results} \label{sec:exp} \begin{table*}[t] \caption{WERs on AMI-IHM Eval Set Using $N$-Best and Lattice Rescoring With Various Combination of RNNLM and Two Label-Synchronous Systems. Lattice Density (Number of Arcs per Second) Is Provided in Square Brackets.} \centering \begin{tabular}{cccccccccc} \toprule \multirow{2}{*}{RNNLM} & \multirow{2}{*}{label-sync-LSTM} & \multirow{2}{*}{label-sync-TFM} && 20-best & 100-best & 500-best && 4-gram lattice & 5-gram lattice\\ & & && [34.4] & [146.9] & [595.2] && [313.3] & [606.0]\\ \midrule \checkmark & & && 16.9 & 16.5 & 16.3 && 16.3 & 16.3 \\ & \checkmark & && 16.2 & 15.8 & 15.5 && 15.3 & 15.3\\ & & \checkmark && 16.3 & 15.7 & 15.4 && 15.3 & 15.3\\ \midrule \checkmark & \checkmark & && 15.4 & 14.7 & 14.3 && 14.1 & 14.1\\ \checkmark & & \checkmark && 15.5 & 14.8 & 14.3 && 14.2 & 14.1\\ \midrule \checkmark & \checkmark & \checkmark && 15.3 & 14.5 & 14.1 && \textbf{13.7} & 13.8\\ \bottomrule \end{tabular} \label{tab:ami_combine} \end{table*} \begin{table*}[t] \caption{WERs on Hub5'00 and RT03 Using 500-Best Rescoring and Lattice Rescoring (5-gram Approximation) With Various Combination of RNNLM and Two Label-Synchronous Systems. Lattice Density (Number of Arcs per Second) Is Provided in Square Brackets.} \centering \begin{tabular}{ccccccccc} \toprule \multirow{2}{*}{RNNLM} &\multirow{2}{*}{label-sync-LSTM} & \multirow{2}{*}{label-sync-TFM} && \multicolumn{2}{c}{Hub5'00 (SWB/CHM)} & & \multicolumn{2}{c}{RT03 (SWBC/FSH)}\\[-0.2em] \cmidrule{5-6}\cmidrule{8-9} & & && 500-best [760.7] & 5-gram lattice [406.8] & & 500-best [721.2] & 5-gram lattice [496.2] \\ \midrule \checkmark & & && 6.8 / 14.3 & 6.8 / 14.7 && 16.1 / 9.4 & 16.2 / 9.6 \\ & \checkmark & && 6.5 / 13.3 & 6.5 / 13.2 && 15.2 / 9.0 & 14.7 / 8.8 \\ & & \checkmark && 6.4 / 12.9 & 6.3 / 12.7 && 14.9 / 8.6 & 14.6 / 8.4\\ \midrule \checkmark & \checkmark & && 5.9 / 12.7 & 5.8 / 12.7 && 14.3 / 8.2 & 13.8 / 7.9\\ \checkmark & & \checkmark && 5.8 / 12.6 & 5.9 / 12.3 && 14.2 / 8.0 & 13.9 / 8.0\\ \midrule \checkmark & \checkmark & \checkmark && 5.8 / 12.4 & \textbf{5.7 / 12.1} && 14.3 / 8.0 & \textbf{13.2 / 7.6}\\ \bottomrule \end{tabular} \label{tab:swbd_combine} \end{table*} \subsection{N-best and Lattice Rescoring} After pruning the lattices generated by frame-synchronous systems by limiting the beam width and the maximum lattice density, 20-best, 100-best and 500-best hypotheses are obtained from these lattices. The $N$-best hypotheses are then forwarded through the RNNLM, label-sync-LSTM and label-sync-TFM. Each hypothesis has five scores, i.e. AM and LM scores, RNNLM score, and scores from two label-sync systems. By following \eqn{score}, the five interpolation coefficients are found by using the covariance matrix adaptation evolution strategy (CMA-ES)~\cite{Hansen2001CompletelyDS} as a black-box optimisation algorithm to minimise the WER on the dev set. By applying the optimal combination coefficients on the test set, the hypotheses with the highest score are picked. For AMI-IHM, by comparing columns 4, 5 and 6 in \tbl{ami_combine} along each row, the WER consistently decreases as the $N$-best list size increases. However, the WER improvement from 100 to 500-best is smaller than from 20 to 100-best, i.e. the gain from increasing $N$ reduces for larger $N$. For lattice rescoring (columns 7 and 8 in \tbl{ami_combine}), the improvement from using a higher-order $n$-gram approximation is marginal whereas the lattice density nearly doubles from 4-gram to 5-gram. For both $N$-best and lattice rescoring, \tbl{ami_combine} shows that the WER is lower when combining scores from more models. Furthermore, if just an additional model is to be used for combination with a frame-synchronous system as in the first block of \tbl{ami_combine}, using a label-synchronous model seems to be more effective than an RNNLM, because a label-synchronous model can be viewed as an audio-grounded language model. The WER of the combined system is 25-29\% relative lower compared to a single system in \tbl{single_wer}. With similar lattice densities, lattice rescoring with a 5-gram approximation has a 2\% relative WER reduction over 500-best rescoring. For SWB-300, 500-best rescoring and lattice rescoring with a 5-gram approximation are reported for Hub5'00 and RT03 sets in \tbl{swbd_combine}. For RT03, the final combined system using lattice rescoring reduces WER by 19-33\% relative compared to single systems in \tbl{single_wer}. Although the 500-best has greater lattice density than the expanded 5-gram lattice, lattice rescoring has 5-8\% relative WER reduction over 500-best rescoring. This is not unexpected because SWB-300 has longer utterances than AMI-IHM on average as shown in \tbl{data}. \subsection{Analysis} \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{fig/werr_by_length.pdf} \vspace{-1em} \caption{Relative WER reduction by utterance length on RT03 for various rescoring methods.} \label{fig:werr} \end{figure} As discussed in \sect{combine}, lattices are a more compact representation of the hypothesis space that scales well with the utterance length. In \fig{werr}, the relative WER reductions (WERRs) by using 20-best rescoring, 500-best rescoring and lattice rescoring with a 5-gram approximation are compared for different utterance lengths measured by the number of words in the reference. As expected, the gap between $N$-best rescoring and lattice rescoring widens as the utterance length increases. The number of alternatives per word represented by $N$-best is smaller for longer utterances, which explains the downward trend in \fig{werr} for 20-best and 500-best. However, the number of alternatives per word for lattices is constant for a given lattice density. Therefore, the WERR from lattice rescoring does not drop even for very long utterances. \sect{lsync} compared the complexities of using RNN and Transformer decoders for label-synchronous models. Based on our implementation, speed disparities between the two types of decoders are significant. For 500-best rescoring, label-sync-TFM is about four times faster than label-sync-LSTM. However, label-sync-LSTM is nearly twice as fast as label-sync-TFM for lattice rescoring with a 5-gram approximation. Explicit comparisons between $N$-best and lattice rescoring, and between RNNLM and label-synchronous system rescoring are not made here as they depend on other factors including implementation, hardware, the degree of parallel computation and the extent of optimisation. For example, representing the $N$-best list in the form of a prefix tree~\cite{Sainath2019TwoPassES} or using noise contrastive estimation~\cite{Chen2016EfficientTA} will significantly accelerate $N$-best rescoring using RNNLM. Under the constraint that no additional acoustic or text data is used, our combined system outperforms various recent results on AMI-IHM~\cite{Kanda2018LatticefreeSM,Sun2021TransformerLM} and SWB-300~\cite{Park2019SpecAugmentAS,Irie2019TrainingLM,Kitza2019CumulativeAF,Wang2020AnIO,Tske2020SingleHA,Saon2021AdvancingRT,Sun2021TransformerLM}. Further improvements are expected if cross-utterance language models or cross-utterance label-synchronous models are used for rescoring~\cite{Irie2019TrainingLM,Tske2020SingleHA,Sun2021TransformerLM}, or combining more and stronger individual systems~\cite{Tuske2021OnTL}. \section{Frame-Synchronous Systems} \label{sec:fsync} Speech recognition can be viewed as a \emph{noisy source-channel} modelling problem~\cite{jelinek1997statistical} where speech is produced and encoded via a noisy channel and the recogniser finds the most probable source text $\mathcal{W}^{*}$ given the observation sequence $\mathcal{O}$. That is, \begin{equation} \mathcal{W}^{*}=\arg\max_{\mathcal{W}}P(\mathcal{W}|\mathcal{O}) \propto \arg\max_{\mathcal{W}}p(\mathcal{O}|\mathcal{W})P(\mathcal{W}), \label{eqn:scmodel} \end{equation} where $p(\mathcal{O}|\mathcal{W})$ is estimated by an AM and $P(\mathcal{W})$ is estimated by an LM \cite{jelinek1997statistical}. With standard HMM AMs, each decoding step consumes one acoustic frame, which makes it frame-synchronous. \subsection{NN-HMM Systems} \label{ssec:hmm} In acoustic modelling, HMMs (as shown in \fig{hmm}) together with NNs, are used to model the generative process of the observation sequences \textit{w.r.t.} the subword units. The model parameters are estimated by maximum likelihood (ML), whose loss function form can be written as \begin{equation} \mathcal{L}_{\text{ML}}=-\ln \sum\nolimits_{\mathcal{S}}\prod\nolimits_{t=1}^{T} P(s_{t+1}|s_t)p(\mathbf{x}_t|s_t), \label{eqn:hmm} \end{equation} where $T$ is the length of the utterance; $\mathcal{S}$ is a collection of all possible HMM state sequences corresponding to the given reference transcription $\mathcal{W}^\text{ref}$, and $[s_{1:T}]\in\mathcal{S}$. $P(s_{t+1}|s_t)$ is the transition probability, which is less important compared to the $p(\mathbf{x}_t|s_t)$ for NN-HMMs. The observation probability $p(\mathbf{x}_t|s_t)$ can be estimated by \begin{equation} p(\mathbf{x}_t|s_t)\propto P(s_t|\mathbf{x}_t)/P(s_t), \label{eqn:convert} \end{equation} where $\mathbf{x}_t$ is the input vector to an NN at time $t$, which is often obtained by stacking a subset of frames in $\mathcal{O}=[\mathbf{o}_{1:T}]$; $P(s_t|\mathbf{x}_t)$ is the output of the NN at $t$ relevant to $s_t$; $P(s_t)$ is the prior probability that can be estimated as the frequency of $s_t$ in the training data. During training, the forward-backward procedure can be used to find the summation in Eqn.~(\ref{eqn:hmm})~\cite{Baum1967FwdBwd}. Alternatively, the summation can be approximated using only one state sequence found by performing state-to-frame alignment with a pre-trained ASR system~\cite{bourlard1994connectionist}. At test-time, by modifying \eqn{scmodel}, the AM and LM scores are combined based on a log-linear interpolation with an LM weight in decoding. \begin{figure}[t] \centering \subfloat[]{\includegraphics[height=0.42in]{fig/HMM.pdf}% \label{fig:hmm}} \hfil \subfloat[]{\includegraphics[height=0.5in]{fig/CTC.pdf}% \label{fig:ctc}} \caption{(a) is a single-state HMM. (b) is a CTC-equivalent HMM topology. The white and grey circles are emission and non-emission states, and $\varnothing$ is the blank symbol.} \label{fig:hmm&ctc} \end{figure} \subsection{CTC and RNN-T} \label{ssec:ctc} CTC trains an NN AM with a blank output symbol at the sequence-level without any explicit HMM structure~\cite{Graves2006CTC}. The training loss function is \begin{align} \mathcal{L}_{\text{CTC}}=-\ln \sum\nolimits_{\mathcal{S}}\prod\nolimits_{t=1}^{T}P(s_t|\mathbf{x}_t), \label{eqn:ctc} \end{align} where $\mathcal{S}$ collects all possible symbol sequences that can map to $\mathcal{W}^\text{ref}$ by removing repeated symbols and blanks. During training, alignments and losses are computed by the forward-backward procedure. By comparing \eqn{ctc} to \eqn{hmm}, CTC is equivalent to a special instantiation of the two-state HMM structure when $P(s_t)$ and ${P(s_{t+1}|s_t)}$ are constant for any state~\cite{Hadian2018EndtoendSR}. As shown in \fig{ctc}, the first emission state of the HMM is the skippable blank state with a self-loop and the second state corresponds to the subword unit. The blank state is shared across all HMMs. Therefore, it is reasonable to view CTC as a special HMM-based AM, which combines the HMM topology, the ML training loss and the forward-backward procedure. Although CTC is often used to model grapheme-based units to avoid using a complex decoder, CTC systems can have much better performance by re-introducing phone-based output units, lexicon and the frame-synchronous decoder~\cite{Li2019IntegratingSA}. To remove the independence assumption across output tokens in CTC, RNN-transducer was proposed~\cite{Graves2012SequenceTW} where a prediction network and a joint network are added such that each non-blank output token depends on the previous ones. Nevertheless, since the underlying loss computation is the same as CTC and the decoding process runs frame-by-frame, RNN-Ts are also frame-synchronous. Recent work has shown that, with a limited history used for the prediction network, RNN-Ts can be decoded similarly as the HMM-based AM with a lexicon and an LM~\cite{Variani2020HybridAT}. \section{Introduction} \label{sec:intro} \IEEEPARstart{A}{utomatic} speech recognition (ASR) is a task that converts a speech sequence to its corresponding text sequence. Since the number of frames in the input sequence is generally different from the number of labels in the output sequence, the decoding procedure that searches for the most probable label sequence given the speech sequence can be performed either on a per-frame or a per-label basis. Based on this criterion, most widely used ASR systems can be divided into \textit{frame-synchronous} and \textit{label-synchronous} categories. Regarding frame-synchronous systems, an extensively studied and widely adopted ASR approach is the noisy source-channel model~\cite{jelinek1997statistical}. It generally consists of an acoustic model (AM) based on hidden Markov models (HMMs), a language model (LM) and a decoder that searches for the most likely word sequence by incorporating acoustic, phonetic, and lexical information. One of the key characteristics of the noisy source-channel model is that the decoding step is synchronous with the input acoustic frames. The emission probabilities of the HMM-based AMs can be modelled by artificial neural networks (NNs)~\cite{bourlard1994connectionist}. Besides the modular HMM-based systems, there is a long history in the investigation of integrated NN-based systems. A popular approach is called connectionist temporal classification (CTC)~\cite{Graves2006CTC}. CTC can be viewed as a variant of the noisy-source channel model and often uses simpler training and decoding procedures than the NN-HMM systems. By allowing the model to produce blank symbols and collapsing repetitive symbols, CTC outputs frame-asynchronous label sequences through a frame-synchronous decoding procedure. More recently, the recurrent neural network transducer (RNN-T)~\cite{Graves2012SequenceTW} was proposed to improve CTC by removing the independence assumption in output labels by incorporating an additional prediction network in the model structure. Inspired by recent advances in machine translation~\cite{Bahdanau2015NeuralMT}, the attention-based encoder-decoder (AED) models have emerged as another integrated NN-based approach for ASR~\cite{Chorowski2015AttentionBasedMF,Lu2015ASO,Chan2016ListenAA}. Since the attention mechanism captures the alignment between the input frames and output labels, the decoder in AED can operate at a per-label basis, which predicts the current label by attending to multiple input frames and taking the previous labels into account, and is, therefore, a label-synchronous method. There are also various other label-synchronous methods studied in the speech community~\cite{Glass2003CSL,Lee2007AnOO,Zweig2009ASC} and algorithms that allow frame-synchronous systems to operate in a label-synchronous fashion~\cite{Jiang2021VFR}. Due to their different modelling approaches, frame-synchronous and label-synchronous systems are highly complementary \cite{Prabhavalkar2017ACO,Li2019IntegratingSA,Dong2020ACO}. On one hand, frame-synchronous systems, such as HMM-based systems, more naturally handle streaming data and incorporate structured knowledge such as lexicons. Label synchronous systems, on the other hand, are designed to model the acoustic and language information jointly and can serve as an audio-grounded LM. In this paper, we propose combining these two distinctive types of systems in a two-pass fashion to leverage the advantages of each. Overall, The contributions of this paper are as follows: \begin{enumerate} \item proposes a simple and effective combination framework for frame-synchronous and label-synchronous systems using $N$-best and lattice rescoring; \item proposes an improved lattice rescoring algorithm for label-synchronous systems; \item achieves competitive word error rates (WERs) on AMI and Switchboard datasets without additional acoustic and text data. \end{enumerate} In the rest of the paper, related work is first described in \sect{related}. Then \sect{fsync} details frame-synchronous systems and draws connections between HMM-based and CTC acoustic models. \sect{lsync} introduces label-synchronous systems and compares between RNN decoders and Transformer decoders. Motivated by the complementarity of two types of system, \sect{combine} presents details of the two-pass combination framework using $N$-best and lattice rescoring. The experimental setup and results are given in \twosect{setup}{exp}, and the paper concludes in \sect{conclusion}. \section{Label-Synchronous Systems} \label{sec:lsync} An AED model maps a $T$-length input sequence $\mathcal{O}$ to an $L$-length output subword sequence $\mathcal{C}=[c_{1:L}]$ where the input length is normally longer than the output length. Instead of decomposing into AMs and LMs as in \sect{fsync}, the AED model computes the posterior distribution $P(\mathcal{C}|\mathcal{O})$ directly following the chain rule of conditional probability: \begin{equation} P(\mathcal{C}|\mathcal{O})=P(c_1|\mathcal{O})\prod\nolimits_{l=2}^{L}P(c_l|c_{1:l-1},\mathcal{O}), \label{eqn:seqprob} \end{equation} The encoder extracts acoustic features while the decoder generates hypotheses, and the acoustic and language information is jointly learned using a single model without making any independence assumption. The neural decoder processes label-by-label, and $P(\mathcal{C}|\mathcal{O})$ is maximised using the per-label cross-entropy loss in training. Hence, AED models are \emph{label-synchronous}. \subsection{Neural Encoder} The neural encoder of an AED model extracts a sequence of hidden representations $\mathcal{E}$ from $\mathcal{O}$ using an NN architecture, \begin{equation} \mathcal{E} = \text{NeuralEncoder}(\mathcal{O}). \end{equation} Commonly used NN architectures for $\text{NeuralEncoder}$ include a stack of RNN layers, such as long short-term memory (LSTM) layers ~\cite{Chorowski2015AttentionBasedMF,Chan2016ListenAA}, Transformer encoder blocks~\cite{Vaswani2017AttentionIA}, and Conformer encoder blocks~\cite{Gulati2020ConformerCT}. \subsection{Neural Decoder} Label-synchronous systems can be categorised into two types according to the decoder architecture. \begin{figure}[t] \centering \subfloat[]{\includegraphics[width=.535\linewidth]{fig/rnn_dec.pdf}% \label{fig:rnn_dec}} \hfil \subfloat[]{\includegraphics[width=0.46\linewidth]{fig/tfm_dec.pdf}% \label{fig:tfm_dec}} \caption{(a) An RNN neural decoder and (b) a Transformer neural decoder predicting the output distribution of subword unit $c_l$ at the $l$-th step.} \label{fig:decs} \end{figure} \subsubsection{RNN Decoder} The first generation of AED models were RNN-based~\cite{Chorowski2015AttentionBasedMF,Chan2016ListenAA}. As shown in \fig{rnn_dec}, at each decoding step $l$, $\mathcal{E}$ is transformed into a context vector $\mathbf{h}_{l}$ based on the annotation vector $\mathbf{a}_{l}$ produced by the attention mechanism. The RNN decoder is assumed to satisfy the first-order Markov property~\cite{Bengio1993}, which assumes $P(c_l|c_{1:l-1},\mathcal{O})= P(c_l|c_{l-1},\mathbf{d}_{l-1},\mathcal{O})$ by conditioning on $\mathcal{O}$ and the previous decoder state $\mathbf{d}_{l-1}$. The decoding procedure stops when the end-of-sentence symbol is generated, which allows output sequences to have variable lengths. More specifically, \begin{align} \mathbf{a}_l & = \text{Attention}(\mathbf{a}_{l-1},\mathbf{d}_{l-1},\mathcal{E})\\ \mathbf{h}_l & = \mathcal{E}\;\mathbf{a}_{l}\\ P(c_l|c_{l-1},\mathbf{d}_{l-1},\mathcal{O}),\mathbf{d}_{l} & = \text{RNNDecoder}(c_{l-1},\mathbf{d}_{l-1},\mathbf{h}_l).\label{eqn:neuraldecoder} \end{align} \subsubsection{Transformer Decoder} Transformers~\cite{Vaswani2017AttentionIA} have recently emerged as another type of AED model without recurrent structures. Instead of the location-based attention-mechanism used by RNN-based AED, Transformers use multi-head scaled dot-product attention and positional encoding to process all time steps in a sequence in parallel while preserving the sequential information, which greatly accelerates training. As illustrated in \fig{tfm_dec}, a Transformer decoder can have multiple blocks, where each block consists of a self-attention layer that performs multi-head attention over the decoder input or the output from the previous decoder block, an encoder-decoder attention layer that performs multi-head attention over the encoder output and a feed-forward module. A Transformer decoder is denoted as \begin{align} P(c_l|c_{1:l-1},\mathcal{O}) & = \text{TransformerDecoder}(c_{1:l-1}, \mathcal{E}). \end{align} At test-time, a sequential step-by-step decoding procedure is still necessary for Transformer decoders. Meanwhile, unlike RNN decoders that employ the first-order Markov property, the hidden representations produced by all blocks in the Transformer decoder at all previous time steps are required to be stored throughout the entire decoding procedure for self-attention, which makes it less memory efficient. \tbl{complexity} shows the complexities of different decoders, which indicates that the RNN decoder is more suitable for long utterances due to its high memory efficiency while Transformer decoders may perform better since they do not make the Markov assumption. \begin{table}[t] \caption{Comparison of the Complexities of RNN and Transformer Decoders with Respect to Output Sequence Length $L$.} \centering \begin{tabular}{cccccc} \toprule \multirow{2}{*}{neural decoder} & \multicolumn{2}{c}{computation complexity} & & \multicolumn{2}{c}{storage complexity}\\ \cmidrule{2-3}\cmidrule{5-6} & training & test & & training & test \\ \midrule RNN & $O(L)$ & $O(L)$ & & $O(L)$ & $O(1)$\\ Transformer & $O(1)$ & $O(L)$ & & $O(L^2)$ & $O(L^2)$\\ \bottomrule \end{tabular} \label{tab:complexity} \end{table} \section{Related Work} \label{sec:related} \subsection{Hybrid CTC and Attention-Based Models} Watanabe \emph{et al.}~\cite{Watanabe2017HybridCA} proposed to augment AED with an extra output layer for CTC in the multi-task training framework. During decoding, beam search is performed based on the label-synchronous decoder of AED, and the frame-synchronous scores produced by the CTC output layer are interpolated with the label-synchronous scores of the current partial hypothesis to improve pruning~\cite{Watanabe2017HybridCA}. There are a few key differences between Watanabe's joint decoding approach~\cite{Watanabe2017HybridCA} and our proposed combination approach. First, our approach performs beam search in a frame-synchronous fashion. It is known that beam search with frame-synchronous systems can more easily handle streaming data and explore a larger search space. Second, although it is possible to incorporate the scores from two types of systems in a single decoding pass, our method achieves this using a frame-synchronous decoding pass followed by a separate rescoring pass by a label-synchronous system. The two-pass approach not only makes it easier to implement by reusing the existing LM rescoring framework, but also allows the two types of system to have different output units. Third, our method is configured to combine multiple systems and can use different model structures with or without sharing the encoder. This offers more flexibility in the choice of the systems and possibly more complementarity. \subsection{Two-Pass End-to-End ASR} Sainath \emph{et al.}~\cite{Sainath2019TwoPassES} proposed using an RNN-T to generate $N$-best hypotheses during the first pass and then rescore them using an AED, which is trained in the multi-task framework by sharing the encoders of RNN-T and AED. Using an adaptive beam and a prefix-tree representation for $N$-best list improves the rescoring results and speed. This is another instance of combining frame-synchronous and label-synchronous systems, which is probably the most similar method to ours. Compared to our approach, Sainath's approach~\cite{Sainath2019TwoPassES} pays more attention to on-device streaming constraints such as model size and latency, and therefore sacrifices the extra complementarity from different encoders. Furthermore, the use of the RNN-T decoder reduced the size of the search space to be explored due to its high computational cost, and thus only the $N$-best hypotheses are used for second pass rescoring. In contrast, our method allows a more flexible choice of the first pass frame-synchronous systems, including NN-HMMs and CTC in addition to RNN-T. Since NN-HMMs and CTC do not have a model-based decoder, it is easier for them to incorporate structured knowledge and construct richer lattices. \subsection{Extending Our Previous Work: Integrating Source-Channel and Attention-Based Models} This paper builds on the techniques and observations in our previous work~\cite{Li2019IntegratingSA}. In that work, we started by comparing Watanabe's joint decoding approach~\cite{Watanabe2017HybridCA} with our proposed two-pass combination approach, termed integrated source-channel and attention-based models (ISCA), under the multi-task training framework. In ISCA, both CTC and NN-HMMs were used as the source-channel model to perform the first pass decoding, and an RNN-based AED model was used to rescore the derived $N$-best hypotheses. It was shown that by using triphone targets together with a lexicon, ISCA outperformed the joint decoding approach. Further improvements were observed when multi-task training was not used, \textit{i.e.} the encoders of the source-channel and AED models are trained separately. Shortly afterwards, an approach similar to ISCA that combines frame-synchronous and label-synchronous systems at the hypothesis level was also studied~\cite{Wong2020CombinationOE}. In this paper, the NN-HMM systems and AED models are built separately to allow them to have different model architectures, output units and training procedures, which maximises their complementarity and the combined performance. Furthermore, not only $N$-best but also lattice rescoring are investigated in this paper, and an improved lattice rescoring algorithm is proposed. For the AED models, Conformer encoders are used in this work instead of the LSTM encoders in \cite{Li2019IntegratingSA} for improved performance. Furthermore, this paper compares an RNN-based decoder with a Transformer-based decoder for $N$-best and lattice rescoring. Lastly, the combination of an NN-HMM system, an RNNLM, and two AED models with different decoders is studied. \section{Experimental Setup} \label{sec:setup} \subsection{Acoustic Data} Two common ASR benchmarks are used for training and evaluation. The Augmented multi-party interaction (AMI)~\cite{Carletta2005TheAM} dataset is relatively small-scale and contains recordings of spontaneous meetings. The individual headset microphone (IHM) channel is used. As shown in \tbl{data}, it has many short utterances as multiple speakers take short turns during meetings. The original corpus has official development (dev) and evaluation (eval) sets. Switchboard-1 Release 2 (SWB-300) is a larger-scale dataset with telephony conversations. Compared to AMI, it has more training data, longer utterances, more speakers and a larger vocabulary. Hub5'00 is used as the development set, which is split into two subsets: Switchboard (SWB) and CallHome (CHM). Note that the SWB subset has overlapping speakers with the SWB-300 training set. RT03 is used as the evaluation set, which is split into Switchboard Cellular (SWBC) and Fisher (FSH) subsets\footnote{The LDC catalogue numbers are LDC97S62 for SWB-300, LDC2002S09 and LDC2002T43 for Hub5'00, and LDC2007S10 for RT03.}. The acoustic data preparation follows the Kaldi recipes~\cite{Povey2011TheKS}. \begin{table}[t] \caption{Two Datasets Used for Experiments.} \centering \begin{tabular}{lcc} \toprule & AMI-IHM & SWB-300 \\ \midrule style & meeting & telephony \\ training data & 78 hours & 319 hours\\ avg. utterance length & 7.4 words & 11.8 words\\ number of speakers & 155 & 520 \\ vocabulary size & 12k & 30k \\ \bottomrule \end{tabular} \label{tab:data} \end{table} \subsection{Text Data and Language Models} \begin{table}[t] \caption{Perplexities of Various Language Models on Two Datasets. Dev and Eval Sets for SWB-300 are Hub5'00 and RT03.} \centering \begin{tabular}{lccc} \toprule dataset & LM & dev & eval\\ \midrule \multirow{3}{*}{AMI-IHM} & 3-gram & 80.2 & 76.7\\ & 4-gram & 79.3 & 75.7\\ & RNNLM & 58.0 & 53.5\\ \midrule \multirow{3}{*}{SWB-300} & 3-gram & 82.8 & 67.7\\ & 4-gram & 80.0 & 65.3\\ & RNNLM & 51.9 & 45.3\\ \bottomrule \end{tabular} \label{tab:ppl} \end{table} For each dataset, its training transcription and Fisher transcription\footnote{The LDC catalogue numbers are LDC2004T19 and LDC2005T19 for Fisher transcription.} are used to train both $n$-gram language models and RNNLMs. Text processing and building $n$-gram LMs for both datasets also follow the Kaldi recipes~\cite{Povey2011TheKS}. RNNLMs are trained using the ESPnet toolkit~\cite{watanabe2018espnet}. The vocabulary used for RNNLMs is the same as for $n$-gram LMs and has 49k words for AMI-IHM and 30k words for SWB-300. RNNLMs have 2-layer LSTMs with 2048 units in each layer. Models are trained with stochastic gradient descent (SGD) with a learning rate of 10.0 and a dropout rate of 0.5. The embedding dimension is 256. Gradient norms are clipped to 0.25 and weight decay is set to $10^{-6}$. Training transcription and Fisher transcription are mixed in a 3:1 ratio. Because of the domain mismatch between AMI and Fisher text data, the RNNLM for AMI is fine-tuned on AMI transcriptions after training using the mixture of data with a learning rate of 1.0. The AMI RNNLM has 161M parameters and the Switchboard RNNLM has 122M parameters. The perplexities of LMs for both datasets are in \tbl{ppl}. \subsection{Acoustic Models and Label-Synchronous Systems} \begin{table}[t] \caption{Single System WERs on AMI-IHM and SWB-300 Datasets. Systems Do Not Use RNNLMs for Rescoring or Decoding.} \centering \begin{tabular}{lccccc} \toprule & \multicolumn{2}{c}{AMI-IHM} & & \multicolumn{2}{c}{SWB-300} \\ \cmidrule{2-3}\cmidrule{5-6} & \multirow{2}{*}{dev} & \multirow{2}{*}{eval} & & Hub5'00 & RT03 \\ & & & & (SWB/CHM) & (SWBC/FSH) \\ \midrule frame-sync & 19.9 & 19.2 & & 8.6 / 17.0 & 18.8 / 11.4\\ label-sync-LSTM & 19.6 & 18.2 & &7.5 / 15.3 & 16.2 / 10.7\\ label-sync-TFM & 19.4 & 19.1 & & 7.8 / 14.4 & 17.5 / 10.4\\ \bottomrule \end{tabular} \label{tab:single_wer} \end{table} Acoustic models of the frame-synchronous systems are factorised TDNNs trained with lattice-free maximum mutual information objective~\cite{Povey2018SemiOrthogonalLM} by following the standard Kaldi recipes~\cite{Povey2011TheKS}. The total numbers of parameters are 10M for AMI-IHM and 19M for SWB-300. Two types of label-synchronous system are trained using the ESPnet toolkit~\cite{watanabe2018espnet} without the CTC branch. Neural encoders are composed of 2 convolutional layers that reduce the frame rate by 4, followed by 16 Conformer blocks~\cite{Gulati2020ConformerCT}. For the Conformer block, the dimension for the feed-forward layer is 2048, the attention dimension is 512 for the model with an RNN decoder and 256 for the model with a Transformer decoder. The number of attention heads is 4 and the convolutional kernel size is 31. For the label-synchronous system with an RNN decoder (label-sync-LSTM), the decoder has a location-aware attention mechanism and 2-layer LSTMs with 1024 units. For the label-synchronous system with the Transformer-based decoder (label-sync-TFM), the decoder has 6-layer Transformer decoder blocks where the attention dimension is 256 and the feed-forward dimension is 2048. Both the label-sync-LSTM and label-sync-TFM are trained using the Noam learning rate scheduler on the Adam optimiser. The learning rate is 5.0 and the number of warmup steps is 25k. Label smoothing of 0.1 and a dropout rate of 0.1 are applied during training. An exponential moving average of all model parameters with a decay factor of 0.999 is used. The label-sync-LSTM has 130M parameters while the label-sync-TFM has 54M parameters. Beam search with a beam-width of 8 is used for decoding. Apart from applying length normalisation for the label-sync-TFM model, other decoding heuristics are not used. SpecAugment~\cite{Park2019SpecAugmentAS} and speed perturbation are applied. Word-piece outputs~\cite{Kudo2018SubwordRI} are used with 200 units for AMI-IHM and 800 units for SWB-300. The single model WERs for the frame-synchronous system and label-synchronous systems are given in \tbl{single_wer}.

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\section{Introduction} \label{1} Military fleet operations take place in a large variety of environments and scenarios resulting in a diverse set of requirements for the fleet mix. The special functionalities of military vehicles and incessantly updated technologies make vehicles hard to reuse after military operations \citep{shinkman2014trashed}. To reduce waste, the US Army requires that fleets of vehicles can be reutilized across a large array of military mission scenarios. Modular vehicles can address this challenge \citep{dasch2016survey}. Modules are assumed to be special types of components which can be easily coupled/decoupled through simple plug-in/pull-out actions on battlefields. This property enables vehicles to be quickly assembled, disassembled and reconfigured (ADR) (as shown in Fig.~\ref{fig_ADR}) on battlefield to react to demands. \begin{figure} \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.50]{ADR.pdf} \end{center} \caption{Operation actions for modularized vehicles: a) assembly, b) disassembly, c) reconfiguration.} \label{fig_ADR} \end{figure} Because of the close connection between the strategy and the fleet performance, researchers have investigated the potential of modularity to boost performance during fleet operations. Bayrak et al. proposed a mathematical model to simulate modular fleet operations in a logistic mission scenario \citep{bayrak2016system}. They noticed a significant operational cost reduction after fleet modularization. Li and Epureanu proposed an intelligent agent-based model in managing modular fleet operations \citep{li2017intelligent, li2018IABS}. Agents were classified into three categories: camp, distributor, and supply. Different types of agents collaboratively and real-timely yielded operational decisions to react to stochastic battlefield demands. Later, Li and Epureanu also modeled the fleet operation as a dynamic system and implemented model predictive control to manage the system dynamics \citep{li2017robustness}. Their results show that modular fleets exhibit a better robustness than conventional fleets as they react to disturbances from the battlefields. However, most previous research focused on a single decision maker who operates against a stochastic demand without intelligence. However, unpredictability from the adversarial reaction is essential and leads to an inability to forecast the outcome of actions or weakly perceived causal links between events on the battlefields \citep{Lynch2014On}. Furthermore, smart systems and artificial intelligence are playing an ever-increasing role. Autonomous vehicles, especially unmanned aerial vehicles, have been widely used to assist military operations \citep{landa1991war,jose2013technology,evers2014online}. It is no surprise that artificial intelligence plays a significant role in management of large-scale fleets of autonomous vehicles. Given an equivalent autonomous decision making system, our goal is to explore the synergy between modularity and autonomy by performing an attacker-defender game between a conventional fleet and a modular fleet. The use of games in modeling the relationship between an attacker and a defender has a long history starting with the work of Dresher \citep{dresher1961games}. The variety of applications and research relate to issues in military operation research and defense studies are rich \citep{kardes2005survey,hausken2009minmax, zhuang2010modeling,paulson2016game}. There are also several studies of attacker-defender games that consider resource-dependent strategies. For example, Powell used game theoretical approach to find a defender's resource allocation strategy for protecting sets from being destroyed by a strategic adversary, i.e., terrorist group \citep{powell2007defending}. The defender is uncertain about the targets that the adversary is likely to strike. Given the distribution of adversary's behaviors, Powell derived a pure strategy for the defender that leads to Bayesian Nash equilibria. Hausken and Zhuang considered a multi-period game where a defender can allocate resources to both defend its own resources and attack the resources of the adversary. Similarly, attacker can also determine the use of their resources for attacking the defender or protecting itself \citep{hausken2011defending}. They adopted a strategy pair which is proven to be a subgame perfect Nash equilibrium for a two-stage game, and they illustrated how the strategy depends on changes in the adversarial resources. Methods based on game theory are popular in the existing literature related to resource-dependent attacker-defender games. However, these applications mainly focus on single-period games or repeated games where most information from previous periods is ignored. Furthermore, strong assumptions on previous approaches, i.e., single resource type, sequential moves, perfect information of the adversary, also make previous research inapplicable to realistic military missions, where the demands and the environment are stochastic and unpredictable \citep{shinkman2014trashed, Lynch2014On, xu2016modeling}. Furthermore, a recent study also shows that the performance of a modular fleet is strongly influenced by the optimality of operation decisions \citep{li2018IABS, li2017intelligent}, which makes the analytical solution intractable. A new method is required to investigate the tactical advantages of fleet modularity. In this study, we go beyond the existing literature and formulate an attacker-defender game using intelligent agent-based modeling techniques \citep{adhitya2007model, yu2009intelligent, onggo2016test, li2017intelligent}. We combine optimization techniques and artificial intelligence to enable each fleet(player) to forecast adversary's behaviors based on experience and to optimize decisions accordingly. By selecting one player as a modular fleet and another as a conventional fleet, we explore the benefits of fleet modularity, including adaptability and unpredictability, when competing in a hostile mission scenario. \section{Game Formation} \label{2} Miltary fleets can face supply shortages due to exogenous supply chain disruptions during armed conflicts \citep{xu2016modeling}. Such events can be explored as a game involving a competition between two military fleets, a red fleet and a blue fleet. The goal of the fleet operation is to satisfy the supply demands which randomly appear at battlefields. Each demand is characterized by a due time, required materials, personnel, and target fleet to accomplish the demand. To satisfy the demand, dispatched convoys have to be formed with vehicles selected from each fleet so that they have enough capacity for delivering the supplies and enough firepower to guarantee the safety of transportation. For convenience, all demands are automatically converted into the attribute requirements for the convoy, i.e., firepower - the ability to deliver effective fire on a target, material capacity - the ability to relocate materials, personnel capacity - the ability to carry personnel, etc. We denote the demands received at time $t$ as $\bm{r}(t)$. According to the due time of demand, attributes required to be satisfied at a future time $t+\tau$ can be obtained as $\bm{d}^x(t+\tau) = [d^x_1(t+\tau),d^x_2(t+\tau),...,d^x_{N_a}(t+\tau)]^T$, where $d^x_e(t+\tau)$ represents the attributes of type $e$ to be satisfied before time $t+\tau$. $N_a$ is the number of attribute types of interest in this game . Matrix $\bm{D}^x(t)$ can also be created and updated to record the demands to be satisfied in the planning horizon $T_p$, i.e., $\bm{D}^x(t) = [\bm{d}^x(t+1),\bm{d}^x(t+2),...,\bm{d}^x(t+T_p)]$. Correspondingly, attributes carried by the dispatched convoy at time $t$ are $\bm{v}^x(t) = [v^x_1(t),v^x_2(t),...,v^x_{N_a}(t)]^T$. The fleet specified as the target fleet by demand becomes defender, with the goal of delivering a qualified convoy to battlefield on time. The other fleet becomes the attacker automatically. The attacker dispatches convoy with the aims to disrupt the fulfillment of the demands by the defender by dispatching an attacker convoy. Thus, each demand initializes a supply task for one fleet and an attack task for another fleet. Based on the demand, the role of the player dynamically changes. The common rules of the game are assumed to be known by the players: delivery of a convoy with capacity of satisfying the demand makes the defender win and the attacker lose. Because the damage from the attacker can reduce the attributes of the defender's convoy, the defender may lose the game even if a convoy with higher attributes than requirements is dispatched by the defender. We denote the conflict between attacker and defender convoys as an event, and assume that the dispatch decisions made by the attacker and the defender are simultaneous. Thus, the game is a simultaneous-move game. To simplify the problem, we assume that both fleets can simultaneously sense the demands regardless of their target. Thus, given the same probability to be selected as the target fleet, each fleet will play equivalent times as attacker and defender to guarantee the fairness of the game. Fig.~\ref{fig_bf} illustrates the convoy competition in a multiple battlefield scenario. \begin{figure} \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.35]{field.pdf} \end{center} \caption{Competition between two fleets at battlefield} \label{fig_bf} \end{figure} Following previous research \citep{azaiez2007optimal, wang2011target}, both attacker and defender are modeled as rational and strategic. Based on the simulation results, we summarized a fixed number of predefined dispatch strategies for the attacker and the defender respectively, as shown in Tab.~\ref{tab_strategies}. For an attacker, the firepower is considered as the only attribute needed to win. Thus, we classify the attack strategy by comparing the firepower to be assigned to attacker convoy to the required firepower for adversary, which is measured by $k_a$. The defender wins when it delivers a convoy which satisfies the demands with consideration of attribute losses due to adversarial disruptions. The strategy is a mixture of decisions in selecting safety coefficients of firepower $k_a$ and capacity(personnel and material) $k_c$. We cluster all the dispatch orders that are less than the requirements in strategy 1, where the defender gives up the game to save resources once a strong adversarial convoy is predicted. \begin{table} \caption{Dispatch strategies for the attacker and the defender fleets} \label{tab_strategies} \begin{center} \begin{tabular}{c l l l l l l l l l l l} & & \\ \hline \hline Attack Strategy & 1 & 2 & 3 & 4 & 5\\ Range of $k_a$ & $[0, 0.5)$ & $[0.5, 1)$ & $[1, 1.5)$ & $[1.5, 2)$ & $[2, 2.5)$ \\ \hline Attack Strategy & 6 & 7 & 8 & 9 & 10\\ Range of $k_a$ & $[2.5, 3)$ & $[3, 3.5)$ & $[3.5, 4)$ & $[4, 4.5)$ & $[4.5, \infty)$\\ \hline \hline Defense Strategy & 1 & 2 & 3 & 4 & 5\\ Range of $k_a$ & $[0, 1)$ & $[1, 1.5)$ & $[1, 1.5)$ & $[1, 1.5)$ & $[1.5, 2)$ \\ Range of $k_c$ & $[0, 1)$ & $[1, 1.5)$ & $[1.5, 2)$ & $[2, \infty)$ & $[1, 1.5)$ \\ \hline Defense Strategy & 6 & 7 & 8 & 9 & 10\\ Range of $k_a$ & $[1.5, 2)$ & $[1.5, 2)$ & $[2, \infty)$ & $[2, \infty)$ & $[2, \infty)$\\ Range of $k_c$ & $[1.5, 2)$ & $[2, \infty)$ & $[1, 1.5)$ & $[1.5, 2)$ & $[2, \infty)$\\ \hline \end{tabular} \end{center} \end{table} In this study, the amount of damage is based on the comparison of the firepower carried by different convoys. The probability of damage of the component of type $i$ of the red fleet $p_{di}^r$ and the blue fleet $p_{di}^b$ shown by \begin{eqnarray} p_{di}^r = \tanh(k_{di}\frac{v^b_{f}}{v^r_{f}})\\ p_{di}^b = \tanh(k_{di}\frac{v^r_{f}}{v^b_{f}}), \end{eqnarray} where $v^x_f$ is the amount of firepower carried by convoy $x$. $k_{di}$ represents the damage factor for a component in type $i$. Each component in the convoy is damaged stochastically based on the calculated probability. To create a fair game, we constrain the amount of supplies, and assume all damaged resources are recoverable. Thus, the amount of resources for both fleets are constant, but the conditions of the resources are dynamic. We penalize damage by requiring a long time to recover. The recovery strategy for damaged vehicles is to replace all the damaged components by healthy ones. Once vehicle modularity is considered, disassembly becomes another option in dealing with a damaged vehicle. Several assumptions are also used to simplify this problem while maintaining a reasonable level of fidelity: \begin{enumerate} \item Each fleet can accurately observe and record the damage that occurs in its convoys. \item Each fleet can accurately observe the composition of the adversarial convoys in every event, i.e., number and types of vehicles. \item Convoys return to base immediately after completing a mission task. \item Mission success is reported to both fleets once it is completed. \item No other type of vehicle damage is considered besides damage caused by the attacker. \item All vehicle components are recoverable. \item Damage occurs independently in vehicles based on the probability of damage. \item The inventory status is updated every hour and accessible to all the agents. \end{enumerate} \section{Problem Statement} \label{problemStatement} To explore the advanced tactics enabled by fleet modularity, we first consider the way operation decisions are made. With simplification, the main procedure involved in decision making \citep{chen2013human} are: 1) perceive battlefield information, 2) analyze adversary's behavior based on received information, 3) optimally schedule the operational actions to accomplish the demands or dispute adversarial actions. Through these procedures, decision makers adaptively adjust their dispatch strategy and operation plan based on the learned adversary's behaviors, i.e., what kind of strategy adversary might adopt in a specific situation. Combined with fleet modularity, a decision making process with guaranteed decision optimality and efficiency is challenge addressed herein. Another existing challenge is the management of an inventory with high diversity. We denote a vehicle with one or more than one damaged components as damaged vehicles, which generates numerous types of damage. For example, a single vehicle with 5 different components has $2^5$ types of vehicle damage, which leads to many distinct recovery strategies. Furthermore, given the limited working capacity that can be used for ADR actions and recovery, it is still challenging to effectively and efficiently schedule operations and select the recovery strategy in reaction to stochastically arrived demands while maintaining reasonable healthy stock levels. \section{Agent-Based Model} {This section presents an agent-based model to automatically yield adaptive tactics and real-timely plan for operational actions accordingly. The model takes into consideration the computational load and the battlefield decision making process. To simplify the notation, we describe the approach from the standpoint of the blue fleet while the red fleet is the adversary. Three types of agents are created to perform different functionalities. The decision-making process is then achieved by the cooporation of these three types of agents. The interconnections are shown in Fig.~\ref{fig_ABM}. \begin{enumerate} \item \textbf{Inference Agent}: analyze adversarial historical behaviors, forecast adversarial future actions. \item \textbf{Dispatch Agent}: optimize dispatch order based on the output of the inference agent. \item \textbf{Base Agent}: optimally plan operation actions to satisfy the dispatch order. \end{enumerate} \begin{figure*} \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.40]{ABM.pdf} \end{center} \caption{Agent-based model in attacker-defender game} \label{fig_ABM} \end{figure*} } \subsection{Inference Agent} The attacker-defender game is a simultaneous-move game. Hence, it is critical to forecast the adversarial actions that used to be accounted for. As combat resources and working capacity are limited, it is possible to possible to find cues from the adversarial historical dispatch actions by inference. For example, if the adversary has dispatched a convoy with a significant amount of vehicles in the near past, it is possible to conclude that the adversary is not capable of grouping up a strong convoy again in the short future. In addition, existing damage in adversary's resources can be analyzed by comparing the firepower of dispatched convoys from historical events. The amount of damage is also useful for the decision maker to infer adversary's available resources. The information that can be used for inference is limited, including demand records $\bm{D}(t)$, previous dispatched vehicles, $\bm{v}^b (\tau)$, and the adversary's previous dispatches $\bm{v}^r(\tau)$. Dispatch decisions depend on an optimization algorithm, on inference of adversarial actions, and on the performances of the decision makers. Together, these lead to a strong nonlinearity in the decision-making process. In addition, the strategy employed has to be adjusted after learning from the adversarial actions. The decision-making process requires a prediction model which is able to update in real-time and capture the causality and correlation from adversary's historical behaviors. We adopt techniques from artificial intelligence to solve this problem. Recurrent neural networks(RNNs) are well known machine learning models that could capture dependencies in time-series data \citep{mikolov2010recurrent}. Compared to conventional neural networks, RNNs can memorize a certain period of historical data and analyze its influence on the future. Long short-term memory model networks is one of the popular RNNs, which is capable of learning long-term dependencies without gradient vanishing problem in RNN. In this study, we implement a variant model of RNN, namely a long short-term memory (LSTM) as our predictive model to capture the correlations in adversarial sequential decisions. The LSTM model is widely used in forecasting based on sequential data, including, stock market prices \citep{chen2015lstm, di2016artificial, fischer2017deep}, traffic \citep{ma2015long, zhao2017lstm}. In this study, we model the inference of the adversary's strategy as a classification problem, where each class corresponds to a strategy. The inputs $\bm{x}^{(m)}$ of training data are the records of each event, including own dispatched convoys, adversary's dispatched convoy and the received demands, which are time-series data recorded during a time horizon $T_b$. The output $y^{(m)}$ of each training sample is the actual dispatch strategy adopted by the adversary. The architecture of LSTM used in this study is shown in Fig.~\ref{fig_training}. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.50]{LSTM.pdf} \end{center} \caption{LSTM architecture designed for inferring the adversary's strategy} \label{fig_training} \end{figure} The status of the LSTM model at time $t$ are described by the input gate $\bm{i}_{t}$, forget gate $\bm{f}_{t}$, output gate $\bm{o}_{t}$, cell state $\bm{c}_{t}$ and activation at $i ^{th}$ hidden layer $\bm{a}^i_{t}$. The forward propagation can be described by \begin{eqnarray} & \bm{i}_{t} = \sigma (\bm{W}_i [\bm{a}_{t-1}, \; \bm{x}^{(m)}_{t}] + \bm{b}_i),\\ & \bm{f}_{t} = \sigma (\bm{W}_f [\bm{a}_{t-1}, \; \bm{x}^{(m)}_{t}] + \bm{b}_f),\\ & \bm{o}_{t} = \sigma (\bm{W}_o [\bm{a}_{t-1}, \; \bm{x}^{(m)}_{t}] + \bm{b}_o),\\ & \bm{c}_{t} = \bm{i}_{t} * \tanh (\bm{W}_c [\bm{a}_{t-1}, \; \bm{x}^{(m)}_{t}] + \bm{b}_c) + \bm{f}_{t} * \tanh (\bm{W}_c [\bm{a}_{t-2}, \; \bm{x}^{(m)}_{t-1}] + \bm{b}_c),\\ & \bm{a}^0_{t} = \bm{o}_{t} * \tanh({\bm{c}}_{t}),\\ & \bm{a}^1_{t} = \bm{W}_{1}\bm{a}^0_{t}+\bm{b}_1,\\ & \bm{a}^2_{t} = \bm{W}_{2}\bm{a}^1_{t}+\bm{b}_2,\\ & \vdots \nonumber \\ & \bm{a}^{N_h}_{t} = \bm{W}_{N_h}\bm{a}^{N_h-1}_{t}+\bm{b}_{N_h},\\ & \bm{p}^{(m)} = \sigma_m(\bm{a}^{N_h}_{t}), \label{eq_LSTM} \end{eqnarray} where $\bm{W}$ and $\bm{b}$ are weights and biases to be obtained through training. $\sigma$ is the sigmoid function. $N_b$ is the number of hidden layer. $\sigma_m$ is the softmax function. $\bm{p}^{(m)}$ records the estimated probability of each class based on inputs from the training sample $m$ and weights of the model, i.e., $\bm{p}^{(m)} = [p_{1}^{(m)}, p_{2}^{(m)},...,p_{N_c}^{(m)}]^T$. The loss function is represented by a cross entropy equation: \begin{equation} \begin{split} E^{m} = -\sum^{N_c}_{i=1} \gamma_{i} \log(p_{i}^{(m)}), \end{split} \end{equation} where $\gamma_i$ is a binary indicator (0 or 1), with value $1$ if class label $i$ is the correct classification and $0$ otherwise. Thus, the training of the model is to minimize the sum of the entropy of the training set to find the best model parameters through backward propagation \citep{hecht1992theory}, i.e., \begin{equation} \min_{\bm{W},\bm{b}} \sum_m E^{m}. \end{equation} Thus, adversary's behavior can be forecasted through \begin{equation} \bar{y}^r = \{ i \; | \; 1 \leq i \leq N_c, \forall 1 \leq j \leq N_c \; : p_i^{(m)} \geq p_j^{(m)}\}, \label{eq_forecast} \end{equation} where $\bar{y}^r$ is the predicted adversary's strategy, which can be used to calculate the possible adversary's dispatch order $\bar{\bm{v}}^r$ by using the upper bounds of the strategy shown in Tab.~\ref{tab_strategies}. \subsection{Dispatch Agent} The goal of the convoy dispatch is to determine the desired attributes that need to be carried by a convoy to maximize the win rate. A convoy with higher attributes, especially in firepower, indicates a higher chance to win. However, as resources are limited, the less vehicles are ordered, the higher the chance that the corresponding order can be achieved by the base agent. Thus, the convoy dispatch should be carefully planned to guarantee the win rate of current mission without overusing resources. To avoid the overuse or underuse of available attributes, it is important to have an accurate evaluation of the available vehicles before ordering vehicle convoy. In addition, time delays exist in operations, the dispatch order needs to be placed ahead of time to provide the base agent with enough time to prepare the order. Nevertheless, the actual planning is determined by base agent and the feasibility of the dispatch order can only be estimated based on the historical behaviors of base agent. This difficulty becomes one of the challenges in placing the dispatch order. Even we ascertain that our convoy order can be achieved by base agent, whether the dispatched convoy can win the event or not is still a question. As damage mechanism is sealed to both fleets, both of them need to speculate the damage mechanism based on the experience and estimate the probability of success accordingly. To resolve this problem, we decouple the estimation of event success in two parts, which are the feasibility of order $p^b_f$ and conditional success rate if order is feasible $p^b_s$. The probability of wining an event for a blue convoy $p^b_{w}$ can be calculated as \begin{equation} \begin{split} p(\text{win}) & = p(\text{win}|\text{order is feasible}) p(\text{order is feasible})\\ p^b_{w} & = p^b_s p^b_f. \end{split} \label{eq_pspf} \end{equation} \subsubsection{Feasibility} The feasibility of the order can be defined as $0$ when infeasible and $1$ when feasible based on a comparison between the dispatch order $\bm{do}^b(t)$ and the actually dispatched convoy $\bm{o}^b_v (t)$. Specifically, an order is feasible if $\bm{do}^b(t) \leq \mathbf{M}_\mathrm{va} \bm{o}^b_v(t)$, and infeasible otherwise, where $\mathbf{M}_\mathrm{va}$ is the mapping between vehicles and attributes and inequality of two vectors refers to the entry-by-entry inequality of the values in the two vectors. The relationship between factors and feasibility is complex and nonlinear because optimizations are implemented in operation planning. We implement a neural network model \citep{hagan1996neural, atsalakis2018forecasting, rezaee2018integrating} to capture these nonlinear inter-connections, as shown in Fig.~\ref{fig_feasNN}. The output of the training set is the feasibility of the order (1 for feasible, 0 for infeasible). \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.35]{feasNN.pdf} \end{center} \caption{Neural network model for feasibility} \label{fig_feasNN} \end{figure} With enough training, the model is capable of evaluating the feasibility of dispatch orders across diverse operation situations. To capture changes in the inventory operation strategy, we periodically re-train the model based on the latest operation information. The relationship between model inputs and feasibility rate can be described by \begin{equation} \begin{aligned} p_f = f_f(\bm{do}^b(t), \bm{s}^b(t), \delta\bm{s}^b(t+1),..., \delta\bm{s}^b(t+T_p), \bm{D}(t)), \end{aligned} \label{eq_f} \end{equation} where, $\bm{s}^b(t)$ records the number of vehicles and component stocks on base at time $t$. $\delta\bm{s}(t+\tau)$ is the changes in inventory stocks at time $t+\tau$ from unfinished operational actions scheduled previously. \subsubsection{Conditional Success Rate} Vehicle damage plays an important role in determining the success of a mission. However, vehicle damage is driven by a stochastic process and varies according to changes in terrain, commander's goal and military tactics. A model is needed to capture the complexity of the damage mechanism. We adopted another neural network model for forecasting the success rate, as shown in Fig.~\ref{fig_succNN}. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.35]{succNN.pdf} \end{center} \caption{Neural network model for success rate} \label{fig_succNN} \end{figure} The output of the training set are success history of previous events (1 for success, 0 for failure). Given forecasted adversary's convoy attributes $\bm{\bar{v}}^r(t)$, the trained model will yields the conditional win rate for a certain dispatch order and mission requirements. The model is capable of capturing changes in the damage mechanisms by continuously feeding in the latest event information and results. By denoting the trained neural network model for success as $f^b_s$, the probability of success can be calculated as \begin{equation} p^b_s = f^b_s(\bm{do}^b(t), \bar{\bm{v}}^r(t), \bm{D}(t)). \label{eq_s} \end{equation} \subsubsection{Optimization} For each dispatch order $\bm{do}^b(t)$, the approach above provides a way to estimate the probability of success and probability of feasibility based on predicted adversary's behavior, demand information and inventory status. An optimization model can be used to seek the optimal dispatch order to maximize the win rate or minimize the failure rate, i.e., $J = 1- p_b^w$. Combining with Eqns.~\ref{eq_f} and \ref{eq_s}, a nonlinear programming model can be formulated to seek the optimal dispatch order as \begin{equation} \begin{aligned} & \min_{\bm{do}^b(t)} & & 1-f_f(\bm{do}^b(t), \bm{s}^b(t), \delta\bm{s}^b(t+1),..., \delta\bm{s}^b(t+T_p), \bm{D}(t)) f^b_s(\bm{do}^b(t), \bar{\bm{v}}^r(t), \bm{D}(t)) \\ & \text{s.t.} & & (a) \; \bm{do}^b(t) \geq 0,\\ \label{eq_opt1} \end{aligned} \end{equation} where $\bm{do}^b(t)$ is the decision variable that specifies the desired attributes to be carried by the blue convoy. Thus, the number of decision variables is the number of attribute types. However, it is intractable to obtain the global optimum by a gradient-based approach because of the non-convexity in objective function. Hence, in this study we implement a pattern search technique to yield optimal dispatch decisions. As the minimized failure rate can be any value in the range of $[0,1]$, the dispatch agent should be capable of giving up the mission once a very high failure rate is predicted. There is also a stream of literature studying risk preferences in repeated and evolutionary games \citep{roos2010risk, lam2006formalizing, zhang2018role}. We define the $\epsilon_f$ as a customizable parameter to represent the minimal failure rate that can be tolerated, which modifies the dispatch order $\bm{do}^b(t)$ as \begin{equation} \bm{do}^b(t)= \left\{ \begin{aligned} &= \; \bm{0}, \;\;\; &\text{for} \;\;\; (1-p_w)>\epsilon_f,\\ &= \; \bm{do}^b(t) &\text{for} \;\;\; (1-p_w)\leq\epsilon_f. \end{aligned} \right. \end{equation} Thus, a convoy can be dispatched only when the probability of success is high enough. Risk aversion behavior is related to $\epsilon$ which is constant during operation. As a future work, it is also interesting to vary $\epsilon_f$ over time to seek an advanced fleet operation strategy, i.e., combination of risk-prone and risk-averse strategies\citep{roos2010risk}. \subsection{Base Agent} The base agent is the one to plan operational actions to accomplish the orders based on the behavior analysis from the inference agent and the dispatch order suggestion from the dispatch agent. Li and Epureanu proposed a model predictive control based approach to real-timely schedule the operation actions in reacting to the received demands\citep{li2018IABS}. However, they did not consider the possible damage that occurs during fleet operation. In this section, we further that research by considering the possible damage during fleet operation, and manage the inventory based on the resulting diverse conditions. For convenience, we simplify the notation of operation actions for fleet $x$ from $\bm{o}^x$ to $\bm{o}$ in this section as no adversary is considered. It is important to schedule the operation actions properly to recover damaged resources and increase utility rate because resources for each player are limited and repairable. It is also essential to allocate the working capacity properly to balance between order satisfaction and damage recovery. In this section, we first model military fleet operation as a time-varying dynamical system. Then, a model predictive control is proposed to manage the system dynamics thus achieving overall operation management. \subsubsection{Dynamical System} The dynamics of the fleet operation is mainly due to the changes of inventory stocks and remaining demands, in terms of \begin{enumerate} \item {Vehicle stocks, $\bm{I}_v = [I_{v_1},I_{v_2},...,I_{N_v}] $}, $N_v$ being the total number of vehicle types, \item {Module/component stocks, $\bm{I}_c = [I_{c_1},I_{c_2},...,I_{N_c}] $}, $N_c$ being the total number of component types, \item {Damaged vehicles, $\bm{I}_{dv} = [I_{dv_1},I_{dv_2},...,I_{N_{dv}(t)}] $}, $N_{dv}(t)$ being the total number of damaged vehicles at time $t$, \item {Damaged components, $\bm{I}_{dc} = [I_{dc_1},I_{dc_2},...,I_{N_{c}}] $}, \item {Unsatisfied demands, $\bm{I}_{a} = [I_{a_1},I_{a_2},...,I_{N_{a}}] $}. \end{enumerate} Although healthy vehicles and damaged vehicles are recorded in a similar vector, their meanings are different. For healthy stocks and damaged components, the subscript of variable is the type of vehicle/component; the value of variable indicates the number. For damaged stocks, the subscript is the index of the damaged vehicle, which is created based on the vehicle receipt date. Binary values are used to represent the status of a vehicle, where 1 represents that the damaged vehicle remains to be repaired; 0 indicates that the damaged stock is recovered or not received yet. For each damaged vehicles, information are recorded as \begin{enumerate} \item{The vehicle type, $\bm{v}_{lt}=[v_{l1t},v_{l2t},...,v_{lN_vt}]$, where $v_{lkt} = 1$ if damaged vehicle of index $l$ is a vehicle in type $k$, and 0 otherwise,} \item {The damaged components, $\bm{v}_{ldc}(t) = [{v}_{ldc_1}(t), {v}_{ldc_2}(t), ..., {v}_{ldc_{N_c}}(t)]$ in a damaged vehicle of index $l$, where ${v}_{ldc_i}$ is the number of damaged components of type $i$ in the damaged vehicle of index $l$,} \item{The healthy components, $\bm{v}_{lc}(t)= [{v}_{lc_1}(t), {v}_{lc_2}(t), ..., {v}_{lc_{N_c}}(t)]$ in a damaged vehicle of index $l$, where ${v}_{lc_i}$ is the number of healthy components of type $i$ in the damaged vehicle of index $l$,.} \end{enumerate} These data are time variant because the number and type of damaged vehicles keep changing with newly occurred damages and vehicle recoveries. We create a state for each newly arrived damaged vehicle and remove the corresponding state once the damaged vehicle is recovered, i.e., its state value changes from 1 to 0. To maintain a desirable resource utility rate, vehicle repair is usually proceeded in a short time after damaged is received. The real-time adjustment of $\bm{I}_{dv}$ keeps the size of vector short and bypass the numerous states incurred from diverse vehicle damage patterns, as mentioned in Section~\ref{problemStatement}. Vehicle conditions are reported to the base agent as one of the inputs. All inputs, $\bm{a}=[\bm{a}_r,\bm{a}_d,\bm{do}]^T$, to the fleet operation system are \begin{enumerate} \item {Returning healthy vehicles, $\bm{a}_r = [a_{r_1},a_{r_2},...,a_{r_{N_v}}] $, where $a_{r_k}$ is the number of returned vehicles of type $k$,} \item {Returning damaged vehicles, $\bm{a}_d = [a_{d_1},a_{d_2},...,a_{d_{N_{dv}}}] $, where $a_{d_l}$ represents a newly arrived damaged vehicles of index $l$,} \item {Dispatch order from dispatch agent, $\bm{do} = [do_1,do_2,...,do_{N_a}]$, where $do_{k}$ is the number of vehicles of type $k$ ordered by the dispatch agent.} \end{enumerate} Based on the characteristics of fleet operation, the operational actions to be determined are also distinct. For conventional fleet, the operation actions include \begin{enumerate} \item {Convoy dispatch, $\bm{o}_v = [o_{v_1},o_{v_2},...,o_{{v_{N_v}}}]$, where $o_{v_k}$ is the number of vehicles of type $k$ in the dispatched convoy,} \item {Recovery of damaged vehicle, $\bm{o}_{dr} = [o_{dr_1},o_{dr_2},...,o_{dr_{N_{dv}}}]$, where $o_{dr_l}$ indicates damaged vehicle of index $l$ to be repaired (1) or not (0),} \item {Recovery of damaged component, $\bm{o}_{c} = [o_{c_1},o_{c_2},...,o_{N_{c}}] $, where $o_{c_i}$ indicates the number of damaged modules of type $i$ to be repaired.} \end{enumerate} Given the time to accomplish the operational action $o_x$ as $\tau_x$, the dynamics of the vehicle stocks $I_{v_k}$ of type $k$, component stocks $I_{c_i}$ of type $i$, damaged vehicles $I_{dv_l}$ of index $l$, damaged components $I_{dc_i}$ of type $i$, and remaining attributes $I_{a_h}$ of type $h$ are governed by \begin{equation} I_{v_k}(t+1) = I_{v_k}(t) - o_{v_k}(t) + a_{r_k}(t) + \sum_{l=1}^{N_{dv}} v_{lkt}(t) o_{dr_l}(t - \tau_{v_l}), \label{eq_conv1} \end{equation} \begin{equation} I_{c_i}(t+1) = I_{c_i}(t) + o_{c_i}(t-\tau_{c_i}) - \sum_{l=1}^{N_{dv}} v_{lc_i}(t) o_{dr_l}(t), \label{eq_conv2} \end{equation} \begin{equation} I_{dv_l}(t+1) = I_{dv_l}(t) - o_{dr_l}(t) + a_{d_l}(t), \label{eq_conv3} \end{equation} \begin{equation} I_{dc_i}(t+1) = I_{dc_i}(t) - o_{c_i}(t) + \sum_{l=1}^{N_{dv}} v_{ldc_i}(t) o_{dr_l}(t-\tau_{dr_l}), \label{eq_conv4} \end{equation} \begin{equation} I_{a_h}(t) = do_{h}(t) - \sum_{k=1}^{N_v} M_{v_k a_h} o_{v_k}(t). \label{eq_conv5} \end{equation} where $M_{v_ka_h}$ represents the amount of attributes of type $h$ carried by vehicle of type $k$. By introducing fleet modularity, several additional operation actions are available, in terms of \begin{enumerate} \item {Vehicle assembly, $\bm{o}_a = [o_{a_1},o_{a_2},...,o_{a_{N_v}}] $, where $o_{a_k}$ indicates the number of vehicles of type $k$ to be assembled,} \item {Vehicle disassembly, $\bm{o}_d = [o_{d_1},o_{d_2},...,o_{d_{N_v}}] $, where $o_{d_k}$ indicates the number of vehicles of type $k$ to be disassembled,} \item {Vehicle reconfiguration, $\bm{o}_r = [o_{1 2},o_{1 3},...,o_{{N_v}{N_v - 1}}] $, where $o_{kk'}$ indicates the number of vehicles of type $k$ to be reconfigured to vehicles of type $k'$,} \item {Damaged vehicle disassembly, $\bm{o}_{dd} = [o_{dd_1},...,o_{dd_{N_{dv}}}]$, where $o_{dl}$ indicates the damaged vehicles of index $l$ to be disassembled.} \end{enumerate} With consideration of these actions, the governing equations of the dynamics become \begin{equation} \begin{split} I_{v_k}(t+1) = & I_{v_k}(t) - o_{v_k}(t) + a_{r_k}(t) + \sum_{{k'}\neq k}^{N_v}o_{k' k}(t-\tau_{k' k}) - \sum_{{k'}\neq k}^{N_v}o_{k k'}(t) \\ & + \sum_{l=1}^{N_{dv}} v_{lkt}(t)o_{dr_l}(t - \tau_{dr_l}) + o_{a_k}(t-\tau_{a_k}) - o_{d_k}(t), \end{split} \label{eq_mod1} \end{equation} \begin{equation} \begin{split} I_{c_i}(t+1) & = I_{c_i}(t) + o_{c_i}(t-\tau_{c_i}) - \sum_{k=1}^{N_v} M_{v_k c_i} [o_{a_k}(t) - o_{d_k}(t-\tau_{d_k}) \\ & + \sum_{{k'}\neq k}^{N_v}o_{k' k}(t) - \sum_{{k'}\neq k}^{N_v}o_{k k'}(t-\tau_{k k'})] + \sum_{l=1}^{N_{dv}} v_{lc_i}(t) o_{dd_l}(t - \tau_{dd_l}), \end{split} \label{eq_mod2} \end{equation} \begin{equation} I_{dv_l}(t+1) = I_{dv_l}(t) - o_{dr_l}(t) - o_{dd_l}(t) + a_{d_l}(t), \label{eq_mod3} \end{equation} \begin{equation} \begin{split} I_{dc_i}(t+1) = I_{dc_i}(t) - o_{c_i}(t) + &\sum_{l=1}^{N_{dv}} v_{ldc_i}(t) o_{dr_l}(t - \tau_{dr_l}) + \sum_{l=1}^{N_{dv}} v_{ldc_i}(t) o_{dd_l}(t - \tau_{dd_l}), \end{split} \label{eq_mod4} \end{equation} \begin{equation} I_{a_h}(t) = do_{h}(t) - \sum_{k=1}^{N_v} M_{v_k a_h} o_{v_k}(t). \label{eq_mod5} \end{equation} where $M_{v_kc_i}$ represents the number of components of type $i$ in vehicle of type $k$. Because of the delays in operation actions, current inventory stocks might be influenced by previously-determined actions. In other words, the current actions may impact the stock level in the future. Thus, we define the state of the system by all inventory statuses, $\bm{I}_{t}(t)=[\bm{I}_v, \bm{I}_c, \bm{I}_{dv}, \bm{I}_{dc}, \bm{I}_a]^T$, that might be influenced by current actions, $\bm{s}(t)$, i.e., \begin{equation} \bm{s}(t) = [\bm{I}_{t}(t), \bm{I}_{t+1}(t), \bm{I}_{t+2}(t), \bm{I}_{t+3}(t)...\bm{I}_{t+\tau_{max}}(t)]^T. \label{state} \end{equation} We use input matrices $\bm{B}_\tau(t)$ to connect the current actions at time $t$ to inventory level at a later time $t+\tau$. Furthermore, the damage in stocks keeps changing over time. The matrices that connect to previous states $\bm{A}(t)$, actions $\bm{B}_\tau(t)$ and inputs $\bm{C}(t)$ are also time-varying matrices. Thus, the system dynamics for both fleets can be written as \begin{equation} \bm{I}(t+1) = \bm{I}(t) + \sum_{\tau=0}^{\tau_{max}} \big[ \bm{B}^c_\tau (t) \bm{o}^c (t - \tau)\big] + \bm{C}^c (t) \mathbf{a} (t- \tau) ,\label{eq_conv} \end{equation} \begin{equation} \bm{I}(t+1) = \bm{I}(t) + \sum_{\tau=0}^{\tau_{max}} \big[ \bm{B}^m_\tau (t) \bm{o}^m (t - \tau)\big] + \bm{C}^m (t) \mathbf{a} (t- \tau), \label{eq_mod} \end{equation} where \begin{equation} \bm{o}^c (t) = [\bm{o}_v(t), \bm{o}_{dr}(t), \bm{o}_c(t)]^T, \end{equation} \begin{equation} \bm{o}^m (t) = [\bm{o}_v(t), \bm{o}_{dr}(t), \bm{o}_c(t), \bm{o}_a(t), \bm{o}_d(t), \bm{o}_r(t), \bm{o}_{dd}(t)]^T. \end{equation} Thus, a state space model can be created to record the influence from the actions at a single time point to the states in the short future. For both fleets, the dynamical system can be represented by \begin{equation} \bm{s}(t+1) = \bm{A}(t)\bm{s}(t) + \bm{B}(t)\bm{o}(t) + \bm{C}(t)\mathbf{a}(t), \label{eq_stockDyn} \end{equation} where \begin{equation} \bm{A}(t) = \begin{bmatrix} \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{I}_{{n_s(t)}\times {n_s(t)}} & \bm{0}_{{n_s(t)}\times {n_s(t)}} & \dots & \bm{0}_{{n_s(t)}\times {n_s(t)}} \\ \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{I}_{{n_s(t)}\times {n_s(t)}} & \dots & \bm{0}_{{n_s(t)}\times {n_s(t)}} \\ \hdotsfor{5} \\ \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{0}_{{n_s(t)}\times {n_s(t)}} & \dots & \bm{I}_{{n_s(t)}\times {n_s(t)}} \\ \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{0}_{{n_s(t)}\times {n_s(t)}} & \bm{0}_{{n_s(t)}\times {n_s(t)}} & \dots & \bm{I}_{{n_s(t)}\times {n_s(t)}} \\ \end{bmatrix}, \end{equation} \begin{equation} \bm{B}(t) = [\bm{B}_0(t),\bm{B}_1(t), \bm{B}_2(t), ... , \bm{B}_{\tau_{max}}(t)]^T, \end{equation} \begin{equation} \bm{C}(t) = [\bm{C}_0(t),\bm{C}_1(t), \bm{C}_2(t), ... , \bm{C}_{\tau_{max}}(t)]^T. \end{equation} \subsubsection{System Control} The goal of system control is to meet the received dispatch orders on time. In the decision-making process, predictions of future system states are always involved. For example, given several dispatch orders, one may want to know what are the influences from satisfying one order on satisfying others. Compared to classical control methodologies, e.g., PID control, model predictive control (MPC) makes better use of future information and adapts to the system changes \citep{li2017robustness}. We separate this section into two parts where we discuss future state prediction first and optimization of operation decisions second. \subsubsection*{Future State Prediction} Because of time delays in the operation actions, the operation decisions made at the current time have to guarantee the match between the attributes of the dispatched convoy and the ordered attributes. Given \begin{enumerate} \item Current system states $\bm{s}(t)= [\bm{s}(t+1), \bm{s}(t+2), \bm{s}(t+3), ... ,\bm{s}(t+t_p)]^T$ , where $t_p$ is the planning horizon, \item Operation actions in the future $\underrightarrow{\bm{o}}_{t} = [\bm{o}(t), \bm{o}(t+1), \bm{o}(t+2), ... ,\bm{o}(t+t_p-1)]^T$, \item System input, $\underrightarrow{\mathbf{a}}_{t} = [\mathbf{a}(t), \mathbf{a}(t+1), \mathbf{a}(t+2), ... ,\mathbf{a}(t+t_p-1)]^T$, \end{enumerate} The future system states $\underrightarrow{\bm{s}}_{t+1}$ are predictable by iteratively using Eq.~\ref{eq_stockDyn}. Thus, one can express $\underrightarrow{\bm{s}}_{t+1}$ as a function of $\underrightarrow{\bm{o}}_{t}$ as \begin{equation} \underrightarrow{\bm{s}}_{t+1} = \bm{P}(t) \bm{s}(t) + \bm{H}(t) \underrightarrow{\bm{o}}_{t} + \bm{G}(t) \underrightarrow{\mathbf{a}}_{t} \label{eq_future} \end{equation} \begin{equation} \bm{P}(t) = [\bm{A}(t),\bm{A}^2(t), \bm{A}^3(t), ... , \bm{A}^n(t)], \end{equation} where \begin{equation} \bm{H}(t)= \begin{bmatrix} \bm{B}(t) & \bm{0}& \dots &\bm{0}\\ \bm{A}(t)\bm{B}(t) & \bm{B}(t) & \dots &\bm{0}\\ \hdotsfor{4}\\ \bm{A}(t)^{t_p-1}\bm{B}(t)&\bm{A}^{t_p-2}(t)\bm{B}(t)& \dots &\bm{B}(t)\\ \end{bmatrix}, \end{equation} \begin{equation} \bm{G}(t)= \begin{bmatrix} \bm{C}(t) & \bm{0}& \dots &\bm{0}\\ \bm{A}(t)\bm{C}(t) & \bm{C}(t) & \dots &\bm{0}\\ \hdotsfor{4}\\ \bm{A}(t)^{t_p-1}\bm{C}(t)&\bm{A}^{t_p-2}(t)\bm{C}(t)& \dots &\bm{C}(t)\\ \end{bmatrix}, \end{equation} with $\bm{P}(t)$ being the matrix that connects the future system outputs with current system states. $\bm{H}(t)$ and $\bm{G}(t)$ being the matrix that connects the system outputs with the future operation actions and inputs respectively. The dynamic model keeps updating to ensure we optimize the operation actions based on the most recent system status. \subsubsection*{Cost Function} The optimization of the fleet operation originates from two facts: 1. a convoy with insufficient attributes suffers a remarkable risk of mission failure; 2. a convoy with redundant attributes can also deteriorate the overall fleet performance from utility reduction. Furthermore, we consider several operational costs that may be significant in realistic fleet operations. As a summary, the costs of interest are \begin{enumerate} \item {Attribute redundancy cost for type $h$ attribute, $c_{o_h}$,} \item {Attribute insufficiency cost for type $h$ attribute, $c_{u_h}$,} \item {ADR action cost: assembly of vehicle of type $k$, $c_{a_k}$, disassembly of vehicle of type $k$, $c_{d_k}$, vehicle reconfiguration from type $k$ to type $k'$, $c_{kk'}$,} \item {Recovery cost: repair a component of type $i$ $c_{c_i}$, repair a damaged vehicle of index $l$, $c_{dr_l}$, disassembly a damaged vehicle of index $l$, $c_{dd_l}$,} \item {Inventory holding cost for holding a healthy/damaged component $c_{hc_i}/c_{hv_k}$, and a healthy/damaged vehicle $c_{hdc_i}, c_{hdv_l}$.} \end{enumerate} The cost function can be expressed as \begin{equation} \begin{aligned} J & & = \sum_{\tau=t}^{\tau=t+T_p-1} & \Big[\sum_{h=1}^{N_a} c_{o_h} I_{a_h}^+(\tau) + \sum_{h=1}^{N_a} c_{u_h} I_{a_h}^-(\tau) + \sum_{l=1}^{N_{dv}} (c_{dr_l} o_{dr_l}(\tau) + c_{dd_l} o_{dd_l}(\tau)) \\ & & & + \sum_{k=1}^{N_v} [c_{a_k} o_{a_k}(\tau) + c_{d_k} o_{d_k}(\tau)) + \sum_{k'\neq k}^{N_v} c_{k k'} o_{k k'}(\tau)] + \sum_{i=1}^{N_c} c_{c_i} o_{c_i}(\tau)\Big]\\ & & + \sum_{\tau=t+1}^{\tau=t+T_p} & [ \sum_{i=1}^{N_c} c_{hc_i} I_{c_i}(\tau) + \sum_{k=1}^{N_v} c_{hv_k} I_{v_k}(\tau)+ \sum_{i=1}^{N_c} c_{hdc_i} I_{dc_i}(\tau) + \sum_{l=1}^{N_{dv}}c_{hdv_l} o_{dv_l}(\tau)], \\ \end{aligned} \end{equation} where $I_{a_h}^+(\tau)$ and $I_{a_h}^-(\tau)$ are non-negative auxiliary variables representing the positive part and negative parts of $I_{a_h}(\tau)$, which satisfies \begin{equation} I_{a_h}(\tau) = I_{a_h}^+(\tau) - I_{a_h}^-(\tau). \end{equation} We record the insufficient and the redundant attributes during the planning horizon as $\underrightarrow{\bm{I}^+_{a}}$ and $\underrightarrow{\bm{I}^-_{a}}$ respectively. The holding costs and the actions-related costs are also aggregated as $\underrightarrow{\bm{c}_h}$ and $\underrightarrow{\bm{c}_a}$. By substituting in Eq.~\ref{eq_future}, we create a mixed-integer programming model to optimize operational decisions as \begin{equation} \begin{aligned} & \min_{\underrightarrow{\bm{I}^+_{a}},\underrightarrow{\bm{I}^-_{a}}, \underrightarrow{\bm{o}}_{t}} & & \bm{c}_{o} \underrightarrow{\bm{I}_{a}}^+ + \bm{c}_{u} \underrightarrow{\bm{I}_{a}}^- + \underrightarrow{\bm{c}_h} \bm{X}_s[\bm{P}(t)\bm{s}(t) + \bm{H}(t) \underrightarrow{\bm{o}}_{t} ] \\ & \text{s.t.} & & (a) \; \underrightarrow{\bm{o}}_{t} \geq 0 \;\; \text{and integer} \\ & & & (b) \; \bm{X}_s \underrightarrow{\bm{s}}_{t+1} \geq 0 \\ & & & (c) \; \sum \bm{o}(t) \leq \bar{P} , \quad \forall t \\ & & & (d) \; \underrightarrow{\bm{I}^+_{a}} - \underrightarrow{\bm{I}^-_{a}} = \bm{X}_a[\bm{P}(t) \bm{s}(t) + \bm{H}(t) \underrightarrow{\bm{o}}_{t} + \bm{G}(t)\underrightarrow{\mathbf{a}}_{t}]\\ & & & (e) \; o_{dd_l}(t) + o_{dr_l}(t) \leq 1 , \quad \forall t,\\ \end{aligned} \end{equation} where $\bm{X}_s$ and $\bm{X}_a$ record indices of inventory stocks and remaining dispatch orders in the created state structure respectively. Constraint (a) ensures that all operational decisions are non-negative and integer; (b) indicates that the amount of inventory stocks are non-negative; (c) ensures that the ADR actions are always constrained by the maximum action capacity $\bar{P}$; (d) preserves the balance between auxiliary variables and remaining orders to be satisfied; (e) specifies that each damaged vehicle can only be recovered by one recovery strategy. As the cost function and constraints are linear and the number of decision variables is large, we first implement a cutting-plan to reduce the decision space and then use an integer programming solver to obtain the solution. The time required for decision-making at each time point is less than 1 second for operating 5 types of modular vehicles with a planning horizon of 12 hours. \section{Numerical Illustrations} In this section, we provide numerical illustrations for a generalized mission scenario to study the different impacts of modularity on fleet performance. In general, it may be difficult to know all fleet parameters accurately. However, it may be possible to obtain reasonable estimates for these parameters. In this study, we assume the resources provided to the fleet operation are fixed and equal, which can be imagined as a competition of two fleets on an isolated island. One of them is a conventional fleet; the other is a modular fleet. Initially, ten of each type of vehicles and components are provided to both fleets. Demands randomly occur at the battlefield based on a Poisson distribution with a time interval of 10 hours. Demands include personnel capacity $d_p$, material capacity $d_m$ and firepower $d_f$, which are generated based on a Gaussian distribution as shown in Eq.~\ref{eq_demand1}, \ref{eq_demand2}, \ref{eq_demand3}. \begin{eqnarray} & d_p \sim \mathcal{N}(40,\,15) \label{eq_demand1}\\ & d_m \sim \mathcal{N}(50,\,20) \label{eq_demand2}\\ & d_f \sim \mathcal{N}(30,\,10) \label{eq_demand3} \end{eqnarray} Because of the lack of diversity in the existing designs of modular vehicles, we borrow five types of modular vehicles as well as six types of modules from \citep{li2017robustness}. The attributes carried by each vehicle are summarized in Tab.~\ref{tab_v2a}. \begin{table}[h!] \caption{Mapping between vehicles and attributes} \label{tab_v2a} \begin{center} \begin{tabular}{c c c c c c} & & \\ \hline \hline Vehicle Type& $1$ & $2$ & $3$ & $4$ & $5$ \\ \hline \hline Firepower & $1$ & $3$ & $8$ & $0$ & $6$ \\ Material capacity & $2$ & $6$ & $2$ & $2$ & $8$ \\ Personnel capacity & $4$ & $1$ & $0$ & $10$ & $5$ \\ \hline \end{tabular} \end{center} \end{table} The costs of insufficiency and redundancy are created based on heuristic estimates. For example, convoys usually suffer a high risk of failure when the attributes of the dispatched convoys are less than the attributes ordered. Thus, the cost for attribute insufficiency is much higher than the cost of attribute redundancy, i.e., $c_{o_h} \gg c_{u_h}, \; \forall h=1,2,...,N_a$. The costs for operation actions are created based on their difficulty and the time required for their execution. We assign the time required for module assembly and disassembly as constant vectors $\mathbf{\tau}_\mathrm{ma}$ and $\mathbf{\tau}_\mathrm{md}$. Vehicle assembly and disassembly times $\mathbf{\tau}_\mathrm{va}$ and $\mathbf{\tau}_\mathrm{vd}$ are calculated by summing up all the times required for assembly of all its components. Similarly, for repair and reconfiguration, we sum all the time needed for actions to process each individual component in the vehicle. We assume that the interfaces between components are well-designed to achieve quick vehicle reconfiguration, where assembly and disassembly time for all types of components are of 1 hour and of 0.5 hour respectively. We assume that ADR actions are executed at working stations. Thus, the number of stations determines the amount of available working capacity. In this study, the number of available work stations for both fleets is 15. We execute a discrete event model to simulate the fleet competition for three years. We separate the 3-year mission into two parts, namely the stochastic stage ($1^{st}$ year), and the learning stage ($2^{nd}$ and $3^{rd}$ year). In the stochastic stage, the dispatch agent randomly chooses a dispatch strategy based on Tab.~\ref{tab_strategies} and passes this decisions to the base agent. First-year operations generate time-series data, including event history and feasibility records, which are important inputs for the learning models. Training of the learning models for each fleet starts at the beginning of the learning stage, where the inference agent and the dispatch agent begin to make decisions based on the historical adversary's behavior. Learning models are also updated monthly to ensure they reflect the adversary's latest behaviors. \section{Fleet comparison} In this section, we compare the performance of the modular fleet and the conventional fleet in the two stages. As one of the important metrics in measuring the fleet performance, we first compare the probability of winning based on the results from multiple simulations. These results are shown in Fig.~\ref{fig_winComp}. Conventional fleet outperforms the modular fleet during the stochastic stage. However, once the intelligence of the agents is introduced, i.e., once both fleets enter the learning stage, the modular fleet wins more often. To explain these results, we first compare the order success, and then compare the feasibility the attributes carried by the actual dispatched convoys of each of the fleets, damage suffered, and the estimation accuracy in inference of the adversary. \begin{figure} [h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.55]{winComp-eps-converted-to.pdf} \end{center} \caption{Comparison of win rate between modular fleet and conventional fleet} \label{fig_winComp} \end{figure} During the stochastic stage, dispatch agents from both fleets place dispatch orders based on a randomly selected strategy. The strategy selection and the order achievement are same for both fleets. To explain the better performance of the conventional fleet, it is necessary to explore the accuracy of both fleets in satisfying the dispatch orders. Mismatched attributes dramatically change the fleet performance: convoys with insufficient attributes may significantly raise the failure rate and the damage; convoys with redundant attributes may increase the win rate slightly, but they can also contribute to insufficiency in the short future because of limited resources. Thus, we calculate the amount of overused and insufficient convoy attributes during every month and compare the dispatch accuracy in Fig.~\ref{fig_errorComp}. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.60]{errorComp-eps-converted-to.pdf} \end{center} \caption{Comparison of total mismatched convoy attributes in all types} \label{fig_errorComp} \end{figure} Compared to the modular fleet, the conventional fleet suffers remarkable redundancy. The higher redundancy comes from the rigidity of the conventional fleet operation. The modular fleet can real-timely reconfigure itself to fit the dispatch orders. However, the conventional fleet can only wait for vehicles to return from the battlefield or for damaged vehicles to be recovered. This limitation hampers the ability of the conventional fleet to satisfy the dispatch orders. Once proper vehicles are scarce, the conventional fleet has to use improper vehicles with little desired attributes to avoid insufficiency. This rigidity in fleet operation is beneficial in improving the success rate during the stochastic stage, because the adversary cannot be aware of the unexpected additional attributes. However, once the adversary starts to learn the behavior of the conventional fleet, this advantage no longer exists. From failures during the stochastic stage, the modular fleet learns that the conventional fleet intents to dispatch convoys with superfluous attributes. As a solution, modular fleet increases the attributes of their dispatch orders correspondingly. These redundant attributes in conventional convoys are powerless in reacting to the intelligent response of the modular fleet and the conventional fleet cannot stay ahead. Besides a better understanding of adversary's behavior, intelligent agents also improve their understanding of the game over time. To explore this process, we first compare the maximum values of convoy attributes dispatched in each month, and compare them of both fleets in Fig.~\ref{fig_attriComp}. After entering the learning stage, both fleets raise their dispatched attributes in all types, especially in firepower. Both fleets learn that the sufficient delivered attributes, especially for firepower, are the key to win the event through the trained success rate model $f_s^b$. With additional flexibility in fleet operation, modular fleet can always form a convoy that could provide required attributes with high safety level, which may not be achievable by the conventional fleet. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.55]{attriComp-eps-converted-to.pdf} \end{center} \caption{Maximum value of attributes carried by convoys in different months} \label{fig_attriComp} \end{figure} The swift reconfiguration of the modular fleet leads to a dramatic increase in the damage to the adversary in the first few months of the learning stage, as shown in Fig.~\ref{fig_dmgComp}. Although the conventional fleet increases the firepower to fight back, the limitation in the vehicle structure results in a lower upper limit in the firepower. Thus, the difference in the fleet ability makes conventional fleet suffer higher damage from more dispatch orders, which forces the conventional fleet to operate in sub-healthy conditions for a long time. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.60]{dmgComp-eps-converted-to.pdf} \end{center} \caption{Module damages occur during each month} \label{fig_dmgComp} \end{figure} The strategies used in the learning stage are also distinct between the two fleets. Fig.~\ref{fig_strategy} compares the proportion of strategies adopted by the two fleets. After learning, both fleets prefer to select the defense strategy with large amount of firepower and a fair amount of capacity, i.e., strategies 8,9. Because of the flexibility of the fleet structure, the modular fleet can more easily adapt to the vehicle damage and the adversary's behavior, which leads to a better balance between different types of vehicles to perform a stronger strategy. The defense strategy selection also impacts the attack strategy. Compared to the modular fleet, the conventional fleet is much more likely to give up missions because of resource insufficiency. This weakness makes the modular fleet confident in dispatching little or even no combat vehicles and win events. As evidence, the proportion of strategy 1 used by the modular fleet is much higher than that by the conventional fleet. In addition, the modular fleet is more capable of performing aggressive strategies, i.e., strategies 8, 9 and 10, more often than the conventional fleet once a strong adversary is learned. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.6]{strategyAllo-eps-converted-to.pdf} \end{center} \caption{Proportion of the attack and defense strategies during different stages} \label{fig_strategy} \end{figure} To further investigate the improved performance of the modularized fleet, we also compared the inference accuracy between the two fleets. We choose the mean square error (MSE) between forecasted and actual convoy attributes as the metric to quantify the inference accuracy. As can be seen from the comparison in Fig.~\ref{fig_infComp}, inference errors are significantly higher at the beginning of the learning stage, because agents are trained by the data from the stochastic dispatch, which contributes little to forecasting the behavior of an intelligent adversary. Once both fleets enter learning stage, the dispatch convoy orders made by trained learning models, $f_f$ and $f_s$, make adversary's behaviors are more explainable. As a result, inference errors are significantly reduced in the following four months. However, the inference errors keep fluctuating during the rest of the learning stage, because both fleets keep checking and countering each other's behavior. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.60]{infComp-eps-converted-to.pdf} \end{center} \caption{Comparison of inference accuracy between the modular fleet and the conventional fleet} \label{fig_infComp} \end{figure} The results also show that it is easier to infer the strategy of the conventional fleet than that of the modular fleet, especially in the attribute of firepower. This phenomenon originates from the higher freedom in decision-making due to modularity. As a defender, a fleet usually needs to prepare a convoy with all types of attributes to satisfy the demands. With limited vehicle stocks, the decision maker of the conventional fleet has constrained choices of strategy. However, for the modular fleet, the decision maker can vary the dispatch strategy by real-time vehicle reconfiguration, i.e., reconfigure vehicles to switch from strategy 4 (higher safety stocks) to strategy 8 (more firepower). However, the burden of modularity is also significant, which is the high acquisition of capacity. According to Fig.~\ref{fig_capaComp}, the modular fleet always requires more working resources, i.e., personnel and assembly machine, than conventional fleet because of additional ADR actions. It can also be observed that the required working resources increase significantly once entering the learning stage, which are due to damages from smarter strikes by the adversary. The higher losses in the conventional fleet also shrink the difference in resource requirement at learning stage. In this study, we only tested the fleet performance at a certain capacity. Studies investigating on the influence of capacity can be found in the literature \citep{li2017robustness, li2018IABS}. \begin{figure}[h!] \begin{center} \setlength{\unitlength}{0.012500in}% \includegraphics[scale=0.60]{capaComp-eps-converted-to.pdf} \end{center} \caption{Comparison of machine requirements between modular fleet and conventional fleet} \label{fig_capaComp} \end{figure} \section{Conclusions} In this paper, we investigated the benefits and burdens from fleet modularization by simulating an attacker-defender game between a modular fleet and a conventional fleet. A noval intelligent agent based approach to proposed for battle-field decision-making process by combining the optimization and machine learning techniques. With continuous retraining, the model is capable to capture the evolution in strategies as a reaction to the adversarial behavior, which reveals game-theoretical behaviors. We simulated the fleet competition for three years which are divided into a stochastic stage and a learning stage. By contrasting the simulation results from the two fleets, we found that the conventional fleet leads when both fleets are selecting strategies stochastically; the modular fleet outperforms the conventional fleet once the intelligence of the decision makers is considered. With additional operational flexibility from assembly, disassembly and reconfiguration actions, the modular fleet exhibits a better adaptability to adversarial actions, a stronger convoy, and a more significant unpredictability from the additional flexibility in operation. \section*{Acknowledgment} This research has been supported by the Automotive Research Center, a US Army Center of Excellence in Modeling and Simulation of Ground Vehicle Systems, headquartered at the University of Michigan. This support is gratefully acknowledged. The authors are solely responsible for opinions contained herein.

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\section{Introduction} This paper deals with the coupling of finite element and boundary element methods to solve the system of the eddy current equations in the whole 3D spatial space and the Landau-Lifshitz-Gilbert equation (LLG), the so-called ELLG system or equations. The system is also called the quasi-static Maxwell-LLG (MLLG) system. The LLG is widely considered as a valid model of micromagnetic phenomena occurring in, e.g., magnetic sensors, recording heads, and magneto-resistive storage device~\cite{Gil,LL,prohl2001}. Classical results concerning existence and non-uniqueness of solutions can be found in~\cite{as,vis}. In a ferro-magnetic material, magnetisation is created or affected by external electro-magnetic fields. It is therefore necessary to augment the Maxwell system with the LLG, which describes the influence of ferromagnet; see e.g.~\cite{cim, KruzikProhl06, vis}. Existence, regularity and local uniqueness for the MLLG equations are studied in~\cite{CimExist}. Throughout the literature, there are various works on numerical approximation methods for the LLG, ELLG, and MLLG equations~\cite{alouges,alok,bako,bapr,cim,ellg,thanh} (the list is not exhausted), and even with the full Maxwell system on bounded domains~\cite{BBP,MLLG}, and in the whole~${\mathbb R}^3$~\cite{CarFab98}. Originating from the seminal work~\cite{alouges}, the recent works~\cite{ellg,thanh} consider a similar numeric integrator for a bounded domain. This work studies the ELLG equations where we consider the electromagnetic field on the whole ${\mathbb R}^3$ and do not need to introduce artificial boundaries. Differently from~\cite{CarFab98} where the Faedo-Galerkin method is used to prove existence of weak solutions, we extend the analysis for the integrator used in~\cite{alouges, ellg, thanh} to a finite-element/boundary-element (FEM/BEM) discretisation of the eddy current part on ${\mathbb R}^3$. This is inspired by the FEM/BEM coupling approach designed for the pure eddy current problem in~\cite{dual}, which allows to treat unbounded domains without introducing artificial boundaries. Two approaches are proposed in~\cite{dual}: the so-called ``magnetic (or $\boldsymbol{H}$-based) approach'' which eliminates the electric field, retaining only the magnetic field as the unknown in the system, and the ``electric (or $\boldsymbol{E}$-based) approach'' which considers a primitive of the electric field as the only unknown. The coupling of the eddy-current system with the LLG dictates that the first approach is more appropriate; see~\eqref{eq:strong}. The main result of this first part is the weak convergence of the discrete approximation towards a weak solution without any condition on the space and time discretisation. This also proves the existence of weak solutions. The remainder of this part is organised as follows. Section~\ref{section:model} introduces the coupled problem and the notation, presents the numerical algorithm, and states the main result of this part of the paper. Section~\ref{sec:pro} is devoted to the proof of this main result. Numerical results are presented in Section~\ref{section:numerics}. The second part of this paper~\cite{FeiTraII} proves a~priori estimates for the proposed algorithm. \section{Model Problem \& Main Result}\label{section:model} \subsection{The problem}\label{subsec:pro} Consider a bounded Lipschitz domain $D\subset {\mathbb R}^3$ with connected boundary $\Gamma$ having the outward normal vector $\boldsymbol{n}$. We define $D^\ast:={\mathbb R}^3\setminus\overline D$, $D_T:=(0,T)\times D$, $\Gamma_T := (0,T)\times\Gamma$, $D_T^{\ast}:=(0,T)\times D^\ast$, and ${\mathbb R}^3_T := (0,T)\times{\mathbb R}^3$ for $T>0$. We start with the quasi-static approximation of the full Maxwell-LLG system from~\cite{vis} which reads as \begin{subequations}\label{eq:strong} \begin{alignat}{2} \boldsymbol{m}_t - \alpha\boldsymbol{m}\times\boldsymbol{m}_t &= -\boldsymbol{m} \times \boldsymbol{H}_{\rm eff} &&\quad\text{in }D_T,\label{eq:llg}\\ \sigma\boldsymbol{E} -\nabla\times\boldsymbol{H}&=0&&\quad\text{in }{\mathbb R}^3_T,\label{eq:MLLG1}\\ \mu_0\boldsymbol{H}_t +\nabla \times\boldsymbol{E} &=-\mu_0\widetilde\boldsymbol{m}_t&&\quad\text{in }{\mathbb R}^3_T,\label{eq:MLLG2}\\ {\rm div}(\boldsymbol{H}+\widetilde\boldsymbol{m})&=0 &&\quad\text{in }{\mathbb R}^3_T,\label{eq:MLLG3}\\ {\rm div}(\boldsymbol{E})&=0&&\quad\text{in }D^\ast_T, \end{alignat} \end{subequations} where $\widetilde \boldsymbol{m}$ is the zero extension of $\boldsymbol{m}$ to ${\mathbb R}^3$ and $\boldsymbol{H}_{\rm eff}$ is the effective field defined by $\boldsymbol{H}_{\rm eff}= C_e\Delta\boldsymbol{m}+\boldsymbol{H}$ for some constant $C_e>0$. Here the parameter $\alpha>0$ and permability $\mu_0\geq0$ are constants, whereas the conductivity $\sigma$ takes a constant positive value in $D$ and the zero value in $D^\ast$. Equation~\eqref{eq:MLLG3} is understood in the distributional sense because there is a jump of~$\widetilde\boldsymbol{m}$ across~$\Gamma$. It follows from~\eqref{eq:llg} that $|\boldsymbol{m}|$ is constant. We follow the usual practice to normalise~$|\boldsymbol{m}|$ (and thus the same condition is required for $|\boldsymbol{m}^0|$). The following conditions are imposed on the solutions of~\eqref{eq:strong}: \begin{subequations}\label{eq:con} \begin{alignat}{2} \partial_n\boldsymbol{m}&=0 && \quad\text{on }\Gamma_T,\label{eq:con1} \\ |\boldsymbol{m}| &=1 && \quad\text{in } D_T, \label{eq:con2} \\ \boldsymbol{m}(0,\cdot) &= \boldsymbol{m}^0 && \quad\text{in } D, \label{eq:con3} \\ \boldsymbol{H}(0,\cdot) &= \boldsymbol{H}^0 && \quad\text{in } {\mathbb R}^3, \\ \boldsymbol{E}(0,\cdot) &= \boldsymbol{E}^0 && \quad\text{in } {\mathbb R}^3, \\ |\boldsymbol{H}(t,x)|&=\mathcal{O}(|x|^{-1}) && \quad\text{as }|x|\to \infty, \end{alignat} \end{subequations} where $\partial_n$ denotes the normal derivative. The initial data~$\boldsymbol{m}^0$ and~$\boldsymbol{H}^0$ satisfy~$|\boldsymbol{m}^0|=1$ in~$D$ and \begin{align}\label{eq:ini} \begin{split} {\rm div}(\boldsymbol{H}^0 + \widetilde \boldsymbol{m}^0)&=0\quad\text{in }{\mathbb R}^3. \end{split} \end{align} Below, we focus on an $\boldsymbol{H}$-based formulation of the problem. It is possible to recover $\boldsymbol{E}$ once $\boldsymbol{H}$ and $\boldsymbol{m}$ are known; see~\eqref{eq:el} \subsection{Function spaces and notations}\label{subsec:fun spa} Before introducing the concept of weak solutions to problem~\eqref{eq:strong}--\eqref{eq:con} we need the following definitions of function spaces. Let $\Ltwo{D}:=L^2(D;{\mathbb R}^3)$ and $\Hcurl{D}:=\set{\boldsymbol{w}\in \Ltwo{D}}{\nabla\times\boldsymbol{w}\in\Ltwo{D}}$. We define $H^{1/2}(\Gamma)$ as the usual trace space of $H^1(D)$ and define its dual space $H^{-1/2}(\Gamma)$ by extending the $L^2$-inner product on $\Gamma$. For convenience we denote \[ {\mathcal X}:=\set{(\boldsymbol{\xi},\zeta)\in\Hcurl{D}\times H^{1/2}(\Gamma)}{\boldsymbol{n}\times\boldsymbol{\xi}|_\Gamma =\boldsymbol{n}\times \nabla_\Gamma\zeta \text{ in the sense of traces}}. \] Recall that~$\boldsymbol{n}\times\boldsymbol{\xi}|_\Gamma$ is the tangential trace (or twisted tangential trace) of~$\boldsymbol{\xi}$, and $\nabla_\Gamma\zeta$ is the surface gradient of~$\zeta$. Their definitions and properties can be found in~\cite{buffa, buffa2}. Finally, if $X$ is a normed vector space then $L^2(0,T;X)$, $H^m(0,T;X)$, and $W^{m,p}(0,T;X)$ denote the usual corresponding Lebesgues and Sobolev spaces of functions defined on $(0,T)$ and taking values in $X$. We finish this subsection with the clarification of the meaning of the cross product between different mathematical objects. For any vector functions $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}$ we denote \begin{gather*} \boldsymbol{u}\times\nabla\boldsymbol{v} := \left( \boldsymbol{u}\times\frac{\partial\boldsymbol{v}}{\partial x_1}, \boldsymbol{u}\times\frac{\partial\boldsymbol{v}}{\partial x_2}, \boldsymbol{u}\times\frac{\partial\boldsymbol{v}}{\partial x_3} \right), \quad \nabla\boldsymbol{u}\times\nabla\boldsymbol{v} := \sum_{i=1}^3 \frac{\partial\boldsymbol{u}}{\partial x_i} \times \frac{\partial\boldsymbol{v}}{\partial x_i} \\ \intertext{and} (\boldsymbol{u}\times\nabla\boldsymbol{v})\cdot \nabla\boldsymbol{w} := \sum_{i=1}^3 \left( \boldsymbol{u}\times\frac{\partial\boldsymbol{v}}{\partial x_i} \right) \cdot \frac{\partial\boldsymbol{w}}{\partial x_i}. \end{gather*} \subsection{Weak solutions}\label{subsec:wea sol} A weak formulation for~\eqref{eq:llg} is well-known, see e.g.~\cite{alouges, thanh}. Indeed, by multiplying~\eqref{eq:llg} by~$\boldsymbol{\phi}\in C^\infty(D_T;{\mathbb R}^3)$, using~$|\boldsymbol{m}|=1$ and integration by parts, we deduce \[ \alpha\dual{\boldsymbol{m}_t}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T} + \dual{\boldsymbol{m}\times\boldsymbol{m}_t}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}+C_e \dual{\nabla\boldsymbol{m}}{\nabla(\boldsymbol{m}\times\boldsymbol{\phi})}_{D_T} = \dual{\boldsymbol{H}}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}. \] To tackle the eddy current equations on ${\mathbb R}^3$, we aim to employ FE/BE coupling methods. To that end, we employ the \emph{magnetic} approach from~\cite{dual}, which eventually results in a variant of the \emph{Trifou}-discretisation of the eddy-current Maxwell equations. The magnetic approach is more or less mandatory in our case, since the coupling with the LLG equation requires the magnetic field rather than the electric field. Multiplying~\eqref{eq:MLLG2} by~$\boldsymbol{\xi}\in C^{\infty}(D,{\mathbb R}^3)$ satisfying~$\nabla\times\boldsymbol{\xi}=0$ in~$D^\ast$, integrating over~${\mathbb R}^3$, and using integration by parts, we obtain for almost all~$t\in[0,T]$ \[ \mu_0 \dual{\boldsymbol{H}_t(t)}{\boldsymbol{\xi}}_{{\mathbb R}^3} + \dual{\boldsymbol{E}(t)}{\nabla\times\boldsymbol{\xi}}_{{\mathbb R}^3} = -\mu_0 \dual{\boldsymbol{m}_t(t)}{\boldsymbol{\xi}}_{D}. \] Using~$\nabla\times\boldsymbol{\xi}=0$ in~$D^\ast$ and~\eqref{eq:MLLG1} we deduce \[ \mu_0 \dual{\boldsymbol{H}_t(t)}{\boldsymbol{\xi}}_{{\mathbb R}^3} + \sigma^{-1}\dual{\nabla\times\boldsymbol{H}(t)}{\nabla\times\boldsymbol{\xi}}_{D} = -\mu_0 \dual{\boldsymbol{m}_t(t)}{\boldsymbol{\xi}}_{D}. \] Since~$\nabla\times\boldsymbol{H}=\nabla\times\boldsymbol{\xi}=0$ in~$D^\ast$, there exists~$\varphi$ and~$\zeta$ such that~$\boldsymbol{H}=\nabla\varphi$ and~$\boldsymbol{\xi}=\nabla\zeta$ in~$D^\ast$. Therefore, the above equation can be rewritten as \[ \mu_0 \dual{\boldsymbol{H}_t(t)}{\boldsymbol{\xi}}_{D} + \mu_0 \dual{\nabla\varphi_t(t)}{\nabla\zeta}_{D^\ast} + \sigma^{-1}\dual{\nabla\times\boldsymbol{H}(t)}{\nabla\times\boldsymbol{\xi}}_{D} = -\mu_0 \dual{\boldsymbol{m}_t(t)}{\boldsymbol{\xi}}_{D}. \] Since~\eqref{eq:MLLG3} implies~${\rm div}(\boldsymbol{H})=0$ in~$D^\ast$, we have~$\Delta\varphi=0$ in~$D^\ast$, so that (formally)~$\Delta\varphi_t=0$ in~$D^\ast$. Hence integration by parts yields \begin{equation}\label{eq:Ht} \mu_0 \dual{\boldsymbol{H}_t(t)}{\boldsymbol{\xi}}_{D} - \mu_0 \dual{\partial_n^+\varphi_t(t)}{\zeta}_{\Gamma} + \sigma^{-1}\dual{\nabla\times\boldsymbol{H}(t)}{\nabla\times\boldsymbol{\xi}}_{D} = -\mu_0 \dual{\boldsymbol{m}_t(t)}{\boldsymbol{\xi}}_{D}, \end{equation} where~$\partial_n^+$ is the exterior Neumann trace operator with the limit taken from~$D^\ast$. The advantage of the above formulation is that no integration over the unbounded domain~$D^\ast$ is required. The exterior Neumann trace~$\partial_n^+\varphi_t$ can be computed from the exterior Dirichlet trace~$\lambda$ of~$\varphi$ by using the Dirichlet-to-Neumann operator~$\dtn$, which is defined as follows. Let $\gamma^-$ be the interior Dirichlet trace operator and $\partial_n^-$ be the interior normal derivative or Neumann trace operator. (The $-$ sign indicates the trace is taken from $D$.) Recalling the fundamental solution of the Laplacian $G(x,y):=1/(4\pi|x-y|)$, we introduce the following integral operators defined formally on $\Gamma$ as \begin{align*} \slp(\lambda):=\gamma^-\overline\slp(\lambda), \quad \dlp(\lambda):=\gamma^-\overline\dlp(\lambda)+\mfrac12, \quad\text{and}\quad \hyp(\lambda):=-\partial_n^-\overline\dlp(\lambda), \end{align*} where, for $x\notin\Gamma$, \begin{align*} \overline\slp(\lambda)(x) := \int_{\Gamma} G(x,y) \lambda(y)\,ds_y \quad\text{and}\quad \overline\dlp(\lambda)(x) :=\int_{\Gamma} \partial_{n(y)}G(x,y)\lambda(y)\,ds_y. \end{align*} Moreover, let $\dlp^\prime$ denote the adjoint operator of $\dlp$ with respect to the extended $L^2$-inner product. Then the exterior Dirichlet-to-Neumann map $\dtn\colon H^{1/2}(\Gamma)\to H^{-1/2}(\Gamma)$ can be represented as \begin{equation}\label{eq:dtn} \dtn = - \slp^{-1}(1/2-\dlp). \end{equation} Another more symmetric representation is \begin{equation}\label{eq:dtn2} \dtn = -(1/2-\dlp^\prime) \slp^{-1}(1/2-\dlp)-\hyp. \end{equation} Recall that~$\varphi$ satisfies~$\boldsymbol{H}=\nabla\varphi$ in~$D^\ast$. We can choose~$\varphi$ satisfying~$\varphi(x)=O(|x|^{-1})$ as $|x|\to\infty$. Now if~$\lambda=\gamma^+\varphi$ then~$\lambda_t=\gamma^+\varphi_t$. Since~$\Delta\varphi=\Delta\varphi_t=0$ in~$D^\ast$, and since the exterior Laplace problem has a unique solution we have~$\dtn\lambda=\partial_n^+\varphi$ and~$\dtn\lambda_t=\partial_n^+\varphi_t$. Hence~\eqref{eq:Ht} can be rewritten as \begin{equation}\label{eq:H lam t} \dual{\boldsymbol{H}_t(t)}{\boldsymbol{\xi}}_{D} - \dual{\dtn\lambda_t(t)}{\zeta}_{\Gamma} + \mu_0^{-1} \sigma^{-1}\dual{\nabla\times\boldsymbol{H}(t)}{\nabla\times\boldsymbol{\xi}}_{D} = - \dual{\boldsymbol{m}_t(t)}{\boldsymbol{\xi}}_{D}. \end{equation} We remark that if~$\nabla_\Gamma$ denotes the surface gradient operator on~$\Gamma$ then it is well-known that $ \nabla_\Gamma\lambda = (\nabla\varphi)|_{\Gamma} - (\partial_n^+\varphi)\boldsymbol{n} = \boldsymbol{H}|_{\Gamma} - (\partial_n^+\varphi)\boldsymbol{n}; $ see e.g.~\cite[Section~3.4]{Monk03}. Hence~$\boldsymbol{n}\times\nabla_\Gamma\lambda = \boldsymbol{n}\times\boldsymbol{H}|_{\Gamma}$. The above analysis prompts us to define the following weak formulation. \begin{definition}\label{def:fembemllg} A triple $(\boldsymbol{m},\boldsymbol{H},\lambda)$ satisfying \begin{align*} \boldsymbol{m} &\in \Hone{D_T} \quad\text{and}\quad \boldsymbol{m}_t|_{\Gamma_T} \in L^2(0,T;H^{-1/2}(\Gamma)), \\ \boldsymbol{H} &\in L^2(0,T;\Hcurl{D})\cap H^1(0,T;\Ltwo{D}), \\ \lambda &\in H^1(0,T;H^{1/2}(\Gamma)) \end{align*} is called a weak solution to~\eqref{eq:strong}--\eqref{eq:con} if the following statements hold \begin{enumerate} \item $|\boldsymbol{m}|=1$ almost everywhere in $D_T$; \label{ite:1} \item $\boldsymbol{m}(0,\cdot)=\boldsymbol{m}^0$, $\boldsymbol{H}(0,\cdot)=\boldsymbol{H}^0$, and $\lambda(0,\cdot)=\gamma^+ \varphi^0$ where~$\varphi^0$ is a scalar function satisfies $\boldsymbol{H}^0=\nabla\varphi^0$ in~$D^\ast$ (the assumption~\eqref{eq:ini} ensures the existence of~$\varphi^0$); \label{ite:2} \item For all $\boldsymbol{\phi}\in C^\infty(D_T;{\mathbb R}^3)$ \label{ite:3} \begin{subequations}\label{eq:wssymm} \begin{align} \alpha\dual{\boldsymbol{m}_t}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T} &+ \dual{\boldsymbol{m}\times\boldsymbol{m}_t}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}+C_e \dual{\nabla\boldsymbol{m}}{\nabla(\boldsymbol{m}\times\boldsymbol{\phi})}_{D_T} \nonumber \\ &= \dual{\boldsymbol{H}}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}; \label{eq:wssymm1} \end{align} \item There holds $\boldsymbol{n}\times \nabla_\Gamma\lambda = \boldsymbol{n}\times \boldsymbol{H}|_{\Gamma}$ in the sense of traces; \label{ite:4} \item For $\boldsymbol{\xi}\in C^\infty(D;{\mathbb R}^3)$ and $\zeta\in C^\infty(\Gamma)$ satisfying $\boldsymbol{n}\times\boldsymbol{\xi}|_{\Gamma}=\boldsymbol{n}\times\nabla_\Gamma\zeta$ in the sense of traces \label{ite:5} \begin{align}\label{eq:wssymm2} \dual{\boldsymbol{H}_t}{\boldsymbol{\xi}}_{D_T} -\dual{\dtn\lambda_t}{\zeta}_{\Gamma_T} + \sigma^{-1}\mu_0^{-1}\dual{\nabla\times\boldsymbol{H}}{\nabla\times\boldsymbol{\xi}}_{D_T} &=-\dual{\boldsymbol{m}_t}{\boldsymbol{\xi}}_{D_T}; \end{align} \end{subequations} \item For almost all $t\in[0,T]$ \label{ite:6} \begin{gather} \norm{\nabla \boldsymbol{m}(t)}{\Ltwo{D}}^2 + \norm{\boldsymbol{H}(t)}{\Hcurl{D}}^2 + \norm{\lambda(t)}{H^{1/2}(\Gamma)}^2 \nonumber \\ + \norm{\boldsymbol{m}_t}{\Ltwo{D_t}}^2 + \norm{\boldsymbol{H}_t}{\Ltwo{D_t}}^2 + \norm{\lambda_t}{H^{1/2}(\Gamma_t)}^2 \leq C, \label{eq:energybound2} \end{gather} where the constant $C>0$ is independent of $t$. \end{enumerate} \end{definition} The reason we integrate over~$[0,T]$ in~\eqref{eq:H lam t} to have~\eqref{eq:wssymm2} is to facilitate the passing to the limit in the proof of the main theorem. The following lemma justifies the above definition. \begin{lemma}\label{lem:equidef} Let $(\boldsymbol{m},\boldsymbol{H},\boldsymbol{E})$ be a strong solution of~\eqref{eq:strong}--\eqref{eq:con}. If $\varphi\in H(0,T;H^1(D^\ast))$ satisfies $\nabla\varphi=\boldsymbol{H}|_{D^\ast_T}$, and if $\lambda:=\gamma^+\varphi$, then the triple $(\boldsymbol{m},\boldsymbol{H}|_{D_T},\lambda)$ is a weak solution in the sense of Definition~\ref{def:fembemllg}. Conversely, let $(\boldsymbol{m},\boldsymbol{H},\lambda)$ be a sufficiently smooth solution in the sense of Definition~\ref{def:fembemllg}, and let~$\varphi$ be the solution of \begin{equation}\label{eq:var phi} \Delta\varphi = 0 \text{ in } D^\ast, \quad \varphi = \lambda \text{ on } \Gamma, \quad \varphi(x) = O(|x|^{-1}) \text{ as } |x|\to\infty. \end{equation} Then $(\boldsymbol{m},\overline\boldsymbol{H},\boldsymbol{E})$ is a strong solution to~\eqref{eq:strong}--\eqref{eq:con}, where~$\overline\boldsymbol{H}$ is defined by \begin{align}\label{eq:repform} \overline\boldsymbol{H}:=\begin{cases} \boldsymbol{H} &\quad\text{in }D_T,\\ \nabla\varphi &\quad\text{in } D_T^\ast, \end{cases} \end{align} and~$\boldsymbol{E}$ is reconstructed by letting $\boldsymbol{E}=\sigma^{-1}(\nabla\times\boldsymbol{H})$ in $D_T$ and by solving \begin{subequations}\label{eq:el} \begin{alignat}{2} \nabla\times\boldsymbol{E} &= -\mu_0\overline\boldsymbol{H}_t &&\quad\text{in } D_T^\ast, \\ {\rm div}(\boldsymbol{E}) &=0 &&\quad\text{in }D_T^\ast, \\ \boldsymbol{n}\times \boldsymbol{E}|_{D_T^\ast} &= \boldsymbol{n}\times \boldsymbol{E}|_{D_T} &&\quad\text{on }\Gamma_T. \end{alignat} \end{subequations} \end{lemma} \begin{proof We follow~\cite{dual}. Assume that $(\boldsymbol{m},\boldsymbol{H},\boldsymbol{E})$ satisfies~\eqref{eq:strong}--\eqref{eq:con}. Then clearly Statements~\eqref{ite:1}, \eqref{ite:2} and~\eqref{ite:6} in Definition~\ref{def:fembemllg} hold, noting~\eqref{eq:ini}. Statements~\eqref{ite:3}, \eqref{ite:4} and~\eqref{ite:5} also hold due to the analysis above Definition~\ref{def:fembemllg}. The converse is also true due to the well-posedness of~\eqref{eq:el} as stated in~\cite[Equation~(15)]{dual}. \end{proof} \begin{remark} The solution~$\varphi$ to~\eqref{eq:var phi} can be represented as $ \varphi = (1/2+\dlp)\lambda - \slp\dtn\lambda. $ \end{remark} The next subsection defines the spaces and functions to be used in the approximation of the weak solution the sense of Definition~\ref{def:fembemllg}. \subsection{Discrete spaces and functions}\label{subsec:dis spa} For time discretisation, we use a uniform partition $0\leq t_i\leq T$, $i=0,\ldots,N$ with $t_i:=ik$ and $k:=T/N$. The spatial discretisation is determined by a (shape) regular triangulation ${\mathcal T}_h$ of $D$ into compact tetrahedra $T\in{\mathcal T}_h$ with diameter $h_T/C\leq h\leq Ch_T$ for some uniform constant $C>0$. Denoting by~${\mathcal N}_h$ the set of nodes of~${\mathcal T}_h$, we define the following spaces \begin{align*} {\mathcal S}^1({\mathcal T}_h)&:=\set{\phi_h\in C(D)}{\phi_h|T \in {\mathcal P}^1(T)\text{ for all } T\in{\mathcal T}_h},\\ {\mathcal K}_{\boldsymbol{\phi}_h}&:=\set{\boldsymbol{\psi}_h\in{\mathcal S}^1({\mathcal T}_h)^3}{\boldsymbol{\psi}_h(z)\cdot\boldsymbol{\phi}_h(z)=0\text{ for all }z\in{\mathcal N}_h}, \quad\boldsymbol{\phi}_h\in{\mathcal S}^1({\mathcal T}_h)^3, \end{align*} where ${\mathcal P}^1(T)$ is the space of polynomials of degree at most~1 on~$T$. For the discretisation of~\eqref{eq:wssymm2}, we employ the space ${\mathcal N}{\mathcal D}^1({\mathcal T}_h)$ of first order N\'ed\'elec (edge) elements for $\boldsymbol{H}$ and and the space ${\mathcal S}^1({\mathcal T}_h|_\Gamma)$ for $\lambda$. Here ${\mathcal T}_h|_{\Gamma}$ denotes the restriction of the triangulation to the boundary~$\Gamma$. It follows from Statement~\ref{ite:4} in Definition~\ref{def:fembemllg} that for each~$t\in[0,T]$, the pair~$(\boldsymbol{H}(t),\lambda(t))\in{\mathcal X}$. We approximate the space~${\mathcal X}$ by \begin{align*} {\mathcal X}_h:=\set{(\boldsymbol{\xi},\zeta)\in {\mathcal N}{\mathcal D}^1({\mathcal T}_h)\times {\mathcal S}^1({\mathcal T}_h|_\Gamma)}{\boldsymbol{n}\times \nabla_\Gamma\zeta = \boldsymbol{n}\times\boldsymbol{\xi}|_\Gamma}. \end{align*} To ensure the condition $\boldsymbol{n}\times \nabla_\Gamma\zeta = \boldsymbol{n}\times \boldsymbol{\xi}|_{\Gamma}$, we observe the following. For any~$\zeta\in {\mathcal S}^1({\mathcal T}_h|_\Gamma)$, if $e$ denotes an edge of ${\mathcal T}_h$ on $\Gamma$, then $\int_e \boldsymbol{\xi}\cdot \boldsymbol{\tau}\,ds = \int_e \nabla \zeta \cdot \boldsymbol{\tau}\,ds = \zeta(z_0)-\zeta(z_1)$, where $\boldsymbol{\tau}$ is the unit direction vector on~$e$, and $z_0,z_1$ are the endpoints of $e$. Thus, taking as degrees of freedom all interior edges of ${\mathcal T}_h$ (i.e. $\int_{e_i}\boldsymbol{\xi}\cdot\boldsymbol{\tau} \,ds$) as well as all nodes of ${\mathcal T}_h|_\Gamma$ (i.e.~$\zeta(z_i)$), we fully determine a function pair $(\boldsymbol{\xi},\zeta)\in{\mathcal X}_h $. Due to the considerations above, it is clear that the above space can be implemented directly without use of Lagrange multipliers or other extra equations. The density properties of the finite element spaces~$\{{\mathcal X}_h\}_{h>0}$ are shown in Subsection~\ref{subsec:som lem}; see Lemma~\ref{lem:den pro}. Given functions $\boldsymbol{w}_h^i\colon D\to {\mathbb R}^d$, $d\in{\mathbb N}$, for all $i=0,\ldots,N$ we define for all $t\in [t_i,t_{i+1}]$ \begin{align*} \boldsymbol{w}_{hk}(t):=\frac{t_{i+1}-t}{k}\boldsymbol{w}_h^i + \frac{t-t_i}{k}\boldsymbol{w}_h^{i+1}, \quad \boldsymbol{w}_{hk}^-(t):=\boldsymbol{w}_h^i, \quad \boldsymbol{w}_{hk}^+(t):=\boldsymbol{w}_h^{i+1} . \end{align*} Moreover, we define \begin{equation}\label{eq:dt} d_t\boldsymbol{w}_h^{i+1}:=\frac{\boldsymbol{w}_h^{i+1}-\boldsymbol{w}_h^i}{k}\quad\text{for all } i=0, \ldots, N-1. \end{equation} Finally, we denote by $\Pi_{{\mathcal S}}$ the usual interpolation operator on ${\mathcal S}^1({\mathcal T}_h)$ We are now ready to present the algorithm to compute approximate solutions to problem~\eqref{eq:strong}--\eqref{eq:con}. \subsection{Numerical algorithm}\label{section:alg} In the sequel, when there is no confusion we use the same notation $\boldsymbol{H}$ for the restriction of $\boldsymbol{H}\colon {\mathbb R}^3_T\to {\mathbb R}^3$ to the domain $D_T$. \begin{algorithm}\label{algorithm} \mbox{} \textbf{Input:} Initial data $\boldsymbol{m}^0_h\in{\mathcal S}^1({\mathcal T}_h)^3$, $(\boldsymbol{H}^0_h,\lambda_h^0)\in{\mathcal X}_h$, and parameter $\theta\in[0,1]$. \textbf{For} $i=0,\ldots,N-1$ \textbf{do:} \begin{enumerate} \item Compute the unique function $\boldsymbol{v}^i_h\in {\mathcal K}_{\boldsymbol{m}_h^i}$ satisfying for all $\boldsymbol{\phi}_h\in {\mathcal K}_{\boldsymbol{m}_h^i}$ \begin{align}\label{eq:dllg} \begin{split} \alpha\dual{\boldsymbol{v}_h^i}{\boldsymbol{\phi}_h}_D &+ \dual{\boldsymbol{m}_h^i\times \boldsymbol{v}_h^i}{\boldsymbol{\phi}_h}_D + C_e\theta k \dual{\nabla \boldsymbol{v}_h^i}{\nabla \boldsymbol{\phi}_h}_D \\ &= -C_e \dual{\nabla \boldsymbol{m}_h^i}{\nabla \boldsymbol{\phi}_h}_D + \dual{\boldsymbol{H}_h^i}{\boldsymbol{\phi}_h}_D. \end{split} \end{align} \item Define $\boldsymbol{m}_h^{i+1}\in {\mathcal S}^1({\mathcal T}_h)^3$ nodewise by \begin{equation}\label{eq:mhip1} \boldsymbol{m}_h^{i+1}(z) =\boldsymbol{m}_h^i(z) + k\boldsymbol{v}_h^i(z) \quad\text{for all } z\in{\mathcal N}_h. \end{equation} \item Compute the unique functions $(\boldsymbol{H}_h^{i+1},\lambda_h^{i+1})\in{\mathcal X}_h$ satisfying for all $(\boldsymbol{\xi}_h,\zeta_h)\in {\mathcal X}_h$ \begin{align}\label{eq:dsymm} \dual{d_t\boldsymbol{H}_h^{i+1}}{\boldsymbol{\xi}_h}_{D} & -\dual{d_t\dtn_h\lambda_h^{i+1}}{\zeta_h}_\Gamma + \sigma^{-1}\mu_0^{-1}\dual{\nabla\times\boldsymbol{H}_h^{i+1}}{\nabla\times\boldsymbol{\xi}_h}_{D} \nonumber \\ &= -\dual{\boldsymbol{v}_h^i}{\boldsymbol{\xi}_h}_{D}, \end{align} where $\dtn_h\colon H^{1/2}(\Gamma)\to {\mathcal S}^1({\mathcal T}_h|_\Gamma)$ is the discrete Dirichlet-to-Neumann operator to be defined later. \end{enumerate} \textbf{Output:} Approximations $(\boldsymbol{m}_h^i,\boldsymbol{H}_h^i,\lambda_h^i)$ for all $i=0,\ldots,N$. \end{algorithm} The linear formula~\eqref{eq:mhip1} was introduced in~\cite{bartels} and used in~\cite{Abert_etal}. Equation~\eqref{eq:dsymm} requires the computation of~$\dtn_h\lambda$ for any~$\lambda\in H^{1/2}(\Gamma)$. This is done by use of the boundary element method. Let~$\mu\in H^{-1/2}(\Gamma)$ and~$\mu_h\in{\mathcal P}^0({\mathcal T}_h|_\Gamma)$ be, respectively, the solution of \begin{align}\label{eq:bem} \slp\mu= (\dlp-1/2)\lambda \quad\text{and}\quad \dual{\slp \mu_h}{\nu_h}_\Gamma = \dual{(\dlp-1/2)\lambda}{\nu_h}_\Gamma\quad\forall\nu_h\in {\mathcal P}^0({\mathcal T}_h|_\Gamma), \end{align} where ${\mathcal P}^0({\mathcal T}_h|_\Gamma)$ is the space of piecewise-constant functions on~${\mathcal T}_h|_\Gamma$. If the representation~\eqref{eq:dtn} of~$\dtn$ is used, then~$\dtn\lambda=\mu$, and we can uniquely define~$\dtn_h\lambda$ by solving \begin{equation}\label{eq:bem1} \dual{\dtn_h\lambda}{\zeta_h}_\Gamma = \dual{\mu_h}{\zeta_h}_\Gamma \quad\forall\zeta_h\in{\mathcal S}^1({\mathcal T}_h|_\Gamma). \end{equation} This is known as the Johnson-N\'ed\'elec coupling. If we use the representation~\eqref{eq:dtn2} for~$\dtn\lambda$ then $\dtn\lambda = (1/2-\dlp^\prime)\mu-\hyp\lambda$. In this case we can uniquely define~$\dtn_h\lambda$ by solving \begin{align}\label{eq:bem2} \dual{\dtn_h\lambda}{\zeta_h}_\Gamma = \dual{(1/2-\dlp^\prime)\mu_h}{\zeta_h}_\Gamma - \dual{\hyp \lambda}{\zeta_h}_\Gamma \quad\forall\zeta_h\in {\mathcal S}^1({\mathcal T}_h|_\Gamma). \end{align} This approach yields an (almost) symmetric system and is called Costabel's coupling. In practice,~\eqref{eq:dsymm} only requires the computation of~$\dual{\dtn_h\lambda_h}{\zeta_h}_\Gamma$ for any~$\lambda_h, \zeta_h\in{\mathcal S}^1({\mathcal T}_h|_\Gamma)$. So in the implementation, neither~\eqref{eq:bem1} nor~\eqref{eq:bem2} has to be solved. It suffices to solve the second equation in~\eqref{eq:bem} and compute the right-hand side of either~\eqref{eq:bem1} or~\eqref{eq:bem2}. It is proved in~\cite[Appendix~A]{afembem} that Costabel's coupling results in a discrete operator which is uniformly elliptic and continuous: \begin{align}\label{eq:dtnelliptic} \begin{split} -\dual{\dtn_h \zeta_h}{\zeta_h}_\Gamma&\geq C_\dtn^{-1}\norm{\zeta_h}{H^{1/2}(\Gamma)}^2\quad\text{for all }\zeta_h\in {\mathcal S}^1({\mathcal T}_h|_\Gamma),\\ \norm{\dtn_h\zeta}{H^{-1/2}(\Gamma)}^2&\leq C_\dtn \norm{\zeta}{H^{1/2}(\Gamma)}^2\quad\text{for all }\zeta\in H^{1/2}(\Gamma), \end{split} \end{align} for some constant $C_\dtn>0$ which depends only on $\Gamma$. Even though the remainder of the analysis works analogously for both approaches, we are not aware of an ellipticity result of the form~\eqref{eq:dtnelliptic} for the Johnson-N\'ed\'elec approach. Thus, from now on $\dtn_h$ is understood to be defined by~\eqref{eq:bem2}. \subsection{Main result} Before stating the main result of this part of the paper, we first state some general assumptions. Firstly, the weak convergence of approximate solutions requires the following conditions on $h$ and $k$, depending on the value of the parameter~$\theta$ in~\eqref{eq:dllg}: \begin{equation}\label{eq:hk 12} \begin{cases} k = o(h^2) \quad & \text{when } 0 \le \theta < 1/2, \\ k = o(h) \quad & \text{when } \theta = 1/2, \\ \text{no condition} & \text{when } 1/2 < \theta \le 1. \end{cases} \end{equation} Some supporting lemmas which have their own interests do not require any condition when $\theta=1/2$. For those results, a slightly different condition is required, namely \begin{equation}\label{eq:hk con} \begin{cases} k = o(h^2) \quad & \text{when } 0 \le \theta < 1/2, \\ \text{no condition} & \text{when } 1/2 \le \theta \le 1. \end{cases} \end{equation} The initial data are assumed to satisfy \begin{equation}\label{eq:mh0 Hh0} \sup_{h>0} \left( \norm{\boldsymbol{m}_h^0}{H^1(D)} + \norm{\boldsymbol{H}_h^0}{\Hcurl{D}} + \norm{\lambda_h^0}{H^{1/2}(\Gamma)} \right) <\infty \quad\text{and}\quad \lim_{h\to0} \norm{\boldsymbol{m}_h^0-\boldsymbol{m}^0}{\Ltwo{D}} = 0. \end{equation} We are now ready to state the main result of this part of the paper. \begin{theorem}[Existence of solutions]\label{thm:weakconv} Under the assumptions~\eqref{eq:hk 12} and~\eqref{eq:mh0 Hh0}, the problem~\eqref{eq:strong}--\eqref{eq:con} has a solution~$(\boldsymbol{m},\boldsymbol{H},\lambda)$ in the sense of Definition~\ref{def:fembemllg}. \end{theorem} \section{Proofs of the main result}\label{sec:pro} \subsection{Some lemmas}\label{subsec:som lem} In this subsection we prove all important lemmas which are directly related to the proofs of the theorem. The first lemma proves density properties of the discrete spaces. \begin{lemma}\label{lem:den pro} Provided that the meshes $\{{\mathcal T}_h\}_{h>0}$ are regular, the union~$\bigcup_{h>0}{\mathcal X}_h$ is dense in~${\mathcal X}$. Moreover, there exists an interpolation operator $\Pi_{\mathcal X}:=(\Pi_{{\mathcal X},D},\Pi_{{\mathcal X},\Gamma})\colon \big({\mathbb{H}}^2(D)\times H^2(\Gamma)\big) \cap {\mathcal X}\to {\mathcal X}_h$ which satisfies \begin{align}\label{eq:int} \norm{(1-\Pi_{\mathcal X})(\boldsymbol{\xi},\zeta)}{\Hcurl{D}\times H^{1/2}(\Gamma) } &\leq C_{{\mathcal X}} h( \norm{\boldsymbol{\xi}}{{\mathbb{H}}^2(D)}+ h^{1/2}\norm{\zeta}{H^2(\Gamma)}), \end{align} where $C_{\mathcal X}>0$ depends only on $D$, $\Gamma$, and the shape regularity of ${\mathcal T}_h$. \end{lemma} \begin{proof} The interpolation operator $\Pi_{\mathcal X}:=(\Pi_{{\mathcal X},D},\Pi_{{\mathcal X},\Gamma})\colon \big({\mathbb{H}}^2(D)\times H^2(\Gamma)\big) \cap {\mathcal X}\to {\mathcal X}_h$ is constructed as follows. The interior degrees of freedom (edges) of $\Pi_{\mathcal X}(\boldsymbol{\xi},\zeta)$ are equal to the interior degrees of freedom of $\Pi_{{\mathcal N}{\mathcal D}}\boldsymbol{\xi}\in{\mathcal N}{\mathcal D}^1({\mathcal T}_h)$, where $\Pi_{{\mathcal N}{\mathcal D}}$ is the usual interpolation operator onto ${\mathcal N}{\mathcal D}^1({\mathcal T}_h)$. The degrees of freedom of $\Pi_{\mathcal X}(\boldsymbol{\xi},\zeta)$ which lie on $\Gamma$ (nodes) are equal to $\Pi_{\mathcal S}\zeta$. By the definition of ${\mathcal X}_h$, this fully determines $\Pi_{\mathcal X}$. Particularly, since $\boldsymbol{n}\times \boldsymbol{\xi}|_\Gamma=\boldsymbol{n}\times\nabla_\Gamma\zeta$, there holds $\Pi_{{\mathcal N}{\mathcal D}}\boldsymbol{\xi}|_\Gamma = \Pi_{{\mathcal X},\Gamma}(\boldsymbol{\xi},\zeta)$. Hence, the interpolation error can be bounded by \begin{align*} \norm{(1-\Pi_{\mathcal X})(\boldsymbol{\xi},\zeta)}{\Hcurl{D}\times H^{1/2}(\Gamma) }&\leq \norm{(1-\Pi_{{\mathcal N}{\mathcal D}})\boldsymbol{\xi}}{\Hcurl{D}}+\norm{(1-\Pi_{\mathcal S})\zeta}{H^{1/2}(\Gamma)}\\ &\lesssim h( \norm{\boldsymbol{\xi}}{{\mathbb{H}}^2(D)}+ h^{1/2}\norm{\zeta}{H^2(\Gamma)}). \end{align*} Since $\big({\mathbb{H}}^2(D)\times H^2(\Gamma)\big)\cap {\mathcal X}$ is dense in ${\mathcal X}$, this concludes the proof. \end{proof} The following lemma gives an equivalent form to~\eqref{eq:wssymm2} and shows that Algorithm~\ref{algorithm} is well-defined. \begin{lemma}\label{lem:bil for} Let $a(\cdot,\cdot)\colon {\mathcal X}\times{\mathcal X}\to {\mathbb R}$, $a_h(\cdot,\cdot)\colon {\mathcal X}_h\times{\mathcal X}_h\to {\mathbb R}$, and $b(\cdot,\cdot)\colon \Hcurl{D}\times\Hcurl{D}\to {\mathbb R}$ be bilinear forms defined by \begin{align*} a(A,B)&:=\dual{\boldsymbol{\psi}}{\boldsymbol{\xi}}_D -\dual{\dtn \eta}{\zeta}_\Gamma, \\ a_h(A_h,B_h) &:= \dual{\boldsymbol{\psi}_h}{\boldsymbol{\xi}_h}_D - \dual{\dtn_h\eta_h}{\zeta_h}_{\Gamma}, \\ b(\boldsymbol{\psi},\boldsymbol{\xi})&:= \sigma^{-1}\mu_0^{-1} \dual{\nabla\times \boldsymbol{\psi}}{\nabla\times \boldsymbol{\xi}}_\Gamma, \end{align*} for all $\boldsymbol{\psi}, \boldsymbol{\xi}\in\Hcurl{D}$, $A:=(\boldsymbol{\psi},\eta)$, $B:=(\boldsymbol{\xi},\zeta)\in{\mathcal X}$, $A_h = (\boldsymbol{\psi}_h,\eta_h), B_h = (\boldsymbol{\xi}_h,\zeta_h) \in {\mathcal X}_h$. Then \begin{enumerate} \item The bilinear forms satisfy, for all $A=(\boldsymbol{\psi},\eta)\in{\mathcal X}$ and $A_h=(\boldsymbol{\psi}_h,\eta_h)\in{\mathcal X}_h$, \begin{align}\label{eq:elliptic} \begin{split} a(A,A)&\geq C_{\rm ell} \big( \norm{\boldsymbol{\psi}}{\Ltwo{D}}^2+\norm{\eta}{H^{1/2}(\Gamma)}^2 \big), \\ a_h(A_h,A_h) &\geq C_{\rm ell} \big( \norm{\boldsymbol{\psi}_h}{\Ltwo{D}}^2+\norm{\eta_h}{H^{1/2}(\Gamma)}^2 \big), \\ b(\boldsymbol{\psi},\boldsymbol{\psi}) &\geq C_{\rm ell}\norm{\nabla\times \boldsymbol{\psi}}{\Ltwo{D}}^2. \end{split} \end{align} \item Equation~\eqref{eq:wssymm2} is equivalent to \begin{equation}\label{eq:bil for} \int_0^T a(A_t(t),B) \, dt + \int_0^T b(\boldsymbol{H}(t),\boldsymbol{\xi}) \, dt = - \dual{\boldsymbol{m}_t}{\boldsymbol{\xi}}_{D_T} \end{equation} for all~$B=(\boldsymbol{\xi},\zeta)\in {\mathcal X}$, where~$A=(\boldsymbol{H},\lambda)$. \item Equation~\eqref{eq:dsymm} is of the form \begin{align}\label{eq:eddygeneral} a_h(d_t A_h^{i+1},B_h) + b(\boldsymbol{H}_h^{i+1},\boldsymbol{\xi}_h) = -\dual{\boldsymbol{v}_h^i}{\boldsymbol{\xi}_h}_\Gamma \end{align} where $A_h^{i+1}:=(\boldsymbol{H}_h^{i+1},\lambda_h^{i+1})$ and $B_h:=(\boldsymbol{\xi}_h,\zeta_h)$. \item Algorithm~\ref{algorithm} is well-defined in the sense that~\eqref{eq:dllg} and~\eqref{eq:dsymm} have unique solutions. \end{enumerate} \end{lemma} \begin{proof} The unique solvability of~\eqref{eq:dsymm} follows immediately from the continuity and ellipticity of the bilinear forms $a_h(\cdot,\cdot)$ and $b(\cdot,\cdot)$. The unique solvability of~\eqref{eq:dllg} follows from the positive definiteness of the left-hand side, the linearity of the right-hand side, and the finite space dimension. \end{proof} The following lemma establishes an energy bound for the discrete solutions. \begin{lemma}\label{lem:denergygen} Under the assumptions~\eqref{eq:hk con} and~\eqref{eq:mh0 Hh0}, there holds for all $k<2\alpha$ and $j=1,\ldots,N$ \begin{align}\label{eq:denergy} \sum_{i=0}^{j-1}& \left( \norm{\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i}{\Ltwo{D}}^2+\norm{\lambda_h^{i+1}-\lambda_h^i}{H^{1/2}(\Gamma)}^2 \right)\notag \\ &+ k \sum_{i=0}^{j-1} \norm{\nabla\times \boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2 + \norm{\boldsymbol{H}_h^j}{\Hcurl{D}}^2 + \norm{\lambda_h^{j}}{H^{1/2}(\Gamma)}^2 + \norm{\nabla \boldsymbol{m}_h^{j}}{\Ltwo{D}}^2\notag \\ &+ \max\{2\theta-1,0\}k^2 \sum_{i=0}^{j-1} \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \\ &+k \sum_{i=0}^{j-1} (\norm{d_t\boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2 +\norm{d_t\lambda_h^{i+1}}{H^{1/2}(\Gamma)}^2)+\sum_{i=0}^{j-1} \norm{\nabla\times (\boldsymbol{H}^{i+1}_h-\boldsymbol{H}^{i}_h)}{\Ltwo{D}}^2 \leq C_{\rm ener}.\notag \end{align} \end{lemma} \begin{proof} Choosing $B_h=A_h^{i+1}$ in~\eqref{eq:eddygeneral} and multiplying the resulting equation by $k$ we obtain \begin{equation}\label{eq:e1} a_h(A_h^{i+1}-A_h^i,A_h^{i+1}) + kb(\boldsymbol{H}_h^{i+1},\boldsymbol{H}_h^{i+1}) = -k\dual{\boldsymbol{v}_h^i}{\boldsymbol{H}_h^i}_D -k\dual{\boldsymbol{v}_h^i}{\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i}_D. \end{equation} On the other hand, it follows from~\eqref{eq:mhip1} and~\eqref{eq:dllg} that \begin{align* \begin{split} \norm{\nabla \boldsymbol{m}_h^{i+1}}{\Ltwo{D}}^2&=\norm{\nabla \boldsymbol{m}_h^{i}}{\Ltwo{D}}^2+k^2\norm{\nabla \boldsymbol{v}_h^{i}}{\Ltwo{D}}^2+2k\dual{\nabla\boldsymbol{m}_h^i}{\nabla\boldsymbol{v}_h^i}_D \\ & = \norm{\nabla\boldsymbol{m}_h^i}{\Ltwo{D}}^2-2(\theta-\tfrac12)k^2\norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 - \frac{2\alpha k}{C_e}\norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + \frac{2k}{C_e}\dual{\boldsymbol{H}_h^i}{\boldsymbol{v}_h^i}_D, \end{split} \end{align*} which implies \begin{align*} k \dual{\boldsymbol{v}_h^i}{\boldsymbol{H}_h^i}_D &= \frac{C_e}{2} \left( \norm{\nabla\boldsymbol{m}_h^{i+1}}{\Ltwo{D}}^2 - \norm{\nabla\boldsymbol{m}_h^{i}}{\Ltwo{D}}^2 \right) + (\theta-\tfrac12)k^2C_e \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + \alpha k \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2. \end{align*} Inserting this into the first term on the right-hand side of~\eqref{eq:e1} and rearranging the resulting equation yield, for any $\epsilon>0$, \begin{align*} &a_h(A_h^{i+1}-A_h^i,A_h^{i+1}) + k b(\boldsymbol{H}_h^{i+1},\boldsymbol{H}_h^{i+1}) \nonumber \\ &+ \frac{C_e}{2} \left( \norm{\nabla \boldsymbol{m}_h^{i+1}}{\Ltwo{D}}^2 - \norm{\nabla\boldsymbol{m}_h^i}{\Ltwo{D}}^2 \right) + (\theta-1/2)k^2 C_e \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + \alpha k \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \nonumber \\ &= -k \dual{\boldsymbol{v}_h^i}{\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^{i}}_D \\ &\leq \frac{\epsilon k}{2} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + \frac{k}{2\epsilon} \norm{\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i}{\Ltwo{D}}^2 \leq \frac{\epsilon k}{2} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + \frac{k}{2\epsilon} a_h(A_h^{i+1}-A_h^i,A_h^{i+1}-A_h^i), \end{align*} where in the last step we used the definition of $a_h(\cdot,\cdot)$ and~\eqref{eq:dtnelliptic}. Rearranging gives \begin{align*} a_h(A_h^{i+1}-A_h^i,A_h^{i+1}) &+ k b(\boldsymbol{H}_h^{i+1},\boldsymbol{H}_h^{i+1}) + \frac{C_e}{2} \left( \norm{\nabla \boldsymbol{m}_h^{i+1}}{\Ltwo{D}}^2 - \norm{\nabla\boldsymbol{m}_h^i}{\Ltwo{D}}^2 \right) \\ &+ (\theta-1/2)k^2 C_e \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + (\alpha-\epsilon /2) k \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \nonumber \\ &\leq \frac{k}{2\epsilon} a_h(A_h^{i+1}-A_h^i,A_h^{i+1}-A_h^i). \end{align*} Summing over $i$ from $0$ to $j-1$ and (for the first term on the left-hand side) applying Abel's summation by parts formula \begin{equation}\label{equ:Abe} \sum_{i=0}^{j-1} (u_{i+1}-u_{i})u_{i+1} = \frac{1}{2} |u_j|^2 - \frac{1}{2} |u_0|^2 + \frac{1}{2} \sum_{i=0}^{j-1} |u_{i+1}-u_{i}|^2, \end{equation} we deduce, after multiplying the equation by two and rearranging, \begin{align* &(1-k/\epsilon) \sum_{i=0}^{j-1} a_h(A_h^{i+1}-A_h^i,A_h^{i+1}-A_h^i) + 2k \sum_{i=0}^{j-1} b(\boldsymbol{H}_h^{i+1},\boldsymbol{H}_h^{i+1}) + a_h(A_h^j,A_h^j) \\ & + C_e \norm{\nabla \boldsymbol{m}_h^{j}}{\Ltwo{D}}^2 + (2\theta-1)k^2 C_e \sum_{i=0}^{j-1} \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + (2\alpha-\epsilon ) k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \\ & \leq C_e \norm{\nabla \boldsymbol{m}_h^{0}}{\Ltwo{D}}^2 + a_h(A_h^0,A_h^0). \end{align*} Since $k<2\alpha$ we can choose $\varepsilon>0$ such that $2\alpha-\epsilon >0$ and $1-k/\epsilon>0$. By noting the ellipticity~\eqref{eq:dtnelliptic}, the bilinear forms $a_h(\cdot,\cdot)$ and $b(\cdot,\cdot)$ are elliptic in their respective (semi-)norms. We obtain \begin{align}\label{eq:e4} \sum_{i=0}^{j-1}& \left( \norm{\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i}{\Ltwo{D}}^2+\norm{\lambda_h^{i+1}-\lambda_h^i}{H^{1/2}(\Gamma)}^2 \right) + k \sum_{i=0}^{j-1} \norm{\nabla\times \boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2 + \norm{\boldsymbol{H}_h^j}{\Ltwo{D}}^2 \notag \\ & + \norm{\lambda_h^{j}}{H^{1/2}(\Gamma)}^2 + \norm{\nabla \boldsymbol{m}_h^{j}}{\Ltwo{D}}^2 + (2\theta-1)k^2 \sum_{i=0}^{j-1} \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \notag \\ & \leq C \left( \norm{\nabla \boldsymbol{m}_h^{0}}{\Ltwo{D}}^2 + \norm{\boldsymbol{H}_h^0}{\Ltwo{D}}^2 + \norm{\lambda_h^{0}}{H^{1/2}(\Gamma)}^2 \right) \leq C, \end{align} where in the last step we used~\eqref{eq:mh0 Hh0}. It remains to consider the last three terms on the left-hand side of~\eqref{eq:denergy}. Again, we consider~\eqref{eq:eddygeneral} and select $B_h=d_tA_h^{i+1}$ to obtain after multiplication by~$2k$ \begin{align*} \begin{split} 2k a_h(d_tA_h^{i+1},d_tA_h^{i+1}) & + 2b(\boldsymbol{H}_h^{i+1},\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i) \\ &= -2k\dual{\boldsymbol{v}_h^i}{d_t\boldsymbol{H}_h^{i+1}}_D \le k \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + k \norm{d_t\boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2, \end{split} \end{align*} so that, noting~\eqref{eq:e4} and~\eqref{eq:elliptic}, \begin{align}\label{eq:e5} \begin{split} k \sum_{i=0}^{j-1} \left( \norm{d_t\boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2 \right. &+ \left. \norm{d_t\lambda_h^{i+1}}{H^{1/2}(\Gamma)}^2 \right) + 2 \sum_{i=0}^{j-1} b(\boldsymbol{H}_h^{i+1},\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i) \\ & \lesssim k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \leq C. \end{split} \end{align} Using Abel's summation by parts formula~\eqref{equ:Abe} for the second sum on the left-hand side, and noting the ellipticity of the bilinear form $b(\cdot,\cdot)$ and~\eqref{eq:mh0 Hh0}, we obtain together with~\eqref{eq:e4} \begin{align}\label{eq:d4} \sum_{i=0}^{j-1}& (\norm{\boldsymbol{H}_h^{i+1}-\boldsymbol{H}_h^i}{\Ltwo{D}}^2+\norm{\lambda_h^{i+1}-\lambda_h^i}{H^{1/2}(\Gamma)}^2)\notag \\ &+ k \sum_{i=0}^{j-1} \norm{\nabla\times \boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2 + \norm{\boldsymbol{H}_h^j}{\Hcurl{D}}^2 + \norm{\lambda_h^{j}}{H^{1/2}(\Gamma)}^2 + \norm{\nabla \boldsymbol{m}_h^{j}}{\Ltwo{D}}^2\notag \\ &+ (2\theta-1)k^2 \sum_{i=0}^{j-1} \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \\ &+k \sum_{i=0}^{j-1} (\norm{d_t\boldsymbol{H}_h^{i+1}}{\Ltwo{D}}^2 +\norm{d_t\lambda_h^{i+1}}{H^{1/2}(\Gamma)}^2)+\sum_{i=0}^{j-1} \norm{\nabla\times (\boldsymbol{H}^{i+1}_h-\boldsymbol{H}^{i}_h)}{\Ltwo{D}}^2 \leq C.\notag \end{align} Clearly, if $1/2\le\theta\le1$ then~\eqref{eq:d4} yields~\eqref{eq:denergy}. If $0\le\theta<1/2$ then since the mesh is regular, the inverse estimate~$\norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}\lesssim h^{-1}\norm{\boldsymbol{v}_h^i}{\Ltwo{D}}$ gives \begin{align*} (2\theta-1)k^2 \sum_{i=0}^{j-1} \norm{\nabla\boldsymbol{v}_h^i}{\Ltwo{D}}^2 + k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 &\gtrsim \left( 1-k^2h^{-1}(1-2\theta) \right) k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \\ &\gtrsim k \sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \end{align*} as $k^2h^{-1}\to0$ under the assumption~\eqref{eq:hk con}. This estimate and~\eqref{eq:d4} give~\eqref{eq:denergy}, completing the proof of the lemma. \end{proof} Collecting the above results we obtain the following equations satisfied by the discrete functions defined from~$\boldsymbol{m}_h^i$, $\boldsymbol{H}_h^i$, $\lambda_h^i$, and $\boldsymbol{v}_h^i$. \begin{lemma}\label{lem:mhk Hhk} Let $\boldsymbol{m}_{hk}^{-}$, $A_{hk}^\pm:=(\boldsymbol{H}_{hk}^\pm,\lambda_{hk}^\pm)$, and $\boldsymbol{v}_{hk}^{-}$ be defined from $\boldsymbol{m}_h^i$, $\boldsymbol{H}_h^i$, $\lambda_h^i$, and $\boldsymbol{v}_h^i$ as described in Subsection~\ref{subsec:dis spa}. Then \begin{subequations}\label{eq:mhk Hhk} \begin{align} \alpha\dual{\boldsymbol{v}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} &+ \dual{(\boldsymbol{m}_{hk}^-\times \boldsymbol{v}_{hk}^-)}{\boldsymbol{\phi}_{hk}}_{D_T} + C_e \theta k \dual{\nabla \boldsymbol{v}_{hk}^-}{\nabla \boldsymbol{\phi}_{hk}}_{D_T} \nonumber \\ &= -C_e \dual{\nabla \boldsymbol{m}_{hk}^-}{\nabla \boldsymbol{\phi}_{hk}}_{D_T} + \dual{\boldsymbol{H}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} \label{eq:mhk Hhk1} \\ \intertext{and with~$\partial_t$ denoting time derivative} \int_0^T a_h(\partial_t A_{hk}(t),B_h) \,dt &+ \int_0^T b(\boldsymbol{H}_{hk}^+(t),\boldsymbol{\xi}_h) \,dt = -\dual{\boldsymbol{v}_{hk}^-}{\boldsymbol{\xi}_h}_{D_T} \label{eq:mhk Hhk2} \end{align} \end{subequations} for all $\boldsymbol{\phi}_{hk}$ and $B_h:=(\boldsymbol{\xi}_h,\zeta_h)$ satisfying $\boldsymbol{\phi}_{hk}(t,\cdot)\in{\mathcal K}_{\boldsymbol{m}_h^i}$ for $t\in[t_i,t_{i+1})$ and $B_h\in{\mathcal X}_h$. \end{lemma} \begin{proof} The lemma is a direct consequence of~\eqref{eq:dllg} and~\eqref{eq:eddygeneral}. \end{proof} The next lemma shows that the functions defined in the above lemma form sequences which have convergent subsequences. \begin{lemma}\label{lem:weakconv} Assume that the assumptions~\eqref{eq:hk con} and~\eqref{eq:mh0 Hh0} hold. As $h$, $k\to0$, the following limits exist up to extraction of subsequences \begin{subequations}\label{eq:weakconv} \begin{alignat}{2} \boldsymbol{m}_{hk}&\rightharpoonup\boldsymbol{m}\quad&&\text{in }\Hone{D_T},\label{eq:wc1}\\ \boldsymbol{m}_{hk}^\pm&\rightharpoonup\boldsymbol{m}\quad&&\text{in } L^2(0,T;\Hone{D}),\label{eq:wc1a} \\ \boldsymbol{m}_{hk}^\pm & \rightarrow\boldsymbol{m} \quad&& \text{in } \Ltwo{D_T}, \label{eq:wc2}\\ (\boldsymbol{H}_{hk},\lambda_{hk})&\rightharpoonup(\boldsymbol{H},\lambda)\quad&&\text{in } L^2(0,T;{\mathcal X}),\label{eq:wc3}\\ (\boldsymbol{H}_{hk}^\pm,\lambda_{hk}^\pm)&\rightharpoonup(\boldsymbol{H},\lambda) \quad&&\text{in } L^2(0,T;{\mathcal X}),\label{eq:wc4}\\ (\boldsymbol{H}_{hk},\lambda_{hk})&\rightharpoonup (\boldsymbol{H},\lambda) \quad&&\text{in } H^1(0,T;\Ltwo{D}\times H^{1/2}(\Gamma)),\label{eq:wc31}\\ \boldsymbol{v}_{hk}^- &\rightharpoonup \boldsymbol{m}_t\quad&&\text{in }\Ltwo{D_T},\label{eq:wc5} \end{alignat} \end{subequations} for certain functions $\boldsymbol{m}$, $\boldsymbol{H}$, and $\lambda$ satisfying $\boldsymbol{m}\in \Hone{D_T}$, $\boldsymbol{H}\in H^1(0,T;\Ltwo{D})$, and $(\boldsymbol{H},\lambda)\in L^2(0,T;{\mathcal X})$. Here~$\rightharpoonup$ denotes the weak convergence and~$\to$ denotes the strong convergence in the relevant space. Moreover, if the assumption~\eqref{eq:mh0 Hh0} holds then there holds additionally $|\boldsymbol{m}|=1$ almost everywhere in~$D_T$. \end{lemma} \begin{proof} Note that due to the Banach-Alaoglu Theorem, to show the existence of a weakly convergent subsequence, it suffices to show the boundedness of the sequence in the respective norm. Thus in order to prove~\eqref{eq:wc1} we will prove that $\norm{\boldsymbol{m}_{hk}}{\Hone{D_T}}\le C$ for all~$h,k>0$. By Step~(3) of Algorithm~\ref{algorithm} and due to an idea from~\cite{bartels}, there holds for all $z\in{\mathcal N}_h$ \begin{align*} \begin{split} |\boldsymbol{m}_h^j(z)|^2 &=|\boldsymbol{m}_h^{j-1}(z)|^2+k^2|\boldsymbol{v}_h^{j-1}(z)|^2=|\boldsymbol{m}_h^{j-2}(z)|^2+k^2|\boldsymbol{v}_h^{j-1}(z)|^2+k^2|\boldsymbol{v}_h^{j-2}(z)|^2\\ &= |\boldsymbol{m}_h^{0}(z)|^2+k^2\sum_{i=0}^{j-1}|\boldsymbol{v}_h^{i}(z)|^2. \end{split} \end{align*} By using the equivalence (see e.g. \cite[Lemma~3.2]{thanh}) \begin{equation}\label{eq:LT13} \norm{\boldsymbol{\phi}}{L^p(D)}^p \simeq h^3 \sum_{z\in{\mathcal N}_h} |\boldsymbol{\phi}(z)|^p, \quad 1 \le p < \infty, \quad \boldsymbol{\phi}\in {\mathcal S}^1({\mathcal T}_h)^3, \end{equation} we deduce that \begin{align}\label{eq:const} \left| \norm{\boldsymbol{m}_h^j}{\Ltwo{D}}^2-\norm{\boldsymbol{m}_h^0}{\Ltwo{D}}^2 \right| & \simeq h^3 \sum_{z\in{\mathcal N}_h} \left( |\boldsymbol{m}_h^j(z)|^2 - |\boldsymbol{m}_h^0(z)|^2 \right) = k^2\sum_{i=0}^{j-1} h^3 \sum_{z\in{\mathcal N}_h} |\boldsymbol{v}_h^i(z)|^2 \nonumber \\ & \simeq k^2\sum_{i=0}^{j-1}\norm{\boldsymbol{v}_h^{i}}{\Ltwo{D}}^2\leq kC_{\rm ener}, \end{align} where in the last step we used~\eqref{eq:denergy}. This proves immediately \begin{align*} \norm{\boldsymbol{m}_{hk}}{\Ltwo{D_T}}^2\simeq k\sum_{i=1}^N \norm{\boldsymbol{m}^i_h}{\Ltwo{D}}^2\leq k\sum_{i=1}^N \big(\norm{\boldsymbol{m}^0_h}{\Ltwo{D}}^2+kC_{\rm ener}\big) \leq C. \end{align*} On the other hand, since $\partial_t\boldsymbol{m}_{hk} = (\boldsymbol{m}_h^{i+1}-\boldsymbol{m}_h^i)/k$ on $(t_i,t_{i+1})$ for $i=0,\ldots,N-1$ and $\boldsymbol{m}_h^{i+1}(z)-\boldsymbol{m}_h^i(z) = k\boldsymbol{v}_h^i(z)$ for all $z\in{\mathcal N}_h$, we have by using~\eqref{eq:denergy} and~\eqref{eq:LT13} \begin{align}\label{eq:dt mhk} \norm{\partial_t\boldsymbol{m}_{hk}}{\Ltwo{D_T}}^2 & = \sum_{i=0}^{N-1} \int_{t_j}^{t_{j+1}} \norm{\partial_t\boldsymbol{m}_{hk}}{\Ltwo{D}}^2 \, dt = k^{-1}\sum_{i=0}^{N-1} \norm{\boldsymbol{m}^{i+1}_h-\boldsymbol{m}^i_h}{\Ltwo{D}}^2 \nonumber \\ &\simeq k^{-1} \sum_{i=0}^{N-1} h^3 \sum_{z\in {\mathcal N}_h} |\boldsymbol{m}^{i+1}_h(z)-\boldsymbol{m}^i_h(z)|^2 = k \sum_{i=0}^{N-1} h^3 \sum_{z\in {\mathcal N}_h} |\boldsymbol{v}^i_h(z)|^2 \nonumber \\ &\simeq k \sum_{i=0}^{N-1} \norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \leq C_{\rm ener}. \end{align} Finally the gradient $\nabla\boldsymbol{m}_{hk}$ is shown to be bounded by using~\eqref{eq:denergy} again as follows: \begin{align*} \norm{\nabla\boldsymbol{m}_{hk}}{\Ltwo{D_T}}^2\simeq k\sum_{i=1}^N \norm{\nabla\boldsymbol{m}^i_h}{\Ltwo{D}}^2\leq C_{\rm ener}kN\leq C_{\rm energy}T. \end{align*} Altogether, we showed that $\{\boldsymbol{m}_{hk}\}$ is a bounded sequence in $\Hone{D_T}$ and thus posesses a weakly convergent subsequence, i.e., we proved~\eqref{eq:wc1}. In particular,~\eqref{eq:mh0 Hh0}, \eqref{eq:denergy}, and~\eqref{eq:const} imply \begin{equation}\label{eq:mhk pm} \norm{\boldsymbol{m}_{hk}^\pm}{L^2(0,T;\Hone{D})} \le \norm{\boldsymbol{m}_{hk}^\pm}{L^\infty(0,T;\Hone{D})} \lesssim C_{\rm ener}, \end{equation} yielding~\eqref{eq:wc1a}. We prove~\eqref{eq:wc2} for $\boldsymbol{m}_{hk}^-$ only; similar arguments hold for $\boldsymbol{m}_{hk}^+$. First, we note that the definition of~$\boldsymbol{m}_{hk}$ and~$\boldsymbol{m}_{hk}^-$, and the estimate~\eqref{eq:dt mhk} imply, for all $t\in[t_j,t_{j+1})$, \begin{align*} \norm{\boldsymbol{m}_{hk}(t,\cdot)-\boldsymbol{m}_{hk}^-(t,\cdot)}{\Ltwo{D}} &= \norm{(t-t_j)\frac{\boldsymbol{m}_{h}^{j+1}-\boldsymbol{m}_{h}^j}{k}}{\Ltwo{D}} \le k\norm{\partial_t\boldsymbol{m}_{hk}(t,\cdot)}{\Ltwo{D}} \lesssim k C_{\rm ener}. \end{align*} This in turn implies \[ \norm{\boldsymbol{m}_{hk}-\boldsymbol{m}_{hk}^-}{\Ltwo{D_T}} \lesssim kT C_{\rm ener} \to 0 \quad\text{as }h,k\to0. \] Thus,~\eqref{eq:wc2} follows from the triangle inequality,~\eqref{eq:wc1}, and the Sobolev embedding. Statement~\eqref{eq:wc3} follows immediately from~\eqref{eq:denergy} by noting that \begin{align*} \norm{(\boldsymbol{H}_{hk},\lambda_{hk})}{L^2(0,T;{\mathcal X})}^2 \simeq k\sum_{i=1}^N \big( \norm{\boldsymbol{H}_h^i}{\Hcurl{D}}^2+\norm{\lambda_{h}^i}{H^{1/2}(\Gamma)}^2 \big) \leq kNC_{\rm ener}\leq TC_{\rm ener}. \end{align*} The proof of~\eqref{eq:wc4} follows analogously. Consequently, we obtain~\eqref{eq:wc31} by using again~\eqref{eq:denergy} and the above estimate as follows: \begin{align*} \norm{\boldsymbol{H}_{hk}}{H^1(0,T;\Ltwo{D})}^2 &\simeq \norm{\boldsymbol{H}_{hk}}{\Ltwo{D_T}}^2 + k\sum_{i=1}^N\norm{d_t\boldsymbol{H}_h^i}{\Ltwo{D}}^2 \leq TC_{\rm ener}+C_{\rm ener}. \end{align*} The convergence of $\lambda_{hk}$ in the statement follows analogously. Finally,~\eqref{eq:wc5} follows from $\partial_t\boldsymbol{m}_{hk}(t)=\boldsymbol{v}^-_{hk}(t)$ and~\eqref{eq:wc1}. To show that $\boldsymbol{m}$ satisfies the constraint $|\boldsymbol{m}|=1$, we first note that \begin{align*} \norm{|\boldsymbol{m}|-1}{L^2(D_T)}\leq \norm{\boldsymbol{m}-\boldsymbol{m}_{hk}}{\Ltwo{D_T}}+\norm{|\boldsymbol{m}_{hk}|-1}{L^2(D_T)}. \end{align*} The first term on the right-hand side converges to zero due to~\eqref{eq:wc1} and the compact embedding of~$\Hone{D_T}$ in~$\Ltwo{D_T}$. For the second term, we note that \begin{align*} \norm{1-|\boldsymbol{m}_{hk}|}{L^2(D_T)}^2 &\lesssim k\sum_{j=0}^N\big(\norm{|\boldsymbol{m}_{h}^j|-|\boldsymbol{m}_h^0|}{L^2(D)}^2 + \norm{1-|\boldsymbol{m}_h^0|}{L^2(D)}^2\big) \nonumber\\ &\leq k\sum_{j=0}^N\big(\norm{|\boldsymbol{m}_{h}^j|^2 - |\boldsymbol{m}_h^0|^2}{L^1(D)}+\norm{|\boldsymbol{m}^0|-|\boldsymbol{m}_h^0|}{L^2(D)}^2\big) \end{align*} where we used $(x-y)^2\leq |x^2-y^2|$ for all $x,y\ge0$. Similarly to~\eqref{eq:const} it can be shown that \begin{equation}\label{eq:mhj mh0} \norm{|\boldsymbol{m}_h^j|^2-|\boldsymbol{m}_h^0|^2}{L^1(D)} \simeq k^2\sum_{i=0}^{j-1} \norm{\boldsymbol{v}_h^{i}}{\Ltwo{D}}^2 \leq kC_{\rm ener}. \end{equation} Hence \[ \norm{1-|\boldsymbol{m}_{hk}|}{L^2(D_T)}^2 \leq kC_{\rm ener} +\norm{\boldsymbol{m}^0-\boldsymbol{m}_h^0}{\Ltwo{D_T}}^2\to 0\quad\text{as }h,k\to 0. \] Altogether, we showed $|\boldsymbol{m}|=1$ almost everywhere in~$D_T$, completing the proof of the lemma. \end{proof} We also need the following strong convergence property. \begin{lemma}\label{lem:str con} Under the assumptions~\eqref{eq:hk 12} and~\eqref{eq:mh0 Hh0} there holds \begin{equation}\label{eq:mhk h12} \norm{\boldsymbol{m}_{hk}^--\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))} \to 0 \quad\text{as } h,k\to 0. \end{equation} \end{lemma} \begin{proof} It follows from the triangle inequality and the definitions of~$\boldsymbol{m}_{hk}$ and~$\boldsymbol{m}_{hk}^-$ that \begin{align*} \norm{\boldsymbol{m}_{hk}^--\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))}^2 &\lesssim \norm{\boldsymbol{m}_{hk}^--\boldsymbol{m}_{hk}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))}^2 + \norm{\boldsymbol{m}_{hk}-\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))}^2 \\ &\leq \sum_{i=0}^{N-1} k^3\norm{\boldsymbol{v}_h^i}{{\mathbb{H}}^{1/2}(D)}^2 + \norm{\boldsymbol{m}_{hk}-\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))}^2 \\ &\leq \sum_{i=0}^{N-1}k^3\norm{\boldsymbol{v}_h^i}{\Hone{D}}^2 + \norm{\boldsymbol{m}_{hk}-\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))}^2. \end{align*} The second term on the right-hand side converges to zero due to~\eqref{eq:wc1} and the compact embedding of \[ \Hone{D_T} \simeq \{\boldsymbol{v} \, | \, \boldsymbol{v}\in L^2(0,T;\Hone{D}), \, \boldsymbol{v}_t\in L^2(0,T;\Ltwo{D}) \} \] into $L^2(0,T;{\mathbb{H}}^{1/2}(D))$; see \cite[Theorem 5.1]{Lio69}. For the first term on the right-hand side, when $\theta>1/2$,~\eqref{eq:denergy} implies $ \sum_{i=0}^{N-1}k^3\norm{\boldsymbol{v}_h^i}{\Hone{D}}^2 \lesssim k \to 0. $ When $0\le\theta\leq 1/2$, a standard inverse inequality,~\eqref{eq:denergy} and~\eqref{eq:hk 12} yield \[ \sum_{i=0}^{N-1}k^3\norm{\boldsymbol{v}_h^i}{\Hone{D}}^2 \lesssim \sum_{i=0}^{N-1}h^{-2}k^3\norm{\boldsymbol{v}_h^i}{\Ltwo{D}}^2 \lesssim h^{-2}k^2 \to 0, \] completing the proof of the lemma. \end{proof} The following lemma involving the ${\mathbb{L}}^2$-norm of the cross product of two vector-valued functions will be used when passing to the limit of equation~\eqref{eq:mhk Hhk1}. \begin{lemma}\label{lem:h} There exists a constant $C_{\rm sob}>0$ which depends only on $D$ such that \begin{equation} \label{eq:l2} \norm{\boldsymbol{w}_{0}\times\boldsymbol{w}_1}{\Ltwo{D}} \leq C_{\rm sob} \norm{\boldsymbol{w}_{0}}{{\mathbb{H}}^{1/2}(D)}\norm{\boldsymbol{w}_1}{\Hone{D}}. \end{equation} for all $\boldsymbol{w}_0\in{\mathbb{H}}^{1/2}(D)$ and $\boldsymbol{w}_{1}\in\Hone{D}$. \end{lemma} \begin{proof} It is shown in~\cite[Theorem~5.4, Part~I]{adams} that the embedding $\iota\colon\Hone{D}\to{\mathbb{L}}^6(D)$ is continuous. Obviously, the identity $\iota\colon \Ltwo{D}\to\Ltwo{D}$ is continous. By real interpolation, we find that $\iota\colon [\Ltwo{D},\Hone{D}]_{1/2}\to [\Ltwo{D},{\mathbb{L}}^6(D)]_{1/2}$ is continuous. Well-known results in interpolation theory show $ [\Ltwo{D},\Hone{D}]_{1/2}= {\mathbb{H}}^{1/2}(D) $ and $ [\Ltwo{D},{\mathbb{L}}^6(D)]_{1/2}={\mathbb{L}}^3(D) $ with equivalent norms; see e.g.~\cite[Theorem~5.2.1]{BL}. By using H\"older's inequality, we deduce \begin{align*} \norm{\boldsymbol{w}_{0}\times\boldsymbol{w}_{1}}{\Ltwo{D}}\leq \norm{\boldsymbol{w}_{0}}{{\mathbb{L}}^3(D)}\norm{\boldsymbol{w}_{1}}{{\mathbb{L}}^6(D)} \lesssim \norm{\boldsymbol{w}_{0}}{{\mathbb{H}}^{1/2}(D)}\norm{\boldsymbol{w}_1}{\Hone{D}}, \end{align*} proving the lemma. \end{proof} Finally, to pass to the limit in equation~\eqref{eq:mhk Hhk2} we need the following result. \begin{lemma}\label{lem:wea con} For any sequence~$\{\lambda_h\}\subset H^{1/2}(\Gamma)$ and any function~$\lambda\in H^{1/2}(\Gamma)$, if \begin{equation}\label{eq:zet con} \lim_{h\to0} \dual{\lambda_h}{\nu}_{\Gamma} = \dual{\lambda}{\nu}_{\Gamma} \quad\forall\nu\in H^{-1/2}(\Gamma) \end{equation} then \begin{equation}\label{eq:dtn zet con} \lim_{h\to0} \dual{\dtn_h\lambda_h}{\zeta}_{\Gamma} = \dual{\dtn\lambda}{\zeta}_{\Gamma} \quad\forall\zeta\in H^{1/2}(\Gamma). \end{equation} \end{lemma} \begin{proof} Let~$\mu$ and~$\mu_h$ be defined by~\eqref{eq:bem} with~$\lambda$ in the second equation replaced by~$\lambda_h$. Then (recalling that Costabel's symmetric coupling is used) $\dtn\lambda$ and~$\dtn_h\lambda_h$ are defined via~$\mu$ and~$\mu_h$ by~\eqref{eq:dtn2} and~\eqref{eq:bem2}, respectively, namely, $\dtn\lambda = (1/2-\dlp^\prime)\mu - \hyp\lambda$ and $ \dual{\dtn_h\lambda_h}{\zeta_h}_\Gamma = \dual{(1/2-\dlp^\prime)\mu_h}{\zeta_h}_\Gamma - \dual{\hyp \lambda_h}{\zeta_h}_\Gamma $ for all~$\zeta_h\in {\mathcal S}^1({\mathcal T}_h|_\Gamma)$. For any~$\zeta\in H^{1/2}(\Gamma)$, let~$\{\zeta_h\}$ be a sequence in~${\mathcal S}^1({\mathcal T}_h|_\Gamma)$ satisfying~$\lim_{h\to0}\norm{\zeta_h-\zeta}{H^{1/2}(\Gamma)}=0$. By using the triangle inequality and the above representations of~$\dtn\lambda$ and~$\dtn_h\lambda_h$ we deduce \begin{align}\label{eq:dtn dtn} \big| \dual{\dtn_h\lambda_h}{\zeta} - \dual{\dtn\lambda}{\zeta}_\Gamma \big| &\leq \big|\dual{\dtn_h\lambda_h-\dtn\lambda}{\zeta_h}_\Gamma\big| + \big|\dual{\dtn_h\lambda_h-\dtn\lambda}{\zeta-\zeta_h}_\Gamma\big| \notag \\ &\le \big| \dual{(\tfrac12-\dlp^\prime)(\mu_h-\mu)}{\zeta_h}_\Gamma \big| + \big| \dual{\hyp (\lambda_h-\lambda)}{\zeta_h}_\Gamma \big| \notag \\ &\quad + \big|\dual{\dtn_h\lambda_h-\dtn\lambda}{\zeta-\zeta_h}_\Gamma\big| \notag \\ &\le \big| \dual{(\tfrac12-\dlp^\prime)(\mu_h-\mu)}{\zeta_h}_\Gamma \big| + \big| \dual{\hyp (\lambda_h-\lambda)}{\zeta}_\Gamma \big| \notag \\ &\quad + \big| \dual{\hyp (\lambda_h-\lambda)}{\zeta_h-\zeta}_\Gamma \big| + \big|\dual{\dtn_h\lambda_h-\dtn\lambda}{\zeta-\zeta_h}_\Gamma\big|. \end{align} The second term on the right-hand side of~\eqref{eq:dtn dtn} goes to zero as~$h\to0$ due to~\eqref{eq:zet con} and the self-adjointness of~$\hyp$. The third term converges to zero due to the strong convergence~$\zeta_h\to\zeta$ in~$H^{1/2}(\Gamma)$ and the boundedness of~$\{\lambda_h\}$ in~$H^{1/2}(\Gamma)$, which is a consequence of~\eqref{eq:zet con} and the Banach-Steinhaus Theorem. The last term tends to zero due to the convergence of~$\{\zeta_h\}$ and the boundedness of~$\{\dtn_h\lambda_h\}$; see~\eqref{eq:dtnelliptic}. Hence~\eqref{eq:dtn zet con} is proved if we prove \begin{equation}\label{eq:muh mu} \lim_{h\to0} \dual{(1/2-\dlp^\prime)(\mu_h-\mu)}{\zeta_h}_{\Gamma} = 0. \end{equation} We have \begin{align}\label{eq:muh mu3} \dual{(\tfrac12-\dlp^\prime)(\mu_h-\mu)}{\zeta_h}_\Gamma = \dual{\mu_h-\mu}{(\tfrac12-\dlp)\zeta}_\Gamma + \dual{\mu_h-\mu}{(\tfrac12-\dlp)(\zeta_h-\zeta)}_\Gamma. \end{align} The definition of~$\mu_h$ implies $ \norm{\mu_h}{H^{-1/2}(\Gamma)} \lesssim \norm{\lambda_h}{H^{1/2}(\Gamma)} \lesssim 1, $ and therefore the second term on the right-hand side of~\eqref{eq:muh mu3} goes to zero. Hence it suffices to prove \begin{equation}\label{eq:muh mu2} \lim_{h\to0} \dual{\mu_h-\mu}{\eta}_\Gamma = 0 \quad\forall\eta\in H^{1/2}(\Gamma). \end{equation} Since~$\slp : H^{-1/2}(\Gamma) \to H^{1/2}(\Gamma)$ is bijective and self-adjoint, for any~$\eta\in H^{1/2}(\Gamma)$ there exists~$\nu\in H^{-1/2}(\Gamma)$ such that \[ \dual{\mu_h-\mu}{\eta}_{\Gamma} = \dual{\mu_h-\mu}{\slp\nu}_{\Gamma} = \dual{\slp(\mu_h-\mu)}{\nu}_{\Gamma} = \dual{\slp(\mu_h-\mu)}{\nu_h}_{\Gamma} + \dual{\slp(\mu_h-\mu)}{\nu-\nu_h}_{\Gamma}, \] where~$\{\nu_h\}\subset{\mathcal P}^0({\mathcal T}_h|\Gamma)$ is a sequence satisfying~$\norm{\nu_h-\nu}{H^{-1/2}(\Gamma)}\to0$. The definitions of~$\mu_h$ and~$\mu$, and the above equation imply \begin{align*} \dual{\mu_h-\mu}{\eta}_{\Gamma} &= \dual{(\dlp-\tfrac12)(\lambda_h-\lambda)}{\nu_h}_{\Gamma} + \dual{\slp(\mu_h-\mu)}{\nu-\nu_h}_{\Gamma} \\ &= \dual{\lambda_h-\lambda}{(\dlp^\prime-\tfrac12)\nu_h}_{\Gamma} + \dual{\slp(\mu_h-\mu)}{\nu-\nu_h}_{\Gamma} \\ &= \dual{\lambda_h-\lambda}{(\dlp^\prime-\tfrac12)\nu}_{\Gamma} + \dual{\lambda_h-\lambda}{(\dlp^\prime-\tfrac12)(\nu_h-\nu)}_{\Gamma} + \dual{\slp(\mu_h-\mu)}{\nu-\nu_h}_{\Gamma}. \end{align*} The first two terms on the right-hand side go to zero due to the convergence of~$\{\lambda_h\}$ and~$\{\nu_h\}$. The last term also approaches zero if we note the boundedness of~$\{\mu_h\}$. This proves~\eqref{eq:muh mu2} and completes the proof of the lemma. \end{proof} \subsection{Proof of Theorem~\ref{thm:weakconv}}\label{section:weak} We are now ready to prove that the problem~\eqref{eq:strong}--\eqref{eq:con} has a weak solution. \begin{proof} We recall from~\eqref{eq:wc1}--\eqref{eq:wc5} that $\boldsymbol{m}\in \Hone{D_T}$, $(\boldsymbol{H},\lambda)\in L^2(0,T;{\mathcal X})$ and $\boldsymbol{H}\in H^1(0,T;\Ltwo{D})$. By virtue of Lemma~\ref{lem:bil for} it suffices to prove that $(\boldsymbol{m},\boldsymbol{H},\lambda)$ satisfies~\eqref{eq:wssymm1} and~\eqref{eq:bil for}. Let $\boldsymbol{\phi}\in C^\infty(D_T)$ and $B:=(\boldsymbol{\xi},\zeta)\in L^2(0,T;{\mathcal X})$. On the one hand, we define the test function $\boldsymbol{\phi}_{hk}:=\Pi_{{\mathcal S}}(\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi})$ as the usual interpolant of~$\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi}$ into ${\mathcal S}^1({\mathcal T}_h)^3$. By definition, $\boldsymbol{\phi}_{hk}(t,\cdot) \in{\mathcal K}_{\boldsymbol{m}_h^j}$ for all $t\in[t_j,t_{j+1})$. On the other hand, it follows from Lemma~\ref{lem:den pro} that there exists~$B_h:=(\boldsymbol{\xi}_h,\zeta_h)\in{\mathcal X}_h$ converging to~$B\in{\mathcal X}$. Equations~\eqref{eq:mhk Hhk} hold with these test functions. The main idea of the proof is to pass to the limit in~\eqref{eq:mhk Hhk1} and~\eqref{eq:mhk Hhk2} to obtain~\eqref{eq:wssymm1} and~\eqref{eq:bil for}, respectively. In order to prove that~\eqref{eq:mhk Hhk1} implies~\eqref{eq:wssymm1} we will prove that as~$h,k\to0$ \begin{subequations}\label{eq:conv} \begin{align} \dual{\boldsymbol{v}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} &\to \dual{\boldsymbol{m}_t}{ \boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}, \label{eq:conv1} \\ \dual{\boldsymbol{m}_{hk}^-\times\boldsymbol{v}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} &\to \dual{\boldsymbol{m}\times\boldsymbol{m}_t}{\boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}, \label{eq:conv2} \\ k\dual{\nabla\boldsymbol{v}_{hk}^-}{\nabla\boldsymbol{\phi}_{hk}}_{D_T} &\to0, \label{eq:conv3} \\ \dual{\nabla\boldsymbol{m}_{hk}^-}{\nabla\boldsymbol{\phi}_{hk}}_{D_T} &\to \dual{\nabla\boldsymbol{m}}{\nabla(\boldsymbol{m}\times\boldsymbol{\phi})}_{D_T}, \label{eq:conv4} \\ \dual{\boldsymbol{H}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} &\to \dual{\boldsymbol{H}}{ \boldsymbol{m}\times\boldsymbol{\phi}}_{D_T}. \label{eq:conv5} \end{align} \end{subequations} Firstly, it can be easily shown that (see~\cite{alouges}) \begin{equation}\label{eq:phi mhk} \norm{\boldsymbol{\phi}_{hk}-\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi}}{L^2(0,T;\Hone{D})} \lesssim h \norm{\boldsymbol{m}_{hk}^-}{L^2(0,T;\Hone{D})} \norm{\boldsymbol{\phi}}{{\mathbb W}^{2,\infty}(D_T)} \lesssim h \norm{\boldsymbol{\phi}}{{\mathbb W}^{2,\infty}(D_T)} \end{equation} and \begin{align}\label{eq:phiinfty} \norm{\boldsymbol{\phi}_{hk}-\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi}}{L^\infty(0,T;\Hone{D})} \lesssim h \norm{\boldsymbol{m}_{hk}^-}{L^\infty(0,T;\Hone{D})} \norm{\boldsymbol{\phi}}{{\mathbb W}^{2,\infty}(D_T)} \lesssim h \norm{\boldsymbol{\phi}}{{\mathbb W}^{2,\infty}(D_T)}, \end{align} where we used~\eqref{eq:mhk pm}. In particular, we have \begin{equation}\label{eq:phi hk inf} \norm{\boldsymbol{\phi}_{hk}}{L^\infty(0,T;\Hone{D})} \lesssim 1. \end{equation} We now prove~\eqref{eq:conv1} and~\eqref{eq:conv5}. With~\eqref{eq:phi mhk}, there holds for $h,k\to0$, \begin{align}\label{eq:phihk mhk} \norm{\boldsymbol{\phi}_{hk}-\boldsymbol{m}\times\boldsymbol{\phi}}{\Ltwo{D_T}} &\le \norm{\boldsymbol{\phi}_{hk}-\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi}}{\Ltwo{D_T}} + \norm{(\boldsymbol{m}_{hk}^--\boldsymbol{m})\times\boldsymbol{\phi}}{\Ltwo{D_T}} \nonumber \\ &\lesssim \big(h + \norm{\boldsymbol{m}_{hk}^--\boldsymbol{m}}{\Ltwo{D_T}}\big) \norm{\boldsymbol{\phi}}{{\mathbb W}^{2,\infty}(D_T)} \to0 \end{align} due to~\eqref{eq:wc2}. Consequently, with the help of~\eqref{eq:wc31} and~\eqref{eq:wc5} we obtain~\eqref{eq:conv1} and~\eqref{eq:conv5}. In order to prove~\eqref{eq:conv2} we note that the elementary identity \begin{equation}\label{eq:abc} \boldsymbol{a}\cdot(\boldsymbol{b}\times\boldsymbol{c})= \boldsymbol{b}\cdot(\boldsymbol{c}\times\boldsymbol{a})= \boldsymbol{c}\cdot(\boldsymbol{a}\times\boldsymbol{b}) \quad\forall\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}\in{\mathbb R}^3 \end{equation} yields \begin{align}\label{eq:idid} \dual{\boldsymbol{m}_{hk}^-\times\boldsymbol{v}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} = \dual{\boldsymbol{v}_{hk}^-}{\boldsymbol{\phi}_{hk}\times\boldsymbol{m}_{hk}^-}_{D_T}. \end{align} It follows successively from the triangle inequality, ~\eqref{eq:l2} and~\eqref{eq:phi hk inf} that \begin{align*} \norm{\boldsymbol{\phi}_{hk}&\times\boldsymbol{m}_{hk}^{-} - (\boldsymbol{m}\times\boldsymbol{\phi})\times\boldsymbol{m}}{\Ltwo{D_T}}\\ &\leq \norm{\boldsymbol{\phi}_{hk}\times(\boldsymbol{m}_{hk}^--\boldsymbol{m})}{\Ltwo{D_T}}+ \norm{(\boldsymbol{\phi}_{hk}-(\boldsymbol{m}\times\boldsymbol{\phi}))\times\boldsymbol{m}}{\Ltwo{D_T}}\\ &\lesssim \Big(\int_0^T\norm{\boldsymbol{\phi}_{hk}(t)}{\Hone{D}}^2\norm{\boldsymbol{m}_{hk}^-(t)-\boldsymbol{m}(t)}{{\mathbb{H}}^{1/2}(D)}^2\,dt\Big)^{1/2}+ \norm{\boldsymbol{\phi}_{hk}-(\boldsymbol{m}\times\boldsymbol{\phi})}{\Ltwo{D_T}}\\ &\leq \norm{\boldsymbol{\phi}_{hk}}{L^\infty(0,T;\Hone{D})} \norm{\boldsymbol{m}_{hk}^--\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))} + \norm{\boldsymbol{\phi}_{hk}-(\boldsymbol{m}\times\boldsymbol{\phi})}{\Ltwo{D_T}} \\ &\lesssim \norm{\boldsymbol{m}_{hk}^--\boldsymbol{m}}{L^2(0,T;{\mathbb{H}}^{1/2}(D))} + \norm{\boldsymbol{\phi}_{hk}-(\boldsymbol{m}\times\boldsymbol{\phi})}{\Ltwo{D_T}}. \end{align*} Thus~\eqref{eq:mhk h12} and~\eqref{eq:phihk mhk} imply $\boldsymbol{\phi}_{hk}\times\boldsymbol{m}_{hk}^-\to (\boldsymbol{m}\times\boldsymbol{\phi})\times\boldsymbol{m}$ in $\Ltwo{D_T}$. This together with~\eqref{eq:wc5} and~\eqref{eq:idid} implies \[ \dual{\boldsymbol{m}_{hk}^-\times\boldsymbol{v}_{hk}^-}{\boldsymbol{\phi}_{hk}}_{D_T} \to \dual{\boldsymbol{m}_t}{(\boldsymbol{m}\times\boldsymbol{\phi})\times\boldsymbol{m}}_{D_T}, \] which is indeed~\eqref{eq:conv2} by invoking~\eqref{eq:abc}. Statement~\eqref{eq:conv4} follows from~\eqref{eq:phi mhk}, \eqref{eq:wc1a}, and~\eqref{eq:wc2} as follows: As $h,k\to 0$, \begin{align*} \dual{\nabla\boldsymbol{m}_{hk}^-}{\nabla\boldsymbol{\phi}_{hk}}_{D_T} &= \dual{\nabla\boldsymbol{m}_{hk}^-}{\nabla(\boldsymbol{\phi}_{hk}-\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi})}_{D_T} + \dual{\nabla\boldsymbol{m}_{hk}^-}{\nabla(\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi})}_{D_T} \\ &= \dual{\nabla\boldsymbol{m}_{hk}^-}{\nabla(\boldsymbol{\phi}_{hk}-\boldsymbol{m}_{hk}^-\times\boldsymbol{\phi})}_{D_T} + \dual{\nabla\boldsymbol{m}_{hk}^-}{\boldsymbol{m}_{hk}^-\times\nabla\boldsymbol{\phi}}_{D_T} \\ &\longrightarrow \dual{\nabla\boldsymbol{m}}{0}_{D_T} + \dual{\nabla\boldsymbol{m}}{\boldsymbol{m}\times\nabla\boldsymbol{\phi}}_{D_T} = \dual{\nabla\boldsymbol{m}}{\nabla(\boldsymbol{m}\times\boldsymbol{\phi})}_{D_T} . \end{align*} Finally, in order to prove~\eqref{eq:conv3} we first note that \eqref{eq:phi mhk} and the boundedness of the sequence~$\{\norm{\boldsymbol{m}_{hk}^-}{L^2(0,T;\Hone{D})}\}$, see~\eqref{eq:mhk pm}, give the boundedness of~$\{\norm{\boldsymbol{\phi}_{hk}}{L^2(0,T;\Hone{D}}\}$, and thus of~$\{\norm{\nabla\boldsymbol{\phi}_{hk}}{\Ltwo{D_T}}\}$. On the other hand, \begin{equation}\label{eq:vhk} \norm{\nabla\boldsymbol{v}_{hk}^-}{\Ltwo{D_T}}^2 = k \sum_{i=0}^{N-1} \norm{\nabla\boldsymbol{v}_{h}^i}{\Ltwo{D}}^2. \end{equation} If $1/2<\theta\le1$ then~\eqref{eq:denergy} and~\eqref{eq:vhk} yield the boundedness of $\{\norm{\nabla\boldsymbol{v}_{hk}^-}{\Ltwo{D_T}}\}$. Hence \[ k\dual{\nabla\boldsymbol{v}_{hk}^-}{\nabla\boldsymbol{\phi}_{hk}}_{D_T} \to0 \quad\text{as }h,k\to 0. \] If $0\le\theta\le1/2$ then the inverse estimate,~\eqref{eq:vhk}, and~\eqref{eq:denergy} yield \[ \norm{\nabla\boldsymbol{v}_{hk}^-}{\Ltwo{D_T}}^2 \lesssim kh^{-2} \sum_{i=0}^{N-1} \norm{\boldsymbol{v}_{h}^i}{\Ltwo{D}}^2 \lesssim h^{-2}, \] so that $ \left| k\dual{\nabla\boldsymbol{v}_{hk}^-}{\nabla\boldsymbol{\phi}_{hk}}_{D_T} \right| \lesssim kh^{-1}. $ This goes to 0 under the assumption~~\eqref{eq:hk 12}. Altogether, we obtain~\eqref{eq:wssymm1} when passing to the limit in~\eqref{eq:mhk Hhk1}. Next, recalling that~$B_h\to B$ in~${\mathcal X}$ we prove that~\eqref{eq:mhk Hhk2} implies~\eqref{eq:bil for} by proving \begin{subequations}\label{eq:convb} \begin{align} \dual{\partial_t\boldsymbol{H}_{hk}}{\boldsymbol{\xi}_h}_{D_T} &\to \dual{\boldsymbol{H}_t}{\boldsymbol{\xi}}_{D_T}, \label{eq:convb1} \\ \dual{\dtn_h\partial_t\lambda_{hk}}{\zeta_h}_{\Gamma_T} &\to \dual{\dtn\lambda_t}{\zeta}_{\Gamma_T}, \label{eq:spec} \\ \dual{\nabla\times\boldsymbol{H}_{hk}^+}{\nabla\times\boldsymbol{\xi}_h}_{D_T} &\to \dual{\nabla\times\boldsymbol{H}}{\nabla\times\boldsymbol{\xi}}_{D_T}, \\ \dual{\boldsymbol{v}_{hk}^-}{\boldsymbol{\xi}_h}_{D_T} &\to \dual{\boldsymbol{v}}{\boldsymbol{\xi}}_{D_T}. \end{align} \end{subequations} The proof is similar to that of~\eqref{eq:conv} (where we use Lemma~\ref{lem:wea con} for the proof of~\eqref{eq:spec}) and is therefore omitted. This proves~(3) and~(5) of Definition~\ref{def:fembemllg}. Finally, we obtain $\boldsymbol{m}(0,\cdot)=\boldsymbol{m}^0$, $\boldsymbol{H}(0,\cdot)=\boldsymbol{H}^0$, and $\lambda(0,\cdot)=\lambda^0$ from the weak convergence and the continuity of the trace operator. This and $|\boldsymbol{m}|=1$ yield Statements~(1)--(2) of Definition~\ref{def:fembemllg}. To obtain~(4), note that $\nabla_\Gamma\colon H^{1/2}(\Gamma)\to {\mathbb{H}}_\perp^{-1/2}(\Gamma)$ and $\boldsymbol{n}\times(\boldsymbol{n}\times(\cdot))\colon \Hcurl{D}\to {\mathbb{H}}_\perp^{-1/2}(\Gamma)$ are bounded linear operators; see~\cite[Section~4.2]{buffa2} for exact definition of the spaces and the result. Weak convergence then proves~(4) of Definition~\ref{def:fembemllg}. Estimate~\eqref{eq:energybound2} follows by weak lower-semicontinuity and the energy bound~\eqref{eq:denergy}. This completes the proof of the theorem. \end{proof} \section{Numerical experiment}\label{section:numerics} The following numerical experiment is carried out by use of the FEM toolbox FEniCS~\cite{fenics} (\texttt{fenicsproject.org}) and the BEM toolbox BEM++~\cite{bempp} (\texttt{bempp.org}). We use GMRES to solve the linear systems and blockwise diagonal scaling as preconditioners. The values of the constants in this example are taken from the standard problem \#1 proposed by the Micromagnetic Modelling Activity Group at the National Institute of Standards and Technology~\cite{mumag}. As domain serves the unit cube $D=[0,1]^3$ with initial conditions \begin{align*} \boldsymbol{m}^0(x_1,x_2,x_3):=\begin{cases} (0,0,-1)&\text{for } d(x)\geq 1/4,\\ (2Ax_1,2Ax_2,A^2-d(x))/(A^2+d(x))&\text{for }d(x)<1/4, \end{cases} \end{align*} where $d(x):= |x_1-0.5|^2+|x_2-0.5|^2$ and $A:=(1-2\sqrt{d(x)})^4/4$ and \begin{align*} \boldsymbol{H}^0= \begin{cases} (0,0,2)&\text{in } D,\\ (0,0,2)-\boldsymbol{m}^0&\text{in } D^\ast. \end{cases} \end{align*} We choose the constants \begin{align*} \alpha=0.5,\quad \sigma=\begin{cases}1&\text{in }D,\\ 0& \text{in }D^\ast,\end{cases}\quad \mu_0=1.25667\times 10^{-6},\quad C_e=\frac{2.6\times 10^{-11}}{\mu_0 \,6.4\times 10^{11}}. \end{align*} For time and space discretisation of $D_T:= [0,5]\times D$, we apply a uniform partition in space ($h=0.1$) and time ($k=0.002$). Figure~\ref{fig:en} plots the corresponding energies over time. Figure~\ref{fig:m} shows a series of magnetizations $\boldsymbol{m}(t_i)$ at certain times $t_i\in[0,5]$. Figure~\ref{fig:h} shows that same for the magnetic field $\boldsymbol{H}(t_i)$. \begin{figure} \psfrag{energy}{\tiny energy} \psfrag{time}{\tiny time $t$} \psfrag{menergy}{\tiny magnetization energy} \psfrag{henergy}{\tiny magnetic energy} \psfrag{sum}{\tiny total energy} \includegraphics[width=0.6\textwidth]{pics/energy2.eps} \caption{The magnetization engergy $\norm{\nabla\boldsymbol{m}_{hk}(t)}{\Ltwo{D}}$ and the energy of the magnetic field $\norm{\boldsymbol{H}_{hk}(t)}{\Hcurl{D}}$ plotted over the time.} \label{fig:en} \end{figure} \begin{figure} \includegraphics[width=0.24\textwidth]{pics/m20.eps} \includegraphics[width=0.24\textwidth]{pics/m21.eps} \includegraphics[width=0.24\textwidth]{pics/m22.eps} \includegraphics[width=0.24\textwidth]{pics/m23.eps} \includegraphics[width=0.24\textwidth]{pics/m24.eps} \includegraphics[width=0.24\textwidth]{pics/m25.eps} \includegraphics[width=0.24\textwidth]{pics/m26.eps} \includegraphics[width=0.24\textwidth]{pics/m27.eps} \includegraphics[width=0.24\textwidth]{pics/m28.eps} \includegraphics[width=0.24\textwidth]{pics/m29.eps} \includegraphics[width=0.24\textwidth]{pics/m210.eps} \includegraphics[width=0.24\textwidth]{pics/colorbar2.eps} \caption{Slice of the magnetization $\boldsymbol{m}_{hk}(t_i)$ at $[0,1]^2\times \{1/2\}$ for $i=0,\ldots,10$ with $t_i=0.2i$. The color of the vectors represents the magnitude $|\boldsymbol{m}_{hk}|$. We observe that the magnetization aligns itself with the initial magnetic field $\boldsymbol{H}^0$ by performing a damped precession.} \label{fig:m} \end{figure} \begin{figure} \includegraphics[width=0.24\textwidth]{pics/h20.eps} \includegraphics[width=0.24\textwidth]{pics/h21.eps} \includegraphics[width=0.24\textwidth]{pics/h22.eps} \includegraphics[width=0.24\textwidth]{pics/h23.eps} \includegraphics[width=0.24\textwidth]{pics/h24.eps} \includegraphics[width=0.24\textwidth]{pics/h25.eps} \includegraphics[width=0.24\textwidth]{pics/h26.eps} \includegraphics[width=0.24\textwidth]{pics/h27.eps} \includegraphics[width=0.24\textwidth]{pics/h28.eps} \includegraphics[width=0.24\textwidth]{pics/h29.eps} \includegraphics[width=0.24\textwidth]{pics/h210.eps} \includegraphics[width=0.24\textwidth]{pics/colorbarh2.eps} \caption{Slice of the magnetic field $\boldsymbol{H}_{hk}(t_i)$ at $[0,1]^2\times \{1/2\}$ for $i=0,\ldots,10$ with $t_i=0.2i$. The color of the vectors represents the magnitude $|\boldsymbol{H}_{hk}|$. We observe only a slight movement in the middle of the cube combined with an overall reduction of field strength.} \label{fig:h} \end{figure} \bibliographystyle{myabbrv}

BkiUerw5qhLBgou5wSXc

\section{Introduction} \label{S1} \setcounter{equation}{0} In this paper, we consider a class of nonlocal feedback controllers with application to the control of certain nonlinear partial differential equations. The research on feedback control laws of this type has become quite active in theoretical physics for stabilizing wave-type solutions of reaction-diffusion systems such as the Schl\"ogl model (also known as Nagumo or Chafee-Infante equation) or the FitzHugh-Nagumo system. The controllers can be characterized as follows: First of all, they are a generalization of Pyragas type controllers that became very popular in the past. We refer to \cite{pyragas1992}, \cite{pyragas2006}, and the survey volume \cite{schoell_schuster2008}. In the simplest form of Pyragas type feedback control, the difference of the current state $u(x,t)$ and the retarded state $u(x,t-\tau)$, multiplied with a real number $\kappa$, is taken as control, i.e. the feedback control $f$ is \[ f(x,t) := \kappa \, (u(x,t) - u(x,t - \tau)), \] where $\tau$ is a fixed time delay and $\kappa$ is the feedback gain. In the nonlocal generalization we consider in this paper, the feedback control is set up by an integral operator of the form \begin{equation} f(x,t) := \kappa \, \left(\int_{0}^{T} g(\tau) u(x,t-\tau)\,d\tau - u(x,t)\right). \label{E1.1} \end{equation} Here, different time delays appear in a distributed way. Depending on the particular choice of the kernel $g$, various spatio-temporal patterns of the controlled solution $u$ can be achieved. We refer to \cite{bachmair_schoell14,loeber_etal2014,siebert_schoell14}, and \cite{siebert_alonso_baer_schoell14} with application to the Schl\"ogl model and to \cite{atay2003,kyrichenko_blyuss_schoell2014,wille_lehnert_schoell2014} with respect to control of ordinary differential equations. Our main goal is the selection of the kernel $g$ in an optimal way. We want to achieve a desired spatio-temporal pattern for the resulting state function and look for an optimal feedback kernel $g$ to approximate this pattern as closely as possible. For this purpose, in the second half of the paper we will concentrate on a particular choice of $g$ as a step function. We are optimizing feedback controllers but we shall apply methods of optimal control to achieve our goal. This leads to new optimal control problems for reaction-diffusion equations containing nonlocal terms with time delay in the state equation. We develop the associated necessary optimality conditions and discuss numerical approaches for solving the problems posed. Working on this class of problems, we observed that standard quadratic tracking type objective functionals are possibly not the right tool for approximating desired time-periodic patterns. We found out that the so-called cross correlation partially better fits to our goals. We report on our numerical tests at the end of this paper. This research contributes results to the optimal control of nonlinear reaction diffusion equations, where wave type solutions such as traveling wave fronts or spiral waves occur in unbounded domains. We mention the papers \cite{borzi_griesse06,brandao_etal08,ckp09} on the optimal control of systems that develop spiral waves or \cite{kun_nag_cha_wag2011,kunisch_wagner2012-3,kunisch_wang12} on systems with heart medicine as background. Moreover, we refer to \cite{buch_eng_kamm_tro2013,casas_ryll_troeltzsch2014b}, where different numerical and theoretical aspects of optimal control of the Schl\"ogl or FitzHugh-Nagumo equations are discussed. It is a characteristic feature of such systems that the computed optimal solutions might be unstable with respect to perturbations in the data, in particular initial data. Feedback control aims at generating stable solutions. Various techniques of feedback control are known, we refer only to the monographies \cite{coron07,lasiecka_triggiani2000a,lasiecka_triggiani2000b,Krstic2010} and to the references cited therein. Moreover, we mention \cite{gugat_troeltzsch2013} on feedback stabilization for the Schl\"ogl model. Pyragas type feedback control is one associated field of research that became very active, cf. \cite{schoell_schuster2008} for an account on current research in this field. In associated publications, the feedback control laws were considered as given. For instance, the kernel in nonlocal delayed feedback was given and it was studied what kind of patterns arise from different choices of the kernel. The novelty of our paper is that we study an associated inverse (say design) problem: Find a kernel such that the associated feedback solution best approximates a desired pattern. \section{Two models of feedback control} \label{S2} \setcounter{equation}{0} We consider the following semilinear parabolic equation with reaction term $R$ and control function (forcing) $f$, \begin{equation} \partial_t u - \Delta u +R(u) = f \label{E:2.1} \end{equation} subject to appropriate initial and boundary conditions in a spatio-temporal domain $\Omega \times (0,T)$. Using a feedback control in the form \eqref{E1.1}, we arrive at the following nonlinear initial-boundary value problem that includes a nonlocal term with time delay, \begin{equation} \label{E:2.2} \begin{array}{rcll} \partial_tu(x,t) - \Delta u(x,t)+R(u(x,t)) &=&\displaystyle \kappa \left( \int_0^T g(\tau) u(x,t-\tau)\,d\tau - u(x,t)\right)& \mbox{in }\Omega, \\[2ex] u(x,s)&=& u_0(x,s)& \mbox{in }\Omega,\\[1ex] \displaystyle {\partial_n u}(x,t) &=& 0&\mbox{on } \Gamma, \end{array} \end{equation} for almost all $t \in (0,T), \, s \in [-T,0]$. \vspace{1ex} Here, $\partial_n$ denotes the outward normal derivative on $\Gamma=\partial \Omega$. We want to determine a feedback kernel $g \in L^\infty(0,T) $ such that the solution $u$ to \eqref{E:2.2} is as close as possible to a desired function $u_d$. The function $g$ will have to obey certain restrictions, namely \begin{eqnarray} 0 \le g(t) &\le& \beta \quad \mbox{ a.e. on } [0,T], \label{E:2.3}\\ \displaystyle \int_0^T g(s)\, ds &=& 1,\label{E:2.4} \end{eqnarray} where $\beta> 0$ is a given (large) positive constant. This upper bound is chosen to have a uniform bound for $g$. It is needed for proving the solvability of the optimal control problem. We shall present the main part of our theory for the general type of $g$ defined above. In our numerical computations, however, we will concentrate on functions $g$ of the following particular form: We select $t_1,\, t_2$ such that $0 \leq t_1 < t_2 \leq T$, $t_2-t_1\ge \delta > 0$ and define \begin{equation} \label{E:2.5} g(t) = \left\{ \begin{array}{cl} \displaystyle \frac{1}{t_2 -t_1}, & t_1 \leq t \leq t_2 \\[2ex] 0, & \mbox{elsewhere}. \end{array} \right. \end{equation} It is obvious that $g$ satisfies the constraints \eqref{E:2.3},\eqref{E:2.4} with $\beta = 1/\delta$. Using this form for $g$, we end up with the particular feedback equation \begin{equation} \label{E:2.6} \partial_tu(x,t) - \Delta u(x,t) +R(u(x,t)) = \kappa \, \left(\frac{1}{t_2-t_1} \int_{t_1}^{t_2} u(x,t-\tau) \, d\tau - u(x,t)\right). \end{equation} In \eqref{E:2.6}, we will also vary $\kappa$ in the state equation as part of the control variables to be optimized. In contrast to this, $\kappa$ is assumed to be fixed in the model with a general control function $g$. In the special model, we have a restricted flexibility in the optimization, because only the real numbers $k, \, t_1, \, t_2$ can be varied. Yet, we are able to generate a class of interesting time-periodic patterns. Throughout the paper we will rely on the following \vspace{1ex} \noindent {\bf Assumptions.} The set $\Omega \subset \mathbb{R}^N$, $N\leq 3$, is a bounded Lipschitz domain; for $N = 1$, we set $\Omega=(a,b)$. By $T > 0$, a finite terminal time is fixed. In theoretical physics, also the choice $T = \infty$ is of interest. However, we do not investigate the associated analysis, because an infinite time interval requires the use of more complicated function spaces. Moreover, the restriction to a bounded interval fits better to the numerical computations. Throughout the paper, we use the notation $Q:= \Omega \times (0,T)$ and $\Sigma = \Gamma \times (0,T)$. for the space-time cylinder. \begin{remark} We will often use the term ''wave type solution'' or ''traveling wave''. This is a function $(x,t) \mapsto u(x,t)$ that can be represented in the form $u(x,t) = v(x-c\,t)$ with some other smooth function $v$. Here, $c$ is the velocity of the wave type solution. Such solutions are known to exist in $\Omega = \mathbb{R}$ but not in in a bounded interval $\Omega=(a,b)$. In our paper, the terms '' wave type solution'' or ''traveling wave'' stand for solutions of the Schl\"ogl model in the bounded domain $(a,b)$. We use these terms, since the computed solutions exhibit a similar behavior as associated solutions in $\Omega = \mathbb{R}$. \end{remark} The reaction term $R$ is defined by \begin{equation} \label{E:2.7} R(u)=\rho\, (u-u_1)(u-u_2)(u-u_3), \end{equation} where $u_1\le u_2\le u_3$ and $\rho > 0$ are fixed real numbers. In our computational examples, we will take $\rho:=1$. The numbers $u_i$, $i = 1,\ldots, 3$, define the fixed points of the (uncontrolled) Schl\"ogl model \eqref{E:2.1}. In view of the time delay, we have to provide initial values $u_0$ for $u$ in the interval $[-T,0]$ for the general model \eqref{E:2.2} and in $[-t_2,0]$ for the special model \eqref{E:2.6}. We assume $u_0 \in C(\bar \Omega \times[-T,0])$ or $u_0 \in C(\bar \Omega \times[-t_2,0])$, respectively. The desired state $u_d$ is assumed to be bounded and measurable on $Q$. \section{Well-posedness of the feedback equation} In this section, we prove the existence and uniqueness of a solution to the general feedback equation \eqref{E:2.2}. To this aim, we first reduce the equation to an inhomogeneous initial-boundary value problem. For $t \in [0,T]$, we write \begin{eqnarray*} \int_0^T g(\tau) u(x,t-\tau) \, d\tau &=& \int_0^t g(\tau) u(x,t-\tau) \, d\tau +\underbrace{\int_t^T g(\tau) u(x,t-\tau) \, d\tau }_{=: U_g(x,t)} \\ &=& \int_0^t g(\tau) u(x,t-\tau) \, d\tau + U_g(x,t). \end{eqnarray*} The function $U_g$ is associated with the fixed initial function $u_0$ and is defined by \[ U_g(x,t) = \int_t^T g(\tau) u_0(x,t-\tau) \, d\tau; \] notice that we have $t-\tau \le 0$ in the integral above. By the assumed continuity of $u_0$, the function $U_g$ belongs to $C(\bar \Omega \times [0,T])$. Next, for given $g \in L^2(0,T)$, we introduce a linear integral operator $K(g): L^2(Q) \to L^2(Q)$ by \begin{equation} \label{E:3.1} (K(g)u)(x,t):= \int_0^t g(\tau) u(x,t-\tau) \, d\tau. \end{equation} Substituting $s = t-\tau$, we obtain the equivalent representation \[ (K(g)u)(x,t)= \int_0^t g(t-s) u(x,s) \, ds. \] Inserting $U_g$ and $K(g)$ in the state equation \eqref{E:2.2}, we obtain the following nonlocal initial-boundary value problem: \begin{equation} \label{E:3.2} \left\{ \begin{array}{rcll} \partial_tu -\Delta u +R(u) + \kappa \, u - \kappa \, K(g) u &=& \kappa \, U_g& \mbox{in }Q, \\[1ex] u(x,0)&=&u_0(x,0)& \mbox{in }\Omega, \\[1ex] \partial_n u &=&0& \mbox{on }\Sigma. \end{array} \right. \end{equation} In the next theorem, we use the Sobolev space \[ W(0,T) = L^2(0,T;H^1(\Omega)) \cap H^1(0,T;L^2(\Omega)). \] \begin{theorem} \label{T:3.1} For all $g \in L^\infty(0,T)$, $U_g \in L^p(Q), \, p > \frac{5}{2},$ and $u_0 \in C(\bar \Omega \times [-T,0])$, the problem \eqref{E:3.2} has a unique solution $u \in W(0,T) \cap C(\bar{Q})$. \end{theorem} \begin{proof} We use the same technique that was applied in \cite{casas_ryll_troeltzsch2014} to show the existence and continuity of the solution to the FitzHugh-Nagumo system. Let us mention the main steps. First, we apply a simple transformation that is well-known in the theory of evolution equations. We set \[ u = e^{\lambda t} v \] with some $\lambda >0$. This transforms the partial differential equation in \eqref{E:3.2} to an equation for the new unknown function $v$, \begin{equation} \label{E:3.3} v_t - \Delta v + e^{-\lambda t}R(e^{\lambda t} v) + (\lambda + \kappa) v = \kappa \, K_\lambda(g) v + e^{-\lambda t} \kappa \, U_g, \end{equation} where the integral operator $K_\lambda(g)$ is defined by \[ (K_\lambda(g) v)(x,t) = \int_0^t e^{-\lambda(t-s) } g(t-s)v(x,s)\, ds. \] If $g \in L^\infty(0,T)$, then both operators $K(g)$ and $K_\lambda(g)$ are continuous linear operators in $L^p(Q)$, for all $p\ge 1$. Moreover, due to the factor $e^{-\lambda(t-s) }$, the norm of $K_\lambda(g): L^2(Q) \to L^2(Q) $ tends to zero as $\lambda \to \infty$. We obtain \begin{equation} \label{E:3.4} \|K_\lambda(g)\|_{\mathcal{L}(L^2(Q))} \le \frac{c}{\sqrt{\lambda}} \|g\|_{L^\infty(0,T)} \end{equation} with some constant $c > 0$. To have this estimate, we assumed in \eqref{E:2.3} that $g$ is uniformly bounded by the constant $\beta$. If $\lambda$ is sufficiently large, then we have \[ \int_Q [e^{-\lambda t}R(e^{\lambda t} v) + (\lambda + \kappa) v - \kappa K_\lambda(g)v] \, v \, dxdt \ge \frac{\lambda}{2} \|v\|^2_{L^2(Q)} \quad \forall v \in L^2(Q), \] because the coercive term $(\lambda + \kappa)\, v$ in the left side is dominating the other terms, cf. \cite{casas_ryll_troeltzsch2014}. With this inequality, an a priori estimate can be derived in $L^2(Q)$ for any solution $v$ of the equation \eqref{E:3.2}. Now, we can proceed as in \cite{casas_ryll_troeltzsch2014}: A fixed-point principle is applied in $L^2(Q)$ to prove the existence and uniqueness of the solution $v$ that in turn implies the same for $u$. For the details, the reader is referred to \cite{casas_ryll_troeltzsch2014}, proof of Theorem 2.1. However, we mention one important idea: Thanks to \eqref{E:3.4}, the term $(\lambda + \kappa)$ absorbes the non-monotone terms in the equation \eqref{E:3.3} so that, in estimations, equation \eqref{E:3.3} behaves like the parabolic equation \[ v_t - \Delta v + \tilde R(v) = F \] with a monotone non-decreasing nonlinearity $\tilde R$ and given right-hand side $F \in L^p(Q)$, $p > 5/2$. This fact can be exploited to verify, for each $r > 0$, the existence of a constant $C_r > 0$ with the following property: If $g \in L^\infty(Q)$ obeys $\|g\|_{ L^\infty(Q)} \le r$ and $u$ is the associated solution to \eqref{E:2.2}, then \begin{equation} \label{E:3.5} \|u\|_{L^\infty(Q)} \le C_r. \end{equation} \hfill $\Box$ \end{proof} \section{Analysis of optimization problems for feedback controllers} \subsection{Definition of two optimization problems} \subsubsection*{General kernel as control} Let a desired function $u_d \in L^\infty(Q)$ be given. In our later applications, $u_d$ models a desired spatio-temporal pattern. Moreover, we fix a non-negative function $c_Q \in L^\infty(Q)$. This function is used for selecting a desired observation domain. We consider the feedback equation \eqref{E:2.2} and want to find a kernel $g$ such that the associated solution $u$ approximates $u_d$ as close as possible in the domain of observation. This goal is expressed by the following functional $j: L^2(Q) \times L^\infty(0,T) \to \mathbb{R}$ that is to be minimized, \[ j(u,g) :=\frac{1}{2} \iint_Q c_Q(u-u_d)^2 \, dxdt + \frac{\nu}{2} \int_0^T g^2(t) \, dt. \] Here, $\nu \ge 0$ is a Tikhonov regularization parameter. The standard choice of $c_Q$ is $c_Q(x,t) = 1$ for all $(x,t) \in Q$. Another selection will be applied for periodic functions $u_d$: $c(x,t) = 1$ for all $(x,t) \in Q$ with $t \ge T/2$ and $c(x,t) = 0$ for all $(x,t) \in Q$ with $t < T/2$. By Theorem \ref{T:3.1}, to each $g \in L^\infty(0,T)$ there exists a unique associated state function $u$ that will be denoted by $u_g$. Then $j$ does only depend on $g$ and we obtain the reduced objective functional $J$, \[ J: g \mapsto j(u_g,g). \] Therefore, our general optimization problem can be formulated as follows: \[ \tag{PG} \min_{g \in C} J(g):= \frac{1}{2} \iint_Q c_Q(u_g-u_d)^2 \, dxdt + \frac{\nu}{2} \int_0^T g^2(t) \, dt, \] where $C \subset L^\infty(0,T)$ is the convex and closed set defined by \[ C:= \left\{ g \in L^\infty(0,T): \ 0 \le g(t) \le \beta \; \mbox{ a.e. in } [0,T] \mbox{ and } \int_0^T g(t) \, dt =1.\right\} \] Notice that $C$ is a weakly compact subset of $L^2(0,T)$. The restrictions on $g$ are motivated by the background in mathematical physics. In particular, the restriction on the integral of $g$ guarantees that \[ \int_0^T g(\tau)u(x,t-\tau)\, d\tau - u(x,t) = 0, \] if $u(x,t-\tau)= u(x,t)$ in Q. By the definition of $u_g$, the optimization is subject to the state equation \eqref{E:2.2}. \subsubsection*{Special kernel as control} The other optimization problem we are interested in, uses the particular form \eqref{E:2.5} of the kernel $g$, \[ \min_{0 \leq t_1 < t_2 \leq T} J_S(\kappa,t_1,t_2):= \frac{1}{2} \iint_Q c_Q(u_{(\kappa,t_1,t_2)}-u_d)^2 \, dxdt + \frac{\nu}{2} (t_1^2+t_2^2 + \kappa^2), \] where $u_{(\kappa,t_1,t_2)}$ is the solution of \eqref{E:2.6} for a given triplet $(\kappa,t_1,t_2)$. This problem might fail to have an optimal solution, because the set of admissible triplets $(\kappa,t_1,t_2)$ is not closed. Notice that we need $t_1<t_2$ in \eqref{E:2.6}. Therefore, we fix $\delta > 0$ and define the slightly changed admissible set \[ C_\delta:= \left\{ (\kappa,t_1,t_2) \in \mathbb{R}^3: \ 0 \leq t_1 < t_2 \leq T, \, t_2 -t_1 \ge \delta, \ \kappa \in \mathbb{R} \right\} \] that is compact. In this way, we obtain the special finite-dimensional optimization problem for step functions $g$, \[ \tag{PS} \min_{(\kappa,t_1,t_2) \in C_\delta} J_S(\kappa,t_1,t_2):= \frac{1}{2} \iint_Q c_Q(u_{(\kappa,t_1,t_2)}-u_d)^2 \, dxdt + \frac{\nu}{2} (t_1^2+t_2^2+ \kappa^2). \] \subsection{Discussion of (PG)} \subsubsection*{The control-to-state mapping $G$} Next, we discuss the differentiability of the control-to-state mappings $g \mapsto u_g$ and $(\kappa,t_1,t_2) \mapsto u(\kappa,t_1,t_2)$. First, we consider the case of the general kernel $g$. The analysis for the particular kernel \eqref{E:2.5} is fairly analogous but cannot deduced as a particular case of (PG). We will briefly sketch it in a separate section. By Theorem \ref{T:3.1}, we know that the mapping $G: g \mapsto u_g$ is well defined from $L^\infty(0,T)$ to $C(\bar Q)$. Now we discuss the differentiability of $G$. To slightly simplify the notation, we introduce an operator $\mathcal{K}: L^\infty(0,T)\times C(\bar Q) \to C(\bar Q)$ by \[ \mathcal{K}(g,u) = K(g)u, \] where $K(g)$ was introduced in \eqref{E:3.1}; notice that $\mathcal{K}$ is bilinear. Let us first show the differentiability for $\mathcal{K}$. We fix $g \in L^\infty(0,T), \, u \in C(\bar Q)$, and select varying increments $h \in L^\infty(0,T)$, $v \in C(\bar Q)$. Then we have \[ \begin{split} &\mathcal{K}(g+h,u+v)= \int_0^T [g(\tau)+h(\tau)] [u(x,t-\tau)+v(x,t-\tau)] \, d\tau \\ &\quad = \int_0^t g(\tau)u(x,t-\tau) \, d\tau + \underbrace{\int_0^t h(\tau)u(x,t-\tau) \, d\tau + \int_0^t g(\tau)v(x,t-\tau)\, d\tau}_{A(g,u)(h,v)}\\ &\quad \qquad + \underbrace{\int_0^t h(\tau)v(x,t-\tau)\, d\tau}_ {R(h,v)} = {\mathcal{K}(g,u)} + A(g,u)(h,v)+ {R(h,v)}, \end{split} \] where $A(g,u): L^\infty(0,T) \times C(\bar Q) \to C(\bar Q)$ is a linear continuous operator and $R: L^\infty(0,T) \times C(\bar Q) \to C(\bar Q)$ is a remainder term. It is easy to confirm that \[ \frac{\|R(h,v) \|_{C(\bar Q)}}{\|(h,v) \|_{L^\infty(0,T)\times C(\bar Q)}} \to 0, \quad \mbox{ if } \|(h,v) \|_{L^\infty(0,T)\times C(\bar Q)} \to 0. \] Therefore, $\mathcal{K}$ is Fr\'{e}chet-differentiable. As a continuous bilinear form, $\mathcal{K}$ is also of class $C^2$. Now, we investigate the control-to-state mapping $G: L^\infty(0,T) \to C(\bar Q)$ defined by $ G: g \mapsto u_g, $ where the state function $u_g$ is defined as the unique solution to \begin{equation} \label{E:4.1} \begin{array}{rcll} \partial_tu - \Delta u + R(u) +\kappa \, u &=& \kappa\,\mathcal{K}(g,u)+ \kappa\,U_g & \text{ in } Q\\ \partial_n u &=& 0&\text{ in } \Sigma\\ u(0)&=& u_0(0)&\text{ in }\Omega . \end{array} \end{equation} In what follows, the initial function $u_0$ will be kept fixed and is therefore not mentioned. Of course, $U_g,\, G$ and some of the operators below depend on $u_0$, but we will not explicitely mention this dependence. To discuss $G$, we need known properties of the following auxiliary mapping $\mathcal{G}: v \mapsto u$, where \[ \begin{array}{rcll} \partial_tu - \Delta u + R(u) +\kappa \, u &=& v & \text{ in } Q\\ \partial_n u &=& 0& \text{ in } \Sigma\\ u(0)&=& u_0(0)& \text{ in } \Omega. \end{array} \] This mapping $\mathcal{G}$ is of class $\mathcal{C}^2$ from $L^p(Q)$ to $W(0,T) \cap \mathcal{C}(\bar{Q})$, if $p>\frac{5}{2}$, in particular from $L^\infty(Q)$ to $L^\infty(Q)$, cf. \cite{casas_ryll_troeltzsch2014} or, for monotone $R$, \cite{cas93}, \cite{rayzid99}, \cite{tro10book}. \smallskip Now (consider $v:= \kappa\,( \mathcal{K}(g,u)+U_g)$ as given and keep the initial function $u_0$ fixed), $u$ solves \eqref{E:4.1} if and only if $u=\mathcal{G}( \kappa\,\mathcal{K}(g,u)+\kappa\, U_g)$, i.e. \begin{equation}\label{E:4.2} u-\mathcal{G}( \kappa\,\mathcal{K}(g,u)+\kappa\, U_g)=0. \end{equation} We introduce a new mapping $\mathcal{F}\,:\,L^\infty(Q)\times L^\infty(0,T) \to L^\infty(Q)$ defined by \[ \mathcal{F}(u,g):=u-\mathcal{G}( \kappa\,\mathcal{K}(g,u)+\kappa\, U_g). \] Then, \eqref{E:4.2} is equivalent to the equation \begin{equation} \label{E:4.3} \mathcal{F}(u,g)=0. \end{equation} We have proved above that the mapping $(g,u) \mapsto \mathcal{K}(g,u)$ is of class $\mathcal{C}^2$ from $L^\infty(0,T) \times L^\infty(Q)$ to $L^\infty(Q)$. Obviously, also the linear mapping $g \mapsto U_g$ is of class $\mathcal{C}^2$ from $L^\infty(0,T)$ to $L^\infty(Q)$. By the chain rule, also $\mathcal{F}$ is $\mathcal{C}^2$ from $L^\infty(Q)\times L^\infty(0,T) \to L^\infty(Q)$ and the mappings $\partial_g \mathcal{F}(\bar{u},\bar{g})$, $\partial_u \mathcal{F}(\bar{u},\bar{g})$ are continuous in the associated pairs of spaces. To use the implicit function theorem, we prove that $\partial_u \mathcal{F}(\bar{u},\bar{g})$ is continuously invertible at any fixed pair $(\bar{u},\bar{g})$. Therefore, we consider the equation \begin{equation}\label{E:4.4} \partial_u \mathcal{F}(\bar{u}, \bar{g})v=z \end{equation} with given right-hand side $z \in L^\infty(Q)$ and show the existence of a unique solution $v \in L^\infty(Q)$. The equation is equivalent with \begin{equation} v- \mathcal{G}'(\underbrace{\kappa \, \mathcal{K}(\bar g,\bar u)+\kappa \, U_g}_{\bar{p}}) \kappa\, \mathcal{K}(g,v)= z. \end{equation} Writing for convenience $\bar p = \kappa\,\mathcal{K}(\bar g,\bar u)+ \kappa\,U_g$, we obtain the simpler form \[ v- \mathcal{G}'(\bar{p})\kappa\,K(\bar{g})v= z. \] A function $z \in L^\infty(Q)$ does not in general belong to $W(0,T)$. To overcome this difficulty, we set $w:=v-z$ and transform the equation to \begin{equation} \label{E:4.6} w=\mathcal{G}'(\bar{p})\underbrace{\kappa\,K(\bar{g})(w+z)}_{q}= \mathcal{G}'(\bar{p})q. \end{equation} where $q := \kappa\,K(\bar{g})(w+z)$. As the next result shows, $w$ is the solution of a parabolic PDE, hence $w \in W(0,T)$. \begin{lem} \label{L5.1} Let $q \in L^p(Q)$ with $p > 5/2$ be given. Then we have $y=\mathcal{G}'(\bar{p})q$ if and only if $y$ solves \[ \begin{array}{rcll} \partial_t y - \Delta y +R'(\bar{u}) y+ \kappa\,y &=&q& \text{ in } Q\\ \partial_n y &=& 0& \text{ in } \Sigma\\ y(0)&=&0& \text{ in } \Omega, \end{array} \] where $\bar{u}$ is the solution associated with $\bar{p}$, i.e. \[ \begin{array}{rcll} \partial_t \bar{u} - \Delta \bar{u} +R(\bar{u})+ \kappa\,\bar{u} &=&\bar{p}& \text{ in } Q\\ \partial_n \bar u &=& 0& \text{ in }\Sigma \\ \bar{u}(0)&=&u_0& \text{ in } \Omega. \end{array} \] \end{lem} We refer to \cite{casas_ryll_troeltzsch2014}. For monotone non-decreasing functions $R$, this result is well known in the theory of semilinear parabolic control problems, see e.g. \cite{cas93}, \cite{rayzid98}, or \cite[Thm. 5.9]{tro10book}. By Lemma \ref{L5.1}, the solution $w$ of \eqref{E:4.6} is the unique solution of the linear PDE \begin{equation} \label{E:4.7} \begin{split} & (\partial_t w- \Delta w +R'(\bar{u})w+ \kappa\,w)(x,t)= q(x,t)\\ & \qquad \qquad = \kappa\int_0^t \bar{g}(\tau) w(x,t-\tau) \, d\tau + \kappa\int_0^t \bar{g}(\tau) z(x,t-\tau) \, d\tau \end{split} \end{equation} subject to $w(0) = 0$ and homogeneous Neumann boundary conditions. By the same methods as above we find that, for all $z \in L^\infty(Q)$, equation $(\ref{E:4.7})$ has a unique solution $w \in W(0,T) \cap L^\infty(Q)$. \smallskip After transforming back by $v = w + z$, we have found that for all $z \in L^\infty(Q)$, \eqref{E:4.4} has a unique solution $v \in L^\infty(Q)$ given by $v = w + z$. Therefore, the inverse operator $\partial_u \mathcal{F}(\bar{u},\bar{g})^{-1}$ exists. The continuity of this inverse mapping follows from a result of \cite{casas_ryll_troeltzsch2014} that the mapping $z \mapsto w$ defined by \eqref{E:4.7} is continuous in $L^\infty(Q)$. Next, we consider the operator $\partial_g \mathcal{F}$. It exists by the chain rule and admits the form \[ \partial_g \mathcal{F}(\bar{u},\bar{g})h=\mathcal{G}'(\kappa\,(K(\bar{g})\bar{u}+U_g))\kappa\,(K(h)\bar{u}+ \partial_gU_g h). \] Setting again $\bar p = \kappa\,(K(\bar{g})\bar{u}+U_g)$ and $q = \kappa\,(K(h)\bar{u}+ \partial_gU_g h)$, we see that \[ \partial_g \mathcal{F}(\bar{u},\bar{g})h=\eta, \] where, by Lemma \ref{L5.1}, $\eta$ solves the equation \[ \partial_t \eta - \Delta \eta + R'(\bar{u})\eta+\kappa\,\eta= q = \kappa\,K(h)\bar{u}+ \kappa\, \partial_gU_g h \] subject to homogeneous initial and boundary conditions. Therefore, $\eta$ is the unique solution to \begin{eqnarray*} ( \partial_t \eta - \Delta \eta + R'(\bar{u})\eta+\kappa\,\eta)(x,t)&=& \kappa\, \int_0^t h(\tau)\bar{u}(x,t-\tau)\, d\tau\\ &&\quad + \kappa\, \int_t^T h(\tau)u_0(x,t-\tau)\, d\tau\\ \eta(x,0)&=&0\\ \partial_n \eta &=& 0. \end{eqnarray*} By $\bar u(x,t) = u_0(x,t)$ for $-T \le t \le 0$, we can re-write this as \begin{eqnarray*} ( \partial_t \eta - \Delta \eta + R'(\bar{u})\eta+\kappa\,\eta)(x,t)&=& \kappa\, \int_0^T h(\tau)\bar{u}(x,t-\tau)\, d\tau\\ \partial_n \eta &=& 0\\ \eta(x,0)&=&0. \end{eqnarray*} Again, the mapping $h \mapsto w$ is continuous from $L^\infty(0,T)$ to $W(0,T) \cap \mathcal{C}(\bar{Q})$. \medskip Collecting the last results, we have the following theorem: \begin{theorem}[Differentiability of $G$] The control-to-state mapping $G: g \mapsto u_g$ associated with equation \eqref{E:3.2} is of class $C^2$. The first order derivative $z:=G'(g)h$ is obtained as the unique solution to \begin{equation}\label{E:4.8} \begin{array}{rcll} (\partial_t z - \Delta z+R'(u_g)z+\kappa \, z)(x,t)&=& \displaystyle \kappa \int_0^T h(\tau) u_g(x,t- \tau) \, d \tau\\ &&\displaystyle \qquad +\kappa \int_0^t g\, (\tau) z(x,t- \tau) \, d \tau&\text{ in }Q\\[1ex] \partial_n z &=& 0&\text{ in } \Sigma\\ z(\cdot,t)&=& 0, \; -T \leq t \leq 0& \text{ in } \Omega. \end{array} \end{equation} \end{theorem} \begin{proof} We already know by Theorem \ref{T:3.1} that, for all $g\in L^\infty(0,T)$, there exists a unique solution $u=G(g) \in W(0,T) \cap \mathcal{C}(\bar{Q})$ solving the equation \[ \mathcal{F}(u,g)=0. \] We discussed above that the assumptions of the implicit function theorem are satisfied. Now this theorem yields that the mapping $g \mapsto G(g)$ is of class $\mathcal{C}^2$. The derivative $G'(g)h$ is obtained by implicit differentiation. By definition of $G(g)$, we have \begin{equation} \label{E:4.9} \begin{array}{rcl} (\partial_t G(g) - \Delta G(g)+R(G(g))+\kappa \, G(g))(x,t)&=& \displaystyle \kappa \int_0^t g(\tau) \, G(g)(x,t- \tau) \, d \tau \\[2ex] && \ \ + \displaystyle \kappa \, \int_t^T g(\tau) \, u_0(x,t- \tau) \, d \tau\\[1ex] \partial_n G(g) &=& 0\\[1ex] G(g)(\cdot,t)&=& u_0(\cdot,t), \; -T \leq t \leq 0. \end{array} \end{equation} Implicit differentiation yields that $z:= G'(g)h$ is the unique solution of \eqref{E:4.8}. Notice that \[ \int_0^t g(\tau) \, G(g)(x,t- \tau) \, d \tau + \int_t^T g(\tau) \, u_0(x,t- \tau) \, d \tau = \int_0^T g(\tau) \, G(g)(x,t- \tau) \, d \tau. \] \hfill $\Box$ \end{proof} \subsection{Existence of an optimal kernel} \begin{theorem} For all $\nu \ge 0$, {\rm (PG)} has at least one optimal solution $\bar{g}$. \end{theorem} \begin{proof} Let $(g_n)$ with $g_n \in C$ for all $n \in \mathbb{N}$ be a minimizing sequence. Since $C$ is bounded, convex, and closed in $L^\infty(0,T)$, we can assume without limitation of generality that $g_n$ converges weakly in $L^2(0,T)$ to $\bar g$, i.e. $g_n \rightharpoonup \bar{g}$, $n \to \infty$. The associated sequence of states $u_n$ obeys the equations \begin{equation} \label{E:4.10} \partial_t u_n - \Delta u_n + \kappa \, u_n = d_n:= -\kappa \, R(u_n) + \kappa \, K(g_n)u_n + \kappa \, U_g. \end{equation} By the principle of superposition, we split the functions $u_n$ as $u_n = \hat{u} + \tilde u_n$, where $\hat{u}$ is the solution of \eqref{E:4.10} with right-hand side $d_n := 0$ and initial value $\hat{u}(0) = u_0(0)$, while $\tilde u_n$ is the solution to the right-hand side $d_n$ defined above and zero initial value. In view of \eqref{E:3.5}, all state functions $u_n$, hence also the functions $\tilde u_n$, are uniformly bounded in $L^\infty(Q)$. Thanks to \cite[Thm. 4]{DiBenedetto1986}, the sequence $(\tilde u_n)$ is bounded in some H\"older space $C^{0,\lambda}(Q)$. By the Arzela-Ascoli theorem, we can assume (selecting a subsequence, if necessary) that $\tilde u_n$ converges strongly in $L^\infty(Q)$. Adding to $\tilde u_n$ the fixed function $\hat u$, we have that $(u_n)$ converges strongly to some $\bar u$ in $L^\infty(Q)$. The boundedness of $(u_n)$ also induces the boundedness of the sequence $(d_n)$ in $L^\infty(Q)$, in particular in $L^2(Q)$. Therefore, we can assume that $d_n$ converges weakly in $L^2(Q)$ to $ \bar d$, $n \to \infty$. Since $(u_n)$ is the sequence of solutions to the ''linear'' equation \eqref{E:4.10} with right-hand side $d_n$, the weak convergence of $(d_n)$ induces the weak convergence of $u_n \rightharpoonup \bar u$ in $W(0,T)$, where $\bar u$ solves \eqref{E:4.10} with right-hand side $\bar d$. Finally, we show that \[ \bar d(t) = -\kappa R(\bar u(t)) + \kappa \, (K(\bar g)\bar u)(t) + \kappa \, U_g(t) \] so that $\bar u$ is the state associated with $\bar g$. Obviously, it suffices to prove that $K(g_n)u_n $ converges weakly to $K(\bar g)\bar u$ in $L^2(Q)$. To this aim, let an arbitrary $\varphi \in L^2(Q)$ be given. Then we have \begin{equation} \label{E:4.11} \begin{split} &\iint_Q \varphi(x,t)\left( \int_0^t g_n(\tau)u_n(x,t-\tau)\, d\tau \right)dx dt \\ &\quad = \int_0^T g_n(\tau) \left(\int_\tau^T \int_\Omega \varphi(x,t)u_n(x,t-\tau) \, dt dx\right) d\tau. \end{split} \end{equation} Clearly, the strong convergence of $(u_n)$ in $L^\infty(Q)$ yields \[ \int_\tau^T \int_\Omega \varphi(x,t)u_n(x,t-\cdot) \, dt dx \to \int_\tau^T \int_\Omega \varphi(x,t)\bar u(x,t-\cdot) \, dt dx \] in $L^2(0,T)$. Along with the weak convergence of $g_n$, this implies \[ \begin{split} & \lim_{n \to \infty} \int_0^T g_n(\tau) \int_\tau^T \int_\Omega \varphi(x,t)u_n(x,t-\tau) \, dt dx d\tau \\ &\qquad = \int_0^T \bar g(\tau) \int_\tau^T \int_\Omega \varphi(x,t)\bar u(x,t-\tau) \, dt dx d\tau . \end{split} \] In view of \eqref{E:4.11}, we finally arrive at \[ \iint_Q \varphi(x,t) \int_0^t g_n(\tau)u_n(x,t-\tau)\, d\tau dx dt \to \iint_Q\varphi(x,t) \int_0^t \bar g (\tau)\bar u(x,t-\tau)\, d\tau dx dt \] as $n \to \infty$. Since this holds for arbitrary $\varphi \in L^2(Q)$, this is equivalent to the desired weak convergence $K(g_n)u_n \rightharpoonup K(\bar g)\bar u$ in $L^2(Q)$. \hfill $\Box$ \end{proof} \subsection{Necessary optimality conditions} \subsubsection{Adjoint equation} In the next step of our analysis, we establish the necessary optimality conditions for a (local) solution $\bar g$ of the optimization problem (PG). This optimization problem is defined by \begin{equation} \label{E:4.12} \left\{ \begin{array}{l} \min J(g), \\[1ex] \displaystyle 0 \le g(t) \le \beta \quad \text{for almost all } t \in [0,T],\\[1ex] \displaystyle \int_0^T g(\tau) \, d\tau =1. \end{array} \right. \end{equation} Although the admissible set belongs to $L^\infty(0,T)$, we consider this as an optimization problem in the Hilbert space $L^2(0,T)$. To set up associated necessary optimality conditions for an optimal solution of \eqref{E:4.12}, we first determine a useful expression for the derivative of the objective functional $J$. We have \[ \begin{split} J(g)&=\displaystyle \frac{1}{2} \iint_Q c_Q(u_g -u_d)^2 \, dxdt + \frac{\nu}{2} \int_0^T g(t)^2 \, dt\\ &\qquad \qquad = \displaystyle \frac{1}{2} \iint_Q c_Q (G(g) -u_d)^2 \, dxdt + \frac{\nu}{2} \int_0^T g(t)^2 \, dt. \end{split} \] Let now be an arbitrary (i.e. not necessarily optimal) $\bar g \in L^\infty(0,T)$ be given and let $\bar u = G(\bar g)$ be the associated state. Then we obtain for $h \in L^\infty(0,T)$ \begin{eqnarray} J'(\bar g)h &=& \nu \int_0^T \bar g(t) \, h(t) \, dt + \iint_Q c_Q(\bar{u}-u_d)(G'(\bar{u})h)\, dxdt \nonumber \\ &=& \int_0^T \nu\bar g(t) h(t) dt + \iint_Q c_Q(x,t)(\bar{u}(x,t) -u_d(x,t)) z(x,t)dxdt \label{E:4.13b} \end{eqnarray} with the solution $z$ to the equation \eqref{E:4.8} for $u_g := u_{\bar g} = \bar u$. The implicit appearance of $h$ via $z$ can be converted to an explicit one by an {\em adjoint equation}. This is the following equation: \begin{equation} \label{E:4.14} \begin{array}{rcl} \begin{array}{rcl} (-\partial_t \varphi - \Delta \varphi +R'(\bar{u})\varphi + \kappa\,\varphi)(x,t) & = & \displaystyle \kappa\int_0^T \bar g(\tau) \varphi(x,t+\tau) \, d\tau \\[2ex] &&\qquad + c_Q(x,t)(\bar{u}(x,t) -u_d(x,t))\\[1ex] && \qquad\qquad \text{ a.e. in } Q,\\[1ex] \partial_n \varphi&=& 0 \quad \text{in } \Sigma,\\[1ex] \varphi(\cdot,t)&=&0\quad t \in [T,2T]. \end{array} \end{array} \end{equation} The solution $\bar \varphi$ of \eqref{E:4.14} is said to be the {\em adjoint state} associated with $\bar g$. \begin{lemma}\label{L:4.3} Let $\bar g,\, \bar h \in L^\infty(0,T)$, and $\bar u = u_{\bar g}$ be given. If $z$ is the solution to the linearized equation \eqref{E:4.8} for $u_g := \bar u$ and $\bar \varphi$ is the unique solution to the adjoint equation \eqref{E:4.14}, then the identity \begin{equation} \label{E:L43} \iint_Q (c_Q\,(\bar{u}-u_d)\, z)(x,t) \, dxdt = \kappa \, \iint_Q \bar \varphi(x,t) \left(\int_0^T h(\tau)\bar{u}(x,t-\tau) \, d\tau \right)\, dxdt \end{equation} is fulfilled: \end{lemma} \begin{proof} We multiply the first equation in \eqref{E:4.8} by the adjoint state $\bar \varphi$ as test function and the first equation in \eqref{E:4.14} by $z$. After integration on $Q$ and some partial integration with respect to $x$, we obtain \[ \begin{split} &\iint_Q \left(\partial_t z \, \bar \varphi + \nabla z \cdot \nabla \bar \varphi + (R'(\bar u) + \kappa)z\, \bar \varphi \right)dxdt \\ &\qquad \qquad = \kappa \, \iint_Q \left(\int_0^T \bar g(\tau) z(x,t-\tau)\, d\tau\right) \bar\varphi(x,t)\, dxdt \\ &\qquad \qquad \qquad + \kappa \iint_Q \left(\int_0^T h(\tau)\bar u(x,t-\tau)\, d\tau\right) \bar\varphi(x,t)\, dxdt \end{split} \] and \[ \begin{split} &\iint_Q \left(- z \, \partial_t \bar\varphi + \nabla z \cdot \nabla \bar\varphi + (R'(\bar u) + \kappa)z\, \bar \varphi \right)dxdt \\ &\qquad = \kappa\,\iint_Q\left( \int_0^T \bar g(\tau) \bar\varphi(x,t+\tau) \, d\tau\right) z(x,t)\, dxdt + \iint_Q c_Q(\bar u - u_d)\, z\, dxdt. \end{split} \] Integrating by parts with respect to $t$, we see that \[ \iint_Q (- z) \, \partial_t \bar\varphi \, dxdt = \iint_Q \bar \varphi \, \partial_t z \, dxdt; \] notice that we have $z(0) = 0$ and $\bar\varphi(T) = 0$. Comparing both weak formulations above, it turns out that we only have to confirm the equation \begin{equation}\label{E:confirm} \iint_Q \int_0^T \bar g(\tau) z(x,t-\tau)\, \bar\varphi(x,t)\, d\tau \, dxdt = \iint_Q\int_0^T \bar g(\tau)\bar \varphi(x,t+\tau) \,z(x,t)\, d\tau \, dxdt. \end{equation} Then the claim of the Lemma follows. To show \eqref{E:confirm} , we proceed as follows: \begin{equation} \label{E:Fubini} \begin{split} &\iint_Q \int_0^T \bar g(\tau) z(x,t-\tau) \varphi(x,t)\, d\tau\, dxdt \hspace{4cm}\\ &\hspace{2cm} =\int_\Omega \int_0^T \int_0^T \bar g(\tau) \, z(x,t-\tau)\, \bar\varphi(x,t)\,d\tau dtdx\\ &\hspace{2cm} =\int_\Omega \int_0^T \int_0^t \bar g(\tau) \, z(x,t-\tau)\, \bar\varphi(x,t)\,d\tau dtdx\\ &\hspace{2cm}=\int_\Omega \int_0^T \int_0^t \bar g(t-\eta) \, z(x,\eta)\, \bar\varphi(x,t)\,d\eta dt\, dx\\ &\hspace{2cm}=\int_\Omega \int_0^T \int_\eta^T \bar g(t-\eta) \, \bar\varphi(x,t)\,dt \, z(x,\eta) \, d\eta\, dx\\ &\hspace{2cm}=\int_\Omega \int_{0}^T \int_0^{T-\eta} \bar g(\sigma) \, \bar\varphi(x,\eta+\sigma)\,d\sigma \, z(x,\eta) \, d\eta\, dx\\ &\hspace{2cm}=\int_\Omega \int_{0}^T \int_0^T \bar g(\sigma) \, \bar\varphi(x,\eta + \sigma)\,d\sigma \, z(x,\eta) \, d\eta\, dx\\ &\hspace{2cm}=\iint_Q \int_{0}^T \bar g(\tau) \, \bar\varphi(x,t + \tau) \, z(x,t)\,d\tau \, dxdt. \end{split} \end{equation} We used $z(x,t-\tau) = 0$ for $\tau > t$ in the second equation, the substitution $\eta = t-\tau$ in the third, the Fubini theorem in the fourth, the substitution $\sigma = t-\eta$ in the fifth, the property $\bar\varphi(x,t) = 0$ for $t \ge T$ in the sixth equation. Finally, we re-named the variables. \hfill $\Box$ \end{proof} \begin{corollary} \label{Cor:4.4} At any $\bar g \in L^\infty(0,T)$, the derivative $J'(\bar g)\,h$ in the direction $h \in L^\infty(0,T)$ is given by \[ J'(\bar g)\,h = \int_0^T \nu \, \bar g(t)\, h(t)\,dt + \kappa \int_0^T h(\tau) \left(\iint_Q \bar \varphi(x,t) \bar{u}(x,t-\tau) \, dxdt\right) d\tau, \] where $\bar \varphi$ is the unique solution of the adjoint equation \eqref{E:4.14}. \end{corollary} This follows immediately by inserting the right-hand side of \eqref{E:L43} in \eqref{E:4.13b} and by interchanging the order of integration with respect to $t$ and $\tau$. \subsubsection{Necessary optimality conditions for (PG)} Let us now establish the necessary optimality conditions for an optimal solution $\bar g$ of \eqref{E:4.12}. They can be derived by the Lagrangian function $L: L^\infty(0,T) \times \mathbb{R} \to \mathbb{R}$, \[ L(g,\mu):=J(g)+\mu\left(\int_0^T g(\tau) \, d\tau -1\right). \] If $\bar g$ is an optimal solution, then there exists a real Lagrange multiplier $\bar{\mu}$ such that the variational inequality \[ J'(\bar{g})(g-\bar{g}) + \bar{\mu} \int_0^T(g-\bar{g}) \, dt \geq 0 \qquad \text{ for all } g \geq 0 \] is satisfied. Inserting the result of Corollary \ref{Cor:4.4} for $h := g - \bar g$, we find \begin{equation} \label{E:4.17} \begin{split} &\int_0^T\left(\nu \bar{g}(t) + \bar \mu +\kappa \iint_Q \bar \varphi(x,s) \bar{u}(x,s-t) \, dxds \right) (g(t)-\bar{g}(t)) \, dt \geq 0 \end{split} \end{equation} for all $0 \le g \le \beta$. \begin{remark} For a Lagrange multiplier rule to hold, a regularity condition must be fulfilled. Here, the constraints are obviously regular at any $\bar g$: Define $F: L^2(0,T) \to \mathbb{R}$ by \[ F(g) = \int_0^T g(\tau)\,d\tau - 1. \] Then \[ F'(g) h = \int_0^T h(\tau)\,d\tau, \] and hence $F'(g): L^2(0,T) \to \mathbb{R}$ is surjective for all $g \in L^2(0,T)$. \end{remark} A simple pointwise discussion of \eqref{E:4.17} leads to the following complementarity conditions for almost all $t \in [0,T]$: \begin{equation} \label{E:4.18} g(t) =\left\{ \begin{array}{rcl} 0 &\mbox{if}&\displaystyle \nu \bar{g}(t) + \bar \mu +\kappa\, \iint_Q \bar \varphi(x,s) \bar{u}(x,s-t) \, dxds > 0\\[2ex] \beta &\mbox{if}&\displaystyle \nu \bar{g}(t) + \bar \mu +\kappa\, \iint_Q \bar \varphi(x,s) \bar{u}(x,s-t) \, dxds < 0. \end{array} \right. \end{equation} If the expression in right-hand side above vanishes, then we obviously have \[ g(t) = -\frac{1}{\nu} \left( \bar \mu +\kappa\, \iint_Q \bar \varphi(x,s) \bar{u}(x,s-t) \, dxds\right). \] In a known way, the last three relations can be equivalently expressed by the projection formula \[ \bar g(t) = \mathbb{P}_{[0,\beta]}\left(-\frac{1}{\nu} \left(\bar \mu +\kappa\, \iint_Q \bar \varphi(s) \bar{u}(x,s-t) \, dxds \right) \right), \] where $\mathbb{P}_{[0,\beta]}: \mathbb{R} \to [0,\beta]$ is defined by \[ \mathbb{P}_{[0,\beta]}(x) = \max(0,\min(\beta,x)). \] \section{Discussion of (PS)} Let us now discuss the changes that are needed to establish the necessary optimality conditions for the problem (PS) with the particular form \eqref{E:2.5} of $g$. Now, $\kappa, t_1,$ and $t_2$ are our control variables. Let us denote by $u_{(\kappa,t_1,t_2)}$ the unique state associated with $(\kappa,t_1,t_2)$. The existence of the derivatives $\partial_{t_i} u_{(\kappa,t_1,t_2)}, \, i=1,2$, and $\partial_{\kappa} u_{(\kappa,t_1,t_2)}$ can be shown again by the implicit function theorem. We omit these details, because one can proceed analogously to the discussion for (PG). To shorten the notation, we write \[ z_i:=\partial_{t_i} u_{(\kappa,t_1,t_2)}, \; i=1,2, \quad z_3 := \partial_{\kappa} u_{(\kappa,t_1,t_2)}. \] By implicit differentiation, we find the functions $z_i$ from linearized equations. Assume that the derivatives have to be determined at the point $(\kappa,t_1,t_2)$ and fix the associated state \ $u:= u_{(\kappa,t_1,t_2)}$ for a while. Then, $z_1$ solves \begin{equation} \label{E:5.1} \begin{split} &\left(\partial_t z_1- \Delta z_1 +R'(u)z_1+\kappa\, z_1\right)(x,t)= \left.\frac{\partial}{\partial t_1} \left[ \frac{\kappa\,}{t_2-t_1} \int_{t_1}^{t_2} u(x,t-\tau) \, d\tau \right]\right.\\ &\quad =\frac{\kappa}{t_2-t_1} \left[\frac{1}{t_2-t_1}\int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau - {u}(x,t-{t_1}) + \int_{{t_1}}^{{t_2}} z_1(x,t-\tau) \, d\tau\right]. \end{split} \end{equation} Analogously, we find for $z_2$ \begin{equation} \label{E:5.2} \begin{split} &(\partial_t z_2- \Delta z_2 +R'(u)z_2+\kappa\, z_2)(x,t)= \\ &\quad=\frac{- \kappa}{{t_2}-{t_1}} \left[\frac{1}{{t_2}-{t_1}}\int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau - {u}(x,t-{t_2}) + \int_{{t_1}}^{{t_2}} z_2(x,t-\tau) \, d\tau\right] \end{split} \end{equation} and for $z_3$ \[ \begin{split} &(\partial_t z_3- \Delta z_3 +R'(u)z_3+\kappa \, z_3 + u)(x,t) = \left.\frac{\partial}{\partial \kappa} \left[ \frac{\kappa\,}{t_2-t_1} \int_{t_1}^{t_2}u(x,t-\tau) \, d\tau \right]\right.\\ &\hspace{0.8cm}=\frac{1}{{t_2}-{t_1}} \left[ \int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau + \kappa \, \int_{{t_1}}^{{t_2}} z_3(x,t-\tau) \, d\tau\right]. \end{split} \] Therefore, the equation for $z_3$ is \begin{equation} \label{E:5.3} \begin{split} &(\partial_t z_3- \Delta z_3 +R'(u)z_3+\kappa \, z_3 )(x,t)- \frac{\kappa}{t_2-t_1} \int_{{t_1}}^{{t_2}} z_3(x,t-\tau) \, d\tau \\ &\hspace{1cm} = \frac{1}{{t_2}-{t_1}} \int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau - u(x,t). \end{split} \end{equation} Again, we introduce an adjoint equation to set up the optimality conditions. To this aim, let $(\kappa,t_1,t_2)$ an arbitrary fixed triplet and $u_{(\kappa,t_1,t_2)}$ be the associated state function. The adjoint equation is \begin{equation} \label{E:5.5} \begin{array}{rcl} \left(-\varphi_t - \Delta \varphi +R'({u_{(\kappa,t_1,t_2)}})\varphi+ \kappa \, \varphi\right)(x,t) &=& \displaystyle \frac{\kappa}{ t_2- t_1}\int_{ t_1}^{ t_2} \varphi(x,t+\tau) \, d\tau\\[3ex] && + c_Q(x,t)({u}_{(\kappa,t_1,t_2)}(x,t) -u_d(x,t)) \\[1ex] &&\hspace{4cm} \text{ in } Q,\\[1ex] \partial_n \varphi&=& 0 \quad \text{in }\Sigma,\\[1ex] \varphi(x,t)&=&0 \quad \text{in } \Omega\times[T,2T]. \\ \end{array} \end{equation} This equation has a unique solution $\varphi \in L^\infty(Q)$ denoted by $\varphi_{(\kappa,t_1,t_2)}$ to indicate the correspondence with $(\kappa,t_1,t_2)$. Existence and uniqueness can be shown in a standard way by the substitution $\tilde t := T-t$ that transforms this equation to a standard forward equation that can be handled in the same way as the state equation. \begin{theorem}[Derivative of $J_S$] \label{T:5.1} Let $(\kappa,t_1,t_2)$ be given, $u:=u_{(\kappa,t_1,t_2)}$ be the associated state, and $\varphi := \varphi_{(\kappa,t_1,t_2)}$ be the associated adjoint state, i.e. the unique solution of the adjoint equation \eqref{E:5.5}. Write for short $\delta := 1/(t_2-t_1)$. Then the partial derivatives of $J_S$ at $(\kappa,t_1,t_2)$ are given by \begin{eqnarray*} \partial_{t_1} J_S&=& \nu t_1+ \frac{\kappa}{\delta} \iint_Q {\varphi}(x,t) \left[ \frac{1}{\delta} \int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau - {u}(x,t- {t_1})\right] \, dxdt, \\[2ex] \partial_{t_2} J_S&=& \nu t_2- \frac{\kappa}{\delta} \iint_Q {\varphi}(x,t) \left[ \frac{1}{\delta} \int_{t_1}^{t_2} {u}(x,t-\tau) \, d\tau - {u}(x,t- {t_2}) \right]\, dxdt\\[2ex] \partial_{\kappa} J_S &=& \nu \kappa + \iint_Q \varphi(x,t) \left[ \frac{1}{\delta} \int_{t_1}^{t_2}{u}(x,t-\tau)\, d\tau - u(x,t)\right]\, dxdt. \end{eqnarray*} \end{theorem} \begin{proof} We verify the expression for $\partial_{t_1} J_S(\kappa,t_1,t_2)$, the other formulas can be shown analogously. To this aim, let $z_1 = \partial_{t_1}u_{(\kappa,t_1,t_2)}$ be the solution of the linearized equation \eqref{E:5.1}. For convenience, we write $z:= z_1$ within this proof. Following the proof of Lemma \ref{L:4.3}, we multiply \eqref{E:5.1} by $\varphi$ and integrate over $Q$. We obtain \begin{equation}\label{E:5.6} \begin{split} &\iint_Q \left(\partial_t z \, \varphi + \nabla z \cdot \nabla \varphi + (R'(u) + \kappa)z\, \varphi \right)dxdt \\[2ex] &\qquad =\frac{\kappa}{\delta} \iint_Q \varphi(x,t) \left[\frac{1}{\delta} \int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau- {u}(x,t-{t_1})\right]\, dxdt\\[1ex] &\qquad \qquad + \frac{\kappa}{\delta} \iint_Q \varphi(x,t) \int_{{t_1}}^{{t_2}} z(x,t-\tau) \, d\tau dxdt. \end{split} \end{equation} Next, we multiply the adjoint equation \eqref{E:5.5} by $z$ and integrate over $Q$ to find \begin{equation} \label{E:5.7} \begin{split} &\iint_Q \left(- z \, \partial_t \varphi + \nabla z \cdot \nabla \varphi + (R'(u) + \kappa)z\, \varphi \right)dxdt \\[2ex] &\qquad = \frac{\kappa}{\delta}\iint_Q \int_{t_1}^{t_2} \varphi(x,t+\tau)\, d\tau \ z(x,t)\, dxdt + \iint_Q c_Q(\bar u - u_d)\, z\, dxdt. \end{split} \end{equation} Now recall that \[ g(t) = \left\{ \begin{array}{cl} \displaystyle \frac{1}{t_2 -t_1}, & t_1 \leq t \leq t_2 \\[2ex] 0, & \mbox{elsewhere}. \end{array} \right. \] Therefore, we can write \[ \begin{split} &\frac{1}{\delta} \iint_Q \varphi(x,t) \int_{{t_1}}^{{t_2}} z(x,t-\tau) \, d\tau dxdt = \iint_Q \int_0^T \varphi(x,t) g(\tau) z(x,t-\tau) \, d\tau dxdt \\ &= \iint_Q \int_{0}^{T} g(\tau) \varphi(x,t+\tau)d\tau z(x,t)dxdt = \frac{1}{\delta}\iint_Q \int_{t_1}^{t_2} \varphi(x,t+\tau)d\tau z(x,t)dxdt, \end{split} \] where the second equation follows from \eqref{E:Fubini}. In view of \[ \iint_Q (- z) \, \partial_t \varphi \, dxdt = \iint_Q \varphi \, \partial_t z \, dxdt, \] a comparison of the equations \eqref{E:5.6} and \eqref{E:5.7} yields \[ \begin{split} & \iint_Q c_Q(u - u_d)\, z\, dxdt = \\ & \qquad \qquad = \frac{\kappa}{\delta}\iint_Q \varphi(x,t) \left[\frac{1}{\delta}\int_{{t_1}}^{{t_2}} {u}(x,t-\tau) \, d\tau- {u}(x,t-{t_1})\right]dxdt. \end{split} \] Since \[ \partial_{t_1} \left[\frac{1}{2}\iint_Q c_Q (u_{(\kappa,t_1,t_2)}-u_d)^2 dxdt \right]= \iint_Q c_Q (u_{(\kappa,t_1,t_2)}-u_d) \, z_1\, dxdt, \] the first claim of the theorem follows immediately. \hfill $\Box$ \end{proof} As a direct consequence of the theorem on the derivative of $J_S$, we obtain the following corollary. \begin{corollary}[Necessary optimality condition for (PS)] Let $(\bar \kappa,\bar t_1,\bar t_2)$ be optimal for the problem {\rm (PS)} and let $\bar u := u_{(\bar \kappa,\bar t_1,\bar t_2)}$ and $\bar \varphi := \varphi_{(\bar \kappa,\bar t_1,\bar t_2)}$ denote the associated state and adjoint state, respectively. Then, with the gradient $\nabla J_S(\bar \kappa,\bar t_1,\bar t_2)$ defined by Theorem \ref{T:5.1} with $\bar \varphi$ and $\bar u$ inserted for $\varphi$ and $u$, respectively, the variational inequality \begin{equation} \nabla J_S(\bar \kappa,\bar t_1,\bar t_2)\cdot (\kappa - \bar \kappa ,t_1 - \bar t_1,t_2 - \bar t_2)^\top \ge 0 \quad \forall (\kappa, t_1, t_2) \in C_\delta \end{equation}\label{E:5.6b} is satisfied. \end{corollary} Since the variable $\kappa$ is unrestricted, the associated part of the variational inequality amounts to \[ \nu \bar \kappa - \iint_Q \bar \varphi(x,t) \bar u(x,t)\, dxdt + \frac{1}{{\bar t_2}-{\bar t_1}} \iint_Q {\bar \varphi}(x,t) {\bar u}(x,t-\bar t_2)\,dxdt = 0. \] If $(\bar t_1,\bar t_2)$ belongs to the interior of the admissible set $C_\delta$, then the associated components of $\nabla J_S$ must vanish as well, hence \begin{eqnarray*} \nu \bar t_1+ \frac{\bar \kappa}{{\bar t_2}-{\bar t_1}} \iint_Q {\bar \varphi}(x,t)\left[ \frac{1}{{\bar t_2}-{\bar t_1}}\int_{{\bar t_1}}^{{\bar t_2}} {\bar u}(x,t-\tau) \, d\tau - {\bar u}(x,t- {\bar t_1}) \right]\, dxdt & =& 0\\ \nu \bar t_2- \frac{\bar \kappa}{{\bar t_2}-{\bar t_1}} \iint_Q {\bar \varphi}(x,t) \left[\frac{1}{{\bar t_2}-{\bar t_1}} \int_{{\bar t_2}}^{{\bar t_2}} {\bar u}(x,t-\tau) \, d\tau- {\bar u}(x,t- {\bar t_2})\right]\, dx dt &=& 0. \end{eqnarray*} \begin{remark}[Application of the formal Lagrange technique] \label{Rem:4.5} To find the form of $\nabla J_S$ and for establishing the necessary optimality conditions, it might be easier to apply the following standard technique that is slightly formal but leads to the same result: We set up the Lagrangian function $\mathcal{L}$, \[ \begin{split} &\mathcal{L}(u,\kappa,t_1,t_2,\varphi)= \frac{1}{2}\iint_Q c_Q (u-u_d)^2 dxdt -\iint_Q \varphi(x,t) \, \cdot \\ &\qquad \cdot \left[(\partial_t u - \Delta u +R(u) +\kappa\,u)(x,t) - \frac{\kappa}{t_2-t_1} \int_{t_1}^{t_2}u(x,t-\tau) \, d\tau \right] dxdt. \end{split} \] If $(\kappa,t_1,t_2)$ is a given triplet, then the adjoint equation for the adjoint state $\varphi_{(\kappa,t_1,t_2)}$ is found by \[ \partial_u \mathcal{L}(u,\kappa,t_1,t_2,\varphi)\, v =0 \quad \forall v \, : \, v(\cdot,t)=0 \mbox{ for } t \leq 0. \] The derivatives of $J_S$ are obtained by associated derivatives of $\mathcal{L}$. For instance, we have \[ \partial_{t_i} J_S(\kappa,t_1,t_2) = \partial_{t_i} \mathcal{L}(u,\kappa,t_1,t_2,\varphi) \] if $u = u_{(\kappa,t_1,t_2)}$ and $\varphi = \varphi_{(\kappa,t_1,t_2)}$ are inserted after having taken the derivative of $\mathcal{L}$ with respect to $t_i$. This obviously yields the first two components of $\nabla J_S$ in Theorem \ref{T:5.1}. For the third component with proceed analogously. \end{remark} \section{Numerical examples for (PS)} \subsection{Introductory remarks} The numerical solution of the problems posed above requires techniques that are adapted to the desired type of patterns $u_d$. In this paper, we concentrate on numerical examples for the simplified problem (PS), where the kernel $g$ is a step function. Although this problem is mathematically equivalent to a nonlinear optimization problem in a convex admissible set of $\mathbb{R}^3$, the obtained patterns are fairly rich and interesting in their own. In a forthcoming paper to be published elsewhere, we will report on the numerical treatment of the more general problem (P), where a kernel function $g$ is to be determined. In this section, we present some results for the problem (PS), where a standard regularized tracking type functional $J$ is to be minimized in the set $C_\delta$. It will turn out that (PS) is only suitable for tracking desired states $u_d$ of simple structure. In all what follows, $\Omega = (a,b)$ is an open interval, i.e. we concentrate on the spatially one-dimensional case. Compared to many optimal control problems for semilinear parabolic equations that were investigated in literature, the numerical solution of the problems posed here is a bit delicate. We are interested in approximating desired states $u_d$ that exhibit certain geometrical patterns. If they have a periodic structure, then the objective function $J$ may exhibit many local minima with very narrow regions of attraction for the convergence of numerical techniques. Therefore, the optimization methods should be started in a sufficiently small neighborhood around the desired optimal solution. Moreover, the standard functional $J$ does not really fit to our needs. We will address the tracking of periodic patterns $u_d$ in Section \ref{S7}. \subsection{Discretization of the feedback system} To discretize the feedback equation \eqref{E:2.5}, we apply an implicit Euler scheme with respect to the time and a finite element approximation by standard piecewise linear and continuous ansatz functions (''hat functions'') with respect to the space variable. For this purpose, the define a time grid by an equidistant partition of $[0,T]$ with mesh size $\tau = T/m$ and node points $t_j = j \, \tau$, $j = 0,\ldots,m$. Associated with this time grid, functions $u_j: \Omega \to \mathbb{R}$ are to be computed that approximate $u(\cdot,t_j)$, $j = 0,\ldots,m$, i.e. $u_j \sim u(\cdot,t_j)$. Based on the functions $u_j$, we define grid functions $u^\tau: Q \to \mathbb{R}$ by piecewise linear approximation, \[ u^\tau(x,t) = \frac{1}{\tau}[(t_{j+1}-t)\, u_j(x) + (t_{j}-t)\, u_{j+1}(x)], \mbox{ if } t \in [t_j,t_{j+1}], \ j = 0,\ldots,m. \] By the implicit Euler scheme, the following system of nonlinear equations is set up, \[ \begin{split} & \frac{u^\tau(x,t_{j+1})-u^\tau(x,t_{j})}{\tau} - \Delta u^\tau(x,t_{j+1})+R(u^\tau(x,t_{j+1})) \\ &\hspace{1cm}= \frac{\kappa}{t_2-t_1} \int_{t_1}^{t_2} u^\tau(x,t_{j+1}-s)\: ds - \kappa \, u^\tau(x,t_{j+1}), \quad j = 0,\ldots,m-1. \end{split} \] The spatial approximation is based on an equidistant partition of $\Omega=(a,b)$ with mesh size $h>0$. Here, we define standard piecewise affine and continuous ansatz functions (hat functions) $\phi_i: \Omega \to \mathbb{R}$ and approximate the grid function $u^\tau$ by $u_h^\tau$, \[ u_h^\tau(x,t_j)=\sum_{i=0}^n u_{ji}\phi_i(x) \] with unknown coefficients $u_{ji} \in \mathbb{R}$, $j = 0,\ldots,m, \, i = 0,\ldots,n$. To set up the discrete system, we define vectors $\mathbf{u}^j \in \mathbb{R}^{n+1}$ by $\mathbf{u}^j = (u_{j0},\ldots,u_{jn})^\top$, $j = 0,\ldots,m$. Moreover, we establish the mass and stiffness matrices \[ M:= \left(\int_\Omega \phi_k(x) \phi_\ell(x) \: dx\right)_{k,\ell=0}^n, \quad A:=\left(\int_\Omega \phi_k'(x) \phi_\ell'(x) \: dx\right)_{k,\ell=0}^n\, , \] and, for $j = 0,\ldots,m$, the vectors \begin{eqnarray*} && R(\mathbf{u}^j):=\left( \int_\Omega R(u_h^\tau(x,t_j))\, \phi_i(x)\: dx\right)_{i=0}^n\\ && K(\mathbf{u}^j):=\frac{\kappa}{t_2-t_1}\left(\int_\Omega \int_{t_1}^{t_2} (u_h^\tau(x,t_j)-s)\,ds \, \phi_i(x) \: dx\right)_{i=0}^n. \end{eqnarray*} \begin{remark}To compute the integrals $R(\mathbf{u}^j)$, we invoke a 4 point Gauss integration. Notice that the functions $x \mapsto R(u_h^\tau(x,t_j))$ are third order polynomials so that the integrand of $R(\mathbf{u}^j)$ is a polynomial of order 4. Here, the 4 point Gauss integration is exact. \end{remark} For the computation of the vectors $K(\mathbf{u}^j)$, we use the trapezoidal rule. Here, for $t_j-s \leq 0$, the values $u_0(x,t_j-s)$ must be inserted. To increase the precision, the primitive of $u_0$ is used. The complete discrete system is \[ M\mathbf{u}^{j+1} - M\mathbf{u}^j+\tau A\mathbf{u}^{j+1} +{\tau}\, R(\mathbf{u}^{j+1}) =\tau K(\mathbf{u}^{j+1})-\tau \kappa M\mathbf{u}^{j+1} \quad j=0,1,\ldots m-1. \] We define \[ F(\mathbf{u}):=[(1+\tau \kappa)M+\tau A]\mathbf{u}+{\tau}\, R({\mathbf{u}})-\tau K(\mathbf{u}) . \] In each time step, we solve the nonlinear equation $F(\mathbf{u})=M \mathbf{u}^{j}$ to obtain $\mathbf{u}^{j+1}$. To this aim, we apply a fixed point iteration. \subsection{Numerical examples for the standard tracking functional} For testing our numerical method, we generate the desired state $u_d$ as solution of the feedback system for a given triplet $(\hat \kappa,\hat t_1,\hat t_2)$, i.e. \[ u_d := u_{(\hat \kappa,\hat t_1,\hat t_2)}. \] If the regularization parameter $\nu$ is small, then the numerical solution of (PS) should return a vector that is close to $(\hat \kappa,\hat t_1,\hat t_2)$. In all of our computational examples, we selected a small number $\delta > 0$ so that the restriction $t_2 -t_1\ge \delta$ was never active. Moreover, the Tikhonov regularization parameter $\nu$ was set to zero, because a regularization was not needed for having numerical stability. In the first numerical examples of this paragraph, the aim is to generate desired wave type solutions that expand with a given velocity. \vspace{1ex} \begin{example}[Desired traveling wave with pre-computed $u_d$] \label{Ex1} We select $\Omega = (-20,20)$, $T = 40$, $u_1 = 0, \, u_2 = 0.25, u_3 = 1$, $\kappa = 0.5$. Moreover, we take as initial function \[ u_0(x,t) := \frac{1}{2}\displaystyle\left(1-\tanh\left(\frac{x-vt}{2\sqrt{2}}\right)\right), \, \quad x \in \Omega,\, t \le 0, \] where $v=(1-2u_2)/\sqrt{2}$ is the velocity of the uncontrolled traveling wave given by $u_0$. Following the strategy explained above, we fix the triplet $(\hat \kappa,\hat t_1,\hat t_2)$ by $(0.5,0.456,0.541)$ and obtain $u_d = u_{(\hat \kappa,\hat t_1,\hat t_2)}.$ This is a traveling wave with a smaller velocity $v_d \approx 0.25$ due to the control term. To test our method, we apply our optimization algorithm to find $(\bar \kappa,\bar t_1,\bar t_2)$ such that the associated state function $u$ coincides with $u_d$. The method should return a result $(\kappa, t_1, t_2)$ that is close to the vector $(0.5,0.456,0.541)$. \end{example} To solve the problem (PS) in Example \ref{Ex1}, we applied the \textsc{Matlab} code \texttt{fmincon}. For the gradient $\nabla J_S(\kappa,t_1,t_2)$, a subroutine was implemented by our adjoint calculus. In this way, we were able to use the differentiable mode of \texttt{fmincon}. During the optimization, we fixed $\kappa = 0.5$ and considered the minimization only with respect to $(t_1,t_2)$. Taking $t_1 =0,\, t_2 = 1$ as initial iterate for the optimization, \texttt{fmincon} returned $t_1 = 0.456, \, t_2 = 0.541$ as solution; notice that we fixed $\nu = 0$. The result is displayed in Fig.\,\ref{fig:Ex1}. \begin{figure}[h]\centering \includegraphics[scale=0.23]{exam1l.jpg}\hspace{20pt} \includegraphics[scale=0.23]{exam1r.jpg} \caption{Example \ref{Ex1}, desired traveling wave $u_d$ (left) and computed optimal state $u$ (right).}\label{fig:Ex1} \end{figure} \begin{example}[Stopping a traveling wave ]\label{Ex2} In contrast to Example \ref{Ex1}, here we fix the desired pattern $u_d$ that is displayed in Fig.\,\ref{fig:Ex2}, left side. This desired pattern was not pre-computed but geometrically designed, i.e. it is a ''synthetic'' pattern that shows a traveling wave stopping at the time $t \approx 16$. The other data are selected as in Example \ref{Ex1}. \end{example} In the optimization process for Example \ref{Ex2}, we fixed $\kappa = -1.5$. The initial iterate was $t_1 = 0, \, t_2 = 1$; \texttt{fmincon} returned $t_1 = 0.05, \, t_2 = 0.94$ as solution. The optimal state is displayed in Fig.\,\ref{fig:Ex2}. \begin{figure}[h]\centering \includegraphics[scale=0.23]{exam2l.jpg}\hspace{20pt} \includegraphics[scale=0.23]{exam2r.jpg} \caption{Example \ref{Ex2}, synthetic desired state $u_d$ (left) and computed optimal state $u$ (right).}\label{fig:Ex2} \end{figure} The next example shows that the applicability of the standard tracking functional $J$ of (PS) is limited to simple patterns $u_d$, e.g. wave type solutions of constant velocity. \begin{example}[Periodic pattern] \label{Ex3} Also here, $u_d$ is a synthetic pattern that was not precomputed. In $\Omega = (-20,20)$, we define \[ u_d(x,t) = 3 \, \sin(t - \cos(\frac{\pi}{40}(x+40))). \] Notice that this function $u_d$ obeys the homogeneous Neumann boundary conditions. It is displayed in Fig.\,\ref{fig:Ex3}, left. Since such a periodic pattern cannot be expected for small times, we consider the tracking only on $[T/2,T]$. Therefore, here we re-define the objective functional $J$ by \[ J_S(\kappa,t_1,t_2) := \int_{T/2}^T\int_\Omega (u_{(\kappa,t_1,t_2)} - u_d)^2 \, dxdt + \frac{\nu}{2} (\kappa^2 + t_1^2+t_2^2). \] \end{example} During the optimization run, we fixed the values $\kappa = -2$ and $t_1 = 0$ and optimized only with respect to $t_2$. Starting from $t_2 = 2$, the code \texttt{fmincon} returned the solution $t_2 = 3.71$. At this point, the Euclidean norm of $\nabla J_S$ is $|\nabla J_S(-2,0,3.71)| \approx 0.045$. This is a fairly good value and indicates that the result should be close to a local minimum. Nevertheless, the computed optimal objective value is very large, \[ J_S(-2,0,3.71) = 1.32\cdot 10^3. \] \begin{remark}In this and in the next examples, we fix $t_1 = 0$. We observed in our computational examples that the optimization with respect to $(\kappa,t_2)$ and $t_1 = 0$ yields sufficiently good results. Moreover, we found examples, where we got the same optimal objective value of $J$ for very different triplets $(\kappa,t_1,t_2)$. \end{remark} \begin{figure}[h]\centering \includegraphics[scale=0.28]{ud.jpg}\hspace{20pt} \includegraphics[scale=0.23]{exam3r.jpg} \caption{Example \ref{Ex3}, Desired periodic pattern $u_d$ (left) and computed optimal state $u$ (right).}\label{fig:Ex3} \end{figure} The computed optimal state for Example \ref{Ex3} is far from the desired one. In particular, the temporal periods are very different. The reason is that the standard quadratic tracking type functional $J$ is not able to resolve the desired periodicity. The main point is that the $L^2$-norm of the difference of a time-periodic function $t \mapsto u_d(t)$ and its phase shifted function $t \mapsto u_d(t-s)$ can be very large, although both functions have the same time period. For instance, in $Q = (-20,20)\times (0,40)$ we have \[ \iint_Q \left(3 \sin(t - \cos(\frac{\pi}{20}(x+20))) - 3 \sin(t - 3 - \cos(\frac{\pi}{20}(x+20)))\right)^2 dxdt = I, \] where $I \approx 1.4374\cdot 10^4$. This brought us to considering another type of objective functionals that is discussed in the next section. \section{Minimizing a cross correlation type functional}\label{S7} \subsubsection*{The cross correlation} As Example \ref{Ex3} showed, we need an objective functional that better expresses our goal of approximating periodic structures. This is the {\em cross correlation} between $u$ and $u_d$. Moreover, in the functional, we have to observe a later part of the time interval, where $u$ already has developed its periodicity. Let us briefly explain the meaning of the cross correlation. Assume that $x_d: \mathbb{R} \to \mathbb{R}$ is a periodic function and $x: [0,T] \to \mathbb{R}$ is another function; think of a function $x$ that is identical with $x_d$ after a time shift. To see, if $x_d$ and $x$ are time shifts of each other, we consider the extremal problem \begin{equation} \label{E:min_s} \min_{s \in \mathbb{R}} \int_0^T (x(t) - x_d(t-s))^2 dt. \end{equation} If $x$ and $x_d$ are just shifted, then the minimal value in \eqref{E:min_s} should be zero by taking the correct time shift $s$. The functional \eqref{E:min_s} can be simplified. To this aim, we expand the integrand, \[ \int_0^T (x(t) - x_d(t-s))^2 dt = \int_0^T x^2(t)\, dt - 2 \int_0^T x(t)x_d(t-s) dt + \int_0^T x_d^2(t-s)\, dt. \] The first integral does not depend on $s$. Since $x_d$ is a periodic function, also the last integral is independent of the shift $s$. Therefore, instead of minimizing the quadratic functional above, we can solve the following problem: \begin{equation} \label{E:7.2} \max_{s \in \mathbb{R}} \frac{\int_0^T x(t)x_d(t-s) dt}{\|x\|_{L^2(0,T)} \|x_d\|_{L^2(0,T)}}, \end{equation} where we additionally introduced a normalization. The result is the so-called {\em cross correlation} between $x$ and $x_d$. The largest possible value in \eqref{E:7.2} is 1; in this case, both functions are collinear. In the application to our control problems, this induces two equivalent objective functionals. The minimization problem \eqref{E:min_s} leads to the optimization problem \begin{equation} \label{E:min_s_b} \min_{(\kappa,t_1,t_2) \in C_\delta} \left( \, \min_{s \in \mathbb{R}} \int_Q(u(x,t) - u_d(x,t-s))^2 dxdt+ \frac{\nu}{2}\, (\kappa^2 + t_1^2 + t_2^2) \right). \end{equation} The other way around is an equivalent problem that uses the cross correlation, \begin{equation} \label{E:min_s_corr} \min_{(\kappa,t_1,t_2) \in C_\delta} J_{corr}(\kappa,t_1,t_2) \end{equation} where \begin{equation} J_{corr}(\kappa,t_1,t_2) := 1 - \max_{s \in \mathbb{R}} \frac{\displaystyle \iint_Q u(x,t)u_d(x,t-s) dxdt}{\displaystyle \|u\|_{L^2(Q)} \|u_d\|_{L^2(Q)}} + \frac{\nu}{2}\, (\kappa^2 + t_1^2 + t_2^2). \end{equation} For solving \eqref{E:min_s_corr}, we applied the \textsc{Matlab} code \texttt{pattern search} that turned out to be fairly efficient in finding global minima for functions that exhibit multiple local minima. In the periodic case, we have to deal with multiple local minima indeed. Here, the cross correlation based functional $J_{corr}$ is more useful, as the next figure shows. \begin{example}[Multiple local minima] \label{Ex4} We pre-compute the desired state by $u_d = u_{(-2,0,2.5)}$ and consider the functions $\kappa \mapsto J_S(\kappa,0,2.5)$ and $ \kappa \mapsto J_{corr}(\kappa,0,2.5)$ for $\kappa$ around the (optimal) parameter $\bar \kappa = -2$. \end{example} The two functions defined in Example \ref{Ex4} are shown in Fig.\,\ref{fig:Ex4}. The function $\kappa \mapsto J_S(\kappa,0,2.5)$ behaves more wildly around $\kappa = -2$ than the function $\kappa \mapsto J_{corr}(\kappa,0,2.5)$ that is based upon the cross correlation. \begin{figure}[h]\centering \includegraphics[scale=0.23]{bild_j_s.jpg}\hspace{20pt} \includegraphics[scale=0.23]{bild_j_corr.jpg} \caption{Example \ref{Ex4}, goal functions $\kappa \mapsto J_S(\kappa,0,2.5)$ (left) and $\kappa \mapsto J_{corr}(\kappa,0,2.5)$ (right).}\label{fig:Ex4} \end{figure} Let us re-consider the optimization problem of Example \ref{Ex3}, but now by the cross correlation based optimization problem \eqref{E:min_s_corr}. Here, we apply the following strategy: We keep $t_1=0$ fixed and optimize only with respect to $(\kappa,t_2)$. Moreover, at the beginning we fixed $\kappa = -2$ and optimized with respect to $\kappa$ in a second run. The computed solution was $t_2 = 2.7631$ with a value $J_{corr}=0.1604, \, (t_1=0, \, \kappa =-2)$; as before the Tikhonov parameter was selected as $\nu = 0$. The computed $u$ of this first step is shown in Fig.\,\ref{fig:Ex5}. Now the agreement, in particular of the temporal period, is much better. Next, we performed an alternating search for $(\kappa,t_2)$ starting with the result obtained in the first step. We obtained $t_2 = 2.7631, \, \kappa = -2.4318$ and the improved objective value $J_{corr} = 0.1229$. This improvement is graphically hardly to distinct from Fig. \ref{fig:Ex5}. \begin{figure}[h]\centering \includegraphics[scale=0.28]{ud.jpg}\hspace{20pt} \includegraphics[scale=0.28]{u31.jpg} \caption{Example \ref{Ex3}, Desired pattern $u_d$ (left) and computed pattern after minimizing $J_{corr}$ with respect to $t_2$ (right) for fixed $\kappa = -2,\, t_1 = 0$.}\label{fig:Ex5} \end{figure} Finally, we consider another example with synthetic $u_d$ that has a larger period than $u_d$ of Example \ref{Ex3}. \begin{example} \label{Ex7}We consider again $\Omega = (-20,20)$ and observe $u_d$ only in the time interval $[20,40]$. For $u_d$ we select \[ u_d(x,t) = 3 \, \displaystyle\sin\left(\frac{t}{2} - \cos\left(\frac{\pi}{20}(x+20)\right)\right). \] Again, $t_1 = 0$ is fixed and the iteration is started with $t_2=2,\, \kappa=-2$. The optimal control parameters are $t_2 = 6.94191, \, \kappa=-2.28$ with computed optimal objective value $J_{corr}=0.1209$. The computed optimal state is shown in Fig.\,\ref{fig:Ex7}. \end{example} \begin{figure}[h]\centering \includegraphics[scale=0.28]{ud4.jpg}\hspace{20pt} \includegraphics[scale=0.28]{exam4r.jpg} \caption{Example \ref{Ex7}, desired pattern $u_d$ (left) and optimal pattern (right) for fixed $t_1 = 0$.}\label{fig:Ex7} \end{figure}

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\section{Conclusion} In this work, we present a referring expression dataset, KB-Ref\xspace, in which the objects are expressed by their visual and non-visual attributes. Such that, it encourages REF algorithms to explore information from images as well as external knowledge bases. The dataset features a large variety of objects ($1,805$ categories) and long expressions ($13.32$ in average). Due to its complexity, directly applying SoTA REF approaches does not achieve promising results. To this end, we propose to tackle the problem with a expression conditioned image and fact attention network (ECIFA\xspace). Experiments show that our proposed model indeed improves the performance on KB-Ref\xspace by a large margin. \section{The KB-Ref Dataset} Different from existing referring expression datasets~\cite{kazemzadeh2014referitgame, liu2019clevr, mao2016generation, yu2016modeling} that mainly cover the visual contents of the referred objects (such as appearance, attributes and relationships), we collect a new dataset called KB-Ref that needs additional commonsense knowledge to identify the referent object. Wikipedia, ConceptNet and WebChild are employed here as knowledge resources. In this section, we will describe our data collection pipeline in detail and give a statistic analysis of the dataset. \subsection{Dataset Collection} \paragraph{Images and Objects} Images of our dataset are sampled from Visual Genome (VG) \cite{krishna2017visual}, which contains over $108$K images with dense annotations on objects, attributes and relationships. There are averagely $36.5$ bounding boxes in each image, which requires complex visual reasoning to localize. Descriptions on object's affordance or other non-visual attributes can help the referring expression comprehension, which, however, usually needs commonsense knowledge to understand. In VG, most objects are canonicalised to a synset ID in WordNet~\cite{miller1995wordnet}. In order to complicate our dataset, we ignore objects that do not belong to any synset. Same as RefCOCO \cite{kazemzadeh2014referitgame} and other REF datasets, objects that appear alone in one image (\textit{i.e.}\xspace, there are no other instances of the same object category within the same image) are also removed. Moreover, we neglect objects whose shorter sides are less than $32$ pixels. Then we eliminate images which has labeled objects less than $5$. With those filtering process, there are $24,453$ images left which have $2,075$ object categories within $208,532$ bounding boxes. They form the basis of our KB-REF dataset. \paragraph{Knowledge Base} In order to aid annotation and evaluation, we construct a knowledge base by collecting facts from three knowledge resources (\textit{i.e.}\xspace, Wikipedia, ConceptNet and WebChild) that are related to the $1,805$ object categories appeared in our dataset. ConceptNet is a graph-structured commonsense knowledge base where facts are represented by triplets of start nodes, relations, and end nodes. There is a closed set of relations in ConceptNet, such as \texttt{IsA}, \texttt{HasA}, \texttt{PartOf}, \texttt{MadeOf}, \texttt{UsedFor}, and \texttt{CapableOf}. WebChild contains fine-grained commonsense knowledge distilled from web-scale amounts of text, in which the facts can be further categorized into properties (\textit{e.g.}\xspace, \texttt{HasShape}, \texttt{HasSzie}, \texttt{HasTaste}), comparative (\textit{e.g.}\xspace, \texttt{FasterThan}, \texttt{SmallerThan}) and part-whole (\textit{e.g.}\xspace, \texttt{PhysicalPartOf}, \texttt{SubstanceOf}, \texttt{MemberOf}). Compared to structured knowledge bases like ConceptNet and WebChild, Wikipedia contains a larger variety of knowledge but in unstructured format. For each object category, we collect the facts in ConceptNet and Webchild whose start nodes or end nodes match the category label, and the Wikipedia article whose theme concept corresponds to this category \footnote{Considering that some object categories have different interpretations in different images, \textit{e.g.}\xspace, a pole may be a ski pole or a bar holding something, in this case, we choose the meaning that appears most frequently in our dataset, and ignore the uncommon ones. But this issue is then fixed in the following human annotation section because we allow human workers to fix or even rewrite the required knowledge.}. To unify the fact format, we translate the triplet fact in ConceptNet and Webchild into sentences, and treat each sentence in Wikipedia articles as a fact. \paragraph{Data Annotation} We ask Amazon Mechanical Turk (MTurk) workers to write down referring expressions for the queried objects. The following requests are put forward in the annotation process. {1)} At least one fact from the constructed knowledge base should be used in referring expression. {2)} The specific object name cannot appear in the expression. Annotators are required to describe the queried object based on the corresponding fact and its visual context. {3)} Multiple auxiliary objects appeared in the image are encouraged to be mentioned in the expression, to aid the search of the target object. To control the dataset bias, we also perform a quality check in background: The frequency of each fact adopted in the expressions cannot exceed $200$. If exceeds, this fact will be removed in the following annotation process. Note that we use TF-IDF (Term Frequency–Inverse Document Frequency) to measure the similarity between facts. If TF-IDF between two facts is larger than $0.5$, they are regarded as the same. The detailed annotating process is as follows. Given the object to be queried in an image (highlighted by a bounding box) and the related facts in our knowledge base, the MTurk worker is asked to generate a unique text description about the object, according to the requests described above. It takes about $2$ minutes to collect one expression. Then another annotator is asked to verify the correctness of the provided expression. The ones that do not conform to the requests will be asked to re-annotate. The annotation interface can be found in the supplementary materials. \subsection{Data Analysis} Totally, we collected $43,284$ expressions for $1,805$ object categories on $16,917$ images, as compared with other datasets listed in Table~\ref{datasetsComp}. Each object instance in one image has a sole referring expression. To be specific, Figure~\ref{statistic}(a) shows the distribution of expression lengths. The average length of referring expression in KB-Ref\xspace is $13.32$ words, which is longer than that in RefCOCOs (including RefCOCO, RefCOCO+ and RefCOCOg) (about $10$). In our dataset, $25,626$, $9,045$ and $8,613$ expressions are generated based on the knowledge facts from Wikipedia, ConceptNet and WebChild, respectively. Figure~\ref{statistic}(b) shows the distribution of the number of objects mentioned in each expression. Averagely, there are $4.34$ objects used per expression, which suggests the complexity of our collected expressions. The distribution of the length of fact sentences is presented in Figure~\ref{statistic}(c), with an average of $16.78$ words per fact, which reflects the rich information recorded in these facts. We also use TF-IDF to calculate the similarity between each expression and the corresponding fact. As shown in Figure~\ref{statistic}(d), most TF-IDFs range from $0.1$ to $0.4$, which illustrates the difference between the expressions and their corresponding facts. Note that our collected expressions not only reflect the knowledge from their corresponding facts but also contain visual information about the target objects. Figure~\ref{statistic}(e) shows the number of instances per image. We can see most of the images include multiple objects ranging from $20$ to $50$. The Figure~\ref{statistic}(f) shows the percentage of knowledge sources of our dataset, most of the facts are from wikipedia. The object category cloud shown in Figure~\ref{statistic}(g) illustrates that our dataset covers a wide range of objects with less bias (the font size in the cloud represents the frequency of the object appeared in our dataset). We split the dataset on the base of images randomly for training, validation and test. There are $31,284$ expressions with $9,925$ images in training set, $4,000$ expressions with $2,290$ images in validation set, and $8,000$ expressions with $4,702$ images in test set. \section{Experiment} In this section, we conduct experiments to analyze the proposed dataset KB-Ref\xspace and baseline model ECIFA\xspace. Firstly, we analyze the bias of our dataset by evaluating our algorithm with different partial input information. Then the proposed ECIFA\xspace is compared with SoTA REF models on our dataset. Lastly, a group of ablation experiments are performed to validate the effectiveness of multi-hop fact attention. Additionally, results of using detected bounding boxes are given. All the experiments are conducted on $8$ Nvidia RTX2080Ti GPUs. The baseline model is implemented with PyTorch, and trained by using SGD optimizer with a learning rate of $1e^{-4}$ initially. The learning rate will decay half if the validation loss does not decrease in consecutive two epochs. we adopt a batch size of $16$, which consists of $16$ expressions and the corresponding object candidates in images, and train the model with $40$ epochs. Same as previous work, we also use accuracy as the evaluation metric, which is calculated by checking whether the target object is correctly selected or not. \begin{table} { \begin{center} \scalebox{0.8}{ \begin{tabular}{l|c|c|c|c} \hline Method & \multicolumn{2}{|c}{Accuracy (\%)} & \multicolumn{2}{|c}{FG Accuracy (\%)} \\ \cline{2-5} & Val & Test & Val & Test\\ \hline Random & $9.93$ & $9.81$ & - & -\\ ECIFA\xspace~ (no image) & $49.73$ & $47.61$ & $41.91$ & $40.35$\\ ECIFA\xspace~ (no facts) & $37.95$ & $35.16$ & - & - \\ ECIFA\xspace~ (partial expression)& $59.07$ & $58.49$ & $48.97$ & $48.61$\\ ECIFA\xspace~ & $59.45$ & $58.97$ & $49.26$ & $48.92$\\ \hline \end{tabular} } \end{center} } \caption{Dataset bias analysis with different settings. The results show that visual and knowledge facts are both important to our dataset. The remove of prepositional phrases and verbs from expressions has a relatively small influence on model performance. FG: fact grounding.} \label{table:bias} \vspace{-0.9cm} \end{table} \subsection{Dataset Bias Analysis} Dataset bias is an important issue of current vision-and-language datasets. In~\cite{cirik2018visual}, Cirik~\textit{et al.}\xspace shows that a system trained and tested on input images without the input referring expression can achieve an accuracy of $71.2 \%$ in top-2 predictions on RefCOCOg~\cite{mao2016generation}, which suggests the significant data bias. Inspired by this work, we analyze our dataset using similar methods. \noindent\textbf{Random} $\,$ The accuracy is obtained by selecting a random object from the candidates in an image. \noindent\textbf{ECIFA\xspace~ (no image)} $\,$ We eliminate all the visual features in ECIFA\xspace, \textit{i.e.}\xspace, $\mathbf{f}_n^a$, $\mathbf{f}_n^g$ and $\mathbf{v}$ are removed from Equation~\ref{eq:fmall} when calculating the matching score. This study is to investigate the importance of visual information in our Kb-Ref. \noindent\textbf{ECIFA\xspace~ (no facts)} $\,$ ECIFA\xspace is re-trained without using knowledge facts, which means that $\mathbf{f}_m^e$ is removed from Equation~\ref{eq:fmall}, so as to study the impact of knowledge facts in our Kb-Ref. \noindent\textbf{ECIFA\xspace~ (partial expression)} $\,$ ECIFA\xspace is re-trained by keeping only nouns and adjectives in the input expression, since description words (\textit{e.g.}\xspace, color, shape) and object categories are basically expressed by adjectives and nouns. It will obscure the relationships between objects, which are usually represented by prepositional phrases and verbs. Table~\ref{table:bias} shows the ablation study results. The \textbf{Random} baseline offers accuracy around $10\%$ on both validation and test sets, as there are around $10$ candidates to be selected for each expression. Without using external knowledge, the accuracy of \textbf{ECIFA\xspace~ (no facts)} drops to $35.16 \%$ on test set. \textbf{ECIFA\xspace~ (no image)} leads to a $11$-percentage drop on the test accuracy. As we can see, the performance drop caused by removing facts is larger than by removing image, which indicates the importance of commonsense knowledge in our REF setting. In addition, by discarding all words except nouns or adjectives, the test accuracy of \textbf{ECIFA\xspace~ (partial expression)} drops slightly by around $0.5$ percentage. This phenomenon indicates that the superior performance of ECIFA\xspace~ does not specifically depend on object relationships. Besides the answering accuracy, we also evaluate the accuracy of `fact grounding' in Table~\ref{table:bias}. A success will be counted if our model gives the highest attention weight to the groundtruth fact. We can see a positive correlation between the answering accuracy and the `fact grounding' accuracy. \subsection{Comparison with State-of-the-art} The following models are evaluated on the KB-Ref and compared with the proposed ECIFA\xspace model. All the models are trained from scratch on the training split of our proposed KB-Ref\xspace dataset, using their own training strategies. \noindent\textbf{CMN} \cite{hu2017modeling} is a modular architecture that utilizes the language attention to parse the input expression into subject, relation and object. The textual components are then aligned with image regions by three modules respectively to calculate the final matching score. \noindent\textbf{SLR} \cite{yu2017joint} is a speaker-listener model that jointly learns for referring expression comprehension and generation. A reinforce module is introduced to guide sampling of more discriminate expressions. \noindent\textbf{MAttNet} \cite{yu2018mattnet} is one of the most popular models for REF. In comparison with CMN which decomposes expression with fixed template, MAttNet propose a generic modular network with three modules for subject, location and relationship to address all kinds of referring expressions. \noindent\textbf{VC} \cite{niu2019variational} is a recent state-of-the-art based on variational Beyesian method, called Variational Context (VC), to exploit the reciprocal relation between the referent and context. \noindent\textbf{LGARNs} \cite{wang2019neighbourhood} is a graph-based reasoning model for referring expression. By building a directed graph over objects in an image and a language-guided graph attention network to highlight the relevant content in the expression, the model can explore the relationships between objects and make the grounding process explainable. \noindent \textbf{Human.} We also test the human performance. In order to reduce inter-human variability, three workers are asked to choose the target object from the candidates, given the referring expression. If at least two of them selected the correct one, then it is regarded as a success. The overall accuracy of all evaluated models with ground truth candidate objects bounding boxes are presented in Table~\ref{table:accuracy_gt}. All the SoTA models show a significant performance drop compared to their performance on RefCOCOs, {{which demonstrates the challenge of our dataset.}} Our model reaches an accuracy of $58.97 \%$ on KB-Ref test set, outperforming all the SoTA models by nearly $12 \%$, which suggests the necessity of exploring external knowledge in our REF setting. In addition, there is still a large gap between our model and the human performance (about $30 \%$ in accuracy). We also visualize some experimental results on Figure~\ref{visualization}. We also add the proposed episodic memory module (EMM) into MAttNet, which improves the test accuracy from $46.03\%$ to $63.57\%$. It further validates the importance of commonsense knowledge integration for our proposed REF task and the effectiveness of EMM. \begin{table} { \begin{center} \scalebox{0.8}{ \begin{tabular}{l|c|c} \hline Method & \multicolumn{2}{|c}{Accuracy (\%)} \\ \cline{2-3} & Val & Test \\ \hline CMN \cite{hu2017modeling} & $41.28$ & $40.03$ \\ SLR \cite{yu2017joint} & $44.03$ & $42.92$ \\ VC \cite{niu2019variational} & $44.63$ & $43.59$ \\ LGARNs \cite{wang2019neighbourhood} & $45.11$ &$44.27$ \\ MAttNet \cite{yu2018mattnet} & $46.86$ & $46.03$ \\ \hline ECIFA\xspace~(Ours) & $59.45$ & $58.97$ \\ MAttNet \cite{yu2018mattnet} + EMM & $64.08$ & $63.57$\\ \hline Human performance & - & $90.13$ \\ \hline \end{tabular} } \end{center} } \caption{Performance (Acc\%) comparison with SoTA REF approaches and our proposed ECIFA\xspace~ on KB-Ref\xspace. Our ECIFA\xspace~ shows the highest accuracy on both validation and test set. All listed models use VGG-16 features.} \label{table:accuracy_gt} \vspace{-0.7cm} \end{table} \begin{figure*}[htbp] \setlength{\abovecaptionskip}{0.3cm} \centering \includegraphics[width=0.85\textwidth]{vis.pdf} \caption{Visualization of some experimental results. Candidate bounding boxes (in blue) are presented in the image. The red one is the object selected by our algorithm. And the green one is chosen by the MAttNet.} \label{visualization} \vspace{-3mm} \end{figure*} \subsection{Ablation Studies} \begin{table} { \begin{center} \scalebox{0.8}{ \begin{tabular}{l|c|c} \hline Method & \multicolumn{2}{|c}{Accuracy (\%)} \\ \cline{2-3} & Val & Test \\ \hline ECIFA\xspace~ (Soft Attention) & $56.03$ & $54.92$ \\ \hline ECIFA\xspace~(EMM, $1$-pass) & $57.49$ & $56.57$ \\ ECIFA\xspace~(EMM, $3$-pass) & $58.52$ & $57.83$ \\ ECIFA\xspace~(EMM, $5$-pass) & $59.45$ & $58.97$ \\ ECIFA\xspace~(EMM, $10$-pass) & $59.47$ & $58.99$ \\ \hline ECIFA\xspace~(EMM, $5$-pass) + Facts Supervision & $59.50$ & $59.01$ \\ ECIFA\xspace~(EMM, $5$-pass)-ResNet-50 & $60.60$ & $60.35$ \\ \hline \end{tabular} } \end{center} } \caption{Ablation study on the facts attention module. Both using $1$ pass, our episodic memory module (EMM) with an attentional LSTM performs better than the naive soft attention, as the former considers the interaction between facts. The EMM performs better when the number of passes increases, which shows the effectiveness of the multi-hop attention strategy used by EMM. The extra fact supervision leads to a marginal accuracy increase. } \label{table:abalation_study} \vspace{-0.9cm} \end{table} \begin{figure*}[htbp] \setlength{\abovecaptionskip}{0.3cm} \centering \includegraphics[width=0.85\textwidth]{mn_att.pdf} \caption{Visualization of the weight calculated by the Two-stage Fact Attention Module. We show the top-$1$ fact of each pass of the ECIFA\xspace.} \label{mn_attention} \vspace{-4mm} \end{figure*} \paragraph{Effectiveness of Episodic Memory Module} The adopted Episodic Memory Module (EMM) performs a multi-hop attention process. To validate it effectiveness, we compare it with a single-pass soft attention module and also evaluate it with different numbers of passes $T = 1, 3, 5, 10$. The single-pass ($T=1$) soft attention module compute the attended facts feature using a weighted sum $\mathbf{f}_n^e = \sum_{k=1}^K \alpha_k \mathbf{s}_k $, instead of using the attentional LSTM as Equation~\ref{mnm_2}. It does not take into consideration the interaction between facts. As shown in Table~\ref{table:abalation_study}, the episodic memory module with one pass already surpasses soft attention by roughly $2.4$ percentages. We also observe that the accuracy is improved with the increase of the number of passes $T$, which validates the advantage of multi-hop attention. As the performance almost saturates at $T=5$, we choose the model ECIFA\xspace~(EMM, $5$-pass) for the following experiments, in order to strike a balance between accuracy and speed. In Figure~\ref{mn_attention}, we also showcase the focused facts over different passes. \paragraph{Impact of Direct Facts Supervision} Note that our episodic memory module is trained in a weakly supervised manner by the remote cross entropy loss for object selection. As the groundtruth supporting fact has been recorded in our dataset, it is straightforward to add a direct supervision on the episodic memory module. To be specific, a target vector is defined where the position corresponding to the ground truth fact is filled with $1$ while others are $0$. A binary cross entropy function is then employed to calculate the loss between the facts attention weights and the target vector, which is applied on the last pass of EMM. The corresponding results in Table~\ref{table:abalation_study} shows that adding direct fact supervision does not yield significantly better performance, which means that the weak supervision is considerably strong for training episodic memory module. \paragraph{Impact of Image feature Extractor} We also try to use ResNet-50 to extract the image feature. As shown in Table~\ref{table:abalation_study}, with replacing the VGG-16 by ResNet-50~\cite{he2016deep}, the performance increases by $1.3$ percentages. \begin{table}[tb] { \begin{center} \scalebox{0.8}{ \begin{tabular}{l|c|c} \hline Method & \multicolumn{2}{|c}{Accuracy (\%)} \\ \cline{2-3} & Val & Test \\ \hline CMN \cite{hu2017modeling} & $20.91$ & $20.46$ \\ SLR \cite{yu2017joint} & $21.33$ & $20.92$ \\ VC \cite{niu2019variational} & $21.68$ & $21.29$ \\ LGARNs \cite{wang2019neighbourhood} & $21.72$ & $21.37$ \\ MAttNet \cite{yu2018mattnet} & $22.04$ & $21.73$ \\ \hline ECIFA\xspace (ours) & $24.11$ & $23.82$ \\ \hline \end{tabular} } \end{center} } \caption{Performance (Acc\%) comparison with SoTA REF approaches and our proposed ECIFA\xspace~ on KB-Ref, using detected bounding boxes.} \label{table:accuracy} \vspace{-1cm} \end{table} \paragraph{Comparison with SOTA using Detected Bounding Boxes} We also evaluate the performance with detected bounding boxes. A $1600$-category Faster-RCNN detector is trained on Visual Genome and then applied on the validation and test images to extract object bounding box and category. The top-$10$ scored bounding boxes are extracted for each image. From Table~\ref{table:accuracy}, we can see that the gap between our model and other counterparts that without using knowledge is roughly $2 \sim 3\%$, which is significantly smaller than the gap achieved using ground-truth bounding boxes. The reason is that the trained detector is far from satisfactory, which only generates low-quality and misleading bounding boxes and labels. A wrong label may lead our proposed model to extract incorrect knowledge from the knowledge base. \section{Introduction} Referring expression comprehension (REF) aims at localizing a specific object in the image, based on an expression in the form of natural language. Several benchmark datasets have been released to test the referring expression comprehension models' ability, such as RefCOCO~\cite{kazemzadeh2014referitgame}, RefCOCOg~\cite{mao2016generation} and CLEVR-Ref+~\cite{liu2019clevr}. Expressions in these existing datasets are usually about the visual appearance and spatial location of the target objects. For example, commonly seen expressions in RefCOCO~\cite{kazemzadeh2014referitgame} mainly include three components that are subject, location and relationship, where the subject component handles the visual categories, colour and other visual attributes; the location phrase handles both absolute and relative location; and the relationship covers subject-object visual relations~\cite{yu2018mattnet}, such as `the second white cup on the table'. Expressions in CLEVR-Ref+~\cite{liu2019clevr} require a longer reasoning chain but only visual attributes (such as size, colour, material) and spatial relationship (such as left, right) are covered. While in practice, humans often use richer knowledge to ask for something they want, not limited to visual information. For example, we commonly use the `affordance' and other non-visual attributes to describe something we want, like `Can you pass me something to knock in this pin' and `I want to eat something low fat'. In this case, one needs to identify the objects in the image in accordance with the commonsense knowledge mentioned in the expression, for example a `rock' in the image can be used to knock in the pin, and `banana' is low fat. Thus, to enable a machine to reason over visual appearance, spatial and semantic relationships and commonsense knowledge is an emerging challenge. Unfortunately, no existing datasets, including the popular RefCOCOg \cite{mao2016generation} and CLEVR-Ref+~\cite{liu2019clevr}, present above features, not to mention a referring expression model that offers this capacity. To this end, we propose a new dataset for referring expression comprehension with commonsense knowledge, {\bf KB-Ref}, collected based on the images from Visual Genome \cite{krishna2017visual} and knowledge facts from Wikipedia, ConceptNet~\cite{speer2017conceptnet} and WebChild~\cite{Tandon2017WebChild}. Similar to RefCOCO dataset family, we ask Amazon Mechanical Turk (MTurk) workers to select an object from the image and use language to describe it so that another person can use it to localize the object. The difference in annotation is that we also provide a list of commonsense facts about the selected object to workers, who must use at least one of the provided knowledge facts together with the visual context to describe the target object. We also ask workers to record the knowledge they used. To verify whether the collected expression is meaningful and whether the recorded knowledge is required to localize the object, we give the annotations to another MTurk group to verify. Only those expressions that need knowledge to solve are kept. This leads to $43,284$ expressions of $1,805$ object categories in $16,917$ images. The average length of the expressions is $13.32$, nearly double the length in RefCOCOg~\cite{mao2016generation}. In our setting, the recorded knowledge facts are provided during the training but are removed during the testing. So the real technical challenge of this new task is how to mine related knowledge and combine it with visual context to find the object that is referred by the expression. To verify whether commonsense knowledge is crucial in our collected dataset, we first evaluate a variety of state-of-the-art (SoTA) referring expression models (such as MattNet~\cite{yu2018mattnet} and LGARNs~\cite{wang2019neighbourhood}) on our KB-Ref dataset, finding that all of them present a large drop compared to their performance on general REF datasets. We then propose an {\bf E}xpression {\bf C}onditioned {\bf I}mage and {\bf F}act {\bf A}ttention network ({\bf ECIFA\xspace}), which uses an top-down attention module to extract expression-related image representations and an episodic memory module to focus attention on a subset of commonsense knowledge facts. The proposed network leads to a significant improvement over SoTA REF methods, on our constructed dataset. Nevertheless, we also evaluate the human performance on the test split and find that there is still a large gap between our baselines and the human accuracy, which suggests that our proposed KB-Ref dataset is considerably challenging. \section{Research Methods} \end{document} \endinput \section{Method} \label{method} In this section, we propose an Expression Conditioned Image and Fact Attention (ECIFA\xspace) network for extra knowledge required referring expression comprehension. Given a natural language expression $q$ and an image $I$, the model is asked to pick the described object $O^*$ from a group of candidates $\{O\}_{n=1}^N$. The bounding boxes of candidate objects are either groundtruth or obtained via off-the-shell detectors. Different from previous settings, extra commonsense knowledge is needed to understand the given expression for object grounding. The overall architecture is illustrated in Figure~\ref{fig:arch}. The model can be generally divided into three components: (1) a top-down image attention module that predicts an attention distribution over the image grids conditioned on the given expression; (2) a multi-hop facts attention module that gather information from a set of related facts in our knowledge base; (3) a matching module which calculates the expression-object matching score for final grounding. We elaborate on each component in the following. It is worth noting that our facts attention module can be plugged into other referring expression models as well. \subsection{Top-down Image Attention Module} Similar to many conventional REF models~\cite{Hu2016cvpr, mao2016generation}, we first represent each word in the given expression $q$ using an one-hot vector, and then encode them iteratively by an LSTM. The hidden states at all time steps are added together\footnote{We also tried to use the last hidden state of the LSTM as the expression feature but the results are slightly worse. We believe the reason is that our expressions are long.} as the holistic representation for the expression, which is denoted as $\mathbf{q}$ with a dimension of $2048$. Meanwhile, the input image is fed into a pre-trained VGG-$16$ net. Feature maps from Conv${5\_}3$ are extracted, denoted as $\mathbf{V}$ of size $7\times7\times512$. A top-down attention mechanism is adopted here to extract information from the image regions that are the most related to the expression, which is formulated as: \begin{flalign}\label{globalatt} \begin{split} {\alpha}_{i,j} &= \mathbf{w}^{\!\top} \tanh(\mathbf{W}_v \mathbf{V}_{i,j} + \mathbf{W}_q \mathbf{q}), \\ \beta_{i,j} &= \exp(\alpha_{i,j}) / (\textstyle{\sum_{k,l}} \exp(\alpha_{k,l})), \, i,j \in \{1,...,7\}\\ \mathbf{v} &= \textstyle{\sum_{k,l}} \beta_{k,l} \mathbf{V}_{k,l}, \end{split} \end{flalign} where $\mathbf{V}_{ij}$ is the local feature vector at position $(i,j)$ in feature maps $\mathbf{V}$; The expression feature $\mathbf{q}$ is used here as the guidance signal; $\mathbf{W}_v$, $\mathbf{W}_q$, and $\mathbf{w}$ are linear transformation weights to be learned; $\beta_{ij}$ is the attention weight at location $(i,j)$. The weighted sum $\mathbf{v}$ is the attended image feature, with the dimension of $512$. It encodes image features that is most relevant to the given expression. \subsection{Two-stage Fact Attention Module} \label{sec:dmn} The distinguishing feature of our proposed dataset is the requirement of commonsense knowledge. In this section, we introduce a two-stage coarse-to-fine fact attention module that distills related information from the massive facts of our constructed knowledge base. For the first stage, we train a Word2Vec~\cite{Mikolov2013Distributed} model with Skip-Gram on the $1,008,406$ facts in our knowledge base. Given a candidate object, we first retrieve its corresponding facts, and then compute the cosine similarity between the averaged Word2Vec word embeddings of each fact and the expression. At most top $K$ facts (denoted as $\{s_k\}, k=1, \dots, K$) are then kept for further processing. At the second stage, inspired by \cite{Xiong2016Dynamic}, we employ an Episodic Memory Module (as shown in Figure \ref{fig:mn}) to focus attention on a subset of the $K$ retrieved facts in the previous stage. Firstly, each fact $s_k$ is encoded by an LSTM with $2048$D hidden states (which does not share parameters with the LSTM encoding expressions), and the averaged hidden states over all time steps (denoted as $\mathbf{s}_k$) is taken as the fact representation, considering that some facts are very long. Next, the episodic memory module is adopted to perform a multi-hop attention over facts $\mathbf{s}_1, \dots, \mathbf{s}_K$ under the guidance of the expression $\mathbf{q}$. At each pass $t$, a set of attention weights are computed as follows: \begin{flalign}\label{mnm_1} \begin{split} \mathbf{z}_{k}^t &= [\mathbf{s}_{k} \circ \mathbf{q};\mathbf{s}_{k} \circ \mathbf{m}^{t-1}; |\mathbf{s}_{k} - \mathbf{q}|;|\mathbf{s}_{k} - \mathbf{m}^{t-1}|], \\ z_{k}^t &= \mathbf{w}_{z}^{{\!\top}} \tanh(\mathbf{W}_{z} \mathbf{z}_{i, k}), \\ {\alpha}_k^t &= \exp({z_k^t}) / (\textstyle{\sum}_{l=1}^K \exp({z_l^t})), \, k = 1, \dots, K, \end{split} \end{flalign} which is then fed into an attentional LSTM to decide how much the hidden state should be updated for each $k$: \begin{flalign}\label{mnm_2} \begin{split} \mathbf{h}_{k}^t &= \alpha_{k}^t \mathrm{LSTM}(\mathbf{s}_{k}, \mathbf{h}_{k-1}^t) + (1-\alpha_{k}^t) \mathbf{h}_{k-1}^t, \, k = 1, \dots, K\\ \end{split} \end{flalign} and the episodic memory for pass $t$ is updated by another LSTM that takes the last hidden state of the attentional LSTM as contextual vector: \begin{flalign}\label{mnm_3} \begin{split} \mathbf{m}^{t} &= \mathrm{LSTM}(\mathbf{h}_{K}^t, \mathbf{m}^{t-1}), \, t = 1, \dots, T.\\ \end{split} \end{flalign} The memory for the last pass $\mathbf{m}^{T}$ is considered as the attended fact feature and fed into the following Matching Module. \begin{comment} There is a number of commonsense knowledge from Wikipedia, ConceptNet and WebChild. To reduce the calculation of our model and remove unrelated knowledge, we train a Word2Vector model with the Skip-Gram on the all the knowledge that we gain from the three knowledge bases. Then for each fact and expression, we gain the vector for each word from the Word2Vector and take the average of the words in the expression or fact as the representation of the corresponding sentences. Third we calculate the cosine between the expression and each knowledge. To enable the commonsense knowledge understanding ability, a Dynamic Memory Network is introduced in our model which explores three knowledge bases for referring expression. To be specific, for each candidate object $O_m$, the top $K = 50$ related sentences compared to the expression in the corresponding knowledge bases\footnote{To find the corresponding commonsense knowledge, we use the object label as a RDF (Resource Description Framework) query to search the corresponding entity from the knowledge bases, which returns us the corresponding knowledge. More details can be found in the supplementary material.} are extracted, denoted as $\{s_k\}, k=1, \dots, K$. We encode each fact $s_k$ by an LSTM, and add the hidden state at each time step as the sentence representation, considering that some supporting facts are very long. The embedded supporting facts for object $O_m$ are denoted as $\mathbf{F}_m^s \in \mathcal{R}^{2048 \times 50}$, where each column vector $ \mathbf{f}_{m,k}^s$ corresponds to one fact. We pad with zero if the number of relevant facts is less than $50$. An Dynamic Memory Network is employed again under the guidance of the expression $q$, as shown in Figure~\ref{fig:mn}, \textit{i.e.}\xspace, \begin{flalign}\label{mn_1} \begin{split} \mathbf{z}_{i, k} &= [\mathbf{F}_{m, k}^s \circ \mathbf{f}^q;\mathbf{F}_{m, k}^s \circ \mathbf{m}_{i}; |\mathbf{F}_{m, k}^s - \mathbf{f}^q|;|\mathbf{F}_{m, k}^s - \mathbf{m}_{i}|] \\ \mathbf{z}_{i, k} &= \tanh(\mathbf{W}_{z} \mathbf{z}_{i, k})\\ \mathbf{g}_{i, k} &= \mathrm{softmax}(\mathbf{W}_{g} \mathbf{z}_{i, k}) \\ \mathbf{h}_{i,k}, \mathbf{c}_{i,k} &= LSTMCell(\mathbf{F}_{m,k}^s,( \mathbf{h}_{i,k-1}, \mathbf{c}_{i,k-1}))\\ \mathbf{h}_{i,k} &= \mathbf{g}_{i, k}\mathbf{h}_{i,k} + (1-\mathbf{g}_{i, k}) \mathbf{h}_{i,k-1}\\ \end{split} \end{flalign} in which, $\mathbf{m}_{i}$ is the memory cell and $\mathbf{m}_{0} &= \mathbf{f}^q$, $\mathbf{h}_{i,0} &= 0, \mathbf{c}_{i,0} &= 0$. We use a LSTMCell to update $\mathbf{m}_{i}$: \begin{flalign}\label{mn_1} \begin{split} \mathbf{m}_{i+1} &= LSTMCell(\mathbf{h}_{i,K}, (\mathbf{m}_{i}, 0))\\ \end{split} \end{flalign} After five iterations, the $\mathbf{f}^e_m &= \mathbf{m}_{5} \in \mathcal{R}^{512}$ is the output of the Dynamic Memory Network and will be fed in the Matching Module. \end{comment} \begin{figure} \centering \includegraphics[width=0.95\linewidth]{MN.pdf} \vspace{-1em} \caption{The computational flow chart of the episodic memory module at pass $t$.} \label{fig:mn} \vspace{-0.5cm} \end{figure} \subsection{Matching Module} The matching module is then used to calculate the matching score between the expression $q$ and each object $O_n$. Specifically, for each candidate object, we calculate its appearance feature by firstly resizing the object region to $224 \times 224$ and then feeding it into a pre-trained VGG-$16$. Feature maps from Conv${5\_3}$ are extracted and averagely pooled. A fully connected layer with $512$ neurons and ReLU are then followed, which results in an appearance feature for object $O_n$ of $\mathbf{f}_n^a \in \mathbb{R}^{512}$. In addition, we also extract the geometric information for each candidate object, $[\frac{x_{tl}}{W}, \frac{y_{tl}}{H}, \frac{x_{br}}{W}, \frac{y_{br}}{H}, \frac{w\cdot h}{W\cdot H}]$, which is a $5$-dimensional vector consisting of four values for top left and bottom right corner coordinates of the object region (normalised between 0 and 1) and one value for its relative area (\textit{i.e.}\xspace, ratio of the bounding box area to the image area, also between 0 and 1). A fully connected layer with $128$ neurons and ReLU are followed, which lead to a geometric feature $\mathbf{f}_n^p \in \mathbb{R}^{128}$. The candidate's appearance feature $\mathbf{f}_n^a$ and geometric feature of $\mathbf{f}_n^g$ are then concatenated with the attended fact feature $\mathbf{f}_n^e$ and the attended image feature $\mathbf{v}$. Another linear transformation is applied to yield a $2048$d feature: \begin{equation} \mathbf{f}_n = \mathbf{W} [\mathbf{f}_n^a; \mathbf{f}_n^g; \mathbf{f}_n^e; \mathbf{v}], \label{eq:fmall} \end{equation} where $\mathbf{W}$ is the parameter to be learned. Finally we calculate the inner product of the expression feature $\mathbf{q}$ and the integrated object feature $\mathbf{f}_n$. Softmax is then applied over all candidate objects, and the object with the highest score will be selected for the expression. During the training, the cross entropy loss is used. \section{Related Work} \subsection{Referring Expression Comprehension} \paragraph{Datasets.} \begin{table*} \newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}} \begin{center} \scalebox{0.8}{ \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline Dataset & Facts & Images & {\tabincell{c}{Objects \\ Categories}} & {\tabincell{c}{Referring \\ Expressions}} & {\tabincell{c}{Objects \\ Instances}} & {\tabincell{c}{Objects Queried \\ Per Image}} & {\tabincell{c}{Average \\ Expression\\ length}}\\ \hline\hline RefCOCO \cite{kazemzadeh2014referitgame} & $\times$ & $19,944$ & $78$ & $142,210$ & $50,000$ & $2.51$ & $10.19$ \\ RefCOCO+ \cite{kazemzadeh2014referitgame} & $\times$ & $19,992$ & $78$ & $141,564$ & $49,856$ & $2.49$ & $10.14$ \\ RefCOCOg \cite{mao2016generation}$^\dag$ & $\times$ & $25,799$ & $78$ & $95,010$ & $49,822$ & $1.93$ & $8.93$ \\ {KB-Ref} (ours) & $\checkmark$ & $16,917$ & $1,805$$^\ddag$ & $43,284$ & $43,284$ & $2.56$ & $13.32$ \\ \hline \end{tabular}} \end{center} \caption{Comparison between different REF datasets. $^\dag$The data of RefCOCOg is based on the revised version. $^\ddag$We use Visual Genome's definition of object categories, which is finer than COCO. For example, `people' in COCO corresponds to `man', `woman', `boy' and `girl' in Visual Genome.} \label{datasetsComp} \vspace{-0.8cm} \end{table*} As presented in Table~\ref{datasetsComp}, commonly used datasets for referring expression comprehension include RefCOCO~\cite{kazemzadeh2014referitgame}, RefCOCO+~\cite{kazemzadeh2014referitgame} and RefCOCOg~\cite{mao2016generation}, all collected on top of MSCOCO images~\cite{MSCOCO}. RefCOCO and RefCOCO+ are collected interactively in a two-player game, with concise phrase descriptions, while RefCOCOg is collected by MTurk workers in a non-interactive setting, using longer declarative sentences. There is no restriction in RefCOCO on language expressions, while RefCOCO+ focuses more on purely appearance descriptions where location words are not allowed. GuessWhat?!~\cite{GuessWhat} is another dataset based on MS-COCO images. Instead of using a single expression, it creates a sequence of sentences (\textit{i.e.}\xspace, dialog) for a given image to perform referring expression comprehension. CLEVR-Ref+~\cite{liu2019clevr} is a recently introduced synthetic dataset, built on the CLEVR environment. { In contrast, our proposed KB-REF is based on images from Visual Genome~\cite{krishna2017visual}, which provides richer annotations including objects, attributes and relationships. The expressions in KB-REF are different with above and need both visual context and commonsense knowledge to resolve.} \paragraph{Approaches.} Referring expression is a visual-linguistic cross-model understanding problem. Some works solve REF jointly with a referring expression generation task~\cite{kazemzadeh2014referitgame, mao2016generation, GroundeR, yu2016modeling}. Some others~\cite{hu2017modeling,Luo2017,niu2019variational,wang2019neighbourhood,yu2018mattnet} propose different types of joint embedding frameworks and directly localize the object which has the highest matching score. The work in~\cite{hu2017modeling,yu2018mattnet} proposes to decompose the expression into sub-phrases, which are then used to trigger separate visual modules to compute matching score. Liu~\textit{et al.}\xspace~\cite{DaqingICCV2019} develop a neural module tree network to regularize the visual grounding along the dependency parsing tree of the sentence. The works in~\cite{Chaoruicvpr2018,SibeiICCV2019,Bohancvpr2018} argue to learn the representations from expression and image regions in a stepwise manner, and perform multi-step reasoning for better matching performance. Wang~\textit{et al.}\xspace~\cite{wang2019neighbourhood} propose a graph-based language-guided attention network to highlight the inter-object and intra-object relationships that are closely relevant to the expression for better performance. Niu~\textit{et al.}\xspace~\cite{niu2019variational} develop a variational Bayesian framework to exploit the reciprocity between the referent and context. { Our proposed baseline model uses a simple visual-expression joint embedding model to calculate the matching score, but we incorporate a knowledge facts attention module to mine external knowledge base for referring expression, which can also be plugged into other REF models for extra knowledge exploring.} \subsection{Visual Understanding and Reasoning with Extra Commonsense Knowledge} Commonsense knowledge has already attracted research attention in visual understanding and reasoning, such as visual relationship detection~\cite{lu2016visual, yu2017visual, xu2019learning}, scene graph generation~\cite{gu2019scene}, visual question answering~\cite{askmeanything, liu2019clevr, Xiong2016Dynamic, FVQA, narasimhan2018straight, Narasimhan2018, KDMN, hudson2019gqa} and zero-shot recognition~\cite{wang2018zero, lee2018multi}. In particular, the incorporation of commonsense knowledge is important for visual question answering (VQA), because a lot of questions are from open-domain that require to perform reasoning beyond the image contents. In~\cite{askmeanything}, attributes extracted from the image are used to query external knowledge based on DBpedia, which enables answer questions beyond the image. Dynamic memory networks are employed in~\cite{VKMN} and~\cite{KDMN} to incorporate structured human knowledge and deep visual features for answer decoding. Wang~\textit{et al.}\xspace~\cite{FVQA} introduce a Fact-Based VQA (FVQA) dataset which requires external knowledge to answer a question. A Graph Convolution Network (GCN) is developed in~\cite{Narasimhan2018} which integrates image, question and all possible facts in an entity graph for answer inferring in FVQA. The work in~\cite{licvpr2019} represents visual information by dense captions and convert VQA as a reading comprehension problem, where extra knowledge is added by text concatenation. In comparison with FVQA where visual reasoning is based on fixed structured knowledge bases, an even larger knowledge based VQA dataset (OK-VQA) is introduced in~\cite{OKVQA} which performs VQA over unstructured open knowledge. The Visual Commonsense Reasoning (VCR)~\cite{zellers2019recognition, hudson2019gqa} dataset contains 290k multiple choice QA problems from 110k movie scenes, which require higher-order cognition and commonsense reasoning. Similarly, as a proxy to evaluate AI systems on both vision and language understanding, REF would require not only object localization based on image appearance, but also a more natural way to achieve human-level semantic understanding. With this requirement, we go one step forward and design a KB-Ref which requires reasoning incorporating external commonsense knowledge. In particular, REF can be regarded as a subtask of VQA with the question as ``\textbf{Where} is \textit{sth. (by referring expression)} in the image?''. However, in VQA the answer is generally open-ended, presented in natural language, while in REF the answer is numerical, chosen from a group of candidate bounding boxes, or directly output a detected one, which makes the evaluation much easier. \section{Introduction} ACM's consolidated article template, introduced in 2017, provides a consistent \LaTeX\ style for use across ACM publications, and incorporates accessibility and metadata-extraction functionality necessary for future Digital Library endeavors. Numerous ACM and SIG-specific \LaTeX\ templates have been examined, and their unique features incorporated into this single new template. If you are new to publishing with ACM, this document is a valuable guide to the process of preparing your work for publication. If you have published with ACM before, this document provides insight and instruction into more recent changes to the article template. The ``\verb|acmart|'' document class can be used to prepare articles for any ACM publication --- conference or journal, and for any stage of publication, from review to final ``camera-ready'' copy, to the author's own version, with {\itshape very} few changes to the source. \section{Template Overview} As noted in the introduction, the ``\verb|acmart|'' document class can be used to prepare many different kinds of documentation --- a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready'' journal article, a SIGCHI Extended Abstract, and more --- all by selecting the appropriate {\itshape template style} and {\itshape template parameters}. This document will explain the major features of the document class. For further information, the {\itshape \LaTeX\ User's Guide} is available from \url{https://www.acm.org/publications/proceedings-template}. \subsection{Template Styles} The primary parameter given to the ``\verb|acmart|'' document class is the {\itshape template style} which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the {\verb|documentclass|} command: \begin{verbatim} \documentclass[STYLE]{acmart} \end{verbatim} Journals use one of three template styles. All but three ACM journals use the {\verb|acmsmall|} template style: \begin{itemize} \item {\verb|acmsmall|}: The default journal template style. \item {\verb|acmlarge|}: Used by JOCCH and TAP. \item {\verb|acmtog|}: Used by TOG. \end{itemize} The majority of conference proceedings documentation will use the {\verb|acmconf|} template style. \begin{itemize} \item {\verb|acmconf|}: The default proceedings template style. \item{\verb|sigchi|}: Used for SIGCHI conference articles. \item{\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles. \item{\verb|sigplan|}: Used for SIGPLAN conference articles. \end{itemize} \subsection{Template Parameters} In addition to specifying the {\itshape template style} to be used in formatting your work, there are a number of {\itshape template parameters} which modify some part of the applied template style. A complete list of these parameters can be found in the {\itshape \LaTeX\ User's Guide.} Frequently-used parameters, or combinations of parameters, include: \begin{itemize} \item {\verb|anonymous,review|}: Suitable for a ``double-blind'' conference submission. Anonymizes the work and includes line numbers. Use with the \verb|\acmSubmissionID| command to print the submission's unique ID on each page of the work. \item{\verb|authorversion|}: Produces a version of the work suitable for posting by the author. \item{\verb|screen|}: Produces colored hyperlinks. \end{itemize} This document uses the following string as the first command in the source file: \begin{verbatim} \documentclass[sigconf]{acmart} \end{verbatim} \section{Modifications} Modifying the template --- including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the \verb|\vspace| command to manually adjust the vertical spacing between elements of your work --- is not allowed. {\bfseries Your document will be returned to you for revision if modifications are discovered.} \section{Typefaces} The ``\verb|acmart|'' document class requires the use of the ``Libertine'' typeface family. Your \TeX\ installation should include this set of packages. Please do not substitute other typefaces. The ``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used, as they will override the built-in typeface families. \section{Title Information} The title of your work should use capital letters appropriately - \url{https://capitalizemytitle.com/} has useful rules for capitalization. Use the {\verb|title|} command to define the title of your work. If your work has a subtitle, define it with the {\verb|subtitle|} command. Do not insert line breaks in your title. If your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The \verb|title| command has a ``short title'' parameter: \begin{verbatim} \title[short title]{full title} \end{verbatim} \section{Authors and Affiliations} Each author must be defined separately for accurate metadata identification. Multiple authors may share one affiliation. Authors' names should not be abbreviated; use full first names wherever possible. Include authors' e-mail addresses whenever possible. Grouping authors' names or e-mail addresses, or providing an ``e-mail alias,'' as shown below, is not acceptable: \begin{verbatim} \author{Brooke Aster, David Mehldau} \email{dave,judy,steve@university.edu} \email{firstname.lastname@phillips.org} \end{verbatim} The \verb|authornote| and \verb|authornotemark| commands allow a note to apply to multiple authors --- for example, if the first two authors of an article contributed equally to the work. If your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last \verb|\author{}| definition: \begin{verbatim} \renewcommand{\shortauthors}{McCartney, et al.} \end{verbatim} Omitting this command will force the use of a concatenated list of all of the authors' names, which may result in overlapping text in the page headers. The article template's documentation, available at \url{https://www.acm.org/publications/proceedings-template}, has a complete explanation of these commands and tips for their effective use. Note that authors' addresses are mandatory for journal articles. \section{Rights Information} Authors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement. Regardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains \LaTeX\ commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document: \begin{itemize} \item the ``ACM Reference Format'' text on the first page. \item the ``rights management'' text on the first page. \item the conference information in the page header(s). \end{itemize} Rights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works. The ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts). \section{CCS Concepts and User-Defined Keywords} Two elements of the ``acmart'' document class provide powerful taxonomic tools for you to help readers find your work in an online search. The ACM Computing Classification System --- \url{https://www.acm.org/publications/class-2012} --- is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the commands to be included in the \LaTeX\ source. User-defined keywords are a comma-separated list of words and phrases of the authors' choosing, providing a more flexible way of describing the research being presented. CCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts). \section{Sectioning Commands} Your work should use standard \LaTeX\ sectioning commands: \verb|section|, \verb|subsection|, \verb|subsubsection|, and \verb|paragraph|. They should be numbered; do not remove the numbering from the commands. Simulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is {\bfseries not allowed.} \section{Tables} The ``\verb|acmart|'' document class includes the ``\verb|booktabs|'' package --- \url{https://ctan.org/pkg/booktabs} --- for preparing high-quality tables. Table captions are placed {\itshape above} the table. Because tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper ``floating'' placement of tables, use the environment \textbf{table} to enclose the table's contents and the table caption. The contents of the table itself must go in the \textbf{tabular} environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on \textbf{tabular} material are found in the \textit{\LaTeX\ User's Guide}. Immediately following this sentence is the point at which Table~\ref{tab:freq} is included in the input file; compare the placement of the table here with the table in the printed output of this document. \begin{table} \caption{Frequency of Special Characters} \label{tab:freq} \begin{tabular}{ccl} \toprule Non-English or Math&Frequency&Comments\\ \midrule \O & 1 in 1,000& For Swedish names\\ $\pi$ & 1 in 5& Common in math\\ \$ & 4 in 5 & Used in business\\ $\Psi^2_1$ & 1 in 40,000& Unexplained usage\\ \bottomrule \end{tabular} \end{table} To set a wider table, which takes up the whole width of the page's live area, use the environment \textbf{table*} to enclose the table's contents and the table caption. As with a single-column table, this wide table will ``float'' to a location deemed more desirable. Immediately following this sentence is the point at which Table~\ref{tab:commands} is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document. \begin{table*} \caption{Some Typical Commands} \label{tab:commands} \begin{tabular}{ccl} \toprule Command &A Number & Comments\\ \midrule \texttt{{\char'134}author} & 100& Author \\ \texttt{{\char'134}table}& 300 & For tables\\ \texttt{{\char'134}table*}& 400& For wider tables\\ \bottomrule \end{tabular} \end{table*} Always use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily. \section{Math Equations} You may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections. \subsection{Inline (In-text) Equations} A formula that appears in the running text is called an inline or in-text formula. It is produced by the \textbf{math} environment, which can be invoked with the usual \texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with the short form \texttt{\$\,\ldots\$}. You can use any of the symbols and structures, from $\alpha$ to $\omega$, available in \LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few examples of in-text equations in context. Notice how this equation: \begin{math} \lim_{n\rightarrow \infty}x=0 \end{math}, set here in in-line math style, looks slightly different when set in display style. (See next section). \subsection{Display Equations} A numbered display equation---one set off by vertical space from the text and centered horizontally---is produced by the \textbf{equation} environment. An unnumbered display equation is produced by the \textbf{displaymath} environment. Again, in either environment, you can use any of the symbols and structures available in \LaTeX\@; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above: \begin{equation} \lim_{n\rightarrow \infty}x=0 \end{equation} Notice how it is formatted somewhat differently in the \textbf{displaymath} environment. Now, we'll enter an unnumbered equation: \begin{displaymath} \sum_{i=0}^{\infty} x + 1 \end{displaymath} and follow it with another numbered equation: \begin{equation} \sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f \end{equation} just to demonstrate \LaTeX's able handling of numbering. \section{Figures} The ``\verb|figure|'' environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below. \begin{figure}[h] \centering \includegraphics[width=\linewidth]{sample-franklin} \caption{1907 Franklin Model D roadster. Photograph by Harris \& Ewing, Inc. [Public domain], via Wikimedia Commons. (\url{https://goo.gl/VLCRBB}).} \Description{A woman and a girl in white dresses sit in an open car.} \end{figure} Your figures should contain a caption which describes the figure to the reader. Figure captions are placed {\itshape below} the figure. Every figure should also have a figure description unless it is purely decorative. These descriptions convey what's in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded. A figure description must be unformatted plain text less than 2000 characters long (including spaces). {\bfseries Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.} For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see \url{https://www.acm.org/publications/taps/describing-figures/}. \subsection{The ``Teaser Figure''} A ``teaser figure'' is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the \verb|\maketitle| command: \begin{verbatim} \begin{teaserfigure} \includegraphics[width=\textwidth]{sampleteaser} \caption{figure caption} \Description{figure description} \end{teaserfigure} \end{verbatim} \section{Citations and Bibliographies} The use of \BibTeX\ for the preparation and formatting of one's references is strongly recommended. Authors' names should be complete --- use full first names (``Donald E. Knuth'') not initials (``D. E. Knuth'') --- and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc. The bibliography is included in your source document with these two commands, placed just before the \verb|\end{document}| command: \begin{verbatim} \bibliographystyle{ACM-Reference-Format} \section{Introduction} ACM's consolidated article template, introduced in 2017, provides a consistent \LaTeX\ style for use across ACM publications, and incorporates accessibility and metadata-extraction functionality necessary for future Digital Library endeavors. Numerous ACM and SIG-specific \LaTeX\ templates have been examined, and their unique features incorporated into this single new template. If you are new to publishing with ACM, this document is a valuable guide to the process of preparing your work for publication. If you have published with ACM before, this document provides insight and instruction into more recent changes to the article template. The ``\verb|acmart|'' document class can be used to prepare articles for any ACM publication --- conference or journal, and for any stage of publication, from review to final ``camera-ready'' copy, to the author's own version, with {\itshape very} few changes to the source. \section{Template Overview} As noted in the introduction, the ``\verb|acmart|'' document class can be used to prepare many different kinds of documentation --- a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready'' journal article, a SIGCHI Extended Abstract, and more --- all by selecting the appropriate {\itshape template style} and {\itshape template parameters}. This document will explain the major features of the document class. For further information, the {\itshape \LaTeX\ User's Guide} is available from \url{https://www.acm.org/publications/proceedings-template}. \subsection{Template Styles} The primary parameter given to the ``\verb|acmart|'' document class is the {\itshape template style} which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the {\verb|documentclass|} command: \begin{verbatim} \documentclass[STYLE]{acmart} \end{verbatim} Journals use one of three template styles. All but three ACM journals use the {\verb|acmsmall|} template style: \begin{itemize} \item {\verb|acmsmall|}: The default journal template style. \item {\verb|acmlarge|}: Used by JOCCH and TAP. \item {\verb|acmtog|}: Used by TOG. \end{itemize} The majority of conference proceedings documentation will use the {\verb|acmconf|} template style. \begin{itemize} \item {\verb|acmconf|}: The default proceedings template style. \item{\verb|sigchi|}: Used for SIGCHI conference articles. \item{\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles. \item{\verb|sigplan|}: Used for SIGPLAN conference articles. \end{itemize} \subsection{Template Parameters} In addition to specifying the {\itshape template style} to be used in formatting your work, there are a number of {\itshape template parameters} which modify some part of the applied template style. A complete list of these parameters can be found in the {\itshape \LaTeX\ User's Guide.} Frequently-used parameters, or combinations of parameters, include: \begin{itemize} \item {\verb|anonymous,review|}: Suitable for a ``double-blind'' conference submission. Anonymizes the work and includes line numbers. Use with the \verb|\acmSubmissionID| command to print the submission's unique ID on each page of the work. \item{\verb|authorversion|}: Produces a version of the work suitable for posting by the author. \item{\verb|screen|}: Produces colored hyperlinks. \end{itemize} This document uses the following string as the first command in the source file: \begin{verbatim} \documentclass[sigconf]{acmart} \end{verbatim} \section{Modifications} Modifying the template --- including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the \verb|\vspace| command to manually adjust the vertical spacing between elements of your work --- is not allowed. {\bfseries Your document will be returned to you for revision if modifications are discovered.} \section{Typefaces} The ``\verb|acmart|'' document class requires the use of the ``Libertine'' typeface family. Your \TeX\ installation should include this set of packages. Please do not substitute other typefaces. The ``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used, as they will override the built-in typeface families. \section{Title Information} The title of your work should use capital letters appropriately - \url{https://capitalizemytitle.com/} has useful rules for capitalization. Use the {\verb|title|} command to define the title of your work. If your work has a subtitle, define it with the {\verb|subtitle|} command. Do not insert line breaks in your title. If your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The \verb|title| command has a ``short title'' parameter: \begin{verbatim} \title[short title]{full title} \end{verbatim} \section{Authors and Affiliations} Each author must be defined separately for accurate metadata identification. Multiple authors may share one affiliation. Authors' names should not be abbreviated; use full first names wherever possible. Include authors' e-mail addresses whenever possible. Grouping authors' names or e-mail addresses, or providing an ``e-mail alias,'' as shown below, is not acceptable: \begin{verbatim} \author{Brooke Aster, David Mehldau} \email{dave,judy,steve@university.edu} \email{firstname.lastname@phillips.org} \end{verbatim} The \verb|authornote| and \verb|authornotemark| commands allow a note to apply to multiple authors --- for example, if the first two authors of an article contributed equally to the work. If your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last \verb|\author{}| definition: \begin{verbatim} \renewcommand{\shortauthors}{McCartney, et al.} \end{verbatim} Omitting this command will force the use of a concatenated list of all of the authors' names, which may result in overlapping text in the page headers. The article template's documentation, available at \url{https://www.acm.org/publications/proceedings-template}, has a complete explanation of these commands and tips for their effective use. Note that authors' addresses are mandatory for journal articles. \section{Rights Information} Authors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement. Regardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains \LaTeX\ commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document: \begin{itemize} \item the ``ACM Reference Format'' text on the first page. \item the ``rights management'' text on the first page. \item the conference information in the page header(s). \end{itemize} Rights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works. The ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts). \section{CCS Concepts and User-Defined Keywords} Two elements of the ``acmart'' document class provide powerful taxonomic tools for you to help readers find your work in an online search. The ACM Computing Classification System --- \url{https://www.acm.org/publications/class-2012} --- is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the commands to be included in the \LaTeX\ source. User-defined keywords are a comma-separated list of words and phrases of the authors' choosing, providing a more flexible way of describing the research being presented. CCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts). \section{Sectioning Commands} Your work should use standard \LaTeX\ sectioning commands: \verb|section|, \verb|subsection|, \verb|subsubsection|, and \verb|paragraph|. They should be numbered; do not remove the numbering from the commands. Simulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is {\bfseries not allowed.} \section{Tables} The ``\verb|acmart|'' document class includes the ``\verb|booktabs|'' package --- \url{https://ctan.org/pkg/booktabs} --- for preparing high-quality tables. Table captions are placed {\itshape above} the table. Because tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper ``floating'' placement of tables, use the environment \textbf{table} to enclose the table's contents and the table caption. The contents of the table itself must go in the \textbf{tabular} environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on \textbf{tabular} material are found in the \textit{\LaTeX\ User's Guide}. Immediately following this sentence is the point at which Table~\ref{tab:freq} is included in the input file; compare the placement of the table here with the table in the printed output of this document. \begin{table} \caption{Frequency of Special Characters} \label{tab:freq} \begin{tabular}{ccl} \toprule Non-English or Math&Frequency&Comments\\ \midrule \O & 1 in 1,000& For Swedish names\\ $\pi$ & 1 in 5& Common in math\\ \$ & 4 in 5 & Used in business\\ $\Psi^2_1$ & 1 in 40,000& Unexplained usage\\ \bottomrule \end{tabular} \end{table} To set a wider table, which takes up the whole width of the page's live area, use the environment \textbf{table*} to enclose the table's contents and the table caption. As with a single-column table, this wide table will ``float'' to a location deemed more desirable. Immediately following this sentence is the point at which Table~\ref{tab:commands} is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document. \begin{table*} \caption{Some Typical Commands} \label{tab:commands} \begin{tabular}{ccl} \toprule Command &A Number & Comments\\ \midrule \texttt{{\char'134}author} & 100& Author \\ \texttt{{\char'134}table}& 300 & For tables\\ \texttt{{\char'134}table*}& 400& For wider tables\\ \bottomrule \end{tabular} \end{table*} Always use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily. \section{Math Equations} You may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections. \subsection{Inline (In-text) Equations} A formula that appears in the running text is called an inline or in-text formula. It is produced by the \textbf{math} environment, which can be invoked with the usual \texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with the short form \texttt{\$\,\ldots\$}. You can use any of the symbols and structures, from $\alpha$ to $\omega$, available in \LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few examples of in-text equations in context. Notice how this equation: \begin{math} \lim_{n\rightarrow \infty}x=0 \end{math}, set here in in-line math style, looks slightly different when set in display style. (See next section). \subsection{Display Equations} A numbered display equation---one set off by vertical space from the text and centered horizontally---is produced by the \textbf{equation} environment. An unnumbered display equation is produced by the \textbf{displaymath} environment. Again, in either environment, you can use any of the symbols and structures available in \LaTeX\@; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above: \begin{equation} \lim_{n\rightarrow \infty}x=0 \end{equation} Notice how it is formatted somewhat differently in the \textbf{displaymath} environment. Now, we'll enter an unnumbered equation: \begin{displaymath} \sum_{i=0}^{\infty} x + 1 \end{displaymath} and follow it with another numbered equation: \begin{equation} \sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f \end{equation} just to demonstrate \LaTeX's able handling of numbering. \section{Figures} The ``\verb|figure|'' environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below. \begin{figure}[h] \centering \includegraphics[width=\linewidth]{sample-franklin} \caption{1907 Franklin Model D roadster. Photograph by Harris \& Ewing, Inc. [Public domain], via Wikimedia Commons. (\url{https://goo.gl/VLCRBB}).} \Description{A woman and a girl in white dresses sit in an open car.} \end{figure} Your figures should contain a caption which describes the figure to the reader. Figure captions are placed {\itshape below} the figure. Every figure should also have a figure description unless it is purely decorative. These descriptions convey what's in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded. A figure description must be unformatted plain text less than 2000 characters long (including spaces). {\bfseries Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.} For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see \url{https://www.acm.org/publications/taps/describing-figures/}. \subsection{The ``Teaser Figure''} A ``teaser figure'' is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the \verb|\maketitle| command: \begin{verbatim} \begin{teaserfigure} \includegraphics[width=\textwidth]{sampleteaser} \caption{figure caption} \Description{figure description} \end{teaserfigure} \end{verbatim} \section{Citations and Bibliographies} The use of \BibTeX\ for the preparation and formatting of one's references is strongly recommended. Authors' names should be complete --- use full first names (``Donald E. Knuth'') not initials (``D. E. Knuth'') --- and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc. The bibliography is included in your source document with these two commands, placed just before the \verb|\end{document}| command: \begin{verbatim} \bibliographystyle{ACM-Reference-Format} \section{Introduction} ACM's consolidated article template, introduced in 2017, provides a consistent \LaTeX\ style for use across ACM publications, and incorporates accessibility and metadata-extraction functionality necessary for future Digital Library endeavors. Numerous ACM and SIG-specific \LaTeX\ templates have been examined, and their unique features incorporated into this single new template. If you are new to publishing with ACM, this document is a valuable guide to the process of preparing your work for publication. If you have published with ACM before, this document provides insight and instruction into more recent changes to the article template. The ``\verb|acmart|'' document class can be used to prepare articles for any ACM publication --- conference or journal, and for any stage of publication, from review to final ``camera-ready'' copy, to the author's own version, with {\itshape very} few changes to the source. \section{Template Overview} As noted in the introduction, the ``\verb|acmart|'' document class can be used to prepare many different kinds of documentation --- a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready'' journal article, a SIGCHI Extended Abstract, and more --- all by selecting the appropriate {\itshape template style} and {\itshape template parameters}. This document will explain the major features of the document class. For further information, the {\itshape \LaTeX\ User's Guide} is available from \url{https://www.acm.org/publications/proceedings-template}. \subsection{Template Styles} The primary parameter given to the ``\verb|acmart|'' document class is the {\itshape template style} which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the {\verb|documentclass|} command: \begin{verbatim} \documentclass[STYLE]{acmart} \end{verbatim} Journals use one of three template styles. All but three ACM journals use the {\verb|acmsmall|} template style: \begin{itemize} \item {\verb|acmsmall|}: The default journal template style. \item {\verb|acmlarge|}: Used by JOCCH and TAP. \item {\verb|acmtog|}: Used by TOG. \end{itemize} The majority of conference proceedings documentation will use the {\verb|acmconf|} template style. \begin{itemize} \item {\verb|acmconf|}: The default proceedings template style. \item{\verb|sigchi|}: Used for SIGCHI conference articles. \item{\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles. \item{\verb|sigplan|}: Used for SIGPLAN conference articles. \end{itemize} \subsection{Template Parameters} In addition to specifying the {\itshape template style} to be used in formatting your work, there are a number of {\itshape template parameters} which modify some part of the applied template style. A complete list of these parameters can be found in the {\itshape \LaTeX\ User's Guide.} Frequently-used parameters, or combinations of parameters, include: \begin{itemize} \item {\verb|anonymous,review|}: Suitable for a ``double-blind'' conference submission. Anonymizes the work and includes line numbers. Use with the \verb|\acmSubmissionID| command to print the submission's unique ID on each page of the work. \item{\verb|authorversion|}: Produces a version of the work suitable for posting by the author. \item{\verb|screen|}: Produces colored hyperlinks. \end{itemize} This document uses the following string as the first command in the source file: \begin{verbatim} \documentclass[sigconf]{acmart} \end{verbatim} \section{Modifications} Modifying the template --- including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the \verb|\vspace| command to manually adjust the vertical spacing between elements of your work --- is not allowed. {\bfseries Your document will be returned to you for revision if modifications are discovered.} \section{Typefaces} The ``\verb|acmart|'' document class requires the use of the ``Libertine'' typeface family. Your \TeX\ installation should include this set of packages. Please do not substitute other typefaces. The ``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used, as they will override the built-in typeface families. \section{Title Information} The title of your work should use capital letters appropriately - \url{https://capitalizemytitle.com/} has useful rules for capitalization. Use the {\verb|title|} command to define the title of your work. If your work has a subtitle, define it with the {\verb|subtitle|} command. Do not insert line breaks in your title. If your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The \verb|title| command has a ``short title'' parameter: \begin{verbatim} \title[short title]{full title} \end{verbatim} \section{Authors and Affiliations} Each author must be defined separately for accurate metadata identification. Multiple authors may share one affiliation. Authors' names should not be abbreviated; use full first names wherever possible. Include authors' e-mail addresses whenever possible. Grouping authors' names or e-mail addresses, or providing an ``e-mail alias,'' as shown below, is not acceptable: \begin{verbatim} \author{Brooke Aster, David Mehldau} \email{dave,judy,steve@university.edu} \email{firstname.lastname@phillips.org} \end{verbatim} The \verb|authornote| and \verb|authornotemark| commands allow a note to apply to multiple authors --- for example, if the first two authors of an article contributed equally to the work. If your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last \verb|\author{}| definition: \begin{verbatim} \renewcommand{\shortauthors}{McCartney, et al.} \end{verbatim} Omitting this command will force the use of a concatenated list of all of the authors' names, which may result in overlapping text in the page headers. The article template's documentation, available at \url{https://www.acm.org/publications/proceedings-template}, has a complete explanation of these commands and tips for their effective use. Note that authors' addresses are mandatory for journal articles. \section{Rights Information} Authors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement. Regardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains \LaTeX\ commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document: \begin{itemize} \item the ``ACM Reference Format'' text on the first page. \item the ``rights management'' text on the first page. \item the conference information in the page header(s). \end{itemize} Rights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works. The ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts). \section{CCS Concepts and User-Defined Keywords} Two elements of the ``acmart'' document class provide powerful taxonomic tools for you to help readers find your work in an online search. The ACM Computing Classification System --- \url{https://www.acm.org/publications/class-2012} --- is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the commands to be included in the \LaTeX\ source. User-defined keywords are a comma-separated list of words and phrases of the authors' choosing, providing a more flexible way of describing the research being presented. CCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts). \section{Sectioning Commands} Your work should use standard \LaTeX\ sectioning commands: \verb|section|, \verb|subsection|, \verb|subsubsection|, and \verb|paragraph|. They should be numbered; do not remove the numbering from the commands. Simulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is {\bfseries not allowed.} \section{Tables} The ``\verb|acmart|'' document class includes the ``\verb|booktabs|'' package --- \url{https://ctan.org/pkg/booktabs} --- for preparing high-quality tables. Table captions are placed {\itshape above} the table. Because tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper ``floating'' placement of tables, use the environment \textbf{table} to enclose the table's contents and the table caption. The contents of the table itself must go in the \textbf{tabular} environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on \textbf{tabular} material are found in the \textit{\LaTeX\ User's Guide}. Immediately following this sentence is the point at which Table~\ref{tab:freq} is included in the input file; compare the placement of the table here with the table in the printed output of this document. \begin{table} \caption{Frequency of Special Characters} \label{tab:freq} \begin{tabular}{ccl} \toprule Non-English or Math&Frequency&Comments\\ \midrule \O & 1 in 1,000& For Swedish names\\ $\pi$ & 1 in 5& Common in math\\ \$ & 4 in 5 & Used in business\\ $\Psi^2_1$ & 1 in 40,000& Unexplained usage\\ \bottomrule \end{tabular} \end{table} To set a wider table, which takes up the whole width of the page's live area, use the environment \textbf{table*} to enclose the table's contents and the table caption. As with a single-column table, this wide table will ``float'' to a location deemed more desirable. Immediately following this sentence is the point at which Table~\ref{tab:commands} is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document. \begin{table*} \caption{Some Typical Commands} \label{tab:commands} \begin{tabular}{ccl} \toprule Command &A Number & Comments\\ \midrule \texttt{{\char'134}author} & 100& Author \\ \texttt{{\char'134}table}& 300 & For tables\\ \texttt{{\char'134}table*}& 400& For wider tables\\ \bottomrule \end{tabular} \end{table*} Always use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily. \section{Math Equations} You may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections. \subsection{Inline (In-text) Equations} A formula that appears in the running text is called an inline or in-text formula. It is produced by the \textbf{math} environment, which can be invoked with the usual \texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with the short form \texttt{\$\,\ldots\$}. You can use any of the symbols and structures, from $\alpha$ to $\omega$, available in \LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few examples of in-text equations in context. Notice how this equation: \begin{math} \lim_{n\rightarrow \infty}x=0 \end{math}, set here in in-line math style, looks slightly different when set in display style. (See next section). \subsection{Display Equations} A numbered display equation---one set off by vertical space from the text and centered horizontally---is produced by the \textbf{equation} environment. An unnumbered display equation is produced by the \textbf{displaymath} environment. Again, in either environment, you can use any of the symbols and structures available in \LaTeX\@; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above: \begin{equation} \lim_{n\rightarrow \infty}x=0 \end{equation} Notice how it is formatted somewhat differently in the \textbf{displaymath} environment. Now, we'll enter an unnumbered equation: \begin{displaymath} \sum_{i=0}^{\infty} x + 1 \end{displaymath} and follow it with another numbered equation: \begin{equation} \sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f \end{equation} just to demonstrate \LaTeX's able handling of numbering. \section{Figures} The ``\verb|figure|'' environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below. \begin{figure}[h] \centering \includegraphics[width=\linewidth]{sample-franklin} \caption{1907 Franklin Model D roadster. Photograph by Harris \& Ewing, Inc. [Public domain], via Wikimedia Commons. (\url{https://goo.gl/VLCRBB}).} \Description{A woman and a girl in white dresses sit in an open car.} \end{figure} Your figures should contain a caption which describes the figure to the reader. Figure captions are placed {\itshape below} the figure. Every figure should also have a figure description unless it is purely decorative. These descriptions convey what's in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded. A figure description must be unformatted plain text less than 2000 characters long (including spaces). {\bfseries Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.} For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see \url{https://www.acm.org/publications/taps/describing-figures/}. \subsection{The ``Teaser Figure''} A ``teaser figure'' is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the \verb|\maketitle| command: \begin{verbatim} \begin{teaserfigure} \includegraphics[width=\textwidth]{sampleteaser} \caption{figure caption} \Description{figure description} \end{teaserfigure} \end{verbatim} \section{Citations and Bibliographies} The use of \BibTeX\ for the preparation and formatting of one's references is strongly recommended. Authors' names should be complete --- use full first names (``Donald E. Knuth'') not initials (``D. E. Knuth'') --- and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc. The bibliography is included in your source document with these two commands, placed just before the \verb|\end{document}| command: \begin{verbatim} \bibliographystyle{ACM-Reference-Format}

BkiUez45qhLBjuJj9wWK

\section{Introduction} The two-dimensional (2D) motion of a single particle in a random magnetic field is attracting strong interest since the model is related both to the slave-boson description of high $T_c$ superconductors\cite{slave} and the Chern-Simons theory of the half-filled Landau level.\cite{chern} The existence of recent experiments measuring transport properties in a static random magnetic field adds considerable interest to this subject.\cite{exp} In principle, all states of a 2D system are exponentially localized according to the scaling theory of localization.\cite{localizacion} The presence of a magnetic field that destroys time-reversal symmetry makes a stand against localization. Nevertheless, perturbative renormalization group calculations show that all states remain localized.\cite{sigma,zhang} While theory points towards localization, numerical work has not led to a consensus of opinion.\cite{numerics} Loosely speaking, states {\it look} extended in finite samples as soon as the edge of the band is left and a definite answer is quite difficult. Therefore, the most extended conclusion is that a mobility edge separates localized states from extended or critical states. The model of 2D random fluxes has been previously studied from a statistical point of view. The analysis of several statistical magnitudes (nearest-neighbor spacing distribution, the probability distribution of wavefunction weights, etc.) shows a systematic flow from the Gaussian Unitary Ensemble (GUE) statistics at small system sizes to Poisson statistics at larger sizes.\cite{verges} If GUE and Poisson statistics are the only possibilities in the macroscopic limit, localization over the whole spectrum follows from the referred numerical results. If, on the contrary, more fixed points are possible, then the flow could eventually start near the metallic universality class and stop at certain {\it new} class characterized probably by some fractal properties (for example, some power law increase of the number of sites visited by an eigenstate). Both the theoretical approaches to the metal-insulator transition and the numerical tools used for the study of localization have a long history. \cite{localizacion,efetov,lee,kramer} The standard numerical method has been really successful describing the critical behavior of the 3D metal-insulator transition of Anderson model.\cite{kramer} Nevertheless, the numerical procedure is somewhat artificial. Let me remind that firstly, localization lengths of 1D systems (strips or bars) of increasing transverse size are calculated and, secondly, extrapolation of the data gives the localization length of infinite 2D or 3D systems. This introduces a fictitious symmetry breaking of the 2D or 3D system since a particular direction is preferentially treated. Moreover, being all 1D states exponentially localized, they become eventually extended after doing finite-size scaling analysis of the data. This last part of the calculation introduces some uncertainties in the procedure. In this work, I return to the original idea of {\it measuring} the conductance of finite samples on the computer. \cite{inicios1,inicios2} Today's use of the older scheme has some advantages: mesoscopic physics has led to a more efficient use of Kubo formula\cite{datta}, computers are much more powerful, there is a large experience in the numerical calculation of localization lengths, etc.. Based on the statistical analysis of conductance data obtained for samples of increasing size, the {\it direct} evaluation of the localization length is possible. I find this approach really appealing for two reasons: i) the {\it exact} static conductance is obtained via Kubo formula (neither a small imaginary part should be added to the energy nor other numerical trick should be used to reach the static limit of the conductance of a finite sample), and ii) the scaling behavior of a {\it measurable} magnitude (conductance) is directly analyzed. The paper is organized as follows. The model that is numerically studied in the paper is defined in Section II, some remarks about the use of the Kubo formula within the present context are given in Section III, the size dependence of the conductance is discussed in Section IV, results for the Anderson model and the model with random fluxes are presented in Section V, and finally, Section VI ends the paper with some remarks. \section{Lattice model} The Hamiltonian describing random magnetic fluxes on a $L \times L$ cluster of the square lattice is : \begin{equation} \hat H = \sum_{l=1}^{L^2} \epsilon_l \hat c_{l}^{\dag} \hat c_{l} - ~t {\sum_{<l l'>}} {e^{2 \pi i \phi_{l l'}} \hat c_{l}^{\dag} \hat c_{l'}} ~~, \label{RMF} \end{equation} \noindent where $\hat c_{l}^{\dag}$ creates an electron on site $l$, $l$ and $l'$ are nearest-neighbor sites, $\epsilon_l$ are random diagonal energies chosen with equal probability from the interval $[-W/2,W/2]$, and $- t$ is the hopping energy (hereafter $t=1$ is chosen). The flux through a given loop $S$ on the lattice is $\Phi_S = \sum_{<l,l'> \in S} \phi_{l l'}$ measured in units of flux quantum $\Phi_0 = {{hc} \over e}$. Link fields satisfy $\phi_{l l'}=-\phi_{l' l}$. Either link fields (Meissner phase) or magnetic fluxes (Debye phase) can be randomly selected.\cite{wheatley} Here, the second possibility has been chosen: uncorrelated fluxes are randomly selected with equal probability from the interval $[- {1 \over 2},{1 \over 2}]$ while $W=0$. Notice that we are considering the model with pure non-diagonal disorder that shows maximum magnetic disorder. This model will be called RMF (Random Magnetic Fluxes or Random Magnetic Field) model hereafter. Some additional calculations have been done for a {\it real} version of Hamiltonian (\ref{RMF}). As before, diagonal disorder is absent ($W=0$) but now link field values are restricted to $0$ and $1/2$ that are randomly chosen with equal probability. The model describes a square lattice in which the hopping integral is constant in absolute value but takes random signs. This model will be shortly called RHS (Random Hopping Signs) model. Finally, a pure Anderson model of real diagonal disorder has also been studied in order to compare with results in the literature. In this case, $W \neq 0$ and all link fields vanish. \section{D.c. conductance via Kubo formula} According to Kubo's linear response theory, the static electrical conductivity is given by:\cite{nozieres} \begin{equation} {\mathrm G} \equiv \sigma_{xx}(0) = - ({{e^2} \over h}) 2 {\mathrm Tr} [ (\hbar \hat v_x) {\mathrm Im} {\mathcal G}(E) (\hbar \hat v_x) {\mathrm Im} {\mathcal G}(E) ] ~~~, \label{Kubo} \end{equation} where the velocity (current) operator $\hat v_x$ is related to the position $\hat x$ operator by the equation of motion: \begin{equation} i \hbar \hat v_x = [\hat H , \hat x] \label{vx} \end{equation} and ${\mathrm Im} {\mathcal G}(E)$ is defined in terms of advanced and retarded Green functions as: \begin{equation} {\mathrm Im} {\mathcal G}(E) \equiv {1 \over {2i}} [{\mathcal G}^+(E) - {\mathcal G}^-(E)] ~~~. \label{img} \end{equation} The conductance is calculated for a system consisting of a $L \times L$ cluster of the square lattice connected to two ideal semiinfinite leads of width $L$. The exact form of the electrical field does not matter in a linear response one-electron theory. Therefore, an abrupt potential drop at one of the two cluster sides provides the simplest numerical use of the Kubo formula. In this case, operator $\hat v_x$ has finite matrix elements on only two adjacent layers (see Eq.(\ref{RMF})) and Green functions are just needed for this considerably restricted subset of sites owing to the trace appearing in Eq.(\ref{Kubo}). Since Green functions are given by: \begin{equation} [E - \hat H - \hat \Sigma_{\mathrm r}(E) - \hat \Sigma_{\mathrm l}(E)] {\mathcal G}(E) = I ~~~, \label{green1} \end{equation} where $\hat \Sigma_{\mathrm r}(E)$ and $\hat \Sigma_{\mathrm l}(E)$ are the selfenergies introduced by the semiinfinite right and left leads, respectively\cite{datta}, the evaluation of the desired elements of ${\mathrm Im} {\mathcal G}(E)$ is efficiently achieved applying an $LU$ decomposition to the band matrix \begin{equation} <l|E - \hat H - \hat \Sigma_{\mathrm r}(E) - \hat \Sigma_{\mathrm l}(E)|l'> ~~~. \label{green2} \end{equation} Selfenergies are troubleless calculated for any energy $E$ using a recurrent algorithm. Retarded or advanced Green functions (${\mathcal G}^+(E)$ or ${\mathcal G}^-(E)$) are obtained using the corresponding selfenergies in Eq.(\ref{green1}). Since lead selfenergies add an imaginary part to some diagonal elements of the Hamiltonian, there is no need to add a small imaginary part to the energy $E$ before solving the set of linear algebraic equations. The discussion of a technical point is here in order. It has been proved that Kubo formalism leads to the more familiar Landauer-B\"uttiker expression of the conductance as a sum of transmission coefficients over channels. \cite{equivalencia} Usually, transmission is obtained within a transfer matrix technique in which the selfenergy produced by the lead at the left is iterated through the disordered part of the sample and matched at the right side to the selfenergy coming from the right lead. In this calculation, matrices of dimension $L$ are inverted $L$ times. It could seem that this method is computationally more appropriate because it deals with smaller matrices (the elements of the Green function within the sample are given by the inverse of matrix (\ref{green2}) which has dimension $L^2$). Nevertheless, since matrix (\ref{green2}) is actually a band matrix of bandwidth $2L+1$ and only a few elements of the Green function are necessary for the evaluation of the conductance via Kubo formula, standard numerical procedures ($LU$ decomposition of a band matrix followed by both a forward substitution and a backsubstitution for each needed Green function element) are quite advantageous. Moreover, canned subroutine packages customized for particular computers can be used. Thus, we end with a very simple computer code written in a high level language that can be advantageously run on any given computer. Typically, 127 seconds are needed for the computation of the conductance of a $100 \times 100$ sample on an alpha DECstation 3000. \section{Scaling of the conductance} Conductance is {\it measured} for a statistical ensemble of $L \times L$ clusters with random disorder. This provides a set $\{ {\mathrm G}_i \}$ of $N$ values of the static conductance that allows the evaluation of both the average value of conductance \begin{equation} {\overline {\mathrm G}} = {1 \over N} \sum_{i=1}^N {\mathrm G}_i \label{mean} \end{equation} and its variance \begin{equation} {\mathrm Var} ({\mathrm G}_1 ...{\mathrm G}_N) = {1 \over N-1} \sum_{i=1}^N ({\mathrm G}_i - \overline {\mathrm G})^2. \label{variance} \end{equation} The error bar of $\overline {\mathrm G}$ is given by its standard deviation \begin{equation} \sqrt{<{\overline {\mathrm G}}^2> - {<\overline {\mathrm G}>}^2} \label{errorbar} \end{equation} which is well approximated by \begin{equation} \sigma(\overline {\mathrm G})= \sqrt {{{\mathrm Var} ({\mathrm G}_1 ...{\mathrm G}_N)} \over N} \label{errorbarbis} \end{equation} as soon as $N$ is large enough. Since conductance shows typical universal fluctuations of order 1 (in units of $e^2/h$), the error of the measured average conductance is $\sim N^{-{1 \over 2}}$. This way of characterizing the whole distribution by just two values ($\overline {\mathrm G}$ and its error bar) works fine for magnitudes showing an almost Gaussian statistics. Nevertheless, the distribution of conductances clearly deviates from normal when the regime of exponential localization is being approached. In this regime both theory and numerics suggest that the distribution of the logarithm of the conductance is the appropriate statistical variable showing a normal distribution around its average value. Therefore, the usual practice is the analysis of $\log {\mathrm G}$ whenever the distribution of $\mathrm G$ deviates significantly from normal. I am not going to follow this standard way of analyzing the conductance for several reasons: i) In principle, since $\mathrm G$ is measured, it seems preferable to work directly with it. Moreover, $\mathrm G$ is both bounded from below ($0$) and from above ($L$, the number of channels). While conductance will vanish in the localization regime, its logarithm will diverge. Consequently, the variance of $\mathrm G$ is bounded but the variance of $\log {\mathrm G}$ is not. Computationally, this means that logarithm averages show more {\it noise} in finite sampling. ii) The statistical characterization of the conductance should work from the metallic regime to the localized one since regime can change as a function of the system size. The switch from $\mathrm G$ to $\log({\mathrm G})$ is somewhat ambiguous. iii) The median plays quite good the role of a central value of some statistical magnitude even when the distribution of this magnitude is not Gaussian. For example, I have checked that the conductance median $\mathrm G_{med}$ gives exactly the same exponentially decaying behavior as $\overline {\log({\mathrm G})}$ in the localization regime. Actually, the median will work for any variable change that preserves ordering among data, i.e., if $f({\mathrm G}_i) < f({\mathrm G}_j)$ whenever ${\mathrm G}_i < {\mathrm G}_j$. This property is quite general and makes the use of the median quite robust. iv) Unfortunately, the median has two statistical disadvantages. First, even for a normal distribution of the original variable, its variance for a finite sampling is larger that the corresponding variance of the mean value.\cite{cramer} Second, the estimation of its error bar is not as straightforward as the calculation of the standard deviation of the mean (Eq.\ref{errorbarbis}). Actually, one is forced to divide the database into subsets, obtain several median estimates and calculate their variance. The result is a bad estimation of the uncertainty of the median. v) In conclusion, I will use the conductance median {\it without} error bar as the magnitude giving the central value of the conductance distribution whenever a transition from the metallic regime to the localization regime takes place somewhere within the computation. On the other hand, the use of the mean value and its error bar will be preferred when the conductance distribution remains normal. From here on, either $\overline {\mathrm G}$ or $\mathrm G_{med}$ will be referred as conductance $\mathrm G$ and will be given in units of the conductance quantum $e^2/h$. Localization is an asymptotic property of the conductance as a function of the linear cluster size $L$. In order to correctly describe this behavior using measurements done on finite samples, finite size corrections should be allowed. Notice that the conductance of a 2D square cluster of side $L$ increases linearly in the absence of disorder; in fact, the conductance is an integer value that gives the total number of channels that are open at the Fermi energy (this number is exactly $L$ at the band center). The following scaling models have been used in this work: \begin{equation} {\mathrm G} (L) = (a+bL^{-1}+cL^{-2}+ ...) \exp (-{{2 L} \over \xi}) ~~~~,\\ \label{fit} \end{equation} and \begin{equation} {\mathrm G} (L) = {L \over {a+bL}} \exp (-{{2 L} \over \xi}) . \label{fit_coc} \end{equation} In both cases, the term multiplying the exponentially decaying factor is designed to describe finite size effects. $\xi$ measures the localization length. The first expression is more intuitive and works always fine (with a large enough number of parameters) while the second expression describes better the metallic regime in which the conductance increases linearly with the system size $L$. All the results presented in the next Section allow a good fit with just three parameters ($a, b, \xi$). The localization length does not depend sensibly on the used model. Finally, let me comment about the reliability of the localization length obtained by the finite scale modeling. Generally speaking, parameters $a, b, c, ...$ are determined by conductance values of small cluster sizes while the localization length $\xi$ is determined by the behavior of conductance at larger system sizes. Since conductance starts decreasing after a region of initial increase, the estimation of the localization lengths is reliable only if the crossover to the region in which conductance decreases can be reached by numerical simulation. According to the data presented in this paper, this happens for localization lengths that are well below ${10}^4$ (lengths are everywhere given in units of $a$, the lattice constant). Notice that the existence of localization at large $L$ is built in the model (see Eqs.(\ref{fit}--\ref{fit_coc})): a decaying conductance does not necessarily mean an exponentially decaying law. In other words, numerical simulation does not prove localization but gives a measure of its characteristic length $\xi$. Nevertheless, after trying a lot of alternative (and worser, i.e., with larger $\chi^2$ deviations from data) models for the description of the conductance dependence on system size, one gets more or less convinced that exponential decay at large $L$ values is quite plausible.\cite{elihu} \section{Results} Although my primary interest is the determination of localization lengths for Hamiltonians describing random magnetic fluxes, the paradigmatic Anderson model of diagonal disorder has also been studied in order to check the numerical tools used in this work. Conductance values are everywhere given in units of $e^2/h$. \subsection{Anderson model} Hamiltonian (\ref{RMF}) without link fields ($\phi_{l l'} = 0$ for all $l, l'$ pairs) describes the Anderson model of diagonal disorder. The width $W$ of the diagonal energies distribution measures the degree of disorder. Exponential localization of all wavefunctions is the accepted state of the art for this model. Nevertheless, measuring the localization length when disorder is weak ($W < 2$) is numerically very difficult. Wavefunctions are clearly localized in the band tails but {\it look} extended well within the band. This fact has originated more than one claim pointing to the existence of a metal-insulator transition at some critical energy (mobility edge) within the band. Following tradition, I have collected conductance data at the band center ($E=0$). It is assumed that states at finite energies are more localized than the band center one. Fig. 1 shows the behavior of both the mean and the median of the conductances samples as a function of the side $L$ of the $L \times L$ disordered cluster in which the current is measured. Since disorder is very large ($W=10$), both measures of a central value show a clear exponential decay. Nevertheless, there are also important differences: localization lengths do not coincide and mean values show more dispersion, i.e., they do not follow a straight line as good as the median does. Looking closer to the data, the reason for the disagreement is obvious: conductance is very small ($G \ll 1$), its distribution is not normal and, consequently, the mean is dominated by outliers conductances of a relatively large value. On the other hand, the logarithm of the conductance seems to behave statistically good. As said in Section IV, the median of $\mathrm G$ gives a good estimation of the central value of its statistical distribution if any monotonic function of $\mathrm G$ behaves normally. Although the statistical distribution of the conductance becomes almost Gaussian for smaller degrees of disorder, I have systematically used the median in the rest of this Section in order to be completely self-consistent. Figures 2 and 3 show the evolution of the scaling behavior of the conductance as disorder diminishes. While finite size effects are still unimportant for $W=5$, a region in which conductance increases linearly with the system size has clearly developed for $W=2$. This region extends more and more as $W$ approaches zero. Actually, the inference of a localization length beyond $W=2$ is not possible analyzing conductance data for sample sizes below $L=100$. As commented before, only systems having a localization length below $\xi \sim {10}^4$ show numerical indications of localization for sample sizes of the order of $L = 100$. In other words, when $\xi \gg L$ one just obtains the linear increase of conductance due to the linear increase of transport channels. Fig. 4 summarizes the main results of this subsection: localization length is plotted as a function of disorder. It seems clear that $\xi$ diverges for vanishing disorder. In fact, a $\chi^2$ fit by power law gives an exponent very close to $-4$ and even the simpler fit by $1+a W^{-4}$ works pretty good (see Fig. 4). Localization length values are also compared with results obtained by finite size scaling based on data collected for strips of increasing width.\cite{anderson1,anderson2,anderson3} Surprisingly, both methods do not give identical results. Present values of $\xi$ are somewhat larger for large disorder but their evolution for smaller disorder values is not so steep. For small disorder values, the scaling of the conductance of square clusters attached to semiinfinite wires yields considerably smaller localization lengths. I have no explanation for this discrepancy. It could be that numerical extrapolation of $\xi$ from finite strips has a larger uncertainty or even that the whole procedure is dominated by the more localized states. Let me repeat here that $\mathrm G$ is a statistical variable and that the scaling behavior of a {\it single} magnitude used to characterize the whole distribution of values can certainly depend on the choice (see Fig. 1 for an example of an extreme discrepancy). \subsection{Random Magnetic Flux} This part presents the main results of the paper. Let me begin with a detailed comparison between statistics based on the mean of the conductance distribution and statistics based on its median. Fig. 5 compares in a logarithmic scale the behavior of both magnitudes at an energy $E=-3.3$ close to the band limit. The values shown for $L=100$ are not so precise as the rest because a sensible percentage of the samples have conductances that are smaller than numerical precision. Apart from minor size effects at small cluster sizes, both magnitudes show exponentially decaying behavior. While localization does not offer doubts, the exact value of localization length depends on the magnitude used to characterize the central value of the distribution of conductances. $\xi$ increases from $20.9$ for the median to $60.6$ for the mean. As said previously, the first value is the correct one since the distribution of the conductance logarithm is indeed normal and gives the same median. Fig. 6 shows that the problem becomes softer as soon as energies well within the band are studied. Although conductance is still below one quantum unit, its distribution comes closer to a Gaussian distribution and therefore, mean and median do not differ as much as in the previous example. Let us analyze now the trend followed by the size dependence of conductance when energy is varied. While localization is clear near the band edges, it is quite difficult to get some insight into the scaling behavior of conductance at energies $|E| < 3$. Fig. 7 shows my larger data collection corresponding to an arbitrary energy $E=0.1751$ close to the band center. 5500 {\it measurements} of the conductance have been necessary to reduce the error bar of the mean to the small values shown in the Figure for $L < 70$. At larger sizes, a still very large number of $500$ conductances of randomly generated samples have been calculated. Error bars are considerably larger but not so much to preclude a precise analysis. In my opinion, the decrease of the conductance for increasing system sizes does not offer any doubt. Moreover, a special calculation at $L=200$ has been carried out using a different algorithm that allows the study of larger systems paying the price of longer computer times.\cite{gauss} The decrease of the conductance below its value at $L \sim 100$ is quite clear: $\mathrm G(200)=1.330 \pm 0.014 < \mathrm G(100)=1.381 \pm 0.005$. The good quality of the database for this energy gives the opportunity of doing a reliable fit by some chosen law. Using Eq.(\ref{fit}) with four parameters ($a,b,c,\xi$), a localization length of $\xi = 5152.8$ is obtained. Fig. 7 shows the fitted model as a thick continuous gray curve. The relatively large value of the localization length is just below the threshold of {\it visibility} ---given the present computer facilities. Let me insist once again, that the data shown in Fig. 7 do not prove exponential decay of conductance although they are nicely described by an exponential law including finite size corrections. On the other hand, it is obvious that any law with properly selected parameters is able to reproduce the roughly linear decrease of conductance for $L>50$. Nevertheless, reproducing the whole scaling behavior is not so trivial. For example, if all conductance values except the one corresponding to $L=200$ are fitted by Eq.(\ref{fit}) and by a rational function model like: \begin{equation} {\mathrm G} (L) = {{aL+bL^2} \over {1+cL+dL^2+eL^3}} ~~~, \label{fit_fraccion} \end{equation} the fit fails to reproduce the last point if the rational function is used ---both adjusting ($a,c,d$) (thin continuous line of Fig. 7) and adjusting ($a,b,c,d,e$) (thick dashed line of Fig. 7)--- whereas it remains almost the same when the exponentially decaying model is employed (thick continuous gray line of the same figure). Actually, a large database has been also collected for the conductance distribution at the band center. Mean conductance steadily increases up to the largest calculated size ($L=200$). A very large localization length $\xi = 16707.5$ is obtained using Eq.(\ref{fit}) with four parameters, although numerical uncertainty is as large as its value. The steady increase of the conductance at the band center of this model was firstly obtained in the seminal work of Lee and Fisher. \cite{inicios2} Although they did, in principle, the same kind of computation, numerical values do not coincide for some unknown reason (the inset of Fig. 2 of the referred paper shows conductances that are about seven times smaller than the ones obtained in this work). In any case, either $\xi$ is infinite just at the band center or larger than any other calculated localization length. Let me point, that the mean conductance has been systematically used to get the localization length of this model for several energies within the band. Numerically, the advantage of the mean relative to the median is its better normal distribution for finite samples.\cite{cramer} Its use is justified as soon as conductance distribution behaves normally (Gaussian distribution). Fig. 8 shows that this is indeed the case at $E=0.1751$ both for $L=64$ and for $L=200$. The small decrease of the mean conductance is not distinguishable in the distribution. This figure also shows the typical width of about one quantum unit that has been observed for the conductance fluctuations in numerous occasions (the standard deviation of the distribution obtained for $L=200$ equals $0.314$). Fig. 9 collects the localization length obtained at several energies. The almost exponential increase of $\xi$ near the band edges perfectly reproduces the findings of previous studies. \cite{numerics} On the other hand, the localization length seems to reach a saturation value below $10^4$ in the larger part of the band being the band center the only possible exception. Two comments are opportune at this point. First, a continuous logarithmic increase of the localization length up to the band center implies a more than macroscopic value of $\xi$ for almost all states in the band. If localization is much larger than the sample size, the own concept of localization looses its meaning. In my opinion, the scenario shown in Fig. 9 is much more attractive: localization length is very large but not {\it infinite}, that is, irrelevant. Second, although renormalization group theories also predict $\xi \sim l \exp (A l^2)$ where $l$ is the mean free path and $A$ some finite constant, my numerical results do not necessarily contradict this behavior because the RMF model gives values of $l$ of order of $1$ when non-diagonal disorder is maximum. \cite{mfp} \subsection{Random Hopping Sign} The last case studied in the paper corresponds to a model of real hopping with a random sign. The Hamiltonian is given by (\ref{RMF}) when link fields are limited to values $0$ and $1 \over 2$. The band extends from $\sim -2 \sqrt 3$ to $\sim 2 \sqrt 3$ in this case. Being the amount of non-diagonal disorder fixed, conductance can be studied as a function of energy. Since this model belongs to a universality class that is different from the previous one (orthogonal instead of unitary), some differences in the scaling behavior of conductance could, in principle, be expected. Nevertheless, numerical results show only minor changes. Scaling of conductance is shown in Fig. 10 for two different energies: $E=0$ (band center) and an arbitrary energy $E=0.1751$ close to it. While localization length is almost infinite ($\xi = 23317$) in the first case, it is not very large in the second one ($\xi = 461.7$). As in the complex model, an infinite value of $\xi$ is compatible with conductance data at the band center but not with the clear decrease shown by the conductance at the second energy. In both cases, the mean and its error bar have been used because the distribution of conductance values follows a Gaussian law in spite of its small average value (Mean conductance is of the order of $1$ quantum unit when the conductance of an ordered sample is as large as $100$ at the band center of a cluster of side $L=100$). Conductance scaling at other energies follows a trend that is quite similar to the one followed by the previously analyzed RMF model. It can be concluded that the universality class of the model does not matter noticeable in the scaling behavior of conductance although numerical values of the localization length are quite different ($\xi=461.7$ ($5152.8$) for RHS (RMF) model at $E=0.1751$). \section{Discussion} Conductance sampling as a function of the system size has been methodically used to infer the localization lengths of a variety of disorder models. Results for the Anderson model of diagonal disorder are globally consistent with the ones obtained using well known approaches. Nevertheless, numerical values of $\xi$ are somewhat surprising: although localization length diverges for vanishing amount of disorder, the divergence is weaker than the one provided by other methods. The origin of the discrepancy is not known although it may be due to the non-normal distribution of conductances when the localization regime is approached. Both the real ---belonging to the Gaussian Orthogonal Ensemble universality class--- and the complex (GUE universality class) models of non-diagonal disorder studied in this work show decaying values of the central conductance as a function of cluster side at all studied energies. When fitted with an exponential law corrected for finite-size effects, values of the localization lengths are obtained. Although large they are typically below $10^4$, a value that is the maximum localization length giving a conductance that diminishes within the size values presently accessible. The behavior of $\xi$ as a function of energy is remarkable. After a region near the band edges in which $\xi$ increases exponentially, a region of almost constant localization length follows. Nevertheless, isolated energies giving diverging values of $\xi$ cannot be discarded by this procedure. Actually, the band center is a candidate for showing a diverging localization length. In my opinion, this work should motivate experimental research in the mesoscopic transport field directed to the determination of the localization length of disordered samples. As shown by the large amount of data presented in this work, the decay of the conductance starts at system sizes well below the localization length. This fact makes the experimental observation of localization relatively easy. Only after having experimentally related values of disorder (or what is equivalent, values of the mean free path) with values of the localization length, could localization theory make a major step from conjecture to fact. On the other hand, the correct microscopic description of the conductance scaling at a finite temperature would greatly expedite a detailed comparison between experiment and numerics. \acknowledgments I acknowledge Luis Mart\'{\i}n-Moreno who helped me checking the computer code used in this work running his own program based on the transfer matrix technique for the same disordered system. I also acknowledge interesting conversations with Luis Brey. This work has been partially supported by Spanish Comisi\'on Interministerial de Ciencia y Tecnolog\'{\i}a (grant MAT94-0058-C02).

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\section{Introduction}\label{sec-1} Numerical simulations of high-speed turbulent flows involving discontinuities such as shockwaves and material interfaces represents a significant computational challenge. High-order central schemes, which are non-dissipative, can resolve smooth and small-scale features but often introduce high-frequency oscillations (Gibbs phenomenon) near discontinuities. In contrast, the nonlinear limiting typically used for capturing discontinuities \textcolor{black}{is} too dissipative in relatively smooth turbulent regions. This poses a significant numerical challenge \textcolor{black}{in designing} a scheme that can effectively treat these contradictory requirements. Considerable efforts have been devoted to the development of numerical schemes that are essentially non-oscillatory near the discontinuities and low-dissipative in the smooth regions. One of the first attempts to develop such methods is the piecewise parabolic method (PPM), proposed by Woodward and Collela \cite{woodward1984numerical}, which uses a four-point centred polynomial to approximate the interface values. Limiters then correct the interface values to achieve non-oscillatory results. A well-known drawback of these slope limiters is that they tend to clip the smooth extrema of the flow and the accuracy degenerates to first-order. Collela and Sekora have proposed an improvement of PPM that can preserve the smooth extrema in Ref. \cite{colella2008limiter}. Suresh and Hyunh \cite{suresh1997accurate} argued that the three-point stencil typically used in total variation diminishing (TVD) methods is not able to distinguish between local extrema and a genuine discontinuity. They have proposed a monotonicity-preserving (MP) scheme that uses a five (or more) point stencil to make this distinction. In contrast, Harten et al. \cite{harten1987uniformly} have proposed an essentially non-oscillatory (ENO) approach that will adaptively choose the appropriate stencil containing the smoothest data, thereby avoiding interpolations across discontinuities. \textcolor{black}{ENO schemes require so many logical conditions for their stencil choosing procedure which make them ill-suited for faster computations.} The weighted essentially non-oscillatory (WENO) schemes, originated by Liu et al. \cite{liu1994weighted} overcame the problems associated with the ENO scheme and were further significantly improved by Jiang and Shu \cite{jiang1995}. WENO schemes achieve high-order accuracy by analyzing all possible stencils through a convex combination enabling the scheme to choose the highest order interpolation for smooth flows and one-sided interpolations in the vicinity of discontinuities. Despite their success, WENO schemes are overly dissipative with regard to resolving small scale solution features. Various modifications of WENO schemes have been proposed over the years to improve their accuracy and reduce their dissipative nature. Henrick et al. \cite{Henrick2005} have noticed that the WENO scheme proposed by Jiang and Shu \cite{jiang1995} losses its accuracy at critical points and proposed the WENO-M scheme as a fix. Borges et al. \cite{Borges2008} proposed another alternative approach for improving the accuracy at these critical points by introducing a global smoothness indicator for the stencils. Pirozzoli \cite{Pirozzoli2002} has proposed a hybrid compact-WENO scheme which combines the conservative compact scheme and the WENO scheme. Compact schemes are implicit in space and have superior dissipation and dispersive properties over explicit schemes of the same order, which is beneficial in smooth regions \cite{lele1992compact}. Ren et al. \cite{ren2003characteristic} have improved the hybrid scheme of Pirozzoli through a continuous weighing function which assisted in a smooth transition from one sub-scheme to the other. Kim and Kwon \cite{kim2005high} proposed a hybrid central-WENO approach through the weighing function of Ren et al.. Their scheme combined a central scheme and the numerical dissipation of the upwind WENO scheme controlled by a weighing function. Hu et al. \cite{Hu2010} proposed a sixth-order adaptive central upwind WENO scheme called WENO-CU6. The scheme adapts between central and upwind schemes smoothly by blending the smoothness indicators of the optimal high-order central stencil and the lower order upwind stencils thereby reducing the inherent numerical dissipation associated with the upwind schemes in the smooth regions. Fu et al. \cite{fu2018new} have developed a new family of schemes called Targeted ENO (TENO) schemes which further improved the capabilities of WENO-CU6, while maintaining the accuracy at first and second-order critical points along with low numerical dissipation. {\color{black} Ghosh and Baeder \cite{ghosh2012compact} developed a class of upwind biased compact-reconstruction finite difference WENO schemes called CRWENO, which combined the compact upwind schemes and WENO schemes. CRWENO method is purely compact, but the scheme is upwind biased and excessively damps the fine-scale structures of turbulence and is computationally expensive as it involves inverting a block-tridiagonal matrix. Weighted Compact Nonlinear Schemes (WCNS) developed by Deng et al. \cite{Deng2000} \textcolor{black}{have} similar discontinuity capturing abilities as that of WENO but are much more flexible than WENO schemes as one can interpolate not only fluxes \cite{Zhang2008}, but also conservative variables \cite{nonomura2012}, primitive or characteristic variables \cite{Wong2017} and still maintain high-order accuracy. Recently, Liu et al. \cite{Liu2015} developed a new class of nonlinear compact schemes with a weighted hybrid interpolation of an upwind and a central interpolation. It was shown that the proposed WCNS with hybrid weighted interpolation displays a more localized dissipation than the classical WENO schemes. Finally, Subramaniam et al. \cite{subramaniam2019high} have proposed an explicit-compact interpolation along with compact finite differences that provides higher resolution and more localized dissipation compared to that of any WCNS methods in the current literature. Despite the apparent advantages of the WENO schemes, the smoothness indicators used for discontinuity detection are expensive to compute, and high-order WENO schemes are not robust enough for effective suppression of numerical oscillations in the presence of discontinuities. Another relevant study by Sun et al. \cite{sun2016boundary} proposed a novel approach for constructing high-fidelity shock-capturing schemes with small numerical dissipation called Boundary Variation Diminishing (BVD) algorithm in a general finite-volume framework. The BVD approach adaptively chooses appropriate reconstruction polynomials from a given set of polynomials to minimize the jumps at the cell interfaces, effectively reducing numerical dissipation in the Riemann solvers. Sun et al. \cite{sun2016boundary} designed a BVD algorithm by hybridizing WENO and THINC (Tangent of Hyperbola for INterface Capturing) schemes, and the resulting scheme maintained the accuracy of WENO in smooth regions and substantially improved solution quality near discontinuities via THINC. Subsequently, various BVD algorithms have been proposed in the literature. Deng et al. \cite{deng2018high} have extended the approach to multi-component flows by hybridizing TVD and THINC schemes leading to significant scheme improvements. In a series of papers, Deng et al. \cite{deng2020implicit,deng2019fifth,deng2020constructing} have also proposed a new BVD algorithm that combines a high-order unlimited linear reconstruction polynomial and THINC schemes. The resulting schemes called $P_{n}T_{m}$, where $n$ is the degree of the unlimited polynomial and $m$ is the number of stages in the THINC scheme. They showed that the BVD principle retrieves the underlying linear scheme for smooth regions while still effectively capturing discontinuities. The $P_{n}T_{m}$ schemes still use upwind biased interpolation, which is a source of numerical dissipation. To address this issue, we present a new algorithm, named HOCUS (High-Order Central Upwind Scheme), which combines the MP scheme and a linear-compact scheme using the BVD principle. Unlike the earlier studies where one of the reconstruction candidates is a non-polynomial function, such as THINC, we consider two fifth-order polynomials. The proposed method has the following advantages: \begin{description} \item[(a)] Unlike the $P_{n}T_{m}$ schemes, which use multiple stages of evaluation of the BVD algorithm, the current approach requires a \textit{single-stage evaluation} \item[(b)] The unlimited linear scheme is evaluated through \textit{compact reconstruction} which has superior dissipation and dispersive properties compared to explicit schemes (see \cite{ghosh2012compact, subramaniam2019high, Pirozzoli2002,lele1992compact, adams1996high} \item[(c)] Finally, since the underlying linear scheme is central, which is non-dissipative, the \textit{inherent numerical dissipation in Riemann solver is reduced in the smooth regions of the flow}. \end{description} The rest of the paper is organized as follows. The new HOCUS with BVD algorithm along with the various reconstruction procedures is presented in Section \ref{sec-2}. Details of time advancement are given in Section \ref{sec3.1}. \textcolor{black}{Several one- and two-dimensional test cases} for linear advection and Euler equations are presented in Section \ref{sec-3}. The numerical experiments clearly demonstrate the new numerical scheme can provide high-order oscillation-free results. Finally, Section \ref{sec-4} summarizes our findings. \section{Numerical method: Spatial discretization}\label{sec-2} Here, we present upwind flux reconstruction using the BVD algorithm in the context of the finite-volume method for conservation laws. For simplicity, we first consider a scalar hyperbolic conservation law represented by the following partial differential equation in one-dimension: \begin{equation} \frac{{\partial U}}{\partial t} + \frac{{\partial F(U)}}{\partial x} = 0 \label{eq:scalar_conservation}, \end{equation} where, $U(x,t)$ is the solution function and $F(U)$ is the physical flux function. \subsection{Finite-volume method} Equation (\ref{eq:scalar_conservation}) is discretized on a uniform grid with $N$ cells on a spatial domain spanning $x \in \left[x_a, x_b \right]$. The cell center locations are at $x_j = x_a + (j - 1/2) \Delta x$, $\forall j \in \{1, \: 2, \: \dots, \: N\}$, where $\Delta x = (x_b - x_a)/N$. The cell interfaces, indexed by half integer values, are at $x_{j+\frac{1}{2}}$, $\forall j \in \{ 0, \: 1, \: 2, \: \dots, \: N \}$. Let $I_j = [x_{j - 1 / 2}, x_{j + 1 / 2}]$ be a control volume (a computational cell) of width $\Delta x_j = x_{j + 1 / 2} - x_{j - 1 / 2}$. Integrating Equation (\ref{eq:scalar_conservation}) over $I_j$, we obtain the following semi-discrete relation, expressed as an ordinary differential equation: \begin{equation}\label{eqn-differencing} \frac{\mathrm{d}}{\mathrm{d} t} \hat{U}_j(t) = - \frac{1}{\Delta x_j} [\hat{F}_{j+ 1 / 2} - \hat{F}_{j - 1 / 2}], \end{equation} where $\hat{U}_j(t)$ is the cell average of the solution in cell $I_j$ at time $t$ and $\hat{F}_{j + 1 / 2}$ is the numerical flux over cell interface, respectively: \begin{equation} \hat{U}_j(t) = \frac{1}{\Delta x} \int_{x_{j - 1 / 2}}^{x_{j + 1 / 2}} U(x,t) \mathrm{d} x \end{equation} \begin{equation}\label{Numerical Flux} \hat{F}_{j+\frac{1}{2}}={F}^{\rm Riemann}_{j+\frac{1}{2}}(U_{j+\frac{1}{2}}^{L},U_{j+\frac{1}{2}}^{R}), \end{equation} where $L$ and $R$ are adjacent values of a cell interface as show in Fig. \ref{fig:interface}. The numerical fluxes at the cell boundaries can be computed by a variety of Riemann solvers which can be written in a canonical form as \begin{equation} {F}^{\rm Riemann}_{j+\frac{1}{2}} = \frac{1}{2}\underbrace{({F^L_{j+\frac{1}{2}}} + {F^R}_{j+\frac{1}{2}})}_{\text{Physical central flux}} - \frac{1}{2} | {A_{j+\frac{1}{2}}}|\underbrace{({U^R_{j+\frac{1}{2}}}-{U^L_{j+\frac{1}{2}}})}_{\text{Numerical dissipation}}, \label{eqn:Riemann} \end{equation} \begin{figure}[H] \centering \includegraphics[width=0.6\textwidth]{Cell.pdf} \caption{Illustration of cell $I_j$ and location of the cell interfaces L, R at $x_{j - 1 / 2}$ and $x_{j + 1 / 2}$.} \label{fig:interface} \end{figure} where $|{A_{j+\frac{1}{2}}}|$ denotes the characteristic signal velocity evaluated at the cell interface, and the Equation (\ref{eqn:Riemann}) clearly indicates the physical central flux and the numerical dissipation terms. The procedure of obtaining the values at the interface, $U_{j + 1 / 2}$, from cell center variables, $\hat{U}_j$, is called reconstruction and then the numerical fluxes at cell interfaces are computed by the Riemann solver. It is obvious from the Equation (\ref{eqn:Riemann}) that our objective is to obtain the values at the left and right interfaces, $U_{i+\frac{1}{2}}^{L}$ and $U_{i+\frac{1}{2}}^{R}$, at cell interfaces $x_{j+\frac{1}{2}}$, $\forall j \in \{ 0, \: 1, \: 2, \: \dots, \: N \}$. We compute these interface values by using two different reconstruction procedures, candidate polynomials, and the BVD algorithm will choose the appropriate value such that the numerical dissipation is minimized. In the following subsections, we provide the details of the calculations of candidate polynomials used for the BVD algorithm. \subsubsection{Linear upwind compact reconstruction} The first candidate considered for the evaluation of the interface states is the linear upwind compact reconstruction scheme. \textcolor{black}{The advantage of the compact schemes is that they yield better accuracy and resolution than the non-compact or explicit schemes despite having a smaller stencil}. These advantages of compact schemes are well known and have been discussed in detail in Ref. \cite{ lele1992compact,adams1996high,Pirozzoli2002,ghosh2012compact,subramaniam2019high}. In the present paper, we consider the fifth-order compact reconstruction, denoted as C5, to compute the interface values at $j+1/2$. The left-biased states are obtained using cell-averaged values, $\hat U$, of the 3-point stencil $(j - 1, j, j+1)$, and right biased states uses $(j, j+1, j+2)$ respectively and the corresponding equations are shown below, \begin{subequations} \begin{alignat}{1} \frac{1}{2} U^{L, C5}_{j-\frac{1}{2}}+ U^{L, C5}_{j+\frac{1}{2}} + \frac{1}{6} U^{L, C5}_{j+\frac{3}{2}}&= \frac{1}{18}\hat U_{j-1}+ \frac{19}{18}\hat U_{j} + \frac{5}{9}\hat U_{j+1} \label{eq:left}\\ \frac{1}{6} U^{R, C5}_{j-\frac{1}{2}}+ U^{R, C5}_{j+\frac{1}{2}} + \frac{1}{2} U^{R, C5}_{j+\frac{3}{2}}&= \frac{5}{9}\hat U_{j}+ \frac{19}{18}\hat U_{j+1} + \frac{1}{18}\hat U_{j+2} \label{eq:CR} \end{alignat} \label{eqn:upwind-compact} \end{subequations} The solution of the above equations requires an inversion of a tridiagonal system of equations due to their inherent, implicit nature. However, the computational complexity of a tridiagonal solution scales linearly with the number of grid points and the inversion can be efficiently carried out by Thomas algorithm. For the implementation of the boundary fluxes, i.e., the first and last interfaces along each grid line, we use the MP5 scheme presented in the next subsection. The resulting tridiagonal system of equations can be represented as \begin{equation} \begin{bmatrix} 1 & 0 & 0 & \cdots & &0 \\ \\ a & b & c & & & & \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ & & & a & b & c\\ \\ 0 & &\dots &0 &0 & 1 \end{bmatrix} \begin{bmatrix} U^{C5}_{\frac{1}{2}} \\ \vdots \\ U^{C5}_{j+\frac{1}{2}} \\ \vdots \\ U^{C5}_{N+\frac{1}{2}} \end{bmatrix} = \begin{bmatrix} d^{MP5}_ {\frac{1}{2}}\\ \vdots \\ d^{C5}_ {j+\frac{1}{2}}\\ \vdots \\ d^{MP5}_ {N+\frac{1}{2}} \end{bmatrix}, \end{equation} where, $N$ is the number of cells, $a$, $b$, and $c$ are the coefficients of the left-hand side and $d$ represents the right-hand side of the system of the Equations (\ref{eqn:upwind-compact}). We use ghost points for the data required by the MP5 scheme at the boundaries. We use the same approach for both periodic and non-periodic test cases, and the numerical tests do not indicate any loss of accuracy or restrictions due to this approach. This scheme is intended to be applied in the smooth regions of the flow field as it is a linear scheme and thereby produces oscillations near discontinuities, and to achieve a shock-capturing capability, and it will be coupled with a shock-capturing scheme through BVD algorithm. In the next subsection, the candidate polynomials used for shock-capturing is explained. \subsubsection{Monotonicity-Preserving scheme} The second candidate considered for the evaluation of the solution vector at the interface is the Monotonicity-Preserving fifth-order (MP5) reconstruction of Suresh and Hyunh \cite{suresh1997accurate}, which uses geometric based approach for shock-capturing. The key advantage of MP5 reconstruction is that it not only captures discontinuities but also preserves the extrema and maintains high-order accuracy in smooth regions. MP5 reconstruction is carried out in two steps. For brevity, we only explain the procedure for the left interface values, $U^{L,MP5}_{j+1/2}$, since the right interface values, $U_{j+1/2}^{R,\rm{MP5}}$ can be obtained via symmetry. In the first step, a fifth-order polynomial is constructed using cells $j-2, j-1, j, j+1,j+2$, to interpolate cell-average values $\hat U_j$ needed to obtain the cell interface values $U^{L}_{j+1/2}$: \begin{equation}\label{eqn:linear5} U^{L,P5}_{j+1/2}=(2\hat U_{j-2} - 13\hat U_{j-1} + 47\hat U_j + 27\hat U_{j+1} - 3\hat U_{j+2})/60 \end{equation} In the second step, interpolated values $U^{L,P5}_{j+1/2}$ are limited based on the following condition: \begin{equation} \label{eq:mp5-cond} (U^{L,P5}_{j+1/2}- \hat U_i)(U^{L,P5}_{j+1/2}-U^{MP}) \leq \epsilon, \\ \end{equation} \begin{equation} \label{eqn:alpha} \begin{aligned} \text{where,} &\ U^{MP} = \hat U_j + \operatorname{minmod}(\hat U_{j+1}-\hat U_{i}, \tilde{\alpha}\,(\hat U_j - \hat U_{j-1}))\,\,,\\ \text{and,} &\operatorname{minmod}(a,b) = \frac{1}{2} \left(\operatorname{sign}(a)+\operatorname{sign}(b)\right)\min(|a|,|b|)\,\,. \end{aligned} \end{equation} The parameter ${\alpha}$ is a constant which will be explained further in Section \ref{sec2.2} - Remark \ref{remark-5} and $\epsilon$ is a small constant set as $\epsilon=10^{-20}$. If the condition given in Equation (\ref{eq:mp5-cond}) is violated then the following algorithm is used. First, we compute the second derivatives: \begin{eqnarray} D_j^{-} &=& \hat U_{j-2} - 2\hat U_{j-1} + \hat U_j\,\,, \\ D_j^{0} &=& \hat U_{j-1} - 2\hat U_{j} + \hat U_{j+1}\,\,, \\ D_j^{+} &=& \hat U_{j} - 2\hat U_{j+1} + \hat U_{j+2}\,\,. \end{eqnarray} Next, we compute: \begin{eqnarray} D^{M4}_{j+1/2} &=& \operatorname{minmod4}(4D_j^0-D_j^+, 4D_j^+-D_j^0, D_j^0, D_j^+)\,\,, \\ D^{M4}_{j-1/2} &=& \operatorname{minmod4}(4D_j^0-D_j^-, 4D_j^--D_j^0, D_j^0, D_j^-)\,\,, \end{eqnarray} where the function $\operatorname{minmod4}$ is given by: \begin{eqnarray} \operatorname{minmod4}(w,a,b,c) &=& 0.125(\operatorname{sign}(w)+\operatorname{sign}(a)) \times \\\nonumber &&|(\operatorname{sign}(w)+\operatorname{sign}(b))(\operatorname{sign(w)}+\operatorname{sign}(c))| \times \\\nonumber &&\min(|w|,|a|,|b|,|c|))\,\,. \end{eqnarray} We then compute \begin{eqnarray} U^{UL} &=& \hat U_j + \alpha(\hat U_j- \hat U_{j-1})\,\,, \\ U^{AV} &=& 0.5(\hat U_j+ \hat U_{j+1})\,\,, \\ U^{MD} &=& U^{AV} - 0.5 D^{M4}_{j+1/2}\,\,,\\ U^{LC} &=& \hat U_j + 0.5(\hat U_j-\hat U_{j-1}) + \frac{4}{3}D^{M4}_{j-1/2}\,\,. \end{eqnarray} Using these expressions, we compute: \begin{eqnarray} U_{\min} = \max(\min(\hat U_j, \hat U_{j+1}, U^{MD}), \min(\hat U_j, U^{UL}, U^{LC}))\,\,,\\ U_{\max} = \min(\max(\hat U_j,\hat U_{j+1}, U^{MD}), \max(\hat U_j, U^{UL}, U^{LC}))\,\,. \end{eqnarray} Finally, a new limited value for the cell interface $U^L_{j+1/2}$ is obtained via \begin{equation}\label{eqn:sc} U_{j+1/2}^{L,\rm{MP5}} = U_{j+1/2}^L + \operatorname{minmod}(U_{\min}-U_{j+1/2}^L, U_{\max}-U_{j+1/2}^L)\,\,. \end{equation} Recently, Zhao et al. \cite{zhao2019general} have evaluated various shock-capturing schemes and found that MP5 scheme is an excellent choice for wave propagation and is the most efficient of all the evaluated fifth-order accurate schemes. They also suggested combining WENO schemes with MP limiters, as in Balsara and Shu \cite{balsara2000monotonicity}, as an effective strategy for shock capturing since the monotonicity preserving WENO schemes gave the lowest overall numerical error. \subsection{Central-upwind scheme with BVD algorithm}\label{sec2.2} In this section, we describe the central-upwind scheme using BVD algorithm. Similar to earlier studies using the BVD algorithm, we consider two-different reconstruction polynomials C5 or C6 (details of C6 scheme are explained below) and MP5 schemes as the possible candidates for the evaluation of the interface states. The BVD algorithm selects the reconstruction polynomial with minimal numerical dissipation from the two candidate reconstructions in a given cell by evaluating the Total Boundary Variation (TBV) at a cell interface, shown in Fig. \ref{fig:BVD_scheme}, given by the following equation: \begin{equation}\label{Eq:TBV} TBV_{j}=\big|U_{j-\frac{1}{2}}^{L}-U_{j-\frac{1}{2}}^{R}\big|+\big|U_{j+\frac{1}{2}}^{L}-U_{j+\frac{1}{2}}^{R} \big| \end{equation} \begin{figure}[H] \centering \includegraphics[width=1.0\textwidth]{BVD-figure.pdf} \caption{Boundary variations at the cell interfaces.} \label{fig:BVD_scheme} \end{figure} The terms on the right-hand side of the Equation (\ref{Eq:TBV}) represent the amount of numerical dissipation introduced in the numerical flux in Equation (\ref{eqn:Riemann}) at each interface of a given cell. When two reconstruction functions of the same data are available at an interface, the BVD algorithm compares the TBVs of the concerned polynomials and chooses the least dissipative one in a given cell. The linear schemes C5 or C6 will be used in the smooth region and the BVD algorithm will switch to the MP5 scheme in the presence of discontinuities. The complete procedure for the new scheme presented in this work is summarized as follows: \begin{description} \item[Step 1.] Evaluate the interface values by using two different reconstruction procedures: \begin{enumerate}[(a)] \item Linear upwind compact reconstruction given by Equation (\ref{eqn:upwind-compact}) and \item MP5 scheme given by Equations (\ref{eqn:linear5}) - (\ref{eqn:sc}). \end{enumerate} \item[Step 2.] Calculate the TBV values for each cell $I_{j}$ by using the compact reconstruction: \begin{equation}\label{Eq:TBVC5} TBV_{j}^{C5}=\big|U_{j-\frac{1}{2}}^{L,C5}-U_{j-\frac{1}{2}}^{R,C5}\big|+\big|U_{j+\frac{1}{2}}^{L,C5}-U_{j+\frac{1}{2}}^{R,C5} \big| \end{equation} and MP5 scheme: \begin{equation}\label{Eq:TBVSC} TBV_{j}^{MP5}=\big|U_{j-\frac{1}{2}}^{L,MP5}-U_{j-\frac{1}{2}}^{R,MP5}\big|+\big|U_{j+\frac{1}{2}}^{L,MP5}-U_{j+\frac{1}{2}}^{R,MP5} \big|. \end{equation} \item[Step 3.] By averaging the left and right interface values computed by the C5 scheme we obtain the central non-dissipative linear scheme. This step is one of the key contributions of this paper in the development of the non-dissipative central scheme, denoted hereafter as HOCUS6. \begin{equation}\label{eqn-add} \begin{cases} U_{{j+\frac{1}{2}}}^{L} = U_{{j+\frac{1}{2}}}^{L, C6} = \frac{1}{2}(U_{{j+\frac{1}{2}}}^{L, C5} +U_{{j+\frac{1}{2}}}^{R, C5})\\ \\ U_{{j+\frac{1}{2}}}^{R}=U_{{j+\frac{1}{2}}}^{R, C6} = \frac{1}{2}(U_{{j+\frac{1}{2}}}^{L, C5} +U_{{j+\frac{1}{2}}}^{R, C5})\\ \end{cases} \rightarrow \textbf{HOCUS6} \end{equation} This step is equivalent to computing the interface fluxes by the following sixth order linear compact reconstruction scheme, denoted as C6 in this paper: \begin{equation}\label{eqn-C6} \frac{1}{3} U^{C6}_{j-\frac{1}{2}}+ U^{C6}_{j+\frac{1}{2}} + \frac{1}{3} U^{C6}_{j+\frac{3}{2}} = \frac{29}{36} ( \hat U_{j+1}+\hat U_{j}) + \frac{1}{36} (\hat U_{j-1}+ \hat U_{j+2}) \end{equation} \item[Step 4.] Now, \textcolor{black}{all the interface values of $j-\frac{3}{2}$, $j-\frac{1}{2}$, $j+\frac{1}{2}$ and $j+\frac{3}{2}$, both $L$ and $R$} are modified according to the following algorithm to obtain non-oscillatory results: \begin{equation}\label{eqn:BVDstep} \begin{aligned} {\rm if } \ \ TBV_{j}^{MP5} < TBV_{j}^{C5} \end{aligned} \end{equation} \item[Step 5.] Finally, evaluate the interface flux $\hat{F}_{j + \frac{1}{2}}$ from ${U}_{j+\frac{1}{2}}^{L,R}$ using Equation (\ref{eqn:Riemann}) \end{description} \begin{remark} \normalfont It is important to note that reconstruction polynomials at the cell interface are first computed by the compact upwind scheme, given by Equations (\ref{eqn:upwind-compact}), and averaged later to obtain the central scheme. Direct evaluation by the central scheme given by Equation (\ref{eqn-C6}) \textcolor{black}{cannot be done as the values of left and right interface values are the same for the central scheme. It will lead to oscillations and failure of the simulation. The total boundary variations of MP5 and C5 schemes should be compared in Equation (\ref{eqn:BVDstep})}. \end{remark} \begin{remark} \normalfont \textcolor{black}{If the underlying solution is smooth on this stencil it results in $U_{{j+\frac{1}{2}}}^{L}=U_{{j+\frac{1}{2}}}^{R}$ and through this step we can \textit{reduce} the numerical dissipation in the Riemann solver shown in Equation (\ref{eqn:Riemann}).} \begin{equation} \text{if} \ U_{{j+\frac{1}{2}}}^{L}=U_{{j+\frac{1}{2}}}^{R} \rightarrow \underbrace{({U^R_{j+\frac{1}{2}}}-{U^L_{j+\frac{1}{2}}})}_{\text{Numerical dissipation}} = 0. \end{equation} \end{remark} \begin{figure}[H] \centering \includegraphics[width=0.6\textwidth]{Explicit-Compact.pdf} \caption{Computational stencils of MP5 and C5 schemes.} \label{fig:scheme} \end{figure} \begin{remark} \normalfont We can also use the left and right interface values computed by the linear fifth order upwind compact scheme, which is denoted as HOCUS5 scheme in this paper, i.e. ${U^{(L,R) C5}}_{j+\frac{1}{2}}$ in Equation (\ref{eqn:BVDstep}), which will also give superior results but the numerical dissipation will still be present as $(U_R - U_L)_{j+\frac{1}{2}} \neq 0$ due to upwinding. The implementation of HOCUS5 scheme is as follows: \begin{equation}\label{eqn-hocus5} \begin{cases} U_{{j+\frac{1}{2}}}^{L} = U_{{j+\frac{1}{2}}}^{L, C5}\\ \\ U_{{j+\frac{1}{2}}}^{R}= U_{{j+\frac{1}{2}}}^{R, C5} \end{cases} \rightarrow \textbf{HOCUS5}, \end{equation} \begin{equation}\label{eqn-blah5} \text{if} ~~ TBV_{j}^{MP5} < TBV_{j}^{C5} ~~~ {U^{L,R}}_{j+\frac{1}{2}}={U^{(L,R)MP5}}_{j+\frac{1}{2}}. \end{equation} \end{remark} \begin{remark} \normalfont \textcolor{black}{The objective of the current paper is not to study various combinations. However, at the same time, we want to present the reasons that motivated us to arrive at the current method, which is a combination of MP5 and C5/C6. In the development of our new scheme, several different combinations of linear and nonlinear schemes were considered, which are given below:} \begin{enumerate} \item For example, it is also possible to use a third-order reconstruction with the \textcolor{black}{minmod limiter \cite{van1979towards} as the shock-capturing scheme in combination with the compact schemes with the present algorithm. The results are shown in Appendix \ref{sec-appb}. } \item \textcolor{black}{We have also considered the THINC scheme in combination with the compact schemes and the results are shown in Appendix \ref{sec-appc}.} \item Finally, by combining the WENO-Z scheme with the compact scheme C6 by using the BVD algorithm, Equation (\ref{eqn:BVDstep}), we observed that the results are too diffusive which motivated us to use monotone schemes as the base shock-capturing scheme. We have also studied a different BVD algorithm proposed in \cite{xie2017hybrid, ivan2014high}, and the results are shown in Appendix C. \end{enumerate} \end{remark} \begin{remark} \normalfont \textcolor{black}{In P4T2 scheme (\cite{deng2019fifth}) the interfaces of the cells $j-1$, $j$ and $j+1$ are corrected by using BVD algorithm and in the current approach the interfaces values of $j-\frac{3}{2}$, $j-\frac{1}{2}$, $j+\frac{1}{2}$ and $j+\frac{3}{2}$ are modified.} \end{remark} \begin{remark}\label{remark-5} \normalfont The value of ${\alpha}$ in Equation (\ref{eqn:alpha}) puts a restriction on the CFL (Courant-Friedrichs-Lewy) number such that CFL $\leq$ 1/(1+${\alpha}$) (see Suresh and Hyunh \cite{suresh1997accurate} for further details). The values considered various schemes for parameter ${\alpha}$ are shown in Table \ref{tab:alpha}. It is the only heuristic parameter in the present approach. For all the simulations in this paper, we considered CFL to be less than or equal to 0.2. \end{remark} \begin{table}[H] \centering \footnotesize \caption{Parameter ${\alpha}$ for MP5, HOCUS5 and HOCUS6 schemes} \begin{tabular}{ccccccc} \hline & MP5 & HOCUS5 & HOCUS6 & \\ \hline ${\alpha}$ & 4 & 7 & 7 \\ \hline \end{tabular}% \label{tab:alpha}% \end{table}% The numerical method described above can be easily extended to multi-dimensional (2D and 3D) problems via dimension by dimension approach using the method of lines. The extension of the scalar conservation equation to Euler equations is discussed in the following section. \subsection{Extension to Euler equations}\label{sec-2.1} The governing equations for the two-dimensional Euler equations are: \begin{equation}\label{eq:2d} \frac{\partial{\mathbf{Q}}}{\partial{t}}+\frac{\partial{\mathbf{F}}}{\partial{x}}+\frac{\partial{\mathbf{G}}}{\partial{y}}=0, \end{equation} \begin{equation} \begin{split} &\mathbf{Q}=\left[\begin{matrix} \rho\\ \rho u\\ \rho v\\ E\\ \end{matrix} \right],\ \mathbf{F}=\left[\begin{matrix} \rho u\\ \rho u^2+p\\ \rho uv\\ (E+p)u\\ \end{matrix} \right],\ \mathbf{G}=\left[\begin{matrix} \rho v\\ \rho uv\\ \rho v^2+p\\ (E+p)v\\ \end{matrix} \right]\textcolor{black}{,} \mathbf{U}=\left[\begin{matrix} \rho\\ u\\ v\\ p\\ \end{matrix} \right], \end{split} \end{equation} where $\mathbf{Q}$, $\mathbf{F}$ and $\mathbf{G}$ are the conservative variables, inviscid flux vectors in $x$ and $y$ directions, respectively. The primitive variables $\rho,u,v\ \text{and} \ p$ represent the density, the $x-$ and $y-$ components of the velocity, and the pressure are stored in the vector $\mathbf{U}$. The total specific energy is defined: \begin{equation} E=\frac{p}{\gamma-1}+\frac{\rho}{2}(u^2+v^2), \end{equation} where $\gamma$ is the constant specific heat ratio. The left and right eigenvectors of the two-dimensional Euler equations, denoted by $\bm{L_n}$ and $\bm{R_n}$, are used for characteristic variable projection are as given as: \begin{align}\label{eqn:leftright} \bm{R_n} = \begin{bmatrix} 1 & 1 & 1 & 0 \\ \\ -\frac{n_xc}{\rho} & 0 & \frac{n_xc}{\rho} & \frac{l_x}{\rho} \\ \\ -\frac{n_yc}{\rho} & 0 & \frac{n_yc}{\rho} & \frac{l_y}{\rho} \\ \\ c^2 & 0 & c^2 & 0 \\ \end{bmatrix}, \enskip \bm{L_n} = \begin{bmatrix} 0 & -\frac{n_x\rho}{2c} &-\frac{n_y\rho}{2c} & \frac{1}{2c^2} \\ \\ 1 & 0 & 0 & -\frac{1}{c^2} \\ \\ 0 & -\frac{n_x\rho}{2c} & \frac{n_y\rho}{2c} & \frac{1}{2c^2} \\ \\ 0 & \rho l_x & \rho l_y & 0 \\ \end{bmatrix}, &&\\\nonumber \end{align} where $\bm{n}$ = $[n_x \ n_y]^t$ and $[l_x \ l_y]^t$ is a tangent vector (perpendicular to $\bm{n}$) such as $[l_x \ l_y]^t$ = $[-n_y \ n_x]^t$. By taking $\bm{n}$ = $[1, 0]^t$ and $[0, 1]^t$ we obtain the corresponding eigenvectors in $x$ and $y$ directions. In the present paper, we use simple dimension-wise implementation for two-dimensional Euler equations which is akin to evaluation of integrals by mid-point rule rather than using high-order quadratures, which is as follows: \begin{equation} \frac{\mathbf{d\hat Q}_{j,i}}{\mathbf{dt}} = -\frac{1}{\Delta x}\left(\mathbf{\hat F_{j + \frac{1}{2}}} - \mathbf{\hat F_{j- \frac{1}{2}}}\right) - \frac{1}{\Delta y} \left(\mathbf{\hat G_{i + \frac{1}{2}}} - \mathbf{\hat G_{i - \frac{1}{2}}}\right), \label{eqn:e6m} \end{equation} where, $\mathbf{\hat Q_{j,i}}$ are cell averaged conservative variables and $\mathbf{\hat{F}}_{}$ and $\mathbf{\hat{G}}_{}$ are interface numerical fluxes computed by an approximate Riemann solver. Here we consider the HLLC (Harten, Leer and Lax with Contact) approximate Riemann solver of Harten et al. \cite{harten1983upstream} and Toro et al. \cite{toro1994restoration}. Considering $x-$direction as an example, the interface flux computed by the HLLC Riemann solver is: \begin{equation} \mathbf{\hat{F}}_{}=\begin{cases} \mathbf{\hat{F}}_{L} & S_{L}\geq0\\ \mathbf{\hat{F}}_{*L} & S_{L}\leq0\leq S_{*}\\ \mathbf{\hat{F}}_{*R} & S_{*}\leq0\leq S_{R}\\ \mathbf{\hat{F}}_{R} & S_{R}\leq0, \end{cases}\label{eqn:hllcformulation} \end{equation} where, $\mathbf{\hat{F}}_{L}=\mathbf{F}\left(\mathbf{\hat{Q}}_{L}\right)\ $ and $\quad\mathbf{\hat{F}}_{R}=\mathbf{F}\left(\mathbf{\hat{Q}}_{R}\right)\ $ are the exact local fluxes at either side of the interface and using the Rankine-Hugoniot jump conditions across the left and right wave: \begin{equation} \mathbf{\hat{F}}_{*L}=\mathbf{\hat{F}}_{L}+S_{L}\left(\mathbf{\hat{Q}}_{*L}-\mathbf{\hat{Q}}_{L}\right)\ \label{eq:FStarL} \end{equation} \begin{equation} \mathbf{\hat{F}}_{*R}=\mathbf{\hat{F}}_{R}+S_{R}\left(\mathbf{\hat{Q}}_{*R}-\mathbf{\hat{Q}}_{R}\right)\,\label{eq:FStarR} \end{equation} where, $S_L$ and $S_R$ are numerical approximations to the speeds of the left most and right most running characteristics that emerge as the solution of the Riemann problem at an interface. It has been shown that under appropriate choice of wave speeds $S_L$ and $S_R$, the HLLC scheme is both positivity preserving and entropy satisfying \cite{batten1997choice}. This choice of wave speeds are \textcolor{black}{as follows}: \begin{align} \nonumber S_L = min(u_{nL}-c_L, \tilde{u}_n-\tilde{c})\\ S_R = max(u_{nR}+c_R, \tilde{u}_n +\tilde{c}) \label{eqn:HLLEwavespeedestimate} \end{align} where $u_{nL,R}$ are the normal velocities across an interface, $c_{L,R}$ are the respective sonic speeds and $\tilde{u}_n,\tilde{c}$ are the standard Roe averaged quantities at the interface. Using the integral form of the conservation laws, closed form expressions for the conserved quantities in the states in the star region $S_*$ can be derived as: \begin{equation} \label{eqn:HLLCstate} \begin{aligned} \mathbf{\hat Q}_{*K}^{HLLC} &= \rho_{K} \left(\frac{S_{K}-u_{nK}}{S_{K}-S_*}\right) \left( \begin{array}{c} 1\\ S_*\\ u_{tK}\\ \frac{(\rho E)_{K}}{\rho_{K}} + (S_*-u_{nK})(S_* + \frac{p_{K}}{\rho_{K}(S_{K}-u_{nK})}) \end{array} \right)\\ \end{aligned} \end{equation} where K =L, R and $u_{tK}$ denote the tangential velocities across an interface. In the above expressions, $S_L$ and $S_R$ can be obtained using Eq.(\ref{eqn:HLLEwavespeedestimate}) and Batten et al. \cite{batten1997choice} provided a closed form expression for $S_*$ which is as follows: \begin{align} \label{eqn:HLLCmiddlewaveestimate} S_* = \frac{p_R - p_L + \rho_Lu_{nL}(S_L - u_{nL}) - \rho_Ru_{nR}(S_R - u_{nR})}{\rho_L(S_L - u_{nL}) -\rho_R(S_R - u_{nR})}. \end{align} Unlike scalar advection, shock-capturing should be carried out using characteristic variables for coupled hyperbolic equations like the Euler equations for optimal \textit{cleanest} results \cite{van2006upwind}. Direct reconstruction of the interfaces values using primitive variables resulted in small oscillations. The complete numerical algorithm for the Euler equations is summarized below, which includes the transforming of primitive variables into characteristics variables necessary for capturing discontinuities. \begin{description} \item[Step 1.] To evaluate the RHS contribution of the advective fluxes in the $x-$ direction from $\mathbf{\hat Q_{j,i}}$ to build $\mathbf{F_{j + \frac{1}{2}}}$ in Equation \ref{eqn:e6m} \\ \begin{enumerate} \item Compute primitive variables $\mathbf{U_{j,i}}$ from $\mathbf{\hat Q_{j,i}}$. \\ \item Interpolate the primitive variables,$\mathbf{U_{j,i}}$, from cell-center to cell interface by compact scheme C5, Equations (\ref{eqn:upwind-compact}), and obtain $\mathbf U_{j+\frac{1}{2},i}^{L,C5}$ and $\mathbf U_{j+\frac{1}{2},i}^{R,C5}$. \\ \item Average the interface values $\mathbf U_{j+\frac{1}{2},i}^{L,C5}$ and $\mathbf U_{j+\frac{1}{2},i}^{R,C5}$ and obtain $\mathbf U_{j+\frac{1}{2},i}^{L}$ and $\mathbf U_{j+\frac{1}{2},i}^{R}$ as shown in Equation (\ref{eqn-add}). \\ \item Interpolate the characteristic variables, $\bm{W}$, projected from the primitive variables by the MP5 scheme, through Equations (\ref{eqn:linear5} - \ref{eqn:sc}). The procedure is as follows, \\ \begin{enumerate} \item Compute the arithmetic averages at the interface $(x_{j+\frac{1}{2}}, y_i)$ by using neighbouring cells, $(x_j, y_i)$ and $(x_{j+1},y_i)$. Compute the left and right eigenvectors $\bm{L_{n}}$ and $\bm{R_{n}}$, Equation (\ref{eqn:leftright}), at the interface using the arithmetic averages, with $\bm{n}$ = $[1, 0]^t$ which corresponds to $x-$ direction. The primitive variables are then transformed to characteristic variables by the following equation: \\ \begin{equation} \bm{W}_{j,i} = \bm{L}_{\bm{n}_{j+\frac{1}{2},i}} \bm{U}_{j,i}. \end{equation} \item Carry out the reconstruction of characteristic variables by MP5 scheme, through Equations (\ref{eqn:linear5} - \ref{eqn:sc}), and obtain left- and right- interface values denoted by $\tilde{\bm{W}}_{j+\frac{1}{2},i}^L$ and $\tilde{\bm{W}}_{j+\frac{1}{2},i}^R$. \\ \item After obtaining $\tilde{\bm{W}}_{j+\frac{1}{2},i}^L$ and $\tilde{\bm{W}}_{j+\frac{1}{2},i}^R$ from the MP5 reconstruction the primitive variables are then recovered by projecting the characteristic variables back to physical fields: \\ \begin{equation} \begin{aligned} {\bm{U}}_{j+\frac{1}{2},i}^{L,MP5} &= \bm{R}_{\bm{n}_{j+\frac{1}{2},i}} \tilde{\bm{W}}_{j+\frac{1}{2},i}^L, \\ {\bm{U}}_{j+\frac{1}{2},i}^{R,MP5} &= \bm{R}_{\bm{n}_{j+\frac{1}{2},i}} \tilde{\bm{W}}_{j+\frac{1}{2},i}^R. \end{aligned} \end{equation} \end{enumerate} \item Modify the interface values by using BVD algorithm, Equation (\ref{eqn:BVDstep}). Primitive variables are used for the BVD algorithm. \item Compute the upwind flux $\mathbf{F_{j + \frac{1}{2}}}$ at the cell interface by using Riemann solver, through Equation (\ref{eqn:hllcformulation}). \\ \end{enumerate} \item[Step 2.] To evaluate the RHS contribution of the advective fluxes in the $y-$ direction to build $\mathbf{G_{i + \frac{1}{2}}}$ repeat steps \textbf{(a)-(f)} in Step 1 with $\bm{n}$ = $[0,1]^t$.\\ \item[Step 3.] Evaluate the right-hand side of the finite-volume equation, Equation (\ref{eqn:e6m}) and perform time integration as described in the next section. \end{description} \section{Numerical method: Temporal discretization}\label{sec3.1} For time integration we use the explicit third-order TVD Runge-Kutta method \cite{jiang1995}: \begin{eqnarray} \label{rk} \mathbf{Q^{(1)}} & = & \mathbf{Q^n} + \Delta t \mathbf{Res}(\mathbf{Q^n}), \nonumber \\ \mathbf{Q^{(2)}} & = & \frac{3}{4} \mathbf{Q^n} + \frac{1}{4} \mathbf{Q^{(1)}} + \frac{1}{4}\Delta t \mathbf{Res}(\mathbf{Q^{(1)}}) ,\\ \mathbf{Q^{n+1}} & = & \frac{1}{3} \mathbf{Q^n} + \frac{2}{3} \mathbf{Q^{(2)}} + \frac{2}{3}\Delta t \mathbf{Res}(\mathbf{Q^{(2)}}), \nonumber \end{eqnarray} where $\mathbf{Res}$ is the residual that is evaluated on the right hand side of the Equation (\ref{eqn:e6m}), the superscripts $\mathbf{n}$ and $\mathbf{n+1}$ denote the current and the subsequent time-steps, and superscripts $\mathbf{(1)-(2)}$ corresponds to intermediate steps respectively. The time step $\Delta t$ is taken as suggested by Titarev and Toro \cite{titarev2004finite}: \begin{equation} \Delta{t}=\text{CFL} \times \text{$min_{j,i}$} (\frac{\Delta{x}}{\left(|u_{j,i}+c_{j,i}|\right)}, \frac{\Delta{y}}{\left(|v_{j,i}+c_{j,i}|\right)}), \end{equation} where $c$ is the speed of sound and given by $c=\sqrt{\gamma{p}/\rho}$. Time integration is performed with a CFL = 0.2 for all the problems unless otherwise stated. Simulations are carried out by MP5, HOCUS5, and HOCUS6 schemes presented in this paper on a uniform grid. We also carried out simulations with a fifth-order WENO spatial reconstruction scheme of Borges et al. \cite{Borges2008}, denoted as WENO-Z, for comparison. A comparison of HOCUS5 and C5 schemes is carried out for the one-dimensional test cases for Euler equations in Section \ref{sec-3.2.1}. \section{Results}\label{sec-3} In this section, the proposed numerical scheme is tested for both the linear advection equation and Euler equations. \subsection{Linear advection equation} \begin{example}\label{ex:adv_comp} Advection of complex waveforms \end{example} \noindent First we consider the linear advection equation: \begin{equation}\label{eq:lin} \dfrac{\partial{u}}{\partial{t}}+\dfrac{\partial{u}}{\partial{x}}=0 \end{equation} with the following initial condition \begin{equation}\label{eq:1st_case} u_0(x)=\left\{ \begin{array}{ll} exp(-log(2)(x+0.7)^2/0.0009 & -0.8\le x\le -0.6,\\ 1 & -0.4\le x \le -0.2,\\ 1-|10(x-0.1)| &\ 0\le x \le 0.2,\\ (1-100(x-0.5))^{0.5} &\ 0.4\le x\le 0.6,\\ 0,& \ otherwise \end{array} \right . \end{equation} The initial condition contains a combination of a square wave, a Gaussian, a sharp triangle and a semi-ellipse. Two sets of results are presented. The first uses a grid size of $200$ cells and results are presented at $t=2$. The second uses a grid size of $400$ cells and results are presented at $t=500$. A CFL of $0.1$ is used for all the simulations. The results are presented in Figs. \ref{fig_advectioncomplex1} and \ref{fig_advectioncomplex2}. Both the HOCUS5 and HOCUS6 schemes produce non-oscillatory results and preserve the initial wave profiles better than the MP5 and WENO-Z schemes. The difference between HOCUS5 and HOCUS6 schemes is marginal. For the results at $t=500$, shown in the second column of Fig. \ref{fig_advectioncomplex1}, HOCUS6 can preserve the waves even after $1 \times 10^6$ time steps similar to that of the $P4T2$ scheme proposed by Deng et al. in \cite{deng2019fifth} (see their Fig. 5). On the other hand, we can see a squaring effect for the MP5 and WENO scheme is completely distorted in Fig. \ref{fig_advectioncomplex2}. \begin{figure}[H] \begin{onehalfspacing} \centering \subfigure[HOCUS5, t=2]{\includegraphics[width=0.48\textwidth]{l_BVD5-2.pdf} \label{fig:HOCUS5_AC}} \subfigure[HOCUS5, t=500]{\includegraphics[width=0.48\textwidth]{l_BVD5-500.pdf} \label{fig:HOCUS5_AC1}} \subfigure[HOCUS6, t=2]{\includegraphics[width=0.48\textwidth]{l_BVD-2.pdf} \label{fig:HOCUS6_AC}} \subfigure[HOCUS6, t=500]{\includegraphics[width=0.48\textwidth]{l_BVD-500.pdf} \label{fig:HOCUS6_AC1}} \caption{Results for HOCUS5 and HOCUS6 schemes for Example \ref{ex:adv_comp}, where solid line: reference solution; \textcolor{black}{cyan stars: HOCUS5}, green circles: HOCUS6.} \label{fig_advectioncomplex1} \end{onehalfspacing} \end{figure} \begin{figure}[H] \begin{onehalfspacing} \centering \subfigure[\textcolor{black}{WENOZ, t=2}]{\includegraphics[width=0.48\textwidth]{l_WENO-2.pdf} \label{fig:WENO-Z_AC}} \subfigure[\textcolor{black}{WENOZ, t=500}]{\includegraphics[width=0.48\textwidth]{l_WENO-500.pdf} \label{fig:WENO-Z_AC1}} \subfigure[MP5, t=2]{\includegraphics[width=0.48\textwidth]{l_MP5-2.pdf} \label{fig:MP5_AC}} \subfigure[MP5, t=500]{\includegraphics[width=0.48\textwidth]{l_MP5-500.pdf} \label{fig:MP5_AC1}} \caption{Results for WENO-Z and MP5 schemes for Example \ref{ex:adv_comp}, where solid line: reference solution; blue squares: WENO-Z; red triangles: MP5.} \label{fig_advectioncomplex2} \end{onehalfspacing} \end{figure} \begin{example}\label{ex:accuracy}{Order of accuracy tests} \end{example} First, we consider the one-dimensional Gaussian pulse advection problem \cite{Yamaleev2009} to verify our scheme's order of accuracy. Here we consider the following initial condition: \begin{equation}\label{eq:26} u_0(x) = exp(-300(x-0.5)^2) \end{equation} on a computational domain of $[0,1]$ subject to periodic boundary conditions. Simulations are performed until $t=1$. To ensure that the effect of time-discretization is negligible a time step of \textcolor{black}{$\Delta t$ = 0.1$\Delta x^{2.0}$} is selected. As can be seen from Table \ref{tab:case1a} the $L_1$ convergence histories and order of accuracies for both HOCUS5 and HOCUS6 are the same as the corresponding linear schemes, C5 and C6, which shows the BVD algorithm is choosing appropriate smoother polynomials resulting in formal order of accuracies. \textcolor{black}{Comparison has also been made with the sixth order explicit central scheme, denoted as E6, which indicates the current approach, HOCUS6, is one order better than the E6 scheme.} \begin{table}[htbp] \centering \footnotesize \caption{\textcolor{black}{$L_1$ errors and numerical orders of accuracy on $u_t$ + $u_x$ = 0 with $u_0$(x) = $exp(-300(x-0.5)^2)$. $N$ is the total number of cells on a uniform mesh and $t = 1$.}} \begin{tabular}{rcccccccr} \hline \hline \multicolumn{1}{c}{N} & MP5 & & WENO-Z & & HOCUS5 & & C5 & \\ \hline & & order & & order & & order & & \multicolumn{1}{c}{order} \\ \hline \multicolumn{1}{c}{40} & 1.04E-02 & & 1.00E-02 & & 5.94E-03 & & 4.95E-03 & \\ \hline \multicolumn{1}{c}{80} & 1.30E-03 & 3.00 & 1.29E-03 & 2.96 & 1.82E-04 & 5.03 & 1.78E-04 & \multicolumn{1}{c}{4.80} \\ \hline \multicolumn{1}{c}{160} & 4.71E-05 & 4.78 & 4.71E-05 & 4.78 & 5.16E-06 & 5.14 & 5.13E-06 & \multicolumn{1}{c}{5.12} \\ \hline \multicolumn{1}{c}{320} & 1.51E-06 & 4.97 & 1.51E-06 & 4.97 & 1.55E-07 & 5.05 & 1.55E-07 & \multicolumn{1}{c}{5.05} \\ \hline \multicolumn{1}{c}{640} & 4.73E-08 & 4.99 & 4.73E-08 & 4.99 & 4.79E-09 & 5.02 & 4.79E-09 & \multicolumn{1}{c}{5.02} \\ \hline \hline & N & HOCUS6 & & C6 & & E6 & & \\ \hline & & & order & & order & & Order & \\ \hline & 40 & 3.28E-03 & & 2.45E-03 & & 2.02E-02 & & \\ \hline & 80 & 3.99E-05 & 6.36 & 3.49E-05 & 6.13 & 4.68E-04 & 5.43 & \\ \hline & 160 & 5.98E-07 & 6.06 & 4.94E-07 & 6.14 & 7.82E-06 & 5.90 & \\ \hline & 320 & 1.05E-08 & 5.83 & 7.19E-09 & 6.10 & 1.26E-07 & 5.96 & \\ \hline & 640 & 1.68E-10 & 5.97 & 1.11E-10 & 6.01 & 1.97E-09 & 5.99 & \\ \hline \end{tabular}% \label{tab:case1a}% \end{table}% \begin{figure}[H] \centering \includegraphics[width=0.45\textwidth]{case-1_accuracy.pdf} \caption{\textcolor{black}{Accuracy plot}} \label{fig_acc} \end{figure} Second, we consider the test case with critical points given in Henrick et al. \cite{Henrick2005}. It is known from the literature that WENO schemes lose their accuracy at critical points where the first and second derivatives go to zero. The initial condition is as follows: \begin{equation}\label{eq:25} u_0(x) = sin(\pi x - sin(\pi x)/\pi ), \end{equation} on a computational domain of $[-1,1]$ subject to periodic boundary conditions. Simulations are performed until \textcolor{black}{$t=8$}. This initial condition has two critical points \cite{Henrick2005}. We chose the same time step as the previous case. Table \ref{tab:case2a} shows the results for the HOCUS5 scheme in comparison with the corresponding linear compact scheme C5. We can see that the HOCUS5 scheme exactly retrieves the linear scheme with fifth-order accuracy, even in the presence of critical points. In Table \ref{tab:case2a} a comparison between HOCUS6 and C6 is shown, where we can see that there is a small difference between the two schemes with HOCUS6 being less accurate. Incorporating additional constraints in the BVD algorithm given by Equation (\ref{eqn:BVDstep}) seems to retrieve the accuracy but these constraints could not be generalized, especially for Euler equations, and are not considered further. \textcolor{black}{Results considering additional constraints are presented in Appendix \ref{sec-appd}}. Nevertheless, the HOCUS6 scheme shows superior absolute accuracy compared to the HOCUS5 scheme. \begin{table}[H] \centering \footnotesize \caption{\textcolor{black} {$L_1$ errors and numerical orders of accuracy on $u_t$ + $u_x$ = 0 with $u_0$(x) = sin($\pi$x - sin($\pi$x)/$\pi$ ). $N$ is the total number of cells on a uniform mesh and $t = 8$.}} \begin{tabular}{rcccccccr} \hline \hline \multicolumn{1}{c}{N} & WENOZ & & MP5 & & HOCUS5 & & \multicolumn{1}{l}{C5} & \\ \hline & & Order & & Order & & Order & & \multicolumn{1}{c}{Order} \\ \hline \multicolumn{1}{c}{20} & \multicolumn{1}{r}{7.89E-03} & & \multicolumn{1}{r}{7.89E-03} & & 1.40E-03 & & 1.40E-03 & \\ \hline \multicolumn{1}{c}{40} & \multicolumn{1}{r}{2.89E-04} & 4.77 & \multicolumn{1}{r}{2.89E-04} & 4.77 & 3.62E-05 & 5.27 & 3.62E-05 & \multicolumn{1}{c}{5.27} \\ \hline \multicolumn{1}{c}{80} & \multicolumn{1}{r}{9.28E-06} & 4.96 & \multicolumn{1}{r}{9.28E-06} & 4.96 & 1.01E-06 & 5.16 & 1.01E-06 & \multicolumn{1}{c}{5.16} \\ \hline \multicolumn{1}{c}{160} & \multicolumn{1}{r}{2.91E-07} & 4.99 & \multicolumn{1}{r}{2.91E-07} & 4.99 & 3.04E-08 & 5.06 & 3.04E-08 & \multicolumn{1}{c}{5.06} \\ \hline \hline & N & HOCUS6 & & C6 & & E6 & & \\ \hline & & & Order & & Order & & Order & \\ \hline & 20 & 7.47E-04 & & 1.30E-04 & & 2.90E-03 & & \\ \hline & 40 & 1.61E-05 & 5.53 & 1.72E-06 & 6.23 & 4.84E-05 & 5.91 & \\ \hline & 80 & 3.13E-07 & 5.69 & 2.74E-08 & 5.97 & 7.84E-07 & 5.95 & \\ \hline & 160 & 4.81E-09 & 6.03 & 5.35E-10 & 5.68 & 1.23E-08 & 5.99 & \\ \hline \end{tabular}% \label{tab:case2a}% \end{table}% \subsection{One-dimensional Euler equations} In this subsection, we consider the test cases for the one-dimensional Euler equations, as outlined in Section 2. \begin{example}\label{Titarev-Toro}{ Titarev-Toro problem} \end{example} In this test case, we consider the shock-entropy wave problem of Titarev-Toro \cite{titarev2004finite}, where a high frequency oscillating sinusoidal wave interacts with a shock wave. It tests the ability of the high-order scheme to capture the extremely high-frequency waves. The initial conditions are given as: \begin{align}\label{shock_tita} (\rho,u,p)= \begin{cases} &(1.515695,\ \ 0.523326,\ \ 1.805),\quad x<-4.5,\\ &(1+0.1\sin(20x\pi),\ \ 0,\ \ 1),\quad x>-4.5, \text{Case-1}\\ &(1+0.1\sin(10x\pi),\ \ 0,\ \ 1),\quad x>-4.5, \text{Case-2}, \end{cases} \end{align} Case-1 corresponds to the original test case proposed in \cite{titarev2004finite} and Case-2 is the modified version with a smaller frequency considered in \cite{deng2019fifth}. Simulations are carried out on a grid with $N=1000$ cells for Case-1 and $N=400$ cells for Case-2. The reference solution is obtained by the WENO-Z scheme on the grid size of $N=3000$ and $N=1600$ for test Case-1 and Case-2, respectively. From Figs.\ \ref{fig_titazoom} and \ref{fig_titaglobal}, we can observe that the HOCUS6 scheme yields the best resolution and captures the fine-scale structures in the solution much better than other schemes. For Case-1, results obtained by the HOCUS6 scheme are similar to that of the those obtained by adaptive upwind-central sixth-order adaptive scheme proposed by Wong and Lele in \cite{Wong2017} (see their Fig. 13) on the same grid size using the same Riemann solver. For Case-2, we obtained superior results in comparison with P4T2 scheme proposed by Deng et al. in \cite{deng2019fifth} (see their Fig. 11) which required $800$ cells to capture the high-frequency waves whereas the current approach requires only $400$ cells. This test shows the capabilities of the adaptive-central scheme as it can capture the high-frequency density fluctuations and demonstrates the capabilities of the proposed HOCUS6 scheme over the upwind HOCUS5 scheme, MP5 and WENO-Z schemes. Also, it can be observed that the compact schemes, HOCUS6 and HOCUS5, are superior to the explicit, MP5 and WENO-Z, schemes for both cases. \begin{figure}[H] \centering \subfigure[Case-1]{\includegraphics[width=0.48\textwidth]{BVD-tita.pdf} \label{fig:tita1}} \subfigure[Case-2]{\includegraphics[width=0.48\textwidth]{BVD-tita2.pdf} \label{fig:tita2}} \caption{Titarev-Toro problem, Example \ref{Titarev-Toro}, at $t=5.0$ by using WENO-Z, MP5, HOCUS5 and HOCUS6 schemes. Fig. (a) Case-1 with N=1000 and Fig. (b) Case-2 with $N=400$, where solid line: reference solution; cyan stars: HOCUS5, green circles: HOCUS6; blue squares: WENO-Z; red triangles: MP5.} \label{fig_titaglobal} \end{figure} \begin{figure}[H] \centering \subfigure[Case-1, local profile]{\includegraphics[width=0.48\textwidth]{BVD-tita-zoom.pdf} \label{fig:tita1local}} \subfigure[Case-2, local profile]{\includegraphics[width=0.48\textwidth]{BVD-tita2-zoom.pdf} \label{fig:tita2local}} \caption{Local density profile of Fig. \ref{fig_titaglobal} for the Titarev-Toro problem, Example \ref{Titarev-Toro}, at $t=5.0$ by using WENO-Z, MP5, HOCUS5 and HOCUS6 schemes. Fig. (a) Case-1 with $N=1000$ and Fig. (b) Case-2 with $N=400$, where solid line: reference solution; cyan stars: HOCUS5, green circles: HOCUS6; blue squares: WENO-Z; red triangles: MP5.} \label{fig_titazoom} \end{figure} \begin{example}\label{sod}{Shock tube problems} \end{example} Here, we present the simulation results for the new algorithm for \textcolor{black}{ three shock-tube problems, namely Sod's problem \cite{sod1978survey}, Lax's problem \cite{lax1954weak} and Le Blanc problem \cite{loubere2005subcell}. For Sod and Lax problems} the specific heat ratio is $\gamma=1.4$ and the computational domain is $[0,1]$. Exact solutions for both cases are presented on a grid size of $1000$ points by an exact Riemann solver \cite{toro2009riemann}. First, Sod's problem is calculated to verify the shock-capturing ability of HOCUS6 scheme. The initial conditions for the Sod problem are given below: \begin{align}\label{sod_prob} (\rho,u,p)= \begin{cases} &(0.125,\ \ 0,\ \ 0.1),\quad 0<x<0.5,\\ &(1,\ \ 0,\ \ 1),\quad 0.5<x<1. \end{cases} \end{align} In the test case, grid points are set to be 100. All results are obtained on a $N=100$ grid and numerical predictions of density and pressure obtained from HOCUS6 scheme at time $t=0.2$ are shown in Fig. \ref{fig_sod}. No overshoots at the discontinuities are observed and the results are in good agreement with the exact solution. Second, Lax's problem is considered with the following initial conditions: \begin{align}\label{lax_prob} (\rho,u,p)= \begin{cases} &(0.445,\ \ 0.698,\ \ 3.528),\quad 0<x<0.5,\\ &(0.5,\ \ 0,\ \ 0.571),\quad 0.5<x<1, \end{cases} \end{align} Numerical results on a $N=200$ grid for density and velocity obtained from HOCUS6 scheme at time $t=0.14$ are shown in Fig.\ \ref{fig_lax}. The HOCUS6 scheme accurately resolves the shock and contact discontinuities without over- and under-shoots. \textcolor{black}{Numerical tests for these test cases using HOCUS5 scheme are shown in \ref{sec-3.2.1}} \textcolor{black}{Third, we consider the Le Blanc problem \cite{loubere2005subcell}, which is an extreme shock-tube problem, with the following initial conditions:} \textcolor{black}{\begin{align}\label{blanc_prob} (\rho,u,p)= \begin{cases} &(1.0,\ \ 0,\ \ \frac{2}{3}\times10^{-1}),\quad 0<x<3.0,\\ &(10^{-3},\ \ 0,\ \ \frac{2}{3}\times10^{-10}),\quad 3.0<x<9, \end{cases} \end{align}} \textcolor{black}{The specific heat ratio for this test case is $\frac{5}{3}$ and the final time is $t=6$ and the exact solution is computed by an exact Riemann solver \cite{toro2009riemann}. Numerical results on a $N=200$ grid for density and pressure obtained from both HOCUS5 and HOCUS6 schemes are shown in Fig.\ \ref{fig_blanc}. There is a small error in the shock position which decreased when the grid resolution is increased (not shown here).} \begin{figure}[H] \centering \subfigure[Density]{\includegraphics[width=0.35\textwidth]{BVD-sod-den.pdf} \label{fig:sod-den}} \subfigure[Pressure]{\includegraphics[width=0.35\textwidth]{BVD-sod-pres.pdf} \label{fig:sod-pres}} \caption{Numerical solution for Sod problem in Example \ref{sod}} \label{fig_sod} \end{figure} \begin{figure}[H] \centering \subfigure[Density]{\includegraphics[width=0.35\textwidth]{BVD-Lax-den.pdf} \label{fig:lax-den}} \subfigure[Velocity]{\includegraphics[width=0.35\textwidth]{BVD-Lax-vel.pdf} \label{fig:lax-pres}} \caption{Numerical solution for Lax problem in Example \ref{sod}} \label{fig_lax} \end{figure} \begin{figure}[H] \centering \subfigure[Density]{\includegraphics[width=0.35\textwidth]{BVD_blanc_density.pdf} \label{fig:blanc-den}} \subfigure[Pressure]{\includegraphics[width=0.35\textwidth]{BVD_blanc_pressure.pdf} \label{fig:blanc-pres}} \caption{\textcolor{black}{Numerical solution for Le Blanc problem in Example \ref{sod}}} \label{fig_blanc} \end{figure} \begin{example}\label{Shu-Osher}{Shu-Osher problem} \end{example} In this test case, the Shu-Osher problem \cite{Shu1988} is computed. This problem simulates the interaction of a right-moving Mach $3$ shock with a perturbed density field. The following initial data is considered: \begin{align}\label{shock_den_prob} (\rho,u,p)= \begin{cases} &(3.857143,\ \ 2.629369,\ \ 10.3333),\quad -5<x<-4,\\ &(1+0.2\sin(5x),\ \ 0,\ \ 1),\quad -4<x<5, \end{cases} \end{align} subject to zero-gradient boundary conditions. Fig. \ref{fig:1d-SO} shows numerical results for the MP5, HOCUS6 and WCNS-Z schemes at time $t=1.8$ on a grid of $N=300$. The \textit{exact} solution is computed using WENO-Z on a much finer grid of $N=1600$. It is shown that all schemes give satisfactory approximations of the wavelike structures behind the shock. But the HOCUS6 scheme displays less dissipation and does a better job resolving the wave phenomenon compared to the WENO-Z and MP5 schemes. \begin{figure}[H] \centering\offinterlineskip \subfigure[Global profile]{\includegraphics[width=0.35\textwidth]{BVD-Shu.pdf} \label{fig:shu-global}} \subfigure[Local profile]{\includegraphics[width=0.375\textwidth]{BVD-Shu-zoom.pdf} \label{fig:shu-local}} \caption{Density profile for Shu-Osher problem, Example \ref{Shu-Osher}, on a $N=300$ points, where solid line: reference solution; green circles: HOCUS6; blue squares: WENO-Z; red triangles: MP5.} \label{fig:1d-SO} \end{figure} \begin{example}\label{blast}{Blast wave problem} \end{example} In this last one-dimensional test case, we consider the problem taken from \cite{woodward1984numerical} which involves the interaction of two blast waves with the following initial condition. \begin{equation} (\rho, u, p)=\left\{\begin{array}{ll} (1.0,0.0,1000) ~~~~ 0.0<x<0.1, \\ (1.0,0.0,0.01) ~ ~~~~0.1<x<0.9.\\ (1.0,0.0,100) ~ ~~~~~ 0.9<x<1.0. \end{array}\right. \end{equation} and Reflective boundary conditions at both ends are used for this simulation. The numerical solutions are computed for time $t=0.038$ over a domain of $[0,1]$ with $N=400$ and $N=800$ cells and depicted in Fig. \ref{fig_blast}. Once again, we have compared the numerical solutions with the reference solution, denoted as \textit{Exact}, which is obtained using WENO-Z scheme on a grid of $1600$ points. It can be seen from Fig. \ref{fig:blast-800} that good agreement is observed compared with the reference solution. HOCUS6 shows an improved solution near the right peak, see the zoomed inset, at about $x = 0.78$, compared to the results of HOCUS5, MP5 and WENO-Z due to reduced numerical dissipation. We also observe no oscillations with the present scheme despite using non-dissipative interpolation for this test case as opposed to the P4T2 scheme proposed by Deng et al. in \cite{deng2019fifth} (see their Fig. 13) which shows small oscillations in density right after the shock. \begin{figure}[H] \centering \subfigure[400-cells]{\includegraphics[width=0.35\textwidth]{BVD-blast-400.pdf} \label{fig:blast-400}} \subfigure[800-cells]{\includegraphics[width=0.35\textwidth]{BVD-blast-compare-800-sub.pdf} \label{fig:blast-800}} \caption{Density profiles obtained by various schemes for Example \ref{blast}. Solution obtained on a grid on $N=400$ and $N=800$ cells are shown in left and right panels, respectively.} \label{fig_blast} \end{figure} \subsubsection{Analysis of BVD algorithm through HOCUS5 and C5 schemes}\label{sec-3.2.1} We carried out further numerical experiments for the test cases \ref{Titarev-Toro}, \ref{sod} and \ref{Shu-Osher} using the linear fifth-order compact scheme C5 given by Equation (\ref{eqn:upwind-compact}), i.e. without any nonlinear shock-capturing mechanism and compared with the HOCUS5 scheme for better understanding of the BVD algorithm. The results are shown in Fig.\ \ref{fig_unlim}. All the numerical simulations \textcolor{black}{are carried out} with a CFL of $0.2$. Primitive variables are directly interpolated to the cell interfaces for the C5 scheme. For Titarev-Toro problem in Example \ref{Titarev-Toro} grid size of $N = 1000$ is used, for Sod and Lax problems in Example \ref{sod} we considered $N=200$ and finally for Shu-Osher problem in Example \ref{Shu-Osher} we carried out the simulations on the grid size of $N=300$. We make the following observations: \begin{itemize} \item In Figs. \ref{fig:unlim-sod} and \ref{fig:unlim-lax} we can see the density profiles for the Sod and Lax problems. We can see that the oscillations are only observed near the discontinuities for the C5 scheme, and it is sufficient to \textit{correct} the cells in that particular region as the linear scheme along with the Riemann solver tolerates the oscillations. We also carried out the same tests with the explicit linear schemes, for example, Equation (\ref{eqn:linear5}), and observed similar results, albeit with a significant numerical dissipation \item Figs. \ref{fig:unlim-shu} and \ref{fig:unlim-tita} show the density profile for the Shu-Osher problem and Titarev-Toro problems. Even though there are small oscillations near the shock-lets for the C5 scheme, the overall wave-like structures are well-preserved and are significantly better than the results obtained by the MP5 and WENO-Z schemes shown previously. The superior dispersion and dissipation properties of the compact schemes are helpful in this regard. Local density profile for the Titarev-Toro test case, shown in Fig.\ref{fig:unlim-tita-compare}, indicates that the results obtained by HOCUS5 scheme and the compact scheme C5 are almost identical. \textcolor{black}{As we can observe from the test cases presented in this section, the HOCUS6 is superior to HOCUS5 and also captures the discontinuities without oscillations.} \begin{figure}[H] \centering\offinterlineskip \subfigure[]{\includegraphics[width=0.33\textwidth]{BVD-unlim-sod.pdf} \label{fig:unlim-sod}} \subfigure[]{\includegraphics[width=0.33\textwidth]{BVD-unlim-lax.pdf} \label{fig:unlim-lax}} \subfigure[]{\includegraphics[width=0.325\textwidth]{BVD-unlim-shu.pdf} \label{fig:unlim-shu}} \subfigure[]{\includegraphics[width=0.325\textwidth]{BVD-unlim-tita.pdf} \label{fig:unlim-tita}} \subfigure[]{\includegraphics[width=0.475\textwidth]{BVD-tita-zoom-unlim-zoom.pdf} \label{fig:unlim-tita-compare}} \caption{Density profiles by C5 and HOCUS5 schemes for various test cases. Figs. (a) and (b) Sod and Lax problem in \ref{sod}, Fig. (b) corresponds to Example \ref{Shu-Osher} and Fig. (d) Titarev-Toro problem case-1 of \ref{Titarev-Toro}. } \label{fig_unlim} \end{figure} \item Nishikawa has shown in \cite{nishikawa2018green} that for a discontinuous solution, linear compact finite-volume schemes converged even without a gradient limiter with mild oscillations even on an unstructured grid, where the other solvers diverged which motivated us to test linear-compact reconstruction scheme for these test cases. Hu et al. also showed it in Ref \cite{hu2015efficient}, although with a different shock-detection algorithm for shock-capturing, that the fifth-order linear explicit upwind scheme tolerates a small overshoot or undershoot. \item These results indicate that the BVD algorithm automatically chooses the highest polynomial for the smooth solution, acts as a capable discontinuity detector, and in turn preserves the wave-like phenomenon. \end{itemize} \subsection{Two-dimensional test cases for Euler equations} In this section we show numerical results of the proposed method for two-dimensional Euler equations. \begin{example}\label{Explosion}{Explosion problem} \end{example} From this test case onwards we consider the two-dimensional Euler equations. In this example, the initial condition consists of two constant states for the flow variables, a circular region of radius $r=0.4$ centered at $(1,1)$ and the region outside of it as mentioned in Toro \cite{toro2009riemann}. The initial conditions are given as: \begin{equation}\label{explosion} (\rho, u,v, p)=\left\{\begin{array}{ll} (1, 0, 0, 1), ~~~~~~~~~ \mbox{If} ~~{x^2+y^2} <r^2, \\ (0.125, 0, 0, 0.1), ~~~~~~~ \mbox{else}. \\ \end{array}\right. \end{equation} In the present case, numerical simulations are carried out over a square domain of size $[0,2]\times [0,2]$ until a final time $t=0.25$ on a uniform grid of resolution $400\times 400$. In Fig.\ \ref{fig:RM-birb}, we can see the density distribution of the simulation computed by the proposed scheme. Cross-sectional slices of density and pressure along the plane $y=0$ computed using the present scheme are shown in Figs.\ \ref{fig:RM-density} and \ref{fig:RM-pressure}, respectively. In Fig.\ \ref{fig:RM-Compare} we compare the density profile obtained by various schemes where the numerical results for the reference solution are computed using WENO-Z scheme over a uniform mesh of resolution $1000\times 1000$. We can observe that the present approach resolves the discontinuities without oscillations. \begin{figure}[H] \centering\offinterlineskip \subfigure[]{\includegraphics[width=0.35\textwidth]{BVD_riemann_2.png} \label{fig:RM-birb}} \subfigure[]{\includegraphics[width=0.35\textwidth]{BVD_explosion_density.pdf} \label{fig:RM-density}} \subfigure[]{\includegraphics[width=0.35\textwidth]{BVD_explosion_den.pdf} \label{fig:RM-Compare}} \subfigure[]{\includegraphics[width=0.35\textwidth]{BVD_explosion_pressure.pdf} \label{fig:RM-pressure}} \caption{\textcolor{black}{Solution for the Example \ref{Explosion}. Fig.\ (a) Bird's eye view of the density distribution, Figs.\ (b) and (d) shows the density and pressure along the plane $y=0$ for HOCUS6 and Fig.\ (c) comparison of all the schemes for the density profile at a local region, where solid line: reference solution; green circles: HOCUS6; cyan stars: HOCUS5; blue squares: WENO-Z; red triangles: MP5.}} \label{fig_explosion} \end{figure} \begin{example}\label{euler-accuracy}{Accuracy of 2D Euler equations} \end{example} In this test case we again evaluate the accuracy of the proposed schemes for the compressible Euler equations considering the following initial conditions: \begin{align}\label{accu-euler} (\rho,u,v,p)= \begin{cases} &(1+0.5 \sin(x+y),\ \ 1.0,\ \, 1.0 \ \ 1.0)\\ \end{cases} \end{align} over the domain $[-1, 1]$ $\times$ $[-1, 1]$. Numerical solutions are computed until a final time $t=2$. The time step is taken as $\Delta t=0.1\Delta x^{2.0}$. Table \ref{tab:case3a} shows the $L_1$-errors and convergence rates of the numerical schemes. \textcolor{black}{We can observe from Table \ref{tab:case3a} that the both HOCUS5 and HOCUS6 schemes have achieved design order of accuracy and convergence rate.} \begin{table}[H] \centering \footnotesize \caption{\textcolor{black}{$L_1$ errors and numerical orders of accuracy for test case in Example \ref{euler-accuracy}. $N$ is the total number of cells on a uniform mesh. t = 2.}} \begin{tabular}{ccccccccc} \hline & WENOZ & & MP5 & & HOCUS5 & & HOCUS6 & \\ \hline & & order & & order & & order & & \\ \hline $20^2$ & 2.27E-04 & & 2.12E-04 & & 5.38E-05 & & 4.20E-05 & \\ \hline $40^2$ & 6.84E-06 & 5.05 & 6.81E-06 & 4.96 & 1.19E-06 & 5.50 & 7.24E-07 & 5.86 \\ \hline $80^2$ & 2.20E-07 & 4.96 & 2.20E-07 & 4.96 & 2.97E-08 & 5.32 & 1.04E-08 & 6.12 \\ \hline $160^2$ & 7.30E-09 & 4.91 & 7.13E-09 & 4.94 & 8.13E-10 & 5.19 & 1.77E-10 & 5.87 \\ \hline \end{tabular}% \label{tab:case3a}% \end{table}% The following points are important to emphasize: \begin{itemize} \item \textcolor{black}{In general, flux integrals should be computed using a high-order Gaussian quadrature with suitable Gaussian integration points over the faces of the control volume to achieve high-order accuracy, third-order or more,} see \cite{titarev2004finite,coralic2014finite,haimovich2017numerical}. Since the flux Jacobian $\partial F/ \partial Q$ is constant for a linear equation, the dimension-by-dimension variable extrapolation also achieves formal order of accuracy without quadratures for a linear problem. The above test case is a linear system, and therefore we can obtain the high-order of accuracy. \item In order to obtain high-order accuracy for a nonlinear problem, a high-order Gaussian quadrature integration or any other suitable approach, has to be used, which is not considered this paper. We are of aware of this as the first author has previously used the same finite-volume schemes in \cite{chamarthi2018high} for the computation of \textit{linear} diffusion and advection-diffusion equations. For nonlinear problems, a finite-difference scheme \cite{chamarthi2019first} has been used to obtain high-order accuracy. \item It is also demonstrated by Zhang et al. in Ref. \cite{zhang2011order} that the finite-volume WENO method with mid-point rule is only second-order accurate for nonlinear systems and Gaussian integral rule is necessary for high-order accuracy. However, they also noted that for flows involving shock waves, which are non-smooth, the resolution characteristics are often comparable despite the obvious difference in the order of accuracy. In the present paper, we are also interested in flows involving discontinuities and we have used the mid-point rule for our computations. In the original BVD algorithm of Sun et al. \cite{sun2016boundary} and the subsequent papers that improved the original scheme, the computations are also based on the finite-volume method with mid-point rule. \item It is important to note that the present scheme interpolates the primitive or characteristic variables to the interfaces, and the BVD algorithm given by Equation (\ref{eqn:BVDstep}) will not work with conservative variables or fluxes. It is also possible to attain high-order accuracy for smooth problems without quadratures for a finite volume approach using the BVD algorithm and is currently being investigated and will be presented elsewhere. \end{itemize} \begin{example}\label{ex:rp}{Riemann Problem} \end{example} The third 2D problem is Riemann problem taken from Schulz-Rinne et al. \cite{schulz1993numerical}, described as configuration 3. The simulation is carried out over a unit square domain $[0,1]\times [0,1]$, with the following initial data: \begin{equation} (\rho, u,v, p)=\left\{\begin{array}{ll} (1.5, 0, 0, 1.5), ~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{if} ~~x > 0.8, ~~y > 0.8, \\ (33/62, 4/\sqrt{11}, 0, 0.3), ~~~~~~~~~~~~~~\mbox{if} ~~ x\leq 0.8, ~~y > 0.8, \\ (77/558, 4/\sqrt{11}, 4/\sqrt{11}, 9/310), ~~\mbox{if} ~~ x \leq 0.8, ~~y\leq 0.8, \\ (33/62, 0, 4/\sqrt{11}, 0.3), ~~~~~~~~~~~~~~\mbox{if} ~~x > 0.8, ~~ y\leq 0.8. \end{array}\right. \end{equation} This initial condition, with constant states of primitive variables along the lines $x=0.8$ and $y=0.8$, produces four shocks at the interfaces of the four quadrants. The small-scale complex structures generated along the slip lines due to Kelvin-Helmholtz instability serves as a benchmark to test numerical dissipation of the scheme. The numerical solutions are computed for time $t=0.8$ on a grid of size $400\times 400$. Non-reflective boundary conditions are applied at all the boundaries for this test case. Figure \ref{fig_riemann} shows the density contours obtained by the various schemes. A closer look at Fig. \ref{fig:HOCUS6_RM} indicates that the roll-up of the slip line forms more vortices for the HOCUS6 scheme in comparison with the WENO-Z, \textcolor{black}{MP5 and HCOUS5 schemes}, and also the various reflected shocks are well captured despite using the non-dissipative scheme. This test case can also be called as double Mach reflection and additional comments regarding roll-up vortices are given in Example \ref{ex:dmr}. \begin{figure}[H] \begin{onehalfspacing} \centering\offinterlineskip \subfigure[WENO-Z]{\includegraphics[width=0.4\textwidth]{Riemann-WENO.pdf} \label{fig:WENO-Z_RM}} \subfigure[MP5]{\includegraphics[width=0.4\textwidth]{Riemann-MP5.pdf} \label{fig:MP5_RM}} \subfigure[\textcolor{black}{HOCUS5}]{\includegraphics[width=0.4\textwidth]{Riemann-BVD5.pdf} \label{fig:HOCUS5_RM}} \subfigure[HOCUS6]{\includegraphics[width=0.4\textwidth]{Riemann-BVD6.pdf} \label{fig:HOCUS6_RM}} \caption{\textcolor{black}{Solution of the Riemann problem, Example \ref{ex:rp}, indicating density contours obtained for WENO-Z, MP5, HOCUS5 and HOCUS6 schemes.}} \label{fig_riemann} \end{onehalfspacing} \end{figure} \begin{example}\label{shock-entropy}{2D shock-entropy wave test} \end{example} In this test case we consider the two-dimensional shock-entropy wave interaction problem proposed in \cite{acker2016improved}. The initial conditions for the test case are as follows, \begin{align}\label{shock_entropy} (\rho,u,p)= \begin{cases} &(3.857143, \ \ 2.629369,\ \ 10.3333),\quad x<-4,\\ &(1+0.2\sin(10x \cos\theta+10y\sin\theta),\ \ 0,\ \ 1),\quad otherwise, \end{cases} \end{align} with $\theta$ = $\pi/6$ over a domain of $[-5,5]\times [-1,1]$. The initial sine waves make an angle of $\theta$ radians with the $x$ axis. Initial conditions are modified as in \cite{deng2019fifth} with a higher frquency for the initial sine waves compared to that of \cite{acker2016improved} to show the benefits of the proposed method. Mesh size of $400\times 80$, which corresponds to $\Delta x= \Delta y =1/40$, is used for the computations. Numerical results in Fig. \ref{fig_shock_entropy} indicate that the resolution of the flow structures is significantly improved by HOCUS6 scheme, Fig. \ref{fig:SE-BVD}, in comparison with the \textcolor{black}{HOCUS5}, WENO-Z and MP5, shown in Figs. \ref{fig:SE-MP5} and \ref{fig:SE-HOCUS5}, respectively. The local density profile in the high-frequency region along $y=0$, shown in Fig. \ref{fig:SSE-Compare}, indicates that the HOCUS6 approach predicts the density amplitudes with lower numerical dissipation compared to the other methods. The reference solution in Fig. \ref{fig:SSE-Compare} is computed on a very fine mesh of $1600 \times 320$ by the WENO-Z scheme. The current approach is also significantly better than the P4T2 scheme proposed by Deng et al. in \cite{deng2019fifth} (see their Fig. 21(d)) which improves only marginally over WENO-Z. These results indicate that the present approach is well suited for compressible shock-turbulence interactions.\\ \begin{figure}[H] \centering\offinterlineskip \subfigure[MP5]{\includegraphics[width=0.48\textwidth]{Entropy-MP5.pdf} \label{fig:SE-MP5}} \subfigure[\textcolor{black}{HOCUS5}]{\includegraphics[width=0.48\textwidth]{Entropy-BVD5.pdf} \label{fig:SE-HOCUS5}} \subfigure[HOCUS6]{\includegraphics[width=0.48\textwidth]{Entropy-BVD6.pdf} \label{fig:SE-BVD}} \subfigure[Local profile]{\includegraphics[width=0.495\textwidth]{Entropy-shock-2D.pdf} \label{fig:SSE-Compare}} \caption{\textcolor{black}{15 density contours for the 2D shock-entropy wave test at $t=1.8$, Example \ref{shock-entropy}, for various schemes are shown in Figs. (a), (b) and (c). Fig. (d) shows the local density in the region with high-frequency waves for all the schemes where solid line: reference solution; green circles: HOCUS6; cyan stars: HOCUS5; blue squares: WENO-Z; red triangles: MP5.}} \label{fig_shock_entropy} \end{figure} \begin{example}\label{ex:rm} {Richtmeyer - Meshkov instability} \end{example} In this example, we consider the two-dimensional single-mode \textcolor{black}{Richtmeyer--Meshkov} instability problem \cite{terashima2009front}. This phenomenon occurs when a shock wave approaches a perturbed interface separating two fluids of different densities. To simplify the physical model we consider that the two different fluids have same specific heat ratio of $\gamma$ =1.4 as is considered in \cite{deng2019fifth}. The computational domain has size of $\left[0, 4 \right] \times \left[ 0, 1 \right]$ and the following initial conditions are used: \begin{equation*} \left( \rho, u, v, p \right) = \begin{cases} \left(1, 0, 0, 1 \right), &\mbox{if x $<$ 3.2}, \\ \left(1.4112, -665/1556, 0, 1.628 \right), &\mbox{if x $<$ 2.9 sin(2$\pi$(y+0.25), perturbed interface}, \\ \left(5.04, 0, 0, 1 \right), &\mbox{for $\mathrm{otherwise}$}. \\ \end{cases} \end{equation*} This initial condition indicates that a shock wave is propagating from the left to the interface imposed minimal disturbance. The computation is conducted until $t=9$, on the uniform mesh size of $320 \times 80$. The upper and lower boundaries were treated as periodic boundary conditions through the use of ghost cells and for the left, and right boundary values are fixed to initial conditions. Fig. \ref{fig_rm} shows the density distribution obtained by various schemes, and we can notice that the HOCUS6 scheme has lower numerical dissipation and produces small scale roll-up vortices due to the instability compared to MP5 and WENO-Z. \textcolor{black}{The difference between HOCUSS5 and HOCUS6 is minimal for this test case.} \begin{figure}[H] \begin{onehalfspacing} \centering\offinterlineskip \subfigure[WENO-Z]{\includegraphics[width=0.6\textwidth]{RT-WENO.pdf} \label{fig:WENO-Z_RT}} \subfigure[MP5]{\includegraphics[width=0.6\textwidth]{RT-MP5.pdf} \label{fig:MP5_RT}} \subfigure[HOCUS5]{\includegraphics[width=0.6\textwidth]{RT-BVD5.pdf} \label{fig:HOCUS5_RT}} \subfigure[HOCUS6]{\includegraphics[width=0.6\textwidth]{RT-BVD.pdf} \label{fig:HOCUS6_RT}} \caption{\textcolor{black}{24 equally spaced contours of density at t = 9 for various schemes for the Richtmeyer-Meshkov instability test case, Example \ref{ex:rm}}} \label{fig_rm} \end{onehalfspacing} \end{figure} \begin{example}\label{ex:rt} {Rayleigh-Taylor instability} \end{example} This type of instability occurs at an interface between fluids of different densities when an acceleration is directed from heavier fluid to lighter fluid. The objective of this test case is to show the feature of dissipation, with bubbles of light fluid rising into the ambient heavy fluid and spikes of heavy fluid falling into the light fluid and resulting in fingering like structure, in two dimensions. The initial conditions of Rayleigh-Taylor instability are \cite{xu2005anti} \begin{equation} \begin{aligned} (\rho,u,v,p)= \begin{cases} &(2.0,\ 0,\ -0.025\sqrt{\frac{5p}{3\rho}\cos(8\pi x)},\ 2y+1.00),\quad 0\leq y< 0.5,\\ &(1.0,\ 0,\ -0.025\sqrt{\frac{5p}{3\rho}\cos(8\pi x)},\ 1y+0.75),\quad 0\leq y\leq 0.5, \end{cases} \end{aligned} \label{eu2D_RT} \end{equation} over the computational domain $[0, 1/4]\times [0,1]$. Reflective boundary conditions are imposed on the right and left boundaries via ghost cells. The flow conditions are set to $\rho=1$, $p=2.5$, and $u=v=$0 on top boundary and $\rho=2$, $p=1.0$, and $u=v=0$ on bottom boundary. The source term $S=(0,0,\rho, \rho v)$ is added to the Euler equations. We performed simulations on a uniform mesh of resolutions $80 \times 320$ and \textcolor{black}{$120 \times 480$} and the computations are conducted until $t = 1.95$. The value of adiabatic constant $\gamma$ is taken to be $5/3$. Figs. \ref{fig:2d-RT} and \ref{fig:2d-RT1} indicates the density distribution of the Rayleigh-Taylor instability problem and it can be seen that HOCUS6 resolved the finer structures in comparison with WENO-Z, \textcolor{black}{MP5 and HOCUS5}. \begin{figure}[H] \begin{onehalfspacing} \centering\offinterlineskip \subfigure[WENO-Z]{% \includegraphics[width=0.20\textwidth]{RYT-WENOZ-80.pdf} \label{fig:RT_Z}} \subfigure[MP5]{% \includegraphics[width=0.20\textwidth]{RYT-MP5-80.pdf} \label{fig:RT_MP5}} \subfigure[HOCUS5]{% \includegraphics[width=0.20\textwidth]{RYT-BVD5-80.pdf} \label{fig:RT_HOCUS5}} \subfigure[HOCUS6]{% \includegraphics[width=0.20\textwidth]{RYT-BVD6-80.pdf} \label{fig:RT_HOCUS6}} \caption{\textcolor{black}{Comparison of density contours obtained by WENO-Z, MP5, HOCUS5 and HOCUS6 on a grid size of 80 $\times$ 320 for the test case in Example \ref{ex:rt}.}} \label{fig:2d-RT} \end{onehalfspacing} \end{figure} \begin{figure}[H] \begin{onehalfspacing} \centering\offinterlineskip \subfigure[WENO-Z]{% \includegraphics[width=0.20\textwidth]{RYT-WENOZ-120.pdf} \label{fig:RT_Z1}} \subfigure[MP5]{% \includegraphics[width=0.20\textwidth]{RYT-MP5-120.pdf} \label{fig:RT_MP51}} \subfigure[HOCUS5]{% \includegraphics[width=0.20\textwidth]{RYT-BVD5-120.pdf} \label{fig:RT_HOCUS51}} \subfigure[HOCUS6]{% \includegraphics[width=0.20\textwidth]{RYT-BVD6-120.pdf} \label{fig:RT_HOCUS61}} \caption{\textcolor{black}{Comparison of density contours obtained by WENO-Z, MP5, HOCUS5 and HOCUS6 on a grid size of $120 \times 480$ for the test case in Example \ref{ex:rt}.}} \label{fig:2d-RT1} \end{onehalfspacing} \end{figure} \begin{example}\label{ex:sb} {Shock-Bubble interaction} \end{example} In this test case, we consider shock-bubble interaction, where a Mach 1.22 shock wave impacts a helium bubble \cite{shankar2010numerical}. For simplicity, both Air (Shock wave) and Helium are treated as an ideal gas. The helium bubble is placed at $x = 3.5$, $y = 0.89$ within in a domain of size $[0, 6.5] \times [0, 1.78]$. The initial radius of the bubble is taken as $0.5$. The shock front is initially placed at $x = 4.5$. The initial condition of the test case is shown in Fig. \ref{fig_initibubble}. Inflow and outflow conditions are applied at the right and left boundary, respectively. Slip-wall boundary conditions are set at the remaining boundaries. The initial conditions for the problem are as follows, which are computed using exact Rankine-Hugoniot jump conditions, \begin{equation} \begin{aligned} (\rho,u,v,p)= \begin{cases} &(1.0,\ 0,\ 0,\ 1),~~~~~~~~~~~~~~~~~~~~~~ \rm{pre-shocked \ air},\\ &(1.3764,\ -0.3947,\ 0,\ 1.5698) ~~~ \rm{post-shocked \ air},\\ &(0.1819,\ 0,\ 0,\ 1)~~~~~~~~~~~~~~~~~~~ \rm{helium \ bubble}. \end{cases} \end{aligned} \label{sb} \end{equation} \textcolor{black}{Simulations are carried out on a mesh resolution of $400 \times 400$, which corresponds to $\Delta x= \Delta y = 0.0025$ until a time of $t=3.25$. Once again, we can see that HOCUS6 resolves more small scale vortices than HOCUS5, WENO-Z and MP5 in Fig.\ref{fig_shockbubble}.} \begin{figure}[H] \centering\offinterlineskip \includegraphics[width=0.7\textwidth]{bubble-init1.pdf} \caption{Initial condition for the shock-bubble interaction} \label{fig_initibubble} \end{figure} \begin{figure}[H] \begin{onehalfspacing} \centering\offinterlineskip \subfigure[WENO-Z]{\includegraphics[width=0.35\textwidth]{2_WENO_local-400.pdf} \label{fig:WENO-Z_BB1}} \subfigure[MP5]{\includegraphics[width=0.35\textwidth]{1_MP5_local-400.pdf} \label{fig:MP5_BB1}} \subfigure[HOCUS5]{\includegraphics[width=0.35\textwidth]{0_BVD5_local-400.pdf} \label{fig:HOCUS5_BB1}} \subfigure[HOCUS6]{\includegraphics[width=0.35\textwidth]{0_BVD6_local-400.pdf} \label{fig:HOCUS6_BB1}} \caption{\textcolor{black}{Density plots for various schemes, WENO-5Z, MP5, HOCUS5 and HOCUS6, at $t$ =3.25 for the Example \ref{ex:sb} on a grid spacing of 0.0025.}} \label{fig_shockbubble} \end{onehalfspacing} \end{figure} \begin{example}\label{ex:dmr}{Double Mach reflection} \end{example} In this test case, introduced by Woodward and Collela \cite{woodward1984numerical}, we simulate an unsteady planar shock wave of Mach $10$ impinging on an inclined surface which is at an angle of $30$ degrees. As typical to the setup of this problem, we tilt the incident shock wave rather than the wall to avoid modelling oblique physical wall boundary. The initial conditions for the problem are as follows: \begin{equation} \begin{aligned} (\rho,u,v,p)= \begin{cases} &(8,\ 8.25 \cos 30^{0},\ -8.25 \sin 30^{0},\ 116.5),\quad x<1/6+\frac{y}{\tan 60^{0}},\\ &(1.4,\ 0,\ 0,\ 1),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ x> 1/6+\frac{y}{\tan 60^{0}}. \end{cases} \end{aligned} \label{eu2D_mach} \end{equation} Post-shock flow conditions are set for the left boundary, and zero-gradient conditions are applied at the right boundary. At the bottom boundary, reflecting boundary conditions are used from $x =1/6 $ to $x=4.0$ and post-shock conditions for $x \in \left[ 0, 1/6 \right]$. The exact solution of the Mach 10 moving oblique shock is imposed at the upper boundary $y = 1$, which is time-dependent. Simulation is carried over the domain $[0,4]\times [0,1]$ and solutions are computed until time $t=0.2$, with a constant CFL number of $0.2$ for MP5, \textcolor{black}{HOCUS5} and HOCUS6 and $0.4$ for WENO-Z scheme respectively, on a grid size of $1024 \times 256$. As there is no physical viscosity in the Euler equations, vortices generated along the slip line due to Kelvin Helmholtz instability and the near-wall jet will indicate the numerical dissipation of a scheme. We can notice from Fig.\ref{fig_doublemach}, the results obtained by the HOCUS6 \textcolor{black}{are better than} WENO-Z and MP5 as it captures more number of vortices along the slip lines and a strong near-wall jet. \textcolor{black}{For this test case the differences between HOCUS5 and HOCUS6 are marginal} Additional simulations are also conducted for this test case, and the observations are as follows: \begin{itemize} \item Simulations are also carried out with the dissipative Global Lax-Friedrich (GLF) Riemann solver, shown in Fig.\ref{fig_doublemach_LF}. It can be seen from Figs. \ref{fig_doublemach} and \ref{fig_doublemach_LF} that WENO-Z is dissipative regardless of the Riemann solver and whereas the adaptive scheme, HOCUS6, has the small scale vortices along the slip line which indicates the advantage of the adaptive non-dissipative scheme. \item Both MP5 and HOCUS6 schemes are found to be preserving positivity of density and pressure for both HLLC and global Lax-Friedrich Riemann solvers for a constant CFL of $0.2$, and stable computations are carried out by WENO-Z by using a CFL of $0.4$ similar to that of Ref. \cite{fu2019low}. Positivity preserving method developed in \cite{hu2013positivity} may be considered to improve robustness. \item HLLC Riemann solver used in this paper will lead to \textit{carbuncle} phenomenon, grid-aligned shock instability, and it can be avoided by using a recently proposed low-dissipation modified component-wise Lax-Friedrichs flux by Fleischmann et al. \cite{fleischmann2020low}. Simulations are also carried out with the proposed method, results not shown here, which further improved the simulations. \item Even though it is not related to the current paper but it is observed that the THINC method used in the original BVD algorithm \cite{sun2016boundary} and the subsequent papers \cite{deng2019fifth, deng2018limiter} with values $\beta=1.2$ or $1.3$ can have compressive or anti-diffusion effect in the numerical results. This anti-diffusion effect improves the roll-up vortices along the slip lines for this test case, see Fig. 11 in Deng et al. \cite{deng2018limiter} and the Riemann problem in Example \ref{ex:rp}. Similar results are also observed for WENO schemes coupled with anti-diffusive flux corrections, see Fig.\ 17 in Xu and Shu \cite{xu2005anti}. A similar approach may be considered for the present HOCUS6 scheme in future. \end{itemize} \begin{figure}[H] \centering\offinterlineskip \subfigure[WENO-Z - HLLC]{\includegraphics[width=0.45\textwidth]{DMR-WENO-local.pdf} \label{fig:WENO-Z_DB}} \subfigure[MP5 - HLLC]{\includegraphics[width=0.45\textwidth]{DMR-MP5-local.pdf} \label{fig:MP5_DB}} \subfigure[HOCUS5 - HLLC]{\includegraphics[width=0.45\textwidth]{DMR-BVD5-local.pdf} \label{fig:HOCUS5_DB}} \subfigure[HOCUS6 - HLLC]{\includegraphics[width=0.45\textwidth]{DMR-BVD-local.pdf} \label{fig:HOCUS6_DB}} \caption{\textcolor{black}{Density contours at $t = 0.2$ for the three numerical schemes in the blown-up region around the Mach stem for Example \ref{ex:dmr} using HLLC Riemann solver on a grid size of $1024 \times 256$.}} \label{fig_doublemach} \end{figure} \begin{figure}[H] \centering \subfigure[WENO-Z - GLF]{\includegraphics[width=0.4\textwidth]{DMR-WENO-laxmax.pdf} \label{fig:WENO-Z_DB-LF}} \subfigure[MP5 - GLF]{\includegraphics[width=0.4\textwidth]{DMR-MP5-laxmax.pdf} \label{fig:MP5_DB-LF}} \subfigure[HOCUS5 - GLF]{\includegraphics[width=0.4\textwidth]{DMR-BVD5-laxmax.pdf} \label{fig:HOCUS5_DB-LF}} \subfigure[HOCUS6 - GLF]{\includegraphics[width=0.4\textwidth]{DMR-BVD-laxmax.pdf} \label{fig:HOCUS6_DB-LF}} \caption{\textcolor{black}{Density contours at $t = 0.2$ for the three numerical schemes in the blown-up region around the Mach stem for Example \ref{ex:dmr} using using GLF Riemann solver on a grid size of $1024 \times 256$.}} \label{fig_doublemach_LF} \end{figure} \begin{figure}[H] \begin{onehalfspacing} \centering\offinterlineskip \subfigure[WENO-Z-HLLC]{\includegraphics[width=0.7\textwidth]{DMR-WENO.pdf} \label{fig:WENO-Z_DB-full}} \subfigure[MP5-HLLC]{\includegraphics[width=0.7\textwidth]{DMR-MP5.pdf} \label{fig:MP5_DB-full}} \subfigure[HOCUS5-HLLC]{\includegraphics[width=0.7\textwidth]{DMR-BVD5.pdf} \label{fig:HOCUS5_DB-full}} \subfigure[HOCUS6-HLLC]{\includegraphics[width=0.7\textwidth]{DMR-BVD.pdf} \label{fig:HOCUS6_DB-full}} \caption{\textcolor{black}{Double Mach reflection, Example \ref{ex:dmr}, $40$ equally spaced contours for the density on a grid size of $1024 \times 256$, $t = 0.2$ by HLLC Riemann solver with various schemes.}} \label{fig_doublemach-full} \end{onehalfspacing} \end{figure} \begin{example}\label{ex:vs}{Viscous Shock tube} \end{example} \textcolor{black}{In this last test case, we consider the viscous shock tube test case to demonstrate the resolution ability of the proposed schemes. This test case simulates the two-dimensional shock-wave and boundary layer interaction. A shock wave is initialized at the mid-location in the domain which interacts with a developing boundary layer, as it reflects from the slip wall at the right boundary. Also, the contact discontinuity and the reflected shock wave interact with each other. The domain for this test case is taken as $x \in [0,1], y \in [0,0.5]$. Viscosity is assumed to be constant, and the Prandtl number is taken as 0.73. A symmetry boundary condition is imposed on the top boundary, and no-slip wall boundary condition is used for other boundaries. The initial conditions are given by equation \eqref{vst} keeping the Mach number of the shock wave at 2.37.} \textcolor{black}{ \begin{equation}\label{vst} \left( {\rho , u,v, p} \right) = \left\{ \begin{array}{l} \left( {120, 0 ,0,120/\gamma } \right), 0 < x < 0.5,\\ \left( {1.2, 0,0, 1.2/\gamma } \right), 0.5 \le x < 1, \end{array} \right. \end{equation}} \textcolor{black}{The problem is solved for a Reynolds number of $Re=500$ on a grid size of $500 \times 200$ cells for time $t=1$. The sixth-order compact scheme is used to compute the velocity and temperature gradients, and the viscous stresses and heat-fluxes are calculated by 6th-order interpolation scheme \cite{lele1992compact,fang2014investigation}. As shown in Fig. \ref{fig_VST}, the results of WENO-Z and MP5 schemes show significant differences from the ``converged'' results in Ref. \cite{daru2009numerical} (see Fig 8b). Both HOCUS5 and HOCUS6 are closer to the reference result despite using a coarse mesh. The primary vortex of HOCUS6 matches well with the results of the reference, obtained on a grid size of 1500 $\times$ 750, whereas that from HOCUS5 is slightly distorted.} \textcolor{black}{As shown in Table \ref{tab:comptime}, it is observed that the HOCUS6 scheme takes more computational time than the WENOZ and MP5 schemes for a given grid size. Concerning the performance, the WENOZ and MP5 schemes need twice the resolution (results not shown here) in order to obtain comparable results as HOCUS6, which leads to two-times higher computing times. It shows that HOCUS schemes do not significantly increase the computational cost.} \begin{table}[H] \centering \footnotesize \caption{\textcolor{black}{Computing time for Viscous shock tube test case, Example \ref{ex:vs}}} \begin{tabular}{lcccc} \hline & & \multicolumn{2}{c}{Numerical method} & \\ \hline & WENOZ & MP5 & HOCUS5 & HOCUS6 \\ \hline Grid & & & & \\ \hline \multicolumn{1}{c}{500 $\times$ 250} & 1240s & 1269s & 1718s & 1704s \\ \hline 750 $\times$ 375 & 4200s & 4312s & 5841s & 5823s \\ \hline \end{tabular}% \label{tab:comptime}% \end{table}% \begin{figure}[H] \centering \subfigure[WENO-Z]{\includegraphics[width=0.45\textwidth]{VST_WENOZ.pdf} \label{fig:WENO-Z_VST}} \subfigure[MP5]{\includegraphics[width=0.45\textwidth]{VST_MP5.pdf} \label{fig:MP5_VSST}} \subfigure[HOCUS5]{\includegraphics[width=0.45\textwidth]{VST_HOCUS5.pdf} \label{fig:HOCUS5_VST}} \subfigure[HOCUS6]{\includegraphics[width=0.45\textwidth]{VST_HOCUS6.pdf} \label{fig:HOCUS6_VST}} \caption{\textcolor{black}{Density contours for different schemes for the Example \ref{ex:vs} with $Re=500$ on a grid size of $500 \times 250$.}} \label{fig_VST} \end{figure} \section{Conclusions}\label{sec-4} We have developed a new hybrid scheme that combines an explicit MP5 scheme and a linear upwind compact reconstruction scheme using a BVD algorithm to enable high-resolution of smooth flow features and low-dissipation non-oscillatory shock capturing. The BVD algorithm, which serves as a discontinuity detector, chooses the highest possible interpolation for the smooth solution, which is central and non-dissipative in the current approach and for the discontinuities, the algorithm prefers the monotone scheme. Primitive variables are interpolated for the compact scheme, and characteristic variable interpolation is \textcolor{black}{carried out for the monotone scheme} for cleaner results. It, therefore, reduces the numerical dissipation inherently present in the upwind schemes and Riemann solvers, which \textcolor{black}{lead} to significant improvements in the numerical solutions for all the test cases considered in this paper. This method provides an alternative to the existing adaptive upwind-central schemes in the literature. It is possible to improve the current approach by optimizing dissipation properties of the MP5 scheme as in \cite{fang2013optimized}. In future, we will extend the methodology to the compressible multi-component flows and unstructured grids. \section*{Acknowledgements} A.S. is supported by Technion fellowship during this work. \section*{Appendix} \renewcommand{\thesubsection}{\Alph{subsection}} \textcolor{black}{In this appendix we present the various combinations of the numerical schemes that are also considered as part of this research work.} \subsection{\textcolor{black}{HOCUS-TVD}} \label{sec-appb} {\textcolor{black}{In this subsection, we present the hybrid approach by combining the compact reconstruction and the MUSCL schemes, named as HOCUS-TVD, with minmod limiter through boundary variation diminishing algorithm. The algorithm is the same as in Section \ref{sec2.2} except for the modification where the MUSCL scheme replaces the MP5 scheme in Equation \ref{Eq:TBVSC}.} {\textcolor{black}{\begin{equation}\label{Eq:TBV-SC} TBV_{j}^{MUSCL}=\big|U_{j-\frac{1}{2}}^{L,MUSCL}-U_{j-\frac{1}{2}}^{R,MUSCL}\big|+\big|U_{j+\frac{1}{2}}^{L,MUSCL}-U_{j+\frac{1}{2}}^{R,MUSCL} \big|. \end{equation} Third order MUSCL scheme is briefly outlined below.} \begin{eqnarray} u_{i+1 / 2}^{R}&=u_{i+1}-\frac{1}{4}\left[(1-\bar{\eta}) \tilde \Delta_{i+3 / 2} u+(1+\bar{\eta}) \ddot \Delta_{i+1 / 2} u\right] \\ u_{i+1 / 2}^{L}&=u_{i+0}+\frac{1}{4}\left[(1-\bar{\eta}) \ddot \Delta_{i-1 / 2} u+(1+\bar{\eta}) \tilde \Delta_{i+1 / 2} u\right] \end{eqnarray} \begin{eqnarray} \tilde\Delta_{i+1 / 2} u &=\operatorname{minmod}\left(\Delta_{i+1 / 2} u, \omega \Delta_{i-1 / 2} u\right) \\ \ddot\Delta_{i+1 / 2} u&=\operatorname{minmod}\left(\Delta_{i+1 / 2} u, \omega \Delta_{i+3 / 2} u\right) \end{eqnarray} {\textcolor{black}{ where, $1 \leq \omega \leq \frac{3-\eta}{1-\eta}$ and minmod function defined as } \[ \mbox{minmod}(a,b) = \left\{ \begin{array}{cl} a & \mbox{if $|a|<|b|$ and $ab>0$} \\ b & \mbox{if $|b|<|a|$ and $ab>0$} \\ 0 & \mbox{if $ab\leq 0$} \end{array} \right. \] {\textcolor{black}{The coefficient $\bar{\eta}$ determines the order of accuracy. A second-order accuracy is achieved by using $\bar{\eta}=-1$ and a third-order accuracy is achieved by using $\bar{\eta}=\frac{1}{3}$. In this paper we considered third order accurate scheme for all our computations. We carried out numerical experiments for the test cases \ref{ex:adv_comp}, \ref{Titarev-Toro}, \ref{Shu-Osher}, \ref{blast}, \ref{ex:dmr} and \ref{ex:vs} using the HOCUS-TVD algorithm. The parameter $\omega$ is taken as 4 for all the simulations. The results are shown in Fig.\ \ref{fig_tvd-1D}. All the numerical simulations \textcolor{black}{are carried out} with a CFL of $0.2$ and the grid sizes are same as that of HOCUS6. We make the following observations:} \begin{itemize} \item \textcolor{black}{Figure \ref{fig:Jiang-tvd} shows the results for the advection of complex waves test case, \ref{ex:adv_comp}. We can see that the HOCUS-TVD is slightly more dissipative than the HCOUS6 method for the semi-elliptical profile. Figures \ref{fig:tita-tvd}, \ref{fig:shu-tvd} and \ref{fig:tita-blast} show density profile for the one-dimensional Euler test cases \ref{Titarev-Toro}, \ref{Shu-Osher}, and \ref{blast} respectively. We can see that in all the cases HCOUS-TVD is more dissipative than the HOCUS6.} \item \textcolor{black}{The two-dimensional test cases are more representative of the properties of the HOCUS-TVD scheme as shown in Figures \ref{fig_tvd-2D}. Fig. \ref{fig:DMR-hllc-tvd} shows the density contours of the Double Mach reflection test case which indicates that the HOCUS-TVD is more dissipative than HOCUS6, shown in \ref{fig:HOCUS-VST-tvd}, as well as the MP5 scheme, shown in Fig. \ref{fig:HOCUS5_DB}. Similar results can be seen for the Viscous shock tube simulation where the primary vortex is distorted compared to the HOCUS6 method, as shown in Figs. \ref{fig:HOCUS-VST-tvd} and \ref{fig:HOCUS6-VST-tvd}.} \item \textcolor{black}{As explained in the introduction, we need at least five points to distinguish between, and extrema and a genuine discontinuity. These results indicate that the combination of compact and MP5 schemes are superior over the combination of compact and TVD schemes at least for the present algorithm.} \end{itemize} \begin{figure}[H] \centering\offinterlineskip \subfigure[]{\includegraphics[width=0.7\textwidth]{HOCUS_TVD_jiang.pdf} \label{fig:Jiang-tvd}} \subfigure[]{\includegraphics[width=0.34\textwidth]{HOCUS_Titarev_TVD.pdf} \label{fig:tita-tvd}} \subfigure[]{\includegraphics[width=0.34\textwidth]{HOCUS_Shu_TVD.pdf} \label{fig:shu-tvd}} \subfigure[]{\includegraphics[width=0.34\textwidth]{HOCUS_Blast_TVD.pdf} \label{fig:tita-blast}} \caption{\textcolor{black}{Density profiles by HOCUS-TVD and HOCUS6 schemes for one-diemnsional test cases. Figure (a) corresponds to Example \ref{ex:adv_comp}, (b) Example \ref{Titarev-Toro}, (c) Example \ref{Shu-Osher} and (d) corresponds to Example \ref{blast} respectively. solid line: reference solution; green circles: HOCUS6; blue squares: HOUCS-TVD.} } \label{fig_tvd-1D} \end{figure} \begin{figure}[H] \centering\offinterlineskip \subfigure[]{\includegraphics[width=0.44\textwidth]{HOCUS_DMR_TVD_HLLC.pdf} \label{fig:DMR-hllc-tvd}} \subfigure[]{\includegraphics[width=0.44\textwidth]{DMR-BVD-local.pdf} \label{fig:DMR-HOCUS6-tvd}} \subfigure[]{\includegraphics[width=0.44\textwidth]{HOCUS_VST_TVD.pdf} \label{fig:HOCUS-VST-tvd}} \subfigure[]{\includegraphics[width=0.44\textwidth]{VST_HOCUS6.pdf} \label{fig:HOCUS6-VST-tvd}} \caption{\textcolor{black}{Figs. (a) and (b) shows the density distribution for Example \ref{ex:dmr} and Figs. (c) and (d) shows the density distribution for Example \ref{ex:vs} using HOCUS-TVD and HOCUS6 respectively.}} \label{fig_tvd-2D} \end{figure} \subsection{\textcolor{black}{C5-T2 scheme}}\label{sec-appc} \textcolor{black}{In this subsection, we present the hybrid approach by combining the compact reconstruction and the THINC schemes, named as C5-T2, using the two-stage boundary variation diminishing algorithm proposed by Deng et al. in \cite{deng2019fifth}. The complete algorithm is as follows.} \begin{description} \item[Step 1.] \textcolor{black}{Evaluate the interface values by using two different reconstruction procedures:} \begin{enumerate}[(a)] \item \textcolor{black}{Linear upwind compact reconstruction given by Equation (\ref{eqn:upwind-compact}) and} \item \textcolor{black}{THINC scheme, which is as follows,} \begin{equation} \textcolor{black}{U_{i+1 / 2}^{L, T H I N C}=\left\{\begin{array}{l} \bar{u}_{min}+\frac{\bar{u}_{\max }}{2}\left(1+\theta \frac{\tanh (\beta)+A}{1+A \tanh (\beta)}\right) \text { if }\left({u}_{i+1}-{u}_{i}\right)\left({u}_{i}-{u}_{i-1}\right)>0 \\ {u}_{i} \text { otherwise } \end{array}\right.} \end{equation} \begin{equation} \textcolor{black}{U_{i-1 / 2}^{R, T H I N C}=\left\{\begin{array}{ll} \bar{u}_{\min }+\frac{\bar{u}_{\max }}{2}(1+\theta A) & \text { if }\left({u}_{i+1}-{u}_{i}\right)\left({u}_{i}-{u}_{i-1}\right)>0 \\ {u}_{i} \text { otherwise }, \end{array}\right.} \end{equation} where, $A=\frac{B / \cosh (\beta)-1}{\tanh (\beta)}$, $B=\exp (\theta \beta(2 C-1))$, $C=\frac{{u}_{i}-\bar{u}_{\min }+\epsilon}{\bar{u}_{\max }+\epsilon}$, $\bar{u}_{\min }=\min \left({u}_{i-1}, {u}_{i+1}\right), {u}_{\max }=\max \left({u}_{i-1}, {u}_{i+1}\right)-{u}_{\min } \text { and } \theta=\operatorname{sgn}\left({u}_{i+1}-{u}_{i-1}\right)$ \end{enumerate} \item[Step 2.] \textcolor{black}{Calculate the TBV values for each cell $I_{j}$ by using the compact reconstruction:} \begin{equation}\label{Eq:TBVC5-1} TBV_{j}^{C5}=\big|U_{j-\frac{1}{2}}^{L,C5}-U_{j-\frac{1}{2}}^{R,C5}\big|+\big|U_{j+\frac{1}{2}}^{L,C5}-U_{j+\frac{1}{2}}^{R,C5} \big| \end{equation} \textcolor{black}{and the THINC scheme:} \begin{equation}\label{Eq:TBV-THINC} TBV_{j}^{THINC}=\big|U_{j-\frac{1}{2}}^{L,THINC}-U_{j-\frac{1}{2}}^{R,THINC}\big|+\big|U_{j+\frac{1}{2}}^{L,THINC}-U_{j+\frac{1}{2}}^{R,THINC} \big|. \end{equation} \textcolor{black}{In this step the parameter $\beta$ in the THINC scheme is taken as 1.1.} \item[Step 4.] \textcolor{black}{Now, \textcolor{black}{all the interface values of the cells $j-1$, $j$, and $j+1$, both $R$ and $L$} are modified according to the following algorithm:} \begin{equation}\label{eqn:BVDstep-1} \begin{aligned} {\rm if } \ \ TBV_{j}^{THINC} < TBV_{j}^{C5} \end{aligned} \end{equation} \item[Step 5.] \textcolor{black}{Denote the interface values, denoted as ${U}_{j+\frac{1}{2}}^{L^{'},R^{'}}$, for stage one of the algorithm, S1. Calculate the TBV values for each cell $I_{j}$ by using the new interface values:} \begin{equation}\label{Eq:TBVS-1} TBV_{j}^{S1}=\big|U_{j-\frac{1}{2}}^{L^{'}}-U_{j-\frac{1}{2}}^{R^{'}}\big|+\big|U_{j+\frac{1}{2}}^{L^{'}}-U_{j+\frac{1}{2}}^{R^{'}} \big| \end{equation} \textcolor{black}{and the THINC scheme:} \begin{equation}\label{Eq:TBV-THINC-2} TBV_{j}^{THINC}=\big|U_{j-\frac{1}{2}}^{L,THINC}-U_{j-\frac{1}{2}}^{R,THINC}\big|+\big|U_{j+\frac{1}{2}}^{L,THINC}-U_{j+\frac{1}{2}}^{R,THINC} \big|. \end{equation} \textcolor{black}{In this step the parameter $\beta$ in the THINC scheme is taken as 1.6.} \item[Step 6.] \textcolor{black}{Now, \textcolor{black}{the interface values of cell $j$, both $R$ and $L$} is modified according to the following algorithm:} \begin{equation}\label{eqn:BVDstep-2} \begin{aligned} {\rm if } \ \ TBV_{j}^{THINC} < TBV_{j}^{S1} \end{aligned} \end{equation} \item[Step 7.] \textcolor{black}{Finally, evaluate the interface flux $\hat{F}_{j + \frac{1}{2}}$ from ${U}_{j+\frac{1}{2}}^{L,R}$ using Equation (\ref{eqn:Riemann})} \end{description} \textcolor{black}{We carried out numerical experiments for the test cases \ref{Titarev-Toro}, \ref{blast}, \ref{ex:dmr} and \ref{ex:vs} using the C5-T2 algorithm. All the numerical simulations \textcolor{black}{are carried out} with a CFL of $0.2$ and the grid sizes are same as that of HOCUS6 with the following observations:} \begin{itemize} \item \textcolor{black}{Figure \ref{fig:blast-thinc} shows the results for the blast wave test case, \ref{fig:tita-blast}. We can see that the C5-T2 has oscillations, unlike the HCOUS6 method. Similar oscillations are also observed for the P4T2 scheme of Deng at al. \cite{deng2019fifth}. For the Titarev-Toro test case, the density profile obtained by C5-T2 is similar to that of HOCUS5 scheme, which is slightly dissipative in comparison with HOCUS6.} \item \textcolor{black}{Fig. \ref{fig:DMR-THINC} shows the density contours of the Double Mach reflection test case. We can see the near-wall jet of C5-T2 scheme is distorted compared to that of HOCUS6. Similar results can be seen for the Viscous shock tube simulation where the primary vortex is distorted along with the vortical structures in the bottom-right corner due to the roll-up of the contact discontinuity and oscillations observed near the shock as shown in Figs. \ref{fig:VST-thinc} and \ref{fig:HOCUS6-VST-tvd}. HOCUS6 is more robust than the C5-T2 scheme.} \begin{figure}[H] \centering\offinterlineskip \subfigure[]{\includegraphics[width=0.44\textwidth]{C5T2_blast.pdf} \label{fig:blast-thinc}} \subfigure[]{\includegraphics[width=0.44\textwidth]{C5T2_titarev.pdf} \label{fig:tita-thinc}} \subfigure[]{\includegraphics[width=0.44\textwidth]{DMR-THINC-HLLC-local.pdf} \label{fig:DMR-THINC}} \subfigure[]{\includegraphics[width=0.44\textwidth]{C5T2_VST.pdf} \label{fig:VST-thinc}} \caption{\textcolor{black}{Density profiles by C5-T2 and HOCUS6 schemes for one and two-diemnsional test cases. Figure (a) corresponds to Example \ref{blast}, (b) Example \ref{Titarev-Toro}, (c) Example \ref{ex:dmr} and (d) corresponds to Example \ref{ex:vs} respectively. solid line: reference solution; green circles: HOCUS6; blue squares: C5-T2. }} \label{fig_thinc-12D} \end{figure} \item \textcolor{black}{Final test case we considered is the following two-dimensional Riemann problem taken from Schulz-Rinne et al. \cite{schulz1993numerical}. The simulation is carried out over a unit square domain $[0,1]\times [0,1]$, with the following initial data:} \begin{equation} \left(\rho, u, v, p_{0}\right)\\ =\left\{\begin{array}{lll} (1.0,-0.6259,0.1,1.0) & x \leq 0.0, y \geq 0.0 \\ (0.8,0.1,0.1,1.0) & x<0.0, y<0.0 \\ (0.5197,0.1,0.1,0.4) & x>0.0, y>0.0 \\ (1.0,0.1,-0.6259,1.0) & x>0.0, y<0.0 \end{array}\right. \end{equation} \textcolor{black}{The small-scale complex structures generated along the slip lines due to Kelvin-Helmholtz instability serves as a benchmark to test numerical dissipation of the scheme. The numerical solutions are computed for time $t=0.25$ on a grid of size $1000\times 1000$. Non-reflective boundary conditions are applied at all the boundaries for this test case. A closer look at Figs. \ref{fig:RR-thinc} and \ref{fig:RR-HOCUS62} indicates that the small-scale structures obtained for the C5-T2 scheme are richer in comparison with the HOCUS6 scheme. Each scheme has its strengths and disadvantages, depending on the problem being addressed.} \end{itemize} \begin{figure}[H] \centering\offinterlineskip \subfigure[C5-T2]{\includegraphics[width=0.44\textwidth]{Riemann-HOCUS_THINC.pdf} \label{fig:RR-thinc}} \subfigure[HOCUS6]{\includegraphics[width=0.44\textwidth]{Riemann-HOCUS6.pdf} \label{fig:RR-HOCUS62}} \caption{Density profiles by C5-T2 and HOCUS6 schemes for Riemann problem} \label{fig_thinc-2D-RR}s \end{figure} \subsection{\textcolor{black}{HOCUS-WENOZ}} In this subsection, we present the hybrid approach by combining the compact reconstruction and the WENO-Z through boundary variation diminishing algorithm. In Fig. \ref{fig:algo1}, we can see the results for algorithm using the Equation (\ref{eqn:BVDstep}) for the advection of complex waves given in Example \ref{ex:adv_comp}. We can notice that the BVD algorithm with WENO-Z and the linear-compact scheme is diffusive and not able to preserve the wave patterns in comparison with the HOCUS6, Fig. \ref{fig:HOCUS6_AC}, which is a combination of MP5 and the linear-compact scheme. Similar results are observed for two-dimensional simulations for the Euler equations. In light of these results, we have also implemented the algorithm given in \cite{xie2017hybrid, ivan2014high} which is as follows \begin{enumerate}[i)] \item Compute the cell-interface value by the linear compact scheme given by Equation (\ref{eqn:upwind-compact}) \begin{equation} \begin{cases} U_{{j+\frac{1}{2}}}^{L} = U_{{j+\frac{1}{2}}}^{L, C6} = \frac{1}{2}(U_{{j+\frac{1}{2}}}^{L, C5} +U_{{j+\frac{1}{2}}}^{R, C5})\\ \\ U_{{j+\frac{1}{2}}}^{R}=U_{{j+\frac{1}{2}}}^{R, C6} = \frac{1}{2}(U_{{j+\frac{1}{2}}}^{L, C5} +U_{{j+\frac{1}{2}}}^{R, C5})\\ \end{cases} \end{equation} \item Compute the smoothness indicator \begin{equation} S = \dfrac{1-TBV_{j}^{(WZ)}}{\max(TBV_{j}^{(WZ)},10^{-20})} , \end{equation} where the total boundary variation of the target cell by WENO-Z reconstruction is given by \begin{equation} TBV_{j}^{WZ}=\frac{\bigg( U^{L, WZ}_{j-\frac{1}{2}}-U^{R, WZ}_{j-\frac{1}{2}}\bigg)^4 + \bigg(U^{L, WZ}_{j+\frac{1}{2}}-U^{R, WZ}_{j+\frac{1}{2}} \bigg)^4}{\big(\hat{U}_{j}-\hat{U}_{j-1}\big)^4+\big(\hat{U}_{j}-\hat{U}_{j+1}\big)^4+10^{-20}}, \end{equation} \item Modify the admissible reconstruction via following equation \begin{equation}\label{eqn:bvd2} \begin{aligned} {\rm if } \ \ S_j < 1\times10^{6} ~~~ {U^{L,R}}_{j+\frac{1}{2}}&={U^{(L,R) WZ}}_{j+\frac{1}{2}} \end{aligned} \end{equation} and the threshold value $1\times10^{6}$ is used to determine the non-smooth solution as considered in \cite{ivan2014high}. The improved results are shown in Fig. \ref{fig:algo2} indicates that we can construct BVD algorithms that can effectively suppress the oscillations and also preserve the smooth solutions. Similar improved results are observed for one-dimensional Euler equations as well, but we noticed that the results are not improved for two-dimensional cases. \end{enumerate} \begin{figure}[H] \begin{onehalfspacing} \centering \subfigure[]{\includegraphics[width=0.48\textwidth]{l_BWENO-500.pdf} \label{fig:algo1}} \subfigure[]{\includegraphics[width=0.48\textwidth]{bv3-weno.pdf} \label{fig:algo2}} \caption{Advection of complex waves in Example \ref{ex:adv_comp} (a) by BVD algorithm with Equation (\ref{eqn:BVDstep}) and (b) with Equation (\ref{eqn:bvd2}), by combining WENO-Z and linear central compact scheme for t = 500.} \label{fig_appen-advec} \end{onehalfspacing} \end{figure} \subsection{\textcolor{black}{Accuracy for critical points test case}}\label{sec-appd} \textcolor{black}{In Step-4 of the HOCUS algorithm discussed in \ref{sec2.2} we have incorporated an additional criteria given by Equation \ref{eqn:BVDstep-extra} and the results are shown in Table \ref{tab:extra-cond}.} \begin{equation}\label{eqn:BVDstep-extra} \begin{aligned} {\rm if } \ \ TBV_{j}^{MP5} < TBV_{j}^{C5} \ \ \& \ \ \left({u}_{i+1}-{u}_{i}\right)\left({u}_{i}-{u}_{i-1}\right)\textcolor{magenta}{<} 0 \end{aligned} \end{equation} \textcolor{black}{ \begin{table}[H] \centering \footnotesize \caption{\textcolor{black} {$L_1$ errors and numerical orders of accuracy on $u_t$ + $u_x$ = 0 with $u_0$(x) = sin($\pi$x - sin($\pi$x)/$\pi$ ). $N$ is the total number of cells on a uniform mesh and $t = 2$.}} \begin{tabular}{ccccccc} \hline N & HOCUS6 & & C6 & & HOCUS6 - Extra & \\ \hline & & order & & order & & order \\ \hline 10 & 6.70E-03 & & 4.02E-03 & & 4.02E-03 & \\ \hline 20 & 1.91E-04 & 5.13 & 3.40E-05 & 6.89 & 3.40E-05 & 6.89 \\ \hline 40 & 3.99E-06 & 5.58 & 8.84E-07 & 5.26 & 8.84E-07 & 5.26 \\ \hline 80 & 7.54E-08 & 5.72 & 2.06E-08 & 5.43 & 2.06E-08 & 5.43 \\ \hline 160 & 1.26E-09 & 5.90 & 4.69E-10 & 5.46 & 4.68E-10 & 5.46 \\ \hline \end{tabular}% \label{tab:extra-cond}% \end{table}} \bibliographystyle{elsarticle-num}

BkiUf8jxK6wB9mn5Br2v

\section{The Gross-Pitaevskii equation} \label{sec:GPE} \index{Gross-Pitaevskii equation} Currently, most theoretical descriptions of exciton-polariton condensates observed \cite{Kasprzak06,Balili07,Lai07,Baumberg08,Wertz09} in incoherently excited semiconductor microcavities are based on the Gross-Pitaevskii equation (GPE). When the polarization of the condensate is of interest, this equation can be generically written as \begin{equation} \label{GPEgener} \mi\hbar\frac{\partial}{\partial t}\vec{\psi}(\vec{r},t) =\frac{\delta H}{\delta\vec{\psi}^*(\vec{r},t)}, \qquad H=\int \mathcal{H}(\vec{\psi}^*,\vec{\psi})d^2r, \end{equation} where the order parameter $\vec{\psi}(\vec{r},t)$ of the condensate is a complex 2D vector function of the 2D position in the microcavity plane $\vec{r}$ and time $t$. Alternatively, one can expand $\vec{\psi}$ on the circular polarization basis \begin{equation} \label{psicirc} \vec{\psi} =\frac{\hat{\vec{x}}+\mi\hat{\vec{y}}}{\sqrt{2}}\psi_{+1} +\frac{\hat{\vec{x}}-\mi\hat{\vec{y}}}{\sqrt{2}}\psi_{-1}, \end{equation} and obtain the coupled GPEs for two circular components $\psi_{\pm1}$ (see Eqs.~(\ref{GM_GPEdetailed}a,b) in Section \ref{sec:LeMonStar}). GPE \eqref{GPEgener} is used in two main flavors, strongly non-equilibrium GPE and quasi-equilibrium GPE, that treat the energy relaxation in two extreme ways. It is completely neglected in the former, and it is considered to be essential in the later. Mathematically, these approaches differ in Hamiltonian density $\mathcal{H}(\vec{\psi},\vec{\psi}^*)$: it is complex in the former and it is real in the later. Each approach has its benefits and drawbacks. The imaginary part of the Hamiltonian for non-equilibrium GPE \cite{Wouters07a,Wouters07b,Keeling08a,Borgh10} is given by the difference of income and escape rates of exciton-polaritons into and out of the condensate. While the escape rate is given by the reciprocal radiative life-time of exciton-polaritons and is independent of the particle density, the income rate is a non-linear function of it. The nonlinearity is essential to stabilize the solution and it appears due to the depletion of an incoherently pumped reservoir. This approach resulted to be quite successful in modeling the experimental data on condensate density profiles for spatially nonuniform exciton-polariton condensates. On the other hand, it cannot describe the spontaneous formation of linear polarization of the condensate---the fact that is quite unfortunate since the observation of spontaneous linear polarization is one of the direct experimental evidences of Bose-Einstein condensation of exciton-polaritons \cite{Kasprzak06,Balili07}. A workaround is to add the Landau-Khalatnikov relaxation into \eqref{GPEgener}, which is equivalent to adding an imaginary part to time $t$. But this relaxation is rather artificial because it changes the number of particles in the condensate. In what follows, we consider the opposite limit assuming a fast relaxation of exciton-polaritons, so that they reach quasi-equilibrium, even with the temperature that can be different from the lattice one. The balance of income and outcome rates produces some steady-state concentration of exciton-polaritons, that can be defined, as usual, by introducing the chemical potential $\mu$. \index{Chemical potential} The coherent fraction of condensed particles can be described by the traditional GPE with real $\mathcal{H}(\vec{\psi},\vec{\psi}^*)$, \begin{equation} \label{Ham} \mathcal{H}=\mathcal{T}-\mu n+\mathcal{H}_\mathrm{int}+\mathcal{H}^\prime. \end{equation} Here $\mathcal{T}$ is the density of the kinetic energy, $\mathcal{H}_\mathrm{int}$ describes interaction between the particles, $\mathcal{H}^\prime$ stands for some possible perturbations, and $n=\vec{\psi}^*\cdot\vec{\psi}$ is the exciton-polariton density. The kinetic energy of exciton-polaritons in planar microcavities depends on the orientation of vector $\vec{\psi}$ with respect to the direction of motion. Near the bottom of lower polariton branch one has \begin{subequations} \label{KinEnergy} \begin{eqnarray} \mathcal{T} &=& \frac{\hbar^2}{2m_l}|\vec{\nabla}\cdot\vec{\psi}|^2 + \frac{\hbar^2}{2m_t}|\vec{\nabla}\times\vec{\psi}|^2 \\ &=& \frac{\hbar^2}{m_l} \left|\frac{\partial\psi_{+1}}{\partial z^*}+\frac{\partial\psi_{-1}}{\partial z}\right|^2 + \frac{\hbar^2}{m_t} \left|\frac{\partial\psi_{+1}}{\partial z^*}-\frac{\partial\psi_{-1}}{\partial z}\right|^2, \end{eqnarray} \end{subequations} where $m_l$ and $m_t$ are the longitudinal and transverse effective masses \index{TE-TM splitting} of polaritons, and the complex derivatives \begin{equation} \label{ComplexDeriv} \frac{\partial}{\partial z}=\frac{1}{2}\left( \frac{\partial}{\partial x}-\mi\frac{\partial}{\partial y} \right), \qquad \frac{\partial}{\partial z^*}=\frac{1}{2}\left( \frac{\partial}{\partial x}+\mi\frac{\partial}{\partial y} \right), \end{equation} are used. The vector $\vec{\psi}$ is proportional to the in-plane electric field vector of exciton-polariton mode. According to \eqref{KinEnergy}, the frequency of transverse electric (TE) mode with in-plane wave vector $\vec{k}\perp\vec{\psi}$ is $\hbar k^2/2m_t$, while for the transverse magnetic (TM) mode with $\vec{k}\|\vec{\psi}$ the frequency is $\hbar k^2/2m_l$. (The same bare frequencies of both modes at $k=0$ are removed from \eqref{KinEnergy}.) The polariton-polariton interaction is also anisotropic: it depends on mutual orientation of $\vec{\psi}$ and $\vec{\psi}^*$. One can construct two quartic invariants from these two vectors and $\mathcal{H}_\mathrm{int}$ is given by \begin{subequations} \label{Hinter} \begin{eqnarray} \mathcal{H}_\mathrm{int} &=&\frac{1}{2}(U_0-U_1)(\vec{\psi}^*\!\cdot\vec{\psi})^2 +\frac{1}{2}U_1|\vec{\psi}^*\times\vec{\psi}|^2 \\ &=&\frac{1}{2}U_0\left(|\psi_{+1}|^4+|\psi_{-1}|^4\right) +(U_0-2U_1)|\psi_{+1}|^2|\psi_{-1}|^2. \end{eqnarray} \end{subequations} It is seen that $U_0$ is the amplitude of interaction of polaritons with the same circular polarization (with the same spin), and $U_0-2U_1$ is the amplitude of interaction of polaritons with opposite circular polarizations (opposite spins). These quantities are denoted by $\alpha_1$ and $\alpha_2$ in some papers. The constant $U_0$ is positive and can be estimated as $\sim\mathcal{E}_ba_B^2$, where $\mathcal{E}_b$ is the exciton binding energy and $a_B$ is the exciton Bohr radius. The interaction of exciton-polaritons with opposite spins depends substantially on the electron-electron and hole-hole exchange processes and is defined by the electron and hole confinement within quantum wells and by the number of quantum wells in the microcavity. As a result, the value of $U_1$ is sensitive to the microcavity geometry. To end this section it is important to mention the limitations of any GPE in application to the condensates of exciton-polaritons in microcavities, or to condensates of any other bosonic excitations that have a finite radiative life-time. Due to interference of light emitted from different parts of condensate there appears dissipative long-range coupling in the system. Most importantly, the escape rate becomes dependent on the symmetry of the condensate wave-function and this favors the formation of particular long-living many-particle states \cite{Aleiner11}. These effects cannot be properly treated in the framework of Gross-Pitaevskii equation \eqref{GPEgener}. \section{Polarization and effects of Zeeman field} \label{sec:Zeeman} \index{Zeeman field} The interaction energy \eqref{Hinter} of the polariton condensate is polarization dependent. While the first term in (\ref{Hinter}a) does not depend on polarization and is simply proportional to the square of the polariton concentration $n=(\vec{\psi}^*\!\cdot\vec{\psi})$, the second term in (\ref{Hinter}a) is sensitive to the degree of the circular polarization of the condensate. For $U_1>0$ the interaction energy is minimized when the second term in (\ref{Hinter}a) is annulated, which is achieved for polarization satisfying $\vec{\psi}^*\times\vec{\psi}=0$, i.e., for the linear polarization. On the other hand, in the case $U_1<0$ the minimum is reached for the circular polarization of the condensate, when $\vec{\psi}^*\times\vec{\psi}=\pm\mi n$. So, there is qualitative change in the ground state of the condensate when $U_1$ changes sign \cite{Shelykh06}. \begin{enumerate} \item[(i)]{$U_1>0$. The ground state is characterized by two angles, the total phase angle $\theta$ and the polarization angle $\eta$. These angles are defined from the Descartes components of the order parameter $\psi_x=\sqrt{n}\,e^{\mi\theta}\cos\eta$ and $\psi_y=\sqrt{n}\,e^{\mi\theta}\sin\eta$. The circular components are then $\psi_{\pm1}=\sqrt{n/2}\,e^{\mi(\theta\mp\eta)}$. There are two broken continuous symmetries and, consequently, the excitation spectrum consists of two Bogoliubov branches. The sound velocities for these branches at $m_l=m_t=m^*$ are $v_0=\sqrt{\mu/m^*}$ and $v_1=\sqrt{nU_1/m^*}$, where $\mu=(U_0-U_1)n$ is the chemical potential. The presence of TE-TM splitting leads to the anisotropy of sound velocities (see \cite{Shelykh06} for details).} \vspace{0.5\baselineskip} \item[(ii)]{$U_1<0$. In this case one of the circular components is zero and the other is $\sqrt{n}\,e^{\mi\theta}$. Since there is only one broken continuous symmetry, the excitation spectrum consists of only one Bogoliubov branch, and the other branch is gaped parabolic with the gap $2|U_1|n$. The chemical potential is $\mu=U_0n$ in this domain, so that the sound velocity for the Bogoliubov excitations is $\sqrt{\mu/m^*}$.} \end{enumerate} \index{Bogoliubov dispersion} \index{Chemical potential} The mean-field theory predicts an arbitrary polarization for $U_1=0$ since in this case the energy of the condensate is polarization independent. In reality, fluctuations destroy the order in this case at any finite temperature $T$. It can be already understood from the excitation spectrum, because, apart from the Bogoliubov branch, there is the gapless parabolic branch with dispersion $\hbar^2k^2/2m^*$ and the condensate would evaporate completely due to excitation of these quasiparticles.\footnote{The concentration of quasiparticles with the energy $\epsilon(k)$ is given by $\int (2\pi)^{-2} [\exp\{\epsilon(k)/T\}-1]^{-1}d^2k$ and the integral diverges logarithmically for small $k$ when $\epsilon(k)\propto k^2$.} One can also map this case to the O(4) nonlinear sigma model, \index{Nonlinear sigma model} where the order is proven to be absent for $T>0$ \cite{Pelissetto02}. Note the similarity between the two-component condensates of exciton-polaritons and three-component condensates of spin-1 cold atoms \cite{Ho98,Ohmi98}. Due to the 3D rotational symmetry, there are also only two interaction constants in the latter case. These constants are defined by the cross-sections of scattering of two atoms with the total spin 0 and 2.\footnote{The case of the total spin 1 is irrelevant since the orbital wave function of colliding bosons is antisymmetric and it cannot be realized within the condensate.} Two different atomic condensates can also be found depending on the sign of the scattering length with the total spin 2: ferromagnetic and anti-ferromagnetic (or polar), which are analogs of circularly and linearly polarized exciton-polariton condensates, respectively. It is the first case, $U_1>0$, that is realized in the exciton-polariton condensates observed so far. The linearly polarized condensate can be seen as composition of equal numbers mutually coherent spin-up and spin-down polaritons. Therefore, it is interesting to study the effect of applied magnetic field to this state \cite{Rubo06}. Considering only weak fields, when the magnetic length is much greater than the exciton Bohr radius, one can study only the effects of Zeeman field, that is described by adding \begin{equation} \label{ZF_Def} \mathcal{H}^\prime=\Omega\,(|\psi_{-1}|^2-|\psi_{+1}|^2) \end{equation} into Hamiltonian \eqref{Ham}. Here the Zeeman field $\Omega$ is given by the half of the Zeeman splitting energy for a single polariton. To find the order parameter for the uniform condensate in the presence of Zeeman field it is convenient to introduce the concentrations of the components $n_{\pm1}=|\psi_{\pm1}|^2$ satisfying $n_{+1}+n_{-1}=n$. Assuming both $n_{+1}$ and $n_{-1}$ to be nonzero, one can take variations of the Hamiltonian \begin{multline} \label{ZF_UniformH} \mathcal{H}_\mathrm{int}+\mathcal{H}^\prime-\mu n \\ =\frac{1}{2}U_0\left(n_{+1}+n_{-1}\right)^2 -2U_1n_{+1}n_{-1}-(\mu+\Omega)n_{+1}-(\mu-\Omega)n_{-1} \end{multline} over $n_{\pm1}$ to obtain \begin{equation} \label{ZF_TwoEqs} -2U_1n_{\pm1}=(\mu-U_0n\mp\Omega). \end{equation} The sum and the difference of Eqs.\ \eqref{ZF_TwoEqs} results in \begin{equation} \label{ZF_SubCrit} \mu=(U_0-U_1)n, \quad n_{\pm1}=\frac{1}{2}\left(n\pm\frac{\Omega}{U_1}\right), \quad \mathrm{for}\;|\Omega|<\Omega_c\equiv nU_1. \end{equation} For higher Zeeman fields, $|\Omega|>\Omega_c$, one of the components becomes empty, $n_{-1}$ for $\Omega>\Omega_c$ and $n_{+1}$ for $\Omega<-\Omega_c$, and in this case $\mu=U_0n-|\Omega|$. Remarkably, for subcritical fields the chemical potential does not change at all, so that there is no change in the position of the emission line. \index{Chemical potential} The only effect of applied Zeeman field is the change of circular polarization degree $\varrho_c=(n_{+1}-n_{-1})/(n_{+1}+n_{-1})=\Omega/nU_1$, that increases linearly with the field. The elliptical polarization of the condensate for subcritical fields is characterized by two angles, and in the the same way as for the linearly polarized condensate, there are two Goldstone modes; only the sound velocities change with the Zeeman field. This implies the full suppression of the Zeeman splitting by polariton-polariton interactions within the condensate \cite{Rubo06}. Note also that for subcritical fields there are two phase transitions in the left and in the right circular component of the condensate, respectively \cite{Keeling08b}. The Zeeman splitting (the gap in the exciton spectrum) appears only for supercritical fields $|\Omega|>\Omega_c$ where the condensate becomes circularly polarized. This effect, observed experimentally by Larionov \emph{et al.} \cite{Larionov10}, allows to measure the spin-dependent interaction constant $U_1$. \index{Zeeman splitting} \section{Vortices in exciton-polariton condensates} \label{sec:Vortices} Vortices play a key role in various physical phenomena both on macroscopic and microscopic level. While the vortex formation is very important for description of different effects in fluid mechanics, in particular, in aerodynamics and turbulent flow motion, the understanding of properties of quantized vortices is crucial for description of phase transitions in condensed matter. The well known examples are the phase transitions in type II superconductors in applied magnetic field, which are related to the formation and melting for vortex lattices\cite{Abrikos57}, and the Berezinskii-Kosterlitz-Thouless (BKT) phase transition\cite{Berez72,KT73,Koster74}. \index{Berezinskii-Kosterlitz-Thouless transition} As it was discussed above, the exciton-polariton condensates possess two-component order parameter \eqref{psicirc} and these condensates allow half-quantum vortices (half-vortices) \cite{Rubo07}. Moreover, the half-vortices are basic topological excitations in this case (see \cite{VolovikBook} for a review on basic properties of half-quantum vortices). In spite of recent observation of both integer \cite{Lagoudakis08} and half-integer \cite{Lagoudakis09} vortices in exciton-polariton condensates, the presence of half-vortices was recently questioned \cite{Flayac10} for the case of two-band dispersion with TE-TM splitting of polariton band given by Eq.\ \eqref{KinEnergy}. In this section I present the details on how the vortex solutions should be found in this case (a short summary of this theory has been given in \cite{Miller10a}). In what follows only the case of zero Zeeman field will be considered. \index{Half-vortex} For a 2D system of radius $R$ the energy of a vortex is finite but logarithmically large, \begin{equation} \label{VortexEng} E_\mathrm{vor}=E_c+E_s\ln(R/a), \end{equation} where $a=\hbar/\sqrt{2m^*\mu}$ is the characteristic radius of vortex core (the effective mass $m^*$ is defined below in Eq.~\eqref{gamma}). The fact that $E_\mathrm{vor}$ diverges logarithmically at $R\rightarrow\infty$ is good: it prevents the single vortices to be excited at low temperatures and thus protects the long-range order of the condensate. Knowledge of prefactor $E_s$ allows to estimate \cite{Koster74} the BKT transition temperature $T_c$. The proliferation of single vortices appears when the free energy $E_\mathrm{vor}-TS$ crosses zero. The vortex core area is $a^2$ and it can be appear in $R^2/a^2$ places, so that the entropy $S=\ln(R/a)^2$ and this gives $T_c=E_s/2$ if one neglects the energy of the core $E_c$ in \eqref{VortexEng}. The energy $E_s\ln(R/a)$ is elaborated on large distances from the vortex core $r\gg a$, which we will refer to as the elastic region, and the study of vortices should begin with establishing the behavior of the order parameter in this region. When TE-TM splitting is present this behavior is, in general, nontrivial. \subsection{The order parameter on large distances} In the elastic region the order parameter changes within the order parameter manyfold, i.e., the polarization of the condensate is linear everywhere in this domain. The circular-polarization components $\psi_{\pm1}$ defined in \eqref{psicirc} can be written in cylindrical coordinates $(r,\phi)$ as \begin{equation} \label{Asymp} \psi_{\pm1}(r\gg a,\phi)=\sqrt{\frac{n}{2}}\,e^{\mi[\theta(\phi)\mp\eta(\phi)]}, \end{equation} where $n$ is the constant concentration of the condensate at large distances, and the phases are written in terms of total phase angle $\theta$ and polarization angle $\eta$. These angles do not depend on the radius $r$ (such dependence would only increase the vortex energy), but they are functions of the azimuthal angle $\phi$. Since the order parameters should be uniquely defined in the whole space, one has \begin{equation} \label{WindNumbers} \eta(\phi+2\pi)-\eta(\phi)=2\pi k, \qquad \theta(\phi+2\pi)-\theta(\phi)=2\pi m. \end{equation} These conditions divide all possible solutions of GPE into topological sectors. Each sector is defined by two topological charges (or winding numbers), $k$ and $m$. The state from one sector cannot be continuously transformed into another sector, or, in other words, any state of the condensate evolves within its own topological sector. The sector $k=m=0$ is the ground state sector; the minimum energy here is reached for position-independent order parameter. By definition, the vortex is the state that minimizes the energy in a topological sector with at least one non-zero winding number. The energy of the $(k,m)$-vortex \eqref{VortexEng} is counted from the ground state energy, i.e., it is the difference between the minimal energy in the $(k,m)$ sector and the minimal energy of $(0,0)$ sector (the ground state energy). Since only the sum and the difference, $\theta\pm\eta$, enter Eq.\ \eqref{Asymp}, the winding numbers can be either both integer or both half-integer, and the corresponding vortices are referred accordingly. Note also that the vortex corresponds to a minimum of Hamiltonian $H$ for specific boundary conditions: $\delta H/\delta\vec{\psi}^*=0$ for the vortex solution and, therefore, it is a static solution of GPE \eqref{GPEgener}.\footnote{Note, however, that this does not imply that a single vortex gives an absolute minimum of the $H$ in the corresponding topological sector. For example, the integer vortex $(1,0)$ can be unstable with respect to decay into the pair of $(\half,\half)$ and $(\half,-\half)$ half-vortices for $m_l<m_t$ (see subsection \ref{sec:Vortices}.2 below).} According to \eqref{WindNumbers} one can add any periodic functions of $\phi$ to $\eta(\phi)$ and $\theta(\phi)$ without changing the topological sector. The proper functions $\eta(\phi)$ and $\theta(\phi)$ for the $(k,m)$-vortex should be found from minimization of Hamiltonian in elastic region. The corresponding part of Hamiltonian is related solely to the kinetic energy term $\int\mathcal{T}d^2r$. After substitution of \eqref{Asymp} into (\ref{KinEnergy}b) and use of the asymptotic behavior of the complex derivative for $r\rightarrow\infty$, \begin{equation} \label{DDzLargeR} \frac{\partial}{\partial z}\rightarrow -\frac{\mi}{2r}e^{-\mi\phi}\frac{\partial}{\partial\phi}, \end{equation} one obtains the product of integrals over $r$ and $\phi$ that results in the second term of Eq.\ \eqref{VortexEng}. The integral over $r$ diverges logarithmically and should be cut by the core size $a$ at small $r$, and by system radius $R$ at large $r$. This gives the factor $\ln(R/a)$. The prefactor is then given by \begin{equation} \label{Prefactor1} E_s=\frac{\hbar^2n}{2m^*}\int_0^{2\pi} \left\{ [1+\gamma\cos(2u)](1+u^\prime)^2+[1-\gamma\cos(2u)]\theta^{\prime2} \right\}d\phi, \end{equation} where the prime denotes the derivative over $\phi$ and \begin{equation} \label{Eta&u} u(\phi)=\eta(\phi)-\phi. \end{equation} The effective mass $m^*$ and the TE-TM splitting parameter $\gamma$ are defined in \eqref{Prefactor1} by \begin{equation} \label{gamma} \frac{1}{m^*}=\frac{1}{2}\left(\frac{1}{m_l}+\frac{1}{m_t}\right), \qquad \gamma=\frac{m_t-m_l}{m_t+m_l}. \end{equation} Variations of the functional \eqref{Prefactor1} over $\theta$ and $u$ lead to the equations \begin{subequations} \label{Eqs_phi} \begin{equation} \left[1-\gamma\cos(2u)\right]\theta^{\prime\prime}+ 2\gamma\sin(2u)u^{\prime}\theta^{\prime}=0, \end{equation} \begin{equation} \left[1+\gamma\cos(2u)\right]u^{\prime\prime} +\gamma\sin(2u)\left( 1-u^{\prime2} -\theta^{\prime2} \right)=0. \end{equation} \end{subequations} In general, the polarization will be radial at least at one specific direction and it is convenient to count the azimuthal angle from this direction and set the total phase to be zero at this direction as well. Then, the solutions of Eqs.~(\ref{Eqs_phi}a,b) for $(k,m)$-vortex should satisfy the boundary conditions \begin{subequations} \label{BoundCond} \begin{equation} u(0)=0, \qquad \theta(0)=0, \end{equation} \begin{equation} u(2\pi)=2(k-1)\pi, \qquad \theta(2\pi)=2m\pi. \end{equation} \end{subequations} The solutions in question are trivial for some particular vortices. \begin{enumerate} \item[(i)]{\emph{Hedgehog vortices.} These are $(1,m)$-vortices having $\theta=m\phi$ and $u\equiv 0$, so that the polarization angle $\eta=\phi$. Polarization points into the radial direction everywhere and these vortices look like hedgehogs. These solutions are similar to magnetic monopoles \cite{Rajaraman}. } \index{Hedgehog vortex} \index{Monopole} \vspace{0.5\baselineskip} \item[(ii)]{\emph{Double-quantized polarization vortex $(2,0)$.} In this special case $\theta\equiv 0$, but $u=\phi$, resulting in $\eta=2\phi$. Polarization rotates twice when one encircles the vortex core. These vortices and experimental possibilities of their excitation in exciton-polaritons fields have been studied by Liew \emph{at el.} \cite{Liew07}. } \end{enumerate} In other cases the solutions should be found numerically. Both Eqs.~(\ref{Eqs_phi}a) and (\ref{Eqs_phi}b) can be integrated once to give \begin{subequations} \label{Eqs_phi1} \begin{equation} \left[1-\gamma\cos(2u)\right]\theta^\prime=C_1, \end{equation} \begin{equation} \left[1+\gamma\cos(2u)\right]u^{\prime2}+ \left[1-\gamma\cos(2u)\right]\theta^{\prime2} -\gamma\cos(2u)=C_2, \end{equation} \end{subequations} and the solutions can be written as integrals of elementary functions. The constants $C_{1,2}$ should then be found, e.g., by shooting, to satisfy the boundary conditions (\ref{BoundCond}). The functions $\theta(\phi)$ and $\eta(\phi)$ are shown in Fig.~\ref{Fig1-Angles} for elementary half-vortices and for two integer vortices $(-1,0)$ and $(0,1)$, that also exhibit nonlinear dependencies of polarization and phase angles. \begin{figure}[t] \centering \includegraphics*[width=0.9\textwidth]{Fig1-Rubo.eps} \caption[]{The dependence of polarization angle $\eta$ (solid lines) and phase angle $\theta$ (dashed lines) on the azimuthal angle $\phi$ for two values of TE-TM splitting parameter: $\gamma=-0.4$ (thin lines) and $\gamma=-0.9$ (thick lines). The panels show the behavior of the angles for the $(\half,\half)$ half-vortex (a), the $(-\half,\half)$ half-vortex (b), the $(-1,0)$ polarization vortex (c), and the $(0,1)$ phase vortex (d). In the last case the periodic function $\eta(\phi)$ has been upscaled for clarity} \label{Fig1-Angles} \end{figure} Fig.~\ref{Fig1-Angles} demonstrates the behavior of angles for negative values of the TE-TM splitting parameter $\gamma$. The functions $\theta(\phi)$ and $\eta(\phi)$ for positive $\gamma$ can be found by the shift. Indeed, the change $u\rightarrow u+(\pi/2)$ in Eqs.~(\ref{Eqs_phi}a,b) results in the change of the sign of $\gamma$. More precisely, to satisfy the boundary conditions \eqref{BoundCond} the transformations can be written as \begin{subequations} \begin{eqnarray} \label{TransSign} && \gamma\rightarrow -\gamma, \\ && u(\phi)\rightarrow u\left(\phi+\frac{\pi}{2(k-1)}\right)+\frac{\pi}{2}, \\ && \theta(\phi)\rightarrow \left(\phi+\frac{\pi}{2(k-1)}\right)-\theta\left(\frac{\pi}{2(k-1)}\right), \end{eqnarray} \end{subequations} and they can be applied to all vortices except the hedgehogs with $k=1$ (and where they are not necessary, of course, since $u(\phi)\equiv0$). The nonlinear change of angles seen in Fig.~\ref{Fig1-Angles} becomes especially evident when $\gamma$ approaches $\pm1$. This limit correspond to a strong inequality between effective masses, e.g., $m_l\gg m_t$ for $\gamma\rightarrow -1$. Qualitatively the strong nonlinearities can be understood if one introduces the effective masses for the phase $m_\theta$ and for the polarization $m_\eta$, \begin{equation} \label{AngleMasses} \frac{1}{m_\theta}=\frac{\cos^2\!u}{m_t}+\frac{\sin^2\!u}{m_l}, \qquad \frac{1}{m_\eta}=\frac{\sin^2\!u}{m_t}+\frac{\cos^2\!u}{m_l}, \end{equation} and writes the energy \eqref{Prefactor1} as \begin{equation} \label{Prefactor2} E_s=\frac{\hbar^2n}{2}\int_0^{2\pi}\left\{ \frac{\eta^{\prime2}}{m_\eta}+\frac{\theta^{\prime2}}{m_\theta} \right\} d\phi. \end{equation} The effective masses \eqref{AngleMasses} depend on the orientation of polarization. Since $u(\phi)$ changes between the values specified by \eqref{BoundCond}, there are sectors where $m_\theta\approx m_l$ and $m_\eta\approx m_t$, and there are sectors where $m_\theta\approx m_t$ and $m_\eta\approx m_l$. To minimize the energy \eqref{Prefactor2} in the case $m_l\gg m_t$, the phase angle changes rapidly and the polarization angle stays approximately constant in the former, while there is the opposite behavior in the latter. \subsection{The energies and interactions of vortices} In the absence of TE-TM splitting the energies $E_s$ of vortices are \begin{equation} \label{Es_gamma0} E_{s0}^{(k,m)}=E_0(k^2+m^2), \qquad E_0=\frac{\pi\hbar^2n}{m^*}, \qquad (\mathrm{for}\;\gamma=0). \end{equation} It is seen that in this case the energy of an elementary half-vortex is exactly half of the energy of an elementary integer vortex. Important consequences can be drawn from this relation concerning the interactions between half-vortices. The four elementary half-vortices can be divided in two kinds, right half-vortices with $k+m=\pm1$, and left ones with $k-m=\pm1$. One can see from \eqref{Asymp} the the right half-vortices possess the vorticity of the left-circular component of the order parameter, the amplitude of this component goes to zero and the phase of this component becomes singular in the vortex core center, and, as a result, the polarization becomes right-circular in the core center. For left half-vortices the picture is opposite. It follows from \eqref{Es_gamma0} that the left and the right half-vortices do not interact with each other. Consider, for example, the $(\half,\half)$ and $(-\half,\half)$ half-vortices. The elastic energy of this pair is $E_0\ln(R/a)$ both when they are far away from each other and when they are in the same place forming the phase vortex $(0,1)$. So, the only possible coupling between the left and the right half-vortices is of short range, related to the overlap of their cores and resulting change of the core energy term $E_c$ in \eqref{VortexEng}. So, the only long-range coupling is present between the half-vortices of the same kind. It can be shown that identical half-vortices repel each other logarithmically, while the half-vortices and anti-half-vortices, $(k,m)$ and $(-k,-m)$, attract each other logarithmically, as it is in the case of vortices and antivortices in one-component condensates \cite{ChaikinLub}. This simple picture is changed in the presence of TE-TM splitting that leads to the long-range interaction between half-vortices of different kind. The logarithmic prefactors $E_s^{(k,m)}$ for elementary half-vortices and elementary integer vortices are shown in Fig.~\ref{Fig2-Es}. For all of them, expect the hedgehog (1,0)-vortex, these energies are even functions of $\gamma$, which can be proven using the transformations \eqref{TransSign}. These energies decrease with increasing $\gamma^2$. The case of hedgehog is special, as it has been discussed above. The hedgehog polarization is radial everywhere and $E_s^{(1,0)}$ is defined purely by the longitudinal effective mass, \begin{equation} \label{Es_HedgV} E_s^{(1,0)}=\pi\hbar^2n/m_l=E_0(1+\gamma). \end{equation} It can be shown that when two vortices with winding numbers $(k_1,m_1)$ and $(k_2,m_2)$ are injected in the condensate and they are separated by distance $r$, such that $a\ll r\ll R$, the energy of the condensate is increased in logarithmic approximation (i.e., omitting the core energies) by \begin{multline} \label{InterGen} E_s^{(k_1+k_2,m_1+m_2)}\ln(R/a) \\ +\left[E_s^{(k_1,m_1)}+E_s^{(k_2,m_2)}-E_s^{(k_1+k_2,m_1+m_2)}\right]\ln(r/a). \end{multline} The second term in \eqref{InterGen} gives the interaction energy of two vortices. The coupling between vortices arising due to TE-TM splitting can be analyzed analytically in the limit of small $\gamma$. The solutions of Eqs.~(\ref{Eqs_phi}a,b) for $k\ne1$ are written as series in $\gamma$, \begin{subequations} \label{SeriesAngles} \begin{multline} \theta(\phi)=m\phi +\frac{m}{2(k-1)}\,\gamma\sin[2(k-1)\phi] \\ +\frac{m[(k-1)^2-(1-m^2)]}{16(k-1)^3}\,\gamma^2\sin[4(k-1)\phi]+\dots, \end{multline} \begin{multline} u(\phi)=(k-1)\phi-\frac{[(k-1)^2+(m^2-1)]}{4(k-1)^2}\,\gamma\sin[2(k-1)\phi] \\ +\frac{[5(k-1)^4+2(k-1)^2(m^2-3)+(m^2-1)^2]}{64(k-1)^4} \,\gamma^2\sin[4(k-1)\phi]+\dots. \end{multline} \end{subequations} Substitution of these expression into \eqref{Prefactor1} gives \begin{equation} \label{SeriesEs} \frac{E_s^{(k,m)}}{E_0}=(k^2+m^2) -\frac{[k^2(k-2)^2+2[2+3k(k-2)]m^2+m^4]}{8(k-1)^2}\,\gamma^2-\dots, \end{equation} and, in particular, \begin{equation} \label{Es_forIntgV} E_s^{(-1,0)}=E_0\left[1-\frac{9}{32}\gamma^2-\dots\right], \qquad E_s^{(0,\pm1)}=E_0\left[1-\frac{5}{8}\gamma^2-\dots\right]. \end{equation} There is no difference between the energies of half-vortices at this order of $\gamma$-series. The difference, however, appears in the next order. The series for the angles up to $\gamma^4$ are rather cumbersome to be presented, but they result in \begin{subequations} \label{Es_forHV} \begin{equation} E_s^{\left(\half,\pm\half\right)}= \frac{E_0}{2}\left[1-\frac{\gamma^2}{2}-\frac{3\gamma^4}{16}-\dots\right], \end{equation} \begin{equation} E_s^{\left(-\half,\pm\half\right)}= \frac{E_0}{2}\left[1-\frac{\gamma^2}{2}-\frac{11\gamma^4}{144}-\dots\right]. \end{equation} \end{subequations} \begin{figure}[t] \centering \includegraphics*[width=0.7\textwidth]{Fig2-Rubo.eps} \caption[]{The logarithmic prefactor of vortex energies $E_s$ (see Eqs.\ \eqref{VortexEng} and \eqref{Prefactor1}) for half-vortices and integer vortices as functions of TE-TM splitting parameter $\gamma$ \eqref{gamma}. The curves are labeled by the winding numbers $(k,m)$ of the vortices, and the energies are given in the units of $E_0=\pi\hbar^2n/m^*$} \label{Fig2-Es} \end{figure} Eqs.\ \eqref{Es_HedgV}, \eqref{Es_forIntgV}, and \eqref{Es_forHV} can be used to find the interactions between half-vortices according to Eq.\ \eqref{InterGen}. Most important interaction that appears due to TE-TM splitting is between the $(\half,\half)$ and $(\half,-\half)$ half-vortices. For small $\gamma$ their coupling constant is linear in $\gamma$ and the interaction energy is \cite{Miller10b} \begin{equation} \label{LRCoupling} V_{(\half,\half),(\half,-\half)}\simeq -\gamma E_0\ln(r/a). \end{equation} It should be noted that the interrelation between $m_t$ and $m_l$, i.e., the sign of $\gamma$, depends on the detuning of the frequency of the cavity photon mode from the center of the stop-band of the distributed Bragg mirror \cite{Panzarini99}. So, one can have both attraction and repulsion of the $(\half,\half)$ and $(\half,-\half)$ half-vortices. The coupling of the other left and right half-vortices is quadratic in $\gamma$. The $(-\half,\half)$ and $(-\half,-\half)$ half-vortices repel each other with the interaction energy being $-(7/32)\gamma^2E_0\ln(r/a)$. The $(-\half,\pm\half)$ and $(\half,\pm\half)$ half-vortices attract each other with the interaction energy being $(1/8)\gamma^2E_0\ln(r/a)$. In the absence of TE-TM splitting there is no coupling between the right half-vortices (with $km>0$) and the left ones (with $km<0$) and there are two decoupled BKT transitions, corresponding to the dissociation of pairs of left and right half-vortices \cite{Rubo07,Keeling08b}. The transition temperature is then estimated from the energy of single half-vortex as $E_0/4$. The TE-TM splitting of polariton bands changes this picture substantially. First, because all four half-vortices become coupled and, secondly, because the energies of a vortex and its antivortex become different, so it is not clear with one should be used in the estimation of critical temperature. One expects qualitative modifications of the BKT transition in the region of $\gamma$ close to $-1$. In this region the attraction of the $(\half,\half)$ and $(\half,-\half)$ half-vortices becomes very strong and, as a result, the hedgehog is the vortex with the smallest energy in the system for $\gamma<\gamma_c\simeq -0.6$ (see Fig.~\ref{Fig2-Es}). It does not mean, however, that the transition temperature can be estimated from the energy of the hedgehog in this region. In fact, the phase transition occurs due to dissociation of vortex-antivortex pairs, and the energy of the $(1,0)$ and $(-1,0)$ pair is still bigger than the energy of the pair of two half-vortices. One expects that when pairs of half-vortices are thermally excited in the system they will tend to form molecules consisting of the hedgehog (formed by merging of the $(\half,\half)$ and $(\half,-\half)$ half-vortices) with the $(-\half,-\half)$ and $(-\half,\half)$ half-vortices being attached to it. The proliferation of these $(-\half,-\half)-(1,0)-(-\half,\half)$ molecules defines the phase transition for $m_l\gg m_t$. \index{Berezinskii-Kosterlitz-Thouless transition} \section{Geometry of the half-vortex fields} \label{sec:LeMonStar} In general, two coupled Gross-Pitaevskii equations for the circular components of the order parameter \index{Gross-Pitaevskii equation} \begin{subequations} \label{GM_GPEdetailed} \begin{multline} \mi\hbar\frac{\partial\psi_{+1}}{\partial t} =-\frac{\hbar^2}{2m^*}\left( \Delta\psi_{+1}+4\gamma\frac{\partial^2\psi_{-1}}{\partial z^2} \right)-\mu\psi_{+1} \\ +\left[U_0(|\psi_{+1}|^2+|\psi_{-1}|^2)-2U_1|\psi_{-1}|^2\right]\psi_{+1}, \end{multline} \begin{multline} \mi\hbar\frac{\partial\psi_{-1}}{\partial t} =-\frac{\hbar^2}{2m^*}\left( \Delta\psi_{-1}+4\gamma\frac{\partial^2\psi_{+1}}{\partial z^{*2}} \right)-\mu\psi_{-1} \\ +\left[U_0(|\psi_{-1}|^2+|\psi_{+1}|^2)-2U_1|\psi_{+1}|^2\right]\psi_{-1}, \end{multline} \end{subequations} are not separated in cylindrical coordinates $(r,\phi)$. The variables are separated only in special cases of hedgehog vortices and the double-quantized polarization vortex discussed in previous section after Eqs.~\eqref{BoundCond}. For other vortices one needs to solve GPEs numerically in two spacial dimensions with the boundary conditions at large distances defined by Eqs.\ \eqref{Asymp} and \eqref{Eqs_phi}. \begin{figure}[t] \centering \includegraphics*[width=0.9\textwidth]{Fig3-Rubo.eps} \caption[]{Showing the geometry of half-vortices for different values of TE-TM splitting parameter $\gamma$ \eqref{gamma}. The plots are obtained from numerical solutions of GPEs \eqref{GM_GPEdetailed} \cite{Miller12}. The interaction constants are related as $U_1=0.55U_0$. The local polarization ellipses are drawn with the thick (red) lines. The streamlines of the current are shown by thin (green) lines. The panels demonstrate the following cases: (a) the half-vortices $(\half,\pm\half)$ for $\gamma=-0.5$ (the lemon morphology); (b) the half-vortices $(\half,\pm\half)$ for $\gamma=0.5$ (the lemon morphology); (c) the half-vortices $(-\half,\pm\half)$ for $\gamma=-0.5$ (the star morphology); (d) the half-vortices $(-\half,\pm\half)$ for $\gamma=0.5$ (the star morphology). } \label{Fig3-LemonStar} \end{figure} Each vortex with nonzero phase winding number $m$ is characterized by a finite superfluid current circulating around its core center. Performing numerical solutions it is important to take into account the fact that streamlines of the current are deformed with respect to perfect circles in the presence of TE-TM splitting. The circular components of the current $\vec{J}$ are given by \begin{multline} \label{GM_Jplus} J_{+1}=\frac{\mi\hbar}{\sqrt{2}\,m^*}\left\{\left( \psi_{+1}\frac{\partial\psi_{+1}^*}{\partial z} +\psi_{-1}\frac{\partial\psi_{-1}^*}{\partial z} -\psi_{+1}^*\frac{\partial\psi_{+1}}{\partial z} -\psi_{-1}^*\frac{\partial\psi_{-1}}{\partial z} \right)\right. \\ +2\gamma\left.\left( \psi_{+1}\frac{\partial\psi_{-1}^*}{\partial z^*} -\psi_{-1}^*\frac{\partial\psi_{+1}}{\partial z^*} \right)\right\}, \end{multline} and $J_{-1}=J_{+1}^*$. They are related to the radial $J_r$ and the azimuthal $J_\phi$ components of the current by \begin{equation} \label{GM_Jr&Jphi} J_r=\frac{1}{\sqrt{2}}\left( e^{\mi\phi}J_{+1}+e^{-\mi\phi}J_{-1} \right), \qquad J_\phi=\frac{\mi}{\sqrt{2}}\left( e^{\mi\phi}J_{+1}-e^{-\mi\phi}J_{-1} \right). \end{equation} At large distances $r\gg a=\hbar/\sqrt{2m^*\mu}$ one can use Eq.~\eqref{Asymp} to find \begin{subequations} \label{Currents} \begin{equation} J_r=\frac{\hbar n}{m^*r}\gamma\sin(2u)\frac{d\theta}{d\phi}, \end{equation} \begin{equation} J_\phi=\frac{\hbar n}{m^*r}\left[1-\gamma\cos(2u)\right]\frac{d\theta}{d\phi}. \end{equation} \end{subequations} Note that the condition of conservation of the total number of polaritons for the static vortex solution of Eqs.~\eqref{GM_GPEdetailed}, \begin{equation} \label{GM_divJ} \mathrm{div}\vec{J}=\frac{1}{r}\left[ \frac{\partial}{\partial r}(rJ_r)+\frac{\partial J_\phi}{\partial\phi} \right]=0, \end{equation} implies $\partial J_\phi/\partial\phi=0$. So, Eq.~(\ref{Eqs_phi}a) obtained in the previous section is in fact the condition of conservation of the azimuthal current. The warping of streamlines of current is shown in Fig.~\ref{Fig3-LemonStar}. The order parameter has been found numerically \cite{Miller12} for different values of TE-TM splitting parameter. To find the static solutions of GPE \eqref{GM_GPEdetailed}, we have been choosing an initial order parameter $\vec{\psi}(\vec{r},t=0)$ satisfying the boundary conditions that follow from Eqs.~\eqref{Eqs_phi} and that are shown in Fig.~\ref{Fig1-Angles} for a given topological sector $(k,m)$. Apart from this the initial functions were continuous but arbitrary. Then the functions were evolved according to \eqref{GM_GPEdetailed} in imaginary time. As a result, the order parameter relaxed to corresponding static half-vortex solution. The resulting half-vortex solutions are found to be independent of the initial shape of $\vec{\psi}(\vec{r},t=0)$. In Fig.~\ref{Fig3-LemonStar} one can see two distinct morphologies of basic half-vortices. The geometry of half-vortex solutions can be discussed in terms of singular optics \cite{Nye99,Soskin01}, where the polarization singularity related to a half-vortex is referred as C-point, to indicate that the polarization is circular at the vortex center and, therefore, the direction of the main axis of polarization ellipse is not defined. The morphologies of the field around C-points are classified by the index of associated real tensor field, and, additionally, by the number of strait polarization lines\footnote{The tangents of polarization lines define by the direction of the main axis of polarization ellipse in each point.} that terminate at C-point \cite{Dennis09}. The tensor index coincides with the polarization winding number $k$, and the number of lines could be either one or three. As a result, three different morphologies can be found. Following Berry and Hannay \cite{Berry77,Nye83}, these morphologies are referred as \emph{lemon}, \emph{star}, and \emph{monstar}. \index{Lemon, star, monstar} The lemon configuration is characterized by $k=\half$ and by only one straight polarization line terminating in the vortex center. This is the morphology of vortices in Fig.~\ref{Fig3-LemonStar}a,b with the straight polarization line being defined by $\phi=0$. The star configuration is characterized by $k=-\half$. In this case there are always three straight lines terminating in the vortex center. The stars are realized in Fig.~\ref{Fig3-LemonStar}c,d and three straight polarization lines are defined by $\phi=0,\pm2\pi/3$. The change of parameter $\gamma$ leads to deformation of polarization texture and to deformation of streamlines of the current, but it does not result in the change of morphologies of half-vortices. In principle, one could expect the transformation of lemon into monstar, since these morphologies possess the same topological index $k=\half$. Contrary to the lemon case, however, the monstar is characterized by three straight polarization lines terminating in the vortex center, similar to the star configuration. So, from geometrical point of view the monstar has got intermediate structure between the lemon and the star, and this is why its name is constructed from ``(le)mon-star''. To have the monstar configuration one needs a special behavior of polarization angle $\eta(\phi)$. Namely, it is necessary to have \begin{equation} \label{GM_MonstarCond} \left.\frac{d\eta}{d\phi}\right|_{\phi=0}>1 \quad \mathrm{for}\;k=\frac{1}{2}. \end{equation} In this case the polarization angle initially rotates faster than the azimuthal angle $\phi$, but since the total rotation of $\eta$ should be still $\pi$ when $\phi$ is changing up to $2\pi$, as it is dictated by the winding number $k=\half$, there will be three roots of the equation $\eta(\phi)=\phi$. These roots, 0 and $\pm\phi_m$, define three straight polarization lines terminating in the vortex center for the monstar geometry.\footnote{Note that for the monstar all polarization lines residing within the sector $-\phi_m<\phi<\phi_m$ terminate in the vortex center, but only three of them are straight, i.e., are having nonzero inclination at $r\rightarrow0$ (see \cite{Dennis09} for the details).} One can see from Fig.~\ref{Fig1-Angles}a and \ref{Fig3-LemonStar}a that when $\gamma$ approaches $-1$ the derivative becomes very close to unity, but it never becomes bigger than 1, so that the monstar is not formed. The reason preventing the appearance of the monstar is that it is not energetically favorable to satisfy the condition \eqref{GM_MonstarCond}. In fact, it is the most energetically favorable to have $\eta=\phi$, as for the hedgehog---the vortex having the smallest energy when $\gamma\rightarrow-1$ (see Fig.~\ref{Fig2-Es}). The rotation of polarization of the half-vortex is also synchronous with the azimuth in rather wide sector, but the polarization never overruns the azimuth. The monstar half-vortices, however, are expected to be found in the exciton-polariton condensates out of equilibrium \cite{Liew08,Borgh10,Keeling11}, where their appearance is not restricted by energetics. \section{Four-component exciton condensates} \label{sec:FourComp} Excitons formed by an electron and a heavy hole in the semiconductor quantum wells can be in four spin states \cite{IvchenkoBook}. The states with the total spin projection $\pm1$ are optically active. These bright excitons are formed by the heavy hole with the spin $+\threehalves$ and the the electron with the spin $-\half$, or by the $-\threehalves$ hole and the $+\half$ electron. The other two states are hidden from the observer and are usually referred to as the dark excitons. The total momentum of these states is $\pm2$ and they are formed either by the $+\threehalves$ hole and the $+\half$ electron, or by the $-\threehalves$ hole and the $-\half$ electron. The exciton-polaritons discussed in the previous sections are coupled states of quantum-well excitons and microcavity photons. Only the bright excitons are involved in this coupling, and the resulting condensates are two-component. Since the frequency of a single exciton-polariton is shifted down with respect to the single exciton frequency by a half of the Rabi frequency, \index{Rabi frequency} the presence of dark excitons is irrelevant in this case provided the exciton-photon coupling is strong enough. Contrary, when pure exciton condensates are of interest, all four exciton spin states should be, in general, taken into account. The formation of exciton condensates is possible for cold indirect excitons in coupled quantum wells \cite{Butov02,Hang95,Butov98,Voros05,Butov07}. The life-time of these excitons is long enough, the excitons can travel coherently over long distances, and the condensates can be formed in quasi-equilibrium conditions. The presence of four-component exciton condensates has also been experimentally demonstrated recently \cite{Leonard09}. The indirect excitons are dipoles oriented along the growth axis of the semiconductor structure, and their main interaction is spin-independent dipole-dipole repulsion. The condensate state, however, is defined by weak spin-dependent interactions arising from electron-electron, hole-hole, and exciton-exciton exchanges \cite{Rubo11}. In what follows, I will assume the signs of these interactions to be such that they favor the distribution of excitons over all four spin states, populating both bright and dark components. This state is similar to the linearly polarized two-component condensates described above, but there is one important qualitative difference between them. The exchange scattering of two excitons can result in transformation of their spin states \cite{Ciuti98}. Namely, two bright excitons can turn into two dark ones after collision and vice versa. These processes are described microscopically by the Hamiltonian \begin{equation} \label{FC-Hmix} \hat{\mathcal{H}}_\mathrm{mix}=W\left[ \hat{\psi}_{+2}^\dag\hat{\psi}_{-2}^\dag \hat{\psi}_{+1}^{\vphantom{\dag}}\hat{\psi}_{-1}^{\vphantom{\dag}} +\hat{\psi}_{+1}^\dag\hat{\psi}_{-1}^\dag \hat{\psi}_{+2}^{\vphantom{\dag}}\hat{\psi}_{-2}^{\vphantom{\dag}} \right]. \end{equation} In mean-field approximation, the creation $\hat{\psi}_{\sigma}^\dag$ and annihilation $\hat{\psi}_{\sigma}$ exciton operators ($\sigma=\pm1,\pm2$) are replaced by the order parameter components, $\psi_{\sigma}^*$ and $\psi_{\sigma}$, respectively. The contribution of the resulting exciton-mixing term $\mathcal{H}_\mathrm{mix}$ into the total energy of the exciton condensate depends on the relative phases of the components. The term of this type is absent in the two-component exciton-polariton case. Remarkably, the mixing of excitons always leads to the decrease of the condensate energy, which is achieved by fixing the proper interrelation between the phases. Denoting by $\theta_\sigma$ the phase of $\psi_\sigma$, one can see that the following relation holds within the order parameter manifold \begin{equation} \label{FC_PhaseLock} \theta_{+2}+\theta_{-2}-\theta_{+1}-\theta_{-1}= \begin{cases} 0 \pmod{2\pi}, & \text{if $W<0$,} \\ \pi \pmod{2\pi}, & \text{if $W>0$.} \end{cases} \end{equation} The mixing term \eqref{FC-Hmix} additionally favors the formation of the four-component exciton condensate with equal occupations of the components. This fact can be seen from different perspective. $\mathcal{H}_\mathrm{mix}$ describes the transformation of \emph{pairs} of excitons, and, in the same way as in the BCS theory of the superconductivity, \index{Bardeen-Cooper-Shrieffer (BCS) theory} this term leads to the pairing of particles. This pairing leads to a decrease in the energy of the system and results in appearance of the gap in the excitation spectrum for one excitation branch. The other three excitation branches are Bogoliubov-like. \index{Bogoliubov dispersion} This follows from the fact that the phase locking condition \eqref{FC_PhaseLock} leaves there angles to be undefined, so that there are three Goldstone modes apart from the gaped mode induced by the mixing. The effect of applied Zeeman field \index{Zeeman field} on four-component exciton condensate is expected to be very spectacular \cite{Rubo11}. The Zeeman splitting is different for dark and bright excitons: the $g$-factor is given by the sum of the electron and hole $g$-factors for the former, and by their difference for the latter. When such a field is applied to the exciton condensate its action is two-fold. On the one hand, it polarizes the bright and dark components with different degrees of circular polarization, and thus reduces the Zeeman energy of the condensate. On the other hand, the induced imbalance in the occupation of the components increases the energy of the mixing term \eqref{FC-Hmix} and suppresses the gap in the spectrum discussed above. The interplay between these two effects can lead to the first-order transitions from the four-component exciton condensate to the two- or the one-component condensates. Note also that due to the presence of $\mathcal{H}_\mathrm{mix}$ the system of equations defining the concentrations of the components of the exciton condensate in the Zeeman field is nonlinear, contrary to the case of two component exciton-polariton condensate (see Eq.~\eqref{ZF_TwoEqs}). Finally, it is important to note that vortices in the four-component exciton condensate in the presence of the mixing of the component are composite: the vorticity of one component should be accompanied by the vorticity of another component to satisfy the the phase-locking condition \eqref{FC_PhaseLock}. As a result one expect twelve elementary vortices. These are four polarization vortices (two in each components), and the eight paired half-vortices in the bright and dark components. \section{Conclusions and perspectives} \label{sec:Concl} The mean-field approximation provides simple and reliable method to study the polarization properties and topological excitations of exciton-polariton and exciton condensates that possess two and four components of the order parameter, respectively. This includes, in particular, the description of the polarization of the ground state and elementary excitations of the condensates and their change in applied Zeeman field, as well as the description of the texture of vortices and vortex interactions. The elementary topological excitations in two-component exciton-polariton condensates are four half-vortices $(k,m)$ with $k,m=\pm\half$, characterized by half-quantum changes of polarization and phase angles. In the absence of transverse-electric-transverse-magnetic (TE-TM) splitting of the lower polariton band there is no coupling between the left half-vortices (with $km<0$) and the right ones (with $km>0$), and one expects two decoupled Berezinskii-Kosterlitz-Thouless (BKT) superfluid transitions happening at the same temperature in the system. The TE-TM splitting results in two qualitative effects. First, the cylindrical symmetry of the half-vortex field is spontaneously broken that leads to warping of the polarization field around a half-vortex and to deviation of the streamlines of the supercurrent from the perfect circles. Secondly, there appears long-range interactions between left and right half-vortices. These interactions are particularly important in the case of large longitudinal polariton mass $m_l$ when it favors the formation of hedgehog (monopole) vortices (1,0) from the $(\half,\half)$ and $(\half,-\half)$ half-vortices. The peculiarities of the superfluid transition in this case and related features of polarization textures of the exciton-polariton condensates are subjects of further studies. In what concerns the geometry of the half-vortex field it is shown that only two configurations, lemon and star, are realized. The monstar configuration is not energetically favorable for any value and sign of TE-TM splitting. The essential feature of four-component exciton condensates is the presence of mixing and related phase locking between dark and bright excitons. One expects a nontrivial Zeeman-field effect resulting in a discontinuous change of the polarization state of the condensate in course of the first-order transition. The presence of composite vortices in different components should lead to the formation of interesting polarization patterns in driven exciton condensates that provide an important topic for investigation, both experimental and theoretical. \section*{Acknowledgements} This work was supported in part by DGAPA-UNAM under the project No. IN112310 and by the EU FP7 IRSES project POLAPHEN. \input{refsrubo} \end{document}

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\section{Multi-Valued Broadcast and Tolerating $t\ge n/3$ Failures} \label{sec:broadcast} Here we briefly discuss the Byzantine {\em broadcast} problem (also known as the ``Byzantine Generals Problem'' \cite{psl82}). Similar to the consensus problem, the broadcast problem also considers achieving agreement among $n$ processors: A designated {\em ``source''} processor tries to broadcast an $L$-bit value to the other processors, while $t<n/3$ processors (probably including the source) may be faulty. Using techniques introduced in this paper, we can achieve error-free multi-valued broadcast with communication complexity $C_{bro}(L) <1.5(n-1)L + \Theta(n^4L^{0.5})$ bits for $t<n/3$ and large $L$ \cite{techreport_BA_complexity}. Notice that the complexity of any broadcast algorithm, even the ones that allow a positive probability of error, is lower bounded by $(n-1)L$. So we can achieve error-free broadcast with complexity within a factor of $1.5+\epsilon$ to the optimal for any constant $\epsilon>0$ and sufficiently large $L$. \section{Tolerating $t\ge n/3$ Failures} Most of our discussion in the previous section is independent of the number of faulty processors. The requirement for $t<n/3$ is needed only for the correctness of the deterministic error-free 1-bit broadcast algorithm \BSB. In practice, it may be desirable to be able to tolerate $t\ge n/3$ failures at the cost of a non-zero probability of error. This need can be met by our algorithm with a small modification: substitute \BSB with any probabilistically correct 1-bit broadcast algorithm that tolerates the desired number of failures (ones with authentication from \cite{Waidner96information-theoreticpseudosignatures,authenticated_BA_Dolev83} for example). With this modification, our algorithm tolerates the same number of failures as the 1-bit broadcast algorithm does, and makes an error only if the 1-bit broadcast algorithm fails. The only difference in the communication complexity is the term {\em sub-linear} in $L$. So for sufficiently large $L$, the complexity of the modified algorithm is also $O(nL)$. \section{Conclusion} \label{sec:conclusion} In this paper, we present efficient error-free Byzantine consensus algorithm for long messages. The algorithm requires $O(nL)$ total bits of communication for messages of $L$ bits for sufficiently large $L$. Our algorithm makes no cryptographic assumption and still is able to always solve the Byzantine consensus problem correctly. \section{Multi-Valued Consensus} \label{sec:consensus} In this section, we describe our consensus algorithm, present a proof of correctness. The $L$-bit input value $v_i$ at each processor is divided into $L/D$ parts of size $D$ bits each, as noted earlier. These parts are denoted as $v_i(1), v_i(2), \cdots , v_i(L/D)$. Our algorithm for achieving $L$-bit consensus consists of $L/D$ sequential executions of Algorithm \ref{alg:consensus} presented in this section (we will discuss the algorithm in detail below). Algorithm \ref{alg:consensus} is executed once for each generation. For the $g$-th generation ($1\le g \le L/D$), each processor $P_i$ uses $v_i(g)$ as its input in Algorithm \ref{alg:consensus}. Each generation of the algorithm results in processor $P_i$ deciding on $g$-th part (namely, $v_i'(g)$) of its final decision value $v_i'$. The value $v_i(g)$ is represented by a vector of $n-2t$ symbols, each symbol represented with $D/(n-2t)$ bits. For convenience of presentation, we assume that $D/(n-2t)$ is an integer. We will refer to these $n-2t$ symbols as the {\em data symbols}. A $(n,n-2t)$ distance-$(2t+1)$ Reed-Solomon code, denoted as $C_{2t}$, is used to encode the $n-2t$ data symbols into $n$ coded symbols. We assume that $D/(n-2t)$ is large enough to allow the above Reed-Solomon code to exist, specifically, $n \le 2^{D/(n-2t)} - 1$. This condition is met only if $L$ is large enough (since $L > D$). We now present some notations to be used in our discussion below. For a $m$-element vector $V$, we denote $V[j]$ as the $j$-th element of the vector, $1\le j \le m$. Given a subset $A \subseteq \{1,\dots,m\}$, denote $V/A$ as the ordered list of elements of $V$ at the locations corresponding to elements of $A$. For instance, if $m=5$ and $A=\{2,4\}$, then $V/A$ is equal to $(V[2],V[4])$. We will say that $V/A \in C_{2t}$ if there exists a codeword $Z\in C_{2t}$ such that $Z/A = V/A$. Otherwise, we will say that $V/A \notin C_{2t}$. Suppose that $Z$ is the codeword corresponding to data $v$. This is denoted as $Z = C_{2t}(v)$, and $v = C_{2t}^{-1}(Z)$. We will extend the definition of the inverse function $C_{2t}^{-1}$ as follows. When set $A$ contains at least $n-2t$ elements, we will define $C_{2t}^{-1}(V/A) = v$, if there exists a codeword $Z\in C_{2t}$ such that $Z/A = V/A$ and $C_{2t}(v) = Z$. Let the set of all the fault-free processors be denoted as $P_{good}$. Algorithm \ref{alg:consensus} for each generation $g$ consists of three stages. We summarize the function of these three stages first, followed by a more detailed discussion: \begin{enumerate} \item Matching stage: Each processor $P_i$ encodes its $D$-bit input $v_i(g)$ for generation $g$ into $n$ coded symbols, as noted above. Each processor $P_i$ sends one of these $n$ coded symbols to the other processors {\bf that it trusts}. Processor $P_i$ trusts processor $P_j$ if and only if the corresponding vertices in the diagnosis graph are connected by an edge. Using the symbols thus received from each other, the processors attempt to identify a ``matching set'' of processors (denoted $P_{match}$) of size $n-t$ such that the fault-free processors in $P_{match}$ are guaranteed to have an identical input value for the current generation. If such a $P_{match}$ is not found, it can be determined with certainty that all the fault-free processors do not have the same input value -- in this case, the fault-free processors decide on a default output value and terminate the algorithm. \item Checking stage: If a set of processors $P_{match}$ is identified in the above matching stage, each processor $P_j \notin P_{match}$ checks whether the symbols received the from processors in $P_{match}$ correspond to a valid codeword. If such a codeword exists, then the symbols received from $P_{match}$ are said to be ``consistent''. If any processor finds that these symbols are not consistent, then misbehavior by some faulty processor is detected. Else all the processors are able to correctly compute the value to be agreed upon in the current generation. \item Diagnosis stage: When misbehavior is detected in the checking stage, the processors in $P_{match}$ are required to {\em broadcast} the coded symbol they sent in the matching stage, using the \BSB algorithm. Using the information received during these broadcasts, the fault-free processors are able to learn new information regarding the potential identity of the faulty processor(s). The {\em diagnosis graph} (called \DG in Algorithm \ref{alg:consensus}) is updated to incorporate this new information. \end{enumerate} In the rest of this section, we discuss each of the three stages in more detail. Note that whenever algorithm \BSB is used, all the fault-free processors will receive the broadcasted information identically. One instance of \BSB is needed for each bit of information broadcasted using \BSB. \begin{algorithm}[!ht] \caption{Multi-Valued Consensus (generation $g$)}\label{alg:consensus} \begin{enumerate*} \item {\bf Matching Stage: }\\ Each processor $P_i$ performs the matching stage as follows: \begin{enumerate*} \item Compute $(S_{i}[1],\dots,S_{i}[n]) = C_{2t}(v_i(g))$, and {\em send} $S_{i}[i]$ to every trusted processor $P_j$ \item \label{step:send_S} $ R_{i}[j] \leftarrow \left\{ \begin{array}{l} \textrm{symbol that $P_i$ receives from $P_j$, if $P_i$ trusts $P_j$;}\\ \perp, \textrm{otherwise} \end{array} \right. $ \item \label{step:M} If $S_{i}[j] = R_{i}[j]$ then $M_{i}[j] \leftarrow $ \TRUE; else $M_{i}[j] \leftarrow $ \FALSE \item $P_i$ broadcasts the vector $M_i$ using \BSB \end{enumerate*} Using the received $M$ vectors: \begin{enumerate*} \setcounter{enumii}{4} \item Find a set of processors $P_{match}$ of size $n-t$ such that \\ \hspace*{0.3in} $M_{j}[k]=M_{k}[j]=$ \TRUE for every pair of $P_j,P_k\in P_{match}$ \item If $P_{match}$ does not exist then decide on a default value and terminate;\\ else enter the Checking Stage \end{enumerate*} \item {\bf Checking Stage:}\\ Each processor $P_j\notin P_{match}$ performs steps 2(a) and 2(b): \begin{enumerate*} \item If $R_{j}/P_{match}\in C_{2t}$ then $Detected_j \leftarrow$ \FALSE; else $Detected_j \leftarrow$ \TRUE. \item Broadcast $Detected_j$ using \BSB. \end{enumerate*} Each processor $P_i$ performs step 2(c): \begin{enumerate*} \setcounter{enumii}{2} \item \label{step:no_detect} Receive $Detected_j$ from each processor $P_j\notin P_{match}$ (broadcasted in step 2(b)).\\ If $Detected_j=$ \FALSE for all $P_j\notin P_{match}$, then decide on $v_i'(g) = C^{-1}_{2t}(R_{i}/P_{match})$; \\ else enter Diagnosis Stage \end{enumerate*} \item {\bf Diagnosis Stage: }\\ Each processor $P_j\in P_{match}$ performs step 3(a): \begin{enumerate*} \item Broadcast $S_{j}[j]$ using \BSB \\ (one instance of \BSB is needed for each bit of $S_j[j]$) \end{enumerate*} Each processor $P_i$ performs the following steps: \begin{enumerate*} \setcounter{enumii}{1} \item $R^\#[j]\leftarrow$ symbol received from $P_j\in P_{match}$ as a result of broadcast in step 3(a) \item For all $P_j\in P_{match}$,\\ \hspace*{0.3in} if $P_i$ trusts $P_j$ and $R_i[j]= R^\#[j]$ then $Trust_i[j]\leftarrow $ \TRUE;\\ \hspace*{0.3in} else $Trust_i[j]\leftarrow $ \FALSE \item Broadcast $Trust_i/P_{match}$ using \BSB \item \label{step:remove_edge} For each edge $(j,k)$ in \DG, \\ \hspace*{0.3in} remove edge $(j,k)$ {\bf if} $Trust_j[k]$ = \FALSE or $Trust_k[j]$ = \FALSE \item \label{step:remove_false_detect} If $R^\#/P_{match}\in C_{2t}$ then \\ \hspace*{0.3in} if for any $P_j\notin P_{match}$, \\ \hspace*{0.6in} $Detected_j =$ \TRUE, but no edge at vertex $j$ was removed in step 3(e) \\ \hspace*{0.3in} then remove all edges at vertex $j$ in \DG \item \label{step:remove_faulty} If at least $t+1$ edges at any vertex $j$ have been removed so far,\\ then processor $P_j$ must be faulty, and all edges at $j$ are removed. \item Find a set of processors $P_{decide} \subset P_{match}$ of size $n-2t$ in the updated $Diag\_Graph$,\\ such that every pair of $P_j,P_k\in P_{decide}$ trust each other. \item Decide on $v_i'(g) = C_{2t}^{-1}(R^\#/P_{decide})$. \end{enumerate*} \end{enumerate*} \end{algorithm} \clearpage \subsection{Matching Stage} The line numbers referred below correspond to the line numbers for the pseudo-code in Algorithm \ref{alg:consensus}. \noindent {\tt Line 1(a):} In generation $g$, each processor $P_i$ first encodes $v_i(g)$, represented by $n-t$ symbols, into a codeword $S_i$ from the code $C_{2t}$. The $j$-th symbol in the codeword is denoted as $S_i[j]$. Then processor $P_i$ sends $S_i[i]$, the $i$-th symbol of its codeword, to all the other processors {\em that it trusts}. Recall that $P_i$ trusts $P_j$ if and only if there is an edge between the corresponding vertices in the diagnosis graph (referred as \DG in the pseudo-code). \noindent {\tt Line 1(b):} Let us denote by $R_i[j]$ the symbol that $P_i$ receives from a trusted processor $P_j$. Processor $P_i$ {\bf ignores} any messages received from untrusted processors, treating the message as a distinguished symbol $\perp$. \noindent {\tt Line 1(c)}: Flag $M_{i}[j]$ is used to record whether processor $P_i$ finds processor $P_j$'s symbol consistent with its own local value. Specifically, the pseudo-code in line 1(c) is equivalent to the following: \begin{itemize} \item When $P_i$ trusts $P_j$: If $R_{i}[j] = S_{i}[j]$, then set $M_{i}[j]=$ \TRUE; else $M_{i}[j]=$ \FALSE. \item When $P_i$ does not trust $P_j$: $M_{i}[j]=$ \FALSE. \end{itemize} \noindent {\tt Line 1(d):} As we will see later, if a fault-free processor $P_i$ does not trust another processor, then the other processor must be faulty. Thus entry $M_i[j]$ in vector $M_i$ is \FALSE if $P_i$ believes that processor $P_j$ is faulty, or that the value at processor $P_j$ differs from the value at $P_i$. Thus, entry $M_i[j]$ being \TRUE implies that, as of this time, $P_i$ believe that $P_j$ is fault-free, and that the value at $P_j$ is possibly identical to the value at $P_i$. Processor $P_i$ uses \BSB to broadcast $M_i$ to all the processors. One instance of \BSB is needed for each bit of $M_i$. \noindent{\tt Lines 1(e) and 1(f):} Due to the use of \BSB, all fault-free processors receive identical vector $M_j$ from each processor $P_j$. Using these $M$ vectors, each processor $P_i$ attempts to find a set $P_{match}$ containing exactly $n-t$ processors such that, for every pair $P_j,P_k\in P_{match}$, $M_{j}[k]=M_{k}[j]=$ \TRUE. Since the $M$ vectors are received identically by all the fault-free processors (using \BSB), they can compute identical $P_{match}$. However, if such a set $P_{match}$ does not exist, then the fault-free processors conclude that all the fault-free processors do not have identical input -- in this case, they decide on some default value, and terminate the algorithm. In the following discussion, we will show the correctness of this step. In the proof of the lemmas \ref{lm:matching_clique} and \ref{lm:matching_same_input}, we assume that the fault-free processors (that is, the processors in set $P_{good}$) always trust each other -- this assumption will be shown to be correct later in Lemma \ref{lm:diagnosis_graph}. \begin{lemma} \label{lm:matching_clique} If for each fault-free processor $P_i\in P_{good}$, $v_i(g) = v(g)$, for some value $v(g)$, then a set $P_{match}$ necessarily exists (assuming that the fault-free processors trust each other). \end{lemma} \begin{proof} Since all the fault-free processors have identical input $v(g)$ in generation $g$, $S_i = C_{2t}(v(g))$ for all $P_i\in P_{good}$. Since these processors are fault-free, and trust each other, they send each other correct messages in the matching stage. Thus, $R_{i}[j] = S_{j}[j] = S_{i}[j]$ for all $P_i,P_j\in P_{good}$. This fact implies that $M_{i}[j]=$ \TRUE for all $P_i,P_j\in P_{good}$. Since there are at least $n-t$ fault-free processors, it follows that a set $P_{match}$ of size $n-t$ must exist. \end{proof} Observe that, although the above proof shows that there exists a set $P_{match}$ containing only fault-free processors, there may also be other such sets that contain some faulty processors as well. That is, all the processors in $P_{match}$ cannot be assumed to be fault-free. Converse of Lemma \ref{lm:matching_clique} implies that, if a set $P_{match}$ does not exist, it is certain that the fault-free processors do not have the same input values. In this case, they can correctly agree on some default value and terminate the algorithm. This proves the correctness of Line 1(f). In the case when a set $P_{match}$ is found, the following lemma is useful. \begin{lemma} \label{lm:matching_same_input} The fault-free processors in $P_{match}$ (that is, all the processors in $P_{match}\cap P_{good}$) have the same input for generation $g$. \end{lemma} \begin{proof} $|P_{match}\cap P_{good}|\ge n-2t$ because $|P_{match}| = n-t$ and there are at most $t$ faulty processors. Consider any two processors $P_i,P_j\in P_{match}\cap P_{good}$. Since $M_i[j]=M_j[i]= \TRUE$, it follows that $S_{i}[i] = S_{j}[i]$ and $S_{j}[j] = S_{i}[j]$. Since there are $n-2t$ fault-free processors in $P_{match}\cap P_{good}$, this implies that the codewords computed by these fault-free processors (in Line 1(a)) contain at least $n-2t$ identical symbols. Since the code $C_{2t}$ has dimension $(n-2t)$, this implies that the fault-free processors in $P_{match}\cap P_{good}$ must have identical input in generation $g$. \end{proof} \subsection{Checking Stage} When $P_{match}$ is found during the matching stage, the checking stage is entered. \noindent{\tt Lines 2(a) and 2(b):} Every fault-free processor $P_j\notin P_{match}$ checks whether the symbols received from the trusted processors in $P_{match}$ are consistent with a valid codeword: that is, check whether $R_{j}/P_{match} \in C_{2t}$. The result of this test is broadcasted as a 1-bit notification $Detected_i$, using \BSB. If $R_{j}/P_{match} \notin C_{2t}$, then processor $P_j$ is said to have detected an {\em inconsistency}. \noindent{\tt Line 2(c):} If no processor announces in Line 2(b) that it has detected an inconsistency, each fault-free processor $P_i$ chooses $C_{2t}^{-1}(R_i/P_{match})$ as its decision value for generation $g$. The following lemma argues correctness of the decision made in Line 2(c). \begin{lemma} \label{lm:checking} If no processor detects inconsistency in Line 2(a), all fault-free processors $P_i\in P_{good}$ decide on the identical output value $v'(g)$ such that $v'(g) = v_j(g)$ for all $P_j\in P_{match}\cap P_{good}$. \end{lemma} \begin{proof} Observe that size of set $P_{match}\cap P_{good}$ is at least $n-2t$, and hence the inverse operations $C_{2t}^{-1}(R_i/P_{match})$ and $C_{2t}^{-1}(R_i/P_{match}\cap P_{good})$ are both defined. Since fault-free processors send correct messages, $R_{i}/P_{match}\cap P_{good}$ are identical for all fault-free processors $P_i\in P_{good}$. Since no inconsistency has been detected by any processor, every fault-free processor $P_i$ decides on $C_{2t}^{-1}(R_i/P_{match})$ as its output. Since $C_{2t}$ has dimension $(n-2t)$, $C_{2t}^{-1}(R_i/P_{match}) = C_{2t}^{-1}(R_i/P_{match}\cap P_{good})$. It then follows that all the fault-free processors $P_i$ decide on the identical value $v'(g) = C_{2t}^{-1}(R_i/P_{match}\cap P_{good})$ in Line 2(c). Since $R_j/P_{match}\cap P_{good} = S_j/P_{match}\cap P_{good}$ for all processors $P_j\in P_{match}\cap P_{good}$, $v'(g) = v_j(g)$ for all $P_j\in P_{match}\cap P_{good}$. \end{proof} \subsection{Diagnosis Stage} When any processor that is not in $P_{match}$ announces that it has detected an inconsistency, the diagnosis stage is entered. The algorithm allows for the possibility that a faulty processor may erroneously announce that it has detected an inconsistency. The purpose of the diagnosis stage is to learn new information regarding the potential identity of a faulty processor. The new information is used to remove one or more edges from the diagnosis graph \DG -- as we will soon show, when an edge $(j,k)$ is removed from the diagnosis graph, at least one of $P_j$ and $P_k$ must be faulty. We now describe the steps in the Diagnosis Stage. \noindent{\tt Lines 3(a) and 3(b):} Every fault-free processor $P_j\in P_{match}$ uses \BSB to broadcast $S_{j}[j]$ to all processors. Let us denote by $R^\#[j]$ the result of the broadcast from $P_j$. Due to the use of \BSB, all fault-free processors receive identical $R^\#[j]$ for each processor $P_j\in P_{match}$. This information will be used for diagnostic purposes. \noindent{\tt Line 3(c) and 3(d):} Every fault-free processor $P_i$ uses flag $Trust_i[j]$ to record whether it ``{\em believes}'', as of this time, that each processor $P_j\in P_{match}$ is fault-free or not. Then $P_i$ broadcasts $Trust_i/P_{match}$ to all processors using \BSB. Specifically, \begin{itemize} \item If $P_i$ trusts $P_j$ {\bf and} $R_i[j] = R^\#[j]$, then set $Trust_i[j]=$ \TRUE; \item If $P_i$ does not trust $P_j$ {\bf or} $R_i[j] \neq R^\#[j]$, then set $Trust_i[j]=$\FALSE. \end{itemize} \noindent{\tt Line 3(e):} Using the $Trust$ vectors, each fault-free processor $P_i$ then removes any edge $(j,k)$ from the diagnosis graph such that $Trust_j[k]$ or $Trust_k[j]=$ \FALSE. Due to the used of \BSB, all fault-free processors receive identical $Trust$ vectors. Hence they will remove the same set of edges and maintain an identical view of the updated \DG. \noindent{\tt Line 3(f):} As we will soon show, in the case $R^\#/P_{match}\in C_{2t}$, a processor $P_j\notin P_{match}$ that announces that it has detected an inconsistency, i.e., $Detected_j =$\TRUE, must be faulty if no edge attached to vertex $j$ was removed in Line 3(e). Such processors $P_j$ are ``{\em isolated}'', by having all edges attached to vertex $j$ removed from \DG, and the fault-free processors will not communicate with it anymore in subsequent generations. \noindent{\tt Line 3(g):} As we will soon show, a processor $P_j$ must be faulty if at least $t+1$ edges at vertex $j$ have been removed. The identified faulty processor $P_j$ is then isolated. \noindent{\tt Lines 3(h) and 3(i):} Since \DG is updated only with information broadcasted with \BSB ($Detected$, $R^\#$ and $Trust$), all fault-free processors maintain an identical view of the updated \DG. Then they can compute an identical set $P_{decide}\subset P_{match}$ containing exactly $n-2t$ processors such that every pair $P_j,P_k\in P_{decide}$ trust each other. Finally, every fault-free processor chooses $C_{2t}^{-1}(R^\#/P_{decide})$ as its decision value for generation $g$. We first prove the following property of the evolution of \DG. \begin{lemma} \label{lm:diagnosis_graph} Every time the diagnosis stage is performed, at least one edge attached to a vertex corresponding to a faulty processor will be removed from $Diag\_Graph$, and only such edges will be removed. \end{lemma} \begin{proof} We prove this lemma by induction. For the convenience of discussion, let us say an edge $(j,k)$ is ``{\em bad}'' if at least one of $P_j$ and $P_k$ is faulty. Consider a generation $g$ starting with any instance of the \DG in which only bad edges have been removed. When the diagnosis stage is performed, there are two possibilities: (1) a fault-free processor $P_i\notin P_{match}$ detects an inconsistency; or (2) a faulty processor $P_j\notin P_{match}$ announces that it has detected an inconsistency. We consider the two possibilities separately: \begin{enumerate} \item A fault-free processor $P_i\notin P_{match}$ detects an inconsistency: In this case, $R_i/P_{match}\notin C_{2t}$. However, according to the definition of $P_{match}$, $R_k/P_{match}=S_k/P_{match}\in C_{2t}$ for every processor $P_k\in P_{match}\cap P_{good}$. This implies that there must be a faulty processor $P_j\in P_{match}$, which is trusted by $P_i$ and $P_k$, has sent different symbols to the fault-free processors $P_i$ and $P_k$ during the matching stage. Thus, the $R^\#[j]$ must be different from at least one of $R_i[j]$ and $R_k[j]$. As a result, $Trust_i[j]=$ \FALSE or $Trust_k[j]=$ \FALSE. Then at least one of the bad edges $(i,j)$ and $(j,k)$ will be removed in Line 3(e). \item A faulty processor $P_j\notin P_{match}$ announces that it detects an inconsistency: Denote by $X\subset P_{match}$ the set of processors $\in P_{match}$ that $P_j$ trusts. According to the algorithm, either an bad edge $(j,k)$ for some $P_k\in X$ was removed in Line 3(e), or none of such edges is removed. In the former case, the bad edge $(j,k)$ is removed. In the later case, there are two possibilities \begin{enumerate} \item $R^\#/{P_{match}}\in C_{2t}$: Given that no edge $(j,k)$ for every $P_k\in X$ was removed in Line 3(e), one can conclude that, {\em if $P_j$ is fault-free}, then $Trust_j[k]=$\TRUE for all $P_k\in X$, and $R_j[k]/X = R^\#[k]/X\in C_{2t}$. On the other hand, observe that $P_j$ computes $Detected_j$ by checking whether $R_j/X\in C_{2t}$, since any message from untrusted processors in $P_{match}$ should have been ignored by $P_j$ in Line 1(b). From $Detected_j=$ \TRUE, one can conclude that, {\em if $P_j$ is fault-free}, $R_j/X\notin C_{2t}$. Now we have a contradiction if $P_j$ is fault-free. So processor $P_j$ must be faulty and all edges at vertex $j$ are bad. These bad edges are removed in Line 3(f). \item $R^\#/{P_{match}}\notin C_{2t}$: In this case, similar to the discussion in case 1, some bad edge connecting two vertices corresponding to processors in $P_{match}$ is removed in Line 3(e). \end{enumerate} \end{enumerate} So by the end of Line 3(f), at least one new bad edge has been removed. Moreover, since $R_i[k] = R^\#[k]$ for all fault-free processors $P_k \in P_{match}\cap P_{good}$, $Trust_{i}[k]$ remains \TRUE for every pair of processors $P_i,P_k\in P_{good}$, which implies that the vertices corresponding to the fault-free processors will remain fully connected, and each will always have at least $n-t-1$ edges. This follows that a processor $P_j$ must be faulty if at least $t+1$ edges at vertex $j$ has been removed. So all edges at $j$ are bad and will be removed in Line 3(g). Now we have proved that for every generation that begins with a $Diag\_Graph$ in which only bad edges have been removed, at least one new bad edge, and only bad edges, will be removed in the updated $Diag\_Graph$ by the end of the diagnosis stage. Together with the fact that $Diag\_Graph$ is initialized as a complete graph, we finish the proof. \end{proof} The above proof of Lemma \ref{lm:diagnosis_graph} shows that all fault-free processors will trust each other throughout the execution of the algorithm, which justifies the assumption made in the proofs of the previous lemmas. The following lemma shows the correctness of Lines 3(h) and 3(i). \begin{lemma} \label{lm:diagnosis_consensus} By the end of diagnosis stage, all fault-free processors $P_i\in P_{good}$ decide on the same output value $v'(g)$, such that $v'(g) = v_j(g)$ for all $P_j\in P_{match}\cap P_{good}$. \end{lemma} \begin{proof} First of all, the set $P_{decide}$ necessarily exists since there are at least $n-2t\ge t+1$ fault-free processors in $P_{match}\cap P_{good}$ that always trust each other. Secondly, since the size of $P_{decide}$ is $n-2t\ge t+1$, it must contain at least one fault-free processor $P_k\in P_{decide}\cap P_{good}$. Since $P_k$ still trusts all processors of $P_{decide}$ in the updated $Diag\_Graph$, $R^\#/{P_{decide}} = R_k/P_{decide}= S_k/P_{decide}$. The second equality is due to the fact that $P_k\in P_{match}$. Finally, since the size of set $P_{decide}$ is $n-2t$, the inverse operation of $C^{-1}_{2t} (R^\#/{P_{decide}})$ is defined, and it equals to $C^{-1}_{2t} (S_k/{P_{decide}}) = v_k(g) = v_j(g)$ for all $P_j\in P_{match}\cap P_{good}$, as per Lemma \ref{lm:matching_same_input}. \end{proof} We can now conclude the correctness of the Algorithm \ref{alg:consensus}. \begin{theorem} \label{thm:consensus} Given $n$ processors with at most $t<n/3$ are faulty, each given an input value of $L$ bits, Algorithm \ref{alg:consensus} achieves consensus correctly in $ L/D$ generations , with the diagnosis stage performed for at most $t(t+1)$ times. \end{theorem} \begin{proof} According to Lemmas \ref{lm:matching_clique} to \ref{lm:diagnosis_consensus}, consensus is achieved correctly for each generation $g$ of $D$ bits. So the termination and consistency properties are satisfied for the $L$-bit outputs after $L/D$ generations. Moreover, in the case all fault-free processors are given an identical $L$-bit input $v$, the $D$ bits output $v'(g)$ in each generation $g$ equals to $v(g)$ as per Lemmas \ref{lm:matching_clique}, \ref{lm:checking} and \ref{lm:diagnosis_consensus}. So the $L$-bit output $v'=v$ and the validity property is also satisfied. According to Lemma \ref{lm:diagnosis_graph} and the fact that a faulty processor $P_j$ will be removed once more than $t$ edges at vertex $j$ have been removed, it takes at most $t(t+1)$ instance of the diagnosis stage before all faulty processors are identified. After that, the fault-free processors will not communicate with the faulty processors. Thus, the diagnosis stage will not be performed any more. So it will be performed for at most $t(t+1)$ times in all cases. \end{proof} \subsection{Complexity} We have discussed the operations of the proposed multi-valued consensus algorithm above. Now let us study the communication complexity of this algorithm. Let us denote by $B$ the complexity of broadcasting 1 bit with one instance of \BSB. In every generation, the complexity of each stage is as follows: \vspace{-5pt} \begin{itemize} \item Matching stage: every processor $P_i$ sends at most $n-1$ symbols, each of $D/(n-2t)$ bits, to the processors that it trusts, and broadcasts $n-1$ bits for $M_i$. So at most $\frac{n(n-1)}{n-2t}D + n(n-1)B$ bits in total are transmitted by all $n$ processors. \item Checking stage: every processor $P_j\notin P_{match}$ broadcasts one bit $Detected_j$ with \BSB, and there are $t$ such processors. So $tB$ bits are transmitted. \item Diagnosis stage: every processor $P_j \in P_{match}$ broadcasts one symbol $S_j[j]$ of $D/(n-2t)$ bits with \BSB; and every processor $P_i$ broadcasts $n-t$ bits of $Trust_i/P_{match}$ with \BSB. So the complexity is $\frac{n-t}{n-2t}DB + n(n-t)B$ bits. \end{itemize} According to Theorem \ref{thm:consensus}, there are $L/D$ generations in total. In the worst case, $P_{match}$ can be found in every generation, so the matching and checking stages will be performed for $L/D$ times. In addition, the diagnosis stage will be performed for at most $t(t+1)$ time. Hence the communication complexity of the proposed consensus algorithm, denoted as $C_{con}(L)$, is then computed as \begin{eqnarray} C_{con}(L) &=&\left(\frac{n(n-1)}{n-2t}D + n(n-1)B + tB\right)\frac{L}{D} + t(t+1)\left(\frac{n-t}{n-2t}D + n(n-t)\right)B \end{eqnarray} For a large enough value of $L$, with a suitable choice of $D = \sqrt{\frac{(n^2-n+t)(n-2t)L}{t(t+1)(n-t)}}$, we have \begin{eqnarray} C_{con}(L) &=& \frac{n(n-1)}{n-2t}L + 2BL^{0.5}\sqrt{\frac{(n^2-n+t)t(t+1)(n-t)}{n-2t}} + t(t+1)n(n-t)B \end{eqnarray} Error-free algorithms that broadcast 1 bit with communication complexity $\Theta(n^2)$ bits are known \cite{bit_optimal_89,opt_bit_Welch92}. So we assume $B=\Theta(n^2)$. Then the complexity of our algorithm for $t<n/3$ becomes \begin{equation} C_{con}(L) = \frac{n(n-1)}{n-2t}L + O(n^4L^{0.5} + n^6) = O(nL + n^4 L^{0.5} + n^6). \end{equation} So for sufficiently large $L$ ($\Omega(n^6)$), the communication complexity approaches $O(nL)$. \section{Introduction} \label{sec:intro} This paper considers the multi-valued Byzantine {\em consensus} problem. The Byzantine consensus problem considers $n$ processors, namely $P_1,...,P_n$, of which at most $t$ processors may be {\em faulty} and deviate from the algorithm in arbitrary fashion. Each processor $P_i$ is given an $L$-bit input value $v_i$, and they want to agree on a value $v$ such that the following properties are satisfied: \begin{itemize} \item {\em Termination}: every fault-free $P_i$ eventually decides on an output value $v_i'$, \item {\em Consistency}: the output values of all fault-free processors are equal, i.e., for every fault-free processor $P_i$, $v_i'=v'$ for some $v'$, \item {\em Validity}: if every fault-free $P_i$ holds the same input $v_i=v$ for some $v$, then $v'=v$. \end{itemize} \noindent Algorithms that satisfy the above properties in all executions are said to be {\bf error-free}. We are interested in the communication complexity of error-free consensus algorithms. {\em Communication complexity} of an algorithm is defined as the maximum (over all permissible executions) of the total number of bits transmitted by all the processors according to the specification of the algorithm. This measure of complexity was first introduced by Yao \cite{Yao_comm_complexity}, and has been widely used by the distributed computing community \cite {authenticated_BA_Dolev83,multi-valued_BA_PODC06, Waidner96information-theoreticpseudosignatures}. \paragraph{System Model:} We assume network and adversary models commonly used in other related work \cite{psl82, bit_optimal_89, opt_bit_Welch92, multi-valued_BA_PODC06,King:PODC2010}. We assume a synchronous fully connected network of $n$ processors, wherein the processor identifiers are common knowledge. Every pair of processors are connected with a pair of directed point-to-point communication channels. Whenever a processor receives a message on such a directed channel, it can correctly assume that the message is sent by the processor at the other end of the channel. We assume a Byzantine adversary that has complete knowledge of the state of the other processors, including the $L$-bit input values. No secret is hidden from the adversary. The adversary can take over up to $t$ processors ($t<n/3$) at any point during the algorithm. These processors are said to be {\em faulty}. The faulty processors can engage in any ``{\em misbehavior}'', i.e., deviations from the algorithm, including sending incorrect messages, and collusion. The remaining processors are {\em fault-free} and follow the algorithm. Finally, we make no assumption of any cryptographic technique, such as authentication and secret sharing. ~ It has been shown that error-free consensus is impossible if $t\ge n/3$ \cite{psl80,psl82}. $\Omega(n^2)$ has been shown to be a lower bound on the number of messages needed to achieve error-free consensus \cite{Bounds_BA_Dolev85}. Since any message must be of at least 1 bit, this gives a lower bound of $\Omega(n^2)$ bits on the communication complexity of any binary (1-bit) consensus algorithm. In practice, agreement is sometimes required for longer messages rather than just single bits. For instance, the ``value'' being agreed upon may be a large file in a fault-tolerant distributed storage system. For instance, as \cite{multi-valued_BA_PODC06} suggests, in a voting protocol, the authorities must agree on the set of all ballots to be tallied (which can be gigabytes of data). Similarly, as also suggested in \cite{multi-valued_BA_PODC06}, multi-valued Byzantine agreement is relevant in secure multi-party computation, where many broadcast invocations can be parallelized and thereby optimized to a single invocation with a long message. The problem of achieving consensus on a single $L$-bit value may be solved using $L$ instances of a 1-bit consensus algorithm. However, this approach will result in communication complexity of $\Omega(n^2L)$, since $\Omega(n^2)$ is a lower bound on communication complexity of 1-bit consensus. In a PODC 2006 paper, Fitzi and Hirt \cite{multi-valued_BA_PODC06} presented a probabilistically correct multi-valued consensus algorithm which improves the communication complexity to $O(nL)$ for sufficiently large $L$, at the cost of allowing a non-zero probability of error. Since $\Omega(nL)$ is a lower bound on the communication complexity of consensus on an $L$-bit value, this algortihm has optimal complexity for large $L$. In their algorithm, an $L$-bit value (or message) is first reduced to a much shorter message, using a universal hash function. Byzantine consensus is then performed for the shorter hashed values. Given the result of consensus on the hashed values, consensus on $L$ bits is then achieved by requiring processors whose $L$-bit input value matches the agreed hashed value deliver the $L$ bits to the other processors jointly. By performing initial consensus only for the smaller hashed values, this algorithm is able to reduce the communication complexity to $O(nL+n^3(n+\kappa))$ where $\kappa$ is a parameter of the algorithm. However, since the hash function is not collision-free, this algorithm is {\bf not} error-free. Its probability of error is lower bounded by the collision probability of the hash function. We improve on the work of Fitzi and Hirt \cite{multi-valued_BA_PODC06}, and present a deterministic error-free consensus algorithm with communication complexity of $O(nL)$ bits for sufficiently large $L$. Our algorithm {\em always} produce the correct result, unlike \cite{multi-valued_BA_PODC06}. For smaller $L$, the communication complexity of our algorithms is $O(nL + n^4 L^{0.5} + n^6)$. To our knowledge, this is the first known error-free multi-valued Byzantine consensus algorithm that achieves, for large $L$, communication complexity linear in $n$. \section*{\hfil #1\hfil}} \renewcommand{\refname}{\hfil References\hfil} \bibliographystyle{plain} \setlength {\parskip}{3pt} \usepackage{appendix} \newcommand{\comment}[1]{} \usepackage{graphicx} \usepackage{latexsym} \usepackage{mdwlist} \usepackage{algorithm} \newcommand{\BSB}{{\em Broadcast\_Single\_Bit~}} \newcommand{\DG}{{\em Diag\_Graph~}} \newcommand{\TRUE}{{\bf true~}} \newcommand{\FALSE}{{\bf false~}} \newcommand{\nchoosek}[2]{{#1 \choose #2}} \newcommand{\BlackBox}{\rule{2.6mm}{2.6mm}} \newenvironment{proof}{\paragraph{\bf Proof:}}{\hspace*{\fill}\(\Box\)} \newtheorem{theorem}{Theorem} \newtheorem{conjecture}{Conjecture} \newtheorem{claim}{Claim} \newtheorem{corollary}{Corollary} \newtheorem{definition}{Definition} \newtheorem{condition}{Condition} \newtheorem{lemma}{Lemma} \newtheorem{example}{Example} \newcommand{\fillblackbox}{\hspace*{\fill}\(\BlackBox\)} \newcommand{\fillbox}{\hspace*{\fill}\(\Box\)} \newcommand{\fig}[1]{Figure~\ref{#1}} \newcommand{\eqn}[1]{Equation~\ref{#1}} \newcommand{\refsec}[1]{Section~\ref{#1}} \newcommand{\num}[1]{(\romannumeral#1)} \def\noflash#1{\setbox0=\hbox{#1}\hbox to 1\wd0{\hfill}} \begin{document} \title{Error-Free Multi-Valued Consensus with Byzantine Failures \footnote{\normalsize This research is supported in part by Army Research Office grant W-911-NF-0710287 and National Science Foundation award 1059540. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the funding agencies or the U.S. government.}} \author{Guanfeng Liang and Nitin Vaidya\\ \normalsize Department of Electrical and Computer Engineering, and\\ \normalsize Coordinated Science Laboratory\\ \normalsize University of Illinois at Urbana-Champaign\\ \normalsize gliang2@illinois.edu, nhv@illinois.edu} \maketitle \date{} \input{abstract} \thispagestyle{empty} \newpage \setcounter{page}{1} \input{intro} \input{nutshell} \input{consensus} \input{broadcast_short} \input{conclusion} \newpage \section{Byzantine Consensus: Salient Features of the Algorithm} \label{sec:nutshell} The goal of our consensus algorithm is to achieve consensus on an $L$-bit value (or message). The algorithm is designed to perform efficiently for large $L$. Consequently, our discussion will assume that $L$ is ``sufficiently large'' (how large is ``sufficiently large'' will become clearer later in the paper). We now briefly describe the salient features of our consensus algorithm, with the detailed algorithm presented later in Section \ref{sec:consensus}. \begin{itemize} \item {\em Algorithm execution in multiple generations:} To improve the communication complexity, consensus on the $L$-bit value is performed ``in parts''. In particular, for a certain integer $D$, the $L$-bit value is divided into $L/D$ parts, each consisting of $D$ bits. For convenience of presentation, we will assume that $L/D$ is an integer. A sub-algorithm is used to perform consensus on each of these $D$-bit values, and we will refer to each execution of the sub-algorithm as a ``generation''. \item {\em Memory across generations:} If during any one generation, misbehavior by some faulty processor is detected, then additional (and expensive) diagnostic steps are performed to gain information on the potential identity of the misbehaving processor(s). This information is captured by means of a {\em diagnosis graph}, as elaborated later. As the sub-algorithm is performed for each new generation, the {\em diagnosis graph} is updated to incorporate any new information that may be learnt regarding the location of the faulty processors. The execution of the sub-algorithm in each generation is adapted to the state of the diagnosis graph at the start of the generation. \item {\em Bounded instances of misbehavior:} With Byzantine failures, it is not always possible to immediately determine the identity of a misbehaving processor. However, due to the manner in which the diagnosis graph is maintained, and the manner in which the sub-algorithm adapts to the diagnosis graph, the $t$ (or fewer) faulty processors can collectively misbehave in at most $t(t+1)$ generations, before all the faulty processors are exactly identified. Once a faulty processor is identified, it is effectively isolated from the network, and cannot tamper with future generations. Thus, $t(t+1)$ is also an upper bound on the number of generations in which the expensive diagnostic steps referred above may need to be performed. \item {\em Low-cost failure-free execution:} Due to the bounded number of generations in which the faulty processors can misbehave, it turns out that the faulty processors do not tamper with the execution in a majority of the generations. We use a low-cost mechanism to achieve consensus in failure-free generations, which helps to achieve low communication complexity. In particular, we use an {\em error detecting code}-based strategy to reduce the amount of information the processors must exchange to be able to achieve consensus in the absence of any misbehavior (the strategy, in fact, also allows detection of potential misbehavior). \item {\em Consistent diagnosis graph maintenance:} A copy of the diagnosis graph is maintained locally by each fault-free processor. To ensure consistent maintenance of this graph, the {\em diagnostic information} (elaborated later) needs to be distributed consistently to all the processors in the network. This operation itself requires a Byzantine broadcast algorithm that solves the ``Byzantine Generals Problem'' \cite{psl82}. With this algorithm, a ``source'' processor broadcasts its message to all other processors reliably, even if some processors (including the source) may be faulty. For this operation we use an error-free 1-bit Byzantine broadcast algorithm that tolerates $t<n/3$ Byzantine failures with communication complexity of $O(n^2)$ bits \cite{opt_bit_Welch92,bit_optimal_89}. This 1-bit broadcast algorithm is referred as \BSB in our discussion. While \BSB is expensive, the cumulative overhead of \BSB is kept low by invoking it a relatively small number of times, when compared to $L$. \end{itemize} We now elaborate on the error detecting code used in our algorithms, and also describe the {\em diagnosis graph} in some more detail. \paragraph{Error detecting code:} We will use Reed-Solomon codes in our algorithms (potentially other codes may be used instead). Consider a $(m,k)$ Reed-Solomon code in Galois Field GF($2^c$), where $c$ is chosen large enough (specifically, $m \leq 2^c - 1$). This code encodes $k$ data symbols from GF($2^c$) into a codeword consisting of $m$ symbols from GF($2^c$). Each symbol from GF($2^c$) can be represented using $c$ bits. Thus, a data vector of $k$ symbols contains $kc$ bits, and the corresponding codeword contains $mc$ bits. Each symbol of the codeword is computed as a linear combination of the $k$ data symbols, such that every subset of $k$ coded symbols represent a set of linearly independent combinations of the $k$ data symbols. This property implies that any subset of $k$ symbols from the $m$ symbols of a given codeword can be used to determine the data vector corresponding to the codeword. Similarly, knowledge of any subset of $k$ symbols from a codeword suffices to determine the remaining symbols of the codeword. So $k$ is also called the {\em dimension} of the code. For a code $C$, let us denote $C()$ as the encoding function, and $C^{-1}()$ as the decoding function. The decoding function can be applied so long as at least $k$ symbols of a codeword are available. \paragraph{Diagnosis Graph:} The fault-free processors' (potentially partial) knowledge of the identity of the faulty processors is captured by a diagnosis graph. A diagnosis graph is an undirected graph with $n$ vertices, with vertex $i$ corresponding to processor $P_i$. A pair of processors are said to ``trust'' each other if the corresponding pairs of vertices in the diagnosis graph is connected with an edge; otherwise they are said to ``accuse'' each other. Before the start of the very first generation, the diagnosis graph is initialized as a fully connected graph, which implies that all the $n$ processors initially trust each other. During the execution of the algorithm, whenever misbehavior by some faulty processor is detected, the diagnosis graph will be updated, and one or more edges will be removed from the graph, using the diagnostic information communicated using the \BSB algorithm. The use of \BSB ensures that the fault-free processors always have a consistent view of the diagnosis graph. As we will show later, the evolution of the diagnosis graph satisfies the following properties: \begin{itemize} \item If an edge is removed from the diagnosis graph, at least one of the processors corresponding to the two endpoints of the removed edge must be faulty. \item The fault-free processors always trust each other throughout the algorithm. \item If more than $t$ edges at a vertex in the diagnosis graph are removed, then the processor corresponding to that vertex must be faulty. \end{itemize} The last two properties above follow directly from the first property, and the assumption that there are at most $t$ faulty processors.

BkiUfIPxK6nrxq6DpXkN

\section{Introduction\label{s:intro}} The large amount of high-precision data already taken at the CERN large hadron collider (LHC) demands theoretical predictions with a corresponding high precision. This situation will be further accentuated with the Run 3, which already started in the spring of 2022. In the context of QCD, the theoretical accuracy is typically increased by performing perturbative calculations of radiative corrections at higher orders in the strong coupling $\alpha_{\mathrm{S}}$. The present high-precision frontier is represented by computations at the next-to-next-to-next-to-leading order (N$^3$LO) in the QCD coupling (see, e.g., Ref.~\cite{Heinrich:2020ybq} and references therein). In theories with massless particles, like QCD, scattering amplitudes lead to infrared (IR) divergent contributions, and finite results are obtained by combining real and virtual radiative corrections in computations of physical observables. The basic property that produces the cancellation of the IR divergences is their universal (i.e., process-independent) structure. The IR singular behaviour of the scattering amplitudes is indeed controlled by universal factorization formulae and by corresponding singular factors for emission of soft and collinear radiation (see, e.g., Ref.~\cite{Agarwal:2021ais} and references therein). The knowledge of these factorization formulae in explicit form is therefore very important to practically organize and greatly simplify the cancellation of the IR divergences in perturbative calculations at various perturbative orders. The cancellation mechanism of the IR divergences produces residual logarithmic contributions that are quantitatively large for a wide class of physical observables which are evaluated in kinematical regions close to the exclusive boundary of the phase space. These large contributions have to be computed at high perturbative orders, and possibly resummed to all orders in perturbation theory (see, e.g., Refs.~\cite{Becher:2014oda, Luisoni:2015xha} and references therein). For instance, QCD resummation for transverse-momentum distributions has reached the next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy \cite{Luo:2019szz, Ebert:2020yqt, Luo:2020epw, Ebert:2020qef}. In general, soft and collinear factorization formulae of scattering amplitudes are important ingredients in the context of QCD computations and resummations of large logarithmic contributions of IR origin. Soft and collinear factorization formulae at ${\cal O}(\alpha_{\mathrm{S}})$ had a key role to devise process-independent and observable-independent methods to perform next-to-leading order (NLO) QCD calculations (see, e.g., Refs.~\cite{Frixione:1995ms, csdip, Frixione:1997np, Catani:2002hc}). Similarly, soft/collinear factorization at ${\cal O}(\alpha_{\mathrm{S}}^2)$ \cite{Campbell:1997hg, Catani:1998nv, Bern:1998sc, Kosower:1999rx, Bern:1999ry, Catani:1999ss, Catani:2000pi, Czakon:2011ve, Bierenbaum:2011gg, Czakon:2018iev, Catani:2011st, Sborlini:2013jba} is used to set up and develop methods (see, e.g., the reviews in Refs.~\cite{Heinrich:2020ybq, Proceedings:2018jsb, Amoroso:2020lgh, TorresBobadilla:2020ekr}) at the next-to-next-to-leading order (NNLO). The knowledge of soft and collinear factorization of scattering amplitudesat ${\cal O}(\alpha_{\mathrm{S}}^3)$ can be exploited in the context of N$^3$LO calculations and of resummed calculations at N$^3$LL accuracy. The singular factors for the various collinear limits at ${\cal O}(\alpha_{\mathrm{S}}^3)$ were presented in Refs.~\cite{DelDuca:1999iql, Birthwright:2005ak, Birthwright:2005vi, DelDuca:2019ggv, DelDuca:2020vst, Catani:2003vu, Sborlini:2014mpa, Sborlini:2014kla, Badger:2015cxa, Czakon:2022fqi, Bern:2004cz, Badger:2004uk, Duhr:2014nda, Catani:2011st}. The study of soft factorization of scattering amplitudes at ${\cal O}(\alpha_{\mathrm{S}}^3)$ involves two-loop, one-loop and tree-level contributions for various soft-parton multiplicities. Single soft-gluon emission at two loop order was examined in Refs.~\cite{Badger:2004uk, Li:2013lsa, Duhr:2013msa, Dixon:2019lnw}. Double soft-parton radiation at one loop level was considered in Refs.~\cite{Zhu:2020ftr,Catani:2021kcy}. Triple soft-gluon emission at the tree level was studied in Ref.~\cite{Catani:2019nqv}. This paper is devoted to study soft gluon-quark-antiquark ($gq{\bar q}$) radiation at the tree level, which has been independently considered very recently in Ref.~\cite{DelDuca:2022noh}. Comments on Ref.~\cite{DelDuca:2022noh} are presented throughout the paper. The outline of the paper is as follows. In Sect.~\ref{s:sfsc} we first introduce our notation and recall the soft factorization formula for scattering amplitudes. Then we present the calculation of the tree-level current for soft $gq{\bar q}$ emission in a generic hard-scattering process. The result for the current has an irreducible correlation component that includes contributions with both abelian and non-abelian characters. In Sect.~\ref{s:tlsc} we consider soft factorization of squared amplitudes and we compute the squared current for soft $gq{\bar q}$ radiation. The squared current leads to irreducible colour dipole and colour tripole interactions. The colour tripole interactions are odd under charge conjugation and they produce charge asymmetry effects between the soft quark and antiquark. In Sect.~\ref{s:e23h} we consider the specific applications to processes with two and three hard partons and, in particular, we discuss the structure of the corresponding charge asymmetry contributions. In Sect.~\ref{s:qed} we generalize our QCD results for soft $gq{\bar q}$ emission to the cases of QED and mixed QCD$\times$QED radiative corrections for soft photon-fermion-antifermion and gluon-fermion-antifermion emissions. A brief summary of our results is presented in Sect.~\ref{s:conc}. In Appendix~\ref{a:tthp} we list the action of colour tripole operators onto scattering amplitudes with two and three hard partons. \section{Soft factorization and soft currents\label{s:sfsc}} In this section we first introduce our notation, mostly following the notation that is also used in Refs.~\cite{Catani:2019nqv, Catani:2021kcy} (more details can be found therein). We also briefly recall the factorization properties of scattering amplitudes in the soft limit and the known tree-level results for the emission of one soft gluon and the emission of a soft quark--antiquark pair. Then we present and discuss our results of the soft current for the emission of a $gq{\bar q}$ system at the tree level. \subsection{Soft factorization of scattering amplitudes\label{s:sfsa}} We study the soft behaviour of a generic scattering amplitude $\M$ whose external-leg particles are on shell and with physical spin polarizations. In our notation all external particles of $\M$ are treated as `outgoing' particles (although they can be initial-state and final-state physical particles), with corresponding outgoing momenta and quantum numbers (e.g., colour, spin and flavour). The perturbative evaluation of $\M$ is performed by using dimensional regularization in $d=4-2\epsilon$ space-time dimensions, and $\mu$ is the dimensional regularization scale. Specifically, we use conventional dimensional regularization (CDR), with $d-2$ spin polarization states for on shell gluons (and photons) and 2 polarization states for on shell massless quarks or antiquarks (and massless leptons). We consider the behaviour of $\M$ in the kinematical configuration where one or more of the momenta of the external-leg massless particles become soft. We denote the soft momenta by $q_\ell^\mu$ ($\ell=1,\dots,N$, and $N$ is the total number of soft particles), while the momenta of the hard particles in $\M$ are denoted by $p_i^\mu$ (in general they are not massless and $p_i^2 \equiv m_i^2 \neq 0$) In this kinematical configuration, $\M(\{q_\ell\}, \{p_i\})$ becomes singular. The dominant singular behaviour is given by the following factorization formula in colour space \cite{Catani:1999ss, Bern:1999ry, Catani:2000pi}: \begin{equation}\label{1gfact} \ket{\M(\{q_\ell\}, \{p_i\})} \simeq \J(q_1,\cdots,q_N) \; \ket{\M (\{p_i\})} \;. \end{equation} Here $\M (\{p_i\})$ is the scattering amplitude that is obtained from the original amplitude $\M(\{q_\ell\}, \{p_i\})$ by simply removing the soft external legs. The factor $\J$ is the soft current for multi-particle radiation from the scattering amplitude. At the formal level the soft behaviour of $\M(\{q_\ell\}, \{p_i\})$ is specified by performing an overall rescaling of all soft momenta as $q_\ell \rightarrow \xi q_l$ (the rescaling parameter $\xi$ is the same for each soft momentum $q_\ell$) and by considering the limit $\xi\rightarrow 0$. In this limit, the amplitude is singular and it behaves as $(1/\xi)^N$ (modulo powers of $\ln \xi$ from loop corrections). This dominant singular behaviour is embodied in the soft current $\J$ on the right-hand side of Eq.~(\ref{1gfact}). In this equation the symbol $\simeq$ means that on the right-hand side we neglect contributions that are less singular than $(1/\xi)^N$ in the limit $\xi\rightarrow 0$. The soft current $\J(q_1,\cdots,q_N)$ in Eq.~(\ref{1gfact}) depends on the momenta, colours and spins of both the soft and hard partons in the scattering amplitude (although the hard-parton dependence is not explicitly denoted in the argument of $\J$). However this dependence entirely follows from the external-leg content of $\M$, and the soft current is completely independent of the internal structure of the scattering amplitude. In particular, we remark that the factorization in Eq.~(\ref{1gfact}) is valid \cite{Bern:1999ry, Catani:2000pi, Feige:2014wja} at arbitrary perturbative orders in the loop expansion of the scattering amplitude. Therefore on both sides of Eq.~(\ref{1gfact}) the scattering amplitudes have the loop expansion ${\ket \M}= {\ket {\M^{(0)}}}+{\ket {\M^{(1)}}}+\dots$, where $\M^{(0)}$ is the contribution to $\M$ at the lowest perturbative order, $\M^{(1)}$ is the one-loop contribution, and so forth. Correspondingly, we have $\J= \J^{(0)}+\J^{(1)}+\dots$, where $\J^{(n)}$ is the contribution to $\J$ at the $n$-th loop accuracy. In the following sections of this paper we limit ourselves to considering explicit expressions of only tree-level currents $\J^{(0)}$ and, for the sake of simplicity, we simply denote them by $\J$ (removing the explicit superscript $(0)$). Considering the emission of soft QCD partons, the all-loop current $\J$ in Eq.~(\ref{1gfact}) is an operator that acts from the colour+spin space of $\M(\{p_i\})$ to the enlarged space of $\M(\{q_\ell\}, \{p_i\})$. In particular, soft radiation produces colour correlations. To take into account the colour structure we use the colour (+ spin) space formalism of Ref.~\cite{csdip}. The scattering amplitude $\M_{s_1 s_2 \dots}^{c_1 c_2\dots}$ depends on the colour ($c_i$) and spin ($s_i$) indices of its external-leg partons. This dependence is embodied in a vector $\ket{\M}$ in colour+spin space through the definition (notation) \begin{equation}\label{Mstate} \M_{s_1 s_2 \dots}^{c_1 c_2 \dots} \equiv \big(\bra{c_1,c_2,\cdots}\otimes\bra{s_1,s_2,\cdots}\big) \; \ket{\M} \;\;, \end{equation} where $\{ \,\ket{c_1,c_2,\cdots}\otimes\ket{s_1,s_2,\cdots} \} = \{ \, \ket{c_1,s_1;c_2,s_2,\cdots} \}$ is an orthonormal basis of abstract vectors in colour+spin space. In colour space the colour correlations produced by soft-gluon emission are represented by associating a colour charge operator $\T_i$ to the emission of a gluon from each parton $i$. If the emitted gluon has colour index $a$ ($a=1,\dots,N_c^2-1$, for $SU(N_c)$ QCD with $N_c$ colours) in the adjoint representation, the colour charge operator is $\T_i \equiv \bra{a} \,T_i^a$ and its action onto the colour space is defined by \begin{equation}\label{defT} \bra{a,c_1,\cdots,c_i,\cdots,c_m}\,\T_i\,\ket{b_1,\cdots,b_i,\cdots,b_m} \equiv \delta_{c_1 b_1} \cdots (T^a)_{c_i b_i} \cdots \delta_{c_m b_m} \;, \end{equation} where the explicit form of the colour matrices $T^a_{c_i b_i}$ depends on the colour representation of the parton $i$: \begin{align} (T^a)_{b c} &= \ui f^{bac} && \text{(adjoint representation) if $i$ is a gluon,} &&\nonumber \\ (T^a)_{\alpha\beta} &= t^a_{\alpha\beta} && \text{(fundamental representation with $\alpha,\beta=1,\dots,N_c$) if $i$ is a quark,} &&\nonumber \\ (T^a)_{\alpha\beta} &= -t^a_{\beta\alpha} && \text{if $i$ is an antiquark.} &&\nonumber &&\label{Tcs} \end{align} We normalize the colour matrices such as $[T_i^a , T_j^b] = i f^{abc} T_i^c \delta_{ij}$ and ${\rm Tr} (t^a t^b) = T_R \,\delta_{ab}$ with $T_R=1/2$. We also use the notation $\sum_a T_i^a T_k^a \equiv \T_i \cdot \T_k$ and $\T_i^2 = C_i$, where $C_i$ is the quadratic Casimir coefficient of the colour representation, with the normalization $C_i=C_A=N_c$ if $i$ is a gluon and $C_i=C_F=(N_c^2-1)/(2N_c)$ if $i$ is a quark or antiquark. Note that each `amplitude vector' $\ket{\M}$ is an overall colour-singlet state. Therefore, colour conservation is simply expressed by the relation \begin{equation} \label{colcons} \sum_i \;\T_i \;\ket{\M} = 0 \;\;, \end{equation} where the sum extends over all the external-leg partons $i$ of the amplitude $\M$. For subsequent use, we also introduce the shorthand notation \begin{equation} \label{csnotation} \sum_i \;\T_i \;\eqcs \;0 \;\;, \end{equation} where the subscript CS in the symbol $\eqcs$ means that the equality between the terms in the left-hand and right-hand sides of the equation is valid if these (colour operator) terms act (either on the left or on the right) onto colour-singlet states. \subsection{Tree-level currents\label{s:tlc}} The tree-level current ${\bom J}(q)$ for the emission of a single soft gluon of momentum $q^\nu$ is well known~\cite{Bassetto:1984ik}: \begin{equation}\label{J1} {\bom J}(q) = \g \,\mu^\epsilon \;\sum_i \;\boldsymbol{b}_i \; \frac{p_i \cdot \varepsilon(q)}{p_i \cdot q} \equiv {\bom J}_\nu(q) \varepsilon^\nu(q)\;\;, \end{equation} where $\g$ is the QCD coupling ($\alpha_{\mathrm{S}}=\g^2/(4\pi)$). The notation $\sum_i$ means that the sum extends over all hard partons (with momenta $p_i$) in ${\cal M}$, $\boldsymbol{b}_i$ is the colour charge of the hard parton $i$, and $\varepsilon^\nu(q)$ is the spin polarization vector of the soft gluon. The current for emission of soft gluons is conserved by acting on colour-singlet states (see Ref.~\cite{Catani:2019nqv} for a general discussion on soft-current conservation). From Eq.~(\ref{J1}) we have \begin{equation}\label{consJ1} q^\nu \,{\bom J}_\nu(q) = \sum_i \boldsymbol{b}_i \;\;, \end{equation} and, therefore, by using colour conservation in Eq.~\eqref{colcons}, the current conservation relation $q^\nu \,{\bom J}_\nu(q) \eqcs 0$ is directly fulfilled. The tree-level currents for emission of two and three soft gluons were computed in Refs.~\cite{Catani:1999ss} and \cite{Catani:2019nqv}, respectively. The emission of a soft quark-antiquark ($q{\bar q}$) pair by tree-level QCD interactions was studied in Ref.~\cite{Catani:1999ss}. Using our notation, the QCD current for radiation of a soft quark and antiquark at the tree level is \cite{Catani:2021kcy} \begin{equation}\label{Jqa} {\bom J}(q_1,q_2) = - \left( \g\,\mu^\epsilon\right)^2 \, \sum_i \,{\bom t}^c \;T^c_i \;\frac{p_i \cdot j(1,2)}{p_i \cdot q_{12}} \;\;, \end{equation} where we have introduced the fermionic current $j^\nu(1,2)$, \begin{equation}\label{fercur} j^\nu(1,2) \equiv \frac{{\overline u}(q_1)\, \gamma^\nu \,v(q_2)}{q_{12}^2} \;\;, \quad \quad \;\;\; \quad q_{12} = q_1+q_2 \;\;. \end{equation} The soft quark and antiquark have momenta $q_1^\nu$ and $q_2^\nu$, respectively, and $u(q)$ and $v(q)$ are the customary Dirac spinors. The spin indices ($s_1$ and $s_2$) and the colour indices ($\alpha_1$ and $\alpha_2$) of the quark and antiquark are embodied in the colour+spin space notation of Eq.~(\ref{Jqa}). Considering the projection $(\bra{\alpha_1,\alpha_2} \otimes \bra{s_1,s_2} \,) \,{\bom J}(q_1,q_2) \equiv J^{\alpha_1,\alpha_2}_{s_1,s_2}(q_1,q_2)$ of the current onto its colour and spin indices, we have $(\bra{\alpha_1,\alpha_2} \otimes \bra{s_1,s_2} \,) \,{\bom t}^c \; {\overline u}(q_1)\, \gamma^\nu \,v(q_2) = t^c_{\alpha_1\alpha_2} \, {\overline u}_{(s_1)}(q_1)\, \gamma^\nu \,v_{(s_2)}(q_2)$. \subsubsection{The tree-level current for soft $gq{\bar q}$ emission\label{s:gqqc}} \begin{figure} \begin{center} \fcolorbox{white}{white} \begin{picture}(300,200) (155,-40) \scalebox{.55}{ \SetColor{Black} \Text(194,140)[]{\Large A)} \Text(370,260)[]{\Large $q_1$} \Text(370,222)[]{\Large $q_2$} \Text(370,158)[]{\Large $q_3$} \Text(240,240)[]{\Large $p_i$} \Text(240,200)[]{\Large $p_j$} \SetWidth{1.0} \Gluon(255,256)(351,256){4.5}{8} \Gluon(255,184)(303,188){4.5}{4} \SetWidth{1.9} \Line(207,244)(303,268) \Line(207,196)(303,172) \SetWidth{1.0} \GOval(183,220)(36,36)(0){0.882} \Gluon(255,256)(351,256){4.5}{8} \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2](303,189)(352,220) \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2,flip](303,188)(351,160) % \Text(488,140)[]{\Large B)} \Text(604,300)[]{\Large $q_1$} \Text(690,300)[]{\Large $q_2$} \Text(690,266)[]{\Large $q_3$} \Text(580,190)[]{\Large $p_i$} \SetWidth{1.0} \GOval(481,222)(36,36)(0){0.882} \SetWidth{1.9} \Line(517,222)(649,222) \SetWidth{1.0} \Gluon(541,222)(601,282){4.5}{7} \Gluon(601,222)(637,257){4.5}{4} \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2](637,258)(673,282) \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2,flip](637,258)(673,258) % \Text(799,140)[]{\Large C)} \Text(977,300)[]{\Large $q_1$} \Text(935,300)[]{\Large $q_2$} \Text(935,266)[]{\Large $q_3$} \Text(910,190)[]{\Large $p_i$} \SetWidth{1.0} \Gluon(850,223)(886,258){4.5}{4} \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2](886,259)(922,283) \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2,flip](886,259)(922,259) \SetWidth{1.0} \GOval(792,222)(36,36)(0){0.882} \SetWidth{1.9} \Line(828,222)(960,222) \SetWidth{1.0} \Gluon(913,223)(973,283){4.5}{7} % \Text(194,-50)[]{\Large D)} \Text(370,120)[]{\Large $q_2$} \Text(370,80)[]{\Large $q_1$} \Text(370,38)[]{\Large $q_3$} \Text(282,-7)[]{\Large $p_i$} \SetWidth{1.0} \GOval(184,31)(36,36)(0){0.882} \SetWidth{1.9} \Line(220,31)(353,31) \SetWidth{1.0} \Gluon(232,31)(280,91){4.5}{6} \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2,flip](280,91)(352,43) \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2](280,91)(352,127) \Gluon(304,103)(352,75){4.5}{4} % \Text(488,-50)[]{\Large E)} \Text(580,-7)[]{\Large $p_i$} \Text(665,95)[]{\Large $q_1$} \Text(665,120)[]{\Large $q_2$} \Text(665,38)[]{\Large $q_3$} \SetWidth{1.0} \GOval(481,30)(36,36)(0){0.882} \SetWidth{1.9} \Line(517,30)(650,30) \SetWidth{1.0} \Gluon(529,30)(577,90){4.5}{6} \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2,flip](577,90)(649,42) \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2](577,90)(649,126) \Gluon(601,75)(649,96){4.5}{4} % \Text(799,-50)[]{\Large F)} \Text(977,50)[]{\Large $q_1$} \Text(977,120)[]{\Large$q_2$} \Text(977,80)[]{\Large $q_3$} \Text(910,-7)[]{\Large $p_i$} \SetWidth{1.0} \GOval(793,31)(36,36)(0){0.882} \SetWidth{1.9} \Line(829,31)(961,31) \SetWidth{1.0} \Gluon(853,31)(877,67){4.5}{3} \Gluon(877,67)(913,103){4.5}{4} \Gluon(877,67)(961,67){4.5}{7} \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2](913,103)(961,126) \Line[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=2,arrowinset=0.2,flip](913,103)(961,79) % } \end{picture} } \caption{ \label{fig:tree} {\em Feynman diagrams that contribute to the current ${\bom J}(q_1,q_2,q_3)$ for soft $gq{\bar q}$ emission. The external-leg hard partons with momenta $p_i$ and $p_j$ are coupled to gluons by using the eikonal approximation. The scattering amplitude $\M(\{p_i\})$ is denoted by the grey circle. }} \end{center} \end{figure} The tree-level current for soft $gq{\bar q}$ emission is denoted by ${\bom J}(q_1,q_2,q_3)$. The soft gluon has momentum $q_1$, while $q_2$ and $q_3$ are the momenta of the soft quark and antiquark, respectively. We compute ${\bom J}(q_1,q_2,q_3)$ by using the method of Ref.~\cite{Catani:1999ss}, namely, we consider eikonal emission of the three soft partons from the external hard partons of the generic scattering amplitude $\M$. The relevant Feynman diagrams are shown in Fig.~\ref{fig:tree}. The external-leg hard partons with momenta $p_i$ and $p_j$ are coupled to gluons by using the eikonal approximation for both vertices and propagators. The remaining contributions to the Feynman diagrams in Fig.~\ref{fig:tree} are treated without any approximations for vertices and propagators. We note that the propagators of the off shell (internal-line) gluons are gauge dependent. We have computed the current by using both axial and covariant gauges for the polarization tensor of the internal-line gluons, and we have checked that the final result for ${\bom J}(q_1,q_2,q_3)$ is explicitly gauge independent. More precisely, the total contribution of the gauge dependent terms vanishes by using the colour conservation relation in Eq.~(\ref{colcons}). We present our result for ${\bom J}(q_1,q_2,q_3)$ in the following form: \begin{equation}\label{J123} {\bom J}(q_1,q_2,q_3) = \sy{{\bom J}(q_1)}{{\bom J}(q_2,q_3)} + \boldsymbol\Gamma(q_1,q_2,q_3) \;\;, \end{equation} where ${\bom J}(q_1)$ and ${\bom J}(q_2,q_3)$ are the currents in Eqs.~(\ref{J1}) and (\ref{Jqa}), and we have introduced the symbol $(\dots)_{sym}$ to denote symmetrized products. The symmetrized product of two colour space operators $A$ and $B$ is defined as \begin{equation}\label{symprod} \sy{A}{B} \equiv \frac12(AB+BA) \;. \end{equation} The right-hand side of Eq.~(\ref{J123}) has the structure of an expansion in irreducible correlations, which is analogous to the structure of the two-gluon and three-gluon soft currents in Refs.~\cite{Catani:1999ss} and \cite{Catani:2019nqv}, respectively. The first term in the right-hand side of Eq.~(\ref{J123}) represents the `independent' (though colour-correlated) emission of the soft gluon and the soft $q{\bar q}$ pair from the hard partons. The term $\boldsymbol\Gamma(q_1,q_2,q_3)$ is definitely an irreducible correlation contribution to soft $gq{\bar q}$ emission. To present our result for the irreducible contribution we consider the projection ${\bom J}^{a_1} \equiv \bra{a_1} {\bom J}$ of the current onto the colour index $a_1$ of the soft gluon. The corresponding projection $\boldsymbol\Gamma^{a_1} \equiv \bra{a_1} \boldsymbol\Gamma$ of the irreducible correlation has the following explicit expression: \begin{equation} \boldsymbol\Gamma^{a_1}(q_1,q_2,q_3) = \left( \g\,\mu^\epsilon\right)^3 \, \sum_i T_i^c \;\boldsymbol\gamma_i^{a_1 c}(q_1,q_2,q_3) \;\;, \label{bfgamma} \end{equation} where \begin{align} \boldsymbol\gamma_i^{a_1 c}(q_1,q_2,q_3) & \equiv \frac12 \{{\bom t}^{a_1},{\bom t}^c\} \gab_i(q_1,q_2,q_3) +\frac12 [ {\bom t}^{a_1},{\bom t}^c ] \gna_i(q_1,q_2,q_3) \;\;, \label{gamma} \\ \gab_i(q_1,q_2,q_3) &= \frac{\varepsilon_\mu(q_1)}{q_{123}^2\;p_i\cdot q_{123}} \ub(q_2)\left( \sla{p}_i\frac{\sla{q}_{13}}{q_{13}^2}\gamma^\mu -\gamma^\mu\frac{\sla{q}_{12}}{q_{12}^2}\sla{p}_i\right)v(q_3) \;\;, \label{psi}\\ \gna_i(q_1,q_2,q_3) &= \frac{\varepsilon_\mu(q_1)}{\,p_i\cdot q_{123}} \; \ub(q_2)\left\{ \frac{p_i^\mu}{q_{23}^2}\left(\frac1{p_i\cdot q_1} -\frac1{p_i\cdot q_{23}}\right)\sla{p}_i\right. \label{chi} \\ &\left. \!\!\! \!\!\!\!\!\! +\frac1{q_{123}^2}\left[\frac1{q_{23}^2} \left(2p_i\cdot(q_{23}-q_1)\gamma^\mu-4q_{23}^\mu\sla{p}_i +4p_i^\mu\sla{q}_1\right) -\gamma^\mu\frac{\sla{q}_{12}}{q_{12}^2}\sla{p}_i -\sla{p}_i\frac{\sla{q}_{13}}{q_{13}^2}\gamma^\mu \right] \right\}v(q_3) \;, \nonumber \end{align} and we have defined $q_{ij}=q_i + q_j$ and $q_{123}=q_1 + q_2 + q_3$. The expression of $\boldsymbol\gamma_i^{a_1 c}$ in Eq.~(\ref{gamma}) involves products (an anticommutator and a commutator) of two matrices ${\bom t}^b$ in the fundamental representation. Considering the projection $J^{a_1}_{\alpha_2 \alpha_3} \equiv \bra{a_1, \alpha_2,\alpha_3} \,{\bom J}$ of the current onto the colour indices $\alpha_2$ and $\alpha_3$ of the soft quark and antiquark, the action of the matrices ${\bom t}^b$ in Eq.~(\ref{gamma}) is $\bra{\alpha_2,\alpha_3} \, {\bom t}^{a} {\bom t}^c = (t^a t^c)_{\alpha_2 \alpha_3}$. The `independent' emission contribution $\sy{{\bom J}(q_1)}{{\bom J}(q_2,q_3)}$ in Eq.~(\ref{J123}) embodies products of the type $T_i^a T_k^b$ of colour charges of two hard partons. In contrast, the irreducible component $\boldsymbol\Gamma$ in Eq.~(\ref{bfgamma}) has a linear dependence on the colour charges $T_i^c$ of the hard partons. This feature of $\boldsymbol\Gamma$ is analogous to the linear dependence on $T_i^c$ of the irreducible component of the currents for double \cite{Catani:1999ss} and triple \cite{Catani:2019nqv} soft-gluon emission. We note that the irreducible component $\boldsymbol\Gamma$ in Eqs.~(\ref{bfgamma}) and (\ref{gamma}) embodies a contribution of abelian type, which is proportional to the kinematical function $\gab_i$, in addition to a purely non-abelian contribution, which is proportional to $\gna_i$. In contrast, in the case of double and triple soft-gluon emission \cite{Catani:1999ss, Catani:2019nqv} the irreducible correlations are maximally non-abelian. The presence of an abelian-type contribution in Eqs.~(\ref{bfgamma}) and (\ref{gamma}) implies corresponding irreducible correlations in the current for soft photon-lepton-antilepton emission in QED (see Sect.~\ref{s:qed}). Writing ${\bom J}(q_1,q_2,q_3) = \varepsilon^\nu(q_1) \,{\bom J}_\nu(q_1,q_2,q_3)$ we make explicit the dependence of the current on the Lorentz index $\nu$ of the soft gluon. It is straightforward to check that the result in Eqs.~(\ref{J123}) and (\ref{bfgamma})--(\ref{chi}) fulfils current conservation, namely, \begin{equation}\label{currCons} q_1^\nu \,{\bom J}_\nu(q_1,q_2,q_3) \eqcs 0 \;. \end{equation} The result of ${\bom J}(q_1,q_2,q_3)$ in Eqs.~(\ref{J123})--(\ref{chi}) fully agrees with the corresponding result of the soft $gq{\bar q}$ current that was first computed in Ref.~\cite{DelDuca:2022noh}. Our results and those in Ref.~\cite{DelDuca:2022noh} formally differ only in the presentation of the expression of $\boldsymbol\gamma_i^{a_1 c}$ in Eq.~(\ref{gamma}). We use the two independent colour structures $\{{\bom t}^{a_1},{\bom t}^c\}$ and $[{\bom t}^{a_1},{\bom t}^c]$, while Ref.~\cite{DelDuca:2022noh} uses the three colour structures ${\bom t}^{a_1} {\bom t}^c$, ${\bom t}^{c} {\bom t}^{a_1}$ and $[{\bom t}^{a_1},{\bom t}^c]= i f^{a_1 c b} {\bom t}^b$, which are linearly dependent. \section{Tree-level squared currents\label{s:tlsc}} Using the colour+spin space notation of Sect.~\ref{s:sfsa}, the squared amplitude $|\M|^2$ (summed over the colours and spins of its external legs) is written as follows \begin{equation} \label{squared} |\M|^2 = \langle{\M} \ket{\,\M} \;\;. \end{equation} Accordingly, the square of the soft factorization formula (\ref{1gfact}) gives \begin{equation} \label{softsquared} | \M(\{q_\ell\}, \{p_i\}) |^2 \simeq \bra{\M (\{p_i\})} \;| \J(q_1,\cdots,q_N) |^2 \;\ket{\M (\{p_i\})} \;, \end{equation} where \begin{equation} \label{spin} | \J(q_1,\cdots,q_N) |^2 = \sum_{\{c_i\}, \{s_i\}} \left[ J^{c_1 \dots c_N}_{s_1 \dots s_N}(q_1,\cdots,q_N) \right]^\dagger J^{c_1 \dots c_N}_{s_1 \dots s_N}(q_1,\cdots,q_N) \;\;, \end{equation} and, analogously to Eq.~(\ref{1gfact}), the symbol $\simeq$ means that we neglect contributions that are subdominant in the soft limit. The squared current $| \J(q_1,\cdots,q_N) |^2 $, which is summed over the colours $c_1 \dots c_N$ and spins $s_1 \dots s_N$ of the soft partons, is still a colour operator that depends on the colour charges of the hard partons in $\M (\{p_i\})$. These colour charges produce colour correlations and, therefore, the right-hand side of Eq.~(\ref{softsquared}) is not proportional to $|\M(\{p_i\})|^2$ in the case of a generic scattering amplitude\footnote{Colour correlations can be simplified in the case of scattering amplitudes with two and three hard partons (see Sect.~\ref{s:e2h} and \ref{s:e3h}).}. The computation of the squared current in Eq.~(\ref{spin}) involves the sum over the physical spin polarization vectors of the soft gluons. These polarization vectors are gauge dependent, but the action of $| \J |^2 $ onto colour singlet states is fully gauge invariant, since the gauge dependent contributions cancel as a consequence of the conservation of the soft current (see, e.g., Ref.~\cite{Catani:2019nqv}). We recall the known results of the squared currents for emission of one soft gluon and of a soft $q{\bar q}$ pair. The square of the soft current ${\bom J}(q_1)$ in Eq.~(\ref{J1}) for single gluon emission is \begin{equation}\label{J1sq} |{\bom J}(q_1)|^2 \equiv {\bom J}(q_1)^\dagger {\bom J}(q_1) \eqcs -\left( \g\,\mu^\epsilon\right)^2 \sum_{i,k} \T_i\cdot \T_k \; \cS_{ik}(q_1) \;\;, \end{equation} where \begin{equation} \label{S1} \cS_{ik}(q) = \frac{p_i\cdot p_k}{p_i\cdot q\; p_k\cdot q} \;\;. \end{equation} The square of the soft current ${\bom J}(q_2,q_3)$ in Eq.~(\ref{Jqa}) for $q{\bar q}$ emission is \cite{Catani:1999ss} \begin{equation}\label{Jqasq} |{\bom J}(q_2,q_3)|^2 \equiv {\bom J}(q_2,q_3)^\dagger\; {\bom J}(q_2,q_3) \eqcs \left( \g\,\mu^\epsilon\right)^4 T_R \sum_{i,k} \T_i\cdot \T_k \; {\cal I}_{ik}(q_2,q_3) \;\;, \end{equation} where \begin{equation} {\cal I}_{ik}(q_2,q_3) = \frac{p_i\cdot q_2\; p_k\cdot q_3+p_i\cdot q_3\; p_k\cdot q_2-p_i\cdot p_k \;q_2\cdot q_3}% {(q_2\cdot q_3)^2\;p_i\cdot q_{23}\; p_k\cdot q_{23}} \;. \label{I23} \end{equation} The colour charge dependence of both squared currents in Eqs.~(\ref{J1sq}) and (\ref{Jqasq}) is given in terms of dipole operators $\T_i\cdot \T_k = \sum_a T_i^a T_k^a$. The insertion of the dipole operators in the factorization formula (\ref{softsquared}) produces colour correlations between two hard partons ($i$ and $k$) in $\M(\{p_i\})$. Using colour charge conservation (see Eq.~\eqref{colcons}), the single-gluon and $q\bar q$ squared currents in Eqs.~(\ref{J1sq}) and (\ref{Jqasq}) can be rewritten as follows \begin{align} |{\bom J}(q_1)|^2 &\eqcs - \left( \g\,\mu^\epsilon\right)^2 \, \frac{1}{2} \sum_{i \neq k} \T_i\cdot \T_k \, w_{ik}(q_1) \;\;,\label{j1q} \\ w_{ik}(q_1) &= \cS_{ik}(q_1)+\cS_{ki}(q_1)-\cS_{ii}(q_1)-\cS_{kk}(q_1) \;\;, \label{w1} \\ |{\bom J}(q_2,q_3)|^2 &\eqcs - \left( \g\,\mu^\epsilon\right)^4 T_R \, \frac{1}{2} \sum_{i \neq k} \T_i\cdot \T_k \, w_{ik}(q_2,q_3) \;\;,\label{j23q} \\ w_{ik}(q_2,q_3) &= {\cal I}_{ii}(q_2,q_3)+{\cal I}_{kk}(q_2,q_3) -{\cal I}_{ik}(q_2,q_3)-{\cal I}_{ki}(q_2,q_3) \;\;. \label{w23} \end{align} As noticed in Refs.~\cite{Catani:2019nqv, Catani:2021kcy}, the expressions in Eqs.~(\ref{j1q}) and (\ref{j23q}) have a more straightforward physical interpretation, since the kinematical functions $w_{ik}(q_1)$ in Eq.~(\ref{w1}) and $w_{ik}(q_2,q_3)$ in Eq.~(\ref{w23}) are directly related to the intensity of soft radiation from two distinct hard partons, $i$ and $k$, in a colour singlet configuration (see Sect.~\ref{s:e2h}). \subsection{The squared current for soft $gq{\bar q}$ radiation}\label{s:gqqsc} The soft-$q\bar q$ kinematical functions ${\cal I}_{ik}(q_2,q_3)$ and $w_{ik}(q_2,q_3)$ in Eqs.~(\ref{I23}) and (\ref{w23}) are {\it symmetric} with respect to the exchange $q_2 \leftrightarrow q_3$ of the quark and antiquark momenta. The evaluation of the one-loop QCD corrections to the soft-$q\bar q$ squared current \cite{Catani:2021kcy} produces also kinematical correlations with an {\it antisymmetric} dependence with respect to $q_2 \leftrightarrow q_3$. As discussed in Ref.~\cite{Catani:2021kcy}, such antisymmetric dependence is related to charge asymmetry effects that are distinct features of the radiation of quarks and antiquarks. An antisymmetric dependence with respect to $q_2 \leftrightarrow q_3$ and ensuing charge asymmetry effects occur also in the tree-level squared current $|{\bom J}(q_1,q_2,q_3)|^2$ for soft $gq\bar q$ radiation, as we discuss in the following. Similar to Ref.~\cite{Catani:2021kcy}, the charge asymmetry effects for soft $gq\bar q$ radiation involve colour correlations that depend on the fully-symmetric colour tensor\footnote{If $N_c=2$ the tensor $d^{abc}$ vanishes and there are no charge asymmetry effects.} $d^{abc}$, \begin{equation} \label{dtens} d^{abc} = \frac{1}{T_R} \,{\rm Tr}\left( \left\{t^a, t^b \right\} t^c \right) \;\;, \end{equation} which is odd under charge conjugation. We compute the square of the tree-level $gq\bar{q}$ current by using the structure in Eq.~\eqref{J123} and we obtain the following three contributions: {\it (I)} the square of the `uncorrelated' current $\sy{{\bom J}(q_1)}{{\bom J}(q_2,q_3)}$, {\it (II)} the product of the `uncorrelated' current and the irreducible current $\boldsymbol\Gamma(q_1,q_2,q_3)$, {\it (III)} the square of the irreducible current. The square of the `uncorrelated' current $\sy{{\bom J}(q_1)}{{\bom J}(q_2,q_3)}$ can be written in terms of the symmetric product of the squared currents $|{\bom J}(q_1)|^2$ and $|{\bom J}(q_2,q_3)|^2$, and an additional irreducible term $W^{(I)}$, which involves colour dipole correlations. We obtain \begin{align}\label{Jsq1} &|\sy{{\bom J}(q_1)}{{\bom J}(q_2,q_3)}|^2 \eqcs \sy{|{\bom J}(q_1)|^2}{|{\bom J}(q_2,q_3)|^2} + W^{(I)}(q_1,q_2,q_3) \;\;, \\ &W^{(I)}(q_1,q_2,q_3) = -\left( \g\,\mu^\epsilon\right)^6 T_R C_A \sum_{i,k} \T_i\cdot \T_k \; \cS^{(I)}_{ik}(q_1,q_2,q_3) \;\;, \label{WI} \end{align} where the momentum function $\cS^{(I)}_{ik}(q_1,q_2,q_3)$ is\footnote{In Eq.~(\ref{SI}) and in the following equations the scalar products $v \cdot u$ between to generic momenta $u^\mu$ and $v^\mu$ are simply denoted by $vu$.} \begin{align} &\cS^{(I)}_{ik}(q_1,q_2,q_3) = \frac{\pq{i}{2}}{4 \, (\qq{2}{3})^2 \, \pq{i}{1} \, \pq{i}{23}} \, \left[\frac{\pp{i}{k}}{\pq{k}{1}} \, \left(\frac{3 \, \pq{k}{3}}{\pq{k}{23}}-\frac{2 \, \pq{i}{3}}{\pq{i}{23}}\right)-\frac{2 \, m_i^2 \, \pq{k}{3}}{\pq{i}{1} \, \pq{k}{23}}\right] \nonumber \\ &+\frac{\pp{i}{k}}{8 \, \qq{2}{3} \, \pq{i}{1} \, \pq{i}{23}} \, \left[\frac{-3 \, \pp{i}{k}}{\pq{k}{1} \, \pq{k}{23}}+2 \, m_i^2 \, \left(\frac{1}{\pq{i}{1} \, \pq{k}{23}}+\frac{1}{\pq{k}{1} \, \pq{i}{23}}\right)\right] +(2\leftrightarrow 3) \;\;. \label{SI} \end{align} The product of the `uncorrelated' current and the irreducible current leads to colour correlations that involve dipoles and tripoles. We have \begin{align} &\sum\sy{{\bom J}(q_1)}{{\bom J}(q_2,q_3)}^\dagger\,\boldsymbol\Gamma(q_1,q_2,q_3) + \text{h.c.} = W^{(II)}(q_1,q_2,q_3) \nonumber \\ &\eqcs -\left( \g\,\mu^\epsilon\right)^6 \, T_R \left[ C_A \sum_{i,k}\T_i\cdot \T_k \; \cS^{(II)}_{ik}(q_1,q_2,q_3) + \sum_{i,k,m} \tri{ikm} \;\cS_{ikm}(q_1,q_2,q_3)\right] \;, \label{W2} \end{align} where `h.c' denotes the hermitian-conjugate contribution, and we have defined the (hermitian) $d$-type colour tripole $\tri{ikm}$ as \begin{equation}\label{tripoli} \tri{ikm} \equiv \sum_{a,b,c} d^{abc} T_i^a T_k^b T_m^c \;. \end{equation} The complete symmetry of $d^{abc}$ causes $\tri{ikm}$ to be completely symmetric in the permutations of its indices $i, k, m$. The kinematical coefficient $\cS^{(II)}_{ik}$ of the dipole contribution to Eq.~(\ref{W2}) reads \begin{align} &\cS^{(II)}_{ik}(q_1,q_2,q_3) = \frac{1}{2\qq{2}{3} \, \pq{i}{123}} \, \left\{\frac{2}{q_{123}^2 \, \qq{2}{3}} \, \left(\frac{\pq{k}{2} \, (2 \, m_i^2 \, \qq{1}{3}-\pq{i}{123} \, \pq{i}{3})}{\pq{i}{1} \, \pq{k}{23}} \right.\right. \nonumber\\ & \qquad\qquad \left. -\frac{\pq{i}{2} \, (2 \, \pp{i}{k} \, \qq{1}{3}-(\pq{i}{1}-\pq{i}{2}+3 \, \pq{i}{3}) \, \pq{k}{3})}{\pq{k}{1} \, \pq{i}{23}}\right)\nonumber\\ &+\frac{1}{q_{123}^2 \, \qq{1}{2}} \, \left(\frac{2 \, \pq{i}{12} \, (\pp{i}{k} \, \qq{2}{3}-\pq{i}{2} \, \pq{k}{3}-\pq{i}{3} \, \pq{k}{2})+m_i^2 \, (\pq{k}{2} \, \qq{1}{3}-\pq{k}{1} \, \qq{2}{3})}{\pq{i}{1} \, \pq{k}{23}}\right.\nonumber\\ &\left.-\frac{m_i^2 \, ((\pq{k}{1}+2 \, \pq{k}{2}) \, \qq{2}{3}+\pq{k}{2} \, \qq{1}{3})-2 \, (\pq{i}{2} \, \pq{k}{1}+(\pq{i}{1}+2 \, \pq{i}{2}) \, \pq{k}{2}) \, \pq{i}{3}}{\pq{k}{1} \, \pq{i}{23}}\right)\nonumber\\ &+\frac{1}{q_{123}^2} \, \left(\frac{\pp{i}{k} \, \pq{i}{123}+m_i^2 \, (\pq{k}{2}-2 \, \pq{k}{1})}{\pq{i}{1} \, \pq{k}{23}}-\frac{3 \, m_i^2 \, \pq{k}{2}-\pp{i}{k} \, \pq{i}{1}}{\pq{k}{1} \, \pq{i}{23}}\right)\nonumber\\ &+\frac{1}{\qq{2}{3}} \, \left(\pq{i}{2} \, \left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, \left(\frac{m_i^2 \, \pq{k}{3}}{\pq{i}{1} \, \pq{k}{23}}-\frac{\pp{i}{k} \, \pq{i}{3}}{\pq{k}{1} \, \pq{i}{23}}\right)\right)\nonumber\\ &\left.+\frac{1}{2} \, m_i^2 \, \pp{i}{k} \, \left(\frac{1}{\pq{i}{23}}-\frac{1}{\pq{i}{1}}\right) \, \left(\frac{1}{\pq{i}{1} \, \pq{k}{23}}-\frac{1}{\pq{k}{1} \, \pq{i}{23}}\right)\right\} + (2\leftrightarrow 3) \;\;. \label{SII} \end{align} The kinematical coefficient $\cS_{ikm}$ of the tripole contribution to Eq.~(\ref{W2}) is \begin{align} &\cS_{ikm}(q_1,q_2,q_3) = \frac{1}{\pq{i}{1} \, \pq{k}{23} \, \pq{m}{123} \, q_{123}^2} \, \left\{\frac{1}{\qq{1}{2} \, \qq{2}{3}} \, \Big[\pq{i}{1} \, (\pq{k}{2} \, \pq{m}{3}+\pq{k}{3} \, \pq{m}{2}) \right. \nonumber\\ & +\pq{i}{2} \, \big(\pq{k}{1} \, \pq{m}{3}+\pq{k}{3} \, \pq{m}{1} +2 \, (\pq{k}{2} \, \pq{m}{3}+\pq{k}{3} \, \pq{m}{2}) \big) \nonumber\\ &+\pq{i}{3} \,(-\pq{k}{1} \, \pq{m}{2}+\pq{k}{2} \, \pq{m}{1})+\qq{1}{3} \, (\pp{i}{k} \, \pq{m}{2}-\pp{i}{m} \, \pq{k}{2}-\pp{k}{m} \, \pq{i}{2})\Big] \nonumber\\ &+\frac{1}{\qq{1}{2}} \, \big(\pp{i}{m} \, \pq{k}{1}-\pp{i}{k} \, \pq{m}{1}-\pp{k}{m} \, (\pq{i}{1}+2 \, \pq{i}{2})\big) \nonumber\\ & \left.+\frac{1}{\qq{2}{3}} \, (\pp{k}{m} \, \pq{i}{3}-\pp{i}{k} \, \pq{m}{3}-\pp{i}{m} \, \pq{k}{3})\right\} - (2\leftrightarrow 3)\;. \label{Sikm} \end{align} Note that, while all dipole kinematical coefficients are symmetric in the exchange $q_2 \leftrightarrow q_3$ of the quark and antiquark momenta (see also $\cS_{ik}^{(III,\ab)}$ and $\cS_{ik}^{(III,\na)}$ in Eqs.~(\ref{sIIIab}) and (\ref{sIIIna})), the coefficient $\cS_{ikm}(q_1,q_2,q_3)$ of the tripole is antisymmetric in such exchange. The square of the irreducible current gives \begin{align} &\boldsymbol\Gamma^\dagger(q_1,q_2,q_3) \, \boldsymbol\Gamma(q_1,q_2,q_3) = W^{(III)}(q_1,q_2,q_3) \nonumber \\ &\eqcs -\left( \g\,\mu^\epsilon\right)^6\, T_R \sum_{i,k} \left\{ \left(C_F-\frac{C_A}{4}\right) \;\cS^{(III,\ab)}_{ik}(q_1,q_2,q_3) + \frac{C_A}{4}\;\cS^{(III,\na)}_{ik}(q_1,q_2,q_3) \right\} \T_i\cdot \T_k \;. \label{Gammasq} \end{align} The kinematical function $\cS^{(III,\ab)}_{ik}$ stems from the abelian-type term proportional to $\gab$ in Eqs.~(\ref{bfgamma}) and (\ref{gamma}), and we obtain \begin{align} &\cS^{(III,\ab)}_{ik}(q_1,q_2,q_3) =\Bigg[ \frac{1}{(q_{123}^2)^2 \, \pq{i}{123} \, \pq{k}{123}} \, \sg \frac{\qq{2}{3}}{\qq{1}{2} \, \qq{1}{3}} \, \pst 2 \, \pp{i}{k} \, \qq{2}{3}+(d-4) \, \pq{i}{1} \, \pq{k}{1}\nonumber \\ &-2 \, \pq{i}{1} \, \pq{k}{23}-4 \, \pq{i}{2} \, \pq{k}{3} \pdt +\frac{1}{\qq{1}{2}} \, \pst \pp{i}{k} \, ((d-2) \, \qq{1}{3}+4 \, \qq{2}{3})+2 \, (4-d) \, \pq{i}{1} \, \pq{k}{2}\nonumber \\ &+2 \, (2-d) \, \pq{i}{1} \, \pq{k}{3}+4 \, \pq{i}{2} \, p_k(q_{2}-q_{3})\pdt \, + (d-4) \, \pp{i}{k}\dg +(2\leftrightarrow 3) \Bigg]+ (i\leftrightarrow k) \;. \label{sIIIab} \end{align} The kinematical function $\cS^{(III,\na)}_{ik}$ comes from the non-abelian term proportional to $\gna$ in Eqs.~(\ref{bfgamma}) and (\ref{gamma}), and we have \begin{align} &\cS^{(III,\na)}_{ik}(q_1,q_2,q_3) =\Bigg[ \frac{1}{\pq{i}{123} \, \pq{k}{123}} \, \sg \frac{1}{(q_{123}^2)^2} \, \sq\frac{8 \, \pp{i}{k} \, \qq{1}{2} \, \qq{1}{3}}{(\qq{2}{3})^2}\nonumber \\ &+\frac{2 \, ((d-6) \, p_i(q_{1}-q_{3})-(2+d) \, \pq{i}{2}) \, \pq{k}{2} \, \qq{1}{3}}{\qq{1}{2} \, \qq{2}{3}}+\frac{\qq{2}{3}}{\qq{1}{2} \, \qq{1}{3}} \, (-2 \, \pp{i}{k} \, \qq{2}{3}\nonumber\\ &+\pq{i}{1} \, ((4-d) \, \pq{k}{1}+2 \, \pq{k}{2})+2 \, \pq{i}{2} \, (\pq{k}{1}+2 \, \pq{k}{3}))+\frac{1}{\qq{2}{3}} \, (8 \, \pp{i}{k} \, \qq{1}{3}+\pq{i}{1} \, ((2-d) \, \pq{k}{1}\nonumber\\ &+2 \, (d-6) \, \pq{k}{2})+8 \, \pq{i}{2} \, p_k(q_{3}-q_{2}))+\frac{1}{\qq{1}{2}} \, (\pp{i}{k} \, ((d-2) \, \qq{1}{3}-4 \, \qq{2}{3})+\pq{i}{1} \, (2 \, (6-d) \, \pq{k}{1}\nonumber\\ &+2 \, (d-4) \, \pq{k}{2}+(10-d) \, \pq{k}{3})+2 \, \pq{i}{2} \, ((d+2) \, \pq{k}{1}-2 \, \pq{k}{2}+6 \, \pq{k}{3})\nonumber\\ &+\pq{i}{3} \, ((d-2) \, \pq{k}{1}+8 \, \pq{k}{2}))\nonumber\\ &+(4-d) \, \pp{i}{k} \dq +\frac{1}{q_{123}^2} \, \sq\frac{1}{(\qq{2}{3})^2} \, \left(\left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, 2 \, \pq{i}{2} \, (2 \, \pp{i}{k} \, \qq{1}{3}-2 \, \pq{i}{2} \, \pq{k}{3}-\pq{i}{3} \, \pq{k}{1})\right)\nonumber\\ &+\frac{1}{\qq{1}{2} \, \qq{2}{3}} \, \left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, ((2 \, \pp{i}{k} \, \pq{i}{2}-m_i^2 \, \pq{k}{2}) \, \qq{1}{3}-2 \, \pq{i}{2} \, (\pq{i}{12} \, \pq{k}{3}+\pq{i}{3} \, \pq{k}{12}))\nonumber\\ &+\frac{1}{\qq{2}{3}} \, \left(\left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, (m_i^2 \, p_k(q_{1}-q_{2})+2 \, \pp{i}{k} \, (2 \, \pq{i}{2}-\pq{i}{1}))\right)\nonumber\\ &+\frac{1}{\qq{1}{2}} \, \left(\left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, (2 \, \pp{i}{k} \, \pq{i}{2}+m_i^2 \, \pq{k}{1})-4 \, \pp{i}{k}\right)\dq\nonumber\\ &+\frac{\pp{i}{k} \, \pq{i}{2} \, \pq{k}{3}}{2 \, (\qq{2}{3})^2} \, \left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, \left(\frac{1}{\pq{k}{1}}-\frac{1}{\pq{k}{23}}\right)\nonumber\\ &+\frac{-(\pp{i}{k})^2}{4 \, \qq{2}{3}} \, \left(\frac{1}{\pq{i}{1}}-\frac{1}{\pq{i}{23}}\right) \, \left(\frac{1}{\pq{k}{1}}-\frac{1}{\pq{k}{23}}\right)\dg +(2\leftrightarrow 3)\Bigg] + (i\leftrightarrow k)\;. \label{sIIIna} \end{align} In summary, the squared current for soft $gq{\bar q}$ emission is obtained by summing the contributions of Eqs.~(\ref{Jsq1}), (\ref{WI}), (\ref{W2}) and (\ref{Gammasq}), and we find \begin{align} \label{J123sq} |{\bom J}(q_1,q_2,q_3)|^2 &= \sy{|{\bom J}(q_1)|^2}{|{\bom J}(q_2,q_3)|^2} + W(q_1,q_2,q_3) \;, \end{align} where \begin{align} W(q_1,q_2,q_3) &= W^{(I)}(q_1,q_2,q_3) +W^{(II)}(q_1,q_2,q_3) +W^{(III)}(q_1,q_2,q_3) \nonumber \\ &\eqcs-\left( \g\,\mu^\epsilon\right)^6 T_R \;\Bigl\{ \;\sum_{i,k} \T_i\cdot \T_k \left[ C_A\,\cS^{(A)}_{ik}(q_1,q_2,q_3) + C_F\,\cS^{(F)}_{ik}(q_1,q_2,q_3) \right] \Bigr. \nonumber \\ & \Bigl. + \sum_{i,k,m} \tri{ikm} \;\cS_{ikm}(q_1,q_2,q_3) \;\Bigr\} \;\;. \label{Wqcd} \end{align} The kinematical function $\cS_{ikm}$ is given in Eq.~(\ref{Sikm}), and the kinematical functions $\cS^{(A)}_{ik}$ and $\cS^{(F)}_{ik}$ of the dipole contributions to Eq.~(\ref{Wqcd}) are \begin{align} \cS^{(A)}_{ik}(q_1,q_2,q_3) &= \cS^{(I)}_{ik}(q_1,q_2,q_3)+\cS^{(II)}_{ik}(q_1,q_2,q_3) \nonumber \\ &+ \frac{1}{4} \left( \cS^{(III,\na)}_{ik}(q_1,q_2,q_3)-\cS^{(III,\ab)}_{ik}(q_1,q_2,q_3) \right) \;\;, \label{SA}\\ \cS^{(F)}_{ik}(q_1,q_2,q_3) &=\cS^{(III,\ab)}_{ik}(q_1,q_2,q_3) \;\;. \label{SF} \end{align} where $\cS^{(I)}_{ik}, \cS^{(II)}_{ik}, \cS^{(III,\ab)}$ and $\cS^{(III,\na)}_{ik}$ are given in Eqs.~(\ref{SI}), (\ref{SII}), (\ref{sIIIab}) and (\ref{sIIIna}). The kinematical functions $\cS^{(A)}_{ik}, \cS^{(F)}_{ik}$ and $\cS_{ikm}$ in Eq.~(\ref{Wqcd}) depend on the hard-parton momenta, and we recall that our results are valid for both massless and massive hard partons (see, e.g., the dependence on $p_i^2=m_i^2$ in Eqs.~(\ref{SI}), (\ref{SII}) and (\ref{sIIIna})). We note that the dipole kinematical functions $\cS^{(A)}_{ik}$ and $\cS^{(F)}_{ik}$ in Eqs.~(\ref{SA}) and (\ref{SF}) (i.e., $\cS^{(III,\ab)}$ and $\cS^{(III,\na)}_{ik}$ in Eqs.~(\ref{sIIIab}) and (\ref{sIIIna})) have an explicit dependence on $\epsilon$ through the number $d=4 - 2\epsilon$ of space-time dimensions. Such $\epsilon$ dependence actually derives from the fact that we use CDR with $h_g= d-2=2(1-\epsilon)$ spin polarization states for the on shell soft gluon with momentum $q_1$. Other versions of dimensional regularizations, such as dimensional reduction (DR) \cite{Siegel:1979wq} and the four-dimensional helicity (4DH) scheme \cite{Bern:1991aq}, use $h_g=2$ spin polarization states. The result for $|{\bom J}(q_1,q_2,q_3)|^2$ in the DR and 4DH schemes is obtained by simply setting $\epsilon=0$ (i.e., $d=4$) in our expressions for $\cS^{(A)}_{ik}$ and $\cS^{(F)}_{ik}$. In Eq.~(\ref{J123sq}) the squared current for the radiation of the three soft partons is decomposed in terms of irreducible correlation contributions, similar to the analogous decomposition of the squared currents for double \cite{Catani:1999ss} and triple \cite{Catani:2019nqv} soft-gluon radiation. The first term in the right-hand side of Eq.~(\ref{J123sq}) represents the `independent' (though colour-symmetrized) radiation of a soft gluon and a soft-$q{\bar q}$ pair according to the corresponding squared currents $|{\bom J}(q_1)|^2$ (see Eq.~(\ref{j1q})) and $|{\bom J}(q_2,q_3)|^2$ (see Eq.~(\ref{j23q})). The term $W(q_1,q_2,q_3)$ in Eq.~(\ref{J123sq}) is an irreducible correlation contribution for soft $gq\bar q$ radiation. The colour structure of $W(q_1,q_2,q_3)$ in Eq.~(\ref{Wqcd}) is given in terms of dipole operators $\T_i\cdot \T_k$ and tripole operators $\tri{ikm}$ (here, two or three of the hard-parton indices $i, k$ and $m$ can also be equal). The presence in the tree-level squared current $|{\bom J}(q_1,q_2,q_3)|^2$ of colour correlations due to the tripole operators $\tri{ikm}$ was first observed in Ref.~\cite{DelDuca:2022noh}. We note that the contributions to Eq.~(\ref{Wqcd}) that are proportional to $\cS^{(A)}_{ik}$ have a purely non-abelian origin. The other contributions to $W$ in Eq.~(\ref{Wqcd}), namely, the dipole terms proportional to $\cS^{(F)}_{ik}$ and the tripole terms, have an `abelian character', since corresponding irreducible correlations also occur for soft photon-lepton-antilepton radiation in QED (see Sect.~\ref{s:qed}). The structure of Eqs.~(\ref{J123sq}) and (\ref{Wqcd}) is identical to that obtained in Ref.~\cite{DelDuca:2022noh}. Exploiting colour conservation and the symmetries of $\T_i\cdot \T_k$ and $\tri{ikm}$ with respect to their parton indices, the kinematical functions $\cS^{(A)}_{ik}, \cS^{(F)}_{ik}$ and $\cS_{ikm}$ can be written in different ways, without affecting the value of $W(q_1,q_2,q_3)$ in Eq.~(\ref{Wqcd}) onto colour singlet states (scattering amplitudes). Our explicit expressions of these kinematical functions appear to be more compact than the related expressions presented in Ref.~\cite{DelDuca:2022noh}. Considering the action of $W$ onto colour singlet states, we have carried out numerical comparisons between our result and the result of Ref.~\cite{DelDuca:2022noh}, and we find complete agreement. The irreducible correlation $W(q_1,q_2,q_3)$ in Eq.~(\ref{Wqcd}) can be rewritten in the following equivalent form: \begin{align} W(q_1,q_2,q_3) &\eqcs-\left( \g\,\mu^\epsilon\right)^6 T_R \;\Bigl\{ \;\frac{1}{2} \sum_{i \neq k} \T_i\cdot \T_k \left[ C_A\,w^{(A)}_{ik}(q_1,q_2,q_3) + C_F\,w^{(F)}_{ik}(q_1,q_2,q_3) \right] \Bigr. \nonumber \\ & \Bigl. + \frac{1}{2} \sum_{i \neq k} \tri{iik}\;w^{(tri)}_{ik}(q_1,q_2,q_3) + \sum_{{\rm dist.}\{i,k,m\}} \tri{ikm} \;w^{(tri)}_{ikm}(q_1,q_2,q_3) \;\Bigr\} \;\;, \label{Wred} \end{align} where the subscript `${\rm dist.}\{i,k,m\}$' in $\sum_{{\rm dist.}\{i,k,m\}}$ denotes the sum over distinct hard-parton indices $i,k$ and $m$ (i.e., $i \neq k, k\neq m, m\neq i$). The dipole kinematical functions $w^{(A)}_{ik}$ and $w^{(F)}_{ik}$ are related to the corresponding functions $\cS^{(A)}_{ik}$ and $\cS^{(F)}_{ik}$ in Eq.~(\ref{Wqcd}), and we have \begin{equation}\label{w3ik} w^{(r)}_{ik}(q_1,q_2,q_3) = \cS^{(r)}_{ik}(q_1,q_2,q_3) + \cS^{(r)}_{ki}(q_1,q_2,q_3) - \cS^{(r)}_{ii}(q_1,q_2,q_3) - \cS^{(r)}_{kk}(q_1,q_2,q_3) \,, \; (r=A,F)\,. \end{equation} The kinematical functions $w^{(tri)}_{ik}$ and $w^{(tri)}_{ikm}$ depend on the tripole functions $\cS_{ikm}$ in Eq.~(\ref{Wqcd}), and we obtain \begin{align} w^{(tri)}_{ik}(q_1,q_2,q_3) &= \Bigl[ \cS_{iik}(q_1,q_2,q_3) + \cS_{iki}(q_1,q_2,q_3) + \cS_{kii}(q_1,q_2,q_3) - \cS_{iii}(q_1,q_2,q_3) \Bigr] \nonumber \\ & - \;( i \leftrightarrow k) \;\;, \label{wiik} \end{align} \begin{align} w^{(tri)}_{ikm}(q_1,q_2,q_3) &= \cS_{ikm}(q_1,q_2,q_3) - \frac{1}{2} \Bigl[ \cS_{iik}(q_1,q_2,q_3) + \cS_{iki}(q_1,q_2,q_3) \Bigr. \nonumber \\ & + \Bigl. \cS_{kii}(q_1,q_2,q_3) - \cS_{iii}(q_1,q_2,q_3) \Bigr] \;\;. \label{wikm} \end{align} The equality of Eqs.~(\ref{Wqcd}) and (\ref{Wred}) follows from colour conservation and the symmetries of the operators $\T_i\cdot \T_k$ and $\tri{ikm}$ with respect to their hard-parton indices. The proof of the equivalence between the two expressions in Eqs.~(\ref{Wqcd}) and (\ref{Wred}) is presented at the end of this section. The expression in Eq.~(\ref{Wred}) involves colour correlations between two and three distinct hard partons. The size of the two hard-parton correlations is controlled by the kinematical functions $w^{(A)}_{ik}, w^{(F)}_{ik}$ and $w^{(tri)}_{ik}$, which are are physically related to the interaction of two hard partons, $i$ and $ k$, in a colour singlet configuration (see Sect.~\ref{s:e2h}). The structure of $W(q_1,q_2,q_3)$ in Eq.~(\ref{Wred}) is almost identical to that of the one-loop QCD corrections to the squared current for soft-$q{\bar q}$ radiation (see Eq.~(61) in Ref.~\cite{Catani:2021kcy}). The only difference is due to the presence of three-parton correlations of the type $f^{abc} T^a_i T^b_k T^c_m$ in the case of the one-loop squared current. Such correlations have a one-loop absorptive origin \cite{Catani:2021kcy} and, therefore, they are absent in the tree-level $gq{\bar q}$ correlation term $W(q_1,q_2,q_3)$. In Ref.~\cite{Catani:2021kcy} the tripole operator with two distinct indices is written as $\tri{iik}= {\widetilde{\bf D}}_i \cdot \T_k = \sum_a D^a_i T^a_k$ in terms of the `$d$-conjugated' charge operator $D^a_i = \sum_{b,c} d^{abc} \,T^b_i \,T^c_i$. We note that the kinematical function $w^{(tri)}_{ik}$ in Eq.~(\ref{wiik}) is antisymmetric under the exchange $i \leftrightarrow k$ of the hard-parton indices. Such antisymmetry of $w^{(tri)}_{ik}$ implies that in the sum over $i$ and $k$ of Eq.~(\ref{Wred}) we can replace $\tri{iik}$ by its antisymmetric component, namely, $\tri{iik} \rightarrow (\tri{iik}- \tri{kki})/2$. Some main features of $W$ in Eq.~(\ref{Wred}) (or, equivalently, Eq.~(\ref{Wqcd})) are fully similar to those of the analogous one-loop corrections in Ref.~\cite{Catani:2021kcy}. The dipole kinematical functions $w^{(r)}_{ik}(q_1,q_2,q_3)$ ($r=A,F$) in Eq.~(\ref{Wred}) (and $\cS^{(r)}_{ik}(q_1,q_2,q_3)$ in Eq.~(\ref{Wqcd})) are {\it symmetric} with respect to the exchange $q_2 \leftrightarrow q_3$ of the quark and antiquark momenta. In contrast, the functions $w^{(tri)}_{ik}(q_1,q_2,q_3)$ and $w^{(tri)}_{ikm}(q_1,q_2,q_3)$ in Eq.~(\ref{Wred}) (and $\cS_{ikm}(q_1,q_2,q_3)$ in Eq.~(\ref{Wqcd})) are {\it antisymmetric} with respect to the exchange $q_2 \leftrightarrow q_3$ and, therefore, they produce a quark--antiquark {\it charge asymmetry} in the tree-level squared current $|{\bom J}(q_1,q_2,q_3)|^2$. We note that such charge asymmetry functions contribute to $W(q_1,q_2,q_3)$ with the associated colour factors $\tri{iik}$ and $\tri{ikm}$ that have a linear dependence on the colour tensor $d^{abc}$. Therefore, since $d^{abc}$ is odd under charge conjugation, the charge asymmetry features of $|{\bom J}(q_1,q_2,q_3)|^2$ are consistent with the charge conjugation invariance of the QCD interactions (see also related comments in Sect.~\ref{s:e2h} and \ref{s:e3h} and in Ref.~\cite{Catani:2021kcy}). In particular, we note \cite{Catani:2021kcy} that the charge asymmetry contributions of $|{\bom J}(q_1,q_2,q_3)|^2$ vanish if the squared current acts on a pure multigluon scattering amplitude $\M(\{p_i\})$, namely, if $\M(\{p_i\})$ has only gluon external lines (with no additional external $q{\bar q}$ pairs or colourless particles). We also note \cite{Catani:2021kcy} that the three-particle correlations of the type $\tri{ikm}$ with three distinct partons contribute only to processes with {\it four} or {\it more} hard partons. General properties of the colour algebra of the $d$-type tripoles $\tri{ikm}$ and their action onto two and three hard-parton states are discussed in Ref.~\cite{Catani:2021kcy} (see also Appendix~\ref{a:tthp}). We conclude this section by proving the equivalence of Eqs.~(\ref{Wqcd}) and (\ref{Wred}). In the case of the dipole contributions proportional to $\T_i\cdot \T_k$, the equivalence directly follows from the colour conservation relation in Eq.~(\ref{colcons}), in the same way as the equivalence between Eqs.~(\ref{J1sq})--(\ref{I23}) and Eqs.~(\ref{j1q})--(\ref{w23}). Considering the tripole contributions to Eq.~(\ref{Wqcd}), we can write \begin{equation} \label{sumtri} \sum_{i,k,m} \tri{ikm} \,\cS_{ikm} = \sum_i \tri{iii} \,\cS_{iii} + \sum_{i \neq k} \tri{iik} \,\left( \cS_{iik} + \cS_{iki} + \cS_{kii}\right) + \sum_{{\rm dist.}\{i,k,m\}} \tri{ikm} \,\cS_{ikm} \;, \end{equation} where we have separated the terms with three equal parton indices, two equal indices and three distinct indices, and we have also used the symmetry property $\tri{iki}=\tri{kii}=\tri{iik}$. Considering the contributions to Eq.~(\ref{Wred}) with three distinct parton indices, we can write \begin{equation} \label{sumtridist} \sum_{{\rm dist.}\{i,k,m\}} \tri{ikm} \,w^{(tri)}_{ikm} \; \eqcs \; \sum_{{\rm dist.}\{i,k,m\}} \tri{ikm} \,\cS_{ikm} + \frac{1}{2} \sum_{i \neq k} \left( \tri{iki} + \tri{ikk} \right) \left( \cS_{iik} + \cS_{iki} + \cS_{kii} - \cS_{iii}\right) \,, \end{equation} where we have used the expression of $w^{(tri)}_{ikm}$ in Eq.~(\ref{wikm}) and we have performed the sum over $m$ in the terms of $w^{(tri)}_{ikm}$ that do not depend on $m$. Exploiting colour conservation, we have used the relation $\sum_{m\neq i,k} \tri{ikm} \,\eqcs\, - (\tri{iki} + \tri{ikk})$ to explicitly carry out the sum over $m$. We can now consider the difference between the terms in the left-hand side of Eqs.~(\ref{sumtri}) and (\ref{sumtridist}), and we obtain \begin{align} \!\! \sum_{i,k,m} \tri{ikm} \,\cS_{ikm} - \sum_{{\rm dist.}\{i,k,m\}} \tri{ikm} \,w^{(tri)}_{ikm} \, &\eqcs \, \frac{1}{2} \sum_{i \neq k} \left( \tri{iik} - \tri{ikk} \right) \left( \cS_{iik} + \cS_{iki} + \cS_{kii} - \cS_{iii}\right) \label{difftri} \\ &= \sum_{i \neq k} \,\tri{iik} \;w^{(tri)}_{ik} \;\;. \label{difffin} \end{align} In Eq.~(\ref{difftri}) we have first inserted the expressions in the right-hand side of Eqs.~(\ref{sumtri}) and (\ref{sumtridist}), and then we have used $\tri{iki}=\tri{iik}$ and the relation $\tri{iii} \eqcs - \sum_{k \neq i} \tri{iik}$ (which follows from colour conservation). In Eq.~(\ref{difffin}) we have first renamed $i \leftrightarrow k$ in the terms of Eq.~(\ref{difftri}) that are proportional to $\tri{ikk}$, and then we have simply used the expression of $w^{(tri)}_{ik}$ in Eq.~(\ref{wiik}). In conclusions, Eq.~(\ref{difffin}) proves the equality of the charge asymmetry contributions in Eqs.~(\ref{Wqcd}) and (\ref{Wred}). \section{Processes with two and three hard partons\label{s:e23h}} The simplest applications of the QCD soft-factorization formula (\ref{softsquared}) regard processes with two and three hard partons. In these processes the structure of the colour correlations produced by soft emission is simplified \cite{Catani:1999ss, Catani:2019nqv, Catani:2021kcy}. In this section we present the explicit expressions of $| \J(q_1,q_2,q_3) |^2$ for $gq{\bar q}$ emission from amplitudes with two and three hard partons and, in particular, we highlight the corresponding charge asymmetry effects. \subsection{Soft $gq{\bar q}$ emission from two hard partons\label{s:e2h}} We consider a generic scattering amplitude $\M_{BC}(\{p_i\})$ whose external legs are two hard partons (denoted as $B$ and $C$), with momenta $p_B$ and $p_C$, and additional colourless particles. The two hard partons can be either a $q{\bar q}$ pair ($\{ BC \} = \{ q{\bar q} \}$) (note the we specify $B=q$ and $C={\bar q}$) or two gluons ($\{ BC \} = \{ gg \}$). There is only {\em one} colour singlet configuration of the two hard partons, and the corresponding one-dimensional colour space is generated by a single colour state vector that we denote as $\ket{B C}$. The colour space amplitude $\ket{\M_{BC}}$ is a colour singlet state and, therefore, it is directly proportional to $\ket{B C}$. The squared current $|\J(q_1,\cdots,q_N) |^2$ in Eq.~(\ref{softsquared}) conserves the colour charge of the hard partons and, consequently, the state $|\J|^2 \,\ket{B C}$ is also proportional to $\ket{B C}$ and we have \begin{equation} \label{jbc} | \J(q_1,\cdots,q_N) |^2 \;\,\ket{B C} = \ket{B C} \;| \J(q_1,\cdots,q_N) |^{2}_{\; BC} \;, \end{equation} \begin{equation} \label{softsquaredAB} \bra{\M_{BC} (\{p_i\})} \;| \J(q_1,\cdots,q_N) |^2 \;\ket{\M_{BC} (\{p_i\})} = | \M_{BC} (\{p_i\}) |^2 \;\, | \J(q_1,\cdots,q_N) |^{2}_{\; BC} \;, \end{equation} where $| \J|^{2}_{\; BC}$ is a $c$-number (it is the eigenvalue of the operator $| \J|^{2}$ onto the colour state $\ket{B C}$). We note that the right-hand side of Eq.~(\ref{softsquaredAB}) is proportional to the squared amplitude $| \M_{BC} (\{p_i\}) |^2$ with no residual colour correlations between the hard partons $B$ and $C$. In this respect, the structure of soft factorization in Eq.~(\ref{softsquaredAB}) is similar to that of soft-photon factorization in QED. We recall \cite{Catani:2019nqv} that Eqs.~(\ref{jbc}) and (\ref{softsquaredAB}) are valid for squared currents $| \J(q_1,\cdots,q_N) |^2$ of an arbitrary number $N$ and an arbitrary type (gluons and quark--antiquark pairs) of soft partons. We also recall \cite{Catani:2019nqv} that Eqs.~(\ref{jbc}) and (\ref{softsquaredAB}) are valid at {\it arbitrary} loop orders in the perturbative expansion of both the squared current and the squared amplitude. We evaluate $| \J(q_1,q_2,q_3) |^{2}_{\; BC}$ for soft $gq{\bar q}$ emission at the tree level by using Eq.~(\ref{J123sq}). The squared current terms $| \J(q_1) |^{2}_{\; BC}$ for single-gluon emission and $| \J(q_2,q_3) |^{2}_{\; BC}$ for soft-$q{\bar q}$ emission are well known \cite{Catani:1999ss}. The correlation term $W(q_1,q_2,q_3)$ of Eq.~(\ref{J123sq}) depends on dipole and tripole colour operators (see Eqs.~(\ref{Wqcd}) or (\ref{Wred})). The action of the dipole operators onto $\ket{B C}$ is elementary \cite{csdip}, and the action of the tripole operators is explicitly evaluated in Ref.~\cite{Catani:2021kcy} (see also Appendix~\ref{a:tthp}). We straightforwardly obtain the following result: \begin{align} \!\! | \J(q_1,q_2,q_3) |^{2}_{\; BC}&= \left( \g\,\mu^\epsilon\right)^6 T_R \,C_F \Bigl\{ \,\Bigl[ C_F \left( w_{BC}(q_1) \;w_{BC}(q_2,q_3) + w^{(F)}_{BC}(q_1,q_2,q_3) \right) \Bigr. \Bigr. \nonumber \\ & + \Bigl. \Bigl. C_A \;w^{(A)}_{BC}(q_1,q_2,q_3) \Bigr] + \frac{1}{2} \,d_A \,w^{(tri)}_{BC}(q_1,q_2,q_3)\Bigr\} \;\;, \;\; ( \{B=q, C={\bar q}\}) \;, \label{BCqq}\\ \!\! | \J(q_1,q_2,q_3) |^{2}_{\; BC}&=\left( \g\,\mu^\epsilon\right)^6 T_R \,C_A \,\Bigl\{ \; C_A \left( w_{BC}(q_1) \;w_{BC}(q_2,q_3) + w^{(A)}_{BC}(q_1,q_2,q_3) \right) \Bigr. \nonumber \\ & + C_F \;w^{(F)}_{BC}(q_1,q_2,q_3) \Bigr\} \;\;, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( \{B C\}=\{gg\}) \;, \label{BCgg} \end{align} where the colour coefficient $d_A$ is related to $d^{abc}$ as follows \begin{equation} \sum_{bc} \,d^{abc} \,d^{dbc} = d_A \;\delta^{ad} \;\;, \quad\quad d_A= \frac{N_c^2-4}{N_c} \;\;. \end{equation} The dipole kinematical functions $w_{BC}(q_1), w_{BC}(q_2,q_3), w^{(A)}_{BC}(q_1,q_2,q_3)$ and $w^{(F)}_{BC}(q_1,q_2,q_3)$ are given in Eqs.~(\ref{w1}), (\ref{w23}) and (\ref{w3ik}), and they are {\it symmetric} under the exchange $B \leftrightarrow C$ of the hard partons (i.e., the exchange $p_B \leftrightarrow p_C$ of the hard-parton momenta). The tripole kinematical function $w^{(tri)}_{BC}(q_1,q_2,q_3)$ is given in Eq.~(\ref{wiik}), and it is {\it antisymmetric} under the exchange $B \leftrightarrow C$. The term that is proportional to $w_{BC}(q_1) w_{BC}(q_2,q_3)$ in Eqs.~(\ref{BCqq}) and (\ref{BCgg}) is due to the independent emission of the soft gluon and of the soft $q{\bar q}$ pair. All the other contributions to Eqs.~(\ref{BCqq}) and (\ref{BCgg}) are due to irreducible correlations for soft $gq{\bar q}$ emission. In the case of soft $gq{\bar q}$ emission form the hard partons $\{BC\}=\{q{\bar q}\}$, the square-bracket contribution in the right-hand side of Eq.~(\ref{BCqq}) is symetric with respect to the exchange $q_2 \leftrightarrow q_3$ of the momenta of the soft quark and antiquark. The kinematical function $w^{(tri)}_{BC}(q_1,q_2,q_3)$ is instead antisymmetric under the exchange $q_2 \leftrightarrow q_3$ and, therefore, the result in Eq.~(\ref{BCqq}) explicitly shows the presence of charge asymmetry effects. Since the function $w^{(tri)}_{BC}(q_1,q_2,q_3)$ is antisymmetric with respect to the separate exchanges $q_2 \leftrightarrow q_3$ and $B \leftrightarrow C$, in Eq.~(\ref{BCqq}) the asymmetry in the momenta of the soft quark and antiquark is correlated with a corresponding asymmetry in the momenta $p_B$ and $p_C$ of the hard quark and antiquark. In particular, $| \J(q_1,q_2,q_3) |^{2}_{\; BC}$ is invariant under the overall exchange of fermions and antifermions (i.e., $\{q_2, p_B \} \leftrightarrow \{q_3, p_C \}$), consistently with charge conjugation invariance. In the case of soft $gq{\bar q}$ radiation form two hard gluons, the tree-level result in Eq.~(\ref{BCgg}) shows no charge asymmetry effects. As argued in Ref.~\cite{Catani:2021kcy} on the basis of charge conjugation invariance, the absence of charge asymmetry effects in $| \J(q_1,q_2,q_3) |^{2}_{\; BC}$ for $\{BC\}=\{gg\}$ persists at arbitrary orders in the QCD loop expansion. \subsection{Soft $gq{\bar q}$ emission from three hard partons\label{s:e3h}} We consider a generic scattering amplitude $\M_{ABC}(\{p_i\})$ whose external legs are three hard partons and additional colourless particles. The three hard partons, which are denoted as $A, B, C$ (with momenta $p_A, p_B ,p_C$), can be either a gluon and a $q{\bar q}$ pair ($\{A B C\}=\{g q {\bar q}\}$) or three gluons ($\{A B C\}=\{g g g\}$). If $\{A B C\}=\{g q {\bar q}\}$, there is only one colour singlet state that can be formed by the three hard partons. If the three hard partons $\{A B C\}$ are gluons, they can form two distinct colour singlet states. The different dimensionality of the colour singlet space for the two cases leads to different features of the associated soft radiation. We discuss the cases $\{A B C\}=\{g q {\bar q}\}$ and $\{A B C\}=\{g g g\}$ in turn. In the case $\{A B C\}=\{g q {\bar q}\}$, we specifically set $A=g$, $B=q$ and $C={\bar q}$. The one-dimensional colour singlet space of the three hard partons is generated by the state vector $\ket{A B C}$, and the colour space amplitude $\ket{\M_{ABC}}$ is directly proportional to $\ket{A B C}$. Since we are dealing with a one-dimensional colour singlet space, we can use the same reasoning as in Sect.~\ref{s:e2h}. The state $\ket{A B C}$ is an eigenstate of the squared current $|\J(q_1,\cdots,q_N) |^2$ in Eq.~(\ref{softsquared}), and we have \begin{equation} \label{jgqq} | \J(q_1,\cdots,q_N) |^2 \;\,\ket{A B C} = \ket{ A B C} \;| \J(q_1,\cdots,q_N) |^{2}_{\; ABC} \;, \quad \quad (\{A B C\}=\{g q {\bar q}\}) \;, \end{equation} \begin{equation} \label{sgqq} \bra{\M_{ABC} (\{p_i\})} \;| \J(q_1,\cdots,q_N) |^2 \;\ket{\M_{ABC} (\{p_i\})} = | \M_{ABC} (\{p_i\}) |^2 \;\, | \J(q_1,\cdots,q_N) |^{2}_{\; ABC} \;, \end{equation} where $| \J|^{2}_{\; ABC}$ is a $c$-number. Similar to Eqs.~(\ref{jbc}) and (\ref{softsquaredAB}), we recall \cite{Catani:2019nqv} that Eqs.~(\ref{jgqq}) and (\ref{sgqq}) are valid for {\em arbitrary} squared currents $| \J(q_1,\cdots,q_N) |^2$ and at {\em arbitrary} loop orders. Considering the tree-level squared current $| \J(q_1,q_2,q_3) |^{2}$ for soft $g g {\bar q}$ emission, the eigenvalue $| \J(q_1,q_2,q_3) |^{2}_{\; ABC}$ in Eqs.~(\ref{jgqq}) and (\ref{sgqq}) can be written in the following form: \begin{align} | \J(q_1,q_2,q_3) |^{2}_{\; ABC} &= \left( \g\,\mu^\epsilon\right)^6 \,T_R \Bigl[ \, F_{ABC}^{({\rm in.em.})}(q_1,q_2,q_3) + W_{ABC}^{({\rm ch.sym.})}(q_1,q_2,q_3) \Bigr. \nonumber \\ &+ \Bigl. W_{ABC}^{({\rm ch.asym.})}(q_1,q_2,q_3) \,\Bigr] \;, \quad \quad \quad \quad (\{A B C\}=\{g q {\bar q}\}) \;, \label{softgqq} \end{align} which directly derives from Eqs.~(\ref{J123sq}) and (\ref{Wred}). The functions $F_{ABC}^{({\rm in.em.})}$ and $W_{ABC}^{({\rm ch.sym.})}$ in Eq.~(\ref{softgqq}) are \begin{align} F_{ABC}^{({\rm in.em.})}(q_1,q_2,q_3) &= \bigl[ C_F \;w_{BC}(q_1) + C_A \;w_{ABC}(q_1) \bigr] \nonumber \\ & \times \bigl[ C_F \;w_{BC}(q_2,q_3) + C_A \;w_{ABC}(q_2,q_3) \bigr] \;, \quad \;\; (\{A B C\}=\{g q {\bar q}\}) \;, \label{fgqq} \end{align} \begin{align} W_{ABC}^{({\rm ch.sym.})}(q_1,q_2,q_3) &= C_F^2 \;w_{BC}^{(F)}(q_1,q_2,q_3) + C_F C_A \left( w_{BC}^{(A)}(q_1,q_2,q_3) + w_{ABC}^{(F)}(q_1,q_2,q_3) \right) \nonumber \\ &+ C_A^2 \;w_{ABC}^{(A)}(q_1,q_2,q_3) \;, \quad \quad \quad \quad \quad \quad \quad \;\;\;\;\; (\{A B C\}=\{g q {\bar q}\}) \;, \label{csgqq} \end{align} where the two hard-parton functions $w_{ik}(q_1), w_{ik}(q_2,q_3), w^{(r)}_{ik}(q_1,q_2,q_3) \,(r=F,A)$ are given in Eqs.~(\ref{w1}), (\ref{w23}), (\ref{w3ik}), and we have used them to define the corresponding three hard-parton functions $w_{ABC}^{(r)}$\footnote{In the cases of one and two soft momenta $(N=1,2)$, the explicit superscripts $(r)$ have to be removed in Eq.~(\ref{wabc}).}: \begin{equation}\label{wabc} w_{ABC}^{(r)}(q_1,\cdots,q_N) \equiv \frac{1}{2} \left[ w_{AB}^{(r)}(q_1,\cdots,q_N) + w_{AC}^{(r)}(q_1,\cdots,q_N) - w_{BC}^{(r)}(q_1,\cdots,q_N) \right] \;. \end{equation} The function $W_{ABC}^{({\rm ch.asym.})}$ in Eq.~(\ref{softgqq}) is \begin{align} W_{ABC}^{({\rm ch.asym.})}(q_1,q_2,q_3) &= \frac{d_A}{4} \Bigl[ - C_A \, \left(w^{(tri)}_{BC}(q_1,q_2,q_3) + w^{(tri)}_{CA}(q_1,q_2,q_3) + w^{(tri)}_{AB}(q_1,q_2,q_3) \right) \Bigr. \nonumber \\ &\Bigl. + \,2 C_F \;w^{(tri)}_{BC}(q_1,q_2,q_3) \Bigr] \;\;, \quad \quad \quad \quad \quad \;\;\;\;\;\; (\{A B C\}=\{g q {\bar q}\}) \;, \label{cagqq} \end{align} where the kinematical function $w^{(tri)}_{ik}$ is given in Eq.~(\ref{wiik}). As discussed below, the symmetry properties of the kinematical functions $w_{ik}(q_1), w_{ik}(q_2,q_3), w^{(r)}_{ik}(q_1,q_2,q_3)$ and $w^{(tri)}_{ik}(q_1,q_2,q_3)$ with respect to their dependence on the hard and soft momenta (see Sect.~\ref{s:gqqsc}) lead to ensuing symmetry properties of the functions $F^{({\rm in.em.})}, W^{({\rm ch.sym.})}$ and $W^{({\rm ch.asym.})}$. The term $F^{({\rm in.em.})}_{ABC}$ in Eq.~(\ref{softgqq}) is the contribution due to the independent emission of the soft gluon and the soft $q{\bar q}$ pair. It originates from the term $\sy{|{\bom J}(q_1)|^2}{|{\bom J}(q_2,q_3)|^2}$ in Eq.~(\ref{J123sq}). We have used the results of the action of both $|{\bom J}(q_1)|^2$ and $|{\bom J}(q_2,q_3)|^2$ onto the three hard-parton state $\ket{ABC}$ \cite{Catani:1999ss}. Such squared currents only depend on colour dipole operators, whose action onto $\ket{ABC}$ is simply given in terms of casimir coeficients $C_F$ and $C_A$ (see, e.g., Ref.~\cite{csdip}). The colour dipole contributions of Eq.~(\ref{Wred}) to the correlation term $W(q_1,q_2,q_3)$ of the squared current in Eq.~(\ref{J123sq}) produce the corresponding irreducible correlation contribution $W^{({\rm ch.sym.})}_{ABC}$ in Eq.~(\ref{softgqq}). We note that both $F^{({\rm in.em.})}_{ABC}(q_1,q_2,q_3)$ and $W^{({\rm ch.sym.})}_{ABC}(q_1,q_2,q_3)$ are symmetric under the exchange $q_2 \leftrightarrow q_3$ (see Eqs.~(\ref{fgqq}) and (\ref{csgqq})) and, therefore, they do not lead to any charge asymmetry of the soft quark and antiquark in $| \J(q_1,q_2,q_3) |^{2}_{\; ABC}$. Both functions $F^{({\rm in.em.})}_{ABC}$ and $W^{({\rm ch.sym.})}_{ABC}$ are also symmetry under the exchange $p_B \leftrightarrow p_C$ of the momenta of the hard quark and antiquark, consistently with the charge conjugation invariance of $| \J(q_1,q_2,q_3) |^{2}_{\; ABC}$. The correlation term $W(q_1,q_2,q_3)$ of Eq.~(\ref{J123sq}) also includes charge asymmetry contributions. In Eq.~(\ref{Wred}) these contributions are proportional to the tripole operators $\tri{iik}$ and $\tri{ikm}$. The action of these tripole operators onto the state $\ket{ABC}$ of the three hard partons $\{A B C\}=\{g q {\bar q}\}$ was evaluated in Ref.~\cite{Catani:2021kcy} (in particular, the operator $\tri{ABC}$ with three distinct indices vanishes). Using the colour algebra results of Ref.~\cite{Catani:2021kcy} (see also Appendix~\ref{a:tthp}), we have computed the charge asymmetry contribution to $| \J(q_1,q_2,q_3) |^{2}_{\; ABC}$, which is given by the function $W_{ABC}^{({\rm ch.asym.})}(q_1,q_2,q_3)$ in Eq.~(\ref{softgqq}). We note the the expression of $W_{ABC}^{({\rm ch.asym.})}$ in Eq.~(\ref{cagqq}) is antisymmetric under the exchange $p_B \leftrightarrow p_C$ of the momenta of the hard quark and antiquark, in complete analogy with the charge asymmetry contribution to Eq.~(\ref{BCqq}), and consistently with the charge conjugation invariance of $| \J(q_1,q_2,q_3) |^{2}_{\; ABC}$ in Eq.~(\ref{softgqq}). We now consider the case $\{A B C\}=\{g g g \}$. The three hard gluons generate a two-dimensional colour singlet space. We choose the basis that is formed by the two colour state vectors $\ket{(ABC)_f \,}$ and $\ket{(ABC)_d \,}$, which are defined as follows \begin{equation} \label{ABCbasis} \bra{\,abc\,} \left(ABC\right)_f \,\rangle \equiv i f^{abc} \,, \;\;\; \bra{\,abc\,} \left(ABC\right)_d \,\rangle \equiv d^{abc} \,, \;\;\; \quad (\{ ABC \} = \{ ggg \}) \;\;, \end{equation} where $a, b, c$ are the colour indices of the three gluons. We note that the two states in Eq.~(\ref{ABCbasis}) are orthogonal and have different charge conjugation. The scattering amplitude $\ket{{\cal M}_{ABC}(\{p_i\})}$ is, in general, a linear combination of the two states in Eq.~(\ref{ABCbasis}), and the action of the squared current $| {\bom J}(q_1, \cdots, q_N) |^2 $ for soft-parton radiation onto $\ket{{\cal M}_{ABC}(\{p_i\})}$ can produce colour correlations between these two states. In general, $| {\bom J}(q_1, \cdots, q_N) |^2 $ can be represented as a $2 \times 2$ correlation matrix that acts onto the two-dimensional space generated by $\ket{\left( ABC \right)_f}$ and $\ket{\left( ABC \right)_d}$. The structure of this correlation matrix is discussed in Refs.~\cite{Catani:2019nqv} and \cite{Catani:2021kcy} for the cases of multiple soft-gluon radiation and soft-$q{\bar q}$ radiation, respectively. Soft $gq{\bar q}$ radiation is discussed in the following. The action of the tree-level squared current $| \J(q_1,q_2,q_3) |^{2}$ for soft $gq{\bar q}$ emission onto the colour singlet states in Eq.~(\ref{ABCbasis}) can be written in the following form: \begin{align} | \J(q_1,q_2,q_3) |^{2} \;\ket{ABC} &= \left( \g\,\mu^\epsilon\right)^6 \,T_R \Bigl[ \, F_{ABC}^{({\rm in.em.})}(q_1,q_2,q_3) + W_{ABC}^{({\rm ch.sym.})}(q_1,q_2,q_3) \Bigr. \nonumber \\ &+ \Bigl. W_{ABC}^{({\rm ch.asym.})}(q_1,q_2,q_3) \,\Bigr] \;\ket{ABC}\;, \quad \quad \quad \quad (\{A B C\}=\{ggg\}) \;, \label{J2ggg} \end{align} where \begin{equation} \label{inem3g} F_{ABC}^{({\rm in.em.})}(q_1,q_2,q_3) = 4 C_A^2 \;E_{ABC}(q_1) \;E_{ABC}(q_2,q_3) \;\;, \quad \quad (\{A B C\}=\{ggg\}) \;, \end{equation} \begin{equation} \label{symggg} W_{ABC}^{({\rm ch.sym.})}(q_1,q_2,q_3) = 2 C_A \!\left[ C_F \,E_{ABC}^{(F)}(q_1,q_2,q_3) + C_A \,E_{ABC}^{(A)}(q_1,q_2,q_3) \right], (\{A B C\}=\{ggg\}) \;, \end{equation} \begin{equation} \label{asymggg} W_{ABC}^{({\rm ch.asym.})}(q_1,q_2,q_3) = 4 \;\tri{BBA} \;E_{ABC}^{(tri)}(q_1,q_2,q_3) \;\;, \quad \quad \;\;\;\; (\{A B C\}=\{ggg\}) \;. \end{equation} The three hard-parton functions $E_{ABC}(q_1), E_{ABC}(q_2,q_3)$ and $E_{ABC}^{(r)}(q_1,q_2,q_3)$ (with $r=F,A,tri$) in Eqs.~(\ref{inem3g})--(\ref{asymggg}) are given as follows\footnote{In the cases of one and two soft momenta $(N=1,2)$, the explicit superscripts $(r)$ have to be removed in Eq.~(\ref{eabc}).} \begin{equation} \label{eabc} E_{ABC}^{(r)}(q_1,\cdots,q_N) \equiv \frac{1}{4} \left[ w_{AB}^{(r)}(q_1,\cdots,q_N) + w_{BC}^{(r)}(q_1,\cdots,q_N) + w_{CA}^{(r)}(q_1,\cdots,q_N) \right] \;, \end{equation} in terms of the corresponding two hard-parton functions $w_{ik}(q_1), w_{ik}(q_2,q_3), w^{(r)}_{ik}(q_1,q_2,q_3)$ in Eqs.~(\ref{w1}), (\ref{w23}), (\ref{w3ik}) and (\ref{wiik}). The symmetry properties of the functions $E_{ABC}^{(r)}$ are the consequence of the corresponding symmetries of the functions $w_{ik}^{(r)}$ in the right-hand side of Eq.~(\ref{eabc}). The function $E_{ABC}^{(tri)}(q_1,q_2,q_3)$ is antisymmetric under the exchange $q_2 \leftrightarrow q_3$ of the momenta of the soft quark and antiquark, and it is also antisymmetric under the exchange of the momenta of two hard gluons (e.g., $p_A \leftrightarrow p_B$). The functions $E_{ABC}^{(r)}$ in Eqs.~(\ref{inem3g}) and (\ref{symggg}) are instead symmetric under the exchange $q_2 \leftrightarrow q_3$ and have a fully symmetric dependence on the hard-gluon momenta $p_A, p_B, p_C$. The result in Eq.~(\ref{J2ggg}) directly derives from Eqs.~(\ref{J123sq}) and (\ref{Wred}), and it has a structure that follows the structure of Eq.~(\ref{softgqq}). The term $F^{({\rm in.em.})}_{ABC}(q_1,q_2,q_3)$ in Eq.~(\ref{J2ggg}) is the contribution of the independent emission of the soft gluon and the soft $q{\bar q}$ pair. The irreducible correlation term $W^{({\rm ch.sym.})}_{ABC}(q_1,q_2,q_3)$ in Eq.~(\ref{J2ggg}) is due to the colour dipole contributions of Eq.~(\ref{Wred}) to $W(q_1,q_2,q_3)$. Both terms $F^{({\rm in.em.})}_{ABC}$ and $W^{({\rm ch.sym.})}_{ABC}$ originate from colour dipole interactions, whose action onto a generic hard-parton state $\ket{ABC}$ are simply proportional to the unit operator in colour space \cite{Catani:1999ss}. The term $W_{ABC}^{({\rm ch.asym.})}$ in Eq.~(\ref{J2ggg}) is due to the colour tripole contributions of Eq.~(\ref{Wred}) to the irreducible correlation operator $W(q_1,q_2,q_3)$ in Eq.~(\ref{J123sq}). The action of the tripole operators onto the three-gluon states in Eq.~(\ref{ABCbasis}) was explicitly evaluated in Ref.~\cite{Catani:2021kcy} (see also Appendix~\ref{a:tthp}). In particular, the tripoles $\tri{ABC}$ with three distinct gluons vanish, while the tripoles with two distinct gluons are proportional to one another (the proportionality factors are $\pm 1$). It turns out that the term $W_{ABC}^{({\rm ch.asym.})}$ in Eq.~(\ref{J2ggg}) is directly proportional to a single tripole operator (e.g., $\tri{BBA}$) as shown in Eq.~(\ref{asymggg}). The tripole operators are odd under charge conjugation and, therefore, they act differently onto the two colour states in Eq.~(\ref{ABCbasis}). Considering the operator $\tri{BBA}$ in Eq.~(\ref{asymggg}), we have \cite{Catani:2021kcy} \begin{equation} \label{triggg} \tri{BBA} \,\ket{\left( ABC \right)_f} = \frac{C_A^2}{4} \;\ket{\left( ABC \right)_d} \;, \;\;\;\; \tri{BBA} \,\ket{\left( ABC \right)_d} = \frac{C_A \,d_A}{4} \;\ket{\left( ABC \right)_f} \;, \end{equation} and we note that the tripole operators produce `pure' transitions between the colour symmetric and colour antisymmetric states $\ket{\left( ABC \right)_f}$ and $\ket{\left( ABC \right)_d}$, which have different charge conjugation. The results in Eqs.~(\ref{J2ggg})--(\ref{asymggg}) can be used to explicitly evaluate the action of the tree-level squared current $| \J(q_1,q_2,q_3) |^{2}$ onto a scattering amplitude $\ket{{\cal M}_{ABC}(\{p_i\})}$ with three hard gluons. The scattering amplitude $\ket{{\cal M}_{ABC}(\{p_i\})}$ is, in general, a linear combination of the two colour states in Eq.~(\ref{ABCbasis}), and we write \begin{equation} \label{3gamp} \ket{{\cal M}_{ABC}(\{p_i\})} = \ket{\left( ABC \right)_f} \;\;{\cal M}_{f}(p_A,p_B,p_C) \,+ \ket{\left( ABC \right)_d} \;\;{\cal M}_{d}(p_A,p_B,p_C) \;\;, \end{equation} where ${\cal M}_f$ and ${\cal M}_d$ are colour stripped amplitudes. Owing to the Bose symmetry of $\ket{{\cal M}_{ABC}}$ with respect to the three gluons, the function ${\cal M}_{f}(p_A,p_B,p_C)$ is antisymmetric under the exchange of two gluon momenta (e.g., $p_A \leftrightarrow p_B$), while ${\cal M}_{d}(p_A,p_B,p_C)$ has a symmetric dependence on $p_A,p_B,p_C$. Using Eqs.~(\ref{J2ggg})--(\ref{asymggg}), (\ref{triggg}) and (\ref{3gamp}) we obtain \begin{align} \label{m23g} &\bra{\M_{ABC} (\{p_i\})} \;| \J(q_1,q_2,q_3) |^2 \;\ket{\M_{ABC} (\{p_i\})} = \left( \g\,\mu^\epsilon\right)^6 \,T_R \nonumber \\ &\;\;\;\; \times \Bigl\{ | \M_{ABC} (\{p_i\}) |^2 \; \Bigl[ \, F_{ABC}^{({\rm in.em.})}(q_1,q_2,q_3) + W_{ABC}^{({\rm ch.sym.})}(q_1,q_2,q_3) \Bigr] \Bigr. \nonumber \\ &\;\;\;\;\;\;\; + \Bigl. C_A^2 d_A (N_c^2-1) \bigl[ \, {\cal M}^\dagger_d(p_A,p_B,p_C) {\cal M}_f(p_A,p_B,p_C) + {\rm h.c.} \bigr] E_{ABC}^{(tri)}(q_1,q_2,q_3) \Bigr\} \;\;, \\ & \quad~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad (\{A B C\}=\{ggg\}) \;, \nonumber \end{align} which is not simply proportional to $| {\cal M}_{ABC}(\{p_i\}) |^2$ (unlike the corresponding result in Eq.~(\ref{sgqq}) for $\{ ABC \} = \{ gq{\bar q} \})$). The contribution to Eq.~(\ref{m23g}) that is proportional to $F_{ABC}^{({\rm in.em.})} + W_{ABC}^{({\rm ch.sym.})}$ is symmetric under the exchange $q_2 \leftrightarrow q_3$ and, therefore, it does not lead to any charge asymmetry of the soft quark and antiquark. The function $E_{ABC}^{(tri)}(q_1,q_2,q_3)$ is instead antisymmetric under the exchange $q_2 \leftrightarrow q_3$. Therefore, in contrast with the case of scattering amplitudes with two hard gluons (see Eq.~(\ref{BCgg})), the expression in Eq.~(\ref{m23g}) involves a charge-asymmetry contribution that is not vanishing, provided the hard-scattering amplitude includes non-vanishing components ${\cal M}_f$ and ${\cal M}_d$ (i.e., $\ket{{\cal M}_{ABC}}$ has no definite charge conjugation). This is the case, for instance, of the amplitude for the decay process $Z \rightarrow ggg$ of the $Z$ boson (see, e.g., Ref.~\cite{vanderBij:1988ac}). We note that the functions $E_{ABC}^{(tri)}$ and $({\cal M}_d^\dagger {\cal M}_f + {\rm h.c.})$ are separately antisymmetric under the exchange of two gluon momenta and, consequently, their product is symmetric. Therefore, the right-hand side of Eq.~(\ref{m23g}) is fully symmetric under permutations of the three hard gluons, as required by Bose symmetry. \section{QED and mixed QCD$\times$QED interactions\label{s:qed}} Our results of Sects.~\ref{s:gqqc} and \ref{s:gqqsc} for soft $gq{\bar q}$ radiation at the tree level in QCD are generalized in this section to deal with soft emission through QED (photon) interactions and mixed QCD$\times$QED (gluon and photon) interactions. We consider a generic scattering amplitude $\M(\{q_\ell\}, \{p_i\})$ whose external soft {\it massless} particles are gauge bosons ($b$), fermions ($f$) and antifermions ($\bar f$). The soft gauge bosons can be gluons ($b=g$) or photons ($b=\gamma$), and the soft massless fermions are quarks ($f=q$) and charged leptons ($f=\ell$). The external massless and massive hard partons in $\M(\{q_\ell\}, \{p_i\})$ are gluons, (anti)quarks and electrically charged particles, such as (anti)leptons and $W^\pm$ bosons. The amplitude $\M$ can also have external particles that carry no colour charge and no electric charge. We formally treat QCD, QED and mixed QCD$\times$QED interactions on equal footing. Therefore, the scattering amplitude $\M$ has a generalized perturbative (loop) expansion in powers of two unrenormalized couplings: the QCD coupling $\g$ and the QED coupling $\gq$ ($\gq^2/(4\pi)=\alpha$ is the fine structure constant at the unrenormalized level). In the soft limit the amplitude $\M(\{q_\ell\}, \{p_i\})$ fulfils the factorization formula (\ref{1gfact}), and the soft current $\J(q_1,\cdots,q_N)$ also has a loop expansion in powers of the two couplings $\g$ and $\gq$. In the following we only consider soft currents at the tree level with respect to both couplings and, therefore, the $N$ parton current $\J(q_1,\cdots,q_N)$ includes all possible contributions that are proportional to $\g^{N-k} \gq^k$ with $0 \leq k \leq N$. The pure QCD and pure QED cases are recovered by setting $\gq=0$ and $\g=0$, respectively. We first recall the known expressions of the soft currents for single-photon and fermion-antifermion emission. The tree-level current ${\bom J}_\gamma(q_1)$ for emission of a single soft photon with momentum $q_1$ is \begin{equation} \label{J1ga} {\bom J}_\gamma(q_1) = \gq \,\mu^\epsilon \sum_{i} e_i \;\frac{p_i \varepsilon(q_1)}{p_i q} \;\;, \end{equation} where $e_i$ is the electric charge (in units of the positron charge $\gq$) of the $i$-th hard parton in $\M(\{p_i\})$. The conservation of the electric charge in $\M(\{p_i\})$ implies that $\sum_i e_i = 0$ (analogously to the colour conservation relation in Eq.~(\ref{colcons})). Note that ${\bom J}_\gamma(q_1)$ is a $c$-number (more precisely, it is proportional to the unit matrix in colour space) since the photon carries no colour charge. The square of the current in Eq.~(\ref{J1ga}) is \begin{equation} \label{J2ga} |{\bom J}_\gamma(q_1)|^2 = - \left(\gq \,\mu^\epsilon \right)^2 \sum_{i,k} e_i\,e_k \;S_{ik}(q_1) \eqcs - \left(\gq \,\mu^\epsilon \right)^2 \frac12 \sum_{i\neq k} e_i\,e_k \;w_{ik}(q_1)\;\;, \end{equation} where the momentum dependent functions $S_{ik}(q_1)$ and $w_{ik}(q_1)$ are given in Eqs.~(\ref{S1}) and (\ref{w1}). The tree-level current ${\bom J}_{f{\bar f}}(q_2,q_3)$ for emission of a soft-$f{\bar f}$ pair is \cite{Catani:2021kcy} \begin{align} \label{J1ff} {\bom J}_{f{\bar f}}(q_2,q_3) &= \delta_{fq} \;{\bom J}(q_2,q_3) - \left( \gq \,\mu^\epsilon \right)^2 \,e_f \;{\bf \Delta}_f \, \sum_{i} e_i \;\frac{p_i j(2,3)}{p_i q_{23}} \;, \end{align} where $q_2$ and $q_3$ are the momenta of the soft fermion $f$ and antifermion $\bar f$, respectively. The first contribution in the right-hand side of Eq.~(\ref{J1ff}) is the QCD current ${\bom J}(q_2,q_3)$ in Eq.~(\ref{Jqa}) (the Kronecker delta symbol $\delta_{fq}$ specifies that the current is not vanishing only if $f=q$), and the second contribution is due to the photon mediated radiation of the $f{\bar f}$ pair. The fermionic current $j^\nu(2,3)$ is given in Eq.~(\ref{fercur}), and $e_f$ is the electric charge of the soft fermion $f$. The factor ${\bf \Delta}_f$ in the right-hand side of Eq.~(\ref{J1ff}) is a colour operator that depends on the type of soft fermion $f$. If $f=\ell$, we simply have ${\bf \Delta}_f= 1$. If $f=q$, ${\bf \Delta}_f$ is the projection operator onto the colour singlet state of the $f{\bar f}$ pair, namely, by using the colour space notation of Sect.~\ref{s:sfsc} we have $\bra{\alpha_2,\alpha_3} \,{\bf \Delta}_f = \delta_{\alpha_2\alpha_3}$, where $\alpha_2$ and $\alpha_3$ are the colour indices of the soft quark and antiquark, respectively. The square of the current ${\bom J}_{f{\bar f}}(q_2,q_3)$ is \cite{Catani:2021kcy} \begin{equation} \label{J2ff} |{\bom J}_{f{\bar f}}(q_2,q_3)|^2 = \delta_{fq} \;|{\bom J}(q_2,q_3)|^2 - \left(\gq \,\mu^\epsilon \right)^4 \,\left( \delta_{f\ell} + N_c \,\delta_{fq} \right) \,e_f^2\,\frac12 \sum_{i\neq k} e_i\,e_k \;w_{ik}(q_2,q_3)\;\;, \end{equation} where $|{\bom J}(q_2,q_3)|^2$ is the QCD squared current in Eq.~(\ref{j23q}) and the function $w_{ik}(q_2,q_3)$ is given in Eqs.~(\ref{I23}) and (\ref{w23}). Similarly to its QCD part, the complete squared current $|{\bom J}_{f{\bar f}}(q_2,q_3)|^2$ is charge symmetric with respect to the exchange $f \leftrightarrow {\bar f}$ (i.e., it is symmetric under $q_2 \leftrightarrow q_3$). We note that the squared current result in Eq.~(\ref{J2ff}) does not include a mixed QCD$\times$QED term proportional to $\g^2 \gq^2$, since such contribution vanishes. In the following we present our results for soft $bf{\bar f}$ radiation at the tree level. The boson $b$ has momentum $q_1$ and the fermion $f$ and antifermion $\bar f$ have momenta $q_2$ and $q_3$, respectively. Similarly to the results in Sect.~\ref{s:gqqc}, we express the current ${\bom J}_{bf\bar{f}}$ for soft $bf{\bar f}$ radiation in terms of an independent emission contribution and an irreducible correlation term $\boldsymbol\Gamma_{bf\bar f}$. In the case of soft $gf{\bar f}$ radiation we obtain \begin{equation}\label{gff} {\bom J}_{gf\bar{f}}(q_1,q_2,q_3) = \sy{{\bom J}(q_1)\,}{{\bom J}_{f\bar f}(q_2,q_3)} + \boldsymbol\Gamma_{gf\bar f}(q_1,q_2,q_3) \;\;, \end{equation} where ${\bom J}(q_1)$ is the QCD soft-gluon current in Eq.~(\ref{J1}) and ${\bom J}_{f\bar f}(q_2,q_3)$ is the soft-$f{\bar f}$ current in Eq.~(\ref{J1ff}). Introducing the explicit dependence on the colour index $a_1$ of the soft gluon, the irreducible correlation $\boldsymbol\Gamma_{gf\bar f}$ is \begin{equation}\label{corrgff} \boldsymbol\Gamma_{gf\bar f}^{a_1}(q_1,q_2,q_3) = \delta_{fq} \, \Bigl[ \boldsymbol\Gamma^{a_1}(q_1,q_2,q_3) + \g\,\gq^2 \mu^{3\epsilon} e_f \sum_i e_i \, {\bom t}^{a_1} \,\gab_i(q_1,q_2,q_3) \Bigr] \;\;, \end{equation} where the term $\boldsymbol\Gamma^{a_1}(q_1,q_2,q_3)$ in the right-hand side is the QCD irreducible correlation in Eq.~(\ref{bfgamma}), and the function $\gab_i(q_1,q_2,q_3)$ is given in Eq.~(\ref{psi}). We note that the correlation $\boldsymbol\Gamma_{gf\bar f}$ is not vanishing only if $f=q$. The second term in the square bracket of Eq.~(\ref{corrgff}) is the mixed QCD$\times$QED correction to the QCD irreducible correlation for soft $gq\bar{q}$ emission. We note that such QCD$\times$QED correlation term is proportional to $\gab_i(q_1,q_2,q_3)$ and, therefore, it has an abelian character. The tree-level current for soft $\gamma f\bar{f}$ emission is \begin{equation}\label{gaff} {\bom J}_{\gamma f\bar{f}}(q_1,q_2,q_3) = {{\bom J}_\gamma(q_1)\,}{{\bom J}_{f\bar f}(q_2,q_3)} + \boldsymbol\Gamma_{\gamma f\bar f}(q_1,q_2,q_3) \;\;, \end{equation} where ${\bom J}_\gamma(q_1)$ and ${\bom J}_{f\bar f}(q_2,q_3)$ are the currents in Eqs.~(\ref{J1ga}) and (\ref{J1ff}). We note that the independent emission contribution in Eq.~(\ref{gaff}) does not require colour symmetrization, since ${\bom J}_\gamma$ and ${\bom J}_{f\bar f}$ commute in colour space. The expression of the irreducible correlation component $\boldsymbol\Gamma_{\gamma f\bar f}$ is \begin{equation}\label{corrgaff} \boldsymbol\Gamma_{\gamma f\bar f}(q_1,q_2,q_3) = \mu^{3\epsilon} e_f \sum_i \left[ \, \delta_{fq}\; \g^2\,\gq \; T_i^c {\bom t}^c + \gq^3 \; e_f\, e_i \; \boldsymbol\Delta_f\right] \,\gab_i(q_1,q_2,q_3)\;\;, \end{equation} where $\gab_i(q_1,q_2,q_3)$ is given in Eq.~(\ref{psi}). The term proportional to $\gq^3$ in Eq.~(\ref{corrgaff}) is entirely due to QED (photon) interactions. We note that even in an abelian gauge theory, like QED, the current for soft $\gamma f\bar{f}$ emission includes an irreducible correlation component, which is due to soft-photon radiation in cascade from soft charged fermions. In contrast, we recall that the current for emission of $N$ soft photons factorizes in terms of $N$ independent emission contributions, with no additional irreducible correlations. The term proportional to $\g^2 \gq$ in Eq.~(\ref{corrgaff}) is the irreducible correlation component that is due to mixed QCD$\times$QED interactions. Also this correlation component is controlled by the abelian function $\gab_i(q_1,q_2,q_3)$. The squared current for soft $g f\bar{f}$ emission is computed by using the expressions in Eqs.~(\ref{gff}) and (\ref{corrgff}). We write the result as follows \begin{equation}\label{J2gff} | {\bom J}_{gf\bar{f}}(q_1,q_2,q_3) |^2 = \sy{\,|{\bom J}(q_1)|^2\,}{|{\bom J}_{f\bar f}(q_2,q_3)|^2} + W_{gf\bar{f}}(q_1,q_2,q_3) \;\;, \end{equation} where $|{\bom J}(q_1)|^2$ is the QCD squared current in Eq.~(\ref{j1q}) and $|{\bom J}_{f\bar f}(q_2,q_3)|^2$ is the squared current in Eq.~(\ref{J2ff}). The irreducible correlation contribution $ W_{gf\bar{f}}$ is not vanishing only if the soft fermion is a quark ($f=q$), and we explicitly have \begin{align}\label{wgff} W_{gf\bar{f}}(q_1,q_2,q_3) &= \delta_{fq} \,\Bigl\{ W(q_1,q_2,q_3) \Bigr. \nonumber \\ & \Bigl. - \,\g^4\,\gq^2\, \mu^{6\epsilon} \,2\,T_R\,e_f \sum_{i,k,m} \T_i\cdot \T_k \;e_m\; \bigr[ \cS_{ikm}(q_1,q_2,q_3)+(k \leftrightarrow m) \bigl] \Bigr. \nonumber \\ &\Bigl. - \,\g^2\,\gq^4\, \mu^{6\epsilon} C_F\,N_c\;e_f^2\sum_{i,k} e_i\, e_k\, \;\cS_{ik}^{(F)}(q_1,q_2,q_3) \Bigr\} \;\;, \end{align} where $W(q_1,q_2,q_3)$ is the QCD term in Eq.~(\ref{Wqcd}), and the momentum dependent functions $\cS_{ikm}(q_1,q_2,q_3)$ and $\cS_{ik}^{(F)}(q_1,q_2,q_3)$ are given in Eqs.~(\ref{Sikm}) and (\ref{SF}). The right-hand side of Eq.~(\ref{wgff}) includes two types of mixed QCD$\times$QED contributions, which both have an abelian character. The contribution proportional to $\g^4 \gq^2$ is controlled by the function $\cS_{ikm}(q_1,q_2,q_3)$ and, therefore, it leads to charge asymmetry in the exchange of the soft quark and antiquark. In contrast, the contribution proportional to $\g^2 \gq^4$ is charge symmetric, since it depends on the function $\cS_{ik}^{(F)}(q_1,q_2,q_3)$. Using Eqs.~(\ref{gaff}) and (\ref{corrgaff}) the squared current for soft $\gamma f{\bar f}$ emission is \begin{equation}\label{J2gaff} | {\bom J}_{\gamma f\bar{f}}(q_1,q_2,q_3) |^2 = {|{\bom J}_\gamma(q_1)|^2\,}{|{\bom J}_{f\bar f}(q_2,q_3)|^2} + W_{\gamma f\bar{f}}(q_1,q_2,q_3) \;\;, \end{equation} where $|{\bom J}_\gamma(q_1)|^2$ is the soft-photon squared current in Eq.~(\ref{J2ga}) and $|{\bom J}_{f\bar f}(q_2,q_3)|^2$ is given in Eq.~(\ref{J2ff}). The irreducible correlation component $W_{\gamma f\bar{f}}$ has the following expression: \begin{align}\label{wgaff} W_{\gamma f\bar{f}}(q_1,q_2,q_3) =& - \,\g^4\,\gq^2\, \mu^{6\epsilon} \,\delta_{fq} \,T_R\,e_f \nonumber \\ & \times \Bigl[ \;\sum_{i,k} e_f \,\T_i\cdot \T_k \;\cS_{ik}^{(F)}(q_1,q_2,q_3) + \sum_{i,k,m} 2 \,e_i \, \T_k\cdot \T_m \;\cS_{ikm}(q_1,q_2,q_3) \Bigr] \nonumber \\ & - \,\left(\gq \,\mu^\epsilon \right)^6 \,\left( \delta_{f\ell} + N_c \,\delta_{fq} \right) \,e_f^3 \nonumber \\ & \times \Bigl[ \;\sum_{i,k} e_f \,e_i\,e_k \;\cS_{ik}^{(F)}(q_1,q_2,q_3) + \sum_{i,k,m} 2 \,e_i \,e_k\,e_m \;\cS_{ikm}(q_1,q_2,q_3) \Bigr] \;\;, \end{align} where the charge symmetric function $\cS_{ik}^{(F)}(q_1,q_2,q_3)$ is given in Eq.~(\ref{SF}) and the charge asymmetry function $\cS_{ikm}(q_1,q_2,q_3)$ is given in Eq.~(\ref{Sikm}). The term proportional to $\gq^6$ in Eq.~(\ref{wgaff}) is entirely due to QED interactions. The term proportional to $\g^4 \gq^2$ is due to mixed QCD$\times$QED interactions, and it is not vanishing only if the soft fermion is a quark. We note that both the QED and QCD$\times$QED contributions to $W_{\gamma f\bar{f}}$ involve charge symmetric and asymmetric effects. We also note that the contribution of ${\cal O}(\g^2 \gq^4)$ to $| {\bom J}_{\gamma f\bar{f}}(q_1,q_2,q_3) |^2$ vanishes. \section{Summary\label{s:conc}} We have considered the radiation of a soft gluon and a soft $q{\bar q}$ pair in QCD hard scattering. The scattering amplitude for soft $gq{\bar q}$ emission in a generic hard-scattering process is singular, and the singular behaviour is controlled in factorized form by a current $\J(q_1,q_2,q_3)$, which has a process-independent structure. We have evaluated the soft $gq{\bar q}$ current $\J(q_1,q_2,q_3)$ at the tree level for a generic scattering amplitude with an arbitrary number and type of external hard partons (gluons and massless and massive quarks and antiquarks). The soft current acts in colour space, and it is written in terms of the colour charges and momenta of the external hard partons. We have expressed the current in terms of two contributions: the contribution of `independent' (and colour symmetrized) emission of the soft gluon and the soft $q{\bar q}$ pair, and an irreducible correlation contribution. The irreducible correlation component of the current includes strictly non-abelian terms (which are analogous to the non-abelian correlations for soft multi-gluon emission) and also terms with an abelian character (analogous correlations appear for soft photon-lepton-antilepton emission in QED). We have computed the tree-level squared current $| \J(q_1,q_2,q_3) |^2$ of soft $gq{\bar q}$ emission for squared amplitudes of generic multiparton hard-scattering processes. We have checked that our result for $| \J(q_1,q_2,q_3) |^2$ numerically agrees with the result obtained in Ref.~\cite{DelDuca:2022noh} in a fully independent way. The irreducible correlation component of $| \J(q_1,q_2,q_3) |^2$ leads to two types of colour interactions between the hard partons: colour dipole interactions (which also appear in the independent emission component) and interactions of tripole type that are proportional to the fully-symmetric tensor $d^{a b c}$. These tripole interactions are the real-emission counterpart of the analogous tripole interactions for soft-$q{\bar q}$ radiation at the one-loop level \cite{Catani:2021kcy}. The tripole correlation contributions to $| \J(q_1,q_2,q_3) |^2$ are antisymmetric under the exchange $q_2 \leftrightarrow q_3$ of the momenta of the soft quark and antiquark and, therefore, they produce charge asymmetry effects in the soft limit of the squared amplitudes. We have explicitly considered the evaluation of $| \J(q_1,q_2,q_3) |^2$ for processes with two and three hard partons, and we have discussed the corresponding charge symmetric and asymmetric contributions. We have finally generalized our QCD study of soft $gq{\bar q}$ emission to the study of QED and mixed QCD$\times$QED interactions in the context of soft gluon-fermion-antifermion and photon-fermion-antifermion radiation. In particular, we have noticed that soft photon-lepton-antilepton emission in QED received (abelian) irreducible correlation contributions due to soft-photon radiation in cascade from soft charged leptons. Both QED and mixed QCD$\times$QED interactions lead to charge asymmetry effects in the exchange of the soft fermion and antifermion. \section*{Acknowledgments} \includegraphics[width=2.2em,angle=90]{eu.eps}~ \begin{minipage}[b]{0.9\linewidth} This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 824093. \end{minipage} LC is supported by the Generalitat Valenciana (Spain) through the plan GenT program (CIDEGENT/2020/011) and his work is supported by the Spanish Government (Agencia Estatal de Investigación) and ERDF funds from European Commission (Grant no. PID2020-114473GB-I00 funded by MCIN/AEI/10.13039/501100011033). \section*{Appendices}

BkiUfKXxK7ICUmfb0mIY

\section{Introduction} It is well known that the entanglement entropy (EE) of conformal field theory (CFT) can calculate in a corresponding gravity theory by the Ryu-Takayanagi (RT) formula \cite{PhysRevLett.96.181602Ryu2006,Ryu:2006ef} in AdS/CFT correspondence \cite{GUBSER1998105,Witten199801}. In general, although the EE of quantum field theory is not easy to calculate, the RT formula tells that the EE $S_A$ of a region $A$ in CFT can calculated as the area of the minimal bulk surface $\mathcal{M}_A$ homologous to $A$ (RT surface): \begin{align} S_A = \frac{\mathrm{Area}\left(\mathcal{M}_A\right)}{4 G_N}, \end{align} where $G_N$ is the Newton constant of gravitation. This relation promotes informational theoretical analysis of AdS/CFT correspondence. By regarding the geometry of a bulk as a tensor network, it implements quantum error correcting code of boundary CFT \cite{Pastawski2015} or MERA \cite{Swingle2012}, subregion subregion correspondence which is proposed for reduced density matrix \cite{Czech_2012}. From this point of view, it is better to regard the RT formula as a flow proposed by the paper \cite{Freedman:2016zud}. The authors introduced ``bit threads'' which are equivalent concept to the RT surface geometrically. The bit threads are defined as a bounded divergenceless vector field $v^\mu$ \begin{align} \nabla_\mu v^\mu=0, \quad \left|v\right| \leq C, \end{align} and it maximizes its flux on a boundary area $A$. The property that the maximal total flux of $v^\mu$ through the area $A$ is equal to the area of the RT surface is proved by the max-flow min-cut theorem \cite{Freedman:2016zud}. The bit threads give an intuitive picture that a vector field carrying information of the boundary propagates in the bulk, and the bulk region stores information of the boundary. Although the concept of bit threads is inspirational, as mentioned by \cite{Freedman:2016zud}, the RT surface has infinitely many equivalent bit threads. Moreover, it is non-trivial task to construct bit threads practically and to calculate the EE of CFT by it due to its dependence of global structure of spacetimes. Therefore, as one of the interpretations of the RT surface respecting the viewpoint of the flow, we propose an interpretation that the RT surfaces are wave fronts of null rays emitted from a point on the AdS boundary. In particular, we prove that in spherical symmetric static spacetimes, owing to its axisymmetry of the configuration, such wave fronts of null rays are extremal surfaces as long as they propagates in the vacuum region. As the RT surface is the extremal surface \cite{Fursaev:2006ih,Lewkowycz2013}, thus the wave front can be considered as the RT surface. On the other hand, we can naturally understand null rays as bit threads. In this picture, we can calculate the EE of CFT by counting the number of such null rays. This method is also valid for wave optical calculation using the flux of a massless scalar field. The flux based calculation method suggests a picture that information prepared on the boundary side spreads to the bulk as null rays. The structure of this paper is as follows. In Section 2, we demonstrate the correspondence between the RT surface and the wave front of the null rays in the BTZ spacetime. In Section 3, we state the detail of our proposal and show it in spherically symmetric static spacetimes with a negative cosmological constant. In Section 4, we introduce the flux formula to calculate the EE of CFT by counting the number of null rays. Finally, Section 5 is devoted to summary. \section{Null wave front and RT surface} In this section, before going to discuss the general situation, we demonstrate that wave fronts of null rays are the RT surface in the BTZ spacetime. \subsection{Ryu-Takayanagi surface} We derive the equation of the RT surface in the BTZ spacetime \cite{PhysRevLett.69.1849Banados1992} \begin{align} ds^2 = - \left(\frac{r^2}{\ell_\text{AdS}^2}-M\right) dt^2 + \left(\frac{r^2}{\ell_\text{AdS}^2}-M\right)^{-1} dr^2 + r^2 d\theta^2,\quad -\pi\le\theta\le\pi, \label{eq:BTZ} \end{align} where $M$ is the mass of the black hole and $\ell_\text{AdS}$ is the AdS radius. We prepare a region (arc) $-\theta_\ell\le\theta\le\theta_\ell$ on the AdS boundary and consider a line anchored to the boundary of this region. The RT surface extremizes the following line area (length) on a constant time slice: \begin{align} \mathrm{Area}\left[r(\cdot), \frac{dr}{d\theta}(\cdot)\right] = \int_{-\theta_\ell}^{\theta_\ell} d\theta \sqrt{ \left(\frac{r^2}{\ell_\text{AdS}^2}-M\right)^{-1} \left( \frac{dr}{d\theta}\right)^2 + r^2 }. \label{line element in Example} \end{align} The equation of the RT surface $r=r_\text{RT}(\theta)$ is the solution of the Euler-Lagrange equation obtained by variation of $\mathrm{Area}[r,dr/d\theta]$ with respect to $r$, and it is \begin{align} r_\text{RT}(\theta) = \frac{\sqrt{M}\, r_{\mathrm{min}}\,\mathrm{sech}\left(\sqrt{M}\, \theta \right)}{\sqrt{M - r_{\mathrm{min}}^2/\ell_\text{AdS}^2 \tanh^2 \left( \sqrt{M}\, \theta \right)}}, \label{eq:RT surface in Ryu-Takayanagi surface} \end{align} where $r_{\mathrm{min}} := r_\text{RT}(\theta=0)$ denotes the minimum of $r$ (see Fig.~\ref{RTsurfaceFig}). Note that $\theta_\ell=\theta(r=\infty)=(1/\sqrt{M})\mathrm{arctanh}(\sqrt{M}\ell_\text{AdS}/r_\text{min})$. \begin{figure}[H] \centering \includegraphics[width=0.45\linewidth,clip]{RTsurfaceBTZ2.pdf} \caption{The RT surface in the BTZ spacetime ($M=\ell_\text{AdS}=1$) with coordinates $(x,y)=(\rho\cos\theta, \rho\sin\theta)$, $\rho :=\ell_\text{AdS}\arctan \left(r/\ell_\text{AdS}\right)$. Each blue lines is parametrized by $r_{\mathrm{min}}=1.01,1.05,1.2,2.0,5.0$. For large interval $2\theta_\ell$ on the AdS boundary, dotted lines become minimal surfaces.} \label{RTsurfaceFig} \end{figure} The entanglement entropy of CFT on the AdS boundary for an arc $|\theta|\le\theta_\ell$ is obtained by substituting \eqref{eq:RT surface in Ryu-Takayanagi surface} into \eqref{line element in Example} : \begin{align} \mathrm{Area}\left(\mathcal{M}_\ell\right) =2\ell_\text{AdS}\log \left( \frac{2 \ell_\text{AdS}}{\epsilon \sqrt{M} } \sinh \left( \sqrt{M}\,\theta_\ell \right) \right) + O(\epsilon), \label{eq:finite-temp} \end{align} where the cutoff is introduced by $\epsilon := \ell_\text{AdS}^2/r~ (r \rightarrow\infty)$. Now let us consider CFT with inverse temperature $\beta$ on $\mathrm{S}^1$. The circumference of the circle is assumed to be $C$ and we prepare an arc $|\theta|\le\theta_\ell$ with the arc length $\ell=C\theta_\ell/\pi$ on it. Then it is possible to write down \eqref{eq:finite-temp} using only CFT quantities. By dividing Eq.~\eqref{eq:finite-temp} with $4G_N$, using the Brown-Henneaux formula $c =3\ell_\text{AdS}/(2G_N)$ \cite{Brown1986} and AdS/CFT dictionary $\beta/C=1/\sqrt{M}$, we obtain the correct EE formula of thermal state of CFT on $\mathrm{S}^1$ ~\cite{Calabrese:2004eu,Calabrese:2009qy} after rescaling the cutoff $\epsilon$: \begin{equation} S_A=\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi\ell}{\beta}\right)\right). \label{eq:EET} \end{equation} \subsection{Null rays and wave front} We consider null rays emitted from a point on the AdS boundary and their wave fronts. Our purpose is to find out the relation between wave fronts of null rays and the RT surface. We consider null rays in the spherically symmetric static spacetime \begin{equation} ds^2 = - f(r) dt^2 + \frac{ dr^2}{f(r)} + r^2 d\Omega_{d-1}^2, \label{eq:metric-Ddim} \end{equation} where $d$ denotes spatial dimension and $d\Omega_{d-1}^2$ is the line element of the unit sphere $\mathrm{S}^{d-1}$. We introduce coordinates on $\mathrm{S}^{d-1}$ as \begin{equation} \begin{bmatrix} x_1\\x_2\\x_3\\ \vdots \\ x_{d-2}\\x_{d-1} \end{bmatrix} = \begin{bmatrix} \cos\psi_1 \hfill\\ \sin\psi_1\cos\psi_2\hfill\\ \sin\psi_1\sin\psi_2 \hfill \\ \vdots \\ \sin\psi_1\sin\psi_2\cdots\sin\psi_{d-2}\cos\psi_{d-1} \\ \sin\psi_1\sin\psi_2\cdots\sin\psi_{d-2}\sin\psi_{d-1} \end{bmatrix} \end{equation} with $0\le\psi_1,\cdots\psi_{d-2}\le\pi,~0\le\psi_{d-1}\le 2\pi$. The line element on $\mathrm{S}^{d-1}$ is \begin{equation} d\Omega_{d-1}^2=d\psi_1^2+\sin^2\psi_1(d\psi_2^2+\sin^2\psi_2(d\psi_3^2+\cdots\cdots)). \end{equation} As is well known, in static spherically symmetric spacetimes, trajectories of null geodesics stay on a spatial two dimensional plane. Thus we can fix coordinate values of $\psi_2,\cdots \psi_{d-1}$ and assume the following (2+1)-dimensional metric to investigate wave fronts of null rays emitted from a point: \begin{equation} ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\theta^2,\quad \theta:=\psi_1. \label{eq:metric3} \end{equation} \begin{figure}[H] \centering \includegraphics[width=0.6\linewidth]{configuration.pdf} \caption{Axisymmetric configuration of congruence of null rays. Coordinates on two dimensional plane are introduced by $(x,y)=(r\cos\theta, r\sin\theta)$. Wave fronts at fixed $t$ are symmetric with respect to rotation about $x$-axis. Projection of null rays on the $(x,y)$-plane are shown as red lines.} \end{figure} \noindent In this metric with coordinates $x^\mu=(t,r,\theta)$, a wave front of null rays emitted from a point source is defined as a $t=$ constant section of null congruences, which forms a $(d-1)$-dimensional surface. Due to the axial symmetry of the configuration, a wave front of null rays is represented as a curve in $(r,\theta)$ space in the present situation. The tangent vector of a null ray is \begin{equation} k^\mu=(k^t, k^r, k^\theta)=\left(\frac{dt}{d\lambda},\frac{dr}{d\lambda},\frac{d\theta}{d\lambda} \right),\quad -f\left(\frac{dt}{d\lambda}\right)^2+\frac{1}{f}\left(\frac{dr}{d\lambda}\right)^2+r^2\left(\frac{d\theta}{d\lambda}\right)^2=0, \label{eq:geod1} \end{equation} where $\lambda$ is the affine parameter. This spacetime has two Killing vectors related to translation of $t$ and $\theta$ directions and there exist two conserved charges $ \omega := f(r)\dfrac{dt}{d\lambda},\ p_\theta := r^2 \dfrac{d\theta}{d\lambda}$. Combining with Eq.~\eqref{eq:geod1}, we obtain a trajectory of a null ray as \begin{align} \theta(r) &= \theta_0 \pm \int_{r_0}^r \frac{b}{r'^2}\left( 1 - f(r') \frac{b^2}{r'^2}\right) ^{-\frac{1}{2}} dr', \label{eq:geo-th} \\ t(r) &= t_0 \pm \int_{r_0}^r \frac{1 }{f(r')} \left(1 - f(r') \frac{b^2}{r'^2}\right) ^{-\frac{1}{2}} dr', \label{eq:geo-t} \\ \lambda(r)&= \lambda_0 \pm \frac{1}{\omega}\int_{r_0}^r \left(1 - f(r') \frac{b^2}{r'^2} \right)^{-\frac{1}{2}} dr', \label{eq:geo-lam} \end{align} where $(t_0,\theta_0,\lambda_0 )= (t(r_0),\theta(r_0),\lambda(r_0) )$ and the impact parameter $b:=p_\theta/\omega$ is introduced. The sign $\pm$ in front of the integral corresponds to the sign of $dr/d\lambda$. For the $(2+1)$-dimensional BTZ spacetime \eqref{eq:BTZ}, we can demonstrate explicitly that wave fronts of null rays are the RT surfaces. We obtain equations of null geodesic from \eqref{eq:geo-th} and \eqref{eq:geo-t} with $(t_0,r_0,\theta_0 )= (0,\infty,0)$: \begin{align} \theta(r) &= \frac{1}{\sqrt{M}} \log \left[ \frac{\sqrt{r^2 - b^2\left(r^2/\ell_\text{AdS}^2 - M\right)} + b\sqrt{M}}{r\sqrt{1-b^2/\ell_\text{AdS}^2}} \right], \label{eq:th}\\ t(\theta) &= \frac{\ell_\text{AdS}}{\sqrt{M}} \mathrm{arctanh}\left( \frac{\ell_\text{AdS}}{b} \tanh \left( \sqrt{M}\, \theta \right)\right). \label{eq:t} \end{align} It is easy to derive a trajectory of a null ray $r=r_\text{NG}(\theta,b)$ with an impact parameter $b$ from \eqref{eq:th}. On the other hand, the equation of a wave front $r=r_\text{WF}(\theta,t)$ at a fixed $t$ is derived by eliminating the parameter $b$ from \eqref{eq:th} and \eqref{eq:t}. After all, \begin{align} r_\text{NG}(\theta,b)&= \frac{\sqrt{M}\, b}{\sqrt{1-b^2/\ell_\text{AdS}^2}}\, \mathrm{csch}\left( \sqrt{M}\, \theta\right), \label{Null geodesic Planar BTZ in Examples} \\ r_\text{WF}(\theta,t)&= \frac{\sqrt{M}\ell_\text{AdS}\, \mathrm{coth} \left(\sqrt{M}\,t/\ell_\text{AdS} \right)\mathrm{sech} \left( \sqrt{M}\, \theta \right)}{\sqrt{ 1-\mathrm{coth}^2\left(\sqrt{M}\,t/\ell_\text{AdS}\right)\mathrm{tanh}^2 \left( \sqrt{M}\, \theta \right)} }. \label{Wave front Planar BTZ in Examples} \end{align} \noindent For the special case $M=-1$, the spacetime reduces to the pure AdS. Figure \ref{fig:nullAdS} and Figure \ref{fig:nullBTZ} show null rays and their wave fronts in the pure AdS spacetime and the BTZ black hole spacetime, respectively. \begin{figure}[H] \centering \includegraphics[width=0.4\linewidth]{AdSWFRT.pdf} \caption{Null rays (dotted red lines) and wave fronts (blue lines) in the pure AdS$_{2+1}$ spacetime $(M=-1, \ell_\text{AdS}=1)$.} \label{fig:nullAdS} \end{figure} \begin{figure}[H] \centering \includegraphics[width=0.4\linewidth]{BTZWFRTM1.pdf} \hspace{1cm} \includegraphics[width=0.4\linewidth]{BTZWFRTM01.pdf} \caption{Null rays (dotted red lines) emitted from $(r,\theta)=(\infty,0)$ and their wave fronts (blue lines) in the BTZ spacetime (left panel: $M=\ell_\text{AdS}=1$, right panel: $M=0.1,\ell_\text{AdS}=1$).} \label{fig:nullBTZ} \end{figure} \noindent Note that Eq.~\eqref{Wave front Planar BTZ in Examples} of the wave front is the same as Eq.~\eqref{eq:RT surface in Ryu-Takayanagi surface} of the RT surface by identifying $r_{\mathrm{min}} = \sqrt{M}\ell_\text{AdS}\coth \left( \sqrt{M} t/\ell_\text{AdS} \right)$, which represents the elapsed time of a null ray traveling from $r = \infty $ to $r = r_{\mathrm {min}} $. Indeed this quantity is obtained by taking $b=0$ in the equation of the null ray \eqref {eq:geo-t}: \begin{align} t = \int_\infty^{r_{\mathrm{min}}} \frac{dr}{r^2/\ell_\text{AdS}^2 - M} = \frac{\ell_\text{AdS}}{\sqrt{M}} \!\ \mathrm{arctanh}\left(\frac{\sqrt{M}\ell_\text{AdS}}{r_{\mathrm{min}}} \right). \end{align} Therefore, we have confirmed that wave fronts of null rays emitted from the AdS boundary coincide with the RT surfaces in the BTZ spacetime. For sufficient elapse of time after emission of null rays, self-intersection of the wave front occurs and identification of the wave front as the RT surface becomes ambiguous. \section{Null wave front and extremal surface} In this section, based on the observation in the previous section for the BTZ spacetime, we show the following proposition for spherically symmetric static spacetimes with a cosmological constant (no matter fields). \vspace{0.2cm} \noindent \textbf{Proposition} ~Wave fronts of null rays emitted from a point are extremal surfaces when the affine parameter of null rays goes to infinity. \vspace{0.2cm} \noindent \textbf{Corollary} ~For spacetimes with a negative cosmological constant, wave fronts of null rays emitted from a point on the AdS boundary are extremal surfaces. \vspace{0.2cm} \noindent We adopt the metric \eqref{eq:metric3} with coordinates $x^\mu=(t,r,\theta,\cdots)$. Let $\xi^\mu = (\partial_t)^\mu$ be the time-like Killing vector, $k^\mu=dx^\mu/d\lambda$ be the tangent vector of null geodesics. We introduce the projection tensor $P^{\mu\nu}=g^{\mu\nu} - \xi^{-2} \xi^\mu \xi^\nu=\mathrm{diag}(0, f,1/r^2,\cdots)$ onto a constant time slice. We denote the tangent vector of null geodesics projected onto the hypersurface as $\tilde{k}^i= P^{i}{}_{j}\, k^j=(k^r, k^\theta,0,\cdots)$. The conserved quantity associated with the Killing vector is $\omega = - \xi^\mu k_\mu=fk^t$ and the norm of the spatial vector $\tilde{k}^i$ is $\tilde{k}^i \tilde{k}_i = \omega^2/f$. We prove the proposition by using the fact that the extremal surface is a surface with zero mean curvature. The mean curvature $H$ of a wave front of null rays on a constant time slice is defined by \begin{equation} H:=D_i \tilde n^i,\quad \tilde n^i=\frac{\tilde k^i}{\tilde k}=\frac{f^{1/2}}{\omega}(k^r,k^\theta,0,\cdots), \end{equation} where $\tilde n^i$ is the unit normal vector of the wave front and $D_i = P_i{}^j\nabla_j=(\nabla_r,\nabla_\theta,\cdots)$ is the covariant derivative on a constant time slice. Then, \begin{align} H&=D_i\left(\frac{f^{1/2}}{\omega}k^i\right)=\frac{1}{f^{-1/2}\sqrt{h}}\partial_i\left(\frac{f^{1/2}}{\omega}f^{-1/2}\sqrt{h}\,k^i\right)\notag\\ &=\frac{f^{1/2}}{\omega\sqrt{h}}\left(\partial_r(\sqrt{h}\,k^r)+\partial_\theta(\sqrt{h}\,k^\theta)\right), \end{align} where $\sqrt{h}=r^{d-1}$ comes from determinant of the metric on $\mathrm{S}^{d-1}$. On the other hand, the expansion of a null congruence is \begin{align} \Theta&=\nabla_\mu k^\mu=\frac{1}{\sqrt{h}}\partial_\mu(\sqrt{h}\,k^\mu) \notag\\ &=\frac{1}{\sqrt{h}}\left(\partial_r(\sqrt{h}\,k^r)+\partial_\theta(\sqrt{h}\,k^\theta)\right). \end{align} Therefore $H=(f^{1/2}/\omega)\Theta$ and the mean curvature $H$ of a wave front is proportional to the expansion of the null geodesic congruence. The expansion $\Theta$ along a null geodesic obeys the Raychaudhuri equation \begin{equation} \frac{d \Theta}{d\lambda} = -\frac{\Theta^2}{d-1}-R_{\mu\nu}k^\mu k^\nu. \label{eq:Raych} \end{equation} In the present case, as the congruence of null geodesics has axial symmetry, the shear and the rotation of the congruence do not appear in this equation. For vacuum spacetimes with a cosmological constant, the term with the Ricci curvature disappears. Then the solution of Eq.~\eqref{eq:Raych} is $\Theta(\lambda) = (d-1)/(\lambda-\lambda_0)$ where $\lambda_0$ is the affine parameter at the source. Thus the expansion goes to zero as the affine parameter goes to infinity, and the mean curvature of the wave front is zero and is the extremal surface. Therefore the proposition is proved. As an example of this proposition, let us consider a wave front in the Minkowski spacetime. A spherical wave front emitted from a point source placed at the spatial infinity becomes plane wave, which is zero mean curvature surface in the Minkowski spacetime. However, in this case, the coordinate time \eqref{eq:geo-t} becomes infinite when a wave front of null rays arrives at an observer. Asymptotically AdS spacetimes are peculiar because they have the timelike boundary. We consider the pure AdS spacetime of which metric function is given by $f(r) = 1+r^2/\ell_\text{AdS}^2$. As $f\approx r^2/\ell_\text{AdS}^2$ in the vicinity of the AdS boundary, the affine parameter of null rays \eqref{eq:geo-lam} from the AdS boundary $r_0=\ell_\text{AdS}^2/\epsilon, \epsilon\rightarrow 0$ diverges as \begin{align} \lambda(r)\approx\frac{1}{\omega}\int_r^{\ell_\text{AdS}^2/\epsilon}\left(1-\frac{b^2}{\ell_\text{AdS}^2}\right)^{-1/2}dr=\frac{\ell_\text{AdS}^2/\epsilon-r}{\omega\sqrt{1-b^2/\ell_\text{AdS}^2}} \to \infty. \end{align} On the other hand, the coordinate time \eqref{eq:geo-t} converges as \begin{align} t(r) \approx \int_r^{\ell_\text{AdS}^2/\epsilon}dr\frac{\ell_\text{AdS}^2}{r^2}\left(1-\frac{b^2}{\ell_\text{AdS}^2}\right)^{-1/2} =\frac{\ell_\text{AdS}^2}{\sqrt{1-b^2/\ell_\text{AdS}^2}}\,\frac{1}{r}. \end{align} This property also holds for general asymptotically AdS spacetimes because they have the same metric in the vicinity of the AdS boundary as the pure AdS spacetime. After all, we conclude that for static spherical symmetric asymptotically AdS spacetimes, wave fronts of a null geodesic congruence emitted from a point source on the AdS boundary are extremal surfaces. \section{Flux formula} Based on the idea presented in the previous section, we can understand null rays as a natural flow characterizing the EE of the dual CFT. Hence a congruence of null rays is one of the bit threads described in Section I. This makes us conceive a picture that null rays propagate in the bulk with information of the AdS boundary. This picture suggests that the EE can be calculable by counting the number of null rays. In this section, we reformulate the RT formula in terms of the wave optics. Concepts of wave fronts and the flux of null rays are naturally derived as the eikonal limit of wave optics. As an application of wave optics to black hole spacetimes, papers \cite{Kanai2013,Nambu2016,Hashimoto2020} investigate image formation of the photon sphere of black holes. In this paper, we focus on the structure of wave fronts of a massless scalar field. For the monochromatic massless scalar field with time dependence $\propto e^{-i\omega t}$, we present wave patterns in Fig.~\ref{fig:wave-ads} and Fig.~\ref{fig:wave-BTZ} (see detail in Appendix). They show wave fronts from a point wave source on the AdS bounary (see Fig.~\ref{fig:nullAdS} and Fig.~\ref{fig:nullBTZ} for corresponding wave fronts in the geometrical optics). \begin{figure}[H] \centering \includegraphics[width=0.4\linewidth,clip]{wave-ads.pdf} \caption{Wave pattern of the monochromatic massless scalar field with $\omega=20$ in the AdS spacetime. Real part of $\phi/|\phi|$ is shown.} \label{fig:wave-ads} \end{figure} \begin{figure}[H] \centering \includegraphics[width=0.4\linewidth,clip]{wave-BTZ1.pdf} \hspace{1cm} \includegraphics[width=0.4\linewidth,clip]{wave-BTZ01.pdf} \caption{Wave pattern of the massless scalar field with $\omega=20$ (Re $\phi/|\phi|$) in the BTZ spacetime with $M=1$ (left panel) and $M=0.1$ (right panel).} \label{fig:wave-BTZ} \end{figure} \noindent For the massless scalar field $\phi(x^\mu)$ obeying the Klein-Gordon equation $\square\phi=(\sqrt{-g})^{-1}\partial_\mu(\sqrt{-g}g^{\mu\nu}\partial_\nu\phi)=0$, the WKB form of the wave function is \begin{equation} \phi(x^\mu)=a(x^\mu)\exp\left[iS(x^\mu)\right], \end{equation} where $a$ and $S$ are real functions. In the eikonal limit, they obey \begin{align} &g^{\mu\nu}\nabla_\mu S \nabla_\nu S=0, \label{eq:HJ} \\ &\nabla_\mu(a^2\nabla^\mu S)=0. \label{eq:KGJ} \end{align} The equation \eqref{eq:HJ} is the Hamilton-Jacobi (HJ) equation and Eq.~\eqref{eq:KGJ} represents conservation of the Klein-Gordon current $J^\mu=(1/2i)(\phi^*\partial^\mu\phi-\phi\partial^\mu\phi^*)$. In terms of the wave vector $k_\mu=\partial_\mu S$, which defines the tangent of null rays, \begin{equation} g_{\mu\nu}k^\mu k^\nu=0,\quad\nabla_\mu(a^2k^\mu)=0. \end{equation} For the stationary case, the phase function $S$ can be written as $S=-\omega t+W(r,\theta)$, \begin{align} &\tilde k^i=(k^r,k^\theta,0,\cdots)=\left(f W_r,\frac{1}{r^2}W_\theta,0,\cdots\right),\quad \tilde k^i\tilde k_i=\frac{\omega^2}{f},\\ & f^{-1/2}D_i\left(a^2 f^{1/2}\tilde k^i\right)=0. \label{eq:KGJ2} \end{align} Here, $\tilde k^i$ represents the tangent vector of null rays projected on a constant time slice. We can write the solution of \eqref{eq:KGJ} as \begin{equation} a(\lambda,\chi)=a(\lambda_0,\chi)\exp\left(-\frac{1}{2}\int_{\lambda_0}^\lambda d\lambda\,\Theta(\lambda)\right),\quad\Theta=\nabla_\mu k^\mu, \end{equation} where the integral is along a null ray (with respect to the affine parameter $\lambda$) and $\chi$ denotes a coordinate distinguishing different geodesics. As the expansion of null congruence from the AdS boundary is zero, the amplitude $a(\lambda,\chi)$ is conserved along a null ray and independent of $\lambda$. Furthermore, for a point source isotropically emitting null rays, $a$ is independent of $\chi$ and can assume to be constant. Thus \eqref{eq:KGJ2} implies \begin{equation} D_i\tilde n^i=0,\quad \tilde n^i=\frac{f^{1/2}}{\omega}\tilde k^i, \quad \tilde n^i\tilde n_i=1, \end{equation} and $\tilde n^i$ is divergenceless normalized vector field. The wave front is the surface with the unit normal $\tilde n^i$, and is the extremal surface. The number of null rays passing through the wave front $\mathcal{E}_A$, which is the extremal surface homologous to the region $A$ on the AdS boundary, is \begin{equation} \mathrm{Area}(\mathcal{E}_A)=\int_{\mathcal{E}_A}\tilde n^i d\Sigma_i,\quad d\Sigma_i=\tilde n_i\sqrt{h}\,d^{d-1}\sigma, \label{eq:RTflux} \end{equation} where $\sqrt{h}$ denotes determinant of the induced metric on $\mathcal{E}_A$. Now let us consider the setup shown as Fig.~\ref{fig:screen-setup}. We prepare a screen $A(\epsilon)$ which is $r=$constant surface in the bulk. For the regularization, the screen is placed at $r=\ell_\text{AdS}^2/\epsilon$ near the AdS boundary. \begin{figure}[H] \centering \includegraphics[width=0.4\linewidth,clip]{screen.pdf} \caption{Null rays (red dotted lines) emitted from a point on the AdS boundary pass through the screen $A(\epsilon)$ placed at $r = \ell_\text{AdS}^2/\epsilon $ (dotted line). As the null rays are orthogonal to the wave front (blue line), the number of null rays is proportional to the area of the RT surface.} \label{fig:screen-setup} \end{figure} \noindent Because $\tilde n^i$ is divergence free vector field, Eq.~\eqref{eq:RTflux} equals to \begin{equation} \mathrm{Area}(\mathcal{E}_A)=\int_{A(\epsilon)}\tilde n^i d\Sigma_i=\frac{1}{\omega}\int_{A(\epsilon)}J^id\Sigma_i. \label{eq:flux-formula} \end{equation} This is a formula for area of the RT surface in terms of flux integration of null rays on the screen $A(\epsilon)$. As the Klein-Gordon current $J^i/\omega=f^{1/2}\tilde k^i/\omega$ represents the number density of null rays, we can regard the Klein-Gordon current as a representation of the amount of information propagating in the bluk from the AdS bounary. As a demonstration, we evaluate the right hand side of this relation for the BTZ spacetime. By fixing the radial coordinate as $r =\ell_\text{AdS}^2 / \epsilon $ in Eq.~\eqref{Null geodesic Planar BTZ in Examples}, the impact parameter $b$ on the screen is \begin{align} b=\frac{ \ell_\text{AdS}^2 \sinh \left( \sqrt{M}\, \theta\right)}{\sqrt{\epsilon^2 M + \ell_\text{AdS}^2 \sinh^2 \left( \sqrt{M}\, \theta\right)}}. \label{eq:impact} \end{align} From Eq.~\eqref{eq:geod1}, the radial component of the tangent vector of the null ray is \begin{align} \frac{\tilde k^r}{\omega} =\left(1-f\,\frac{b^2}{r^2}\right)^{1/2} = \sqrt{1- \left(\frac{r^2}{\ell_\text{AdS}^2}-M\right)\frac{b^2}{r^2}}, \label{kr on screen in Geodesical viewpoint} \end{align} and on the screen, \begin{equation} \left.\frac{\tilde k^r}{\omega}\right|_{A(\epsilon)}=\frac{ \sqrt{M} \cosh \left( \sqrt{M}\, \theta\right)}{\sqrt{ M + (\ell_\text{AdS}^2/\epsilon^2) \sinh^2 \left( \sqrt{M}\, \theta\right)}}. \end{equation} The area element on the screen is \begin{align} \left. d \Sigma_r\right|_{A(\epsilon)} = \left.r\,n_r\,d\theta\right|_{A(\epsilon)} = r f^{-1/2}d\theta, \end{align} where $n_r$ is the radial component of the unit normal to the screen. Thus \begin{equation} f^{1/2}\frac{\tilde k^r}{\omega} \left.d\Sigma_r\right|_{A(\epsilon)} =r\left(1-f\, \frac{b^2}{r^2}\right)^{1/2}\!\!\!\!d\theta =\frac{\sqrt{M} \ell_\text{AdS} \cosh(\sqrt{M}\theta)d\theta}{\sqrt{\sinh^2(\sqrt{M}\theta)+M\epsilon^2/\ell_\text{AdS}^2}}. \end{equation} Therefore, \eqref{eq:flux-formula} become \begin{align} \int_{-\theta_\ell}^{\theta_\ell} \frac{f^{1/2}\tilde k^i d\Sigma_i }{\omega} &= \int_{-\theta_\ell}^{\theta_\ell} d\theta \frac{\sqrt{M} \ell_\text{AdS} \cosh \left( \sqrt{M}\, \theta \right)}{\sqrt{M\epsilon^2/\ell_\text{AdS}^2 + \sinh^2 \left( \sqrt{M}\, \theta \right)}} \nonumber\\ & = \ell_\text{AdS} \log \left[ \frac{ \sqrt{\sinh^2(\sqrt{M}\, \theta_\ell)+M\epsilon^2/\ell_\text{AdS}^2}+\sinh(\sqrt{M}\theta_\ell)}{\sqrt{\sinh^2(\sqrt{M}\, \theta_\ell)+M\epsilon^2/\ell_\text{AdS}^2}-\sinh(\sqrt{M}\theta_\ell)} \right] \nonumber\\ & = 2 \ell_\text{AdS} \log\left[ \frac{\ell_\text{AdS}}{(\epsilon/2) \sqrt{M}} \sinh \left( \sqrt{M}\, \theta_\ell\right) \right]+O(\epsilon), \end{align} and reproduces the ``area'' of the RT surface~\eqref{eq:finite-temp}. Dividing by $4G_N$, this result correctly reproduces the EE of CFT \eqref{eq:EET}. Therefore, we can regard such a null geodesic congruence as one realization of the bit threads. \section{Summary} In this paper, we show that wave fronts of null rays emitted from a point on the AdS boundary are extremal surfaces in static spherical symmetric spacetimes. Thus the RT surface can be understood as a wave front, and null rays naturally define a flow characterizing the amount of the EE of CFT. Hence such a flow can be regarded as the bit threads. As we assumed a point source on the AdS boundary, the shape of a region on the AdS boundary (entangling surface) becomes spherical because the boundary of the region is a wave front on the AdS boundary. However, by superposing point sources, it is possible to construct an extremal surface homologous to a region with arbitrary shapes on the AdS boundary by considering the envelope of wave fronts from each point sources. Thus the method presented in this paper may be applicable to the plateaux problem~\cite{Hubeny2013, Freivogel2015} with non-trivial shapes of an entangling surface and to further understanding of property of the holographic EE. \acknowledgments{ Y. N. was supported in part by JSPS KAKENHI Grant Number 19K03866.}

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\section{Introduction} \label{sec_intro} \vspace{-0.1in} The method of {\em stable random projections}\cite{Article:Indyk_JACM06,Proc:Li_SODA08,Proc:Li_Hastie_NIPS07}, as an efficient tool for computing pairwise distances in massive high-dimensional data, provides a promising mechanism to tackle some of the challenges in modern machine learning. In this paper, we provide an easy-to-implement algorithm for {\em stable random projections} which is both statistically accurate and computationally efficient. \vspace{-0.15in} \subsection{Massive High-dimensional Data in Modern Machine Learning} \vspace{-0.05in} We denote a data matrix by $\mathbf{A}\in\mathbb{R}^{n\times D}$, i.e., $n$ data points in $D$ dimensions. Data sets in modern applications exhibit important characteristics which impose tremendous challenges in machine learning \cite{Book:Bottou_07}: \begin{itemize} \item Modern data sets with $n=10^5$ or even $n=10^6$ points are not uncommon in supervised learning, e.g., in image/text classification, ranking algorithms for search engines, etc. In the unsupervised domain (e.g., Web clustering, ads clickthroughs, word/term associations), $n$ can be even much larger. \item Modern data sets are often of ultra high-dimensions ($D$), sometimes in the order of millions (or even higher), e.g., image, text, genome (e.g., SNP), etc. For example, in image analysis, $D$ may be $10^3\times 10^3 = 10^6$ if using pixels as features, or $D = 256^3 \approx 16$ million if using color histograms as features. \item Modern data sets are sometimes collected in a dynamic streaming fashion. \item Large-scale data are often heavy-tailed, e.g., image and text data. \end{itemize} Some large-scale data are dense, such as image and genome data. Even for data sets which are sparse, such as text, the absolute number of non-zeros may be still large. For example, if one queries ``machine learning" (a not-too-common term) in Google.com, the total number of pagehits is about 3 million. In other words, if one builds a term-doc matrix at Web scale, although the matrix is sparse, most rows will contain large numbers (e.g., millions) of non-zero entries. \vspace{-0.1in} \subsection{Pairwise Distances in Machine Learning} \vspace{-0.05in} Many learning algorithms require a similarity matrix computed from pairwise distances of the data matrix $\mathbf{A}\in\mathbb{R}^{n\times D}$. Examples include clustering, nearest neighbors, multidimensional scaling, and kernel SVM (support vector machines). The similarity matrix requires $O(n^2)$ storage space and $O(n^2D)$ computing time. This study focuses on the $l_\alpha$ distance ($0<\alpha\leq 2$). Consider two vectors $u_1$, $u_2\in\mathbb{R}^D$ (e.g., the leading two rows in $\mathbf{A}$), the $l_\alpha$ distance between $u_1$ and $u_2$ is\vspace{-0.15in} {\small\begin{align} d_{(\alpha)} = \sum_{i=1}^D|u_{1,i}-u_{2,i}|^\alpha. \end{align}}\vspace{-0.15in} Note that, strictly speaking, the $l_\alpha$ distance should be defined as {\small$d^{1/\alpha}_{(\alpha)}$}. Because the power operation {\small$(.)^{1/\alpha}$} is the same for all pairs, it often makes no difference whether we use {\small$d^{1/\alpha}_{(\alpha)}$} or just $d_{(\alpha)}$; and hence we focus on $d_{(\alpha)}$. The radial basis kernel (e.g., for SVM) is constructed from $d_{(\alpha)}$ \cite{Article:Chapelle_99,Book:Scholkopf_02}:\vspace{-0.15in} {\small\begin{align}\label{eqn_rbs_kernel} \mathbf{K}(u_1,u_2) = \exp\left(-\gamma\sum_{i=1}^D\left|u_{1,i} - u_{1,i}\right|^\alpha\right), \hspace{0.2in} 0<\alpha\leq 2. \end{align}}\vspace{-0.15in} When $\alpha=2$, this is the Gaussian radial basis kernel. Here $\alpha$ can be viewed as a {\em tuning} parameter. For example, in their histogram-based image classification project using SVM, \cite{Article:Chapelle_99} reported that $\alpha = 0$ and $\alpha = 0.5$ achieved good performance. For heavy-tailed data, tuning $\alpha$ has the similar effect as term-weighting the original data, often a critical step in a lot of applications \cite{Article:Leopold_ML02,Proc:Rennie_ICML03}. For popular kernel SVM solvers including the {\em Sequential Minimal Optimization (SMO}) algorithm\cite{Proc:Platt_NIPS98}, storing and computing kernels is the major bottleneck. Three computational challenges were summarized in \cite[page 12]{Book:Bottou_07}: \begin{itemize} \item \textbf{\em Computing kernels is expensive} \item \textbf{\em Computing full kernel matrix is wasteful} \hspace{0.3in} Efficient SVM solvers often do not need to evaluate all pairwise kernels. \item \textbf{\em Kernel matrix does not fit in memory}\hspace{0.3in} Storing the kernel matrix at the memory cost {\small$O(n^2)$} is challenging when {\small $n>10^5$}, and is not realistic for {\small$n>10^6$}, because {\small$O\left(10^{12}\right)$} consumes at least $1000$ GBs memory. \end{itemize} A popular strategy in large-scale learning is to evaluate distances \textbf{on the fly}\cite{Book:Bottou_07}. That is, instead of loading the similarity matrix in memory at the cost of $O(n^2)$, one can load the original data matrix at the cost of $O(nD)$ and recompute pairwise distances on-demand. This strategy is apparently problematic when $D$ is not too small. For high-dimensional data, either loading the data matrix in memory is unrealistic or computing distances on-demand becomes too expensive. Those challenges are not unique to kernel SVM; they are general issues in distanced-based learning algorithms. The method of {\em stable random projections} provides a promising scheme by reducing the dimension $D$ to a small $k$ (e.g., $k = 50$), to facilitate compact data storage and efficient distance computations. \vspace{-0.1in} \subsection{Stable Random Projections} \vspace{-0.05in} The basic procedure of {\em stable random projections} is to multiply $\mathbf{A}\in\mathbb{R}^{n\times D}$ by a random matrix $\mathbf{R}\in\mathbb{R}^{D\times k}$ ($k\ll D$), which is generated by sampling each entry $r_{ij}$ i.i.d. from a symmetric stable distribution $S(\alpha,1)$. The resultant matrix $\mathbf{B} = \mathbf{A\times R}\in\mathbb{R}^{n\times k}$ is much smaller than $\mathbf{A}$ and hence it may fit in memory. Suppose a stable random variable $x \sim S(\alpha, d)$, where $d$ is the scale parameter. Then its characteristic function (Fourier transform of the density function) is\vspace{-0.15in} {\small\begin{align}\notag \text{E}\left(\exp\left(\sqrt{-1}x t\right)\right) = \exp\left(-d|t|^\alpha\right), \end{align}}\vspace{-0.15in} \noindent which does not have a closed-form inverse except for $\alpha = 2$ (normal) or $\alpha = 1$ (Cauchy). Note that when $\alpha = 2$, $d$ corresponds to ``$\sigma^2$'' (not ``$\sigma$'') in a normal. Corresponding to the leading two rows in $\mathbf{A}$, $u_1$, $u_2\in\mathbb{R}^D$, the leading two rows in $\mathbf{B}$ are $v_1 = \mathbf{R}^\text{T}u_1$, $v_2 = \mathbf{R}^\text{T}u_2$. The entries of the difference, \vspace{-0.15in} {\small\begin{align}\notag x_j = v_{1,j} - v_{2,j} = \sum_{i=1}^D r_{ij}\left(u_{1,i} -u_{2,i}\right) \sim S\left(\alpha, d_{(\alpha)} = \sum_{i=1}^D|u_{1,i} - u_{2,i}|^\alpha\right), \end{align}}\vspace{-0.1in} \noindent for $j = 1$ to $k$, are i.i.d. samples from a stable distribution with the scale parameter being the $l_\alpha$ distance $d_{(\alpha)}$, due to properties of Fourier transforms. For example, when $\alpha = 2$, a weighted sum of i.i.d. standard normals is also normal with the scale parameter (i.e., variance) being the sum of squares of all weights. Once we obtain the stable samples, one can discard the original matrix $\mathbf{A}$ and the remaining task is to estimate the scale parameter $d_{(\alpha)}$ for each pair.\\ Some applications of {\em stable random projections} are summarized as follows:\vspace{-0.05in} \begin{itemize} \item \textbf{\em Computing all pairwise distances} \hspace{0.3in} The cost of computing all pairwise distances of {\small$\mathbf{A}\in\mathbb{R}^{n\times D}$, $O(n^2D)$}, is significantly reduced to {\small$O(nDk + n^2k)$}. \item \textbf{\em Estimating $l_\alpha$ distances online} \hspace{0.3in} For {\small$n>10^5$}, it is challenging or unrealistic to materialize all pairwise distances in $\mathbf{A}$. Thus, in applications such as online learning, databases, search engines, and online recommendation systems, it is often more efficient if we store $\mathbf{B}\in\mathbb{R}^{n\times k}$ in the memory and estimate any distance {\em on the fly} if needed. Estimating distances online is the standard strategy in large-scale kernel learning\cite{Book:Bottou_07}. With {\em stable random projections}, this simple strategy becomes effective in high-dimensional data. \item \textbf{\em Learning with dynamic streaming data} \hspace{0.3in} In reality, the data matrix may be updated overtime. In fact, with streaming data arriving at high-rate\cite{Article:Indyk_JACM06,Proc:Babcock_PODS02}, the ``data matrix'' may be never stored and hence all operations (such as clustering and classification) must be conducted on the fly. The method of {\em stable random projections} provides a scheme to compute and update distances on the fly in one-pass of the data; see relevant papers (e.g., \cite{Article:Indyk_JACM06}) for more details on this important and fast-developing subject. \item \textbf{\em Estimating entropy} \hspace{0.08in} The entropy distance {\small$\sum_{i=1}^D|u_{1,i} - u_{2,i}|\log |u_{1,i} - u_{2,i}|$} is a useful statistic. A workshop in NIPS'03 {\small(\url{www.menem.com/~ilya/pages/NIPS03})} focused on entropy estimation. A recent practical algorithm is simply using the difference between the $l_{\alpha_1}$ and $l_{\alpha_2}$ distances\cite{Proc:Zhao_IMC07}, where {\small$\alpha_1 = 1.05$, $\alpha_2=0.95$}, and the distances were estimated by {\em stable random projections}. \end{itemize} If one tunes the $l_\alpha$ distances for many different $\alpha$ (e.g., \cite{Article:Chapelle_99}), then {\em stable random projections} will be even more desirable as a cost-saving device. \vspace{-0.1in} \section{The Statistical Estimation Problem} \vspace{-0.05in} Recall that the method of {\em stable random projections} boils down to a statistical estimation problem. That is, estimating the scale parameter $d_{(\alpha)}$ from $k$ i.i.d. samples $x_j \sim S(\alpha, d_{(\alpha)})$, $j = 1$ to $k$. We consider that a good estimator $\hat{d}_{(\alpha)}$ should have the following desirable properties: \begin{itemize} \item (Asymptotically) unbiased and small variance. \item Computationally efficient. \item Exponential decrease of error (tail) probabilities. \end{itemize} The {\em arithmetic mean} estimator {\small$\frac{1}{k}\sum_{j=1}^k|x_j|^2$} is good for $\alpha =2$. When $\alpha<2$, the task is less straightforward because (1) no explicit density of $x_j$ exists unless $\alpha = 1$ or $0+$; and (2) $\text{E}(|x_j|^t)<\infty$ only when $-1<t<\alpha$. \vspace{-0.1in} \subsection{Several Previous Estimators} Initially reported in arXiv in 2006, \cite{Proc:Li_SODA08} proposed the {\em geometric mean} estimator \vspace{-0.05in} {\small\begin{align}\notag \hat{d}_{(\alpha),gm} &= \frac{\prod_{j=1}^k |x_j|^{\alpha/k}}{\left[\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k}. \end{align}} where $\Gamma(.)$ is the Gamma function, and the {\em harmonic mean} estimator\vspace{-0.06in} {\small\begin{align}\notag \hat{d}_{(\alpha),hm} &=\frac{-\frac{2}{\pi}\Gamma(-\alpha)\sin\left(\frac{\pi}{2}\alpha\right)}{\sum_{j=1}^k|x_j|^{-\alpha}}\left(k - \left(\frac{-\pi\Gamma(-2\alpha)\sin\left(\pi\alpha\right)}{\left[\Gamma(-\alpha)\sin\left(\frac{\pi}{2}\alpha\right)\right]^2}-1 \right)\right). \end{align}} More recently, \cite{Proc:Li_Hastie_NIPS07} proposed the {\em fractional power} estimator\vspace{-0.1in} {\small\begin{align}\notag &\hat{d}_{(\alpha),fp} = \left(\frac{1}{k}\frac{\sum_{j=1}^k|x_j|^{\lambda^* \alpha}}{\frac{2}{\pi} \Gamma(1-\lambda^*)\Gamma(\lambda^*\alpha)\sin\left(\frac{\pi}{2}\lambda^*\alpha\right)} \right)^{1/\lambda^*} \times \\\notag &\hspace{0.in} \left(1-\frac{1}{k}\frac{1}{2\lambda^*}\left(\frac{1}{\lambda^*}-1\right)\left(\frac{\frac{2}{\pi} \Gamma(1-2\lambda^*)\Gamma(2\lambda^*\alpha)\sin\left(\pi\lambda^*\alpha\right)}{\left[\frac{2}{\pi} \Gamma(1-\lambda^*)\Gamma(\lambda^*\alpha)\sin\left(\frac{\pi}{2}\lambda^*\alpha\right)\right]^2}-1 \right) \right), \end{align}} \noindent where\vspace{-0.2in} {\small\begin{align}\notag \lambda^* = \underset{-\frac{1}{2\alpha}\lambda<\frac{1}{2}}{\text{argmin}}\ \ \frac{1}{\lambda^2}\left(\frac{\frac{2}{\pi} \Gamma(1-2\lambda)\Gamma(2\lambda\alpha)\sin\left(\pi\lambda\alpha\right)}{\left[\frac{2}{\pi} \Gamma(1-\lambda)\Gamma(\lambda\alpha)\sin\left(\frac{\pi}{2}\lambda\alpha\right)\right]^2}-1 \right). \end{align}}\vspace{-0.1in} All three estimators are unbiased or asymptotically (as {\small$k\rightarrow\infty$}) unbiased. Figure \ref{fig_efficiency} compares their asymptotic variances in terms of the Cram\'er-Rao efficiency, which is the ratio of the smallest possible asymptotic variance over the asymptotic variance of the estimator, as {\small$k\rightarrow\infty$}. \begin{figure}[h]\vspace{-0.12in} \begin{center} \includegraphics[width = 2.5in]{efficiency_all.eps} \end{center} \vspace{-0.3in} \caption{The Cram\'er-Rao efficiencies (the higher the better, max = $100\%$) of various estimators, including the {\em optimal quantile} estimator proposed in this study. }\label{fig_efficiency}\vspace{-0.15in} \end{figure} The {\em geometric mean} estimator, $\hat{d}_{(\alpha),gm}$ exhibits tail bounds in exponential forms, i.e., the errors decrease exponentially fast: \vspace{-0.15in} {\small\begin{align}\notag \mathbf{Pr}\left(|\hat{d}_{(\alpha),gm} - d_{(\alpha)}| \geq \epsilon d_{(\alpha)}\right) \leq 2\exp\left(-k\frac{\epsilon^2}{G_{gm}}\right). \end{align}}\vspace{-0.15in} The {\em harmonic mean} estimator, $\hat{d}_{(\alpha),hm}$, works well for small $\alpha$, and has exponential tail bounds for $\alpha=0+$. The {\em fractional power} estimator, $\hat{d}_{(\alpha),fp}$, has smaller asymptotic variance than both the {\em geometric mean} and {\em harmonic mean} estimators. However, it does not have exponential tail bounds, due to the restriction {\small$-1<\lambda^*\alpha<\alpha$} in its definition. As shown in \cite{Proc:Li_Hastie_NIPS07}, it only has finite moments slightly higher than the $2nd$ order, when $\alpha$ approaches 2 (because {\small$\lambda^*\rightarrow 0.5$}), meaning that large errors may have a good chance to occur. We will demonstrate this by simulations. \vspace{-0.1in} \subsection{The Issue of Computational Efficiency} \vspace{-0.05in} In the definitions of $\hat{d}_{(\alpha),gm}$, $\hat{d}_{(\alpha),hm}$ and $\hat{d}_{(\alpha),fp}$, all three estimators require evaluating fractional powers, e.g., $|x_j|^{\alpha/k}$. This operation is relatively expensive, especially if we need to conduct this tens of billions of times (e.g., $n^2 =10^{10}$). For example, \cite{Article:Chapelle_99} reported that, although the radial basis kernel (\ref{eqn_rbs_kernel}) with $\alpha = 0.5$ achieved good performance, it was not preferred because evaluating the square root was too expensive. \vspace{-0.1in} \subsection{Our Proposed Estimator} \vspace{-0.05in} We propose the {\em optimal quantile} estimator, using the $q^*$th smallest {\small$|x_j|$: \begin{align}\label{eqn_d_oq} \hat{d}_{(\alpha),oq}\propto \left(q^*\text{-quantile}\{|x_j|,j=1, 2, ..., k\}\right)^\alpha, \end{align} where $q^*=q^*(\alpha)$} is chosen to minimize the asymptotic variance. This estimator is computationally attractive because \textbf{selecting} should be much less expensive than evaluating fractional powers. If we are interested in {\small$d_{(\alpha)}^{1/\alpha}$} instead, then we do not even need to evaluate any fractional powers. As mentioned, in many cases using either $d_{(\alpha)}$ or {\small$d_{(\alpha)}^{1/\alpha}$} makes no difference and $d_{(\alpha)}$ is often preferred because it avoids taking {\small$(.)^{1/\alpha}$} power. The radial basis kernel (\ref{eqn_rbs_kernel}) requires $d_{(\alpha)}$. Thus this study focuses on $d_{(\alpha)}$. On the other hand, if we can estimate {\small$d_{(\alpha)}^{1/\alpha}$} directly, for example, using (\ref{eqn_d_oq}) without the $\alpha$th power, we might as well just use {\small$d_{(\alpha)}^{1/\alpha}$} if permitted. In case we do not need to evaluate any fractional power, our estimator will be even more computationally efficient. In addition to the computational advantages, this estimator also has good theoretical properties, in terms of both the variances and tail probabilities: \begin{enumerate} \item Figure \ref{fig_efficiency} illustrates that, compared with the {\em geometric mean} estimator, its asymptotic variance is about the same when $\alpha <1$, and is considerably smaller when $\alpha>1$. Compared with the {\em fractional power} estimator, it has smaller asymptotic variance when $1<\alpha \leq 1.8$. In fact, as will be shown by simulations, when the sample size $k$ is not too large, its mean square errors are considerably smaller than the {\em fractional power} estimator when $\alpha >1$. \item The {\em optimal quantile} estimator exhibits tail bounds in exponential forms. This theoretical contribution is practically important, for selecting the sample size $k$. In learning theory, the generalization bounds are often loose. In our case, however, the bounds are tight because the distribution is specified. \end{enumerate} The next section will be devoted to analyzing the {\em optimal quantile} estimator. \vspace{-0.15in} \section{The Optimal Quantile Estimator} \vspace{-0.05in} Recall the goal is to estimate $d_{(\alpha)}$ from {\small$\{x_j\}_{j=1}^k$}, where {\small$x_j\sim S(\alpha,d_{(\alpha)})$}, i.i.d. Since the distribution belongs to the scale family, one can estimate the scale parameter from quantiles. Due to symmetry, it is natural to consider the absolute values:\vspace{-0.1in} {\small\begin{align}\label{eqn_quantile} \hat{d}_{(\alpha),q} = \left(\frac{q\text{-Quantile}\{|x_j|, j = 1, 2, ..., k\}}{q\text{-Quantile}\{|S(\alpha,1)|\}}\right)^\alpha, \end{align}\vspace{-0.1in} } \noindent which is best understood by the fact that if {\small$x \sim S(\alpha, 1)$}, then {\small$d^{1/\alpha}x \sim S(\alpha, d)$}, or more obviously, if $x\sim N(0,1)$, then {\small$\left(\sigma^2\right)^{1/2} x\sim N\left(0,\sigma^2\right)$}. By properties of order statistics \cite{Book:David}, any $q$-quantile will provide an asymptotically unbiased estimator. Lemma \ref{lem_var_q} provides the asymptotic variance of $\hat{d}_{(\alpha),q}$. \begin{lemma}\label{lem_var_q} Denote {\small$f_X\left(x; \alpha,d_{(\alpha)}\right)$} and {\small$F_X\left(x; \alpha,d_{(\alpha)}\right)$} the probability density function and the cumulative density function of {\small$X\sim S(\alpha, d_{(\alpha)})$}, respectively. The asymptotic variance of $\hat{d}_{(\alpha),q}$ defined in (\ref{eqn_quantile}) is {\small\begin{align}\label{eqn_var_q} \text{Var}\left(\hat{d}_{(\alpha),q}\right) =&\frac{1}{k}\frac{(q-q^2)\alpha^2/4}{f^2_X\left(W;\alpha,1\right) W^2 } d_{(\alpha)}^2+O\left(\frac{1}{k^2}\right) \end{align}} \noindent where {\small$W = F_X^{-1}\left((q+1)/2;\alpha,1\right)=q\text{-Quantile}\{|S(\alpha,1)|\}$}. \textbf{Proof:} See Appendix \ref{proof_lem_var_q}. $\Box$. \end{lemma} \vspace{-0.15in} \subsection{Optimal Quantile $q^*(\alpha)$} \vspace{-0.05in} We choose $q = q^*(\alpha)$ so that the asymptotic variance (\ref{eqn_var_q}) is minimized, i.e.,\vspace{-0.15in} {\small\begin{align}\label{eqn_g} q^*(\alpha) = \underset{q}{\text{argmin}} \ g(q;\alpha), \ \ \ \ \ \ g(q;\alpha) = \frac{q-q^2}{f^2_X\left(W;\alpha,1\right) W^2 }. \end{align}}\vspace{-0.1in} The convexity of $g(q;\alpha)$ is important. Graphically, $g(q;\alpha)$ is a convex function of $q$, i.e., a unique minimum exists. An algebraic proof, however, is difficult. Nevertheless, we can obtain analytical solutions when $\alpha = 1$ and $\alpha = 0+$. \begin{lemma}\label{lem_convexity} When $\alpha = 1$ or $\alpha = 0+$, the function $g(q;\alpha)$ defined in (\ref{eqn_g}) is a convex function of $q$. When $\alpha = 1$, the optimal $q^*(1) = 0.5$. When $\alpha =0+$, $q^*(0+)=0.203$ is the solution to $-\log q^* + 2q^* -2 = 0$. \textbf{Proof:} See Appendix \ref{proof_lem_convexity}. $\Box$. \end{lemma} It is also easy to show that when $\alpha = 2$, $q^*(2) = 0.862$. We denote the {\em optimal quantile} estimator by $\hat{d}_{(\alpha),oq}$, which is same as $\hat{d}_{(\alpha),q^*}$. For general $\alpha$, we resort to numerical solutions, as presented in Figure \ref{fig_opt_quantile}. \vspace{-0.2in} \begin{figure}[h] \begin{center} \mbox{\hspace{-0.in} \subfigure[$q^*$]{\includegraphics[height = 1.5in]{opt_quantile.eps}}\hspace{0.2in} \subfigure[$W^\alpha(q^*)$]{\includegraphics[height = 1.5in]{opt_q_const.eps}} } \end{center} \vspace{-0.3in} \caption{(a) The optimal values for $q^*(\alpha)$, which minimizes asymptotic variance of $\hat{d}_{(\alpha),q}$, i.e., the solution to (\ref{eqn_g}). (b) The constant $W^\alpha(q^*) = \{q^*\text{-quantile}\{|S(\alpha,1)|\}\}^\alpha$. }\label{fig_opt_quantile}\vspace{-0.2in} \end{figure} \vspace{-0.15in} \subsection{Bias Correction} \vspace{-0.05in} Although $\hat{d}_{(\alpha),oq}$ (i.e., $\hat{d}_{(\alpha),q^*}$) is asymptotically (as $k\rightarrow\infty$) unbiased, it is seriously biased for small $k$. Thus, it is practically important to remove the bias. The unbiased version of the {\em optimal quantile} estimator is\vspace{-0.2in} {\small \begin{align}\label{eqn_oqc} \hat{d}_{(\alpha),oq,c} = \hat{d}_{(\alpha),oq}/B_{\alpha,k}, \end{align}}\vspace{-0.2in} \noindent where $B_{\alpha,k}$ is the expectation of $\hat{d}_{(\alpha),oq}$ at $d_{(\alpha)} =1$. For $\alpha = 1$, $0+$, or $2$, we can evaluate the expectations (i.e., integrals) analytically or by numerical integrations. For general $\alpha$, as the probability density is not available, the task is difficult and prone to numerical instability. On the other hand, since the Monte-Carlo simulation is a popular alternative for evaluating difficult integrals, a practical solution is to simulate the expectations, as presented in Figure \ref{fig_bias}. \begin{figure}[h]\vspace{-0.2in} \begin{center} \mbox{\hspace{-0.2in} {\includegraphics[height = 1.5in]{bias_p.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{bias_k.eps}} } \end{center} \vspace{-0.3in} \caption{The bias correction factor $B_{\alpha,k}$ in (\ref{eqn_oqc}), obtained from $10^8$ simulations for every combination of $\alpha$ (spaced at 0.05) and $k$. {\small$B_{\alpha,k} = \text{E}\left(\hat{d}_{(\alpha),oq}; d_{(\alpha)}=1\right)$}. }\label{fig_bias} \end{figure} Figure \ref{fig_bias} illustrates that $B_{\alpha,k}>1$, meaning that this correction also reduces variance while removing bias (because {\small $\text{Var}(x/c)= \text{Var}(x)/c^2$}). For example, when $\alpha = 0.1$ and $k = 10$, $B_{\alpha,k}\approx 1.24$, which is significant, because $1.24^2 = 1.54$ implies a $54\%$ difference in terms of variance, and even more considerable in terms of the mean square errors MSE = variance + bias$^2$. $B_{\alpha,k}$ can be tabulated for small $k$, and absorbed into other coefficients, i.e., this does not increase the computational cost at run time. We fix $B_{\alpha,k}$ as reported in Figure \ref{fig_bias}. The simulations in Section \ref{sec_simulations} directly used those fixed $B_{\alpha,k}$ values. \vspace{-0.1in} \subsection{Computational Efficiency} \vspace{-0.05in} Figure \ref{fig_compu_ratio} compares the computational costs of the {\em geometric mean}, the {\em fractional power}, and the {\em optimal quantile} estimators. The {\em harmonic mean} estimator was not included as it costs very similarly to the {\em fractional power} estimator. We used the build-in function ``pow''in gcc for evaluating the fractional powers. We implemented a ``quick select'' algorithm, which is similar to quick sort and requires on average linear time. For simplicity, our implementation used recursions and the middle element as pivot. Also, to ensure fairness, for all estimators, coefficients which are functions of $\alpha$ and/or $k$ were pre-computed. \vspace{-0.2in} \begin{figure}[h] \begin{center} \mbox{\hspace{-0.in} {\includegraphics[height = 1.5in]{compu_ratio_p.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{compu_ratio_k.eps}} } \end{center} \vspace{-0.3in} \caption{Relative computational cost ({\small$\hat{d}_{(\alpha),gm}$} over {\small$\hat{d}_{(\alpha),oq,c}$} and {\small$\hat{d}_{(\alpha),gm}$} over {\small$\hat{d}_{(\alpha),fp}$}), from $10^6$ simulations at each combination of $\alpha$ and $k$. The left panel averages over all $k$ and the right panel averages over all $\alpha$. Note that the cost of {\small$\hat{d}_{(\alpha),oq,c}$} includes evaluating the $\alpha$th moment once. }\label{fig_compu_ratio}\vspace{-0.15in} \end{figure} Normalized by the computing time of {\small$\hat{d}_{(\alpha),gm}$}, we observe that relative computational efficiency does not strongly depend on $\alpha$. We do observe that the ratio of computing time of {\small$\hat{d}_{(\alpha),gm}$} over that of {\small$\hat{d}_{(\alpha),oq,c}$} increases consistently with increasing $k$. This is because in the definition of {\small$\hat{d}_{(\alpha),oq}$} (and hence also {\small$\hat{d}_{(\alpha),oq,c}$}), it is required to evaluate the fractional power once, which contributes to the total computing time more significantly at smaller $k$. Figure \ref{fig_compu_ratio} illustrates that, (A) the {\em geometric mean} estimator and the {\em fractional power} estimator are similar in terms of computational efficiency; (B) the {\em optimal quantile} estimator is nearly one order of magnitude more computationally efficient than the {\em geometric mean} and {\em fractional power} estimators. Because we implemented a ``na\'ive'' version of ``quick select'' using recursions and simple pivoting, the actual improvement may be more significant. Also, if applications require only {\small$d_{(\alpha)}^{1/\alpha}$}, then no fractional power operations are needed for {\small$\hat{d}_{(\alpha),oq,c}$} and the improvement will be even more considerable. \vspace{-0.1in} \subsection{Error (Tail) Bounds} Error (tail) bounds are essential for determining $k$. The variance alone is not sufficient for that purpose. If an estimator of $d$, say $\hat{d}$, is normally distributed, {\small$\hat{d} \sim N\left(d, \frac{1}{k}V\right)$}, the variance suffices for choosing $k$ because its error (tail) probability {\small $\mathbf{Pr}\left(|\hat{d}-d|\geq \epsilon d\right) \leq 2\exp\left(-k\frac{\epsilon^2}{2V}\right)$} is determined by $V$. In general, a reasonable estimator will be asymptotically normal, for small enough $\epsilon$ and large enough $k$. For a finite $k$ and a fixed $\epsilon$, however, the normal approximation may be (very) poor. This is especially true for the {\em fractional power} estimator, {\small$\hat{d}_{(\alpha),fp}$.} Thus, for a good motivation, Lemma \ref{lem_bounds} provides the error (tail) probability bounds of $\hat{d}_{(\alpha),q}$ for any $q$, not just the optimal quantile $q^*$. \begin{lemma}\label{lem_bounds} Denote {\small$X \sim S(\alpha,d_{(\alpha)})$} and its probability density function by {\small$f_X(x;\alpha,d_{(\alpha)})$} and cumulative function by {\small$F_X(x;\alpha,d_{(\alpha)})$}. Given {\small$x_j\sim S(\alpha,d_{(\alpha)})$}, i.i.d., $j = 1$ to $k$. Using {\small$\hat{d}_{(\alpha),q}$} in (\ref{eqn_quantile}), then\vspace{-0.15in} {\small\begin{align} &\mathbf{Pr}\left( \hat{d}_{(\alpha),q} \geq(1+\epsilon) d_{(\alpha)}\right) \leq \exp\left(-k\frac{\epsilon^2}{G_{R,q}}\right), \hspace{0.in} \epsilon >0, \\ &\mathbf{Pr}\left( \hat{d}_{(\alpha),q} \leq(1-\epsilon) d_{(\alpha)}\right) \leq \exp\left(-k\frac{\epsilon^2}{G_{L,q}}\right), \hspace{0.in} 0< \epsilon <1, \end{align}}\vspace{-0.4in} {\small\begin{align}\label{eqn_G_R} &\frac{\epsilon^2}{G_{R,q}} = -(1-q)\log\left(2-2F_R\right) - q \log(2F_R-1) + (1-q)\log (1-q) + q\log q, \\\label{eqn_G_L} &\frac{\epsilon^2}{G_{L,q}} = -(1-q)\log\left(2-2F_L\right)- q \log (2F_L-1)+ (1-q)\log (1-q) + q\log q,\\\notag\vspace{0.1in} &W = F_X^{-1}((q+1)/2;\alpha,1) = q\text{-quantile}\{|S(\alpha,1)|\},\\\notag &F_R = F_X\left((1+\epsilon)^{1/\alpha}W;\alpha,1\right), \hspace{0.2in} F_L = F_X\left((1-\epsilon)^{1/\alpha}W;\alpha,1\right). \end{align}}\vspace{-0.15in} As $\epsilon\rightarrow 0+$\vspace{-0.1in} {\small\begin{align}\label{eqn_G_RL_limit} &\underset{\epsilon\rightarrow0+}{\lim} G_{R,q} = \underset{\epsilon\rightarrow0+}{\lim} G_{L,q} = \frac{q(1-q)\alpha^2/2}{f_X^2\left(W;\alpha,1\right)W^2}. \end{align}} \textbf{Proof:} \ \ See Appendix \ref{proof_lem_bounds}. $\Box$ \end{lemma} The limit in (\ref{eqn_G_RL_limit}) as $\epsilon\rightarrow 0$ is precisely twice the asymptotic variance factor of $\hat{d}_{(\alpha),q}$ in (\ref{eqn_var_q}), consistent with the normality approximation mentioned previously. This explains why we express the constants as {\small$\epsilon^2/G$}. (\ref{eqn_G_RL_limit}) also indicates that the tail bounds achieve the ``optimal rate'' for this estimator, in the language of large deviation theory. By the Bonferroni bound, it is easy to determine the sample size $k$ {\small\begin{align}\notag\vspace{-0.15in} &\mathbf{Pr}\left(|\hat{d}_{(\alpha),q} - d_{(\alpha)}|\geq \epsilon d_{(\alpha)}\right) \leq 2 \exp\left(-k\frac{\epsilon^2}{G}\right) \leq \delta/(n^2/2 \Longrightarrow k \geq \frac{G}{\epsilon^2}\left(2\log n - \log \delta\right). \end{align}} \vspace{-0.2in} \begin{lemma}\label{lem_JL_quantile} Using $\hat{d}_{(\alpha),q}$ with {\small$k\geq \frac{G}{\epsilon^2}\left(2\log n - \log \delta\right)$}, any pairwise $l_\alpha$ distance among $n$ points can be approximated within a $1\pm\epsilon$ factor with probability $\geq1-\delta$. It suffices to let {\small$G = \max\{G_{R, q}, G_{L, q}\}$}, where {\small$G_{R, q}$, $G_{L, q}$} are defined in Lemma \ref{lem_bounds}. \end{lemma} The Bonferroni bound can be unnecessarily conservative. It is often reasonable to replace {\small$\delta/(n^2/2)$} by {\small$\delta/T$}, meaning that except for a $1/T$ fraction of pairs, any distance can be approximated within a $1\pm\epsilon$ factor with probability $1-\delta$. Figure \ref{fig_bounds} plots the error bound constants for $\epsilon<1$, for both the recommended {\em optimal quantile} estimator $\hat{d}_{(\alpha),oq}$ and the baseline {\em sample median} estimator {\small$\hat{d}_{(\alpha),q=0.5}$}. Although we choose {\small$\hat{d}_{(\alpha),oq}$} based on the asymptotic variance, it turns out {\small$\hat{d}_{(\alpha),oq}$} also exhibits (much) better tail behaviors (i.e., smaller constants) than {\small$\hat{d}_{(\alpha),q=0.5}$}, at least in the range of $\epsilon<1$. \begin{figure}[h]\vspace{-0.2in} \begin{center} \mbox{\hspace{-0.in} {\includegraphics[height = 1.5in]{G_R_opt.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{G_L_opt.eps}} }\vspace{-0.1in} \mbox{\hspace{-0.in} {\includegraphics[height = 1.5in]{G_R_me.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{G_L_me.eps}} } \end{center} \vspace{-0.3in} \caption{Tail bound constants for quantile estimators; the lower the better. Upper panels: optimal quantile estimators $\hat{d}_{(\alpha),q^*}$. Lower panels: median estimators $\hat{d}_{(\alpha),q=0.5}$. }\label{fig_bounds}\vspace{-0.2in} \end{figure} Consider {\small$k = \frac{G}{\epsilon^2}\left(\log 2T - \log \delta\right)$} (recall we suggest replacing $n^2/2$ by $T$), with $\delta = 0.05$, $\epsilon = 0.5$, and $T = 10$. Because $G_{R,q^*}\approx 5\sim 9$ around $\epsilon=0.5$, we obtain $k \approx 120\sim215$, which is still a relatively large number (although the original dimension $D$ might be $10^6$). If we choose $\epsilon = 1$, then approximately $k \approx 40\sim 65$. It is possible $k = 120\sim 215$ might be still conservative, for three reasons: (A) the tail bounds, although ``sharp," are still upper bounds; (B) using $G = \max\{G_{R,q^*}, G_{L,q^*}\}$ is conservative because $G_{L,q^*}$ is usually much smaller than $G_{R,q^*}$; (C) this type of tail bounds is based on relative error, which may be stringent for small ($\approx 0$) distances. In fact, some earlier studies on {\em normal random projections} (i.e., $\alpha =2$) \cite{Proc:Bingham_kdd01,Proc:Fradkin_KDD03} empirically demonstrated that $k\geq50$ appeared sufficient. \vspace{-0.15in} \section{Simulations}\label{sec_simulations} \vspace{-0.05in} We resort to simulations for comparing the finite sample variances of various estimators and assessing the more precise error (tail) probabilities. One advantage of {\em stable random projections} is that we know the (manually generated) distributions and the only source of errors is from the random number generations. Thus, we can simply rely on simulations to evaluate the estimators without using real data. In fact, after projections, the projected data follow exactly the stable distribution, regardless of the original real data distribution. Without loss of generality, we simulate samples from $S(\alpha, 1)$ and estimate the scale parameter (i.e., 1) from the samples. Repeating the procedure $10^7$ times, we can reliably evaluate the mean square errors (MSE) and tail probabilities. \vspace{-0.15in} \subsection{Mean Square Errors (MSE)} \vspace{-0.05in} As illustrated in Figure \ref{fig_simu_mse}, in terms of the MSE, the {\em optimal quantile} estimator $\hat{d}_{(\alpha),oq,c}$ outperforms both the {\em geometric mean} and {\em fractional power} estimators when $\alpha >1$ and $k\geq 20$. The {\em fractional power} estimator does not appear to be very suitable for $\alpha >1$, especially for $\alpha$ close to 2, even when the sample size $k$ is not too small (e.g., $k = 50$). For $\alpha <1$, however, the {\em fractional power} estimator has good performance in terms of MSE, even for small $k$. \begin{figure}[h] \begin{center} \mbox{\hspace{-0.in} {\includegraphics[height = 1.5in]{mse_k10.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{mse_k20.eps}} }\vspace{-0.1in} \mbox{\hspace{-0.in} {\includegraphics[height = 1.5in]{mse_k50.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{mse_k100.eps}} } \end{center} \vspace{-0.3in} \caption{Empirical mean square errors (MSE, the lower the better), from $10^7$ simulations at every combination of $\alpha$ and $k$. The values are multiplied by $k$ so that four plots can be at about the same scale. The MSE for the {\em geometric mean} (gm) estimator is computed exactly since closed-form expression exists. The lower dashed curves are the asymptotic variances of the {\em optimal quantile} (oq) estimator. }\label{fig_simu_mse}\vspace{-0.2in} \end{figure} \vspace{-0.2in} \subsection{Error(Tail) Probabilities} \vspace{-0.05in} Figure \ref{fig_simu_tail} presents the simulated right tail probabilities, {\small$\mathbf{Pr}\left(\hat{d}_{(\alpha)} \geq (1+\epsilon)d_{(\alpha)}\right)$}, illustrating that when $\alpha>1$, the {\em fractional power} estimator can exhibit very bad tail behaviors. For $\alpha<1$, the {\em fractional power} estimator demonstrates good performance at least for the probability range in the simulations. Thus, Figure \ref{fig_simu_tail} demonstrates that the {\em optimal quantile} estimator consistently outperforms the {\em fractional power} and the {\em geometric mean} estimators when $\alpha >1$. \begin{figure}[h] \begin{center} \mbox {\includegraphics[height = 1.5in]{tail_R_50.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{tail_R_150.eps}} }\vspace{-0.1in} \mbox{\hspace{-0.1in} {\includegraphics[height = 1.5in]{tail_R_175.eps}}\hspace{0.2in} {\includegraphics[height = 1.5in]{tail_R_195.eps}} } \end{center} \vspace{-0.3in} \caption{The right tail probabilities (the lower the better), from $10^7$ simulations at each combination of $\alpha$ and $k$. }\label{fig_simu_tail}\vspace{-0.2in} \end{figure} \vspace{-0.1in} \section{The Related Work} \vspace{-0.05in} There have been many studies of {\em normal random projections} in machine learning, for dimension reduction in the $l_2$ norm, e.g., \cite{Proc:Fradkin_KDD03}, highlighted by the Johnson-Lindenstrauss (JL) Lemma \cite{Article:JL84}, which says $k = O\left(\log n/\epsilon^2\right)$ suffices when using normal (or normal-like, e.g., \cite{Article:Achlioptas_JCSS03}) projection methods. The method of {\em stable random projections} is applicable for computing the $l_\alpha$ distances ($0<\alpha \leq 2$), not just for $l_2$. \cite[Lemma 1, Lemma 2, Theorem 3]{Article:Indyk_JACM06} suggested the {\em median} (i.e., $q = 0.5$ quantile) estimator for $\alpha =1$ and argued that the sample complexity bound should be $O\left(1/\epsilon^2\right)$ ($n = 1$ in their study). Their bound was not provided in an explicit form and required an ``$\epsilon$ is small enough'' argument. For $\alpha \neq 1$, \cite[Lemma 4]{Article:Indyk_JACM06} only provided a conceptual algorithm, which ``is not uniform.'' In this study, we prove the bounds for any $q$-quantile and any $0<\alpha\leq 2$ (not just $\alpha=1$), in explicit exponential forms, with no unknown constants and no restriction that ``$\epsilon$ is small enough.'' The quantile estimator for stable distributions was proposed in statistics quite some time ago, e.g., \cite{Article:Fama_71,Article:McCulloch_86}. \cite{Article:Fama_71} mainly focused on $1\leq \alpha \leq 2$ and recommended using $q=0.44$ quantiles (mainly for the sake of smaller bias). \cite{Article:McCulloch_86} focused on $0.6\leq \alpha \leq 2$ and recommended $q=0.5$ quantiles This study considers all $0<\alpha \leq 2$ and recommends $q$ based on the minimum asymptotic variance. Because the bias can be easily removed (at least in the practical sense), it appears not necessary to use other quantiles only for the sake of smaller bias. Tail bounds, which are useful for choosing $q$ and $k$ based on confidence intervals, were not available in \cite{Article:Fama_71,Article:McCulloch_86}. Finally, one might ask if there might be better estimators. For $\alpha = 1$, \cite{Article:Chernoff_67} proposed using a linear combination of quantiles (with carefully chosen coefficients) to obtain an asymptotically optimal estimator for the Cauchy scale parameter. While it is possible to extend their result to general $0<\alpha<2$ (requiring some non-trivial work), whether or not it will be practically better than the {\em optimal quantile} estimator is unclear because the extreme quantiles severely affect the tail probabilities and finite-sample variances and hence some kind of truncation (i.e., discarding some samples at extreme quantiles) is necessary. Also, exponential tail bounds of the linear combination of quantiles may not exist or may not be feasible to derive. In addition, the {\em optimal quantile} estimator is computationally more efficient. \vspace{-0.15in} \section{Conclusion} \vspace{-0.05in} Many machine learning algorithms operate on the training data only through pairwise distances. Computing, storing, updating and retrieving the ``matrix'' of pairwise distances is challenging in applications involving massive, high-dimensional, and possibly streaming, data. For example, the pairwise distance matrix can not fit in memory when the number of observations exceeds $10^6$ (or even $10^5$). The method of {\em stable random projections} provides an efficient mechanism for computing pairwise distances using low memory, by transforming the original high-dimensional data into {\em sketches}, i.e., a small number of samples from $\alpha$-stable distributions, which are much easier to store and retrieve. This method provides a uniform scheme for computing the $l_\alpha$ pairwise distances for all $0<\alpha\leq 2$. Choosing an appropriate $\alpha$ is often critical to the performance of learning algorithms. In principle, we can tune algorithms for many $l_\alpha$ distances; and {\em stable random projections} can provide an efficient tool. To recover the original distances, we face an estimation task. Compared with previous estimators based on the {\em geometric mean}, {\em the harmonic mean}, or the {\em fractional power}, the proposed {\em optimal quantile} estimator exhibits two advantages. Firstly, the {\em optimal quantile} estimator is nearly one order of magnitude more efficient than other estimators (e.g., reducing the training time from one week to one day). Secondly, the {\em optimal quantile} estimator is considerably more accurate when $\alpha >1$, in terms of both the variances and error (tail) probabilities. Note that $\alpha \geq1$ corresponds to a convex norm (satisfying the triangle inequality), which might be another motivation for using $l_\alpha$ distances with $\alpha \geq1$. One theoretical contribution is the explicit tail bounds for general quantile estimators and consequently the sample complexity bound $k = O\left(\log n/\epsilon^2\right)$. Those bounds may guide practitioners in choosing $k$, the number of projections. The (practically useful) bounds are expressed in terms of the probability functions and hence they might be not as convenient for further theoretical analysis. Also, we should mention that the bounds do not recover the optimal bound of the {\em arithmetic mean} estimator when $\alpha = 2$, because the {\em arithmetic mean} estimator is statistically optimal at $\alpha =2$ but the {\em optimal quantile} estimator is not. While we believe that applying {\em stable random projections} in machine learning has become straightforward, there are interesting theoretical issues for future research. For example, how theoretical properties of learning algorithms may be affected if the approximated (instead of exact) $l_\alpha$ distances are used?

BkiUfOk5qhDCZ_S7ea24

\section{Introduction and preliminaries} In this paper we consider a Brownian motion with drift, and we present some asymptotic results for the last zero crossing time (in a time interval $[0,t]$) as the drift tends to infinity. In our proofs we handle some formulas presented in \cite{IafrateOrsingher}, and we refer to the theory of large deviations (see \cite{DemboZeitouni} as a reference on this topic). We recall some results in \cite{IafrateOrsingher}. We consider a Brownian motion (starting at the origin at time zero), with drift $\mu\neq 0$. Moreover, for some $t>0$, we consider the last zero crossing time $T_{\mu,t}$ of this Brownian motion in the time interval $[0,t]$. Then (see Theorem 2.1 in \cite{IafrateOrsingher}) the distribution function of $T_{\mu,t}$ is defined by \begin{equation}\label{eq:distribution-function} P(T_{\mu,t}\leq a)=1-\frac{2}{\pi}\int_0^{\sqrt{\frac{t-a}{a}}}\frac{e^{-\frac{\mu^2}{2}a(1+y^2)}}{1+y^2}dy\ (\mbox{for}\ a\in[0,t]), \end{equation} and its density is \begin{equation}\label{eq:density} f_{\mu,t}(a):=\left(\frac{e^{-\frac{\mu^2}{2}t}}{\pi\sqrt{a(t-a)}} +\frac{\mu^2}{2\pi}\int_a^t\frac{e^{-\frac{\mu^2}{2}y}}{\sqrt{a(y-a)}}dy\right)1_{[0,t]}(a). \end{equation} In this paper we also consider the probability that this Brownian motion crosses zero in the time interval $[a,b]$ for some $b>a>0$ (see the proof of Theorem 3.1 in \cite{IafrateOrsingher}): \begin{equation}\label{eq:crossing-probability} \Psi_{[a,b]}(\mu)=\frac{1}{\pi}\int_0^{b-a}\frac{e^{-\frac{\mu^2}{2}(a+s)}}{a+s}\sqrt{\frac{a}{s}}ds. \end{equation} We also recall the concept of large deviation principle (LDP for short) for a family of random variables $\{W(r):r>0\}$ defined on the same probability space $(\Omega,\mathcal{F},P)$. In view of what follows we assume that these random variables are real valued. Then $\{W(r):r>0\}$ satisfies the large deviation principle (LDP from now on) with speed $v_r$ and rate function $I$ if the following conditions hold: $\lim_{r\to\infty}v_r=\infty$, the function $I:\mathbb{R}\to [0,\infty]$ is lower semi-continuous; \begin{equation}\label{eq:UB} \limsup_{r\to\infty}\frac{1}{v_r}\log P(W(r)\in C)\leq-\inf_{w\in C}I(w)\ \mbox{for all closed sets}\ C; \end{equation} \begin{equation}\label{eq:LB} \liminf_{r\to\infty}\frac{1}{v_r}\log P(W(r)\in G)\geq-\inf_{w\in G}I(w)\ \mbox{for all open sets}\ G. \end{equation} Moreover, a rate function $I$ is said to be good if all its level sets $\{\{w\in\mathbb{R}:I(w)\leq\eta\}:\eta\geq 0\}$ are compact. Now we present a list of the results proved in this paper. \begin{itemize} \item The LDP for $W(r)=T_{\mu\sqrt{r},t}$ with $v_r=r$ and a rate function $J$ (Proposition \ref{prop:LD}). \item For every choice of positive numbers $\{\gamma_r:r>0\}$ such that \begin{equation}\label{eq:conditions-MD} \lim_{r\to\infty}\gamma_r=0\ \mbox{and}\ \lim_{r\to\infty}r\gamma_r=\infty, \end{equation} the LDP for $W(r)=r\gamma_rT_{\mu\sqrt{r},t}$ with $v_r=1/\gamma_r$ and the same rate function $\tilde{J}$ (Proposition \ref{prop:MD}). It will be explained that this class of LDPs is inspired by the results in the literature on moderate deviations. \item We study the asymptotic behavior, as $r\to\infty$, for the probability in \eqref{eq:crossing-probability} with $\mu\sqrt{r}$ in place of $\mu$ (Proposition \ref{prop:crossing-probability}). We also highlight some similar aspects with asymptotic estimates in insurance literature (see Remark \ref{rem:analogies-with-ruin-probability}). \end{itemize} Throughout the paper, we consider the notation $$\varphi(x;\mu,\sigma^2)=\frac{e^{-\frac{(x-\mu)^2}{2\sigma^2}}}{\sqrt{2\pi\sigma^2}}$$ for the well-known density of a Normal distribution with mean $\mu$ and variance $\sigma^2$. We conclude with the outline of the paper. In Section \ref{sec:results}, we prove the results. Finally, in Section \ref{sec:MD-comments}, we compare some aspects of the classical moderate deviation results, and the results in this paper; in particular we discuss some common features and differences. \section{Results}\label{sec:results} We start with the first result, i.e. a LDP with $v_r=r$. We remark that, since we deal with random variables $\{W(r):r>0\}$ which take values on a compact interval $K$ of the real line (i.e. $K=[0,t]$), we can refer to a useful consequence of Theorem 4.1.11 in \cite{DemboZeitouni}. In fact, we can say that we have the LDP if the two following conditions hold for all $w\in\mathbb{R}$: \begin{equation}\label{eq:forLB} \lim_{\varepsilon\to 0}\liminf_{r\to\infty}\frac{1}{r}\log P(W(r)\in (w-\varepsilon,w+\varepsilon))\geq-I(w); \end{equation} \begin{equation}\label{eq:forUB} \lim_{\varepsilon\to 0}\limsup_{r\to\infty}\frac{1}{r}\log P(W(r)\in (w-\varepsilon,w+\varepsilon))\leq-I(w). \end{equation} Actually, when $w\notin K$, these two bounds can be easily checked with $I(w)=\infty$ (\eqref{eq:forLB} trivially holds; moreover, if we take $\varepsilon>0$ small enough to have $(w-\varepsilon,w+\varepsilon)\cap K=\emptyset$, we have $P(W(r)\in(w-\varepsilon,w+\varepsilon))=0$ for all $r>0$, which yields \eqref{eq:forUB}); so we can consider only the case $w\in K$. \begin{Proposition}\label{prop:LD} The family of random variables $\{T_{\mu\sqrt{r},t}:r>0\}$ satisfies the LDP with speed $v_r=r$ and good rate function $J$ defined by $$J(b):=\left\{\begin{array}{ll} \frac{\mu^2}{2}b&\ if\ b\in[0,t]\\ \infty&\ otherwise. \end{array}\right.$$ \end{Proposition} \begin{proof} We prove the LDP by referring to the consequence of Theorem 4.1.11 in \cite{DemboZeitouni} recalled above; so, for all $b\in [0,t]$ (here $[0,t]$ plays the role of compact interval $K$), we have to check \eqref{eq:forLB} and \eqref{eq:forUB} with $T_{\mu\sqrt{r},t}$ and $J(b)$ in place of $W(r)$ and $I(w)$, respectively. We have three different cases. \paragraph{Case $b\in(0,t)$.} Without loss of generality we can take $\varepsilon>0$ small enough to have $(b-\varepsilon,b+\varepsilon)\subset(0,t)$. Then, there exists $\tilde{a}_r=\tilde{a}_r(\varepsilon,b)\in(b-\varepsilon,b+\varepsilon)$ such that \begin{equation}\label{eq:mean-value-theorem} P(T_{\mu\sqrt{r},t}\in(b-\varepsilon,b+\varepsilon))=\int_{b-\varepsilon}^{b+\varepsilon}f_{\mu\sqrt{r},t}(a)da =f_{\mu\sqrt{r},t}(\tilde{a}_r)2\varepsilon. \end{equation} For the proof of \eqref{eq:forLB} we take into account \eqref{eq:density}, and after some manipulations, we get \begin{multline*} f_{\mu\sqrt{r},t}(\tilde{a}_r)\geq\frac{1}{\pi\sqrt{(b+\varepsilon)(t-(b-\varepsilon))}}\int_{\tilde{a}_r}^t\frac{\mu^2r}{2}e^{-\frac{\mu^2r}{2}y}dy\\ =\frac{e^{-\frac{\mu^2r}{2}\tilde{a}_r}-e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{(b+\varepsilon)(t-(b-\varepsilon))}} =\frac{e^{-\frac{\mu^2r}{2}\tilde{a}_r}(1-e^{-\frac{\mu^2r}{2}(t-\tilde{a}_r)})}{\pi\sqrt{(b+\varepsilon)(t-(b-\varepsilon))}} \geq\frac{e^{-\frac{\mu^2r}{2}(b+\varepsilon)}(1-e^{-\frac{\mu^2r}{2}(t-(b+\varepsilon))})}{\pi\sqrt{(b+\varepsilon)(t-(b-\varepsilon))}}; \end{multline*} thus, by \eqref{eq:mean-value-theorem}, we have $$\liminf_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (b-\varepsilon,b+\varepsilon))\geq-\frac{\mu^2}{2}(b+\varepsilon),$$ and therefore $$\lim_{\varepsilon\to 0}\liminf_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (b-\varepsilon,b+\varepsilon))\geq-\frac{\mu^2}{2}b=-J(b).$$ For the proof of \eqref{eq:forUB} we take into account \eqref{eq:density} and, after some manipulations, we get \begin{multline*} f_{\mu\sqrt{r},t}(\tilde{a}_r)\leq \frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{(b-\varepsilon)(t-(b+\varepsilon))}} +\frac{\mu^2r}{2\pi}\int_{\tilde{a}_r}^t\frac{e^{-\frac{\mu^2r}{2}y}}{\sqrt{\tilde{a}_r(y-\tilde{a}_r)}}dy\\ \leq\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{(b-\varepsilon)(t-(b+\varepsilon))}} +\frac{\mu^2r}{2\pi}\frac{e^{-\frac{\mu^2r}{2}\tilde{a}_r}}{\sqrt{b-\varepsilon}}\int_{\tilde{a}_r}^t\frac{1}{\sqrt{y-\tilde{a}_r}}dy\\ \leq\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{(b-\varepsilon)(t-(b+\varepsilon))}} +\frac{\mu^2r}{2\pi}\frac{e^{-\frac{\mu^2r}{2}(b-\varepsilon)}}{\sqrt{b-\varepsilon}}2\sqrt{t-(b-\varepsilon)}; \end{multline*} thus, by \eqref{eq:mean-value-theorem} and by Lemma 1.2.15 in \cite{DemboZeitouni}, we have $$\limsup_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (b-\varepsilon,b+\varepsilon)) \leq\max\left\{-\frac{\mu^2}{2}t,-\frac{\mu^2}{2}(b-\varepsilon)\right\}=-\frac{\mu^2}{2}(b-\varepsilon),$$ and therefore $$\lim_{\varepsilon\to 0}\limsup_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (b-\varepsilon,b+\varepsilon))\leq-\frac{\mu^2}{2}b=-J(b).$$ \paragraph{Case $b=0$.} We can take $\varepsilon>0$ small enough to have $(0,\varepsilon)\subset(0,t)$. Then, there exists $\tilde{a}_r=\tilde{a}_r(\varepsilon)\in(0,\varepsilon)$ such that \begin{equation}\label{eq:mean-value-theorem-b=0} P(T_{\mu\sqrt{r},t}\in(0,\varepsilon))=\int_0^\varepsilon f_{\mu\sqrt{r},t}(a)da =f_{\mu\sqrt{r},t}(\tilde{a}_r)\varepsilon. \end{equation} For the proof of \eqref{eq:forLB} we take into account \eqref{eq:density}, and after some manipulations, we get \begin{multline*} f_{\mu\sqrt{r},t}(\tilde{a}_r)\geq\frac{1}{\pi\sqrt{\varepsilon t}}\int_{\tilde{a}_r}^t\frac{\mu^2r}{2}e^{-\frac{\mu^2r}{2}y}dy\\ =\frac{e^{-\frac{\mu^2r}{2}\tilde{a}_r}-e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{\varepsilon t}} =\frac{e^{-\frac{\mu^2r}{2}\tilde{a}_r}(1-e^{-\frac{\mu^2r}{2}(t-\tilde{a}_r)})}{\pi\sqrt{\varepsilon t}} \geq\frac{e^{-\frac{\mu^2r}{2}\varepsilon}(1-e^{-\frac{\mu^2r}{2}(t-\varepsilon)})}{\pi\sqrt{\varepsilon t}}; \end{multline*} thus, by \eqref{eq:mean-value-theorem-b=0}, we have $$\liminf_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (0,\varepsilon))\geq-\frac{\mu^2}{2}\varepsilon,$$ and therefore, since $P(T_{\mu\sqrt{r},t}\in (0,\varepsilon))=P(T_{\mu\sqrt{r},t}\in (-\varepsilon,\varepsilon))$, $$\lim_{\varepsilon\to 0}\liminf_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (-\varepsilon,\varepsilon))\geq 0=-J(0).$$ The proof of \eqref{eq:forUB} is immediate; in fact we have $$\lim_{\varepsilon\to 0}\limsup_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (-\varepsilon,\varepsilon))\leq 0=-J(0),$$ noting that $\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (-\varepsilon,\varepsilon))\leq 0$ for all $r>0$. \paragraph{Case $b=t$.} We can take $\varepsilon>0$ small enough to have $(t-\varepsilon,t)\subset(0,t)$. Then, by \eqref{eq:density}, we have \begin{equation}\label{eq:starting-point-b=t} P(T_{\mu\sqrt{r},t}\in(t-\varepsilon,t))=\int_{t-\varepsilon}^tf_{\mu\sqrt{r},t}(a)da =\int_{t-\varepsilon}^t\left(\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{a(t-a)}} +\frac{\mu^2r}{2\pi}\int_a^t\frac{e^{-\frac{\mu^2r}{2}y}}{\sqrt{a(y-a)}}dy\right)da. \end{equation} For the proof of \eqref{eq:forLB} we take into account \eqref{eq:starting-point-b=t}, and after some manipulations, we get $$P(T_{\mu\sqrt{r},t}\in(t-\varepsilon,t))\geq\int_{t-\varepsilon}^t\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{a(t-a)}}da \geq\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{t}}\int_{t-\varepsilon}^t\frac{1}{\sqrt{t-a}}da =\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{t}}2\sqrt{\varepsilon};$$ thus $$\liminf_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t))\geq-\frac{\mu^2}{2}t,$$ and therefore, since $P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t))=P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t+\varepsilon))$, $$\lim_{\varepsilon\to 0}\liminf_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t+\varepsilon))\geq-\frac{\mu^2}{2}t=-J(t).$$ For the proof of \eqref{eq:forUB} we take into account \eqref{eq:starting-point-b=t} and, after some manipulations, we get \begin{multline*} P(T_{\mu\sqrt{r},t}\in(t-\varepsilon,t))=\frac{e^{-\frac{\mu^2r}{2}t}}{\pi}\int_{t-\varepsilon}^t\frac{1}{\sqrt{a(t-a)}}da +\frac{\mu^2r}{2\pi}\int_{t-\varepsilon}^tda\int_a^tdy\frac{e^{-\frac{\mu^2r}{2}y}}{\sqrt{a(y-a)}}\\ \leq\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{t-\varepsilon}}\int_{t-\varepsilon}^t\frac{1}{\sqrt{t-a}}da +\frac{\mu^2r}{2\pi}\int_{t-\varepsilon}^tda\frac{e^{-\frac{\mu^2r}{2}a}}{\sqrt{a}}\int_a^tdy\frac{1}{\sqrt{y-a}}\\ \leq\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{t-\varepsilon}}2\sqrt{\varepsilon} +\frac{\mu^2r}{2\pi}\int_{t-\varepsilon}^tda\frac{e^{-\frac{\mu^2r}{2}a}}{\sqrt{a}}2\sqrt{t-a} \leq\frac{e^{-\frac{\mu^2r}{2}t}}{\pi\sqrt{t-\varepsilon}}2\sqrt{\varepsilon} +\frac{\mu^2r}{2\pi}\frac{e^{-\frac{\mu^2r}{2}(t-\varepsilon)}}{\sqrt{t-\varepsilon}}2\sqrt{\varepsilon}; \end{multline*} thus, by Lemma 1.2.15 in \cite{DemboZeitouni}, we have $$\limsup_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t)) \leq\max\left\{-\frac{\mu^2}{2}t,-\frac{\mu^2}{2}(t-\varepsilon)\right\}=-\frac{\mu^2}{2}(t-\varepsilon),$$ and therefore, since $P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t))=P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t+\varepsilon))$, $$\lim_{\varepsilon\to 0}\limsup_{r\to\infty}\frac{1}{r}\log P(T_{\mu\sqrt{r},t}\in (t-\varepsilon,t+\varepsilon))\leq-\frac{\mu^2}{2}t=-J(t).$$ \end{proof} Now we present the class of LDPs inspired by moderate deviations. \begin{Proposition}\label{prop:MD} For every choice of positive numbers $\{\gamma_r:r>0\}$ such that \eqref{eq:conditions-MD} holds, the family of random variables $\{r\gamma_rT_{\mu\sqrt{r},t}:r>0\}$ satisfies the LDP with speed $v_r=1/\gamma_r$ and good rate function $\tilde{J}$ defined by $$\tilde{J}(b):=\left\{\begin{array}{ll} \frac{\mu^2}{2}b&\ if\ b\in[0,\infty)\\ \infty&\ otherwise. \end{array}\right.$$ \end{Proposition} \begin{proof} We can restrict the attention on the case $b\geq 0$ because we deal with nonnegative random variables (and $[0,\infty)$ is a closed set). We assume for the moment that we have \begin{equation}\label{eq:half-lines} \lim_{r\to\infty}\gamma_r\log P(r\gamma_rT_{\mu\sqrt{r},t}\geq z)=-\frac{\mu^2}{2}z\ (\mbox{for all}\ z\geq 0). \end{equation} We have to check the upper bound \eqref{eq:UB} and the lower bound \eqref{eq:LB} with $v_r=1/\gamma_r$ and $W(r)=r\gamma_rT_{\mu\sqrt{r},t}$. \begin{itemize} \item The upper bound \eqref{eq:UB} trivially holds if $C\cap [0,\infty)$ is empty. On the contrary, there exists $z_C:=\min(C\cap [0,\infty))$ (the existence of $z_C$ is guaranteed because $C\cap [0,\infty)$ is a non-empty closed set), and we have $$P(r\gamma_rT_{\mu\sqrt{r},t}\in C)\leq P(r\gamma_rT_{\mu\sqrt{r},t}\geq z_C);$$ thus, by \eqref{eq:half-lines} and by the monotonicity of $\tilde{J}(b)$ on $[0,\infty)$, we get $$\limsup_{r\to\infty}\frac{1}{1/\gamma_r}\log P(r\gamma_rT_{\mu\sqrt{r},t}\in C)\leq \limsup_{r\to\infty}\gamma_r\log P(r\gamma_rT_{\mu\sqrt{r},t}\geq z_C)=-\frac{\mu^2}{2}z_C=-\inf_{b\in C}\tilde{J}(b).$$ \item It is known (see e.g. \cite{DemboZeitouni}, condition (b) with equation (1.2.8)) that the lower bound \eqref{eq:LB} holds if and only if, for all $b\geq 0$ and for all open sets $G$ such that $b\in G$, we have \begin{equation}\label{eq:LB-equivalent} \liminf_{r\to\infty}\frac{1}{1/\gamma_r}\log P(r\gamma_rT_{\mu\sqrt{r},t}\in G)\geq-\tilde{J}(b). \end{equation} In order to get this condition we remark that there exists $\varepsilon>0$ such that $(b-\varepsilon,b+\varepsilon)\subset G$, and we have $$P(r\gamma_rT_{\mu\sqrt{r},t}\in G)\geq P(r\gamma_rT_{\mu\sqrt{r},t}\geq b-\varepsilon)-P(r\gamma_rT_{\mu\sqrt{r},t}\geq b+\varepsilon).$$ Then, by a suitable application of Lemma 19 in \cite{GaneshTorrisi} (and by taking into account \eqref{eq:half-lines}), we get \begin{multline*} \liminf_{r\to\infty}\frac{1}{1/\gamma_r}\log P(r\gamma_rT_{\mu\sqrt{r},t}\in G)\\ \geq\liminf_{r\to\infty}\gamma_r\log (P(r\gamma_rT_{\mu\sqrt{r},t}\geq b-\varepsilon)-P(r\gamma_rT_{\mu\sqrt{r},t}\geq b+\varepsilon)) \geq-\frac{\mu^2}{2}(b-\varepsilon). \end{multline*} Thus we get \eqref{eq:LB-equivalent} by letting $\varepsilon$ go to zero. \end{itemize} In conclusion, we complete the proof showing that \eqref{eq:half-lines} holds. The case $z=0$ is trivial, and therefore we take $z>0$. We take $r$ large enough such that $\frac{z}{r\gamma_r}\in[0,t]$ (we recall that $\lim_{r\to\infty}r\gamma_r=\infty$) and, by \eqref{eq:distribution-function}, we have $$P(r\gamma_rT_{\mu\sqrt{r},t}\geq z)=P\left(T_{\mu\sqrt{r},t}\geq\frac{z}{r\gamma_r}\right)= \frac{2}{\pi}\int_0^{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}\frac{e^{-\frac{\mu^2z}{2\gamma_r}(1+y^2)}}{1+y^2}dy.$$ Then $$P(r\gamma_rT_{\mu\sqrt{r},t}\geq z)= \frac{e^{-\frac{\mu^2z}{2\gamma_r}}}{\pi}\int_{-\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}^{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}} \frac{e^{-\frac{\mu^2z}{2\gamma_r}y^2}}{1+y^2}dy =\sqrt{\frac{2\gamma_r}{\pi\mu^2z}}e^{-\frac{\mu^2z}{2\gamma_r}} \int_{-\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}^{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}} \frac{\varphi(y;0,\gamma_r/(\mu^2z))}{1+y^2}dy,$$ and $$\gamma_r\log P(r\gamma_rT_{\mu\sqrt{r},t}\geq z) =\gamma_r\log\sqrt{\frac{2\gamma_r}{\pi\mu^2z}}-\frac{\mu^2}{2}z+\gamma_r\log \int_{-\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}^{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}} \frac{\varphi(y;0,\gamma_r/(\mu^2z))}{1+y^2}dy;$$ thus we complete the proof of \eqref{eq:half-lines} showing that \begin{equation}\label{eq:final-step} \lim_{r\to\infty}\int_{-\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}^{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}} \frac{\varphi(y;0,\gamma_r/(\mu^2z))}{1+y^2}dy=1. \end{equation} In order to do that we remark that, by the triangular inequality and after some easy manipulations, we get $$\left|\int_{-\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}^{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}} \frac{\varphi(y;0,\gamma_r/(\mu^2z))}{1+y^2}dy-1\right|\leq A_1(r)+A_2(r)$$ where $$A_1(r):=\left|\int_{-\infty}^\infty\frac{\varphi(y;0,\gamma_r/(\mu^2z))}{1+y^2}dy-1\right|$$ and $$A_2(r):=2\int_{\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}}^\infty\frac{\varphi(y;0,\gamma_r/(\mu^2z))}{1+y^2}dy.$$ Then \eqref{eq:final-step} holds (and this completes the proof) noting that: \begin{itemize} \item $\lim_{r\to\infty}A_1(r)=0$ by the weak convergence of the centered Normal distribution with variance $\frac{\gamma_r}{\mu^2z}$ to zero (as $r\to\infty$); \item $\lim_{r\to\infty}A_2(r)=0$ by taking into account that $$0\leq A_2(r)\leq\frac{2\frac{\gamma_r}{\mu^2z}}{\sqrt{2\pi}\sqrt{\frac{t-z/(r\gamma_r)}{z/(r\gamma_r)}}} \exp\left(-\frac{t-z/(r\gamma_r)}{2z/(r\gamma_r)}\frac{\mu^2z}{\gamma_r}\right)$$ by a well-known estimate of the tail of Gaussian distribution. \end{itemize} \end{proof} We conclude with the minor result for the crossing probability in \eqref{eq:crossing-probability}. \begin{Proposition}\label{prop:crossing-probability} We consider $b>a>0$. Then $$\lim_{r\to\infty}\frac{1}{r}\log\Psi_{[a,b]}(\mu\sqrt{r})=-\frac{\mu^2}{2}a\ \mbox{and}\ \lim_{r\to\infty}e^{\frac{\mu^2ra}{2}}\sqrt{r}\Psi_{[a,b]}(\mu\sqrt{r})=\sqrt{\frac{2}{\pi\mu^2a}}.$$ \end{Proposition} \begin{proof} We have $$\Psi_{[a,b]}(\mu\sqrt{r})=\frac{1}{\pi}\int_0^{b-a}\frac{e^{-\frac{\mu^2r}{2}(a+s)}}{a+s}\sqrt{\frac{a}{s}}ds =\frac{2}{\pi}\int_0^{\sqrt{\frac{b-a}{a}}}\frac{e^{-\frac{\mu^2ra}{2}(1+y^2)}}{1+y^2}dy$$ (in the second equality we take into account the change of variable $s=ay^2$). Thus $$\Psi_{[a,b]}(\mu\sqrt{r})=\frac{2}{\pi}e^{-\frac{\mu^2ra}{2}}\int_0^{\sqrt{\frac{b-a}{a}}}\frac{e^{-\frac{\mu^2ra}{2}y^2}}{1+y^2}dy =\sqrt{\frac{2}{\pi\mu^2ra}}e^{-\frac{\mu^2ra}{2}}\int_{-\sqrt{\frac{b-a}{a}}}^{\sqrt{\frac{b-a}{a}}}\frac{\varphi(y;0,1/(\mu^2ra))}{1+y^2}dy.$$ Since $$\lim_{r\to\infty}\int_{-\sqrt{\frac{b-a}{a}}}^{\sqrt{\frac{b-a}{a}}}\frac{\varphi(y;0,1/(\mu^2ra))}{1+y^2}dy=1,$$ then, by the weak convergence of the centered Normal distribution with variance $\frac{1}{\mu^2ra}$ to zero (as $r\to\infty$), we can get the desired limits with some easy computations. \end{proof} \begin{Remark}\label{rem:analogies-with-ruin-probability} There are some similarities between the limits in Proposition \ref{prop:crossing-probability} and some asymptotic estimates of level crossing probabilities in the literature. For instance, if we denote the level crossing probability by $\psi(r)$ (here $r>0$ is the level), under suitable conditions (see e.g. \cite{DuffyLewisSullivan}) we have \begin{equation}\label{eq:DLS} \lim_{r\to\infty}\frac{1}{h(r)}\log\psi(r)=-w \end{equation} for some $w>0$ and some scaling function $h(\cdot)$. For instance here we recall the case of the Cram\'er-Lundberg model in insurance (see e.g. \cite{AlbrecherAsmussen}), where $\psi(r)$ is interpreted as the ruin probability and $r$ as the initial capital. Then, under suitable hypotheses, we have the two following statements: \begin{itemize} \item for some $w>0$ and some $c_1,c_2\in(0,1]$ with $c_1\leq c_2$ (see e.g. Theorem 6.3 in \cite{AlbrecherAsmussen}, Chapter IV), we have $$c_1e^{-wr}\leq\psi(r)\leq c_2e^{-wr},$$ which yields \eqref{eq:DLS} with $h(r)=r$; \item for some $c>0$ (see e.g. (4.3) in \cite{AlbrecherAsmussen}, Chapter I; see also Theorem 1.2.2(b) in \cite{EmbrechtsKluppelbergMikosch}) we have \begin{equation}\label{eq:CL-approximation} \lim_{r\to\infty}e^{wr}\psi(r)=c. \end{equation} \end{itemize} So the limits in Proposition \ref{prop:crossing-probability} for $\Psi_{[a,b]}(\mu\sqrt{r})$ have some relationship with the limits here for $\psi(r)$. However the scaling factor $e^{\frac{\mu^2ra}{2}}\sqrt{r}$ (see the second limit in Proposition \ref{prop:crossing-probability}) is different from $e^{wr}$ in \eqref{eq:CL-approximation}. \end{Remark} \section{Comparison with moderate deviation results in the literature}\label{sec:MD-comments} The term moderate deviations is used in the literature for a class of LDPs for suitable centered random variables, and governed by the same quadratic rate function (here we restrict the attention on real valued random variables for simplicity but, actually, a similar concept can be given for vector valued random variables). Proposition \ref{prop:MD} also provides a class of LDPs: the random variables are not centered, but they converge to zero because the rate function $J$ in Proposition \ref{prop:LD} uniquely vanishes at zero. So in this section we want to discuss analogies and differences between the moderate deviations results in the literature, usually related to the use of G\"{a}rtner Ellis Theorem (see e.g. Theorem 2.3.6 in \cite{DemboZeitouni}), and the results in this paper. We start with Claim \ref{claim:LDMD}, which provides the usual framework for both large and moderate deviations for a family of random variables $\{W(r):r>0\}$. There is an initial LDP, and a class of LDPs which concerns moderate deviations. One can immediately see the analogies with the statements of Propositions \ref{prop:LD} and \ref{prop:MD} in this paper; in particular \eqref{eq:conditions-MD-gen} below plays the role of \eqref{eq:conditions-MD} in Proposition \ref{prop:MD}. \begin{Claim}\label{claim:LDMD} We assume that, for all $\theta\in\mathbb{R}$, $$\Lambda(\theta):=\lim_{r\to\infty}\frac{1}{v_r}\log\mathbb{E}[e^{v_r\theta W(r)}]$$ exists as an extended real number. Then, under suitable hypotheses (see e.g. part (c) of Theorem 2.3.6 in \cite{DemboZeitouni}) the LDP holds with speed $v_r\to\infty$ and good rate function $\Lambda^*$ defined by $$\Lambda^*(w):=\sup_{\theta\in\mathbb{R}}\left\{\theta w-\Lambda(\theta)\right\}.$$ Furthermore, we set $\tilde{\Lambda}(\theta):=\frac{\theta^2}{2}\Lambda^{\prime\prime}(0)$, where $\Lambda$ is the function above, and $\Lambda^{\prime\prime}$ is its second derivative (note that $\Lambda^{\prime\prime}(0)\geq 0$ because $\Lambda$ is a convex function). Then, for every choice of positive numbers $\{\gamma_r:r>0\}$ such that \begin{equation}\label{eq:conditions-MD-gen} \lim_{r\to\infty}\gamma_r=0\ \mbox{and}\ \lim_{r\to\infty}v_r\gamma_r=\infty, \end{equation} we can prove that $$\lim_{r\to\infty}\frac{1}{1/\gamma_r}\log\mathbb{E}[e^{(1/\gamma_r)\theta\sqrt{v_r\gamma_r}(W(r)-\mathbb{E}[W(r)])}]= \tilde{\Lambda}(\theta)$$ for all $\theta\in\mathbb{R}$; thus $\left\{\sqrt{v_r\gamma_r}(W(r)-\mathbb{E}[W(r)]):r\geq 1\right\}$ satisfies the LDP with speed $1/\gamma_r$ and good rate function $\tilde{\Lambda}^*$ defined by $$\tilde{\Lambda}^*(w):=\sup_{\theta\in\mathbb{R}}\left\{\theta w-\tilde{\Lambda}(\theta)\right\}= \left\{\begin{array}{ll} \frac{w^2}{2\Lambda^{\prime\prime}(0)}&\ \mbox{if}\ \Lambda^{\prime\prime}(0)>0\\ \left\{\begin{array}{ll} 0&\ \mbox{if}\ w=0\\ \infty&\ \mbox{if}\ w\neq 0 \end{array}\right. &\ \mbox{if}\ \Lambda^{\prime\prime}(0)=0. \end{array}\right.$$ \end{Claim} \begin{Remark}\label{rem:Taylor-formula-order-2} When $\Lambda^{\prime\prime}(0)>0$, the Taylor formula of order 2 of $\Lambda^*$, and initial point $\Lambda^\prime(0)$, is $\tilde{\Lambda}^*(w-\Lambda^\prime(0))$. A similar relationship concerns the Mac Laurin formula of order 2 of $\Lambda$, that is $\theta\Lambda^\prime(0)+\tilde{\Lambda}(\theta)$. \end{Remark} In the framework of Claim \ref{claim:LDMD} we also have the following typical features. \begin{Claim}\label{claim:mean-variance-fill-the-gap} Firstly we have \begin{equation}\label{eq:mean-variance-formula} \lim_{r\to\infty}\mathbb{E}[W(r)]=\Lambda^\prime(0)\ \mbox{and}\ \lim_{r\to\infty}v_r\mathrm{Var}[W(r)]=\Lambda^{\prime\prime}(0). \end{equation} Moreover, we can say that moderate deviations fill the gap between two different regimes (as $r\to\infty$): \begin{itemize} \item the convergence of $W(r)-\mathbb{E}[W(r)]$ to zero (case $\gamma_r=1/v_r$; note that only the first condition in \eqref{eq:conditions-MD-gen} holds), which is equivalent to the convergence of $W(r)$ to $\Lambda^\prime(0)$; \item the weak convergence of $\sqrt{v_r}(W(r)-\mathbb{E}[W(r)])$ to the centered Normal distribution with variance $\Lambda^{\prime\prime}(0)$ (case $\gamma_r=1$; note that only the second condition in \eqref{eq:conditions-MD-gen} holds). \end{itemize} \end{Claim} We present an illustrative example. We consider the case where $$W(r):=\frac{X_1+\cdots+X_r}{r}$$ (here $r$ is an integer) and $\{X_n:n\geq 1\}$ are i.i.d. real valued random variables; moreover we assume that $\mathbb{E}[e^{\theta X_1}]$ is finite in a neighborhood of the origin $\theta=0$, and therefore all the (common) moments of the random variables $\{X_n:n\geq 1\}$ are finite. We denote the common mean by $\mu$ and the common variance by $\sigma^2$. Then: \begin{itemize} \item as far as Claim \ref{claim:LDMD} is concerned, we have the initial LDP $v_r=r$ and $\Lambda(\theta):=\log\mathbb{E}[e^{\theta X_1}]$ (in this case G\"{a}rtner Ellis Theorem is not needed because we can refer to the Cram\'er Theorem, see e.g. Theorem 2.2.3 in \cite{DemboZeitouni}), and the class of LDPs (see e.g. Theorem 3.7.1. in \cite{DemboZeitouni}); \item as far as Claim \ref{claim:mean-variance-fill-the-gap} is concerned, the limits in \eqref{eq:mean-variance-formula} trivially hold with $\mathbb{E}[W(r)]=\mu=\Lambda^\prime(0)$ and $v_r\mathrm{Var}[W(r)]=\sigma^2=\Lambda^{\prime\prime}(0)$ for all integer $r\geq 1$; moreover moderate deviations fill the gap between the regimes of the classical Law of Large Numbers (for centered random variables) and Central Limit Theorem. \end{itemize} Finally, we concentrate the attention on the cases studied in Propositions \ref{prop:LD} and \ref{prop:MD} in this paper. We already remarked that the random variables in Proposition \ref{prop:LD} are not centered, but they converge to zero as $r\to\infty$ because the rate function $J$ uniquely vanishes at zero. The Mac Laurin formula of order 2 of $J$ (for nonnegative arguments) is $\tilde{J}$, and therefore we have some analogy with what happens for $\Lambda^*$ and $\tilde{\Lambda}^*$ in Remark \ref{rem:Taylor-formula-order-2}; in fact $J$ is the restriction of $\tilde{J}$ on $[0,t]$. Furthermore, one could investigate if we have limits as the ones in \eqref{eq:mean-variance-formula}. Here we take into account Theorem 2.2 in \cite{IafrateOrsingher} (see also Remark 2.4 in \cite{IafrateOrsingher} for the mean value). Firstly, we have $$\mathbb{E}[T_{\mu\sqrt{r},t}]=\frac{1-e^{-\frac{\mu^2r}{2}t}}{\mu^2r},$$ and therefore $\lim_{r\to\infty}\mathbb{E}[T_{\mu\sqrt{r},t}]=0$. Moreover \begin{multline*} \mathrm{Var}[T_{\mu\sqrt{r},t}]=\mathbb{E}[T_{\mu\sqrt{r},t}^2]-\mathbb{E}^2[T_{\mu\sqrt{r},t}] =\frac{3}{4}\int_0^tae^{-\frac{\mu^2r}{2}a}da-\left(\frac{1-e^{-\frac{\mu^2r}{2}t}}{\mu^2r}\right)^2\\ =\frac{3}{4}\left(-\frac{2t}{\mu^2r}e^{-\frac{\mu^2r}{2}t}+\left(\frac{2}{\mu^2r}\right)^2(1-e^{-\frac{\mu^2r}{2}t})\right) -\frac{(1-e^{-\frac{\mu^2r}{2}t})^2}{\mu^4r^2}\\ =-\frac{3t}{2\mu^2r}e^{-\frac{\mu^2r}{2}t}+\frac{3}{\mu^4r^2}(1-e^{-\frac{\mu^2r}{2}t})-\frac{(1-e^{-\frac{\mu^2r}{2}t})^2}{\mu^4r^2}; \end{multline*} so, in order to have a finite and positive limit, we have to take a different scaling, i.e. \begin{equation}\label{eq:variance-limit} \lim_{r\to\infty}r^2\mathrm{Var}[T_{\mu\sqrt{r},t}]=\frac{2}{\mu^4} \end{equation} (on the contrary the limit for the variance with the same scaling as in \eqref{eq:mean-variance-formula}, and therefore with the speed $v_r=r$ as in Proposition \ref{prop:LD}, is equal to zero). We can also say that, as happens for Theorem 3.7.1 in \cite{DemboZeitouni}, Proposition \ref{prop:MD} fill the gap between two regimes (as $r\to\infty$). We have again a convergence to zero for $T_{\mu\sqrt{r},t}$ (case $\gamma_r=1/r$) but, on the contrary, $rT_{\mu\sqrt{r},t}$ (case $\gamma_r=1$) converges weakly to the distribution of a nonnegative random variable $Y$ with distribution function $G$ defined by $$G(a)=1-\frac{2}{\pi}\int_0^\infty\frac{e^{-\frac{\mu^2a}{2}(1+y^2)}}{1+y^2}dy,\ \mbox{for all}\ a\geq 0.$$ Finally we compute the variance of $Y$. We have \begin{multline*} \mathrm{Var}[Y]=\mathbb{E}[Y^2]-\mathbb{E}^2[Y] =\int_0^\infty \underbrace{P(Y^2>a)}_{=P(Y>\sqrt{a})}da-\left(\int_0^\infty P(Y>a)da\right)^2\\ =\int_0^\infty\frac{2}{\pi}\int_0^\infty\frac{e^{-\frac{\mu^2\sqrt{a}}{2}(1+y^2)}}{1+y^2}dyda -\left(\int_0^\infty\frac{2}{\pi}\int_0^\infty\frac{e^{-\frac{\mu^2a}{2}(1+y^2)}}{1+y^2}dyda\right)^2; \end{multline*} moreover, after some manipulations (in particular, for the second equality we consider the change of variable $\bar{a}=\sqrt{a}$ in the first term and, later, we put $a$ in place of $\bar{a}$), we obtain \begin{multline*} \mathrm{Var}[Y]=\frac{2}{\pi}\int_0^\infty\frac{1}{1+y^2}\int_0^\infty e^{-\frac{\mu^2\sqrt{a}}{2}(1+y^2)}dady -\frac{4}{\pi^2}\left(\int_0^\infty\frac{1}{1+y^2}\int_0^\infty e^{-\frac{\mu^2a}{2}(1+y^2)}dady\right)^2\\ =\frac{2}{\pi}\int_0^\infty\frac{1}{1+y^2}\int_0^\infty e^{-\frac{\mu^2a}{2}(1+y^2)}2adady -\frac{4}{\pi^2}\left(\int_0^\infty\frac{1}{1+y^2}\int_0^\infty e^{-\frac{\mu^2a}{2}(1+y^2)}dady\right)^2\\ =\frac{4}{\pi}\int_0^\infty\frac{1}{1+y^2}\frac{\Gamma(2)}{(\frac{\mu^2}{2}(1+y^2))^2}dy -\frac{4}{\pi^2}\left(\int_0^\infty\frac{1}{\frac{\mu^2}{2}(1+y^2)^2}dy\right)^2\\ =\frac{16}{\pi\mu^4}\int_0^\infty\frac{1}{(1+y^2)^3}dy-\frac{16}{\pi^2\mu^4}\left(\int_0^\infty\frac{1}{(1+y^2)^2}dy\right)^2; \end{multline*} finally we consider the change of variable $y=\tan\theta$ in both integrals and we get \begin{multline*} \mathrm{Var}[Y]=\frac{16}{\pi\mu^4}\int_0^{\pi/2}\cos^4\theta d\theta-\frac{16}{\pi^2\mu^4}\left(\int_0^{\pi/2}\cos^2\theta d\theta\right)^2\\ =\frac{16}{\pi\mu^4}\underbrace{\left(\left[\frac{\cos^3\theta\sin\theta}{4} +\frac{3}{4}\left(\frac{\theta+\sin\theta\cos\theta}{2}\right)\right]_{\theta=0}^{\theta=\pi/2} -\frac{1}{\pi}\left(\left[\frac{\theta+\sin\theta\cos\theta}{2}\right]_{\theta=0}^{\theta=\pi/2}\right)^2\right)}_{=\frac{3\pi}{16} -\frac{1}{\pi}\frac{\pi^2}{16}}=\frac{2}{\mu^4}. \end{multline*} Thus, in some sense, we have some analogy with the classical moderate deviation results, because the limit of rescaled variance in \eqref{eq:variance-limit} coincides with the variance of the weak limit $Y$ of $rT_{\mu\sqrt{r},t}$ (concerning the case $\gamma_r=1$). \paragraph{Acknowledgements.} We thank an anonymous referee for the careful reading of the first version of the manuscript. His/her comments allowed us to correct two inaccuracies.

BkiUfRPxK6wB9lIs-6d4

\section{Introduction} \label{sec:Introduction} Brane configurations in string/M-theory have offered a new avenue for producing wide classes of solutions of physical interest in lower dimensions. The attempts of an early date have mainly aimed at the construction of various kinds of black holes~\cite{GM,Strominger:1996sh,Maldacena:1996ky,Tanabe} via the Kaluza-Klein compactification of (intersecting) branes~\cite{ Gueven:1992hh,Argurio:1997gt,Ohta:1997wp, Aref'eva:1997nz,Ohta:1997gw}. An interesting application of this idea is to dynamically realize our 4D universe by incorporating time-dependence~\cite{Lu:1995cs,Lu2,Lu3}. An alternate mechanism that provides lower-dimensional spacetimes is the brane world~\cite{RS,BW}, in which our 4D world is regarded as the hypersurface embedded in the bulk. Cosmological evolutionary scenarios based on the dynamically moving brane or colliding branes give a significant modification from the standard cosmology in high energy r\'egime, still consistent with the present day observations~\cite{ Kraus:1999it,Ida:1999ui,Khoury:2001wf}. One can extend these studies further into the case where the brane involves a nontrivial space-and time-coordinate dependence. Correspondingly, the 4D reduced solution becomes spatially inhomogeneous and evolving in time. A preliminary result was presented in~\cite{Gibbons:2005rt}, where colliding D3 branes were discussed within the framework of type IIB supergravity (see~\cite{Chen:2005jp} for analysis of the Ho\v{r}awa-Witten domain wall). Lower-dimensional solutions obtained by compactifying extended directions of these moving branes or dynamically intersecting branes have much richer properties than those obtained from static counterparts. Recently, the authors of~\cite{MOU} have studied dynamical solutions describing intersecting branes in more general settings and obtained a number of intriguing solutions with wide potential applications. Among other things, their tantalizing findings are the ``cosmological black holes'' in the expanding FLRW universe, that is, black holes in a non-isolated system which asymptotically approaches to the homogeneous and isotropic cosmology. In this paper, we are concerned with this ``candidate'' black hole spacetime obtained in~\cite{MOU}. Studies of black holes in our universe have been primarily focused upon the stationary spacetimes in the literature~\cite{Carter}. Such approaches are definitely the first step because we can anticipate that dynamic variations will die away and the system will settle down to equilibrium states if sufficiently long time passed after the formation of a black hole. A number of physical properties of stationary black holes have been elucidated by many people. Specifically, black hole uniqueness theorem in stationary spacetime is the major triumph of mathematical relativity and established that the Kerr solution describes all vacuum black holes in isolated systems~\cite{Carter}. An essential crux toward the uniqueness proof is the demonstration that the event horizon in stationary spacetime is a Killing horizon~\cite{HE}. Since a Killing horizon is a null surface to which the Killing vector is normal, we can identify locus of a black hole simply by the local spacetime symmetry. Furthermore, it has been revealed that Killing horizons admit three laws of black hole mechanics which bear an amazing resemblance to ordinary thermodynamics~\cite{ BHTD1,BHTD2,Hawking1974,Wald:1975kc,Wald:1993nt,Iyer:1994ys,Iyer:1995kg, Gao:2001ut}. This implies a deep association between classical gravity, statistical physics and quantum mechanics, so that black hole thermodynamics is expected to have a key r\^ole for understanding quantum aspects of gravity. A notable progression in string theory is the microscopic derivation of black hole entropy in the perspective of intersecting brane configurations~\cite{Strominger:1996sh,Maldacena:1996ky}. If we get rid of the stationarity assumptions to discuss dynamics, uniqueness theorem no longer holds. Accordingly, a variety of black hole solutions are likely to exist. However, very little has been known concerning the exact solutions of Einstein's equations that describe growing black holes interplaying with surroundings. A novel aspect of non-isolated system is that it generally possesses a time-dependence and need not be asymptotically flat. The background fluid distributions filling the universe become inhomogeneous due to the presence of the black hole, while the black hole grows by swallowing the ambient matters. Such a complexity has rendered the system considerably elusive. A large amount of effort has been devoted thus far to attempt to obtain black holes in the FLRW universe. An initiated work is the simplest model invented by Einstein and Straus~\cite{Einstein:1945id}, which is often refereed to as a ``Swiss-Cheese Universe.'' They matched the Schwarzschild solution with an FLRW universe by means of a ``cut and paste'' method. So the black-hole part metric still maintains a time-symmetry, and then it appears that this model does not capture realistic situation of dynamic phase. If there is a positive cosmological constant, we have the Schwarzschild-de Sitter (SdS) or the Reissner-Nordstr\"om-de Sitter (RNdS) solution~\cite{SdS,Carter}. Those spacetime can be redescribed by the coordinate transformation in the form of a black hole in the exponentially expanding universe~\cite{BrillHayward}. However they have a ``timelike'' Killing vector and are essentially static. In recent years, Sultana and Dyer have constructed a more sophisticated black hole solution in a dynamical background by a conformal technique~\cite{SultanaDyer}. The matter content is composed of null dust and usual dust fluids, and the solution tends to an Einstein-de Sitter spacetime asymptotically. This model, however, suffers from the issue of violating energy conditions: energy densities of both fluids become negative at late times. One of the other widely known black hole candidate in FLRW universe is the McVittie solution~\cite{Mcvittie1933}, which is a spherically symmetric, expanding solution of the Einstein equations sourced by a perfect fluid. Taking asymptotic limits, the solution looks like an FLRW universe at ``infinity,'' and like a black hole near the ``horizon,'' whence one might be led to conclude that the McVittie spacetime might describe a black hole in the expanding FLRW universe. Attractive as this might be, however, such an optimistic outlook would jump to a hasty conclusion. As asserted in~\cite{Nolan}, the McVittie solution is disqualified as a black hole in an FLRW universe. Since our spacetime metric is in appearance quite similar to the McVittie solution in several respects, the above concrete example motivates us to explore the global structure of time-dependent black holes with enough care. In this paper, we intend to provide a comprehensive account of the global picture of dynamical solutions obtained in~\cite{MOU}. Another interesting issue of a time-dependent spacetime containing black holes is the collision of black holes. Kastor and Traschen found the collection of extremely charged black holes in the de Sitter universe and discussed their collision~\cite{KT,BHKT}. This solution is a time-dependent generalization of its celebrated cousin, the Majumbdar-Papapetrou solution, which describes extremely charged RN black holes~\cite{Hartle:1972ya}. In spite of the lack of Bogomol'nyi-Prasad-Sommerfield (BPS) states for $\Lambda >0$, it is somewhat astonishing that the superposition of RNdS black holes is possible. The same situation happens in our spacetime obtained by the time-dependent intersecting brane system in which no BPS states are preserved. Making use of this exact solution, we can discuss the collision of black holes in the power-law contracting universe just as the brane collision~\cite{Gibbons:2005rt}. The plan of the rest paper is as follows. We shall begin by reviewing the dynamical ``black hole solution'' derived in~\cite{MOU}. Section~\ref{sec:matter} involves a detailed examination about the properties of matter fields. Our main result is contained in Section~\ref{sec:spacetime_structure}, where we will elucidate spacetime structures based on local and global perspectives. We draw the conformal diagram that allows us to pictorially identify a black hole embedded in an expanding universe. Section~\ref{conclusion} summarizes our conclusive results with several future outlook. In order to keep the mainstream of the text, we relegate some issues to Appendixes. We shall work in the units of $c=\hbar=1$ and retain the gravitational constant $\kappa^2=8\pi G$. We pertain to the notation $R$ to denote the circumference radius, so that we use the script throughout the paper for the spacetime curvature ${\cal R^\mu}_{\nu\rho\sigma}V^\nu:=2\nabla_{[\rho }\nabla_{\sigma ]}V^\mu$, ${\cal R}_{\mu \nu }:={\mathcal R^{\rho }}_{\mu \rho \nu }$ and ${\cal R}:={\cal R^\mu }_\mu $. \section{Time-dependent solutions from the intersecting brane system} \label{sec:review} The authors of~\cite{MOU} have classified the possible time-dependent intersecting brane systems in M-theory by assuming the metric form, and presented some interesting solutions in lower dimensions by compactification. Among other things, one of the most interesting solutions is a ``black hole'' in the expanding universe. In the case where the all branes are at rest, i.e., spacetime is static, the 4D reduced solution indeed describes a black hole, which is obtained from M2-M2-M5-M5 brane system (four brane charges) or from M2-M5-W-KK brane system (two brane charges plus a Brinkmann wave and a Kaluza-Klein monopole). The 5D black hole solution is similarly derived from M2-M2-M2 brane system (three brane charges) or from M2-M5-W brane system (two branes plus a Brinkmann wave). Time-dependent extensions of those lower-dimensional solutions are produced from the dynamical intersecting brane systems, in which only single brane is time-dependent~\cite{MOU}. These 11D solutions and the procedure of the dimensional reduction are shortly summarized in Appendix~\ref{Intersecting_brane}. After the toroidal compactification of 11D intersecting brane solutions, we have a 4D ``black hole solution,'' which is spherically symmetric, time and radial coordinate-dependent (we relegate the extension into the case of non-spherical, multiple black holes to Appendix~\ref{multi_BH}). The 4D metric in the Einstein frame is given by \begin{eqnarray} {\rm d} s^2=-\Xi {\rm d} t^2+{1\over \Xi}\left({\rm d} r^2+r^2 {\rm d} \Omega_2^2\right)\,, \label{4D_metric} \end{eqnarray} with \begin{eqnarray} \Xi &=& \nonumber \left[ \left(\frac{t}{t_0}+\frac{Q_T}{r}\right) \left(1+\frac{Q_S}{r}\right) \right. \\ && \left. ~~ \left(1+\frac{Q_{S'}}{r}\right) \left(1+\frac{Q_{S''}}{r}\right) \right]^{-1/2} \label{Xi} \,, \end{eqnarray} where ${\rm d} \Omega_2^2={\rm d} \theta^2+\sin^2\theta {\rm d}\phi ^2$ denotes the line element of a unit round two-sphere. The constants $Q_T$ and $Q_S, Q_{S'}, Q_{S'\hspace{-.1em}'}$ are charges of one time-dependent brane and three static branes, respectively. Here and hereafter, the script ``$T$'' and ``$S$'' are understood to trace their origin to time-dependent and static branes. The above metric manifests that the conditions of stationarity and asymptotic flatness were both relaxed. Assuming $t/t_0>0 $ and changing to the new time slice $\bar t$ defined by \begin{eqnarray} {\bar t \over \bar t_0} =\left ({t \over t_0} \right)^{3/4} ~~~~~{\rm with}~~~ \bar t_0=\frac{4}{3}t_0 \,, \end{eqnarray} we are able to put the solution (\ref{4D_metric}) into a more suggestive form, \begin{eqnarray} {\rm d} s^2&=&-\bar \Xi \, {\rm d} \bar t^2+{a^2\over \bar \Xi} \left({\rm d} r^2+r^2 {\rm d} \Omega_2^2\right)\,, \label{4D_metric1} \end{eqnarray} where \begin{eqnarray} \bar \Xi &=& \left[ \left(1+{Q_T\over a^4 r}\right) \left(1+{Q_{S}\over r}\right) \nonumber \right. \\ && \left. ~~ \left(1+{Q_{S'}\over r}\right) \left(1+{Q_{S'\hspace{-.1em}'}\over r}\right)\right]^{-1/2 } \label{Xi1} \,, \end{eqnarray} and \begin{eqnarray} a &=&\left({\bar t\over \bar t_0}\right)^{1/3} \label{scale_factor_4} \,. \end{eqnarray} When we take the limit of $r\rightarrow \infty$, we can find that the metric~(\ref{4D_metric1}) asymptotically tends to a flat FLRW spacetime, \begin{eqnarray} {\rm d} s^2&=&-{\rm d} \bar t^2+a^2 \left({\rm d} r^2+r^2 {\rm d} \Omega_2^2\right)\,. \end{eqnarray} Here, the scale factor expands as $a\propto \bar t^{1/3}$, which is the same as the expansion law of the universe filled by a stiff matter. Hence we expect that this spacetime is asymptotically FLRW universe with the equation of state $P=\rho$. On the other hand, taking the limit $r\rightarrow 0$ with $t$ being finite, the time-dependence turns off and the metric~(\ref{4D_metric1}) reduces to the direct product of 2D anti-de Sitter (AdS$_2$) space with round sphere, \begin{eqnarray} {\rm d} s^2&=&-{r^2\over \bar Q^2}{\rm d} t^2 +{\bar Q^2\over r^2} \left({\rm d} r^2+r^2{\rm d} \Omega^2_2\right)\,, \label{near_horizon} \end{eqnarray} where $\bar Q:=(Q_TQ_{S}Q_{S'}Q_{S'\hspace{-.1em}'})^{1/4}$ plays the r\^ole of curvature radii of AdS$_2$ and S$^2$. This is a typical ``throat'' geometry of an extreme black hole. To take the case of an extreme RN spacetime with the Arnowitt-Deser-Misner (ADM) mass $M=\bar Q$, the metric in the isotropic coordinates reads \begin{eqnarray} {\rm d} s^2 &=&-\left(1+{\bar Q \over r}\right)^{-2}{\rm d} t^2 \nonumber \\ && +\left(1+{\bar Q \over r}\right)^{2}\left({\rm d} r^2+r^2 {\rm d} \Omega^2_2\right) \label{4D_RNBH} \,, \end{eqnarray} which indeed asymptotes to the spacetime~(\ref{near_horizon}) in the limit $r\rightarrow 0$ with $t$ kept finite.\footnote{ It should be emphasized that the AdS$_2 \times$ S$^2$ geometry~(\ref{near_horizon}) indeed approximates the whole portion of the near-horizon geometry--the three dimensional null surface--of an extremal RN solution (and more generally, of any kinds of an extremal black hole~\cite{Kunduri:2007vf}), not restricted to the ``throat.'' To the contrary, the spacetime~(\ref{near_horizon}) fails to describe the near-horizon geometry of the event horizon of our time-dependent solution. As proof of the incident, we will show in later section that the temperature of a dynamical black hole does not vanish in the present case, unlike the extremal RN black hole having a zero Hawking temperature. } Thus, we may speculate that there might exist a degenerate event horizon at $r=0$. For this reason stated above, it might be tempted to regard the present spacetime~(\ref{4D_metric}) as a degenerate black hole with radius $\bar Q$ residing at the center $r=0$ of the expanding FLRW universe. However, there is {\it a prioi} no guarantee that the global pictures of the solution is obtainable simply from the asymptotic considerations, just as we inferred the causal structure of the Schwarzschild-AdS spacetime from those of the Schwarzschild and the AdS metrics. Furthermore, it is far from clear to what extent the spacetime shares the physical properties as its limiting counterparts. In fact, a more detailed argument casts doubt on the above na\"ive expectation. To see this, it is instructive to consider the McVittie spacetime~\cite{Mcvittie1933}, \begin{align} {\rm d} s^2=& -\left({1-{M/ 2ar}\over 1+{M/ 2ar}}\right)^2{\rm d} t^2 \nonumber \\ &+ a^2\left(1+{M\over 2ar}\right)^4\left({\rm d} r^2+r^2 {\rm d} \Omega^2_2\right) \,, \label{McVittie} \end{align} where $a=a(t)$. The metric (\ref{4D_metric1}) with (\ref{Xi1}) and~(\ref{scale_factor_4}) looks quite similar to the McVittie spacetime in appearance. In the limit of $r\rightarrow \infty$, the McVittie spacetime asymptotes to the flat FLRW universe with the scale factor $a$. If $a$ is set to be constant ($a\equiv 1$), we recover the Schwarzschild spacetime with the ADM mass $M$ written in the isotropic coordinates. So one might deduce that the McVittie spacetime~(\ref{McVittie}) describes a black hole immersed in the FLRW universe. However, it turns out that this exemplifies that our na\"ive estimate ceases to be true.\footnote{The case $a=e^{Ht}$ with constant $H$ is exceptional for which the metric describes an SdS spacetime. But we are reluctant to refer to this as a black hole in expanding universe, since the metric is rewritten in a static form thanks to Birkhoff's theorem. } When the scale factor obeys the power-law form $a\propto t^p~ (p >0)$, a curvature singularity appears at $r=M/(2a)$. As Nolan pointed out~\cite{Nolan}, the spacetime events described by $r=M/(2a)$ in part consist of a shell-crossing spacelike singularity lying a future of a big bang singularity $a=0$. The surface $r=M/(2a)$ fails to describe a (regular) horizon of a black hole in FLRW universe. Inferring from the ``monopole term'' $m/r$, does this spacetime instead describe a point mass (singularity) at $r=0$ embedded in the expanding universe? The answer is NO. It turns out that $r=0$ corresponds to infinity, rather than the locus of a point particle in the universe. Besides that, according to the quasi-local definition of horizon, the McVittie spacetime may serve as a ``white hole'' in an expanding FLRW universe. As a good lesson of above, we are required to take special care to conclude what the present spacetime describes. In this paper, we study the above spacetime~(\ref{4D_metric}) more thoroughly [we are working mainly in Eq.~(\ref{4D_metric}) rather than Eq.~(\ref{4D_metric1}), because the former coordinates cover wider range than the latter]. We assume $t_0>0$, viz, the background universe is expanding. For simplicity and definiteness of our argument, we will specialize to the case in which all charges are equal, i.e., $Q_T=Q_{S}=Q_{S'}=Q_{S'\hspace{-.1em}'}\equiv Q~(>0)$.\footnote{ If $Q_T$ is different from other three same charges $Q_S$, the present result still holds. It is because such a difference amounts to the trivial conformal change $$ {\rm d} s^2=(Q_T/Q_S)^{1/2}\left[ -\Xi_* {\rm d} t_*^2+\Xi_*^{-1}({\rm d} r^2+{\rm d} \Omega_2^2)\right] $$ with simple parameter redefinitions: $\Xi_*=[(t_*/t_{*0}+Q_S/r)(1+Q_S/r)^3]^{-1/2}$, $t_*=(Q_T/Q_S)^{-1/2}t$, and $t_{*0}=(Q_T/Q_S)^{1/2}t_0$.} To be specific, we will analyze the spacetime metric \begin{eqnarray} {\rm d} s^2 =- \Xi\, {\rm d} t^2 + \Xi^{-1}\left( {\rm d} r^2+ r^2 {\rm d} \Omega_2^2\right) \label{metric} \,, \end{eqnarray} whose component $\Xi $, Eq.~(\ref{Xi}), is simplified to \begin{eqnarray} \Xi =\left(H_TH_S^3\right)^{-1/2}\,, \end{eqnarray} with \begin{eqnarray} H_T=\frac{t}{t_0}+\frac{Q}{r}, \qquad H_S=1+\frac{Q}{r} \label{metric_Xi} \,. \end{eqnarray} A more general background with distinct charges are yet to be investigated. The result for the 5D solution will be given in Appendix~\ref{5DBH}. \section{Matter fields and their properties} \label{sec:matter} It is a good starting point to draw our attention toward the matter fields. Since we know explicitly the 4D metric components, we can read off the total energy-momentum tensor of matter fluid(s) from the 4D Einstein equations, \begin{eqnarray} \kappa^2T_{\mu\nu}=\mathcal G_{\mu\nu} \,, \end{eqnarray} where $\mathcal G_{\mu\nu}=\mathcal R_{\mu\nu}-(\mathcal R/2)g_{\mu\nu}$ is the Einstein tensor. What kind of matter fluids we expect? There may appear at lest two fluid components: one is a scalar field and the other is a U(1) gauge field. This is because we compactify seven spaces and we have originally 4-form field in 11D supergravity theory. The torus compactification gives a set of scalar fields and the 4-form field behaves as a U(1) gauge field in 4D. In our solution, we assume four branes, which give rise to four U(1) gauge fields. As shown in Appendix~\ref{Intersecting_brane}, we can derive the following effective 4D action from 11D supergravity solution via compactification, \begin{eqnarray} {\cal S}&=&\int {\rm d} ^4x\sqrt{-g}\left[ {1\over 2\kappa^2}{\cal R}-{1\over 2}(\nabla\Phi)^2 \right. \nonumber \\ &&~~~~~~~~~ \left. -{1\over 16\pi}\sum_{A} e^{\lambda_A\kappa\Phi} (F_{\mu\nu}^{(A)})^2\right] \,, \label{4D_action} \end{eqnarray} where $\Phi$, $F_{\mu\nu}^{(A)}$, and $\lambda_A$ ($A=T,S,S',S'\hspace{-.1em}'$) are a scalar field, four U(1) fields, and coupling constants, respectively. The above action yields the following set of basic equations, \begin{eqnarray} && \mathcal G_{\mu\nu}=\kappa^2\left(T_{\mu\nu}^{(\Phi)} +T_{\mu\nu}^{\rm (em)}\right)\,, \label{4D_Einstein} \\ && \Box \Phi -{\kappa \over 16\pi}\sum_{A} \lambda_A e^{\lambda_A\kappa\Phi} (F_{\mu\nu}^{(A)})^2=0\,, \label{eq_dilaton} \\ && \nabla^\nu \left(e^{\lambda_A\kappa\Phi} F_{\mu\nu}^{(A)}\right)=0 \label{eq_U(1)} \,, \end{eqnarray} where \begin{align} T_{\mu\nu}^{(\Phi)}&= \nabla_\mu \Phi\nabla_\nu \Phi -{1\over 2}g_{\mu\nu}(\nabla\Phi)^2\,, \\ T_{\mu\nu}^{\rm (em)} &= {1\over 4\pi}\sum_{A} e^{\lambda_A\kappa\Phi}\left[F_{\mu\rho}^{(A)}F_{~~\nu}^{(A)\rho} -{1\over 4}g_{\mu\nu}(F_{\alpha\beta}^{(A)})^2\right] \,. \end{align} For the present case with all the same charges, two different coupling constants appear. A simple calculation shows that our spacetime metric~(\ref{metric}) satisfies the above basic equations (\ref{4D_Einstein}), (\ref{eq_dilaton}), and (\ref{eq_U(1)}), provided the dilaton profile \begin{align} \kappa\Phi&= {\sqrt{6}\over 4}\ln\left({H_T \over H_S}\right) \label{Phi}\,, \end{align} and four electric gauge-fields \begin{align} \kappa F_{01}^{(T)} &= - {\sqrt{2\pi}Q\over r^2H_T^2}\,, \nonumber \\ \kappa F_{01}^{(S)} &=\kappa F_{01}^{(S')}= \kappa F_{01}^{(S'\hspace{-.1em}')}= - {\sqrt{2\pi}Q\over r^2H_S^2} \label{form_field} \,, \end{align} with the coupling constants \begin{align} \lambda_T=\sqrt{6}\,,\qquad \lambda_{S}\equiv \lambda_{S'}\equiv \lambda_{S'\hspace{-.1em}'}=-\sqrt{6}/3. \end{align} The U(1) fields are expressed in terms of the electrostatic potentials $F_{\mu \nu }^{(A)}=\nabla_\mu A^{(A)}_\nu - \nabla_\nu A_\mu^{(A)}$ as, \begin{eqnarray} \kappa A_{0}^{(T)}&=&{\sqrt{2\pi}\over H_T}\,, \nonumber \\ \kappa A_{0}^{(S)}&=&\sqrt{2\pi}{\left({1\over H_S}-1\right)} \label{U(1)_potential} \,, \end{eqnarray} where we have tuned $A_0^{(S)}$ to assure $A_0^{(S)}\rightarrow 0$ as $r\rightarrow\infty$ using a gauge freedom. Therefore the present spacetime~(\ref{metric}) is the exact solution of the Einstein-Maxwell-dilaton system~(\ref{4D_action}). One may verify that $Q$ is the (electric) charge satisfying \begin{align} \frac{Q}{\sqrt G}=\frac{1 }{4\pi} \int_S e^{\lambda_A\kappa \Phi}F_{\mu \nu }^{(A)} {\rm d} S^{\mu \nu }, \end{align} where $S$ is a round sphere surrounding the source. This expression is obtainable by the first integral of Eq.~(\ref{eq_U(1)}). Note that one can also find magnetically charged solution instead of (\ref{form_field}). However, this can be realized by a duality transformation \begin{align} \Phi \to -\Phi, \qquad F_{\mu \nu }^{(A)}\to \frac 12e^{\lambda_A\kappa \Phi}\epsilon_{\mu\nu\rho\sigma}F^{(A)\rho \sigma }\,, \end{align} which is a symmetry involved in the action~(\ref{4D_action}). Henceforth, we will make our attention only to the electrically charged case. This restriction does not affect the global spacetime picture. \subsection{Energy density and pressure} Using our solution (\ref{4D_metric}), we can evaluate the components of the energy-momentum tensors, i.e., the energy density and pressures for each field ( the dilaton $\Phi$ and U(1) fields $F_{\mu \nu }^{(A)}$). They are given by \begin{widetext} \begin{eqnarray} \rho^{(\Phi)}=P^{(\Phi)}_r ={1\over 2}\left(\Xi^{-1}\,\dot{\Phi}^2+\Xi\,\Phi'^2\right) &=&{3\over 16\kappa^2}\left[ {1\over t_0^2}\left({H_S\over H_T}\right)^{3/2}+ {Q^2\over r^4}{(H_T-H_S)^2\over (H_T^5H_S^7)^{1/2}} \right] >0\,, \label{Phi_energy} \\ P^{(\Phi)}_\theta=P^{(\Phi)}_\phi ={1\over 2}\left(\Xi^{-1}\,\dot{\Phi}^2-\Xi\,\Phi'^2\right) &=&{3\over 16\kappa^2}\left[ {1\over t_0^2}\left({H_S\over H_T}\right)^{3/2}- {Q^2\over r^4}{(H_T-H_S)^2\over (H_T^5H_S^7)^{1/2}} \right]\,, \label{Phi_pressure} \\ \rho^{\rm (em)}=-P^{\rm (em)}_r=P^{\rm (em)}_\theta=P^{\rm (em)}_\phi &=& {1\over 8\pi}\left[ e^{\lambda_T\kappa\Phi} (F_{01}^{(T)})^2 +3e^{\lambda_S\kappa\Phi} (F_{01}^{(S)})^2 \right]\,, \nonumber \\ &=&{Q^2\over 4\kappa^2r^4}\left[ {1\over H_T^4}\left({H_T\over H_S}\right)^{3/2}+ {3\over H_S^4}\left({H_T\over H_S}\right)^{-1/2} \right]>0 \label{em_energy} \,. \end{eqnarray} \end{widetext} The time $t=t_0$ is special at which the energy density of the scalar field is uniform $\kappa^2 \rho^{(\Phi)} (t_0)=3/(16t_0^2)$, and soon after it becomes gradually inhomogeneous. We shall refer to $t_0$ as the {\it fiducial time}. As the universe expands, the energy density of the scalar field at infinity behaves $\rho^{(\Phi)}\propto t^{-3/2}\propto \bar t^{-2}\propto a^{-6}$, as expected for the FLRW universe with stiff matter or a massless scalar field. The energy density of the U(1) fields evaluated at the fiducial time is $\kappa^2\rho^{\rm (em)} (t_0)=Q^2/(rH_S)^4=Q^2/(r+Q)^4$, which is the same as that of the extreme RN spacetime, and decreases in time as $\kappa^2\rho^{\rm (em)}(t, r)\propto t^{-1/2}\propto\bar t^{-2/3}\propto a^{-2}$ near infinity as in the same manner for the FLRW universe. We can also find the energy flux $J^\mu :=T^{\hat 0 \mu }$ whose spatial component ${\cal F}$ is given by \begin{align} \kappa^2{\cal F} =\kappa^2 T^{\hat 0}_{~\hat 1}=\kappa^2 {T^{(\Phi)\hat 0}}_{\hat 1} =-{3Q\over {8}t_0} {(H_T-H_S)\over r^2 H_T^2 H_S} \,, \label{flux} \end{align} where a hat denotes the tetrad component. Only the scalar field contributes to the energy flux, since there exist no magnetic fields, i.e., no Poynting flux. One can find that the $\mathcal F$ becomes negative for $t>t_0$, implying that the scalar field energy is falling toward the black hole. However, as it turns out later, the flux never gets into the black hole. This may be attributed to that the repulsive force caused by the U(1) fields becomes strong near the horizon and finely balances the attractive gravitational force of the dilaton field. It is worthwhile here to discuss the issue of the energy conditions. Many candidates of ``black hole solutions'' in expanding universe found in the literature (McVittie's solution~\cite{Mcvittie1933}, Sultana-Dyer solution~\cite{SultanaDyer}, etc) do not respect energy conditions in the whole of spacetime. Now the present system--U$(1)$ gauge fields coupled to dilaton--apparently satisfies the energy condition and hence it provides us a nontrivial example. We may verify this explicitly as follows. Inspecting Eqs. (\ref{Phi_energy})--(\ref{em_energy}), we notice $\rho^{(\Phi)}= P_r^{(\Phi)} \ge P_\theta^{(\Phi)}$ and $\rho ^{(\rm em)}=-P_r^{(\rm em)} \ge P_r^{(\rm em)}$, from which we obtain \begin{align} \rho =& \rho^{(\Phi)}+\rho^{\rm (em)} =P_r^{(\Phi)}-P_r^{(\rm em)}\nonumber \\ \ge & P_r^{(\Phi)}+P_r^{(\rm em)}=P_r, \nonumber \\ \rho =&\rho^{(\Phi)}+\rho^{\rm (em)}\ge P_\theta^{(\Phi)}+P_\theta^{(\rm em)}=P_\theta \,, \end{align} and \begin{align} \rho+P_r&= ( \rho^{(\Phi)}+P_r^{(\Phi)} )+(\rho^{(\rm em)}+P_r^{(\rm em)})>0\,, \nonumber \\ \rho+P_\theta &=(\rho^{(\Phi)}+P_\theta^{(\Phi)}) +(\rho^{(\rm em)}+P_\theta^{(\rm em)})>0\,. \end{align} These equations mean that $\rho\ge |P_i| ~(i=1,2,3)$ is satisfied anywhere. The energy flux $J^\mu =T^{\hat 0\mu }$ satisfies \begin{align} J^\mu J_\mu &=-\rho^2+\mathcal F^2 \nonumber \\ &=-(\rho^{({\rm em })})^2 -2\rho^{({\rm em})}\rho^{(\Phi)}-(P_\theta^{(\Phi)})^2 <0\,, \end{align} where we have used the relation $(\rho^{(\Phi)})^2-\mathcal F^2=(P_\theta^{(\Phi)})^2$ at the second equality. Hence, the energy current $J^\mu $ is a timelike vector everywhere. It then follows that the spacetime~(\ref{4D_metric}) satisfies the dominant energy condition [$\rho\ge |P_i| ~(i=1,2,3)$ and $J^\mu $ is non-spacelike]. \bigskip \subsection{Misner-Sharp energy} Another useful quantity to characterize matter fields is the quasilocal energy, which is defined on the closed two-surfaces. If the spacetime has an SO($3$)-symmetry, we are able to give a physically satisfactory quasilocal energy introduced by Misner and Sharp~\cite{MS1964}. The utility of spherical symmetry lies in the fact that we can covariantly employ the circumference radius, \begin{align} R(t, r):= |r| \Xi^{-1/2}=|r|(H_TH_S^3)^{1/4}, \label{R} \end{align} in terms of which the area of metric sphere is given by $4\pi R^2$. $R$ is a geometrical quantity and has an invariant meaning. It is occasionally of great advantage, instead of the comoving coordinate $r$, to make use of $R$. Using the circumference radius, the Misner-Sharp quasilocal energy is defined by~\cite{MS1964}, \begin{align} m(t,r) :=\frac{4\pi R}{\kappa^2} \left[1-g^{\mu \nu }(\nabla_\mu R)(\nabla_\nu R)\right]\,. \end{align} This quantity is a useful local measure to demonstrate geometric properties of spacetime~\cite{Hayward1994,hideki,nozawa}. The Misner-Sharp energy represents a mass energy contained inside the surface of radius $R$. Once the compact surface is specified, the Misner-Sharp mass is given without any ambiguity. Such a quasi-localization is possible because of spherical symmetry, in which no gravitational wave exists. By definition, it is characterized by geometric structure (metric components and its first derivatives) and does not require the premise of asymptotic structure of spacetime. So, it is considerably advantageous for the analysis of local spacetime structure. Physical interpretation of the Misner-Sharp energy has been further backed by various desirable properties: it satisfies the first law of thermodynamics~\cite{Hayward:1997jp}, and it shows the properties of positivity and monotonicity under the dominant energy conditions, and it reduces to ADM mass in the asymptotically flat spatial infinity. Reference~\cite{hideki} reinterpreted it by the integral of a locally conserved energy current coming from the symplectic structure of spherical symmetry. One may find the superiority of the use of Misner-Sharp energy in the next section. The present metric (\ref{metric}) with (\ref{metric_Xi}) gives rise to \begin{widetext} \begin{align} \kappa^2 m = \frac{\pi |r|[r^2H_TH_S^5+16t_0^2H_T^2H_S^2-(H_St+3H_Tt_0)^2]}{4t_0^2 (H_T^{7}H_S^{5})^{1/4}}. \label{MSenergy} \end{align} Physical meaning of each term in this equation is best understood at the fiducial time $t=t_0$, at which the Misner-Sharp mass is expressed as \begin{align} \kappa^2 m(t_0,R) = {\pi(|r|H_S)^3\over 4 t_0^2} +4\pi{Q\over H_S}+4\pi Q =\kappa^2\left( {4\pi\over 3}R^3\,\rho^{(\Phi)}(t_0) +4\pi \int_Q^R R^2 {\rm d} R \rho^{\rm (em)}(t_0,R)\right) +4\pi Q \,. \label{MSenergy_0} \end{align} \end{widetext} The first term corresponds to the energy of the scalar field, and the second and the last terms to the U(1) energies outside of and inside of the black hole, respectively. If we set $Q=0$, the second and third terms in Eq. (\ref{MSenergy}) are combined to cancel, and $m$ is expressed by the coordinates~(\ref{4D_metric1}) as \begin{align} m=\frac{a r^3}{2 G} \left(\frac{{\rm d} a}{{\rm d} \bar t}\right)^2 =\frac{4\pi }{3} (ar)^3 \rho^{(\Phi)}, \end{align} as expected for the background FLRW universe. This also justifies the first term in (\ref{MSenergy}) to be the contribution of the scalar field. \section{Spacetime structure} \label{sec:spacetime_structure} Let us now turn to the main task of revealing spacetime structure of the solution~(\ref{metric}). In order to address this issue, it is of significance to discuss the followings: \begin{itemize} \item Singularity \item Trapped surface \item Event horizon \item Asymptotic structure \end{itemize} The first two topics are associated with the local character of spacetime, whereas the last two require global considerations. These are elementary issues to be explored in order to characterize the spacetime. We wish to show that our metric~(\ref{metric}) describes a black hole in the FLRW universe. To this end, we need to establish in the first place that the far outside region from the central inhomogeneous domain behaves non-pathologically. If the spacetime admits a naked singularity in the asymptotic region--other than the initial big bang singularity--that are not covered by the event horizon, the solution would not gain a popularity as a black hole. According to a series of theorems due to Penrose and Hawking~\cite{HE}, the appearance of spacetime singularity is closely associated to the presence of trapped surfaces. Thus, the examination of trapping property may provide us useful information of local spacetime geometry. In particular, we have two competing effects due to the black hole and the expanding cosmology: the former tends to focus light rays back into the hole while the latter tends to spread it out to infinity. At first sight, one might expect to gain only a limited perception from the local point of view, even though the curvature singularity and local horizons are indeed of importance. Nevertheless, under physically reasonable circumstances, the existence of the local horizon implies that the event horizon lies outside it~\cite{HE}. To be more precise, all trapped surfaces are contained within black holes under the null energy condition provided the spacetime is asymptotically flat with some additional technical assumptions (see Proposition 9.2.8 in~\cite{HE} for the proof). So, this criterion is of use for our study. In order to define a black hole as a ``region of no escape,'' the spacetime must allow null infinity as an idealization of an observer sufficiently far in the distance from some central region. Thus the asymptotic analysis should also be put on an emphasis. In particular, we will take a close look at null geodesic motions, since the null rays play a privileged r\^ole in the black hole geometry. Above listed considerations are sufficient to provide us with insight into the global pictures of our dynamical spacetime~(\ref{4D_metric1}). In the remainder of this paper, we shall simplify our notations by using dimensionless variables $\tilde t:={t/t_0}$ and $\tilde r := {r/ Q}$. The metric is also rewritten into dimensionsless form ${\rm d} \tilde s_4^2=Q^{-2}\,{\rm d} s_4^2 $ as \begin{eqnarray} {\rm d} \tilde s_4^2 =-\tau^2 \, \tilde\Xi \, {\rm d} \tilde t^2 +\tilde \Xi^{-1}\left({\rm d} \tilde r^2+\tilde r^2 {\rm d} \Omega_2^2\right) \,, \label{4D_metric0} \end{eqnarray} with \begin{eqnarray} \tau :={t_0\over Q} \,,\qquad \tilde\Xi :={\tilde r^2\over [ (1+\tilde t\tilde r)(1+\tilde r)^3]^{1/2}} \,. \label{tau_Xi} \end{eqnarray} In this form, the metric involves only one dimensionless parameter $\tau=t_0/Q$, the physical meaning of which will be revealed below. We will affix ``tilde'' to denote dimensionless quantities in what follows. \subsection{Singularity} \label{sec:singularity} The scalar curvature and the Kretschmann invariant scalar are given by \begin{widetext} \begin{align} {\cal R}=& \kappa^2 \left(P_\theta^{(\Phi)}+P_\phi^{(\Phi)}\right) ={3\over 8Q^2}\left[ {1\over \tau^2}\left({H_T\over H_S}\right)^{3/2}- {1\over \tilde r^4}{(H_T-H_S)^2\over (H_T^{5}H_S^{7})^{1/2}} \right], \\ {\cal R}_{\mu\nu\rho\sigma}{\cal R}^{\mu\nu\rho\sigma} =& {1\over 64 \tau^4 Q^4 H_T^5 H_S^7 } \Biggl[15H_T^2H_S^{10} +6\left({\tau^2 \over \tilde r^4}\right)H_T H_S^5 \left(7H_T^2+10 H_T H_S-H_S^2\right) \nonumber \\ &+ \left({\tau^4 \over \tilde r^6}\right)\left\{ 9(31H_S^2+2H_S+159)H_T^4 +12(7H_S^2+ 9 H_S+ 96)H_T^3H_S \right. \nonumber \\ &+ \left. 6(15H_S^2-126H_S+143)H_T^2H_S^2 -12(H_S-1)(H_S+15)H_TH_S^3 +71(H_S-1)^2H_S^4 \right\} \Biggr]\,. \label{Kretschmann} \end{align} \end{widetext} These curvature invariants diverge when $H_T=0$ and $H_S=0$, that is, at \begin{eqnarray} \tilde t=\tilde t_s(\tilde r ):=-1/\tilde r, \qquad {\rm and } \qquad \tilde r=-1 \,. \end{eqnarray} At these spacetime points, the circumference radius $R$, Eq.~(\ref{R}), vanishes, i.e., they are central shell-focusing singularities. Thus, around infinity is free from singularities and is well-behaved. It deserves to observe that $\tilde t=0$ surface, where the scale factor $a(\bar t)$ appearing in the metric (\ref{4D_metric1}) vanishes, is not singular at all since the curvature invariants remain finite therein. It follows that the big bang singularity $\tilde t=0 $ is smoothed out due to a nonvanishing Maxwell charge $Q~(>0)$. Hence, one has also to consider the $\tilde t<0$ region in the coordinates (\ref{metric}). In addition, we find that the $\tilde r=0$ surface is neither singular, thereby we may extend the spacetime across the $\tilde r=0$ surface to $\tilde r<0$. Since the allowed region is where $H_TH_S^3 >0$ is satisfied, we shall focus attention to the coordinate domain \begin{align} \tilde t\ge \tilde t_s(\tilde r),\qquad \tilde r\ge -1\,, \end{align} in the subsequent analysis. Another permitted region $\tilde t>\tilde t_s$ and $\tilde r<-1$ is not our immediate interest here, since it turns out to be causally disconnected to the outside region, as we shall show below. Possible allowed coordinate ranges are depicted in Figure~\ref{fig:allowed}. \begin{figure}[h] \begin{center} \includegraphics[width=6.5cm]{allowed.eps} \caption{ Allowed coordinate ranges. The grey zone denotes the forbidden region, and the dashed curves correspond to curvature singularities.} \label{fig:allowed} \end{center} \end{figure} Since our spacetime is spherically symmetric, electromagnetic and gravitational fields do not radiate. Thereby, it is more advantageous to concentrate on their ``Coulomb components.'' For this purpose, let us introduce the Newman-Penrose null tetrads by \begin{align} & l_\mu{\rm d} \tilde x^\mu=\sqrt{\frac{\Xi}{2}}(-\tau {\rm d} \tilde t+\Xi^{-1}{\rm d} \tilde r), \nonumber \\ & n_\mu{\rm d} \tilde x^\mu= \sqrt{\frac{\Xi}{2}} (-\tau {\rm d} \tilde t-\Xi^{-1}{\rm d} \tilde r), \label{NPtetrads} \\ & m_\mu{\rm d} \tilde x^\mu =\frac {\tilde r}{\sqrt{2\Xi}} ({\rm d} \theta + i \sin \theta {\rm d} \phi ). \nonumber \end{align} with $\bar m_\mu$ being a complex conjugate of $m_\mu$. They satisfy the orthogonality conditions $l^\mu n_\mu=-1=-m^\mu\bar m_\mu$ and $l^\mu l_\mu=n^\mu n_\mu =m^\mu m_\mu=\bar m^\mu\bar m_\mu=0$. Since $\tilde t$ is a timelike coordinate everywhere, $l^\mu $ and $n^\mu$ are both future-directed null vector orthogonal to metric spheres. The only nonvanishing Maxwell and Weyl scalar are their ``Coulomb part,'' $\phi_1^{(A)}:=-\frac 12F_{\mu\nu}^{(A)}(l^\mu n^\nu +\bar m^\mu m^\nu)$ and $\Psi_2 :=-C_{\mu\nu\rho\sigma}l^\mu m^\nu \bar m^\rho n^\sigma$, both of which are invariant under the tetrad transformations due to the type D character. It is readily found that \begin{align} \phi^{(T)}_1 =\frac{\sqrt \pi }{\sqrt 2 \kappa Q\tilde r^2 H_T^2}, \qquad \phi^{(S)}_1 =\frac{\sqrt \pi }{\sqrt 2 \kappa Q\tilde r^2 H_S^2}\,, \end{align} and \begin{align} \Psi_2 =\frac{\Xi_{,\tilde r}-\tilde r\Xi_{,\tilde r\tilde r}}{6Q^2 \tilde r} =\frac{6 \tilde r H_T^2+(H_T-H_S)^2+2\tilde t \tilde r H_S^2 }{8Q^2 \tilde r^4(H_T^{5}H_S^{7})^{1/2}}\,. \end{align} The loci of singularities at which these quantities diverge are the same as the positions of the above singularities. One may also recognize that at the fiducial time $\tilde t=1$, above curvature invariants are the same as the extremal RN solution, as expected. Let us next look into the causal structure of singularities. Since the Misner-Sharp mass~(\ref{MSenergy}) becomes negative as approaching these singularities [the third term in Eq.~(\ref{MSenergy}) begins to give a dominant contribution], we speculate from our rule of thumb that these singularities are both contained in the untrapped region and possess the timelike structure. To see this more rigorously, let us consider radial null geodesics \begin{align} \left. \frac{{\rm d} \tilde r}{{\rm d} \tilde t}\right|_\pm =\pm \tau \tilde \Xi \,, \label{radial_null} \end{align} in the neighborhood of these singularities. Here, the upper (lower) sign corresponds to the geodesics along the direction $l^\mu$ ($n^\nu$), which we shall refer to as outgoing (ingoing). If we can find an infinite number of null geodesics that emanate from and terminate into the singularity, the singularity turns out to be timelike. If there exists a unique geodesic terminating into (emanating from) the singularity and an infinite number of geodesics emanating from (terminating into) it, the singularity has an ingoing (outgoing) null structure. Whereas, if there exist an infinite number of ingoing and outgoing geodesics terminating into or emanating from the singularity, we can conclude that it is spacelike. We begin by the analysis of the singularity at $ \tilde r=-1$. In the vicinity of the singularity $ \tilde r=-1$, we suppose that the null geodesics have the following asymptotic solution, \begin{align} \tilde r+1 = C_1 \left| \tilde t-\tilde t_1 \right|^p, \label{asy_geodesics1} \end{align} where $C_1 $, $\tilde t_1$ and $p$ are constants. $C_1$ is taken to be positive since we are concerned with the region $ \tilde r>-1$. $\tilde t_1$ can be regarded as the arrival time of the ingoing null geodesics at the singularity, or the departure time of the outgoing null geodesics from the singularity. Consider first the $ \tilde t>\tilde t_1$ case, that is, the geodesics emanating from the singularity. Substituting the assumed form~(\ref{asy_geodesics1}) into~Eq. (\ref{radial_null}), we find that only the outgoing null geodesics have the solution, for which \begin{align} p=\frac{2}{5}, \qquad C_1^{5/2} = \frac 52\tau (1-\tilde t_1)^{-1/2} \,. \label{Cpt1} \end{align} This reveals that for $ \tilde t>\tilde t_1$ there exist radial null geodesics that departed the singularity at $ \tilde t=\tilde t_1$. Following the identical procedure, it can be shown that for $ \tilde t<\tilde t_1$ only the ingoing null geodesics have the solution for which $C_1$ and $p$ are given by~(\ref{Cpt1}). This means that there exist radial null geodesics that will get to the singularity at $ \tilde t=\tilde t_1$. Therefore, we establish that the singularity $ \tilde r=-1$ is truly timelike, that is, a locally naked singularity, since we have only one set of outgoing and ingoing null solutions parametrized by their arrival or emanating time. It is also obvious from Eq.~(\ref{Cpt1}) that the singularity $ \tilde r=-1$ exists only for $ \tilde t<1$. In an analogous fashion, assume the asymptotic form of the null geodesics near the singularity $ \tilde r=\tilde r_s (=-1/ \tilde t)$ as \begin{align} \tilde r+\frac{1}{ \tilde t} =C_2\left| \tilde t-\tilde t_2\right|^q\,, \label{asy_geodesics2} \end{align} where $C_2$, $\tilde t_2$ and $q$ are constants. In the present case, $C_2$ takes positive (negative) value for $\tilde t_2 > 1 ~(\tilde t_2< 1) $. \begin{widetext} Plugging this into Eq.~(\ref{radial_null}), we obtain $q=2/3$ and \begin{subequations} \label{Cqt2} \begin{align} C _2 ^{3/2} &= \frac{3}{2}\tau \left[\tilde t_2^2 (\tilde t_2-1)^3\right]^{-1/2}, \qquad \textrm{for outgoing null with $\tilde t>\tilde t_2$}\,,\label{Cqt2a}\\ (-C _2) ^{3/2}&= \frac{3}{2}\tau \left[\tilde t_2^2 (1-\tilde t_2)^3\right]^{-1/2}, \qquad \textrm{for ingoing null with $\tilde t>\tilde t_2$}\,, \label{Cqt2b}\\ C _2 ^{3/2}&= \frac{3}{2}\tau \left[\tilde t_2^2 (\tilde t_2-1)^3\right]^{-1/2}, \qquad \textrm{for ingoing null with $\tilde t<\tilde t_2$}\,, \label{Cqt2c}\\ (-C _2) ^{3/2}&= \frac{3}{2}\tau \left[\tilde t_2^2 (1-\tilde t_2)^3\right]^{-1/2}, \qquad \textrm{for outgoing null with $\tilde t<\tilde t_2$}\,. \label{Cqt2d} \end{align} \end{subequations} \end{widetext} Eqs.~(\ref{Cqt2b}) and (\ref{Cqt2d}) indicate that there exist null geodesics that originate from and come to the singularity at any time $\tilde t_2<0$. Thus, the singularity $\tilde t=-1/\tilde r ~(<0)$ appearing in the $ \tilde r>0$ region is also timelike. Whereas, Eqs.~(\ref{Cqt2a}) and (\ref{Cqt2c}) show that the timelike singularity $ \tilde r=-1/ \tilde t ~(<0)$ occurs only for $ \tilde t >1$. Although we have not dwelt on the existence proof of the asymptotic solutions [Eqs~(\ref{asy_geodesics1}) with (\ref{Cpt1}) and (\ref{asy_geodesics2}) with (\ref{Cqt2})] of the radial null geodesics, the proof can be obtained via the contraction mapping method following the argument in e.g., the appendix of~\cite{Nozawa}. It is also noted that it is sufficient to focus attention on the radial null geodesics in the vicinity of central singularities: causal geodesics excluding radial null geodesics will fail to emanate from singularity if radial null geodesics do not arise (see also the appendix in~\cite{Nozawa}). \subsection{Trapping horizons} \label{sec:TH} Our primary concern in this article is to reveal that the spacetime describes a black hole. However, this is not only technically but also conceptually difficult. Since a black hole is by definition a ``region of no escape,'' the locus of event horizon as its boundary has a teleological meaning. Unless we know the entire future of our universe, we are unable to determine whether a ``black hole candidate'' is qualified as a black hole. The concept of the black hole is considerably messy in practical point of view. In physically acceptable situations, however, we can rely on the the notion of {\it trapped surface}, first introduced by Penrose~\cite{Penrose:1964wq}. The concept of trapped surface is inherently local, such a difficulty does not arise. Imagine a massive star undergoing a gravitational collapse to form a black hole. There appears a region for which even ``outgoing'' null rays are dragged back due to strong gravity and have a negative expansion. For each time slice, this defines an {\it apparent horizon}~\cite{HE} as an outermost boundary of the trapped region in the asymptotically flat spacetimes. Hayward generalized these quasilocal concepts to define a class of {\it trapping horizons}~\cite{Hayward1993}. One strength of the use of trapping horizons is just to encompass various types of horizons associated not only with black holes but also with white holes and cosmological ones. As commented at the beginning of this section, the underlying aim of this direction is to gain useful guide for event horizon from these local analysis. Let us consider a compact spacelike orientable surface $S$. We take $S$ as a metric sphere, respecting an SO(3)-symmetry of background universe. It then follows that the Newman-Penrose tetrads $(l^\mu, n^\mu)$ defined in Eq.~(\ref{NPtetrads}) are normal to $S$ (i.e., they are radial) and future-directed null vectors. Due to the spherical symmetry, they are shear-free and rotation-free. Define the associated null expansions $\theta_\pm $ by \begin{align} \theta _+ :=2m^{(\mu}\bar m^{\nu)}\nabla_\mu l_\nu, \qquad \theta _- :=2m^{(\mu}\bar m^{\nu)}\nabla_\mu n_\nu. \end{align} In the coordinates~(\ref{4D_metric0}), they are expressed as \begin{align} \tilde \theta_\pm =\frac{\tilde rH_S(H_TH_S^3)^{1/2} \pm \tau(3 H_T+H_S\tilde t)} {2\sqrt 2 \tau \tilde r (H_T^{5}H_S^{7})^{1/4}}. \label{expansions} \end{align} Note that the signs of $\theta_\pm $ have an invariant meaning, but each value of $\theta _\pm$ is not a universal quantity due to the ``class III'' tetrad rotations. An invariant combination is their product $\theta_+\theta_-=-2R^{-2}(\nabla_\mu R)(\nabla^\mu R)$. Expansions $\theta_\pm$ characterize the extent to which the light rays are diverging or converging, or equivalently the rate at which the area of metric sphere is increasing or decreasing in the null directions. In terms of null expansions $\theta_\pm$, a metric sphere is said to be {\it trapped} ({\it untrapped}) if $\theta_+\theta_- > 0$ ($\theta_+\theta_- < 0$), and {\it marginal} when $\theta _+\theta_- =0$. A marginal surface is said to be {\it future} ({\it past}) if $\theta _+=0~ (\theta_-=0)$. A future marginal surface is further classified into {\it outer } ({\it inner}) if $n^\mu \nabla_\mu \theta _+< 0$ ($n^\mu \nabla_\mu \theta _+> 0$) and similarly a past marginal surface is called {\it outer } ({\it inner}) if $ l^\mu \nabla_\mu \theta _-> 0$ ($ l^\mu \nabla_\mu \theta _-<0$). A {\it trapping horizon} is the closure of a hypersurface foliated by future or past, and outer or inner marginal surfaces~\cite{Hayward1993}. In terms of the Misner-Sharp energy, $R< 2Gm$ ($R>2Gm$) defines the trapped (untrapped) region and the marginal surface ($\theta _+\theta_- =0$) is positioned at the ``Schwarzschild radius'' $R=2Gm$. A ``normal'' spacetime region as occurred in the flat space is composed of untrapped surfaces on which outgoing rays have positive expansions while the ingoing rays have negative expansions. Among these classes of trapping horizons, the future-outer trapping horizons turn out to be most relevant in the context of black holes. The future outer trapping horizon properly captures the intuitive idea that the ingoing null rays are converging with the outgoing null ray being instantaneously parallel on the horizon, diverging outside and converging inside. Inner trapping horizons are associated with cosmological horizons, and interior horizons of black holes. The past trapping horizons arise when discussing white holes and cosmological ones. Since the concept of trapping horizons is sufficiently general, it is considerably useful for the analysis of black holes especially in the non-asymptotically flat spacetimes. The expansions are intimately associated to the variation of the Misner-Sharp energy through the first law~\cite{Hayward1994}, \begin{align} l^\mu \nabla_\mu m&=2\pi R^3(T_{\mu \nu }l^\mu n^\nu \theta_ + -T_{\mu \nu }l^\mu l^\nu \theta_- ), \nonumber \\ n^\mu \nabla_\mu m&=2\pi R^3(T_{\mu \nu }l^\mu n^\nu \theta_ - -T_{\mu \nu }n^\mu n^\nu \theta_+ )\,. \label{variationFormula} \end{align} The present system satisfies the dominant energy condition, which implies $T_{\mu \nu }l^\mu l^\nu \ge 0$, $T_{\mu \nu }n^\mu n^\nu \ge 0$ and $T_{\mu \nu }l^\mu n^\nu \ge 0$. Hence, Eq.~(\ref{variationFormula}) establishes that $m$ is not decreasing (not increasing) along the $l^\mu $-direction ($n^\mu $-direction) in the untrapped region of $\theta_+>0$ and $\theta_-<0$. This illustrates that the Misner-Sharp energy is a monotonically increasing function toward outwards in the ordinary region where $\theta_+>0$ and $\theta_-<0$ are satisfied, which accords with our intuition. Let us now investigate the properties of trapping horizons more closely. From Eq.~(\ref{expansions}), trapping horizons occur at \begin{align} \tilde rH_S(H_TH_S^{3})^{1/2}\pm \tau(3 H_T+H_S\tilde t)=0, \label{THeq} \end{align} where the upper (lower) sign corresponds to $\tilde \theta_+=0$ ($\tilde \theta_-=0$). Noticing that $\tilde rH_S\ge 0$, Eq.~(\ref{THeq}) is solved as $\tilde t=\tilde t^{(\mp)}_{\rm TH}(\tilde r)$, where \begin{widetext} \begin{align} \label{TH} \tilde t_{\rm TH}^{(\mp)}(\tilde r):=& \frac{\tilde r^2}{2\tau^2(H_S+3)^2} \left[H_S^5-6\tau^2(H_S+3)\tilde r^{-3} \mp H_S^3 \sqrt{H_S^4+4\tau^2(H_S+3)\tilde r^{-3}}\right]. \end{align} \end{widetext} Here, $\tilde \theta_\pm=0$ holds at $\tilde t=\tilde t^{(\mp)}_{\rm TH}(\tilde r)$. In the pursuing subsection, we shall separately analyze the $\tilde r> 0$ and $\tilde r<0$ cases corresponding to outside and inside the black-hole event horizon. Before going into the detailed argument, we pause for a moment to discuss the behavior of $\tilde R(\tilde t, \tilde r)$. We have taken $l^\mu$ (and correspondingly $n^\mu$) in such a way that $\tilde r$ increases (decreases) along $l^\mu ~(n^\mu )$. From \begin{align} \tilde R_{,\tilde t} =\frac{|\tilde r| }{4} \left(\frac{H_S^3}{H_T^3}\right)^{1/4},\quad \tilde R_{,\tilde r} =\frac{|\tilde r|[3H_T+H_S\tilde t]} {4\tilde r(H_T^{3}H_S)^{1/4}}, \end{align} one finds that $\tilde R_{,\tilde r}> 0$ ($\tilde R_{,\tilde r}< 0$) holds for $\tilde t > \tilde t_{c}(\tilde r)$ ($\tilde t < \tilde t_{c}(\tilde r)$), where \begin{align} \tilde t_{c}(\tilde r):=-\frac{3}{4\tilde r+1 }. \end{align} This means that $\tilde R $ increases as $\tilde r$ grows for $\tilde t>\tilde t_c$, deserving $l^\mu $ to be called ``outgoing.'' However, this is no longer true for $\tilde t<\tilde t_c (\tilde r)$. This means that there exists a maximum value of $\tilde R(\tilde t, \tilde r)$ for given time (see Figure~\ref{fig:Sr}). It should be also remarked that the invariant scalar curvatures are all finite at this surface $\tilde t=\tilde t_c(\tilde r)$, this is not the shell-crossing singularity. \begin{figure}[t] \begin{tabular}{cc} \includegraphics[width=6cm]{Sr.eps} \end{tabular} \caption{ Typical behaviors of $\tilde R=[(\tilde t\tilde r+1)(\tilde r+1)^3]^{1/4}$ as a function of $\tilde r$ with fixed time. For fixed $\tilde t<0$, the circumference radius $\tilde R$ ceases to increase monotonically with $\tilde r$, but has a maximum at $\tilde t=\tilde t_c(\tilde r)$. For $\tilde t>0$, $\tilde R$ turns to be monotonic in $\tilde r$. } \label{fig:Sr} \end{figure} \subsubsection{Trapping horizons in the region of $\tilde r> 0$} Let us begin by the case of $\tilde r> 0$. After simple calculations, one obtains \begin{widetext} \begin{align} \tilde t_{\rm TH}^{(+)}-\tilde t_{c}(\tilde r)&= \frac{\tilde r^2}{2\tau (H_S+3)^2}\left(H_S^3\sqrt{H_S^4+{4\tau^2}{\tilde r^{-3}}(H_S+3)} +H_S^5\right)> 0\,, \\ \tilde t_{c}(\tilde r)-\tilde t_{\rm TH}^{(-)}&= \frac{r^2}{2\tau (H_S+3)^2}\left(H_S^3\sqrt{H_S^4+{4\tau^2}{\tilde r^{-3}}(H_S+3)} -H_S^5\right)> 0\,, \label{THminusTc}\\ \tilde t_{\rm TH}^{(-)}-\tilde t_s (\tilde r)&= \frac{\tilde r^2}{2\tau (H_S+3)^2} \left[ \frac 12H_S\left(H_S^2-\sqrt{H_S^4+4\tau^2\tilde r^{-3}(H_S+3)}\right)^2 +\frac 12 H_S^5+\frac{\tau^2}{\tilde r^2}H_S(H_S+3) \right]> 0\,, \end{align} \end{widetext} for $\tilde r> 0$. Figure~\ref{fig:TH} shows the typical curves of trapping horizons $\tilde t^{(\pm )}_{\rm TH}$. The region $\tilde t_{\rm TH}^{(-)}<\tilde t<\tilde t^{(+)}_{\rm TH}$ denotes a past trapped region in which even ingoing light rays are diverging due to the cosmic expansion. \begin{center} \begin{figure}[t] \includegraphics[width=5.5cm]{TH.eps} \caption{Typical curves of trapping horizons $\tilde t^{(\pm)}_{\rm TH}(\tilde r)$ for $\tau <\tau _{\rm crit}$ outside the event horizon $\tilde r>0$. The plot is $\tau =1$. $\tilde t^{(+)}_{\rm TH}$ first occurs at some time (in this case $\tilde t=\tilde r=1$) and bifurcates into two branches. The black dashed line and the grey dotted line denote $\tilde t_s(\tilde r)$ and $\tilde t_{c}(\tilde r)$, respectively.} \label{fig:TH} \end{figure} \end{center} Next, let us delve deeper into the type of trapping horizon. It is straightforward to find that \begin{align} l^\mu\tilde \nabla_\mu \tilde \theta_- =\left.\frac{F}{4\tilde r^4(H_T^5H_S^{7})^{1/2}} \right|_{\tilde t=\tilde t_{\rm TH}^{(+)}}, \label{THtype1} \end{align} along the trapping horizon with $\tilde \theta_-=0$. Here, we have introduced a function \begin{align} F:= H_S^2-(4\tilde r^2+8 \tilde r+1)H_T^2\,, \end{align} the sign of which controls the type of trapping horizon. For $\tilde r>0$, the inequality $\tilde r+1<\sqrt{4\tilde r^2 +8 \tilde r+1}$ holds, from which we obtain \begin{align} F <- \tilde t (H_T+1/r)(4\tilde r^2+8 \tilde r+1)<0\,. \end{align} Hence, for the trapping horizon with $\tilde t^{(+)}_{\rm TH}>0$, $l^\mu \tilde \nabla_\mu \tilde \theta_- <0$ is satisfied. That is to say, the past trapping horizon $\tilde t^{(+)}_{\rm TH}$ occurred in the $\tilde r>0, \tilde t>0$ region is always of outer-type. This may be ascribed to the cosmological origin since a past-outer trapping horizon develops when the background FLRW universe is filled with a stiff matter. More stringent bound for the condition $F>0$ is numerically found to be $\tau < \tau_{\rm crit} \sim 5.444$. In the case of $\tau> \tau_{\rm crit}$, a part of the past trapping horizon near $\tilde r=0$ becomes inner rather than outer. Similarly, we obtain \begin{align} n^\mu\tilde \nabla_\mu \tilde \theta_+ =\left.\frac{F}{4 \tilde r^4(H_T^5H_S^{7})^{1/2}} \right|_{\tilde t=\tilde t_{\rm TH}^{(-)}}\,, \label{THtype2} \end{align} along the trapping horizon with $\tilde \theta_+=0$. From Eq. (\ref{THminusTc}), we find \begin{align} F|_{\tilde t=\tilde t^{(-)}_{\rm TH}(\tilde r)} &>\left.H_S^2-(4\tilde r^2+8 \tilde r+1) H_T^2\right|_{\tilde t=\tilde t_c}\nonumber \\ &=12 H_S^2(3+H_S)^{-2}>0\,. \end{align} Hence, the trapping horizon $\tilde t^{(-)}_{\rm TH}$ is necessarily of future-inner type. The appearance of untrapped region $\tilde \theta_+<0 $ and $\tilde \theta_->0$ is due to the repulsive nature of the timelike naked singularity. From the general argument, the outer (inner) trapping horizons must be non-timelike (non-spacelike) under the null energy condition~\cite{Hayward1994,Nozawa}. So the above analysis asserts that the trapping horizon $\tilde t^{(+)}_{\rm TH}$ never becomes timelike for $\tau < \tau_{\rm crit} $, while $\tilde t^{(-)}_{\rm TH}$ cannot be spacelike anywhere. Here, it is worthwhile to remark the behavior of trapping horizons in various asymptotic limits. In the $\tilde r\to 0$ limit, Eq.~(\ref{TH}) yields that $\tilde r \tilde t^{(\pm)}_{\rm TH}$ is finite and given by \begin{align} &\tilde r \tilde t^{(+)}_{\rm TH} = \frac{1+ \sqrt{1+4\tau ^2}}{ 2\tau^2 }> 0\,, \label{r0limit1} \\ & \tilde r \tilde t^{(-)}_{\rm TH} = \frac{1- \sqrt{1+4\tau ^2}}{ 2\tau^2 }< 0\,, \label{r0limit2} \end{align} as $\tilde r\to 0$. Using this equation, one can find that the circumference radii of the trapping horizons $\tilde t^{(\pm)}_{\rm TH}$ respectively approach to some constants $\tilde R_\pm $ as $\tilde r\to 0$, where \begin{align} \tilde R_\pm := \left(\frac{\sqrt{1+4\tau^2 }\pm 1}{2\tau }\right)^{1/2}\,. \label{R_plusminus} \end{align} These surfaces correspond to the infinite redshift $(\tilde t\to \infty) $ and blueshift $(\tilde t\to -\infty )$ surfaces with respect to an asymptotic observer. We will see in the next section that these surfaces represent the black hole and white hole horizons. Notice that $\tilde R_+ $ (and respectively $\tilde R_-$) is a monotonically decreasing (increasing) function of $\tau $ and they behave as $\tilde R_+ \to \infty $ and $\tilde R_-\to 0$ in the limit $\tau\to 0$, while they asymptotically tend to unity as $\tau \to \infty $. According to Eq. (\ref{R_plusminus}), $Q$ and $\tau$ are expressed in terms of $R_\pm$ as \begin{align} Q=\sqrt{R_+R_-}, \qquad \tau =\frac{R_+R_-}{R_+^2-R_-^2}\,. \label{Qtau} \end{align} Hence, we find that the charge $Q$ sets the geometrical mean of horizon radii and their relative ratio is encoded in the parameter $\tau $. The physical meaning of $\tau$ is found by evaluating the energy densities of the dilaton field and U(1) field at the horizon $R_+$ as follows: Those densities are given by \begin{align} &\kappa^2\rho^{(\Phi)}|_{R_+} ={3\over 8t_0^2\tilde R_+^6}\,, \\ &\kappa^2\rho^{\rm (em)}|_{R_+} ={1+3\tilde R_+^{8}\over 4Q^2\tilde R_+^{10}} \,. \end{align} Then, the ratio is found to be \begin{align} {\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+} ={2\tau^2 (1+3\tilde R_+^{8})\over 3\tilde R_+^4} \,. \end{align} From the expression of $R_+$, we find that $\tau $ has a one-to-one correspondence to $\rho ^{(\rm em)}/\rho^{(\Phi)}|_{R_+}$ and is given by \begin{align} \tau^2&={1\over 8}\left[3\left({\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+}-1\right)-\sqrt{3\left(2\,{\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+}-1\right)} \right]\,, \nonumber \\ &\approx \left\{ \begin{array}{ccc} {3\over 8} \,{\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+} & {\rm for} & {\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+} \gg 2\,, \\ {1\over 4}\left({\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+}-2\right) & {\rm for} & {\rho^{\rm (em)} \over \rho^{(\Phi)}}\Big{|}_{R_+} \sim 2\,. \end{array} \right. \nonumber \\~ \end{align} $\tau$ is a monotonic function of the ratio, and it vanishes when the ratio approaches $2$. The ratio $(\rho^{\rm (em)}/\rho^{(\Phi)})|_{R_+}$, which must be larger than 2 for $\tau^2>0$, corresponds to $\tau$ by a one-to-one mapping. Hence $\tau$ is related to the ratio of two densities at the horizon. In the limit $\tilde r\to 0$, one finds that $\tilde \theta_+ $ and $\tilde \theta _-$ both vanish, implying that the $\tilde r=0$ surface becomes degenerate into an ingoing and outgoing null structure. On the other hand, taking $\tilde r\to \infty $, $\tilde t^{(+)}_{\rm TH}$ diverges as $(\tilde r^2/4\tau)^2$, whence $\tilde R (\tilde t^{(+)}_{\rm TH} (\tilde r), \tilde r) \to \tilde r^3/4\tau \to \infty $. While we have $\tilde t^{(-)}_{\rm TH}\to -1/\tilde r= \tilde t_{s}(\tilde r)$ as $\tilde r\to \infty $. But this does not mean that the trapping horizon $\tilde t^{(-)}_{\rm TH}$ plunges into the singularity $\tilde t_s$ as $\tilde r\to \infty$. One can verify by taking higher order terms into account that $\tilde t^{(-)}_{\rm TH}$ tends to have a constant radius $\tilde R (\tilde t^{(-)}_{\rm TH} (\tilde r), \tilde r)\to \sqrt{\tau}$ in this limit. \subsubsection{Trapping horizons in the region of $r<0$} For negative values of $\tilde r$, $\tilde t^{(-)}_{\rm TH}>\tilde t^{(+)}_{\rm TH}$ holds, in contrast to the $\tilde r>0$ case. Two trapping horizons $\tilde t^{(\pm)}_{\rm TH}$ develop only in the region $\tilde r_0<\tilde r< 0 $ and they coincide at the point $\tilde r=\tilde r_0$, where the square-root of Eq.~(\ref{TH}) vanishes. Namely, $\tilde r_0$ satisfies \begin{align} I(\tilde r_0):= H_S^4(\tilde r_0)+4\tau^2(H_S(\tilde r_0)+3)\tilde r_0^{-3}=0\,. \end{align} Since $I(-1)=-12\tau^2 <0$ and $I(-1/4)=81 >0$, $-1<\tilde r_0<-1/4$ is concluded. We can also find that the trapping horizon $\tilde t^{(+)}_{\rm TH}$ negatively diverges at $\tilde r=-1/4$. Figure~\ref{TH_negativer} plots typical curves of trapping horizons occurred in $\tilde r<0$. Equations (\ref{THtype1}) and (\ref{THtype2}) continue to be true for $\tilde r<0$. We can find numerically that at $\tilde t^{(+)}_{\rm TH}(\tilde r)$ $F<0$ holds around $\tilde r\sim 0$ implying that it is spacelike. The future trapping horizon $\tilde t^{(-)}_{\rm TH}(\tilde r)$ and other portion of past trapping horizon $\tilde t^{(+)}_{\rm TH}(\tilde r)$ become timelike. In the limit $\tilde r\to 0$, the trapping horizons $\tilde t^{(\pm)}_{\rm TH}(\tilde r)$ have constant circumference radii $\tilde R_\pm$ as outside. \begin{figure}[t] \bigskip \begin{center} \includegraphics[width=5.5cm]{TH_negativer.eps} \caption{Typical curves of trapping horizons $\tilde t^{(\pm)}_{\rm TH}(\tilde r)$ for $\tau<\tau_{\rm crit}$ inside the event horizon $\tilde r<0$. The plot is for $\tau=1$, for which the trapping horizon $\tilde t^{(+)}_{\rm TH}$ has a maximum at $\tilde r\simeq -0.1080$, spacelike for $-0.1299\lesssim \tilde r<0$ and merge with $\tilde t^{(-)}_{\rm TH}$ at $\tilde r_0\simeq -0.2679$. } \label{TH_negativer} \end{center} \end{figure} \subsubsection{Constant $R$ surfaces} The trapping horizons occur at $(\tilde \nabla_\mu \tilde R)(\tilde \nabla^\mu \tilde R)=0$, i.e., the surfaces of $\tilde R={\rm constant}$ become null. Then the contours of circumference radius will help us to recognize the positions of trapping horizons in terms of circumference radius. Solving Eq.~(\ref{R}) with respect to $\tilde t$, we obtain \begin{eqnarray} \tilde t={1\over \tilde r}\left[{\tilde R^4\over (1+\tilde r)^3}-1\right] \,. \end{eqnarray} Taking the derivative of this equation with fixing $\tilde R$, we find the relation between ${\rm d} \tilde t$ and ${\rm d} \tilde r$. Inserting this to the metric~(\ref{4D_metric0}), the line element of $\tilde R={\rm constant}$ surface is given by \begin{widetext} \begin{eqnarray} {\rm d} \tilde s^2=-{\tau^2(1+4\tilde r)^2\over \tilde r^2 \tilde R^2 (1+\tilde r)^8} \left(\tilde R^2-\tilde R_1^2\right) \left(\tilde R^2+\tilde R_1^2\right) \left(\tilde R^2-\tilde R_2^2\right) \left(\tilde R^2+\tilde R_2^2\right) {\rm d} \tilde r^2\,, \end{eqnarray} where \begin{align} \tilde R_1^2&=\tilde R^2 (\tilde t_{\rm TH}^{(+)}(\tilde r), \tilde r)= {(1+\tilde r)^4\over 2\tau |1+4\tilde r|}\left[1 +\sqrt{1+{4\tau^2(1+4\tilde r) \over (1+\tilde r)^4}}\right]\,, \\ \tilde R_2^2&=\tilde R^2 (\tilde t_{\rm TH}^{(-)}(\tilde r), \tilde r) ={(1+\tilde r)^4\over 2\tau(1+4\tilde r)}\left[-1 +\sqrt{1+{4\tau^2(1+4\tilde r) \over (1+\tilde r)^4}}\right] \,. \end{align} \end{widetext} One finds that $\tilde R_1\to \tilde R_+$ and $\tilde R_2\to \tilde R_-$ in the limit $\tilde r \to 0$, and $\tilde R_1^2\to \infty$ and $\tilde R_2^2 \to \tau $ as $\tilde r\to \infty $. It is notable that $\tilde R_1$ and $\tilde R_2$ are not independent but fulfills the constraint \begin{align} \tau =\frac{\tilde R_1^2 \tilde R_2^2} {\tilde R_1^2-\tilde R_2^2}\,. \end{align} From $\tilde R_1^2>\tilde R_2^2>0>-\tilde R_2^2>-\tilde R_1^2$, we conclude that if $\tilde R^2>\tilde R_1^2$ or $\tilde R^2<\tilde R_2^2$, the $\tilde R={\rm constant}$ curves are timelike, while they are spacelike when $\tilde R_2^2<\tilde R^2<\tilde R_1^2$. In the region where $\tilde R={\rm constant}$ curves are timelike (spacelike), a metric sphere is untrapped (trapped). We sketch in Figure~\ref{fig:constantR} the region where $\tilde R={\rm constant}$ curves are timelike (grey region), and spacelike (white region). As $\tilde r\to 0$, we find that $\tilde R_\pm$ are null surfaces. \begin{figure}[t] \includegraphics[width=7cm]{spacelike.eps} \caption{Signature of $\tilde R={\rm constant}$ surfaces for $\tau =1$. $\tilde R={\rm constant}$ curve is timelike (grey region) and spacelike (white region), which are separated by trapping horizons $\tilde t^{(\pm)}_{\rm TH}$} \label{fig:constantR} \end{figure} \subsection{Event horizons} \label{sec:EH} Let us proceed to demonstrate the structure of the future and past event horizons, utilizing several results obtained in the previous sections. Before embarking on this programme, let us recapitulate basic features of event horizons. The future event horizon is defined by a future boundary of the causal past of future null infinity. The past event horizon is similarly defined by interchanging the r\^ole of future and past. These event horizons are by definition the achronal 3D null surfaces. In addition, the future (past) event horizon is generated by null geodesic generators which have no future (past) endpoint in $(M, g_{\mu \nu })$~\cite{HE}. If the spacetime is stationary, the black hole event horizon must be a Killing horizon~\cite{HE} (see~\cite{Hollands:2008wn} for the degenerate case). This theorem is considerably significant because it enables us to identify the locus of event horizon simply from the spacetime symmetry. It is {\it a priori} no interrelationship between the event horizon and the Killing horizon. Such a lucky consequence is exceptional to the stationary case. In the dynamical case, on the other hand, we have no specific guidelines for identifying the future event horizon but to evolve the spacetime into the infinite future.\footnote{ The Sultana-Dyer solution~\cite{SultanaDyer} is an exceptional instance: it is conformal to the Schwarzschild metric hence its causal picture is extracted in a simple fashion~\cite{Saida:2007ru}. } Nevertheless, we can say, regardless of this adversity, that the black hole event horizon has to cover the trapped surfaces, provided the outside region of a black hole is sufficiently well-behaved~\cite{HE}. Inspecting that the spacetime~(\ref{4D_metric0}) appears to have a good behavior at least for $\tilde r>0$ and that the $\tilde \theta_\pm =0$ surfaces comprise null surfaces~(\ref{R_plusminus}) in the limit $\tilde r\to 0$ and $\tilde t\to \pm \infty $, it may be reasonable to consider these null surfaces as the possible candidates of black and white hole horizons. Analyzing the near-horizon geometry and behaviors of null geodesics, we shall see below that this expectation is indeed true. \subsubsection{Near horizon geometry} From the behavior of trapped surfaces, we can deduce that the null surface $\tilde r=0$ is a plausible horizon candidate. We shall scrutinize the structure of this surface in detail. We first look at the ``throat'' geometry~(\ref{near_horizon}). Taking the fiducial time $\tilde t=1$ for simplicity (the same conclusion is derived for any value of finite $\tilde t$), the proper distance $\tilde s$ from the spacetime point $(1, \tilde r, \tilde\theta , \tilde\phi)$ to $(1, \tilde r=0, \tilde \theta , \tilde\phi)$ is given by \begin{align} \tilde s&= \lim_{\tilde r_0 \to 0 } \int ^{\tilde r}_{\tilde r_0} \sqrt{\tilde g_{\tilde r\tilde r} (1, \tilde r)} {\rm d} \tilde r \nonumber \\ &=\lim_{\tilde r_0 \to 0 } \biggl[\tilde r +\ln \tilde r\biggl]_{\tilde r_0}^{\tilde r} \to \infty \,. \label{distance} \end{align} This implies that the point $\tilde r=0$ corresponds not to the regular origin of polar coordinates but to ``spatial infinity,'' as in the extremal RN spacetime. In the extremal RN case, the future (past) event horizon is a null surface generated by $\tilde r=0$ and $\tilde t= \infty $ ($t=-\infty $), with its infinite ``throat'' at $\tilde r=0$ with $\tilde t$ being finite. Analogously, the event horizon of present spacetime, if it exists, should have $\tilde r=0$ with finite $\tilde t$ as its ``throat.'' So we deduce that the only candidate of event horizons in the present spacetime is $\tilde r=0$ and $\tilde t=\pm \infty$. Therefore, the most convenient way to see the structure of these candidate horizons is to take the {\it near-horizon limit}, defined by \begin{align} \tilde t\ \to \frac{\tilde t}{\epsilon }, \qquad \tilde r\to \epsilon \tilde r, \qquad \epsilon \to 0\,, \label{rescaling} \end{align} where $\epsilon $ is a positive constant. Other conceivable limits fail to produce any sensible results. After the rescaling, the coordinate ranges of $\tilde t$ and $\tilde r$ are free from a restriction other than $\tilde r>0$. Taking the limit (\ref{rescaling}) in Eq.~(\ref{4D_metric0}), one obtains the near-horizon metric: \begin{align} {\rm d} \tilde s_{\rm NH}^2 =& -{\tau^2 \tilde r^2 (1+\tilde t \tilde r)^{-1/2}} {\rm d} \tilde t^2 \nonumber \\ & +\tilde r^{-2}{(1+\tilde t\tilde r)^{1/2}} \left({\rm d} \tilde r ^2+\tilde r^2 {\rm d} \Omega_2^2 \right)\,, \label{NHmetric} \end{align} which does not involve the parameter $\epsilon $. As a direct consequence of (\ref{rescaling}), the near-horizon metric~(\ref{NHmetric}) is invariant under the flow \begin{align} \xi^\mu =\tilde t\left(\frac{\partial }{\partial \tilde t}\right)^\mu -\tilde r\left(\frac{\partial }{\partial \tilde r}\right)^\mu \,. \label{horizonKilling} \end{align} Namely, $\xi^\mu $ is a Killing vector in the spacetime~(\ref{NHmetric}). Changing the coordinate $\tilde r $ to the circumference radius $\tilde R = \left(1+ \tilde t\tilde r \right)^{1/4}$, the near-horizon metric~(\ref{NHmetric}) transforms into \begin{align} {\rm d} \tilde s_{\rm NH}^2 =& -f(\tilde R){{\rm d} \tilde t^2\over \tilde t^2 \tilde R^2} -8{\tilde R^5\over \tilde t(\tilde R^4-1)}{\rm d} \tilde t {\rm d} \tilde R \nonumber \\ &+{16\tilde R^8\over (\tilde R^4-1)^2}{\rm d} \tilde R^2+\tilde R^2 {\rm d} \Omega_2^2\,, \label{NH_metric_tR} \end{align} where \begin{align} f(\tilde R):=&\tau^2(\tilde R^4-1)^2-\tilde R^4 \nonumber \\ =& \tau^2 (\tilde R^4-\tilde R_+^4)(\tilde R^4-\tilde R_-^4 ) \,. \label{4D_metric4} \end{align} Here, $\tilde R_+$ and $\tilde R_-$ have been defined in Eq.~(\ref{R_plusminus}). In the coordinates~(\ref{NH_metric_tR}), we have $\xi^\mu =\tilde t(\partial /\partial \tilde t)^\mu $. Apart from $\tilde R=0$ (which is indeed a curvature singularity) and the points at $\theta=0, \pi $ (which are north and south poles of 2-sphere), there appear additional coordinate singularities at $\tilde t=0$ and $\tilde R=1$ in the metric~(\ref{NH_metric_tR}). Although the metric~(\ref{NH_metric_tR}) is time-dependent, we can eliminate the time-dependence of the metric~(\ref{NH_metric_tR}) by changing to the time slice, \begin{eqnarray} \eta_\pm := \ln (\pm \tilde t)\,,~~~ {\rm for }~~ \tilde t\gtrless 0, \end{eqnarray} in terms of which the Killing vector is written as $\xi^\mu =(\partial /\partial \eta _\pm)^\mu $ and the near-horizon metric (\ref{NH_metric_tR}) is given by \begin{align} {\rm d} \tilde s^2_{\rm NH}=&-{f(\tilde R)\over \tilde R^2}\left[{\rm d} \eta_\pm +{4\tilde R^7\over (\tilde R^4-1)f(\tilde R)}{\rm d} \tilde R\right]^2 \nonumber \\& +{16\tau^2 \tilde R^8\over f(\tilde R)}{\rm d} \tilde R^2+\tilde R^2 {\rm d} \Omega_2^2 \,. \label{4D_metric3} \end{align} The sign of $\eta_\pm$ has been chosen in such a way that $\eta_+ ~(\eta_-) $ increases (decreases) as $\tilde t$ increases. Performing a further coordinate transformation, \begin{align} \tilde T _\pm = \eta_\pm + \int ^{\tilde R}{4\tilde R^7\over (\tilde R^4-1)f(\tilde R)}{\rm d} \tilde R\,, \label{tilde_T} \end{align} the near-horizon metric~(\ref{4D_metric3}) is brought into a familiar form, \begin{align} {\rm d} \tilde s^2_{\rm NH} = -{f(\tilde R)\over \tilde R^2} {\rm d} \tilde T^2_\pm +{16\tau^2 \tilde R^8\over f(\tilde R)}{\rm d} \tilde R^2+\tilde R^2 {\rm d} \Omega_2^2 \,. \label{NHmetric2} \end{align} This metric describes a static black hole whose horizons occur where the lapse function vanishes $f(\tilde R)=0$, i.e., where the Killing field $\xi^\mu =(\partial /\partial \tilde T_\pm)^\mu $ becomes null. The condition $f(\tilde R)=0$ gives two roots $\tilde R=\tilde R_\pm$ given by Eq.~(\ref{R_plusminus}), which coincide with the trapping horizons in the $\tilde r\to 0$ limit taken. Thus, we conclude that the null surfaces $\tilde R=\tilde R_\pm$ in the original spacetime are locally isometric to the Killing horizons in the static spacetime (\ref{NHmetric2}). Reminding the fact that the outside domain of the original spacetime~(\ref{4D_metric0}) is highly dynamical and hence is lack of non-spacelike Killing field, it comes out a novel surprise for us that the near-horizon metric~(\ref{NHmetric2}) permits the unexpected symmetry~(\ref{horizonKilling}). Observe that the vector field (\ref{horizonKilling}) satisfies the Killing equation in the original spacetime (\ref{4D_metric0}) {\it only at the horizon $\tilde R=\tilde R_\pm$}. This may be ascribed to the fact that the 11D solution is supersymmetric if all branes are at rest. The supersymmetry does not allow energy inflow, consistent with the property that the Killing horizon is totally geodesic. This may be clear by considering the Raychaudhuri equation: $T_{\mu \nu }\xi^\mu \xi^\nu \to 0$ is indeed satisfied in the limit (\ref{r0limit1}) or (\ref{r0limit2}). As far as the authors know, this is a first realization of asymptotic symmetry appearance at the black-hole horizon under the dynamical circumstance. Let us devote some space here to discuss the near-horizon static metric~(\ref{NHmetric2}) in more detail. In this limit, the dilaton~(\ref{Phi}) and the Maxwell fields~(\ref{form_field}) are reduced to \begin{align} \kappa \Phi =\sqrt{6} \ln \tilde R\,, \label{NHdilaton} \end{align} and \begin{align} \kappa \tilde F^{(T)} &= -4\sqrt{2\pi}\tau\tilde R^{-5}~ {\rm d} \tilde T\wedge {\rm d} \tilde R\,, \nonumber \\ \kappa \tilde F^{(S)} &= -{4\sqrt{2\pi}\tau}{\tilde R^3}~ {\rm d} \tilde T\wedge {\rm d} \tilde R\,. \label{NHMaxwell} \end{align} We can confirm Eqs.~(\ref{NHmetric2}), (\ref{NHdilaton}) and (\ref{NHMaxwell}) still satisfy the original field equations~(\ref{4D_Einstein}), (\ref{eq_dilaton}) and (\ref{eq_U(1)}), which justifies that the near-horizon limit (\ref{rescaling}) is well-defined. It is obvious that the near-horizon metric (\ref{NHmetric2}) describes a static black hole, whose asymptotic structure is neither flat nor AdS. Such an unusual asymptotic structure is, however, that one commonly encounters in Einstein-Maxwell-dilaton gravity (see e.g.,~\cite{Chan:1995fr,Yazadjiev:2005du}). Albeit this peculiar asymptotics, it is easy to find that the causal structure is akin to that of the nonextremal RN-AdS solution (see Figure~\ref{PD_static}). The spatial infinity $\tilde R\to \infty $ consists of a timelike boundary $\mathcal I$, a timelike singularity resides at the center $\tilde R=0$, and the two distinct outer and inner horizons $\tilde R_\pm$ of black hole and white hole arise. \begin{figure}[t] \begin{center} \includegraphics[width=5cm]{PD_staticnew.eps} \caption{Conformal diagram of a maximally extended near-horizon metric~(\ref{NHmetric2}). The black hole has outer and inner horizons, whose radii are respectively given by $\tilde R_+$ and $\tilde R_-$. Infinity consists of a timelike surface denoted by $\mathcal I$. The white circles mark points at infinity ($i^0$, $i^\pm$ or the ``throat'') and should not be regarded as spacetime events. The filled circles $B$ represent bifurcation surfaces {\it for the metric}~(\ref{NHmetric2}) at which $\xi^\mu =(\partial/\partial T_\pm)^\mu $ vanishes, i.e., $B$'s are the fixed points under this isometry. Thick red lines correspond to $\tilde t ={\rm constant}$ surfaces, and blue dotted lines denote $\tilde r={\rm constant}$ surfaces. $\tilde R=1$ is a coordinate singularity for the metric~(\ref{NH_metric_tR}) corresponding to $\tilde t=0$. The shaded regions approximate our dynamical metric~(\ref{4D_metric0}) in the neighborhood of horizons. } \label{PD_static} \end{center} \end{figure} Since $\tilde R_+$ is strictly larger than $\tilde R_-$ for finite $\tau $ [see Eq.~(\ref{Qtau})], we find that the horizons are not degenerate. From the general formula $\kappa^2_{\pm } = \mp (1/2)(\nabla_\mu \xi_\nu )\nabla^\mu \xi^\nu $, one obtains the surface gravities of these horizons,\footnotetext{It deserves to mention that the values of surface gravities are sensitive to the norm of the generator $\xi^\mu $ of the Killing horizon. The spacetime~(\ref{NHmetric2}) is not asymptotically flat, hence there exists no meaningful way to fix the values. (In the asymptotically flat case, we usually require at infinity $\xi_\mu\xi^\mu \to -1$, which eliminates the ambiguity). However, this ambiguity causes no harm in our present discussion. The important point to note here is that they take nonvanishing positive values. We will revisit this issue when we discuss thermodynamics in the ensuing section.} \begin{align} \kappa _\pm =\pm \frac{f' (\tilde R_\pm ) }{8\tau \tilde R_\pm ^5} =\frac{\sqrt{1+4\tau^2 }}{\sqrt{1+4\tau^2}\pm 1}\,, \label{kappa_pm} \end{align} where $\kappa_+$ and $\kappa_- $ are the surface gravity of the outer and inner event horizons, respectively. The surface gravity of the Killing horizon is constant over the horizon, illustrating the equilibrium state. The nonvanishing surface gravity might be seemingly puzzling, in light of the situation that the 11D solution was ``maximally charged'' because of the supersymmetry in the static limit. Introduce the null coordinates $u_\pm$ by \begin{align} u_\pm &=\tilde T_\pm -\int^{\tilde R}\frac{4\tau \tilde R^5}{f(\tilde R)}{\rm d} \tilde R \nonumber \\&= \ln \left[\pm \tilde t|\tilde R^4-1|^{-1}|\tilde R^2+ \tilde R_+^2|^{1/\kappa _+}|\tilde R^2- \tilde R_-^2|^{1/\kappa_- }\right], \end{align} where the integration follows from a direct calculation by Eqs.~(\ref{4D_metric4}), (\ref{tilde_T}) and (\ref{kappa_pm}). The coordinates $u_\pm$ are well-defined at $\tilde R=\tilde R_+$. Using $u_\pm$, the metric~(\ref{NHmetric2}) is translated into the single null form, \begin{align} {\rm d} \tilde s^2_{\rm NH} =-\frac{f(\tilde R)}{\tilde R^2 } {\rm d} u_\pm ^2- 2 {\rm d} u_\pm {\rm d} \tilde R +\tilde R^2 {\rm d} \Omega_2^2\,. \label{NHmetric_null} \end{align} Let us consider the plus-coordinate in Eq.~(\ref{NHmetric_null}) and discuss the outer white hole horizon (the boundary of I' and II). It is immediate to find that the null generator of the Killing horizon $\xi^\mu $ is expressed in this coordinates as $\xi^\mu =(\partial/\partial u_+)^\mu $. $u _+$ is the Killing parameter ($\xi^\mu\nabla_\mu u_+ =1$) of null geodesic generators. It then follows that the renormalized tangent vector \begin{align} k^\mu =\frac{1}{\kappa_+ }e^{\kappa_+ u_+ }\left( \frac{\partial }{\partial u_+ }\right)^\mu , \end{align} is affinely parametrized, i.e., $k^\mu $ satisfies the zero-acceleration geodesic equation $k^\nu \nabla_\nu k^\mu =0$ on the horizon. (Note that we are considering a white hole so that $\xi ^\nu \nabla_\nu \xi^\mu =-\kappa_+ \xi^\mu $.) This means that the affine parameter $\lambda $ [$k^\mu =(\partial /\partial \lambda )^\mu $] is related to the Killing parameter $u_+$ as \begin{align} \lambda =-e^{-\kappa_+ u_+}. \end{align} This manifests that the affine parameter $\lambda $ values from $-\infty $ to $0 $ as $u_+$ ranges from $-\infty $ to $\infty $, implying the bifurcation surface--a closed surface at which $\xi^\mu $ vanishes. This can be verified by noticing $\xi^\mu =(\partial/\partial \tilde T_+)=\kappa_+ \lambda (\partial/\partial \lambda )^\mu \to 0$ as $\lambda \to 0$. The similar argument goes through to $u_-$. Therefore, the nondegenerate Killing horizon (i.e., a Killing horizon with nonvanishing surface gravity) is incomplete either into the past or future. This is a general consequence of a Killing horizon~\cite{Carter}. However, the above discussion does not mean that the horizon in our original spacetime is the bifurcate Killing horizon. This is consistent with results in~\cite{Racz} which asserts that the nondegenerate Killing horizon is the bifurcate Killing horizon. In their proof, it is assumed that the horizon is smooth ($C^\infty$-class), while the horizon in the present case is only finite times differentiable ($C^k$-class with $k$ finite) (see Section~\ref{sec:asysol} below). To summarize this subsection, we can expect that the original spacetime~(\ref{4D_metric0}) would have the future-event horizon at $\tilde r\to 0$ with $\tilde t\to \infty $, and the past-event horizon at $\tilde r\to 0$ with $\tilde t\to -\infty $. We have found that the point at $\tilde r=0$ with $\tilde t$ being finite corresponds to ``throat infinity'' just as that of the extreme RN spacetime, at which future and past event horizons should intersect. In the neighborhood of these horizon candidates, the spacetime~(\ref{4D_metric0}) is approximated by the near-horizon geometry~(\ref{NHmetric2}) with Killing horizons, in which several portions of Killing horizons with radii $\tilde R_\pm$ appear. What portion of Killing horizons in Figure~\ref{PD_static} does it correspond to the horizon in our original spacetime? The answer is obvious: the ``white hole portion'' [grey-colored line segment encompassing blocks I, II and III' in Figure~\ref{PD_static}] only satisfies the above criteria. \subsubsection{Null geodesics} We marked out $\tilde R_+$ as a black hole horizon in the spacetime~(\ref{4D_metric0}). To conclude this more rigorously, we face up the problem of solving geodesic motions. Since the present spacetime is spherically symmetric, it suffices us to focus on radial null geodesics to argue causal structures. Although examinations of nonradial and/or timelike geodesic motions are important issues in order to clarify the detailed physical properties of the solution, we will not discuss these since behaviors of radial null geodesics are sufficient to determine the causal structure. The radial null geodesic equations are governed by \begin{align} \ddot{\tilde t}-\frac{1}{4H_T}\dot{\tilde t}^2+\frac{(H_S+3H_T)}{2\tilde r^2H_TH_S}\dot{\tilde t}\dot{\tilde r}+\frac{H_S^3}{4\tau^2 }\dot{\tilde r}^2 &=0\,, \label{geo1}\\ \ddot{\tilde r}+\frac{\tau^2 (H_S+3H_T)}{4\tilde r^2H_T^2H_S^4}\dot{\tilde t}^2+\frac 1{2 H_T}\dot{\tilde t}\dot{\tilde r}& \nonumber \\ -\frac{(H_S+3H_T)}{4\tilde r^2H_TH_S}\dot{\tilde r}^2&=0\,,\label{geo2}\\ -\tau^2 \dot{\tilde t}^2+H_TH_S^3 \dot{\tilde r}^2&=0\,,\label{geo3} \end{align} where the dot denotes a differentiation with respect to an affine parameter $\lambda $. These equations are combined to give \begin{align} \ddot{\tilde t} \pm \frac{ \tau (H_S+3H_T)}{2\tilde r^2(H_T^3H_S^5)^{1/2}} \dot{\tilde t}^2&=0\,,\label{geo4} \\ \ddot{\tilde r}\pm \frac{(H_TH_S^3)^{1/2}}{2\tau H_T}\dot{\tilde r}^2&=0\,, \label{geo5} \end{align} where the plus (minis) sign refers to the outgoing (ingoing) geodesics. Unfortunately, the radial null geodesics do not appear to admit a first integral other than Eq. (\ref{geo3}), so it is not amendable to analytic study. Instead, we try to solve numerically Eqs.~(\ref{geo4}) and (\ref{geo5}) subjected to the initial constraint (\ref{geo3}). Making use of the degrees of freedom of the affine parameter $\lambda \to a \lambda+b$, we are able to choose $\lambda=0$ at the starting point of the geodesics and set $\dot{\tilde t}(0) $ at any values we wish. Fixing the orientation of future-directed geodesics to be $\dot{\tilde t}>0$, and past-directed to be $\dot{\tilde t}<0$, we choose $\dot{\tilde t}(0) \equiv 1~(-1)$ for future (past) directed radial null geodesics without loss of generality. Hence the residual freedoms that distinguish different geodesics are two, corresponding to the initial values [$\tilde t(0)$ and $\tilde r(0)$] for each $\tau>0 $. Let us begin our consideration by the geodesics in the outside region $\tilde r>0$. Taking the representative spacetime events $p_I~(I=1,2,3)$ such that $ \tilde t_1>\tilde t_{\rm TH}^{(+)}$, $\tilde t_{\rm TH}^{(-)}<\tilde t_2< \tilde t_{\rm TH}^{(+)}$ and $\tilde t_3<\tilde t_{\rm TH}^{(-)}$ where $\tilde t_I\equiv \tilde t|_{p_I}$ (see Figure~\ref{fig:TH}), we have examined behaviors of geodesics starting from $\tilde t=\tilde t_I$. We call the geodesics emanating from the event $p_I$ as Class-$I$. Since $\tilde R_1>\tilde R_+$ ($\tilde R_2<\tilde R_-$), Class-1 (Class-3) geodesics initially have a circumference radius larger (smaller) than $\tilde R_+ $ ($\tilde R_-$). We depict several typical geodesic curves emanating from $(\tilde t_I, \tilde r(0)=1)$ for $\tau=1$ in Figure~\ref{geodesics_figures}. This is a representative figure for $\tau<\tau_{\rm crit}$. Qualitative behavior of geodesics seems not so sensitive to the initial radial position $\tilde r(0)$. The numerical results are summarized as follows: \begin{widetext} \begin{center} \begin{figure}[h] \includegraphics[width=9cm]{geodesics_figures.eps} \caption{Radial null geodesics in the outside region ($\tilde r>0$) emanating from $\tilde t(0)=\tilde t_I$ and $\tilde r(0)=1$ for $\tau =1$. The diagrams in the top, upper middle, lower middle, bottom rows correspond to future-directed ingoing null geodesics, future-directed outgoing null geodesics, past-directed ingoing null geodesics and past-directed outgoing null geodesics. The red and blue lines denote $\tilde R_+$ and $\tilde R_-$, respectively. } \label{geodesics_figures} \end{figure} \end{center} \end{widetext} \bigskip\noindent $\bullet$ {\it Future-directed ingoing null geodesics:} Class 1 geodesics monotonically decrease the circumference radius and arrive at $\tilde R_+$ within a finite affine time. Class 2 geodesics first increase circumference radius since they are originally in the trapped region $\tilde \theta_->0$. But they always move across $\tilde t^{(+)}_{\rm TH}$, and finally reach $\tilde R_+$ with decreasing area. The qualitative behavior of Class 3 are the same as that of Class 2, except that Class 3 geodesics always cross $\tilde R=\tilde R_+$ twice, and if $\tilde t_3$ is sufficiently close to $\tilde t_{s}$, they may cross $\tilde R_-$. All classes of geodesics have infinite redshift $\tilde t(\lambda )\to +\infty $ when they finally arrive at $\tilde R_+$. \bigskip\noindent $\bullet$ {\it Future-directed outgoing null geodesics:} Class 1 geodesics necessarily go out to infinity $\tilde R\to \infty $. Class 2 geodesics may extend out to infinity or arrive at the singularity $\tilde t=\tilde t_s$ if $\tilde t_2$ is small. Class 3 geodesics inevitably plunge into the singularity $\tilde t=\tilde t_{s}$ within a finite affine time. \bigskip\noindent $\bullet$ {\it Past-directed ingoing null geodesics:} Class 1 geodesics originate from $\tilde R>\tilde R_+$ and their radii monotonically decrease toward $\tilde R_-$. Class 2 geodesics have qualitatively the same behavior. Class 3 geodesics start from $\tilde R<\tilde R_-$ with increasing area, then cross $\tilde R_-$ (with finite $\dot{\tilde t}$), attain the maximum radius and get back to $\tilde R_-$ again with undergoing infinite blueshift. \bigskip\noindent $\bullet$ {\it Past-directed outgoing null geodesics:} Class 1 geodesics may initially increase the area, but all geodesics unavoidably terminate into the singularity $\tilde t=\tilde t_s$ within a finite affine time. From these results, we conclude that the null surfaces $\tilde R=\tilde R_\pm$ locate within a finite affine time from outside spacetime events. Behaviors of future-directed outgoing null rays of Class 2 geodesics imply that there exists a critical null curve $\tilde t=\tilde t_*(\tilde r)$ such that outgoing rays emanating from $\tilde t>\tilde t_*$ can get to infinity, whereas outgoing rays emanating from $\tilde t<\tilde t_*$ fall into the singularity. \begin{widetext} \begin{center} \begin{figure}[h] \includegraphics[width=9cm]{geodesics_negativer.eps} \caption{Radial null geodesics in the inside region ($\tilde r<0$) emanating from $\tilde t(0)=\tilde t_I$ and $\tilde r(0)=-1/10$ for $\tau =1$. The diagrams in the top, upper middle, lower middle, bottom rows correspond to future-directed ingoing null geodesics, future-directed outgoing null geodesics, past-directed ingoing null geodesics and past-directed outgoing null geodesics. The red and blue lines denote $\tilde R_+$ and $\tilde R_-$, respectively. } \label{geodesics_figures2} \end{figure} \end{center} \end{widetext} Let us discuss next the geodesics inside the horizon. We call Class 1 as $\tilde t >\tilde t_1>\tilde t^{(-)}_{\rm TH}$, Class 2 as $\tilde t^{(-)}_{\rm TH} >\tilde t_2>\tilde t^{(+)}_{\rm TH}$ and Class 3 as $\tilde t^{(+)}_{\rm TH}>\tilde t_3$ (see Figure~\ref{TH_negativer}). Figure~\ref{geodesics_figures2} plots the geodesic curves emanating from the spacetime event $(t_I, \tilde r(0)=-1/10)$ with $\dot{\tilde t}(0)=\pm 1$. Geodesics starting from $\tilde r(0)<\tilde r_0$ show the same behavior as Class 1. The result is: \bigskip \noindent $\bullet$ {\it Future-directed ingoing null geodesics:} Class 1 geodesics and Class 2 geodesics starting from not so small $\tilde t_2$ eventually fall into the singularity $\tilde t=\tilde t_s$. Class 2 geodesics with $\tilde t_2$ close to $\tilde t^{(+)}_{\rm TH}(\tilde r)$ fall into the singularity $\tilde r=-1$ within a finite affine time. Class 3 geodesics initially increase area, but they eventually plunge into the singularity at $\tilde r=-1$. \bigskip\noindent $\bullet$ {\it Future-directed outgoing null geodesics:} Class 1 geodesics increase area and approach $\tilde R=\tilde R_-$ with infinite redshift $\tilde t(\lambda ) \to + \infty $. Class 2 and Class 3 geodesics decrease area and arrive at $\tilde R=\tilde R_-$ with infinite redshift $\tilde t(\lambda ) \to + \infty $. \bigskip\noindent $\bullet$ {\it Past-directed ingoing null geodesics:} Class 1 geodesics monotonically reduce area and fall into the singularity $\tilde t=\tilde t_s $. Class 2 and Class 3 geodesics initially grow the area and then turn to decrease, and finally fall into the singularity $\tilde r=-1$. \bigskip\noindent $\bullet$ {\it Past-directed outgoing null geodesics:} All geodesics eventually approach $\tilde R=\tilde R_+$ with infinite blueshift $\tilde t (\lambda )\to -\infty $. \bigskip From these results, the null surface $\tilde R=\tilde R_-$ has an ingoing null structure, analogous to the white hole horizon or the black hole inner horizon. \subsubsection{Asymptotic solutions of geodesics} \label{sec:asysol} We have numerically established that the radial null geodesics are incomplete at the null surfaces $\tilde R=\tilde R_\pm$. We shall look into the asymptotic geodesic solutions and discuss further the horizon structure. If a null geodesic is known as $\tilde t=\tilde t(\tilde r)$, the affine parameter $\lambda $ is obtained by a simple quadrature~\cite{BHKT} \begin{align} \lambda =\int {\rm d} \tilde r \exp\left[ \pm \int U[\tilde t(\tilde r'), \tilde r']{\rm d} r' \right]\,, \label{affine_sol} \end{align} where we have used the shorthand notation \begin{align} U = \frac{(H_TH_S^3)^{1/2}}{2\tau H_T}\,. \end{align} In the event horizon limit (\ref{r0limit1}), we have \begin{align} U~ \to ~ \frac{1}{\tilde r (1+\sqrt{1+4\tau^2})}\,. \end{align} Substituting this into Eq.~(\ref{affine_sol}) and solving with respect to the radial coordinate, we obtain the asymptotic solution of the future-directed null geodesic around the horizon $\tilde R=\tilde R_+$ as \begin{align} \tilde r=c_1^{(+)} (\lambda -\lambda_+)^{1/\kappa_+}\,, \quad \tilde t=c_2^{(+)} (\lambda -\lambda_+)^{-1/\kappa_+}\,, \label{asysol_futureEH} \end{align} where $\lambda_+$ corresponds to the arrival time for the geodesics at the horizon, and $c_1^{(+)}$ and $c_2^{(+)}$ are positive constants satisfying $c_1^{(+)}c_2^{(+)} =(1+\sqrt{1+4\tau^2})/(2\tau^2)$. $\kappa _+$ has been given in Eq.~(\ref{kappa_pm}). We can find from Eq.~(\ref{asysol_futureEH}) that the radial geodesics indeed reach the horizon within a finite affine time~\cite{BHKT}. Equation (\ref{asysol_futureEH}) implies that $\tilde r$ and $\tilde t$ are not smooth functions of $\lambda $ [note that $1/\kappa_+$ never takes an integer]. Similarly, we obtain \begin{align} \tilde r=c_1^{(-)} (\lambda -\lambda_-)^{1/\kappa_-}\,, \quad \tilde t=c_2^{(-)} (\lambda -\lambda_-)^{-1/\kappa_-}\,, \end{align} for an outgoing null geodesic near the horizon $\tilde R=\tilde R_-$. Constants $c_1^{(-)}$ and $c_2^{(-)}$ satisfy $c_1^{(-)}c_2^{(-)} =(1-\sqrt{1+4\tau^2})/(2\tau^2)$. \subsection{Carter-Penrose diagram} We are now in a position to discuss global causal structures of spacetime, by assembling considerations hitherto obtained. The optimal way to appreciate the large scale causal structure is to draw the Carter-Penrose conformal diagram, which enables us to visually capture the global light-cone fabric. We first notice the followings: \bigskip \begin{itemize} \item[(i)] The only candidate of future and past event horizons are $\tilde r=0$ and $\tilde t\to \pm \infty$, which are joined at the ``throat'' at $\tilde r=0$ and $\tilde t$ being finite. \item[(ii)] The near-horizon geometry of the event horizons is locally isometric to that of the static black hole~(\ref{NHmetric2}). The white-hole portion corresponds to the horizon in the original spacetime~(\ref{4D_metric}) with the bifurcation surface replaced by a smooth surface. \item[(iii)] There are the curvature singularities at $\tilde t=\tilde t_s(r)=-1/\tilde r$ and $\tilde r=-1$. These singularities are timelike (section~\ref{sec:singularity}). The time-dependent singularity $\tilde t_s(\tilde r)$ present in the $\tilde r<0 $ domain exists for $\tilde t>1$, whereas $\tilde t_s(\tilde r)$ lying in the $\tilde r>0$ region and the other singularity $\tilde r=-1$ exist for eternity. \end{itemize} These observations prompt us to imagine the positional relation between singularities and the horizons. Figure~\ref{fg:fig2} describes the conformal diagram of our dynamical black hole. From properties (i) and (ii), we can depict the identical horizon structure as in Figure~\ref{PD_static}. Since the $\tilde t= {\rm constant}(<\infty) $ lines are everywhere spacelike, each slice originates from the ``throat'' $\tilde r=0$. For negative values of $\tilde t$, $\tilde t={\rm constant}$ surfaces must intersect the singularity $\tilde t_s(\tilde r)$ at finite $\tilde r$ (see Figure~\ref{fig:TH}). Considering property (iii) that the singularity outside the horizon is only $\tilde t=\tilde t_s (\tilde r)<0$, the right side dotted portion of grey line can be drawn in Figure~\ref{fg:fig2}. Outside the horizon $\tilde r>0$, one can depict the contours of $\tilde t={\rm constant}$ and $\tilde r={\rm constant}$ family of surfaces, both of them to be orthogonal~(Figure~\ref{fg:fig2}). These aspects are all consistent with our numerical survey of geodesics. We thus conclude that the spacetime metric~(\ref{4D_metric0}) indeed describes a black hole in the FLRW universe [aside from the undesirable timelike naked singularity $\tilde t_s(\tilde r)$]. Although the null surface $\tilde R_-$ is a one way membrane of ``region of no entrance,'' it does not deserve to be a white hole horizon in a mathematical sense since the spacetime does not possess the past null infinity. Inside the event horizon $\tilde R<\tilde R_+$, the timelike singularities are vertically joined at $\tilde t=-1$. The past boundary $\tilde R_+$ can be matched to the black hole horizon. We can find as sketched in Figure~\ref{fg:fig2} that these patches are infinitely arrayed vertically. It should be emphasized, however, that this is only a possible extension, since the horizon is not analytic in the present case. One may glue the near-horizon geometry (\ref{NHmetric}) to the spacetime (\ref{4D_metric0}) across the horizon. \begin{widetext} \begin{center} \begin{figure}[h] \begin{tabular}{ cc } \includegraphics[width=4.2cm]{PD_const_t.eps} ~~~~~~~~~~~& \includegraphics[width=4.2cm]{PD_const_r.eps} \\ (a) The contour of $\tilde t={\rm constant}$.~~~~~~~~~~~~~& (b) The contour of $\tilde r={\rm constant}$. \\ \end{tabular} \caption{ Conformal diagram of the spacetime~(\ref{4D_metric0}). We draw (a) the contour (spacelike) curves of $\tilde t={\rm constant}$ by red lines and (b) the contour (timelike) curves of $\tilde r={\rm constant}$ by blue lines. The singularity of $\tilde R=0$ consists of three parts: a black-hole singularity at $\tilde r=-1$ and at $\tilde r=\tilde r_s(\tilde t)=-1/\tilde t$ inside the horizon, and a ``big bang singularity'' $\tilde t=\tilde t_s(\tilde r)=-1/\tilde r$ outside the horizon. } \label{fg:fig2} \end{figure} \end{center} \end{widetext} Next, we wish to fill in the trapping horizons into this diagram (we only consider the $\tau=1$ case). We should remind the following remarks: \begin{itemize} \item[(i)] There are trapping horizons $\tilde t=\tilde t_{\rm TH}^{(\pm)}$ at which $\tilde \theta_\mp =0$. Outside the horizon $(\tilde r>0)$, the whole portion of trapping horizon $\tilde t^{(+)}_{\rm TH}$ is past-outer for $\tau <\tau_{\rm crit}$ hence always spacelike, analogous to FLRW universe filled by a stiff matter. While the trapping horizon $\tilde t^{(-)}_{\rm TH}$ is always timelike. Inside the trapping horizon ($\tilde r<0$), $\tilde t^{(\pm)}_{\rm TH}$ coincide at $\tilde r=\tilde r_0$. $\tilde t^{(-)}_{\rm TH}$ and a part of $\tilde t^{(+)}_{\rm TH}$ near $r\simeq r_0$ are timelike. Other portion of $\tilde t=\tilde t_{\rm TH}^{(+)}$ changes signature near $\tilde r= 0$ into spacelike. \item[(ii)] Trapping horizons occur where $\tilde R={\rm constant}$ surfaces becomes null. The contour curve of circumference radius is spacelike for $\tilde R_2 <\tilde R<\tilde R_1$ (see Figure~\ref{fig:constantR}). As approaching the event horizon, the trapping horizons $\tilde t^{(\pm)}_{\rm TH}$ tend to have constant radii $\tilde R_\pm$. \item[(iii)] For $\tilde r<0$, the circumference radius $\tilde R=[(1+\tilde t\tilde r)(1+\tilde r)^3]^{1/4} $ becomes infinitely large as $\tilde t\to-\infty $ with $\tilde r$ staying constant. We can show following the same argument in Eq.~(\ref{distance}) that this is a ``past timelike infinity.'' \footnote{ This is slightly different from the extremal RN geometry, for which it takes an infinite affine time to reach the corresponding point from inside the black hole, but the point locates at the finite circumference radius. The reason why the point in the present spacetime has an infinite circumference radius inside the horizon might be due to the cosmic expansion. } \end{itemize} Outside the horizon $\tilde r>0$, the $\tilde R={\rm constant}~(>\tilde R_+)$ surfaces are the same as the FLRW cosmology: there exist a spacelike past trapping horizon $\tilde t_{\rm TH}^{(+)}(\tilde r)$, above which $\tilde R={\rm constant}$ surfaces are timelike and below which $\tilde R={\rm constant}$ surfaces are spacelike (see Figure.~\ref{fg:fig1}). For $\sqrt \tau \equiv 1<\tilde R<\tilde R_+$, $\tilde R={\rm constant}$ curves are everywhere spacelike and lie in the future of a critical null curve $\tilde t=\tilde t_*(\tilde r)$. For $\tilde R_-<\tilde R<1\equiv \sqrt \tau$, $\tilde R={\rm constant}$ curves cross the future trapping horizon $\tilde t^{(-)}_{\rm TH}(\tilde r)$ and change signature. For $\tilde R<\tilde R_-$, $\tilde R={\rm constant}$ curves are always timelike. Inside the horizon $(\tilde r<0)$, $\tilde R={\rm constant}$ curves are the same as outside for $\tilde R<\tilde R_-$ and $\sqrt \tau\equiv 1<\tilde R<\tilde R_+$. Whereas, $\tilde R={\rm constant}$ curves for $\tilde R>\tilde R_+$ quite differ from those in the outside. They cross the trapping horizons twice. \begin{figure}[t] \includegraphics[width=4.2cm]{PD_const_S.eps} \caption{ A conformal diagram of the black hole in the expanding universe. The curves of $\tilde R={\rm constant}$ are shown by green lines. Null curve $\tilde R_+$ is the future event horizon, and ${\cal I}^+$ is the future null infinity. The trapping horizons $\tilde t=\tilde t^{(+)}_{\rm TH}(\tilde r)$ and $\tilde t=\tilde t^{(-)}_{\rm TH}(\tilde r)$ are also shown by red and blue lines. The diagram can be extended beyond the upper $\tilde R_-$ null curve (possibly with the contracting patch $\tau<0$ that turns the above figure upside-down) in a continuous but a nonanalytic manner. } \label{fg:fig1} \end{figure} \section{Black hole thermodynamics} Black hole thermodynamics has been established as rigorous mathematical laws of black holes with Killing horizons~\cite{BHTD1,BHTD2,Hawking1974,Wald:1975kc,Wald:1993nt,Iyer:1994ys,Iyer:1995kg, Gao:2001ut}. Since the three laws of black hole thermodynamics interrelate the classical gravity, quantum mechanics and statistical mechanics, they are likely to have a key r\^ole toward quantum laws of gravity. Lastly, we discuss the thermodynamic properties of the present time-dependent black hole. Since the present spacetime~(\ref{4D_metric0}) possesses a Killing horizon, the thermodynamic laws continue to hold without change. It turned out that the Killing horizon has nonvanishing surface gravities~(\ref{kappa_pm}). However, their values are dependent on the normalization of the Killing field generators of the horizon. Here, we intend to obtain the temperature associated to the time translation in the FLRW universe. To this end, we resort to the laws of trapping horizons. This is a hotly-discussed issue in recent years~\cite{Hayward1993,Hayward:1997jp,nozawa}. Laws of trapping horizons are local extensions of black hole thermodynamics. When discussing the dynamical aspects of black holes, a major obstacle for extending the black hole thermodynamics to a non-stationary setting is how to define a surface gravity. In the case of a spherically symmetric spacetime, a natural time direction is specified by the {\it Kodama flow}~\cite{Kodama,Hayward:1997jp,nozawa}. Write the spherically symmetric metric as, \begin{align} {\rm d} s^2 =g_{AB}(x){\rm d} x^A {\rm d} x^B+R^2 (x){\rm d} \Omega_2^2\,, \end{align} where $g_{AB}(x)~(A, B=1,2)$ is the two-dimensional Lorentz manifold $(M^2, g_{AB})$ perpendicular to the metric sphere. The coordinate, $x^A$, on $M^2$ corresponds to $t$ and $r$. Using this coordinate patch, the Kodama vector is defined by~\cite{Kodama} \begin{align} K^A =-\epsilon ^{AB}D _B R, \label{kodama} \end{align} where $\epsilon_{AB}$ and $D_A$ are the volume element and the covariant derivative of $g_{AB}$. We may view $K^A$ as a spacetime vector by $K^\mu =K^A(\partial_A)^\mu $. It follows immediately from the orthogonal property $K^\mu \nabla_\mu R=0$ that $K^\mu $ is divergence-free, $\nabla_\mu K^\mu =R^{-2}D_A(R^{-2}K^A)=0$. Another key property comes from the relation $K^\mu K_\mu =-(\nabla _\mu R)(\nabla^\mu R)$, so that $K^\mu $ is timelike (spacelike) in the untrapped (trapped) region, i.e., $K^\mu $ defines a preferred timelike direction in the untrapped region, irrespective of the non-stationarity of spacetime. Specifically, the Kodama vector becomes null at the trapping horizon, just as in the same way the Killing vector becomes null at the Killing horizon. It is enlightening here to look into the relation between $K^\mu $ and the Misner-Sharp energy, which is also characteristic to spherically symmetric spacetimes~\cite{hideki}. Inspecting $\nabla_\mu K_\nu =D_A K_B(\nabla_\mu x^A)(\nabla_\nu x^B)$, a simple calculation shows that ${\cal G}_{\mu \nu }\nabla^\mu K^\nu =0$ holds in any spherical spacetimes. Hence we can define a divergence-free vector field $\kappa ^2 J^\mu =-{\cal G^\mu }_\nu K^\nu $, representing an energy current due to the Einstein equations. Integration of $J^\mu $ over the volume $V$ with exterior boundary $S$ yields the Misner-Sharp energy $m=-\int_V J^\mu {\rm d} \Sigma_\mu $. To summarize, the Misner-Sharp energy is a charge associated with the locally conserved current. As seen above, the Kodama vector in the spherical spacetime plays a r\^ole similar to the Killing field in stationary spacetime. One can speculate that laws of trapping horizons are related to an observer along the Kodama flow. \subsection{Temperature: 0th law} A na\"ive definition of the surface gravity for the trapping horizon is to replace the Killing field by the Kodama vector in the definition of surface gravity of a Killing horizon. This prescription does not work, since the trapping horizon is not the null surface generated by the Kodama vector. A proposed definition of surface gravity for the trapping horizon~\cite{Hayward:1997jp,nozawa} is given by the ``equilibrium part'' \begin{align} K^\nu \nabla_{[\nu }K_{\mu ]}=\kappa _{\rm TH} K_\mu \,, \end{align} where the equality is evaluated at the trapping horizon. After some amount of algebra, one finds that the surface gravities of trapping horizons $\tilde t=\tilde t^{(\pm )}_{\rm TH}$ are given by \begin{align} \tilde \kappa^{(+)}_{\rm TH} = \frac{(1+\tilde r)^6(1+4\tilde r)^3}{8\tilde R_1^9}\left[\frac{4\tilde r^2+8 \tilde r+1}{(1+\tilde r)^8} \tilde R_1^8-1\right] \,, \\ \tilde \kappa^{(-)}_{\rm TH} = \frac{(1+\tilde r)^6(1+4\tilde r)^3}{8\tilde R_2^9}\left[\frac{4\tilde r^2+8\tilde r+1}{(1+\tilde r)^8}\tilde R_2^8-1\right] \,. \end{align} Taking the event horizon limit $\tilde r \to 0$ [see Eqs.~(\ref{r0limit1}) and (\ref{r0limit2})] in the above equations, we obtain the black hole temperature \begin{align} T_{\rm BH}^{(\pm )}:=&\left.{\tilde \kappa ^{(\pm)}_{\rm TH} \over 2\pi Q}\right|_{\tilde r\to 0} ={\sqrt{1+4\tau^2}\over 16\pi\tau^2 \tilde R_\pm^5 Q}\,. \label{temperature} \end{align} Comparing this with Eq.~(\ref{kappa_pm}), these are equivalent to surface gravities associated with renormalized generator of the horizon, \begin{align} \xi ^\mu =\left(\frac{\partial }{\partial \tilde T_\pm}\right)^\mu \ \to \ \frac{1}{4\tau \tilde R_\pm^3} \left(\frac{\partial }{\partial \tilde T_\pm}\right)^\mu \,, \end{align} which coincides with the Kodama vector evaluated on the horizon for the near-horizon metric~(\ref{NHmetric2}). For the future horizon $\tilde R_+$, the temperature $T_{\rm BH}^{~(+)}$ takes the maximum value \begin{eqnarray} T_{\rm BH (max)}^{(+)}={3^{-{9\over 4}}\over 2\pi Q}\approx 0.00213856 \, Q^{-1} \end{eqnarray} at $\tau=\sqrt{3}/2$ for fixed charge. The temperature vanishes in both limits of $\tau\rightarrow \infty$ (degenerate horizon) and $\tau\rightarrow 0$ (no horizon). The former recovers the result for the extremal RN black hole. The temperature at the past horizon $\tilde R_-$, on the other hand, has no maximum value. It monotonically increases to infinity as $\tau\rightarrow 0$. $T_{\rm BH}^{(-)}$ is always higher than $T_{\rm BH}^{(+)}$. \subsection{Energy balance: 1st law} It is a widely accepted criterion that a well-defined energy should satisfy an energy balance law. The Misner-Sharp energy indeed fulfills this, as in Eq.~(\ref{variationFormula}). We can rewrite each term in this equation into more recognizable form. Defining 2D quantities, \begin{align} P_{\rm eff}=-\frac 12 {T^A}_A, \qquad \psi_A = T_{AB}D^B R+PD_A R\,, \end{align} and making use of the Einstein equations, one arrives at the {\it unified first law}~\cite{Hayward:1997jp}, \begin{align} D_A m =A\psi_A+P_{\rm eff}D_A V\,, \label{unfied1stlaw} \end{align} where $A=4\pi R^2$ and $V=\frac{4\pi}3R^3$ denote the area and volume of the metric sphere. This equation illustrates that the mass variation is supplied by an injection of energy current and the external work term. The expression of $\psi_A$ is comparatively messy, but it is straightforward to obtain. Projecting Eq.~(\ref{unfied1stlaw}) along the generator , $\zeta ^\mu =\zeta^A (\partial_A)^\mu $, of the trapping horizon and noticing the fact that $\zeta^\mu \nabla_\mu (m/R)=0$, one obtains \begin{align} A \zeta^A \psi_A =\frac{\kappa _{\rm TH}}{8\pi} \zeta^A D_A A\,. \end{align} This is an energy balance law of a trapping horizon (see \cite{Hayward:1997jp,hideki} for detailed derivation). Here, along the trapping horizon with $\theta_+=0$, $\zeta^A$ is obtained as $\zeta^A=\pm \epsilon ^{AB}D_B\theta _+$ (where the sign should be appropriately chosen in such a way that it is outgoing in the spacelike case or future-directed in the timelike case). Using Einstein's equations, we find that the surface gravity is expressed in terms of the Misner-Sharp energy and the pressure as \begin{align} \kappa_{\rm TH} =\frac{m}{R^2} -4\pi R P_{\rm eff}\,, \end{align} where equality is understood at the trapping horizon. This is the Newtonian analogue definition of acceleration. \subsection{Entropy: 2nd law} It follows from the first law of a trapping horizon that we can identify the entropy by one quarter of the area of the trapping horizon, i.e., it accords with the ``Bekenstein-Hawking formula,'' \begin{align} S_{\rm TH}^{(+)}=\frac{A_1}{4G}=\frac{\pi R_1^2}{G}\,, \qquad S_{\rm TH}^{(-)}=\frac{A_2}{4G}=\frac{\pi R_2^2}{G}. \end{align} Taking the event horizon limit~(\ref{r0limit1}) or (\ref{r0limit2}), we recover the well known result~\cite{Hawking1974,Wald:1993nt} \begin{align} S_{\rm BH}^{(\pm)}={{A}_{\rm BH}^{(\pm )}\over 4G}= {\pi Q^2 \over 2G\tau}\left(\pm 1+\sqrt{1+4\tau^2}\right) \,. \end{align} In the limit $\tau\to \infty $ with fixed charge, the above entropy reduces to that of the extremal RN black hole, $S_{\rm BH}^{(\pm)}={\pi Q^2 / G}$. \section{Concluding Remarks} \label{conclusion} In this paper, we have made a thorough discussion about the causal structure and physical properties of the spacetime derived from intersecting M-branes. We have found that the solution indeed describes a black hole embedded in the FLRW cosmology filled with fluid obeying the stiff equation of state. The global causal structure is displayed in Figures~\ref{fg:fig2} and \ref{fg:fig1}. Since the solution is approximated by the extreme RN solution near the ``throat'' and the flat FLRW universe with $P=\rho $ at infinity, one might first envisage that the causal structure is obtainable by patching these two limiting spacetimes. That is to say, according to our first intuition, one might have expected that spacetime should possess a spacelike big-bang singularity at $t=0$, there should exist a degenerate event horizon, and the timelike singularity should appear only inside the hole. However, our careful analysis revealed that the global causal structure is completely different from the above rough estimate. Our solution satisfies the dominant energy condition, so that the energy densities are always positive, absolute value of principal pressures do not exceed the energy density for respective fluids and the energy flux current is always causal. This desirable property is not seen in the solutions found in the literature. Hence the results presented in this paper open up new avenues for further research on black holes surrounded by usual matters in the expanding universe from higher-dimensional point of view. Our solution, however, may not have a direct astrophysical relevance because of nonzero electromagnetic charge. The charge is probably also responsible for the timelike singularity $t=t_s(r)$ outside the horizon. The timelike singularity does not develop as the big-bang singularity in the usual FLRW cosmology with fluid $P=w \rho ~(-1\le w\le 1)$. Unfortunately, the construction of black hole solution without charge may be beyond the intersecting brane picture. In the process for obtaining the global structure, we gave a coherent description concerning the trapping horizons. The main idea on which our discussion is based is that the trapping horizons reflect the physical situations of marginal surfaces on which the either of expansions of light ray vanishes. This local character enables us to relate it to the curvature singularity and the Misner-Sharp energy. A more important belief to which we resort is that the trapping horizon with negative outgoing expansion does not occur outside the horizon. The present spacetime indeed has this property (except in the neighborhood of singularity). We confirm the infinite redshift (blueshift) surface as a black hole horizon (white hole horizon in a quoted sense) combining the analysis of near-horizon geometry and the behaviors of null geodesics. It was somewhat surprising that the solution admits a nondegenerate Killing horizon. The Killing horizon is usually associated with symmetry of time-translation and angular-rotation. The black hole remains the same size and fails to grow, although the the black hole is surrounded with fluid. This characteristic property may be ascribed to the fact that the 11D solution was supersymmetric in the static limit. Although the dynamically brane intersecting solution breaks supersymmetry, it still maintains a part of the BPS characters. The same takes place in the Kastor-Traschen black hole. In this paper, we have taken a particular notice on the solution, whose 11D ``oxidized'' solution has four kinds of harmonics of spherical symmetry. A more general non-spherical spacetime is of course more complicated. Still, the profound understanding of spherically symmetric case will of substantial aid in exposing more complex structures of dynamical black holes. \acknowledgments We would like to thank Gary W. Gibbons, Hideki Maeda, Nobuyoshi Ohta and Harvey S. Reall for valuable comments and discussions. KM would acknowledge hospitality of DAMTP and the Centre for Theoretical Cosmology, Cambridge University during his stay in September, 2009. This work was partially supported by the Grant-in-Aid for Scientific Research Fund of the JSPS (No.19540308) and for the Japan-U.K. Joint Research Project, and by the Waseda University Grants for Special Research Projects.

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\section*{I. Introduction} \hskip 20pt Recently, the theories of quantum enveloping algebras and quantum groups have drawn wide concerns, and made great progress $^{[1,2]}$. For a generic $q$, there have been standard methods for calculating the highest weight representations and Clebsch-Gordan coefficients of quantum enveloping algebras $^{[3,4]}$. Based on them, a series of solutions $R_{q}$ of the simple Yang-Baxter equation (without spectral parameter) and solutions $R_{q}(x)$ of the Yang-Baxter equation $^{[5]}$ were found. In terms of the quantum double, the general form of universal ${\cal R}$ matrix were obtained in principle $^{[3]}$. For $q$ being a root of unity, the representation theory was studied, too $^{[6]}$. The representation of quantum affine algebras and their applications to the solvable lattice models has been made great advance $^{[7]}$. On the other hand, following the general idea of Connes on the non-commutative geometry $^{[8]}$, Woronowicz $^{[9]}$ elaborated the framework of the non-commutative differential calculus. He introduced the bimodule over the quantum groups and presented varies theorems concerning the differential forms and exterior derivative. The $q$-deformed gauge theory was studied from several different viewpoints $^{[8-10]}$. A quantum group is introduced as a Hopf algebra ${\cal A}=Fun_{q}(G)$ which is both non-commutative and non-cocommutative. It is the continuous deformation of the Hopf algebra of the functions on a Lie group, and is freely generated by the non-commutative matrix elements $T^{a}_{~b}$: $$(R_{q})^{ca}_{~rs}T^{r}_{~b}T^{s}_{~d}~=~ T^{a}_{~r}T^{c}_{~s}(R_{q})^{sr}_{~bd} \eqno (1.1) $$ \noindent where the matrix $T$ takes value in the minimal representation, and $R_{q}$ is the solution of the simple Yang-Baxter equation corresponding to the minimal representation. Faddeev-Reshetikhin-Takhtajan $^{[11]}$ introduced two linear functionals $L^{\pm}$ which belong to the dual Hopf algebra ${\cal A'}$. They are defined by their values on the elements $T^{a}_{~b}$: $$\left(L^{+}\right)^{a}_{~b}\left(T^{c}_{~d}\right)~=~ \left(R_{q}^{-1}\right)^{ac}_{~bd},~~~~~ \left(L^{-}\right)^{a}_{~b}\left(T^{c}_{~d}\right)~=~ \left(R_{q}\right)^{ca}_{~db} \eqno (1.2) $$ As is well known, the quantum enveloping algebra is also a Hopf algebra dual to ${\cal A}$, and the eq.(1.1) implies, in fact, this dual relation (see Sec.II), so it must tightly connect with the operators $L^{\pm}$. Ref.[11] gave their relations for $U_{q}A_{1}$, but their generalization is not trivial. Some authors tried to make clear the profound relations between quantum groups and quantum enveloping algebras $^{[12]}$. In this paper, after a further discussion on those relations we carefully study the concrete expressions of the operators $L^{\pm}$ by the generators in a quantum enveloping algebra, and explicitly give the expressions in two typical examples: $U_{q}A_{N}$ and $U_{q}G_{2}$ algebras. \vspace{10mm} \section*{II. The dual relation} \hskip 20pt The generators $t_{j}$, $e_{j}$ and $f_{j}$ of a quantum enveloping algebra $U_{q}{\cal G}$ satisfy the following algebraic relations: $$\begin{array}{l} t_{i}t_{j}~=~t_{j}t_{i},~~~~t_{j}^{-1}t_{j}~=~t_{j}t_{j}^{-1}~=~ {\bf 1},\\ t_{i}e_{j}~=~q_{i}^{a_{ij}}e_{j}t_{i},~~~~ t_{i}f_{j}~=~q_{i}^{-a_{ij}}f_{j}t_{i},~~~~q_{i}~=~q^{d_{i}}\\ {}~[e_{i},f_{j}]~=~\delta_{ij}\omega_{j}^{-1}\left( t_{j}-t_{j}^{-1}\right),~~~~~~~~\omega_{j}~=~q_{i}-q_{i}^{-1}\\ \displaystyle \sum_{n=0}^{1-a_{ij}}~(-1)^{n}\left[\begin{array}{c} 1-a_{ij}\\[-2mm] n\end{array} \right]_{q_{i}}~e^{1-a_{ij}-n} e_{j}e_{i}^{n}~=~0,~~~~i\neq j \\ \displaystyle \sum_{n=0}^{1-a_{ij}}~(-1)^{n}\left[\begin{array}{c} 1-a_{ij}\\[-2mm] n\end{array} \right]_{q_{i}}~f^{1-a_{ij}-n} f_{j}f_{i}^{n}~=~0,~~~~i\neq j \end{array} \eqno (2.1) $$ \noindent where $a_{ij}$ is the Cartan matrix element of a Lie algebra ${\cal G}$, and $d_{j}$ is half-length of the simple root. For the longer simple root, $d_{j}=1$. As a Hopf algebra, the coproduct $\Delta$, the counit $\varepsilon$ and the antipode $S$ of the generators are defined as follows: $$\begin{array}{c} \Delta(t_{j})~=~t_{j}\otimes t_{j},~~~~ \Delta(e_{j})~=~e_{j}\otimes {\bf 1}+t_{j}\otimes e_{j}, \\ \Delta(f_{j})~=~f_{j}\otimes t_{j}^{-1}+{\bf 1}\otimes f_{j}, \\ \varepsilon(t_{j})~=~1,~~~~\varepsilon(e_{j})~=~\varepsilon(f_{j})~=~0,\\ S(t_{j})~=~t_{j}^{-1},~~~~S(e_{j})~=~-t_{j}^{-1}e_{j},~~~~ S(f_{j})~=~-f_{j}t_{j}, \end{array} \eqno (2.2) $$ In fact, the representation matrix elements $D^{[\lambda]}_{q}(\alpha)_{\mu \nu}$ of a quantum enveloping algebra give the dual relation between the generators $\alpha\in U_{q}{\cal G}$ and the non-commutative quantity $\left(T^{[\lambda]}\right)^{\mu}_{~\nu}$ of the quantum group in the corresponding representation: $$\begin{array}{c} \left\langle \alpha, \left(T^{[\lambda]}\right)^{\mu}_{~\nu} \right\rangle~=~D^{[\lambda]}_{q}(\alpha)_{\mu \nu},~~~~ \alpha \in U_{q}{\cal G} \end{array} \eqno (2.3) $$ \noindent When $[\lambda]$ is the minimal representation, $\left(T^{[\lambda]}\right)^{\mu}_{~\nu}$ becomes $T^{a}_{~b}$. The universal ${\cal R}$ matrix can be expressed as $${\cal R}~=~\displaystyle \sum_{i}~u^{i}\otimes v_{i}~\in~ U_{q}{\cal G}\otimes U_{q}{\cal G} \eqno (2.4) $$ \noindent where $u^{i}$ and $v_{i}$ are the dual bases of up- and down-Borel subalgebras, respectively. Take the value of ${\cal R}$ matrix in the minimal representation, we get the $R_{q}$ matrix: $$\langle {\cal R}, T^{c}_{~d}\otimes T^{a}_{~b}\rangle {}~=~\left(R_{q}\right)^{ca}_{~db} \eqno (2.5) $$ The universal ${\cal R}$ matrix satisfies $^{[3,4]}$: $${\cal R}\Delta(\alpha)~=~\left(P\circ \Delta(\alpha) \right){\cal R}, {}~~~~\alpha\in U_{q}{\cal G} \eqno (2.6) $$ $${\cal R}^{-1}~=~\left({\rm id}\otimes S^{-1}\right){\cal R} \eqno (2.7) $$ \noindent where $P$ is the space transposition operator. Substituting eq.(2.6) into eq.(2.5), we get $$\begin{array}{l} R^{ca}_{~rs}~\langle \Delta(\alpha), T^{r}_{~b}\otimes T^{s}_{~d}\rangle {}~=~\langle {\cal R}\Delta(\alpha), T^{c}_{~b}\otimes T^{a}_{~d}\rangle \\ {}~=~\left\langle \left(P\circ \Delta(\alpha)\right){\cal R}, T^{c}_{~b}\otimes T^{a}_{~d}\right\rangle {}~=~\langle \Delta(\alpha), T^{a}_{~s}\otimes T^{c}_{~r}\rangle R^{rs}_{~bd} \end{array} \eqno (2.8) $$ \noindent Due to arbitrariness of $\alpha$, one can immediately get eq.(1.1) from eq.(2.8). Similarly, it is easy to show that equation (1.1) holds for any other representation. So the consistent condition (1.1) is just the direct reflection of the dual relation between a quantum group and a quantum enveloping algebra. Now we rewrite eq.(1.2) as the form of dual relation: $$\left\langle \left(L^{-}\right)^{a}_{~b}, T^{c}_{~d}\right\rangle {}~=~\left(R_{q}\right)^{ca}_{~db} $$ \noindent Comparing it with eq.(2.5), we have $$\left(L^{-}\right)^{a}_{~b}~=~ \langle {\cal R}, {\rm id} \otimes T^{a}_{~b}\rangle \eqno (2.9)$$ \noindent Similarly, $$\left(L^{+}\right)^{a}_{~b}~=~ \langle {\cal R}^{-1}, T^{a}_{~b}\otimes {\rm id}\rangle $$ \noindent {}From eq.(2.7) we have: $$\left(SL^{+}\right)^{a}_{~b}~=~ \langle {\cal R}, T^{a}_{~b}\otimes {\rm id}\rangle \eqno (2.10)$$ \noindent In a different notation, the analogous relations were also given in Ref.[13]. By making use of the general form of the universal ${\cal R}$ matrix given by Jimbo $^{[3]}$, one can calculate the explicit relations between $\left(L^{\pm}\right)^{a}_{~b}$ and the generators in a quantum enveloping algebra from eqs.(2.9) and (2.10). Denote by $Q_{+}$ the set of all the non-negative integral combinations of simple roots, $$Q_{+}~=~\left\{\beta|\beta=\displaystyle \sum_{j} m_{j} {\bf r}_{j}, m_{j}\in {\bar {\bf Z}_{-}} \right\} $$ \noindent For a given $\beta \in Q_{+}$, we define two finite-dimensional spaces $N_{\beta}^{\pm}$: $$\begin{array}{ll} N_{\beta}^{+}:~~~~&\left\{e_{i_{1}}e_{i_{2}}\cdots e_{i_{m}}| \displaystyle \sum_{j}{\bf r}_{i_{j}}=\beta\right\}\\ N_{\beta}^{-}:~~~~&\left\{f_{i_{1}}f_{i_{2}}\cdots f_{i_{m}}| \displaystyle \sum_{j}{\bf r}_{i_{j}}=\beta\right\} \end{array} $$ \noindent where ${\bf r}_{i_{j}}$ is the simple root corresponding to the generators $e_{i_{j}}$ and $f_{i_{j}}$. Those two spaces both are Hopf algebras, and they are associated and dual to each other $^{[4]}$. In these dual spaces we can adopt the following dual bases: $$u^{i}\in N_{\beta}^{+},~~~~v_{i}\in N_{\beta}^{-},$$ \noindent so that the general form of universal ${\cal R}$ matrix can be formally expressed as $^{[2,4]}$: $${\cal R}~=~\left(\displaystyle \sum_{\beta \in Q_{+}}~L_{\beta} \right) {}~q^{-H},~~~~ L_{\beta}~=~\displaystyle \sum_{i}~u^{i}\otimes v_{i}. \eqno (2.11) $$ \noindent A representation is called integrable if: i) $V=\displaystyle \oplus_{\mu} V_{\mu}$ $$V_{\mu}~=~\left\{v_{\mu}\in V|t_{j}v_{\mu}=q^{\mu_{j}}_{j} v_{\mu}, {}~~\mu =\sum \mu_{j} \lambda_{j},~~1\leq j \leq N\right\}$$ ii) For any $v \in V$, there exists a common $M$ such that: $$e_{j}^{M}~v~=~0,~~~~f_{j}^{M}~v~=~0. $$ \noindent And $q^{-H}$ is given by $$q^{-H} \left(v_{\mu}\otimes v_{\nu}\right)~=~ q^{-\displaystyle \sum_{ij}~d_{i}(a^{-1})_{ij}\mu_{i}\nu_{j}} \left(v_{\mu}\otimes v_{\nu}\right) \eqno (2.12) $$ \noindent where $(a^{-1})$ is the inverse Cartan matrix of the Lie algebra ${\cal G}$. When calculating $\left(L^{-}\right)^{a}_{~b}$ by eq.(2.9), the action on the first subspace should keep its operator form, but for the second subspace, it should be taken value in the minimal representation, i.e., expressed as matrix form. By making use of this method, we are able to express $\left(L^{\pm}\right)^{a}_{~b}$ operators by the generators in any quantum enveloping algebra in principle. In the rest of this paper we will compute the explicit expressions for two typical quantum enveloping algebras as examples. \vspace{10mm} \section*{III. $U_{q}A_{N}$ algebra} \hskip 20pt The reason for choosing $U_{q}A_{N}$ algebra as the first example is that it is the simplest and the most useful qutantum enveloping algebra. Jimbo $^{[2]}$ pointed out that for $U_{q}A_{N}$ any generator corresponding to a non-simple root can be expressed as a linear combination of the generators $e_{j}$ (or $f_{j}$) that correspond to the simple roots: $$\begin{array}{l} E_{j(j+1)}~=~e_{j},~~~~E_{ij}~=~E_{ik} E_{kj}-q E_{kj} E_{ik},\\ E_{(j+1)j}~=~f_{j},~~~~E_{ji}~=~E_{jk} E_{ki}-q^{-1} E_{ki} E_{jk}, \end{array},~~~~~~i<j<k \eqno (3.1) $$ \noindent Through calculation, we find that $\left( L^{\pm} \right)^{a}_{~b}$ are directly related with these generators. In the minimal representation of $U_{q}A_{N}$, the bases can be represented by only one index $a$, corresponding to the following weight: $$a \longrightarrow {\bf \lambda}_{a}-{\bf \lambda}_{a-1},~~~~ {\bf \lambda}_{0}={\bf \lambda}_{N+1}=0,~~~~1\leq a \leq N+1 \eqno (3.2) $$ \noindent where ${\bf \lambda}_{a}$ are the fundamental dominant weights of $U_{q}A_{N}$. The matrix forms of the generators in these bases are given as follows: $$\begin{array}{l} t_{j}~|a\rangle~=~q^{\left(\delta_{aj}-\delta_{(a-1)j}\right)}~|a\rangle\\ e_{j}~|a\rangle~=~|(a+1)\rangle,~~~~ f_{j}~|a\rangle~=~|(a-1)\rangle \end{array}~~~~1\leq a \leq N+1 \eqno (3.3) $$ {}From eq.(2.12), we have $$\begin{array}{l} q^{-H}~\left(|a\rangle \otimes |b\rangle \right)~=~ \displaystyle \left\{\prod_{j}~t_{j}^{\tau_{jb}} \otimes {\bf 1} \right\}~\left(|a\rangle \otimes |b\rangle \right) \\ \tau_{jb}~=~(a^{-1})_{j(b-1)}-(a^{-1})_{jb}~=~\left\{\begin{array}{ll} \displaystyle {j \over N+1}-1,~~~~~&j\geq b \\ \displaystyle {j \over N+1},~~~~~&j< b \end{array} \right. \end{array} $$ \noindent or briefly express it as: $$q^{-H}~\left(|a\rangle \otimes |b\rangle \right)~=~\left\{ \displaystyle \left(\prod_{j=1}^{N}~t_{j}^{j/(N+1)}\right)~ \displaystyle \left(\prod_{k=b}^{N}~t_{k}^{-1} \right)\otimes {\bf 1} \right\}~\left(|a\rangle \otimes |b\rangle \right) \eqno (3.4) $$ Now we use eq.(2.9) to calculate $\left(L^{-}\right)^{a}_{~b}$. If $a<b$, $\left(L^{-}\right)^{a}_{~b}$ vanishes. If $a=b$, $\left(L^{-}\right)^{a}_{~b}$ only contains factor $q^{-H}$ given in eq.(3.4). If $a>b$, there is only one term in the summation of ${\cal R}$ that has nonvanishing contribution to $\left(L^{-}\right)^{a}_{~b}$, where $v_{i}$ is: $$v_{i}~=~f_{a-1}~f_{a-2}~\cdots~f_{b},~~~~a\geq b \eqno (3.5) $$ \noindent The key to the problem is to find $u^{i}$ dual to the above $v_{i}$. In the calculation the following dual relations $^{[3,4]}$ are used: $$\begin{array}{l} \langle~t_{i}^{n}~,~t_{j}^{m}~\rangle~=~q_{i}^{-a_{ij}},~~~~ \langle~e_{i}~,~f_{j}~\rangle~=~- \delta_{ij}~\omega_{i}^{-1},\\ \langle~e_{i}~,~t_{j}^{m}~\rangle {}~=~\langle~t_{i}^{n}~,~f_{j}~\rangle~=~0, \end{array} \eqno (3.6) $$ \noindent When $(a-b)$ is small, the dual operator $u^{i}$ is easy to calculate. We can drew the general form of $u^{i}$ from these $u^{i}$, then prove it by induction. Noting that for $U_{q}A_{N}$ all $\omega_{j}$ are equal to each other and can be denoted by one symbol $\omega$. The result is: $$\begin{array}{rl} u^{i}&=~-\omega q^{a-b-1} \displaystyle \sum_{P}~ (-q)^{-n(P)}e_{p_{1}} e_{p_{2}} \cdots e_{p_{a-b}} \\ &=~(-1)^{a-b} \omega E_{ba} \end{array}~~~~~a>b \eqno (3.7) $$ \noindent where $P$ is a certain permutation of $(a-b)$ objects, that can be expressed by a product of some transpositions, and each transposition moves a smaller number from the right of a bigger one to the left. $$P~=~\left(\begin{array}{cccc} (a-1)&(a-2)&\cdots&b\\ p_{1}&p_{2}&\cdots&p_{a-b} \end{array} \right) $$ \noindent In such a multiplication expression, each transposition appears at most once. We are only interested in the "neighboring transposition" that permutes the places of two neighboring numbers $c$ and $(c-1)$. Two $P$ are called equivalent if their multiplication expressions contain same neighboring transpositions without concerning their order. The summation in eq.(3.6) runs over all inequivalent permutations include the unit element. $n(P)$ is the number of neighboring transpositions contained in $P$. Substituting (3.4), (3.7) into (2.9) and (2.11), we finally get $$\left(L^{-}\right)^{a}_{~b}~=~\left\{ \begin{array}{ll} 0,&a<b \\ \left\{\displaystyle \prod_{j=1}^{N} t_{j}^{j/(N+1)}\right\} {}~\left\{\displaystyle \prod_{k=b}^{N} t_{k}^{-1}\right\}, &a=b \\ (-1)^{a-b} \omega E_{ba} \left\{\displaystyle \prod_{j=1}^{N} t_{j}^{j/(N+1)}\right\} {}~\left\{\displaystyle \prod_{k=b}^{N} t_{k}^{-1}\right\},~~~~&a>b \end{array} \right. \eqno (3.8) $$ \noindent Similarly $$\left(L^{+}\right)^{a}_{~b}~=~\left\{ \begin{array}{ll} (-1)^{b-a+1} \omega \left\{\displaystyle \prod_{j=1}^{N} t_{j}^{-j/(N+1)}\right\} {}~\left\{\displaystyle \prod_{k=a}^{N} t_{k}\right\} E_{ba},~~~~&a<b\\ \left\{\displaystyle \prod_{j=1}^{N} t_{j}^{-j/(N+1)}\right\} {}~\left\{\displaystyle \prod_{k=a}^{N} t_{k}\right\}, &a=b \\ 0, &a>b \end{array} \right. \eqno (3.9) $$ Note that it is only for $U_{q}A_{N}$ where the generators corresponding to non-simple roots can be expressed as (3.1), so that the expressions of $\left(L^{\pm}\right)^{a}_{~b}$ become so simple. For other algebras, the expressions become much more complicated. $U_{q}G_{2}$ algebra is an example that will be discussed in the next section. \vspace{10mm} \section*{IV. $U_{q}G_{2}$ algebra} \hskip 20pt $U_{q}G_{2}$ is another typical example where the lengths of two simple roots are different. Let $$\begin{array}{l} q_{1}~=~q,~~~~q_{2}~=~q^{1/3},~~~~[m]~=~\displaystyle {q_{2}^{m}-q_{2}^{-m} \over q_{2}-q_{2}^{-1} },\\ \omega_{2}~=~q_{2}-q_{2}^{-1},~~~~\omega_{1}~=~q_{1}-q_{1}^{-1} {}~=~[3]\omega_{2} \end{array} \eqno (4.1) $$ \noindent The minimal representation of $U_{q}G_{2}$ algebra is given as follows: $$\begin{array}{l} D_{q}(t_{1})~=~{\rm diag}\left\{1,~q,~q^{-1},1,~q,~q^{-1},~1\right\},\\ D_{q}(t_{2})~=~{\rm diag}\left\{q_{2},~q_{2}^{-1},~q_{2}^{2},1,~ q_{2}^{-2},~q_{2},~q_{2}^{-1}\right\},\\ D_{q}(e_{1})_{23}~=~D_{q}(f_{1})_{32}= D_{q}(e_{1})_{56}~=~D_{q}(f_{1})_{65}=1,\\ D_{q}(e_{2})_{12}~=~D_{q}(f_{2})_{21}= D_{q}(e_{2})_{67}~=~D_{q}(f_{2})_{76}=1,\\ D_{q}(e_{2})_{34}~=~D_{q}(f_{2})_{43}= D_{q}(e_{2})_{45}~=~D_{q}(f_{2})_{54}=[2]^{1/2} \end{array} \eqno (4.2) $$ \noindent The rest of elements are vanishing. Through tedious calculation, we obtain: $$\begin{array}{ll} \left(L^{-}\right)^{1}_{~1}~=~t_{1}^{-1}t_{2}^{-2},~~~~~ &\left(L^{-}\right)^{2}_{~2}~=~t_{1}^{-1}t_{2}^{-1},\\ \left(L^{-}\right)^{3}_{~3}~=~t_{2}^{-1},~~~~~ &\left(L^{-}\right)^{4}_{~4}~=~1,\\ \left(L^{-}\right)^{5}_{~5}~=~t_{2},~~~~~ &\left(L^{-}\right)^{6}_{~6}~=~t_{1}t_{2},\\ \left(L^{-}\right)^{7}_{~7}~=~t_{1}t_{2}^{2},~~~~~ & \end{array} \eqno (4.3) $$ $$\begin{array}{ll} \left(L^{-}\right)^{2}_{~1}~=~-\omega_{2} e_{2} t_{1}^{-1}t_{2}^{-2},~~~~~ &\left(L^{-}\right)^{3}_{~2}~=~-\omega_{1} e_{1} t_{1}^{-1}t_{2}^{-1},\\ \left(L^{-}\right)^{4}_{~3}~=~-\omega_{2} [2]^{1/2} e_{2} t_{2}^{-1},~~~~~ &\left(L^{-}\right)^{5}_{~4}~=~-\omega_{2} [2]^{1/2} e_{2} ,\\ \left(L^{-}\right)^{6}_{~5}~=~-\omega_{1} e_{1} t_{2},~~~~~ &\left(L^{-}\right)^{7}_{~6}~=~-\omega_{2} e_{2} t_{1}t_{2}, \end{array} \eqno (4.4) $$ $$\begin{array}{l} \left(L^{-}\right)^{3}_{~1}~=~\omega_{2} \left(q e_{1} e_{2}-e_{2} e_{1}\right)t_{1}^{-1}t_{2}^{-2},\\ \left(L^{-}\right)^{4}_{~2}~=~- \omega_{2} [2]^{1/2} \left(e_{1} e_{2}-q e_{2} e_{1}\right) t_{1}^{-1}t_{2}^{-1},\\ \left(L^{-}\right)^{5}_{~3}~=~ \omega_{2}^{2} q_{2}^{-1} e_{2}^{2} t_{2}^{-1},\\ \left(L^{-}\right)^{6}_{~4}~=~\omega_{2} [2]^{1/2} \left(q e_{1} e_{2}-e_{2} e_{1}\right) ,\\ \left(L^{-}\right)^{7}_{~5}~=~- \omega_{2} \left(e_{1} e_{2}-q e_{2} e_{1}\right) t_{2}, \end{array} \eqno (4.5) $$ $$\begin{array}{l} \left(L^{-}\right)^{4}_{~1}~=~\omega_{2} q_{2}^{2} [2]^{-1/2} \left\{e_{1}e_{2}^{2}-([6]/[3])e_{2}e_{1}e_{2}+e_{2}^{2}e_{1}\right\} t_{1}^{-1}t_{2}^{-2},\\ \left(L^{-}\right)^{5}_{~2}~=~- \omega_{2} [2]^{-1} \left\{e_{1}e_{2}^{2}-q_{2}^{2}[2]e_{2}e_{1}e_{2} +q_{2}^{4}e_{2}^{2}e_{1}\right\} t_{1}^{-1}t_{2}^{-1},\\ \left(L^{-}\right)^{6}_{~3}~=~- \omega_{2} [2]^{-1} \left\{q_{2}^{4}e_{1}e_{2}^{2}-q_{2}^{2}[2]e_{2}e_{1}e_{2} +e_{2}^{2}e_{1}\right\} t_{2}^{-1},\\ \left(L^{-}\right)^{7}_{~4}~=~ \omega_{2} q_{2}^{2} [2]^{-1/2} \left\{e_{1}e_{2}^{2}-([6]/[3])e_{2}e_{1}e_{2}+e_{2}^{2}e_{1}\right\}, \end{array} \eqno (4.6) $$ $$\begin{array}{l} \left(L^{-}\right)^{5}_{~1}~=~\omega_{2} q_{2} [2]^{-1} \left\{e_{1}e_{2}^{3} -([4]-q_{2}^{-1})e_{2}e_{1}e_{2}^{2}+q_{2}([4]-q_{2}) e_{2}^{2}e_{1}e_{2}-q_{2}e_{2}^{3}e_{1}\right\} t_{1}^{-1}t_{2}^{-2},\\ \left(L^{-}\right)^{6}_{~2}~=~- \omega_{1}\omega_{2}q_{2}^{2}[6]^{-1} \left\{e_{1}^{2}e_{2}^{2}+[3]e_{2}e_{1}^{2}e_{2}-([6][5]/[3][2]) e_{1}e_{2}^{2}e_{1}+e_{2}^{2}e_{1}^{2}\right)t_{1}^{-1}t_{2}^{-1},\\ \left(L^{-}\right)^{7}_{~3}~=~-\omega_{2} q_{2}[2]^{-1} \left(q_{2}e_{1}e_{2}^{3} -q_{2}([4]-q_{2})e_{2}e_{1}e_{2}^{2}+([4]-q_{2}^{-1}) e_{2}^{2}e_{1}e_{2}-e_{2}^{3}e_{1}\right\}t_{2}^{-1}, \end{array} \eqno (4.7) $$ $$\begin{array}{l} \left(L^{-}\right)^{6}_{~1}~=~\omega_{1} q ([6][2])^{-1} \left\{ q_{2}^{2}e_{1}^{2}e_{2}^{3}+q_{2}^{-2}e_{2}^{3}e_{1}^{2} +q_{2}([4]-q_{2}^{-1})e_{2}e_{1}^{2}e_{2}^{2}\right. \\ {}~~~~~+q_{2}^{-1}([4]-q_{2})e_{2}^{2}e_{1}^{2}e_{2} -q_{2}^{2}([4]-q_{2})([6]/[3])e_{1}e_{2}^{2}e_{1}e_{2}\\ {}~~~~~\left.-q_{2}^{-2}([4]-q_{2}^{-1})([6]/[3])e_{2}e_{1}e_{2}^{2}e_{1} +([6][4]/[3][2])e_{1}e_{2}^{3}e_{1} \right\}t_{1}^{-1}t_{2}^{-2},\\ \left(L^{-}\right)^{7}_{~2}~=~\omega_{1} q ([6][2])^{-1} \left\{ q_{2}^{-2}e_{1}^{2}e_{2}^{3}+q_{2}^{2}e_{2}^{3}e_{1}^{2} +q_{2}^{-1}([4]-q_{2})e_{2}e_{1}^{2}e_{2}^{2}\right. \\ {}~~~~~+q_{2}([4]-q_{2}^{-1})e_{2}^{2}e_{1}^{2}e_{2} -q_{2}^{-2}([4]-q_{2}^{-1})([6]/[3])e_{1}e_{2}^{2}e_{1}e_{2}\\ {}~~~~~\left.-q_{2}^{2}([4]-q_{2})([6]/[3])e_{2}e_{1}e_{2}^{2}e_{1} +([6][4]/[3][2])e_{1}e_{2}^{3}e_{1} \right\}t_{1}^{-1}t_{2}^{-1},\\ \left(L^{-}\right)^{7}_{~1}~=~\omega_{1}\omega_{2} q ([6][2])^{-1}\left\{([6][4]^{2}/[3][2])e_{2}e_{1}e_{2}^{2}e_{1}e_{2} -([6]/[3][2])e_{1}e_{2}^{4}e_{1}\right.\\ {}~~~~~\left.-([4][2]-1)e_{2}^{2}e_{1}^{2}e_{2}^{2} +(e_{2}^{4}e_{1}^{2}+e_{1}^{2}e_{2}^{4}) -([6][5]/[3][2])(e_{1}e_{2}^{2}e_{1}e_{2}^{2} +e_{2}^{2}e_{1}e_{2}^{2}e_{1}) \right\}t_{1}^{-1}t_{2}^{-2}, \end{array} \eqno (4.8) $$ $\left(L^{+}\right)^{a}_{~b}$ can be obtained from $\left(L^{-}\right)^{b}_{~a}$ by following transformations: replace $q_{j}$ by $q_{j}^{-1}$ (so $\omega_{j}$ becomes $-\omega_{j}$), $e_{j}$ by $f_{j}$, $t_{i}$ by $t_{i}^{-1}$, and reverse the product order of operators in each term. For example, $$\left(L^{+}\right)^{1}_{~4}~=~-\omega_{2} q_{2}^{-2} [2]^{-1/2} t_{1}t_{2}^{2} \left\{f_{2}^{2}f_{1}-([6]/[3])f_{2}f_{1}f_{2}+f_{1}f_{2}^{2}\right\}$$ \noindent It seems to us that this relation between $\left(L^{+}\right)^{a}_{~b}$ and $\left(L^{-}\right)^{b}_{~a}$ holds for all quantum enveloping algebras. \vspace{10mm} {\bf References} \noindent [1] V. G. Drinfel'd, Quantum Group, Proceedings of the International Congress of Mathematicians, Berkeley, 1986, Vol.1, p.798. \noindent [2] M. Jimbo, Lett. Math. Phys. {\bf 10}(1985)63; Commun. Math. Phys. {\bf 102}(1986)537. \noindent [3] M. Jimbo, Topics from representations of $U_{q}(g)$, - An introductory guide to physicists, 1991, Nankai Lectures on Mathematical Physics, World Scientific, Singapore, p.1. \noindent [4] Zhong-Qi Ma, Yang-Baxter Equation and Quantum Enveloping Algebras, World Scientific, Singapore, 1993. \noindent [5] C. N. Yang, Phys. Rev. Lett. {\bf 19}(1967)1312; R. J. Baxter, Ann. 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Phys. {\bf 157}(1993)305. \end{document}

BkiUfW05qhDCjWqnQ12X

\section{Introduction} \label{sec:intro} \emph{This Game Is Not Going To Load Itself} (TGINGTLI{}) \cite{tgingtli} is a free game created in 2015 by Roger ``atiaxi'' Ostrander for the Loading Screen Jam, a game jam hosted on \protect\url{itch.io}, where it finished $7$th overall out of $46$ entries. This game jam was a celebration of the expiration of US Patent 5,718,632 \cite{US5718632}, which covered the act of including mini-games during video game loading screens. In this spirit, TGINGTLI{} is a real-time puzzle game themed around the player helping a game load three different resources of itself --- save data, gameplay, and music, colored red, green, and blue --- by placing arrows on the grid cells to route data entering the grid to a corresponding sink cell. Figure~\ref{fig:realplay} shows an example play-through. \begin{figure} \centering \includegraphics[scale=0.63]{figs/realplay_input}\hfill \includegraphics[scale=0.63]{figs/realplay} \caption{% Left: The (eventual) input for a real-world Level 16 in TGINGTLI{}. Right: A successful play-through that routes every packet to its corresponding sink.} \label{fig:realplay} \end{figure} We formalize TGINGTLI{} as follows. You are given an $m \times n$ grid where each unit-square cell is either empty, contains a data sink, or contains an arrow pointing in one of the four cardinal directions. (In the implemented game, $m=n=12$ and no arrows are placed initially.) Each data sink and arrow has a color (resource) of red, green, or blue; and there is exactly one data sink of each color in the grid. In the online version (as implemented), sources appear throughout the game; in the offline version considered here, all sources are known a priori. Note that an outer edge of the grid may have multiple sources of different colors. Finally, there is an loading bar that starts at an integer $k_0$ and has a goal integer~$k^*$. During the game, each source periodically produces data packets of its color, which travel at a constant speed into the grid. If a packet enters the cell of an arrow of the same color, then the packet will turn in that direction. (Arrows of other colors are ignored.) If a packet reaches the sink of its color, then the packet disappears and the loading bar increases by one unit of data. If a packet reaches a sink of the wrong color, or exits the grid entirely, then the packet disappears and the loading bar decreases by one unit of data, referred to as taking damage. Packets may also remain in the grid indefinitely by going around a cycle of arrows; this does not increase or decrease the loading bar. The player may at any time permanently fill an empty cell with an arrow, which may be of any color and pointing in any of the four directions. If the loading bar hits the target amount $k^*$, then the player wins; but if the loading bar goes below zero, then the player loses. In Section~\ref{sec:NP-hardness}, we prove NP-hardness of the {TGINGTLI} decision problem: given a description of the grid (including sources, sinks, and preplaced arrows), can the player place arrows to win? This reduction works even for just six sources and three colors; it introduces a new problem, \defn{3DSAT}, where variables have three different colors and each clause mixes variables of all three colors. In Section~\ref{sec:sigma-2}, we introduce more detailed models for the periodic behavior of sources, and show that many sources of differing periods enable both NP- and coNP-hardness of winning the game, \emph{even without player input} (just simulating the game). On the positive side, we prove that these problems are in $\Sigma_2^P$; and in NP when the source periods are all equal, as in our first NP-hardness proof, so this case is in fact NP-complete. In Section~\ref{sec:perfect-layouts}, we consider how levels start in the implemented game: a grid with placed sinks but no preplaced arrows. We give a full characterization of when there is a \defn{perfect layout} of arrows, where all packets are guaranteed to route to the correct sink, \emph{no matter where sources get placed}. In particular, this result provides a winning strategy for most sink arrangements in the implemented game. Notably, because this solution works independent of the sources, it works in the online setting. \xxx{We should perhaps point to some of the other hardness of videogames literature and if there is literature on solving small instances of puzzle games through math and exhaustive search.} \section{NP-Hardness for Three Colors and Six Sources} \label{sec:NP-hardness} We first prove that TGINGTLI{} is NP-hard by reducing from a new problem called \defn{3-Dimensional SAT (3DSAT)}, defined by analogy to 3-Dimensional Matching (3DM). 3DSAT is a variation of 3SAT where, in addition to a 3CNF formula, the input specifies one of three colors (red, green, or blue) to each variable of the CNF formula, and the CNF formula is constrained to have trichromatic clauses, i.e., to have exactly one variable (possibly negated) of each color. \begin{lemma} 3DSAT is NP-complete. \end{lemma} \begin{proof} We reduce from 3SAT to 3DSAT by converting a 3CNF formula $F$ into a 3D CNF formula~$F'$. For each variable $x$ of~$F$, we create three variables $x^{(1)}, x^{(2)}, x^{(3)}$ in $F'$ (intended to be equal copies of $x$ of the three different colors) and add six clauses to $F'$ to force $x^{(1)} = x^{(2)} = x^{(3)}$: % \def\halfup#1{\raisebox{2.5ex}{\smash{$#1$}}} \begin{align*} \lnot x^{(1)} \lor x^{(2)} \lor x^{(3)} &\iff (x^{(1)} \to x^{(2)}) \lor x^{(3)} \\ \lnot x^{(1)} \lor x^{(2)} \lor \lnot x^{(3)} &\iff (x^{(1)} \to x^{(2)}) \lor \lnot x^{(3)} ~\halfup{\Bigg\rbrace \iff x^{(1)} \to x^{(2)}} \\ x^{(1)} \lor \lnot x^{(2)} \lor x^{(3)} &\iff (x^{(2)} \to x^{(3)}) \lor x^{(1)} \\ \lnot x^{(1)} \lor \lnot x^{(2)} \lor x^{(3)} &\iff (x^{(2)} \to x^{(3)}) \lor \lnot x^{(1)} ~\halfup{\Bigg\rbrace \iff x^{(2)} \to x^{(3)}} \\ x^{(1)} \lor x^{(2)} \lor \lnot x^{(3)} &\iff (x^{(3)} \to x^{(1)}) \lor x^{(2)}\\ x^{(1)} \lor \lnot x^{(2)} \lor \lnot x^{(3)} &\iff (x^{(3)} \to x^{(1)}) \lor \lnot x^{(2)} ~\halfup{\Bigg\rbrace \iff x^{(3)} \to x^{(1)}} \end{align*} % Thus the clauses on the left are equivalent to the implication loop $x^{(1)} \implies x^{(2)} \implies x^{(3)} \implies x^{(1)}$, which is equivalent to $x^{(1)} = x^{(2)} = x^{(3)}$. For each clause $c$ of $F$ using variables $x$ in the first literal, $y$ in the second literal, and $z$ in the third literal, we create a corresponding clause $c'$ in $F'$ using $x^{(1)}$, $y^{(2)}$, and $z^{(3)}$ (with the same negations as in~$c$). All clauses in $F'$ (including the variable duplication clauses above) thus use a variable of the form $x^{(i)}$ in the $i$th literal for $i \in \{1,2,3\}$, so we can 3-color the variables accordingly. \end{proof} \begin{theorem} \label{thm:NP-hard36} {TGINGTLI} is NP-hard, even with three colors and six sources of equal period. \end{theorem} \begin{proof} Our reduction is from 3DSAT. Figure~\ref{fig:reduction-sketch} gives a high-level sketch: each variable gadget has two possible routes for a packet stream of the corresponding color, and each clause gadget allows at most two colors of packets to successfully route through. When a clause is satisfied by at least one variable, the clause gadget allows the other variables to successfully pass through; otherwise, at least one of the packet streams enters a cycle. Variables of the same color are chained together to re-use the packet stream of that color. \begin{figure} \centering \subcaptionbox{Satisfied clause}{\includegraphics[scale=0.5]{figs/reduction-sketch-sat}}\hfil \subcaptionbox{Unsatisfied clause}{\includegraphics[scale=0.5]{figs/reduction-sketch-unsat}} \caption{Sketch of our NP-hardness reduction.} \label{fig:reduction-sketch} \end{figure} In detail, most cells of the game board will be prefilled, leaving only a few empty cells (denoted by question marks in our figures) that the player can fill. For each color, say red, we place a red source gadget on the left edge of the construction. Then, for each red variable $x$ in sequence, we place a variable gadget of Figure~\ref{fig:gadget:variable} at the end of the red stream. To prevent the packets from entering a loop, the player must choose between sending the stream upward or downward, which results in it following one of the two rightward paths, representing the literals $x$ and $\overline x$ respectively. The path followed by the packet stream is viewed as \emph{false}; an empty path is viewed as \emph{true}. \begin{figure}[t] \centering \begin{minipage}[b]{0.51\linewidth} \centering \includegraphics[scale=0.9]{figs/clause.pdf} \caption{Clause gadget. At most two streams of data, representing false literals, can pass through the gadget (by placing upwards arrows in the ``?'' cells) without entering a cycle. Placing any other direction of arrow also puts the stream into a cycle.} \label{fig:gadget:clause} \end{minipage}\hfill \begin{minipage}[b]{0.22\linewidth} \centering \includegraphics[scale=0.9]{figs/variable.pdf} \caption{\centering Variable gadget lets the player route packets from a source to one of two literal paths.} \label{fig:gadget:variable} \end{minipage}\hfill \begin{minipage}[b]{0.22\linewidth} \centering \includegraphics[scale=0.9]{figs/merge.pdf} \caption{\centering Merge gadget combines two literal paths back into one.\linebreak~} \label{fig:gadget:merge} \end{minipage} \bigskip \begin{minipage}[b]{0.5\linewidth} \centering \includegraphics[scale=0.9]{figs/crossover.pdf} \caption{Crossover gadget between two literal paths of same or different colors. (The center cell is colored different from both paths.)} \label{fig:gadget:crossover} \end{minipage}\hfill \begin{minipage}[b]{0.46\linewidth} \includegraphics[scale=0.9]{figs/death.pdf} \caption{Damage gadget forces damage at a unit rate after a desired start delay.\hspace{\fill}\linebreak~} \label{fig:gadget:damage} \end{minipage} \end{figure} Then we route each literal path to sequentially visit every clause containing it. Figure~\ref{fig:gadget:crossover} shows a crossover gadget to enable such routing. (Note that it still works if both lines are the same color.) Figure~\ref{fig:gadget:clause} shows the clause gadget, which allows at most two packet streams to pass through it. If all three literals are false then at least one stream of data must be placed into a cycle. On the other hand, if the clause is satisfied, then the literal paths carrying data can pass their data on to the next clause, and so on. Note that the length of the diverted path through the clause is the same for all three colors. After all uses of the red variable \(x\), we lengthen the two red literal paths corresponding to \(x\) and \(\overline{x}\) to have the same length, then combine them back together using the merge gadget of Figure~\ref{fig:gadget:merge}. We then route this red path into the variable gadget for the next red variable, and so on. Finally, after all red variables, we connect the red stream to a red sink. We lengthen the red, green, and blue streams to all have the same length~$\ell$. If the player successfully satisfies all clauses, then they will increase the loading bar by $3$ units ($1$~per color) after an initial delay of~$\ell$. We set the parameters so that the player wins in this case: $k^* - k_0 = 3$. Otherwise, the loading rate is at most~$2$. To ensure that the player loses in this case, we add $3$ damage gadgets of Figure~\ref{fig:gadget:damage}, each incurring damage at a rate of $1$ after an initial delay of $\ell+1$. Thus we obtain a net of $-1$ per period, so the player eventually loses even if $k_0$ is large. \end{proof} This NP-hardness result does not need a very specific model of sources and how they emit packets. To understand whether the problem is in NP, we need a more specific model, which is addressed in the next section. \section{Membership in $\Sigma_2^P$ and Hardness from Source Periodicity} \label{sec:sigma-2} \iffalse membership in sigma 2 - algorithm check if you didn't lose check if you didn't win skip to the next event co-np hard: checking if you didn't win/lose \fi In this section, we consider the effect of potentially differing periods for different sources emitting packets. Specifically, we show that carefully setting periods together with the unbounded length of the game results in both NP- and coNP-hardness of determining the outcome of {TGINGTLI}, \emph{even when the player is not making moves}. Conversely, we prove that the problem is in~$\Sigma_2^P$, even allowing player input. \subsection{Model and Problems} More precisely, we model each source $s$ as emitting data packets of its color into the grid with its own period $p_s$, after a small warmup time $w_s$ during which the source may emit a more specific pattern of packets. {TGINGTLI} as implemented has a warmup behavior of each source initially (upon creation) waiting 5 seconds before the first emitted packet, then emitting a packet after 2 seconds, after 1.999 seconds, after 1.998 seconds, and so on, until reaching a fixed period of 0.5 seconds. This is technically a warmup period of 1881.25 seconds with 1500 emitted packets, followed by a period of 0.5 seconds. In the \defn{simulation} problem, we are given the initial state of the grid, a list of timestamped \defn{events} for when each source emits a packet during its warmup period, when each source starts periodic behavior, and when the player will place each arrow. We assume that timestamps are encoded in binary but (to keep things relatively simple) periods and warmup times are encoded in unary. The problem then asks to predict whether the player wins; that is, the loading bar reaches~$k^*$ before a loss occurs. In the \defn{game} problem, we are not given the player's arrow placements. If we allow nondeterministic algorithms, the game problem reduces to the simulation problem: just guess what arrows we place, where, and at what times. A natural approach to solving the simulation problem is to simulate the game from the initial state to each successive event. Specifically, given a state of the game (a grid with sinks, sources of varying periods and offsets, placed arrows, and the number of in-flight packets at each location) and a future timestamp \(t\), we wish to determine the state of the game at time \(t\). Using this computation, we can compute future states of the game quickly by ``skipping ahead'' over the time between events. On an $m \times n$ grid, there are $O(m n)$ events, so we can determine the state of the game at any time \(t\) by simulating polynomially many phases between events. This computation is easy to do. Given the time \(t\), we can divide by each source's period and the period of each cycle of arrows to determine how many packets each source produces and where the arrows route them --- either to a sink which affects loading, off the grid, stuck in a cycle, or in-flight outside a cycle --- and then sum up the effects to obtain the new amount loaded and the number of packets at each location. \xxx{Be more specific?} However, being able to compute future states does not suffice to solve the simulation and game problems because there might be an intermediate time where the loading amount drops below $0$ or reaches the target amount~$k^*$. Nonetheless, this suffices to show that the problems are in \(\Sigma_2^P\), by guessing a win time and verifying there are no earlier loss times: \begin{lemma} The simulation and game problems are in \(\Sigma_2^P\). \end{lemma} \begin{proof} The player wins if there exists a time with a win such that all smaller times are not losses. To solve the simulation problem, nondeterministically guess the winning time and verify that it is a win by computing the state at that time. Then check using a coNP query that there was no loss before that time, again using the ability to quickly compute states at individual timestamps. To solve the game problem, we first existentially guess the details of the arrow placements, then solve the resulting simulation problem as before. \end{proof} An easier case is when the source periods are all the same after warmup, as implemented in the real game. Theorem~\ref{thm:NP-hard36} proved this version of the game NP-hard, and we can now show that it is NP-complete: \begin{lemma} If all sources have the same polynomial-length period after a polynomial number $t_p$ of time steps, then the simulation problem is in P and the game problem is in NP. \end{lemma} \begin{proof} In this case, we can check for wins or losses in each phase between events by explicitly simulating longer than all packet paths, at which point the loading bar value becomes periodic with the common source period. (Cycles of arrows may have different periods but these do not affect the loading bar, and thus do not matter when checking for wins and losses.) We skip over each phase, checking for win or loss along the way. If the game continues past the last event, we measure the sign of the net score change over the period. If it is positive, the player will eventually win; if it is negative, the player will eventually lose; and if it is zero, the game will go on forever. \end{proof} In the remainder of this section, we consider the case where each source can be assigned any integer period, and the period does not change over time. \subsection{Periodic Sum Threshold Problem} With varying source periods, the challenge is that the overall periodic behavior of the game can have an extremely large (exponential) period. We can model this difficulty via the \defn{Periodic Sum Threshold Problem}, defined as follows. We are given a function $f(x) = \sum_{i} g_i(x)$ where each $g_i$ has unary integer period $T_i$ and unary maximum absolute value $M_i$. In addition, we are given a unary integer $\tau > 0$ and a binary integer time $x^*$. The goal is to determine whether there exists an integer $x$ in $[0, x^*)$ such that $f(x) \geq \tau$. (Intuitively, reaching $\tau$ corresponds to winning.) \begin{theorem} \label{thm:pstp-np-complete} The Periodic Sum Threshold Problem is NP-complete, even under the following restrictions: \begin{enumerate} \item \label{prop:one-hot} Each $|g_i|$ is a one-hot function, i.e., $g_i(x) = 0$ everywhere except for exactly one $x$ in its period where $g_i(x) = \pm 1$. \item \label{prop:lambda} We are given a unary integer $\lambda < 0$ such that $f(x) > \lambda$ for all $0 \leq x < x^*$ and $f(x^*) \leq \lambda$. (Intuitively, dipping down to $\lambda$ corresponds to losing.) \end{enumerate} \end{theorem} \begin{proof} First, the problem is in NP: we can guess $x \in [0,x^*)$ and then evaluate whether $f(x) \leq c$ in polynomial time. For NP-hardness, we reduce from 3SAT. We map each variable $v_i$ to the $i$th prime number $p_i$ excluding~$2$. Using the Chinese Remainder Theorem, we can represent a Boolean assignment $\phi$ as a single integer $0 \le x < \prod_i p_i$ where $x \equiv 1 \mod p_i$ when $\phi$ sets $v_i$ to true, and $x \equiv 0 \mod p_i$ when $\phi$ sets $v_i$ to false. (This mapping does not use other values of $x$ modulo~$p_i$. In particular, it leaves $x \equiv -1 \mod p_i$ unused, because $p_i \geq 3$.) Next we map each clause such as $C = (v_i \vee v_j \vee \neg v_k)$ to the function % $$g_C(x) = \max\{[x \equiv 1 \mod p_i], [x \equiv 1 \mod p_j], [x \equiv 0 \mod p_k]\},$$ % i.e., positive literals check for $x \equiv 1$ and negated literals check for $x \equiv 0$. This function is $1$ exactly when $x$ corresponds to a Boolean assignment that satisfies~$C$. This function has period $p_i p_j p_k$, whose unary value is bounded by a polynomial. Setting $\tau$ to the number of clauses, there is a value $x$ where the sum is $\tau$ if and only if there is a satisfying assignment for the 3SAT formula. (Setting $\tau$ smaller, we could reduce from Max 3SAT.) To achieve Property~\ref{prop:one-hot}, we split each $g_C$ function into a sum of polynomially many one-hot functions (bounded by the period). In fact, seven functions per clause suffice, one for each satisfying assignment of the clause. To achieve Property~\ref{prop:lambda}, for each prime $p_i$, we add the function $h_i(x) = -[x \equiv -1 \mod p_i]$. This function is $-1$ only for unused values of \(x\) which do not correspond to any assignment \(\phi\), so it does not affect the argument above. Setting $-\lambda$ to the number of primes (variables) and $x^* = \prod_i p_i - 1$, we have $\sum_i h_i(x^*) = \lambda$ because $h_i(x^*) \equiv -1 \mod p_i$ for all~$i$, while $\sum_i h_i(x) > \lambda$ for all $0 \leq x < x^*$. All used values \(x\) are smaller than \(x^*\). In total, $f(x)$ is the sum of the constructed functions and we obtain the desired properties. \end{proof} \subsection{Simulation Hardness for Two Colors} We can use our hardness of the Periodic Sum Threshold Problem to prove hardness of simulating {TGINGTLI}, even without player input. \begin{theorem} \label{thm:sim NP-hard} Simulating {TGINGTLI} and determining whether the player wins is NP-hard, even with just two colors. \end{theorem} \begin{proof} We reduce from the Periodic Sum Threshold Problem proved NP-complete by Theorem~\ref{thm:pstp-np-complete}. For each function $g_i$ with one-hot value $g_i(x_i) = 1$ and period~$T_i$, we create a blue source $b_i$ and a red sources $r_i$, of the same emitting period~$T_i$, and route red and blue packets from these sources to the blue sink. By adjusting the path lengths and/or the warmup times of the sources, we arrange for a red packet to arrive one time unit after each blue packet which happens at times $\equiv x_i \mod T_i$. Thus the net effect on the loading bar value is $+1$ at time $x_i$ but returns to $0$ at time $x_i + 1$. Similarly, for each function $g_i$ with one-hot value $g_i(x_i) = -1$, we perform the same construction but swapping the roles of red and blue. Setting $k_0 = -\lambda-1 \geq 0$, the loading bar goes negative (and the player loses) exactly when the sum of the functions $g_i$ goes down to~$\lambda$. Setting $k^* = k_0 + \tau$, the loading bar reaches $k^*$ (and the player wins) exactly when the sum of the functions $g_i$ goes up to~$\tau$. \end{proof} This NP-hardness proof relies on completely different aspects of the game from the proof in Section~\ref{sec:NP-hardness}: instead of using player input, it relies on varying (but small in unary) periods for different sources. More interesting is that we can also prove the same problem coNP-hard: \begin{theorem} Simulating {TGINGTLI} and determining whether the player wins is coNP-hard, even with just two colors. \end{theorem} \begin{proof} We reduce from the complement of the Periodic Sum Threshold Problem, which is coNP-complete by Theorem~\ref{thm:pstp-np-complete}. The goal in the complement problem is to determine whether there is \emph{no} integer $x$ in $[0, x^*)$ such that $f(x) \geq \tau$. The idea is to negate all the values to flip the roles of winning and losing. For each function $g_i$, we construct two sources and wire them to a sink in the same way as Theorem~\ref{thm:sim NP-hard}, but negated: if $g_i(x_i) = \pm 1$, then we design the packets to have a net effect of $\mp 1$ at time $x_i$ and $0$ otherwise. Setting $k_0 = \tau-1$, the loading bar goes negative (and the player loses) exactly when the sum of the functions $g_i$ goes up to~$\tau$, i.e., the Periodic Sum Threshold Problem has a ``yes'' answer. Setting $k^* = k_0 - \lambda$, the loading bar reaches $k^*$ (and the player wins) exactly when the sum of the functions $g_i$ goes down to $\lambda$, i.e., the Periodic Sum Threshold Problem has a ``no'' answer. \end{proof} \section{Characterizing Perfect Layouts} \label{sec:perfect-layouts} Suppose we are given a board which is empty except for the location of the three data sinks. Is it possible to place arrows such that all possible input packets get routed to the correct sink? We call such a configuration of arrows a \defn{perfect layout}. In particular, such a layout guarantees victory, regardless of the data sources. In this section, we give a full characterization of boards and sink placements that admit a perfect layout. Some of our results work for a general number $c$ of colors, but the full characterization relies on $c=3$. \subsection{Colors Not Arrows} We begin by showing that we do not need to consider the directions of the arrows, only their colors and locations in the grid. Let \(B\) be a board with specified locations of sinks, and let \(\partial B\) be the set of edges on the boundary of \(B\). Suppose we are given an assignment of colors to the cells of \(B\) that agrees with the colors of the sinks; let \(C_i\) be the set of grid cells colored with color \(i\). We call two cells of \(C_i\), or a cell of \(C_i\) and a boundary edge $e \in \partial B$, \defn{visible} to each other if and only if they are in the same row or the same column and no sink of a color other than \(i\) is between them. Let \(G_i\) be the graph whose vertex set is \(C_i \cup \partial B\), with edges between pairs of vertices that are visible to each other. \begin{lemma} \label{lem:perfect-colors} Let \(B\) be a board with specified locations of sinks. Then \(B\) admits a perfect layout if and only if it is possible to choose colors for the remaining cells of the grid such that, for each color~\(i\), the graph \(G_i\) is connected. \end{lemma} \begin{proof} ($\Longrightarrow$) Without loss of generality, assume that the perfect layout has the minimum possible number of arrows. Color the cells of the board with the same colors as the sinks and arrows in the perfect layout. (If a cell is empty in the perfect layout, then give it the same color as an adjacent cell; this does not affect connectivity.) Fix a color \(i\). Every boundary edge is connected to the sink of color \(i\) by the path a packet of color \(i\) follows when entering from that edge. (In particular, the path cannot go through a sink of a different color.) By minimality of the number of arrows in the perfect layout, every arrow of color \(i\) is included in such a path. Therefore \(G_i\) is connected. ($\Longleftarrow$) We will replace each cell by an arrow of the same color to form a perfect layout. Namely, for each color~\(i\), choose a spanning tree of \(G_i\) rooted at the sink of color~\(i\), and direct arrows from children to parents in this tree. By connectivity, any packet entering from a boundary edge will be routed to the correct sink, walking up the tree to its root. \end{proof} \subsection{Impossible Boards} \label{sec:impossible-boards} Next we show that certain boards cannot have perfect layouts. First we give arguments about boards containing sinks too close to the boundary or each other. Then we give an area-based constraint on board size. \begin{lemma} \label{lem:sink-distance} If there are fewer than $c-1$ blank cells in a row or column between a sink and a boundary of the grid, then there is no perfect layout. \end{lemma} \begin{proof} A perfect layout must prevent packets of the other \(c-1\) colors entering at this boundary from reaching this sink; this requires enough space for \(c-1\) arrows. \end{proof} \begin{lemma} \label{lem:impossible-c3} For $c=3$, a board has no perfect layout if either (as shown in Figure~\ref{fig:impossible}) \begin{enumerate}[(a)] \item a data sink is two cells away from three boundaries and adjacent to another sink; \item a data sink is two cells away from two incident boundaries and is adjacent to two other sinks; \item a data sink is two cells away from two opposite boundaries and is adjacent to two other sinks; or \item a data sink is two cells away from three boundaries and is one blank cell away from a pair of adjacent sinks. \end{enumerate} \end{lemma} \begin{figure} \centering \subcaptionbox{}{\includegraphics[scale=0.75]{figs/impossible1}}\hfil \subcaptionbox{}{\includegraphics[scale=0.75]{figs/impossible2}}\hfil \subcaptionbox{}{\includegraphics[scale=0.75]{figs/impossible3}}\hfil \subcaptionbox{}{\includegraphics[scale=0.75]{figs/impossible4}} \caption{Sink configurations with no perfect layout. Dots indicate arrows of forced colors (up to permutation within a row or column).} \label{fig:impossible} \end{figure} \begin{proof} Assume by symmetry that, in each case, the first mentioned sink is red. Cases (a), (b), and (c): The pairs of cells between the red sink and the boundary (marked with dots in the figure) must contain a green arrow and a blue arrow to ensure those packets do not reach the red sink. Thus there are no available places to place a red arrow in the same row or column as the red sink, so red packets from other rows or columns cannot reach the red sink. Case (d): The pairs of cells between the red sink and the boundary (marked with green and blue dots in the figure) must contain a green arrow and a blue arrow to ensure those packets do not collide with the red sink. Thus the blank square between the red sink and the other pair of sinks must be a red arrow pointing toward the red sink, to allow packets from other rows and columns to reach the red sink. Assume by symmetry that the sink nearest the red sink is green. As in the other cases, the pairs of cells between the green sink and the boundary must be filled with red and blue arrows. Thus there are no green arrows to route green packets from other rows or columns to the green sink. \end{proof} We now prove a constraint on the sizes of boards that admit a perfect layout. \begin{lemma} \label{lem:size-constraint} Let \(c\) be the number of colors. Suppose there is a perfect layout on a board where \(m\) and \(n\) are respectively the number of rows and columns, and \(p\) and \(q\) are respectively the number of rows and columns that contain at least one sink. Then \begin{equation} \label{eqn:size-constraint} c(m + n) + (c-2)(p + q) \le m n - c. \end{equation} \end{lemma} \begin{proof} Each of the \(m - p\) unoccupied rows must contain \(c\) vertical arrows in order to redirect packets of each color out of the row. Each of the \(p\) occupied rows must contain \(c-1\) vertical arrows to the left of the leftmost sink in order to redirect incorrectly colored packets from the left boundary edge away from that sink; similarly, there must be \(c-1\) vertical arrows to the right of the rightmost sink. Thus we require \(c(m - p) + 2(c - 1)p = c m + (c - 2)p\) vertical arrows overall. By the same argument, we must have \(c n + (c - 2)q\) horizontal arrows, for a total of \(c(m + n) + (c - 2)(p + q)\) arrows. There are \(m n - c\) cells available for arrows, which proves the claim. \end{proof} Up to insertion of empty rows or columns, rotations, reflections, and recolorings, there are six different configurations that $c=3$ sinks may have with respect to each other, shown in Figure~\ref{fig:3-sink-configs}. We define a board's \defn{type} according to this configuration of its sinks (C, I, J, L, Y, or /). \begin{figure}[htbp] \centering \hfil \subcaptionbox{C}{\includegraphics[scale=0.9]{figs/config-C}}\hfil \subcaptionbox{I}{\includegraphics[scale=0.9]{figs/config-I}}\hfil \subcaptionbox{J}{\includegraphics[scale=0.9]{figs/config-J}}\hfil \subcaptionbox{L}{\includegraphics[scale=0.9]{figs/config-L}}\hfil \subcaptionbox{Y}{\includegraphics[scale=0.9]{figs/config-Y}}\hfil \subcaptionbox{/}{\includegraphics[scale=0.9]{figs/config-slash}}% \hfil \caption{The six possible configurations of three sinks up to rotations, reflections, recolorings, and removal of empty rows.} \label{fig:3-sink-configs} \end{figure} A board's type determines the values of \(p\) and \(q\) and thus the minimal board sizes as follows. Define a board to have size \defn{at least} $m \times n$ if it has at least $m$ rows and at least $n$ columns, or vice versa. \begin{lemma} \label{lem:size-constraint-c3} For a perfect layout to exist with \(c=3\), it is necessary that: \begin{itemize} \item Boards of type Y or / have size at least $7\times8$. \item Boards of type C or J have size at least $7\times8$ or $6\times9$. \item Boards of type L have size at least $7\times7$ or $6\times9$. \item Boards of type I have size at least $7\times7$, $6\times9$, or $5\times11$. \end{itemize} \end{lemma} \begin{proof} These bounds follow from Lemma~\ref{lem:size-constraint} together with the requirement from Lemma~\ref{lem:sink-distance} that it be possible to place sinks at least two cells away from the boundary. \end{proof} \subsection{Constructing Perfect Layouts} In this section, we complete our characterization of boards with perfect layouts for \(c=3\). We show that Lemmas \ref{lem:sink-distance}, \ref{lem:impossible-c3}, and \ref{lem:size-constraint-c3} are the only obstacles to a perfect layout: \begin{theorem} \label{thm:characterization-c3} A board with $c=3$ sinks has a perfect layout if and only if the following conditions all hold: \begin{enumerate} \item All sinks are at least two cells away from the boundary (Lemma~\ref{lem:sink-distance}). \item The board does not contain any of the four unsolvable configurations in Figure~\ref{fig:impossible} (Lemma~\ref{lem:impossible-c3}). \item The board obeys the size bounds of Lemma~\ref{lem:size-constraint-c3}. \end{enumerate} \end{theorem} We call a board \defn{minimal} if it has one of the minimal dimensions for its type as defined in Lemma~\ref{lem:size-constraint-c3}. Our strategy for proving Theorem~\ref{thm:characterization-c3} will be to reduce the problem to the finite set of minimal boards, which we then verify by computer. We will accomplish this by removing empty rows and columns from non-minimal boards to reduce their size, which we show can always be done while preserving the above conditions. \begin{lemma} \label{lem:characterization-c3-minimal} All minimal boards satisfying the three conditions of Theorem~\ref{thm:characterization-c3} have a perfect layout. \end{lemma} \begin{proof} \renewcommand{\qedsymbol}{\(\blacksquare\)} The proof is by exhaustive computer search of all such minimal boards. We wrote a Python program to generate all possible board patterns, reduce each perfect layout problem to Satisfiability Modulo Theories (SMT), and then solve it using Z3 \cite{z3}. The results of this search are in Appendix~\ref{apx:computer-solutions}. \end{proof} If \(B_0\) and \(B_1\) are boards, then we define \defn{\(B_0 \pmb{\lessdot} B_1\)} to mean that \(B_0\) can be obtained by removing a single empty row or column from \(B_1\). \begin{lemma} \label{lem:add-row} If \(B_0 \lessdot B_1\) and \(B_0\) has a perfect layout, then \(B_1\) also has a perfect layout. \end{lemma} \begin{proof} By symmetry, consider the case where $B_1$ has an added row. By Lemma~\ref{lem:perfect-colors}, it suffices to show that we can color the cells of the new row while preserving connectivity in each color. We do so by duplicating the colors of the cells (including sinks) in an adjacent row. Connectivity of the resulting coloring follows from that of the original. \end{proof} \begin{lemma} \label{lem:characterization-c3-nonminimal} Let \(B_1\) be a non-minimal board satisfying the three conditions of Theorem~\ref{thm:characterization-c3}. Then there exists a board \(B_0\) that also satisfies all three conditions and such that \(B_0 \lessdot B_1\). \end{lemma} \begin{proof} By symmetry, suppose $B_1$ is non-minimal in its number $m$ of rows. By removing a row from $B_1$ that is not among the first or last two rows and does not contain a sink, we obtain a board \(B'_0\) satisfying conditions (1) and (3) such that \(B'_0 \lessdot B_1\). If \(B'_0\) also satisfies condition (2), then we are done, so we may assume that it does not. Then \(B'_0\) must contain one of the four unsolvable configurations, and \(B_1\) is obtained by inserting a single empty row or column to remove the unsolvable configuration. Figure~\ref{fig:perfect-reductions} shows all possibilities for \(B'_0\), as well as the locations where rows or columns may be inserted to yield a corresponding possibility for \(B_1\). (\(B'_0\) may have additional empty rows and columns beyond those shown, but this does not affect the proof.) For each such possibility, Figure~\ref{fig:perfect-reductions} highlights another row or column which may be deleted from \(B_1\) to yield \(B_0 \lessdot B_1\) where \(B_0\) satisfies all three conditions. \end{proof} \begin{figure} \centering \hfil \subcaptionbox{$L$, $7\times7$}{\includegraphics[scale=0.6]{solver/reductions/L7_7_reduction}}\hfill \subcaptionbox{$L$, $6\times9$}{\includegraphics[scale=0.6]{solver/reductions/L6_9_reduction}}\hfill \subcaptionbox{\label{fig:perfect-reductions-row-choice}$I$, $5\times11$}{\includegraphics[scale=0.6]{solver/reductions/I5_11_0_reduction}}\hfill \subcaptionbox{$I$, $5\times11$}{\includegraphics[scale=0.6]{solver/reductions/I5_11_1_reduction}\vspace{0.2in}}\hfill \subcaptionbox{$I$, $5\times11$}{\includegraphics[scale=0.6]{solver/reductions/I5_11_2_reduction}}% \hfil \caption{ All boards satisfying conditions (1) and (3) but not (2), up to rotations, reflections, and recolorings. An empty row or column may be inserted in any of the locations marked ``$+$'' to yield a board satisfying all three conditions. Removing the row or column marked ``$-$'' then preserves the conditions. In case~(c), remove a row that does not contain the blue sink. In case~(d), $\vcenter{\hbox{\includegraphics[scale=0.6]{figs/dotdotdot}}}$ denotes zero or more rows. } \label{fig:perfect-reductions} \end{figure} \begin{proof}[Proof of Theorem~\ref{thm:characterization-c3}] It follows from Lemmas \ref{lem:sink-distance}, \ref{lem:impossible-c3}, and \ref{lem:size-constraint-c3} that all boards with perfect layouts must obey the three properties of the theorem. We prove that the properties are also sufficient by induction on the size of the board. As a base case, the claim holds for minimal boards by Lemma~\ref{lem:characterization-c3-minimal}. For non-minimal boards \(B_1\), Lemma~\ref{lem:characterization-c3-nonminimal} shows that there is a smaller board \(B_0\) that satisfies all three conditions and such that \(B_0 \lessdot B_1\). By the inductive hypothesis, \(B_0\) has a perfect layout. Lemma~\ref{lem:add-row} shows that \(B_1\) also has a perfect layout. \end{proof} \section{Open Questions} The main complexity open question is whether {TGINGTLI} is $\Sigma_2^P$-complete. Given our NP- and coNP-hardness results, we suspect that this is true. One could also ask complexity questions of more restrictive versions of the game. For example, what if the board has a constant number of rows? When characterizing perfect layouts, we saw many of our lemmas generalized to different numbers of colors. It may be interesting to further explore the game and try to characterize perfect layouts with more than three colors. A related problem is which boards and configurations of sinks admit a \defn{damage-free} layout, where any packet entering from the boundary either reaches the sink of the correct color or ends up in an infinite loop. Such a layout avoids losing, and in the game as implemented, such a layout actually wins the game (because the player wins if there is ever insufficient room for a new source to be placed). Can we characterize such layouts like we did for perfect layouts? Perfect and damage-free layouts are robust to any possible sources. However, for those boards that do not admit a perfect or damage-free layout, it would be nice to have an algorithm that determines whether a given set of sources or sequence of packets still has a placement of arrows that will win on that board. Because the board starts empty except for the sinks, our hardness results do not apply. Having a unique solution is often a desirable property of puzzles. Thus it is natural to ask about ASP-hardness and whether counting the number of solutions is \#P-hard. \section*{Acknowledgments} This work was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.892) taught by Erik Demaine in Spring 2019. We thank the other participants of that class for related discussions and providing an inspiring atmosphere. In particular, we thank Quanquan C. Liu for helpful discussions and contributions to early results. Most figures of this paper were drawn using SVG Tiler [\url{https://github.com/edemaine/svgtiler}]. Icons (which match the game) are by looneybits and released in the public domain [\url{https://opengameart.org/content/gui-buttons-vol1}]. \bibliographystyle{alpha}

BkiUfXY5qhLBrnqN0VT4

\section{Introduction} In the earliest prediction of the hidden-charm baryon $N_{c\bar c}$ states above 4~GeV~\cite{Wu:2010jy}, which are now known as the $P_c$ states and have been reported by the LHCb Collaboration~\cite{Aaij:2015tga,Aaij:2019vzc}, the~$\Lambda_{c\bar c}$ states are also predicted with different masses and spin-parities. Ten $\Lambda_{c\bar c}$ states are obtained in a coupled-channel model in Ref.~\cite{Xiao:2019gjd} and their couplings to $J/\psi \Lambda$ are found to be sufficiently large. Similar to the discovery of the $P_c$ states in the $J/\psi p$ channel~\cite{Xiao:2013yca}, the~$\Lambda_{c\bar c}$ states could be sought in the $J/\psi\Lambda$ invariant mass distributions. In Ref.~\cite{Wang:2019nvm}, ten isoscalar molecular pentaquarks in the $\bar{D}^{(*)} \Xi_c$, $\bar{D}^{(*)} \Xi_c^{\prime}$ and $\bar{D}^{(*)} \Xi_c^*$ systems are predicted by calculating the effective potentials with the chiral effective field theory up to the next-to-leading order. These hidden-charm pentaquark states with strangeness are investigated in the quark model including the color octet type and color singlet type~\cite{Irie:2017qai} and the one-boson exchange model~\cite{Chen:2016ryt}. The decay behaviors of the $\Lambda_{c \bar c}$ states in various hadronic molecular assumptions with different spin-parities are studied in Ref.~\cite{Shen:2019evi} and these decay patterns are suggested to help understanding the internal structures after the experimental searches. Although there are various predictions for such strangeness baryons within hidden charm theoretically, there is no experimental evidence for them until~now. Since the first prediction of the $\Lambda_{c\bar c}$ states, there arises various suggestions of searching for these states experimentally. The decay of $\Lambda_{b}$ into $J/\psi K^{0} \Lambda$ states is studied in Ref.~\cite{Lu:2016roh} and a clear peak is predicted in the $J/\psi \Lambda$ invariant mass distribution. In Ref.~\cite{Feijoo:2015kts}, the~authors propose the $\Lambda _b \to J/\psi \eta \Lambda$ decay and find that a resonance signal with similar strength as the case of $P_c$ should be observed in the $J/\psi \Lambda$ invariant mass spectrum. The $\Xi_b^{-} \to J/\psi K^{-} \Lambda$ reaction is discussed in Ref.~\cite{Chen:2015sxa} and a hidden-charm pentaquark state with strangeness is expected to be clearly seen. In the last process, the~$K^-$ meson in the final state is a charged meson, and~it makes this process suitable for experimental measurement. Actually, the~possible peak structures in the $J/\psi \Lambda$ invariant mass spectrum in the $\Xi_b^{-} \to J/\psi K^{-} \Lambda$ decay have already been sought by the LHCb Collaboration~\cite{Aaij:2017bef}, while the results of these data should still be under analysis. At this point, it is worthwhile to mention that the $\Lambda_b^0\to J/\psi\Lambda \pi^+\pi^-$ should also be a promising process to seek for the $\Lambda_{c\bar c}$ states. In this paper, we will study the $\Xi_b^{-} \to K^{-} J/\psi \Lambda$ decay and focus on another mechanism, triangle singularity, which could also produce peak structures in the $J/\psi \Lambda$ invariant mass~spectrum. A triangle singularity is due to the simultaneous on-shellness of three intermediate particles in a loop diagram, and~at the same time the interactions at all vertices can happen classically~\cite{Coleman:1965xm}. It is a logarithmic singularity, and~is able to produce peaks in energy distributions. It has been known for a long time~\cite{Karplus:1958zz,Landau:1959fi,Bjorken:1959fd}, and~there were debates in the 1960s on whether it could produce observable effects~\cite{Schmid:1967ojm}. Nevertheless, it was difficult to find reactions that satisfy the rigorous kinematic conditions and at the same time can evade the subtle interfering cancellation between the tree-level and triangle diagram, dictated by the Schmid theorem at that time~\cite{Schmid:1967ojm}. \textls[-15]{In the last 10 years or so, suggestions were proposed that many reactions which can be (or have been) measured at modern high-energy experiments could match the kinematic conditions and might receive important contributions from triangle singularities; then the study of triangle singularity started to bring a lot interest until now~\cite{Wu:2011yx, Aceti:2012dj, Wu:2012pg, Wang:2013cya,Wang:2013hga, Ketzer:2015tqa, Achasov:2015uua, Liu:2015taa, Liu:2015fea, Guo:2015umn, Szczepaniak:2015eza, Guo:2016bkl, Bayar:2016ftu, Wang:2016dtb, Pilloni:2016obd, Xie:2016lvs, Szczepaniak:2015hya, Roca:2017bvy, Debastiani:2017dlz, Samart:2017scf, Sakai:2017hpg, Pavao:2017kcr, Xie:2017mbe, Bayar:2017svj, Liang:2017ijf, Oset:2018zgc, Dai:2018hqb, Dai:2018rra, Guo:2019qcn, Liang:2019jtr, Nakamura:2019emd, Du:2019idk, Liu:2019dqc, Jing:2019cbw, Braaten:2019gfj, Aaij:2019vzc, Sakai:2020ucu, Sakai:2020fjh, Molina:2020kyu, Braaten:2020iye, Alexeev:2020lvq, Ortega:2020ayw}.~For a recent review, we refer to Ref.~\cite{Guo:2019twa} (see Tables~\ref{table:TS} and \ref{table:TSbc} therein for reactions proposed since the 1990s)}. The fact that triangle singularities may affect the extraction of the $P_c$ resonance parameters from the $\Lambda_b \to J/\psi K^- p$ reaction was immediately noticed~\cite{Guo:2015umn,Liu:2015fea}. Later, the~$\Lambda(1890)\chi_{c1}p$ triangle was identified to be the most special from an analysis of many triangle diagrams with various possible combinations of a a charmonium, a~proton and a $\Lambda^*$~\cite{Bayar:2016ftu}. In this case, both triangle singularity and threshold effect occur at about 4.45~GeV as already pointed out in Ref.~\cite{Guo:2015umn}. Once the $\chi_{c1} p$ is in an $S$-wave, such that the quantum numbers of the $J/\psi p$ could be $J^P=1/2^+$ or $3/2^+$, it would generate a narrow peak in the $J/\psi p$ invariant mass distribution. The updated LHCb measurement~\cite{Aaij:2019vzc} does not have a peak at 4.45~GeV, but~there is a dip at this energy between the two narrow $P_c(4440)$ and $P_c(4457)$ peaks. Thus, the~possibility of interference of such a triangle singularity with pentaquark signals is still there, and~might be partly responsible for the special production properties of the $P_c$ states~\cite{Sakai:2019qph}. However, for~the cases that the spin and parity of the $P_c$ are $3/2^-$ or $5/2^+$, as~considered in, e.g.,~Refs.~\cite{Chen:2015moa, Roca:2015dva, He:2015cea, Lin:2017mtz, He:2016pfa, Chen:2016otp, Xiang:2017byz}, the~triangle singularity contribution from the $\Lambda^*(1890)\chi_{c1}p$ loop would be severely suppressed. In the present work, we will investigate possible triangle singularities in the $\Xi_b^{-} \to J/\psi K^{-} \Lambda$ reaction in a similar way to Ref.~\cite{Bayar:2016ftu}. This work is organized as follows. After the introduction, we first show the theoretical framework and an analysis of possible triangle singularities in Section~\ref{sec:theory}, and~the corresponding discussions are presented. Then in Section~\ref{sec:analysis}, a~detailed analysis and discussions of the amplitude with $\Xi(2120)\chi_{c1}\Lambda$ are given, which is followed by a brief summary in Section~\ref{sec:summary}. \begin{table}[tb] \caption{\label{table:TS}The charmonium and $\Xi^{* -}$ states in Fig.~\ref{Fig:1stdiag} that could generate triangle singularities and the positions of the corresponding singularities. Among the listed hyperons, $\Xi(2030)$ is a three-star state; $\Xi(2250)$ and $\Xi(2370)$ are two-star states; $\Xi(2120)$ and $\Xi(2500)$ are one-star states~\cite{Zyla:2020}. } \renewcommand\arraystretch{1.5} \begin{tabular}{p{1.0cm}<{\centering}p{2.0cm}<{\centering}p{2.0cm}<{\centering}p{3.0cm}<{\centering}} \hline No. & $c \bar c$ & $\Xi^{* -}$ & Position of triangle singularity (MeV) \\ \hline 1 & $J/\psi$ & $\Xi(2500)$ & 4232 \\ 2 & $\chi_{c0}$ & $\Xi(2250)$ & 4546 \\ 3 & $\chi_{c0}$ & $\Xi(2370)$ & 4665 \\ 4 & $\chi_{c1}$ & $\Xi(2120)$ & 4628 \\ 5 & $\chi_{c1}$ & $\Xi(2250)$ & 4696 \\ 6 & $\chi_{c2}$ & $\Xi(2120)$ & 4680 \\ 7 & $h_c$ & $\Xi(2120)$ & 4644 \\ 8 & $h_c$ & $\Xi(2250)$ & 4730 \\ 9 & $\eta_c(2S)$ & $\Xi(2030)$ & 4754 \\ 10 & $\eta_c(2S)$ & $\Xi(2120)$ & 4797 \\ 11 & $\psi(2S)$ & $\Xi(2030)$ & 4810 \\ \hline \end{tabular} \end{table} \unskip \begin{table}[tb] \caption{\label{table:TSbc}The $D_s^{* -}$, $\Xi_c^{* 0}$ and intermediate particle 3 in Fig.~\ref{Fig:2nddiag} and Fig.~\ref{Fig:3rddiag} that could generate triangle singularities and the positions of the corresponding singularities. } \renewcommand\arraystretch{1.5} \begin{tabular}{p{0.3cm}<{\centering}p{1.9cm}<{\centering}p{1.6cm}<{\centering}p{1.7cm}<{\centering}p{2.3cm}<{\centering}} \hline No. & $D_s^{* -}$ & $\Xi_c^{* 0}$ & Intermediate particle 3 & Position of triangle singularity (MeV) \\ \hline 1 & $\bar D_{s1/s3}^*(2860)$ & $\Xi_c(2930)$ & $\bar{D}^0$ & 4838 \\ 2 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3055)$ & $\bar{D}^0$ & 4922 \\ 3 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3080)$ & $\bar{D}^0$ & 4957 \\ 4 & $\bar D_{s1/s3}^*(2860)$ & $\Xi_c(2930)$ & $\bar{D}^{* 0}$ & 4959 \\ 5 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3080)$ & $\bar{D}^{* 0}$ & 5089 \\ 6 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3080)$ & $\Lambda_c^+$ & 4990 \\ 7 & $\bar D_{s1/s3}^*(2860)$ & $\Xi_c(2930)$ & $\Lambda_c^+$ & 5149 \\ \hline \end{tabular} \end{table} \section{Theoretical Framework and Analysis Results of All Possible Triangle~Singularities} \label{sec:theory} In the process of a triangle loop diagram, the~initial particle first decays to two intermediate particles $A$ and $B$. Then particle $A$ continues to decay into two particles, one of which is a final state and the other is the intermediate particle $C$. This intermediate particle $C$ could interact with particle $B$ to generate final state particles. The kinematic conditions for a triangle singularity in the physical region are~\cite{Schmid:1967ojm,Bayar:2016ftu}: (1) the three intermediate particles $A$, $B$ and $C$ are all on the mass shell at the same time; (2) the velocity of particle $C$ is larger than that of particle $B$ and is in the same direction in the rest frame of the initial state. In other words, triangle singularity corresponds to the situation that the loop reaction has actually occurred as a classical process~\cite{Coleman:1965xm}, rather than a virtual process. We are interested in the possible triangle singularity effects in the $J/\psi\Lambda$ invariant mass distribution. The~region of triangle singularity peak is limited to a small region starting from the threshold of two particles that rescatter into the final-state $J/\psi\Lambda$ (for a detailed analysis, see Ref.~\cite{Guo:2019twa}). There are four possible kinds of triangle diagrams for the $\Xi_b^- \to K^- \Lambda J/\psi$ decay that may have a triangle singularity in the physical region (when the widths of the intermediate particles are neglected) as shown in Figure~\ref{Fig:feyndiag4}. Using the intuitive equation derived in Ref.~\cite{Bayar:2016ftu}, we find that the physical-region triangle singularity emerges in the process of Figure~\ref{Fig:feyndiag4}a--c. The $\Xi^{* -}$ in Figure~\ref{Fig:feyndiag4}a represents all the ten $\Xi$ resonances in the Review of Particle Physics (RPP)~\cite{Zyla:2020}, including $\Xi(1530)$, $\Xi(1620)$, $\Xi(1690)$, $\Xi(1820)$, $\Xi(1950)$, $\Xi(2030)$, $\Xi(2120)$, $\Xi(2250)$, $\Xi(2370)$ and $\Xi(2500)$. For the charmonium states $c \bar c$, we take $\eta_c$, $J/\psi$, $\chi_{c 0}$, $\chi_{c 1}$, $\chi_{c 2}$, $h_c$, $\eta_c(2S)$ and $\psi(2S)$ into consideration. We take all the listed $\Xi_c^{* 0}$ and $D_s^{* -}$ in RPP~\cite{Zyla:2020} into account for the exchanged particles in Figure~\ref{Fig:feyndiag4}b,c. \begin{figure}[h] \centering \subfigure[]{\label{Fig:1stdiag} \includegraphics[width=0.5\linewidth]{figures/feynman1}} \subfigure[]{\label{Fig:2nddiag} \includegraphics[width=0.45\linewidth]{figures/feynman2}} \\ \ \ \subfigure[]{\label{Fig:3rddiag} \includegraphics[width=0.45\linewidth]{figures/feynman3}} \ \ \subfigure[]{ \includegraphics[width=0.45\linewidth]{figures/feynman4}} \caption{Four possible kinds of triangle diagrams for the $\Xi_b^- \to K^- \Lambda J/\psi$ decay that may have a triangle singularity of interest. The~momentum of each particle and the numbering of the exchanged particles used here are given in the brackets in the first diagram. \label{Fig:feyndiag4}} \end{figure} Through checking the two conditions of triangle singularity for all possible combinations of the considered intermediate particles mentioned above, we find that there are in total 11 cases for Figure~\ref{Fig:feyndiag4}a and 7 cases for Figure~\ref{Fig:feyndiag4}b,c that would have singularities in the physical region. The charmonium and $\Xi^{* -}$ states in Figure~\ref{Fig:feyndiag4}a, which could generate peak structures in the $J/\psi \Lambda$ mass spectrum, and~the corresponding positions of these triangle singularities are given in Table~\ref{table:TS}. In Table~\ref{table:TSbc}, we present the three exchanged particles of Figure~\ref{Fig:feyndiag4}b,c together with the corresponding triangle singularity positions, which are generally higher than those from Figure~\ref{Fig:feyndiag4}a listed in Table~\ref{table:TS}. Note that there is another combination, $\bar D_{sJ}(3040)\Xi_c(3645)\bar D^0$, which can give rise to a triangle singularity at 4511~MeV. However, so far the $D_{sJ}(3040)$ has only been observed in $D^*K$~\cite{Zyla:2020}, suggesting that it might have an unnatural parity and thus could not decay into $\bar D K$. Consequently, this combination is not listed in the table. One~notices that both the $D_{s1}^*(2700)$ was observed in $B\to D\bar D K$~\cite{Brodzicka:2007aa,Lees:2014abp} and the $D_{s1/s3}^*(2860)$ were seen in the decays of $B_s\to \bar D \bar K \pi$~\cite{Aaij:2014xza}. Similarly, the~$D_{s1}^*(2700)$ should be easily produced from the decays of the $\Xi_b$ associated with a $\Xi_c$ baryon, as~shown in the first vertex in Figure~\ref{Fig:feyndiag4}b,c. Thus, among~the combinations listed in Table~\ref{table:TSbc}, the~ones with the $D_{s1}^*(2700)$ are expected to be more important than the others. Once a structure is observed in the $J/\psi\Lambda$ spectrum above 4.9~GeV in a future experiment, such~triangle singularities need to be considered. In the rest of this paper, we will focus on the singularities in Table~\ref{table:TS}, whose situation is more~involved. Since the triangle singularity is caused by the on-shellness of the intermediate particles, we first consider the scalar loop integral as follows, \begin{eqnarray} I&=&i \int \frac{d^4 q}{(2\pi)^4} \frac1{(q^2-m_1^2+i m_1\Gamma_1)[(P-q)^2-m_2^2+i m_2\Gamma_2]} \nonumber\\ &&\times \frac1{[(k_1-q)^2-m_3^2+i \epsilon]}, \label{Eq:I} \end{eqnarray} \noindent where the momenta and the numbering of particle masses are as given in the first diagram in Figure~\ref{Fig:feyndiag4}, and~$\Gamma_1$ and $\Gamma_2$ are the widths of intermediate particles 1 and 2, respectively. We can use $|I|^2$ to analyze the singular behavior of the diagrams in Figure~\ref{Fig:feyndiag4}. It should be noted that the interactions of all the related vertices are not included here. We should consider these interactions from a physical point of view to draw further~conclusions. Let us turn to the three vertices in Figure~\ref{Fig:feyndiag4}a. For the first vertex $\Xi_b \to \Xi^{*} c \bar c$, it is a weak decay and involves the Cabibbo--Kobayashi--Maskawa (CKM) matrix elements ${V_{bc}^*V_{cs}}$. The~strength should be comparable with that of $B \to K c \bar c$ for the same charmonium. The branching ratios for different charmonia in the latter process, which can be found in the RPP~\cite{Zyla:2020}, are given in Table~\ref{table:1vertex}. It can be seen that for the $\chi_{c2}$ and $h_c$, the~branching ratios are one-order-of-magnitude smaller than that of the other charmonia. This indicates that the cases of the charmonium being the $\chi_{c2}$ or $h_c$ would not provide a strong signal of triangle singularity, and~can be safely neglected. For $\chi_{c0}$, the~decay rates from the neutral and charged $B$ mesons are very different. Both the two branching ratios of the $\chi_{c0}$ are smaller than the rest, although~in the charged case it is of the same order of magnitude. We conclude that, for~the $\Xi_b \to \Xi^* c\bar{c}$ decay, the~cases where the intermediate charmonium is one of $J/\psi$, $\chi_{c1}$, $\eta_c(2S)$ and $\psi(2S)$ are expected to be the main decay~channels. \begin{table}[tb] \caption{\label{table:1vertex}The branching ratios for different charmonia in $B \to K c \bar c$.} \renewcommand\arraystretch{1.5} \begin{tabular}{p{2.0cm}<{\centering}p{3.0cm}<{\centering}p{3.0cm}<{\centering}} \hline $c \bar c$ & $BR(B^0 \to c \bar{c} K^0)$ & $BR(B^+ \to c \bar{c} K^+)$ \\ \hline $J/\psi$ & $8.73 \times 10^{-4}$ & $1.01 \times 10^{-3}$ \\ $\chi_{c0}$ & $1.11 \times 10^{-6}$ & $1.49 \times 10^{-4}$ \\ $\chi_{c1}$ & $3.93 \times 10^{-4}$ & $4.84 \times 10^{-4}$ \\ $\chi_{c2}$ & $<1.5 \times 10^{-5}$ & $1.1 \times 10^{-5}$ \\ $h_c$ & & $<3.8 \times 10^{-5}$ \\ $\eta_c(2S)$ & & $4.4 \times 10^{-4}$ \\ $\psi(2S)$ & $5.8 \times 10^{-4}$ & $6.21 \times 10^{-4}$ \\ \hline \end{tabular} \end{table} For the vertex of $c \bar c \Lambda \to J/\psi \Lambda$, the~coupling strength is affected from the spin interactions between the charm quark and anticharm quark. Once the $c \bar c$ has a spin flip, this vertex will be much suppressed due to breaking the heavy quark spin symmetry. The quantum numbers of the final state $J/\psi$ are $1^-$, and~the spins of charm and anticharm quarks are in the same direction. However, in~the charmonia $h_c$ and $\eta_c(2S)$, the~total spin of charm and anticharm quarks is $0$. Therefore, the~$h_c \Lambda$ or $\eta_c(2S) \Lambda \to J/\psi \Lambda$ process would be suppressed because of the spin flip of the charm and anticharm quarks. For the other charmonia considered here, the~couplings of $c \bar c \Lambda \to J/\psi \Lambda$ could be of the same order of magnitude but are still not determined. We will write the amplitudes with different $J^P$ assumptions that are subject to the Lorentz covariant orbital-spin coupling scheme~\cite{Zou:2002yy} to study their behavior in the next~section. For the $\Xi^* \bar{K} \Lambda$ vertex, we can infer its strength from the $\Xi^* \to \Lambda \bar K$ process for each considered $\Xi^*$ state. For the $\Xi(2030)$, the~$\Lambda \bar K$ state occupies about 20\% of all possible decay rates. In the RPP, the~$\Lambda \bar K$ channel is the only seen and listed decay mode for the $\Xi(2120)$~\cite{Zyla:2020}, although~its exact ratio has not been measured yet. The $\Lambda \bar K$ channel is not listed in the decay modes of the $\Xi(2250)$ and $\Xi(2370)$, and~actually all the decay channels for the former are three-body states, meaning that the $\Lambda \bar K$ mode might be suppressed for these two states. For the $\Xi(2500)$, the~$\Lambda \bar K$ mode is listed, with~the ratio of branching fractions $\Gamma(\Lambda \bar K)/\left(\Gamma(\Xi\pi)+\Gamma(\Lambda \bar K)+\Gamma(\Sigma \bar K)+ \Gamma(\Xi(1530)\pi)\right) = 0.5\pm0.2$ measured in Ref.~\cite{Alitti:1969rb}, but~it is not clear whether the peak observed therein corresponds to the $\Xi(2500)$ as commented in RPP~\cite{Zyla:2020}. Through the analysis of the three vertices above, for~the 11 cases listed in Table~\ref{table:TS}, nos.~2 and 3 and nos.~5--10 are suppressed; only nos.~1, 4 and 11 are left. Now let us turn to the character of the intermediate particles in the loop, especially their quantum numbers, since they actually affect a lot not only the magnitude but also the line shape of the singularity. The quantum numbers are quite important as they directly determine the partial waves of the interaction, and~high partial wave interactions should be suppressed in the near-threshold region. For the $\Xi(2030)$ state, its spin is equal to or greater than $5/2$~\cite{Hemingway:1977uw}. It leads to high partial-wave ($l \ge 2$) interaction at the $\Xi(2030) \bar{K} \Lambda$ vertex, and~will not be considered further. {We will discuss that how the high partial-wave interactions in the triangle loop would suppress the contribution of the triangle singularity later.} For the other two $\Xi^*$ states, their quantum numbers are not determined yet, and~we will consider various $J^P$ possibilities~later. The $m_{J/\psi \Lambda}$ distribution for $|I|^2$ with an arbitrary normalization factor of these three cases is shown in Figure~\ref{Fig:4TS}. For the widths of the involved charmonium, we quote their central values given in the RPP~\cite{Zyla:2020}. While for some of the $\Xi^*$ states like the $\Xi(2120)$ and $\Xi(2500)$, the~widths are not precisely determined. For the width of the former, we take the upper bound of the given range, which is $20$ MeV. In the latter case, two values were reported by different experimental groups, which are $150$~\cite{Alitti:1969rb} and $59$~MeV~\cite{Bartsch:1969qs}. We give results using these two values. As shown in Figure~\ref{Fig:4TS}, the~triangle singularity from the $J/\psi \Xi(2500) \Lambda$ loop diagram is much broader than the other two cases. As discussed above, the~triangle loop diagram that involves the $\Xi(2030) \bar{K} \Lambda$ vertex should be severely suppressed because of the existence of the $D$- or higher wave interaction, while for the $\Xi(2120)$ involved vertices, there~could be lower partial-wave interactions. The contribution from the $\Xi(2120)\chi_{c1}\Lambda$ loop (the blue dotdashed line in Figure~\ref{Fig:4TS}) is expected to be much larger than that from the $\Xi(2030)\psi(2S)\Lambda$ (the green dotted line) once all the interactions at the vertices are~included. \begin{figure}[h] \centering \includegraphics[width=0.95\linewidth]{figures/4TS} \caption{The value of $|I|^2$ in the $m_{J/\psi \Lambda}$ invariant mass distribution. The~red solid (orange dashed), green dotted and blue dotdashed lines correspond to the Nos.~1, 4 and 11 cases in Table~\ref{table:TS}, respectively. For~the case of No.1, the~red solid and orange dashed lines represent the different choices of the $\Xi(2500)$ width: the narrower and wider peaks correspond to taking 59 and 150~MeV for the width, respectively. } \label{Fig:4TS} \end{figure} Combining all the analysis above, the~$\Xi(2120)\chi_{c1}\Lambda$ loop is found to be the most promising process to generate an observable triangle singularity in the $\Xi^-_{b} \to K^- J/\psi \Lambda$ decay ({We should also note that the $\Xi(2120)$ is just a one-star state~\cite{Zyla:2020}, and~its existence needs further confirmation}). We will calculate and discuss the structures from this triangle loop diagram in detail in the next~section. \section{Detailed Analysis of the Amplitudes for the Diagram with \texorpdfstring{$\bm{\chi_{c1} \Xi(2120) \Lambda}$}{chic1 Xi Lambda} Loop} \label{sec:analysis} We will focus on the triangle diagram involving the $\chi_{c1}$ and the $\Xi(2120)$ in this section, as~it is the most promising diagram to have a sizeable triangle singularity contribution. The specific interactions of vertices are taken into account in this section, here we are going to study them by constructing effective Lagrangians with the lowest number of derivatives. At present, the~quantum numbers of the $\Xi(2120)$ state and the $J/\psi \Lambda$ system are both flexible, and~the former affects the $\Xi_b \Xi^* \chi_{c1}$ and $\Xi^* \Lambda K^-$ vertices, while the later influences the $\chi_{c1} \Lambda J/\psi \Lambda$ vertex. Let us discuss them one by~one. Firstly, let us consider different choices of the spin-parity quantum numbers of the $\Xi(2120)$, which~are still unknown. The two vertices involving the $\Xi(2120)$ in this diagram are $\Xi_b \Xi(2120) \chi_{c1}$ and $\Xi(2120) \Lambda \bar K$. In Table~\ref{table:PW}, the~assumed spin-parity of $\Xi(2120)$ and the corresponding lowest allowed partial waves of these two vertices are listed. {In the Lorentz covariant orbital-spin coupling scheme, the~interaction of a vertex is directly related to its partial wave $L$ and the relative momentum of the two final particles, i.e.,~the strength of interaction is proportional to $q^L$. Therefore, the~$D$- or higher partial waves are neglected since the momentum is very small around where the triangle singularity happens.} Thus, $J^P=1/2^+$, $1/2^-$ and $3/2^+$ possibilities are not suppressed by high partial waves. Among~these three cases, the~$J^P=1/2^-$ possibility is singled out as both vertices are in $S$ waves, and~the strength of the amplitude in this case should be stronger than that of the other cases. Therefore, we will calculate the amplitude with $J^P(\Xi^*)=1/2^-$ to check the structure behavior from the triangle~singularity. \begin{table}[tb] \caption{\label{table:PW}The interactions of the $\Xi_b \Xi(2120) \chi_{c1}$ and $\Xi(2120) \Lambda \bar K$ vertices with different spin-parities of $\Xi(2120)$.} \renewcommand\arraystretch{1.5} \begin{tabular}{p{2.0cm}<{\centering}p{3.0cm}<{\centering}p{3.0cm}<{\centering}} \hline $J^P$ & $\Xi_b \Xi(2120) \chi_{c1}$ & $\Xi(2120) \Lambda \bar K$ \\ \hline $1/2^+$ & $S$-wave & $P$-wave \\ $1/2^-$ & $S$-wave & $S$-wave \\ $3/2^+$ & $S$-wave & $P$-wave \\ $3/2^-$ & $S$-wave & $D$-wave \\ $5/2^+$ & $P$-wave & $F$-wave \\ $5/2^-$ & $P$-wave & $D$-wave \\ \hline \end{tabular} \end{table} We then consider the quantum numbers of the $J/\psi \Lambda$ in the final state for fixed spin-parity of the $\Xi(2120)$. For the same reason mentioned before, the~$D$- or higher partial wave interactions are not considered in the present discussion. If the quantum numbers of the $J/\psi \Lambda$ system are $1/2^+$ or $3/2^+$, the~$\chi_{c1} \Lambda$ system can be in an $S$-wave and the $J/\psi \Lambda$ in a $P$-wave. On the contrary, if~the quantum numbers of the $J/\psi \Lambda$ system are $1/2^-$ or $3/2^-$, there is a $P$-wave in the $\chi_{c1} \Lambda$ system and an $S$-wave in the $J/\psi \Lambda$ system. In both cases, there are an $S$-wave and a $P$-wave for this four-particle-vertex, but~the corresponding behaviors of these two triangle loop integrations are different. { As discussed in the previous paragraph, the~effective Lagrangian of a $P$-wave interaction is proportional to the relative momentum of the involved particle. Meanwhile, at~the singularity point, the~invariant mass $m_{J/\psi \Lambda}$ is not far away from the $\chi_{c1}\Lambda$ threshold, and~both the $\chi_{c1}$ and the $\Lambda$ are on-mass-shell. It tells that the momenta of the $\chi_{c1}$ and $\Lambda$ are small. Thus, for~the negative parity case, the~$P$-wave is in the $\chi_{c1}\Lambda$ system, which provides a loop momentum $q$ to the loop integral and suppresses the singular behavior around the $\chi_{c1}\Lambda$ threshold, while the $P$-wave interaction of the $J/\psi \Lambda$ system for the positive parity case is independent of the loop integration.} In the next step, we will take the quantum numbers of the $J/\psi \Lambda$ system as $1/2^+$ and $1/2^-$, respectively, to~investigate how the finial results are~affected. We first take the quantum numbers of the $J/\psi \Lambda$ system being $1/2^+$ and the spin-parity of the $\Xi(2120)$ are assumed to be $1/2^-$. The amplitudes for the vertices are constructed as \begin{eqnarray} t_{\Xi_b \Xi^* \chi_{c1}} &=& g_a \bar{u}(P-q) \gamma^\mu u(P) \epsilon^*_\mu(q), \nonumber \\ t_{\Xi^* \Lambda \bar K} &=& g_b \bar{u}(k_1-q) u(P-q), \nonumber \\ t_{\Lambda \chi_{c1} \Lambda J/\psi} &=& g_c \epsilon_\mu(q) \bar{u}(k_1-k_2) ({\Slash k_1}+\sqrt{k_1^2}) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) \nonumber \\ && \times u(k_1-q) \epsilon_\nu^*(k_2) k_1^\nu \frac{k_1^2+m_{J/\psi}^2-m_{\Lambda}^2}{k_1^2}, \end{eqnarray} here we follow the momentum conventions given in Figure~\ref{Fig:feyndiag4}a. The amplitude of this triangle diagram~reads \begin{eqnarray} \mathcal{M} &=& g \int \frac{d^4 q}{(2\pi)^4} \bar{u}(k_1-k_2) ({\Slash k_1}+\sqrt{k_1^2}) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) \nonumber\\ && \times \frac{{\Slash k_1}-{\Slash q}+m_3}{(k_1-q)^2-m_3^2} \frac{{\Slash P}-{\Slash q}+m_2}{(P-q)^2-m_2^2+i m_2 \Gamma_2} \nonumber\\ && \times \frac{-\gamma_{\mu}+\frac{q_\mu {\Slash q}}{m_1^2}}{q^2-m_1^2+i m_1 \Gamma_1} u(P) \frac{k_1^2+m_{J/\psi}^2-m_{\Lambda}^2}{k_1^2} \nonumber\\ && \times \epsilon_\nu^*(k_2) k_1^\nu. \end{eqnarray} Then we calculate the decay with the quantum numbers of the $J/\psi \Lambda$ system being $1/2^-$. The~amplitude for the $\Lambda \chi_{c1} \Lambda J/\psi$ vertex reads \begin{eqnarray} t_{\Lambda \chi_{c1} \Lambda J/\psi}&=&g_c \bar{u}(k_1-k_2) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) ({\Slash k_1}+\sqrt{k_1^2}) \nonumber\\ && \times u(k_1-q) \epsilon_\mu^*(k_2) \epsilon_\nu(q) k_1^\nu \frac{2k_1 \cdot q}{k_1^2}, \end{eqnarray} and the amplitude in this case is \begin{eqnarray} \mathcal{M} &=& g \int \frac{d^4 q}{(2\pi)^4} \bar{u}(k_1-k_2) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) ({\Slash k_1}+\sqrt{k_1^2}) \nonumber\\ && \times \frac{{\Slash k_1}-{\Slash q}+m_3}{(k_1-q)^2-m_3^2} \frac{{\Slash P}-{\Slash q}+m_2}{(P-q)^2-m_2^2+i m_2 \Gamma_2} \nonumber\\ && \times \frac{-{\Slash k_1}+\frac{\Slash q}{m_1^2} k_1 \cdot q}{q^2-m_1^2+i m_1 \Gamma_1} u(P) \frac{2k_1 \cdot q}{k_1^2} \epsilon_\mu^*(k_2). \label{eq:m12m} \end{eqnarray} It should be noticed that the absolute values of the coupling constants in both cases are unknown. Therefore, our results are normalized to 1 at the maximum. The differential decay width in the rest frame of the initial particle $\Xi_b$ is \begin{equation} {\mathrm{d} \Gamma = \frac1{(2\pi)^3} \frac1{32m_{\Xi_b}^3} \overline{{\left| \mathcal M \right|}^2} \mathrm{d}m_{K \Lambda}^2 \mathrm{d}m_{J/\psi \Lambda}^2.} \label{eq:dGamma} \end{equation} {After integrating $\mathrm{d}m_{K \Lambda}^2$, t}he normalized differential decay widths in the $J/\psi \Lambda$ invariant mass distribution in these two cases are shown in Figure~\ref{Fig:dGamma} with the widths of the exchanged particles taken into~consideration. \begin{figure}[h] \centering \subfigure []{\label{Fig:dGamma1} \includegraphics[width=0.95\linewidth]{figures/dGamma}} \\ \subfigure[] {\label{Fig:dGamma2} \includegraphics[width=0.95\linewidth]{figures/dGamma2}} \caption{The $m_{J/\psi \Lambda}$ invariant mass distribution for the $\Xi^-_{b} \to K^- J/\psi \Lambda$ process via the $\chi_{c1} \Xi(2120) \Lambda$ loop with the quantum numbers of the $J/\psi \Lambda$ system being (\textbf{a}) $1/2^+$, and~(\textbf{b}) $1/2^-$, respectively. \label{Fig:dGamma}} \end{figure} It can be seen that in the $1/2^+$ case the peak structure from the triangle singularity emerges at $m_{J/\psi \Lambda}=4.628$~GeV and is distinct enough comparing to the background. However, this structure could not be observed any more in the $1/2^-$ case. Naively one would expect that the peak should still be there since the numerator of the integral would not diminish the singularity generated from the denominator. Then here arises a question why the $P$-wave $\chi_{c1} \Lambda$ in the four-particle vertex can suppress the singular behavior so much. When constructing the effective Lagrangian of the four-particle vertex in the Lorentz covariant orbital-spin coupling scheme~\cite{Zou:2002yy}, the~$P$-wave interaction in the $\chi_{c1} \Lambda$ system introduces a factor of the momentum difference of the exchanged $\Lambda$ and $\chi_{c1}$. Thus, the~amplitude is directly proportional to the velocity difference between these two particles in the $1/2^-$ case. As~discussed above, when the triangle singularity happens, all the three intermediate particles are on the mass shell and the process of each vertex actually happens classically. In this $\chi_{c1} \Xi(2120) \Lambda$ loop diagram, we~find that the velocity of the intermediate $\Lambda$ is very close to that of the $\chi_{c1}$ when the singularity occurs. In~other words, their relative velocity is quite small in the rest frame of $\Xi_b$, meaning that the singularity is located very close to the $\chi_{c1}\Lambda$ threshold. More specifically, the~factor $\left(-{\Slash k_1}+\frac{\Slash q}{m_1^2} k_1 \cdot q\right)$ in Equation~\eqref{eq:m12m} from the $P$-wave $\chi_{c1} \Lambda$ system in the triangle singularity region can be derived as~follows, \begin{equation} -{\Slash k_1}+\frac{\Slash q}{m_1^2} k_1 \cdot q \sim \frac{|\vec{q}|}{q^0}-\frac{|\vec{k}_1|}{k_1^0} \sim v_{\Lambda}-v_{\chi_{c1}}\sim 0, \end{equation} explaining the suppression of the singular behavior in this~case. Since the amplitude is actually related to the partial wave interactions of the $\chi_{c1} \Lambda \to J/\psi \Lambda$ vertex, the~$3/2^+$ and $3/2^-$ cases should be similar to the $1/2^+$ and $1/2^-$ cases, respectively. This~indicates that this $\Xi(2120) \chi_{c1} \Lambda$ loop diagram could generate a peak in the $m_{J/\psi \Lambda}$ mass spectrum for the $\Xi^-_{b} \to K^- J/\psi \Lambda$ decay when the quantum numbers of the $J/\psi \Lambda$ system are $1/2^+$ or $3/2^+$. We~then note that the threshold of the $\chi_{c1}\Lambda$ is at 4.626~GeV, which gives rise to a cusp (for the $S$-wave) in the distribution of the differential decay width. As the positions of the threshold and the triangle singularity (at about 4.628~GeV) are quite close, these two structures would lead to only one peak around this energy as shown in Figure~\ref{Fig:dGamma}a. Thus, if~a structure is observed in this region in future experiments, the~triangle singularity effects need to be taken into account for the $J^P=\frac12^+$ or $\frac32^+$ quantum~numbers. There are a lot of similarities between the $\Lambda_b \to K^- J/\psi p$ and $\Xi_b \to K^- J/\psi \Lambda$ through triangle loop diagrams. The $\Lambda_b \Lambda^* \chi_{c1}$ vertex, as~considered in Refs.~\cite{Guo:2015umn,Bayar:2016ftu}, and~the $\Xi_b \Xi^* \chi_{c1}$ vertex involve the same product of CKM matrix elements and should share a similar coupling. The strength of the two four-particle-vertices $\chi_{c1} p \to J/\psi p$ and $\chi_{c1} \Lambda \to J/\psi \Lambda$ should also be similar due to the approximate light-flavor SU(3) symmetry. For the two rest vertices $\Lambda(1890) K^- p$ and $\Xi(2120) K^- \Lambda$, the~$\bar K N$ channel occupies a rate of 20--35\% among all the decay modes of $\Lambda(1890)$, while the branching ratio of $\Xi(2120)$ is still unknown and $\bar K \Lambda$ is the only observed decay channel. We can roughly regard the strengths of these two vertices to be of the same order of magnitude. Thus, studying possible structures in the $\Xi_b \to K^- J/\psi \Lambda$ decay around 4.63~GeV should also provide useful insights into the possible role of triangle singularity in the $\Lambda_b \to K^- J/\psi p$ decay around 4.45~GeV. Yet, the~existence of one-star $\Xi(2120)$ state needs to be confirmed, which can be studied by analyzing the $\Lambda \bar K$ invariant mass distribution of the process under~discussion. \section{Summary} \label{sec:summary} Inspired by the analysis of the triangle singularities in the $\Lambda_b \to K^- J/\psi p$ process in Ref.~\cite{Bayar:2016ftu}, we~apply the same method to the $\Xi_b \to K^- J/\psi \Lambda$ process via charmonium-$\Xi^*$-$\Lambda$ intermediate states to check whether there exist possible triangle singularities. We find that there are 11 and 7 possible triangle singularities with various combinations of charmonium-$\Xi^*$ states and $D_S^*$-$\Xi_c^*$ states, respectively. For~the former, the~$\chi_{c1} \Xi(2120) \Lambda$ loop diagram is expected to be the most promising process to observe the singularity structure; for the latter, those with $\bar D_{s1}^*(2700)$ listed in Table~\ref{table:TSbc} should be more important than the others. Nevertheless, the~absolute values of all these amplitudes could not be precisely evaluated without knowing the amplitudes for all the involved vertices. For the $\chi_{c1} \Xi(2120) \Lambda$ diagram, when the spin-parity quantum numbers of the $J/\psi\Lambda$ system are $1/2^+$ or $3/2^+$, the~joint effects of the triangle singularity and the $\chi_{c1}\Lambda$ threshold would produce a narrow peak around 4.63~GeV in the $J/\psi\Lambda$ invariant mass distribution. A study of such effects needs to be taken into account in the search of hidden-charm strange pentaquarks, and~a study of the $\Xi_b \to K^- J/\psi \Lambda$ will also shed light on the role of triangle singularities in the $\Lambda_b\to K^- J/\psi p$ process. \section*{Acknowledgments} We are grateful to Eulogio Oset for a careful reading of our~manuscript. This work is supported in part by the Fundamental Research Funds for the Central Universities, by~the National Natural Science Foundation of China (NSFC) under Grants No.~11835015, No.~11947302 and No.~11961141012, by~the Chinese Academy of Sciences (CAS) under Grants No.~XDB34030303 and No.~QYZDB-SSW-SYS013, by~NSFC and the Deutsche Forschungsgemeinschaft (DFG) through the funds provided to the Sino-German Collaborative Research Center ``Symmetries and the Emergence of Structure in QCD'' (NSFC~Grant No. 11621131001, DFG Grant No. CRC110), and~by the CAS Center for Excellence in Particle Physics~(CCEPP). \section{Introduction} In the earliest prediction of the hidden-charm baryon $N_{c\bar c}$ states above 4~GeV~\cite{Wu:2010jy}, which are now known as the $P_c$ states and have been reported by the LHCb Collaboration~\cite{Aaij:2015tga,Aaij:2019vzc}, the~$\Lambda_{c\bar c}$ states are also predicted with different masses and spin-parities. Ten $\Lambda_{c\bar c}$ states are obtained in a coupled-channel model in Ref.~\cite{Xiao:2019gjd} and their couplings to $J/\psi \Lambda$ are found to be sufficiently large. Similar to the discovery of the $P_c$ states in the $J/\psi p$ channel~\cite{Xiao:2013yca}, the~$\Lambda_{c\bar c}$ states could be sought in the $J/\psi\Lambda$ invariant mass distributions. In Ref.~\cite{Wang:2019nvm}, ten isoscalar molecular pentaquarks in the $\bar{D}^{(*)} \Xi_c$, $\bar{D}^{(*)} \Xi_c^{\prime}$ and $\bar{D}^{(*)} \Xi_c^*$ systems are predicted by calculating the effective potentials with the chiral effective field theory up to the next-to-leading order. These hidden-charm pentaquark states with strangeness are investigated in the quark model including the color octet type and color singlet type~\cite{Irie:2017qai} and the one-boson exchange model~\cite{Chen:2016ryt}. The decay behaviors of the $\Lambda_{c \bar c}$ states in various hadronic molecular assumptions with different spin-parities are studied in Ref.~\cite{Shen:2019evi} and these decay patterns are suggested to help understanding the internal structures after the experimental searches. Although there are various predictions for such strangeness baryons within hidden charm theoretically, there is no experimental evidence for them until~now. Since the first prediction of the $\Lambda_{c\bar c}$ states, there arises various suggestions of searching for these states experimentally. The decay of $\Lambda_{b}$ into $J/\psi K^{0} \Lambda$ states is studied in Ref.~\cite{Lu:2016roh} and a clear peak is predicted in the $J/\psi \Lambda$ invariant mass distribution. In Ref.~\cite{Feijoo:2015kts}, the~authors propose the $\Lambda _b \to J/\psi \eta \Lambda$ decay and find that a resonance signal with similar strength as the case of $P_c$ should be observed in the $J/\psi \Lambda$ invariant mass spectrum. The $\Xi_b^{-} \to J/\psi K^{-} \Lambda$ reaction is discussed in Ref.~\cite{Chen:2015sxa} and a hidden-charm pentaquark state with strangeness is expected to be clearly seen. In the last process, the~$K^-$ meson in the final state is a charged meson, and~it makes this process suitable for experimental measurement. Actually, the~possible peak structures in the $J/\psi \Lambda$ invariant mass spectrum in the $\Xi_b^{-} \to J/\psi K^{-} \Lambda$ decay have already been sought by the LHCb Collaboration~\cite{Aaij:2017bef}, while the results of these data should still be under analysis. At this point, it is worthwhile to mention that the $\Lambda_b^0\to J/\psi\Lambda \pi^+\pi^-$ should also be a promising process to seek for the $\Lambda_{c\bar c}$ states. In this paper, we will study the $\Xi_b^{-} \to K^{-} J/\psi \Lambda$ decay and focus on another mechanism, triangle singularity, which could also produce peak structures in the $J/\psi \Lambda$ invariant mass~spectrum. A triangle singularity is due to the simultaneous on-shellness of three intermediate particles in a loop diagram, and~at the same time the interactions at all vertices can happen classically~\cite{Coleman:1965xm}. It is a logarithmic singularity, and~is able to produce peaks in energy distributions. It has been known for a long time~\cite{Karplus:1958zz,Landau:1959fi,Bjorken:1959fd}, and~there were debates in the 1960s on whether it could produce observable effects~\cite{Schmid:1967ojm}. Nevertheless, it was difficult to find reactions that satisfy the rigorous kinematic conditions and at the same time can evade the subtle interfering cancellation between the tree-level and triangle diagram, dictated by the Schmid theorem at that time~\cite{Schmid:1967ojm}. \textls[-15]{In the last 10 years or so, suggestions were proposed that many reactions which can be (or have been) measured at modern high-energy experiments could match the kinematic conditions and might receive important contributions from triangle singularities; then the study of triangle singularity started to bring a lot interest until now~\cite{Wu:2011yx, Aceti:2012dj, Wu:2012pg, Wang:2013cya,Wang:2013hga, Ketzer:2015tqa, Achasov:2015uua, Liu:2015taa, Liu:2015fea, Guo:2015umn, Szczepaniak:2015eza, Guo:2016bkl, Bayar:2016ftu, Wang:2016dtb, Pilloni:2016obd, Xie:2016lvs, Szczepaniak:2015hya, Roca:2017bvy, Debastiani:2017dlz, Samart:2017scf, Sakai:2017hpg, Pavao:2017kcr, Xie:2017mbe, Bayar:2017svj, Liang:2017ijf, Oset:2018zgc, Dai:2018hqb, Dai:2018rra, Guo:2019qcn, Liang:2019jtr, Nakamura:2019emd, Du:2019idk, Liu:2019dqc, Jing:2019cbw, Braaten:2019gfj, Aaij:2019vzc, Sakai:2020ucu, Sakai:2020fjh, Molina:2020kyu, Braaten:2020iye, Alexeev:2020lvq, Ortega:2020ayw}.~For a recent review, we refer to Ref.~\cite{Guo:2019twa} (see Tables~\ref{table:TS} and \ref{table:TSbc} therein for reactions proposed since the 1990s)}. The fact that triangle singularities may affect the extraction of the $P_c$ resonance parameters from the $\Lambda_b \to J/\psi K^- p$ reaction was immediately noticed~\cite{Guo:2015umn,Liu:2015fea}. Later, the~$\Lambda(1890)\chi_{c1}p$ triangle was identified to be the most special from an analysis of many triangle diagrams with various possible combinations of a a charmonium, a~proton and a $\Lambda^*$~\cite{Bayar:2016ftu}. In this case, both triangle singularity and threshold effect occur at about 4.45~GeV as already pointed out in Ref.~\cite{Guo:2015umn}. Once the $\chi_{c1} p$ is in an $S$-wave, such that the quantum numbers of the $J/\psi p$ could be $J^P=1/2^+$ or $3/2^+$, it would generate a narrow peak in the $J/\psi p$ invariant mass distribution. The updated LHCb measurement~\cite{Aaij:2019vzc} does not have a peak at 4.45~GeV, but~there is a dip at this energy between the two narrow $P_c(4440)$ and $P_c(4457)$ peaks. Thus, the~possibility of interference of such a triangle singularity with pentaquark signals is still there, and~might be partly responsible for the special production properties of the $P_c$ states~\cite{Sakai:2019qph}. However, for~the cases that the spin and parity of the $P_c$ are $3/2^-$ or $5/2^+$, as~considered in, e.g.,~Refs.~\cite{Chen:2015moa, Roca:2015dva, He:2015cea, Lin:2017mtz, He:2016pfa, Chen:2016otp, Xiang:2017byz}, the~triangle singularity contribution from the $\Lambda^*(1890)\chi_{c1}p$ loop would be severely suppressed. In the present work, we will investigate possible triangle singularities in the $\Xi_b^{-} \to J/\psi K^{-} \Lambda$ reaction in a similar way to Ref.~\cite{Bayar:2016ftu}. This work is organized as follows. After the introduction, we first show the theoretical framework and an analysis of possible triangle singularities in Section~\ref{sec:theory}, and~the corresponding discussions are presented. Then in Section~\ref{sec:analysis}, a~detailed analysis and discussions of the amplitude with $\Xi(2120)\chi_{c1}\Lambda$ are given, which is followed by a brief summary in Section~\ref{sec:summary}. \begin{table}[tb] \caption{\label{table:TS}The charmonium and $\Xi^{* -}$ states in Fig.~\ref{Fig:1stdiag} that could generate triangle singularities and the positions of the corresponding singularities. Among the listed hyperons, $\Xi(2030)$ is a three-star state; $\Xi(2250)$ and $\Xi(2370)$ are two-star states; $\Xi(2120)$ and $\Xi(2500)$ are one-star states~\cite{Zyla:2020}. } \renewcommand\arraystretch{1.5} \begin{tabular}{p{1.0cm}<{\centering}p{2.0cm}<{\centering}p{2.0cm}<{\centering}p{3.0cm}<{\centering}} \hline No. & $c \bar c$ & $\Xi^{* -}$ & Position of triangle singularity (MeV) \\ \hline 1 & $J/\psi$ & $\Xi(2500)$ & 4232 \\ 2 & $\chi_{c0}$ & $\Xi(2250)$ & 4546 \\ 3 & $\chi_{c0}$ & $\Xi(2370)$ & 4665 \\ 4 & $\chi_{c1}$ & $\Xi(2120)$ & 4628 \\ 5 & $\chi_{c1}$ & $\Xi(2250)$ & 4696 \\ 6 & $\chi_{c2}$ & $\Xi(2120)$ & 4680 \\ 7 & $h_c$ & $\Xi(2120)$ & 4644 \\ 8 & $h_c$ & $\Xi(2250)$ & 4730 \\ 9 & $\eta_c(2S)$ & $\Xi(2030)$ & 4754 \\ 10 & $\eta_c(2S)$ & $\Xi(2120)$ & 4797 \\ 11 & $\psi(2S)$ & $\Xi(2030)$ & 4810 \\ \hline \end{tabular} \end{table} \unskip \begin{table}[tb] \caption{\label{table:TSbc}The $D_s^{* -}$, $\Xi_c^{* 0}$ and intermediate particle 3 in Fig.~\ref{Fig:2nddiag} and Fig.~\ref{Fig:3rddiag} that could generate triangle singularities and the positions of the corresponding singularities. } \renewcommand\arraystretch{1.5} \begin{tabular}{p{0.3cm}<{\centering}p{1.9cm}<{\centering}p{1.6cm}<{\centering}p{1.7cm}<{\centering}p{2.3cm}<{\centering}} \hline No. & $D_s^{* -}$ & $\Xi_c^{* 0}$ & Intermediate particle 3 & Position of triangle singularity (MeV) \\ \hline 1 & $\bar D_{s1/s3}^*(2860)$ & $\Xi_c(2930)$ & $\bar{D}^0$ & 4838 \\ 2 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3055)$ & $\bar{D}^0$ & 4922 \\ 3 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3080)$ & $\bar{D}^0$ & 4957 \\ 4 & $\bar D_{s1/s3}^*(2860)$ & $\Xi_c(2930)$ & $\bar{D}^{* 0}$ & 4959 \\ 5 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3080)$ & $\bar{D}^{* 0}$ & 5089 \\ 6 & $\bar D_{s1}^*(2700)$ & $\Xi_c(3080)$ & $\Lambda_c^+$ & 4990 \\ 7 & $\bar D_{s1/s3}^*(2860)$ & $\Xi_c(2930)$ & $\Lambda_c^+$ & 5149 \\ \hline \end{tabular} \end{table} \section{Theoretical Framework and Analysis Results of All Possible Triangle~Singularities} \label{sec:theory} In the process of a triangle loop diagram, the~initial particle first decays to two intermediate particles $A$ and $B$. Then particle $A$ continues to decay into two particles, one of which is a final state and the other is the intermediate particle $C$. This intermediate particle $C$ could interact with particle $B$ to generate final state particles. The kinematic conditions for a triangle singularity in the physical region are~\cite{Schmid:1967ojm,Bayar:2016ftu}: (1) the three intermediate particles $A$, $B$ and $C$ are all on the mass shell at the same time; (2) the velocity of particle $C$ is larger than that of particle $B$ and is in the same direction in the rest frame of the initial state. In other words, triangle singularity corresponds to the situation that the loop reaction has actually occurred as a classical process~\cite{Coleman:1965xm}, rather than a virtual process. We are interested in the possible triangle singularity effects in the $J/\psi\Lambda$ invariant mass distribution. The~region of triangle singularity peak is limited to a small region starting from the threshold of two particles that rescatter into the final-state $J/\psi\Lambda$ (for a detailed analysis, see Ref.~\cite{Guo:2019twa}). There are four possible kinds of triangle diagrams for the $\Xi_b^- \to K^- \Lambda J/\psi$ decay that may have a triangle singularity in the physical region (when the widths of the intermediate particles are neglected) as shown in Figure~\ref{Fig:feyndiag4}. Using the intuitive equation derived in Ref.~\cite{Bayar:2016ftu}, we find that the physical-region triangle singularity emerges in the process of Figure~\ref{Fig:feyndiag4}a--c. The $\Xi^{* -}$ in Figure~\ref{Fig:feyndiag4}a represents all the ten $\Xi$ resonances in the Review of Particle Physics (RPP)~\cite{Zyla:2020}, including $\Xi(1530)$, $\Xi(1620)$, $\Xi(1690)$, $\Xi(1820)$, $\Xi(1950)$, $\Xi(2030)$, $\Xi(2120)$, $\Xi(2250)$, $\Xi(2370)$ and $\Xi(2500)$. For the charmonium states $c \bar c$, we take $\eta_c$, $J/\psi$, $\chi_{c 0}$, $\chi_{c 1}$, $\chi_{c 2}$, $h_c$, $\eta_c(2S)$ and $\psi(2S)$ into consideration. We take all the listed $\Xi_c^{* 0}$ and $D_s^{* -}$ in RPP~\cite{Zyla:2020} into account for the exchanged particles in Figure~\ref{Fig:feyndiag4}b,c. \begin{figure}[h] \centering \subfigure[]{\label{Fig:1stdiag} \includegraphics[width=0.5\linewidth]{figures/feynman1}} \subfigure[]{\label{Fig:2nddiag} \includegraphics[width=0.45\linewidth]{figures/feynman2}} \\ \ \ \subfigure[]{\label{Fig:3rddiag} \includegraphics[width=0.45\linewidth]{figures/feynman3}} \ \ \subfigure[]{ \includegraphics[width=0.45\linewidth]{figures/feynman4}} \caption{Four possible kinds of triangle diagrams for the $\Xi_b^- \to K^- \Lambda J/\psi$ decay that may have a triangle singularity of interest. The~momentum of each particle and the numbering of the exchanged particles used here are given in the brackets in the first diagram. \label{Fig:feyndiag4}} \end{figure} Through checking the two conditions of triangle singularity for all possible combinations of the considered intermediate particles mentioned above, we find that there are in total 11 cases for Figure~\ref{Fig:feyndiag4}a and 7 cases for Figure~\ref{Fig:feyndiag4}b,c that would have singularities in the physical region. The charmonium and $\Xi^{* -}$ states in Figure~\ref{Fig:feyndiag4}a, which could generate peak structures in the $J/\psi \Lambda$ mass spectrum, and~the corresponding positions of these triangle singularities are given in Table~\ref{table:TS}. In Table~\ref{table:TSbc}, we present the three exchanged particles of Figure~\ref{Fig:feyndiag4}b,c together with the corresponding triangle singularity positions, which are generally higher than those from Figure~\ref{Fig:feyndiag4}a listed in Table~\ref{table:TS}. Note that there is another combination, $\bar D_{sJ}(3040)\Xi_c(3645)\bar D^0$, which can give rise to a triangle singularity at 4511~MeV. However, so far the $D_{sJ}(3040)$ has only been observed in $D^*K$~\cite{Zyla:2020}, suggesting that it might have an unnatural parity and thus could not decay into $\bar D K$. Consequently, this combination is not listed in the table. One~notices that both the $D_{s1}^*(2700)$ was observed in $B\to D\bar D K$~\cite{Brodzicka:2007aa,Lees:2014abp} and the $D_{s1/s3}^*(2860)$ were seen in the decays of $B_s\to \bar D \bar K \pi$~\cite{Aaij:2014xza}. Similarly, the~$D_{s1}^*(2700)$ should be easily produced from the decays of the $\Xi_b$ associated with a $\Xi_c$ baryon, as~shown in the first vertex in Figure~\ref{Fig:feyndiag4}b,c. Thus, among~the combinations listed in Table~\ref{table:TSbc}, the~ones with the $D_{s1}^*(2700)$ are expected to be more important than the others. Once a structure is observed in the $J/\psi\Lambda$ spectrum above 4.9~GeV in a future experiment, such~triangle singularities need to be considered. In the rest of this paper, we will focus on the singularities in Table~\ref{table:TS}, whose situation is more~involved. Since the triangle singularity is caused by the on-shellness of the intermediate particles, we first consider the scalar loop integral as follows, \begin{eqnarray} I&=&i \int \frac{d^4 q}{(2\pi)^4} \frac1{(q^2-m_1^2+i m_1\Gamma_1)[(P-q)^2-m_2^2+i m_2\Gamma_2]} \nonumber\\ &&\times \frac1{[(k_1-q)^2-m_3^2+i \epsilon]}, \label{Eq:I} \end{eqnarray} \noindent where the momenta and the numbering of particle masses are as given in the first diagram in Figure~\ref{Fig:feyndiag4}, and~$\Gamma_1$ and $\Gamma_2$ are the widths of intermediate particles 1 and 2, respectively. We can use $|I|^2$ to analyze the singular behavior of the diagrams in Figure~\ref{Fig:feyndiag4}. It should be noted that the interactions of all the related vertices are not included here. We should consider these interactions from a physical point of view to draw further~conclusions. Let us turn to the three vertices in Figure~\ref{Fig:feyndiag4}a. For the first vertex $\Xi_b \to \Xi^{*} c \bar c$, it is a weak decay and involves the Cabibbo--Kobayashi--Maskawa (CKM) matrix elements ${V_{bc}^*V_{cs}}$. The~strength should be comparable with that of $B \to K c \bar c$ for the same charmonium. The branching ratios for different charmonia in the latter process, which can be found in the RPP~\cite{Zyla:2020}, are given in Table~\ref{table:1vertex}. It can be seen that for the $\chi_{c2}$ and $h_c$, the~branching ratios are one-order-of-magnitude smaller than that of the other charmonia. This indicates that the cases of the charmonium being the $\chi_{c2}$ or $h_c$ would not provide a strong signal of triangle singularity, and~can be safely neglected. For $\chi_{c0}$, the~decay rates from the neutral and charged $B$ mesons are very different. Both the two branching ratios of the $\chi_{c0}$ are smaller than the rest, although~in the charged case it is of the same order of magnitude. We conclude that, for~the $\Xi_b \to \Xi^* c\bar{c}$ decay, the~cases where the intermediate charmonium is one of $J/\psi$, $\chi_{c1}$, $\eta_c(2S)$ and $\psi(2S)$ are expected to be the main decay~channels. \begin{table}[tb] \caption{\label{table:1vertex}The branching ratios for different charmonia in $B \to K c \bar c$.} \renewcommand\arraystretch{1.5} \begin{tabular}{p{2.0cm}<{\centering}p{3.0cm}<{\centering}p{3.0cm}<{\centering}} \hline $c \bar c$ & $BR(B^0 \to c \bar{c} K^0)$ & $BR(B^+ \to c \bar{c} K^+)$ \\ \hline $J/\psi$ & $8.73 \times 10^{-4}$ & $1.01 \times 10^{-3}$ \\ $\chi_{c0}$ & $1.11 \times 10^{-6}$ & $1.49 \times 10^{-4}$ \\ $\chi_{c1}$ & $3.93 \times 10^{-4}$ & $4.84 \times 10^{-4}$ \\ $\chi_{c2}$ & $<1.5 \times 10^{-5}$ & $1.1 \times 10^{-5}$ \\ $h_c$ & & $<3.8 \times 10^{-5}$ \\ $\eta_c(2S)$ & & $4.4 \times 10^{-4}$ \\ $\psi(2S)$ & $5.8 \times 10^{-4}$ & $6.21 \times 10^{-4}$ \\ \hline \end{tabular} \end{table} For the vertex of $c \bar c \Lambda \to J/\psi \Lambda$, the~coupling strength is affected from the spin interactions between the charm quark and anticharm quark. Once the $c \bar c$ has a spin flip, this vertex will be much suppressed due to breaking the heavy quark spin symmetry. The quantum numbers of the final state $J/\psi$ are $1^-$, and~the spins of charm and anticharm quarks are in the same direction. However, in~the charmonia $h_c$ and $\eta_c(2S)$, the~total spin of charm and anticharm quarks is $0$. Therefore, the~$h_c \Lambda$ or $\eta_c(2S) \Lambda \to J/\psi \Lambda$ process would be suppressed because of the spin flip of the charm and anticharm quarks. For the other charmonia considered here, the~couplings of $c \bar c \Lambda \to J/\psi \Lambda$ could be of the same order of magnitude but are still not determined. We will write the amplitudes with different $J^P$ assumptions that are subject to the Lorentz covariant orbital-spin coupling scheme~\cite{Zou:2002yy} to study their behavior in the next~section. For the $\Xi^* \bar{K} \Lambda$ vertex, we can infer its strength from the $\Xi^* \to \Lambda \bar K$ process for each considered $\Xi^*$ state. For the $\Xi(2030)$, the~$\Lambda \bar K$ state occupies about 20\% of all possible decay rates. In the RPP, the~$\Lambda \bar K$ channel is the only seen and listed decay mode for the $\Xi(2120)$~\cite{Zyla:2020}, although~its exact ratio has not been measured yet. The $\Lambda \bar K$ channel is not listed in the decay modes of the $\Xi(2250)$ and $\Xi(2370)$, and~actually all the decay channels for the former are three-body states, meaning that the $\Lambda \bar K$ mode might be suppressed for these two states. For the $\Xi(2500)$, the~$\Lambda \bar K$ mode is listed, with~the ratio of branching fractions $\Gamma(\Lambda \bar K)/\left(\Gamma(\Xi\pi)+\Gamma(\Lambda \bar K)+\Gamma(\Sigma \bar K)+ \Gamma(\Xi(1530)\pi)\right) = 0.5\pm0.2$ measured in Ref.~\cite{Alitti:1969rb}, but~it is not clear whether the peak observed therein corresponds to the $\Xi(2500)$ as commented in RPP~\cite{Zyla:2020}. Through the analysis of the three vertices above, for~the 11 cases listed in Table~\ref{table:TS}, nos.~2 and 3 and nos.~5--10 are suppressed; only nos.~1, 4 and 11 are left. Now let us turn to the character of the intermediate particles in the loop, especially their quantum numbers, since they actually affect a lot not only the magnitude but also the line shape of the singularity. The quantum numbers are quite important as they directly determine the partial waves of the interaction, and~high partial wave interactions should be suppressed in the near-threshold region. For the $\Xi(2030)$ state, its spin is equal to or greater than $5/2$~\cite{Hemingway:1977uw}. It leads to high partial-wave ($l \ge 2$) interaction at the $\Xi(2030) \bar{K} \Lambda$ vertex, and~will not be considered further. {We will discuss that how the high partial-wave interactions in the triangle loop would suppress the contribution of the triangle singularity later.} For the other two $\Xi^*$ states, their quantum numbers are not determined yet, and~we will consider various $J^P$ possibilities~later. The $m_{J/\psi \Lambda}$ distribution for $|I|^2$ with an arbitrary normalization factor of these three cases is shown in Figure~\ref{Fig:4TS}. For the widths of the involved charmonium, we quote their central values given in the RPP~\cite{Zyla:2020}. While for some of the $\Xi^*$ states like the $\Xi(2120)$ and $\Xi(2500)$, the~widths are not precisely determined. For the width of the former, we take the upper bound of the given range, which is $20$ MeV. In the latter case, two values were reported by different experimental groups, which are $150$~\cite{Alitti:1969rb} and $59$~MeV~\cite{Bartsch:1969qs}. We give results using these two values. As shown in Figure~\ref{Fig:4TS}, the~triangle singularity from the $J/\psi \Xi(2500) \Lambda$ loop diagram is much broader than the other two cases. As discussed above, the~triangle loop diagram that involves the $\Xi(2030) \bar{K} \Lambda$ vertex should be severely suppressed because of the existence of the $D$- or higher wave interaction, while for the $\Xi(2120)$ involved vertices, there~could be lower partial-wave interactions. The contribution from the $\Xi(2120)\chi_{c1}\Lambda$ loop (the blue dotdashed line in Figure~\ref{Fig:4TS}) is expected to be much larger than that from the $\Xi(2030)\psi(2S)\Lambda$ (the green dotted line) once all the interactions at the vertices are~included. \begin{figure}[h] \centering \includegraphics[width=0.95\linewidth]{figures/4TS} \caption{The value of $|I|^2$ in the $m_{J/\psi \Lambda}$ invariant mass distribution. The~red solid (orange dashed), green dotted and blue dotdashed lines correspond to the Nos.~1, 4 and 11 cases in Table~\ref{table:TS}, respectively. For~the case of No.1, the~red solid and orange dashed lines represent the different choices of the $\Xi(2500)$ width: the narrower and wider peaks correspond to taking 59 and 150~MeV for the width, respectively. } \label{Fig:4TS} \end{figure} Combining all the analysis above, the~$\Xi(2120)\chi_{c1}\Lambda$ loop is found to be the most promising process to generate an observable triangle singularity in the $\Xi^-_{b} \to K^- J/\psi \Lambda$ decay ({We should also note that the $\Xi(2120)$ is just a one-star state~\cite{Zyla:2020}, and~its existence needs further confirmation}). We will calculate and discuss the structures from this triangle loop diagram in detail in the next~section. \section{Detailed Analysis of the Amplitudes for the Diagram with \texorpdfstring{$\bm{\chi_{c1} \Xi(2120) \Lambda}$}{chic1 Xi Lambda} Loop} \label{sec:analysis} We will focus on the triangle diagram involving the $\chi_{c1}$ and the $\Xi(2120)$ in this section, as~it is the most promising diagram to have a sizeable triangle singularity contribution. The specific interactions of vertices are taken into account in this section, here we are going to study them by constructing effective Lagrangians with the lowest number of derivatives. At present, the~quantum numbers of the $\Xi(2120)$ state and the $J/\psi \Lambda$ system are both flexible, and~the former affects the $\Xi_b \Xi^* \chi_{c1}$ and $\Xi^* \Lambda K^-$ vertices, while the later influences the $\chi_{c1} \Lambda J/\psi \Lambda$ vertex. Let us discuss them one by~one. Firstly, let us consider different choices of the spin-parity quantum numbers of the $\Xi(2120)$, which~are still unknown. The two vertices involving the $\Xi(2120)$ in this diagram are $\Xi_b \Xi(2120) \chi_{c1}$ and $\Xi(2120) \Lambda \bar K$. In Table~\ref{table:PW}, the~assumed spin-parity of $\Xi(2120)$ and the corresponding lowest allowed partial waves of these two vertices are listed. {In the Lorentz covariant orbital-spin coupling scheme, the~interaction of a vertex is directly related to its partial wave $L$ and the relative momentum of the two final particles, i.e.,~the strength of interaction is proportional to $q^L$. Therefore, the~$D$- or higher partial waves are neglected since the momentum is very small around where the triangle singularity happens.} Thus, $J^P=1/2^+$, $1/2^-$ and $3/2^+$ possibilities are not suppressed by high partial waves. Among~these three cases, the~$J^P=1/2^-$ possibility is singled out as both vertices are in $S$ waves, and~the strength of the amplitude in this case should be stronger than that of the other cases. Therefore, we will calculate the amplitude with $J^P(\Xi^*)=1/2^-$ to check the structure behavior from the triangle~singularity. \begin{table}[tb] \caption{\label{table:PW}The interactions of the $\Xi_b \Xi(2120) \chi_{c1}$ and $\Xi(2120) \Lambda \bar K$ vertices with different spin-parities of $\Xi(2120)$.} \renewcommand\arraystretch{1.5} \begin{tabular}{p{2.0cm}<{\centering}p{3.0cm}<{\centering}p{3.0cm}<{\centering}} \hline $J^P$ & $\Xi_b \Xi(2120) \chi_{c1}$ & $\Xi(2120) \Lambda \bar K$ \\ \hline $1/2^+$ & $S$-wave & $P$-wave \\ $1/2^-$ & $S$-wave & $S$-wave \\ $3/2^+$ & $S$-wave & $P$-wave \\ $3/2^-$ & $S$-wave & $D$-wave \\ $5/2^+$ & $P$-wave & $F$-wave \\ $5/2^-$ & $P$-wave & $D$-wave \\ \hline \end{tabular} \end{table} We then consider the quantum numbers of the $J/\psi \Lambda$ in the final state for fixed spin-parity of the $\Xi(2120)$. For the same reason mentioned before, the~$D$- or higher partial wave interactions are not considered in the present discussion. If the quantum numbers of the $J/\psi \Lambda$ system are $1/2^+$ or $3/2^+$, the~$\chi_{c1} \Lambda$ system can be in an $S$-wave and the $J/\psi \Lambda$ in a $P$-wave. On the contrary, if~the quantum numbers of the $J/\psi \Lambda$ system are $1/2^-$ or $3/2^-$, there is a $P$-wave in the $\chi_{c1} \Lambda$ system and an $S$-wave in the $J/\psi \Lambda$ system. In both cases, there are an $S$-wave and a $P$-wave for this four-particle-vertex, but~the corresponding behaviors of these two triangle loop integrations are different. { As discussed in the previous paragraph, the~effective Lagrangian of a $P$-wave interaction is proportional to the relative momentum of the involved particle. Meanwhile, at~the singularity point, the~invariant mass $m_{J/\psi \Lambda}$ is not far away from the $\chi_{c1}\Lambda$ threshold, and~both the $\chi_{c1}$ and the $\Lambda$ are on-mass-shell. It tells that the momenta of the $\chi_{c1}$ and $\Lambda$ are small. Thus, for~the negative parity case, the~$P$-wave is in the $\chi_{c1}\Lambda$ system, which provides a loop momentum $q$ to the loop integral and suppresses the singular behavior around the $\chi_{c1}\Lambda$ threshold, while the $P$-wave interaction of the $J/\psi \Lambda$ system for the positive parity case is independent of the loop integration.} In the next step, we will take the quantum numbers of the $J/\psi \Lambda$ system as $1/2^+$ and $1/2^-$, respectively, to~investigate how the finial results are~affected. We first take the quantum numbers of the $J/\psi \Lambda$ system being $1/2^+$ and the spin-parity of the $\Xi(2120)$ are assumed to be $1/2^-$. The amplitudes for the vertices are constructed as \begin{eqnarray} t_{\Xi_b \Xi^* \chi_{c1}} &=& g_a \bar{u}(P-q) \gamma^\mu u(P) \epsilon^*_\mu(q), \nonumber \\ t_{\Xi^* \Lambda \bar K} &=& g_b \bar{u}(k_1-q) u(P-q), \nonumber \\ t_{\Lambda \chi_{c1} \Lambda J/\psi} &=& g_c \epsilon_\mu(q) \bar{u}(k_1-k_2) ({\Slash k_1}+\sqrt{k_1^2}) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) \nonumber \\ && \times u(k_1-q) \epsilon_\nu^*(k_2) k_1^\nu \frac{k_1^2+m_{J/\psi}^2-m_{\Lambda}^2}{k_1^2}, \end{eqnarray} here we follow the momentum conventions given in Figure~\ref{Fig:feyndiag4}a. The amplitude of this triangle diagram~reads \begin{eqnarray} \mathcal{M} &=& g \int \frac{d^4 q}{(2\pi)^4} \bar{u}(k_1-k_2) ({\Slash k_1}+\sqrt{k_1^2}) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) \nonumber\\ && \times \frac{{\Slash k_1}-{\Slash q}+m_3}{(k_1-q)^2-m_3^2} \frac{{\Slash P}-{\Slash q}+m_2}{(P-q)^2-m_2^2+i m_2 \Gamma_2} \nonumber\\ && \times \frac{-\gamma_{\mu}+\frac{q_\mu {\Slash q}}{m_1^2}}{q^2-m_1^2+i m_1 \Gamma_1} u(P) \frac{k_1^2+m_{J/\psi}^2-m_{\Lambda}^2}{k_1^2} \nonumber\\ && \times \epsilon_\nu^*(k_2) k_1^\nu. \end{eqnarray} Then we calculate the decay with the quantum numbers of the $J/\psi \Lambda$ system being $1/2^-$. The~amplitude for the $\Lambda \chi_{c1} \Lambda J/\psi$ vertex reads \begin{eqnarray} t_{\Lambda \chi_{c1} \Lambda J/\psi}&=&g_c \bar{u}(k_1-k_2) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) ({\Slash k_1}+\sqrt{k_1^2}) \nonumber\\ && \times u(k_1-q) \epsilon_\mu^*(k_2) \epsilon_\nu(q) k_1^\nu \frac{2k_1 \cdot q}{k_1^2}, \end{eqnarray} and the amplitude in this case is \begin{eqnarray} \mathcal{M} &=& g \int \frac{d^4 q}{(2\pi)^4} \bar{u}(k_1-k_2) \gamma_5 (\gamma^\mu-\frac{k_1^\mu {\Slash k_1}}{k_1^2}) ({\Slash k_1}+\sqrt{k_1^2}) \nonumber\\ && \times \frac{{\Slash k_1}-{\Slash q}+m_3}{(k_1-q)^2-m_3^2} \frac{{\Slash P}-{\Slash q}+m_2}{(P-q)^2-m_2^2+i m_2 \Gamma_2} \nonumber\\ && \times \frac{-{\Slash k_1}+\frac{\Slash q}{m_1^2} k_1 \cdot q}{q^2-m_1^2+i m_1 \Gamma_1} u(P) \frac{2k_1 \cdot q}{k_1^2} \epsilon_\mu^*(k_2). \label{eq:m12m} \end{eqnarray} It should be noticed that the absolute values of the coupling constants in both cases are unknown. Therefore, our results are normalized to 1 at the maximum. The differential decay width in the rest frame of the initial particle $\Xi_b$ is \begin{equation} {\mathrm{d} \Gamma = \frac1{(2\pi)^3} \frac1{32m_{\Xi_b}^3} \overline{{\left| \mathcal M \right|}^2} \mathrm{d}m_{K \Lambda}^2 \mathrm{d}m_{J/\psi \Lambda}^2.} \label{eq:dGamma} \end{equation} {After integrating $\mathrm{d}m_{K \Lambda}^2$, t}he normalized differential decay widths in the $J/\psi \Lambda$ invariant mass distribution in these two cases are shown in Figure~\ref{Fig:dGamma} with the widths of the exchanged particles taken into~consideration. \begin{figure}[h] \centering \subfigure []{\label{Fig:dGamma1} \includegraphics[width=0.95\linewidth]{figures/dGamma}} \\ \subfigure[] {\label{Fig:dGamma2} \includegraphics[width=0.95\linewidth]{figures/dGamma2}} \caption{The $m_{J/\psi \Lambda}$ invariant mass distribution for the $\Xi^-_{b} \to K^- J/\psi \Lambda$ process via the $\chi_{c1} \Xi(2120) \Lambda$ loop with the quantum numbers of the $J/\psi \Lambda$ system being (\textbf{a}) $1/2^+$, and~(\textbf{b}) $1/2^-$, respectively. \label{Fig:dGamma}} \end{figure} It can be seen that in the $1/2^+$ case the peak structure from the triangle singularity emerges at $m_{J/\psi \Lambda}=4.628$~GeV and is distinct enough comparing to the background. However, this structure could not be observed any more in the $1/2^-$ case. Naively one would expect that the peak should still be there since the numerator of the integral would not diminish the singularity generated from the denominator. Then here arises a question why the $P$-wave $\chi_{c1} \Lambda$ in the four-particle vertex can suppress the singular behavior so much. When constructing the effective Lagrangian of the four-particle vertex in the Lorentz covariant orbital-spin coupling scheme~\cite{Zou:2002yy}, the~$P$-wave interaction in the $\chi_{c1} \Lambda$ system introduces a factor of the momentum difference of the exchanged $\Lambda$ and $\chi_{c1}$. Thus, the~amplitude is directly proportional to the velocity difference between these two particles in the $1/2^-$ case. As~discussed above, when the triangle singularity happens, all the three intermediate particles are on the mass shell and the process of each vertex actually happens classically. In this $\chi_{c1} \Xi(2120) \Lambda$ loop diagram, we~find that the velocity of the intermediate $\Lambda$ is very close to that of the $\chi_{c1}$ when the singularity occurs. In~other words, their relative velocity is quite small in the rest frame of $\Xi_b$, meaning that the singularity is located very close to the $\chi_{c1}\Lambda$ threshold. More specifically, the~factor $\left(-{\Slash k_1}+\frac{\Slash q}{m_1^2} k_1 \cdot q\right)$ in Equation~\eqref{eq:m12m} from the $P$-wave $\chi_{c1} \Lambda$ system in the triangle singularity region can be derived as~follows, \begin{equation} -{\Slash k_1}+\frac{\Slash q}{m_1^2} k_1 \cdot q \sim \frac{|\vec{q}|}{q^0}-\frac{|\vec{k}_1|}{k_1^0} \sim v_{\Lambda}-v_{\chi_{c1}}\sim 0, \end{equation} explaining the suppression of the singular behavior in this~case. Since the amplitude is actually related to the partial wave interactions of the $\chi_{c1} \Lambda \to J/\psi \Lambda$ vertex, the~$3/2^+$ and $3/2^-$ cases should be similar to the $1/2^+$ and $1/2^-$ cases, respectively. This~indicates that this $\Xi(2120) \chi_{c1} \Lambda$ loop diagram could generate a peak in the $m_{J/\psi \Lambda}$ mass spectrum for the $\Xi^-_{b} \to K^- J/\psi \Lambda$ decay when the quantum numbers of the $J/\psi \Lambda$ system are $1/2^+$ or $3/2^+$. We~then note that the threshold of the $\chi_{c1}\Lambda$ is at 4.626~GeV, which gives rise to a cusp (for the $S$-wave) in the distribution of the differential decay width. As the positions of the threshold and the triangle singularity (at about 4.628~GeV) are quite close, these two structures would lead to only one peak around this energy as shown in Figure~\ref{Fig:dGamma}a. Thus, if~a structure is observed in this region in future experiments, the~triangle singularity effects need to be taken into account for the $J^P=\frac12^+$ or $\frac32^+$ quantum~numbers. There are a lot of similarities between the $\Lambda_b \to K^- J/\psi p$ and $\Xi_b \to K^- J/\psi \Lambda$ through triangle loop diagrams. The $\Lambda_b \Lambda^* \chi_{c1}$ vertex, as~considered in Refs.~\cite{Guo:2015umn,Bayar:2016ftu}, and~the $\Xi_b \Xi^* \chi_{c1}$ vertex involve the same product of CKM matrix elements and should share a similar coupling. The strength of the two four-particle-vertices $\chi_{c1} p \to J/\psi p$ and $\chi_{c1} \Lambda \to J/\psi \Lambda$ should also be similar due to the approximate light-flavor SU(3) symmetry. For the two rest vertices $\Lambda(1890) K^- p$ and $\Xi(2120) K^- \Lambda$, the~$\bar K N$ channel occupies a rate of 20--35\% among all the decay modes of $\Lambda(1890)$, while the branching ratio of $\Xi(2120)$ is still unknown and $\bar K \Lambda$ is the only observed decay channel. We can roughly regard the strengths of these two vertices to be of the same order of magnitude. Thus, studying possible structures in the $\Xi_b \to K^- J/\psi \Lambda$ decay around 4.63~GeV should also provide useful insights into the possible role of triangle singularity in the $\Lambda_b \to K^- J/\psi p$ decay around 4.45~GeV. Yet, the~existence of one-star $\Xi(2120)$ state needs to be confirmed, which can be studied by analyzing the $\Lambda \bar K$ invariant mass distribution of the process under~discussion. \section{Summary} \label{sec:summary} Inspired by the analysis of the triangle singularities in the $\Lambda_b \to K^- J/\psi p$ process in Ref.~\cite{Bayar:2016ftu}, we~apply the same method to the $\Xi_b \to K^- J/\psi \Lambda$ process via charmonium-$\Xi^*$-$\Lambda$ intermediate states to check whether there exist possible triangle singularities. We find that there are 11 and 7 possible triangle singularities with various combinations of charmonium-$\Xi^*$ states and $D_S^*$-$\Xi_c^*$ states, respectively. For~the former, the~$\chi_{c1} \Xi(2120) \Lambda$ loop diagram is expected to be the most promising process to observe the singularity structure; for the latter, those with $\bar D_{s1}^*(2700)$ listed in Table~\ref{table:TSbc} should be more important than the others. Nevertheless, the~absolute values of all these amplitudes could not be precisely evaluated without knowing the amplitudes for all the involved vertices. For the $\chi_{c1} \Xi(2120) \Lambda$ diagram, when the spin-parity quantum numbers of the $J/\psi\Lambda$ system are $1/2^+$ or $3/2^+$, the~joint effects of the triangle singularity and the $\chi_{c1}\Lambda$ threshold would produce a narrow peak around 4.63~GeV in the $J/\psi\Lambda$ invariant mass distribution. A study of such effects needs to be taken into account in the search of hidden-charm strange pentaquarks, and~a study of the $\Xi_b \to K^- J/\psi \Lambda$ will also shed light on the role of triangle singularities in the $\Lambda_b\to K^- J/\psi p$ process. \section*{Acknowledgments} We are grateful to Eulogio Oset for a careful reading of our~manuscript. This work is supported in part by the Fundamental Research Funds for the Central Universities, by~the National Natural Science Foundation of China (NSFC) under Grants No.~11835015, No.~11947302 and No.~11961141012, by~the Chinese Academy of Sciences (CAS) under Grants No.~XDB34030303 and No.~QYZDB-SSW-SYS013, by~NSFC and the Deutsche Forschungsgemeinschaft (DFG) through the funds provided to the Sino-German Collaborative Research Center ``Symmetries and the Emergence of Structure in QCD'' (NSFC~Grant No. 11621131001, DFG Grant No. CRC110), and~by the CAS Center for Excellence in Particle Physics~(CCEPP).

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\section{Introduction: the Current State of Peer Reviews in Machine Learning} Machine learning as a field has been undergoing incredible growth in recent years in all dimensions, including new findings and levels of participation, but perhaps most striking is the growth in conference paper submissions. In the last year alone, the number of conference paper submissions needing review has increased by 47\% for ICML, by 50\% for NeurIPS, and by almost 100\% for ICLR. (See Figure~\ref{neurips}.) This exponential growth rate has undoubtedly strained the community, particularly in its capacity to perform timely and thorough peer review \citep{lipton2018troubling,sculley2018winners}. This is alarming: peer review is the primary mechanism by which we enforce rigorous standards for our scientific community. But the increased strain on experienced reviewers recalls the {\em tragedy of the commons}, in which a limited communal resource is laid waste from overuse. Indeed, we may already be at a breaking point. In mid-2018 we looked at hundreds of past NeurIPS and ICLR reviews\footnote{Reviews at: \url{https://openreview.net/group?id=ICLR.cc/2018/Conference} and \url{http://papers.nips.cc/book/advances-in-neural-information-processing-systems-30-2017}} to find high quality reviews to use as exemplars in an updated reviewer guide.\footnote{Guide at: \url{https://nips.cc/Conferences/2018/PaperInformation/ReviewerACSACGuidelines}} We had hoped to find many reviews that evaluated \emph{quality}, \emph{clarity}, \emph{originality} and \emph{significance} in a clear and thorough way, but were surprised at the difficulty of finding individual reviews that satisfied all three. Instead we ended up using review snippets highlighting just one of of these areas at a time. Overall, this proved to be a sobering exercise, but one that gave a realistic view of the current state of ML reviewing. This is far from an isolated anecdote. The NeurIPS Experiment of 2014, conducted by Fawcett and Cortes,\footnote{See Eric Price's summary at \url{http://blog.mrtz.org/2014/12/15/the-nips-experiment.html}} found that when papers were randomly re-reviewed by independent committees of reviewers within the NeurIPS reviewing process, almost 60\% of the papers accepted by the first committee were rejected by the second. The dramatic increase in submissions and the corresponding reliance on less experienced reviewers is likely to have only exacerbated this effect. We believe the the key issues here are structural. Reviewers donate their valuable time and expertise anonymously as a service to the community with no compensation or attribution, are increasingly taxed by a rapidly increasing number of submissions, and are held to no enforced standards. Thus, we propose a structural solution to this issue, with two key components: \begin{itemize} \item Reviewers should be held to objective standards for review quality, as determined by a rubric that can be applied consistently by area chairs and other senior committee members. \item Reviewers should be incentivized and rewarded for their work appropriately, commensurate with market rates for their time and level of expertise. \end{itemize} We give an example rubric for consideration in Section 2, and Section 3 provides a detailed look at financial compensation as one possible mechanism to allow reviewers to prioritize this important work ahead of competing demands on their time. \begin{figure} \centering \label{neurips} \caption{{\bf Submissions Per Year for NeurIPS (left) and ICLR (right).} Both conferences have seen exponential growth in submissions per year, straining the pool of experienced reviewers.} \includegraphics[width=1.8in]{neurips.png} \includegraphics[width=1.8in]{iclr.png} \vskip-0.25in \end{figure} \section{A Rubric for Evaluating Peer Reviews} Rubrics have long been used in education~\cite{wolf2007rubric} to evaluate student performance since they can make evaluation more objective, consistent and fair. They also help students understand what is expected of them by making the evaluation criteria explicit~\cite{andrade2005} and thus can act as a guide to communicate expectations to reviewers. We believe such a rubric, detailing specific expectations, with gradations of review quality would help evaluate incentivized reviews and improve review quality overall. As a starting point, we provide a seven point rubric for review quality in Figure~\ref{rubric} with a checklist for components of high quality reviews. We believe this quick checklist could be applied with low effort by area chairs to provide a reasonably objective assessment of review quality. \section{Incentives and Compensation} The obvious cost of enforcing the high standards from the rubric in Figure~\ref{rubric} is one of time and added effort. Proper incentives are important if this cost is to be borne in a sustainable way. While other strong incentives such as free or reserved registrations are reasonable \citep{sculley2018winners}, here we examine the idea of using direct financial incentives to make clear that such an approach is indeed feasible. Regardless of the specific mechanism, we do view this professional service as one worthy of professional compensation. The goal is to provide sufficient reward to empower reviewers to prioritize their reviewing responsibility over other competing demands on their time, while equally empowering area chairs and other senior committee members or chairs with an appropriate mechanism to motivate compliance with high standards of reviewing. \paragraph{What is a reasonable level of compensation?} To make this proposal concrete, we would suggest a figure of USD 1000 per paper per review as a reasonable value of compensation. We arrive at this figure by noting that a complete review including reading, understanding, verifying, and writing should take at least half a day (4 hours) of attention from a qualified expert in the field. Assuming a rate of USD 250 per hour as a conservative estimate for consulting work done by experts in machine learning\footnote{One estimate: \url{https://towardsdatascience.com/how-to-price-an-ai-project-f7270cb630a4}} in the current market gives a value of USD 1000 per review per paper. As with other honoraria, some reviewers will likely decline, especially those with industrial research affiliations. \paragraph{Would every review qualify?} We do not propose financial compensation for its own sake, but rather as an important incentive for review excellence. Thus, we suggest that area chairs use a rubric like that in Section 2 to assess the quality of each review on an objective scale, and that only those reviews passing a high bar for excellence be rewarded with payment. Area chairs would be encouraged to iterate with reviewers when necessary to help reviews meet the bar. The ability to withhold compensation would be an explicit and important dis-incentive against hasty, incomplete, late, or otherwise low quality reviews. \paragraph{Alternate incentives.} Financial compensation is not the only form of reasonable incentive. We note with encouragement other forms of incentive such as reserved registrations that have been made available to the top 1000 reviewers for NeurIPS 2018, and ``top reviewer'' awards at ICML, both of which are good examples of non-monetary incentives. However, an ideal incentive is one that rewards additional work in a smoothly increasing way. For example, a reviewer who does a great job with 10 reviews should be rewarded more than one who does a great job with 5, and either reviewer should ideally see additional benefit if they were to do one more. Discrete incentives such as reserved registrations introduce a step function that detracts from this effect, while financial incentives increase smoothly as reviewers contribute additional value. \paragraph{Related compensation practices.} It is worth noting other fields do incorporate compensation into their peer review and publication processes. For instance, in fields where publication is mainly journal-based, professional editors are employed to oversee the initial screening of submitted work, selecting individual reviewers, and adjudicating disputes in the review process itself. These editors usually have PhDs in the journal's field. Although compensation for journal article reviewing is rare, the National Institute of Health (NIH), a major government funding agency for research in biology and medicine, pays an honorarium for peer review activities for grants \cite{nih-honorarium}. Grant review at the NIH occurs over a short and concentrated time-period, resembling the conference review process in machine learning. In-person, electronic and mail-in participation in the review process are all compensated with an honorarium. Neither the presence of paid editorial oversight nor compensated peer review have been raised as impediments to objectivity in the fields that employ them. \begin{figure} \caption{{\bf A 7 point rubric for evaluating review quality.} This checklist can be used by area chairs or other senior committee members.} \begin{itemize} \sc \small \item[$\square$] Review establishes the relevant expertise of the reviewer \item[$\square$] Review demonstrates understanding of the contribution via concise summary \item[$\square$] Review evaluates the quality of the writing and composition of the work \item[$\square$] Review places the contribution in current context of prior publications and evaluates novelty \item[$\square$] For each proof or formal claim, review demonstrates checked correctness or demonstrates flaw \item[$\square$] For each empirical result, review evaluates experimental protocols, verifies baselines and evaluates statistical significance \item[$\square$] Review evaluates actual significance of the work and interest to the community \end{itemize} \vskip-0.2in \label{rubric} \end{figure} \section{Funding and Budgeting} Compensating USD 1000 per review per paper creates a material cost that must be funded, on the order of USD 15 million for a conference like NeurIPS 2018. Here we give several alternatives by which this could be supported, along with ideas for reducing the overall figure. \paragraph{Differential pricing for conference registration.} NeurIPS 2018 famously sold out within roughly 6 minutes of registration opening. From an economic point of view, it appears that demand far exceeds supply, and the classic response would be to raise prices. However, we do not believe that it is appropriate to make it more difficult for students or academics to register; promoting an inclusive community is critical to the long-term vitality of the field. Instead, we suggest that it may be reasonable to offer {\em differential pricing}, for example charging significantly more for industry-affiliated attendees than for academic researchers or current students. To begin the conversation, imagine a world in which industry-affiliated attendees were charged a figure on the order of USD 5000. This would provide a budget of the correct magnitude. As beneficial side effects, it may also mitigate some of the registration crunches and provide a mechanism to further promote inclusion and diversity at registration time. \paragraph{Co-pay with submission.} A co-pay is a classic solution to help ensure that consumers do not abuse low cost resources.\footnote{See \url{https://en.wikipedia.org/wiki/Copayment}} In this setting, paper submission would entail a fee which would both be used both to collect funds to help offset reviewing cost, while also providing a disincentive for authors to submit work that is not yet fully ready for review. Such methods should obviously not be used in ways that would deter participation from authors who would be overly burdened by such a cost; as suggested by \citep{sculley2018winners} we would recommend that a payment-in-kind mechanism by which authors can provide paper reviews in place of a monetary co-pay be made available as an additional alternative. \paragraph{Explicit sponsorship.} Industry sponsorship currently provides more than USD 1.6 million to NeurIPS to fund various events. Given the value of machine learning research to the industrial world, this may be seen as a surprisingly low figure. We believe it may be reasonable to create sponsorship opportunities to help fund review costs, and to highlight these as the most prestigious sponsors. Such framing may be sufficient to create a sufficient budget. However, because of the possibility of perceived bias toward sponsoring institutions, we consider direct sponsorship a less desirable option. \paragraph{Incentivizing efficiency.} One benefit of providing compensation for reviews is that it concretely highlights the true value of reviewer time, and may motivate measures to ensure that this time is used as efficiently as possible. Conference review tools are often seen as difficult to work with, but the question of whether to invest in improving them may become more clear if the consideration is a 10\% improvement in reviewer time on a multi-million dollar budget. \paragraph{Reducing time spent on low-quality papers.} Currently, a large fraction of reviewer time is spent on providing high quality reviews for papers that are far below the acceptance threshold for a given conference. In 2018, there were 1366 NeurIPS submissions averaging an overall score of 4 or below. These consumed more than 4000 reviews, or the full review capacity of nearly 700 reviewers, despite the fact that these papers are essentially certain of rejection. We suggest dispensing with the custom that all papers receive the same number of reviews. Multi-stage triage that quickly removes papers far below the bar may reduce the reviewing burden substantially, while improving the ability of reviewers to dive deeply into the details of higher quality papers. This is common practice in journal-centric fields like biology and medicine, where professional editors perform triage for the review process and filter out papers clearly below the bar from further review. \section{Possible Long Term Effects} In current practice, peer reviews are unfunded and are often distributed to exactly those people who are least economically advantaged in the current research community, such as students nearing completion of their PhDs who are currently required by tradition to provide this value for free. We believe that compensating reviewers for their reviews will help such members of the community by providing an important additional source of funding which rewards expertise and may also help enable such researchers to take greater risks and pursue more ambitious directions. Additionally, current practice is to involve a large number of reviewers, and cap review load (typically at six papers), in order to avoid overly taxing any individual reviewer. This load capping makes it difficult for individual reviewers to be well calibrated to the standards of a given conference or to compare a range of similar submissions in a hot topic for a given year, and actively requires broadening the reviewer pool to less expert reviewers if all relevant experts are fully capped. In a system in which reviewers may actually prefer to review more papers in order to increase their compensation, we may end up with reviewers deciding to take on 10 or 20 reviews. Such reviewers may have better perspective than is currently typical, and may be more efficient on a per-review basis when reviewing many papers on a similar topic. \section*{Acknowledgements} We thank Samy Bengio, George Dahl, Hugo Larochelle, Jamie Morgenstern, and especially the anonymous reviewers for this paper for their helpful discussions and feedback. We also thank Tara Sainath for providing the ICLR submissions-per-year data and for her permission to reproduce the ICLR chart from her slides. Finally, after initial publication we found an earlier independent use of the phrase ``tragedy of the commons'' in conjunction with thoughts on peer review process in the field of Ecology from \citep{tragedyecology}, focusing on problems connected with authors overloading the journal review process with papers that may not be not fully ready for full review. It is perhaps some comfort to know that machine learning is not alone as a field in facing such considerations. \bibliographystyle{unsrt}

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\section{Introduction} The purpose of this paper is to develop numerical schemes for the solution of Mean Field Games (MFGs) and Mean Field Control (MFC) problems. The mathematical theory of these problems has attracted a lot of attention in the last decade (see e.g.~\cite{MR2295621,Cardaliaguet-2013-notes,MR3134900,MR3752669,MR3753660}), and from the numerical standpoint several methods have been proposed, see e.g.~\cite{MR2679575,MR2888257,MR3148086,BricenoAriasetalCEMRACS2017,MR3914553} and~\cite{MR3501391,MR3575615,MR3619691,balata2019class} for finite time horizon MFG and MFC respectively, and~\cite{MR3698446,MR3882530,MR3772008} for stationary MFG. However, despite recent progress, the numerical analysis of these problems is still lagging behind because of their complexity, in particular when the dimension is high or when the dynamics is affected by a source of common noise. In~\cite{CarmonaLauriere_DL_periodic}, we have studied the ergodic problems, whose mathematical analysis led to an infinite dimensional optimization problem for which we identified and implemented numerical schemes capable of providing stable numerical solutions. Here, we consider mean field control problems in finite time horizon. The thrust of our approach is the numerical solution of Forward-Backward Stochastic Differential Equations (FBSDEs) of the McKean-Vlasov type. Indeed, the well established probabilistic approach to MFGs and MFC posits that the search for Nash equilibria for MFGs, as well as the search for optimal controls for MFC problems, can be reduced to the solutions of FBSDEs of this type. See for example the books \cite{MR3752669,MR3753660} for a comprehensive expos\'e of this approach. Our mathematical analysis of the model leads to an optimization problem for a large number of agents, for which we can identify and implement numerical schemes capable of providing stable numerical solutions. We prove the theoretical convergence of these approximation schemes and we demonstrate the efficiency of their implementations by comparing their outputs to solutions of benchmark models obtained either by analytical formulas or by deterministic schemes for Partial Differential Equations (PDEs). In particular, we solve numerically two examples with common noise. Although the idea of using machine learning and neural networks for control problems is not new (see e.g.~\cite{MR2137498}), the core of our numerical schemes is based on a generalization to the mean field setting of the deep learning method used very recently e.g. in~\cite{han2016deep-googlecitations,MR3736669} for control problems and partial differential equations. Similar ideas have been used by Fouque and Zhang in~\cite{FouqueZhang}. However, their work is restricted to a specific linear-quadratic problem and does not provide any theoretical convergence result. The present paper is structured as follows. In Section~\ref{sec:pb-assumptions}, we introduce useful notations, provide the standing assumptions and state our main results (see for example Theorem~\ref{thm:main-thm-discreteMKV}), including rates of convergence. In Section~\ref{sec:proof-main}, we outline the key steps for the proof of our main result. In Section~\ref{sec:two-num-meth}, we present two numerical methods for mean field type problems, the first one tackling directly an optimal control of MKV dynamics, and second one dealing with MKV FBSDEs. Last, numerical results are presented in Section~\ref{sec:numres}. \section{Formulation of the problem, standing assumptions and main results} \label{sec:pb-assumptions} Given a probability measure $\mu_0$ on $\RR^d$, a $d$-dimensional Wiener process $\bW = (W_t)_{t \ge 0}$ and a class $\mathbb{A}$ of admissible controls taking values in a closed convex subset $A$ of $\RR^k$, the problem can be stated in the following way. \begin{problem} \label{pb:MFC1-def} Minimize over $\boldsymbol{\alpha} = (\alpha_t)_{t \ge 0} \in \mathbb{A}$ the quantity \begin{equation} \label{eq:MFC1-cost} J(\boldsymbol{\alpha}) = \EE\left[\int_0^T f(t, X_t, \cL(X_t), \alpha_t)dt + g(X_T,\cL(X_T))\right] \, , \end{equation} under the constraint that the process $\bX = (X_t)_{t \ge 0}$ solves the Stochastic Differential Equation (SDE) \begin{equation} \label{eq:MFC1-dyn} dX_t = b(t, X_t, \cL(X_t), \alpha_t) dt + \sigma(t, X_t, \cL(X_t)) dW_t \, , t \ge 0, \quad X_0 \sim \mu_0, \end{equation} \end{problem} \noindent where we use the notation $\cL(V)$ for the law of a random variable $V$. For the sake of definiteness we choose $\mathbb{A} = \HH^{2,k}$, the set of $\RR^k$-valued progressively-measurable square-integrable processes defined as: $$ \HH^{2,d} = \left\{ Z \in \HH^{0,k} ;\; \EE \int_0^T |Z_s|^2 ds < \infty \right\}, $$ where $\HH^{0,k}$ denotes the set of $\RR^k$-valued progressively measurable processes on $[0,T]$. These control processes are often called \emph{open loop}. When it is helpful to stress which control is being used, we shall denote by $\bX^{\boldsymbol{\alpha}} = (X^{\boldsymbol{\alpha}}_t)_{t \ge 0}$ the solution to~\eqref{eq:MFC1-dyn} when the control $\boldsymbol{\alpha}$ is used. In equation \eqref{eq:MFC1-dyn}, the drift and volatility coefficients $b$ and $\sigma$ are functions on $[0,T]\times \RR^d\times\cP_2(\RR^d)\times A$ with values in $\RR^d$ and $\RR^{d\times d}$ respectively. We shall assume that the drift and volatility functions $b$ and $\sigma$ are regular enough so that for each admissible control $\boldsymbol{\alpha}$, existence and uniqueness of a process $\bX^{\boldsymbol{\alpha}}$ satisfying \eqref{eq:MFC1-dyn} hold. Specific assumptions under which this is indeed the case are given in Appendix \ref{ap:assumptions}. We explain below how to understand the notion of regularity with respect to the measure argument $\mu$. \subsection{Definitions, notations and background} For any $p\geq 1$, $\cP_p(\RR^d)$ is the set of probability measures of order $p$ on $\RR^d$, namely those probability measures on $\RR^d$ for which the $p$-th moment $M_p(\mu)$ defined in \eqref{fo:M_p_mu} below is finite. $W_p$ is the $p$-Wasserstein distance defined, for $\mu,\mu' \in \cP_p(\RR^d)$ by $$ W_p(\mu, \mu') = \inf_{\pi \in \Pi(\mu,\mu')} \left[\int_{\RR^d \times \RR^d} |x-y|^p \pi(dx,dy)\right]^{1/p} $$ where $\Pi(\mu,\mu')$ denotes the set of probability measures on $\RR^d\times \RR^d)$ with marginals $\mu$ and $\mu'$. For $\mu \in \cP_p(\RR^d)$, the $p$-th moment is denoted by \begin{equation} \label{fo:M_p_mu} M_p(\mu) = \left( \int_{\RR^d} |x|^p d \mu(x) \right)^{1/p}. \end{equation} For an integer $r \geq 0$ and a domain $D \subseteq \RR^d$, we denote by $\cC^r(D; \RR^{d'})$ the set of functions on $D$ taking values in $\RR^{d'}$ which are continuously differentiable up to order $r$ (included). It is endowed with the usual norm: for $r=0$, this is the sup norm and for $r > 0$, it is the sum of the sup norms of the derivatives up to order $r$. For $K>0$, we will denote by $\cC^0_K(D; \RR^{d'})$ the subset of continuous functions with (sup) norm bounded by $K$. For $L>0$, we denote by $Lip_L(D; \RR^{d'})$ the set of Lipschitz functions on $D$ with Lipschitz constant at most $L$. When we consider real valued functions, i.e. $d'=1$, we shall write simply $\cC^r(D), \cC^0_K(D)$ and $Lip_L(D)$. \subsection{Standing assumptions} \label{sub:assumptions} In this subsection we introduce the assumptions under which existence of optimal controls holds, and under which we prove convergence of the numerical algorithms we propose to compute the solutions. Some of the assumptions are stated at a high level, and the reader may wonder for which classes of coefficients these assumptions are satisfied. In Appendix \ref{ap:assumptions}, we give low level explicit conditions under which all of our assumptions hold. Roughly speaking, for the state dynamics and the cost functions, we use slight variations on assumption ``\textnormal{\textbf{Control of MKV Dynamics}}'' from~\cite[p. 555]{MR3752669}. Throughout the paper, we assume that the volatility is not controlled and that the initial distribution $\mu_0$ is in $\cP_4(\RR^d)$. Still, in order to justify and quantify the approximation of the optimal control by a neural network, we shall need extra regularity assumptions. To formulate them, we introduce more notations. The Hamiltonian of the system is the function $H$ defined by \begin{equation} \label{eq:def-Hamiltonian} H(t, x, \mu, y, z, \alpha) = \, b(t, x, \mu, \alpha) \cdot y + \sigma(t, x, \mu) \cdot z + f(t, x, \mu, \alpha) \end{equation} for $t \in [0,T]$, $x,y \in \RR^d$, $z \in \RR^{d \times d}$, $\mu \in \cP_2(\RR^d)$ and $\alpha \in A$, and since the volatility $\sigma$ does not depend on the control variable in our setting, we will also make use of the notation $\tilde H$ for the reduced Hamiltonian defined by \begin{equation} \label{eq:def-Hamiltonian-red} \tilde H(t, x, \mu, y, \alpha) = \, b(t, x, \mu, \alpha) \cdot y + f(t, x, \mu, \alpha). \end{equation} We shall assume that given any $(t, x, \mu, y) \in [0,T] \times \RR^d \times \cP_2(\RR^d) \times \RR^d$, the function $A \ni \alpha \mapsto \tilde H(t, x, \mu, y, \alpha)$ has a unique minimizer $\hat{\alpha}(t, x, \mu, y)$: \begin{equation} \label{eq:def-hat-alpha} \hat{\alpha}(t, x, \mu, y) = \argmin_{\alpha \in A} \tilde H(t, x, \mu, y, \alpha) \end{equation} being (jointly) Lipschitz in all its variables. We shall also assume that the coefficients $b$ and $\sigma$, as well as the cost functions $f$ and $g$, are differentiable with respect to the variables $x$ and $\mu$. The partial derivatives with respect to the argument $\mu$ have to be understood in the Wasserstein sense, or in the Lions sense (L-derivatives). See \cite[Chapter 5]{MR3752669} for details. The forward-backward system of SDEs (FBSDE for short) associated to the control problem is (see e.g.~\cite[Section 6.4.2]{MR3752669}) \begin{equation} \label{eq:MKV-FBSDE} \left\{ \begin{aligned} d X_t = & b\bigl(t,X_t,\cL(X_t),\hat{\alpha}(t, X_t, \cL(X_t), Y_t)\bigr) dt +\sigma\bigl(t,X_t,\cL(X_t),\hat{\alpha}(t, X_t, \cL(X_t), Y_t)\bigr) dW_t \\ d Y_t = & - \partial_x H(t, X_t, \cL(X_t), Y_t,Z_t,\hat{\alpha}(t, X_t, \cL(X_t), Y_t)) dt \\ & \quad - \tilde{\EE}\left[\partial_\mu H(t, \tilde{X}_t, \cL(X_t), \tilde Y_t, \tilde Z_t, \hat{\alpha}(t, \tilde{X}_t, \cL(X_t), \tilde{Y}_t))(X_t)\right] dt + Z_t d W_t, \end{aligned} \right. \end{equation} with initial condition $X_0 = \xi \in L^4(\Omega, \cF_0, \PP; \RR^d)$ and terminal condition $Y_T = \partial_x g(X_T, \cL(X_T)) + \tilde{\EE}\left[\partial_\mu g(\tilde{X}_T, \cL(X_T))(X_T)\right]$. The tilde $\tilde{}$ over random variables means that these random variables are copies in the sense that they have the same distributions as the original variables, but are possibly defined on a different probability space over which the expectation is denoted by $\tilde \EE$. We shall assume that the system \eqref{eq:MKV-FBSDE} is uniquely solvable and that there exists a function $\cU$, called master field of the FBSDE~\eqref{eq:MKV-FBSDE}, such that the process $(Y_t)_{t \in [0,T]}$ can be represented as \begin{equation} \label{eq:link-Y-cU} Y_t = \cU(t, X_t, \mu_t), \qquad t \in [0,T], \end{equation} where $\mu_t = \cL(X_t)$. Next, we introduce the decoupling field \begin{equation} \label{eq:def-decouplingfield} V(t, x) = \cU(t, x, \mu_t), \end{equation} which we assume to be jointly Lipschitz in its variables, and to be differentiable with respect to $x$ with $\partial_x V$ being Lipschitz in its variables. The optimal control $\hat{\boldsymbol{\alpha}} = (\hat{\alpha}_t)_{t \in [0,T]}$ can be rewritten in the feedback form \begin{equation} \label{eq:hat-alpha-feedbackform} \hat{\alpha}_t = \hat{\alpha}(t, X_t, \mu_t, V(t, X_t)). \end{equation} \vskip 2pt\noindent For the sake of the analysis of the time discretization that we will use in our numerical scheme, we shall also assume that, at each $t \in [0,T]$, the feedback control $x \mapsto \hat{\alpha}(t, x, \mu_t, V(t, x))$ is twice differentiable with second order derivatives which are Lipschitz continuous. We refer the reader to Appendix \ref{ap:assumptions} where we articulate explicitly a specific sets of assumptions under which all the properties stated above hold. Unless otherwise specified, the constants depend only on the data of the problem ($T$, $\mu_0$, $d$, $k$, and the constants appearing in the assumptions), and $C$ denotes a generic constant whose value might change from one line to the next. \subsection{Approximation results} \label{sec:approx-result} Next, we propose a new optimization problem, amenable to numerical computations (see Section~\ref{sec:two-num-meth}), which serves as a proxy for the original MKV control problem, and for which we quantify the approximation error. The rationale behind this new problem is encapsulated in the following three steps: \begin{itemize} \item the distribution $\cL(X_t)$ of the state $X_t$ is approximated by the empirical distribution of $N$ agents; \item the set $\mathbb{A}$ of controls by a set of controls in feedback form, the feedback function being given by neural nets with a fixed architecture; \item the time variable is discretized. \end{itemize} Before defining the problem, we first introduce notations pertaining to neural networks. \subsection{Neural networks} We denote by: $$ \mathbf{L}^\psi_{d_1, d_2} = \left\{ \phi: \RR^{d_1} \to \RR^{d_2} \,\Big|\, \exists \beta \in \RR^{d_2}, \exists w \in \RR^{d_2 \times d_1}, \forall i \in \{1,\dots,d_2\}, \;\phi(x)_i = \psi\left(\beta_i + \sum_{j=1}^{d_1} w_{i,j} x_j\right) \right\} $$ the set of layer functions with input dimension $d_1$, output dimension $d_2$, and activation function $\psi: \RR \to \RR$ with at most linear growth. For the sake of definiteness, we shall assume that the activation function $\psi:\RR \to \RR$ is a $2\pi-$periodic function of class $\cC^3$ satisfying: \begin{equation} \label{eq:cond-sigma-Fourier} \hat\psi_1:=\int_{-\pi}^\pi \psi(x) e^{-i x} dx \neq 0. \end{equation} More general activation functions could be accommodated at the expense of additional technicalities. The choice of this class of activation functions is motivated by the fact that we want to find a neural network which approximates, on a compact set, the optimal feedback control while being Lipschitz continuous and whose Lipschitz constant can be related to the one of the optimal control. Building on this notation we define: \begin{equation} \label{fo:N_psi} \bN^\psi_{d_0, \dots, d_{\ell+1}} = \left\{ \varphi: \RR^{d_0} \to \RR^{d_{\ell+1}} \,\Big|\, \forall i \in \{0, \dots, \ell-1\}, \exists \phi_i \in \mathbf{L}^\psi_{d_i, d_{i+1}}, \exists \phi_\ell \in \mathbf{L}_{d_{\ell}, d_{\ell+1}}, \varphi = \phi_\ell \circ \phi_{\ell-1} \circ \dots \circ \phi_0 \right\} \, \end{equation} the set of regression neural networks with $\ell$ hidden layers and one output layer, the activation function of the output layer being the identity $\psi(x)=x$. We shall not use the superscript $\psi$ when the activation function is the identity. The number $\ell$ of hidden layers, the numbers $d_0$, $d_1$, $\cdots$ , $d_{\ell+1}$ of units per layer, and the activation functions (one single function $\psi$ in the present situation), are what is usually called the architecture of the network. Once it is fixed, the actual network function $\varphi\in \bN^\psi_{d_0, \dots, d_{\ell+1}} $ is determined by the remaining parameters: $$ \theta=(\beta^{(0)}, w^{(0)},\beta^{(1)}, w^{(1)},\cdots\cdots,\beta^{(\ell-1)}, w^{(\ell-1)},\beta^{(\ell)}, w^{(\ell)}) $$ defining the functions $\phi_0$, $\phi_1$, $\cdots$ , $\phi_{\ell-1}$ and $\phi_\ell$ respectively. Their set is denoted by $\Theta$. For each $\theta\in\Theta$, the function $\varphi$ computed by the network will sometimes be denoted by $\varphi_\theta$. We will work mostly with the case $d_0 = d+1$ and $d_{\ell+1} = k$, dimensions of $(t,x)$ and of the control variable respectively. As implied by the above discussion, the search for optimal controls in the general class $\mathbb{A}$ of open loop controls eventually leads to controls in feedback form given by \eqref{eq:hat-alpha-feedbackform}. This fact is the rationale for the second step announced earlier, namely the search for approximately optimal controls among the controls given in feedback form by neural network functions $\varphi$. Notice that if $\boldsymbol{\alpha}$ is such a control given in the form $\alpha_t = \varphi(t, X^\alpha_t)$ for some $ \varphi \in \bN^\psi_{d_0, \dots, d_{\ell+1}}$, then such an $\boldsymbol{\alpha}$ is admissible, i.e. $\boldsymbol{\alpha}\in\mathbb{A}$, because of standard properties of solutions of MKV stochastic differential equations with Lipschitz coefficients, and the fact that the function $\varphi$ is at most of linear growth. Last, we stress that the elements of $\bN^\psi_{d_0, \dots, d_{\ell+1}}$ have the same regularity as the activation function $\psi$, namely $\cC^3$. \subsection{New optimization problem and main result} For the discretization of the time interval, we will use a grid $t_0=0 < t_1 < \dots < t_{N_T} = T$ where $N_T$ is a positive integer. For simplicity we consider a uniform grid, that is, $t_n = n \Delta t$, with $\Delta t = T/N_T$. The three approximation steps described at the beginning of this section lead to the following minimization problem. The latter can be viewed as the problem of a central planner trying to minimize the social cost of $N$ agents using a decentralized control rule in feedback form given by a neural network. \begin{problem}\label{pb:discrete-MKV} Minimize the quantity \begin{equation} \label{eq:def-checkJN} \check J^N(\boldsymbol{\varphi}) = \EE\left[\frac{1}{N}\sum_{i=1}^N \Bigl(\Delta t\sum_{n=0}^{N_T-1} f\left(t_n, \check X^i_{t_n}, \check\mu_{t_n}, \varphi(t_n, \check X^i_{t_n})\right)+ g(\check X^i_{t_{N_T}},\check\mu_{t_{N_T}})\Bigr)\right] \, , \end{equation} over $\boldsymbol{\varphi} \in \bN^\psi_{d+1, d_2, \dots, d_{\ell+1}, k}$, under the dynamic constraint: \begin{equation} \label{eq:discrete-dyn} \check X^i_{t_{n+1}} = \check X^i_{t_n} + b\left(t_n, \check X^i_{t_n}, \check\mu_{t_n}, \varphi(t_n, \check X^i_{t_n})\right) \Delta t + \sigma\left(t_n, \check X^i_{t_n}, \check\mu_{t_n}\right) \Delta \check W^i_n \, , \quad n \in \{0,\dots, N_T-1\}, i \in \{1,\dots,N\} \, , \end{equation} where the $(\check X^i_0)_{i \in \{1, \dots, N\}}$ are i.i.d. with common distribution $\mu_0$, $\check\mu_{t_n} = \frac{1}{N} \sum_{i=1}^N \delta_{\check X^i_{t_n}}$, and the $(\Delta \check W_n^i)_{i,n}$ are i.i.d. random variables with distribution $\mathcal N(0, \Delta t)$. \end{problem} We show that solving Problem~\ref{pb:discrete-MKV} provides an approximate solution to Problem~\ref{pb:MFC1-def}, and we quantify the accuracy of the approximation. The main theoretical result of the paper is the following. \begin{theorem} \label{thm:main-thm-discreteMKV} There exists a constant $C$ depending only on the data of the problem such that: $$ \inf_{\boldsymbol{\alpha} \in \mathbb{A}} J(\boldsymbol{\alpha}) \ge \inf_{\boldsymbol{\varphi} \in \bN^{\psi}_{d+1, n_{\mathrm{in}}, k}} \check J^N(\boldsymbol{\varphi}) - C \epsilon(N, n_{\mathrm{in}}, \Delta t) \, $$ where \begin{align*} \epsilon(N,n,\Delta t) &= \epsilon_1(N) + \epsilon_2(n) + \epsilon_3(\Delta t) \end{align*} with $$ \epsilon_1(N) \in O\left( N^{-1/\max(d, 4)} \sqrt{1 + \ln (N) \mathbf{1}_{\{d=4\}}} \right), \qquad \epsilon_2(n) \in O\left( n^{-\frac{1}{3(d+1)}} \right), \qquad \epsilon_3(\Delta t) \in O\left( \sqrt{\Delta t} \right), $$ the constants in the big Bachmann - Landau terms $O(\cdot)$ depending only on the data of the problem and on the activation function $\psi$ through $\hat\psi_1$. \end{theorem} The proof is provided in Section~\ref{sec:proof-main}. \begin{remark} The dependence of the error on the the number $n_{\mathrm{in}}$ of units in the layer could certainly be improved by looking at multilayer neural networks. Unfortunately, quantifying the performance of these architectures is much more difficult and we could not find in the function approximation literature suitable rate of convergence results which could be useful in our setting. Also, the error from the Euler scheme could probably be improved to order $1$ instead of $1/2$ (i.e., the last term $\epsilon_3(\Delta t)$ should be of the order $\Delta t$ instead of $\sqrt{\Delta t}$). However, we are only able to reach this order at the expense of extra regularity properties of the decoupling field. Since we did not want to add (and prove) extra smoothness assumptions on the decoupling field, we only claim order $1/2$ in the mesh of the time discretization. \end{remark} \begin{remark} Theorem~\ref{thm:main-thm-discreteMKV} provides a bound on the approximation error. In the numerical implementation, the expectation in $\check J^N$ over the $N-$agent population is replaced by an empirical average over a finite number of populations, which leads to an estimation (or ``generalization'') error. In the ergodic, setting we have analyzed this type of error in~\cite{CarmonaLauriere_DL_periodic}. In the finite horizon setting considered here, similar techniques can be used to bound the generalization error. We refrain from doing so for the sake of brevity, leaving it for a future work. \end{remark} \section{Proof of the main result} \label{sec:proof-main} We split the proof into three steps presented in separate subsections, each one consisting in the control of the approximation error associated one of the steps described in the bullet points at the beginning of the previous section. As explained above, the first step consists in approximating the mean field problem with open loop controls by a problem with $N$-agents using \emph{distributed} closed-loop controls (see subsection~\ref{sec:approx-Nagents-feedback}); in the second step, we replace general closed-loop controls by the subclass of closed-loop controls that can be represented by neural networks (see subsection~\ref{sec:approx-NN}); in the third step, we discretize time (see subsection~\ref{sec:approx-discretetime}). \subsection{Problem with $N$ agents and closed-loop controls} \label{sec:approx-Nagents-feedback} Because of \eqref{eq:hat-alpha-feedbackform}, we expect the optimal control to be in feedback form. So in the sequel, we restrict our attention to closed-loop controls that are deterministic functions of $t$ and $X_t$, i.e. of the form $\alpha_t = v(t, X_t)$ for some deterministic function $v : [0,T] \times \RR^d \to A\subset\RR^k$. We denote by $\VV$ the class of admissible feedback functions, i.e., the set of measurable functions $v : [0,T] \times \RR^d \to A$ such that the control process $\boldsymbol{\alpha}$ defined by $\alpha_t = v(t, X^\alpha_t)$ for all $t$ is admissible, i.e. $\boldsymbol{\alpha} \in \mathbb{A}$. Note that if the feedback function $v$ is Lipschitz in $x$ uniformly in $t$, then the state SDE \eqref{eq:MFC1-dyn} is well posed, and since $v$ is at most of linear growth, standard stability estimates for solutions of Lipschitz SDE of McKean-Vlasov type guarantee that $\boldsymbol{\alpha}\in\mathbb{A}$ or equivalently $\bv\in\VV$. \vskip 2pt According to the first step described above, we recast Problem~\ref{pb:MFC1-def} as the limiting problem for the optimal control of a large number of agents by a \emph{central planner} as the number of agents tends to infinity. In the model with $N<\infty$ agents, the goal is to minimize the average (``social'') cost, when all the agents are using the same control rule, namely the same feedback function of their individual states. The resulting optimization problem is: \begin{problem} \label{pb:MKV-Nagents} Minimize the quantity \begin{equation} \label{eq:def-JN-closedloop} J^N(\bv) = \frac{1}{N}\sum_{i=1}^N \EE\left[\int_0^T f(t, X^i_t, \mu^N_t, v(t, X^i_t))dt + g(X^i_T,\mu^N_T)\right] \, , \end{equation} over $\bv = (v(t, \cdot))_{0\le t \le T} \in \VV$ under the constraint: \begin{equation} \label{eq:evolXi-closedloop} dX^i_t = b(t, X^i_t, \mu^N_t, v(t,X^i_t)) dt + \sigma(t, X^i_t, \mu^N_t) dW^i_t \, , \quad t \ge 0, \, i \in \{1,\dots,N\} \, , \end{equation} where the $\bW^i=(\bW^i)_{i=1, \dots, N}$ are independent $d$-dimensional Wiener processes, the $(X^i_0)_{i \in \{1, \dots, N\}}$ are i.i.d. with common distribution $\mu_0$ and are independent of the Wiener processes, and where $\mu^N_t = \frac{1}{N} \sum_{i=1}^N \delta_{X^i_t}$ is the empirical distribution of the population of the $N$ agents at time $t$. \end{problem} We have the following result. \begin{proposition} \label{prop:approx-feedback-N} If we define the feedback function $\hat\bv$ by: \begin{equation} \label{fo:vhat} \hat v(t,x) = \hat \alpha\left( t, x, \mu_t, V(t, x) \right), \end{equation} then $$ \inf_{\boldsymbol{\alpha} \in \mathbb{A}} J(\boldsymbol{\alpha}) \ge J^N(\hat \bv) - \epsilon_1(N) \, , $$ with $$ \epsilon_1(N) = c_1 \sqrt{N^{-2/\max(d, 4)} \left(1 + \ln (N) \mathbf{1}_{\{d=4\}}\right)} \, , $$ for some constant $c_1$ depending only on the data of the problem. \end{proposition} Recall that $\hat\alpha$ is the minimizer of the Hamiltonian defined by~\eqref{eq:def-hat-alpha}, $\mu_t$ and $V$ are respectively the marginal distribution of $X_t$ and the decoupling field of the associated MKV FBSDE system~\eqref{eq:MKV-FBSDE} defined by~\eqref{eq:def-decouplingfield}, and as explained above $\hat\bv$ is admissible under the standing assumptions. \begin{proof} The result follows from a slight modification of the proof of~\cite[Theorem 6.17]{MR3753660}. Note that our standing assumptions ensure that the assumption ``\textnormal{\textbf{Control of MKV Dynamics}}'' from~\cite[p. 555]{MR3752669} holds, that $\sigma$ does not depend on the control, and that $\mu_0 \in \cP_4(\RR^d)$. We can thus reuse directly the last inequality of the proof of~\cite[Theorem 6.17]{MR3753660}, and we obtain: \begin{equation} \label{eq:last-ineq-thm617} J^N(\hat \boldsymbol{\alpha}^{(N)}) \leq J + \epsilon_1(N) \end{equation} where $J = \inf_{\boldsymbol{\alpha} \in \mathbb{A}} J(\boldsymbol{\alpha})$ is the objective function minimized over all admissible open loop controls, and $\hat{\boldsymbol{\alpha}}^N$ is the distributed control given in feedback form by~\eqref{eq:hat-alpha-feedbackform}, namely \begin{equation} \label{eq:def-hatbalpha-feedback} \hat \alpha^{N,i}_t = \hat v (t, X^{N,i}_t) = \hat \alpha\left( t, X^{N,i}_t, \mu_t, V(t, X^{N,i}_t) \right) \, . \end{equation} Notice that thanks to the symmetry between the agents, $J^N(\hat \bv)$ defined by~\eqref{eq:def-JN-closedloop}, can be viewed as the cost of a typical player. In other words, for each $j \in \{1,\dots,N\}$, $$ J^N(\boldsymbol{\alpha}) = J^{N,j}(\boldsymbol{\alpha}) $$ where $$ J^{N,j}(\boldsymbol{\alpha}) = \EE\left[\int_0^T f(X^j_t, \mu^N_t, \alpha^j_t)dt + g(X^j_T,\mu^N_T)\right] $$ under the constraint: \begin{equation*} dX^i_t = b(t, X^i_t, \mu^N_t, \alpha^i_t) dt + \sigma(t, X^i_t, \mu^N_t) dW^i_t \, , \quad t \ge 0, \, i \in \{1,\dots,N\} \, , \end{equation*} where the $(X^i_0)_{i \in \{1, \dots, N\}}$ are i.i.d. with distribution $\mu_0$, also independent of the Wiener processes $\bW^i, i=1, \dots, N$. Since~\eqref{eq:def-hatbalpha-feedback} says that $\hat \alpha^{N,i}_t$ is given by an admissible feedback function, $$ J^N(\hat \boldsymbol{\alpha}^{(N)}) \geq \inf_{\bv \in \VV} J^N(\bv). $$ As a consequence, inequality~\eqref{eq:last-ineq-thm617} yields $$ \inf_{\bv \in \VV} J^N(\bv) \leq \inf_{\boldsymbol{\alpha} \in \mathbb{A}} J(\boldsymbol{\alpha}) + \epsilon_1(N), $$ which concludes the proof. \end{proof} \subsection{Problem with neural networks as controls} \label{sec:approx-NN} Next, we show that the feedback function $\hat\bv$ used in Proposition~\ref{prop:approx-feedback-N}, can be approximated by a neural network in such a way that minimizing over neural networks ends up being not much worse than minimizing over feedback controls, and we quantify the loss due to this approximation. The main result of this section is the following. \begin{proposition} \label{prop:approx-NN-cmpJ-epsilon} There exists two positive constants $K_1$ and $K_2$ depending on the data of the problem and $\psi$ through $\hat\psi(1)$, $\|\psi'\|_{\cC^0(\TT)}$, $\|\psi''\|_{\cC^0(\TT)}$ and $\|\psi'''\|_{\cC^0(\TT)}$, such that for each integer $n_{\mathrm{in}} \ge 1$, there exists $\hat\boldsymbol{\varphi} \in \bN^{\psi}_{d+1,n_{\mathrm{in}},k}$ such that the Lipschitz constants of $\hat\boldsymbol{\varphi}$, $\partial_x\hat\boldsymbol{\varphi}$ and $\partial^2_{x,x}\hat\boldsymbol{\varphi}$ are bounded by $K_1$, and which satisfies $$ J^N(\hat \bv) \ge J^N(\hat\boldsymbol{\varphi}) - K_2 n_{\mathrm{in}}^{- \frac{1}{3(d+1)}}. $$ \end{proposition} Given the Lipschitz continuity of the optimal feedback control $\hat{\bv}$ w.r.t. $(t,x)$ under the standing assumptions, and given the result of Proposition~\ref{prop:approx-fct-NN} below quantifying the rate at which one can approximate Lipschitz functions by neural networks over compact sets, the proof of the above result relies on the regularity property of the objective functional provided by Proposition~\ref{lem:var-J-feedback-compact}: if two controls are close enough, the objective value does not vary too much. The proof of Proposition~\ref{prop:approx-fct-NN} relies on a special case of~\cite[Theorems 2.3 and 6.1]{MhaskarMicchelli} which we state below for the sake of completeness. It forces us to work with periodic functions, but in return, it provides a neural network approximation for a function and its derivative. This is important in an optimal control setting as it provides needed bounds on the Lipschitz constant of the control. First we recall a standard notation. For a positive integer $m$, $\TT^m$ denotes the $2\pi-$torus in dimension $m$. For positive integers $n$ and $m$, and a function $g \in \cC^0(\TT^m) $, $E_n^m(g)$ denotes the trigonometric degree of approximation of $g$ defined by: $$ E_n^m(g) = \inf_{T} \|g - T\|_{\cC^0(\TT^m)} $$ where the infimum is over trigonometric polynomials of degree at most $n$ in each of its $m$ variables. While \cite[Theorems 2.3 and 6.1]{MhaskarMicchelli} does not differentiate between a time and a space variable, and the same regularity is assumed for all the components of the variables, the proofs of these theorems actually give the following result. \begin {theorem}[Theorems 2.3 and 6.1 in~\cite{MhaskarMicchelli}] \label{th:MM} There exists an absolute constant $C_{mm}$ such that for every positive integers $d, n$ and $N$, there exists a positive integer $n_{\mathrm{in}} \le C_{mm} N n^d$ with the following property. For any $f:\TT\times \TT^{d} \to \RR$ of class $\cC^{1,3}(\TT\times\TT^d)$, there exists $\varphi_f \in \bN^\psi_{d+1, n_{\mathrm{in}}, 1}$ such that: \begin{align} \label{eq:MhaskarMicchelli-bdd-f} \|f - \varphi_f\|_{\cC^0(\TT^{d+1})} &\leq c\left[E^{d+1}_n(f) + E^1_N( \psi) n^{(d+1)/2} \|f\|_{\cC^0(\TT^{d+1})} \right], \\ \label{eq:MhaskarMicchelli-bdd-grad-f} \|\partial_i f - \partial_i \varphi_f\|_{\cC^0(\TT^{d+1})} &\leq c\left[E^{d+1}_n(\partial_i f) + E^1_N(\psi') n^{(d+1)/2} \|\partial_i f\|_{\cC^0(\TT^{d+1})} \right], \quad i = 2,\dots,d+1, \\ \label{eq:MhaskarMicchelli-bdd-grad2-f} \|\partial_{i,j} f - \partial_{i,j} \varphi_f\|_{\cC^0(\TT^{d+1})} & \leq c\left[E^{d+1}_n(\partial_{i,j} f) + E^1_N(\psi'') n^{(d+1)/2} \|\partial_{i,j} f\|_{\cC^0(\TT^{d+1})} \right], \quad i,j = 2,\dots,d+1, \end{align} where $\partial_i$ denotes the partial derivative with respect to the $i^{th}$ variable and the constant $c$ depends only on $d$ and $\hat\psi(1)$. \end{theorem} In the sequel, we denote by $\cC(R, K, L_1, L_2, L_3)$ the class of functions $f: [0,T] \times \bar{B}_d(0,R) \ni (t,x) \mapsto f(t,x) \in \RR^{k}$ such that $f$ is Lipschitz continuous in $(t,x)$ with $\cC^0([0,T] \times \bar{B}_d(0,R))-$norm bounded by $K$ and Lipschitz constant bounded by $L_1$, and $f$ is twice differentiable w.r.t. $x$ such that for every $i=1,\dots,d$, $\partial_{x_i} f$ is Lipschitz continuous w.r.t. $(t,x)$ with Lipschitz constant bounded by $L_2$ and for every $i,j=1,\dots,d$, $\partial_{x_i, x_j} f$ is Lipschitz continuous w.r.t. $(t,x)$ with Lipschitz constant bounded by $L_3$. Note that this class of functions implicitly depends on $d,k,T$, which are part of the \emph{data of the problem}. We will sometimes use the notation $\nabla f = (\partial_{x_1} f, \dots, \partial_{x_d} f)$ and $\nabla_{(t,x)} f = (\partial_t f, \partial_{x_1} f, \dots, \partial_{x_d} f)$. \vskip 2pt The workhorse of our control of the approximation error is the following. \begin{proposition} \label{prop:approx-fct-NN} For every real numbers $K>0$, $L_1>0, L_2>0$ and $L_3>0$, there exists a constant $C$ depending only on the above constants, on $d,k,T,$ and on the activation function through $\hat\psi(1)$, $\|\psi'\|_{\cC^0}$, $\|\psi''\|_{\cC^0}$, and $\|\psi'''\|_{\cC^0}$, and there exists a constant $n_0$ depending only on $C_{mm}$ and $d$ with the following property. For every $R>0$, for every $f \in \cC(R, K, L_1, L_2, L_3)$ and for every integer $n_{\mathrm{in}}>0$, there exists a one-hidden layer neural network $\varphi_f \in \bN^{\psi}_{{d + 1}, n_{\mathrm{in}}, {k}}$ such that $$ \|f - \varphi_f\|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{k})} \le C (1+R)n_{\mathrm{in}}^{-1/(2 (d+1))}, $$ and such that the Lipschitz constants of $\varphi_f$, $\partial_x \varphi_f$ and $\partial^2_{x,x} \varphi_f$ are at most $C (1+Rn_{\mathrm{in}}^{-1/(2 (d+1))})$. \end{proposition} We stress that the constant $C$ in the above statement does not depend on $R$. \begin{remark} The exponent $-1/(2(d+1))$ in the statement of the proposition is what is blamed for the so-called curse of dimensionality. \end{remark} \begin{remark} \label{re:C2+C3} The Lipschitz continuity of $\partial_{x_i, x_j} f$ and the bound on its Lipschitz constant will only be used in the analysis of one of the contributions of the Euler scheme to the error due to the time discretization (see Section~\ref{sec:approx-discretetime}). There, in order to apply the above result to the optimal control $\hat\bv$ we shall require the extra Assumptions~\ref{hyp:euler-extra-assumption-gradV} and~\ref{hyp:euler-extra-assumption-gradAlpha} added to the standing assumptions in Apppendix \ref{ap:assumptions}. We stress that these assumptions can be omitted if one is not interested in the analysis of the error due to time discretization. \end{remark} \begin{proof} Let $f \in \cC(R, K, L_1, L_2, L_3)$ be as in the statement. Proving the desired bound for each component of $f$ separately, we may assume that $k=1$ without any loss of generality. Let $n_0 = \lceil C_{mm} 2^{2(d+1)} \rceil$, and fix the integers $n_{\mathrm{in}}$ and $n$ so that $n_{\mathrm{in}} \ge n_0$ and $n = \lfloor (n_{\mathrm{in}}/C_{mm})^{1/(2(d+1))} \rfloor$. As a result, \begin{equation} \label{eq:nin_n} C_{mm} n^{2(d+1)} \le n_{\mathrm{in}}, \qquad \hbox{ and } \qquad n^{-1} \le 2 \left( n_{\mathrm{in}} / C_{mm} \right)^{-1/(2(d+1))}. \end{equation} For $\sigma>0$, we denote by $g_\sigma$ the density of the mean-zero Gaussian distribution on $\RR^{d+1}$ with variance/covariance matrix $\sigma^2 I_{d+1}$ where $I_{d+1}$ denotes the $(d+1)\times(d+1)$ identity matrix, and we define $\tilde f=f*g_\sigma: \RR^{d+1} \to \RR$ which is $\cC^\infty(\RR^{d+1})$. We choose $\sigma>0$ small enough to ensure that $$ \|f - \tilde f\|_{\cC^0([0,T] \times \bar{B}_d(0,R))}\le n^{-1}, \quad \| \tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R))} \le K, \quad \| \nabla_{(t,x)} \tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d})} \le L_1, $$ and for $i,j=1,\ldots,d$ $$ \| \nabla_{(t,x)} \partial_{x_i}\tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d})} \le L_2, \qquad \hbox{and} \qquad \| \nabla_{(t,x)} \partial_{x_i, x_j}\tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d})} \le L_3. $$ We then choose a constant $K_0$ depending only on $T$ and $L_2$, and a function $\zeta:[0,\infty)\to [0,1]$ with the following properties: $\zeta$ is $\cC^\infty$ and non-increasing, and there exists $R' \in (\sqrt{T^2+R^2}+1, K_0(1+R))$ such that $\zeta(r)=1$ if $r \le\sqrt{T^2+R^2}$ and $\zeta(r)=0$ if $r>R'-1$, and $|\zeta'(r)|\le L_2, |\zeta''(r)|\le L_3$ for all $r\ge 0$. We now consider the function $\xi$ defined on $[-R', R']^{d+1}$ by $\xi(t,x)=\tilde f(t,x)\zeta(|(t,x)|)$. Note that $\xi\in \cC^\infty$, that $\xi(t,x)$ coincides with $\tilde f(t,x)$ for $t \in (0,T) $ and $|x|\le R$, and that $\xi(t,x)=0$ if $|(t,x)|\ge R'-1$. In particular, we have \begin{align} \label{eq:approx-f-tildef-xi} \|f - \xi\|_{\cC^0([0,T] \times \bar{B}_d(0,R))} = \|f - \tilde f\|_{\cC^0([0,T] \times \bar{B}_d(0,R))} &\le n^{-1}, \\ \label{eq:approx-nabla-f-tildef-xi} \| \nabla_{(t,x)} \xi \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d+1})} = \| \nabla_{(t,x)} \tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d+1})} &\le L_1, \\ \label{eq:approx-nabla-partial-f-tildef-xi} \| \nabla_{(t,x)} \partial_{x_i} \xi \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d+1})} = \| \nabla_{(t,x)} \partial_{x_i} \tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d+1})} &\le L_2, \qquad i=1,\dots,d, % \\ \label{eq:approx-nabla-partial2-f-tildef-xi} \| \nabla_{(t,x)} \partial_{x_i,x_j} \xi \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d+1})} = \| \nabla_{(t,x)} \partial_{x_i,x_j} \tilde f \|_{\cC^0([0,T] \times \bar{B}_d(0,R); \RR^{d+1})} & \le L_3, \qquad i,j=1,\dots,d. \end{align} We now apply the result of Theorem \ref{th:MM} to the function $\tilde \xi$ defined by $$ [-\pi, \pi]^{d+1} \ni (t,x) \mapsto \tilde\xi(t,x) := \xi\left( R' t/\pi , R' x/\pi \right) $$ extended into a $2\pi$-periodic function in each variable, and we bound from above the right hand sides of~\eqref{eq:MhaskarMicchelli-bdd-f},~\eqref{eq:MhaskarMicchelli-bdd-grad-f} and~\eqref{eq:MhaskarMicchelli-bdd-grad2-f}. We use a Jackson type result, namely the fact that the trigonometric degree of approximation of a function of class $\cC^r$ is of order $O(n^{-r})$ when using polynomials of degree at most $n$. More precisely, by~\cite[Theorem 4.3]{MR0251410}, if $F:\RR^k \to \RR$ is an $r$-times continuously differentiable function which is $2\pi$-periodic in each variable, then for every positive integer $m$, there exists a trigonometric polynomial $T_m$ of degree at most $m$ such that $$ |F(x) - T_m(x)| \leq C m^{-r} \left(\sum_{i=1}^k (M_r)^{k-i} \omega_i\left(\frac{\partial^r F}{\partial x_i^r}; \frac{1}{m}\right)\right), \qquad x \in [-\pi, \pi]^k, $$ where $C$ is an absolute constant, $M_r = \max_{1 \le s \le r+1} \begin{pmatrix} r+1 \\ s \end{pmatrix}$, and $\omega_i$ is the modulus of continuity defined, for a function $F \in \cC^0(\RR^k)$ and $h>0$, as $$ \omega_i (F; h) = \max_{x \in \RR^k,\, x'\in\RR\,|x_i - x'| \le h} |F(x_1, \dots, x_i, \dots, x_k) - F(x_1, \dots, x', \dots, x_k)|. $$ In particular, if $F$ is $L-$Lipschitz, then $\omega_i (F; h) \le L h $ and $\omega_i (F; h) \le 2\|F\|_\infty $ when $F$ is merely bounded. We apply this result to $F = \tilde \xi$, $F = \partial_{x_i} \tilde \xi(x)$ and $F = \partial_{x_i,x_j} \tilde \xi(x)$, with $m=n$, $k = d+1$ and $r = 1$, since $\tilde \xi$ is at least of class $\cC^{2}$. Note that, by the definition of $\tilde \xi$, by~\eqref{eq:approx-nabla-f-tildef-xi},~\eqref{eq:approx-nabla-partial-f-tildef-xi} and~\eqref{eq:approx-nabla-partial2-f-tildef-xi}, $\tilde \xi$, $\partial_{x_i} \tilde \xi$ and $\partial_{x_i,x_j} \tilde \xi$ have derivatives with respect to all variables which are uniformly bounded from above by $\frac{R'}{\pi} L_1$, $\left(\frac{R'}{\pi}\right)^2 L_2$ and $\left(\frac{R'}{\pi}\right)^3 L_3$ respectively. Thus, we obtain $$ E^{d+1}_n( \tilde \xi )\le c_0 R' n^{-1}, \qquad E^{d+1}_n( \partial_i \tilde \xi )\le c_0 (R')^2 n^{-1}, \qquad\text{and}\qquad E^{d+1}_n( \partial_{i,j} \tilde \xi )\le c_0 (R')^3 n^{-1}, $$ where $c_0$ depends only on $L_1$, $L_2$, $L_3$ and $d$. Note that we could have expected an upper bound of the order of $n^{-2}$ but this does not seem achievable with the above arguments because we do not have a control of the modulus of continuity of $\partial_t\xi$ since we do not know that $\xi$ is twice differentiable in time. Moreover, applying~\cite[Theorem 4.3]{MR0251410} to $\psi$, $\psi'$ and $\psi''$, which are $\cC^1$, we obtain that for any integer $N$, $$ E^1_N( \psi)\le c_1 N^{-1}, \qquad E^1_N( \psi' )\le c_1 N^{-1}, \qquad\text{and}\qquad E^1_N( \psi'' )\le c_1 N^{-1}, $$ where $c_1$ depends only on $\|\psi'\|_{\cC^0(\TT)}$, $\|\psi''\|_{\cC^0(\TT)}$ and $\|\psi'''\|_{\cC^0(\TT)}$. We use the above inequalities with $N = n^{1+(d+1)/2}$, so that $N^{-1} n^{(d+1) / 2} = n^{-1}$. By Theorem~\ref{th:MM}, we obtain that there exist $\tilde n_{\mathrm{in}} \le C_{mm} N n^{d+1}$ and $\tilde \varphi\in \bN^\psi_{d+1, \tilde n_{\mathrm{in}}, 1}$ such that, \begin{align*} \|\tilde \xi - \tilde \varphi\|_{\cC^0(\TT^{d+1})} &\leq c_2 (1 + R') n^{-1}, \\ \|\partial_{x_i} \tilde \xi - \partial_{x_i} \tilde \varphi\|_{\cC^0(\TT^{d+1})} &\leq c_2 (1 + (R')^2) n^{-1}, \qquad i=1,\dots,d, \\ \|\partial_{x_i,x_j} \tilde \xi - \partial_{x_i,x_j} \tilde \varphi\|_{\cC^0(\TT^{d+1})} & \leq c_2 (1 + (R')^3) n^{-1}, \qquad i,j=1,\dots,d, \end{align*} where $c_2$ depends only on $\hat \psi(1)$, $d$, $K$, $c_0$ and $c_1$. Notice that, by our choice of $N$ and~\eqref{eq:nin_n}, it holds $\tilde n_{\mathrm{in}} \le C_{mm} N n^{d+1} \le C_{mm} n^{2(d+1)} \le n_{\mathrm{in}}$ so that, up to adding dummy neurons to $\tilde \varphi$, we obtain the existence of $\varphi\in \bN^\psi_{d+1, n_{\mathrm{in}}, 1}$ such that, \begin{align} \label{eq:bdd-tildexi-varphi} \|\tilde \xi - \varphi\|_{\cC^0(\TT^{d+1})} &\leq c_2(1 + R') n^{-1}, \\ \label{eq:bdd-grad-tildexi-varphi} \|\nabla_{(t,x)} \tilde \xi - \nabla_{(t,x)} \varphi\|_{\cC^0(\TT^{d+1}; \RR^{d})} &\leq c_2 (1 + (R')^2) n^{-1}, \\ \label{eq:bdd-grad2-tildexi-varphi} \|\nabla_{(t,x)} \partial_{x_i} \tilde \xi - \nabla_{(t,x)} \partial_{x_i} \varphi\|_{\cC^0(\TT^{d+1}; \RR^{d})} & \leq c_2 (1 + (R')^3) n^{-1}, \qquad i=1,\dots,d. \end{align} Consider the function $\varphi_f$ defined by $\varphi_f: [0,T] \times \bar{B}_d(0,R) \ni (t,x) \mapsto \varphi (\pi t / R', \pi x / R' )$. Combining~\eqref{eq:approx-f-tildef-xi},~\eqref{eq:bdd-tildexi-varphi} and~\eqref{eq:nin_n}, yields \begin{align*} \| f - \varphi_f \|_{\cC^0([0,T] \times \bar{B}_{d}(0,R))} &\le \| f - \xi \|_{\cC^0([0,T] \times \bar{B}_{d}(0,R))} + \| \xi - \varphi_f \|_{\cC^0([0,T] \times \bar{B}_{d}(0,R))} \\ &\le \| f - \xi \|_{\cC^0([0,T] \times \bar{B}_{d}(0,R))} + \| \tilde \xi - \varphi \|_{\cC^0(\TT^{d+1})} \\ &\le n^{-1} + c_2(1 + R') n^{-1} \\ &\le c_3 (1 + R) n_{\mathrm{in}}^{-1/(2(d+1))}, \end{align*} where $c_3$ depends only on $c_2, C_{mm}, d$ and $K_0$. Moreover, from the definition of $\varphi_f$, ~\eqref{eq:bdd-grad-tildexi-varphi}, and~\eqref{eq:approx-nabla-f-tildef-xi}, we obtain \begin{align*} \| \nabla_{(t,x)} \varphi_f \|_{\cC^0( [0,T] \times \bar{B}_{d}(0,R); \RR^{d})} &= \frac{\pi}{R'} \| \nabla_{(t,x)} \varphi \|_{\cC^0([0,\pi T / R'] \times \bar{B}_{d}(0, \pi R/ R'); \RR^{d})} \\ &\le \frac{\pi}{R'} \left( \|\nabla_{(t,x)} \tilde \xi \|_{\cC^0([0,\pi T / R'] \times \bar{B}_{d}(0, \pi R/ R'); \RR^{d})} + c_2 (1 + (R')^2) n^{-1} \right) \\ &= \frac{\pi}{R'} \left( \frac{R'}{\pi} \|\nabla_{(t,x)} \xi \|_{\cC^0([0, T] \times \bar{B}_{d}(0, R); \RR^{d})} + c_2 (1 + (R')^2) n^{-1} \right) \\ &\le c_4 \left(1 + R n_{\mathrm{in}}^{-1/(2(d+1))} \right), \end{align*} where $c_4$ depends only on $c_2$, $L_1$, $C_{mm}, d$ and $K_0$. Similarly, for each $i=1\dots,d$, from the definition of $\varphi_f$, ~\eqref{eq:bdd-grad2-tildexi-varphi}, and~\eqref{eq:approx-nabla-partial-f-tildef-xi}, we obtain \begin{align*} \| \nabla_{(t,x)} \partial_{x_i} \varphi_f \|_{\cC^0( [0,T] \times \bar{B}_{d}(0,R); \RR^{d})} &= \left(\frac{\pi}{R'}\right)^2 \| \nabla_{(t,x)} \partial_{x_i} \varphi \|_{\cC^0([0,\pi T / R'] \times \bar{B}_{d}(0, \pi R/ R'); \RR^{d})} \\ &\le \left(\frac{\pi}{R'}\right)^2 \left( \|\nabla_{(t,x)} \partial_{x_i} \tilde \xi \|_{\cC^0([0,\pi T / R'] \times \bar{B}_{d}(0, \pi R/ R'); \RR^{d})} + c_2 (1 + (R')^3) n^{-1} \right) \\ &= \left(\frac{\pi}{R'}\right)^2 \left( \left(\frac{R'}{\pi}\right)^2 \|\nabla_{(t,x)} \partial_{x_i} \xi \|_{\cC^0([0, T] \times \bar{B}_{d}(0, R); \RR^{d})} + c_2 (1 + (R')^3) n^{-1} \right) \\ &\le c_5 \left(1 + R n_{\mathrm{in}}^{-1/(2(d+1))} \right), \end{align*} where $c_5$ depends only on $c_2$, $L_3$, $C_{mm}, d$ and $K_0$. This completes the proof. \end{proof} We then turn our attention to how the objective functional changes when evaluated on two controls that are similar on a compact set. More precisely, we show the following result. Since, under our assumptions, both the (approximately optimal) feedback control $\hat\bv$ and the control $\hat\boldsymbol{\varphi}$ produced by the neural network are Lipschitz continuous, we restrict our attention to feedback controls with this regularity. The following proposition provides a control of the variation of the objective function when approximating a feedback control on a compact set. \begin{proposition} \label{lem:var-J-feedback-compact} Let $\bv$ and $\bw \in \VV$ be two Lipschitz continuous feedback controls (i.e., Lipschitz continuous functions of $(t,x)$). There exists a constant $C$ depending only on the data of the problem and the Lipschitz constant of the controls $\bv$ and $\bw$ such that for all $\Gamma > 0$ and $R > 0$, if \begin{equation} \label{eq:varJN-bnd-bv-bw} \| \bv_{|[0,T] \times \bar{B}_{d}(0,R)} - \bw_{|[0,T] \times \bar{B}_{d}(0,R)} \|_{\cC^0([0,T] \times \bar{B}_{d}(0,R))} \leq \Gamma \, , \end{equation} then $$ \left | J^N(\bv) - J^N(\bw) \right | \leq C \left(\Gamma^2 + \frac{1}{R}\right)^{1/2} \, . $$ \end{proposition} The arguments required for the proof are rather standard. However, we could not find a close enough version of this result in the literature. So for the sake of completeness we provide the proof in Appendix~\ref{app:proof-approx-NN}. \begin{proof}[Proof of Proposition \ref{prop:approx-NN-cmpJ-epsilon}] Let $n_{\mathrm{in}}\ge 1$ be an integer, and let $R = n_{\mathrm{in}}^{1/3(d+1)}$. As explained in Subsection \ref{sub:assumptions} on our standing assumptions, the function $(t,x) \mapsto \hat{\bv}(t,x)$ defined by~\eqref{fo:vhat} is in the class $\cC(R, K, L_1, L_2, L_3)$ for some constants $K, L_1, L_2, L_3$ depending only on the data of the problem. So by Proposition~\ref{prop:approx-fct-NN} applied to $f = \hat{\bv}$, there exists a neural network $\hat{\boldsymbol{\varphi}} \in \bN^{\psi}_{d+1, n_{\mathrm{in}}, k}$, such that $$ \| \hat{\bv} - \hat{\boldsymbol{\varphi}} \|_{\cC^0([0,T] \times \bar{B}_{d}(0,R))} \le \Gamma, $$ with $\Gamma = \Gamma(R, n_{\mathrm{in}}) = C_0 (1 + R) n_{\mathrm{in}}^{-1/2(d+1)}$ for some $C_0$ depending only on the data of the problem and on $\hat\psi(1), \|\psi'\|_{\cC^0}, \|\psi''\|_{\cC^0}, \|\psi'''\|_{\cC^0}$. Moreover, still by Proposition~\ref{prop:approx-fct-NN}, the Lipschitz constants of $\hat\boldsymbol{\varphi}$ and $\partial_{x_i} \hat\boldsymbol{\varphi}$ can be bounded from above by $C_0 (1 + R n_{\mathrm{in}}^{-1/2(d+1)})$, which is itself bounded by $2 C_0$ thanks to our choice of $R$. Next, applying Proposition~\ref{lem:var-J-feedback-compact} yields $$ J^N(\hat{\bv}) \geq J^N(\hat{\boldsymbol{\varphi}}) - C_2 \left(\Gamma^2 + \frac{1}{R}\right)^{1/2} \,, $$ for some constant $C_2$ depending only on the data of the problem and $C_0$. Recalling that $R = n_{\mathrm{in}}^{1/3(d+1)}$, \begin{align*} \Gamma^2 + \frac{1}{R} &= C_0^2 \left(1 + n_{\mathrm{in}}^{1/3(d+1)} \right)^2 n_{\mathrm{in}}^{-1/(d+1)} + n_{\mathrm{in}}^{-1/3(d+1)} \\ &\le C_1 n_{\mathrm{in}}^{-1/3(d+1)}, \end{align*} where $C_1$ depends only on $C_0$. This completes the proof. \end{proof} \subsection{Problem in discrete time} \label{sec:approx-discretetime} We now prove that the objective values of the discrete and continuous time problems (respectively Problem~\ref{pb:discrete-MKV} and Problem~\ref{pb:MKV-Nagents} when minimizing over neural networks) are close. One possibility is to view the interacting particle dynamics as a system of SDEs in dimension $N \times d$ and apply known results about the Euler scheme for systems of SDEs. However such results usually involve constants which depend upon the dimension. This is unsatisfactory as we want the constants to be independent of $N$. Viewing the problem as the optimal control of a finite number of particles is reminiscent of the particle method for MKV equations studied by Bossy and Talay e.g.~\cite{MR1370849}. However their bound for the error induced by the time discretization (see~\cite[Lemma 2.7]{MR1370849}) is not applicable directly to our setting. For these reasons and for the sake of completeness, we provide in Appendix~\ref{app:proof-discreteT} a detailed proof, which relies on the following estimate for the strong error rate also, proved in the appendix under the assumptions and notations we now specify. $\varphi$ is a Lipschitz continuous feedback control function of $t$ and $x$, and $(X^i_t)_{i=1,\ldots,N,\;0\le t\le T}$ is the $N$-agent continuous-time state process satisfying the MKV dynamics given by equation \eqref{eq:evolXi-closedloop} in the statement of Problem \ref{pb:MKV-Nagents} with feedback function $\varphi$ instead of $v$. $N_T$ is an integer giving the number of time steps in the subdivision $t_n=nT/N_T$ for $n=0,\ldots,N_T$. We denote by $\Delta t=T/N_T$ the mesh of this subdivision. and $(\check X^i_{t_n})_{i=1,\ldots,N,\;n=0,\ldots,N_T}$ is the $N$-agent discrete-time state process satisfying the MKV dynamics given by equation \eqref{eq:discrete-dyn} in the statement of Problem \ref{pb:discrete-MKV} with the current Lipschitz feedback function $\varphi$ not necessarily given by a neural network. Under these conditions we have \begin{lemma}[Strong error for the Euler scheme] \label{lem:discreteT-strongerr} For all $n=0,\dots,N_T$, $$ \sum_{i=1}^N \EE |\check X^i_{t_n} - X^i_{t_n}|^2 \leq C N\Delta t $$ where $C$ depends only on the data of the problem, the Lipschitz constant of $\varphi$ and $\varphi(0,0)$. \end{lemma} Using this estimate, we can prove the following bound. \begin{proposition} \label{prop:approx-discreteT} Assume that, for every $(t,x) \in [0,T] \times \RR^d$, \begin{align} \label{eq:hyp-vaphi-Euler} |\partial_t\varphi(t,x)| \le C_1 |(t, x)|, \qquad |\partial_x\varphi(t,x)| \le C_1, \qquad |\partial^2_{xx} \varphi(t,x)| \le C_1. \end{align} There exists a constant $C$ depending only on the data of the problem and on the control $\varphi$ its Lipschitz constant, its value at $(0,0)$ and $C_1$ introduced above, such that for all $N$ and $t\in[0,T]$, we have $$ |J^N(\varphi) - \check J^N(\varphi)| \leq C \sqrt{\Delta t}. $$ \end{proposition} In particular, for the same $\hat\boldsymbol{\varphi}$ as the one constructed in Proposition~\ref{prop:approx-NN-cmpJ-epsilon}, the constant $C$ depends only on the data of the problem and on the activation function through $\hat\psi(1)$. \section{Two numerical methods} \label{sec:two-num-meth} In this section, we describe two numerical methods for mean field type problems. The first method is a \emph{straightfoward} implementation of the optimization problem underpinning the optimal control of MKV dynamics. The second method is geared to the solution of forward backward systems of stochastic differential equations of the McKean-Vlasov type (MKV FBSDEs). As a result, it can be applied to the solutions of both mean field control (MFC) problems and mean field games (MFGs). \subsection{Method 1: minimization of the MFC cost function} \label{sec:method-1} The goal is to minimize $\check J^N$ defined by~\eqref{eq:def-checkJN} by searching for the right function $\boldsymbol{\varphi}$ in a family of functions $x\mapsto \boldsymbol{\varphi}_\theta(x)$ parameterized by the parameter $\theta\in\Theta$, the desired parameter $\theta$ minimizing the functional: $$ \JJ^N(\theta) = \check J^N(\boldsymbol{\varphi}_\theta). $$ This cost function can be interpreted as a loss function and viewed as an expectation. Its minimization is \emph{screaming} for the use of the Robbins-Monro procedure. Moreover, if we use the family $(\varphi_\theta)_{\theta\in\Theta}$ given by a feed-forward neural network, this minimization can be implemented efficiently with the powerful tools based on the so-called Stochastic Gradient Descent (SGD) developed for the purpose of machine learning. Recall the notation $\bN^\psi_{d_0, \dots, d_{\ell+1}}$ introduced in \eqref{fo:N_psi} for a set of neural networks with a specific architecture. In the implementation of the minimization, we replace the expectation in~\eqref{eq:def-checkJN} by an empirical average over a finite number of sample populations each of size $N$. At each iteration $m = 1, 2, \dots,$ of the SGD, we pick $N$ samples $x^{i}_0,$ $i=1,\dots,N,$ i.i.d. with distribution $\mu_0$, and $N \times N_T$ samples $\epsilon^{i}_{n},$ $i=1,\dots,N, 0 = 1, \dots, N_T-1$, i.i.d. with distribution $\cN(0, \sqrt{\Delta t})$, and we denote by $\theta_m$ the parameters of the neural network at the $m$-th iteration. We then consider the following loss function which depends on the sample $S = ((x^{i}_0)_{i=1,\dots,N}, (\epsilon^{i}_n)_{i=1,\dots,N, n=0,\dots,N_T-1})$: \begin{equation} \label{eq:bbJ-S-thetam} \JJ_{S}(\theta_{m}) = \frac{1}{N}\sum_{i=1}^N \sum_{n=0}^{N_T-1} f\left(t_n, \check X^i_{t_n}, \check\mu_{t_n}, \varphi_{\theta_m}(t_n, \check X^i_{t_n})\right) \Delta t + g\left(\check X^i_{t_{N_T}},\check\mu_{t_{N_T}}\right) \, , \end{equation} where $\check\mu_{t_n} = \frac{1}{N} \sum_{i=1}^N \delta_{\check X^i_{t_n}}$, under the constraints: \begin{equation*} \check X^i_{t_{n+1}} = \check X^i_{t_n} + b\left(t_n, \check X^i_{t_n}, \check\mu_{t_n}, \varphi_{\theta_m}(t_n, \check X^i_{t_n})\right) \Delta t + \sigma\left(t_n, \check X^i_{t_n}, \check\mu_{t_n}\right) \epsilon^i_n \, , \end{equation*} for $n \in \{0,\dots, N_T-1\}, i \in \{1,\dots,N\}$, with $(\check X^i_0)_{i \in \{1, \dots, N\}} = (x^i_0)_{i \in \{1, \dots, N\}}$. \begin{algorithm} \DontPrintSemicolon \KwData{An initial parameter $\theta_0\in\Theta$. A sequence $(\alpha_m)_{m \geq 0}$ of learning rates.} \KwResult{Approximation of $\theta^*$} \Begin{ \For{$m = 0, 1, 2, \dots$}{ Pick $S = ((x^{i}_0)_{i=1,\dots,N}, (\epsilon^{i}_n)_{i=1,\dots,N, n=1,\dots,N_T})$ where $x^i_0$ are i.i.d. with distribution $\mu_0$, and $\epsilon^{i}_n$ are i.i.d. with distribution $\cN(0, \sqrt{\Delta t})$\; Compute the gradient $\nabla \JJ_{S}(\theta_{m})$ of $\JJ_{S}(\theta_{m})$ defined by~\eqref{eq:bbJ-S-thetam} \; If $|\nabla \JJ_{S}(\theta_{m})|$ is small enough, stop; otherwise continue \; Set $\theta_{m+1} = \alpha_m \theta_{m} + (1-\alpha_m) \nabla \JJ_{S}(\theta_{m})$ \; } \KwRet{ $\theta_{m}$} } \caption{SGD for Mean Field Control\label{algo:SGD-MFC}} \end{algorithm} Details on the implementation and numerical examples are given in Section \ref{sec:numres} \subsection{Method 2: MKV FBSDE} We now propose a method for a general system of forward-backward SDEs of McKean-Vlasov type (or MKV FBSDE for short). For simplicity, we restrict our attention to the case of a non-controlled volatility. Such a system then takes the following generic form \begin{equation} \label{eq:MKV-FBSDE-general} \left\{ \begin{aligned} d X_t = & B\left(t, X_t, \cL(X_t), Y_t \right) dt + \sigma\left(t, X_t, \cL(X_t) \right) d W_t, \\ d Y_t = & - F\left(t, X_t, \cL(X_t), Y_t, \sigma\left(t, X_t, \cL(X_t) \right)^\dagger Z_t \right) dt + Z_t d W_t, \end{aligned} \right. \end{equation} with initial condition $X_0 = \xi \in L^2(\Omega, \cF_0, \PP; \RR^d)$ and terminal condition $Y_T = G(X_T, \cL(X_T))$. The system~\eqref{eq:MKV-FBSDE} derived from the application of the Pontryagin stochastic maximum principle applied to our MKV control problem is an instance of the above general MKV FBSDE. As argued in \cite{MR3752669}, it turns out that the solution of a MFG can also be captured by a FBSDE system of the form given by~\eqref{eq:MKV-FBSDE-general}. The strategy of the method is to replace the backward equation forced on us by the optimization, by a forward equation and treat its initial condition, which is what we are looking for, as a control for a new optimization problem. This strategy has been successfully applied to problems in economic contract theory where it is known as Sannikov's trick. See for example \cite{MR2963805, MR3738664}. Translated into the present context, this strategy allows us to turn the search for a solution of~\eqref{eq:MKV-FBSDE-general} into the following optimization procedure. We let the controller choose the initial point and the volatility of the $Y$ process, and penalize it proportionally to how far it is from matching the terminal condition. More precisely, the problem is defined as: Minimize over $y_0 : \RR^d \to \RR^d, z: \RR_+ \times \RR^d \to \RR^{d \times d}$ the cost functional $$ J_{FBSDE}(y_0, z) = \EE \left[ \, \left| Y^{y_0,z}_T - G(X^{y_0,z}_T, \cL(X^{y_0,z}_T)) \right|^2 \, \right] $$ where $(X^{y_0,z}, Y^{y_0,z})$ solve \begin{equation} \label{eq:MKV-FBSDE-2forward} \left\{ \begin{aligned} d X^{y_0,z}_t = & B\left(t, X^{y_0,z}_t, \cL(X^{y_0,z}_t), Y^{y_0,z}_t \right) dt + \sigma\left(t, X^{y_0,z}_t, \cL(X^{y_0,z}_t) \right) d W_t, \\ d Y^{y_0,z}_t = & - F\left(t, X^{y_0,z}_t, \cL(X^{y_0,z}_t), Y^{y_0,z}_t, \sigma\left(t, X^{y_0,z}_t, \cL(X^{y_0,z}_t) \right)^\dagger z(t, X^{y_0,z}_t) \right) dt + z(t, X^{y_0,z}_t) d W_t, \end{aligned} \right. \end{equation} with initial condition $$ X^{y_0,z}_0 = \xi \in L^2(\Omega, \cF_0, \PP; \RR^d), \qquad Y^{y_0,z}_0 = y_0 (X_0). $$ Note that the above problem is an optimal control problem of MKV dynamics, if we view $(X^{y_0,z}_t,Y^{y_0,z}_t)$ as the state and $(y_0,z)$ as the control. It is of a peculiar form, since the control is the initial value and the volatility of the second component of the state as functions of the first component of the state. Under suitable conditions, the optimally controlled process $(X_t,Y_t)_t$ solves the FBSDE system~\eqref{eq:MKV-FBSDE-general} and vice-versa. Due to its special structure, the above MFC problem does not fit in the framework we have analyzed in Section~\ref{sec:approx-result}. However, for numerical purposes, we can still apply the first method described above, see \S~\ref{sec:method-1}: we consider a finite-size population, replace the the controls (namely, $y_0$ and $z$) by neural networks, discretize time, and then use SGD to perform the optimization. In Section \ref{sec:numres} below, we illustrate the performance of this method on MKV FBSDEs coming from MFGs, game models for which we cannot use the first method, see Section~\ref{sec:numres}. \section{Numerical results} \label{sec:numres} In this section, we provide numerical results obtained using implementations of the two numerical methods proposed above. Algorithm 1 refers to the first method, based on direct minimization of the cost function for a mean field control problem; Algorithm 2 refers to the second method, which solves a control problem encoding the solution to an FBSDE system of MKV type. Some of the test cases (see in particular test cases 5 and 6 below) do not fall in the scope of the theory presented in the previous sections. However the numerical results show that the algorithms are robust enough and work well even in these cases. As a benchmark, we shall use either the solution provided by an analytical formula, or the solution computed by a deterministic method based on finite differences for the corresponding PDE system, see e.g.~\cite{MR2679575}. We recall here the PDE system for the sake of completeness and for future reference. For simplicity, in the numerical examples considered below, we take $\sigma$ to be constant and we assume that the initial distribution $\mu_0$ has a density denoted by $m_0$. In \cite{MR3134900}, A. Bensoussan, J. Frehse and P. Yam have proved that a necessary condition for the existence of a smooth feedback function $v^* $ achieving $J( v^*)= \min J(v)$ is that \begin{displaymath} v^*(t,x)={\rm{argmin}}_{v} \Bigl( f(x, m(t,\cdot), v)+ \nabla u(t,x)\cdot b(x,m(t,\cdot),v)\Bigr), \end{displaymath} where $J$ is defined by~\eqref{eq:MFC1-cost} (with a slight abuse of notation we view $J$ as a function of a feedback control) and where $(m,u)$ solve the following system of partial differential equations \begin{equation} \label{eq:MFC-PDEsystem} \left\{ \begin{split} 0&= \frac{\partial u} {\partial t} (t,x) + \frac{\sigma^2}{2} \Delta u(t,x) + \tilde H( x, m(t,\cdot), \nabla u(t,x)) + \int \frac{\partial \tilde H} {\partial m}(\xi, m(t,\cdot), \nabla u(t, \xi))(x) m(t,\xi) d\xi , \\ 0&= \frac{\partial m} {\partial t} (t,x) - \frac{\sigma^2}{2} \Delta m(t,x) + \mathrm{div}\Bigl( m(t,\cdot) \frac{\partial \tilde H} {\partial p} (\cdot,m(t,\cdot),\nabla u(t,\cdot))\Bigr)(x) , \end{split} \right. \end{equation} where $\tilde H$ is defined by~\eqref{eq:def-Hamiltonian-red}, with the initial and terminal conditions \begin{equation*} m(0,x)=m_0(x)\quad \hbox{and}\quad u(T,x) = g (x, m(T,\cdot)) + \int \frac{\partial g} {\partial m} (\xi, m(T,\cdot))(x) m(T,\xi) d\xi. \end{equation*} Here $\frac{\partial}{\partial m}$ denotes a Fr\'echet derivative. This PDE system has been analyzed and solved numerically e.g. in~\cite{MR3392611} and we use a similar numerical method below to compute our benchmark solutions. \vskip 12pt \noindent \textbf{Test case 1:} As a first testbed, let us consider a linear-quadratic mean field control problem, that is, the dynamics is linear and the cost quadratic in the state, the control and the expectation of the state. For simplicity, we consider a one-dimensional problem. To wit, recalling the notations~\eqref{eq:MFC1-cost}--\eqref{eq:MFC1-dyn}, we take \begin{align*} &b(x, \mu, v) = Ax + \bar A \bar \mu + B v, \\ &f(x, \mu, v) = Q x^2 + \bar Q (\bar \mu - Sx)^2 + R v^2, \\ &g(x, \mu) = Q_T x^2 + \bar Q_T(\bar \mu - S_Tx)^2 \end{align*} for $x, v\in \RR, \mu \in \cP_2(\RR), \bar \mu = \int \xi \mu(d\xi) \in \RR$, where $A,\bar A, B,$ $Q, \bar Q, R, S,$ $Q_T, \bar Q_T, S_T$ are real numbers. Numerical results obtained with Algorithm 1 are presented in Figure~\ref{fig:ex-lq-benchmark}. We recall that in this method, the cost function is also the loss function minimized by the SGD. The value of this loss w.r.t. the number of iterations is displayed in Figure~\ref{fig:ex-lq-benchmark-cost}, for various various number time steps (denoted by $Nt$), number of samples in one population (denoted by $Ns$), and two different activation functions (ReLU or sigmoid). In this simple mean field LQ model, one has access to an analytical solution via Riccati equations. This allows us to compare the learned control with the actual optimal control. As shown in Figure~\ref{fig:ex-lq-benchmark-errcontrol}, the $L^2$ distance between them decreases as the number of iterations of SGD increases. As expected, increasing the number of time steps and the number of samples in a population improves the result. Moreover, even though the optimal control is linear in the state and hence the ReLU activation function seems to be better suited for this problem, using the sigmoid activation function seems to provide comparable or even slightly better results. For each line, the results have been obtained by averaging over $5$ different runs of SGD. \begin{figure} \centering \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-MFC_iter20000_NEW_lq-mfc_control_loss-eps-converted-to}.pdf} \caption{Cost for learned control} \label{fig:ex-lq-benchmark-cost} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-MFC_iter20000_NEW_lq-mfc_control_l2err-eps-converted-to}.pdf} \caption{$L^2$ error on the control} \label{fig:ex-lq-benchmark-errcontrol} \end{subfigure} \caption{Test case 1. Solution computed by Algorithm 1.} \label{fig:ex-lq-benchmark} \end{figure} \vskip 12pt \noindent \textbf{Test case 2:} Our second example is inspired by the min-LQG problem of~\cite{salhab2015a}. We consider, as in the first test case, a linear dynamics and a quadratic running cost, but we modify the terminal cost to encourage each agent to end up close to one of two targets at the final time. More precisely, considering again a one-dimensional case, with $\xi_1, \xi_2 \in \RR$, we take \begin{align*} &g(x,m) = g(x) = \min \left\{ |x - \xi_1 |, | x - \xi_2 | \right\} . \end{align*} Notice that $g$ is differentiable everywhere except at the point $\frac{1}{2}(\xi_1 + \xi_2)$, and hence the gradient of $g$ is discontinuous. Numerical results obtained using Algorithm 2 are presented in Figure~\ref{fig:ex-minlq-benchmark} for an example where the two targets are $\xi_1 = 0.25$, $\xi_2 = 0.75$, and the initial distribution is Gaussian with mean $1$. We recall that with this method, we learn $(y_0(\cdot), z(\cdot))$, and then $(Y_{t_n})_{n > 0}$ can be obtained by Monte-Carlo simulations, which yield the blue points. To assess the accuracy of our method, we compare the result with the solution obtained by a deterministic PDE method based on a finite difference scheme (red line). From the perspective of the coupled system~\eqref{eq:MFC-PDEsystem} of Hamilton-Jacobi-Bellman and Fokker Planck (HJB-FP system for short) characterizing the optimal solution, $Y_t$ correspond to the gradient $\partial_x u(t, X_t)$ of the adjoint function $u$, solution to the HJB equation. We see respectively on Figure~\ref{fig:ex-minlq-benchmark-cost-Y-t0} and Figure~\ref{fig:ex-minlq-benchmark-cost-Y-tT} that $Y_0$ is better approximated than $Y_T$, which is consistent with the fact that $Y_0$ is directly learned under the form $y_0(X_0)$ where $y_0$ is a neural network, whereas $Y_T$ accumulates the errors made on all intermediate time steps until the terminal time. Note however that the algorithm manages to learn the jump at $x=\frac{1}{2}(\xi_1+\xi_2) = 1.0$. \begin{figure} \centering \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_MINLQG_Nt200_lq-mfg_decoupling_ONLY_Nt200_t0-eps-converted-to}.pdf} \caption{$Y_t$ at time $t=0$} \label{fig:ex-minlq-benchmark-cost-Y-t0} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_MINLQG_Nt200_lq-mfg_decoupling_ONLY_Nt200_t20-eps-converted-to}.pdf} \caption{$Y_t$ at time $t=T = 0.2$} \label{fig:ex-minlq-benchmark-cost-Y-tT} \end{subfigure} \caption{Test case 2. Solution computed by Algorithm 2.} \label{fig:ex-minlq-benchmark} \end{figure} \vskip 12pt \noindent \textbf{Test case 3:} For the sake of comparison with the existing literature, we now turn our attention to examples considered in~\cite{MR3914553}. We first consider the problem presented in their section 4.2.2, which consists in solving the FBSDE (without MKV dynamics) \begin{align*} &d X_t = \rho \cos(Y_t) dt + \sigma dW_t, \qquad X_0 = x_0, \\ & Y_t = \EE_t[\sin(X_T)]. \end{align*} Here, $\rho$ is seen as a coupling parameter and, in particular, for $\rho=0$, the forward equation does not depend on the backward component so the equations are decoupled. The system is solved for various values of $\rho$ and for each value of $\rho$, we look at the value of the backward process $Y$ at time $0$. The existence and uniqueness of a solution is discussed in~\cite{MR3914553}, where the authors also point out that the numerical method they propose suffers from a bifurcation phenomenon for large values of $\rho$. Results obtained by our Algorithm 2 are presented in Figure~\ref{fig:ex-CCD72-Y0}. For the sake of comparison, we also display the values obtained using a deterministic method based on a finite difference scheme for the corresponding PDE system~\eqref{eq:MFC-PDEsystem}. In this example, the value of $Y_0$ corresponds to the value of $\partial_x u(0, x_0)$. These results are to be compared with~\cite[Figure 2]{MR3914553}. \begin{figure} \centering \includegraphics[width=0.5\linewidth]{{FIG_noMKV_eq72_XT_solverZ_res-y0-rho_NEW-eps-converted-to}.pdf} \caption{Test case 3. Value of $Y_0$ as a function of the parameter $\rho$, computed by Algorithm 2 (in blue) and benchmark values via PDE method (in red).} \label{fig:ex-CCD72-Y0} \end{figure} \vskip 12pt \noindent \textbf{Test case 4:} We next turn our attention to a FBSDE system arising from a MFG. Let us recall the example considered in~\cite[Section 4.2.3]{MR3914553}. The problem consists in solving the MFG in which the dynamics of a typical agent is $d X^\alpha_t = \alpha_t dt + d W_t$ and her cost is $$ J^{MFG}(\mu, \alpha) = \EE\left[ g(X^\alpha_T) + \int_0^T \left(\frac{1}{2\rho} \alpha^2_t + X^\alpha_t \mathrm{atan}\left(\int x \mu_t(dx)\right) \right) dt \right] $$ when she uses control $\alpha$ and the flow of distribution of the population is given by $\mu = (\mu_t)_t$, and where the terminal cost $g$ is such that $g'(x) = \arctan(x)$. As in the previous test case, $\rho >0$ is a parameter. Since this is a MFG, Algorithm 1 can not be employed so we resort to Algorithm 2 in order to solve the associated MKV FBSDE system, namely: \begin{align*} &d X_t = - \rho Y_t dt + \sigma dW_t, \qquad X_0 = x_0, \\ & d Y_t = \arctan(\EE[X_t]) dt + Z_t dW_t, \qquad Y_T = g'(X_T) := \arctan(X_T). \end{align*} Here again, we solve the problem for various values of $\rho$ and, for each of them, we compute the value of $Y_0$. The numerical results are displayed in Figure~\ref{fig:ex-CCD73-Y0}. We see that the solution is very close to the benchmark, obtained by a deterministic method for the corresponding PDE system. In particular, a noticeable difference with the results of~\cite[Figure 3]{MR3914553} is that our method does not suffer from a bifurcation phenomenon. \begin{figure} \centering \includegraphics[width=0.5\linewidth]{{FIG_solverZ_empirical_NoBN_sigma0_larger_corrB_res-y0-rho_NEW-eps-converted-to}.pdf} \caption{Test case 4. Value of $Y_0$ as a function of the parameter $\rho$, computed by Algorithm 2 (in blue) and benchmark values via PDE method (in red).} \label{fig:ex-CCD73-Y0} \end{figure} \vskip 12pt \noindent \textbf{Test case 5:} We present an example of mean field control problem with a simple source of common noise. For simplicity, we consider the one-dimensional case. The idea is to consider a terminal cost which penalizes the distance to one of two targets, the relevant target being revealed only at time $T/2$. The model is inspired by~\cite{YAJML}, where a more complex model has been studied for crowd motion. Here, we represent the common noise by a random process $\epsilon^0 = (\epsilon^0_t)_{t \in [0,T]}$ which takes value $0$ for $t \in [0, T/2)$ and at time $T/2$ it jumps to value $-c_T$ or $+c_T$, each with probability $1/2$, and then it keeps this value until time $T$. Here $c_T>0$ is a constant and the two targets are $\pm c_T$. For simplicity, we restrict our attention to controls which are feedback functions of $t, X_t$ and $\epsilon^0_t$. The problem is: Minimize over $v: [0,T] \times \RR \times \{-c_T,0,c_T\} \to \RR$ the quantity \begin{equation*} J(v) = \EE\left[\int_0^T |v(t, X_t, \epsilon^0_t)|^2 dt + (X_T - \epsilon^0_T)^2 + K_T(X_T - \EE[ X_T | \epsilon^0])^2 \right] \, , \end{equation*} under the constraint that the process $\bX = (X_t)_{t \ge 0}$ solves the SDE \begin{equation*} dX_t = v(t, X_t, \epsilon^0_t) dt + \sigma dW_t \, , t \ge 0, \quad X_0 \sim \mu_0. \end{equation*} Here $K_T$ is a positive constant parameterizing the weight of the cost to pay for being far from the mean at time $T$. Note that the dynamics and the cost functions are assumed to depend on the \emph{conditional} distribution of $X_t$ given the common noise $\epsilon^0$. The solution can be characterized through a system of $6$ PDEs (three coupled systems of HJB and FP equations). We use this to have a benchmark, and compare with the result obtained by Algorithm 1 suitably generalized to handle the common noise. More precisely, at each iteration of SGD, we sample the initial positions of $N$ agents, the $N$ idiosyncratic noises appearing in the dynamics, as well as one path of the common noise. The neural network representing the control function takes as inputs the current time, the current state and the current value of the common noise. Figure~\ref{fig:ex-mfc-commonnoise} displays the evolution of the density in the scenario where $\epsilon^0_T = -c_T$ (in blue) and in the scenario where it is $\epsilon^0_T = c_T$ (in red), with $c_T = 1.5$ and the initial distribution is centered around $0$. The solution obtained by the PDE method is shown with lines, whereas the solution obtained by Algorithm~1 is shown with a histogram obtained by Monte Carlo simulations. We see that, until time $T/2$, the distribution concentrates around $0$, because the agents do not know the final target but they know that the final cost will penalize distance to the mean position. After time $T/2$, the valid target is revealed and the distribution moves toward this target. Note however that due to the running cost, which penalizes displacement, the distribution does not reach exactly the target. \begin{figure} \centering \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-CN_test10i_density_overlay_t0}.pdf} \caption*{$t=0$} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-CN_test10i_density_overlay_t2}.pdf} \caption*{$t=0.2$} \end{subfigure}% \\ \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-CN_test10i_density_overlay_t4}.pdf} \caption*{$t=0.5$} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-CN_test10i_density_overlay_t6}.pdf} \caption*{$t=0.6$} \end{subfigure}% \\ \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-CN_test10i_density_overlay_t8}.pdf} \caption*{$t=0.8$} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_LQ-CN_test10i_density_overlay_t10}.pdf} \caption*{$t=1.0$} \end{subfigure}% \caption{Test case 5. Distribution computed by Algorithm 1 (histograms) and by deterministic method for the PDE system (full and dashed lines). Blue (resp. red) corresponds to the scenario where $\epsilon^0$ takes the value $-1.5$ (resp. $+1.5$) at time $T/2 = 0.5$.} \label{fig:ex-mfc-commonnoise} \end{figure} \vskip 12pt \noindent \textbf{Test case 6:} Here we consider the example of mean field game for systemic risk introduced in~\cite{MR3325083}. This model involves common noise in the form of a Brownian motion $\bW^0 = (W^0_t)_{t \ge 0}$. Given a flow of conditional mean positions $\bar {\mathbf m} = (\bar m_t)_{t \in [0,T]}$ adapted to the filtration generated by $\bW^0$, the cost function of a representative player is $$ J^{MFG}(\bar m, \alpha) = \EE\left[\int_0^T \left( \frac{1}{2}\alpha_t^2 - q \alpha_t (\bar m_t - X_t) + \frac{\epsilon}{2} (\bar m_t - X_t)^2 \right) dt + \frac{c}{2} (\bar m_T - X_T)^2\right] $$ and the dynamics are $$ dX_t = [a (\bar m_t - X_t) + \alpha_t] dt + \sigma \left( \rho \, dW^0_t + \sqrt{1 - \rho^2} dW_t\right) $$ Here $\rho \in [0, 1]$ is a constant parameterizing the correlation between the noises, and $q, \epsilon, c, a, \sigma$ are positive constants. We assume that $q \le \epsilon^2$ so that the running cost is jointly convex in the state and the control variables. For the interpretation of this model in terms of systemic risk, the reader is referred to~\cite{MR3325083}. The model is of linear-quadratic type and hence has an explicit solution through a Riccati equation, which we use as a benchmark. Since this example is a mean field game, we used Algorithm 2 to solve the appropriate FBSDE system. Figure~\ref{fig:ex-mfg-sysrisk-traj} displays, for one realization of the common noise, sample trajectories of $X^i$ and $Y^i$ for two different values of $i$, namely $i=1,2$. The $L^2$ error, averaged over $5$ different runs of SGD, is shown in Figure~\ref{fig:ex-mfg-sysrisk-error}. Clearly, the error decreases as the number of time steps and the number of elements in the population increases. \begin{figure} \centering \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_DEEPSOLVER-FBSDE_SYSRISK_test1o_cmp_trajX-eps-converted-to}.pdf} \caption*{Trajectory of $X^i, i=1,2$} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_DEEPSOLVER-FBSDE_SYSRISK_test1o_cmp_trajY-eps-converted-to}.pdf} \caption*{Trajectory of $Y^i, i=1,2$} \end{subfigure}% \caption{Test case 6. Sample trajectories: solution computed by Algorithm 2 (full lines, in cyan and blue) and by analytical formula (dashed lines, in orange and red).} \label{fig:ex-mfg-sysrisk-traj} \end{figure} \begin{figure} \centering \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_DEEPSOLVER-FBSDE_SYSRISK_errL2_X_cmpNt}.pdf} \caption*{$L^2$ error on $X$} \end{subfigure}% \begin{subfigure}{.45\textwidth} \centering \includegraphics[width=\linewidth]{{FIG_DEEPSOLVER-FBSDE_SYSRISK_errL2_Y_cmpNt}.pdf} \caption*{$L^2$ error on $Y$} \end{subfigure}% \caption{Test case 6. $L^2$ error for the solution computed by Algorithm 2 compared with the benchmark solution. The number of time steps is denoted by Nt and the number of elements in the population sampled at each iteration of SGD is denoted by Ns.} \label{fig:ex-mfg-sysrisk-error} \end{figure} \bibliographystyle{plain} \small

BkiUfa425V5jayWgFLSe

\section*{ORCID information} \section{Introduction} \label{key} The thriving field of reticular chemistry provides many options for framework materials with high porosity \cite{Xu.2020}. Various metal-organic frameworks (MOFs) \cite{Yaghi:1995gp,yaghi2003reticular,Zhou:2014fc}, covalent-organic frameworks (COFs) \cite{Cote:2005bf,ding2013covalent} and porous organic frameworks (POFs) \cite{Budd:2004cz, Jiang:2007jt} have been synthesised with a range of pore sizes and structures. These structures show great potential for use in gas adsorption \cite{Pei:2015hv, han2008covalent}, separation \cite{JianRongLi:2012kw} and catalyst carriers \cite{Rogge:2017ds}. That said, the major challenge for commercial applications are the issues of physical and chemical stability \cite{das2017porous}. One route to stability is to focus on tetrahedral networks. Elemental carbon and silicon form stable crystalline and amorphous tetrahedral networks, and similar crystalline and amorphous tetrahedral networks are found in silica (including the diamond-like $\beta$-cristobalite form \cite{Schmahl:1992vp,Tucker:2001vg}). Other tetrahedral crystalline networks include the negative thermal expansion materials zinc cyanide \cite{Goodwin:2005ea}, Cu$_2$O \cite{Schafer:2002ec,Artioli:2006fs} and silicon dicarbodiimide \cite{Li:2020hc}, and the zeolitic imidazolate frameworks (ZIFs) \cite{Park:2006gh}. Amorphous tetrahedral networks can form what are called ``continuous random networks'' (CRNs) \cite{Zachariasen:1932eo,Steinhardt:1974ba,Tu:1998dh,Barkema:2000cs}, which are effectively infinite in extent and in principle may be free of defects with dangling or non-bridging bonds. Amorphous silicon \cite{Laaziri:1999ia,Barkema:2000cs,Cliffe:2017cd} and silica \cite{Wright:1994ck,Tucker:2005hn} are two well-known examples, but others include a wide range of other amorphous aluminosilicates \cite{Wright:2020bq} and amorphous ZIF \cite{bennett2010structure}. In this paper we focus on a recently-discovered new amorphous material, carbon di-4,4$^\prime$-biphenyl, which has been designated as a porous amorphous framework with the name PAF-1 \cite{Ben:2009ie,Pei:2015hv}. PAF-1 has an ultra-high surface area (BET surface area 5600 m$^{2}$g$^{-1}$) with uniform pore size, and excellent physicochemical stability \cite{ben2013porous}. It too may have a tetrahedral network, but its atomic structure has not previously been modelled. In this paper we address the question of the atomic structure of PAF-1 through a combination of neutron total scattering and molecular dynamics simulations. We conclude that the evidence from these two approaches is fully consistent with the atomic structure of PAF-1 being a CRN with tetrahedral C sites connected by the biphenyl moieties. That is, the structure is directly analogous to those of amorphous silicon, silica and zinc imidazolate. This is consistent with a previous suggestion \cite{trewin2010porous,structuraldisorder2013andrew} based on porosity measurements \cite{Thomas:2014if}, but is now confirmed directly by diffraction measurements and characterised through the MD simulations. A special point of interest is that in the total scattering spectrum we see the same first sharp diffraction peak as seen in amorphous silica, but shifted to lower scattering vector, consistent with the larger length scale associated with distance between tetrahedral sites. This may open up a wider understanding of tetrahedral networks in general. \section{Experimental details} \subsection{Sample synthesis} Terkis(4-bromophenyl)methane (H-TBM) and deuterated terkis(4-bromophenyl)methane-d16 (D-TBM) were prepared (Supporting Information, Figures S1--S2). For preparation of hydrogenous PAF-1 and nominally deuterated PAF-1, here denoted as H-PAF-1 and D-PAF-1 respectively, a nickel(0)-catalysed Yamamoto-type Ullmann coupling reaction was used with H-TBM and D-TBM as monomers respectively. The nominally matched-contrast material containing both hydrogen and deuterium, denoted as H/D-PAF-1, was prepared with mixture of H-TBM and D-TBM in mole ratio of 1.7834:1. This ratio target was chosen to give a mean neutron coherent scattering length for the deuterium/hydrogen species of zero. In this case the H/D sites would be invisible to neutrons, and only carbon atoms would be visible in the neutron total scattering experiments. As we see below, the characterisation shows that the target compositions were not achieved. Nevertheless, the use of three samples with different H:D ratios gives a contrast in the total scattering experiments against which the models of the atomic structure can be judged. Full details of sample preparation are given in the Supporting Information. \subsection{Materials characterisation} The structure of the monomers were examined by $^{13}$C NMR, $^{1}$H NMR and mass spectroscopy studies (Supporting Information, Figures S8--S10). Complete (H-PAF-1) or nearly-complete (D-PAF-1, H/D-PAF-1) disappearance of the peak in the FT-IR spectra (Supporting Information Figure S11) corresponding to stretching of the C--Br bond confirmed the efficient polymerisation. The lack of long-range crystallographic order in the product structures was confirmed by measuring the powder x-ray diffraction pattern prior to the neutron total scattering measurements. X-ray powder diffraction data are shown in the Supporting Information, Figure S12. It had previously been shown that H-PAF-1 is stable in air \cite{Ben:2009ie}. Thermogravimetric analysis showed that both D-PAF-1 and H/D-PAF-1 show good thermal stabilities on heating to more than 400 $^\circ$C in air (Supporting Information Figure S13). Low pressure N$_{2}$ sorption measurement at 77 K shows that H-PAF-1, D-PAF-1 and H/D-PAF-1 each show exceptionally high N$_{2}$ uptake and typically microporous characterisation. Calculated BET surface areas and pore size distributions are similar for all three samples, and their values indicate nearly complete polymerisation of the monomers. Data are presented in the Supporting Information (Figure S14). Although the synthesis had aimed at one sample being nearly 100\% deuterated, and another having a balance of H and D to give zero coherent scattering from the H/D sites, some exchange of H for D is always highly likely during the polymerisation and purification procedures. Mass spectrometer measurements were used to obtain the accurate H/D ratio. The results showed that D content in nominally D-PAF-1 and H/D-PAF-1 were $70.2 \pm 0.3$\% and $21.2 \pm 0.4$\%, respectively. The D content in H-PAF-1 was detected to be natural abundance. In the synthesis of the target mixed H/D composition, the starting materials had either fully-deuterated or fully-hydrogenated phenyl rings, so that the distribution of H/D atoms is not completely random. It is assumed that the subsequent exchange of deuterium for hydrogen atoms as discussed above occurred at random on the deuterated phenyl rings. In principle the correlation between H isotopes on the same phenyl ring only affects the conversion of the simulated pair distribution functions (Section \ref{sec:PDFs}) to the scattering functions (Section \ref{sec:scatteringfunctions}), and was fully taken into account in the data presented below. However, in practice it made almost no difference because of the dominance of the correlations between pairs of carbon atoms. \subsection{Neutron scattering experiments} \subsubsection{Total scattering experiments at ISIS} Neutron total scattering experiments were performed on the NIMROD diffractometer \cite{Bowron:2010cs} at the ISIS spallation neutron source. NIMROD is designed to perform measurements of total scattering in the forward scattering direction, thereby minimising the effects of inelastic scattering in data corrections associated with light atoms. For measurements at room temperature (300 K) the samples were loaded into a flat cell of thickness 4 mm with thin walls of vanadium. For low-temperature measurements, the samples were loaded into thin-walled cylindrical vanadium cans of diameter 6 mm, and mounted within a closed-cycle refrigerator (CCR). Measurements were performed at temperatures of 10, 150 K, counting for 6 hours of beam time with a proton current of 40 $\mu$A. In addition to the measurements on the samples, additional measurements were performed of the empty instrument, the empty CCR, and the empty sample can inside the CCR for data correction, and of a vanadium rod for normalisation. The data were processed and corrected using the GUDRUN tool \cite{Soper:2012vs}, including converting between scattering angles and neutron flight time to scattering vector $Q$ -- where $\hbar \mathbf{Q}$ is the change in momentum of the scattered neutron, and $Q = | \mathbf{Q} |$ -- and placing the measured scattering function $S(Q)$ onto an absolute scale. Inelasticity effects were accounted for using an iterative method developed at ISIS \cite{Soper.2013}. As will be explained below, we did not convert the total scattering data to the pair distribution function. \subsubsection{Small angle scattering experiments at CSNS} Further experiments to obtain measurements of the small angle scattering down to lower values of $Q$ were performed on the Small Angle Neutron Scattering (SANS) instrument at China Spallation Neutron Source (CSNS) \cite{Ke:2018gs}. The sample was mounted in a standard rectangular sample container made of thin silica glass. The incident neutron beam had wavelengths in the range of 1--10 \AA\ defined by a double-disc bandwidth chopper, and was collimated to the sample by a pair of apertures. Each measurement took two hours. Data were corrected for background scattering, sources of beam attenuation, and detector efficiencies. The final data set extended the NIMROD data down to $Q = 0.01$ \AA$^{-1}$. \subsection{Building the random network}\label{sec:PAFmaker} Here we describe the method we used to build atomic models of PAF-1. To start we created a CRN of points with tetrahedral coordination to neighbour pooints, created using the WWW method \cite{Wooten:1985fp} starting from a random arrangement of points \cite{Barkema:2000cs}. The task of decorating this arrangement of points was tackled using software written by one of the authors (MTD; available on request). The network was expanded to give an appropriate distance between the neighbouring points, and each connection between neighbouring network points was decorated with a planar 4,4$^\prime$-biphenyl moiety. A Monte Carlo procedure was used to rotate the biphenyl moieties about their long axes in order to maximise the distances between the closest H atoms of neighbouring moieties. After this, a simple relaxation procedure was used to continue to maximise the distances between the closest H atoms, allowing individual phenyl rings to rotate about the long axis of the biphenyl moiety. The final configuration was then relaxed further using molecular dynamics simulations, as described below. \begin{figure*}[t] \includegraphics[width=0.35\textwidth]{amorphous_PAF.png} \hspace{1cm} \includegraphics[width=0.35\textwidth]{amorphous_PAF_tubes.png} \caption{Atomic structure of amorphous PAF-1 from our model as relaxed by MD simulation at a temperature of 300 K, shown as atoms on the left and as highlighting the porosity on the right, calculated as a solvent-accessible surface using a test sphere of radius 1.4 \AA. The images were constructed using the CrystalMaker visualisation tool \cite{Palmer:2015hja}.} \label{fig:structure} \end{figure*} The configuration used in this work contains 10496 atoms (256 tetrahedral sites). One snapshot of the configuration after being relaxed by the simulation is shown in Figure \ref{fig:structure}. This shows two views, one with the atoms and bonds, and the other showing the pores in the structure. From the simulation at a temperature of 300 K, the mean configuration edge length is 72.80~\AA, giving a number density of 0.663 formula units per nm$^3$. \subsection{Molecular dynamics simulations} Classical molecular dynamics (MD) simulations using force fields were performed using the software DL\_POLY version 4.08 \cite{Todorov:2006ee}. The phenyl rings were treated as rigid bodies. All atomic charges were set equal to zero; whilst this is clearly an approximation, it is less important for a covalent network material than it would be for a crystal containing discrete molecules. Interatomic potentials were taken from three sources: the MM3 potential for internal forces \cite{Allinger:1989em}, the Williams potential for distance interactions \cite{Williams:2001tp}, and taking a potential for the biphenyl torsion angle from recent ab initio calculations \cite{Johansson:2008fz}. Key equations and parameter values are given in the SI. The MD simulations were performed initially in the constant-temperature \cite{Nose:1984bf, Hoover:1985cu} constant-stress \cite{Parrinello:1980kx, Melchionna:2006fi} ensemble, using a time step of 0.001 ps. The configurations were equilibrated for a simulated time of 20 ps and then run for a simulated time of 30 ps to obtain data for analysis (note that we are only interested in fast dynamics, with 30 ps giving a frequency-resolution of 0.033 THz) with configurations saved periodically for analysis. For a study of the atomic dynamics the simulations were perform in the constant-energy constant-volume ensemble. \section{From atomic structure to the neutron total scattering functions} \begin{figure*}[!t] \begin{center} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{partialD_lowr.pdf} \subcaption{Partial functions, low $r$} \label{fig:partialD1} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{partialD.pdf} \subcaption{Partial functions, wide $r$} \label{fig:partialD2} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{combinedD.pdf} \subcaption{Combined functions} \label{fig:Dcombined} \end{subfigure} \caption{\label{fig:Dcomparison} a,b) Comparison of the partial PDF functions $D_\mathrm{norm}(r)$ for each element pair, defined here as $r\left( g_{mn}(r)-1 \right)$, seen for the lower-$r$ range of data (a) and across the full range of distances (b); c) Comparison of the neutron PDF functions $D(r)$ for each sample, formed from equation \ref{eq:Dr}. The PDFs were calculated from configurations obtained from the MD simulations performed at a temperature of 300 K.} \end{center} \end{figure*} \subsection{Pair distribution functions} \label{sec:PDFs} From the MD configurations, as shown in Figure \ref{fig:structure}, a set of partial pair distribution functions (partial PDFs) $g_{mn}(r)$ were formed, where $m$ and $n$ represent two atomic species. We used the standard definition of $g_{mn}(r)$, with limiting values $g_{mn}(r \rightarrow 0) = 0$ and $g_{mn}(r \rightarrow \infty) = 1$. The definition is discussed in more detail in the SI (section S5). For reasons that will become clear later, the $g_{mn}(r)$ functions are converted here for presentation to new functions of the form $d_{mn}(r) = r\left( g_{mn}(r)-1 \right)$. These are shown for the different atom pairs over two distance ranges in Figures \ref{fig:partialD1} and \ref{fig:partialD2}. The peaks for atomic pairs within the rigid phenyl rings were calculated as Dirac $\delta$ functions, and for presentation and subsequent analysis they were convolved with Gaussian functions to reflect thermal broadening. For distances above 10 \AA, all partial PDFs show a slow oscillation with wavelength of around 14 \AA. These oscillations are more-or-less equal for each atom pair. We will discuss these oscillations in some detail later at various points in this paper (Section \ref{ref:SANS}, Section \ref{sec:cubic} and, particularly, Section \ref{sec:FSDP}). To provide a direct connection with the experimental neutron scattering data, the partial $d_{mn}(r)$ functions are combined to give \begin{equation} \label{eq:Dr} D(r) = 4\pi\rho \sum_{m,n}c_m c_n \overline{b}_m \overline{b}_n d_{mn}(r) \end{equation} where $\overline{b}_m$ is the coherent neutron scattering length for atomic species $m$, with values taken from the standard work of Sears \cite{Sears:1992}. The $D(r)$ functions were calculated for the three experimental cases of nominally deuterated, partially deuterated, and hydrogenous. The results are shown in Figure \ref{fig:Dcombined}. The key point from Figure \ref{fig:Dcombined} is that the oscillations in the PDF at higher $r$ are enhanced in the nominally-deuterated case, and significantly reduced in the hydrogenous case. We can interpret this by considering the factors $c_m c_n \overline{b}_m \overline{b}_n$ in equation \ref{eq:Dr}. From Figure S15 in the SI we see that in cases of high D concentration the C--H/D contribution to $D(r)$ is similar in value to that of the constant C--C contribution, whereas at high H concentration the C--H/D contribution has a value almost opposite to that of the C--C contribution. In all cases the product H/D--H/D contribution is much smaller, being zero for the perfect null composition. Since the longer-$r$ oscillations are present in each partial PDF, the effect of changing the hydrogen/deuterium ratio is to go from the case where all three PDFs have positive weighting to the case where the C--H PDF has a negative weighting, and thus in the combination the amplitude of the oscillations are diminished. This point is important when we consider the experimental scattering functions. \subsection{Neutron scattering functions} \label{sec:scatteringfunctions} \begin{figure*}[t] \begin{center} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{Comparison_D_1.pdf} \subcaption{Nominally deuterated} \label{fig:DQiQ} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{Comparison_null_1.pdf} \subcaption{Nominally null H/D composition} \label{fig:nullQiQ} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{Comparison_H_1.pdf} \subcaption{Hydrogenated} \label{fig:HQiQ} \end{subfigure} \caption{\label{fig:allQiQ} Comparison of $Qi(Q)$ functions calculated from MD at 300 K (red curves) and measured from total scattering experiments performed at 300 K (black points)} \end{center} \end{figure*} The overall scattering function can be decomposed into three parts: \begin{equation} S(Q) = i(Q) + \sum_m c_m \overline{b_m^2} + S_0 \delta(Q) \label{eq:SQi} \end{equation} The important first term is obtained from the relation \begin{equation} \label{eq:D2QiQ} Qi(Q) = \int_0^\infty D(r) \sin(Qr) \, \mathrm{d}r \end{equation} and is the function we will consider here. In principle an experimental form of $D(r)$ could have been obtained using a reverse transformation of the measured function $Qi(Q)$, but we do not do that here because of difficulty accounting properly for the small-angle scattering (discussed below). The calculated $Qi(Q)$ functions are compared with the experimental data in Figure \ref{fig:allQiQ}. The only adjustable parameter was a single scale factor applied to each data set equally to bring the calculated functions in line with the experimental data. We consider that the agreement is excellent, particularly -- and importantly -- considering that nothing about the model has been adjusted to give this agreement other than the value of the single overall scale factor. What is also significant about the agreement between the calculated and experimental $Qi(Q)$ functions is that the calculation has reproduced the variation between the three different samples. Whilst broadly the peaks in the scattering functions of the three samples are in similar places, their relative heights and depths between them differ between the different samples. The only discrepancy between model calculation and experiment is in the case of the fully deuterated sample around $Q \sim 10$ \AA$^{-1}$, but in the other two samples the agreement there is excellent. \subsection{Small angle scattering}\label{ref:SANS} \begin{figure*}[t] \begin{center} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{SANSlog.pdf} \subcaption{Experimental, ISIS and CSNS} \label{fig:AllSANS} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{SANSlin.pdf} \subcaption{Experimental, ISIS} \label{fig:iQ_lowQ_exp} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{iQpeak_calc.pdf} \subcaption{Calculated} \label{fig:iQ_lowQ_calc} \end{subfigure} \caption{\label{fig:lowQ} a) Comparison of the low-$Q$ regions of the measured $i(Q)$ function for each of the three compositions of amorphous PAF-1 at a temperature of 300 K, plotted logarithmically to highlight the small angle regime. Here we compare data from ISIS (continuous curves, $0.02 < Q < 10$ \AA$^{-1}$) and from CSNS (dashed curves, $0.01 < Q < 0.2$ \AA$^{-1}$). The black dot-dash curve is of the form $Q^{-4}$ representing Porod scattering from particle surfaces. b) Comparison of the low-$Q$ regions of the experimental $i(Q)$ for each of the three compositions, highlighting better the peak at around $Q \sim 0.4$ \AA$^{-1}$. c) Comparison of the low-$Q$ regions of the calculated $i(Q)$ from the MD simulations for each of the three samples at 300 K, showing the variation of the calculated peak at $Q \sim 0.45$ \AA$^{-1}$ with composition.} \end{center} \end{figure*} Figure \ref{fig:lowQ} shows in more detail the scattering function $i(Q)$ at lower values of $Q$. In Figure \ref{fig:AllSANS} the data for $0.01 < Q < 10$ \AA$^{-1}$ from both ISIS and CSNS are plotted on logarithmic scales for both $Q$ and $i(Q)$. The CSNS data have been scaled to put them onto the same scale as the ISIS data, but no attempt has been made to account for the levels of background scattering in the CSNS data. Figure \ref{fig:AllSANS} also shows a line of the form $Q^{-4}$ to show that this closely represents the variation of $i(Q)$ for $Q$ below 0.05 \AA$^{-1}$. This form of scattering is characteristic of Porod scattering arising from interference due to a discontinuous surface \cite{Ciccariello:1988ex}, which in this case means the particle surface. There is no sign of Guinier scattering associated with particle size in our data for $Q > 0.01$ \AA$^{-1}$. The low-$Q$ region of the $i(Q)$ data shown in Figure \ref{fig:lowQ} also shows a strong feature at $Q \sim 0.45$~\AA$^{-1}$ around the onset of the rise of the Porod scattering, as highlighted in the linear plot of the ISIS data in Figure \ref{fig:iQ_lowQ_exp}. A peak at this value of $Q$ corresponds to an oscillation in the PDF with period of around 14 \AA, exactly as was seen in Figure \ref{fig:Dcomparison}. Indeed, this peak is reproduced, without the low-$Q$ Porod scattering, in the simulations as the Fourier transform of the computed $D(r)$ functions, as shown in Figure \ref{fig:iQ_lowQ_calc}. The agreement with experiment is good in two regards. First, the position of the peak in $Q$ in the simulation matches the experimental position well. Secondly, the variation of the peak intensity with changing composition from nominally-pure deuteration to pure hydrogenation in the calculation is consistent with the experimental data in Figure \ref{fig:iQ_lowQ_exp}. \section{Analysis of the local atomic structure} \begin{figure*}[t] \begin{center} \begin{subfigure}[b]{0.33\textwidth} \includegraphics[width=0.99\textwidth]{CC_300K.pdf} \subcaption{C--C distances} \label{fig:CCdistances} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{End_HH_distances.pdf} \subcaption{End H--H distances} \label{fig:HHdistances1} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{Middle_HH_distances.pdf} \subcaption{Inter-phenyl H--H distances} \label{fig:HHdistances2} \end{subfigure} \caption{\label{fig:atomdistances} Distribution of interatomic distances in amorphous PAF-1 at 300 K from MD simulations. a) Distribution of C--C bond distances. b) Distribution of H--H distances between the end hydrogens of different biphenyl moieties around the tetrahedral carbon side c) Distribution of H--H distances between the hydrogens of different rings around the central C--C bond of the biphenyl moieties. The curves to act as guides to the eye were obtained by smoothing the actual data points.} \end{center} \end{figure*} The comparison between the calculated and experimental scattering factors for the three samples of PAF-1 presented in the previous section gives us confidence that our model based on a CRN of connected tetrahedral sites is a reasonable representation of the real amorphous structure. We now report an analysis of some short-range aspects of the atomic structure -- what is often called the ``local structure'' -- from the MD simulations. \subsection{Interatomic distances} In our simulation we have two different types of C--C distances outside the rigid phenyl rings, namely the distance between the two bonded carbon atoms linking the phenyl rings in a biphenyl moiety, and between the tetrahedral carbon atom and the carbon atoms in the linked biphenyl ring; since the phenyl rings were kept rigid in our MD simulations, we do not consider the intra-phenyl C--C distances. We expect that the inter-phenyl distance is the shorter of the two for the chemical reason that the $\pi$-electron delocalisation across the biphenyl bond gives a higher bond order than the tetrahedral bond. This is reflected in the choice of parameters in the interatomic potentials, and from Figure \ref{fig:CCdistances}, which shows the distributions of these two critical C--C distances, we can see that the simulation has given the expected result. We now consider the two types of H--H distance distributions. The first, shown in Figure \ref{fig:HHdistances1}, concerns the H atoms at the ends of the biphenyl moieties closest to the tetrahedral carbon, and consider the distances between these two H atoms in each moiety and the pairs of H atoms in their four neighbours. The first peak in the distribution function, around 2.5 \AA, represents the closest approach of H atoms in neighbouring phenyl rings. The position of this peak is consistent with a similar peak seen in the simulation of a crystalline phase, discussed below. The distribution shows a satisfactory relaxation of the local structure around the tetrahedral site, with no crowding of the H atoms that would lead to some H...H distances being unreasonably short. Figure \ref{fig:HHdistances2} shows the second interesting H--H distances, namely between closest H atoms in the two phenyl rings in the middle of the biphenyl moiety. The first peak in this distribution function has the same mean distance as for the distribution of H--H distances about the tetrahedral carbon. This mean distance of around 2.5 \AA\ corresponds to a $45^\circ$ torsion angle between two phenyl rings rotating about the long axis of the biphenyl moiety. For comparison, the value for a planar biphenyl moiety is 1.7 \AA, and for a $90^\circ$ torsion angle it is 3.6 \AA. This mean distance of 2.5 \AA\ is very similar to the mean distance between H atoms on different molecules around the tetrahedral site discussed immediately above (Figure \ref{fig:HHdistances1}). Finally we want to make a comment about the widths of the peaks in the distribution functions shown in Figure \ref{fig:atomdistances}. In an ordered material these widths would be related to the size of thermal fluctuations of the atomic structure. However, in an amorphous phase these widths may more strongly reflect static structural disorder, and in Section \ref{sec:temperature_atomic_distributions} below we will see that the peak widths in the bond distance distributions are mostly due to structural disorder. \begin{figure*}[t] \begin{center} \begin{subfigure}[b]{0.33\textwidth} \includegraphics[width=0.99\textwidth]{Angle_300K.pdf} \subcaption{C--C--C angles} \label{fig:angles} \end{subfigure} \hspace{1cm} \begin{subfigure}[b]{0.33\textwidth} \includegraphics[width=0.99\textwidth]{Torsion_300K.pdf} \subcaption{Torsion angles} \label{fig:torsions} \end{subfigure} \caption{\label{fig:anglestorsions} Distributions of a) C--C--C bond angles, and b) phenyl torsion angles, obtained from the MD simulations of amorphous PAF-1 at a temperature of 300 K. The curves are shown as smoothed data.} \end{center} \end{figure*} \subsection{Bond and torsion angles} There are three important C--C--C angles, namely the angle subtended on the tetrahedral site (nominally $109.47^\circ$), the angle subtended on the carbon atom bonded to the tetrahedral site (nominally around $120^\circ$), and the angle subtended on the carbon atom on one phenyl ring that is linked to the neighbouring phenyl ring within the same biphenyl moiety (also nominally around $120^\circ$). As for the bond distances, we exclude the angles within the phenyl rings since these are fixed by the rigid bodies. Figure \ref{fig:angles} shows the distribution functions for all three angles, which are defined graphically as \textsf{A}, \textsf{B} and \textsf{C} in an inset to the figure. The distributions of the second two angles are almost identical, with same average angle and similar widths. On the other hand, the distribution of tetrahedral angles is much broader, by around a factor of 2, and the distribution is not quite symmetric. Figure \ref{fig:torsions} shows the distribution of biphenyl torsion angles. This is centred around an angle of $45^\circ$, with a width of around $20^\circ$. This is consistent with the results of a DFT study of the torsion angle in the biphenyl molecule \cite{Johansson:2008fz}, and fully consistent with the discussion of inter-phenyl H--H distances shown in Figure \ref{fig:HHdistances2} as discussed above. In each case in Figure \ref{fig:anglestorsions} the widths of the peaks may be due to the effects of thermal fluctuations of the atomic structure or due to structural disorder, as discussed above with regard to the bond distance distributions. As for bond lengths, analysis of the effects of temperature (Section \ref{sec:temperature_atomic_distributions}) show that the peak widths in the bond angle distributions are mostly due to structural disorder. \section{Comparison with a hypothetical diamond-like crystal structure} \subsection{Model} Given the close similarities between the short-range atomic structures of amorphous silica and the diamond-like cristobalite crystal structure \cite{Keen:1999cb}, it is useful to explore a crystalline version based on an initial diamond network of tetrahedral sites. A diamond-like PAF structure for use in MD simulations was constructed using the same method as used to construct an amorphous structure, as discussed in section \ref{sec:PAFmaker}. The sample contained 512 tetrahedral sites as a $4 \times 4 \times 4$ supercell of the basic diamond-type unit cell, with a total of 20992 atoms. We performed two types of simulation, first constraining the cubic metric while allowing the volume to change in a constant-temperature constant-pressure ensemble (NPT), and second allowing the simulation box to change shape as well as size in a constant-temperature constant-stress ensemble (NST). \begin{figure*}[!t] \begin{center} \begin{subfigure}[b]{0.45\textwidth} \includegraphics[height=6cm]{CubicCrystal.png} \subcaption{} \label{fig:crystal1} \end{subfigure} \begin{subfigure}[b]{0.45\textwidth} \includegraphics[height=6cm]{REVCON_collapsed.png} \subcaption{} \label{fig:crystal2} \end{subfigure} \caption{\label{fig:crystal} Projection of the simulated crystal configuration of cubic metric at a temperature of 300 K, a) viewed down one of the cube axes, and b) showing a view of the configuration collapsed into one unit cell, with a fading based on distance from the viewer. Carbon and hydrogen atoms are shown as black and pink spheres respectively.} \end{center} \end{figure*} \subsection{Atomic structure with the cubic metric} One of the NPT configurations is shown in Figure \ref{fig:crystal}, both as a simple projection and with all atoms collapsed back into one unit cell. Both highlight the diamond-type network in different ways. Calculated diffraction patterns for both the whole configurations and collapsed configurations are consistent with space group $Fd\overline{3}m$, the same as the cubic $\beta$-cristobalite phase of silica. The first Bragg peak has index 111, and it is noticeably the strongest Bragg peak. Of the next six Bragg peaks, namely 220, 311, 222, 400, 331 and 422 consistent with the systematic absences of space group $Fd\overline{3}m$, the sequence of intensities are alternatively strong and weak. This pattern loosely reflects the systematic order of strong and weak reflections in $\beta$-cristobalite. The $Fd\overline{3}m$ symmetry implies orientational disorder of the phenyl groups around the long axis of the biphenyl moiety. \begin{figure*}[!t] \begin{center} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{partialD_lowr_cubic_crystal.pdf} \subcaption{Partial functions, low $r$} \label{fig:partialD1_crystal} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{partialD_cubic_crystal.pdf} \subcaption{Partial functions, wide $r$} \label{fig:partialD2_crystal} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{combinedD_cubic_crystal.pdf} \subcaption{Combined functions} \label{fig:Dcombined_crystal} \end{subfigure} \caption{\label{fig:Dcomparison_crystal} a,b) Comparison of the partial PDF functions $D(r)$ for each element pair in the crystalline phase of PAF-1 of cubic metric, defined here as $r\left( g_{mn}(r)-1 \right)$, as calculated from a MD simulation performed at a temperature of 300 K. Results are shown for the lower-$r$ range of data (a) and across the full range of distances (b). Panel c) gives a comparison of the neutron PDF functions $D(r)$ for each sample, formed from equation \ref{eq:Dr}. These data should be compared with the corresponding functions shown in Figure \ref{fig:Dcomparison}. } \end{center} \end{figure*} The calculated PDFs for the NPT ensemble are shown in Figure \ref{fig:Dcomparison_crystal}, for comparison with the corresponding PDFs from the amorphous phase shown previously in Figure \ref{fig:Dcomparison}. Over the range of data $r < 8$~\AA\ (Figure \ref{fig:partialD1_crystal}) the partial PDFs are almost identical to those seen in the amorphous phase, Figure \ref{fig:partialD1}. Whilst a large part of the PDF over this range includes intra-biphenyl distance correlations, there are also important inter-phenyl correlations within the biphenyl moieties and between neighbouring moieties around the tetrahedral sites. We can conclude clearly from this analysis that the short-range structure of the cubic crystalline phase is very similar to that of the amorphous phase, which is consistent with the discussion above about there being some degree of orientational disorder consistent with the apparent space group symmetry. Figures \ref{fig:partialD2_crystal} and \ref{fig:Dcombined_crystal} show the partial and combined PDFs of the crystalline phase over a wider range of distance, up to 35 \AA. The existence of an oscillation in each of the PDFs with a period of around 14 \AA\ is similar to what was seen in the PDFs of the amorphous phase, Figures \ref{fig:partialD2} and \ref{fig:Dcombined}. The key difference is that the PDFs are more attenuated at higher-$r$ in the amorphous phase. This is reminiscent of the comparison of the PDFs in amorphous silica and crystalline $\beta$-cristobalite \cite{Keen:1999cb}. We calculated the same atomic distribution functions for both crystalline phases as shown for the amorphous phase in Figures \ref{fig:atomdistances} and \ref{fig:anglestorsions}; the results are given in the Supporting Information as Figures S18b and S20b for the bond lengths and S21b and S22b for the bond and torsion angles. The results are virtually the same as for the amorphous phase. This is entirely consistent with the point made in the discussion here regarding the PDF, namely that the local structure of the crystalline phase shows similar disorder as the amorphous phase. \subsection{Scattering functions with the cubic metric}\label{sec:cubic} \begin{figure*}[!t] \begin{center} \begin{subfigure}[b]{0.33\textwidth} \includegraphics[width=0.99\textwidth]{QiQ_crystal.pdf} \subcaption{} \label{fig:crystal_QiQ} \end{subfigure} \hspace{1cm} \begin{subfigure}[b]{0.33\textwidth} \includegraphics[width=0.99\textwidth]{Crystal_Bragg.pdf} \subcaption{} \label{fig:crystal_bragg} \end{subfigure} \caption{\label{fig:Crystal_scattering} a) Comparison of the scattering functions $Qi(Q)$ for the three crystalline materials, calculated from the MD simulations performed at a temperature of 300 K; b) Comparison of the corresponding functions $i(Q)$ for lower values of $Q$ highlighting the Bragg reflections (broadened by the finite range of the PDF transformed to give $Qi(Q)$).} \end{center} \end{figure*} The scattering functions $Qi(Q)$ calculated for the crystal phase are shown in Figure \ref{fig:crystal_QiQ} for the same D/H compositions as used in the experimental study of the amorphous phase. They are very similar to the corresponding functions calculated for the amorphous phase shown in Figure \ref{fig:allQiQ}. This is not surprising given the similarities of the PDFs in both cases (Figures \ref{fig:Dcomparison} and \ref{fig:Dcomparison_crystal}). The only differences in the PDFs, at higher $r$, feed through to the difference between a broad peak and a Bragg peak, where the latter should be a Dirac $\delta$-function. In practice the finite range of $r$ used in the Fourier transform, equation \ref{eq:D2QiQ}, means that the Bragg peaks will be broadened by an amount of order $2\pi/r_\mathrm{max}$. The low-$Q$ part of the scattering function $i(Q)$ is shown in Figure \ref{fig:crystal_bragg}, which can be compared with the corresponding function for the amorphous phase in Figure \ref{fig:iQ_lowQ_calc}. The scattering from the crystalline phase is sharper than for the amorphous phase (as noted, broadening by the finite range of $r$ in the PDF), and the peaks can be associated with Bragg reflections. We have indicated the Miller indices of the prominent peaks in the scattering function. The strongest Bragg peak, which has Miller indices 111, is of particular interest. The position of this peak in $Q$ is the same as the strong peak from the amorphous phase as shown in Figure \ref{fig:lowQ}, and both arise from the strong oscillations in the PDFs as seen in Figure \ref{fig:Dcomparison} for the amorphous phase and Figure \ref{fig:Dcomparison_crystal} for the crystalline phase. This is interesting in comparison with silica, SiO$_2$. The scattering data for amorphous silica shows a strong first peak at $Q = 1.53 \pm 0.02$ \AA$^{-1}$ \cite{Wright:1994ck}, which corresponds closely to the value of $Q$ for the 111 Bragg peak in the $\beta$-cristobalite crystalline phase (lattice parameter $a = 7.13$ \AA\ \cite{Schmahl:1992vp}, giving the Bragg peak at $Q = \sqrt{3} \times 2 \pi/a = 1.53$ \AA$^{-1}$). This similarity is remarkable, and we return to this point in Section \ref{sec:FSDP}. \begin{figure*}[!t] \begin{center} \begin{subfigure}[b]{0.45\textwidth} \includegraphics[height=6cm]{REVCON_tet_100.png} \subcaption{Projection down [100]} \label{fig:crystal3} \end{subfigure} \begin{subfigure}[b]{0.45\textwidth} \includegraphics[height=6cm]{REVCON_tet_001.png} \subcaption{Projection down [001]} \label{fig:crystal4} \end{subfigure} \caption{\label{fig:tetragonal_crystal} Projection of the simulated crystal configuration relaxed without imposition of the cubic metric at a temperature of 300 K, viewed down the a) $[100]$ and b) $[001]$ axes, and showing only the carbon atoms with a fading based on distance from the viewer.} \end{center} \end{figure*} \subsection{Structure without the constraint of the cubic metric} Unexpectedly, without the constraint of the cubic metric, allowing changes to both the shape and size of the atomic configuration during the simulation, we found that the crystal structure transforms to one that is metrically tetragonal, apparently with a disordered structure as in the metrically-cubic phase. Projections of the atomic structure collapsed onto one unit cell are shown in Figure \ref{fig:tetragonal_crystal}. This structure has a density lower than that of the metrically cubic crystal by around 2.5\%. There is an expansion along the $[001]$ direction by around 11\%, and a contraction in the perpendicular plane by about 6\%. The local atomic structure of this tetragonal phase is remarkably similar to that of the crystal phase with a cubic metric, which as we noted is very similar to that of the amorphous phase. The distributions of various bond lengths and angles of the two crystalline phases are compared in the Supporting Information, Figures S18b and S20b for the bond lengths and S21b and S22b for the bond and torsion angles respectively. The agreement is always close, and in some case nearly identical. Clearly the structure is able to flex easily like a traditional wine rack \cite{Cairns.2011}, without significant distortions around the network joints (the tetrahedral sites). \section{First diffraction peak} \label{sec:FSDP} There has been a long-standing controversy within the scientific literature regarding the nature of the first peak in the diffraction patterns of amorphous materials, particularly because this peak is usually the sharpest \cite{Salmon:1994jx,Elliott:1999dv,Massobrio:2001gr,Du:2005dv,Lucovsky:2010es,Micoulaut:2013ex,Shatnawi:2016hy,Shi:2019go,Shi:2019kx}. The best known example is the first sharp diffraction peak (FSDP) in amorphous silica \cite{Wright:1994ck}, as mentioned above. A corresponding peak can be seen in the scattering data in amorphous ZIF at $Q \sim 1$~\AA$^{-1}$ \cite{bennett2010structure}. As we noted earlier, amorphous silica has an atomic structure based on that of a tetrahedral CRN similar to PAF-1, as does our proposed model for amorphous ZIF \cite{bennett2010structure}. Given the link between the value of $Q$ for the 111 Bragg reflection in $\beta$-cristobalite and of the FSDP in silica, we can define a characteristic length scale from the density to correspond to the value of the $d$-spacing of the 111 peak of a diamond-type crystal structure of the same atomic density: $\ell = (8/\rho_\mathrm{t})^{1/3}/\sqrt{3}$, where $\rho_\mathrm{t}$ is the number of tetrahedral sites per unit volume. For silica, amorphous ZIF, and PAF-1 we have the following results for $2 \pi / \ell$ and the position of the FSDP, $Q_\mathrm{FSDP}$: \vspace{0.5cm} \setlength{\tabcolsep}{1.5mm} \begin{tabular}{@{}lcccc} \hline \hline Material & $\ell$ (\AA) & $2 \pi/\ell$ (\AA$^{-1}$) & $Q_\mathrm{FSDP}$ (\AA$^{-1}$) & Ref. \\ \hline Silica & 4.117 & 1.526 & 1.53 & \onlinecite{Wright:1994ck} \\ ZIF & 7.787 & 0.807 & 1.075 & \onlinecite{bennett2010structure} \\ PAF-1 & 13.242 & 0.474 & $\sim 0.45$ & --\\ \hline \hline \end{tabular} \setlength{\tabcolsep}{-1.5mm} \vspace{0.5cm} \noindent It can be seen the position of the FSDP scales closely with the density of tetrahedral sites, as given by $2 \pi / \ell$. The agreement is not as good for ZIF as it is for silica and PAF-1, but in these other two cases it is almost exact. The FSDP, as has been pointed out before \cite{Wright:1994ck}, represents not a distance in the glass (it isn't a correlation length) but a period of oscillation in the atomic density. In the case of silica and ZIF, the oscillations with wavelength $\ell$ are not very different from interatomic separation and therefore are not clearly visible in the PDF, but in the case of PAF-1 the oscillations extend beyond the range of the strong and identifiable interatomic peaks in the PDF over the range 0--10 \AA, Figure \ref{fig:Dcomparison}. As noted, these oscillations are stronger still in the crystal phase, Figure \ref{fig:Dcomparison_crystal}. The oscillations occur in all three types of atom pair (C--C, C--H and H--H). We consider this to be an intriguing result, given the importance of the debate around the origin of the FSDP in amorphous silica. Whilst the atomic structures of amorphous silica and PAF-1 are based on similar tetrahedral networks, the length scales associated with the FSDP in PAF-1 are much longer than for amorphous silica. This suggests the possibility of resolving the controversy of the FSDP with further research starting from porous framework materials as the end-point of a spectrum of tetrahedral amorphous networks with different connectors and densities, rather than silica as the other end point. \section{Effect of temperature} \subsection{Thermal expansion}\label{sec:thermal_expansion} \begin{figure}[t] \includegraphics[width=0.45\textwidth]{volume.pdf} \caption{Volume of the PAF-1 network as a function of temperature obtained from the MD simulations reported in this paper. The curve is a guide to an eye, obtained as a fitted second-order polynomial.} \label{fig:thermal_expansion} \end{figure} Simulations performed at several temperatures show that the PAF-1 network undergoes negative thermal expansion. The volume per formula unit is plotted as a function of temperature in Figure \ref{fig:thermal_expansion}. At the higher temperatures the coefficient of thermal expansivity is equal to $-24$ MK$^{-1}$. which is comparable to that of many significant NTE materials \cite{Dove:2016bv}. It is not surprising that the PAF-1 network shows NTE, because in its idealised form it is fully expanded, so fluctuations in local structure involving rotations and displacements of the biphenyl moieties will necessarily give rise to NTE through the tension mechanism. This has been discussed in several places \cite{Barrera:2005dt,Miller:2009kt,Romao:2013ch,Dove:2016bv}, and particularly pertinent is the discussion of networks where the linkers are rod-like molecular ions, such as Zn(CN)$_2$ \cite{Goodwin:2005ea,Fang:2013ji} or Cd(CN)$_2$ \cite{phillips2008nanoporosity}, and Si(NCN)$_2$ \cite{Li:2020hc}. \subsection{Atomic distributions} \label{sec:temperature_atomic_distributions} The PDF $D(r)$ functions calculated from the MD simulations for temperatures of 50 K and 150 K are given in the Supporting Information as Figure S16, and they can be compared with the data for 300 K given here in Figure \ref{fig:Dcomparison}. There is actually very little difference between any of the PDFs at different temperatures. The similarities arise partly from the fact that the network configurations are the same in each case, although below we will explore some differences in local structure that are not reflected strongly in the PDFs. The effects of thermal expansion on the PDF are slight, giving a linear contraction of 0.4 \% over the temperature change from 50 to 300 K, and affecting only the part of the PDF for distances larger than the molecular length which only show the broad oscillation. Given that the PDFs hardly change with temperature, we might expect that the corresponding scattering functions $Qi(Q)$ also barely change with temperature. The calculated and measured scattering functions shown at 300 K in Figure \ref{fig:allQiQ} can be compared with corresponding data for 50 K and 150 K in Figure S17 in the Supporting Information (actually Figure S17 includes a separate measurement of the data at 300 K of corresponding quality as the data at low temperatures, slightly lower than the quality presented in Figure \ref{fig:allQiQ}). It is seen in Figure S17 that neither experimental nor calculated scattering functions show significant variation with temperature. The PDF is in fact a fairly blunt probe of local structure, and more detail can be obtained by comparing calculations of the distributions of bond lengths and bond angles from the results of the MD simulations. Figure S18a in the Supporting Information compares the distribution of C--C bond lengths at different temperatures, corresponding to the data shown for 300 K in Figure \ref{fig:CCdistances}. The distributions for both the inter-phenyl C--C distances and for the distances bonding the phenyl group to the tetrahedral carbon atoms maintain the same average distance and show a narrowing of the distributions on cooling, consistent with the increased size of thermal fluctuations on heating. On the other hand, there is rather less temperature variation of the two types of close H..H distances as shown in Figures \ref{fig:HHdistances1} and \ref{fig:HHdistances2}. The comparisons for different temperatures are shown in Figure S19. There is some variation of the close H...H distances between phenyl rings with temperature, but the local structure around the tetrahedral site is primarily determined by the structural disorder and not by thermal fluctuations. Next we consider the distributions of C--C--C bond angles, as shown and defined in Figure \ref{fig:angles} for the simulation at 300 K. The variation with temperature is shown as Figure S21a in the Supporting Information. The angle subtended by the bond between two phenyl groups, with a mean angle of around $120^\circ$ and labelled \textsf{C} in Figure \ref{fig:angles}, shows a variation with temperature suggesting that the primary effect is from thermal fluctuations. The other angle with a mean of around $120^\circ$, namely between the phenyl group and the bond to the tetrahedral carbon and labelled \textsf{B} in Figure \ref{fig:angles}, shows a slightly more complicated variation with temperature. Whilst the distribution narrows with cooling, it also develops a small bifurcation, with the two maxima in the distribution function at 50 K separated by around $2.5^\circ$. Finally the tetrahedral angle, labelled \textsf{A} in Figure \ref{fig:angles}, shows an even more complicated behaviour with temperature. In Figure \ref{fig:angles} the distribution has a peak at the ideal tetrahedral angle, $109.5^\circ$, but it is not a symmetrical distribution, with a significant tail in the distribution to lower angles. On cooling it is seen that with lower thermal motion this is actually a two-peak function, with a second maximum at around $103.5^\circ$. This distortion presumably arises from the disordered packing of the biphenyl moieties around the tetrahedral site. The distribution of biphenyl torsion angles is shown in Figure S22. Compared to the distribution shown in Figure \ref{fig:torsions}, the distributions at lower temperatures are similar in shape without a significant change in the mean of the distribution and with a sharpening associated with lower thermal fluctuations. \subsection{Analysis of the network dynamics} \begin{figure*}[t] \begin{center} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{VACF.pdf} \subcaption{} \label{fig:VACF} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{WideDOS.pdf} \subcaption{} \label{fig:wideDOS} \end{subfigure} \begin{subfigure}[b]{0.32\textwidth} \includegraphics[width=0.99\textwidth]{NarrowDOS.pdf} \subcaption{} \label{fig:narrowDOS} \end{subfigure} \caption{\label{fig:dynamics} a) Velocity autocorrelation functions for C (black) and H (magneta) atoms in the MD simulation of 300 K. b,c) Associated power spectra of the two velocity autocorrelation functions over the full range of frequencies (b) and over the range of low frequencies highlighting better the zero-frequency limit (c). Note that, as discussed in the text, these results do not include internal vibrations of the phenyl rings.} \end{center} \end{figure*} The time-dependent velocity autocorrelation functions for the C and H atoms were calculated separately, and are shown in Figure \ref{fig:VACF}. One expects the correlation functions for the two atom types to be broadly similar given that the phenyl rings are held rigid in the simulation, with the differences coming from the terminal 4,4$^\prime$ C atoms in the biphenyl moieties, and from the tetrahedral C atoms. The Fourier transforms of the velocity autocorrelation functions give power spectra that are exactly the vibrational density of states \cite{Dove:1993}. The power spectra for carbon and hydrogen atoms are shown in Figures \ref{fig:wideDOS} and \ref{fig:narrowDOS}. Because the simulation used rigid phenyl rings, the density of states does not include internal vibrations of the phenyl groups. To understand the vibrational density of states better, we performed a vibrational analysis of a simulated carbon tetra-4-biphenyl cluster performed using the same potentials.The highest frequency group of modes in the density of states is between 35--40 THz, and it is strong for the carbon atoms and very weak for the hydrogen atoms. The cluster showed clearly that these modes correspond to vibrations of the tetrahedral carbon atoms with variation of the C--C bond length. Furthermore, the feature around 18 THz corresponds to motions of the phenyl groups that bend the tetrahedral C--C--C angles. One the other hand, the lowest-frequency peak extending below 0.5 THz corresponds to whole-body rotations of the biphenyl moieties. These are the modes that will give rise to the negative thermal expansion seen above (Section \ref{sec:thermal_expansion}), corresponding to flexing of the network without distortions of the CC$_4$ tetrahedra or flexing of the biphenyl moieties. \section{Conclusion} The main achievement in this paper has been to demonstrate that an atomic model for amorphous PAF-1 based on a continuous random network of connected tetrahedral sites provides a good description of the neutron scattering function. The atomic structure was relaxed by molecular dynamics simulations, and this allowed a detailed characterisation of the atomic structure. The neutron scattering experiments were performed using isotopic substitution of deuterium for hydrogen, which includes the case where the scattering is primarily from C--C correlations, and in the cases of pure hydrogen and pure deuterium gives a different balance of scattering from C--C and H--H correlations, but more importantly changes the sign of the contribution from the C--H correlation. We obtain good agreement for all three H/D ratios used in this study. The atomic structure given by the MD simulations has been characterised by considering distributions of interatomic distances, bond angles, and torsional angles. The atomic structure is able to relax well in two regards, firstly to maintain a reasonable distribution of biphenyl torsion angles, and secondly by reasonably avoidance of close H\dots H contacts around the tetrahedral site. The simulations do show some angular distortions around the tetrahedral site. Comparison with a hypothetical crystal based on the diamond structure -- analogous to the $\beta$-cristobalite form of silica -- suggests that if there is a crystalline form it is likely to have orientational disorder of the phenyl groups about the long axis of the biphenyl moiety. Atomic distribution functions for the crystalline phase are remarkably similar to those of the amorphous phase. One very interesting result is the presence of long-period oscillations in the pair distribution functions for all three types of atomic correlations, which give rise to a first sharp diffraction peak in both the calculation and measurement of the scattering function. The calculation of the how the peak varies with isotopic substitution is consistent with the experimental results. This peak corresponds to the strong $111$ Bragg reflection from the crystalline phase in a way that is exactly analogous with the correspondence between the first sharp diffraction peak in amorphous silica and the $111$ Bragg peak in $\beta$-cristobalite. However, unlike other glasses such as silica glass and amorphous ZIF, the oscillations in the PDF of PAF-1 that correspond to the first sharp diffraction peak extend far beyond the low-$r$ peaks representing short-range order. This material thus provides a unique opportunity to disambiguate these properties of amorphous frameworks. The samples we studied show very strong small angle scattering with a classical Porod form proportional to $Q^{-4}$, consistent with scattering from particulate surfaces. The existence of strong scattering made it hard to separate the small-angle scattering from the wide-angle scattering, which meant that accurate extraction of the pair distribution function was not possible. Hence our comparison between simulation and experiment has been with the scattering function only. The combination of neutron total scattering on samples of different isotopic composition with molecular dynamics simulation has successfully confirmed and characterised the proposed atomic structure of the amorphous phase of PAF-1. This approach can be commended for application to other amorphous organic network materials. \section*{Associated Content} \subsection*{Supporting Information} The Supporting Information is available free. \begin{itemize} \item Document containing information on sample synthesis; sample characterisation using NMR spectroscopy, mass spectrometry, Fourier transform infrared spectroscopy, powder x-ray diffraction, thermogravimetric analysis, mass spectrometer coupled thermogravimetric analysis, N$_2$ sorption isotherm measurements; molecular dynamics simulations; variation of the PDF with varying H/D ratio; definition of partial pair distribution functions; variation of pair distribution function and scattering function with temperature; local structure from molecular dynamics simulations with varying temperature and crystallinity. The document contains Figures S1--S22, Table S1, and includes seven references \cite{ben2009targeted,kildahl2007longer,wang2018molecular,Williams:2001tp,Allinger:1989em,Johansson:2008fz,Sears:1992}. \item Compressed folder containing the input files for the molecular dynamics simulations and final configurations \item CIF file of the PAF-1 configuration \item Software to build the initial PAF-1 configuration \item Compressed folder containing the calculated PDFs and experimental neutron scattering data, together with Matlab scripts for processing the data \end{itemize} \subsection*{Data availability} The original ISIS data can be obtained from the ISIS repository with DOI 10.5286/ISIS.E.RB1820458. \section*{Author Information} \subsection*{Corresponding Author} \begin{description} \item [Martin T Dove] \textit{College of Computer Science, Sichuan University, Chengdu, Sichuan 610065, China; Department of Physics, School of Sciences, Wuhan University of Technology, 205 Luoshi Road, Hongshan district, Wuhan, Hubei, 430070, China; School of Mechanical Engineering Dongguan University of Technology, Dongguan, Guangdong, 523830 China; School of Physical and Chemical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK}; Email: martin.dove@icloud.com; ORCID: orcid.org/0000-0002-8030-1457 \end{description} \subsection*{Authors} \begin{description} \item [Guanqun Cai] \textit{School of Physical and Chemical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom}; ORCID: orcid.org/0000-0001-8144-5084 \item [He Lin] \textit{Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences, 99 Haike Road, Shanghai, 201210, China}; Email: linhe@zjlab.org.cn; ORCID: orcid.org/0000-0002-9907-8494 \item [Ziqiang Zhao] \textit{Institute of Molecules Plus, Tianjin University, Tianjin 300072, China}; Email: zqzhao19@tju.edu.cn; ORCID: orcid.org/0000-0002-5527-0820 \item [Jiaxun Liu] \textit{School of Physical and Chemical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom}; ORCID: orcid.org/0000-0001-5708-9278 \item [Anthony E Phillips] \textit{School of Physical and Chemical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom}; Email: a.e.phillips@qmul.ac.uk; ORCID: orcid.org/0000-0003-4225-0158 \item [Thomas F Headen] \textit{ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX, United Kingdom}; Email: tom.headen@stfc.ac.uk; ORCID: orcid.org/0000-0003-0095-5731 \item [Tristan G A Youngs] \textit{ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX, United Kingdom}; Email: tristan.youngs@stfc.ac.uk; ORCID: orcid.org/0000-0003-3538-5572 \item [Yang Hai] \textit{School of Mechanical Engineering Dongguan University of Technology, Dongguan, Guangdong, 523830 China}; Email: haiyang@dgut.edu.cn; ORCID: orcid.org/0000-0002-5340-9008 \item [Haolai Tian] \textit{Spallation Neutron Source Science Center, Dongguan 523803, Guangdong, China; Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; University of Chinese Academy of Sciences, Beijing 100049, China}; Email: tianhl@ihep.ac.cn; ORCID: orcid.org/0000-0003-0585-8279 \item [Chunyong He] \textit{Spallation Neutron Source Science Center, Dongguan 523803, Guangdong, China; Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; University of Chinese Academy of Sciences, Beijing 100049, China}; Email: hechunyong@ihep.ac.cn; ORCID: orcid.org/0000-0002-0412-1550 \item [Yubin Ke] \textit{Spallation Neutron Source Science Center, Dongguan 523803, Guangdong, China; Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; University of Chinese Academy of Sciences, Beijing 100049, China}; Email: keyb@ihep.ac.cn; ORCID: orcid.org/0000-0002-4983-7755 \item [Juzhou Tao] \textit{Spallation Neutron Source Science Center, Dongguan 523803, Guangdong, China; Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; University of Chinese Academy of Sciences, Beijing 100049, China}; Email: taoj@ihep.ac.cn; ORCID: orcid.org/0000-0003-2807-8486 \item [Teng Ben] \textit{Zhejiang Engineering Laboratory for Green Syntheses and Applications of Fluorine-Containing Specialty Chemicals, Institute of Advanced Fluorine-Containing Materials, Zhejiang Normal University, Jinhua 321004, China; Key Laboratory of the Ministry of Education for Advanced Catalysis Materials, Institute of Physical Chemistry, Zhejiang Normal University, Jinhua 321004, China}; Email: tengben@zjnu.edu.cn; ORCID: orcid.org/0000-0002-0847-330X \end{description} \subsection*{Author Contributions} GC and HL contributed equally. \subsection*{Notes} The authors declare no competing financial interests. \section*{Acknowledgments} HL thanks the National Key R\&D Program of China (2017YFA0403801) and the National Natural Science Foundation of China (U1732120) for financial support. GC and JL are grateful for funding from the China Scholarship Council and Queen Mary University of London. YK is grateful to the funding from Natural Science Foundation of Guangdong Province (grant number 2018A030313728). Neutron beam time was provided by ISIS under project number RB1820432, and by CSNS. Simulations were performed on the HPC Midlands Plus tier-2 system supported by EPSRC (EP/P020232/1) (MTD co-investigator). The authors acknowledge Dr Wen Lei and Jie Pan from the Shanghai Research Institute of Chemical Industry Co., Ltd. for their help in the analysis of different isotopologues in this study. \newpage

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\section{Helicity components of diffractive DIS} The detection of leading protons $p'$ from diffractive DIS $ep\rightarrow e'p'X$ gives access to several new observables: the diffraction slope $B_{D}$ which quantifies the impact parameter properties of diffractive DIS and new helicity structure functions. In this talk we report predictions~\cite{NPZ1} for the nontrivial $\beta$-dependence of the diffraction slope and suggest a new method~\cite{NPZ2} for measuring $R^{D}=d\sigma^{D}_{L}/d\sigma^{D}_{T}$ for diffractive DIS based on the azimuthal correlation of the $(e,e')$ and $(p,p')$ scattering planes. The differential cross-section of the diffractive process $ep \rightarrow e'p'X$ reads \begin{eqnarray} &&Q^{2}y{d\sigma(ep \rightarrow ep'X)\over dQ^2 dy dM^2 dp_{\perp}^2 d\phi} = {1\over 2\pi}{\alpha_{em}\over \pi}\left[\left(1-y+{y^2\over 2}\right){d\sigma^{D}_{T} \over dM^2 dp_{\perp}^2} +\left(1-y\right){d\sigma^{D}_{L} \over dM^2 dp_{\perp}^2} \right. \nonumber \\ &&\left. +\left(1-y\right){d\sigma^{D}_{TT'} \over dM^2 dp_{\perp}^2}\cdot\cos{(2\phi)} +\left(2-y\right)\sqrt{1-y}\;{d\sigma^{D}_{LT} \over dM^2 dp_{\perp}^2}\cdot\cos{(\phi)}\right], \end{eqnarray} where $p_{\perp}$ is the (p,p') momentum transfer, $\phi$ is the azimuthal angle between (e,e') and (p,p') scattering planes. We focus on the $q\bar{q}$ excitation which dominates at large $\beta$. Following the technique developed in~\cite{NZ}, we find (for the kinematical variables see Fig.~1) \begin{equation} {d\sigma_{i}^{D} \over dM^2 dp_{\perp}^2}=\frac{\alpha_{em}}{24\pi^2} \sum_{f} Z_f^2 \int d^2\vec{k}{1-J^2 \over 4J}\alpha_{S}^2 [h_i(z_{+})+h_i(z_{-})], \end{equation} where $z_{\pm}={1\over 2}(1\pm J)$, $J=\sqrt{1-4{k^2+m_f^2\over M^2}}$, $h_{T}=[1-2z(1-z)]\vec{\Phi}_1^2+m_f^2 \Phi_2^2$ , $h_{LT}\cdot\cos{(\phi)}=2z(1-z)(1-2z)Q(\vec{\Phi}_1\vec{t})\Phi_2$ , $h_{L}=4z^2(1-z)^2 Q^2 \Phi_2^2$ , $h_{TT'}\cdot\cos{(2\phi)}=2z(1-z)[\vec{\Phi}_1^2-2(\vec{\Phi}_1\vec{t})^2]$ and $m_{f}$ is the quark mass. The helicity amplitudes $\vec{\Phi_{1}}$, $\Phi_{2}$ equal \begin{equation} \Phi_j=\int d^2\vec{\kappa}\frac{f\left(x_{\Pom},\vec{\kappa},\vec{p}_{\perp}\right)}{\kappa^4} [D_{j}(\vec{r}+\vec{\kappa})+D_{j}(\vec{r}-\vec{\kappa}) -D_{j}(\vec{r}+{\vec{p}_{\perp}\over 2})-D_{j}(\vec{r}-{\vec{p}_{\perp}\over 2})], \end{equation} where $\vec{D}_{1}\left(\vec{r}\right)=\vec{r}\cdot D_{2}\left(\vec{r}\right)=\vec{r}/(\vec{r}^2+\epsilon^2)$, $\vec{r}=\vec{k}-({1\over 2}-z)\vec{p}_{\perp}$ and $f\left(x_{\Pom},\vec{\kappa},\vec{p}_{\perp}\right)$ is the off-forward unintegrated gluon density. \begin{figure} \epsfxsize=4.5in \epsffile{fig1.eps} \caption{a) Diffractive DIS, b) The definition of the azimuthal angle $\phi$, c) One of the pQCD diagrams for diffractive excitation of the $q\bar{q}$ state. \label{fig:diag}} \end{figure} Following~\cite{NZ,NNPZ,GNZ} we find \begin{equation} \Phi_j \propto \int^{\bar{Q}^2} {d\kappa^{2} \over \kappa^2} f(x_{\Pom},\kappa^2,p_{\perp}^2) = G(x_{\Pom},\bar{Q}^2,p_{\perp}^2)\approx G(x_{\Pom},\bar{Q}^2)\cdot (1-{1\over 2}B_{3\Pom} p_{\perp}^2). \end{equation} where the pQCD hardness scale equals \begin{equation} \bar{Q}^2=(k^2+m_f^2)\left(1+{Q^2\over M^2}\right)={k^2+m_f^2 \over 1-\beta}. \end{equation} In Eq.~(4) we parameterize the small-$p_{\perp}^2$ dependence by the diffraction slope $B_{3\Pom}$ which comes from the proton vertex and gluon propagation effects (see Fig.~1c). We emphasize that $B_{3\Pom}$ depends neither on $\beta$ nor flavor. For excitation of heavy flavours we have the fully analytic results (for the discussion of twist-4 $F_T$ and $F_{TT'}$ see~\cite{NPZ2,BGNPZ}) \begin{eqnarray} &&F_{T}^{D(4)}= \frac{2\pi e_f^2}{9\sigma_{tot}^{pp}} \frac{\beta(1-\beta)^2}{m_f^2}[ (1-B_{3\Pom}p_{\perp}^2)(3+4\beta+8\beta^2) \nonumber \\ &&+\frac{p_{\perp}^2}{m_f^2}\frac{1}{10}(5-16\beta-7\beta^2-78\beta^3+126\beta^4)] \bar{G}_{T}^2 \nonumber \\ &&F_{L}^{D(4)}= \frac{2\pi e_f^2}{9\sigma_{tot}^{pp}} \frac{12\beta^3}{Q^2}[(1-B_{3\Pom}p_{\perp}^2) 2(1-2\beta)^2 \bar{G}_{L}^2 \nonumber \\ &&+\frac{p_{\perp}^2}{m_f^2}(1-\beta)(1-7\beta+23\beta^2-21\beta^3) \bar{G}_{T}^2 ] \nonumber \\ &&F_{LT}^{D(4)}={p_{\perp}\over Q}\cdot \frac{2\pi e_f^2}{9\sigma_{tot}^{pp}} \frac{\beta^2 (1-\beta)}{m_f^2}[(1-B_{3\Pom}p_{\perp}^2)12\beta^2 (2-3\beta) \nonumber \\ &&+\frac{p_{\perp}^2}{m_f^2}\frac{1}{20}(1-\beta)(2+7\beta+12\beta^2-483\beta^3+672\beta^4)] \bar{G}_{T}^2. \end{eqnarray} where $\bar{G}_{T,L}= \alpha_s(\bar{Q}^2_{T,L}) G(x_{\Pom},\bar{Q}^2_{T,L})$ and $\bar{Q}^2_T ={m_f^2\over 1-\beta}$, $\bar{Q}^2_L = {Q^2\over 4\beta}$. The principal point is that $F_T$ and $F_{LT}$ are dominated by the aligned jet configurations, $k^2 \sim m_f^2$, whereas $F_L$ comes from the large $k^2$ jets, $k^2 \sim {M^2\over 4}$. In our calculations for light flavours we use the parameterization for the soft glue from~\cite{BGNPZ}. \section{Azimuthal asymmetry and $L/T$ separation} In contrast to the inclusive DIS where $R=\sigma_L/\sigma_T \ll 1$, in diffractive DIS pQCD predicts~\cite{GNZ} $R^{D} \gg 1$ for $\beta > 0.9$, despite the fact that $F_L$ is of higher twist. Because neither the proton nor electron energy will be changed at HERA, one must exploit the azimuthal asymmetry $A_{LT}^{D(4)}=F_{LT}^{D(4)}/(F_{T}^{D(4)}+F_{L}^{D(4)})$. The key observation is that both $F_{LT}$ and $F_{T}$ come from the aligned jet configurations and the $LT/T$ ratio is model independent \begin{equation} R^{D(4)}_{LT/T}=\frac{F_{LT}^{D(4)}}{F_{T}^{D(4)}}={p_{\perp}\over Q}\cdot\frac{12\beta^3\left(2-3\beta\right)}{\left(1-\beta\right)\left(3+4\beta+8\beta^2\right)}. \end{equation} Consequently, the measurement of $A_{LT}^{D(4)}$ amounts to the measurement of $R^{D(4)}\equiv F_{L}^{D(4)}/F_{T}^{D(4)}=R^{D(4)}_{LT/T}/A_{LT}^{D(4)}-1$. The predicted asymmetry is quite substantial in the interesting region of $\beta \sim 0.9$ (Fig.~2), can be measured with the ZEUS and H1 leading proton spectrometers and one can test the pQCD result $R^{D}\gg 1$ experimentally. \begin{figure} \hspace{1.4in} \epsfxsize=2.3in \epsffile{fig2.eps} \caption{Our prediction for the azimuthal asymmetry $A_{LT}^{D(4)}$ at $Q^2=100GeV^2$ and $x_{\Pom}=0.001$. The solid and dashed lines are for the GRV and MRS gluon structure functions, respectively. \label{fig:asym}} \end{figure} \section{Peculiarities of the diffraction cone for diffractive DIS} The diffraction slope $B_D$ is defined by the formula $d\sigma(ap\rightarrow XY) \propto \exp{(-B_D p_{\perp}^2)}$. The experience with diffraction of hadrons and real photons can be summarized as follows. One can write down an essentially model-independent decomposition $B_{D}=\Delta B_{aX} +\Delta B_{pY} + \Delta B_{int}$ in which $\Delta B_{int}$ interaction (exchange) range, and $\Delta B_{aX}$ and $\Delta B_{pY}$ come from the (transverse) size of the $aX$ and $pY$ transition vertices. These contributions $\Delta B_{aX,pY}$ depend strongly on the excitation energy in the $i\rightarrow j$ transition $\Delta M^{2} =m_{j}^{2}-m_{i}^{2}$ and vanish for excitation of the continuum~\cite{HAG}, $\Delta M^{2} \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 1\div 2 GeV^2$. The experimental data on double diffraction dissociation $pp \rightarrow XY$ into high mass states $X,Y$ give $B_{D}\approx \Delta B_{int} \sim 1.5-2 GeV^{-2}$. In single diffraction $ap\rightarrow Xp'$ into high mass states $B_{D} \approx \Delta B_{int}+\Delta B_{pp} \sim 6-7 GeV^{-2}$ is independent of the projectile $a=p, \pi, K, \gamma$. It has been argued~\cite{NZ} that in the triple-pomeron regime of diffractive DIS, $\beta \ll 1$, one must find $B_{D} \approx B_{3\Pom} \sim 6 GeV^{-2}$, which has indeed been confirmed by the ZEUS collaboration~\cite{ZEUS}. Hereafter we focus on finite $\beta$, dominated by the $q\bar q$ excitation, $X=q\bar q$. For diffractive DIS at finite $\beta$ the excited mass is large, $M^2={1-\beta \over \beta}Q^2 \gg m_V^2$, hence the continuum is excited and naively one would expect $\Delta B_{\gamma*X} \approx 0$, and $B_{D}\approx B_{3\Pom}$ independently of $\beta$. Our principal finding is that this is not the case, $\Delta B_{\gamma*X}$ is large and varies substantially with $\beta$. Our results for the small-$p_{\perp}^2$ of diffractive structure functions are given by Eqs.(6). We focus on the transverse cross section which dominates at $\beta < 0.9$. The component $\Delta B_{\gamma*X}$ comes from the term $\propto {p_{\perp}^2\over m_{f}^2}$. These formulas are directly applicable for heavy flavour excitation. In the diffraction excitation of light flavours there is a sensitivity to the gluon structure function in the soft region, and the rate of variation of the gluon structure function in the soft region emerges as a scale instead of ${1\over m_{f}^2}$. However, the qualitative features of the $\beta$ dependence do not change from heavy to light flavours. The numerical results are shown in Fig.~3. The most striking prediction is the rise of $B_D$ when $\beta$ decreases from $\beta \sim 1$ to $\beta \sim {1\over 2}$. This rise can be related to the rise of the scanning radius discussed in~\cite{GNZ}: $r_S^2 \sim {1\over \bar{Q}^2_T} \sim {1-\beta \over m_f^2}$. Numerically, at $\beta \sim 1/2$ we have $\Delta B_{\gamma*X} \sim {1\over 10m_f^2}$. In excitation of small masses, $4m_f^2 \ll M^2 \ll Q^2$, i.e., $\beta \rightarrow 1$, we predict a substantial drop of $B_{D}$, because here $\Delta B_{\gamma*X} \sim -{1\over 5m_f^2}$. This is a legitimate pQCD domain because $\bar{Q}^2_T={Q^2\over 4}$ for $M^2 \sim 4m_f^2$. Very close to the threshold \begin{equation} F_{T}^{D(4)}(p_{\perp}^2,x_{\Pom},v,Q^{2})=\frac{128\pi e_f^2}{3\sigma_{tot}^{pp}} {m_f^2 \over Q^4} v [(1-B_{3\Pom}p_{\perp}^2)+ {p_{\perp}^2 \over 6m_f^2}v^2] \bar{G}_{T}^2, \end{equation} where $v=\sqrt{1-{4m_f^2\over M^2}}$. In the spirit of exclusive-inclusive duality, $B_D=B_{3\Pom}$ can be related to the diffraction slope for V(1S) vector meson production, whereas the drop of $B_D$ for somewhat higher masses correlates nicely with the prediction~\cite{NNPZ} $B_{V'(2S)} \ll B_{V(1S)}$. The experimental observation of this large-$\beta$ drop of $B_D$ is not easy because of the masking effect of the longitudinal cross section for which $B_D \approx B_{3\Pom}$. \begin{figure} \hspace{0.3in} \epsfxsize=4.0in \epsffile{fig3.eps} \caption{Our predictions for the diffraction slope $B_{D}$ for the transverse structure function at $Q^2=100GeV^2$ and $x_{\Pom}=0.001$. \label{fig:slope}} \end{figure} \section*{References}

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\section{Introduction} This paper presents a learning algorithm to address both \emph{unsupervised}~\citep{Gong_CVPR12,Basura_ICCV13,SUN_Return15} and \emph{semi-supervised}~\citep{Hoffman_IJCV14,Duan_ICML12,Tsai_CVPR16} domain adaptation problems. Our goal here is to learn a latent space in which domain disparities are minimized. We show such a space can be learned by first matching the statistical properties of the projected domains (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, covariance matrices), and then adapting the Mahalanobis metric of the latent space to the labeled data, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, minimizing the distances between pairs sharing the same class label while pulling away samples with different class labels. We develop a geometrical solution to jointly learn projections onto the latent space and the Mahalanobis metric there by making use of the concepts of Riemannian geometry. Thanks to deep learning, we are witnessing rapid growth in classification accuracy of the imaging techniques if substantial amount of labeled data is provided~\citep{Krizhevsky_NIPS12,Simonyan_VGG14,He_CVPR16,Herath_Survey17}. However, harnessing the attained knowledge into a new application with limited labeled data (or even without having labels) is far beyond clear~\citep{Koniusz_CVPR16,Long_Deep16,Ganin_ICML15,Tzeng_Deep14,Qiang_CVPR15}. To make things even more complicated, due to the inherit \emph{bias} of datasets~\citep{Torralba_CVPR11,Shimodaira_JSPI00}, straightforward use of large amount of auxiliary data does not necessarily assure improved performances. For example, the ImageNet~\citep{ILSVRC15} data is hardly useful for an application designed to classify images captured by a mobile phone camera. Domain adaptation is the science of reducing such undesired effects in transferring knowledge from the available auxiliary resources to a new problem. The most natural solution to the problem of DA is by identifying the structure of a common space that minimizes a notion of domain mismatch. Once such a space is obtained, one can design a classifier in it, hoping that the classifier will perform equally well across the domains as the domain mismatched is minimized. Towards this end, several studies assume that either \textbf{1.} a subspace of the target\footnote{In DA terminology target domain refers to the data directly related to the task. Source domain data is used as the auxiliary data for knowledge transferring.} domain is the right space to perform DA and learn how the source domain should be mapped onto it~\citep{Saenko_ECCV10,Tsai_CVPR16}, or \textbf{2.} subspaces obtained from both source and target domains are equally important for classification, hence trying to either learn their evolution~\citep{Gopalan_ICCV11,Gong_CVPR12} or similarity measure~\citep{Shi_ICDM10,Wang_IJCAI11,Duan_ICML12}. Objectively speaking, a common practice in many solutions including the aforementioned methods, is to simplify the learning problem by separating the two elements of it. That is, the algorithm starts by fixing a space (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, source subspace of~\cite{Basura_ICCV13,Tsai_CVPR16}), and learns how to transfer the knowledge from domains accordingly. A curious mind may ask why should we resort to a predefined and fixed space in the first place. In this paper, we propose a learning scheme that avoids such a separation. That is, we do not assume that a space or a transformation, apriori is known and fixed for DA. In essence, we propose to learn the structure of a Hilbert space (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, its metric) along the transformations required to map the domains onto it jointly. This is achieved through the following contributions, \begin{itemize} \item [(i)] We propose to learn the structure of a latent space, along its associated mappings from the source and target domains to address both problems of unsupervised and semi-supervised DA. \item [(ii)] Towards this end, we propose to maximize a notion of discriminatory power in the latent space. At the same time, we seek the latent space to minimize a notion of statistical mismatch between the source and target domains (see Fig.~\ref{fig:mainDiag} for a conceptual diagram). \item [(iii)] Given the complexity of the resulting problem, we provide a rigorous mathematical modeling of the problem. In particular, we make use of the Riemannian geometry and optimization techniques on matrix manifolds to solve our learning problem\footnote{Our implementation is available on \url{https://sherath@bitbucket.org/sherath/ils.git}.}. \item [(iv)] We extensively evaluate and contrast our solution against several baseline and state-of-the-art methods in addressing both unsupervised and semi-supervised DA problems. \end{itemize} \begin{figure*}[!t] \centering \includegraphics[width=0.9\textwidth]{mainDiag.jpg} \caption{\textbf{A conceptual diagram of our proposal.} The marker shape represents the instance labels and color represents their original domains. Both source and target domains are mapped to a latent space using the transformations $\Mat{W}_s$ and $\Mat{W}_t$. The metric, $\Mat{M}$ defined in the latent space is learned to maximize the discriminative power of samples in it. Indicated by dashed ellipsoids are the domain distributions. The statistical loss of our cost function aims to reduce such discrepancies within the latent space. Our learning scheme identifies the transformations $\Mat{W}_s$ and $\Mat{W}_t$ and the metric $\Mat{M}$ jointly. This figure is best viewed in color.} \label{fig:mainDiag} \end{figure*} \section{Proposed Method} \label{sec:proposed_method} In this work, we are interested in learning an \textbf{I}nvariant \textbf{L}atent \textbf{S}pace (ILS) to reduce the discrepancy between domains. We first define our notations. Bold capital letters denote matrices (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\Mat{X}$) and bold lower-case letters denote column vectors (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\Vec{x}$). $\mathbf{I}_n$ is the $n \times n$ identity matrix. $\SPD{n}$ and $\ST{p}{n}$ denote the SPD and Stiefel manifolds, respectively, and will be formally defined later. We show the source and target domains by $\mathcal{X}_s \subset \mathbb{R}^{s}$ and $\mathcal{X}_t \subset \mathbb{R}^{t}$. The training samples from the source and target domains are shown by $\{\Vec{x}^s_i,y^s_i\}_{i = 1}^{n_s}$ and $\{\Vec{x}^t_i\}_{i = 1}^{n_t}$, respectively. For now, we assume only source data is labeled. Later, we discuss how the proposed learning framework can benefit form the labeled target data. Our idea in learning an ILS is to determine the transformations $\mathbb{R}^{s \times p} \ni \Mat{W}_s: \mathcal{X}_s \to \mathcal{H}$ and $\mathbb{R}^{t \times p} \ni \Mat{W}_t: \mathcal{X}_t \to \mathcal{H}$ from the source and target domains to a latent p-dimensional space $\mathcal{H} \subset \mathbb{R}^p$. We furthermore want to equip the latent space with a Mahalanobis metric, $\Mat{M} \in \SPD{p}$, to reduce the discrepancy between projected source and target samples (see Fig.~\ref{fig:mainDiag} for a conceptual diagram). To learn $\Mat{W}_s$, $\Mat{W}_t$ and $\Mat{M}$ we propose to minimize a cost function in the form \begin{equation} \label{eqn:generalCost} \mathcal{L} = \mathcal{L}_{d} + \lambda \mathcal{L}_{u}\;. \end{equation} In Eq.~\ref{eqn:generalCost}, $\mathcal{L}_{d}$ is a measure of dissimilarity between labeled samples. The term $\mathcal{L}_{u}$ quantifies a notion of statistical difference between the source and target samples in the latent space. As such, minimizing $\mathcal{L}$ leads to learning a latent space where not only the dissimilarity between labeled samples is reduced but also the domains are matched from a statistical point of view. The combination weight $\lambda$ is envisaged to balance the two terms. The subscripts ``d'' and ``u'' in Eq.~\ref{eqn:generalCost} stand for ``Discriminative'' and ``Unsupervised''. The reason behind such naming will become clear shortly. Below we detail out the form and properties of $\mathcal{L}_{d}$ and $\mathcal{L}_{u}$. \subsection{Discriminative Loss} The purpose of having $\mathcal{L}_{d}$ in Eq.~\ref{eqn:generalCost} is to equip the latent space $\mathcal{H}$ with a metric to \textbf{1.} minimize dissimilarities between samples coming from the same class and \textbf{2.} maximizing the dissimilarities between samples from different classes. Let $\Mat{Z} = \{\Vec{z}_j\}_{j = 1}^n$ be the set of labeled samples in $\mathcal{H}$. In unsupervised domain adaptation $\Vec{z}_j = \Mat{W}_s^T\Vec{x}^s_j$ and $n = n_s$. In the case of semi-supervised domain adaptation, \begin{equation*} \Mat{Z} = \Big\{\Mat{W}_s^T\Vec{x}^s_j \Big\}_{j=1}^{n_s} \bigcup \Big\{\Mat{W}_t^T\Vec{x}^t_j \Big\}_{j=1}^{n_{tl}}, \end{equation*} where we assume $n_{tl}$ labeled target samples are provided (out of available $n_t$ samples). From the labeled samples in $\mathcal{H}$, we create {$N_p$} pairs in the form $(\Vec{z}_{1,k},\Vec{z}_{2,k}),~k = 1,2,\cdots,{N_p}$ along their associated label $y_k \in \{-1,1\}$. Here, $y_k = 1$ iff label of $\Vec{z}_{1,k}$ is similar to that of $\Vec{z}_{2,k}$ and $-1$ otherwise. That is the pair $(\Vec{z}_{1,k},\Vec{z}_{2,k})$ is similar if $y_k = 1$ and dissimilar otherwise. To learn the metric $\Mat{M}$, we deem the distances between the similar pairs to be small while simultaneously making the distances between the dissimilar pairs large. In particular, we define $\mathcal{L}_{d}$ as, \begin{align} \label{eqn:distCost} \mathcal{L}_{d} &= \frac{1}{N_p} \sum\limits_{k=1}^{N_p} \ell_\beta\big(\Mat{M},y_k,\Vec{z}_{1,k} - \Vec{z}_{2,k},1 \big) + r(\Mat{M}), \end{align} with \begin{align} \label{eqn:ell_b} \hspace{-2ex} \ell_\beta\big(\Mat{M},y,\Vec{x},u \big) = \frac{1}{\beta}\log\Big(1+\exp\big(\beta y(\Vec{x}^T\Mat{M}\Vec{x} -u )\big)\Big). \end{align} Here, $\ell_\beta$ is the generalized logistic function tailored with large margin structure (see Fig.~\ref{fig:logisticLoss}) having a margin of $u$\footnote{For now we keep the margin at $u=1$ and later will use this to explain the soft-margin extension.}. First note that the quadratic term in Eq.~\ref{eqn:ell_b} (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\Vec{x}^T\Mat{M}\Vec{x}$) measures the Mahalanobis distance between $\Vec{z}_{1,k}$ and $\Vec{z}_{2,k}$ if used according to Eq.~\ref{eqn:distCost}. Also note that $\ell_\beta\big(\cdot,\cdot,\Vec{x},\cdot \big) = \ell_\beta\big(\cdot,\cdot,-\Vec{x},\cdot \big)$, hence how samples are order in the pairs is not important. To better understand the behavior of the function $\ell_\beta$, assume the function is fed with a similar pair, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $y_k = 1$. For the sake of discussion, also assume $\beta = 1$. In this case, $\ell_\beta$ is decreased if the distance between $\Vec{z}_{1,k}$ and $\Vec{z}_{2,k}$ is reduced. For a dissimilar pair (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $y_k = -1$), the opposite should happen to have a smaller objective. That is, the Mahalanobis distance between the samples of a pair should be increased. The function $\ell_\beta\big(\cdot,\cdot,\Vec{x},\cdot \big)$ can be understood as a smooth and differentiable form of the hinge-loss function. In fact, $\ell_\beta\big(\cdot,\cdot,\Vec{x},\cdot \big)$ asymptotically reaches the hinge-loss function if $\beta\rightarrow\infty$. The smooth behavior of $\ell_\beta\big(\cdot,\cdot,\Vec{x},\cdot \big)$ is not only welcomed in the optimization scheme but also avoids samples in the latent space to collapse into a single point. \begin{figure}[h!] \centering \includegraphics[width=0.7\textwidth]{Gen_Logistic_Func.jpg} \caption{The behavior of the proposed $\ell_\beta$(~\ref{eqn:ell_b}) with $u=1$ for various values of parameter $\beta$. Here, the horizontal axis is the value of the Mahalanobis distance and the function is plotted for $y = +1$. When $\beta\rightarrow\infty$, the function approaches the hinge-loss. An example of the soft-margin case (see Eq.~\ref{eqn:soft_margin_cost}), is also plotted for $\beta=5$ curve. The figure is best seen in color.} \label{fig:logisticLoss} \end{figure} Along the general practice in metric learning, we regularize the metric $\Mat{M}$ by $r(\Mat{M})$. The divergences derived from the $\log\det(\cdot)$ function are familiar faces for regularizing Mahalanobis metrics in the litrature~\citep{Davis_ICML07,Saenko_ECCV10}. Among possible choices, we make use of the Stein divergence~\citep{Anoop_TPAMI13} in this work. Hence, \begin{equation} r(\Mat{M}) = \frac{1}{p}\delta_s(\Mat{M},\Mat{I_p}). \end{equation} \noindent Where, \begin{equation} \delta_s(\Mat{P},\Mat{Q}) = \log\det\Big(\frac{\Mat{P}+\Mat{Q}}{2}\Big) - \frac{1}{2}\log\det \big(\Mat{P}\Mat{Q}\big), \end{equation} for $\Mat{P},\Mat{Q} \in \SPD{}$. Our motivation in using the Stein divergence stems from its unique properties. Among others, Stein divergence is symmetric, invariant to affine transformation and closely related to geodesic distances on the SPD manifold~\citep{Anoop_TPAMI13,Harandi_SPD16,Cherrian_LogDet16}. \subsection*{Soft Margin Extension} For large values of $\beta$, the cost in Eq.~\ref{eqn:distCost} seeks the distances of similar pairs to be less than 1 while simultaneously it deems the distances of dissimilar pairs to exceed 1. This hard-margin in the design of $\ell_\beta\big(\cdot,\cdot,\Vec{x},\cdot \big)$ is not desirable. For example, with a large number of pairs, it is often the case to have outliers. Forcing outliers to fit into the hard margins can result in overfitting. As such, we propose a soft-margin extension of Eq.~\ref{eqn:ell_b}. The soft-margins are implemented by associating a non-negative slack variable $\epsilon_k$ to a pair according to \begin{align} \label{eqn:soft_margin_cost} \mathcal{L}_{d} &= \frac{1}{N_p} \sum\limits_{k=1}^{N_p} \ell_\beta\big(\Mat{M},y_k,\Vec{z}_{1,k} - \Vec{z}_{2,k},1+y_k\epsilon_k \big) + r(\Mat{M}) + \frac{1}{N_p} \sqrt{\sum \epsilon_k^2}, \end{align} where a regularizer on the slack variables is also envisaged. \subsection*{Matching Statistical Properties} A form of incompatibility between domains is due to their statistical discrepancies. Matching the first order statistics of two domains for the purpose of adaptation is studied in~\cite{Pan_TNN11,Masha_ICCV14,Tsai_CVPR16}\footnote{ The use of Maximum Mean Discrepancy (MMD)~\citep{Borgwardt_Bio06} for domain adaptation is a well-practiced idea in the literature (see for example~\cite{Pan_TNN11,Masha_ICCV14,Tsai_CVPR16}). Empirically, determining MMD boils down to computing the distance between domain averages when domain samples are lifted to a reproducing kernel Hilbert space. Some studies claim matching the first order statistics is a weaker form of domain adaptation through MMD. We do not support this claim and hence do not see our solution as a domain adaptation method by minimizing the MMD.}. In our framework, matching domain averages can be achieved readily. In particular, let $\bar{\Vec{x}}^s_i = \Vec{x}^s_i - \Vec{m}_s$ and $\bar{\Vec{x}}^t_j = \Vec{x}^t_j - \Vec{m}_t$ be the centered source and target samples with $\Vec{m}_s$ and $\Vec{m}_t$ being the mean of corresponding domains. It follows easily that the domain means in the latent space are zero\footnote{We note that $\sum \Mat{W}_s^T\bar{\Vec{x}}^s_i = \Mat{W}_s^T\sum\bar{\Vec{x}}^s_i = \Vec{0}$. This holds for the target domain as well.} and hence matching is achieved. To go beyond first order statistics, we propose to match the second order statistics (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, covariance matrices) as well. The covariance of a domain reflects the relationships between its features. Hence, matching covariances of source and target domains in effect improves the cross feature relationships. We capture the mismatch between source and target covariances in the latent space using the $\mathcal{L}_{u}$ loss in Eq.~\ref{eqn:generalCost}. Given the fact that covariance matrices are points on the SPD manifold, we make use of the Stein divergence to measure their differences. This leads us to define $\mathcal{L}_{u}$ as \begin{equation} \label{eqn:statCost} \begin{split} \mathcal{L}_{u} = \frac{1}{p}\delta_s(\Mat{W}_s^T \Mat{\Sigma}_s\Mat{W}_s,\Mat{W}_t^T\Mat{\Sigma}_{t}\Mat{W}_t), \end{split} \end{equation} with $\Mat{\Sigma}_s \in \SPD{s}$ and $\Mat{\Sigma}_t \in \SPD{t}$ being the covariance matrices of the source and target domains, respectively. We emphasize that matching the statistical properties as discussed above is an unsupervised technique, enabling us to address unsupervised DA. \subsection{Classification Protocol} Upon learning $\Mat{W}_s$, $\Mat{W}_t$, $\Mat{M}$, training samples from the source and target (if available) domains are mapped to the latent space using $\Mat{W}_s\Mat{M}^{\frac{1}{2}}$ and $\Mat{W}_t\Mat{M}^{\frac{1}{2}}$, respectively. For a query from the target domain $\Vec{x}_q^t$, $\Mat{M}^{\frac{1}{2}}\Mat{W}_t^T\Vec{x}_q^t$ is its latent space representation which is subsequently classified by a nearest neighbor classifier. \section{Optimization} \label{sec:opt} The objective of our algorithm is to learn the transformation parameters ($\Mat{W}_s$ and $\Mat{W}_t$), the metric $\Mat{M}$ and slack variables $\epsilon_1,\epsilon_2,...\epsilon_{N_p}$ (see Eq.~\ref{eqn:soft_margin_cost} and Eq.~\ref{eqn:statCost}). Inline with the general practice of dimensionality reduction, we propose to have orthogonality constraints on $\Mat{W}_s$ and $\Mat{W}_t$. That is $\Mat{W}_s^T\Mat{W}_s = \Mat{W}_t^T\Mat{W}_t = \mathbf{I}_p$. We elaborate how orthogonality constraint improves the discriminatory power of the proposed framework later in our experiments. The problem depicted in Eq.~\ref{eqn:generalCost} is indeed a non-convex and constrained optimization problem. One may resort to the method of Projected Gradient Descent (PGD)~\citep{Boyd_Convex04} to minimize Eq.~\ref{eqn:generalCost}. In PGD, optimization is proceed by projecting the gradient-descent updates onto the set of constraints. For example, in our case, we can first update $\Mat{W}_s$ by ignoring the orthogonality constraint on $\Mat{W}_s$ and then project the result onto the set of orthogonal matrices using eigen-decomposition. As such, optimization can be performed by alternatingly updating $\Mat{W}_s$, $\Mat{W}_t$, the metric $\Mat{M}$ and slack variables using PGD. In PGD, to perform the projection, the set of constraints needs to be closed though in practice one can resort to open sets. For example, the set of SPD matrices is open though one can project a symmetric matrix onto this set using eigen-decomposition. Empirically, PGD showed an erratic and numerically unstable behavior in addressing our problem. This can be attributed to the non-linear nature of Eq.~\ref{eqn:generalCost}, existence of open-set constraints in the problem or perhaps the combination of both. To alleviate the aforementioned difficulty, we propose a more principle approach to minimize Eq.~\ref{eqn:generalCost} by making use of the Riemannian optimization technique. We take a short detour and briefly describe the Riemannian optimization methods below. \subsection*{Optimization on Riemannian manifolds.} Consider a non-convex constrained problem in the form \begin{align} \mathrm{minimize}~f(\Vec{x}) \notag\\ \mathrm{s.t.}~~~\Vec{x} \in \mathcal{M}\;, \label{eqn:opt_riemannian_manifold} \end{align} where $\mathcal{M}$ is a Riemannian manifold, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, informally, a smooth surface that locally resembles a Euclidean space. Optimization techniques on Riemannian manifolds (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, Conjugate Gradient) start with an initial solution $\Vec{x}^{(0)} \in \mathcal{M}$, and iteratively improve the solution by following the geodesic identified by the gradient. For example, in the case of Riemannian Gradient Descent Method (RGDM), the updating rule reads \begin{equation} \Vec{x}^{(t+1)} = \tau_{\Vec{x}^{(t)}} \big( -\alpha~\mathrm{grad}~f(\Vec{x}^{(t)})\big)\;, \end{equation} with $\alpha > 0$ being the algorithm's step size. Here, $\tau_{\Vec{x}} ( \cdot): T_\Vec{x}\mathcal{M} \to \mathcal{M}$, is called the retraction\footnote{Strictly speaking and in contrast with the exponential map, a retraction only guarantees to pull a tangent vector on the geodesic locally, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, close to the origin of the tangent space. Retractions, however, are typically easier to compute than the exponential map and have proven effective in Riemannian optimization~\citep{Absil_Book}.} and moves the solution along the descent direction while assuring that the new solution is on the manifold $\mathcal{M}$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, it is within the constraint set. $T_\Vec{x}\mathcal{M}$ is the tangent space of $\mathcal{M}$ at $\Vec{x}$ and can be thought of as a vector space with its vectors being the gradients of all functions defined on $\mathcal{M}$. We defer more details on Riemannian optimization techniques to the \textbf{Appendix A}. As for now, it suffices to say that to perform optimization on the Riemannian manifolds, the form of Riemannian gradient, retraction and the gradient of the objective with respect to its parameters (shown by $\nabla$) are required. The constraints in Eq.\ref{eqn:generalCost} are orthogonality (transformations $\Mat{W}_s$ and $\Mat{W}_t$) and p.d. for metric $\Mat{M}$. The geometry of these constraints are captured by the Stiefel~\citep{James_Stiefel76,Harandi_GB16} and SPD~\citep{Harandi_SPD16,Cherrian_SPD2016} manifolds, formally defined as \begin{definition}[The Stiefel Manifold] The set of ($n \times p$)-dimensional matrices, $p \leq n$, with orthonormal columns endowed with the Frobenius inner product\footnote{% Note that the literature is divided between this choice and another form of Riemannian metric. See~\cite{Edelman_1998} for details.} forms a compact Riemannian manifold called the Stiefel manifold $\ST{n}{p}$~\citep{Absil_Book}. \begin{equation} \ST{n}{p} \triangleq \{\Mat{W} \in \mathbb{R}^{n \times p}: \Mat{W}^T\Mat{W} = \mathbf{I}_p\}\;. \label{eqn:stiefel_manifold} \end{equation} \end{definition} \begin{definition}[The SPD Manifold] The set of ($p \times p$) dimensional real, SPD matrices endowed with the Affine Invariant Riemannian Metric (AIRM)~\citep{Pennec_IJCV06} forms the SPD manifold $\SPD{p}$. \begin{equation} \hspace{-0.5ex}\SPD{p} \triangleq \{\Mat{M} \in \mathbb{R}^{p \times p} \hspace{-0.5ex}: \Vec{v}^T\Mat{M}\Vec{v} >0,~\hspace{-0.5ex}\forall \Vec{v} \in \mathbb{R}^p-\{\Vec{0}_p\}\}. \label{eqn:spd_manifold} \end{equation} \end{definition} Updating $\Mat{W}_s$, $\Mat{W}_t$ and $\Mat{M}$ and slacks can be done alternatively using Riemannian optimization. As mentioned above, the ingredients for doing so are \textbf{1.} the Riemannian tools for the Stiefel and SPD manifolds along \textbf{2.} the form of gradients of the objective with respect to its parameters. To do complete justice, in Table.~\ref{tab:ReimannianGrads} we provide the Riemmanian metric, form of Riemannian gradient and retraction for the Stiefel and SPD manifolds. In Table.~\ref{tab:gradientsComputed}, the gradient of Eq.~\ref{eqn:generalCost} with respect to $\Mat{W}_s$, $\Mat{W}_t$ and $\Mat{M}$ and slacks is provided. The detail of derivations can be found in the \textbf{Appendix B}. A tiny note about the slacks worth mentioning. To preserve the non-negativity constraint on $\epsilon_k$, we define $\epsilon_k = e^{v_k}$ and optimize on $v_k$ instead. This in turn makes optimization for the slacks an unconstrained problem. \begin{remark} From a geometrical point of view, we can make use of the product topology of the parameter space to avoid alternative optimization. More specifically, the set \begin{equation} \label{e:prodTopology} \mathcal{M}_{prod.} = \ST{s}{p} \times \ST{t}{p} \times \SPD{p} \times \mathbb{R}^{N_p}, \end{equation} can be given the structure of a Riemannian manifold using the concept of product topology~\citep{Absil_Book}. \end{remark} \begin{remark} In Fig.~\ref{fig:OptimizationCost}, we compare the convergence behavior of PGD, alternating Riemannian optimization and optimization using the product geometry. While optimization on $\mathcal{M}_{prod.}$ convergences faster, the alternating method results in a lower loss. This behavior resembles the difference between the stochastic gradient descent compared to its batch counterpart. \end{remark} \begin{remark} The complexity of the optimization depends on the number of labeled pairs. One can always resort to a stochastic solution~\cite{Song_CVPR16,Sa_ICML15,Bonnabel_Stoch13} by sampling from the set of similar/dissimilar pairs if addressing a very large-scale problem. In our experiments, we did not face any difficulty optimizing with an i7 desktop machine with 32GB of memory. \end{remark} \begin{table*}[t] \caption{Riemannian metric, gradient and retraction on $\ST{n}{p}$ and $\SPD{p}$. Here, $\mathrm{uf}(\Mat{A}) = \Mat{A}(\Mat{A}^T\Mat{A})^{-1/2}$, which yields an orthogonal matrix, $\mathrm{sym}(\Mat{A}) = \frac{1}{2}(\Mat{A} + \Mat{A}^T)$ and $\mathop{\rm expm}\nolimits(\cdot)$ denotes the matrix exponential.} \label{tab:ReimannianGrads} \small \centering \begin{tabular}{|c|c|c|} \hline {\bf } &{\bf $\ST{n}{p}$} &{\bf $\SPD{p}$}\\ \hline Matrix representation &$\Mat{W} \in \mathbb{R}^{n \times p}$ &$\Mat{M} \in \mathbb{R}^{p \times p}$\\ Riemannian metric &$g_\nu(\xi,\varsigma) = \mathop{\rm Tr}\nolimits(\xi^T\varsigma)$ &$g_\mathcal{S}(\xi,\varsigma) = \mathop{\rm Tr}\nolimits \left( \Mat{M}^{-1} \xi \Mat{M}^{-1}\varsigma\right)$\\ Riemannian gradient &$\nabla_\Mat{W}(f) - \Mat{W}\mathrm{sym}\Big(\Mat{W}^T \nabla_\Mat{W}(f)\Big)$ &$\Mat{M}\mathrm{sym}\Big(\nabla_\Mat{M}(f)\Big)\Mat{M}$\\ Retraction &$\mathrm{uf}(\Mat{W} + \xi)$ &$\Mat{M}^\frac{1}{2}\mathop{\rm expm}\nolimits(\Mat{M}^{-\frac{1}{2}}\xi\Mat{M}^{-\frac{1}{2}})\Mat{M}^\frac{1}{2}$ \\ \hline \end{tabular} \label{tab:riemannian_tools} \end{table*} \begin{figure}[t!] \centering \includegraphics[width=0.52\textwidth]{Optimization.jpg} \caption{Optimizing Eq.~\ref{eqn:generalCost} using PGD (red curve), Riemannian gradient descent using alternating approach (blue curve) and product topology (green curve). Optimization using the product topology converges faster but a lower cost can be attained using alternating Riemannian optimization. } \label{fig:OptimizationCost} \end{figure} \begin{table*}[] \centering \caption{Gradients of soft-margin $\ell_\beta$ and $\mathcal{L}_{u}$ w.r.t. the model parameters and slack variables. Without less of generality we only consider a labeled similar ($y_k = +1$) pair $\Vec{x}^s_i$ and $\Vec{x}^t_j$. Here, $r = \exp{\big(\beta \big((\Mat{W}_s^T\Vec{x}^s_i - \Mat{W}_t^T\Vec{x}^t_i)^T\Mat{M}(\Mat{W}_s^T\Vec{x}^s_i - \Mat{W}_t^T\Vec{x}^t_i) - 1 -e^{v_k}\big)}\big)$.} \label{tab:gradientsComputed} \scalebox{1}{ \begin{tabular}{|c|l|} \hline $ \nabla_{\Mat{W}_s}\ell_\beta $ & $\frac{2}{N_p}(1 + r^{-1})^{-1}\Vec{x}^s_i({\Vec{x}^s_i}^T\Mat{W}_s - {\Vec{x}^t_j}^T\Mat{W}_t)\Mat{M} $ \\ \hline $\nabla_{\Mat{W}_t}\ell_\beta$ & $\frac{2}{N_p}(1 + r^{-1})^{-1}\Vec{x}^t_j({\Vec{x}^t_j}^T\Mat{W}_t - {\Vec{x}^s_i}^T\Mat{W}_s)\Mat{M} $ \\ \hline $\nabla_{\Mat{M}}\ell_\beta$ & $\frac{1}{N_p}(1 + r^{-1})^{-1}\big(\Mat{W}_s^T{\Vec{x}^s_i}-\Mat{W}_t^T\Vec{x}^t_j\big) \big({\Vec{x}^s_i}^T\Mat{W}_s - {\Vec{x}^t_j}^T\Mat{W}_t\big)$ \\ \hline $\nabla_{v_k}\ell_\beta$ & $\frac{-1}{N_p}e^{v_k}(1 + r^{-1})^{-1}$ \\ \hline $\nabla_{\Mat{W}_s}\mathcal{L}_{u}$ & $ \frac{1}{p}\Mat{\Sigma}_s\Mat{W}_s \Big(2\big(\Mat{W}_s^T\Mat{\Sigma}_s\Mat{W}_s + \Mat{W}_t^T\Mat{\Sigma}_t\Mat{W}_t\big)^{-1} - \big( \Mat{W}_s^T\Mat{\Sigma}_s\Mat{W}_s \big)^{-1}\Big)$ \\ \hline \end{tabular}} \end{table*} \section{Related Work} \label{sec:related_work} The literature on domain adaptation spans a very broad range (see~\cite{Patel_Survey15} for a recent survey). Our solution falls under the category of domain adaptation by subspace learning (DA-SL). As such, we confine our review only to methods under the umbrella of DA-SL. One notable example of constructing a latent space is the work of~\cite{Daume_DA10}. In particular, the authors propose to use two fixed and predefined transformations to project source and target data to a common and higher-dimensional space. As a requirement, the method only accepts domains with the same dimensionality and hence cannot be directly used to adapt heterogeneous domains. \cite{Gopalan_ICCV11} observed that the geodesic connecting the source and target subspaces conveys useful information for DA and proposed the Sampling Geodesic Flow (SGF) method~\citep{Gopalan_ICCV11}. The Geodesic Flow Kernel (GFK) is an improvement over the SGF technique where instead of sampling a few points on the geodesic, the whole curve is used for domain adaptation~\citep{Gong_CVPR12}. In both methods, the domain subspaces are fixed and obtained by Principal Component Analysis (PCA) or Partial Least Square regression (PLS)~\citep{Krishnan_PLS11}. In contrast to our solution, in SGF and GFK learning the domain subspaces is disjoint from the knowledge transfer algorithm. In our experiments, we will see that the subspaces determined by our method can even boost the performance of GFK, showing the importance of joint learning of domain subspaces and the knowledge transfer scheme. In~\cite{Ni_Subspace13} dictionary learning is used for interpolating the intermediate subspaces. Domain adaptation by fixing the subspace/representation of one of the domains is a popular theme in many recent works, as it simplifies the learning scheme. Examples are the max-margin adaptation~\citep{Hoffman_IJCV14,Duan_ICML12}, the metric/similarity learning of~\cite{Saenko_ECCV10} and its kernel extension~\citep{Kulis_CVPR11}, the landmark approach of~\cite{Tsai_CVPR16}, the alignment technique of~\cite{Basura_ICCV13,Fernando_PRL15}, correlation matching of~\cite{SUN_Return15} and methods that use maximum mean discrepancy (MMD)~\citep{Borgwardt_Bio06} for domain adaptation~\citep{Pan_TNN11,Masha_ICCV14}. In contrast to the above methods, some studies opt to learn the domain representation along the knowledge transfer method jointly. Two representative works are the HeMap~\citep{Shi_ICDM10} and manifold alignment~\citep{Wang_IJCAI11}. The HeMap learns two projections to minimize the instance discrepancies~\citep{Shi_ICDM10}. The problem is however formulated such that equal number of source and target instances is required to perform the training. The manifold alignment algorithm of~\cite{Wang_IJCAI11} attempts to preserve the label structure in the latent space. However, it is essential for the algorithm to have access to labeled data in both source and target domains. Our solution learns all transformations to the latent space. We do not resort at subspace representations learned disjointly to the DA framework. With this use of the latent space, our algorithm is not limited for applications where source and target data have similar dimensions or structure. \section{Experimental Evaluations} \label{sec:experiments} We run extensive experiments on both semi-supervised and unsupervised settings, spanning from the handcrafted features (SURF) to the current state-of-the-art deep-net features (VGG-Net). For comparisons, we use the implementations made available by the original authors. Our method is denoted by \textbf{ILS}. \subsection{Implementation Details} Since the number of dissimilar pairs is naturally larger than the number of similar pairs, we randomly sample from the different pairs to keep the sizes of these two sets equal. We initialize the projection matrices $\Mat{W_s}$, $\Mat{W_t}$ with PCA, following the transductive protocol~\cite{Gong_CVPR12, Basura_ICCV13, Hoffman_IJCV14, Tsai_CVPR16}. For the semi-supervised setting, we initialize $\Mat{M}$ with the Mahalanobis metric learned on the similar pair covariances~\citep{Koestinger_CVPR12}, and for the unsupervised setting, we initialize it with the identity matrix. For all our experiments we have $\lambda = 1$. We include an experiment showing our solution's robustness to $\lambda$ in the supplementary material. We use the toolbox provided by~\cite{manopt} for our implementations. \begin{remark} To have a simple way of determining $\beta$ in Eq.~\ref{eqn:ell_b}, we propose a heuristic which is shown to be effective in our experiments. To this end, we propose to set $\beta$ to the reciprocal of the standard deviation of the similar pair distances. \end{remark} \subsection{Semi-supervised Setting} \begin{figure}[t!] \centering \includegraphics[width=0.8\textwidth]{AWCD2.jpg} \caption{Samples from the Office$+$Caltech10 dataset~\citep{Saenko_ECCV10, Gong_CVPR12}. Although the DSLR and Webcam images depict similar content, they have considerably different resolutions.} \label{fig:AWDC} \end{figure} In our semi-supervised experiments, we follow the standard setup on the Office$+$Caltech10 dataset with the train/test splits provided by~\cite{Hoffman_ICLR2013}. The Office+Caltech10 dataset contains images collected from 4 different sources (see Fig.~\ref{fig:AWDC}) and 10 object classes. The corresponding domains are \textbf{A}mazon, \textbf{W}ebcam, \textbf{D}SLR, and \textbf{C}altech. We use a subspace of dimension 20 for DA-SL algorithms. We employ SURF~\citep{Bay_SURF06} for the handcrafted feature experiments. We extract VGG-Net features with the network model of~\citep{Simonyan_VGG14} for the deep-net feature experiments\footnote{The same SURF and VGG-Net features are used for the unsupervised experiments as well.}. We compare our performance with the following benchmarks: \noindent\textbf{1-NN-t and SVM-t :} Basic Nearest Neighbor (1-NN) and linear SVM classifiers trained only on the target domain. \noindent\textbf{HFA~\citep{Duan_ICML12} :} This method employs latent space learning based on the max-margin framework. As in its original implementation, we use the RBF kernel SVM for its evaluation. \noindent\textbf{MMDT~\citep{Hoffman_IJCV14} :} This method jointly learns a transformation between the source and target domains along a linear SVM for classification. \noindent\textbf{CDLS~\citep{Tsai_CVPR16} :} This is the cross-domain landmark search algorithm. We use the parameter setting ($\delta = 0.5$ in the notation of~\cite{Tsai_CVPR16}) recommended by the authors. Table~\ref{tab:SemiSuper_SURF}, Table~\ref{tab:SemiSuper_VGGFC6} and Table~\ref{tab:Super_FC7} report the performances using the handcrafted SURF, VGG-FC6 and VGG-FC7 features, respectively. For the SURF features our solution achieves the best performance in 7 out 12 cases. For the VGG-FC6 and VGG-FC7 features, our solution tops in 9 and 7 sets respectively. It seems that in comparison to VGG-FC6, VGG-FC7 features are less discriminative for all the DA algorithms. We notice the \textbf{1-NN-t} baseline performs the worst for both SURF and the VGG-Net features. Hence, it is clear that the used features do not favor the nearest neighbor classifier. We observe that \textbf{C}altech and \textbf{A}mazon domains contain the largest number of test instances. Although the performances of all tested methods decrease on these domains, particularly on \textbf{C}altech, our method achieves the top rank in almost all domain transformations. \begin{table*}[!t] \centering \small \caption{Semi-supervised domain adaptation results using SURF features on Office$+$Caltech10~\cite{Gong_CVPR12} dataset with the evaluation setup of~\cite{Hoffman_IJCV14}. The best score (in bold blue), the second best (in blue). } \label{tab:SemiSuper_SURF} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & A$\rightarrow$W & A$\rightarrow$D & A$\rightarrow$C & W$\rightarrow$A & W$\rightarrow$D & W$\rightarrow$C & D$\rightarrow$A & D$\rightarrow$W & D$\rightarrow$C & C$\rightarrow$A & C$\rightarrow$W & C$\rightarrow$D \\ \hline 1-NN-t & 34.5 & 33.6 & 19.7 & 29.5 & 35.9 & 18.9 & 27.1 & 33.4 & 18.6 & 29.2 & 33.5 & 34.1 \\ SVM-t & 63.7 & {\color{Blue}57.2} & 32.2 & 46.0 & 56.5 & 29.7 & 45.3 & 62.1 & 32.0 & 45.1 & 60.2 & 56.3 \\ HFA & 57.4 & 55.1 & 31.0 & \textbf{\color{blue}56.5} & 56.5 & 29.0 & 42.9 & 60.5 & 30.9 & 43.8 & 58.1 & 55.6 \\ MMDT & {\color{Blue}64.6} & 56.7 & {\color{Blue}36.4} & 47.7 & {\color{Blue}67.0} & 32.2 & 46.9 & {\color{Blue}74.1} & 34.1 & 49.4 & {\color{Blue}63.8} & {\color{Blue}56.5} \\ CDLS & \textbf{\color{blue}68.7} & \textbf{\color{blue}60.4} & 35.3 & 51.8 & 60.7 & {\color{Blue}33.5} & {\color{Blue}50.7} & 68.5 & {\color{Blue}34.9} & {\color{Blue}50.9} & \textbf{\color{blue}66.3} & \textbf{\color{blue}59.8} \\ \textbf{ILS} (1-NN) & 59.7 & 49.8 & \textbf{\color{blue}43.6} & {\color{Blue}54.3} & \textbf{\color{blue}70.8} & \textbf{\color{blue}38.6} & \textbf{\color{blue}55.0} & \textbf{\color{blue}80.1} & \textbf{\color{blue}41.0} & \textbf{\color{blue}55.1} & 62.9 & 56.2 \\ \hline \end{tabular}} \vspace{-1mm} \end{table*} \begin{table*}[!t] \centering \small \caption{Semi-supervised domain adaptation results using VGG-FC6 features on Office$+$Caltech10~\cite{Gong_CVPR12} dataset with the evaluation setup of~\cite{Hoffman_IJCV14}. The best (in bold blue), the second best (in blue).} \label{tab:SemiSuper_VGGFC6} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & A$\rightarrow$W & A$\rightarrow$D & A$\rightarrow$C & W$\rightarrow$A & W$\rightarrow$D & W$\rightarrow$C & D$\rightarrow$A & D$\rightarrow$W & D$\rightarrow$C & C$\rightarrow$A & C$\rightarrow$W & C$\rightarrow$D \\ \hline 1-NN-t & 81.0 & 79.1 & 67.8 & 76.1 & 77.9 & 65.2 & 77.1 & 81.7 & 65.6 & 78.3 & 80.2 & 77.7 \\ SVM-t & 89.1 & \textbf{\color{blue}88.2} & 77.3 & 86.5 & 87.7 & 76.3 & 87.3 & 88.3 & 76.3 & 87.5 & 87.8 & 84.9 \\ HFA & 87.9 & 87.1 & 75.5 & 85.1 & 87.3 & 74.4 & 85.9 & 86.9 & 74.8 & 86.2 & 86.0 & \textbf{\color{blue}87.0} \\ MMDT & 82.5 & 77.1 & {\color{Blue}78.7} & 84.7 & 85.1 & 73.6 & 83.6 & 86.1 & 71.8 & 85.9 & 82.8 & 77.9 \\ CDLS & \textbf{\color{blue}91.2} & 86.9 & 78.1 & {\color{Blue}87.4} & {\color{Blue}88.5} & {\color{Blue}78.2} & {\color{Blue}88.1} & {\color{Blue}90.7} & {\color{Blue}77.9} & {\color{Blue}88.0} & {\color{Blue}89.7} & 86.3 \\ \textbf{ILS} (1-NN) & {\color{Blue}90.7} & {\color{Blue}87.7} & \textbf{\color{blue}83.3} & \textbf{\color{blue}88.8} & \textbf{\color{blue}94.5} & \textbf{\color{blue}82.8} & \textbf{\color{blue}88.7} & \textbf{\color{blue}95.5} & \textbf{\color{blue}81.4} & \textbf{\color{blue}89.7} & \textbf{\color{blue}91.4} & {\color{Blue}86.9} \\ \hline \end{tabular}} \vspace{-1mm} \end{table*} \smallskip \begin{table*}[!t] \small \centering \caption{Semi-supervised domain adaptation results using VGG-FC7 features on Office$+$Caltech10~\cite{Gong_CVPR12} dataset with the evaluation setup of~\cite{Hoffman_IJCV14}. The best (in bold blue), the second best (in blue).} \label{tab:Super_FC7} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & A$\rightarrow$W & A$\rightarrow$D & A$\rightarrow$C & W$\rightarrow$A & W$\rightarrow$D & W$\rightarrow$C & D$\rightarrow$A & D$\rightarrow$W & D$\rightarrow$C & C$\rightarrow$A & C$\rightarrow$W & C$\rightarrow$D \\ \hline 1-NN-t & 81.8 & 78.2 & 68.3 & 77.8 & 77.6 & 67.4 & 78.1 & 81.5 & 66.9 & 79.0 & 80.6 & 77.4 \\ SVM-t & 87.5 & \textbf{{\color{blue}85.4}} & 76.8 & 86.2 & 85.6 & 75.8 & {\color{blue}87.0} & 87.1 & 76.0 & 87.1 & 86.4 & 84.4 \\ HFA & 86.6 & {\color{blue}85.3} & 75.2 & 84.9 & 85.5 & 74.8 & 85.8 & 86.5 & 75.1 & 86.0 & 85.3 & \textbf{{\color{blue}84.8}} \\ MMDT & 76.9 & 73.3 & 78.1 & 83.6 & 79.5 & 72.2 & 82.3 & 83.8 & 71.7 & 85.3 & 77.8 & 72.6 \\ CDLS & {\color{blue}\textbf{90.0}} & 85.0 & {\color{blue}78.5} & {\color{blue}87.2} & {\color{blue}86.5} & {\color{blue}79.0} & \textbf{{\color{blue}87.7}} & {\color{blue}89.5} & \textbf{{\color{blue}78.8}} & {\color{blue}87.8} & \textbf{{\color{blue}89.7}} & {\color{blue}84.6} \\ \textbf{ILS} (1-NN) & {\color{blue}89.3} & 84.0 & \textbf{{\color{blue}81.9}} & \textbf{{\color{blue}88.4}} & \textbf{{\color{blue}91.0}} & \textbf{{\color{blue}80.8}} & 86.9 & \textbf{{\color{blue}94.4}} & \textbf{{\color{blue}78.8}} & \textbf{{\color{blue}88.9}} & {\color{blue}88.7} & 83.3 \\ \hline \end{tabular}} \vspace{-1mm} \end{table*} \subsection{Unsupervised Setting} In the unsupervised domain adaptation problem, only labeled data from the source domain is available~\citep{Basura_ICCV13, Gong_CVPR12}. We perform two sets of experiments for this setting. (1) We evaluate the object recognition performance on the Office+Caltech10 dataset. Similar to the semi-supervised settings, we use the SURF and VGG-Net features. Our results demonstrate that the learned transformations by our method are superior domain representations. (2) We analyze our performance when the domain discrepancy is gradually increased. This experiment is performed on the PIE-Face dataset. We compare our method with the following benchmarks: \smallskip \noindent\textbf{1-NN-s and SVM-s :} Basic 1-NN and linear SVM classifiers trained only on the source domain. \noindent \textbf{GFK-PLS~\citep{Gong_CVPR12} :} The geodesic flow kernel algorithm where partial least squares (PLS) implementation is used to initialize the source subspace. Results are evaluated on kernel-NNs. \noindent \textbf{SA~\citep{Basura_ICCV13} :} This is the subspace alignment algorithm. Results are evaluated using 1-NN. \noindent \textbf{CORAL~\citep{SUN_Return15} :} The correlation alignment algorithm that uses a linear SVM on the similarity matrix formed by correlation matching. \subsubsection{Office+Caltech10 (Unsupervised)} We follow the original protocol provided by~\cite{Gong_CVPR12} on Office+Caltech10 dataset. Note that several baselines, determine the best dimensionality per domain to achieve their maximum accuracies on SURF features. We observed that a dimensionality in the range [20,120] provides consistent results for our solution using SURF features. For VGG features we empirically found the dimensionality of 20 suits best for the compared DA-SL algorithms. Table.~\ref{tab:Unsuper_SURF}, Table.~\ref{tab:Unsuper_VGG} and Table.~\ref{tab:Unsuper_FC7} present the unsupervised setting results using the SURF, VGG-FC6 and VGG-FC7 features. For all feature types, our solution yields the best performance in 8 domain transformations out of 12. Similarly to the semi-supervised experiments we notice the VGG-FC7 is less favorable for DA algorithms. \smallskip \begin{table*}[!t] \small \centering \caption{Unsupervised domain adaptation results using SURF features on Office$+$Caltech10~\cite{Gong_CVPR12} dataset with the evaluation setup of~\cite{Gong_CVPR12}.The best (in bold blue), the second best (in blue).} \label{tab:Unsuper_SURF} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & A$\rightarrow$W & A$\rightarrow$D & A$\rightarrow$C & W$\rightarrow$A & W$\rightarrow$D & W$\rightarrow$C & D$\rightarrow$A & D$\rightarrow$W & D$\rightarrow$C & C$\rightarrow$A & C$\rightarrow$W & C$\rightarrow$D \\ \hline 1-NN-s & 23.1 & 22.3 & 20.0 & 14.7 & 31.3 & 12.0 & 23.0 & 51.7 & 19.9 & 21.0 & 19.0 & 23.6 \\ SVM-s & 25.6 & 33.4 & 35.9 & 30.4 & 67.7 & 23.4 & 34.6 & 70.2 & 31.2 & 43.8 & 30.5 & 40.3 \\ GFK-PLS & 35.7 & 35.1 & {\color{Blue}37.9} & 35.5 & 71.2 & 29.3 & 36.2 & 79.1 & 32.7 & 40.4 & 35.8 & {\color{Blue}41.1} \\ SA & 38.6 & 37.6 & 35.3 & 37.4 & {\color{Blue}80.3} & 32.3 & 38.0 & {\color{Blue}83.6} & 32.4 & 39.0 & 36.8 & 39.6 \\ CORAL & {\color{Blue}38.7} & {\color{Blue}38.3} & \textbf{\color{blue}40.3} & {\color{Blue}37.8} & \textbf{\color{blue}84.9} & \textbf{\color{blue}34.6} & {\color{Blue}38.1} & \textbf{\color{blue}85.9} & {\color{Blue}34.2} & {\color{Blue}47.2} & {\color{Blue}39.2} & 40.7 \\ \textbf{ILS} (1-NN) & \textbf{\color{blue}40.6} & \textbf{\color{blue}41.0} & 37.1 & \textbf{\color{blue}38.6} & 72.4 & {\color{Blue}32.6} & \textbf{\color{blue}38.9} & 79.1 & \textbf{\color{blue}36.9} & \textbf{\color{blue}48.6} & \textbf{\color{blue}42.0} & \textbf{\color{blue}44.1} \\ \hline \end{tabular}} \vspace{-1mm} \end{table*} \begin{table*}[!t] \centering \small \caption{Unsupervised domain adaptation results using VGG-FC6 features on Office$+$Caltech10~\cite{Gong_CVPR12} dataset with the evaluation setup of~\cite{Gong_CVPR12}.The best (in bold blue), the second best (in blue).} \label{tab:Unsuper_VGG} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & A$\rightarrow$W & A$\rightarrow$D & A$\rightarrow$C & W$\rightarrow$A & W$\rightarrow$D & W$\rightarrow$C & D$\rightarrow$A & D$\rightarrow$W & D$\rightarrow$C & C$\rightarrow$A & C$\rightarrow$W & C$\rightarrow$D \\ \hline 1-NN-s & 60.9 & 52.3 & 70.1 & 62.4 & 83.9 & 57.5 & 57.0 & 86.7 & 48.0 & 81.9 & 65.9 & 55.6 \\ SVM-s & 63.1 & 51.7 & 74.2 & 69.8 & 89.4 & 64.7 & 58.7 & 91.8 & 55.5 & 86.7 & 74.8 & 61.5 \\ GFK-PLS & 74.1 & 63.5 & 77.7 & 77.9 & \textbf{\color{blue}92.9} & 71.3 & 69.9 & 92.4 & 64.0 & 86.2 & 76.5 & 66.5\\ SA & {\color{Blue}76.0} & 64.9 & 77.1 & 76.6 & 90.4 & 70.7 & 69.0 & 90.5 & 62.3 & 83.9 & 76.0 & 66.2 \\ CORAL & 74.8 & {\color{Blue}67.1} & \textbf{\color{blue}79.0} & {\color{Blue}81.2} & {\color{Blue}92.6} & {\color{Blue}75.2} & {\color{Blue}75.8} & \textbf{\color{blue}94.6} & {\color{Blue}64.7} & \textbf{\color{blue}89.4} & {\color{Blue}77.6} & {\color{Blue}67.6} \\ \textbf{ILS} (1-NN) & \textbf{\color{blue}82.4} & \textbf{\color{blue}72.5} & {\color{Blue}78.9} & \textbf{\color{blue}85.9} & 87.4 & \textbf{\color{blue}77.0} & \textbf{\color{blue}79.2} & {\color{Blue}94.2} & \textbf{\color{blue}66.5} & {\color{Blue}87.6} & \textbf{\color{blue}84.4} & \textbf{\color{blue}73.0} \\ \hline \end{tabular}} \vspace{-1mm} \end{table*} \begin{table*}[!t] \centering \small \caption{Unsupervised domain adaptation results using VGG-FC7 features on Office$+$Caltech10~\cite{Gong_CVPR12} dataset with the evaluation setup of~\cite{Gong_CVPR12}.The best (in bold blue), the second best (in blue).} \label{tab:Unsuper_FC7} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & A$\rightarrow$W & A$\rightarrow$D & A$\rightarrow$C & W$\rightarrow$A & W$\rightarrow$D & W$\rightarrow$C & D$\rightarrow$A & D$\rightarrow$W & D$\rightarrow$C & C$\rightarrow$A & C$\rightarrow$W & C$\rightarrow$D \\ \hline 1-NN-s & 64.0 & 50.8 & 72.6 & 64.5 & 83.1 & 60.2 & 61.2 & 88.2 & 52.8 & 82.6 & 65.3 & 54.9 \\ SVM-s & 68.0 & 51.8 & 76.2 & 70.1 & 87.4 & 65.5 & 58.7 & 91.2 & 56.0 & 86.7 & 74.8 & 61.3 \\ GFK-PLS & 74.0 & 57.6 & 76.6 & 75.0 & {\color{blue}89.6} & 62.1 & 67.5 & \textbf{{\color{blue}91.9}} & 62.9 & 84.1 & 73.6 & 63.4 \\ SA & {\color{blue}75.0} & 60.7 & 76.2 & 74.6 & {88.8} & 67.5 & 66.0 & 89.5 & 59.4 & 82.6 & 73.6 & 63.2 \\ CORAL & 71.8 & {\color{blue}61.3} & \textbf{{\color{blue}78.6}} & {\color{blue}81.4} & \textbf{{\color{blue}90.1}} & {\color{blue}73.6} & {\color{blue}71.2} & 93.5 & {\color{blue}63.0} & {\textbf{\color{blue}88.6}} & {\color{blue}76.0} & {\color{blue}63.8} \\ \textbf{ILS} (1-NN) & \textbf{{\color{blue}80.9}} & \textbf{{\color{blue}71.3}} & {\color{blue}78.4} & \textbf{{\color{blue}85.7}} & 84.8 & \textbf{{\color{blue}75.1}} & \textbf{{\color{blue}76.5}} & {\color{blue}91.8} & \textbf{{\color{blue}66.2}} & {\color{blue}87.1} & \textbf{{\color{blue}80.1}} & \textbf{{\color{blue}67.1}} \\ \hline \end{tabular}} \end{table*} \noindent \textbf{Learned Transformations as Subspace Representations}: We consider both GFK~\citep{Gong_CVPR12} and SA~\citep{Basura_ICCV13} as DA-SL algorithms. Both these methods make use of PCA subspaces to adapt the domains. To the best of our knowledge, there exists no through studies claiming that PCA is the method of choice to for GFK and SA. As a matter of fact, Gong \emph{et al}\onedot show that the performance of GFK can be improved if PLS\footnote{Despite using labeled data, this method falls under the unsupervised setting since it does not use the labeled target data.} algorithm is employed to define the source subspace~\citep{Gong_CVPR12}. Whether PCA or PLS is used to define the subspaces, identification of the subspaces is disjoint from the domain adaptation technique in GFK and SA. In contrast, transformations in the ILS algorithm are linked to the adaptation process. This makes a curious mind wondering whether the learned transformations in the ILS algorithm capture better structures for adaptation or not. We empirically show that this is indeed the case by using the learned $\Mat{W}_s$ as the source subspace in GFK and SA. \begin{figure}[t] \centering \includegraphics[width=0.7\textwidth]{AccuracyGain_GFK_SA.jpg} \caption{The accuracy gain on Office-Caltech dataset for GFK~\citep{Gong_CVPR12} and SA~\citep{Basura_ICCV13} when their initial PCA subspaces are replaced with PLS and our $W_s$ transformation matrices.} \label{fig:AccGain} \end{figure} Figure~\ref{fig:AccGain} compares the accuracy gains over PCA spaces by using PLS and our $\Mat{W}_s$ initialization. It is clear that the highest classification accuracy gain is obtained by our $\Mat{W}_s$ initialization. This proves that $\Mat{W}_s$ is capable to learn a more favorable subspace representation for domain adaptation. \subsubsection{PIE-Multiview Faces} The PIE Multiview dataset includes face images of 67 individuals captured from different views, illumination conditions, and expressions. In this experiment, we use the views $C27$ (looking forward) as the source domain and $C09$ (looking down), and the views $C05$, $C37$, $C02$, $C25$ (looking towards left in an increasing angle, see Fig.~\ref{fig:pie-faces}) as target domains. We expect the face inclination angle to reflect the complexity of transfer learning. We normalize the images to 32$\times$ 32 pixels and use the vectorized gray-scale images as features. Empirically, we observed that the GFK~\citep{Gong_CVPR12} and SA~\citep{Basura_ICCV13} reach better performances if the features are normalized to have unit $\ell_2$ norm. We therefore use $\ell_2$ normalized features in our evaluations. The dimensionality of the subspaces for all the subspace based methods (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,~\cite{Gong_CVPR12,Basura_ICCV13}) including ours is 100. Table.~\ref{tab:PIE_Results} lists the classification accuracies with increasing angle of inclination. Our solution attains the best scores for 4 views and the second best for the $C09$. With the increasing camera angle, the feature structure changes up to a certain extent. In other words, the features become heterogeneous. However, our algorithm boosts the accuracies even under such challenging conditions. \begin{figure}[t!] \centering \scalebox{0.6}{ \includegraphics[width=0.8\textwidth]{PIE-FACES.jpg}} \caption{Two instances of the PIE-Multiview face data. Here, the view from $C27$ is used as the source domain. Remaining views are considered to be the target for each transformation.} \label{fig:pie-faces} \end{figure} \begin{table}[t!] \centering \small \caption{PIE-Multiview results. The variation of performance w.r.t. face orientations when frontal face images are considered as the source domain } \label{tab:PIE_Results} \scalebox{0.85}{ \begin{tabular}{|r|c|c|c|c|c|} \hline \multicolumn{1}{|c|}{camera pose$\rightarrow$} & C09 & C05 & C37 & C25 & C02 \\ \hline 1-NN-s & 92.5 & 55.7 & 28.5 & 14.8 & 11.0 \\ SVM-s & 87.8 & 65.0 & 35.8 & 15.7 & {\color{Blue}16.7} \\ GFK-PLS & 92.5 & 74.0 & 32.1 & 14.1 & 12.3 \\ SA & \textbf{\color{blue}97.9} & {\color{Blue}85.9} & {\color{Blue}47.9} & {\color{Blue}16.6} & 13.9 \\ CORAL & 91.4 & 74.8 & 35.3 & 13.4 & 13.2 \\ \textbf{ILS} (1-NN) & {\color{Blue}96.6} & \textbf{\color{blue}88.3} & \textbf{\color{blue}72.9} & \textbf{\color{blue}28.4} & \textbf{\color{blue}34.8} \\ \hline \end{tabular}} \end{table} \section{Parameter Sensitivity and Orthogonality} In all the above experiments, we keep $\lambda = 1$ (see Eq.~\ref{eqn:generalCost}). To analyze the sensitivity of our method to the changes in parameter $\lambda$, we performed an experiment using the unsupervised protocol. This is because the statistical loss plays a significant role in establishing the correspondence between the source and the target in the unsupervised DA. We consider two random splits from each of the Office+Caltech10 dataset along VGG-FC6 features here. Our results are shown in Fig.~\ref{fig:ParamSensitivity}. When $\lambda = 0$, no statistical loss term is considered. It is clear that for this case the performance drops considerably. For other values of $\lambda$, the performance is superior and there is little variation in performance. In other words, our method remains robust. We further investigate the benefit of orthogonality constraint on $\Mat{W}_s$ and $\Mat{W}_t$ against free-form and unconstrained transformations. Using the orthogonality constraint provides a considerable performance gain as shown in Fig.~\ref{fig:ParamSensitivity}. While orthogonality makes the optimization more complicated, it seems it guides the learning to better uncovering the form of adaptation. \begin{figure}[!t] \centering \scalebox{0.6}{ \includegraphics[width=1\textwidth]{Sensitivity_Plot.jpg}} \caption{Sensitivity to $\lambda$ in the unsupervised DA.} \label{fig:ParamSensitivity} \end{figure} \section*{Conclusion} In this paper, we proposed a solution for both semi-supervised and unsupervised Domain Adaptation (DA) problems. Our solution learns a latent space in which domain discrepancies are minimized. We showed that such a latent space can be obtained by \textbf{1.} minimizing a notion of discriminatory power over the available labeled data while simultaneously \textbf{2.} matching statistical properties across the domains. To determine the latent space, we modeled the learning problem as a minimization problem on Riemannian manifolds and solved it using optimization techniques on matrix manifolds. Empirically, we showed that the proposed method outperformed state-of-the-art DA solutions in semi-supervised and unsupervised settings. With the proposed framework we see possibilities of extending our solution to large scale datasets with stochastic optimization techniques, multiple source DA and for domain generalization~\citep{Ghifary_TPAMI16,Gan_CVPR16}. In terms of algorithmic extensions we look forward to use dictionary learning~\citep{Koniusz_SparseCode16} and higher order statistics matching.

BkiUfk7xK02iP4WkBtPB

\section{Introduction} \label{Section1} Feedback control laws are ubiquitous in modern science and engineering and can be found in autonomous vehicles, fluid flow control, and network dynamics, among many others. A distinctive feature of feedback laws is their ability to compensate external perturbations in real time. While an offline training phase is often affordable, an operational feedback law must be able to provide a control signal at a rate that is determined by the underlying time scales of the physical system to be controlled. This requirement poses a formidable computational constraint for the synthesis of feedback controls which require an online optimization procedure. A natural approach to generate an optimal feedback law for real-time control is to follow a dynamic programming approach. Here, we solve a nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) for the value function of the optimal control problem under study. This is done in an offline phase, and once the value function has been computed, a feedback law is obtained as by-product. Once online, the cost of implementing an HJB-based control in real time, assuming the current state of the system is available, is reduced to the evaluation of a nonlinear feedback map. Unfortunately, the dynamic programming approach is not suitable for systems described by large-scale dynamics, as the computational complexity of approximating the associated high-dimensional HJB PDEs goes beyond the reach of traditional computational methods. Only very recently, the use of effective computational approaches such as sparse grids~\cite{axelsparse,kang1}, tree structure algorithms~\cite{allatree}, polynomial approximation ~\cite{poli1,KK17,AKK21} tensor decomposition methods~\cite{tensor2,tensor3,tensor4,tensor5}, and representation formulas \cite{chow1,chow3} have addressed the solution of high-dimensional HJB PDEs. Recent works making use of deep learning ~\cite{ml1,ml2,ml3,ml4,ml5,nakazim,nakazim2} anticipate that the synthesis of optimal feedback laws for large-scale dynamics can be a viable path in the near future. An alternative to the dynamic programming approach is the use of a Nonlinear Model Predictive Control (NMPC) framework \cite{larsnmpc}. Here, an optimal open-loop control law is computed at every sampling instant of the dynamics. However, the control action is optimized over a prediction horizon, which is sufficiently large to guarantee asymptotic stability of the closed-loop, but short enough to ensure a computing time that is compatible with the rate at which the control law is called. The NMPC framework embodies the trade-off between optimality and real-time computability. It has been shown \cite{GR2008} that the NMPC corresponds to a relaxation of the dynamic programming approach, in the sense that the NMPC law is suboptimal with respect to the optimal stabilizing feedback law provided by the dynamic programming approach, although its suboptimality can be controlled by increasing the prediction horizon. There exists a third synthesis alternative, which incorporates elements from both dynamic programming and NMPC, known as the State-Dependent Riccati Equation (SDRE) approach \cite{sdrefirst1,sdrefirst2}. The SDRE method originates from the dynamic programming and the HJB PDE associated to infinite horizon optimal stabilization, however, it circumvents its solution by reformulating the feedback synthesis as the sequential solution of state-dependent Algebraic Riccati Equations (ARE), which are updated online along a trajectory. The SDRE feedback is implemented similarly as in NMPC, but the online solution of an optimization problem is replaced by a nonlinear matrix equation. Alternatives to the online formulation include the use of neural networks \cite{WANG98,ABK21} in supervised learning, and reformulating the SDRE synthesis as an optimization problem \cite{Astolfi2020}. However, in this work we will focus on addressing the SDRE synthesis from a numerical linear algebra perspective. The efficient solution of matrix equations arising in feedback control has been subject of extensive research over the last decades, leading to the design of solvers which can effectively be applied to large-scale dynamics such as those arising in the control of systems governed by PDEs (see, e.g., \cite{Antoulas.05,Benner2005a,Benner.Saak.survey13,Binietal.book.12,Simoncini.survey.16,Slowik2007}), and agent-based models \cite{HK18}. Moreover, under certain stabilizability conditions, this feedback law generates a locally asymptotically stable closed-loop and approximates the optimal feedback law from the HJB PDE. The purpose of the present work is to study the design of SDRE feedback laws for the control of nonlinear PDEs, including nonlinear reaction and transport terms. This is a class of problems that are inherently large-scale, and where the presence of nonlinearities renders linear controllers underperformant. In this framework, the use of dynamic programming or NMPC methods is often prohibitively expensive, as for instance, in feedback control for multi-dimensional PDEs. Here instead, we propose and assess different alternatives for control design based on the SDRE approach which can be effectively implemented for two and three dimensional PDEs. The methodology studied in this paper is based on the parametrization of the SDRE synthesis proposed in \cite{BTB00,BLT07}. In these works, different SDRE feedback laws are developed based on the representation of the nonlinear terms in the system dynamics. This is particularly relevant to PDE dynamics, as unlike the nonlinear ODE world, there exists a clear taxonomy of physically meaningful nonlinearities. In some particular cases, the SDRE approach is reduced to a series of offline computations, and a real-time controller only requires a nonlinear feedback evaluation. We explore the limitations of such a parametrization. In the more general case, the SDRE synthesis requires the sequential solution of AREs at a high rate, a task that can is demanding for large-scale dynamics. Here, we propose a variant requiring the solution of a Lyapunov equation instead, thus mitigating the computational cost associated to the online synthesis. The paper is structured as follows. In section~\ref{sec:review_synth} and its subsections we review the $\cH_2/\cH_{\infty}$ optimal feedback synthesis problem. In section~\ref{sec:suboptimal} and its subsections we present the suboptimal approximation to these feedback laws by means of the SDRE approach, including offline, online, and hybrid offline-online implementations, illustrated with an application to the control of the Sine-Gordon equation. Section~\ref{sec:nla} discusses numerical linear algebra aspects of the solution of the Algebraic Riccati and Lyapunov equations which constitute the core building blocks of the SDRE feedback synthesis. Finally, in section~\ref{numerics} we perform a computational assessment of the proposed methodology on the two-dimensional degenerate Zeldovich and viscous Burgers PDEs. \section{Optimal feedback synthesis for nonlinear dynamics} \label{sec:review_synth} In this section we revisit the use of dynamic programming and Hamilton-Jacobi-Bellman/Isaacs PDEs for the computation of optimal feedback controls for nonlinear dynamics. We begin by stating the problem of optimal feedback stabilization via $\cH_2$ synthesis, to then focus on robustness under perturbation in the framework of $\cH_{\infty}$ control. \subsection{The $\cH_2$ synthesis and the Hamilton-Jacobi-Bellman PDE} We consider the following infinite horizon optimal control problem: \[\underset{u(\cdot)\in\mathcal{U}}{\min}\;\mathcal{J}(u(\cdot);x):=\int\limits_0^\infty \Big(\norm{y(t)}_Q^2+\norm{u(t)}_R^2\Big)\, dt \] subject to the nonlinear dynamical constraint \begin{equation}\label{dynamics} \dot y(t)= f(y(t))+g(y(t))u(t)\,,\quad y(0)=x, \end{equation} where $y(t)=(y_1(t),\ldots,y_d(t))^{\top}\in\mathbb{R}^d$ denotes the state of the system, and the control signal $u(\cdot)$ belongs to $\mathcal{U}:=L^{\infty}(\mathbb{R}_+;\mathbb{R}^m)$. The running cost is given by $\norm{y}_Q^2:=y^{\top}Qy$ with $Q\in\mathbb{R}^{d\times d},\,Q\succ0$, and $\norm{u}_R^2=u^{\top}Ru$ with $R\in\mathbb{R}^{m\times m},\,R\succ0$. We assume the system dynamics $f(y):\mathbb{R}^d\rightarrow\mathbb{R}^d$ to be such that $f(0)=0$, and the control operator {$g(y):\mathbb{R}^d\rightarrow\mathbb{R}^{d\times m}$} to be $\mathcal{C}^1(\mathbb{R}^d)$. This formulation of the $\cH_2$ synthesis corresponds to the asymptotic stabilization of nonlinear dynamics towards the origin. By the application of the Dynamic Programming Principle, it is well-known that the optimal value function \[V(x)=\underset{u(\cdot)\in\mathcal{U}}{\inf} \mathcal{J}(u(\cdot);x)\] characterizing the solution of this infinite horizon control problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation \begin{equation}\label{hjb} \underset{u\in \mathbb{R}^m}{\min}\;\{ \nabla V(x)^{\top}(f(x)+g(x) u)+ \norm{x}_Q^2+\norm{u}_R^2\}=0\,,\quad V(0)=0\,. \end{equation} The explicit minimizer $u^*$ of eq. \eqref{hjb} is given by \begin{equation}\label{optc} u^*(x)=-\frac{1}{2}R^{-1} g(x)^{\top} \nabla V(x)\,. \end{equation} By inserting \eqref{optc} into \eqref{hjb}, we obtain the HJB equation \begin{equation}\label{hjb2} \nabla V(x)^{\top} f(x)-\frac{1}{4}\nabla V(x)^{\top}g(x)R^{-1}g(x)^{\top}\nabla V(x)+x^{\top}Qx=0\,,\tag{HJB} \end{equation} to be understood in the classical sense. We recall that under the additional linearity assumptions $f(x)=Ax$ with $A\in\mathbb{R}^{d\times d}$ and $g(x)=B\in\mathbb{R}^{d\times m}$, the value function is known to be a quadratic form, $V(x)=x^{\top}\Pi x$, with $\Pi\in\mathbb{R}^{d\times d}$ positive definite, and eq. \eqref{hjb2} becomes an Algebraic Riccati Equation for $\Pi$ \begin{align}\label{are} A^{\top}\Pi+\Pi A -\Pi B R^{-1}B^{\top}\Pi+Q=0\,.\tag{ARE} \end{align} Solving for $V(x)$ in eq. \eqref{hjb2} globally in $\mathbb{R}^d$ allows an online synthesis of the optimal feedback law \eqref{optc} by evaluating the gradient $\nabla V(x)$ and $g(x)$ at the current state $x$. This leads to an inherently robust control law in the sense that if the state of the system is perturbed to $x'=x+\delta x$, there still exists a stabilizing feedback action departing from the perturbed state, namely, $u^*(x')$. However, this control design neglects the modelling of the disturbance/uncertainties affecting the dynamics, with no general stabilization guarantees. We overcome this limitation by formulating an $\cH_{\infty}$ synthesis, which we describe in the following. \subsection{The $\cH_{\infty}$ problem and the Hamilton-Jacobi-Isaacs PDE} We extend the previous formulation by considering a nonlinear dynamical system of the form \begin{equation}\label{dynamicsp} \dot y(t)= f(y(t))+g(y(t))u(t)+h(y(t))w(t)\,,\quad y(0)=x\,, \end{equation} where an additional disturbance signal $w(\cdot)\in\cW$, with $\cW=L^2(\mathbb{R}_+;\mathbb{R}^p)$ enters the system through $h(y):\mathbb{R}^d\rightarrow\mathbb{R}^{d\times p}$. We assume that $y=0$ is an equilibrium of the system for $u=w=0$. The $\cH_{\infty}$ control objective is to achieve both internal stability of the closed-loop dynamics and disturbance attenuation through the design of a feedback law $u=u(y)$ such that for a given $\gamma>0$, and for all $T\geq 0$ and $w\in L_2([0,T[,\mathbb{R}^p)$, the inequality \begin{equation}\label{eq:kk10} \int\limits_{0}^{T}\Big(\norm{y}^2_{Q}+\norm{u(t)}^2_R\Big)\, dt\leq\gamma^2\int\limits_{0}^{T}\norm{w(t)}^2_{P}\;dt \end{equation} holds. Here, $P\in\mathbb{R}^{p\times p},\, P\succ0\,,$ and $y$ is the solution to \eqref{dynamicsp}. The parameter $\gamma$ plays a crucial role in $\cH_{\infty}$ control design. We say that the control system \eqref{dynamicsp} has ${\cL}_2$-gain not greater than $\gamma$, if \eqref{eq:kk10} holds. Finding the smallest $\gamma^*$ for which this inequality is verified, also known as the $\cH_{\infty}$-norm of the system, is a challenging problem in its own right. The simplest yet computationally expensive method to find the $\cH_{\infty}$-norm of a system is through a bisection algorithm, as described in \cite{BBK89}. Applying dynamic programming techniques leads to a characterization of the value function for this problem in terms of the solution of a Hamilton-Jacobi-Isaacs PDE \begin{equation}\label{hji} \underset{u\in\mathbb{R}^m}{\min}\,\,\underset{w\in \mathbb{R}^p}{\max}\{ \nabla V(x)^{\top}(f(x)+g(x) u+h(x)w)+ \norm{x}^2_Q+\norm{u}_R^2-\gamma^2\|w\|_P^2\}=0\,, \end{equation} valid for all $\gamma \geq \gamma^*$. The unconstrained minimizer and maximizer of \eqref{hji}, $u^*_{\gamma}$ and $w^*_{\gamma}$ respectively, are explicitly computed as \begin{align} u^*_\gamma(x)&=-\frac12 R^{-1}g(x)^{\top}\nabla V_\gamma(x)\,,\label{uopt}\\ w^*_\gamma(x)&=\frac{1}{2\gamma^2} P^{-1}h(x)^{\top}\nabla V_\gamma(x)\,,\label{wopt} \end{align} where $ V_\gamma(x)$ solves the Hamilton-Jacobi-Isaacs equation \begin{align}\label{hjie} \nabla V_\gamma(x)^{\top}f(x)+ \frac14 \nabla V_\gamma(x)^{\top}S(x)\nabla V_\gamma(x)+x^{\top}Qx=0 \,,\tag{HJI} \end{align} with \begin{align} S(x)=\frac{1}{\gamma^2}h(x)P^{-1}h(x)^{\top}-g(x)R^{-1}g(x)^{\top}, \end{align} and $V_\gamma(0)=0$. For an initial condition which is not a steady state we have \begin{equation} \int\limits_{0}^{\top}\Big(\norm{y}+\norm{u(t)}^2_R\Big)\, dt\leq\gamma^2\int\limits_{0}^{\top}\norm{w(t)}^2_{P}\;dt + V_\gamma(x), \end{equation} see e.g. \cite[Theorem 16]{V92}. When there is no confusion, we denote $V_\gamma(x)$ by $V$. Note that if the disturbance attenuation is neglected by taking $\gamma\to\infty$, we recover the solution of \eqref{hjb2} instead. Moreover, under the linearity assumptions $f(x)=Ax$ with $A\in\mathbb{R}^{d\times d}$, $g(x)=B\in \mathbb{R}^{d\times m}$, and $h(x)=H\in \mathbb{R}^{d\times p}$, the value function is a quadratic form $V_{\gamma}(x)=x^{\top}\Pi x$, where $\Pi\in\mathbb{R}^{d\times d}$ is positive definite, and eq. \eqref{hjie} becomes the following Algebraic Riccati Equation for $\Pi$ \begin{equation}\label{arehinf} A^\top\Pi+\Pi A-\Pi\left( BR^{-1}B^\top- \dfrac{1}{2\gamma^2}HP^{-1}H^\top\right)\Pi+Q=0\,,\tag{\mbox{$\text{ARE}_{\infty}$}} \end{equation} solving the $\cH_{\infty}$ problem for full state feedback. In the following section we discuss the construction of computational methods to synthesize nonlinear feedback laws inspired by the solution of HJB/HJBI PDEs. \section{Sub-optimal feedback control laws}\label{sec:suboptimal} \renewcommand{\thealgorithm}{3.\arabic{algorithm}} \setcounter{algorithm}{0} Despite the extensive parametrization of the control problem, eqns. \eqref{hjb2} and \eqref{hjie} remain first-order, stationary nonlinear PDEs whose numerical approximation is a challenging task. These nonlinear PDEs are cast over $\mathbb{R}^d$, where $d$ relates to the dimension of the state space of the dynamics, which can be arbitrarily large. In particular, if the system dynamics correspond to nonlinear PDEs, the direct solution of the resulting infinite-dimensional eqns. \eqref{hjb2} and \eqref{hjie} remains an open computational problem. We explore different alternatives to circumvent the solution of high-dimensional HJB PDEs by resorting to the sequential solution of the Algebraic Riccati Equations \eqref{are} and \eqref{arehinf} respectively, providing a tractable alternative for feedback synthesis in large-scale nonlinear dynamics. The different techniques we propose trade the optimality associated to the HJB-based control for computability, and hence will be referred to as \textsl{suboptimal} feedback laws. For the sake of simplicity, we focus on the $\mathcal{H}_\infty$ synthesis. The $\mathcal{H}_2$ feedback follows directly choosing $h(x(t)):=0$ in \eqref{dynamicsp}. We assume a quadratic cost, i.e. $\ell(x) = x^\top Qx$ with $Q\in\mathbb{R}^{d\times d}$ symmetric positive semidefinite. \subsection{Linear-Quadratic Regulator (LQR) control law} The simplest suboptimal control law for nonlinear systems uses the optimal feedback law for the linear-quadratic control problem arising from linearization around an equilibrium, which we assume to be $x=0$. For $\gamma>\gamma^*$, we solve eq. \eqref{arehinf} with $A_{ij}=\frac{\partial f_i(x)}{\partial x_j}|_{x=0}$, $B=g(0)$, and $H=h(0)$. From the solution $\Pi$, we obtain the linear feedback law \begin{equation}\label{lqr} u(x)=-R^{-1}B^\top\Pi x\,. \end{equation} Such a feedback law, applied to the original nonlinear system, can only guarantee stabilization in a neighbourhood around the origin. \subsection{State-Dependent Riccati Equation (SDRE)}\label{secsdre} If we express the system dynamics through a space-dependent representation of the form \begin{equation}\label{sdress1} \dot x=A(x)x+B(x)u(t) +H(x)\omega(t)\,, \end{equation} we can synthesize a suboptimal feedback law inherited from \eqref{hjie} by following an approach known as the State-dependent Riccati Equation (SDRE). This method has been thoroughly analyzed in the literature, see, e.g.\cite{C97,BLT07}, and despite being purely formal, it is extremely effective for stabilizing nonlinear dynamics. The SDRE approach is based on the idea that infinite horizon optimal feedback control for systems of the form \eqref{sdress1} are linked to a state-dependent ARE: \begin{equation}\label{sdre1} A^\top(x)\Pi(x)+\Pi(x) A(x)-\Pi(x)S(x)\Pi(x)+Q=0\,, \end{equation} where \[ S(x)=B(x)R^{-1}B^\top(x)- \dfrac{1}{2\gamma^2}H(x)P^{-1}H(x)^\top\,. \] Solving this equation leads to a state-dependent Riccati operator $\Pi(x)$, from where we obtain a nonlinear feedback law given by \begin{equation}\label{sdref1} u(x):=-R^{-1}B^\top(x)\Pi(x)x\,. \end{equation} It is important to observe that the operator equation \eqref{sdre1} admits an analytical solution only in a limited number of cases. More importantly, even if this solution is computed for every state $x$, the closed-loop differs from the optimal feedback obtained from solving \eqref{hjie}, as the SDRE approach assumes that the value function is always locally approximated as $V(x)=x^{\top}\Pi(x)x$. From an optimal control perspective the SDRE can be interpreted as a model predictive control loop where at a given instant, the dynamics $(A(x),B(x),H(x))$ are frozen at the current state and an LQR feedback is numerically approximated. The procedure is illustrated in Algorithm \ref{Alg_sdre}. The resulting feedback is naturally different from the optimal nonlinear feedback law, and will remain different regardless of how fast the SDRE feedback is updated along a trajectory. The latter is also observed in \cite{Astolfi2020}, where it is shown that a direct derivation of the SDRE approach departing from the HJB PDE would lead to additional terms in \eqref{sdress1}. Notwithstanding, it is possible to show local asymptotic stability for the SDRE feedback. We recall the following result \cite{BLT07} concerning asymptotic stability of the SDRE closed-loop in the $\mathcal{H}_2$ case. \begin{proposition}Assume a nonlinear system \begin{equation} \dot x(t)=f(x(t))+B(x(t))u(t)\,, \end{equation} where $f(\cdot)$ is $\mathcal{C}^1$ for $\|x\|\leq\delta$, and $B(\cdot)$ is continuous. If $f$ is parametrized in the form $f(x)=A(x)x$, and the pair $(A(x),B(x))$ is stabilizable for every $x$ in a non-empty neighbourhood of the origin. Then, the closed-loop dynamics generated by the feedback law \eqref{sdref1} are locally asymptotically stable. \end{proposition} \begin{algorithm}[ht] \caption{SDRE-MPC loop} \label{Alg_sdre} \begin{algorithmic}[1] \REQUIRE $\{t_0,t_1,\ldots\},$ model \eqref{sdress1}, $R,Q$, \FOR{$k=0,1,\ldots$} \STATE Compute $\Pi(x(t_k))$ from \eqref{sdre1} \STATE Set $K(x(t_k)) := R^{-1}B^\top(x(t_k))\Pi(x(t_k))$ \STATE Set $u(t):=-K(x(t_k))x(t)$ \STATE Integrate system dynamics to $x(t_{k+1})$ \ENDFOR \end{algorithmic} \end{algorithm} Assuming the stabilizability hypothesis above, the main bottleneck in the implementation of Algorithm \ref{Alg_sdre} is the high rate of calls to an ARE solver for \eqref{sdre1}. Moreover, these ARE calls are expected to be sufficiently fast for real-time feedback control. This is a demanding computational task for the type of large-scale dynamics arising in optimal control of PDEs. In the following, we discuss two variations the SDRE-MPC algorithm which mitigate this limitation by resorting to offline and more efficient online computations. \subsection{Offline approximation of the SDRE} A first alternative for more efficient SDRE computations was proposed in \cite{BTB00}, inspired by a power series argument first discussed in \cite{Wernli1975}. Assuming that the state operator $A(x)$ can be decomposed into $A(x)=A_0+f_1(x)A_1\,,$ where $A_0$ and $A_1$ are constant matrices and $f(x)$ is a scalar function, $B(x)=B$ and $H(x)=H$, then the Riccati operator $\Pi(x)$ solving \eqref{sdre1} is approximated by \[ \Pi(x)=\sum_{n=0}^{\infty}(f_1(x))^nL_n\,, \] where the matrices $L_n\in\mathbb{R}^{d\times d}$ solve \begin{align} L_{0} A_{0}+A_{0}^{\top} L_{0}-L_{0} S L_{0}+Q&=0\,,\label{are0}\\ L_{1}\left(A_{0}-S L_{0}\right)+\left(A_{0}^{\top}-L_{0} S\right) L_{1}+L_{0} A_1+A_1^{\top} L_{0}&=0\,,\\ L_{n}\left(A_{0}-S L_{0}\right)+\left(A_{0}^{\top}-L_{0} S\right)L_{n}+Q_n&=0\,,\\ Q_n:=L_{n-1}A_1+A_1^{\top}L_{n-1}-\sum_{k=1}^{n-1}L_{k} SL_{n-k}&\notag\,. \end{align} After solving a single ARE and $N$ Lyapunov equations, an $N-$order approximation of $\Pi(x)$ yields the feedback \begin{equation} u^N(x)=-R^{-1}B^\top\left(\sum\limits_{n=0}^N(f_1(x))^nL_n\right)x \end{equation} Unfortunately, the reduction above is only possible for a scalar nonlinearity. If the nonlinearity that can be expressed as \begin{equation}\label{sdrea} A(x)=A_0+\sum_{j=1}^r f_j(x)A_j\,, \end{equation} where $A_j \in R^{d\times d}$ and the state dependence is restricted to $r$ scalar functions $f_j(x):R^d\rightarrow\mathbb{R}$, then a first order approximation of $\Pi(x)$ is given by \begin{equation}\label{pbar} \Pi(x)\approx \tilde\Pi(x)=\Pi_0+\sum_{j=1}^r\Pi_jf_j(x) \end{equation} where $\Pi_0$ solves \eqref{arehinf} for $A_0$ and the remaining $\Pi_j$ satisfy the Lyapunov equations \begin{equation}\label{lyap_off} \Pi_jC_0+C_0^\top\Pi_j+Q_j=0\,,\quad j=1,\ldots, r\,, \end{equation} with $C_0=A_0-S\Pi_0$, and $Q_j=\Pi_0A_j+A_j\Pi_0$. The resulting feedback law is given by \begin{equation}\label{cont_off} u(x)=-R^{-1}B^\top\tilde\Pi(x)x\,. \end{equation} Overall, this approach requires the computation of the ARE associated to $A_0$ in addition to $r$ Lyapunov equations whose solution is fully parallelizable. Its implementation is summarized below. \begin{algorithm}[H] \caption{Offline SDRE} \label{Alg_sdreoff} \begin{algorithmic}[1] \STATE Offline computations: \STATE Compute $\Pi_0$ from \eqref{arehinf} \FOR{$j=1,\ldots,r$} \STATE Compute $\Pi_j$ from the Lyapunov equation \eqref{lyap_off} \ENDFOR \STATE Online process: \STATE Set $u(x(t)):= -R^{-1}B^\top\tilde\Pi(x(t))x(t)$ \STATE Integrate system dynamics for $x(t)$ \end{algorithmic} \end{algorithm} \paragraph{\bf An offline-online SDRE approach} Although the previous approach is a valid variant to circumvent the online solution of Riccati equations at a high rate, it becomes unfeasible in cases where both $d$ and $r$ are large, as it requires the solution of $r$ Lyapunov equations \eqref{lyap_off} and storage of the solution matrix with $d^2$ entries each. Such a large-scale setting arises naturally in feedback control of dynamics from semidiscretization of nonlinear PDEs and agent-based models. We present a variant of the offline SDRE approach which circumvents this limitation by resorting to an online phase requiring the solution of a single Lyapunov equation per step. Let us define the quantity $W(x) = \sum\limits_{j=1}^r\Pi_jf_j(x)$. Multiplying each equation in \eqref{lyap_off} by its corresponding $f_j(x)$, it follows that $W(x)$ satisfies the Lyapunov equation \begin{equation}\label{lyap_on} W(x) C_0+C_0^\top W(x) +\sum_{j=1}^r Q_jf_j(x) = 0. \end{equation} Therefore, the feedback law can be expressed as \begin{equation}\label{con:var} u(x)=-R^{-1}B^\top\left(\Pi_0+\sum_{j=1}^r\Pi_jf_j(x)\right)x = -R^{-1}B^\top\left(\Pi_0+W(x) \right)x . \end{equation} The feedback law \eqref{con:var} can be computed by solving an offline Riccati equation for $\Pi_0$ and an online Lyapunov equation for $W(x)$; {see section~\ref{sec:nla} for a discussion on the computational costs of the two approaches.} The offline-online SDRE approach is summarized in Algorithm \ref{Alg_sdreon} below. \begin{algorithm}[H] \caption{Offline-online SDRE} \label{Alg_sdreon} \begin{algorithmic}[1] \STATE Offline phase: \STATE Compute $\Pi_0$ from \eqref{arehinf} \STATE Online phase: \FOR{$k=0,1,\ldots$} \STATE Compute $W(x(t_k))$ from the Lyapunov equation \eqref{lyap_on} \STATE Set $K(x(t_k)) := R^{-1}B^\top\left(\Pi_0 +W(x(t_k))\right)$ \STATE Set $u(t):=-K(x(t_k))x(t)$ \STATE Integrate system dynamics to $x(t_{k+1})$ \ENDFOR \end{algorithmic} \end{algorithm} \begin{remark} The approximation of the state space solution $x(t_{k+1})$ at time $t_{k+1}$ can be performed with both explicit or implicit time-stepping schemes. In both cases, the feedback gain remains frozen at $K(x(t_n))$. \end{remark} \subsection{A preliminary test: the damped Sine-Gordon equation}\label{sinegordon} We present a preliminary assessment of the effectiveness of Algorithm \ref{Alg_sdre} and Algorithm \ref{Alg_sdreon} for the $\mathcal{H}_2$ case. In section \ref{numerics} we will focus on large-scale, two-dimensional PDEs and $\mathcal{H}_\infty$ control. Given a domain $\Omega\subset\mathbb{R}$, we consider {the control of} the damped Sine-Gordon equation {(see e.g. \cite{MR04})} with homogeneous Dirichlet boundary conditions over $\Omega\times\mathbb{R}^+_0$: \begin{align}\label{ex:wave} \begin{aligned} \partial_{tt} X(\xi,t)&= -\alpha \pt X(\xi,t) + \partial_{\xi\xi} X(\xi,t)-\beta \sin X(\xi,t)+\chi_{\omega_c}(\xi)u(t)\\ X(\xi,t)&=0 \quad\xi\in\partial\Omega, t>0\,,\\ X(\xi,0)&={\mathbf x}_0(\xi)\,,\;\xi\in\Omega\,,\\ \pt X (\xi,0)&={\mathbf x}_1(\xi)\,,\;\xi\in\Omega\,, \end{aligned} \end{align} where the control variable $u(t)$ acts through an indicator function $\chi_{\omega_c}(\xi)$ supported over $\omega_c\subset\Omega$. {The cost functional to be minimized is given by: \begin{equation}\label{costex_wave} \mathcal{J}(u(\cdot);X(\cdot,0)):=\int\limits_0^{\infty} \sum_{i=1}^z \dfrac{1}{|\omega_{o_i}|}\left(\int_{\omega_{o_i}} X(\xi,t)\, d\xi\right)^2+R\,|u(t)|^2 \, dt\, \end{equation} where $\omega_o:=\cup_{i = 1}^z \omega_{o_i} \subset\Omega$ represents a collection of local patches where we average the state. } Defining $y(t)=(X(\cdot,t), \dot X(\cdot,t))^{\top}$, we write the dynamics as a first-order abstract evolution system $$ \dot{y}(t)=Ay(t) + f(y(t))+ Bu(t)\;, $$ where \begin{align}\label{ops} A= \begin{bmatrix} 0 & I \\ \partial^2_{\xi\xi} & -\alpha I \end{bmatrix} ,\,\,\, f(y(t))= \begin{bmatrix} 0 \\ -\beta \sin(X(\cdot,t)) \end{bmatrix}, \,\,\, Bu(t)= \begin{bmatrix} 0\\ \chi_{\omega_c}(\xi) u(t) \end{bmatrix}. \end{align} We approximate the operators above using a finite difference discretization in space. Given the domain $\Omega=[\xi_L,\xi_R]$, we construct the uniform grid $\xi_i=\xi_L+(i-1)\Delta \xi$ with $\Delta\xi=(\xi_R-\xi_L)/(d-1)$. We define the discrete state $X_i(t):=X(\xi_i,t)$, and the discrete augmented state $Y(t)=(X_1(t),\ldots,X_d(t),\dot X_1(t),\ldots, \dot X_d(t))^{\top}$. The discrete operators read \begin{align}\label{opsd} A_d= \begin{bmatrix} \mathbf{0}_{d\times d} \;\; \mathbf{I}_{d} \\ -\Delta_d \;\; \alpha \mathbf{I}_{d} \end{bmatrix} ,\quad Bu(t)= \begin{bmatrix} \mathbf{0}_{d\times 1}\\ \{\chi_{\omega_c}(\xi_i)\}_{i=1}^d \end{bmatrix}u(t)\,, \end{align} where $\mathbf{I}_d$ is the $d\times d$ identity matrix, and $\Delta_d$ is the discrete Laplace operator\footnote{The notation $\texttt{tridiag}([\begin{array}{ccc}a&b&c\end{array}],d)$ stands for a tridiagonal $d\times d$ matrix having the constant values $b\in\mathbb{R}$ on the main diagonal, $a\in\mathbb{R}$ on the lower diagonal and $c\in\mathbb{R}$ on the upper diagonal.}, $\Delta_d:=\Delta\xi^{-2}\texttt{tridiag}([\begin{array}{ccc}1&-2&1\end{array}],d)\in\mathbb{R}^{d\times d}$. {The quantity $\sum_{i=1}^z \dfrac{1}{|\omega_{o_i}|}\left(\int_{\omega_{o_i}} X(\xi,t)\, d\xi\right)^2$ in \eqref{costex_wave} is approximated by $X(t)^{\top}Q X(t)$ where $Q= C^{\top}C\in\mathbb{R}^{d\times d}$ and $$ C^{\top}=\Delta\xi\, \begin{bmatrix} \dfrac{\{\chi_{\omega_{o_1}}(\xi_i)\}_{i=1}^d}{|\omega_{o_1}|}, \ldots, \dfrac{\{\chi_{\omega_{o_z}}(\xi_i)\}_{i=1}^d}{|\omega_{o_z}|} \end{bmatrix}\in\mathbb{R}^{d\times z}\,. $$} To express the nonlinearity in a semilinear form consistent with \eqref{sdress1} we define \begin{equation} \tilde f(Y(t))= -\beta \begin{pmatrix} \mathbf{0}_{d\times 1} \\ \left\{\dfrac{\sin(X_i(t))}{X_i(t)}\right\}_{i=1}^d \end{pmatrix}Y(t)\,. \end{equation} In our test we consider the following values for the given parameters, $\Omega = [-10,10],\, t\in[0,10],\, \alpha =0.05,\, \beta = 2$, $\omega_c = [-1,1]$, and $$X(\xi,0) = 0,\quad \partial_t X(\xi,0) =\dfrac{8}{3}\mbox{sech}\left(\dfrac{2}{3\xi}\right).$$ In the cost functional {\eqref{costex_wave}} we set $R=1$, $\omega_o(x) = [-2.5,-1.5] \cup [1.5, 2.5]$ and $z=2$. In this test we take $d=402$ nodes in the finite difference discretization. Time-stepping is performed with an implicit Euler method with a time step of $0.1$. The small size Riccati and Lyapunov equations are solved using the Matlab functions {\tt icare} and {\tt lyap}, respectively. Controlled dynamics with different feedback controls are shown in Figure~\ref{fig1:wave}. The presence of the damping term $\alpha \pt X(\xi,t)$ generates a stable trajectory for both the uncontrolled and LQR-controlled dynamics using the linearized feedback \eqref{lqr}. However, we still observe differences in the state and control variables with respect to the SDRE controllers, SDRE-MPC and SDRE offline-online, described in Algorithms \ref{Alg_sdre} and \ref{Alg_sdreon}, respectively. The accumulated running costs in Figure \ref{fig1:wave} (top-left) indicate that the SDRE-MPC implementation achieves the best closed-loop performance, followed by SDRE offline-online, both outperforming linearized LQR and uncontrolled trajectories. However, the main difference between the SDRE-MPC and SDRE offline-online closed-loops is related to computational time. The SDRE-MPC solver requires the solution of multiple AREs in sequence, taking a total of 23 minutes of CPU time for this test, whereas the online solution of Lyapunov equations of the SDRE offline-online implementation reduces this computation time to 45 seconds. A deeper study on the methods performance in the large scale setting will be reported on in section~\ref{numerics}, while implementation aspects associated with the solution of these algebraic equations are discussed in the next section. \begin{figure}[htbp] \centering \includegraphics[scale=0.3]{plot_waves/cost.eps} \includegraphics[scale=0.3]{plot_waves/contr.eps} \includegraphics[scale=0.29]{plot_waves/unc.eps} \includegraphics[scale=0.29]{plot_waves/lqr.eps} \includegraphics[scale=0.29]{plot_waves/sdre.eps} \includegraphics[scale=0.29]{plot_waves/lyap.eps} \caption{Section \ref{sinegordon}. Damped sine-Gordon equation, controlled trajectories. Top: accumulated running cost with different $\mathcal{H}_2-$control (left) and corresponding control inputs (right). Middle: trajectories, uncontrolled (left), and linearized (right). Bottom: trajectories, $\mathcal{H}_2$-controlled solution with SDRE-MPC Algorithm \ref{Alg_sdre} (left), and $\mathcal{H}_2$-controlled solution with SDRE offline-online Algorithm \ref{Alg_sdreon} (right). } \label{fig1:wave} \end{figure} \section{Solving large-scale Algebraic Riccati/Lyapunov equations} \label{sec:nla} The time steps discussed in Section~\ref{secsdre} all use matrices that stem from the solution of algebraic matrix equations, and more precisely the (quadratic) Riccati and the (linear) Lyapunov equations. The past few years have seen a dramatic improvement in the effectiveness of numerical solution strategies for solving these equations in the large scale setting. For a survey in the linear case we refer the reader to the recent article \cite{Simoncini.survey.16}, while for the algebraic Riccati equation we point the reader to, e.g., \cite{B08,SSM14,Binietal.book.12}. In our derivation we found projection methods to be able to adapt particularly well to the considered setting, with a similar reduction framework for both linear and quadratic problems; other approaches are reviewed for instance in \cite{Benner.Saak.survey13,Simoncini.survey.16}. We emphasize that all considered methods require that the zero order term in the matrix equation, e.g., matrix $Q$ in the Riccati equation \eqref{sdre1}, be low rank. The general idea consists of first determining an approximation space ${\cal K}_k$ that can be naturally expanded if needed, and then seeking an approximate solution in this space, by imposing a Galerkin condition on the matrix residual for computing the projected approximate solution. Let ${\cal V}_k$ be such that ${\cal K}_k={\rm range}({\cal V}_k)$, with ${\cal V}_k$ having orthonormal columns. Recalling that in both the Riccati and Lyapunov equations the solution matrix is symmetric, the approximate solution can be written as ${\cal V}_k Y {\cal V}_k^\top$. Consider the Riccati equation \eqref{sdre1} for a fixed $x=x(t_*)$, so that we set $A(x(t_*))=A_*$ and $B(x(t_*))=B_*$. Let ${\cal R} = A_*^\top {\cal V}_k Y_X {\cal V}_k^\top + {\cal V}_k Y_X {\cal V}_k^\top A_* - {\cal V}_k Y_X {\cal V}_k^\top B_* {\cal V}_k Y_X {\cal V}_k^\top + Q$. Imposing the Galerkin condition on ${\cal R}$ means that the residual matrix ${\cal R}$ be orthogonal to the approximation space, in the matrix sense, that is \begin{equation}\label{eqn:reduced_Riccati} {\cal V}_k^\top {\cal R} {\cal V}_k =0 \quad \Leftrightarrow \quad {\cal V}_k^\top A_*^\top {\cal V}_k Y_X + Y_X {\cal V}_k^\top A_* {\cal V}_k - Y_X ({\cal V}_k^\top B_* {\cal V}_k) Y_X + {\cal V}_k^\top Q {\cal V}_k=0, \end{equation} where the orthogonality of the columns of ${\cal V}_k$ was used. If ${\cal V}_k$ has small dimensions, the {\it reduced} Riccati matrix equation on the right also has small dimensions and can be solved by a ``dense'' method to determine $Y_X$; see, e.g., \cite{Binietal.book.12}. The cost of solving the reduced quadratic equation with coefficient matrices of size $\hat k$ is at least $63 \hat k^3$ floating point operations with an invariant subspace approach \cite{Laub.79}. Note that the large matrix ${\cal V}_k Y_X {\cal V}_k^\top$ is never constructed explicitly, since it would be dense even for sparse data. Analogously, for the Lyapunov equation in \eqref{lyap_on}, we can write $W \approx {\cal V}_k Y {\cal V}_k^\top$ for some $Y=Y_W$ to be determined. Let ${\cal Q}_* = \sum_k Q_k f_k(x(t_*))$. As before, letting $\widetilde{\cal R} = C_0^\top {\cal V}_k Y {\cal V}_k^\top + {\cal V}_k Y {\cal V}_k^\top C_0 + {\cal Q}_*$, the Galerkin condition leads to $$ {\cal V}_k^\top \widetilde{\cal R} {\cal V}_k =0 \quad \Leftrightarrow \quad ({\cal V}_k^\top C_0^\top {\cal V}_k) Y + Y {\cal V}_k^\top C_0 {\cal V}_k + {\cal V}_k^\top{\cal Q}_* {\cal V}_k = 0. $$ This {\it reduced} Lyapunov equation can be solved by means of a ``dense'' method at a cost of about 15 $\hat k^3$ floating point operations for coefficient matrices of size $\hat k$, if the real Schur decomposition is employed; see, e.g., \cite{Simoncini.survey.16}. Note that the computational cost is significantly lower than that of solving the reduced Riccati equation with matrices of the same size. \subsection{On the selection of the approximation space} Choices as approximation space explored in the literature include polynomial and rational Krylov subspaces \cite{Simoncini.survey.16}. They both enjoy the property of being nested as they enlarge, that is ${\cal K}_k \subseteq {\cal K}_{k+1}$ where $k$ is associated with the space dimension. Rational Krylov subspaces have emerged as the key choice because they are able to deliver accurate approximate solutions with a relatively small space dimension, compared with polynomial spaces. Given a starting tall matrix $V_0$ and an invertible stable coefficient matrix $A_*$, we have used two distinct rational spaces: the Extended Krylov subspace, $$ {\cal E}{\cal K}_k = {\rm range}([V_0, A_*^{-1} V_0, A_* V_0, A_*^{-2} V_0, A_*^2 V_0, \ldots, A_*^{k-1} V_0, A_*^{-k} V_0]), $$ which only involves matrix-vector products and solves with $A_*$, and the (fully) Rational Krylov subspace, $$ {\cal R}{\cal K}_ k = {\rm range}([V_0, (A_*- \sigma_2 I)^{-1} V_0, \ldots, \prod_{j=2}^k (A_*- \sigma_j I)^{-1} V_0]) . $$ where $\sigma_j$ can be computed a-priori or adaptively. In both cases, the space is expanded iteratively, one block of vectors at the time, and systems with $A_*$ or with $(A_*- \sigma_j I)$ are solved by fast sparse methods. For $A_*$ real valued and stable, the $\sigma_j$'s are selected to be in ${\mathbb C}^+$, so that $A_*-\sigma_jI$ is nonsingular. The actual choice of the shifts is a key step and a rich literature is available, yielding theoretically grounded effective strategies; see, e.g., the discussion in \cite{Simoncini.survey.16}. In our implementation we used the Extended Krylov subspace for solving the Lyapunov equation in Algorithm \ref{Alg_sdreon}, which has several advantages, such as the computation of the sparse Cholesky factorization of $A_0$ once for all. On the other hand, we used the Rational Krylov subspace for the Riccati equation, which has been shown to be largely superior over the Extended Krylov on this quadratic equation, in spite of requiring the solution of a different (shifted) sparse coefficient matrix at each iteration \cite{Benner.Saak.survey13,SSM14}. Nonetheless, in Section~\ref{sec:feedback_matrix} we report an alternative approach that makes the Extended Krylov subspace competitive again for the Riccati problem with $A_0$ symmetric. Except for the operations associated with the reduced problems, the computational costs per iteration of the Riccati and Lyapunov equation solvers are very similar, if the same approximation space is used. Although we refer to the specialized literature for the algorithmic details\footnote{See {\tt www.dm.unibo.it/$\widetilde{\mbox{ }}$simoncin/software} for some related software.}, we would like to include some important implementation details that are specific to our setting. In particular, the matrix $C_0$ employed in Algorithm \ref{Alg_sdreon} is given by $C_0 = A_0 - \left(BR^{-1} B^\top - \frac 1 {2\gamma^2} H P^{-1} H^\top\right) \Pi_0$, which is not easily invertible if explicitly written, since it is dense in general. Note that $A_0$ is in general sparse, as it stems from the discretization of a partial differential operator. We can write $$ C_0 = A_0 - \left([B, H] G [B,H]^\top\right) \Pi_0, \quad {\rm with} \quad G = \begin{pmatrix} R^{-1}& 0\\ 0 & - \frac 1 {2\gamma^2} P^{-1} \end{pmatrix}. $$ Using the classical Sherman-Morrison-Woodbury formula, the product $C_0^{-1} V$ for some tall matrix $V$ can be obtained as $$ W:=C_0^{-1} V = A_0^{-1} V - A_0^{-1} [B, H] G_1^{-1} [B,H]^\top \Pi_0 A_0^{-1}V, $$ with $G_1 = I + G [B,H]^\top \Pi_0 A_0^{-1} [B, H]$, which is assumed to be nonsingular. Therefore, $C_0^{-1} V$ can be obtained by first solving sparse linear systems with $A_0$, and then using matrix-matrix products. More precisely, the following steps are performed: \vskip 0.1in - Solve $A_0 W_1 = V$ - Solve $A_0 W_2 = [B, H]$ - Compute $G_1 = I + G [B,H]^\top \Pi_0 W_2$ - Compute $W = W_1 - W_2 ( \, G_1^{-1} ( [B,H]^\top \Pi_0 W_1) \, )$ \vskip 0.1in We also recall that $\Pi_0$, the solution to the initial Riccati equation, is stored in factored form, and this should be taken into account when computing matrix-matrix products with $\Pi_0$. While trying to employ the Rational Krylov space, we found that the structure of $C_0$ made the selection of optimal shifts $\{\sigma_j\}$ particularly challenging, resulting in a less effective performance of the method. Hence, our preference went for the Extended Krylov space above, with the above enhancement associated with solves with $C_0$. \subsection{Feedback matrix oriented implementation}\label{sec:feedback_matrix} In Algorithm \ref{Alg_sdre} the Riccati equation needs to be solved at each time step $t_n$. However, its solution $\Pi(x_n)$ is only used to compute the feedback matrix $K(x_n) : = - R^{-1} B^\top \Pi(x_n)$. Hence, it would be desirable to be able to immediately compute $K(x_n)$ without first computing $\Pi(x_n)$. This approach has been explored in the Riccati equation literature but also for other problems based on Krylov subspaces, see, e.g., \cite{Palitta.Simoncini.18a,Kressner.proc.2008}. In this section, in the case where the matrix $A(x_n)$ is symmetric, we describe the implementation of one of the projection methods described above, that is able to directly compute $K(x_n)$ without computing $\Pi(x_n)$ (in factored form), and more importantly, without storing and computing the whole approximation basis. The latter feature is particularly important for large scale problems, for which dealing with the orthogonal approximation basis represents one of the major computational and memory costs. To the best of our knowledge, this variant of the Riccati solver is new, while it is currently explored in \cite{Palitta.Pozza.Simoncini.21} for related control problems and the rational Krylov space. Here we consider the Extended Krylov subspace. For $A(x_n)$ symmetric, the orthonormal basis of ${\cal E}{\cal K}_k$ can be constructed by explicitly orthogonalizing only with respect to the previous two basis blocks. Hence, only two previous blocks of vectors need to be stored in memory, and require explicit orthogonalization when the new block of vectors is added to the basis \cite{Palitta.Simoncini.18a}. This is also typical of polynomial Krylov subspaces constructed for symmetric matrices, giving rise to the classical Lanczos three-term recurrence \cite{Kressner.proc.2008}. With this procedure, in the reduced equation \eqref{eqn:reduced_Riccati} the matrices $A_k := {\cal V}_k^\top A_* {\cal V}_k$, $B_k := {\cal V}_k^\top B$ and $Q_k:={\cal V}_k^\top Q {\cal V}_k$ are computed as $k$ grows by updating the new terms at each iteration, and the solution $Y_X$ can be obtained without the whole matrix ${\cal V}_k$ being available. Note that the stopping criterion does not require the computation of the whole residual matrix, so that also in the standard solver the full matrix ${\cal V}_k Y_X {\cal V}_k^\top$ is never explicitly accessed. However, to be able to compute $$ K = -R B^\top {\cal V}_k Y_X {\cal V}_k^\top = -R B_k^\top Y_X {\cal V}_k^\top, $$ the basis ${\cal V}_k$ appearing on the right still seems to be required. As already done in the literature, this problem can be overcome by a so-called ``two-step'' procedure: at completion, once the final $Y_X$ is available, the basis ${\cal V}_k$ is computed again one block at the time, and the corresponding terms in the product $Y_X {\cal V}_k^\top$ are updated. Since $A(x_n)$ is already factorized and the orthogonalization coefficients are already available (they correspond to the non-zero entries of $A_k$), then the overall computational cost is feasible; we refer the reader to \cite{Palitta.Simoncini.18a} and its references for additional details for the two-step procedure employed for different purposes. \section{Large-scale nonlinear dynamical systems}\label{numerics} In this section we present a numerical assessment of the proposed methodology applied to the synthesis of feedback control for two-dimensional nonlinear PDEs. The first test is a nonlinear diffusion-reaction equation, known as the degenerate Zeldovich equation, where the origin is an unstable equilibrium and traditional linearization-based controllers fail. The second test studies the viscous Burgers' equation with a forcing term. We discretize the control problem in space using finite differences, similarly as in Section \ref{sinegordon}. Controlled trajectories are integrated in time using an implicit Euler method, which is accelerated using a Jacobian--Free Newton Krylov method (see e.g. \cite{KK04}). The goal of all our tests is the optimal and robust stabilization of the dynamics to the origin, encoded in the optimization of the following cost: \begin{align}\label{costex1} \begin{aligned} \mathcal{J}(u(\cdot),w(\cdot);X(\cdot,0))&:=\int\limits_0^{\infty} \sum_{i=1}^z \dfrac{1}{|\omega_{o_i}|}\left(\int_{\omega_{o_i}} X(\xi,t)\, d\xi\right)^2\\ &\quad\quad+R|u(t)|^2 -\gamma^2 P|w(t)|^2\, \, dt\,. \end{aligned} \end{align} {This expression is similar to \eqref{costex_wave}, but includes the $\mathcal{H}_\infty$ term $-\gamma^2 P|w(t)|^2$.} The reported numerical simulations were performed on a MacBook Pro with CPU Intel Core i7-6, 2,6GHz and 16GB RAM, using Matlab \cite{matlab7}. \begin{figure}[htbp] \centering \begin{tikzpicture}[scale=4] \draw (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0); \fill[black] (0.1,0.1) -- (0.3,0.1) -- (0.3,0.3) -- (0.1,0.3) -- (0.1,0.1); \fill[black] (0.7,0.7) -- (0.9,0.7) -- (0.9,0.9) -- (0.7,0.9) -- (0.7,0.7); \fill[black] (0.1,0.7) -- (0.3,0.7) -- (0.3,0.9) -- (0.1,0.9) -- (0.1,0.7); \fill[black] (0.7,0.1) -- (0.9,0.1) -- (0.9,0.3) -- (0.7,0.3) -- (0.7,0.1); \fill[blue] (0.1,0.4) -- (0.3,0.4) -- (0.3,0.6) -- (0.1,0.6) -- (0.1,0.4); \fill[blue] (0.1,0.4) -- (0.3,0.4) -- (0.3,0.6) -- (0.1,0.6) -- (0.1,0.4); \fill[blue] (0.7,0.4) -- (0.9,0.4) -- (0.9,0.6) -- (0.7,0.6) -- (0.7,0.4); \fill[blue] (0.4,0.1) -- (0.6,0.1) -- (0.6,0.3) -- (0.4,0.3) -- (0.4,0.1); \fill[blue] (0.4,0.7) -- (0.6,0.7) -- (0.6,0.9) -- (0.4,0.9) -- (0.4,0.7); \draw (0,0) node[below left] {$0$}; \draw (0.1,0) node[below ] {$0.1$}; \draw (0.3,0) node[below ] {$0.3$}; \draw (0.4,0) node[below ] {$\,\,0.4$}; \draw (0.6,0) node[below ] {$0.6$}; \draw (0.7,0) node[below ] {$\,\,0.7$}; \draw (0.9,0) node[below ] {$0.9$}; \draw (1,0) node[below ] {$1$}; \draw (0,0.1) node[left ] {$0.1$}; \draw (0,0.3) node[left ] {$0.3$}; \draw (0,0.4) node[left ] {$0.4$}; \draw (0,0.6) node[left] {$0.6$}; \draw (0,0.7) node[left] {$0.7$}; \draw (0,0.9) node[left ] {$0.9$}; \draw (0,1) node[left ] {$1$}; \end{tikzpicture} \caption{Locations of the inputs $\omega_c(\xi)=\omega_d(\xi)$ (black) and outputs $\omega_o(\xi)$ (blue) in the region $\Omega$ for the degenerate Zeldovich equation.} \label{fig:bc_coll} \end{figure} \subsection{Case study 1: the degenerate Zeldovich equation}\label{sec:test1} We consider the control of a Zeldovich-type equation arising in combustion theory \cite{zeldo} over $\Omega\times\mathbb{R}^+_0$, with $\Omega\subset \mathbb{R}^2$ and Neumann boundary conditions: \begin{align}\label{ex1} \begin{aligned} \pt X(\xi,t)&=\epsilon\Delta X(\xi,t)+\nu X(\xi,t)+\mu(X^2(\xi,t)-X^3(\xi,t))\\ &\qquad +\chi_{\omega_c}(\xi)u(t)+\chi_{\omega_d}(\xi)w(t)\\ \pxi X(\xi,t)&=0\,, \quad\xi\in\partial\Omega,\, t>0\,,\\ X(\xi,0)&={\mathbf x}_0(\xi)\,,\quad \xi\in\Omega\,. \end{aligned} \end{align} The scalar control and disturbance act, respectively, through functions $\chi_{\omega_c}(\xi)$ and $\chi_{\omega_d}(\xi)$ with support $\omega_c,\omega_d\subset\Omega$. The uncontrolled dynamics have three equilibrium points: $X\equiv 0$, $X\equiv\frac12\left(1\pm\sqrt{1+\frac{\nu}{\mu}}\right)$. Our goal is to stabilize the system to $X\equiv 0$, which is an unstable equilibrium point. A first step towards the application of the proposed framework is the space discretization of the system dynamics, leading to a finite-dimensional state-space representation. Following the setting presented in section~\ref{sinegordon}, using a finite difference discretization leads to \begin{equation}\label{pol:sd} \dot X(t) = \epsilon\Delta_dX(t)+\nu X(t)+\mu X(t)\circ X(t) \circ \left(\textbf{1}_{d\times 1}-X(t)\right) +B u(t) + H w(t)\,, \end{equation} where the discrete state $X(t)=(X_1(t),\ldots,X_d(t))^\top\in\mathbb{R}^d$ corresponds to the approximation of $X(\xi, t)$ at the grid points and the symbol $\circ$ denotes the Hadamard or component-wse product. The matrix $\Delta_d\in\mathbb{R}^{d\times d}$ is the finite difference approximation of the Neumann Laplacian and the matrices $B,H\in \mathbb{R}^d$ are the discretization of the indicator functions supported over $\omega_c$ and $\omega_d$, respectively. {The discretization of \eqref{costex1} follows similarly as in section~\ref{sinegordon}.} Once the finite-dimensional state-space representation is obtained, we proceed to express the system in semilinear form \eqref{sdrea} and implement the proposed algorithms. To set Algorithm \ref{Alg_sdre} and Algorithm \ref{Alg_sdreon}, from \eqref{pol:sd} we define $$A(X) := \epsilon\Delta_d +\nu\mathbf{I}_{d}+ \mu \texttt{diag}(X(t)-X(t)\circ X(t)),$$ where $\texttt{diag}(v)$ denotes a diagonal matrix with the components of the vector $v$ on the main diagonal, and decompose $A(X)$ as $$ A_0 = \epsilon\Delta_d +\nu\mathbf{I}_{d},\qquad [A_j]_{k,l}=\delta_{k,j}\delta_{l,j},\qquad f_j(X)=\mu(X_j-X_j^2),\qquad j=1\ldots,d, $$ where $\Delta_d$ is the two-dimensional discrete Laplacian and $\delta_{i,j}$ denotes the Kronecker delta. In this test we set $$\Omega = [0,1]\times[0,1],\,\, \epsilon=0.2,\,\, \nu = 0.1,\,\, \mu=10,\,\, R = 0.1,$$ and the initial condition ${\mathbf x}(\xi,0) = \sin (\xi_1)\sin(\xi_2)$, on a discretized space grid of $n_{\xi_1}\times n_{\xi_2}$ nodes with $n_{\xi_1} = n_{\xi_2} = 101$ ($d=10201$). For the matrices $B, H$ and $C$ we considered a collection of patches depicted in Figure \ref{fig:bc_coll}, and given by $$ \omega_d(\xi) = [0.1,0.3]^2 \cup [0.7,0.9]^2 \cup \left([0.1,0.3]\times[0.7,0.9]\right) \cup \left( [0.7,0.9]\times[0.1,0.3]\right), $$ $\omega_c(\xi)=\omega_d(\xi)$, and \begin{align*} \begin{aligned} \omega_o(\xi) =& \left([0.1,0.3]\times[0.4, 0.6]\right) \cup \left([0.4,0.6]\times[0.1, 0.3]\right) \\ &\cup \left([0.4,0.3]\times[0.7,0.9]\right) \cup \left( [0.7,0.9]\times[0.4,0.6]\right)\,. \end{aligned} \end{align*} In the following, we analyse results for the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ controls, i.e. $\gamma = 0$ and $\gamma\neq 0$, respectively, in \eqref{costex1}. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new/uncontrolled.eps} \includegraphics[scale=0.29]{test_new/lqr.eps} \includegraphics[scale=0.29]{test_new/sdre_1.eps} \includegraphics[scale=0.29]{test_new/lyap_1.eps} \caption{Test \ref{sec:test1h2}: state of the system \eqref{ex1} at time $t=3$. Top: uncontrolled solution (left), $\mathcal{H}_2$-solution with LQR control (right). Bottom: $\mathcal{H}_2$-controlled solution with Algorithm \ref{Alg_sdre} (left), $\mathcal{H}_2$-controlled solution with Algorithm \ref{Alg_sdreon} (right).} \label{fig1:sol} \end{figure} \begin{figure}[htbp] \centering \includegraphics[scale=0.3]{test_new/cost.eps} \includegraphics[scale=0.3]{test_new/contr.eps} \includegraphics[scale=0.3]{test_new/cpu_eff.eps} \includegraphics[scale=0.3]{test_new/cost_eff.eps} \caption{Test \ref{sec:test1h2}. Top: Cumulative cost functional functional (left) with $\mathcal{H}_2-$control and corresponding control inputs (right). Bottom: CPU time for Algorithm \ref{Alg_sdre} and Algorithm \ref{Alg_sdreon} (left), and convergence of the cost functional with respect to the dimension of the problem $d$ ($x$-axis) (right).} \label{fig1:cost} \end{figure} \begin{example}{\it Experiments for $\mathcal{H}_2$-control.}\label{sec:test1h2} We start by presenting results for $\mathcal{H}_2$-control, i.e. $P \equiv 0$ in \eqref{costex1} and $H\equiv 0$ in \eqref{pol:sd}. In Figure \ref{fig1:sol} we show a snapshot of the controlled trajectories at $t=3$, a horizon sufficiently large for the dynamics to approach a stationary regime. In the top-left panel the uncontrolled problem reaches the stable equilibrium $X\approx 1.02$. In the top-right panel we show the results of the LQR control computed by linearizing equation \eqref{pol:sd} around the origin, which also fails to stabilize around the unstable equilibrium $X \equiv 0$. The controlled solutions with Algorithm \ref{Alg_sdre} and Algorithm \ref{Alg_sdreon} are shown at the bottom of the same figure: both algorithms reach the desired configuration. The corresponding control input is shown in the top-right panel of Figure \ref{fig1:cost}. We observe that the LQR control has a completely different behavior with respect to the control computed by Algorithm \ref{Alg_sdre} or Algorithm \ref{Alg_sdreon}. The performance results of the different controlled trajectories is presented in Figure \ref{fig1:cost} where we show the evaluation of the cumulative cost functional in the top-left panel. As expected, Algorithm \ref{Alg_sdre} provides the best closed-loop performance among the proposed algorithms. However, in terms of efficiency Algorithm \ref{Alg_sdreon} is faster than Algorithm \ref{Alg_sdre} when increasing the dimension of the problem as shown in the bottom-left panel of Figure \ref{fig1:cost}. When the dimension $d$ increases ($x$-axis in the plot), the cost functional converges to $0.6$ for Algorithm \ref{Alg_sdre} and to $1$ for Algorithm \ref{Alg_sdreon}. Both methods are able to stabilize the problem. \end{example} \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new/lyap_2_20.eps} \includegraphics[scale=0.29]{test_new/lyap_3_20.eps} \includegraphics[scale=0.29]{test_new/sdre_2_20.eps} \includegraphics[scale=0.29]{test_new/sdre_3_20.eps} \caption{Test \ref{sec:ex2}: state of the system at time $t=3$ with $\mathcal{H}_2-$ and $\mathcal{H}_\infty-$controls and disturbance $w(t)=0.1\sin(2t)$. Top: $\mathcal{H}_2$-control using Algorithm \ref{Alg_sdre} (left) and Algorithm \ref{Alg_sdreon} (right). Bottom: $\mathcal{H}_\infty$-control using Algorithm \ref{Alg_sdre} (left) and Algorithm \ref{Alg_sdreon} (right).} \label{fig1:dist1} \end{figure} \begin{example}{\it Experiments for $\mathcal{H}_\infty$-control.}\label{sec:ex2} We next show the results of the optimal solution under disturbances with the following configuration: $$P = 1,\,\, \gamma = 0.5,\,\, w_1(t)=0.1\sin(40t),\,\, w_2(t)=0.1\sin(2t).$$ We omit reporting the behavior of the LQR-based control as it fails to stabilize the dynamics. To compare the proposed approaches we compute the $\mathcal{H}_2-$ and $\mathcal{H}_\infty-$ controls with the same disturbance using both Algorithm \ref{Alg_sdre} and Algorithm \ref{Alg_sdreon}. The results are presented in Figure \ref{fig1:dist1} and Figure \ref{fig1:dist2}. In every test case, the SDRE-based methodologies effectively stabilize the perturbed dynamics to a small neighbourhood around $X \equiv 0$. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new/lyap_2_40.eps} \includegraphics[scale=0.29]{test_new/lyap_3_40.eps} \includegraphics[scale=0.29]{test_new/sdre_2_40.eps} \includegraphics[scale=0.29]{test_new/sdre_3_40.eps} \caption{Test \ref{sec:ex2}: Solutions at time $t=3$ with $\mathcal{H}_2-$ and $\mathcal{H}_\infty-$control and disturbance $w(t)=0.1\sin(40t)$. Top: $\mathcal{H}_2$-control using Algorithm \ref{Alg_sdre} (left) and Algorithm \ref{Alg_sdreon} (right). Bottom: $\mathcal{H}_\infty$-control using Algorithm \ref{Alg_sdre} (left) and Algorithm \ref{Alg_sdreon} (right).} \label{fig1:dist2} \end{figure} A quantitative study is proposed in Figure \ref{fig1:an1} where we show the evaluation of the cumulative $\mathcal{H}_2$ cost functional for both disturbances in the left panels. As expected, Algorithm \ref{Alg_sdre} with $\mathcal{H}_\infty-$control exhibits the best performance, closely followed by Algorithm \ref{Alg_sdreon}. The right panels of Figure \ref{fig1:an1} show the different control inputs, which reflect the observed differences in closed-loop performance. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new/cost_dist_20.eps} \includegraphics[scale=0.29]{test_new/contr_dist_20.eps} \includegraphics[scale=0.29]{test_new/cost_dist_40.eps} \includegraphics[scale=0.29]{test_new/contr_dist_40.eps} \caption{Test \ref{sec:ex2}: evaluation of the cumulative cost functional $\mathcal{H}_2$ (left) and control inputs (right) for $w(t)=0.1\sin(2t)$ (top) and $w(t)=0.1\sin(40t)$ (bottom). } \label{fig1:an1} \end{figure} \end{example} \begin{example}{\it On the use of a feedback matrix oriented implementation}\label{sec:ex3} To conclude the first case study we provide a numerical example where we synthesize the feedback operator $K(x)=-R^{-1}B^{\top}\Pi(x)$ directly, circumventing the computation of $\Pi(x)$, as explained in Section \ref{sec:feedback_matrix}. For this test we introduce the following changes: $\Omega = [0,1]\times[0,1],\,\, \epsilon=0.1,\,\, \nu = 0,\,\, \mu=8$, and initial condition ${\mathbf x}_0(\xi) = \sin (\xi_1)\sin(\xi_2)$, on a discretized space grid of $n_{\xi_1}\times n_{\xi_2}$ nodes with $n_{\xi_1} = n_{\xi_2} = 101$. We replace Neumann boundary conditions with zero Dirichlet boundary conditions. For this test, we only consider the $\mathcal{H}_2-$control case since many considerations are similar to the previous part of this section. {We use Algorithm \ref{Alg_sdre} with the Extended Krylov subspace as in Section \ref{sec:feedback_matrix} and compare the performances of Algorithm \ref{Alg_sdre} using the Rational Krylov subspaces and Extended Krylov subspaces.} Figure \ref{fig3:sol} reports the solution at time $t=3$. In the top-left panel the uncontrolled state grows in time. In the top-right panel we show the results of the LQR control computed by linearizing equation \eqref{pol:sd} around the desired configuration. The control steers the solution to the origin at a very slow rate. The controlled solution with Algorithm \ref{Alg_sdre} {using an Extended Krylov subspace} and Algorithm \ref{Alg_sdreon} are shown at the bottom of the same figure. Both algorithms reach the desired configuration. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new/dirichlet/uncontrolled.eps} \includegraphics[scale=0.29]{test_new/dirichlet/lqr.eps} \includegraphics[scale=0.29]{test_new/dirichlet/sdre.eps} \includegraphics[scale=0.29]{test_new/dirichlet/lyap.eps} \caption{Test \ref{sec:ex3}: controlled trajectories \eqref{ex1} at time $t=3$ with zero Dirichlet boundary conditions. Top: Uncontrolled solution (left), LQR control (right). Bottom: $\mathcal{H}_2$-controlled solution with Algorithm \ref{Alg_sdre} with Extended Krylov subspaces (left), $\mathcal{H}_2$-controlled solution with Algorithm \ref{Alg_sdreon} (right).} \label{fig3:sol} \end{figure} \begin{figure}[htbp] \centering \includegraphics[scale=0.3]{test_new/dirichlet/cost.eps} \includegraphics[scale=0.3]{test_new/dirichlet/contr.eps} \includegraphics[scale=0.3]{test3/cpu.eps} \includegraphics[scale=0.3]{test3/cost.eps} \caption{Test \ref{sec:ex3}: cumulative functional (top-left) with $\mathcal{H}_2-$control and corresponding control input (top-right). Evaluation of the cost functional with respect to the dimension of the problem $d$ (x-axis) (bottom-right) and CPU time (bottom-left) for both Algorithm \ref{Alg_sdre} {using Rational Krylov subspaces and Extended Krylov subspaces} and Algorithm \ref{Alg_sdreon}.} \label{fig3:cost} \end{figure} \end{example} The performance of the different feedback synthesis methods is presented in Figure \ref{fig3:cost}. Here we show the evaluation of the cumulative cost functional in the top-left panel. For completeness we provide the control inputs on the top-right panel. We compare the performance of Algorithm \ref{Alg_sdre} using Rational Krylov (RK) and Extended Krylov (EKSM) subspaces as the problem dimension increases. Specifically, we recall that the Extended Krylov subspace computes directly the gain matrix. As expected, Algorithm \ref{Alg_sdre} provides the best closed-loop performance among the proposed algorithms. However, in terms of CPU time Algorithm~\ref{Alg_sdreon} is faster than Algorithm~\ref{Alg_sdre} when increasing the problem dimension $d$ as shown in the bottom-left panel of Figure~\ref{fig3:cost}. When the problem dimension $d$ increases, the cost functional converges to $0.9$ for Algorithm~\ref{Alg_sdre} and to $1.3$ for Algorithm \ref{Alg_sdreon}. The two different Krylov subspace methods in Algorithm \ref{Alg_sdre} lead exactly to the same solution. {In the plot legend we refer to {\it Alg 3.1 RK} for Rational Krylov subspaces and to {\it Alg 3.1 EKSM} for Extended Krylov subspaces.} \subsection{Case study 2: the viscous Burgers equation with exponential forcing term} The second experiment deals with the control of a viscous Burgers equation with exponential forcing term over $\Omega\times\mathbb{R}^+_0$, with $\Omega\subset\mathbb{R}^2$ and Dirichlet boundary conditions: \begin{align}\label{ex2} \begin{aligned} \pt X(\xi,t)&=\epsilon\Delta X(\xi,t)- X(\xi,t)\cdot \nabla X(\xi,t)+1.5X(\xi,t)e^{-0.1X(\xi,t)}\\ &\qquad+\chi_{\omega_c}(\xi)u(t)+\chi_{\omega_d}(\xi)w(t)\\ X(\xi,t)&=0\,, \quad\xi\in\partial\Omega,\\ X(\xi,0)&={\mathbf x}_0(\xi)\,,\quad\xi\in\Omega\,. \end{aligned} \end{align} In this case, the scalar control and disturbance act, respectively, through the indicator function $\chi_{\omega_c}(\xi), \chi_{\omega_d}(\xi)$ with $\omega_c,\omega_d\subset\Omega$. A finite difference discretization of the space of the system dynamics leads to a state space representation of the form \begin{equation}\label{burg:sd} \dot X(t) = \Delta_d X(t)- X(t)\circ (DX(t) + 1.5e^{-0.1X(t)}) +B u(t) + H w(t)\,, \end{equation} where the matrices $\Delta_d,D\in\mathbb{R}^{d\times d}$ and $B,H\in \mathbb{R}^d$ are finite-dimensional approximations of the Laplacian, gradient, control and disturbance operators, respectively, and the exponential term is understood component-wise. {In particular, $D$ is obtained using a backward finite difference discretization $$D:=-\Delta\xi^{-1}( B_{n_{\xi_2}}\otimes \mathbf{I}_{n_{\xi_1}} + \mathbf{I}_{n_{\xi_2}}\otimes B_{n_{\xi_1}})\,,$$ where $B_n:=\texttt{tridiag}([\begin{array}{ccc}-1 &1& 0\end{array}],n)$, and $\otimes$ denotes the Kronecker product.} We proceed to express the semi-discretized dynamics in semilinear form. For this, we define $$A(X) := \epsilon\Delta_d -\tilde D(X) + \texttt{diag}(1.5e^{-0.1X(t)})\,,\quad [\tilde D(X)]_{k,l}=D_{k,l}X_k\,,$$ and then \begin{align*} A_0 = \epsilon\Delta_d,\quad [A_j]_{k,l}=D_{j,l}\delta_{k,l},\quad f_j(X)=X_j,\quad j=1,\ldots,d\,,\\ [A_j]_{k,l}=\delta_{k,j-d}\delta_{l,j-d},\quad f_j(X)=1.5e^{-0.1X_{j-d}},\quad j=d+1,\ldots,2d\,. \end{align*} For our numerical experiments we set $\Omega = [0,1]\times[0,1]$, $\epsilon=0.1$, $R = 0.05$, and initial condition ${\mathbf x}_0(\xi) = \sin (\xi_1)\sin(\xi_2)$, on a discretized space grid of $n_{\xi_1}\times n_{\xi_2}$ nodes with $n_{\xi_1} = n_{\xi_2} = 101$ ($d=10201$). The matrices $B$ and $C$ are defined as in the previous study case (see Figure \ref{fig:bc_coll}). In the following we discuss the results for $\mathcal{H}_2$ and $\mathcal{H}_\infty$ control, i.e., $P = 0$ and $P\neq 0$, respectively, in \eqref{costex1}. \begin{example}{\it Experiments for $\mathcal{H}_2-$control.}\label{sec2:ex1} The trajectories of the controlled system problem with $P = 0$ in \eqref{costex1} are shown in Figure \ref{fig2:sol}. The uncontrolled solution tends to move towards the top-right corner of $\Omega$. All algorithms tend to control the solution to zero for large time, but at a different rate. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new_burgers/uncontrolled.eps} \includegraphics[scale=0.29]{test_new_burgers/lqr.eps} \includegraphics[scale=0.29]{test_new_burgers/sdre_1.eps} \includegraphics[scale=0.29]{test_new_burgers/lyap_1.eps} \caption{Test \ref{sec2:ex1}: controlled dynamics \eqref{ex2} at time $t=3$. Top: uncontrolled solution (left), $\mathcal{H}_2$-solution with LQR control (right). Bottom: $\mathcal{H}_2$-controlled solution with Algorithm \ref{Alg_sdre} (left), $\mathcal{H}_2$-controlled solution with Algorithm \ref{Alg_sdreon} (right). The controllers stabilize the dynamics to $X\equiv 0$ at a different rate.} \label{fig2:sol} \end{figure} The control inputs are then shown in the top-right of Figure \ref{fig2:cost}. Algorithm \ref{Alg_sdre} has the largest control input, leading to the smallest cumulative cost functional as shown in the top-left panel of Figure \ref{fig2:cost}. We also observe that the values of the cost functional are very similar for Algorithm~\ref{Alg_sdre} and Algorithm~\ref{Alg_sdreon}. This is also confirmed for different discretization of increasing dimensions as shown in the bottom-left panel of Figure \ref{fig2:cost}. However, the time needed for Algorithm \ref{Alg_sdreon} to compute the solution is lower than for Algorithm~\ref{Alg_sdre} as depicted in the bottom-right panel of Figure \ref{fig2:cost}. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new_burgers/cost.eps} \includegraphics[scale=0.29]{test_new_burgers/contr.eps} \includegraphics[scale=0.29]{test_new_burgers/cost_dim.eps} \includegraphics[scale=0.29]{test_new_burgers/cpu_dim.eps} \caption{Test \ref{sec2:ex1}. Top: evaluation of the cumulative cost functional with $\mathcal{H}_2-$control (left) and control inputs (right). Bottom: cost functional for dynamics of increasing dimension for different algorithms (left) and CPU time for both Algorithm \ref{Alg_sdre} and Algorithm \ref{Alg_sdreon} (right).} \label{fig2:cost} \end{figure} \end{example} \begin{example}{\it Experiments for $\mathcal{H}_\infty-$control.}\label{sec2:ex2} Finally, we discuss the results for $P=1$, $\gamma=0.1$ and disturbance $w(t) = \{0.1\sin(2t)\}$ in \eqref{ex2}. The results presented in Figure \ref{fig2:an1} are in line with our first case study. In this example, Algorithm \ref{Alg_sdre} stabilizes the solution faster. Since it is difficult to visualize differences in the controlled state variables, we provide a qualitative analysis through Figure \ref{fig2:an1}. We show the evaluation of the cost functional \eqref{costex1} on the left panel and the control inputs on the right panel. Again, we find that Algorithm \ref{Alg_sdre} with $\gamma\neq0$ has the lowest values for the cost functional. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{test_new_burgers/cost_dist.eps} \includegraphics[scale=0.29]{test_new_burgers/contr_dist.eps} \caption{Test \ref{sec2:ex2}: evaluation of the cost functional $\mathcal{H}_2$ (left) and control input( right) for $w(t)=0.1\sin(2t)$.} \label{fig2:an1} \end{figure} \end{example} \section{Conclusions and future work} In this work we have discussed different alternatives for the synthesis of feedback laws for stabilizing nonlinear PDEs. In particular, we have studied the use of state-dependent Riccati equation methods, both for $\mathcal{H}_2$ and $\mathcal{H}_\infty$ synthesis. Implementing an SDRE feedback law requires expressing the dynamics in semilinear form and the solution of algebraic Riccati equations at an arbitrarily high rate. This is a stringent limitation in PDE control, where high-dimensional dynamics naturally emerge from space discretization. Hence, we study offline and offline-online synthesis alternatives which circumvent or mitigate the computational effort required in the SDRE synthesis. Most notably, we have proposed an offline-online method which replaces the sequential solution of algebraic Riccati equations by Lyapunov equations. Through extensive computational experiments, including two-dimensional nonlinear PDEs, we have assessed that the SDRE offline-online method provides a reasonable approximation of purely online SDRE synthesis, yielding similar performance results at a reduced computational cost. Moreover, the nonlinearities arising in nonlinear reaction and nonlinear advection PDE models can be easily represented within the semilinearization framework required by SDRE methods. In conclusion, SDRE-based feedback laws constitute a reasonable alternative for suboptimal feedback synthesis for large-scale, but well structured, nonlinear dynamical systems. Future research directions include the study of the SDRE methodology for high-dimensional systems arising from interacting particle systems, and the interplay with deep learning techniques to lower the computational burden associated to a real-time implementation. \bibliographystyle{spmpsci} \section{Solving large-scale Algebraic Riccati/Lyapunov equations} The time steps discussed in Section~\ref{secsdre} all use matrices that stem from the solution of algebraic matrix equations, and more precisely the (quadratic) Riccati and the (linear) Lyapunov equations. The past few years have seen a dramatic improvement in the effectiveness of numerical solution strategies for solving these equations in the large scale setting. For a survey in the linear case we refer the reader to the recent article \cite{Simoncini.survey.16}, while for the algebraic Riccati equation we point the reader to, e.g., \cite{B08,SSM14,Binietal.book.12}. In our derivation we found projection methods to be able to adapt particularly well to the considered setting, with a similar reduction framework for both linear and quadratic methods, other approaches are reviewed for instance in \cite{Benner.Saak.survey13,Simoncini.survey.16}. We emphasize that all considered methods require that the zero order term in the matrix equation, e.g., matrix $Q$ in the Riccati equation \eqref{sdre1}, be low rank. The general idea consists of first determining an approximation space ${\cal K}_k$ that can be naturally expanded if needed, and then seeking an approximate solution in this space, by imposing a Galerkin condition on the matrix residual for computing the projected approximate solution. Let ${\cal V}_k$ be such that ${\cal K}_k={\rm range}({\cal V}_k)$, with ${\cal V}_k$ having orthonormal columns. Recalling that in both the Riccati and Lyapunov equations the solution matrix is symmetric, the approximate solution can be written as ${\cal V}_k Y {\cal V}_k^\top$. Consider the Riccati equation \eqref{sdre1} for a fixed $x=x(t_*)$, so that we set $A(x(t_*))=A_*$ and $B(x(t_*))=B_*$. Let ${\cal R} = A_*^\top {\cal V}_k Y_X {\cal V}_k^\top + {\cal V}_k Y_X {\cal V}_k^\top A_* - {\cal V}_k Y_X {\cal V}_k^\top B_* {\cal V}_k Y_X {\cal V}_k^\top + Q$. Imposing the Galerkin condition on ${\cal R}$ means that the residual matrix ${\cal R}$ be orthogonal to the approximation space, in the matrix sense, that is \begin{equation}\label{eqn:reduced_Riccati} {\cal V}_k^\top {\cal R} {\cal V}_k =0 \quad \Leftrightarrow \quad {\cal V}_k^\top A_*^\top {\cal V}_k Y_X + Y_X {\cal V}_k^\top A_* {\cal V}_k - Y_X ({\cal V}_k^\top B_* {\cal V}_k) Y_X + {\cal V}_k^\top Q {\cal V}_k=0, \end{equation} where the orthogonality of the columns of ${\cal V}_k$ was used. If ${\cal V}_k$ has small dimensions, the {\it reduced} Riccati matrix equation on the right also has small dimensions and can be solved by a ``dense'' method to determine $Y_X$; see, e.g., \cite{Binietal.book.12}. The cost of solving the reduced quadratic equation with coefficient matrices of size $\hat k$ is at least $63 \hat k^3$ floating point operations with an invariant subspace approach \cite{Laub.79}. Note that the large matrix ${\cal V}_k Y_X {\cal V}_k^\top$ is never constructed explicitly, since it would be dense even for sparse data. Analogously, for the Lyapunov equation in \eqref{lyap_on}, we can write $W \approx {\cal V}_k Y {\cal V}_k^\top$ for some $Y=Y_W$ to be determined. Let ${\cal Q}_* = \sum_k Q_k f_k(x(t_*))$. As before, letting $\widetilde{\cal R} = C_0^\top {\cal V}_k Y {\cal V}_k^\top + {\cal V}_k Y {\cal V}_k^\top C_0 + {\cal Q}_*$, the Galerkin condition leads to $$ {\cal V}_k^\top \widetilde{\cal R} {\cal V}_k =0 \quad \Leftrightarrow \quad ({\cal V}_k^\top C_0^\top {\cal V}_k) Y + Y {\cal V}_k^\top C_0 {\cal V}_k + {\cal V}_k^\top{\cal Q}_* {\cal V}_k = 0. $$ This {\it reduced} Lypaunov equation can be solved by means of a ``dense'' method at a cost of about 15 $\hat k^3$ floating point operations for coefficient matrices of size $\hat k$, if the real Schur decomposition is employed; see, e.g., \cite{Simoncini.survey.16}. Note that the computational cost is significantly lower than that of solving the reduced Riccati equation with matrices of the same size. \subsection{On the selection of the approximation space} Choices as approximation space explored in the literature include polynomial and rational Krylov subspaces \cite{Simoncini.survey.16}. They both enjoy the property of being nested as they enlarge, that is ${\cal K}_k \subseteq {\cal K}_{k+1}$ where $k$ is associated with the space dimension. Rational Krylov subspaces have emerged as the key choice because they are able to deliver accurate approximate solutions with a relatively small space dimension, compared with polynomial spaces. Given a starting tall matrix $V_0$ and an invertible stable coefficient matrix $A_*$, we have used two distinct rational spaces: the Extended Krylov subspace, $$ {\cal E}{\cal K}_k = {\rm range}([V_0, A_*^{-1} V_0, A_* V_0, A_*^{-2} V_0, A_*^2 V_0, \ldots, A_*^{k-1} V_0, A_*^{-k} V_0]), $$ which only involves matrix-vector products and solves with $A_*$, and the (fully) Rational Krylov subspace, $$ {\cal R}{\cal K}_ k = {\rm range}([V_0, (A_*- \sigma_2 I)^{-1} V_0, \ldots, \prod_{j=2}^k (A_*- \sigma_j I)^{-1} V_0]) . $$ where $\sigma_j$ can be computed a-priori or adaptively. In both cases, the space is expanded iteratively, one block of vectors at the time, and systems with $A_*$ or with $(A_*- \sigma_j I)$ are solved by fast sparse methods. In our implementation we used the Extended Krylov subspace for solving the Lyapunov equation in Algorithm \ref{Alg_sdreon}, which has several advantages, such as the computation of the sparse Cholesky factorization of $A_0$ once for all. On the other hand, we used the Rational Krylov subspace for the Riccati equation, which has been shown to be largely superior over the Extended Krylov on this quadratic equation, in spite of requiring the solution of a different (shifted) sparse coefficient matrix at each iteration \cite{Benner.Saak.survey13,SSM14}. Nonetheless, in Section~\ref{sec:feedback_matrix} we report an alternative approach that makes the Extended Krylov subspace competitive again for the Riccati problem with $A_0$ symmetric. Except for the operations associated with the reduced problems, the computational costs per iteration of the Riccati and Lyapunov equation solvers are very similar, if the same approximation space is used. Although we refer to the specialized literature for the algorithmic details\footnote{See {\tt www.dm.unibo.it/$\widetilde{\mbox{ }}$simoncin/software} for some related software.}, we would like to include some important implementation details that are specific to our setting. In particular, the matrix $C_0$ employed in Algorithm \ref{Alg_sdreon} is given by $C_0 = A_0 - \left(BR^{-1} B^\top - \frac 1 {2\gamma^2} H P^{-1} H^\top\right) \Pi_0$, which is not easily invertible if explicitly written, since it is dense in general. Note that $A_0$ is in general sparse, as it stems from the discretization of a partial differential operator. We can write $$ C_0 = A_0 - \left([B, H] G [B,H]^\top\right) \Pi_0, \quad {\rm with} \quad G = \begin{pmatrix} R^{-1}& 0\\ 0 & - \frac 1 {2\gamma^2} P^{-1} \end{pmatrix}. $$ Using the classical Sherman-Morrison-Woodbury formula, the product $C_0^{-1} V$ for some tall matrix $V$ can be obtained as $$ W:=C_0^{-1} V = A_0^{-1} V - A_0^{-1} [B, H] G_1^{-1} [B,H]^\top \Pi_0 A_0^{-1}V, $$ with $G_1 = I + G [B,H]^\top \Pi_0 A_0^{-1} [B, H]$, which is assumed to be nonsingular. Therefore, $C_0^{-1} V$ can be obtained by first solving sparse linear systems with $A_0$, and then using matrix-matrix products. More precisely, the following steps are performed: \vskip 0.1in - Solve $A_0 W_1 = V$ - Solve $A_0 W_2 = [B, H]$ - Compute $G_1 = I + G [B,H]^\top \Pi_0 W_2$ - Compute $W = W_1 - W_2 ( \, G_1^{-1} ( [B,H]^\top \Pi_0 W_1) \, )$ \vskip 0.1in We also recall that $\Pi_0$, the solution to the initial Riccati equation, is stored in factored form, and this should be taken into account when computing matrix-matrix products with $\Pi_0$. While trying to employ the Rational Krylov space, we found that the structure of $C_0$ made the selection of optimal shifts $\{\sigma_j\}$ particularly challenging, resulting in a less effective performance of the method. Hence, our preference went for the Extended Krylov space above, with the above enhancement associated with solves with $C_0$. \subsection{Feedback matrix oriented implementation}\label{sec:feedback_matrix} In Algorithm \ref{Alg_sdre} the Riccati equation needs to be solved at each time step $t_n$. However, its solution $\Pi(x_n)$ is only used to computed the feedback matrix $K(x_n) : = - R^{-1} B^\top \Pi(x_n)$. Hence, it would be desirable to be able to immediately compute $K(x_n)$ without first computing $\Pi(x_n)$. This approach has been explored in the Riccati equation literature but also for other problems based on Krylov subspaces, see, e.g., \cite{Palitta.Simoncini.18a,Kressner.proc.2008}. In this section, in the case where the matrix $A(x_n)$ is symmetric, we describe the implementation of one of the projection methods described above, that is able to directly compute $K(x_n)$ without computing $\Pi(x_n)$ (in factored form), and more importantly, without storing and computing the whole approximation basis. The latter feature is particularly important for large scale problems, for which dealing with the orthogonal approximation basis represents one of the major computational and memory costs. To the best of our knowledge, this variant of the Riccati solver is new, while it is currently explored in \cite{Palitta.Pozza.Simoncini.21} for the rational Krylov space. Here we consider, the Extended Krylov subspace. For $A(x_n)$ symmetric, the orthonormal basis of ${\cal E}{\cal K}_k$ can be constructed by explicitly orthogonalizing only with respect to the previous two basis blocks. Hence, only two previous blocks of vectors need to be stored in memory, and require explicitly orthogonalization when the new block of vectors is added to the basis \cite{Palitta.Simoncini.18a}. This is also typical of polynomial Krylov subspaces costructed for symmetric matrices, giving rise to the classical Lanczos three-term recurrence \cite{Kressner.proc.2008}. With this procedure, in the reduced equation \eqref{eqn:reduced_Riccati} the matrices $A_k := {\cal V}_k^\top A_* {\cal V}_k$, $B_k := {\cal V}_k^\top B$ and $Q_k:={\cal V}_k^\top Q {\cal V}_k$ are computed as $k$ grows by updating the new terms at each iteration, and the solution $Y_X$ can be obtained without the whole matrix ${\cal V}_k$ being available. Note that the stopping criterion does not require the computation of the whole residual matrix, so that also in the standard solver the matrix ${\cal V}_k Y_X {\cal V}_k^\top$ is never required until completion. However, to be able to compute $$ K = -R B^\top {\cal V}_k Y_X {\cal V}_k^\top = -R B_k^\top Y_X {\cal V}_k^\top, $$ the basis ${\cal V}_k$ appearing on the right still seems to be required. As already done in the literature, this problem can be overcome by a so-called ``two-step'' procedure: at completion, once the final $Y_X$ is available, the basis ${\cal V}_k$ is computed again one block at the time, and the corrisponding terms in the product $Y_X {\cal V}_k^\top$ are updated. Since $A(x_n)$ is already factorized and the orthogonalization coefficients are already available (they correspond to the nonzero entries of $A_k$), then the overall computational cost is accessible; we refer the reader to \cite{Palitta.Simoncini.18a} and its references for additional details for the two-step procedure employed for different purposes.

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\section{Introduction}\label{sec:intro} \subsection{Overview and statement of main result} One of the most powerful tools in the study of the Langlands program is the \emph{Taylor--Wiles--Kisin patching method} which, famously, was originally introduced by Taylor and Wiles \cite{Wiles,TaylorWiles} to prove Fermat's Last Theorem, via proving a special case of Langlands reciprocity for $\GL_2$. In its modern formulation (due to Kisin \cite{Kisin} and others) this method considers a ring $R_\infty$, which can be determined explicitly from local Galois theoretic data, and constructs a maximal Cohen--Macaulay module $M_\infty$ over $R_\infty$ by gluing together various cohomology groups. Due to its construction, $M_\infty$ is closely related to certain automorphic representations, and so determining its structure has many applications in the Langlands program beyond simply proving reciprocity. A few years after Wiles' proof, Diamond \cite{DiamondMult1} and Fujiwara \cite{Fujiwara} discovered that patching can also be used to prove mod $\ell$ multiplicity one statements in cases where the $q$-expansion principle does not apply. In this argument they consider a case when the ring $R_\infty$ is formally smooth, and so the Auslander-Buchsbaum formula allows him to show that $M_\infty$ is free over $R_\infty$, a fact which easily implies multiplicity one. There are however, many situations arising in practice in which $R_\infty$ is not formally smooth, and so this method cannot be used to determine multiplicity one statements. In this paper, we introduce a new method for determining the structure of a patched module $M_\infty$ arising from the middle degree cohomology of certain Shimura varieties, which applies in cases when $R_\infty$ is Cohen--Macaulay, but not necessarily formally smooth. Using this, we are able to compute the multiplicities for Shimura curves over totally real number fields in the minimal level case, under some technical hypotheses. Our main result is the following (which we state here, using some notation and terminology which we define later): \begin{thm}\label{Mult 2^k} Let $F$ be a totally real number field, and let $D/F$ be a quaternion algebra which is ramified at all but at most one infinite place of $F$. Take some irreducible Galois representation $\overline{\rho}:G_F\to \GL_2(\overline{\F}_\ell)$, where $\ell>2$ is a prime which is unramified in $F$, and prime to the discriminant of $D$. Assume that: \begin{enumerate} \item $\overline{\rho}$ is automorphic for $D$. \label{automorphic} \item $\overline{\rho}|_{G_v}$ is finite flat for all primes $v|\ell$ of $F$. \label{finite flat} \item If $v$ is an prime of $F$ at which $\overline{\rho}$ ramifies and $\Nm(v)\equiv -1\pmod{\ell}$, then either $\overline{\rho}|_{I_v}$ is irreducible or $\overline{\rho}|_{G_v}$ is absolutely reducible. \label{vexing} \item If $v$ is any prime of $F$ at which $D$ ramifies, then $\overline{\rho}$ is Steinberg at $v$ and $\Nm(v)\not\equiv -1\pmod{\ell}$. \label{-1modl} \item The restriction $\overline{\rho}|_{G_{F(\zeta_\ell)}}$ is absolutely irreducible. \label{TW conditions} \end{enumerate} Let $K^{\min}\subseteq D^\times(\mathbb{A}_{F,f})$ be the \emph{minimal} level at which $\overline{\rho}$ occurs, and let $X^D(K^{\min})$ be the Shimura variety (of dimension either 0 or 1) associated to $K^{\min}$. Then the \emph{multiplicity}\footnote{In the case when $D$ unramified at exactly one infinite place of $F$, $X^D(K^{\min})$ is an algebraic curve, and so this multiplicity is just the number of copies of $\overline{\rho}$ which appear in the {\'etale}\ cohomology group $H^1_{\et}(X^D(K^{\min}),\mu_\ell)$. In the case when $D$ is ramified at all infinite pales of $F$, $X^D(K^{\min})$ is just a discrete set of points and so $\overline{\rho}$ does not actually appear in the cohomology. In this case, by the multiplicity we just mean the dimension of the eigenspace $H^0(X^D(K^{\min}),\mathbb{F}_\ell)[m]$ for $m$ the corresponding maximal ideal of the Hecke algebra.} with which $\overline{\rho}$ occurs in the mod $\ell$ cohomology of $X^D(K^{\min})$ is $2^k$, where \[k = \#\bigg\{v \bigg| D \text{ ramifies at } v, \overline{\rho} \text{ is unramified at } v \text{ and, } \overline{\rho}(\Frob_v) \text{ is a scalar} \bigg\}.\] \end{thm} Note that the conditions that $D$ ramifies at $v$, $\overline{\rho}$ is unramified at $v$ and, $\overline{\rho}(\Frob_v)$ is a scalar imply that $\Nm(v)\equiv 1\pmod{\ell}$. \subsection{Some history of higher multiplicity results} A special case Theorem \ref{Mult 2^k} was first proven by Ribet \cite{RibetMult2} in the case when $F=\mathbb{Q}$, $D = D_{pq}$ is the indefinite quaternion algebra ramified at two primes, $p$ and $q$, and $K^{\min}$ is the group of units in a maximal order of $D_{pq}(\mathbb{A}_{F,f})$, i.e. the ``level one'' case (in which case $k$ is forced to be either $0$ or $1$, as $\overline{\rho}$ is necessarily ramified at at least of of $p$ and $q$). He also proved a more general result in the case when $F=\mathbb{Q}$ and $D = D_p$ is the definite quaternion algebra ramified at one prime, $p$. Yang \cite{Yang} gave a (non-sharp) upper bound on the multiplicity in the case where $F=\mathbb{Q}$ and $\overline{\rho}$ was ramified in at least half of the primes in the discriminant of $D$, and also showed that multiplicities of at least 4 are achievable. Helm \cite{Helm} strengthened this result to prove the optimal upper bound of $2^k$ on the multiplicity, again in the case of $F = \mathbb{Q}$, but without the ramification condition for $\overline{\rho}$. Cheng and Fu \cite{ChengFu} generalized Ribet's multiplicity $2$ results to the case when $F$ was a totally real number field, and also generalized this somewhat to the higher weight case. More precisely, they consider two indefinite quaternion algebras $D_0$ and $D$ (called $D$ and $D'$ in their paper), where $D$ is ramified precisely at the primes where $D_0$ is ramified as well as two additional primes $\mathfrak{p}$ and $\mathfrak{q}$. Provided that $\overline{\rho}$ is ramified at $\mathfrak{q}$ and that the Shimura curve associated to $D_0$ satisfies an appropriate multiplicity result one at \emph{non-minimal} level (which holds by the $q$-expansion principle in the case when $F=\mathbb{Q}$ and $D_0 = M_2(\mathbb{Q})$, and follows in many other cases by known cases of Ihara's Lemma), they prove Theorem \ref{Mult 2^k} for $D$. In this case $k$ is again forced to be either $0$ or $1$, as $\mathfrak{p}$ is the only prime in the discriminant of $D'$ at which $\overline{\rho}$ can possibly be unramified and scalar. They further proved (see Remark 3.9 in their paper) cases of Theorem \ref{Mult 2^k} in which $k$ is allowed to be arbitrary, showing unconditionally that a multiplicity of $2^k$ is possible for any $k$. These results require that $\overline{\rho}$ ramifies at at least $k$ primes in the discriminant of $D$, and still rely on certain non-minimal multiplicity one results. Their arguments rely heavily on the assumption that $\overline{\rho}$ ramifies at certain primes in the discriminant of $D$ (as well as on the non-minimal multiplicity one results) so it is very unlikely that their approach could be used to prove results in the same generality as this paper (even if the needed multiplicity one results were proven in full generality). In particular, their method cannot handle the case when $\overline{\rho}$ is unramified at every prime in the discriminant, or even more so when $\overline{\rho}$ is unramified and scalar at every prime in the discriminant, whereas the methods of this paper cover those cases with no added difficulty. Based on the results over $\mathbb{Q}$, Buzzard, Diamond and Jarvis \cite{BDJ} formulated a mod $\ell$ local-global compatibility conjecture, which gives a conjectural description of the multiplicity for arbitrary $F$, $D$ and (prime to $\ell$) level. Theorem \ref{Mult 2^k} is a special case of this conjecture. \subsection{Overview of our method} The previous results relied heavily on facts about integral models of Shimura curves, as well as other results such as mod $\ell$ multiplicity one statements for modular curves (arising from the $q$-expansion principle) and Ihara's Lemma. Our approach is entirely different, and does not rely on any such statements about Shimura curves. Our method relies on the natural self-duality of the module $M_\infty$, combined with an explicit calculation of the ring $R_\infty$ arising in the patching method, together with its Weil class group $\Cl(R_\infty)$ and dualizing module $\omega_{R_\infty}$. The fact that $M_\infty$ is self-dual, and of generic rank $1$, implies that $M_\infty$ corresponds to an element of $[M_\infty] \in \Cl(R_\infty)$ satisfying $2[M_\infty] = [\omega_{R_\infty}]$. Provided that $\Cl(R_\infty)$ is $2$-torsion free (which it is in our situation) this uniquely determines $M_\infty$ up to isomorphism. Thus determining the structure of $M_\infty$ is simply a matter of computing everything explicitly enough to determine the unique module $M_\infty$ satisfying $2[M_\infty] = [\omega_{R_\infty}]$. While these computations may be quite difficult in higher dimensions, all of the relevant local deformation rings have been computed by Shotton \cite{Shotton} in the $\GL_2$ case, and moreover his computations show that the ring $R_\infty/\lambda$ is (the completion of) the ring of functions on a toric variety. This observation makes it fairly straightforward to apply our method in the $\GL_2$ case, and hence to precisely determine the structure of the patched module $M_\infty$. Additionally, our explicit description of the patched module $M_\infty$ allows us to extract more refined data about the Hecke module structure of the cohomology groups, beyond just the multiplicity statements (see Theorem \ref{R=T}, below). This has potential applications to the integral Eichler basis problem and the study of congruence ideals. \subsection{An alternate approach to the multiplicity $2$ result of \cite{CG18}} Another approach to proving a multiplicity $2$ statement by analyzing the structure of $R_\infty$, in a slightly different context, was given by Calegari and Geraghty in \cite[Section 4]{CG18}. Their approach again relies on explicit computations of local deformation rings, done in \cite{Snowden} in their context, as well as a computation of the dualizing module $\omega_{R_\infty}$ (which is also necessary for our method). Unlike the method described in this paper however, their method also crucially uses the work of \cite{Gross} on the $p$-divisible group of $J_1(N)$, and its Hecke module structure, which was ultimately proved using the $q$-expansion principle. This makes it somewhat difficult to generalize their approach. Our method can be used to provide an alternate proof of Calegari and Geraghty's multiplicity $2$ result, without the reliance on the results of \cite{Gross}. In fact one can check that (in the notation of \cite[Section 4]{CG18}) that $(\varpi,\beta)$ is a regular sequence for the ring $\widetilde{R}^\dagger$, and that $\widetilde{R}^\dagger/(\varpi,\beta)$ is isomorphic to the ring $\overline{S}$ considered in Section \ref{sec:toric} of this paper, which proves their multiplicity $2$ result without needing any computations beyond the ones already carried out in this paper (and also gives an analogue of Theorem \ref{R=T}). Since our method does not rely on the work of \cite{Gross}, it would automatically give a generalization of Calegari and Geraghty's result to Shimura curves over $\mathbb{Q}$. It's also likely that this could be extended to totally real number fields by analyzing the appropriate local deformation rings considered in \cite{Snowden}. \subsection{Explanations for the conditions in Theorem \ref{Mult 2^k}} Many of the conditions in the statement of Theorem \ref{Mult 2^k} were included primarily to simplify the proof and exposition, and are not fundamental limitations on our method. Condition (\ref{finite flat}) is essentially an assumption that the minimal level of $\overline{\rho}$ is prime to $\ell$. It, together with the earlier assumption that $\ell$ does not ramify in $F$, is included to ensure that the local deformation rings $R_v^{\square,\fl,\psi}(\overline{\rho}|_{G_v})$ considered in Section \ref{sec:deformation} are formally smooth. As the local deformation rings at $v|\ell$ are known to be formally smooth in more general situations, this condition can likely be relaxed somewhat with only minimal modifications to our method. Even more generally, it is likely that our techniques can be extended to certain other situations in which the local deformation rings at $v|\ell$ are \emph{not} formally smooth, provided we can still explicitly compute these rings. Condition (\ref{vexing}) rules out the so-called `vexing' primes. It is mainly for convenience, to allow us to phrase our argument in terms of a `minimal level' $K^{\min}$ for $\overline{\rho}$ and avoid a discussion of types. The treatment of minimally ramified deformation conditions in \cite{CHT} should allow this restriction to be removed without much added difficulty. Condition (\ref{-1modl}) ensures that the Steinberg deformation ring, $R^{\square,\st,\psi}(\overline{\rho}|_{G_v})$ from Section \ref{sec:deformation} is a domain. In the case when $R^{\square,\st,\psi}(\overline{\rho}|_{G_v})$ fails to be a domain, Thorne \cite{ThorneDih} has defined new local deformation conditions which pick out the individual components. It's likely this could allow this condition to be removed as well, with only small modifications to our arguments. The restriction to the minimal level is similarly intended to ensure that the deformation rings considered will be domains. It is possible this restriction can be relaxed in certain cases, particularly in cases when Ihara's Lemma is known. We intend to explore the possibility of relaxing or removing some of these conditions in future work. Lastly, condition (\ref{TW conditions}) is the classical ``Taylor--Wiles condition''\footnote{Experts will note that there is also another Taylor--Wiles condition one must assume in the case when $\ell=5$ and $\sqrt{5}\in F$. In our case however, this situation is already ruled out by the assumption that $\ell$ is unramified in $F$, and so we do not need to explicitly rule it out.}, which is a technical condition necessary for our construction in Section \ref{sec:patching}. It is unlikely that this condition can be removed without a significant breakthrough. \subsection{Definitions and Notation}\label{ssec:notation} Let $F$ be a totally real number field, with ring of integers $\O_F$. We will always use $v$ to denote a finite place $v\subseteq \O_F$. For any such $v$, let $F_v$ be the completion of $F$ and let $\O_{F,v}$ be its ring of integers. Let $\varpi_v$ be a uniformizer in $\O_{F,\mathfrak{p}}$ and let $\k_v = \O_{F,v}/\varpi_v = \O_F/v$ be the residue field. Let $\Nm(v) = \#\k_v$ be the \emph{norm} of $v$. Let $D$ be a quaternion algebra over $F$ with discriminant $\mathfrak{D}$ (i.e. $\mathfrak{D}$ is the product of all finite primes of $F$ at which $D$ is ramified). Assume that $D$ is either ramified at all infinite places of $F$ (the \emph{totally definite} case), or split at exactly one infinite place (the \emph{indefinite} case). Fix a maximal order of $D$, so that we may regard $D^\times$ as an algebraic group defined over $\O_F$. Now fix a prime $\ell> 2$ which is relatively prime to $\mathfrak{D}$ and does not ramify in $F$. For the rest of this paper we will fix a finite extension $E/\mathbb{Q}_\ell$. Let $\O$ be the ring of integers of $E$, $\lambda\in \O$ be a uniformizer and $\mathbb{F} = \O/\lambda$ be its residue field. For any $\lambda$-torsion free $\O$-module $M$, we will write $M^\vee = \Hom_\O(M,\O)$ for its dual. We define a \emph{level} to be a compact open subgroup \[K = \prod_{v\subseteq \O_F} K_v \subseteq \prod_{v\subseteq \O_F} D^\times(\O_{F,v})\subseteq D^\times(\mathbb{A}_{F,f})\] where we have $K_v = D^\times(\O_{F,v})$ for each $v|\mathfrak{D}$. We say that $K$ is \emph{unramified} at some $v\nmid \mathfrak{D}$ if $K_v = \GL_2(\O_{F,v})$. Note that $K$ is necessarily unramified at all but finitely many $v$. Write $N_K$ for the product of all places $v\nmid \mathfrak{D}$ where $K$ is ramified. If $D$ is totally definite, let \[S^D(K) = \left\{f: D^\times(F)\backslash D^\times(\mathbb{A}_{F,f})/K \to \O\right\}.\] If $D$ is indefinite, let $X^D(K)$ be the Riemann surface $D^\times(F)\backslash \left(D^\times(\mathbb{A}_{F,f})\times \mathcal{H}\right)/K$ (where $\mathcal{H}$ is the complex upper half plane). Give $X^D(K)$ its canonical structure as an algebraic curve over $F$, and let $S^D(K) = H^1(X^D(K),\mathbb{Z})\otimes_\mathbb{Z}\O$. Also define $\overline{S^D}(K) = S^D(K)\otimes_\O\mathbb{F}$. For any finite prime ideal $v$ of $F$ for which $v\nmid\mathfrak{D} N_K$, consider the double-coset operators $T_v,S_v:S^D(K)\to S^D(K)$ given by \begin{align*} T_v &= \left[K\begin{pmatrix}\varpi_v&0\\0&1\end{pmatrix}K\right],& S_v &= \left[K\begin{pmatrix}\varpi_v&0\\0&\varpi_v\end{pmatrix}K\right]. \end{align*} Let \begin{align*} \mathbb{T}^D(K) &= \O\bigg[ T_v,S_v, S_v^{-1}\bigg|v\subseteq \O_F, v\nmid\mathfrak{D} N_K\bigg] \subseteq \End_{\O}(S^D(K)) \end{align*} be the (anemic) Hecke algebra. It will sometimes be useful to treat the $\mathbb{T}^D(K)$'s as quotients of a fixed ring $\mathbb{T}^{\univ}_S=\O\left[T_v,S_v^{\pm1}\right]_{v\not\in S}$, where $S$ is a finite set of primes, containing all primes dividing $\mathfrak{D} N_K$ (here, $T_v$ and $S_v$ are treated as commuting indeterminants). We can thus think of any maximal ideal $m\subseteq \mathbb{T}^D(K)$ as being a maximal ideal of $\mathbb{T}^{\univ}_S$, and hence as being a maximal ideal of $\mathbb{T}^D(K')$ for all $K'\subseteq K$. Now let $G_F = \Gal(\overline{\Q}/F)$ be the absolute Galois group of $F$. For any $v$, let $G_v = \Gal(\overline{F_v}/F_v)$ be the absolute Galois group of $F_v$, and let $I_v\unlhd G_v$ be the inertia group. Fix embeddings $\overline{\Q}\hookrightarrow \overline{F_v}$ for all $v$, and hence embeddings $G_v\hookrightarrow G$. Let $\Frob_v\in G_v$ be a lift of (arithmetic) Frobenius. Let $\varepsilon_\ell:G_F\to \O^\times$ be the cyclotomic character (given by $\sigma(\zeta) = \zeta^{\varepsilon_\ell(\sigma)}$ for any $\sigma\in G_F$ and $\zeta\in \mu_{\ell^\infty}$), and let $\overline{\e}_\ell:G_F\to \mathbb{F}^\times$ be its mod $\ell$ reduction. Now take a maximal ideal $m\subseteq \mathbb{T}^D(K)$, and note that $\mathbb{T}^D(K)/m$ is a finite extension of $\mathbb{F}$. It is well known (see \cite{Carayol1}) that the ideal $m$ corresponds to a two-dimensional semisimple Galois representation $\overline{\rho}_m:G_F\to \GL_2(\mathbb{T}^D(K)/m)\subseteq \GL_2(\overline{\F}_\ell)$ satisfying: \begin{enumerate} \item $\overline{\rho}_m$ is \emph{odd}. \item If $v\nmid \mathfrak{D},\ell,N_K$, then $\overline{\rho}_m$ is unramified at $v$ and we have \begin{align*} \tr(\overline{\rho}_m(\Frob_v)) &\equiv T_v\pmod{m}\\ \det(\overline{\rho}_m(\Frob_v)) &\equiv \Nm(v)S_v\pmod{m}. \end{align*} \item If $v|\ell$ and $v\nmid \mathfrak{D},N_K$, then $\overline{\rho}_m$ is finite flat at $v$. \item If $v|\mathfrak{D}$ then \[\overline{\rho}_m|_{G_v} \sim \begin{pmatrix} \overline{\chi}\,\overline{\e}_\ell&*\\ 0 & \overline{\chi} \end{pmatrix}.\] where $\overline{\chi}:G_v\to \overline{\F}_\ell^\times$ is an unramified character. \end{enumerate} We say that $m$ is \emph{non-Eisenstein} if $\overline{\rho}_m$ is absolutely irreducible. In keeping with property (4) above, we will say that a local representation $r:G_v\to \GL_2(\overline{\F})$ (resp. $r:G_v\to \GL_2(\overline{E})$) is \emph{Steinberg} if it can be written (in some basis) as \begin{align*} & r = \begin{pmatrix} \overline{\chi}\,\overline{\e}_\ell&*\\ 0 & \overline{\chi} \end{pmatrix} & &\left(\text{resp. } r = \begin{pmatrix}\chi \varepsilon_\ell& *\\0&\chi\end{pmatrix}\right) \end{align*} for some unramified character $\overline{\chi}:G_v\to \overline{\F}$ (resp. $\chi:G_v\to \overline{E}$). We say that a global representation $r:G_F\to \GL_2(\overline{\F})$ (resp. $r:G_F\to \GL_2(\overline{E})$) is \emph{Steinberg at $v$} if $r|_{G_v}$ is Steinberg. Now if $\psi:G_F\to \O^\times$ is a character for which $\psi\varepsilon_\ell^{-1}$ has finite image, define the \emph{fixed determinant} Hecke algebra $\mathbb{T}^D_{\psi}(K)$ to be quotient of $\mathbb{T}^D(K)$ on which $\Nm(v)S_v=\psi(\Frob_v)$ for all $v\nmid \mathfrak{D},\ell$ at which $K$ is unramified. Note that by Chebotarev density, a maximal ideal $m\subseteq \mathbb{T}^D(K)$ is in the support of $\mathbb{T}^D_{\psi}(K)$ if and only if $\overline{\rho}_m$ has a lift $\rho:G_F\to \GL_2(\O)$ which is modular of level $K$ with $\det\rho = \psi$ (which in particular implies that $\det\overline{\rho}_m\equiv \psi\pmod{\lambda}$). Now for any continuous \emph{absolutely irreducible} representation $\overline{\rho}:G_F\to \GL_2(\overline{\F}_\ell)$, define: \[\mathcal{K}^D(\overline{\rho}) = \left\{K\subseteq D^\times(\mathbb{A}_{F,f})\middle| \overline{\rho}\sim \overline{\rho}_m \text{ for some } m\subseteq \mathbb{T}^D(K)\right\}\] (that is, $\mathcal{K}^D(\overline{\rho})$ is the set of levels $K$ at which the representation $\overline{\rho}$ can occur.) From now on, fix an absolutely irreducible Galois representation $\overline{\rho}:G_F\to \GL_2(\overline{\F}_\ell)$ for which $\mathcal{K}^D(\overline{\rho})\ne\ensuremath{\varnothing}$ (i.e. $\overline{\rho}$ is ``automorphic for $D$''). In particular, this implies that $\overline{\rho}$ is \emph{odd}, and satisfies the numbered conditions in Section \ref{ssec:notation}. Also assume that $\overline{\rho}|_{G_v}$ is finite flat at all primes $v|\ell$ of $F$. We will say that a \emph{minimal level} of $\overline{\rho}$ is an element of $\mathcal{K}^D(\overline{\rho})$ which is maximal under inclusion. The assumption that $\overline{\rho}|_{G_v}$ is finite flat for all $v|\ell$ implies that we may pick a minimal level $K^{\min} = \prod_{v\subseteq\O_F}K^{\min}_v$ of $\overline{\rho}$ for which $K^{\min}_v = D^\times(\O_v)$ for all $v|\ell$. From now on fix such a $K^{\min}$. Given any level $\displaystyle K = \prod_{v\subseteq\O_F}K_v\subseteq K^{\min}$, we say that $K$ is of \emph{minimal level} at some $v\subseteq \O_F$ if $K_v = K^{\min}_v$. Now given $K\in \mathcal{K}^D(\overline{\rho})$ and $m\subseteq \mathbb{T}^D(K)$ for which $\overline{\rho} \sim \overline{\rho}_m$ we define the number: \[\nu_{\overline{\rho}}(K) = \begin{cases} \dim_{\mathbb{T}^D(K)/m} \overline{S^D}(K)[m]& \text{ if } D \text{ is totally definite}\\ \frac{1}{2}\dim_{\mathbb{T}^D(K)/m} \overline{S^D}(K)[m]& \text{ if } D \text{ is indefinite} \end{cases}\] called the \emph{multiplicity} of $\overline{\rho}$ at level $K$. This number is closely related to the mod $\ell$ local-global compatibility conjectures given in \cite{BDJ}. Note that $\nu_{\overline{\rho}}(K)$ does not depend on the choice of coefficient ring $\O$. Theorem \ref{Mult 2^k} is precisely the assertion that $\nu_{\overline{\rho}}(K^{\min}) = 2^k$. \subsection{Endomorphisms of Hecke modules} We close this section by stating another result of our work: \begin{thm}\label{R=T} Let $\overline{\rho}$ satisfy the conditions of Theorem \ref{Mult 2^k}. If $D$ is totally definite then trace map $S^D(K^{\min})_m\otimes_{\mathbb{T}^{D}(K^{\min})_m}S^D(K^{\min})_m\to \omega_{\mathbb{T}^{D}(K^{\min})_m}$, induced by the self-duality of $S^D(K^{\min})_m$ is surjective (where $\omega_{\mathbb{T}^{D}(K^{\min})_m}$ is the dualizing sheaf\footnote{Which exists as $\mathbb{T}^{D}(K^{\min})_m$ is a $\lambda$-torsion free local $\O$-module of Krull dimension 1, by definition.} of $\mathbb{T}^{D}(K^{\min})_m$), and moreover the natural map $\mathbb{T}^{D}(K^{\min})_m\to \End_{\mathbb{T}^{D}(K^{\min})_m}(S^D(K^{\min})_m)$ is an isomorphism. If $D$ is indefinite, then the natural map $\mathbb{T}^{D}(K^{\min})_m\to \End_{\mathbb{T}^{D}(K^{\min})_m[G_F]}(S^D(K^{\min})_m)$ is an isomorphism. \end{thm} As explained in \cite{EmTheta}, this statement has applications towards the integral Eichler basis problem, so can likely be used to strengthen the results of Emerton \cite{EmTheta}. Also the work of \cite{BKM} shows that this statement has an important application to the study of congruence ideals, where it can serve as something of a substitute for a multiplicity one statement. \section*{Acknowledgments} I would like to thank Matt Emerton for suggesting this problem, and for all of his advice. I would also like to specifically thank him for pointing out that Theorem \ref{R=T} followed from my work. I also thank Jack Shotton, Frank Calegari, Toby Gee and Florian Herzig for their comments on earlier drafts of this paper, and for many helpful discussions. I would also like to thank an anonymous reviewer for their helpful suggestions. \section{Galois Deformation Rings}\label{sec:deformation} In this section we will define the various Galois deformation rings which we will consider in the rest of the paper, and review their relevant properties. \subsection{Local Deformation Rings}\label{ssec:local def} Fix a finite place $v$ of $F$ and a representation $\overline{r}:G_v\to \GL_2(\mathbb{F})$. Let $\mathcal{C}_\O$ (resp. $\mathcal{C}_\O^\wedge$) be the category of Artinian (resp. complete Noetherian) local $\O$-algebras with residue field $\mathbb{F}$. Consider the (framed) deformation functor $\mathcal{D}^\square(\overline{r}):\mathcal{C}_\O\to {\bf Set}$ defined by \begin{align*} \mathcal{D}^\square(\overline{r})(A) &= \{r:G_v\to \GL_2(A), \text{ continuous lift of }\overline{r} \}\\ &= \left\{(M,r,e_1,e_2)\middle|\parbox{3.5in}{$M$ is a free rank 2 $A$-module with a basis $(e_1,e_2)$ and $r:G_v\to \Aut_A(M)$ such that the induced map $G\to \Aut_A(M)= \GL_2(A)\to \GL_2(\mathbb{F})$ is $\overline{r}$} \right\}_{/\sim} \end{align*} It is well-known that this functor is \emph{pro-representable} by some $R^\square(\overline{r})\in \mathcal{C}^\wedge_\O$, in the sense that $\mathcal{D}^\square(\overline{r}) \equiv \Hom_\O(R^\square(\overline{r}),-)$. In particular, $\overline{r}$ admits a universal lift $r^\square:G_v\to \GL_2(R^\square(\overline{r}))$. For any continuous homomorphism, $x:R^{\square}(\overline{r})\to \overline{E}$, we obtain a Galois representation $r_x:G_v\to \GL_2(\overline{E})$ lifting $\overline{r}$, from the composition $G_v\xrightarrow{r^\square} \GL_2(R^\square(\overline{r}))\xrightarrow{x} \GL_2(\overline{E})$. Now for any character $\psi: G_v\to \O^\times$ with $\psi\equiv \det \overline{r} \pmod{\lambda}$ define $R^{\square,\psi}(\overline{r})$ to be the quotient of $R^\square(\overline{r})$ on which $\det r^{\square}(g) = \psi(g)$ for all $g\in G_v$. Equivalently, $R^{\square,\psi}(\overline{r})$ is the ring pro-representing the functor of deformations of $\overline{r}$ with determinant $\psi$. Given any two characters $\psi_1,\psi_2:G_v\to \O^\times$ with $\det \overline{r}\equiv \psi_1\equiv \psi_2 \pmod{\lambda}$ we have $\psi_1\psi_2^{-1} \equiv 1 \pmod{\lambda}$, and so (as $1+\lambda\O$ is pro-$\ell$ and $\ell\ne 2$) there is a unique $\chi:G_v\to \O^\times$ with $\psi_1 = \psi_2\chi^2$. But now the map $r\mapsto r\otimes \chi$ is an automorphism of the functor $D^\square(\overline{r})$ which can be shown to induce a natural isomorphism $R^{\square,\psi_1}(\overline{r})\cong R^{\square,\psi_2}(\overline{r})$. Thus, up to isomorphism, the ring $R^{\square,\psi}(\overline{r})$ does not depend on the choice of $\psi$. We call $R^{\square}(\overline{r})$ (respectively $R^{\square,\psi}(\overline{r})$) the \emph{deformation ring} (respectively the \emph{fixed determinant deformation ring}) of $\overline{r}$. In order to prove our main results, we will also need to consider various deformation rings with \emph{fixed type}. Instead of defining these in general, we will consider only the specific examples which will appear in our arguments. If $v|\ell$ and $\overline{r}$ and $\psi$ are both \emph{flat}, define $R^{\square,\fl,\psi}(\overline{r})$ to be the ring pro-representing the functor of (framed) flat deformations of $\overline{r}$ with determinant $\psi$. We will refrain from giving a precise definition of this, as it is not relevant to our discussion. We will refer the reader to \cite{Kisin}, \cite{FontLaff}, \cite{Ramakrishna} and \cite{CHT} for more details, and use only the following result from \cite[Section 2.4]{CHT}: \begin{prop} If $F_v/\mathbb{Q}_\ell$ is unramified, then $R^{\square,\fl,\psi}(\overline{r})\cong \O[[X_1,\ldots,X_{3+[F_v:\mathbb{Q}_\ell]}]]$. \end{prop} Also if $v\nmid \ell$, we recall the notion of a ``minimally ramified'' deformation given in \cite[Definition 2.4.14]{CHT} (which we again refrain from precisely defining). If $\psi$ is a minimally ramified deformation of $\overline{\psi}$, we will let $R^{\square,\min,\psi}(\overline{r})$ be the maximal reduced $\lambda$-torsion free quotient of $R^{\square,\psi}(\overline{r})$ with the property that if $x:R^{\square,\min,\psi}(\overline{r})\to \overline{E}$ is a continuous homomorphism, then the corresponding lift $r_x:G_v\to \GL_2(\overline{E})$ of $\overline{r}$ is minimally ramified. Again, we will refrain from giving a more detailed description of this, and instead we will use only the following well-known result (cf \cite{Shotton,CHT}): \begin{prop}\label{prop:R^min} $R^{\square,\min,\psi}(\overline{r})\cong \O[[X_1,X_2,X_3]]$. \end{prop} \begin{rem} It is a well-known result (see for instance \cite{DiamondVexing}) that if $\overline{\rho}:G_F\to \GL_2(\overline{\F}_\ell)$ is absolutely irreducible and automorphic for $D$ and $v\nmid\ell,\mathfrak{D}$ is a prime of $F$ satisfying condition (\ref{vexing}) of Theorem \ref{Mult 2^k} (i.e. either $\Nm(v)\not\equiv -1\pmod{\ell}$, $\overline{\rho}|_{I_v}$ is irreducible or $\overline{\rho}|_{G_v}$ is absolutely reducible) then an \emph{automorphic} lift $\rho:G_F\to \GL_2(\overline{E})$ is minimally ramified at $v$ if and only if $\rho$ is automorphic of level $K$ for some level $K$ which is minimal at $v$. This was the primary reason for including condition (\ref{vexing}) in Theorem \ref{Mult 2^k}. \end{rem} Now assume that $v\nmid \ell$ and $\overline{r}$ is \emph{Steinberg} (in the sense of Section \ref{ssec:notation}). We define $R^{\square,\st}(\overline{r})$ (called the \emph{Steinberg deformation ring}) to be the maximal reduced $\lambda$-torsion free quotient of $R^{\square}(\overline{r})$ for which $r_x:G_v\to \GL_2(\overline{E})$ is Steinberg for every continuous homomorphism $x:R^{\square,\st}(\overline{r})\to \overline{E}$. Similarly if $\psi:G_v\to \O^\times$ is an \emph{unramified} character with $\psi\equiv \det\overline{r}\pmod{\lambda}$ (by assumption, $\overline{r}$ is Steinberg, and hence $\det\overline{r}$ is unramified), we define $R^{\square,\st,\psi}(\overline{r})$ (called the \emph{fixed determinant Steinberg deformation ring}) to be the maximal reduced $\lambda$-torsion free quotient of $R^{\square,\psi}(\overline{r})$ for which $r_x:G_v\to \GL_2(\overline{E})$ is Steinberg for every continuous homomorphism $x:R^{\square,\st,\psi}(\overline{r})\to \overline{E}$. It follows from our definitions that $R^{\square,\st,\psi}(\overline{r})$ is the maximal reduced $\lambda$-torsion free quotient of $R^{\square,\st}(\overline{r})$ on which $\det\rho^{\square}(g) = \psi(g)$ for all $g\in G_v$. \subsection{Global Deformation Rings}\label{ssec:global def} Now take a representation $\overline{\rho}:G_F\to \GL_2(\mathbb{F})$ satisfying: \begin{enumerate} \item $\overline{\rho}$ is absolutely irreducible. \item $\overline{\rho}$ is odd. \item For each $v|\ell$, $\overline{\rho}|_{G_v}$ is finite flat. \item For each $v|\mathfrak{D}$, $\overline{\rho}$ is Steinberg at $v$. \item $\mathcal{K}^D(\overline{\rho})\ne \ensuremath{\varnothing}$. \end{enumerate} Let $\Sigma_\ell^D$ be a set of finite places of $F$ containing: \begin{itemize} \item All places $v$ at which $\overline{\rho}$ is ramified \item All places $v|\mathfrak{D}$ (i.e. places at which $D$ is ramified) \item All places $v|\ell$ \end{itemize} (we allow $\Sigma_\ell^D$ to contain some other places in addition to these), and let $\Sigma\subseteq \Sigma_\ell^D$ consist of those $v\in \Sigma_\ell^D$ with $v\nmid \ell,\mathfrak{D}$. Now as in \cite{Kisin} define $R^{\square}_{F,S}(\overline{\rho})$ (where $\Sigma_\ell^D\subseteq S$) to be the $\O$-algebra pro-representing the functor $\mathcal{D}^{\square}_{F,S}(\overline{\rho}):\mathcal{C}_\O\to {\bf Set}$ which sends $A$ to the set of tuples $\big(\rho:G_{F,S}\to \End_A(M), \{(e^v_1,e^v_2)\}_{v\in\Sigma_\ell^D} \big)$, where $M$ is a free rank $2$ $A$-module with an identification $M/m_A = \mathbb{F}^2$ sending $\rho$ to $\overline{\rho}$, and for each $v\in \Sigma_\ell^D$, $(e^v_1,e^v_2)$ is a basis for $M$, lifting the standard basis for $M/m_A= \mathbb{F}^2$, up to equivalence. Also define the \emph{unframed} deformation ring $R_{F,S}(\overline{\rho})$ to be the $\O$-algebra pro-representing the functor $\mathcal{D}_{F,S}(\overline{\rho}):\mathcal{C}_\O\to {\bf Set}$ which sends $A$ to the set of free rank $2$ $A$ modules $M$ with action $\rho:G_{F,S}\to \End_A(M)$ for which $\rho\equiv\overline{\rho}\pmod{m_A}$, up to equivalence. This exists because $\overline{\rho}$ is absolutely irreducible. We will let $\rho^{\univ}:G_F\to \GL_2(R_{F,S}(\overline{\rho}))$ denote the universal lift of $\overline{\rho}$. Now take any character $\psi:G_F\to \O^\times$ for which: \begin{enumerate} \item $\psi \equiv \det \overline{\rho} \pmod\lambda$. \item $\psi$ is unramified at all places outside of $\Sigma_\ell^D$, \emph{and} all places dividing $\mathfrak{D}$. \item $\psi$ is flat at all places dividing $\ell$. \item $\psi\varepsilon_\ell^{-1}$ has finite image. \end{enumerate} Note that as $\psi\varepsilon_\ell^{-1}$ has finite image, condition (3) is equivalent to the assertion that $\psi\varepsilon_\ell^{-1}$ is unramified at all places dividing $\ell$. Let $\mathcal{D}^{\square,\psi}_{F,S}\subseteq \mathcal{D}^{\square}_{F,S}$ be the subfunctor of $\mathcal{D}^{\square}_{F,S}$ which sends $A$ to the set of tuples $\big(\rho:G_{F,S}\to \End_A(M), \{(e^v_1,e^v_2)\}_{v\in\Sigma_\ell^D} \big)$ (up to equivalence) in $\mathcal{D}^{\square}_{F,S}(A)$ for which $\det \rho = \psi$. Define $\mathcal{D}^{\psi}_{F,S}\subseteq \mathcal{D}_{F,S}$ similarly. Let $R^{\square,\psi}_{F,S}(\overline{\rho})$ and $R^{\psi}_{F,S}(\overline{\rho})$ to be the rings pro-representing $\mathcal{D}^{\square,\psi}_{F,S}$ and $\mathcal{D}^{\psi}_{F,S}$. Equivalently, these are the quotients of $R^{\square}_{F,S}(\overline{\rho})$ and $R_{F,S}(\overline{\rho})$, respectively, on which $\det\rho = \psi$. Now note that the morphism of functors \[ \left(\rho,\{(e^v_1,e^v_2)\}_{v\in \Sigma_\ell^D}\right) \mapsto \bigg(\rho|_{G_v}:G_v\to \End_A(M),(e^v_1,e^v_2)\bigg)_{v\in\Sigma_\ell^D} \] induces a map: \[\pi:R_{\loc}=\hat{\bigotimes_{v\in \Sigma_\ell^D}}R^{\square}(\overline{\rho}|_{G_v}) \to R^{\square}_{F,S}.\] Now consider the ring \[R^{\square}_{\Sigma,\mathfrak{D},\ell} = \left[\hat{\bigotimes_{v|\ell}}R^{\square,\fl}(\overline{\rho}|_{G_v})\right]\widehat{\otimes} \left[ \hat{\bigotimes_{v\in\Sigma}}R^{\square,\min}(\overline{\rho}|_{G_v}) \right]\widehat{\otimes} \left[\hat{\bigotimes_{v|\mathfrak{D}}}R^{\square,\st}(\overline{\rho}|_{G_v})\right],\] so that $R^{\square}_{\Sigma,\mathfrak{D},\ell}$ is a quotient of $R_{\loc}$. Using the map $\pi$ above, we may now define $R^{\square,D}_{F,S}(\overline{\rho})= R^{\square}_{F,S}\otimes_{R_{\loc}}R^{\square}_{\Sigma,\mathfrak{D},\ell}$. We may also define $R_{\loc}^\psi$, $R^{\square,\psi}_{\Sigma,\mathfrak{D},\ell}$ and $R^{\square,D,\psi}_{F,S}(\overline{\rho})$ analogously, by adding superscripts of $\psi$ to all of the rings used in the definitions. Also note that the morphism of functors $\big(\rho, \{(e^v_1,e^v_2)\}_{v\in\Sigma_\ell^D} \big)\mapsto \rho$ induces a map $R^{\psi}_{F,S}(\overline{\rho}) \to R^{\square,\psi}_{F,S}(\overline{\rho})$. As in \cite[(3.4.11)]{Kisin} this maps is formally smooth of dimension $j = 4|\Sigma_\ell^D|-1$, and so we may identify $R^{\square,\psi}_{F,S}(\overline{\rho}) = R^\psi_{F,S}(\overline{\rho})[[w_1,\ldots,w_j]]$. We can now define a map $R^{\square,\psi}_{F,S}(\overline{\rho})\to R^{\psi}_{F,S}(\overline{\rho})$ by sending each $w_i$ to $0$. Using this map, we may now define a unframed version of $R^{\square,D,\psi}_{F,S}(\overline{\rho})$ via \[R^{D,\psi}_{F,S}(\overline{\rho})= R^{\square,D,\psi}_{F,S}(\overline{\rho})\otimes_{R^{\square,\psi}_{F,S}(\overline{\rho})} R^{\psi}_{F,S}(\overline{\rho}) = R^{\square,D,\psi}_{F,S}(\overline{\rho})/(w_1,\ldots,w_j).\footnote{ This definition may seem somewhat strange and ad-hoc. We have defined it this way because it will be convenient for our argument to view $R^{D,\psi}_{F,S}(\overline{\rho})$ as a quotient of $R^{\square,D,\psi}_{F,S}(\overline{\rho})$. Fortunately we will see later (in the remark following Lemma \ref{gen free}) that $R^{D,\psi}_{F,S}(\overline{\rho})$ is actually a fairly nicely behaved ring in the cases relevant to us, and so this definition will likely coincide with most ``natural'' ones.}\] It follows from these definitions that the maps $x:R^\psi_{F,S}(\overline{\rho})\to \overline{E}$ that factor through $R^{D,\psi}_{F,S}(\overline{\rho})$ are precisely those for which the induced representation $\rho_x:G_F\to \GL_2(R^{\psi}_{F,S}(\overline{\rho}))\to \GL_2(\overline{E})$ satisfies: \begin{itemize} \item $\overline{\rho}_x|_{G_v}$ is flat at all $v|\ell$ \item $\overline{\rho}_x|_{G_v}$ is Steinberg at all $v|\mathfrak{D}$ \item $\overline{\rho}_x|_{G_v}$ is minimally ramified at all $v\in\Sigma$. \end{itemize} (This is simply because the definition of $R^{D,\psi}_{F,S}(\overline{\rho})$ is such that any map $x:R^\psi_{F,S}(\overline{\rho})\to \overline{E}$ factors through $R^{D,\psi}_{F,S}(\overline{\rho})$ if and only if the corresponding map $x:R^{\square,\psi}\twoheadrightarrow R^\psi_{F,S}(\overline{\rho})\to \overline{E}$ factors through $R^{\square,D,\psi}_{F,S}(\overline{\rho})$.) In order to prove Theorem \ref{R=T} we will need slightly more refined information about the relationship between $R_{F,S}(\overline{\rho})$ and $R^{\psi}_{F,S}(\overline{\rho})$. Let $\overline{\psi} = \det\overline{\rho}:G_{F}\to \mathbb{F}^\times$ be the reduction of $\psi$. Let $\mathcal{D}_{F,S}(\overline{\psi}):\mathcal{C}_\O\to{\bf Set}$ be the functor which sends $A\in\mathcal{C}_\O$ to the set of maps $\chi:G_{F,S}\to A^\times$ satisfying $\chi\equiv \overline{\psi}\pmod{m_A}$, up to equivalence. Let $R_{F,S}(\overline{\psi})$ be the ring pro-representing $\mathcal{D}_{F,S}(\overline{\psi})$. Now for any $A\in\mathcal{C}_\O$, $A$ is a finite ring of $\ell$-power order, and so $m_A\subseteq A$ also has $\ell$-power order. It follows that $(1+m_A,\times)$ is an abelian multiplicative group of $\ell$-power order. In particular, as $\ell$ is odd, the map $x\mapsto x^2$ is an automorphism of $(1+m_A,\times)$, and hence it has an inverse $\sqrt{\cdot}: (1+m_A,\times)\to (1+m_A,\times)$. It is easy to see that $x\mapsto x^2$, and hence $x\mapsto \sqrt{x}$, commutes with morphisms in $\mathcal{C}_\O$, and is thus an automorphism of the functor $A\mapsto (1+m_A,\times)$ from $\mathcal{C}_\O$ to $\mathbf{Ab}$. Now consider any $\rho:G_{F,S}\to \End(M)$ in $\mathcal{D}_{F,S}(\overline{\rho})(A)$. By definition we have $\det \rho\equiv \det\overline{\rho} \equiv \psi \pmod{m_A}$, and so $(\det \rho)^{-1}\psi\equiv 1\pmod{m_A}$. That is, the image of $(\det \rho)^{-1}\psi:G_{F,S}\to A^\times$ lands in $1+m_A$. By the above work, it follows that there is a unique character $\sqrt{(\det \rho)^{-1}\psi}:G_{F,S}\to 1+m_A\subseteq A^\times$ with $(\sqrt{(\det \rho)^{-1}\psi})^2 = (\det \rho)^{-1}\psi$. Thus we may define a representation $\rho^{\psi}:G_{F,S}\to \End(M)$ by $\rho^{\psi}= (\sqrt{(\det \rho)^{-1}\psi})\rho$. Notice that $\rho^\psi\in\mathcal{D}_{F,S}(\overline{\rho})(A)$ and we have $\det\rho^\psi = \psi$, so that $\rho^\psi\in\mathcal{D}^\psi_{F,S}(\overline{\rho})(A)$. It is easy to see that the map $\rho\mapsto \rho^\psi$ is a natural transformation $\mathcal{D}_{F,S}\to \mathcal{D}^\psi_{F,S}$. We now claim that the map $\mathcal{D}_{F,S}(\overline{\rho})\to \mathcal{D}_{F,S}(\overline{\psi})\times \mathcal{D}^{\psi}_{F,S}(\overline{\rho})$ given by $\rho\mapsto (\det \rho,\rho^\psi)$ is an isomorphism of functors. Indeed, it has an inverse given by $(\chi,\rho)\mapsto\sqrt{\chi\psi^{-1}}\rho$. Looking at the rings these functors represent gives the following: \begin{lemma}\label{lem:fixed determinant} There is a natural isomorphism $R_{F,S}(\overline{\psi})\widehat{\otimes}_\O R^\psi_{F,S}(\overline{\rho})\xrightarrow{\sim} R_{F,S}(\overline{\rho})$ of $\O$-algebras, induced by the natural transformation $\rho\mapsto (\det\rho,\rho^{\psi})$. \end{lemma} Lemma \ref{lem:fixed determinant} may be though of as giving a natural way of separating the determinant of a representation $\rho:G_{F,S}\to \GL_2(A)$ from the rest of the representation. \subsection{Two Lemmas about Deformation Rings} We finish this section by stating two standard results (cf. \cite{Kisin}) which will be essential for our discussion of Taylor--Wiles--Kisin patching in Section \ref{sec:patching}. The first concerns the existence of an ``$R\to \mathbb{T}$'' map: \begin{lemma}\label{R->T} Assume that $\overline{\rho}$ satisfies all of the numbered conditions listed in Section \ref{ssec:global def}. Take $K=\prod_vK_v\in \mathcal{K}^D(\overline{\rho})$ and let $S$ be a set of finite places of $F$ containing $\Sigma_\ell^D$ such that $K$ is unramified outside of $S$. Then there is a surjective map $R_{F,S}(\overline{\rho})\twoheadrightarrow \mathbb{T}^{D}(K)_m$, which induces a surjective map $R^{\psi}_{F,S}(\overline{\rho})\twoheadrightarrow \mathbb{T}^{D}_{\psi}(K)_m$ for any character $\psi:G_F\to \O^\times$ lifting $\det\overline{\rho}$ for which $m$ is the support of $\mathbb{T}^{D}_{\psi}(K)$. If $K_v$ is maximal for all $v|\mathfrak{D}$ and all $v|\ell$ and $K_v = K_v^{\min}$ for all $v\in \Sigma$, then this map factors through $R^{\psi}_{F,S}(\overline{\rho})\twoheadrightarrow R^{D,\psi}_{F,S}(\overline{\rho})$. \end{lemma} Note that (as mentioned in the remark following Proposition \ref{prop:R^min}) condition (\ref{vexing}) of Theorem \ref{Mult 2^k} is needed to ensure that $R_{F,S}(\overline{\rho})\twoheadrightarrow \mathbb{T}^{D}(K)_m$ does indeed factor through $R^{\psi}_{F,S}(\overline{\rho})\twoheadrightarrow R^{D,\psi}_{F,S}(\overline{\rho})$. The second concerns the existence of ``Taylor--Wiles'' primes: \begin{lemma}\label{TW primes} Assume that $\overline{\rho}$ satisfies all of the numbered conditions listed in Section \ref{ssec:global def} \emph{and} condition (\ref{TW conditions}) of Theorem \ref{Mult 2^k}. Let $S$ be a set of finite places of $F$ containing $\Sigma_\ell^D$, such for any prime $v\in S\smallsetminus \Sigma^D_\ell$, $\Nm(v)\not\equiv1\pmod{\ell}$ and the ratio of the eigenvalues of $\overline{\rho}(\Frob_v)$ is not equal to $\Nm(v)^{\pm 1}$ in $\overline{\F}_\ell^\times$. Then there exist integers $r,g\ge 1$ such that for any $n\ge 1$, there is a finite set $Q_n$ of primes of $F$ for which: \begin{itemize} \item $Q_n\cap S=\ensuremath{\varnothing}$. \item $\# Q_n = r$. \item For any $v\in Q_n$, $\Nm(v)\equiv 1\pmod{\ell^n}$. \item For any $v\in Q_n$, $\overline{\rho}(\Frob_v)$ has distinct eigenvalues. \item There is a surjection $R^{\square,\psi}_{\Sigma,\mathfrak{D},\ell}[[x_1,\ldots,x_g]]\twoheadrightarrow R^{\square,D,\psi}_{F,S\cup Q_n}(\overline{\rho})$. \end{itemize} Moreover, we have $\dim R^{\square,\psi}_{\Sigma,\mathfrak{D},\ell} = r+j-g+1$. \end{lemma} From now on we will write $R_\infty$ to denote $R^{\square,\psi}_{\Sigma,\mathfrak{D},\ell}[[x_1,\ldots,x_g]]$ so that $\dim R_\infty = r+j+1$. By the results of Section \ref{ssec:local def} we have \[R_\infty= \left[\hat{\bigotimes_{v|\mathfrak{D}}}R^{\square,\st,\psi}(\overline{\rho}|_{G_v})\right][[x_1,\ldots,x_{g'}]] \] for some integer $g'$. In Section \ref{sec:toric} below, we will use the results of \cite{Shotton} to explicitly compute the ring $R_\infty$, and then use the theory of toric varieties to study modules over $R_\infty$. In Chapter \ref{sec:patching}, we will use Lemma \ref{R->T} and \ref{TW primes} to construct a particular module $M_\infty$ over $R_\infty$ out of a system of modules over the rings $\mathbb{T}^D(K)$, and then use the results of Chapter \ref{sec:toric} to deduce the structure of $M_\infty$. This will allow us to prove Theorems \ref{Mult 2^k} and \ref{R=T}. \section{Class Groups of Local Deformation Rings}\label{sec:toric} In our situation, all of the local deformation rings which will be relevant to us were computed in \cite{Shotton}. In this section, we will use this description to explicitly describe the ring $R_\infty$, and to study its class group (or rather, the class group of a related ring). We first introduce some notation which we will use for the rest of this paper. If $R$ is any Noetherian local ring, we will always use $m_R$ to denote its maximal ideal. If $M$ is a (not necessarily free) finitely generated $R$-module, we will say that the \emph{rank} of $M$, denoted by $\rank_RM$ is the cardinality of its minimal generating set. If $R$ is a domain we will write $K(R)$ for its fraction field. If $M$ is a finitely generated $R$-module, then we we will say that the \emph{generic rank} of $M$, denoted $\genrank_RM$ is the $K(R)$-dimension of $M\otimes_RK(R)$ (that is, the rank of $M$ at the generic point of $R$). From now on assume that $R$ is a (not necessarily local) normal Cohen--Macaulay \emph{domain} with a dualizing sheaf\footnote{Which will be the case for all Cohen--Macaulay rings we will consider.}, we will use $\omega_R$ to denote the dualizing sheaf of $R$. For any finitely generated $R$-module $M$, we will let $M^* = \Hom_R(M,\omega_R)$. We say that $M$ is \emph{reflexive} if the natural map\footnote{As is it fairly easy to show that the dual of a finitely generated $R$-module is reflexive (cf \cite[\href{http://stacks.math.columbia.edu/tag/0AV2}{Tag 0AV2}]{stacks-project}) this definition is equivalent to simply requiring that there is \emph{some} isomorphism $M\xrightarrow{\sim}M^{**}$. In particular, if $M\cong M^*$ then $M$ is automatically reflexive.} $M\to M^{**}$ is an isomorphism. We will let $\Cl(R)$ denote the \emph{Weil divisor class group} of $R$, which is isomorphic (cf \cite[\href{http://stacks.math.columbia.edu/tag/0EBM}{Tag 0EBM}]{stacks-project}) to the group of generic rank $1$ reflexive modules over $R$. For any generic rank $1$ reflexive sheaf $M$, let $[M]\in \Cl(R)$ denote the corresponding element of the class group. The group operation is then defined by $[M]+[N] = [(M{\otimes_R}N)^{**}]$. Note that $[\omega_R]\in\Cl(R)$ and we have $[M^*] = [\omega_R]-[M]$ for any $[M]\in\Cl(R)$. Lastly, given any reflexive module $M$, the natural perfect pairing $M^*\times M\to\omega_R$ gives rise to a natural map $\tau_M:M^*\otimes_RM\to \omega_R$ (defined by $\tau_M(\varphi\otimes x) = \varphi(x)$) called the \emph{trace map}. Also we will let \[k = \#\bigg\{v|\mathfrak{D} \bigg|\overline{\rho} \text{ is unramified at } v \text{ and, } \overline{\rho}(\Frob_v) \text{ is a scalar} \bigg\}.\] as in Theorem \ref{Mult 2^k}. Our main result of this section is the following: \begin{thm}\label{self-dual} If $M_\infty$ is a finitely-generated module over $R_\infty$ satisfying: \begin{enumerate} \item $M_\infty$ is maximal Cohen--Macaulay over $R_\infty$; \item we have $M_\infty^*\cong M_\infty$ (and hence $M_\infty$ is reflexive); \item $\genrank_{R_\infty}M_\infty = 1$; \end{enumerate} then $\dim_\mathbb{F} M_\infty/m_{R_\infty} = 2^k$. Moreover, the trace map $\tau_{M_\infty}:M_\infty\otimes_{R_\infty}M_\infty\to \omega_{R_{\infty}}$ is surjective. \end{thm} Thus, to prove Theorem \ref{Mult 2^k}, it will suffice to construct a module $M_\infty$ over $R_\infty$ satisfying the conditions of Theorem \ref{self-dual} with $\dim_\mathbb{F} M_\infty/m_{R_\infty} = \nu_{\overline{\rho}}(K^{\min})$. The last statement, that $\tau_{M_\infty}$ is surjective, will be used to prove Theorem \ref{R=T} (see the end of Section \ref{sec:patching}). Our primary strategy for proving Theorem \ref{self-dual} is to note that conditions (1) and (3) imply that $M_\infty$ is the module corresponding to a Weil divisor on $R_\infty$, and condition (2) implies that we have $2[M_\infty] = [\omega_{R_\infty}]$ in $\Cl(R_\infty)$. Provided that $\Cl(R_\infty)$ is $2$-torsion free, this means that conditions (1), (2) and (3) uniquely characterize the module $M_\infty$. Proving the theorem would thus simply be a matter of computing the unique module $M_\infty$ satisfying the conditions of the theorem explicitly enough. Unfortunately, while we can give a precise description of the ring $R_\infty$ in our situation, it is difficult to directly compute $\Cl(R_\infty)$ from that description. Instead, we will first reduce the statement of Theorem \ref{self-dual} to a similar statement over the ring $\overline{R}_\infty=R_\infty/\lambda$, and then to a statement over a finitely generated graded $\mathbb{F}$-algebra $\mathcal{R}$ with $\widehat{\mathcal{R}}\cong \overline{R}_\infty$ (see Theorems \ref{self-dual mod l} and \ref{self-dual decompleted} below). We will then be able to directly compute $\Cl(\mathcal{R})$, and the unique module $\Mc$ with $2[\Mc] = [\omega_{\mathcal{R}}]$ in $\Cl(\mathcal{R})$, by using the theory of toric varieties. In Section \ref{ssec:R_infty} we summarize the computations in \cite{Shotton} to explicitly describe the rings $R_\infty$ and $\overline{R}_\infty$, and reduce Theorem \ref{self-dual} to the corresponding statement over $\overline{R}_\infty$ (Theorem \ref{self-dual mod l}). In Section \ref{ssec:toric varieties} we introduce the ring $\mathcal{R}$, and show that it is the coordinate ring of an affine toric variety. Finally in Section \ref{ssec:class group} we use the theory of toric varieties to compute $\Cl(\mathcal{R})$, which allows us to prove a ``de-completed'' mod $\lambda$ version of Theorem \ref{self-dual}. In Section \ref{ssec:decompletion} we adapt the method of Danilov \cite{Danilov} for computing the class groups of completions of graded rings to show that $\Cl(\mathcal{R})\cong \Cl(\overline{R}_\infty)$, from which we deduce Theorem \ref{self-dual mod l} and hence Theorem \ref{self-dual}. \subsection{Explicit Calculations of Local Deformation Rings}\label{ssec:R_infty} In order to prove Theorem \ref{self-dual}, it will be necessary to first compute the ring $R_\infty$, or equivalently to compute $R^{\square,\st,\psi}(\overline{\rho}|_{G_v})$ for all $v|\mathfrak{D}$. These computations were essentially done by Shotton \cite{Shotton}, except that he considers the non fixed determinant version, $R^{\st,\square}(\overline{\rho}|_{G_v})$ instead of $R^{\st,\square,\psi}(\overline{\rho}|_{G_v})$. Fortunately, it is fairly straightforward to recover $R^{\st,\square,\psi}(\overline{\rho}|_{G_v})$ from $R^{\st,\square}(\overline{\rho}|_{G_v})$. Specifically, we get: \begin{thm}\label{R^st} Take any place $v|\mathfrak{D}$. Recall that we have assumed that $\Nm(v)\not\equiv -1\pmod{\ell}$. If the residual representation $\overline{\rho}|_{G_v}:G_v\to \GL_2(\mathbb{F})$ is not scalar, then $R^{\square,\st,\psi}(\overline{\rho}|_{G_v})\cong\O[[X_1,X_2,X_3]]$. If $\overline{\rho}|_{G_v}:G_v\to \GL_2(\mathbb{F})$ is scalar then \[R^{\square,\st,\psi}(\overline{\rho}|_{G_v})\cong S_v= \O[[A,B,C,X,Y,Z]]/\mathcal{I}_v\] where $\mathcal{I}_v$ is the ideal generated by the $2\times 2$ minors of the matrix \[\begin{pmatrix} A& B&X&Y\\ C&A&Z&X+2\frac{\Nm(v)-1}{\Nm(v)+1} \end{pmatrix}.\] The ring $S_v$ is a Cohen--Macaulay and non-Gorenstein domain of relative dimension $3$ over $\O$. $(\lambda,C,Y,B-Z)$ is a regular sequence for $S_v$. Moreover, $S_v[1/\lambda]$ is formally smooth of dimension $3$ over $E$. \end{thm} \begin{proof} For convenience, let $R_{\st} = R^{\st,\square}(\overline{\rho}|_{G_v})$ and $R_{\st}^{\psi} = R^{\st,\square,\psi}(\overline{\rho}|_{G_v})$. By definition, $R_{\st}^{\psi}$ is the maximal reduced $\lambda$-torsion free quotient of $R_{\st}$ on which $\det\rho^{\square}(g) = \psi(g)$ for all $g\in G_v$. Now let $I_v/\tilde{P}_v\cong \mathbb{Z}_\ell$ be the maximal pro-$\ell$ quotient of $I_v$, so that $\tilde{P}_v\unlhd G_v$ and $T_v=G_v/\tilde{P}_v\cong \mathbb{Z}_\ell\rtimes\widehat{\mathbb{Z}}$. Now let $\sigma,\phi\in T_v$ be topological generators for $\mathbb{Z}_\ell$ and $\widehat{\mathbb{Z}}$, respectively (chosen so that $\phi$ is a lift of arithmetic Frobenius, so that $\phi\sigma\phi^{-1} = \sigma^{\Nm(v)}$). Now as in \cite{Shotton}, we may assume that the universal representation $\rho^\square:G_v\to\GL_2(R_{\st})$ factors through $T_v$. As we already have $\det\rho^\square(\sigma) = 1 = \psi(\sigma)$, it follows that $R_{\st}^\psi$ is the maximal reduced $\lambda$-torsion free quotient of $R$ on which $\det\rho^\square(\phi) = \psi(\phi)$. As explained in Section \ref{sec:deformation}, up to isomorphism the ring $R^\psi_{\st}$ is unaffected by the choice of $\psi$, so it will suffices to prove the claim for a particular choice of $\psi$. Thus from now on we will assume that $\psi$ is unramified and $\displaystyle\psi(\phi) = \frac{\Nm(v)}{(\Nm(v)+1)^2}t^2$ where \[ t = \begin{cases} \Nm(v)+1 & \Nm(v)\not\equiv \pm 1\pmod{\ell}\\ 2 & \Nm(v)\equiv 1\pmod \ell \end{cases} \] so that $t\equiv \Nm(v)+1\equiv \tr\overline{\rho}(\phi)\pmod\ell$ (this particular choice of $t$ is made to agree with the computations of \cite{Shotton}). But now by the definition of $R_{\st} = R^{\st,\square}(\overline{\rho}|_{G_v})$ we have that $\Nm(v)\left(\tr\rho^\square(\phi)\right)^2 = (\Nm(v)+1)^2\det \rho^\square(\phi)$ and so \[\det\rho^\square(\phi) = \frac{\Nm(v)}{(\Nm(v)+1)^2}\left(\tr \rho^\square(\phi)\right)^2\] (where we have used the fact that $\Nm(v)\not\equiv -1\pmod \ell$, and so $\Nm(v)+1$ is a unit in $\O$). It follows that \begin{align*} \det\rho^\square(\phi)-\psi(\phi) &= \frac{\Nm(v)}{(\Nm(v)+1)^2}\left(\tr \rho^\square(\phi)\right)-\frac{\Nm(v)}{(\Nm(v)+1)^2}t^2\\ &= \frac{\Nm(v)}{(\Nm(v)+1)^2}(\tr \rho^\square(\phi)+t)(\tr \rho^\square(\phi)-t). \end{align*} But now \[\frac{\Nm(v)}{(\Nm(v)+1)^2}\left(\tr \rho^\square(\phi)+t\right) \equiv \frac{2\Nm(v)}{(\Nm(v)+1)^2}\tr\overline{\rho}(\phi) \equiv \frac{2\Nm(v)}{\Nm(v)+1}\pmod{m_R}\] and so as $\ell\nmid 2,\Nm(v),\Nm(v)+1$ we get that $\displaystyle \frac{\Nm(v)}{(\Nm(v)+1)^2}\left(\tr \rho^\square(\phi)+t\right)$ is a unit in $R_{\st}$. It follows that $R_{\st}^\psi$ is the maximal reduced $\lambda$-torsion free quotient of \[R_{\st}^{\psi,\circ} = \frac{R_{\st}}{\left(\det\rho^\square(\phi)-\psi(\phi) \right)} = \frac{R_{\st}}{\tr \rho^\square(\phi)-t}\] It now follows immediately from Shotton's computations that in each case $R_{\st}^{\psi,\circ}$ is already reduced and $\lambda$-torsion free (and so $R_{\st}^\psi = R_{\st}^{\psi,\circ}$) and has the form described in the statement of Theorem \ref{R^st} above. Indeed, first assume that $\Nm(v)\not\equiv \pm 1\pmod{\ell}$. By \cite[Proposition 5.5]{Shotton} we may write $R_{\st} = \O[[B,P,X,Y]]$ with \begin{align*} \rho^\square(\sigma) &= \begin{pmatrix}1&X\\y&1\end{pmatrix}^{-1} \begin{pmatrix}1&x+B\\0&1\end{pmatrix} \begin{pmatrix}1&X\\y&1\end{pmatrix}\\ \rho^\square(\phi) &= \begin{pmatrix}1&X\\y&1\end{pmatrix}^{-1} \begin{pmatrix}\Nm(v)(1+P)&0\\0&1+P\end{pmatrix} \begin{pmatrix}1&X\\y&1\end{pmatrix}, \end{align*} for some $x\in \O$. Thus we have \[\tr \rho^\square(\phi) = (\Nm(v)+1)(1+P) = t+tP\] and so (as $t=\Nm(v)+1\in\O$ is a unit), $R_{\st}^{\psi,\circ} = R_{\st}/(tP) \cong \O[[B,P,X,Y]]/(P)\cong \O[[B,X,Y]]$, as desired. Now assume that $\Nm(v)\equiv 1\pmod{\ell}$. Again, following the computations of \cite[Proposition 5.8]{Shotton} we can write \begin{align*} \rho^\square(\sigma) &= \begin{pmatrix} 1+A&x+B\\mathbb{C}&1-A \end{pmatrix}\\ \rho^\square(\phi) &= \begin{pmatrix} 1+P&y+R\\S&1+Q \end{pmatrix} \end{align*} (for $x,y\in \O$) where $A,B,C,P,Q,R$ and $S$ topologically generate $R_{\st}$. Now following Shotton's notation, let $T = P+Q$, so that $\tr \rho^\square(\phi) = 2+T= t+T$ and thus $R_{\st}^{\psi,\circ} = R_{\st}/(T)$. In both cases ($\overline{\rho}|_{G_v}$ non-scalar and scalar) Shotton's computations immediately give the desired description of $R_{\st}^\psi$.\footnote{In Shotton's notation, when $\overline{\rho}|_{G_v}$ is scalar $R_{\st}^\psi$ would be cut out by the $2\times 2$ minors of the matrix $\begin{pmatrix}X_1&X_2&X_3&X_4\\Y_1&-X_1&Y_3&X_3+2\frac{\Nm(v)-1}{\Nm(v)+1}\end{pmatrix}$. This is equivalent to the form stated in Theorem \ref{R^st} via the variable substitutions $A = X_1$, $B = -X_2$, $C = Y_1$, $X = X_3$, $Y = X_4$ and $Z = Y_3$.} Moreover, Shotton shows that $S_v$ is indeed Cohen--Macaulay and non Gorenstein of relative dimension $3$ over $\O$, and that $S_v[1/\lambda]$ is formally smooth of dimension $3$ over $E$ . As $S_v$ is Cohen--Macaulay, the claim that $(\lambda,C,Y,B-Z)$ is a regular sequence simply follows by noting that \[S_v/(\lambda,C,Y,B-Z)\cong \mathbb{F}[[A,B,X]]/(A^2,AB,AX,B^2,BX,X^2) = \mathbb{F}\oplus\mathbb{F} A\oplus \mathbb{F} B\oplus \mathbb{F} X\] is a zero dimensional ring, and so $(\lambda,C,Y,B-Z)$ is a system of parameters. \end{proof} Thus letting $\mathfrak{D}_1|\mathfrak{D}$ be the product of the places $v|\mathfrak{D}$ at which $\overline{\rho}|_{G_v}:G_v\to \GL_2(\mathbb{F})$ is scalar, we have \[R_\infty= \left[\hat{\bigotimes_{v|\mathfrak{D}_1}}S_v\right][[x_1,\ldots,x_{s}]] \] for some integer $s$. As the rings $S_v$ are all Cohen--Macaulay by Theorem \ref{R^st}, it follows that $R_\infty$ is as well. Now note that the description of $S_v$ in Theorem \ref{R^st} becomes much simpler if we work in characteristic $\ell$. Indeed, if $\overline{\rho}|_{G_v}$ is scalar then $\Nm(v)\equiv 1\pmod{\ell}$ and so, $\overline{S}=S_v/\lambda$ is an explicit graded ring not depending on $v$. Specifically, we have $\overline{S} = \mathbb{F}[[A,B,C,X,Y,Z]]/\overline{\I}$ where $\overline{\I}$ is the (homogeneous) ideal generated by the $2\times 2$ minors of the matrix $\begin{pmatrix} A&B&X&Y\\ C&A&Z&X \end{pmatrix}$. It thus follows that \[\overline{R}_\infty = R_\infty/\lambda \cong \overline{S}^{\hat{\otimes} k}[[x_1,\ldots,x_{s}]],\] which will be much easier to work with than $R_\infty$. In particular, note that $\overline{R}_\infty$ is still Cohen--Macaulay, as $R_\infty$ is $\lambda$-torsion free by definition. It will thus be useful to reduce Theorem \ref{self-dual} the following ``mod $\lambda$'' version: \begin{thm}\label{self-dual mod l} If $\overline{M}_\infty$ is a finitely-generated module over $\overline{R}_\infty$ satisfying: \begin{enumerate} \item $\overline{M}_\infty$ is maximal Cohen--Macaulay over $\overline{R}_\infty$ \item We have $\overline{M}_\infty^*\cong \overline{M}_\infty$. \item $\genrank_{\overline{R}_\infty}\overline{M}_\infty=1$. \end{enumerate} then $\dim_\mathbb{F} \overline{M}_\infty/m_{\overline{R}_\infty} = 2^k$. Moreover, the trace map $\tau_{\overline{M}_\infty}:\overline{M}_\infty\otimes_{\overline{R}_\infty}\overline{M}_\infty\xrightarrow{\sim} \omega_{\overline{R}_\infty}$ is surjective. \end{thm} \begin{proof}[Proof that Theorem \ref{self-dual mod l} implies \ref{self-dual}] Assume that Theorem \ref{self-dual mod l} holds, and that $M_\infty$ satisfies the hypotheses of Theorem \ref{self-dual}. As $R_\infty$ is flat over $\O$, it is $\lambda$-torsion free and thus $\lambda$ is not a zero divisor on $M_\infty$ (by condition (1)). It follows that $\overline{M}_\infty= M_\infty/\lambda$ is maximal Cohen--Macaulay over $\overline{R}_\infty$ and $\genrank_{\overline{R}_\infty}\overline{M}_\infty = \genrank_{R_\infty}M_\infty = 1$. (In general, if $R$ is Cohen--Macaulay and $M$ is maximal Cohen--Macaulay over $R$, then for any regular element $x\in R$, $\genrank_{R/x}(M/x) = \genrank_RM$, provided $R/x$ is also a domain.) Moreover, as $M_\infty$ is maximal Cohen--Macaulay over $R_\infty$, \cite[Proposition 21.12b]{Eisenbud} gives \[\overline{M}_\infty = M_\infty/\lambda = \Hom_{R_\infty}(M_\infty,\omega_{R_\infty})/\lambda = \Hom_{R_\infty/\lambda}(M_\infty/\lambda,\omega_{R_\infty}/\lambda) = \Hom_{\overline{R}_\infty}(\overline{M}_\infty,\omega_{\overline{R}_\infty}),\] where we have used the fact that $\omega_{R_\infty}/\lambda\cong \omega_{\overline{R}_\infty}$, by \cite[Chapter 21.3]{Eisenbud}. Thus $\overline{M}_\infty$ is self-dual. Thus $\overline{M}_\infty$ satisfies all of the hypotheses of Theorem \ref{self-dual mod l}, and so $\dim_\mathbb{F} \overline{M}_\infty/m_{\overline{R}_\infty} = 2^k$ and $\tau_{\overline{M}_\infty}$ is surjective. Now we obviously have that $\overline{M}_\infty/m_{\overline{R}_\infty}\cong M_\infty/m_{R_\infty}$, so the first conclusion of Theorem \ref{self-dual} follows. Also, the trace map $\tau_{\overline{M}_\infty}:\overline{M}_\infty\otimes_{\overline{R}_\infty}\overline{M}_\infty\to \omega_{\overline{R}_\infty}$ is just the mod-$\lambda$ reduction of the map $\tau_{M_\infty}:M_\infty\otimes M_\infty\to \omega_{R_\infty}$, so it follows that $\tau_{\overline{M}_\infty}$ is surjective if and only if $\tau_{M_\infty}$ is. Thus the second conclusion of Theorem \ref{self-dual} follows. \end{proof} As hinted above, we will prove Theorem \ref{self-dual mod l} by computing the class group of $\overline{R}_\infty$. We finish this section by proving the following lemma, which will make the second conclusion of Theorem \ref{self-dual mod l} easier to prove (and will also be useful in the proof of Theorem \ref{R=T}): \begin{lemma}\label{trace} If $R$ is a Cohen--Macaulay ring with a dualizing sheaf $\omega_R$, and $M$ is a reflexive $R$-module, then the trace map $\tau_M:M^*\otimes_RM\to \omega_R$ is surjective if and only if there \emph{exists} an $R$-module surjection $M^*\otimes_RM\twoheadrightarrow \omega_R$. \end{lemma} \begin{proof} Assume that $f:M^*\otimes_RM\to \omega_R$ is a surjection. Take any $\alpha\in\omega_R$. Then we can write \[\alpha = f\left(\sum_{i\in I}b_i\otimes c_i\right) = \sum_{i\in I}f(b_i\otimes c_i)\] for some finite index set $I$ and some $b_i\in M^*$ and $c_i\in M$. For each $i\in I$, consider the $R$-linear map $\varphi_i:M\to \omega_R$ defined by $\varphi_i(c) = f(b_i\otimes c)$. Then we have $\varphi_i\in M^*$ for all $i$ and so \begin{align*} \alpha &= \sum_{i\in I}f(b_i\otimes c_i) = \sum_{i\in I}\varphi_i(c_i) = \sum_{i\in I}\tau_M(\varphi_i\otimes c_i) = \tau_M\left(\sum_{i\in I}\varphi_i\otimes c_i \right). \end{align*} Thus $\tau_M$ is surjective. \end{proof} \subsection{Toric Varieties}\label{ssec:toric varieties} For the remainder of this section we will consider the rings $\mathcal{S} = \mathbb{F}[A,B,C,X,Y,Z]/\overline{\I}$, where again $\overline{\I}$ is the ideal generated by the $2\times 2$ minors of the matrix $\begin{pmatrix} A&B&X&Y\\ C&A&Z&X \end{pmatrix}$, and $\mathcal{R} = \mathcal{S}^{\otimes k}[x_1,\ldots,x_s]$. Note that $\mathcal{S}$ and $\mathcal{R}$ are naturally finitely generated graded $\mathbb{F}$-algebras. Let $m_{\mathcal{S}}$ and $m_{\mathcal{R}}$ denote their irrelevant ideals, and note $\overline{S}$ and $\overline{R}_\infty$ are the completions of $\mathcal{S}$ and $\mathcal{R}$ at these ideals. The goal of this subsection and the next one is to prove the following ``de-completed'' version of Theorem \ref{self-dual mod l}. In Section \ref{ssec:decompletion} we will show that this implies Theorem \ref{self-dual mod l}, and hence Theorem \ref{self-dual}. \begin{thm}\label{self-dual decompleted} If $\Mc$ is a finitely-generated module over $\mathcal{R}$ satisfying: \begin{enumerate} \item $\Mc$ is maximal Cohen--Macaulay over $\mathcal{R}$ \item We have $\Mc^*\cong \Mc$. \item $\genrank_{\mathcal{R}}\Mc=1$. \end{enumerate} then $\dim_\mathbb{F} \Mc/m_{\mathcal{R}} = 2^k$. Moreover, the trace map $\tau_{\Mc}:\Mc\otimes_{\mathcal{R}}\Mc\xrightarrow{\sim} \omega_{\mathcal{R}}$ is surjective. \end{thm} As outlined above, we will prove this theorem by computing $\Cl(\mathcal{R})$. The key insight that allows us to preform this computation is that $\mathcal{S}$, and hence $\mathcal{R}$, is the coordinate ring of an affine \emph{toric} variety. In this section, we review the basic theory of toric varieties and show that $\mathcal{S}$ and $\mathcal{R}$ indeed correspond to toric varieties. We shall primarily follow the presentation of toric varieties from \cite{CLS}. Unfortunately \cite{CLS} works exclusively with toric varieties over $\mathbb{C}$, whereas we are working in positive characteristic. All of the results we will rely on work over arbitrary base field, usually with identical proofs, so we will freely cite the results of \cite{CLS} as if they were stated over arbitrary fields. We refer the reader to \cite{MS} and \cite{DanilovToric} for a discussion of toric varieties over arbitrary fields. We recall the following definitions. For any integer $d\ge 1$, let $T_d = \mathbb{G}_m^d = (\mathbb{F}^\times)^d$, thought of as a group variety. Define the two lattices \begin{align*} M &= \Hom(T_d,\mathbb{G}_m) & N &= \Hom(\mathbb{G}_m,T_d), \end{align*} called the \emph{character lattice} and the \emph{lattice of one-parameter subgroups}, respectively. Note that $M\cong N\cong \mathbb{Z}^d$. We shall write $M$ and $N$ additively. For $m\in M$ and $u\in N$, we will write $\chi^m:T_d\to \mathbb{G}_m$ and $\lambda^u:\mathbb{G}_m\to T_d$ to denote the corresponding morphisms. First note that there is a perfect pairing $\langle\ ,\ \rangle:M\times N\to \mathbb{Z}$ given by $t^{\langle m,u\rangle} = \chi^m(\lambda^u(t))$. We shall write $M_\mathbb{R}$ and $N_\mathbb{R}$ for $M\otimes_\mathbb{Z}\mathbb{R}$ and $N\otimes_\mathbb{Z}\mathbb{R}$, which are each $d$-dimensional real vector spaces. We will extend the pairing $\langle\ ,\ \rangle$ to a perfect pairing $\langle\ ,\ \rangle:M_\mathbb{R}\times N_\mathbb{R}\to \mathbb{R}$. For the rest of this section, we will (arbitrarily) fix a choice of basis $e_1,\ldots,e_d$ for $M$, and so identify $M$ with $\mathbb{Z}^d$. We will also identify $N$ with $\mathbb{Z}^d$ via the dual basis to $e_1,\ldots,e_d$. Under these identifications, $\langle\ ,\ \rangle$ is simply the usual (Euclidean) inner product on $\mathbb{Z}^d$. We can now define: \begin{defn}\label{def:toric} An \emph{(affine) toric variety} of dimension $d$ is a pair $(X,\iota)$, where $X$ is an affine variety $X/\mathbb{F}$ of dimension $d$ and $\iota$ is an open embedding $\iota:T_d\hookrightarrow X$ such that the natural action of $T_d$ on itself extends to a group variety action of $T_d$ on $X$. We will usually write $X$ instead of the pair $(X,\iota)$. For such an $X$, we define the \emph{semigroup} of $X$ to be \[\mathsf{S}_X=\{m\in M|\chi^m:T_d\to \mathbb{G}_m \text{ extends to a morphism }X\to \mathbb{A}^1\}\subseteq M\] For convenience, we will also say that a finitely generated $\mathbb{F}$-algebra $R$ (together with an inclusion $R\hookrightarrow \mathbb{F}[M]$) is \emph{toric} if $\Spec R$ is toric. \end{defn} The primary significance of affine toric varieties is that they are classified by their semigroups. Specifically: \begin{prop}\label{semigroup} If $X$ is an affine toric variety of dimension $d$, then $X = \Spec \mathbb{F}[\mathsf{S}_X]$, and the embedding $\iota:T_d\hookrightarrow X$ is induced by $\mathbb{F}[\mathsf{S}_X]\hookrightarrow \mathbb{F}[M]$ (using the fact that $T_d = \Spec\mathbb{F}[M]$). Moreover we have \begin{enumerate} \item The semigroup $\mathsf{S}_X$ spans $M$ (that is, $\mathbb{Z}\mathsf{S}_X$ has rank $d$). \item If $\mathsf{S}_X$ is \emph{saturated} in $M$ (in the sense that $km\in\mathsf{S}_X$ implies that $m\in \mathsf{S}_X$ for all $k> 0$ and $m\in M$) then $X$ is a \emph{normal} variety. \end{enumerate} Conversely, if $\mathsf{S}\subseteq M$ is a finitely generated semigroup spanning $M$ then the inclusion $\mathbb{F}[\mathsf{S}]\hookrightarrow \mathbb{F}[M]$ gives $\Spec \mathbb{F}[\mathsf{S}]$ the structure of a $d$-dimensional affine toric variety. \end{prop} \begin{proof} cf. \cite{CLS} Proposition 1.1.14 and Theorems 1.1.17 and 1.3.5. \end{proof} If $R$ is a toric $\mathbb{F}$-algebra, we will write $\mathsf{S}_R$ to mean $\mathsf{S}_{\Spec R}$. While it can be difficult to recognize toric varieties directly from Definition \ref{def:toric}, the following Proposition makes it fairly easy to identify toric varieties in $\mathbb{A}^s$. \begin{prop}\label{toric ideal} Fix an integer $h\ge 1$ and let $\Phi:\mathbb{Z}^h\to M$ be any homomorphism with finite cokernel, and let $L = \ker \Phi$. Let $\mathsf{S}\subseteq M$ be the semigroup generated by $\Phi(e_1),\ldots,\Phi(e_h)\in M$. Then we have an isomorphism $\mathbb{F}[z_1,\ldots,z_h]/I_L\cong\mathbb{F}[\mathsf{S}]$ given by $z_i\mapsto \Phi(e_i)$, where \[I_L = \left(z^{\alpha}-z^{\beta}\middle|\alpha,\beta\in \mathbb{Z}_{\ge 0}^h \text{ such that } \alpha-\beta\in L\right)\subseteq \mathbb{F}[z_1,\ldots,z_h].\] (Where, for any $\alpha = (\alpha_1,\ldots,\alpha_h)\in \mathbb{Z}_{\ge 0}^h$, we write $z^\alpha = z_1^{\alpha_1}\cdots z_h^{\alpha_h}\in \mathbb{F}[z_1,\ldots,z_h]$.) Moreover $I_L$ can be explicitly computed as follows: Assume that $\L = (\ell^1,...,\ell^r)$ is a $\mathbb{Z}$-basis for $L$, with $\ell^i = (\ell^i_1,\ldots,\ell^i_h)\in \mathbb{Z}^h$. Write each $\ell^i$ as $\ell^i = \ell^i_+-\ell^i_i$ where \begin{align*} \ell_+^i &= (\max\{\ell^i_1,0\},\ldots,\max\{\ell^i_h,0\})\in \mathbb{Z}_{\ge0}^h\\ \ell_-^i &= (\max\{-\ell^i_1,0\},\ldots,\max\{-\ell^i_h,0\})\in \mathbb{Z}_{\ge0}^h. \end{align*} Then if $I_{\L} = \left(z^{\ell^1_+}-z^{\ell^1_-},\ldots,z^{\ell^h_+}-z^{\ell^h_-}\right)\subseteq \mathbb{F}[z_1,\ldots,z_h]$, $I_L$ is the \emph{saturation} of $I_{\L}$ with respect to $z_1\cdots z_h$, that is: \begin{align*} I_L &= \left(I_{\L}:(z_1\cdots z_h)^\infty \right) = \bigg\{f\in \mathbb{F}[z_1,\ldots,z_h]\bigg|(z_1\cdots z_h)^mf\in I_{\L}\text{ for some } m\ge 0\bigg\}\\ &= \bigg(z^{\alpha}-z^{\beta}\bigg|\alpha,\beta\in \mathbb{Z}^h_{\ge0}\text{ such that }(z_1\cdots z_h)^m\left(z^{\alpha}-z^{\beta}\right)\in I_{\L}\text{ for some } m\ge 0\bigg). \end{align*} Conversely, if $I\subseteq \mathbb{F}[z_1,\ldots,z_h]$ is any \emph{prime} ideal which can be written in the form $I = \left(z^{\alpha_i}-z^{\beta_i}\middle| i\in \mathscr{A} \right)$ for a finite index set $\mathscr{A}$ and $\alpha_i,\beta_i\in\mathbb{Z}_{\ge 0}^h$, then $I = I_L$ for some $L$ and so $\mathbb{F}[z_1,\ldots,z_h]/I$ can be given the structure of a toric $\mathbb{F}$-algebra. \end{prop} \begin{proof} This mostly follows from \cite[Propositions 1.1.8, 1.1.9 and 1.1.11]{CLS}. The statement that $I_L = \left(I_{\L}:(z_1\cdots z_h)^\infty \right)$ is \cite[Exercise 1.1.3]{CLS} or \cite[Lemma 7.6]{MS} \end{proof} Now applying the above results to the $\mathbb{F}$-algebra $\mathcal{S}$, we get: \begin{prop}\label{S toric} $\mathcal{S}$ may be given the structure of a $3$-dimensional toric $\mathbb{F}$-algebra, with semigroup \[\mathsf{S}_\mathcal{S}=\{(a,b,c)\in \mathbb{Z}^3|a,b,c\ge 0, 2a+2b\ge c\}\subseteq M\] under some choice of basis $e_1,e_2,e_3$ for $M$. Moreover: \begin{enumerate} \item $\Spec\mathcal{S}$ is the affine cone over a surface $\mathcal{V}\subseteq\P^5$ isomorphic to $\P^1\times \P^1$. The embedding $\P^1\times\P^1\xrightarrow{\sim}\mathcal{V}\subseteq \P^5$ corresponds to the very ample line bundle $\O_{\P^1}(2)\boxtimes\O_{\P^1}(1)$ on $\P^1\times\P^1$. \item $\mathcal{S}$ is isomorphic to the ring $\mathbb{F}[x,xz,xz^2,y,yz,yz^2]\subseteq \mathbb{F}[x,y,z]$. \item $\Spec \mathcal{S}$ is a \emph{normal} variety. \item $\mathcal{R}$ is toric of dimension $3k+s$. \end{enumerate} \end{prop} \begin{proof} Take $d=3$ in the above discussion, and fix an isomorphism $M\cong \mathbb{Z}^3$. Write $\mathsf{S} = \{(a,b,c)\in \mathbb{Z}^3|a,b,c\ge 0, 2a+2b\ge c\}$. We will first show that $\mathcal{S}\cong \mathbb{F}[\mathsf{S}]$. Note that $\mathsf{S}$ is generated by the (transposes of) the columns of the matrix \[\Phi = \begin{pmatrix} 1&1&1&0&0&0\\ 0&0&0&1&1&1\\ 1&2&0&1&2&0 \end{pmatrix},\] which in particular gives an isomorphism $\mathbb{F}[\mathsf{S}]\cong \mathbb{F}[x,xz,xz^2,y,yz,yz^2]\subseteq \mathbb{F}[x,y,z]$. Let $L = \ker \Phi$. By Proposition \ref{toric ideal} it follows that $\mathbb{F}[\mathsf{S}] \cong \mathbb{F}[A,B,C,X,Y,Z]/I_L$ (where we have identified the ring $\mathbb{F}[z_1,z_2,z_3,z_4,z_5,z_6]$ with $\mathbb{F}[A,B,C,X,Y,Z]$ in the obvious way, in order to keep are notation consistent). But now note that $L$ is a rank $3$ lattice with basis $\L = (\ell^1,\ell^2,\ell^3)$ given by the vectors: \[\ell^1= \begin{pmatrix} 1\\0\\-1\\-1\\0\\1 \end{pmatrix}, \ell^2= \begin{pmatrix} 1\\0\\-1\\1\\-1\\0 \end{pmatrix}, \ell^3= \begin{pmatrix} 1\\-1\\0\\1\\0\\-1 \end{pmatrix}.\] It follows that \[I_{\L} = (AZ-CX,AX-CY,AX-BZ)\] whence it is straightforward to compute that \begin{align*} I_L &= \left(I_{\L}:(ABCXYZ)^\infty \right)\\ &=\left(A^2-BC,AZ-CX,AX-CY,AX-BZ,AY-BX,X^2-YZ \right) = \overline{\I}. \end{align*} Thus $\mathcal{S} = \mathbb{F}[A,B,C,X,Y,Z]/\overline{\I}$ is indeed toric with $\mathsf{S}_{\mathcal{S}} = \mathsf{S}$, and we have \[\mathcal{S}\cong \mathbb{F}[\mathsf{S}_\mathcal{S}]\cong \mathbb{F}[x,xz,xz^2,y,yz,yz^2].\] Moreover, this isomorphism sends the ideal $(A,B,C,X,Y,Z)\subseteq \mathcal{S}$ to the ideal\\ $(x,xz,xz^2,y,yz,yz^2)$, from which (2) easily follows. Now the semigroup $\mathsf{S}_\mathcal{S} = \{(a,b,c)\in \mathbb{Z}^3|a,b,c\ge 0,2a+2b\ge c\ge 0\}$ is clearly saturated in $M$, and so $\Spec \mathcal{S}$ is indeed normal, proving (3). Now note that $\O_{\P^1}(2)\boxtimes\O_{\P^1}(1)$ is indeed a very ample line bundle on $\P^1\times\P^1$ and corresponds to the (injective) morphism $f:\P^1\times \P^1 \to \P^5$ defined by \[f([s:t],[u:v]) = [s^2u:stu:t^2u:s^2v:stv:t^2v].\] It thus follows that the coordinate ring on the cone over the image of $f$ is isomorphic to \[\mathbb{F}[s^2u,stu,t^2u,s^2v,stv,t^2v]\cong\mathbb{F}[x,xz,xz^2,y,yz,yz^2],\] proving (1). Lastly, recalling that $M = \mathbb{Z}^3$, let $M' = M = M^{\oplus k}\oplus \mathbb{Z}^s = \mathbb{Z}^{3k+s}$ and define: \[\mathsf{S}' = (\mathsf{S})^{\otimes k}\otimes \mathbb{Z}_{\ge 0}^s\] so that \[\mathbb{F}[\mathsf{S}'] = \mathbb{F}[\mathsf{S}]^{\otimes k}\otimes \mathbb{F}[x]^{\otimes s} \cong \mathcal{S}^{\otimes k}[x_1,\ldots,x_s] = \mathcal{R}\] so that $\mathcal{R}$ is indeed toric of dimension $3k+s$, proving (4). \end{proof} We will now restrict our attention to \emph{normal} affine toric varieties. The advantage to doing this is that Proposition \ref{semigroup} has a refinement (see Proposition \ref{toric cone} below) that allows us to characterize normal toric varieties much more simply, using cones instead of semigroups. We now make the following definitions: \begin{defn}\label{def:cone} A \emph{convex rational polyhedral cone} in $N_\mathbb{R}$ is a set of the form: \[\sigma = \Cone(S) = \left\{\sum_{\lambda\in S}x_\lambda\lambda\middle|x_\lambda\ge 0 \text{ for all }\lambda\in S \right\}\subseteq N_\mathbb{R}\] for some finite subset $S\subseteq N$. A \emph{face} of $\sigma$ is a subset $\tau\subseteq \sigma$ which can be written as $\tau = \sigma\cap H$ for some hyperplane $H\subseteq N_\mathbb{R}$ which does not intersect the interior of $\sigma$. We write $\tau\preceq\sigma$ to say that $\tau$ is a face of $\sigma$. It is clear that any face of $\sigma$ is also a convex rational polyhedral cone. We say that $\sigma$ is \emph{strongly convex} if $\{0\}$ is a face of $\sigma$. We write $\mathbb{R}\sigma$ for the subspace of $N_\mathbb{R}$ spanned by $\sigma$, and we will let the dimension of $\sigma$ be $\dim \sigma = \dim_\mathbb{R} \mathbb{R}\sigma$. We make analogous definitions for cones in $M_\mathbb{R}$. For a convex rational polyhedral cone $\sigma\subseteq N_\mathbb{R}$ (or similarly for $\sigma\subseteq M_\mathbb{R}$), we define its \emph{dual} cone to be: \[\sigma^\vee=\{m\in M_\mathbb{R}|\langle m,u\rangle\ge 0 \text{ for all } u\in \sigma\}.\] \end{defn} It is easy to see that $\sigma^\vee$ is also convex rational polyhedral cone. If $\sigma$ is strongly convex and $\dim\sigma = d$ then the same is true of $\sigma^\vee$. Moreover, for any $\sigma$ we have $\sigma^{\vee\dual} = \sigma$. We now have the following: \begin{prop}\label{toric cone} If $X$ is a normal affine toric variety of dimension $d$, then there is a (uniquely determined) strongly convex rational polyhedral cone $\sigma_X\subseteq N_\mathbb{R}$ for which $\sigma_X^\vee\cap M = \mathsf{S}_X$ and \begin{align*} \sigma_X\cap N&=\{u\in N|\lambda^u:\mathbb{G}_m\to T_d \text{ extends to a morphism }\mathbb{A}^1\to X\}\\ &=\left\{u\in N\middle|\lim_{t\to 0}\lambda^u(t)\text{ exists in }X\right\}. \end{align*} We call $\sigma_X$ the \emph{cone associated to }$X$. Again, if $R$ is a toric $\mathbb{F}$-algebra, then we write $\sigma_R$ for $\sigma_{\Spec R}$. \end{prop} \begin{proof} This follows from \cite{CLS} Theorem 1.3.5 (for $\sigma_{X}^\vee\cap M$) and Proposition 3.2.2 (for $\sigma_X\cap N$). Note that it is clear from our definitions that a convex rational polyhedral cone $\sigma\subseteq N_\mathbb{R}$ is uniquely determined by $\sigma\cap N$. \end{proof} \begin{rem} Based on the statement of Proposition \ref{toric cone}, it would seem more natural to simply define the cone associated to $\mathcal{X}$ to be $\sigma_\mathcal{X}^\vee$, and not mention the lattice $N$ at all. The primary reason for making this choice in the literature is to simplify the description of non-affine toric varieties, which is not relevant to our applications. Nevertheless we shall use the convention established in Proposition \ref{toric cone} to keep our treatment compatible with existing literature, and specifically to avoid having to reformulate Theorem \ref{class group}, below. \end{rem} Rephrasing the statement of Proposition \ref{S toric} in terms of cones, we get: \begin{cor}\label{S cone} We have $\sigma_{\mathcal{S}} = \Cone(e_1,e_2,e_3,2e_1+2e_2-e_3)$. \end{cor} \begin{proof} The description of $\mathsf{S}_\mathcal{X}$ in Proposition \ref{S toric} immediately implies that \[\sigma_\mathcal{X}^\vee = \{(x,y,z)\in \mathbb{R}^3|x,y,x\ge 0, 2x+2y\ge z\} = \Cone(e_1,e_2,e_1+2e_3,e_2+2e_3).\] Thus we get \begin{align*} \sigma_\mathcal{X} &= \sigma_\mathcal{X}^{\vee\dual} = \{u\in N|\langle m,u\rangle\ge 0 \text{ for all } m\in\sigma_\mathcal{X}^\vee\}\\ &= \{u\in N|\langle m,e_1\rangle,\langle m,e_2\rangle,\langle m,e_1+2e_3\rangle,\langle m,e_2+2e_3\rangle\ge 0\}\\ &= \{(x,y,z)\in \mathbb{R}^3|x,y\ge 0,x+2z\ge 0, y+2z\ge 0\}\\ &=\Cone(e_1,e_2,e_1+2e_3,e_2+2e_3). \end{align*} \end{proof} \subsection{Class Groups of Toric Varieties}\label{ssec:class group} The benefit of this entire discussion is that Weil divisors on toric varieties are much easier work with than they are for general varieties. In order to explain this, we first introduce a few more definitions. For any variety $X$, we will let $\Div(X)$ denote the group of Weil divisors of $X$. If $X = \Spec R$ is normal, affine and toric of dimension $d$, then the torus $T_d$ acts on $X$, and hence acts on $\Div(X)$. We say that a divisor $D\in \Div(X)$ is \emph{torus-invariant} if it is preserved by this action. We will write $\Div_{T_d}(X)\subseteq \Div(X)$ for the group of torus invariant divisors. Now consider the (strongly convex, rational polyhedral) cone $\sigma_R\subseteq N_\mathbb{R}$. We will let $\sigma_R(1)$ denote the set of \emph{edges} (1 dimensional faces) of $\sigma_R$. For any $\rho\in\sigma_R(1)$, note that $\rho\cap N$ is a semigroup isomorphic to $\mathbb{Z}_{\ge 0}$, and so there is a unique choice of generator $u_\rho\in \rho\cap N$ (called a minimal generator). By Proposition \ref{toric cone}, the limit $\displaystyle\gamma_\rho = \lim_{t\to 0}\lambda^{u_\rho}(t)\in X$ exists. Thus we may consider its orbit closure $D_\rho= \overline{T_d\cdot \gamma_\rho}\subseteq X$. The following theorem allows us to characterize $\Cl(R)$, and $[\omega_R]\in\Cl(R)$, entirely in terms of the set $\sigma_R(1)$. \begin{thm}\label{class group} Let $X = \Spec R$ be a normal affine toric variety, with cone $\sigma_R\subseteq N_\mathbb{R}$. We have the following: \begin{enumerate} \item For any $\rho\in\sigma_R(1)$, $D_\rho\subseteq X$ is a torus-invariant prime divisor. Moreover, $\displaystyle\Div_{T_d}(X) = \bigoplus_{\rho\in\sigma_X(1)}\mathbb{Z} D_\rho$. \item Any divisor $D\in \Div(X)$ is rationally equivalent to a torus-invariant divisor. \item For any $m\in M$, the rational function $\chi^m\in K(X)$ has divisor $\displaystyle\div(\chi^m) = \sum_{\rho\in\sigma_X(1)}\langle m,u_\rho\rangle D_\rho$. \item For any torus-invariant divisor $D$, \[\O(D) = \{f\in K(X)| \div(f)+D\ge 0\} = \bigoplus_{\chi^m\in\O(D)}\mathbb{F} \chi^m = \bigoplus_{\div(\chi^m)+D\ge 0}\mathbb{F} \chi^m \subseteq K(X)\] \item There is an exact sequence \[M\to\Div_{T_d}(X)\to \Cl(R)\to 0\] where the first map is $m\mapsto \div(\chi^m)$ and the second map is $D\mapsto \O(D)$. \item $R$ is Cohen--Macaulay and we have $\displaystyle\omega_R\cong \O\left(-\sum_{\rho\in\sigma_X(1)}D_\rho\right)$ \end{enumerate} \end{thm} \begin{proof} By the orbit cone correspondence (\cite{CLS} Theorem 3.2.6), it follows that each $D_\rho$ is a torus-invariant prime divisor, and moreover that these are the only torus-invariant prime divisors. The rest of (1) follows easily from this (cf. \cite{CLS} Exercise 4.1.1). (3) is just \cite{CLS} Proposition 4.1.2. (4) follows from \cite{CLS} Proposition 4.3.2. (5) is \cite{CLS} Theorem 4.1.3, and (2) is an immediate corollary of (5). Lastly (6) is \cite{CLS} Theorems 8.2.3 and 9.2.9. \end{proof} But now Corollary \ref{S cone} and Theorem \ref{class group} make it straight-forward to compute $\Cl(\mathcal{S})$ and $[\omega_{\mathcal{S}}]$: \begin{prop}\label{Cl(S)} Let $e_0 = 2e_1+2e_2-e_3$, so that $\sigma_{\mathcal{X}} = \Cone(e_0,e_1,e_2,e_3)$. For each $i$, let $\rho_i = \mathbb{R}_{\ge 0}e_i$ and $D_i = D_{\rho_i}$, so that $u_{\rho_i} = e_i$ and $\sigma_{\mathcal{X}}(1) = \{\rho_0,\rho_1,\rho_2,\rho_3\}$. Then: \begin{enumerate} \item We have an isomorphism $\Cl(\mathcal{S})\cong \mathbb{Z}$ given by $k\mapsto \O(kD_0)$. \item $\omega_{\mathcal{S}}\cong \O(2D_0)$. \item If $\Mc$ is a generic rank 1 reflexive, self-dual module over $\mathcal{S}$, then $\Mc\cong \O(D_0)$. \item Identifying $\mathcal{S}$ with $\mathbb{F}[x,xz,xz^2,y,yz,yz^2]\subseteq \mathbb{F}[x,y,z]$ as in Proposition \ref{S toric} we get \begin{align*} \O(D_0) &\cong \mathcal{S}\cap xz\mathbb{F}[x,y,z] = (xz,xz^2)\subseteq\mathcal{S}\\ \omega_{\mathcal{S}} = \O(2D_0) &\cong \mathcal{S}\cap x\mathbb{F}[x,y,z] = (x,xz,xz^2)\subseteq\mathcal{S}, \end{align*} so in particular, $\dim_\mathbb{F}\O(D_0)/m_{\mathcal{S}}=2$. \item There is a surjection $\O(D_0)\otimes_{\mathcal{S}}\O(D_0)\twoheadrightarrow \omega_{\mathcal{S}}$. \end{enumerate} \end{prop} \begin{proof} Write $x = \chi^{e_1}, y = \chi^{e_2}$ and $z = \chi^{e_3}$, so that $\mathbb{F}[M] = \mathbb{F}[x^{\pm 1},y^{\pm 1},z^{\pm 1}]$. By Theorem \ref{class group}(3) we get that \begin{align*} \div(x) &= 2D_0+D_1,& \div(y) &= 2D_0+D_2,& \div(z) &= D_3-D_0. \end{align*} It follows that $D_1\sim -2 D_0$, $D_2\sim -2 D_0$ and $D_3\sim D_0$, and so from Theorem \ref{class group}(5) we get that $\Cl(\mathcal{S})$ is generated by $\O(D_0)$. Moreover, the exactness in Theorem \ref{class group}(5) gives that the above relations are the only ones between the $D_i$'s, and so $\O(D_0)$ is non-torsion in $\Cl(\mathcal{S})$, indeed giving the isomorphism $\Cl(\mathcal{S})\cong \mathbb{Z}$. (Alternatively, Theorem \ref{class group}(5) implies that the $\mathbb{Z}$-rank of $\Cl(\mathcal{S})$ is at least $\rank \div_{T_3}(\mathcal{X})-\rank M = 4-3=1$.) This proves (1). For (2), we simply use Theorem \ref{class group}(6): \[\omega_{\mathcal{S}}\cong \O(-D_0-D_1-D_2-D_3) \cong \O\big(-D_0-(-2D_0)-(-2D_0)-D_0\big) = \O(2D_0).\] Now by (1), any generic rank 1 reflexive $\mathcal{S}$-module is in the form $\O(kD_0)$ for some $k\in \mathbb{Z}$, and by (2) $\O(kD_0)^*\cong \O((2-k)D_0)$. Thus if $\O(kD_0)$ is self-dual then $k=1$, giving (3). By the above computations, we get that \[\div(x^ay^bz^c) = (2a+2b-c)D_0+2aD_1+2bD_2+cD_3\] for any $(a,b,c)\in \mathbb{Z}^3$. Now note that $\O(D_0) \cong \O(-D_1-D_3)$ and $\O(2D_0) \cong \O(-D_1)$ which are both ideals of $\mathcal{S}$. But now by Theorem \ref{class group}(4) as ideals of $\mathcal{S}$ we have \begin{align*} \O(D_0) &\cong \O(-D_1-D_3) = \left(x^ay^bz^c\middle| 2a+2b-c\ge 0, 2a\ge 1,2b\ge 0, c\ge 1 \right)\\ &= \left(x^ay^bz^c\middle| 2a+2b-c\ge 0, a\ge 1,b\ge 0, c\ge 1 \right)\\ &= \left(x^ay^bz^c\middle|x^ay^bz^c\in\mathcal{S}, xz|x^ay^bz^c\right)\\ &=\mathcal{S}\cap xz\mathbb{F}[x,y,z] = (xz,xz^2)\\ \O(2D_0) &\cong \O(-D_1) = \left(x^ay^bz^c\middle| 2a+2b-c\ge 0, 2a\ge 1,2b\ge 0, c\ge 0 \right)\\ &= \left(x^ay^bz^c\middle| 2a+2b-c\ge 0, a\ge 1,b\ge 0, c\ge 0 \right)\\ &= \left(x^ay^bz^c\middle|x^ay^bz^c\in\mathcal{S}, x|x^ay^bz^c\right)\\ &=\mathcal{S}\cap x\mathbb{F}[x,y,z] = (x,xz,xz^2), \end{align*} proving (4). Now identify $\O(D_0)$ with $(xz,xz^2)\subseteq \mathcal{S}$ and $\omega_{\mathcal{S}}$ with $(x,xz,xz^2)\subseteq \mathcal{S}$. Notice that \[\O(D_0)\O(D_0) = (xz,xz^2)(xz,xz^2) = (x^2z^2,x^2z^3,x^2z^4) = xz^2(x,xz,xz^2) = xz^2\omega_{\mathcal{S}}.\] Thus we can define a surjection $f:\O(D_0)\otimes_{\mathcal{S}}\O(D_0)\twoheadrightarrow \omega_{\mathcal{S}}$ by $\displaystyle f(\alpha\otimes \beta) = \frac{1}{xz^2}\alpha\beta$, proving (5). \end{proof} We can now compute $\Cl(\mathcal{R})$ and $\omega_{\mathcal{R}}$, by using the following lemma: \begin{lemma}\label{toric product} For any normal, affine toric varieties $X$ and $Y$ the natural map $\Cl(X)\oplus\Cl(Y)\to \Cl(X\times Y)$ given by $([A],[B])\mapsto [A\boxtimes B]$ is an isomorphism which sends $([\omega_X],[\omega_Y])$ to $\omega_{X\times Y}$. \end{lemma} \begin{proof} By \cite{CLS} Proposition 3.1.14, $X\times Y$ is a toric variety with cone $\sigma_{X\times Y} = \sigma_X\times \sigma_Y$. It follows that $\sigma_{X\times Y}(1) = \sigma_X(1)\sqcup \sigma_Y(1)$. The claim now follows immediately from Theorem \ref{class group}. \end{proof} Thus we have: \begin{cor}\label{Cl(R)} The map $\varphi:\Cl(\mathcal{S})^k\to \Cl(\mathcal{R})$ given by \[([A_1],\ldots,[A_k])\mapsto \big[(A_1\boxtimes A_2\boxtimes \cdots\boxtimes A_a)[x_1,\ldots,x_s]\big]\] is an isomorphism which sends $([\omega_\mathcal{S}],\ldots,[\omega_\mathcal{S}])$ to $[\omega_{\mathcal{R}}]$. Consequently there is a unique self-dual generic rank $1$ reflexive module $\Mc$ over $\mathcal{R}$, which is the image of $([\O(D_0)],\ldots,[\O(D_0)])$. We have that $\dim _\mathbb{F}\Mc/m_{\mathcal{R}} = 2^k$ and there is a surjection $\Mc\otimes_{\mathcal{R}}\Mc\twoheadrightarrow \omega_{\mathcal{R}}$. \end{cor} \begin{proof} The isomorphism follows immediately from Corollary \ref{toric product} (noting that $\mathbb{A}^1$ is a toric variety with $\Cl(\mathbb{A}^1) = 0$ and $\omega_{\mathbb{A}^1} = \mathbb{A}^1$). Now for any self-dual generic rank 1 reflexive module $\Mc$ over $\mathcal{R}$, it follows that $[\Mc] = \varphi([A_1],\ldots,[A_a])$ where each $A_i$ is self-dual. Proposition \ref{Cl(S)} implies that each $A_i$ is isomorphic to $\O(D_0)$, as claimed. For this $\Mc$ we indeed have \begin{align*} \Mc/m_{\mathcal{R}} &= \frac{\O(D_0)^{\boxtimes k}[x_1,\ldots,x_s]}{m_{\mathcal{S}}^{\boxtimes k}\boxtimes(x_1,\ldots,x_s)} \cong \left(\frac{\O(D_0)}{m_{\mathcal{S}}}\right)^{\boxtimes k} = \left(\mathbb{F}^2\right)^{\boxtimes k} = \mathbb{F}^{2^k}. \end{align*} Also, the surjection $\O(D_0)\otimes_{\mathcal{S}}\O(D_0)\twoheadrightarrow \omega_{\mathcal{S}}$ from Proposition \ref{Cl(S)} indeed gives a surjection \begin{align*} \Mc\otimes_{\mathcal{R}}\Mc &=\left(\O(D_0)^{\boxtimes k}[x_1,\ldots,x_s] \right)\otimes_{\mathcal{R}}\left(\O(D_0)^{\boxtimes k}[x_1,\ldots,x_s]\right)\\ &\cong\left(\O(D_0)\otimes\O(D_0)\right)^{\boxtimes k}[x_1,\ldots,x_s]\twoheadrightarrow\omega_{\mathcal{S}}^{\boxtimes k}[x_1,\ldots,x_s]\cong \omega_{\mathcal{R}}. \end{align*} \end{proof} Which completes the proof of Theorem \ref{self-dual decompleted}. \begin{rem} In our proof of Theorem \ref{self-dual decompleted}, we never actually used the first condition, namely that $\Mc$ was maximal Cohen--Macaulay over $\mathcal{R}$. We only used the (strictly weaker) assumption that $\Mc$ was reflexive, which, combined with the fact that $\Mc$ was self-dual, was enough to uniquely determine the structure of $\Mc$. In most situations, the modules $M_\infty$ produced by the patching method will be maximal Cohen--Macaulay, but it is possible that they might fail to be self-dual (e.g. if they arise from the cohomology of a non self-dual local system). In this situation it is possible to formulate a weaker version of Theorem \ref{self-dual}, where one drops the self-duality assumption. Specifically one can show (in the notation of Proposition \ref{Cl(S)}) that the only Cohen--Macaulay generic rank one modules over the ring $\mathcal{S}$ are the 5 modules: \begin{align*} \O(-D_0) &= (xz,xz^2,yz,yz^2)\\ \O &= \overline{S}\\ \O(D_0) &= (xz,xz^2)\\ \O(2D_0) &= (x,xz,xz^2)\\ \O(3D_0) &= (x^2,x^2z,x^2z^2,x^2z^3). \end{align*} This can be done quite simply by first completing at $m_{\mathcal{S}}$, and noting that if $(t_1,t_2,t_3)$ is a regular sequence for $\overline{S}$ then it must also be a regular sequence for $\Mc_{m_{\mathcal{S}}}$ over $\overline{S}$ where $\Mc$ is any maximal Cohen--Macaulay module over $\mathcal{S}$, which implies that $\overline{S}$ and $\Mc_{m_{\mathcal{S}}}$ are both finite free $\mathbb{F}[[t_1,t_2,t_3]]$-modules. Moreover if $\genrank_{\mathcal{S}}\Mc = 1$ then $\overline{S}$ and $\Mc_{m_{\mathcal{S}}}$ have the same rank over $\mathbb{F}[[t_1,t_2,t_3]]$, and so \[\dim_{\mathbb{F}}\Mc/m_{\mathcal{S}}\Mc = \dim_{\mathbb{F}}\Mc_{m_{\mathcal{S}}}/m_{\mathcal{S}}\Mc_{m_{\mathcal{S}}} \le \dim_{\mathbb{F}}\Mc/(t_1,t_2,t_3) = \dim_{\mathbb{F}}\mathcal{S}/(t_1,t_2,t_3).\] Thus using the regular sequence $(x,yz^2,y-xz^2)$ for $\mathcal{S}$, we see that if $\Mc$ is a maximal Cohen--Macaulay $\mathcal{S}$-module of generic rank $1$, then $\dim_{\mathbb{F}}\Mc/m_{\mathcal{S}}\Mc\le \dim_{\mathbb{F}}\mathcal{S}/(x,yz^2,y-xz^2) = 4$. It is easy to verify from the description of $\Cl(\mathcal{S})$ given in Proposition \ref{Cl(S)}, and the description of $\O(D)$ from Theorem \ref{class group} that the five modules listed above are the only generic rank $1$ reflexive $\mathcal{S}$-modules $\Mc$ with $\dim_{\mathbb{F}}\Mc/m_{\mathcal{S}}\Mc\le 4$, and all of these can directly be shown to be maximal Cohen--Macaulay. This unfortunately does not allow us to uniquely deduce the structure of $\Mc$ and hence of $M_\infty$, but it does give us the bound $\dim_\mathbb{F} M_\infty/m_{R_\infty} \le 4^k$, and could potentially lead to more refined information about $M_\infty$, which may be of independent interest. \end{rem} \subsection{Class groups of completed rings}\label{ssec:decompletion} The goal of this section is to prove that Theorem \ref{self-dual decompleted} implies Theorem \ref{self-dual mod l}. We shall do this by proving that the natural map $\Cl(\mathcal{R})\to \Cl(\overline{R}_\infty)$ given by $[\Mc]\mapsto [\Mc\otimes_{\mathcal{R}}\overline{R}_\infty]$ is an isomorphism. First note that the Theorem \ref{self-dual mod l} will indeed follow from this. Assume that $\overline{M}_\infty$ is an $\overline{R}_\infty$-module satisfying the conditions of Theorem \ref{self-dual mod l}. Then in particular it corresponds to an element of $\Cl(\overline{R}_\infty)$, and so there is some reflexive generic rank $1$ $\mathcal{R}$-module $\Mc$ with $\overline{M}_\infty\cong \Mc\otimes_{\mathcal{R}}\overline{R}_\infty = \varprojlim \Mc/m_{\mathcal{R}}^n\Mc$. We claim that $\Mc$ satisfies conditions (1) and (2) of Theorem \ref{self-dual decompleted} (by assumption, it satisfies condition (3)). Showing that $\Mc$ is self-dual is equivalent to showing that $2[\Mc] = [\omega_{\mathcal{R}}]$ in $\Cl(\mathcal{R})$, which follows from the fact that $2[\Mc\otimes_{\mathcal{R}}\overline{R}_\infty] = 2[\overline{M}_\infty] = [\omega_{\overline{R}_\infty}]$ in $\Cl(\overline{R}_\infty)$ and the fact that $\omega_{\overline{R}_\infty}\cong \omega_{\mathcal{R}}\otimes_{\mathcal{R}}\overline{R}_\infty$ (cf \cite[Corollaries 21.17 and 21.18]{Eisenbud}). We now observe that $\Mc$ is maximal Cohen--Macaulay over $\mathcal{R}$.\footnote{Strictly speaking it is not necessary to show this, as condition (1) was never used in the proof of Theorem \ref{self-dual decompleted}, but we will still show it for the sake of completeness.} By Theorem \ref{R^st}, $(C,Y,B-Z)$ is a regular sequence for $\overline{S}$ consisting of homogeneous elements. It follows that this is also a regular sequence for $\mathcal{S}$, and so $\mathcal{R}$ also has a regular sequence $(z_1,\ldots,z_{3k+s})$ consisting entirely of homogeneous elements. Now it follows that this regular sequence is also regular for $\overline{R}_\infty$, and hence for $\overline{M}_\infty$. But now as the $z_i$'s are all homogeneous it follows that $\Mc/(z_1,\ldots,z_i)\hookrightarrow \overline{M}_\infty/(z_1,\ldots,z_i)$ for all $i$ and so $(z_1,\ldots,z_{3k+s})$ is also a regular sequence for $\Mc$. Hence $\Mc$ is maximal Cohen--Macaulay over $\mathcal{R}$. Hence $\Mc$ satisfies the conditions of Theorem \ref{self-dual decompleted}, so we get that $\dim_{\mathbb{F}}\Mc/m_{\mathcal{R}}\Mc = 2^k$ and $\tau_{\Mc}:\Mc\otimes_{\mathcal{R}}\Mc\to\omega_{\mathcal{R}}$ is surjective. Now as $\overline{M}_\infty/m_{\overline{R}_\infty}\overline{M}_\infty\cong \Mc/m_{\mathcal{R}}\Mc$, we indeed get $\dim_{\mathbb{F}}\overline{M}_\infty/m_{\overline{R}_\infty}\overline{M}_\infty = 2^k$. To show that $\tau_{\overline{M}_\infty}$ is surjective, note that \[(\Mc\otimes_{\mathcal{R}}\Mc)\otimes_{\mathcal{R}}\overline{R}_\infty \cong (\Mc\otimes_{\mathcal{R}}\overline{R}_\infty)\otimes_{\overline{R}_\infty}(\Mc\otimes_{\mathcal{R}}\overline{R}_\infty) \cong \overline{M}_\infty\otimes_{\overline{R}_\infty}\overline{M}_\infty\] so as $\omega_{\mathcal{R}}$ is a quotient of $\Mc\otimes_{\mathcal{R}}\Mc$ it follows that $\omega_{\overline{R}_\infty}\cong \omega_{\mathcal{R}}\otimes_{\mathcal{R}}\overline{R}_\infty$ is a quotient of $\overline{M}_\infty\otimes_{\overline{R}_\infty}\overline{M}_\infty$ and so $\tau_{\overline{M}_\infty}$ is indeed surjective by Lemma \ref{trace}, which completes the proof of Theorem \ref{self-dual mod l}. Unfortunately, it is not true in general that if $R$ is a graded $\mathbb{F}$-algebra and $\widehat{R}$ is the completion at the irrelevant ideal then the map $\Cl(R)\to \Cl(\widehat{R})$ is an isomorphism. However Danilov \cite{Danilov} has shown that this is true in certain cases: \begin{thm}[Danilov]\label{Danilov} Let $V$ be a smooth projective variety with a very ample line bundle $\mathcal{L}$ giving an injection $V\hookrightarrow \P^N$. Let $\Spec S\subseteq \mathbb{A}^{N+1}$ be the affine cone on $V$, so that $S$ is a graded $\mathbb{F}$-algebra, and let $\widehat{S}$ be the completion of $R$ at the irrelevant ideal. Then: \begin{enumerate} \item The natural map $\Cl(S)\to \Cl(\widehat{S})$ is an isomorphism if and only if $H^1(V,\mathcal{L}^{\otimes i}) = 0$ for all $i\ge 1$. \item For $s>0$, the natural map $\Cl(S)\to \Cl(\widehat{S}[[x_1,\ldots,x_s]])$ is an isomorphism if and only if $H^1(V,\mathcal{L}^{\otimes i}) = 0$ for all $i\ge 0$. \end{enumerate} \end{thm} We now make the following observation: \begin{lemma}\label{H^d(S)=0} There exists a smooth projective variety $\mathcal{V}$ and an ample line bundle $\mathcal{L}$ on $\mathcal{V}$ such that $\Spec \mathcal{S}$ is the affine cone over $\mathcal{V}$, under the projective embedding induced by $\mathcal{L}$. We have $H^d(\mathcal{V},\mathcal{L}^{\otimes i}) = 0$ for all $d\ge 1$ and $i\ge 0$. \end{lemma} \begin{proof} This is largely a restatement of Proposition \ref{S toric}(1). Specifically we have $\mathcal{V} = \P^1\times\P^1$ and $\mathcal{L} = \O_{\P^1}(2)\boxtimes\O_{\P^1}(1)$. To prove the vanishing of cohomology, we simply note that $H^d(\P^1,\O_{\P^1}(i)) = 0$ for all $d\ge 1$ and $i\ge 0$, and so \[H^d(\P^1\times\P^1,\O_{\P^1}(2i)\boxtimes\O_{\P^1}(i)) = \bigoplus_{e=0}^dH^e(\P^1,\O_{\P^1}(2i))\otimes H^{d-e}(\P^1,\O_{\P^1}(i)) = 0\] for any $d\ge 1$ and $i\ge 0$. \end{proof} \begin{rem} By \cite[Excercise 18.16]{Eisenbud}, the conclusion of Lemma \ref{H^d(S)=0} about the vanishing of the cohomology groups $H^d(\mathcal{V},\mathcal{L}^{\otimes i})$ holds whenever $\mathcal{S}$ is Cohen--Macaulay and $d<\dim \mathcal{V}$. This means the results of this section will be applicable in most cases where the ring $R_\infty$ is Cohen--Macaulay, and so this does not impose a fundamental limitation on our method. \end{rem} It follows that $\Cl(\mathcal{S})\cong \Cl(\overline{S})$. In fact (as the natural map $\Cl(\mathcal{S})\to \Cl(\mathcal{S}[x_1,\ldots,x_s])$ is an isomorphism) it follows that the natural map $\Cl(\mathcal{S}[x_1,\ldots,x_s])\to \Cl(\overline{S}[[x_1,\ldots,x_s]])$ is an isomorphism, and so Theorem \ref{self-dual mod l} follows in the case when $k=1$. When $k>1$ however, we cannot directly appeal to Theorem \ref{Danilov}, as $\Spec \mathcal{R}$ is no longer the cone over a smooth projective variety, and in fact $\Spec \mathcal{R}$ does not have isolated singularities. Fortunately it is fairly straightforward to adapt the method of \cite{Danilov} to our situation. Specifically, we will prove the following (which obviously applies to the ring $\mathcal{R}$): \begin{prop}\label{Danilov for products} Let $\mathcal{V}_1,\ldots,\mathcal{V}_k$ be a collection of smooth projective varieties of dimension at least $1$, and for each $j$, let $\mathcal{L}_j$ be a very ample line bundle on $\mathcal{V}_j$ giving an injection $\mathcal{V}_j\hookrightarrow \P^{N_j}$. Let $\Spec S_j\subseteq \mathbb{A}^{N_j+1}$ be the affine cone on $V_j$. Let \[R = \left[\bigotimes_{j=1}^kS_j\right][x_1,\ldots,x_s]\] (for some $s\ge 0$), so that $R$ is a graded $\mathbb{F}$-algebra. Let $\widehat{R}$ be the completion of $R$ at the irrelevant ideal. If we have that $H^d(\mathcal{V}_j,\mathcal{L}_j^{\otimes i}) = 0$ for all $d = 1,2$, $j=1,\ldots,k$ and $i\ge 0$, then the natural map $\Cl(R)\to \Cl(\widehat{R})$ is an isomorphism. \end{prop} \begin{proof} For simplicity, we first reduce to the case $s=0$. If $s\ge 2$, then we may simply let $\mathcal{V}_{k+1} = \P^{s-1}$, $\mathcal{L} = \O_{\P^{s-1}}(1)$ and note that we still have the cohomology condition $H^d(\mathcal{V}_{k+1},\mathcal{L}_{k+1}^{\otimes i}) = H^d(\P^{s-1},\O_{\P^{s-1}}(i)) = 0$ for all $d\ge 1$ and $i\ge 0$. So the $s\ge 2$ case follows from the $s=0$ case. The $s=1$ case now follows from the $s=0$ and $s\ge 2$ cases by letting $\displaystyle R_0 = \bigotimes_{j=1}^kS_j$ and considering the commutative diagram: \begin{center} \begin{tikzpicture} \node(11) at (0,2) {$\Cl(R_0)$}; \node(21) at (4,2) {$\Cl(R_0[x_1])$}; \node(31) at (8,2) {$\Cl(R_0[x_1,x_2])$}; \node(10) at (0,0) {$\Cl(\widehat{R}_0)$}; \node(20) at (4,0) {$\Cl(\widehat{R}_0[[x_1]])$}; \node(30) at (8,0) {$\Cl(\widehat{R}_0[[x_1,x_2]])$}; \draw[->] (11)--(10); \draw[->] (21)--(20); \draw[->] (31)--(30); \draw[->] (11)--(21); \draw[->] (21)--(31); \draw[->] (10)--(20); \draw[->] (20)--(30); \end{tikzpicture} \end{center} and noting that the maps on the top row are isomorphisms by standard properties of the class groups of varieties, and the maps on the bottom row are injective (since if $M$ is a reflexive $\widehat{R}_0$ module and $M[[x_1]] = M\otimes_{\widehat{R}_0}\widehat{R}_0[[x_1]]$ is a free $\widehat{R}_0[[x_1]]$-module, then $M/m_{\widehat{R}_0}\cong M[[x_1]]/m_{\widehat{R}_0[[x_1]]}\cong \mathbb{F}$, and so $M$ is a cyclic, and thus a free $\widehat{R}$-module). So from now on, we shall assume $s=0$. We first introduce some notation. For each $j$, let $Y_j = \Spec S_j$. Let $\displaystyle X = \prod_{j=1}^k\mathcal{V}_j$ and $\displaystyle Y = \prod_{j=1}^kY_j = \Spec R$. Also let $\displaystyle Z_j = \prod_{j'\ne j}Y_{j'}\subseteq Y$ and $Z = Z_1\cup Z_2\cdots\cup Z_k\subseteq Y$. Note that each $Z_j$ is irreducible subscheme of $Y$ of codimension at least $2$. Write $Z_j = \Spec R/I_j$ and $Z = \Spec R/I$. Note that $I_j = m_jR$, where $m_j$ is the irrelevant ideal of $S_j$, and $I = I_1I_2\cdots I_k$. In particular, $I_j$ and $I$ are homogeneous ideals of $R$. Now let $\widehat{Y} = \Spec \widehat{R}$, $\widehat{I}_j = I_j\widehat{R}$, $\widehat{I} = I\widehat{R}$, $\widehat{Z}_j = \Spec \widehat{R}/\widehat{I}_j$ and $\widehat{Z} = \Spec \widehat{R}/\widehat{I}$. Note that the $\widehat{Z}_j$'s are still irreducible, and we have $\widehat{Z} = \widehat{Z}_1\cup\widehat{Z}_2\cup\cdots\cup\widehat{Z}_k$. Now let $\displaystyle C = \Proj\left(\bigoplus_{n=0}^\infty I^n\right)$ and $\displaystyle \widetilde{C} = \Proj\left(\bigoplus_{n=0}^\infty \widehat{I}^n\right)$ be the blowups of $Y$ and $\widehat{Y}$ along $Z$ and $\widehat{Z}$ and let $p:C\to Y$ and $\widetilde{p}:\widetilde{C}\to \widehat{Y}$ be the projection maps. Let $E_j = p^{-1}(Z_j)$, $\widetilde{E}_j = \widetilde{p}^{-1}(\widehat{Z}_j)$, $E = p^{-1}(Z)$ and $\widetilde{E} = \widetilde{p}^{-1}(\widehat{Z})$. Note that the $E_j$'s and $\widetilde{E}_j$'s are irreducible and we have $E = E_1\cup E_2\cup\cdots\cup E_k$ and $\widetilde{E} = \widetilde{E}_1\cup \widetilde{E}_2\cup\cdots\cup \widetilde{E}_k$. Let $m_R\subseteq R$ denote the irrelevant ideal and let $\widehat{m}_R = m_R\widehat{R}\subseteq R$ be its completion. Notice that we have natural isomorphisms $p^{-1}(\{m_R\}) = E_1\cap E_2\cap\cdots \cap E_k \cong X$ and $\widetilde{p}^{-1}(\{\widehat{m}_R\}) = \widetilde{E}_1\cap \widetilde{E}_2\cap\cdots \cap \widetilde{E}_k \cong X$. Identify $X$ with its images in both $C$ and $\widetilde{C}$. We will let $\widehat{C}$ and $\widehat{\Ct}$ denote the formal completions of $C$ and $\widetilde{C}$ along the subvariety $X$. Lastly, we define a rank $k$ vector bundle $\xi$ on $X$ as follows. For each $j$, let $\pi_jX\to \mathcal{V}_j$ be the projection map, so that $\pi_j^* \mathcal{L}_j = \O_{\mathcal{V}_1}\boxtimes\cdots\boxtimes \mathcal{L}_j\boxtimes\cdots\boxtimes \O_{\mathcal{V}_k}$ is a line bundle on $X$. We will let $\xi = \pi_1^* \mathcal{L}_1\oplus \pi_2^* \mathcal{L}_2\oplus\cdots\oplus \pi_k^* \mathcal{L}_k$. We first observe the following: \begin{lemma}\label{V(xi)} There is an isomorphism $C\cong V(\xi)$, where $V(\xi)$ is the total space of the vector bundle $\xi$ over $X$. This isomorphism is compatible with the inclusions $X\hookrightarrow C$ and $X\hookrightarrow V(\xi)$. Moreover we have isomorphisms of formal schemes $\widehat{C}\cong \widehat{V}(\xi)\cong \widehat{\Ct}$, where $\widehat{V}(\xi)$ is the completion of $V(\xi)$ along the zero section $X\hookrightarrow V(\xi)$. These isomorphisms are again compatible with the natural inclusions of $X$. \end{lemma} \begin{proof} Letting $V(\mathcal{L}_j)$ be the total space of $\mathcal{L}_j$ over $\mathcal{V}_j$ we see that \begin{align*} V(\xi) &= V(\pi_1^*\mathcal{L}_1\oplus \pi_2^*\mathcal{L}_2\oplus\cdots\pi_k^*\mathcal{L}_k) \cong V(\pi_1^*\mathcal{L}_1)\times_XV(\pi_2^*\mathcal{L}_2)\times_X\cdots\times_XV(\pi_k^*\mathcal{L}_k)\\ & \cong V(\mathcal{L}_1)\times V(\mathcal{L}_2)\times\cdots V(\mathcal{L}_k). \end{align*} Now as in \cite[Lemma 1(3)]{Danilov}, each $V(\mathcal{L}_j)$ is the blowup of $\Spec S_j$ at the point $m_j$. Now using this and the fact that $I = I_1I_2\cdots I_k = m_1\otimes m_2\otimes \cdots \otimes m_k$ we indeed get \begin{align*} V(\xi) &\cong \prod_{j=1}^kV(\mathcal{L}_j) \cong \prod_{j=1}^k\Proj\left(\bigoplus_{n=0}^\infty m_j^n \right) \cong \Proj\left(\bigoplus_{n=0}^\infty \left(m_1^n\otimes m_2^n\otimes\cdots\otimes m_k^n\right)\right) = \Proj\left(\bigoplus_{n=0}^\infty I^n\right) = C, \end{align*} where we used the fact that $\displaystyle\Proj\left(\bigoplus_{n=0}^\infty A_n \right)\times \Proj\left(\bigoplus_{n=0}^\infty B_n \right) \cong \Proj\left(\bigoplus_{n=0}^\infty A_n\otimes B_n \right)$ for finitely generated graded $R$-algebras $\displaystyle\bigoplus_{n=0}^\infty A_n$ and $\displaystyle\bigoplus_{n=0}^\infty B_n$. (See for instance, \cite[Exercise 9.6.D]{RisingSea}) It is easy to check that these isomorphisms are compatible with the embeddings $X\hookrightarrow C, V(\xi)$. This automatically gives $\widehat{C} \cong \widehat{V}(\xi)$. Now notice that the subscheme $X = p^{-1}(\{m_R\})\subseteq C$ is cut out by the ideal sheaf $\mathscr{I}=p^*(m_R)$ and similarly the subscheme $X = \widetilde{p}^{-1}(\{\widehat{m}_R\})\subseteq \widetilde{C}$ is cut out by the ideal sheaf $\widehat{\Is} = \widetilde{p}^*(\widehat{m}_R)$. But now using the fact that $\widehat{m}^a_R/\widehat{m}^b_R = m^a_R/m^b_R$ for all $a>b$, as in \cite[Section 4]{Danilov} we get that \begin{align*} \widehat{\Ct} &= \varinjlim_n \left(X,\O_{\widetilde{C}}/\widehat{\Is}^{n+1}\right) = \varinjlim_n \widetilde{p}^{-1}\left(\Spec\left(\widehat{R}/\widehat{m}_R^{n+1}\right) \right) = \varinjlim_n \Proj\left(\bigoplus_{i=0}^\infty \widehat{m}_R^i/\widehat{m}_R^{n+i+1}\right)\\ &= \varinjlim_n \Proj\left(\bigoplus_{i=0}^\infty m_R^i/m_R^{n+i+1}\right) = \varinjlim_n p^{-1}\left(\Spec\left(R/m_R^{n+1}\right)\right) = \varinjlim_n \left(X,\O_{C}/\mathscr{I}^{n+1}\right) = \widehat{C}, \end{align*} completing the proof of the lemma. \end{proof} We next note the following analogue of \cite[Section 2]{Danilov}: \begin{lemma}\label{Cl to Pic} Assume the setup of Proposition \ref{Danilov for products}, but without the assumption about the vanishing of the cohomology groups. There is a commutative diagram with exact rows: \begin{center} \begin{tikzpicture} \node(01) at (0,2) {$0$}; \node(11) at (2,2) {$\mathbb{Z}^k$}; \node(21) at (4,2) {$\Pic(X)$}; \node(31) at (6,2) {$\Cl(R)$}; \node(41) at (8,2) {$0$}; \node(00) at (0,0) {$0$}; \node(10) at (2,0) {$\mathbb{Z}^k$}; \node(20) at (4,0) {$\Pic(\widehat{V}(\xi))$}; \node(30) at (6,0) {$\Cl(\widehat{R})$}; \node(40) at (8,0) {$0$}; \draw[->] (11)--(10) node[midway,left]{$=$}; \draw[->] (21)--(20); \draw[->] (31)--(30); \draw[->] (01)--(11); \draw[->] (11)--(21); \draw[->] (21)--(31); \draw[->] (31)--(41); \draw[->] (00)--(10); \draw[->] (10)--(20); \draw[->] (20)--(30); \draw[->] (30)--(40); \end{tikzpicture} \end{center} Where the map $\Cl(R)\to \Cl(\widehat{R})$ is the natural completion map, and the map $\Pic(X)\to \Pic(\widehat{V}(\xi))$ is the pullback along the projection map $\widehat{V}(\xi)\to X$. \end{lemma} \begin{proof} Let $U= Y\smallsetminus Z$ and $\widetilde{U}=\widehat{Y}\smallsetminus \widehat{Z}$, and note that $p$ and $\widetilde{p}$ induce isomorphisms $C\smallsetminus E\cong U$ and $\widetilde{C}\smallsetminus\widetilde{E}\cong\widetilde{U}$. Thus we will also regard $U$ and $\widetilde{U}$ as being open subschemes of $C$ and $\widetilde{C}$. As in \cite[Lemma 3]{Danilov} we get that $U$ and $\widetilde{U}$ are both regular. Now as $Z\subseteq Y$ and $\widehat{Z}\subseteq \widehat{Y}$ have codimension at least two, the restriction maps $\Cl(R) = \Cl(Y)\to \Cl(U)$ and $\Cl(\widehat{R})=\Cl(\widehat{Y})\to \Cl(\widetilde{U})$ are isomorphisms. Now as each $E_j\subseteq C$ is an irreducible subvariety of codimension 1, and $E = E_1\cup E_2\cup \cdots\cup E_k$, we get the the restriction map $\Cl(C)\to \Cl(C\smallsetminus E) = \Cl(U)$ is a surjection, with kernel equal to the $\mathbb{Z}$-span of $[E_1],\ldots,[E_k]$ (cf \cite[Proposition II.6.5]{Hartshorne}). We claim that $[E_1],\ldots,[E_k]\in \Cl(C)$ are $\mathbb{Z}$-linearly independent. Assume not. Then there exists some non-unit rational function $g$ on $C$ for which $\div g = n_1[E_1]+\cdots+n_k[E_k]$, and so in particular, $\supp g\subseteq E$. But then writing $g = p^*(g')$ for some rational function on $Y$, we get that $\supp g' \subseteq p(E) = Z$, which implies that $g'$, and hence $g$, is a unit as $Z\subseteq Y$ has codimension at least two, a contradiction. Thus we have an exact sequence $0\to \mathbb{Z}^k\to \Cl(C)\to \Cl(R)\to 0$. Similarly we have a surjection $\Cl(\widetilde{C})\to \Cl(\widehat{R})$ with kernel spanned by $[\widetilde{E}_1],\ldots,[\widetilde{E}_k]\in \Cl(\widetilde{C})$, which are also $\mathbb{Z}$-linearly independent. This gives the exact sequence $0\to \mathbb{Z}^k\to \Cl(\widetilde{C})\to \Cl(\widehat{R})\to 0$. It remains to give isomorphisms $\Pic(X)\cong \Cl(C)$ and $\Pic(\widehat{V}(\xi))\cong \Cl(\widetilde{C})$ compatible with the other maps. First, as $C$ and $\widetilde{C}$ are locally factorial, we get that $\Cl(C)\cong \Pic(C)$ and $\Cl(\widetilde{C})\cong \Pic(\widetilde{C})$. By \cite[Proposition 3]{Danilov}, the zero section $X\hookrightarrow V(\xi)$ gives an isomorphism $\Pic(V(\xi))\cong \Pic(X)$, so by Lemma \ref{V(xi)}, $\Pic(X)\cong \Pic(V(\xi))\cong \Pic(C)\cong \Cl(C)$. Now as $\widehat{R}$ is an \emph{adic} Noetherian ring with ideal of definition $\widehat{m}_R$, the morphism $\widetilde{p}:\widetilde{C}\to \widehat{Y}=\Spec \widehat{R}$ is projective, and $\widehat{\Ct}$ is the completion of $\widetilde{C}$ along $X = \widetilde{p}^{-1}(\{m_R\})$, the argument of \cite[Proposition 4]{Danilov} implies that $\Pic(\widetilde{C})\cong \Pic(\widehat{\Ct})$ is an isomorphism. Thus Lemma \ref{V(xi)} gives $\Pic(\widehat{V}(\xi))\cong\Pic(\widehat{\Ct})\cong \Pic(\widetilde{C})$, establishing the desired commutative diagram. \end{proof} Thus it will suffice to show that the map $\Pic(X)\to \Pic(\widehat{V}(\xi))$ induced by the projection $\widehat{V}(\xi)\to X$ is an isomorphism. Now write $\displaystyle C_n = \underline{\Spec}_X\left(\bigoplus_{i=0}^n\xi^{\otimes i}\right)$ (where $\underline{\Spec}_X$ denotes the relative $\Spec$ over $X$), so that $\displaystyle\widehat{V}(\xi) = \varinjlim_n C_n$. As in \cite[Proposition 5]{Danilov} we have $\displaystyle\Pic(\widehat{V}(\xi))\cong \varprojlim_n\Pic(C_n)$. Now for each $n$, let $\pr_n:C_n\to X$ be the projection, and let $i_n:X\to C_n$ be the zero section. Note that we canonically have $C_0 = X$ and $i_0$ and $\pr_0$ are just the identity map. We have that $\pr_n\circ i_n = \id_X$ and so $i_n^*\circ \pr_n^* = \id_{\Pic(X)}$. Hence $\pr_n^*:\Pic(X)\to \Pic(C_n)$ is an injection (and in fact, $\Pic(X)$ is a direct summand of $\Pic(C_n)$). It follows that the map $\displaystyle\pr^* = (\pr_n^*):\Pic(X)\to \varprojlim_n\Pic(C_n)\cong \Pic(\widehat{V}(\xi))$ is injective. In particular this means that $\Cl(R)\to\Cl(\widehat{R})$ is injective. Now for each $n$ we have $\Pic(C_n) = H^1(X,\O^*_{C_n})$. As in \cite[Section 3]{Danilov}, we consider the exact sequence of sheaves on $X$: \[0\to \xi^{\otimes (n+1)} \to \O^*_{C_{n+1}} \to \O^*_{C_n} \to 1,\] where the first map sends $s\in \Gamma(W,\xi^{\otimes (n+1)})$ to $1+s\in \Gamma(W,\O^*_{C_{n+1}})$. Then the long exact sequence of cohomology gives an exact sequence: \[H^1(X,\xi^{\otimes(n+1)}) \to \Pic(C_{n+1})\to \Pic(C_n)\to H^2(X,\xi^{\otimes(n+1)}).\] We now claim that $H^d(X,\xi^{\otimes i}) = 0$ for all $d=1,2$ and $i\ge 0$. First note that \begin{align*} \xi^{\otimes i} &= \left(\bigoplus_{j=1}^k\pi_j^* \mathcal{L}_j \right)^{\otimes i} = \bigoplus_{\substack{i_1+\cdots+i_k = i\\ i_1,\ldots,i_k\ge 0}}\left(\pi_1^*\mathcal{L}_1^{\otimes i_1}\otimes\cdots\otimes \pi_k^*\mathcal{L}_k^{\otimes i_k}\right) = \bigoplus_{\substack{i_1+\cdots+i_k = i\\ i_1,\ldots,i_k\ge 0}}\mathcal{L}_1^{\otimes i_1}\boxtimes\cdots\boxtimes \mathcal{L}_k^{\otimes i_k} \end{align*} but now for any $i_1,\ldots,i_k\ge 0$ and any $d=1,2$ we get: \begin{align*} H^d(X,\mathcal{L}_1^{\otimes i_1}\boxtimes\cdots\boxtimes \mathcal{L}_k^{\otimes i_k}) &=H^d(\mathcal{V}_1\times\cdots\times \mathcal{V}_k,\mathcal{L}_1^{\otimes i_1}\boxtimes\cdots\boxtimes \mathcal{L}_k^{\otimes i_k})\\ &= \bigoplus_{\substack{d_1+\cdots+d_k = d\\ d_1,\ldots,d_k\ge 0}}\ \bigotimes_{j=1}^k H^{d_j}(\mathcal{V}_j,\mathcal{L}_j^{\otimes i_j}) = 0, \end{align*} since for any $k$-tuple $(d_1,\ldots,d_k)$ with $d_1+\cdots+d_k=d\in\{1,2\}$ and $d_1,\ldots,d_k\ge 0$, there must be some index $j$ for which $d_j \in \{1,2\}$, and so $H^{d_j}(\mathcal{V}_j,\mathcal{L}^{\otimes i_j}) = 0$ by assumption. Thus for any $n\ge 0$, we indeed get that $H^1(X,\xi^{\otimes (n+1)}) = H^2(X,\xi^{\otimes (n+1)}) = 0$, and so we have $\Pic(C_{n+1})\cong \Pic(C_n)$. Thus as $\pr_0^*:\Pic(X)\to \Pic(C_0)$ is an isomorphism, it follows by induction that $\pr_n^*:\Pic(X)\to \Pic(C_n)$ is an isomorphism for all $n$, and so $\displaystyle\pr:\Pic(X)\to \varinjlim_n \Pic(C_n) = \Pic(\widehat{V}(\xi))$ is an isomorphism. Hence the completion map $\Cl(R)\to \Cl(\widehat{R})$ is indeed an isomorphism, completing the proof. \end{proof} So indeed, $\Cl(\mathcal{R})\to \Cl(\overline{R}_\infty)$ is an isomorphism. As noted above, this completes the proof of Theorem \ref{self-dual mod l}, and hence of Theorem \ref{self-dual}. \section{The construction of $M_\infty$}\label{sec:patching} From now on assume that $\overline{\rho}:G_F\to \GL_2(\mathbb{F})$ satisfies condition (\ref{TW conditions}) of Theorem \ref{Mult 2^k} (i.e. the ``Taylor-Wiles'' condition). The goal of this section is to construct a module $M_\infty$ over $R_\infty$ satisfying the conditions of Theorem \ref{self-dual}. We shall construct $M_\infty$ by applying the Taylor--Wiles--Kisin patching method \cite{Wiles,TaylorWiles,Kisin} to a natural system of modules over the rings $\mathbb{T}^D(K)$. For convenience we will follow the ``Ultrapatching'' construction introduced by Scholze in \cite{Scholze}. The primary advantage to doing this is that Scholze's construction is somewhat more ``natural'' than the classical construction, and thus it will be easier to show that $M_\infty$ satisfies additional properties (in our case, that it is self-dual). \subsection{Ultrapatching}\label{ssec:ultrapatching} In this subsection, we briefly recall Scholze's construction (while introducing our own notation). From now on, fix a nonprincipal ultrafilter $\mathfrak{F}$ on the natural numbers $\mathbb{N}$ (it is well known that such an $\mathfrak{F}$ must exist, provided we assume the axiom of choice). For convenience, we will say that a property $\mathcal{P}(n)$ holds for \emph{$\mathfrak{F}$-many $i$} if there is some $I\in \mathfrak{F}$ such that $\mathcal{P}(i)$ holds for all $i\in I$. For any collection of sets $\mathscr{A} =\{A_n\}_{n\ge 1}$, we define their \emph{ultraproduct} to be the quotient \[\uprod{\mathscr{A}} = \left(\prod_{n=1}^\infty A_n\right)/\sim\] where we define the equivalence relation $\sim$ by $(a_n)_n\sim (a_n')_n$ if $a_i = a_i'$ for $\mathfrak{F}$-many $i$. If the $A_n$'s are sets with an algebraic structure (eg. groups, rings, $R$-modules, $R$-algebras, etc.) then $\uprod{\mathscr{A}}$ naturally inherits the same structure. Also if each $A_n$ is a finite set, and the cardinalities of the $A_n$'s are bounded (this is the only situation we will consider in this paper), then $\uprod{\mathscr{A}}$ is also a finite set and there are bijections $\uprod{\mathscr{A}}\xrightarrow{\sim} A_i$ for $\mathfrak{F}$-many $i$. Moreover if the $A_n$'s are sets with an algebraic structure, such that there are only finitely many distinct isomorphism classes appearing in $\{A_n\}_{n\ge 1}$ (which happens automatically if the structure is defined by \emph{finitely} many operations, eg. groups, rings or $R$-modules or $R$-algebras over a \emph{finite} ring $R$) then these bijections may be taken to be isomorphisms. This is merely because our conditions imply that there is some $A$ such that $A\cong A_i$ for $\mathfrak{F}$-many $i$ and hence $\uprod{\mathscr{A}}$ is isomorphic to the ``constant'' ultraproduct $\uprod{\{A\}_{n\ge 1}}$ which is easily seen to be isomorphic to $A$, provided that $A$ is finite. Lastly, in the case when each $A_n$ is a module over a \emph{finite} local ring $R$, there is a simple algebraic description of $\uprod{\mathscr{A}}$. Specifically, the ring $\displaystyle\mathcal{R} = \prod_{n=1}^\infty R$ contains a unique maximal ideal $\mathfrak{Z}_\mathfrak{F}\in \Spec\mathcal{R}$ for which $\mathcal{R}_{\mathfrak{Z}_\mathfrak{F}}\cong R$ and $\displaystyle\left(\prod_{n=1}^\infty A_n \right)_{\mathfrak{Z}_\mathfrak{F}}\cong \uprod{\mathscr{A}}$ as $R$-modules. This shows that $\uprod{-}$ is a particularly well-behaved functor in our situation. In particular, it is exact. For the rest of this section, fix a power series ring $S_\infty = \O[[z_1,\ldots,z_t]]$ and consider the ideal $\mathfrak{n} = (z_1,\ldots,z_n)$. We can now make our main definitions: \begin{defn}\label{def:patching} Let $\mathscr{M} = \{M_n\}_{n\ge 1}$ be a sequence of finite type ${S_\infty}$-modules. \begin{itemize} \item We say that $\mathscr{M}$ is a \emph{weak patching system} if the $S_\infty$-ranks of the $M_n$'s are uniformly bounded. \item We say that $\mathscr{M}$ is a \emph{patching system} if it is a weak patching system, and for any open ideal $\a\subseteq S_\infty$, we have $\Ann_{S_\infty}(M_i)\subseteq \a$ for all but finitely many $n$. \item We say that $\mathscr{M}$ is \emph{free} if $M_n$ is free over ${S_\infty}/\Ann_{S_\infty}(M_n)$ for all but finitely many $n$. \end{itemize} Furthermore, assume that $\mathscr{R} = \{R_n\}_{n\ge1}$ is a sequence of finite type local ${S_\infty}$-\emph{algebras}. \begin{itemize} \item We say that $\mathscr{R} = \{R_n\}_{n\ge1}$ is a \emph{(weak) patching algebra}, if it is a (weak) patching system. \item If $M_n$ is an $R_n$-module (viewed as an ${S_\infty}$-module via the ${S_\infty}$-algebra structure on $R_n$) for all $n$ we say that $\mathscr{M} = \{M_n\}_{n\ge1}$ is a \emph{(weak) patching $\mathscr{R}$-module} if it is a (weak) patching system. \end{itemize} Now for any weak-patching system $\mathscr{M}$, we define its patched module to be the $S_\infty$-module \[\mathscr{P}(\mathscr{M}) = \varprojlim_{\a}\uprod{\mathscr{M}/\a},\] where the inverse limit is taken over all open ideals of $S_\infty$. If $\mathscr{R}$ is a (weak) patching algebra and $\mathscr{M}$ is a (weak) patching $\mathscr{R}$-module, then $\mathscr{P}(\mathscr{R})$ inherits a natural $S_\infty$-algebra structure, and $\mathscr{P}(\mathscr{M})$ inherits a natural $\mathscr{P}(\mathscr{R})$-module structure. \end{defn} In the above definition, the ultraproduct essentially plays the role of pigeonhole principal in the classical Taylor-Wiles construction, with the simplification that is is not necessary to explicitly define a ``patching datum'' before making the construction. Indeed, if one were to define patching data for the $M_n/\a$'s (essentially, imposing extra structure on each of the modules $M_n/\a$) then the machinery of ultraproducts would ensure that the patching data for $\uprod{\mathscr{M}/\a}$ would agree with that of $M_n/\a$ for infinitely many $n$. It is thus easy to see that our definition agrees with the classical construction (cf. \cite{Scholze}). Thus the standard results about patching (cf \cite{Kisin}) may be rephrased as follows: \begin{thm}\label{patching} Let $\mathscr{R}$ be a weak patching algebra, and let $\mathscr{M}$ be a \emph{free} patching $\mathscr{R}$-module. Then: \begin{enumerate} \item $\mathscr{P}(\mathscr{R})$ is a finite type $S_\infty$-algebra. $\mathscr{P}(\mathscr{M})$ is a finitely generated \emph{free} $S_\infty$-module. \item The structure map $S_\infty\to \mathscr{P}(\mathscr{R})$ (defining the $S_\infty$-algebra structure) is injective, and thus $\dim \mathscr{P}(\mathscr{R}) = \dim S_\infty$. \item The module $\mathscr{P}(\mathscr{M})$ is maximal Cohen--Macaulay over $\mathscr{P}(\mathscr{R})$. $(\lambda,z_1,\ldots,z_t)$ is a regular sequence for $\mathscr{P}(\mathscr{M})$. \item Let $\mathfrak{n} = (z_1,\ldots,z_t)\subseteq S_\infty$, as above. Let $R_0$ be a finite type local $\O$-algebra, and let $M_0$ be a finitely generated $R_0$-module. If, for each $n\ge 1$, there are isomorphisms $R_n/\mathfrak{n}\cong R_0$ of $\O$-algebras and $M_n/\mathfrak{n}\cong M_0$ of $R_n/\mathfrak{n}\cong R_0$-modules, then we have $\mathscr{P}(\mathscr{R})/\mathfrak{n}\cong R_0$ as $\O$-algebras and $\mathscr{P}(\mathscr{M})/\mathfrak{n}\cong M_0$ as $\mathscr{P}(\mathscr{R})/\mathfrak{n}\cong R_0$-modules. \end{enumerate} \end{thm} From the set up of Theorem \ref{patching} there is very little we can directly conclude about the ring $\mathscr{P}(\mathscr{R})$. However in practice one generally takes the rings $R_n$ to be quotients of a fixed ring $R_\infty$ (which in our case will be a result of Lemma \ref{TW primes}) of the same dimension as $S_\infty$ (and thus as $\mathscr{P}(\mathscr{R})$). Thus we define a \emph{cover} of a weak patching algebra $\mathscr{R} = \{R_n\}_{n\ge 1}$ to be a pair $(R_\infty,\{\varphi_n\}_{n\ge 1})$ (which we will denote by $R_\infty$ when the $\varphi_n$'s are clear from context), where $R_\infty$ is a complete, local, topologically finitely generated $\O$-algebra of Krull dimension $\dim S_\infty$ and $\varphi_n:R_\infty\to R_n$ is a surjective $\O$-algebra homomorphism for each $n$. We have the following: \begin{thm}\label{surjective cover} If $(R_\infty,\{\varphi_n\})$ is a cover of a weak patching algebra $\mathscr{R}$, then the $\varphi_n$'s induce a natural continuous surjection $\varphi_\infty:R_\infty\twoheadrightarrow \mathscr{P}(\mathscr{R})$. If $R_\infty$ is a domain then $\varphi_\infty$ is an isomorphism. \end{thm} \begin{proof} The $\varphi_n$'s induce a continuous map $\displaystyle\Phi = \prod_{n\ge 1}\varphi_n:R_\infty\to \prod_{n\ge 1}R_n$, and thus induce continuous maps \[\Phi_\a:R_\infty\xrightarrow{\Phi}\prod_{n\ge 1}R_n\twoheadrightarrow \prod_{n\ge 1}(R_n/\a)\twoheadrightarrow \uprod{\mathscr{R}/\a}\] for all open $\a\subseteq {S_\infty}$. Hence they indeed induce a continuous map \[\varphi_\infty=(\Phi_\a)_\a:R_\infty\to \varprojlim_\a \uprod{\mathscr{R}/\a} = \mathscr{P}(\mathscr{R}).\] Now as $R_\infty$ is complete and topologically finitely generated, it is compact, and thus $\varphi_\infty(R_\infty)\subseteq \mathscr{P}(\mathscr{R})$ is closed. So to show that $\varphi_\infty$ is surjective, it suffices to show that $\varphi_\infty(R_\infty)$ is dense, or equivalently that each $\Phi_\a$ is surjective. Now for any $n$ and any open $\a\subseteq$, $R_n/\a$ is a finite set with the structure of a continuous $R_\infty$ algebra (defined by the continuous surjection $\varphi_n:R_\infty\twoheadrightarrow R_n\twoheadrightarrow R_n/\a$) and the cardinalities of the $R_n/\a$'s are bounded. As noted above, this implies that $\uprod{\mathscr{R}/\a}$ also has the structure of an $R_\infty$-algebra (which is just the structure induced by $\Phi_\a$). As $R_\infty$ is \emph{topologically} finitely generated, there are only finitely many distinct isomorphism classes of $R_\infty$-algebras in $\{R_n/\a\}_{n\ge 1}$. By the above discussion of ultraproducts, this implies that $R_i/\a\cong \uprod{\mathscr{R}/\a}$ as $R_\infty$-algebras for $\mathfrak{F}$-many $i$. But now taking any such $i$, as the structure map $R_\infty\to R_i/\a$ is surjective, and so the structure map $\Phi_\a:R_\infty\to \uprod{\mathscr{R}/\a}$ is as well. The final claim simply follows by noting that if $R_\infty$ is a domain and $\varphi_\infty$ is \emph{not} injective, then $\mathscr{P}(\mathscr{R})\cong R_\infty/\ker\varphi_\infty$ would have Krull dimension strictly smaller than $R_\infty$, contradicting our assumption that $\dim R_\infty = \dim S_\infty = \dim\mathscr{P}(\mathscr{R})$. \end{proof} In order to construct the desired module $M_\infty$ over $R_\infty$ satisfying the conditions of Theorem \ref{self-dual}, we will construct a weak patching algebra $\mathscr{R}^\square$ covered by $R_\infty$, and a free patching $\mathscr{R}^\square$-module $\mathscr{M}^\square$, and then define $M_\infty = \mathscr{P}(\mathscr{M}^\square)$. \subsection{Spaces of automorphic forms}\label{ssec:M_psi} In this section, we will construct the spaces of automorphic forms $M(K)$ and $M_\psi(K)$ that will be used in Section \ref{ssec:Ms^sqaure} to construct the patching system $\mathscr{M}^\square$, producing $M_\infty$. Recall that $\overline{\rho}:G_F\to \GL_2(\overline{\F}_\ell)$ is assumed to be a Galois representation satisfying all of the conditions of Theorem \ref{Mult 2^k}. In particular $\mathcal{K}^D(\overline{\rho})\ne\ensuremath{\varnothing}$, so that $\overline{\rho} = \overline{\rho}_m$ for some $K\in\mathcal{K}^D(\overline{\rho})$ and some $m\subseteq \mathbb{T}^D(K)$. By enlarging $\O$ if necessary, assume that $\mathbb{T}^D(K)/m = \mathbb{F}$ and $\mathbb{F}$ contains all eigenvalues of $\overline{\rho}(\sigma)$ for all $\sigma\in G_F$. Since the results of Theorems \ref{Mult 2^k} and \ref{R=T} are known classically in the case when $F = \mathbb{Q}$ and $D = \GL_2$, we will exclude this case for convenience. Thus we will assume that $D(F)$ is a division algebra. For any $K\in \mathcal{K}^D(\overline{\rho})$, define $M(K)= S^D(K)_m^\vee$ if $D$ is totally definite and \[M(K)= \Hom_{R_{F,S}(\overline{\rho})[G_F]}(\rho^{\univ},S^D(K)_m^\vee)\] if $D$ is indefinite. Note that this definition depends only on the $\mathbb{T}^D(K)_m$-module structure of $S^D(K)^\vee_m$, and not on the specific choice of $S$ in $R_{F,S}(\overline{\rho})$. Give $M(K)$ its natural $\mathbb{T}^D(K)_m$-module structure. \begin{rem} The purpose of the definition of $M(K)$ in the indefinite case is to ``factor out'' the Galois action on $S^D(K)^\vee$. This construction was described by Carayol in \cite{Carayol2}. As in \cite{Carayol2} we have that the natural evaluation map $M(K)\otimes_{R_{F,S}(\overline{\rho})}\rho^{\univ}\to S^D(K)_m^\vee$ is an isomorphism, and so $S^D(K)_m^\vee\cong M(K)^{\oplus 2}$ as $\mathbb{T}^D(K)_m$-modules. If we did not do this, and only worked with $S^D(K)^\vee_m$, then the module $M_\infty$ we will construct would have generic rank 2 instead of generic rank 1, and so we would not be able to directly apply Theorem \ref{self-dual}. \end{rem} Note that it follows from the definitions that $\dim_{\mathbb{F}}M(K)/m = \nu_{\overline{\rho}}(K)$ for all $K$. For technical reasons (related to the proof of Lemma \ref{TW primes}) we cannot directly apply the patching construction to the modules $M(K)$. Instead, it will be necessary to introduce ``fixed-determinant'' versions of these spaces, $M_\psi(K)$. We now make the following definition: If $D$ is definite, a level $K\subseteq D^\times(\mathbb{A}_{F,f})$ is \emph{sufficiently small} if for all $t\in D^\times(\mathbb{A}_{F,f})$ we have $K\mathbb{A}_{F,f}^\times\cap (t^{-1}D^\times(F)t) = F^\times$. This is condition (2.1.2) in \cite{KisinFM}. If $D$ is indefinite, a level $K\subseteq D^\times(\mathbb{A}_{F,f})$ is \emph{sufficiently small} if for all $t\in D^\times(\mathbb{A}_{F,f})$ the action of $(K\mathbb{A}_{F,f}^\times\cap (t^{-1}D^\times(F)t)/F^\times$ on $\mathcal{H}$ is free. Note that this implies that the Shimura variety $X_K$ does not contain any elliptic points. The importance of considering sufficiently small levels is the following standard lemma: \begin{lemma}\label{sufficiently small} Let $K\subseteq D^\times(\mathbb{A}_{F,f})$ be a level, and let $K'\unlhd K$ be level which is a normal subgroup of $K$. Consider a finite subgroup $G \le K\mathbb{A}_{F,f}^\times/K'F^\times$, and let $\O[G]$ be its group ring and $\a_G\subseteq \O[G]$ be the augmentation ideal. Also let $m$ be a non-Eisenstein maximal ideal of $\mathbb{T}^D_K$. If $K$ is sufficiently small then: \begin{enumerate} \item If $D$ is totally definite (resp. indefinite) then $G$ acts freely on the double quotient $D^\times(F)\backslash D^\times(\mathbb{A}_{F,f})/K'$ (resp. the Riemann surface $X^D(K') = D^\times(F)\backslash \left(D^\times(\mathbb{A}_{F,f})\times \mathcal{H}\right)/K'$) by right multiplication. \item $S^D(K')_m^\vee$ is a finite projective $\O[G]$-module. \item If $G = KF^\times/K'F^\times$ then the operator $\displaystyle\sum_{g\in G}g$ induces an isomorphism $S^D(K')_m^\vee/\a_GS^D(K')_m^\vee\xrightarrow{\sim} S^D(K)_m^\vee$. \end{enumerate} \end{lemma} \begin{proof} The definite case essentially follows from the argument of \cite[Lemma (2.1.4)]{KisinFM}. In the indefinite case, (1) is true by definition, and the argument of \cite[Lemme 3.6.2]{BD} shows that (2) and (3) follow from (1). \end{proof} This lemma will allow us to construct the desired free patching system $\mathscr{M}^\square$. However, in order to use this lemma it will be necessary to first restrict our attention to sufficiently small levels $K$. First by the conditions on $\overline{\rho}$ and \cite[Lemma 4.11]{DDT} we may pick a prime $w\not\in \Sigma^D_\ell$ satisfying \begin{itemize} \item $\Nm(w)\not\equiv 1\pmod{\ell}$ \item The ratio of the eigenvalues of $\overline{\rho}(\Frob_w)$ is not equal to $\Nm(w)^{\pm 1}$ in $\overline{\F}_\ell^\times$. \item For any nontrivial root of unity $\zeta$ for which $[F(\zeta):F]\le 2$, $\zeta+\zeta^{-1} \not\equiv 2\pmod{w}$. \end{itemize} Define \begin{align*} U_w &= \left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \GL_2(\O_{F,w})\middle| \begin{pmatrix}a&b\\c&d\end{pmatrix} \equiv\begin{pmatrix}1&*\\0&1\end{pmatrix}\pmod{w}\right\}\subseteq \GL_2(F_w)\\ U_w^{-} &= \left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \GL_2(\O_{F,w})\middle| \begin{pmatrix}a&b\\c&d\end{pmatrix} \equiv\begin{pmatrix}*&*\\0&*\end{pmatrix}\pmod{w}\right\}\subseteq \GL_2(F_w)\\ \end{align*} Let $K_0$ (resp. $K_0^-$) be the preimage of $U_w$ (resp. $U_w^-$)under the map $K^{\min}\hookrightarrow D^{\times}(\mathbb{A}_{F,f})\twoheadrightarrow \GL_2(F_w)$. We then have the following: \begin{lemma}\label{lem:aux prime} $K_0$ is sufficiently small, and we have compatible isomorphisms $\mathbb{T}^D(K^{\min})_m\cong \mathbb{T}^D(K_0)_m$, $S^D(K^{\min})_m\cong S^D(K_0)_m$ and $M(K^{\min})\cong M(K_0)$. \end{lemma} \begin{proof} The fact that $K_0$ is sufficiently small follows easily from last hypothesis on $w$ (cf \cite[(2.1.1)]{Kisin}). As in \cite[Section 4.3]{DDT}, the first two conditions on $w$ imply that $w$ is not a level-raising prime for $\overline{\rho}$ and so we obtain natural isomorphisms $\mathbb{T}^D(K^{\min})_m\cong \mathbb{T}^D(K_0^{-})_m\cong \mathbb{T}^D(K_0)_m$. By the definition of $M(K)$, the isomorphism $M(K^{\min})\cong M(K_0)$ will follow from $S^D(K^{\min})_m\cong S^D(K_0)_m$, so it suffices to prove this isomorphism. It follows from the argument of \cite[Lemma 2.2]{TaylorMero}\footnote{Note that this proof does not rely on Taylor's assumption that $\Nm(w)\equiv 1$, only on the assumption that the ratio of the eigenvalues of $\overline{\rho}(\Frob_w)$ is not $\Nm(w)^{\pm1}$. Also by the definition of $U_w^-$, no lift of $\overline{\rho}$ occurring in $S^D(K_0^-)$ can have determinant ramified at $w$, and so the fact that we have not yet fixed determinants does not affect the argument.} that there is an isomorphism $S^D(K^{\min})_m\cong S^D(K_0^-)_m$. Now the argument of \cite[Lemma 4.11]{BDJ} implies that the map $S^D(K_0^-)_m\to S^D(K_0)^{K_0/K_0^-}_m$ is an isomorphism (as the assumptions that $\ell>2$ and $\overline{\rho}|_{G_{F(\zeta_\ell)}}$ is absolutely irreducible imply that $\overline{\rho}$ is not ``badly dihedral'', in the sense defined in that argument). Finally, as $K_0/K_0^-\cong ((\O_F/w)^\times)^2$ has prime to $\ell$ order, we get that $S^D(K_0)^{K_0/K_0^-}_m\cong S^D(K_0)_m$, giving the desired isomorphism. \end{proof} It now follows that $\nu_{\overline{\rho}}(K^{\min}) = \nu_{\overline{\rho}}(K_0)$. Also be definition, $K_0$ and $K^{\min}$ agree at all places besides $w$, and hence at all places in $\Sigma^D_\ell$. Thus Lemma \ref{R->T} gives a surjection $R^{D,\psi}_{F,S}(\overline{\rho})\twoheadrightarrow \mathbb{T}^{D}(K_0)_m$ (note that we are using condition (\ref{vexing}) of Theorem \ref{Mult 2^k} here). We will now restrict our attention to levels contained in $K_0$. For any level $K\subseteq K_0$, let $C_K = F^\times\backslash \mathbb{A}_{F,f}^\times/(K\cap \mathbb{A}_{F,f}^\times)$ denote the image of $\mathbb{A}_{F,f}^\times$ in the double quotient $D^\times(F)\backslash D^\times(\mathbb{A}_{F,f})/K$. Note that this is a finite abelian group. For any finite place $v$ of $F$, let $\overline{\varpi}_v$ denote the image of the uniformizer $\varpi_v\in F_v^\times\subseteq\mathbb{A}_{F,f}^\times$ in $C_K$. By the definition of $S_v$, we see that $\overline{\varpi}_v$ acts on $S^D(K)^\vee$ as $S_v$ for all $v\not\in S$, and so we may identify $\O[C_K]$ with a subring of $\mathbb{T}^D(K)$. Specifically, it is the $\O$-subalgebra generated by the Hecke operators $S_v$ for $v\not\in S$. Now the action of $C_K$ on $D^\times\backslash D^\times(\mathbb{A}_{F,f})/K$ induces an action of $\O[C_K]$ on $M(K)$. By Lemma \ref{sufficiently small} (with $K'=K$ and $G=C_K\hookrightarrow K\mathbb{A}_{F,f}^\times/K$) $S^D(K)_m^\vee$ is a finite projective $\O[C_K]$-module. Let $m' = m\cap \O[C_K]$, so that $m'$ is a maximal ideal of $\O[C_K]$. It follows that $S^D(K)^\vee_m$ is a finite free $\O[C_K]_{m'}$-module. Let $C_{K,\ell}\le C_K$ be the Sylow $\ell$-subgroup. Since $C_K$ is abelian, we have $C_K\cong C_{K,\ell}\times (C_K/C_{K,\ell})$ and so $\O[C_K] \cong \O[C_{K,\ell}]\otimes_\O\O[C_K/C_{K,\ell}]$. Now as $C_K/C_{K,\ell}$ has prime to $\ell$ order, by enlarging $\O$ if necessary, we may assume that $\O[C_K/C_{K,\ell}]\cong \O^{\oplus \#(C_K/C_{K,\ell})}$ as an $\O$-algebra, and so $\O[C_K] \cong \O[C_{K,\ell}]^{\oplus \#(C_K/C_{K,\ell})}$. But now as $\O[C_{K,\ell}]$ is a complete local $\O$-algebra (as $\O[\mathbb{Z}/\ell^n\mathbb{Z}]\cong \O[x]/((1+x)^{\ell^n}-1)$ is for any $n$, and $C_{K,\ell}$ is a finite abelian $\ell$-group), it follows that $\O[C_K]_{m'}\cong \O[C_{K,\ell}]$ for any maximal ideal $m'$. Hence there is an embedding $\O[C_{K,\ell}]\hookrightarrow \mathbb{T}^D(K)_m$ which makes $S^D(K)^\vee_m$ into a finite projective (and hence free) $\O[C_{K,\ell}]$-module. It follows that $M(K)$ is also a finite free $\O[C_{K,\ell}]$-module. Indeed, this is simply by definition in the case when $D$ is definite. If $D$ is indefinite, this follows from the fact that $M(K)^{\oplus 2} \cong S^D(K)^\vee_m$ is free, and direct summands of free $\O[C_{K,\ell}]$-modules are projective and hence free. Now fix a character $\psi:G_F\to\O^\times$ for which $m$ is in the support of $\mathbb{T}^D_\psi(K^{\min})$. For any level $K\subseteq K_0$, define an ideal $\mathfrak{I}_\psi = \left(\Nm(v)[\overline{\varpi}_v]-\psi(\Frob_v)\middle| v\not\in S \right)\subseteq \O[C_{K,\ell}]$. As $m$ is also in the support of $\mathbb{T}^D_\psi(K)$, it follows that $\mathfrak{I}_\psi$ contained in the kernel of some map $\mathbb{T}^D_\psi(K)\to \O$ (corresponding to some lift of $\rho:G_F\to \GL_2(\O)$ of $\overline{\rho}$ which is modular of level $K$ and has $\det \rho = \psi$), and so we can deduce that $\O[C_{K,\ell}]/\mathfrak{I}_\psi\cong \O$. We may now define $M_\psi(K)=M(K)/\mathfrak{I}_\psi M(K)$. It follows that $M_\psi(K)$ is a finite free $\O$-module. Moreover, by definition it follows that $\mathbb{T}^D_\psi(K)_m$ is exactly the image of $\mathbb{T}^D(K)_m$ in $\End_\O(M_\psi(K))$, and $M_\psi(K)= M(K)\otimes_{\mathbb{T}^D(K)_m}\mathbb{T}^D_\psi(K)_m$. It is necessary to consider the modules $M_\psi(K)$ instead of $M(K)$, because the patching argument requires us to work with fixed-determinant deformation rings. Fortunately, as $\mathfrak{I}_\psi\subseteq m$, we get $\dim_\mathbb{F} M_\psi(K_0)/m = \dim_\mathbb{F} M(K_0)/m=\nu_{\overline{\rho}}(K_0) = \nu_{\overline{\rho}}(K^{\min})$, and so considering the $M_\psi(K)$'s instead of the $M(K)$'s will still be sufficient to prove Theorem \ref{Mult 2^k}. \subsection{A Patching System Producing $M_\infty$}\label{ssec:Ms^sqaure} For the rest of this paper, we will take the ring $S_\infty$ from the Section \ref{ssec:ultrapatching} to be $\O[[y_1,\ldots,y_r,w_1,\ldots,w_j]]$, where $r$ is an in Lemma \ref{TW primes} and $j = 4|\Sigma_\ell^D| - 1$ is as in Section \ref{ssec:global def}, and let $\mathfrak{n} = (y_1,\ldots,y_r,w_1,\ldots,w_j)$ as before. Note that $\dim S_\infty = r+j+1 = \dim R_\infty$ by Lemma \ref{TW primes}. We will construct a weak patching algebra $\mathscr{R}^\square$ covered by $R_\infty$ using the deformation rings $R^{\square,D,\psi}_{F,S\cup Q_n}(\overline{\rho})$, and construct a free patching $\mathscr{R}^\square$-module $\mathscr{M}^\square$ using the spaces $M_\psi(K)$ constructed above. We then take $M_\infty = \mathscr{P}(\mathscr{M}^\square)$. By Theorems \ref{patching} and \ref{surjective cover} it will then follow that $M_\infty$ is maximal Cohen--Macaulay over $R_\infty$. In Section \ref{ssec:prop M_infty}, we will show that $M_\infty$ satisfies the remaining conditions of Theorem \ref{self-dual}. From now on, fix $S=\Sigma^D_\ell\cup\{w\}$, where $w$ is the prime chosen in Section \ref{ssec:M_psi} above, and fix a collection of \emph{sets} of primes $\mathcal{Q} = \{Q_n\}_{n\ge 1}$ satisfying the conclusion of Lemma \ref{TW primes}. For any $n$, let $\Delta_n$ be the maximal $\ell$-power quotient of $\displaystyle\prod_{v\in Q_n}(\O_F/v)^\times$. Consider the ring $\Lambda_n=\O[\Delta_n]$, and note that: \[\Lambda_n\cong\frac{\O[[y_1,\ldots,y_r]]}{\left((1+y_1)^{\ell^{e(n,1)}}-1,\ldots, (1+y_r)^{\ell^{e(n,r)}}-1\right)}\] where $\ell^{e(n,i)}$ is the $\ell$-part of $\Nm(v)-1 = \# (\O_F/v)^\times$, so that $e(n,i)\ge n$ by assumption. Let $\a_n = (y_1,\ldots,y_r)\subseteq \Lambda_n$ be the augmentation ideal. Also let $\displaystyle H_n = \ker\left(\prod_{v\in Q_n}(\O_F/v)^\times\twoheadrightarrow\Delta_n\right)$. For any finite place $v$ of $F$, consider the compact open subgroups of $\GL_2(\O_{F,v})$: \begin{align*} \Gamma_0(v) &=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \GL_2(\O_{F,v})\middle| \begin{pmatrix}a&b\\c&d\end{pmatrix} \equiv \begin{pmatrix}*&*\\0&*\end{pmatrix}\pmod{v}\right\}\subseteq \GL_2(\O_{F,v})\\ \Gamma_1(v) &=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \Gamma_0(v)\middle| a \equiv d\pmod{v}\right\}\subseteq \Gamma_0(v)\subseteq\GL_2(\O_{F,v}). \end{align*} Notice that $\Gamma_1(v)\unlhd \Gamma_0(v)$ and we have group isomorphisms \[\Gamma_0(v)/\Gamma_1(v)\cong \Gamma_0(v)F^\times/\Gamma_1(v)F^\times\xrightarrow{\sim} (\O_F/v)^\times\] given by $\begin{pmatrix}a&b\\c&d\end{pmatrix}\mapsto ad^{-1}\pmod{v}$. Now let $\displaystyle\Gamma_H(Q_n)\subseteq \prod_{v\in Q_n}\Gamma_0(v)$ be the preimage of $\displaystyle H_n\subseteq \prod_{v\in Q_n}(\O_F/v)^\times$ under the map \[\prod_{v\in Q_n}\Gamma_0(v)\twoheadrightarrow \prod_{v\in Q_n}\Gamma_0(v)/\Gamma_1(v) \cong \displaystyle\prod_{v\in Q_n}(\O_F/v)^\times\] finally let $K_n\subseteq K_0$ be the preimage of $\Gamma_H(Q_n)$ under \[K_0\hookrightarrow \prod_{v\subseteq \O_F}D^\times(\O_{F,v})\twoheadrightarrow \prod_{v\in Q_n}D^\times(\O_{F,v}).\] Now for any $n\ge 1$, any $v\in Q_n$ and any $\delta\in (\O_F/v)^\times$, consider the double coset operators $U_v,\langle \delta\rangle_v:S^D(K_n)\to S^D(K_n)$ defined by \begin{align*} U_v &= \left[K\begin{pmatrix}\varpi_v&0\\0&1\end{pmatrix}K\right],& \langle\delta\rangle_v &= \left[K\begin{pmatrix}d&0\\0&1\end{pmatrix}K\right], \end{align*} where $d\in \O_F$ is a lift of $\delta\in (\O_F/v)^\times$. Note that $\langle\delta\rangle_v$ does not depend on the choice of $d$. In fact, if $\delta,\delta'\in (\O_F/v)^\times$ have the same image under $(\O_F/v)^\times \to \prod_{v\in Q_n}(\O_F/v)^\times\twoheadrightarrow \Delta_n$, then $\langle\delta\rangle_v = \langle\delta'\rangle_v$. Define \[\overline{\T}^D(K_n) = \mathbb{T}^D(K_n)\bigg[U_v,\langle \delta\rangle_v\bigg|v\in Q_n,\delta\in(\O_F/v)^\times \bigg] \subseteq \End_\O(S^D(K_n)),\] and note that this is a commutative $\O$-algebra extending $\mathbb{T}^D(K_n)$, which is finite free as an $\O$-module. Also for convenience set $\overline{\T}^D(K_0) = \mathbb{T}^D(K_0)$. Note that the double coset operators $U_v$ and $\langle \delta\rangle_v$ commute with the action of $\mathbb{T}^D(K_n)$ and (in the case when $D$ is indefinite) $G_F$ on $S^D(K_n)$, and thus they descend to maps $U_v,\langle \delta\rangle_v:M_\psi(K_n)\to M_\psi(K_n)$. Let $\overline{\T}^D_\psi(K_n)_m$ denote the image of $\overline{\T}^D(K_n)_m$ in $\End_\O(M_\psi(K_n))$. As in \cite[(3.4.5)]{Kisin}, $\overline{\T}^D_\psi(K_n)_m$ has $2^{\#Q_n}$ different maximal ideals (corresponding to the different possible choices of eigenvalue for $U_v$ for $v\in Q_n$). Fix any such maximal ideal $m_{Q_n}\subseteq\overline{\T}^D_\psi(K_n)_m$. Let $\overline{\T}_n=(\overline{\T}^D_\psi(K_n)_m)_{m_{Q_n}}$ and $M_n = M_\psi(K_n)_{m_{Q_n}}$ and $R_n = R^{\psi}_{F,S\cup Q_n}(\overline{\rho})$. Also define $M_0 = M_\psi(K_0)$, $\overline{\T}_0 = \mathbb{T}_0= \mathbb{T}^D_\psi(K_0)_m$ and $R_0 = R^{\psi}_{F,S}(\overline{\rho})$ (note that $M_n$, $\mathbb{T}_n$ and $R_n$ all have fixed determinant $\psi$, but we are suppressing this in our notation). We now have the following standard result (cf \cite{deShalit,DDT,TaylorMero,Kisin}, and also Lemma \ref{sufficiently small} above): \begin{prop}\label{aug ideal} For any $n\ge 1$, there is a surjective map $R_n\twoheadrightarrow \overline{\T}_n$ giving $M_n$ the structure of a $R_n$-module. There exists an embedding $\Lambda_n\hookrightarrow R_n$ under which $M_n$ is a finite rank free $\Lambda_n$-module. Moreover, we have $R_n/\a_n\cong R_0$ and $M_n/\a_n\cong M_0$ (so in particular, $\rank_{\Lambda_n}M_n = \rank_\O M_0$) \end{prop} Now let $R_n^\square = R^{\square,\psi}_{F,S\cup Q_n}(\overline{\rho})$, and recall from Section \ref{ssec:global def} that $R_n^\square = R_n[[w_1,\ldots,w_j]]$ for some integer $j$. Using this, we may define framed versions of $\overline{\T}_n$ and $M_n$. Namely \begin{align*} \overline{\T}^{\square}_n&= R^{\square}_{n}\otimes_{R_{n}}\overline{\T}_n\cong \overline{\T}_n[[w_1,\ldots,w_j]]\\ M^{\square}_n&=R^{\square}_{n}\otimes_{R_n}M_n\cong M_n[[w_1,\ldots,w_j]] \end{align*} so that $M^{\square}_n$ inherits a natural $\overline{\T}^{\square}_n$-module structure, and we still have a surjective map $R^{\square}_{n}\twoheadrightarrow \overline{\T}^{\square}_n$ (and so $M_n^\square$ inherits a $R_n^\square$-module structure). Note that the ring structure of $\overline{\T}^{\square}_n$ and the $\overline{\T}^{\square}_n$-module structure of $M^\square_n$ do not depend on the choice of the set $S$, and so we may define this without reference to a specific $S$. Also for any $n$, consider the ring $\Lambda_n^\square = \Lambda_n[[w_1,\ldots,w_j]] = \O[\Delta_n][[w_1,\ldots,w_j]]$, which we will view as a quotient of the ring $S_\infty = \O[[y_1,\ldots,y_r,w_1,\ldots,w_j]]$ from above. Rewriting Proposition \ref{aug ideal} in terms of the framed versions of $R_n$ and $M_n$, we get: \begin{prop}\label{aug ideal framed} There exists an embedding $\Lambda_n^\square\hookrightarrow R_n^\square$ under which $M_n^\square$ is a finite rank free $\Lambda_n^\square$-module. Moreover, we have $R_n^\square/\mathfrak{n}\cong R_0$ and $M_n^\square/\mathfrak{n}\cong M_0$ (so in particular, $\rank_{S_\infty}M_n^\square = \rank_{\Lambda_n^\square}M_n^\square = \rank_\O M_0$). \end{prop} So in particular, the rings $R_n^\square$ are $S_\infty$-algebras and the modules $M_n^\square$ are $S_\infty$-modules. Lastly, for any $n\ge 0$, define $R^{\square,D}_n = R^{\square,D,\psi}_{F,S\cup Q_n}(\overline{\rho})$ and $R_n^D = R^{D,\psi}_{F,S\cup Q_n}(\overline{\rho})$ (where we take $Q_0=\ensuremath{\varnothing}$). Note that the actions of $R_n^\square$ and $R_0$ on $M_n^\square$ and $M_0$ factor through $R^{\square,D}_n$ and $R_0^D$, respectively. Temporarily writing $R^{\square,\psi}_{\Sigma,\mathfrak{D},\ell} = R_{\loc}^{\psi}/J$, so that $R^{\square,D}_n = R^{\square}_n/J$ for all $n\ge 0$, we also see that \[R^{\square,D}_n/\mathfrak{n}\cong (R^\square_n/\mathfrak{n})/J\cong (R^\square_0/(w_1,\ldots,w_j))/J\cong (R^\square_0/J)/(w_1,\ldots,w_j) \cong R^{\square,D}_0/(w_1,\ldots,w_j)\cong R^D_0.\] Now we claim that the collections $\mathscr{R}^{\square,D}=\{R_n^{\square,D}\}_{n\ge 1}$ and $\mathscr{M}^\square = \{M_n^\square\}_{n\ge 1}$ satisfy the necessary conditions to apply Theorem \ref{patching}. Specifically we have: \begin{lemma}\label{M is patching} $\mathscr{R}^{\square,D}$ is a weak patching algebra over $R_0^D$ and $\mathscr{M}^\square$ is a free patching $\mathscr{R}^{\square,D}$-module over $M_0$. Moreover, the surjections $R_\infty\twoheadrightarrow R_n^{\square,D}$ from Theorem \ref{TW primes} induce an isomorphism $R_\infty\cong \mathscr{P}(\mathscr{R}^{\square,D})$. \end{lemma} \begin{proof} Let $S_\infty' = \O[[y_1,\ldots,y_r]]\subseteq S_\infty$ with ideal $\mathfrak{n}' = (y_1,\ldots,y_r) = \mathfrak{n}\cap S_\infty'$, so that $R_n$ is a $S_\infty'$-algebra and $M_n$ is a $S_\infty'$-module. By definition, we have $R_n^\square = R_n\otimes_{S_\infty'}S_\infty$ and $M_n^\square = M_n\otimes_{S_\infty'}S_\infty$. Thus by Proposition \ref{aug ideal}, \begin{align*} \rank_{S_\infty}R_n^\square &= \rank_{S_\infty'}R_n = \rank_\O R_0\\ \rank_{S_\infty}M_n^\square&= \rank_{S_\infty'}M_n = \rank_\O M_0. \end{align*} Also as $R^{\square,D}_n$ is a quotient of $R^\square_n$, we get that $\rank_{S_\infty}R^{\square,D}_n\le \rank_{S_\infty}R_n^\square = \rank_\O R_0$. Thus the $S_\infty$-ranks of the $R_n^\square$'s and $M_n^\square$'s are bounded so $\mathscr{R}^{\square,D}$ is a weak patching algebra and $\mathscr{M}^\square$ is a weak patching $\mathscr{R}$-module. Also as noted above $R_n^{\square,D}/\mathfrak{n}\cong R_0^D$ and $M_n^\square/\mathfrak{n}\cong M_0$, so $\mathscr{R}^{\square,D}$ and $\mathscr{M}^\square$ are over $R_0^D$ and $M_0$, respectively. Now by Proposition \ref{aug ideal framed}, for any $n$ we have, \[I_n=\Ann_{S_\infty} M_n^\square = \Ann_{S_\infty}\Lambda_n^\square = \left(\Ann_{S_\infty'}M_n\right) = \left((1+y_1)^{\ell^{e(n,1)}},\ldots,(1+y_r)^{\ell^{e(n,r)}}\right)\subseteq S_\infty\] (where as above, $e(n,i)\ge n$ for each $i$) and $M_n^\square$ is free over $S_\infty/\Ann_{S_\infty}M_n^\square = \Lambda_n^\square$. It remains to show that $\mathscr{M}^\square$ is a patching system, i.e. that for any open $\a\subseteq S_\infty$, $I_n = \Ann_{S_\infty}M_n^\square\subseteq \a$ for all but finitely many $n$. But as ${S_\infty}/\a$ is finite, and the group $1+m_{S_\infty}$ is pro-$\ell$, the group $(1+m_{S_\infty})/\a = \im(1+m_{S_\infty}\hookrightarrow {S_\infty}\twoheadrightarrow {S_\infty}/\a)$ is a finite $\ell$-group. Since $1+y_i\in 1+m_{S_\infty}$ for all $i$, there is an integer $K\ge 0$ such that $(1+y_i)^{\ell^K}\equiv 1 \pmod{\a}$ for all $i=1,\ldots,d'$. Then for any $n\ge K$, $e(n,i)\ge n\ge K$ for all $i$, and so indeed $I_n\subseteq \a$ by definition. The final statement follows from Lemma \ref{surjective cover} after noting that $R_\infty$ is a domain (by Theorem \ref{R^st} and the discussion following it) and $\dim R_\infty = \dim S_\infty$. \end{proof} Thus we may define $M_\infty = \mathscr{P}(\mathscr{M}^\square)$. By Theorem \ref{patching} and Lemma \ref{M is patching} we get that $M_\infty$ is maximal Cohen--Macaulay over $\mathscr{P}(\mathscr{R}^\square)\cong R_\infty$ and \[\dim_\mathbb{F} M_\infty/m_{R_\infty} = \dim_\mathbb{F} (M_\infty/\mathfrak{n})/m_{R_\infty} = \dim_\mathbb{F} M_0/m_{R_0} = \nu_{\overline{\rho}}(K^{\min}).\] \subsection{The Properties of $M_\infty$}\label{ssec:prop M_infty} We shall now show that $M_\infty$ satisfies the remaining conditions of Theorem \ref{self-dual}. We start by showing $\genrank_{R_\infty}M_\infty = 1$. First, the fact that $R_\infty[1/\lambda]$ is formally smooth implies that: \begin{lemma}\label{gen free} $M_0[1/\lambda]$ is free of rank $1$ over $R_0^D[1/\lambda]$. In particular, the natural map $R_0^D[1/\lambda]\twoheadrightarrow \mathbb{T}_0[1/\lambda]$ is an isomorphism. \end{lemma} \begin{proof} As $M_\infty$ is maximal Cohen--Macaulay over $R_\infty$, $M_\infty[1/\lambda]$ is also maximal Cohen--Macaulay over $R_\infty[1/\lambda]$. Since $R_\infty[1/\lambda]$ is a formally smooth domain, this implies that $M_\infty[1/\lambda]$ is \emph{locally} free over $R_\infty[1/\lambda]$ of some rank, $d$. Now quotienting by $\mathfrak{n}$ we get that $M_\infty[1/\lambda]/\mathfrak{n} \cong M_0[1/\lambda]$ is locally free over $R_\infty[1/\lambda]/\mathfrak{n}\cong R_0^D[1/\lambda]$ of constant rank $d$. But now $R_0^D[1/\lambda]$ is a finite dimensional commutative $E$-algebra, and hence is a product of local rings. Thus as $M_0[1/\lambda]$ is locally free of rank $d$, it must actually be free of rank $d$. But now by classical generic strong multiplicity $1$ results we get that $M_0[1/\lambda]$ is free of rank $1$ over $\mathbb{T}_0[1/\lambda]$ (recalling that $\mathbb{T}_0 = \mathbb{T}^D_\psi(K_0)\cong \mathbb{T}^D_\psi(K^{\min})$ and $M_0 = M_\psi(K_0)\cong M_\psi(K^{\min})$), which is a quotient of $R_0^D[1/\lambda]$. Thus $d=1$ and hence $M_0[1/\lambda]\cong R_0^D[1/\lambda]$. Lastly, as the action of $R_0^D[1/\lambda]$ on $M_0[1/\lambda]$ is free \emph{and} factors through $R_0^D[1/\lambda]\twoheadrightarrow \mathbb{T}_0[1/\lambda]$, we get that $R_0^D[1/\lambda]\twoheadrightarrow \mathbb{T}_0[1/\lambda]$ is an isomorphism. \end{proof} It is now straightforward to compute $\genrank_{R_\infty}M_\infty$. Let $K(R_\infty)$ and $K(S_\infty)$ be the fraction fields of $R_\infty$ and $S_\infty$, respectively. As $R_\infty$ is a finite type free $S_\infty$-algebra, $K(R_\infty)$ is a finite extension of $K(S_\infty)$. It follows that \[M_\infty\otimes_{R_\infty}K(R_\infty) \cong M_\infty\otimes_{S_\infty}K(S_\infty).\] Since $R_\infty$ and $M_\infty$ are both finite free $S_\infty$-modules, we thus get \begin{align*} \genrank_{R_\infty}M_\infty &= \dim_{K(R_\infty)}\left[M_\infty\otimes_{R_\infty}K(R_\infty)\right] = \dim_{K(R_\infty)}\left[M_\infty\otimes_{S_\infty}K(S_\infty)\right]\\ &= \frac{\dim_{K(S_\infty)}\left[M_\infty\otimes_{S_\infty}K(S_\infty)\right]}{\dim_{K(S_\infty)}K(R_\infty)}= \frac{\rank_{S_\infty}M_\infty}{\rank_{S_\infty}R_\infty} \end{align*} But now for any finite free $S_\infty$ module $A$ we have \[\rank_{S_\infty}A = \rank_{S_\infty/\mathfrak{n}}A/\mathfrak{n} = \rank_{\O}A/\mathfrak{n} = \dim_E(A/\mathfrak{n})[1/\lambda]\] and so the fact that $M_\infty[1/\lambda]/\mathfrak{n}\cong M_0[1/\lambda]\cong R^D_0[1/\lambda]\cong R_\infty[1/\lambda]/\mathfrak{n}$ implies that $\rank_{S_\infty}M_\infty = \rank_{S_\infty}R_\infty$, giving $\genrank_{R_\infty}M_\infty = 1$. \begin{rem} It is worth mentioning here that Shotton's computations of local deformation rings \cite{Shotton} (particularly the fact that $R_\infty$ is Cohen--Macaulay, by Theorem \ref{R^st}) actually imply an integral ``$R=\mathbb{T}$'' theorem (see for instance, Section 5 of \cite{Snowden}). Specifically one considers the surjection $f:R_0^D\twoheadrightarrow \mathbb{T}_0$. As shown in Lemma \ref{gen free} (see also \cite{Kisin}), $f$ is an isomorphism after inverting $\lambda$ (i.e. $R_0^D[1/\lambda]\cong \mathbb{T}_0[1/\lambda]$). This means that $\ker f\subseteq R_0^D$ is a torsion $\O$-module. But now $R_\infty$ is Cohen--Macaulay, and $M_\infty$ is a maximal Cohen--Macaulay module over $R_\infty$. Since $(\lambda,y_1,\ldots,y_r,w_1,\ldots,w_j)$ is a regular sequence for $M_\infty$ (by Theorem \ref{patching}(3)) it follows that it is also a regular sequence for $R_\infty$. Thus $R_0^D\cong R_\infty/\mathfrak{n} = R_\infty/(y_1,\ldots,y_r,w_1,\ldots,w_j)$ is Cohen--Macaulay and $\lambda$ is a regular element on $R_0^D$ (i.e. a non zero divisor). But this implies that $R_0^D$ is $\lambda$-torsion free, giving that $\ker f = 0$, so indeed $f:R_0^D\twoheadrightarrow \mathbb{T}_0$ is an isomorphism. \end{rem} It remains to show the following: \begin{prop}\label{M*} $M_\infty\cong M_\infty^*= \Hom_{R_\infty}(M_\infty,\omega_{R_\infty})$ as $R_\infty$-modules. \end{prop} This will ultimately follow from the fact that the modules $M(K)$ were naturally self-dual: \begin{lemma}\label{pairing} For any $n\ge 1$, there is a $\overline{\T}^D(K_n)_m$-equivariant perfect pairing $M(K_n)\times M(K_n)\to \O$. This induces a $\overline{\T}^D_\psi(K_n)_m$-equivariant perfect pairing $M_\psi(K_n)\times M_\psi(K_n)\to \O$, and thus a $\overline{\T}_n$-equivariant perfect pairing $M_n\times M_n\to \O$. \end{lemma} \begin{proof} First note that there is a $\overline{\T}^D(K_n)$-equivariant perfect pairing $S^D(K_n)\times S^D(K_n) \to \O$. In the totally definite case, this is the monodromy pairing, in the indefinite case it is Poincar\'e duality (although this must be modified slightly in order to make the pairing $\overline{\T}^D(K_n)$-equivariant, see \cite[3.1.4]{Carayol2}). Completing and dualizing gives a $\mathbb{T}^D(K_n)_m$-equivariant perfect pairing $S^D(K_n)_m^\vee\times S^D(K_n)_m^\vee \to \O$. In the totally definite case, this is already the desired pairing $M(K_n)\times M(K_n)\to \O$. In the indefinite case, it follows from \cite[3.2.3]{Carayol2} that the self-duality on $S^D(K_n)^\vee_m$ implies that $M(K_n)= \Hom_{R_n[G_F]}(\rho^{\univ},S^D(K_n)_m^\vee)$ is also $\mathbb{T}^D(K_n)_m$-equivariantly self-dual. Now in the notation above we have $\O[C_{K,\ell}]=\O[S_v]_{v\not\in S}\subseteq \mathbb{T}^D(K_n)_m\subseteq \overline{\T}^D(K_n)_m$, $M(K_n)$ is a finite free $\O[C_{K,\ell}]$-module and $M_\psi(K_n)=M(K_n)/\mathfrak{I}_\psi M(K_n)$. To deduce the pairing on $M_\psi(K_n)$, it will suffice to show that $M(K)/\mathfrak{I}_\psi M(K)\cong M(K)[\mathfrak{I}_\psi]$ as $\mathbb{T}^D(K)_m$-modules. Consider the character $\phi:C_K\to \O^\times$ defined $\phi(\overline{\varpi}_v)=\psi(\Frob_v)/\Nm(v)$ for all $v\not\in S$ (this is well defined by the construction of $C_K$), and note that $\mathfrak{I}_\psi$ is generated by the elements $g-\phi(g)$ for all $g\in C_K$. Let $\displaystyle a_\psi = \sum_{g\in C_K}\phi(g)^{-1}g\in \O[C_K]\subseteq \mathbb{T}^D(K)_m$. The standard theory of group rings implies that multiplication by $a_\psi$ induces a short exact sequence \[0\to \mathfrak{I}_\psi\to \O[C_K]\xrightarrow{a_\psi} (\O[C_K])[\mathfrak{I}_\psi]\to 0\] Since $M(K)$ is a free $\O[C_K]$-module, this implies the multiplication by $a_\psi$ induces an isomorphism $a_\psi:M(K)/\mathfrak{I}_\psi M(K)\xrightarrow{\sim} M(K)[\mathfrak{I}_\psi]$. As $a_\psi\in\O[C_K]\subseteq \mathbb{T}^D(K)_m$, this is the desired isomorphism of $\mathbb{T}^D(K)_m$-modules, and so we indeed have a $\mathbb{T}^D_\psi(K)_m$-equivariant perfect pairing $M_\psi(K)\times M_\psi(K)\to \O$. The final statement, about the pairing $M_n\times M_n\to \O$ follows by localizing at $m_{Q_n}$. \end{proof} To deduce Proposition \ref{M*} from Lemma \ref{pairing}, we shall make use of the following lemma: \begin{lemma}\label{change of ring} If $A$ is a local Cohen--Macaulay ring and $B$ is a local $A$-algebra which is also Cohen--Macaulay with $\dim A = \dim B$, then for any $B$-module $M$, \[\Hom_A(M,\omega_A)\cong \Hom_B(M,\omega_B)\] as left $\End_B(M)$-modules. \end{lemma} \begin{proof} By \cite[\href{http://stacks.math.columbia.edu/tag/08YP}{Tag 08YP}]{stacks-project} there is an isomorphism \[\Hom_A(M,\omega_A)\cong \Hom_B(M,\Hom_A(B,\omega_A))\] sending $\alpha:M\to \omega_A$ to $\alpha':m\mapsto (b\mapsto \alpha(bm))$, which clearly preserves the action of $\End_B(M)$ (as $(\alpha\circ \psi)(bm) = \alpha(b\psi(m))$ for any $\psi\in\End_B(M)$). It remains to show that $\Hom_A(B,\omega_A) \cong \omega_B$, which is just Theorem 21.15 from \cite{Eisenbud} in the case $\dim A = \dim B$. \end{proof} \begin{proof}[Proof of Proposition \ref{M*}] By Lemma \ref{pairing}, we have $M_n\cong \Hom_\O(M_n,\O)$ as $R_n^D$-modules for all $n\ge 1$. Now as $\Delta_n$ is a finite group, the ring $\Lambda_n = \O[\Delta_n]$ has Krull dimension 1. Moreover as in the proof of Lemma \ref{M is patching}, $\Lambda_n = \O[[y_1,\ldots,y_r]]/I_n$, where $I_n$ is generated by $r$ elements. Thus $\Lambda_n$ is a complete intersection, and so $\omega_{\Lambda_n} = \Lambda_n$. Thus by Lemma \ref{change of ring} we have $M_n\cong \Hom_{\Lambda_n}(M_n,\Lambda_n)$, again as $R_n^D$-modules. Tensoring with $\O[[w_1,\ldots,w_j]]$, this implies $M_n^\square\cong \Hom_{\Lambda_n^\square}(M_n^\square,\Lambda_n^\square)$ as $R_n^\square$-modules (and hence as $R_\infty$-modules). Moreover, by Lemma \ref{aug ideal}, $M_n^\square$ is finite free over $\Lambda_n^\square$. Now take any open ideal $\a\subseteq S_\infty$. Letting $\Lambda^\square = S_\infty/I_n$ as in the proof of Lemma \ref{M is patching} we have that $I_n\subseteq \a$ for all but finitely many $n$. For any such $n$, we now have: \begin{align*} M_n^\square/\a&\cong \Hom_{\Lambda_n^\square}(M_n^\square,\Lambda_n^\square)/\a \cong \Hom_{\Lambda_n^\square}(M_n^\square,\Lambda_n^\square/\a) = \Hom_{\Lambda_n^\square}(M_n^\square,S_\infty/\a)\\ &= \Hom_{S_\infty/\a}(M_n^\square/\a,S_\infty/\a) \end{align*} as $R_n^{\square,D}/\a$-modules. Now as noted above, we have that $\uprod{\mathscr{R}^{\square,D}/\a}\cong R_i^{\square,D}/\a$ and $\uprod{\mathscr{M}^\square/\a}\cong M_i^\square/\a$ for $\mathfrak{F}$-many $i$. Taking any such $i$, the above computation gives that: \[\uprod{\mathscr{M}^\square/\a} \cong \Hom_{{S_\infty}/\a}\left(\uprod{\mathscr{M}^\square/\a},{S_\infty}/\a\right)\] as $\uprod{\mathscr{R}^{\square,D}/\a}$-modules. Taking inverse limits, it now follows that: \[M_\infty = \mathscr{P}(\mathscr{M}^\square)\cong \varprojlim_\a\Hom_{{S_\infty}/\a}\left(\uprod{\mathscr{M}^\square/\a},{S_\infty}/\a\right)\] as $\mathscr{P}(\mathscr{R}^{\square,D})$-modules. Now we claim that the right hand side is just $\Hom_{S_\infty}(M_\infty,{S_\infty})$. Using the fact that $M_\infty$, and thus $\Hom_{S_\infty}(M_\infty,{S_\infty})$, is a finite free ${S_\infty}$-module (and thus is $m_{S_\infty}$-adically complete) we get that: \[\Hom_{S_\infty}(M_\infty,{S_\infty}) \cong \varprojlim_\a \Hom_{S_\infty}(M_\infty,{S_\infty})/\a\] as $\displaystyle\mathscr{P}(\mathscr{R}^{\square,D}) = \varprojlim_\a\mathscr{P}(\mathscr{R}^{\square,D})/\a$-modules. But now for any $\a$, as $M_\infty\cong\mathscr{P}(\mathscr{M}^\square)$ is a projective ${S_\infty}$-module: \[\Hom_{S_\infty}(M_\infty,{S_\infty})/\a\cong \Hom_{{S_\infty}/\a}(M_\infty/\a,{S_\infty}/\a)\cong \Hom_{{S_\infty}/\a}\left(\uprod{\mathscr{M}^\square/\a},{S_\infty}/\a\right)\] as $\mathscr{P}(\mathscr{R}^{\square,D})/\a = \uprod{\mathscr{R}^{\square,D}/\a}$-modules. So indeed: \[\Hom_{S_\infty}(M_\infty,{S_\infty}) \cong \varprojlim_\a\Hom_{{S_\infty}/\a}\left(\uprod{\mathscr{M}^\square/\a},{S_\infty}/\a\right) \cong M_\infty\] as $\mathscr{P}(\mathscr{R}^{\square,D})$-modules, and hence as $R_\infty$-modules. But now as $\dim R_\infty = \dim {S_\infty}$ and $S_\infty$ is regular (and thus Gorenstein), Lemma \ref{change of ring} implies that \[M_\infty\cong \Hom_{S_\infty}(M_\infty,{S_\infty}) \cong \Hom_{R_\infty}(M_\infty,\omega_{R_\infty})\] as $R_\infty$-modules, as claimed. \end{proof} This shows that $M_\infty$ indeed satisfies the conditions of Theorem \ref{self-dual}, and so completes the proof of Theorem \ref{Mult 2^k}. \subsection{Endomorphisms of Hecke modules} It remains to show Theorem \ref{R=T}. We first note that Theorem \ref{R=T} can be restated in terms of the objects considered the previous section as follows: \begin{prop}\label{prop:R=T M(K_0)} The trace map $M(K_0)\otimes_{\mathbb{T}^D(K_0)_m} M(K_0) \to \omega_{\mathbb{T}^D(K_0)_m}$ induced by the perfect pairing from Lemma \ref{pairing} is surjective, and the natural map $\mathbb{T}^D(K_0)_m\to \End_{\mathbb{T}^D(K_0)_m}(M(K_0))$ is an isomorphism. \end{prop} \begin{proof}[Proof that Proposition \ref{prop:R=T M(K_0)} implies Theorem \ref{R=T}] First note that by Lemma \ref{lem:aux prime} it suffices to prove Theorem \ref{R=T} with all of the levels $K^{\min}$ replaced by $K_0$. If $D$ is definite, then self-duality and the definition of $M(K_0)$ implies that $M(K_0)\cong S^D(K_0)_m^\vee\cong S^D(K_0)_m$, and so the statement of Theorem \ref{R=T} is immediate. Now assume that $D$ is indefinite. To simply notation, write $\mathbb{T} = \mathbb{T}^D(K_0)_m$, $M = M(K_0)$, $S_{K_0} = S^D(K_0)_m$ and $R = R_{F,S}(\overline{\rho})$. As noted above, we have an isomorphism \[M\otimes_{R} \rho^{\univ} = \Hom_{R[G_F]}(\rho^{\univ},S_{K_0}^\vee)\otimes_{R}\rho^{\univ}\xrightarrow{\sim}S_{K_0}^\vee\] by \cite{Carayol2}. It now follows that \begin{align*} \End_{R[G_F]}(S_{K_0}) &\cong \End_{R[G_F]}(S_{K_0}^\vee)\\ &\cong \Hom_{R[G_F]}(M\otimes_{R}\rho^{\univ},S_{K_0}^\vee)\\ &\cong \Hom_{R[G_F]}\left((M\otimes_{R}R[G_F])\otimes_{R[G_F]}\rho^{\univ},S_{K_0}^\vee\right)\\ &\cong \Hom_{R[G_F]}(M\otimes_{R}R[G_F],\Hom_{R[G_F]}(\rho^{\univ},S_{K_0}^\vee))\\ &\cong \Hom_{R}(M,\Hom_{R[G_F]}(\rho^{\univ},S_{K_0}^\vee))\\ &\cong \Hom_{R}(M,M) = \End_{R}(M) = \End_{\mathbb{T}}(M) = \mathbb{T} \end{align*} as $\mathbb{T}$-algebras (where we used \cite[\href{http://stacks.math.columbia.edu/tag/00DE}{Tag 00DE}, \href{http://stacks.math.columbia.edu/tag/05DQ}{Tag 05DQ}]{stacks-project}). \end{proof} The above work implies the following ``fixed-determinant'' version of Proposition \ref{prop:R=T M(K_0)}: \begin{prop}\label{prop:R=T M_psi(K_0)} The trace map $M_\psi(K_0)\otimes_{\mathbb{T}_\psi^D(K_0)_m} M_\psi(K_0) \to \omega_{\mathbb{T}_\psi^D(K_0)_m}$ induced by the perfect pairing from Lemma \ref{pairing} is surjective, and the natural map $\mathbb{T}_\psi^D(K_0)_m\to \End_{\mathbb{T}_\psi^D(K_0)_m}(M_\psi(K_0))$ is an isomorphism. Moreover, the surjection $R^{D,\psi}_{F,S}(\overline{\rho})\twoheadrightarrow \mathbb{T}_\psi^D(K_0)_m$ from Lemma \ref{R->T} is an isomorphism. \end{prop} \begin{proof} Recall that in the notation of Section \ref{ssec:Ms^sqaure}, $M_\psi(K_0) = M_0$, $\mathbb{T}_\psi^D(K_0)_m = \mathbb{T}_0$ and $R^{D,\psi}_{F,S}(\overline{\rho}) = R_0^D$. As shown above, $M_\infty$ satisfies the hypotheses of Theorem \ref{self-dual}, so by the last conclusion of Theorem \ref{self-dual}, we get that the trace map $\tau_{M_\infty}:M_\infty\otimes_{R_\infty} M_\infty \to \omega_{R_\infty}$ is surjective. As noted above, $(y_1,\ldots,y_r,w_1,\ldots,w_j)$ is a regular sequence for $M_\infty$, and hence for $R_\infty$. It follows that $R_0^D \cong R_\infty/\mathfrak{n}$ is Cohen--Macaulay and $M_0 \cong M_\infty/\mathfrak{n}$ is maximal Cohen--Macaulay over $R_0^D$. Moreover, we get that the dualizing module of $R_0^D$ is just $\omega_{R_0^D}\cong \omega_{R_\infty}/\mathfrak{n}$. But now quotienting out by $\mathfrak{n}$, we thus get a surjective map $M_0\otimes_{R_0^D}M_0\twoheadrightarrow \omega_{R_0^D}$, which (by Lemma \ref{trace}) implies that the trace map $\tau_{M_0}:M_0\otimes_{R_0^D}M_0\to \omega_{R_0^D}$ is also surjective. But now, as in \cite[Lemmas 2.4 and 2.6]{EmTheta}, we have the following commutative algebra result: \begin{lemma}\label{endomorphism} Let $B$ be an $\O$-algebra and let $U$ and $V$ be $B$-modules. Assume that $B,U$ and $V$ are all finite rank free $\O$-modules, and we have a $B$-bilinear perfect pairing $\langle\ ,\ \rangle:V\times U\to \O$. Moreover, assume that $U[1/\lambda]$ is free over $B[1/\lambda]$. Define $\phi:U\otimes_BV\to \Hom_{\O}(B,\O)$ by $\phi(u\otimes v)(b) = \langle bu,v\rangle = \langle u,bv\rangle$. Then $\phi$ is surjective if and only if the natural map from $B$ to the center of $\End_B(U)$ is an isomorphism. \end{lemma} Applying this with $B = R_0^D$, $U = M_0$, $V = M_0^*$ and $\langle\ ,\ \rangle:M_0\times M_0\to \O$ being the perfect pairing from Lemma \ref{pairing} implies that the natural map $R_0^D\to Z(\End_{R_0^D}(M_0))$ is an isomorphism. (Here we have used the fact that $\omega_{R_0^D}\cong \Hom_{\O}(R_0^D,\O)$ as in the proof of Lemma \ref{change of ring}, and $M_0[1/\lambda]\cong R_0^D[1/\lambda]$ by Lemma \ref{gen free}.) But now as $M_0$ is free over $\O$, we get that \[\End_{R_0^D}(M_0)\hookrightarrow \End_{R_0^D[1/\lambda]}(M_0[1/\lambda]) \cong \End_{R_0^D[1/\lambda]}(R_0^D[1/\lambda]) = R_0^D[1/\lambda]\] and so $\End_{R_0^D}(M_0)$ is commutative. Hence the natural map $R_0^D\to \End_{R_0^D}(M_0)$ is an isomorphism. As the action of $R_0^D$ on $M_0$ factors through $R_0^D\twoheadrightarrow \mathbb{T}_0$, this implies that this map and the map $\mathbb{T}_0\to\End_{\mathbb{T}_0}(M_0)$ are isomorphisms.\footnote{As noted in the remark following Lemma \ref{gen free}, the fact that $R_0^D\twoheadrightarrow\mathbb{T}_0$ is an isomorphism follows more simply from the fact that $R_\infty$ is a Cohen-Macaulay domain, flat over $\O$, and that $R_\infty[1/\lambda]$ is formally smooth (as shown in \cite{Shotton}).} \end{proof} It remains to deduce Proposition \ref{prop:R=T M(K_0)} from Proposition \ref{prop:R=T M_psi(K_0)}. As in the proof of Proposition \ref{prop:R=T M(K_0)}, write $\mathbb{T} = \mathbb{T}^D(K_0)_m$, $M = M(K_0)$ and $R = R_{F,S}(\overline{\rho})$. Also write $\mathbb{T}_\psi = \mathbb{T}_\psi^D(K_0)_m$, $M_\psi = M_\psi(K_0)$ and $R_\psi = R^{\psi}_{F,S}(\overline{\rho})$ and $R^D_\psi = R^{D,\psi}_{F,S}(\overline{\rho})\cong \mathbb{T}_\psi$. Lastly recall the group $C_{K_0,\ell}$ from Section \ref{ssec:M_psi} above, and abbreviate this as $C = C_{K_0,\ell}$. Recall that $\O[C]$ is a subalgebra of $\mathbb{T}$ and $M$ is free over $\O[C]$ with we have $M_\psi= M/\mathfrak{I}_\psi$. Our argument will hinge on the following key fact about the structure of $\mathbb{T}$: \begin{lemma} There is an isomorphism $\mathbb{T}\cong \mathbb{T}_\psi\otimes_\O\O[C]$. \end{lemma} \begin{proof} As in Lemma \ref{gen free} we have $\mathbb{T}_\psi[1/\lambda]\cong M_\psi[1/\lambda]$ by classical generic multiplicity 1 results. An analogous argument gives $\mathbb{T}[1/\lambda]\cong M[1/\lambda]$. As $\mathbb{T},\mathbb{T}_\psi,M$ and $M_\psi$ are all finite rank free $\O$-modules, this gives $\rank_\O \mathbb{T}_\psi = \rank_\O M_\psi$ and $\rank_\O \mathbb{T} = \rank_\O M$. Also as $M$ is free over $\O[C]$, $M/\mathfrak{I}_\psi\cong M_\psi$ and $\O[C]/\mathfrak{I}_\psi\cong \O$, we get that $\rank_\O M = \left(\rank_\O M_\psi \right)\left(\rank_\O\O[C]\right)$. It follows that $\rank_\O \mathbb{T} = \rank_\O(\mathbb{T}_\psi\otimes_\O\O[C])$. As all of these rings are free over $\O$, it will thus suffice to give a surjection $ \mathbb{T}_\psi\otimes_\O\O[C]\twoheadrightarrow \mathbb{T}$. Now recall the isomorphism $R_{F,S}(\overline{\psi})\widehat{\otimes}_\O R_\psi\xrightarrow{\sim} R$ from Lemma \ref{lem:fixed determinant}. This gives us maps $\alpha:R_{F,S}(\overline{\psi})\to R\twoheadrightarrow \mathbb{T}$ and $\beta:R_\psi\to R\twoheadrightarrow \mathbb{T}$ such that $(\im\alpha)(\im\beta) = \mathbb{T}$. It will thus suffice to show that $\alpha$ and $\beta$ factor through surjections $R_{F,S}(\overline{\psi})\twoheadrightarrow \O[C]$ and $R_\psi\twoheadrightarrow \mathbb{T}_\psi$, respectively. Note that the map $R\twoheadrightarrow \mathbb{T}$ from Lemma \ref{R->T} is induced by a representation $\rho^{\mod}:G_{F,S}\to \GL_2(\mathbb{T})$ lifting $\overline{\rho}$, and we have $\tr(\rho^{\mod}(\Frob_v)) = T_v$ and $\det(\rho^{\mod}(\Frob_v)) = \Nm(v)S_v$ for all $v\not\in S$ (cf \cite{Kisin}). By Lemma \ref{lem:fixed determinant}, the map $R_{F,S}(\overline{\psi})\to R$ is characterized by the morphism of functors $\rho\mapsto\det\rho$, and so satisfies $\psi^{\univ}(g)\mapsto \det\rho^{\univ}(g)$ for all $g\in G_{F,S}$ (where $\psi^{\univ}$ is the universal lift of $\overline{\psi}$). Thus for any $v\not\in S$, we have $\alpha(\psi^{\univ}(\Frob_v)) = \det\rho^{\mod}(\Frob_v) = \Nm(v)S_v$. By Chebotarev density, it follows that $\im\alpha = \O[S_v]_{v\not\in S} = \O[C]\subseteq \mathbb{T}$. Now for any map $x:\mathbb{T}\to\overline{E}$, let $\rho_x:G_{F,S}\to \GL_2(\mathbb{T})\xrightarrow{x} \GL_2(\overline{E})$ be the modular lift of $\overline{\rho}$ corresponding to $x$. Let $\rho^{\psi}_{x\circ\beta}:G_{F,S}\to \GL_2(R_\psi)\xrightarrow{x\circ\beta}\GL_2(\overline{E})$ be the lift of $\overline{\rho}$ with determinant $\psi$ corresponding to $x\circ\beta:R_\psi\to R\to\mathbb{T}\to\overline{E}$. From the construction of the map $R_\psi\to R$, we see that $\rho^{\psi}_{x\circ\beta}$ is the (unique) twist of $\rho_x$ with determinant $\psi$. In particular, $\rho^{\psi}_{x\circ\beta}$ is also modular of level $K_0$. It follows that $\rho^{\psi}_{x\circ\beta}$ is flat at every $v|\ell$, Steinberg at every $v|\mathfrak{D}$ and minimally ramified at every $v\in\Sigma^D_\ell$, $v\nmid\ell,\mathfrak{D}$. As remarked in Section \ref{ssec:global def}, this implies that $x\circ\beta:R_\psi\to\mathbb{T}\to \overline{E}$ factors through $R^D_\psi\cong \mathbb{T}_\psi$ for any $x:\mathbb{T}\to\overline{E}$. But now as $\mathbb{T}$ is a finite free $\O$-module, we have an injection $\mathbb{T}\hookrightarrow \mathbb{T}\otimes_\O\overline{E}$, and as $\mathbb{T}$ is reduced, $\mathbb{T}\otimes_\O\overline{E}$ is a finite product of copies of $\overline{E}$. Thus by the above, the map $R_\psi\xrightarrow{\beta}\mathbb{T}\hookrightarrow\mathbb{T}\otimes_\O\overline{E}$ factors through $R_\psi\twoheadrightarrow R_\psi^D\cong \mathbb{T}_\psi$, and so $\beta:R_\psi\to \mathbb{T}$ does as well. So indeed $\beta\otimes\alpha$ induces a surjection (and hence an isomorphism) $\mathbb{T}_\psi\otimes\O[C]\to \mathbb{T}$. \end{proof} Now as $\O[C]$ is a complete intersection, we have $\omega_{\O[C]}\cong \O[C]$. Thus using Lemma \ref{change of ring} we get \[\omega_{\mathbb{T}} \cong \Hom_{\O[C]}(\mathbb{T},\O[C])\cong \Hom_{\O[C]}(\mathbb{T}_\psi\otimes_\O\O[C],\O[C])\cong \Hom_{\O}(\mathbb{T}_\psi,\O)\otimes_\O\O[C]\cong \omega_{\mathbb{T}_\psi}\otimes_{\O}\O[C].\] In particular, $\omega_\mathbb{T}/\mathfrak{I}_\psi\cong \omega_{\mathbb{T}_\psi}$. Now consider the trace map $\tau_M:M\otimes_\mathbb{T} M\to \omega_\mathbb{T}$, and notice that the induced map $\tau_M\otimes_{\O[C]}(\O[C]/\mathfrak{I}_\psi):(M/\mathfrak{I}_\psi)\otimes_\mathbb{T}(M/\mathfrak{I}_\psi)\to \omega_\mathbb{T}/\mathfrak{I}_\psi$ can be identified with $\tau_{M_\psi}:M_\psi\otimes_{\mathbb{T}_\psi}M_\psi\twoheadrightarrow \omega_{\mathbb{T}_\psi}$. Since $-\otimes_{\O[C]}(\O[C]/\mathfrak{I}_\psi)$ is right-exact and $\tau_{M_\psi}$ is surjective, it follows that $\tau_M$ is surjective as well. It now follows by the argument in the proof of Proposition \ref{prop:R=T M_psi(K_0)} that the map $\mathbb{T}\to \End_\mathbb{T}(M)$ is an isomorphism. This completes the proof the Proposition \ref{prop:R=T M(K_0)}, and hence of Theorem \ref{R=T}. \bibliographystyle{amsalpha} {\footnotesize

BkiUfmI5qhLBs2ux8gaO

\section{Introduction} \label{sec:introduction} In the standard one-electron picture of linear photoeffect the kinetic energy of electrons photoemitted from the system scales with the energy of the quanta of applied electromagnetic (EM) field following Einstein's relation \begin{equation} \epsilon_f=\hbar\omega_x+\epsilon_B. \label{eq:Einstein} \end{equation} Here $\epsilon_f$ and $\epsilon_B$ are the final kinetic and the initial binding energy of the electron, respectively, and $\hbar\omega_x$ is the photon energy. In nonlinear multiphoton photoemission ($m$PP, $m\geq 2$) that measures the dynamics of photoexcitation and unoccupied electronic structure of solids\cite{ReutzelPRX} $\epsilon_f$ scales with multiples $p\hbar\omega_x$ ($1\leq p \leq m$) of the photon energy, with $p$ depending on (i.e. decreasing with) the number of resonant intermediate states partaking in the transition (see Fig.\,2 in Ref. [\onlinecite{TM}]). Novel measurements of two-photon photoemission (2PP) spectra from Ag(111), Ag(100) and Ag(110) surfaces\cite{MarcelPRL,AndiNJP,ACSPhotonics} have confirmed with high resolution the obeyance of such scalings in one-electron transitions which broadly determine the 2PP spectra from metallic band states.\cite{Adawi,FausterSteinmann,PetekOgawa,Weinelt,TimmBennemann,UebaGumhalter} However, besides this, they have also revealed peaks in photoemission yield at electron kinetic energies which do not scale with the photon energy of applied EM field as would be expected from the extension of Einsteinian relation (\ref{eq:Einstein}) to 2PP. Instead, these peaks occur as resonances pinned to the double bulk plasmon energy $2\hbar\omega_p$ above the Fermi level $E_F$ and their energy appears independent of the energy of excitation source as long as $\hbar\omega_x\geq \hbar\omega_p\sim 3.8$ eV. This peculiar behaviour is illustrated in Fig.\,\ref{AllAg}. It can not be interpreted as a standard $n$-th plasmon satellite whose energy would exhibit a combined ($p,n$) scaling: \begin{equation} \epsilon_{sat}=p\hbar\omega_{x}+\epsilon_{B}-n\hbar\omega_{p}. \label{eq:satellite} \end{equation} Identical finding, although differently interpreted, was reported at the dawn of 2PP spectroscopy for the Ag(111) surface.\cite{Steinmann85} Analogous feature is also implicit in one-photon photoemission (1PP) spectra from submonolayer sodium coated Ag(100) recorded in the constant initial state (CIS) mode which exhibit peaks pinned to $\hbar\omega_p$ above $E_F$.\cite{RefHorn,Horn} These intriguing "non-Einsteinian" photoemission modes call for interpretations going beyond the sole effect of electron excitation by dielectrically screened EM fields\cite{Feibelman,Forstmann,Krasovskii,Reshef} which oscillate with the frequencies of the applied external fields. Evidently, the novel paradigm should involve real plasmon excitation and energy exchange in the intermediate and final stages of one-electron emission processes. The nonthermally excited plasmons whose energy largely exceeds the thermal energy $kT$ may like electrons\cite{encyclopedia} also be termed "hot". Hence a proof of concept is needed to ascertain as how the primary optically induced electronic polarization may generate hot plasmons that via subsequent absorption can supply energy for non-Einsteinian electron emission from the irradiated system. \begin{figure}[tb] \rotatebox{0}{\epsfxsize=8 cm \epsffile{plas2PP_Fig1}} \caption{Panel (a): Selection of 2PP spectra from three low index surfaces of Ag for excitation field energies $\hbar\omega_x$ in the ENZ region where the real part of Ag dielectric function $\varepsilon(\omega)$ is near zero.[\onlinecite{ACSPhotonics}] The manifolds of all the dots shown above the four representative plasmonic peak maxima make the $2\hbar\omega_p$ scaling curves in panel (b) below. Panel (b): Dependence of the measured final state energies of 2PP electron yields from Ag on the variation of energy $\hbar\omega_x$. The data shown by triangles were recorded in the constant inital state mode (CIS) involving the surface state (SS) and the first image potential state (IP1) on Ag(111), the IP1 and the Fermi level $E_F$ on Ag(100), and $E_F$ on Ag(110). They exhibit the standard scaling of 2PP yield energy with $2\hbar\omega_x$ in emission from SS and $E_F$, and with $\hbar\omega_x$ in emission from IP1. In contrast to this, there also appear yields with energy $\sim 2\hbar\omega_p$ above $E_F$ (dots) which do not scale with the multiples of the radiation field energy $p\hbar\omega_x$. This $2\hbar\omega_p$ emission extends far beyond the ENZ region denoted by vertical shading.} \label{AllAg} \end{figure} The possibility that real plasmons appear as intermediaries in photoemission and photoabsorption was alluded to long ago.\cite{Hopfield1965,Tzoar,Gornyi,Kliewer} More recently, and in the context of ultrashort pulse photoexcitations, it was noted in Ref. [\onlinecite{Schueler}]. Expanding on this we first elaborate a microscopic mechanism of generation of real plasmons by light induced electronic polarization, i.e. by the primary optically excited electron-hole pairs. This mechanism is complementary to direct plasmon excitation by local EM fields in microscopically inhomogeneous media.\cite{Feibelman,Forstmann,Podbiel} Once being excited in the system, real plasmons may support additional excitation pathways for the electrons that could sustain preferential narow band optical energy conversion.\cite{Linic,Boriskina,Narang2016} Such consecutive processes may become effective in linear and nonlinear photoemission by supplementing the primary hot electron excitation channels,\cite{Marini2002,Atwater14,Nordlander15,Louie15,Halas15,deAbajo16,Atwater16,Khurgin,Govorov17} and thereby modify the distributions of photoemission yields from metals.\cite{MarcelPRL,ACSPhotonics,Horn} To demonstrate this we first define a calculable measure of the spectral distribution of hot plasmons excitable in interactions of the electron system with external EM fields. Establishment of a distribution of real plasmons in the system enables us to exploit the analogy with $m$PP theory\cite{UebaGumhalter} and formulate the plasmon generated electron yields that supplement the standard $m$PP ones. In Sec.\,\ref{sec:model} we introduce a minimal model for description of electron gas in interaction with plasmons and pulsed EM field, viz. with only two members of the plethora of excitation modes in polarizable media.\cite{Rivera} Using this model we demonstrate in Sec. \ref{sec:spectrum} a construction of quantum mechanical expression describing the spectrum of real plasmons generated in the system by optically induced electronic polarization. The construction is explicitly shown for 1PP in which case it combines the quadratic response to EM field\cite{Ashcroft,Mahan1PP,Caroli} with the nonperturbative scattering formalism for description of multiboson excitation processes.\cite{PhysRep} They are respectively used to formulate the preparation of electronic polarization excited by EM field and subsequent polarization-induced generation of plasmonic coherent states over the electronic state of the system. Hot electrons arising from spontaneous decay of excited plasmons complement the preferential photoelectron yield from the Fermi level\cite{Hopfield1965} at the multiples of plasmon energy and thereby explain the non-Einsteinian features of $m$PP spectra reported in Refs. [\onlinecite{MarcelPRL,ACSPhotonics,Steinmann85,Horn}]. Special limits of the obtained results that are of general interest are explored and plasmon generation probabilities derived and discussed in Sec. \ref{sec:application}. Expanding on this we propose in Sec. \ref{sec:plasmonpump} a theoretical framework in which plasmons excited in multiphoton driven polarization processes may give rise to plasmon frequency-locked electron emission from irradiated solids. In Sec. \ref{sec:results} we employ the density functional theory to calculate the relevant rates of consecutive plasmon and hot electron generation in Ag bands that illustrate the scenarios for non-Einsteinian channels of 1PP and 2PP from metals. This provides essential prerequisites for the studies of plasmonically assisted channeling of broad band optically induced electronic excitations into a narrow energy interval spanned only by the range of Ag plasmon dispersion. \section{Excitation of plasmons by optically induced electronic polarization} \label{sec:System} \subsection{Description of the model} \label{sec:model} We start with a quantum description of light-matter interaction\cite{josab} based on a simple model Hamiltonian $H$ comprising the component that describes the unperturbed system, $H^{syst}$, and coupling $W(t)$ of the system to EM field \begin{equation} H=H^{syst}+W(t). \label{eq:H} \end{equation} Here $H^{syst}$ is composed of $H_{0}^{e}$ describing electrons in crystal band(s), $H_{0}^{pl}$ describing unperturbed plasmons, and $V^{e-pl}$ describing their interaction, viz. \begin{equation} H^{syst}=H_{0}^{e}+H_{0}^{pl}+V^{e-pl}=H_{0}^{syst}+V^{e-pl}. \label{eq:Hsyst} \end{equation} In the second quantization they are respectively given by \begin{eqnarray} H_{0}^{e} &=& \sum_{\bf k}\epsilon_{\bf k}c_{\bf k}^{\dag}c_{\bf k}, \label{eq:allHel}\\ H_{0}^{pl} &=& \sum_{\bf q}\hbar\omega_{\bf q}a_{\bf q}^{\dag}a_{\bf q}, \label{eq:allHpl}\\ V^{e-pl} &=& g\sum_{\bf k,\bar{k},q}V_{\bf \bar{k},k}^{\bf q}c_{\bf \bar{k}}^{\dag}c_{\bf k}a_{\bf q}^{\dag} +h.c., \label{eq:allHint} \end{eqnarray} where $c_{\bf k}^{\dag}$ and $c_{\bf k}$ ($a_{\bf q}^{\dag}$ and $a_{\bf q}$) denote the electron (plasmon) creation and destruction operators, respectively, which obey the fermion (boson) commutation rules. $\epsilon_{\bf k}$ ($\hbar\omega_{\bf q}$) stand for the electron (plasmon) energy associated with the quantum numbers ${\bf k}$ (${\bf q})$ which may include the Bloch momentum, band index, etc., describing the unperturbed dynamics of quasiparticles in the system. In this general notation $V_{\bf \bar{k},k}^{\bf q}$ is the electron-plasmon coupling matrix element and $g$ is the coupling constant eventually to be set equal to unity. For later convenience we assume that this interaction is switched on and off adiabatically, viz. $V^{e-pl}\propto e^{-\eta|t|}$ where $\eta\rightarrow 0^{+}$. The electron coupling to the EM field is described by \begin{eqnarray} W(t)&=&\sum_{i}{\cal E}_{i}(\omega_{x,i},t,\sigma_{x,i})e^{-i(\omega_{x,i}t+\varphi_{i})} \nonumber\\ &\times&\sum_{\bf k_{1},k_{2}}W_{\bf k_{1},k_{2}}c_{\bf k_{1}}^{\dag}c_{\bf k_{2}}+h.c. \label{eq:W} \end{eqnarray} Here ${\cal E}_{i}(\omega_{x,i},t,\sigma_{x,i})>0$, $\omega_{x,i}>0$ and $\varphi_{i}$ are the amplitude modulation of the $i$-th pulse constituting the EM field (pulse profile centered at $t_{x,i}$ and of duration $\sigma_{x,i}$) and the pulse carrier frequency and phase in the crystal, respectively, and $W_{\bf k_1,k_2}$ denote the dipole matrix elements of the electron interaction with the field. Quite generally, the field amplitudes ${\cal E}_{i}(\omega_{x,i},t,\sigma_{x,i})$ and the matrix elements $W_{\bf k_1,k_2}$ depend on the dielectric properties and the symmetry of the system that affect selection rules for ${\bf k_2\leftrightarrow k_1}$ field-induced electronic transitions. Thus, the effective ${\cal E}_{i}(\omega_{x,i},t,\sigma_{x,i})$ would incorporate local and nonlocal Fresnel or Feibelman\cite{Feibelman} corrections for the EM field in the sample. It may also be strongly enhanced for metals at epsilon-near-zero\cite{Reshef} (ENZ) where the real component of linear permitivity is zero. Effective $W_{\bf k_1,k_2}$ may also embody the surface enhanced\cite{Walden} and Drude-assisted\cite{KliewerBennemann} transitions in a phenomenological fashion. In the following we shall restrict our discussions to the quadratic and quartic response of the electronic system to action of one pulse of the EM field. In this $i=1$ limit of (\ref{eq:W}) the formal derivations of the various 1PP- and 2PP-related transition rates can be demonstrated in a conscpicuous fashion. Nonlinear responses induced by two pulses ($i=1,2$) and leading to 2PP and 3PP photoemission yields that are unaffected by real plasmon excitations were studied numerically in the Supplementary Material of Ref. [\onlinecite{TM}]. \subsection{Formulation of the spectrum of hot plasmons generated by photoexcited quasiparticles} \label{sec:spectrum} We seek a scenario based on the minimal electron-plasmon model outlined in Sec. \ref{sec:model} that may enable interpretation of the plasmon energy locked photoelectron yields for $\omega_x\geq \omega_p$ reported previously\cite{MarcelPRL,ACSPhotonics,Steinmann85,Horn} and here in Fig. \ref{AllAg} for three low index Ag surfaces. Such yields occuring at multiples $n\hbar\omega_p$ of plasmon energy indicate the pumping of electrons by hot plasmons generated in the course of photoemission process. An example of lowest order process of this kind ($n=1$), which starts from the electron-plasmon system in the ground state that is linearly perturbed by the EM field (\ref{eq:W}), can be symbolically represented by the flow scheme \begin{widetext} \begin{eqnarray} & &n=1: \nonumber\\ & &|\mbox{electronic grd. state}\rangle|\rm{photon}\rangle {\stackrel{\it W}{\scriptscriptstyle \longrightarrow}} |\rm{polarization}\rangle {\stackrel{\it V^{e-pl}}{\scriptscriptstyle \longrightarrow}} |\rm{polarization'}\rangle|\rm{plasmon}_{\hbar\omega_p}\rangle {\stackrel{\it V^{pl-e}}{\scriptscriptstyle \longrightarrow}} |\rm{polarization''}\rangle|\rm{electron_{\hbar\omega_p}}\rangle \label{eq:sequence} \end{eqnarray} \end{widetext} with symbols over arrows denoting the interactions in (\ref{eq:H}) that drive particular transitions. Here $|\rm{polarization}\rangle $, $|\rm{polarization'}\rangle$ and $|\rm{polarization''}\rangle$ denote the excited states of the electronic system with (i) one primary photoexcited electron above and a primary hole below $E_F$, (ii) primary electron scattered to a secondary state by emission of a real plasmon and the unperturbed primary hole, and (iii) ionized state of the system with secondary electron, primary hole and secondary hole created by plasmon absorption that gives rise to electron excitation to an emanating state $|{\rm electron_{\hbar\omega_p}}\rangle$ whose energy exceeds the initial one by $\hbar\omega_p$. This propagation sequence can be also illustrated by open diagram\cite{DuBois} shown in Fig. \ref{sequence}. For secondary holes created at the Fermi surface, the energy of the secondary emanating electrons is $\hbar\omega_p$-locked above $E_F$. Analogous sequence can be established for the case of plasmon generation by primary hole scattering. As is evident from the sequence (\ref{eq:sequence}) and the ensuing description, the key intermediate events determining the final $\hbar\omega_p$-locked electron yield are the emissions of real plasmons which likewise photons can convey their energy to excite electrons emanating from the system. Therefore, we need as a prerequisite a rigorous assesment of multiple excitation of real plasmons using the above framework for light-matter interaction. To this end we define the multiplasmon spectrum ${\cal S}^{pl}(\varepsilon,t)$ as composed of energies $\varepsilon$ of all real plasmons generated during the propagation of electronic polarization excited by the interaction of electrons with the pulsed EM field (\ref{eq:W}). This is obtained from the expresssion \begin{equation} {\cal S}^{pl}(\varepsilon,t)=\langle\Psi(t)|\delta(\varepsilon - H_{0}^{pl})|\Psi(t)\rangle, \label{eq:N} \end{equation} in which the function $\delta(\varepsilon - H_{0}^{pl})$ acts as a "plasmon counter" operator that projects from the wavefunction $|\Psi(t)\rangle$ of the laser-perturbed system all the states with multiply excited bosonic energies $n\hbar\omega_{\bf q}$, each component being weighted by the corresponding boson excitation probability. This expression for plasmon counting has an advantage over the form % \begin{equation} {\cal N}^{pl}(N,t)=\langle\Psi(t)|\delta(N - \sum_{\bf q} a_{\bf q}^{\dag}a_{\bf q})|\Psi(t)\rangle, \label{eq:Ndirect} \end{equation} which directly gives the number $N=\sum_{\bf q}n_{\bf q}$ of excited bosons. The latter may diverge in the case of linear boson dispersion and transient and localized perturbations.\cite{Hopfield} By contrast, expression (\ref{eq:N}) renders regular results also for linearly dispersing bosons. \begin{figure}[tb] \rotatebox{0}{\epsfxsize=7.5cm \epsffile{plas2PP_Fig2}} \caption{Diagrammatic representation of the sequence of excitation events (\ref{eq:sequence}) linearly induced in the electron-plasmon system by pulsed EM field. Full red, dashed blue and purple wiggly lines denote electron, hole and plasmon propagation, respectively. Blue and red circles denote matrix elements $V_{\bf \bar{k_{1}},k_{1}}$ and $W_{\bf k_{1},k_{2}}$ defined in Eqs. (\ref{eq:allHint}) and (\ref{eq:W}), respectively. The difference of quasiparticle energies of the secondary electron and hole arising from real plasmon absorption is $\Delta\epsilon \simeq \hbar\omega_p$. } \label{sequence} \end{figure} To derive an explicit form of (\ref{eq:N}) we need to find a convenient and realistic representation of the excited state wavefunction $|\Psi(t)\rangle$ and to facilitate this we introduce several simplifying assumptions. The preparation of excited state starts from the initial unperturbed ground state of the system $|0\rangle$ at some instant $t_{0}$ in the remote past before the perturbation $W(t)$ has been switched on. Real plasmon generation driven by one photon- and two photon-induced polarization are illustrated with open diagrams\cite{DuBois} in Figs. \ref{Psi} and \ref{Psi2P}. However, since the "non-Einsteinian peaks" have been observed already in 1PP spectra,\cite{Horn} in order to present a proof of concept for their occurence it suffices to employ the general framework of linear photoemission theory\cite{Ashcroft} and calculate (\ref{eq:N}) in the quadratic response with respect to perturbation (\ref{eq:W}). This amounts to acting at the time $t'<t$ with $W(t')$ on the state $U^{syst}(t',t_{0})|0\rangle$ only once (one photon absorption driven processes) and then letting the thus excited system to evolve under the action of the Hamiltonian (\ref{eq:Hsyst}). This interaction sequence is symbolically represented by the diagram in Fig. \ref{Psi}. This produces $|\Psi(t)\rangle$ in the form \begin{equation} |\Psi(t)\rangle= \int_{t_0}^{t}\frac{dt'}{i\hbar}U^{syst}(t,t')W(t')U^{syst}(t',t_{0})|0\rangle, \label{eq:Psifull} \end{equation} where $U^{syst}(t,t')$ is the evolution operator in the Schr\"{o}dinger picture of the system in the absence of interaction $W(t')$, viz. \begin{equation} U^{syst}(t,t')= \exp[-\frac{i}{\hbar}H^{syst}(t-t')]. \label{eq:Usyst} \end{equation} Note in passing that in accord with Fig. \ref{Psi} expression (\ref{eq:Psifull}) defines the wave function of an open system describing the processes of photon absorption and subsequent real plasmon generation and not the steady state evolution of electron-plasmon system discussed in Ref. [\onlinecite{PS}]. Expression (\ref{eq:N}) with substitution (\ref{eq:Psifull}) is exact within the quadratic response but still operationally too complicated. Hence, at this stage we shall invoke the next simplifying assumption. As we are interested only in real plasmon generation we shall neglect the fluctuations of the ground state (i.e. the Fermi sea) induced by electron-plasmon interaction $V^{e-pl}$ in $U^{syst}(t',t_{0})$ on the RHS of the integrand in (\ref{eq:Psifull}), and replace it by \begin{equation} U_{0}^{syst}(t,t')= \exp[-\frac{i}{\hbar}H_{0}^{syst}(t-t')], \label{eq:U0syst} \end{equation} where $H_{0}^{syst}=H_{0}^{e}+H_{0}^{pl}$. This yields a simpler and operationally more convenient expression \begin{equation} |\Psi(t)\rangle= \int_{t_0}^{t} \frac{dt'}{i\hbar}U^{syst}(t,t')W(t')U_{0}^{syst}(t',t_{0})|0\rangle, \label{eq:Psi} \end{equation} in which $U_{0}^{syst}(t',t_{0})$ is diagonal on $|0\rangle$, viz. $U_{0}^{syst}(t',t_{0})|0\rangle=e^{-\frac{i}{\hbar}E_{0}(t'-t_{0})}|0\rangle$ where $E_{0}$ is the ground state energy, hereafter set equal to zero. Introducing the Fourier transform \begin{equation} \delta(\varepsilon - H_{0}^{pl})=(2\pi\hbar)^{-1}\int_{-\infty}^{\infty}d\tau e^{\frac{i}{\hbar}(\varepsilon -H_{0}^{pl})\tau} \label{eq:delta} \end{equation} we can bring (\ref{eq:N}) after suppressing index $i$ in (\ref{eq:W}) to the form \begin{widetext} \begin{eqnarray} {\cal S}^{pl}(\varepsilon,t)&=&\frac{1}{2\pi\hbar^3}\int_{-\infty}^{\infty}d\tau e^{i\varepsilon\tau}\int_{t_0}^{t} dt''\int_{t_0}^{t} dt'\langle 0|W(t'')U^{syst}(t,t'')^{\dag}e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}U^{syst}(t,t')W(t')|0\rangle \label{eq:Npl}\\ &=& \frac{1}{2\pi\hbar^3}\int_{-\infty}^{\infty}d\tau e^{i\varepsilon\tau}\int_{t_0}^{t}{\cal E}(\omega_x,t'',\sigma_{x})e^{i\omega_{x}t''} dt''\int_{t_0}^{t}{\cal E}(\omega_x,t',\sigma_{x})e^{-i\omega_{x}t'} dt'\nonumber\\ &\times& \sum_{\bf k_{1}'',k_{2}'',k_{1}',k_{2}'} W_{\bf k_{1}'',k_{2}''}W_{\bf k_{2}',k_{1}'}\langle 0|c_{\bf k_{2}''}^{\dag}c_{\bf k_{1}''}U^{syst}(t,t'')^{\dag}e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}U^{syst}(t,t')c_{\bf k_{1}'}^{\dag}c_{\bf k_{2}'}|0\rangle. \label{eq:Nplkk} \end{eqnarray} \end{widetext} Note here that $\tau$ appearing in $e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}$ is not associated with any time evolution of the system but only plays the role of integration variable conjugate to the projected energy $\varepsilon$, whereas the real evolution times are $t,t',t''$. \begin{figure}[tb] \rotatebox{0}{\epsfxsize=8 cm\epsffile{plas2PP_Fig3}} \caption{Diagrammatic representation of linearly induced plasmon generation processes described by (\ref{eq:N}) with $|\Psi(t)\rangle$ defined in (\ref{eq:Psi}) and leading to (\ref{eq:Nplkk}). Meaning of the symbols is same as in Fig. \ref{sequence}. The time evolution after the application of pulsed perturbation (\ref{eq:W}) is governed by the Hamiltonian (\ref{eq:Hsyst}). Plasmon counting implemented through the operator $\delta(\varepsilon-H_{0}^{pl})$ is denoted symbolically by ${\cal S}(\varepsilon)$. Real multiplasmon excitations may occur for excited quasiparticle energies exceeding the multiexcitation threshold $n\omega_p$.} \label{Psi} \end{figure} \begin{figure}[tb] \rotatebox{0}{\epsfxsize=6cm\epsffile{plas2PP_Fig4}} \caption{Diagrammatic representation of multiplasmon generation driven by electronic polarizarion excited by double action of the laser pulse. All symbols have the same meaning as in Fig. \ref{Psi}. Electron induced plasmon generation may take place either in the intermediate and final electron states (upper panel), or in the final state of the two photon-driven process (lower panel).} \label{Psi2P} \end{figure} The main complications in evaluating (\ref{eq:Nplkk}) arise with the treatment of matrix elements of the products of three generalized exponential operators in the second line of (\ref{eq:Nplkk}). This is addressed in the next subsection. \subsection{Approximations and operator algebra in the calculations of hot plasmon spectrum in Eq. (\ref{eq:Nplkk})} \label{sec:approximations} We seek a nonperturbative solution in terms of $V^{e-pl}$ to expression in the second line on the RHS of (\ref{eq:Nplkk}). Here the sum $\sum_{\bf k_{1}'',k_{2}'',k_{1}',k_{2}'}...$ comprises coherent (${\bf k_{1'}=k_{1''}}$ and ${\bf k_{2'}=k_{2''}}$) and incoherent components (${\bf k_{1'}\neq k_{1''}}$ and ${\bf k_{2'}\neq k_{2''}}$) arising in the calculation of the ground state average in (\ref{eq:Nplkk}). The evaluation of all such components represents a formidable task but simplifications which enable grasping the most salient features of hot plasmon generation are possible. For this sake we shall restrict the plasmon generation to processes in which inelastic electron scattering above the Fermi level is dominant (i.e. $\epsilon_{\bf k}>E_{F}, \epsilon_{\bf \bar{k}}>E_{F}$ after the action of $V^{e-pl}$ on the photoexcited state), and neglect plasmon generation by holes excited below $E_{F}$. This is justified for the CIS mode photoyield measured from the vicinity of the Fermi surface (cf. Fig. 7 of Ref. [\onlinecite{Horn}]) in which case the processes involving the hole recoil (i.e. ${\bf k_{2}'\neq k_{2}''}$) due to real plasmon emission must be excluded. This is also consistent with the earlier exclusion of Fermi sea fluctuations in the interval $(t_0,t')$ in Eq. (\ref{eq:Psi}). A quantitative support for such approximation will be provided in Sec. \ref{sec:results} by the calculations of plasmon generation by electrons and holes propagating in Ag bands. By restricting the plasmon coupling to only one type of quasiparticle we omit all plasmon mediated excitonic effects associated with the propagation of primary photoexcited e-h pair\cite{Mahan,Gavoret,Combescot,Marini,PSS} but this restriction does not invalidate the desired proof of concept of hot plasmon generation in solids. In the opposite limit of dominant plasmon-hole coupling a completely analogous procedure of neglecting electron-induced plasmon generation can be followed.\cite{Reining,FCaruso,GW+C,Reining2018} Moreover, for $\omega_p \leq\omega_x<2\omega_p$ (experimental conditions in Fig. \ref{AllAg}) only one of these two hot plasmon generation channels can be effective. In the next step, and in accord with Ref. [\onlinecite{Ashcroft}], we only consider such coherent contributions from (\ref{eq:Nplkk}) which give rise to steady state quantities in the long time limit, here effective already past the duration of the pulse. This renders ${\bf k_{1}''=k_{1}'=k_{1}}$, ${\bf k_{2}''=k_{2}'=k_{2}}$ and reduces the calculation of (\ref{eq:Nplkk}) to a much simpler quantity in which, as regards the interaction with plasmons, the electronic states above and below $E_F$ are decoupled except for the relation between ${\bf k_{1}}$ and ${\bf k_{2}}$ imposed by the selection rules contained in the optical transition matrix elements $W_{\bf k_{1},k_{2}}$. In this case the ground state average in (\ref{eq:Nplkk}) factorizes into diagonal electron and hole components \begin{eqnarray} & &\langle 0|c_{\bf k_{2}}^{\dag}c_{\bf k_{1}}U^{syst}(t,t'')^{\dag}e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}U^{syst}(t,t')c_{\bf k_{1}}^{\dag}c_{\bf k_{2}}|0\rangle\nonumber\\ &=& \langle 0|c_{\bf k_{2}}^{\dag}(t'')c_{\bf k_{2}}(t')|0\rangle\nonumber\\ &\times& \langle 0|c_{\bf k_{1}}U^{syst}(t,t'')^{\dag}e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}U^{syst}(t,t')c_{\bf k_{1}}^{\dag}|0\rangle. \label{eq:Ndiag} \end{eqnarray} Here $\langle 0|c_{\bf k_{2}}^{\dag}(t'')c_{\bf k_{2}}(t')|0\rangle=n_{\bf k_{2}}\exp[i\epsilon_{\bf k_{2}}(t''-t')]$ is the unperturbed hole propagator or correlation function with $n_{\bf k_{2}}=\langle 0|c_{\bf k_{2}}^{\dag}c_{\bf k_{2}}|0\rangle$, and the electron component appears in the form of an excited state average of a product of generalized exponential operators. The coupled photoexcited electron-plasmon entity is usually termed plasmonic polaron or plasmaron.\cite{Tediosi,Chis} The electron component on the RHS of (\ref{eq:Ndiag}),viz. \begin{eqnarray} & &\langle 0|c_{\bf k_{1}}U^{syst}(t'',t)^{\dag}e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}U^{syst}(t,t')c_{\bf k_{1}}^{\dag}|0\rangle\nonumber\\ &=& \langle{\bf k_{1}}|U^{syst}(t'',t)^{\dag}e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}U^{syst}(t,t')|{\bf k_{1}}\rangle \label{eq:electronpart} \end{eqnarray} acquires the form of the counting-intercepted (via the action of $e^{-\frac{i}{\hbar}H_{0}^{pl}\tau}$) plasmaron propagator which describes spontaneous emissions of plasmons\cite{PS} by the primary electron photoexcited to the state $|{\bf k_{1}}\rangle$. Its evaluation proceeds by formal manipulations of the three generalized exponential operators partaking in this excited state average. The details of corresponding calculations are presented in Appendix \ref{sec:ElectronPart}. Making use of expression (\ref{eq:<eWfinal>}) derived therein we obtain the leading contribution to (\ref{eq:Nplkk}) that is quadratic (i.e. perturbative) in the EM field, while the effect of electron-plasmon coupling $V_{\bf \bar{k}_{1},k_{1}}^{\bf q}$ is formally included to all orders (i.e. nonperturbatively). Introducing the transient integral \begin{equation} \Delta_{\bf \bar{k},k}^{\bf q}(t,t')=\frac{1}{\hbar} \int_{t'}^{t}e^{-\frac{i}{\hbar}(\epsilon_{\bf k}-\epsilon_{\bf \bar{k}}-\hbar\omega_{\bf q})t_{1}}dt_{1} \label{eq:DeltaText} \end{equation} the afore mentioned leading contribution reads \begin{widetext} \begin{eqnarray} {\cal S}^{pl}(\varepsilon,t) &=& \frac{1}{\hbar^2}\int_{t_0}^{t}{\cal E}(\omega_x,t'',\sigma_{x}) dt''\int_{t_0}^{t}{\cal E}(\omega_x,t',\sigma_{x}) dt' \sum_{\bf k_{1},k_{2}} |W_{\bf k_{1},k_{2}}|^{2}(1-n_{\bf k_{1}})n_{\bf k_{2}}e^{\frac{i}{\hbar}(\hbar\omega_{x} -\epsilon_{\bf k_{1}}+\epsilon_{\bf k_{2}})(t''-t')}\nonumber\\ &\times& \frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}d\tau e^{\frac{i}{\hbar}\varepsilon\tau}\exp\left[-\sum_{\bf \bar{k}_1,q}\left|V_{\bf \bar{k}_1,k_{1}}^{\bf q}\Delta_{\bf k_{1},\bar{k}_1}^{\bf q}(t,(t'+t'')/2)\right|^{2}(1-n_{\bf \bar{k}_1})(1-e^{-i\omega_{\bf q}\tau})\right], \label{eq:Nplexp} \end{eqnarray} \end{widetext} with a proviso $\epsilon_{\bf k_{1}},\epsilon_{\bf \bar{k}_1}>E_{F}$. This and the symmetry selection rules contained in $V_{\bf \bar{k}_1,k_1}^{\bf q}$ restrict the ${\bf q}$-summation in the last line on the RHS of (\ref{eq:Nplexp}). The quadratic response formalism here applied to description of electron photoexcitation temporally restricted to the pulse duration, i.e. short times around $t_{x}$, required the introduction of finite times $t$, $t'$ and $t''$ in the general expression (\ref{eq:Nplkk}). Such dependence on the intermediate propagation times obviously renders the spectra (\ref{eq:Nplexp}) of plasmons emitted by photoexcited electrons much more complicated than would have been obtained within the standard scattering formalism.\cite{PhysRep,GW} The same formulation can be extended to quartic response appropriate to 2PP\cite{UebaGumhalter,encyclopedia} but this is technically, although not so much conceptually, more demanding. Lastly, we observe that in the absence of electron-plasmon coupling in (\ref{eq:Hsyst}) one obtains $w_{\bf k_{1}}(t,t',t'')=0$, the last line on the RHS of (\ref{eq:Nplexp}) is replaced by $\delta(\varepsilon)$ and the remainder yields the transient electronic polarization spectrum shown in Fig. 3 of Ref. [\onlinecite{PSS}]. \section{Applications of the obtained results in special limits} \label{sec:application} In this section we illustrate the behaviour of expression (\ref{eq:Nplexp}) in simple limits of physical relevance that all demonstrate the formation of plasmonic coherent states\cite{Glauber} in electronic systems perturbed by ultrashort laser pulses. \subsection{Illustration of generation of plasmonic coherent states in the limit of nondispersive plasmons } \label{sec:nondispersive} The multiplasmon features of (\ref{eq:Nplkk}) and (\ref{eq:Nplexp}) obtained from the plasmon counting formula (\ref{eq:N}) can now be best illustrated by assuming nondispersive plasmon frequency $\omega_{\bf q}=\omega_{p}$ in the second line on the RHS of (\ref{eq:Nplexp}). This allows the ${\bf q}$-summations to be readily carried out to give \begin{eqnarray} w_{\bf k_{1}}(t,t',t'')=\sum_{\bf \bar{k}_1,q}\left|V_{\bf \bar{k}_1,k_{1}}^{\bf q}\Delta_{\bf \bar{k}_1,k_1}^{\bf q}(t,t'+t'')/2)\right|^{2}(1-n_{\bf \bar{k}_1})\nonumber\\ \rightarrow \sum_{\bf \bar{k}_1}(1-n_{\bf \bar{k}_1}){\cal V}_{\bf \bar{k}_{1},k_{1}}^{2}\left|\int_{\frac{(t'+t'')}{2}}^{t}e^{-\frac{i}{\hbar}(\epsilon_{\bf k_{1}}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{p})t_{1}}\frac{dt_{1}}{\hbar}\right|^{2}. \label{eq:w} \end{eqnarray} where ${\cal V}_{\bf \bar{k}_{1},k_{1}}^{2}=\sum_{\bf q}\left|V_{\bf \bar{k}_1,k_{1}}^{\bf q}\right|^2$. Using this, expanding the exponential function in the last line on the RHS of (\ref{eq:Nplexp}) into a power series and taking its Fourier transform term by term yields (with $t_{0}\rightarrow -\infty$) \begin{widetext} \begin{eqnarray} {\cal S}^{pl}(\varepsilon,t)&=& \frac{1}{\hbar^2}\int_{-\infty}^{t}{\cal E}(\omega_x,t'',\sigma_{x}) dt''\int_{-\infty}^{t}{\cal E}(\omega_x,t',\sigma_{x}) dt' \sum_{\bf k_{1},k_{2}} |W_{\bf k_{1},k_{2}}|^{2}(1-n_{\bf k_{1}})n_{\bf k_{2}}e^{-\frac{i}{\hbar}(\epsilon_{\bf k_{1}}-\epsilon_{\bf k_{2}}-\omega_{x})(t''-t')}\nonumber\\ &\times& \sum_{n=0}^{\infty}e^{-w_{\bf k_{1}}(t,t',t'')}\frac{\left[w_{\bf k_{1}}(t,t',t'')\right]^{n}}{n!}\delta(\varepsilon-n\omega_{p}). \label{eq:Nplexpanded} \end{eqnarray} \end{widetext} The last line on the RHS of (\ref{eq:Nplexpanded}) clearly demonstrates the signature of plasmon counting effectuated in ${\cal S}^{pl}(\omega,t)$ by the operator (\ref{eq:delta}) and manifesting through the integer $n$ ($0\leq n\leq \infty$) in the expansion of the diabatically developing time dependent plasmonic coherent state \begin{equation} |w_{\bf k_{1}}(t,t',t'')\rangle=\sum_{n=0}^{\infty}e^{-w_{\bf k_{1}}(t,t',t'')/2}\sqrt{\frac{w_{\bf k_{1}}(t,t',t'')^{n}}{n!}}|n\rangle, \label{eq:cohstate} \end{equation} where $|n\rangle$ is an eigenstate of the plasmon number operator, viz. $a_{p}^{\dag}a_{p}|n\rangle=n_p|n\rangle$. The time dependent excitation probabilities $w_{\bf k_{1}}(t,t',t'')$ incorporate time-energy uncertainities arising from finite temporal integration boundaries on the RHS of (\ref{eq:w}). Additional uncertainities may arise in (\ref{eq:Nplexpanded}) from integrations of pulse profiles over the final observation intervals $(t_0,t)$. Expression (\ref{eq:cohstate}) bears analogy with the solutions for boson fields driven by external currents\cite{GlauberPR} or energetic particles\cite{Ritchie}, and the plasmonic polaron model\cite{FCaruso} in which the expression preceding each $\delta(\varepsilon-n\hbar\omega_{p})$ on the RHS of (\ref{eq:Nplexpanded}) gives the probability for $n$-plasmon excitation, in the present case caused by the electron component of laser pulse-induced polarization.\cite{satellites} Note also that in this nondispersive plasmon limit we readily obtain from (\ref{eq:N}), (\ref{eq:Ndirect}) and (\ref{eq:Nplexpanded}) the equivalence \begin{equation} {\cal N}^{pl}(N,t)\leftrightarrow \hbar\omega_{p}{\cal S}^{pl}(\varepsilon,t) \label{eq:NS} \end{equation} implementable through $\delta(N-n)\leftrightarrow\hbar\omega_{p}\delta(\varepsilon-n\hbar\omega_{p})$. \subsection{Scattering boundary conditions limit of Eq. (\ref{eq:Nplexp})} \label{sec:SBC} \begin{figure}[tb] \rotatebox{0}{ \epsfxsize=8cm \epsffile{plas2PP_Fig5}} \caption{Schematic illustration of the temporal behaviour of the external EM field amplitude ${\cal E}(\omega_x,t',\sigma_{x})$ (red curve) and $V^{e-pl}(t')=V^{e-pl}e^{-\eta|t'|}$ (blue curve, $\eta\rightarrow 0^{+}$) appropriate to the scattering boundary conditions for preparation of electrons in excited states $|{\bf k_1}\rangle$ above the Fermi level and their subsequent interaction with plasmons. The horizontal axis illustrates the hierarchy of times established in expression (\ref{eq:hierarchy}).} \label{SBC} \end{figure} The scattering boundary conditions (SBC) limit of Eq. (\ref{eq:Nplexp}) is particularly illustrative as it allows to trace the energy conservation during the primary photoexcitation of electronic polarization and subsequent plasmon generation. To illustrate this we follow the standard collision theory approach\cite{GW} and explore the consequences of adiabatic switching on and off of the electron-plasmon interaction in Eq. (\ref{eq:allHint}) relative to the action of the pulse (\ref{eq:W}). Here we assume that the pulse centered at $t_x$ and with amplitude ${\cal E}(\omega_x,t,\sigma_{x})$ acts for sufficiently long preparation time $\sigma_{x}$ in the interval $t_0<t_x<-\eta^{-1}$ that is embedded in the much longer interval $(t_0\rightarrow -\infty,t\rightarrow\infty)$. This establishes the hierarchy among the various characteristic times\cite{Sakaue} appearing in expression (\ref{eq:Nplexp}), viz. \begin{equation} t_0\ll(t_x,t',t'')\ll-\eta^{-1} \ll 0\ll \eta^{-1}\ll t, \label{eq:hierarchy} \end{equation} with $t_0\rightarrow -\infty$ and $t\rightarrow\infty$. This is sketched schematically in Fig. \ref{SBC}. Application of these SBC to definition (\ref{eq:DeltaText}) enables a standard substitution \begin{equation} \Delta_{\bf \bar{k}_1,k_1}^{\bf q}(t,(t'+t'')/2)\left|_{\rm SBC}\right.\rightarrow 2\pi\delta(\epsilon_{\bf k_1}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{\bf q}) \label{eq:Deltaonshell} \end{equation} into Eq. (\ref{eq:Nplexp}) which puts plasmon emission by primary photoexcited electron on the energy shell. This makes expressions in the first and second line on the RHS of (\ref{eq:Nplexp}) effectively factorizable in the time domain. Consequently, the integrals over $t'$ and $t''$ in the first line can be readily carried out to lead to an expression describing the role of experimental apparatus in the energy conserving preparation of quasiparticles in the excited states $|{\bf k_{1}}\rangle$. \subsubsection{Factorization of the "apparatus function"} \label{sec:apparatus} The compliance of expression (\ref{eq:Nplexp}) with conditions (\ref{eq:hierarchy}) and $\omega_{x}^{-1}\ll \sigma_{x}<\eta^{-1}<t$ leads in (\ref{eq:Nplexpanded}) to the factorization of the "apparatus function" containing the effective fields \begin{eqnarray} {\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x})&=&|W_{\bf k_{1},k_{2}}|^{2}(1-n_{\bf k_{1}})n_{\bf k_{2}}\nonumber\\ &\times& \left|\int_{-\infty}^{\infty}\frac{dt'}{\hbar}{\cal E}(\omega_x,t',\sigma_{x}) e^{-\frac{i}{\hbar}(\hbar\omega_{x} -\epsilon_{\bf k_{1}}+\epsilon_{\bf k_{2}})t'}\right|^{2}, \label{eq:apparatus} \end{eqnarray} which exhibits the property \begin{equation} {\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x})\propto\sigma_{x}\delta(\hbar\omega_x-\epsilon_{\bf k_{1}}+\epsilon_{\bf k_{2}}). \label{eq:calA} \end{equation} Combining (\ref{eq:apparatus}) with the second line in Eq. (\ref{eq:Nplexp}) the latter can be represented in the form of on-the-energy-shell spectrum ${\cal S}^{pl}(\varepsilon,t)$ with minimal time-energy uncertainities during both stages of primary photoexcitation of electronic polarization and sequential plasmon emission \begin{widetext} \begin{eqnarray} \lim_{t\rightarrow\infty}{\cal S}^{pl}(\varepsilon,t)&=&\sum_{\bf k_{1},k_{2}} {\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x})\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}d\tau e^{\frac{i}{\hbar}\varepsilon\tau}\exp\left[-\sum_{\bf \bar{k}_1,q}w_{\bf \bar{k}_1,k_{1}}^{\bf q}(1-e^{-i\omega_{\bf q}\tau})\right] \label{eq:NSBC}\\ &=& \sum_{\bf k_{1},k_{2}} {\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x})e^{-w_{\bf k_{1}}} \left[\delta(\varepsilon)+\sum_{\bf \bar{k}_1,q}w_{\bf \bar{k}_1,k_{1}}^{\bf q}\delta(\varepsilon-\omega_{\bf q})+\sum_{\bf \bar{k}_1,q;{\bf \bar{k}_{1}',q'}}w_{\bf \bar{k}_1,k_{1}}^{\bf q}w_{\bf \bar{k}_{1}',k_{1}}^{\bf q'}\delta(\varepsilon-\omega_{\bf q}-\omega_{\bf q'})+\cdots\right]\nonumber\\ \label{eq:NSBCexpanded} \end{eqnarray} \end{widetext} Here \begin{equation} w_{\bf \bar{k}_1,k_{1}}^{\bf q}=\left|2\pi V_{\bf \bar{k}_1,k_{1}}^{\bf q}\delta(\epsilon_{\bf k_{1}}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{\bf q})\right|^{2}(1-n_{\bf \bar{k}_1}), \label{eq:wscatt} \end{equation} is a dimensionless transition probability in which the divergent square of the $\delta$-function can be regularized following the procedures appropriate to the treatments of explicitly time-dependent quantum transition probabilities.\cite{PhysRep,Messiah,FarisPRL} This is described in Sec.\,S1 of Ref. [\onlinecite{SM}]. As shown therein, from the on-the-energy-shell limit (\ref{eq:wscatt}) directly follows Fermi's golden rule expression for the differential transition rate describing electron transition ${\bf k_1}\rightarrow {\bf \bar{k}_1}$ through the energy conserving emission of one plasmon of wavevector ${\bf q}$, viz. \begin{equation} \Gamma_{\bf \bar{k}_1,k_1}=2\pi\sum_{\bf q}|V_{\bf \bar{k}_1,k_1}^{\bf q}|^2(1-n_{{\bf \bar{k}_1}})\delta(\epsilon_{\bf k_{1}}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{\bf q}). \label{eq:Gammaplasdiff} \end{equation} The saturation limits (\ref{eq:wscatt}) and (\ref{eq:Gammaplasdiff}) become effective already few femtoseconds after plasmon excitation.\cite{FarisPRL,LazicPRL} The total transition rate per unit time, $1/\tau_{\bf k_1}=\Gamma_{\bf k_1}^{>}/\hbar$, for one plasmon excitation by an electron scattered from the primary photoexcited state $|{{\bf k}_{1}}\rangle$ {\it above} $E_F$ to all unoccupied final band states $|{\bf \bar{k}_1}\rangle$ satisfying energy conservation, is obtained from\cite{SS1975} \begin{equation} \Gamma_{\bf k_1}^{>}=\sum_{\bf \bar{k}_1}\Gamma_{\bf \bar{k}_1,k_1}=2\pi\sum_{\bf \bar{k}_1,q}|V_{\bf \bar{k}_1,k_1}^{\bf q}|^2(1-n_{{\bf \bar{k}_1}})\delta(\epsilon_{\bf k_{1}}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{\bf q}). \label{eq:Gammaplas} \end{equation} Here the summation over the final electron quantum numbers ${\bf \bar{k}_1}$ introduces the one-plasmon excitation threshold $\theta(\epsilon_{\bf k_{1}}-\hbar\omega_{\bf q}-E_F)$. The quantity ({\ref{eq:Gammaplas}) depends solely on the intrinsic properties of the electron-plasmon system. Expressions (\ref{eq:Gammaplasdiff}) and (\ref{eq:Gammaplas}) will serve in Sec. \ref{sec:pumpingplasmon} for carrying out the calculations of primary hot electron decay caused by plasmon emision. It is notable that within the SBC the total energy conservation over the temporally nonoverlapping interactions $W(t')$ and $V^{e-pl}(t')$ in the interval $(t_0,t)$ is linked only via the primary excited quasiparticle energy $\epsilon_{\bf k_1}$. This is easily seen by combining energy conservations from the two nonoverlapping events of electron photoexcitation and plasmon emission. This yields for one-plasmon excitation the condition $\hbar\omega_x=\epsilon_{\bf \bar{k}_1}+\hbar\omega_{\bf q}-\epsilon_{\bf k_2}$ where $\epsilon_{\bf k_2}$ is the primary photoexcited hole energy. Furthermore, since in the preparation of momentum and energy resolved scattering experiments the primary quasiparticle momentum and energy are kept fixed, the "apparatus function" in (\ref{eq:NSBC}) obeying this condition will be effectively restricted to a single factorizable ${\bf k_1}$-component $\sum_{\bf k_{2}} {\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x})$. This leads to the formulation of the plasmon emission event in accord with the standard collision theory developed in Chs. 3-5 of Ref. [\onlinecite{GW}]. The coherent state related representation of (\ref{eq:NSBC}) is again most easily visualized in the simple nondispersive limit $\omega_{\bf q}=\omega_p$ employed in Sec. \ref{sec:nondispersive} which together with (\ref{eq:apparatus}) and (\ref{eq:w1}) yields a Poissonian distribution spectrum \begin{eqnarray} \lim_{t\rightarrow\infty}{\cal S}^{pl}(\varepsilon,t)&=&\sum_{\bf k_{1},k_{2}} {\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x}) \nonumber\\ &\times& \sum_{n=0}^{\infty}e^{-w_{\bf k_{1}}}\frac{\left(w_{\bf k_{1}}\right)^{n}}{n!}\delta(\varepsilon-n\hbar\omega_p), \label{eq:NSBCcoh} \end{eqnarray} with \begin{equation} w_{\bf k_{1}}=\sum_{\bf \bar{k}_1,q}w_{\bf \bar{k}_1,k_1}^{\bf q} \label{eq:w1} \end{equation} In this limit expressions (\ref{eq:Psi}) and (\ref{eq:cohstate}) tend to a superposition of electronic and quasistationary coherent plasmonic states that within the SBC acquire a compact form \begin{eqnarray} |\Psi_{\rm SBC}\rangle&=& \sum_{{\bf k_{1},k_{2}},n=0}^{n=\infty} \sqrt{{\cal A}_{\bf k_{1},k_{2}}(\omega_x,\sigma_{x})}\nonumber\\ &\times& e^{-\frac{w_{\bf k_{1}}}{2}}\sqrt{\frac{\left(w_{\bf k_{1}}\right)^{n}}{n!}}|{\bf k_{1},k_{2}}\rangle|n\rangle. \label{eq:cohscatt} \end{eqnarray} In the opposite case of temporally overlapping interactions $W(t')$ and $V^{e-pl}(t')$, the time dependences in the first and second line on the RHS of (\ref{eq:Nplexp}) do not factorize and simple closed form solutions are possible only for special pulse shapes (see Sec. \ref{sec:cohfinstate}). \subsubsection{Truncated plasmonic coherent states} \label{sec:truncated} The plasmonic excitation spectrum given by the expression following the apparatus function on the RHS of (\ref{eq:NSBC}), (\ref{eq:NSBCexpanded}) and (\ref{eq:NSBCcoh}) is unitary by construction. However, as it was derived from second order cumulant expansion which approximates the higher order plasmon emission events by a succession of uncorrelated first order ones (cf. Appendix \ref{sec:ElectronPart}), the excitation threshold condition $\epsilon_{\bf k_1}-E_F\geq n\hbar\omega_p$ for $n$-plasmon emission ($n\geq 2$) does not explicitly appear in the third and higher order terms in the square brackets in (\ref{eq:NSBCexpanded}). To remedy this defficiency of second order cumulant approximation all the terms on the RHS of (\ref{eq:NSBCexpanded}) beyond the second one should be consecutively multiplied by the factors $\theta(\epsilon_{\bf k_{1}}-n\hbar\omega_{p}-E_{F})$ in order to remove excess weights that violate energy conservation. Since the thus modified spectrum violates unitarity it must be reunitarized with respect to the $n=0$ elastic line $e^{-w_{\bf k_{1}}}\delta(\varepsilon)$. One such procedure was outlined in the last two paragraphs in Sec. III.C of Ref. [\onlinecite{chopoff}] and amounts to multiplying the elastic line $\delta(\varepsilon)$ in expression (\ref{eq:NSBCexpanded}) by the sum of all removed ${\bf k_1}$-dependent weights \begin{equation} \sum_{n=2}^{\infty}\sum_{\bf \bar{k}_1,q;...;\bar{k}_{1}',q'}w_{\bf \bar{k}_1,k_{1}}^{\bf q}\cdots w_{\bf \bar{k}_{1}',k_{1}}^{\bf q'}\theta(n\hbar\omega_p +E_F-\epsilon_{\bf k_1}). \label{eq:chopoff} \end{equation} This procedure ensures that the sum of energies of excited plasmons can not exceed the absorbed photon energy and leaves the common Debye-Waller factor $e^{-w_{\bf k_{1}}}$ from (\ref{eq:NSBCexpanded}) unchanged. Besides, constraining $n$ by energy conservation transforms (\ref{eq:NSBCcoh}) into a sub-Poissonian distribution and the plasmonic wavefunction (\ref{eq:cohscatt}) into a truncated coherent state. \subsection{Final plasmonic state in the limit of excitation by extremely short pulses} \label{sec:cohfinstate} \begin{figure}[tb] \rotatebox{0}{ \epsfxsize=9cm \epsffile{plas2PP_Fig6}} \caption{Diagrammatic illustration of perturbative contributions to plasmon pumped electron yield following the procedures outlined in Refs. [\onlinecite{TimmBennemann}] and [\onlinecite{Caroli}]. Positive time is running upward. Sketch of Keldysh diagram for plasmonically induced electron yield in (a) quadratic response (analogous to 1PP yield), and in (b) quartic response (analogous to 2PP yield). Full red and dashed blue lines denote propagation of electrons and holes above and below the Fermi level, respectively. Purple wiggly lines indicate plasmons giving rise to electron excitation via electron-plasmon interaction matrix elements $V_{\bf \bar{k}_1,k_1}^{\bf q}$ denoted by blue dots. Vertical full black arrows symbolize time dependent electron yield captured by the electron counter. In the processes depicted in panels (a) and (b) the wiggly lines can be any real plasmon emanating from the primary photoinduced polarization vertices shown in Figs. \ref{Psi} and \ref{Psi2P} or their complex conjugates. (c) Illustration of the mechanism of plasmonic filter for electron yield obtained through temporal juxtaposition of the processes depicted in Fig. \ref{Psi} and the herein panel (a). This particular yield channel is open for $\hbar\omega_x>\hbar\omega_p>\phi$. } \label{plasmonyield} \end{figure} In this example we consider the boundary conditions pertaining to extremely short driving field pulse duration in comparison with the switching on time of the electron-plasmon interaction, viz. the limit $1/\eta\gg\sigma_{x}$. This is the situation opposite to the one sketched in Fig. \ref{SBC}, so that now a narrow pulse envelope with $t_x$ close to $0$ is embedded within the adiabatic development of the electron-plasmon interaction governed by $e^{-\eta|t|}$. To explore the inferences of the very short pulse limit in a tractable calculation we follow Keldysh\cite{Keldysh2017} and assume a transient profile ${\cal E}(\omega_x,t,\sigma_{x})\rightarrow{\cal E}_0\sigma_{x}\delta(t-t_x)$. Using this Ansatz in the simple example (\ref{eq:Nplexpanded}) we find for $t>t_x$ \begin{eqnarray} {\cal S}^{pl}(\varepsilon,t)&=& \left(\frac{{\cal E}_0\sigma_{x}}{\hbar}\right)^2 \sum_{\bf k_{1},k_{2}} |W_{\bf k_{1},k_{2}}|^{2}(1-n_{\bf k_{1}})n_{\bf k_{2}}\nonumber\\ &\times& \sum_{n=0}^{\infty}e^{-w_{\bf k_{1}}(t)}\frac{\left[w_{\bf k_{1}}(t)\right]^{n}}{n!}\delta(\varepsilon-n\hbar\omega_{p}), \label{eq:Nplcoh} \end{eqnarray} with \begin{eqnarray} w_{\bf k_{1}}(t)&=& \sum_{\bf \bar{k}_1,q}(1-n_{\bf \bar{k}_1})\left|V_{\bf \bar{k}_1,k_{1}}^{\bf q}\right|^{2}\nonumber\\ &\times& 2\left(\frac{1-\cos[(\epsilon_{\bf k_{1}}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{p})(t-t_x)]}{(\epsilon_{\bf k_{1}}-\epsilon_{\bf \bar{k}_1}-\hbar\omega_{p})^{2}}\right). \label{eq:v} \end{eqnarray} Observe in (\ref{eq:Nplcoh}) the modified "apparatus function" $({\cal E}_0\sigma_{x}/\hbar)^2 |W_{\bf k_{1},k_{2}}|^{2}(1-n_{\bf k_{1}})n_{\bf k_{2}}$ relative to expression (\ref{eq:apparatus}). Also notable is the absence of a monochromatic energy conservation in the primary excitation process [otherwise arising from the upper line in (\ref{eq:Nplexp})] which is here triggered by the white spectrum of $\delta$-function pulse. In the case of subcycle pump pulses the physically relevant limit of (\ref{eq:Nplcoh}) is for $t\rightarrow\infty$.\cite{Keldysh2017} Employing this limit expression (\ref{eq:v}) tends to (\ref{eq:wscatt}) in which the $\delta$-functions should be converted to nondiverging expressions via the appropriate regularization procedures described in Sec.\,S1 of Ref. [\onlinecite{SM}]. Hence, both extreme regimes of optical excitation considered in the previous and present subsections support the formation of coherent plasmonic states (\ref{eq:cohstate}) with a common distributional probability (\ref{eq:wscatt}). However, the distinct temporal boundary conditions imposed on the optical pumping processes give rise to different "apparatus functions" in (\ref{eq:NSBCcoh}) and (\ref{eq:Nplcoh}), and thereby to the pulse-specific weights of the respective plasmonic spectra $S^{pl}(\varepsilon,t)$. \section{Secondary hot plasmon-driven pumping of nonlinear electron yield} \label{sec:plasmonpump} The presence of a distribution of real plasmons in the system, either in the form of "coherent states", (\ref{eq:cohstate}) and (\ref{eq:cohscatt}), or a more general one, may have interesting implications on the dynamics of electronic excitations in the valence and conduction bands. Like the photons of applied EM field, real plasmons propagating in the system may also act, via the interaction $V^{e-pl}$ in (\ref{eq:Hsyst}), as a secondary pump field for excitation of electrons to states above the vacuum level. These electrons would emanate from the system with kinetic energies solely determined by the multiples of plasmon energy. This is illustrated by the diagrams (a) and (b) in Fig. \ref{plasmonyield}. The essential difference between the plasmon and photon drivings appears first in the meaning of wiggly lines denoting the exciting boson fields (plasmon vs. photon), and second, in the interaction vertices describing their coupling to electrons ($V_{\bf \bar{k}_1,k_1}^{\bf q}$ vs. $W_{\bf k_1,k_2}$, respectively). Another difference arises in temporal constraints on the two boson fields. In the case of photons they are imposed by the modulation ${\cal E}(\omega_x,t,\sigma_{x})$ of ultrashort pulses. By contrast, once the "hot" plasmons have been excited by the primary photoinduced electronic polarization they can be regarded as a steady continuous wave (cw) field whose action on electrons is limited only by the plasmon lifetime.\cite{Marini2002} The transient secondary plasmon-induced electron population $\langle N_{\bf \bar{k}}(t)\rangle$ in an outgoing state $|{\bf \bar{k}}\rangle$ can be most directly demonstrated by taking the mean value of the ${\bf \bar{k}}$-resolved electron occupation operator $\hat{N}_{\bf \bar{k}}=c_{\bf \bar{k}}^{\dag}c_{\bf \bar{k}}$ over the quasistationary multiplasmonic state $|\Psi(0)\rangle$ reached at $t=0$ (which requires $t_x+\sigma <0$) and subsequently developed by the long time limit $t\gg 0$ of the evolution operator (\ref{eq:Usyst}), viz. \begin{equation} \langle N_{\bf \bar{k}}(t)\rangle=\langle\Psi(0)|U^{syst}(t,0)^{\dag}\hat{N}_{\bf \bar{k}}U^{syst}(t,0)|\Psi(0)\rangle \label{eq:J_k} \end{equation} Now, in analogy with the $m$PP formalism (cf. Eqs. (4) and (10) in Ref. [\onlinecite{encyclopedia}]) expanding $U^{syst}(t,0)$ in powers of $V^{e-pl}$ gives the quadratic, quartic, etc. response expressions for plasmon-driven electron yields described next. \subsubsection{One plasmon-assisted electron yield} \label{sec:1yield} We shall formulate the rate of creation of hot electron population in bulk bands through one plasmon-driven processes within the quadratic response approach adopted from the theory of cw-driven 1PP.\cite{Ashcroft,Mahan1PP,Caroli} This is illustrated diagrammatically in Fig. \ref{plasmonyield}(a) and can be studied in two complementary usages of state-to-state resolved electron excitation rates \begin{equation} \gamma_{\bf k',k}^{(\hbar\omega_p)}=2\pi\sum_{\bf q}|V_{\bf k',k}^{\bf q}|^{2}n_{\bf k}(1-n_{\bf k'})\delta(\epsilon_{\bf k'}-\epsilon_{\bf k}-\hbar\omega_{\bf q}). \label{eq:gamma1resolved} \end{equation} First, one may be interested as how a particular initial electron state $|{\bf k}\rangle$ below $E_F$ contributes to the total hot electron production in the states $|{\bf k'}\rangle$. This would correspond to the CIS mode and the pertinent electron excitation rate is given by $\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}/\hbar$ where \begin{equation} \gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}=\sum_{\bf k'}\gamma_{\bf k',k}^{(\hbar\omega_p)}. \label{eq:GammaCIS} \end{equation} Alternatively, one may be interested in how much the manifold of initial states $\{|\bf k\rangle\}$ gives rise to plasmon-induced excitation of electrons into a particular state $|{\bf k'}\rangle$ above $E_F$. The corresponding constant final state (CFS) rate is given by $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}/\hbar$ where \begin{equation} \gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}=\sum_{\bf k}\gamma_{\bf k',k}^{(\hbar\omega_p)}. \label{eq:GammaCFS} \end{equation} Then, the expression \begin{equation} \gamma^{(\hbar\omega_p)}=\sum_{\bf k}\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}=\sum_{\bf k'}\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}, \label{eq:gamma1tot} \end{equation} gives the total electron excitation rate induced by one plasmon absorption. The transitions rates $\Gamma_{\bf k_1}^{>}$, $\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}$ and $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ are appropriate for description of electron-plasmon scattering dynamics subject to SBC defined in (\ref{eq:hierarchy}). Note also that the excitation rates (\ref{eq:GammaCIS}) or (\ref{eq:GammaCFS}), on the one hand, and (\ref{eq:Gammaplas}) on the other hand, are not simple time reversals of each other because in the discussed experiment they involve different subspaces of initial and final electron states. \subsubsection{Two plasmon-assisted electron yield} \label{sec:2yield} Multiplasmon-assisted electron yields require the existence of more than one real plasmon in the system. Once the real plasmons have been excited their annihilation can give rise to $2\hbar\omega_p$-features presented in Fig. \ref{AllAg}. This process can be visualized as a quartic response of the electron system to the plasmon field and is illustrated diagrammatically in Fig. \ref{plasmonyield}(b). Its description can be brought into a full analogy with the earlier formulated theory of 2PP processes presented in Sec. 2.1 of Ref. [\onlinecite{UebaGumhalter}], but with real plasmons instead of photons acting as the pump and probe fields. Hence, in the long time limit expression (\ref{eq:J_k}) produces the excitation rate describing the total energy-resolved two plasmon-induced electron yield \begin{widetext} \begin{equation} P(\epsilon)=\sum_{\bf k'',k}\left|2\pi\sum_{\bf k',q',q}\frac{V_{\bf k'',k'}^{\bf q'}V_{\bf k',k}^{\bf q}}{\epsilon_{\bf k'}-\epsilon_{\bf k}-\hbar\omega_{\bf q}+i\delta}\delta(\epsilon_{\bf k''}-\epsilon_{\bf k}-\hbar\omega_{\bf q}-\hbar\omega_{\bf q'}) \right|^{2}\delta(\epsilon-\epsilon_{\bf k''}), \label{eq:2plasPE} \end{equation} \end{widetext} where ${\bf k}$, ${\bf k'}$ and ${\bf k''}$ denote the quantum numbers of initial occupied, and intermediate and final unoccupied electron states, respectively. Expression on the RHS of (\ref{eq:2plasPE}) considerably simplifies in the case of weakly dispersive plasmons for which $\omega_{\bf q}\simeq\omega_p$. The most notable feature of the above described plasmon pumping mechanism is its monochromaticity fixed at the multiples of plasmon frequency $n\omega_p$, irrespective of the value of primary photon pump frequency $\omega_x$. Thus, as long as $\omega_x\geq \omega_p$ in 2PP the secondary plasmon-driven pumping of electronic excitations may become effective for the times $t>t_x+\sigma_{x}$. This enables the non-Einsteinian plasmonic CIS mode of photoelectron yield from the Fermi level at kinetic energies $2\hbar\omega_p-\phi>0$. \subsubsection{Plasmonic filter of electron yield} \label{sec:filter} The locking of electron emission energy at the plasmon energy can take place also in a temporal succession of the primary photon-induced plasmon generation shown schematically in Fig. \ref{Psi}, and the secondary plasmon-driven pumping of electronic polarization shown in Fig. \ref{plasmonyield}(a). One of the lowest order contributions (i.e. quadratic in photon and plasmon fields) to this combined process is sketched in Fig. \ref{plasmonyield}(c). This mechanism becomes effective provided the condition $\hbar\omega_x>\hbar\omega_p>\phi$ holds and the limit of electron detection times exceeding the characteristic times of the system is reached. The latter is required because here plasmons appear in the intermediate states past the primary optically induced electronic transitions and not as a "preexistent" distribution of real excitations already available in (\ref{eq:cohscatt}) for generation of electron yield via the processes sketched in panels (a) and (b) of Fig. \ref{plasmonyield}. Hence, for the plasmonic energy filter to be effective the interval between the plasmon creation times [the times of lower plasmon vertices in Fig. \ref{plasmonyield}(c)] and the electron detection time (the time of topmost vertex in the same plot) must be sufficiently long to allow for establishment of energy conservation between the successive processes. Most importantly, this does not imply the resonance condition $\hbar\omega_{x}\simeq \hbar\omega_{p}$, as dominantly discussed in plasmonics,\cite{Linic,Boriskina,Narang2016,Nordlander15,deAbajo16} but only the requirement $\hbar\omega_{x}\geq \hbar\omega_{p}$.\cite{RefCaroli} The same applies also to higher order processes [e.g. involving two-photon pumping succeeded by the subprocess illustrated in Fig. \ref{plasmonyield}(b)] albeit in a more complicated fashion. Thus, for CIS mPP from the states near the Fermi surface the electron yield may exhibit secondary peak structures located at multiples of the plasmon energy $n\hbar\omega_p$ above $E_F$ and their interpretation is possible only beyond the standard Einstein picture of photoeffect quantified through Eq. (\ref{eq:Einstein}). However, the coherence of pumping processes producing the electron yield can be quite different for photon and plasmon driven emissions. \section{Results and discussion} \label{sec:results} In Refs.\,[\onlinecite{MarcelPRL,AndiNJP,ACSPhotonics}] we have presented exhaustive descriptions of the Einsteinian one-electron aspects of 2PP spectra from three low index surfaces of Ag that is consistent with the common understanding of such processes. In this section we focus on applications of concepts and results obtained in Secs.\,\ref{sec:System}-\ref{sec:plasmonpump} to study the concomitantly discovered non-Einsteinian plasmon-induced electron yield from Ag surfaces reported and heuristically interpreted in Ref.\,[\onlinecite{ACSPhotonics}]. Accordingly, we shall consider primary electronic excitations induced by photon energies in the range shown on the horizontal axes in Fig.\,\ref{AllAg}. Such low photon energies limit the truncated coherent plasmonic states generated in quadratic response by one photon-induced electronic polarization to contain maximum one real plasmon [cf. discussion preceding Eq. (\ref{eq:chopoff})]. Multiplasmon states are then generated in higher order response, i.e. quartic, etc (cf. Fig. \ref{Psi2P}). Also, as the on-the-energy-shell limits of transition rates (\ref{eq:Gammaplasdiff}) and (\ref{eq:Gammaplas}) are established in few femtoseconds past the instant of plasmon creation\cite{FarisPRL,LazicPRL,PSS2012} we shall henceforth consider only the results from this limit. \subsection{Band structure and dielectric properties of Ag} \label{sec:bandstructure} Calculations of the various quasiparticle excitation rates (\ref{eq:Gammaplas}), (\ref{eq:GammaCIS}), (\ref{eq:GammaCFS}) and (\ref{eq:2plasPE}) require realistic input quantities for the underlying minimal plasmon model. This specifically concerns the Ag electronic band structure and the ensuing dielectric properties. Figure \ref{preliminaries} illustrates the computed electronic band structure over the first Brillouin zone (BZ) of bulk Ag and the thereof derived dielectric function (for details of ensuing calculation procedures see Ref. [\onlinecite{ACSPhotonics}] and Sec. S2 of Supplementary Material\cite{SM} building on Refs. [\onlinecite{bib:qe1,bib:pseudo,bib:pbe,bib:novko1,bib:novko2,bib:caruso1,bib:epw,bib:caruso2,bib:wan90,bib:autowan90,bib:gw1,bib:gw2}]). The extrema and kinks in the band dispersion curves around the endpoints of arrows denoting interband transitions (ib) in Fig. \ref{preliminaries}(a) give rise to maxima in the electronic density of states (DOS). Such maxima can strongly affect optical and plasmonic transition matrix elements in specific energy regions (see Sec. S2.C in Ref. [\onlinecite{SM}]) and thereby also the excitation rates (\ref{eq:Gammaplas}), (\ref{eq:GammaCIS}) and (\ref{eq:GammaCFS}). \begin{figure*}[tb] \rotatebox{0}{ \epsfxsize=14cm \epsffile{plas2PP_Fig7}} \caption{(a): First Brillouin zone of \emph{fcc} Ag lattice. Green dots denote the high symmetry points of the zone and the red rectangle its section in the (110) plane. (b): Band structure of Ag along the high symmetry points in the first Brillouin zone computed within the GW-framework. The most prominent interband transitions (ib) in the energy range $\sim$ 4 eV are denoted by vertical arrows. Earlier GW results (circles) are taken from Ref. [\onlinecite{Louie15}]. Experimental points are from Refs. [\onlinecite{exp1}] and [\onlinecite{exp2}]. (c): Real and imaginary parts of the RPA dielectric function for ${\bf q=0}$ calculated for the band structure displayed in (a). (d): Comparison of the thereof calculated loss function with the experimental data from Ref. [\onlinecite{Palik}]. (e): Bulk plasmon dispersion in Ag calculated in RPA for the band structure shown in (b). Circles and diamonds denote experimental points of plasmon dispersion.[\onlinecite{lindau,zacharias}]} \label{preliminaries} \end{figure*} \subsection{Hot plasmon generation} \label{sec:pumpingplasmon} Figure \ref{wk1}(a) shows the locations of ${\bf k}$-states along the high symmetry directions in the first BZ of Ag from which the decay of hot electrons gives rise to significant plasmon generation rates $\Gamma_{\bf k}^{>}$ defined in Eq. (\ref{eq:Gammaplas}). Depending on the initial electron ${\bf k}$-state these values can be as large as $\sim 120$ meV. This implies that plasmons can be excited already within $5-6$ femtoseconds past the primary optical polarization excitation by the pump pulse, i.e. well within the duration of laser pulses used in experiments.\cite{MarcelPRL,ACSPhotonics} Analogous calculations of $\Gamma_{\bf k}^{<}$ corresponding to plasmon generation by holes scattered from the initial $s,p$-band states $|{\bf k}\rangle$ extending to more than 4 eV below $E_F$ give negligible values. This finding provides {\it a fortiori} support to the neglect of plasmon emission by holes photoinduced in $s,p$-bands that was assumed in the derivation of expression (\ref{eq:Ndiag}). Good agreement between the angular integrated $\Gamma_{\bf k}^{>}$ shown in Fig. \ref{wk1}(b) and the analogous GW-derived quantity\,\cite{SM} justifies the use of minimal electron-plasmon model in the present calculations. \subsection{$\hbar\omega_p$-yield} \label{sec:plasmon1} \begin{figure}[tb] \rotatebox{0}{ \epsfxsize=8cm \epsffile{plas2PP_Fig8}} \caption{ (a): Electronic band structure along the high symmetry points of the first Brillouin zone in Ag. The magnitude of the band- and momentum-resolved photoexcited quasiparticle decay rate due to one plasmon emission $\Gamma_{\bf k}$, Eq. (\ref{eq:Gammaplas}), is quantified by color bar in the inset. (b): Angle-integrated $\Gamma_{\bf k}$ as a function of energy. Note the threshold at $\sim E_F+\hbar\omega_p$ and very low values below $E_F$ and the $d$-band edge. Yellow line denotes the result of analogous GW-based calculation which incorporates the full linear electronic density response of the system.} \label{wk1} \end{figure} (i) {\it Bulk transitions.} Figure \ref{hotelectron} illustrates the rates $\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}$ and $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ of creation of hot electron population in bulk Ag through the plasmon driven processes sketched in Fig. \ref{plasmonyield}(a) (for methods of calculation see Sec.\,S2 in Ref.\,[\onlinecite{SM}]). Panel (a) shows the locations of ${\bf k}$-states in the BZ for which $\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}$ describing electron excitation from a fixed occupied state $|{\bf k}\rangle$ below $E_F$ to the states $|{\bf k'}\rangle$ above $E_F$ is large. Panel (b) shows equivalent information on the energy scale of the occupied bands. Here two domains with maximum electron excitation rates are clearly discernible. They correlate with the enhanced DOS at the upper edges of occupied $d$-bands and the very narrow energy window just below $E_F$.\cite{Phillips} The latter case of plasmon-induced electron excitation from $E_F$ which promotes electrons to the states of energy $\sim E_F+\hbar\omega_p$ is overwhelmingly dominant and offers explanation of the $\hbar\omega_p$ peak in the CIS 1PP spectra of Ref. [\onlinecite{Horn}]. This also supports the conjectures of dominant plasmon-induced electron excitation from $E_F$ made in Ref. [\onlinecite{Hopfield1965}]. Panel (c) of Fig. \ref{hotelectron} shows a complementary picture in which $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ describing electron excitation from all occupied states $|{\bf k}\rangle$ below $E_F$ to a fixed final state $|{\bf k'}\rangle$ above $E_F$ is maximum. Again, two distinct types of plasmon-induced interband transitions are dominant. The first are from the upper edge of $d$-band to $E_F$, and thus unobservable in photoemission. The second are from $E_F$ to the states with energy $\sim E_F+\hbar\omega_p$, and thus observable in the experiment described in Ref. [\onlinecite{Horn}]. The results presented in Figs. \ref{preliminaries}-\ref{hotelectron} above quantify the sequence (\ref{eq:sequence}) leading to the energy-locked excitation of electrons to energy $\hbar\omega_p$ above the Fermi level of the irradiated system. Essentially, this sequence appears on the LHS, and its conjugate on the RHS, of the diagram shown in Fig. \ref{plasmonyield}(c). Here the vertex encompassing one incoming and one outgoing red electron line and one wavy plasmon line can be identified with the plasmaron vertex.\cite{Chis} Note that the two-hole state $|\rm{polarization}''\rangle$ in (\ref{eq:sequence}) is mapped in the appearance of two hole dashed lines in Fig. \ref{plasmonyield}(c). The measures of durations of pumping of the involved excited states are $\sigma_{x}$, $\hbar/\Gamma_{\bf k}^{>}$ and $\hbar/\gamma_{\bf k'}^{CFS}$, respectively. The occurence of $\hbar\omega_p$ component of 1PP spectrum from Ref. [\onlinecite{Horn}], that can not be explained by relations (\ref{eq:Einstein}) or (\ref{eq:satellite}), is now interpretable by the scenario (\ref{eq:sequence}) and supported by the results of calculations presented in Figs. \ref{preliminaries}-\ref{hotelectron}. \begin{figure}[tb] \rotatebox{0}{ \epsfxsize=8cm \epsffile{plas2PP_Fig9}} \caption{(a): Band structure of Ag with electronic transitions ib2 and ib1 contributing to CIS resolved rates $\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}$ [Eq. (\ref{eq:GammaCIS})] for plasmon-induced hot electron generation processes sketched in Fig. \ref{plasmonyield}(a). The magnitudes of CIS and CFS transition rates are quantified by color bars in the insets. (b): Angle-integrated $\gamma_{\bf k}^{{\rm CIS}(\hbar\omega_p)}$ shown on the energy scale. The CIS amplitudes for transitions from the occupied states closely below $E_F$ to unoccupied bands above $E_F$ are dominant. (c): Electronic transitions ib2 and ib1 contributing to CFS resolved plasmon-induced hot electron generation rates $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ [Eq. (\ref{eq:GammaCFS})] from all initial to a fixed final ${\bf k}$-state at $\sim E_F$ and final ${\bf k'}$-state at $\sim E_F+\hbar\omega_p$, respectively. (d): Angle integrated $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ shown on the energy scale. Here the dominant amplitudes are for transitions to $E_F$ and $\sim E_F+\hbar\omega_p$. } \label{hotelectron} \end{figure} (ii) {\it Bulk and surface transitions.} To investigate and illustrate the effects of the surface on plasmon-assisted electron yield we carry out calculations analogous to those in (i) above by projecting the bulk band states onto the Ag(110) crystal plane (for calculational details see Sec.\,S2 E in Ref.\,[\onlinecite{SM}]). Figure \ref{hotelectron(110)}(a) illustrates as how due to the breakdown of translational symmetry perpendicular to the surface new states appear in the gaps of the bulk band structure. The concomitant relaxation of momentum conservation perpendicular to the surface opens new channels of hot electron excitation by plasmon absorption. This gives rise to the enhancements of hot electron excitation rates $\gamma_{\bf k}^{CIS(\hbar\omega_p)}$ and $\gamma_{\bf k'}^{CFS(\hbar\omega_p)}$, as can be deduced from the intensities of $\gamma_{\bf k'}^{CFS(\hbar\omega_p)}$ shown in Fig. \ref{hotelectron(110)}(a). The contributions of surface-induced channels to plasmon-assisted hot electron generation can be best visualized from the comparison of angular integrated surface enhanced $\gamma_{\bf k'}^{CFS(\hbar\omega_p)}$ shown in Fig. \ref{hotelectron(110)}(b) with its bulk counterpart shown in Fig. \ref{hotelectron}(d). The different vertical scales in the two plots reveal different magnitudes of the respective quantities. Furthermore, the surface-induced channels give rise to the nonvanishing intensity of $\gamma_{\bf k'}^{CFS(\hbar\omega_p)}$ also in the intermediate interval $(E_F<\epsilon<E_F+\hbar\omega_p)$ but the largest contributions at $E_F$ and $E_F+\hbar\omega_p$ persist, in accord with experimental trends observed for the other low index Ag surfaces.\cite{Horn} The comparison between bulk and surface results nicely illustrates how surface effects can be of paramount importance in descriptions of hot carrier generation in plasmonic materials. \begin{figure}[tb] \rotatebox{0}{ \epsfxsize=7.5cm \epsffile{plas2PP_Fig10}} \caption{(a): Band structure of Ag across the (110)-surface projected bulk Brillouin zone showing the intensities of surface enhanced $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ quantified by the color bar. (b): Angle integrated surface enhanced $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ from (a) shown on the energy scale. Shaded area spans the width of $E_F\rightarrow E_F+\hbar\omega_p$ plasmon-induced electronic transitions. Notable is the surface induced enhancement of all displayed quantities relative to the purely bulk induced values shown in Fig. \ref{hotelectron}. (c): Isoenergetic distribution of surface enhanced $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ averaged around $\epsilon_{\bf k'}\simeq E_F+\hbar\omega_p$[\onlinecite{averaged}] and shown across the projected BZ ($a=4.06\,{\rm \AA}$). The highest intensities are around $\bar{\Gamma}$, $\bar{\rm Y}$ and $\bar{S}$ points. } \label{hotelectron(110)} \end{figure} \subsection{$2\hbar\omega_p$-yield} \label{sec:plasmon2} The 2PP features shown in Fig. \ref{AllAg}, recorded with pulse frequencies $\omega_x$ obeying $\omega_p\leq\omega_x\leq 2\omega_p$, can not be visualized within a unique excitation sequence of the type (\ref{eq:sequence}) because the excited states giving rise to the $2\hbar\omega_p$-yield may be reached via several different excitation paths involving two plasmons. This is symbolically represented by Fig. \ref{Psi2P} after exclusion of the plasmon lines emanating from the holes. However, concepts analogous to those employed in the previous subsection to interpret the $\hbar\omega_p$-yield may be extended to interpret the $2\hbar\omega_p$ features as well. The point of departure is the polarization-induced formation of plasmonic coherent states governed by $\Gamma_{\bf k_1}^{>}$ illustrated in Fig. \ref{wk1}. The thus generated real plasmons may give rise to primary and secondary plasmon pumping over virtual and real intermediate states, i.e. nonresonant and resonant intermediate excitation paths. \begin{figure}[tb] \rotatebox{0}{\epsfxsize=8cm \epsffile{plas2PP_Fig11}} \caption{(a): Band structure of Ag projected onto (110) surface showing the points with maximal intensities of $\gamma_{\bf k''}^{CFS(2\hbar\omega_p)}$ (see color bar for computed values). The second plasmon-induced electronic transitions start from resonant intermediate states lying in the orange shaded energy window at $\sim E_F+\hbar\omega_p$ that have been occupied in one-plasmon absorption processes illustrated in panels (c) and (d) of Fig. \ref{hotelectron(110)}. (b): Angle integrated $\gamma_{\bf k''}^{CFS(2\hbar\omega_p)}$ from (a) shown on the energy scale. Shaded area straddles the width of plasmonically-induced electronic transitions to $\sim E_F+2\hbar\omega_p$. (c): Isoenergetic distribution of $\gamma_{\bf k''}^{{\rm CFS}(2\hbar\omega_p)}$ averaged around $\epsilon_{\bf k''}\simeq E_F+2\hbar\omega_p$[\onlinecite{averaged}] and shown across the projected BZ ($a=4.06\,{\rm \AA}$). The accumulation of highest intensities is in the vicinity of $\bar{\Gamma}$ and $\bar{\rm Y}$ points. } \label{hotelectron2(110)} \end{figure} In the calculations of the one plasmon-driven electronic excitations reported in subsection \ref{sec:plasmon1} above we have identified a rich phase space of resonant states available for the one-plasmon pumping stage [cf. Fig. \ref{hotelectron}(c)]. These states also support resonant intermediate channels in the two-plasmon driven pumping of electrons from $E_F$. Since dominant contributions to dissipative processes come from on-the-energy-shell transitions we shall henceforth consider only the manifold of sequential electron excitation paths via the resonant intermediate states.\cite{KeldyshIonization} Therefore, the second plasmonically induced electronic excitation from an intermediate resonant state $|{\bf k'}\rangle$ to a final state $|{\bf k''}\rangle$ is characterized by the state-to-state excitation rate \begin{equation} \gamma_{\bf k'',k'}^{(2\hbar\omega_p)}=2\pi\sum_{\bf q'}|V_{\bf k'',k'}^{\bf q'}|^{2}n_{\bf k'}(1-n_{\bf k''})\delta(\epsilon_{\bf k''}-\epsilon_{\bf k'}-\hbar\omega_{\bf q'}). \label{eq:gamma2resolved} \end{equation} Here the energies $\epsilon_{\bf k'}\simeq E_F+\hbar\omega_p$ of the intermediate resonant states $|{\bf k'}\rangle$ occupied after the first plasmon absorption lie within the orange horizontal stripe shown in Fig. \ref{hotelectron2(110)}(a). Expression (\ref{eq:gamma2resolved}) is an analog of the one-plasmon induced transition rate $\gamma_{\bf k',k}^{(\hbar\omega_p)}$ defined in (\ref{eq:gamma1resolved}). For the second plasmon absorption processes the CFS rate reads \begin{equation} \gamma_{\bf k''}^{{\rm CFS}(2\hbar\omega_p)}=\sum_{\bf k'}\gamma_{\bf k'',k'}^{(2\hbar\omega_p)}. \label{eq:gamma2} \end{equation} This description is akin to treating the second step of 2PP proposed in Ref. [\onlinecite{Freericks}], but with hot plasmons instead of photons playing the role of probe field in the middle vertices in Fig. \ref{plasmonyield}(b). Panel (a) in Fig. \ref{hotelectron2(110)} shows the values of excitation rates $\gamma_{\bf k''}^{{\rm CFS}(2\hbar\omega_p)}$ (\ref{eq:gamma2}) for the second plasmon-induced electronic transitions in the projection of bulk BZ of Ag onto the (110) crystal surface. Here all transitions commence in the ${\bf k'}$-states within the orange stripe at the energy $\epsilon_{\bf k'}\simeq E_F+\hbar\omega_p$ occupied in one-plasmon absorption processes described by $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$, and terminate in the ${\bf k''}$-states with $\epsilon_{\bf k''}\simeq E_F+2\hbar\omega_p$. Notable are the magnitudes of $\gamma_{\bf k''}^{{\rm CFS}(2\hbar\omega_p)}$ in the final electronic states with pronounced surface character (see surface state bands between 7.1\,eV and 7.3\,eV in Fig.\,S1 of Ref.\,[\onlinecite{SM}]). Panel (b) shows the values of angle integrated $\gamma_{\bf k''}^{{\rm CFS}(2\hbar\omega_p)}$ on the energy scale, displaying a prominent peak at the energy $\sim E_F+2\hbar\omega_p$. Panel (c) shows the isoenergetic distribution of the values of $\gamma_{\bf k''}^{{\rm CFS}(2\hbar\omega_p)}$ across the (110) surface projected BZ obtained from averaging around $\epsilon_{\bf k''}\simeq E_F+2\hbar\omega_p$.\cite{averaged} The accumulations of significant contributions around the $\bar{\Gamma}$ and $\bar{\rm Y}$ points indicate a narrower phase space of final states than in the case of one plasmon-induced transitions displayed in Fig. \ref{hotelectron(110)}(c). Taking into account only resonant processes the {\it joint} CFS rate $\gamma_{\bf k''}^{{\rm res}(2\hbar\omega_p)}$ of two plasmon-induced electron excitations is in the present situation obtained from (\ref{eq:2plasPE}) by retaining only the on-the-energy shell components of intermediate state propagations contributing to this expression.\cite{NozieresI} This gives \begin{equation} \gamma_{\bf k''}^{{\rm res}(2\hbar\omega_p)} \simeq \frac{\pi}{2}\rho_{f}(\epsilon_{\bf k''})\sum_{\bf k'} \gamma_{\bf k'',k'}^{(2\hbar\omega_p)}\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}. \label{eq:gamma2omega_p} \end{equation} Here $\rho_{f}(\epsilon_{\bf k''})$ is the density of electron states around the final ${\bf k''}$-vector for which $\epsilon_{\bf k''}\simeq E_F+2\hbar\omega_p$, and $\gamma_{\bf k'}^{{\rm CFS}(\hbar\omega_p)}$ and $\gamma_{\bf k'',k'}^{(2\hbar\omega_p)}$ are defined in Eqs. (\ref{eq:GammaCFS}) and (\ref{eq:gamma2resolved}), respectively. The peculiar prefactor $\frac{\pi}{2}$ arises from the combination of different temporal boundary conditions governing the intermediate and final state electron propagation (i.e. over semi-infinite vs. infinite time interval, respectively). Using (\ref{eq:gamma2omega_p}) we can pinpoint the essential features of two plasmon-induced electron emission illustrated diagramatically in Fig. \ref{plasmonyield}(b) and leading to non-Einsteinian electron yields. We first observe that the intensity bottleneck of resonant $2\hbar\omega_p$-electron emission described by the rate (\ref{eq:gamma2omega_p}) is the overlap of the phase space of intermediate resonant states $|{\bf k'}\rangle$ with the phase space of final states $|{\bf k''}\rangle$. This overlap strongly depends on the magnitude of plasmon wave vector cutoff. In the case of Ag(110) this is illustrated in Figs. \ref{hotelectron(110)}(c) and \ref{hotelectron2(110)}(c) and favours the $2\hbar\omega_p$-emission in the directions of ${\bf k''}$ whose components $k_{x}''$ and $k_{y}''$ are restricted to the vicinity of $\bar{\Gamma}$ point. Here the trends noted in Fig. \ref{hotelectron2(110)} are further corroborated by the evaluation of consecutive two plasmon-induced resonant electron excitation rate (\ref{eq:gamma2omega_p}). The results are presented in Fig. \ref{res2wp_den}. Panel (a) shows the distribution of the values of $\gamma_{\bf k''}^{{\rm res}(2\hbar\omega_p)}$ within the (110)-surface projected BZ of Ag obtained from averaging around $\epsilon_{\bf k''}\simeq E_F+2\hbar\omega_p$.\cite{averaged} Panels (b) and (c) show histograms of the same data along the rectangular paths marked in the projected zone. All panels illustrate anisotropic accumulation of the largest values of $\gamma_{\bf k''}^{{\rm res}(2\hbar\omega_p)}$ around the $\bar{\Gamma}$ point, thereby favouring the majority of two plasmon-assisted photoemission intensity around the normal to the Ag(110) surface. The corresponding magnitudes of the rates characterizing the two plasmon-induced electron excitations indicate that these events can take place within the duration of the ultrashort laser pulses. A good agreement between these theoretical predictions and experimental findings presented in Refs. [\onlinecite{MarcelPRL,ACSPhotonics}] and Fig. \ref{AllAg} substantiates the mechanism of channeling of broad band optically induced electronic excitations into electron yield "monochromatized" to within the narrow range of plasmon dispersion. \begin{figure}[tb] \rotatebox{0}{\epsfxsize=7.5cm \epsffile{plas2PP_Fig12}} \caption{(a): Distribution of the magnitudes of resonant two plasmon-induced electron excitation rates (\ref{eq:gamma2omega_p}) averaged around $\epsilon_{\bf k''}\simeq E_F+2\hbar\omega_p$ [\onlinecite{averaged}] and shown across the (110) surface-projected bulk BZ of Ag (see color bar for absolute values). (b) and (c): Histograms of the same data along the blue and red rectangular paths marked in (a).} \label{res2wp_den} \end{figure} \section{Summary and outlook} \label{sec:summary} Following the above theoretical tour de force that establishes a mechanism for non-Einsteinin plasmonic photoemission, it is important to consider its practical implications. The decay of plasmons into hot electrons is an intensely invoked but hardly directly investigated theme in physics and chemistry.\cite{Narang2016,Marini2002,Atwater14,Nordlander15,Louie15,Halas15,deAbajo16,Atwater16,Khurgin} The experimental measurement of plasmonic photoemission\cite{MarcelPRL,ACSPhotonics} and its theoretical interpretation presented herein imply that the excitation of the bulk plasmonic mode in the single crystal Ag samples may lead to transfer of the entire plasmon energy to selectively promote electrons from the Fermi level to $E_F+n\hbar\omega_p$, rather than to a distribution that is defined by the density of states and energy conservation, as is commonly believed.\cite{Narang2016,Marini2002,Atwater14,Nordlander15,Louie15,Halas15,deAbajo16,Atwater16,Khurgin} There has been evidence that this selectivity occurs in alkali metals\cite{Gornyi,SmithSpicer,Smith,SmithFisher} and we have demonstrated that it also occurs in single crystalline Ag. If indeed the plasmonic response of metals can transfer energy entirely to hot electrons, rather than distributing it also among the holes, implies that plasmonic energy harvesting can be far more economical than commonly assumed. Therefore, it is important to consider whether the reported plasmonic photoemission is a peculiarity of the studied physical system or a common response of metals. The evidence so far suggests that it is indeed a plasmonic response of metals, based on what has been reported for alkali metals,\cite{Gornyi,SmithSpicer,Smith,SmithFisher} single crystalline Ag films,\cite{Horn} and Ag(111), (100), and (110) crystals.\cite{MarcelPRL,ACSPhotonics} Our starting model, however, is not constrained by momentum conservation, so it does not preclude similar plasmonic decay into hot electrons for surface plasmon polaritons or bulk plasmons in nanostructured Ag. Indeed, a recent report on 2PP from size selected Ag clusters, where momentum conservation is relaxed,\cite{Douketis} confirms our claim of plasmonic photoemission from Ag.\cite{Shibuta} Moreover, our previous studies of two-photon photoemission from silver nanoclusters on TiO$_2$ and graphite surfaces, which were interpreted as originating from interface states,\cite{TanArgondizzo,TanLiu,TanDai} most likely include strong enhancement from plasmonic excitation of electrons from $E_F$. Thus, our experimental and theoretical evidence for plasmonic photoemission invites a broader reinterpretation of ultrafast electron dynamics in metals. To this end, we leave the open question whether an unresolved anomaly in hot electron dynamics in Cu\cite{HertelKnoesel,OgawaNagano,Pawlik,KnoeselHotzel,Weida} can be attributed to processes that could be interpreted as interband polarization having a similar role as plasmonic photoemission in Ag. In the case of Cu, the apparent hot electron lifetimes for photon excitation above the interband threshold have appeared to be anomalous, because they do not follow the Fermi liquid theory.\cite{OgawaNagano,Pawlik} Wolf and coworkers have attributed this anomaly to the d- to sp-band excitation,\cite{KnoeselHotzel} which enables the d-band holes to decay by Auger recombination to generate hot electrons having energies of $>2$ eV above $E_F$. While this can explain some results, there has never been clear evidence that there is a delayed rise of hot electrons reflecting the lifetimes of d-band holes.\cite{Weida} One can conceive that as the $d-$ to $sp$-band excitation channel becomes energetically accessible, the interband polarization of $d-$holes decays by a coherent process, analogous to plasmonic photoemission in Ag, by transferring the hole energy to electrons at $E_F$, rather than through an incoherent Auger process. To summarize, we have studied the response of a system of coupled electrons and plasmons in a metal to perturbations exerted by a laser pulse on the electrons. We demonstrate that the wavefunction of the pulse-driven system incorporates the truncated plasmonic coherent states whose decay may generate complementary channels of plasmonically assisted electron emission from metal bands. This new paradigm enables us to elucidate the origin and quantify the intensity of "non-Einsteinian" plasmon energy-locked electron yield from the Fermi surface observed in photoemission from Ag(110).\cite{ACSPhotonics} This reveals an as yet unexplored mode of "monochromatized" electron emission in optical energy conversion and harvesting\cite{Boriskina,Narang2016} from broad band radiation that so far has escaped the deserved scrutiny. This work is dedicated to the memory of Hiromu Ueba, scientist and friend, who pioneered the theory of 2PP from surfaces.\cite{UebaSS} \acknowledgments D. N. acknowledges the financial support from the Croatian Science Foundation (Grant no. UIP-2019-04-6869) and from the European Regional Development Fund for the ``Center of Excellence for Advanced Materials and Sensing Devices'' (Grant No. KK.01.1.1.01.0001). A.L. and H.P. acknowledge the financial support from DOE-BES Division of Chemical Sciences, Geosciences, and Biosciences Grant No. DE-SC0002313. M.R. acknowledges funding from the Alexander von Humboldt Foundation. Computational resources provided by the DIPC Computing Center are also acknowledged.

BkiUfmo5qg5Ds8lAVh6X

\section{Introduction} Reinforcement learning (RL) algorithms have been studied extensively in the past decade. When the state and action spaces are large or even infinite, traditional tabular RL algorithms \citep[e.g.,][]{watkins1989learning, jaksch2010near, azar2017minimax} become computationally inefficient or even intractable. To overcome this limitation, modern RL algorithms with function approximation are proposed, which often make use of feature mappings to map states and actions to a low-dimensional space. This greatly expands the application scope of RL. While RL can provide personalized service such as online recommendation and personalized advertisement, existing algorithms rely heavily on user's sensitive data. Recently, how to protect sensitive information has become a central research problem in machine learning. For example, in online recommendation systems, users want accurate recommendation from the online shopping website to improve their shopping experience while preserving their personal information such as demographic information and purchase history. Differential privacy (DP) is a solid and highly successful notion of algorithmic privacy introduced by \citet{dwork2006calibrating}, which indicates that changing or removing a single data point will have little influence on any observable output. However, DP is vulnerable to membership inference attacks \citep{shokri2017membership} and has the risk of data leakage. To overcome the limitation of DP, a stronger notion of privacy, \textit{local differential privacy} (LDP), was introduced by \citet{kasiviswanathan2011can,duchi2013local}. Under LDP, users will send privatized data to the server and each individual user maintains its own sensitive data. The server, on the other hand, is totally agnostic about the sensitive data. (Local) differential privacy has been extensively studied in multi-armed bandit problems, which can be seen as a special case of MDPs with unit planing horizon and without state transition. Nevertheless, \citet{shariff2018differentially} proved that the standard DP is incompatible in the contextual bandit setting, which will yield a linear regret bound in the worst case. Therefore, they studied a relaxed version of DP named \textit{jointly differentially private} (JDP), which basically requires that changing one data point in the collection of information from previous users will not have too much influence on the decision of the future users. Recently, LDP has attracted increasing attention in multi-armed bandits. \citet{gajane2018corrupt} are the first to study LDP in stochastic multi-armed bandits (MABs). \citet{chen2020locally} studied combinatorial bandits with LDP guarantees. \citet{zheng2020locally} studied both MABs and contextual bandits, and proposed a locally differentially private algorithm for contextual linear bandits. However, differentially private RL is much less studied compared with bandits, even though MDPs are more powerful since state transition is rather common in real applications. For example, a user may click a link provided by the recommendation system to visit a related webpage, which can be viewed as state transition. In tabular RL, \citet{vietri2020private} proposed a $\varepsilon$-JDP algorithm and proved an $\tilde{\cO}(\sqrt{H^4SAT} + SAH^3(S + H)/\varepsilon)$ regret, where $H$ is the length of the planning horizon, $S$ and $A$ are the number of states and actions respectively, $K$ is the number of episodes, and $T = KH$ is the number of interactions with the MDP. Recently, \citet{garcelon2020local} designed the first LDP tabular RL algorithm with an $\tilde{\cO}(\max\cbr{H^{3/2}S^2A\sqrt{T}/\varepsilon, HS\sqrt{AT}})$ regret. However, as we mentioned before, tabular RL algorithms suffer from computational inefficiency when applied to large state and action spaces. Therefore, a natural question arises: \begin{center} \textit{Can we design a privacy-preserving RL algorithm with linear function approximation while maintaining the statistical utility of data?} \end{center} In this paper, we answer this question affirmatively. More specifically, we propose a locally differentially private algorithm for learning linear mixture MDPs \citep{jia2020model, ayoub2020model,zhou2020provably} (See Definition~\ref{definMDP} for more details.), where the transition probability kernel is a linear function of a predefined $d$-dimensional feature mapping over state-action-next-state triple. The key idea is to inject Gaussian noises into the sensitive information in the UCRL-VTR backbone, and the main challenge is how to balance the tradeoff between the Gaussian perturbations for privacy preservation and the utility to learn the optimal policy. Our contributions are summarized as follows. \begin{itemize}[leftmargin = *] \item We propose a novel algorithm named LDP-UCRL-VTR to learn the optimal value function while protecting the sensitive information. We show that our algorithm guarantees $\rbr{\varepsilon, \delta}$-LDP and enjoys an $\tilde{\cO}( d^{5/4}H^{7/4}T^{3/4}\rbr{\log(1/\delta)}^{1/4}\sqrt{1/\varepsilon})$ regret bound, where $T$ is the number of rounds and $H$ is the length of episodes. To our knowledge, this is the first locally differentially private algorithm for RL with linear function approximation. \item We prove a $\Omega(dH\sqrt{T}/\rbr{e^{\varepsilon}(e^{\varepsilon}-1)})$ lower bound for learning linear mixture MDPs under $\varepsilon$-LDP constraints. Our lower bound suggests that the aforementioned upper bound might be improvable in some parameters (i.e., $d,H,T$). As a byproduct, our lower bound also implies $\Omega(d\sqrt{T}/\rbr{e^{\varepsilon}(e^{\varepsilon}-1)})$ lower bound for $\varepsilon$-LDP contextual linear bandits. This suggests that the algorithms proposed in \citet{zheng2020locally} might be improvable as well. \end{itemize} \paragraph{Notation} We use lower case letters to denote scalars, lower and upper case bold letters to denote vectors and matrices. For a vector $\xb\in \RR^d$ , we denote by $\|\xb\|_1$ the Manhattan norm and denote by $\|\xb\|_2$ the Euclidean norm. For a semi-positive definite matrix $\bSigma$ and any vector $\xb$, we define $\nbr{\xb}_{\bSigma}:=\nbr{\bSigma^{1/2}\xb}_2 = \sqrt{\xb^{\top} \bSigma \xb}$. $\mathds{1}(\cdot)$ is used to denote the indicator function. For any positive integer $n$, we denote by $[n]$ the set $\cbr{1, \dots, n}$. For any finite set $\cA$, we denote by $\abr{\cA}$ the cardinality of $\cA$. We also use the standard $\cO$ and $\Omega$ notations, and the notation $\tilde{\cO}$ is used to hide logarithmic factors. We denote $D_{1:h} = \cbr{D_1, \dots, D_h}$. For two distributions $p$ and $p^{\prime}$, we define the Kullback–Leibler divergence (KL-divergence) between $p$ and $p^{\prime}$ as follows: \[\text{KL}(p, p^{\prime}) = \int p(\zb)\log \frac{p(\zb)}{p^{\prime}(\zb)}\ud \zb.\] \section{Related Work} \textbf{Reinforcement Learning with Linear Function Approximation} Recently, there have been many advances in RL with function approximation, especially in the linear case. \citet{jin2020provably} considered linear MDPs where the transition probability and the reward are both linear functions with respect to a feature mapping $\bphi: \cS\times \cA \rightarrow \RR^d$, and proposed an efficient algorithm for linear MDPs with $\tilde{\cO}(\sqrt{d^3H^3T})$ regret. \citet{yang2019sample} assumed the probabilistic transition model has a linear structure. They also assumed that the features of all state-action pairs can be written as a convex combination of the anchoring features. \citet{wang2019optimism} designed a statistically and computationally efficient algorithm with generalized linear function approximation, which attains an $\tilde{\cO}(H\sqrt{d^3T})$ regret bound. \citet{zanette2020learning} proposed RLSVI algorithm with $\tilde{\cO}(d^2\sqrt{H^4T})$ regret bound under the linear MDPs assumption. \citet{jiang2017contextual} studied a larger class of MDPs with low Bellman rank and proposed an OLIVE algorithm with polynomial sample complexity. Another line of work considered linear mixture MDPs (a.k.a., linear kernel MDPs) \citep{jia2020model, ayoub2020model,zhou2020provably}, which assumes the transition probability function is parameterized as a linear function of a given feature mapping on a triplet $\bpsi:\cS \times \cA \times \cS \to \RR^d$. \citet{jia2020model} proposed a model-based RL algorithm, UCRL-VTR, which attains an $\tilde{\cO}(d\sqrt{H^3T})$ regret bound. \citet{ayoub2020model} considered the same model but with general function approximation, and proved a regret bound depending on Eluder dimension \citep{russo2013eluder}. \citet{zhou2020nearly} proposed an improved algorithm which achieves the nearly minimax optimal regret. \citet{he2020logarithmic} showed that logarithmic regret is attainable for learning both linear MDPs and linear mixture MDPs. \noindent\textbf{Differentially Private Bandits} The notion of \textit{differential privacy} (DP) was first introduced in \citet{dwork2006calibrating} and has been extensively studied in both MAB and contextual linear bandits. \citet{basu2019differential} unified different privacy definitions and proved an $\Omega(\sqrt{KT}/\rbr{e^{\varepsilon}(e^{\varepsilon}-1)})$ regret lower bound for locally differentially private MAB algorithms, where $K$ is the number of arms. \citet{shariff2018differentially} derived an impossibility result for learning contextual bandits under DP constraint by showing an $\Omega(T)$ regret lower bound for any $(\varepsilon, \delta)$-DP algorithms. Hence, they considered the relaxed \textit{joint differential privacy} (JDP) and proposed an algorithm based on Lin-UCB \citep{abbasi2011improved} with $\tilde{\cO}(\sqrt{T}/\varepsilon)$ regret while preserving $\varepsilon$-JDP. Recently, a stronger definition of privacy, \textit{local differential privacy} \citep{duchi2013local,kasiviswanathan2011can}, gained increasing interest in bandit problems. Intuitively, LDP ensures that each collected trajectory is differentially private when observed by the agent, while DP requires the computation on the entire set of trajectories to be DP. \citet{zheng2020locally} proposed an LDP contextual linear bandit algorithm with $\tilde{\cO}(d^{3/4}T^{3/4})$ regret. \textbf{Differentially Private RL} In RL, \citet{balle2016differentially} is the first to propose a private algorithm for policy evaluation with linear function approximation. In the tabular setting, \citet{vietri2020private} designed a $\varepsilon$-JDP algorithm for regret minimization which attains an $\tilde{\cO}(\sqrt{H^4SAT} + SAH^3(S + H)/\varepsilon)$ regret. Recently, \citet{garcelon2020local} presented an optimistic algorithm with LDP guarantees. Their algorithm enjoys an $\tilde{\cO}(\max\cbr{H^{3/2}S^2A\sqrt{T}/\varepsilon, HS\sqrt{AT}})$ regret upper bound. They also provided a $\tilde{\Omega}(\sqrt{HSAT}/\min\cbr{\exp(\varepsilon)-1,1})$ regret lower bound. However, all these private RL algorithms are in the tabular setting, and private RL algorithms with linear function approximation remain understudied. \section{Preliminaries} In this paper, we study locally differentially private RL with linear function approximation for episodic MDPs. In the following, we will introduce the necessary background and definitions. \subsection{Markov Decision Processes} \paragraph{Episodic Markov Decision Processes} We consider the setting of an episodic time-inhomogeneous Markov decision process \citep{puterman2014markov}, denoted by a tuple $M = M(\cS, \cA, H, \cbr{r_h}^H_{h=1}, \cbr{\PP_h}_{h=1}^H)$, where $\cS$ is the state space, $\cA$ is the action space, $H$ is the length of each episode (planning horizon), $r_h: \cS\times \cA\rightarrow [0,1]$ is the deterministic reward function and $\PP_h(s'|s,a) $ is the transition probability function which denotes the probability for state $s$ to transfer to state $s'$ given action $a$ at stage $h$. A policy $\pi = \cbr{\pi_h}_{h=1}^H$ is a collection of $H$ functions, where $\pi_h(s)$ denote the action that the agent will take at stage $h$ and state $s$. Moreover, for each $h\in [H]$, we define the value function $V^{\pi}_h: \cS\rightarrow \RR$ that maps state $s$ to the expected value of cumulative rewards received under policy $\pi$ when starting from state $s$ at the $h$-th stage. We also define the action-value function $Q_h^{\pi}: \cS\times \cA\rightarrow \RR$ which maps a state-action pair $(s,a)$ to the expected value of cumulative rewards when the agent starts from state-action pair $(s,a)$ at the $h$-th stage and follows policy $\pi$ afterwards. More specifically, for each state-action pair $(s,a) \in \cS \times \cA$, we have \begin{align*} &Q_h^{\pi}(s,a)=r_h(s,a)+\EE\sbr{\sum_{h^{\prime}=h+1}^H r_{h^{\prime}}\big(s_{h^{\prime}},\pi_{h^{\prime}}(s_h^{\prime})\big)}\,, V_h^{\pi}(s) = Q_h^{\pi}\big(s, \pi_h(s)\big)\,, \end{align*} where $s_h = s, a_h = a$ and $s_{h^{\prime}+1}\sim \PP_{h'}(\cdot\given s_{h^{\prime}}, a_{h^{\prime}})$. For each function $V:\cS\rightarrow \RR$, we further denote $\sbr{\PP_h V}(s,a) = \EE_{s^{\prime}\sim \PP_h(\cdot\given s,a)}V(s^{\prime})$. Using this notation, the Bellman equation with policy $\pi$ can be written as \begin{align*} Q^{\pi}_h(s,a) = r_h(s,a)+\sbr{\PP_{h} V^{\pi}_{h+1}}(s,a),\qquad V_h^{\pi}(s) = Q_h^{\pi}(s, \pi_h(s)), \qquad V_{H+1}^{\pi}(s)=0, \end{align*} We define the optimal value function $V_h^*$ as $V_h^*(s) = \max_{\pi}V_h^{\pi}(s)$ and the optimal action-value function $Q_h^*$ as $Q_h^*(s,a) = \max_{\pi}Q_h^{\pi}(s,a)$. With this notation, the Bellman optimality equation can be written as follows \begin{align*} Q^{*}_h(s,a) = r_h(s,a)+\sbr{\PP_{h} V^{*}_{h+1}}(s,a), \qquad V^{*}_{h+1}(s) = \max_{a\in \cA}Q^{*}_h(s,a),\qquad V^{*}_{H+1}(s) = 0\,. \end{align*} In the setting of an episodic MDP, an agent aims to learn the optimal policy by interacting with the environment and observing the past information. At the beginning of the $k$-th episode, the agent chooses the policy $\pi_k$ and the adversary picks the initial state $s_1^k$. At each stage $h\in[H]$, the agent observes the state $s_h^k$, chooses an action following the policy $a_h^k=\pi_h^k(s_h^k)$ and observes the next state with $s_{h+1}^k \sim \PP_h(\cdot|s_h^k,a_h^k)$. The difference between $V_1^{*}(s_1^k)$ and $V_1^{\pi^k}(s_1^k)$ represents the expected regret in the $k$-th episode. Thus, the total regret in first $K$ episodes can be defined as \[\text{Regret}(K)=\sum_{k=1}^{K} \rbr{V_1^{*}(s_1^k)-V_1^{\pi_k}(s_1^k)}\,.\] \paragraph{Linear Function Approximation} In this work, we consider a class of MDPs called \textit{linear mixture MDPs} \citep{jia2020model, ayoub2020model,zhou2020provably}, where the transition probability function can be represented as a linear function of a given feature mapping $\bphi(s^\prime, s,a): \cS\times\cA\times\cS\to \RR^d$ satisfying that for any bounded function $V: S \to [0, H]$ and any tuple $(s, a) \in S \times A$, we have \begin{equation} \left\|\boldsymbol{\phi}_{V}(s, a)\right\|_{2} \leq H, \text { where } \boldsymbol{\phi}_{V}(s, a)=\sum_{s^{\prime} \in \mathcal{S}} \boldsymbol{\phi}(s^{\prime} \given s, a) V(s^{\prime})\,.\label{linearMixure} \end{equation} Formally, we have the following definition: \begin{definition}[\citealt{jia2020model, ayoub2020model,zhou2020provably}]\label{definMDP} An MDP $(\cS, \cA, H, \cbr{r_h}^H_{h=1}, \cbr{\PP_h}_{h=1}^H)$ is an inhomogeneous, episodic bounded linear mixture MDP if there exist vectors $\btheta_h\in \RR^d$ with $\nbr{\btheta^*_h}\le \sqrt{d}$ and a feature map $\bphi: \cS\times \cA \times \cS\rightarrow \RR^d$ satisfying \eqref{linearMixure} such that $\PP_h(s^\prime\given s, a) = \left<\bphi(s^{\prime}\given s, a), \btheta^*_h\right>$ for any state-action-next-state triplet $(s, a, s^\prime) \in S \times A \times S$ and stage $h$. \end{definition} Therefore, learning the underlying $\btheta_h^*$ can be regarded as solving a ``linear bandit'' problem (Part V, \citet{lattimore2020bandit}), where the context is $\bphi_{V_{k,h+1}}(s^k_h, a^k_h) \in \RR^d$, and the noise is $V_{k,h+1}(s^k_{h+1}) - [\PP_h V_{k,h+1}](s^k_h, a^k_h)$. \subsection{(Local) Differential Privacy}\label{sseDP} In this subsection, we introduce the standard definition of differential privacy \citep{dwork2006calibrating} and local differential privacy \citep{kasiviswanathan2011can, duchi2013local}. We also present the definition of Gaussian mechanism. \paragraph{Differential Privacy} Differential privacy is a mathematically rigorous notion of data privacy. In our setting, DP considers that the information collected from all the users can be observed and aggregated by a server. It ensures that the algorithm's output renders neighboring inputs indistinguishable. Thus, we formalize the definition as follows: \begin{definition}[Differential Privacy]\label{definitionDP} For any user $k\in [K]$, let $D_k$ be the information sent to a privacy-preserving mechanism from user $k$ and the collection of data from all the users can be written as $\cbr{D_{k}}_{1:K}= \{D_1,\dots, D_k, \dots, D_K\}$. For any $\varepsilon\ge0$ and $\delta\ge 0$, a randomized mechanism $\cM$ preserves $\rbr{\varepsilon, \delta}$-differential privacy if for any two neighboring datasets $\cbr{D_{k}}_{1:K},\cbr{D_{k}^{\prime}}_{1:K} \in \cZ$ which only different at one entry, and for any measurable subset $U\in \cZ$, it satisfies \[\PP\rbr{\cM\rbr{\cbr{D_{k}}_{1:K}} \in U} \leq e^{\varepsilon}\PP\rbr{\cM\rbr{\cbr{D_{k}^{\prime}}_{1:K}} \in U} +\delta\,, \] where $\cbr{D_k}_{1:K} = \{D_1,\dots, D_k, \dots, D_K\}, \cbr{D_k^{\prime}}_{1:K} = \{D_1,\dots, D_{k}^\prime, \dots, D_K\}$. \end{definition} \paragraph{Local Differential Privacy} In online RL, we view each episode $k \in [K]$ as a trajectory associated to a specific user. A natural way to conceive LDP in RL setting is to guarantee that for any user, the information send to the server has been privatized. Thus, LDP ensures that the server is totally agnostic to the sensitive data, and we are going to state the following definition: \begin{definition}[Local Differential Privacy]\label{definitionLDP} For any $\varepsilon\ge0$ and $\delta\ge 0$, a randomized mechanism $\cM$ preserves $\rbr{\varepsilon, \delta}$-local differential privacy if for any two users $u$ and $u^{\prime}$ and their corresponding data $D_{u}, D_{u^{\prime}}\in \cU$, it satisfies: \[\PP\rbr{\cM\rbr{D_{u}} \in U} \leq e^{\varepsilon}\PP\rbr{\cM\rbr{D_{u^\prime}} \in U} +\delta,\quad U\in \cU\,. \] \end{definition} \begin{remark} The dataset $\cbr{D_k}_{1:K}$ in DP is a collection of information from users $1, \dots, K$, where the subscript indicates the $k$-th user. Post-processing theorem implies that LDP is a more strict notion of privacy than DP. \end{remark} Now we are going to introduce the Gaussian mechanism which is widely used as a privacy-preserving mechanism to ensure DP/LDP property. \begin{lemma}\label{gaussian mechanism} (The Gaussian Mechanism, \citealt{dwork2014algorithmic}). Let $ f : \NN^{\cX} \mapsto \RR^d $ be an arbitrary d-dimensional function (a query), and define its $\ell_2$ sensitivity as $\Delta_2f=\max _{\text{adjacent(x,y)}} \nbr{f(x)-f(y)}_{2} $, where $\text{adjacent}(x,y)$ indicates that $x,y$ are different only at one entry. For any $0\leq \varepsilon \leq 1 $ and $c^2>2\log(1.25/\delta)$, the Gaussian Mechanism with parameter $\sigma \geq c\Delta_2f/\varepsilon$ is $(\varepsilon,\delta)$-differentially private. \end{lemma} \section{Algorithm}\label{sectionAlgorithm} We propose LDP-UCRL-VTR algorithm as displayed in Algorithm \ref{algorithm}, which can be regarded as a variant of UCRL-VTR algorithm proposed in \citet{jia2020model} with $\rbr{\varepsilon, \delta}$-LDP guarantee. Algorithm \ref{algorithm} takes the privacy parameters $\varepsilon, \delta$ as input (Line \ref{1}). For the first user $k = 1$, we simply have $\bLambda_{1, h} = \bSigma_{1,h} = \lambda \Ib$ and $\hat{\btheta}_{1, h} = \mathbf{0}$ (Line \ref{4}). For local user $k$ and received information $\bLambda_h^k, \ub_h^{k}$, the optimistic estimator of the optimal action-value function is constructed with an additional UCB bonus term (Line \ref{6}), \begin{align*} Q_{k,h}(\cdot, \cdot)&\gets \min\left\{H, r_h(\cdot, \cdot) + \left<\hat{\btheta}_{k,h}, \bphi_{V_{k, h+1}}(\cdot, \cdot)\right> + \beta \nbr{\bSigma_{k,h}^{-1/2}\bphi_{V_{k,h+1}}(\cdot, \cdot)}_2\right\}\,, \end{align*} and $\beta$ is specified as $c d^{3/4}(H-h+1)^{3/2}k^{1/4}\log(dT/\alpha)\rbr{\log((H-h+1)/\delta}^{1/4}\sqrt{1/\varepsilon}$, where $c$ is an absolute constant. From the previous sections, we know that learning the underlying $\btheta_h^*$ can be regarded as solving a ``linear bandit'' problem, where the context is $\bphi_{V_{k,h+1}}(s^k_h, a^k_h) \in \RR^d$, and the noise is $V_{k,h+1}(s^k_{h+1}) - [\PP_h V_{k,h+1}](s^k_h, a^k_h)$. Therefore, to estimate $Q_h^*$, it suffices to estimate the vector $\btheta_h^*$ by ridge regression with input $\bphi_{V_{k, h+1}}(s_h^k,a_h^k)$ and output $V_{k,h+1}(s_{h+1}^k)$. In order to implement the ridge regression, the server should collect the information of $\bphi_{V_{k, h+1}}(s_h^k,a_h^k)\bphi_{V_{k, h+1}}(s_h^k,a_h^k)^\top$ and $V_{k,h+1}(s_{h+1}^k)$ from each user $k$ (Line \ref{14}). Thus, we need to add noises to privatize the data before sending these information to the server in order to kept user's information private. In LDP-UCRL-VTR, we attain LDP by adding a symmetric Gaussian matrix and $d$-dimensional Gaussian noise (Line \ref{11} and Line \ref{12}). For simplicity, we denote the original information (without noise) $\Delta \tilde{\bLambda}_h^{k}=\bphi_{V_{k, h+1}}(s_h,a_h)\bphi_{V_{k, h+1}}(s_h,a_h)^\top, \Delta\tilde{\ub}_h^{k}= \bphi_{V_{k, h+1}}(s_h^k,a_h^k)V_{k, h+1}(s_{h+1}^k)$, where $k$ indicates the user and $h$ indicates the stage. Since the input information to the server is kept private by the user, it is easy to show that LDP-UCRL-VTR algorithm satisfies $(\varepsilon, \delta)$-LDP. After receiving the information from user $1$ to user $k$, the server aggregates information $\bLambda_{k,h}, \ub_{k,h}$, and maintains them for $H$ stages separately (Line \ref{18}). Besides, since the Gaussian matrix may not preserve the PSD property, we adapt the idea of shifted regularizer in \citet{shariff2018differentially} and shift this matrix $\bLambda_{k,h}$ by $r\Ib$ to guarantee PSD (Line \ref{19}). We then calculate $\hat{\btheta}_{k+1,h}$ and send $\bSigma_{k+1,h}, \hat{\btheta}_{k+1,h}$ back to $k+1$-th user in order to get a more precise estimation of $\btheta_h^*$ for better exploration. \paragraph{Comparison with related algorithms.} We would like to comment on the difference between our LDP-UCRL-VTR and other related algorithm. The key difference between our LDP-UCRL-VTR and UCRL-VTR \citep{jia2020model}, which is the most related algorithm, is that we add additive noises to the contextual vectors and the optimistic value functions in order to guarantee privacy. Then the server collects privatized information from different users and update $\bLambda, \ub$ for ridge regression. A shifted regularizer designed in \citet{shariff2018differentially} is used to guarantee PSD property of the matrix. It is easy to show that if we add no noise to user's information, our LDP-UCRL-VTR algorithm degenerates to inhomogeneous UCRL-VTR. Another related algorithm is the Contextual Linear Bandits with LDP in \citet{zheng2020locally}, which is an algorithm designed for contextual linear bandits. Setting $H= 1$, our LDP-UCRL-VTR will degenerate to Contextual Linear Bandits with LDP in \citet{zheng2020locally}. \begin{algorithm}[t!] \caption{LDP-UCRL-VTR} \label{algorithm} \begin{algorithmic}[1] \Require privacy parameters $\varepsilon, \delta$, failure probability $\alpha$, parameter $\lambda$ \State Set $\sigma=4H^3\sqrt{2\log(2.5H/\delta)}/\varepsilon$ \label{1} \For{user $k = 1,\dots, K$} \State\textbf{For the local user $k$:} \State Receive $\{\bSigma_{k, 1},\dots, \bSigma_{k,H},\hat{\btheta}_{k, 1},\dots,\hat{\btheta}_{k, H}\}$ from the server\label{4} \For{$h = H, \dots, 1$} \State $Q_{k,h}(\cdot, \cdot)\gets \min\left\{H - h + 1, r_h(\cdot, \cdot)+ \left<\hat{\btheta}_{k,h}, \bphi_{V_{k, h+1}}(\cdot, \cdot)\right> +\beta_h \nbr{\bSigma_{k,h}^{-1/2}\bphi_{V_{k,h+1}}(\cdot, \cdot)}_2\right\}$\label{6} \State $V_{k,h}(\cdot)\gets \max_{a} Q_{k,h}(\cdot,a).$ \EndFor \State Receive the initial state $s_1^k$ \For{$h = 1, \dots, H$} \State Take action $a_h^k \gets \argmax_{a\in \cA} Q_{k,h}(s_h^k)$, and observe $s_{h+1}^k$\label{10} \State Set $\Delta \bLambda_h\gets\bphi_{V_{k, h+1}}(s_h,a_h)\bphi_{V_{k, h+1}}(s_h,a_h)^\top+\Wb_{k,h}$, where $\Wb_{k,h}(i,j) = \Wb_{k,h}(j,i)$ and $\Wb_{k,h}(i, j)\overset{\text{i.i.d}}{\sim} \mathcal{N}(0, \sigma^2), \forall i \le j$\label{11} \State Set $\Delta \ub_{h}\gets\bphi_{V_{k, h+1}}(s_h^k,a_h^k)V_{k, h+1}(s_{h+1}^k)+\bxi_h $, where $\bxi_h \sim \mathcal{N}(\mathbf{0}_d, \sigma^2 \Ib_{d\times d})$\label{12} \EndFor \State Send $D^k=\{ \Delta \bLambda_1,\dots,\Delta \bLambda_H,\Delta\ub_1,\dots,\Delta\ub_H\}$ to the server\label{14} \Statex \State\textbf{For the server:} \For{$h = 1, \dots, H$} \State $\bLambda_{k+1, h} \gets \bLambda_{k,h} + \Delta \bLambda_h, \ub_{k+1,h} \gets \ub_{k,h} + \Delta \ub_h$\label{18} \State $\bSigma_{k+1,h} \gets \bLambda_{k+1,h} + r\Ib$\label{19} \State $\hat{\btheta}_{k+1,h} \gets \rbr{\bSigma_{k+1,h}}^{-1}\ub_{k+1,h}$ \EndFor \State Send $\{\bSigma_{k+1,1},\dots, \bSigma_{k+1,H},\hat{\btheta}_{k+1, 1},\dots,\hat{\btheta}_{k+1, H}\}$ to user $k+1$ \EndFor \end{algorithmic} \end{algorithm} \section{Main Results} In this section, we provide both privacy and regret guarantees for Algorithm \ref{algorithm}. The detailed proofs of the main results are deferred to the appendix. \subsection{Privacy Guarantees} Recall that in Algorithm \ref{algorithm}, we use Gaussian mechanism to protect the private information of the contextual vectors and the optimistic value functions. Based on the property of Gaussian mechanism, we can show that our algorithm is $(\varepsilon, \delta)$-LDP. \begin{theorem}\label{LDPtheorem} Algorithm \ref{algorithm} preserves $(\varepsilon,\delta)$-LDP. \end{theorem} The privacy analysis relies on the fact that if the information from each user satisfies $(\varepsilon, \delta)$-DP, then the whole algorithm is $(\varepsilon, \delta)$-LDP. \subsection{Regret Upper Bound} The following theorem states the regret upper bound of Algorithm \ref{algorithm}. \begin{theorem}\label{mainTheorem} For any fixed $\alpha \in (0, 1)$, for any privacy parameters $\varepsilon > 0$ and $\delta>0$, if we set the parameters $\lambda = 1$ and $\beta_h = \tilde{\cO}( d^{3/4}(H-h+1)^{3/2}k^{1/4}\log(dT/\alpha)\rbr{\log((H-h+1)/\delta)}^{1/4}\sqrt{1/\varepsilon})$ for user $k$, with probability at least $1-\alpha$, the total regret of Algorithm \ref{algorithm} in the first $T$ steps is at most $\tilde{O}(d^{5/4}H^{7/4}T^{3/4}\log(dT/\alpha)\rbr{\log(H/\delta)}^{1/4}\sqrt{1/\varepsilon})$, where $T = KH$ is the number of interactions with the MDP. \end{theorem} \begin{remark} By setting $\alpha = \delta$, our regret bound can be written as $\tilde{\cO}(d^{5/4}H^{7/4}T^{3/4}\rbr{\log(H/\delta)}^{1/4}\sqrt{1/\varepsilon})$. Compared with UCRL-VTR, which enjoys an upper bound of $\tilde{\cO}(d\sqrt{H^3T})$, our bound suggests that learning the linear mixture MDP under the LDP constraint is inherently harder than learning it non-privately. \end{remark} \subsection{Regret Lower Bounds} \label{lower} In this subsection, we present a lower bound for learning linear mixture MDPs under the $\varepsilon$-LDP constraint. We follow the idea firstly developed in \citet{zhou2020nearly}, which basically shows that learning a linear mixture MDP is no harder than learning $H/2$ linear bandit problems. As a byproduct, we also derived the regret lower bound for learning $\varepsilon$-LDP contextual linear bandits. In detail, in order to prove the regret lower bound for MDPs under $\varepsilon$-LDP constraint, we first prove the lower bound for learning $\varepsilon$-LDP linear bandit problems. We adapted the proof techniques in \citet[Theorem 24.1]{lattimore2020bandit} and \citet{basu2019differential}. In the non-private setting, the observed history of a contextual bandit algorithm in the first $T$ rounds can be written as $\cH_T = \cbr{\xb_t, y_t}_{t = 1}^T$. Given history $\cH_{t-1}$, the contextual linear bandit algorithm chooses action $\xb_t$, and the reward is generated from a distribution $f_{\btheta}(\cdot | \xb_t)$, which is conditionally independent of the previously observed history. We use $\PP_{\pi,{\btheta}}^T$ to denote the distribution of observed history up to time $T$, which is induced by $\pi$ and $f_{\btheta}$. Hence, we have \begin{align} \PP^T_{\pi, {\btheta}} = \PP_{\pi, {\btheta}}(\cH_T) = \prod^T_{t=1}\pi (\xb_t\given \cH_{t-1})f_{\btheta}(y_t\given \xb_t)\,,\notag \end{align} where $\pi$ is the stochastic policy (the distribution over an action set induced by a bandit algorithm) and $f_{\btheta}(\cdot\given \xb_t)$ is the reward distribution given action $\xb_t$, which is conditionally independent of the previously observed history $\cH_{t-1}$. In the LDP setting, the privacy-preserving mechanism $\cM$ generates the privatized version of the context $\xb_t$, denoted by $\tilde \xb_t = \cM(\xb_t)$, to the contextual linear bandit algorithm. For simplicity, we denote $\cM_{\pi}$ as the distribution (stochastic policy) by imposing a locally differentially private mechanism $\cM$ on the distribution (policy) $\pi$. Also, we use $f^{\cM}_{\btheta}$ to denote the conditional distribution of $\tilde y_t$ parameterized by $\btheta$, where $\tilde y_t$ is the privatized version of $y_t$ obtained by the privacy-preserving mechanism $\cM$. We denote the observed history by $\tilde{\cH}_T:=\cbr{\rbr{\tilde{\xb}_t, \tilde{y}_t}}_{t=1}^{T}$, where $\tilde{\xb}_t$, $\tilde{y}_t$ are the privatized version of contexts and rewards. Similarly, we have\ \begin{align} \tilde{\PP}_{\pi, {\btheta}}^T:=\PP_{\pi, {\btheta}}(\tilde{\cH}_T) = \prod_{t=1}^T\cM_{\pi}(\tilde{\xb}_t\given \tilde{\cH}_{t-1}) f^{\cM}_{\btheta}(\tilde{y}_t\given \xb_t)\,.\label{eq:definition of P} \end{align} With the formulation above, we proved the following key lemma for $\varepsilon$-LDP contextual linear bandits. \begin{lemma}\label{KLdecom} (Locally Differentially Private KL-divergence Decomposition) We denote the reward generated by user $t$ for action $\xb_t$ as $y_t = \xb_t^\top \btheta + \eta_t$, where $\eta_t$ is a zero-mean noise. If the reward generation process is $\varepsilon$-locally differentially private for both the bandits with parameters $\btheta_1$ and $\btheta_2$, we have, \begin{align*} \text{KL}(\tilde{\PP}_{\pi,{\btheta_1}}^T,\tilde{\PP}_{\pi,{\btheta_2}}^T)&\le 2\min\cbr{4, e^{2\varepsilon}}(e^{\varepsilon}-1)^2 \cdot \sum_{t=1}^T \EE_{\pi,{\btheta_1}}\sbr{\text{KL}(f^{\cM}_{\btheta_1}(\tilde{y}_t\given \xb_t), f^{\cM}_{\btheta_2}(\tilde{y}_t\given \xb_t))}, \end{align*} where $\tilde{\xb}_t$, $\tilde{y}_t$ are the privatized version of contexts and rewards. \end{lemma} Lemma \ref{KLdecom} can be seen as an extension of Lemma 3 in \citet{basu2019differential} from multi-armed bandits to contextual linear bandits. Equipped with Lemma \ref{KLdecom}, the KL-divergence of privatized history distributions can be decomposed into the distributions of rewards. We construct a contextual linear bandit with Bernoulli reward. In detail, for an action $\xb_t \in \cA \subseteq \RR^d$, the reward follows a Bernoulli distribution $y_t \sim B(\left<\btheta,\xb_t\right> + \delta)$, where $0\le\delta\le 1/3$. We first derive a regret lower bound of learning contextual bandits under the LDP constraint in the following lemma. \begin{lemma}[Regret Lower Bound for LDP Contextual Linear Bandits]\label{banditLowerBound} Given an $\varepsilon$-locally differentially private reward generation mechanism with $\varepsilon$ and a time horizon $T$, for any environment with finite variance, the pseudo regret of any algorithm $\pi$ satisfies \[\text{Regret}(T) \ge \frac{c}{\min\cbr{2, e^{\varepsilon}}(e^{\varepsilon}-1)}d\sqrt{T}\,.\] \end{lemma} Since the distribution of rewards will only influence the KL-divergence by an absolute constant, the lower bound we obtained is similar to that in \citet[Theorem 24.1]{lattimore2020bandit}, which assumes that the reward follows a normal distribution. According to the proof of Lemma \ref{banditLowerBound}, the only difference between our hard-to-learn MDP instance and that in \citet{zhou2020nearly} is that we need to specify $\Delta = \sqrt{\delta}/\rbr{\min\cbr{2, e^{\varepsilon}}(e^{\varepsilon}-1)\sqrt{T}}$. We then utilize the hard-to-learn MDPs constructed in \citet{zhou2020nearly} and obtain the following lower bound for learning linear mixture MDPs with $\varepsilon$-LDP guarantee: \begin{theorem}\label{RLlower} For any $\varepsilon$-LDP algorithm there exists a linear mixture MDP parameterized by $\bTheta = \rbr{\btheta_1, \dots, \btheta_H}$ such that the expected regret is lower bounded as follows: \[\EE_{\bTheta}\text{Regret}(M_{\bTheta}, K) \ge \Omega\rbr{\frac{1}{\min\cbr{2,e^{\varepsilon}}(e^{\varepsilon}-1)}dH\sqrt{T}},\] where $T = KH$ and $\EE_{\bTheta}$ denotes the expectation over the probability distribution generated by the interaction of the algorithm and the MDP. \end{theorem} \begin{remark} Compared with the upper bound $\tilde{\cO}(d^{5/4}H^{7/4}T^{3/4}\rbr{\log(H/\delta)}^{1/4}\sqrt{1/\varepsilon})$ in Theorem~\ref{mainTheorem}, it can be seen that there is a $d^{1/4}T^{1/4}H^{3/4}$ gap between our upper bound and lower bound if treating $\varepsilon$ as a constant. It is unclear if the upper bound and/or the lower bound are not tight \end{remark} \begin{remark} Notice that $\varepsilon$-LDP is a special case of $(\varepsilon, \delta)$-LDP, where $\delta = 0$. Thus our lower bound for $\varepsilon$-LDP linear mixture MDPs is also a valid lower bound for learning $(\varepsilon, \delta)$-LDP linear mixture MDPs. \end{remark} \begin{figure*}[!ht] \centering \begin{tikzpicture \node[circle, minimum width =35pt , minimum height =35pt ,draw=blue!70, fill=blue!20, very thick] (0) at(0,0){$S_1$} ; \node[circle, minimum width =35pt , minimum height =35pt ,draw=blue!70, fill=blue!20, very thick] (1) at(2.5,0){$S_2$}; \node[circle, minimum width =25pt , minimum height =25pt ,draw=blue!0, fill=blue!0, very thick] (2) at(5,0){$\dots$}; \node[circle, minimum width =35pt , minimum height =35pt ,draw=blue!70, fill=blue!20, very thick] (3) at(7.5,0){$S_{S -1}$}; \node[circle, minimum width =35pt , minimum height =35pt ,draw=blue!70, fill=blue!20, very thick] (4) at(10,0){$S_{S}$}; \path[->, bend angle=25, shorten >=1pt, semithick] (0) edge[bend left] node[above]{$ap_h$} (1) edge[loop left] node{$1-ap_h$} (0) (1) edge[bend left] node[below]{$(1-ap_h)/2$} (0) edge[bend left] node[above]{$ap_h$} (2) edge[loop above] node[above]{$(1-ap_h)/2$} (1) (2) edge[bend left] node[below]{$(1-ap_h)/2$} (1) edge[bend left] node[above]{$ap_h$} (3) (3) edge[bend left] node[below]{$(1-ap_h)/2$} (2) edge[bend left] node[above]{$ap_h$} (4) edge[loop above] node[above]{$(1-ap_h)/2$} (3) (4) edge[bend left] node[below]{$(1-ap_h)/2$} (3) edge[loop right] node{$1-ap_h$} (4); \end{tikzpicture} \caption{The transition kernel $\PP_h$ of inhomogeneous ``RiverSwim'' MDP instance.} \label{fig:mdp} \end{figure*} \begin{figure*}[!ht] \subfigure[$S=3, A=2, H = 6, d=18$, homogeneous]{% \includegraphics[width=0.44\linewidth]{1.pdf} } \hfill \subfigure[$S=5, A = 2, H = 10, d=50$, homogeneous]{% \includegraphics[width=0.44\linewidth]{2.pdf} }\ \subfigure[$S=3, A=2, H = 6, d=18$, inhomogeneous]{% \includegraphics[width=0.44\linewidth]{3.pdf} } \hfill \subfigure[$S=5, A = 2, H = 10, d=50$, inhomogeneous]{% \includegraphics[width=0.44\linewidth]{4.pdf} } \caption{Evaluation of the algorithms in two ``RiverSwim'' MDPs. Results are averaged over $10$ runs and the standard deviations are calculated to plot the confidence band. These results show that the cumulative regret of LDP-UCRL-VTR is sublinear in $K$, and its performance is getting closer to that of UCRL-VTR while the privacy guarantee becomes weaker, i.e., choosing a larger $\varepsilon$.} \label{fig:result} \end{figure*} \section{Experiments} In this section, we carry out experiments to evaluate the performance of LDP-UCRL-VTR, and compare with its non-private counterpart UCRL-VTR \citep{jia2020model}. \subsection{Experimental Setting} We tested LDP-UCRL-VTR on a benchmark MDP instance named ``RiverSwim'' \citep{strehl2008analysis,ayoub2020model}. The purpose of this MDP is to tempt the agent to go left while it is hard for a short sighted agent to go right since $r(0, 0) \ne 0, r(s, 1) = 0, 0 \le s \le \abr{\cS}-1$. Therefore, it is hard for the agent to decide which direction to choose. In our experiment, the reward in each stage is normalized by $H$, e.g., $r(0, 0) = 5/(1000H), r(S, 1) = 1/H$. Our LDP-UCRL-VTR is also tested on the time-inhomogeneous ``RiverSwim'', where for each $h \in [H]$, the transition probability $p_h$ is sampled from a uniform distribution $U(0.8, 1)$. We also choose $H = 2S$. Figure \ref{fig:mdp} shows the state transition graph of this MDP. \subsection{Results and Discussion} We evaluate LDP-UCRL-VTR with different privacy budget $\varepsilon$ and compare it with UCRL-VTR on both homogeneous and inhomogeneous ``RiverSwim''. For UCRL-VTR, we set $\sqrt{\beta} = c\sqrt{d_1} + (H - h + 1) \sqrt{2\log(1/K)+\log \det \Mb}$, where $d_1 = SA$ with $S$ being the number of states and $A$ being the number of actions, $\Mb$ is the covariance matrix in Algorithm 3 \citep{jia2020model}. We fine tune the hyper parameter $c$ for different experiments. For LDP-UCRL-VTR, we choose $\delta = 0.1, \alpha = 0.01$. Since $\delta$ and $\alpha$ are prefixed for all experiments, they can be treated as constants. Thus, we can choose $\beta$ in the form $cd^{3/4}(H - h + 1)^{3/2}k^{1/4}$ and only fine tune $c$. The results for each $\varepsilon$ are averaged over $10$ runs. In our experiments, since the reward is normalized by $H$, we need to recompute $\sigma$ for the Gaussian mechanism. Recall that $\sigma = 2\Delta f H\sqrt{2\log(2.5H/\delta)}/\varepsilon$, where $\Delta f$ represents the $\ell_2$ sensitivity of $\Delta\ub$ in Algorithm \ref{algorithm}. In our setting, $\abr{Q}\le 1$ and therefore $\Delta f\le 1$. Thus, we set $\sigma = 4 H\sqrt{2\log(2.5H/\delta)}/\varepsilon$. In addition, we set $K = 10000$ for all experiments. To fine tune the hyper parameter $c$, we use grid search and select the one which attains the best result. The experiment results are shown in Figure \ref{fig:result}. From Figure \ref{fig:result}, we can see that the cumulative regret of LDP-UCRL-VTR is indeed subliear in $K$. In addition, it is not surprising to see that LDP-UCRL-VTR incurs a larger regret than UCRL-VTR. The performance of LDP-UCRL-VTR with larger $\varepsilon$ is closer to that of UCRL-VTR as the privacy guarantee becomes weaker. Our results are also greatly impacted by $H$ and $d$, as the convergence (learning speed) slows down as we choose larger $H$ and smaller $\varepsilon$. The experiments are consistent with our theoretical results. \section{Conclusion and Future Work}\label{sectionDiscussion} In this paper, we studied RL with linear function approximation and LDP guarantee. To the best of our knowledge, our designed algorithm is the first provable privacy-preserving RL algorithm with linear function approximation. We proved that LDP-UCRL-VTR satisfies $(\varepsilon, \delta)$-LDP. We also show that LDP-UCRL-VTR enjoys an $\tilde{\cO}( d^{5/4}H^{7/4}T^{3/4}\rbr{\log(1/\delta)}^{1/4}\sqrt{1/\varepsilon})$ regret. Besides, we proved a lower bound $\Omega(dH\sqrt{T}/\rbr{e^{\varepsilon}(e^{\varepsilon}-1)})$ for $\varepsilon$-LDP linear mixture MDPs. We also provide experiments on synthetic datasets to corroborate our theoretical findings. In our current results, there is still a gap between the regret upper bound and the lower bound. We conjecture the gap to be a fundamental difference between learning linear mixture MDPs and tabular MDPs. In the future, it remains to study if this gap could be eliminated.

BkiUg0_xK7IACwaaBys5

\section{Introduction} The Randall-Sundrum (RS) model \cite{Randall:1999ee} was originally proposed to resolve the hierarchy between the scales of weak and gravitational interactions, $m_W \sim 10^2$~GeV and ${\bar M}_P \sim 10^{18}$~GeV, respectively. The RS model is based on a truncated AdS$_5$ spacetime, bounded by two 4D Minkowski walls, often called UV (Planck) and IR (TeV) branes. The curvature in 5D induces a warp factor in the metric which redshifts scales of order ${\bar M}_P$ at the UV brane to scales of order $m_W$ at the IR brane. Since the metric depends exponentially on the 5$^{th}$ coordinate, explaining $m_W/{\bar M}_P \sim 10^{-16}$ does not require hierarchic parameters. The requisite brane-separation was shown to be easily accommodated early on \cite{Goldberger:1999uk}. Initially, it was assumed that all Standard Model (SM) fields reside at the IR brane. The most striking and distinct signature of this model would then be weak scale spin-2 Kaluza-Klein (KK) excitations of the graviton, appearing as resonances in high energy collisions \cite{Davoudiasl:1999jd}. It was soon realized that resolving the hierarchy only required the Higgs to be localized near the IR brane \cite{Goldberger:1999wh} and SM gauge \cite{Davoudiasl:1999tf,Pomarol:1999ad} and fermion \cite{Grossman:1999ra} fields could propagate in the 5D bulk. It was shown that placing the fermions in the bulk provides a natural mechanism for generation of SM fermion masses and also suppression of unwanted 4-fermion operators \cite{Gherghetta:2000qt}. This is achieved by a mild modulation of bulk fermion masses that control the exponential localization of fermion zero modes. As the Higgs is kept near the IR brane, small 4D Yukawa couplings are naturally obtained, if light flavor zero modes are UV brane localized. Given the correspondence between location in the bulk and scale in warped backgrounds, operators containing light flavors are suppressed by scales much larger than $m_W$. The above setup, an RS-type geometry with a flavored bulk, offers an attractive simultaneous resolution of hierarchy and flavor puzzles. However, the experimental signals of these warped models are now much more elusive. This is because spreading the gauge fields over the bulk and localizing the light fermions near the UV brane suppresses their couplings to IR-brane-localized KK modes, the main signatures of warped models. Therefore, their production via and decay into light SM fermions and gauge fields are suppressed. This feature is generic to warped models of hierarchy and flavor, largely independently of their details. Recently, various studies have been performed to assess the prospects for discovering the new warped scenarios, given that the old set of signatures are now mostly inaccessible. Precision data require the new KK states to be heavier than roughly 2-3~TeV \cite{Carena:2007ua}, even assuming new custodial symmetries \cite{custodial}. Generally speaking, it has been shown that the most likely new state in these models to be discovered at the LHC is the first KK gluon \cite{KKgluon1,KKgluon2}. The analysis of Refs.~\cite{KKgluon1,KKgluon2} suggests that KK gluons as heavy as 4~TeV will be within the reach of the LHC. However, for the KK modes of the weak sector, the corresponding reach is in the 2-3~TeV range \cite{Agashe:2007ki}. For gauge KK masses in the above ranges, the graviton KK modes are most likely not accessible, even with an upgraded LHC luminosity \cite{KKgraviton1,KKgraviton2}. Thus, typically, the gauge KK modes may be discovered, while the KK gravitons which are the most distinct RS-type signature will be out of reach, at the LHC. In this work, we examine the discovery prospects for the KK modes of the SM fermion sector. Observation of these states will provide direct evidence for the presence of SM fermions in the 5D bulk, a necessary ingredient of the warped flavor scenarios. Briefly put, we find that with gauge KK masses set at 3~TeV, a currently acceptable value, and for generic zero-mode profiles that yield a realistic flavor hierarchy, the SM KK fermions are not accessible at the LHC, even after a luminosity upgrade. Hence, we are faced with a situation in which the KK modes of the graviton and SM fermions are not discovered during the LHC program. However, it seems reasonable to require direct observation of these KK states in order to establish an RS-type warped model as a theory of hierarchy and flavor. Given our conclusion that the LHC is unlikely to establish realistic warped models, we set out to determine the minimum requirements that a future machine needs to meet in order to make this task possible. We take 3~TeV to be a reference mass for the lowest gauge KK mode. This sets the mass scales of all other KK states, in the simplest generic warped models. What we find is that, typically, in order to have firm evidence for the lowest KK states with spins 1/2, 1, and 2, we need center of mass energies ${\sqrt s}\approx 60$~TeV and integrated luminosities $L \sim 1$~ab$^{-1}$! This suggests that future {\it luminosity and energy} upgrades of the LHC will most likely be insufficient to verify all the essential features of realistic warped models. Therefore, if these models do describe Nature, the LHC will likely find evidence for them. However, the Next Hadron Collider (NHC), defined to have ${\sqrt s}\approx 60$~TeV and $L \sim 1$~ab$^{-1}$, must be part of the post-LHC experimental program aimed at establishing the underlying theory. We also show that if these states are not observed at the LHC the eventual reach of NHC for the gluon KK states is typically in excess of 10~TeV. In the next section we review key aspects of typical Warped Hierarchy-and-Flavor Models (WHFM). In section 3, we study the prospects for verification of realistic WHFM at colliders and estimate the required parameters of the NHC. Our conclusions are presented in section 4. \section{Warped Hierarchy-and-Flavor Models} Here, we will briefly describe the generic properties of WHFM. Much of what will follow is well-known from previous works and is mainly included to provide some background for our further discussions. \subsection{General Features} The RS metric is given by \cite{Randall:1999ee} \begin{equation} ds^2 = e^{-2 \sigma} \eta_{\mu\nu} dx^\mu dx^\nu - r_c^2 d\phi^2, \label{metric} \end{equation} where $\sigma = k r_c |\phi|$, $k$ is the 5D curvature scale, $r_c$ is the radius of compactification, $-\pi\leq\phi\leq\pi$, and a $\mathbb Z_2$ orbifolding of the 5$^{\rm th}$ dimension is assumed. To solve the hierarchy problem, the Higgs is assumed to be localized near the TeV-brane, where the reduced metric ``warps" $\vev{H}_5 \sim {\bar M}_P$ down to the weak scale: $\vev{H}_4 = e^{-kr_c \pi} \vev{H}_5$. For $kr_c \approx 11.3$, we then get $\vev{H}_{\tt SM} \equiv \vev{H}_4 \sim 100$~GeV. Originally, it was assumed that all SM content resides at the IR-brane \cite{Randall:1999ee}. However, as the cutoff scale in the 4D effective theory is also red-shifted to near the weak-scale, this would lead to unsuppressed higher dimensional operators that result in large violations of experimental bounds on various effects, such as those on flavor-changing neutral currents. This problem can be solved by realizing that points along the warped 5$^{\rm th}$ dimension correspond to different effective 4D scales. In particular, by localizing first and second generation fermions away from the IR-brane, the effective scale that suppresses higher dimensional operators made up of these fields is pushed to much higher scales \cite{Gherghetta:2000qt}. In the process of suppressing the dangerous operators, this setup also leads to a natural mechanism for obtaining small fermion masses. The above localization is achieved by introducing a 5D mass term in the bulk for each fermion field $\Psi$ \cite{Grossman:1999ra}. Let $c \equiv m_\Psi/k$, where $m_\Psi$ is the 5D mass of the fermion. Each 5D fermion $\Psi$ has left- and right-handed components $\Psi_{L,R}$ which can be expanded in KK modes \begin{equation} \Psi_{L,R}(x, \phi) = \sum_{n=0}^{\infty}\psi_{L,R}^{(n)}(x) \frac{e^{2 \sigma}}{\sqrt{r_c}} f^{(n)}_{L,R}(\phi). \label{KKmodes} \end{equation} The KK wavefunctions $f^{(n)}_{L,R}$ are orthonormalized \begin{equation} \int d\phi \, e^\sigma f^{(m)}_{L,R}\, f^{(n)}_{L,R} = \delta_{m n}. \label{orthonormal} \end{equation} One can then show that the $n\neq 0$ modes are given by \cite{Grossman:1999ra} \begin{equation} f^{(n)}_{L,R} = \frac{e^{\sigma/2}}{N^{L,R}_n}\, Z_{\frac{1}{2}\pm c} (z_n), \label{fKK} \end{equation} where the normalization $N^{L,R}_n$ is fixed by Eq.(\ref{orthonormal}); throughout our work, $Z_a = J_a + b_n Y_a$ denotes a linear combination of Bessel functions of order $a$. In Eq.(\ref{fKK}), $z_n \equiv (m_n/k) e^\sigma$ and $m_n$ is the KK mass. The zero-mode wavefunction is given by \begin{equation} f^{(0)}_{L,R} = \frac{e^{\mp c \sigma}}{N^{L,R}_0}, \label{f0} \end{equation} with the normalization \begin{equation} N^{L,R}_0 = \left[\frac{e^{k r_c \pi(1\mp 2c)} - 1}{k r_c(1/2\mp c)}\right]^{1/2}. \label{N0} \end{equation} Note that in our convention, the singlet (right-handed) and doublet (left-handed) zero mode wavefunctions are defined with opposite signs for mass parameters $c^S$ and $c^D$, respectively. Hence, for example, UV-localization for the singlet and doublet zero modes correspond to $c^S < -1/2$ and $c^D > 1/2$, respectively. In the SM, all $SU(2)$ doublets are left-handed, while the singlets are right-handed. One can impose a ${\mathbb Z_2}$ parity on bulk fermion fields so that only the doublets have left-handed zero modes and only the singlets have right-handed zero modes. However, both the doublets and singlets have left- and right-handed higher KK modes. To project a particular 4D zero mode chirality, Neumann-like boundary conditions are chosen for the corresponding field at $\phi=0, \pi$; the other chirality will then obey Dirichlet boundary conditions. For example, if $\Psi^D$ represents a weak SM doublet, we require \begin{equation} (\partial_\phi + r_c m_\Psi)f^{(n)}_{L} = 0 \;\; ; \;\; f^{(n)}_{R} = 0 \quad {\rm at}\;\; \phi = 0,\pi. \label{bc} \end{equation} The above choice results in a left-handed zero mode, given by Eq.(\ref{f0}). We also find \begin{equation} b_n^{L,R} = - \frac{J_{\pm(c - 1/2)}(z_n)}{Y_{\pm(c - 1/2)}(z_n)} \quad {\rm at}\;\; \phi = 0,\pi. \label{bn} \end{equation} The above equations fix the wavefunctions and the mass eigenvalues, once $c$ is specified for any $SU(2)$ doublet $\Psi^D$; $(L,R)$ equations result in the same KK mass spectrum. Similar equations can also be derived for singlets $\Psi^S$. The above fermion profiles lead to a natural scheme for SM fermion masses \cite{Grossman:1999ra,Gherghetta:2000qt}. We will assume that the Higgs is on the IR-brane; this is a very good approximation since the Higgs must be highly IR-localized. Then, a typical Yukawa term in the 5D action will take the form \begin{equation} S^5_{\tt Y} = \int \!d^4x \; d\phi \sqrt{-g} \,\frac{\lambda_5}{k} H(x) \Psi^D_L \Psi^S_R \delta(\phi-\pi), \label{S5Y} \end{equation} where $\lambda_5 \sim 1$ is a dimensionless 5D Yukawa coupling and $\Psi^{D,S}$ are doublet left- and singlet right-handed 5D fermions, respectively. After the rescaling $H \rightarrow e^{k r_c \pi} H$, the 4D action resulting from Eq.(\ref{S5Y}) is \begin{equation} S^4_{\tt Y} = \int \!d^4x \; \sqrt{-g} \,\lambda_4 H(x) \psi^{(D,0)}_L \psi^{(S,0)}_R + \ldots, \label{S4Y} \end{equation} where the 4D Yukawa coupling for the corresponding zero-mode SM fermion is given by \cite{Gherghetta:2000qt} \begin{equation} \lambda_4 = \frac{\lambda_5}{k r_c}\left[\frac{e^{(1-c^D+c^S)kr_c \pi}} {N^{D,L}_0 N^{S,R}_0}\right], \label{lam4} \end{equation} Thus, in the quark sector, there are, in general, 9 different values for $c^{D,S}$: 3 for the doublets and 6 for the singlets. One can see that the exponential form of the effective Yukawa coupling $\lambda_4$ can accommodate a large hierarchy of values without the need for introducing unnaturally small 5D parameters. With the fermions in the bulk, the gauge fields must also follow. A 5D gauge field $A_M$ has scalar and vector projections in 4D. The scalar zero mode corresponding to $A_\phi$ is projected out using $\mathbb Z_2$ parity or a Dirichlet boundary condition. As is well-known, the remaining projections $A_\mu$ can be expanded in KK modes as \begin{equation} A_\mu = \sum_{n=0}^\infty A_\mu^{(n)}(x) \frac{\chi^{(n)}(\phi)}{\sqrt{r_c}}. \label{AKK} \end{equation} The gauge field KK wavefunctions are given by \cite{Davoudiasl:1999tf,Pomarol:1999ad} \begin{equation} \chi^{(n)}_A = \frac{e^{\sigma}}{N_n^A} Z_{1}(z_n) \label{chiA} \end{equation} subject to the orthonormality condition \begin{equation} \int_{-\pi}^{\pi} d\phi \, \chi^{(m)}_A \chi^{(n)}_A = \delta_{m n}. \label{ortho-gauge} \end{equation} The above equation fixes the normalization $N_n^A$ and Neumann boundary conditions at $\phi=0,\pi$ fix the wavefunctions and yield the KK masses. The 5D action for the coupling of the bulk fermion $\Psi$ to the gauge field $A_M$ is given by \begin{equation} S_{\Psi A} = g_5\int \!d^4x \; d\phi V \left[V_l^M {\bar \Psi \gamma^l A_M \Psi}\right], \label{PsiA} \end{equation} where $g_5$ is the 5D gauge coupling, $V$ is the determinant of the f\"{u}nfbein $V_l^M$, with $l=0,\ldots,3$, $V_\lambda^M = e^\sigma \delta^M_\lambda$, and $V_4^4 = -1$; $\gamma^l = (\gamma^\lambda, i \gamma^5)$. Dimensional reduction of the action (\ref{PsiA}) then yields the couplings of the fermion and gauge KK towers. With our conventions, the 4D gauge coupling is given by $g_4 = g_5/\sqrt{2 \pi r_c}$, which fixes the couplings of all the other modes. For completeness, we also write down the wavefunction of the graviton KK modes \cite{Davoudiasl:1999jd} \begin{equation} \chi^{(n)}_G = \frac{e^{2\sigma}}{N_n^G} Z_{2}(z_n) \label{chiG} \end{equation} which obey Neumann boundary conditions and are orthonormalized according to \begin{equation} \int_{0}^{\pi} d\phi \, e^{-2\sigma} \chi^{(m)}_G \chi^{(n)}_G = \delta_{m n}. \label{ortho-grav} \end{equation} The graviton wavefunctions are fixed and the corresponding KK masses are obtained, via the boundary conditions, as before. WHFM are subject to various experimental constraints, including those from precision electroweak \cite{Carena:2007ua} and flavor data \cite{Agashe:2004cp,Agashe:2006iy}. A number of models with a custodial $SU(2)_L\times SU(2)_R$ bulk symmetry have been proposed to address these constraints \cite{Agashe:2003zs,Agashe:2004rs,Agashe:2006at}. In the following, we will limit the scope of our study to bulk SM without specifying a particular framework for such constraints. We will not discuss the phenomenology of the extra exotica in these models, as they do not change the qualitative picture for the SM KK partners that we present here. This will suffice to demonstrate our key observations. For a study of possible light exotic quarks in some warped scenarios see Ref.~\cite{Dennis:2007tv}. \subsection{Reference Parameters} To comply with precision constraints, we will choose the mass of the first KK mode of gauge fields to be 3~TeV \cite{Carena:2007ua}. This implies that we have assumed extra custodial symmetries, as explained in the above. We ignore brane localized kinetic terms for various bulk fields, as they are most naturally loop suppressed. Then, the RS geometry fixes the ratios of all KK masses. In what follows, we will also ignore potential mixing among various KK modes that can induce small shifts in their masses. These considerations do not change our main conclusions regarding discovery reach for WHFM, at colliders. The masses of various KK modes are given by \begin{equation} m_n = x_n \, k e^{-kr_c \pi}, \label{mKK} \end{equation} where for gauge fields $x_n = 2.45, 5.56, 8.70, \ldots$ and for the graviton $x^G_n = 3.83, 7.02, 10.17,\ldots$. The values of KK masses for SM fermions depends on the bulk mass parameter $c$. In the fermion sector, we will only discuss the reach for SM KK quarks, since we will concentrate on multi-TeV hadron colliders. Here, we choose $c^D \approx - c^S \approx 0.6$ for {\it light} quarks. This choice results in masses of order 10-100~MeV for ${\cal O}(1)$ bulk Yukawa couplings. This range roughly covers the quark flavors $(u,d,s)$ which constitute the dominant quark initial states for collider production of new physics. It turns out that the KK modes of all quarks, except for the singlet top quark, are roughly degenerate in mass with KK modes of the gauge fields. To get a reasonable top quark mass, we choose $c^D_t \approx c^S_t \approx 0.4$, giving a singlet top first KK mass roughly 1.5 times the first gauge KK mass. To assess the relative significance of various production channels, we must know the relevant couplings that enter the calculations (an earlier qualitative discussion in a somewhat different context can be found in Ref.\cite{Davoudiasl:2007zx}). Again, we only discuss typical values to keep our discussion more general and less parameter-specific. We adopt the notation $g^l_{mn}$ to denote the coupling of the $l^{th}$ gauge KK mode to fermions of the $m^{th}$ and $n^{th}$ KK levels; $g^0_{00} \equiv g_{SM}$. We will focus on the gluon and quark sectors, so $g_{SM} = g_s$, the strong coupling constant. First, we will consider single production of KK quarks. This cannot proceed via fusion of quark and gluon zero modes, since this vertex is zero by orthogonality of fermion KK wavefunctions. Next, is the possibility of production in association with a zero mode quark. The gluon-mediated diagram again gives zero, by orthogonality. However, KK-gluon-mediation gives a non-zero result. Given that KK gluons are IR-localized, the only feasible channels involve the third generation zero modes. We find that the coupling $g^1_{00}\sim g_s/5$, for light quarks. Note that the initial states cannot be gluons, by orthogonality of the gluon KK modes. For $(t, b)_L$ and $t_R$, we get $g^1_{01} \sim \sqrt{2\pi}g_s$. Hence, production in these channels is roughly proportional to $(\sqrt{2 \pi}g_s^2/5)^2\sim g_s^4/4$. Given that the KK modes of the singlet top are about 1.5 heavier than the doublets, the most promising channel is single production of a third generation doublet KK mode in association with a $t$ or a $b$. Next, let us examine pair production of KK quarks. Gluon mediated production can come from zero mode quark and gluon initial states; each of these amplitudes is proportional to $g_s^2$. There is also a KK gluon mediated channel. Here, the amplitude is roughly proportional to $g_s^2$; this is due to an approximate cancellation of volume suppression and enhancement at the two vertices for this process. Therefore, we see that pair-production has a number of SM-strength channels available to it, unlike the single production. This can offset the kinematic suppression from producing two heavy states. In fact we will later see that pair-production will dominate single-production with our typical choices of parameters. Finally, since we will also discuss discovery reach for KK gravitons, we will briefly review their relevant couplings. As is well-known, light quark zero modes have a negligible coupling to KK gravitons. This is because of the extreme IR-localization of these KK gravitons, compared with KK gauge fields. Thus, in generic WHFM, only the coupling $C^{ A A G }_{ 0 0 n }$ of two gluons to the KK graviton is important for its collider production \cite{KKgraviton1,KKgraviton2}. In units of the graviton zero mode coupling, $1/{\bar M}_P$, we have \cite{Davoudiasl:2000wi} \begin{equation} C^{A A G}_{0 0 n} = e^{k r_c \pi} \frac{2 \left[1 - J_0 \left(x_n^G \right) \right]}{k r_c \pi \left(x_n^G \right)^2 |J_2 \left(x_n^G \right)|}, \label{ggG} \end{equation} where the first few $x^G_n$ were given above. \section{Experimental Prospects for Verification of WHFM} Given the discussion in the previous section, the first thing we need to do is to demonstrate that single (associated) production of KK fermions together with their corresponding zero modes is relatively suppressed in comparison to the production of KK fermion pairs. We will concentrate on the quark and gluon sectors. This reaction is dominated by the subprocess $q\bar q \rightarrow g^{(1)}\rightarrow q^{(1)}\bar q^{(0)}+h.c.$, where the first gluon KK mode $g^{(1)}$ is somewhat off-shell. (We assume the width to mass ratio of this gluon KK to be 1/6 in our analysis, following Ref.~\cite{KKgluon1}.) To be concrete, we focus our attention on the excitations of the left-handed third generation quark doublet $Q_3=(t,b)_L^T$ since it has a large coupling to $g^{(1)}$, as discussed above. For the LHC, the results of these calculations for single production can be seen in Fig.~\ref{fig1} while generalization to higher energy colliders can be seen in Fig.~\ref{fig2}. Note that no cuts or branching fractions have been included in these calculations, in order to avoid a model-specific analysis. These rates will go down if we assume a more massive gluon KK state. In this work, we will generally assume that ${\cal O}(100)$ events will be sufficient to establish the discovery of a KK mode. The results in Figs. (\ref{fig1}) and (\ref{fig2}) represent the case with the product of the entrance and exit channel couplings equal to $g_s^2$, the SM value. However, from our discussion in the previous section, we expect that a more realistic value, for these processes, is $g_s^2/2$, suppressing the production by factor of about 4. The rates, as can be seen from the figures, are not very impressive, even with this optimistic assumption. It is clear that this process will be unobservable at the LHC even with a luminosity upgrade. Further, note that the cross section only increases by a factor of order $\sim 20$ in the peak region when going from $\sqrt s=$14 to 60 TeV. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{singleKKsplit.ps}} \vspace*{0.1cm} \caption{Rates for the associated production of first $t_L$ and $b_L$ KK excitations together with their corresponding zero modes at the LHC; the results for $t(b)_L$ correspond to to the higher (lower) member of each histogram pair, respectively. Here the first gluon KK mass is fixed at 3~TeV and we show the results as a function of the fermion pair mass. The three sets of histograms assume that the $b_L$ KK mass is 100 $(200, 300)$~GeV heavier than the gauge KK whereas the threshold for $t_L$ is somewhat higher due to the larger zero mode top mass.} \label{fig1} \end{figure} \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{singleKKdeg.ps}} \vspace*{0.1cm} \caption{Same as the last figure but now for different values of $\sqrt s $ and taking the first gluon KK and fermion KK masses to be degenerate at 3~TeV. From bottom to top the histograms correspond to $\sqrt s=14$, 21, 28 and 60 TeV, respectively.} \label{fig2} \end{figure} We now turn to the possibility of fermion KK pair production which can arise from both $q\bar q$ and $gg$ initial states as discussed above and is mediated by the entire gluon KK tower, including the zero mode. (Note that in our calculations we include only the first three gluon KK excitations as well as the SM gluon.) The results of these calculations are shown in Fig.~\ref{fig3} and \ref{fig4} with no cuts or branching fractions applied. Again we see that for the LHC the rates are far too small to be useful but they grow quite rapidly as the collider energy increases; without much effort we see that $\sqrt s=28$ GeV is perhaps the bare minimum requirement to observe these KK states and an even higher value is likely to be necessary if efficiencies and branching fractions are suitably accounted for. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{ferKKpair=1mA.ps}} \vspace*{0.1cm} \caption{Fermion KK pair production rate at various collider energies as a function of the fermion pair invariant mass assuming that the first gluon and fermion KK states are degenerate with a mass of 3 TeV. From bottom to top the histograms correspond to $\sqrt s=14$, 21, 28 and 60 TeV, respectively. No cuts have been applied. The dotted histogram shows the result of only including zero mode gluon exchange.} \label{fig3} \end{figure} \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{ferKKpair=1.1mA.ps}} \vspace*{0.1cm} \caption{Same as the previous figure but now assuming that the KK fermion is $10\%$ more massive than the first gauge KK state.} \label{fig4} \end{figure} We will now discuss the likely dominant decay channels for KK quarks, without entering into a detailed analysis. We will consider the gauge eigen-basis picture, for simplicity. Given that the KK quarks are somewhat heavier than the gluon KK modes, $q_{KK} \rightarrow q \, g_{KK}$, with $q$ a zero-mode quark, is a possible decay channel; let us denote this possibility as channel A. Another potentially important decay mode is through the brane-localized Yukawa coupling to the Higgs: $q^i_{KK} \rightarrow H \, q^j_{KK}$ where the final quark KK mode is lighter than the one in the initial sate; we will refer to this as channel B. Channel C in the following will refer to $q^i_{KK} \rightarrow H \, q^j$ with $q^j$ denoting another zero-mode quark. (The weak-sector analog of channel A will be suppressed by the weak coupling constant so we will ignore it in the discussion which follows.) Note that here if $q^i$ is a weak doublet then $q^j$ is a weak singlet, and vice versa. The subsequent decay of the gluon KK state in channel A produces the final state $q\, t{\bar t}$. In channel B, depending on whether $q^j_{KK}$ decays through channels A or C, we get either $H\,q\,t{\bar t}$ or $H\, H\, q$, respectively as a final state. The couplings in channels A and C are controlled by the overlap of the zero mode quark with the IR brane states, $g_{KK}$ or $H$. For channel B this coupling is ${\cal O}(1)$, when allowed, since it involves three IR-brane states (CFT composites, in the dual picture). For example, if a singlet quark is much more UV-localized than its doublet counterpart, the doublet KK mode will decay through channel A, whereas the singlet KK mode will decay mostly through channel B (since it is more massive than the doublet KK mode). With our reference values, the singlet and doublet KK modes of light flavors are taken to be degenerate, which essentially eliminates channel B. However, this is not typically expected to be the case in a realistic setup, where doublet and singlet quarks have differing values of profile parameter $c$. We have also ignored the effect of mixing in the mass eigen-basis, where one expects that the degenerate KK modes are split by off-diagonal Yukawa couplings \cite{Agashe:2006iy}. Exactly what decay modes will dominate for each quark KK state depends on the $c$ values and possible mixing angles. Nonetheless, given the above three channels, the typical final states corresponding to the decay of the pair-produced quark KK modes are expected to be given by $2\times [q\, t{\bar t}]$, $2\times [H\,q\,t{\bar t}]$, or $2\times [H\,H\,q]$. Needless to say, these are complicated final states and require further study regarding their reconstruction and possible backgrounds. A complete verification of WHFM, based on the RS picture, would require the observation of the first graviton KK state. This is known to be difficult at LHC energies if the gauge KK mass is 3 TeV; the graviton KK mass in this case is $\simeq 4.7$ TeV. A promising search mode is to look for the process $gg\rightarrow G^{(1)} \rightarrow Z_LZ_L$ \cite{KKgraviton2} which is rather clean; here $Z_L$ denotes a longitudinal $Z$ which is IR-localized. The SM background arises from the conventional tree-level process $q\bar q \rightarrow ZZ$ via $t-$ and $u-$ channel diagrams which is highly peaked in the forward and backward directions and is reducible by strong rapidity cuts. Note that as the $\sqrt s$ of the collider increases the average collision energy also increases. This leads to a stronger peaking of the SM $ZZ$ background in both the forward and backward directions. Since the decay products of the $Z$'s essentially follow those of their original parent particle a tightening of the rapidity cuts will be necessary as $\sqrt s$ increases to maintain a reasonable signal to background ratio. In performing these calculations, given our assumptions, the only free parameter is the ratio $k/{\bar M}_P$. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{graviton_c=0.5.ps}} \vspace*{0.1cm} \caption{Production rate for the first graviton KK excitation decaying into two $Z$ bosons assuming a rapidity cut $|y|<2(1)$ on the $Z$'s corresponding to the dotted(solid) histograms. The histograms correspond, from bottom to top, to collider energies of $\sqrt s=14$, 21, 28 and 60 TeV, respectively, $Z$ branching fractions are not included and $k/{\bar M}_P=0.5$ has been assumed.} \label{fig5} \end{figure} Figs.~\ref{fig5},~\ref{fig6} and ~\ref{fig7} show the results of these calculations for three different values of the ratio $k/{\bar M}_P=0.5,1.0$ and 0.1, respectively; as expected, in all cases the event rate is far too low at the LHC to be observable. For $k/{\bar M}_P=0.5$, the graviton KK is a reasonably well-defined resonance structure which grows quite wide (narrow) when this same ratio equals 1 (0.1). For a fixed rapidity cut, the signal over the background is is seen to diminish as the center of mass energy of the collider grows larger. If we identify the graviton through the decay $ZZ \rightarrow jj \ell^+ \ell^-$, which has a branching fraction of a few percent, it is clear that a $\sqrt s \sim 60$ TeV collider will be necessary to observe this state. Here, we ignore detection issues related to having collimated jets from highly boosted $Z$'s. A realistic jet resolution could make these conclusions less optimistic. However, we refrain from making assumptions about the detector capabilities and data analysis methods of future NHC experiments. Note that if $k/{\bar M}_P$ is sufficiently small even this large collider energy will be insufficient as to discover the graviton KK resonance as it becomes quite narrow. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{graviton_c=1.0.ps}} \vspace*{0.1cm} \caption{Same as the previous figure but now with $k/{\bar M}_P=1.0$.} \label{fig6} \end{figure} \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{graviton_c=0.1.ps}} \vspace*{0.1cm} \caption{Same as the previous figure but now with $k/{\bar M}_P=0.1$.} \label{fig7} \end{figure} It is also of some interest to examine the possibility of searching for gluon KK states via pair production now that we are considering higher energy colliders. The production of resonant single gluon KK states is made difficult since it can only occur through the light initial state partons which have suppressed couplings to the KK gluons. The possibility of pair production avoids these issues as it arises from both $gg$ as well as $q\bar q$ initial states and occurs through the exchange of the complete gluon KK tower; we include the first three gluon KK states in the calculations below in addition to the SM gluon zero mode. Once produced, gluon KK states almost uniquely decay to top quark pairs, in typical scenarios \cite{KKgluon1, KKgluon2}. Since 2 KK gluon states are made, the final state consists of two pairs of top quarks. Fig.~\ref{fig8} shows the results of these calculations for the same collider energies as above, {\it i.e.}, $\sqrt s=14$, 21, 28 and 60 TeV. As can be seen here, these signal rates can be quite significant once we go to energies substantially above that of the LHC. In order to determine how significant the resulting signal from these rates is we need to have an estimate of the SM background. We will not consider this background here and only provide the production rate. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{gaugepair.ps}} \vspace*{0.1cm} \caption{Same as Fig.~\ref{fig3} except now for the pair production of the lightest gluon KK state.} \label{fig8} \end{figure} \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{glu1top_2.ps}} \vspace*{0.1cm} \caption{The resonant production rate for the first gluon KK state in the $t\bar t$ channel assuming $|y|<2.5$ for the dotted case and $|y|<1$ for the solid one. From bottom to top the histograms correspond to $\sqrt s=14$, 21, 28 and 60 TeV respectively. No efficiencies or branching fractions are included.} \label{fig9} \end{figure} The $q\bar q \rightarrow g^{(1)} \rightarrow t\bar t$ gluon KK channel essentially sets the discovery reach for the RS scenario at hadron colliders \cite{KKgluon1, KKgluon2}. At the LHC, a 3 TeV KK gluon should be visible above the usual top quark pair SM background, with suitable rapidity cuts, after branching fractions and tagging efficiencies are accounted for. Unfortunately, as in the graviton KK $ZZ$ mode above, as the center of mass energy of the collider increases the forward/backward peaked SM $gg,q\bar q \rightarrow t\bar t$ process grows quite rapidly relative to the KK gluon signal for fixed rapidity cuts. (Recall that at these energies the decay products of the top quark will essentially follow the original top flight direction.) This result can be seen quite explicitly in Fig.~\ref{fig9}. Here we see that the obvious gluon KK peak structure slowly disappears with increasing collider energy. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{gluKKreach_S.ps}} \vspace*{0.1cm} \caption{Signal rate for a possible gluon KK resonance as a function of the collider energy employing the cuts described in the text. Branching fractions and efficiencies have been neglected. From top to bottom the results are shown for gluon KK masses in the range from 3 to 12 TeV in steps of 1 TeV.} \label{fig10} \end{figure} If we want to ensure a significant signal to background ratio for gluon KK states at higher energy colliders we need to tighten our rapidity cuts from the usual `central detector' requirements, $|y|<2.5$. In our analysis below we will assume that $|y|<1$ to increase the S/B ratio. Furthermore we will define the signal region to be in the $t\bar t$ invariant mass range within $\pm \Gamma_{KK}$, the gluon KK width, of the gluon KK mass with $\Gamma_{KK}/M_{KK}=1/6$ assumed in our analysis below. Fig.~\ref{fig10} shows the resulting signal rates following this procedure for a range of KK gluon masses as a function of the collider energy; branching fractions and efficiencies have been ignored in obtaining these results. \begin{figure}[htbp] \centerline{ \includegraphics[width=7.5cm,angle=90]{gluKKreach_signi.ps}} \vspace*{0.1cm} \caption{Same as the previous figure but now showing the signal significance. Again, branching fractions and efficiencies have been neglected.} \label{fig11} \end{figure} Fig.~\ref{fig11} shows the signal significance for gluon KK production for a range of masses as a function of the hadron collider center of mass energy. In a more realistic calculation which includes top quark branching fractions $f$ and $b$-tagging efficiencies $\epsilon$, the results shown here must be scaled by $\sqrt {f \epsilon} \sim 0.15$ \cite{KKgluon1}. Taking this factor into account we see that, {\it e.g.}, at a $\sqrt s= 21\, (28, 60)$ TeV collider KK gluon masses as large as 5.5 (7, 12) TeV may become accessible. One could say that this covers the entire `natural' parameter space for WHFM. \section{Conclusions} Amongst models of physics beyond the Standard Model, WHFM are rather unique to the extent that they are capable of providing a simultaneous natural resolution of two important puzzles, namely the hierarchy and the flavor problems. In particular, this is in sharp contrast to supersymmetry, which is otherwise an extremely interesting theoretical construct. Thus, it is clearly important to establish the requirements for direct experimental verification of WHFM. These more recent developments of the original RS model \cite{Randall:1999ee}, which explain gauge and flavor hierarchies in one framework, have KK particles in the few TeV range. Low energy precision tests suggest that gauge KK masses in WHFM are likely to be heavier than about 3 TeV. Perhaps the most compelling and unique signature of these models is the spin-2 KK graviton. Unfortunately, it seems at the LHC, even with an upgraded luminosity, KK gravitons, expected here to lie above 4.7 TeV, are very challenging to observe and likely inaccessible. Prospects for KK gluons up to masses around 4 TeV seem brighter at the LHC. Thus it may well be that some early indication of the underlying RS idea may find support at the LHC. However, though considerable work exists in the literature on warped {\it bosonic} KK modes, not much has been done for the SM fermion counterparts. The observation of these {\it fermionic} KK modes can in essence be taken as direct experimental evidence for warped bulk flavor. With that perspective in mind, in this work, we set out to provide an exploratory study of the parameters needed for the next generation of machines that could provide significant experimental support for the WHFM, and especially for generation of flavor through bulk localization. We concentrate on hadron colliders only, as they are expected to yield the largest kinematic reach. For definiteness, throughout our numerical study here, we take the lightest gauge KK mass to be 3 TeV; SM fermionic KK modes are always as massive or heavier. First, we studied single KK fermion production. The most promising candidate in this regard is the KK mode of the third generation doublet produced in association with a $t$ or a $b$ quark. We found that at the LHC the prospects for finding this KK fermion through single production are rather grim. In fact, we showed that even a 28 TeV machine is only likely to see at most a handful of candidate events of this category. The prospects improve significantly for a 60 TeV machine wherein a few tens of events are possible. Pair production of KK fermions typically seems to have over an order of magnitude larger cross section compared to the single-production, for large $\sqrt{s}$. For pair production, a 28 TeV machine can give several tens of events and 60 TeV produces hundreds of such candidate events. One may expect that a sample of this latter size is needed for a reliable verification of the bulk flavor scenarios, after cuts and efficiencies are taken into account; we do not delve into these issues in this exploratory work. We have revisited the earlier study of the KK graviton through the ``gold plated" $ZZ$ mode \cite{KKgraviton2}. We find that for this unique channel, only a 60 TeV machine can provide ${\cal O}(100)$ events over a plausible range of $k/{\bar M}_P$ values. Thus, it seems that verification of the WHFM, in the sense of directly measuring the properties of signature KK modes, requires what we refer to as the Next Hadron Collider (NHC) with $\sqrt{s} \approx 60$~TeV and ${\cal O} (1)$~ab$^{-1}$ of integrated luminosity. We have also extended earlier studies of the resonance production of the first KK gluon state and its detection through a $t \bar t$ pair. We find that colliders with energies 14, 21, 28 and 60 TeV can allow detection of the first KK gluon up to masses 4, 5.5, 7, and 12 TeV, respectively. We hence conclude that an NHC-class machine must be an integral part of the high energy experimental program if hints of WHFM are discovered at the LHC. The same conclusion holds for 4D models that explain hierarchy and flavor in a similar fashion and constitute dual dynamical scenarios, according to AdS/CFT correspondence \cite{Maldacena:1997re,ArkaniHamed:2000ds}. \acknowledgments H.D. and A.S. are supported in part by the DOE grant DE-AC02-98CH10886 (BNL).

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\section*{Preface} \section{Introduction} \label{sec.introduction} Let us define the FCFS infinite bipartite matching model formally, following \cite{caldentey-kaplan-weiss:09,adan-weiss:11}. \begin{definition}\label{de-ckw} Consider two finite sets of {\em types} ${\mathcal{C}}=\{c_1,\ldots,c_I\}$ and $\S=\{s_1,\ldots,s_J\}$ and a bipartite {\em compatibility} graph $G=({\mathcal{C}},\S,{\mathcal{E}})$, with ${\mathcal{E}}\subset {\mathcal{C}}\times \S$, assumed to be connected. Consider two independent random sequences $(c^m)_{m\in T}$ and $(s^n)_{n\in T}$ which are chosen respectively i.i.d. from ${\mathcal{C}}$ with probabilities $\alpha=\{\alpha_{c_1},\ldots,\alpha_{c_I}\}$, and i.i.d. from $\S$ with probabilities $\beta=\{\beta_{s_1},\ldots,\beta_{s_J}\}$. The parameter set $T$ can be finite, for instance $T={\mathbb{I}}_N=0,1,2,\ldots,N$, one sided infinite, $T={\mathbb{N}}=0,1,2,3,\ldots$, or two sided infinite, $T={\mathbb{Z}}=\ldots,-2,-1,0,1,2,3\ldots$. We assume that $\alpha_i\neq 0$ for all $i$ and $\beta_j\neq 0$ for all $j$. The triple $(G,\alpha,\beta)$ defines an {\em infinite bipartite matching model}. \end{definition} In the compatibility graph, $(c_i,s_j)\in {\mathcal{E}}$ denotes that the types $c_i$ and $s_j$ are compatible and can be matched together. If $(c_i,s_j) \not\in {\mathcal{E}}$, we say that $c_i$ and $s_j$ are incompatible and they cannot be matched to each other. We study {\em first come first served (FCFS) matchings} between the two sequences, as defined precisely in \cite{adan-weiss:11}, the exact definition is repeated in Section \ref{sec.FCFS}. Loosely speaking it means that an item from $S$ is matched to the earliest possible compatible item from $C$, and vice versa. This is unambiguously defined when $T$ is finite or when $T={\mathbb{N}}$. In the case $T={\mathbb{Z}}$, defining precisely the meaning of the FCFS matching is the purpose of Section \ref{sec.loynes}. Figure \ref{fig.match1} illustrates FCFS matching of two sequences, for a given compatibility graph. Items are browsed from left to right. For instance, $c^1$ cannot be matched with $s^1$ nor $s^2$ since $c^1=c_1, s^1=s^2=s_2$ and $(c_1,s_2) \not\in {\mathcal{E}}$, but it will be matched with $s^3=s_3$ which is the earliest possible match. \begin{figure}[htb] \begin{center} \includegraphics[width=5.25in]{FigMatch1.pdf} \end{center} \caption{FCFS matching of two sequences (left), for the compatibility graph $G$ (right)} \label{fig.match1} \end{figure} {\bf Notations}: Throughout the paper we shall use the following notations, terms, and illustrations to accompany our mathematical analysis. For convenience we shall refer to $s^n$ as servers, to $c^m$ as customers (we could equally well call them, according to the context, donors and patients, calls and agents, males and females, buy bids and sell bids), and we shall refer to $T$ as time (equivalently we could call it location, position, event). We use $c_i,\,i=1,\ldots,I$ to denote the different types of customers, and we use $c^m$ to denote the type of the $m$-th customer. Similarly we use $s_j, \, j =1, \ldots, J$ to denote the types of servers, and $s^n$ denotes the type of the $n$-th server. In the accompanying figures, we will simplify the notation by writing $c^m=i$ if $c^m=c_i$, and $s^n=j$ if $s^n=s_j$, and we will arrange the sequences in two lines, the top one containing the ordered customers, and the bottom one containing the ordered servers. At a later stage, we will sometimes swap items between the top and the bottom lines in a way to be explained then. In figures, we put an edge between $s^n$ and $c^m$ if they are matched, and we call this edge a {\em link} in the matching, characterized by the pair of times $(m,n)$. \medskip {\bf Motivations and previous works:} In 1984, Ed Kaplan \cite{kaplan:84} asked the following question when studying public housing in Boston: Entitled families list those projects where they wish to live, and when a home becomes available in a project, it is assigned by law to the longest waiting family that listed it. The question then was what fraction of families with a specific list of preferences is actually assigned to each of the projects in their list. Kaplan called these fractions the {\em matching rates}. The same mechanism may also model child adoptions, blood banks, organ transplants, and the operation of web servers and cloud computing. Kaplan's problem is related to the queueing model of parallel server scheduling, where there is a stream of arriving customers of several types ${\mathcal{C}}=\{c_1,\ldots,c_I\}$ which are served by several pools of parallel servers, with $n_1,\ldots,n_J$ servers of types $\S=\{s_1,\ldots,s_J\}$, and compatibility graph $G=({\mathcal{C}},\S,{\mathcal{E}})$, where the policy is FCFS, that is, whenever a server is available he attends to the longest waiting compatible customer. This queueing model is notoriously hard to analyze, as demonstrated for instance in a paper by Foss and Chernova \cite{foss-chernova:98}, where it is proved that, with general service time distributions, the stability of this model cannot be determined by the fluid model alone. Motivated by Kaplan's problem, Whitt and Talreja \cite{talreja-whitt:07} studied a formal fluid model of the overloaded parallel server queueing model under FCFS policy, and derived explicit matching rates in the special cases of bipartite graphs that are trees or can be decomposed as a tree of complete bipartite graphs. However, the parallel server queueing model may be inappropriate for the housing scenario, as apartments do not act as servers. A more appropriate model is to assume that houses as well as families arrive in random streams, independent of each other, and, when a match is made, they depart the system. This led to the formulation, recalled in Definition \ref{de-ckw}, of the FCFS bipartite infinite matching model by Caldentey, Kaplan and Weiss \cite{caldentey-kaplan-weiss:09}, which makes the further simplification of avoiding all consideration of timing of the independent arrival streams, and considers only the i.i.d. types of the ordered arrivals of customers and servers. In \cite{caldentey-kaplan-weiss:09}, a necessary condition for stability was derived, and the FCFS matching rates were calculated for almost complete bipartite graphs. The following questions were left open: Is the condition for stability also sufficient, and can we calculate the matching rates for general bipartite graphs. Turning back to the parallel server queueing model, it turned out that the special case where arrivals are Poisson, service time distributions are exponential, and the service rates depend only on the type of server, and not on the type of customer that is served, the model under FCFS policy is tractable. Indeed, by specifying how customers are assigned to servers when there is a choice of several compatible available servers, it was shown in \cite{visschers-adan-weiss:12} that there exists a random assignment regime under which the queue lengths are Markovian, satisfy partial balance equations, and have a product form stationary distribution. This result was extended to a loss model in \cite{adan-hurkens-weiss:10,adan-weiss:12b}. Admittedly, the very special form of the random assignment of customers to a choice of servers made this result of limited usefulness. But a similar product form result was later obtained in \cite{adan-weiss:14} for the parallel server queueing model with a more natural policy: the FCFS-ALIS policy where arriving customers that have a choice of several free compatible servers are assigned to the longest idle server (ALIS). An unexpected connection between the parallel server queueing model and the bipartite matching model was found in \cite{adan-weiss:11} and provided a breakthrough for the bipartite matching model. Indeed, it was proved in \cite{adan-weiss:11} that a Markov chain associated with the bipartite matching model, and directly inspired by the queue length process of the parallel server queueing model, enjoys the same remarkable properties as its source of inspiration: it satisfies partial balance, and has a product form stationary distribution. As a by-product, it was obtained that the necessary conditions for stability were also sufficient, and, furthermore, if was possible to derive exact formula for the computation of the matching rates. Let us mention that further research on infinite bipartite matching under policies different from FCFS, and on infinite matching in graphs that are not bipartite, are described in \cite{busic-gupta-mairesse:10} and \cite{mairesse-moyal:14}. To complete this review of previous work, let us mention another conjectural connection between the bipartite matching model and the parallel server queueing model. In \cite{adan-boon-weiss:13,adan-boon-weiss:14}, it is conjectured that in a parallel server queueing model, stabilized by abandonments, and operated under FCFS-ALIS, matching rates of the queueing system under many server scaling equal the matching rates of the FCFS infinite bipartite matching model. This conjecture is as yet not proved, but simulations indicate that it may be correct, and that it can be used to evaluate performance and assist in design of such systems. \medskip {\bf New contributions of the paper}: Consider the FCFS infinite bipartite matching model. The following questions were left unanswered by the different contributions mentioned above. First, can one construct FCFS matching of two sequences that are defined for all the integers, $\mathbb{Z} = \ldots,-2,-1,0,1,\ldots$? Second, partial balance usually indicates reversibility, can we show that the FCFS infinite matching model is time reversible in some sense? Third, the Markov chain that was defined and analyzed in \cite{adan-weiss:11} is one of several Markov chains that can be associated with the FCFS infinite matching model, can we obtain stationary distributions for all of them (e.g. for the chain defined in \cite{caldentey-kaplan-weiss:09})? We answer all these questions in the current paper. Indeed, we uncover the fundamental structure of the model in the following three steps: \begin{compactitem} \item We derive a Loynes' scheme, which enables to get to stationarity through sample path dynamics, and to prove the existence of a unique FCFS matching over $\mathbb{Z}$. \item We define a pathwise transformation in which we interchange the positions of the two items in a matched pair, see Figure \ref{fig.match2}, and we prove the ``dynamic reversibility'' of the model under this transformation. \item We construct ``primitive'' Markov chains whose product form stationary distributions are obtained directly from the dynamic reversibility. Using these as building blocks, we derive product form stationary distributions for multiple ``natural'' Markov chains associated with the model, and we compute various non-trivial performance measures as a by-product. \end{compactitem} \medskip \begin{figure}[htb] \begin{center} \includegraphics[width=5.25in]{FigMatch2.pdf} \end{center} \caption{The matched sequences of Figure \ref{fig.match1}, after the exchange transformation} \label{fig.match2} \end{figure} {\bf Organization}: We now briefly describe the content of the sections of the paper. In Section \ref{sec.FCFS}, we state the definition of FCFS matching and summarize some results from \cite{adan-weiss:11}. These include uniqueness of the FCFS matching of two sequences over ${\mathbb{N}}$, and two specific mechanisms to construct the matching. We define two Markov chains associated with these two mechanisms, which we call {\em the natural Markov chains}, and state the necessary and sufficient condition for their ergodicity, given by {\em complete resource pooling}. In Section \ref{sec.loynes}, we use a Loynes' type construction to obtain FCFS matching of two sequences over ${\mathbb{Z}}$. We show that if complete resource pooling holds then the Loynes' construction, matching all customers and servers in positions $n > -k$ and letting $k\to\infty$, converges almost surely to a unique FCFS matching. This also yields the unique stationary version of the Markov chains associated with the construction of the matching. In Section \ref{sec.reversibility}, we discuss the {\em exchange transformation}, in which for a matched pair $c^m,s^n$, we exchange the customer and the server, so that we now have a customer in position $n$, denoted by ${\tilde{c}}^n$ (on the servers line) and a server in position $m$, denoted by ${\tilde{s}}^m$ (on the customers line), as illustrated in Figure \ref{fig.match2}. We show the following reversibility result: starting from two independent i.i.d. sequences over ${\mathbb{Z}}$ with FCFS matching between them, and performing the exchange transformation on all the links, we obtain two sequences of exchanged customers and servers and a matching between them. It is then true that the sequences are again independent i.i.d., and the matching between them is FCFS in the reversed time direction. In Section \ref{sec.stationary}, we consider several Markov chains that accompany the construction of FCFS matching. As suggested by the reversibility, these also have product from stationary distributions. The stationary probabilities have an extremely simple structure, they behave like finite sequences of multi-Bernoulli trials. We also derive an expression for the normalizing constant of the stationary distributions. As is often the case, its computation is not easy, and we conjecture it is $\sharp P$. We illustrate these results for the ``NN'' example that was described in \cite{caldentey-kaplan-weiss:09}, and that could not be fully analyzed at the time. Finally, in Section \ref{sec.perfomance}, we consider some performance measures for a stationary matching. First, we recall the results from \cite{adan-weiss:11} on the computation of the matching rates $r_{c_i,s_j}$, that is, the frequency of matches of customer type $c_i$ with server type $s_j$. Then we obtain novel results on another performance measure that is the random link length, $L_{c_i,s_j}$, given by $m-n$ for a match of $c^m$ with $s^n$, where $c^m=c_i,\,s^n=s_j$. We obtain the stationary distribution of the link length, which is a mixture of convolutions of Geometric random variables. \section{FCFS bipartite matching over ${\mathbb{N}}$} \label{sec.FCFS} In this section we provide a precise definition of bipartite FCFS matching, quote the result of uniqueness of the matching for $T={\mathbb{N}}$, describe two Markov chains to construct this unique matching, define complete resource pooling, and quote the result that these Markov chains are ergodic if and only if complete resource pooling holds. \begin{definition} \label{def.fcfs} Let $c=(c^m)_{m\in T_1}$ and $s=(s^n)_{n\in T_2} $ be some fixed ordered sequences of customers and servers. Here $T_1$ and $T_2$ are index sets which are either finite, one-sided infinite or bi-infinite. \begin{compactitem} \item[-] A \emph{partial matching} of $c$ and $s$ is a set $A\subset T_1 \times T_2$ corresponding to customer-server pairs satisfying: \begin{eqnarray*} &(i) & (m,n) \in A \implies (c^m,s^n)\in {\mathcal{E}}\\ & (ii) &\forall m , \ \# \{n \, : \, (m,n)\in A\} \leq 1, \ \forall n, \ \#\{m \, : \, (m,n) \in A\} \leq 1 \:. \end{eqnarray*} For a partial matching $A$, we denote \[ A_c=\{m\,:\, \exists n, \, (m,n)\in A\}, \quad A_s=\{n\,:\, \exists m, \,(m,n)\in A\}\:. \] \item[-] A partial matching $A$ is a \emph{(complete) matching} if there are no unmatched compatible pairs left outside of $A$, that is: \[ m\not\in A_c, \ n\not\in A_s \quad \implies \quad (c^m,s^n) \not\in {\mathcal{E}}. \] \item[-] A matching $A$ is FCFS if for every $(m,n)\in A$, \begin{eqnarray*} && \mbox{if $l < n,\, (c^m,s^l)\in {\mathcal{E}}$, then there exists $k<m$ such that $(k,l)\in A$ and } \\ && \mbox{if $k < m,\, (c^k,s^n)\in {\mathcal{E}}$, then there exists $l<n$ such that $(k,l)\in A$.} \end{eqnarray*} Note that the condition simply checks for every match, that all earlier compatible items have been matched already, which is the FCFS property. \item[-] A matching is \emph{perfect} if $T_1=T_2$, and all customers and servers are matched. \end{compactitem} \end{definition} The following Lemma was proved by induction in \cite{adan-weiss:11}. \begin{lemma}[Adan \& Weiss, 2011 \cite{adan-weiss:11}] \label{thm.unique} For every finite index sets ${\mathbb{I}}_M,{\mathbb{I}}_N$, there exists a complete FCFS matching, and it is unique. \end{lemma} \begin{definition} \label{def.fair} The sequences $(c^m)_{m\in{\mathbb{N}}}$ and $(s^n)_{n\in {\mathbb{N}}}$ are {\em matchable} if the number of type $c_i$ customers and the number of type $s_j$ servers is infinite for all $i=1\ldots,I$ and $j=1,\ldots,J$. The sequences $(c^m)_{m\in{\mathbb{Z}}}$ and $(s^n)_{n\in {\mathbb{Z}}}$ are {\em matchable} if the same also holds for times $-1,-2,\ldots$. \end{definition} The following theorem was proved by Adan and Weiss \cite{adan-weiss:11}, we provide a sketch of the proof, for completeness. \begin{theorem} \label{thm.oneside} For any two matchable sequences $(c^m)_{m\in{\mathbb{N}}}$ and $(s^n)_{n\in {\mathbb{N}}}$, there exists a unique perfect FCFS matching. The matching can be obtained in a constructive way up to arbitrary length. \end{theorem} \begin{proof} The uniqueness follows from Lemma \ref{thm.unique}. If we matched all servers up to server $s^n$, then we can find an unmatched customer that will match $s^{n+1}$, because there is an infinite number of customers of each type, and similarly for $c^m,\,c^{m+1}$. By the uniqueness stated in Lemma \ref{thm.unique}, any construction in which complete $\mathbb{I}_M,\mathbb{I}_N$ matchings are extended by letting $M,N$ increase to infinity in any order will lead to the same infinite perfect matching over ${\mathbb{N}}$ (we present three constructive methods in the next paragraphs). \end{proof} We consider three methods of constructing a FCFS matching step by step, and define three Markov chains associated with them: \begin{description} \item[(i)] Matching pair by pair: Proceeding from a complete FCFS matching of $(c^m,s^n)_{m,n\le N}$, we add the pair $c^{N+1},s^{N+1}$, and match them FCFS to compatible previously unmatched servers and customers if possible, or to each other if possible, or else leave one or both unmatched. With each step we associate a state that consists of the ordered lists of the unmatched servers and customers. It is easy to see that the step by step evolution of the state defines a countable state discrete time irreducible and aperiodic Markov chain. We denote it by $O=(O_N)_{N\in {\mathbb{N}}}$, this is the `natural' pair by pair FCFS Markov chain. \item[(ii)] Matching server by server: Proceeding from the FCFS matching of all the servers $s^n,\, n\le N$, we add the next server $s^{N+1}$, and match it to the first compatible customer that has not yet been matched. With each step we associate a state that consists of the ordered list of skipped customers. This again defines a countable state discrete time irreducible and aperiodic Markov chain. We denote it by $Q^s=(Q^s_N)_{N\in {\mathbb{N}}}$, this is the `natural' server by server FCFS Markov chain. \item[(iii)] Matching customer by customer is analogous, resulting in a `natural' customer by customer FCFS Markov chain $Q^c=(Q^c_N)_{N\in {\mathbb{N}}}$. \end{description} All three Markov chains have a common state if at time $N$ a perfect matching of all previous customers and servers is reached, in which case the state consists of empty lists and is denoted by $\emptyset$ or by $0$. Because these Markov chains have a common state, they will be transient, null recurrent, or ergodic together. We will define several additional Markov chains associated with the construction of FCFS matching in Section \ref{sec.stationary}. All of them will be transient, null recurrent, or ergodic together, and have a common empty state. \begin{definition} \label{def.ergodic} We will say that the FCFS bipartite matching for a given $(G,\alpha,\beta)$ is {\em ergodic} if the corresponding Markov chains are ergodic. \end{definition} We introduce some further notation to formulate the condition for ergodicity: We let $\S(c_i)$ be the set of server types compatible with customer type $c_i$, and ${\mathcal{C}}(s_j)$ be the set of customer types compatible with server type $s_j$. For $C\subset {\mathcal{C}}$ and $S\subset \S$, we define \[ \S(C) = \bigcup_{c_i\in C} \S(c_i), \qquad {\mathcal{C}}(S) = \bigcup_{s_j\in S} {\mathcal{C}}(s_j) \:. \] We also define \[ {\mathcal{U}}(S) = \overline{{\mathcal{C}}(\overline{S})} = {\mathcal{C}} \setminus {\mathcal{C}}(\S \setminus \ S) \] (where $\;\overline{\;\cdot\;}\;$ denotes the complement) which is the set of customer types that can only be served by the servers in $S$. We call these the {\em unique customer types} of $S$. Finally, we let \[ \alpha_{C} = \sum_{c_i\in C} \alpha_{c_i}, \qquad \beta_{S} = \sum_{s_j\in S} \beta_{s_j}\:. \] The following lemma follows immediately from the definitions. \begin{lemma}\label{le-equiv} The following three conditions are equivalent: \begin{eqnarray} \label{eqn.pooling} &\forall C\subset {\mathcal{C}}, C\ne\emptyset, C\ne {\mathcal{C}}, & \alpha_C < \beta_{\S(C)} \nonumber \\ &\forall S\subset \S, S\ne\emptyset, S\ne \S, & \beta_S < \alpha_{C(S)} \\ &\forall S\subset \S, S\ne\emptyset, S\ne \S, &\beta_S > \alpha_{{\mathcal{U}}(S)}. \nonumber \end{eqnarray} \end{lemma} In words, these three conditions ensure that there is enough service capacity for every subset of customer types, there is enough demand for service to ensure that every subset of servers can be fully utilized, and every subset of server types has enough capacity to serve customer types which are unique to this subset. \begin{definition} \label{def.crp} We say that the infinite bipartite matching model $(G,\alpha,\beta)$ satisfies {\em complete resource pooling}, if the equivalent conditions in Lemma \ref{le-equiv} hold. \end{definition} \begin{theorem}[Adan \& Weiss 2011 \cite{adan-weiss:11}]\label{th-ergodic} The FCFS bipartite matching for $(G,\alpha,\beta)$ is ergodic if and only if complete resource pooling holds. \end{theorem} We will derive a simple confirmation of this result in Section \ref{sec.marginals}. \section{Loynes' construction of FCFS bipartite matching over ${\mathbb{Z}}$} \label{sec.loynes} In this section we show that if complete resource pooling holds, then for two independent bi-infinite sequences of i.i.d. customers and servers, $(s^n,c^n)_{n\in {\mathbb{Z}}}$ there exists almost surely a unique FCFS matching. This matching coincides with the matchings obtained from the stationary versions of the various Markov chains described above. The matching is obtained using a Loynes' type scheme (see \cite{loynes:62} for the original Loynes' construction). \begin{figure}[htb] \begin{center} \includegraphics[width=2.5in]{FigLoynes.pdf} \end{center} \caption{Backward step changes the matching} \label{fig.Loynes} \end{figure} Consider the (unique by Theorem \ref{thm.oneside}) matching of $(s^n,c^n)_{n > -K}$, and let $K\to \infty$. At first glance we notice that as we let $K$ increase the matching changes. This is illustrated in the example of Figure \ref{fig.Loynes}. In this example, if $s^{-K-1}=s_2$ and $c^{-K-1}=c_2$, then starting empty at $-K-1$, the state at $-K$ cannot be empty, so the matching starting empty from $-K$ is different from the matching starting from $-K-1$. Nevertheless, in this simple example, classical results can be used to prove that the matching does converge as $K\to\infty$. The condition for that is that complete resource pooling should hold, which in this example happens when $\alpha_2 < \beta_1$. We sketch the argument to provide some intuition for the general proof to come. Denote by $(O_n^{[-K]})_{n\geq -K}$, the pair by pair FCFS Markov chain associated with the sequence $(c^n,s^n)_{n\geq -K}$ (see the definition above). Denote by $X_n^{[-K]}$ the number of unmatched customers, all of them of type $c_2$, (or of unmatched servers, all of them of type $s_2$) in $O_n^{[-K]}$. Observe that: \[ X_{n+1}^{[-K]} = \begin{cases} X_{n}^{[-K]} + 1& \mathrm{if} \quad (c^{n+1},s^{n+1}) = (c_2,s_2) \\ \max(X_{n}^{[-K]} - 1,0) & \mathrm{if} \quad (c^{n+1},s^{n+1}) = (c_1,s_1) \\ X_{n}^{[-K]} & \mathrm{if} \quad (c^{n+1},s^{n+1}) = (c_1,s_2)\ \mathrm{or} \ (c_2,s_1) \:. \end{cases} \] So $(X_n^{[-K]})_{n\ge -K}$ can be interpreted as the queue-length process of a discrete-time version of an M/M/1 queue. Complete resource pooling then implies that the drift of this queue is negative. A straightforward adaptation of the original Loynes argument (designed for a continuous time G/G/1 queue with negative drift), yields the existence of a limiting process for $(X_n^{[-K]})_{n\geq -K}$ when $K\rightarrow +\infty$, see for instance Chapter 2.1 in \cite{BaBr}. Let us denote the limiting process by $(X_n^{\infty})_{n\in {\mathbb{Z}}}$. Again, the original Loynes argument shows that the set of indices $\{n\in{\mathbb{Z}}, \ X_n^{\infty} = 0\}$ is a.s. infinite. These indices can be viewed as regeneration points for the processes $(X_n^{[-K]})_{n\ge -K}$. In particular, if $X_k^{\infty}=0$ then $X_k^{[-K]}=0$ for all $-K\leq k$. Now, observe that these regeneration points are also regeneration points for the processes $(O_n^{[-K]})_{n\geq -K}$, that is, $[X_k^{\infty}=0]\implies [\forall K, \ -K\leq k, \ O_k^{[-K]}=\emptyset]$. Between two regeneration points, we have a finite complete matching of the sequences of customers and of servers, and this matching will not change over time. Therefore, there exists a limiting bi-infinite matching which is obtained by simply concatenating the finite matchings between regeneration points. \medskip The goal is to prove the same thing in general. We will prove the following result: \begin{theorem} \label{thm.co-loynes} For two independent i.i.d. sequences $(s^n,c^n)_{n\in {\mathbb{Z}}}$, if complete resource pooling holds, there exists almost surely a unique FCFS matching over all of ${\mathbb{Z}}$, and it can be obtained by Loynes' scheme, of constructing a FCFS matching from $-K$ to $\infty$, and letting $K \to \infty$. \end{theorem} We prove the Theorem in several steps, requiring three lemmas and two propositions. The next two lemmas are pathwise results which do not depend on any probabilistic assumption. \begin{lemma}[Monotonicity] \label{thm.monotonicity} Consider $c^1,\ldots,c^M$ and $s^1,\ldots,s^N$, and complete FCFS matching between them. Assume there are $K$ customers and $L$ servers left unmatched. Consider now an additional customer $c^0$, and the complete FCFS matching between $c^0,c^1,\ldots,c^M$ and $s^1,\ldots,s^N$. Then this matching will have no more than $K+1$ customers and $L$ servers unmatched. \end{lemma} \begin{proof} Denote $A=(c^1,\ldots,c^M)$ and $B=(s^1,\ldots,s^N)$. In the matching of $(c^0,A)$ and $B$, if $c^0$ has no match, then all the other links in the matching are the same as in the matching of $A$ and $B$, so the total number of unmatched customers is $K+1$ and unmatched servers is $L$. If $c^0$ is matched to $s^n$ and $s^n$ is unmatched in the matching of $A$ and $B$, then $(c^0,s^n)$ is a new link and all the other links in the matching of $(c^0,A)$ and $B$ are the same as in the matching of $A,B$, so the total number of unmatched customers is $K$ and unmatched servers is $L-1$. If $c^0$ is matched to $s^{n_1}$ and $s^{n_1}$ was matched to $c^{m_1}$ in the $A,B$ matching, then $(c^0,s^{n_1})$ is a new link, and the link $(s^{n_1},c^{m_1})$ in the $A,B$ matching is disrupted. We now look for a match for $c^{m_1}$ in the matching of $(c^0,A)$ and $B$. Clearly, $c^{m_1}$ is not matched to any of $s^j,\,j<n_1$, since in the construction of the $A,B$ matching $c^{m_1}$ was not matched to any of those. So $c^{m_1}$ will either remain unmatched, or it will be matched to some $s^{n_2}$, where $n_2>n_1$. In the former case, all the links of the $A,B$ matching except $(s^{n_1},c^{m_1})$ remain unchanged in the $(c^0,A)$ and $B$ matching, and so the numbers of unmatched items remain $K+1$ and $L$. In the latter case, there are again two possibilities: If $s^{n_2}$ is unmatched in the $A,B$ matching, then the $(c^0,A),B$ matching will have disrupted one link and added 2 links retaining all other links of the $A,B$ matching, so the numbers of unmatched items are $K$ and $L-1$. If $s^{n_2}$ is matched to $c^{m_2}$ in the $A,B$ matching, then the link $s^{n_2},c^{m_2}$ is disrupted, and we now look for a match for $c^{m_2}$ in the $(c^0,A),B$ matching. Similar to $c^{m_1}$, either $c^{m_2}$ remains unmatched, resulting in $K+1$ and $L$ unmatched items in the $(c^0,A),B$ matching, or, by the same argument as before, $c^{m_2}$ will be matched to $s^{n_3}$, where $n_3>n_2$. Repeating these arguments for any additional disrupted links, we conclude that we either end up with one more link, so the number of unmatched items are $K$ and $L-1$, or we have the same number of links and the number of unmatched items are $K+1$ and $L$. \end{proof} \begin{lemma}[Subadditivity] \label{thm.subadditivity} Let $A'=(c^1,\ldots,c^m),\,A''=(c^{m+1},\ldots,c^M)$ and $B'=(s^1,\ldots,s^n)$, $B''=(s^{n+1},\ldots,s^N)$ and let $A=(c^1,\ldots,c^M)$, $B=(s^1,\ldots,s^N)$. Consider the complete FCFS matching of $A',B'$, of $A'',B''$, and of $A,B$ and let $K',K'',K$ be the number of unmatched customers and $L',L'',L$ be the number of unmatched servers in these three matchings. Then $K \le K'+K''$ and $L\le L'+L''$. \end{lemma} \begin{proof} Let $\hat{A}'=(\hat{c}^1,\ldots,\hat{c}^{K'})$ and $\hat{B}'=(\hat{s}^1\ldots,\hat{s}^{L'})$ be the ordered unmatched customers and servers from the complete FCFS matching of $A',B'$. Then the FCFS matching of $(\hat{A}',A'')$ and $(\hat{B}',B'')$ will have exactly the same ordered unmatched customers and servers as the FCFS matching of $A$ and $B$. We now construct the matching of $(\hat{A}',A'')$ and $(\hat{B}',B'')$ in steps. We look first at the matching of $(\hat{c}^{K'},A'')$ with $B''$, next the matching of $(\hat{c}^{K'},A'')$ with $(\hat{s}^{L'},B'')$, and so on, pre-pending each of $(\hat{c}^1,\ldots,\hat{c}^{K'})$ and $(\hat{s}^1\ldots,\hat{s}^{L'})$ in reverse order, to the current matching. At each step, by Lemma \ref{thm.monotonicity}, the number of links is unchanged or increased by 1, and the number of unmatched items goes down by one or is increased by 1. It follows that the total number unmatched customers is $\le K'+K''$ and of unmatched servers is $\le L'+L''$. \end{proof} \begin{lemma} \label{thm.decrease} Consider an incompatible pair $c^0,s^0$. Then there exists an $h$ and a sequence $c^1,\ldots,c^h$, $s^1,\ldots,s^h$ with $h\leq \min\{I,J\}-1$, where $(s^i,c^i),\,i=1,\ldots,h$ are compatible, such that the FCFS matching of $c^0,c^1,\ldots,c^h,s^0,s^1,\ldots,s^h$ is perfect. \end{lemma} Clearly, the probability of occurrence of such a sequence is strictly positive and lower bounded by: $\delta = \prod_{(c,s)\in {\mathcal{E}}} \alpha_{c} \beta_{s}$. \begin{proof} Because the bipartite graph is connected, and there is no direct edge between $c^0,s^0$, there exists a simple path (i.e. with no repeated nodes), $c^0\to s_{j_1} \to c_{i_1} \cdots \to s_{j_h} \to c_{i_h} \to s^0$ which connects them, with $1\le h \le \min\{I,J\}-1$. Clearly, the FCFS matching of $c^0,c^1,\ldots,c^h,s^0,s^1,\ldots,s^h$, where $c^l=c_{i_l},s^l=s_{j_l},\,l=1,\ldots,h$ is perfect, with exactly the links of the path, where $c^0$ is matched to $s^1$, and $s^0$ is matched to $c^h$. Note that FCFS matching of $c^1,\ldots,c^h,s^1,\ldots,s^h$ consists of $h$ perfectly matched blocks of length one. \end{proof} We assume from now on in this section that complete resource pooling holds. By Theorem \ref{th-ergodic}, the pair by pair matching Markov chain $(O_N)_{N\in {\mathbb{N}}}$ is ergodic. Using the Kolmogorov extension theorem \cite{oksendal:03}, we may define (in a non-constructive way) a stationary version $O^*=(O^*_N)_{N\in {\mathbb{Z}}}$ of the Markov chain. Define also $O^{[k]}=(O^{[k]}_N)_{N \ge -k}$ the realization of the Markov chain that starts at $O^{[k]}_{-k}=\emptyset$. Our first task is to show forward coupling, namely that $O^*$ and $O^{[0]}$ coincide after a finite time $\tau$ with $E(\tau)<\infty$. Following that we use standard arguments to show backward coupling and convergence to a unique matching. \begin{proposition}[Forward coupling] \label{thm.forward} The two processes $(O^*_n)_{n\in {\mathbb{N}}}$ and $(O^{[0]}_n)_{n\in {\mathbb{N}}}$ will couple after a finite time $\tau$, with $E(\tau)<\infty$. \end{proposition} \begin{proof} Denote by $|z|$ the number of unmatched customers (equal to the number of unmatched servers), for any state $z$ of the pair by pair matching process, we refer to it as the length of the state. Consider the sequence of times $0 \le M_0 < M_1 < \cdots < M_\ell,\cdots$ at which $O^*_{M_\ell} = \emptyset$. This sequence is infinite with probability 1, and $E(M_\ell) = E(M_0)+ \ell E(M_1-M_0) < \infty, \ell\ge 0$ by the ergodicity. Consider the state $z_0=O^{[0]}_{M_0}$. Then $|z_0| \le M_0$. By the monotonicity result of Lemma \ref{thm.monotonicity} and Lemma \ref{thm.subadditivity}, the states of $O^{[0]}$ satisfy $|z_0| \ge |O^{[0]}_{M_1}| \ge \ldots \ge |O^{[0]}_{M_\ell}|$, i.e. the length of the state of $O^{[0]}$ at the times $M_0,M_1,\ldots$ is non-increasing. This is because each block of customers and servers in times between $M_{\ell-1}$ and $M_\ell$ on its own has 0 unmatched. Furthermore, there is a positive probability greater or equal to $\delta >0$ that every $\bar{h}=\min\{I,J\}-1$ steps along this sequence the length will actually decrease by at least 1, as shown in Lemma \ref{thm.decrease}. This will happen if the blocks that follow $M_\ell$ are blocks of length 1 forming a path in the bipartite graph that connects an unmatched pair in $O^{[0]}_{M_\ell}$. Such a sequence of blocks will consist of $\le \bar{h}$ blocks, and it will occur with probability $\ge \delta$. Hence, there will be coupling after at most $\sum_{j=1}^{|z_0|} L_j \bar{h}$ perfect matching blocks of $O^*$, where $L_j$ are i.i.d. geometric random variables with $\delta$ probability of success. So coupling occurs almost surely, and the coupling time $\tau$ satisfies $E(\tau) \le E(M_0) \left( 1 + \frac{1}{\delta} \bar{h} E(M_1-M_0) \right)$. \end{proof} Note that once $O^{[0]}$ and $O^*$ couple, they stay together forever. We now need to show backward coupling. \begin{proposition}[Backward coupling] \label{thm.backcoupling} Let $O^*$ be the stationary pair by pair FCFS matching process, and let $O^{[-k]}$ be the process starting empty at time $-k$. Then $\lim_{k\to\infty} O^{[-k]}_N = O^*_N$ for all $N\in{\mathbb{Z}}$ almost surely. \end{proposition} \begin{proof} The statement of almost surely refers to the product measure $(\alpha\otimes\beta)^{\otimes {\mathbb{Z}}}$ on the product space $(C\times S)^{{\mathbb{Z}}}$. Define $T_k = \inf \{N\ge -k: O^{[-k]}_N = O^*_N \}$. By the forward coupling Proposition \ref{thm.forward}, we get that $T_k$ is almost surely finite. Let $\hat{T}_K = \max_{0\le k \le K} T_k$. It is $\ge 0$, and is also almost surely finite for any $K$. $\hat{T}_K$ is the time at which all the processes starting empty at time $-k$, where $0\le k\le K$, couple with $O^*$, and remain merged forever. Define the event $E_K = \{ \omega : \forall n \ge 0,\, O^{[-n]}_{\hat{T}_K} = O^*_{\hat{T}_K} \}$, in words, those $\omega$ for which the process starting empty at any time before $0$, will merge with $O^*$ by time $\hat{T}_K$. We claim that $P(E_K)>0$. We evaluate $P(\overline{E}_K)$. For any fixed $n \geq 0$, call $E_{n,K}$ the event that $O^{[-n]}$ couples with $O^*$ by time $\hat{T}_K$. We have $E_K = \bigcap_{n \geq 0} E_{n,K} = \bigcap_{n > K} E_{n,K}$ (by definition of $\hat{T}_K$, $E_{n,K}$ is always true for $n\leq K$, so we only need to consider $n>K$), so $\overline{E_K} = \bigcup_{n > K} \overline{E_{n,K}}$. The event $\overline{E_{n,K}}$ will happen if starting at the last time prior to $-K$ at which the process $O^{[-n]}$ was empty, the next time that it is empty is after time $0$. The reason for that is that otherwise the process $O^{[-n]}$ reaches state $\emptyset$ at some time $k\in [-K,0]$ and from that time onwards it is coupled with $O^{[-k]}$, and will couple with $O^*$ by time $\hat{T}_K$. Define for $n>K$, $D_n =\{\omega : O^{[-n]}_m \ne \emptyset \mbox{ for } -n < m \le 0\}$. Clearly by the above, $\bigcup_{n > K} \overline{E_{n,K}} \subseteq \bigcup_{n > K} D_n$. Let $\tau$ denote the recurrence time of the empty state. Then: \[ P(\overline{E_K}) = P(\bigcup_{n > K} \overline{E_{n,K}}) \le P(\bigcup_{n > K} D_n) \le \sum_{n > K} P(\tau > n). \] By the ergodicity $\sum_{l=0}^\infty P(\tau> l) =E(\tau)<\infty$. Hence we have that $P(\overline{E_K}) \to 0$ as $K\to\infty$, and therefore $P(E_K)>0$ for large enough $K$, and $P(E_K) \to 1$ as $K\to \infty$. Note also that $E_K \subseteq E_{K+1}$. Define now $\hat{T}= \sup_{k\ge 0} T_k$. We claim that $\hat{T}$ is finite a.s. Consider any $\omega$. Then by $P(E_K) \to 1$ as $K\to \infty$ and by the monotonicity of $E_K$, almost surely for this $\omega$ there exists a value $\ell$ such that $\omega \in E_\ell$. But if $\omega \in E_l$, then $\hat{T}(\omega) \le \hat{T}_\ell < \infty$. So, all processes starting empty before time $0$ will couple with $O^*$ by time $\hat{T}$. By the stationarity of the sequences $(s^n,c^n)_{n\in{\mathbb{Z}}}$ and of $O^*$, we then also have that all processes $O^{[-k]}_N$ starting empty before $-k$ will couple with $O^*$ by time 0, if $k\ge \hat{T}$. Hence using the Loynes' scheme of starting empty at $-k$ and letting $k\to\infty$ the constructed process will merge with $O^*$ at time 0. But the same argument holds not just for 0, but for any negative time $-N$. Hence $O^{[-k]}$ and $O^*$ couple at $-N$ (and stay coupled) for any $k > N+ \hat{T}$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm.co-loynes}] We saw that $\lim_{k\to\infty} O^{[-k]}_N=O^*_N$ for all $N$ almost surely. Each process $O^{[-k]}_N$ determines matches uniquely for all $N> - k$, so if we fix $N$, matches from $N$ onwards are uniquely determined by $\lim_{k\to\infty} O^{[-k]}_N$. Hence $(O^*_N)_{N\in{\mathbb{Z}}}$ determines for every server $s^n$ and every customer $c^n$ his match, uniquely, almost surely. This proves the theorem. \end{proof} \section{Exchange Transformation and Dynamic Reversibility} \label{sec.reversibility} \begin{figure}[htb] \begin{center} \includegraphics[width=2.5in]{FigFCFS.pdf} \end{center} \caption{FCFS is not preserved when time is reversed} \label{fig.noreverse} \end{figure} The FCFS matching depends on the time direction in which it is constructed. The simple example in Figure \ref{fig.noreverse} shows that FCFS is not preserved if the time direction is reversed. Nevertheless, this model has an elegant reversibility result. In this section we introduce the exchange transformation, in which we switch the positions of each matched pair of customer and server. Figures \ref{fig.match1} and \ref{fig.match2} illustrate the exchange transformation. We show that the exchanged sequences are independent i.i.d. and that the matching is FCFS in reversed time. \begin{definition} Consider a FCFS bipartite matching of sequences $(s^n,c^n)_{n\in T}$. The exchange transformation of the matched pair $s^n,c^m$, is the matched pair ${\tilde{s}}^m,{\tilde{c}}^n$ where ${\tilde{s}}^m=s^n$ and ${\tilde{c}}^n=c^m$. \end{definition} \begin{lemma} \label{thm.reverse} Let $A$ be a perfect matching of $c^1,\ldots,c^M$, and $s^1,\ldots,s^M$. Let ${\tilde{c}}^1,\ldots,{\tilde{c}}^M$, ${\tilde{s}}^1,\ldots,{\tilde{s}}^M$ be the sequences obtained by the exchange transformation, retaining the same links of the matched pairs. The resulting matching of ${\tilde{c}}^1,\ldots,{\tilde{c}}^M$, ${\tilde{s}}^1,\ldots,{\tilde{s}}^M$ is the unique FCFS matching in reversed time. \end{lemma} \begin{proof} Denote by $\tilde{A}$ the matching of ${\tilde{c}}^1,\ldots,{\tilde{c}}^M$, ${\tilde{s}}^1,\ldots,{\tilde{s}}^M$ obtained by retaining the links of $A$. Then $(n,m)\in \tilde{A}$ if and only if $(m,n)\in A$. Take $(n,m)\in \tilde{A}$, i.e. ${\tilde{c}}^n=c'$ and ${\tilde{s}}^m=s'$ are matched (and of course, $(s',c')\in {\mathcal{E}}$). To show that $\tilde{A}$ is FCFS in reversed time, we need to show that: \begin{eqnarray*} && \mbox{if $k > m,\, (c',{\tilde{s}}^k)\in {\mathcal{E}}$, then there exists $l > n$ such that $(k,l)\in \tilde{A}$ and } \\ && \mbox{if $l > n, \, ({\tilde{c}}^l,s')\in {\mathcal{E}}$, then there exists $k > m$ such that $(k,l)\in \tilde{A}$ }. \end{eqnarray*} Assume to the contrary that there exists $k>m$ such that ${\tilde{s}}^k=s''$, and $(c',s'') \in {\mathcal{E}}$, and there is no $l>n$ such that $(k,l)\in \tilde{A}$. Because this is a complete matching, there exists an $l \le n$ such that ${\tilde{c}}^l$ is matched to ${\tilde{s}}^k$, and because ${\tilde{c}}^n$ is matched to ${\tilde{s}}^m$, we actually have $l< n$. So we have that ${\tilde{s}}^k=s''$ is matched to ${\tilde{c}}^l=c''$, where $k>m$ and $l<n$. Consider now the original matching. We have that $c^m=c'$ is matched to $s^n=s'$, and we have $s^l={\tilde{s}}^k=s''$ and $c^k={\tilde{c}}^l=c''$ which are matched. But, $(c',s'')\in {\mathcal{E}}$, and $l<n$ while $k>m$. This contradicts the FCFS property of the matching $A$ of the unexchanged sequences. A similar contradiction is obtained if we assume the second part of the definition is violated. The proof is illustrated in the following figure: \begin{center} \includegraphics[width=3.5in]{FigReversibility.pdf} \end{center} \end{proof} \begin{lemma} \label{thm.blocks} Consider the FCFS matching of two i.i.d. sequences, and let $\O_{(m+1,m+M)}$ be the block of customers and servers for the times $[m+1,m+M]$. Then the conditional probability of observing values $\O_{(m+1,m+M)} = \big((c^{m+1},\ldots,c^{m+M})$, $(s^{m+1},\ldots,s^{m+M})\big)$ conditional on the event that the FCFS matching of these values is a perfect match is: \begin{eqnarray*} && P\Big(\O_{m+1,M+M}= \big((c^{m+1},\ldots,c^{m+M}),\, (s^{m+1},\ldots,s^{m+M})\big) \\ && \qquad \Big| \, \O_{m+1,M+M} \mbox{ forms a perfect FCFS match} \Big) \\ && \quad = \kappa_M \prod_{i=1}^{I} {\alpha_{c_i}}^{\# c_i} \prod_{j=1}^{J} {\beta_{s_j}}^{\# s_j} \end{eqnarray*} where $\kappa_M$ is a constant that may depend on $M$, and $\# c_i$, $\# s_j$ count the number of type $c_i$ customers and type $s_j$ servers in the block. \end{lemma} \begin{proof} The conditional probability is calculated using Bayes formula: \begin{eqnarray*} && \quad P(\mbox{seeing $c^{m+1},\ldots,c^{m+M}, s^{m+1},\ldots,s^{m+M}$ }| \mbox{ having a perfect match}) \\ && \quad = \frac{P(\mbox{having a perfect match }| \mbox{ seeing $c^{m+1},\ldots,c^{m+M}, s^{m+1},\ldots,s^{m+M}$})} {P( \mbox{having a perfect match of length $M$})} \\ && \qquad \times P(\mbox{seeing $c^{m+1},\ldots,c^{m+M}, s^{m+1},\ldots,s^{m+M}$}) \\ && \quad = \kappa_M \times \mathbf{1}_{\{c^{m+1},\ldots,c^{m+M}, s^{m+1},\ldots,s^{m+M} \mbox{ is a perfect match\}}} \times \prod_{i=1}^{I} {\alpha_{c_i}}^{\# c_i} \prod_{j=1}^{J} {\beta_{s_j}}^{\# s_j} \end{eqnarray*} where $\kappa_M=1\big/ P( \mbox{having a perfect match of length $M$})$. \end{proof} \begin{corollary} \label{thm.symmetric} Let $\O_{m+1,M+M}$ be a FCFS perfectly matched block, and let $\overleftarrow{\O}_{m+1,M+M}$ be obtained from $\O_{m+1,M+M}$ by performing the exchange transformation and time reversal. Then $\overleftarrow{\O}^M$ is a FCFS perfectly matched block, and \[ P(\overleftarrow{\O}_{m+1,M+M}) = P(\O_{m+1,M+M}) \] \end{corollary} \begin{proof} That $\overleftarrow{\O}_{m+1,M+M}$ is a FCFS perfectly matched block follows from Lemma \ref{thm.reverse} and that $P(\overleftarrow{\O}_{m+1,M+M}) = P(\O_{m+1,M+M})$ follows from Lemma \ref{thm.blocks}. \end{proof} We now assume that the system is ergodic, i.e. the Markov chains $O_N,Q^s_N,Q^c_N$ are ergodic (which holds if and only if complete resource pooling holds). We have shown in Theorem \ref{thm.co-loynes} that for independent i.i.d sequences of servers and customers over ${\mathbb{Z}}$, under ergodicity, there exists a.s. a unique FCFS matching, which corresponds to the stationary version of the Markov chains associated with the FCFS matching (generated by the Loynes' construction). Let $(c^n,s^n)_{n\in {\mathcal{Z}}}$ be the sequences of customers and servers, let $A$ be the FCFS matching. Define $A_n = A \cap \{(\cdot,n), (n,\cdot)\}$, that is, $A_n$ gives the matchings of the customer $c^n$ and of the server $s^n$. Consider the quadruple $(O_n,c^n,s^n,A_n)_{n\in {\mathcal{Z}}}$, denote by ${\mathfrak{p}}=(O_n,c^n,s^n,A_n)_{n\in {\mathcal{Z}}}$ the sample path corresponding to $(c^n,s^n)_{n\in{\mathcal{Z}}}$, and by $\mathfrak{P}$ its probability distribution. Build the exchange transformation on ${\mathfrak{p}}$ and consider it in reverse time. We obtain a new quadruple based on the two doubly infinite sequences of customers and servers ${\tilde{c}}^n,{\tilde{s}}^m,\,m,n \in {\mathbb{Z}}$, and a matching between them. Denote the transformed path by $\psi{\mathfrak{p}}$. Then $\psi{\mathfrak{p}}$ retains the links of ${\mathfrak{p}}$. Denote the probability distribution of the new quadruple by $\psi{\mathfrak{P}}$. Our goal is to prove that ${\mathfrak{P}}=\psi{\mathfrak{P}}$. Let $\mathfrak{P}^0$ be the Palm version of the measure $\mathfrak{P}$ with respect to $(O_n)_{n\in {\mathcal{Z}}}$, that is, $\mathfrak{P}^0$ is the law of $(O_n,c^n,s^n,A_n)_{n\in {\mathcal{Z}}}$ conditioned on the event $\{O_0=0\}$. A realization of a process of law $\mathfrak{P}^0$ can be obtained by considering a bi-infinite sequence $\O=(\O_M)_{M\in {\mathbb{Z}}}$ of perfectly matched blocks of i.i.d customers and servers. Let us perform the exchange transformation and time-reversal on the sequence $\O=(\O_M)_{M\in {\mathbb{Z}}}$, and let $\psi{\mathfrak{P}}^0$ be the probability distribution of the result. Clearly, $\psi{\mathfrak{P}}$ is the stationary version of $\psi{\mathfrak{P}}^0$. Now according to Lemma \ref{thm.reverse}, we have $\psi{\mathfrak{P}}^0 ={\mathfrak{P}}^0$. Since $\psi{\mathfrak{P}}$ is the stationary version of $\psi{\mathfrak{P}}^0$ and since ${\mathfrak{P}}$ is the stationary version of ${\mathfrak{P}}^0$, we deduce that $\psi{\mathfrak{P}}={\mathfrak{P}}$. The key point in the argument above is the link between time-stationarity and event-stationarity. For general background on Palm calculus, see for instance Chapter 1 in \cite{BaBr}. So we obtain the following result. \begin{theorem} Consider a bipartite matching model under complete resource pooling. Let $(c^n,s^n)_{n\in {\mathbb{Z}}}$ be the independent i.i.d. sequences of customers and servers, with the unique FCFS matching between them. Then the exchanged sequences $({\tilde{c}}^n,{\tilde{s}}^n)_{n\in {\mathbb{Z}}}$ are independent i.i.d. of the same law as $(c^n,s^n)_{n\in {\mathbb{Z}}}$. The FCFS matching in reverse time between them (using Loynes' construction in reversed time) consists of the same links as the matching between $(c^n,s^n)_{n\in {\mathbb{Z}}}$. \end{theorem} \begin{proof} That $({\tilde{c}}^n,{\tilde{s}}^n)_{n\in {\mathbb{Z}}},$ are independent i.i.d. sequences follows from the identity of $\psi{\mathfrak{P}}$ and ${\mathfrak{P}}$. That the Loynes' construction in reversed time will use the same links follows, since the links of $(c^n,s^n)_{n\in {\mathbb{Z}}}$ are the FCFS matchings in reversed time between $({\tilde{c}}^n,{\tilde{s}}^n)_{n\in {\mathbb{Z}}}$, and by Theorem \ref{thm.co-loynes} this matching is unique. \end{proof} \section{Stationary Distributions} \label{sec.stationary} We have found that for any two independent i.i.d. sequences of customers $C=(c^m)_{m\in{\mathbb{Z}}}$ and of servers $S=(s^n)_{n\in{\mathbb{Z}}}$, under complete resource pooling, there is a unique FCFS matching almost surely. Furthermore, if we exchange every matched pair $(c^m,s^n)$ of customers and servers and retain the matching, we obtain two permuted sequences, of matched and exchanged customers ${\tilde{C}}=({\tilde{c}}^n)_{n\in{\mathbb{Z}}}$, and of matched and exchanged servers ${\tilde{S}}=({\tilde{s}}^m)_{m\in{\mathbb{Z}}}$. These new sequences are again independent and i.i.d., and the retained matching between them is FCFS in reversed time direction, and it is the unique FCFS matching of ${\tilde{C}},\,{\tilde{S}}$ in reversed time. \begin{figure}[htb] \begin{center} \includegraphics[scale=0.35]{FigMarkov5.pdf} \end{center} \caption{Four mechanisms of FCFS matching} \label{fig.Markov} \end{figure} \medskip In this section we consider the Markovian evolution of the stationary FCFS matching on ${\mathbb{Z}}$, and derive stationary distributions of several Markov chains associated with it. The FCFS matching of $C,\,S$ evolves moving step by step from the past up to position $N$, where we add matches and perform exchanges at each step from $N$ to $N+1$. Four ways in which this can be done are illustrated in Figure \ref{fig.Markov} where light circles represent customers and dark circles represent servers (in original or exchanged positions). In each of these, if we reverse the time direction we get a Markovian construction of FCFS matches between ${\tilde{C}},\, {\tilde{S}}$ that moves from $N+1$ to $N$ and at each of these steps adds matches for elements ${\tilde{c}}^n,{\tilde{s}}^m$ and exchanges them back to $c^m,s^n$. We exploit this reversibility to derive the stationary distributions. The outline of this section is as follows: In Section \ref{sec.mechanism} we first describe in more details the four mechanisms and define a Markov chain associated with each. The states of these processes consist of the ordered lists of items in the region encircled by a dashed ellipse in each of the four panels in Figure \ref{fig.Markov}. We call these the detailed Markov chains. In Section \ref{sec.reversal} we formulate Lemmas \ref{thm.timereversal}, \ref{thm.timereversal2} on time reversal, that associates each Markov chain in the forward time direction, with a corresponding Markov chain in the reversed time direction. The proof of these Lemmas is immediate. Following this, in Section \ref{sec.distributions}, we use time reversal to derive in Theorem \ref{thm.stationary} the stationary distributions of the detailed Markov chains. These are, up to a normalizing constant, simply the distributions of a finite sequence of multi-Bernoulli trials. We also show in Theorem \ref{thm.stationary} that all these distributions possess the same normalizing constant. In Section \ref{sec.marginals} we define a Markov chain with an augmented state description, and obtain its stationary distribution as a corollary to Theorem \ref{thm.stationary}. The advantage of this augmented chain is that its state can be re-interpreted as the state of a queue with parallel servers which is overloaded, as described in \cite{visschers-adan-weiss:12,adan-weiss:11,adan-weiss:14}. Under this interpretation it is possible to sum over the detailed states and to obtain the stationary distribution of a host of other processes associated with FCFS matching. Furthermore, by summing over all the states we obtain the normalizing constant for the stationary distributions of Theorem \ref{thm.stationary}. We conjecture that its calculation is $\sharp$-P hard. Finally, in Section \ref{sec.natural} we again sum over states to obtain the stationary distribution of the `natural' Markov chains. We illustrate this in Section \ref{sec.example} for the FCFS matching model of the ``NN''-system, that was only partly analyzed in \cite{caldentey-kaplan-weiss:09}, because its analysis could not be completed at the time. \subsection{Mechanisms for evolution of FCFS matching and detailed Markov chains} \label{sec.mechanism} We consider four mechanisms for the Markovian evolution of the stationary FCFS matching, and define an associated Markov chain for each. \subsubsection*{Server by server matching} At time $N$ all servers $s^n,\,n\le N$ have been matched and exchanged with the customers to which they were matched, as illustrated in panel (i) of Figure \ref{fig.Markov}. At this point the servers line has ${\tilde{c}}^n,\,n\le N$ customers that matched and replaced servers $s^n,\,n\le N$, and servers $s^n,\,n>N$ are still unmatched. On the customers line there is a position ${\underline{M}}$ such that all the customers $c^m,\,m<{\underline{M}}$ have been matched and replaced by ${\tilde{s}}^m$, and $c^{{\underline{M}}}$ is the first unmatched customer, and there is a position ${\overline{M}}$ such that all customers $c^m,\,m>{\overline{M}}$ have not yet been matched, and $c^{\overline{M}}$ is the last customer that has been matched, so that now ${\tilde{s}}^{\overline{M}}$ is the matched and exchanged server in position ${\overline{M}}$. If the matching for $n\le N$ is perfect then ${\overline{M}}={\underline{M}}-1=N$, otherwise $L={\overline{M}}-{\underline{M}}+1\ge 2$. We let ${\mathfrak{z}}=0$ in the former case (sometimes we write ${\mathfrak{z}}=\emptyset$), and in the latter case we let ${\mathfrak{z}}=(z^1,\ldots,z^L)$ be the ordered sequence of unmatched customers and of matched and exchanged servers so that ${\mathfrak{z}}^1=c^{\underline{M}}$, ${\mathfrak{z}}^L={\tilde{s}}^{\overline{M}}$ and $z^l,\,1<l<L$ is either $c^{{\underline{M}}+l-1}$ if unmatched or ${\tilde{s}}^{{\underline{M}}+l-1}$ if matched and exchanged. We define the {\em server by server FCFS detailed matching process} $Z^s=(Z^s_N)_{N\in {\mathbb{Z}}}$ with $Z^z_N={\mathfrak{z}}$. It is a Markov chain where the transition from $Z^s_N$ to $Z^s_{N+1}$ depends on the current state ${\mathfrak{z}}$, and on the innovation variables which are the types of server $s^{N+1}$ and of customers $c^m,\,m>{\overline{M}}$. \subsubsection*{Customer by customer matching} Similar to server by server matching, at time $M$ all customers $c^m,\,m\le M$ have been matched and exchanged with servers, as illustrated in panel (ii) of Figure \ref{fig.Markov}. We define a {\em customer by customer FCFS detailed matching process}, $Z^c=(Z^c_M)_{M\in {\mathbb{Z}}}$ so that the state $Z^c_M={\mathfrak{y}}$ is ${\mathfrak{y}}=0$ for perfect match, and otherwise ${\mathfrak{y}}=(z^1,\ldots,z^L)$ where $z^1=s^{{\underline{N}}}$ is the first unmatched server on the servers line, $z^L={\tilde{c}}^{\overline{N}}$ is the last matched and exchanged customer, and $z^l,\,1<l<L$ is either $s^{{\underline{N}}+l-1}$ if unmatched or ${\tilde{c}}^{{\underline{N}}+l-1}$ if matched and exchanged. It is a Markov chain where the transition from $Z^c_M$ to $Z^c_{M+1}$ depends on the current state ${\mathfrak{y}}$, and on the innovation variables which are the types of customer $c^{M+1}$ and of servers $s^n,\,n>{\overline{N}}$. \subsubsection*{Pair by pair backward matching} For pair by pair backward FCFS matching (illustrated in panel (iii) of Figure \ref{fig.Markov}) we assume that all possible FCFS matches between $s^n,c^m,\,m,n \le N$ have been made and exchanged, and in step $N+1$ we add the pair $s^{N+1},c^{N+1}$, and if possible match and exchange each of them FCFS to previous unmatched items or to each other. We define the pair by pair backwards detailed FCFS matching process $D=(D_N)_{N\in {\mathbb{Z}}}$ as $D_N=({\mathfrak{z}},{\mathfrak{y}})$, where ${\mathfrak{z}}=(z^1,\ldots,z^L)$ describes the customers line and ${\mathfrak{y}}=(y^1,\ldots,y^K)$ describes the server line. Here $z^1$ is the first unmatched customer, in position $N-L+1$, and the remaining items of ${\mathfrak{z}}$ are either unmatched customers or matched and exchanged servers, $y^1$ is the first unmatched server, in position $N-K+1$, and the remaining items of ${\mathfrak{y}}$ are either unmatched servers or matched and exchanged customers. The number of unmatched customers in ${\mathfrak{z}}$ needs to be equal to the number of unmatched servers in ${\mathfrak{y}}$. We may have ${\mathfrak{z}}={\mathfrak{y}}=0$ if there is a perfect match, otherwise both $L\ge 1$ and $K\ge 1$. This is a Markov chain, whose next state depends on the current state and the random innovation consists of the types of $s^{N+1},c^{N+1}$. \subsubsection*{Pair by pair forward matching} For pair by pair forward FCFS matching (illustrated in panel (iv) of Figure \ref{fig.Markov}) we assume all customers $s^n,c^m,\,m,n \le N$ have been matched and exchanged. After step $N$ we consider the pair in position $N+1$, which may contain either items which were matched and exchanged already, or items which are still unmatched, and then in step $N+1$ the items which are still unmatched after step $N$ are matched and exchanged with each other or with items in positions $> N+1$. We define the pair by pair forward FCFS matching process $E=(E_N)_{N\in T}$ as $E_N = ({\mathfrak{y}},{\mathfrak{z}})$, where ${\mathfrak{y}}=(y^1,\ldots,y^K)$ lists items in positions $N+1,\ldots,N+K$ on the customer line, where $y^K$ is the last matched and exchanged server ${\tilde{s}}^{N+K}$, and $y^k,\,1\le k<K$ is either an unmatched customer or a matched and exchanged server in position $N+k$, and where $(z^1,\ldots,z^L)$ lists items in positions $N+1,\ldots,N+L$ on the server line, where $z^L$ is the last matched and exchanged customer ${\tilde{c}}^{N+L}$ and $z^l,\,1\le l<L$ is either an unmatched server, or a matched and exchanged customer in position $N+k$. $E_N=\emptyset$ after a perfect match, otherwise $K,L \ge 1$. $E_N$ is a Markov chain, whose next state depends on the current state, and the random innovation consists of the $c^m,\,m>N+L$ and $s^n,\,n>N+K$. It is useful at this point to write down some of the transition probabilities for the detailed Markov chains. We start with three types of transitions for $Z^s$: \begin{description} \item If $Z_N^s = {\mathfrak{z}} =( c_{i_0},\ldots,c_i,\ldots,{\tilde{s}}_{j_0})$ and $s^{N+1}=s_j$, such that $c_i$ is the first customer compatible with $s_j$, then $Z_{N+1} = {\mathfrak{z}}' = (c_{i_0},\ldots,{\tilde{s}}_j,\ldots,{\tilde{s}}_{j_0})$, and the probability of this transition is $P({\mathfrak{z}}\to {\mathfrak{z}}') = \beta_{s_j}$. \item If $Z_N^s = {\mathfrak{z}} = (c_i,{\tilde{s}}_{j_1},\ldots,c_{i_0},\ldots,{\tilde{s}}_{j_0})$ and $s^{N+1}$ is compatible with $c_i$, and $c_{i_0}$ is the first unmatched customer following $c_i$, then $Z_{N+1} = {\mathfrak{z}}' = (c_{i_0},\ldots,{\tilde{s}}_{j_0})$, and the probability of this transition is $P({\mathfrak{z}}\to {\mathfrak{z}}') = \beta_{\S(c_i)}$. \item If $Z_N^s = {\mathfrak{z}} = (c_{i_0},\ldots,{\tilde{s}}_{j_0})$ and $s^{N+1}=s_j$, and $s_j$ is incompatible with all the unmatched servers in ${\mathfrak{z}}$, then we may have $Z_{N+1}^s = {\mathfrak{z}}' = (c_{i_0},\ldots,{\tilde{s}}_{j_0},c_{i_1},\ldots,c_{i_k},{\tilde{s}}_j)$, where $c_{i_1},\ldots,c_{i_k}$ are incompatible with $s_j$, with probability: $P({\mathfrak{z}}\to {\mathfrak{z}}') = \beta_{s_j} \alpha_{c_{i_1}},\ldots,\alpha_{c_{i_k}} \alpha_{{\mathcal{C}}(s_j)}$. \end{description} The first type of transition of $Z^s$ is a simple exchange of elements, in the second type some elements $z_1,\ldots$ are eliminated, and this may also be a transition to the empty state, and in the third type of transition elements are added after $z_L$. Transitions for $Z^c$ are similar to those of $Z^s$. Transitions of the pair by pair process $D$ are of more types, we list just two of them, which represent most of the possibilities: \begin{description} \item If $D_N = ({\mathfrak{z}},{\mathfrak{y}}) = \big((c_{i_0},\ldots,c_i,\ldots),(s_{j_0},\ldots )\big)$, and $s^{N+1}=s_k,c^{N+1}=c_l$, and if $s_k$ is compatible with $c_i$ but not with any earlier unmatched customer in ${\mathfrak{z}}$, while $c_l$ does not match any of the unmatched servers in ${\mathfrak{y}}$ then $D_{N+1}=({\mathfrak{z}}',{\mathfrak{y}}') = \big((c_{i_0},\ldots,{\tilde{s}}_k,\ldots,c_l),(s_{j_0},\ldots,{\tilde{c}}_i )\big)$, and the transition probability for this is $P(({\mathfrak{z}},{\mathfrak{y}})\to ({\mathfrak{z}}',{\mathfrak{y}}')) = \alpha_{c_l}\beta_{s_k}$. \item If $D_N = ({\mathfrak{z}},{\mathfrak{y}}) = \big((c_{i},\ldots,c_{i_0},\ldots),(s_j,\ldots,s_{j_0},\ldots )\big)$, and $s^{N+1}$ is compatible with $c_i$, and $c^{N+1}$ is compatible with $s_j$, and the next unmatched customer and server are $c_{i_0},s_{j_0}$, then $D_{N+1}=({\mathfrak{z}}',{\mathfrak{y}}') = \big((c_{i_0},\ldots,{\tilde{s}}_j),(s_{j_0},\ldots,{\tilde{c}}_i )\big)$, and the probability of this transition is $P(({\mathfrak{z}},{\mathfrak{y}})\to ({\mathfrak{z}}',{\mathfrak{y}}')) = \alpha_{{\mathcal{C}}(s_j)} \beta_{\S(c_i)}$. \end{description} We see here that in each transition of $D$ a single element is added after $z_K$ and after $y_L$, and sometimes elements $z_1,\ldots$ and/or $y_1,\ldots$ are eliminated. In the latter case the transition may be to the empty state. Transitions of the pair by pair process $E$ are also of several types. Some of those are similar to transitions of $D$, others are somewhat different. We again list just a few of them, which represent most of the possibilities: \begin{description} \item If $E_N = ({\mathfrak{y}},{\mathfrak{z}}) = \big(({\tilde{s}}_l,y_1,\ldots,c_i,\ldots,y_k),(s_j,z_1,\ldots,z_l)\big)$, so that $c^{N+1}$ has already been matched and exchanged with ${\tilde{s}}_l$ which was in an earlier position, but $s_j$ has not yet been matched, and $c_i$ is the first compatible unmatched customer in ${\mathfrak{y}}$, then $E_{N+1}= ({\mathfrak{y}}',{\mathfrak{z}}') = \big((y_1,\ldots,{\tilde{s}}_j,\ldots,y_k)$, $(z_1,\ldots,z_l)\big)$, in words, ${\tilde{s}}_j$ replaces $c_i$, and the first elements in $({\mathfrak{y}},{\mathfrak{z}})$, are erased. This transition is deterministic, and happens with probability $P(({\mathfrak{y}},{\mathfrak{z}})\to ({\mathfrak{y}}',{\mathfrak{z}}')) = 1$. \item If $E_N = ({\mathfrak{y}},{\mathfrak{z}}) = \big(({\tilde{s}}_l,y_1,\ldots,y_k),(s_j,z_1,\ldots,z_l)\big)$, so that $c^{N+1}$ has already been matched and exchanged with ${\tilde{s}}_l$ which was in an earlier position, but $s_j$ has not yet been matched, and if none of the unmatched customers in ${\mathfrak{y}}$ are compatible with $s_j$, then $E_{N+1}= ({\mathfrak{y}}',{\mathfrak{z}}') = \big((y_1,\ldots,y_k,c_{i_1},\ldots,c_{i_m},{\tilde{s}}_j),(z_1,\ldots,z_l)\big)$, where $s_j$ skipped all of ${\mathfrak{y}}$ as well as $c_{i_1},\ldots,c_{i_m}$, before finding a match. The probability of this transition is $P(({\mathfrak{y}},{\mathfrak{z}})\to ({\mathfrak{y}}',{\mathfrak{z}}')) = \alpha_{c_{i_1}}\cdots\alpha_{c_{i_m}}\alpha_{{\mathcal{C}}(s_j)}$. \item If $E_N = \emptyset$, then the next state can be $E_{N+1} = \emptyset$. This happens if and only if the next pair, $c^{N+1}=c_i,\,s^{N+1}=s_j$ are compatible. The probability of this transition is $P(({\mathfrak{y}},{\mathfrak{z}})\to ({\mathfrak{y}}',{\mathfrak{z}}')) = \sum_{(c_i,s_j)\in G} \alpha_{c_i}\beta_{s_j}$ \item If $E_N = \emptyset$, then the next state can be of the form: $E_{N+1} = ({\mathfrak{y}}',{\mathfrak{z}}') =\big((c_{i_1},\ldots,c_{i_{k-1}},{\tilde{s}}_j)$, $(s_{i_1},\ldots,s_{i_{l-1},}{\tilde{c}}_i)\big)$, $k,l\ge 1$, where in the transition from $N$ to $N+1$, the new pair is $c^{N+1}=c_i,\,s^{N+1}=s_j$ which are incompatible. In matching them, they are matched and exchanged with some ${\tilde{s}}_{i_l},{\tilde{c}}_{i_k}$, after skipping incompatible $s_{i_1},\ldots,s_{i_{l-1}}$ and $c_{i_1},\ldots,c_{i_{k-1}}$. The probability of this transition is: $P(({\mathfrak{y}},{\mathfrak{z}})\to ({\mathfrak{y}}',{\mathfrak{z}}')) =\alpha_{c_i} \beta_{s_j} \beta_{s_{i_1}}\cdots\beta_{s_{i_{l-1}}} \alpha_{c_{i_1}}\cdots\alpha_{c_{i_{k-1}}} \alpha_{{\mathcal{C}}(s_j)} \beta_{\S(c_i)}$. \end{description} We see here that in each transition of $E$, the first element of the state is deleted, and there may be an exchange of elements from ${\mathfrak{y}},{\mathfrak{z}}$, or a geometric number of new elements is added. Exceptional is the empty state, where transition can be to the empty state, or to a state with a geometric number of new elements in ${\mathfrak{y}}',{\mathfrak{z}}'$. \subsection{Time reversal of the detailed Markov chains} \label{sec.reversal} Examining panel (i) of Figure \ref{fig.Markov} we see that it illustrates FCFS matching and exchange of server and customer lines $C$ and $S$ all the way from $-\infty$ up to position $N$, and at the same time it also illustrates matching and exchange of server and customer lines ${\tilde{C}}$ and ${\tilde{S}}$ FCFS in reversed time, all the way from $\infty$ to $N+1$. Our main observation now is that if $Z^s_N={\mathfrak{z}}=(z^1,\ldots,z^L)$, then $(z^L,\ldots,z^1)$ is exactly the state of the customer by customer FCFS matching in reversed time of the sequences ${\tilde{C}},\,{\tilde{S}}$, when all customers ${\tilde{c}}^n,n\ge N+1$ have been matched to some ${\tilde{s}}^m$, and exchanged back to a customer $c^m$ and server $s^n$. For ${\mathfrak{z}}=(z^1,\ldots,z^L)$ we denote $\overleftarrow{{\mathfrak{z}}}=(z^L.\ldots,z^1)$. We denote the Markov chain of customer by customer FCFS matching of ${\tilde{C}},\,{\tilde{S}}$ in reversed time by $\overleftarrow{Z}$, so that $\overleftarrow{Z}^c_N$, is the state where all ${\tilde{c}}^n,\,n > N$ have been matched. We then state formally (see Figure \ref{fig.reversal1}): \begin{lemma}[Time reversal] \label{thm.timereversal} The Markov chain $\overleftarrow{Z}^c_N$ of customer by customer FCFS matching of ${\tilde{C}},\,{\tilde{S}}$ in reversed time, is the time reversal of the Markov chain $Z^s_N$ of server by server FCFS matching of $C,\,S$, in the sense that \begin{equation} \label{eqn.reversal} Z^s_N = {\mathfrak{z}}, \; Z^s_{N+1} = {\mathfrak{z}}' \mbox{ if and only if } \overleftarrow{Z}^c_{N+1} = \overleftarrow{{\mathfrak{z}}'}, \; \overleftarrow{Z}^c_N = \overleftarrow{{\mathfrak{z}}}. \end{equation} This implies that the reversal of the transition $Z^s_N={\mathfrak{z}} \to Z^s_{N+1}={\mathfrak{z}}'$ is exactly the transition $\overleftarrow{Z}^c_{N+1}=\overleftarrow{{\mathfrak{z}}'} \to \overleftarrow{Z}^c_N=\overleftarrow{{\mathfrak{z}}}$. In other words, if the transition of $Z^s_N \to Z^s_{N+1}$ matches and exchanges $s^n$ with $c^m$, then the transition of $\overleftarrow{Z}^c_{N+1} \to \overleftarrow{Z}^c_N$ matches and exchanges ${\tilde{c}}^n$ with ${\tilde{s}}^m$. \end{lemma} \begin{figure}[htb] \begin{center} \includegraphics[scale=0.30]{FigMarkov3.pdf} \end{center} \caption{Single match and exchange of customer server pair and its reversal} \label{fig.reversal1} \end{figure} \begin{proof} The only thing to note is that if $s^n,\,c^m$ are matched in the forward FCFS direction, then ${\tilde{s}}^m,\,{\tilde{c}}^n$ are matched in the reversed FCFS direction. This is so by the almost sure uniqueness of the matching as shown in Theorem \ref{thm.co-loynes}, and by Lemma \ref{thm.reverse}. \end{proof} A similar observation on time reversal holds also for the pair by pair backward and forward detailed Markov chains. Examining panel (iii) of Figure \ref{fig.Markov} we again see that it illustrates FCFS matching and exchange of server and customer lines $C$ and $S$ all the way from $-\infty$ up to position $N$, and at the same time it also illustrates matching and exchange of server and customer lines ${\tilde{C}}$ and ${\tilde{S}}$ FCFS in reversed time, all the way from $\infty$ to $N+1$. Our main observation now is that if $Z^s_N=({\mathfrak{z}},{\mathfrak{y}})=\big((z^1,\ldots,z^L),(y^1,\ldots,y^K)\big)$, then $\big((y^K,\ldots,y^1),(z^L,\ldots,z^1)\big)=(\overleftarrow{{\mathfrak{y}}},\overleftarrow{{\mathfrak{z}}})$ is exactly the state of the pair by pair forward detailed FCFS matching in reversed time, of the sequences ${\tilde{C}},\,{\tilde{S}}$, when all customers and servers ${\tilde{c}}^n,{\tilde{s}}^m,\,m,n > N$ have been matched and exchanged back to a customer $c^m$ and server $s^n$. We denote the pair by pair forward detailed FCFS matching of ${\tilde{C}},\,{\tilde{S}}$ in reversed time by $\overleftarrow{E}_N$. We then state formally (see Figure \ref{fig.reversal2}): \begin{lemma}[Time reversal] \label{thm.timereversal2} The Markov chain $\overleftarrow{E}_N$ of pair be pair forward FCFS matching of ${\tilde{C}},\,{\tilde{S}}$ in reversed time, is the time reversal of the Markov chain $D_N$ of pair by pair backward FCFS matching of $C,\,S$, in the sense that \begin{equation} \label{eqn.reversal3} D_N = ({\mathfrak{z}},{\mathfrak{y}}), \; D_{N+1} = ({\mathfrak{z}}',{\mathfrak{y}}') \mbox{ if and only if } \overleftarrow{E}_{N+1} = (\overleftarrow{{\mathfrak{y}}'},\overleftarrow{{\mathfrak{z}}'}), \; \overleftarrow{E}_N = (\overleftarrow{{\mathfrak{y}}},\overleftarrow{{\mathfrak{z}}}). \end{equation} This implies that the reversal of the transition $D_N={\mathfrak{z}} \to Z_{N+1}={\mathfrak{z}}'$ is exactly the transition $\overleftarrow{E}_{N+1}=(\overleftarrow{{\mathfrak{y}}'},\overleftarrow{{\mathfrak{z}}'}) \to \overleftarrow{E}_N=(\overleftarrow{{\mathfrak{y}}},\overleftarrow{{\mathfrak{z}}})$. In other words, if the transition of $D_N \to D_{N+1}$ looks for matches for $c^{N+1},s^{N+1}$ and exchanges $s^n$ with $c^m$, then the transition of $\overleftarrow{E}_{N+1} \to \overleftarrow{E}_N$ considers the elements in position $N$, and transforms them back. \end{lemma} \begin{figure}[htb] \begin{center} \includegraphics[scale=0.30]{FigMarkov4.pdf} \end{center} \caption{Adding pair backward and its reversal adding pair forward} \label{fig.reversal2} \end{figure} \begin{proof} The only thing to note, as in the proof of Lemma \ref{fig.reversal1}, is that if $s^n,\,c^m$ are matched in the forward FCFS direction, then ${\tilde{s}}^m,\,{\tilde{c}}^n$ are matched in the reversed FCFS direction. \end{proof} \subsection{Stationary distributions of the detailed Markov chains} \label{sec.distributions} We are now ready to derive the stationary distributions of the detailed Markov chains. \begin{theorem} \label{thm.stationary} (i) The stationary distribution of $Z^s_N$ and of $Z^c_M$ is given, up to a normalizing constant, by \begin{equation} \label{eqn.stationrydist} \pi_{Z^s}({\mathfrak{z}}) = \pi_{Z^c}(\overleftarrow{{\mathfrak{z}}}) = \prod_{i=1}^{I} {\alpha_{c_i}}^{\# c_i} \prod_{j=1}^{J} {\beta_{s_j}}^{\# s_j}, \qquad \end{equation} where $\# c_i$ is the number of customers of type $c_i$, and $\# s_j$ is the number of servers of type $s_j$, as they appear in the state ${\mathfrak{z}}$. (ii) The stationary distribution of $D_N$ and of $E_N$ is given, up to a normalizing constant, by \begin{equation} \label{eqn.pairdist} \pi_D({\mathfrak{z}},{\mathfrak{y}}) = \pi_E(\overleftarrow{{\mathfrak{y}}},\overleftarrow{{\mathfrak{z}}}) = \prod_{i=1}^{I} {\alpha_{c_i}}^{\# c_i} \prod_{j=1}^{J} {\beta_{s_j}}^{\# s_j}, \end{equation} where $\# c_i$, $\# s_j$ count customers and servers as they appear in the state $({\mathfrak{z}},{\mathfrak{y}})$. (iii) The normalizing constant is the same in all four distributions. \end{theorem} \begin{proof} We prove (i) using Kelly's Lemma, and the time reversal result. The proof of (ii) is similar and in included in Appendix \ref{sec.appendix} for completeness. To show (iii) we show that there is a 1-1 correspondence between states of $Z^s_N$ and $D_N$. {\em Proof of (i)} We use Kelly's Lemma (\cite{kelly:79}, Section 1.7): For a Markov chain $X_t$, if we can find non-negative $\pi(i)$ and $p_{j\to i}$ such that \begin{equation} \label{eqn.conditions} \sum_i p_{j\to i} =1 \mbox{ for all $j$}, \; \mbox{ and }\quad \pi(i) P(X_{t+1}=j\,|\,X_t=i) = \pi(j) p_{j\to i} \mbox{ for all $i,j$ } \end{equation} then $\pi$ is the stationary distribution of $X_t$, and $p_{j\to i}$ are the transition rates of the reversed stationary process, $p_{j\to i}= P(X_t=i\,|\,X_{t+1}=j)$. Lemma \ref{thm.timereversal} shows that the time reversed transition probabilities of $Z^s_N$ are \begin{equation} \label{eqn.reversal2} P(Z^s_N={\mathfrak{z}} \,|\,Z^s_{N+1}={\mathfrak{z}}') = P(\overleftarrow{Z}^c_N=\overleftarrow{{\mathfrak{z}}} \,|\, \overleftarrow{Z}^c_{N+1}=\overleftarrow{{\mathfrak{z}}'}) = P(Z^c_{M+1} =\overleftarrow{{\mathfrak{z}}} \,|\, Z^c_M =\overleftarrow{{\mathfrak{z}}'}). \end{equation} The condition of summation to 1 in (\ref{eqn.conditions}) is satisfied since $P(Z^c_{M+1} ={\mathfrak{y}}' \,|\, Z^c_M ={\mathfrak{y}} )$ are transition probabilities. To check the condition of detailed balance in (\ref{eqn.conditions}) all we need to do is check that the conjectured form of $\pi_{Z^s}$ in (\ref{eqn.stationrydist}) satisfies \begin{equation} \label{eqn.checking1} \pi_{Z^s}({\mathfrak{z}}) P(Z^s_{N+1}={\mathfrak{z}}' \,|\,Z^s_N={\mathfrak{z}}) = \pi_{Z^s}({\mathfrak{z}}') P(Z^s_n={\mathfrak{z}} \,|\,Z^s_{N+1}={\mathfrak{z}}') = \pi_{Z^c}(\overleftarrow{{\mathfrak{z}}'}) P(Z^c_{M+1} =\overleftarrow{{\mathfrak{z}}} \,|\, Z^c_M =\overleftarrow{{\mathfrak{z}}'}). \end{equation} There are three types of transitions of $Z^s$ (see Figure \ref{fig.serverbyserver}), with analogous transitions for $Z^c$ \begin{figure}[htb] \begin{center} \includegraphics[scale=0.30]{FigServerbyServer.pdf} \end{center} \caption{Possible transitions of the server by server FCFS Markov chain $Z^s_N$} \label{fig.serverbyserver} \end{figure} We check (\ref{eqn.checking1}) for each of them. {\em (a) Simple exchange:} If $z^l = c_i$ and ${z'}^l = s_j$, then according to (\ref{eqn.stationrydist}) we have \[ \pi_{Z^c}(\overleftarrow{{\mathfrak{z}}'}) = \pi_{Z^s}({\mathfrak{z}}') = \pi_{Z^s}({\mathfrak{z}}) \frac{\beta_{s_j}}{\alpha_{c_i}} \] and the direct and reversed transition rates are: \[ P(Z^s_{N+1} = {\mathfrak{z}}' | Z^s_N = {\mathfrak{z}}) = \beta_{s_j}, \qquad P(Z^c_{M+1} = \overleftarrow{{\mathfrak{z}}} | Z^c_M = \overleftarrow{{\mathfrak{z}}'}) = \alpha_{c_i}. \] {\em (b) Deletion of elements from start of ${\mathfrak{z}}$}: If ${\mathfrak{z}} = z^1,\ldots,z^L$, and ${\mathfrak{z}}' = z^{k+1},\ldots,z^L$, and the deleted part is $c_i, s_{j_1}, \ldots, s_{j_{k-1}}$, then according to (\ref{eqn.stationrydist}) we have \[ \pi_{Z^s}({\mathfrak{z}}') = \pi_{Z^s}({\mathfrak{z}}) \frac{1}{\alpha_{c_i} \beta_{s_{j_1}} \cdots \beta_{s_{j_{k-1}}}} \] and the direct and reversed transition rates are: \[ P(Z^s_{N+1} = {\mathfrak{z}}' | Z^s_N = {\mathfrak{z}}) = \beta_{\S(c_i)}, \qquad P(Z^c_{M+1} = \overleftarrow{{\mathfrak{z}}} | Z^c_M = \overleftarrow{{\mathfrak{z}}'}) = \alpha_{c_i} \beta_{s_{j_1}} \cdots \beta_{s_{j_{k-1}}} \beta_{\S(c_i)}. \] {\em (c) Addition of elements to ${\mathfrak{z}}$}: If ${\mathfrak{z}}=z^1,\ldots,z^l$ and ${\mathfrak{z}}'=z^1,\ldots,z^l,c_{i_1},\ldots,c_{i_k},s_j$, then according to (\ref{eqn.stationrydist}) we have \[ \pi_{Z^s}({\mathfrak{z}}') = \pi_{Z^s}({\mathfrak{z}}) \alpha_{c_{i_1}} \cdots \alpha_{c_{i_k}} \beta_{s_j} \] and the direct and reversed transition rates are: \[ P(Z^s_{N+1} = {\mathfrak{z}}' | Z^s_N = {\mathfrak{z}}) = \beta_{s_j} \alpha_{c_{i_1}} \cdots \alpha_{c_{i_k}} \alpha_{{\mathcal{C}}(s_j)} , \qquad P(Z^c_{M+1} = \overleftarrow{{\mathfrak{z}}} | Z^c_M = \overleftarrow{{\mathfrak{z}}'}) = \alpha_{{\mathcal{C}}(s_j)} . \] It is now immediate to check that the balance condition of (\ref{eqn.conditions}) holds. This also proves the form of $\pi_{Z^c}$, as the reversed process. \medskip {\em Proof of (ii)} This is similar, and we included it in Appendix \ref{sec.appendix} \medskip {\em Proof of (iii)} We show that there is a one to one correspondence between the states of $Z^s_N$ and of $D_N$, such that the stationary probabilities of corresponding states are equal up to the normalizing constants. This implies equality of the normalizing constants of $\pi_D$ and $\pi_{Z^s}$. The other processes have the same normalizing constants by the time reversibility. We note first that a sequence of servers and customers $(z^1,\ldots,z^L)$ is a possible state ${\mathfrak{z}}$ of $Z^s$ if and only if $z^l=c_i,z^k=s_j,l<k$ implies $(c_i,s_j)\not\in{\mathcal{E}}$, and similarly it is a possible state ${\mathfrak{y}}$ of $Z^c$ if and only if $z^l=c_i,z^k=s_j,l > k$ implies $(c_i,s_j)\not\in{\mathcal{E}}$. Necessity follows from FCFS, sufficiency follows by constructing sequences $(c^1,\ldots,c^L)$ and $(s^1,\ldots,s^L)$ whose matching yields $(z^1,\ldots,z^l)$ on the customer line for ${\mathfrak{z}}$, or on the server line for ${\mathfrak{y}}$. To show the correspondence consider a state $D_N=({\mathfrak{z}},{\mathfrak{y}})=\big((z^1,\ldots,z^L),(y^1,\ldots,y^K)\big)$. Then ${\mathfrak{z}}'=(z^1,\ldots,z^L,y^K,\ldots,y^1)$ is a possible state of $Z^s$. We need to show that any customer (unmatched, or matched and exchanged) in the sequence $(z^1,\ldots,z^L,y^K,\ldots,y^1)$ is incompatible with any server (unmatched, or matched and exchanged) that appears later in the sequence. Consider $z^l=c^l$, i.e. $c^l$ is unmatched. Then it is incompatible with any ${\tilde{s}}^j=z^j,\,l<j\le L$, because it was skipped by them in the matching of the customer line, and it is incompatible with any of $y^k=s^k$ because all unmatched customers in ${\mathfrak{z}}$ are incompatible with all unmatched servers in ${\mathfrak{y}}$. Consider then $y^k={\tilde{c}}^k$. It is incompatible with any $s^j=y^j,\,1\le j <k$, because it was skipped by ${\tilde{c}}^k$ in the matching of the server line. This shows the correspondence in one direction. Consider now a state of $Z^s$, ${\mathfrak{z}}=(z^1,\ldots,z^K)$. If we break it into $(z^1,\ldots,z^L)$ and $(y^{K-L}=z^{L+1},\ldots,y^1=z^K)$, we get two subsequences, ${\mathfrak{z}}'=(z^1,\ldots,z^L)$ and ${\mathfrak{y}}=(y^1,\ldots,y^{K-L})$. By the above arguments, any matched and exchanged ${\tilde{s}}$ in ${\mathfrak{z}}$ cannot be matched to any earlier unmatched customer in the sequence ${\mathfrak{z}}'$ or to any customer in ${\mathfrak{y}}$, and any matched and exchanged ${\tilde{c}}$ in ${\mathfrak{y}}$ cannot be matched to any earlier unmatched server in ${\mathfrak{y}}$ or to any matched and exchanged server in ${\mathfrak{z}}'$. So it remains to choose $L$ in such a way that the number of unmatched customers in ${\mathfrak{z}}'$ is equal to the number of unmatched servers in ${\mathfrak{y}}$. We do that as follows: we count the same number of customers $c^l$ from the left of ${\mathfrak{z}}$ as of servers ${\tilde{s}}^k$ from the right of ${\mathfrak{z}}$, for as long as $l<k$. The final pair will either be $l=k-1$ in which case we take $L=l$, or $l < k-1$. In that case all of $z^{l+1},\ldots,z^{k-1}$ must all be either servers, in which case we let $L=k-1$, or all of them are customers, in which case we let $L=l$. This shows the correspondence in the other direction. We note that two corresponding states have the same set of customers and the same set of servers. Hence their steady state probabilities are equal up to the normalizing constant. This shows that they have the same normalizing constant, and completes the proof. \end{proof} \subsection{Augmented state, marginals and the normalizing constant} \label{sec.marginals} The extreme simplicity of the stationary probabilities obtained in Theorem \ref{thm.stationary} is deceptive, since it does not indicate which states are possible, according to the compatibility graph and the FCFS matching policy. In particular, there seems to be no simple way of deciding what are all the possible states of the four detailed Markov chains. As a result it is not at all obvious how to calculate the normalizing constant for the distributions (\ref{eqn.stationrydist}), (\ref{eqn.pairdist}). To allow us to classify states in a convenient way, and thus to allow us to count them and to add up their stationary probabilities, we define an augmented detailed Markov chain. It also describes the server by server FCFS matching mechanism, but its states are augmentations of the states of $Z^s$, in that they are even more detailed. Consider server by server FCFS matching when all servers up to position $N$ have been matched and exchanged. For the server by server FCFS detailed matching process we defined $Z^s_N$ as the sequence of elements from position ${\underline{M}}$ of the first unmatched customer to position ${\overline{M}}$ of the last matched and exchanged server on the customer line. We now consider positions ${\underline{N}} < {\underline{M}}$ and ${\overline{N}}> {\overline{M}}$ such that the interval of positions ${\underline{N}}$ to ${\overline{N}}$ contains for each server type at least one matched and exchanged server, and it contains for each customer type at least one unmatched customer, and the interval is minimal. Let ${\mathfrak{z}}=(z^1,\ldots,z^K)$ where $K={\overline{N}}-{\underline{N}}+1$, $z^1={\tilde{s}}^{\underline{N}}$, $z^K=c^{\overline{N}}$, and for ${\underline{N}}<l<{\overline{N}}$, $z^l$ is either an unmatched customer or a matched and exchanged server in position ${\underline{N}}+l-1$. We consider the process $\o Z^s=(\o Z^s_N)_{N\in {\mathbb{Z}}}$ where $\o Z^s_N={\mathfrak{z}}$. Note that $\o Z^s_N={\mathfrak{z}}$ differs from $Z^s_N$ by the addition of some servers before $c^{\underline{M}}$ and some customers after ${\tilde{s}}^{\overline{M}}$. We always have $K\ge I+J$. We define also the {\em server by server FCFS augmented matching process} ${\mathcal{Z}}=({\mathcal{Z}}_N)_{N\in T}$ with state ${\mathfrak{z}}=(z^1,\ldots,z^L)$ with $L={\overline{M}}-{\underline{N}}+1 \ge J$, which includes elements from positions ${\underline{N}}$ to ${\overline{M}}$ on the customer line, starting with $z^1={\tilde{s}}^{\underline{N}}$ and ending with $z^L={\tilde{s}}^{\overline{M}}$ \begin{corollary} \label{thm.stationary2} The stationary distributions of $\o Z^s$ and of ${\mathcal{Z}}$ are given by \begin{eqnarray} \label{eqn.stationrydist3} \pi_{\o Z^s}({\mathfrak{z}}) &=& B \prod_{i=1}^{I} {\alpha_{c_i}}^{\# c_i} \prod_{j=1}^{J} {\beta_{s_j}}^{\# s_j}, \\ \label{eqn.stationrydist4} \pi_{{\mathcal{Z}}}({\mathfrak{z}}) &=& B \prod_{i=1}^{I} {\alpha_{c_i}}^{\# c_i} \prod_{j=1}^{J} {\beta_{s_j}}^{\# s_j}, \end{eqnarray} where $\# c_i$ is the number of customers of type $c_i$ in ${\mathfrak{z}}$, and $\# s_j$ is the number of servers of type $s_j$ in ${\mathfrak{z}}$. \end{corollary} \begin{proof} The proof of the form of $\pi_{\o Z^s}({\mathfrak{z}})$, up to the normalizing constant, is similar to the proof of Theorem \ref{thm.stationary}, since the reversed process corresponds to the customer by customer matching of ${\tilde{C}},{\tilde{S}}$. The proof of the form of $\pi_{{\mathcal{Z}}}({\mathfrak{z}})$ is obtained by summing $\pi_{\o Z^s}(\cdot)$ over all sequences of customers $c^{{\overline{M}}+1},\ldots,c^{\overline{N}}$, which sum up to 1, because we are summing over all outcomes of one or more geometric distributions. Finally, the proof that the normalizing constant is again equal to $B$ is obtained by summing also over all sequences ${\tilde{s}}^{\underline{N}},\ldots,{\tilde{s}}^{{\underline{M}}-1}$. Summation over all added servers and customers returns us to $\pi_{Z^s}(\cdot)$. \end{proof} The motivation for considering the augmented process ${\mathcal{Z}}$ is that each state ${\mathcal{Z}}_N={\mathfrak{z}}$ can be written in a different form, and in that form we can actually enumerate all the possible states. This enables us to obtain stationary distributions of various marginal processes, and finally to derive an explicit expression for the normalizing constant $B$. We now rewrite the state ${\mathfrak{z}}=z^1,\ldots,z^L$ as follows: Let $S_J$ be the type of server $z^L={\tilde{s}}^{\overline{M}}$. Define recursively, for $1 \le j < J$, $S_j$ as the type of the last server in the sequence $z^1,\ldots,z^L$ which is different from $S_{j+1},\ldots,S_J$. Then $R=(S_1,\ldots,S_J)$ is a permutation of the server types $s_1,\ldots,s_J$. Let ${\mathfrak{w}}_1,\ldots,{\mathfrak{w}}_{J-1}$ be the subsequences of customer and server types between the locations of $S_1,\ldots,S_J$ in ${\mathfrak{z}}$. We will then write the state as ${\mathfrak{z}}=(S_1,{\mathfrak{w}}_1,\ldots,{\mathfrak{w}}_{J-1},S_J)$. The idea of presenting the state in this form stems from \cite{visschers:00,visschers-adan-weiss:12} and was used in \cite{adan-weiss:11,adan-weiss:14}. The main feature of ${\mathfrak{z}}=(S_1,{\mathfrak{w}}_1,\ldots,{\mathfrak{w}}_{J-1},S_J)$ is that all the customers in ${\mathfrak{w}}_\ell$ are of types in ${\mathcal{U}}(S_1,\ldots,S_\ell)$ and all the servers in ${\mathfrak{w}}_\ell$ are of types in $\{S_{\ell+1},\ldots,S_J\}$. Of course we can write the stationary distribution of states of ${\mathcal{Z}}$, given in (\ref{eqn.stationrydist4}), also as: \begin{equation} \label{eqn.stationaryZ} \pi_{{\mathcal{Z}}}(S_1,{\mathfrak{w}}_1,\ldots,{\mathfrak{w}}_{J-1},S_J) = B \prod_{j=1}^J \beta_{s_j} \prod_{\ell=1}^{J-1} \left( \prod_{c_i\in {\mathcal{U}}\{S_1,\ldots,S_\ell\}} \alpha_{c_i}^{\#(c_i,{\mathfrak{w}}_\ell)} \prod_{s_j\in \{S_{\ell+1},\ldots,S_J\}} \beta_{s_j}^{\#(s_j,{\mathfrak{w}}_\ell)} \right), \end{equation} where $\#(c_i,{\mathfrak{w}}_\ell)$, $\#(s_j,{\mathfrak{w}}_\ell)$ count the number of type $c_i$ customers and of type $s_j$ servers in ${\mathfrak{w}}_\ell$. We will use the notation $B^s=B \prod_{j=1}^J \beta_{s_j} $. We now consider several processes, some of them Markovian and some of them not Markovian, which are derived by aggregating the states of the detailed augmented Markov chain ${\mathcal{Z}}$. \begin{description} \item The process $W_N = (S_1,w_1,\ldots,w_{J-1},S_J)$, where $w_\ell$ is obtained from ${\mathfrak{w}}_\ell$ by replacing each $c^m$ with a $0$, and each ${\tilde{s}}^n$ with a $1$, so each $w_\ell$ is a sequence of $0$ and $1$. \item The process $X_N=(S_1,n_1,\ldots,n_{J-1},S_J)$ where $n_\ell = \sharp 0(w_\ell)$ is the number of unmatched customers between $S_\ell$ and $S_{\ell+1}$. \item The process $Y_N=(S_1,m_1,\ldots,m_{J-1},S_J)$ where $m_\ell = \sharp 1(w_\ell)$ is the number of matched and exchanged servers between $S_\ell$ and $S_{\ell+1}$. \item The process $U_N=(S_1,n_1,m_1,\ldots,n_{J-1},m_{J-1},S_J)$ which records both the number of unmatched customers and matched and exchanged servers between $S_\ell$ and $S_{\ell+1}$. \item The process $V_N =( S_1,n_1+m_1,\ldots,n_{J-1}+m_{J-1},S_J)$ which records the lengths of the $w_\ell$. \item The process $R_N=(S_1,\ldots,S_N)$ which is the permutation of server types after the $N$th match. \end{description} \begin{theorem} The stationary distributions of $W,X,Y,U,V,R$ are given by: \begin{eqnarray} \label{eqn.stationaryW} \hspace{-0.4in} &&\pi_W(S_1,w_1,\ldots,w_{J-1},S_J) = B^s \prod_{\ell=1}^{J-1} \alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}}^{\#0(w_\ell)} \beta_{\{S_{\ell+1},\ldots,S_J\}}^{\#1(w_\ell)}, \\ \label{eqn.stationaryU} \hspace{-0.4in} &&\pi_U(S_1,n_1,m_1,\ldots,S_J) = B^s \prod_{\ell=1}^{J-1} \binom{n_\ell+m_\ell}{n_\ell} \left(\alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}}\right)^{n_\ell} \left(\beta_{\{S_{\ell+1},\ldots,S_J\}}\right)^{m_\ell}, \\ \label{eqn.stationaryX} \hspace{-0.4in} &&\pi_X(S_1,n_1,\ldots,n_{J-1},S_J) = B^s \prod_{\ell=1}^{J-1} \frac{ \left(\alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}}\right)^{n_\ell} } { \left(\beta_{\{S_1,\ldots,S_\ell\}}\right)^{n_\ell+1} }, \\ \label{eqn.stationaryY} \hspace{-0.4in} &&\pi_Y(S_1,m_1,\ldots,m_{J-1},S_J) = B^s \prod_{\ell=1}^{J-1} \frac{ \left(\beta_{\{S_{\ell+1},\ldots,S_J\}}\right)^{m_\ell} } { \left(\alpha_{{\mathcal{C}}\{S_{\ell+1},\ldots,S_J\}}\right)^{m_\ell+1} }, \\ \label{eqn.stationaryV} \hspace{-0.4in} && \pi_V(S_1,r_1,\ldots,r_{J-1},S_J) = B^s \prod_{\ell=1}^{J-1} \left( \alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}} + \beta_{\{S_{\ell+1},\ldots,S_J\}}\right)^{r_\ell}, \\ \label{eqn.stationaryR} \hspace{-0.4in} && \pi_R(S_1,\ldots,S_J) = B^s \prod_{\ell=1}^{J-1} \left( \beta_{\{S_1,\ldots,S_\ell\}} - \alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}} \right)^{-1}. \end{eqnarray} \end{theorem} \begin{proof} To obtain (\ref{eqn.stationaryW}) we sum (\ref{eqn.stationaryZ}), where each 0 in $w_\ell$ can be any $c_i\in {\mathcal{U}}(S_1,\ldots,S_\ell)$, and each 1 in $w_\ell$ can be any $s_j\in \{S_{\ell+1},\ldots,S_J\}$. To obtain (\ref{eqn.stationaryU}) we sum (\ref{eqn.stationaryW}) over the different locations of the 0's in each $w_\ell$. To obtain (\ref{eqn.stationaryX}) we sum (\ref{eqn.stationaryU}) where each $m_\ell$ needs to range from 0 to $\infty$. We use Newton's binomial formula for $(1-\beta_{\{S_{\ell+1},\ldots,S_J\}})^{-(n_\ell+1)}$, and replace that by $(\beta_{\{S_1,\ldots,S_\ell\}})^{-(n_\ell+1)}$. (\ref{eqn.stationaryY}) is obtained similar to (\ref{eqn.stationaryX}) by summing (\ref{eqn.stationaryU}) over the $n_\ell$. To obtain (\ref{eqn.stationaryV}) we sum (\ref{eqn.stationaryU}), where $n_\ell$ ranges from 0 to $r_\ell=n_\ell+m_\ell$. We use Newton's binomial formula for $\left( \alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}} + \beta_{\{S_{\ell+1},\ldots,S_J\}}\right)^{r_\ell}$. Finally, to obtain (\ref{eqn.stationaryR}) we sum (\ref{eqn.stationaryV}) over each $r_\ell$ ranging form 0 to $\infty$, and replace $(1-\beta_{\{S_{\ell+1},\ldots,S_J\}})$ by $\beta_{\{S_1,\ldots,S_\ell\}}$. \end{proof} We are now ready to obtain the normalizing constant $B$ (see \cite{adan-weiss:11}). \begin{theorem} The normalizing constant $B$ is given by: \begin{eqnarray} \label{eqn.B} B &=& \left( \prod_{j=1}^J \beta_{s_j} \times \sum_{(S_1,\ldots,S_J)\in \P_J} \prod_{\ell=1}^{J-1} \left( \beta_{\{S_1,\ldots,S_\ell\}} - \alpha_{{\mathcal{U}}(\{S_1,\ldots,S_\ell\})} \right)^{-1} \right)^{-1} \\ &=& \left( \prod_{i=1}^I \alpha_{c_i} \times \sum_{(C_1,\ldots,C_I)\in \P_I} \prod_{\ell=1}^{I-1} \left( \beta_{\S(\{C_1,\ldots,C_\ell\})} - \alpha_{\{C_1,\ldots,C_\ell\}} \right)^{-1} \right)^{-1}. \end{eqnarray} where the summation is over all permutations of $s_1,\ldots,s_J$ in the first expression, and over all permutations of $c_1,\ldots,c_I$ in the second expression. \end{theorem} \begin{proof} We obtain $B^s$ by summation of (\ref{eqn.stationaryR}) over all permutations of $s_1,\ldots,s_J$, and obtain $B$ by dividing it by $\prod_{j=1}^J \beta_{s_j}$. The second expression is obtained by using the symmetric derivation for customer by customer matching. \end{proof} By observing when $B$ is finite we obtain: \begin{corollary} A necessary and sufficient condition for ergodicity of all the FCFS matching Markov chains is complete resource pooling (\ref{eqn.pooling}). \end{corollary} \begin{corollary} The conditional distributions of the numbers of unmatched customers and of matched and exchanged servers, given the permutation is a product of geometric probabilities: \begin{eqnarray} \label{eqn.geometric-c} \hspace{-0.5in} && P(X_N=(S_1,n_1,\ldots,n_{J-1},S_J)\,|\, S_1,\ldots,S_J)= \prod_{\ell=1}^{J-1} \left( \frac{ \alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}} } { \beta_{\{S_1,\ldots,S_\ell\}} }\right)^{n_\ell} \left( 1 - \frac{\alpha_{{\mathcal{U}}\{S_1,\ldots,S_\ell\}}}{\beta_{\{S_1,\ldots,S_\ell\}}} \right), \\ \label{eqn.geometric-s} \hspace{-0.5in} && P(Y_N=(S_1,m_1,\ldots,m_{J-1},S_J)\,|\,S_1,\ldots,S_J ) = \prod_{\ell=1}^{J-1} \left( \frac{ \beta_{\{S_{\ell+1},\ldots,S_J\}} } { \alpha_{{\mathcal{C}}\{S_{\ell+1},\ldots,S_J\}} }\right)^{m_\ell} \left( 1 - \frac{\beta_{\{S_{\ell+1},\ldots,S_J\}}}{\alpha_{{\mathcal{C}}\{S_{\ell+1},\ldots,S_J\}}} \right). \end{eqnarray} \end{corollary} \subsection{Stationary distribution of the `natural' Markov chains} \label{sec.natural} We now consider the `natural' Markov chains $O$ of pair by pair and $Q^s$, $Q^c$ of server by server and customer by customer FCFS matching. The state consists of the ordered list of unmatched customers and/or servers. \begin{theorem} The stationary distributions for $Q^s=(Q^s_N)_{N\in {\mathbb{Z}}}$, for $Q^c=(Q^c_N)_{N\in {\mathbb{Z}}}$ and for $O=(O_N)_{N\in {\mathbb{Z}}}$ are given by \begin{equation} \label{eqn.stationaryQs} \pi_{Q^s}(c^1,\ldots,c^L) = B \big(1- \beta_{\S(\{c^1,\ldots,c^L\})}\big) \prod_{\ell=1}^L \frac{\alpha_{c^\ell}}{\beta_{\S(\{c^1,\ldots,c^\ell\})}}, \end{equation} \begin{equation} \label{eqn.stationaryQc} \pi_{Q^c}(s^1,\ldots,s^K) = B \big(1- \beta_{{\mathcal{C}}(\{s^1,\ldots,s^K\})}\big) \prod_{k=1}^K \frac{\beta_{s^k}}{\alpha_{{\mathcal{C}}(\{s^1,\ldots,s^k\})}}. \end{equation} \begin{equation} \label{eqn.stationaryO} \pi_{O}(c^1,\ldots,c^L,s^1,\ldots,s^L) = B \prod_{\ell=1}^L \frac{\alpha_{c^\ell}}{\beta_{\S(\{c^1,\ldots,c^\ell\})}} \frac{\beta_{s^\ell}}{\alpha_{{\mathcal{C}}(\{s^1,\ldots,s^\ell\})}}. \end{equation} \end{theorem} \begin{proof} To prove (\ref{eqn.stationaryQs}), we need to sum $\pi_{Z^s}={\mathfrak{z}}$ over all states ${\mathfrak{z}}$ of the form: \[ c^1,s^{1,1},\ldots,s^{1,j_1},c^2,\ldots,c^\ell,s^{\ell,1},\ldots,s^{\ell,j_\ell},\ldots,c^L,s^{L,1},\ldots,s^{L,j_L}, \] where for $1\le \ell <L$ the range of $j_\ell$ is $0 \le j_\ell < \infty$ and for $j_L$ the range is $1 \le j_L < \infty$, and where each of $s^{\ell,m}$ ranges over possible values $s_j \not\in \S(\{c^1,\ldots,c^\ell\})$. We therefore have, by (\ref{eqn.stationrydist}), that \begin{eqnarray} \label{eqn.summation} && \pi_{Q^s}(c^1,\ldots,c^L) = B \sum \alpha_{c^1}\beta_{s^{1,1}}\cdots\beta_{s^{1,j_1}}\alpha_{c^2}\cdots\alpha_{c^\ell} \beta_{s^{\ell,1}}\cdots\beta_{s^{\ell,j_\ell}}\cdots\alpha_{c^L}\beta_{s^{L,1}}\cdots\beta_{s^{L,j_L}} \nonumber\\ && \qquad = B \sum \alpha_{c^1}\big(1- \beta_{\S(c^1)}\big)^{j_1} \alpha_{c^2} \cdots \alpha_{c^\ell} \big(1 - \beta_{\S(\{c^1,\ldots,c^\ell\})}\big)^{j_\ell} \cdots \alpha_{c^L} \big(1 - \beta_{\S(\{c^1,\ldots,c^L\})}\big)^{j_L} \nonumber \\ && \qquad = B \frac{\alpha_{c^1}}{\beta_{\S(c^1)}} \cdots \frac{\alpha_{c^\ell}}{\beta_{\S(\{c^1\ldots,c^\ell\})}} \cdots \frac{\alpha_{c^\ell}}{\beta_{\S(\{c^1\ldots,c^\ell\})}} (1 - \beta_{\S(\{c^1,\ldots,c^L\})}), \end{eqnarray} which proves formula (\ref{eqn.stationaryQs}). Formula (\ref{eqn.stationaryQc}) is proved in the same way. To prove (\ref{eqn.stationaryO}), we need to sum $\pi_D=({\mathfrak{z}},{\mathfrak{y}})$ over all states $({\mathfrak{z}},{\mathfrak{y}})$ of the form: \[ c^1,s^{1,1},\ldots,s^{1,j_1},c^2,\ldots,c^L,s^{L,1},\ldots,s^{L,j_L} s^1,c^{1,1},\ldots,c^{1,k_1},s^2,\ldots,s^L,c^{L,1},\ldots,c^{L,k_L}, \] where for $1 \le \ell \le L$ the range of $j_\ell,k_\ell$ is $0 \le j_\ell,k_\ell < \infty$, and where each of $s^{\ell,m}$ ranges over possible values $s_j \not\in \S(\{c^1,\ldots,c^\ell\})$, and each of $c^{\ell,m}$ ranges over possible values $c_i \not\in {\mathcal{C}}(\{s^1,\ldots,s^\ell\})$. Performing the summation as in (\ref{eqn.summation}) we obtain (\ref{eqn.stationaryO}). \end{proof} \subsubsection{An example: the "NN" system} \label{sec.example} We consider the ``NN'' system, which is illustrated in Figure \ref{fig.NNsystem}. This system was studied in \cite{caldentey-kaplan-weiss:09}, \begin{figure}[htb] \begin{center} \includegraphics[scale=0.30]{FigNN-system.pdf} \end{center} \caption{Compatibility graph and probabilities of the ``NN'' system} \label{fig.NNsystem} \end{figure} where ergodicity under complete resource pooling was demonstrated but stationary probabilities could not at that time be obtained. We can easily calculate stationary probabilities of the pair by pair, server by server and customer by customer `natural' FCFS processes, using formulae (\ref{eqn.stationaryQs})--(\ref{eqn.stationaryO}), (\ref{eqn.B}). The conditions for stability, i.e. for complete resource pooling, are: \[ \beta_1 > \alpha_3, \qquad \alpha_1 > \beta_3, \qquad \alpha_1 + \beta_1 <1. \] Some examples are (we write $\alpha_i,\beta_j$ for $\alpha_{c_i},\beta_{s_j}$): \begin{eqnarray*} && P(Q^s_N=c_1,c_1,c_1,c_1) = B \beta_1 \left( \frac{\alpha_1}{\beta_2+\beta_3} \right)^4, \\ && P(Q^s_N=c_3,c_3,c_3,c_3,c_3) = B (1-\beta_1) \left( \frac{\alpha_3}{\beta_1} \right)^5, \\ && P(Q^c_N= s_3,s_3,s_3,s_2,s_3,s_2) = B \alpha_3 \left( \frac{\beta_3}{\alpha_1} \right)^3 \left( \frac{\beta_2}{\alpha_1+\alpha_2} \right)^2 \frac{\beta_3}{\alpha_1+\alpha_2}, \\ && P\big(O_N = (c_3,c_3,c_2,c_3,c_2,c_3),(s_3,s_3,s_3,s_3,s_3,s_3)\big) = B \left( \frac{\alpha_3}{\beta_1} \right)^2 \left( \frac{\alpha_2}{\beta_1+\beta_2} \right)^2 \left( \frac{\alpha_3}{\beta_1+\beta_2} \right)^2 \left( \frac{\beta_3}{\alpha_1} \right)^6. \end{eqnarray*} The value of the normalizing constant is: \[ B = \frac{(\alpha_1-\beta_3)(\beta_1-\alpha_3)(1 -\alpha_1 - \beta_1)}{\alpha_1 \alpha_2 \beta_1 \beta_2}. \] \section{Calculation of performance measures} \label{sec.perfomance} \subsection{Matching rates} \label{sec.matchingrates} Assume ergodicity (complete resource pooling) holds. The {\em matching rate} $r_{c_i,s_j}$ is the a.s. limit of the fraction of matches of customer of type $c_i$ with server of type $s_j$, in the complete FCFS matching of $(s^n,c^m)_{0\le m,n \le N}$, as $N\to\infty$. An expression for $r_{c_i,s_j}$ was derived in \cite{adan-weiss:11}. We include this expression here for completeness and also because of its close similarity to the derivation of the distribution of {\em link lengths} $L_{s_j},\,L_{c_i,s_j}$, in Section \ref{sec.linklengths}. Both $r_{c_i,s_j}$ and the distribution of $L_{c_i,s_j}$ are obtained by considering the state of the process $^o Z^s$ which is ${\mathfrak{s}}=(S_1,{\mathfrak{w}}_1,S_2,{\mathfrak{w}}_2,\ldots,S_J,{\mathfrak{w}}_J)$ (or equivalently, of the process ${\mathcal{Z}}$ with the addition of ${\mathfrak{w}}_J$, where ${\mathfrak{w}}_J$ is the sequence of $c^{{\overline{M}}+1},\ldots,c^{\overline{N}}$). The final expressions include summation over all the permutations $S_1,\ldots,S_J \in \P_J$ of the servers $s_1,\ldots,s_J$. For convenience we use the following notations relative to each permutation $S_1,\ldots,S_J$: \[ \alpha_{(k)} = \alpha_{U\{S_1,\ldots,S_k\}}, \qquad \beta_{(k)} = \beta_{\{S_1,\ldots,S_k\}}=\beta_{S_1}+\cdots+\beta_{S_k} . \] Note that if $U\{S_1,\ldots,S_k\}=\emptyset$ then $\alpha_{(k)}=0$. Further, \[ \phi_k = \frac{\alpha_{U\{S_1,\ldots,S_k\}\cap \{c_i\}}}{\alpha_{U\{S_1,\ldots,S_k\}}}, \qquad \psi_k = \frac{\alpha_{U\{S_1,\ldots,S_k\}\cap (C(s_j)\backslash\{c_i\})}}{\alpha_{U\{S_1,\ldots,S_k\}}}, \qquad \chi_k=1-\phi_k-\psi_k , \] where $\phi_k,\psi_k,\chi_k$ express the conditional probability that $s^{N+1}=s_j$ and $c^m\in {\mathfrak{w}}_k$ form an $(s_j,c_i)$ match, or an $(s_j,c_k),\,c_k\ne c_i$ match, or no match at all, respectively. By convention $0/0=0$. The expression for the matching rate is: \begin{eqnarray} \label{eqn.matches} r_{c_i,s_j} &=& \beta_{s_j} \sum_{(S_1,\ldots,S_J)\in \P_J} \pi_R( S_1,\ldots,S_J) \nonumber \\ && \left( \sum_{k=1}^{J-1} \phi_k \frac{\alpha_{(k)}}{\beta_{(k)}-\alpha_{(k)} \chi_{k}} \prod_{l=1}^{k-1} \frac{\beta_{(l)}-\alpha_{(l)}}{\beta_{(l)}-\alpha_{(l)} \chi_{l}} + \frac{\phi_J}{\phi_J+\psi_J} \prod_{l=1}^{J-1} \frac{\beta_{(l)}-\alpha_{(l)}}{\beta_{(l)}-\alpha_{(l)} \chi_{l}} \right) . \end{eqnarray} \subsection{Link lengths} \label{sec.linklengths} Assume ergodicity (equivalently, complete resource pooling) holds, and consider a stationary FCFS matching over ${\mathbb{Z}}$. For each server $s^n$, if $s^n$ is matched to $c^m$ we let $L(s^n,c^m)=m-n$ denote the link length. We define the random variable $L_{s_j}$ to have the stationary distribution of link lengths of servers of type $s_j$. We define the random variable $L_{s_j,c_i}$ to have the stationary distribution of link lengths of matches between servers of type $s_j$ and customers of type $c_i$. In this section we derive the distributions of $L_{s_j}$ and of $L_{s_j,c_i}$. They are mixtures of convolutions of some positively signed and some negatively signed geometric random variables. Consider the system following server by server FCFS matching and exchanging of all servers up to $N$. The state is ${\mathfrak{s}}= {\tilde{s}}^{\underline{N}},\ldots,c^{\overline{N}} = S_1,{\mathfrak{w}}_1,\ldots,S_J,{\mathfrak{w}}_J$. Let $s^{N+1}=s_j$. $s^{N+1}$ will be matched to one of the elements of ${\mathfrak{s}}$, since ${\mathfrak{s}}$ contains customers of all types. Say it is matched to $c^m$. We are interested in $L_{s_j} = m-(N+1)$. We are also interested in the special case that $c^m=c_i$. The conditional random variable, conditional on $c^m = c_i$ is then $L_{c_i,s_j}=m-(N+1)$. We introduce the following additional notation. For the state ${\mathfrak{s}}$, we let $\sharp c$ and $\sharp {\tilde{s}}$ count respectively the number of unmatched customers and the number of matched exchanged servers in ${\mathfrak{s}}$. Also for $c^m$ we let $\sharp c_L$, $\sharp c_R$, $\sharp {\tilde{s}}_L$, $\sharp {\tilde{s}}_R$ count respectively the number of customers and of servers, to the left (preceding) and to the right (succeeding) of $c^m$ in ${\mathfrak{s}}$. \begin{proposition} If $s^{N+1}$ is matched to $c^m$, then $L(s^{N+1},c^m)=m-(N+1)$ is equal to the total number of unmatched customers preceding $c^m$ minus the total number of matched exchanged servers following $c^m$, that is: \begin{equation} \label{eqn.linkcount} L(s^{N+1},c^m)=m-(N+1) =\sharp c_L - \sharp {\tilde{s}}_R. \end{equation} \end{proposition} \begin{proof} Figure \ref{fig.link} describes a link from $s^{N+1}$ to $c^m$. \begin{figure}[htb] \begin{center} \includegraphics[scale=0.30]{FigLinks.pdf} \end{center} \caption{The length of the link $L(s^{N+1},c^m)$, from $s^{N+1}$ to $c^m$} \label{fig.link} \end{figure} In this figure we see the position of $N$ relative to ${\underline{N}}$ and ${\overline{N}}$, and the position $m$. Conditional on $S_1,\ldots,S_J$ we have the words inbetween these last servers of each type, ${\mathfrak{w}}_1,\ldots,{\mathfrak{w}}_J$. The match occurs in position $m$ in the middle of the word ${\mathfrak{w}}_\ell$. $G_k$ denotes the random number of customers in the word ${\mathfrak{w}}_k$ skipped by $s^{N+1}$ in the search for a match, where $k=1,\ldots,\ell$. We will see that $G_k$ are independent and geometrically distributed. $H_k$ is the random number of matched and exchanged servers, occurring in the word ${\mathfrak{w}}_k$ after the position $m$, for $k=\ell,\ldots,J$. We will see that $H_k$ are also independent and geometrically distributed, and $G_1,\ldots,G_\ell$ and $H_\ell,\ldots,H_J$ are independent. In this figure, $\sharp c_L = G_1 + \cdots +G_\ell$ and $\sharp s_R = H_\ell + \cdots +H_J$ The lemma follows from: \begin{eqnarray*} && N - {\underline{N}} +1 = \sharp {\tilde{s}}, \qquad \sharp {\tilde{s}} = \sharp {\tilde{s}}_L + \sharp {\tilde{s}}_R, \qquad m- {\underline{N}} = \sharp c_L + \sharp {\tilde{s}}_L, \\ && L(s^{N+1},c^m) = m - (N+1) = m - {\underline{N}} - \sharp {\tilde{s}} = \sharp c_L - \sharp {\tilde{s}}_R. \end{eqnarray*} \end{proof} We now need to count the number of matched customers preceding $c^m$, and matched and exchanged servers succeeding $c^m$. We condition on $s^{N+1}=s_j$, and on $R_N=(S_1,\ldots,S_J)$. Then ${\mathfrak{w}}_k$ consists of sequences of skipped customers of type ${\mathcal{U}}\{S_1,\ldots,S_k\}$, and exchanged servers of type $S_{k+1},\ldots,S_J$, and their number is geometrically distributed (with values $0,1,\ldots)$ as in (\ref{eqn.geometric-c}), (\ref{eqn.geometric-s}): \[ \sharp(c^r,{\mathfrak{w}}_k)\sim Geom_0\left(1 - \frac{\alpha_{(k)}}{\beta_{(k)}}\right), \qquad \sharp({\tilde{s}}^h,{\mathfrak{w}}_k)\sim Geom_0\left(1 - \frac{1 - \beta_{(k)}}{1-\alpha_{(k)}}\right) \] \begin{figure}[htb] \begin{center} \includegraphics[scale=0.40]{FigTrees.pdf} \end{center} \caption{Probabilities in searching for a match in ${\mathfrak{w}}_k$} \label{fig.trees} \end{figure} We now count $\sharp c_L$, the number of skipped customers prior to a match $c^m$. Within the word ${\mathfrak{w}}_k$, following each skipped customer or at the beginning of the word, we can encounter either a next skipped customer, or the end of the word, or a match for $s^{N+1}=s_j$. When a match occurs it can be with $c^m=c_i$, or it can be with $c^m \in {\mathcal{C}}(s_j)\backslash c_i$. The probabilities for these outcomes are given in Figure \ref{fig.trees}. Consider the event that $s^{N+1}$ finds a match of any type within ${\mathfrak{w}}_\ell$, after skipping $r_1,\ldots,r_{\ell-1},r_\ell$ customers in ${\mathfrak{w}}_1,\ldots,{\mathfrak{w}}_\ell$. The probability of this event is: \[ \left[ \prod_{k=1}^{\ell-1} \left( \frac{\alpha_{(k)}}{\beta_{(k)}} \chi_k \right)^{r_k} \left(1 - \frac{\alpha_{(k)}}{\beta_{(k)}} \right) \right] \left( \frac{\alpha_{(\ell)}}{\beta_{(\ell)}} \chi_\ell \right)^{r_\ell} \frac{\alpha_{(\ell)}}{\beta_{(\ell)}} (\phi_\ell + \psi_\ell ), \] Dividing and multiplying by $1 - \frac{\alpha_{(k)}}{\beta_{(k)}} \chi_k,\; k=1,\ldots,\ell$ and rearranging we obtain: \[ \left( \frac{ \alpha_{(\ell)} (\phi_\ell + \psi_\ell ) }{\beta_{(\ell)} -\alpha_{(\ell)} \chi_\ell } \prod_{k=1}^{\ell-1} \frac{ \beta_{(k)} - \alpha_{(k)} }{ \beta_{(k)} - \alpha_{(k)} \chi_k } \right) \prod_{k=1}^\ell \left(\frac{\alpha_{(k)}}{\beta_{(k)}} \chi_k \right)^{r_k} \left(1 - \frac{\alpha_{(k)}}{\beta_{(k)}} \chi_k \right), \] here the $r_k$ are seen to have a geometric distribution, and the probability multiplying the geometric distribution terms is the probability that the match will occur in the word ${\mathfrak{w}}_\ell$. Consider next the same event conditional on the match being of type $c_i$. We need to divide the probabilities in each step by $1- \frac{\alpha_{(k)}}{\beta_{(k)}} \psi_k$. The resulting probability is: \[ \left[ \prod_{k=1}^{\ell-1} \left( \frac{\alpha_{(k)} \chi_k}{\beta_{(k)} - \alpha_{(k)} \psi_k} \right)^{r_k} \left(\frac{\beta_{(k)} - \alpha_{(k)}}{\beta_{(k)} - \alpha_{(k)} \psi_k} \right) \right] \left( \frac{\alpha_{(\ell)} \chi_\ell}{\beta_{(\ell)} - \alpha_{(\ell)} \psi_\ell} \right)^{r_\ell} \frac{\alpha_{(\ell)} \phi_\ell}{\beta_{(\ell)} - \alpha_{(\ell)} \psi_\ell}, \] Again, dividing and multiplying by $1 - \frac{\alpha_{(k)} \chi_k }{\beta_{(k)}- \alpha_{(k)} \psi_k }\; k=1,\ldots,\ell$ and rearranging we obtain: \[ \left( \frac{ \alpha_{(\ell)} \phi_\ell }{ \beta_{(\ell)} - \alpha_{(\ell)} (\psi_\ell + \chi_\ell) } \prod_{k=1}^{\ell-1} \frac{ \beta_{(k)} - \alpha_{(k)} }{ \beta_{(k)} - \alpha_{(k)} (\psi_k + \chi_k) } \right) \prod_{k=1}^\ell \left(\frac{\alpha_{(k)} \chi_k }{\beta_{(k)}- \alpha_{(k)} \psi_k } \right)^{r_k} \left(1 - \frac{\alpha_{(k)} \chi_k }{\beta_{(k)}- \alpha_{(k)} \psi_k } \right), \] where again we have $r_k$ with geometric distributions, multiplied by the probability that the match with $c_i$ happens in word ${\mathfrak{w}}_\ell$. We see that conditional on $S_1,\ldots,S_J$ the value of $\sharp c_L$ in the match of $s^{N+1}=s_j$ with $c^m$, is a mixture of convolutions of $\ell$ geometric random variables, where the mixture is over the value of $\ell$. We use $\bigstar _{k=1}^\ell$ to denote convolution of $\ell$ random variables indexed by $k$: \begin{equation} \label{eqn.convolution1} \sharp c_L \sim G_1\star\cdots\star G_\ell = \bigstar _{k=1}^\ell Geom_0 \left (1 - \frac{\alpha_{(k)}}{\beta_{(k)}} \chi_k \right) \mbox{ w.p. } \frac{ \alpha_{(\ell)} (\phi_\ell + \psi_\ell ) }{\beta_{(\ell)} -\alpha_{(\ell)} \chi_\ell } \prod_{k=1}^{\ell-1} \frac{ \beta_{(k)} - \alpha_{(k)} }{ \beta_{(k)} - \alpha_{(k)} \chi_k } \end{equation} and in the match of $s^{N+1}=s_j$ with $c^m = c_i $: \begin{equation} \label{eqn.convolution2} \sharp c_L \sim G_1\star\cdots\star G_\ell = \bigstar _{k=1}^\ell Geom_0 \left (1 - \frac{\alpha_{(k)}\chi_k}{\beta_{(k)} -\alpha_{(k)}\psi_k } \right) \mbox{ w.p. } \frac{ \alpha_{(\ell)} \phi_\ell }{ \beta_{(\ell)} - \alpha_{(\ell)} (\psi_\ell + \chi_\ell) } \prod_{k=1}^{\ell-1} \frac{ \beta_{(k)} \alpha_{(k)} }{ \beta_{(k)} - \alpha_{(k)} (\psi_k + \chi_k) } \end{equation} The number of matched and exchanged servers succeeding the match, $\sharp {\tilde{s}}_R$, is independent of the number of skipped customers prior to the match. Also, if the match occurs in ${\mathfrak{w}}_\ell$, the number of matched and exchanged servers after $c^m$ is geometric by the memoryless property of geometric random variables. Hence, conditional on $S_1,\ldots,S_J$ in the match of $s^{N+1}=s_j$ with $c^m$, $\sharp s_R$ is a mixture of convolutions of geometric random variables, plus the succeeding servers $S_{\ell+1},\ldots,S_J$ \[ \sharp {\tilde{s}}_R \sim H_\ell\star\cdots\star H_J + (J-\ell) = (J-\ell)+ \bigstar _{k=\ell}^J Geom_0 \left (1 - \frac{1-\beta_{(k)}}{1-\alpha_{(k)}} \right) \begin{array}{c} \mbox{with appropriate probabilities}\\ \mbox{as in (\ref{eqn.convolution1}), (\ref{eqn.convolution2})}\end{array} \] {\bf Note}: In all these equations, empty sums, empty products, and 0 probabilities occur when necessary. We summarize these results in terms of generating functions. \begin{theorem} The generating functions of the distributions of $L_{s_j},\,L_{c_i,s_j}$ are: \begin{eqnarray} \label{eqn.generating1} && E({\sf{Z}}^{L_{s_j}}) = \sum_{(S_1,\ldots,S_J)\in \P_J} \pi_R( S_1,\ldots,S_J) \sum_{\ell = 1}^J \frac{ \alpha_{(\ell)} (\phi_\ell + \psi_\ell ) }{\beta_{(\ell)} -\alpha_{(\ell)} \chi_\ell } \prod_{k=1}^{\ell-1} \frac{ \beta_{(k)} - \alpha_{(k)} }{ \beta_{(k)} - \alpha_{(k)} \chi_k } \nonumber \\ &&\qquad \times \prod_{k=1}^\ell \Big( \frac{ \beta_{(k)} - \alpha_{(k)} \chi_k } { \beta_{(k)} - \alpha_{(k)} \chi_k {\sf{Z}} } \Big) \times \prod_{k=\ell}^J \Big( \frac{ \beta_{(k)} - \alpha_{(k)} } { 1-\alpha_{(k)} - (1-\beta_{(k)}) {{\sf{Z}}}^{-1} } \Big) \times \frac{1}{{\sf{Z}}^{J-\ell}} \end{eqnarray} \begin{eqnarray} \label{eqn.generating2} && E({\sf{Z}}^{L_{s_j,c_i}}) = \sum_{(S_1,\ldots,S_J)\in \P_J} \pi_R( S_1,\ldots,S_J) \sum_{\ell = 1}^J \frac{ \alpha_{(\ell)} \phi_\ell }{ \beta_{(\ell)} - \alpha_{(\ell)} (\psi_\ell + \chi_\ell) } \prod_{k=1}^{\ell-1} \frac{ \beta_{(k)} - \alpha_{(k)} }{ \beta_{(k)} - \alpha_{(k)} (\psi_k + \chi_k) } \nonumber \\ && \qquad \times \prod_{k=1}^\ell \Big( \frac{ \beta_{(k)} - \alpha_{(k)}(\psi_k + \chi_k) } { \beta_{(k)} -\alpha_{(k)} \psi_k - \alpha_{(k)} \chi_k {\sf{Z}} } \Big) \times \prod_{k=\ell}^J \Big( \frac{ \beta_{(k)} - \alpha_{(k)} } { 1-\alpha_{(k)} - (1-\beta_{(k)}) {{\sf{Z}}}^{-1} } \Big) \times \frac{1}{{\sf{Z}}^{J-\ell}} \end{eqnarray} \end{theorem}

BkiUgCQ5qhDCuPMnCmzS

\section{Introduction} Monopole has been known to exist in non-abelian gauge theory. One of the main developments was given by 't Hooft in~\cite{tHooft:1974kcl}, and in parallel with a work by Polyakov in~\cite{Polyakov:1974ek}, in which he showed that monopole could arise as soliton in a Yang-Mills-Higgs theory, without introducing Dirac's string~\cite{Dirac:1931kp}, by spontaneously breaking the symmetry of $SO(3)$ gauge group into $U(1)$ gauge group. Later on, Julia and Zee showed that a more general configuration of soliton called dyon may exist as well within the same model~\cite{Julia:1975ff}. Futhermore, the exact solutions were given by Prasad and Sommerfiled in~\cite{Prasad:1975kr} by taking some limit where $V\to 0$. These solutions were proved by Bogomolnyi in~\cite{Bogomolny:1975de} to be solutions of the first-order differential equations which turn out to be closely related with the study of supersymmetric system~\cite{Witten:1978mh}\footnote{In this article, we shall call the limit $V\to0$ as BPS limit and the first-order differential equations as BPS equations.}. At high energy the Yang-Mills theory may receive contributions from higher derivative terms. This can be realized in string theory in which the effective action of open string theory may be described by the Born-Infeld type of actions~\cite{Abouelsaood:1986gd}. However, there are several ways in writing the Born-Infeld action for non-abelian gauge theory because of the ordering of matrix-valued field strength~\cite{Abouelsaood:1986gd,Tseytlin:1997csa,Hashimoto:1997gm,Gross:1997mx,Brecher:1998tv,Gonorazky:1998ay}. Further complications appear when we add Higgs field into the action. One of examples has been given by Nakamula and Shiraishi in which the action exhibits the usual BPS monopole and dyon~\cite{Nakamula:2013rfa}. Unfortunately, the resulting BPS equations obviously do not capture essensial feature of the Born-Infeld action namely there is no dependency over the Born-Infeld parameter. Other example such as in~\cite{Grandi:1999rv}, the monopole's profile depends on the Born-Infeld parameter, but the BPS equations are not known so far. In this article, we would like to derive the well-known BPS equations of monopole and dyon in the $SU(2)$ Yang-Mills-Higgs model and their Born-Infeld type extensions, which we shall call them Nakamula-Shiraishi models, using a procedure called BPS Lagrangian method developed in~\cite{Atmaja:2015umo}. We then extend those models to their Generalized versions by adding scalar-dependent couplings to each of the kinetic terms and derive the BPS equations for monopole and dyon. In section 2, we will first discuss in detail about the BPS Lagrangian method. The next section 3, we describe how to get the BPS equations for monopole(dyon) from energy density of the $SU(2)$ Yang-Mills-Higgs model using Bogomolny's trick. We write explicitly its effective action and effective actions of the Nakamula-Shiraishi models by taking the 't Hooft-Polyakov(Julia-Zee) ansatz for monopole(dyon). In section 4, we use the BPS Lagrangian approach to reproduce the BPS equations for monopole and dyon in the $SU(2)$ Yang-Mills-Higgs model and Nakamula-Shiraishi models. Later, in section 5, we generalize the $SU(2)$ Yang-Mills-Higgs model by adding scalar-dependent couplings to scalar and gauge kinetic terms and derive the corresponding BPS equations. We also generalize Nakamula-Shiraishi models in section 6 and derive their corresponding BPS equations. We end with discussion in section 7. \section{BPS Langrangian Method} In deriving BPS equations of a model, we normally use so called Bogomolnyi's trick by writing the energy density into a complete square form~\cite{Bogomolny:1975de}. However, there are several rigorous methods have been developed in doing so. The first one is based on the Bogomolnyi's trick by assuming the existance of a homotopy invariant term in the energy density that does not contribute to Euler-Lagrange equations~\cite{Adam:2013hza}. The second method called first-order formalism works by solving a first integral of the model, together with stressless condition,~\cite{Bazeia:2005tj,Bazeia:2006mh,Bazeia:2007df}. The third method called On-Shell method works by adding and solving auxiliary fields into the Euler-Lagrange equations and assuming the existance of BPS equations within the Euler-Lagrange equations~\cite{Atmaja:2014fha,Atmaja:2015umo}. The forth method called First-Order Euler-Lagrange(FOEL) formalism, which is generalization of Bogomolnyi decomposition using a concept of strong necessary condition developed in~\cite{Sokalski:2001wk}, works by adding and solving a total derivative term into the Lagrangian~\cite{Adam:2016ipc}\footnote{In our opinion the procedure looks similar to the On-Shell method by means that adding total derivative terms into the Lagrangian is equivalent to introducing auxiliary fields in the Euler-Lagrange equations. However, we admit that the procedure is written in a more covariant way.}. The last method, which we shall call BPS Lagrangian method, works by identifying the (effective)Lagrangian with a BPS Lagrangian such that its solutions of the first-derivative fields give out the desired BPS equations~\cite{Atmaja:2015umo}. This method was developed based on the On-Shell method by one of the author of this article and it is much easier to execute compared to the On-Shell method. We chose to use the BPS Lagrangian method to find BPS equations of all models considered in this article. The method is explained in the following paragraphs. In general the total static energy of N-fields system, $\vec{\phi}=(\phi_1,\ldots,\phi_N)$, with Lagrangian density $\mathcal{L}$ is defined by $E_\text{static}=-\int d^d x~ \mathcal{L}$. The Bogomolnyi's trick explains that the static energy can be rewritten as \begin{equation} E_\text{static}=\left(\int d^dx \sum_{i=1}^N \Phi_i(\vec{\phi},\partial\vec{\phi})\right)+E_\text{BPS}, \label{Bogomolnyi trick} \end{equation} with $\{\Phi_i\}$ is a set of positive-semidefinite functions and $E_\text{BPS}$ is the boundary contributions defined by $E_\text{BPS}=-\int d^dx~ \mathcal{L}_\text{BPS}$. Neglecting the contribution from boundary terms in $\mathcal{L}_\text{BPS}$, as they do not affect the Euler-Lagrange equations, configurations that minize the static energy are also solutions of the Euler-Lagrange equations and they are given by $\{\Phi_i =0\}$ known as BPS equations. Rewriting the static energy to be in the form of equation (\ref{Bogomolnyi trick}) is not always an easy task. However it was argued in~\cite{Atmaja:2015umo} that one does not need to know the explicit form of equation (\ref{Bogomolnyi trick}) in order to obtain the BPS equations. By realizing that in the BPS limit, where the BPS equations are assumed to be exist, remaining terms in the total static energy are in the form of boundary terms, $E_\text{static}=E_\text{BPS}$. Therefore we may conclude that BPS equations are solutions of $\mathcal{L}-\mathcal{L}_{BPS}=\sum_{i=1}^N \Phi_i(\vec{\phi},\partial\vec{\phi})=0$. Now let us see in detail what is inside $\mathcal{L}_\text{BPS}$. Suppose that in spherical coordinates the system effectively depends on only radial coordinate $r$. As shown by the On-Shell method on models of vortices~\cite{Atmaja:2014fha}, the total static energy in the BPS limit can be defined as \begin{equation} E_\text{BPS}=Q(r\rightarrow\infty)-Q(r\rightarrow 0)=\int^{r\rightarrow\infty}_{r\rightarrow 0} dQ, \end{equation} where $Q$ is called BPS energy function. The BPS energy function $Q$ does not depend on the coordinate $r$ explicitly however in general it can also depend on $r$ explicitly in accordance with the choosen ansatz. In most of the cases if we choose the ansatz that does not depend explicitly on coordinate $r$ then we would have $Q\neq Q(r)$. Hence, with a suitable ansatz, we could write $Q=Q(\tilde{\phi}_1,\ldots,\tilde{\phi}_N)$ in which $\tilde{\phi}_i$ is the effective field of $\phi_i$ as a function of coordinate $r$ only. Assume that $Q$ can be treated with separation of variables \begin{equation} Q\equiv \prod_{i=1}^N Q_i(\tilde{\phi}_i), \label{eq:guessQ} \end{equation} this give us a pretty simple expression of $E_\text{BPS}$, i.e. \begin{equation} E_\text{BPS}=\int \sum_{i=1}^N {\partial Q\over \partial \tilde{\phi}_i} {d \tilde{\phi}_i \over dr} dr, \end{equation} and we could obtain $\mathcal{L}_\text{BPS}$ in terms of the effective fields and their first-derivative. Now we proceed to find the $\Phi_i$s from $\mathcal{L}-\mathcal{L}_{BPS}=\sum_{i=1}^N \Phi_i(\vec{\phi},\partial\vec{\phi})$. As we mentioned $\Phi_i$ must be positive-semidefinite function and we restric it has to be a function of $\vec{\tilde{\phi}}$ and $\partial_r\tilde{\phi}_i$ for each $i=1,\ldots,N$. The BPS equation $\Phi_i=0$ gives solutions to $\partial_r\tilde{\phi}_i$ as follows \begin{equation} \partial_r\tilde{\phi}_i= \left\{F^{(1)}_i,F^{(2)}_i,...,F^{(m)}_i\right\}, \end{equation} with $F^{(k)}_i=F^{(k)}_i(\vec{\tilde{\phi}};r)$ ($k=1,...,m$). Positive-semidefinite condition fixes $m$ to be an even number and further there must be even number of equals solutions in $\{F^{(k)}_i\}$. As an example if $m=2$ for all $i$ then $\Phi_i=0$ is a quadratic equation in $\partial_r\tilde{\phi}_i$ and so we will have $F^{(1)}_i=F^{(2)}_i$. The restriction on $\Phi_i\equiv\Phi_i(\partial_r\tilde{\phi}_i)$ forces us to rewrite the function $\mathcal{L}-\mathcal{L}_{BPS}$ into partitions $\sum_{i=1}^N\Phi_i$ explicitly. This is difficult to apply on more general forms of Lagrangian, since there exists a possibility that there are terms with $\partial\tilde{\phi}_i\partial\tilde{\phi}_j$ where $i\neq j$. Another problem is ambiguity in choosing which terms contain non-derivative of fields that should belong to which partitions $\Phi_i$. For more general situations, the BPS equations can be obtained by procedures explained in~\cite{Atmaja:2015umo} which we describe below. On a closer look, we can consider $\mathcal{L}-\mathcal{L}_\text{BPS}=0$ as a polynomial equation of first-derivative fields. Seeing it as the polynomial equation of $\partial_r\tilde{\phi}_1$, whose maximal power is $m_1$, its roots are \begin{equation} \partial_r\tilde{\phi}_1= \left\{G^{(1)}_1,G^{(2)}_1,\ldots,G^{(m_1)}_1\right\}, \end{equation} with $G^{(k)}_1=G^{(k)}_1(\vec{\tilde{\phi}},\partial_r\tilde{\phi}_2,\ldots,\partial_r\tilde{\phi}_N;r)$ and $k=1,...,m_1$. Then we have \begin{equation} \Phi_1\propto\left(\partial_r\tilde{\phi}_1-G^{(1)}_1\right)\left(\partial_r\tilde{\phi}_1-G^{(2)}_1\right)\ldots\left(\partial_r\tilde{\phi}_1-G^{(m_1)}_1\right). \end{equation} As we mentioned before here $m_1$ must be an even number and to ensure positive-definiteness at least two or more even number of roots must be equal. This will result in some constraint equations that are polynomial equations of the remaining first-derivative fields $(\partial_r\tilde{\phi}_2,\ldots,\partial_r\tilde{\phi}_N)$. Repeating the previous procedures for $\partial_r\tilde{\phi}_2$ until $\partial_r\tilde{\phi}_N$ whose $\Phi_N$ is \begin{equation} \Phi_N\propto\left(\partial_r\tilde{\phi}_N-G^{(1)}_N\right)\left(\partial_r\tilde{\phi}_N-G^{(2)}_N\right)...\left(\partial_r\tilde{\phi}_N-G^{(m_N)}_N\right), \end{equation} with $m_N$ is also an even number. Now all $G^{(k)}_N$ are only functions of $\vec{\tilde{\phi}}$ and equating some of the roots will become constraint equations that we can solve order by order for each power series of $r$. As an example let take $N=2$ and $m_1,m_2=2$. Then the constraint $G^{(1)}_1-G^{(2)}_1=0$ can be seen as a quadratic equation of $\partial_r\tilde{\phi}_2$. This give us the last constraint $G^{(1)}_2-G^{(2)}_2=0$. Since the model is valid for all $r$, we could write the constraint as $G^{(1)}_2-G^{(2)}_2=\sum_{n}a_n r^n$, where all $a_n$s are independent of $\partial_r\tilde{\phi}_1$ and $\partial_r\tilde{\phi}_2$. Then all $a_n$s need to be zero and from them we can find each $Q_i(\tilde{\phi}_i)$. Then the BPS equations for $\partial_r\tilde{\phi}_i$ can be found. We can see that this more general method is straightforward for any Lagrangian. This will be used throughout this paper, since we will later use some DBI-type Lagrangian that contains terms inside square root which is not easy to write the partitions explicitly. In \cite{Atmaja:2015umo}, with particular ansatz for the fields, writing $Q=2\pi F(f) A(a)$ is shown to be adequate for some models of vortices. Here, we show that the method is also able to do the job, at least for some known models of magnetic monopoles and dyons, using the well-known 't Hooft-Polyakov ansatz. \section{The 't Hooft-Polyakov Monopole and Julia-Zee Dyon} The model is described in a flat $(1+3)$-dimensional space-time whose Minkowskian metric is $\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$. The standard Lagrangian for BPS monopole, or the $SU(2)$ Yang-Mills-Higgs model, has the following form~\cite{Polyakov:1974ek,tHooft:1974kcl} \begin{equation}\label{eq:canonicalLagrangian} \mathcal{L}_\text{s}={1\over 2} \mathcal{D}_\mu \phi^a \mathcal{D}^\mu \phi^a -{1\over 4} F^{a\mu\nu} F^{a}_{~\mu\nu} - V(\vert\phi\vert), \end{equation} with $SU(2)$ gauge group symmetry and $\phi^a$, $a=1,2,3$, is a triplet real scalar field in adjoint representation of $SU(2)$. The potensial $V$ is a function of $|\phi|=\phi^a\phi^a$ which is invariant under $SU(2)$ gauge transformations. Here we use Einstein summation convention for repeated index. The definitions of covariant derivative and field strength tensor of the $SU(2)$ Yang-Mills gauge field are as follows \begin{subequations} \begin{align} \mathcal{D}_\mu \phi^a=&\partial_\mu \phi^a + e~\epsilon^{abc}A^b_\mu \phi^c,\\ F^a_{\mu\nu}=&\partial_\mu A^a_\nu - \partial_\nu A^a_\mu +e~\epsilon^{abc} A^b_\mu A^c_\nu, \end{align} \end{subequations} with $e$ is the gauge coupling and $\epsilon^{abc}$ is the Levi-Civita symbol. The latin indices $(a,b,c)$ denote the ``vector components'' in the vector space of $SU(2)$ algebra with generators $T_a={1\over2}\sigma_a$, where $\sigma_a$ is the Pauli's matrix. The generators satisfy commutation relation $[T_a,T_b]=i\epsilon_{abc}T_c$ and their trace is $\mathrm{tr}(T_aT_b)={1\over2}\delta_{ab}$. With these generators, the scalar field, gauge field, adjoint covariant derivative and field strength tensor can then be re-written in a compact form, respectively, as $\phi=\phi^a T_a$, $A_\mu =A^a_\mu T_a$, \begin{subequations}\label{eq:derivedLagrangian} \begin{align} \mathcal{D}_\mu \phi=&\mathcal{D}_\mu \phi^a T_a=\partial_\mu \phi -ie[A_\mu, \phi],\\ F_{\mu\nu}=&F_{\mu\nu}^aT_a=\partial_\mu A_\nu-\partial_\nu A_\mu-ie[A_\mu,A_\nu]. \end{align} \end{subequations} These lead to the Lagrangian \begin{equation} \mathcal{L}_\text{s}=\mathrm{tr} \left(\mathcal{D}_\mu \phi \mathcal{D}^\mu \phi-{1\over 2}F^{\mu\nu} F_{\mu\nu} \right)-V(|\phi|). \end{equation} Variying \eqref{eq:canonicalLagrangian} with respect to the scalar field and the gauge field yields \begin{subequations} \begin{align} \mathcal{D}_\mu\left(\mathcal{D}^\mu\phi^b\right)=&-{\partial V\over \partial \phi^b},\label{eq:EoMscalar}\\ \mathcal{D}_\nu F^{b\mu\nu}=&e\epsilon^{bca}\phi^c\mathcal{D}^\mu\phi^a,\label{eq:EoMgauge} \end{align} \end{subequations} with additional Bianchi identity \begin{equation} \mathcal{D}_\mu \tilde{F}^{a\mu\nu}=0, \end{equation} where $\tilde{F}^{a\mu\nu}={1\over2}\epsilon^{\mu\nu\kappa\lambda}{F}^a_{\kappa\lambda}$. Through out this paper, we will consider only static configurations. The difference between monopole and dyon is whether $A^a_0$ is zero or non-zero, respectively. For monopole, the Bianchi identity becomes \begin{equation} \mathcal{D}_iB^a_i=0.\label{Bianchi} \end{equation} Here $B^a_i={1\over 2}\epsilon_{ijk}F_{jk}$ and $i,j,k=1,2,3$ are the spatial indices. For dyon, $A^a_0\neq0$, there are additional equations of motion for ``electric'' part since the Gauss law is non-trivial, \begin{equation} \mathcal{D}_i E^b_i=-e~\epsilon^{bca}\phi^c\mathcal{D}_0\phi^a,\label{Gauss} \end{equation} where $E^a_i=F^a_{0i}$. We could write the energy-momentum tensor $T_{\mu\nu}$ by varying the action with respect to the space-time metric. The energy density is then given by $T_{00}$ component, \begin{align} T_{00}={1\over2}\left(\mathcal{D}_0\phi^a\mathcal{D}_0\phi^a+\mathcal{D}_i\phi^a\mathcal{D}_i\phi^a+E^a_iE^a_i+B^a_iB^a_i\right)+V(|\phi|). \end{align} In \cite{Prasad:1975kr}, it is possible to obtain the exact solutions of the Euler-Lagrange equations in the BPS limit, i.e. $V=0$ but still maintaining the asymptotic boundary conditions of $\phi$, and we define a new parameter $\alpha$ such that \begin{align} T_{00}=&{1\over2}\left(\mathcal{D}_0\phi^a\mathcal{D}_0\phi^a+\mathcal{D}_i\phi^a\mathcal{D}_i\phi^a\sin^2\alpha+E^a_iE^a_i+\mathcal{D}_i\phi^a\mathcal{D}_i\phi^a\cos^2\alpha+B^a_iB^a_i\right)\nonumber\\ =&{1\over2}\left((\mathcal{D}_0\phi^a)^2+(\mathcal{D}_i\phi^a\sin\alpha\mp E^a_i)^2+(\mathcal{D}_i\phi^a\cos\alpha\mp B^a_i)^2\right) \pm E^a_i \mathcal{D}_i\phi^a\sin\alpha\pm B^a_i \mathcal{D}_i\phi^a\cos\alpha.\label{eq:alpha} \end{align} The last two terms can be converted to total derivative \begin{subequations} \begin{align} E^a_i \mathcal{D}_i\phi^a=\partial_i(E^a_i\phi^a)-(\mathcal{D}_iE^a_i) \phi^a=\partial_i(E^a_i\phi^a),\\ B^a_i \mathcal{D}_i\phi^a=\partial_i(B^a_i\phi^a)-(\mathcal{D}_iB^a_i) \phi^a=\partial_i(B^a_i\phi^a), \end{align} \end{subequations} after employing the Gauss law (\ref{Gauss}) and Bianchi identity (\ref{Bianchi}). They are related to the ``Abelian'' electric and magnetic fields identified in~\cite{tHooft:1974kcl}, respectively. Since the total energy is $E=\int d^3x~ T_{00}$, the total derivative terms can be identified as the electric and magnetic charges accordingly \begin{subequations} \begin{align} \mathcal{Q}_E=\int dS^i E^a_i \phi^a,\\ \mathcal{Q}_B=\int dS^i B^a_i \phi^a, \end{align} \end{subequations} with $dS^i$ denoting integration over the surface of a 2-sphere at $r\to\infty$. Therefore the total energy is $E\geq\pm\left(\mathcal{Q}_E\sin\alpha+\mathcal{Q}_B\cos\alpha\right)$ since the other terms are positive semi-definite. The total energy is saturated if the BPS equations are satisfied as folows~\cite{Coleman:1976uk} \begin{subequations} \label{BPS dyons} \begin{align} \mathcal{D}_0\phi^a=0,\\ \mathcal{D}_i\phi^a\sin\alpha=&E^a_i,\\ \mathcal{D}_i\phi^a\cos\alpha=&B^a_i. \end{align} Solutions to these equations are called BPS dyons; they are particullary called BPS monopoles for $\alpha=0$. The energy of this BPS configuration is simply given by \begin{equation} E_{BPS}=\pm\left(\mathcal{Q}_E\sin\alpha+\mathcal{Q}_B\cos\alpha\right). \end{equation} \end{subequations} Adding the constant $\alpha$ contained in $\sin\alpha$ and $\cos\alpha$ is somehow a bit tricky. We will show later using BPS Lagrangian method that this constant comes naturally as a consequence of idenfitiying two of the effective fields. Employing the 't Hooft-Polyakov, together with Julia-Zee, ansatz~\cite{Polyakov:1974ek,tHooft:1974kcl,Julia:1975ff} \begin{subequations}\label{eq:ansatz} \begin{align} \phi^a&= f(r) {x^a\over r},\\ A^a_0&={j(r)\over e} {x^a\over r},\\ A^a_i&={1-a(r)\over e} \epsilon^{aij} {x^j\over r^2}, \end{align} \end{subequations} where $x^a\equiv(x,y,z)$, and $x^i\equiv(x,y,z)$ as well, denotes the Cartesian coordinate. Notice that the Levi-Civita symbol $\epsilon^{aij}$ in (\ref{eq:ansatz}) mixes the space-index and the group-index. Substituting the ansatz (\ref{eq:ansatz}) into Lagrangian \eqref{eq:canonicalLagrangian} we can arrive at the following effective Lagrangian \begin{align} \mathcal{L}_\text{s} =&- {f'^2\over 2} -\left({ af\over r}\right)^2 + {j'^2\over 2e^2} +\left({aj\over er}\right)^2 - \left({a'\over er}\right)^2 -{1\over 2} \left({a^2-1\over er^2}\right)^2 - V(f), \label{eq:effcanLag} \end{align} where $'\equiv {\partial \over \partial r}$ otherwise it means taking derivative over the argument. As shown in the effective Lagrangian above there is no dependency over angles coordinates $\phi$ and $\theta$ despite the fact that the ansatz (\ref{eq:ansatz}) depends on $\phi$ and $\theta$. Thus we may derive the Euler-Lagrange equations from the effective Lagrangian (\ref{eq:effcanLag}) which are given by \begin{subequations}\label{eq:EoMstd} \begin{align} -{1\over r^2}(r^2 f')'+{2 a^2f\over r^2}=&-V'(f),\\ -{(r^2 j')'\over er^2}+{2a^2j\over er^2}=&0,\\ {a(a^2-1)\over r^2}+a(e^2f^2-j^2)-a''=&0. \end{align} \end{subequations} Later we will also consider the case for generalize Lagrangian of (\ref{eq:canonicalLagrangian}) by adding scalar-dependent couplings to the kinetic terms as follows~\cite{Casana:2012un} \begin{equation}\label{eq:LagcanG} \mathcal{L}_\text{G}=-{1\over 4}w(|\phi|)F^a_{\mu\nu}F^{a\mu\nu}+{1\over 2}G(|\phi|)\mathcal{D}_\mu\phi^a\mathcal{D}^\mu\phi^a-V(|\phi|). \end{equation} The equations of motions are now given by \begin{subequations} \begin{align} \mathcal{D}_\mu\left(G~\mathcal{D}^\mu\phi^b\right)=&-{\partial V\over \partial \phi^b}+{1\over 2}{\partial G\over \partial \phi^b}\mathcal{D}_\mu\phi^a\mathcal{D}^\mu\phi^a-{1\over 4}{\partial w\over \partial \phi^b}F^a_{\mu\nu}F^{a\mu\nu},\label{EoMscalarG}\\ \mathcal{D}_\nu\left(w~F^{b\mu\nu}\right)=&e\epsilon^{bca}\phi^cG~\mathcal{D}^\mu\phi^a.\label{EoMgaugeG} \end{align} \end{subequations} In \cite{Casana:2012un,Casana:2013lna}, they found BPS monopole equations and a constraint equation $G=w^{-1}$. Using our method in the following sections, we obtain the similar BPS monopole equations and constraint equation. Furthermore, we generalize it to BPS dyon equations with a more general constraint equation. There are other forms of Lagrangian for BPS monopole and dyon which were presented in the Born-Infeld type of action by Nakamula and Shiraishi in \cite{Nakamula:2013rfa}. The Lagrangian for BPS monopole is different from the BPS dyon. The Lagrangians are defined such that the BPS equations (\ref{BPS dyons}) satisfy the Euler-Lagrange equations in the usual BPS limit. The Lagrangian for monopole and dyon are given respectively by\cite{Nakamula:2013rfa} \begin{align} \mathcal{L}_\text{NSm}=-b^2~\mathrm{tr}&\left( \sqrt{1-{2\over b^2} \mathcal{D}_\mu \phi \mathcal{D}^\mu \phi} \sqrt{1+{1\over b^2} F_{\mu\nu} F^{\mu\nu}} -1 \right) -V(|\phi|),\label{eq:NSmonopole}\\ \mathcal{L}_\text{NSd}=-b^2~\mathrm{tr}&\left( \left\{1-{2\over b^2} \mathcal{D}_\mu \phi \mathcal{D}^\mu \phi +{1\over b^2} F_{\mu\nu} F^{\mu\nu} -{1\over 4b^4}\left(F_{\mu\nu} \tilde{F}^{\mu\nu}\right)^2\right.\right.\nonumber\\ &\left.\left.+{4\over b^4}\tilde{F}_\mu^{~\nu} \tilde{F}^{\mu\lambda}\mathcal{D}_\nu\phi\mathcal{D}_\lambda\phi\right\}^{1/2} -1\right) -V(|\phi|),\label{eq:NSdyon} \end{align} with $b^2$ is the Born-Infeld parameter and the potential $V$ is taken to be the same as in (\ref{eq:canonicalLagrangian}). It is apparent that, even though $E^a_i=0$, $\mathcal{L}_\text{NSd}\neq\mathcal{L}_\text{NSm}$. Using the ansatz (\ref{eq:ansatz}), both Lagrangians can be effectively writen as \begin{align} \mathcal{L}_\text{NSm}=-2b^2&\left( \sqrt{1+{1\over 2b^2} \left(f'^2+{2a^2f^2\over r^2}\right)} \sqrt{1+{1\over 2b^2} \left({2a'^2\over e^2r^2}+{(a^2-1)^2\over e^2r^4}\right)} -1 \right) \nonumber\\&-V(f),\label{eq:effNSm}\\ \mathcal{L}_\text{NSd} =-2b^2&\left(\left\{ 1+{1\over 2b^2}\left( f'^2+{2a^2f^2\over r^2} +{2a'^2\over e^2r^2}+{(a^2-1)^2\over e^2r^4} -{j'^2\over e^2}-{2a^2j^2\over e^2 r^2}\right) \right.\right.\nonumber\\&\left.\left. +{1\over4b^4}\left( -\left(-{(a^2-1)j'\over e^2r^2}-{2aja'\over e^2r^2}\right)^2 +\left({(a^2-1)f'\over er^2}+{2 afa'\over er^2}\right)^2 \right)\right\}^{1/2} \right.\nonumber\\&\left.-1\right)-V(f).\label{eq:effNSd} \end{align} We can see immeditely that $\mathcal{L}_\text{NSd}(j=0)\neq\mathcal{L}_\text{NSm}$. However, by assuming the BPS equations $B^a_i=\pm\mathcal{D}_i\phi^a$ is valid beforehand we would get $\mathcal{L}_\text{NSd}=\mathcal{L}_\text{NSm}$. Hence from both Lagrangians, we could obtain the same BPS equations when we turn off the ``electric'' part for monopole. \section{BPS Equations in $SU(2)$ Yang-Mills-Higgs and Nakamula-Shiraishi Models} Here we will show that the BPS Lagrangian method~\cite{Atmaja:2015umo} can also be used to obtain the known BPS equations for monopole and dyon in the $SU(2)$ Yang-Mills model \eqref{eq:canonicalLagrangian}, and the Nakamula-Shiraishi models, \eqref{eq:NSmonopole} and \eqref{eq:NSdyon}. To simplify our calculations, from here on we will set the gauge coupling to unity, $e=1$. \subsection{BPS monopole and dyon in $SU(2)$ Yang-Mills-Higgs model} Writing the ansatz \eqref{eq:ansatz} in spherical coordinates, \begin{subequations}\label{spherical ansatz} \begin{align} \phi^a &\equiv f (\cos\varphi\sin\theta,\sin\varphi\sin\theta,\cos\theta),\\ A^a_0 &\equiv {j} (\cos\varphi\sin\theta,\sin\varphi\sin\theta,\cos\theta),\\ A^a_r &\equiv (0,0,0),~\\ A^a_\theta&\equiv {(1-a)}(\sin\varphi,-\cos\varphi,0),\\ A^a_\varphi &\equiv {(1-a)} \sin\theta (\cos\varphi\cos\theta,\sin\varphi\cos\theta,-\sin\theta), \end{align} \end{subequations} we find that there is no explicit $r$ dependent in all fields above. Therefore we propose that the BPS energy function for the case of monopole, where $j=0$, should take the following form \begin{equation} Q(a,f)=4\pi F(f)A(a).\label{eq:Q} \end{equation} Since $\int d^3x~ \mathcal{L}_{\text{BPS}}=-\int dQ$, we have the BPS Lagrangian \begin{equation} \mathcal{L}_{\text{BPS}}=-{FA'(a)\over r^2}a'-{F'(f)A\over r^2}f'.\label{eq:LagBPS} \end{equation} Before showing our results, for convenience we define through all calculations in this article $x=f',~y=a',~ Q_a=F~A'(a),$ and $Q_f=F'(f)A$. Employing $\mathcal{L}_\text{s}-\mathcal{L}_\text{BPS}=0$, where $\mathcal{L}_\text{s}$ is \eqref{eq:effcanLag} and $\mathcal{L}_\text{BPS}$ is \eqref{eq:LagBPS}, we can consider it as a quadratic equation of either $a'$ or $f'$. Here we show the roots of $f'$ (or $x$) first which are \begin{equation}\label{eq:YMH monopole 1} f'_\pm=\frac{Q_f \pm\sqrt{ Q_f^2-a^4-2 a^2 \left(f^2 r^2-1\right)-2 r^2 y (y-Q_a)-2 r^4 V-1}}{r^2}. \end{equation} The two roots will be equal, $f_+=f_-$, if the terms inside the square-root is zero, which later can be considered as a quadratic equation for $a'$ (or $y$) with roots \begin{equation}\label{eq:YMH monopole 2} a'_\pm= {1\over 2}Q_a \pm{1\over 2r}\sqrt{-2 a^4+a^2 \left(4-4 f^2 r^2\right)+Q_a^2 r^2+2 Q_f^2-4 r^4 V-2}. \end{equation} Again, we need the terms inside the square-root to be zero for two roots to be equal, $a_+=a_-$. The last equation can be written in power series of $r$, \begin{equation} \left(2 Q_f^2-2 \left(a^2-1\right)^2\right)+ \left(Q_a^2-4 a^2 f^2\right)r^2-4 V~r^4=0, \end{equation} Demanding it is valid for all values of $r$, we may take $V=0$, which is just the same BPS limit in~\cite{Prasad:1975kr}. From the terms with quadratic and zero power of $r$, we obtain \begin{align} FA'(a)=& \pm 2 a f,\\ F'(f)A=& \pm(a^2-1), \end{align} which implies \begin{equation} FA=\pm(a^2-1)f. \end{equation} Inserting this into equations (\ref{eq:YMH monopole 1}) and (\ref{eq:YMH monopole 2}), we reproduce the known BPS equations for monopole, \begin{subequations}\label{eq:knownBPS} \begin{align} f'=&\pm\frac{a^2-1}{r^2},\\ a'=&\pm af. \end{align} \end{subequations} Now let us take $j(r)\neq0$ and consider the BPS limit, $V\to 0$. In this BPS limit, we can easly see from the effective Lagrangian (\ref{eq:effcanLag}), the Euler-Lagrange equations for both fields $f$ and $j$ are equal. Therefore it is tempted to identify $j\propto f$. Let us write it explicitly as \begin{equation} j(r)=\beta f(r),\label{eq:beta} \end{equation} where $\beta$ is a real valued constant. With this identification, we can again use \eqref{eq:Q} as the BPS energy function for dyon and hece give the same BPS Lagrangian (\ref{eq:LagBPS}). Now the only difference, from the previous monopole case, is the effective Lagrangian (\ref{eq:effcanLag}) which takes a simpler form \begin{equation} \mathcal{L}_\text{s} =- \left(1-\beta^2\right)\left({f'^2\over 2} +\left({ af\over r}\right)^2\right) - \left({a'\over r}\right)^2 -{1\over 2} \left({a^2-1\over r^2}\right)^2-V. \end{equation} Here we still keep the potential $V$ and we will show later that $V$ must be equal to zero in order to get the BPS equations using the BPS Lagrangian method. Applying \eqref{eq:beta} and solving $\mathcal{L}_\text{s}-\mathcal{L}_\text{BPS}=0$ as quadratic equation for $f'$ (or $x$) give us two roots \begin{equation} f'_\pm=\frac{Q_f\pm\sqrt{\text{DD}}}{\left(1-\beta ^2\right) r^2}, \end{equation} with \begin{equation} \text{DD}=\left(\beta ^2-1\right) \left(a^4-2 a^2 \left(\left(\beta ^2-1\right) f^2 r^2+1\right)+2 r^2 y (y-Q_a)+2 r^4 V+1\right)+Q_f^2. \end{equation} Next, requiring $f'_+=f'_-$, we obtain \begin{equation} a'_\pm=\frac{1}{2} \left(Q_a\pm\sqrt{\frac{\text{DDD}}{\left(\beta ^2-1\right) r^2}}\right), \end{equation} where we arrange $\text {DDD}$ in power series of $r$, i.e. \begin{align} \text {DDD}=2\left(\left(1-a^2\right)^2\left(1-\beta^2\right)-Q_f^2\right)+\left(1-\beta ^2\right)\left(4 a^2 \left(1-\beta ^2\right) f^2-Q_a^2\right) r^2+4 V\left(1-\beta ^2\right)r^4. \end{align} Again, $a'_-=a'_+$, we get $DDD=0$. Solving the last equation, which must be valid for all values of $r$, we conclude $V=0$ from $r^4$-terms, for non-trivial solution, and from the remaining terms we have \begin{subequations} \begin{align} FA'(a)=& \pm 2 af \sqrt{1-\beta ^2},\\ F'(f)A=& \pm \left(a^2-1\right) \sqrt{1-\beta ^2}, \end{align} \end{subequations} which give us \begin{equation} FA=\pm \left(a^2-1\right)f \sqrt{1-\beta ^2}. \end{equation} The BPS equations are then \begin{subequations}\label{eq:knownBPSdyon} \begin{align} f'\sqrt{1-\beta ^2}=&\pm\frac{a^2-1}{r^2},\\ a'=&\pm af\sqrt{1-\beta ^2}. \end{align} \end{subequations} Since $f'$ and $a'$ are real-valued, $\beta$ should take values $|\beta|< 1$. They become the BPS equations for monopole \eqref{eq:knownBPS} when we set $\beta=0$. We can see that this constant is analogous to the constant $\alpha$, or precisely $\beta=-\sin\alpha$, in \eqref{eq:alpha}, see~\cite{weinberg2012classical} for detail. Substituting $\beta=-\sin\alpha$ into equations \eqref{eq:knownBPSdyon}, we get the same BPS equations as in \cite{Prasad:1975kr,Coleman:1976uk}. Here we can see the constant $\beta$ is naturally bounded as required by the BPS equations (\ref{eq:knownBPSdyon}). \subsection{BPS monopole and dyon in Nakamula-Shiraishi model} In this subsection we will show that the Lagrangians (\ref{eq:NSmonopole}) and (\ref{eq:NSdyon}) of Nakamula-Shiraishi model do indeed posses the BPS equations \eqref{eq:knownBPS} (and \eqref{eq:knownBPSdyon}) respectively after employing the BPS Lagrangian method. Substituting \eqref{eq:effNSm} and \eqref{eq:Q} into $\mathcal{L}_\text{NSm}-\mathcal{L}_\text{BPS}=0$ and following the same procedures as the previous subsection give us the roots of $a'$, \begin{align} a'_\pm=\frac{Q_a r^4 \left(2 b^2-V\right)+Q_a Q_f r^2 x\pm\sqrt{r^2 \left(2 a^2 f^2+r^2 \left(2 b^2+x^2\right)\right) \text{DD}}}{r^2 \left(4 a^2 f^2+2 r^2 \left(2 b^2+x^2\right)-Q_a^2\right)}, \end{align} where \begin{align} \text{DD}=&-4 a^6 f^2+a^4 \left(-2 r^2 \left(2 b^2+x^2\right)+8 f^2+Q_a^2\right)\nonumber\\ &-2 a^2 \left(f^2 \left(4 b^2 r^4+2\right)-2 r^2 \left(2 b^2+x^2\right)+Q_a^2\right)\nonumber\\ &+2 b^2 r^4 \left(Q_a^2+4 Q_f x-2 r^2 \left(2 V+x^2\right)\right)-4 b^2 r^2+Q_a^2\nonumber\\ &+2 r^2 \left(-Q_f x+r^2 V+x\right) \left(r^2 V-(Q_f+1) x\right). \end{align} Solving $\text{DD}=0$ give us \begin{align} f'_\pm=\frac{2 Q_f r^4 \left(2 b^2-V\right)\pm \sqrt{-2r^2 \left(\left(a^2-1\right)^2+2 b^2 r^4\right) \text{DDD}}}{2 r^2 \left(a^4-2 a^2+2 b^2 r^4-Q_f^2+1\right)}, \end{align} where \begin{align}\label{4b^2} \text{DDD}=&2 b^2 r^4 \left(4 a^2 f^2-Q_a^2\right)+4 b^2 r^2 \left(\left(a^2-1\right)^2-Q_f^2\right)\nonumber\\ &+\left(\left(a^2-1\right)^2-Q_f^2\right) \left(4 a^2 f^2-Q_a^2\right)+2 r^6 V \left(4 b^2-V\right). \end{align} Then the last equation $\text{DDD}=0$ give us $V=0$, or $V=4b^2$, \begin{subequations} \begin{align} FA'(a)=&\pm 2af,\\ F'(f)A=&\pm (a^2-1), \end{align} \end{subequations} which again give us \begin{equation} FA=\pm(a^2-1)f, \end{equation} and thus we have $a'$ and $f'$, with $V=0$, \begin{subequations} \begin{align} f'=&\pm\frac{a^2-1}{r^2},\\ a'=&\pm af. \end{align} \end{subequations} the same BPS equations \eqref{eq:knownBPS} for monopole. The other choice of potential $V=4b^2$ will result the same BPS equations with opposite sign relative to the BPS equations of $V=0$, \begin{subequations} \begin{align} f'=&\mp\frac{a^2-1}{r^2},\\ a'=&\mp af. \end{align} \end{subequations} For dyon, using the same identification \eqref{eq:beta}, we have the effective Lagrangian \eqref{eq:effNSd} shortened to \begin{align} \mathcal{L}_\text{NSd}=-2b^2&\left(\left\{ 1+{1-\beta^2\over 2b^2}\left( f'^2+{2a^2f^2\over r^2}\right) +{1\over 2b^2}\left({2a'^2\over r^2}+{(a^2-1)^2\over r^4}\right) \right.\right.\nonumber\\&\left.\left. +{1-\beta^2\over4b^4}\left({(a^2-1)f'\over r^2}+{2afa'\over r^2}\right)^2 \right\}^{1/2} -1\right)-V. \end{align} Equating the above effective Lagrangian with $\mathcal{L}_\text{BPS}$, using the same BPS energy density \eqref{eq:Q}, and solving this for $a'$ give us \begin{align} a'_\pm=\frac{-2 a^3 \beta ^2 f x+2 a^3 f x+2 a \beta ^2 f x-2 a f x-2 b^2 Q_a r^2-Q_a Q_f x+Q_a r^2 V\pm\sqrt{\text{DD}}}{4 a^2 \left(\beta ^2-1\right) f^2-4 b^2 r^2+Q_a^2}, \end{align} where \begin{align} \text{DD}=&\left(2 a \left(a^2-1\right) \left(\beta ^2-1\right) f x+2 b^2 Q_a r^2+Q_a \left(Q_f x-r^2 V\right)\right)^2\nonumber\\ +&\left(4 a^2 \left(\beta ^2-1\right) f^2-4 b^2 r^2+Q_a^2\right) \times\nonumber\\ &\left[2 b^2 \left(a^4-2 a^2 \left(\left(\beta ^2-1\right) f^2 r^2+1\right)-2 Q_f r^2 x+r^4 \left(2 V-\beta ^2 x^2+x^2\right)+1\right) \right.\nonumber\\ &\left.-x^2 \left(a^4 \left(\beta ^2-1\right)-2 a^2 \left(\beta ^2-1\right)+\beta ^2+Q_f^2-1\right)+2 Q_f r^2 V x-r^4 V^2\right]. \end{align} Solving $\text{DD}=0$ for $f'$ give us \begin{align} f'_\pm=&\frac{\text{K}\pm\sqrt{2}\sqrt{\text{M}~\text{DDD}}}{\text{L}} \end{align} where \begin{align} \text{K}=&2 a^3 \left(\beta ^2-1\right) f Q_a r^2 \left(2 b^2-V\right)-4 a^2 \left(\beta ^2-1\right) f^2 Q_f r^2 \left(2 b^2-V\right) \nonumber\\& -2 a \left(\beta ^2-1\right) f Q_a r^2 \left(2 b^2-V\right)+8 b^4 Q_f r^4-4 b^2 Q_f r^4 V,\\ \text{L}=&-4 a \left(a^2-1\right) \left(\beta ^2-1\right) f Q_a Q_f\nonumber\\ &-4 \left[a^4 b^2 \left(\beta ^2-1\right) r^2-a^2 \left(\beta ^2-1\right) \left(f^2 \left(2 b^2 \left(\beta ^2-1\right) r^4+Q_f^2\right)+2 b^2 r^2\right) \right.\nonumber\\&\left. +b^2 r^2 \left(\left(\beta ^2-1\right) \left(2 b^2 r^4+1\right)+Q_f^2\right)\right] \nonumber\\ &+\left(\beta ^2-1\right) Q_a^2 \left(a^4-2 a^2+2 b^2 r^4+1\right),\\ \text{M}=&-b^2 \left(a^4-2 a^2 \left(\left(\beta ^2-1\right) f^2 r^2+1\right)+2 b^2 r^4+1\right) \left(4 a^2 \left(\beta ^2-1\right) f^2-4 b^2 r^2+Q_a^2\right),\\ \text{DDD}=&-2 r^4 \left(b^2 \left(\beta ^2-1\right) \left(4 a^2 \left(\beta ^2-1\right) f^2+Q_a^2\right)\right)-\left(\beta ^2-1\right) \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)^2\nonumber\\ &+4 b^2 r^2 \left(a^4 \left(\beta ^2-1\right)-2 a^2 \left(\beta ^2-1\right)+\beta ^2+Q_f^2-1\right)+2 \left(\beta ^2-1\right) r^6 V \left(4 b^2-V\right). \end{align} We may set $M=0$, but this will imply $b^2=0$ which is not what we want. Requiring $\text{DDD}=0$ valid for all values of $r$, the terms with $r^6$ give us $V=0$ or $V=4b^2$. The terms with $r^0$ imply \begin{equation} FA'(a)= \frac{2 a f F'(f)A}{a^2-1}. \end{equation} This is indeed solved by the remaining terms which imply \begin{align} FA'(a)=&\pm 2af \sqrt{1-\beta ^2},\\ F'(f)A=&\pm \left(a^2-1\right) \sqrt{1-\beta ^2}. \end{align} This again give us \begin{equation} FA=\pm \left(a^2-1\right)f \sqrt{1-\beta ^2}, \end{equation} hence, for $V=0$, \begin{subequations} \begin{align} f'=&\pm\frac{a^2-1}{\sqrt{1-\beta ^2} r^2},\\ a'=&\pm af\sqrt{1-\beta ^2}, \end{align} \end{subequations} the same BPS equations \eqref{eq:knownBPSdyon} for dyon. Similar to the monopole case choosing $V=4b^2$ will switch the sign in the BPS equations. It is apparent that it the limit of $\beta\to0$, the BPS equations for dyon becomes the ones for monopole. This indicates that in the BPS limit and $\beta\to0$, $\mathcal{L}_\text{NSd}\to\mathcal{L}_\text{NSm}$, since in general, even though in the limit of $\beta\to0$, $\mathcal{L}_\text{NSd}\slashed{\to}\mathcal{L}_\text{NSm}$. Now we know that the method works. In the next sections, we use it in some generalized Lagrangian whose BPS equations, for monopole or dyon, may or may not be known. \section{BPS Equations in Generalized $SU(2)$ Yang-Mills-Higgs Model} In this section, we use the Lagrangian \eqref{eq:LagcanG} whose its effective Lagrangian is given by \begin{equation}\label{eq:gencanLag} \mathcal{L}_\text{G}=-G \left(\frac{f'^2}{2}+\frac{a^2 f^2}{r^2}\right)+w \left(\frac{j'^2}{2}+\frac{a^2 j^2}{r^2}\right)-w \left(\frac{a'^2}{r^2}+\frac{\left(a^2-1\right)^2}{2 r^4}\right)-V. \end{equation} We will see later it turns out that $G$ and $w$ are related to each other by some constraint equations. \subsection{BPS monopole case} In this case, the BPS equations are already known \cite{Casana:2012un,Casana:2013lna}. Setting $j=0$ and employing $\mathcal{L}_\text{G}-\mathcal{L}_\text{BPS}=0$ we get \begin{equation} f'_\pm=\frac{Q_f r^2\pm\sqrt{-r^4 \left(G \left(a^4 w+2 a^2 \left(f^2 G r^2-w\right)+2 r^2 y (w y-Q_a)+2 r^4 V+w\right)-Q_f^2\right)}}{G r^4}, \end{equation} and from $f'_+=f'_-$ we have the roots of $a'$ (or $y$) \begin{align} a'_\pm=&\frac{G Q_a r^2-\sqrt{G r^2 \left\{G \left(Q_a^2 r^2-2 w \left(2 r^2 \left(a^2 f^2 G+r^2 V\right)+\left(a^2-1\right)^2 w\right)\right)+2 Q_f^2 w\right\}}}{2 G r^2 w}. \end{align} The terms inside the curly bracket in the square root must be zero in which after rearranging in power series of $r$ \begin{equation} 2 w \left(Q_f^2-\left(a^2-1\right)^2 G w\right)+G r^2 \left(Q_a^2-4 a^2 f^2 G w\right)-4 r^4 (G V w)=0, \end{equation} we obtain $V=0$, \begin{align} FA'(a)=&\pm 2af \sqrt{Gw},\\ F'(f)A=&\pm \left(a^2-1\right) \sqrt{Gw}. \end{align} These imply \begin{equation} {\partial\over \partial f}\left( f \sqrt{G w}\right)=\sqrt{G w} \end{equation} , and hence \begin{equation} w={c\over G},\label{monopole constraint} \end{equation} where $c$ is a positive constant. The BPS equations are given by \begin{subequations}\label{eq:GenBPSm} \begin{align} f'=&\pm\frac{\left(a^2-1\right)}{r^2}\sqrt{w\over G},\\ a'=&\pm a f\sqrt{G\over w}, \end{align} \end{subequations} with a constraint equation $w~G=c$, where $c$ is a positive constant. This constant can be fixed to one, $c=1$, by recalling that in the corresponding non-generalized version, in which $G=w=1$, we should get back the same BPS equations of (\ref{eq:knownBPS}). \subsection{BPS dyon case} As previously setting $j=\beta f$ and employing $\mathcal{L}_\text{G}-\mathcal{L}_\text{BPS}=0$ we get \begin{equation} f'_\pm=\frac{Q_f r^2\pm\sqrt{r^4 \text{DD}}}{r^4 \left(G-\beta ^2 w\right)}, \end{equation} with \begin{equation} \text{DD}=Q_f^2-\left(G-\beta ^2 w\right) \left(a^4 w+2 a^2 \left(f^2 r^2 \left(G-\beta ^2 w\right)-w\right)+2 r^2 y (w y-Q_a)+2 r^4 V+w\right), \end{equation} and from $\text{DD}=0$ we have the roots of $a'$ \begin{align} a'_\pm=&{Q_a\pm \sqrt{\text{DDD}\over r^2 \left(G-\beta ^2 w\right)}\over 2w}, \end{align} where \begin{align} \text{DDD}=&r^2 \left(G-\beta ^2 w\right) \left(4 a^2 f^2 w \left(\beta ^2 w-G\right)+Q_a^2\right)\nonumber\\ &+2 w \left(\left(a^2-1\right)^2 w \left(\beta ^2 w-G\right)+Q_f^2\right)+4 r^4 V w \left(\beta ^2 w-G\right). \end{align} Requiring $\text{DDD}=0$ we obtain $V=0$, \begin{align} FA'(a)=&\pm 2 a f \sqrt{w\left(G-\beta ^2 w\right)},\\ F'(f)A=&\pm \left(a^2-1\right) \sqrt{w\left(G-\beta ^2 w\right)} . \end{align} Similar to the monopole case these imply \begin{equation} {w(G-\beta ^2 w)}=c,\label{dyon constraint} \end{equation} where $c$ is a positive constant and it can also be fixed to $c=1-\beta^2$ demanding that at $G=w=1$ we should get the same BPS equations (\ref{eq:knownBPSdyon}). At $\beta\to0$, we get back the constraint equation (\ref{monopole constraint}) for monopole case. These give us the BPS equations \begin{subequations}\label{eq:GenBPSd} \begin{align} f'=&\pm\frac{\left(a^2-1\right)}{r^2}\sqrt{w\over G-\beta ^2 w},\\ a'=&\pm a f \sqrt{G-\beta ^2 w\over w}, \end{align} \end{subequations} in which at $\beta\to0$ we again get back the BPS equations for monopole case \eqref{eq:GenBPSm}. \section{BPS Equations in Generalized Nakamula-Shiraishi Model} Here we present the generalized version of the Nakamula-Shiraishi models (\ref{eq:NSmonopole}) and (\ref{eq:NSdyon}) for both monopole and dyon respectively. \subsection{BPS monopole case} For a generalized version of (\ref{eq:NSmonopole}) is defined by \begin{align} \label{NSMonopole} \mathcal{L}_\text{NSmG}=&-b^2\mathrm{tr}\left( \sqrt{1-{2\over b^2}G(|\phi|) \mathcal{D}_\mu \phi \mathcal{D}^\mu \phi} \sqrt{1+{1\over b^2}w(|\phi|) \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu}} -1 \right) -V(|\phi|), \end{align} where after inserting the ansatz, we write its effective Lagrangian as \begin{align} \label{NSMonopoleEff} \mathcal{L}_\text{NSmG}=&-2b^2\left( \sqrt{1+{G\over 2b^2} \left(f'^2+{2a^2f^2\over r^2}\right)} \sqrt{1+{w\over 2b^2} \left({2a'^2\over r^2}+{(a^2-1)^2\over r^4}\right)} -1\right) -V. \end{align} Using the similar BPS Lagrangian (\ref{eq:LagBPS}), we solve $\mathcal{L}_\text{NSmG}-\mathcal{L}_{\text{BPS}}=0$ as a quadratic equation of $a'$ (or $y$) first as such the roots are given by \begin{align} a'=\frac{Q_a r^4 \left(2 b^2-V\right)+Q_a Q_f r^2 x\pm\sqrt{r^2 \left(2 a^2 f^2 G+r^2 \left(2 b^2+G x^2\right)\right) \text {DD}}}{r^2 \left(2 w \left(2 a^2 f^2 G+r^2 \left(2 b^2+G x^2\right)\right)-Q_a^2\right)}, \end{align} with \begin{align} \text {DD}=&w \left(\left(a^2-1\right)^2 \left(Q_a^2-2 G w \left(2 a^2 f^2+r^2 x^2\right)\right)+2 \left(r^3 V-Q_f r x\right)^2\right) \nonumber\\ &+2 b^2 r^2 \left(Q_a^2 r^2-2 w \left(a^4 w+2 a^2 \left(f^2 G r^2-w\right)+r^4 \left(G x^2+2 V\right)-2 Q_f r^2 x+w\right)\right). \end{align} Taking $\text {DD}=0$, we obtain the roots for $f'$, \begin{align} f'=\frac{2Q_f r^4 w \left(2 b^2-V\right)\pm \sqrt{2r^2 w \left(\left(a^2-1\right)^2 w+2 b^2 r^4\right) \text {DDD}}}{2 r^2 w \left(\left(a^2-1\right)^2 G w+2 b^2 G r^4-Q_f^2\right)}, \end{align} with \begin{align} \text {DDD}=&2 b^2 G r^4 \left(Q_a^2-4 a^2 f^2 G w\right)+4 b^2 r^2 w \left(Q_f^2-\left(a^2-1\right)^2 G w\right)\nonumber\\ &+\left(Q_f^2-\left(a^2-1\right)^2 G w\right) \left(4 a^2 f^2 G w-Q_a^2\right)+2 G r^6 V w \left(V-4 b^2\right). \end{align} Requiring $\text {DDD}=0$, we obtain from the terms with $r^6$ that $V=0$ or $V=4b^2$. The remaining terms are also zero if $Q_a=\pm 2 af \sqrt{G w}$ and $Q_f=\pm\left(a^2-1\right) \sqrt{G w}$. These again imply \begin{equation} G={1\over w}, \end{equation} which is equal to the constraint equation (\ref{monopole constraint}) for monopole in Generalized $SU(2)$ Yang-Mills-Higgs model. Then the BPS equations, with $V=0$, are \begin{subequations} \begin{align} f'=&\pm\frac{\left(a^2-1\right)}{r^2} \sqrt{w\over G},\\ a'=&\pm a f \sqrt{G\over w}, \end{align} \end{subequations} which are equal to BPS equations \eqref{eq:GenBPSm} for monopole in the Generalized $SU(2)$ Yang-Mills-Higgs model. \subsection{BPS dyon case} The generalization of Lagrangian (\ref{eq:NSdyon}) is defined as \begin{align} \label{NSDyon} \mathcal{L}_\text{NSdG}=-b^2\mathrm{tr}&\left( \left\{1-{2\over b^2}G(|\phi|) \mathcal{D}_\mu \phi \mathcal{D}^\mu \phi +{1\over b^2}w(|\phi|) \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu} -{1\over 4b^4}G_1(|\phi|) \left(\mathcal{F}_{\mu\nu} \tilde{\mathcal{F}}^{\mu\nu}\right)^2\right.\right.\nonumber\\ &\left.\left.+{4\over b^4}G_2(|\phi|) \tilde{\mathcal{F}}_\mu^{~\nu} \tilde{\mathcal{F}}^{\mu\lambda}\mathcal{D}_\nu\phi\mathcal{D}_\lambda\phi\right\}^{1/2} -1\right) -V(|\phi|). \end{align} Employing the relation $j(f)=\beta f$, its effective Lagrangian is \begin{align} \label{NSDyonEff} \mathcal{L}_\text{NSdG}=-2b^2&\left(\left\{ 1+{G- w\beta^2\over 2b^2}\left( f'^2+{2a^2f^2\over r^2}\right) +{w\over 2b^2}\left({2a'^2\over r^2}+{(a^2-1)^2\over r^4}\right) \right.\right.\nonumber\\&\left.\left. +{G_2-G_1\beta^2\over4b^4}\left({(a^2-1)f'\over r^2}+{2afa'\over r^2}\right)^2 \right\}^{1/2} -1\right)-V. \end{align} Employing $\mathcal{L}_\text{NSdG}-\mathcal{L}_{\text{BPS}}=0$, with the same BPS Lagrangian (\ref{eq:LagBPS}), and solving it as a quadratic equation of $a'$ first we get \begin{align} a'=\frac{2 a^3 \beta ^2 f G_1 x-2 a^3 f G_2 x-2 a \beta ^2 f G_1 x+2 a f G_2 x+2 b^2 Q_a r^2+Q_a Q_f x-Q_a r^2 V \pm\frac{1}{2} \sqrt{\text{DD}}}{4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)+4 b^2 w r^2-Q_a^2}, \end{align} where \begin{align} \text{DD}=&\left(-4 a \left(a^2-1\right) f x \left(G_2-\beta ^2 G_1\right)+4 b^2 Q_a r^2+Q_a \left(2 Q_f x-2 r^2 V\right)\right)^2 \nonumber\\ &-4 \left(-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)-4 b^2 w r^2+Q_a^2\right)\text{H},\\ \text{H}=&x^2 \left(\left(a^2-1\right)^2 \beta ^2 G_1-\left(a^2-1\right)^2 G_2+Q_f^2\right)-2 b^2 \text{J}-2 Q_f r^2 V x+r^4 V^2,\\ \text{J}=&r^2 \left(2 a^2 f^2 G-2Q_f x+r^2 \left(2 V+G x^2\right)\right)+w \left(a^4-2 a^2 \left(\beta ^2 f^2 r^2+1\right)-\beta ^2 r^4 x^2+1\right). \end{align} Then from $\text {DD}=0$ we have a quadratic equation of $f'$ whose roots are \begin{align} f'=\frac{\text {K}\pm {1\over 2}\sqrt{\text {DDD}}}{\text {L}}, \end{align} where \begin{align} \text {K}=&-4 a^3 b^2 \beta ^2 f G_1 Q_a r^2+4 a^3 b^2 f G_2 Q_a r^2+2 a^3 \beta ^2 f G_1 Q_a r^2 V-2 a^3 f G_2 Q_a r^2 V\nonumber\\ &+8 a^2 b^2 \beta ^2 f^2 G_1 Q_f r^2-8 a^2 b^2 f^2 G_2 Q_f r^2-4 a^2 \beta ^2 f^2 G_1 Q_f r^2 V+4 a^2 f^2 G_2 Q_f r^2 V\nonumber\\ &+4 a b^2 \beta ^2 f G_1 Q_a r^2-4 a b^2 f G_2 Q_a r^2-2 a \beta ^2 f G_1 Q_a r^2 V+2 a f G_2 Q_a r^2 V\nonumber\\ &-8 b^4 w Q_f r^4+4 b^2 w Q_f r^4 V,\\ \text {L}=&-\beta ^2 G_1 \left(a^2 (-Q_a)+2 a f Q_f+Q_a\right)^2+2 b^2 r^2 \text {M}+G_2 ~\text {N}+8 b^4 w r^6 \left(\beta ^2 w-G\right), \end{align} where in L we define M and N as \begin{align} \text {M}=&r^2 G \left(4 a^2 \beta ^2 f^2 G_1+Q_a^2\right)+w \left(2 Q_f^2-\beta ^2 \left(Q_a^2 r^2-2 G_1 \left(a^4-2 a^2 \left(\beta ^2 f^2 r^2+1\right)+1\right)\right)\right),\\ \text {N}=&-4 a^4 b^2 w r^2+4 a^2 \left(f^2 \left(Q_f^2-2 b^2 r^4 \left(G-\beta ^2 w\right)\right)+2 b^2 w r^2\right)-4 \left(a^2-1\right) a f Q_a Q_f\nonumber\\ &+\left(a^2-1\right)^2 Q_a^2-4 b^2 w r^2, \end{align} and \begin{subequations}\label{eq:complicatedDDD} \begin{align} \text {DDD}=T_0-8T_2r^2+8T_4r^4+16b^2T_6r^6+T_8r^8-32T_{10}r^{10}+T_{12}r^{12}, \end{align} where \begin{align} T_0=&8 \left(a^2-1\right)^2 b^2 w \left(G_2-\beta ^2 G_1\right) \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)^2 \left(4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)-Q_a^2\right), \end{align} \begin{align} T_2=&-4 \left(a^2-1\right)^2 b^4 w^2 \left(\left(a^2-1\right)^2 \beta ^2 G_1-\left(a^2-1\right)^2 G_2+Q_f^2\right) \left(4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)-Q_a^2\right)\nonumber\\ &-b^2 \left(G_2-\beta ^2 G_1\right) \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)^2 \times\nonumber\\ &\left(4 \left(a^2-1\right)^2 b^2 w^2-2 a^2 f^2 \left(G-\beta ^2 w\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right)\right), \end{align} \begin{align} T_4=&2 \left(a^2-1\right)^2 b^4 w \left(\beta ^2 w-G\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right)^2\nonumber\\ &+2 a^2 f^2 \left(V-2 b^2\right)^2 \left(G_2-\beta ^2 G_1\right)^2 \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)^2\nonumber\\ &+4 b^4 w \left(\left(a^2-1\right)^2 \beta ^2 G_1-\left(a^2-1\right)^2 G_2+Q_f^2\right)\times\nonumber\\ &\left(4 \left(a^2-1\right)^2 b^2 w^2-2 a^2 f^2 \left(G-\beta ^2 w\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right)\right)\nonumber\\ &-2 \left(G_2-\beta ^2 G_1\right) \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)^2 \times\nonumber\\ &\left(b^4 \left(4 a^2 f^2 w \left(\beta ^2 w-G\right)+Q_a^2\right)-4 a^2 b^2 f^2 V \left(G_2-\beta ^2 G_1\right)+a^2 f^2 V^2 \left(G_2-\beta ^2 G_1\right)\right), \end{align} \begin{align} T_6=&-4 \left(a^2-1\right)^2 b^4 w^2 \left(G-\beta ^2 w\right) \left(4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)-Q_a^2\right)\nonumber\\ &+b^2 \left(G-\beta ^2 w\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right) \times\nonumber\\ &\left(4 \left(a^2-1\right)^2 b^2 w^2-2 a^2 f^2 \left(G-\beta ^2 w\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right)\right)\nonumber\\ &-4 a f w Q_f \left(V-2 b^2\right)^2 \left(G_2-\beta ^2 G_1\right) \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)\nonumber\\ &+w V \left(4 b^2-V\right) \left(G_2-\beta ^2 G_1\right) \left(\left(a^2-1\right) Q_a-2 a f Q_f\right)^2\nonumber\\ &+4 w \left(\left(a^2-1\right)^2 \beta ^2 (-G_1)+\left(a^2-1\right)^2 G_2-Q_f^2\right) \times\nonumber\\ &\left(b^4 \left(4 a^2 f^2 w \left(\beta ^2 w-G\right)+Q_a^2\right)-4 a^2 b^2 f^2 V \left(G_2-\beta ^2 G_1\right)+a^2 f^2 V^2 \left(G_2-\beta ^2 G_1\right)\right), \end{align} \begin{align} T_8=&64 b^4 w^2 Q_f^2 \left(V-2 b^2\right)^2-8 \left(8 b^6 w \left(G-\beta ^2 w\right) \left(4 \left(a^2-1\right)^2 b^2 w^2 \right.\right.\nonumber\\&\left.\left. -2 a^2 f^2 \left(G-\beta ^2 w\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right)\right) \right.\nonumber\\&\left. +4 b^2 \left(G-\beta ^2 w\right) \left(Q_a^2-4 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right) \left(b^4 \left(4 a^2 f^2 w \left(\beta ^2 w-G\right)+Q_a^2\right) \right.\right.\nonumber\\&\left.\left. -4 a^2 b^2 f^2 V \left(G_2-\beta ^2 G_1\right)+a^2 f^2 V^2 \left(G_2-\beta ^2 G_1\right)\right) \right.\nonumber\\&\left. +8 b^4 w^2 V \left(4 b^2-V\right) \left(\left(a^2-1\right)^2 \beta ^2 (-G_1)+\left(a^2-1\right)^2 G_2-Q_f^2\right)\right), \end{align} \begin{align} T_{10}=&b^4 w \left(\beta ^2 w-G\right) \left(4 b^4 \left(4 a^2 f^2 w \left(\beta ^2 w-G\right)+Q_a^2\right)+4 b^2 V \left(Q_a^2-8 a^2 f^2 \left(G_2-\beta ^2 G_1\right)\right) \right.\nonumber\\&\left. +V^2 \left(8 a^2 f^2 \left(G_2-\beta ^2 G_1\right)-Q_a^2\right)\right), \end{align} \begin{align} T_{12}=128 b^6 w^2 V \left(4 b^2-V\right) \left(\beta ^2 w-G\right). \end{align} \end{subequations} in order to find $V,Q_f,Q_a,G_1$ and $G_2$, we have to solve equation $\text {DDD}=0$. Since the model is valid for all $r$, then each $T_0$ until $T_{12}$ must be equal to zero. From $T_{12}=0$ we need either $V=0$ or $V=4b^2$. This verify that the BPS limit is indeed needed to obtain the BPS equations. Putting $V=0$ into $\text {DDD}$ we simplify a little the equations \eqref{eq:complicatedDDD} above. From $T_0=0$ we have \begin{equation} Q_f= \frac{\left(a^2-1\right) Q_a}{2 a f}, \end{equation} which we input into $\text {DDD}$ again. From $T_2=0$ we obtain $Q_a= \pm 2 a f \sqrt{G_2-\beta ^2 G_1}$. Now we will input each into twos separate cases. \begin{enumerate} \item Setting $Q_a= - 2 a f \sqrt{G_2-\beta ^2 G_1}$, only $T_8$ and $T_{10}$ are not vanished. Both can vanish if $\beta ^2 (w^2- G_1)+(G_2-w G)=0$ hence we have $G_2-\beta ^2G_1=wG-w^2\beta^2$. \item Setting $Q_a= 2 a f \sqrt{G_2-\beta ^2 G_1}$, we also arrive at the same destination. \end{enumerate} From these steps, we obtain that \begin{align} FA'(a)=&\pm2 a f \sqrt{w\left(G-\beta ^2 w\right)},\\ F'(f)A=&\pm(a^2-1) \sqrt{w\left(G-\beta ^2 w\right)}, \end{align} which again imply that \begin{equation} w\left(G-\beta ^2 w\right)=1-\beta^2, \end{equation} which is equal to the constraint equation (\ref{dyon constraint}) for dyon in the Generalized $SU(2)$ Yang-Mills-Higgs model. Substituting everything, we obtain the BPS equations, with $V=0$, \begin{align} f'&=\pm\frac{\left(a^2-1\right)}{r^2}\sqrt{w\over G-\beta ^2 w},\\ a'&=\pm a f \sqrt{G-\beta ^2 w\over w}, \end{align} which is again equal to the BPS equations \eqref{eq:GenBPSd} for dyon in the Generalized $SU(2)$ Yang-Mills-Higgs model. \section{Discussion} We have shown that the BPS Lagrangian method, which was used before in~\cite{Atmaja:2015umo} for BPS vortex, can also be applied to the case of BPS monopole and dyon in $SU(2)$ Yang-Mills-Higgs model (\ref{eq:canonicalLagrangian}). One main reason is because the effective Lagrangian (\ref{eq:effcanLag}) only depends on radial coordinate similar to the case of BPS vortex. We also took similar ansatz for the BPS Lagrangian (\ref{eq:LagBPS}) in which the BPS energy function $Q$ (\ref{eq:Q}) does not depend on the radial coordinate explicitly and it is a separable function of $f$ and $a$. This due to no explicit dependent over radial coodinate on the ansatz for the fields written in spherical coordinates as in (\ref{spherical ansatz}). The BPS dyon could be obtained by identifiying the effective field of the time-component gauge fields $j$ to be propotional with the effective field of the scalars $f$ by a constant $\beta$, $j=\beta f$. This identification seems natural by realizing that both effective fields give the same Euler-Lagrange equation in the BPS limit. Fortunately we found that the BPS Lagrangian method forced us to take this limit when solving the last equation with explicit power of radial coodinate order by order, which are also the case for all other models considered in this article. In this article we used this simple identification which gives us the known result of BPS dyon~\cite{Prasad:1975kr}. It turns out that the constant $\beta$ takes values $|\beta|< 1$ and it will be equal to BPS dyon in~\cite{Prasad:1975kr,Coleman:1976uk} if we set $\beta=-\sin{\alpha}$, with $\alpha$ is a constant. There is also a possibility where the both effective fields are independent, or having no simple relation, but this will be discussed elsewhere. Appliying the BPS Lagrangian method to Born-Infeld extensions of the $SU(2)$ Yang-Mills-Higgs model, which is called Nakumula-Shiraishi models, we obtained the same BPS equations as shown in~\cite{Nakamula:2013rfa}. Those BPS equations switch the sign if we shift the potential to a non-zero constant $4b^2$, $V\to V+ 4b^2$ in which the BPS limit now becomes $V\to 4b^2$, as shown below the equation (\ref{4b^2}). Therefore adding the energy density to a constant $4b^2$ seems to be related to a transition from monopole(dyon) to anti-monopole(anti-dyon) and vice versa. Since this transition is between BPS monopoles, or dyons, it would be interesting to study continuous transitions by adding the energy density slowly from $0$ to $4b^2$, which we would guest to be transition from BPS monopole(dyon) to Non-BPS monopole and then to the corresponding BPS anti-monopole(anti-dyon) with higher energy. This transition also appears in all Born-Infeld type of action discussed in this article and we wonder if this transition is generic in all other type of Born-Infeld actions at least with the ones posses BPS monopole(dyon) in the BPS limit. However, this kind of transition does not appear in $SU(2)$ Yang-Mills-Higgs model and its Generalized version since it would correspond to taking $b\to\infty$ in the Nakamula-Shiraishi models, which means adding an infinite potential energy to the Lagrangians. In particular case of monopole Lagrangian (\ref{eq:NSmonopole}), we might try to use the identification $j=\beta f$, as previuosly, into the Lagrangian (\ref{eq:NSmonopole}) and look for the BPS equations for dyon from it. However, there is no justification for this identification because the Euler-Lagrange equations for $f$ and $j$ are not identical even after substituting $j=\beta f$ into both Euler-Lagrange equations in the BPS limit. We might also try to consider $f$ and $j$ independently by adding a term that is proportional to $j'$ in the BPS Lagrangian (\ref{eq:LagBPS}), but it will turn out that this term must be equal to zero and thus forces us to set $j=0$. Suprisingly, for the case of dyon Lagrangian (\ref{eq:NSdyon}), the effective action (\ref{eq:effNSd}) gives the identical Euler-Lagrange equations for $f$ and $j$ upon substituting $j=\beta f$ in the BPS limit. Therefore it is valid to use this identification for particular Born-Infeld type action of (\ref{eq:NSdyon}) for dyon. We also applied the BPS Lagrangian method to the Generalized version of $SU(2)$ Yang-Mills-Higgs model (\ref{eq:LagcanG}) in which the effective action is given by (\ref{eq:gencanLag}). For monopole case, we found there is a constraint between the scalar-dependent couplings of gauge kinetic term $w$ and of scalar kinetic term $G$, which is $G=1/w$, similar to the one obtained in~\cite{Casana:2012un}. The BPS equations are also modified and depend explicitly on these scalar-dependent couplings. For dyon case, the constraint is generalized to $w(G-\beta^2w)=1-\beta^2$, with $\beta<|1|$, and the BPS equations are modified as well. This is relatively new result compared to~\cite{Casana:2012un,Casana:2013lna} in which they did not discussed about dyon. As previously assumed $w,G>0$, the constraint leads to $w_\pm={1\over 2\beta^2}\left(G\pm\sqrt{G^2-4\beta^2\left(1-\beta^2\right)}\right)$. Reality condition on $w_\pm$ gives lower bound to $G$ as such $G\geq|2\beta\sqrt{1-\beta^2}|$ in all values of radius $r$. The Generalized version of Nakamula-Shiraishi model for monopole, with Lagrangian (\ref{NSMonopole}) and effective Lagrangian (\ref{NSMonopoleEff}), has also been computed. The results are similar to the Generalized version of $SU(2)$ Yang-Mills-Higgs model for monopole in the BPS limit. In the case for Generalized version of Nakamula-Shiraishi model for dyon, with Lagrangian (\ref{NSDyon}) and effective Lagrangian (\ref{NSDyonEff}), the results are similar to the Generalized version of $SU(2)$ Yang-Mills-Higgs model for dyon, eventhough there are two additional scalar-dependent couplings $G_1$ and $G_2$. These additional couplings are related to the kinetic terms's couplings by $G_2-\beta^2 G_1=w(G-\beta^2 w)$. In the appendix, based on our results, we write down explicitly the complete square-forms of all effective Lagragians (\ref{eq:effNSm}), (\ref{eq:effNSd}), (\ref{NSMonopoleEff}), and (\ref{NSDyonEff}). \acknowledgments We would like to thank Handhika Satrio Ramadhan during the initial work of this article. A.N.A would like to thank CERN for hospitality during the visit, that was supported by RISET-PRO Non-degree 2017 program, where the initial writing of this article has been done.

BkiUgDLxK6nrxq6DqSMQ

\section*{1.\,\, Frumkin-Derjaguin Theory of Wetting Phenomena} The thermodynamics of interfacial phase transitions (IPT) was founded over 70 years ago by Frumkin \cite{frumkin} and combined with the concept of the `disjoining pressure' introduced even earlier by Derjaguin in the same context \cite{frumderj} and in the context of adsorption in porous media \cite{derj1,derj2}. Frumkin was directly concerned with the stability of a film of liquid adsorbed at a planar solid-gas interface, which he treated as the direct generalisation of van der Waals's molecular theory of bulk phase transitions to interfacial phase transitions. In particular, he considered the thermodynamic stability of a planar fluid film as a function of its thickness by implicitly assuming that over-saturated gas can exist in the absence of bulk liquid whenever required to enable the system to traverse a van der Waals loop within, for example, an adsorption isotherm. The interfacial thermodynamic free-energy is the surface-excess grand potential of the liquid film per unit area, $\gamma_{SG}$, historically known as surface tension. When the bulk free energy terms are subtracted via appropriate choices of Gibbs dividing surfaces, the conjugate pair of thermodynamic variables for isothermal processes (which in bulk are chemical potential $\mu$ and number density $\rho$) translate to (field, density) = ($\mu$, $\Gamma$), where $\Gamma$ denotes the adsorption (excess amount of molecules adsorbed per unit area). In the context of wetting films, as considered by Frumkin, it is natural to use a suitably defined film thickness $\ell$ (rather than $\Gamma$) as the effective order parameter, which can be done rigorously by defining the thickness to be the adsorption divided by a number density that is fixed for a given isotherm; the obvious choice suitable for all but unphysically thick unsaturated films is $\ell \equiv \Gamma/\Delta\rho$ where for any particular isotherm $\Delta\rho \equiv \rho_L-\rho_V$ is the difference between the saturated liquid density and the density of coexisting vapour at the chosen temperature $T$. A transformation from the field $\mu$ to the order parameter $\ell$ is then via the usual transformation between the grand ensemble and the canonical ensemble: \begin{equation} \gamma_{SG}(\mu) = f(\ell)-\mu\,\Delta\rho\,\ell\,\,, \label{trans} \end{equation} where $f(\ell)$ denotes the interfacial excess Helmholtz free energy per unit area. This statistical-thermodynamics immediately yields the second-law of thermodynamics in two equivalent forms: \begin{eqnarray} \partial\gamma_{SG}(\mu) &=& -\Delta\rho\,\,\ell\,\partial\mu \label{2ndmu}\\ \partial f(\ell) &=& \mu\,\Delta\rho\,\partial\ell\,\,. \label{2ndell} \end{eqnarray} The first of these results (\ref{2ndmu}) is just the Gibbs adsorption equation. Frumkin and Derjaguin use the mean-field quantity $\mu(\ell)$ to define a disjoining pressure \begin{equation} \Pi(\ell) \equiv \left[\mu_o-\mu(\ell)\right]\Delta\rho\,\,, \label{pimu} \end{equation} where $\mu_o$ denotes the chemical potential corresponding to an equilibrium film thickness $\ell_{o}$. Note that by construction one can read off the disjoining pressure, for any chosen $\mu_o$, directly from an experimental adsorption isotherm (i.e. $\ell$ versus $\mu$) and hence we can hereafter write $\mu(\ell)$ as introduced in Eq.~(\ref{pimu}). It is also possible to introduce a mean-field potential $\psi(\ell)$, an example of what we now call a Landau potential or in this specific context the interface potential, by imagining a constraint sufficient to stabilize planar adsorbed films away from equilibrium: \begin{eqnarray} \psi(\ell;\mu_o) &\equiv& f(\ell)-\mu_o\,\Delta\rho\,\ell\,\,,\label{vee}\\ \Pi(\ell) &=& -\frac{\partial \psi(\ell)}{\partial\ell}\,\,. \label{dispot} \end{eqnarray} Once a specific $\mu_o$ has been chosen, all values of the film thickness except that for which $\mu(\ell)=\mu_o$ represent the free-energy of metastable or unstable planar films. At $\mu(\ell)=\mu_o$ there is always at least one value of $\ell$ corresponding to the absolute minimum of $\psi(\ell)$, where the interface potential equals the equilibrium surface tension $\gamma_{SG}(\mu_o)$. \\ The Russian School summarise the above statistical-thermodynamics with the following transformation of the second law of interfacial thermodynamics: \begin{equation} \gamma_{SG} = \psi + \ell\Pi\,;\,\,\,\,\partial \psi = -\Pi\partial\ell\,;\,\,\,\,\partial\gamma_{SG} = \ell\partial\Pi\,\,. \label{russ} \end{equation} Here one can note three distinct but equivalent approaches to applying Frumkin-Derjaguin theory in practice. Frumkin emphasised plots of the thermodynamic potential versus the inverse of the order parameter; i.e. $\gamma_{SG}$ versus $1/\ell$ which is the direct equivalent of a $p-V$ isotherm in van der Waals' theory of bulk phase transitions. Derjaguin preferred to analyse the field as a function of its conjugate order parameter; i.e. $\mu(\ell)$ or equivalently adsorption isotherms $\ell(\mu)$. In either case, a first-order phase transition between equilibrium films of different thickness $\ell_{o}$ and $\ell_{1}$ would appear as a van der Waals loop associated with an equal-area Maxwell construction: \begin{eqnarray} \int_{1/\ell_{1}}^{1/\ell_{o}}d\frac{1}{\ell}\gamma_{SG}(\ell) &=& \gamma_{SG}(\mu_o)\left(\frac{1}{\ell_{o}}-\frac{1}{\ell_{1}}\right)\,\,, \label{frumderj2}\\ \int_{\ell_{o}}^{\ell_{1}}d\ell\Pi(\ell) &=& 0\,\,, \label{fderj3} \end{eqnarray} respectively \cite{footeq}. The final route, based on identifying $\psi(\ell)$, was adopted by physicists as the work-horse of density functional theory (DFT) of inhomogeneous fluids \cite{evansrev}, although the nature and significance of the constraint needed to stabilise a film out of equilibrium has typically been treated rather vaguely \cite{jrhdis}. In the vicinity of a first-order interfacial phase transition $\psi(\ell)$ develops a second minimum directly related to the Maxwell construction for a van der Waals loop in $\Pi(\ell)$: \begin{equation} \psi(\ell_{o})=\psi(\ell_{1})=\gamma_{SG}(\mu_o)\,\,. \label{landth} \end{equation} In the absence of a first-order phase transition in the approach to saturation ($\mu = \mu_{sat}$) every chosen value of $\mu_o < \mu_{sat}$ corresponds exclusively to an equilibrium system ($\ell_{o} = \ell(\mu=\mu_o$)). Then when the above statistical thermodynamics is applied to partial wetting at saturation (defining an equilibrium planar film surrounding an adsorbed drop of excess liquid with a non-zero value of Young's contact angle) one obtains probably the most well-known expression from Frumkin-Derjaguin theory \cite{derj1} \begin{equation} \int_{\ell_{o}}^{\infty}d\ell\Pi(\ell) = \psi(\ell_{o})-\psi(\infty) = \gamma_{LV}(\cos\theta -1)\,\,, \label{frumderj} \end{equation} where the final equality has introduced Young's equation \cite{young} for the contact angle $\theta$. The above result is a direct expression of the thermodynamic work needed to quasi-statically thin a macroscopically thick film of excess liquid down to its equilibrium thickness. \\ Frumkin-Derjaguin theory was used as the thermodynamic basis of the famous analysis of Dzyaloshinskii, Lifshitz and Pitaevskii (DLP), who adapted high-temperature quantum field-theory to directly calculate the dispersion interaction contribution to the normal component of the pressure tensor and hence obtain the excess pressure on the surface due to the presence of an adsorbed film. By identifying this quantity with Derjaguin's disjoining pressure DLP were able for the first time to show that one could explicitly implement Frumkin-Derjaguin theory of interfacial phase transitions within theoretical physics \cite{DLP1,DLP2}. This work was adopted as a foundation stone of modern colloid and interface science, despite the clear warning of DLP that their field-theory could only be trusted for macroscopically thick films (films surrounding drops with a Young's contact angle of no more than about 0.1 degrees \cite{DLP2}). Much harder to explain is why it was more than 15 years after DLP before the interfacial phase transitions of Frumkin were `rediscovered' and named wetting transitions \cite{cahn,ebsaam,Nak-Fish}. Direct use of Frumkin-Derjaguin theory constitutes a mean-field theory of interfacial phase transitions (MFIPT), because of the constraint required to define a van der Waals loop through values of the film thickness where a planar film would be metastable or unstable to long-wavelength capillary wave fluctuations (eventually broadening into coexistence between films of different thicknesses). Interestingly, the upper critical dimension at and below which $\psi(\ell)$ is renormalised by long-wavelength capillary-wave fluctuations away from planarity is known to be $d=3$ for short-ranged models \cite{lip83} and in the presence of dispersion forces naturally occurring IPT are believed to be mean-field \cite{lip84}. \\ \section*{2. Physical interpretation of the mean-field disjoining pressure} In his monograph, Frenkel \cite{Frenkel} reports that Derjaguin's disjoining pressure was apparently first discovered ``by D. Talmud and S. Bresler, who noticed in 1931 that two drops of mercury freely moving on the bottom of a glass vessel filled with molten paraffin did not coalesce but remained separated from each other by a very thin paraffin film'', \cite{Bresler}. Frenkel chooses to review this work in the context of the ``specific elasticity of thin films''. However, the name ``disjoining pressure'' indicates that the Russian School understood that Eq.~(\ref{pimu}) must represent the force per unit area exerted by the fluid film on the substrate, minus that which would be present for the equilibrium film at $\mu=\mu_o$. In fact, DLP appear to assume that the normal component of the pressure tensor is directly proportional to the film chemical potential \cite{DLP1}. In which case one could envisage direct measurements of the disjoining pressure by placing pressure sensors just behind the substrate surface. Mean-field DFT has been used to successfully test this deep connection within statistical mechanics \cite{jrhdis}. In particular, for a wall-fluid model in which the substrate field is represented as a one-body external field $v^{ext}(z)$ the statistical mechanical definition of the disjoining pressure must be \begin{equation} \Pi(\ell) = -\int_{-\infty}^{\infty}dz\left[\rho(z;\ell)-\rho(z;\ell_o)\right]v^{ext\,'}(z)\,\,, \label{pism} \end{equation} where $\rho(z;\ell_o)$ is the density profile of the adsorbed film at equilibrium (at chemical potential $\mu_o$). At saturation ($\mu_o=\mu_{sat}$) one can replace $\rho(z;\ell_o)$ with the wall-liquid density profile $\rho_{SL}(z)$, for the purpose of calculating the disjoining pressure from Eq.~(\ref{pism}). DLP express Eq.~(\ref{pism}) in terms of the normal component of a pressure tensor. The full tensor can be defined to within an arbitrary path integral (or gauge) from the many-body expression for the total momentum $\dot{J}({\mbox{\bf r}},t)$ \cite{hensch}. The conservation of momentum $\langle\dot{J}({\mbox{\bf r}},t)\rangle=0$ leads to a sum rule for the equilibrium pressure tensor: \begin{equation} \nabla^{\beta}p^{\alpha\beta}({\mbox{\bf r}})=-\rho({\mbox{\bf r}})\nabla^{\alpha}v^{ext}({\mbox{\bf r}})\,\,. \label{pten} \end{equation} In planar symmetry this reduces to a sum rule for the normal component of the pressure tensor, as required to justify Eq.~(\ref{pism}): \begin{equation} p_{N}\,'(z) = -\rho(z)v^{ext\,'}(z)\,\,. \label{pn} \end{equation} The above can be regarded as an expression of Newton's third law; i.e. the force of the fluid on the wall is equal and opposite to that of the wall on the fluid. Thus, there are two equivalent ways of expressing this force balance, either in terms of the intermolecular forces (known as a virial theorem) or, equivalently, in terms of the wall-fluid force as in Eq.~(\ref{pism}), \cite{jrhrev}. \\ \begin{figure}[tbh] \center \includegraphics[width=14cm,height=8cm,angle=0]{fig1.eps} \caption{Planar mean-field interfacial theory defined from confinement of a liquid film in an asymmetric planar slit (see text).} \end{figure} A potential illustration of the connection between the statistical mechanical disjoining pressure and the mean-field film chemical potential is sketched in Fig.~1. Here a liquid film is depicted confined between the substrate of interest and a hard-wall boundary. If sum rule (\ref{pn}) is integrated over the hard-wall boundary one finds a conceptually important result: \begin{equation} p_{N}(L) = kT\rho_{HW}(L)\,\,, \label{pnhw} \end{equation} where $k$ denotes Boltzmann's constant and the notation highlights the fact that the right-side of (\ref{pnhw}) only applies at the hard-wall boundary; i.e. $\rho(z=L_-) \equiv \rho_{HW}(L)$. If even a nm-thin film of liquid is adsorbed at a substrate then one must be close to saturation and thus one will be able to gently confine the film at a pressure $p_{N}(L)$ close to the vapour pressure. Since the second virial coefficient is negative along the liquid-vapour coexistence curve, one must have a very low value for the density at the hard-wall (provided of course that the substrate field has a range smaller than $L$). Namely, an incipient drying layer must be present at the hard-wall \cite{jrhvs}, as sketched in Fig.~1. In this manner the hard-wall is able to suppress capillary-wave fluctuations away from planar symmetry, without significantly altering the thermodynamic state of the liquid film. In effect, the pore width $L$ is being used to set the thickness $\ell$ of a constrained liquid film. Now consider an infinitesimal change of pore width induced by moving one of boundary walls while keeping the other fixed. If the wall that is moved is the hard-wall, then standard analysis yields \cite{jrhrev,jrhsolv} \begin{eqnarray} -\left(\frac{\partial\Omega}{\partial L}\right)_{T,\mu_o} &=& -\int d^3r \rho(z)\frac{\partial v_{HW}(L-z)}{\partial L} \label{solv1}\\ &=& p_{N}(L)A\,\,, \label{solvf} \end{eqnarray} where the integral is over all space (i.e. extends across the hard-wall of area $A$) and sum rule (\ref{pnhw}) has been inserted to obtain the second line. If the substrate wall was moved instead, then a result of identical form to (\ref{solv1}) is obtained, except that the hard-wall field $v_{HW}$ is replaced by the substrate field (note that the sensible way to carry out this alternative derivation is to swap the direction of the $z$ axis so that the substrate is defined to lie at $z=L$ until the final result can be re-expressed back in the geometry of Fig.~1): \begin{equation} -\left(\frac{\partial\Omega}{\partial L}\right)_{T,\mu_o} = -A\int_{-\infty}^{L}\rho(z)v^{ext\,'}(z)\,\,. \label{pnsub} \end{equation} Inserting Eq.~(\ref{pn}) then gives back the previous result (\ref{solvf}) since one need only integrate out to the hard-wall or to when the substrate field has fallen to zero (whichever is the least). By defining $\psi \equiv (\Omega + p_oV)/A$, one can rewrite sum rule (\ref{solvf}) in a form directly analogous to (\ref{dispot}): \begin{equation} p_{N}(L)-p_o = -\frac{\partial \psi}{\partial L}\,\,, \label{solvpi} \end{equation} except that now the disjoining pressure really is defined in terms of the force per unit area on the hard-wall; $p_o$ is the value $p_{N}(L\rightarrow \infty)$. In this interpretation the disjoining pressure $\Pi(L)$ is the extra pressure on the film needed to maintain a planar film of width $\ell\approx L$ at fixed $T$ and fixed $\mu=\mu_o$. Alternatively, the result can be expressed in terms of the conservation of momentum normal to the substrate wall; i.e. from (\ref{pnsub}). It is this alternative route that is directly expressed in the definition (\ref{pism}). To within the validity of replacing $L$ with $\ell$, we may therefore conclude that the disjoining pressure of Frumkin-Derjaguin theory, defined as a chemical potential difference (\ref{dispot}), is equivalent to the extra pressure on a substrate wall that would be induced by an adsorbed planar film of metastable or unstable thickness. This deep and confusing correspondence is of great significance theoretically, because expressing the disjoining pressure in terms of the conservation of momentum normal to the substrate (\ref{pism}) allows for alternative developments in terms of exact physics \cite{DLP1,mik-weeks,adamjrh}. In principal, experiment has direct access to the local normal force on the substrate wall and so by expressing inhomogeneous fluid phenomena in terms of sum rules involving the exact disjoining pressure, as we shall do in the following section, one would be able to directly analyse experimental systems. \section*{3.\,\, Beyond mean-field interfacial phenomena} The physics discussed so far has either implicitly or explicitly assumed the presence of constraints sufficient to prevent large scale fluctuations away from planar symmetry. Thus, any prediction of interfacial phase transitions will be mean-field (MFIPT). To go beyond mean-field one must include these fluctuations. There are standard procedures for doing this in theoretical physics, which essentially boil down to extending the class of constraints to include all physically appropriate classes of symmetry (known as summing over `all' fluctuations). For example, given sufficient numerical resources, one could use DFT to search for solutions of the class $\rho(x,z)$ and $\rho(x,y,z)$, in addition to $\rho(z)$. Frumkin-Derjaguin theory would have to be similarly extended, since Eq.~(\ref{dispot}) implicitly assumes density profiles of the class $\rho(z)$ stabilized by imposing a non-equilibrium chemical potential $\mu(\ell\neq\ell_{o})$. Generalised Frumkin-Derjaguin Theory (GFDT) would need to be based on instantaneous disjoining potential profiles $\hat{\Pi}(x)$ and $\hat{\Pi}(x,y)$, with the sum over fluctuations embodied in formal path-integral definitions defining the full statistical mechanical averages \cite{mik-weeks} \begin{eqnarray} \Pi(\ell_o) &=& \langle\hat{\Pi}\rangle \label{dispothat}\\ \ell_o &=& \langle\hat{\ell}\rangle \,\,, \label{lhat} \end{eqnarray} where the instantaneous film profile is denoted $\hat{\ell}(x)$ or $\hat{\ell}(x,y)$, depending on the class of broken symmetry embodied in the fluctuations. The physical significance of fluctuation physics to interfacial phase transitions is heavily dependent on geometry. If the overall geometry is planar then three-dimensional systems are not strongly affected by thermal fluctuations, although for short-range models (such as Ising models) fluctuations do generate higher-order contributions to interfacial critical phenomena needed to fully explain the observed physics \cite{parry}. For quasi two-dimensional systems however, even in overall `planar' geometry, interfacial fluctuations are sufficiently dominant to invalidate MFIPT. Let us therefore restrict further discussion to the geometry of Fig.~2, while bearing in mind the obvious generalisation to higher-dimensional geometry in special cases. \begin{figure}[tbh] \center \includegraphics[width=14cm,height=4cm,angle=0]{fig2.eps} \caption{A thermal fluctuation of an adsorbed film away from planar symmetry in quasi two-dimensional geometry. The instantaneous height of the liquid film along the substrate is denoted $h(x)\equiv\hat{\ell}(x)$.} \end{figure} Figure~2 depicts a region of the interface at which a molecularly thin film has fluctuated away from the substrate, known as a `strong' fluctuation regime. Collective fluctuations beyond relevant molecular correlation lengths (typically less than a nm) are much slower than molecular time scales and so one expects local mechanical equilibrium. Thus, it is no accident that the sketch in Fig.~2 reminds readers of Young's picture of an adsorbed drop with a well-defined contact angle arising from the cylindrical (or spherical) cap geometry away from the substrate. Laplace's formulation of local mechanical equilibrium demands that there be an excess pressure inside the drop required to maintain the non-zero curvature $\kappa$ of the film profile. Hence, one can immediately appreciate that a fluctuation of this nature is directly linked to an interfacial pressure wave (surface sound) embodied in the $x$-dependence of the instantaneous disjoining pressure $\hat{\Pi}(x)$. The lowest level of theoretical physics sufficient to describe strong fluctuations requires the replacement of the interface potential $\psi(\ell)$ with an interfacial Hamiltonian functional: \begin{equation} H[\hat{\ell}]=\gamma_{LV}\int_{-\infty}^{\infty}dx\left\{\sqrt{1+(\hat{\ell}')^2}-1\right\}+W[\hat{\ell}]\,\,. \label{iham} \end{equation} The first term in the interfacial Hamiltonian defines the cost of distorting the outer surface of the film away from planar symmetry. It might be thought that the use of $\gamma_{LV}$ as the restoring force is problematic for nanoscopic fluctuations but one should note that in this description the film profile $\hat{\ell}(x)$ is already coarse-grained over molecular scales and thus its gradient (the dash denotes $d/dx$) is zero everywhere except where a significant deviation has occurred. The final term in (\ref{iham}) denotes the generalisation of the mean-field interface potential to non-planar geometry; i.e. we must use a non-local functional of $\hat{\ell}(x)$ instead of simply integrating over a mean-field $\psi(\ell(x))$, as emphasised by Parry and coworkers \cite{parry}. $W[\hat{\ell}]$ is known as the binding potential (functional).\\ It is actually possible to use the fundamental statistical mechanics of DFT to regard (\ref{iham}) and hence $W[\hat{\ell}]$ as formally exact functionals defined by the unique relationship between the field $\phi\equiv \mu_o-v^{ext}(x,z)$ and the instantaneous density profile $\rho(x,z)$, \cite{evansrev}. The latter can be used to self-consistently define the instantaneous coarse-grained profile $\hat{\ell}(x)$, as discussed below. That is, $\phi(x,z)$ is the formally exact constraint needed to define an arbitrary film profile. The final result will then be expressed as a formal sum over all physically realisable configurations of the film profile. Note that from this point of view our choice of $\gamma_{LV}$ as the restoring force for bending the film profile is a formal definition, in the same class as choosing a Gibbs dividing surface consistent with macroscopic physics; it effects only the distribution of free-energy between interfacial free energy and higher-order line tension contributions (which it is not particularly informative to include explicitly in the macroscopic Young-Laplace force-balance below). The film profile, for a particular constraint $\phi(x,z)$, is formally given by minimization of the functional $H[\hat{\ell}]$; i.e. $\delta H/\delta \hat{\ell} =0$. The final step is to note that conservation of momentum perpendicular to the substrate requires us to identify the instantaneous disjoining pressure as the functional derivative \begin{equation} \hat{\Pi}[\hat{\ell}] \equiv -\frac{\delta W[\hat{\ell}]}{\delta\hat{\ell}}\,\,, \label{insdp} \end{equation} which in any case is immediately recognisable as the required generalisation of Eq.~(\ref{dispot}) in Frumkin-Derjaguin theory. Namely, with this identification, setting the functional derivative of (\ref{iham}) to zero leads directly to the sum rule \begin{equation} \hat{\Pi}(x) = -\gamma_{LV}\frac{d}{dx}\left(\frac{\hat{\ell}'}{\sqrt{1+(\hat{\ell}')^2}}\right) \equiv -\gamma_{LV}\hat{\kappa}(x)\,\,, \label{dpfin} \end{equation} where $\hat{\kappa}(x)$ is the coarse-grained interfacial curvature profile. That this realisation of Laplace's force balance is a direct consequence of the conservation of momentum normal to the substrate can be seen in Fig.~2 if we regard the fluctuation as the formation of a cylindrical-cap section of a drop of radius $R$ and in-plane radius $r = R\sin\theta$, defining Young's contact angle $\theta$. We note immediately from (\ref{dpfin}) that at the top of the drop $\hat{\Pi}$ reduces to $\hat{\Pi}_c=\gamma_{LV}/R$, the Laplace pressure difference across the curved film profile. That is, the pressure under the drop is bigger than outside of it by the well-known amount. If sum rule (\ref{dpfin}) is integrated from outside the fluctuation to the top of the cylindrical cap then one obtains zero, which is an expression of force-balance between the bending of the film profile near the three-phase contact line and the `macroscopic' curvature of the cylindrical cap; i.e. \begin{equation} -\int_{-\infty}^{x_c}dx\left[\hat{\Pi}(x)-\hat{\Pi}_c\right] = r\hat{\Pi}_c = \gamma_{LV}\sin\theta\,\,, \label{cmsr} \end{equation} where $x_c$ denotes the $x$-coordinate of the centre of the cylindrical-cap portion. In the macroscopic limit where the three-phase contact line can be treated as the intersection (at angle $\theta$) of a planar liquid-vapour interface with a planar substrate then Eq.~(\ref{cmsr}) reduces to an integral over the three-phase contact region alone \begin{equation} -\int_{-\infty}^{\infty}dx\hat{\Pi}(x) = \gamma_{LV}\sin\theta\,\,, \label{jrhfb} \end{equation} which is the normal component of Young's force balance through the three-phase contact line \cite{adamjrh,unbalanced,derj1,shandeG,jrhpre}. Here, the disjoining pressure is balanced by the distortion of the film surface as one crosses the three-phase contact line. In colloid and interface science this contribution is often equated with the disjoining pressure for planar films (of varying thickness) and is separated out from the macroscopic contribution $\hat{\Pi}_c$; see e.g. \cite{velarde}. When the above integral is taken over the entire surface, in overall planar geometry (Fig.~2), the implied force balance is equivalent to full thermodynamic equilibrium; i.e. $\langle\hat{\Pi}\rangle =0$ as required for thermodynamic equilibrium of a planar film. When the equilibrium geometry is non-planar (e.g. in the presence of a macroscopic adsorbed drop) both (\ref{dpfin}) and (\ref{jrhfb}) average to sum rules of identical form but now involving $\langle\hat{\Pi}\rangle(x)$ and $\langle\hat{\kappa}\rangle(x)$.\\ From the above we can conclude that by construction the interface Hamiltonian theory and associated disjoining pressure is fully consistent with the conservation of momentum perpendicular to the substrate. In field theory, sum rules of this class are known as a Ward identity and are equivalent to the condition of translational invariance (of the entire system) in the $z$-direction \cite{mik-weeks}. It is this physics that enables us to equate the exact disjoining pressure profile with the excess force on the substrate surface underneath the film: \begin{equation} \hat{\Pi}[\phi](x) = -\int_{-\infty}^{\infty}dz\left[\rho(x,z;\hat{\ell})-\rho(z;\ell_o)\right]\frac{dv^{ext}_{sub}(z)}{dz}\,\,, \label{fbgen} \end{equation} where the subscript on the external field emphasises that this is the true substrate field without an additional constraint. Note that (\ref{fbgen}) reduces to (\ref{pism}) when the fluctuations are restricted to planar `breathing modes', in contrast to (\ref{dpfin}). Thus, the general statistical mechanical result (\ref{fbgen}) contains more physics than (\ref{iham}). In principle, sensors could be placed just under the substrate surface to measure the fluctuating disjoining pressure profile $\hat{\Pi}(x)$, provided the fluctuations were sufficiently slow (to enable a local time average before the fluctuation disappeared). It is thus of some interest to note the intimate connection between a measurement of $\hat{\Pi}(x)$ and the associated $\hat{\ell}$. First let us define a local contact angle profile (not to be confused with Young's contact angle): \begin{eqnarray} \tan\theta(x) &\equiv& \hat{\ell}'(x)\,\,,\label{tanthx}\\ \sin\theta(x) &\equiv& \frac{\hat{\ell}'}{\sqrt{1+(\hat{\ell}')^2}}\,\,. \label{sinthx} \end{eqnarray} Then the first integral of (\ref{dpfin}) allows us to define $\theta(x)$ in terms of the disjoining pressure profile: \begin{equation} \theta(x) = \sin^{-1}\left\{-\frac{1}{\gamma_{LV}}\int_{-\infty}^{x}d\tilde{x}\hat{\Pi}(\tilde{x})\right\}\,\,, \label{thetasr} \end{equation} where the local profile is assumed to be planar at the left boundary. Finally, the integral of (\ref{tanthx}) shows that the local fluctuating profile is fully determined by the disjoining pressure profile defining $\theta(x)$: \begin{equation} \hat{\ell}(x) = \int_{-\infty}^{x}d\tilde{x}\tan\theta(\tilde{x})\,\,, \label{fprof} \end{equation} with the zero of film height defined at the far left boundary. In non-planar geometry (such as a system with an equilibrium three-phase contact line) the above results are immediately applicable to the final equilibrium profile \cite{adamjrh}. For overall planar geometry, the true equilibrium film width requires a path integral (\ref{lhat}) over all fluctuations. In principal either of the profiles $\hat{\ell}(x)$ or $\hat{\Pi}(x)$ could be directly measured, but these are not independent measurements because they determine each other through the conservation of momentum normal to the substrate. In a fully molecular system, this link allows one to uniquely define what is meant by the coarse-grained profile $\hat{\ell}(x)$, at least in the absence of overhangs \cite{adamjrh}. One can go further and define correlation function hierarchies from the fluctuating disjoining pressure \cite{mik-weeks}. It has taken the present author his entire working lifetime to grasp the true significance of Derjaguin's disjoining pressure to the statistical mechanics of inhomogeneous fluids. \section*{Acknowledgments} I am grateful to A. O. Parry and M. G. Velarde for insightful discussion.

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\section{Introduction} Previous works have shown that there is a significant performance drop when applying models trained on cased data to uncased data and vice-versa \cite{wang-etal-2006-capitalizing}. Since capitalization is not always available due to real world constraints, there have been some methods trying to use casing prediction (called \textit{truecasing}) to tackle this trade-off. The work we reproduce tries to battle this issue in two popular NLP tasks by lowercasing 50\% of the original datasets. \section{Contributions} This paper effectively shows how well work from \cite{ner-and-pos-original} can be reproducede and how well it applies to a few other settings.\\ The original paper does show how casing issue in NLP can be effectively solved through a method which requires close to none overhead in terms of development time, and no additional overhead in runtime, especially when compared to methods such as truecasing (This is true for mixing 50\% cased and 50\% uncased, if we mix full datasets of both then training will take approximately twice as long).\\ It also serves as a reproducing work to show that truecasing performance can be reproduced. \subsection{Hypotheses from original paper} \label{sec:original_hypotheses} Original paper proposes following hypotheses: \begin{itemize} \item Truecasing fits strongly to data it is presented leading to performance drops on different datasets. \item Mixing cased and uncased data provides the best performance in NER task on CoNLL 2003 dataset. This is not due to larger size of the dataset, since the same method works if we only lowercase 50\% of the original dataset. \item Such techinque generalizes well on noisy datasets such as Twitter data. \item Mixing cased and uncased data provides the best performance in POS task on Penn Treebank dataset. This is not due to larger size of the dataset, since the same method works if we only lowercase 50\% of the original dataset. \end{itemize} \subsection{Hypotheses addressed in this work} \label{sec:current_hypotheses} In addition to hypotheses tested in Section~\ref{sec:original_hypotheses} we also tested: \begin{itemize} \item POS: \begin{itemize} \item Mixing both cased and uncased data leads to the best performance regardless of word \textbf{embedding} used. \item Mixing both cased and uncased data leads to the best performance regardless of word \textbf{dataset} used. \item Using ELMo model with CRF layer in POS task to outperform the one without, regardless of data casing technique. This is based on \cite{BiLSTM-CRF}. \end{itemize} \end{itemize} \subsection{Experiments} We conducted three main experiments from the paper, which were done roughly in parallel to each other. Truecasing is a task where given all lowercase sentences we have to predict correct casing. Here we trained a simple Bidirectional LSTM with a logical layer on top, as described in both \cite{ner-and-pos-original} and \cite{susanto-etal-2016-learning}. Since the former paper, does a great job of mentioning hyper-parameters used in the network, the hyperparameter search is not required. Part of Speech tagging is a task in which given a sentence we have to predict part of speech for each word. It is our the most in depth experiment, which lead a lot of additional hypotheses which got tested. Firstly, we had to find optimal hyper-parameter setting. The search was not too large, since a few of the parameters (such as number of layers) were known, but it still took a significant amount of time. After finding such setting we compared its results with ones reported the original paper. Then we investigated whether claims from this paper are applicable to other datasets and encodings. Lastly, we looked at the CRF layer, and what its impact on performance on the whole model is. Named Entity Recognition is a task in which given a sentence we have to predict which parts of it describe some entity. It had the most sophisticated model out of all the experiments we tried reproducing. Fortunately we did not have to do a hyperparameter search since \cite{DBLP:journals/corr/LampleBSKD16} provided a similar model to that of the original paper. \section{Code} There are no public repos which do mention the project in their \textit{READMEs}. However, in case of truecasing, primary author of \cite{ner-and-pos-original} has two repositories: \href{https://github.com/mayhewsw/truecaser}{truecaser} and \href{https://github.com/mayhewsw/pytorch-truecaser}{python-truecaser}. The former one refers to the original implementation from \cite{susanto-etal-2016-learning}, while the python one is a port to python. The (probably) primary author of \cite{susanto-etal-2016-learning} also has a \href{https://github.com/raymondhs/char-rnn-truecase}{github repository} possibly related to truecaser we are trying to reproduce. However, to validate results and test ease of application we decided to not use either of these resources and focus on custom implementation. This also removes dependence on Andrej Karpathy's \href{https://github.com/karpathy/char-rnn}{char-rnn} all of the above mentioned repositories are forks of. We were not able to find publicly available code for either NER or POS parts of the original paper. Hence we needed to reimplement the code from scratch. All our work is in \href{https://github.com/andreaskuster/uw-nlp}{public github repository}. We tried to make all results as easily accessible as possible, which means that there is significant overlap between this report and the \textit{README} on that repository. Truecasing and NER were implemented using PyTorch \cite{pytorch}, while POS was implemented in Keras \cite{keras}, on top of TensorFlow \cite{tensorflow2015-whitepaper}. \section{Experimental setup and results} \subsection{Datasets} \label{sec:datasets} Datasets were a common resource about all three parts of experiments. Hence we will describe them separate here, and for each experiment specify which exact dataset was used: \begin{itemize} \item CoNLL2003 - From \cite{conll2003}. Not publicly available. Shared as part of class. \item Peen Tree Bank (PTB) - From \cite{penn-treebank}. Not publicly available. Shared as part of class. We used a loader from \href{https://www.nltk.org/}{nltk} (after appending nonpublic data to it). In particular: \begin{itemize} \item Sections 0-18 are used for training \item Sections 19-21 are used for development, i.e. choosing hyperparameters \item Sections 22-24 are used for test, i.e. reporting accuracy \end{itemize} \item Twitter - From \cite{twitter-dataset}. Specifically we found \href{https://github.com/GateNLP/broad_twitter_corpus}{authors github repository} with data available. \item Wikipedia - From \href{https://github.com/raymondhs/char-rnn-truecase/tree/master/data/wiki}{github repository}, which we suspect is authored by primary author of \cite{susanto-etal-2016-learning}. This data was used for truecasing experiment. \end{itemize} Additionally, in case of NER and POS each of these datasets occurs in 5 different experimental flavors: \begin{itemize} \item Cased (C) - Standard dataset, as downloaded. \item Uncased (U) - Uncased dataset. It's a standard dataset, with an additional step of converting everything to lowercase. \item Cased + Uncased (C + U) - Combination of both cased and uncased flavors. Hence it is twice the size of either cased or uncased. \item Half Mixed (C + U 50) - Combination of cased and uncased flavors. However, only 50\% of data is lowercase, and remaining half is as-is. \item Truecase Test (TT) - Truecased Test dataset. Training data is the same as cased one. Test is converted to uncased version, and then a truecaser is run on it. The Truecaser is not trained on any data, and is provided (in case of this paper, truecaser trained on Wikipedia dataset is used). \item Truecase All (TA) - Truecased Train and Test dataset. The transformation described for test in TT is applied to both train and test. The Truecaser is not trained on any data, and is provided (in case of this paper, truecaser trained on Wikipedia dataset is used). \end{itemize} \subsection{Padding} To remove repetition in description of experiments we also want to discuss padding and how results should be understood. All experiments described below used padding to the maximum sentence size for training. However, for results on both validation (development) and test such padding was either removed or ignored, and hence it is not affecting the results reported in this paper. \subsection{Truecasing} \label{sec:exp-truecase} \subsubsection{Model description} We used the exact same model as described in \cite{susanto-etal-2016-learning}, a 2 layer, bidirectional LSTM. Since encoding was not specified, nor any character level encoding was mentioned in class we used a PyTorch \cite{pytorch} Encoder layer, which learned weights from bag-of-words to a feature-sized word representation. Note that in this model size of word embeddings is equal to hidden state size (both 300 dimensional). Then on top of these two layer a binary linear layer was applied. Hence output of this model is 2 dimensional, one specifying that character should be upper cased (true), other that character should be left as is. Implementation of this model took 2 hours, mostly due to transposes required by LSTM layers in PyTorch framework. It also helped that assignment 2 required us to implement LSTM model, and we could reuse parts of it. \subsubsection{Hyperparameters} We used a 2 layer, bidirectional LSTM with hidden size of 300. On top of that a linear layer, with 2 classes was applied, resulting in binary output. We used Adam \cite{adam} optimizer with default settings (learning rate of 0.001, betas of 0.9 and 0.999), and batch size of 100, as suggested in the \cite{susanto-etal-2016-learning}. Since these were the hyperparameters specified, we did not perform hyperparameter search. We also used an Out-Of-Vocabulary (OOV) rate of 0.5\%. Due to initial mistake we first defined out-of-vocabulary rate in following manner. When each token was drawn it had a 0.5\% chance to be masked with OOV token. However, after clarification from TA we switched OOV to be done at the initialization of tokens. In particular sorted tokens in order of increasing occurrence, and masked all least occurring tokens (such that their sum contained at least 0.5\% of total amount of tokens) with OOV token. This skewed our OOV distribution towards uncommon tokens. This is a desired behavior, due to the fact that the differences between datasets will be in these rare-occurring tokens. For wikipedia dataset, with first approach we got a dictionary of size 64, and with second approach it was 41. \subsubsection{Results} We trained these models for 30 epochs, with crossentropy loss. We recorded both training and validation loss for both, and chose the models with lowest validation loss for each of OOV settings. In both cases, such point was reached after around 7-8 epochs. The performance of the model matched one claimed both by original paper, and original truecasing paper on Wikipedia dataset. However, there was a significant mismatch in performance on other datasets provided in the original paper. This can be seen in Table \ref{tab:truecase-result}. A drop-off between datasets is expected, because as explained in original paper, various sources will use various casing schemes specific to the domain. \begin{table}[h] \centering \begin{tabular}{|l|c|c|} \hline Dataset & Initial OOV & Proper OOV \\ \hline Wikipedia & 92.65 & 92.71\\ \hline \hline CoNLL Train & 66.03 & 65.32 \\ \hline CoNLL Test & 63.49 & 63.28 \\ \hline PTB 01-18 & 78.53 & 78.73 \\ \hline PTB 22-24 & 78.47 & 78.69 \\ \hline \end{tabular} \caption{F1 scores of best performing model (chosen on validation set of Wikipedia dataset) for various datasets. Initial OOV stands for at-the-read techinque of creating OOV tokens, while proper is the former one of two described above.} \label{tab:truecase-result} \end{table} For reference Table \ref{tab:trucase-original-result} contains results from investigated papers. We can see that there is not a high difference in performance between results for Wikipedia dataset, but there is a 10\% difference in both CoNLL and PTB between our results and original ones. The possible reason for it might be the fact that original paper used fork of, before mentioned, \href{https://github.com/karpathy/char-rnn}{char-rnn}. We looked at it, searching for differences between our intuition and actual implementation, but could not find any. \begin{table}[h] \centering \begin{tabular}{|l|c|} \hline Dataset & Original \\ \hline Wikipedia (Susanto) & 93.19 \\ \hline Wikipedia & 93.01 \\ \hline \hline CoNLL Train & 78.85 \\ \hline CoNLL Test & 77.35 \\ \hline PTB 01-18 & 86.91 \\ \hline PTB 22-24 & 86.22 \\ \hline \end{tabular} \caption{F1 scores from reference papers. All entries but ``Wikipedia (Susanto)" are from \cite{ner-and-pos-original}.} \label{tab:trucase-original-result} \end{table} This leaves us in conclusion that, in terms of performance, this experiment can be only partially reproduced, and it is possible that additional techniques (such as dropout, etc.) were applied during training of the model. However, in terms of between-datasets trends the paper describes, we have confirmed them, and \textbf{were able to (partially) reproduce the first hypothesis of the paper}. \subsection{Part of Speech Tagging} \label{sec:exp-pos} \subsubsection{Model description} Model for POS task is not well described in the original paper, and rather references other papers for implementation details \cite{ma-hovy-2016-end}. This left us with only a rough sketch of a model, and a hyperparameter search for most of variables. Model used here is a single BiLSTM layer, followed by a Time Distributed Dense layer, followed by an output CRF layer. We used ELMo as the encoding for the input to BiLSTM layer. \subsubsection{Hyperparameters} \label{sec:exp-pos-hyper} As mentioned before, the task required a hyperparameter search. It was performed on standard (Cased) version of PTB dataset. Due to time constraints we opted for grid search with following settings: \begin{table}[h] \centering \begin{tabular}{|l|l|l|} \hline Hyperparameter & Values & Best \\ \hline LSTM Hidden Units & $2^{\{0, 1, 2, 3, 5, 7, 9\}}$ & 512 \\ \hline LSTM Drop. & 0.0, 0.2, 0.4 & 0.0 \\ \hline LSTM Recurr. Drop. & 0.0, 0.2, 0.4 & 0.0 \\ \hline learning rate & 0.1, 0.001 & 0.001 \\ \hline \end{tabular} \caption{Hyperparamter Search for POS experiment. Values investigated are in the middle column, while optimal combination of hyperparameters is in the right one.} \label{tab:pos-hypersearch} \end{table} One of the surprises of optimal setting is that both of dropout's are zero. However, as seen on Fig. \ref{fig:pos-hyperdrop}, we can see that all settings converge to roughly the same performance on validation set, with $0.0$ having the fastest convergence. \begin{figure}[h] \centering \includegraphics[scale=0.3]{pos_hypersearch_drop.png} \caption{Performance of various settings for different dropout values in LSTM layer.} \label{fig:pos-hyperdrop} \end{figure} Sizes of Time Distributed Dense layer, and CRF layer are the same, and equal to number of classes given dataset. Additionally size of ELMo encoding (and hence size of input to BiLSTM layer) is 1024. Due to time constraints we also used early stopping. Training was stopped either after 40 epochs or if validation accuracy improvement was smaller than 0.001 over 4 epochs. Lastly, for simplicity of this paper, we are not discussing impact of all hyperparameters here. However, they are discussed in depth, with plots in the \href{https://github.com/andreaskuster/uw-nlp/tree/master/pos}{experiment on the github repository}. \subsubsection{Results} We tested the optimal model on 5 different setups, as described in list of flavors in Sec. \ref{sec:datasets}. The dataset used in this case was PTB, with splits as described before. \begin{table}[h] \centering \begin{tabular}{|l|c|c|c|} \hline Exp. & Test (C) & Test (U) & Avg \\ \hline C & 97.30 & 88.29 & 92.78 \\ U & 96.51 & 96.51 & 96.51 \\ C + U & 97.51 & 96.59 & \textbf{97.05} \\ C + U 50 & 97.12 & 96.19 & \textbf{96.66} \\ TT & 95.04 & 95.04 & 95.04 \\ TA & 96.61 & 96.61 & 96.61 \\ \hline \end{tabular} \caption{Accuracies for the POS task, in our setup. Averages of two best performing flavors are emboldened for readability.} \label{tab:pos-results-our} \end{table} \begin{table}[h] \centering \begin{tabular}{|l|c|c|} \hline Exp. & Our & Original \\ \hline C & 92.78 & 93.26 \\ U & 96.51 & 97.45 \\ C + U & \textbf{97.05} & \textbf{97.57} \\ C + U 50 & \textbf{96.66} & \textbf{97.61} \\ TT & 95.04 & 95.21 \\ TA & 96.61 & 97.38 \\ \hline \end{tabular} \caption{Comparison of average accuracies for the POS task. Results on the left are using our code, while ones of the left are reported in the original paper. For both sources 2 best results are emboldened.} \label{tab:pos-results-comp} \end{table} We can see that we agree with the hypothesis that mixing cased and uncased provides the best performance on the Penn Tree Bank as can be seen in Table \ref{tab:pos-results-our}. This confirms the hypothesis. However, as in case of Truecasing, we conclude that in terms of absolute performance we did get different results as reference paper. This can be seen in Table \ref{tab:pos-results-comp}, where there is about 1\% difference in accuracy between our implementation and the original one. In addition to this the gain due to mixing cased and uncased data is sometimes smaller than the difference between implementations, such as in case of Uncased flavor, where reference has higher accuracy than our C + U flavor. Overall, \textbf{we confirm the hypothesis}, but with a grain of salt due to lack of transparency in implementation in the original paper. \subsection{Named Entity Recognition} \label{sec:exp-ner} \subsubsection{Model description} Since the model wasn't described well in the original paper we used a fork of \href{https://github.com/ZhixiuYe/NER-pytorch}{NER-pytorch} which implements ~\cite{DBLP:journals/corr/LampleBSKD16}. This implementation has a similar performance to that of the original paper. This model uses pre-trained GloVe embeddings concatenated with character embeddings trained on the training data using a bi-directional LSTM. The character embeddings length is one of the hyper-parameters for the model. Similar to the pos model, it uses a single BiLSTM followed by a highway layer and and output CRF layer. \subsubsection{Hyperparameters} The character embedding size and number of hidden units in LSTM served as hyper-parameters for the model. We evaluated the model on the validation set for character embeddings size 25 and 40 along with LSTM hidden layer size of 200, 300 and 400 and found that (25,200) performed slightly better. LSTM Dropout and learning rate could have been other hyper-parameters we could've optimized but we maintained them at 0.5 and 0.015 due to time constraints. \subsubsection{Results} We tested on the same 5 variants as in case of POS experiment. Dataset used was CoNLL2003, similarly to the original paper. Table \ref{tab:ner-results-our} shows results of these variants using our code. We can see that, mixing cased and uncased data is again a very good data pre-processing step, which provides respectively first and third highest F1 score. \begin{table}[h] \centering \begin{tabular}{|l|c|c|c|} \hline Exp. & Test (C) & Test (U) & Avg \\ \hline C & 90.63 & 81.47 & 86.05 \\ U & 89.72 & 89.72 & \textbf{89.72} \\ C + U & 90.10 & 88.65 & 89.38 \\ C + U 50 & 90.84 & 89.54 & \textbf{90.19} \\ TT & 80.89 & 80.89 & 80.89 \\ TA & 88.43 & 88.43 & 88.43 \\ \hline \end{tabular} \caption{F1 scores for NER task, in our setup. Averages of two best performing variants are emboldened.} \label{tab:ner-results-our} \end{table} Table \ref{tab:ner-results-comp} compares scores from our implementation to ones from \cite{ner-and-pos-original}. We can see there is a 1-2\% gap in absolute results between our implementation and the original. This can happen due to minor differences in the implementations. \begin{table}[h] \centering \begin{tabular}{|l|c|c|} \hline Exp. & Our & Original \\ \hline C & 86.05 & 63.46 \\ U & \textbf{89.72} & 89.32 \\ C + U & 89.38 & \textbf{90.49} \\ C + U 50 & \textbf{90.19} & \textbf{90.37} \\ TT & 80.89 & 82.93 \\ TA & 88.43 & 90.25 \\ \hline \end{tabular} \caption{Comparison of average F1 scores for the NER task. Results on the left are using our code, while ones of the left are reported in the original paper. For both sources 2 best results are emboldened.} \label{tab:ner-results-comp} \end{table} However, the most interesting difference in the cased variant, where there is a above 30\% gap between our and the original implementation. After closer investigation we discovered that the reason for it is huge different in its performance on uncased data (81.47 in our implementation vs 34.46 in original one). We do not have a firm intuition on why this is happening. However, it might be the case that models trained on cased dataset are highly unstable when tested on uncased data. Overall, however we can see that relative performance of results is similar, and mixing cased and uncased data provides the best performance with our implementation. Because of this we believe \textbf{our results support second hypothesis of the paper}. Another experiment that \cite{ner-and-pos-original} performed is testing how well their model from NER task transfers to other datasets. In particular they used the Twitter dataset, since it has very different properties than CoNLL2003. Table \ref{tab:ner-twitter-comp} compares performance of our implementation to the original one when transferred to the Twitter dataset. We see that our implementation performs much worse than the original one (by 30\% or more in each case). \begin{table}[h] \centering \begin{tabular}{|l|c|c|} \hline Exp. & Our & Original \\ \hline C & 33.24 & 58.63 \\ U & 14.54 & 53.13 \\ C + U & 31.31 & 66.14 \\ C + U 50 & 32.94 & 64.69 \\ TT & 23.45 & 58.22 \\ TA & 29.19 & 62.66 \\ \hline \end{tabular} \caption{Comparison of average F1 scores for the NER task, when transferred to the Twitter dataset. Results on the left are using our code, while ones of the left are reported in the original paper. Since, it is a direct comparison we do not embold any results.} \label{tab:ner-twitter-comp} \end{table} Comparison here is not relative, but direct, on absolute values, due to the fact that we are testing our performance on a new dataset. This means, that our model did much worse than the original one. This is very counterintuitive, when we consider the fact that in the original dataset, our cased experiment generalized much better. Our intuition is that there was a dropout used in the original implementation, which was undocumented in the paper. Overall, we \textbf{cannot support the thrid hypothesis from original paper}. \section{Experiments beyond the original paper} \subsection{Datasets} In addition to the datasets used in original paper, we additionally used: \begin{itemize} \item Brown - From \href{http://korpus.uib.no/icame/brown/bcm.html}{Brown University}. We used a loader from \href{https://www.nltk.org/}{nltk}. \item CoNLL2000 - From \cite{conll2000}. We used a loader from \href{https://www.nltk.org/}{nltk}. \item PTB Reduced (PTB R) - Same as Penn Tree Bank described in Section \ref{sec:datasets}, but: \begin{itemize} \item Train consists of sections 0-4 \item Validation consists of sections 5-6 \item Test consists of sections 7-8 \end{itemize} \end{itemize} \subsection{Part of Speech Tagging} The aim of the additional experiments is to find out if the hypothesis from the paper is more generally applicable. We run the same experiments on LSTM models with different word embeddings, with or without the CRF layer, and on different datasets. \subsubsection{Hyperparameters} Due to results in Section \ref{sec:exp-pos-hyper} the same model was used as in the original experiment. One important note for encodings is that they change number of inputs to BiLSTM layer, thus slightly modifying its size. In particular both GloVe and word2vec have size of 300, which is much smaller than 1024 in ELMo. \subsubsection{Results} Since there were 3 experiments roughly separate experiments let us present them separately. \begin{table}[h] \centering \begin{tabular}{|l|c|c|c|} \hline Exp. & word2vec & GloVe & ELMo \\ \hline C & 83.71 & 91.01 & 92.78 \\ U & 80.97 & 94.67 & 96.51 \\ C + U & \textbf{86.15} & \textbf{96.36} & \textbf{97.05} \\ C + U 50 & 85.01 & \textbf{95.35} & \textbf{96.66} \\ TT & 85.74 & 93.82 & 95.04 \\ TA & \textbf{86.64} & 95.20 & 96.61 \\ \hline \end{tabular} \caption{Average accuracies for different encodings. Note that ELMo encodings are the exact copy of results from \ref{tab:pos-results-our}. Similarly to other tables 2 best scores are emboldened for readability.} \label{tab:epos-encodings} \end{table} Table \ref{tab:epos-encodings} shows results for various encodings. We see that we consistently mixed dataset appears to be in the top 2 results. We can also see that encoding has a strong effect on absolute performance of the model, especially in case of word2vec which has a 10\% drop in accuracy relative to GloVe and ELMo. This is an expected behavior as ELMo is known to outperform both word2vec and GloVe in many cases. There is an interesting notion that both of Truecasing flavors tend to perform better than Half Mixed one. However, due to significant drop in word2vec with respect to other encodings we believe that it rather an artifact of the encoding itself rather than a theme. Additionally, performance of all three scenarios is rather close further pointing to encoding specific problem. Overall, we believe that the first additional hypothesis is \textbf{supported by the above results}. \begin{table}[h] \centering \begin{tabular}{|l|c|c|c|} \hline Exp. & PTB R & Brown & CoNLL2000 \\ \hline C & 92.36 & 89.50 & 92.86\\ U & 95.48 & 92.91 & 96.83 \\ C + U & \textbf{96.22} & \textbf{96.47} & \textbf{99.23} \\ C + U 50 & \textbf{95.71} & \textbf{93.92} & \textbf{97.16} \\ TT & 94.62 & 92.11 & 95.40 \\ TA & 95.35 & 92.62 & 96.79 \\ \hline \end{tabular} \caption{Average accuracies for different datasets.} \label{tab:epos-results-datasets} \end{table} Table \ref{tab:epos-results-datasets} shows results for different dataset. Here results are clear, mixing cased and uncased datasets provides the best result regardless of a dataset. In fact for Brwon CoNLL2000 gaps between this technique and others is even larger than on the original PTB dataset. Hence, these results \textbf{strongly support our second additional hypothesis}. \begin{table}[h] \centering \begin{tabular}{|l|c|c|} \hline Exp. & No CRF & CRF \\ \hline C & 92.88 & 92.78 \\ U & 96.52 & 96.51 \\ C + U & 97.02 & 97.05 \\ C + U 50 & 96.83 & 96.66 \\ TT & 94.78 & 95.04 \\ TA & 96.56 & 96.61 \\ \hline \end{tabular} \caption{Average accuracies for the original model with CRF and without CRF.} \label{tab:epos-results-crf} \end{table} Table \ref{tab:epos-results-crf} compares performance of the original model with or without the last layer. Considering that most of results are within a few hundredths between two models, with TT being 0.26\% away from the original implementation, we conclude that CRF can be eliminated from the model without major differences in performance. This is very beneficial on the implementation and runtime side, since CRF is a non-standard library, making model are complicated, and does not have multi-gpu support, causing it to run much longer. These results, \textbf{strongly support our third additional hypothesis}, and we recommend potential future users of BiLSTM to not include CRF layer (at least for POS task). \section{Computational requirements} Due to relative lack of overlap between three sections each of us used different computational resources: \begin{itemize} \item Truecasing: \begin{itemize} \item GPU: NVIDIA RTX 2080 Super \item CPU: AMD Ryzen 7 3700x \item Runtime: 10h (Original Paper) \end{itemize} \item POS: \begin{itemize} \item GPU: 2x NVIDIA V100 \item CPU: AMD EPYC 7501 \item Runtime: 100h (Original Paper + Additional Experiments) \end{itemize} \item NER: \begin{itemize} \item GPU: NVIDIA GTX 1080 \item CPU: Intel Core i7-6850K \item Runtime: 26h (Original Paper + Twitter Evaluation) \end{itemize} \end{itemize} This is required for every report. Include every item listed in the syllabus, plus any other relevant statistics (e.g. if you have multiple model sizes, report info for each). Include the information listed in the syllabus, including the \textbf{total} number of GPU hours used for all experiments, and the number of GPU hours for \textbf{each} experiment. \section{Discussion and recommendations} \cite{ner-and-pos-original} is mostly reproducible, with the three out of four original hypothesis reproduced in our report. In particular all hypotheses which were not involving transferring task to a new dataset were successful. We believe that transferring NER to Twitter is also a feasible claim, but involving a lot of additional tunning to the model, which should be described in the original paper. This is because of our additional experiments in POS, which shown that at least in that task, the same type of model is transferrable to other datasets. Most of the elements of this paper worked surprisingly well when reproduced, and implementation was not a significant struggle, due to libraries such as PyTorch and Keras. Our main concern is that if we were not provided access to datasets by TAs, it would be a much harder task, since two datasets used in the original paper were not available publicly. Our suggestion to future researches is to use publicly available datasets, if possible. Overall, we would suggest future researchers to use a mixed cased and uncased data in their work, since it provide a little development time overhead and no additional runtime. \bibliographystyle{acl_natbib}

BkiUgMvxK6wB9lIs_2gj

\section{Introduction} The classic particle-vortex duality~\cite{Peskin1978,Dasgupta1981} of bosonic systems revealed that different Lagrangians can secretly correspond to the same conformal field theory in the infrared limit. Recently the boson-fermion dualities~\cite{Polyakov1988,wufisher,Giombi2012,Aharony2012_1,Aharony2012_2,seiberg1}, fermion-fermion dualities~\cite{Son2015,MetlitskiDualDirac,WangDualDirac,mrossduality}, and many descendant 2+1D field theory dualities~\cite{WangO5,SSWW2016,Hsin2016,YouXu2015,Hsin2017,Aharony2016,Karch2016,Metlitski2017,Gaiotto2018,Gomis2017,KarchRobinson2017,GurAri2015,Benini2018,Jensen2017,Jensen2018,Cordova2017_1,Cordova2017_2,mengxu} have attracted a lot of attentions. The importance of these dualities are not just limited to their own theoretical interests. They also have close connections to gauge-gravity duality \cite{Giombi2012,Aharony2012_1,Aharony2012_2}, mirror symmetry in supersymmetric field theory~\cite{Kachru2017_1,Kachru2017_2,Karch2018}, fractional quantum Hall systems\cite{Fradkin1996,Son2015,Radicevic2016,Hui2018,WangSenthil2016,Sodemann2017} as well as deconfined quantum critical points~\cite{WangO5,dualnumerical}. These dualities have tremendously advanced our understandings of 2+1D conformal field theories. In fact, a large class of them are weaved into a duality web through SL$(2,\mathbb{Z})$ transformations on U$(1)$ currents and on the Chern-Simons terms of the U$(1)$ gauge fields\cite{SSWW2016,Witten2003}. While the duality web itself provides interesting cross-check among different dualities with U$(1)$ gauge fields, analytical derivation for certain members of the web are also obtained using wire constructions \cite{Mross2016,Mross2017}, deformation of exact supersymmetric dualities\cite{Kachru2017_2,Karch2018,GurAri2015} and lattice constructions \cite{Dasgupta1981,Chen2018} of Euclidean path integrals. Although some of the descendant dualities mentioned above have received rather positive numerical evidences~\cite{dualnumerical,karthik2017}, rigorously speaking most of the dualities are still conjectures. Apart from the consistency check on global symmetries, anomalies and matching global phase diagrams~\cite{Hsin2016,Hsin2017,Aharony2016,Karch2016,Metlitski2017,Gaiotto2018,Gomis2017,KarchRobinson2017,GurAri2015,Benini2018,Jensen2017,Jensen2018,Cordova2017_1,Cordova2017_2}, rigorous analytical results are much harder to obtain for dualities in the presence of non-Abelian gauge fields. Exact results, say, on partition functions, operator scaling dimensions, etc. in these dualities mostly rely on the large-$N$ limit\cite{Giombi2012,Aharony2012_1,Aharony2012_2,Banerjee2014,Jain2013,Maldacena2013} or well-established level-rank dualities of Chern-Simons gauge theories without dynamical matter fields \cite{Naculich1990,Mlawer1991,Naculich2007}. Derivation of dualities with non-Abelian gauge group of finite rank and with dynamical matter fields is certainly of great importance. Among all the approaches mentioned above, deriving the dualities via the lattice construction of the Euclidean path integral\cite{Dasgupta1981,Chen2018} stands out for its exactness and explicitness in connecting the degrees of freedom on the two sides of a duality without using any large-$N$ limit or supersymmetry. This method often only relies on the mild assumption that the theory defined on the lattice indeed flows under renormalization group to its natural continuum limit in the infrared (IR). Previously, the lattice construction has only been carried out for dualities with U(1) gauge fields\cite{Dasgupta1981,Chen2018}. In this paper, we generalize the lattice construction to dualities with orthogonal nonabelian gauge groups. We focus on the duality between a critical $\textrm{SO}(3)$ vector boson coupled a $\textrm{SO}(3)_1$ Chern-Simons gauge theory and a free massless Majorana fermion in 2+1D, which was proposed in Ref.~\onlinecite{Metlitski2017,Hsin2017}. We first construct the Euclidean spacetime lattice path integral of a theory with $\textrm{SO}(3)$ vector bosons and Majorana fermions coupled to a $\textrm{SO}(3)$ gauge field, which is connected to the theory of a single flavor critical boson coupled to a $\textrm{SO}(3)_1$ Chern-Simons term in the IR. We perform an {\it exact} mapping of this spacetime lattice path integral to that of a free Majorana fermion whose continuum limit agrees with the expected dual of the critical bosons. Generalizing this construction, we obtain a slightly different lattice duality between an interacting model with O(3) vector bosons and Majorana fermions coupled to O(3) gauge fields and the theory of free Majorana fermions. \section{Lattice Duality with \textrm{SO}(3) gauge fields} \label{Sec:SO(3)Duality} In Ref. \onlinecite{Metlitski2017,Hsin2017}, a duality between the $\textrm{SO}(3)$ critical boson coupled to $\textrm{SO}(3)_1$ Chern-Simons gauge theory and the free gapless Marjorana fermions in 2+1D was proposed. The continuum description of the boson side of the duality is given by the Lagrangian: \begin{align} \mathcal{L}_b = & |(\partial_\mu-ia_\mu )\phi|^2 + r |\phi|^2 + g|\phi|^4 \nonumber \\ & + \frac{i}{2\cdot 4\pi} \text{tr}_{\textrm{SO}(3)} \left( a \wedge da - \frac{2i}{3}a \wedge a \wedge a \right), \label{Eq:BosonContinuumTheory} \end{align} where $\phi$ is a 3-component real vector field coupled to an $\textrm{SO}(3)$ gauge field $a$. The $\textrm{SO}(3)$ gauge field $a$ is subject to a Chern-Simons term at level-1 in which the trace ``$\text{tr}$" is taken in the vector representation of $\textrm{SO}(3)$. This theory has an dual description of a single two-component Majorana fermion $\xi$ in 2 + 1D given by the simple Lagrangian \begin{align} \mathcal{L}_f = & \bar{\xi} \gamma^\mu\partial_\mu\xi - m\bar{\xi}\xi. \label{Eq:FermionContinuumTheory} \end{align} This duality is directly shown to hold in the gapped phases with $r$ and $m$ carrying the same sign\cite{Metlitski2017,Hsin2017}. It is further conjectured to be valid even at the critical point where $r=m=0$. In this section, we will start with formulating a lattice theory in close connection to the critical boson theory and constructing its path integral on the Euclidean spacetime lattice. We will then map this path integral exactly to that of a free Majorana fermion. As we will see later this exact mapping is valid regardless of whether the resulting phases are gapped or at the critical point. We will also refer to this type of exact mapping between two lattice theories as a lattice duality. After establishing the lattice duality, we will discuss the correlation functions and the $Z_2$ global symmetry across the dualtiy. \subsection{Euclidean Path Integral on the Lattice} We start with the basic ingredients for constructing the spacetime lattice path integral for the boson side of the duality. We consider a discretized 3D Euclidean spacetime lattice, which is taken to be a cubic lattice for simplicity. On each site $n$ of the spacetime lattice, there is a 3-component unit vector $v_n = (v_{n,1},v_{n,2},v_{n,3})^\intercal$ that represents the SO$(3)$ vector boson fields. The vector boson fields couple to their nearest neighbors and to the $\textrm{SO}(3)$ gauge field residing on the links that connect them, leading to the following contribution to the Euclidean action: \begin{align} S_{\rm bg}[v,O] = \sum_n \sum_{\mu=x,y,z} -J~ v_{n+\mu,i} O_{ij}^{n\mu} v_{n,j}, \label{Eq: Action_SO(3)_vectors} \end{align} where $\mu=x,y,z$ is summed over the unit lattice vector along the positive $x,y,$ and $z$ direction. $J$ is the coupling constant of the SO$(3)$ vector bosons, which we assume to be always positive. $O_{ij}^{n\mu}$ is an $\textrm{SO}(3)$ matrix that represents the $\textrm{SO}(3)$ gauge connection along the links between sites $n$ and $n+\mu$. The repeated $\textrm{SO}(3)$ vector/matrix indices $i,j$ are implicitly assumed to be summed automatically from 1 to 3. According to the Lagrangian Eq. \ref{Eq:BosonContinuumTheory}, we also need to introduce a Chern-Simons term for the gauge field $O^{n\mu}$. While it is difficult to directly write down the Chern-Simons term on the lattice, we can circumvent the difficulty by coupling the $\textrm{SO}(3)$ gauge field to a massive Majorana fermion. This method is a direct generalization of the construction of the U(1) Chern-Simons term in a lattice duality studied in Ref. \onlinecite{Chen2018}. In this method, the Chern-Simons term can be viewed as the outcome of integrating out the massive Majorana fermions. Following Wilson's approach\cite{Wilson1974,Wilson}, we can write down the action for lattice Majorana fermions coupled to the $\textrm{SO}(3)$ gauge field: \begin{align} & S_{\rm fg}[\chi,O] = \nonumber \\ & \sum_n \sum_{\mu = x,y,z } \bar{\chi}_{n+\mu,i} (\sigma^\mu -R) O^{n\mu}_{ij} \chi_{n,j} + M \sum_n \bar{\chi}_{n,i} \chi_{n,i} , \label{Eq: Action_SO(3)_Majorana} \end{align} Again, the repeated $\textrm{SO}(3)$ indices $i,j$ are automatically summed over. For each site $n$ and each $\textrm{SO}(3)$ color index $i$, the fermion field $\chi_{n,i} = (\chi_{n,i,1}, \chi_{n,i,2})^{\intercal}$ is a 2-component spinor consists of two real Grassmann numbers. $\sigma^\mu$ stands for the Pauli matrices. $\bar{\chi}_{n+\mu,i}$ is defined as $\bar{\chi}_{n+\mu,i} = \chi^{\intercal} \sigma^y$. When we turn off the gauge field $O^{n\mu}$, the action Eq. \ref{Eq: Action_SO(3)_Majorana} alone gives rise to $2^3=8$ Majorana fermions in the IR whose masses are controlled by the parameters $R$ and $M$. The mass configuration of the massive Majorana fermions determines the Chern number $C$ of the occupied bands of the Majorana fermions. To be more precise, by the band structure of Majorana fermions, we refer to the band structure obtained from quantizing the $8$ Majorana fermions. And by occupied bands, we mean all the negative energy states associated to the $8$ IR Majorana fermions after the quantization. The Chern number also serves as the level of the Chern-Simons term of the $\textrm{SO}(3)$ gauge field when the fermion field $\chi_n$ is integrated out in the action $S_{\rm fg}[\chi,O]$. The relation between the Chern number $C$ and parameters $R$ and $M$ that control the bare band structure of the Majorana fermions field $\chi_n$ is given by\cite{Golterman1993,Coste1989} \begin{align} C=\begin{cases} 2 {\rm sgn}(R) & 0 <|M| < |R|, \\ -{\rm sgn}(R) & |R| <|M| < 3|R|, \\ 0 & |M| > 3|R|. \\ \end{cases} \label{Eq:Majorana_Chern_Number} \end{align} Naively, by combining the action Eq. \ref{Eq: Action_SO(3)_vectors} and Eq. \ref{Eq: Action_SO(3)_Majorana}, one would have already had all the essential ingredients for a lattice version of the field theory Eq. \ref{Eq:BosonContinuumTheory}. However, for reasons that will become clear later, we would also like to include the interaction between the vector bosons and the Majorana fermions: \begin{align} & S_{\rm int}[\chi, v] = \frac{U_1}{4} \sum_{n,\mu} \varepsilon_{ii'i''}\varepsilon_{jj'j''} \mathcal{J}^{n\mu}_{ij} \mathcal{J}^{n\mu}_{i'j'} v_{n+\mu,i''} v_{n, j''} \nonumber \\ & ~~~~ + \frac{U_2}{36} \sum_{n,\mu} \varepsilon_{ii'i''}\varepsilon_{jj'j''} \mathcal{J}^{n\mu}_{ij} \mathcal{J}^{n\mu}_{i'j'} \mathcal{J}^{n\mu}_{i''j''} , \label{Eq:S_SO3_int_def} \end{align} where we've introduced the notation $\mathcal{J}^{n\mu}_{ij} \equiv \bar{\chi}_{n+\mu,i} (\sigma^\mu -R) \chi_{n,j} $ for the fermion current. $\varepsilon_{ii'i''}$ and $\varepsilon_{jj'j''}$ represent the totally-anti-symmetric tensor. $U_1$ and $U_2$ are coupling constants of these interactions in $S_{\rm int}$. It is straightforward to verify that $S_{\rm int}$ is invariant under the $\textrm{SO}(3)$ gauge transformations. Now, we are ready to introduce our Euclidean spacetime lattice path integral for one side of the duality: \begin{align} Z_b = \int D[O^{n\mu}] \int D[\chi_{n,i}] \int D[v_n] e^{-S_{\rm bg} - S_{\rm fg} - S_{\rm int }}, \label{Eq:Lattice_Boson_Fermion_theory} \end{align} where the integration of the gauge field $\int D[O^{n\mu}]$ is implemented as the integration of the matrix $O^{n\mu}$ on each link over the Haar measure of $\textrm{SO}(3)$. The boson field $v_n$ on each site is integrated over the unit 2-sphere $S^2$, while the fermion fields $\chi_{n,i}$ are integrated as real Grassmann numbers. As we discussed, when the coupling constants $U_1$ and $U_2$ are set to zero, after the Majorana fermions are integrated out, this model naturally realizes the theory of $\textrm{SO}(3)$ vector bosons coupled to a $\textrm{SO}(3)$ gauge field with the Chern-Simons level given by Eq. \ref{Eq:Majorana_Chern_Number}. When $U_1$ and $U_2$ are finite, the relation between the Chern-Simons level generated by the Majorana fermions and parameters $M$ and $R$ is expected to be modified. We will address the effect of $S_{\rm int}$ and how it alters the interpretation of the lattice path integral $Z_b$ as we proceed in obtaining its dual theory. \subsection{Exact Mapping of Lattice Path Integral} \label{Sec:ExactMap} We notice that the $\textrm{SO}(3)$ gauge field $O^{n\mu}$ in Eq. \ref{Eq:Lattice_Boson_Fermion_theory} can be integrated out analytically. In order to do so, we only need to consider the $S_{\rm bg}[v,O] $ and $S_{\rm fg}[\chi,O] $ parts of the action: \begin{align} & \int D[O^{n\mu}] ~ e^{-S_{\rm bg}[v, O]} e^{ - S_{\rm fg}[\chi, O] } \nonumber \\ & = \exp \left( -\sum_n M \bar{\chi}_{n,i} \chi_{n,i}\right) \times \prod_{n,\mu} \left[ \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{J^l}{l!} \frac{(-1)^m}{m!} \right. \nonumber \\ & ~~~~ \left. \int dO^{n\mu} \left(v_{n+\mu,i} O^{n\mu}_{ij} v_{n,j} \right)^l \left( \bar{\chi}_{n+\mu,i} (\sigma^\mu -R) O^{n\mu}_{ij} \chi_{n,j} \right)^m \right], \end{align} From now on, we will set $R=-1$ which renders the matrices $\sigma^\mu-R$ rank 1 and which combined with the fermionic statistics of the Grassmann numbers leads to vanishing contributions for all the terms with $m>3$ (see Appendix \ref{App:IntegrationSO(3)} for a more detailed explanation). Notice that the integration over all gauge field configurations $\int D[O^{n\mu}]$ factorizes into the integration of SO(3) matrices $O^{n\mu}$ under the Haar measure on each link. After conducting a term by term integration, we obtain that \begin{align} & \int D[O^{n\mu}] ~ e^{-S_{\rm bg}} e^{ - S_{\rm fg}} = \left(\frac{\sinh J}{J}\right)^{3N_s} e^{-S_{\rm bfg}[v,\chi]} \end{align} where $N_s$ is the number of sites in the Euclidean spacetime lattice and the effective action $S_{\rm bfg}[v,\chi]$ takes the form: \begin{align} S_{\rm bfg}[v,\chi] = & \sum_{n} M \bar{\chi}_{n,i} \chi_{n,i} \nonumber \\ &+\sum_{n,\mu} \Big\{ K (\bar{\chi}_{n+\mu,i} v_{n+\mu,i}) (\sigma^\mu -R) (\chi_{n,j} v_{n,j}) \nonumber \\ &- \frac{K}{4} \varepsilon_{ii'i''}\varepsilon_{jj'j''} \mathcal{J}^{n\mu}_{ij} \mathcal{J}^{n\mu}_{i'j'} v_{n+\mu,i''} v_{n, j''} \nonumber \\ & \left. + \frac{1-K^2}{36} \varepsilon_{ii'i''}\varepsilon_{jj'j''} \mathcal{J}^{n\mu}_{ij} \mathcal{J}^{n\mu}_{i'j'} \mathcal{J}^{n\mu}_{i''j''} \right\}. \label{Eq:Eff_Action_From_SO3} \end{align} with $K = \frac{J \cosh J - \sinh J}{J \sinh J}$ which is a positive number between 0 and 1 for all $J>0$. The details of the derivation of Eq. \ref{Eq:Eff_Action_From_SO3} can be found in Appendix \ref{App:IntegrationSO(3)}. Notice that the effective action $S_{\rm bfg}[v,\chi]$ contains similar interactions to the ones we introduce in $S_{\rm int}$. We will choose \begin{align} U_1 = K, ~~ U_2 = -1 + K^2, \label{Eq:Cancellation_Condition} \end{align} so that the interactions in $S_{\rm bfg}[v,\chi]$ and in $S_{\rm int}[v,\chi]$ exactly cancel off each other. For every fixed configuration of the vector field $v_n$, $S_{\rm bfg} + S_{\rm int}$ contains one and only one propagating Majorana fermion field \begin{align} \xi_n \equiv \sqrt{ K } \sum_i \chi_{n,i} v_{n,i}. \label{Eq:Fermion_Rewriting} \end{align} There are also two other Majorana fermion fields, denoted as $\xi'_n$ and $\xi''_n$, orthogonal to $\xi_n$ in the $\textrm{SO}(3)$ ``color space". For a fixed configuration of the vector boson $v_n$, we can perform a basis rotation from the $\chi_n$ field to $\xi_n$, $\xi_n'$ and $\xi_n''$ in the path integral. Notice that the fermion fields $\xi_n'$ and $\xi_n''$ only have local mass terms but not any kinetic terms (i.e. the $(\sigma^\mu-R)$ term). They can be directly integrated out producing a factor that only depends on the mass parameter $M$. Therefore, after integrating out $\xi_n'$ and $\xi_n''$ , we can write \begin{align} & \int D[O^{n\mu}] \int D[\chi_{n,i}] e^{-S_{\rm bg} - S_{\rm fg} - S_{\rm int }} \nonumber \\ & = \mathcal{N} \int D[\xi_n] \exp\left(-\sum_{n,\mu} \bar{\xi}_{n+\mu} (\sigma^\mu -R) \xi_n - \sum_n M' \bar{\xi}_{n} \xi_{n}\right), \label{Eq:Dual_Action_SO3} \end{align} where $\mathcal{N}$ is an overall normalization constant and the mass parameter $M'$ for the Majorana fermion mode $\xi_n$ is given by \begin{align} M' = M/K. \end{align} Now, we've arrived at the dual theory which exactly describes free Majorana fermions on the Euclidean spacetime lattice. Although Eq. \ref{Eq:Dual_Action_SO3} is derived on fixed configuration of the boson field $v_n$, its right hand side does not depend on $v_n$. The independence on the vector boson configuration $v_n$ is expected also from the fact that any fixed vector boson configuration $v_n$ can be connected to any other by $\textrm{SO}(3)$ gauge transformations before we perform any integration on the left hand side. Therefore, further integration over the boson field $v_n$ in Eq. \ref{Eq:Dual_Action_SO3} will only introduce an extra overall multiplicative factor and will not change the nature of the duality. At this point, we have constructed an exact mapping between the theory in Eq. \ref{Eq:Lattice_Boson_Fermion_theory} and free Majorana fermions described by the action in Eq. \ref{Eq:Dual_Action_SO3}. This exact mapping is valid regardless of whether or not the model $Z_b$ (and its dual Majorana fermion theory) is at the critical point or not. Having already set $R=-1$, if we further take $M'=3$, the dual Majorana fermion $\xi_n$, based on the change of Chern number for $M'>3$ and $M'<3$ givein in Eq. \ref{Eq:Majorana_Chern_Number}, is exactly at a critical point described by a single free gapless Majorana fermion in the IR. It implies that the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} with the following choices of parameter is also critical: \begin{align} M = 3 K, ~~~ U_1 = K, ~~ U_2 = -1 + K^2 . \label{Eq:critical_coupling} \end{align} Now, we would like to return to the discussion of the continuum limit of the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} at the criticality. First of all, $0<K<1$ for any positive coupling $J$. Hence, $M$ is always smaller than 3 at the critical point. If we choose $J$ such that $1<M<3$ and neglect the effect of the interaction in $S_{\rm int}$ as we first integrate out the fermions $\chi_n$ in the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory}, we would naturally identify the resulting model as a $\textrm{SO}(3)$ critical bosons coupled to an $\textrm{SO}(3)_1$ Chern-Simons gauge theory. The presence of finite $U_{1,2}$ complicates the integration over the fermions field $\chi_n$ in Eq. \ref{Eq:Lattice_Boson_Fermion_theory}. However, in a naive continuum limit of Eq. \ref{Eq:Lattice_Boson_Fermion_theory}, the interactions appear in $S_{\rm int}$ all contain high powers of spacetime derivatives and may be considered irrelevant in a continuum field theory. Furthermore, when the deviation of $M$ from 3 (as well as 1) where the bare mass of the Majorana fermion $\chi_n$ vanishes, is of order 1, the bare mass of the Majorana fermion $\chi_n$ in Eq. \ref{Eq:Lattice_Boson_Fermion_theory} is of order 1. Although Eq. \ref{Eq:critical_coupling} will require $U_{1,2}$ to be also of order 1 at the same time, the small prefactor in $S_{\rm int}$ in defining the coupling constant, as well as the high power of spacetime derivatives, may only lead to a small correction to the integration over $\chi_n$ in Eq. \ref{Eq:Lattice_Boson_Fermion_theory} compared to the case with vanishing $U_{1,2}$. Since the level of the Chern-Simons term has to be quantized, corrections induced by the interaction $S_{\rm int}$, if small, will not be able to change it. Hence, we speculate that the Euclidean spacetime lattice path integral Eq.~\ref{Eq:Lattice_Boson_Fermion_theory} still corresponds to the field theory Eq.~\ref{Eq:BosonContinuumTheory} in the IR with $g$ flows to a fixed point value. \subsection{Correlation Functions and Global Symmetry} \label{Sec:Global_Sym} Regardless of the interpretation of the theory Eq. \ref{Eq:Lattice_Boson_Fermion_theory} in the continuum limit, the mapping discussed above is explicit and exact, which allow us to identify not only the partition functions on both sides of the duality but also the correlation functions. In particular, any correlation functions of the Majorana fermion $\xi_n$ can be exactly reproduced in the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} by the operators defined in Eq. \ref{Eq:Fermion_Rewriting}. The exact mapping also helps us keep track of the global symmetry on the two sides of the duality. The model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} possesses a global $Z_2$ symmetry: \begin{align} Z_2: v_n \rightarrow - v_n,~~~ \chi_n \rightarrow -\chi_n, \end{align} while it is evident from Eq. \ref{Eq:Fermion_Rewriting} that the free Majorana fermion $\xi_n$ is neutral under this $Z_2$ symmetry. This apparent mismatch of the $Z_2$ global symmetry on the two lattice theories raises an potential subtlety in the IR duality of their speculated continuum counterparts Eq. \ref{Eq:BosonContinuumTheory} and Eq. \ref{Eq:FermionContinuumTheory}, where the $Z_2$ symmetry in the continuum field theories acts as $Z_2: \phi \rightarrow -\phi,~ \xi \rightarrow \xi$. This subtlety is not induced by the lattice construction and was already noticed in Ref. \onlinecite{Metlitski2017} for the continuum field theories. The conjectured resolution of this symmetry mismatch is that the $Z_2$ symmetry is not spontaneously broken and the energy gap of the $Z_2$ charged excitations remains finite across the critical point on the boson side of the duality. If we assume that the lattice theory Eq. \ref{Eq:Lattice_Boson_Fermion_theory} correctly captures in the continuum boson theory Eq. \ref{Eq:BosonContinuumTheory} in the IR, we can direct test the aforementioned conjecture. In the lattice theory Eq. \ref{Eq:Lattice_Boson_Fermion_theory}, there are two $\textrm{SO}(3)$ gauge-invariant $Z_2$-charged (local) operators: \begin{align} \epsilon_{ii'i''} v_{n,i} \chi_{n,i'} \chi_{n,i''}~~\text{and}~~\epsilon_{ii'i''} \chi_{n,i} \chi_{n,i'} \chi_{n,i''}, \end{align} where we have suppressed the spinor indices of the fermion field $\chi_n$. Being an bosonic operator that is $\textrm{SO}(3)$ gauge-invariant and odd under the $Z_2$ symmetry, $\epsilon_{ii'i''} v_{n,i} \chi_{n,i'} \chi_{n,i''}$ can serve as an order parameter for any potential spontaneous breaking of the global $Z_2$ symmetry in the theory Eq. \ref{Eq:Lattice_Boson_Fermion_theory}. Interestingly, we notice that the operator $\epsilon_{ii'i''} v_{n,i} \chi_{n,i'} \chi_{n,i''}$ is always proportional to the product of the fermion fields $\xi_n'$ and $\xi_n''$ that are introduced after we integrate out the $\textrm{SO}(3)$ gauge field in Eq. \ref{Eq:Eff_Action_From_SO3} and that are orthogonal to the low-energy fermion field $\xi_n$ defined in Eq. \ref{Eq:Fermion_Rewriting}. As stated previously, unlike the fermion field $\xi_n$, the fields $\xi_n'$ and $\xi_n''$ only have local mass terms but no kinetic term. Therefore, any correlation of the fields $\xi_n'$ and $\xi_n''$ is short-range, which implies that the correlation of the operator $\epsilon_{ii'i''} v_{n,i} \chi_{n,i'} \chi_{n,i''}$ is also short-range and, consequently, that the $Z_2$ global symmetry is unbroken. This statement is in agreement with Ref. \onlinecite{Metlitski2017}. Furthermore, it is evident directly from the derivation of the exact mapping in the previous subsection, that the only low-energy field at the critical point is the fermion field $\xi_n$ which is neutral under the global symmetry $Z_2$. That in turn implies that all the $Z_2$ charged excitations in the lattice theory Eq. \ref{Eq:Lattice_Boson_Fermion_theory} are gapped across the critical point. This observation offers another strong evidence for the conjecture on the $Z_2$ global symmetry proposed in Ref. \onlinecite{Metlitski2017}. \subsection{An Alternative Dual Model} \label{Sec:Alternative_Dual} In Sec. \ref{Sec:ExactMap}, we enforce the condition Eq. \ref{Eq:Cancellation_Condition} to ensure that the dual Majorana fermion $\xi_n$ is exactly free. In this subsection, we proceed with the lattice duality without this condition Eq. \ref{Eq:Cancellation_Condition}. In fact, for generic values of $U_{1,2}$ with $U_1 < K$, the path integral $e^{-S_{\rm bfg}-S_{\rm int}}$ (in a fixed background of vector boson configuration $v_n$) can be viewed as containing an $\textrm{SO}(2)$ gauge field in disguise on the dual side (see Appendix \ref{App:Derivation_Dual}). We will discuss the physical meaning of this $\textrm{SO}(2)$ gauge field later. To elucidate the $\textrm{SO}(2)$ gauge structure, we first introduce an $\textrm{SO}(2)$ gauge field represented by a $2\times 2$ orthogonal matrix $U^{n\mu}_{ab}$ (with $a,b=1,2$) on each link (connecting site $n$ and site $n+\mu$). Then, we construct the Majorana field $\eta_{n,a}$ that is charged under the $\textrm{SO}(2)$ gauge field: \begin{align} \eta_{n,1} = (K- U_1)^{\frac{1}{4}} \sum_i w_{n,i}' \chi_{n,}, \nonumber \\ \eta_{n,2} = (K- U_1)^{\frac{1}{4}} \sum_i w_{n,i}'' \chi_{n,i}, \label{Eq:eta_fermion_Def} \end{align} where $w_{n}'$ and $w_{n}''$ are two mutually orthogonal 3-component unit vectors that are both orthogonal to the fixed vector bosons field background $v_n$. The subscript $a$ in $\eta_{n,a}$ represents the SO(2) ``color" index. In fact, the fermion fields $\eta_{n,a}$ are essential the same as the fermion fields $\xi_n'$ and $\xi_n''$ discussed in Sec. \ref{Sec:ExactMap} but with an extra $(K-U_1)^{\frac{1}{4}}$ prefactor that makes the fields $\eta_{n,a}$ well-defined only when $K>U_1$. As we will see, unlike the non-propagating fermion fields $\xi_n'$ and $\xi_n''$ in Sec. \ref{Sec:ExactMap}, $\eta_{n,a}$ can be interpreted as propagating fermion fields in dual theory thanks to $K>U_1$. Having introduced these ingredients, we can rewrite $e^{-S_{\rm bfg}-S_{\rm int}}$ as (see Appendix \ref{App:Derivation_Dual} for detailed derivation) \begin{widetext} \begin{align} e^{- S_{\rm bfg} - S_{\rm int}} & = \exp \left[- \left( \sum_{n} M' \bar{\xi}_{n} \xi_{n} + \sum_{n,\mu} \bar{\xi}_{n+\mu} (\sigma^\mu -R) \xi_{n} \right) \right] \nonumber \\ & \times \int D[U^{n\mu}] \exp\left[-\left(\sum_n M'' \bar{\eta}_{n,a} \eta_{n,a} +\sum_{n,\mu}\bar{\eta}_{n+\mu,a} (\sigma^\mu -R) U^{n\mu}_{ab} \eta_{n,b} \right) \right] \nonumber \\ & \times \exp \left[ -V \sum_{n,\mu} \left( \bar{\xi}_{n+\mu} (\sigma^\mu -R) \xi_{n} \right) \left( \bar{\eta}_{n+\mu,1} (\sigma^\mu -R) \eta_{n,1} \right) \left( \bar{\eta}_{n+\mu,2} (\sigma^\mu -R) \eta_{n,2} \right) \right], \label{Eq:Alternative_Dual} \end{align} \end{widetext} where the repeated $\textrm{SO}(2)$ indices $a,b$ are automatically summed from 1 to 2. The integration of the SO(2) gauge configuration $\int D[U^{n\mu}] $ should be understood as the integration of the matrix $U^{n\mu}$ over the Haar measure of $\textrm{SO}(2)$ on every link of the spacetime lattice. The mass parameters $M'$, $M''$ of the fermion $\xi_n$ and $\eta_n$ are given by \begin{align} M' = M/K, ~~ M'' = M (K- U_1)^{-\frac{1}{2}}, \end{align} and the coupling constant $V$ by \begin{align} V= \frac{1+U_2 -K^2}{(K-U_1)K} \end{align} We can see that the right hand side of Eq. \ref{Eq:Alternative_Dual} is a theory that describes a Majorana fermion $\xi_n$ and Majorana fermions $\eta_{n,a}$ coupled to an $\textrm{SO}(2)$ gauge field. The two types of fermions interact with each other via the $\textrm{SO}(2)$ gauge-invariant interaction given in the third line of Eq. \ref{Eq:Alternative_Dual}. Physically, the $\textrm{SO}(2)$ gauge field introduced in the dual theory in Eq. \ref{Eq:Alternative_Dual} essentially recovers the residue $\textrm{SO}(2)$ subgroup of the $\textrm{SO}(3)$ gauge group in the original model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} in the presence of a fixed vector boson configuration $v_n$. The way to understand it is to notice that the fermions $\eta_{n,1}$ and $\eta_{n,2}$ that are charged under the $\textrm{SO}(2)$ gauge field are essentially the two fermion fields orthogonal to $\xi_n$ (as well as the fixed vector field value $v_n$) is the $\textrm{SO}(3)$ color space. The $\textrm{SO}(2)$ gauge group can be viewed as the residue gauge group of $\textrm{SO}(3)$ that preserves the vector field configuration $v_n$. As we tune to the parameter regime where $M'$ is close to 3, $K$ is close to $U_1$ and $V$ is small. The Majorana fermion $\chi_n$ will have a small bare mass, while the mass parameter $M''$ of the Majorana fermion $\eta_n$ can be set to be $M''\gg 3$ which ensures not only a big energy gap but also a trivial bare band stucture of $\eta_n$. Furthermore, if $V$ is tuned to a small value by tuning $U_2$, the dual theory will only have an ``almost free" (or weakly-interacting) Majorana fermion $\xi_n$ at low energy. Since the dual low-energy Majorana fermion $\xi_n$ is now weakly interacting, the exact position of the critical point of the dual theory Eq. \ref{Eq:Alternative_Dual} (as well as the original model Eq. \ref{Eq:Lattice_Boson_Fermion_theory}) is expected to be shifted from $M'=3$. \section{Lattice Duality with O(3) gauge fields} \label{Sec:O(3)Duality} In this section, we generalize the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} studied in the previous section to the case with $\textrm{O}(3)$ gauge fields. The action $S_{\rm bg}[v, O]$ given in Eq. \ref{Eq: Action_SO(3)_vectors} and $S_{\rm fg}[\chi, O]$ in Eq. \ref{Eq: Action_SO(3)_Majorana} can be directly promoted to accommodate $\textrm{O}(3)$ gauge field $O^{n\mu}$. We will rename them as $S_{\rm bg}'[v, O]$ and $S_{\rm fg}'[\chi, O]$ in order to distinguish them from their $\textrm{SO}(3)$-gauge-group counterpart. All the interactions in $S_{\rm int}$ are not invariant under the $\textrm{O}(3)$ gauge transforms. Although one can always consider a ``O(3)-gauged" version of $S_{\rm int}$, we will exclude them from the generalized model for simplicity. The Euclidean lattice path integral of the generalized model is given by \begin{align} Z_b' = \int D[O^{n\mu}] \int D[\chi_{n,i}] \int D[v_n] e^{-S'_{\rm bg} - S'_{\rm fg}}, \label{Eq:Lattice_Boson_Fermion_theory_O3} \end{align} where the integration of the boson field $v_n$ and the fermion field $\chi_n$ follow the same rules as before. The integration of the gauge field $\int D[O^{n\mu}]$ is now carried over the Haar measure of $\textrm{O}(3)$ instead of $\textrm{SO}(3)$. However, since we can write $\textrm{O}(3) = \textrm{SO}(3)\times Z_2$, the integration over $\textrm{O}(3)$ can be viewed as the integration over the Haar measure of $\textrm{SO}(3)$ together with an ``integration" over the $Z_2$ subgroup. Similar to the treatment of the previous section, we can first integrate out the $\textrm{O}(3)$ gauge field \begin{align} & \int D[O^{n\mu}] ~ e^{-S'_{\rm bg}} e^{ - S'_{\rm fg}} = \left(\frac{2\sinh J}{J}\right)^{3N_s} e^{-S'_{\rm bfg}[v,\chi]} \end{align} where the effective action $S'_{\rm bfg}[v,\chi]$ takes the form \begin{align} & S'_{\rm bfg}[v,\chi] = \nonumber \\ & \sum_{n} M \bar{\chi}_{n,i} \chi_{n,i} + \sum_{n,\mu} K (\bar{\chi}_{n+\mu,i} v_{n+\mu,i}) (\sigma^\mu -R) (\chi_{n,j} v_{n,j}), \label{Eq:Eff_Action_From_O3} \end{align} which is much simpler compared to Eq. \ref{Eq:Eff_Action_From_SO3} of the $\textrm{SO}(3)$ case. Again, we introduce the fermion field $\xi_n$ given in Eq. \ref{Eq:Fermion_Rewriting} and integrate out the orthogonal fermion fields $\xi_n'$ and $\xi_n''$ to obtain the following exact mapping \begin{align} & \int D[O^{n\mu}] ~ e^{-S'_{\rm bg}} e^{ - S'_{\rm fg}} = \nonumber \\ & \mathcal{N}' \int D[\xi] \exp\left(-\sum_{n,\mu} \bar{\xi}_{n+\mu} (\sigma^\mu -R) \xi_n - \sum_n M' \bar{\xi}_{n} \xi_{n}\right), \label{Eq:Dual_Fermion_O3} \end{align} where $\mathcal{N}'$ is an overall normalization constant and the mass parameter $M'$ given by \begin{align} M' = M/K. \end{align} Now, we've established an exact mapping from the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory_O3} to a model with the free Majorana freemion $\xi_n$ on the dual side. Again, we emphasize that this lattice duality is exact and is valid for any choice of coupling constant $J$ and mass parameter $M$. Based on the exact mapping, by setting $M'=M/K = 3$, we can tune both sides of the duality to the critical point where one side of the duality is captured by a massless free Majorana fermion in the IR. While one side of the duality is naturally identified with the free massless Majorana fermion in the continuum limit, the IR nature of the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory_O3} remains unclear. One may follow the same strategy as the previous section to integrate over the fermion field $\chi$ first and interpret the continuum theory as a vector boson coupled to a Chern-Simons term. However, it is unclear how the $Z_2$ part of the $\textrm{O}(3)$ gauge field behaves after we integrate out the fermions field $\chi_n$. A possible reconciliation with the duality studied in the previous section and the IR duality between the continuum field theories Eq. \ref{Eq:BosonContinuumTheory} and Eq. \ref{Eq:FermionContinuumTheory} is that only the $\textrm{SO}(3)$ subgroup of the $\textrm{O}(3)$ is deconfined. Another puzzling feature of this lattice duality with $\textrm{O}(3)$ gauge field is that while the dual Majorana fermion stays at the critical point, we can freely tune the parameters $K$ such that the bare band structure of the fermions $\chi_n$ in the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory_O3} can have different Chern numbers (and the naively expected Chern-Simons levels): $C=1$ when $1<M<3$ and $C=-2$ when $M<1$. We will defer resolution of these puzzles to future studies. Similar boson-fermion lattice duality with different gauge groups can also be constructed following the same method we discussed here. For example, as detailed in Appendix \ref{sec:Z2 gauge}, when the gauge group is reduced to $Z_2$, we can obtain a lattice duality between a free massless Majorana fermion and a $Z_2$ matter field coupled to a $Z_2$ gauge field. The lattice duality constructed in this section has an $\textrm{O}(3)$ gauge field on one side and a single Majorana fermion on the other. Interestingly, Ref. \onlinecite{Cordova2017} propose an IR duality with similar but yet distinct features. The proposed IR duality is between a critical vector boson coupled to an $\textrm{O}(3)_{1,1}^0$ Chern-Simons gauge theory (following Ref. \onlinecite{Cordova2017}'s notation) and a single Majorana fermion plus a decoupled $Z_2$ gauge theory. In fact, as detailed in Appendix \ref{App:Majorana_Z2}, a modified version of the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory_O3}, can be exactly mapped to a free Majorana fermion plus a decoupled $Z_2$ gauge theory on the lattice. The modified model is on its own is speculated to be connected to in the IR the critical vector boson coupled to an $\textrm{O}(3)_{1,1}^0$ Chern-Simons gauge theory. \section{Summary and Discussion} In this work, we construct the Euclidean spacetime lattice path integral of a theory with strongly interacting bosons and Majorana fermions coupled to $\textrm{SO}(3)$ gauge field and exactly map this theory to the theory of free Majorana fermions. This exact mapping or lattice duality is argued to be connected to the IR duality between critical bosons coupled to a $\textrm{SO}(3)_1$ Chern-Simons term and a single free Majorana fermion proposed in Ref.~\onlinecite{Metlitski2017,Hsin2017}. This lattice duality provides an exact mapping of the Euclidean path integrals and correlation functions between both sides of the duality, which allows us to obtain a strong evidence for the conjecture in Ref. \onlinecite{Metlitski2017} regarding the apparent mismatch of a global $Z_2$ symmetry on the two sides of the IR duality. This model is also generalized to lattice theories with $\textrm{O}(3)$ gauge fields and with $Z_2$ gauge fields respectively, which are both again exactly dualized to a free Majorana fermion. A different generalization of the lattice model with $\textrm{O}(3)$ gauge fields is shown to be dual to a free Majorana fermion plus a decoupled $Z_2$ gauge field. While the lattice dualities are exact, the fate of these strongly interacting lattice theories in the IR still needs more attention. In Sec. \ref{Sec:SO(3)Duality}, we argue that the model Eq. \ref{Eq:Lattice_Boson_Fermion_theory} with strongly-interacting boson and fermions coupled to $\textrm{SO}(3)$ gauge field should be related to the critical boson coupled to a Chern-Simons term in the IR. To solidify this argument, we need to understand the effect of $S_{\rm int}$, in particular, on the Chern-Simons level when we integrate out the fermion fields $\chi_n$ in Eq. \ref{Eq:Lattice_Boson_Fermion_theory}. Knowing that the gauge invariant fermion in the lattice duality has a critical point, the bosonic side of the duality should certainly also have a critical point, but whether this critical point really corresponds to the one described by the field theory Eq.~\ref{Eq:BosonContinuumTheory} is difficult to prove. This is certainly a question to be answered in the future. Similarly, as we discussed in Sec. \ref{Sec:O(3)Duality}, the IR nature of the theory of strongly-interacting boson and fermions coupled to the $\textrm{O}(3)$ gauge field given in Eq. \ref{Eq:Lattice_Boson_Fermion_theory_O3} also remains unclear. In particular, the role of the $Z_2$ subgroup of the $\textrm{O}(3)$ gauge group and the Chern number of the bare band structure of the fermion fields $\chi_n$ both require further investigation. In a broader picture, it is an interesting direction to generalize the lattice construction of dualities to interacting theories with more general non-Abelian gauge groups and more flavors of interacting bosons or fermions. While being exact without the use of any large-$N$ limit, holography or supersymmetry, the lattice construction is potentially a powerful tool in deriving previously proposal IR dualities with non-Abelian gauge fields as well as discovering new ones. Once the lattice construction of these duality is constructed, one should be able to read out the correspondence between the operators on both sides of the duality. {\it Note added:} After the completion of this work, we learned of Ref. \onlinecite{ChenDual2018}, which is a similar and independent attempt to generalize the lattice construction to tackle dualities with non-Abelian gauge fields. \begin{acknowledgements} The authors thank Cenke Xu for insightful discussions. CMJ thanks Jing-Yuan Chen for private communication on Ref. \onlinecite{ChenDual2018} prior to its arXiv posting. ZB acknowledges support from the Pappalardo fellowship at MIT. \end{acknowledgements}

BkiUa-s5qg5A50uBvmNF

\section{Introduction} The holographic principle in general~\cite{Maldacena:1997re,Erlich:2005qh}, and the D4-D8-D$\bar 8$ holographic set-up in particular~\cite{Sakai:2004cn} provide a framework for addressing QCD in the infrared in the double limit of a large number of colors and strong $^\prime$t Hooft gauge coupling $\lambda=g_{YM}^2N_c$. It is confining and exhibits spontaneous chiral symmetry breaking geometrically. The light meson sector is well described by an effective action with manifest chiral symmetry and very few parameters, yet totally in line with more elaborate effective theories of QCD~\cite{Fujiwara:1984mp}. The same set-up can be minimally modified to account for the description of heavy-light mesons, with manifest heavy quark symmetry~\cite{Liu:2016iqo,Liu:2017xzo,Liu:2017frj,Li:2017dml,Fujii:2020jre}. Light and heavy-light baryons are dual to instantons and instanton-heavy meson bound states in bulk~\cite{Hata:2007mb,Hashimoto:2008zw,Kim:2008pw,Hata:2007tn,Hashimoto:2009st,Lau:2016dxk}, providing a robust geometrical approach to the multi-body bound state problem. The holographic construction provides a dual realization of the chiral soliton approach and its bound states variants~\cite{Zahed:1986qz,Rho:1992yy}, without the shortcomings of the derivative expansion. It is a geometrical realization of the molecular approach ~\cite{Wu:2010jy,Karliner:2015ina}, without the ambiguities of the nature of the meson exchanges, and the arbitrariness in the choice of the many couplings and form factors~\cite{Lin:2019qiv}. Alternative holographic models for the description of heavy hadrons, have been developed in~\cite{Dosch:2015bca,Sonnenschein:2018fph}. Chiral symmetry constrains the light quark interactions, while heavy quark symmetry restricts the spin interactions between heavy quarks~\cite{Shuryak:1981fza,Isgur:1991wq}. Both symmetries are inter-twined by the phenomenon of chiral doubling~\cite{Nowak:1992um,Bardeen:1993ae,Nowak:2004jg} as shown experimentally in~\cite{Aubert:2003fg,Besson:2003cp}. A theoretical approach to the multiquark states should have manifest chiral and heavy quark symmetry, a clear organizational principle in the confining regime, and should address concisely the multi-body bound state problem. The holographic construction provides this framework. In~\cite{Liu:2017frj} two of us have analyzed the holographic baryon spectrum by considering three massless lavors $u,d,s$. The strange quark mass was introduced through a bulk instanton holonomy, assumed small and treated in perturbation theory. However, the strange quark mass is intermediate between heavy and light, and may require a treatment beyond perturbation theory. In this work, we will propose such a treatment. We will consider the kaon mass as large, and identify the strangeness brane as heavy in the formulation outlined in~\cite{Liu:2017xzo,Liu:2017frj,Liu:2021tpq}. Hyperons (baryons with strangeness $-1$) will be sought as bound states of massive kaons, to a bulk flavor instanton made of only the light $u,d$ flavors. The ensuing modular Hamiltonian for the hyperons, will be diagonalized without recourse to perturbation theory. This construction bears much in common with the revised bound state approach for hyperons in the Skyrme model~\cite{Callan:1985hy,Callan:1987xt}. The result is a much improved holographic hyperon spectrum, in comparison to the one in~\cite{Liu:2017frj}. The organization of the paper is as follows: In section~\ref{sec_modular}, we detail the modular Lagrangian for hyperons (baryons with strangeness $-1$) in leading order in the heavy meson mass expansion. We retain the exact kaon mass contribution, and the non-local contributions stemming from the Coulomb back-reaction and the bulk form of Gauss law. In section~\ref{sec_spectrum1} we show that the modular Hamiltonian maps onto the Landau problem of a particle in a magnetic field in 2-dimensions, plus spin-orbit coupling. In section~\ref{sec_spectrum2} we detail the hyperon spectra, including the strange pentaquark exotic $\Theta_s^+$, for two different approximations of the Gauss law contribution. Our conclusions are in~\ref{sec_conclusions}. A number of Appendices are added to complement the derivations in the text. \section{The Modular Lagrangian}~\label{sec_modular} The modular Lagrangian for the holographic description of heavy-light mesons bound to a bulk flavor instanton, has been discussed in~\cite{Liu:2017xzo,Liu:2017frj} for standard baryons, and for their exotics in~\cite{Liu:2019mxw,Liu:2019yye,Liu:2021tpq,Liu:2021ixf}. Here we propose to use it for kaons, assuming the strangeness to be a heavy flavor. In brief, we identify the strange heavy-light flavor field $\Phi$ in bulk with the $(0^-,1^-)$ kaon multiplet as $\Phi=(K,K^*)$, and proceed to bind it to a light flavor instanton as in~\cite{Liu:2017xzo,Liu:2021tpq}. The ensuing modular Lagrangian is composed of the collective variables $(X_i, a_I, \rho)$ for the instanton collective position, SU(2) orientation and size. In addition, when binding to the core instanton in bulk, the kaon multiplet transmutes to a 2-component complex modular coordinate $\chi$, via a zero-mode. In the analysis to follow for the hyperon spectrum, this coordinate will be quantized as a boson, much in the spirit of the bound state approach in the dual analysis in~\cite{Callan:1985hy,Callan:1987xt}. The fermionic statistics was considered in~\cite{Liu:2017xzo,Liu:2021tpq}, in the analysis of the much heavier baryons and their exotics, as it captures the key features of the heavier quark, and heavy quark symmetry. \subsection{General} The full holographic modular Lagragian for heavy-light kaons bound to a bulk flavor instanton, is given by (see Eq. 23 in~\cite{Liu:2021tpq}) \begin{align} \label{FULL} {\cal L}=&+\frac{1}{2}\dot \chi^{\dagger}\dot \chi+\frac{3i}{\tilde \rho^2}\chi^{\dagger}\dot \chi-\frac{37+12\frac{Z^2}{\rho^2}}{192}\chi^{\dagger}\chi\nonumber\\ &+\frac{78i}{5\tilde \rho^2}\chi^{\dagger}\tau^a\chi \chi^a -\frac{12}{5\tilde \rho^4} (\chi^\dagger \tau^a\chi)^2 \nonumber \\ &+\left(\frac{1}{4 }\frac{\dot\rho^2}{\rho^2}+\frac{ \dot a_I^2}{4}+\frac{\dot X^2}{4\rho^2}\right)\chi^{\dagger}\chi-\frac{1}{2}m_H^2\chi^{\dagger}\chi \nonumber\\ &+{\cal L}_{\Phi_0} [m_H] +{\cal L}_{\rm Coulomb}\,, \end{align} where $\rho$ is the size of the instanton with $\tilde \rho=16\pi^2a N_c \rho$. The rescaling $\chi\rightarrow e^{im_Ht}\sqrt{m_H}\chi$ with the heavy and bare mass $m_H$ of the $(K,K^*)$ multiplet, subsumed. The moduli of $SU(2)$ rotation reads $$\chi^a=Tr(\tau^a{a}_I^{-1}\dot{a}_I)\ .$$ The first two lines are standard, with the first term in the third line following from the coupling ${\rm tr } \Phi^2 \chi^{\dagger}\chi$, and leading to a non-vanishing correction to the metric in the space $y_I=(\rho, \rho a_I)$. The constraint field contribution ${\cal L}_{\Phi_0}[m_H]$ in the last line was analyzed in Appendices A.3 and A.4 of Ref.~\cite{Liu:2021tpq}, and is given by \begin{align}\label{eq:phi0finalX} {\cal L}_{\Phi_0}[m_H]= -\frac{1}{8}J_0^\dagger\frac{1}{-D_M^2+m_H^2} J_0\ , \end{align} with the {\it non-local} source $J_0$ given in (\ref{J0X}). It follows from the Gauss constraint on the flavor gauge field in bulk, and is by far the most involved to unravel. For convenience, we detail its analysis in Appendix~\ref{app_phi0},\ref{app_expansion}. Aside from the explicit mass dependence in (\ref{FULL}), there is an implicit mass dependence in ${\cal L}_{\Phi_0}[m_H]$ which we have noted in the argument. Since the strange mass is intermediate between light $u,d$ and heavy $c,b$, we will address the implicit mass dependence in ${\cal L}_{\Phi_0}[m_H]$ both in the light $m_H\rightarrow 0$, and heavy $m_H\rightarrow \infty$. The Coulomb contribution ${\cal L}_{\rm Coulomb}$ was originally detailed in Appendix~B in~\cite{Liu:2021tpq}, and for convenience, briefly reviewed in Appendix~\ref{app_coulomb}, with the result \begin{align}\label{eq:coulombfinalX} {\cal L}_{\rm Coulomb}=-J^\dagger_C\frac{1}{2\left(-aN_c \nabla^2+f^2\chi^{\dagger}\chi\right)}J_C \ , \end{align} The {\it non-local} source $J_C=\left(\rho^{cl}+\rho\right)$ is given in (\ref{RHOCL}-\ref{RHO}). Throughout, the Coulomb contribution which is small, will be mostly ignored. It is a correction to be added in perturbation theory to the modular Hamiltonian, and assessed only at the end. The holographic heavy kaon mass in the large mass limit, is given by~\cite{Liu:2016iqo} \begin{align} \label{KKX} M_K=m_H+\frac{M_{\rm KK}}{2\sqrt{2}} \ , \end{align} with $m_H$ the bare mass of the kaon doublet, and the Kaluza-Klein scale $M_{KK}=475$ MeV. In what will follow, $m_H\sim M_K$, unless specified otherwise. Note that a naive expansion of the Coulomb and Gauss constraint contributions in (\ref{FULL}) as shown in Appendix~\ref{app_naive}, leads to a degenerate but stable hyperon spectrum to order $m_H^0$, but unstable at sub-leading order. The unexpanded constraints produce a stable hyperon spectrum as we detail below. \subsection{ ${\cal L}_{\Phi_0}[0]$ and no Coulomb} We start the analysis of (\ref{FULL}) by considering the simple case with $m_H=0$ {\it only} in the Gauss constraint or ${\cal L}_{\Phi_0}[0]$, and no Coulomb back-reaction. Both approximations will be revisited below. With this in mind, the modular Lagrangian simplifies \be \label{QUAD1} {\cal L}_{\rm qua}=&&\frac{1}{2}\dot \chi^{\dagger}\dot \chi+\frac{3i}{\tilde \rho^2}\chi^{\dagger}\dot \chi-\frac{37+12\frac{Z^2}{\rho^2}}{192}\chi^{\dagger}\chi\nonumber\\ &&+\frac{99i}{40\tilde \rho^2}\chi^{\dagger}\tau^a\chi \chi^a-\frac{75}{8\tilde \rho^4}\chi^{\dagger}\chi \ . \ee Note that without the spin-orbit coupling, we have \begin{align} \label{SP0} {\cal L}_0=\frac{1}{2}\dot \chi^{\dagger}\dot \chi+\frac{3i}{\tilde \rho^2}\chi^{\dagger}\dot \chi-\frac{37+12\frac{Z^2}{\rho^2}}{192}\chi^{\dagger}\chi-\frac{75}{8\tilde \rho^4}\chi^{\dagger}\chi \ . \end{align} By setting the kaon modular variable $\chi$ as \begin{align} \label{chi12} \chi=\left( \begin{array}{c} x_1+iy_1 \\ x_2+iy_2 \end{array}\right) \ , \end{align} (\ref{SP0}) can be written as two harmonic oscillators coupled to magnetic field \be \label{QUAD2} {\cal L}_0=&&\frac{1}{2}(\dot {\vec{x_1}}^2+\dot {\vec{x_2}}^2)+\omega_c (y_1\dot x_1-x_1\dot y_1+y_2\dot x_2-x_2\dot y_2)\nonumber\\ &&-\frac{m_H^2+\Omega^2}{2}(\vec{x_1}^2+\vec{x_2}^2) \ , \ee where we have defined \begin{align} \Omega^2=\frac{75}{4\tilde \rho^4}+\frac{37+6\sqrt{6}\frac{1}{\tilde \rho^2}}{96} \ \ , \ \ \omega_c=\frac{3}{\tilde \rho^2} \ . \end{align} This observation will be exploited next. \section{Hyperon spectrum}~\label{sec_spectrum1} Following on the preceding arguments, we now analyze the modular Hamiltonian stemming from (\ref{QUAD1}). Without the spin-orbit contributions as we noted in (\ref{QUAD2}), it maps on the well-known Landau problem in 2-dimensions. In this regime, the hyperons are stable but degenerate. The spin-orbit contribution modifies the potential in the holographic $\rho$-direction, and lifts the hyperon degeneracy. \subsection{Landau problem} For the modular Lagrangian (\ref{QUAD2}), the pertinent Schroedinger equation reads \begin{align} H\phi_n(x)=E\phi_n \ , \end{align} with the modular Hamiltonian \begin{align} H=\frac{1}{2}D_i^{\dagger}D_i+\frac{\omega^2}{2}\bar z z \ , \end{align} with $z=x+iy$. The long derivative is $D_i=\partial_i-iA_i$, with the U(1) gauge field $A_i=\omega_c(y,-x)$. We now define the operators \be a&=&\frac{i}{2\sqrt{\omega_c}}(D_x-iD_y)\nonumber\\ &=&\frac{i}{2\sqrt{\omega_c}}(\partial_x-i\partial_y+\omega_c(x-iy)) \ , \nonumber\\ b&=&\frac{-i}{2\sqrt{\omega_c}}(-\partial_x-i\partial_y- \omega_c(x+iy)) \ , \ee which diagonalizes the kinetic contribution \begin{align} \frac{1}{2}D_i^{\dagger}D_i=\omega_c (2a^{\dagger}a+1) \ , \end{align} For the harmonic contribution, we note that \begin{align} b^{\dagger}-a=-i\sqrt{\omega_c}(x-iy) \ , \end{align} hence the Hamiltonian can be written as \begin{align} \label{HX1} H=\omega_c (2a^{\dagger}a+1)+\frac{\omega^2}{2\omega_c}(b^{\dagger}-a)(b-a^{\dagger}) \ . \end{align} The Hamiltonian (\ref{HX1}) can then be diagonalized with the help of the following Bogoliubov transformation \begin{align} a^{\dagger}=\cosh \theta A^{\dagger}+\sinh \theta B \ , \\ b^{\dagger}= \cosh \theta B^{\dagger}+\sinh \theta A \ . \end{align} Using $[A,A^{\dagger}]=[B,B^{\dagger}]=1$ and $[A,B]=[A,B^{\dagger}]=0$, which preserves the commutation relations, we can fix the value of $\theta$ as \begin{align} \tanh 2\theta=\frac{2\alpha}{1+2\alpha}\ , \,\,\, \alpha=\frac{\omega^2}{4\omega_c^2} \ . \end{align} The modular Hamiltonian without spin-orbit coupling is then diagonalized as \begin{align} \label{HX2} H=\frac{\Omega_++\Omega_-}{2} + \Omega_+A^{\dagger}A+\Omega_{-}B^{\dagger}B \ , \end{align} with \begin{align} \Omega_{\pm}=\sqrt{m_H^2+\Omega^2 +\omega_c^2}\pm \omega_c \ . \end{align} In the next subsection we will explore the spin-orbital contribution. \subsection{Spin-orbit} For fixed modular variable $\tilde\rho$, the holographic spectrum without spin-orbit following from (\ref{HX2}) is harmonic. Since the modular coordinate $\chi$ is quantized as a boson, the net spin and isospin of the hyperon core is determined by the instanton quantum moduli with $[IJ^P]=[\frac12\frac 12^+]$ assignment, in the absence of spin-orbit effects. With spin-orbit contributions, the resulting hyperon states carry $[\frac 12\oplus \frac 12,\big( \frac 12\oplus l\big)^+]$ assignments, for even $l$. We now proceed to analyze the dynamical effects of the spin-spin and spin-orbit effects. \subsubsection{The $l=0$ state} For $l=0$ and $J=\frac 12$, the energy level with $n$ $B^{\dagger}$ excitations is \begin{align} E_n-E_0=n\left(\sqrt{m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{ Z^2}{\rho^2}}{96}}-\frac{3}{\tilde \rho^2}\right) \ , \end{align} and the lowest one is archived for $n=1$. To proceed we need to fix the $\rho$ wave function. For that, the induced potential is given by $\Omega_-$ to which we add the harmonic oscillator potential term $\frac{1}{2}\omega_\rho^2\tilde \rho^2$, plus the quartic term $-\frac{3i}{5\tilde \rho^4 }\chi^{\dagger}\tau^a\dot \chi \chi^{\dagger}\tau^a \chi$ as in Eq.~43 in~\cite{Liu:2021tpq}. The result is \begin{widetext} \begin{align} \label{POTX1} V(\tilde \rho)=\frac{1}{2}\omega_\rho^2\tilde \rho^2+\sqrt{m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{Z^2}{\rho^2}}{96}}-\frac{3}{\tilde \rho^2}+\frac{9(\sqrt{m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{Z^2}{ \rho^2}}{96}}-\frac{3}{\tilde \rho^2})}{5(m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{\tilde Z^2}{\tilde \rho^2}}{96})\tilde \rho^4} \ . \end{align} \end{widetext} We note that (\ref{POTX1}) is stable for small $\rho$. The additional parameter $\delta$ captures a spin-spin ordering ambiguity to be discussed below. With this in mind, and using the estimate \begin{align} \frac{Z^2}{\rho^2} \approx \sqrt{\frac{3}{2}}\frac{1}{\tilde \rho^2} \end{align} the splitting between $\Lambda^0$ and nucleon, can be solved numerically for $\delta=1$. The result is \begin{align} M_{\Lambda^0}-M_{N}=0.237M_{\rm KK} \ , \end{align} For $M_{\rm KK}=0.475$ GeV, the splitting is about $112.7$ MeV, smaller than the empirical splitting of $177$ MeV. This is reasonable, since the omitted Coulomb back-reaction is positive (see below). \subsubsection{The $l=2$ state} For the $l\ne 0$ cases, the quantization needs to be considered more carefully, as operator ordering issues arise. Indeed, we note that the spin operator in the Bogoliubov transformed basis, reads \begin{align} \chi^{\dagger}\tau^a \chi =\frac{1}{\sqrt{m_H^2+\Omega^2+\omega_c^2}}(A^{\dagger}-B)_i\tau^a_{ij} (A-B^{\dagger})_j \ . \end{align} The spin-spin and spin-orbit effects will be treated in first order perturbation theory. When evaluating the average of $\chi^{\dagger}\tau^a \chi$, one recovers the standard Schwinger representation of a $\frac{1}{2}$-spin \begin{align} S^a=\frac{1}{2}B^{\dagger}_i\tau^a_{ij} B_j \ , \end{align} with $A_1, A_2$ constructed using $(x_1,y_1)$ and $(x_2, y_2)$, respectively. When evaluating $(\chi^{\dagger}\tau^a\chi)^2$, without normal ordering, gives \be &&\langle 0|B_i (A^{\dagger}-B)\tau^a (B^{\dagger}-A^{\dagger}) \nonumber\\ &&\times(A^{\dagger}-B)\tau^a (B^{\dagger}-A) B_i^{\dagger}|0\rangle =12 \ , \ee for $i=1,2$. With normal ordering, the result is different \be &&\langle 0|B_i :(A^{\dagger}-B)\tau^a (B^{\dagger}-A^{\dagger})\nonumber\\ &&\times (A^{\dagger}-B)\tau^a (B^{\dagger}-A): B_i^{\dagger}|0\rangle =6 \ . \ee The normal ordering ambiguity is captured by a c-number $\delta$. The third and perhaps most physical choice, amounts to dropping the anti-particle contribution to the spin through $A,A^\dagger$, \begin{align} \langle 0|B_i B\tau^aB^{\dagger} B\tau^a B^{\dagger} B_i^{\dagger}|0\rangle\nonumber =3 \ . \end{align} which is ordering free. This choice corresponds to $\delta=1$, and will be subsumed throughout. For $l=2,4,..$, one has $J=(l\pm 1)/2$. We first consider the $J=(l-1)/2$ case. Following our recent arguments in~\cite{Liu:2021tpq} (e.g. Eqs. 44-45), the effective potential reads \begin{widetext} \be && V\bigg(J=\frac{l-1}{2},\tilde \rho\bigg)=\frac{1}{2\tilde \rho^2}\left(l(l+2)-\frac{(l+2)\alpha N_c}{\sqrt{m_H^2+\Omega^2+\omega_c^2}\tilde \rho^2}+\frac{3 \alpha^2N_c^2}{4(m_H^2+\Omega^2+\omega_c^2)\tilde \rho^4}\right)\nonumber \\ &+&\frac{\omega_\rho^2}{2}\tilde \rho^2+\sqrt{m_H^2+\Omega^2+\omega_c^2}-\omega_c +\frac{9(\sqrt{m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{Z^2}{ \rho^2}}{96}}-\frac{3}{\tilde \rho^2})}{5(m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{ Z^2}{\rho^2}}{96})\tilde \rho^4} \ , \ee \end{widetext} with $\alpha=\frac{33}{10}$. The ${1}/{m_H^2}$ term due to the spin-orbit coupling is kept to maintain stability at small $\rho$. The change of the potential as one increases $m_H$ tends to decrease for larger $l$. For $l=2$, the the potentials at $m_H=2$ and $m_H=\infty$ differ moderately, but the net difference is small. Similarly, in the $J=\frac{l+1}{2}$ case the effective potential is \begin{widetext} \be &&V\bigg(J=\frac{l+1}{2},\tilde \rho\bigg)=\frac{1}{2\tilde \rho^2}\left(l(l+2)+\frac{l\alpha N_c}{\sqrt{m_H^2+\Omega^2+\omega_c^2}\tilde \rho^2}+\frac{3 \alpha^2N_c^2}{4(m_H^2+\Omega^2+\omega_c^2)\tilde \rho^4}\right)\nonumber \\ &&+\frac{\omega_\rho^2}{2}\tilde \rho^2+\sqrt{m_H^2+\Omega^2+\omega_c^2}-\omega_c+ \frac{9(\sqrt{m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{ Z^2}{ \rho^2}}{96}}-\frac{3}{\tilde \rho^2})}{5(m_H^2+\frac{111}{4\tilde \rho^4}+\frac{37+12\frac{Z^2}{\rho^2}}{96})\tilde \rho^4} \ . \ee \end{widetext} For $\delta=1$ and $m_H=M_{KK}$, a numerical analysis for the hyperon states gives \be &&[J=\frac{1}{2}, l=2, I=1]: M(\Sigma_{s}(1\frac{1}{2}^+))-M_N=302{\text MeV} \nonumber \\ &&[J=\frac 32, l=2, I=1]: M(\Sigma_{s}(1\frac{3}{2}^+))-M_N=501{\text MeV} \nonumber\\ \ee which are to be compared to the measured values of $254$ MeV and $444$ MeV. The splitting between the centroid is much more accurate \begin{align} M(\Sigma_{s}(1\frac{3}{2}^+))-M(\Sigma_{s}(1\frac{1}{2}^+))=199 {\text MeV} \ , \end{align} compared to $191$ MeV, empirically. \begin{widetext} \section{Hyperon spectrum revisited}~\label{sec_spectrum2} We now consider the hyperon spectrum with spin-orbit effect, but with ${\cal L}_{\Phi_0}[m_H]$ in the opposite limit of large $m_H$ for comparison. The details of ${\cal L}_{\Phi_0}[m_H]$ are presented in Appendix~\ref{app_phi0}, including its closed form results in the heavy mass limit. \subsection{Without Coulomb} In this case the potentials in the holographic $\rho$-direction are modified as follows \be V_{l=0}(\tilde \rho)=&&\frac{1}{2}\omega_\rho^2\tilde \rho^2+\bigg(m_H^2+\frac{9}{\tilde \rho^4}(1+\frac{4.11}{m_H^2\tilde \rho^2})+\frac{37+12\frac{ Z^2}{ \rho^2}}{96}\bigg)^{\frac 12}-\frac{3}{\tilde \rho^2}\nonumber\\ &&+\frac{9\bigg(\bigg({m_H^2+\frac{9}{\tilde \rho^4}(1+\frac{4.11}{m_H^2\tilde \rho^2})+\frac{37+12\frac{ Z^2}{ \rho^2}}{96}}\bigg)^{\frac 12}-\frac{3}{\tilde \rho^2}\bigg)}{5(m_H^2+\frac{9}{\tilde \rho^4}(1+\frac{4.11}{m_H^2\tilde \rho^2})+\frac{37+12\frac{ Z^2}{\rho^2}}{96})\tilde \rho^4} \ . \ee for $l=0$, and for $l=2$ \be &&V_{l}(J=\frac{l-1}{2},\tilde \rho)=V_{l=0}(\tilde \rho)+\frac{1}{2(1+\frac{1}{2m_H\tilde \rho^2})\tilde \rho^2}\left(l(l+2)-\frac{(l+2)\tilde \alpha N_c}{\sqrt{m_H^2+\tilde \Omega^2+\omega_c^2}\tilde \rho^2}+\frac{3\tilde \alpha^2N_c^2}{4(m_H^2+\tilde \Omega^2+\omega_c^2)\tilde \rho^4}\right) \nonumber\\ &&V_{l}(J=\frac{l+1}{2},\tilde \rho)=V_{l=0}(\tilde \rho)+\frac{1}{2(1+\frac{1}{2m_H\tilde \rho^2})\tilde \rho^2}\left(l(l+2)+\frac{(l)\tilde \alpha N_c}{\sqrt{m_H^2+\tilde \Omega^2+\omega_c^2}\tilde \rho^2}+\frac{3 \tilde \alpha^2N_c^2}{4(m_H^2+\tilde \Omega^2+\omega_c^2)\tilde \rho^4}\right) \nonumber\\ \ee \begin{table}[h] \caption{Hyperon and exotic spectrum} \begin{center} \begin{tabular}{cccccccccc} \hline \hline $B$ & $IJ^P$ & $l$ & $n_\rho$ & $n_z$ & Mass(small)& Mass(small with Coulomb)& Mass(large)&Mass(large with Coulomb) & Exp-MeV \\ \hline \hline $\Lambda_s$ &$0{\frac 12}^+$ & 0 & 0& 0& 962 & 1182 & 974 & 1152 & 1115 \\ $\Sigma_s$ &$1{\frac 12}^+$ & 2 & 0& 0& 1134 & 1315 & 1149 & 1306 & 1192 \\ &$1{\frac 32}^+$ & 2 & 0& 0& 1346 & 1472 & 1254 & 1398 & 1387 \\ $\Theta_s^+$ &$0{\frac 12}^+$ & 0 & 0& 0& - & 1617 & - & 1599 & \\ \hline \hline \end{tabular} \end{center} \label{tab_bindtetb} \end{table}% \end{widetext} with \be \tilde \alpha=&&\frac{13}{10}+\frac{162}{35\tilde m_H^2 \tilde \rho^4 } \ , \nonumber\\ \tilde \Omega^2=&&\frac{37+6\sqrt{6}\frac{1}{\tilde \rho^2}}{96}+\frac{9*4.11}{m_H^2\tilde \rho^2} \ . \ee Also, there is a modification to the curvature in the $\rho$ direction. We should also include the leading warping contribution at large $m_H$, the details of which are identical to those presented in~\cite{Liu:2021tpq}. With this in mind, the hyperon spectrum is now given by \be &&[J=\frac{1}{2}, l=0, I=0]: M(\Lambda)-M_N=68.1{\text MeV} \ ,\nonumber \\ &&[J=\frac{1}{2}, l=2, I=1]: M(\Sigma_{s}(1\frac{1}{2}^+))-M_N=289{\text MeV} \nonumber\\ &&[J=\frac{3}{2}, l=2 , I=1]: M(\Sigma_{s}(1\frac{3}{2}^+))-M_N=400{\text MeV} \nonumber\\ \ee The $J=\frac{1}{2}$ $\Sigma$ state is pushed up, and the $J=\frac{3}{2}$ $\Sigma$ state is pushed down, with a split in the centroid \begin{align} \frac{M_{\Sigma}(1\frac{1}{2}^+)+M_{\Sigma}(1\frac{3}{2}^+)}{2}-M_N=344{\text MeV} \ , \end{align} which is close to the empirical value of 349 MeV. \subsection{With Coulomb} As we indicated earlier, throughout we assumed $m_H\sim M_{KK}$ in (\ref{KKX}). Here we correct for this shortcoming, with \begin{align} m_H=0.68 M_{\rm KK} \ . \end{align} and $M_{\rm KK}=475$ MeV fixed by the light baryon spectrum~\cite{Hashimoto:2009st}. Also, the neglected Coulomb contribution can be estimated in perturbation theory, and in the heavy meson mass limit, it is about \begin{align} V_{\rm C}\approx \frac{83}{30 \tilde \rho^2} \ . \end{align} This provides for an upper bound estimate. With in mind, the modified hyperon masses are \be &&[J=\frac{1}{2}, l=0, I=0]: M(\Lambda)-M_N=214{\text MeV} \ , \nonumber\\ &&[J=\frac{1}{2},l=2, I=1]: M(\Sigma_{s}(1\frac{1}{2}^+))-M_N=368{\text MeV}\nonumber \\ &&[J=\frac{3}{2}, l=2 , I=1]: M(\Sigma_{s}(1\frac{3}{2}^+))-M_N=460{\text MeV}\nonumber\\ \ee The experimental values are $177$ MeV, $254$ MeV and $440$ MeV respectively, with $37$ MeV, $133$ MeV and $20$ MeV differences. Using the corrected value of $m_H$ above, and the upper estimate for the Coulomb correction, in Table~\ref{tab_bindtetb}, we collect all hyperon masses for the three approximations presented earlier. The chief observation is that the large mass analysis without Coulomb corrections appear closer to the empirical values of the lowest three empirical hyperons, without any adjustable parameter. These results are to be compared to those reported by Callan and Klebanov using the Skyrme model~\cite{Callan:1985hy, Callan:1987xt}, with also no Coulomb corrections. We recall that in the present holographic construction, the relative splitting between the hyperons, and the splitting of the hyperon centroid from the nucleon, which eliminate much of the uncertainty in $M_{KK}$, are in remarkable agreement with the empirically reported splittings. \subsection{Exotics} This approach extends to light multiquark exotics with open or hidden strangeness, much like the heavier multiquark exotics with open or hidden charm and bottom discussed in~\cite{Liu:2019mxw,Liu:2019yye,Liu:2021tpq,Liu:2021ixf}. In particular, an estimate of the mass of the strange pentaquark $\Theta_s^+$ (the exotic $uudd\bar s$), is given in Table~\ref{tab_bindtetb}. The mass of about 1600 MeV, stems mainly from the $\Omega_+$ frequency (anti-particle) which is $\frac{6}{\tilde \rho^2}$ higher than the $\Omega_-$ (particle). (Recall that the effective magnetic field induced by the bulk Chern-Simons interaction, is repulsive for particles, and attractive for anti-particles). An additional repulsion of about $\frac{3}{\tilde \rho^2}$ stems from the Coulomb back-reaction in the heavy mass estimate. A $\Theta_s^+$ of about 1600 MeV lies above the $nK^+$ threshold of 1434 MeV, and is unstable against strong decay. This result is consistent with the fact that the proposed $\Theta_s^+$ state~\cite{Jaffe:1976ii,Praszalowicz:1987em,Diakonov:1997mm,Praszalowicz:2003ik}, is in so far unaccounted for experimentally. \section{Conclusions}~\label{sec_conclusions} In the holographic construction presented in~\cite{Liu:2016iqo,Liu:2017xzo,Liu:2017frj}, heavy hadrons are described in bulk using a set of degenerate $N_f$ light D8-D$\bar 8$ branes plus one heavy probe brane in the cigar-shaped geometry that spontaneously breaks chiral symmetry. This construction enforces both chiral and heavy-quark symmetry and describes well the low-lying heavy-light mesons, baryons and multi-particle exotics~\cite{Liu:2019mxw,Liu:2019yye,Liu:2021tpq,Liu:2021ixf}. Heavy hadrons whether standard or exotics, are composed of heavy-light mesons bound to a core instanton in bulk. In~\cite{Liu:2017frj} the analysis of the hyperon spectrum was carried to order $m_H^0$ where spin effects are absent. In this analysis, we have now carried the analysis at next to leading order in $1/m_H$ where the spin-orbit and spin corrections are manifest. In contrast to~\cite{Liu:2017frj}, the modular fields were quantized as bosons and not fermions. The quantized Hamiltonian describes a particle in an external 2-dimensional magnetic field, with spin-orbit coupling. The hyperon spectrum with the Gauss constraint treated in both the heavy and light kaon mass limit, shows very small changes. It is in overall agreement with the empirical hyperon spectrum, and is much improved in comparison to the analysis in~\cite{Liu:2017frj}, where the strange mass was analysed perturbatively. This construction allows for the description of multiquark exotics with strangeness, and shows that the contentious exotic $\Theta_s^+$ is unbound. In a way, this construction should be regarded as the dual of the improved Callan-Klebanov construction for hyperons, as bound kaon-Skyrmions~\cite{Callan:1985hy,Callan:1987xt}. \vskip 1cm {\bf Acknowledgements} This work is supported by the Office of Science, U.S. Department of Energy under Contract No. DE-FG-88ER40388, by the Polish National Science Centre (NCN) Grant UMO-2017/27/B/ST2/01139, and by the 2021-N17/MNS/35 grant of the Faculty of Physics, Astronomy and Applied Computer Science of the Jagiellonian University.

BkiUa2fxK6EuNCvenjdh

\section*{Acknowledgments} The authors would like to thank Allianz, Amsterdam Institute for Advanced Metropolitan Solutions, Brose, Cisco, Ericsson, Fraunhofer Institute, Liberty Mutual Institute, Kuwait-MIT Center for Natural Resources and the Environment, Shenzhen, Singapore- MIT Alliance for Research and Technology (SMART), UBER, Vitoria State Government, Volkswagen Group America, and all the members of the MIT Senseable City Lab Consortium for supporting this research. Research of S.H.S. was supported by NSF Grants DMS-1513179 and CCF-1522054. \section*{Methods} We wish derive an expression for the sensing power of a vehicle fleet. We quantify this by their covering fraction $\langle C \rangle_{N_V}$, the average fraction of street segments covered at least once when $N_V$ vehicles move on the street network $S$ according to the urban explorer process, during a reference period $\mathcal{T}$. Given the non-trivial topology of $S$ and the non-markovian nature of the urban explorer process, it is difficult to solve for $\langle C \rangle_{N_V}$ exactly. We can however derive a good approximation. It turns out that it is easier to first solve for the `trip-level' $\langle C \rangle_{N_T}$ metric, that is, when $N_T$, the number of trips in the dependent variable, so we begin with this case (the `vehicle-level' expression $\langle C \rangle_{N_V}$ then follows naturally). Imagine we have a population $\mathcal{P}$ of taxi trajectories (defined, recall, as a sequence of street segments). The source of this population $\mathcal{P}$ is unimportant for now; it could come from a taxi (or fleet of taxis) moving according to the urban explorer process, or from empirical data, as we later discuss. Given $\mathcal{P}$, our strategy to find $\langle C \rangle_{N_T}$ is to map to a ``ball-in-bin process'': we imagine street segments as bins into which balls are added when they are traversed by a trajectory taken from $\mathcal{P}$. Note that, in contrast to the traditional ball-in-bin process, a random number of balls are added at each step, since taxis trajectories have random length. \textbf{Trajectories with unit length}. Let $L$ be the random length of a trajectory. The special case of $L = 1$ is easily solved, because then drawing $N_{T}$ trips at random from $\mathcal{P}$ is equivalent to placing $N_B$ balls into $N_S$ bins, where $N_S$ is the number of segments, and each bin is selected with probability $p_i$. As indicated by the notation, we estimate these with the segment popularities discussed in the main text (we discuss this more later). Let $\vec{M} = (M_1, M_2, \dots, M_{N_S})$, where $M_i$ is the number of balls in the $i$-th bin. It is well known that the $M_i$ are multinomial random variables, \begin{equation} \vec{M} \sim \text{Multi}(N_{T}, \vec{p}) \end{equation} where $\vec{p} = (p_1, p_2, \dots p_{N_S})$. The (random) fraction of segments covered is \begin{equation} C = \frac{1}{N_S} \sum_{i=1}^{N_S} 1_{(M_i \geq 1)} \end{equation} where $1_A$ represents the indicator function of random event $A$. The expectation of this quantity is \begin{equation} \langle C \rangle_{(N_T, L=1)} = \frac{1}{N_S} \sum_{i=1}^{N_S} \mathbb{P}_{N_T}(M_i \geq 1) \label{tempC} \end{equation} \noindent (note we introduce $L$ as a subscript for explanatory purposes). The number of balls in each bin is binomially distributed $M_i \sim Bi(N_B, p_i)$. The which has survival function $\mathbb{P}(M_i \geq 1) = (1 - (1-p_i)^{N_B})$. Substituting this into \eqref{tempC} gives the result \begin{equation} \langle C \rangle_{(N_B, L=1)} = 1- \frac{1}{N_S} \sum_{i=1}^{N_S} (1-p_i)^{N_{B}}. \label{C_Lone} \end{equation} \textbf{Trajectories with fixed length}. Trajectories of fixed (i.e. non-random) length $L > 1$ impose \textit{spatial} correlations between the bins $M_i$ (recall that in the classic ball and bin problem, the $M_i$ are already correlated, since their sum is constant and equal to the total number of balls added $N_B$). This is because trajectories are contiguous in space; a trajectory that covers a given segment is more likely to cover neighboring segments. Given the non-trivial topology of the street network $S$, the correlations between bins are hard to characterize. To get around this, we make the strong assumption that for $N_T \gg 1 $ the spatial correlations between bins are asymptotically zero. This assumption greatly simplifies our analysis. It lets us re-imagine the ball-in-bin process so that adding a trajectory of length $L$ is equivalent to adding $L$ balls into \textit{non-contiguous} bins chosen randomly according to $p_i$. Then, selecting $N_T$ trajectories of length $L$ from $\mathcal{P}$ is equivalent to throwing $N_B = L*N_T$ balls into $N_S$ bins $\langle C \rangle_{(N_T,L_{fixed})} = \langle C \rangle_{(N_T*L,L = 1)}$. Hence the expected coverage is a simple modification of \eqref{C_Lone}: \begin{equation} \langle C \rangle_{(N_B, L=1)} = 1- \frac{1}{N_S} \sum_{i=1}^{N_S} (1-p_i)^{ L* N_T}. \label{C_Lfixed} \end{equation} Assuming neighboring segments are spatially uncorrelated is a drastic simplification, and effectively removes the spatial dimension from our model. Yet surprisingly, as we will show, it leads to predictions that agree well with data. \textbf{Trajectories with random lengths}. Generalizing to random $L$ is straightforward. Let $S_{N_T} = \sum_{i=1}^{N_T} L_i$ be the number of segments covered by $N_T$ trajectories. By the law of total expectation \begin{equation} \langle C \rangle_{(N_T,L)} = \sum_{n=0}^{\infty} \langle C \rangle_{(n,L_{fixed})}\mathbb{P}(S_{N_T} = n). \label{temp} \end{equation} The first term in the summand is given by \eqref{C_Lfixed}. For the second term we need to know how the trajectory lengths are distributed. In Supplementary Figure 4 we show $L \sim Lognormal(\tilde{\mu}, \tilde{\sigma}^2) $. It is known that a sum of lognormal random variables is itself approximately lognormal $S_{N_T} \sim Lognormal(\mu_S, \sigma_S^2)$, for some $\mu_S$ and $\sigma_S$. There are many different choices for $\mu_S, \sigma_S$; for a review see \cite{lognormal}. We follow the Fenton-Wilkinson method, in which $\sigma_S^2 = \ln{( \frac{\exp{{\tilde{\sigma}^2} -1}}{N_T} + 1 )}$ and $\mu_S = \ln{ (N_T \exp( \tilde{\mu}))} + (\tilde{\sigma}^2 - \sigma_S^2) / 2$. Then, \begin{equation} \mathbb{P}(S_{N_T} = n) = \frac{1}{ n \sigma_S \sqrt{2 \pi}} e^{ - \frac{ (\ln{n} - \mu_S)^2}{2 \sigma_S^2} }. \end{equation} Substituting this into \eqref{temp} gives \begin{equation} \langle C \rangle_{(N_T, L)} = \frac{1}{N_S n \sigma_S \sqrt{2 \pi}} \sum_{n=0}^{\infty} \sum_{i=1}^{N_S} \big( 1 - (1-p_i)^n \big) e^{ - \frac{ (\ln{n} - \mu_S)^2}{2 \sigma_S^2} }. \label{semifinal_C} \end{equation} The above equation fully specifies the desired $\langle C \rangle_{(N_T,L)}$. It turns out however that the sum over $n$ is dominated by its expectation, so we collapse it, replacing $n$ by its expected value $\langle L \rangle*N_T$. This yields the much simpler expression $ \langle C \rangle_{(N_T,L)} = \langle C \rangle_{(N_T*\langle L \rangle, L = 1)}$, or \begin{equation} \langle C \rangle_{N_T} \approx 1 - \frac{1}{N_S} \sum_{i=1}^{N_S} (1-p_i)^{\langle L \rangle*N_T} \label{final_C_mbar1_trip} \end{equation} which appears in the main text. \textbf{Extension to vehicle level}. Translating our analysis to the level of vehicles is straightforward. Let $B$ be the random number of segments that a random vehicle in $\mathcal{V}$ covers in the reference period $\mathcal{T}$ (in Supplementary Figure 4 we show how $B$ are distributed in our data sets). Then we simply replace $\langle L \rangle$ with $\langle B \rangle$ in the expression for $\langle C \rangle_{N_T}$ to get $\langle C \rangle_{N_V}$, \begin{equation} \langle C \rangle_{N_V} \approx 1 - \frac{1}{N_S}\sum_{i=1}^{N_S} (1-p_i)^{\langle B \rangle*N_V}. \label{final_C_mbar1_vehicle} \end{equation} \noindent \textbf{Model parameters}. The parameters $\langle L \rangle, \langle B \rangle$ in \eqref{final_C_mbar1_vehicle} as easily estimated from our data sets (see Supplementary Note 2). The bin probabilities $p_i$ are trickier. They have a clear definition in the ball-in-bin formalism, but in our model, the interpretation is not as clean; they represent the probability that a \textit{subunit} of a trajectory taken at random from $\mathcal{P}$ covers the $i$-th segment $S_i$. As mentioned above, we estimate these with the segment popularities, which we calculate in two ways: (i) deriving them directly from our data sets; or (ii) from the urban explorer process (recall these methods led to similar distributions of $p_i$ as shown in Fig.~\ref{schematic}(e)). \bibliographystyle{naturemag} \section*{Supplementary Figures} \begin{figure}[H] \centering \includegraphics[width= 0.6 \columnwidth]{all_ps.png} \caption{\textbf{Empirical segment popularities.} Log log plot of the distributions of segment popularities for each city. The curves are similar, but not universal as discussed in Supplementary Note 2. Note each curve has a non-straight tail, indicating a deviation from Zipf's law.} \label{ps_all_cities} \end{figure} \begin{figure}[H] \centering \includegraphics[width= 0.75 \columnwidth]{ue_versus_data_all_cities.png} \caption{\textbf{Urban explorer segment popularities versus data.} Segment popularities $p_i$ derived from the urban explorer process (blue) and empirical data sets (orange) for all cities. $10^7$ timesteps were after which the distribution of $p_i$ were stationary. The bias parameters were (a) $\beta = 2.75$ (b) $\beta = 1.5$ (c) $\beta = 3.0$ (d) $\beta = 0.25$, (e) $\beta = 0.25$ (f) $\beta = 1.0$ (g) $\beta = 1.0$ (h) $\beta = 1.75$ (i) $\beta = 1.25$ (j) $\beta = 0.75$.} \label{ue_versus_data} \end{figure} \clearpage \begin{figure}\centering \includegraphics[width= 0.5 \linewidth]{survival_prob.png} \caption{\textbf{Approximation of multinomial survival function}. Survival probability for multinomial distribution estimated from \eqref{final}, and via Monte Carlo. $10^5$ trials were used in each Monte Carlo approximation. 50 bins were used, with $p_i = 1/50$. The survival probability is defined as $\mathbb{P}(M_1 > b, M_2 > b, \dots)$. Here we took $b = 5$. Note the excellent agreement between theory and simulation (both curves lie on top of each other)} \label{approx} \end{figure} \begin{figure} [H] \centering \includegraphics[width= 0.75 \linewidth]{daily_length_trip_all_cities_fitted.png} \caption{\textbf{Distributions of trajectories lengths for the trip-level datasets}. Histograms of the trajectory lengths $L$ during a given day for city. Red dotted lines show lognormal curves of best fit. We list the parameters of best fit $\mu, \sigma$, the sample mean $\langle L \rangle $, and the day the data were taken from for each subplot. Notice Chicago appears to have two humps. Data taken from other days are qualitatively similar. (a) Yangpu, 04/02/15, $(\mu, \sigma, \langle L \rangle) = (3.36, 0.52, 29.6)$ (b) NYC, 01/05/11, $(\mu, \sigma, \langle L \rangle) = (3.37, 0.57, 30.8)$ (c) Chicago, 05/21/14, $(\mu, \sigma, \langle L \rangle) = (3.36, 0.98, 51.02)$ (d) Vienna, 03/25/11, $(\mu, \sigma, \langle L \rangle) = (3.91, 0.51, 45.54)$ (e) San Fransisco, 05/24/08, $(\mu, \sigma, \langle L \rangle) = (2.99, 0.87, 28.90)$ (f) Singapore, 02/16/11, $(\mu, \sigma, \langle L \rangle) = (3.97, 0.68, 60.78)$.} \label{daily_L} \end{figure} \begin{figure} [H] \centering \includegraphics[width= 0.75 \linewidth]{daily_length_vehicles_all_cities_fitted.png} \caption{\textbf{Distributions of distance traveled by taxis for vehicle-level datasets}. Histograms of $B$, the distance traveled (measured in segments) by a taxi in a day for each city. Red dotted lines show lognormal curves of best fit. We list the parameters of best fit $\mu, \sigma$, the sample mean $\langle B \rangle $, and the day the data were taken from for each subplot (a) Beijing 03/02/13, $(\mu, \sigma, \langle B \rangle) = (5.56, 0.65, 245)$ (b) Changsha 03/02/14 $(\mu, \sigma, \langle B \rangle) = (5.46, 0.31, 131)$ (c) Hangzshou 04/22/14 $(\mu, \sigma, \langle B \rangle) = (5.13, 0.18, 366)$ (d) Shanghai 03/02/14 $(\mu, \sigma, \langle B \rangle) = (5.41, 0.35, 270)$.} \label{daily_L_vehicles} \end{figure} \begin{figure}[H] \centering \includegraphics[width= 0.75 \columnwidth]{temporal_fluctuations1.png} \caption{\textbf{Temporal fluctuations of trip-level datasets}. Generally speaking little daily variation in each quantity. (a) Number of scannable street segments (b) Best fit exponent in truncated power law $\alpha$. (c) Average length of trajectory (d) Total number of trips.} \label{temp_fluct} \end{figure} \begin{figure}[H] \centering \includegraphics[width= 0.70 \columnwidth]{temporal_fluctuations_vehicles.png} \caption{\textbf{Temporal fluctuations of vehicle-data}. Generally speaking little daily variation in each quantity. (a) Number of scannable street segments (b) Best fit exponent in truncated power law $\alpha$. (c) Average daily distance traveled by a taxi (d) Total number of trips.} \label{temp_fluct_vehicles} \end{figure} \begin{figure} [H] \centering \includegraphics[width= 0.75 \columnwidth]{model_versus_theory_all_cities1.png} \caption{\textbf{Minimum street sampling problem}. Analytic prediction versus trip-level data. The red curve shows theoretical results, while the black curve shows probabilities estimated from data. The parameters for each subplot were $\bar{C} = 0.5$, $m = 1$. The number of trials used in the Monte Carlo estimate of $\mathbb{P}(C)$ was 1000. (a) Yangpu on 04/02/15 (b) NYC on 01/05/11 (c) Chicago on 05/21/14 (d) Vienna on 03/25/11 (e) San Fransisco on 05/24/08 (f) Singapore on 02/16/11} \label{P_all_cities} \end{figure} \begin{figure} [H] \centering \includegraphics[width= 0.75 \columnwidth]{scaling_collapse_vehicles_data_all_days.png} \caption{\textbf{Scaling collapse of vehicle-level data on different days} Counterpart of Figure 3 in the main text. As can be seen, a close approximation to a true scaling collapse is achieved only on Tuesday. Note the Hangzhou dataset has strong variations. This is not surprising, since as shown in Figure~\ref{temp_fluct_vehicles}, this dataset has strong temporal variations. In particular, $\langle B \rangle$ varies much more than the other datasets.} \label{scaling_collapse_vehicles} \end{figure} \begin{figure} [H] \centering \includegraphics[width= 0.6 \columnwidth]{scaling_collapse_trips_greater_range_clean.png} \caption{\textbf{Scaling collapse of trip-level data}. In contrast to vehicle-level datasets -- Supplementary Figure~\ref{scaling_collapse_vehicles} -- the trip-level datasets do not show universal behavior. There are however some trends. As can be seen the Chicago, San Francsico, and Yangpu datasets collapse to a common curve, where the other datasets do not. The data for each city are the same as those used in Figure 2 (main text). Trip data on different days show the same trends.} \label{scaling_collapse_trips} \end{figure} \begin{figure} [H] \centering \includegraphics[width= 0.6 \columnwidth]{segments_manhattan.png} \caption{\textbf{Average segment coverage versus number of sensor-equipped taxis in Manhattan on 03/08/2011}. Different colors show results for different scanning thresholds. That is, the \% of segment at least $m$ times, where $m = 1,2,3,4$. Black lines show one standard deviation away from mean value. Notice that just $10$ vehicles scan more than a third of segments, while 30 scan more than half.} \label{manhattan_segments} \end{figure} \section*{Supplementary Note 1} \textbf{Data sets}. We have 10 real-world data sets from 9 cities: New York, Chicago, Vienna, San Francisco, Singapore, Beijing, Changsha, Hangszhou, and Shanghai. We had two independent data sets for Shanghai, independent in the sense they occurred on different years (2014 and 2015). For 2015 dataset, we selected only those trips starting and ending in the subcity ''Yangpu'', and hereafter consider it a separate city. The datasets were collected from various sources. Those from Beijing, Changsha, and, Hangszhou were provided by a third-party organization that collected driving data from taxi operation companies. The Shanghai datasets were provided by the `'1st Shanghai Open Data Apps 2015" (an annual competition).The New York dataset has been obtained from the New York Taxi and Limousine Commission for the year 2011 via a Freedom of Information Act request. The Vienna and Singapore datasets were provided to the MIT SENSEable City Lab by AIT and the Singapore government, respectively. The San Francisco and Chicago data sets were publicly available \cite{sanfran_dataset}, \cite{chicago_dataset}. Note the NYC, Vienna, San Francisco, and Singapore datasets were the same as used in previous studies \cite{santi2014}, \cite{scaling_law}. The four data sets from Chinese cities were very large ($\sim$ GB worth of data per day). For computational convenience, we therefore subsampled the datasets by selecting only those trips which occurred in a $20$ km box surrounding the city center. The city center was found using OpenStreetMap, the GPS coordinates of which were $(39.9059631, 116.3912480)$, $(28.1979483, 112.9713300)$, $(30.2489634, 120.2052342)$, $(31.2253441, 121.4888922)$ respectively. The temporal range of the data sets was not uniform. NYC was the most comprehensive, consisting of a years worth of taxi trips in Manhattan. The remaining data sets were for one week. The sizes of the cities was also different. We show this in Figures~\ref{temp_fluct}(a) and~\ref{temp_fluct_vehicles}(a) by showing $N_S$, the number of scannable segments, for each city over the course of a week. \begin{table}[] \centering \label{data_table} \caption{Properties of data sets} \label{data} \begin{tabular}{|p{2.2 cm}|p{2.0cm}|p{1.5 cm}|p{5.0cm}|p{1.1 cm}|p{0.9 cm}|p{1.0 cm}|} \hline \textbf{City} & \textbf{Trajectories} & \textbf{Taxi Ids} & \textbf{Temporal range} & $N_{S,total}$ & $N_S$ & $\frac{N_{S,total}}{N_S}$ \\[0.15 cm] \hline Yangpu & Real (GPS) & Yes & 1 Week: \; 04/01/15 -- 04/04/15 & 2919 & 2657 & 0.94 \\ NYC & Generated & Yes & 1 Year: \; 12/31/10 -- 12/31/11 & 7954 & 7265 & 0.91\\ Chicago & Generated & No & 1 Week: \; 06/23/14 -- 06/30/14 & 24054 & 12492 & 0.52 \\ Vienna & Generated & No & 1 Week: \; 03/07/11 -- 10/07/11 & 24054 & 15775 & 0.66 \\ San Francisco & Generated & No & 1 Week: \; 05/21/08 -- 05/28/08 & 15453 & 11708 & 0.76 \\ Singapore & Generated & No & 1 Week: \; 02/21/11 -- 02/28/11 & 32362 & 25255 & 0.78 \\ Beijing & Real (GPS) & Yes & 1 Week: \; 03/01/14 -- 03/07/14 & 54665 & 27024 & 0.49 \\ Changsha & Real (GPS) & Yes & 1 Week: \; 03/01/14 -- 03/07/14 & 18067 & 9882 & 0.55 \\ Hangszhou & Real (GPS) & Yes & 1 Week: \; 04/21/15 -- 04/28/15 & 39056 & 16125.0 & 0.41 \\ Shanghai & Real (GPS) & Yes & 1 Week: \; 03/01/14 -- 03/07/14 & 49899 & 21002 & 0.49 \\[0.1 cm] \hline \end{tabular} \end{table} Each data set consists of a set of taxis trips. The representation of these trips differs by data set. For the Chinese cities, a trip is the set of GPS coordinates of the taxis position as its serves its passenger. Since in our model we represent cities by a street \textit{networks}, we convert the set of GPS coordinates to a \textit{trajectory}. (Recall in the main text we defined a trajectory as a sequence of street segments $T_r = (S_{i_1}, S_{i_2}, \dots)$). We matched the taxi trajectories to OpenStreetMap (driving networks) following the idea proposed in \cite{GPS_convert}, which using a Hidden Markov Model to find the most likely road path given a sequence of GPS points. The HMM algorithm overcomes the potential mistakes raised by nearest road matching, and is more robust when GPS points are sparse. For the remaining data sets, each trip $i$ is represented by a GPS coordinate of pickup location $O_i$ and dropoff location $D_i$ (as well as the pickup times and dropoff times). As for the Chinese cities, we snap these GPS coordinates to the nearest street segments using OpenStreetMap. We do not however have details on the trajectory of each taxi (the intermediary path taken by the taxi when brining the passenger from $O_i$ to $D_i$.) We thus needed to approximate trajectories. We had two methods for this, one sophisticated, one simple. The sophisticated method was for the Manhattan dataset. Here, as was done in \cite{santi2014}, we used hour-by-hour variability in the traffic congestion, we did X. For the remaining cities, we used the simple method of finding the weighted shortest path between $O_i$ and $D_i$ (where segments were weighted by their length). As we will show, in spite of the different representations of trajectories, the sensing properties of the taxi fleets from each city are very similar. This gives us confidence in X. Lastly, for five of the nine cities -- the Chinese cities plus NYC --- taxi trips are recorded with the ID of the taxi which completed that trip. Hence for these `vehicle-level' datasets we can calculate $\langle C \rangle_{N_V}$ -- the sensing potential of a fleet as a function of the number of constituent vehicles $N_V$ directly. For the remaining cities, it is unknown which taxis completed which trips. Hence for these 'trip-level' data sets, we can solve only for $\langle C \rangle_{N_T}$. Hence we hereafter divide our datasets into these two categories -- `vehicle-level' and 'trip-level' -- and use these terms throughout the paper. For the sake of comparison, we decided to consider NYC and Yangpu part of the trip-level datasets. That way, the three different representation of trajectories feature in the trip levels datasets, giving more confidence in their results. We summarize all the properties of dataset discussed above in Supplementary Table 1 In the main text, we noted that we analyzed subsets of the real-world street networks, those sub-networks containing only 'potentially scannable' edges. We had two reasons for doing this. First, there are some streets which are never traversed by taxis in our data sets; since there are permanently out of reach of taxi-based sensing, we do not consider them. Second, since city borders aren't well defined, the total number of streets in a city $N_{S, total}$ is also ill-defined (potential exceptions being Manhattan and Singapore, which have sharp borders). In light of these two complications, we consider only those streets which were traversed at least once by taxis in our dataset $N_S$, which in general is different to $N_{S,total}$. \section*{Supplementary Note 2} \textbf{Estimation of parameters from data sets.} There are three parameters in our model: $p_i$, the segment popularities, $B$, the random distance traveled by a taxi randomly selected from $\mathcal{V}$, and $L$, the random length of a trip (recall $B$ is needed for the vehicle-level data for which $\langle C \rangle$ is a function of $N_V$, the number of vehicles, and $L$ is needed for trip-level data, for which $\langle C \rangle$ function of $N_T$.) Supplementary Figures~\ref{daily_L} and~\ref{daily_L_vehicles} shows the distributions $\mathbb{P}(L)$ and $\mathbb{P}(B)$ for each city on a given day. Note that while $L$, the random length of a trajectory (measured in segments), can be directly found from our datasets, $B$, the total distance traveled (again measured in segments) by a taxi during the reference period $T$ is not. This is because our data sets contain taxi \textit{trips} only -- a trip implying a passenger is on board -- and do not include the distance traveled by taxis when they are empty. Hence, estimating $B$ from our datasets constitutes a \textit{lower} bound for the true $B$. Coming back to Supplementary Figures~\ref{daily_L} and~\ref{daily_L_vehicles}, we see the distribution $\mathbb{P}(L), \mathbb{P}(B)$ are well fit by lognormals (shown as red curves in the figures). The lognormal fits well in all cases, with exceptions being Chicago, and to a lesser extent, San Francisco (which is contrast to the others appears to be monotonically decreasing). In panel (c) of Supplementary Figures~\ref{temp_fluct} and~\ref{temp_fluct_vehicles} we show how $\langle L \rangle$ varies by day. There is little variation. In the other panels of these two figures, we show how the number of scannable street segments $N_S$, the total number of trips, the number of active taxis, as well as $\alpha$ (a parameter characterizing the distribution of the segment popularities $p_i$ -- we will discuss this parameter shortly) vary by day. In most cases, there is also little variation. These are encouraging findings, since they indicate the behavior of our model is general, not giving (significantly) different results on different days of the week. To test the universalities in $p_i$, we fit each dataset to various heavy tailed distributions, listed in equation~\eqref{heavy_tailed}. \begin{align} P_{exponential}(x) &= \lambda e^{- \lambda (x - x_{min})} \nonumber \\ P_{power \; law}(x) &= (\alpha-1) x_{min}^{\alpha-1} x^{-\alpha} \nonumber \\ P_{log \; normal}(x) &= x^{-1} \exp(-\frac{(\log x - \mu)^2}{2 \sigma^2} ) \nonumber \\ P_{stretched \; exponential}(x) &= \beta \lambda x^{\beta -1} e^{-\lambda (x^{\beta}- x_{min}^{\beta})} \nonumber \\ P_{truncated \; power law}(x) &= \frac{\lambda^{1-\alpha}}{\Gamma(1-\alpha,\lambda x_{min})} x^{-\alpha} e^{- \lambda x} \label{heavy_tailed} \end{align} We performed the fitting using the python package `powerlaw'. By default this package determines a minimum value $p_{min}$ below which data are discarded. Since we want to model the full $\mathbb{P}(p)$ (and not just the tail), we set this equal to the minimum value in our datasets. We show the results of the fittings in Table 2. For each city, either a truncated power laws or stretched exponentials was selected as the distribution which fit the data best. Thus, we only report the parameters for those two distributions (the parameters are defined by equations~\eqref{heavy_tailed}). As detailed in documentation of `powerlaw', parameters of best fit are found by maximum liklihood estimation. We estimated errors in these parameters by bootstrapping: new data sets $(p_i^*)_{i^*=1}^{N_S}$ were drawn uniformly at random from the original data set $(p_i)_{i=1}^{N_s}$ 1000 times, best fit parameters were found for each of these 1000 realization, the standard deviation of which was taken as the standard error in each parameter. The `goodness of fit' measure for each distribution is quantified by the KS (kolmogorov-smirnoff) parameter $D$, defined by \begin{equation} D = \max_x \Big| CDF_{empirical}(x) - CDF_{theoretical}(x) \Big| \end{equation} \noindent where smaller $D$ values indicate better fits, and where $CDF$ denotes the cumulative density function. Finally, the liklihood-ratio test was used to compare the distribution of one fit to another. This has two parameters $\Lambda, p_1$. The sign of $\Lambda$ tells which distribution is more likely to have generated the data (positive means the first, negative means the second), while the $p_1$-value gives a measure of the confidence in the value of $\Lambda$ (the smaller, the more confident). We adopt the convention that $\Lambda > 0$ indicates the stretched exponential is preferred over the truncated power law (and $\Lambda < 0$ indicates the opposite). As can be seen in Supplementary Table 2, the tests tell us $\mathbb{P}(p)$ of three of cities are best modeled by stretched exponentials, while the others are best modeled by truncated power laws. The values for $p_i$ were all $< O(10^{-26})$ (and as small as $O(10^{-222})$), so we truncated all values to zero. There are some mild similarities in the best fit parameters, but no evidence of a convincing trend. Hence, we conclude that the segment popularity distributions $\mathbb{P}(p)$ are not universal. Like $\mathbb{P}(L)$ and $\mathbb{P}(B)$, there is little daily variation in $\mathbb{P}(p)$. We demonstrate this in Supplementary Figure~\ref{temp_fluct}(b) and~\ref{temp_fluct_vehicles}(b) where we show the maximum liklihood exponent $\alpha$ of the truncated power law fit measured day-by-day (for clarity, we do not display the $\beta$ parameter of the stretched exponential, but they show the same trends). \begin{widetext}[h!] \vspace{1cm} \centering \begin{tabular}{ |P{2.5cm}|P{6cm}|P{5 cm}|P{2 cm}|} \multicolumn{4}{c}{\textbf{Maximum liklihood parameters}} \\ \hline & $(\lambda, \beta, D)_{\text{ \color{red}{stretched \; exponential}}}$ & $(\lambda, \alpha, D)_{\text{trunc. power law}}$ & $(\Lambda,p_1)$ \\ \hline Yangpu & $\Big( (1.3 \pm 0.1)*10^{6}, 0.266 \pm 0.003, 0.08 \Big)$ & $(5830 \pm 10, 1.132 \pm 0.004, 0.07)$ & $(-1327,0)$ \\ NYC & $\Big((15 \pm 4)*10^{3}, 0.499 \pm 0.005, 0.03 \Big)$ & $(780 \pm 20, 1.00 \pm 10^{-6},0.25)$ & $(1600,0)$ \\ Singapore & $\Big( (591 \pm 8)*10^{3}, 0.499 \pm 0.004, 0.02 \Big)$ & $(3400 \pm 200, 1 \pm (6*10^{-8}), 0.2)$ & $(3282,0)$ \\ Chicago & $\Big( (3.4 \pm 0.9)*10^{6}, 0.187 \pm 0.005, 0.04 \Big)$ & $(650 \pm 30, 1.170 \pm 0.006, 0.03)$ & $(-208,0)$ \\ San Francisco & $\Big( (47 \pm 7)*10^{5},0.257 \pm 0.005, 0.03 \Big)$ & $(1330 \pm 50, 1.156 \pm 0.008, 0.04)$ & $(-218,0)$ \\ Vienna & $\Big( (41 \pm 0.5)*10^{5}, 0.293 \pm 0.006, 0.04 \Big)$ & $(2420 \pm 80, 1.196 \pm 0.008, 0.05)$ & $(-278,0)$ \\ Beijing & $\Big( (1.2 \pm 0.4)*10^{5}, 0.42 \pm 0.002, 0.06 \Big)$ & $(5940 \pm 10, 1.00 \pm 10^{-6}, 0.08)$ & $(824,0)$ \\ Changsha & $\Big( (7.5 \pm 0.2)*10^{5}, 0.34 \pm 0.003, 0.04 \Big)$ & $(1750 \pm 10, 1.02 \pm 0.02, 0.04)$ & $(248,0)$ \\ Hangzhou & $\Big( (1.3 \pm 0.2)*10^{6}, 0.23 \pm 0.003, 0.05 \Big)$ & $(1770 \pm 20, 1.16 \pm 0.004, 0.04)$ & $(560,0)$ \\ Shanghai & $\Big( (7.8 \pm 0.4)*10^{5}, 0.43 \pm 0.004, 0.05 \Big)$ & $(4970 \pm 10, 1.00 \pm 10^{-6}, 0.06)$ & $(564,0)$ \\ \hline \label{MLE_estimates} \end{tabular} \vspace{1cm} \end{widetext} \textbf{Compare $C_{model}$ and $C_{data}$}. In the main text we compare our expression for $\langle C \rangle$ against data for a given reference period of a day. The empirical $\langle C \rangle$ were found by subsampling the datasets on a given day; random subsets were drawn from a day's worth of trips, and the average fraction of segments covered by those subsets was computed. As mentioned in the main text, we tested the analytic prediction two ways: using $p_i$ estimated by the stationary distributions of the urban explorer process (dashed line), and also directly from our datasets (thick line). In the latter case we calculated the distribution of $p_i$ for each day of the week (excluding Sunday), then used those to calculate six separate $\langle C \rangle$, the average of which is shown. This way, both temporal fluctuations and the bias of using the same datasets to estimate $p_i$ and the empirical $\langle C \rangle$ (which recall was calculated for a \textit{single} day) were minimized. We discuss this in more detail in Supplementary Note 2. For both these cases, the parameter $\langle B \rangle$ was estimated from datasets. In Supplementary Figures 4 and 5 we show the empirical distributions of $p_i$, $B$, and $L$, and show in Supplementary Figures 6 and 7 that they do not vary significantly on different days of the week. \section*{Supplementary Note 3} \textbf{Scaling Collapse}. We first discuss the vehicle-level data. In the main text we derived \begin{equation} \langle C \rangle_{(N_V)} = 1 - \frac{1}{N_S}\sum_{i=1}^{N_S} (1-p_i)^{\langle B \rangle*N_V}. \label{temp} \end{equation} which contains the parameters $p_i, \langle B \rangle$ and $N_S$. Since $p_i$ and $N_S$ specify the distribution of $P(p_i)$, and since the $\mathbb{P}(p)$ are approximately universal across cities (see Supplementary Figure~\ref{ps_all_cities}), we only need to remove the parameter $\langle B \rangle$ from \eqref{temp} to make it city independent. Thus the simple rescaling $N_V \rightarrow N_V / \langle B \rangle$ gives the city-independent quantity \begin{equation} \langle C \rangle_{(N_V / \langle B \rangle)} = 1 - \frac{1}{N_S}\sum_{i=1}^{N_S} (1-p_i)^{N_V}. \end{equation} We plot this in Supplementary Figure~\ref{scaling_collapse_vehicles} for different days. As can be seen, the quality of the data collapse varies by day. Hangzhou varies the most, which is to be expected, since it is this dataset set which has the highest temporal variation, as shown in Supplementary Figure~\ref{temp_fluct_vehicles}. We apply the same procedure to the trip-level data, except now the scaling is $N_T \rightarrow N_T / \langle L \rangle$. Figure~\ref{scaling_collapse_trips} shows the result. A universal scaling collapse is absent, although there is some similarities between the data sets; Chicago, Yangpu, and San Francisco and nearly coincident. The lack of full universal behavior is perhaps due to the inferior quality of the trip-level datasets (inferior because the trajectories are inferred). \section*{Supplementary Note 4} We here give explicit values for $N_T^*$ and $N_V^*$ the numbers of trips and vehicles needed to cover half of the city's scannable street segments, i.e. the solutions to $\langle C \rangle(N_T^*) = 0.5$ and $\langle C \rangle(N_V^*) = 0.5$. \begin{table}[h!] \centering \label{data_table_Nstart} \caption{Coverage statistics. $N_{T,total}$ refers to the total number of trips occurring on the specified day.} \label{N_star_data} \begin{tabular}{|p{1.5 cm}|p{1.5cm}|p{1.5 cm}|P{1.75cm}|P{1.5cm}|} \hline \textbf{City} & $N_T^*$ & $N_{T,total}$ & $N_T^* / N_{T,total}$ & Date \\[0.15 cm] \hline Yangpu & 947 & 17571 & 5.40 \% & 04/02/15 \\ New York & 1179 & 466237 & 0.25 \% & 01/05/11 \\ Chicago & 2619 & 67848 & 3.86 \% & 05/21/14 \\ Vienna & 1010 & 10948 & 9.23 \% & 03/25/11 \\ San fran & 1923 & 36089 & 5.33 \% & 05/24/08 \\ Singapore & 1782 & 401879 & 0.44 \% & 02/16/11 \\[0.1 cm] \hline \end{tabular} \end{table} \begin{table}[h!] \centering \label{data_table_Nstart} \caption{Coverage statistics. $N_{V,total}$ refers to the total number of taxis on the specified day.} \label{N_star_data} \begin{tabular}{|p{1.5 cm}|p{1.5cm}|p{1.5 cm}|P{1.75cm}|P{1.5cm}|} \hline \textbf{City} & $N_V^*$ & $N_{V,total}$ & $N_V^* / N_{V,total}$ & Date \\[0.15 cm] \hline Beijing & 211 & 4000 & 5.28 \% & 03/01/14 \\ Changsha & 227 & 4300 & 5.28 \% & 03/01/14 \\ Hangzhou & 132 & 2500 & 5.28 \% & 04/21/15 \\ Shanghai & 148 & 2800 & 5.29 \% & 03/01/14 \\[0.1 cm] \hline \end{tabular} \end{table} As previously discussed, while we consider Manhattan part of the trip-level datasets, taxi trips are recorded along with taxi IDs. This means we can find $N_V^*$ for this data set (as opposed to only $N_T^*$). The average number of trips per taxi is X, when remarkably translates to $N_V^* = 30$: just 30 random taxis cover half of the scannable street segments. Even more remarkably, one-third of the scannable street segments are scanned by just five random taxis! Supplementary Figure~\ref{manhattan_segments} displays these figures. \section*{Supplementary Note 5} \textbf{Minimum street sampling problem}. In the main text we quantified the sensing potential of a vehicle fleet by $\langle C \rangle_{(N_V, m)}$, the average number of segments covered $m$ times when $N_V$ randomly selected vehicles were equipped with a sensor. Note that here the number of vehicles $N_V$ was the independent variable. In some contexts, it might be advantageous to have the reverse scenario, in which $C$ is the independent variable. That is, given a target coverage $\bar{C}$, how many vehicles are needed to ensure this coverage is attained (with a target probability $\bar{p}$). We can this `minimum street sampling' problem, and define and solve it below. \begin{definition} (MINIMUM STREET SAMPLING): Given a street network $S$, an observation period $T$, a minimum sampling requirement $m$, and a collection $\mathcal{V}$ of vehicles moving in $S$ during $T$, where vehicle trajectories are taken from $\mathcal{P}$ according to a given probability distribution \textbf{P}; what is the minimum number $N_V^*$ of vehicles randomly selected from $\mathcal{V}$ such that $\mathbb{P}(C(N_V,m) \geq \bar{C}) \geq \bar{p}$, where $0 < \bar{C} \leq 1$ is the target street coverage and $\bar{p}$ is a target probabilistic sampling guarantee? \end{definition} This formulation of the sensing potential problem is likely of more utility for urban managers, wishing to know how many vehicles to equip to sensors to guarantee a certain coverage. The minimum street sampling problem is harder to solve that the 'sensing potential of a fleet' problem. This is because it requires the survival function of the multinomial distribution $\mathbb{P}_{N_T}(M_1 \geq m_1, M_2 \geq m_2 \dots,)$, which to our knowledge has no known closed form. We here adapt a technique used in \cite{levin1981representation} to derive an excellent approximation to this survival function. \textbf{Approximation of survival function}. The pdf for the multinomial distribution is \begin{equation} \mathbb{P}_{N_T}(M_1 = m_1, M_2 = m_2, \dots) = \frac{N_T!}{m_1! \dots m_{N_S}!} \prod_k^{N_S} p_k^{m_k} \end{equation} where $N_T$ is the number of balls which have been dropped, $N_S$ is the number of bins, $M_i$ is the random number of balls in bin $i$, and $p_i$ is the probability of selecting bin $i$. We seek the survival function \begin{equation} \mathbb{P}_{N_T}(M_1 \geq m_1, M_2 \geq m_2, \dots). \end{equation} The idea is to represent each $M_i$ as an independent Poisson random variable, conditional on their sum being fixed (this is a well known identity between the Multinomial and Poisson distributions). First let $A_i$ be the event $X_i \geq m_i$, where $X_i \sim Poi(s p_i)$, where $s$ is a real number (we will explain its significance later). Using Bayes' Theorem, we express the survival function as \begin{equation} \mathbb{P}_{N_T} \Big( A_1,\dots, A_{N_S}| \sum_{i=1}^{N_S} X_i = N_T \Big) = \frac{\mathbb{P}(A_1 \dots, A_{N_S})}{\mathbb{P}(\sum_{i=1}^{N_S} X_i = N_T )} \mathbb{P} \Big(\sum_{i=1}^{N_S} X_i = N | A_1, \dots, A_{N_S} \Big). \label{bayes} \end{equation} The numerator in the first term is easily found, since the events $A_i$ are independent Poisson random variables. Recalling that if $X_i \sim Poi(\lambda_i)$ then $\mathbb{P}(X_i \geq m_i) = 1-\Gamma (m_i,\lambda_i )/\Gamma (m_i) $, where $\Gamma(n,x) = \int_x^{\infty} t^{n-1} e^{-t} dt$ is the upper incomplete gamma function, we find \begin{equation} \mathbb{P}(A_1 \dots, A_{N_S}) = \prod_{i=1}^{N_S} \Bigg( 1-\frac{\Gamma (m_i, s p_i )}{\Gamma (m_i)} \Bigg). \end{equation} The denominator is also easy to find. Since $X_i \sim Poi(s p_i)$ and $\sum_i p_i = 1$, we see $\sum_i X_i \sim Poi(s)$ (sums of poissons are also poisson). Then \begin{equation} \mathbb{P} \Big( \sum_{i=1}^{N_S} X_i = N_T \Big) = \frac{s^N_T e^{-s}}{N_T!} \end{equation} For the second term in \eqref{bayes}, we note that conditioning on the joint event $A_1, A_2, \dots$ means the range of the summands are constrained to $[a_i, \infty]$. Hence the summands, which we call $Y_i$, are truncated Poisson random variables, which we denote by $ Y_i \sim Poi_{[a_i, \infty]}(s p_i)$. We note that the mean of a truncated Poisson random variable is not the same as an untruncated one. In particular, if $W_i \sim Poi_{[a,\infty]}(\lambda)$, then \begin{align} \mathbb{E} (W_i) &= \lambda \frac{q_{a-1}}{q_{a}} \\ Var(W_i) &= \lambda^2 \frac{ q_{a-2} q_{a} - q_{a-1}^2 }{q_{a}^2} + \lambda \frac{q_{a-1}}{q_{a}} \end{align} where \begin{align} q_{a} = \left\{ \begin{array}{ll} 1 - \frac{\Gamma(a,\lambda)}{\Gamma(a)} & \quad a \geq 1 \\ 0 & \quad a < 1 \end{array} \right. \end{align} \noindent Returning to the second term in \eqref{bayes}, we find \begin{equation} \mathbb{P} \Big( \sum_{i=1}^{N_S} X_i = N_T | A_1, \dots, A_{N_S} \Big) = \mathbb{P} \Big( \sum_{i=1}^{N_S} X_i = N_T | A_1, \dots, A_{N_S} \Big) = \mathbb{P} \Big( \sum_{i=1}^{N_S} Poi_{[a_i, \infty]}(s p_i) = N_T \Big). \end{equation} We were unable to find an analytic form for the above sum. Instead, we used a first order normal approximation. This states that for a sequence of random variables $(W_i)_i$ with mean $\mu_i$ and variance $\sigma_i^2$, \begin{equation} \sum_{i=1}^{N_S} W_i \xrightarrow[]{d} N(s_{\mu},s_{\sigma}) \end{equation} as $N_s \rightarrow \infty$, where \begin{align} s_{\mu} = \sum_i \mu_i \\ s_{\sigma}^2 = \sum_i \sigma_i^2. \end{align} Then the term becomes \begin{equation} \mathbb{P} \Big( \sum_{i=1}^{N_S} X_i = N_T | A_1, \dots, A_{N_S} \Big) = \frac{1}{\sqrt{2 \pi} s_{\sigma}} e^{ - \frac{(N_T - s_{\mu})^2}{2 s_{\sigma}^2} } \end{equation} Pulling all this together \begin{equation} \mathbb{P}_{N_T}(M_1 \geq m_1, M_2 \geq m_2, \dots) \approx \frac{N_T!}{s^{N_T} e^{-s}} \frac{1}{\sqrt{2 \pi} s_{\sigma}} e^{ - \frac{(N_T - s_{\mu})^2}{2 s_{\sigma}^2} } \prod_{i=1}^{N_S} \Bigg( 1-\frac{\Gamma (m_i, s p_i )}{\Gamma (m_i)} \Bigg)\label{temp2} \end{equation} \noindent Now, the variable $s$ is a free parameter. Determining the optimal $s$ is an open problem. Following \cite{levin1981representation} we use $s = N_T$, which, when inserted into \eqref{temp2}, along with Stirling's approximation $\frac{N_T!}{N_T^{N_T} e^{-N_T}} \approx \sqrt{2 \pi N_T}$ yields our final expression \begin{equation} \mathbb{P}_{N_T}(M_1 \geq m_1, M_2 \geq m_2, \dots) \approx \sqrt{\frac{N_T}{s_{\sigma}^2}} e^{ - \frac{(N_T - s_{\mu})^2}{2 s_{\sigma}^2} } \prod_{i=1}^{N_S} \Bigg( 1-\frac{\Gamma (m_i, N_T p_i )}{\Gamma (m_i)} \Bigg). \label{final} \end{equation} \noindent To test the accuracy of the above approximation to the survival function, we compared it to Monte Carlo estimates. The results are shown in Figure~\ref{approx}, in which excellent agreement is evident. \textbf{Solve minimum street sampling}. We leverage the survival function \eqref{final} to solve the minimum street sampling problem in the same way as we did to solve for $C$ in the main text: we assume placing $N_T$ trajectories of random length $L$ into $N_S$ bins is the same as placing $L*N_T$ balls into $N_S$ bins, \begin{align} \mathbb{P}_{(N_T, L)}(M_1 \geq m_1, \dots) = \sum_{n=0}^{\infty} & \mathbb{P}_{(N_T, L=1)}(M_1 \geq m_1, \dots) \mathbb{P}(S_{N_T} = n) \label{final_P1} \end{align} \noindent where $\mathbb{P}_{(N_T,L=1)}(M_1 \geq m_1, \dots)$ is given by equation \eqref{final}. As for the expression for $C$, this can be extended to the vehicle level by replacing $L$ by $B$. Also as in the main text, this sum is well dominated by its average, leading to the simpler expression \begin{align} \mathbb{P}_{(N_T, L)}(M_1 \geq m_1, \dots) = \mathbb{P}_{(\langle L \rangle *N_T, L=1)}(M_1 \geq m_1, \dots) \label{final_P} \end{align} \noindent When full coverage $\bar{C} = 1$ is desired, equation \eqref{final_P} solves the minimum street sampling problem. However, when less than full coverage $C < 1$ is desired, we must marginalize over all combinations of $N_S*C$ segments above threshold. This is because in our formulation of the minimum street sampling problem we require just a bare fraction $\bar{C}$ of segments be covered, which is achievable by a large number of combinations of segments. Of course if \textit{targeted} coverage were desired (i.e were specific street segments were desired to be senses with specific sensing requirements $m$), then \eqref{final_P} could be used. Staying within our current formulation however, an enumeration of all $C N_S$ combinations of bins is required to marginalize $\mathbb{P}(M_1 \geq m_1, \dots)$. For large $N_S$ enumerating these combinations is infeasible. To get around this, we instead estimate $\mathbb{P}_{(\langle L \rangle *N_T, L=1)}(M_1 \geq m_1, \dots)$ by Monte Carlo; we draw samples of size $\langle L \rangle *N_T$ from a multinomial distribution 1000 times, and count the fraction of times at least $\bar{C}$ of the $N_S$ bins are above the threshold $m$. This lets us estimate $\mathbb{P}(C > \bar{C})(N_T)$, from which we can read off the desired $N_T^*(\bar{P})$ solving the minimum street sampling problem. In Figure~\ref{P_all_cities} we compare our predictions versus data for a target coverage of $\bar{C} = 0.5$. While the precise shapes of the theoretical and empirical curves do not agree, our model correctly captures the right range of variation. In particular, the error $N_{T,model}(\bar{P} \approx 1)$ - $N_{T,data}(\bar{P} \approx 1)$ is $\approx 200$. Expressed relative to the total number of trips, this is $ \sim 10^{-4}$ for the NYC and Singapore datasets, and $\sim 10^-2$ for the other datasets, which is good accuracy.

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\section{Introduction} \label{sec:intro} \newcommand{\mypara}[1]{{\smallskip\noindent {\bf #1}}\quad} \mypara{The problem.} We study a geometric optimization problem that arises in wireless network design, as well as in robotics and various facility location problems. The task is to select a number of locations $t_j$ for the base station antennas (\emph{servers}), and assign a transmission range $r_j$ to each $t_j$, in order that each $p_i\in Y$ for a given set $Y=\{p_1,\ldots,p_n\}$ of $n$ demand points (\emph{clients}) is covered. We say that client $p_i$ is covered if and only if $p_i$ is within range of some transmission point $t_{j_i}$, i.e., $d(t_{j_i},p_i)\leq r_{j_i}$. The resulting cost per server is some known function $f$, such as $f(r)=r^\alpha$. The goal is to minimize the total cost, $\sum_j f(r_j)$, over all placements of at most $k$ servers that cover the set $Y$ of clients. In the context of modeling the energy required for wireless transmission, it is common to assume a superlinear ($\alpha>1$) dependence of the cost on the radius; in fact, physically accurate simulation often requires superquadratic dependence ($\alpha>2$). A quadratic dependence ($\alpha=2$) models the total area of the served region, an objective arising in some applications. A linear dependence ($\alpha=1$) is sometimes assumed, as in Lev-Tov and Peleg\cite{lp-ptasb-05}, who study the base station coverage problem, minimizing the sum of radii. The linear case is important to study not only in order to simplify the problem and gain insight into the general problem, but also to address those settings in which the linear cost model naturally arises\cite{cps-opapr-00,210191}. For example, the model may be appropriate for a system with a narrow-angle beam whose direction can either rotate continuously or adapt to the needs of the network. Another motivation for us comes from robotics, in which a robot is to map or scan an environment with a laser scanner\cite{fkn-osar-05,fkn-sar-04}: For a fixed spatial resolution of the desired map, the time it takes to scan a circle corresponds to the number of points on the perimeter, i.e., is proportional to the radius. Our problem is a type of clustering problem, recently named \emph{min-size $k$-clus\-ter\-ing} by Bil\`o et al.\cite{ESA05}. Clustering problems tend to be NP-hard, so most efforts, including ours, are aimed at devising an approximation algorithm or a polynomial-time approximation scheme (PTAS). We also introduce a new problem, which we call \emph{minimum cost covering tour} (MCCT), in which we combine the problem of finding a short tour and placing covering disks centered along it. The objective is to minimize a linear combination of the tour length and the transmission/covering costs. The problem arises in the autonomous robot scanning problem\cite{fkn-osar-05,fkn-sar-04}, where the covering cost is linear in the radii of the disks, and the overall objective is to minimize the total time of acquisition (a linear combination of distance travelled and sum of scan radii). Another motivation is the distribution of a valuable or sensitive resource: There is a trade-off between the cost of broadcasting from a central location (thus wasting transmission or risking interception) and the cost of travelling to broadcast more locally, thereby reducing broadcast costs but incurring travel costs. \mypara{Location Constraints.} In the absence of constraints on the server locations, it may be optimal to place one server at each demand point. Thus, we generally set an upper bound, $k$, on the number of servers, or we restrict the possible locations of the servers. Here, we consider two cases of location constraints: (1) Servers are restricted to lie in a discrete set $\{t_1,\ldots,t_m\}$; or (2) Servers are constrained to lie on a line (which may be fully specified, or may be selected by the optimization). \mypara{Our results.} We provide a number of new results, some improving previous work, some giving the first results of their kind. In the discrete case studied by Lev-Tov and Peleg\cite{lp-ptasb-05}, and Bil\'o et al.\cite{ESA05}, we give improved results. For the discrete 1D problem where $Y\subseteq\mathbb{R}$, we improve their 4-approximation to a linear-time 3-approximation by using a ``Closest Center with Growth'' (CCG) algorithm, and, as an alternative to the previous $O((n+m)^3)$ algorithm\cite{lp-ptasb-05}, we give a near-linear-time 2-approximation that uses a ``Greedy Growth'' (GG) algorithm. Unfortunately, we cannot extend our proofs to the 1$\frac12$D problem. Intuitively, greedy growth works as follows: start with a disk with center at each server, each disk of radius zero; among all clients, find one that requires the least radial disk growth to capture it; repeat until all clients are covered. Note that for $\alpha\geq 2$ the 2D variants of the problem are already proved to be NP-Hard and to have a PTAS\cite{ESA05}. In the general 2D case with clients $Y\subset \R^2$, we strengthen the hardness result of Bil{\'o} et al.~\cite{ESA05} by showing that the discrete problem is already hard for any superlinear cost function, i.e., $f(r)=r^{\alpha}$ with $\alpha>1$. Furthermore, we generalize the min-size clustering problem in two new directions. On the one hand, we consider less restrictive server placement policies. For instance, if we only restrict the servers to lie on a given fixed line, we give a dynamic programming algorithm that solves the problem exactly, in time $O(n^2\log n)$ for any $L_p$ metric in the linear cost case, and in time $O(n^4\log n)$ in the case of superlinear non-decreasing cost functions. For simple approximations, our algorithm ``Square Greedy'' (SG) gives in time $O(n \log n)$ a $3$-approximation to the square covering problem with any linear or superlinear cost function. A small variation, ``Square Greedy with Growth'' (SGG), gives a 2-approximation for a linear cost function, also in time $O(n\log n)$. The results are also valid for covering by $L_p$ disks for any~$p$, but with correspondingly coarser approximation factors. A practical example in which servers are restricted to lie along a line is that of a highway that cuts through a piece of land, and the server locations are restricted to lie along the highway. The line location problem arises when one not only needs to locate the servers, but also needs to select an optimal corridor for the placement of the highway. Other relevant examples may include devices powered by a microwave or laser beam lining up along the beam. If the servers are restricted to lie on a horizontal line, but the location of this line may be chosen freely, then we show that the exact optimal position (with $\alpha=1$) is not computable by radicals, using an approach similar to that of Bajaj\cite{b-pgans-86,b-adgop-88} in addressing the unsolvability of the Fermat-Weber problem. On the positive side, we give a fully polynomial-time approximation scheme (FPTAS) requiring time $O((n^3/\varepsilon) \log n)$ if $\alpha=1$ and time $O((n^5/\varepsilon) \log n)$ if $\alpha>1$. For servers on an unrestricted line, of any slope, and $\alpha=1$, we give $O(1)$-approximations (4-approximation in $O(n^4\log n)$ time, or $8\sqrt{2}$-approximation in $O(n^3\log n)$ time) and an FPTAS requiring time $O((n^5/\varepsilon^2)\log n)$. We give the first algorithmic results for the new problem, minimum cost covering tour (MCCT), which we introduce. Given a set $Y\subseteq\mathbb{R}^2$ of $n$ clients, our goal is to determine a polygonal tour $T$ and a set $X$ of $k$ disks of radii $r_j$ centered on $T$ that cover $Y$ while minimizing the cost $\mathrm{length}(T)+C\sum r_i^\alpha$. Our results are for $\alpha=1$. The ratio $C$ represents the relative cost of touring versus transmitting. We show that MCCT is NP-hard if $C$ is part of the input. At one extreme, if $C$ is small then the optimum solution is a single server placed at the circumcenter of $Y$ (we can show this to be the case for $C\leq 4$). At the other extreme (if $C$ very large), the optimum solution is a TSP among the clients. For any fixed value of $C>4$, we present a PTAS for MCCT, based on a novel extension of the $m$-guillotine methods of~\cite{m-gsaps-99}. \mypara{Related work.} There is a vast family of clustering problems, among which are the \emph{$k$-center} problem in which one minimizes $\max_j r_j$, the \emph{$k$-median} problem in which one minimizes $\sum_i d(p_i,t_{j_i})$, and the \emph{$k$-clustering} problem in which one minimizes the maximum over all clusters of the sum of pairwise distances between points in that cluster. For the geometric instances of these related clustering problems, refer to the survey by Agarwal and Sharir\cite{299918}. When $k$ is fixed, the optimal solution can be found in time $O(n^k)$ using brute force. In the plane, one of the only results for the min-size clustering problem is a small improvement for $k=2$ by Hershberger\cite{160550}, in subquadratic time $O(n^2/\log\log n)$. Approximation algorithms and schemes have been proposed, particularly for geometric instances of these problems (e.g., \cite{Arora:1998tw}). Clustering for minimizing the sum of radii was studied for points in metric spaces by Charikar and Panigrahy\cite{985670}, who present an $O(1)$-approximation algorithm using at most $k$ clusters. For the linear-cost model ($\alpha=1$), our problem has been considered recently by Lev-Tov and Peleg\cite{lp-ptasb-05} who give an $O((n+m)^3)$ algorithm when the clients and servers all lie on a given line (the 1D problem), and a linear-time 4-approximation in that case. They also give a PTAS for the two-dimensional case when the clients and servers can lie anywhere in the plane. Bil\`o et al.\cite{ESA05} show that the problem is NP-hard in the plane for the case $f(r)=r^\alpha$, $\alpha\geq 2$, either when the sets $X$ and $Y$ are given and $k$ is left unspecified ($k=n$), or when $k$ is fixed but then $X=Y$. They give a PTAS for the linear cost case ($\alpha=1$) and a slightly more involved PTAS for a more general problem in which the cost function is superlinear, there are fixed additive costs associated with each transmission server and there is a bound $k$ on the number of servers. There are many problems dealing with covering a set of clients by disks of {\em given} radius. Hochbaum and Maass\cite{hm-ascpp-85} give a PTAS for covering with a minimum number of disks of fixed radius, where the disk centers can be taken anywhere in the plane. They introduce a ``grid-shifting technique,'' which is used and extended by Erlebach~et~al.\cite{ejs-ptasg-05}. Lev-Tov and Peleg\cite{lp-ptasb-05} and Bil\`o et al.\cite{ESA05} extend the method further in obtaining their PTAS results for the discrete version of our problem. When a discrete set $X$ of potential server locations is given, Gonzalez\cite{Gonzalez:1991bb} addresses the problem of maximizing the number of covered clients while minimizing the number of servers supplying them, and he gives a PTAS for such problems with constraints such as bounded distance between any two chosen servers. In\cite{bg-aoscf-95}, a polynomial-time constant approximation is obtained for choosing a subset of minimum size that covers a set of points among a set of candidate disks (the radii can be different but the candidate disks must be given). The closest work to our combined tour/transmission cost (MCCT) is the work on covering tours: the ``lawn mower'' problem\cite{Arkin:2000bd}, and the TSP with neighborhoods\cite{ah-aagcs-94,dm-aatsp-03}, each of which has been shown to be NP-hard and has been solved with various approximation algorithms. In contrast to the MCCT we study, the radius of the ``mower'' or the radius of the neighborhoods to be visited is specified in advance. \section{Scenario (1): Server Locations Restricted to a Discrete Set} \label{sec:scenario1a} \subsection{The one-dimensional discrete problem with linear cost} Consider the case of $m$ fixed server locations $X=\{t_1,...,t_m\}$, $n$ client locations $Y=\{p_1,...,p_n\}$, and a linear ($\alpha=1$) cost function, with clients and servers all located along a fixed line. Without loss of generality, we may assume that $X$ and $Y$ are sorted in the same direction, at an extra cost of $O((n+m)\log(n+m))$. Lev-Tov and Peleg\cite{lp-ptasb-05} give an $O((n+m)^3)$ dynamic programming algorithm for finding an exact solution. Bil\`o et al.\cite{ESA05} show that the problem is solvable in polynomial time for any value of $\alpha$ by reducing it to an integer linear program with a totally unimodular constraint matrix. The complexities of these algorithms, while polynomial, is high. Lev-Tov and Peleg also give a simple ``closest center'' algorithm (CC) that gives a linear-time $4$-approximation. We improve to a $3$-approximation in linear time, and a 2-approximation in $O(m+n\log m)$ time. We now describe an algorithm which also runs in linear time, but achieves an approximation factor of $3$. \smallskip \noindent \textbf{Closest Center with Growth (CCG) Algorithm}: Process the clients $\{p_1,...,p_n\}$ from left to right keeping track of the rightmost extending disk. Let $\omega_R$ denote the rightmost point of the rightmost extending disk, and let $R$ denote the radius of this disk. (In fact the rightmost extending disk will always be the last disk placed.) If $\omega_R$ is equal to, or to the right of the next client processed, $p_i$, then $p_i$ is already covered so ignore it and proceed to the next client. If $p_i$ is not yet covered, consider the distance of $p_i$ to $\omega_R$ compared with the distance of $p_i$ to its closest center $\hat{t_i}$. If the distance of $p_i$ to $\omega_R$ is less than or equal to the distance of $p_i$ to its closest center $\hat{t_i}$, then grow the rightmost extending disk just enough to capture $p_i$. Otherwise use the disk centered at $\hat{t_i}$ of radius $|p_i - \hat{t_i}|$ to cover $p_i$. \begin{lemma} For $\alpha=1$, CCG yields a $3$-approximation to OPT in $O(n+m)$ time. \end{lemma} The proof is similar to that of the next lemma, and omitted in this version. If we consider a single disk $D$ with clients $p_L$ and $p_R$ on the left and right edges of $D$, associated centers $\hat{x}_L$, $\hat{x}_R$ at distances respectively radius($D$)$- \epsilon$ to the left and radius($D$)$- \epsilon$ to the right, along with a dense set of clients in the left hand half of $D$ we see that $3$ is the best possible constant for CCG. Finally we offer an algorithm that achieves a $2$-approximation but runs in time $O(m+n\log{m})$. \smallskip \noindent \textbf{Greedy Growth (GG) Algorithm}: Start with a disk with center at each server all of radius zero. Now, amongst all clients, find the one which requires the least radial disk growth to capture it. Repeat until all clients are covered. An efficient implementation uses a priority queue to determine the client that should be captured next. One can set up the priority queue in $O(m)$ time. Note that the priority queue will never have more than $2m$ elements, and that each $p_i$ eventually gets captured, either from the right or from the left. Each capture can be done in time $O(\log{m})$ for a total running time of $O(m+n\log{m})$. \begin{lemma} \label{l:2-approx} For $\alpha=1$, GG yields a $2$-approximation to OPT in $O(m+n\log{m})$ time. \end{lemma} \begin{proof} Define intervals $J_i$ as follows: when capturing a client $p_i$ from a server $t_j$ whose current radius (prior to capture) is $r_j$, let $J_i=(t_j+r_j,p_i]$ if $p_i>t_j$, and $J_i=[p_i,t_j-r_j)$ otherwise. Our first trivial yet crucial observation is that $J_i\cap J_k=\emptyset$ if $i\neq k$. Also note that the sum of the lengths of the $J_i$ is equal to the sum of the radii in the GG cover. Consider now a fixed disk $D$ in OPT, centered at $t_D$, and the list of intervals $J_i$ whose $p_i$ is inside $D$. As before, at most one such $J_i$ extends outward to the right from the right edge of $D$. If so, call it $J_R$, and define $J_L$ symmetrically. If $J_R$ exists, it cannot extend more than radius($D$) to the right of $D$. Let $\lambda=\mathrm{length}(J_R)$. We argue that there is an interval of length $\lambda$ in $D$, to the right of $t_D$, which is free of $J_i$'s. It follows that there is at most radius($D$) worth of segments to the right of $t_D$. Of course, this is also true if $J_R$ does not exist. By symmetry, there is also radius($D$) worth of segments to the left of $t_D$, whether $J_L$ exists or not, yielding the claimed $2$-approximation. Assume $J_R$ exists. Then the algorithm successively extends $J_R$ by growth to the left up to some maximum point (possibly stopping right at $p_R$). Since the growth could have been induced by clients to the right of $J_R$, that maximum point is not necessarily a client. There is, however, some client inside $D$ that is captured last in this process. This client $p_i$ (possibly $p_R$) cannot be within $\lambda$ of $t_D$, since otherwise it would have been captured prior to the construction of~$J_R$. If there is no client between $t_D$ and $p_i$ we are done, since then there could be no interval $J_k$ in between. Thus consider the client $p_{i-1}$ just to the left of $p_i$. Suppose $d(p_{i-1}, p_i) \geq \lambda$. Then, if $p_{i-1}$ is eventually captured from the left, we would have the region between $p_{i-1}$ and $p_i$ free of $J_k$'s and be done. On the other hand, if $p_{i-1}$ is captured from the right, it must be captured by a server between $p_{i-1}$ and $p_i$, and that server is at least $\lambda$ to the left of $p_i$ since otherwise $p_i$ would be captured by that server prior to $p_R$. This leaves the distance from the server to $p_i$ free of $J_k$'s. Hence the only case of concern is if $d(p_{i-1}, p_i) < \lambda$. Clearly $p_{i-1}$ must not have been captured at the time when $p_R$ is captured since otherwise $p_i$ would have been captured before $p_R$, contradicting the assumption that $p_i$ is captured by growth leftward from $p_R$. Similarly, there cannot be a server between $p_{i-1}$ and $p_i$, since otherwise both $p_{i-1}$ and $p_i$ would be captured before $p_R$. Together with the definition of $p_i$, this implies that $p_{i-1}$ is captured from the left. Therefore, to the left of $p_{i-1}$, there must be one or more intervals $\{J_{l_i}\}$ whose length is at least $\lambda$ that are constructed before $p_{i-1}$ is captured. Similarly, to the right of $p_i$, there must be some one or more intervals $\{J_{r_j}\}$ whose length is at least $\lambda$, constructed before $p_i$ is captured. However, either the last $J_{l_i}$ is placed before the last $J_{r_j}$ or vice versa. In the first case, there are no $\lambda$ length obstructions left in the left-hand subproblem, so $p_{i-1}$ will be covered, and with $\lambda$ length obstructions remaining in the right subproblem, $p_i$ will be captured by growth rightward. The second case is symmetrical to the first. In either case we have a contradiction. \end{proof} To see that the factor $2$ is tight, just consider servers at $-2+\epsilon, 0$ and $2-\epsilon$ and clients at $-1$ and $1$. \subsection{Hardness of the two-dimensional discrete problem with superlinear cost} In 2D, we sketch an NP-hardness proof, for any $\alpha>1$. This strengthens the NP-hardness proof of\cite{ESA05}, which only works in the case $\alpha\geq 2$. \begin{theorem} \label{thm:hard} For any a fixed $\alpha>1$, let the cost function of a circle of radius $r$ be $f(r)=r^\alpha$. Then it is NP-hard to decide whether a discrete set of $n$ clients in the plane, and a discrete set of $m$ potential transmission points allow a cheap set of circles that covers all demand points. \end{theorem} \begin{proofsketch} Let $I$ be an instance of {\sc Planar 3Sat}, and let $G_I$ be the corresponding variable-clause incidence graph. After choosing a suitable layout of this planar graph, resulting in integer variables with coordinates bounded by a polynomial in the size of $G_I$ for all vertices and edges, we replace each the vertex representing any particular variable by a closed loop, using the basic idea shown in the left of Figure~\ref{fig:clause}; this allows two fundamentally different ways of covering those points cheaply (using the ``odd'' or the ``even'' circles), representing the two truth assignments. For each edge from a vertex to a variable, we attach a similar chain of points that connects the variable loop to the clause gadget; the parity of covering a variable loop necessarily assigns a parity to all incident chains. Note that choosing sufficiently fine chains guarantees that no large circles can be used, as the overall weight of all circles in a cheap solution will be less than 1. (It is straightforward to see that for any fixed $\alpha>1$, this can be achieved by choosing coordinates that are polynomial in the size of $G_I$, with the exponent being $O(1/(\alpha-1))$.) \begin{figure}[htbp] \centering{{\includegraphics[width=0.59\columnwidth]{images/variable.eps}} {\includegraphics[width=0.39\columnwidth]{images/clause.eps}}} \caption{ (Left) The switch structure of a variable gadget. Note how there are two fundamentally different ways to cover all points cheaply. (Right) The structure of a clause gadget. One small circle is needed for picking up the client point at the center of the gadget.} \label{fig:clause} \end{figure} For the clauses choose a hexagonal arrangement as shown in the right of Figure~\ref{fig:clause}: There is one central point that must be covered somehow; again, $\alpha>1$ guarantees that it is cheaper to do this from a nearby transmission point, rather than increasing the size of a circle belonging to a chain gadget. Now it is straightforward to see that there is a cheap cover, using only the forced circles, iff the truth assignment corresponding to the covering of variabe loops assures that each clause has at least one satisfying variable. \qed \end{proofsketch} \section{Scenario (2): Server Locations\\ Restricted to a Line} \label{sec:scenario1} \subsection{Servers along a fixed horizontal line} \subsubsection{Exact solutions} \label{sec:1b-exact} Suppose that the servers are required to lie on a fixed horizontal line, which we take without loss of generality to be the $x$-axis. Such a restriction could arise naturally (e.g., the servers must be connected to a power line, must lie on a highway, or in the main corridor in a building). In addition, this case must be solved first before attempting to solve the more general problem---along a polygonal curve. In this section, we describe dynamic programming algorithms to compute a set of server points of minimum total cost. For notational convenience, we assume that the clients $Y$ are indexed in left-to-right order. Without loss of generality, we also assume that all the clients lie on or above the $x$-axis, and that no two clients have the same $x$-coordinate. (If a client $p_i$ lies directly above another client $p_j$, then any circle enclosing $p_i$ also encloses $p_j$, so we can remove $p_j$ from $Y$ without changing the optimal cover.) Let us call a circle $C$ \emph{pinned} if it is the leftmost smallest axis-centered circle enclosing some fixed subset of clients. Equivalently, a circle is pinned if it is the leftmost smallest circle passing through a chosen client or a chosen pair of clients. Under any $L_p$ metric, there are at most $O(n^2)$ pinned circles. As long as the cost function $f$ is non-decreasing, there is a minimum-cost cover consisting entirely of pinned circles. \mypara{Linear Cost.} If the cost function $f$ is linear (or sublinear), we easily observe that the circles in any optimum solution must have disjoint interiors. (If two axis-centered circles of radius $r_i$ and $r_j$ intersect, they lie in a larger axis-centered circle of radius at most $r_i+r_j$.) In this case, we can give a straightforward dynamic programming algorithm that computes the optimum solution under any $L_p$ metric. The algorithm given in Figure~\ref{2alg} (left) finds the minimum-cost cover by disjoint pinned circles, where distance is measured using any $L_p$ metric. We call the rightmost point enclosed by any pinned circle $C$ the \emph{owner} of~$C$. If we use brute force to compute the extreme points enclosed by each pinned circle and to test whether any points lie directly above a pinned circle, this algorithm runs in $O(n^3)$ time. With some more work, however, we can improve the running time by nearly a linear factor. This improvement is easiest in the $L_\infty$ metric, in which circles are axis-aligned squares. Each point $p_i$ is the owner of exactly $i$ pinned squares: the unique axis-centered square with $p_i$ in the upper right corner, and for each point $p_j$ to the left of $p_i$, the leftmost smallest axis-centered square with $p_i$ and $p_j$ on its boundary. We can easily compute all these squares, as well as the leftmost point enclosed by each one, in $O(i\log i)$ time. (To simplify the algorithm, we can actually ignore any pinned square whose owner does not lie on its right edge.) If we preprocess $P$ into a priority search tree in $O(n\log n)$ time, we can test in $O(\log n)$ time whether any client lies directly above a horizontal line. The overall running time is now $O(n^2\log n)$. For any other $L_p$ metric, we can compute the extreme points enclosed by all $O(n^2)$ pinned circles in $O(n^2)$ time using the following duality transformation. If $C$ is a circle centered at $(x,0)$ with radius $r$, let $C^*$ be the point $(x,r)$. For each client $p_i$, let $p_i^* = \{ C^* | \mbox{$C$ is centered on the $x$-axis and } p_i \in C \}$, and let $Y^* = \{p_i^*\mid p_i\in Y\}$. We easily verify that each set $p_i^*$ is an infinite $x$-monotone curve. (Specifically, in the Euclidean metric, the dual curves are hyperbolas with asymptotes of slope $\pm 1$.) Moreover, any two dual curves $p_i^*$ and $p_j^*$ intersect exactly once; i.e., $Y^*$ is a set of pseudo-lines. Thus, we can compute the arrangement of $Y^*$ in $O(n^2)$ time. For each pinned circle $C$, the dual point $C^*$ is either one of the clients $p_i$ or a vertex of the arrangement of dual curves $Y^*$. A circle $C$ encloses a client $p_i$ if and only if the dual point $C^*$ lies on or above the dual curve $p_i^*$. After we compute the dual arrangement, it is straightforward to compute the leftmost and rightmost dual curves below every vertex in $O(n^2)$ time by depth-first search. Finally, to test efficiently whether any points lie directly above an axis-centered ($L_p$) circle, we can use the following two-level data structure. The first level is a binary search tree over the $x$-coordinates of~$Y$. Each internal node $v$ in this tree corresponds to a canonical vertical slab $S_v$ containing a subset $p_v$ of the clients. For each node $v$, we partition the $x$-axis into intervals by intersecting it with the furthest-point Voronoi diagram of $p_v$, in $O(\abs{p_v}\log \abs{p_v})$ time. To test whether any points lie above a circle, we first find a set of $O(\log n)$ disjoint canonical slabs that exactly cover the circle, and then for each slab $S_v$ in this set, we find the furthest neighbor in $p_v$ of the center of the circle by binary search. The region above the circle is empty if and only if all $O(\log n)$ furthest neighbors are inside the circle. Finally, we can reduce the overall cost of the query from $O(\log^2 n)$ to $O(\log n)$ using fractional cascading. The total preprocessing time is $O(n\log^2 n)$. \begin{theorem}\label{thm:DPlinear} Given $n$ clients in the plane, we can compute in $O(n^2\log n)$ time a covering by circles (in any fixed $L_p$ metric) centered on the $x$-axis, such that the sum of the radii is minimized. \end{theorem} \begin{figure*}[t] \centerline{\small\begin{algorithm} $\underline{\textsc{MinSumOfRadiusCircleCover}(Y):}$\+ \\ for every pinned circle $C$\+ \\ find the leftmost and rightmost points enclosed by $C$\- \\ $Cost[0] \gets 0$ \\ for $i \gets 1$ to $n$ \+ \\ $Cost[i] \gets \infty$ \\ for each pinned circle $C$ owned by $p_i$\+ \\ if no points in $P$ lie directly above $C$\+ \\ $p_j \gets$ leftmost point enclosed by $C$ \\ $Cost[i] \gets \min\{Cost[i],\, Cost[j-1] + radius(C))\}$\-\-\- \\ return $Cost[n]$ \end{algorithm}\hfill \small\begin{algorithm} $\underline{\textsc{MinSuperlinearCostCircleCover}(Y,f):}$\+ \\ sort the pinned circles from left to right by their centers \\ $Cost[0] \gets 0$ \\ for $j\gets 1$ to $p+1$\+ \\ $Cost[j] \gets \infty$ \\ for $i\gets 1$ to $j-1$\+ \\ if $C_i$ and $C_j$ exclude each other's apices \\ and $B(C_i, C_j)$ is empty\+ \\ $Cost[j] \gets \min\{Cost[j],\, Cost[i] + f(radius(C_i)))\}$\-\-\- \\ return $Cost[p+1]$ \\ \quad \end{algorithm}} \caption{The dynamic programming algorithm: Left: linear cost; Right: superlinear cost function.} \label{2alg} \end{figure*} \mypara{Superlinear Cost.} A similar dynamic programming algorithm computes the optimal covering under any superlinear (in fact, any \emph{non-decreasing}) cost function~$f$. As in the previous section, our algorithm works for any $L_p$ metric. For the moment, we will assume that $p$ is finite. Although two circles in the optimal cover need not be disjoint, they cannot overlap too much. Clearly, no two circles in the optimal cover are nested, since the smaller circle would be redundant. Moreover, the highest point (or \emph{apex}) of any circle in the optimal cover must lie outside all the other circles. If one circle~$A$ contains the apex of a smaller circle $B$, then the lune $B\setminus A$ is completely contained in an even smaller circle~$C$ whose apex is the highest point in the lune; it follows that $A$ and $B$ cannot both be in the optimal cover. See Figure \ref{F:overlap}(a). \begin{figure}[ht] \centering\footnotesize\sf \begin{tabular}{c@{\qquad}c@{\qquad}c} \scalebox{0.70}{\includegraphics[height=1.1in]{images/L2overlap.eps}} & \scalebox{0.70}{\includegraphics[height=1in]{images/L2recurse.eps}} & \scalebox{0.70}{\includegraphics[height=1in]{images/L2between.eps}} \\ (a) & (b) & (c) \end{tabular} \caption{(a) The apex of each circle in the optimal cover lies outside the other circles. (b) The points $Y_C$ lie in the shaded region. (c) If $A$ and $C$ are adjacent circles in the optimal covering, the shaded region $B(A,C)$ is empty.} \label{F:overlap} \end{figure} To compute the optimal cover of $Y$, it suffices to consider subproblems of the following form. For each pinned circle $C$, let $Y_C$ denote the set of clients outside $C$ and to the left of its center; see Figure \ref{F:overlap}(b). Then for each pinned circle $C$, we have $cost(Y_C) = \min_{A} (f(radius(A)) + cost(Y_A))$, where the minimum is taken over all pinned circles $A$ satisfying the following conditions: (1) The center of $A$ is left of the center of $C$; (2) the apex of $A$ is outside $C$; (3) the apex of $C$ is outside $A$; and (4) $A$ encloses every point in $Y_C\setminus Y_A$. The last condition is equivalent to there being no clients inside the region $B(A,C)$ bounded by the $x$-axis, the circles $A$ and $C$, and vertical lines through the apices of $A$ and $C$; see Figure \ref{F:overlap}(c). Our dynamic programming algorithm (Figure~\ref{2alg} (right)) considers the pinned circles $C_1, C_2, \dots, C_p$ in left to right order by their centers; that is, the center of $C_i$ is left of the center of $C_j$ whenever $i<j$. To simplify notation, let $Y_i = Y_{C_i}$. For convenience, we add two circles $C_0$ and $C_{p+1}$ of radius zero, centered far to the left and right of $Y$, respectively, so that $Y_0= \varnothing$ and $Y_{p+1} = Y$. Implementing everything using brute force, we obtain a running time of $O(n^5)$. However, we can improve the running time to $O(n^4\log n)$ using the two-level data structure described in the previous section, together with a priority search tree. The region $B(C_i, C_j)$ can be partitioned into two or three three-sided regions, each bounded by two vertical lines and either a circular arc or the $x$-axis. We can test each three-sided region for emptiness in $O(\log n)$ time. \begin{theorem}\label{thm:DPsuperlinear} Let $f:\R_+\to\R$ be a fixed non-decreasing cost function. Given $n$ clients in the plane, we can compute in $O(n^4\log n)$ time a covering by circles (in any fixed $L_p$ metric) centered on the $x$-axis, such that the sum of the costs of the circles is minimized. \end{theorem} The algorithm is essentially unchanged in the $L_\infty$ metric, except now we define the apex of a square to be its upper right corner. It is easy to show that there is an optimal square cover in which no square contains the apex of any other square. Equivalently, we can assume without loss of generality that if two squares in the optimal cover overlap, the larger square is on the left. To compute the optimal cover, it suffices to consider subsets $Y_C$ of points either directly above or to the right of each pinned square $C$. For any two squares $A$ and~$C$, the region $B(A,C)$ is now either a three-sided rectangle or the union of two three-sided rectangles, so we can use a simple priority search tree instead of our two-level data structure to test whether $B(A,C)$ is empty in $O(\log n)$ time. However, one further observation does improve the running time by a linear factor: Without loss of generality, the rightmost box in the optimal cover of $Y_C$ has the rightmost point of $Y_C$ on its right edge. Thus, there are at most $n$ candidate boxes $C_i$ to test in the inner loop; we can easily enumerate these candidates in $O(n)$ time. \begin{theorem} Let $f:\R_+\to\R$ be a fixed non-decreasing cost function. Given $n$ clients in the plane, we can compute in $O(n^3\log n)$ time a covering by axis-aligned squares centered on the $x$-axis, such that the sum of the costs of the squares is minimized. \end{theorem} \subsubsection{Fast and simple solutions} \label{sec:1b-cheap} In this section we describe simple and inexpensive algorithms that achieve constant factor approximations for finding a minimum-cost cover with disks centered along a fixed horizontal line $L$, using any $L_p$ metric. The main idea for the proofs of this section is to associate with a given disk $D$ in OPT, a set of disks in the approximate solution and argue that the set of associated disks cannot be more than a given constant factor cover of $D$, in terms of cumulative edge length, cumulative area, and so forth. As in the previous section, the case of $L_\infty$ metric is the easiest to handle. By equivalence of all the $L_p$ metrics, constant-factor $c$-approximations for squares will extend to constant-factor $c'$-approximations for $L_p$ disks. \smallskip \noindent \textbf{Square Greedy Cover Algorithm (SG):} Process the client points in order of decreasing distance from the line $L$. Find the farthest point $p_1$ from $L$; cover $p_1$ with a square $S_1$ exactly of the same height as $p_1$ centered at the projection of $p_1$ on $L$. Remove all points covered by $S_1$ from further consideration and recurse, finding the next farthest point from $L$ and so forth. In the case where two points are precisely the same distance from $L$, break ties arbitrarily. Obviously, SG computes a valid covering of $Y$ by construction.We begin the analysis with a simple observation. \begin{lemma} \label{overlay_lemma} In the SG covering, any point in the plane (not necessarily a client) cannot be covered by more than two boxes. \end{lemma} \begin{proof} Suppose $S_i$ and $S_j$ are two squares placed during the running of SG and that $i < j$ so that $S_i$ was placed before $S_j$. Then $S_i$ cannot contain the center point of $S_j$ since then $S_j$ would not have had the opportunity to be placed, and similarly $S_j$ cannot contain the center point of $S_i$. Now consider a point $p \in S_i \cap S_j$. If $p$ were covered by a third square $S_k$ then either one of $\{S_i, S_j\}$ would contain the center of $S_k$, or $S_k$ would contain the center of one of $\{S_i, S_j\}$, neither of which is possible. \end{proof} \begin{theorem} \label{SG_approx_lemma} Given a set $Y$ of $n$ clients in the plane and any $\alpha\geq 1$, SG computes in time $O(n \log n)$ a covering of $Y$ by axis-aligned squares centered on the $x$-axis whose cost is at most three times the optimal. \end{theorem} \begin{proof} Let $Y=\{p_1,\ldots,p_n\}$ and consider a square $S$ in OPT. We consider those squares $\{S_{i_j}\}$ selected by SG corresponding to points $\{p_{i_j} : p_{i_j} \in S\}$, see Figure \ref{fig:protrusion_arg}, \begin{figure} [h] \begin{center} \epsfxsize=10cm \scalebox{0.70}{\epsfbox{images/cover.eps}} \caption{Squares of the SG algorithm inside a square of the optimal solution.}\label{fig:protrusion_arg} \end{center} \end{figure} and argue that these squares cannot have more than three times the total edge length of $S$. The same will then follow for all of SG and all of OPT. The argument, without modification, covers the case of cost measured in terms of the sum of edge length raised to an arbitrary positive exponent $\alpha \ge 1$. Arguing as in Lemma \ref{overlay_lemma} it is easy to see that at most two boxes $S_{i_j}$ associated with points $p_{i_j} \in S$ processed by SG actually protrude outside of $S$, one on the left and one on the right. Denote by $r$ the total horizontal length of these protruding parts of squares, then $r \leq s$, the side length of $S$, since the side length of each protruding square is at most $s$ and at most half of each square is protruding. Because of Lemma \ref{overlay_lemma} the total horizontal length of all nonprotruding parts of the squares $S_{i_j}$ is at most $2s$, consequently all points covered by $S$ in OPT are covered by a set of squares $S_{i_j}$ in SG whose total (horizontal) edge length $\sum_j s_{i_j}$ is at most $3s$. For exponents $\alpha > 1$ observe that $\sum_j s_{i_j} \le 3s$ and $0 \le s_{i_j} \le s$ for all $j$ implies that $\sum_j {s_{i_j}}^\alpha \le 3s^\alpha$. To analyze the running time of the algorithm we need some more details about the data structures used: Initially, sort the points by $x$-coordinate and separately by distance from the line $L$ in time $O(n \log{n})$ and process the points in order of decreasing distance from $L$. As the point $p_i$ at distance $d_i$ from $L$ is processed, we throw away points which are within horizontal distance $d_i$ from $p_i$. This takes time $O(\log{n} + k_i)$ time where $k_i$ is the number of points within $d_i$ from $p_i$. Since we do this up to $n$ times with $k_1 + \cdots + k_l = n$ the total running time is $O(n \log{n})$. \end{proof} For the linear cost function, it is easy to modify the SG algorithm to get a $2$-approximation algorithm. \smallskip \noindent \textbf{Square Greedy with Growth Algorithm (SGG):} Process the points as in SG. However, if capturing a point $p_i$ by a square $S_i$ would result in an overlap with already existing square $S_j$ then, rather than placing $S_i$, grow $S_j$ just enough to capture $p_i$, keeping the vertical edge furthest from $p_i$ at the same point on $L$. If placing $S_i$ would overlap two squares, grow the one which requires the smallest edge extension. Break ties arbitrarily. A proof somewhat similar to that of Lemma~\ref{l:2-approx} shows that: \begin{theorem} \label{thm:SGG} Given $n$ clients in the plane, SGG computes in time $O(n \log n)$ a covering by axis-aligned squares centered on the $x$-axis whose cumulative edge length is at most twice the optimal. \end{theorem} \begin{proof} As we process points $p_i$ using SGG, attribute to each point $p_i$ a line segment $s_i$ along $L$ as follows. If processing $p_i$ resulted in the placement of a square $S_i$ centered at the projection of $p_i$ in $L$ then attribute to $p_i$ the projection on $L$ of a horizontal edge of $S_i$ (Case 1). If, on the other hand, processing of $p_i$ resulted in the growing of a prior square $S_j$ to just capture $p_i$, attribute to $p_i$ the projection on $L$ of the portion of the horizontal edge of the expanded $S_j$ needed to capture $p_i$ (Case 2). (This amount is at most the distance of $p_i$ to $L$ since otherwise $p_i$ would have been fallen into case 1.) We must show that the lengths of the segments is no more than twice the edge lengths of squares in OPT. It suffices to show that for any square $S$ in OPT, the segments $s_i$ associated with points $p_i \in S$ processed by SGG cannot have total edge length which exceeds twice the edge length $s$ of $S$. To see this observe that the sum of the lengths of those $s_i$ lying completely inside $S$ does not exceed $s$ since they are nonoverlapping. In addition, each of the parts of the at most two segments protruding from $S$ can have length at most $s/2$, in case 1 for the same reason as in the SG algorithm, in case 2 since the total length of the segment is at most $s/2$. In order to make SGG efficient, we proceed as in SG. In addition, we maintain a balanced binary search tree containing the $x$-coordinates of the vertical sides of the squares already constructed. For each new point $p_i$ to be processed we locate its $x$-coordinate within this structure to obtain its neighboring squares and to decide whether case 1 or case 2 applies. This can be done in time $O(\log n)$ just as adding a new square in case 1 or updating an existing square in case 2. Removing points covered by the new or updated square is done as in SG, so that the total runtime remains $O(n \log n)$. \end{proof} Unlike SG, SGG is not a constant factor approximation for area. Consider $n$ consecutive points at height $1$ separated one from the next by distance of $1 + \epsilon$. Processing the points left to right using SGG covers all points with one square of edge length $n + (n-1)\epsilon$, and so area $O(n^2)$, while covering all points with $n$ overlapping squares each of edge length $2$, uses total area~$4n$. Finally, extending these results from squares to disks in any $L_p$ metric is not difficult. Enclosing each square in the algorithm by an $L_p$ disk leads to an approximation factor $3c^2$ for GG and $2c^2$ for SGG, where $c=p^{\alpha/p}$. In particular, for $L_2$ disks, this yields a $2\sqrt2$-approximation for $\alpha=1$ and a $4$-approximation for $\alpha=2$. \subsection{Finding the best axis-parallel line} When the horizontal line $\ell$ is not given but its orientation is fixed, we first prove that finding the best line, even for $\alpha = 1$, is uncomputable, then in this linear case give a simple approximation, and finally a PTAS. \subsubsection{A hardness result -- uncomputability by radicals} Our approach is similar to the approach used by Bajaj on the unsolvability of the Fermat-Weber problem and other geometric optimization problems\cite{b-pgans-86,b-adgop-88}. \begin{theorem}\label{thm:radicals} Let $c(t)=\sum_i{r_i}$ denote the minimum cost of a cover whose centers lie on the line of equation $y=t$. There exists a set $Y$ of clients such that, if $t_0$ is the value that minimizes $c(t)$, then $t_0$ is uncomputable by radicals. \end{theorem} The proof proceeds by exhibiting such a point set and showing by differentiating $c(t)$ that $t_0$ is the root of a polynomial which is proven not to be solvable by radicals. The following definitions and facts can be found in a standard abstract algebra reference; see, for example, Rotman\cite{rotman-2002}. A polynomial with rational coefficients is \emph{solvable by radicals} if its roots can be expressed using rational numbers, the field operations, and taking $k$th roots. The \emph{splitting field} of a polynomial $f(x)$ over the field of rationals $\Q$ is the smallest subfield of the complex numbers containing all of the roots of $f(x)$. The \emph{Galois group} of a polynomial $f(x)$ with respect to the coefficient field $\Q$ is the group of automorphisms of the splitting field that leave $\Q$ fixed. If the Galois group of $f(x)$ over $\Q$ is a symmetric group on five or more elements, then $f(x)$ is not solvable by radicals over $\Q$. Consider the following set of points:$\{(3,4), (-3,-2), (102,2),$ $(98,-2), (200,-2)\}$. By exhaustive case analysis, we can show that the optimal solution must consist of one circle through the first two points, a second circle through the next two points, and a third circle touching the last point, and the optimal horizontal line must lie in the range $-2\le y \le 2$. For a given value of $y$ in this range, the cost of the best cover is $$ c(y) = \sqrt{2(y-1)^2 + 18} + \sqrt{2y^2 + 8} + (2-y). $$ Therefore, in order to find the best horizontal line, we must minimize $c(y)$. Setting the derivative to zero, we obtain the equation $$ c'(y) = \frac{2(y-1)}{\sqrt{2(y-1)^2 + 18}} +\frac{2y }{\sqrt{2y^2 + 8}} - 1 = 0. $$ We easily verify that $c''(y)$ is always positive. The minimum value $c(y) \approx 8.3327196$ is attained at $y\approx 1.4024709$, which is a root of the following polynomial: \begin{eqnarray*} f(y)& = & 1024+512 y-1600y^2+1536y^3-960y^4\\ & & +368y^5 -172y^6+28y^7 -7y^8. \end{eqnarray*} Using the computational system GAP\cite{GAP4}, we compute that the Galois group of $f(y)$ is the symmetric group~$S_8$, so the polynomial is not solvable by radicals. \subsubsection{Fast and simple constant-factor approximations} The simple constant factor approximations for a fixed line can be extended to the case of approximations to the optimal solution on an arbitrary axis-parallel line with the same constant factors, though with a multiplicative factor of $O(n^2)$ increase in running time. \subsubsection{An FPTAS for finding the best horizontal line} We begin with the case $\alpha = 1$. Let $d$ denote the distance between the highest and lowest point. Clearly, $d/2 \leq$ OPT $\leq n d/2$. Partition the horizontal strip of height $d$ that covers the points into $2n/\epsilon$ horizontal strips, each of height $\delta = d\epsilon/2n$, using $2n/\epsilon -1$ regularly-spaced horizontal lines, $\ell_i$. For each line $\ell_i$, we run the exact dynamic programming algorithm, and keep the best among these solutions. Consider the line, $\ell^*$, that contains OPT. We can shift line $\ell^*$ to the nearest $\ell_i$, while increasing the radius of each disk of OPT by at most $\delta$, to obtain a covering of the points by disks centered on some $\ell_i$; the total increase in cost is at most $\delta n = d\epsilon/2 \leq \epsilon$ OPT. Thus, our algorithm computes a $(1+\epsilon)$-approximation in time $O((n^3/\epsilon)\log{n})$. In order to generalize this result to the case $\alpha>1$, let us write PSEUDO-OPT for the lowest cost of a solution on any of the horizontal lines $\ell_i$, SHIFT for the result of shifting OPT to the closest of these lines, and $r_1,...,r_m$ for the radii of the optimal set of disks. For an arbitrary power $\alpha \ge 1$, we have \begin{eqnarray*} \mbox{PSEUDO-OPT} & \le & \mbox{SHIFT} \le \sum_{i=1}^{m} (r_i+\delta)^\alpha \\ & \le & \sum_{i=1}^{m}r_i^\alpha + \delta \alpha \sum_{i=1}^{m} (r_i+\delta)^{\alpha-1} \\ & \le & \mbox{OPT}(1 + \delta \alpha 2^{2\alpha-1} n /d). \end{eqnarray*} The last line uses $\delta \le d, r_i \le d$ and $\mbox{OPT} \ge (d/2)^\alpha$. Choosing $\delta = \varepsilon d /(\alpha 2^{2\alpha-1}n)$ gives the desired $(1+\varepsilon)$-approximation. Together with the results from previous sections we have: \begin{theorem} Given $n$ clients in the plane and a fixed $\alpha \geq 1$, there exists an FPTAS for finding an optimally positioned horizontal line and a minimum-cost covering by disks centered on that line. It runs in time $O((n^3/\epsilon) \log n)$ for the linear cost case ($\alpha=1$) and $O((n^5/\epsilon) \log n)$ for $\alpha>1$. \end{theorem} \subsection{Approximating the best line --\\ any orientation} Finally, we sketch approximation results for selecting the best line whose orientation is not given. We give both a constant factor approximation and a PTAS for the linear cost case ($\alpha=1$). \subsubsection{Fast and simple constant-factor approximations} Given a line $\ell$, we say that a set $\mathcal{D}$ of disks $D_1$,\ldots,$D_k$ is \emph{$\ell$-centered} if the centers of every disk $C_j$ in $\mathcal{D}$ belongs to $\ell$. Recall that the cost of $\mathcal{D}$ is the sum of all its radii. \begin{lemma}\label{lem:1} Given $k\geq 1$, a line $\ell$, an $\ell$-centered set $\mathcal{D}$ of $k$ disks that cover $Y$, and any point $p_0$ on $\ell$, there exist $p'\in Y$ and an $\ell'$-centered set $\mathcal{D}'$ of $k$ disks that cover $Y$, where $\ell'$ is the line that joins $p_0$ and $p'$, such that the cost of $\mathcal{D}'$ is at most $2^\alpha$ times the cost of $\mathcal{D}$. \end{lemma} \begin{proof} We will assume without loss of generality that $\ell$ is the $x$-axis, $p_{0}$ is the origin and that no other point in $Y$ lies on the $y$-axis. The latter restriction can easily be enforced by a small perturbation. Let the coordinates of $p_i$ be $x_i$ and $y_i$, and let $m_{i}$ denote the slope $y_{i}/x_{i}$ of the line $\ell_i$ for $1\leq i\leq n$. First, we reorder $Y$ so that $|m_{1}| \leq \dots \leq |m_{n}|$. In what follows we assume that $x_{1}> 0$ and $y_{1}\geq 0$. The other cases can be treated analogously. For each disk $D_j=D(t_j,r_j)$ in $\mathcal{D}$, we construct a disk $D'_j$ whose radius is $r'_{j} = 2r_{j}$ and center $t'_j$ is obtained from $t_{j}$ by rotating it around the origin counterclockwise by an angle $\tan^{-1}(m_{1})$. The set $\mathcal{D}'$ of $k$ disks thus defined is $\ell'$-centered, where $\ell' = \{(x,y) \in \R^{2} \mid y=m_{1}x\}$ and $p_{1} \in \ell'$. To see that $\mathcal{D}'$ covers $Y$, simply observe that $d(t_{j},t'_{j}) \leq r_{j}$ for all $1 \leq j \leq k$ and apply the triangle inequality: any point in $D_j$ must be at distance at most $2r_j$ of $t'_j$. The cost of this new solution is clearly at most $2^\alpha$ times that of $\mathcal{D}$ in the linear cost case. \end{proof} By a double appplication of this lemma, first about an arbitrary $p_0$ yielding a point $p'=p_i$, then about $p_i$ yielding another $p'=p_j$, it is immediate that any $\ell$-centered cover of $Y$ can be transformed into an $\ell_{i,j}$-centered cover whose cost is increased at most four-fold, where $\ell_{i,i}$ is the line joining $p_i$ and $p_j$. By computing (exactly or approximately) the optimal set of disks for all $O(n^{2})$ lines defined by two different points of $Y$, we conclude: \begin{theorem} \label{thm:1} Given $n$ clients in the plane and a fixed $\alpha \geq 1$, in $O(n^{4} \log n)$ time, we can find a collinear set of disks that cover $P$ at cost at most $4^\alpha OPT$, and for $\alpha=1$, in $O(n^{3} \log n)$ time, we can find a collinear set of disks that cover $P$ at cost at most $8\sqrt{2}OPT$. \end{theorem} \subsubsection{A PTAS for finding the best line with\\ unconstrained orientation} We now prove that finding the best line with unconstrained orientation and a minimum-cost covering with disks whose centers are on that line admits a PTAS. \begin{theorem} \label{thm:any-slope} Let $Y$ be a set of $n$ clients in the plane that can be covered by an optimal collinear set of disks at linear cost $OPT$ (i.e., $\alpha=1$), and $\epsilon>0$. In $O((n^{4}/\epsilon^2) \log n)$ time, we can find a collinear set of disks that cover $Y$ at cost at most $(1+\epsilon)OPT$. \end{theorem} \begin{proof} Let $H$ be a strip of minimal width $h$ that contains $Y$. Using a rotating calipers approach, $H$ can be computed in $O(n \log n)$ time. If $h=0$, we can conclude that $OPT=0$ and we are done. Otherwise, we can assume wlog that $H$ is horizontal and that its center line is the $x$-axis. Let $R$ denote the smallest enclosing axis-parallel rectangle $R$ of $Y$, $w$ its width, and $h$ its height, Then $h \le w$ and, moreover, $h/2 \leq OPT$. Let $\ell^*$ be the optimal line. We now distinguish two cases: {\bf Case 1. $w \geq 2h$:} Observe that both vertical sides $v_1,v_2$ of $R$ contain a point of $Y$. Therefore, $\ell^*$ must have distance at most $OPT$ to $v_1$ and $v_2$. A straightforward calculation shows that then $\ell^*$ must intersect the lines $\ell_1$ and $\ell_2$ extending $v_1$ and $v_2$ at a distance of at most $4 OPT$ from the $x$-axis. The idea is now to put points on those parts of $\ell_1$ and $\ell_2$ which are equally spaced at distance $\delta = \varepsilon OPT / n$. Then we consider all lines passing through one of these points on $\ell_1$ and one on $\ell_2$. For each such line we find the optimal covering of $P$ by circles centered on it using the algorithm of Theorem~\ref{thm:DPlinear}, and give out the best one as an approximation for the optimum. Observe, that there is one of the lines checked, $\hat{\ell}$, whose intersection points with $\ell_1$ and $\ell_2$ are at distance at most $\delta/2$ from the ones of $\ell^*$. Elementary geometric considerations show that to any point $p$ in $\ell^*$ closest to some point of $P$ there is a point in $\hat{\ell}$ within distance at most $\delta$. Consequently, to any circle of radius $r$ of the optimal covering centered on $\ell^*$, there is a circle on $\hat{l}$ of radius $r+\delta$ covering the same set of points (or more). Thus, $\hat{l}$ has a covering that differs by at most $ n \delta$ from the optimal one. By the choice of $\delta$ we have a $1+\varepsilon$-approximation to the optimum. Observe, that we chose $O(OPT/\delta)=O(n/\varepsilon)$ points on $\ell_1$ and $\ell_2$, so we are checking $O((n/\varepsilon)^2)$ lines. For each of them, we apply the algorithm of Theorem~\ref{thm:DPlinear} which has runtime $O(n^2 \log n)$ yielding a total runtime of $O(n^4 / {\varepsilon^2} \log n)$. {\bf Case 2. $w < 2h$:} In this case the optimal line $\ell^*$ can have a steeper slope and even be vertical. Of course, it must intersect $R$ and we expand $R$ to a cocentric rectangle $R'$ such that the footpoint of any point in $Y$ on $\ell^*$ must lie inside $R'$. An easy geometric consideration shows that extending the width of $R$ by $h$ and its height by $w$ will suffice, so $R'$ is a square of side length $w+h$. Then we put equally spaced points of distance $\delta = \varepsilon OPT / n$ on the whole boundary of $R'$, apply the algorithm of Theorem~\ref{thm:DPlinear} to all lines passing through any two of these points, and return the one giving the smallest covering as an approximation to the optimum. The same consideration as in the first case shows that this is indeed a $(1+\varepsilon)$-approximation. Since the length of the boundary of $R'$ is $4(w+h) \leq 12h \leq 24 OPT$, we obtain the desired runtime in this case, as well. For both cases it remains to show how to obtain a suitable value of $\delta$, since we do not know the value of $OPT$. Since any value below $OPT$ suffices, we simply run a constant factor $c$ approximation algorithm of Theorem~\ref{thm:DPlinear} and take $1/c$ times the value it returns instead of $OPT$ in the definition of $\delta$. \end{proof} \section{Minimum-Cost Covering Tours} \label{sec:scenario3} We now consider the minimum cost covering tour (MCCT) problem: Given $k\geq 1$ and a set $Y=\{p_1,\ldots,p_n\}$ of $n$ clients, determine a cover of $Y$ by (at most) $k$ disks centered at $X=\{t_1,\ldots,t_k\}$ with radii $r_j$ and a tour $T$ visiting $X$, such that the cost $\mathrm{length}(T)+C\sum r_i^\alpha$ is minimized. We refer to the tour $T$, together with the disks centered on $X$, as a \emph{covering tour} of $Y$. Our results are for the case of linear transmission costs ($\alpha=1$). We first show a weak hardness result, then characterize the solution for $C\leq 4$, and finally give a PTAS for a fixed $C>4$. \subsection{A hardness result} We prove the NP-hardness of MCCT where $C$ is also part of the input. Note that this does not prove the NP-hardness of MCCT where $C$ is a fixed constant, which is the problem for which we give a PTAS below. Note also that $C$ appears in the run time exponent of that PTAS, and so the PTAS no longer runs in polynomial time if $C$ is not a fixed constant. \begin{theorem} \label{thm:hard-MCCT} MCCT with linear cost is NP-hard if the ratio $C$ is part of the input. \end{theorem} \begin{proofsketch} We show a reduction from {\sc Hamilton cycle in grid graphs}. Given a set of $n$ points on a grid, we construct an instance of MCCT in which each of the given points is a client. We set $C$ to be larger than $2n$. We claim that the grid graph has a Hamilton cycle if and only if there is a tour $T$ visiting a set of disk centers with radii $r_i$ whose cost is at most~$n$. Clearly a Hamilton cycle in the grid graph yields a tour of cost $n$ with each client contained in a disk of radius 0 centered at that point. Conversely, suppose we have a tour whose cost (length plus sum of radii) is at most $n$. Note that no two clients can be contained in a single disk, as such a disk must have radius at least 0.5, and thus its contribution to the cost $C\cdot r_i >2n\cdot 0.5 =n$ contrary to our assumption. Next we want to show that each disk in an optimal solution is centered at the client it covers. Suppose this is not the case, there is some client $j$ which is covered by a disk centered at $c_j\not= j$. Let the distance between client $j$ and the center of the disk covering it be $d$. Now consider an alternate feasible solution in which the tour visits $c_j$ then $j$ then back to $c_j$, covering $j$ with a disk of radius 0. No other client is affected by this change, as the disk only covers point $j$. The cost of the new solution is the cost of the original (optimal) solution $+2d-Cd$ as we add $2d$ to the length of the tour, but decrease $C\sum r_i$ by $Cd$. Since $C>2$ the new solution is better than the original optimal solution, a contradition.\qed \end{proofsketch} \subsection{The case {\Large $C\leq 4$}: The exact solution is\\ a single circle} \begin{theorem} \label{c=4_lemma} In the plane, with a cost function of $\mathrm{length}(T)+C \sum r_i$ and $C \leq 4$, the minimum-cost solution is to broadcast to all clients from the circumcenter of the client locations and no tour cost. \end{theorem} The proof rests on the following elementary geometry lemma (whose proof is omitted here). \begin{lemma} \label{3_point_lemma} For three points $p$, $q$ and $r$ in the plane, such that the triangle $pqr$ contains its own circumcenter, the length of a trip from $p$ to $q$ to $r$ and back to $p$ is at least $4r$ where $r$ is the circumradius of the points. \end{lemma} \begin{proofof}{Theorem \ref{c=4_lemma}} Let $r(X)$ and $r(Y)$ denote the minimum radius of a circle enclosing $X$ or $Y$, respectively. Let $T$ be a covering tour of $Y$, $X\subseteq T$ be the set of disk centers and $r_j$ their radii. Finally, let $r_{\mathrm{max}}=\max_j r_j$. By the triangle inequality, Lemma \ref{3_point_lemma} implies that the $\mathrm{length}(T)\geq 4r(X)$. Since the tour visits all the centers in $X$ and the disks centered at $X$ cover $Y$, we have $r(Y)\leq r(X)+ r_{\mathrm{max}}$. By definition, the cost of $T$ is $\mathrm{length}(T) + C\sum_j{r_j}$, which by the observation above is at least $4 r(X) + C\sum_j{r_j} \geq 4r(X) + C r_{\mathrm{max}}$. The assumption $C\leq 4$ then implies that it be at least $C(r(X)+ r_{\mathrm{max}})\geq Cr(Y)$, which is the cost of covering by a single disk with a zero-length tour. \qed \end{proofof} \subsection{The case {\Large $C>4$}: A PTAS} We distinguish between two cases for the choice of transmission points: they may either be arbitrary points in the plane (selected by the algorithm) or they may be constrained to lie within a discrete set ${\cal T}$ of candidate locations. The constant $C$ specifies the relative weight associated with the two parts of the cost function -- the length of the tour, and the sum of the disk radii. If $C$ is very small ($C\leq 4$), then the solution is to cover the set $Y$ using a single disk (the minimum enclosing disk), and a corresponding tour of length 0 (the singleton point that is the center of the disk). If $C$ is very large, then the priority is to minimize the sum of the radii of the $k$ disks. Thus, the solution is to compute a covering of $Y$ by $k$ disks that minimizes the sum of radii (as in\cite{lp-ptasb-05}), and then link the resulting disk centers with a traveling salesman tour (TSP). (In the case that $k\geq n$, the disks in the covering will be of radius 0, and the problem becomes that of computing a TSP tour on~$Y$.) Note that our algorithm gives an alternative to the Lev-Tov and Peleg PTAS\cite{lp-ptasb-05} for coverage alone. Our algorithm is based on applying the $m$-guillotine method\cite{m-gsaps-99}, appropriately adapted to take into account the cost function and coverage constraint.\footnote{The ``$m$'' in this section refers to a parameter, which is $O(1/\varepsilon)$, not the number of servers.} We need several definitions; we largely follow the notation of \cite{m-gsaps-99}. Let $G=(V,E)$ be an embedding of a connected planar graph, of total Euclidean edge-length $L$. Let ${\cal D}$ be a set of disks centered at each vertex $v$ of $G$ of radius $r_v$. We refer to the pair $(G,{\cal D})$ as a {\em covering network} if the union $\cup_{v\in V} D_v$ of the disks covers the clients $Y$. We can assume without loss of generality that $G$ is restricted to the unit square $B$, i.e., $\cup_{e\in E} e\subset int(B)$. Our algorithm relies on there being a polynomial-size set of candidate locations for the transmission points that will serve as the vertices of the covering tour we compute. In the case that a set ${\cal T}$ of candidate points is given, this is no issue; however, in the case that the transmission points are arbitrary, we appeal to the following grid-rounding lemma (proved in the full paper). \begin{lemma} \label{lem:grid} One can perturb any covering network $(G,{\cal D})$ to have its vertices all at grid points on a regular grid of spacing $\delta=O(\epsilon \cdot diam(S)/n)$, while increasing the total cost by at most a factor of $(1+\epsilon)$. \end{lemma} An axis-aligned rectangle, $W\subseteq B$, is called a {\em window}; rectangle $W$ will correspond to a subproblem in a dynamic programming algorithm. An axis-parallel line $\ell$ that intersects $W$ is called a {\em cut}. For a covering network with edge set $E$ and a set of disks ${\cal D}$, we say that $(E,{\cal D})$ satisfies the {\em $m$-guillotine property with respect to window $W$} if either (1) all clients $Y\subset W$ lie within disks of ${\cal D}$ that intersect the boundary of $W$; or (2) there exists a cut $\ell$ with certain properties (an \emph{$m$-good cut with respect to $W$}) that splits $W$ into $W_1$ and $W_2$, and $(E,{\cal D})$ recursively satisfies the $m$-guillotine property with respect to both $W_1$ and~$W_2$. Due to the lack of space, we cannot give the full definition of an $m$-good cut (see the full paper). The crux of the method is a structural theorem, which shows how to convert any covering network $(G,{\cal D})$ into another covering network $(G',{\cal D}')$, such that the new graph $G'$ satisfies the $m$-guillotine property, and that the total cost of the new instance $(G',{\cal D}')$ is at most $O((L+CR)/m)$ times greater than the original instance $(G,{\cal D})$, where $L$ is the total edge length of $G$ and $R$ the sum of the radii of ${\cal D}$. The construction is recursive: at each stage, we show that there exists a cut with respect to the current window $W$ (which initially is the unit square $B$), such that we can ``afford'' (by means of a charging scheme) to add short horizontal/vertical edges in order to satisfy the $m$-guillotine property, without increasing the total edge length too much. We then apply a dynamic programming algorithm, running in $O(n^{O(m)})$ time, to compute a minimum-cost covering network having a prescribed set of properties: (1) it satisfies the $m$-guillotine property (with respect to $B$), which is necessary for the dynamic program to have the claimed efficiency; (2) its disks cover the clients $Y$; and (3) its edge set contains an Eulerian subgraph. This third condition allows us to extract a tour in the end. In the proof of the following theorem (see the full paper), we give the details of the dynamic programming algorithm that yields: \begin{theorem} \label{cor:ptas} The min-cost covering tour problem has a PTAS that runs in time $O(n^{O(1/\epsilon)})$. \end{theorem} \section*{Acknowledgments} We thank all of the participants of the McGill-INRIA International Workshop on Limited Visibility, at the Bellairs Research Institute of McGill University, where this research was originated. We acknowledge valuable conversations with Nancy Amato, Beppe Liotta, and other workshop participants and heartily thank the organizers Sue Whitesides and Hazel Everett for facilitating and enabling a wonderful working environment. \balance \edef\baselinestretch{0.93} \bibliographystyle{abbrv}

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\section{Introduction} Determinantal point processes (DPP) define distributions over sets of points in a metric space, e.g. Euclidean $\mathbb{R}^d$, which are characterised by repulsion between points. An archetypal example from random matrix theory are the eigenvalues of the Gaussian unitary ensemble (GUE) \cite{mehta2004random}. Their joint p.d.f. can be shown to be \cite{livan2018introduction,mehta2004random} \begin{align} p(\lambda_1, \ldots, \lambda_N) \propto \prod_{i < j} |\lambda_i - \lambda_j|^2 \prod_{i=1}^N \frac{e^{-\frac{\lambda_i^2}{2}}}{\sqrt{2\pi}}. \end{align} One sees immediately the competing effects of the repulsive terms $|\lambda_i-\lambda_j|^2$ which encourage configurations of eigenvalues with large pairwise separations and the confining Gaussian potentials $e^{-\frac{\lambda_i^2}{2}}$ which encourage eigenvalues to stray not too far from the origin. The eigenvalues of a GUE and other random matrix ensembles can indeed be seen quite precisely to be distributed as certain determinantal point processes \cite{mehta2004random} \medskip In general a DPP on locally compact Polish space\footnote{Recall that a Polish space is separable and completely metrizable. Separability (i.e. the existence of some dense countable subset) is crucial for a well-behaved product measure in this context. Being completely metrizable makes speaking of convergence much simpler. Local compactness is a natural condition when one is speaking of repulsion between points.} $\mathbb{X}$ is defined by a measure $\mu$ and a measurable \emph{kernel} $K$ such that, for any $n\in\mathbb{N}$ \begin{align} p(x_1, \ldots x_n)dx_1\ldots dx_n = \frac{1}{N!}\det\left(K(x_i, x_j)\right)_{i,j=1}^n \prod_{i=1}^n \mu(dx_i). \end{align} The repulsive nature of the DPP is encoded in the determinantal structure. Intuitively, the determinant measures the volume occupied by the points $x_i$ embedded in some space induced by the kernel and so samples are biased towards configurations with larger volumes, meaning greater separation between points. More precisely, consider a \emph{projection kernel} \cite{hough2006determinantal} defined as \begin{align} K_N(x, x') = \frac{1}{N} \sum_{i=1}^N\phi_i(x)\phi_i(x') \end{align} where $\phi_i$ are orthonormal functions on $\mathbb{X}$ with respect to the chosen measure $\mu$, i.e. \begin{align}\label{eq:projection_def} \int_{\mathbb{X}} d\mu(x) \phi_i(x)\phi_j(x) = \delta_{ij}. \end{align} With this construction \begin{align} \det\left(K(x_i, x_j)\right)_{i,j=1}^n = \left(\det\vec{\Phi}(x_{1:N})\right)^2 \end{align} where the matrix $\vec{\Phi}(x_{1:N})_{ij} = \phi_j(x_i)$ and so we see that the DPP density favours configurations with large separation of the $x_i$ in the embedding space induced by the $\phi_j$. Projection DPPs can be viewed as a truncation of DPPs. Indeed by Mercer's theorem there exists some eigenfunction expansion \begin{align} K(x, y) = \sum_{i=1}^{\infty} \lambda_i \phi_i(x)\phi_i(y). \end{align} Truncating this expansion and rescaling appropriately to absorb the eigenvalues produces the projection DPP expression (\ref{eq:projection_def}). A DPP so defined is referred to as $\text{DPP}(\mu, K_N)$. Samples from $\text{DPP}(\mu, K_N)$ have the convenient property that their cardinality is almost surely $N$ \cite{hough2006determinantal,NEURIPS2019_1d54c76f} \medskip DPPs have several applications in machine learning and statistics \cite{kulesza2012determinantal}, being the natural class of distributions for any point process with repulsion. Recently \cite{NEURIPS2019_1d54c76f} proposed using certain DPPs to construct Monte Carlo (MC) integration estimators that can outperform na\"{i}ve MC integration. In particular, the authors discuss the Bardenet-Hardy (BH) estimator \cite{bardenet2020monte} and the Ermakov-Zolotukhin (EZ) estimator \cite{ermakov1960polynomial}. Both estimators rely on on the repulsive nature of projection DPPs to provide super-na\"{i}ve MC convergence. Indeed, \cite{NEURIPS2019_1d54c76f} establish a central limit theorem for BH with decay of speed $\sqrt{N^{1 + 1/d}}$; similar results for EZ are not yet available. \medskip \cite{NEURIPS2019_1d54c76f} focusses specifically on the case $\mathbb{X}=[-1, 1]^d$ with measure \begin{align} \mu(\bm{x}) = \prod_{i=1}^d (1-x)^{a_i}(1+x)^{b_i}dx_i \end{align} in which case the appropriate orthogonal functions $\phi_i$ are multivariate Jacobi polynomials. \medskip In this paper we extend approach of \cite{NEURIPS2019_1d54c76f} to Gaussian measure on $\mathbb{R}^d$. In machine learning applications, accurate MC estimation of integration against Gaussian measure is arguably the most common case, and certainly more common than Jacobi measure. Our techniques also open the way for applications of Gaussian DPPs on $\mathbb{R}^d$ in statistical models. The central contribution we present is a novel rejection sampler for Gauss-Hermite projection DPPs based on a factorisation trick and a random matrix theory connection. Alongside the development of this sampler, we also present experimental results using it in BH and EZ estimation, as well as timing and efficiency results for the sampler. \section{Multivariate Hermite polynomials and a factorised DPP sampler} \paragraph{Special case.} In the one dimensional case, the procedure in this section is not required. Rather one can sample a $N\times N$ GUE matrix $X$ and compute its eigenvalues $\lambda_1,\ldots, \lambda_N$. The eigenvalues are then distributed as $\text{DPP}(\mu, K_N)$. \subsection{Factorised sampler in special cases} Let $\{H_j\}_{j=0}^{\infty}$ denote the Physicist's Hermite polynomials defined by the orthogonality condition on $\mathbb{R}$ \begin{align} \int_{\mathbb{R}} dx ~ e^{-x^2} H_j(x)H_k(x) = \sqrt{\pi}2^j j! \delta_{jk}. \end{align} We define from these the following orthonormal polynomials \begin{align} \psi_j = \frac{H_j(x/\sqrt{2})}{\sqrt{\sqrt{2\pi}2^j j!}} \end{align} which obey \begin{align} \int_{\mathbb{R}} dx ~ e^{-\frac{x^2}{2}} \psi_j(x)\psi_k(x) = \delta_{jk}. \end{align} From these $\psi_j$ it is simple to define multivariate polynomials orthonormal with respect to Gaussian measure on $\mathbb{R}^d$ \begin{align} \phi_{\vec{i}}(\vec{x}) = \prod_{\ell = 1}^d \psi_{i_{\ell}}(x_{\ell}) \end{align} where $\vec{i}$ is a multi-index of length $d$ and indeed \begin{align} \int_{\mathbb{R}^d} d\bm{x} ~ e^{-\frac{\bm{x}^2}{2}} \phi_{\vec{j}}(\bm{x})\phi_{\vec{k}}(\bm{x}) = \prod_{\ell=1}^d \int_{\mathbb{R}} dx_{\ell} ~ e^{-\frac{x_{\ell}^2}{2}} \psi_{j_{\ell}}(x_{\ell})\psi_{k_{\ell}}(x_{\ell}) = \prod_{\ell=1}^d\delta_{j_{\ell}k_{\ell}} = \delta_{\vec{j}\vec{k}}. \end{align} Following \cite{NEURIPS2019_1d54c76f} we can place a natural ordering $\mathfrak{b}:\mathbb{N}^d\rightarrow\mathbb{N}$ on the $\{\phi_{\vec{i}}\}$ sorting first by the maximum degree of $\vec{i}$ and within a given maximum degree sorting lexicographically. Again following \cite{NEURIPS2019_1d54c76f,hough2006determinantal} we construct a kernel from a finite dimensional subspace of $\text{span}(\{\phi_{\vec{i}}\}_{\mathfrak{b}(\vec{i})=0}^{\infty})$: \begin{align} K_N(\vec{x}, \vec{y}) = \sum_{\mathfrak{b}(\vec{i})=0}^{N-1} \phi_{\vec{i}}(\vec{x})\phi_{\vec{i}}(\vec{y}). \end{align} We can now define a projection DPP with respect to un-normalised standard Gaussian base measure on $\mathbb{R}^d$, denoted by $\mu$, i.e. $d\mu(\bm{x}) = e^{-\frac{\bm{x}^2}{2}}d\bm{x}$. \begin{align} \{\bm{x}_1, \ldots, \bm{x}_N\}\sim\text{DPP}(\mu, K_N) ~ \iff ~ p(\bm{x}_1, \ldots, \bm{x}_N)d\bm{x} = \frac{1}{N!}\det\left(K_N(\bm{x}_p, \bm{x}_q)\right)_{p,q=1}^N d\mu(\bm{x}). \end{align} To sample from $\text{DPP}(\mu, K_N)$ we can follow a procedure similar to that used in \cite{NEURIPS2019_1d54c76f} for DPPs with respect to Jacobi base measure. \cite{NEURIPS2019_1d54c76f} gives the follows factorisation \begin{align}\label{eq:dpp_factorisation} p(\bm{x}_1, \ldots, \bm{x}_N)d\bm{x}_1\ldots d\bm{x}_N = \frac{K_N(\bm{x}_1, \bm{x}_1)}{N} \omega(\bm{x}_1) d\bm{x}_1 \prod_{n=2}^N \omega(\bm{x}_n)d\bm{x}_n \frac{K_N(\bm{x}_n, \bm{x}_n) - \vec{k}_{n-1}(\bm{x}_n)^T\vec{K}_{n-1}^{-1}\vec{k}_{n-1}(\bm{x}_n)}{N - (n-1)}, \end{align} where $\omega(\bm{x}) = e^{-\frac{\bm{x}^2}{2}}$, $\vec{k}_{n-1}(\bm{x}) = (K_N(\bm{x}_1, \bm{x}), \ldots, K_N(\bm{x}_{n-1}, \bm{x}))^T$ and $(\vec{K}_{n})_{ij} = K_N(\vec{x}_i, \vec{x}_j)$ for $1 \leq i,j \leq n$. This factorisation follows from writing the $N\times N$ kernel matrix as a $( (N-1) + 1) )\times ( (N-1) + 1) )$ block matrix, separating out $\vec{K}_{N-1}$ from $K_N(\vec{x}_N, \vec{x}_N)$, using the Schur formula for block matrix determinants, and finally iterating this procedure. Thus for a given $N$, points are sampled sequentially using a chain rule scheme i.e. $\bm{x}_j$ is sampled from $p(\bm{x}_j \mid \bm{x}_1,\ldots, \bm{x}_{j-1})$ by exploiting the above factorisation. The artificial induced order can then been thrown away to give a sample from the DPP. The problem is thus reduced too sampling from the conditional densities given in the product in (\ref{eq:dpp_factorisation}). A rejection sampler with good rejection bound can be used for the conditionals $p(\bm{x}_j \mid \bm{x}_1,\ldots, \bm{x}_{j-1})$ using as proposal distribution \begin{align} f_N(\bm{x})d\bm{x} = \frac{1}{N} d\mu(\bm{x})\sum_{\mathfrak{b}(\vec{i})=0}^{N-1} \phi_{\vec{i}}(\bm{x})^2 = \frac{1}{N} d\mu(\bm{x}) K_N(\bm{x}, \bm{x}). \end{align} Indeed, one has \begin{align} \frac{K_N(\bm{x}_n, \bm{x}_n) - \vec{K}_{n-1}(\bm{x}_n)^T\vec{K}_{n-1}^{-1}\vec{K}_{n-1}(\bm{x}_n)}{N - (n-1)} \left(f(\bm{x})\right)^{-1} \leq \frac{N}{N - (n-1)} \end{align} which follows from the positive definiteness of kernel matrices and holds for any choice of kernel and any base measure $\mu$. \medskip At this point \cite{NEURIPS2019_1d54c76f} sample from $f_N(\bm{x})d\bm{x}$ by viewing it as a mixture and constructing efficient rejection samplers for the mixture components with distributions $d\mu(\bm{x})\phi_{\vec{i}}(\bm{x})^2$. This approach cannot be adapted to the Gauss-Hermite case considered here, as it relies on good uniform bounds which are not available for Hermite polynomials\footnote{See \cite{krasikov2004new} for the best bounds on Hermite polynomials. Roughly speaking, the problem is that $\psi_n(x)^2$ has very sharp peak values which grow with $n$. See also Appendix \ref{app:spherical} for an alternative Gaussian orthonormal function basis that also suffers from bad bounds.} but are for Jacobi polynomials. We propose an alternative sampler for $f_N(\bm{x})d\bm{x}$ that side-steps the need for uniform bounds on Hermite polynomials and instead exploits a connection to random matrix theory. Note that our approach could be applied, with appropriate modifications, to other DPPs with different base measures, including the Jacobi DPP studied in \cite{bardenet2020monte}. Initially we specialise to the case $N = n^d$ where $n$ is a positive integer. With this assumption one can make the following factorisation \begin{align} f_N(\bm{x}) = \prod_{\ell=1}^d \left(\frac{1}{n} e^{-\frac{x_{\ell}^2}{2}} \sum_{i_{\ell}=0}^{n} \psi_{i_{\ell}}(x_{\ell})^2\right) \end{align} and so sampling from $f_N(\bm{x})d\bm{x}$ can be achieved by sampling the entries of $\bm{x}$ \emph{independently} from \begin{align} \rho_{n}(x_{\ell})dx_{\ell} = \frac{1}{n} e^{-\frac{x_{\ell}^2}{2}} \sum_{i_{\ell}=0}^{n} \psi_{i_{\ell}}(x_{\ell})^2 ~ dx_{\ell}. \end{align} To sample from the univariate distribution $\rho_{n}(x)dx$ we draw on a result from random matrix theory, namely that $\rho_{n}$ is the spectral density of an $n\times n$ Gaussian unitary ensemble (GUE) matrix \cite{livan2018introduction}. An Hermitian $n\times n$ random matrix $X$ is GUE if it has density proportional to $e^{-\frac{1}{2}\text{Tr} X^2}$. The eigenvalues and eigenvectors of a GUE matrix are independent so let us denote the density of its unordered eigenvalues by $p(\lambda_1, \ldots,\lambda_n)$. The \emph{spectral density} is then simply the single eigenvalue marginal \begin{align} \int dx_2\ldots dx_{n} ~ p(x, x_2, \ldots, x_n). \end{align} This suggests the following sampling scheme: \begin{enumerate} \item Sample $X$ as a $n\times n$ GUE matrix. \item Compute the eigenvalues $\lambda_1, \ldots, \lambda_{n}$ of $X$. \item Select $x$ uniformly at random from $\{\lambda_1, \ldots, \lambda_{n}\}$. \item $x$ has distribution $\rho_{n}(x)dx$. \end{enumerate} While elegant, this scheme is quite inefficient. Sampling an $n\times n$ GUE matrix requires sampling $n^2$ independent standard univariate Gaussians. Computing the spectrum of $X$ is an $\mathcal{O}(n^3)$ operation. So overall each sample drawn from $\rho_{n}(x)dx$ by this method has time complexity $\mathcal{O}(n^3) + n^2$. The random matix theory interpretation of $\rho_{n}$ does however provide an alternative sampling approach. The following result is a special case of Wigner's theorem \cite{livan2018introduction}: \begin{align}\label{eq:wigner_thm} \lim_{n\rightarrow\infty} \sqrt{2n}\rho(\sqrt{2n}x) = \frac{1}{\pi} \sqrt{2 - x^2} ~\mathbb{I}\{x^2 \leq 2\} \equiv \rho_{SC}(x) \end{align} where the density on the right hand side is Wigner's semi-circle. The error terms in (\ref{eq:wigner_thm}) are actually known to be $\mathcal{O}(n^{-1})$, which can be seen easily from the Christoffel-Darboux expansions in Section 10.4 of \cite{livan2018introduction}. We can thus say \begin{align}\label{eq:semicircle_approx} \rho_{n}(x) = \frac{1}{2 n\pi} \sqrt{4n - x^2} ~ \mathbb{I}\{ x^2 \leq 4n\} + \mathcal{O}(n^{-3/2}) \end{align} and the error terms are non-vacuous in the bulk of the support excluding regions of width $\mathcal{O}(n^{-3/2})$ at the edges of the support. This approximation suggests a rejection sampling approach. Consider the following class of mixture distributions with densities \begin{align} h_{p, \nu}(x) = pg_{\nu}(x) + (1-p)\rho_{SC}(x) \end{align} where $g_{\nu}$ is Student's $t$ density with $\nu$ degrees of freedom. We will use $h_{p, \nu}$ as the proposal distribution for a rejection sampler of $\rho_n(x)dx$. The reasoning behind this choice is the following: \begin{itemize} \item (\ref{eq:semicircle_approx}) suggests using a semi-circle as proposal, i.e. propose $x = (\epsilon + \sqrt{2n})y$ where $y\sim \rho_{SC}(x)dx$ and $\epsilon>0$ is small and included to inflate the semi-circle slightly so that it encompasses $\rho_n$. \item This approach fails however, as $\rho_n$ is in fact supported on all of $\mathbb{R}$. \item To ensure that any rejection bound holds in the tails, we seek to modify the semi-circle to endow it with heavy tails. For large $n$, the largest eigenvalue of a $n\times n$ GUE follows the Tracy-Widom tail distribution which has a right tail like $e^{-cx^{3/2}}$ \cite{anderson2010introduction}. This heuristic suggests that a proposal with Gaussian tails may not have sufficient tail weight. \item Hence the mixture of a semi-circle and a Student's $t$. It may be possible to use a lighter-tailed distribution, however we find this is a stable and reliable choice, ensuring that the rejection sampler bound is satisfied everywhere. \end{itemize} \paragraph{Sampling from the semi-circle.}To use $h_{p, \nu}$ as proposal density, we must be able to sample from it. Sampling from Student's $t$ is standard, so we need only address sampling from the semi-circle. The semi-circle is compactly supported and has a simple unimodal shape, so it is easy to hand-engineer an appropriate rejection sampler. We choose a proposal density \begin{align} r(x) = \frac{1}{2}(\mathcal{N}(x; -1, (4/5)^2) + \mathcal{N}(x; 1, (4/5)^2)) \end{align} and then numerically find the bound \begin{align} \frac{\rho_{SC}(x)}{r(x)} \leq 1.42. \end{align} \medskip We of course require a bound $\rho_{n}/h_{p, \nu} \leq M_{n} ~ \forall~x$ so that rejection sampling can be used. Such a bound does not appear analytically forthcoming, however since we are a considering only simple univariate densities, it is quite practical to hand-craft a bound empirically. Indeed, it is a simple matter numerically to compute near-optimal bounds for any $n, p$ and $\nu$. The acceptance rate can then be estimated and finally the best values $p_{n}, \nu_{n}$ along with near-optimal bounds $M_{n}$ can be found for each $n$ to give near optimal acceptance rates. \medskip Following the above empirical procedure, we conclude the degrees of freedom $\nu$ is not particularly important to the efficiency of the sampler, so we set it to 10. The mixture probability $p$ is important, however, and we fit a simple parametric curve to it: \begin{align}\label{eq:mixture_p} 0.100 - \frac{0.486}{n} + \frac{0.647}{n^{1/2}} + \frac{0.272}{n^{1/4}}. \end{align} The empirical results and this simple fit are shown in Figure \ref{fig:optimal_p}. Note that our parametric fit is monotonically decreasing, which is intuitively appropriate as the tails of $\rho_n$ lighten as $n$ increases. \begin{figure} \centering \includegraphics[width=0.4\linewidth]{figures/optimal_mixture.pdf} \caption{Empirical results for the optimal mixture probability with respect to the best attainable acceptance probability in the rejection sampler for $\rho_{n}$.} \label{fig:optimal_p} \end{figure} \medskip Now fixing the form of the mixture probability in (\ref{eq:mixture_p}), we numerically find approximations to the tightest bounds $M_{n}$. Practically, this is done by evaluating the ratio $\rho_{n}/h_{p, \nu}$ over $(-2\sqrt{n} - 10, 2\sqrt{n} + 10)$ on a finely-spaced grid and taking the maximum. We then fit a parametric form to the resulting data, building in some extra tolerance to ensure that our parametric fit is certainly a uniform upper bound for $\rho_{n}/f_{p, \nu}$. The results are shown in Figure \ref{fig:optimal_m} and the parametric fit is \begin{align}\label{eq:reject_bound} 0.492 + \frac{1.058}{n} - \frac{3.352}{n^{1/2}} + \frac{3.308}{n^{1/4}}. \end{align} \begin{figure} \centering \includegraphics[width=0.4\linewidth]{figures/optimal_bound.pdf} \caption{Empirical results for the tightest uniform rejection bound on $\rho_{n}/f_{p, \nu}$.} \label{fig:optimal_m} \end{figure} Figure \ref{fig:rejection_sampler_bound} shows the target density $\rho_n$ and the scaled proposal density $M_nh_{p_n, \nu}$. We show some results of using our rejection sampler for $\rho_n$ for various $n$ in Figure \ref{fig:optimal_m}. \begin{figure} \centering \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/densities_n3.pdf} \caption{$n=3$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/densities_n5.pdf} \caption{$n=5$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/densities_n10.pdf} \caption{$n=10$} \end{subfigure} \caption{Comparison of the target density $\rho_n$ and the scaled proposal density $M_nf_{p_n, \nu}$ for various $n$ values.} \label{fig:rejection_sampler_bound} \end{figure} \begin{figure} \centering \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/sample_n3.pdf} \caption{$n=3$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/sample_n5.pdf} \caption{$n=5$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/sample_n10.pdf} \caption{$n=10$} \end{subfigure} \caption{Results of using our rejection sampler for $\rho_n$ for a few different $n$ values. We used 10000 samples and 50 histogram bins.} \label{fig:optimal_m} \end{figure} Finally, fixing our parametric fits, we can estimate the actual obtained acceptance probability. The results from 10000 trials for each $n$ value are shown in Figure \ref{fig:accept_prob}. Note that the acceptance probability is generally increasing in $n$ and never less that $0.7$. \begin{figure} \centering \includegraphics[width=0.4\textwidth]{figures/accept_prob.pdf} \caption{The empirically obtained acceptance probability for the rejection sampler of $\rho_n$ for various $n$ values. Calculated from 10000 trials.} \label{fig:accept_prob} \end{figure} \medskip \subsection{Extension to rejection sampler in the general case}\label{subsec:general_N} In the previous subsection we restricted to an extremely special case $N=n^d$. In this section we leverage the sampler we constructed in that case to construct one for general $N$. \medskip Given $N\in\mathbb{N}$, and dimension $d>1$, define $n = \lceil N^{1/d} \rceil$ and define also $m = n^d - N$. Note that $0 \leq m \leq n^d - (n-1)^d - 1$. Define \begin{align} q_n(\bm{x}) = \frac{1}{n^d} d\mu(\bm{x})\sum_{\mathfrak{b}(\vec{i})=0}^{n^d-1} \phi_{\vec{i}}(\bm{x})^2 \end{align} and then we can immediately write down the bound \begin{align}\label{eq:gen_N_reject} \frac{f_N(\bm{x})}{q_n(\bm{x})} = \frac{\frac{1}{N} d\mu(\bm{x})\sum_{\mathfrak{b}(\vec{i})=0}^{N-1} \phi_{\vec{i}}(\bm{x})^2}{\frac{1}{n^d} d\mu(\bm{x})\sum_{\mathfrak{b}(\vec{i})=0}^{n^d-1} \phi_{\vec{i}}(\bm{x})^2} \leq \frac{n^d}{n^d - m}. \end{align} Sampling from $q_n$ can be achieved using the factorised rejection sampler derived in the previous subsection, so we have all that is required for a rejection sampler for $f_N$ for general $N$. \medskip Figure \ref{fig:accept_prob_general_N} shows empirical acceptance probabilities for dimension $2, 3$ and $4$ over a range of $N$ values. Note that these values are just for the rejection sampler described by (\ref{eq:gen_N_reject}), treating the rejection sampler for $q$ as a black-box. Unsurprisingly the acceptance probability has some oscillatory behaviour in $N$, with the peaks corresponding to perfect powers of $d$, followed by sharp drops as $m$ jumps from its best value of $0$ to its worst of $n^d - 1$. That being said, the acceptance probabilities are still very good for $d=2$ above around $N=100$. For smaller values of $N$ the acceptance probabilities are still good enough, particularly as the other components of the DPP sampler become cheaper with decreasing $N$. $d=3, 4$ also appear feasible, though clearly less efficient that $d=2$. Higher values of $d$ are also plausible, though the scaling with $d$ is clearly poor (c.f. \cite{NEURIPS2019_1d54c76f,bardenet2020monte} where the acceptance probability scales like $2^{-d}$.) \begin{figure} \centering \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/accept_prob_bootstrap_d2.pdf} \caption{$d=2$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/accept_prob_bootstrap_d3.pdf} \caption{$d=3$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/accept_prob_bootstrap_d3.pdf} \caption{$d=4$} \end{subfigure} \caption{Empirical acceptance probabilities for the general $N$ sampler constructed in Section \ref{subsec:general_N}. 1000 trials are used for each $N$ values.} \label{fig:accept_prob_general_N} \end{figure} \medskip Figure \ref{fig:2d_dpp_sample} show a sample from the constructed Gauss-Hermite DPP in 2 dimension and with 500 points. For comparison we also show a sample from the base measure, a standard Gaussian in 2 dimensions, and from a Poisson process on $(-2N^{1/4}, 2N^{1/4})$ (the domain in which the DPP has the vast majority of its support). The effect of the DPP repulsion is evident when comparing it to the base measure, as the points have filled a much larger domain. The repulsion is also evident when comparing the DPP to the Poisson process. The two processes have produced points in very similar domains, but the DPP shows much more uniformity of placement, with the Poisson process showing lots of characteristic clustering of points. \begin{figure} \centering \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/2d_500pt_dpp_sample.pdf} \caption{$\text{DPP}(\mu, K_N)$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/2d_500pt_iid_sample.pdf} \caption{i.i.d. Gaussian} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/2d_500pt_iid_uniform_sample.pdf} \caption{Poisson process} \end{subfigure} \caption{Samples in 2 dimension with $N=500$. We show the DPP and for comparison i.i.d. samples from the base Gaussian measure and also a Poisson process on $(-2N^{1/4}, 2N^{1/4})$.} \label{fig:2d_dpp_sample} \end{figure} \medskip Figure \ref{fig:timing} shows some timing results for the DPP sampler in dimensions $d=1,2,3,4$. The results are broadly similar to those of \cite{NEURIPS2019_1d54c76f}, which is unsurprising since the top-level chain rule step of the sampler is the same and the lower-level rejection samplers are efficient in both cases for small $d$. Our rejection sampler for $f_N$ introduces $r_{N,d}^{-1}ds^{-1}_{\lceil N^{1/d}\rceil}$, where $s_n$ is the acceptance probability for our sampler of $\rho_{n}$, so $s_n \geq 0.7$, and $r_{N, d}$ is the acceptance probability for our overall rejection sampler for $f_N$, so it appears $r_{N,d} \geq 0.4$ for $N\geq 200$ and with most $N$ values having much better values. Comparing the figure to the corresponding one in \cite{NEURIPS2019_1d54c76f} does suggest that for $d=4$ our rejection sampler for $f_N$ is introducing extra inefficiency beyond that of the top-level chain-rule sampler. However, recalling the acceptance probabilities in Figure \ref{fig:accept_prob_general_N}, it appears that the poorer performance of for $d=4$ is caused by the complexity of computing many more Hermite polynomial evaluations, namely $(\lceil N^{1/d}\rceil)^d$ rather than just $N$. This is encouraging, as the best case scenario (as achieved by \cite{NEURIPS2019_1d54c76f} in the Jacobi case) requires evaluation of the first $N^{1/d}$ Hermite polynomials. Our sampling strategy required $\lceil N^{1/d} \rceil$ which is of the same order of complexity for large $N$. \begin{figure} \centering \includegraphics[width=0.4\textwidth]{figures/timing.pdf} \caption{The wall-time of sampling from $\text{DPP}(\mu, K_N)$ for various dimensions $1,2,3,4$ and a range of $N$ values. Shown are means and one standard deviation either side over 30 repeated samplings for each $N$ and $d$.} \label{fig:timing} \end{figure} \section{Application to Monte Carlo integration} In \cite{NEURIPS2019_1d54c76f} the authors present two DPP-based Monte Carlo (MC) integration schemes (BH and EZ) for integrals of the from \begin{align} \int_{[-1, 1]^d} d\mu(\bm{x}) f(\bm{x}) \end{align} where $\mu$ is multivariate Jacobi measure on $[-1, 1]^d$. Armed with our multivariate Gauss-Hermite DPP, we can also apply their methods to integrals of the form \begin{align} \int_{\mathbb{R}^d} d\mu(\bm{x}) f(\bm{x}) \end{align} where recall $d\mu(\bm{x}) = e^{-\frac{\bm{x}^2}{2}}d\bm{x}$. Moreover, given some mean $\vec{\mu}$ and covariance matrix $\vec{\Sigma}$ we have \begin{align} \int_{\mathbb{R}^d}d\bm{x} ~ \mathcal{N}(\bm{x}; \vec{\mu}, \vec{\Sigma}) f(\bm{x}) &= \frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}d\mu(\bm{x})f(\vec{\Sigma}^{1/2}\bm{x} + \vec{\mu}) \end{align} and so estimation of integrals against \emph{any Gaussian probability measure} is possible by the reparametrisation trick. Other than switching to our Gauss-Hermite DPP from the Jacobi DPP, we make no other changes to the BH and EZ approaches. \subsection{Perfect esimation of polynomials} Consider functions $\mathbb{R}^d\rightarrow\mathbb{R}$ of the form \begin{align} f(x) = \sum_{i_1, \ldots, i_d=1}^p a_{\vec{i}} \prod_{k=1}^d x_k^{i_k} \end{align} where $a_{\vec{i}}$ are coefficients. The integral of $f$ against Gaussian measure is simple to compute: \begin{align}\label{eq:poly_true} \int_{\mathbb{R}^d} d\bm{x} ~ \mathcal{N}(\bm{x}; \vec{\mu}, \vec{\Sigma}) f(\bm{x}) = \sum_{i_1, \ldots, i_d=1}^p a_{\vec{i}} \prod_{k=1}^d \int_{\mathbb{R}} dx ~ \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}} x_k^{i_k} = \sum_{i_1, \ldots, i_d=1}^p a_{\vec{i}} \prod_{k=1}^d m(i_k) \end{align} where $m(i) = (i-1)!!$ if $i$ even and $m(i) = 0$ otherwise. Figure \ref{fig:poly_results} shows the results of using the BH, EZ and na\"{i}ve MC integration approaches. We show the mean and standard deviation over 30 repeated samples. The polynomial are selected at random but are kept fixed throughout. The degree of the polynomials is 10 for $d=1$, 5 for $d=2, 3$. The reduction in degree for higher dimension is necessary to prevent the true values given by (\ref{eq:poly_true}) from becoming unreasonably large. \begin{figure} \centering \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/poly1d-mean.pdf} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/poly2d-mean.pdf} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/poly3d-mean.pdf} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/poly1d-std.pdf} \caption{$d=1$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/poly2d-std.pdf} \caption{$d=2$} \end{subfigure} \begin{subfigure}[b]{0.3\textwidth} \centering \includegraphics[width=\textwidth]{figures/poly3d-std.pdf} \caption{$d=3$} \end{subfigure} \caption{Result of MC integration on multivariate polynomials using Gauss-Hermite DPPs and comparing the na\"{i}ve MC approach to the BH and EZ approaches. The top row shows the sample means over 30 repetitions and the bottom row shows the sample standard deviation.} \label{fig:poly_results} \end{figure} The results generally accord with those of \cite{NEURIPS2019_1d54c76f}, with the the EZ estimator performing best but become less reliable with increasing $d$, in particular prone to ill-conditioning leading to enormous outliers. The BH estimator becomes more favourable with increasing dimension. All estimators benefit from increasing sample size $N$, but the EZ estimator benefits most dramatically. For example in the $d=2$ case the EZ estimator is very poor until around $N=30$ when, beyond which it recovers the true value with zero variance (perfect estimation). This can be explained as follows: we chose a degree $5$ polynomial. For $N < 25$, the kernel $K_N$ includes only multivariate Hermite polynomials of at most degree $4$, hence the chosen polynomial is not in the span of the RKHS basis. \subsection{Marginalised Gaussian process posteriors}\label{subsec:gp} Here we will consider a Gaussian process (GP) posterior on a simple synthetic dataset. The synthetic data are just draw from a one dimensional sine wave and are shown in Figure \ref{fig:gp_posterior_pnt}. We will use a simple scaled radial basis function (RBF) kernel without automatic relevance determination: $$ k(x, x') = v \exp\left(-\frac{(x - x)^2}{l^2}\right)$$ where $v>0$ and $l>0$ are hyperparameters. In standard GP practice, the hyperparameters $v,l$ are optimised as point estimates by maximising the log marginal likelihood of the training data under the model. To enforce the positivity constrains on $v,l$ we adopt the standard practice (used e.g. by GPyTorch \cite{gardner2018gpytorch}) of writing them as softplus transforms of raw hyperparameters: \begin{align} v = \log(\exp(\theta_1) + 1), ~~~l = \log(\exp(\theta_2) + 1). \end{align} In Bayesian treatments of GP hyperparameters, one promotes a point estimate of $\theta$ to a full posterior distribution (see e.g. \cite{lalchand2020approximate}). Further pursuit of such ideas is well outside the scope of this paper, however it suffices to note that a Gaussian form of the posterior is the most practical choice (e.g. as in Variational inference approaches \cite{lalchand2020approximate}). We will place a Gaussian over the raw hyperparameters $\theta$. For demonstration purposes, we will take a unit variance factorised Gaussian centred on the point-estimate raw-hyperparameters from applying marginal likelihood maximisation.\footnote{This is not dissimilar to a Laplace approximation of a posterior over the hyperparameters, though the variance is certainly not correct.} We will consider a simple 1-dimensional index space and a simple single-output regression problem on synthetic data. \medskip We now seek the hyperparameter-marginalised GP posterior confidence intervals. The point-wise CDF is \begin{align}\label{eq:psi_cdf} \Psi(y) = \int d\theta ~ \Phi\left(\frac{y - \mu(\theta + \theta_*)}{\sigma(\theta + \theta_*)}\right) \frac{e^{-\frac{\theta^2}{2}}}{2\pi} \end{align} where $\theta_*$ is the centre of the raw-hyperparameter Gaussian obtained from MLL maximisation, $\mu,\sigma$ are the point-wise posterior mean and standard deviation of the GP posterior as a function of $\theta$, and $\Phi$ is the standard Gaussian CDF. There is an implicit dependence throughout on the index point $x$. We then seek $y_2 = \Psi^{-1}(0.975), ~ y_1 = \Psi^{-1}(0.025)$, the boundaries of a confidence interval. Rather than embarking on numerical inversion of $\Psi$, we will opt for simpler heuristics that nevertheless demonstrate the DPP MC integration methods, which is our objective. We compute \begin{align} \bar{\mu} = \int d\theta ~ \frac{e^{-\frac{\theta^2}{2}}}{2\pi} \mu(\theta + \theta_*) \end{align} and \begin{align} \bar{\sigma} = \int d\theta ~ \frac{e^{-\frac{\theta^2}{2}}}{2\pi} \sigma(\theta + \theta_*). \end{align} These yield confidence intervals $(\bar{\mu} - 1.96\bar{\sigma}, \bar{\mu}+1.96\bar{\sigma})$, which are certainly not the posterior $95\%$ confidence intervals of the marginalised GP, but are perhaps not dissimilar in some cases. Figure \ref{fig:gp_posterior} shows the point estimate posterior on a test set alongside the same EZ marginalised pseudo posterior. We note merely that the increased posterior uncertainty appears reasonable and also that the BH and na\"{i}ve plots are very similar, but with slightly higher variance in the na\"{i}ve case. \begin{figure} \centering \begin{subfigure}[b]{0.4\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp2d_point_est_posterior.pdf} \caption{Point estimate.} \label{fig:gp_posterior_pnt} \end{subfigure} \begin{subfigure}[b]{0.4\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp2d_ez_posterior.pdf} \caption{EZ marginalised.} \end{subfigure} \caption{Marginalised GP posterior using EZ MC integration to compute $\bar{\mu}$ and $\bar{\sigma}$. The shaded region in (b) shows $(\bar{\mu} - 1.96\bar{\sigma}, \bar{\mu}+1.96\bar{\sigma})$, whereas the shaded region in (a) is the standard posterior $95\%$ confindence interval for the point estimate GP.} \label{fig:gp_posterior} \end{figure} \medskip To really test the utility of our DPP MC integration approaches in this case, we must assess the accuracy of estimation of the CDF $\Psi$ defined in (\ref{eq:psi_cdf}). To this end, for all $x$ values in the test set, we compute $\Psi(y)$ for each \begin{align} y \in \{\bar{\mu} - 4\bar{\sigma}, \bar{\mu} - 1.96\bar{\sigma}, \bar{\mu} - \bar{\sigma}, \bar{\mu}, \bar{\mu} + \bar{\sigma}, \bar{\mu} + 1.96\bar{\sigma}, \bar{\mu} + 4\bar{\sigma}\}. \end{align} Accurate estimation of such values is critical for accurate estimation of confidence intervals. We repeat this estimation 30 times (resampling the DPP/i.i.d. sample points) leading to 30 values for each of the 7 $y$ values at each of the 50 test points. To visualise, we then compute the sample standard deviation over the 30 re-samplings, leading to $7\times 50$ standard deviation values of which we take the sample mean over the 50 points. Figure \ref{fig:gp_posterior_cdfs} shows the resulting standard deviation values for each of the 7 $y$ values and each of the estimators. The results show, perhaps surprisingly, EZ outperforming BH and na\"{i}ve in all cases. This indicates that the GP posterior CDF has rapidly decaying coefficients in an Hermite polynomial expansion \cite{NEURIPS2019_1d54c76f}. \begin{figure} \centering \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_0.pdf} \caption{$y=\bar{\mu} - 4\bar{\sigma}$} \end{subfigure} \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_1.pdf} \caption{$y=\bar{\mu} - 1.96\bar{\sigma}$} \end{subfigure} \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_2.pdf} \caption{$y=\bar{\mu} - \bar{\sigma}$} \end{subfigure} \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_3.pdf} \caption{$y=\bar{\mu}$} \end{subfigure} \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_4.pdf} \caption{$y=\bar{\mu} + \bar{\sigma}$} \end{subfigure} \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_5.pdf} \caption{$y=\bar{\mu} + 1.96\bar{\sigma}$} \end{subfigure} \begin{subfigure}[b]{0.24\textwidth} \centering \includegraphics[width=\textwidth]{figures/gp_quant_6.pdf} \caption{$y=\bar{\mu} + 4\bar{\sigma}$} \end{subfigure} \caption{Estimating the GP CDF $\Psi$ defined in (\ref{eq:psi_cdf}). Each plot shows results for na\"{i}ve, BC and EZ estimators using a range of sample sizes. We plot the standard deviation of the estimates over resampling of the DPP/i.i.d. points. The error bars show standard deviation over the 50 GP evaluation points. Each subfigure considers a different $y$ value in $\Psi(y)$.} \label{fig:gp_posterior_cdfs} \end{figure} \section{Conclusions and future work} We have considered multivariate Hermite polynomials for the construction of determinantal points processes with Gaussian base measure. To enable their practical use, we have developed a novel multi-stage sampler for such Gauss-Hermite DPPs that is quite different from prior analogous samplers for other DPPs. Our experimental timing and efficiency results demonstrate that the sampler is practically feasible and in-line with performance obtained by prior DPP samplers. We have presented some experimental results using Gauss-Hermite DPPs in BH and EZ approaches to MC integration, showing strong performance in a variety of tasks. In particular, we have presented some novel experimental results using Gauss-Hermite DPPs and BH/EZ estimation to marginalise the hyperparameters of a Gaussian process regression posterior. The excellent results of the EZ estimator in this case suggest that the GP posterior, as functions of the raw hyperparameters, are well adapted to the Hermite basis. We remark that the factorised form (\ref{eq:dpp_factorisation}) of the DPP posterior is clearly related to the posterior covariance of a Gaussian process with kernel $K_N$. It would be interesting to consider if this similarity could be in part responsible for the excellent EZ performance, perhaps suggesting that EZ MC integration is particularly suited to GP posteriors. Future work should also consider optimising the implementations to avoid repeated Hermite polynomials evaluations, in particular evaluations used in sampling from the top-level proposal density $f_N$ could be recycled for use in evaluating the acceptance ratio itself. Finally, we constructed an empirically optimised rejection sampler for the univariate truncated Hermite mixtures $\rho_n$ being inspired by the convergence of such distributions to the Wigner semi-circle law. This leaves open the question of obtaining analytically rejection bounds, and in particular bounds that are good enough for practical use in the rejection sampler. We have not been able to derive such bounds, but the semi-circle convergence does suggest that they might be available.

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\section{Introduction} \begin{figure}[tbp] \centering \subfloat[Sparse kernels of layer $l+1$ which are connected to the unnecessary output channel of layer $l$]{\includegraphics[scale=0.25]{filterPruning1.pdf} \label{filterpruning1}} \hfil \subfloat[Pruning the filter of layer $l$ and sparse kernels of layer $l+1$, which are connected to the unnecessary output channel of layer $l$ keeping the same output channel of layer $l+1$]{\includegraphics[scale=0.25]{filterPruning2.pdf} \label{filterpruning2}} \caption[]{An illustration of filter pruning via Hierarchical sparse group regularization based on the feature-wise grouping. In convolutional layer, each filter makes one output channel (activation), these colors are the same. For example, the filter of orange of layer $l$ makes orange output channel of layer $l$. \subref{filterpruning1} The Hierarchical sparse group regularization based on the feature-wise grouping make the weights of the unnecessary kernels to be almost zero. Since the output of convolution from the input channel connected to the unnecessary kernels will be zero in the layer $l+1$, the output channels are not influenced by the pruning the unnecessary kernel. \subref{filterpruning2} If the increase of the classification loss of the network after pruning the filters connected to the unnecessary output channel of layer $l$ is very small, we can prune the filters of layer $l$ and kernels of layer $l+1$ connected to the unnecessary output channel of layer $l$. Then the output channels of layer $l+1$ are almost the same as the output channels before pruning. } \label{FPHGSR} \end{figure} Deep convolutional neural networks (CNNs) have been successful with excellent performance in computer vision tasks such as image classification \cite{krizhevsky2012imagenet, szegedy2015going}. But along with it, the network becomes deeper and wider, requiring excessive parameters\cite{he2016deep,huang2017densely}, which increases the computational cost. To solve this problem, many approaches have been proposed such as knowledge distillation \cite{hinton2015distilling, romero2014fitnets} and network pruning \cite{han2015learning, liu2017learning, he2019filter}, that can be reduced redundant parameters of large networks while preserving accuracy. Recent work on network pruning, there are 2 types of methods by using sparse regularization to obtain a sparse network whose unnecessary connections are pruned. One of the methods is unstructured pruning \cite{han2015learning,louizos2017learning}, which prune individual weights by using L0 regularization, L1 regularization\cite{tibshirani1996regression} or L2 regularization. These unstructured sparse regularizations enforce unnecessary individual weights of a large network set to be 0. However, such unstructured pruning cannot prune whole kernels or filters (that is a subset of kernels connected to a channel) in CNNs, which does not speed up computations or reduce memory weight without special libraries and hardware. In contrast, structured pruning enables to reduce the number of kernels or filters from CNNs \cite{wen2016learning, alvarez2016learning, zhou2016less} by using group sparse regularization such as group lasso\cite{yuan2006model,schmidt2010graphical}, whose method is widely used to pruning unnecessary kernels or filters keeping high performance. The weights of the convolutional layer are structured as convolution kernels. When the filters connected to a channel are considered as a group, group sparse regularization treats a subset of kernels as individual weight in the same group, mutual interaction between kernels in the group is not taken into account. In our previous work, we proposed the concept of the hierarchical group sparse regularization \cite{mitsuno2020hierarchical} to introduce such interactions in the structured sparse regularization criterion. The hierarchical group sparse regularization can treat the filter for the input channel or the output channel as a group and convolutional kernels as a group in the same group to prune the unnecessary subsets of weights. As a result, we can prune the weights more adequately depending on the structure of the network and the number of channels keeping high performance. In this paper, we propose a filter pruning method with the hierarchical group sparse regularization based on the feature-wise grouping as illustrated in Fig. \ref{FPHGSR}, whose grouping consider as a group the kernels connected to a input channel. By this grouping, we can prune unnecessary output channels in layer $l-1$ (the input channels in layer $l$). After training with the hierarchical group sparse regularization, we calculate the influence of each channel on the classification loss of the randomly selected training samples, and prune the channels based on the increase of the classification low. After we obtain a compact network by the filter pruning, the parameters of the compact network are retrained from scratch. To confirm the effectiveness of the proposed method, we have performed experiments with different network architectures (VGG and ResNet) on different data sets (CIFAR-10, CIFAR-100, and TinyImageNet-200). The results show that we can obtain the compact networks with about 50\% less parameters without decrease of the classification accuracy. The contributions of this paper are summarized as follows: \begin{itemize} \item We propose a filter pruning method with the hierarchical group sparse regularization based on the feature-wise grouping, the regularization can prune filters more adequately depending on the structure of the network and the number of channels than non-hierarchical sparse regularization. \item The feature-wise grouping can prune the filters connected to unnecessary input channels by removing the channels with low influence on the classification loss. \item The effectiveness of the proposed pruning method is confirmed through experiments with different network architectures (VGG and ResNet) on different data sets (CIFAR-10, CIFAR-100, and TinyImageNet-200). \end{itemize} \section{Related Work} \subsection{Network Pruning} Network pruning can efficiently prune redundant weights or filters to compress deep CNN while maintaining accuracy. Various method have been proposed. There is two methods of network pruning in which unstructured pruning and structured pruning. \subsubsection{Unstructured Pruning} The way of unstructured pruning reduces the individual weight of neural networks \cite{lecun1990optimal,hassibi1993second,han2015learning,han2015deep,srinivas2015data,louizos2017learning}. Optimal brain damage \cite{lecun1990optimal} and optimal brain surgeon \cite{hassibi1993second} prune unimportant weight from a network to compute the influence of each weight on the training loss based on Hessian matrix. S.~Han et al. \cite{han2015learning} and C.~Louizos et al.\cite{louizos2017learning} utilized unstructured sparse regularizations such as L1 regularization to make a sparse networks, which reduce unnecessarily individual weights by enforcing to be 0. These unstructured pruning method makes network weights sparse, but it can only achieve speedup and compression with dedicated libraries and hardware. \subsubsection{Structured Pruning} Structured pruning \cite{li2016pruning,hu2016network,wen2016learning, alvarez2016learning, zhou2016less,he2017channel,molchanov2016pruning,peng2019collaborative,molchanov2019importance,liu2017learning,ye2018rethinking,huang2018data,luo2017thinet,yu2018nisp,suau2018principal,huang2018learning,chin2018layer,lin2018accelerating,he2018soft,wang2018skipnet,liu2018rethinking,peng2019collaborative,lin2019towards,he2019filter} has been exploited by many researchers. Pruning methods using group lasso as structured sparse regularization were proposed by many researchers \cite{wen2016learning, alvarez2016learning, zhou2016less}. These methods enforce unnecessary subset of weights to be zero to obtain sparse networks at group levels such as kernels or filters. Also, Y.~He et al. \cite{he2017channel} proposed channels selection based on lasso regression. Pruning of unnecessary filters with the scaling parameter of batch normalization layers were also proposed by enforcing sparsity of the parameter \cite{liu2017learning,ye2018rethinking,huang2018data}. In the papers \cite{li2016pruning,he2018soft}, filters with relatively low weight magnitudes were pruned based on the norm of the weights. Y.~He et al. \cite{he2019filter} proposed to prune the most replaceable filters containing redundant information and the relatively less important filters using the norm-based criterion. Some works \cite{molchanov2016pruning,lin2018accelerating,peng2019collaborative,molchanov2019importance} used Taylor expansions to evaluate the influence of pruned filters to the classification loss. \subsection{Sparse Regularization} In this section, we review structured sparse regularization criteria for filter pruning of deep neural networks. We assume that the objective function of the optimization for determining the trainable weights is given by \begin{equation} \label{training objective} J(W) = \mathbb{L}(f(x,W)|y) + \lambda \sum^{L}_{l=1}R(W^l) \end{equation} where $(x,y)$ denotes the pair of the input and target, $W$ is a set of all trainable weights of all the $L$ layers in the CNN, $\mathbb{L}(\cdot)$ is the standard loss for the CNN, and $R(W^l)$ is the regularization term at layer $l$. The parameter $\lambda$ is used to balances the loss and the regularization term. Also, we assume the weight in the layer $l$ as $W^l \in \mathbb{R}^{C_l\times C_{l-1}\times K_l \times K_l}$, where $C_l$ and $C_{l-1}$ are the number of output channels and input channels, $K_l$ is the kernel size of the layer $l$ respectively. In the fully connected layers, $K_l = 1$ The structured sparse regularization such as group lasso regularization\cite{yuan2006model,schmidt2010graphical}, exclusive sparsity \cite{zhou2010exclusive,kong2014exclusive} and group $L_{1/2}$ regularization criterion \cite{li2018smooth,alemu2019group} can reduce unnecessary kernels or filters on CNN. The structured sparse regularization is defined as \begin{equation} \label{structured sparse regularization} R(W^l) = \sum_{g \in G} r(W^l_g), \end{equation} where $g \in G$ is a group in the set of groups $G$, $W^l_g$ is subset of weights for the group $g$ such as a filter that connected to a channel, and $r(\cdot)$ is the function to calculate the sparseness of the structured sparse regularization. In the case of group lasso, the calculate of the regularization is written as $r(W^l_g)=\|W^l_g\|_2$. To prune filters that connected unnecessary channels, there are two types of grouping, namely the neuron-wise grouping and the feature-wise grouping. The way of grouping for convolutional kernels are shown in Fig.~\ref{the way of grouping}. In these groupings, structured sparse regularization treats subset of kernels as individual weight in the same group. So, mutual interaction between kernels in the group is not taken into account. In our previous work \cite{mitsuno2020hierarchical}, we proposed the concept of the hierarchical group sparse regularization to introduce such interactions in the structured sparse regularization criterion. For example, the hierarchical group sparse regularization criterion is defined by taking the square of the sub-groups as \begin{equation} \label{hierarchical group Lasso regularization square} r_{SQ}(W^l_g) = \left(\sum_{k \in K}r(W^l_{g,k})\right)^2, \end{equation} where $k \in K$ is a kernel in a filter (a set of kernels) $K$ and $w^l_{g,k}$ is a kernel in the group $g$ and $r(\cdot)$ is non-hierarchical group sparse regularization such as group lasso, exclusive sparsity and group $L_{1/2}$. \begin{figure}[tbp] \centering \subfloat[the neuron-wise]{\includegraphics[scale=0.2]{neuronwise.pdf} \label{neuron-wise grouping}} \hfil \subfloat[the feature-wise]{\includegraphics[scale=0.2]{featurewise.pdf} \label{feature-wise grouping}} \caption[]{The way of grouping for convolutional kernels. \subref{neuron-wise grouping} The filters connected to a output channel are consider as a group. We call this grouping the neuron-wise grouping. By this grouping, we can prune unnecessary output channels. \subref{feature-wise grouping} The filters connected to a input channel are considered as a group. We call this grouping the feature-wise grouping. By this grouping, we can prune unnecessary of output channels in layer $l-1$ (the input channels in layer $l$).} \label{the way of grouping} \end{figure} \section{Proposed Method} We propose the feature-wise filter pruning algorithm for deep convolutional neural networks. It consists of the following steps; 1) Train the large network as the initial network. 2) Train the network with the structured sparse regularization based on the feature-wise grouping to find unnecessary filters connected to input channels by enforcing the weights of unnecessary filters to be zero. 3) Prune the filters with smaller influence on the classification loss. 4) Train the obtained compact network from scratch. The details of these steps are explained in the next sub-sections. \subsection{Training with The Hierarchical Sparse Regularization Based on The Feature-Wise Grouping} First, we train the network as the initial network without sparse regularization. After obtaining the initial network, we train the network with the structured sparse regularization based on the feature-wise grouping such as group lasso, exclusive sparsity, group L$_{1/2}$ regularization, and the hierarchical group sparse regularization to get the sparse network. In this paper, we propose to use the hierarchical group sparse regularization, which is proposed by Mitsuno et al. \cite{mitsuno2020hierarchical}, and show that the hierarchical group sparse regularization performs better than the non-hierarchical regularization. The feature-wise group sparse regularization is defined as \begin{equation} \label{feature-wise grouping criteria} R(W^l) = \sum^{c_{l-1}}_{j=1}r(W^l_{,j,,}) . \end{equation} The structured sparse regularization criterion based on the feature-wise grouping enforces the filters connected to unnecessary input channels of the convolutional layer to be zero. As a result, we can remove unnecessary filters connected to output channels in layer $l-1$ by forcing the unnecessary filters connected to the input channels in layer $l$ to be zero. Since the unnecessary input channels connected to the filters don't have influences to the outputs of the layer and final loss of the network, we can prune the filters connected to the output channels of the layer $l-1$ (the unnecessary input channels of the layer $l$). \subsection{Filter Pruning} After training with the structured sparse regularization based on the feature-wise grouping, we obtain sparse network in which all weights of the filters connected to the unnecessary input channels are close to zero. Thus we can prune the filters connected to the unnecessary output channels of layer $l-1$ (the unnecessary input channels of the layer $l$) if the filters have less influence to the classification loss after pruning. We can implement the filter selection method for pruning as shown in Algorithm. \ref{Pruning selection algorithm}. \begin{algorithm} \caption{Filter Pruning (backward filter selection)} \label{Pruning selection algorithm} \begin{algorithmic}[1] \renewcommand{\algorithmicrequire}{\textbf{Input:}} \renewcommand{\algorithmicensure}{\textbf{Output:}} \REQUIRE filter of the sparse model $W\in \mathbb{R}^{L}$, the number of conv layer $L^c$, the number of all output channel of the conv layer $C^c$, filter mask $m^{l}_{c^{l}} \in M^l$, training sample $s \in x$, loss $\mathbb{L}(\cdot)$, pruning rate $P$ \ENSURE weight of pruned model $W^*\in \mathbb{R}^{L}$ \STATE $M = 1$ \FOR {$n$ in 1 \ldots $C^{c}P$} \FOR {$l$ in 1 \ldots $L^{c}$} \FOR {$c$ in 1 \ldots $C^{c}_{l}$} \STATE $M^* = M$ \STATE $m^{*l}_{c} = 0$ \STATE $e_{l,c} = \mathbb{L}(f(s,W\odot M^*)|y)$ \ENDFOR \ENDFOR \STATE $\mathop{\rm arg~min}\limits_{l^*\in L^{c},c^* \in C^{c}_{l}}$ $e_{l^{*},c^{*}}$ s.t. $m^{l^{*}}_{c^{*}} == 1$ \STATE $m^{l^{*}}_{c^{*}} = 0$ \ENDFOR \STATE pruning $W^* = W[M == 1]$ \RETURN $W^*$ \end{algorithmic} \end{algorithm} Molchanov et al. \cite{molchanov2016pruning} proposed to use a criteria based on Taylor expansion for ranking and pruning one filter at a time. Chin et al. \cite{chin2018layer} also introduced a global ranking approach. Here we use the global ranking for filter pruning to automatically obtain the pruned network architecture. We introduce a binary mask $m^{l}_{c^{l}} \in M^l $ for each output channel of the layer $l$, $m^{l}_{i} \in \{0,1\}$. By using these masks, we can prune $i^{th}$ filter of the layer $l$ by making $m^{l}_{i} = 0$. We search the filter which has the minimum loss increase after the filter is pruned. The increase of the classification loss after pruning is represented as \begin{eqnarray} \mathop{\rm arg~min}\limits_{l^*\in L^{c},i^* \in C^{c}_{l}} |\mathbb{L}(f(s,W)|y) - \mathbb{L}(f(s,W\odot M)|y)| \end{eqnarray} where $W\odot M$ denotes the element-wise product. The classification loss $\mathbb{L}$ are evaluated by using randomly selected training samples. By updating the mask and repeating the filter selection algorithm until the number of the masked channels are $C^{c}P$ channels. After the filters for pruning are selected, we prune the filters from the network, where $m^{l}_{i} = 0$. We also prune the batch normalization layers connected to the pruned output channels at the same time. Then we can obtain the compact network architecture with small parameters and less computational operations. The obtained compact network after pruning can achieve almost the same accuracy with the original large network by fine-tuninig. But Liu et al. \cite{liu2018rethinking} examined and showed that the fine-tuning of the pruned model can only give comparable or worse performance than the training of the compact model with randomly initialized weights. So we trained the obtained compact network from scratch. \section{Experiments} To confirm the effectiveness of the proposed pruning method, we have performed experiments using different data sets (CIFAR-10, CIFAR-100, and TinyImageNet-200) and different network architectures (VGG nets \cite{simonyan2014very} and ResNet \cite{he2016deep}). CIFAR-10 contains 60,000 color images of ten different animals and vehicles. They are divided into 50,000 training images and 10,000 testing images. The size of each image is $32 \times 32$ pixels. CIFAR-100 also contains 60,000 color images of 100 different categories and 50,000 images are used for training and the remainings are used for test. The size of each image is also $32 \times 32$ pixels. TinyImageNet-200 contains 110,000 color images of 200 different categories and 100,000 images are used for training and 10,000 images for test. The size of the image is $64 \times 64$ pixels. In the following experiments, the number of channels of the network at each layer is adjusted to prevent overfitting, depending on each dataset. \subsection{Pruning of VGG Nets} To confirm the effectiveness of the proposed pruning method, we have performed experiments using VGG14 (13-conv + 1-fc layers) with batch normalization layers for different data sets (CIFAR-10, CIFAR-100, and TinyImageNet-200). All the initial networks are trained from scratch by using SGD optimizer with a momentum of $0.9$. We used the weight decay with the strength of $5*10^{-4}$ to prevent overfitting. The mini-batch size for CIFAR-10/100 was set to $128$ and the network was trained for $200$ epochs. For TinyImageNet-200, the mini-batch size was set to $256$ and the network was trained for $200$ epochs. The initial learning rate was set to $0.1$ and it was divided by $0.2$ after $[60,120,160]$ training epochs. Then the hierarchical sparse regularization was applied to the weights except for the bias term in all convolutional layers. All the networks were trained by using SGD optimizer with a momentum of $0.9$. For CIFAR-10/100, the mini-batch size was set to $128$ and the network was trained for $100$ epochs. For TinyImageNet-200, the mini-batch size was set to $256$ and the network was trained for $100$ epochs. The initial learning rate of $0.01$ which is divided by $0.1$ after $1/3$ and $2/3$ training epochs. The hyper-parameter $\lambda$, which balances the cross-entropy loss and the hierarchical sparse regularization criterion, was experimentally determined in the range from $10^{-1} $ to $10^{-7}$. We used hierarchical squared GL$_{1/2}$ regularization (HSQ-GL12)\cite{mitsuno2020hierarchical}, which has shown outstanding performance compare to various sparse regularization criteria in our previous work. To evaluate the sparsity of the trained network, the ratio of the zero weights was calculated by assuming that the weights whose absolute value is less than $10^{-3}$ are zero. After training the networks with sparse regularization, we selected one best trained network from the trained networks with the various hyper-parameter $\lambda$ in sparse regularization. To obtain the compact network, the channels at the convolutional layers except for the output channel of the final convolutional layer were pruned based on the influence of the pruning of each channel to the classification loss. To evaluate the influence of the pruning, we calculate the increase of the classification loss for the $128$ randomly selected samples from the training samples. The pruning rate $P$, which is a ratio of the pruned channels in the whole channels of the networks, was changed from $0.1$ to $0.9$. When the number of channels of a layer becomes $0$, the experiments are stopped. After pruning, we can obtain compact networks. The parameters of the obtained compact network were trained from scratch using the same parameter settings as the training of the initial networks. \begin{figure}[tbp] \centering \subfloat[VGG14 on CIFAR-10]{ \includegraphics[scale=0.28]{cifar10_vgg14_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{cifar10_vgg14_number_of_channels.png} \label{VGG14 on CIFAR-10}} \hfil \subfloat[VGG14 on CIFAR-100] { \includegraphics[scale=0.28]{cifar100_vgg14_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{cifar100_vgg14_number_of_channels.png} \label{VGG14 on CIFAR-100} } \hfil \subfloat[VGG14 on TinyImageNet-200] { \includegraphics[scale=0.28]{tinyimagenet200_vgg14_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{tinyimagenet200_vgg14_number_of_channels.png} \label{VGG14 on TinyImageNet-200} } \caption[]{The results with VGG14 on CIFAR-10/100 and TinyImageNet-200. Comparison of test accuracy of the pruned networks (left). For CIFAR-10/100, the average of three trials are shown. The numbers of the pruned channels in each layer is shown in the right figure. Each line shows the numbers of the pruned channels at each layer. Different colors denote the results with different pruning rates $P$. } \label{VGG nets} \end{figure} The results for VGG14 are shown in Fig. \ref{VGG nets}. For CIFAR-10, the propose method is scceeded to prune 85\% of parameters with only 0.53\% drop of the test accuracy. It is impressive that the network in which 41\% parameters are pruned achieves better test accuracy than the baseline network. Similarly, we can prune 41\% of parameters with only 0.85\% drop of the test accuracy for CIFAR-100. For TinyImageNet-200, 37\% of parameters can be pruned with only 0.70\% drop of the test accuracy. For all datasets, we observe similar tendency of pruning such that the parameters in the deep layer are pruned much more than the shallow layer. \subsection{Pruning of ResNet} We also have performed experiments with ResNet for CIFAR-10, CIFAR-100, and TinyImageNet-200. For CIFAR-10/100, we trained ResNet20 (19 convolution(conv) + one fully connected(fc) layers) and ResNet32 (31-conv + 1-fc layers) with batch normalization layers. For TinyImageNet-200, ResNet18 (17 convolution(conv) + one fully connected(fc) layers) and ResNet34 (33-conv + 1-fc layers) were trained with batch normalization layers. Parameter settings are the same with the experiments for VGG net. \begin{figure}[tbp] \centering \subfloat[ResNet20 on CIFAR-10]{ \includegraphics[scale=0.28]{cifar10_resnet20_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{cifar10_resnet20_number_of_channels.png} \label{ResNet20 on CIFAR-10}} \hfil \subfloat[ResNet20 on CIFAR-100]{ \includegraphics[scale=0.28]{cifar100_resnet20_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{cifar100_resnet20_number_of_channels.png} \label{ResNet20 on CIFAR-100}} \hfil \subfloat[ResNet18 on TinyImageNet-200]{ \includegraphics[scale=0.28]{tinyimagenet200_resnet18_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{tinyimagenet200_resnet18_number_of_channels.png} \label{ResNet18 on TinyImageNet-200}} \caption[]{The results with ResNet20 and ResNet18. Comparison of test accuracy of the pruned networks (left). The average of three trials is shown. The numbers of the pruned channels in each layer of the networks are shown in the right figure. } \label{ResNet20 and ResNet18} \end{figure} \begin{figure}[htbp] \centering \subfloat[ResNet32 on CIFAR-10]{ \includegraphics[scale=0.28]{cifar10_resnet32_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{cifar10_resnet32_number_of_channels.png} \label{ResNet32 on CIFAR-10}} \hfil \subfloat[ResNet32 on CIFAR-100]{ \includegraphics[scale=0.28]{cifar100_resnet32_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{cifar100_resnet32_number_of_channels.png} \label{ResNet32 on CIFAR-100}} \hfil \subfloat[ResNet34 on TinyImageNet-200]{ \includegraphics[scale=0.28]{tinyimagenet200_resnet34_graph_of_prune_ratio.png} \includegraphics[scale=0.28]{tinyimagenet200_resnet34_number_of_channels.png} \label{ResNet34 on TinyImageNet-200}} \caption[]{The results with ResNet32 and ResNet34. Comparison of test accuracy of pruned networks (left). For CIFAR-10/100, the average of three trials is shown. Right figure shows the numbers of pruned channels in each layer of the networks. } \label{ResNet32 and ResNet34} \end{figure} The results with ResNet are shown in Fig. \ref{ResNet20 and ResNet18} and Fig. \ref{ResNet32 and ResNet34}. By using the proposed pruning method, we can obtain the compact network which has only 54\% of the parameters with the baseline network but can achieve almost same test accuracy with the baseline network (only 0.31\% drop) for ResNet20 on CIFAR-10. The pruned network with 16\% parameters gives better test accuracy than the baseline network. Similaly, for ResNet20 on CIFAR-100, the network with 15\% parameters gives almost same test accuracy with the baseline network (only 0.85\% drop). The network pruned 34\% of the parameters for ResNet18 on TinyImageNet-200 achieved higher test accuracy than the baseline network. The network pruned 54\% of the parameters for ResNet32 on CIFAR-10 and the network pruned 13\% of the parameters for ResNet32 on CIFAR-100 are also gives the almost same test accuracy (only 0.73\% drop and only 1.2\% drop). For ResNet32 on TinyImageNet-200, we can prune the network for 16\% of the parameters with only 3.8\% drop of the test accuracy. From the right figures, it is noticed that the parameters in the deep layer of each residual block are pruned more. \subsection{Comparison with The State-of-the-art Method} The proposed method was compared with one of the state-of-the-art filter pruning methods FPGM-mix. The filter pruning via geometric median (FPGM) \cite{he2019filter} is one of the state-of-the-art method and FPGM-mix is a mixture of FPGM and their previous norm-based method \cite{he2018soft}. We have performed experiments with the same pruning rate of $P$. The ratio of FPGM and the norm-based method is determined according to the paper \cite{he2018soft}. Namely, $3/4$ of the filters are selected with FPGM and the remaining $1/4$ filters are selected with the norm-based criterion. The results of the comparisons are shown in Fig.\ref{VGG nets}, Fig.\ref{ResNet20 and ResNet18} and Fig.\ref{ResNet32 and ResNet34}. For VGG nets, ResNet20 and ResNet18, the performance of the pruned network with the proposed method is better than the FPGM-mix for all pruning ratios. For ResNet32 and ResNet34, the test accuracy of the pruned network with the proposed method gives better than the network obtained by FPGM-mix when the pruning rate is larger than 60\%. Our proposed method consider the structure of the networks to prune unnnecessary parameters. So that, the proposed method gives better performance than the FPGM-mix when the pruning rate is high. These results show the effectiveness of the proposed method compares to the state-of-the-art method, especially when the network is pruned more than 50\% of the parameters. \section{Conclusion} We propose a filter pruning method with the hierarchical group sparse regularization based on the feature-wise grouping for pruning filters, which connected to unnecessary input channels, using the influence of the classification loss. At first the network was trained with the hierarchical sparse regularization. We take the strategy of the step-wise pruning of the filters by searching the filter with the minimum loss increase. Then the obtained compact network was retrained from scratch. Experiments using CIFAR-10/100 and TinyImageNet-200 datasets show the outstanding performance than the state-of-the-art pruning method. Especially the performance of the pruned network is better than the state-of-the-art pruning method when more than 50\% of the parameters are pruned. \bibliographystyle{unsrt}

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\section{\label{sec:level1}First-level heading:\protect\\ The line break was forced \lowercase{via} \textbackslash\textbackslash} This sample document demonstrates proper use of REV\TeX~4.2 (and \LaTeXe) in mansucripts prepared for submission to APS journals. Further information can be found in the REV\TeX~4.2 documentation included in the distribution or available at \url{http://journals.aps.org/revtex/}. When commands are referred to in this example file, they are always shown with their required arguments, using normal \TeX{} format. In this format, \verb+#1+, \verb+#2+, etc. stand for required author-supplied arguments to commands. For example, in \verb+\section{#1}+ the \verb+#1+ stands for the title text of the author's section heading, and in \verb+\title{#1}+ the \verb+#1+ stands for the title text of the paper. Line breaks in section headings at all levels can be introduced using \textbackslash\textbackslash. A blank input line tells \TeX\ that the paragraph has ended. Note that top-level section headings are automatically uppercased. If a specific letter or word should appear in lowercase instead, you must escape it using \verb+\lowercase{#1}+ as in the word ``via'' above. \subsection{\label{sec:level2}Second-level heading: Formatting} This file may be formatted in either the \texttt{preprint} or \texttt{reprint} style. \texttt{reprint} format mimics final journal output. Either format may be used for submission purposes. \texttt{letter} sized paper should be used when submitting to APS journals. \subsubsection{Wide text (A level-3 head)} The \texttt{widetext} environment will make the text the width of the full page, as on page~\pageref{eq:wideeq}. (Note the use the \verb+\pageref{#1}+ command to refer to the page number.) \paragraph{Note (Fourth-level head is run in)} The width-changing commands only take effect in two-column formatting. There is no effect if text is in a single column. \subsection{\label{sec:citeref}Citations and References} A citation in text uses the command \verb+\cite{#1}+ or \verb+\onlinecite{#1}+ and refers to an entry in the bibliography. An entry in the bibliography is a reference to another document. \subsubsection{Citations} Because REV\TeX\ uses the \verb+natbib+ package of Patrick Daly, the entire repertoire of commands in that package are available for your document; see the \verb+natbib+ documentation for further details. Please note that REV\TeX\ requires version 8.31a or later of \verb+natbib+. \paragraph{Syntax} The argument of \verb+\cite+ may be a single \emph{key}, or may consist of a comma-separated list of keys. The citation \emph{key} may contain letters, numbers, the dash (-) character, or the period (.) character. New with natbib 8.3 is an extension to the syntax that allows for a star (*) form and two optional arguments on the citation key itself. The syntax of the \verb+\cite+ command is thus (informally stated) \begin{quotation}\flushleft\leftskip1em \verb+\cite+ \verb+{+ \emph{key} \verb+}+, or\\ \verb+\cite+ \verb+{+ \emph{optarg+key} \verb+}+, or\\ \verb+\cite+ \verb+{+ \emph{optarg+key} \verb+,+ \emph{optarg+key}\ldots \verb+}+, \end{quotation}\noindent where \emph{optarg+key} signifies \begin{quotation}\flushleft\leftskip1em \emph{key}, or\\ \texttt{*}\emph{key}, or\\ \texttt{[}\emph{pre}\texttt{]}\emph{key}, or\\ \texttt{[}\emph{pre}\texttt{]}\texttt{[}\emph{post}\texttt{]}\emph{key}, or even\\ \texttt{*}\texttt{[}\emph{pre}\texttt{]}\texttt{[}\emph{post}\texttt{]}\emph{key}. \end{quotation}\noindent where \emph{pre} and \emph{post} is whatever text you wish to place at the beginning and end, respectively, of the bibliographic reference (see Ref.~[\onlinecite{witten2001}] and the two under Ref.~[\onlinecite{feyn54}]). (Keep in mind that no automatic space or punctuation is applied.) It is highly recommended that you put the entire \emph{pre} or \emph{post} portion within its own set of braces, for example: \verb+\cite+ \verb+{+ \texttt{[} \verb+{+\emph{text}\verb+}+\texttt{]}\emph{key}\verb+}+. The extra set of braces will keep \LaTeX\ out of trouble if your \emph{text} contains the comma (,) character. The star (*) modifier to the \emph{key} signifies that the reference is to be merged with the previous reference into a single bibliographic entry, a common idiom in APS and AIP articles (see below, Ref.~[\onlinecite{epr}]). When references are merged in this way, they are separated by a semicolon instead of the period (full stop) that would otherwise appear. \paragraph{Eliding repeated information} When a reference is merged, some of its fields may be elided: for example, when the author matches that of the previous reference, it is omitted. If both author and journal match, both are omitted. If the journal matches, but the author does not, the journal is replaced by \emph{ibid.}, as exemplified by Ref.~[\onlinecite{epr}]. These rules embody common editorial practice in APS and AIP journals and will only be in effect if the markup features of the APS and AIP Bib\TeX\ styles is employed. \paragraph{The options of the cite command itself} Please note that optional arguments to the \emph{key} change the reference in the bibliography, not the citation in the body of the document. For the latter, use the optional arguments of the \verb+\cite+ command itself: \verb+\cite+ \texttt{*}\allowbreak \texttt{[}\emph{pre-cite}\texttt{]}\allowbreak \texttt{[}\emph{post-cite}\texttt{]}\allowbreak \verb+{+\emph{key-list}\verb+}+. \subsubsection{Example citations} By default, citations are numerical\cite{Beutler1994}. Author-year citations are used when the journal is RMP. To give a textual citation, use \verb+\onlinecite{#1}+: Refs.~\onlinecite{[][{, and references therein}]witten2001,Bire82}. By default, the \texttt{natbib} package automatically sorts your citations into numerical order and ``compresses'' runs of three or more consecutive numerical citations. REV\TeX\ provides the ability to automatically change the punctuation when switching between journal styles that provide citations in square brackets and those that use a superscript style instead. This is done through the \texttt{citeautoscript} option. For instance, the journal style \texttt{prb} automatically invokes this option because \textit{Physical Review B} uses superscript-style citations. The effect is to move the punctuation, which normally comes after a citation in square brackets, to its proper position before the superscript. To illustrate, we cite several together \cite{[See the explanation of time travel in ]feyn54,*[The classical relativistic treatment of ][ is a relative classic]epr,witten2001,Berman1983,Davies1998,Bire82}, and once again in different order (Refs.~\cite{epr,feyn54,Bire82,Berman1983,witten2001,Davies1998}). Note that the citations were both compressed and sorted. Futhermore, running this sample file under the \texttt{prb} option will move the punctuation to the correct place. When the \verb+prb+ class option is used, the \verb+\cite{#1}+ command displays the reference's number as a superscript rather than in square brackets. Note that the location of the \verb+\cite{#1}+ command should be adjusted for the reference style: the superscript references in \verb+prb+ style must appear after punctuation; otherwise the reference must appear before any punctuation. This sample was written for the regular (non-\texttt{prb}) citation style. The command \verb+\onlinecite{#1}+ in the \texttt{prb} style also displays the reference on the baseline. \subsubsection{References} A reference in the bibliography is specified by a \verb+\bibitem{#1}+ command with the same argument as the \verb+\cite{#1}+ command. \verb+\bibitem{#1}+ commands may be crafted by hand or, preferably, generated by Bib\TeX. REV\TeX~4.2 includes Bib\TeX\ style files \verb+apsrev4-2.bst+, \verb+apsrmp4-2.bst+ appropriate for \textit{Physical Review} and \textit{Reviews of Modern Physics}, respectively. \subsubsection{Example references} This sample file employs the \verb+\bibliography+ command, which formats the \texttt{\jobname .bbl} file and specifies which bibliographic databases are to be used by Bib\TeX\ (one of these should be by arXiv convention \texttt{\jobname .bib}). Running Bib\TeX\ (via \texttt{bibtex \jobname}) after the first pass of \LaTeX\ produces the file \texttt{\jobname .bbl} which contains the automatically formatted \verb+\bibitem+ commands (including extra markup information via \verb+\bibinfo+ and \verb+\bibfield+ commands). If not using Bib\TeX, you will have to create the \verb+thebibiliography+ environment and its \verb+\bibitem+ commands by hand. Numerous examples of the use of the APS bibliographic entry types appear in the bibliography of this sample document. You can refer to the \texttt{\jobname .bib} file, and compare its information to the formatted bibliography itself. \subsection{Footnotes}% Footnotes, produced using the \verb+\footnote{#1}+ command, usually integrated into the bibliography alongside the other entries. Numerical citation styles do this% \footnote{Automatically placing footnotes into the bibliography requires using BibTeX to compile the bibliography.}; author-year citation styles place the footnote at the bottom of the text column. Note: due to the method used to place footnotes in the bibliography, \emph{you must re-run Bib\TeX\ every time you change any of your document's footnotes}. \section{Math and Equations} Inline math may be typeset using the \verb+$+ delimiters. Bold math symbols may be achieved using the \verb+bm+ package and the \verb+\bm{#1}+ command it supplies. For instance, a bold $\alpha$ can be typeset as \verb+$\bm{\alpha}$+ giving $\bm{\alpha}$. Fraktur and Blackboard (or open face or double struck) characters should be typeset using the \verb+\mathfrak{#1}+ and \verb+\mathbb{#1}+ commands respectively. Both are supplied by the \texttt{amssymb} package. For example, \verb+$\mathbb{R}$+ gives $\mathbb{R}$ and \verb+$\mathfrak{G}$+ gives $\mathfrak{G}$ In \LaTeX\ there are many different ways to display equations, and a few preferred ways are noted below. Displayed math will center by default. Use the class option \verb+fleqn+ to flush equations left. Below we have numbered single-line equations; this is the most common type of equation in \textit{Physical Review}: \begin{eqnarray} \chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2} \left( \begin{array}{c} |{\bf p}|+p_z\\ px+ip_y \end{array}\right)\;, \\ \left\{% \openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta \frac{1\sum^{a}_{b}}{A^2}% \right\}% \label{eq:one}. \end{eqnarray} Note the open one in Eq.~(\ref{eq:one}). Not all numbered equations will fit within a narrow column this way. The equation number will move down automatically if it cannot fit on the same line with a one-line equation: \begin{equation} \left\{ ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta \frac{1\sum^{a}_{b}}{A^2}% \right\}. \end{equation} When the \verb+\label{#1}+ command is used [cf. input for Eq.~(\ref{eq:one})], the equation can be referred to in text without knowing the equation number that \TeX\ will assign to it. Just use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in the \verb+\label{#1}+ command. Unnumbered single-line equations can be typeset using the \verb+\[+, \verb+\]+ format: \[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow q^+g^+g^+ \dots ~. \] \subsection{Multiline equations} Multiline equations are obtained by using the \verb+eqnarray+ environment. Use the \verb+\nonumber+ command at the end of each line to avoid assigning a number: \begin{eqnarray} {\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1} \delta_{\sigma_1,-\sigma_2} (g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\ &&\times [\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1), \end{eqnarray} \begin{eqnarray} \sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2} (N^2-1)\nonumber \\ & &\times \left( \sum_{i<j}\right) \sum_{\text{perm}} \frac{1}{S_{12}} \frac{1}{S_{12}} \sum_\tau c^f_\tau~. \end{eqnarray} \textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline equation if \verb+\nonumber+ is also used on that line. Incorrect cross-referencing will result. Notice the use \verb+\text{#1}+ for using a Roman font within a math environment. To set a multiline equation without \emph{any} equation numbers, use the \verb+\begin{eqnarray*}+, \verb+\end{eqnarray*}+ format: \begin{eqnarray*} \sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2} (N^2-1)\\ & &\times \left( \sum_{i<j}\right) \left( \sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}} \right) \frac{1}{S_{12}}~. \end{eqnarray*} To obtain numbers not normally produced by the automatic numbering, use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired equation number. For example, to get an equation number of (\ref{eq:mynum}), \begin{equation} g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum} \end{equation} \paragraph{A few notes on \texttt{tag}s} \verb+\tag{#1}+ requires the \texttt{amsmath} package. Place the \verb+\tag{#1}+ command before the \verb+\label{#1}+, if any. The numbering produced by \verb+\tag{#1}+ \textit{does not affect} the automatic numbering in REV\TeX; therefore, the number must be known ahead of time, and it must be manually adjusted if other equations are added. \verb+\tag{#1}+ works with both single-line and multiline equations. \verb+\tag{#1}+ should only be used in exceptional cases---% do not use it to number many equations in your paper. Please note that this feature of the \texttt{amsmath} package is \emph{not} compatible with the \texttt{hyperref} (6.77u) package. Enclosing display math within \verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce a set of equations that are labeled with letters, as shown in Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below. You may include any number of single-line and multiline equations, although it is probably not a good idea to follow one display math directly after another. \begin{subequations} \label{eq:whole} \begin{eqnarray} {\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1} (g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\ &&\times [\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2} \end{eqnarray} \begin{equation} \left\{ abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta \frac{1\sum^{a}_{b}}{A^2} \right\},\label{subeq:1} \end{equation} \end{subequations} Giving a \verb+\label{#1}+ command directly after the \verb+\begin{subequations}+, allows you to reference all the equations in the \texttt{subequations} environment. For example, the equations in the preceding subequations environment were Eqs.~(\ref{eq:whole}). \subsubsection{Wide equations} The equation that follows is set in a wide format, i.e., it spans the full page. The wide format is reserved for long equations that cannot easily be set in a single column: \begin{widetext} \begin{equation} {\cal R}^{(\text{d})}= g_{\sigma_2}^e \left( \frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2} +\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2} \right) + x_WQ_e \left( \frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2} +\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2} \right)\;. \label{eq:wideeq} \end{equation} \end{widetext} This is typed to show how the output appears in wide format. (Incidentally, since there is no blank line between the \texttt{equation} environment above and the start of this paragraph, this paragraph is not indented.) \section{Cross-referencing} REV\TeX{} will automatically number such things as sections, footnotes, equations, figure captions, and table captions. In order to reference them in text, use the \verb+\label{#1}+ and \verb+\ref{#1}+ commands. To reference a particular page, use the \verb+\pageref{#1}+ command. The \verb+\label{#1}+ should appear within the section heading, within the footnote text, within the equation, or within the table or figure caption. The \verb+\ref{#1}+ command is used in text at the point where the reference is to be displayed. Some examples: Section~\ref{sec:level1} on page~\pageref{sec:level1}, Table~\ref{tab:table1},% \begin{table}[b \caption{\label{tab:table1}% A table that fits into a single column of a two-column layout. Note that REV\TeX~4 adjusts the intercolumn spacing so that the table fills the entire width of the column. Table captions are numbered automatically. This table illustrates left-, center-, decimal- and right-aligned columns, along with the use of the \texttt{ruledtabular} environment which sets the Scotch (double) rules above and below the alignment, per APS style. } \begin{ruledtabular} \begin{tabular}{lcdr} \textrm{Left\footnote{Note a.}}& \textrm{Centered\footnote{Note b.}}& \multicolumn{1}{c}{\textrm{Decimal}}& \textrm{Right}\\ \colrule 1 & 2 & 3.001 & 4\\ 10 & 20 & 30 & 40\\ 100 & 200 & 300.0 & 400\\ \end{tabular} \end{ruledtabular} \end{table} and Fig.~\ref{fig:epsart}.% \begin{figure}[b] \includegraphics{plots/D1_comparison.pdf \caption{\label{fig:epsart} A figure caption. The figure captions are automatically numbered.} \end{figure} \section{Floats: Figures, Tables, Videos, etc.} Figures and tables are usually allowed to ``float'', which means that their placement is determined by \LaTeX, while the document is being typeset. Use the \texttt{figure} environment for a figure, the \texttt{table} environment for a table. In each case, use the \verb+\caption+ command within to give the text of the figure or table caption along with the \verb+\label+ command to provide a key for referring to this figure or table. The typical content of a figure is an image of some kind; that of a table is an alignment.% \begin{figure*} \includegraphics{fig_2 \caption{\label{fig:wide}Use the figure* environment to get a wide figure that spans the page in \texttt{twocolumn} formatting.} \end{figure*} \begin{table*} \caption{\label{tab:table3}This is a wide table that spans the full page width in a two-column layout. It is formatted using the \texttt{table*} environment. It also demonstates the use of \textbackslash\texttt{multicolumn} in rows with entries that span more than one column.} \begin{ruledtabular} \begin{tabular}{ccccc} &\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\ Ion&1st alternative&2nd alternative&lst alternative &2nd alternative\\ \hline K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\ Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.} &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\ Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnotemark[1] &$(4e)^{\text{a}}$\\ He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\ Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\ \end{tabular} \end{ruledtabular} \end{table*} Insert an image using either the \texttt{graphics} or \texttt{graphix} packages, which define the \verb+\includegraphics{#1}+ command. (The two packages differ in respect of the optional arguments used to specify the orientation, scaling, and translation of the image.) To create an alignment, use the \texttt{tabular} environment. The best place to locate the \texttt{figure} or \texttt{table} environment is immediately following its first reference in text; this sample document illustrates this practice for Fig.~\ref{fig:epsart}, which shows a figure that is small enough to fit in a single column. In exceptional cases, you will need to move the float earlier in the document, as was done with Table~\ref{tab:table3}: \LaTeX's float placement algorithms need to know about a full-page-width float earlier. Fig.~\ref{fig:wide} has content that is too wide for a single column, so the \texttt{figure*} environment has been used.% \begin{table}[b] \caption{\label{tab:table4}% Numbers in columns Three--Five are aligned with the ``d'' column specifier (requires the \texttt{dcolumn} package). Non-numeric entries (those entries without a ``.'') in a ``d'' column are aligned on the decimal point. Use the ``D'' specifier for more complex layouts. } \begin{ruledtabular} \begin{tabular}{ccddd} One&Two& \multicolumn{1}{c}{\textrm{Three}}& \multicolumn{1}{c}{\textrm{Four}}& \multicolumn{1}{c}{\textrm{Five}}\\ \hline one&two&\mbox{three}&\mbox{four}&\mbox{five}\\ He&2& 2.77234 & 45672. & 0.69 \\ C\footnote{Some tables require footnotes.} &C\footnote{Some tables need more than one footnote.} & 12537.64 & 37.66345 & 86.37 \\ \end{tabular} \end{ruledtabular} \end{table} The content of a table is typically a \texttt{tabular} environment, giving rows of type in aligned columns. Column entries separated by \verb+&+'s, and each row ends with \textbackslash\textbackslash. The required argument for the \texttt{tabular} environment specifies how data are aligned in the columns. For instance, entries may be centered, left-justified, right-justified, aligned on a decimal point. Extra column-spacing may be be specified as well, although REV\TeX~4 sets this spacing so that the columns fill the width of the table. Horizontal rules are typeset using the \verb+\hline+ command. The doubled (or Scotch) rules that appear at the top and bottom of a table can be achieved enclosing the \texttt{tabular} environment within a \texttt{ruledtabular} environment. Rows whose columns span multiple columns can be typeset using the \verb+\multicolumn{#1}{#2}{#3}+ command (for example, see the first row of Table~\ref{tab:table3}).% Tables~\ref{tab:table1}, \ref{tab:table3}, \ref{tab:table4}, and \ref{tab:table2}% \begin{table}[b] \caption{\label{tab:table2} A table with numerous columns that still fits into a single column. Here, several entries share the same footnote. Inspect the \LaTeX\ input for this table to see exactly how it is done.} \begin{ruledtabular} \begin{tabular}{cccccccc} &$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$& &$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\ \hline Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1] & 0.680 & 1.870 & 3.700 \\ Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2] & 0.450 & 1.930 & 3.760 \\ Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3] & 0.750 & 2.170 & 3.560 \\ Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4] & 0.900 & 2.370 & 3.720 \\ Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2] & 0.380 & 1.730 & 2.830 \\ Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5] & 0.760 & 2.110 & 3.120 \\ Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5] & 1.120 & 2.620 & 3.480 \\ Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3] & 1.330 & 2.800 & 3.590 \\ Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4] & 1.420 & 3.030 & 3.740 \\ In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5] & 0.960 & 2.460 & 3.780 \\ Tl& 0.480 & 18.90 & 3.550 & & & & \\ \end{tabular} \end{ruledtabular} \footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.} \footnotetext[2]{Here's the second.} \footnotetext[3]{Here's the third.} \footnotetext[4]{Here's the fourth.} \footnotetext[5]{And etc.} \end{table} show various effects. A table that fits in a single column employs the \texttt{table} environment. Table~\ref{tab:table3} is a wide table, set with the \texttt{table*} environment. Long tables may need to break across pages. The most straightforward way to accomplish this is to specify the \verb+[H]+ float placement on the \texttt{table} or \texttt{table*} environment. However, the \LaTeXe\ package \texttt{longtable} allows headers and footers to be specified for each page of the table. A simple example of the use of \texttt{longtable} can be found in the file \texttt{summary.tex} that is included with the REV\TeX~4 distribution. There are two methods for setting footnotes within a table (these footnotes will be displayed directly below the table rather than at the bottom of the page or in the bibliography). The easiest and preferred method is just to use the \verb+\footnote{#1}+ command. This will automatically enumerate the footnotes with lowercase roman letters. However, it is sometimes necessary to have multiple entries in the table share the same footnote. In this case, there is no choice but to manually create the footnotes using \verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+. \texttt{\#1} is a numeric value. Each time the same value for \texttt{\#1} is used, the same mark is produced in the table. The \verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular} environment. Examine the \LaTeX\ source and output for Tables~\ref{tab:table1} and \ref{tab:table2} for examples. Video~\ref{vid:PRSTPER.4.010101} illustrates several features new with REV\TeX4.2, starting with the \texttt{video} environment, which is in the same category with \texttt{figure} and \texttt{table}.% \begin{video} \href{http://prst-per.aps.org/multimedia/PRSTPER/v4/i1/e010101/e010101_vid1a.mpg}{\includegraphics{vid_1a}}% \quad \href{http://prst-per.aps.org/multimedia/PRSTPER/v4/i1/e010101/e010101_vid1b.mpg}{\includegraphics{vid_1b}} \setfloatlink{http://link.aps.org/multimedia/PRSTPER/v4/i1/e010101}% \caption{\label{vid:PRSTPER.4.010101}% Students explain their initial idea about Newton's third law to a teaching assistant. Clip (a): same force. Clip (b): move backwards. }% \end{video} The \verb+\setfloatlink+ command causes the title of the video to be a hyperlink to the indicated URL; it may be used with any environment that takes the \verb+\caption+ command. The \verb+\href+ command has the same significance as it does in the context of the \texttt{hyperref} package: the second argument is a piece of text to be typeset in your document; the first is its hyperlink, a URL. \textit{Physical Review} style requires that the initial citation of figures or tables be in numerical order in text, so don't cite Fig.~\ref{fig:wide} until Fig.~\ref{fig:epsart} has been cited. \begin{acknowledgments} We wish to acknowledge the support of the author community in using REV\TeX{}, offering suggestions and encouragement, testing new versions, \dots. \end{acknowledgments} \section{Introduction} \label{sec:introduction} The flexibility and expressiveness of deep learning (DL) models are attractive features, which have led to their application to a variety of scientific problems (see e.g.\ \citet{Raghu20} for a recent review). Despite this recent progress, deep neural networks remain opaque models, and their power as universal approximators \cite{Cybenko89, Hornik89, Hornik91} comes at the expense of interpretability \cite{Molnar22}. Many techniques have been developed to gain insight into such black-box models \cite{Zeiler14, Simonyan14, Zhou16, Ribeiro16, Selvaraju17, Shrikumar17, Lundberg17, Chattopadhay18}. These solutions vary in their computational efficiency and in the range of tasks to which they can be applied; however, there is no consensus as to which method provides the most trustworthy interpretations, and a general framework to interpret deep neural networks is still an avenue of active investigation (see e.g.\ \citet{Li21, Linardatos21} for recent reviews). DL models are also widely used in representation learning, where a high-dimensional dataset is compressed to a smaller set of variables; this latent representation should contain all the relevant information for downstream tasks such as reconstruction, classification or regression \cite{Schmidhuber92, Bengio13}. Disentanglement of these compressed variables is also often imposed, in order to associate each latent to a physical quantity of domain interest \cite{Bengio13, Louizos15, Chen16, Lample17, Higgins17, Jha18, Locatello19, Lezama19}. However, how best to access the information captured by these latent vectors and connect it to the relevant factors remain open questions. In this work, we focus on representation learning and link the latent variables to relevant physical quantities by estimating their mutual information (MI), a well-established information-theoretic measure of the relationship between two random variables. MI allows us to interpret what the DL model has learned about the domain-specific parameters relevant to the problem: by interrogating the model through MI, we aim to discover what information is used by the model in making predictions, thus achieving the interpretation of its inner workings. We also use MI to quantify the level of disentanglement of the latent variables. MI has found applications in a variety of scientific fields, including astrophysics \cite{Pandey17, Sarkar20, Bhattacharjee20, Upham21, Malz21, Sarkar21, Jeffrey21, LucieSmith22, Sarkar22}, biophysics \cite{Fairhall12, Charzynska16, Tkacik16, Levchenko16, Wegner18, Holmes19, Uda20}, and dynamical complex systems \cite{Wicks07, Dunleavy12, Runge15, Myers19, Svenkeson19, Diego19, Jiang19, Jia20}, to name a few. However, estimating the mutual information $I (X, Y)$ between two random variables $X$ and $Y$, given samples from their joint distribution $p_{(X, Y)}$, remains a long-standing challenge, since it requires $p_{(X, Y)}$ to be known or estimated accurately \citep{Paninski03, Vergara15}. When $X$ and $Y$ are continuous variables with values over $\mathcal{X} \times \mathcal{Y}$, $I (X, Y)$ is defined as: \begin{align} I (X, Y) \equiv \int_{\mathcal{X} \times \mathcal{Y}} p_{(X, Y)}(x, y) \ln{\frac{p_{(X, Y)}(x, y)}{ p_X(x) p_Y(y)}} \mathop{}\!{d} x \mathop{}\!{d} y \ , \label{eq:MI} \end{align} where $p_X$ and $p_Y$ are the marginal distributions of $X$ and $Y$, respectively, and $\ln$ refers to the natural logarithm, so that MI is measured in natural units (nat). $I (X, Y)$ represents the amount of information one gains about $Y$ by observing $X$ (or vice versa): it captures the full dependence between two variables going beyond the Pearson correlation coefficient, since $I (X, Y) = 0$ if and only if $X$ and $Y$ are statistically independent \cite{Cover06}. A comprehensive summary of MI and its properties can be found in \citet{Vergara15}. The most straightforward estimator of $I (X, Y)$ given samples of $p_{(X, Y)}$ consists of binning the data and approximating Eq.~(\ref{eq:MI}) with a finite sum over the bins. This approach is heavily dependent on the binning scheme, and is prone to systematic errors \cite{Fraser86, Moon95, Darbellay99, Kwak02, Kraskov04, Suzuki08, Saxe19, Holmes19, Pichler22}. \citet{Kraskov04} proposed an estimator (hereafter referred to as KSG), based on $k$-nearest neighbors, which rewrites $I (X, Y)$ in terms of the Shannon entropy, and then applies the Kozachenko-Leonenko entropy estimator \cite{Kozachenko87} to calculate each term (see Sect.~\ref{sec:ksg_mine} for more details). However, the KSG estimator only returns a point estimate, is strongly dependent on the number of chosen neighbors, and does not scale well with sample size \cite{Gao14}. Bayesian approaches to obtain the full distribution of MI have also been discussed \cite{Hutter01, Hutter05, Archer13}, but they are not easily applicable to continuous data, and have been shown to be strongly dependent on the chosen prior \cite{Archer13}. More recently, MI estimators based on bounds approximated by neural networks have gained interest \cite{Tishby15, Chen16, Alemi16, Brakel17, Kolchinsky19, Belghazi18, Oord18, Moyer18, Poole19, Peng19, Hjelm19, Song20, Gokmen21}. In particular, \citet{Belghazi18} proposed a neural estimator of $I (X, Y)$ (hereafter referred to as MINE) rewriting it as a Kullback-Leibler (KL) divergence \cite{Kullback51}, and considering its Donsker-Varadhan representation \cite{Donsker83} (see Sect.~\ref{sec:ksg_mine} for more details). While yielding differentiable MI estimates (essential e.g.\ for backpropagation when training DL models), such neural-network-based estimators do not necessarily return an accurate estimate of Eq.~(\ref{eq:MI}), are heavily dependent on the training hyperparameters, and have been shown to suffer from a poor variance-bias tradeoff \cite{Song20}. The use of MI estimates for interpreting deep representation learning has recently been investigated as well \citep{Chen18, Chen16, LucieSmith22, Sedaghat21}; however, exploiting MI to interpret deep representation learning requires the uncertainties on the MI estimate to be quantified, ensuring that any trends in MI are statistically significant. To address these requirements, we present GMM-MI (pronounced ``Jimmie''), an algorithm to estimate the full distribution of $I (X, Y)$ based on fitting samples drawn from the distribution with Gaussian mixture models (GMMs). While the use of GMMs to estimate MI is not new \cite{Kerroum10, Eirola14, Lan06, Leiva04, Nilsson02, Polo22}, these previous works only considered MI in the context of feature selection, and did not carry out uncertainty quantification on the relevant MI estimates. GMM-MI has been designed to be a flexible tool that can be applied to multiple settings where MI estimation is required. Crucially, it also returns error estimates which we verified to be statistically correct on test datasets including bivariate distributions of various shapes and non-linear transformations of Gaussian samples. After extensive validation of GMM-MI on these toy data for which the ground truth MI is known, including comparisons to the KSG and MINE estimators in terms of both efficiency and accuracy, we additionally train representation-learning models on high-dimensional datasets. These real-world case studies include simulations of dark matter halos formed through non-linear physical processes, real astrophysical spectra and synthetic shape images with known labels; we demonstrate the use of GMM-MI to achieve the interpretability of such models. The paper is structured as follows. In Sect.~\ref{sec:proposed_procedure} we describe GMM-MI, and recall the essential details of the KSG and MINE estimators in Sect.~\ref{sec:ksg_mine}. In Sect.~\ref{sec:validation}, we present extensive experiments where we first validate our MI estimator on toy data, and then in Sect.~\ref{sec:results} we use MI to interpret the latent space of DL models trained on synthetic and real data. We conclude in Sect.~\ref{sec:conclusion}, including an outlook over planned extensions of our algorithm. \section{Method} \label{sec:method} \subsection{Estimation procedure (GMM-MI)} \label{sec:proposed_procedure} Our algorithm uses a GMM with $c$ components to obtain a fit of the joint distribution $p_{(X, Y)}$: \begin{equation} p_{(X, Y)}(x, y | \theta) = \sum_{i=1}^c w_i \mathcal{N} (x, y| \mu_i, \Sigma_i) \ , \end{equation} where $\theta$ is the set of weights $w_{1 : c}$, means $\mu_{1:c}$ and covariance matrices $\Sigma_{1:c}$. With this choice, the marginals $p(x)$ and $p(y)$ are also GMMs, with parameters determined by $\theta$. Our procedure for estimating MI and its associated uncertainty is as follows. \begin{enumerate \item For a given number of GMM components $c$, we randomly initialize $n_{\rm{init}}$ different GMM models. Each set of initial GMM parameters is obtained by first randomly assigning the responsibilities, namely the probabilities that each point belongs to a component $i$, sampling from a uniform distribution. The starting values of each $\mu_i$ and $\Sigma_i$ are calculated as the sample mean and covariance matrix of all points, weighted by the responsibilities, while each $w_i$ is initialized as the average responsibility across all points. Having multiple initializations is crucial to reduce the risk of stopping at local optima during the optimization procedure \cite{Ueda98, Bovy11, Emilie16, Baudry15, Melchior18}. \item We fit the data using $k$-fold cross-validation: this means that we train a GMM on $k-1$ subsets of the data (or ``folds''), and evaluate the trained model on the remaining validation fold. Each fit is performed with the expectation-maximization algorithm \cite{Dempster77}, and terminates when the change in log-likelihood on the training data is smaller than a chosen threshold. We also add a small regularization constant $\omega$ to the diagonal of each covariance matrix, as described e.g.\ in \citet{Melchior18}, to avoid singular covariance matrices. \item We select the model with the highest mean validation log-likelihood across folds $\hat{\ell}_c$, since it has the best generalization performance. Among the $k$ models corresponding to $\hat{\ell}_c$, we also store the final GMM parameters with the highest validation log-likelihood on a single fold: these will be used to initialize each bootstrap fit in step 5, thus reducing the risk of stopping at local optima and significantly accelerating convergence. \item We repeat steps 1--3 iteratively increasing the number of GMM components from $c=1$. We stop when $\hat{\ell}_{c+1} - \hat{\ell}_{c}$ is smaller than a user-specified positive threshold, and select this value of $c$ as the optimal number of GMM components to fit. In this way, we avoid overfitting the training data and adding too many components, which would considerably slow down the procedure while not significantly improving the density estimation. \item We bootstrap the data $n_{\rm{b}}$ times, and fit a GMM to each bootstrapped realization. Each fit is initialized with the set of parameters selected in step 3, and with the number of components found in step 4. We use bootstrap to capture not just a point estimate of MI, but its full distribution. \item For each fitted model, we calculate MI by solving the integral in Eq.~(\ref{eq:MI}) using Monte Carlo (MC) integration over $M$ samples. \item We return the sample mean and standard deviation of the distribution of MI values. \end{enumerate} We choose the initialization procedure described in step 1 for its speed, but in our implementation of GMM-MI other initialization procedures are also available and could be alternatively used. For instance, it is possible that the random initialization we set as default returns overlapping components which inhibit the optimization procedure; in those cases, we recommend switching to an initialization based on $k$-means \citep{Lloyd82}. On the other hand, $k$-means itself is known to only guarantee convergence to local optima \citep{Arthur07}; for this reason, we also provide the possibility to perturb the means by a user-specified scale after an initial call to $k$-means. We call this approach ``randomized $k$-means'', and offer full flexibility to select the most appropriate initialization type based on the data being analyzed. Our implementation also allows the user to set a higher patience, i.e.\ consider more than one additional component in step 4 after the validation loss has started to decrease; alternatively, it is possible to select the number of components yielding the lowest Akaike information criterion (AIC, \cite{Akaike74}) or Bayesian information criterion (BIC, \cite{Schwarz78}), with details in Appendix~\ref{app:AIC_BIC}. The number of folds ($k$) should be set based on the number of available samples, so that each fold is representative of the data. The number of initializations ($n_{\rm{init}}$), bootstrap realizations ($n_{\rm{b}}$), and MC samples ($M$) should be chosen based on the available computational budget. In many instances, the factors of variation that are used to generate the data are discrete variables; in these cases, we will need to estimate MI between a continuous variable $X$ and a categorical variable $F$ which can take $v$ different values $f_{1:v}$. In this case, assuming the $v$ values have equal probability (as will be the case when considering the 3D shapes dataset in Sect.~\ref{sec:3D_shapes}), the mutual information $I(X, F)$ can be expressed as: \begin{align} \nonumber I(X, F) = &\frac{1}{v} \sum_{i=1}^{v} \int_{\mathcal{X}} \mathop{}\!{d} x \ p_{(X|F)}(x|f_i) \times \\ & \times \left [ \ln p_{(X|F)}(x|f_i) - \ln \frac{1}{v} \sum_{j=1}^{v} p_{(X|F)}(x|f_j) \right ] \ , \label{eq:MI_cat} \end{align} where we use a GMM to fit each conditional probability $p_{(X|F)}(x|f_i)$. The full derivation of Eq.~(\ref{eq:MI_cat}) can be found in Appendix~\ref{app:MI_der}. \subsection{Alternative estimators} \label{sec:ksg_mine} In order to validate our algorithm, we compare it with two established estimators of MI. The KSG estimator, first proposed in \citet{Kraskov04}, rewrites MI as: \begin{align} I(X, Y) = H(X) + H(Y) - H(X, Y) \ , \label{eq:MI_ksg} \end{align} where $H(\cdot)$ refers to the Shannon entropy, defined for a single variable as: \begin{align} \label{eq:entropy} H (X) \equiv - \int_{\mathcal{X}} p_X(x) \ln{p_X(x)} \ . \end{align} The Kozachenko-Leonenko estimator \cite{Kozachenko87} is then used to evaluate the entropy in Eq.~(\ref{eq:entropy}): \begin{align} \label{eq:kl_entropy} \widehat{H} (X) = - \psi(k) + \psi(N) + \ln{c_{\rm{d}}} + \frac{d}{N} \sum_{i=1}^{N} \ln {\epsilon^{(k)}} (i) \ , \end{align} where $\psi(\cdot)$ is the digamma function, $k$ is the chosen number of nearest neighbors, $N$ is the number of available samples, $d$ is the dimensionality of $X$, $c_{\rm{d}}$ is the volume of the unit ball in $d$ dimensions, and $\epsilon^{(k)}$ is twice the distance between the $i^{\rm{th}}$ data point and its $k^{\rm{th}}$ neighbor. Applying Eq.~(\ref{eq:kl_entropy}) to each term in Eq.~(\ref{eq:MI_ksg}) would lead to biased estimates of MI \cite{Kraskov04, Holmes19}; for this reason, the KSG estimator actually considers a ball containing the $k$-nearest neighbors around each sample, and counts the number of points within it in both the $x$ and $y$ direction. The resulting estimator of MI then becomes \cite{Kraskov04, Holmes19}: \begin{align} \widehat{I} (X, Y) = \psi(k) + \psi(N) - \frac{1}{k} - \langle \psi(n_x^{(k)}) + \psi(n_y^{(k)}) \rangle \ , \end{align} where $n_x^{(k)}$ $(n_y^{(k)})$ represents the number of points in the $x$ $(y)$ direction, and $\langle \cdot \rangle$ indicates the mean over the available samples. In our experiments, we consider the implementation of the KSG estimator available from \textsc{sklearn} in this \href{https://scikit-learn.org/stable/modules/generated/sklearn.feature_selection.mutual_info_regression.html}{https link}. We also compare our algorithm against the MINE estimator proposed in \citet{Belghazi18}. MI as defined in Eq.~(\ref{eq:MI}) can be interpreted as the KL divergence $D_{\rm{KL}}$ between the joint distribution and the product of the marginals: \begin{align} I (X, Y) = D_{\rm{KL}} \left[ p_{(X, Y)} || p_X p_Y \right] \ , \end{align} where the KL divergence between two generic probability distributions $p_X$ and $q_X$ defined over $\mathcal{X}$ is defined as: \begin{align} D_{\rm{KL}} \left[ p_X || q_X \right] \equiv \int_{\mathcal{X}} \mathop{}\!{d} x \ p_X(x) \ln{\frac{p_X(x)}{q_X(x)}} \ . \end{align} The MINE estimator then considers the Donsker-Varadhan representation \cite{Donsker83} of the KL divergence: \begin{align} D_{\rm{KL}} \left[ p_X || q_X \right] = \sup_{T} \mathbb{E}_{p_X} \left [ T \right] - \ln{\mathbb{E}_{q_X} \left [ e^T \right]} \ , \end{align} where the supremum is taken over all the functions $T$ such that the expectations $\mathbb{E}\left[ \cdot \right ]$ are finite, and parameterizes $T$ with a neural network. In our experiments, we consider the implementation available in this \href{https://github.com/gtegner/mine-pytorch}{https link}, which includes the mitigation of the gradient bias through the use of an exponential moving average, as suggested in \citet{Belghazi18}. \subsection{Representation learning} \label{sec:bvae} We apply our MI estimator GMM-MI to interpret the latent space of representation-learning models. Specifically, we consider $\beta$-variational autoencoders ($\beta$-VAEs, \cite{Kingma14, Higgins17}), where one neural network is trained to encode high-dimensional data $D$ into a distribution over disentangled latent variables $\mathbf{z}$, and a second network decodes samples of the latent distribution back into data points $\widetilde{D}$. The two networks are trained together to minimize the following loss function: \begin{align} \mathcal{L} = \rm{MSE}(D, \widetilde{D}) + \beta D_{\rm{KL}} \left[p_{\phi}(\mathbf{z} | D) || p(\mathbf{z})\right] \ , \label{eq:VAE} \end{align} where $\rm{MSE}$ indicates the mean squared error, $p_{\phi}(\mathbf{z} | D)$ represents the encoder parameterized by a set of weights $\phi$, $p(\mathbf{z})$ is the prior over the latent variables $\mathbf{z}$, and $\beta$ is a regularization constant which controls the level of disentanglement of $\mathbf{z}$. We will also reproduce the results of \citet{LucieSmith22} in Sect.~\ref{sec:dm_haloes}, for which the architecture is slightly different: the latent samples are combined with a given query (the radius $r$) and fed through the decoder to predict dark matter halo density profiles at each given $r$. This model is referred to as the interpretable variational encoder (IVE), with an analogous loss function to Eq.~(\ref{eq:VAE}). \section{Validation} \label{sec:validation} In this section, we validate GMM-MI on toy data for which the ground truth MI is known. We run all the MI estimations on a single CPU node with 40 2.40GHz Intel Xeon Gold 6148 cores using no more than 300 MB of RAM, reporting the speed performance in each case. \begin{figure}[ht!] \subfloat[Bivariate Gaussian distribution.]{% \includegraphics[trim={0 0.1cm 0 0.1cm},clip, width=\columnwidth]{plots/D1_comparison.pdf}% } \subfloat[Gamma-exponential distribution.]{% \includegraphics[trim={0 0.1cm 0 0.1cm},clip, width=\columnwidth]{plots/gamma_exp_comparison.pdf}% } \subfloat[Ordered Weinman exponential distribution.]{% \includegraphics[trim={0 0.1cm 0 0.1cm},clip, width=\columnwidth]{plots/exp_exp_comparison.pdf}% } \caption{Estimates of mutual information (MI) for: (a) a bivariate Gaussian distribution with varying correlation coefficient $\rho$; (b) a gamma-exponential distribution with varying $\alpha$, as in Eq.~(\ref{eq:gammaexp}); (c) an ordered Weinman exponential distribution with varying $\alpha$, as in Eq.~(\ref{eq:weinman}). The dashed black line indicates the ground truth MI. We compare the KSG estimator (\cite{Kraskov04}, solid red line), the MINE estimator (\cite{Belghazi18}, dotted-dash green line), and our estimator GMM-MI, indicated with the gray shaded area (mean $\pm$ two standard deviations). The numbers in parentheses indicate the time to obtain a single estimate with KSG and MINE, and the full distribution of MI in the case of GMM-MI, for each $\rho$ or $\alpha$. These estimates are obtained from $N=200$ samples, and with hyperparameters reported in Sect.~\ref{sec:validation}.}\label{fig:D1_compare} \end{figure} \begin{figure*}[ht!] \subfloat[$f(y) = y$.]{% \includegraphics[width=0.69\columnwidth]{plots/identity_transformation.pdf}% } \subfloat[$f(y) = y + 0.5y^3$.]{% \includegraphics[width=0.69\columnwidth]{plots/cube_transformation.pdf}% } \subfloat[$f(y) = \ln{(y + 5.5)}$.]{% \includegraphics[width=0.69\columnwidth]{plots/log_transformation.pdf}% } \caption{\textit{Left panel}: violin plots of the bias and standard deviation returned by GMM-MI, when applied to samples from a bivariate Gaussian distribution with $\rho=0.6$, with varying number of samples $N$. The bias is defined as the difference between the MI estimate returned by GMM-MI and the ground truth MI. The black crosses represent the mean of each distribution, while the grey line in the bottom plots indicates the expected power law $\propto \frac{1}{\sqrt{N}}$ passing through the estimated mean MI value at $N=20\,000$. \textit{Middle panel}: same as in the left panel, but all marginal $y$ samples are mapped through a cubic function, to demonstrate the invariance of MI under invertible transformations. \textit{Right panel}: same as in the middle panel, but with a logarithmic function.} \label{fig:invariance_compare} \end{figure*} We first consider a bivariate Gaussian distribution with unit variance of each marginal and varying level of correlation $\rho \in \left[ -1, 1 \right]$, following \citet{Belghazi18}. In this case, the true value of I(X, Y) can be obtained analytically by solving the integral in Eq.~(\ref{eq:MI}), yielding: \begin{align} I(X, Y)_{\rm{true}} = -\frac{1}{2} \ln {\left(1-\rho^2 \right)} \ . \label{eq:mi_true} \end{align} We consider two additional bivariate distributions, the gamma-exponential distribution \cite{Darbellay99, Darbellay00, Kraskov04, Haeri14}, with density ($\alpha>0$ is a free parameter): \begin{align} \label{eq:gammaexp} p_{(X, Y)} (x, y | \alpha) = \begin{cases} \frac{1}{\Gamma(\alpha)} x^{\alpha} e^{-x-xy} & x>0, y>0 \\ 0 & \mbox{otherwise} \end{cases}\ , \end{align} where $\Gamma$ is the gamma function, and the ordered Weinman exponential distribution \cite{Darbellay99, Darbellay00, Kraskov04, Haeri14}, with density: \begin{align} \label{eq:weinman} p_{(X, Y)} (x, y | \alpha) = \begin{cases} \frac{2}{\alpha} e^{-2x-\frac{y-x}{\alpha}} & y>x>0 \\ 0 & \mbox{otherwise} \end{cases}\ . \end{align} The true value of $I(X, Y)$ for these distributions can be obtained analytically, and is reported in Appendix~\ref{app:true_mi}. Since $I(X, Y)$ is invariant under invertible transformations of each random variable \citep{Kraskov04}, we consider $\ln(X)$ and $\ln{(Y)}$ when estimating MI in the case of the last two distributions \cite{Kraskov04}. To demonstrate the power of our estimator, we restrict ourselves to the case with only $N=200$ samples. To estimate MI, we consider the KSG estimator with 1 neighbor (to minimize the bias, and following \citet{Kraskov04}), the MINE estimator trained for 50 epochs with a learning rate of $10^{-3}$ and a batch size of 32, and our estimator GMM-MI with $k=2$ folds, $n_{\rm{init}} =3$ different initializations, a log-likelihood threshold on each individual fit of $10^{-5}$, a threshold on the mean validation log-likelihood to select the number of GMM components of $10^{-5}$, $n_{\rm{b}} = 100$ bootstrap realizations, $M=10^4$ MC samples, and a regularization scale of $\omega=10^{-12}$. The results are reported in Fig.~\ref{fig:D1_compare}. The KSG estimator is the fastest, and yields MI values closely matching the ground truth, but returns biased estimates around e.g.\ $|\rho| = 0.4$ in the bivariate Gaussian case, and $\alpha\simeq 1$ in the ordered Weinman case. The MINE estimator is more computationally expensive and shows a relatively high variance, which is expected since MINE has been shown to be prone to variance overestimation due to the use of batches \cite{Poole19}. GMM-MI, on the other hand, returns a distribution of MI in good agreement with the ground truth in $\mathcal{O}(1)$ s, and provides an uncertainty estimate due to the finite sample size. We also found the results of GMM-MI to be robust to the choice of hyperparameters: changing the values of the likelihood threshold, MC samples, bootstrap realizations or regularization scale by one order of magnitude, or doubling the number of folds and initializations, did not significantly change the results obtained with GMM-MI. We further validate GMM-MI by testing that it is unbiased as the number of available samples $N$ increases, as well as by showing that it satisfies the MI property of invariance under invertible non-linear transformations \citep{Kraskov04}. We consider a bivariate Gaussian distribution with $\rho=0.6$, and three different functions applied to one marginal variable $Y$: $f(y) = y$ (identity), $f(y) = y+0.5y^3$ (cubic) and $f(y) = \ln{(y+5.5)}$ (logarithmic). To deal with these datasets, we change the GMM-MI hyperparameters to $k=3$, $n_{\rm{init}} =5$, and $M=10^5$; however, we find no significant variations in the results even with different sets of hyperparameters. We repeat the estimation procedure of MI 500 times, drawing $N$ samples with a different seed every time, and considering $N=200$, $N=2\,000$ and $N=20\,000$. For each estimate, we calculate the bias, i.e.\ the difference between the estimated value of MI and the ground truth. We report violin plots of the bias and of the MI standard deviation as returned by GMM-MI across the 500 trials in Fig.~\ref{fig:invariance_compare}. The mean bias, indicated as a black cross, converges to 0 as $N$ grows, and it is always well below the typical value of the standard deviation, thus demonstrating that GMM-MI is unbiased. This is true even when considering the cubic and the logarithmic transformations, further confirming that GMM-MI correctly captures the invariance property of MI. Moreover, in all cases the standard deviations returned by GMM-MI follow a power law $\propto \frac{1}{\sqrt{N}}$ as expected, represented as a grey line in the bottom plots. Remarkably, we found that even with very low numbers of samples ($N=50$), GMM-MI returns MI values consistent with the ground truth, even when applying the non-linear transformations considered in this section. \begin{figure}[ht!] \centering \includegraphics[trim={0 0.1cm 0 0.1cm},clip, width=\columnwidth]{plots/MI_bootstrap.pdf} \caption{Mutual information distributions (means $\pm$ one standard deviation) when bootstrapping $N=200$ samples from a bivariate Gaussian distribution using three different estimators. As reported in \citet{Holmes19}, the KSG estimator (purple line) returns a biased estimate of MI. On the other hand, the MINE estimator (green line) and our algorithm based on Gaussian mixture models (red line) all agree with the ground truth (dashed black line). However, MINE takes two orders of magnitude more time than our estimator, and returns a higher-variance estimate since it includes the variability due to the neural network initialization.} \label{fig:bs_compare} \end{figure} \subsubsection{A note on bootstrap} As reported in \citet{Holmes19}, using bootstrap to associate an error bar to MI estimates can lead to catastrophic failures: duplicate points can be interpreted as fine-scale features, introducing spurious extra MI. In this section, we address this concern and empirically show that, despite including a bootstrap step, our procedure does not lead to biased estimates of MI. We consider the same experiment described in \citet{Holmes19}, where a single data set of $N=200$ bivariate Gaussian samples with $\rho=0.6$ is bootstrapped 20 times. We apply the KSG (with 3 neighbors, following \citet{Holmes19}) and MINE estimators to each bootstrapped realization, and compare it against our estimator with $n_{\mathrm{b}} = 20$. The results are reported in Fig.~\ref{fig:bs_compare}. The KSG estimator returns a mean MI biased by a factor of 4, while both MINE and our procedure return an accurate estimate. However, MINE is two orders of magnitude more computationally demanding, and returns an error bar which is larger than with our procedure, since it tends to overestimate the variance, as discussed in Sect.~\ref{sec:validation}. \section{Results} \label{sec:results} In this section, we apply our estimator to interpret the latent space of representation-learning models trained on three different datasets, ranging from synthetic images to cosmological simulations. We use our MI estimator to quantify the level of disentanglement of latent variables, and link them to relevant physical parameters. In the following experiments, we consider $k=3$ folds, $n_{\rm{init}} =5$ different initializations, a log-likelihood threshold on each individual fit of $10^{-5}$, $n_{\rm{b}} = 100$ bootstrap realizations, $M=10^5$ MC samples, and a regularization scale of $\omega=10^{-15}$; as in the experiments described in Sect.~\ref{sec:validation}, we found GMM-MI to be robust to the hyperparameter choices. Obtaining the full distribution of MI with our algorithm typically takes $\mathcal{O}(10)$ s on the datasets we analyze, using the same hardware described in Sect~\ref{sec:validation}. \subsubsection{3D Shapes} \label{sec:3D_shapes} We consider the 3D Shapes dataset \cite{Kim18, Burgess18}, which consists of images of various shapes that were generated by the following factors: shape (4 values), scale (8 values), orientation (15 values), floor color (10 values), wall color (10 values), and object color (10 values). Each combination of factors is included in the dataset exactly once, for a total of 480\,000 images. We train a $\beta$-VAE, as described in Sect.~\ref{sec:bvae}, on this dataset, using a 6-dimensional latent space and setting the value of $\beta$ using cross-validation. After training, we encode 10\% of the data, which were not used for training or validation, and sample one point from each latent distribution. To interpret what each latent variable $z_i$ has learned about each generative factor of variation $f_j$, we measure the mutual information $I(z_i, f_j)$ using Eq.~(\ref{eq:MI_cat}). In Fig.~\ref{fig:MI_s} we report the MI values for all latents and factors using GMM-MI: except for scale and shape, each latent variable carries information about a single factor of variation. The difficulty in disentangling scale and shape was also reported in \citet{Kim18}. To assess the level of dependence between latent variables, we calculate $I(z_i, z_j)$: these values are below $10^{-4}$ nat for all pairs, except for the one carrying information about both scale and shape, i.e.\ $I(z_2, z_4) = 0.04 \pm 0.01$ nat. \begin{figure} \centering \includegraphics[width=\columnwidth]{plots/MI_shapes3D_hm.pdf} \caption{Mutual information (MI) between each ground truth factor and each latent variable of the $\beta$-VAE model trained on the 3D Shapes dataset. The values are obtained using GMM-MI, and the associated error estimates (not shown) are typically $\mathcal{O}(10^{-3})$--$\mathcal{O}(10^{-4})$ nat. All zeros indicate values of MI below 0.01 nat. Each latent is dependent upon only a single factor, except for $z_2$ and $z_4$, which appear entangled with scale and shape, as also found in \citet{Kim18}.} \label{fig:MI_s} \end{figure} \subsubsection{Dark matter halo density profiles} \label{sec:dm_haloes} \begin{figure} \centering \includegraphics[width=\columnwidth]{plots/MI_trend.pdf} \caption{Mutual information (MI) between each latent variable and dark matter halo density $\rho_{\rm{truth}}$ in each radial bin for the $\mbox{IVE}_{\rm{infall}}$ model \cite{LucieSmith22}. The points with darker error bars correspond to the sample mean and standard deviation obtained with our algorithm (GMM-MI). The striped areas indicate the kernel density estimate (KDE) values shown in \citet{LucieSmith22}, with bandwidths of 0.3 (lower limit) and 0.1 (upper limit). There is good agreement between the two approaches; in particular, the GMM-MI estimates overlap with the KDE estimates at lower (higher) bandwidth when the MI estimates are higher (lower), due to the different KDE bandwidths sometimes underfitting and sometimes overfitting the data. GMM-MI is designed to avoid such cases, as we explain in Sect.~\ref{sec:dm_haloes}. } \label{fig:MI_trend} \end{figure} In the standard model of cosmology, only 5\% of our Universe consists of baryonic matter, while the remainder consists of dark matter (25\%) and dark energy (70\%) \cite{Dodelson03}. In particular, dark matter only interacts via the gravitational force, and gathers into stable large-scale structures, called `halos', where galaxy formation typically occurs. Given the highly non-linear physical processes taking place during the formation of such structures, a common tool to analyze dark matter halos are cosmological $N$-body simulations, where particles representing the dark matter are evolved in a box under the influence of gravity \cite{Navarro96, Tormen97, Jenkins98}. Dark matter halos forming within such simulations exhibit a universal spherically-averaged density profile as a function of their radius \cite{Navarro97, Huss99, Wang09}; this universality encompasses a huge range of halo masses and persists within different cosmological models. While the universality of the density profile is still not fully understood, \citet[][LS22 hereafter]{LucieSmith22} showed that it is possible to train a deep representation learning model to compress raw dark matter halo data into a compact disentangled representation that contains all the information needed to predict dark matter density profiles. Following LS22, we consider 4332 dark matter halos from a single $N$-body simulation, and encode them using their $\mbox{IVE}_{\rm{infall}}$ model with 3 latent variables. The latent representation is used to predict the dark matter halo density profile in 13 different radial bins. We calculate the MI between the ground-truth halo density in each radial bin and each latent variable, aiming to reproduce the middle panel of fig. 4 in LS22, where further details can be found. We show the trend of MI for all radial bins and latent variables in Fig.~\ref{fig:MI_trend}. We compare the estimates from GMM-MI with those obtained using kernel density estimation (KDE) with different bandwidths, as done in LS22. A major difference between the two approaches is that our bands indicate the error coming from the limited sample size, while their bands represent the sensitivity of the KDE to different bandwidths. The results are in good agreement: in particular, GMM-MI returns estimates closer to the KDE approach with smaller bandwidth when MI is high; in this case, the higher KDE bandwidth value underfits the data. On the other hand, for lower values of MI, GMM-MI yields estimates consistent with the KDE ones at higher bandwidth, since the lower bandwidth overfits the data. This confirms that GMM-MI avoids underfitting and overfitting of the data by design. We also checked that the latent variables of the $\mbox{IVE}_{\rm{infall}}$ model are independent: as in LS22, the MI between each pair of latents is $\mathcal{O}(10^{-2})$ nat. \begin{figure}[htb] \subfloat[Our results, using GMM-MI.]{% \includegraphics[width=\columnwidth]{plots/Sedaghat_MI_1.pdf}% } \subfloat[Results from \citet{Sedaghat21}, using histograms.]{% \includegraphics[width=\columnwidth]{plots/Sedaghat_MI_2.pdf}% } \caption{\textit{Top panel}: Mutual information (MI) between the six most-informative latent variables (based on the median absolute deviation) and six astrophysical parameters related to the stellar spectra described in Sect.~\ref{sec:stellar_spectra}, calculated using our algorithm GMM-MI; the corresponding error bars (not shown) are typically $\mathcal{O}(10^{-2})$ nat. \textit{Bottom panel}: Same as the top panel, using the MI estimator of \citet{Sedaghat21} based on histograms. The results agree overall: the 124$^{\rm{th}}$ variable is highly correlated with the radial velocity, while the effective temperature $T_{\rm{eff}}$ and mass seem mostly correlated with the 85$^{\rm{th}}$ latent. The metallicity $\left[\mbox{M}/ \mbox{H}\right]$, the atmospheric air mass and the signal-to-noise ratio (SNR) do not display strong dependence on any of the selected latent variables.}\label{fig:MI_sed} \end{figure} \subsubsection{Stellar spectra} \label{sec:stellar_spectra} We consider the model presented in \citet[][S21 hereafter]{Sedaghat21}, where a $\beta$-VAE is trained on about $7\,000$ real unique spectra with a 128-dimensional latent space. These spectra were collected by the High-Accuracy Radial-velocity Planet Searcher instrument (HARPS, \cite{Pepe02, Mayor03}) in the spectral range 378--691 nm, and include mainly stellar spectra, even though Jupiter and asteroid contaminants are present in the dataset. All details about the data, the preprocessing steps and the training procedure can be found in S21. To select the most informative latent variables, the median absolute deviation (MAD) is calculated for each of them; the rest of the analysis is carried out on the six most informative latents only. We calculate MI between each of these six variables and six known physical factors, all treated as continuous variables. These are the star radial velocity, its effective temperature $T_{\rm{eff}}$, its mass, its metallicity $\left[\mbox{M}/ \mbox{H}\right]$, the atmospheric air mass and the signal-to-noise ratio (SNR). The MI estimates obtained with GMM-MI are shown in the top panel of Fig.~\ref{fig:MI_sed}: the 124$^{\rm{th}}$ latent variable shows high dependence on the radial velocity, while the 85$^{\rm{th}}$ latent appears entangled with both the effective temperature and the mass. The other physical parameters do not show a dependence on a particular latent amongst the ones with the highest MAD, even though in S21 a more complete analysis exploring latent traversals and investigating subsets of data is presented. The bottom panel of the same figure shows the results obtained with the procedure outlined in S21, which uses histograms with a certain number of bins (40 in this case) as density estimators. The trend agrees with our results, even though the particularly high number of bins chosen might overfit the data and overestimate MI (compare e.g.\ the $\left[\mbox{M}/ \mbox{H}\right]$ MI estimates), analogously to the KDE results in Fig.~\ref{fig:MI_trend}. On the other hand, our algorithm provides a robust way to select the hyperparameters, thus avoiding underfitting or overfitting the samples. \section{Conclusions} \label{sec:conclusion} We presented GMM-MI (pronounced ``Jimmie''), an efficient and robust algorithm to estimate the mutual information (MI) between two random variables given samples of their joint distribution. Our algorithm uses Gaussian mixture models (GMMs) to fit the available samples, and returns the full distribution of MI through bootstrapping, thus including the uncertainty on MI due to the finite sample size. GMM-MI is demonstrably accurate, and benefits from the flexibility and computational efficiency of GMMs. Moreover, it can be applied to both discrete and continuous settings, and is robust to the choice of hyperparameters. We extensively validated GMM-MI on toy datasets for which the ground truth MI is known, showing equal or better performance with respect to established estimators like KSG \cite{Kraskov04} and MINE \cite{Belghazi18}. We demonstrated the application of our estimator to interpret the latent space of three different deep representation-learning models trained on synthetic shape images, large-scale structure in cosmological simulations and real spectra of stars. We calculated both the MI between latent variables and physical factors, and the MI between the latent variables themselves, to investigate their degree of disentanglement, reproducing MI estimates obtained with various techniques, including histograms and kernel density estimators. These results further validate the accuracy of GMM-MI and confirm the power of MI for gaining interpretability of deep learning models. We plan to extend our work by improving the density estimation with more flexible tools such as normalizing flows (NFs, \cite{Dinh14, Rezende15}), which can be seamlessly integrated into neural network-based settings and can benefit from graphics processing unit (GPU) acceleration. Moreover, combining NFs with a differentiable numerical integrator would make our estimator amenable to backpropagation, thus allowing its use in the context of MI optimization. We will explore this avenue in future work. \section*{Software and Data} The code and data to reproduce this work are provided as Supplemental Material. GMM-MI is publicly available in this GitHub repository (https://github.com/dpiras/GMM-MI, also accessible by clicking the icon \href{https://github.com/dpiras/GMM-MI}{\faicon{github}}), together with all results from the paper. \begin{acknowledgments} We thank Nima Sedaghat, Martino Romaniello and Vojtech Cvrcek for sharing the stellar spectra model and data. We are also grateful to Justin Alsing for useful discussions about initialization procedures for GMM fitting. DP was supported by the UCL Provost's Strategic Development Fund. The work of HVP was supported by the G\"{o}ran Gustafsson Foundation for Research in Natural Sciences and Medicine and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 101018897 CosmicExplorer). HVP and LLS acknowledge the hospitality of the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. The participation of HVP and LLS at the Aspen Center for Physics was supported by the Simons Foundation. This study was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 818085 GMGalaxies. AP is additionally supported by the Royal Society. NG was funded by the UCL Graduate Research Scholarship (GRS) and UCL Overseas Research Scholarship (ORS). This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. This work used computing equipment funded by the Research Capital Investment Fund (RCIF) provided by UKRI, and partially funded by the UCL Cosmoparticle Initiative. This work used facilities provided by the UCL Cosmoparticle Initiative. \end{acknowledgments} \section*{Author contributions} \textbf{D.P.}: formal analysis; investigation; validation; software; writing -- original draft preparation, review \& editing; visualization. \textbf{H.V.P.}: conceptualization; methodology; validation; writing -- review \& editing; funding acquisition. \textbf{A.P.}: conceptualization; methodology; validation; writing -- review \& editing; funding acquisition. \textbf{L.L.-S}: methodology; validation; resources; writing -- review \& editing. \textbf{N.G.}: software; validation; writing -- review \& editing. \textbf{B.N.}: writing -- review \& editing.

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\section{Introduction} \label{sec:introduction} Recent few years have witnessed rapid progress in large vision-language (VL) pre-trained models~\cite{clip,align,declip,filip,alayrac2022flamingo,yuan2021florence} as well as their remarkable performance on downstream vision tasks. A VL pre-trained model generally involves data encoders and it is becoming increasingly popular to exploit image-test contrastive loss~\cite{clip} to align the embedding of images and texts into a shared space. When adapting to downstream tasks in relatively data-limited or label-limited settings, it is often ineffective to fine-tune the entire model, due to its high complexity. Then, prompt tuning as a representative parameter-efficient learning paradigm has emerged as an efficient way to adapt VL model to downstream tasks. Albeit considerable achievements have been made, existing prompt tuning methods generally require visual data to learn prompts (as shown in Fig.~\ref{fig:intro}(a)). For example, CoOp~\cite{coop} learns from annotated images. CoCoOp~\cite{cocoop} further introduces generalizable input-conditional prompts. DualCoOp~\cite{dualcoop} adapts CLIP to multi-label recognition tasks by training pairs of positive and negative prompts with partial-labeled images. Nonetheless, the performance of these prompting methods may be limited when it is infeasible to obtain sufficient image data or annotate the required images. In this paper, we advocate treating \textbf{T}exts \textbf{a}s \textbf{I}mages for prompt tuning, \ie, TaI prompting. It is considered feasible as the image encoder and text encoder in many pre-trained VL models~\cite{clip,align} encode images and texts into a shared space. Given an image and its caption, the visual features produced by the image encoder will be close to the text feature of the caption produced by the text encoder. Therefore, in addition to extracting visual features from images, it is also feasible to extract text features as alternatives form, for example, descriptive sentences and captions, for prompt tuning (see Fig.~\ref{fig:intro}(b)). TaI prompting has several interesting properties and merits. Taking a downstream image recognition task as an example, given a set of object categories, one can easily crawl a large set of text descriptions that contain object names from these categories. Text descriptions are easily accessible in this way, and class labels can be directly derived from text descriptions, which means, in contrast to prompting from images, TaI prompting may suffer less from the data-limited and label-limited issues. We use multi-label image recognition~\cite{voc2007,coco,nuswide,chen2019multi,ye2020add} to verify the effectiveness of our TaI prompting in this paper. To begin with, we crawl the captions from public image caption datasets (\eg, MS-COCO~\cite{coco}) and localized narratives from object detection datasets (\eg, Open Images~\cite{openimages}) to form the training set of text descriptions. For any specific multi-label recognition task, we adopt a noun filter to map the nouns in the text descriptions to the corresponding object categories, and then only keep the text descriptions that contain one or more classes of target objects. To better cope with multi-label classification, we introduce double-grained prompt tuning (\ie, TaI-DPT) which involves: (i) a set of global prompts to generate embeddings for classifying whole sentences or images, and (ii) a set of local prompts to extract embeddings for discriminating text tokens or image patches. Given a set of text descriptions, global and local prompts can be tuned by minimizing the ranking loss~\cite{rankingloss}. Note that, though these prompts are learned from text descriptions solely, they can be readily deployed to classify whole images as well as image patches during testing (see Fig.~\ref{fig:intro}(c)). Experimental results show that, without using any labeled images, our TaI prompting surpasses zero-shot CLIP~\cite{clip} by a large margin on multiple benchmarks, \eg, MS-COCO, VOC2007, and NUS-WIDE. Moreover, when images are also available during training, our TaI prompting can be combined with existing methods of prompting from images to improve its performance. In particular, given a few annotated images, our TaI-DPT can be integrated with CoOp as a prompt ensemble for improving classification accuracy. With partially labeled training data being provided, we may also combine TaI-DPT and DualCoOp~\cite{dualcoop} to improve multi-label recognition accuracy consistently. Extensive results verify the effectiveness of our TaI-DPT, using in isolation or in combination, in comparison to state-of-the-arts. To sum up, the contribution of this work include: \begin{itemize} \setlength{\itemsep}{0pt} \setlength{\parsep}{0pt} \setlength{\parskip}{0pt} \item We propose Texts as Images in prompt tuning (\ie, TaI prompting) to adapt VL pre-trained models to multi-label image recognition. Text descriptions are easily accessible and, in contrast to images, their class labels can be directly derived, making our TaI prompting very compelling in practice. \item We present double-grained prompt tuning (\ie TaI-DPT) to extract both coarse-grained and fine-grained embeddings for enhancing multi-label image recognition. % Experiments on multiple benchmarks show that TaI-DPT achieves comparable multi-label recognition accuracy against state-of-the-arts. \item The prompts learned by TaI-DPT can be easily combined with existing methods of prompting from images in an off-the-shelf manner, further improving multi-label recognition performance. \end{itemize} \begin{figure*} \centering \includegraphics[width=1.0\linewidth]{cvpr2023figbig.pdf} \vspace{-7.5mm} \caption{Training and testing pipeline of our proposed Text-as-Image (TaI) prompting, where we use text descriptions instead of labeled images to train the prompts. (a) During training, we use two identical text encoders from pre-trained CLIP to extract the global \& local class embeddings ($\mathbf{G} \& \mathbf{L}$) and overall \& sequential text embeddings ($\mathbf{h} \& \mathbf{H}$) respectively from the prompts and text description. The corresponding cosine similarity ($\mathbf{p} \& \mathbf{P}$) between the embeddings are guided by the derived pseudo labels with ranking loss. (b) During testing, we replace the input from text descriptions to images. The global and local class embeddings can discriminate target classes from global \& local image features ($\mathbf{f} \& \mathbf{F}$). The final classification results are obtained by merging the scores of the two branches.} \label{fig:pipeline} \vskip -0.1in \end{figure*} \section{Related Work} \label{sec:relatedworks} \subsection{Multi-Label Image Recognition} Multi-label image recognition~\cite{ye2020add,benbaruch2020asymmetric,chen2019multi,chen2019multi2,guo2019multi,gao2020multi,zhang2018multilabel,he2018reinforced,wei2015hcp} aims to recognize all the object categories~\cite{coco,voc2007} or concepts~\cite{nuswide} in an input image. To cope with multi-label images that are content-rich, various modules~\cite{wang2016cnn,chen2019learning} have been introduced to better represent the inter-class relationships and modern classification losses~\cite{benbaruch2020asymmetric,rankingloss} have been used to make model learning easier. To model the label dependencies, CNN-RNN~\cite{wang2016cnn} introduces recurrent neural networks, \eg, RNN and LSTM, to predict appeared classes in a sequential manner. \cite{chen2019multi,chen2019learning,ye2020add,wang2020fast} use graph convolution modules to learn the correlation between class labels. CHAMP~\cite{vaswani2022all} measures the severity of misclassification by building a domain-specific hierarchy tree according to the relation of categories, where each class are related to a tree node, to improve the robustness of the model. Albeit effective, these methods requires a considerable number of labeled images to let the models learn the category relationships sufficiently. While in data-limited or label-limited regimes, \eg, few-shot or partial-label data, it will be difficult for these models to learn well as expected. Specifically designed loss functions also struggle to obtain significant improvements when learning with limited data. \noindent \textbf{Multi-Label Recognition from Few-shot Samples.} To better exploit the small number of samples, LaSO~\cite{alfassy2019laso} synthesizes samples by manipulates the features of paired training images. Different ways of manipulating label sets are used to train the model, resulting in generalizable discriminative features. \cite{simon2022meta} introduces a meta-learning framework for better learning of past tasks and generalization to new tasks, and leverages the number of labels as useful information for learning. \noindent \textbf{Multi-Label Recognition from Partial-label Data.} Partial-label refers to the scenarios where some labels are unknown. \cite{durand2019learning} propose a normalized BCE loss to balance the proportion of known labels. \cite{chen2022structured} learns to complement unknown labels by utilizing within-image and cross-image semantic correlations. \cite{pu2022semantic} blends the representation of training images and class proxies to compensate the loss of information due to unknown labels. Albeit significant progress has been made, it remains a challenging issue for learning multi-label image recognition in image-limited or label-limited regimes. Built upon VL pre-trained models, this paper suggests to generate prompts from text descriptions instead of images, thereby offering a novel yet complementary perspective for handling low resource multi-label image recognition. \subsection{Prompt Tuning for Vision-Language Models} To transfer pre-trained knowledge to downstream tasks in data-limited settings, prompt tuning~\cite{coop,zhu2022prompt,vpt,cpt,nps,ge2022domain} has become a popular parameter-efficient way to achieve the goal, due to its flexibility and ease of use. CoOp~\cite{coop} learns the prompts by using (a few) annotated images of each class from target dataset. CoCoOp~\cite{cocoop} further proposes to improve CoOp~\cite{coop} by formulating the prompts in an image-conditional way to maintain better generalization to unseen classes. To avoid overfitting, ProGrad~\cite{zhu2022prompt} leverages predictions from zero-shot CLIP to regularize gradients in prompt learning process. TPT~\cite{shu2022test} suggests to optimize test-time prompts by promoting the consistency of augmented test images. ProDA~\cite{lu2022prompt} uses multiple pieces of prompts to estimate the distribution of classifier weights for better handle of varying visual features. DualCoOp~\cite{dualcoop} firstly adapts CLIP to multi-label image recognition with partially labeled data by learning pairs of positive and negative prompts for each class to ensure independent binary classification for each class. Albeit existing prompt tuning approaches have achieved significant improvements in downstream tasks, images as well as a portion of class labels are prerequisite to supervise the optimization of the learnable prompts. In this paper, we propose to treat texts as images in prompt tuning, which, compared to labeled images, are much easier to collect with existing caption datasets and modern search engines. Our proposed TaI-DPT surpasses zero-shot CLIP by a large margin, and can be combined with the prompts learned by existing methods of prompting from images to further boost recognition performance. \section{Proposed Method} \label{sec:method} In this section, we present our proposed Text-as-Image prompting, \ie, TaI prompting, for adapting pre-trained VL models to multi-label image recognition. Our TaI prompting uses only easily-accessed free-form texts as training data to learn effective prompts for downstream multi-label recognition tasks. To begin with, We present an overview of TaI prompting in \cref{methods0}. Then, we introduce our preparation of training texts in \cref{methods1}. We further explain the design of the double-grained prompt tuning (\ie, TaI-DPT) and the training and testing procedure in \cref{methods2}, and provide the loss function used to train the model in \cref{methods3}. Finally, we combine TaI-DPT with the existing methods of prompting from images to improve multi-label recognition performance further. CLIP is used to introduce our methdod. \subsection{Overview of Our Method} \label{methods0} ~\cref{fig:pipeline} illustrates the design of our proposed TaI-DPT framework, including the training and testing phases. During training, we learn prompts with only supervision from texts. Two identical copies of the text encoder ${\rm Enc_T}$ from the pre-trained CLIP are used to encode the prompts and text data, respectively. We introduce two sorts of trainable prompts (\ie, the global prompts and local prompts) to obtain global and regional class embeddings. A noun filtering strategy is used to generate classification pseudo-labels for each text description, which is applied to supervise the classification scores obtained by calculating the cosine similarity of class embeddings and text features. Only the parameters in prompts are optimized in the training phase, while the text encoders are both kept frozen. During testing, the class embeddings are obtained by encoding the two sets of learned prompts with the text encoder ${\rm Enc_T}$ as in training, while the other input source changes from text descriptions to test images. Pre-trained image encoder ${\rm Enc_I}$ from CLIP is used to extract global and dense features of each test image, then computing global and local classification scores with class embeddings generated by the global prompts and the local prompts via cosine similarity. The final classification result is obtained by fusing the global and local classification scores. In the following, we explain the details of the main components of our proposed method. \begin{figure*} \centering \includegraphics[width=1.0\linewidth]{cvpr2023_texthm.pdf} \caption{Visualization of correlations between the local class embedding $\boldsymbol L$ and sequential token feature from texts. % Each class embedding clearly correlates to words that describe the corresponding class (shown in highlight regions) rather than the global \textless EOS\textgreater~token. } \label{fig:texthm} \end{figure*} \begin{figure} \centering \includegraphics[width=1.0\linewidth]{cvpr2023_imghm.pdf} \caption{Visualization of correlations between the local class embedding $\boldsymbol L$ and dense image feature. The learned class embeddings can focus on the location of the object effectively } \vskip -0.18in \label{fig:imghm} \end{figure} \subsection{Preparation of Text Descriptions} \label{methods1} To obtain sufficient category information from the language that helps in image recognition, we have to ensure that: 1) the collected text descriptions should contain rich contents that describe a relatively complete scene of an image, and 2) the contents of all text descriptions need to cover the category set of the target dataset so that the prompts can learn the discriminative features of each class well and thus obtain better recognition performance. With an aim of ensuring reproducibility, we use captions from public image caption datasets (\emph{e.g.}, MS-COCO~\cite{coco}) and localized narratives from object detection datasets (\emph{e.g.}, OpenImages~\cite{openimages}) as our language data source, while avoiding the workloads associated with randomly crawling texts from the Internet in this paper. Note that although each caption is paired with a corresponding image and human-annotated labels, we only use the captions, and no information from the pictures and labels are disclosed during training. For a target multi-label recognition dataset $\mathcal{X}$ that has a category set $\mathcal{S}$ = \{$\rm s_1$, $\rm s_2$, $\rm s_3$, ..., $\rm s_{\textit {C}}$\}, where \textit{C} denotes the number of categories and $\rm s_i$ denotes particular class name like ``dog'', ``plane'', etc., we search for sentences that contain at least one class name $\rm s_i$ in $\mathcal{S}$. Since multiple words or phrases usually exist to represent the same meaning for each class, searching solely for exact match of category names in texts may lead to many false negatives in the obtained pseudo ground-truth labels, which is harmful to prompt tuning. Towards tackling this issue, we introduce a noun filter to map nouns with similar meanings into the corresponding class label. Specifically, we construct a synonym dictionary $\mathcal{D}$ by including common synonyms of each class name in the target dataset. If a word in a text description matches any synonym of a specific class name, it is considered to contain a description of that category. Several examples of synonyms are shown as follows: {\small\begin{verbatim} {'dog','pup','puppy','doggy'} {'person','people','man','woman','human'} {'bicycle','bike','cycle'} {'car','taxi','automobile'} {'boat','raft','dinghy'} ... \end{verbatim}} \noindent More details of the synonym dictionary $\mathcal{D}$ are provided in the \emph{Suppl}. Then we conduct noun filtration by the following steps. First, for each text description, we use the tokenizer and lemmatizer from NLTK~\cite{bird2009natural} to recover the stem of each word in the sentences. Next, for all keywords in $\mathcal{D}$, which contains all synonyms of the category set $\mathcal{S}$, we search in our language data source for sentences that contains at least one class name. For the text descriptions that do not match any synonym of any class name, we simply drop it away to ensure each piece of data has at least one concerned label. Finally, for each retained text description, we convert the class names it contains into binary pseudo-ground-truth vectors by setting classes that appear as positive and other classes as negative, following the order of class labels in the target dataset $\mathcal{X}$. The word-level filtered labels may not be precisely correct since our searching strategy mentioned above is rather simple considering the diversity of free-form texts, where complex paraphrases and misspellings that widely exist in the corpus are not fully addressed. However, such a simple noun filtration can guarantee reproducibility of this work and already leads to satisfactory results of our TaI, as will be shown. And our experiments also demonstrate that this simple and efficient data preparation lead to practical prompt tuning and compelling multi-label recognition accuracy. \subsection{Text-as-Image for Dual-grained Prompt Tuning} \label{methods2} Following \cite{coop}, a prompt is defined as: \begin{equation} \boldsymbol t_i = [\boldsymbol v_1, \boldsymbol v_2, \boldsymbol v_3, ..., \boldsymbol v_M, \boldsymbol s_i] \label{eq:0} \end{equation} where $i\in\{1, 2, ..., C\} $ is the class index, $\boldsymbol s_i$ denotes word embedding of the $i$-th class name $\rm s_i$. For $j \in\{1, \ldots, M\}$, $\boldsymbol v_j$ is a learnable word embedding whose dimension is the same as the dimension of normal word embeddings in the vocabulary. Just like in previous methods, \eg CoOp~\cite{coop}, the prompts are learned by maximizing the probability of classifying each image into its ground-truth class: \begin{equation} p(y = i| \boldsymbol x) = \frac{\exp(\langle {\rm Enc_T}(\boldsymbol t_i),{\rm Enc_I}(\boldsymbol x)\rangle /\tau)}{{\textstyle \sum_{j=1}^{C}}\exp(\langle{\rm Enc_T}(\boldsymbol t_j), {\rm Enc_I}(\boldsymbol x)\rangle /\tau)} \label{eq:1} \end{equation} \noindent where $ \boldsymbol x $ denotes the image and $\langle \cdot, \cdot \rangle$ calculates the cosine similarity. After large-scale pre-training with image-text contrastive loss, text features have been well-aligned to the image features of the same semantic meanings. Therefore, based on the aligned VL representation, we advocate considering the feature of a piece of text description that describes a specific category, as an alternative to an image feature. Given a piece of text description, optimizing the similarity between its feature representation produced by a VL model and some class embeddings is considered, for guiding the learning of prompts towards achieving categorical discriminative information. Apart from using the global sentence representation (\ie, the coarsest-grained text feature), we find that the sequential feature of word tokens from CLIP also possesses rich fine-grained information which is very similar to the region feature of dense image feature. In CLIP~\cite{clip}, cosine similarity between global image features, obtained by visual attention pooling, and global text features, obtained by projecting the feature of the last \textless EOS\textgreater~token, are directly supervised with contrastive loss. In general, the global feature is sufficient for single-label classification because the target object usually is prominent in the picture. However, in multi-label recognition, the global feature is usually dominated by major objects, suppressing the recognition of non-significant objects concurrently existing in the image. Thus, it motivates us to explore fine-grained features and avoid the domination of the overly prominent object. To achieve this goal, we propose double-grained prompt tuning (\ie, TaI-DPT) that uses two sets of prompts to handle global (\ie, the coarsest-grained level) and local (\ie, the fine-grained level) features, respectively, in two parallel branches. The global prompts achieve discrimination by learning from the global feature directly learned in CLIP, while the local prompt learns from localized features. Formally, the double-grained prompt is defined as follows: \begin{equation} \begin{split} \boldsymbol t^G_i = [\boldsymbol v_1, \boldsymbol v_2, \boldsymbol v_3, ..., \boldsymbol v_M, \boldsymbol s_i], \\ \boldsymbol t^L_i = [\boldsymbol v_1', \boldsymbol v_2', \boldsymbol v_3', ..., \boldsymbol v_M', \boldsymbol s_i], \label{eq:dpt} \end{split} \end{equation} \noindent where $\boldsymbol v_j$ and $\boldsymbol v_j', j \in\{1, \ldots, M\}$ are learnable embeddings that are concatenated with word embedding $ \boldsymbol s_i$ of the $i$-th class to obtain the global prompt $\boldsymbol t^G_i$ and local prompt $\boldsymbol t^L_i$, respectively. The sequences in Eq.~\eqref{eq:dpt} are fed to a copy of the text encoder $\rm Enc_T$ of CLIP to generate global and local class embeddings for each class, \ie $\boldsymbol G_i = {\rm Enc_T}(\boldsymbol t^G_i)$ and $\boldsymbol L_i = {\rm Enc_T}(\boldsymbol t^L_i)$, $\boldsymbol G =\{\boldsymbol G_i\}^C_{i=1}$ and $\boldsymbol L =\{\boldsymbol L_i\}^C_{i=1}$ are encouraged to be correlated with global and local features, respectively. Note that the proposed double-grained prompts are different from dual prompts~\cite{dualcoop}, which include a pair of contrastive positive and negative prompts for each class (More discussion about the differences between our method and DualCoOp is provided in the \emph{Suppl}). To preserve the fine-grained region features for the input image, we maintain the feature map before attention pooling layer of CLIP. As for the input text description, we preserve the sequential token features of the entire sentence instead of only the \textless EOS\textgreater token features. So we have: \begin{equation} \begin{split} \{\boldsymbol f, \boldsymbol F \} = {\rm Enc_I}(\boldsymbol x), \\ \{\boldsymbol h, \boldsymbol H \} = {\rm Enc_T}(\boldsymbol r), \label{eq:enct} \end{split} \end{equation} \noindent where $\boldsymbol r$ denotes a piece of training text description. $\boldsymbol f, \boldsymbol h \in \mathbb{R}^D$ are the extracted global image and text features. $\boldsymbol F \in \mathbb{R}^{N_1\times D}$ and $\boldsymbol H \in \mathbb{R}^{N_2\times D}$ are the flattened dense image features and sequential token features, respectively, where $ N_1 = H \times W $ denotes the flattened spatial dimension of visual feature and $ N_2$ denotes the length of text tokens. Then, the global and local similarities are computed by: \begin{equation} \begin{split} \boldsymbol p_i = \langle \boldsymbol u, \boldsymbol G_i \rangle , \boldsymbol P_{ij} = \langle \boldsymbol U_j, \boldsymbol L_i \rangle \label{eq:cos} \end{split} \end{equation} \noindent where $\boldsymbol u$ denotes either language feature $\boldsymbol h$ in training or visual feature $\boldsymbol f$ in testing, and $\boldsymbol U$ denotes $\boldsymbol H$ or $\boldsymbol F$ coordinately. Information in local branch $\boldsymbol P$ (visualized in \cref{fig:texthm} and \cref{fig:imghm}) can be aggregated in a spatially weighted manner: \begin{equation} \begin{split} \boldsymbol p_{i}' =\textstyle \sum_{j=1}^{N} \displaystyle \frac{\exp(\boldsymbol P_{ij} / \tau_s)}{ {\textstyle \sum_{j=1}^{N}}\exp(\boldsymbol P_{ij} / \tau_s) } \cdot \boldsymbol P_{ij} \label{eq:spatial} \end{split} \end{equation} \noindent where $\tau_s$ accommodates the extent of focusing on a specific location. $\boldsymbol p_i$ and $\boldsymbol p_i'$ are optimized by the loss terms $\mathcal{L}_{global}$ and $\mathcal{L}_{local}$, respectively, which we will discuss in ~\cref{methods3}. And in the testing phase, $\boldsymbol p$ and $\boldsymbol p'$ are combined to obtain the final classification score. The visualization results in \cref{fig:texthm} and \cref{fig:imghm} show that the learned local class embedding $\boldsymbol L$ can focus on each specific location where corresponding class appears, both in text descriptions and images, even if the fine-grained visual and language features are not explicitly supervised in the training of CLIP. \subsection{Learning Objective} \label{methods3} We briefly discuss the loss terms used during the training of TaI-DPT. The overall learning objective is defined as $\mathcal{L} =\mathcal{L}_{global}+\mathcal{L}_{local}$, where $\mathcal{L}_{global}$ and $\mathcal{L}_{local}$ are loss terms for global text embedding and local text tokens, respectively. We adopt the ranking loss~\cite{rankingloss} to measure the discrepancy between classification scores and ground-truth labels, instead of a commonly used binary cross-entropy loss. The binary cross-entropy loss is generally accompanied with a sigmoid function $\boldsymbol \sigma(x) = 1/(1 + \exp(-x)) $ to convert model outputs to probabilities. Nevertheless, we observe that the value of cosine similarities between image and text CLIP features $\boldsymbol p$ are not evenly distributed on either side of 0. Directly constraining the probability $\boldsymbol \sigma(\boldsymbol p)$ makes the optimization more difficult in this case, and this is why we employ a different ranking loss function~\cite{rankingloss}. There may exist other options, \eg, the asymmetric loss as in ~\cite{dualcoop}. Specifically, $\mathcal{L}_{global}$ and $\mathcal{L}_{local}$ are formulated as follows: \begin{equation} \begin{split} \mathcal{L}_{global} &\!=\! { \sum\nolimits_{i \in \{{c^+}\}}} { \sum\nolimits_{j\in \{{c^-}\}}} \max(0, m \!-\!\boldsymbol p_i \!+\! \boldsymbol p_j),\\ \mathcal{L}_{local} &\!=\! { \sum\nolimits_{i \in \{{c^+}\}}} { \sum\nolimits_{j \in \{{c^-}\}}} \max(0, m \!-\! \boldsymbol p_i'\!+\! \boldsymbol p_j'), \label{eq:rl} \end{split} \end{equation} \noindent where $\boldsymbol p$ and $\boldsymbol p'$ are global and aggregated local similarities described in \cref{methods2}, $m$ is the margin controlling how much higher the similarity score with the positive classes is than with the negative classes. During training, we minimize the overall objective $\mathcal{L}$ with frozen text encoders, by optimizing the global and local prompts. \begin{figure} \centering \includegraphics[width=0.9\linewidth]{cvpr2023_ensemble.pdf} \caption{Our learned double-grained prompt tuning is easy to combine with existing prompt tuning methods with ensemble.} \label{fig:ensemble} \end{figure} \subsection{Incorporating with Prompting from Images} \label{methods4} Though our TaI-DPT is very different from existing methods of prompting from images, it is also complementary to them. To show this, we utilize an off-the-shelf prompt ensemble strategy to combine our TaI-DPT with existing methods in this section. As illustrated in \cref{fig:ensemble}, using CoOp~\cite{coop} as an example, we can simply combine the scores of CoOp~\cite{coop} and that of our TaI-DPT in a weighted sum manner. In particular, our TaI-DPT can be integrated with CoOp~\cite{coop} when a few annotated images are provided and integrated with DualCoOp~\cite{dualcoop} when partially labeled training data are available. We ensemble prompts by fusing the predicted scores, rather than averaging the class embeddings generated by different prompts, since the image encoder used in different methods may be different (\eg we conduct our experiments with ResNet50, while DualCoOp uses ResNet101 for partial-label prompting). So ensembling with the classification score is more convenient. In \cref{e:3}, we also empirically show that our prompt ensemble strategy is effective in advancing multi-label recognition performance in the few-shot and partially labeled settings. \begin{figure*}[h] \begin{minipage}[t]{0.65\linewidth} \centering \includegraphics[width=1.0\textwidth]{cvpr2023_fewshot_margin.pdf} \vspace{-8mm} \caption{Comparison of different methods in few-shot multi-label recognition on VOC2007 and MS-COCO. Our zero-shot TaI-DPT can achieve comparable results with methods trained by 16-shot labeled image samples. And learned prompt ensemble proofs the complementarity between images and texts.} \vspace{-6mm} \label{fig:fewshot} \end{minipage} \hfill \begin{minipage}[t]{0.32\linewidth} \centering \includegraphics[width=1.0\textwidth]{result_ablation.png} \vspace{-6mm} \caption{Ablation experiment on number of texts and performance of TaI prompting on VOC2007.} \vspace{-6mm} \label{fig:ablation} \end{minipage} \end{figure*} \section{Experiments} \label{sec:experiments} \subsection{Implementation Details} \label{e:1} \noindent \textbf{Architecture.} In our experiments, we adopt CLIP ResNet-50~\cite{clip} as the visual encoder, and use the corresponding CLIP Transformer as the text encoder. During training, the parameters of both the two encoders are kept frozen, and only learnable prompts are optimized. \noindent \textbf{Learnable Prompts.} Our learnable prompts are shared among classes of all datasets. Class-specific prompting~\cite{coop} (\ie, an individual set of parameters for each category) has also been explored, but brings limited benefits. Hence, we adopt the shared prompts and initialize the value of each parameter with the Gaussian noise sampled from $\mathcal{N}(0, 0.02)$. In our experiments, the length of both the global prompts and local prompts are set to $M$ = 16, while a longer sequence brings trivial improvements. \noindent \textbf{Datasets.} To evaluate our TaI-DPT, we conduct the experiments on VOC2007~\cite{voc2007}, MS-COCO~\cite{coco}, and NUS-WIDE~\cite{nuswide}. VOC2007 contains 20 common categories, and following~\cite{chen2019multi,chen2019learning,dualcoop}, we form the training/test set based on the official \texttt{trainval}/\texttt{test} split (5,011 images/4,952 images). MS-COCO includes 80 categories, and following the official split, we take 82,081 images to form the training set and 40,504 images to form the validation set. NUS-WIDE includes 81 concepts, which have certain inclusion relationships. We adopt its test set (107,859 images) to evaluate our method. For zero-shot experiments in \cref{e:2}, the training sets of the datasets are not used, and we use only text data to learn the prompts as mentioned in \cref{methods1}. Besides, for VOC2007 and MS-COCO, the language data sources are captions from MS-COCO. For NUS-WIDE, we introduce localized narratives from OpenImages~\cite{openimages}, which have a broader range of content, to cover all the concepts in NUS-WIDE. In \cref{e:3} and \cref{e:4}, for each dataset, the corresponding training data is used to conduct the experiments of partial-label and few-shot multi-label classification. \noindent \textbf{Training Details.} We adopt SGD optimizer to learn our prompts, and the training epochs is set to 20 for all datasets. The learning rates for MS-COCO, VOC2007, and NUS-WIDE are empirically initialized with 1e-4, 1e-4, and 1e-3, and decay by the cosine annealing rule during training. For ranking loss, we choose $m=1$, and scale the $\boldsymbol p$ and $\boldsymbol p'$ by a factor of 4. $\tau_s$ is set as 0.02 via validation. \begin{table} [!h] \centering \caption{Comparison with zero-shot methods on VOC2007, MS-COCO, and NUS-WIDE. Our proposed TaI-DPT outperforms CLIP~\cite{clip} by a large margin on all datasets.} \vspace{-2mm} \setlength{\tabcolsep}{1.8mm} \begin{tabular}{c|c|ccc} \toprule Method & DPT & VOC2007 & MS-COCO & NUSWIDE \\ \hline \multirow{2}{*}{ZSCLIP} & \ding{55} & 76.2 & 47.3 & 36.4\\ & \checkmark & 77.3 & 49.7 & 37.4 \\ \hline \multirow{2}{*}{TaI} & \ding{55} & 86.0 & 61.1 & 44.9 \\ & \checkmark & \textbf{88.3} & \textbf{65.1} & \textbf{46.5} \\ \bottomrule \end{tabular} \vskip -0.1in \label{tab:zclip} \end{table} \begin{table*} [!ht] \centering \caption{Results of integrating our TaI-DPT with partial-label multi-label recognition method based on pre-trained CLIP. Our approach further improves the frontier performance of DualCoOp~\cite{dualcoop}. $*$ indicates the results of our reproduction.} \vspace{-2mm} \setlength{\tabcolsep}{2.3mm} \begin{tabular}{c|c|ccccccccc|c} \toprule Datasets & Method & 10\% &20\% & 30\% & 40\% & 50\% & 60\% & 70\% & 80\% & 90\% & Avg. \\ \midrule \multirow{4}{*}{MS-COCO} & SARB~\cite{pu2022semantic} & 71.2 & 75.0 & 77.1 & 78.3 & 78.9 & 79.6 & 79.8 & 80.5 & 80.5 & 77.9\\ & DualCoOp~\cite{dualcoop} & 78.7 & 80.9 & 81.7 & 82.0 & 82.5 & 82.7 & 82.8 & 83.0 & 83.1 & 81.9\\ & DualCoOp* & 81.0 & 82.3 & 82.9 & 83.4 & 83.5 & 83.9 & 84.0 & 84.1 & 84.3 & 83.3\\ & +TaI-DPT & \textbf{81.5} & \textbf{82.6} & \textbf{83.3} & \textbf{83.7} & \textbf{83.9} & \textbf{84.0} & \textbf{84.2} & \textbf{84.4} & \textbf{84.5} & \textbf{83.6}\\ \hline \multirow{4}{*}{PascalVOC 2007} & SARB~\cite{pu2022semantic} & 83.5 & 88.6 & 90.7 & 91.4 & 91.9 & 92.2 & 92.6 & 92.8 & 92.9 & 90.7 \\ & DualCoOp~\cite{dualcoop} & 90.3 & 92.2 & 92.8 & 93.3 & 93.6 & 93.9 & 94.0 & 94.1 & 94.2 & 93.2\\ & DualCoOp* & 91.4 & 93.8 & 93.8 & 94.3 & 94.6 & 94.7 & 94.8 & 94.9 & 94.9 & 94.1\\ & +TaI-DPT & \textbf{93.3} & \textbf{94.6} & \textbf{94.8} & \textbf{94.9} & \textbf{95.1} & \textbf{95.0} & \textbf{95.1} & \textbf{95.3} & \textbf{95.5} & \textbf{94.8}\\ \hline \multirow{2}{*}{NUS-WIDE} & DualCoOp* & 54.0 & 56.2 & 56.9 & 57.4 & 57.9 & 57.9 & 57.6 & 58.2 & 58.8 & 57.2\\ & +TaI-DPT & \textbf{56.4} & \textbf{57.9} & \textbf{57.8} & \textbf{58.1} & \textbf{58.5} & \textbf{58.8} & \textbf{58.6} & \textbf{59.1} & \textbf{59.4} & \textbf{58.3}\\ \bottomrule \end{tabular}\vspace{-3mm} \label{tab:partial} \end{table*} \subsection{Comparison with Zero-Shot Methods} \label{e:2} To demonstrate the effectiveness of our proposed TaI and DPT, we first compare it with the zero-shot CLIP (ZSCLIP). For fair comparison, we also introduce the DPT to ZSCLIP. Specifically, we adopt two identical default prompts ``a photo of a [CLASS]'' to separately deal with global and local features as DPT does. \Cref{tab:zclip} lists the comparison results on VOC2007~\cite{voc2007}, MS-COCO~\cite{coco}, and NUS-WIDE\cite{nuswide} datasets. From the table, our TaI prompting surpasses ZSCLIP by a large margin of 9.8\%, 13.8\%, and 8.5\% mAP on VOC2007, MS-COCO, and NUS-WIDE, respectively, showing the effectiveness of our TaI. Furthermore, after training with fine-grained token features extracted from texts, our proposed DPT demonstrates a more powerful capability of discriminating local features than the default hand-crafted prompts and single global prompts. \begin{table}[h] \centering \caption{Comparison with existing multi-label few-shot learning methods on MS-COCO. The evaluation is based on mAP for zero-shot, 1-shot and 5-shot with 16 novel classes.} \vspace{-2mm} \setlength{\tabcolsep}{3.8mm} \begin{tabular}{c|ccc} \toprule Method & \textbf{0-shot} & 1-shot & 5-shot \\ \midrule LaSO~\cite{alfassy2019laso} & - & 45.3 & 58.1 \\ ML-FSL~\cite{simon2022meta} & - & 54.4 & 63.6 \\ TaI-DPT & 59.2 & - & - \\ \bottomrule \end{tabular}\vspace{-5mm} \label{tab:laso} \end{table} \subsection{Comparison with Few-Shot Methods} \label{e:3} We further compare with multi-label few-shot learning methods to verify the effectiveness of our TaI-DPT. In contrast to the well-studied single-label few-shot classification problem, few works tackle the multi-label few-shot scenario. Existing methods \cite{alfassy2019laso,simon2022meta} often deploy models trained on seen classes to few-shot novel classes. In \Cref{tab:laso}, we compare our zero-shot TaI-DPT to few-shot methods on 16 novel classes (we refer readers to \cite{alfassy2019laso} for details about data split). Our TaI-DPT is comparable to the methods trained on 5-shot samples. Besides, we consider a new multi-label few-shot setting where all the classes are regarded as novel classes. We select 1, 2, 4, 8, and 16-shot samples for each category following the strategy in \cite{alfassy2019laso}. For fair comparison, we train CoOp~\cite{coop} and our TaI in the same settings, and we also extend them with DPT for a more comprehensive comparison. For CoOp-DPT, we set two sets of learnable prompts, to deal with global and local features, respectively. The results are illustrated in \cref{fig:fewshot}. One can see that, even without any image information regarding novel classes, our TaI can achieve comparable results to CoOp trained on 16-shot. Similar trends with the MS-COCO dataset and the DPT setting support our observation that the discriminative feature of text data can be used as images for prompting. Moreover, benefiting from the flexibility of prompts, we can easily integrate our TaI-DPT with CoOp-DPT by utilizing prompt ensembles. As illustrated in \cref{fig:fewshot}, though CoOp-DPT has achieved a high accuracy, combining our prompts learned with text data still brings further improvement on recognition performance. This also proves that texts and images are complementary to each other to some extent. \subsection{Integration with Partially Labeled Methods} \label{e:4} Following~\cite{dualcoop}, we conduct the experiments of multi-label recognition with partial-labeled images. We reproduce DualCoOp on partial-labeled VOC2007 and MS-COCO with the same experimental setting as reported (reproduced results are marked with *) and explore the enhancement brought by integration with TaI-DPT. The results are reported in \Cref{tab:partial}. With no prior knowledge from pre-trained models, previous forefront method like SARB~\cite{pu2022semantic} struggles to learn from incomplete labels. While DualCoOp~\cite{dualcoop} achieves promising performance by prompting with images, TaI-DPT can still bring further improvements. \subsection{Ablation Study} \label{e:ablation} To thoroughly investigate the effect of each component, we conduct a series of ablation studies on the quantity of texts, training loss, ensemble weight, and texts \emph{v.s.} images for prompting. More details are shown in the \emph{Suppl}. \noindent\textbf{Quantity of texts. } Here, we mainly discuss the the effect of the number of text descriptions used in training on the performance of TaI-DPT on VOC2007. Following the data preparation procedure in \cref{methods1}, we end up with a total number of 66087 pieces of text that contain descriptions for 20 categories involved in VOC2007. We test the performance of TaI-DPT with different numbers of randomly selected texts, and the results are shown in \cref{fig:ablation}. When no collected texts are available, 80 templates of hand-crafted prompts from \cite{clip}, like ``a cropped photo of a [CLASS]'', are used for training (all templates are shown in the \emph{Suppl}), and each template sentence correlates with one positive label corresponding to the class name inserted in [CLASS]. The increasing number of texts gradually forms a complete description of target categories, and the relationship between classes is also better characterized, which results in ascending performance. \section{Conclusion} \label{sec:conclusion} In this paper, we propose a new view of treating texts as images in prompt tuning (\ie TaI), which learns the prompt from discriminative features of text descriptions. Compared to prior prompt tuning methods trained with images, our TaI benefits from the easy accessibility of scalable content-rich texts, which enables prompt tuning for vision tasks (\eg, multi-label image recognition) even without downstream image data. Double-grained prompting is further introduced to utilize both the global and fine-grained features for better multi-label recognition ability. Nonetheless, when few-shot image samples or partial-labeled images are available, our TaI-DPT can conveniently integrate with existing prompting methods. Experiments on MS-COCO, VOC2007, and NUS-WIDE show the validity of our proposed method. {\small \bibliographystyle{ieee_fullname}

BkiUaO7xK02iP15vdlEc

\section{Introduction}\label{Sec: Intro} Given a four dimensional $\mathcal{N}=2$ superconformal field theory (SCFT) with marginal deformations, it is interesting to write down its weakly coupled gauge theory descriptions. In such descriptions, gauge couplings take the role of the coordinate on the conformal manifold and the gauge theory is interpreted as conformal gauging of various strongly coupled isolated SCFTs \cite{Argyres:2007cn}. It is quite common to find more than one weakly coupled descriptions, and they are S-dual to each other as the gauge couplings are often related by $e.g.$, $\tau\propto -{1\over \tau}$. Finding all weakly coupled gauge theory descriptions is often very difficult for a generic strongly coupled $\mathcal{N}=2$ SCFT. The above questions are solved for class ${\cal S}$ theory where the Coulomb branch spectrum has integral scaling dimensions: one represents our theory by a Riemann surface $\Sigma$ with \textit{regular} singularity so that S-duality is interpreted as different degeneration limits of $\Sigma$ into three punctured sphere \cite{Gaiotto:2009we}; Once a degeneration is given, the remaining task is to identify the theory corresponding to a three punctured sphere, as well as the gauge group associated to the cylinder connecting those three punctured spheres. In class ${\cal S}$ theory framework, $\Sigma$ appears naturally as the manifold on which we compactify 6d $(2,0)$ theory. Certain $\cN=2$ SCFTs and their S-duality can be studied via geometric engineering, see \cite{DelZotto:2015rca}. There is a different type of $\mathcal{N}=2$ SCFT called Argyres-Douglas (AD) theories \cite{Argyres:1995jj, Xie:2012hs}. The Coulomb branch spectrum of these theories has fractional scaling dimension and they also admit marginal deformations. Again, one can engineer such AD theories by using $(2,0)$ theory on Riemann spheres $\Sigma_{g = 0}$ with \textit{irregular} singularity\footnote{We will henceforth drop the subscript $g = 0$ in what follows to denote the Riemann sphere}. Since we can not interpret the exact marginal deformations as the geometric moduli of $\Sigma$, there is no clue how weakly coupled gauge theory descriptions can be written down in general, besides some simple cases where one can analyze the Seiberg-Witten curve directly \cite{Buican:2014hfa}. It came as quite a surprise that one can still interpret S-duality of $A_{N-1}$-type AD theory in terms of an auxiliary punctured Riemann surface \cite{Xie:2017vaf}. The main idea of \cite{Xie:2017vaf} is giving a map from $\Sigma$ with irregular singularities to a punctured Riemann sphere $\Sigma^{'}$, and then find weakly coupled gauge theory as the degeneration limit of $\Sigma^{'}$ into three punctured sphere. The main purpose of this paper is to generalize the idea of \cite{Xie:2017vaf} to AD theories engineered using general 6d $(2,0)$ theory of type $\mfg$. The major results of this paper are \begin{itemize} \item We revisit the classification of irregular singularity of class $(k, b)$ in \cite{Xie:2012hs, Wang:2015mra}: \be \Phi \sim \frac{T_k}{z^{2+\frac{k}{b}}} + \sum_{-b \leq l < k} \frac{T_l}{z^{2+\frac{l}{b}}} \ee and find new irregular singularity which gives SCFT in four dimensions. Briefly, they are the configuration for which \begin{enumerate} \item[(\rnum{1})] $T_k$ is regular-semisimple, whose classification was studied in \cite{Wang:2015mra}. \item[(\rnum{2})] The new cases are that $T_k$ is semisimple. \item[(\rnum{3})] Fix a pair $(k,b)$ and type $T_k$, we can consider the degeneration of $T_k$ and the crucial constraint is that the corresponding Levi subalgebra has to be the same for $T_l$, $l > -b$. \end{enumerate} \item We successfully represent our theory by an auxiliary punctured sphere from the data defining our theory from 6d (2,0) SCFT framework, and we then find weakly coupled gauge theory descriptions by studying degeneration limit of new punctured sphere. \end{itemize} For instance, we find that for $\mfg = D_N$, $b = 1$ and large $k$ and all coefficient matrices regular semisimple, one typical duality frame looks like \vspace{6pt} \begin{adjustwidth}{0cm}{} \begin{tikzpicture \draw (16.5,0) node[anchor=west] (1) {$\cT_{N-1}$,}; \draw (13.8,0.6) node[anchor=west] (2) {$SO(2N-2)$}; \draw (12.4,0) node[anchor=west] (3) {$ \cT_{N-2}$}; \draw (10.3,0.6) node[anchor=west] (4) {$ \dots \dots$}; \draw (8.9,0) node[anchor=west] (5) {$ \cT_3$}; \draw (7.0,0.6) node[anchor=west] (6) {$SO(6)$}; \draw (5.6,0) node[anchor=west] (7) {$ \cT_2$}; \draw (3.6,0.6) node[anchor=west] (8) {$SO(4)$}; \draw (2.2,0) node[anchor=west] (9) {$\cT_1$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) (7) edge[sedge] (8) (8) edge[sedge] (9) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} where ${\cT}_i$ is given by $D_{i+1}$ theory $\pbra{\Rnum{3}_{k,1}^{[1; 2i]^{\times (k+1)}, [1^{i+1}; 0]}, [1^{2i+2}]}$. The notation we use to label the AD theories is \be \pbra{\Rnum{3}_{k,b}^{ \{ \mfl_i \} },\ \ \ Q}, \ee where $\Rnum{3}$ means type-\Rnum{3} singularity in the sense of \cite{Xie:2012hs}, and $\{ \mfl_i \}$ are Levi subalgebra for each coefficient matrix $T_i$ and $Q$ is the label for regular puncture. Each notation will be explained in the main text. The same theory has a second duality frame, given by \vspace{6pt} \begin{adjustwidth}{0cm}{} \begin{tikzpicture \draw (16.5,0) node[anchor=west] (1) {${\widehat \cT}'_{N}$,}; \draw (13.8,0.6) node[anchor=west] (2) {$SU(N)$}; \draw (12.4,0) node[anchor=west] (3) {$ {\widehat \cT}_{N-1}$}; \draw (10.3,0.6) node[anchor=west] (4) {$ \dots \dots$}; \draw (8.9,0) node[anchor=west] (5) {$ {\widehat \cT}_3$}; \draw (7.0,0.6) node[anchor=west] (6) {$SU(3)$}; \draw (5.6,0) node[anchor=west] (7) {$ {\widehat \cT}_2$}; \draw (3.6,0.6) node[anchor=west] (8) {$SU(2)$}; \draw (2.2,0) node[anchor=west] (9) {${\widehat \cT}_1$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) (7) edge[sedge] (8) (8) edge[sedge] (9) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} where ${\widehat \cT}_i$, $1 \leq i \leq N-1$ is given by $\pbra{\Rnum{3}_{k,1}^{[i,1]^{\times (k+1)}, [1^{i+1}]}, [1^{i+1}]}$, and ${\widehat \cT}'_{N}$ is given by $\pbra{\Rnum{3}_{k,1}^{[N; 0]^{\times (k+1)}, [1^{2N}; 0]}, Q}$. An unexpected corollary is that the quiver with $SO(2n)$ gauge groups are dual to quivers with $SU(n)$ gauge groups, and each intermediate matter content does not have to be engineered from the same $\mfg$-type in 6d. Similar feature appears when $\mfg = E_{6,7,8}$, as will be demonstrated in this work. The paper is organized as follows. In section \ref{Sec: PunctureClass} we briefly review regular punctures and their associated local data, and then proceed to classify (untwisted) irregular punctures for $\mfg = D_N$ and $\mfg = E_{6,7,8}$ theories. We give relevant Coulomb branch spectrum. The map from $\Sigma$ to $\Sigma'$ is described in Section \ref{Sec: mappToRiemannSphere}. Section \ref{Sec: S-duality-DN} is devoted to study the duality frames for $D_N$ theories. We consider both untwisted and twisted theories. Finally, we study S-duality frame for $E_{6,7,8}$ theories in section \ref{Sec: S-duality-E}. We conclude in section \ref{Sec: conclusion}. \section{SCFTs from M5 branes}\label{Sec: PunctureClass} M5 brane compactifications on Riemann surface $\Sigma$ provide a large class of $\cN = 2$ superconformal theories in four dimensions. To characterize the theory, one needs to specify a Lie algebra $\mfg$ of ADE type, the genus $g$ of the Riemann surface, and the punctures on $\Sigma$. Regular punctures are the loci where the Higgs field $\Phi$ has at most simple poles; while irregular punctures are those with $\Phi$ having higher order poles. The class $\cS$ theories developed in \cite{Gaiotto:2009we} are SCFTs with $\Sigma$ of arbitrary genus and arbitrary number of regular punctures, but no irregular puncture. Later, it was realized that one may construct much larger class of theories by utilizing irregular punctures \cite{Nanopoulos:2010zb, Bonelli:2011aa, Xie:2012hs}. However, in this case the Riemann surface is highly constrained. One may use either \begin{enumerate} \item[$\bullet$] A Riemann sphere with only one irregular puncture at the north pole; \item[$\bullet$] A Riemann sphere with one irregular puncture at the north pole and one regular puncture at the south pole. \end{enumerate} where the genus $g = 0$ condition is to ensure the $\mC^*$ action on the Hitchin system, which guarantees $U(1)_r$ R-symmetry and superconformality. This reduces classification of theories into classification of punctures. In this section we revisit the classification and find new irregular singularity which will produce new SCFTs. \subsection{Classification of punctures} \subsubsection{Regular punctures}\label{subsubsec: regular puncture} Near the regular puncture, the Higgs field takes the form \be \Phi \sim \frac{\Lambda}{z} + M, \label{regularLocal} \ee and classification of regular puncture is essentially classification of nilpotent orbits. The puncture itself is associated with the \textit{Nahm label}, while $\Lambda$ is given by the \textit{Hitchin label}. They are related by the \textit{Spaltenstein map}. We now briefly review the classification. \bigskip {\bf Lie algebra} $\mfg = A_{N-1}.$ The nilpotent orbit is classified by the partition $Y=\bbra{n_1^{h_1},\ldots, n_r^{h_r}}$, where $n_i$ are column heights, and the flavor symmetry is \cite{Gaiotto:2009we, Chacaltana:2010ks} \begin{equation} G_{\rm flavor}=S\pbra{\prod_{i=1}^rU(h_i)}. \end{equation} The spectral curve is \begin{equation} \det(x-\Phi(z))=0\rightarrow x^N+\sum_{i=2}^N\phi_i(z) x^{N-i}=0. \end{equation} Each $\phi_i$ is the meromorphic differentials on the Riemann surface, living in the space $H^0(\Sigma, K^{\otimes i})$. The order of pole $p_i$ of the regular puncture at $\phi_i$ determines the local dimension of Coulomb branch spectrum with scaling dimension $\Delta = i$. It is given by $p_i=i-s_i$ where $s_i$ is the height of $i$-th box of the Young Tableaux $Y$; here the labeling is row by row starting from bottom left corner. \bigskip {\bf Lie algebra} $\mfg = D_{N}.$ We now review classification of regular punctures of $D_N$ algebra. For a more elaborated study, the readers may consult \cite{Chacaltana:2012zy, Chacaltana:2011ze}. A regular puncture of type $\mfg = D_N$ is labelled by a partition of $2N$, but not every partition is valid. It is a requirement that the even integers appear even times, which we will call a $D$\textit{-partition}. Moreover, if all the entries of the partition are even, we call it \textit{very even} D-partition. The very even partition corresponds to two nilpotent orbit, which we will label as $\cO^{\Rnum{1}}_{\bbra{\cdot}}$ and $\cO^{\Rnum{2}}_{\bbra{\cdot}}$. We again use a Young tableau with decreasing column heights to represent such a partition, and we call it a \textit{Nahm partition}. Given a Nahm partition, the residual flavor symmetry is given by \be G_{\rm flavor} = \prod_{h\ {\text{odd}}} \text{Spin}(n^h) \times \prod_{h\ {\text{even}}} {\rm Sp}\pbra{n^h}. \ee We are interested in the contribution to the Coulomb branch dimension from each puncture. When $\mfg = A_{N-1}$ case we simply take transpose and obtain a \textit{Hitchin partition} \cite{Chacaltana:2010ks}. However, for $\mfg = D_N$ the transpose does not guarantee a valid Young tableaux. Instead it must be followed by what is called \textit{D-collapse}, denoted as $(\cdot)_D$, which is described as follows: \begin{enumerate} \item[(\rnum{1})] Given a partition of $2N$, take the longest even entry $n$, which occurs with odd multiplicity (if the multiplicity is greater than 1, take the last entry of that value), then picking the largest integer $m$ which is smaller than $n-1$ and then change the two entries to be $(n,m) \rightarrow (n-1, m+1)$. \item[(\rnum{2})] Repeat the process for the next longest even integer with odd multiplicity. \end{enumerate} The \textit{Spaltenstein map} $\mfS$ of a given partition $d$ is given by $(d^{\mathsf{T}})_D$ and we obtain the resulting \textit{Hitchin partition} or \textit{Hitchin diagram}\footnote{Unlike \cite{Chacaltana:2011ze}, here we define the Hitchin diagram to be the one after transpose. So that when reading Young diagram one always reads column heights.}. The Spaltenstein map is neither one-to-one nor onto; it is not an involution as the ordinary transpose either. The set of Young diagram where $\mfS$ is an involution is called \textit{special}. More generally, we have $\mfS^3 = \mfS$. Given a regular puncture data, one wishes to calculate its local contribution to the Coulomb branch. We begin with the special diagram. Using the convention in \cite{Chacaltana:2011ze}, we can construct the local singularity of Higgs field in the Hitchin system as \eqref{regularLocal} where $\Lambda$ is an $\mathfrak{so}(2N)$ nilpotent matrix associated to the Hitchin diagram and $M$ is a generic $\mathfrak{so}(2N)$ matrix. Then, the spectral curve is identified as the SW curve of the theory, which takes the form \be \det(x - \Phi(z)) = x^{2N} + \sum_{i = 1}^{N-1}x^{2(N-i)} \phi_{2i}(z) + {\tilde \phi}(z)^2. \label{D_SW curve} \ee We call ${\tilde \phi}$ the Pfaffian. This also determines the order of poles for each coefficient $\phi_{2i}$ and ${\tilde \phi}$. We will use $p_{2i}^{\alpha}$ to label the order of poles for the former, and ${\tilde p}^{\alpha}$ to label the order of poles for the latter. The superscript $\alpha$ denotes the $\alpha$-th puncture. The coefficient for the leading order singularity for those $\phi$'s and ${\tilde \phi}$ are not independent, but satisfy complicated relations \cite{Tachikawa:2009rb, Chacaltana:2011ze}. Note that, the Coulomb branch dimensions of $D_N$ class $\cS$ theory are not just the degrees for the differentials; in fact the Coulomb branch is the subvariety of \be V_{C} = \bigoplus_{k=1}^{N-1} H^0(\Sigma, K^{2k}) \oplus \bigoplus_{k=3}^{N-1}W_k \oplus H^0(\Sigma, K^{N}) \ee where $W_k$'s are vector spaces of degree $k$. If we take $c^{(k)}_l$ to be the coefficients for the $l$-th order pole of $\phi_k$, then the relation will be either polynomial relations in $c^{(k)}_l$ or involving both $c^{(k)}_l$ and $a^{(k)}$, where $a^{(k)}$ is a basis for $W_k$. For most of the punctures, the constraints are of the form \be c^{(k)}_l = \dots, \ee while for certain very even punctures, as ${\tilde \phi}$ and $\phi_N$ may share the same order of poles, the constraints would become \be c^{(N)}_l \pm 2{\tilde c}_l = \dots. \ee For examples of these constraints, see \cite{Chacaltana:2011ze}. When the Nahm partition $d$ is non-special, one needs to be more careful. The pole structure of such a puncture is precisely the same as taking $d_s = \mfS^2(d)$, but some of the constraints imposed on $d_s$ should be relaxed. In order to distinguish two Nahm partitions with the same Hitchin partition, one associates with the latter a discrete group, and the map \be d_{{\rm Nahm}} \rightarrow \pbra{\mfS(d_{{\rm Nahm}}), \cC(d_{{\rm Nahm}})} \ee makes the Spaltenstein dual one-to-one. This is studied by Sommers and Achar \cite{sommers2001lusztig, achar2002local, achar2003order} and introduced in the physical context in \cite{Chacaltana:2012zy}. Now we proceed to compute the number of dimension $k$ operators on the Coulomb branch, denoted as $d_{k}$. We have \be d_{2k} = (1-4k)(1-g) + \sum_{\alpha}(p_{2k}^{\alpha} - s_{2k}^{\alpha} + t_{2k}^{\alpha}), \ee where $g$ is the genus of Riemann surface, $s_{2k}^{\alpha}$ is the number of constraints of homogeneous degree $2k$, and $t_{2k}^{\alpha}$ is the number of $a^{(2k)}$ parameters that give the constraints $c^{(4k)}_l = \pbra{a^{(2k)}}^2$. For $d_{2k+1}$, since there are no odd degree differentials, the numbers are \be d_{2k+1} = \sum_{\alpha} t_{2k+1}^{\alpha}, \ee which is independent of genus. Finally, we take special care for $d_N$. When $N$ is even, it receives contributions from both $\phi_N$ and the Pfaffian ${\tilde \phi}$. We have \be d_{N} = 2(1-2N)(1-g) + \sum_{\alpha}(p_{N}^{\alpha} - s_{N}^{\alpha}) + \sum_{\alpha}{\tilde p}^{\alpha}. \ee When $N$ is odd, it only receives contribution from the Pfaffian: \be d_{N} = (1-2N)(1-g) + \sum_{\alpha}{\tilde p}^{\alpha}. \ee \bigskip {\bf Lie algebra} $\mfg = E_{6,7,8}.$ Unlike classical algebras, Young tableau are no longer suitable for labelling those elements in exceptional algebras. So we need to introduce some more mathematical notions. Let $\mfl$ be a Levi subalgebra, and $\cO^{\mfl}_e$ is the distinguished nilpotent orbit in $\mfl$. We have \vspace{5pt} {\bf Theorem} \cite{collingwood1993nilpotent}. There is one-to-one correspondence between nilpotent orbits of $\mfg$ and conjugacy classes of pairs $(\mathfrak{l}, \cO^{\mfl}_e)$ under adjoint action of $G$. \vspace{5pt} The theorem provides a way to label nilpotent orbits. For a given pair $(\mathfrak{l}, \cO^{\mfl}_e)$, let $X_N$ denote the Cartan type of semi-simple part of $\mathfrak{l}$. $\cO^{\mfl}_e$ in $\mfl$ gives a weighted Dynkin diagram, in which there are $i$ zero labels. Then the nilpotent orbit is labelled as $X_N(a_i)$. In case there are two orbits with same $X_N$ and $i$, we will denote one as $X_N(a_i)$ and the other as $X_N(b_i)$. Furthermore if $\mfg$ has two root lengths and one simple component of $\mathfrak{l}$ involves short roots, then we put a tilde over it. An exception of above is $E_7$, where it has one root length, but it turns out to have three pairs of nonconjugate isomorphic Levi-subalgebras. We will use a prime for one in a given pair, but a double prime for the other one. Such labels are \textit{Bala-Carter labels}. The complete list of nilpotent orbits for $E_6$ and $E_7$ theory are given in \cite{Chacaltana:2014jba, Chacaltana:2017boe}. We will examine them in more details later in this section and in section \ref{Sec: S-duality-E}. \subsection{Irregular puncture}\label{subsubsec: gradingKac} \subsubsection{Grading of the Lie algebra} We now classify irregular punctures of type $\mfg$. We adopt the Lie-algebraic techniques reviewed in the following. Recall that for an irregular puncture at $z \sim 0$, the asymptotic solution for the Higgs field $\Phi$ looks like \cite{Xie:2012hs, Wang:2015mra, Bonelli:2011aa, Nanopoulos:2010zb} \be \Phi \sim \frac{T_k}{z^{2+\frac{k}{b}}} + \sum_{-b \leq l < k} \frac{T_l}{z^{2+\frac{l}{b}}}, \label{HiggsPole} \ee where all $T_l$'s are semisimple elements in Lie algebra $\mfg$, and we also require that $(k, b)$ are coprime. The Higgs field shall be singled valued when $z$ circles around complex plane, $z \rightarrow z e^{2\pi i}$, which means the resulting scalar multiplication of $T_l$ comes from gauge transformation: \be T_l \rightarrow e^{\frac{2\pi i l}{b}}\, T_l = \sigma\, T_l\, \sigma^{-1} \ee for $\sigma$ a $G$-gauge transformation. This condition can be satisfied provided that there is a finite order automorphism (torsion automorphism) that gives grading to the Lie algebra: \be \mathfrak{g} = \bigoplus_{j \in \mathbb{Z}_b} \mathfrak{g}^j. \label{Zb grading} \ee All such torsion automorphisms are classified in \cite{kats1969auto, reeder2010torsion, reeder2012gradings}, and they admit a convenient graphical representation called \textit{Kac diagrams}. A Kac diagram $\mD$ for $\mfg$ is an extended Dynkin diagram of $\mfg$ with labels $(s_0, s_1, \dots s_r)$ on each nodes, called \textit{Kac coordinates}, where $r$ is the rank of $\mfg$. Here $s_0$ is always set to be $2$. Let $(\alpha_1, \dots, \alpha_r)$ be simple roots, together with the highest root $-\alpha_0 = \sum_{i=1}^r a_i \alpha_i$ where $(a_1, \dots, a_r)$ are the mark. We also define the zeroth mark $a_0$ to be $1$. Then the torsion automorphism associated with $\mD$ has order $m = \sum_{i=0}^{r} a_i s_i$ and acts on an element associated with simple root $\alpha_i$ as \be \sigma: \ \ \ \ \mfg_{ \alpha_i} \rightarrow \epsilon^{ s_i} \mfg_{\alpha_i}, \ \ \ \ \ i = 1,2, \dots r, \ee and extend to the whole algebra $\mfg$ via multiplication. Here $\epsilon$ is the $m$th primitive root of unity. It is a mathematical theorem \cite{dynkin1972semisimple} that all $s_i$ can only be $0,1$ or $2$. We call $\mD$ \textit{even} if all its Kac coordinates are even, otherwise $\mD$ is called \textit{odd}. For even diagrams, we may divide the coordinate and the order $m$ by $2$ since the odd grading never shows up in \eqref{Zb grading}. We will adopt this convention in what follows implicitly\footnote{This convention would not cause any confusion because if even diagrams are encountered, the label $s_0$ would be reduced to $1$; for odd diagrams this label remains to be $2$, so no confusion would arise.}. There are two quantities in the grading of special physical importance. The \textit{rank} of the $G^0$ module $\mfg^j$, denoted as ${\rm rank}(G^0 | \mfg^j)$, is defined as the dimension of a maximal abelian subspace of $\mfg^j$, consisting of semisimple elements \cite{elashvili2013cyclic}. We are interested in the case where $\mfg^1$ has positive rank: $r = {\rm rank}(G^0 | \mfg^1) > 0$. Another quantity is the intersection of centralizer of semi-simple part of $\mfg^1$ with $\mfg^0$, and this will give the maximal possible flavor symmetry. As we get matrix $T_j$ out of $\mfg^j$, we are interested in the case where $\mfg^j$ generically contains regular semisimple element. We call such grading \textit{regular semisimple}. A natural way to generate regular semisimple grading is to use nilpotent orbits. For $\mfg = A_{N-1}$ it is given in \cite{Xie:2017vaf}. We give the details of $D_N$ and $E_{6,7,8}$ in Appendix \ref{nilpoGrading}. Note when coefficient matrices are all regular semisimple, the AD theory with only irregular singularity can be mapped to type \Rnum{2}B string probing three-fold compound Du Val (cDV) singularities \cite{Shapere:1999xr}, which we review in Appendix \ref{type2B}. We list the final results in table \ref{table:isolatedsingularitiesALEfib}. \begin{table}[!htb] \begin{center} \begin{tabular}{ |c|c|c| } \hline $\mfg$& Singularity & $b$ \\ \hline $A_{N-1}$ &$x_1^2+x_2^2+x_3^N+z^{k N / b}=0$& $ b | N$ \\ \hline $~$& $x_1^2+x_2^2+x_3^N+x_3 z^{k (N-1) / b}=0$ & $b | (N-1)$\\ \hline $D_N$ & $x_1^2+x_2^{N-1}+x_2x_3^2+z^{k (2N-2) / b}=0$ & $b | (2N-2)$ \\ \hline $~$ &$x_1^2+x_2^{N-1}+x_2x_3^2+z^{k N / b} x_3=0$& $b | N$ \\ \hline $E_6$ & $x_1^2+x_2^3+x_3^4+z^{12k/b} =0$&$b | 12$ \\ \hline $~$ & $x_1^2+x_2^3+x_3^4+z^{9k / b} x_3=0$ &$b | 9$ \\ \hline $~$ & $x_1^2+x_2^3+x_3^4+z^{8k / b} x_2=0$ &$b |8$ \\ \hline $E_7$ & $x_1^2+x_2^3+x_2x_3^3+z^{18k / b}=0$& $b | 18$ \\ \hline $~$ & $x_1^2+x_2^3+x_2x_3^3+z^{14k / b}x_3=0$ &$b | 14$ \\ \hline $E_8$ & $x_1^2+x_2^3+x_3^5+z^{30k/b}=0$& $b | 30$ \\ \hline $~$ & $x_1^2+x_2^3+x_3^5+z^{24k/b} x_3=0$ & $b | 24$ \\ \hline $~$ & $x_1^2+x_2^3+x_3^5+z^{20k/b} x_2=0$ & $b | 20$ \\ \hline \end{tabular} \end{center} \caption{Classification of irregular singularities with regular semisimple coefficient matrices and the 3-fold singularities corresponding to them. In the table, $b | N$ means that $b$ is a divisor of $N$.} \label{table:isolatedsingularitiesALEfib} \end{table} This is a refinement and generalization of the classification done in \cite{Wang:2015mra, Xie:2017vaf}. We emphasize here that the grading when $\mfg^j$ generically contain semisimple elements are also crucial for obtaining SCFTs; here $b$ may be more arbitrary. Such grading will be called \textit{semisimple}. In classical Lie algebra, semisimple element $T_i$ can be represented by the matrices. In order for the spectral curve $\det(x - \Phi(z))$ to have integral power for monomials, the matrices for leading coefficient $T_k$ is highly constrained. In particular, when $\mfg = A_{N-1}$, we have \be T = \left( \begin{array}{cccc} a_1 \Xi & & & \\ & \ddots & & \\ & & a_r \Xi & \\ & & & 0_{(N-r b)}\end{array} \right). \label{Adegenerate} \ee Here $\Xi$ is a $b \times b$ diagonal matrix with entries $\{ 1, \omega, \omega^2, \dots, \omega^{b-1} \}$ for $\omega$ a $b$-th root of unity $\exp\pbra{2\pi i / b}$. For $\mfg = D_N$, things are more subtle and $T$ depends on whether $b$ is even or odd. A representative of Cartan subalgebra is \be \left( \begin{array}{cc} Z & 0 \\ 0 & - Z^{\mathsf{T}} \end{array} \right), \label{so(2N)matrix} \ee where $Z \in {\rm Mat}_{N\times N}(\mathbb{C})$. When $b$ is odd, we have \be Z = \left( \begin{array}{cccc} 0_{N-b r} & & & \\ & a_1 \Xi & & \\ & & \ddots & \\ & & & a_{r} \Xi \end{array} \right). \ee When $b$ is even, we define $\Xi' = \{ 1, \omega^2, \omega^4, \dots, \omega^{b-2} \}$, then $\Xi = \Xi' \cup (-\Xi')$. Then the coefficient matrix take the form \be Z = \left( \begin{array}{cccc} 0_{N-r b/2} & & & \\ & a_1 \Xi' & & \\ & & \ddots & \\ & & & a_{r} \Xi' \end{array} \right). \ee Counting of physical parameters in two cases are different, as we will see in section \ref{subsec: field data} momentarily. In particular, the allowed mass parameters are different for these two situations. \subsubsection{From irregular puncture to parameters in SCFT}\label{subsec: field data} We have classified the allowed order of poles for Higgs field in \eqref{HiggsPole}, and write down in classical algebras the coefficient matrix $T_i$. The free parameters in $T_i$ encode exact marginal deformations and number of mass parameters. Based on the discussion above and the coefficient matrix, we conclude that the number of mass parameters is equal to ${\rm rank}(\mfg^0)$ and the number of exact marginal deformation is given by ${\rm rank}(G^0 | \mfg^k) - 1$ if the leading matrix is in $\mfg^k$. we may list the maximal number of exact marginal deformations and number of mass parameters in tables \ref{table:ANpara} - \ref{table:E8para}. We focus here only in the case when $T$'s are regular semisimple, while for semisimple situation the counting is similar. \begin{table} \centering \begin{tabular}{|c|c|c|} \hline order of singularity $b$ & mass parameter & exact marginal deformations\\ \hline $b | N$ & $N / b - 1$ & $N/b -1$ \\ $b | (N-1)$ & $(N-1) / b$ & $(N-1) / b - 1$ \\ \hline \end{tabular} \caption{\label{table:ANpara}Summary of mass parameters and number of exact marginal deformations in $A_{N-1}$. } \end{table} \begin{table} \centering \begin{tabular}{|c|c|c|} \hline order of singularity $b$ & mass parameter & exact marginal deformations\\ \hline odd, $b | N$ & $N / b$ & $N/b -1$ \\ even, $b| N$ & $0$ & $2N / b - 1$ \\ odd, $b | (2N-2)$ & $(N-1) / b + 1$ & $ (N-1) / b - 1$ \\ even, $b | (2N-2)$ & $1$ or $0$ & $(2N-2) / b - 1$ \\ \hline \end{tabular} \caption{\label{table:DNpara}Summary of mass parameters and number of exact marginal deformations in $D_N$. Note when $b$ is even divisor of $2N-2$ but not a divisor of $N-1$, the number of mass parameter is zero, otherwise it is one.} \end{table} \begin{table} \centering \begin{tabular}{|c|c|c|} \hline order of singularity $b$ & mass parameter & exact marginal deformations\\ \hline $12$ & $0$ & $0$ \\ $9$ & $0$ & $0$ \\ $8$ & $1$ & $0$ \\ $6$ & $0$ & $1$ \\ $4$ & $2$ & $1$ \\ $3$ & $0$ & $2$ \\ $2$ & $2$ & $3$ \\ \hline \end{tabular} \caption{\label{table:E6para}Summary of mass parameters and number of exact marginal deformations in $E_6$. } \end{table} \begin{table} \centering \begin{tabular}{|c|c|c|} \hline order of singularity $b$ & mass parameter & exact marginal deformations\\ \hline $18$ & $0$ & $0$ \\ $14$ & $0$ & $0$ \\ $9$ & $1$ & $0$ \\ $7$ & $1$ & $0$ \\ $6$ & $0$ & $2$ \\ $3$ & $1$ & $2$ \\ $2$ & $0$ & $6$ \\ \hline \end{tabular} \caption{\label{table:E7para}Summary of mass parameters and number of exact marginal deformations in $E_7$. } \end{table} \begin{table} \centering \begin{tabular}{|c|c|c|} \hline order of singularity $b$ & mass parameter & exact marginal deformations\\ \hline $30$ & $0$ & $0$ \\ $24$ & $0$ & $0$ \\ $20$ & $0$ & $0$ \\ $15$ & $0$ & $0$ \\ $12$ & $0$ & $1$ \\ $10$ & $0$ & $1$ \\ $8$ & $0$ & $1$ \\ $6$ & $0$ & $3$ \\ $5$ & $0$ & $1$ \\ $4$ & $0$ & $3$ \\ $3$ & $0$ & $3$ \\ $2$ & $0$ & $7$ \\ \hline \end{tabular} \caption{\label{table:E8para}Summary of mass parameters and number of exact marginal deformations in $E_8$. } \end{table} \vspace{6pt} $\bullet$ {\bf Argyres-Douglas matter}. We call the AD theory without any marginal deformations the \textit{Argyres-Douglas matter}. They are isolated SCFTs and thus are the fundamental building blocks in S-duality. In the weakly coupled description, we should be able to decompose the theory into Argyres-Douglas matter connected by gauge groups. \subsubsection{Degeneration and graded Coulomb branch dimension}\label{subsubsec: irregDegeneration} Our previous discussion focused on the case where we choose generic regular semisimple element for a given positive rank grading. More generally, we may consider $T_k$ semisimple. We first examine the singularity where $b = 1$: \be \Phi \sim \frac{T_{\ell}}{z^{\ell}} + \frac{T_{\ell-1}}{z^{\ell-1}} + \dots + \frac{T_{1}}{z^1}, \label{MaximalIrregSing} \ee with $T_{\ell} \subset \dots \subset T_2 \subset T_1$ \cite{Witten:2007td}. For this type of singularity, the \textit{local} contribution to the dimension of Coulomb branch is \be \dim^{\rho}_{\mathbb{C}} {\rm Coulomb} = \frac{1}{2} \sum_{i=1}^{\ell} \dim (\cO_{T_i}). \label{CBdimension} \ee This formula indicates that the Coulomb branch dimensions are summation of each semisimple orbit in the irregular singularity. It is reminiscent of the regular puncture case reviewed in section \ref{subsubsec: regular puncture}, where the local contribution to Coulomb branch of each puncture is given by half-dimension of the nilpotent orbits, $\dim^{\rho}_{\mathbb{C}} {\rm Coulomb} = \frac{1}{2} \dim \mfS(\cO_\rho)$ \cite{Chacaltana:2012zy}. To label the degenerate irregular puncture, one may specify the centralizer for each $T_{\ell}$. Given a semisimple element $x \in \mfg$, the centralizer $\mfg^x$ is called a \textit{Levi subalgebra}, denoted as $\mfl$. In general, it may be expressed by \be \mfl = \mfh \oplus \sum_{\Delta' \subset \Delta} \mfg_{\alpha}, \ee where $\mfh$ is a Cartan subalgebra and $\Delta'$ is a subset of the simple root $\Delta$ of $\mfg$. We care about its semisimple part, which is the commutator $\bbra{\mfl, \mfl}$. The classification of the Levi subalgebra is known. For $\mfg$ of ADE type, we have \begin{enumerate} \item[$\bullet$] $\mfg = A_{N-1}$: $\mfl = A_{i_1} \oplus A_{i_2} \oplus \dots A_{i_k}$, with $(i_1 + 1) + \dots + (i_k+1) = N$. \item[$\bullet$] $\mfg = D_{N}$: $\mfl = A_{i_1} \oplus A_{i_2} \oplus \dots A_{i_k} \oplus D_j$, with $(i_1 + 1) + \dots + (i_k+1) + j = N$. \item[$\bullet$] $\mfg = E_6$: $\mfl = E_6,~D_5, ~A_5, ~A_4+A_1, ~2A_2+A_1,~D_4,~A_4, ~A_3+A_1, ~2A_2,~A_2+2A_1, ~A_3, ~A_2+A_1,~3A_1, ~A_2, ~2A_1, ~A_1,~0$. \item[$\bullet$] $\mfg = E_7$: $E_7, ~E_6,~ D_6, ~D_5+A_1, ~A_6, ~A_5+A_1,~ A_4+A_2, ~A_3+A_2+A_1, ~D_5, ~D_4+A_1, ~A_5^{'}, ~A_5^{''}, ~A_4+A_1, ~A_3+A_2,~A_3+2A_1,~2A_2+A_1,~A_2+3A_1,~D_4,~A_4,~(A_3+A_1)^{'},~ (A_3+A_1)^{''}, ~2A_2, ~A_2+2A_1, ~4A_1, ~A_3, ~A_2+A_1, ~(3A_1)^{'}, ~(3A_1)^{''}, ~A_2, ~2A_1, ~A_1,~0$. \item[$\bullet$] $\mfg = E_8$: $E_8, ~E_7, ~E_6+A_1, ~D_7, ~D_5+A_2, ~A_7, ~A_6+A_1, ~A_4+A_3, ~A_4+A_2+A_1, ~E_6, ~D_6, ~D_5+A_1, ~D_4+A_2, ~A_6, ~A_5+A_1,~ A_4+A_2, ~A_4+2A_1, ~2A_3, ~A_3+A_2+A_1, ~2A_2+2A_1, ~D_5, ~D_4+A_1, ~A_5, ~A_4+A_1,~ A_3+A_2, ~A_3+2A_1,~ 2A_2+A_1, ~A_2+A_1, ~D_4, ~A_4, ~A_3+A_1, ~2A_2, ~A_2+2A_1, ~4A_1, ~A_3, ~A_2+A_1, ~3A_1, ~A_2, ~2A_1, ~A_1, ~0$. \end{enumerate} Fixing the Levi subalgebra for $T_i$, the corresponding dimension for the semisimple orbit is given by \be \dim (\cO_{T_i}) = \dim G - \dim L_i. \ee We emphasize here that Levi subalgebra itself completely specify the irregular puncture. However, they may share the semisimple part $\bbra{\mfl, \mfl}$. The SCFTs defined by them can be very different. Motivated by the similarity between \eqref{CBdimension} and that of regular punctures, we wish to use nilpotent orbit to label the semisimple orbit ${\cal O}_{T_i}$, so that one can calculate the graded Coulomb branch spectrum. The correspondence lies in the theorem we introduced in section \ref{subsubsec: regular puncture}: there is a one-to-one correspondence between the nilpotent orbit $\cO^{\mfg}_{\rho}$ and the pair $(\mfl, \cO^{\mfl}_e)$. Moreover, we only consider those nilpotent orbit with principal $\cO^{\mfl}_e$. For $\mfg = A_{N-1}$, principal orbit is labelled by partition $[N]$, while for $D_N$, it is the partition $[2N-1, 1]$. Then, given a Nahm label whose $\cO^{\mfl}_e$ is principal, we take the Levi subalgebra piece $\mfl$ out of the pair $(\mfl, \cO^{\mfl}_e)$; we use the Nahm label $\rho$ as the tag such $T_i$. We conjecture that this fully characterize the coefficients $T_i$. To check the validity, we recall orbit induction \cite{kempken1983induced, de2009induced}. Let $\cO^{\mfl}_{\bar e}$ be an arbitrary nilpotent orbit in $\mfl$. Take a generic element $m$ in the center $\mfz$ of $\mfl$. We define \be {\rm Ind}^{\mfg}_{\mfl} \cO^{\mfl}_{\bar e} : = \lim_{m \rightarrow 0} {\cal O}_{m+{\bar e}}, \ee which is a nilpotent orbit in $\mfg$. It is a theorem that the induction preserves codimension: \be \dim G - \dim_{\mC} {\rm Ind}^{\mfg}_{\mfl} \cO^{\mfl}_{\bar e} = \dim L - \dim_{\mC} \cO^{\mfl}_{\bar e}. \label{inducedCodimThm} \ee In particular, when $\cO^{\mfl}_{\bar e}$ is zero orbit in $\mfl$, from \eqref{inducedCodimThm} we immediately conclude that \be \dim \cO_{T} = \dim G - \dim L = \dim_{\mC} {\rm Ind}^{\mfg}_{\mfl} \cO^{\mfl}_0, \ee for $T$ the semisimple orbit fixed by $L$. The Bala-Carter theory is related to orbit induction via \cite{collingwood1993nilpotent} \be \dim \mfS(\cO_{\rho}) = \dim_{\mC} {\rm Ind}^{\mfg}_{\mfl} \mfS(\cO^{\mfl}_{\rm principal}) = \dim_{\mC} {\rm Ind}^{\mfg}_{\mfl} \cO^{\mfl}_0 = \dim \cO_{T}. \ee Therefore, treating each semisimple orbit $\cO_{T}$ as a nilpotent orbit $\cO_{\rho}$, their local contribution to Coulomb branch is exactly the same. In the $A_{N-1}$ case, Levi subalgebra contains only $A_{i}$ pieces; the distinguished nilpotent orbit in it is unique, which is $[i+1]$. Therefore, we have a one-to-one correspondence between Nahm partitions and Levi subalgebra. More specifically, a semisimple element of the form \be x = \text{diag}(a_1, \dots, a_1, a_2, \dots, a_2, \dots, a_k, \dots, a_k), \ee where $a_i$ appears $r_i$ times, has Levi subgroup \be L = S\bbra{U(r_1) \times U(r_2) \times \dots \times U(r_k)}. \label{AtypeLevi} \ee whose Nahm label is precisely $[r_1, r_2, \dots, r_k]$. For $D_N$ case, if the semisimple element we take looks like \be x = \text{diag}(a_1, \dots, a_1, \dots, a_k, \dots, a_k, -a_1, \dots, -a_1, \dots, -a_k, \dots, -a_k, 0,\dots, 0 ), \ee where $a_i$ appears $r_i$ times and $0$ appears ${\tilde r}$ times with $\sum 2r_i + {\tilde r} = 0$, the Levi subgroup is given by \be L = \prod_i U(r_i) \times SO({\tilde r}). \ee We call $L$ of type $[r_1, \dots, r_k; {\tilde r}]$. Here we see clearly the ambiguity in labelling the coefficient $T_i$ using Levi subalgebra. For instance, when $\mfg = D_4$, we have $[1;6]$ and $[4;0]$ having the same Levi subalgebra, but clearly they are different type of matrices and the SCFT associated with them have distinct symmetries and spectrum. We will examine them in more detail in section \ref{Sec: S-duality-DN}. With Nahm labels for each $T_i$, we are now able to compute the graded Coulomb branch spectrum. For each Nahm label, we have a collection of the pole structure $\{ p^{\alpha}_{i_1}, \dots, p^{\alpha}_{i_r} \}$ for $i_k$ the degrees of differentials. There are also constraints that reduce or modifies the moduli. Then we conjecture that, at differential of degree $k$ the number of graded moduli is given by \be d_k = \sum_{\alpha} \pbra{p_k^{\alpha} - s_k^{\alpha} + t_k^{\alpha}} - 2k + 1. \ee They come from the term $u_i$ in $(u_0+u_1z+\ldots+u_{d_k-1}z^{d_k-1})x^{h^{\vee}-k}$, with $h^{\vee}$ the dual Coxeter number. However, it might happen that there are constraints of the form $c^{(2k)} = \pbra{a^{(k)}}^2$ in which $k$ is not a degree for the differentials. In this case, $t_k$ should be added to the some $k' > k$ such that $d^{\rm local}_{k'} < k' - 1$. When a regular puncture with some Nahm label is added to the south pole, one may use the same procedure to determine the contributions of each differential to the Coulomb branch moduli. We denote them as $\{ d^{(\rm reg)}_{k}\}$. Then, we simply extend the power of $z_\beta x^{2(N-k)}$ to $-d^{(\rm reg)}_{k} < \beta < d_{k}$. \vspace{6pt} $\bullet$ \textit{Example}: let us consider an $E_6$ irregular puncture of class $(k,1)$ where $k$ is very large. Take $T_{\ell} = \dots = T_2$ with Levi subalgebra $D_5$, and $T_1$ with Levi subalgebra $0$. We associate to $T_i$ with $i \geq 2$ Nahm label $D_5$. As a regular puncture, it has pole structure $\{1, 2,3,4,4,6 \}$ with complicated relations \cite{Chacaltana:2014jba}: \be & c_3^{(6)} = \frac{3}{2} c_1^{(2)} a_2^{(4)}, \ \ \ \ \ c_4^{(8)} =3\pbra{a_2^{(4)}}^2, \\[0.5em] & c_4^{(9)} = -\frac{1}{4} c_2^{(5)} a_2^{(4)}, \ \ \ \ \ c_6^{(12)} =\frac{3}{2} \pbra{a_2^{(4)}}^3, \\[0.5em] & c_5^{(12)} = \frac{3}{4} c_3^{(8)} a_2^{(4)}. \ee After subtracting it we have pole structure $\{1, 2,2,3,3,4 \}$. There is one new moduli $a^{(4)}$, and we add it to $\phi_5$. The Nahm label $0$ has pole structure $\{ 1,4,5,7,8,11 \}$. Then we have the Coulomb branch spectrum from such irregular puncture as \be & \phi_2: \frac{2k}{k+1}, \dots, \frac{k+2}{k+1}, \ \ \ \ \ \phi_5: \frac{5k}{k+1}, \dots, \frac{2k+3}{k+1}, \\[0.5em] & \phi_6: \frac{6k}{k+1}, \dots, \frac{4k+5}{k+1}, \ \ \ \ \ \phi_8: \frac{8k}{k+1}, \dots, \frac{5k+6}{k+1}, \\[0.5em] & \phi_9: \frac{9k}{k+1}, \dots, \frac{6k+7}{k+1}, \ \ \ \ \ \phi_{12}: \frac{12k}{k+1}, \dots, \frac{8k+9}{k+1}. \ee \bigskip One can carry out similar analysis for general irregular singularity of class $(k,b)$. The idea is to define a cover coordinate $\omega$ and reduce the problem to integral order of pole. Consider an irregular singularity defined by the following data $\Phi= T / z^{2+{k\over b}}+\ldots$; we define a cover coordinate $z=\omega^b$ and the Higgs field is reduced to \begin{equation} \Phi= {T' \over \omega^{k+b+1}} + \ldots \end{equation} Here $T'$ is another semisimple element deduced from $T$, see examples in section \ref{sec: S-duality-(k,b)}. Once we go to this cover coordinate, we can use above study of degeneration of irregular singularity with integral order of pole. We emphasize here that not all degeneration are allowed due to the specific form of $T$. \subsubsection{Constraint from conformal invariance} As we mentioned, not all choices of semisimple coefficient $T_i$ define SCFTs. Consider the case $b=1$, and the irregular singularity is captured by by a sequence of Levi subgroup $\mfl_{\ell}\supset \mfl_{\ell-1} \supset \ldots \supset \mfl_1$. The necessary condition is that the number of parameters in the leading order matrix $T_k$ should be no less than the number of exact marginal deformations. As will be shown later, it turns out that this condition imposes the constraint that \begin{equation} \mfl_{\ell}= \mfl_{\ell-1}\ldots= \mfl_2 = \mfl, \end{equation} with $\mfl_1$ arbitrary. Then we have following simple counting rule of our SCFT: \begin{itemize} \item The maximal number of exact marginal deformation is equal to $r-r_{\mfl}-1$, where $r$ the rank of $\mfg$ and $r_{\mfl}$ the rank of semi-simple part of $\mfl$. The extra minus one comes from scaling of coordinates. \item The maximal flavor symmetry is $G_{\mfl}\times U(1)^{r-r_{\mfl}}$, here $G_{\mfl}$ is the semi-simple part of $\mfl$. \end{itemize} Similarly, for $b\neq 1$, the conformal invariance implies that all the coefficients except $T_1$ should have the same Levi subalgebra. This is automatic when the grading is regular semisimple, but it is an extra restriction on general semi-simple grading. For example, consider $A_{N-1}$ type $(2,0)$ theory with following irregular singularity whose leading order matrix takes the form: \be T = \left( \begin{array}{cccc} a_1 \Xi & & & \\ & \ddots & & \\ & & a_r \Xi & \\ & & & 0_{(N-r b)}\end{array} \right). \label{Adegenerate} \ee When the subleading term in \eqref{HiggsPole} has integral order, the corresponding matrix can take the following general form: \be T^{'} = \left( \begin{array}{cccc} a'_1 {\mathbb{I}}_b & & & \\ & \ddots & & \\ & & a'_r {\mathbb{I}}_b & \\ & & & {\mathbb{K}}_{(N-r b)}\end{array} \right). \ee Here ${\mathbb{I}}_b$ is the identify matrix with size $b$, and ${\mathbb{K}}_{N-rb}$ is a generic diagonal matrix. However, due to the constraints, only for ${\mathbb{K}}_{N-rb}= \kappa\, {\mathbb{I}}_{N-rb}$, $T^{'}$ has the same Levi-subalgebra as $T$. This situation is missed in previous studies \cite{Xie:2017vaf}. \subsection{SW curve and Newton polygon} Recall that the SW curve is identified as the spectral curve $\det(x - \Phi(z))$ in the Hitchin system. For regular semisimple coefficient $T_i$ without regular puncture, we may map the curve to the mini-versal deformation of three fold singularity in type \Rnum{2}B construction. For given Lie algebra $\mfg$, we have the deformed singularity: \begin{align} &A_{N-1}:~x_1^2+x_2^2+ x_3^N+\phi_2(z)x_3^{N-2}+\ldots+\phi_{N-1}(z)x_3+\phi_{N}(z) =0, \nonumber\\ &D_N:~x_1^2+x_2^{N-1}+x_2 x_3^2+\phi_2(z)x_2^{N-2}+\ldots+\phi_{2N-4}(z)x_2+\phi_{2N-2}(z)+\tilde{\phi}_N(z)x_3=0, \nonumber\\ &E_6:~x_1^2+x_2^3+x_3^4+\phi_2(z)x_2x_3^2+\phi_5(z)x_2x_3+\phi_6(z)x_3^2+\phi_8(z)x_2+\phi_9(z)x_3+\phi_{12}(z)=0, \nonumber\\ &E_7:~x_1^2+x_2^3+x_2x_3^3+\phi_2(z)x_2^2x_3+\phi_{6}(z) x_2^2+\phi_{8}(z)x_2x_3+\phi_{10}(z)x_3^2 \nonumber\\ &~~~~~~+\phi_{12}(z)x_2+\phi_{14}(z)x_3+\phi_{18}(z)=0, \nonumber\\ & E_8:~x_1^2+x_2^3+x_3^5+\phi_2(z)x_2x_3^3+\phi_8(z)x_2x_3^2+\phi_{12}(z)x_3^3+ \nonumber\\ &~~~~~~~\phi_{14}(z)x_2x_3+\phi_{18}(z)x_3^2+\phi_{20}(z)x_2+\phi_{24}(z)x_3+\phi_{30}(z)=0, \label{3foldforms} \end{align} and $\phi_i$ is the degree $i$ differential on Riemann surface. A useful diagrammatic approach to represent SW curve is to use Newton polygon. When irregular singularity degenerates, the spectrum is a subset of that in regular semisimple $T_i$'s, so understanding Newton polygon in regular semisimple case is enough. The rules for drawing and reading off scaling dimensions for Coulomb branch spectrum is explained in \cite{Xie:2012hs, Wang:2015mra}. In particular, the curve at the conformal point determines the scaling dimension for $x$ and $z$, by requiring that the SW differential $\lambda = x dz$ has scaling dimension $1$. \bigskip $\bullet$ $\mfg = A_{N-1}$. The Newton polygon for regular semisimple coefficient matrices is already given in \cite{Xie:2012hs} and we do not repeat here. Here we draw the polygon when $T$ is semisimple for some semisimple grading, in the form \eqref{Adegenerate}. We give one example See figure \ref{ANnewton}. \begin{figure}[htbp] \centering \begin{adjustwidth}{5.0cm}{} \includegraphics[width=5cm]{ANNewton.png} \end{adjustwidth} \caption[Newton polygon for $A_{N-1}$ type Argyres-Douglas theories]{An example of Newton polygon for $A_5$ theory with semisimple grading. Each black dot represents a monomial in SW curve. The white dots mean that the monomials are omitted. The letters have scaling dimension $\bbra{x} = 3/5$, $\bbra{z} = 2/5$. In general, if the vertex at the top has coordinate $(a,b)$, then we have the relation $(N-a)\bbra{x} = b \bbra{z}$ and $[x] + [z] = 1$.} \label{ANnewton} \end{figure} \bigskip $\bullet$ $\mfg = D_N$. There are two types of Newton polygon, associated with Higgs field \be \Phi \sim \frac{T}{z^{2+\frac{k}{N}}}, \ \ \ \ \ \ \Phi \sim \frac{T}{z^{2+\frac{k}{2N-2}}}, \ee We denote two types and their SW curves at conformal point as \be & D_N^{(N)}[k]: \ \ \ x^{2N} + z^{2k} = 0, \\[0.5em] & D_N^{(2N-2)}[k]: \ \ \ x^{2N} + x^2 z^{k} = 0. \ee The full curve away from conformal point, and with various couplings turned on, is given by \eqref{D_SW curve}. In figure \ref{DNnewton}, we list examples of such Newton polygon. \begin{figure}[htbp] \centering \includegraphics[width=12cm]{DNnewton.png} \caption[Newton polygon for $D_N$ type Argyres-Douglas theories]{A collection of Newton polygon for examples of SCFT with $\mfg = D_N$. Each black dot represents a monomial in SW curve in the form of $x^{\alpha} z^{\beta}$; except that for the $x^0$ axis, each term represents the Pfaffian ${\tilde \phi}$, so we shall read it as $\sqrt{z^{\beta}}$. The white dots mean that the monomials are omitted. The upper left diagram gives $D_4^{(4)}[3]$ theory, while the upper right diagram gives $D_4^{(6)}[5]$. The two lower diagrams represent the same irregular puncture, but with an additional regular puncture ($e.g.$ maximal) at south pole. We denote them as $(D_4^{(4)}[3], F)$ and $(D_4^{(6)}[5], F)$ theory, respectively.} \label{DNnewton} \end{figure} \bigskip $\bullet$ $\mfg = E_{6,7,8}$. We can consider Newton polygon from the 3-fold singularities. In this way we may draw the independent differentials unambiguously. We give the case for $E_6$ with $b = 8, 9, 12$ in figure \ref{E6newton}. The other two exceptional algebras are similar. \begin{figure}[htbp] \centering \begin{adjustwidth}{-0.3cm}{} \includegraphics[width=16cm]{E6Newton.png} \end{adjustwidth} \caption[Newton polygon for $E_6$ type Argyres-Douglas theories]{A collection of Newton polygons for examples of SCFT with $\mfg = E_6$. Each black dot represents a monomial in SW curve in the three fold form. The white dots mean that the monomials are omitted. The upper left diagram gives $b= 12$, $k = 6$ theory, while the upper middle diagram gives $b = 9$, $k = 6$ theory and the upper right gives $b = 8$, $k = 6$ theory. The three lower diagrams represent the same irregular puncture, but with an additional regular puncture ($e.g.$ maximal) at south pole. } \label{E6newton} \end{figure} \section{Mapping to a punctured Riemann surface}\label{Sec: mappToRiemannSphere} As we mentioned in section \ref{Sec: Intro}, to generate S-duality we construct an auxiliary Riemann sphere $\Sigma'$ from the initial Riemann sphere $\Sigma$ with irregular punctures. We now describe the rules. The motivation for such construction comes from 3d mirror in class $\cS$ theory \cite{Benini:2010uu, Xie:2016uqq, Nanopoulos:2010bv}. To recapitulate the idea, from 3d mirror perspective we may interpret the Gaiotto duality as splitting out the quiver theories with three quiver legs. Each quiver leg carries a corresponding flavor symmetry on the Coulomb branch and can be gauged. The 3d mirror of $A_{N-1}$ type Argyres-Douglas theories are know and they are also constructed out of quiver legs. We then regard each quiver leg as a ``marked points'' on the Riemann sphere $\Sigma'$. Unlike the class $\cS$ counterpart, now there will be more types of marked points with different rank. Recall our setup is that the initial Riemann sphere $\Sigma$ is given by one irregular singularity of class $(k,b)$, with coefficient satisfying \be T_{\ell} = T_{\ell-1} = \dots = T_3 = T_2, \ \ \ \ \ T_1\ \text{arbitrary}, \ \ \ \ \ \ell = k + b + 1, \ee possibly with a regular puncture $Q$. We denote it as $\pbra{\Rnum{3}_{k,b}^{ \{ \mfl_i \}_{i=1}^{\ell} },\ Q}$, where $\mfl_i$ is the Levi subalgebra for the semisimple element $T_i$. We now describe the construction of $\Sigma'$. \vspace{8pt} $\bullet$ {\bf Lie algebra} $\mfg = A_{N-1}$. A generic matrix looks like \begin{equation} T_i={\rm diag} \pbra{\underbrace{a_1\Xi_b,\ldots, a_1\Xi_b}_{r_1},\ldots,\underbrace{a_s \Xi_b,\ldots, a_s \Xi_b}_{r_s},\underbrace{0,\ldots,0}_{N-(\sum r_j) b} }, \ \ \ \ \ 2 \leq i \leq \ell, \end{equation} The theory is represented by a sphere with one red marked point (denoted as a cross \textcolor{red}{$\times$}) representing regular singularity; one blue marked point (denoted as a square \legendbox{blue}) representing $0$'s in $T_i$, which is further associated with a Young tableaux with size $N-(\sum r_j)b$ to specify its partition in $T_1$. There are $s$ black marked points (denoted as black dots $\bullet$) with size $r_j$, $j=1,\ldots, s$ and each marked point carrying a Young tableaux of size $r_j$. Notice that there are $s-1$ exact marginal deformations which is the same as the dimension of complex structure moduli of punctured sphere. There are two exceptions: if $b=1$, the blue marked point is just the same as the black marked point. If $k=1, b=1$, the red marked point is the same as the black marked point as well \cite{Xie:2017vaf}. \vspace{8pt} $\bullet$ {\bf Lie algebra} $\mfg = D_{N}$. We have the representative of Cartan subalgebra as \eqref{so(2N)matrix} and when $b$ is odd, \be Z = {\rm diag}(\underbrace{a_1 \Xi_b,\ldots, a_1 \Xi_b}_{r_1},\ldots,\underbrace{a_s \Xi_b,\ldots, a_s \Xi_b}_{r_s},\underbrace{0,\ldots,0}_{N-(\sum r_j) b}), \ee while when $b$ is even, \be Z = {\rm diag}(\underbrace{a_1 \Xi'_{b/2},\ldots, a_1 \Xi'_{b/2}}_{r_1},\ldots,\underbrace{a_s \Xi'_{b/2},\ldots, a_s \Xi'_{b/2}}_{r_s},\underbrace{0,\ldots,0}_{N-(\sum r_j) b / 2}). \ee The theory is represented by a Riemann sphere with one red cross representing regular singularity, one blue puncture representing $0$'s in $T_i$; we also have a D-partition of $2\bbra{N-(\sum r_j)b}$ to specify further partition in $T_1$. Moreover, there are $s$ black marked point with size $r_j$, $j=1,\ldots, s$ and each marked point carrying a Young tableaux of size $r_j$ (no requirement on the parity of entries). \vspace{8pt} $\bullet$ {\bf Lie algebra} $\mfg=E_{6,7,8}$: Let us start with the case $b=1$, and the irregular puncture is labelled by Levi-subalgebra $L_l=\ldots=L_2=\mfl$ and a trivial Levi-subalgebra $L_1$. We note that there is at most one non-$A$ type Lie algebra for $\mfl$: $\mfl=A_{i_1}+\ldots+A_{i_k}+\mfh$; Let's define $a={\rm rank}(\mfg)-{\rm rank}(\mfh)-\sum_{j=1}^k (i_j+1)$, we have the following situations: \begin{itemize} \item $a\geq 0$: we have $k$ black punctures with flavor symmetry $U(i_j+1),~j=1,\ldots,k$, and $a$ more black marked point with $U(1)$ flavor symmetry; we have a blue puncture with $H$ favor symmetry ($\mfh = {\rm Lie}(H)$), and finally a red puncture representing the regular singularity. \item $a<0$: When there is a $2A_1$ factor in $\mfl$, we regard it as $D_2$ group and use a blue puncture for it; when the rank of $\mfl$ is ${\rm rank}(\mfg)-1$, we put all $A$-type factor of $\mfl$ in a single black marked point. \end{itemize} The $b\neq 1$ case can be worked out similarly. \subsection{AD matter and S-duality} We now discuss in more detail about the AD matter for $b=1$. Recall that the number of exact marginal deformations is equal to $r-r_{\mfl}-1$, where $r = {\rm rank}(\mfg)$, and $r_\mfl = {\rm rank}(\mfl)$. The AD matter is then given by the Levi subalgebra with rank $r-1$. We can list all the possible Levi subalgebra for AD matters in table \ref{table:AD matter}. \begin{table} \centering \begin{tabular}{|c|c|} \hline Lie algebra $\mfg$ & Levi subalgebra associated to AD matter\\ \hline $A_{N-1}$ & $A_{n}+A_{m},~~(n+1)+(m+1)=N$ \\ $D_N$ & $A_{n}+D_m,~~n+1+m=N$ \\ $E_6$ & $D_5,~A_5,~A_4+A_1,~2A_2+A_1$ \\ $E_7$ & $E_6,~D_6,~D_5+A_1,~A_6,~A_5+A_1,~A_4+A_2,~A_3+A_2+A_1$\\ $E_8$ & $E_7,~E_6+A_1, ~D_7,~D_5+A_2,~A_7,~A_6+A_1, ~A_4+A_3, ~A_4+A_2+A_1$\\ \hline \end{tabular} \caption{\label{table:AD matter}Possible Levi subalgebra for $T_{\ell}$ that corresponds to AD matter without exact marginal deformations.} \end{table} \vspace{8pt} {\bf S-duality frames}. With the auxiliary Riemann sphere $\Sigma'$, we conjecture that the S-duality frame is given by different degeneration limit of $\Sigma'$; the quiver theory is given by gauge groups connecting Argyres-Douglas matter without exact marginal deformations. For AD theories of type $\mfg$, the AD matter is given by three punctured sphere $\Sigma'$: one red cross, one blue square and one black dot. The rank of black dot plus the rank of blue square should equal to the rank of the red cross. See figure \ref{ADmatter} for an illustration. Each marked points carry a flavor symmetry. Their flavor central charge is given by \cite{Xie:2013jc, Xie:2017vaf} \be k_{G}^{\rm red} = h^{\vee} - \frac{b}{k+b}, \ \ \ \ k_{G}^{\rm black / blue} = h^{\vee} + \frac{b}{k+b}, \ee where $h^{\vee}$ is the dual Coxeter number of $G$. This constraints the configuration such that one can only connect black dot and red cross, or blue square with red cross to cancel one-loop beta function. \begin{figure}[htbp] \centering \begin{adjustwidth}{5cm}{} \includegraphics[width=4cm]{ADmatter.png} \end{adjustwidth} \caption[AD matter]{An example of Argyres-Douglas matter of type $\mfg$. The theory has no exact marginal deformations, and in the meantime the punctured Riemann sphere $\Sigma'$ has no complex structure moduli.} \label{ADmatter} \end{figure} \subsection{Central charges}\label{subsec: central charges} The central charges $a$ and $c$ can be computed as follows \cite{Shapere:2008zf, Xie:2013jc}: \be 2a - c = \frac{1}{4} \sum \pbra{2[u_i] - 1}, \ \ \ \ a - c = -\frac{1}{24} \dim_{\mathbb{H}} {\rm Higgs}, \label{centralCharge2} \ee This formula is valid for the theory admits a Lagrangian 3d mirror. We know how to compute the Coulomb branch spectrum, and so the only remaining piece is to the dimension of Higgs branch which can be read from the mirror. For theories with $b=1$, the local contribution to the Higgs branch dimension with flavor symmetry $G$ for red marked point is \begin{equation} \dim^{\rm red}_{\mathbb{H}} {\rm Higgs} = {1\over 2}(\dim G- {\rm rank}(G)), \end{equation} while for blue and black marked point, we have \begin{equation} \dim^{\rm blue / black}_{\mathbb{H}} {\rm Higgs} = {1\over 2}(\dim G + {\rm rank}(G)). \end{equation} The total contribution to the Higgs branch is the summation of them, except that for $A_{N-1}$, we need to subtract one. \section{S-duality for $D_N$ theory}\label{Sec: S-duality-DN} \subsection{Class $(k, 1)$}\label{sec: S-duality-(k,1)} In this section we first consider $\mfg = D_N$, and the irregular singularity we take will be \be \Phi = \frac{T_{\ell}}{z^{\ell}} + \frac{T_{\ell-1}}{z^{\ell-1}} + \dots + \frac{T_1}{z} + T_{\rm reg}, \label{MaximalIrreg} \ee where $T_{\rm reg}$ is the regular terms. This amounts to take $k = \ell - 2$, $b = 1$\footnote{Careful readers may wonder whether $n_1 = 1$ comes from $D_N^{(N)}[k']$ or $D_N^{(2N-2)}[k']$, as their relevant matrices in section \ref{subsec: field data} are different. However, in the case $n_1 = 1$, leaving two diagonal entries to be zero has the same Levi subgroup ($SO(2)$) as that of leaving it to be diag$(a, -a)$, which is $U(1)$. So two cases actually coincide.}. We settle the questions raised in previous sections: (\rnum{1}) we show which choices of $T_i$'s give legitimate deformation for SCFT; (\rnum{2}) we illustrate how to count graded Coulomb branch spectrum and (\rnum{3}) how to obtain its S-dual theory. In dealing with these questions, we first utilize the case $D_3 \simeq A_3$, where we already know the results \cite{Xie:2017vaf}. \subsubsection{Coulomb branch spectrum} Recall that in section \ref{subsubsec: irregDegeneration}, one maps each semisimple orbit $\cO_{T_i}$ to a nilpotent orbit with the same dimension. We may use the recipe of section \ref{subsubsec: regular puncture} to calculate the Coulomb branch spectrum. Let us see how this works. \vspace{6pt} \textit{Example 1: non-degenerating $D_4$ theory of class $(1,1)$.} As we have $\ell = 3$, there are three regular punctures whose labels are $\bbra{1^8}$. For such a maximal puncture, the pole structure for the differential is $\{p_2, p_4, p_6; {\tilde p}\} = \{1,3,5; 3 \}$ and there are no relations. Then, the total contributions to the moduli are $\{ d_2, d_4, d_6; {\tilde d}_4 \} = \{ 0, 2, 4; 2 \}$. This is consistent with the Newton polygon of $D_4^{(4)}[4]$. \vspace{6pt} \textit{Example 2: degenerating $D_4$ theory of class $(1,1)$.} In this example we take $T_3$ and $T_2$ to be labelled by Levi subalgebra of type $[1, 1, 1; 2]$, while $T_1$ is still of type $[1,1,1,1; 0]$. For the former, we see that it is the same as the Levi subalgebra $[1,1,1,1;0]$. Then we are back to the previous example. This is indeed the same spectrum as indicated by Newton polygon of $D_4^{(6)}[6]$. \vspace{6pt} \textit{Example 3: degenerating $D_3$ theory of class $(1,1)$.} We take $T_3$ and $T_2$ to have Levi subalgebra of type $[2,1;0]$, giving a regular puncture labelled by Nahm partition $[2,2,1,1]$. In terms of Nahm partition for $A_3$, they are equivalent to $[2,1,1]$. We also take $T_1$ to be maximal. From $A_3$, the algorithm in \cite{Xie:2012hs} determines the set of Coulomb branch operators to be $\{3/2 \}$. In the language of $D_3$, the partition $[2,2,1,1]$ gives the pole structure $\{1, 2; 2 \}$, while the maximal puncture has pole structure $\{1,3;2\}$; both of them have no constraints. Then, $\{d_2, d_4; {\tilde d}_3 \} = \{ 0,0; 1\}$, giving a Coulomb branch moduli with dimension $3/2$. So we see two approaches agree. \subsubsection{Constraints on coefficient matrices} As we mentioned before, not every choice of $\{T_{\ell}, T_{\ell - 1}, \dots, T_1 \}$ is allowed for the SCFT to exist. Those which are allowed must have $T_{\ell} = \dots = T_2$, and $T_1$ is a further partition of them. In this section we show why this is so. The idea of our approach is that, the total number of exact marginal deformations shall not exceed the maximum determined by the leading matrix $T_{\ell}$. We examine it on a case by case basis. \vspace{6pt} \textit{$D_3$}. In this case we may directly use the results of \cite{Xie:2017vaf}. Our claim holds. \vspace{6pt} \textit{$D_4$}. First of all we list the correspondence between Nahm label of the regular puncture and the Levi subalgebra in table \ref{D4Induction}. The regular puncture data are taken from \cite{Chacaltana:2011ze}. There are several remarks. For very even partitions, we have two matrix representation for two nilpotent orbits; they cannot be related by Weyl group actions\footnote{The Weyl group acts on entries of $Z = {\rm diag}(a_1,a_2,\dots,a_N)$ by permuting them or simultaneously flip signs of even number of elements.}. Moreover, we also see that there are multiple coefficient matrices sharing the same Levi subalgebra; $e.g.$ $[4;0]$ and $[1;6]$. Therefore, we do need regular puncture and Nahm label to distinguish them. Finally, we need to exclude orbit which is itself distinguished in $D_4$, as their Levi subalgebra is maximal, meaning we have zero matrix. \begin{table} \centering \begin{adjustwidth}{-1.7cm}{} \begin{tabular}{|c|c|c|c|c|c|} \hline Levi subalgebra & matrix $Z$ & regular puncture & pole structure & constraints & flavor symmetry \\ \hline $\bbra{1,1,1,1;0}$ & $\pbra{\begin{array}{cccc} a & & & \\ & b & &\\ & & c &\\ & & & d \end{array}}$ & $[1^8]$ & $\{1,3,5;3\}$ & $ - $ & $- $ \\ \hline $\bbra{2,1,1;0}$ & $\pbra{\begin{array}{cccc} a & & & \\ & a & &\\ & & b &\\ & & & c \end{array}}$ & $\bbra{2^2, 1^4}$ & $\{1,3,4;3\} $ & $ - $ & $SU(2)$ \\ \hline $\bbra{1,1;4}$ & $\pbra{\begin{array}{cccc} 0 & & & \\ & 0 & &\\ & & b &\\ & & & c \end{array}}$ & $\bbra{3, 1^5}$ & $\{1,3,4;2\}$ & $ - $ & $SO(4)$ \\ \hline $\bbra{2,2;0}$ & $\pbra{\begin{array}{cccc} a & & & \\ & a & &\\ & & b &\\ & & & \pm b \end{array}}$ & $\bbra{2^4}^{\Rnum{1}, \Rnum{2}}$ & $\{1,3,4;3\}$ & $ c_3^{(4)} \pm 2{\tilde c}_3 = 0$ & $SU(2) \times SU(2)$ \\ \hline $\bbra{3,1;0}$ & $\pbra{\begin{array}{cccc} a & & & \\ & a & &\\ & & a &\\ & & & b \end{array}}$ & $\bbra{3,3,1,1}$ & $\{1,2,4;2\}$ & $ c^{(6)}_4 = (a_3)^2 $ & $SU(3)$\\ \hline $\bbra{2; 4}$ & $\pbra{\begin{array}{cccc} a & & & \\ & a & &\\ & & 0 &\\ & & & 0 \end{array}}$ & $\bbra{3,2,2,1}^*$ & $\{1,2,4;2\}$ & $ - $ & $SU(2) \times SO(4)$ \\ \hline $\bbra{1;6}$ & $\pbra{\begin{array}{cccc} 0 & & & \\ & 0 & &\\ & & 0 &\\ & & & a \end{array}}$ & $\bbra{5,1,1,1}$& $\{1,2,2;1\}$ & $ - $ & $SO(6)$ \\ \hline $\bbra{4;0}$ & $\pbra{\begin{array}{cccc} a & & & \\ & a & &\\ & & a &\\ & & & \pm a \end{array}}$ & $\bbra{4,4}^{\Rnum{1}, \Rnum{2}}$& $\{1,2,3;2\}$ & \begin{tabular}[x]{@{}c@{}}$c_2^{(4)} \pm 2{\tilde c}_2 = (c_1^{(2)})^2 / 4, $ \\[0.3em] $c_3^{(6)} = \mp {\tilde c}_2 c_1^{(2)}$ \end{tabular} & $SU(4)$ \\ \hline \end{tabular} \end{adjustwidth} \caption{\label{D4Induction}Association of a nilpotent orbit to a Levi subalgebra for $D_4$. Here $Z$ follows the convention in \eqref{so(2N)matrix}. The partition $[3,2,2,1]$ is non-special, and we use the * to mark it. In the last column we list the semisimple part of maximal possible flavor symmetry. The partition $[5,3]$ and $[7,1]$ are excluded; the first one is non-principal in $\mathfrak{so}(8)$ while the second gives trivial zero matrix.} \end{table} Now consider $\ell = 3$, and $T_3$ has the Levi subalgebra $[1,1;4]$, with one exact marginal deformation. One can further partition it into the orbit with Levi subalgebra $[2,1,1;0]$ and $[1,1,1,1;0]$. If we pick $T_2$ to be $[2,1,1;0]$, then no matter what we choose for $T_1$, there will be two dimension $2$ operators, this is a contradiction. So $T_2$ must be equal to $T_3$. The second example has $\ell = 3$, but $T_3$ now is associated with $[3,3,1,1]$. This puncture has a relation $c_4^{(6)} = (a^{(3)})^2$, so we remove one moduli from $\phi_6$, and add one moduli to $\phi_4$. The possible subpartitions are $[2^2,1^4]$, $[1^8]$. If $T_2 \neq T_3$ then there will be two exact marginal deformations from $\phi_4$ and ${\tilde \phi}$. This is a contradiction, so we must have $T_2 = T_3$. As a third example, we may take $\ell = 4$ and $T_4$ corresponding to the regular punctures $[2^4]$, whose pole structure is $\{ 1,3,4; 3 \}$, with one constraints $c^{(4)}_3 \pm 2 {\tilde c}_3 = 0$. Then each of the local contribution to Coulomb moduli is $\{d_2, d_4, d_6; d_3\} = \{ 1,2,4; 3 \}$. From the matrix representation we know there is one exact marginal coupling. If we pick $T_3$ to be $[2^2, 1^4]$, then by simple calculation we see that there are two dimension $2$ operators. So we have to pick $T_3 = T_4$. Similarly, we have to pick $T_2 = T_3 = T_4$. Therefore, we again conclude that we must have $T_4 = T_3 = T_2$, while $T_1$ can be arbitrary. \vspace{6pt} \textit{$D_5$}. We now check the constraints for the Lie algebra $D_5$. To begin with, we list the type of Levi-subgroup and its associated regular puncture in table \ref{D5Induction}. Now we examine the constraints on coefficient matrices. We first take $\ell = 3$, and pick $T_3$ to be of the type $[3,2;0]$ whose associated regular puncture is $[3,3,2,2]$. There is a constraint $c_6^{(8)} = \pbra{c_3^{(4)}}^2 / 4$, so the local contribution to Coulomb branch is $\{d_2, d_4, d_6, d_8; d_5\} = \{1,3,4,5;3 \}$. If we take $T_2$ to be $e.g.$, $[2^4, 1^2]$, then the moduli from ${\tilde \phi}$ contribute one more exact marginal deformations other than $\phi_4$, which is a contradiction. Therefore, we again conclude that we must have $T_3 = T_2$, with arbitrary subpartition $T_1$. \begin{table} \centering \begin{adjustwidth}{-2.2cm}{} \begin{tabular}{|c|c|c|c|c|c|} \hline Levi subalgebra & matrix $Z$ & regular puncture & pole structure & constraints &flavor symmetry\\ \hline $\bbra{1,1,1,1,1;0}$ & diag$(a,b,c,d,e)$ & $[1^{10}]$ & $\{1,3,5,7;4\}$ & $-$ & $-$\\ \hline $\bbra{2,1,1,1;0}$ & diag$(a,a,b,c,d)$ & $\bbra{2^2, 1^6}$ & $\{1,3,5,6;4\}$ & $-$ & $SU(2)$ \\ \hline $\bbra{1,1,1;4}$ & diag$(0,0,a,b,c)$ & $\bbra{3, 1^7}$ & $\{1,3,5,6;3\}$ & $-$ & $SO(4)$ \\ \hline $\bbra{2,2,1;0}$ & diag$(a,a,b,b,c)$ & $\bbra{2^4,1^2}$ & $\{1,3,4,6;4\}$ & $-$ & $SU(2) \times SU(2)$ \\ \hline $\bbra{3,1,1;0}$ & diag$(a,a,a,b,c)$ & $\bbra{3^2,1^4}$ & $\{1,3,4,6;3\}$ & $c_6^{(8)} = \pbra{a^{(4)}}^2$ & $SU(3)$ \\ \hline $\bbra{2,1;4}$ & diag$(a,a,b,0,0)$ & $\bbra{3,2^2,1^3}^*$ & $\{1,3,4,6;3\}$ & $-$ & $SU(2) \times SO(4)$ \\ \hline $\bbra{3,2;0}$ & diag$(a,a,a,b,b)$ & $\bbra{3,3,2,2}$& $\{1,3,4,6;3\}$ & $c_6^{(8)} = \pbra{c_3^{(4)}}^2 / 4$ & $SU(3) \times SU(2)$ \\ \hline $\bbra{3;4}$ & diag$(0,0,a,a,a)$ & $\bbra{3,3,3,1}$& $\{1,2,4,5;3\}$ & $-$ & $SU(3) \times SO(4)$ \\ \hline $\bbra{1,1;6}$ & diag$(0,0,0,a,b)$ & $\bbra{5,1^5}$& $\{1,3,4,4;2\}$ & $-$ & $SO(6)$ \\ \hline $\bbra{4,1;0}$ & diag$(a,a,a,a,b)$ & $\bbra{4,4,1,1}$& $\{1,2,4,5;3\}$ & \begin{tabular}[x]{@{}c@{}}$c_4^{(6)} = (a^{(3)})^2, $ \\ $c_5^{(8)} = 2 a^{(3)} {\tilde c}_{3}$ \end{tabular} & $SU(4)$ \\ \hline $\bbra{2;6}$ & diag$(0,0,0,a,a)$ & $\bbra{5,2,2,1}^*$& $\{1,2,4,4;2\}$ & $-$ & $SU(2) \times SO(6)$ \\ \hline $\bbra{5;0}$ & diag$(a,a,a,a,a)$ & $\bbra{5,5}$& $\{1,2,3,4;2\}$ & \begin{tabular}[x]{@{}c@{}}${c'}_2^{(4)} \equiv c_2^{(4)} - (c_1^{(2)})^2 / 4, $ \\ $c_3^{(6)} = c_1^{(2)} {c'}_2^{(4)} / 2$, \\ $c_4^{(8)} = \pbra{{c'}_2^{(4)}}^2$ \end{tabular} & $SU(5)$ \\ \hline $\bbra{1;8}$ & diag$(0,0,0,0,a)$ & $\bbra{7,1,1,1}$& $\{1,2,2,2;1\}$ & $-$ & $SO(8)$ \\ \hline \end{tabular} \end{adjustwidth} \caption{\label{D5Induction}Association of a nilpotent orbit to a Levi subalgebra for $D_5$. $Z$ is the convention taken in \eqref{so(2N)matrix}. the Nahm partition $[5,3,1,1]$, $[7,3]$ and $[9,1]$ are excluded. } \end{table} \vspace{6pt} Based on the above examples and analogous test for other examples, we are now ready to make a conjecture about the classification of SCFT for degenerating irregular singularities: $\bullet$ {\bf Conjecture.} In order for the maximal irregular singularity \eqref{MaximalIrreg} of type $D$ to define a viable SCFT in four dimensions, we must have $T_{\ell} = T_{\ell - 1} = \dots = T_2\ (\ell \geq 3)$, while $T_1$ can be arbitrary subpartition of $T_i$. We emphasize at last that when $\ell = 2$, the scaling for $x$ in SW curve is zero. Therefore, we may have arbitrary partition $T_2$ and $T_1$, so that $\cO_{T_2} \subset \cO_{T_1}$. \subsubsection{Generating S-duality frame}\label{subsec: S-duality} With the above ingredients in hand, we are now ready to present an algorithm that generates S-duality for various Argyres-Douglas theories of $D$ type. This may subject to various consistency checks. For example, the collection of Coulomb branch spectrum should match on both sides; the conformal anomaly coefficients (central charges) $(a, c)$ should be identical. The latter may be computed from \eqref{centralCharge2}. \bigskip {\bf Duality at large $k$.} For such theories with $\ell = k + 2$, if we take the Levi subalgebra of $T_{\ell} = \dots = T_2$ to be of type $\bbra{r_1, \dots r_n; {\tilde r}}$, then there are $n - 1$ exact marginal couplings. For each $r_i$, $1 \leq i \leq n$ as well as ${\tilde r}$ there is further partition of it in $T_1$: \be & \bbra{r_i; 0} \rightarrow \bbra{m_1^{(i)}, \dots, m_{s_i}^{(i)}}, \ \ \ \ \sum_{j=1}^{s_i} m^{(i)}_j = r_i, \\[0.5em] & \bbra{0; {\widetilde r}} \rightarrow \bbra{{\widetilde m}_1, \dots, {\widetilde m}_{s}; {\widetilde r}'}, \ \ \ \ \ \ 2\sum_{j=1}^s {\widetilde m}_j + {\widetilde r}'= {\widetilde r}. \ee The Argyres-Douglas matter is given by $Z$ in \eqref{so(2N)matrix} of the leading coefficient matrix $T_{\ell}$: \be Z_1 = \pbra{\begin{array}{cccccc} a & & & & &\\ & \ddots & & & & \\ & & a & & & \\ & & & 0 & &\\ & & & & \ddots & \\ & & & & & 0 \end{array}}. \ee They are given by a three-punctured sphere with one black dot of type $[r_1, \dots, r_m]$ with $\sum r_i = n$ for $n$ being the number of $a$'s, one blue square which is degeneration of $[0; 2N-2n]$ and one red cross. However, we note the exception when $N = 2$: in this case, since the theory is in fact given by two copies of $SU(2)$ group, so the Argyres-Douglas matter is represented differently. We will see this momentarily. \vspace{8pt} \textit{Example 1: $D_3 \simeq A_3$}. This case can be analyzed from either Lie algebra perspective. Let us take $T_{\ell}$ to be regular semisimple. We also add a regular puncture labelled by a red cross. One duality frame is given in the first line of figure \ref{D3Sduality}. \begin{figure}[htbp] \centering \includegraphics[width=15cm]{A3D3Sdual.png} \caption[Compare A3 and D3 S duality]{Comparison of S duality from $A_3$ (upper half) and $D_3$ (lower half) perspective. From the $A_3$ point of view, each black dot is given by $[1]$, and the new red marked point after degeneration is given by $SU(2)$ puncture $[1,1]$. The two theories on the left and right sides are $(A_1, D_{2k+2})$ theory, which is given by irregular puncture whose $T_{k+2}, \dots, T_1 = [1,1]$, and one regular puncture. The theory in the middle is $(III_{k,1}^{[2,2]^{\times (k+1)},[1,1,1,1]}, F)$ theory. Here $F$ denotes maximal puncture. From the $D_3$ point of view, two $(A_1, D_{2k+2})$ theories combine together and form a $D_2$ type theory. The theory on the right is $(III_{k,1}^{[1;4]^{\times (k+1)},[1^3; 0]}, F)$. } \label{D3Sduality} \end{figure} We can perform various checks for this duality. First of all, $(A_1, D_{2k+2})$ theory has Coulomb branch spectrum \be \Delta(\cO_i) = 2 - \frac{i}{k+1}, \ \ \ \ i = 1,2,\dots, k. \ee For the middle theory, for simplicity we focus on the case where the regular puncture is maximal, but replacing it with any regular puncture does not affect the result. The Coulomb branch spectrum for this theory is \be \Delta(\cO) = & \frac{2k+3}{k+1}, \ \frac{2k+4}{k+1}, \dots, \frac{4k+4}{k+1}, \\[0.5em] & \frac{2k+3}{k+1}, \ \frac{2k+4}{k+1}, \dots, \frac{3k+3}{k+1}, \\[0.5em] &\ \frac{k+2}{k+1}, \ \ \frac{k+3}{k+1}, \dots, \ \frac{2k+2}{k+1}. \ee We see that along with two $SU(2)$ gauge groups, the combined Coulomb branch spectrum nicely reproduces all the operators of the initial theory. Secondly, we may calculate the central charge. We know the central charges for $(A_1, D_{2k+2})$ theory are \be a = \frac{k}{2} + \frac{1}{12}, \ \ \ \ c = \frac{k}{2} + \frac{1}{6}. \ee The central charges for the initial theory are, with the help of \eqref{centralCharge2} and three dimensional mirror, \be a = 5k + \frac{55}{8}, \ \ \ \ c = 5k + \frac{58}{8}. \ee The central charges for the middle theory are obtained similarly: \be a = 4k + \frac{131}{24}, \ \ \ \ c = 4k + \frac{142}{24}. \ee We find that \be & a_{(I_{4,4k}, F)} = 2 a^V_{SU(2)} + 2 a_{(A_1, D_{2k+2})} + a_{(III_{k,1}^{[2,2]^{\times (k+1)},[1,1,1,1]}, F)}, \\[0.5em] & c_{(I_{4,4k}, F)} = 2 c^V_{SU(2)} + 2 c_{(A_1, D_{2k+2})} + c_{(III_{k,1}^{[2,2]^{\times (k+1)},[1,1,1,1]}, F)}. \ee Here $a^V$ and $c^V$ denote the contribution from vector multiplet. Finally, we may check the flavor central charge and beta functions for the gauge group. The flavor central charge for $SU(2)$ symmetry of $(A_1, D_{2k+2})$ theory is $(2k+1) / (k+1)$. The middle theory has flavor symmetry $SU(2)^2 \times SU(4)$. Each $SU(2)$ factor has flavor central charge $2 + 1/(k+1)$, so we have a total of $4$, which exactly cancels with the beta function of the gauge group. Now we use $D_3$ perspective to analyze the S-duality. See the second line of figure \ref{D3Sduality} for illustration. It is not hard to figure out the correct puncture after degeneration of the Riemann sphere. To compare the Coulomb branch spectrum, we assume maximal regular puncture. For the theory on the left hand side, using Newton polygon we have \be \Delta(\cO) = & \frac{k+2}{k+1}, \ \frac{k+3}{k+1}, \ \dots, \ \frac{2k+1}{k+1}, \\[0.5em] & \frac{k+2}{k+1}, \ \frac{k+3}{k+1}, \ \dots, \ \frac{2k+1}{k+1}. \\[0.5em] \ee We see it is nothing but the two copy of $(A_1, D_{2k+2})$ theories. For the theory on the right hand side, the spectrum is exactly the same as the $A_3$ theory $(III_{k,1}^{[2,2]^{\times (k+1)},[1,1,1,1]}, F)$. We thus conjecture that: \be a_{(III_{k,1}^{[1;4]^{\times (k+1)},[1^3; 0]}, F)} = 4k + \frac{131}{24}, \ \ \ \ c_{(III_{k,1}^{[1;4]^{\times (k+1)},[1^3; 0]}, F)} = 4k + \frac{142}{24}. \label{D3centralCharge1} \ee This is the same as computed by the recipe in section \ref{subsec: central charges}. There is another duality frame described in figure \ref{A3SdualityFrame2}. From $D_3$ perspective, we get another type of Argyres-Douglas matter and the flavor symmetry is now carried by a black dot, which is in fact $SU(3)$. It connects to the left to an $A_2$ theory with all $T_i$'s regular semisimple. This theory can further degenerate according to the rules of $A_{N-1}$ theories, and we do not picture it. We conjecture that the central charges for the theory $\pbra{III_{k,1}^{[3;0]^{\times (k+1)},[1,1,1;0]}, F}$ are \be a_{\pbra{III_{k,1}^{[3;0]^{\times (k+1)},[1,1,1;0]}, F}} = 3k + \frac{17}{4}, \ \ \ \ c_{\pbra{III_{k,1}^{[3;0]^{\times (k+1)},[1,1,1;0]}, F}} = 3k + \frac{19}{4}. \ee \begin{figure}[htbp] \centering \includegraphics[width=15cm]{A3Frame2.png} \caption[Invisible A3 S duality]{Another S-duality frame. The upper one is from $A_3$ perspective. Here in the weakly coupled description, the rightmost theory is still $(A_1, D_{2k+2})$, the middle theory is given by $\pbra{III_{k,1}^{[2,1]^{\times (k+1)},[1,1,1]}, F}$, and the leftmost theory is given by $\pbra{III_{k,1}^{[3,1]^{\times (k+1)},[1,1,1,1]}, F}$. The lower one is from the $D_3$ perspective. The left theory without blue marked points should be understood as $A_2$ theory. The right hand theory is given by $\pbra{III_{k,1}^{[3;0]^{\times (k+1)},[1,1,1;0]}, F}$. All the computation can be done similarly by replacing full puncture $F$ to be other arbitrary regular puncture $Q$.} \label{A3SdualityFrame2} \end{figure} \vspace{8pt} \textit{Example 2: $D_4$.} Now we consider a more complicated example. Let us take a generic large $\ell > 3$ and all the coefficient matrices to be regular semisimple, $T_{\ell} = \dots = T_1 = [1^4; 0]$. There are several ways to get weakly coupled duality frame, which is described in figure \ref{D4SdualityFrame}. The regular puncture can be arbitrary. We have checked their Coulomb branch spectrum matches with the initial theory, as well as the fact that all gauge couplings are conformal. For $(a)$ in figure \ref{D4SdualityFrame}, we can compute the central charges for the theory $\pbra{III_{k,1}^{[1;6]^{\times (k+1)},[1^4;0]}, Q}$ when $Q$ is a trivial regular puncture. Recall the initial theory may be mapped to hypersurface singularity in type \Rnum{2}B construction: \be a_{\pbra{III_{k,1}^{[1^4;0]^{\times (k+2)}},\, S}} = \frac{84k^2 - 5k - 5}{6(k+1)}, \ \ \ \ c_{\pbra{III_{k,1}^{[1^4;0]^{\times (k+2)}},\, S}} = \frac{42k^2 - 2k - 2}{3(k+1)}, \ee while we already know the central charges for $(A_1, D_{2k+2})$ and $\pbra{III_{k,1}^{[1;4]^{\times (k+1)},[1^3;0]}, F}$ theory in \eqref{D3centralCharge1}. Therefore we have \be a_{\pbra{III_{k,1}^{[1;6]^{\times (k+1)},[1^4;0]},\, S}} = \frac{54k^2 - 95k - 65}{6(k+1)}, \ \ \ \ c_{\pbra{III_{k,1}^{[1;6]^{\times (k+1)},[1^4;0]},\, S}} = \frac{108k^2 - 185k - 125}{12(k+1)}. \ee This is the same as computed from \eqref{centralCharge2}. \begin{figure}[htbp] \centering \includegraphics[width=15.0cm]{D4Sdual.png} \caption[D4 S duality]{The weakly coupled duality frame for $D_4$ theory of class $(k,1)$. For $(a)$, the leftmost theory is two copies of $(A_1, D_{2k+2})$, the middle theory is given by $\pbra{III_{k,1}^{[1;4]^{\times (k+1)},[1^3;0]}, F}$, and the rightmost theory is given by $\pbra{III_{k,1}^{[1;6]^{\times (k+1)},[1^4;0]}, Q}$ where $Q$ is any $D_4$ regular puncture. For $(b)$, the leftmost theory is $(A_1, D_{2k+2})$, followed by the theory $\pbra{III_{k,1}^{[2,1]^{\times (k+1)},[1^3]}, F}$. This is then followed by $\pbra{III_{k,1}^{[3,1]^{\times (k+1)},[1^4;0]}, F}$, and the rightmost theory is still $\pbra{III_{k,1}^{[1;6]^{\times (k+1)},[1^4;0]}, Q}$. For $(c)$ and $(d)$, the rightmost theory is given by $\pbra{III_{k,1}^{[4;0]^{\times (k+1)},[1^4;0]}, Q}$. Then there are two different ways the theory $\pbra{III_{k,1}^{[1^4]^{\times (k+2)}}, F}$ can be further degenerated. Finally for $(e)$, the leftmost theory is again two copies of $(A_1, D_{2k+2})$ theory. The middle theory is $D_4$ theory $\pbra{III_{k,1}^{[2; 4]^{\times (k+1)}, [1^4]}, F}$, and the rightmost theory is given by $(A_1, D_{2k+2})$.} \label{D4SdualityFrame} \end{figure} Notice that in $(a)$ of figure \ref{D4SdualityFrame}, the leftmost and middle theory may combine together, which is nothing but the theory $\pbra{III_{k,1}^{[1^3;0]^{\times (k+2)}},\, F}$. We can obtain another duality frame by using an $SU(3)$ gauge group. See $(b)$ of figure \ref{D4SdualityFrame}. We can try to split another kind of Argyres-Douglas matter, and use the black dot to carry flavor symmetry. The duality frames are depicted in $(c)$ and $(d)$ in figure \ref{D4SdualityFrame}. Again, we can compute the central charges for the Argyres-Douglas matter $\pbra{III_{k,1}^{[4;0]^{\times (k+1)},[1^4;0]}, S}$: \be a_{\pbra{III_{k,1}^{[4;0]^{\times (k+1)},[1^4;0]},\, S}} = \frac{108k^2 - 145k - 85}{12(k+1)}, \ \ \ \ c_{\pbra{III_{k,1}^{[4;0]^{\times (k+1)},[1^4;0]},\, S}} = \frac{27k^2 - 35k - 20}{3(k+1)}, \ee same as computed from \eqref{centralCharge2}. By comparing the duality frames, we see a surprising fact in four dimensional quiver gauge theory. In particular, $(a)$ in figure \ref{D4SdualityFrame} has $SO(2n)$ gauge groups while $(c)$ in figure \ref{D4SdualityFrame} has $SU(n)$ gauge groups. The Argyres-Douglas matter they couple to are completely different, and our prescription says they are the same theory! \vspace{8pt} \textit{General $D_N$.} Based on the above two examples, we may conjecture the S-duality for $D_N$ theories of class $(k,1)$ for large. The weakly coupled description can be obtained recursively, by splitting Argyres-Douglas matter one by one. See figure \ref{DNSdualityFrame} for illustration of two examples of such splitting. In the first way we get the Argyres-Douglas matter $\pbra{III_{k,1}^{[1;2N-2]^{\times (k+1)},[1^{N};0]}, Q}$, with remaining theory $\pbra{III_{k,1}^{[1^{N-1};0]^{\times (k+2)}}, F}$. The gauge group in between is $SO(2N-2)$. In the second way, we get the Argyres-Douglas matter $\pbra{III_{k,1}^{[N;0]^{\times (k+1)},[1^{N};0]}, Q}$, with remaining theory $\pbra{III_{k,1}^{[1^N]^{\times (k+1)}}, F}$. The gauge group is $SU(N)$. The central charges $(a, c)$ for special cases of regular puncture can be computed similarly. \begin{figure}[htbp] \centering \begin{adjustwidth}{-0.5cm}{} \includegraphics[width=16cm]{DNSdual.png} \end{adjustwidth} \caption[DN S duality]{The weakly coupled duality frame for $D_N$ theory of class $(k,1)$. One starts with maximal irregular puncture and a regular puncture, and recursively degenerate a sequence of Argyres-Douglas matter. The first line gives Argyres-Douglas matter $\pbra{III_{k,1}^{[1;2N-2]^{\times (k+1)},[1^{N};0]}, Q}$ and the second line gives $\pbra{III_{k,1}^{[N;0]^{\times (k+1)},[1^{N};0]}, Q}$. We get in general a quiver with $SU$ and $SO$ gauge groups.} \label{DNSdualityFrame} \end{figure} \bigskip {\bf Duality at small $k$}. We see previously that when $k$ is large enough, new punctures appearing in the degeneration limit are all full punctures. We argue here that when $k$ is small, this does not have to be so. In this section, we focus on $D_5$ theory, with coefficient matrices $T_{\ell} = \dots = T_1 = [1,\dots, 1; 0]$ and one trivial regular puncture. The auxiliary Riemann sphere is given by five black dots of type $[1]$, one trivial blue square and one trivial red cross. We will focus on the linear quiver only. \vspace{6pt} \textit{$D_5$ theory}. The linear quivers we consider are depicted in figure \ref{D5Linear}. \begin{figure}[htbp] \centering \begin{adjustwidth}{0cm}{} \includegraphics[width=15cm]{D5Special.png} \end{adjustwidth} \caption[D5 Linear Quiver]{The linear quiver that we will examine for $k$ small, when $\mfg = D_5$.} \label{D5Linear} \end{figure} After some lengthy calculations, we find that, for the first quiver (where red crosses are all connected with blue squares), when $k = 1$, the quiver theory is \vspace{6pt} \begin{adjustwidth}{-0.5cm}{} \begin{tikzpicture \draw (18,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{1,1}^{[1; 8]^{\times 2}, [1^2;6]}, [9,1]}$.}; \draw (17.2,1) node[anchor=west] (2) {$SO(3)$}; \draw (13.5,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{1,1}^{[1; 6]^{\times 2}, [1^4;0]}, [5,1^3]}$}; \draw (12.6,1) node[anchor=west] (4) {$SO(5)$}; \draw (9,0) node[anchor=west] (5) {$ \pbra{\Rnum{3}_{1,1}^{[1; 4]^{\times 2}, [1^3; 0]}, [1^6]}$}; \draw (8.5,1) node[anchor=west] (6) {$SO(4)$}; \draw (5.5,0) node[anchor=west] (7) {$ \pbra{\Rnum{3}_{1,1}^{[1^2; 0]^{\times 3}}, [1^4]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} In particular, we have checked the central charge and confirm that the middle gauge group is indeed $SO(5)$. Moreover, its left regular puncture is superficially $[1^6]$ but only $SO(5)$ symmetry remains, similar for the right blue marked points $[; 6]$\footnote{We could imagine a similar situation of three hypermultiplets with $SO(6)$ symmetry for six half-hypermultiplets. We then only gauge five of them with $SO(5)$ gauge group. In this way, one mass parameter is frozen, so we get a total of two mass parameters.}. For $k = 2$, we have the quiver \vspace{6pt} \begin{adjustwidth}{-0.5cm}{} \begin{tikzpicture \draw (18,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{2,1}^{[1; 8]^{\times 3}, [1^3;4]}, [9,1]}$.}; \draw (17.2,1) node[anchor=west] (2) {$SO(5)$}; \draw (13.5,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{2,1}^{[1; 6]^{\times 3}, [1^4;0]}, [3,1^5]}$}; \draw (12.5,1) node[anchor=west] (4) {$SO(6)$}; \draw (9,0) node[anchor=west] (5) {$ \pbra{\Rnum{3}_{2,1}^{[1; 4]^{\times 3}, [1^3; 0]}, [1^6]}$}; \draw (8.5,1) node[anchor=west] (6) {$SO(4)$}; \draw (5.5,0) node[anchor=west] (7) {$ \pbra{\Rnum{3}_{2,1}^{[1^2; 0]^{\times 4}}, [1^4]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} For $k = 3$, we have the quiver \begin{adjustwidth}{-0.5cm}{} \begin{tikzpicture \draw (18,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{3,1}^{[1; 8]^{\times 4}, [1^5;0]}, [9,1]}$.}; \draw (17.2,1) node[anchor=west] (2) {$SO(8)$}; \draw (13.5,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{3,1}^{[1; 6]^{\times 4}, [1^4;0]}, [1^8]}$}; \draw (12.5,1) node[anchor=west] (4) {$SO(6)$}; \draw (9,0) node[anchor=west] (5) {$ \pbra{\Rnum{3}_{3,1}^{[1; 4]^{\times 4}, [1^3; 0]}, [1^6]}$}; \draw (8.5,1) node[anchor=west] (6) {$SO(4)$}; \draw (5.5,0) node[anchor=west] (7) {$ \pbra{\Rnum{3}_{3,1}^{[1^2; 0]^{\times 5}}, [1^4]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} Finally, for $k > 3$ we reduce to the case in previous section. It is curious to see that some of the gauge group becomes smaller and smaller when $k$ decreases, due to appearance of next-to-maximal puncture. Moreover, there are theories ($i.e.$ $ \pbra{\Rnum{3}_{1,1}^{[1; 8]^{\times 2}, [1^2;6]}, [9,1]}$) whose Coulomb branch spectrum is empty. When this happens, the theory is in fact a collection of free hypermultiplets. The same situation happens for the second type of $D_5$ quiver. When $k$ starts decreasing, the sizes of some gauge groups for the quiver theory decrease. When $k = 1$ we get: \vspace{6pt} \begin{adjustwidth}{-1.7cm}{} \begin{tikzpicture \draw (19,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{1,1}^{[5; 0]^{\times 2}, [2,2,1;0]}, [9,1]}$.}; \draw (18.3,1) node[anchor=west] (2) {$SU(2)$}; \draw (14.9,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{1,1}^{[4,1]^{\times 2}, [1^5]}, [2,2,1]}$}; \draw (14.2,1) node[anchor=west] (4) {$SU(4)$}; \draw (11.2,0) node[anchor=west] (5) {$ \pbra{\Rnum{3}_{1,1}^{[3,1]^{\times 2}, [1^4]}, [1^4]}$}; \draw (10.6,1) node[anchor=west] (6) {$SU(3)$}; \draw (7.5,0) node[anchor=west] (7) {$ \pbra{\Rnum{3}_{1,1}^{[2,1]^{\times 2}, [1^3]}, [1^3]}$}; \draw (6.9,1) node[anchor=west] (8) {$SU(2)$}; \draw (4.4,0) node[anchor=west] (9) {$ \pbra{\Rnum{3}_{1,1}^{[1,1]^{\times 3}}, [1^2]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) (7) edge[sedge] (8) (8) edge[sedge] (9) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} When $k = 2$, we have the quiver \vspace{6pt} \begin{adjustwidth}{-1.7cm}{} \begin{tikzpicture \draw (19,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{2,1}^{[5; 0]^{\times 3}, [1^5;0]}, [9,1]}$.}; \draw (18.3,1) node[anchor=west] (2) {$SU(5)$}; \draw (15.1,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{2,1}^{[4,1]^{\times 3}, [1^5]}, [1^5]}$}; \draw (14.2,1) node[anchor=west] (4) {$SU(4)$}; \draw (11.2,0) node[anchor=west] (5) {$ \pbra{\Rnum{3}_{2,1}^{[3,1]^{\times 3}, [1^4]}, [1^4]}$}; \draw (10.6,1) node[anchor=west] (6) {$SU(3)$}; \draw (7.5,0) node[anchor=west] (7) {$ \pbra{\Rnum{3}_{2,1}^{[2,1]^{\times 3}, [1^3]}, [1^3]}$}; \draw (6.9,1) node[anchor=west] (8) {$SU(2)$}; \draw (4.4,0) node[anchor=west] (9) {$ \pbra{\Rnum{3}_{2,1}^{[1,1]^{\times 4}}, [1^2]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) (5) edge[sedge] (6) (6) edge[sedge] (7) (7) edge[sedge] (8) (8) edge[sedge] (9) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} Finally when $k > 2$, all the gauge groups do not change anymore and stay as those in previous section. We can carry out similar analysis for all $D_N$ theory when $k$ is small. This indicates that as we vary the external data, the new punctures appearing in the degeneration limit vary as well. \subsection{Class $(k,b)$}\label{sec: S-duality-(k,b)} For general $b > 1$ and $(k, b)$ coprime, we need to classify which irregular punctures engineer superconformal theories, and study its duality as before. One subtlety that appears here is that, unlike $b = 1$ case in previous section, here we need to carefully distinguish between whether $b$ is an odd/even divisor of $N$/$2N-2$, as their numbers of exact marginal deformations are different. See section \ref{subsec: field data} for details. \subsubsection{Coulomb branch spectrum and degenerating coefficient matrices}\label{subsec: Coulomb spectrum-(k,b)} We have mentioned in section \ref{subsubsec: irregDegeneration} how to count graded Coulomb branch dimension for general $b > 1$. We elaborate the procedure here. \vspace{6pt} (\rnum{1}) \textit{$b$ is an odd divisor of $N$}. We may label the degenerating matrices similar to labelling the Levi subgroup: $[r_1, \dots, r_{n}; {\tilde r}]$, where $\sum 2 b r_i + {\tilde r} = 2N$, and there are $n - 1$ exact marginal deformations. To calculate the Coulomb branch spectrum, we first introduce a covering coordinate $z = w^b$, such that the pole structure becomes: \be \frac{T_{\ell}}{z^{2+\frac{k}{b}}} \rightarrow \frac{T'_{\ell}}{w^{k+b+1}}, \ee and $T'_{\ell}$ is given by Levi subgroup of type $[r_1, \dots, r_1, \dots, r_n, \dots r_n; {\tilde r}]$, where $r_i$ is repeated $b$ times. Then we are back to the case $b = 1$ and we can repeat the procedure in section \ref{subsubsec: irregDegeneration}. This would give the maximal degree ${d_{2i}}$ in the monomial $w^{d_{2i}} x^{2N-2i}$ that gives Coulomb branch moduli. The monomial corresponds to the degree $2i$ differential $\phi_{2i}$, and after converting back to coordinate $z$, we have the degree of $z$ in $z^{d'_{2i}} x^{2N-2i}$ as: \be d'_{2i} \leq \left \lfloor \frac{d_{2i} - 2i(b-1)}{b} \right \rfloor, \ee and similar for the Pfaffian ${\tilde \phi}$. \vspace{6pt} (\rnum{2}) \textit{$b$ is an even divisor of $N$}. We can label the matrix $T_{\ell}$ as $[r_1, r_2, \dots, r_n; {\tilde r}]$ such that $\sum b r_i + {\tilde r} = 2N$. Then, we take the change of variables $z = w^b$, and $T'_{\ell}$ is given by repeating each $r_i$ $(b/2)$ times, while ${\tilde r}$ is the same. This reduces to the class $(k,1)$ theories. \vspace{6pt} (\rnum{3}) \textit{$b$ is an odd divisor of $2N-2$}. We use $[r_1, \dots, r_n; {\tilde r}]$ to label the Levi subgroup, which satisfies $2b\sum r_i + {\tilde r} = 2N-2$. To get the Coulomb branch spectrum, we again change the coordinates $z = w^b$, and the new coefficient matrix $T'_{\ell}$ is now given by Levi subgroup of type $[r_1, \dots, r_1, \dots, r_n, \dots, r_n; {\tilde r}]$, where each $r_i$ appears $b$ times. This again reduces to the class $(k,1)$ theories. \vspace{6pt} (\rnum{4}) \textit{$b$ is an even divisor of $2N-2$}. This case is similar once we know the procedure in cases (\rnum{2}) and (\rnum{3}). We omit the details. The above prescription also indicates the constraints on coefficient matrices in order for the resulting 4d theory is a SCFT. We conclude that $T_i$ should satisfy $T_{\ell} = \dots = T_2$, $T_1$ is arbitrary. To see our prescription is the right one, we can check the case $D_4$. As an example, we can consider the Higgs field \be \Phi \sim \frac{T_{\ell}}{z^{2+\frac{1}{4}}} + \dots, \ \ \ \ \ell = 6, \ee and all $T_i$ to be $[1,1; 0]$. Using the above procedure, we know that at $\phi_6$ there is a nontrivial moduli whose scaling dimension is $6/5$. This is exactly the same as that given by hypersurface singularity in type \Rnum{2}B construction. Similarly, we may take $D_5$ theory: \be \Phi \sim \frac{T_{\ell}}{z^{2+\frac{1}{4}}} + \dots, \ \ \ \ \ell = 6, \ee and all $T_i$'s given by $[1,1;2]$. After changing variables we have $T'_i$ given by $[1,1,1,1;2]$, which is the same as $[1^5; 0]$. Then we have two Coulomb branch moduli with scaling dimension $\{ 6/5, 8/5 \}$, same as predicted by type \Rnum{2}B construction. \subsubsection{Duality frames} Now we study the S-duality for these theories. As one example, we may consider $D_4$ theory of class $(k,b) = (3,2)$, and $T_{\ell}$ is given by $[1,1,1,1;0]$. We put an extra trivial regular puncture at the south pole. This theory has Coulomb branch spectrum \be \Delta(\cO) = \left \{ \pbra{\frac{6}{5}}^{\times 4}, \pbra{\frac{8}{5}}^{\times 3}, (2)^{\times 3}, \pbra{\frac{12}{5}}^{\times 3}, \frac{14}{5}, \frac{16}{5}, \frac{18}{5} \right \}. \ee In the degeneration limit, we get three theories, described in figure \ref{D4(3,2)Sduality}. The middle theory $\pbra{\Rnum{3}_{3,2}^{[1;4]^{\times 5}, [1,1,1;0]}, [3,1,1,1]}$ gets further twisted in the sense mentioned in next subsection \ref{Sec: S-duality-twisted}, and has Coulomb branch spectrum $\{ 6/5, 8/5, 12/5, 12/5, 14/5, 16/5, 18/5 \}$. Besides it, the far left theory is two copies of $(A_1, D_5)$ theory with Coulomb branch spectrum $\{8/5, 6/5\}$ each. The far right theory is an untwisted theory, given by $\pbra{\Rnum{3}_{3,2}^{[1;6]^{\times 5}, [1,1;4]}, S}$, giving spectrum $\{12/5, 6/5 \}$. Along with the $SO(4)$ and $SU(2)$ gauge group, we see that the Coulomb branch spectrum nicely matches together. We conjecture that this is the weakly coupled description for the initial Argyres-Douglas theory. \begin{figure}[htbp] \centering \begin{adjustwidth}{0cm}{} \includegraphics[width=15cm]{32D4Sdual.png} \end{adjustwidth} \caption[D4 S duality for (3,2) class]{S-duality for $D_4$ theory of class $(3,2)$. Here we pick the coefficient matrices to be of type $[1,1,1,1;0]$, with a trivial regular puncture (this setup can be relaxed to general $D_4$ regular punctures). In the degeneration limit, we get $SO(4) \times SU(2)$ gauge group plus three Argyes-Douglas matter. The leftmost theory is in fact two copies of $(A_1, D_5)$ theory, while the middle theory is given by twisted $D_3$ theory, given by twisting the theory $\pbra{\Rnum{3}_{3,2}^{[1;4]^{\times 5}, [1,1,1;0]}, [3,1,1,1]}$. The rightmost theory is $\pbra{\Rnum{3}_{3,2}^{[1;6]^{\times 5}, [1,1;4]}, S}$ theory.} \label{D4(3,2)Sduality} \end{figure} In this example, each gauge coupling is exactly conformal as well. As a second example, we consider $D_3$ theory of class $(3,2)$. The coefficient matrices are given by $T_{6} = \dots = T_2 = T_1 = [1,1; 2]$. We put a trivial regular puncture at the south pole. This theory has Coulomb branch spectrum \be \Delta(\cO) = \left \{ \frac{6}{5}, \frac{6}{5}, \frac{7}{5}, \frac{8}{5}, \frac{9}{5}, 2, \frac{12}{5} \right \}, \ee and is represented by an auxiliary Riemann sphere with two black dots of type $[1]$, one blue square of size $2$ and one trivial red cross. See figure \ref{D3(3,2)Sduality}. After degeneration, we get two theories. We compute that the first theory is a twisting of $\pbra{\Rnum{3}^{[1;2]^{\times 5}, [1,1;0]}, [1^4]}$, having spectrum $\{ 6/5, 7/5, 8/5, 9/5 \}$. The second theory $\pbra{\Rnum{3}_{3,2}^{[1;4]^{\times 5}, [1,1,1; 0]}, S}$ has spectrum $\{12/5, 6/5 \}$. The middle gauge group is $SO(3)$, although the two sides superficially have $SO(4)$ symmetry. \begin{figure}[htbp] \centering \begin{adjustwidth}{0cm}{} \includegraphics[width=14cm]{32D3Sdual.png} \end{adjustwidth} \caption[D3 S duality for (3,2) class]{S-duality for $D_3$ theory of class $(3,2)$. Here we pick the coefficient matrices to be of type $[1,1;2]$, with a trivial regular puncture (this setup can be relaxed to general $D_3$ regular punctures). } \label{D3(3,2)Sduality} \end{figure} \subsection{$\mathbb{Z}_2$-twisted theory}\label{Sec: S-duality-twisted} If the Lie algebra $\mfg$ has a nontrivial automorphism group ${\rm Out}(\mfg)$, then one may consider \textit{twisted} punctures. This means as one goes around the puncture, the Higgs field undergoes an action of nontrivial element $o \in {\rm Out}(\mfg)$: \be \Phi(e^{2\pi i} z) = h \bbra{o\pbra{\Phi(z)}} h^{-1}, \ee where $h \in \mfg / \mfj^{\vee}$ with $\mfj^{\vee}$ the invariant subalgebra under ${\rm Out}(\mfg)$. Let us denote $\mfj$ the Langlands dual of $\mfj^{\vee}$. In this section we solely consider $D_N$ theory with automorphism group $\mathbb{Z}_2$. It has invariant subalgebra $\mfj^{\vee}= B_{N-1}$ whose Langlands dual is $\mfj = C_{N-1}$. For more details of other Lie algebra $\mfg$, see \cite{Chacaltana:2012zy, Chacaltana:2012ch, Chacaltana:2013oka, Chacaltana:2014nya, Chacaltana:2015bna, Chacaltana:2016shw}. We review some background for twisted regular punctures as in \cite{Chacaltana:2013oka}, and then proceed to understand twisted irregular punctures and their S-duality. For previous study of S-duality for twisted theory, see \cite{Argyres:2007tq, Tachikawa:2010vg}. \subsubsection{Twisted regular punctures} Following \cite{Chacaltana:2013oka}, a regular twisted $D_N$ punctures are labelled by nilpotent orbit of $C_{N-1}$, or a $C$-\textit{partition} $d$ of $2N-2$, where all odd parts appear with even multiplicity. To fix the local Higgs field, note that $\mathbb{Z}_2$ automorphism group split the Lie algebra $\mfg$ as $\mfg = \mfj_{1} \oplus \mfj_{-1}$, with eigenvalue $\pm1$ respectively. Apparently, $\mfj_1 = B_{N-1}$. The Higgs field behaves as \be \Phi \sim \frac{\Lambda}{z} + \frac{\Lambda'}{z^{1/2}} + M, \ee where $\Lambda'$ is a generic element of $\mfj_{-1}$ and $M$ is a generic element of $\mfj_1$. $\Lambda$ is an element residing in the nilpotent orbit of $B_{N-1}$, which is given by a $B$-\textit{partition} of $2N-1$, where all even parts appear with even multiplicity. It is again related to the C-partition $d$ via the Spaltenstein map $\mfS$. To be more specific, we have $\mfS(d) = \pbra{d^{+\mathsf{T}}}_B$: \begin{enumerate} \item[$\bullet$] First, ``$+$'' means one add an entry $1$ to the C-partition $d$; \item[$\bullet$] Then, perform transpose of $d^{+}$, corresponding to the superscript $\mathsf{T}$; \item[$\bullet$] Finally, $(\cdot)_B$ denotes the $B$-\textit{collapse}. The procedure is the same as D-collapse in section \ref{subsubsec: regular puncture}. \end{enumerate} For later use we will also introduce the action $\mfS$ on a B-partition $d'$. This should give a C-partition. Concretely, we have $\mfS(d') = ({d'}^{\mathsf{T}-})_C$: \begin{enumerate} \item[$\bullet$] First, ``$\mathsf{T}$'' means one take transpose of $d'$; \item[$\bullet$] Then, perform reduction of ${d'}^{\mathsf{T}}$, corresponding to subtract the last entry of ${d'}^{\mathsf{T}}$ by $1$; \item[$\bullet$] Finally, $(\cdot)_C$ denotes the $C$-\textit{collapse}. The procedure is the same as B- and D-collapse except that it now operates on the odd part which appears even multiplicity. \end{enumerate} Given a regular puncture with a C-partition, we may read off its residual flavor symmetry as \be G_{\rm flavor} = \prod_{h\ \text{even}} SO(n^h) \times \prod_{h\ \text{odd}} Sp\pbra{n^h}. \ee We may also calculate the pole structure of each differential $\phi_{2i}$ and the Pfaffian ${\tilde \phi}$ in the Seiberg-Witten curve \eqref{D_SW curve}. We denote them as $\{ p_2, p_4, \dots, p_{2N-2}; {\tilde p} \}$; in the twisted case, the pole order of the Pfaffian ${\tilde \phi}$ is always half-integer. As in the untwisted case, the coefficient for the leading singularity of each differential may not be independent from each other. There are constraints for $c^{(2k)}_l$, which we adopt the same notation as in section \ref{subsubsec: regular puncture}. The constraints of the form \be c^{(2k)}_{l} = \pbra{a^{(k)}_{l/2}}^2 \ee effectively remove one Coulomb branch moduli at degree $2k$ and increase one Coulomb branch moduli at degree $k$; while the constraints of the form \be c^{(2k)}_{l} = \dots \ee only removes one moduli at degree $2k$. For the algorithm of counting constraints for each differentials and complete list for the pole structures, see reference \cite{Chacaltana:2013oka}. After knowing all the pole structures and constraints on their coefficients, we can now compute the graded Coulomb branch dimensions exactly as those done in section \ref{subsubsec: regular puncture}. We can also express the local contribution to the Coulomb branch moduli as \be \dim^{\rho}_{\mathbb{C}}\text{Coulomb} = \frac{1}{2} \bbra{\dim_{\mathbb{C}} \mfS(\cO_{\rho}) + \dim \mfg / \mfj^{\vee}}, \ee here $\cO_{\rho}$ is a nilpotent orbit in $C_{N-1}$ and $\mfS(\cO_{\rho})$ is a nilpotent orbit in $B_{N-1}$. \subsubsection{Twisted irregular puncture} Now we turn to twisted irregular puncture. We only consider the ``maximal twisted irregular singularities''. The form of the Higgs field is, in our $\mathbb{Z}_2$ twisting, \be \Phi \sim \frac{T_{\ell}}{z^{\ell}} + \frac{U_{\ell}}{z^{\ell-1/2}} + \frac{T_{\ell-1}}{z^{\ell-1}} + \frac{U_{\ell-1}}{z^{\ell-3/2}} + \dots + \frac{T_1}{z} + \dots. \label{twistedDNhiggs} \ee Here all the $T_{i}$'s are in the invariant subalgebra $\mathfrak{so}(2N-1)$ and all $U_i$'s are in its complement $\mfj_{-1}$. To get the Coulomb branch dimension, note that the nontrivial element $o \in \rm{Out}(\mfg)$ acts on the differentials in the SW curve as \be o: \ \ \ \ & \phi_{2i} \rightarrow \phi_{2i}\ \ \text{for}\ 1\leq i \leq N-1, \\[0.5em] & {\tilde \phi}_N \rightarrow -{\tilde \phi}_N. \ee Then, the Coulomb branch dimension coming from the twisted irregular singularities can be written as \cite{Wang:2015mra}: \be \dim^{\rho}_{\mathbb{C}}\text{Coulomb} = \frac{1}{2} \bbra{\sum_{i=1}^{\ell} \dim T_i + \sum_{i=2}^{\ell} (\dim \mfg / \mfj^{\vee} - 1) + \dim \mfg / \mfj^{\vee} }. \ee In the above formula, the $-1$ term in the middle summand comes from treating $U_i, 2 \leq i \leq \ell$ as parameter instead of moduli of the theory. It corresponds to the Pfaffian ${\tilde \phi}_N$ which switches sign under $o \in \rm{Out}(\mfg)$. As in the untwisted case, we are also interested in the degeneration of $T_i$ and the graded Coulomb branch dimension. First of all, we know that as an $\mathfrak{so}(2N-1)$ matrix, $T_i$ can be written down as \be \pbra{\begin{array}{ccc} 0 & u & v \\[0.5em] -v^{\mathsf{T}}& Z_1 & Z_2 \\[0.5em] -u^{\mathsf{T}}& Z_3 & -Z_1 \end{array}}, \ee with $Z_{1,2,3}$ $(N-1) \times (N-1)$ matrices, and $Z_{2,3}$ are skew symmetric; while $u$, $v$ are row vectors of size $N-1$. After appropriate diagonalization, only $Z_1$ is nonvanishing. So a Levi subalgebra can be labelled by $[r_1, \dots, r_n; {\tilde r}+1]$, with ${\tilde r} + 1$ always an odd number. The associated Levi subgroup is \be L = \prod_i U(r_i) \times SO({\tilde r}+1). \ee \iffalse Next we seek a nilpotent orbit $\cO_{\rho}$ in $C_{N-1}$ such that $\dim \cO_T = \dim \mfS(\cO_{\rho})$. This can be done precisely in the same way as in section \ref{subsubsec: irregDegeneration}: namely we induce a nilpotent orbit of $\mfg = \mathfrak{so}(2N-1)$ from the zero orbit in $\mfl$. To achieve this, we first uplift $L$ to the Levi subalgebra ${\tilde L}$ of $\mathfrak{sl}(2N-1)$, given by $d_L = [r_1, r_1, \dots, r_n, r_n, {\tilde r}+1]$. The induced nilpotent orbit ${\rm{Ind}}^{\mathfrak{sl}_{2N-1}}_{\tilde \mfl} \cO^{\tilde \mfl}_0$ is labelled by partition $d_L^{\mathsf{T}}$. Therefore we conclude that \be {\rm Ind}^{\mfg}_{\mfl} \cO^{\mfl}_0 = \cO_{\pbra{d_L^{\mathsf{T}}}_B}. \ee This makes it clear that we may pick a nilpotent orbit $\cO_{\rho}$ in $C_{N-1}$ to label each coefficient matrix $T_i$, so long as $\mfS (\cO_{\rho}) = \cO_{\pbra{d_L^{\mathsf{T}}}_B}$. We list such correspondence in for $D_4$ theory in table \ref{D4TwistedCorr}. \begin{table} \centering \begin{adjustwidth}{-0.7cm}{} \begin{tabular}{|c|c|c|c|c|} \hline Levi subalgebra of $\mathfrak{so}(7)$ & matrix $Z_1$ & regular puncture of $\mathfrak{sp}(6)$ & pole structure & constraints\\ \hline $\bbra{1,1,1;1}$ & $\pbra{\begin{array}{ccc} a & & \\ & b &\\ & & c \end{array}}$ & $[1,1,1,1,1,1]$ & $\left \{1,3,5; \frac{7}{2} \right\}$ & $ - $\\ \hline $\bbra{2,1;1}$ & $\pbra{\begin{array}{ccc} a & & \\ & a &\\ & & c \end{array}}$ & $\bbra{2,2,1,1}$ & $\left \{1,3,5; \frac{5}{2} \right \} $ & $ c_5^{(6)} = (a^{(3)})^2 $ \\ \hline $\bbra{3;1}$ & $\pbra{\begin{array}{ccc} a & & \\ & a &\\ & & a \end{array}}$ & $\bbra{3, 3}$ & $\left \{1,3,4; \frac{5}{2} \right \}$ & \begin{tabular}[x]{@{}c@{}}$c_3^{(4)} = (a^{(2)})^2, $ \\ $c_4^{(6)} = 2 a^{(2)} {\tilde c}_{5/2}$ \end{tabular}\\ \hline $\bbra{1,1; 3}$ & $\pbra{\begin{array}{ccc} a & & \\ & b &\\ & & 0 \end{array}}$ & $\bbra{2, 1,1,1,1}^*$ & $\left \{1,3,5; \frac{5}{2} \right \}$ & $-$\\ \hline $\bbra{2 ; 3}$ & $\pbra{\begin{array}{ccc} a & & \\ & a &\\ & & 0 \end{array}}$ & $\bbra{2,2,2}$ & $\left \{1,3,4; \frac{5}{2} \right \}$ & $-$\\ \hline $\bbra{1; 5}$ & $\pbra{\begin{array}{ccc} a & & \\ & 0 &\\ & & 0 \end{array}}$ & $\bbra{4,1,1}^*$ & $\left \{1,3,3;\frac{3}{2} \right\}$ & $ - $ \\ \hline \end{tabular} \end{adjustwidth} \caption{\label{D4TwistedCorr}The correspondence between Levi subalgebra and the twisted regular puncture in $D_4$ theory. The contribution of the Pfaffian to pole structure has been shifted by $1/2$. } \end{table} \fi Now we state our proposal for whether a given twisted irregular puncture defines a SCFT in four dimensions. Similar to untwisted case, we require that $T_{\ell} = T_{\ell - 1} = \dots = T_2$ and $T_1$ can be further arbitrary partition of $T_{i \geq 2}$. When all the $T_i$'s are regular semisimple, we can draw Newton polygon for these theories. They are the same as untwisted case, except that the monomials living in the Pfaffian ${\tilde \phi}_N$ get shift down one half unit \cite{Wang:2015mra}. \vspace{6pt} \textit{Example: $D_4$ maximal twisted irregular puncture with $\ell = 3$.} We consider all $T_i$ to be regular semisimple $\mathfrak{so}(7)$ element $[1,1,1; 1]$, plus a trivial twisted regular puncture. From Newton polygon, we know the spectrum for this theory is $\{ 2, 3/2, 3, 5/2, 2, 3/2, 7/4, 5/4 \}$. \subsubsection{S-duality for twisted $D_N$ theory of class $(k,1)$} Having all the necessary techniques at hand, we are now ready to apply the algorithm previously developed and generate S-duality frame. We state our rules as follows for theory of class $(k,1)$ with $k = \ell - 2$. \begin{enumerate} \item[$\bullet$] Given coefficient matrices $T_{\ell} = \dots = T_2 = [r_1, \dots, r_n; {\tilde r}+1]$, and $T_1$ being further partition of $T_i$, we represent the theory on an auxiliary Riemann sphere with $n$ black dots with size $r_i, 1 \leq i \leq n$, a blue square with size ${\tilde r}$, and a red cross representing the regular puncture, labelled by a C-partition of $2N-2$. \item[$\bullet$] Different S-duality frames are given by different degeneration limit of the auxiliary Riemann sphere. \item[$\bullet$] Finally, one needs to figure out the newly appeared punctures. The gauge group can only connect a red cross and a blue square (${Sp}$ gauge group). This is different from untwisted case we considered before. \end{enumerate} Let us proceed to examine examples. We first give a comprehensive discussion of $D_4$ theory. \vspace{6pt} {\bf Duality at large $k$}. We have initially three black dots of type $[1]$, a trivial blue square and an arbitrary red cross representing a regular puncture. This theory has a part of Coulomb branch spectrum coming from irregular puncture: \be \Delta(\cO) & = \frac{k+2}{k+1}, \dots, \frac{2k}{k+1}, \\[0.5em] & = \frac{k+2}{k+1}, \dots, \frac{4k}{k+1}, \\[0.5em] & = \frac{k+2}{k+1}, \dots, \frac{6k}{k+1}, \\[0.5em] & = \frac{k+3/2}{k+1}, \dots, \frac{4k - 1/2}{k+1}. \ee The S-duality frame for this theory is given in figure \ref{D4TwistedSduality}. \begin{figure}[htbp] \centering \begin{adjustwidth}{0cm}{} \includegraphics[width=14.6cm]{D4Twisted.png} \end{adjustwidth} \caption[D4 S duality for twisted punctures]{S-duality for twisted $D_4$ theory of class $(k,1)$ with large $k$. Each Argyres-Douglas matter is connected with $Sp$ gauge group. Assembling the black dot and the blue square we can read off the data for the irregular puncture and thus identify the theory.} \label{D4TwistedSduality} \end{figure} The duality frame in figure \ref{D4TwistedSduality} tells us the Coulomb branch spectrum of each piece. The leftmost theory $\pbra{\Rnum{3}_{k,1}^{[1;1]^{\times (k+2)}}, F}$ has the spectrum \be \Delta_1(\cO) & = \frac{k+2}{k+1}, \dots, \frac{2k+1}{k+1}, \\[0.5em] & = \frac{k+3/2}{k+1}, \dots, \frac{2k+3/2}{k+1}. \ee The rightmost theory is given by $\pbra{\Rnum{3}_{k,1}^{[1; 5]^{\times (k+1)}, [1,1,1;1]}, Q}$ whose spectrum comes from the irregular part is \be \Delta_2(\cO) & = \frac{k+2}{k+1}, \dots, \frac{2k}{k+1}, \\[0.5em] & = \frac{2k+3}{k+1}, \dots, \frac{4k}{k+1}, \\[0.5em] & = \frac{4k+5}{k+1}, \dots, \frac{6k}{k+1}, \\[0.5em] & = \frac{3k+7/2}{k+1}, \dots, \frac{4k - 1/2}{k+1}. \ee Finally, the middle theory is $\pbra{\Rnum{3}_{k,1}^{[1; 3]^{\times (k+1)}, [1,1;1]}, F}$. It contributes to the Coulomb branch spectrum coming from the irregular puncture \be \Delta_3(\cO) & = \frac{k+2}{k+1}, \dots, \frac{2k+1}{k+1}, \\[0.5em] & = \frac{2k+3}{k+1}, \dots, \frac{4k+3}{k+1}, \\[0.5em] & = \frac{2k+5/2}{k+1}, \dots, \frac{3k + 5/2}{k+1}. \ee These three pieces nicely assemble together and form the total spectrum of original theory. We thus have $Sp(2) \times Sp(4)$ gauge groups. \vspace{6pt} {\bf Duality at small $k$}. Similar to the untwisted case, we expect that some of the gauge group would be smaller. We now focus on a trivial twisted regular puncture in figure \ref{D4TwistedSduality}. Analysis for other twisted regular punctures are analogous. We find that for $k = 1$, \vspace{6pt} \begin{adjustwidth}{0cm}{} \begin{tikzpicture \draw (13,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{1,1}^{[1; 5]^{\times 2}, [1,1;3]}, [6]}$.}; \draw (11.5,1) node[anchor=west] (2) {$Sp(2)$}; \draw (7.1,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{1,1}^{[1;3]^{\times 2}, [1,1;1]}, [2,1,1]}$}; \draw (5.9,1) node[anchor=west] (4) {$Sp(2)$}; \draw (2.2,0) node[anchor=west] (5) {$ \pbra{\Rnum{3}_{1,1}^{[1; 1]^{\times 3}}, [1,1]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) (3) edge[sedge] (4) (4) edge[sedge] (5) ; \end{tikzpicture} \end{adjustwidth} When $k \geq 2$, the second $Sp(2)$ gauge group becomes $Sp(4)$ and we reduce to the large $k$ calculations. \vspace{6pt} \textit{S-duality of $D_N$ theory}. When $k$ is large, the intermediate gauge group in the degeneration limit does not depend on which twisted regular puncture one puts, and they are all full punctures. To obtain the duality frames, we can again follow the recursive procedure by splitting the Argyres-Douglas matter one by one. See the example of such splitting in figure \ref{DNTwistedSduality}. Again, due to twisting things become more constraining, and all matter should have a blue square on its auxiliary Riemann sphere. \begin{figure}[htbp] \centering \begin{adjustwidth}{0cm}{} \includegraphics[width=14.6cm]{DNTwisted.png} \end{adjustwidth} \caption[DN S duality for twisted punctures]{S-duality for twisted $D_N$ theory of class $(k,1)$ with large $k$. Here we present the duality frame recursively by splitting the Argyres-Douglas matter. In the first line we split a theory $\pbra{\Rnum{3}_{k,1}^{[1;1]^{\times (k+2)}}, F}$ with $F$ a full $D_2$ twisted puncture; in the second line we split a theory $\pbra{\Rnum{3}_{k,1}^{[1;2N-3]^{\times (k+1)}, [1^{N-1};1]}, Q}$ with original regular puncture $Q$.} \label{DNTwistedSduality} \end{figure} When $k$ is small, some of the intermediate puncture would be smaller. One needs to figure out those punctures carefully. We leave the details to interested readers. \section{Comments on S-duality for $E$-type theories}\label{Sec: S-duality-E} Finally, we turn to the duality frames for $\mfg = \mathfrak{e}_{6,7,8}$. We focus on the Lie algebra $\mathfrak{e}_6$ while state our conjecture for $\mathfrak{e}_7$ and $\mathfrak{e}_8$ case. A complete list of all the relevant data for regular punctures can be found in \cite{Chacaltana:2014jba, Chacaltana:2017boe, Chacaltana:2015bna}. We will use some of their results here for studying irregular puncture. \subsection{Irregular puncture and S-duality for $E_6$ theory} We focus on the irregular singularity \eqref{MaximalIrreg}. The first task is to characterize the degeneration of coefficient matrices. Those matrices $T_i$, $1 \leq i \leq \ell$ shall be represented by a Levi subalgebra $\mfl$. See section \ref{subsubsec: irregDegeneration} for the list of conjugacy classes. For each Levi subalgebra $\mfl$, we associate a nilpotent orbit with Nahm label. Since we are already using Bala-Carter's notation, we can directly read of $\mfl$. See table \ref{table:E6 orbit}. Here we exclude Bala-Carter label of the form $E_6(\cdot)$, as it gives maximal Levi subalgebra so the irregular puncture is trivial. \begin{table} \centering \begin{adjustwidth}{4.0cm}{} \begin{tabular}{|c|c|} \hline Levi subalgebra $\mfl$ & Nahm Bala-Carter label \\ [0.1em] \hline $0$ & $0$\\ $A_1$ & $A_1$\\ $2A_1$ & $2A_1$\\ $3A_1$ & $(3A_1)^*$\\ $A_2$ & $A_2$\\ $A_2 + A_1$ & $A_2+A_1$\\ $2A_2$ & $2A_2$\\ $A_3$ & $A_3$\\ $2A_2 + A_1$ & $(2A_2 + A_1)^*$\\ $A_2 + 2A_1$ & $A_2 + 2A_1$\\ $A_3 + A_1$ & $(A_3+A_1)^*$\\ $D_4$ & $D_4$ \\ $A_4$ & $A_4$ \\ $A_4 + A_1$ & $A_4+A_1$ \\ $A_5$ & $(A_5)^*$\\ $D_5$ & $D_5$ \\ \hline \end{tabular} \end{adjustwidth} \caption{\label{table:E6 orbit}The correspondence between Nahm label and the Levi subalgebra. The Levi subalgebra $E_6$ is omitted as it does not give any irregular puncture. We use $*$ to denote the non-special nilpotent orbit. The pole structure and constraints can be found in \cite{Chacaltana:2014jba}. Again, we exclude those with non-principal orbit in the Levi subalgebra.} \end{table} We are now ready to count the Coulomb branch spectrum for a given $E_6$ irregular puncture of class $(k,1)$, were $\ell = k + 2$. We use the SW curve from type \Rnum{2}B construction, whose isolated singularity has the form\footnote{As we consider $(k,1)$ theory, there is no distinction between whether it comes from $b = 8, 9$ or $12$. We can simply pick anyone of them.} \be x_1^2 + x_2^3 + x_3^4 + z^{12k} = 0, \ee whose deformation looks like \be x^2_1 + x^3_2 + x^4_3 + \phi_2(z) x_2 x^2_3 + \phi_5(z) x_2 x_3 + \phi_6(z) x^2_3 + \phi_8 (z) x_2 + \phi_9 (z) x_3 + \phi_{12}(z) = 0, \label{E_6 deform sing} \ee where at the singularity $\phi_{12} = z^{12k}$. The Coulomb branch spectrum is encoded in these Casimirs. For example, when $k$ = 1 and regular semisimple coefficients, we know the scaling dimensions for each letter are \be \bbra{x_1} = 3, \ \ [x_2] = 2, \ \ [x_3] = \frac{3}{2}, \ \ [z] = \frac{1}{2}. \ee By enumerating the quotient algebra generator of this hypersurface singularity we know that the number of moduli for each differential is $\{ d_2, d_5, d_6, d_8, d_9, d_{12} \} = \{ 0, 3, 4, 6, 7, 10\}$. This is consistent with adding pole structures and subtract global contribution of three maximal $E_6$ regular punctures. \subsubsection{S-duality for $E_6$ theory} We now study the S-duality for $E_6$ theory of class $(k,1)$, with coefficient all regular semisimple. From the $D_N$ S-duality, we know that the Levi subalgebra directly relates to the flavor symmetry. If we take the coefficient matrix to be regular semisimple, then our initial theory is given by a sphere with six black dots, one trivial blue square and one red cross (which is an arbitrary $E_6$ regular puncture. We only consider large $k$ situation. In type \Rnum{2}B construction \eqref{E_6 deform sing}, the scaling dimensions for each letter are \be \bbra{x_1} = \frac{6k}{k+1}, \ \ [x_2] = \frac{4k}{k+1}, \ \ [x_3] = \frac{3k}{k+1}, \ \ [z] = \frac{1}{k+1}. \ee So we have the spectrum of initial theory coming from irregular puncture as: \be & \phi_2: \frac{2k}{k+1}, \dots, \frac{k+2}{k+1}, \ \ \ \ \ \phi_5: \frac{5k}{k+1}, \dots, \frac{k+2}{k+1}, \\[0.5em] & \phi_6: \frac{6k}{k+1}, \dots, \frac{k+2}{k+1}, \ \ \ \ \ \phi_8: \frac{8k}{k+1}, \dots, \frac{k+2}{k+1}, \\[0.5em] & \phi_9: \frac{9k}{k+1}, \dots, \frac{k+2}{k+1}, \ \ \ \ \ \phi_{12}: \frac{12k}{k+1}, \dots, \frac{k+2}{k+1}. \ee There are several ways to split Argyres-Douglas matter. For example, we may pop out two black dots and one trivial blue square. We get the duality frame \vspace{6pt} \begin{adjustwidth}{3cm}{} \begin{tikzpicture \draw (7.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{[1,1;0]^{\times k+2}}, [1^4]}$,}; \draw (5.9,1) node[anchor=west] (2) {$SO(4)$}; \draw (2.2,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{(2A_1)^{\times (k+1)}, 0}, Q_{E_6}}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} and here the right hand side theory is two copies of $(A_1, D_{2k+2})$ theory. This duality frame persists to $k = 1$. We have checked that the central charge matches. The second way is to pop out a trivial black dot and the $E_6$ regular puncture. This results in $D_5$ gauge group: \vspace{6pt} \begin{adjustwidth}{3cm}{} \begin{tikzpicture \draw (7.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{(D_5)^{\times (k+1)}, 0}, Q}$,}; \draw (5.9,1) node[anchor=west] (2) {$SO(10)$}; \draw (2.2,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{[1^5; 0]^{\times (k+2)}}, [1^{10}]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} where the theory $\pbra{\Rnum{3}_{k,1}^{[1^5; 0]^{\times (k+2)}}, [1^{10}]}$ can be further degenerate according to $D_N$ type rules. The spectrum counting is explained in the example in section \ref{subsubsec: irregDegeneration}. We see it correctly reproduces $SO(10)$ flavor symmetry. We have also checked that the central charge matches. Another way is to give $SU(6)$ gauge group in the degeneration limit, by poping out a trivial blue puncture and red cross. \vspace{6pt} \begin{adjustwidth}{3cm}{} \begin{tikzpicture \draw (7.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{(A_5)^{\times (k+1)}, 0}, Q_{E_6}}$,}; \draw (5.9,1) node[anchor=west] (2) {$SU(6)$}; \draw (2.2,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{[1^6]^{\times (k+2)}}, [1^{6}]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} We find that the central charges match as well. \subsection{$E_7$ and $E_8$ theory} Finally, we turn to $E_7$ and $E_8$ Argyres-Douglas theories. Tinkertoys for $E_7$ theories have been worked out in \cite{Chacaltana:2017boe}. Similar ideas go through and we will outline the steps here. The key ingredient is to use type \Rnum{2}B construction to count the moduli. For $E_7$ theory, the deformed singularity has the form \be x_1^2 & + x_2^3 + x_2 x_3^3 + \phi_2(z) x_2^2 x_3 + \phi_6(z) x_2^2 + \phi_8(z) x_2 x_3\\[0.5em] & + \phi_{10}(z)x_3^2 + \phi_{12}(z)x_2 + \phi_{14}(z) x_3 + \phi_{18}(z) = 0, \ee where $\{ \phi_2, \phi_6, \phi_8, \phi_{10}, \phi_{12}, \phi_{14}, \phi_{18} \}$ are independent differentials. For $E_8$ theory, the deformed hypersurface singularity has the form: \be x_1^2 & + x_2^3 + x_3^5 + \phi_2(z) x_2 x^3_3 + \phi_8(z) x_2 x^2_3 + \phi_{12}(z) x^3_3\\[0.5em] & + \phi_{14}(z)x_2 x_3 + \phi_{18}(z)x^2_3 + \phi_{20}(z) x_2 + \phi_{24}(z)x_3 + \phi_{30}(z) = 0, \ee where $\{ \phi_2, \phi_8, \phi_{12}, \phi_{14}, \phi_{18}, \phi_{20}, \phi_{24}, \phi_{30} \}$ are independent differentials. The regular puncture for these two exceptional algebras are again given the Bala-Carter label. One can read off the Levi subalgebra similar as before. This then provides the way of counting Coulomb branch spectrum. The duality frame can then be inferred by comparing the spectrum in the degeneration limit, and checked with central charge computation \eqref{centralCharge2}. For example, we have in $\mfe_7$ theory one duality frame which looks like \vspace{6pt} \begin{adjustwidth}{1.8cm}{} \begin{tikzpicture \draw (9.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{(E_6)^{\times (k+1)}, 0}, Q}$,}; \draw (6.4,1) node[anchor=west] (2) {$E_6$}; \draw (1.4,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{(0)^{\times (k+2)}}, F_{\mfe_6}}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} where $F_{\mfe_6}$ is the full $E_6$ regular puncture. Another duality frame is \vspace{6pt} \begin{adjustwidth}{1.8cm}{} \begin{tikzpicture \draw (9.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{(A_6)^{\times (k+1)}, 0}, Q}$.}; \draw (6.4,1) node[anchor=west] (2) {$SU(7)$}; \draw (1.4,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{[1^7]^{\times (k+2)}}, [1^7]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} For $\mfe_8$ theory, we have the duality frames \vspace{6pt} \begin{adjustwidth}{1.8cm}{} \begin{tikzpicture \draw (9.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{(E_7)^{\times (k+1)}, 0}, Q}$,}; \draw (6.4,1) node[anchor=west] (2) {$E_7$}; \draw (1.4,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{(0)^{\times (k+2)}}, F_{\mfe_7}}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} and \vspace{6pt} \begin{adjustwidth}{1.8cm}{} \begin{tikzpicture \draw (9.1,0) node[anchor=west] (1) {$ \pbra{\Rnum{3}_{k,1}^{(A_7)^{\times (k+1)}, 0}, Q}$,}; \draw (6.4,1) node[anchor=west] (2) {$SU(8)$}; \draw (1.4,0) node[anchor=west] (3) {$ \pbra{\Rnum{3}_{k,1}^{(0)^{\times (k+2)}}, [1^8]}$}; \path (1) edge[sedge] (2) (3) edge[sedge] (2) ; \end{tikzpicture} \end{adjustwidth} \vspace{6pt} We have checked that the central charges and the Coulomb branch spectrum matches. The left hand theory of each duality frames can be further degenerated according to known rules for lower rank ADE Lie algebras, and we do not picture them anymore. Here we see the interesting duality appears again: the quivers with $E_N$ type gauge group is dual to quivers with $A_{N-1}$ type quivers. \section{Conclusion and discussion}\label{Sec: conclusion} In this paper, we classified the Argyres-Douglas theory of $D_N$ and $E_{6,7,8}$ type based on classification of irregular punctures in the Hitchin system. We developed a systematic way of counting graded dimension. Generalizing the construction in \cite{Xie:2017vaf}, we also obtained duality frames for these AD theories, and find a novel duality between quivers with $SO / E_N$ gauge groups and quivers with $SU$ gauge groups. An interesting question to ask is whether one can understand the duality from geometry. In other words, whether one can engineer these quiver theories in string theory, and the duality is interpreted as operations on the geometry side. A related question would be whether such exotic duality exist in three dimensions. In $A_{N-1}$-type AD theories, we can perform dimensional reduction and mirror symmetry to get a Lagrangian theory, which is in general a quiver with $SU$ gauge groups \cite{boalch2008irregular}. One expects that such mirror theory also exists for $D_N$ and $E$-counterpart. Then, the three dimensional mirror of the above duality would be a natural construction. S-duality in four dimensional superconformal theories sometimes facilitate the calculation of partition functions \cite{Gadde:2010te}. It will be interesting to see the duality frames obtained for AD theories can give partition function of some of them. Partition functions of certain $A_{N-1}$ type AD theories were recently computed in \cite{Cordova:2015nma, Song:2015wta, Buican:2015ina, Buican:2017uka}. In particular, the Schur index encodes two dimensional chiral algebra \cite{Beem:2013sza, Beem:2017ooy} while Coulomb branch index gives geometric quantization of Hitchin moduli space \cite{Gukov:2016lki, Fredrickson:2017yka, Fredrickson:2017jcf} and new four manifold invariants \cite{Gukov:2017zao}. As we mentioned in section \ref{subsubsec: regular puncture}, there are more fundamental invariants arise for $D_N$ Hitchin system, so one may wonder its Hitchin fibration structure, as well as its fixed point under $U(1)$ action. In our construction, we have obtained many AD theories whose coefficient matrices in the Higgs field degenerate. Then one can try to study their chiral algebra, characters and representations. A useful approach is taken in \cite{Buican:2017fiq}. Study of the associated chiral algebra would have further implication on the dynamics of the theory, for instance chiral ring structure, symmetries and the presence of a decoupled free sector. Furthermore, one may explore if there are corresponding $\cN = 1$ Lagrangian theories that flows to $D_N$ type AD theory, following the construction in \cite{Maruyoshi:2016tqk, Maruyoshi:2016aim, Agarwal:2016pjo, Agarwal:2017roi}. Our study of S-duality may have many implication for the general investigation of conformal manifold for four dimensional $\cN = 2$ superconformal theories. In particular, with those duality frames, one can ask if they exhaust all the possible frames, what is the group action on the conformal manifold and how the cusps look like. There are progress in computing S-duality group from homological algebra point of view \cite{Caorsi:2016ebt, Caorsi:2017bnp}. We hope to better understand these structures in future publications. \acknowledgments{The authors wish to thank Sergei Gukov and Yifan Wang for discussions and comment. K.Y. would like to thank 2017 Simons Summer Workshop where part of the work is done. K.Y. is supported by DOE Grant DE-SC0011632, the Walter Burke Institute for Theoretical Physics, and Graduate Fellowship at Kavli Institute for Theoretical Physics. D.X. is supported by Center for Mathematical Sciences and Applications at Harvard University, and in part by the Fundamental Laws Initiative of the Center for the Fundamental Laws of Nature, Harvard University. }

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\section{Introduction} Since its introduction by Akaike in the early seventies~\cite{Akaike:73}, the celebrated Akaike's Information Criterion (AIC) has been an essential tool for the statistician and its use is almost systematic in problems of model selection and estimator selection for prediction. By choosing among estimators or models constructed from finite degrees of freedom, the AIC recommends more specifically to maximize the log-likelihood of the estimators penalized by their corresponding degrees of freedom. This procedure has found pathbreaking applications in density estimation, regression, time series or neural network analysis, to name a few (\cite{ClaeskensHjort:08}). Because of its simplicity and negligible computation cost---whenever the estimators are given---, it is also far from outdated and continues to serve as one of the most useful devices for model selection in high-dimensional statistics. For instance, it can be used to efficiently tune the Lasso (\cite{ZouHastieTib:07}). Any substantial and principled improvement of AIC is likely to have a significant impact on the practice of model choices and we bring in this paper an efficient and theoretically grounded solution to the problem of overfitting that can occur when using AIC on small to medium sample sizes. The fact that AIC tends to be unstable and therefore perfectible in the case of small sample sizes is well known to practitioners and and has long been noted. Suguira~\cite{Sugiura:78} and Hurvich and Tsai~\cite{HurTsai:89} have proposed the so-called AICc (for AIC corrected), which tends to penalize more than AIC. However, the derivation of AICc comes from an asymptotic analysis where the dimension of the models are considered fixed relative to the sample size. In fact, such an assumption does not fit the usual practice of model selection, where the largest models are of dimensions close to the sample size. Another drawback related to AICc is that it has been legitimated through a mathematical analysis only in the linear regression model and for autoregressive models (\cite{ClaeskensHjort:08}). However, to the best of our knowledge, outside these frameworks, there is no theoretical ground for the use of AICc. Building on considerations from the general nonasymptotic theory of model selection developed during the nineties (see for instance \cite{BarBirMassart:99} and \cite{Massart:07}) and in particular on Castellan's analysis~\cite{Castellan:03}, Birg\'{e} and Rozenholc~\cite{BirRozen:06} have considered an AIC modification specifically designed for the selection of the bin size in histogram selection for density estimation. Indeed, results of \cite{Castellan:03}---and more generally results of \cite{BarBirMassart:99}---advocate to take into account in the design of penalty the number of models to be selected. The importance of the cardinality of the collection of models for model selection is in fact a very general phenomenon and one of the main outcomes of the nonasymptotic model selection theory. In the bin size selection problem, this corresponds to adding a small amount to AIC. Unfortunately, the theory does not specify uniquely the term to be added to AIC. On the contrary, infinitely many corrections are accepted by the theory and in order to choose a good one, intensive experiments were conducted in~\cite{BirRozen:06}. The resulting AIC correction therefore always has the disadvantage of being specifically designed for the task on which it has been tested. We propose a general approach that goes beyond the limits of the unbiased risk estimation principle. The latter principle is indeed at the core of Akaike's model selection procedure and is more generally the main model selection principle, which underlies procedures such as Stein's Unbiased Risk Estimator (SURE, \cite{MR630098}) or cross-validation (\cite{ArlotCelisse:10}). We point out that it is more efficient to estimate a quantile of the risk of the estimators---the level of the quantile depending on the size of the collection of models---than its mean. We also develop a (pseudo-)testing point of view, where we find that unbiased risk estimation does not in general allow to control the sizes of the considered tests. This is thus a new, very general model selection principle that we put forward and formalize. We call it an over-penalization procedure, because it systematically involves adding small terms to traditional penalties such as AIC. Our procedure consists in constructing an estimator from multiple pseudo-tests, built on some random events. From this perspective, our work shares strong connections recent advances in robust estimation by designing estimators from tests (\cite{MR2449129,MR2219712,MR3186748,BarBirgSart:17,lecue2017learning}). However, our focus and perspectives are significantly different from this existing line of works. Indeed, estimators such as T-estimators (\cite{MR2219712}) or $\rho$-estimators (\cite{BarBirgSart:17}) are quasi-universal estimators in the sense that they have very strong statistical guarantees, but they have the drawback to be very difficult---if feasible---to compute. In particular, \cite{MR2449129} also builds estimators from frequency histograms, but to our knowledge no implementation of such estimators exists and it seems to be an open question whether a polynomial time algorithm can effectively compute them or not. Here, we rather keep the tractability of AIC procedure, but we don't look particularly at robustness properties (for instance against outliers). We focus on improving AIC in the nonasymptotic regime. In addition, it is worth noting that several authors have examined some links between multiple testing and model selection, in particular by making some modifications to classical criteria (see for instance \cite[Chapter 7]{MR3184277}). But these lines of research differ significantly from our approach. Indeed, the first and main use in the literature of multiple testing point of view for model selection concerns variable selection, i.e. the identification of models, particularly in the context of linear regression. (\cite{MR921623, MR2280619,MR2183942,MR2823520,MR2281879,MR2668704}). It consists in considering simultaneously the testing of each variable being equal to zero or not. Instead, we consider model selection from a predictive perspective and do not focus on model identification. It should also be noted that multiple tests may be considered after model selection or at the same time as selective inference (\cite{MR2395714,MR3174645,MR2156820}), but these questions are not directly related to the scope of our paper. Lets us now detail our contributions. \begin{itemize} \item We propose a general formulation of the model selection task for prediction in terms of a (pseudo-)test procedure (understood in a non classical way which will be detailed in the Section~\ref{ssection_testing}), thus establishing a link between two major topics of contemporary research. In particular, we propose a generic property that the pseudo-tests collection should satisfy in order to ensure an oracle inequality for the selected model. We call this property the ``transitivity property'' and show that it generalizes penalization procedures together with T-estimators and $\rho $-estimators. \item Considering the problem of density estimation by selecting a histogram, we prove a sharp and fully nonasymptotic oracle inequality for our procedure. Indeed, we describe a control of Kullback-Leibler (KL) divergence---also called excess risk---of the selected histogram as soon as we have one observation. We emphasize that this very strong feature may not be possible when considering AIC. We also stressed that up to our knowledge, our oracle inequality is the first nonasymptotic result comparing the KL divergence of the selected model to the KL divergence of the oracle in an unbounded setting. Indeed, oracle inequalities in density estimation are generally expressed in terms of Hellinger distance---which is much easier to handle than the KL divergence, because it is bounded---for the selected model. \item In order to prove our oracle inequality, we improve upon the previously best known concentration inequality for the chi-square statistics (Castellan \cite{Castellan:03}, Massart \cite{Massart:07}) and this allows us to gain an order of magnitude in the control of the deviations of the excess risks of the estimators. Our result on the chi-square statistics is general and of independent interest. \item We also prove new Bernstein-type concentration inequalities for log-densities that are unbounded. Again, these probabilistic results, which are naturally linked to information theory, are general and of independent interest. \item We generalize previous results of Barron and Sheu \cite{BarSheu:91} regarding the existence of margin relations in maximum likelihood estimation (MLE). Indeed, related results of \cite{BarSheu:91} where established under boundedness of the log-densities and we extend them to unbounded log-densities with moment conditions. \item Finally, from a practical point of view, we bring a nonasymptotic improvement of AIC that has, in its simplest form, the same computational cost as AIC. Furthermore, we show that our over-penalization procedure largely outperforms AIC on small and medium sample sizes, but also surpasses existing AIC corrections such as AICc or Birg\'{e}-Rozenholc's procedure. \end{itemize} Let us end this introduction by detailing the organization of the paper. We present our over-penalization procedure in Section~\ref{section_framework_and_notations_MLE}. More precisely, we detail in Sections~\ref{section_max_vrais_hist_MLE} and \ref{section_histo} our model selection framework related to MLE via histograms. Then in Section~\ref{section_overpen} we define formally over-penalization procedures and highlight their generality. We explain the ideas underlying over-penalization from three different angles: estimation of the ideal penalty, pseudo-testing and a graphical point of view. Section~\ref{section_results} is devoted to statistical guarantees related to over-penalization. In particular, as concentration properties of the excess risks are at the heart of the design of an over-penalization, we detail them in Section~\ref{section_risks_bounds_MLE}. We then deduce a general and sharp oracle inequality in Section~\ref{section_results_bounded_setting} and highlight the theoretical advantages compared to an AIC analysis. New mathematical tools of a probabilistic and analytical nature and of independent interest are presented in Section~\ref{section_prob_tools}. Section~\ref{section_expe} contains the experiments, with detailed practical procedures. We consider two different practical variations of over-penalization and compare them with existing penalization procedures. The superiority of our method is particularly transparent. The proofs are gathered in Section~\ref{section_proofs_MLE}, as well as in the Appendix~\ref{sec:appendix} which provides further theoretical developments that extend the description of our over-penalization procedure. \section{Statistical Framework and Notations\label% {section_framework_and_notations_MLE}} The over-penalization procedure, described in Section~\ref{section_overpen}, is legitimated at a heuristic level within a generic M-estimator selection framework. We put to emphasis in Section~\ref{section_max_vrais_hist_MLE} on maximum likelihood estimation (MLE) since, as proof of concept for our over-penalization procedure, our theoretical and experimental results will address the case of bin size selection for maximum likelihood histogram selection in density estimation. In order to be able to discuss in Section~\ref{section_overpen} the generality of our approach in an M-estimation setting, our presentation of MLE brings notations which extend directly to M-estimation with a general contrast. \subsection{Maximum Likelihood Density Estimation} \label{section_max_vrais_hist_MLE} We are given $n$ independent observations $\left( \xi_{1},\ldots,\xi_{n}\right)$ with unknown common distribution $P$ on a measurable space $\left( \mathcal{Z},\mathcal{T}\right) $. We assume that there exists a known probability measure $\mu$ on $\left(\mathcal{Z},\mathcal{T}\right) $ such that $P$ admits a density $f_{\ast }$ with respect to $\mu$: $f_{\ast}=dP/d\mu $. Our goal is to estimate the density $f_{\ast } $. For an integrable function $f$ on $\mathcal{Z}$, we set $Pf=P\left( f\right) =\int_{\mathcal{Z}}f\left( z\right) dP\left( z\right) $ and $\mu f=\mu \left( f\right) =\int_{\mathcal{Z}}f\left( z\right) d\mu \left( z\right) $. If $P_{n}=1/n\sum_{i=1}^{n}\delta _{\xi _{i}}$ denotes the empirical distribution associated to the sample $\left( \xi _{1},\ldots ,\xi _{n}\right) $, then we set $P_{n}f=P_{n}\left( f\right) =1/n\sum_{i=1}^{n}f\left( \xi _{i}\right) $. Moreover, taking the conventions $\ln 0=-\infty $, $0\ln 0=0$ and defining the positive part as $% \left( x\right) _{+}=x\vee 0$, we set \begin{equation*} \mathcal{S}=\left\{ f:\mathcal{Z}\longrightarrow \mathbb{R}_{+}\text{; }% \int_{\mathcal{Z}}fd\mu =1\text{ and }P\left( \ln f\right) _{+}<\infty \right\} \text{ .} \end{equation*}% We assume that the unknown density $f_{\ast }$ belongs to $\mathcal{S}$. Note that since $P\left(\ln f_{\ast }\right) _{-}=\int f_{\ast }\ln f_{\ast}% \mathbbm{1} _{f_{\ast }\leq 1}d\mu <\infty $, the fact that $f_{\ast }$ belongs to $\mathcal{S}$ is equivalent to $\ln (f_{\ast })\in L_{1}\left(P\right)$, the space of integrable functions on $\mathcal{Z}$ with respect to $P$. We consider the MLE of the density $f_{\ast }$. To do so, we define the maximum likelihood contrast $\gamma $ to be the following functional,% \begin{equation*} \gamma :f\in \mathcal{S}\longmapsto \left( z\in \mathcal{Z\longmapsto }-\ln \left( f\left( z\right) \right) \right) \text{ .} \end{equation*}% Then the risk $P\gamma \left( f\right) $ associated to the contrast $\gamma $ on a function $f\in \mathcal{S}$ is the following, \begin{equation*} P\gamma \left( f\right) =P\left( \ln f\right) _{-}-P\left( \ln f\right) _{+}\in \mathbb{R\cup \left\{ +\infty \right\} }\text{ .} \end{equation*} Also, the excess risk of a function $f$ with respect to the density $f_{\ast }$ is classically given in this context by the KL divergence of $f$ with respect to $f_{\ast }$. Recall that for two probability distributions $P_{f}$ and $P_{g}$ on $\left( \mathcal{Z},\mathcal{T}\right) $ of respective densities $f$ and $g$ with respect to $\mu $, the KL divergence of $P_{g}$ with respect to $P_{f}$ is defined to be \begin{equation*} \mathcal{K}\left( P_{f},P_{g}\right) =% \begin{cases} \int_{\mathcal{Z}}\ln \left( \frac{dP_{f}}{dP_{g}}\right) dP_{g}=\int_{% \mathcal{Z}}f\ln \left( \frac{f}{g}\right) d\mu & \text{if}\quad P_{f}\ll P_{g} \\ \infty & \text{otherwise.}% \end{cases}% \end{equation*}% By a slight abuse of notation we denote $\mathcal{K}\left( f,t\right) $ rather than $\mathcal{K}\left( P_{f},P_{g}\right) $ and by the Jensen inequality we notice that $\mathcal{K}\left( f,g\right) $ is a nonnegative quantity, equal to zero if and only if $f=g$ $\ \mu $-$a.s.$ Hence, for any $% f\in \mathcal{S}$, the excess risk of a function $f$ with respect to the density $f_{\ast }$ satisfies \begin{equation} P\left( \gamma (f))-P(\gamma \left( f_{\ast }\right) \right) =\int_{\mathcal{% Z}}\ln \left( \frac{f_{\ast }}{f}\right) f_{\ast }d\mu =\mathcal{K}\left( f_{\ast },f\right) \geq 0 \label{def_excess_risk_kullback} \end{equation}% and this nonnegative quantity is equal to zero if and only if $f_{\ast }=f$ $% \ \mu -a.s.$ Consequently, the unknown density $f_{\ast }$ is uniquely defined by \begin{align*} f_{\ast }& =\arg \min_{f\in \mathcal{S}}\left\{ P\left( -\ln f\right) \right\} \\ & =\arg \min_{f\in \mathcal{S}}\left\{ P\gamma \left( f\right) \right\} \text{ .} \end{align*}% For a model $m$, that is a subset $m\subset \mathcal{S}$, we define the maximum likelihood estimator on $m$, whenever it exists, by \begin{align} \hat{f}_{m}& \in \arg \min_{f\in m}\left\{ P_{n}\gamma \left( f\right) \right\} \label{def_estimator_MLE} \\ & =\arg \min_{f\in m}\left\{ \frac{1}{n}\sum_{i=1}^{n}-\ln \left( f\left( \xi _{i}\right) \right) \right\} \text{ }. \notag \end{align} \subsection{Histogram Models\label{section_histo}} The models $m$ that we consider here to define the maximum likelihood estimators as in (\ref{def_estimator_MLE}) are made of histograms defined on a fixed partition of $\mathcal{Z}$. More precisely, for a finite partition $% \Lambda _{m}$ of $\mathcal{Z}$ of cardinality $\left\vert \Lambda _{m}\right\vert =D_{m}+1$, $D_{m}\in \mathbb{N}$, we set \begin{equation*} m=\left\{ f=\sum_{I\in \Lambda _{m}}\beta _{I}\mathbbm{1}_{I}\text{ ; }% \left( \beta _{I}\right) _{I\in \Lambda _{m}}\in \mathbb{R}_{+}^{D_{m}+1},% \text{ }f\geq 0\text{ and }\sum_{I\in \Lambda _{m}}\beta _{I}\mu \left( I\right) =1\right\} \text{ .} \end{equation*}% Note that the smallest affine space contained in $m$ is of dimension $D_{m}$% . The quantity $D_{m}$ can thus be interpreted as the number of degrees of freedom in the (parametric) model $m$. We assume that any element $I$ of the partition $\Lambda _{m}$ is of positive measure with respect to $\mu $: for all $I\in \Lambda _{m}$, \ \ $\mu \left( I\right) >0$ . As the partition $% \Lambda _{m}$ is finite, we have $P\left( \ln f\right) _{+}<\infty $ for all $f\in m$ and so $m\subset \mathcal{S}$. We state in the next proposition some well-known properties that are satisfied by histogram models submitted to the procedure of MLE (see for example \cite[Section 7.3]{Massart:07}). \begin{proposition} Let \begin{equation*} f_{m}=\sum_{I\in \Lambda _{m}}\frac{P\left( I\right) }{\mu \left( I\right) }% \mathbf{1}_{I}\text{ }. \end{equation*}% Then $f_{m}\in m$ and $f_{m}$ is called the KL projection of $f_{\ast }$ onto $m$. Moreover, it holds \begin{equation*} f_{m}=\arg \min_{f\in m}P\left( \gamma (f)\right) \text{ .} \end{equation*}% The following Pythagorean-like identity for the KL divergence holds, for every $f\in m$, \begin{equation} \mathcal{K}\left( f_{\ast },f\right) =\mathcal{K}\left( f_{\ast },f_{m}\right) +\mathcal{K}\left( f_{m},f\right) \text{ .} \label{pythagore_relation_MLE} \end{equation}% The maximum likelihood estimator on $m$ is well-defined and corresponds to the so-called frequencies histogram associated to the partition $\Lambda _{m}$. We also have the following formulas,% \begin{equation*} \hat{f}_{m}=\sum_{I\in \Lambda _{m}}\frac{P_{n}\left( I\right) }{\mu \left( I\right) }\mathbbm{1}_{I}\text{ and } P_n(\gamma(f_m)-\gamma(\hat{f}_m))=\mathcal{K}(\hat{f}_m,f_m)\text{ }. \end{equation*}% \end{proposition} \begin{remark} Histogram models are special cases of general exponential families exposed for example in Barron and Sheu \cite{BarSheu:91} (see also Castellan \cite% {Castellan:03} for the case of exponential models of piecewise polynomials). The projection property (\ref{pythagore_relation_MLE}) can be generalized to exponential models (see \cite[Lemma 3]{BarSheu:91} and Csisz\'ar \cite% {Csiszar:75}). \end{remark} \begin{remark} As by \eqref{def_excess_risk_kullback} we have \begin{equation*} P\left( \gamma (f_{m})-\gamma \left( f_{\ast }\right) \right) =\mathcal{K}% \left( f_{\ast },f_{m}\right) \end{equation*} and for any $f\in m$, \begin{equation*} P\left( \gamma (f)-\gamma \left( f_{\ast }\right) \right) =\mathcal{K}\left( f_{\ast },f\right) \end{equation*}% we easily deduce from (\ref{pythagore_relation_MLE}) that the excess risk on $m$ is still a KL divergence, as we then have for any $f\in m$,% \begin{equation*} P\left( \gamma (f)-\gamma (f_{m})\right) =\mathcal{K}\left( f_{m},f\right) \text{ }. \end{equation*} \end{remark} \subsection{Over-Penalization} \label{section_overpen} Now let's define our model selection procedure. We propose three ways to understand the benefits of over-penalization. Of course, the three points of view are interrelated, but they provide different and complementary insights on the behavior of over-penalization. \subsubsection{Over-Penalization as Estimation of the Ideal Penalty}\label{ssection_ideal_pen} We are given a collection of histogram models denoted $\mathcal{M}_{n}$, with finite cardinality depending on the sample size $n$, and its associated collection of maximum likelihood estimators $\left\{ \hat{f}_{m};m\in \mathcal{M}_{n}\right\} $. By taking a (nonnegative) penalty function $\pen$ on $\mathcal{M}_{n}$, \begin{equation*} \pen:m\in \mathcal{M}_{n}\longmapsto \pen\left( m\right) \in \mathbb{R}^{+}% \text{ ,} \end{equation*}% the output of the penalization procedure (also called the selected model) is by definition any model satisfying, \begin{equation} \widehat{m}\in \arg \min_{m\in \mathcal{M}_{n}}\left\{ P_{n}(\gamma (\hat{f}% _{m}))+\pen\left( m\right) \right\} \text{ .} \label{def_proc_2_MLE} \end{equation}% We aim at selecting an estimator $\hat{f}_{\widehat{m}}$ with a KL divergence, pointed on the true density $f_{\ast }$, as small as possible. Hence, we want our selected model to have a performance as close as possible to the excess risk achieved by an oracle model (possibly non-unique), defined to be, \begin{align} m_{\ast }\in & \arg \min_{m\in \mathcal{M}_{n}}\left\{ \mathcal{K}(f_{\ast },% \hat{f}_{m})\right\} \label{def_oracle} \\ =& \arg \min_{m\in \mathcal{M}_{n}}\left\{ P(\gamma (\hat{f}_{m}))\right\} \text{ .} \label{def_oracle_2} \end{align}% From (\ref{def_oracle_2}), it is seen that an ideal penalty in the optimization task (\ref{def_proc_2_MLE}) is given by% \begin{equation*} \pen_{\mathrm{id}}\left( m\right) =P(\gamma (\hat{f}_{m}))-P_{n}(\gamma (% \hat{f}_{m}))\text{ ,} \end{equation*}% since in this case, the criterion $\crit_{\mathrm{id}}\left( m\right) =P_{n}(\gamma (\hat{f}_{m}))+\pen_{\text{id}}\left( m\right) $ is equal to the true risk $P(\gamma (\hat{f}_{m}))$. However $\pen_{\mathrm{id}}$ is unknown and, at some point, we need to give some estimate of it. In addition, $\pen_{\mathrm{id}}$ is random, but we may not be able to provide a penalty, even random, whose fluctuations at a fixed model $m$ would be positively correlated to the fluctuations of $\pen_{\text{id}}\left( m\right) $. This means that we are rather searching for an estimate of a \textit{deterministic functional} of $\pen_{\mathrm{id}}$. But which functional would be convenient? The answer to this question is essentially contained in the solution of the following problem. \medskip \noindent \textbf{Problem 1. }\textit{For any fixed} $\beta \in \left( 0,1\right) $ \textit{find the} deterministic penalty $\pen_{\mathrm{id}% ,\beta }:\mathcal{M}_{n}\rightarrow R_{+}$, \textit{that minimizes the value of} $C$, \textit{among constants} $C>0$ \textit{which satisfy the following oracle inequality}, \begin{equation} \mathbb{P}\left( \mathcal{K}(f_{\ast },\hat{f}_{\widehat{m}})\leq C\inf_{m\in \mathcal{M}_{n}}\left\{ \mathcal{K(}f_{\ast },\hat{f}% _{m})\right\} \right) \geq 1-\beta \text{ .} \label{problem_beta} \end{equation} \noindent The solution---or even the existence of a solution---to the problem given in (\ref{problem_beta}) is not easily accessible and depends on assumptions on the law $P$ of data and on approximation properties of the models, among other things. In the following, we give a reasonable candidate for $\pen_{\mathrm{id},\beta }$. Indeed, let us set $\beta _{\mathcal{M}% }=\beta /$Card$\left( \mathcal{M}_{n}\right) $ and define \begin{equation} \pen_{\mathrm{opt},\beta }\left( m\right) =q_{1-\beta _{\mathcal{M}}}\left\{ P(\gamma (\hat{f}_{m})-\gamma (f_{m}))+P_{n}(\gamma (f_{m})-\gamma (\hat{f}% _{m}))\right\} \text{ ,} \label{pen_opt_quantile} \end{equation}% where $q_{\lambda }\left\{ Z\right\} =\inf \left\{ q\in \mathbb{R};\mathbb{P}% \left( Z\leq q\right) \geq \lambda \right\} $ is the quantile of level $% \lambda $ for the real random variable $Z$. Our claim is that $\pen_{\text{% opt,}\beta }$ gives in (\ref{problem_beta}) a constant $C$ which is close to one, under some general assumptions (see Section \ref{section_results} for precise results). Let us explain now why $\pen_{\mathrm{opt},\beta }$ should lead to a nearly optimal model selection. We set% \begin{equation*} \Omega _{0}=\bigcap_{m\in \mathcal{M}_{n}}\left\{ P(\gamma (\hat{f}% _{m})-\gamma (f_{m}))+P_{n}(\gamma (f_{m})-\gamma (\hat{f}_{m}))\leq \pen_{% \mathrm{opt},\beta }\left( m\right) \right\} \text{ .} \end{equation*}% We see, by definition of $\pen_{\mathrm{opt},\beta }$ and by a simple union bound over the models $m\in \mathcal{M}_{n}$, that the event $\Omega _{0}$ is of probability at least $1-\beta $. Now, by definition of $\widehat{m}$, we have, for any $m\in \mathcal{M}_{n}$, \begin{equation} P_{n}(\gamma (\hat{f}_{\widehat{m}}))+\pen_{\mathrm{opt},\beta }(\widehat{m}% )\leq P_{n}(\gamma (\hat{f}_{m}))+\pen_{\mathrm{opt},\beta }\left( m\right) \text{ .} \label{def_min_crit} \end{equation}% By centering by $P\left( \gamma \left( f_{\ast }\right) \right) $ and using simple algebra, Inequality (\ref{def_min_crit}) can be written as, \begin{align*} & P(\gamma (\hat{f}_{\widehat{m}})-\gamma \left( f_{\ast }\right) )+\left[ % \pen_{\mathrm{opt},\beta }(\widehat{m})-(P(\gamma (\hat{f}_{\widehat{m}% })-\gamma (f_{\widehat{m}}))+P_{n}(\gamma (f_{\widehat{m}})-\gamma (\hat{f}_{% \widehat{m}})))\right] \\ \leq & P(\gamma (\hat{f}_{m})-\gamma \left( f_{\ast }\right) )+\left[ \pen_{% \mathrm{opt},\beta }\left( m\right) -(P(\gamma (\hat{f}_{m})-\gamma (f_{m}))+P_{n}(\gamma (f_{m})-\gamma (\hat{f}_{m})))\right] \\ & +\left( P_{n}-P\right) \left( \gamma (f_{m})-\gamma (f_{\widehat{m}% })\right) \text{ .} \end{align*}% Now, on $\Omega _{0}$, we have $\pen_{\mathrm{opt},\beta }(\widehat{m}% )-(P(\gamma (\hat{f}_{\widehat{m}})-\gamma (f_{\widehat{m}}))+P_{n}(\gamma (f_{\widehat{m}})-\gamma (\hat{f}_{\widehat{m}})))\geq 0$, so we get on $% \Omega _{0}$, \begin{eqnarray*} &&P(\gamma (\hat{f}_{\widehat{m}})-\gamma \left( f_{\ast }\right) ) \\ &\leq &P(\gamma (\hat{f}_{m})-\gamma \left( f_{\ast }\right) )+\left[ \pen_{% \mathrm{opt},\beta }\left( m\right) -(P(\gamma (\hat{f}_{m})-\gamma (f_{m}))+P_{n}(\gamma (f_{m})-\gamma (\hat{f}_{m})))\right] \\ &&+\left( P_{n}-P\right) \left( \gamma (f_{m})-\gamma (f_{\widehat{m}% })\right) \text{ .} \end{eqnarray*}% Specifying to the MLE context, the latter inequality writes,% \begin{align*} \mathcal{K(}& f_{\ast },\hat{f}_{\widehat{m}}) \\ \leq & \mathcal{K(}f_{\ast },\hat{f}_{m})+\underset{(a)}{\underbrace{\left[ % \pen_{\mathrm{opt},\beta }\left( m\right) -(\mathcal{K(}f_{m},\hat{f}_{m})+% \mathcal{K(}\hat{f}_{m},f_{m}))\right] }}+\underset{(b)}{\underbrace{\left( P_{n}-P\right) \left( \gamma (f_{m})-\gamma (f_{\widehat{m}})\right) }}\text{ .} \end{align*} In order to get an oracle inequality as in (\ref{problem_beta}), it remains to control $(a)$ and $(b)$ in terms of the excess risks $\mathcal{K(}f_{\ast },\hat{f}_{m})$ and $\mathcal{K(}f_{\ast },\hat{f}_{\widehat{m}})$. Quantity $(a)$ is related to deviations bounds for the true and empirical excess risks of the M-estimators $\hat{f}_{m}$ and quantity $(b)$ is related to fluctuations of empirical bias around the bias of the models. Suitable controls of these quantities (as achieved in our proofs) will give sharp oracle inequalities (see Section \ref{section_results} below). \begin{remark} Notice that our reasoning is not based on the particular value of the contrast, so that to emphasize this point we choose to keep $\gamma$ in most of our calculations rather than to specify to the KL divergence related to the MLE case. As a matter of fact, the penalty $\pen_{\mathrm{opt},\beta } $ given in (\ref{pen_opt_quantile}) is a good candidate in the general context of M-estimation. \end{remark} We define an over-penalization procedure as follows. \begin{definition} A penalization procedure as defined in (\ref{def_proc_2_MLE}) is said to be an over-penalization procedure if the penalty $\pen$ that is used satisfies $% \pen\left( m\right) \geq \pen_{\mathrm{opt},\beta }\left( m\right) $ for all $m\in \mathcal{M}_{n}$ and for some $\beta \in \left( 1/2,1\right) $. \end{definition} Based on concentration inequalities for the excess risks (see Section \ref% {section_risks_bounds_MLE}) we propose the following over-penalization penalty for histogram selection,% \begin{equation} \pen_{+}\left( m\right) =\left( 1+C\max \left\{ \sqrt{\frac{D_{m}\ln (n+1)}{n% }};\sqrt{\frac{\ln (n+1)}{D_{m}}};\frac{\ln (n+1)}{D_{m}}\right\} \right) \frac{D_{m}}{n}\text{ ,} \label{pen_a} \end{equation}% where $C$ is a constant that should be either fixed a priori ($C=1$ or $2$ are typical choices) or estimated using data (see Section \ref{section_expe} for details about the choice of $C$). The logarithmic terms appearing in (\ref{pen_a}) are linked to the cardinal of the collection of models, since in our proofs we take a constant $\alpha$ such that $\ln\rm{Card} (\mathcal{M}_n) + 2\ln(n+1) \leq \alpha \ln(n+1)$. The constant $\alpha$ then enters in the constant $C$ of (\ref{pen_a}). We show below nonasymptotic accuracy of such procedure, both theoretically and practically. \subsubsection{Over-Penalization through a pseudo-testing approach\label% {ssection_testing}} Let us now present a multiple test point of view on the model selection problem. The goal is to infer the oracle model (\ref{def_oracle}) so that an oracle inequality of the type of (\ref{problem_beta}) is ensured. This task can be formulated by solving iterative pseudo-tests. Indeed, set the following collection of null and alternative hypotheses indexed by pairs of models: for $\left( m,m^{\prime }\right)\in\mathcal{M}_{n}^{2}$, \begin{equation*} \left\{ \begin{array}{c} H_{0}\left( m,m^{\prime }\right) :\mathcal{K}(f_{\ast },\hat{f}% _{m})>C_{n}\cdot \mathcal{K}(f_{\ast },\hat{f}_{m^{\prime }}) \\ H_{1}\left( m,m^{\prime }\right) :\mathcal{K}(f_{\ast },\hat{f}_{m})\leq C_{n}\cdot \mathcal{K}(f_{\ast },\hat{f}_{m^{\prime }})% \end{array}% \right. \text{ ,} \end{equation*}% where the constant $C_{n}>1$ is uniform in $m,m^{\prime }\in \mathcal{M}_{n}$. To each pair $\left( m,m^{\prime }\right) \in \mathcal{M}_{n}^{2}$, let us assume that we are given a test $T\left( m,m^{\prime }\right) $ that is equal to one if $H_{0}\left( m,m^{\prime }\right) $ is rejected and zero otherwise. It should be noted at this stage that what we have just called ``pseudo-test'' does not enter directly into the classical theory of statistical tests, since the null and alternative hypotheses that we consider are random events. However, as we will see, the only notion related to our pseudo-tests needed for our model selection study is the notion of the ``size'' of a pseudo-test, which we will give in the following and which will provide a mathematically consistent analysis of the statistical situation. Moreover,it seems that random assumptions naturally arise in some statistical frameworks. For instance, in the context of variable selection along along the Lasso path, Lockhart \textit{et al.}\cite{MR3210970} consider sequential random null hypotheses based on the active sets of the variable included up to a step on the Lasso path (see especially \cite[Section 2.6]{MR3210970} as well as the discussion of B\"{u}hlmann \textit{et al.} \cite[Section 2]{MR3210971} and the rejoinder \cite[Section 2]{MR3210977}). Finally, if the testing of random hypotheses disturbs the reader, we suggest taking our multiple pseudo-testing interpretation of the model selection task in its minimal sense, that is, as a mathematical description aiming at investigating tight conditions on the penalty, that would allow for near-optimal oracle guarantees for the penalization scheme (\ref{def_proc_2_MLE}). In order to ensure an oracle inequality such as in (\ref{problem_beta}), we want to avoid as far as possible selecting a model whose excess risk is far greater than the one of the oracle. In terms of the preceding tests, we will see that this exactly corresponds to controlling the ``size'' of the pseudo-tests $T\left( m,m^{\prime }\right) $. Let us note $\mathcal{R}\left( m,m^{\prime }\right) =\left\{ T\left( m,m^{\prime }\right) =1\right\} $ the event where the pseudo-test $T\left( m,m^{\prime }\right) $ rejects the null hypothesis $H_{0}\left( m,m^{\prime }\right) $ and $\mathcal{T}\left( m,m^{\prime }\right) =\left\{ \mathcal{K}% (f_{\ast },\hat{f}_{m})>C_{n}\cdot \mathcal{K}(f_{\ast },\hat{f}_{m^{\prime }})\right\} $ the event where the hypothesis $H_{0}\left( m,m^{\prime }\right) $ is true. By extension with the classical theory of statistical testing, we denote $\alpha \left( m,m^{\prime }\right) $ the size of the pseudo-test $T\left( m,m^{\prime }\right) $, given by% \begin{equation*} \alpha \left( m,m^{\prime }\right) =\mathbb{P}\left( \mathcal{R}\left( m,m^{\prime }\right) \left\vert \mathcal{T}\left( m,m^{\prime }\right) \right. \right) \text{ .} \end{equation*} To explain how we select a model $\widehat{m}$ that is close to the oracle model $m_{\ast }$, let us enumerate the models of the collection: $\mathcal{M% }_{n}=\left\{ m_{1},...,m_{k}\right\} $. Note that we do not assume that the models are nested. Now, we test $T\left( m_{1},m_{j}\right) $ for $j$ increasing from $2$ to $k$. If there exists $i_{1}\in \left\{ 2,..,k\right\} $ such that $T\left( m_{1},m_{i_{1}}\right) =0$, then we perform the pseudo-tests $% T\left( m_{i_{1}},m_{j}\right) $ for $j$ increasing from $i_{1}+1$ to $k$ or choose $m_{k}$ as our oracle candidate if $i_{1}=k$. Otherwise, we choose $% m_{1}$ as our oracle candidate. In general, we can thus define a finite increasing sequence $m_{i_{l}}$ of models and a selected model $\widehat{m}=% \widehat{m}\left( T\right) $ through the use of the collection of pseudo-tests $% \left\{ T\left( m,m^{\prime }\right) ;m,m^{\prime }\in \mathcal{M}% _{n}\right\} $. Also, the number of pseudo-tests that are needed to define $% \widehat{m}$ is equal to $% \card% \left( \mathcal{M}_{n}\right) -1$. \begin{figure}[tbp] \centering \includegraphics[width=0.7\textwidth]{arbre} \caption{Iteration of the pseudo-tests along the collection of models.} \end{figure} Let us denote $\mathcal{P}$ the set of pairs of models that have effectively been tested along the iterative procedure. We thus have $% \card% (\mathcal{P})=% \card% \left( \mathcal{M}_{n}\right) -1$. Now define the event $\Omega _{T}$ under which there is a first kind error along the collection of pseudo-tests in $\mathcal{% P}$,% \begin{equation*} \Omega _{T}=\bigcup_{\left( m,m^{\prime }\right) \in \mathcal{P}}\left[ \mathcal{R}\left( m,m^{\prime }\right) \bigcap \mathcal{T}\left( m,m^{\prime }\right) \right] \end{equation*}% and assume that the sizes of the pseudo-tests are chosen such that \begin{equation} \beta =\sum_{\left( m,m^{\prime }\right) \in \mathcal{M}^2_n}\alpha \left( m,m^{\prime }\right) <1\text{ .} \label{control_sizes} \end{equation} Assume also that the selected model $\widehat{m}$ has the following Transitivity Property (\textbf{TP}):\medskip \begin{description} \item[(TP)] If for any $(m,m^{\prime })\in \mathcal{M}_{n}^{2},$ $T\left( m,m^{\prime }\right) =1$ then $T\left( \widehat{m},m^{\prime }\right) =1$% .\medskip \end{description} We believe that the transitivity property (\textbf{TP}) is intuitive and legitimate as it amounts to assuming that there is no contradiction in choosing $\widehat{m}$ as a candidate oracle model. Indeed, if a model $m$ is thought to be better than a model $m^{\prime }$, that is $T\left( m,m^{\prime }\right) =1$, then the selected model $\widehat{m}$ should also be thought to be better that $m^{\prime }$, $T\left( \widehat{m},m^{\prime }\right) =1$. The power of the formalism we have just introduced lies in the fact that the combination of Assumptions (\ref{control_sizes}) and (\textbf{TP}) automatically implies that an oracle inequality as in (\ref{problem_beta}) is satisfied. Indeed, property (\textbf{TP}) ensures that $\mathcal{T}\left( \widehat{m},m_{\ast }\right) \subset \mathcal{R}\left( \widehat{m},m_{\ast }\right) $ because on the event $\mathcal{T}\left( \widehat{m},m_{\ast }\right) $, we have $\widehat{m}\neq m_{\ast }$, so there exists $m$ such that $\left( m,m_{\ast }\right) \in \mathcal{P}$ and $T\left( m,m_{\ast }\right) =1$ (otherwise $\widehat{m}=m_{\ast }$) which in turn ensures $% T\left( \widehat{m},m_{\ast }\right) =1$. Consequently, $\mathcal{T}\left( \widehat{m},m_{\ast }\right) =\mathcal{T}\left( \widehat{m},m_{\ast }\right) \cap \mathcal{R}\left( \widehat{m},m_{\ast }\right) \subset \Omega _{T}$, which gives% \begin{equation} \mathbb{P}\left( \mathcal{K}(f_{\ast },\hat{f}_{\widehat{m}})>C_{n}\cdot \mathcal{K}(f_{\ast },\hat{f}_{m_{\ast }})\right) =\mathbb{P}\left( \mathcal{% T}\left( \widehat{m},m_{\ast }\right) \right) \leq \mathbb{P}\left( \Omega _{T}\right) \leq \beta \text{ ,} \label{oracle_test} \end{equation}% that is equivalent to (\ref{problem_beta}). Let us turn now to a specific choice of pseudo-tests corresponding to model selection by penalization. If we define for a penalty $% \pen% $, the penalized criterion $% \crit% _{% \pen% }\left( m\right) =P_{n}(\gamma (\hat{f}_{m}))+% \pen% \left( m\right) $, $m\in \mathcal{M}_{n}$ and take the following pseudo-tests% \begin{equation*} T\left( m,m^{\prime }\right) =T_{% \pen% }\left( m,m^{\prime }\right) =\mathbbm{1} _{\left\{ \crit% _{% \pen% }\left( m\right) \leq \crit% _{% \pen% }\left( m^{\prime }\right) \right\} }\text{ ,} \end{equation*}% then it holds% \begin{equation*} \widehat{m}(T_{% \pen% })\in \arg \min_{m\in \mathcal{M}_{n}}\left\{ P_{n}(\gamma (\hat{f}_{m}))+% \pen% \left( m\right) \right\} \end{equation*}% and property (\textbf{TP}) is by consequence satisfied. It remains to choose the penalty $% \pen% $ such that the sizes of the pseudo-tests $T_{% \pen% }\left( m,m^{\prime }\right) $ are controlled. This is achieved by taking $% \pen% =% \pen% _{\text{opt},\tilde{\beta} /2}$ as defined in (\ref{pen_opt_quantile}), with $\tilde{\beta}=\beta/\text{Card}(\mathcal{M}_n)$. Indeed, in this case,% \begin{align} \alpha \left( m,m^{\prime }\right) =&\mathbb{P(}P_{n}(\gamma (\hat{f}_{m}))+% \pen\left( m\right) \leq P_{n}(\gamma (\hat{f}_{m^{\prime }}))+\pen\left( m^{\prime }\right) \left\vert \mathcal{T}\left( m,m^{\prime }\right)\right. ) \notag \\ \approx&\mathbb{P(}\mathcal{K}(f_{\ast },\hat{f}_{m})+\left[ \pen_{\text{opt,% }\tilde{\beta} /2}\left( m\right) -(\mathcal{K}(f_{m},\hat{f}_{m})+\mathcal{K}(\hat{f% }_{m},f_{m}))\right] \label{approx} \\ &\leq \mathcal{K}(f_{\ast },\hat{f}_{m^{\prime }})+\left[ \pen_{\mathrm{opt}% ,\tilde{\beta}/2}\left( m^{\prime }\right) -(\mathcal{K}(f_{m^{\prime }},\hat{f}% _{m^{\prime }})+\mathcal{K}(\hat{f}_{m^{\prime }},f_{m^{\prime }}))\right] \left\vert \mathcal{T}\left( m,m^{\prime }\right) \right. ) \notag \\ \leq & \frac{\beta }{2\text{Card}\left( \mathcal{M}^2_{n}\right)} +\mathbb{P}% \left(\mathcal{K}(f_{\ast },\hat{f}_{m}) \leq \mathcal{K}(f_{\ast },\hat{f}% _{m^{\prime }})\right. \notag \\ &\hspace{2.8cm}\left.+\left[ \pen_{\mathrm{opt},\tilde{\beta} /2}\left( m^{\prime}\right) -(\mathcal{K}(f_{m^{\prime }},\hat{f}_{m^{\prime }})+% \mathcal{K}(\hat{f}_{m^{\prime }},f_{m^{\prime }}))\right] \left\vert \mathcal{T}\left(m,m^{\prime }\right)\right.\right)\text{ .} \notag \end{align}% In line (\ref{approx}), the equality is only approximated since we neglected the centering of model biases by their empirical counterparts, as these centered random variables should be small compared to the other quantities for models of interest. Now assume that \begin{equation*} \mathbb{P}\left( \pen_{\mathrm{opt},\tilde{\beta} /2}\left( m^{\prime }\right) -(% \mathcal{K}(f_{m^{\prime }},\hat{f}_{m^{\prime }})+\mathcal{K}(\hat{f}% _{m^{\prime }},f_{m^{\prime }}))\geq \varepsilon _{n}\mathcal{K}(f_{\ast },% \hat{f}_{m^{\prime }})\right) \leq \frac{\beta }{2\text{Card}\left( \mathcal{% M}^2_{n}\right) }\text{ ,} \end{equation*}% for some deterministic sequence $\varepsilon _{n}$ not depending on $% m^{\prime }$. Such result is obtained in Section~\ref% {section_risks_bounds_MLE} and is directly related to the concentration behavior of the true and empirical excess risks. Then we get \begin{align*} &\mathbb{P}\left( \mathcal{K}(f_{\ast },\hat{f}_{m})\leq \mathcal{K}(f_{\ast },\hat{f}_{m^{\prime }})+\left[ \pen_{\mathrm{opt},\tilde{\beta}/2 }\left( m^{\prime }\right) -(\mathcal{K}(f_{m^{\prime }},\hat{f}_{m^{\prime }})+\mathcal{K}(% \hat{f}_{m^{\prime }},f_{m^{\prime }}))\right] \left\vert \mathcal{T}\left( m,m^{\prime }\right) \right. \right) \\ \leq &\frac{\beta }{2\text{Card}\left( \mathcal{M}^2_{n}\right) }+\mathbb{P}% \left( \mathcal{K}(f_{\ast },\hat{f}_{m})\leq (1+\varepsilon _{n})\mathcal{K}% (f_{\ast },\hat{f}_{m^{\prime }})\left\vert \mathcal{K}(f_{\ast },\hat{f}% _{m})>C_{n}\mathcal{K}(f_{\ast },\hat{f}_{m^{\prime }})\right. \right) \\ =&\frac{\beta }{2\text{Card}\left( \mathcal{M}^2_{n}\right) }\text{ ,} \end{align*}% where the last equality is valid if $C_{n}\geq 1+\varepsilon _{n}$. In this case,% \begin{equation*} \alpha \left( m,m^{\prime }\right) \leq \frac{\beta }{\text{Card}\left( \mathcal{M}^2_{n}\right) } \end{equation*}% and inequality (\ref{control_sizes}) is satisfied and so is the oracle inequality (\ref{oracle_test}). \begin{remark} There is a gap between the penalty $\pen_{\mathrm{opt},\beta}$ considered in Section \ref{ssection_ideal_pen} to ensure an oracle inequality and the penalty $\pen_{\mathrm{opt},\tilde{\beta}/2}$ defined in this section. This comes from the fact that using the pseudo-testing framework, we aim at controlling the probability of the event $\Omega_T$ under which there is a first kind error along the pseudo-tests performed in $\mathcal{P}$. Despite the fact that the set $\mathcal{P}$ consists in $\rm{Card}(\mathcal{M}_n)-1$ pseudo-tests, we give a bound that takes into account the $\rm{Card}(\mathcal{M}^2_n)$ pseudo-tests defined from the pairs of models. There is a possible loss here that consists in inflating the set $\mathcal{P}$ in order to make the union bound valid. However, such loss would only affect the constant $C$ in our over-penalization procedure (\ref{pen_a}) by a factor $2$, since the modification of the order of the quantile affects the penalty through a logarithmic factor. \end{remark} \begin{remark} The Transitivity Property (\textbf{TP}) allows to unify most the selection rules. Indeed, as soon as we want to select an estimator $\tilde{f}$ (or a model) that optimizes a criterion, \begin{equation*} \tilde{f}\in \arg \min_{f\in \mathcal{F}}\left\{ \crit% \left( f\right) \right\} \text{ ,} \end{equation*}% where $\mathcal{F}$ is a collection of candidate functions (or models), then $\tilde{f}$ is also defined by the collection of tests $T\left( f,f^{\prime }\right) =\mathbf{1}_{\left\{ \crit% \left( f\right) \leq \crit% \left( f^{\prime }\right) \right\} }$. In particular, in T-estimation (\cite% {MR2219712}) as well as in $\rho $-estimation \cite{BarBirgSart:17}) the estimator is indeed a minimizer of a criterion that is interpreted as a diameter of a subset of functions in $\mathcal{F}$. Note that in T-estimation, the criterion is itself constructed through the use of some (robust) tests, that by consequence do not act at the same level as our tests in this case. \end{remark} \subsubsection{Graphical insights on over-penalization} Finally, let us provide a graphic perspective on our over-penalization procedure. If the penalty $% \pen% $ is chosen accordingly to the unbiased risk estimation principle, then it should satisfy, for any model $m\in \mathcal{M}_{n}$,% \begin{equation*} \mathbb{E}\left[ P_{n}(\gamma (\hat{f}_{m}))+% \pen% \left( m\right) \right] \sim \mathbb{E}\left[ P\left( \gamma (\hat{f}% _{m})\right) \right] \text{ .} \end{equation*}% In other words, the curve $\mathcal{C}_{n}:m\mapsto P_{n}(\gamma (\hat{f}% _{m}))+% \pen% \left( m\right) $ fluctuates around its mean, which is essentially the curve $\mathcal{C}_{P}:m\mapsto \mathbb{E}\left[ P\left( \gamma (\hat{f}% _{m})\right) \right] $, see Figure~\ref{fig:fig1}. Furthermore, the largest is the model $m$, the largest are the fluctuations of $P_{n}(\gamma (\hat{f}% _{m}))=\mathcal{K}\left( \hat{f}_{m},f_{m}\right) +P_{n}(\gamma (f_{m}))$. This is seen for instance through the concentration inequalities established in Section \ref{section_deviation_bounds}\ below for the empirical excess risk $\mathcal{K}\left( \hat{f}_{m},f_{m}\right) $. Consequently, it can happen that the curve $\mathcal{C}_{n}$ is rather flat for the largest models and that the selected model is among the largest of the collection, see Figure~\ref{fig:fig1}. \begin{figure}[tbp] \centering \includegraphics[width=0.65\textwidth]{dessin-1} \caption{A schematic view of the situation corresponding to a selection procedure based on the unbiased risk principle. The penalized empirical risk (in red) fluctuates around the expectation of the true risk. The size of the deviations typically increase with the model size, making the shape of the curves possibly flat for the largest models of the collection. Consequently, the chosen model can potentially be very large and lead to overfitting.} \label{fig:fig1} \end{figure} By using an over-penalization procedure instead of the unbiased risk estimation principle, we will compensate the deviations for the largest models and thus obtain a thinner region of potential selected models, see Figures~\ref{fig:fig2} and \ref{fig:fig3}. In other words, we will avoid overfitting and by doing so, we will ensure a reasonable performance of our over-penalization procedure in situations where unbiased risk estimation fails. This is particularly the case when the amount of data is small to moderate. \begin{figure}[tbp] \centering \includegraphics[width=0.65\textwidth]{dessin-2-2} \caption{The correction that should be applied to an unbiased risk estimation procedure would ideally be of the size of the deviations of the risk for each model of the collection.} \label{fig:fig2} \end{figure} \begin{figure}[tbp] \centering \includegraphics[width=0.65\textwidth]{dessin-3-3} \caption{After a suitable correction, the minimum of the red curve has a better shape. In addition, the region of models that can be possibly selected is substantially smaller and in particular avoids the largest models of the collection.} \label{fig:fig3} \end{figure} \section{Theoretical Guarantees} \label{section_results} We state here our theoretical results related to the behavior of our over-penalization procedure. As explained in Section~\ref{section_overpen}, concentration inequalities for true and empirical excess risks are essential tools for understanding our model selection problem. As the theme of concentration inequalities for the excess risk constitutes a very recent and exciting area of research, these inequalities also have an interest in themselves and we state them out in Section~\ref{section_risks_bounds_MLE}. In Section~\ref{section_results_bounded_setting}, we give a sharp oracle inequality proving the optimality of our procedure. We also compare our result to what would be obtained for AIC, which suggests the superiority of over-penalization in the nonasymptotic regime, i.e. for a small to medium sample size. \subsection{True and empirical excess risks' concentration\label% {section_risks_bounds_MLE}} In this section, we fix the linear model $m$ made of histograms and we are interested by concentration inequalities for the true excess risk $P\left( \gamma (\hat{f}_{m})-\gamma (f_{m})\right) $ on $m$ and for its empirical counterpart $P_{n}\left( \gamma (f_{m})-\gamma (\hat{f}_{m})\right) .$ \begin{theorem} \label{opt_bounds}Let $n\in \mathbb{N}$, $n\geq 1$ and let $\alpha ,A_{+},A_{-}$ and $A_{\Lambda }$ be positive constants. Take $m$ a model of histograms defined on a fixed partition $\Lambda _{m}$ of $\mathcal{Z}$. The cardinality of $\Lambda _{m}$ is denoted by $D_{m}.$ Assume that $% 1<D_{m}\leq A_{+}n/(\ln (n+1))\leq n$ and% \begin{equation} 0<A_{\Lambda }\leq D_{m}\inf_{I\in \Lambda _{m}}\left\{ P\left( I\right) \right\} \text{ .} \label{lower_reg-1} \end{equation}% If $\left( \alpha +1\right) A_{+}/A_{\Lambda }\leq \tau =\sqrt{3}-\sqrt{2}< 0.32$, then a positive constant $A_{0}$ exists, only depending on $\alpha ,A_{+}$ and $A_{\Lambda }$, such that by setting% \begin{equation} \varepsilon _{n}^{+}\left( m\right) =\max \left\{ \sqrt{\frac{D_{m}\ln (n+1)% }{n}};\sqrt{\frac{\ln (n+1)}{D_{m}}};\frac{\ln (n+1)}{D_{m}}\right\} \label{def_A_0_MLE-1} \end{equation}% and \begin{equation*} \varepsilon _{n}^{-}\left( m\right) =\max \left\{ \sqrt{\frac{D_{m}\ln (n+1)% }{n}};\sqrt{\frac{\ln (n+1)}{D_{m}}}\right\} \text{ ,} \end{equation*}% we have, on an event of probability at least $1-4(n+1)^{-\alpha }$, \begin{equation} \left( 1-A_{0}\varepsilon _{n}^{-}\left( m\right) \right) \frac{D_{m}}{2n}% \leq \mathcal{K}\left( f_{m},\hat{f}_{m}\right) \leq \left( 1+A_{0}\varepsilon _{n}^{+}\left( m\right) \right) \frac{D_{m}}{2n}\text{ ,} \label{upper_true_risk} \end{equation}% \begin{equation} \left( 1-A_{0}\varepsilon _{n}^{-}\left( m\right) \right) \frac{D_{m}}{2n}% \leq \mathcal{K}\left( \hat{f}_{m},f_{m}\right) \leq \left( 1+A_{0}\varepsilon _{n}^{+}\left( m\right) \right) \frac{D_{m}}{2n}\text{ .} \label{upper_emp_risk} \end{equation} \end{theorem} In the previous theorem, we obtain sharp upper and lower bounds for true and empirical excess risk on $m$. They are optimal at the first order since the leading constants are equal in the upper and lower bounds. They show the concentration of the true and empirical excess risks around the value $D_{m}/(2n)$. Moreover, Theorem \ref{opt_bounds} establishes equivalence with high probability of the true and empirical excess risks for models of reasonable dimension. Concentration inequalities for the excess risks as in Theorem\ref{opt_bounds} is a new and exciting direction of research related to the theory of statistical learning and to high-dimensional statistics. Boucheron and Massart \cite{BouMas:10} obtained a pioneering result describing the concentration of the empirical excess risk around its mean, a property that they call a high-dimensional Wilks phenomenon. Then a few authors obtained results describing the concentration of the true excess risk around its mean \cite{saum:12}, \cite{chatterjee2014}, \cite{MurovandeGeer:15} or around its median \cite{bellec2016bounds}, \cite{bellec2017towards} for (penalized) least square regression\ and in an abstract M-estimation framework \cite% {vandeGeerWain:16}. In particular, recent results of \cite{vandeGeerWain:16} include the case of MLE on exponential models and as a matter of fact, on histograms. Nevertheless, Theorem \ref{opt_bounds} is a valuable addition to the literature on this line of research since we obtain here nonetheless the concentration around a fixed point, but an explicit value $D_{m}/2n$ for this point. On the contrary, the concentration point is available in \cite% {vandeGeerWain:16} only through an implicit formula involving local suprema of the underlying empirical process. The principal assumption in Theorem \ref{opt_bounds} is inequality (\ref% {lower_reg-1}) of lower regularity of the partition with respect to $P$. It is ensured as soon as the density $f_{\ast }$ is uniformly bounded from below and the partition is lower regular with respect to the reference measure $\mu $ (which will be the Lebesgue measure in our experiments). No restriction on the largest values of $f_{\ast }$ are needed. In particular, we do not restrict to the bounded density estimation setting. Castellan \cite{Castellan:99} proved related, but weaker inequalities than in Theorem \ref{opt_bounds}\ above. She also asked for a lower regularity property of the partition, as in Proposition 2.5 \cite{Castellan:99}, where she derived a sharp control of the KL divergence of the histogram estimator on a fixed model. More precisely, Castellan assumes that there exists a positive constant $B$ such that \begin{equation} \inf_{I\in \Lambda _{m}}\mu \left( I\right) \geq B\frac{\left( \ln (n+1)\right) ^{2}}{n}\text{ }. \label{castellan_regularity} \end{equation}% This latter assumption is thus weaker than (\ref{lower_reg-1}) for the considered model as its dimension $D_{m}$ is less than the order $n\left( \ln (n+1)\right) ^{-2}$. We could assume (\ref{castellan_regularity}) instead of (\ref{lower_reg-1}) in order to derive Theorem \ref{opt_bounds}. This would lead to less precise results for second order terms in the deviations of the excess risks but the first order bounds would be preserved. More precisely, if we replace assumption (\ref{lower_reg-1}) in Theorem \ref{opt_bounds} by Castellan's assumption (\ref% {castellan_regularity}), a careful look at the proofs show that the conclusions of Theorem \ref{opt_bounds} are still valid for $\varepsilon _{n}=A_{0}\left( \ln (n+1)\right) ^{-1/2}$, where $A_{0}$ is some positive constant. Thus assumption (\ref{lower_reg-1}) is not a fundamental restriction in comparison to Castellan's work \cite{Castellan:99}, but it leads to more precise results in terms of deviations of the true and empirical excess risks of the histogram estimator. The proof of Theorem \ref{opt_bounds}, that can be found in Section \ref% {section_deviation_bounds}, is based on an improvement of independent interest of the previously best known concentration inequality for the chi-square statistics. See Section \ref{section_chi_square}\ for the precise result. \subsection{An Oracle Inequality} \label{section_results_bounded_setting} First, let us state the set of five structural assumptions required to establish the nonasymptotic optimality of the over-penalization procedure. These assumptions will be discussed in more detail at the end of this section, following the statement of a sharp oracle inequality. \underline{Set of assumptions (\textbf{SA})} \begin{description} \item[(P1)] Polynomial complexity of $\mathcal{M}_{n}$: $\card\left( \mathcal{M}_{n}\right) \leq n^{\alpha _{\mathcal{M}}}.$ \item[(P2)] Upper bound on dimensions of models in $\mathcal{M}_{n}$: there exists a positive constant $A_{\mathcal{M},+}$ such that for every $m\in \mathcal{M}_{n},$ \begin{equation*} D_{m}\leq A_{\mathcal{M},+}\frac{n}{\left( \ln (n+1)\right) ^{2}}\leq n\text{ }. \end{equation*} \item[(Asm)] The unknown density $f_{\ast }$ satisfies some moment condition and is uniformly bounded from below: there exist some constants $A_{\min }>0 $ and $p>1$ such that, \begin{equation*} \int_{\mathcal{Z}}f_{\ast }^{p}\left[ (\ln f_{\ast })^{2}\vee 1\right] d\mu <+\infty \end{equation*}% and \begin{equation} \inf_{z\in \mathcal{Z}}f_{\ast }\left( z\right) \geq A_{\min }>0\text{ }. \label{lower_target} \end{equation} \item[(Alr)] Lower regularity of the partition with respect to $\mu $: there exists a positive finite constant $A_{\Lambda }$ such that, for all $m\in \mathcal{M}_{n}$, \begin{equation*} D_{m}\inf_{I\in \Lambda _{m}}\mu \left( I\right) \geq A_{\Lambda }\geq A_{\mathcal{M},+}(\alpha _{\mathcal{M}}+6)/\tau\text{ }, \end{equation*} where $\tau=\sqrt{3}-\sqrt{2}>0$. \item[(\textbf{Ap)}] The bias decreases like a power of $D_{m}$: there exist $\beta _{-}\geq \beta _{+}>0$ and $C_{+},C_{-}>0$ such that% \begin{equation*} C_{-}D_{m}^{-\beta _{-}}\leq \mathcal{K}\left( f_{\ast },f_{m}\right) \leq C_{+}D_{m}^{-\beta _{+}}\text{ }. \end{equation*} \end{description} We are now ready to state our main theorem related to optimality of over-penalization. \begin{theorem} \label{theorem_opt_pen_MLE}Take $n\geq 1$ and $r\in \left( 0,p-1\right) $. For some $\Delta >0$, consider the following penalty,% \begin{equation*} \pen% \left( m\right) =\left( 1+\Delta \varepsilon _{n}^{+}\left( m\right) \right) \frac{D_{m}}{n}\text{ , \ \ \ \ \ \ \ \ \ \ for all\ }m\in \mathcal{M}_{n}% \text{ .} \end{equation*}% Assume that the set of assumptions (\textbf{SA}) holds and that \begin{equation} \beta _{-}<p\left( 1+\beta _{+}\right) /(1+p+r)\text{ \ \ or \ \ }% p/(1+r)>\beta _{-}+\beta _{-}/\beta _{+}-1\text{ }. \label{condition_beta} \end{equation}% Then there exists an event $\Omega _{n}$ of probability at least $% 1-(n+1)^{-2}$ and some positive constant $A_{1}$ depending only on the constants defined in (\textbf{SA}) such that, if $\Delta \geq A_{1}>0$ then we have on $\Omega _{n}$,% \begin{equation} \mathcal{K}\left( f_{\ast },\hat{f}_{\widehat{m}}\right) \leq (1+\delta _{n})\inf_{m\in \mathcal{M}_{n}}\left\{ \mathcal{K}\left( f_{\ast },\hat{f}% _{m}\right) \right\} \text{ ,} \label{oracle_um_opt} \end{equation}% where $\delta _{n}=L_{\text{(\textbf{SA})},\Delta ,r}\left( \ln (n+1)\right) ^{-1/2}$ is convenient. \end{theorem} We derive in Theorem \ref{theorem_opt_pen_MLE} a pathwise oracle inequality for the KL excess risk of the selected estimator, with constant almost one. Our result thus establishes the nonasymptotic quasi-optimality of over-penalization with respect to the KL divergence. It is worth noting that three very strong features related to oracle inequality (\ref{oracle_um_opt}) significantly improve upon the literature. Firstly, inequality (\ref{oracle_um_opt}) expresses the performance of the selected estimator through its KL divergence and compare it to the KL divergence of the oracle. Nonasymptotic results pertaining to (robust) maximum likelihood based density estimation usually control the Hellinger risk of the estimator, \cite{Castellan:03}, \cite{Massart:07}, \cite% {BirRozen:06}, \cite{MR2219712}, \cite{BarBirgSart:17}. The main reason is that the Hellinger risk is much easier to handle that the KL divergence from a mathematical point of view. For instance, the Hellinger distance is bounded by one while the KL divergence can be infinite. However, from a M-estimation perspective, the natural excess risk associated to likelihood optimization is indeed the KL divergence and not the Hellinger distance. These two risks are provably close to each other in the bounded setting \cite% {Massart:07}, but may behave very differently in general. Second, nonasymptotic results describing the performance of procedures based on penalized likelihood, by comparing more precisely the (Hellinger) risk of the estimator to the KL divergence of the oracle, all deal with the case where the log-density to be estimated is bounded (\cite{Castellan:03}, \cite{Massart:07}). Here, we substantially extend the setting by considering only the existence of a finite polynomial moment for the large values of the density to be estimated. Finally, the oracle inequality (\ref{oracle_um_opt}) is valid with positive probability---larger than $3/4$---as soon as one data is available. To our knowledge, any other oracle inequality describing penalization performance for maximum likelihood density estimation is valid with positive probability only when the sample size $n$ is greater than an integer $n_{0}$ which depends on the constants defining the problem and that is thus unknown. We emphasize that we control the risk of the selected estimator \textit{for any sample size} and that this property is essential in practice when dealing with small to medium sample sizes. Based on the arguments developed in Section \ref{section_overpen},\ we believe that such a feature of Theorem \ref{theorem_opt_pen_MLE} is accessible only through the use of over-penalization and we conjecture in particular that \textit{it is impossible} \textit{using AIC} \textit{to achieve such a control the KL divergence of the selected estimator for any sample size}. The oracle inequality (\ref{oracle_um_opt}) is valid under conditions (\ref% {condition_beta}) relating the values of the bias decaying rates $\beta _{-}$% and $\beta _{+}$ to the order $p$ of finite moment of the density $f_{\ast }$ and the parameter $r$. In order to understand these latter conditions, let us assume for simplicity that $\beta _{-}=\beta _{+}=:\beta $. Then the conditions (\ref{condition_beta}) both reduce to $\beta <p/(1+r).$ As $r$ can be taken as close to zero as we want, the latter inequality reduces to $% \beta <p$. In particular, if the density to be estimated is bounded ($% p=\infty $), then conditions (\ref{condition_beta}) are automatically satisfied. If on the contrary the density $f_{\ast }$ only has finite polynomial moment $p$, then the bias should not decrease too fast. In light of the following comments, if $f_{\ast }$ is assumed to be $\alpha $-H\"{o}lderian, $\alpha \in (0,1]$, then $\beta \leq 2\alpha \leq 2$ and the conditions (\ref% {condition_beta}) are satisfied, in the case where $\beta _{-}=\beta _{+}$, as soon as $p\geq 2$. To conclude this section, let's now comment on the set of assumptions (\textbf{SA}). Assumption (\textbf{P1}) indicates that the collection of models has increasing polynomial complexity. This is well suited to bin size selection because in this case we usually select among a number of models which is strictly bounded from above by the sample size. In the same manner, Assumption (\textbf{P2}) is legitimate enough for us and corresponds to practice, where we aim at considering bin sizes for which each element of the partition contains a few sample points. Assumption (\textbf{Asm}) imposes conditions on the density to be estimated. More precisely, Assumption (\ref{lower_target}) stating that the unknown density is uniformly bounded from below can also be found in the work of Castellan \cite{Castellan:99}. The author assumes moreover in Theorem 3.4 where she derives an oracle inequality for the (weighted) KL excess risk of the histogram estimator, that the target is of finite sup-norm. Furthermore, from a statistical perspective, the lower bound (\ref{lower_target}) is legitimate since, by Assumption (\textbf{Alr}), we use models of lower-regular partitions with respect to the Lebesgue measure. In the case where Inequality (\ref{lower_target}) would not hold, one would typically have to consider exponentially many irregular histograms to take into account the possibly vanishing mass of some elements of the partitions (for more details on this aspect that goes beyond the scope of the present paper, see for instance \cite{Massart:07}). We require in (\textbf{Ap}) that the quality of the approximation of the collection of models is good enough in terms of bias. More precisely, we require a polynomially decreasing of excess risk of KL projections of the unknown density onto the models. For a density $f_{\ast }$ uniformly bounded away from zero, the upper bound on the bias is satisfied when for example, $% \mathcal{Z}$ is the unit interval, $\mu =\leb$ is the Lebesgue measure on the unit interval, the partitions $\Lambda _{m}$ are regular and the density $f_{\ast }$ belongs to the set $\mathcal{H}\left( H,\alpha \right) $ of $% \alpha $-h\"{o}lderian functions for some $\alpha \in \left( 0,1\right] $: if $f\in \mathcal{H}\left( H,\alpha \right) $, then for all $\left( x,y\right) \in \mathcal{Z}^{2}$ \begin{equation*} \left\vert f\left( x\right) -f\left( y\right) \right\vert \leq H\left\vert x-y\right\vert ^{\alpha }\text{ .} \end{equation*}% In that case, $\beta _{+}=2\alpha $ is convenient and AIC-type procedures are adaptive to the parameters $H$ and $\alpha $, see Castellan \cite% {Castellan:99}. In assumption (\textbf{Ap}) of Theorem \ref{theorem_opt_pen_MLE} we also assume that the bias $\mathcal{K}\left( f_{\ast },f_{m}\right) $ is bounded from below by a power of the dimension $D_{m}$ of the model $m$. This hypothesis is in fact quite classical as it has been used by Stone \cite% {Stone:85} and Burman \cite{Burman:02}\ for the estimation of density on histograms and also by Arlot and Massart \cite{ArlotMassart:09} and Arlot \cite{Arl:2008a}, \cite{Arl:09} in the regression framework. Combining Lemma 1 and 2 of Barron and Sheu \cite{BarSheu:91} -\ see also inequality (\ref% {margin_like_opt}) of Proposition \ref{lemma_margin_like_bounded_below}\ below -- we can show that \begin{equation*} \frac{1}{2}e^{-3\left\Vert \ln \left( \frac{f_{\ast }}{f_{m}}\right) \right\Vert _{\infty }}\int_{\mathcal{Z}}\frac{\left( f_{m}-f_{\ast }\right) ^{2}}{f_{\ast }}d\mu \leq \mathcal{K}\left( f_{\ast },f_{m}\right) \end{equation*}% and thus assuming for instance that the target is uniformly bounded, $% \left\Vert f_{\ast }\right\Vert _{\infty }\leq A_{\ast }$, we get \begin{equation*} \frac{A_{\min }^{3}}{2A_{\ast }^{4}}\int_{\mathcal{Z}}\left( f_{m}-f_{\ast }\right) ^{2}d\mu \leq \mathcal{K}\left( f_{\ast },f_{m}\right) \text{ .} \end{equation*}% Now, since in the case of histograms the KL projection $f_{m}$ is also the $% L_{2}\left( \mu \right) $ projection of $f_{\ast }$ onto $m$, we can apply Lemma 8.19 in Section 8.10 of Arlot \cite{Arlot:07} to show that assumption (% \textbf{Ap}) is indeed satisfied for $\beta _{-}=1+\alpha ^{-1}$, in the case where $\mathcal{Z}$ is the unit interval, $\mu =\leb$ is the Lebesgue measure on the unit interval, the partitions $\Lambda _{m}$ are regular and the density $f_{\ast }$ is a non-constant $\alpha $-h\"{o}lderian function. The proof of Theorem \ref{theorem_opt_pen_MLE} and further descriptions of the behavior of the procedure can be found in the supplementary material, Section \ref{section_oracle_proofs}. \section{Probabilistic and Analytical Tools} \label{section_prob_tools} In this section we set out some general results that are of independent interest and serve as tools for the mathematical description of our statistical procedure. The first two sections contain new or improved concentration inequalities, for the chi-square statistics (Section \ref{section_chi_square}) and for general log-densities (Section \ref{section_concentration_inequalities}). We established in Section \ref{section_margin_like_relations} some results that are related to the so-called margin relation in statistical learning and that are analytical in nature. \subsection{Chi-Square Statistics' Concentration} \label{section_chi_square} The chi-square statistics plays an essential role in the proofs related to Section \ref{section_risks_bounds_MLE}. Let us recall its definition. \begin{definition} Given some histogram model $m$, the statistics $\chi _{n}^{2}\left( m\right) $ is defined by% \begin{equation*} \chi _{n}^{2}\left( m\right) =\int_{\mathcal{Z}}\frac{\left( \hat{f}% _{m}-f_{m}\right) ^{2}}{f_{m}}d\mu =\sum_{I\in m}\frac{\left( P_{n}\left( I\right) -P\left( I\right) \right) ^{2}}{P\left( I\right) }\text{ .} \end{equation*} \end{definition} The following proposition provides an improvement upon the previously best known concentration inequality for the right tail of the chi-square statistics, available in \cite{Castellan:03}---see also \cite[Proposition 7.8]{Massart:07} and \cite[Theorem 12.13]{BouLugMas:13}. \begin{proposition} \label{prop_upper_dev_chi_2} For any $x,\theta >0$, it holds \begin{equation} \mathbb{P}\left( \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) }\geq \sqrt{\frac{D_{m}}{n}}+\left( 1+\sqrt{2\theta }+\frac{% \theta }{6}\right) \sqrt{\frac{2x}{n}}\right) \leq \exp \left( -x\right) \text{ ,} \label{upper_chi_part} \end{equation}% where we set $\Omega _{m}\left( \theta \right) =\bigcap_{I\in m}\left\{ \left\vert P_{n}\left( I\right) -P\left( I\right) \right\vert \leq \theta P\left( I\right) \right\} $. More precisely, for any $x,\theta >0$, it holds with probability at least $1-e^{-x}$,% \begin{equation} \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) }<% \sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{n}}+2\sqrt{\frac{\theta }{n}}\left( \sqrt{x}\wedge \left( \frac{xD_{m}}{2}\right) ^{1/4}\right) +\frac{\theta }{3% }\sqrt{\frac{x}{n}}\left( \sqrt{\frac{x}{D_{m}}}\wedge \frac{1}{\sqrt{2}}% \right) \text{ .} \label{upper_chi_gene} \end{equation} \end{proposition} The proof of Theorem \ref{prop_upper_dev_chi_2} can be found in Section \ref% {ssection_chisquare_proof}. Let us detail its relationship with the bound in Proposition 7.8 in \cite% {Massart:07}, which is: for any $x,\varepsilon >0$,% \begin{equation} \mathbb{P}\left( \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \varepsilon ^{2}/\left( 1+\varepsilon /3\right) \right) }\geq \left( 1+\varepsilon \right) \left( \sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{n}}% \right) \right) \leq \exp \left( -x\right) \text{ }. \label{bound_Massart} \end{equation}% By taking $\theta =\varepsilon ^{2}/\left( 1+\varepsilon /3\right) >0$, we get $\varepsilon =\theta /6+\sqrt{\theta ^{2}/36+\theta }>\theta /6+\sqrt{% \theta }>0$. Assume that $D_{m}\geq 2x$. We obtain by (\ref{upper_chi_part}% ), with probability at least $1-\exp \left( -x\right) $,% \begin{align*} \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) }<& \sqrt{\frac{D_{m}}{n}}+\left( 1+\sqrt{2\theta }+\frac{\theta }{6}\right) \sqrt{\frac{2x}{n}} \\ \leq & \sqrt{\frac{D_{m}}{n}}+\left( 1+\sqrt{2}\varepsilon \right) \sqrt{% \frac{2x}{n}} \\ <& \left( 1+\varepsilon \right) \left( \sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x% }{n}}\right) \text{ .} \end{align*}% So in this case, inequality (\ref{upper_chi_part}) improves upon (\ref% {bound_Massart}). Notice that in our statistical setting (see Theorem \ref% {opt_bounds}) the restriction on $D_{m}$ is as follows $D_{m}\leq A_{+}n\left( \ln (n+1)\right) ^{-1}$. Furthermore, in our proofs we apply (% \ref{upper_chi_part}) with $x$ proportional to $\ln (n+1)$ (see the proof of Theorem \ref{opt_bounds}). Hence, for a majority of models, we have $x\ll D_{m}$ and so \begin{equation*} \sqrt{\frac{2x}{n}}\ll \sqrt{\frac{D_{m}}{n}}\text{ .} \end{equation*}% As a result, the bounds that we obtain in Theorem \ref{opt_bounds} by the use of Inequality (\ref{upper_chi_part}) are substantially better than the bounds we would obtain by using Inequality (\ref{bound_Massart}) of \cite% {Massart:07}. More precisely, the term $\sqrt{D_{m}\ln (n+1)/n}$ in (\ref% {def_A_0_MLE-1}) would be replaced by $\left( D_{m}\ln (n+1)/n\right) ^{1/4}$% , thus changing the order of magnitude for deviations of the excess risks. Now, if $D_{m}\leq 2x$ then (\ref{upper_chi_part}) gives that with probability at least $1-e^{-x}$,% \begin{align*} \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) } <&\sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{n}}+2\sqrt{\frac{\theta }{n}}\left( \frac{xD_{m}}{2}\right) ^{1/4}+\frac{\theta }{6}\sqrt{\frac{2x}{n}} \\ \leq &\left( 1+\frac{\sqrt{\theta }}{2^{1/4}}\right) \sqrt{\frac{D_{m}}{n}}% +\left( 1+\frac{\sqrt{\theta }}{2^{3/4}}+\frac{\theta }{6}\right) \sqrt{% \frac{2x}{n}} \\ <&\left( 1+\varepsilon \right) \left( \sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{% n}}\right) \text{ .} \end{align*}% So in this case again, inequality (\ref{upper_chi_gene}) improves upon (\ref% {bound_Massart}), with an improvement that can be substantial depending on the value of the ratio $x/D_{m}$. The following result describes the concentration from the left of the chi-square statistics and is proved in Section \ref{ssection_chisquare_proof}% . \begin{proposition} \label{prop_dev_gauche_chi}Let $\alpha ,$ $A_{\Lambda }>0$. Assume $% 0<A_{\Lambda }\leq D_{m}\inf_{I\in m}\left\{ P\left( I\right) \right\} $. Then there exists a positive constant $A_{g}$ depending only on$\,A_{\Lambda }$ and$\,\alpha $ such that \begin{equation*} \mathbb{P}\left( \chi _{n}\left( m\right) \leq \left( 1-A_{g}\left( \sqrt{% \frac{\ln (n+1)}{D_{m}}}\vee \frac{\sqrt{\ln (n+1)}}{n^{1/4}}\right) \right) \sqrt{\frac{D_{m}}{n}}\right) \leq \left( n+1\right) ^{-\alpha }\text{ }. \end{equation*} \end{proposition} \subsection{Bernstein type concentration inequalities for log-densities\label% {section_concentration_inequalities}} The following proposition gives concentration inequalities for the empirical bias at the right of its mean. \begin{proposition} \label{delta_bar_dev_droite_MLE} Consider a density $f\in \mathcal{S}$. We have, for all $z\geq 0$, \begin{equation} \mathbb{P}\left( P_{n}\left( \ln \left( \left. f\right/ f_{\ast }\right) \right) \geq \frac{z}{n}\right) \leq \exp \left( -z\right) \text{ .} \label{dev_droite_gene_2} \end{equation}% Moreover, if we can take a finite quantity $v$ which satisfies $v\geq \int \left( f\vee f_{\ast }\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu $, we have for all $z\geq 0$,% \begin{equation} \mathbb{P}\left( \left( P_{n}-P\right) \left( \ln \left( \left. f\right/ f_{\ast }\right) \right) \geq \sqrt{\frac{2vz}{n}}+\frac{2z}{n}\right) \leq \exp \left( -z\right) \text{ .} \label{bernstein_dev_droite_2} \end{equation} \end{proposition} One can notice, with Inequality (\ref{dev_droite_gene_2}), that the empirical bias always satisfies some exponential deviations at the right of zero. In the Information Theory community, this inequality is also known as the ``No Hyper-compression Inequality'' (\cite{grunwald2007minimum}). Inequality (\ref{bernstein_dev_droite_2}) seems to be new and takes the form of a Bernstein-like inequality, even if the usual assumptions of Bernstein's inequality are not satisfied. In fact, we are able to recover such a behavior by inflating the usual variance to the quantity $v$. We now turn to concentration inequalities for the empirical bias at the left of its mean. \begin{proposition} \label{delta_bar_dev_gauche_MLE}Let $r>0$. For any density $f\in \mathcal{S}$ and for all $z\geq 0$, we have% \begin{equation} \mathbb{P}\left( P_{n}\left( \ln \left( \left. f\right/ f_{\ast }\right) \right) \leq -z/nr-\left( 1/r\right) \ln \left( P\left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] \right) \right) \leq \exp \left( -z\right) \text{ }. \label{delta_dev_gauche_2} \end{equation}% Moreover, if we can set a quantity $w_{r}$ which satisfies $w_{r}\geq \int \left( \frac{f_{\ast }^{r+1}}{f^{r}}\vee f_{\ast }\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu $ , then we get, for all $z\geq 0 $,% \begin{equation} \mathbb{P}\left( \left( P_{n}-P\right) \left( \ln \left( \left. f\right/ f_{\ast }\right) \right) \leq -\sqrt{\frac{2w_{r}z}{n}}-\frac{2z}{nr}\right) \leq \exp \left( -z\right) \text{ }. \label{delta_dev_gauche_gauss_2} \end{equation} \end{proposition} \subsection{Margin-Like Relations} \label{section_margin_like_relations} Our objective in this section is to control the variance terms $v$ and $w_{r}$, appearing respectively in Lemma \ref{delta_bar_dev_droite_MLE}\ and \ref% {delta_bar_dev_gauche_MLE}, in terms of the KL divergence pointed on the target $f_{\ast }$. This is done in Lemma \ref{lem:margin_like_unbounded}\ below under moment assumptions for $f_{\ast }$. \begin{proposition} \label{lem:margin_like_unbounded}Let $p>1$ and $c_{+},c_{-}>0$.\textup{\ }% Assume that the density $f_{\ast }$ satisfies% \begin{equation} J:=\int_{\mathcal{Z}}f_{\ast }^{p}\left( \left( \ln \left( f_{\ast }\right) \right) ^{2}\vee 1\right) d\mu <+\infty \text{ and }Q:=\int_{\mathcal{Z}}% \frac{\left( \ln \left( f_{\ast }\right) \right) ^{2}\vee 1}{f_{\ast }^{p-1}}% d\mu <+\infty \label{moment_p_bis} \end{equation}% Take a density $f$ such that $0<c_{-}\leq {\inf_{z\in \mathcal{Z}}\left\{ f\left( z\right) \right\} \leq \sup_{z\in \mathcal{Z}}}\left\{ f\left( z\right) \right\} \leq c_{+}<+\infty $. Then, for some $A_{MR,d}>0$ only depending on $J,Q,p,c_{+}$ and $c_{-}$, it holds% \begin{equation} P\left[ \left( \frac{f}{f_{\ast }}\vee 1\right) \left( \ln \left( \frac{f}{% f_{\ast }}\right) \right) ^{2}\right] \leq A_{MR,d}\times \mathcal{K}\left( f_{\ast },f\right) ^{1-\frac{1}{p}}\text{ .} \label{margin_like_soft_droite_gene} \end{equation}% More precisely, $A_{MR,d}=\left( 4c_{-}^{1-p}\left( \left( \ln c_{-}\right) ^{2}\vee 1\right) J+4c_{+}^{p}\left( \left( \ln c_{+}\right) ^{2}\vee 1\right) Q\right) ^{1/p}$ holds. For any $0<r\leq p-1$, we have the following inequality,% \begin{equation} P\left[ \left( \frac{f_{\ast }}{f}\vee 1\right) ^{r}\left( \ln \left( \frac{f% }{f_{\ast }}\right) \right) ^{2}\right] \leq A_{MR,g}\times \mathcal{K}% \left( f_{\ast },f\right) ^{1-\frac{r+1}{p}}\text{ ,} \label{margin_like_soft_gauche_gene} \end{equation}% available with $A_{MR,g}=\left( 4c_{-}^{1-p}\left( \left( \ln c_{-}\right) ^{2}\vee 1\right) J+2\left( \left( \ln \left( c_{+}\right) \right) ^{2}+J+Q\right) \right) ^{\frac{r+1}{p}}$. \end{proposition} Theorem \ref{lem:margin_like_unbounded} states that the variance terms, appearing in the concentration inequalities of Section~\ref% {section_concentration_inequalities}, are bounded from above, under moment restrictions on the density $f_{\ast }$, by a power less than one of the KL divergence pointed on $f_{\ast }$. The stronger are the moment assumptions, given in (\ref{moment_p_bis}), the closer is the power to one. One can notice that $J$ is a restriction on large values of $f_{\ast}$, whereas $Q$ is related to values of $f_{\ast }$ around zero. We call these inequalities ``margin-like relations'' because of their similarity with the margin relations known first in binary classification (% \cite{MamTsy:99}, \cite{Tsy:04a}) and then extended to empirical risk minimization (see \cite{MassartNedelec:06}\ and \cite{Arl_Bar:2008}\ for instance). Indeed, from a general point of view, margin relations relate the variance of contrasted functions (logarithm of densities here) pointed on the contrasted target to a function (in most cases, a power) of their excess risk. Now we're tightening the restrictions on the values of $f_{\ast }$ around zero. Indeed, we ask in the following lemma that the target is uniformly bounded away from zero. \begin{proposition} \label{lemma_margin_like_bounded_below}Let $p>1$ and $A_{\min },c_{+},c_{-}>0 $.\textup{\ }Assume that the density $f_{\ast }$ satisfies% \begin{equation*} J:=\int_{\mathcal{Z}}f_{\ast }^{p}\left( \left( \ln \left( f_{\ast }\right) \right) ^{2}\vee 1\right) d\mu <+\infty \text{ and }0<{A_{\min }\leq \inf_{z\in \mathcal{Z}}f_{\ast }\left( z\right) }\text{ \textup{.}} \end{equation*}% Then there exists a positive constant $A_{MR,-}$ only depending on $A_{\min },J,r$ and $p$ such that, for any $m\in \mathcal{M}_{n}$,% \begin{equation} P\left[ \left( \frac{f_{m}}{f_{\ast }}\vee 1\right) \left( \ln \left( \frac{% f_{m}}{f_{\ast }}\right) \right) ^{2}\right] \leq A_{MR,-}\times \mathcal{K}% \left( f_{\ast },f_{m}\right) ^{1-1/p} \label{margin_like_soft} \end{equation}% and for any $0<r\leq p-1$, \begin{equation} P\left[ \left( \frac{f_{\ast }}{f_{m}}\vee 1\right) ^{r}\left( \ln \left( \frac{f_{m}}{f_{\ast }}\right) \right) ^{2}\right] \leq A_{MR,-}\times \mathcal{K}\left( f_{\ast },f_{m}\right) ^{1-\frac{r+1}{p}}\text{ .} \label{margin_like_gauche_soft} \end{equation}% If moreover $\ln \left( f_{\ast }\right) \in L_{\infty }\left( \mu \right) $% , i.e. $0<{A_{\min }\leq \inf_{z\in \mathcal{Z}}}$\textup{$f_{\ast }$}${% \left( z\right) \leq \left\Vert f_{\ast }\right\Vert _{\infty }<+\infty }$, then there exists $\tilde{A}>0$ only depending on $r,{A_{\min }}$ and ${% \left\Vert f_{\ast }\right\Vert _{\infty }}$ such that, for any $m\in \mathcal{M}_{n}$,% \begin{equation} P\left[ \left( \frac{f_{m}}{f_{\ast }}\vee 1\right) \left( \ln \left( \frac{% f_{m}}{f_{\ast }}\right) \right) ^{2}\right] \vee P\left[ \left( \frac{% f_{\ast }}{f_{m}}\vee 1\right) ^{r}\left( \ln \left( \frac{f_{m}}{f_{\ast }}% \right) \right) ^{2}\right] \leq \tilde{A}\times \mathcal{K}\left( f_{\ast },f_{m}\right) \text{ }. \label{margin_like_opt} \end{equation} \end{proposition} It is worth noting that Lemma \ref{lemma_margin_like_bounded_below} is stated only for projections $f_{m}$ because we actually take advantage of their special form (as local means of the target) in the proof of the lemma. The benefit, compared to results of Lemma \ref{lem:margin_like_unbounded}, is that Inequalities (\ref{margin_like_soft}), (\ref{margin_like_gauche_soft}% ) and (\ref{margin_like_opt}) do not involve assumptions on the values of $% f_{m}$ (and in particular they do not involve the sup-norm of $f_{m}$). \section{Experiments} \label{section_expe} A simulation study is conducted to compare the numerical performance of the model selection procedures we discussed. We demonstrate the usefulness of our procedure on simulated data examples. The numerical experiments were performed using R. \begin{figure}[t] \centering \includegraphics[width=0.4\textwidth]{test} \includegraphics[width=0.4\textwidth]{dim} \caption{Estimation of the over-penalization constant.} \label{fig:proportion} \end{figure} \subsection{Experimental Setup} We have compared the numerical performance of our procedure with the classic methods of penalisation of the literature on several densities. In particular, we consider the estimator of \cite{BirRozen:06} and AICc (\citep{Sugiura:78, HurTsai:89}). We also report on AIC's behaviour. In the following, we name the procedure of \cite{BirRozen:06} by BR, and our criterion AIC$_1$ when the constant $C=1$ in \eqref{pen_a} and AIC$_a$ for a fully adaptive procedure which will be detailed below. More specifically, the performance of the following four model selection methods were compared: \begin{enumerate} \item AIC: \begin{equation*} \widehat{m}_{\mathrm{AIC}}\in\arg\min_{m\in \mathcal{M}_{n}}\left\{\mathrm{crit_{\mathrm{AIC}}}(m)\right\}, \end{equation*} with \[ \mathrm{crit_{\mathrm{AIC}}}(m) = P_{n}(\gamma(\widehat{f}_{m}))+\frac{D_m}{n}. \] \item AICc: \begin{equation*} \widehat{m}_{\mathrm{AICc}}\in\arg\min_{m\in \mathcal{M}_{n}}\left\{P_{n}(\gamma(\widehat{f}_{m}))+\pen_{\mathrm{AICc}}(m)\right\}, \end{equation*} with \[ \pen_{\mathrm{AICc}}(m) = \frac{D_m}{n-D_m-1}. \] \item BR: \begin{equation*} \widehat{m}_{\mathrm{BR}}\in\arg\min_{m\in \mathcal{M}_{n}}\left\{P_{n}(\gamma(\widehat{f}_{m}))+\pen_{\mathrm{BR}}(m)\right\}, \end{equation*} with \[ \pen_{\mathrm{BR}}(m) = \frac{ (\log{D_m})^{2.5}}{n} \] \item AIC$_1$: \begin{equation*} \widehat{m}_{\mathrm{AIC}_1}\in\arg\min_{m\in \mathcal{M}_{n}}\left\{P_{n}(\gamma(\widehat{f}_{m}))+\pen_{\mathrm{AIC}_1}(m)\right\}, \end{equation*} with \[ \pen_{\mathrm{AIC}_1}(m) = 1\times\max \left\{ \sqrt{\frac{D_{m}\ln (n+1)}{n% }};\sqrt{\frac{\ln (n+1)}{D_{m}}};\frac{\ln (n+1)}{D_{m}}\right\} \frac{D_{m}}{n}\text{ ,} \] \item AIC$_a$: \begin{equation*} \widehat{m}_{\mathrm{AIC}_a}\in\arg\min_{m\in \mathcal{M}_{n}}\left\{P_{n}(\gamma(\widehat{f}_{m}))+\pen_{\mathrm{AIC}_a}(m)\right\}, \end{equation*} with \[ \pen_{\mathrm{AIC}_a}(m) = \hat{C}\max \left\{ \sqrt{\frac{D_{m}\ln (n+1)}{n% }};\sqrt{\frac{\ln (n+1)}{D_{m}}};\frac{\ln (n+1)}{D_{m}}\right\} \frac{D_{m}}{n}\text{ ,} \] where $\hat{C}={\rm median}_{\alpha \in \mathcal{P}} \hat{C}_{\alpha}$, with $\hat{C}_{\alpha}={\rm median}_{m\in \mathcal{M}_{\alpha}} |\hat{C}_m|$, where \begin{equation*} \hat{C}_m=\frac{\Delta_m}{\max \left\{ \sqrt{\frac{D_m}{n}};\sqrt{\frac{1}{D_{m}}}\right\}\frac{D_m}{2}} \text{ ,} \end{equation*} $\Delta_m$ is the least-squares distance between the opposite of the empirical risk $-P_{n}(\gamma(\widehat{f}_{m}))$ and a fitted line of equation $y=xD_m/n+\hat{a}$ (see Figure \ref{fig:proportion}), $\mathcal{P}$ is the set of proportions $\alpha$ corresponding to the longest plateau of equal selected models when using penalty \eqref{pen_a} with constant $C=\hat{C}_{\alpha}$ and $\mathcal{M}_{\alpha}$ is the set of models in the collection associated to the proportion $\alpha$ of the largest dimensions. \end{enumerate} \medskip Let us briefly explain the ideas underlying the design of the fully adaptive AIC$_a$ procedure. According to the definition of penalty $\rm{pen}_{\rm{opt},\beta}$ given in \eqref{pen_opt_quantile} the constant $C$ in our over-penalization procedure \eqref{pen_a} should ideally estimate some normalized deviations of the sum of the excess risk and the empirical excess risk on the models of the collection. Based on Theorem \ref{opt_bounds},we can also assume that the deviations of excess risk and excess empirical risk are of the same order. Moreover, considering the largest models in the collection neglects questions of bias and, therefore, the median of the normalized deviations of the empirical risk around its mean for the largest models should be a reasonable estimator of the $C$ constant. Now the problem is to give a tractable definition to the "largest models" in the collection. To do this, we choose a proportion $\alpha$ of the largest dimensions of the models at hand and calculate using these models an estimator $\hat{C}_{\alpha}$ of the constant $C$ in \eqref{pen_a}. We then proceed for each $\alpha$ in a grid of values between $0$ and $1$ to a model selection step by over-penalization using the constant $C=\hat{C}_{\alpha}$. This gives us a graph of the selected dimensions with respect to the proportions (see Figure~\ref{fig:proportion}). Finally we define our over-penalization constant $\hat{C}$ as the median of the values of the constants $\hat{C}_{\alpha}$, $\alpha \in \mathcal{P}$ where $\mathcal{P}$ is the largest plateau in the graph of the selected dimensions with respect to proportions $\alpha$. The models that we used along the experiments are made of histograms densities defined a on a regular partition of the interval $[0,1]$ (with the exception of the density Isosceles triangle which is supported on $[-1,1]$). We show the performance of the proposed method for a set of four test distributions (see Figure~\ref{fig:orig}) and described in the \textit{benchden}\footnote{Available on the CRAN \url{http://cran.r-project.org}.} R-package \cite{Mildenberger-Weinert2012} which provides an implementation of the distributions introduced in \cite{berlinet1994comparison}. \begin{figure}[t] \centering \subfigure[Isosceles triangle]{\includegraphics[width=0.24\textwidth]{density16}} % \subfigure[Bilogarithmic peak]{\includegraphics[width=0.24\textwidth]{density28}} % \subfigure[Beta (2,2)]{\includegraphics[width=0.24\textwidth]{density17}} % \subfigure[Infinite peak]{\includegraphics[width=0.24\textwidth]{density8}} \caption{Test densities $f$.} \label{fig:orig} \end{figure} \subsection{Results} We compared procedures on $N=100$ independent data sets of size $n$ ranging from $50$ to $1000$. We estimate the quality of the model-selection strategies using the median KL divergence. Boxplots were made of the KL risk over the $N$ trials. The horizontal lines of the boxplot indicate the $5\%, 25\%, 50\%, 75\%$, and $95\%$ quantiles of the error distribution. The median value of AIC (horizontal black line) is also superimposed for visualization purposes. It can be seen from Figure~\ref{fig:result} that, as expected, for each method and in all cases, the KL divergence decreases as the sample size increases. We also see clearly that there is generally a substantial advantage to modifying AIC for sample sizes smaller than $1000$. Moreover, none of the methods is more effective than the others in all cases. However, in almost all cases, AIC$_1$ always had a lower median KL than the others. We therefore recommend using AIC$_1$ rather than AIC$_c$. Indeed, compared to AIC$_1$, it seems that AIC$_c$ is not penalizing enough,which translates into a worse performance for samples equal to $50$ and $100$. Moreover, both procedures have exactly the same calculation costs. On the contrary, it seems that the BR criterion penalizes too much. As a result, its performance deteriorates relative to other methods as the sample size increases. Finally, the behavior of AIC$_a$ is quite good in general, but it is often a little less efficient than AIC$_1$. We interpret this fact by postulating that the extra computations that we engage in AIC$_a$ to remove any hyper-parameter from the procedure actually induces extra variance in the estimation. \begin{figure}[t!] \centering \subfigure[Isosceles triangle]{ \includegraphics[width=0.24\textwidth]{plot50density_16MC100_KL} \includegraphics[width=0.24\textwidth]{plot100density_16MC100_KL} \includegraphics[width=0.24\textwidth]{plot500density_16MC100_KL} \includegraphics[width=0.24\textwidth]{plot1000density_16MC100_KL} } \subfigure[Bilogarithmic peak]{ \includegraphics[width=0.24\textwidth]{plot50density_28MC100_KL} \includegraphics[width=0.24\textwidth]{plot100density_28MC100_KL} \includegraphics[width=0.24\textwidth]{plot500density_28MC100_KL} \includegraphics[width=0.24\textwidth]{plot1000density_28MC100_KL} } \subfigure[Beta (2,2)]{ \includegraphics[width=0.24\textwidth]{plot50density_17MC100_KL} \includegraphics[width=0.24\textwidth]{plot100density_17MC100_KL} \includegraphics[width=0.24\textwidth]{plot500density_17MC100_KL} \includegraphics[width=0.24\textwidth]{plot1000density_17MC100_KL} } \subfigure[Infinite peak]{ \includegraphics[width=0.24\textwidth]{plot50density_8MC100_KL} \includegraphics[width=0.24\textwidth]{plot100density_8MC100_KL} \includegraphics[width=0.24\textwidth]{plot500density_8MC100_KL} \includegraphics[width=0.24\textwidth]{plot1000density_8MC100_KL} } \includegraphics[width=0.45\textwidth]{legende} \caption{KL divergence results. Box plots of the KL divergence to the true distribution for the estimated distribution. The solid black line corresponds to the AIC KL divergence median. The term inside the box is the number of time the KL divergence equals $\infty$ out of 100.} \label{fig:result} \end{figure} \subsection*{Acknowledgements} The first author warmly thanks Matthieu Lerasle for instructive discussions on the topic of estimation by tests and Alain C{\'e}lisse for a nice discussion at a early stage of this work. He is also grateful to Pascal Massart for having pushed him towards obtaining better oracle inequalities than in a previous version of this work. Finally, we owe thanks to Sylvain Arlot and Amandine Dubois for a careful reading that helped to correct some mistakes and improve the presentation of the paper. \section{Proofs} \label{section_proofs_MLE} \subsection{Proofs Related to Section \protect\ref{section_chi_square}\label% {ssection_chisquare_proof}} \begin{proof}[Proof of Theorem \protect\ref{prop_upper_dev_chi_2}] We fix $x,\theta >0$ and we set $z>0$ to be chosen later. Let us set for any $I\in m$, $\varphi _{I}=\left( P\left( I\right) \right) ^{-1/2}\mathbbm{1} _{I}$. The family of functions $\left( \varphi _{I}\right) _{I\in m}$ forms an orthonormal basis of $\left( m,\left\Vert \cdot \right\Vert _{2}\right) $% . By Cauchy-Schwarz inequality, we have% \begin{equation*} \chi _{n}\left( m\right) =\sup_{\left( a_{I}\right) _{I\in m}\in B_{2}}\left\vert \left( P_{n}-P\right) \left( \sum_{I\in m}a_{I}\varphi _{I}\right) \right\vert \text{ ,} \end{equation*}% where $B_{2}:=\left\{ \left( a_{I}\right) _{I\in m}\text{ };\text{ }% \sum_{I\in m}a_{I}^{2}\leq 1\right\} $. Furthermore, the case of equality in Cauchy-Schwarz inequality gives us% \begin{equation*} \chi _{n}\left( m\right) =\left( P_{n}-P\right) \left( \sum_{I\in m}a_{I}^{\infty }\varphi _{I}\right) \text{ with }a_{I}^{\infty }=\frac{% \left( P_{n}-P\right) \left( \varphi _{I}\right) }{\chi _{n}\left( m\right) }% \text{ .} \end{equation*}% Hence, by setting $\mathcal{A}\left( s\right) :=B_{2}\bigcap \left\{ \left( a_{I}\right) _{I\in m}\text{ };\text{ }\sup_{I\in m}\left\vert a_{I}\left( P\left( I\right) \right) ^{-1/2}\right\vert \leq s\right\} $ for any $s\geq 0 $, we get \begin{equation} \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) }% \boldsymbol{1}_{\left\{ \chi _{n}\left( m\right) \geq z\right\} }\leq \sup_{\left( a_{I}\right) _{I\in m}\in \mathcal{A}\left( \theta /z\right) }\left\vert \left( P_{n}-P\right) \left( \sum_{I\in m}a_{I}\varphi _{I}\right) \right\vert \text{ .} \label{upper_sup} \end{equation}% By applying Bousquet's inequality (\cite{Bousquet:02}) to the supremum in the right-hand side of (\ref{upper_sup}) then gives for any $\delta >0$,% \begin{align} &\mathbb{P}\left( \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) }\boldsymbol{1}_{\left\{ \chi _{n}\left( m\right) \geq z\right\} }\geq \left( 1+\delta \right) E_{m}+\sqrt{\frac{2\sigma _{m}^{2}x}{% n}}+\left( \frac{1}{\delta }+\frac{1}{3}\right) \frac{b_{m}x}{n}\right) \label{bousquet_z_1} \\ \leq &\mathbb{P}\left( \sup_{\left( a_{I}\right) _{I\in m}\in \mathcal{A}% \left( \theta /z\right) }\left\vert \left( P_{n}-P\right) \left( \sum_{I\in m}a_{I}\varphi _{I}\right) \right\vert \geq \left( 1+\delta \right) E_{m}+% \sqrt{\frac{2\sigma _{m}^{2}x}{n}}+\left( \frac{1}{\delta }+\frac{1}{3}% \right) \frac{b_{m}x}{n}\right) \notag \\ \leq &\exp \left( -x\right) \text{ ,} \notag \end{align}% with% \begin{equation*} E_{m}\leq \mathbb{E}\left[ \chi _{n}\left( m\right) \right] \leq \mathbb{E}% ^{1/2}\left[ \chi _{n}^{2}\left( m\right) \right] =\sqrt{\frac{D_{m}}{n}}% \text{ \ \ };\text{ \ \ }\sigma _{m}^{2}\leq \sup_{\left( a_{I}\right) _{I\in m}\in B_{2}}\mathbb{V}\left( \sum_{I\in m}a_{I}\varphi _{I}\left( \xi _{1}\right) \right) \leq 1 \end{equation*}% and% \begin{equation*} b_{m}=\sup_{\left( a_{I}\right) _{I\in m}\in \mathcal{A}\left( \theta /z\right) }\left\Vert \sum_{I\in m}a_{I}\varphi _{I}\right\Vert _{\infty }\leq \frac{\theta }{z}\text{ .} \end{equation*}% Injecting the latter bounds in (\ref{bousquet_z_1}) and taking $z=\sqrt{% \left( D_{m}\right) /n}+\sqrt{2x/n}$, we obtain that with probability at least $1-\exp \left( -x\right) $,% \begin{equation} \chi _{n}\left( m\right) \boldsymbol{1}_{\Omega _{m}\left( \theta \right) }<\left( 1+\delta \right) \sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{n}}+\left( \frac{1}{\delta }+\frac{1}{3}\right) \frac{\theta x}{\left( \sqrt{D_{m}}+% \sqrt{2x}\right) \sqrt{n}}\text{ .} \label{upper_chi} \end{equation}% By taking $\delta =\sqrt{\theta x}\left( D_{m}+\sqrt{2xD_{m}}\right) ^{-1/2}$% , the right-hand side of (\ref{upper_chi}) becomes% \begin{align*} &\sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{n}}+2\sqrt{\frac{\theta x}{n}}\frac{% \sqrt{D_{m}}}{\sqrt{D_{m}+\sqrt{2xD_{m}}}}+\frac{\theta x}{3\left( \sqrt{% D_{m}}+\sqrt{2x}\right) \sqrt{n}} \\ \leq &\sqrt{\frac{D_{m}}{n}}+\sqrt{\frac{2x}{n}}+2\sqrt{\frac{\theta }{n}}% \left( \sqrt{x}\wedge \left( \frac{xD_{m}}{2}\right) ^{1/4}\right) +\frac{% \theta }{3}\sqrt{\frac{x}{n}}\left( \sqrt{\frac{x}{D_{m}}}\wedge \frac{1}{% \sqrt{2}}\right) \text{ ,} \end{align*}% which gives (\ref{upper_chi_gene}). Inequality (\ref{upper_chi_part}) is a direct consequence of (\ref{upper_chi_gene}). \end{proof} \begin{proof}[Proof of Theorem \protect\ref{prop_dev_gauche_chi}] Let us set for any $I\in m$, $\varphi _{I}=\left( P\left( I\right) \right) ^{-1/2}\mathbbm{1}_{I}$. The family of functions $\left( \varphi _{I}\right) _{I\in m}$ forms an orthonormal basis of $\left( m,\left\Vert \cdot \right\Vert _{2}\right) $. By Cauchy-Schwarz inequality, we have% \begin{equation*} \chi _{n}\left( m\right) =\sup_{\left( a_{I}\right) _{I\in m}\in B_{2}}\left\vert \left( P_{n}-P\right) \left( \sum_{I\in m}a_{I}\varphi _{I}\right) \right\vert \text{ ,} \end{equation*}% where $B_{2}=\left\{ \left( a_{I}\right) _{I\in m};\sum_{I\in m}a_{I}^{2}\leq 1\right\} $. As \begin{equation*} \sigma _{m}^{2}:=\sup_{\left( a_{I}\right) _{I\in m}\in B_{2}}\mathbb{V}% \left( \sum_{I\in m}a_{I}\varphi _{I}\left( \xi _{1}\right) \right) \leq 1 \end{equation*}% and \begin{equation*} b_{m}=\sup_{\left( a_{I}\right) _{I\in m}\in B_{2}}\left\Vert \sum_{I\in m}a_{I}\varphi _{I}\right\Vert _{\infty }\leq \sup_{I\in m}\left\Vert \varphi _{I}\right\Vert _{\infty }=\sup_{I\in m}\frac{1}{\sqrt{P\left( I\right) }}\text{ ,} \end{equation*}% we get by Klein-Rio's inequality (see \cite{Klein_Rio:05}), for every $% x,\delta >0$, \begin{equation} \mathbb{P}\left( \chi _{n}\left( m\right) \leq \left( 1-\delta \right) \mathbb{E}\left[ \chi _{n}\left( m\right) \right] -\sqrt{\frac{2x}{n}}% -\left( \frac{1}{\delta }+1\right) \frac{b_{m}x}{n}\right) \leq \exp \left( -x\right) \text{ }. \label{dev_left_chi} \end{equation}% Note that we have $b_{m}\leq \sqrt{D_{m}A_{\Lambda }^{-1}}$. Now, we bound $% \mathbb{E}\left[ \chi _{n}\left( m\right) \right] $ by below. By Theorem 11.10 in \cite{BouLugMas:13} applied to $\chi _{n}\left( m\right) $, we get, for any $\zeta >0$,% \begin{align} \mathbb{E}\left[ \chi _{n}^{2}\left( m\right) \right] -\mathbb{E}^{2}\left[ \chi _{n}\left( m\right) \right] \leq & \frac{\sigma _{m}^{2}}{n}+4\frac{% b_{m}}{n}\mathbb{E}\left[ \chi _{n}\left( m\right) \right] \notag \\ \leq & \frac{1}{n}+4\frac{\sqrt{D_{m}A_{\Lambda }^{-1}}}{n}\mathbb{E}\left[ \chi _{n}\left( m\right) \right] \notag \\ \leq & \frac{1}{n}+4\zeta ^{-1}\frac{D_{m}A_{\Lambda }^{-1}}{n^{2}}+\zeta \mathbb{E}^{2}\left[ \chi _{n}\left( m\right) \right] \text{ .} \label{variance_bound} \end{align}% The latter inequality results from $2ab\leq \zeta a^{2}+\zeta ^{-1}b^{2}$ applied with $a=\mathbb{E}\left[ \chi _{n}\left( m\right) \right] $ and $b=2% \sqrt{D_{m}A_{\Lambda }^{-1}}/n$. As $\mathbb{E}\left[ \chi _{n}^{2}\left( m\right) \right] =D_{m}/n$, (\ref{variance_bound}) applied with $\zeta =n^{-1/2}$ gives% \begin{align} \mathbb{E}\left[ \chi _{n}\left( m\right) \right] \geq & \sqrt{\frac{1}{% 1+n^{-1/2}}\left( \frac{D_{m}}{n}-4A_{\Lambda }^{-1}\frac{D_{m}}{n^{3/2}}-% \frac{1}{n}\right) _{+}} \notag \\ \geq & \sqrt{\frac{D_{m}}{n}}\left( 1-L_{A_{\Lambda }}D_{m}^{-1/2}\vee n^{-1/4}\right) \text{ .} \label{lower_mean} \end{align}% Hence, by using (\ref{lower_mean}) and taking $x=\alpha \ln (n+1)$ and $% \delta =n^{-1/4}\sqrt{\ln (n+1)}$ in (\ref{dev_left_chi}), we obtain with probability at least $1-(n+1)^{-\alpha }$,% \begin{align*} & \chi _{n}\left( m\right) \\ \geq & \left( 1-\frac{\sqrt{\ln (n+1)}}{n^{1/4}}\right) \sqrt{\frac{D_{m}}{n}% }\left( 1-L_{A_{\Lambda }}\frac{1}{\sqrt{D_{m}}}\vee \frac{1}{n^{1/4}}\right) \\ {}& -\sqrt{\frac{2\alpha \ln (n+1)}{n}}-\left( \frac{n^{1/4}}{\sqrt{\ln (n+1)% }}+1\right) \frac{\alpha \sqrt{D_{m}}\ln (n+1)}{n} \\ \geq & \sqrt{\frac{D_{m}}{n}}\left( 1-L_{A_{\Lambda }}\frac{\sqrt{\ln (n+1)}% }{n^{1/4}}\vee \frac{1}{\sqrt{D_{m}}}-\sqrt{\frac{2\alpha \ln (n+1)}{D_{m}}}% -L_{\alpha }\frac{\sqrt{\ln (n+1)}}{n^{1/4}}\right) \\ \geq & \sqrt{\frac{D_{m}}{n}}\left( 1-L_{A_{\Lambda },\alpha }\frac{\sqrt{% \ln (n+1)}}{n^{1/4}}\vee \sqrt{\frac{\ln (n+1)}{D_{m}}}\right) \text{ ,} \end{align*}% which concludes the proof. \end{proof} \subsection{Proofs related to Section \protect\ref% {section_concentration_inequalities}} In this section, our aim is to control for any histogram model $m$, the deviations of the empirical bias from its mean, which is the true bias. Concentration inequalities for the centered empirical bias are provided in Section~\ref{section_concentration_inequalities}. In order to prove oracle inequalities for the KL divergence, we relate the magnitude of these deviations to the true bias. This is done in Section~\ref% {section_margin_like_relations}. Since we believe that the results in this section are of interest in themselves, we have stated them, when possible, for a general density in $\mathcal{S}$ rather for the projections $f_{m}$. The results in this section are based on the Cram\`{e}r-Chernoff method (see \cite{BoucheronLugosiMassart:2013} for instance). Let us recall that if we set $S:=\sum_{i=1}^{n}X_{i}-\mathbb{E}\left[ X_{i}\right] $, where $\left( X_{i}\right) _{i=1}^{n}$ are $n$ i.i.d. real random variables, and for any $% \lambda \geq 0$ and $y\in \mathbb{R}_{+}$, \begin{equation*} \varphi _{S}\left( \lambda \right) :=\ln \left( \mathbb{E}\left[ \exp \left( \lambda S\right) \right] \right) =n\left( \ln \left( \mathbb{E}\left[ \exp \left( \lambda X_{1}\right) \right] \right) -\lambda \mathbb{E}\left[ X_{1}% \right] \right) \end{equation*}% and \begin{equation*} \varphi _{S}^{\ast }\left( y\right) :=\sup_{\lambda \in \mathbb{R}% _{+}}\left\{ \mathbb{\lambda }y\mathbb{-}\varphi _{S}\left( \lambda \right) \right\} \text{ ,} \end{equation*}% then \begin{equation} \mathbb{P}\left( S\geq y\right) \leq \exp \left( -\varphi _{S}^{\ast }\left( y\right) \right) \text{ .} \label{markov} \end{equation} \begin{proof}[Proof of Proposition \protect\ref{delta_bar_dev_droite_MLE}] We first prove concentration inequality (\ref{dev_droite_gene_2}). We set $% X_{i}:=\ln \left( \left. f\right/ f_{\ast }\right) \left( \xi _{i}\right) $ and use Inequality (\ref{markov}). For $\lambda \in \left[ 0,1\right] $, as $% \mathbb{E}\left[ X_{1}\right] =-\mathcal{K}\left( f_{\ast },f\right) $, we have% \begin{align*} \left. \varphi _{S}\left( \lambda \right) \right/ n=& \ln \left( P\left[ \left( \left. f\right/ f_{\ast }\right) ^{\lambda }\right] \right) +\lambda \mathcal{K}\left( f_{\ast },f\right) \notag \\ \leq & \lambda \ln \left( P\left[ \left. f\right/ f_{\ast }\right] \right) +\lambda \mathcal{K}\left( f_{\ast },f\right) =\lambda \mathcal{K}\left( f_{\ast },f\right) \text{ ,} \end{align*}% where the inequality derives from the concavity of the function $x\mapsto x^{\lambda }$. By setting $\mathcal{K=K}\left( f_{\ast },f\right) $, we thus get% \begin{equation*} \varphi _{S}^{\ast }\left( y\right) \geq \sup_{\lambda \in \left[ 0,1\right] }\left\{ \mathbb{\lambda }y\mathbb{-}\varphi _{S}\left( \lambda \right) \right\} \geq \left( y-n\mathcal{K}\right) _{+}\text{ ,} \end{equation*}% which implies,% \begin{equation} \mathbb{P}\left( (P_{n}-P)\left( \ln \left( f/f_{\ast }\right) \right) \geq x\right) \leq \exp \left( -n(x-\mathcal{K})_{+}\right) \text{ }. \label{ineq_right_zero} \end{equation}% Inequality (\ref{dev_droite_gene_2}) is a direct consequence of (\ref% {ineq_right_zero}). Moreover, we notice that for any $u\in \mathbb{R}$, $% \exp \left( u\right) \leq 1+u+\frac{u^{2}}{2}\exp \left( u_{+}\right) $ and $% \ln \left( 1+u\right) \leq u$, where $u_{+}=u\vee 0$. By consequence, for $% \lambda \in \left[ 0,1\right] $, it holds% \begin{gather*} \left. \varphi _{S}\left( \lambda \right) \right/ n=\ln \left( \mathbb{E}% \left[ \exp \left( \lambda X_{1}\right) \right] \right) -\lambda \mathbb{E}% \left[ X_{1}\right] \\ \leq \ln \left( 1+\lambda \mathbb{E}\left[ X_{1}\right] +\frac{\lambda ^{2}}{% 2}\mathbb{E}\left[ X_{1}^{2}\exp \left( \lambda \left( X_{1}\right) _{+}\right) \right] \right) -\lambda \mathbb{E}\left[ X_{1}\right] \\ \leq \frac{\lambda ^{2}}{2}\mathbb{E}\left[ X_{1}^{2}\exp \left( \lambda \left( X_{1}\right) _{+}\right) \right] \leq \frac{\lambda ^{2}}{2}\mathbb{E}% \left[ X_{1}^{2}\exp \left( \left( X_{1}\right) _{+}\right) \right] \leq \frac{\lambda ^{2}}{2}v\text{ .} \end{gather*}% Now, we get, for any $y\geq 0$,% \begin{equation} \varphi _{S}^{\ast }\left( y\right) \geq \sup_{\lambda \in \left[ 0,1\right] }\left\{ \mathbb{\lambda }y\mathbb{-}\varphi _{S}\left( \lambda \right) \right\} \geq \sup_{\lambda \in \left[ 0,1\right] }\left\{ \mathbb{\lambda }y% \mathbb{-}n\frac{\lambda ^{2}v}{2}\right\} =\left( \frac{y^{2}}{2nv}\mathbf{1% }_{y\leq nv}+\left( y-\frac{nv}{2}\right) \mathbbm{1}_{y>nv}\right) \text{ .} \label{lower_phi_star_droite} \end{equation}% So, by using (\ref{markov}) with (\ref{lower_phi_star_droite}) taken with $% x=y/n$, it holds% \begin{equation} \mathbb{P}\left( \left( P_{n}-P\right) \left( \ln \left( \left. f\right/ f_{\ast }\right) \right) \geq x\right) \leq \exp \left( -n\left( \frac{x^{2}% }{2v}\mathbbm{1}_{x\leq v}+\left( x-\frac{v}{2}\right) \mathbbm{1}% _{x>v}\right) \right) \text{ .} \label{dev_droite_prelim} \end{equation}% To obtain (\ref{bernstein_dev_droite_2}), we notice that Inequality (\ref% {dev_droite_prelim}) implies by simple calculations, for any $z\geq 0$,% \begin{equation*} \mathbb{P}\left( \left( P_{n}-P\right) \left( \ln \left( \left. f\right/ f_{\ast }\right) \right) \geq \sqrt{\frac{2vz}{n}}\mathbbm{1}_{z\leq nv/2}+\left( \frac{z}{n}+\frac{v}{2}\right) \mathbbm{1}_{z>nv/2}\right) \leq \exp \left( -z\right) \text{ .} \end{equation*}% To conclude the proof, it suffices to remark that $\sqrt{2vz/n}\mathbbm{1}% _{z\leq nv/2}+\left( z/n+v/2\right) \mathbbm{1}_{z>nv/2}\leq \sqrt{2vz/n}% +2z/n$. \end{proof} \begin{proof}[Proof of Proposition~\ref{delta_bar_dev_gauche_MLE}] Let us first prove the inequality of concentration (\ref{delta_dev_gauche_2}). We set $X_{i}:=\ln \left( \left. f_{\ast }\right/ f\right) \left( \xi _{i}\right) $ and use (\ref{markov}). For $\lambda \in \left[ 0,r\right] $, we have by H\"{o}lder's inequality, $P\left[ \left( \left. f_{\ast }\right/ f\right) ^{\lambda }\right] \leq P\left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] ^{\lambda /r}$. Hence,% \begin{align*} \left. \varphi _{S}\left( \lambda \right) \right/ n=& \ln \left( P\left[ \left( \left. f_{\ast }\right/ f\right) ^{\lambda }\right] \right) -\lambda \mathcal{K}\left( f_{\ast },f\right) \\ \leq & \lambda \left( \frac{1}{r}\ln \left( P\left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] \right) -\mathcal{K}\left( f_{\ast },f\right) \right) \text{ .} \end{align*}% Let us notice that by concavity of $\ln $, we have $\frac{1}{r}\ln \left( P% \left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] \right) -\mathcal{% K}\left( f_{\ast },f\right) \geq 0$. Now we get, for any $y\geq 0$,% \begin{align*} \varphi _{S}^{\ast }\left( y\right) \geq & \sup_{\lambda \in \left[ 0,r% \right] }\left\{ \mathbb{\lambda }y\mathbb{-}\varphi _{S}\left( \lambda \right) \right\} \\ \geq & \sup_{\lambda \in \left[ 0,r\right] }\left\{ \mathbb{\lambda }\left( y% \mathbb{-}n\left( \frac{1}{r}\ln \left( P\left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] \right) -\mathcal{K}\left( f_{\ast },f\right) \right) \right) \right\} \\ =& \left( ry\mathbb{-}n\left( \ln \left( P\left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] \right) -r\mathcal{K}\left( f_{\ast },f\right) \right) \right) _{+}\text{ .} \end{align*}% Using (\ref{markov}), we obtain, for any $x\geq 0$,% \begin{equation} \mathbb{P}\left( (P_{n}-P)(\ln \left( f/f_{\ast }\right) )\leq -x\right) \leq \exp \left( -n(rx-\ln \left[ \left( \left. f_{\ast }\right/ f\right) ^{r}\right] +r\mathcal{K}\left( f_{\ast },f\right) )_{+}\right) \text{ .} \label{delta_dev_gauche_0} \end{equation}% Inequality (\ref{delta_dev_gauche_2}) is a straightforward consequence of (% \ref{delta_dev_gauche_0}). As in the proof of Theorem \ref% {delta_bar_dev_droite_MLE}, we notice that for any $u\in \mathbb{R}$, $\exp \left( u\right) \leq 1+u+\frac{u^{2}}{2}\exp \left( u_{+}\right) $ and $\ln \left( 1+u\right) \leq u$, where $u_{+}=u\vee 0$. By consequence, for $% \lambda \in \left[ 0,r\right] $, it holds% \begin{gather*} \left. \varphi _{S}\left( \lambda \right) \right/ n=\lambda \mathbb{E}\left[ X_{1}\right] +\ln \left( \mathbb{E}\left[ \exp \left( -\lambda X_{1}\right) % \right] \right) \\ \leq \ln \left( 1-\lambda \mathbb{E}\left[ X_{1}\right] +\frac{\lambda ^{2}}{% 2}\mathbb{E}\left[ X_{1}^{2}\exp \left( \lambda \left( -X_{1}\right) _{+}\right) \right] \right) +\lambda \mathbb{E}\left[ X_{1}\right] \\ \leq \frac{\lambda ^{2}}{2}\mathbb{E}\left[ X_{1}^{2}\exp \left( \lambda \left( -X_{1}\right) _{+}\right) \right] \leq \frac{\lambda ^{2}}{2}\mathbb{E% }\left[ X_{1}^{2}\exp \left( r\left( -X_{1}\right) _{+}\right) \right] \leq \frac{\lambda ^{2}}{2}w_{r}\text{ .} \end{gather*}% Now we get, for any $y\geq 0$,% \begin{gather*} \varphi _{S}^{\ast }\left( y\right) \geq \sup_{\lambda \in \left[ 0,r\right] }\left\{ \mathbb{\lambda }y\mathbb{-}\varphi _{S}\left( \lambda \right) \right\} \\ \geq \sup_{\lambda \in \left[ 0,r\right] }\left\{ \mathbb{\lambda }y\mathbb{-% }\frac{n\lambda ^{2}w_{r}}{2}\right\} =\frac{y^{2}}{2nw_{r}}\mathbbm{1}% _{y\leq rnw_{r}}+r\left( y-\frac{rnw_{r}}{2}\right) \mathbbm{1}_{y>rnw_{r}}% \text{ ,} \end{gather*}% which gives \begin{equation} \mathbb{P}\left( (P_{n}-P)(\ln \left( f/f_{\ast }\right) )\leq -x\right) \leq \exp \left( -n\left( \frac{x^{2}}{2nw_{r}}\mathbbm{1}_{x\leq rnw_{r}}+r\left( x-\frac{rnw_{r}}{2}\right) \mathbbm{1}_{x>rnw_{r}}\right) \right) \label{delat_dev_gauche_gauss_0} \end{equation}% Inequality (\ref{delta_dev_gauche_gauss_2}) is again a consequence of (\ref% {delat_dev_gauche_gauss_0}), by the same kind of arguments as those involved in the proof of \ref{bernstein_dev_droite_2}\ in Lemma \ref% {delta_bar_dev_droite_MLE}. \end{proof} \subsection{Proofs related to Section \protect\ref% {section_margin_like_relations}} \begin{proof}[Proof of Proposition \protect\ref{lem:margin_like_unbounded}] Let us take $q>1$ such that $1/p+1/q=1$. It holds% \begin{align*} P\left[ \left( \frac{f}{f_{\ast }}\vee 1\right) \left( \ln \left( \frac{f}{% f_{\ast }}\right) \right) ^{2}\right] =& \int \left( f\vee f_{\ast }\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu \\ =& \int \frac{f_{\ast }\vee f}{f_{\ast }\wedge f}\left( \left( f_{\ast }\wedge f\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}\right) ^{\frac{1}{p}+\frac{1}{q}}d\mu \\ =& \int \left( \frac{f_{\ast }\vee f}{\left( f_{\ast }\wedge f\right) ^{% \frac{1}{q}}}\left\vert \ln \left( \frac{f}{f_{\ast }}\right) \right\vert ^{% \frac{2}{p}}\right) \left( \left( f_{\ast }\wedge f\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}\right) ^{\frac{1}{q}}d\mu \\ \leq & \left( \int \left( f_{\ast }\wedge f\right) \left( \ln \left( \frac{f% }{f_{\ast }}\right) \right) ^{2}d\mu \right) ^{\frac{1}{q}}\left( \underset{% :=I}{\underbrace{\int \frac{\left( f_{\ast }\vee f\right) ^{p}}{\left( f_{\ast }\wedge f\right) ^{p-1}}\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu }}\right) ^{\frac{1}{p}} \end{align*}% where in the last step we used H\"{o}lder's inequality. Now, by Lemma 7.24 of Massart \cite{Massart:07}, it also holds% \begin{equation*} \frac{1}{2}\int \left( f_{\ast }\wedge f\right) \left( \ln \left( \frac{f}{% f_{\ast }}\right) \right) ^{2}d\mu \leq \mathcal{K}\left( f_{\ast },f\right) \text{ .} \end{equation*}% In order to prove (\ref{margin_like_soft_droite_gene}), it thus remains to bound $I$ in terms of $p,c_{+}$ and $c_{-}$ only. First, we decompose $I$ into two parts,% \begin{equation} \int \frac{\left( f_{\ast }\vee f\right) ^{p}}{\left( f_{\ast }\wedge f\right) ^{p-1}}\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu =\int \frac{f_{\ast }^{p}}{f^{p-1}}\left( \ln \left( \frac{f_{\ast }% }{f}\right) \right) ^{2}\mathbbm{1}_{f_{\ast }\geq f}d\mu +\int \frac{f^{p}}{% f_{\ast }^{p-1}}\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}% \mathbbm{1}_{f\geq f_{\ast }}d\mu\text{.} \label{ineq_0_gene} \end{equation}% For the first term in the right-hand side of (\ref{ineq_0_gene}), we get% \begin{gather} \int \frac{f_{\ast }^{p}}{f^{p-1}}\left( \ln \left( \frac{f_{\ast }}{f}% \right) \right) ^{2}\mathbbm{1}_{f_{\ast }\geq f}d\mu \leq c_{-}^{1-p}\int f_{\ast }^{p}\left( \ln \left( \frac{f_{\ast }}{c_{-}}\right) \right) ^{2}d\mu \notag \\ \leq 4c_{-}^{1-p}\left( \left( \ln c_{-}\right) ^{2}\vee 1\right) \int f_{\ast }^{p}\left( \left( \ln f_{\ast }\right) ^{2}\vee 1\right) d\mu <+\infty \text{ ,} \label{ineq_1_gene} \end{gather}% where in the second inequality we used the following fact: $\left( a+b\right) ^{2}\leq 4\left( a^{2}\vee 1\right) \left( b^{2}\vee 1\right) $, for any real numbers $a$ and $b$. The finiteness of the upper bound is guaranteed by assumption (\ref{moment_p_bis}). For the second term in the right-hand side of (\ref{ineq_0_gene}), it holds by same kind of arguments that lead to (\ref{ineq_1_gene}),% \begin{gather} \int \frac{f^{p}}{f_{\ast }^{p-1}}\left( \ln \left( \frac{f}{f_{\ast }}% \right) \right) ^{2}\mathbbm{1}_{f\geq f_{\ast }}d\mu \leq c_{+}^{p}\int \frac{1}{f_{\ast }^{p-1}}\left( \ln \left( \frac{c_{+}}{f_{\ast }}\right) \right) ^{2}d\mu \notag \\ \leq 4c_{+}^{p}\left( \left( \ln c_{+}\right) ^{2}\vee 1\right) \int f_{\ast }^{1-p}\left( \left( \ln f_{\ast }\right) ^{2}\vee 1\right) d\mu <+\infty \text{ ,} \label{ineq_2_gene} \end{gather}% where in the last inequality we used the following fact: $\left( a-b\right) ^{2}\leq 4\left( a^{2}\vee 1\right) \left( b^{2}\vee 1\right) $. Again, the finiteness of the upper bound is guaranteed by (\ref{moment_p_bis}). Inequality (\ref{margin_like_soft_droite_gene}) then follows from combining (% \ref{ineq_0_gene}), (\ref{ineq_1_gene}) and (\ref{ineq_2_gene}). Inequality (\ref{margin_like_soft_gauche_gene}) follows from the same kind of computations. Indeed, we have by the use of H\"{o}lder's inequality,% \begin{gather*} P\left[ \left( \frac{f_{\ast }}{f}\vee 1\right) ^{r}\left( \ln \left( \frac{f% }{f_{\ast }}\right) \right) ^{2}\right] =\int \left( \frac{f_{\ast }^{r+1}}{% f^{r}}\vee f_{\ast }\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu \\ =\int \left( \frac{f_{\ast }^{r+1}\vee f_{\ast }f^{r}}{f^{r}\left( f_{\ast }\wedge f\right) ^{1-\frac{r+1}{p}}}\left\vert \ln \left( \frac{f}{f_{\ast }}% \right) \right\vert ^{\frac{2\left( r+1\right) }{p}}\right) \left( \left( f_{\ast }\wedge f\right) \left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}\right) ^{1-\frac{r+1}{p}}d\mu \\ \leq \left( \int \left( f_{\ast }\wedge f\right) \left( \ln \left( \frac{f}{% f_{\ast }}\right) \right) ^{2}d\mu \right) ^{1-\frac{r+1}{p}}\left( \underset% {:=I_{r}}{\underbrace{\int \frac{f_{\ast }^{p}\vee f_{\ast }^{\frac{p}{r+1}% }f^{\frac{rp}{r+1}}}{f^{\frac{rp}{r+1}}f_{\ast }^{\frac{p}{r+1}-1}\wedge f^{p-1}}\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu }}% \right) ^{\frac{r+1}{p}}\text{ .} \end{gather*}% In order to prove (\ref{margin_like_soft_gauche_gene}), it thus remains to bound $I_{r}$ in terms of $p,c_{+}$ and $c_{-}$ only. Again, we split $I_{r}$ into two parts,% \begin{equation} \int \frac{f_{\ast }^{p}\vee f_{\ast }^{\frac{p}{r+1}}f^{\frac{rp}{r+1}}}{f^{% \frac{rp}{r+1}}f_{\ast }^{\frac{p}{r+1}-1}\wedge f^{p-1}}\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}d\mu =\int \frac{f_{\ast }^{p}}{% f^{p-1}}\left( \ln \left( \frac{f_{\ast }}{f}\right) \right) ^{2}\mathbbm{1}% _{f_{\ast }\geq f}d\mu +\int f_{\ast }\left( \ln \left( \frac{f}{f_{\ast }}% \right) \right) ^{2}\mathbbm{1}_{f\geq f_{\ast }}d\mu \text{ .} \label{ineq_0_gauche_gene} \end{equation}% The first term in the right-hand side of (\ref{ineq_0_gauche_gene}) is given by (\ref{ineq_1_gene}) above. For the second term in the right-hand side of (% \ref{ineq_0_gauche_gene}), it holds% \begin{align} \int f_{\ast }\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}% \mathbbm{1}_{f\geq f_{\ast }}d\mu \leq & \int f_{\ast }\left( \ln \left( \frac{c_{+}}{f_{\ast }}\right) \right) ^{2}d\mu \notag \\ \leq & 2\left( \ln \left( c_{+}\right) \right) ^{2}+2P\left( \left( \ln f_{\ast }\right) ^{2}\right) \text{ .} \label{ineq_2_gene_bis} \end{align}% Furthermore we have $f_{\ast }^{p-1}+f_{\ast }^{-p}\geq 1$, so $P\left( \left( \ln f_{\ast }\right) ^{2}\right) \leq P\left( f_{\ast }^{p-1}\left( \ln f_{\ast }\right) ^{2}\right) +P\left( f_{\ast }^{-p}\left( \ln f_{\ast }\right) ^{2}\right) \leq J+Q$ and by (\ref{ineq_2_gene_bis}), \begin{equation*} \int f_{\ast }\left( \ln \left( \frac{f}{f_{\ast }}\right) \right) ^{2}% \mathbbm{1}_{f\geq f_{\ast }}d\mu \leq 2\left( \left( \ln \left( c_{+}\right) \right) ^{2}+J+Q\right) <+\infty \text{ ,} \end{equation*}% where the finiteness of the upper bound comes from (\ref{moment_p_bis}). Inequality (\ref{margin_like_soft_gauche_gene}) then easily follows. \end{proof} \bigskip \begin{proof}[Proof of Proposition \protect\ref% {lemma_margin_like_bounded_below}] Let us first prove Inequality (\ref{margin_like_soft}). Considering the proof of Inequality (\ref{margin_like_soft_droite_gene}) of Lemma \ref% {lem:margin_like_unbounded} given above, we see that it is sufficient to bound the second term in the right-hand side of (\ref{ineq_0_gene}), applied with $f=f_{m}$, in terms of $A_{\min },J$ and $p$ only. It holds% \begin{equation} \int \frac{f_{m}}{f_{\ast }^{p-1}}\left( \ln \left( \frac{f_{m}}{f_{\ast }}% \right) \right) ^{2}\mathbbm{1}_{f_{m}\geq f_{\ast }}d\mu \leq A_{\min }\int \left( \frac{f_{m}}{A_{\min }}\right) ^{p}\left( \ln \left( \frac{f_{m}}{% A_{\min }}\right) \right) ^{2}d\mu \text{ .} \label{ineq_2_lem} \end{equation}% Now, we set $h$ an auxilliary function, defined by $h\left( x\right) =x^{p}\left( \ln x\right) ^{2}$ for any $x\geq 1$. It is easily seen that $h$ is convex. So it holds, for any $I\in \Lambda _{m}$,% \begin{equation*} h\left( \frac{f_{m}\left( I\right) }{A_{\min }}\right) =h\left( \int_{I}% \frac{f_{\ast }}{A_{\min }}\frac{d\mu }{\mu \left( I\right) }\right) \leq \int_{I}h\left( \frac{f_{\ast }}{A_{\min }}\right) \frac{d\mu }{\mu \left( I\right) }\text{ .} \end{equation*}% From the latter inequality and from (\ref{ineq_2_lem}), we deduce% \begin{equation*} \int \frac{f_{m}}{f_{\ast }^{p-1}}\left( \ln \left( \frac{f_{m}}{f_{\ast }}% \right) \right) ^{2}\mathbbm{1}_{f_{m}\geq f_{\ast }}d\mu \leq A_{\min }\int h\left( \frac{f_{m}}{A_{\min }}\right) d\mu \leq A_{\min }\int h\left( \frac{% f_{\ast }}{A_{\min }}\right) d\mu \leq 4A_{\min }^{1-p}\left( \left( \ln A_{\min }\right) ^{2}\vee 1\right) J\text{ .} \end{equation*}% Inequality (\ref{margin_like_soft}) is thus proved. In the same manner, to establish Inequality (\ref{margin_like_gauche_soft}) it suffices to adapt the proof of inequality (\ref% {margin_like_soft_gauche_gene}) given above by controlling the second term in the right-hand side of (\ref{ineq_0_gauche_gene}), applied with $f=f_{m}$% , in terms of $A_{\min },p$ and $P\left( \ln f_{\ast }\right) ^{2}$ only. Let us notice that the function $f$ defined on $\left[ 1,+\infty \right) $ by $f\left( x\right) =x\left( \ln x\right) ^{2}$ is convex. We have% \begin{equation*} \int f_{\ast }\left( \ln \left( \frac{f_{m}}{f_{\ast }}\right) \right) ^{2}% \mathbbm{1}_{f_{m}\geq f_{\ast }}d\mu \leq \int f_{m}\left( \ln \left( \frac{% f_{m}}{A_{\min }}\right) \right) ^{2}d\mu =A_{\min }\int f\left( \frac{f_{m}% }{A_{\min }}\right) d\mu \text{ .} \end{equation*}% Now, for any $I\in \Lambda _{m}$, it holds $f\left( \frac{f_{m}\left( I\right) }{A_{\min }}\right) =f\left( \int_{I}\frac{f_{\ast }}{A_{\min }}% \frac{d\mu }{\mu \left( I\right) }\right) \leq \int_{I}f\left( \frac{f_{\ast }}{A_{\min }}\right) \frac{d\mu }{\mu \left( I\right) }$. Hence,% \begin{equation*} \int f_{\ast }\left( \ln \left( \frac{f_{m}}{f_{\ast }}\right) \right) ^{2}% \mathbbm{1}_{f_{m}\geq f_{\ast }}d\mu \leq A_{\min }\int f\left( \frac{% f_{\ast }}{A_{\min }}\right) d\mu \leq 2P\left( \ln f_{\ast }\right) ^{2}+2\left( \ln A_{\min }\right) ^{2}\text{ ,} \end{equation*}% which gives the desired upper-bound and proves (\ref{margin_like_gauche_soft}% ). In the event that $f_{\ast }\in L_{\infty }\left( \mu \right) $, we have to prove (\ref{margin_like_opt}). We have ${\inf_{z\in \mathcal{Z}}}$\textup{$% f_{\ast }$}${\left( z\right) \leq \inf_{z\in \mathcal{Z}}}$\textup{$f_{m}$}${% \left( z\right) \leq \left\Vert f_{m}\right\Vert _{\infty }\leq \left\Vert f_{\ast }\right\Vert _{\infty }}$, so it holds% \begin{equation*} P\left[ \left( \frac{f_{m}}{f_{\ast }}\vee 1\right) \left( \ln \left( \frac{% f_{m}}{f_{\ast }}\right) \right) ^{2}\right] \vee P\left[ \left( \frac{% f_{\ast }}{f_{m}}\vee 1\right) ^{r}\left( \ln \left( \frac{f_{m}}{f_{\ast }}% \right) \right) ^{2}\right] \leq \left( \frac{{\left\Vert f_{\ast }\right\Vert _{\infty }}}{A_{\min }}\right) ^{r\vee 1}P\left[ \left( \ln \left( \frac{f_{m}}{f_{\ast }}\right) \right) ^{2}\right] \text{ .} \end{equation*}% Now, Inequality (\ref{margin_like_opt}) is a direct consequence of Lemma 1 of Barron and Sheu \cite{BarSheu:91}, which contains the following inequality, \begin{equation*} P\left[ \left( \ln \left( \frac{f_{m}}{f_{\ast }}\right) \right) ^{2}\right] \leq 2\exp \left( \left\Vert \ln \left( \frac{f_{m}}{f_{\ast }}\right) \right\Vert _{\infty }\right) \mathcal{K}\left( f_{\ast },f_{m}\right) \text{ }. \end{equation*}% This finishes the proof of Lemma \ref{lemma_margin_like_bounded_below}. \end{proof} \bibliographystyle{abbrv}

BkiUac7xK4tBVhvvpon_

\section{Introduction} \label{sec:intro} Over the last couple of years, large pretrained models have achieved human performance on a large share of established natural language understanding benchmark datasets \citep{devlin2018BERT}. Recent results report a surge in performance to near-human levels on the Winograd Schema Challenge \citep[WSC;][]{levesque2011WSC} in particular\citep{liu2019RoBERTa}. However, variations in task formulation across papers and evaluations makes it hard to understand the true degree of recent progress. The WSC is an English commonsense reasoning evaluation that requires a model to resolve carefully-constructed ambiguous pronouns. For example, in the sentence ``\textit{\underline{Jim} yelled at Kevin because \textbf{he} was so upset.}'' the reader will likely have to consider the motivation of the query \underline{noun phrase} (NP) to recognize whether the pronoun \textit{\textbf{he}} refers to \textit{\underline{Jim}}. The accuracy of WSC has seen an abrupt increase from 64\% to 89\% on the SuperGLUE \citep{wang2019superglue} leaderboard upon the release of RoBERTa \citep{liu2019RoBERTa}. While many works \citep{kocijan2019WSCTrick, liu2019RoBERTa, raffel2019t5} attribute such improvements to improved pretraining and the use of auxiliary training datasets, the impact of the task formalization---the combination of input specification, task specific layer design, and loss function---has not yet been seriously studied. The SuperGLUE WSC baseline with BERT (64\%) resolves pronoun references for individual examples by concatenating the pronoun and query NP embeddings and making a binary prediction for the NP span. Meanwhile, RoBERTa (89\%) uses a pretrained masked language modeling (MLM) head as part of the output layer and treats the task as a multiple-choice (MC) decision between candidate NPs. We refer to these two task formalizations as pointwise span (\textbf{P-Span}) and multiple choice masked language modeling (\textbf{MC-MLM}). In our work, we interpolate between P-Span and MC-MLM using both BERT and RoBERTa to understand tasks' sensitivity to formalization and the components contributing to MC-MLM's improvement. We find MC-MLM outperforms P-Span and reduces sensitivity to hyperparameters and random restarts. We also see large variances of scores spanning random guessing to state-of-the-art (SotA) performance. The biggest gain comes from including MC inference. Further, paired training with query and candidate NPs, using a softmax over candidates, and using a pretrained MLM head all lead to reductions in variance. We show that these formalization choices impact performance differences between the BERT and RoBERTa approaches on SuperGLUE WSC, with validation accuracy increasing between P-Span and MC-MLM by 21.1\% using RoBERTa and 10.5\% using BERT. The effect of task formalization may incentivize gains from supplemental MC options and aggressive hyperparameter tuning. To avoid this, we suggest future benchmarks impose more structure, such as in this case either explicitly distributing gold candidate NPs or enforcing rules against their use. For system developers, this result highlights the value of fine-tuning pretrained language modeling heads to target tasks in low-resource settings \citep{raffel2019t5}, at least where the task format makes this an option. \section{Related Work} \label{sec:related} \paragraph{WSC Datasets} \citet{levesque2012WSC} launch the WSC with 108 handbuilt question-answer pairs, which has since grown to 273 examples, often called WSC273. Since then, several similar or derived datasets have emerged \citep{kocijan2020review}. The SuperGLUE version of the task recasts examples from WSC273 and the Pronoun Disambiguation Problems dataset \citep{morgenstern2016pdp} into 554 training and 104 validation binary classification problems. 146 test examples are derived from fiction books and handcrafted by the original WSC authors. \citet{sakaguchi2019winogrande} collect a larger dataset of fill-in-the-blank-format problems, called WinoGrande, via crowdsourcing with adversarial filtering. The dataset has five training sets, ranging from 160 to 41k examples, and shared validation and test sets with 1.3k and 1.8k examples, respectively. \paragraph{Approaches} \citet{trinh2018lmWSC} generate inputs for their recurrent neural network language model by replacing the \textsc{Pron} with either the query or candidate NP and compare the probability of the two sentences, yielding 64\% accuracy on WSC273. \citet{radford2019gpt2} use this method with a transformer language model, boosting accuracy to 71\%. \citet{ruan2019bertWSC} fine-tune BERT in a similar way, reaching 71\% as well. \citet{kocijan2019WSCTrick} also use BERT, but include additional Winograd-like training data and use the model's pretrained MLM head to achieve 74\% accuracy. In another style of approach, \citet{klein2019attention} experiment with BERT pretrained \textit{attention} weights without fine-tuning, and achieve an accuracy of 60\%. For SuperGLUE WSC, the official baseline uses BERT and a linear classifier on BERT's output embeddings for the \texttt{[CLS]} token, pronoun token, and query NP span representations but fail to exceed the majority-class baseline, only matching it at 64\%. \citet{liu2019RoBERTa} use the newer RoBERTa and adapt the \citet{kocijan2019WSCTrick} approach with cross entropy loss to raise this accuracy to 89\%. T5 \citep{raffel2019t5} marks the pronoun in the input and fine-tune a transformer encoder-decoder model to generate the target NP, achieving the current state of the art at 94\%. Looking to WinoGrande, \citet{sakaguchi2019winogrande} adapt \citet{ruan2019bertWSC}'s method with RoBERTa as the baseline model, achieving 68\% accuracy on WinoGrande-Medium and 79\% accuracy on the full test set. \section{Methods under Study} \label{sec:method} We evaluate six formalizations---three existing ones and three that we introduce---to interpolate between P-Span and MC-MLM. These all use an output layer on top of an MLM pretrained transformer model, but differ in the input specification, loss function, prediction method, contextual embeddings used by the output layer, and label type. Table \ref{tab:formalizations} presents an overview. \paragraph{MC-MLM} This approach follows that of \citet{liu2019RoBERTa} in the introduction of RoBERTa. Here, the pronoun in the input is replaced by \texttt{[MASK]}. The model then uses its pretrained MLM head to evaluate the probability $\text{NP}_i$ should replace \texttt{[MASK]} and uses a softmax over the log probabilities. For multi-token NPs the model compares the \textit{geometric mean} of these probabilities. \paragraph{MC-Sent} This approach follows the WinoGrande baselines. Here, we specify the inputs by replacing the pronoun with an NP candidate and marking it with an additional \texttt{[SEP]} token. The output head feeds each option's \texttt{[CLS]} embedding into a linear layer and applies a softmax over the outputs. MC-Sent trains a linear layer from scratch, while MC-MLM may take advantage of the embedding model's MLM pretraining \paragraph{MC-Sent-NoSoftmax} MC-Sent-NoSoftmax only differs from MC-Sent by replacing the final softmax with a sigmoid and computes the probabilities of whether each input sequence is correct. Without softmax, MC-Sent-NoSoftmax is unable to provide larger gradients for examples with smaller margins between candidates. We refer to this as softmax scaling. \paragraph{MC-Sent-NoPairLoss} MC-Sent-NoPairLoss and MC-Sent-NoSoftmax differ by loss function, where MC-Sent-NoPairLoss only considers the query input. MC-Sent-NoPairLoss is unable to use gradients from multiple candidates to neutralize signals from shared words and focus on NP options. We refer to this as paired training. \paragraph{P-Sent} In P-Sent, we further remove \textit{MC evaluation} by restricting the model to a single binary classification question. This forces P-Sent to resolve pronoun references without implicitly learning to detect and eliminate NPs. \paragraph{P-Span} Instead of replacing \textsc{Pron} with NPs to determine the validity of the input sentence, P-Span follows the SuperGLUE baseline to determine whether the NP reference is correct. It first averages over the representations from the \textsc{Pron} and NP spans to create span representations. The span representations are then concatenated with the \texttt{[CLS]} token embedding and used by a logistic regression classifier. \begin{figure*}[t] \centering \subfloat{ \includegraphics[width=1\textwidth]{figures/wsc-roberta.pdf} \label{subfig:wsc} } \vspace{-5mm} \subfloat{ \includegraphics[width=1\textwidth]{figures/wsc-bert.pdf} \label{subfig:bert_results} } \vspace{-5mm} \subfloat{ \includegraphics[width=1\textwidth]{figures/winogrande-roberta.pdf} \label{subfig:winogrande} } \caption{Plots of validation accuracy from 60 runs on each corpus. The orange line marks the median number and label marks 75\textsuperscript{th} percentiles.} \label{fig:results} \end{figure*} \section{Experiments} \label{sec:experiments} \paragraph{Implementation} Our code\footnote{\url{https://github.com/nyu-mll/wsc-formalizations/tree/code_release}} builds on Huggingface Transformers \citep{wolf2019huggingface} and fairseq \citep{ott2019fairseq}. All our experiments use either pretrained RoBERTa-large or BERT-large-cased models. We evaluate on the validation set every epoch with early stopping. We conduct a random hyperparameter search of 60 trials over the space of learning rate \{1e-5, 2e-5, 3e-5\}, epoch limit \{10, 20, 40\}, batch size \{8, 16, 32, 64\}, and random seed. \paragraph{Datasets} We run experiments on SuperGLUE WSC and WinoGrande-Medium. We do not cover larger WinoGrande sizes due to computation constraints. Each WSC example includes a sentence, a \textsc{Pron} span, and a \textit{single} marked NP span. Following \citeauthor{liu2019RoBERTa}, for MC-based formalizations, we mine candidate NPs with spaCy\footnote{\url{https://spacy.io/}} and only keep one example from the group of examples that only differ by query NP to avoid redundancy. For WinoGrande, each example provides a sentence with two marked NP spans and a fill-in-the-blank to represent the \textsc{Pron}. When using asymmetric formalizations like P-Sent, we duplicate each example, making one option the query and the other the candidate. For P-Span, we use the first appearance of query or candidate NP in the sentence as the NP span and use the blank as \textsc{Pron} span. \begin{table}[t] \centering \resizebox{.45\textwidth}{!}{ \begin{tabular}{l r r r r r} \toprule & \multicolumn{3}{c}{WSC} & \multicolumn{2}{c}{WinoGrande} \\ Formalization & \multicolumn{1}{l}{Test} & \multicolumn{1}{l}{Std} & \multicolumn{1}{l}{Kurt} & \multicolumn{1}{l}{Std} & \multicolumn{1}{l}{Kurt} \\ \midrule MC-MLM & 86 & 2 & 3 & 3 & 13\\ MC-Sent & 77 & 7 & -1 & 5 & 4 \\ MC-Sent-NoSoftmax & 77 & 8 & -1 & 4 & 3 \\ MC-Sent-NoPairLoss & 86 & 6 & 5 & 1 & 1 \\ P-Sent & 67 & 5 & 24 & 2 & 28 \\ P-Span & 80 & 4 & 0 & 7 & -2 \\ \bottomrule \end{tabular}} \caption{Validation accuracy standard deviation (\textit{Std}) and excess kurtosis (\textit{Kurt}) for WSC and WinoGrande and test accuracy for WSC using RoBERTa. Test results are from an ensemble of the top five models.} \label{tab:stats} \end{table} \paragraph{RoBERTa Results} Figures \ref{subfig:wsc} and \ref{subfig:winogrande} and Table \ref{tab:stats} show the distribution over validation accuracies from 60 runs with each formalization using RoBERTa. We do not report WinoGrande test results since submissions require test set predictions from all five training sets and we only train using WinoGrande-Medium. We also include the majority-class baseline and human performance. From the WSC test results, we find MC-MLM outperforms P-Span. The 6\% gain between P-Sent and MC-Sent-NoPairLoss indicates MC evaluation alone may improve accuracy. However, we also find most formalizations are sensitive to hyperparameter choices and random seeds. Given the small size of the SuperGLUE WSC test set at 146 examples, we find it more informative to focus on the distribution of validation results. In both datasets, we see three main changes. First, including paired training with MC-Sent-NoSoftmax increases performance variance by adding more weight to the tail of higher scores. Second, we see the weight of higher performances increase even more with softmax scaling in MC-Sent. In WSC, the higher scores become the body of the distribution with smaller variance. In WinoGrande, the distribution of MC-Sent has an increased excess kurtosis indicating the tail of higher scores occur more frequently. Finally, the model achieves higher scores with significantly lower variance using MLM in MC-MLM. This may be a result of fine-tuning the pretrained MLM head rather than a new initialization. We see two main differences between WSC and WinoGrande results. First, P-Sent performs significantly worse than P-Span on WinoGrande. We suspect this is due to WinoGrande's adversarial filtering removing examples that are easy to classify from sentence representations. Second, MC-Sent-NoPairLoss does not benefit WinoGrande and may indicate the benefit from MC evaluation may not extend to other Winograd like corpora. \paragraph{BERT Results} Figure \ref{subfig:bert_results} shows ablation results using BERT and WSC. We find that RoBERTa outperforms BERT with both models using the same MC-MLM formalization, which is in line with leaderboard performances. We also find similar trends across task formalizations in Figure \ref{subfig:wsc}, further highlighting the impact of formalization decisions on performance gains. Most formalizations are still sensitive to hyperparameter choices and random seed, MC evaluation alone provides a benefit over P-Span at the 75\textsuperscript{th} percentile of roughly 6\%, and incorporating MLM provides additional benefits in performance. However, we also find that using BERT's pretrained MLM head does not provide the lower variance displayed with RoBERTa. Comparing the performances of intermediate formalizations, we see that BERT generally performs worse than RoBERTa. This is consistent with the findings from \citet{tenney2019probing} that show BERT embeddings encode information less suited for coreference resolution during pretraining. Consequently, BERT's pretrained MLM head would be less optimized for a coreference resolution task like WSC than RoBERTa's and may not provide the same stability benefits. \section{Conclusion} \label{sec:Conclusion} By only varying task formalization, we observe a wide range of results among reasonable task formalizations on WSC and WinoGrande evaluations. Having access to candidate NPs during inference alone improves the performance on SuperGLUE WSC. However, models with MC evaluation are highly sensitive to hyperparameters and fail to perform better on WinoGrande. We find training with paired inputs, using a softmax over candidates, and reusing a pretrained MLM head all help to learn commonsense reasoning and reduce this sensitivity. While we find evidence that these formalization choices can largely influence WSC performance, we do not see obvious evidence of similar occurrences on other task comparisons with RoBERTa. For MC formalizations, we follow \citeauthor{liu2019RoBERTa} for WSC and use spaCy to mine candidate NPs. This extrinsic preprocessing step yields dramatic gains without significantly changing the reasoning ability of the model. We view such gains as orthogonal to the intent of the task and urge benchmark creators to minimize the opportunity for these insubstantial improvements by imposing as much structure as is possible in the released data, for example, by providing candidate NPs explicitly. We also encourage future reports of system performances to use the same task formalization whenever possible. At a minimum, greater emphasis should be given to task formalization decisions when they deviate from the prevailing standard. We believe this will help disentangle gains due to models' reasoning abilities, especially in situations where these decisions significantly impact performance, such as in WSC. Finally, we find that differences between reasonable formalizations can have big impacts on performance with our case study using WSC. For example, using a pretrained MLM task head as the basis for a downstream task classifier yields strong results with very little hyperparameter sensitivity. This echoes the strong results seen with T5 and offers further motivation to explore these kinds of design decisions in other tasks. \section*{Acknowledgements} We thank Vid Kocijan, Ethan Perez, Jason Phang, Aishwarya Kamath, Iacer Calixto, Clara Vania, Rasika Bhalerao, Alex Warstadt, Richard Pang, He He, and Ernie Davis for their helpful feedback. This project has benefited from financial support to SB by Eric and Wendy Schmidt (made by recommendation of the Schmidt Futures program), by Samsung Research (under the project \textit{Improving Deep Learning using Latent Structure}), by Intuit, Inc., and in-kind support by the NYU High-Performance Computing Center and by NVIDIA Corporation (with the donation of a Titan V GPU). This material is based upon work supported by the National Science Foundation under Grant No. 1922658. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. \section{Introduction} \label{sec:intro} In recent years, large pre-trained models \citep{devlin2018BERT} based on Transformer architecture \citep{vaswani2017transformer} have achieved super-human performance on natural language understanding (NLU) tasks. However, these models still fall short on the Winograd Schema Challenge (WSC) \citep{levesque2011WSC}. This task requires the reader to use contextual semantics to disambiguate pronoun reference. For example, in the sentence \vspace{1mm} \textit{``The trophy didn't fit into the brown suitcase because it was too small.''} \vspace{1mm} \noindent the reader must understand how a \textit{trophy} can fit into a \textit{suitcase} to determine the correct antecedent of the pronoun \textit{it}. The path to human performance exhibits a dichotomy on the SuperGLUE \citep{wang2019superglue} leaderboard\footnote{\url{https://super.gluebenchmark.com/leaderboard}} with scores skyrocketing from 64.4\% to 89.0\% or higher. One common theme among high performing models \citep{liu2019RoBERTa} is a formalization of the task with masked language modeling (MLM). \citet{kocijan2019WSCTrick} show that using additional data and this formalization help BERT achieve an 8.8\% gain in performance. This begs the question how can changing formalization improve performance. We refer to these two task formalizations as pointwise span (\textbf{P-SPAN}) and multiple choice masked language modeling (\textbf{MC-MLM}). P-SPAN uses a sigmoid span classifier to determine if the pronoun (PRON) refers to the noun phrase (NP) query. MC-MLM uses a softmax classifier to determine the reference of the masked PRON by comparing likelihoods among NP candidates. We investigate how reformatting WSC from P-SPAN to MC-MLM results in better performance. By comparing results from six formalizations, we seek to understand (1) the task's sensitivity to formalization and (2) what components contribute to MC-MLM's improved performance. In this work, we reaffirm the impact of formalizations on WSC and find MC-MLM outperforms P-SPAN while both formalizations use the same contextual respresentations. From our ablation study, we find that MC-MLM's improvement is mainly due to multiple choice (MC) inference, paired training with query and candidate NPs, higher weighting of hard examples, and using an MLM head. While some of these components, like MC evaluation, may seem shallower in nature, we find deeper reasons that allow MC-MLM to better learn common sense reasoning in WSC. \section{Related Works} \label{sec:related} Since the introduction of WSC, several datasets have been released to train and evaluate models \citep{sakaguchi2019winogrande}. The original dataset, which we refer to as WSC-273, consists of 273 hand-crafted problems \citep{levesque2011WSC}. SuperGLUE \citep{wang2019superglue} contains a dataset that expands upon the original WSC-273 by including additional pronoun disambiguation problems \citep{morgenstern2016pdp}. The data recasts these problems into binary True/False tasks. For the remainder of our work, we refer to the SuperGLUE dataset as WSC. \citet{sakaguchi2019winogrande} present a large-scale Winograd-like dataset called WinoGrande. While similar to WSC, WinoGrande formats its examples as fill-in-the-blank problems. Its default formalization forms inputs by replacing the blank with each NP option and trains the model to choose the correct answer among all inputs. \citet{trinh2018lmWSC} first propose the use of neural network language models for WSC-273. They formalize the task by creating sentences with candidate NPs in place of the PRON. Through this approach \citet{trinh2018lmWSC} achieve an accuracy of 63.7\% on WSC-273. \citet{wang2019superglue} use a similar approach with P-SPAN and BERT \citep{devlin2018BERT} as a baseline for the SuperGLUE challenge. This formalization, inspired by \citet{tenney2019probing}, generates a span representation from the BERT embeddings of the PRON and candidate NP. This span representation is then used by a logistic regression classifier. Through this formalization, BERT achieves 64.3\% accuracy on WSC. \citet{kocijan2019WSCTrick} also fine-tune BERT, but formalize the task using MC-MLM. They mask the PRON and use BERT for MLM to evaluate the probability of candidate NP reference conditioned on the training sentence. They train their model using a margin loss. \citet{kocijan2019WSCTrick} use this formalization to fine-tune BERT on additional Winograd-like data to achieve 72.5\% accuracy on WSC-273. Later contextual embedding models like RoBERTa \citep{liu2019RoBERTa} adopt the MC-MLM formalization, but replace the margin loss with a cross entropy loss. Combined with refined pre-training, RoBERTa outperforms BERT and achieves 89.0\%\footnote{Scores are as reported on the SuperGLUE leader board.} on WSC. \section{Methodology} \label{sec:method} We use the following formalizations to interpolate between P-SPAN and MC-MLM, isolating components of MC-MLM that contribute to its improved performance. The formalizations differ by the input formatting, loss function used for training, inference with multiple choice, contextual embeddings used by the final layer, and label type. Table \ref{tab:formalizations} presents an overview of the formalizations. All formalizations use RoBERTa \citep{liu2019RoBERTa} contextual embeddings. \paragraph{P-SPAN} We follow \citet{wang2019superglue}'s approach with P-SPAN and use RoBERTa embeddings. The model uses the contextual representations from the PRON and NP to generate a span representation. The span representation is then concatenated with the \texttt{[CLS]} embedding and used by a logistic regression classifier to determine whether the input NP reference is correct. We use this formalization as our baseline. \paragraph{P-SENT} In P-SENT, we follow the input formatting of \citet{sakaguchi2019winogrande} by replacing the pronoun with a NP candidate and marking it with an additional \texttt{[SEP]} token. The following interpolating formalizations use the same formatting. With the change of input formatting, P-SENT removes the potential difficulty of predicting based on a pointwise span window. Instead, the model simply predicts whether the sentence is correct using the \texttt{[CLS]} token embedding. \paragraph{MC-SENT-PLOSS} MC-SENT-PLOSS first applies P-SENT to all candidate inputs for a given example. The model then compares the logits in an MC fashion during inference. Losses and training are independent for each candidate. We only differ MC-SENT-PLOSS from P-SENT by using MC evaluation. Despite the two models' shared training procedure, MC evaluation may improve performance by allowing the model to use process of elimination rather than independently resolving the pronoun reference. \paragraph{MC-SENT-PAIR} The first three formalizations train the model using independent inputs with different candidate NPs. In MC-SENT-PAIR, the model aggregates the losses of all candidate inputs to train with paired examples. We refer to this as paired training. Training the model with highly similar query and candidate choices increases the signal-to-noise ratio, making learning easier. \paragraph{MC-SENT} In MC-SENT-PLOSS and MC-SENT-PAIR, the model directly compares outputs from P-SENT during MC evaluation. By rescaling these outputs using softmax, the model is able to provide larger gradients to uncertain examples with smaller relative margins. We refer to this as scaling. In MC-SENT, the model feeds the contextual representation of each inptus' \texttt{[CLS]} token into a linear layer followed by a softmax. Through this scaling, the model is able to give higher training weight to difficult examples. \paragraph{MC-MLM} We use an approach similar to \citet{kocijan2019WSCTrick} for MC-MLM. In this formalization, the PRON in the input is first replaced by \texttt{[MASK]}. The model then uses MLM to evaluate $\mathbb{P}(\text{NP}_i\vert s)$: the probability $\text{NP}_i$ should replace \texttt{[MASK]}. The model then uses a softmax and compares these probabilities. By introducing the MLM component, the model may take advantage of RoBERTa's MLM pre-training to boost its performance. \section{Experiments} \label{sec:experiments} \subsection{Implementation} \label{sec:data} Our code\footnote{*ANONYMIZED*} uses PyTorch \citep{paszke2019pytorch}, Huggingface Transformers \citep{wolf2019huggingface}, and FAIRSEQ \citep{ott2019fairseq}. All our experiments use pre-trained RoBERTa-large model. Following \citet{liu2019RoBERTa}, we use AdamW \cite{loshchilov2018adamw} optimizer with (0.9, 0.999) $\beta$'s and 0.001 weight decay. We use a linear learning rate scheduler with 6\% warm up steps. We evaluate over validation set every epoch, and early stop if the performance does not improve for 20 validations. We conduct a random hyperparameter search of 60 trials over the space of learning rate \{1e-5, 2e-5, 3e-5\}, number of epochs \{10, 20, 40\}, batch size \{8, 16, 32, 64\}, and random seeds. \subsection{Data} \label{sec:data} We use WSC data from SuperGLUE\footnote{\url{https://super.gluebenchmark.com/tasks}\label{wsc}} and WinoGrande \citep{sakaguchi2019winogrande}. Each example in WSC consist of a sentence, a PRON span, and an NP span. Following \citep{liu2019RoBERTa} for MC-based formalizations, we mine candidate NPs with spaCy\footnote{\url{https://spacy.io/}}. We only keep one example from the group of examples that share the same sentence and PRON and only differ by query NP. WinoGrande examples provide a sentence with two NP options. In the sentence, an underscore is placed at the location of the PRON. When using asymmetrical formalizations like P-SENT, we duplicates each example into two and in turn make one of the option being the query NP, and the other being the candidate NP. In order to use P-SPAN formalization, we use the first appearance of query NP or candidate NP in the sentence as the NP span, and use the underscore as PRON span. Except for the special cases above, other formalization can be applied seamlessly to both data sets. \begin{figure*}[t] \centering \subfloat[WSC]{ \includegraphics[width=0.8\textwidth]{figures/wsc_violin.pdf} \label{subfig:wsc} }\hfill \subfloat[WinoGrande]{ \includegraphics[width=0.8\textwidth]{figures/winogrande_violin.pdf} \label{subfig:wsc} }\hfill \caption{Boxplot of validation accuracy from 60 runs on WSC. The red line marks the median number and higher edge of the rectangle marks the 3\textsuperscript{rd} quartile. The numbers indicate the 3\textsuperscript{rd} quartile values.} \label{fig:results} \end{figure*} \begin{table}[t] \centering \resizebox{.45\textwidth}{!}{ \begin{tabular}{l l l l l} \toprule Formalization & Top & 3\textsuperscript{rd} Quartile & Difference & Test\\ \midrule MC-MLM & 92.3\% & 86.1\% & 6.3\% & 86.3\%\\ MC-SENT & 87.5\% & 82.7\% & 4.8\% & \\ MC-SENT-PAIR & 87.5\% & 78.9\% & 8.7\% & \\ MC-SENT-PLOSS & 86.5\% & 67.3\% & 19.2\% & 86.3\%\\ P-SENT & 77.9\% & 63.7\% & 14.2\% & 67.1\%\\ P-SPAN & 76.0\% & 65.4\% & 10.6\% & 80.1\%\\ \bottomrule \end{tabular}} \caption{Statistics of validation accuracy. Difference is between top and 3\textsuperscript{rd} quartile values. Test results are from an ensemble of top 5 models.} \label{tab:stats} \end{table} \subsection{WSC Results} \label{sec:WSC results} Figure \ref{fig:results} shows the validation accuracy from 60 runs of each formalization. We also include the majority class baseline and human performance. Table \ref{tab:stats} shows the max results, 3\textsuperscript{rd} quartile results, and the difference between them. We first observe that most formalizations are sensitive to hyperparameter choices and random seeds. Across all formalizations, the gap between the top value and 3\textsuperscript{rd} quartile is fairly large with a minimum of 4.8\% (Table \ref{tab:stats}). Comparing 3\textsuperscript{rd} quartile values in Figure \ref{fig:results}, we see MC-MLM outperforms P-SPAN on the validation set by 21.2\%. The largest improvement between MC-SENT-PLOSS and MC-SENT-PAIR of 10.8\% indicates paired training with both candidate and query NPs makes learning the task easier. The remaining 10.4\% increase is distributed roughly evenly among the other changes in formalization, excluding P-SPAN to P-SENT. When looking at the distribution of points, we see three main changes. First, including MC evaluation between P-SENT and MC-SENT-PLOSS extends the tail of higher scores and allows the model to achieve better accuracy. However, the body of both formalizations remains relatively unchanged suggesting that this may be due to favorable hyperparameters. We then see the tail of higher scores fattens between MC-SENT-PLOSS and MC-SENT-PAIR and ultimately becomes the body with MC-SENT. This suggests that paired training and scaling with softmax help the model more reliably achieve better performance. Finally, we see that the model achieves higher scores with less variance with the introduction of MLM between MC-SENT and MC-MLM. This may be a result of fine-tuning the existing MLM head rather than training a linear layer from scratch. \subsection{WinoGrande Results} \label{sec:Winogrande results} We run the experiments on the medium size Winogrande dataset and include the results in the Appendix \ref{appendix:A}. Similar to WSC, we observe MC-SENT-PAIR and MC-SENT incrementally making high performance easier to achieve, MC-MLM reducing the variance of the accuracy. However, there are two main differences. First, P-SPAN is much higher than its sentence-based counterpart P-SENT. We suspect this is because when collecting Winogrande, adversarial filtering deliberately removes examples easy to classify from sentence representation, while P-SPAN formalization bypass this since it does not rely on sentence representation. Second, the tail of higher scores are less common on Winogrande than WSC, this suggests Winogrande is more sensitive to hyperparameters and random seeds. \section{Discussion} \label{sec:discussion} By using the same contextual embedding network and simply changing task formalization, we reaffirm that MC-MLM largely outperforms P-SPAN on WSC. We learn that having access to candidate NPs during inference alone improves the performance on WSC by a large margin. However, this shallow change only contributes to part of the improvement. We find deeper reasons behind MC formalizations that allow pre-trained transformers to better learn commonsense reasoning. Our ablation study shows that training with paired candidate and query NPs, higher weighting of difficult edge examples, and reusing MLM from pre-training also benefit learning. Potentially expanding upon these ideas may lead to even better methods to learning commonsense reasoning tasks. We recommend that researchers provide all options while collecting Winograd-like datasets. \section{Winogrande} \label{appendix:A} \begin{table}[h] \centering \resizebox{.45\textwidth}{!}{ \begin{tabular}{l l l l} \toprule Formalization & Top & 3\textsuperscript{rd} Quartile & Difference\\ \midrule MC-MLM & 69.5\% & 67.5\% & 1.9\% \\ MC-SENT & 68.0\% & 53.1\% & 14.8\%\\ MC-SENT-PAIR & 66.4\% & 53.9\% & 12.5\%\\ MC-SENT-PLOSS & 54.5\% & 53.0\% & 1.6\% \\ P-SENT & 63.2\% & 50.4\% & 12.8\% \\ P-SPAN & 67.9\% & 65.9\% & 1.9\% \\ \bottomrule \end{tabular}} \caption{Statistics of validation accuracy. Difference is between top and 3\textsuperscript{rd} quartile values.} \label{tab:stats} \end{table} \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/winogrande_violin.pdf} \caption{Boxplot of validation accuracy from 60 runs on WinoGrande. The red line marks the median number and higher edge of the rectangle marks the 3\textsuperscript{rd} quartile. The numbers indicate the 3\textsuperscript{rd} quartile values.} \label{fig:results} \end{figure*} \end{appendices}

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\section{Introduction} \setcounter{equation}{0} Around 20 years ago, Ruijsenaars \cite{SR-CMP} undertook the task of constructing action-angle variables for the non-relativistic and relativistic Calogero models of type $A_n$ (for reviews of these models, see e.g.~\cite{OP-Rep,SR-Kup,Cal,Suth}). In each case he made use of a commutation relation satisfied by the Lax matrix of the model under study and another matrix function of the phase-space variables, which he exhibited directly. By conjugating these matrices so as to make the Lax matrix diagonal, he noticed that the conjugate of the other matrix becomes the Lax matrix of another Calogero type model whose particle-position variables are furnished by the eigenvalues of the original Lax matrix, i.e., the action variables of the original model. The so-obtained duality between model-1 and model-2 is thus characterized by the fact that the action variables of model-1 are the particle-position variables of model-2, and vice versa. This observation was used in \cite{SR-CMP} to derive integration algorithms for the commuting flows and to calculate the scattering data. The simplest manifestation of the duality occurs in the rational Calogero model, which is actually self-dual \cite{SR-CMP, AKM}. The self-duality of this model admits a nice geometric `explanation' in terms of the symplectic reduction due to Kazhdan, Kostant and Sternberg \cite{KKS,OP-Rep}. As it will serve as a paradigm motivating our considerations, we next outline this explanation in some detail. In fact, Kazhdan, Kostant and Sternberg reduced the cotangent bundle \begin{equation} T^* u(n) \simeq u(n) \times u(n) = \{ (x,y)\} \label{1.1}\end{equation} by means of the adjoint action of the group $U(n)$, imposing the moment map constraint \begin{equation} [x,y]= \ri \kappa (\1_n- w w^\dagger):= \mu_\kappa, \label{1.2}\end{equation} where all $n$ components of the column vector $w$ are equal to $1$ and $\kappa$ is a real constant. The evaluation functions $X(x,y):=\ri x$ and $Y(x,y):=\ri y$ can be viewed as `unreduced Lax matrices' since $\{ \tr ( X^k)\}$ and $\{\tr ( Y^k)\}$ form two Abelian subalgebras in the Poisson algebra $C^\infty(T^*u(n))$. These Abelian algebras survive the reduction, because their elements are $U(n)$ invariants. If one describes the reduced phase space in terms of a gauge slice where $X$ is diagonal, then -- by solving the moment map constraint -- $Y$ becomes the Lax matrix of the rational Calogero model whose action variables are the eigenvalues of $Y$. If one describes the reduced phase space in terms of a gauge slice where $Y$ is diagonal, then $X$ becomes the Lax matrix of the `dual Calogero model'. The correspondence between the variables of the two Calogero models is obviously a symplectomorphism, for it represents the transformation between two gauge slices realizing the \emph{same} reduced phase space. The self-duality stems from the symmetrical roles of $x$ and $y$, and the commutation relation of the Lax matrices is just the constraint (\ref{1.2}) in disguise. Ruijsenaars hinted in \cite{SR-CMP, RIMS94, RIMS95} that there might exist a similar geometric picture behind the duality in other cases as well, which he left as a problem for `the aficionados of Lie theory'. Later Gorsky and coworkers \cite{GN,Nekr,Fock+} (see also \cite{AF,FR}) introduced interesting new ideas and confirmed this expectation in several cases. In particular, they derived the local version of the so-called $\mathrm{III}_{\mathrm{b}}$ trigonometric Ruijsenaars-Schneider model \cite{SR-Kup,RIMS95,DV}, by reducing a Hamiltonian system on the magnetic cotangent bundle of the loop group of $U(n)$ \cite{GN}\footnote{It is worth noting that besides the $\mathrm{III}_{\mathrm{b}}$ model there exist also other important, physically different real forms \cite{SR-Kup,RIMS95} of the complex trigonometric Ruijsenaars-Schneider model.}. The investigations in \cite{GN,Nekr,Fock+,AF,FR} focused on the local aspects and did not touch on the quite tricky global definition of the pertinent gauge slices, which is necessary to obtain complete commuting flows. We believe, however, that the reduction approach works also {\it globally} and that it is possible to characterize the Ruijsenaars duality in a {\it finite dimensional} symplectic reduction picture in {\it all} cases studied in \cite{SR-CMP,RIMS94,RIMS95}. We plan to explore this issue systematically in a series of papers, and here we report the first results of our analysis. In this paper we study a case of the duality which has not been previously described in the symplectic reduction framework. Namely, we expound the geometric picture that links together the dual pair consisting of the hyperbolic Sutherland model and the rational Ruijsenaars-Schneider model \cite{SR-CMP}. In Sect.~2, we start with two sets of `canonical integrable systems' on the cotangent bundle of the real Lie group $GL(n,\bC)$, whose Hamiltonians span two commutative families, $\{H_j\}$ and $\{\hat H_k\}$. By `canonical integrability' we simply mean that one can directly write down the Hamiltonian flows. Then, in Sect.~3, we describe a symplectic reduction of $T^*GL(n,\bC)$ for which our canonical integrable systems descend to the reduced phase space. By using the shifting trick of symplectic reduction, we exhibit two distinguished cross sections of the orbits of the gauge group that define two models of the reduced phase space. In terms of cross section $S_1$, the family $\{ H_j\}$ translates into the action variables of the Sutherland model and the family $\{\hat H_k\}$ becomes equivalent to the Sutherland particle coordinates. Cross section $S_1$ is described by Theorem 1, which summarizes well-known results \cite{OP-Rep}. In terms of cross section $S_2$, the family $\{H_j\}$ translates into the coordinate variables of the rational Ruijsenaars-Schneider model and the family $\{\hat H_k\}$ gives the action variables of this model. We call cross section $S_2$ the `Ruijsenaars gauge slice'. Its characterization by Theorem 2 below is our principal technical result. The duality symplectomorphism between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models will be realized as the gauge transformation between the cross sections $S_1$ and $S_2$ mentioned above. In addition, analogously to the case of the rational Calogero model, the symplectic reduction immediately yields integration algorithms for the commuting flows of the dual pair of models, and permits to recognize the commutation relations of the Lax matrices used by Ruijsenaars as equivalents to the moment map constraint of the reduction. These consequences of Theorem 1 and Theorem 2 are developed in Sect.~4. The self-contained presentation of the relatively simple example of the Ruijsenaars duality that follows may also facilitate the geometric understanding of this remarkable phenomenon in more complicated cases. \section{Canonical integrable systems on $T^* GL(n,\bC)$} \setcounter{equation}{0} Here we describe the two families of canonical integrable systems and their symmetries that will be used to derive the dual pair of integrable many-body models by symplectic reduction. For a general reference on symplectic reduction, we mention the textbook \cite{LM}. Consider the real Lie algebra $\G:= gl(n,\bC)$ and equip it with the invariant bilinear form \begin{equation} \langle X, Y\rangle := \Re \tr(XY) \qquad \forall X,Y\in \G, \label{2.1}\end{equation} which permits to identify $\G$ with $\G^*$ by the map $\jmath: \G^* \to \G$ as \begin{equation} \langle \jmath(\alpha), X\rangle = \alpha(X) \qquad \forall \alpha\in \G^*,\, X\in \G. \label{2.2}\end{equation} Then use left-trivialization to obtain a model of the cotangent bundle of the real Lie group $G:= GL(n,\bC)$ as \begin{equation} T^* G \simeq G \times \G = \{ (g, J^R)\,\vert\, g\in G,\,\, J^R \in \G\}, \label{2.3}\end{equation} where $\alpha_g \in T_g^* G$ is represented by $(g, \jmath\circ L_g^*(\alpha_g))\in G\times \G$ with the left-translation $L_g\in \mathrm{Diff}(G)$. In terms of this model, the canonical symplectic form $\Omega$ of $T^*G$ takes the form \begin{equation} \Omega = d \langle J^R, g^{-1} dg\rangle. \label{2.4}\end{equation} Next introduce the matrix functions $\cL_1$ and $\cL_2$ on $T^*G$ by the definitions \begin{equation} \cL_1(g,J^R):= J^R \quad\hbox{and}\quad \cL_2(g,J^R):= g g^\dagger. \label{2.5}\end{equation} We may think of these as `unreduced Lax matrices' since they generate the Hamiltonians \begin{equation} H_j := \frac{1}{j} \Re\tr(\cL_1^j), \qquad j=1,\ldots, n, \label{2.6}\end{equation} and \begin{equation} \hat H_k:= \frac{1}{2k} \tr(\cL_2^k), \qquad k=\pm 1,\ldots, \pm n, \label{2.7}\end{equation} so that both $\{ H_j\}$ and $\{ \hat H_k\}$ form commuting sets\footnote{We could have admitted also the imaginary part of the trace in (\ref{2.6}), and the Hamiltonians in (\ref{2.7}) are not functionally independent; but these are the definitions that will prove convenient for us.} with respect to the Poisson bracket on the phase space $T^*G$. It is easy to determine the flows of the Hamiltonians introduced above, through any initial value $(g(0), J^R(0))$. In fact, the flow belonging to $H_j$ (\ref{2.6}) is given by \begin{equation} g(t) = g(0) \exp(t (J^R(0))^{j-1}), \qquad J^R(t)=J^R(0). \label{2.8}\end{equation} The flow generated by $\hat H_k$ (\ref{2.7}) reads as \begin{equation} J^R(t)= J^R(0) - t \left(g^\dagger(0) g(0)\right)^k, \quad g(t)=g(0). \label{2.9}\end{equation} We are going to reduce the phase space $T^*G$ by using the symmetry group \begin{equation} K:=U(n)^L \times U(n)^R, \label{2.11}\end{equation} where the notation reflects the fact that the two $U(n)$ factors operate by left- and right-multiplications, respectively. This means that an element $(\eta_L, \eta_R) \in K$ (with $\eta_{L,R}\in U(n)$) acts by the symplectomorphism $\Psi_{\eta_L,\eta_R}$ defined by \begin{equation} \Psi_{\eta_L,\eta_R}(g,J^R) := (\eta_L g \eta_R^{-1}, \eta_R J^R \eta_R^{-1}). \label{2.12}\end{equation} This action is generated by an equivariant moment map. To describe this map, let us note that every $X\in \G$ can be uniquely decomposed (the Cartan decomposition) as \begin{equation} X= X_+ + X_- \quad\hbox{with}\quad X_+ \in u(n),\, X_- \in \ri u(n), \label{2.13}\end{equation} i.e., into anti-Hermitian and Hermitian parts. Identify $u(n)$ with $u(n)^*$ by the `scalar product' $\langle . , . \rangle$ restricted to $u(n) \subset \G$. Then the moment map $\Phi: T^* G \to u(n)^L \oplus u(n)^R$ reads \begin{equation} \Phi(g, J^R)= ( (gJ^R g^{-1})_+, - J^R_+). \label{2.14}\end{equation} The Hamiltonians $H_j$ and $\hat H_k$ are invariant under the symmetry group $K$. Hence the commutative character of the families $\{ H_j\}$ and $\{ \hat H_k\}$ survives any symplectic reduction based on this symmetry group. It is also clear that the flows of the reduced Hamiltonians will be provided as projections of the above given obvious flows to the reduced phase space. However, in general it is a matter of `art and good luck' to find a value of the moment map that leads to interesting reduced systems. In the present case it is well-known \cite{OP-Rep} that by setting the $u(n)^R$-component of the moment map $\Phi$ to zero, and by setting the $u(n)^L$-component equal to the constant $\mu_\kappa$ defined in (1.2), one obtains the hyperbolic Sutherland model from the Hamiltonian system $(T^*G, \Omega, H_2)$. Our goal is to characterize the reduced Hamiltonian systems coming from $(T^*G, \Omega, H_j)$ and from $(T^*G, \Omega, \hat H_k)$. To describe the latter systems, it will be technically very convenient to make use of the so-called shifting trick of symplectic reduction. This means that before performing the reduction we extend the phase space by a coadjoint orbit. In the present case, we consider the $U(n)$ orbit through $-\mu_\kappa$ given by \begin{equation} \cO^L_{\kappa}:= \{ \ri\kappa (v v^\dagger - \1_n) \,\vert\, v\in \bC^n,\, \vert v \vert^2 = n\,\}. \label{2.15}\end{equation} The orbit carries its own (Kirillov-Kostant-Souriau) symplectic form, which we denote by $\Omega^\cO$. The vector $v$ matters only up to phase and $(\cO^L_\kappa, \Omega^\cO)$ can be identified as a copy of $\bC P_{n-1}$ endowed with a multiple of the K\"ahler form defined by the Fubini-Study metric. \section{Symplectic reduction of the extended phase space} \setcounter{equation}{0} The extended phase space to consider now is \begin{equation} T^*G \times \cO_\kappa^L = \{ (g,J^R, \xi) \} \label{3.1}\end{equation} with the symplectic form \begin{equation} \Omega^\ext = \Omega + \Omega^\cO. \label{3.2}\end{equation} The symmetry group $K$ acts by the symplectomorphisms $\Psi_{\eta_L,\eta_R}^\ext$ given by \begin{equation} \Psi^\ext_{\eta_L,\eta_R}(g,J^R,\xi) := (\eta_L g \eta_R^{-1}, \eta_R J^R \eta_R^{-1}, \eta_L \xi \eta_L^{-1}), \label{3.3}\end{equation} and the corresponding moment map $\Phi^\ext$ is \begin{equation} \Phi^\ext(g, J^R,\xi)= ( (gJ^R g^{-1})_+ +\xi , - J^R_+). \label{3.4}\end{equation} We are going to reduce at the zero value of the extended moment map, i.e., we wish to describe the reduced phase space \begin{equation} T^*G \times \cO_\kappa^L//_0 K. \label{3.5}\end{equation} In our case this space of $K$-orbits is a smooth manifold, as will be seen from its models. It is equipped with the reduced symplectic form, $\Omega^\red$, which is characterized by the equality \begin{equation} \pi^* \Omega^\red = \iota^* \Omega^\ext. \label{3.5+}\end{equation} Here $\pi$ is the submersion from $(\Phi^\ext)^{-1}(0)$ to the space of orbits in (\ref{3.5}), and $\iota$ is the injection from $(\Phi^\ext)^{-1}(0)$ into $T^*G\times \cO^L_\kappa$. Before turning to the details, a few remarks are in order. First, note that one can reduce in steps, initially implementing only the reduction by the factor $U(n)^R$ of $K$ (\ref{2.11}). This leads to the equality \begin{equation} T^*G \times \cO_\kappa^L//_0 K = [T^* (G/U(n)^R) \times \cO^L_\kappa ]//_0 U(n)^L, \label{3.6}\end{equation} where $G/U(n)^R$ can be viewed as the symmetric space of positive definite Hermitian matrices. It is for computational convenience that we start from the larger phase space $T^* GL(n,\bC)$. Second, the advantage of the shifting trick is that convenient models of the reduced phase space may become available as cross sections of the $K$-orbits in $(\Phi^\ext)^{-1}(0)$, which are more difficult to realize without using the auxiliary orbital degrees of freedom. But in principle one can always do without the shifting trick: in our case we have $T^*G \times \cO_\kappa^L//_0 K \equiv T^*G //_{(\mu_\kappa,0)} K$, with the reduced phase space defined by (Marsden-Weinstein) point reduction on the right-hand-side of the equality. Third, if we define \begin{equation} \xi(v):= \ri \kappa ( v v^\dagger - \1_n) \qquad \forall v\in \bC^n \quad\hbox{with}\quad \vert v \vert^2 =n, \label{3.7}\end{equation} then we have \begin{equation} \eta \xi(v) \eta^{-1} = \xi(\eta v) \qquad \forall \eta \in U(n). \label{3.8}\end{equation} This means that the obvious action of $U(n)$ on $\bC^n$ underlies the coadjoint action on the orbit $\cO^L_\kappa$, and we shall use this to transform the $\bC^n$-vectors $v$ that correspond to the elements $\xi(v)$. Fourth, we define the extended Hamiltonians $H_j^\ext$ and $\hat H_k^\ext$ by \begin{equation} H_j^\ext(g,J^R,\xi):= H_j(g,J^R), \qquad \hat H^\ext_k(g,J^R,\xi):= \hat H_k(g,J^R). \label{3.9}\end{equation} The flows on $T^*G \times \cO^L_\kappa$ are obtained from the flows of the unextended Hamiltonians on $T^*G$ simply by adding the relation $\xi(t)=\xi(0)$. We may also define the matrix functions $\cL_1^\ext$ and $\cL_2^\ext$ on $T^*G \times \cO^L_\kappa$ by \begin{equation} \cL_1^\ext(g,J^R,\xi) = J^R\quad\hbox{and}\quad \cL_2^\ext(g, J^R,\xi) = g g^\dagger, \label{3.10+}\end{equation} whereby we can write \begin{equation} H_j^\ext = \frac{1}{j} \Re\tr((\cL_1^\ext)^j) \quad\hbox{and}\quad \hat H_k^\ext= \frac{1}{2k} \tr((\cL_2^\ext)^k). \label{3.11+}\end{equation} Now we turn to the description of two alternative models of the reduced phase space that will be shown to carry a dual pair of integrable many-body models. As a preparation, we associate to any vector $q \in \bR^n$ the diagonal matrix \begin{equation} \mathbf{q}:= \diag(q^1,\ldots, q^n). \label{3.10}\end{equation} We let $\C$ denote the open domain (Weyl chamber) \begin{equation} \C:= \{\, q\in \bR^n \,\vert\, q^1 > q^2 >\cdots > q^n\,\}, \label{3.11}\end{equation} and equip \begin{equation} T^* \C \simeq \C \times \bR^n \label{3.12}\end{equation} with the Darboux form \begin{equation} \Omega_{T^*\C}(q,p) := \sum_k dp_k \wedge dq^k. \label{3.13}\end{equation} \subsection{The Sutherland gauge slice $S_1$} Let us define the $\ri u(n)$-valued matrix function $L_1$ on $T^*\C$ (cf.~(\ref{3.12})) by the formula \begin{equation} L_1(q,p)_{jk}:= p_j \delta_{jk} - \ri (1-\delta_{jk}) \frac{\kappa}{ \sinh(q^j - q^k)}. \label{3.14}\end{equation} This is just the standard Lax matrix of the hyperbolic Sutherland model \cite{Mos,CRM,OP-Rep}. The following result is well-known \cite{KKS,OP-Rep}, but for readability we nevertheless present it together with a proof. \medskip \noindent \textbf{Theorem 1.} \emph{ The manifold $S_1$ defined by \begin{equation} S_1:= \{\, (e^{\mathbf{q}}, L_1(q,p), -\mu_\kappa)\,\vert\, (q,p)\in \C\times \bR^n\, \} \label{3.15}\end{equation} is a global cross section of the $K$-orbits in the submanifold $(\Phi^\ext)^{-1}(0)$ of $T^* G \times \cO^L_\kappa$. If $\iota_1: S_1 \to T^*G \times \cO_\kappa^L$ is the obvious injection, then in terms of the coordinates $q$, $p$ on $S_1$ one has \begin{equation} \iota_1^*(\Omega^\ext) = \sum_k dp_k \wedge dq^k. \label{3.16}\end{equation} Therefore, the symplectic manifold $(S_1, \sum_k dp_k\wedge dq^k)\simeq (T^*\C, \Omega_{T^*\C})$ is a model of the reduced phase space defined by (\ref{3.5}).} \medskip \noindent {\bf Proof.} Our task is to bring every element of $(\Phi^\ext)^{-1}(0)$ to a unique normal form by the `gauge transformations' provided by the group $K$. For this purpose, let us introduce the submanifold $\mP$ of $GL(n,\bC)$ given by the positive definite Hermitian matrices. Recall that the exponential map from $\ri u(n)$ to $\mP$ is a diffeomorphism. By the polar (Cartan) decomposition, every $g\in GL(n,\bC)$ can be uniquely written as \begin{equation} g= g_- g_+ \quad\hbox{with}\quad g_-\in \mP,\, g_+ \in U(n). \label{3.17}\end{equation} It is readily seen that $g$ can be transformed by the $K$-action into an element for which \begin{equation} g_+ = \1_n \quad\hbox{and}\quad g_-= e^{\mathbf{q}} \quad \hbox{with}\quad q\in \bR^n,\quad q^1 \geq q^2 \geq \cdots \geq q^n. \label{3.18}\end{equation} The moment map constraint requires $J_+^R=0$ and, by (\ref{3.4}), for triples of the form $(e^{\mathbf{q}}, J_-^R, \xi(v))$ we are left with the condition \begin{equation} \left(e^{\mathbf q} J_-^R e^{-\mathbf{q}}\right)_+ + \xi(v)=0. \label{3.19}\end{equation} This implies that the diagonal entries of the Hermitian matrix $J_-^R$ are arbitrary and the diagonal entries of the anti-Hermitian matrix $\xi(v)$ vanish. By (\ref{3.7}), it follows from $\xi(v)_{jj}=0$ that \begin{equation} v_j = e^{\ri \theta_j}, \qquad \forall j, \label{3.20}\end{equation} with some phase factors $\theta_j$. The off-diagonal components of the constraint (\ref{3.19}) are \begin{equation} (J_-^R)_{jk} \sinh (q^j - q^k) + \ri \kappa v_j \bar v_k =0, \qquad \forall j\neq k. \label{3.21}\end{equation} Since $v_j \bar v_k\neq 0$, we see from (\ref{3.18}) and (\ref{3.21}) that $q$ must belong to the open Weyl chamber $\C$ (\ref{3.11}). Then the residual gauge transformations permitted by the partial gauge fixing condition (\ref{3.18}) are given precisely by the maximal torus of $\bT_n \subset U(n)$, diagonally embedded into $K=U(n)^L \times U(n)^R$. They operate according to \begin{equation} (e^{\mathbf{q}}, J^R_-, v) \mapsto (e^{\mathbf{q}}, \tau J^R_- \tau^{-1}, \tau v) \qquad \forall \tau\in \bT_n. \label{3.22}\end{equation} Hence we can completely fix the residual gauge freedom by transforming the vector $v$ into the representative $w$, whose components are all equal to $1$. At the same time, by (\ref{3.21}), $J_-^R$ becomes equal to $L_1(q,p)$, where $\mathbf{p}$ is the arbitrary diagonal part of $J_-^R$. The calculation of $\iota_1^* (\Omega^\ext)$ as well as the rest of the statements of the theorem is now obvious. \emph{Q.E.D.} \medskip \noindent \textbf{Remark 1.} If we spell out the moment map constraint (\ref{3.19}) for the solution $J_-^R= L_1$, $v= w$ and also multiply this equation both from the left and from the right by $e^{\mathbf{q}}$, then we obtain \begin{equation} [ e^{2\mathbf{q}}, L_1(q,p)] + 2 i\kappa\left( (e^{\mathbf{q}} w) (e^{\mathbf{q}} w)^\dagger - e^{2\mathbf{q}}\right)=0. \end{equation} This is the commutation relation for the Lax matrix (\ref{3.14}) used in \cite{SR-CMP}. \subsection{The Ruijsenaars gauge slice $S_2$} In this subsection we denote the elements of $T^*\C=\C\times \bR^n$ as pairs $(\hat p, \hat q)$. We introduce the $\mP$-valued matrix function $L_2$ on $T^*\C$ by the formula \begin{equation} L_2(\hat p,\hat q)_{jk}=u_j(\hat p,\hat q) \left[ \frac{2\ri \kappa}{2 \ri \kappa + (\hat p^j-\hat p^k)} \right] u_k(\hat p,\hat q) \label{3.23}\end{equation} with the $\bR_+$-valued functions \begin{equation} u_j(\hat p, \hat q):= e^{-\hat q_j} \prod_{m\neq j} \left[ 1 + \frac{4 \kappa^2}{(\hat p^j - \hat p^m)^2}\right]^\frac{1}{4}. \label{3.24}\end{equation} One can calculate the principal minors of $L_2$ with the help of the Cauchy determinant formula, and thereby confirm that $L_2$ is indeed a positive definite matrix. Therefore it admits a unique positive definite square root, and we use it to define the $\bR^n$-valued function \begin{equation} v(\hat p, \hat q):= L_2(\hat p, \hat q)^{-\frac{1}{2}} u(\hat p,\hat q), \label{3.25}\end{equation} where $u=(u_1,\ldots, u_n)^T$. It can be verified directly, and can be seen also from the proof below, that $\vert v(\hat p, \hat q)\vert^2 =n$. Thus, by using (\ref{3.7}), we have the $\cO_\kappa^L$-valued function \begin{equation} \xi(\hat p, \hat q):= \xi(v(\hat p, \hat q)). \label{3.26}\end{equation} The function $L_2$ (\ref{3.23}) in nothing but the standard Lax matrix of the rational Ruijsenaars-Schneider model \cite{RS}, with variables denoted by somewhat unusual letters. Using the above notations, we now state the main technical result of the present paper. \medskip \noindent \textbf{Theorem 2.} \emph{ The manifold $S_2$ defined by \begin{equation} S_2:= \{\, ( L_2(\hat p, \hat q)^\frac{1}{2},\mathbf{\hat p}, \xi(\hat p,\hat q))\,\vert\, (\hat p,\hat q)\in \C\times \bR^n\, \} \label{3.27}\end{equation} is a global cross section of the $K$-orbits in the submanifold $(\Phi^\ext)^{-1}(0)$ of $T^* G \times \cO^L_\kappa$. If $\iota_2: S_2 \to T^*G \times \cO_\kappa^L$ is the obvious injection, then in terms of the coordinates $\hat p$, $\hat q$ on $S_2$ one has \begin{equation} \iota_2^*(\Omega^\ext) = \sum_k d \hat q_k \wedge d\hat p^k. \label{3.28}\end{equation} Therefore, the symplectic manifold $(S_2, \sum_k d\hat q_k\wedge d\hat p^k)\simeq (T^*\C, \Omega_{T^*\C})$ is a model of the reduced phase space defined by (\ref{3.5}).} \medskip \noindent {\bf Proof.} Let us begin by noting that by means of the $K$-action we can transform each element of $(\Phi^\ext)^{-1}(0)$ into an element $(g, J^R, \xi(v))$ that satisfies \begin{equation} g=g_- \in \mP \quad\hbox{and}\quad J^R = \mathbf{\hat p} \quad \hbox{with}\quad \hat p \in \bR^n,\quad \hat p^1 \geq \hat p^2 \geq \cdots \geq \hat p^n. \label{3.29}\end{equation} After this partial gauge fixing the moment map constraint becomes \begin{equation} g_-^{-1} \mathbf{\hat p} g_- - g_- \mathbf{\hat p} g_-^{-1} = 2 \xi(v). \label{3.30}\end{equation} In order to solve this equation, we multiply it both from the left and from the right by $g_-$, which gives \begin{equation} [\mathbf{\hat p} , g_-^2] = 2\ri \kappa ( u u^\dagger - g_-^2), \label{3.31}\end{equation} where we have combined the unknowns $g_-$ and $v$ to define \begin{equation} u:= g_- v. \label{3.32}\end{equation} This equation then permits us to express $g_-^2$ in terms of $\mathbf{\hat p}$ and $u$ as \begin{equation} (g_-^2)_{jk} = u_j \left[ \frac{2\ri \kappa}{2 \ri \kappa + (\hat p^j-\hat p^k)} \right] \bar u_k. \label{3.33}\end{equation} By calculating the determinant from the last relation, we obtain \begin{equation} \det(g_-^2) = \left(\prod_m \vert u_m\vert^2\right) \prod_{j<k} \frac{(\hat p^j - \hat p^k)^2}{(\hat p^j - \hat p^k)^2 + 4 \kappa^2}. \label{3.34}\end{equation} Since $\det(g_-^2) \neq 0$, we must have \begin{equation} \hat p^1 > \hat p^2 >\cdots > \hat p^n \quad \hbox{and}\quad u_j \neq 0 \quad \forall j. \label{3.35}\end{equation} In particular, $\hat p$ must belong to the open Weyl chamber $\C$. This implies that the residual gauge transformations permitted by our partial gauge fixing are generated by the maximal torus $\bT_n$ of $U(n)$, diagonally embedded into $K$, which act according to \begin{equation} (g_-, \mathbf{\hat p}, v) \mapsto (\tau g_- \tau^{-1}, \mathbf{\hat p}, \tau v) \qquad \forall \tau\in \bT_n . \label{3.36}\end{equation} By applying these gauge transformations to $u$ in (\ref{3.32}) we have \begin{equation} \tau: u \mapsto \tau u. \label{3.37}\end{equation} It follows that we obtain a complete gauge fixing by imposing the conditions \begin{equation} u_j > 0 \qquad \forall j. \label{3.38}\end{equation} For fixed $\hat p$, the positive vector $u$ remains arbitrary, and thus it can be uniquely parametrized by introducing a new variable $\hat q \in \bR^n$ via equation (\ref{3.24}). The outcome of the above discussion is that the manifold $S_2$ given by (\ref{3.27}) is a global cross section of the $K$-orbits in $(\Phi^\ext)^{-1}(0)$, which provides a model of the reduced phase space (\ref{3.5}). Indeed, the formula (\ref{3.33}) of $g_-^2$ becomes identical to the formula (\ref{3.23}) of $L_2$ if we take into account the gauge fixing conditions (\ref{3.38}) and the parametrization (\ref{3.24}). At the same time, the inversion of the relation (\ref{3.32}) yields $v(\hat p, \hat q)$ in (\ref{3.25}). It remains to show that the variables $(\hat p, \hat q)$ that parametrize $S_2$ are Darboux coordinates on the reduced phase space. Direct substitution into the symplectic form $\Omega^\ext$ (\ref{3.2}) is now cumbersome, since we do not have $L_2(\hat p, \hat q)^{\frac{1}{2}}$ explicitly. We circumvent this problem by proceeding as follows. We define the smooth functions $F^a$ and $E^a$ on $T^*G \times \cO_\kappa^L$ by \begin{equation} F^a(g, J^R, \xi):= \frac{1}{a} \tr [(J_-^R)^a] \quad\hbox{and}\quad E^a(g,J^R,\xi):= \frac{1}{a} \tr [(g g^\dagger)^a], \qquad a=1,\ldots, n. \label{3.39}\end{equation} We also define the functions $\cF^a$ and $\cE^a$ on the gauge slice $S_2$ by \begin{equation} \cF^a(\hat p, \hat q):= \frac{1}{a}\tr [\mathbf{\hat p}^a] \quad\hbox{and}\quad \cE^a(\hat p, \hat q):= \frac{1}{a}\tr [L_2(\hat p, \hat q)^a], \qquad a=1,\ldots, n. \label{3.40}\end{equation} It is clear that $F^a$ and $E^a$ are $K$-invariant functions, and by means of the injection map $\iota_2: S_2 \to T^*G \times \cO_\kappa^L$ we have \begin{equation} \iota_2^* F^a = \cF^a, \qquad \iota_2^* E^a = \cE^a \qquad \forall a=1,\ldots, n. \label{3.41}\end{equation} Therefore, we can determine the induced Poisson brackets of these functions by two methods. First, denote the Poisson bracket on the extended phase space by $\{.,.\}^\ext$. The Poisson bracket of $K$-invariant functions is again $K$-invariant and a straightforward calculation gives (for any $1\leq a,b \leq n$) \begin{equation} \iota_2^* \{ E^a, E^b\}^\ext = \iota_2^* \{ F^a, F^b\}^\ext =0, \qquad \iota_2^* \{ E^a, F^b\}^\ext = 2 \,\tr[(\mathbf{\hat p})^{b-1}L_2^a]. \label{3.42}\end{equation} Second, denote by $\{.,.\}^\red$ the Poisson bracket on $C^\infty(S_2)$ induced by the reduced symplectic structure. We wish to show that \begin{equation} \{ \hat p^j, \hat p^k\}^\red = \{ \hat q_j, \hat q_k\}^\red =0 \quad\hbox{and}\quad \{ \hat p^j, \hat q_k\}^\red = \delta^j_k. \label{3.43}\end{equation} Now, if we assume that the last relation holds, then it can be verified by direct calculation\footnote{Only the relation $\{ \cE^a, \cE^b\}^\red=0$ requires non-trivial effort, but this was established in \cite{RS} as well as in the more recent papers dealing with the dynamical $r$-matrix structure of the Ruijsenaars-Schneider models \cite{Avan}.} (for any $1\leq a,b \leq n$) that \begin{equation} \{ \cE^a, \cE^b\}^\red = \{ \cF^a, \cF^b\}^\red =0, \qquad \{\cE^a, \cF^b\}^\red = 2 \,\tr[(\mathbf{\hat p})^{b-1}L_2^a]. \label{3.44}\end{equation} From general principles, the restriction of the Poisson bracket of $K$-invariant functions to a gauge slice always yields the induced Poisson bracket of the restricted functions. By taking into account (\ref{3.41}), we conclude from the comparison of the equations (\ref{3.42}) and (\ref{3.44}) that the Poisson bracket on $C^\infty(S_2)$ that arises from the symplectic reduction coincides with the Poisson bracket given in coordinates by (\ref{3.43}), at least if we restrict our attention to the collection of the functions $\cF^a$, $\cE^a$. Thus it remains to prove that the functions $\cF^a$, $\cE^a$ can serve as local coordinates around any point from a dense open submanifold of $S_2$. Indeed, the symplectic form on this dense submanifold could be then written as $\sum_k d\hat q_k\wedge d\hat p^k$. The smoothness of the symplectic form would then allow us to conclude the same on the whole of $S_2$. The map from $\C \times \bR^n$ to $\C\times (\bR_+)^n$ given by $(\hat p, \hat q)\mapsto (\hat p, u(\hat p, \hat q))$ is clearly a diffeomorphism, and thus we can use $\hat p^k, u_k$ ($k=1,\ldots, n$) as coordinates on $S_2$. When expressed in these new coordinates, we denote our functions of interest as $\tilde \cF^a$, $\tilde \cE^a$ and $\tilde L_2$: \begin{equation} \tilde \cF^a(\hat p, u)= \cF^a(\hat p, \hat q), \quad \tilde \cE^a(\hat p, u)= \cE^a(\hat p, \hat q), \quad \tilde L_2(\hat p, u)= L_2(\hat p, \hat q) \label{tilded-funct}\end{equation} if $u=u(\hat p, \hat q)$. To finish the proof, it is sufficient to show that the Jacobian determinant of the map $\C \times (\bR_+)^n \to \bR^{n}\times \bR^n$ given in coordinates by $ \hat p^k, u_k \mapsto \tilde \cF^a(\hat p,u), \tilde \cE^a(\hat p, u)$ is non-vanishing on a dense open subset of $\C \times (\bR_+)^n$. For this, notice from (\ref{3.23}) and (\ref{3.40}) that $\tilde \cF^a$ and $\tilde \cE^a$ are rational functions of the variables $\hat p^k, u_k$, and thus the Jacobian \begin{equation} \det \frac{\partial (\tilde \cF^a, \tilde \cE^b)}{\partial (\hat p^j, u_k)} \label{3.45}\end{equation} is also a rational function of the same variables (that is, the quotient of two polynomials in the $2n$-variables $\hat p^k, u_k$). This implies that the Jacobian (\ref{3.45}) either vanishes identically or is non-vanishing on a dense open subset of $\C\times (\bR_+)^n\simeq S_2$. Now we show that the Jacobian (\ref{3.45}) does not vanish identically. First of all, from the fact that $\tilde \cF^a$ does not depend on $u_k$, we see easily that \begin{equation} \det \frac{\partial (\tilde \cF^a, \tilde \cE^b)}{\partial (\hat p^j, u_k)}= \det\biggl[\frac{\partial \tilde \cF^a}{\partial \hat p^j}\biggr] \det\biggl[\frac{\partial \tilde \cE^b}{\partial u_k}\biggr]. \label{dcp}\end{equation} The first determinant on the r.h.s. is the Vandermonde one: \begin{equation} \det\biggl[\frac{\partial \tilde \cF^a}{\partial \hat p^j}\biggr]=\prod_{i<m}(\hat p^m-\hat p^i), \label{vdm}\end{equation} which never vanishes on $S_2$ due to (\ref{3.35}). The second determinant on the r.h.s of (\ref{dcp}) parametrically depends on $\hat p$. In particular, if we take a real parameter $s>0$ and consider the curve $\hat p(s)$ defined by $\hat p^j(s):=e^{(n+1-j)s}$ for $j=1,\ldots,n$, then we observe that in the limit $s\to\infty$ the matrix $\tilde L_2$ (\ref{tilded-funct}) becomes diagonal, \begin{equation} \lim_{s \to \infty} \tilde L_2( \hat p(s), u)=\diag(u_1^2,...,u_n^2). \end{equation} Hence, in the limit $s\to\infty$, we encounter again a Vandermonde determinant: \begin{equation} \lim_{s\to\infty} \det\biggl[\frac{\partial \tilde \cE^b}{\partial u_k}\biggr]( \hat p(s), u) = 2^n \left(\prod_j u_j \right) \prod_{i<m}(u_m^2-u_i^2). \label{abc}\end{equation} Obviously, we can choose $u$ in such a way that the r.h.s. of equation (\ref{abc}) does not vanish. Then the Jacobian determinant (\ref{3.45}) does not vanish at $(\hat p(s), u)$ for large enough $s$. \emph{Q.E.D.} \medskip \noindent \textbf{Remark 2.} The consequence (\ref{3.31}) of the moment map constraint yields the commutation relation satisfied by the Lax matrix $L_2$ (\ref{3.23}) and $u(\hat p, \hat q)$ (\ref{3.24}): \begin{equation} [ L_2(\hat p, \hat q), \mathbf{\hat p}] + 2\ri \kappa \left( u(\hat p,\hat q)u(\hat p,\hat q)^\dagger - L_2(\hat p, \hat q) \right) =0, \end{equation} which played a crucial role in the analysis in \cite{SR-CMP}. \section{The dual pair of many-body models} \setcounter{equation}{0} We now enumerate important consequences of the results presented in the preceding sections. \begin{enumerate} \item{Since $S_1$ (\ref{3.15}) and $S_2$ (\ref{3.27}) are two models of the same reduced phase space (\ref{3.5}), there exists a \emph{natural symplectomorphism} between these two models: \begin{equation} (S_1, \sum_k dp_k \wedge dq^k)\equiv (T^*G \times \cO_\kappa^L//_0 K, \Omega^\red) \equiv (S_2, \sum_k d\hat q_k \wedge d\hat p^k). \label{4.1}\end{equation} By definition, a point of $S_1$ is related to that point of $S_2$ which represents the same element of the reduced phase space.} \item{The pull-backs of the `unreduced Lax matrices' (\ref{3.10+}) to $S_1$ and $S_2$ satisfy, respectively, \begin{equation} \iota_1^* \cL_1^\ext = L_1 \quad\hbox{and}\quad \iota_2^* \cL_2^\ext =L_2. \label{4.2}\end{equation} By the symplectic reduction, the $K$-invariant Hamiltonians (\ref{3.11+}) descend to the families of Hamiltonians defined on $(S_1, \sum_k dp_k \wedge dq^k)$ and on $(S_2, \sum_k d\hat q_k \wedge d\hat p^k)$, respectively, by \begin{equation} H_j^\red =\frac{1}{j} \tr(L_1^j), \quad (j=1,\ldots, n) \quad\hbox{and}\quad \hat H_k^\red = \frac{1}{2k} \tr(L_2^k), \quad (k=\pm 1,\ldots, \pm n). \label{4.3}\end{equation} The commutativity of $\{ H_j^\red\}$ and $\{ \hat H_k^\red\}$ (where the elements with fixed sign of $k$ form a complete set) is inherited from the commutativity of the unreduced Hamiltonians in (\ref{3.11+}). (Note that $\tr (L_1^j)=\Re \tr( L_1^j)$ since $L_1$ (\ref{3.14}) is a Hermitian matrix.) } \item{According to classical results, $L_1$ (\ref{3.14}) is the Lax matrix of the hyperbolic Sutherland model and $L_2$ (\ref{3.23}) is the Lax matrix of the rational Ruijsenaars-Schneider model. The basic Hamiltonians of these many-body models are indeed reproduced as \begin{equation} H_{\mathrm{hyp-Suth}}(q,p) \equiv \frac{1}{2}\sum_k p_k^2 + \frac{\kappa^2}{2}\sum_{j\neq k} \frac{1}{\sinh^2(q^j-q^k)}= \frac{1}{2}\tr(L_1(q,p)^2), \label{4.4}\end{equation} \begin{equation} H_\mathrm{rat-RS}(\hat p, \hat q) \equiv \sum_k \cosh(\hat q_k) \prod_{j\neq k}\left[ 1+ \frac{4\kappa^2}{(\hat p^k - \hat p^j)^2}\right]^\frac{1}{2} = \frac{1}{2}\tr(L_2(\hat p, \hat q) + L_2(\hat p, \hat q)^{-1}). \label{4.5} \end{equation} The Lax matrices themselves arose naturally (\ref{4.2}) by means of the symplectic reduction.} \item{Consider two points of $S_1$ and $ S_2$ that are related by the symplectomorphism (\ref{4.1}). Suppose that these two points are parametrized by $(q,p)\in \C\times \bR^n$ and by $(\hat p, \hat q)\in \C\times \bR^n$ according to (\ref{3.15}) and (\ref{3.27}), respectively. The fact that they lie on the same $K$-orbit means, since $g_+$ in $g=g_- g_+$ (\ref{3.17}) is fixed to $\1_n$ in both gauges, that there exists some $\eta\in U(n)$ (actually unique up to the center of $U(n)$ that acts trivially) for which \begin{equation} (\eta e^{\mathbf{q}} \eta^{-1}, \eta L_1(q,p) \eta^{-1}, -\eta \mu_\kappa \eta^{-1}) = (L_2(\hat p, \hat q)^\frac{1}{2},\mathbf{\hat p}, \xi(\hat p,\hat q)). \label{4.6} \end{equation} This shows that the matrix $\mathbf{\hat p}$, which encodes the coordinate-variables of the rational Ruijsenaars-Schneider model, results by diagonalizing the Sutherland Lax matrix $L_1(q,p)$. The same formula shows that $e^{2 \mathbf{q}}$, which encodes the coordinate-variables of the hyperbolic Sutherland model, results by diagonalizing the Ruijsenaars-Schneider Lax matrix $L_2(\hat p, \hat q)$. The original direct construction \cite{SR-CMP} of the map between the phase spaces of the two many-body models relied on diagonalization of the Lax matrices, but in that approach it was quite difficult to prove the canonicity of the map, which comes for free in the symplectic reduction framework. } \item{ It is obvious from the above observations that the two many-body models characterized by the Hamiltonians (\ref{4.4}) and (\ref{4.5}) are dual to each other in the sense described in the Introduction. On the one hand, the Ruijsenaars-Schneider particle-coordinates $\hat p^1,\dots, \hat p^n$ \emph{regarded as functions on $S_1$} define action variables for the hyperbolic Sutherland model. On the other, the Sutherland particle coordinates $q^1,\ldots, q^n$ \emph{regarded as functions on $S_2$} can serve as action variables for the rational Ruijsenaars-Schneider model. } \item{The well-known solution algorithms \cite{OP-Rep,RS} for the commuting Hamiltonians exhibited in (\ref{4.3}) can be viewed as byproducts of the geometric approach. First of all, it should be noted that all the flows generated by the reduced Hamiltonians are \emph{complete}, since this is true for the unreduced Hamiltonians whose flows stay in $(\Phi^\ext)^{-1}(0)$. By taking an initial value on the gauge slice $S_1$ and projecting the flow (\ref{2.8}) back to $S_1$ we obtain that the reduced Hamiltonian $H_j^\red$ (\ref{4.3}) generates the following evolution for the Sutherland coordinate-variables: \begin{equation} e^{2 \mathbf{q}(t)} = \cD[ e^{\mathbf{q}(0)}\exp( 2 t L_1(0)^{j-1}) e^{\mathbf{q}(0)}], \label{4.7}\end{equation} where the zero argument refers to the $t=0$ initial value, and $\cD$ denotes the operator that brings its Hermitian matrix-argument to diagonal form with eigenvalues in non-increasing order. Similarly, we obtain from (\ref{2.9}) that the reduced Hamiltonian $\hat H_k^\red$ (\ref{4.3}) generates the following flow for the Ruijsenaars-Schneider coordinate variables: \begin{equation} \mathbf{\hat p}(t)= \cD[\mathbf{\hat p}(0) - t L_2(0)^k]. \label{4.8}\end{equation} We here used that $(g^\dagger(0)g(0))^k= L_2(0)^k$ for any initial value on $S_2$. } \end{enumerate} To summarize, we presented the duality between the hyperbolic Sutherland model (\ref{4.4}) and the rational Ruijsenaars-Schneider model (\ref{4.5}) in the framework of symplectic reduction. In this way we obtained a Lie theoretic understanding of results due to Ruijsenaars \cite{SR-CMP}, who originally discovered and investigated the duality by direct means. Our approach also simplifies a considerable portion of the original technical arguments. The general line of reasoning that we followed may be adapted to explore more complicated cases of the duality in the future, too. For example, it will be demonstrated in \cite{FK1,FK3} that the reduction approach works in a conceptually very similar manner for the dualities concerning trigonometric Ruijsenaars-Schneider models. Finally, let us mention that the dual pairs of models studied by Ruijsenaars at the classical level \cite{SR-CMP,RIMS94,RIMS95} are associated at the quantum mechanical level with so-called bispectral problems \cite{DG}, as was first conjectured in \cite{SR-Kup} and later confirmed in several papers. Concerning the bispectral property, and in particular the bispectral interpretation of the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models, the reader may consult \cite{Chal,Haine} and references therein. We expect that this intriguing phenomenon could be understood also in terms of a quantum Hamiltonian reduction counterpart of our approach. \bigskip \bigskip \noindent{\bf Acknowledgements.} L.F. was partially supported by the Hungarian Scientific Research Fund (OTKA grant T049495) and by the EU network `ENIGMA' (MRTN-CT-2004-5652). He wishes to thank to I. Marshall and V. Rubtsov for useful discussions and remarks.

BkiUb7025V5hYDkHGird

\section{Introduction} In $d$-dimensional conformal field theory (CFT), a conformal symmetry is $SO(d,2)$ and two-point functions and three-point functions are completely determined up to prefactors. Four-point functions are not completely determined by the conformal symmetry and expressed as a function depending on two conformal cross-ratios. The function depending on conformal cross-ratios can be decomposed by conformal blocks or conformal partial waves which are fundamental objects in CFT. In the context of the AdS/CFT correspondence, correlation functions are calculated by Witten diagrams \cite{Witten}. Recently, holographic duals of the conformal blocks, which are called by ``geodesic Witten diagrams," are proposed \cite{Hijano}. Intermediate points among a bulk-to-bulk propagator and two bulk-to-boundary propagators in Witten diagrams are integrated in a whole region of anti-de Sitter (AdS) spacetime, but similar points in geodesic Witten diagrams are integrated along geodesics connecting two boundary points. We can confirm that the geodesic Witten diagrams are actually a holographic dual of conformal partial waves by explicit computations. More elegant way is to check that the geodesic Witten diagrams are eigenfunctions of a Casimir operator with a suitable boundary condition. The story is changed when boundaries or defects exist in CFT \cite{McAvity}. The boundaries or defects reduces its conformal symmetry from $SO(d,2)$ to $SO(d-1,2)$ when the boundaries or defects preserve the conformal symmetry. In this case, one-point functions do not vanish, and two-point functions become non-trivial. Hence, the two-points functions will be decomposed by conformal blocks instead of four-point functions. There are two distinct decompositions in boundary CFTs (bCFTs) or defect CFTs (dCFTs); an ambient channel decomposition and a boundary channel decomposition. In the ambient channel, its decompositions are based on operator product expansion (OPE) in a standard ambient spacetime\footnote{As in \cite{ADFK,KS} we use the term ambient spacetime for the $d$-dimensional spacetime whose coordinates are expressed by $x$ labelled by $\mu ,\nu$. The $(d-1)$-dimensional boundary or defect is embedded in the $d$-dimensional spacetime. The direction transverse to the boundary or defect is called $w$ and the directions parallel to them are called $\vec{x}$ labelled by $i,j$. That is, $x = (\vec{x},w)$. The term ``bulk" will be reserved for the $(d+1)$-dimensional spacetime of the holographic dual, whose coordinates are $X$ labelled by $M,N$. We do not use the term ``boundary" as the boundary of AdS spacetime.}. That is, two-point functions can be expanded by one-point functions of primary operators and their descendants. As mentioned before, one-point functions do not vanish in general because of the presence of the boundary or defect. The primary operators in the ambient spacetime can be also expanded by local operators localized on the boundary or defect. Hence, two-point functions are decomposed by two-point functions of boundary operators. This decomposition is called the boundary channel decomposition. \begin{figure} \begin{center} \begin{tikzpicture} \fill (0,0) circle (3pt); \draw[line width=1pt] (-3.5,0) --node[above]{$r=+\infty$} (0,0) --node[above]{$r=-\infty$} (3.5,0); \draw[dashed,line width=1pt] (0,0) -- (0,-3.5); \draw[dashed,line width=1pt] (0,0) to [out=330,in=175] (340:3.5); \draw[dashed,line width=1pt] (0,0) to [out=210,in=5] (200:3.5); \draw[dashed,line width=1pt] (0,0) to [out=300,in=165] (310:3.5); \draw[dashed,line width=1pt] (0,0) to [out=240,in=15] (230:3.5); \draw[->] (350:3.8) arc [start angle = 350, end angle = 330, radius = 3.8] ; \node (r) at (4,-1.5) [above] {$r$}; \end{tikzpicture} \caption{A schematic figure of the Janus geometry \eqref{metric}. The dashed lines are at constant $r$ and represent sliced AdS$_d$ spacetimes.} \label{fig:metric} \end{center} \end{figure} Simple holographic models of bCFTs or dCFTs are described by a $(d+1)$-dimensional metric \begin{equation} \dd s^2 = \dd r^2+\mathrm{e}^{2A(r)}\dd s_{\mathrm{AdS}_d}^2 \qquad \text{with} \qquad \dd s_{\mathrm{AdS}_d}^2 =L^2\frac{\dd w^2+\dd\vec{x}^2 }{w^2} \label{metric} \end{equation} where $r$ is a radial coordinate and $r\to \pm \infty$ corresponds to an AdS boundary (see Fig. \ref{fig:metric}). When $\mathrm{e}^{A(r)}=\cosh (r/L)$ (hereafter we set the AdS radius $L=1$), the geometry becomes a pure $(d+1)$-dimensional AdS spacetime. If one define new coordinates $z$ and $x_d$ as \begin{equation} z=\frac{w}{\cosh r} \,, \quad x_d= w \tanh r \,, \end{equation} one recover a Poincar\'{e} AdS coordinate. Examples described by this metric \eqref{metric} include AdS sliced Randall-Sundrum models \cite{KR1,KR2}, the AdS/bCFT proposal by Takayanagi \cite{Takayanagi} and Janus solutions of type IIB supergravity \cite{Janus}. It is possible to generalize \eqref{metric} to a higher dimensional internal space but we do not consider this situation in this paper. It is naturally expected that the above two decompositions can be realized as geodesic Witten diagrams. The first attempt has been done by Rastelli and Zhou \cite{RZ}. They only addressed a simple situation where the defect is a probe brane at $r=0$. Our previous paper \cite{KS} discussed geodesic Witten diagrams in situations where boundaries or defects are introduced by non-trivial profiles of backgrounds fields on the pure AdS background. Our situations include the probe brane case \cite{RZ} but addressed the special case where two operators are identical and have the same conformal dimensions for simplicity. In this paper we generalize our previous work to the case where two operators are different. The organization of this paper is as follows. The next section is devoted to preliminaries for the following sections. We review two distinct decompositions of two-point functions in CFT side and a holographic dual of boundary OPE. In section \ref{sec3} we decompose of two-point functions with different conformal dimensions into geodesic Witten diagrams. The ambient channel decomposition and the boundary channel decomposition are treated in subsections \ref{subsec31} and \ref{subsec32} respectively. In section \ref{sec4} we explain how our prescription works in some holographic BCFTs and dCFTs using some examples. We will conclude in section \ref{con}. \section{Preliminaries} \subsection{CFT side} In this subsection we mainly focus on two-point functions and boundary operator product expansion from the viewpoint of a CFT side. In CFTs with boundaries or defects, two-point functions are expressed as \begin{equation} \langle \mathcal{O}_1(\vec{x}_1,w_1) \mathcal{O}_2 (\vec{x}_2,w_2) \rangle = \frac{f(\eta)}{(2w_1)^{\Delta_1}(2w_2)^{\Delta_2}} \label{eq21} \end{equation} where $\eta$ is a conformal cross-ratio, \begin{equation} \eta = \frac{(\vec{x}_1-\vec{x}_2)^2+(w_1-w_2)^2}{w_1w_2} \,. \end{equation} Note that the two-point functions do not vanish even if the two conformal dimensions $\Delta_1$ and $\Delta_2$ are different. In the ambient spacetime, the two-point functions can be expanded by a sum of one-point functions using a standard OPE, \begin{equation} \mathcal{O}_1(x_1) \mathcal{O}_2 (x_2) = \sum_k \frac{\lambda_k}{(x_1-x_2)^{\Delta_1+\Delta_2-\Delta_k}}\mathcal{O}_k (x_2) \,. \end{equation} This gives us a form of $f$ as \begin{equation} f(\eta) = \lambda_{\bf{1}} \left( \frac{4}{\eta} \right)^{\Delta_1}\delta_{\Delta_1,\Delta_2}+ \sum_N \lambda_N a_N f_{\mathrm{ambient}} (\Delta_N, \eta) \label{24} \end{equation} where the sum is over all primary fields and $a$ is determined by an expectation value of a one-point function, \begin{equation} \langle \mathcal{O}(\vec{x},w)\rangle =\frac{a_\mathcal{O}}{(2w)^\Delta} \,. \end{equation} We extract a contribution of an identity operator explicitly. This contribution vanishes when the two conformal dimensions are different. The contribution of the $N$-th block, $f_{\mathrm{ambient}} (\Delta_N, \eta) $, is given by \cite{Liendo} \begin{equation} f_{\mathrm{ambient}} (\Delta_N, \eta)= \left( \frac{\eta}{4} \right) ^{\frac{\Delta_N-\Delta_1-\Delta_2}{2}} {}_2F_1 \left( \frac{\Delta_N+\Delta_1-\Delta_2}{2},\frac{\Delta_N-\Delta_1+\Delta_2}{2},\Delta_N-\frac{d}{2}+1,-\frac{\eta}{4} \right) \,. \label{26} \end{equation} Let us move on to a discussion on the boundary channel. Primary operators in the ambient spacetime can also be expanded in terms of boundary localized operators $\mathcal{\hat{O}}(\vec{x})$, \begin{equation} \mathcal{O}(\vec{x},w) = \frac{1}{(2w)^\Delta} \sum_k c^\mathcal{O}_{k} (2w)^{\Delta_k} \mathcal{\hat{O}}_{k}(\vec{x}) \label{bope} \end{equation} where $\mathcal{\hat{O}}(\vec{x})$ includes both of primaries and their descendants. This expansion is called boundary operator product expansion (BOPE). Using BOPE \eqref{bope}, we obtain the boundary channel expansion, \begin{equation} f(\eta )= \sum_{n,m} c_{n}^{\mathcal{O}_1} c_{m}^{\mathcal{O}_2} f_\partial (\Delta_n, \eta) \delta_{\Delta_n,\Delta_m} \end{equation} where the conformal block in the boundary channel is given by \cite{Liendo} \begin{equation} f_\partial (\Delta_n, \eta) = \left( \frac{\eta}{4}\right)^{-\Delta_n} {}_2F_1 \left( \Delta_n,\Delta_n-\frac{d}{2} +1, 2\Delta_n-d+2,-\frac{4}{\eta} \right)\,. \label{29} \end{equation} Each conformal blocks \eqref{26} and \eqref{29} satisfy a Casimir equation with a suitable boundary condition. \subsection{AdS side} The BOPE can be understood as a mode-decomposition of fields living in the bulk holographically \cite{ADFK}. A scalar field $\phi_{d+1}(r,x)$ dual to an ambient space operator $\mathcal{O}$ of dimension $\Delta$ satisfies the following equation of motion\footnote{When the scalar field couples to a dilaton field, a term of the dilaton field appears in front of the kinetic term and the equation of motion will be changed. However, a main story does not so changed. For example, see our previous paper \cite{KS}.}, \begin{equation} \label{eom} (\Box - M^2(r)) \phi_{d+1} (r,x) = ( {\cal D}^2_r + \mathrm{e}^{-2 A} \partial_d^2 - M^2(r) ) \phi_{d+1} (r,x)= 0 \,. \end{equation} Here $\Box$ is the Laplacian in the full Janus background geometry \eqref{metric} and $\partial_d^2$ stands for the AdS$_d$ Laplacian on the sliced AdS$_d$ spacetime. The radial operator ${\cal D}^2_r$ is defined as \begin{equation} {\cal D}^2_r := \partial_r^2 + d A'(r) \, \partial_r \,. \end{equation} The potential term $M^2(r)$ includes not only a scalar mass $M_0^2$ but also contributions from background fields $X(r)$. According to the standard AdS/CFT dictionary, the scalar mass is related to the conformal dimension as \begin{equation} \Delta (\Delta -d)=M_0^2 \,. \end{equation} We make a separation of variables ansatz to the bulk field, \begin{equation} \label{sov} \phi_{d+1} (r,x) = \sum_n \psi_n(r) \phi_{d,n} (x) \,. \end{equation} The separated fields on sliced AdS$_d$ spacetime satisfy \begin{equation} \label{onslicefree} \partial_d^2 \phi_{d,n} = m_n^2 \phi_{d,n} \,. \end{equation} The eigenvalues $m_n^2$ are then determined by the internal mode equation: \begin{equation} \label{215} {\cal D}^2_r \psi _n(r) - M^2(r) \psi_n(r) = - \, \, \mathrm{e}^{-2 A(r)} m_n^2 \psi_n(r) \, . \end{equation} This second differential equation can be converted to a standard Schr\"odinger equation by a change of variables and a redefinition of the field. A change of the coordinate from $r$ to a conformal coordinate $z$ with $\mathrm{d}r=\mathrm{e}^A \mathrm{d}z$ removes the $\mathrm{e}^{-2A}$ factor in front of the eigenvalues $m_n^2$ and a redefinition of the field $\psi_n = \mathrm{e}^{-(d-1)A/2} \Psi_n$ removes the first derivatives acting on the modefunction. Finally the differential equation reduces to a standard Schr\"odinger equation for $\Psi_n(z)$ together with an effective potential \cite{DFGK} \begin{equation} \label{potential} V(z) = \frac{1}{2} \left [ \left ( \frac{d-1}{2} \frac{\mathrm{d}A}{\mathrm{d}z} \right )^2 + \frac{d-1}{2} \frac{\mathrm{d}^2A}{\mathrm{d}z^2} + M^2 \mathrm{e}^{2A} \right ]\, . \end{equation} A completeness relation and an orthogonality relation for the original field $\psi_n(r)$, \begin{equation} \sum_n \psi_n(r) \psi_n(r') = \mathrm{e}^{-(d-2)A(r)} \delta(r-r') \, , \qquad \int \! \mathrm{d}r \, \mathrm{e}^{(d-2) A(r)} \, \psi_m \psi_n = \delta_{mn}\,, \end{equation} become standard ones for the rescaled field $\Psi _n(z)$. According to \cite{ADFK}, the primaries appearing in the BOPE \eqref{bope} are related to the modes $\phi_{d,n}$ in one-to-one correspondence and their conformal dimensions are related with eigenvalues $m_n^2$ as \begin{equation} \Delta_n [ \Delta_n-(d-1) ] = m_n^2 \,. \end{equation} A boundary condition of the modefunctions at $r \to \pm \infty$ requires \begin{equation} \Delta_n=\Delta +n \,. \end{equation} Finally we summarize a bulk-to-bulk propagator and a bulk-to-boundary propagator to fix our notation and also give a useful decomposition of the bulk-to-bulk propagator. The bulk-to-bulk propagator is a Green function of the differential operator $\Box -M^2(r)$, \begin{equation} (\Box -M^2(r_1)) G_{\Delta,d+1} (r_1,\vec{x}_1,w_1,r_2,\vec{x}_2,w_2)= \frac{1}{\sqrt{-g}} \delta (r_1-r_2) \delta (\vec{x}_1-\vec{x}_2) \delta (w_1-w_2) \,. \label{221} \end{equation} The bulk-to-boundary propagator is defined as a solution of the following differential equation, \begin{equation} (\Box -M^2(r)) K_{\Delta,d+1} (r,\vec{x}_1,w_1,\vec{x}_2,w_2) =0 \end{equation} and is required to approach an appropriate delta function, \begin{equation} \lim_{r\to \infty } \mathrm{e}^{(d-\Delta)r} K_{\Delta,d+1} (r,\vec{x}_1,w_1,\vec{x}_2,w_2)= \delta (\vec{x}_1-\vec{x}_2) \delta (w_1-w_2) \,. \end{equation} The bulk-to-boundary propagator is related with the bulk-to-bulk propagator, \begin{equation} K_{\Delta,d+1}(r_1,\vec{x}_1,w_1,\vec{x}_2,w_2)= \lim_{r_2 \to \infty} (2\Delta -d) \frac{\mathrm{e}^{\Delta r_2}}{(2w_2)^{\Delta}} G_{\Delta,d+1} (r_1,\vec{x}_1,w_1,r_2,\vec{x}_2,w_2) \,. \label{bulkboundary} \end{equation} The bulk-to-bulk propagator in the Janus metric \eqref{metric} can be decomposed into a sum of products of modefunctions and bulk-to-bulk propagators on the sliced AdS$_d$ spacetime, \begin{equation} G_{\Delta,d+1}(X_1,X_2)=\sum_n \psi_n(r_1) \psi_n(r_2) G_{\Delta_{n},d} (x_1,x_2) \label{224} \end{equation} with \begin{equation} \Delta_{n}=\Delta+n\,, \quad n \in \mathbb{N}\,. \end{equation} $G_{\Delta_{n},d}$ is a bulk-to-bulk propagator on sliced AdS$_d$ spacetime, \begin{equation} G_{\Delta_{n},d} (x_1,x_2) =\frac{C_{\Delta_n ,d}}{2^{\Delta_n}(2\Delta_n-(d-1))}\xi^{\Delta_n} {}_2F_1 \left( \frac{\Delta_n}{2},\frac{\Delta_n+1}{2},\Delta_n-\frac{d-1}{2}+1,\xi^2 \right) \end{equation} with \begin{equation} C_{\Delta_n ,d}=\frac{\Gamma (\Delta_n)}{\pi^{(d-1)/2}\Gamma (\Delta_n-(d-1)/2)} \end{equation} and $\xi$ the chordal coordinate, \begin{equation} \xi = \frac{2w_1 w_2}{w_1^2+w_2^2+(\vec{x}_1-\vec{x}_2)^2} \,. \end{equation} Using \eqref{bulkboundary}, similar decompositions can be applied to bulk-to-boundary propagators. Note that the bulk-to-bulk propagator on sliced AdS$_d$ spacetime is equal to conformal blocks in boundary channel up to prefactor. \section{Decompositions of two-point function} \label{sec3} In this section we decompose two-point functions into geodesic Witten diagrams in a situation where a boundary or a defect is introduced weakly so that the decompositions into geodesic Witten diagrams make sense. This situation is achieved by small background fields $X(r)$, \begin{equation} X(r) = \varepsilon \, \delta X(r) \end{equation} where $\varepsilon$ a small parameter. Since an energy-momentum tensor constructed from the background fields is corrected at second order of $\varepsilon$, the Janus metric \eqref{metric} can be expanded as \begin{equation} g =g_{0}+\varepsilon ^2 \, \delta g \end{equation} where $g_0$ represents a metric of the pure AdS$_{d+1}$ geometry. Hence, we can treat the geometry as the pure AdS at leading order of $\varepsilon$. In the following sections, bulk-to-bulk and bulk-to-boundary propagators are those of the pure AdS. First, let us consider a situation where the boundary or defect is absent. In this case, the two-point function is determined by the conformal symmetry \begin{equation} \langle \mathcal{O}_1 (\vec{x}_1,w_1) \mathcal{O}_2 (\vec{x}_2,w_2) \rangle = \frac{\mathcal{N} \delta_{\Delta_1,\Delta_2} }{[(\vec{x}_1-\vec{x}_2)^2+(w_1-w_2)^2]^{\Delta_1}} \end{equation} up to a prefactor $\mathcal{N}$. Comparing with \eqref{eq21}, we obtain \begin{equation} f (\eta)=\mathcal{N} \left( \frac{\eta}{4}\right)^{-\Delta_1} \delta_{\Delta_1,\Delta_2} \,. \label{35} \end{equation} Equation \eqref{35} corresponds to the ambient channel expansion because one-point functions vanish except the identity operator and \eqref{24} reduces to \eqref{35}. The two distinct decompositions should be same, then the following relation must hold, \begin{equation} \mathcal{N} \left( \frac{\eta}{4}\right)^{-\Delta} = \sum_{n=0}^\infty (c_n^{\mathcal{O}})^2 f_\partial (\Delta +n, \eta) \,. \end{equation} The BPOE coefficients are obtained as \cite{KS} \begin{equation} (c_n^\mathcal{O})^2=(C_n)^2 (2\Delta -d)^2 \frac{C_{\Delta_n,d}}{2\Delta_n-(d-1)}\frac{1}{4^{\Delta_n}} \end{equation} where $C_n$ is an asymptotic value of the modefunction, \begin{equation} \psi_n (r) = C_n \mathrm{e}^{-\Delta r} +\mathcal{O} (\mathrm{e}^{-(\Delta+2)r}) \,. \end{equation} This identity is a special case of (A.7b) in \cite{Hogervorst} because (A.7b) in \cite{Hogervorst} with $h=d/2,\ell_1=\ell_2 =-\Delta$ and $\rho =\eta /(\eta+4)$ reduces to this identity. Similar identities have also been used in \cite{HH}. Summarizing the above discussion, two decompositions are the same when the boundary or defect is absent. Next, we add the boundary or defect as a perturbation and decompose two-point functions in two different ways. Our claim is that a leading correction to the two-point functions with different conformal dimensions is given by \begin{equation} \begin{split} &\delta \langle \mathcal{O}_{\Delta_1} (\vec{x}_1,w_1) \mathcal{O}_{\Delta_2} (\vec{x}_2,w_2) \rangle \\ & = 2 \varepsilon \int \! \dd ^{d+1}Y' \, \sqrt{-g_0}\, \mathrm{e}^{-2A} K_{\Delta_1,d+1} (\vec{x}_1,w_1,Y') K_{\Delta_2,d+1} (\vec{x}_2,w_2,Y') \, \delta V_{\Delta_1,\Delta_2} (r')\,. \label{2pt} \end{split} \end{equation} Here $K_{\Delta_1,d+1}$ and $K_{\Delta_2,d+1}$ are bulk-to-boundary propagators between boundary points inserted the operators and a bulk point where a source term $2 \varepsilon \, \mathrm{e}^{-2A} \delta V_{\Delta_1,\Delta_2}$ is inserted. The potential is obtained from a variation of the action\footnote{The awkward factor $2 \mathrm{e}^{-2A}$ is included so that this potential reduces a leading correction of \eqref{potential} for $\Delta_1=\Delta_2$. Note that the change of variables and rescaling of the field are required.} \begin{equation} 2 \varepsilon \, \mathrm{e}^{-2A} \delta V_{\Delta_1,\Delta_2} = -\frac{\delta^2 S}{\delta \phi_{\Delta_1} \delta \phi_{\Delta_2} } \,. \end{equation} For $\Delta_1=\Delta_2$, \eqref{2pt} can be obtained explicitly by expanding the differential equation for the bulk-to-bulk propagator \eqref{221} around backgrounds, multiplying it by another bulk-to-bulk propagator and integrating the intersect \cite{KS}. We expect that the similar equation of \eqref{2pt} with the same conformal dimensions can be generalized to a case with different conformal dimensions. In the following two subsections, we give two different decompositions of the two-point function starting from \eqref{2pt}. \subsection{Ambient channel} \label{subsec31} In the ambient channel, we can use similar techniques used in geodesic Witten diagrams without boundary or defect. The two bulk-to-boundary propagators intersected at a same bulk point can be decomposed as \cite{Hijano} \begin{equation} \begin{split} &K_{\Delta_1,d+1}(\vec{x}_1,w_1,Y') K_{\Delta_2,d+1}(Y',\vec{x}_2,w_2) \\ &=\sum_N b_N \int _{\gamma} \! \mathrm{d} \lambda \, K_{\Delta_1,d+1}(\vec{x}_1,w_1,Y(\lambda)) K_{\Delta_2,d+1}(Y(\lambda ),\vec{x}_2,w_2) G_{\Delta_N,d+1}(Y(\lambda),Y') \label{39} \end{split} \end{equation} with coefficients \begin{equation} b_N= \frac{2\Gamma (\Delta_N)}{\Gamma ((\Delta_N +\Delta_1-\Delta_2 )/2)\Gamma ((\Delta_N -\Delta_1+\Delta_2 )/2)} \frac{(-1)^N}{N!} \frac{(\Delta_1)_N(\Delta_2)_N}{(\Delta_1+\Delta_2 +N-d/2)_N} \end{equation} where $\Delta_N=2\Delta +2N$ and $(x)_n=\Gamma (x+n)/\Gamma (x)$ is the Pochhammer symbol. $\gamma$ parameterized by $\lambda$ represents a geodesic anchored to two boundary points $(\vec{x}_1,w_1)$ and $(\vec{x}_2,w_2)$. Inserting this decomposition \eqref{39} into \eqref{2pt}, the two-point function is decomposed as a sum of geodesic Witten diagrams, \begin{equation} \begin{split} & \delta \langle \mathcal{O}_{\Delta_1} (\vec{x}_1,w_1) \mathcal{O}_{\Delta_2} (\vec{x}_2,w_2) \rangle = 2 \varepsilon \sum_N b_N \int \! \dd ^{d+1}Y' \, \sqrt{-g_0}\, \mathrm{e}^{-2A} \\ & \times \int _{\gamma} \! \mathrm{d} \lambda \, K_{\Delta_1,d+1}(\vec{x}_1,w_1,Y(\lambda)) K_{\Delta_2,d+1}(Y(\lambda ),\vec{x}_2,w_2) G_{\Delta_N,d+1}(Y(\lambda),Y') \, \delta V_{\Delta_1,\Delta_2} (r') \,. \label{ambient} \end{split} \end{equation} This is just the ambient channel decomposition (See Fig. \ref{fig:ambient}). \begin{figure} \begin{center} \begin{tikzpicture} \fill (0,0) circle (3pt); \fill[green] (-1.5,0) circle (4pt); \fill[green] (-3.5,0) circle (4pt); \node[yshift=2pt,above] (O1) at (-1.3,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] (O1) at (-3.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw[line width=1pt] (-4.5,0) -- (0,0) -- (2.5,0); \draw[snake it,line width=1pt] (-1.5,0) -- (0,-3); \draw[snake it,line width=1pt] (-3.5,0) -- (0,-3); \node at (0.9,-1) {$K_{\Delta_1,d+1}(\vec{x}_1,w_1,Y')$}; \node at (-2.9,-2.4) {$K_{\Delta_2,d+1}(\vec{x}_2,w_2,Y')$}; \node at (0,-3.5) {$2\varepsilon \, \mathrm{e}^{-2A} \delta V(r')$}; \fill (0,-3) circle (3pt); \node at (2.9,-2) {\scalebox{1.2}{$=\displaystyle \sum_N$}}; \fill (8,0) circle (3pt); \fill[green] (6.5,0) circle (4pt); \fill[green] (4.5,0) circle (4pt); \node[yshift=2pt,above] (O1) at (6.7,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] (O1) at (4.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw[line width=1pt] (3.5,0) -- (8,0) -- (10.5,0); \draw[dashed] (4.5,0) arc [start angle = 180, end angle = 360, radius = 1]; \draw[snake it,line width=1pt] (6.5,0) -- (5.5,-1); \draw[snake it,line width=1pt] (4.5,0) -- (5.5,-1); \draw[snake it,line width=1pt] (5.5,-1) -- (8,-3); \node at (7.1,-0.8) {$K_{\Delta_1,d+1}$}; \node at (4.1,-0.8) {$K_{\Delta_2,d+1}$}; \node at (5.1,-1.4) {$Y(\lambda)$}; \node at (8.7,-2) {$G_{\Delta_N,d+1}(Y(\lambda),Y')$}; \node at (8,-3.5) {$2\varepsilon \, \mathrm{e}^{-2A} \delta V(r')$}; \fill (8,-3) circle (3pt); \end{tikzpicture} \caption{Schematic picture of the ambient channel decomposition. The dashed line is a geodesic. The point $r=r'$ where the source is inserted is integrated.} \label{fig:ambient} \end{center} \end{figure} It is difficult to prove these geodesic Witten diagrams are conformal blocks by explicit computations. However, we can easily confirm this using Casimir equation. In fact a combination \begin{equation} \int _{\gamma} \! \mathrm{d} \lambda \, K_{\Delta_1,d+1}(\vec{x}_1,w_1,Y(\lambda)) K_{\Delta_2,d+1}(Y(\lambda ),\vec{x}_2,w_2) G_{\Delta_N,d+1}(Y(\lambda),Y') \end{equation} satisfy Casimir equation with a suitable boundary condition \cite{RZ}. \subsection{Boundary channel} \label{subsec32} For a purpose of a decomposition in the boundary channel, the decomposition of the bulk-to-bulk propagator on AdS$_{d+1}$ \eqref{224} is useful, \begin{align} G_{\Delta_1,d+1}(X_1,Y')&=\sum_n \psi_n^{(1)}(r_1) \psi_n^{(1)}(r') G_{\Delta_{1,n},d} (x_1,y') \,, \\ G_{\Delta_2,d+1}(X_2,Y')&=\sum_m \psi_m^{(2)}(r_2) \psi_m^{(2)}(r') G_{\Delta_{2,m},d} (x_2,y') \end{align} with \begin{equation} \Delta_{1,n}=\Delta_1+n\,, \quad \Delta_{2,m}=\Delta_2+m \quad \mbox{with} \quad n,m \in \mathbb{N}\,. \end{equation} To remind us that each mode functions depend on the conformal dimension, the superscripts $(1)$ and $(2)$ are added. Plugging the corresponding decompositions of the bulk-to-boundary propagators obtained from the relation \eqref{bulkboundary} into \eqref{2pt}, the decomposition of the two-point function includes a sum, \begin{equation} \sum_{n,m} C_n^{(1)}C_m^{(2)} \delta V_{\Delta_1,\Delta_2}(r') \psi_n^{(1)}(r') \psi_m^{(2)}(r') G_{\Delta_{1,n},d} (x_1,y') G_{\Delta_{2,m},d} (x_2,y') \end{equation} where $C_n^{(1)}$ and $C_m^{(2)}$ are asymptotic values of the modefunctions. We decompose this sum whether $\Delta_{1,n}=\Delta_{2,m}$ or not, \begin{align} &\sum_{n,m} C_n^{(1)}C_m^{(2)} \delta V_{\Delta_1,\Delta_2}(r') \psi_n^{(1)}(r') \psi_m^{(2)}(r') G_{\Delta_{1,n},d} (x_1,y') G_{\Delta_{2,m},d} (x_2,y') \notag \\ &=\! \sum_{\substack{n,m\\ \Delta_{1,n} \neq \Delta_{2,m} }} \! \! \! \! \! C_n^{(1)}C_m^{(2)} \frac{\delta V_{\Delta_1,\Delta_2}(r') \psi_n^{(1)}(r') \psi_m^{(2)}(r') }{(m_n^{(1)})^2-(m_m^{(2)})^2} \left( (m_n^{(1)})^2-(m_m^{(2)})^2 \right) G_{\Delta_{1,n},d} (x_1,y') G_{\Delta_{2,m},d} (x_2,y') \notag \\ &+\! \sum_{\substack{n\\ \Delta_{1,n} = \Delta_{2,m} }} \! \! \! \! \! C_n^{(1)}C_m^{(2)} \delta V_{\Delta_1,\Delta_2}(r') \psi_n^{(1)}(r') \psi_m^{(2)}(r') G_{\Delta_{1,n},d} (x_1,y') G_{\Delta_{2,m},d} (x_2,y') \,. \end{align} The first sums over $n$ and $m$ such that $\Delta_{1,n} \neq \Delta_{2,m}$ while the second term sums over $n$ such that $\Delta_{1,n} = \Delta_{2,m}$ (Since $m$ is determined by $n$ from the condition $\Delta_{1,n} = \Delta_{2,m}$, a sum over $m$ does not appear). Hence, the second term appears only in the case where $\Delta_1-\Delta_2$ is integer. By integrating the first term about $y'$, it becomes \begin{equation} \begin{split} &\int \! \dd ^d y' \sqrt{-g_{\mathrm{AdS}_d}} \left( (m_n^{(1)})^2-(m_m^{(2)})^2 \right) G_{\Delta_{1,n},d} (x_1,y') G_{\Delta_{2,m},d} (x_2,y') \\ &= G_{\Delta_{1,n},d} (x_1,x_2) - G_{\Delta_{2,m},d} (x_1,x_2) \end{split} \end{equation} where we use a relation \begin{equation} (m_n^{(1)})^2 G_{\Delta_{1,n},d} (x_1,y') = \partial_d^2 G_{\Delta_{1,n},d} (x_1,y') -\frac{1}{\sqrt{-g_{\mathrm{AdS}_d}}} \delta (x_1-y') \end{equation} and a similar relation for $G_{\Delta_{2,m},d}$. \begin{figure}[t] \begin{center} \begin{tikzpicture} \fill (0,0) circle (3pt); \fill[green] (-1.5,0) circle (4pt); \fill[green] (-3.5,0) circle (4pt); \node[yshift=2pt,above] (O1) at (-1.3,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] (O1) at (-3.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw[line width=1pt] (-4.5,0) -- (0,0) -- (2.5,0); \draw[snake it,line width=1pt] (-1.5,0) -- (0,-3); \draw[snake it,line width=1pt] (-3.5,0) -- (0,-3); \node at (0.9,-1) {$K_{\Delta_1,d+1}(\vec{x}_1,w_1,Y')$}; \node at (-2.9,-2.4) {$K_{\Delta_2,d+1}(\vec{x}_2,w_2,Y')$}; \node at (0,-3.5) {$2\varepsilon \, \mathrm{e}^{-2A} \delta V(r')$}; \fill (0,-3) circle (3pt); \node at (2.9,-2) {\scalebox{1.2}{$=\displaystyle \sum_n$}}; \fill (8,0) circle (3pt); \fill[green] (6.5,0) circle (4pt); \fill[green] (4.5,0) circle (4pt); \node[yshift=2pt,above] (O1) at (6.7,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] (O1) at (4.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw[line width=1pt] (3.5,0) -- (8,0) -- (10.5,0); \draw[dashed] (4.5,0) arc [start angle = 180, end angle = 310, radius = 3.5]; \draw[dashed] (6.5,0) arc [start angle = 180, end angle = 360, radius = 1.5]; \draw[snake it,line width=1pt] (6.5,0) -- (7.25,-1.3); \draw[snake it,line width=1pt] (4.5,0) -- (6.25,-3.03); \draw[snake it,line width=1pt] (6.25,-3.03) -- (7.25,-1.3); \fill (7.25,-1.3) circle (2pt); \fill (6.25,-3.03) circle (2pt); \node at (8,-0.7) {$K_{\Delta_1,d+1}$}; \node at (4.6,-1.8) {$K_{\Delta_2,d+1}$}; \node at (7.8,-2.4) {$G_{\Delta_n,d}$}; \node at (8.5,-1.3) {$(\vec{x}_1,w_1,r')$}; \node at (5.8,-3.5) {$(\vec{x}_2,w_2,r')$}; \end{tikzpicture} \caption{Schematic picture of the boundary channel decomposition. The dashed lines are geodesics. We need to integrate about the point $r=r'$ where the source is inserted. Anomalous dimension terms are omitted.} \label{fig:boundary} \end{center} \end{figure} Summarizing the above equations, the two-point function can be decomposed as \begin{align} &\delta \langle \mathcal{O}_1 (\vec{x}_1,w_1) \mathcal{O}_2 (\vec{x}_2,w_2) \rangle =2 \varepsilon \frac{(2\Delta_1 -d)(2\Delta_2-d)}{(2w_1)^{\Delta_1}(2w_2)^{\Delta_2}} \notag \\ &\times \left[ \sum_{\substack{n,m\\ \Delta_{1,n} \neq \Delta_{2,m} }} \! \! \! \! \! C_n^{(1)}C_m^{(2)} \int \! \dd r' \, \mathrm{e}^{(d-2)A} \frac{\delta V_{\Delta_1,\Delta_2}(r') \psi_n^{(1)}(r') \psi_m^{(2)}(r') }{(m_n^{(1)})^2-(m_m^{(2)})^2} \left( G_{\Delta_{1,n},d} (x_1,x_2)- G_{\Delta_{2,m},d} (x_1,x_2)\right) \right. \notag \\ &\left. \qquad \qquad +\! \sum_{\substack{n\\ \Delta_{1,n} = \Delta_{2,m} }} \! \! \! \! \! C_n^{(1)}C_m^{(2)} \int \! \dd r' \, \mathrm{e}^{(d-2)A}\delta V_{\Delta_1,\Delta_2}(r') \psi_n^{(1)}(r') \psi_m^{(2)}(r') \right. \notag \\ & \left. \qquad \qquad \qquad \qquad \times \int \! \dd^d y' \, \sqrt{-g_{\mathrm{AdS}_d}} G_{\Delta_{1,n},d} (x_1,y') G_{\Delta_{2,m},d} (x_2,y') \, \, \right] \,. \label{boundary} \end{align} The first term just represents a sum of conformal blocks or conformal partial waves because the propagator $G_{\Delta_{n},d}$ is equal to $f_\partial (\Delta_n, \eta)$ up to normalization. We can also confirm this by Casimir equation. By using a following relation, \begin{equation} K_{\Delta,d+1} (r,\vec{x},w,\vec{x},w) = \frac{ \Gamma (\Delta ) (2\Delta -d)}{\pi ^{d/2} \Gamma (\Delta -d/2)} \left( \frac{\cosh r}{w}\right)^{\Delta} \,, \end{equation} the terms in front of the parenthesis in \eqref{boundary} can be converted into bulk-to-boundary propagators. Then, the two-point function is decomposed into geodesic Witten diagrams as Fig. \ref{fig:boundary}. The second term represents an anomalous dimension of operators on the boundary or defect. In the Feynman diagram language, this term corresponds to a mass shift. Similar terms have appeared in geodesic Witten diagrams without boundary or defect \cite{Hijano} but there is difference. In the case without boundary or defect, four-point functions are non-trivial and decomposed into geodesic Witten diagrams. For $\Delta_1+\Delta_2-\Delta_3-\Delta_4 \in 2 \mathbb{Z} $ in s-channel, anomalous dimensions of operators $\mathcal{O}_i$ appear \cite{ano1,ano2}. On the other hand, in the presence of the boundary or defect, the anomalous dimension is associated with boundary operators $\hat{\mathcal{O}}$ not ambient operators $\mathcal{O}$. In conclusion we confirmed that the two-point function can be decomposed into conformal blocks in the boundary channel, and the anomalous conformal dimension terms appeared because the boundary operators receive corrections to their conformal dimensions. \section{Examples} \label{sec4} In the previous section we obtain two different decompositions \eqref{ambient} and \eqref{boundary}. In this section we explain how our prescription works using a D3/D5 brane system \cite{KR1,KR2} and also comment on the Takayanagi's proposal \cite{Takayanagi} and the Janus solutions \cite{Janus}. Correlation functions in the D3/D5 model have been discussed holographically in \cite{DFO,DFGK} and also geodesic Witten diagrams have already obtained in \cite{RZ}. However we would like to stress that our prescription can concretely determine coefficients in front of geodesic Witten diagrams, appeared in two-point functions. The D3/D5 brane system is constructed from a stuck of $N$ D3-branes and a stuck of $N_{\mathrm{f}}$ D5-branes. The D3-branes extend $0,1,2,3$-directions, while the D5-branes extend $0,1,2,4,5,6$-directions. We assume that $N$ and 't Hooft coupling $\lambda$ are large and the number of flavour branes $N_\mathrm{f}$ is much smaller than $N$, $N_\mathrm{f}/N \ll 1$, in order to treat the D5-branes as probes. The dual field theory of this model is the $d=4$, $\mathcal{N}=4$ supersymmetric gauge theory coupled to flavour fields on a defect extending $0,1,2$-directions. The action of the bulk constructed from type IIB supergravity action for the D3-branes $S_{\mathrm{IIB}}$, a DBI action $S_{\mathrm{DBI}}$, and a Chern-Simons action $S_{\mathrm{CS}}$, for D5-branes, \begin{equation} S=S_{\mathrm{IIB}}+S_{\mathrm{DBI}}+S_{\mathrm{CS}} \,. \end{equation} The type IIB supergravity action compactified on $S^5$ sphere contains \begin{equation} S_{\mathrm{bulk}}= -N^2 \int \! \dd ^{5}X \, \sqrt{-g_0} \left[ \sum_I \left( \frac{1}{2} (\partial_M \phi_I )^2 +\frac{M^2}{2}\phi_I^2 \right) +\sum_{I,J,K}\alpha_3 \phi_I \phi_J \phi_K +\cdots \right] \end{equation} where $g_0$ is AdS$_5$ metric and $\phi_I$ represents bulk fields. In this section, we do not care about coefficients of interaction terms which is independent on $N$, $\lambda$ and $N_\mathrm{f}$ and do not give explicit expressions for such coefficients. The action for the defect is \begin{equation} \begin{split} S_{\mathrm{defect}} &= -N \lambda^{1/2}N_\mathrm{f}\int \dd ^{4}x \, \sqrt{-g_{\mathrm{AdS}_4}} \\ & \qquad \qquad \times \left[ \sum_I \frac{1}{2}(\partial_i \psi_I)^2 + \sum_{m,n\geq 0} \sum_{I,J}\beta_{m,n} \phi_{I_1}\cdots \phi_{I_m} \psi_{J_1}\cdots \psi_{J_n} +\cdots \right] \end{split} \end{equation} where $\psi_I$ defect scalar fields. To canonically normalize kinetic terms, we define rescaled fields, \begin{equation} \phi' :=N\phi \,, \quad \psi' :=\sqrt{NN_\mathrm{f}\lambda^{1/2}}\psi \,, \end{equation} then the actions become schematically \begin{align} S_{\mathrm{bulk}} &= - \int \! \dd ^{5}X \, \sqrt{-g_{5}} \left( \frac{1}{2} (\partial_M \phi')^2 +\frac{M^2}{2}\phi'^2 +\frac{\alpha_3}{N} \phi'^3 +\cdots \right)\,, \\ S_{\mathrm{defect}} &= -\int \dd ^{4}x \, \sqrt{-g_{4}} \left( \frac{1}{2}(\partial_m \psi')^2 + \sum_{m,n\geq 0} \beta_{m,n} N^{1-m-n/2}\lambda^{1/2-n/4}N_\mathrm{f}^{1-n/2} \phi'^m \psi'^n +\cdots \right) \,. \end{align} The potential is determined from the interaction term in $S_{\mathrm{defect}}$, \begin{align} 2 \mathrm{e}^{-2A}\varepsilon \, \delta V_{\Delta_1,\Delta_2} (r) &= - \frac{\delta^2 S_{\mathrm{defect}}}{\delta \phi_{\Delta_1} \delta \phi_{\Delta_2}} \notag \\ &=\beta_{2,0}\sqrt{\lambda}\frac{N_\mathrm{f}}{N}\delta (r) \,. \label{deltapot} \end{align} This potential is proportional to the small parameter $N_{\mathrm{f}}/N$ and the delta function as expected. Inserting this potential \eqref{deltapot} into obtained decompositions \eqref{ambient} and \eqref{boundary}, we recover the same geodesic Witten diagrams in \cite{RZ}. However, our prescription determines also coefficients coming from vertices. The ambient channel decomposition involves the bulk $\phi^3$ interaction and the defect $\phi \psi$ interaction. Each vertices have orders $1/N$ and $\sqrt{\lambda} N_\mathrm{f}$ and totally gives an order $\sqrt{\lambda}N_\mathrm{f}/N$ as expected. On the other side, the boundary channel decomposition involves the defect $\phi \psi$ interaction. The order of the vertex is $(\sqrt{\lambda}N_\mathrm{f}/N)^{1/2}$ as expected. Our prescription works well in the D3/D5 brane system. See Fig. \ref{fig:D3D5}. \begin{figure}[t] \begin{minipage}{0.5\hsize} \begin{center} \begin{tikzpicture} \fill (0,0) circle (3pt); \fill[green] (-1.5,0) circle (4pt); \fill[green] (-3.5,0) circle (4pt); \node[yshift=2pt,above] at (-1.3,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] at (-3.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw[line width=1pt] (-4.5,0) -- (0,0) -- (2.5,0); \draw[dotted] (0,0) -- (0,-4); \draw[dashed] (-3.5,0) arc [start angle = 180, end angle = 360, radius = 1]; \draw[snake it,line width=1pt] (-1.5,0) -- (-2.5,-1); \draw[snake it,line width=1pt] (-3.5,0) -- (-2.5,-1); \draw[snake it,line width=1pt] (-2.5,-1) -- (0,-3); \node at (-0.8,-0.8) {$K_{\Delta_1,d+1}$}; \node at (-3.8,-0.8) {$K_{\Delta_2,d+1}$}; \node at (-2.1,-2.3) {$G_{\Delta_N,d+1}$}; \fill (0,-3) circle (3pt); \node at (0,-4.5) {D5 brane}; \end{tikzpicture} \end{center} \subcaption{The ambient channel} \end{minipage} \begin{minipage}{0.5\hsize} \begin{center} \begin{tikzpicture} \fill (8,0) circle (3pt); \fill[green] (6.5,0) circle (4pt); \fill[green] (4.5,0) circle (4pt); \node[yshift=2pt,above] at (6.7,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] at (4.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw[line width=1pt] (3.5,0) -- (8,0) -- (10.5,0); \draw[dotted] (8,0) -- (8,-4); \draw[dashed] (4.5,0) arc [start angle = 180, end angle = 310, radius = 3.5]; \draw[dashed] (6.5,0) arc [start angle = 180, end angle = 360, radius = 1.5]; \draw[snake it,line width=1pt] (6.5,0) -- (8,-1.5); \draw[snake it,line width=1pt] (4.5,0) -- (8,-3.5); \draw[snake it,line width=1pt] (8,-3.5) -- (8,-1.5); \node at (6.8,-1.1) {$K_{\Delta_1,d+1}$}; \node at (5.9,-2.3) {$K_{\Delta_2,d+1}$}; \node at (8.8,-2.5) {$G_{\Delta_n,d}$}; \node at (8,-4.5) {D5 brane}; \end{tikzpicture} \end{center} \subcaption{The boundary channel} \label{fig:two} \end{minipage} \caption{Two distinct decompositions in the D3/D5 brane system.} \label{fig:D3D5} \end{figure} Comments on other examples are in order. The Takayanagi's proposal is a holographic model of boundary CFTs. In this proposal, an additional boundary is introduced in the bulk and the spacetime is restricted in the region $r_\ast <r< \infty$ (See Fig. \ref{fig:takayana}). Matter fields are localized on the additional boundary. When the backreaction coming from the boundary can be ignored, the bulk is described by pure AdS. The main difference between the Takayanagi's proposal and the D3/D5 model is spacetime is terminated at $r=r_\ast$ or not. Then if we restrict the integration region in \eqref{2pt} appropriately, we can apply our prescription to the Takayanagi's proposal. \begin{figure}[t] \begin{center} \begin{tikzpicture} \fill (0,0) circle (3pt); \fill[green] (-1.5,0) circle (4pt); \fill[green] (-3.5,0) circle (4pt); \node[yshift=2pt,above] (O1) at (-1.3,0) {$\mathcal{O}_1(\vec{x}_1,w_1)$}; \node[yshift=2pt,above] (O1) at (-3.5,0) {$\mathcal{O}_2(\vec{x}_2,w_2)$}; \draw (-4.5,0) -- (0,0); \draw[loosely dotted] (0,0) -- (2.5,0); \draw (0,0) -- (295:3.8); \draw[snake it,line width=1pt] (-1.5,0) -- (295:2.7); \draw[snake it,line width=1pt] (-3.5,0) -- (295:2.7); \end{tikzpicture} \caption{A leading correction of two-point functions in the Takayanagi's proposal.} \label{fig:takayana} \end{center} \end{figure} Finally we comment on the Janus solutions. In previous our paper \cite{KS}, we treated the $d=4$ Janus solution as an example and an extension to different operators are straightforward. Hence we give a brief comment on differences between the D3/D5 model and the Janus solutions here. The dual CFTs of the Janus solutions are so called interface CFTs and the coupling constant jumps at the interface. This is achieved by a nontrivial profile of the dilaton field. The potential of the D3/D5 system is localized on the D5 branes and is proportional to the delta function while that of the Janus is not localized and spreads in whole of AdS. Since interaction terms also comes from kinetic terms coupled to the dilaton, small modifications are needed. Our prescription also works nicely in Janus solutions. \section{Conclusion and Discussion} \label{con} In the presence of boundaries or defects in CFTs, two-point functions are still nontrivial and are functions of the conformal cross-ratio. Furthermore the two-point functions do not vanish even if conformal dimensions are different. In BCFTs or dCFTs, the two-point functions can be decomposed into conformal blocks or conformal partial waves in two distinct ways; ambient channel decomposition and boundary channel decomposition. The holographic dual of conformal blocks in dCFT was initiated by Rastelli and Zhou \cite{RZ}. They discussed a very simple model, a D3/D5 brane model, and obtained geodesic Witten diagrams with general conformal dimensions. In our previous work \cite{KS}, we gave decompositions of two-point functions into geodesic Witten diagrams in general holographic setups with boundaries or defects. Our work includes results in \cite{RZ} as a special case but considered only the case with two identical operators. In this paper we generalized our previous paper to a situation where two operators are different and have different conformal dimensions. The leading correction of the two-point functions is given by two bulk-to-boundary propagators coupled to a source term which represents an existence of boundaries or defects \eqref{2pt}. We derived the ambient channel decomposition \eqref{ambient} and the boundary channel decomposition from \eqref{2pt}. Note that these two different decompositions are equivalent since we derived them from \eqref{2pt}. In the boundary channel, the decomposition of the two-point functions includes not only conformal partial waves but also anomalous dimensions if the conformal dimensions satisfy $\Delta_1-\Delta_2 \in \mathbb{Z}$. To present that our prescription works, we applied it to a D3/D5 brane system. Defect matter fields localize on the D5 brane and the potential is proportional to a delta function. When the probe D5-branes sit on $r=0$, our geodesic Witten diagrams reduce to that of \cite{RZ}. Recently, two-point functions in dCFT dual to the D3/D5 system with flux have been discussed \cite{deLeeuw:2017dkd}. It would be nice to compare their work with our prescription. In addition, we commented on other examples, the Takayanagi's proposal and the Janus soutions. We confirmed that our prescription works well in both examples. In this paper we only considered scalar operators to simplify our discussion. Geodesic Witten diagrams with an external spin was initiated in \cite{NT} and extended to arbitrary external and internal spins in \cite{Castro,Dyer,Hideki}. It is possible to generalize our discussion to two-point functions with spins. Our starting equation \eqref{2pt} involves bulk-to-boundary propagators, $K_{\Delta,d+1}$. If one wants to consider two-point functions with spins, one should replace the bulk-to-boundary propagators without spins by that with spins. However, we need an equation similar to \eqref{39} for the ambient channel decomposition and a decomposition similar to \eqref{224} for the boundary channel decomposition. These are not easy tasks because a decomposition of two-point functions may include various representations. We leave this problem as a future work. \section*{Acknowledgements} The author would like to thank Andreas Karch, Tatsuma Nishioka and Zhu Rui Dong for fruitful discussions. The work is supported by the Grant-in-Aid for Japan Society for the Promotion of Science Fellows, No. 16J01567.

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\section{Introduction} Suppose that an extended quantum system in $d$ dimensions (for example a quantum spin system), is prepared at time $t=0$ in a pure state $|\psi_0\rangle$ which is the ground state of some hamiltonian $H_0$ (or, more generally, in a thermal state at a temperature less than the mass gap to the first excited state). For times $t>0$ the system evolves \em unitarily \em according to the dynamics given by a \em different \em hamiltonian $H$, which may be related to $H_0$ by varying a parameter such as an external field. This variation, or quench, is supposed to be carried out over a time scale much less than the inverse mass gap. How does the state $|\psi(t)\rangle=e^{-iHt} |\psi_0\rangle$ evolve? For a finite number of degrees of freedom, the system generically shows a periodic (or quasiperiodic) behavior with a period that typically increases when the number of degrees of freedom grows. This is the well-known phenomenon of \em quantum recurrence\em. However, in the thermodynamic limit this is no longer necessarily the case, and the natural question arises as to whether the system (or, rather, a macroscopically large subsystem) reaches a stationary state for very large times. To attack this question we consider the simpler question of how the correlation functions, expectation values of products of local observables, evolve and whether they reach constant values for large times. This problem has its own theoretical interest, being a first step towards the understanding of equilibration in quantum systems. On the same fundamental level we can also ask whether the approach can provide a new tool for the characterization of the collective excitations in strongly correlated systems. However, until recently it has been considered a largely academic question, because the time scales over which most condensed matter systems can evolve coherently without coupling to the local environment are far too short, and the effects of dissipation and noise are inescapable. Recent developments of experimental tools for studying the behavior of ultra-cold atoms have revised completely this negative attitude. In fact, thanks to the phenomenon of Feshbach resonance it is possible to tune the coupling of an interacting system to essentially any value and in extremely short times. In addition, the coupling to dissipative degrees of freedom can be much weaker than in ordinary condensed matter systems. This has allowed the observation, among other things, of the collapse and revival of a Bose-Einstein condensate \cite{uc} and of the quenching of a spinor condensate \cite{spinor}. Furthermore, using highly anisotropic optical lattices, it has been possible to build and study essentially one-dimensional systems \cite{1dol}, and, of direct interest for this paper, the coherent non-equilibrium dynamics of these integrable models has been measured \cite{kww-06}. (Other interesting experiments are considered in Refs. \cite{uc2}.) These striking experimental results have largely motivated the development of new numerical methods to study non-equilibrium dynamics, the most successful one being the time-dependent density matrix renormalization group (DMRG) \cite{tdmrg}. On the purely theoretical side these kind of questions were first considered (as far as we are aware) in the seventies in the context of the quantum Ising-XY model in Refs.~\cite{bm1,bm2} (see also \cite{ir-00,ao-03,sps-04,cl-05,dssc-06,cc-07} for recent developments). However, in the very last few years, after the previously mentioned experimental progress, the study of quantum quenches has been pursued in a systematic manner and a large number of results is nowadays available. To quote only a few examples, the models considered include several realizations of one-dimensional (1D) Bose gases \cite{gg,bilbao,rmrnm-06,rmo-06,topk-05,bbis-04,mg-05,ksdz-05,kla,cdeo-06,aa-02}, Luttinger liquids \cite{c-06,p-06}, coupled 1D condensates \cite{gdlp-07}, strongly correlated 1D fermions \cite{mwnm-06}, and mean field fermion condensates \cite{bcs,df}. A closely related topic, that will not be considered here, concerns the formation of defects when crossing a critical point with changing the external parameter at a fixed rate, i.e. the so-called quantum Kibble-Zurek mechanism \cite{qzm}. Most of these papers concern the exact or approximate solutions of very specific models. These are often strongly relevant, since in many cases they can be directly compared with existing experimental results or give very accurate predictions for future investigations. However, from the study of specific models it is difficult to draw general conclusions on the physics of quantum quenches. There have been at least two notable exceptions. In Ref. \cite{gg} it was conjectured that the asymptotic state at very large times can be described by a generalized Gibbs ensemble (for more details we refer the reader to section \ref{phint} where this conjecture will be explicitly discussed in relation to some of our findings). In Ref.~\cite{cc-06} we studied the time evolution after a quench in general, exploiting the path integral approach and the well-known mapping of the quantum problem to a classical one in $d+1$ dimensions. The translational invariance in the (imaginary) time direction is explicitly broken and thus the initial state plays the role of a boundary condition. This limits the range of applicability of field-theoretical methods, but when the hamiltonian $H$ is at or close to a quantum critical point we could use the renormalization group (RG) theory of boundary critical behavior (see, e.g., \cite{dd}). From this point of view, particularly powerful analytic results are available for $d=1$ because then the $1+1$-dimensional problem is described asymptotically by a boundary conformal field theory (BCFT)\cite{cardy-84,cardy-05}. The aim of this paper is twofold: on the one hand we give a detailed description of the methods and the results reported briefly in Ref. \cite{cc-06}, and on the other we generalize these results and give some new physical insights. Several results, in fact, appear here for the first time. The paper is organized as follows. In Sec.~\ref{secpath} we introduce the path-integral formalism that is applied in the following section \ref{secCFT} to one-dimensional critical systems by means of CFT. In Sec.~\ref{secint} we consider two simple 1D models whose non-equilibrium dynamics is exactly solvable to check the correctness of the previous results and to extend our findings to gapped and lattice systems. In Sec.~\ref{HD} we generalize the method to higher dimensions. In the last section \ref{phint} we analyze all our results and explain most of the general findings in quantum quenches by means of simple physical arguments. We broadly discuss some open questions and possible future developments. \section{Path integral formulation and surface criticality} \label{secpath} Let us consider a lattice quantum theory in $d$ space dimensions. The lattice spacing is $a$, and the lattice variables are labelled by a discrete vector variable ${\bf r}$. Time is considered to be continuous. The dynamics of the theory is described by the hamiltonian $H$. Suppose we prepare this system in a state $|\psi_0\rangle$ that is not an eigenstate of $H$ and unitarily evolve it according to $H$. The expectation value of a local operator ${\cal O}(\{ {\bf r}_i\})$ at time $t$ is \begin{equation} \langle {\cal O}(t,\{ {\bf r}_i\})\rangle= \langle \psi_0 | e^{i H t} {\cal O}(\{ {\bf r}_i\})e^{-i H t}|\psi_0 \rangle\,. \end{equation} We modify this time-dependent expectation value as \begin{equation} \langle {\cal O}(t,\{ {\bf r}_i\})\rangle= Z^{-1} \langle \psi_0 | e^{i H t-\epsilon H} {\cal O}(\{ {\bf r}_i\}) e^{-i H t-\epsilon H}| \psi_0 \rangle\,, \label{Oexp} \end{equation} where we have included damping factors $e^{-\epsilon H}$ in such a way as to make the path integral representation of the expectation value absolutely convergent. The normalization factor $Z=\langle\psi_0|e^{-2\epsilon H}|\psi_0 \rangle$ ensures that the expectation value of the identity is one. At the end of the calculation we shall set $\epsilon$ to zero. Eq. (\ref{Oexp}) may be represented by an analytically continued path integral in imaginary time over the field variables $\phi(\tau,{\bf r})$, with initial and final values weighted by the matrix elements with $|\psi_0\rangle$: \begin{equation} \int[d\phi(\tau,{\bf r})]\langle\psi_0|\phi(\tau_2,{\bf r})\rangle \langle\phi(\tau_1,{\bf r})|\psi_0\rangle\,e^{-\int_{\tau_1}^{\tau_2}L[\phi]d\tau} \end{equation} where $\int_{\tau_1}^{\tau_2}L[\phi]d\tau$ is the (euclidean) action. The operator ${\cal O}$ is inserted at $\tau=0$, and the width of the slab is $2\epsilon$. $\tau_1$ and $\tau_2$ should be considered as real numbers during the calculation, and only at the end should they be continued to their effective values $\pm\epsilon-i t$. In this way we have reduced the real-time non-equilibrium evolution of a $d$ dimensional systems to the thermodynamics of a $d+1$ field theory in a slab geometry with the initial state $|\psi_0\rangle$ playing the role of boundary condition at both the borders of the slab. The validity of our results relies on the technical assumption that the leading asymptotic behavior given by field theory, which applies to the Euclidean region (large imaginary times), may simply be analytically continued to find the behavior at large real time. Eq. (\ref{Oexp}) can be in principle used to describe the time evolution in any theory and for all possible initial conditions. Unfortunately, in confined geometries only a few field theories with very specific boundary conditions can be solved analytically in such a way to have results that can be continued from real to complex values. Some examples of these will be given in Sec. \ref{HD}. (An interesting case concerns integrable massive boundary field theories as for example the recent application to the dynamics of coupled 1D condensates in Ref. \cite{gdlp-07}.) However, the treatment greatly simplifies if a system is at or close to a (quantum) phase transition. In fact, in this case, we can use the powerful tools of Renormalization Group (RG) theory of boundary critical phenomena (see, e.g. \cite{dd}). For example, in the case of a scalar order parameter (corresponding to the Ising universality class), the slab geometry of above is usually described by the action \cite{dd} \begin{equation} S(\phi)=\int d^d r \int_{\tau_1}^{\tau_2} d\tau \left( \frac{1}{2}(\partial \phi)^2+\frac{m^2}{2} \phi^2+\frac{g}{4!}\phi^4\right)+ \int d^d r \,{c}[\phi(\tau=\tau_1,r)+\phi(\tau=\tau_2,r)]\,, \label{LGH} \end{equation} where $m^2$ measures the distance from criticality, $g>0$ ensures the stability in the broken-symmetry phase, and $c$ denotes the surface enhancement, i.e. the difference of the interactions on the surface with respect to the bulk. The value of $c$ depends on the boundary conditions. For example, the choice $c=+\infty$ forces the field to vanish on the boundary, thus corresponding to Dirichlet boundary conditions. On the other hand $c=-\infty$ forces the field to diverge at the boundary. Finite values of $c$ correspond to intermediate boundary conditions. According to the RG theory, the different boundary universality classes are characterized by the fixed point of $c$. A complete RG analysis (see, e.g.,\cite{dd}) shows that the possible fixed points of $c$ are $c^*=\infty,0,-\infty$, with $c^*=\pm\infty$ stable and $c^*=0$ unstable. Any other value of $c$ flows under RG transformations to $\pm \infty$. These universality classes are called ordinary ($c^*=\infty$), special ($c^*=0$), and extraordinary ($c^*=-\infty$) \cite{dd}. Thus, for the purpose of extracting the asymptotic behavior, as long as $|\psi_0\rangle$ is translationally invariant, we may replace it by the appropriate RG-invariant boundary state $|\psi_0^*\rangle$ to which it flows. The difference may be taken into account, to leading order, by assuming that the RG-invariant boundary conditions are not imposed at $\tau=\tau_1$ and $\tau_2$ but at $\tau=\tau_1-\tau_0$ and $\tau=\tau_2+\tau_0$. In the language of boundary critical behavior, $\tau_0$ is called the extrapolation length \cite{dd}. It characterizes the RG distance of the actual boundary state from the RG-invariant one. It is always necessary because scale-invariant boundary states are not in fact normalizable\cite{cardy-05}. It is expected to be of the order of the typical time-scale of the dynamics near the ground state of $H_0$, that is the inverse gap $m_0^{-1}$. (This can be checked explicitly for a free field theory, see later). The effect of introducing $\tau_0$ is simply to replace $\epsilon$ by $\epsilon+\tau_0$. The limit $\epsilon\to0+$ can now safely be taken, so the width of the slab is then taken to be $2\tau_0$. For simplicity in the calculations, in the following we will consider the equivalent slab geometry between $\tau=0$ and $\tau =2\tau_0$ with the operator ${\cal O}$ inserted at $\tau=\tau_0+it$. This is illustrated in the left part of Fig.~\ref{figpiO}. In fact, in this geometry we can consider products of operators at different times $t_j$, by analytically continuing their labels to $\tau_0+it_j$. Let us discuss to what initial conditions the fixed points correspond. In the case of free boundary conditions, when the order parameter is unconstrained, the continuum limit forces it to vanish there, so it corresponds to the ordinary transition ($c=\infty$). For a lattice model, this corresponds to a completely disordered initial state (e.g. for the Ising chain in a transverse field corresponds to infinite field). In contrast for non-vanishing fixed boundary conditions (e.g. + or $-$ for Ising-like systems), the continuum limit makes the order parameter to diverge at the boundary, thus corresponding to the extraordinary transition $(c=-\infty)$. \begin{figure}[b] \centering \includegraphics[width=16cm]{piO.eps} \caption{Left: Space-imaginary time region in (\ref{Oexp}). ${\rm Im} w_i= \tau$, that will be analytically continued to $\tau\to \tau_0+it$. Right: Conformal mapping of the left geometry to the upper half-plane (c.f. Eq. (\ref{logmap})). Note that ${\rm arg} z_i= \theta=\pi \tau/2\tau_0$.} \label{figpiO} \end{figure} \section{One space dimension and Conformal Field Theory} \label{secCFT} In this section we specialize the methods just introduced to the case when $H$ is at a quantum critical point whose long-distance behavior is given by a 1+1-dimensional CFT, with dispersion relation $\Omega_k=v|k|$. We set $v=1$ in the following. RG-invariant boundary conditions then correspond to conformally invariant boundary states. In this case the correlation functions are accessible through the powerful tools of boundary CFT. The main property we will repeatedly use in the following is the relation of correlation functions of (primary) operators among two geometries connected by a conformal transformation. For example, the slab geometry of above is just a two-dimensional strip whose points are labelled by a complex number $w=r+i\tau$ with $0<{\rm Im} w<2\tau_0$. The strip can be obtained from the upper half-plane (UHP) ${\rm Im} z>0$ by the conformal mapping \begin{equation} w(z)=\frac{2\tau_0}{\pi}\log z\,, \label{logmap} \end{equation} with the images of points at the same imaginary time on the strip lying along ${\rm arg} z_i= \theta=\pi \tau/2\tau_0$. In the case where $\cal O$ is a product of local \em primary \em scalar operators $\Phi_i(w_i)$, the expectation value in the strip is related to the one in the UHP by the standard transformation \begin{equation} \langle\prod_i\Phi_i(w_i)\rangle_{\rm strip} =\prod_i|w'(z_i)|^{-x_i} \langle\prod_i\Phi_i(z_i)\rangle_{\rm UHP}\,, \end{equation} where $x_i$ is the bulk scaling dimension of $\Phi_i$. Note that the (eventual) expectation values of the $\Phi_i$ in the ground state of $H$ are supposed to have been subtracted off. The asymptotic real time dependence is obtained via the analytic continuation $\tau\to\tau_0+it$, and taking the limit $t,r_{ij}\gg \tau_0$. In the following subsections we apply these methods to some specific cases. \subsection{The one-point function} In the UHP, the one-point function of a scalar primary field with bulk scaling dimension $x$ is $\langle\Phi(z)\rangle_{\rm UHP}= A_b^\Phi [2{\rm Im}(z)]^{-x}$, as a simple consequence of scaling invariance. The normalization factor $A^\Phi_b$ is a non-universal amplitude. In CFT the normalizations are chosen in such a way that the bulk two-point functions have unit amplitude (i.e. $\langle\Phi(z_1) \Phi(z_2)\rangle_{\rm bulk}=|z_2-z_1|^{-2x}$). This choice fixes unambiguously the amplitude $A^\Phi_b$ that turns out to depend both on the considered field $\Phi$ and on the boundary condition on the real axis $b$. It vanishes if $\Phi$ corresponds to an operator whose expectation value in $|\psi_0\rangle$ vanishes, and thus $\langle\Phi(t)\rangle=0$, for all times. When the primary field is not vanishing on the boundary, performing the conformal mapping (\ref{logmap}) we obtain \begin{equation} \langle \Phi(w)\rangle_{\rm strip}= |w'(z)|^{-x} \langle \Phi(z(w))\rangle_{\rm UHP}= A^\Phi_b \left[\frac{\pi}{4\tau_0} \frac{1}{\sin(\pi \tau/(2\tau_0))}\right]^{x} \label{phistrip} \end{equation} that continued to real time $\tau=\tau_0+i t$ gives \begin{equation} \langle\Phi(t)\rangle= A^\Phi_b \left[\frac{\pi}{4\tau_0} \frac{1}{\cosh(\pi t/(2\tau_0))}\right]^{x} \simeq A^\Phi_b \left(\frac{\pi}{2\tau_0}\right)^{x} e^{-x\pi t/2\tau_0}\,. \label{onepoint} \end{equation} Thus the order parameter (and any other observables described by a primary field) decays exponentially in time to the ground-state value, with a non-universal relaxation time $t_{\rm rel}^{\cal O}=2\tau_0/x_{\cal O} \pi$. The ratio of the relaxation times of two different observables equals the inverse of the ratio of their scaling dimensions and it is then {\it universal}. The normalization factor $A^\Phi_b$ is known for the simplest boundary universality classes \cite{cl-91}. In the case of $\Phi$ being the order parameter and the boundary condition is fixed ($\psi_0(x)=\infty$) $A^\Phi_b$ is $1$ for the free boson and $2^{1/4}$ for the Ising model. An important exception to this law is the local energy density (or any piece thereof). This corresponds to the $tt$ component of the energy-momentum tensor $T_{\mu\nu}$. In CFT this is not a primary operator. Indeed, if it is normalized so that $\langle T_{\mu\nu}\rangle_{\rm UHP}=0$, in the strip \cite{cardybn-86} $\langle T_{tt}({\bf r},\tau)\rangle=\pi c/24(2\tau_0)^2$ (where $c$ is the central charge of the CFT) so that it does not decay in time. Of course this is to be expected since the dynamics conserves energy. A similar feature is expected to hold for other local densities corresponding to globally conserved quantities which commute with $H$, for example the total spin in isotropic models. \subsection{The two-point function} In the case of the one-point function, scaling invariance was enough to fix the functional dependence on the position in the UHP. However, the form of the two-point function depends explicitly on the boundary universality class and on the operator considered. In the following subsections we will consider the equal-time correlation function for the order parameter in the gaussian and in the Ising universality classes that are easily treated in full generality. At the quantum level they describe (among the other things) a chain of harmonic oscillators (explicitly considered in Sec. \ref{secharm}) and the Ising model in a transverse field (whose real time evolution has been considered in Refs. \cite{bm1,bm2,ir-00,sps-04} and is briefly reviewed in Sec. \ref{secIs}). Finally we will discuss the general form of the two-point function for asymptotically large time and distance, that can be obtained from general CFT arguments. \subsubsection{The gaussian model} The content of this subsection has been already reported in Ref. \cite{cc-05}, during the study of the time evolution of the entanglement entropy, that transforms like the two-point function of a primary field in a boundary gaussian theory \cite{cc-04}. We report it here for sake of completeness. For a free boson the two-point function in the UHP is \cite{cardy-84} \begin{equation} \langle\Phi(z_1) \Phi(z_2) \rangle_{\rm UHP}=\left (\frac{z_{1\bar2}z_{2\bar1}}{z_{12}z_{\bar1\bar2}z_{1\bar1}z_{2\bar2}}\right)^x\,, \label{bosUHP} \end{equation} with $z_{ij}=|z_i-z_j|$ and $z_{\bar k}={\overline{z_k}}$. Note that $\Phi$ is not the gaussian field $\theta(z)$, but its exponential $\Phi(z)=e^{i\theta(z)}$ (see Sec. \ref{secharm}). Under the conformal mapping (\ref{logmap}) we obtain the two-point function on the strip at imaginary time $\tau$, at distance $r$ apart \begin{eqnarray} \langle\Phi(r,\tau) \Phi(0,\tau) \rangle_{\rm strip}&=& |w'(z_1)|^{-x} |w'(z_2)|^{-x} \langle \Phi(z_1(w))\langle \Phi(z_2(w))\rangle_{\rm UHP}= \nonumber \\ &=& \left[\left(\frac{\pi}{2\tau_0}\right)^2 \frac{\cosh (\pi r/2\tau_0)- \cos (\pi \tau/\tau_0)}{8 \sinh^2 (\pi r/4\tau_0) \sin^2(\pi \tau/2\tau_0)}\right]^x, \label{2ptstrip} \end{eqnarray} that continued to real time $\tau=\tau_0+i t$ gives \begin{equation} \langle\Phi(r,t) \Phi(0,t) \rangle= \left[\left(\frac{\pi}{2\tau_0}\right)^2 \frac{\cosh(\pi r/2\tau_0)+\cosh(\pi t/\tau_0)} {8 \sinh^2(\pi r/4\tau_0) \cosh^2(\pi t/2\tau_0)}\right]^x\,. \end{equation} In the case where $r/\tau_0$ and $t/\tau_0$ are large this simplifies to \begin{equation} \langle\Phi(r,t) \Phi(0,t) \rangle= (\pi/2\tau_0)^{2x}\left(\frac{e^{\pi r/2\tau_0} +e^{\pi t/\tau_0}} {e^{\pi r/2\tau_0}\cdot e^{\pi t/\tau_0}}\right)^{x}\propto \begin{cases} e^{-x \pi t/\tau_0} \qquad &{\rm for}\; t<r/2\\ e^{-x \pi r/2\tau_0} \qquad &{\rm for}\; t>r/2\\ \end{cases}\,. \label{freeboson} \end{equation} i.e. the two point function at fixed $r$ decays exponentially in time up to $t^*=r/2$ and then saturates to a value that depends exponentially on the separation. In the case of fixed initial condition, with one-point function given by Eq. (\ref{onepoint}), the connected correlation function is \begin{equation} \langle\Phi(r,t) \Phi(0,t) \rangle_{\rm conn}= \langle\Phi(r,t) \Phi(0,t) \rangle- \langle \Phi(0,t)\rangle^2\propto \begin{cases} 0 & {\rm for}\; t<r/2\,,\\ e^{-x\pi r/2\tau_0}- e^{-x\pi t/\tau_0} & {\rm for}\; t>r/2\,, \end{cases} \end{equation} i.e. correlations start developing at $t^*=r/2$ and, being $t\gg \tau_0$, at $t^*$ the connected two-point function almost immediately jumps to its asymptotic value. In the case of disordered initial conditions ($\psi_0(r)=0$), connected and full correlation functions are equal. \subsubsection{The Ising universality class} For the Ising model the two-point function in the UHP is \cite{cardy-84} \begin{equation} \langle\Phi(z_1) \Phi(z_2) \rangle_{\rm UHP}=\left (\frac{z_{1\bar2}z_{2\bar1}}{z_{12}z_{\bar1\bar2}z_{1\bar1}z_{2\bar2}}\right)^{1/8}F(\eta)\,, \end{equation} where $F(\eta)$ is given by \begin{equation} F(\eta)=\frac{\sqrt{1+\eta^{1/2}}\pm \sqrt{1-\eta^{1/2}}}{\sqrt{2}}\,, \label{defF} \end{equation} and $\eta$ is the four-point ratio \begin{equation} \eta= \frac{z_{1\bar1}z_{2\bar2}}{z_{1\bar 2}z_{2\bar1}}\,. \end{equation} The sign $\pm$ depends on the boundary conditions. $+$ corresponds to fixed boundary conditions and and $-$ to disordered ones. The only difference with respect to the gaussian case is that we have also to map $F(\eta)$ according to the conformal transformation (\ref{logmap}). After simple algebra we have \begin{equation} \eta=\frac{2 \sin^2 (\pi\tau/2\tau_0)}{\cosh(\pi r/2\tau_0)-\cos(\pi \tau/\tau_0)}\,, \end{equation} and so \begin{eqnarray} &&\langle\Phi(r,\tau) \Phi(0,\tau) \rangle_{\rm strip}= \left[\left(\frac{\pi}{2\tau_0}\right)^2 \frac{\cosh (\pi r/2\tau_0)- \cos (\pi \tau/\tau_0)}{8 \sinh^2 (\pi r/4\tau_0) \sin^2(\pi \tau/2\tau_0)}\right]^{1/8}\frac{1}{\sqrt2}\nonumber\\ &&\times \left[\sqrt{1+ \frac{\sqrt2 \sin (\pi \tau/2\tau_0)}{\sqrt{\cosh(\pi r/2\tau_0)-\cos(\pi \tau/\tau_0)}}}\pm \sqrt{1- \frac{\sqrt2 \sin (\pi \tau/2\tau_0)}{\sqrt{\cosh(\pi r/2\tau_0)-\cos(\pi \tau/\tau_0)}}} \right]\,. \end{eqnarray} Analytically continuing to real time $\tau=\tau_0+i t$ we obtain \begin{eqnarray} &&\langle\Phi(r,t) \Phi(0,t) \rangle= \left[\left(\frac{\pi}{2\tau_0}\right)^2 \frac{\cosh(\pi r/2\tau_0)+\cosh(\pi t/\tau_0)} {8 \sinh^2(\pi r/4\tau_0) \cosh^2(\pi t/2\tau_0)}\right]^{1/8}\frac{1}{\sqrt2}\nonumber\\ &&\times \left[\sqrt{1+ \frac{\sqrt2 \cosh(\pi t/2\tau_0)}{\sqrt{\cosh(\pi r/2\tau_0)+\cosh(\pi t/\tau_0)}}}\pm \sqrt{1- \frac{\sqrt2 \cosh(\pi t/2\tau_0)}{\sqrt{\cosh(\pi r/2\tau_0)+\cosh(\pi t/\tau_0)}}} \right]\,, \end{eqnarray} that for $r/\tau_0$ and $t/\tau_0$ much larger than $1$ simplifies to \begin{equation} (\pi/2\tau_0)^{1/4}\frac{1}{\sqrt2}\left(\frac{e^{\pi r/2\tau_0} +e^{\pi t/\tau_0}} {e^{\pi r/2\tau_0}\cdot e^{\pi t/\tau_0}}\right)^{1/8} \left[\sqrt{1+ \frac{ e^{\pi t/2\tau_0}}{\sqrt{e^{\pi r/2\tau_0}+e^{\pi t/\tau_0}}}}\pm \sqrt{1- \frac{ e^{\pi t/2\tau_0}}{\sqrt{e^{\pi r/2\tau_0}+e^{\pi t/\tau_0}}}} \right]\,. \end{equation} Note that the exponential terms in the square root are always $\ll 1$. Thus for fixed boundary condition we get the free boson result Eq. (\ref{freeboson}) with $x=1/8$. For the connected part we need to subtract $\langle \Phi(0,t)\rangle^2$ given by Eq. (\ref{onepoint}) with $A^\Phi_+=2^{1/4}$. We finally obtain \begin{equation} \langle\Phi(r,t) \Phi(0,t) \rangle_{\rm conn}= \langle\Phi(r,t) \Phi(0,t) \rangle- \langle \Phi(0,t)\rangle^2\propto \begin{cases} 0 & {\rm for}\; t<r/2\,,\\ e^{-\pi r/16\tau_0}- e^{- \pi t/8\tau_0} & {\rm for}\; t>r/2\,. \end{cases} \end{equation} Thus, also for the Ising model with fixed boundary conditions, connected correlations start developing at $t=t^*=r/2$. In the case of disordered initial condition, we have \begin{equation} \langle\Phi(r,t) \Phi(0,t) \rangle \propto \left(\frac{e^{\pi r/2\tau_0} +e^{\pi t/\tau_0}} {e^{\pi r/2\tau_0}\cdot e^{\pi t/\tau_0}}\right)^{1/8} \frac{ e^{\pi t/2\tau_0}}{\sqrt{e^{\pi r/2\tau_0}+e^{\pi t/\tau_0}}}\sim \begin{cases} e^{-\pi (r-3/2 t)/4\tau_0 } &\quad {\rm for}\, t<r/2\,, \\ e^{-\pi r /16\tau_0 } &\quad {\rm for}\, t>r/2\,, \end{cases} \end{equation} resulting in an exponential space dependence even for $t<r/2$ (clearly in this case the connected correlation function equals the full one). \subsubsection{The general two point-function} From the results reported for the gaussian and Ising models, it is now relatively simple to understand the general properties of the time dependence of the two-point function in the very general case. The two-point function in the half-plane has the general form \cite{cardy-84} \begin{equation} \langle\Phi(z_1) \Phi(z_2) \rangle_{\rm UHP}=\left (\frac{z_{1\bar2}z_{2\bar1}}{z_{12}z_{\bar1\bar2}z_{1\bar1}z_{2\bar2}}\right)^x F(\eta)\,, \label{2ptgen} \end{equation} where the function $F(\eta)$ depends explicitly on the considered model. Under the conformal map to the strip we know that the first part of Eq. (\ref{2ptgen}) transforms according to Eq. (\ref{2ptstrip}). Thus we need only to map $F(\eta)$ that, in the general case, is an unknown function. However, during the study of the Ising model we showed that the analytical continuation of $\eta$ for $t,r\gg\tau_0$ is \begin{equation} \eta \sim \frac{e^{\pi t/\tau_0}}{e^{\pi r/2\tau_0}+ e^{\pi t/\tau_0}}\,. \end{equation} Thus for $t<r/2$ we have $\eta\sim e^{\pi (t-r/2)/\tau_0}\ll 1$ and in the opposite case $t>r/2$ we have $\eta\sim1$. As a consequence to have the asymptotic behavior of the two-point function we only need to know the behavior close to $\eta\sim0$ (i.e. the behavior close to the surface) and for $\eta\sim 1$ (i.e. deep in the bulk). Fortunately they are both exactly known. Indeed when $\eta\sim1$ the two points are deep in the bulk, meaning $F(1)=1$. Instead for $\eta\ll 1$, from the short-distance expansion, we have \begin{equation} F(\eta)\simeq (A^\Phi_b)^2\eta^{x_b}, \end{equation} where $x_b$ is the boundary scaling dimension of the leading boundary operator to which $\Phi$ couples and $A^\Phi_b$ is the bulk-boundary operator product expansion coefficient that equals the one introduced in Eq. (\ref{onepoint}) [see e.g. Ref. \cite{cl-91}]. All the previous observations and the explicit calculations of the previous sections lead for $t>r/2$ to \begin{equation} \langle\Phi(r,t)\Phi(0,t)\rangle \propto e^{-x\pi r/2\tau_0}\,, \end{equation} while for $t<r/2$ we get \begin{equation} \langle\Phi(r,t)\Phi(0,t)\rangle \propto (A^\Phi_b)^2 e^{-x\pi t/\tau_0} \times e^{\pi x_b(t-r/2)/\tau_0}\,. \end{equation} Note that if $\langle\Phi\rangle\neq0$, $x_b=0$ and the last factor is absent. The leading term is then just $\langle\Phi\rangle^2$. Thus the leading term in the connected two-point function vanishes for $t<r/2$, and its first non-vanishing contributions is given by subleading terms either in $F$ or in the bulk-boundary short-distance expansion. It is very interesting that we only have to know the behavior as $\eta\to0$ and $1$ to get the results we need for large $r$ and $t$. However, we stress that only a complete calculation (as those performed in the preceding sections) gives the full analytic structure of the CFT result needed to justify the analytical continuation from imaginary to real time. Moreover, the behavior within a distance $O(\tau_0)$ of the horizon $r=2t$ depends on the detailed form of $F$. \subsection{Correlations functions at different times} \begin{figure}[t] \centerline{\epsfig{width=14cm,file=fig2.eps}} \caption{ Left: Space-time region for the correlation functions at different times. Right: Conformal mapping to the upper half-plane.} \label{fig2} \end{figure} Let us consider the case of the two-point function calculated at different real times $\langle\Phi(r,t) \Phi(0,s)\rangle$. This is again obtained by mapping the imaginary time strip to the UHP, but in this case the two points are $w_1=r+i\tau_1$ and $w_2=0+i\tau_2$, that, at the end of the calculation, must be analytically continued to $\tau_1=\tau_0+i t$ and $\tau_2=\tau_0+i s$. See Fig. \ref{fig2} for a pictorial representation of the space-time domain in the strip and the resulting mapping to the UHP. Let us start the discussion with the free boson. The distances appearing in Eq. (\ref{bosUHP}) are \begin{eqnarray} z_{12}^2&=&1+e^{\pi r/\tau_0}-2e^{\pi r/2\tau_0} \cos (\theta_1-\theta_2)\,,\qquad z_{1\bar1}=2\sin\theta_1\,,\nonumber\\ z_{1\bar2}^2&=&1+e^{\pi r/\tau_0}-2e^{\pi r/2\tau_0} \cos (\theta_1+\theta_2)\,, \qquad z_{2\bar2}=2 e^{\pi r/2\tau_0}\sin\theta_2\,, \end{eqnarray} where $\theta_i=\pi \tau_i/2\tau_0$. Thus the correlation function on the strip is \begin{eqnarray} &&\langle\Phi(r,\tau_1) \Phi(0,\tau_2) \rangle_{\rm strip}= |w'(z_1)|^{-x} |w'(z_2)|^{-x} \langle \Phi(z_1(w))\langle \Phi(z_2(w))\rangle_{\rm UHP}= \\ &&= \left[\left(\frac{\pi}{2\tau_0}\right)^2 \frac{\cosh (\pi r/2\tau_0)- \cos (\pi (\tau_1+\tau_2)/2\tau_0)}{4 \sin(\pi \tau_1/2\tau_0)\sin(\pi \tau_2/2\tau_0) (\cosh (\pi r/2\tau_0)- \cos (\pi (\tau_1-\tau_2)/2\tau_0)) }\right]^{x}\nonumber , \end{eqnarray} that for $\tau_1=\tau_2$ reduces to Eq. (\ref{2ptstrip}) as it should. Continuing to real times and considering $r,t,s,|t-s| \gg\tau_0$ we obtain \begin{equation} \displaystyle \left(\frac{\pi}{2\tau_0}\right)^{2\Delta} \frac{e^{\frac{\Delta \pi r}{\tau_0}} +e^{\frac{\Delta \pi}{\tau_0} (t+s)}}{ e^{\frac{\Delta \pi}{\tau_0} (t+s)}(e^{\frac{\Delta \pi r}{\tau_0}} +e^{\frac{\Delta \pi}{\tau_0} |t-s|})} = \begin{cases} e^{-x\pi(t+s)/4\tau_0}\quad &{\rm for}\, r>t+s\,,\\ e^{-x\pi r /4\tau_0}\quad &{\rm for}\, t-s<r<t+s\,,\\ e^{-x\pi|t-s|/4\tau_0}\quad &{\rm for}\, r<|t-s|\,. \end{cases} \end{equation} Following the line sketched in the previous subsection, it is easy to generalize this result to the most general CFT. In the case of a theory with fixed initial conditions (i.e., $\langle\Phi\rangle\neq0$) the asymptotic result is the same as before, with only the crossover points being affected by the precise expression for $F(\eta)$. Instead, in the case where $\langle\Phi\rangle=0$, the first case gains an additional factor $e^{-\pi x_b(t+s-r)/4\tau_0}$. Note that the autocorrelation function (i.e. $r=0$) has only an exponential dependence on the time separation $t-s$ and does not exhibit aging in this regime. \subsection{Evolution with boundaries} \label{evbou} We now consider the case of time evolution of a half-chain with some boundary condition at $r=0$. For simplicity we assume that the (conformal) boundary condition is of the same kind of the initial boundary condition (for example we fix all the spins at $t=0$ and at the boundary $r=0$ to point in the same direction). The space-time region we have to consider is depicted in Fig. \ref{fig3}. If different initial and boundary were considered, one needs to insert boundary conditions changing operators at the corners of the figure. The $w$ plane is mapped into the UHP by \begin{equation} z(w)=\sin \frac{\pi w}{2\tau_0}\,, \label{mapsin} \end{equation} with the corners at $\pm \tau_0$ mapped to $\pm1$. The mapping of $w_1$ is \begin{equation} z_1\equiv z(w_1)= z(-\tau_0+\tau_1+i r)= -\cos (\pi \tau_1/2\tau_0)\cosh (\pi r/2\tau_0) +i \sin (\pi \tau_1/2\tau_0)\sinh (\pi r/2\tau_0)\,. \end{equation} In the $z$ plane the 1-point function is \begin{equation} \langle \Phi (z_1)\rangle_{\rm UHP}\propto |{\rm Im} {z_1}|^{-x}\to \left[\sin (\pi \tau_1/2\tau_0)\sinh (\pi r/2\tau_0) \right]^{-x}\,, \end{equation} and \begin{equation} |w'(z_1)|^2= \left(\frac{2\tau_0}{\pi}\right)^2 \frac{1}{|1-z^2|}\propto \frac{1}{\cosh (\pi r/\tau_0) -\cos(\pi \tau_1/\tau_0) }\,. \end{equation} Thus on the strip we have \begin{equation} \langle \Phi (w_1)\rangle_{\rm strip}= |w'(z_1)|^{-x} \langle \Phi (w(z_1))\rangle_{\rm UHP}\propto \left[ \frac{\sin^2 (\pi \tau_1/2\tau_0)\sinh^2(\pi r/2\tau_0)}{ \cosh (\pi r/\tau_0) -\cos(\pi \tau_1/\tau_0) }\right]^{-x/2}\,, \end{equation} that continued to real time $\tau_1=i t$ is \begin{equation} \langle \Phi (t,r)\rangle\propto \left[ \frac{\cosh (\pi t/\tau_0)+\cosh (\pi r/\tau_0)}{ \cosh(\pi t/2\tau_0)^2\sinh^2 (\pi r/2\tau_0)} \right]^{x/2}, \end{equation} and for $t,r\gg\tau_0$ simplifies to \begin{equation} \langle \Phi (t,r) \propto \left[ \frac{e^{\pi r/\tau_0}+e^{\pi t/\tau_0}}{e^{\pi r/\tau_0}\cdot e^{\pi t/\tau_0}} \right]^{x/2}= \begin{cases} e^{-\pi xt/2\tau_0}\quad &{\rm for}\, t<r\,,\\ e^{-\pi xr/2\tau_0}\quad &{\rm for}\, t>r\,. \end{cases} \end{equation} Note that in this case the characteristic time is $t^*=r$ and not $r/2$. This explains also why the entanglement entropy of a semi-infinite chain with free boundary condition at $x=0$, has characteristic time $t^*=r$ as firstly noted in Ref. \cite{dmcf-06}. \begin{figure}[t] \centerline{\epsfig{width=14cm,file=fig3.eps}} \caption{Left: Space-time region for the one-point function in a boundary (at $r=0$) geometry. Note $w=\tau+ir$. Right: Conformal mapping to the upper-half plane, c.f. Eq. (\ref{mapsin}).} \label{fig3} \end{figure} \subsection{Discussion and interpretation of the CFT results} All the correlation functions calculated so far display two very general features: first there is a sharp horizon (or light-cone) effect at $t=t^*=r/2$ (or $r$) resulting in a behavior before and after $t^*$ completely different; second the asymptotic long-time correlation functions are the same as those at finite temperature $\beta_{\rm eff}=4\tau_0$. The light-cone effect is a very general phenomenon and and will be discussed in section \ref{phint}. In the following sections we will point out that also the ``effective temperature'' is a general phenomenon, but it has some specific CFT features that are worthy of comment. In fact, it is very easy to understand the \em technical \em reason why we find an effective temperature despite the fact that we are studying a pure state at $T=0$. The finite temperature correlations can be calculated by studying the field theory on a cylinder of circumference $\beta=1/T$. In CFT a cylinder can be obtained by mapping the complex plane with the logarithmic transformation $\beta/(2\pi)\log z$. The form for the two-point function in the slab depends in general on the function $F(\eta)$ --cf. Eq. (\ref{defF})-- but when we analytically continue and take the limit of large real time, we find that effectively the points are far from the boundary, i.e. at $\eta=1$. Thus we get the same result as we would get if we conformally transformed from the full plane to a cylinder, and from Eq. (\ref{logmap}) the effective temperature is $\beta_{\rm eff}=4 \tau_0$. A similar argument can be worked out for the multi-point functions as well. An addtional comment concerns what we expect for correlation functions of general operators, not only primary. We have clearly seen that the local energy density does not relax, of course, and this is consistent with its not being primary. But, we also know that at finite temperature $\langle T_{tt}\rangle_\beta= \pi c/6 \beta^2$, that is perfectly compatible with the previous result with $\beta_{\rm eff}=4\tau_0$. Furthermore, for large real times, the two-point function of $T_{tt}-\langle T\rangle_{tt}$ does behave as though it was at finite temperature. This means, in particular, that the energy fluctuations in a large but finite volume (the specific heat) are the same as those at finite temperature. Thus one is tempted to extend this finite temperature interpretation to non-primary operators, and indeed it is the case. In fact, in the argument of above for the equivalence of the long-time correlations and finite $T$, there are essentially three steps. First, we need to write down the form of the correlation function in the half-plane, but this only depends on special conformal transformation and so is valid for any quasi-primary operator like $T_{\mu\nu}$. Second we have to transform from the slab to the half-plane: for non-primary operators this can have some anomalous term. Finally we need to compare the large limit of this to what one would get transforming directly from the cylinder to the full plane. However, the two conformal transformations we are comparing are both logarithmic (in one case $2\tau_0/\pi\log z$, in the other case $\beta/2\pi\log z$), so the anomalous terms should be the same. Thus the two correlations functions are the same. This argument works for all quasi-primary operators. Then, since we can get all the non-quasiprimaries by considering successive operator product expansions with the stress tensor, it also works for all operators. \section{Exact real-time dynamics in simple integrable chains} \label{secint} The results of the previous section rely on the technical assumption that the leading asymptotic behavior given by CFT, which applies to the euclidean region (large imaginary times), may simply be analytically continued to find the behavior at large real time. While such procedures have been shown to give the correct behavior for the time-dependent correlations in equilibrium, it is important to check them in specific solvable cases for non-equilibrium evolution. Thus, as a complement to the CFT calculations, in this section we consider the \em real-time \em evolution of two simple analytically tractable models. We solve the dynamics of a chain of coupled harmonic oscillators and we review and re-analyze some known results for the Ising-XY chain in a transverse magnetic field. Beyond providing examples of the CFT results (with central charge $c=1$ and $1/2$ respectively) in the critical case, these models allow us to take into account the effects of a finite mass-gap and of the lattice. \subsection{The chain of harmonic oscillators} \label{secharm} The simplest model with an exactly solvable non-equilibrium dynamics is surely a chain of coupled harmonic oscillators with hamiltonian \begin{equation} H=\frac12\sum_r\Big[\pi_r^2+m^2\phi_r^2+\sum_j\omega_j^2(\phi_{r+j}-\phi_r)^2 \Big]\,. \label{Hchaingen} \end{equation} We introduce a coupling more general than simple nearest-neighbor hopping so as to allow for a general dispersion relation below. For simplicity we also assume periodic boundary conditions. $\varphi_n$ and $\pi_n$ are the position and the momentum operators of the $n$-th oscillator, with equal time commuting relations \begin{equation} [\varphi_m,\pi_n]=i\delta_{nm}\,\qquad [\varphi_n,\varphi_m]=[\pi_n,\pi_m]=0\,. \end{equation} The hamiltonian can be written in diagonal form $H(m)=\sum_k \Omega_k A^\dag_k A_k$ with modes \begin{eqnarray} A_k &=&\frac{1}{\sqrt{2 \Omega_k}}(\Omega_k \varphi_k+i \pi_k)\,,\\ A_k^\dag&=&\frac{1}{\sqrt{2 \Omega_k}}(\Omega_k \varphi_{-k}-i \pi_{-k})\,,\\ \Omega_k^2&=&m^2+2\sum_j\omega_j\left(1-\cos(2\pi kj/N)\right)\,. \end{eqnarray} Note that we use the same symbols for the operators and their Fourier transforms ($\varphi_k=1/\sqrt{N} \sum_{n=0}^{N-1} e^{2\pi ikn/N} \varphi_n$ and analogously for $\pi_k$). We consider the scenario in which the system is prepared in a state $|\psi_0\rangle$, that is ground-state of $H(m_0)$, and at the time $t=0$ the mass is quenched to a different value $m\neq m_0$. We use the notation $\Omega_{0k}$ for the dispersion relation for $t<0$ and the $\Omega_k$ for the one for $t>0$. Since $\langle\psi_0| \varphi_n |\psi_0\rangle=0$, the expectation value of the field $\varphi_n$ vanishes at any time. This example in fact corresponds to the quench from the disordered phase in the language of the previous section. Thus we concentrate our attention on the two-point function \begin{equation} \langle \psi(t)| \varphi_n \varphi_0|\psi(t) \rangle= \langle\psi_0| \varphi^H_n(t) \varphi^H_0(t) |\psi_0\rangle \,, \end{equation} where we introduced the operator in the Heisenberg picture $\varphi^H_n(t)$, whose time evolution is given by \begin{equation} \varphi^H_n(t)=\sum_{k=0}^{N-1} \sqrt{\frac{2}{N \Omega_k}} \left(e^{i(p_k n-\Omega_k t)} A_k+e^{-i(p_k n-\Omega_k t)} A_k^\dag\right)\,, \end{equation} where $p_k=2\pi k/N$. Accordingly, the product of the two fields is \begin{equation} \varphi^H_n(t)\varphi^H_0(t)=\frac{2}{N}\sum_{k,k'}\frac{1}{\sqrt{\Omega_k\Omega_{k'}}} \left(e^{i(p_k n-\Omega_k t)} A_k+e^{-i(p_k n-\Omega_k t)} A_k^\dag\right) \left(e^{-i\Omega_{k'} t} A_{k'}+e^{i\Omega_{k'} t} A_{k'}^\dag\right)\,. \end{equation} In order to have the time dependent two-point function we need the expectation values of the bilinear combinations of $A$'s on the initial state, that is annihilated by the $A_{0k}$. Thus it is enough to write $A$'s as functions of $A_0$'s, i.e. \begin{eqnarray} \displaystyle A_k&=&\frac{1}{2}\left[ A_{0k} \left(\sqrt{\frac{\Omega_k}{\Omega_{0k}}}+\sqrt{\frac{\Omega_{0k}}{\Omega_k}}\right)+ A_{0-k}^\dag\left(\sqrt{\frac{\Omega_k}{\Omega_{0k}}}-\sqrt{\frac{\Omega_{0k}}{\Omega_k}}\right) \right]\equiv c_k A_{0k} +d_k A^\dag_{0-k} \,, \\ A^\dag_k&=&\frac{1}{2}\left[ A_{0k}^\dag\left(\sqrt{\frac{\Omega_k}{\Omega_{0k}}}+\sqrt{\frac{\Omega_{0k}}{\Omega_k}}\right)+ A_{0-k} \left(\sqrt{\frac{\Omega_k}{\Omega_{0k}}}-\sqrt{\frac{\Omega_{0k}}{\Omega_k}}\right) \right]\equiv c_k A^\dag_{0k} +d_k A_{0-k} \,, \end{eqnarray} leading to (we understand $\langle\cdot \rangle=\langle \psi_0|\cdot|\psi_0\rangle$) \begin{eqnarray} \langle A_k A_{k'}\rangle &=& \langle(c_k A_{0k} +d_k A^\dag_{0-k})(c_{k'} A_{0k'} +d_{k'} A^\dag_{0-k'})\rangle= c_k d_{k'} \langle A_{0k} A^\dag_{0-k'}\rangle= c_k d_k \delta_{k,-k'}\,, \\ \langle A_k A_{k'}^\dag\rangle &=& \langle(c_k A_{0k} +d_k A^\dag_{0-k})(c_{k'} A^\dag_{0k'} +d_{k'} A_{0-k'})\rangle= c_k c_{k'} \langle A_{0k} A^\dag_{0k'}\rangle= c_k^2 \delta_{k,k'}\,,\\ \langle A_k^\dag A_{k'}\rangle &=& \langle(c_k A^\dag_{0k} +d_k A_{0-k})(c_{k'} A_{0k'} +d_{k'} A^\dag_{0-k'})\rangle= d_k d_{k'} \langle A_{0-k} A^\dag_{0-k'}\rangle= d_k^2 \delta_{k,k'}\,, \label{dk}\\ \langle A_k^\dag A_{k'}^\dag\rangle &=& \langle(c_k A^\dag_{0k} +d_k A_{0-k})(c_{k'} A^\dag_{0k'} +d_{k'} A_{0-k'})\rangle= d_k c_{k'} \langle A_{0-k} A^\dag_{0k'}\rangle= c_k d_k \delta_{k,-k'}\,. \end{eqnarray} Finally we arrive at \begin{eqnarray}\displaystyle \langle\varphi^H_n(t)\varphi^H_0(t)\rangle&=&\frac{2}{Na}\sum_k \frac{1}{\Omega_k}\left[ c_k d_k e^{i(p_k n-2\Omega_k t)}+c_k^2 e^{ip_k n}+d_k^2 e^{-i p_kn} +c_k d_k e^{-i(p_k n-2\Omega_k t)}\right] \,,\label{finlat} \end{eqnarray} which, in the thermodynamic limit ($N\to\infty$), may be written as \begin{equation} \label{2ptgaussian} \langle\varphi_r^H(t)\varphi^H_0(t)\rangle- \langle\varphi^H_r(0)\varphi^H_0(0)\rangle =\int_{\rm BZ} \frac{dk}{(2\pi)} e^{ikr} \frac{(\Omega_{0k}^2-\Omega_k^2)(1-\cos(2\Omega_kt))} {\Omega_k^2\Omega_{0k}}\,, \end{equation} where the integral is on th first Brillouin zone $|k|<\pi/a$. Note that for $t=0$ and for $m=m_0$ this two-point function reduces to the static one, as it should. This result can also be found by integrating the Heisenberg equations of motion for each mode. For future reference it is also useful to write down explicitly the Fourier transform known as momentum distribution function \begin{equation} \rho(k)= \frac{(\Omega_{0k}^2+\Omega_k^2)-(\Omega_{0k}^2-\Omega_k^2) \cos(2\Omega_kt)}{\Omega_k^2\Omega_{0k}}\,. \label{MDF} \end{equation} Note that when considering correlation functions in momentum space, a long-time limit does not exist and we need to take the time average, in contrast to what happens in real space. \subsubsection{The continuum limit} In Eq. (\ref{finlat}) everything is completely general and applies to any chain with finite lattice spacing. Let us know discuss the continuum limit that is achieved by sending $N\to\infty$ in such a way that $(1/N) \sum_k \to \int_{-\infty}^{\infty} d p/(2\pi)$, $p_n\to p$, and $\Omega_k^2 \to \Omega_p^2=m^2+p^2$. In this limit the correlation function becomes ($\Delta m^2=m_0^2-m^2$) \begin{equation} G(r,t)\equiv \langle\varphi_r^H(t)\varphi_0^H(t)\rangle= \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{i p r} \frac{-\Delta m^2 \cos(2 \sqrt{p^2+m^2} t) + m^2+m_0^2 +2 p^2}{{(m^2+p^2)\sqrt{m_0^2+p^2}}} \,. \label{Gcont} \end{equation} Since a closed form for such integral is quite difficult to write down in the most general case, we will only consider some particular cases. Let us first consider the conformal evolution $(m=0)$ from an initial state with a very large mass $m_0\to\infty$. This should reproduce the CFT result previous section for all the time $t$, since the correlations of the initial state shrink to a point. We have \begin{equation} G(r,t)= m_0\int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{i p r} \frac{1-\cos(2 p t)}{p^2}=m_0 \begin{cases} 0 \;&{\rm for}\, t<r/2\,, \\\displaystyle \frac{1}{2}(2t-r)\; &{\rm for}\, t>r/2\,. \end{cases} \label{Gconf} \end{equation} To compare such result with the conformal result given by Eq. (\ref{freeboson}), we have to keep in mind that the primary field is not the gaussian one $\varphi^H(r,t)$, but its imaginary exponential. Thus we need $\langle e^{i q \varphi^H(r,t)} e^{-i q \varphi^H(0,t)}\rangle$, with an arbitrary $q$. Despite of the apparent complexity of such correlator, it is very simple to obtain it using the standard property of gaussian integrals \begin{equation} \langle e^{i q\varphi^H(r,t)} e^{-iq \varphi^H(0,t)}\rangle= e^{-q^2\langle(\varphi^H(r,t)-\varphi^H(0,t))^2\rangle/2}= e^{q^2(G(r,t)-G(0,t))}\,, \end{equation} leading to \begin{equation} \langle e^{i q\varphi^H(r,t)} e^{-i q\varphi^H(0,t)}\rangle= \begin{cases} e^{-q^2 m_0 t}\qquad & {\rm for } \;t<r/2\,,\\ e^{-q^2 m_0 r/2}\qquad & {\rm for } \;t>r/2\,, \end{cases} \end{equation} that is exactly the same of Eq. (\ref{freeboson}) with $x_\Phi\propto q^2$ and $\tau_0\propto m_0^{-1}$, confirming that $\tau_0$ is just proportional to the correlation in the initial state, as its interpretation in terms of the extrapolation length suggests. The case $m=0$ and $m_0$ finite corresponds to a conformal evolution from a generic state with correlations proportional to $m_0^{-1}$. Thus we expect the CFT result Eq. (\ref{freeboson}) to be true for asymptotic large times and separations. \begin{figure}[t] \centering \includegraphics[width=\textwidth]{fig1.eps} \caption{Left: $G(r,t)/m_0$ given by Eq. (\ref{Gxt}) as function of $t$, at fixed $r=1$. Three different values of $m_0=10,3,1$ (from the bottom to the top) are shown. Inset: Lattice effects showing the $\cos 4t$ oscillations on top of the continuum result. Right: $G(r,t)/m_0$ given by the numerical integral of Eq. (\ref{Gmfin}) as function of $t$, at fixed $m,r=1$. It is compared with the asymptotic behavior for $0<2t-r\ll m^{-1}$ and for $t\gg r$. } \label{figGG} \end{figure} From Eq. (\ref{Gcont}), the two-point function of the gaussian field is \begin{equation} G(r,t)= \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{ipr} \frac{-m_0^2 \cos(2 p t) +m_0^2 +2 p^2}{{p^2\sqrt{m_0^2+p^2}}}\,, \end{equation} that can be written as \begin{equation} 2 \pi G(r,t)=2 K_0(m_0 r)+f(m_0r)-\frac{f(m_0(r-2t))+f(m_0(r+2t))}{2}\,, \label{Gxt} \end{equation} where $K_0(y)$ is the modified Bessel function and \begin{equation} f(y)= 1+\frac{1}{2}G^{21}_{13}\left(\frac{ y^2}{4}\left| \begin{matrix} & 3/2 & \\ 0 &1 &1/2 \end{matrix}\right. \right)\,, \end{equation} with $G^{21}_{13}$ the Meijer $G$-function (see e.g. \cite{math}). Note that $f(x)$ is characterized by $f''(x)=-K_0(x)$ and $f(0)=f'(0)=0$. In the limit $m_0\to\infty$, it is easy to show that $G(r,t)$ reduces to the previous result. In the left panel of Fig. \ref{figGG} , we report $G(r,t)$ as function of $t$ at fixed $r=1$ for $m_0=10,3,1$. It is evident that a finite $m_0$ results in smoothing the curve close to $t=r/2$ and giving an offset in zero. Both the effects are more pronounced as $m_0$ decreases. For large $t$, independently on $m_0\neq 0$, we have $G(r,t)= t+O(t^0)$, confirming that the CFT result is correct for asymptotic large times. The case with arbitrary $m$ and $m_0$ is quite cumbersome to be worked out analytically and not really illuminating. For this reason we concentrate here on the massive evolution from a state with $m_0\to\infty$. In this case the correlation function reads \begin{equation} G(r,t)=m_0 \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{ipr} \frac{1-\cos(2 \sqrt{p^2+m^2} t)}{{(m^2+p^2)}} \,. \label{Gmfin} \end{equation} In the limit $t\to \infty$ the cosine term averages to zero, giving \begin{equation} G(r,t=\infty)=m_0 \int_{-\infty}^{\infty} \frac{dp}{2\pi} \frac{ e^{ipr}}{{(m^2+p^2)}} =\frac{m_0}{m} \frac{e^{-mr}}{2}\,. \label{genmtinf} \end{equation} To understand the time dependence, let us first note that, despite the presence of a square-root, the integrand of Eq. (\ref{Gmfin}) is analytic in $p$, since the square-root is the argument of the even cosine function. Thus we can make the integral in the complex plane and use the Cauchy theorem. As long as $r>2t$, the behavior for $p\to i\infty$ is dominated by $e^{i p r}$ and we can safely close the contour path in the upper half-plane where the residue at $p=i m$ is zero. As a consequence $G(r,t<r/2)=0$. For $t>r/2$, the integral is more difficult, since we can not close the contour in the upper half-plane, because the cosine is ``larger'' than $e^{ipr}$ for $p\to i\infty$. However, the approach to the asymptotic value is easily worked out. In fact, for $t\gg r$, the term $e^{ipr}$ in the integral (\ref{Gmfin}) can be approximated with $1$, since it is slowly oscillating. Thus we have \begin{equation} G(r,t\gg r)-G(r,t=\infty) \simeq -m_0 \int_{-\infty}^{\infty} \frac{dp}{2\pi} \frac{\cos(2 \sqrt{p^2+m^2} t)}{{(m^2+p^2)}} = - m_0 f_m(t)\,, \end{equation} where $f_m(t)=1/(2m)-t _1F_2(1/2; 1,3/2;-m^2t^2)$, that satisfies $f'_m(t)=-J_0(2mt)$ and $f_m(0)=1/(2m)$. Note that $f_m(t)$ is just the result for $r=0$. In the complementary region $0<2t-r\ll m^{-1}$, the integral in Eq. (\ref{Gmfin}) is dominated by the modes with $p\gg m$, and so it can be described by the conformal result. In the right panel of Fig. \ref{figGG}, we plot the time-dependence of $G(r,t)$ at fixed $r$ as obtained by numerically integrating Eq. (\ref{Gmfin}) for $m=r=1$. The plot shows that the conformal result $t-r/2$ describes the behavior close to $t=r/2$, while for larger times the asymptotic expression gives an excellent approximation. We finally note that $G(r,t)$ at fixed $t$ displays spatial oscillations (for $r<2t$, else it vanishes), as it can be simply realized by a stationary phase argument. \subsubsection{Lattice effects} \label{lateff} Another advantage of this simple model is that the effects of the lattice can be easily understood. In fact, by a stationary phase argument, the dominant contribution to (\ref{2ptgaussian}) in the limits of large $t$ and $r$ comes from where, the group velocity $v_k\equiv\Omega'_k=r/2t$, independently of the explicit form of $\Omega_k$. Consequently, the two-point function of a gapless model with dispersion relation $\Omega_k=2 \sin k/2$ differs from the continuum limit previously derived for $t>r/2v_m$, where $v_m$ is the maximum group velocity. In particular $G(r,t)$ receives a contribution from the slowest mode ($k=\pi$ with $v_\pi=0$) whose effect is to add to Eq. (\ref{Gconf}) fast oscillations going as $\cos(2\Omega_\pi t)= \cos 4t$. The resulting $G(r,t)$ is plotted in the inset of Fig. \ref{figGG} in the gapless case. Thus, lattice effects play a more important role in the cases where the asymptotic result vanishes. A typical example is the energy density that in this model is proportional to $(\varphi_{r+1}-\varphi_r)^2$. The continuum calculation would just give zero (the mean-energy is conserved), but the approach to this value is governed by lattice effects and is dominated by the smallest group velocity that comes from the zone boundary at $|k|=\pi$. In fact, just with a trivial calculation we have \begin{eqnarray} \langle(\varphi_{r+1}-\varphi_r)^2\rangle&=& 2(\langle\varphi_0^2\rangle-\langle\varphi_0\varphi_1\rangle)= \int\frac{dk}{2\pi} (1-\cos k) \frac{(1-\cos2\Omega_k t)}{2\sin^2k/2}= \nonumber\\&=& \int\frac{dk}{2\pi} (1-\cos2\Omega_k t)\propto J_0(4t)\sim t^{-1/2} \cos 4t\,, \end{eqnarray} that displays the typical $\cos 4t$ from the zone boundary at $|k|=\pi$. The lattice dispersion relations are sensitive to the microscopical details of the model and consequently quantities like the energy-density show a dependence on this. For example, a lattice massless fermion has dispersion relation $\Omega_k=2\sin k/2$ and energy density given by $\sin^2 k$. In this case the time evolution of the energy density is \begin{equation} \int\frac{dk}{2\pi} \sin^2k \frac{(1-\cos2\Omega_k t)}{2\sin^2k/2}\,, \end{equation} that for large times goes like $t^{-3/2}\cos 4t$, resulting in a different power law compared to before. If the system is quenched to a gapped $H$ with $\Omega_k^2=m^2+2(1-\cos k)$ the maximum group velocity corresponds to a non-zero wave number. This gives rise to spatial oscillations in the correlation function (this is true also in the continuum limit as already discussed in the previous subsection). \subsection{The Ising-XY chain in a transverse magnetic field} \label{secIs} The most studied one dimensional quantum spin model is the so-called XY chain in a transverse field, defined by the hamiltonian \begin{equation} H=\sum_{j=1}^N \left[\frac{(1+\gamma)}{2} \sigma^x_j \sigma_{j+1}^x +\frac{(1-\gamma)}{2}\sigma^y_j \sigma_{j+1}^y- h\sigma_z\right]\,, \end{equation} where $\sigma_i^{x,y,z}$ are the Pauli matrices, $\gamma$ is called anisotropy parameter, and $h$ is the applied external transverse field. It is well known that for any $\gamma\neq0$ the model undergoes a phase transition at $h=1$, that is in the universality class of the Ising model (defined by $\gamma=1$). For simplicity we will just consider the Ising case, other values of $\gamma\neq 0$ being equivalent. We consider the non-equilibrium unitary dynamics that follows from a quench of the magnetic field at $t=0$ from $h_0$ to $h_1\neq h_0$. Earlier works on this subject by Barouch et al. date back to seventies \cite{bm1,bm2}. Exploiting the mapping of this model onto a free fermion, the time evolution of the transverse magnetization (that is {\it not} the order parameter) was obtained exactly \cite{bm1}: \begin{equation} m_z(t)=\frac{1}{\pi} \int_0^\pi dk \frac{(h_0-h_1)\sin^2k\cos(2\epsilon_1 t)- (\cos k-h_1)[(\cos k-h_0)(\cos k-h_1)+\sin^2 k]}{\epsilon_0 \epsilon_1^2}\,, \end{equation} where $\epsilon_i=\epsilon(h_i)$ with $\epsilon(h)=\sqrt{\sin^2 k+(h-\cos k)^2}$. In Ref. \cite{bm1} it was shown that (for non-exceptional parameters) the approach to the asymptotic value for $t\to\infty$ is of the form $t^{-3/2}\cos 4t$. This is simply shown in the case of $h_0=\infty$ and $h_1=1$, when the integral simplifies and we obtain (see also \cite{ir-00}) \begin{equation} m_z(t)=\frac{1}{2}+\frac{1}{\pi} \int_0^\pi dk \frac{\sin^2k\cos(2\epsilon_1 t)}{4\sin^2 k/2}= \frac{1}{2}+\frac{J_1(4t)}{4t}\,, \end{equation} where $J_1$ is the Bessel function of the first kind whose asymptotic expansion for large argument is $J_1(x)\sim \sqrt{2/\pi x} \cos (x+\pi/4)$. To understand this result we should keep in mind that the transverse magnetization is {\it not} the order parameter: it corresponds to the product of two ``disorder parameters'' at neighbor sites on the dual lattice. Thus it must have the symmetry of an energy operator, for which CFT just predicts a constant asymptotic result. The power law term $t^{-3/2}\cos(4t)$ is just the lattice correction to the asymptotic. This correction clearly shows the fermionic nature of the model on the basis of what discussed in the previous section. The asymptotic result of the two-point function for $t\to\infty$ has been studied in Ref. \cite{sps-04}. Calling $G_n=\langle\Phi(n,\infty) \Phi(0,\infty)\rangle$, in the case $h_0=\infty$ it has been found \begin{equation} G_n=\begin{cases}\displaystyle \frac{1}{2^nh_1^n}\,,&{\rm for}\, h_1\geq1\,,\\ \frac{1}{2^n} \cos[n\arccos(h_1)] \,,&{\rm for}\, h_1\leq1\,, \end{cases} \end{equation} instead for $h_0=0$ \begin{equation} G_n=\begin{cases} \frac{1}{2^n}\,,&{\rm for}\, h_1\geq1\,,\\ \frac{h_1^{n+1}}{2^n}\cosh\left[(n+1)\ln \left(\frac{1+\sqrt{1-h_1^2}}{h_1} \right)\right]\,,& {\rm for} h_1\leq1\,, \end{cases} \end{equation} that for large $n$ decay exponentially with $n$. It has been shown that $G_n$ is decaying exponentially with $n$ for general $h_0$ and $h_1$ \cite{sps-04}, although closed forms are not available. The exponential decay is the prediction of CFT that (maybe surprisingly) applies even far from the critical point $h_1=1$. The time dependence of two-point function has been studied in Ref. \cite{ir-00} by means of exact diagonalization of the model with open boundary condition at the two ends $r=0,L$. The results of interest for this paper are \begin{itemize} \item The connected two-point function of $\sigma_z$ in the thermodynamic limit and at the critical point is \begin{equation} \langle \sigma_r^z(t)\sigma_0^z(t)\rangle_c =\left[\frac{r}{2t}J_{2r}(4t)\right]^2-\frac{r^2-1}{4t^2} J_{2r+1}(4t)J_{2r-1}(4t)\,, \end{equation} which is valid both for $h_0=0,\infty$. Neglecting fast oscillations, $\langle \sigma_r^z(t)\sigma_0^z(t)\rangle_c$ increases as $r^2$ for $r<2t$ and then it drops almost immediately to $0$. \item The autocorrelation function (not connected) of $\sigma_z$ at different times is \begin{equation} \langle \sigma_0^z(t_1)\sigma_0^z(t_2)\rangle =J_0^2(2t_2-2t_1)-\frac{1}{4}[f(t_2+t_1)\pm g(t_2-t_1)]\,, \end{equation} where $f(x)=J_2(2x)+J_0(2x)$, $g(x)=J_2(2x)-J_0(2x)$ and the sign $+$ ($-$) refers to $h_0=0$ ($h_0=\infty$). \end{itemize} Let us comment on these results in view of the general understanding we found so far. For $t<t^*$ all the connected correlation functions are zero, in agreement with CFT. All the oscillation terms of the asymptotic form $\cos (4t)$ are, as we discussed in the previous section, a lattice effect. Concerning the $\sigma_z$ correlator, the $r^2$ dependence for $t>r/2$ is a consequence of the fact that $\sigma_z$ is not primary. The same is true for the two-time correlations function of $\sigma_z$ that also decays as a power law of $t-s$ for large times, instead of the exponential prediction by CFT for primary field. \section{Higher dimensions} \label{HD} Until now we just considered one-dimensional systems. Despite the fact that in low dimensions the effect of fluctuations is more pronounced making the physics highly non-trivial it is desirable to have results in higher dimensions as well. The method presented in Sec. \ref{secpath} to obtain the non-equilibrium dynamics of a quantum model close to a critical point from the critical behavior of a system confined in a slab geometry applies to generic dimension $d$ through the study of the hamiltonian (\ref{LGH}). Its analysis proceeds via field-theoretical RG that may provide the all scaling quantities of the model in an expansion close to the upper critical dimensions (u.c.d.), that is $D=d+1=4$. Above the u.c.d. mean-field (or gaussian) results are exact, with logarithmic correction at the u.c.d.. For dimensions lower than the u.c.d., the scaling quantities are obtained as series in $\epsilon=4-(d+1)$. Thus for the time-evolution problem the simple mean-field solution represents an exact scaling result (a part log corrections) for the physically relevant three-dimensional case. An alternative method to attack analytically the hamiltonian (\ref{LGH}) is to consider an $N$ component field $\phi$ and taking the limit $N\to\infty$, but this will not be employed here. \subsection{Dirichlet boundary conditions: the two-point function} The $D$-dimensional slab geometry with Dirichlet boundary conditions has been the subject of several investigations. The two-point function has been calculated at the first order in $\epsilon$ expansion in Ref. \cite{kd-99}. The gaussian two-point function, with partial Fourier transform in the parallel directions reads \cite{kd-99} \begin{equation} G(p,z_1,z_2)=\frac{1}{2b}\left( e^{-b|z_1-z_2|}-e^{-b(z_1+z_2)}+\frac{e^{-b(z_1-z_2)}+e^{-b(z_2-z_1)}-e^{-b(z_1+z_2)}-e^{b(z_1+z_2)}}{e^{2bL}-1} \right)\,, \label{Ggau} \end{equation} with $b=\sqrt{p^2+m^2}$. We are interested in the case where $L=2\tau_0$, $z_1=z_2=\tau$ that we will analytically continue to $\tau\to \tau_0+i t$, and for computational simplicity we will restrict to the massless case $m=0$. Thus in real space and imaginary time, we have ($p=|{\mathbf p}|$) \begin{eqnarray} G(r,\tau)&=& \int \frac{d^3p}{(2\pi)^3} e^{-i {\mathbf{p\cdot r}}}\frac{1}{2p}\left( 1-e^{-2p\tau}+\frac{2(1-\cosh(2p\tau))}{e^{4p\tau_0}-1}\right)=\nonumber\\ &=& \frac{1}{(2\pi)^2} \int_0^\infty p^2 dp \frac{1}{2p}\left( 1-e^{-2p\tau}+\frac{2(1-\cosh(2p\tau))}{e^{4p\tau_0}-1}\right) \int_{-1}^1 d(\cos\theta)e^{ipr \cos\theta}=\nonumber\\ &=&\frac{1}{(2\pi)^2 r} \int_0^\infty dp \sin{pr}\left( 1-e^{-2p\tau}+\frac{2(1-\cosh(2p\tau))}{e^{4p\tau_0}-1}\right)\,. \label{Gr3D} \end{eqnarray} This integral can be performed by making the sum over all the residues coming from the denominator $e^{4p\tau_0}-1$. The calculation is rather involved, but the final result is very simple: \begin{equation} G(r,\tau)= \frac{1}{\pi r\tau_0}\,\frac{\coth(\pi r/4\tau_0) \sin^2(\pi\tau/2\tau_0)}{ \cosh(\pi r /2\tau_0)-\cos(\pi\tau/\tau_0)}\,. \end{equation} Continuing to real time $\tau=\tau_0+i t$ we obtain \begin{equation} G(r,t)= \frac{1}{\pi r\tau_0}\,\frac{\coth(\pi r/4\tau_0) \cosh^2(\pi t/2\tau_0)}{ \cosh(\pi r /2\tau_0)+\cosh(\pi t/\tau_0)}\,, \end{equation} that for $t,r\gg \tau_0$ simplifies to \begin{equation} G(r,t)\simeq \frac{1}{r\tau_0}\, \frac{ e^{\pi t/\tau_0}}{ e^{\pi t/\tau_0}+ e^{\pi r/2\tau_0}}\propto \begin{cases} e^{\pi (t-r/2)/2\tau_0}/r &{\rm for }\; t<r/2\,,\\ 1/r &{\rm for }\; t>r/2\,. \end{cases} \end{equation} We recall that this result in $d=3$ is exact a part log corrections. Thus the basic structure of the two-point function in 3D is the same as in 1D, with a characteristic time $t^*=r/2$. Using the result in Ref. \cite{kd-99} it is in principle possible to calculate the correlation functions for $d<3$ in the $\epsilon$ expansion framework. However, this requires the analytical continuation of complicated functions, resulting in a quite cumbersome algebra, as we shall see in the following for a simpler observable. Eq. (\ref{Ggau}) can be used in principle to determine the gaussian behavior in any $d<3$. Unfortunately the integral one gets is not analytically tractable for $d\neq1,3$. A possible strategy would be to perform the analytic continuation before of the integral and then evaluate it through a saddle-point approximation. It is straightforward to show that the result obtained in this way is equivalent to what we discuss in section \ref{rtsm} where we remand for the analysis of $1<d<3$. \subsection{Dirichlet boundary conditions: a non-trivial one-point function} In the case of Dirichlet boundary conditions, the order parameter profile in the slab geometry is trivially vanishing. However not all the one-point expectation values are zero. Let us consider as a typical example the operator ${\cal O}=\phi^2$ that has been calculated at the first order in $\epsilon$ expansion in Ref. \cite{ked-95}. It has the scaling form \begin{equation} \langle {\cal O}(z,L)\rangle \simeq L^{-d+1/\nu} H(z/L)\,, \end{equation} where $\nu$ is the correlation length exponent. In $D=4$, the function $H(x)$ is \cite{ked-95} \begin{equation} H(x)=\frac{\pi^2}{\sin^2(\pi x)}-\frac{\pi^2}{3}\,, \end{equation} that, using $L=2\tau_0$ and continuing to $z=\tau=\tau_0+it$, leads to the real time evolution \begin{equation} \langle {\cal O}(t)\rangle -\langle {\cal O}(t=\infty)\rangle \simeq \tau_0^{-2}\frac{\pi^2}{\cosh^2(\pi t/2\tau_0)}\sim \tau_0^{-2} e^{-\pi t/2\tau_0}\,, \end{equation} i.e. $\langle {\cal O}(t)\rangle$ approaches its asymptotic value exponentially, as it as been found for all primary operators in $d=1$. The one-loop result for the $O(N)$ model in $D=4-\epsilon$ dimensions is \cite{ked-95} \begin{equation} H(x)=\left[\frac{\pi^2}{\sin^2(\pi x)}-\frac{\pi^2}{3}\right] \left(1+\epsilon\frac{N+2}{N+8}\ln \frac{\pi}{\sin\pi x}\right)- \epsilon [\zeta'(2,x)+\zeta'(2,1-x)]+{\rm const}\,, \end{equation} where const stands for terms do not depend on $x$ and $\zeta'(\alpha,x)=\partial_\alpha \zeta(\alpha,x)$ and $\zeta$ is the generalized Riemann function. This example clearly shows that within the $\epsilon$ expansion, the scaling functions contain logarithmic contributions that originate from the expansion of power law as e.g. $\sin^{\epsilon} \pi x=1+\epsilon\ln\sin \pi x$. To have a function with good analytical structure to perform the real time continuation, it is desirable to ``exponentiate'' such logarithms. In Ref. \cite{ked-95} the exponentiation procedure leaded to \begin{multline} H(x)=\left(\frac{\pi}{\sin(\pi x)}\right)^{2-1/\nu} [\zeta(D-2,x)+\zeta(D-2,1-x)-2\zeta(D-2)]\\- \frac{\pi^2}{6}(2-1/\nu) \left(\frac{\pi}{\sin(\pi x)}\right)^{D-2-1/\nu}\,, \end{multline} with $\nu=1/2 +(N+2)/(N+8) \,\epsilon/4$. Performing the analytical continuation we have the sum of two exponentials, and in any dimension $d<3$ the second one has a largest ``relaxation time'' that hence is dominating for large $t$. Considering only the second term we have \begin{equation} \langle {\cal O}(t\gg\tau_0)\rangle-\langle {\cal O}(t=\infty)\rangle \propto e^{-(d-1-1/\nu)\pi t/2\tau_0}\,. \end{equation} Note that the subleading term $ e^{-(2-1/\nu)\pi t/2\tau_0}$ is multiplied by a $\log t$ term arising from the $\zeta$ function. This example put forward the idea that (at least for Dirichlet boundary conditions) the exponential relaxation of the one-point functions is not only a property of one-dimensional systems, but holds in any dimension (with eventually $\log$ corrections) with relaxation times related to the scaling dimensions of the operator. \subsection{Fixed boundary conditions: the one-point function} The case of the extraordinary transition, that corresponds to fixed boundary conditions, has been considered in Ref. \cite{k-97}. The magnetization profile can be written as \cite{k-97} \begin{equation} \phi(z)=\frac{2K_4}{L} \frac{{\rm dn}(2K_4 z/L)}{{\rm sn}(2K_4 z/L)}\,, \end{equation} where $K_4=K(k_4)$, where $K(k)$ is the elliptic integral, $k_4$ the elliptic modulus that in terms of the parameter of the model is $\phi^2 L^2=(2K_4)^2(2 k_4^2-1)$, and sn$(x)$ and dn$(x)$ and the Jacobi functions. Continuing to real time, and using the properties of the Jacobi functions we obtain \begin{equation} \phi(t)=\sqrt{1-k_4} {\rm cn} (K_4 t/\tau_0,1-k_4)\,, \end{equation} that, contrarily to all the other cases we have considered, is oscillating and not exponentially decreasing. To our knowledge there are no result in the $\epsilon$-expansion for the magnetization profile. However, using ``local-functional methods'' \cite{bu-98} it has been obtained an approximate profile in $D=3$ that involves, as in mean-field theory, Jacobi elliptic functions. It can be easily continued to real time via $z\to \tau_0+i t$ and again one finds an oscillating behavior with time. The method exploited in Ref. \cite{bu-98} can be used in any dimensions, obtaining an always an oscillating behavior with a period that diverges as $D$ approaches 2, recovering Eq. (\ref{phistrip}). All these calculations are essentially mean field and we do not know how the inclusion of fluctuations changes them. It can possible that for the extraordinary transition (i.e. fixed initial conditions), the exponential decay founds at $D=2$ is more an exception rather than a rule, because of the simple analytic structure of the trigonometric functions in the complex plane. Another possibility is that fluctuations destroy these oscillations. Only a complete analytical calculation (e.g. in large $N$) can help in understanding this point. \subsection{A real-time solvable model} \label{rtsm} As for the one-dimensional case, it is worth to check the results coming from the analytical continuation of large imaginary-time with exactly solvable models. The simplest (and probably one of the few) model solvable in generic dimension is the generalization of the hamiltonian (\ref{Hchaingen}) to a $d$-dimensional hypercubic lattice. The solution of such model proceeds via Fourier transform as in one dimension. The final result is simply give by Eq. (\ref{2ptgaussian}) with the replacement $dk\to d^dk$, i.e. \begin{equation} \label{2ptgaussiand} \langle\varphi_r(t)\varphi_0(t)\rangle- \langle\varphi_r(0)\varphi_0(0)\rangle =\int_{\rm BZ} \frac{d^dk}{(2\pi)^d}e^{i\bf{k \cdot r}} \frac{(\Omega_{0k}^2-\Omega_k^2)(1-\cos(2\Omega_kt))}{\Omega_k^2\Omega_{0k}}\,. \end{equation} First of all let us note that this expression in \em virtually \em identical to Eq. (\ref{Gr3D}) if we take $\tau_0\propto m_0^{-1}\to 0$. In fact, taking $\tau_0\to 0$ in Eq. (\ref{Gr3D}), only the third term matters, since it is $O(\tau_0^{-1})$ relative to the first two. Taking $z=r+it$ and $b=\Omega_p$ we get $G(p)\propto \tau_0^{-1}\int \Omega_p^{-2} (1-\cos2\Omega_p t)$ that is Eq. (\ref{2ptgaussiand}) for $m_0\to\infty$. To understand the general features let us consider in details the conformal evolution ($\Omega_k=v |{\bf k}|$) from a disordered state ($\Omega_{0k}= m_0$). The derivative of the two-point function is ($k=|{\bf k}|$) \begin{equation} \partial_t \langle\varphi_r(t)\varphi_0(t)\rangle=2m_0 {\rm Im} \int \frac{d^dk}{(2\pi)^d} \frac{e^{i({\bf k \cdot r}-2k v t)}}{k}\,. \end{equation} Except for $d=1$, this can be done analytically only in $d=3$, where we can write it as \begin{equation} \sim \int k dk \int d\theta \sin\theta e^{ik(r \cos\theta-2v t)}\sim \int dk \big(\sin(kr)/r\big) e^{2ikvt}\sim (1/r) \delta(vt-r/2)\,. \end{equation} Integrating with respect to $t$ we get zero for $t<r/2v$ and $1/r$ for $t>r/2v$. For general $d$ we have \begin{equation} \partial_t \langle\varphi_r(t)\varphi_0(t)\rangle\propto 2m_0 {\rm Im} \int k^{d-2}dk\int d\theta (\sin \theta)^{d-2} e^{ik(r\cos\theta-2vt)}\,. \end{equation} By a saddle point argument, we can assume that the dominant behavior comes from $\theta$ close to zero, so the $\theta$ integral gives \begin{equation} \int d\theta \theta^{d-2}e^{-ikr\theta^2/2}\sim (kr)^{-(d-1)/2}\,, \end{equation} leading to \begin{equation} \int dk \frac{k^{d-2}}{(kr)^{(d-1)/2}} e^{i(k(r-2vt))}\sim r^{-(d-1)/2}(2vt-r)^{-(d-1)/2}\Theta(vt-r/2)\,. \end{equation} Integrating with respect to $t$, we get zero for $t<r/2v$, as expected by causality, and \begin{equation} r^{-(d-1)/2} (2vt-r)^{(3-d)/2}\,, \end{equation} for $t>r/2v$. It is interesting that this gaussian correlation function blows up at large $t$ only for $d<3$, when we expect the fluctuations to become important. It would be interesting to study this in the $\phi^4$ theory for large $N$, by replacing $\phi^4$ by $3\langle\phi^2\rangle\phi^2$ in the usual way, where now $\langle\phi^2\rangle$ depends on $t$ and is calculated self-consistently. \section{Physical interpretation and discussion} \label{phint} In this paper we studied in general the non-equilibrium unitary dynamics that follows a sudden quantum quench. We showed that if the hamiltonian $H$ governing the time evolution is at a critical point, while $H_0$ (i.e. the one for $t<0$) is not, the expectation value of a class of operators (primary ones in CFT) relaxes to the ground-state value exponentially in time with universal ratio of decaying constants. We also found that connected two-point functions of operators at distance $r$ are vanishing for $t<r/2v$, while for $t>r/2v$ reach exponentially fast a value that depends \em exponentially \em on the separation, in contrast with the power laws typical of equilibrium configuration. We also considered the real-time dynamics of simple exactly solvable models and we found that several of the typical characteristics of the critical points still hold. Roughly speaking, critical points are not special as far as quenching dynamics is concerned. In fact, also for gapped systems, connected correlation functions vanish (or are strongly suppressed) for $t<r/2v$ and for asymptotic large times resemble those at finite temperature despite the fact that the whole system is in a pure state. Several other examples in the recent literature (see e.g. \cite{ir-00,cc-05,dmcf-06,c-06,gg,cdeo-06}) gives further evidence that these two effects are actually true in general, at least in the realm of exactly solvable models considered so far. In the following we give a simple interpretation of these two features separately trying to understanding their physical origin. \subsection{The horizon effect} The qualitative, and many of the quantitative, features found for the time evolution of correlation functions may be understood physically on the basis of a picture we first introduced in Ref. \cite{cc-05} to describe the time evolution of the entanglement entropy. Later we generalized it to correlation functions in Ref. \cite{cc-06} and it has been largely adopted thereafter \cite{dmcf-06,c-06,spinor,cdeo-06}. We emphasize that such scheme is not an ab initio calculation but rather a simplified picture which allows us to explain physically our findings. The initial state $|\psi_0\rangle$ has an (extensively) high energy relative to the ground state of the hamiltonian $H$ which governs the subsequent time evolution, and therefore acts as a source of quasiparticle excitations. Those quasi-particles originating from closely separated points (roughly within the correlation length $\xi_0=m_0^{-1}$ of the ground state of $H_0$) are quantum entangled and particles emitted from far different points are incoherent. If the quasiparticle dispersion relation is $E=\Omega_k$, the classical velocity is ${\bf v}_k=\nabla_k\Omega_k$. We assume that there is a maximum allowed speed $v_m=\max_{\bf k} |{\bf v_k}|$. A quasiparticle of momentum $k$ produced at ${\bf r}$ is therefore at ${\bf r}+{\bf v}_k t$ at time $t$, ignoring scattering effects. This is the only physical assumption of the argument. Scattering effects are not present in the theories considered so far, but as evident from the argument outlined below they can play a role for only $t>r/2v_m$ (allowing, perhaps, for a renormalization of $v_m$ by the interactions). These \em free \em quasi-particles have two distinct effects. Firstly, incoherent quasi-particles arriving a given point $\bf r$ from well-separated sources cause relaxation of (most) local observables at $\bf r$ towards their ground state expectation values. (An exception is the local energy density which of course is conserved.) Secondly, entangled quasi-particles arriving at the same time $t$ at points with separation $|{\bf r}|\gg\xi_0$ induce quantum correlations between local observables. In the case where they travel at a unique speed $v$ (as in CFT), therefore, there is a sharp ``horizon'' or light-cone effect: the connected correlations do not change from their initial values until time $t\sim |{\bf r}|/2v$. In the CFT case this horizon effect is rounded off in a (calculable) manner over the region $t-|{\bf r}|/2v\sim\tau_0$, since quasi-particles remain entangled over this distance scale. After this they rapidly saturate to \em time-independent \em values. For large separations (but still $\ll 2vt$), these decay \em exponentially \em $\sim\exp(-\pi x|{\bf r}| /2\tau_0)$. Thus, while the generic one-point functions relax to their ground-state values (we recall in CFT this relaxation is exponential $\sim\exp(-\pi xvt/\tau_0)$), the correlation functions do not, because, at quantum criticality, these would have a power law dependence. Of course, this is to be expected since the mean energy is much higher than that of the ground state, and it does not relax. This simple argument also explains why for the case of a semi-infinite chain the relevant time scale is $r/v$ rather than $r/2v$, since one of the two particles arriving in $r$ has been reflected from the end of the chain. This has been also stressed in Ref. \cite{dmcf-06}, in the study of the time dependence of entanglement entropy of finite chains with open boundary conditions. All our results are consistent with this picture as long as the quasi-particles are assumed to all propagate at the same speed, resulting from a ``conformal'' dispersion relation $\Omega_k=v |k|$. However, it is very simple to generalize this picture to different dispersion relations, taking into account that each particle propagates at group velocity $v_k\equiv\Omega'_k$ appropriate to the wave number $k$. In this case the horizon effect first occurs at time $t\sim |{\bf r}|/2v_m$, where $v_m$ is the maximum group velocity. If $v_m$ occurs at a non-zero wave number, it gives rise to spatial oscillations in the correlation function. Thus again connected correlation function are expected to be strongly suppressed for $t<t^*$ and start developing only after $t^*$. In the case of a general dispersion relation we do not have a proof, beyond the stationary phase approximation, that the connected correlation functions remain constant up to this time, but one would expect it on the grounds of causality. (The proof in the case of a relativistic dispersion relation uses Lorentz invariance in an essential way.) However, because there are also quasi-particles moving at speeds less than $v_m$, the approach to the asymptotic behavior at late times is less abrupt. In fact, for a lattice dispersion relation where $\Omega'_k$ vanishes at the zone boundary, the approach to the limit is slow, as an inverse power of $t$. A similar result applies to the $1$-point functions. This is consistent with the exact results obtained here and elsewhere. \subsection{The large time limit and the generalized Gibbs ensemble} The existence and the understanding of the asymptotic state resulting from the evolution from an arbitrary state is one of the most-interesting problem in statistical mechanics. A robust theory able to predict this state ab-initio still does not exist. A currently popular idea is that for late times the system (or rather macroscopically large subsystems) `look like' they are in a thermal state, despite the fact that the actual state of the whole system is pure. A common belief is that a region of dimension $r$ can be thermalized by the infinitely large rest of the system which acts as a bath (see e.g. \cite{cc-05,eo-06,cdeo-06}). But this intriguing idea is not sufficient to give the value of the resulting effective temperature. A major step toward the clarification of the properties of the asymptotic state has been made by Rigol et al. \cite{gg}. In fact, it was conjectured that if the asymptotic stationary state exists, it is given by a generalized Gibbs ensemble obtained by maximizing the entropy $S=-{\rm Tr}\, \rho \log \rho$, subject to all the constraints imposed by the dynamics \cite{gg}. Consequently, denoting with $I_m$ a maximal set of commuting and linearly independent integrals of motion, the density matrix is \begin{equation} \rho=Z^{-1} e^{-\sum_m \lambda_m I_m},\qquad Z= {\rm Tr}\,\, e^{-\sum_m \lambda_m I_m}. \label{gge} \end{equation} We note that such a density matrix describes a pure state only if the model under consideration is integrable, i.e. if the number of integral of motions equals the number degrees of freedom. If there are not enough integrals of motion, $\rho$ corresponds to a mixed state and it is not clear to us to which extent it can describe the pure state resulting from the time-evolution. The values of the Lagrange multipliers $\lambda_m$ are fixed by the initial conditions: \begin{equation} {\rm Tr}\,\,I_m \rho= \langle I_m\rangle_{t=0}\,. \end{equation} In the following we will take this generalized Gibbs ensemble as a postulate and we will show how it nicely and naturally explains the ``effective temperature'' effect observed for large times. However we stress that there is still no proof for this assumption that, to our opinion, cannot be considered on the same fundamental level as the thermal Gibbs ensemble. Let us consider the chain of harmonic oscillators of the previous section as a typical example. We will soon see that most of the features are quite general. In this case the natural choice for an infinite set of integral of motion is the number of particles with momentum $k$, i.e. $n_k=A^\dag_k A_k$. Most other observables can be written in terms of these, i.e. $H=\sum_k \Omega_k n_k$. Consequently the expectation value is given by \begin{equation} \langle {\cal O} \rangle_{t=\infty}= {\rm Tr}\, {\cal O} \rho= {\rm Tr}\, {\cal O} Z^{-1} e^{-\sum_k \lambda_k n_k}\,, \end{equation} that can be seen as a thermal density matrix with a $k$ dependent effective temperature given by \begin{equation} \beta_{\rm eff}(k)\Omega_k = \lambda_k\,. \end{equation} Thus an effective temperature already appeared. Note that this state can still be pure because such a temperature is $k$ dependent. To fix $\lambda_k$ we need $\langle n_k\rangle_{t=0}$. From Eq. (\ref{dk}) we get \begin{equation} \langle n_k\rangle_{t=0}= \langle A^\dag_k A_k\rangle_{t=0}=d_k^2= \frac14\left(\frac{\Omega_k}{\Omega_{0k}}+\frac{\Omega_{0k}}{\Omega_k}\right)-\frac12\,. \end{equation} The calculation then proceeds as for a thermal distribution \begin{equation} \langle n_k\rangle_\rho= {\rm Tr}\, n_k \rho=-\frac{\partial}{\partial\lambda_k} \ln Z\,, \qquad {\rm with} \quad Z= {\rm Tr}\, e^{-\sum_k \lambda_k n_k}=\prod_k \sum_{n_k=0}^\infty e^{-\lambda_k n_k}= \prod_k\frac1{1-e^{-\lambda_k}}\,, \end{equation} so that \begin{equation} \langle n_k\rangle_\rho= \frac{\partial}{\partial\lambda_k}\sum_k \ln (1-e^{-\lambda_k})= \frac1{e^{\lambda_k}-1}\,. \end{equation} From this $e^{\lambda_k}=1+d_k^{-2}$ and \begin{equation} \beta_{\rm eff}(k)= \frac{\ln (1+d_k^{-2})}{\Omega_k}\,. \end{equation} At finite temperature the correlation function in momentum space is \begin{equation} \langle \varphi_k \varphi_{-k}\rangle_\beta= \langle\frac2{\Omega_k} (A_k+A^\dag_k) (A_{-k}+A^\dag_{-k})\rangle_\beta= \frac2{\Omega_k}\left(\langle A^\dag_k A_{-k} \rangle_\beta +\langle A_kA^\dag_{-k}\rangle_\beta\right)= \frac2{\Omega_k}\frac{1+e^{-\beta \Omega_k}}{1-e^{-\beta \Omega_k}}\,, \end{equation} that substituting the previous result for $\beta_{\rm eff}$ gives \begin{equation} \rho(k)_{t=\infty}=\langle \varphi_k \varphi_{-k}\rangle_{t=\infty}= \frac2{\Omega_k}(1+2d_k^2)=\frac{\Omega_k^2+\Omega_{0k}^2}{\Omega_k^2\Omega_{0k}}\,, \end{equation} that is exactly the time-average of Eq. (\ref{MDF}) reproducing, after Fourier transforming, the well defined correlation in real space for $t\to\infty$. So this generalized Gibbs ensemble correctly reproduces the exact diagonalization of the model and gives also few insights more. In fact, in the limit $m_0\to\infty$ the effective temperature is independent from $k$ and $m$ obtaining $\beta_{\rm eff}=4/m_0$, explaining a posteriori the simplicity of the results in this case. Note instead that for arbitrary $m_0$ and $m=0$, i.e. conformal evolution, $\beta_{\rm eff}(k)$ is a function of $k$. In this case, the large distance properties of correlation functions are described by the mode with $k=0$, for which, independently of $m_0$ we get $\beta_{\rm eff}(k=0,m=0)=4/m_0$, consistently with the previous findings and general expectations. Furthermore we can conclude that the large $r$ asymptotic behavior is always governed by the effective temperature $\beta_{\rm eff}(k=0)=2(\log(|m_0-m|/(m_0+m))/m$. Another interesting feature is that $\beta_{\rm eff}(k=0)$ gives the asymptotic behavior of the correlation functions of all those observables that are effectively coupled with the zero-mode. In the opposite case the relevant temperature is the largest $\beta_{\rm eff}(k)$ with the $k$ mode coupled with the observable. Another interesting feature is that on a very rough basis one can be tempted to assume that that $\beta_{\rm eff}(k)$ is directly related to the excess of energy of the mode $k$ generated by the quench, but this is not the case. In fact $\langle H\rangle_{t=0}=\sum_k \Omega_k d_k^2$ that is different from $\beta_{\rm eff}^{-1}$, being the same only in the limit $d_k^{-2}\to0$, i.e. $m_0\to\infty$. Clearly the same reasoning of before applies every time we consider a model that can be diagonalized in momentum space with a proper choice of the quasi-particles, i.e. every time that $H=\sum_k \Omega_k A_k^\dag A_k$ for some $A_k$. This means that the excitations are non-interacting. As far as we are aware all the applications of the generalized Gibbs ensemble to the date only concern this kind of models \cite{gg,c-06}, but there are few numerical hints suggesting that can be true more generically \cite{mwnm-06}. Now we outline how we imagine the generalized Gibbs ensemble given by Eq. (\ref{gge}) could be used to justify the effective temperature scenario for large time for any integrable system, i.e. with a complete set of integrals of motion. The hamiltonian can be written as $H=\sum_m a_m I_m$, with some $a_m$ eventually zero. Thus one can think to an $m$-dependent effective temperature $\beta_{\rm eff}(m)=\lambda_m/a_m$, but this temperature does not give directly the behavior of the correlation function for large distance because in general the integral of motions are not diagonal in $k$-space. Thus we can only conjecture that the correlation functions of a given operator $O(r)$ are governed by the largest $\beta_{\rm eff}(m)$ with $m$ among the integral of motions to which $O(r)$ effectively couples. We stress that this is only a crude argument and we are still not able to put it on a firmer basis. \subsection{Discussion and open questions} We presented a quite complete picture on the time evolution of a quantum system after a sudden quench of one hamiltonian parameter. Despite the fact that a lot of work has been done, still more is left for future investigation. The first problem that must be addressed is the real-time dynamics of effectively interacting systems. In this direction Bethe Ansatz solvable models like the Lieb-Liniger gas and Heisenberg spin chains are among the best candidates for an analytical approach. This would clarify how a non-trivial two-body scattering matrix can modify the time-evolution (only inside the light-cone) of the correlation functions and whether the crude argument we outlined for an ``effective temperature'' for the long-time state is valid. To clarify this point also the study of (boundary) integrable massive field theories can be of some help. Another approach, currently under investigation\cite{sotcardy}, is the direct analysis of the perturbative expansion for the correlation functions in a $\lambda\phi^4$ field theory. This has some simplifying features in the large $m_0$ limit which may allow it to be resummed to all orders. It is very important to get non-trivial results for such a non-integrable interacting theory. Numerical computations for non-integrable systems should also be performed to understand if and eventually to which extent the generalized Gibbs ensemble picture is valid beyond integrability. Some numerics concerning this point are already available \cite{ksdz-05,kla,mwnm-06} but still a clear scenario is not emerged. Among non-integrable models special care must be given to disordered systems because of the non existence of a speed of sound as a consequence of Anderson localization \cite{bo-07}. Some insights can be obtained from those models whose equilibrium behavior can be analytically obtained by means of the strong-disorder renormalization-group \cite{sdrg}. Even in this case time-dependent DMRG can help a general understanding. The time-evolution of the entanglement entropy \cite{dmcf-06} already revealed that in these systems there is no sign of the light-cone (as easily predictable because of the absence of speed of sound) and the effective motion of ``quasi-particles'' is more diffusive rather than ballistic. Furthermore the results at the largest available times does not look thermal at all, but this can be also due to the time window accessible with numerics. Another very interesting question is what happens at finite temperature and under what conditions a system can equilibrate. It should be relatively simple to generalize the results for all the ``quasi-free'' models already considered at $T=0$. Work in this direction is in progress, but it will be almost impossible to consider the same problem for more complicated models. Finally it is also important to understand the role played by the initial state. We assumed always a translational invariant one with short range correlations. Thus one natural modification consists in taking a state that is only locally different from the actual ground-state. This problem is known in the literature as a ``local quench'' and it has already been considered to some extent (see e.g. \cite{aop-03,ksz-05,ep-07}), but still a general picture as the one outlined here for global quenches does not exist (this is also interesting for eventual connections with quantum impurity problems, see \cite{s-98} as a review). Another natural modification of the initial state is one with long-range correlations, such as a critical one and let it evolve with another critical hamiltonian. This has been done for the Luttinger liquid \cite{c-06} and the results show the typical functional dependence of a light-cone scenario (i.e. everything depends only on $x\pm 2vt$) but the long-time correlations decays as power-laws, with exponents that are different from equilibrium ones and can be predicted by the generalized Gibbs ensemble. \section*{Acknowledgments} This work was supported in part by EPSRC grants GR/R83712/01 and EP/D050952/1. This work has been finished when PC was a guest of the Institute for Theoretical Physics of the Universiteit van Amsterdam. This stay was supported by the ESF Exchange Grant 1311 of the INSTANS activity.

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\section{Introduction}\label{Introduction} Realizations of internet of things (IoT) is a hot and developing topic for both industry and academia \cite{Chettri2020Compre, Palattella_IoT_2016, Akpakwu_Sruvey_2018}. With the huge benefits of IoT, a whole set of distinctive challenges is also exposed to the wireless physical layer design. These challenges include hyper-connectivity and low latency (the number of wirelessly connected IoT devices will increase exponentially fast and the number of devices served simultaneously is also large), high energy efficiency (EE) and spectral efficiency (SE) (the served devices are in low transmit power regime while having possible high mobility) as well as sporadic transmission. Significant work was conducted on physical layer optimization of IoT networks. In particular, in~\cite{Wang2019User} the authors proposed a hybrid nonorthogonal multiple access (NOMA) framework to support the tele-traffic demand of IoT, in~\cite{Yang2018Energy} and \cite{Ansere2020Optimal} the energy efficient resource allocation problems for IoT networks are studied, and in ~\cite{chu2017wireless}, the authors investigated a wirelessly powered sensor network and maximized the system sum throughput via energy beamforming and time allocation designs. For drastic improvements of IoT network physical layer performance, Massive multiple-input and multiple-output (mMIMO) technique is expected to be one of the best candidates to tackle the challenges faced by IoT systems due to its distinctive advantages such as high SE, high EE, and scalability~\cite{Ngo_2013_Energy, marzetta2016fundamentals}. Some works on cellular mMIMO supported IoT systems are reported and summarized in \cite{bana2019massive} and an analysis on wirelessly powered cell-free (CF) IoT is provided in~\cite{Wang2020Wirelessly} where a joint optimization of uplink (UL) and downlink (DL) power control is also introduced. In \cite{Liu2018Massive1} and \cite{Liu2018Massive2}, the authors provide a framework for user activity detection and channel estimation with a cellular base station (BS) equipped with a large number of antennas, and characterize the achievable uplink rate. The user activity detection is based on assigning to each user a unique pilot, which serves as the user identifier. These pilots then are used as columns of a sensing matrix. Active users synchronously send their pilots and a BS receives a linear combination of these pilots. Next, the BS runs a compressive sensing detection algorithm for the sensing matrix and identifies the pilots that occur as terms in the linear combination. These pilots, in their turn, reveal the active users. In this paper, we propose to support IoT systems with CF mMIMO~\cite{Ngo_17_Cellfree}. We adopt the pilot assignment method introduced in \cite{Liu2018Massive1} and \cite{Liu2018Massive2} in our work (i.e., each IoT device is assigned a unique pilot) and assume perfect detection of active devices. The contributions of this work are the following. For UL transmission, linear minimum mean square error (LMMSE) channel estimation and MIMO MMSE receiver are adopted in our design to adapt to IoT scenarios where devices usually have a small transmit power (e.g., 20 mW). Accurate and simple UL signal-to-noise-plus-interference ratio (SINR) approximations incorporating LMMSE channel estimation and MIMO MMSE receiver are derived based on random matrix (RM) theory. Efficient and flexible max-min power control algorithms are designed for both exact and RM SINRs, which can achieve both high SE and EE. To further increase the EE of IoT systems, target rate power control algorithms are designed for both exact and RM SINRs, where a predefined UL common per-device rate can be achieved by all served devices. Simulation results show that the designs proposed can obtain huge SE and EE improvements compared with sub-optimal designs and full power transmission schemes, respectively. For DL CF mMIMO IoT systems, a neural network (NN) approach is introduced to simplify DL max-min power control. By predicting the normalized transmit power for every access point (AP) under optimal max-min power control, DL max-min power control is converted from a high-complexity quasi-concave problem to a low-complexity convex optimization problem. With the aid of the NN prediction, we further develop a scalable power control algorithm that works for very large areas and has very low complexity. Simulation results show high prediction accuracy of the proposed NN approach and significant EE improvements by the scalable power control algorithm compared with full power transmission schemes. The organization of this paper is as follows. A CF mMIMO supported IoT system model incorporating the unique pilot assignment and LMMSE channel estimations is given in Section \ref{System Model}. Section \ref{Uplink Transmission} considers UL transmission under a linear optimal MIMO receiver and two RM SINR approximations are presented. UL Max-min and target rate power control algorithms are introduced in Section \ref{Uplink Power Control} and Section \ref{Uplink Simulation Results} shows the UL simulation results. DL transmission under LMMSE channel estimation and conjugate beamforming (CB) precoding is considered in Section \ref{Downlink Transmission}. Section \ref{Downlink Power Control} shows simplified DL max-min power control and scalable DL power control algorithms which are both aided by NN. DL simulation results are also provided in this Section. The conclusion of the paper is provided in Section \ref{Conclusion}. Notation: boldface upper- and lower-case letters denote matrices and column vectors, respectively. $(\cdot)^{T}$ $(\cdot)^{*}$, and $(\cdot)^{H}$ denote transpose, conjugate, and conjugate transpose operations, respectively. $z \sim \mathcal{CN}(0, \sigma^2)$ denotes a circularly symmetric complex Gaussian random variable with zero mean and variance $\sigma^2$. $\mathbf{I}_M$ denotes an $M \times M$ identity matrix, $\text{tr}(\cdot)$ denotes trace operator of a matrix, and $\|\cdot\|_2$ denotes the Euclidean norm operation. $\mathop{\mathrm{diag}}\{\mathbf{v}\}$ denotes generating a square diagonal matrix with the elements of vector $\mathbf{v}$ on the main diagonal and $\mathop{\mathrm{diag}}\{\mathbf{A}\}$ denotes generating a column vector of the main diagonal elements of $\mathbf{A}$. \section{System Model}\label{System Model} We consider an IoT system supported by CF mMIMO. $M$ APs are uniformly distributed in a wide serving area, they cooperate with each other to serve $\bar{K}$ IoT devices. We denote the number of active devices at one moment as $K$ and assume that the $K$ active devices are randomly chosen from all $\bar{K}$ IoT devices with $M >> K$. Note that $\bar{K}$ is the total number of IoT devices served in this area and is much larger than $K$ (i.e., $\bar{K} >> K$). In our system, orthogonal frequency-division multiplexing (OFDM) is used and we assume a flat-fading channel model for each OFDM subcarrier. For a given subcarrier, the {\em channel coefficient} $g_{mk}$ between the $m$-th AP and the $k$-th device is modeled as: \begin{equation} \label{eq:ch_coeffi} g_{mk} = \sqrt{\beta_{mk}}h_{mk}, \end{equation} where $\beta_{mk}, m = 1,..., M, k = 1,..., K$ are the {\em large-scale fading coefficients} which include path loss and shadow fading. $h_{mk}, m = 1,..., M, k = 1,..., K$ are {\em small-scale fading coefficients} with $i.i.d.\ \mathcal{CN}(0,1)$ and stay constant during the coherence interval of length $\tau_c$ measured in the number of OFDM symbols. As in \cite{Liu2018Massive1} and \cite{ Liu2018Massive2}, we assign a unique pilot $\boldsymbol{\psi}_k$ to each IoT device. During the pilot transmission, pilots are synchronously transmitted by active devices with the {\em pilot length} being $\tau$. Denote $\sqrt{\tau}\boldsymbol{\psi}_{k} \in \mathbb{C}^{\tau\times 1}$ as the pilot transmitted by the $k$-th device, where $||\boldsymbol{\psi}_{k}||_{2} = 1$, the received pilot signal at AP sides are given as: \begin{equation} \label{eq:re_pilot} \mathbf{Y} = [\mathbf{y}_{1}\,\mathbf{y}_{2}\, ...\, \mathbf{y}_{m}] = \sqrt{\tau\rho_{p}}\boldsymbol{\Psi}\mathbf{G}^{T} + \mathbf{W}, \end{equation} \begin{equation} \mathbf{y}_{m} = \sqrt{\tau\rho_{p}}\boldsymbol{\Psi}\mathbf{g}_{[m]} + \mathbf{w}_{m}, \end{equation} where $\rho_{p}$ is the normalized {\em pilot signal-to-noise ratio} (SNR) of each pilot symbol, $\boldsymbol{\Psi} = [\boldsymbol{\psi}_{1}\,\boldsymbol{\psi}_{2}\,...\,\boldsymbol{\psi}_{K}] \in \mathbb{C}^{\tau \times K}$ are {\em pilot matrix} for $K$ devices, $\mathbf{G}^{T} = [\mathbf{g}_{[1]}\,\mathbf{g}_{[2]}\,...\,\mathbf{g}_{[M]}] \in \mathbb{C}^{K\times M}$ are the {\em channel coefficient matrix}, here $\mathbf{g}_{[m]} = [g_{m1},g_{m2},...,g_{mK}]^{T} \in \mathbb{C}^{K\times 1}$. $\mathbf{W}\in \mathbb{C}^{\tau \times M}$ are noise matrix with $i.i.d.\ \mathcal{CN}(0,1)$ components. Optimal channel estimation (i.e., LMMSE) are applied at the AP sides and the estimated channel coefficients at the $m$-th AP have the form: \begin{equation} \begin{aligned} \label{eq:est_g} \hat{\bf g}_{[m]} = \sqrt{\tau\rho_{p}}\mathbf{B}_{m}\boldsymbol{\Psi}^{H}\left(\tau\rho_{p}\boldsymbol{\Psi}\mathbf{B}_{m}\boldsymbol{\Psi}^{H} + \mathbf{I}_{\tau}\right)^{-1}\mathbf{y}_{m},\\ \hat{g}_{mk} = \sqrt{\tau\rho_{p}}\beta_{mk}\boldsymbol{\psi}_{k}^{H}\left(\tau\rho_{p}\boldsymbol{\Psi}\mathbf{B}_{m}\boldsymbol{\Psi}^{H} + \mathbf{I}_{\tau}\right)^{-1}\mathbf{y}_{m}, \end{aligned} \end{equation} where $\mathbf{B}_{m} = \mathop{\mathrm{diag}}\{[\beta_{m1},\beta_{m2},...,\beta_{mK}]\}$. Then the variance of the estimated channel coefficient $\hat{g}_{mk}$ is equal to \begin{equation} \begin{aligned} \label{eq:gamma} \gamma_{mk} \triangleq \mathbb{E}\{|\hat{g}_{mk}|^{2}\} = \sqrt{\tau\rho_{p}}\beta_{mk}\boldsymbol{\psi}_{k}^{H}\mathbf{a}_{mk}, \end{aligned} \end{equation} where $\mathbf{a}_{mk} = \sqrt{\tau\rho_{p}}\beta_{mk}\left(\tau\rho_{p}\boldsymbol{\Psi}\mathbf{B}_{m}\boldsymbol{\Psi}^{H} + \mathbf{I}_{\tau}\right)^{-1}\boldsymbol{\psi}_{k}$. Let $\tilde{g}_{mk}={ g}_{mk}-\hat{g}_{mk}$ be the channel estimation error. Since the estimation error and the estimate are orthogonal under LMMSE estimation, the variance of $\tilde{g}_{mk}$ is given by \begin{equation} \label{eq:est_err_g} \mathbb{E}\{|\tilde{g}_{mk}|^{2}\} = \beta_{mk} - \gamma_{mk}. \end{equation} \section{Uplink Transmission}\label{Uplink Transmission} \subsection{Uplink Data Transmission}\label{sec:UL_Data_Trans} Define $\eta_{k}$ as the {\em UL power coefficient} for the $k$-th device. The signals received by the APs are \begin{equation} \label{eq:rece_sig_UL} \mathbf{y}_{u} = \sqrt{\rho_{u}}\sum_{k=1}^{K}\sqrt{\eta_{k}}\mathbf{g}_{k}s_{k} + \mathbf{w}_{u}, \end{equation} where $s_{k}$ is the {\em data symbol} transmitted by the $k$-th device, which satisfies $\mathbb{E}\{|s_k|^2\} = 1$, $\rho_{u}$ is the {\em normalized UL SNR}, $\mathbf{g}_{k} = [g_{1k}, g_{2k},..., g_{Mk}]^{T} \in \mathbb{C}^{M\times 1}$ is the channel vector between the $k$-th device and all APs, and $\mathbf{w}_{u} \in \mathbb{C}^{M\times 1}$ is the noise vector with $i.i.d.\ \mathcal{CN}(0,1)$ components. We assume that APs cooperate to estimate $s_k$ by using a linear MIMO receiver, $\mathbf{v}_k$, and computing: \begin{equation} \label{eq:decode_symbol} \begin{aligned} \hat{s}_{k} = \mathbf{v}_{k}^{H}\mathbf{y}_{u} = \mathbf{v}_{k}^{H}\bigg(&\sqrt{\rho_{u}\eta_{k}}\hat{\mathbf{g}}_{k}s_{k} + \sqrt{\rho_{u}}\sum_{k'\neq k}\sqrt{\eta_{k'}}\hat{\mathbf{g}}_{k'}s_{k'} +\\ &\sqrt{\rho_{u}}\sum_{k'=1}^{K}\sqrt{\eta_{k'}}\tilde{\mathbf{g}}_{k'}s_{k'} + \mathbf{w}_{u}\bigg). \end{aligned} \end{equation} Based on (\ref{eq:est_err_g}) and (\ref{eq:decode_symbol}), the UL SINR expression for the $k$-th data symbol is given as: \begin{equation} \label{eq:SINR_k_UL} \text{SINR}_{k}^{u}(\boldsymbol{\eta}) = \frac{\rho_{u}\eta_{k}\mathbf{v}_{k}^{H}\hat{\mathbf{g}}_{k}\hat{\mathbf{g}}_{k}^{H}\mathbf{v}_{k}}{\mathbf{v}_{k}^{H}\left(\rho_{u}\sum_{k'\neq k}\eta_{k'}\hat{\mathbf{g}}_{k'}\hat{\mathbf{g}}_{k'}^{H} + \mathbf{D}\right)\mathbf{v}_{k}}, \end{equation} where \begin{equation} \label{eq:def_D} \mathbf{D} = \rho_{u}\sum_{k'=1}^{K}\eta_{k'}(\mathbf{B}_{k'}-\boldsymbol{\Gamma}_{k'}) + \mathbf{I}_{M}, \end{equation} $\mathbf{B}_{k'} \triangleq \mathop{\mathrm{diag}}\{[\beta_{1k'}, \beta_{2k'},..., \beta_{Mk'}]\}$, and $\mathbf{\Gamma}_{k'} \triangleq \mathop{\mathrm{diag}}\{[\gamma_{1k'}, \gamma_{2k'},...,\gamma_{Mk'}]\}$. Based on (\ref{eq:decode_symbol}) and (\ref{eq:SINR_k_UL}), and using Rayleigh-Ritz theorem, we find the optimal (LMMSE) choice of ${\bf v}_k$: \begin{equation} \label{eq:decoder_MMSE} \mathbf{v}_{k}^{\text{MMSE}} = \sqrt{\rho_{u}\eta_{k}}\left(\rho_{u}\sum_{k'=1}^{K}\eta_{k'}\hat{\mathbf{g}}_{k'}\hat{\mathbf{g}}_{k'}^{H} + \mathbf{D}\right)^{-1}\hat{\mathbf{g}}_{k}. \end{equation} Substituting (\ref{eq:decoder_MMSE}) into (\ref{eq:SINR_k_UL}), we then obtain the corresponding SINR expression: \begin{equation} \begin{aligned} \label{eq:SINR_k_UL_MMSE} &\text{SINR}_{k}^{u, \text{MMSE}}(\boldsymbol{\eta}) = \\ &\frac{\rho_{u}\eta_{k}\hat{\mathbf{g}}_{k}^{H}\left(\rho_{u}\sum_{k'=1}^{K}\eta_{k'}\hat{\mathbf{g}}_{k'}\hat{\mathbf{g}}_{k'}^{H} + \mathbf{D}\right)^{-1}\hat{\mathbf{g}}_{k}}{1-\rho_{u}\eta_{k}\hat{\mathbf{g}}_{k}^{H}\left(\rho_{u}\sum_{k'=1}^{K}\eta_{k'}\hat{\mathbf{g}}_{k'}\hat{\mathbf{g}}_{k'}^{H} + \mathbf{D}\right)^{-1}\hat{\mathbf{g}}_{k}}. \end{aligned} \end{equation} \subsection{RM based SINR Approximations} In this section, two SINR approximations of (\ref{eq:SINR_k_UL_MMSE}) are derived based on random matrix (RM) theory \cite{Hoydis_13_UL/DL}, \cite{Wagner_12_MISO}. \subsubsection{RM Approximation 1} \begin{equation} \label{eq:RMT_Approx1} \text{SINR}_{k}^{u,\text{AP1}}(\boldsymbol{\eta}) = \frac{\rho_{u}\eta_{k}\text{tr}\left(\boldsymbol{\Gamma}_{k}\mathbf{T}\right)}{M}, \end{equation} where \begin{equation} \label{eq:T_Approx1} \mathbf{T} = \left(\frac{\rho_{u}}{M}\sum_{k'=1}^{K}\frac{\eta_{k'}\mathbf{\Gamma}_{k'}}{1 + e_{k'}} + \frac{\mathbf{D}}{M}\right)^{-1}, \end{equation} \begin{equation} \label{eq:e_k} e_{k'} = \lim_{t\rightarrow \infty} e_{k'}^{(t)}\ \text{with}\ e_{k'}^{(0)} = M, \;\;\forall k', \end{equation} \begin{equation} \label{eq:e_k_t} e_{k'}^{(t)} = \frac{\rho_{u}\eta_{k'}}{M}\text{tr}\left(\mathbf{\Gamma}_{k'}\Bigg(\frac{\rho_{u}}{M}\sum_{j=1}^{K}\frac{\eta_{j}\mathbf{\Gamma}_{j}}{1 + e_{j}^{(t-1)}} + \frac{\mathbf{D}}{M}\Bigg)^{-1}\right), \forall k'. \end{equation} \subsubsection{RM Approximation 2} \begin{equation} \label{eq:RMT_Approx2} \text{SINR}_{k}^{u,\text{AP2}}(\boldsymbol{\eta}) = \frac{\rho_{u}\eta_{k}\text{tr}\left(\mathbf{\Gamma}_{k}\mathbf{T}_{k}\right)}{M}, \end{equation} where \begin{equation} \label{eq:T_k} \mathbf{T}_{k} = \left(\frac{\rho_{u}}{M}\sum_{k'\neq k}^{K}\frac{\eta_{k'}\mathbf{\Gamma}_{k'}}{1 + e_{k,k'}} + \frac{\mathbf{D}}{M}\right)^{-1}, \end{equation} \begin{equation} e_{k,k'} = \lim_{t\rightarrow \infty}e_{k,k'}^{(t)}\ \text{with}\ e_{k,k'}^{(0)} = M, \;\; \forall k', \end{equation} \begin{equation} e_{k,k'}^{(t)}=\frac{\rho_{u}\eta_{k'}}{M}\text{tr}\left(\mathbf{\Gamma}_{k'}\Bigg(\frac{\rho_{u}}{M}\sum_{j\neq k}^{K}\frac{\eta_{j}\mathbf{\Gamma}_{j}}{1 + e_{k,j}^{(t-1)}} + \frac{\mathbf{D}}{M}\Bigg)^{-1}\right), \forall k'. \end{equation} The derivations of RM Approximation 1 and 2 are given in Appendix \ref{Proof for RM Approximations}. It is noted that both RM Approximation 1 and 2 involve only large-scale fading coefficients. It is also important to note that compared with the RM approximation derived in \cite{nayebi2016MMSE} for the case of reuse of orthogonal pilots, RM Approximation 1 and 2 have much simpler form, which allows for low complexity and infrequent power control algorithms presented in the next section. \section{Uplink Power Control}\label{Uplink Power Control} In this section, we consider UL power control with per-device power constraint. The UL power coefficients defined in Section \ref{sec:UL_Data_Trans} should satisfy the following power constraint: $0 \leq \mathbb{E}\{|x_k|^2\} \leq \rho_u, k = 1,2,...,K$ where $x_k = \sqrt{\rho_u}\eta_ks_k$ is the signal transmitted by the $k$-th device. Since $\mathbb{E}\{|s_k|^2\} = 1$, the UL power constraint can be rewritten as $0 \leq \eta_k \leq 1, \forall k$. \subsection{Max-min power control by exact SINR}\label{sec:max_min_exact_SINR} Based on the power constraint given above, basic UL max-min power control with random pilots under an IoT system can be formulated as the problem shown below: \begin{equation} \label{eq:max_min_prbm_UL} \begin{aligned} &\max_{\boldsymbol{\eta}}\min_{k=1,...,K}\text{SINR}^{u}_{k}\\ &\text{s.t.}\; 0 \leq \eta_{k} \leq 1,\, k = 1,..., K, \end{aligned} \end{equation} where $\text{SINR}^u_k$ in (\ref{eq:max_min_prbm_UL}) can be replaced by both $\text{SINR}^{u,\text{MMSE}}_k$ given in (\ref{eq:SINR_k_UL_MMSE}) and the RM SINR Approximations. In this section, we introduce an iterative weighted max-min power control algorithm designed based on exact SINR given in (\ref{eq:SINR_k_UL_MMSE}). In the algorithm, a {\em rate weighting} vector $\mathbf{u} = [u_{1}, u_{2},..., u_{K}]^{T} \in \mathbb{R}_{0^{+}}^{K}$ constrained by $\|\mathbf{u}\|_{2} = 1$ is incorporated. Vector $\mathbf{u}$ can be used to drop some devices under poor channel condition by assigning very small weights to these devices. If all the devices are required to achieve the same rate, $u_{k} = 1/\sqrt{K},\ \forall k$. On the other hand, a {\em power weighting} vector $\boldsymbol{\nu} = [\nu_{1}, \nu_{2}, ..., \nu_{K}]^{T} \in \mathbb{R}_{0^{+}}^{K}$ is also included and the weighted normalized maximum transmit power of the $k$-th device is defined as $\rho'_{u,k} \triangleq \rho_{u}\nu_{k}$. From (\ref{eq:SINR_k_UL_MMSE}), we observed that the task of achieving a uniform data rate for all served devices was equivalent for the term, $\rho_{u}\eta_{k}\hat{\mathbf{g}}_{k}^{H}\left(\rho_{u}\sum_{k'=1}^{K}\eta_{k'}\hat{\mathbf{g}}_{k'}\hat{\mathbf{g}}_{k'}^{H} + \mathbf{D}\right)^{-1}\hat{\mathbf{g}}_{k}$, in (\ref{eq:SINR_k_UL_MMSE}) having the same value for every served device. It is also observed that the term, $\rho_{u}\sum_{k'=1}^{K}\eta_{k'}\hat{\mathbf{g}}_{k'}\hat{\mathbf{g}}_{k'}^{H} + \mathbf{D}$, is included in the SINR expression for each active device (i.e., independent with the device index). With these two observations, an iterative max-min power control algorithm based on exact SINR can be designed. Here, we define matrix $\mathbf{J}_k,\ k = 1, 2,..., K$ as \begin{equation} \label{eq:J_matrix} \mathbf{J}_{k} \triangleq \hat{\mathbf{g}}_{k}\hat{\mathbf{g}}_{k}^{H} + \mathbf{B}_{k} - \boldsymbol{\Gamma}_{k}. \end{equation} The details of the algorithm incorporating vectors $\mathbf{u}$ and $\boldsymbol{\nu}$ are given below in {\bf Algorithm} \ref{Algorithm 1} where (\ref{eq:eta_1}) guarantees that the power constraints are satisfied. \begin{algorithm} \caption{Max-min Power Control - Exact SINR} \label{Algorithm 1} \begin{algorithmic}[1] \item Initialize vectors $\mathbf{u}$ and $\boldsymbol{\nu}$ with predefined setting. Initialize $\eta_{k}^{(0)} = 1, \forall k$, and $d_{k}^{(0)}, \forall k$ as \begin{equation} \label{eq:d_k_0} d_{k}^{(0)} = \rho'_{u,k}\hat{\mathbf{g}}_{k}^{H}\left(\sum_{k'=1}^{K}\rho'_{u,k'}\eta_{k'}^{(0)}\mathbf{J}_{k'} + \mathbf{I}_{M}\right)^{-1}\hat{\mathbf{g}}_{k}. \end{equation} Set $n = 0$ and choose a tolerance $\epsilon > 0$. \item Compute $\alpha = \min_{k'}(d_{k'}^{(n)}/u_{k'}), k' = 1,..., K$ and Update $d_{k}^{(n+1)}, \forall k$ as below \begin{equation} \begin{aligned} \label{eq:d_updation} d_{k}^{(n+1)} = \rho'_{u,k}\hat{\mathbf{g}}_{k}^{H}\left(\alpha\sum_{k'=1}^{K}\frac{\rho'_{u,k'}u_{k'}}{d_{k'}^{(n)}}\mathbf{J}_{k'} + \mathbf{I}_{M}\right)^{-1}\hat{\mathbf{g}}_{k}. \end{aligned} \end{equation} \item Stop if $\max_{k}|d_{k}^{(n+1)} - d_{k}^{(n)}|\leq \epsilon, k = 1,..., K$, set $d_{k} = d_{k}^{(n+1)}, \forall k$, and the power control coefficients, $\eta_{k}, \forall k$ are given by \begin{equation} \label{eq:eta_1} \eta_{k} = \frac{\min_{k'}\left(d_{k'}/u_{k'}\right)}{d_{k}/u_{k}}, \, k' = 1,...,K. \end{equation} Otherwise, set $n = n + 1$ and go to step 2. \end{algorithmic} \end{algorithm} \begin{theorem}\label{converg_algm1} Given $\{\hat{g}_{mk}\}$, $\{\beta_{mk}\}$, $\{\gamma_{mk}\}$, $\rho_u$, $\mathbf{u}$, and $\boldsymbol{\nu}$, if a max-min power control is feasible, the iteration in Step 2 of Algorithm {\ref{Algorithm 1}} converges to a unique solution. The $\{\eta_{k}\}$ given by step 3 of Algorithm \ref{Algorithm 1} realize this max-min power control. \end{theorem} \begin{proof} The proof of Theorem \ref{converg_algm1} is given in Appendix \ref{Convergence of Algorithm 1 and 3}. \end{proof} Note that each element of vector $\mathbf{u}$ can have different values. As mentioned earlier, when we would like to drop some devices under poor channel condition, vector $\mathbf{u}$ can be designed as follows. {\bf Algorithm} \ref{Algorithm 1} will be firstly run with $u_k=1/\sqrt{K},\ \forall k$, to obtain $\eta_{k}, \forall k$. We find $K_p$ devices with the largest power coefficients. These devices consume most of the energy and drag the rates of other devices down. Thus it looks natural to lower their rates by assigning their $u_k$ to some small value $u_p$. The value of $u_{k}$ for the remaining devices is set as $u_{g} > u_{p}$. The relationship between $u_{g}$ and $u_{p}$ is given by \begin{equation} \label{eq:relation_up_ug} u_{p}^{2}K_{p} + u_{g}^{2}\left(K- K_{p}\right) = 1. \end{equation} If we would like to virtually drop $K_p$ devices, we set $u_p$ as a very small number (e.g. $10^{-8}$). \subsection{Max-min power control by RM Approximation 1} For RM Approximation 1 given in (\ref{eq:RMT_Approx1}) - (\ref{eq:e_k_t}), matrix $\mathbf{T}$ is included in the SINR expression for every active device and is independent with the device index, thus a max-min power control algorithm based on RM Approximation 1 can be designed. Its detailed form incorporating the rate and power weighting vectors is given in {\bf Algorithm} \ref{Algorithm 2} below. \begin{algorithm} \caption{Max-min Power Control - RM SINR} \label{Algorithm 2} \begin{algorithmic}[1] \item Initialize $\mathbf{u}$, $\boldsymbol{\nu}$ as predetermined, $\eta_{k}^{(0)} = 1, \forall k$, $\mathbf{D}^{(0)} = \sum_{k'=1}^{K}\rho'_{u,k'}\eta_{k'}^{(0)}(\mathbf{B}_{k'}-\boldsymbol{\Gamma}_{k'}) + \mathbf{I}_{M}$. Initialize $\mathbf{T}^{(0)} = \left(\frac{1}{M}\sum_{k'=1}^{K}\frac{\eta_{k'}^{(0)}\rho'_{u,k'}\mathbf{\Gamma}_{k'}}{1+e_{k'}} + \frac{\mathbf{D}^{(0)}}{M}\right)^{-1}$ where $e_{k'}, \forall k'$ are computed by (\ref{eq:e_k}) and (\ref{eq:e_k_v_t}) below \begin{equation} \label{eq:e_k_v_t} e_{k'}^{(t)} = \frac{\rho'_{u,k'}\eta_{k'}^{(0)}}{M}\text{tr}\Bigg(\mathbf{\Gamma}_{k'}\Bigg(\frac{1}{M}\sum_{j=1}^{K}\frac{\eta_{j}^{(0)}\rho'_{u,j}\mathbf{\Gamma}_{j}}{1 + e_{j}^{(t-1)}} + \frac{\mathbf{D}^{(0)}}{M}\Bigg)^{-1}\Bigg) \end{equation} Set $n = 0$ and choose a tolerance $\epsilon > 0$. \item Compute $\alpha = \min_{k'}\text{tr}\left(\nu_{k'}\mathbf{\Gamma}_{k'}\mathbf{T}^{(n)}\right)/u_{k'},\ k' = 1, ..., K$, Update $\mathbf{T}^{(n+1)}$ as \begin{equation} \begin{aligned} \label{eq:T_update} &\mathbf{T}^{(n+1)} = \\ &\left(\frac{\alpha}{M}\sum_{k'=1}^{K}\frac{\rho_{u}u_{k'}}{\text{tr}\left(\mathbf{\Gamma}_{k'}\mathbf{T}^{(n)}\right)}\left(\mathbf{B}_{k'} - \frac{\xi_{k'}\mathbf{\Gamma}_{k'}}{1 + \xi_{k'}}\right) + \frac{\mathbf{I}_{M}}{M}\right)^{-1}, \end{aligned} \end{equation} where $\xi_{k'} = \frac{\rho_{u}\alpha u_{k'}}{M}$. \item Stop if $\|\mathbf{T}^{(n+1)} - \mathbf{T}^{(n)}\|_{2} \leq \epsilon$. Set $\mathbf{T} = \mathbf{T}^{(n+1)}$ and the power control coefficients, $\eta_{k}, \forall k$ are given by \begin{equation} \label{eq:eta_al_AP1} \eta_{k} = \frac{\min_{k'}\left(\text{tr}\left(\nu_{k'}\mathbf{\Gamma}_{k'}\mathbf{T}\right)/u_{k'}\right)}{\text{tr}\left(\nu_{k}\mathbf{\Gamma}_{k}\mathbf{T}\right)/u_{k}},\ k' = 1,..., K. \end{equation} Otherwise, set $n = n + 1$ and go to step 2. \end{algorithmic} \end{algorithm} \begin{theorem}\label{theorem: converg_algm2} Given $\{\beta_{mk}\}$, $\{\gamma_{mk}\}$, $\rho_u$, $\mathbf{u}$, and $\boldsymbol{\nu}$, if a max-min power control is feasible, the iteration in Step 2 of Algorithm \ref{Algorithm 2} converges to a unique solution. The $\{\eta_{k}\}$ given by step 3 of Algorithm \ref{Algorithm 2} realize this max-min power control. \end{theorem} \begin{proof} The proof of Theorem \ref{theorem: converg_algm2} is given in Appendix \ref{Convergence of Algorithgm 2 and 4}. \end{proof} \subsection{Power Control with Target Rate by Exact SINR}\label{sec:target_exact_SINR} Under IoT systems, the IoT devices will use energy harvesting and/or infrequently replaced batteries. Thus, high EE of the designed system is highly desirable to support IoT systems. Here we define the {\em UL EE} of a system as \begin{equation} \label{eq:EE} E_{u} \triangleq \frac{\sum_{k = 1}^{K}R_{k}^{u}}{P_{u}\sum_{k=1}^{K}\eta_{k}}, \end{equation} where $R_{k}^{u}$ is the UL rate for the $k$-th device, $P_{u}$ is the UL maximum transmit power per data symbol. Since in some IoT application scenarios, the data rate requirement for each device is not relatively high, a target rate power control which can achieve a predetermined target rate for every served device while keeping high EE is desired. A basic UL target rate power control problem with random pilots under an IoT system can be formulated as shown below: \begin{equation} \label{eq:target_probm} \begin{aligned} &\text{Find} \;\eta_k, k = 1,2,...,K\\ &\text{s.t.}\; \text{SINR}_k^u = S_t,\, k = 1, 2, ..., K,\\ & \quad\;\; 0 \leq \eta_k \leq 1,\; k = 1,2, ..., K, \end{aligned} \end{equation} where $S_t$ is the predetermined target SINR value and $\text{SINR}_k^u$ can also be replaced by $\text{SINR}_k^{u, \text{MMSE}}$ or RM SINR Approximations. Note that max-min power control can be regarded as a kind of target rate power control where the target rate is the maximum uniform data rate achieved by all served devices. Thus, the design approach for the max-min power control algorithm mentioned in Section \ref{sec:max_min_exact_SINR} can be applied for the target rate power control. The difference is that for max-min power control, $\alpha$ given in step 2 of Algorithm \ref{Algorithm 1} needs to be updated in iteration so that $\alpha$ will go towards the target value where all served devices can achieve the maximum uniform data rate, while in target rate power control, $\alpha$ is a fixed value dependent on $S_t$. In this section, we introduce a target rate power control algorithm incorporating rate and power weighting vectors. Before going to the details of the algorithm, we first compute the per-device rate under full power condition using (\ref{eq:SINR_k_UL_MMSE}) for all devices. We regard the devices as {\em poor devices} if their per-device rates under full power are smaller than the target rate and regard the remaining devices as {\em good devices}. We set $u_k = u_p$ and $u_k = u_g$ for poor and good devices, respectively. The algorithm details are then given in {\bf Algorithm} \ref{Algorithm 3}. \begin{algorithm} \caption{Target Rate Power Control - Exact SINR} \label{Algorithm 3} \begin{algorithmic}[1] \item Initialize $\boldsymbol{\nu}$, $\eta_{k}^{(0)}, \forall k$, and $d_{k}^{(0)}, \forall k$ as in step 1 of Algorithm \ref{Algorithm 1}. With a target SINR, $S_{t}$, compute $\alpha = S_{t}/(1 + S_{t})$. Set $n = 0$ and choose a tolerance $\epsilon > 0$. \item Update $d_{k}^{(n+1)}, \forall k$ using (\ref{eq:d_updation}) where $\rho'_{u,k'}u_{k'}, \forall k'$ in (\ref{eq:d_updation}) are substituted by $\rho'_{u,k'}, \forall k'$. \item Stop if $\max_{k}|d_{k}^{(n+1)}-d_{k}^{(n)}| \leq \epsilon, k = 1,..., K$ and set $d_{k} = d_{k}^{(n+1)}, \forall k$. Otherwise, set $n = n + 1$ and go to step 2. \item The power control coefficients are computed as $\eta_{k} = \alpha/d_{k}, \forall k$. If the constraints $0 \leq \eta_{k} \leq 1, \forall k$ are satisfied, then Algorithm \ref{Algorithm 3} ends. Otherwise, initialize $\eta_{k}^{(0)}$, and $d_{k}^{(0)}, \forall k$ as in step 1 of Algorithm \ref{Algorithm 1}. Assign the value of each element of vector $\mathbf{u}$ by $u_{g}$ or $u_{p}$ according to the per-device rate under full power case. Set $n = 0$, $\alpha = \alpha/u_{g}$, and go to step 5. \item Update $d_{k}^{(n+1)}, \forall k$ using (\ref{eq:d_updation}). \item Stop if $\max_{k}|d_{k}^{(n+1)} - d_{k}^{(n)}| \leq \epsilon, k = 1,..., K$, set $d_{k} = d_{k}^{(n+1)}, \forall k$, and compute power control coefficients as $\eta_{k} = \alpha u_{k}/d_{k}, \forall k$. Otherwise, set $n = n + 1$ and go to step 5. \end{algorithmic} \end{algorithm} \begin{theorem}\label{theorem:converg_algm3} Given $\{\hat{g}_{mk}\}$, $\{\beta_{mk}\}$, $\{\gamma_{mk}\}$, $\rho_u$, $\mathbf{u}$, and $\boldsymbol{\nu}$, if a target-rate power control with target SINR, $S_t$, is feasible, the iterations in step 2 and step 5 of Algorithm \ref{Algorithm 3} converge separately to their unique solutions. The $\{\eta_{k}\}$ given by step 6 of Algorithm \ref{Algorithm 3} achieves this target-rate power control. \end{theorem} \begin{proof} The proof of Theorem \ref{theorem:converg_algm3} is given in Appendix \ref{Convergence of Algorithm 1 and 3}. \end{proof} \subsection{Power Control with Target Rate by RM Approximation 1} In this section we consider the same settings as in Section \ref{sec:target_exact_SINR}, but use RM Approximation 1. The detailed form is given in {\bf Algorithm} \ref{Algorithm 4}. \begin{algorithm} \caption{Target Rate Power Control - RM SINR} \label{Algorithm 4} \begin{algorithmic}[1] \item Initialize $\boldsymbol{\nu}$, $\eta_{k}^{(0)}, \forall k$, $\mathbf{D}^{(0)}$ and $\mathbf{T}^{(0)}$ as in step 1 of Algorithm \ref{Algorithm 2}. With a target SINR denoted as $S_{t}$, compute $\alpha = S_{t}M/\rho_{u}$. Set $n = 0$ and choose a tolerance $\epsilon > 0$. \item Update $\mathbf{T}^{(n+1)}$ using (\ref{eq:T_update}) where $\rho_{u}u_{k'}, \forall k'$ are substituted by $\rho_{u}$ and $\xi_{k'} = S_t,\ \forall k'$. \item Stop if $\|\mathbf{T}^{(n+1)} - \mathbf{T}^{(n)}\|_2 < \epsilon$ and set $\mathbf{T} = \mathbf{T}^{(n+1)}$. Otherwise, set $n = n + 1$ and go to step 2. \item The power control coefficients are computed as $\eta_{k} = \alpha/\text{tr}\left(\nu_{k}\mathbf{\Gamma}_{k}\mathbf{T}\right), \forall k$. If the constraints $0 \leq \eta_{k} \leq 1, \forall k$ are satisfied, Algorithm \ref{Algorithm 4} ends. Otherwise, Initialize $\eta_{k}^{(0)}, \forall k$, $\mathbf{D}^{(0)}$ and $\mathbf{T}^{(0)}$ as in step 1 of Algorithm \ref{Algorithm 2}. Assign the value of each element of vector $\mathbf{u}$ by $u_{g}$ or $u_{p}$ according to the per-device rate under full power case. Set $n = 0, \alpha=\alpha/u_{g}$, and go to step 5. \item Update $\mathbf{T}^{(n+1)}$ using (\ref{eq:T_update}) where $\xi_{k'} = \frac{S_{t}u_{k'}}{u_g}$. \item Stop if $\|\mathbf{T}^{(n+1)} - \mathbf{T}^{n}\|_2 \leq \epsilon$, set $\mathbf{T} = \mathbf{T}^{(n+1)}$, and compute power control coefficients by $\eta_{k} = \alpha u_{k}/\text{tr}\left(\nu_{k}\mathbf{\Gamma}_{k}\mathbf{T}\right),\ \forall k$. Otherwise, set $n = n + 1$ and go to step 5. \end{algorithmic} \end{algorithm} \begin{theorem}\label{theorem:converg_algm4} Given $\{\beta_{mk}\}$, $\{\gamma_{mk}\}$, $\rho_u$, $\mathbf{u}$, and $\boldsymbol{\nu}$, if a target-rate power control with target SINR, $S_t$, is feasible, the iterations given by step 2 and step 5 of Algorithm \ref{Algorithm 4} converge separately to their unique solutions. The $\{\eta_{k}\}$ given by step 6 of Algorithm \ref{Algorithm 4} achieves this target-rate power control. \end{theorem} \begin{proof} The proof of Theorem \ref{theorem:converg_algm4} is given in Appendix \ref{Convergence of Algorithgm 2 and 4}. \end{proof} \subsection{Algorithm Complexity Comparison} It is noted that in Algorithms \ref{Algorithm 1} and \ref{Algorithm 3} the computation of (\ref{eq:d_updation}) involves an inverse operation of an $M \times M$ non-sparse matrix whose complexity is $\sim\mathcal{O}(M^{3})$. On the other hand, matrix $\mathbf{T}$ is a diagonal matrix, so the computation complexity of (\ref{eq:T_update}) in Algorithms \ref{Algorithm 2} and \ref{Algorithm 4} is $\sim \mathcal{O}(MK)$. In addition, exact SINR given in (\ref{eq:SINR_k_UL_MMSE}) involves both small-scale and large-scale fading coefficients, frequent updates of the power control coefficients are required. On the contrary, RM Approximation 1 involves only large-scale fading coefficients, power control coefficients can be updated in a much slower rate. \section{Uplink Simulation Results}\label{Uplink Simulation Results} \subsection{Setup and Parameters for Numerical Simulations}\label{Setup and Parameters for Numerical Simulations} We consider networks where $M$ APs and $\bar{K}$ IoT devices are uniformly distributed in a $D \times D \ m^{2}$ square area and $K$ out of $\bar{K}$ devices are active at one moment. The serving area is wrapped around to avoid boundary effects. The large-scale fading coefficients $\beta_{mk}, \forall m, \forall k$ are products of $\text{PL}_{mk}$ and $\text{SF}_{mk}$: \begin{equation} \label{eq:large_fading} \beta_{mk} = \text{PL}_{mk}\text{SF}_{mk},\ \text{with}\ \text{SF}_{mk} = 10^{\frac{\sigma_{\text{sh}}z_{mk}}{10}}, \end{equation} where $z_{mk} \sim \mathcal{N}(0,1)$. The path loss is generated as in \cite{Ngo_17_Cellfree} where a three-slope model \cite{Tang_01_Pathloss} and the Hata-Cost 231 propagation model \cite{3GPP_ETSI} are used. Shadow fading coefficients are generated based on \cite{Wang08_Joint_Shadow}. The detailed simulation setup and parameters are given in Table \ref{tbl:1}. \begin{table} \renewcommand{\arraystretch}{1.4} \begin{center} \caption{Simulation Setup and Parameters.}~\label{tbl:1} \fontsize{8}{8}\selectfont \begin{tabular}{|>{\centering\arraybackslash}m{6cm}|>{\centering\arraybackslash}m{1.5cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline \textbf{Parameter} & \textbf{Value} \\ \hline $f_c$ (Carrier frequency) & 1.9 GHz \\ \hline BW (Bandwidth) & 20 MHz \\ \hline Noise figure & 9 dB \\ \hline $\tau_c$ & 200 \\ \hline $P_{u}$ (UL maximum transmit power per data symbol) & 20 mW \\ \hline $P_{p}$ (UL maximum transmit power per pilot symbol) & 20 mW \\ \hline $P_{d}$ (DL maximum transmit power per AP) & 200 mW \\ \hline \end{tabular} \end{center} \end{table} As performance measures, exact achievable rate for the $k$-th device, $R_{k}^{u, \text{MMSE}}$, and its corresponding throughput, $U_{k}^{u, \text{MMSE}}$, are given by \begin{equation} \label{eq:exact_rate} R_{k}^{u,\text{MMSE}} = \mathbb{E}\left[\log_{2}\left(1 + \text{SINR}_{k}^{u,\text{MMSE}}\right)\right], \end{equation} \begin{equation}\label{eq:throughput} U_{k}^{u, \text{MMSE}} = \text{BW}\frac{\tau_c - \tau}{2\tau_c}R_{k}^{u, \text{MMSE}}, \end{equation} where the expectation in (\ref{eq:exact_rate}) is over small-scale fading. On the other hand, the approximated achievable rate of the $k$-th device by RM Approximation 1 is given by $R_{k}^{u,\text{AP1}} = \log_{2}\left(1 + \text{SINR}_{k}^{u,\text{AP1}}\right)$. Throughout all our simulations we assume that pilots $\boldsymbol{\psi}_k$ are generated random $\tau$-tuples with uniform distribution over the surface of a complex unit sphere. \subsection{Results and Discussions} The approximation accuracy of RM Approximation 1 and 2 under full power case in terms of per-device rate is given in Fig. \ref{fig:AP_Exact_Comparison}. \begin{figure} \begin{center} \includegraphics [width=0.48\textwidth]{AP_Exact_Comparison_2.png} \caption{Per-device rate comparison between exact SINR, RM Approximation 1, and RM Approximation 2. Here, $M = 1024$, $K = 256$, $\tau = 256$, and area = 1 $\text{km}^{2}$.}\label{fig:AP_Exact_Comparison} \end{center} \end{figure} For both correlated and i.i.d. shadow fading, the per-device rates obtained using RM Approximation 1 and 2 are quite close to those obtained by exact SINR. This observation verifies Theorem \ref{Theorem 1} in Appendix \ref{Theorems}. The per-device rate performance comparison under max-min power control based on both exact SINR and RM Approximation 1 is given in Fig. \ref{fig:max_min_power_control}. \begin{figure} \begin{center} \includegraphics [width=0.5\textwidth]{Max_min_Power_Comparison_M160_K40_tau40_D1000_P20_N100_inde_corr.png} \caption{Per-device rate comparison for max-min power control algorithms based on exact SINR and RM Approximation 1. Here, $M = 160$, $K = 40$, $\tau = 40$, and area = 1 $\text{km}^{2}$.}\label{fig:max_min_power_control} \end{center} \end{figure} Note that the RM AP 1 curves are obtained as follows. The max-min power control coefficients obtained by Algorithm \ref{Algorithm 2} are substituted into (\ref{eq:exact_rate}) to compute the per-device rates of RM AP 1. Under such a computation, we observe that the per-device rates achieved based on RM AP1 are equivalent to or even better than the performance obtained based on exact SINR. One explanation for this phenomenon is that in order to obtain a uniform service to every device, this uniform rate achieved by max-min power control is limited. Under some realizations of small-scale fading, this rate can be small enough to reduce the expected per-device rate calculated by (\ref{eq:exact_rate}). On the other hand, the power control coefficients used to obtain the curves of RM AP 1 are based on large-scale fading, and as shown in Fig. \ref{fig:max_min_power_control}, better performance can be achieved. Fig \ref{fig:Opt_subopt_max_min_comparison} shows the per-device throughput comparison of CF IoT systems with different settings. We consider the cases of i.i.d. and correlated fading cases. \begin{figure} \begin{center} \includegraphics [width=0.51\textwidth]{Opt_subopt_max_min_comparison_M128_K40_tau60_D100_P20_N300.png} \caption{Performance comparison between optimal and sub-optimal CF IoT systems. Here, $M = 128$, $K = 40$, $\tau = 60$, and area = 0.01 $\text{km}^{2}$.}\label{fig:Opt_subopt_max_min_comparison} \end{center} \end{figure} In Fig. \ref{fig:Opt_subopt_max_min_comparison}, `Subopt' denotes sub-optimal channel estimation applied in \cite{Ngo_17_Cellfree} and MR is short for maximum-ratio MIMO receiver. It is observed that around 7 times performance improvement is achieved by our system with optimal channel estimation and MMSE MIMO receiver compared with systems with sub-optimal channel estimation and/or MR MIMO receiver. We see, however, that power control does not lead to a significant increase in data rates for both of these cases. However, as it is shown below, power control does lead to a large gain in terms of EE, which is crucial for IoT systems. The energy efficiencies of the full power transmission case, Algorithm \ref{Algorithm 3} and Algorithm \ref{Algorithm 4} with different target rates are shown in Fig. \ref{fig:EE_Comparison}. \begin{figure} \begin{center} \includegraphics [width=0.52\textwidth]{EE_Comparison_inde_M160_K40_tau40_D1000_Pu20.png} \caption{Energy efficiency comparison among full power case and different target per-device rates under i.i.d. shadow fading. Here, $M = 160$, $K = 40$, $\tau = 40$, and area = 1 $\text{km}^{2}$.} \label{fig:EE_Comparison} \end{center} \end{figure} We see that the power control gives large gain over the full power transmission. In particular, for target rates of 0.01 and 0.1 bits/s/Hz we obtain 17-fold and 9-fold improvements, respectively. It is also observed that higher EE is obtained by Algorithm \ref{Algorithm 4} (i.e., based on RM Approximation 1) than Algorithm \ref{Algorithm 3} (i.e., based on exact SINR). \section{Downlink Transmission}\label{Downlink Transmission} In this section, we consider DL transmission under optimal channel estimation and CB precoding for CF mMIMO IoT systems. \subsection{Downlink Data Transmission} Define $\eta_{mk}, m = 1,2,...,M, k = 1,2,..., K$ as the {\em DL power coefficient} for the data symbol transmitted by the $m$-th AP for the $k$-th device. The transmitted signal from the $m$-th AP is given by \begin{equation} \label{eq:x_m_trans} x_m = \sqrt{\rho_{d}}\sum_{k=1}^{K}\sqrt{\eta_{mk}}\hat{g}_{mk}^*s_k. \end{equation} where $s_k$ is the symbol intended for the $k$-th active device and satisfies $\mathbb{E}\{|s_k|^2\} = 1$. Then the received signal at the $k$-th device under CB precoding is given by \begin{equation} \label{eq:y_d_k} \begin{aligned} y_{k}^{d} = &\sqrt{\rho_{d}}\sum_{m=1}^{M}\sqrt{\eta_{mk}}\mathbb{E}[\hat{g}_{mk}^{*}g_{mk}]s_{k} \\ &+\sqrt{\rho_{d}}\sum_{m=1}^{M}\sqrt{\eta_{mk}}\left(\hat{g}_{mk}^{*}g_{mk} - \mathbb{E}[\hat{g}_{mk}^{*}g_{mk}]\right)s_{k} \\ & + \sqrt{\rho_{d}}\sum_{k'\neq k}\sum_{m=1}^{M}\sqrt{\eta_{mk'}}\hat{g}_{mk'}^{*}g_{mk}s_{k'} + w_{k}^{d}. \end{aligned}. \end{equation} Based on (\ref{eq:y_d_k}), a closed-form expression for the DL SINR is derived by \cite{Rao_Internet_2019, Rao_cellfree_2018} using the technique in \cite{marzetta2016fundamentals}. This closed-form SINR expression with optimal channel estimation and CB precoding is given in (\ref{eq:SINR_d_k_radom}). \begin{figure*} \begin{equation} \label{eq:SINR_d_k_radom} \begin{aligned} \text{SINR}_{k}^{\text{IoT}} = \frac{\rho_{d}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}}\gamma_{mk}\right)^{2}}{\splitfrac{1 + \rho_{d}\sum_{m=1}^{M}\eta_{mk}\gamma_{mk}\beta_{mk} + \rho_{d}\sum_{k'\neq k}\Bigg(\sum_{m=1}^{M}\eta_{mk'}\beta_{mk}||\mathbf{a}_{mk'}||_{2}^{2} + } {\tau\rho_{p}\left(\left|\sum_{m=1}^{M}\sqrt{\eta_{mk'}}\beta_{mk}\boldsymbol{\psi}_{k}^{H}\mathbf{a}_{mk'}\right|^{2} + \sum_{m=1}^{M}\eta_{mk'}\sum_{j=1}^{K}\beta_{mk}\beta_{mj}\left|\boldsymbol{\psi}_{j}^{H}\mathbf{a}_{mk'}\right|^{2}\right)\Bigg)}} \end{aligned}. \end{equation} \hrulefill \end{figure*} When the devices are assigned orthonormal pilots, it can be verified that $\text{SINR}_{k}^{\text{IoT}}$ is converted to $\text{SINR}_{k}^{\text{orth}}$, \begin{equation} \label{eq:SINR_d_k_orth} \text{SINR}_{k}^{\text{orth}} = \frac{\rho_{d}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}\gamma_{mk}}\right)^{2}}{1 + \rho_{d}\sum_{k'=1}^{K}\sum_{m=1}^{M}\eta_{mk'}\gamma_{mk'}\beta_{mk}}, \end{equation} which coincides with the SINR derived in \cite{Ngo_17_Cellfree}. \section{Downlink Power Control}\label{Downlink Power Control} In the DL transmission, we consider the following per-AP power constraint: \begin{equation} \label{eq:pow_constr_DL} \mathbb{E}\{|x_m|^2\} \leq \rho_{d},\, m = 1,2,...,M. \end{equation} With the model defined in (\ref{eq:ch_coeffi}), (\ref{eq:pow_constr_DL}) can be rewritten as \begin{equation} \label{eq:pow_constr_DL_2} \sum_{k=1}^{K}\eta_{mk}\gamma_{mk} \leq 1, m = 1,2,...,M. \end{equation} \subsection{Optimal Power Control} According to the power constraint given in (\ref{eq:pow_constr_DL_2}), DL max-min power control based on random pilots under an IoT system can be formulated as the problem shown below: \begin{equation} \begin{aligned} \label{eq:max_min_dl_IoT} &\max_{\boldsymbol{\eta}}\min_{k = 1,..., K}\text{SINR}_{k}^{\text{IoT}}\\ &\text{s.t.} \sum_{k' = 1}^{K}\eta_{mk'}\gamma_{mk'} \leq 1,\ m = 1,2,..., M\\ &\;\;\;\;\;\eta_{mk'} \geq 0,\ m = 1,2,..., M, k' = 1,2,..., K. \end{aligned} \end{equation} The optimization problem (\ref{eq:max_min_dl_IoT}) is quasi-concave \cite{Ngo_17_Cellfree}. It can be solved by performing a bisection search and solving a convex feasibility problem in each step \cite{boyd2004convex}. However, as the number of APs and IoT devices increases, the bisection search becomes too complex, and a more simple power control algorithm is required. For orthonormal pilots, the max-min power control is also a quasi-concave problem and is shown below: \begin{equation} \begin{aligned} \label{eq:max_min_dl_orth} &\max_{\boldsymbol{\eta}}\min_{k}\text{SINR}_{k}^{\text{orth}}(\boldsymbol{\eta}) = \frac{\rho_{d}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}}\gamma_{mk}\right)^{2}}{1 + \rho_{d}\sum_{m = 1}^{M}\beta_{mk}\sum_{k'=1}^{K}\eta_{mk'}\gamma_{mk'}}\\ &\qquad\qquad\text{s.t.} \sum_{k' = 1}^{K}\eta_{mk'}\gamma_{mk'} \leq 1, m = 1,..., M\\ & \qquad\qquad\quad\;\eta_{mk'} \geq 0, m = 1,..., M, k' = 1,..., K. \end{aligned} \end{equation} \subsection{Power Control using Neural Network} As we noticed above, the complexity of finding the optimal solution of (\ref{eq:max_min_dl_IoT}) or (\ref{eq:max_min_dl_orth}) is too high for any practical applications. In this Section we suggest to use Neural Networks for finding low complexity suboptimal power control. We first let $p_m= \sum_{k=1}^K \eta_{mk} \gamma_{mk}$ be the normalized transmit power of the $m$-th AP and let $p_m^{\text{opt}}$ be the optimal value of $p_m$ with respect to the optimization problem (\ref{eq:max_min_dl_orth}). It is noticeable that if we can find $p_{m}^{\text{opt}},\ \forall m$, the problem in (\ref{eq:max_min_dl_orth}) becomes equivalent to \begin{equation} \begin{aligned} \label{eq:max_min_dl_orth_convex} &\max_{\boldsymbol{\eta}}\min_{k}\text{SINR}_{k}^{\text{orth}}(\boldsymbol{\eta}) = \frac{\rho_{d}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}}\gamma_{mk}\right)^{2}}{1 + \rho_{d}\sum_{m=1}^{M}p_{m}^{\text{opt}}\beta_{mk}}\\ &\qquad\qquad\text{s.t.} \sum_{k'=1}^{K} \eta_{mk'}\gamma_{mk'} = p_{m}^{\text{opt}}\\ & \qquad\qquad\quad\;\eta_{mk'} \geq 0, m = 1,..., M, k' = 1,..., K, \end{aligned} \end{equation} which is a convex problem \cite{Nayebi2017Precoding} and has significant smaller complexity compared with the quasi-concave problem (\ref{eq:max_min_dl_orth}). To convert problem (\ref{eq:max_min_dl_orth}) to a convex problem, $p_{m}^{\text{opt}}, \forall m$, need to be found. It is observed in \cite{Nayebi2017Precoding} that an exponential relation often approximately holds between $\beta_{m}^{\text{max}}$ and $p_{m}^{\text{opt}}$ where $\beta_{m}^{\text{max}}$ is defined as the largest large-scale fading coefficient between the $m$-th AP and its serving devices, i.e., $\beta_{m}^{\text{max}} = \max_{k=1,.., K}\beta_{mk}$. An exponential regression can then be implemented to predict $p_{m}^{\text{opt}}, \forall m$. We denote the outputs of the exponential regression as $p_{m}(\beta_{m}^{\text{max}}), \forall m$, and these outputs can be used in (\ref{eq:max_min_dl_orth_convex}). In addition, we find that as the length of random pilots goes to infinity and assume that random pilots $\boldsymbol{\psi}_k$ are used in IoT systems, the value of $\text{SINR}_{k}^{\text{IoT}}$ will approach to the value of $\text{SINR}_{k}^{\text{orth}}$, i.e., \begin{equation} \label{eq:IoT_to_orth} \lim_{\tau\rightarrow \infty} \text{SINR}_{k}^{\text{IoT}} \longrightarrow \text{SINR}_{k}^{\text{orth}}. \end{equation} It is also noted that even with finite, but reasonably large $\tau$, $\text{SINR}_{k}^{\text{IoT}} \approx \text{SINR}_{k}^{\text{orth}}$. Based on these observations, low complexity power control for DL IoT systems with random pilots can be implemented as follows. First, $p_{m}^{\text{opt}},\, \forall m$, are approximated using exponential regression based on $\beta_{m}^{\text{max}},\, \forall m$, as in \cite{Nayebi2017Precoding}. The outputs, $p_{m}(\beta_{m}^{\text{max}}),\, \forall m$, are then substituted into (\ref{eq:max_min_dl_orth_convex}), and solving the obtained convex optimization problem, we find the power coefficients $\eta_{mk}$. However, the performance achieved using $p_{m}(\beta_{m}^{\text{max}})$ is not close enough to the optimal performance achieved by $p_{m}^{\text{opt}}$. What is even more important is that the generality of this method is limited. In particular, if we found a function $p_m(\beta_m^{\text{max}})$ that matches well $p_m^{\text{opt}}$ for one network, typically this function is not accurate for another network. Moreover, for some networks, no exponential relationship can be found between $p_{m}^{\text{opt}}$ and $\beta_{m}^{\text{max}}$, e.g., this is the case for high density networks. To overcome the above problems, we propose to use a simple fully connected NN to approximate $p_{m}^{\text{opt}}, \forall m$. As shown in Fig.~\ref{fig:NN}, the structure of the NN we used includes three hidden fully connected layers and each layer has four neurons. \begin{figure*} \begin{center} \includegraphics [width=0.75\textwidth]{NN_1.png} \caption{Neural network for predicting $p_{m}^{\text{NN}}, \forall m$.}\label{fig:NN} \end{center} \hrulefill \end{figure*} Tangent-sigmoid activation function is used for the three hidden layers and rectified linear unit (ReLU) activation function is used in the output layer. The input vector of the NN $\bar{\boldsymbol{\beta}}_m = [\bar{\beta}_{m1}, \bar{\beta}_{m2},...,\bar{\beta}_{m\widehat{K}}]^T \in \mathbb{R}^{\widehat{K} \times 1}$ consists of $\widehat{K}$ largest large-scale fading coefficients from the $m$-th AP to its nearby serving devices. For example, $\bar{\beta}_{m1} = \max_{k=1,.., K}\beta_{mk}$, $\bar{\beta}_{m2}$ is the second largest large-scale fading coefficient and so on. By $p_m^\text{NN}$ we denote the approximation of $p_m^{\text{opt}}$ predicted by the NN for the $m$-th AP. \subsection{Scalable Power Control with High Energy Efficiency}\label{sec:Scalable_pow_control} In real-life applications, we expect that the areas covered by CF networks will be most likely very large. Such large areas will contain so large number of APs and IoT devices that even the computation complexity of solving the convex problem (\ref{eq:max_min_dl_orth_convex}) would be too high. Thus a scalable power control algorithm whose complexity grows linearly with $M$ and $K$ is required. Here we propose a sub-optimal scalable power control algorithm that achieves high EE at the same time. We first define the {\em density of a network} as \begin{equation} \label{eq:density_network} \text{Density} = \frac{\text{Number of APs}}{\text{Serving Area}}. \end{equation} We also assume that the ratio $M/K$ is fixed. As mentioned above, for very large networks even the convex problem (\ref{eq:max_min_dl_orth_convex}) becomes too heavy. For such networks we replace the max-min power optimization with the uniform power control, in which the power coefficients are given by \begin{equation} \label{eq:uniform_power_control} \eta_{m} = \frac{p_{m}^{\text{NN}}}{\sum_{k=1}^{K}\gamma_{mk}},\ \eta_{m} = \eta_{mk},\, \forall k, \end{equation} where $p_m^\text{NN}$ is also obtained using the NN structure given in Fig.~\ref{fig:NN}. It is important to note that the computation complexity of predicting every $p_m^{\text{NN}}$ by the NN is not only very low due to the simple structure of the NN, but also keeps constant as $M$ and $K$ increase. This means that we obtain a scalable network since the amount of computations conducted by each AP does not depend on the number of APs in the network. At the same time, it will be shown later that (\ref{eq:uniform_power_control}) provides much higher EE compared with the full power case. \subsection{Neural Network Training} In the process of training, $\widehat{K}$ is set as four and Levenberg-Marquardt algorithm \cite{levenberg1944method}, \cite{marquardt1963algorithm} is used to train the NN. We use $10^4$ training samples $p_m^\text{opt}$ obtained by finding the optimal solutions of (\ref{eq:max_min_dl_orth}). After the training we find the NN weights $\boldsymbol{\theta}_{j}^{n_j}, \forall n_j, \forall j$ with $\boldsymbol{\theta}_{j}^{n_j}$ being the weight vector of the $n_j$-th neuron in $j$-th layer and these weights are applied for online prediction of $p_m^\text{NN}$. This approach is used for producing results presented in Fig.~\ref{fig:NN_Any_Area}. For training the NN for scalable power control, the same training parameters and algorithm mentioned above are adopted. The idea is to train NN for small areas, which have a relatively small number of APs and therefore afford solving (\ref{eq:max_min_dl_orth}) and finding $p_m^{\text{opt}}$ needed for training. The area is wrapped around, see details in \cite{Ngo_17_Cellfree}, in order to mimic an infinite size network. Next, we use {\em the same NN} for large areas that have the same density and are not wrapped around. This means that we use the same weights $\boldsymbol{\theta}_{j}^{n_j}, \forall n_j, \forall j$, and each AP, say AP $m$, uses input vector $\bar{\boldsymbol{\beta}}_m = [\bar{\beta}_{m1}, \bar{\beta}_{m2},...,\bar{\beta}_{m\widehat{K}}]^T$ composed by $\widehat{K}$ largest coefficients $\beta_{mk}$. Note that the NN is trained offline before online application. Thus, the online complexity of the trained NN only comes from the inference stage (i.e., the prediction of $p_m^{\text{opt}}$) which is very low due to the simple NN structure applied. \subsection{Simulation Results and Discussions} The same simulation setup and parameters given in Section \ref{Setup and Parameters for Numerical Simulations} are used here. {\bf Experiment 1: } In this experiment, we would like to train a NN so that it can be applied to any given serving area with a fixed number of APs and devices. In our experiment we used $M=128$ and $K=4$. Since the number of APs and devices is relatively small, for given powers $p_m^\text{opt}$, we can find power coefficients by solving the problem (\ref{eq:max_min_dl_orth_convex}). We train our NN for several areas and next we use the obtained NN for an area that was not used for training. The results of this approach are shown in Fig. \ref{fig:NN_Any_Area}. \begin{figure} \begin{center} \includegraphics [width=0.5\textwidth]{ML_Prediction_K4_Different_D_Modify.png} \caption{Performance comparison between NN based and optimal power control algorithms of DL CF IoT systems. Here, $M = 128$, $K = 4$.}\label{fig:NN_Any_Area} \end{center} \end{figure} The NN used in Fig. \ref{fig:NN_Any_Area} is trained by squares with areas: $0.016, 0.063, 0.25, 0.56, 1, 4\ \text{km}^2$ and then the obtained NN is used for squares of sizes $0.72$ and $2.25\ \text{km}^2$. We see that this NN provides the performance which is close to the performance obtained with the optimal power control. {\bf Experiment 2:} In this experiment, we fixed the density of the serving area as $\frac{M = 64}{0.03\,km^2}$ and train the NN with $M = 64$ and $K = 16$. Then we use the trained NN for finding $p_m^\text{NN}$ for networks covering large areas with the same density and the same ratio $M/K$, see Fig.~\ref{fig:NN_fix_density_large}. First, we compare our NN approach with other approaches for a small area and a small $M$ and $K$, in Fig.~\ref{fig:NN_fix_density}. We use this small area, $M$, and $K$ since this allows us to find the solution of (\ref{eq:max_min_dl_orth}) and (\ref{eq:uniform_power_control}), which are necessary for the comparisons in Fig.~\ref{fig:NN_fix_density}. \begin{figure} \begin{center} \includegraphics [width = 0.5\textwidth]{NN_Fix_Density_M64_K16_tau64_D177_Modify_2.png} \caption{Spectral efficiency comparison among full power, uniform power control, and optimal power control cases for DL CF IoT systems. Here, $M = 64$, $K = 16$, $\text{Area} = 0.03\ \text{km}^{2}$.}\label{fig:NN_fix_density} \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics [width=0.5\textwidth]{NN_Fix_Density_M4096_K1024_tau64_D1414_1.png} \caption{Energy efficiency comparison between full power and uniform power control for DL CF IoT systems. Here, $M = 4096$, $K = 1024$, $\text{Area} = 2\ \text{km}^{2}$.}\label{fig:NN_fix_density_large} \end{center} \end{figure} Note that the vertical appearance of ``Max-min Opt" is due to equalization of the data rates of all $K$ devices. We present rates when we use (\ref{eq:uniform_power_control}) with maximal $p_m$ (Uniform Full), optimal $p_m^{\text{opt}}$ (Uniform Opt), and NN produced $p_m^{\text{NN}}$ (Uniform NN). It is observed from Fig.~\ref{fig:NN_fix_density} that under uniform power control, the SE performance is almost the same using either $p_{m}^{\text{NN}}, \forall m$, or $p_{m}^{\text{opt}}, \forall m$, which suggests accurate predictions of the proposed NN. Although the performance gap between uniform power control and full power case is not big under current relatively high area density case, we found that the gap would increase in low area density case. More importantly, the NN power control allows us to significantly improve EE compared with the full power transmission as it is shown in Fig.~\ref{fig:NN_fix_density_large}. Note that EE is crucially important for IoT networks. Similar to UL transmission, we define DL EE as $$E_d \triangleq \frac{\sum_{k=1}^{K}R_{k}^{d}}{\sum_{m=1}^{M}P_m}$$ where $P_m$ can be the maximum transmit power of the $m$-th AP (i.e, $P_d$) or $p_m^{\text{NN}}P_d$. Fig.~\ref{fig:NN_fix_density_large} shows the EE comparison between uniform power control and full power case for a network with the same density $\frac{M = 64}{0.03\,km^2}$ and $M=4096$ and $K=1024$. We observe that our NN power control leads to 4-fold improvement of EE. \section{Conclusion}\label{Conclusion} In this work, we proposed IoT systems supported by CF mMIMO with optimal components - LMMSE channel estimation and MMSE MIMO receiver. We derived two random matrix approximations for device's UL SINR and used one of them for efficient and low complexity power control algorithms that give large EE gains, which is very important for low powered IoT devices. For DL transmission a NN aided max-min power control algorithm is proposed. Comparing with the optimal max-min power control algorithm, it has significantly lower complexity and achieves comparable performance. We further proposed a scalable NN algorithm for transmit power control. This algorithm, though sub-optimal, incorporates a simple fully connected NN and obtains power coefficients with very low complexity. The scalability of this algorithm is also important for IoT systems since it works for a very large service area (i.e., covering a large number of IoT devices). Multifold gain in EE is obtained by this algorithm compared with the full power transmission approach.

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\section{Conclusion} We design and evaluate a novel moving target defense framework, MPD, which aims to generate customized dialects during communication and dynamically select different dialects in a self-synchronous manner. Our experiment results using FTP and MQTT indicate that MPD is able to harden the security while incurring low execution overhead effectively. \newpage \section{Evaluation} \label{sec:eval} \begin{table*}[t] \caption{Examples of byte-shuffling dialect evolution pattern, including dialect index and parameters for the dialect generating function, computed for different FTP GET commands.} \label{tab:syn_ftp} \centering \begin{threeparttable} \begin{tabular}{c|c|c|ccc} \Xhline{1pt} Original packet&Customized packet&Index&Position&Length&Offset\\[1pt] \Xhline{.5pt} rget,sample.txt&r,etgsample.txt&8&1&1&3\\[1pt] rget,sample.txt&tger,sample.txt&7&0&1&3\\[1pt] rget,blog.css&etrg,blog.css&4&0&2&2\\[1pt] rget,template.pdf&etrg,template.pdf&4&0&2&2\\[1pt] \Xhline{1pt} \end{tabular} \end{threeparttable} \end{table*} \begin{table*}[t] \caption{Examples of packet-splitting dialect evolution pattern, including dialect index and parameters for the dialect generating function, computed for different FTP GET commands.} \label{tab:syn_ftp2} \centering \begin{threeparttable} \begin{tabular}{c|c|c|cccc} \Xhline{1pt} \multirow{2}*{Original packet}&\multirow{2}*{Customized sub-packets}&\multirow{2}*{Index}&\multicolumn{4}{c}{Length}\\ &&&Pkt1&Pkt2&Pkt3&Pkt4\\[1pt] \Xhline{.5pt} rget,sample.txt&[r][ge][t,][sample.txt]&6&1&2&2&10\\[1pt] rget,sample.txt&[r][ge][t][,sample.txt]&2&1&2&1&11\\[1pt] rget,blog.css&[r][ge][t][,blog.css]&2&1&2&1&9\\[1pt] rget,template.pdf&[rg][et][,][template.pdf]&4&2&2&1&12\\[1pt] \Xhline{1pt} \end{tabular} \end{threeparttable} \end{table*} In this section, we evaluate the performance of MPD and the effectiveness of the proposed moving target defense. \textbf{Experiment Setup:} Our experiments are conducted on a 2.60GHz Intel(R) Core(TM) CPU i7-9750H machine with 16 Gigabytes of memory. The operating system is Ubuntu 18.04 LTS. We perform MTC, SSM, and PTD modules on the selected communication protocol. \begin{figure}[t] \centering \includegraphics[scale=0.5]{media/timing.png} \caption{Timing diagram of GET command in FTP (a) and CONNECT action in MQTT (b).} \label{fig:timing} \end{figure} \begin{figure}[t] \centering \includegraphics[width=0.5\textwidth]{media/cmp.png} \caption{Results of establishing connection with server by genuine FTP client (a) and attacker FTP client (b).} \label{fig:cmp} \end{figure} \textbf{Target Protocol: FTP } FTP is a standard network protocol used for the transfer of computer files between the client and the server. Many FTP clients and automated utilities have been developed for desktops, servers, mobile devices, and hardware. As a target protocol for our proof of concept evaluation, FTP offers two main benefits: (i) It is a lightweight protocol having better performance and easier to test, and (ii) it does not have many complex features, making it easier to customize and analyze. FTP packet format contains IP header, TCP header, and FTP message contents. We programmed a standard FTP client and server and applied our implementation to them. For instance, when a GET command is received on the server, the corresponding file (if file exists) or rejection message (if the file does not exist) will be sent to the client via four handshakes, as shown in Figure~\ref{fig:timing} (a). After adding the MTC module on both server and client, the packet for transferring the GET command will be cast into a proper dialect during each handshake. Table~\ref{tab:syn_ftp} and \ref{tab:syn_ftp2} show byte-shuffling dialect and packet-splitting dialect as well as the evolution process of key parameters during four consecutive file transfers. Due to the different pseudo-random numbers being generated for each message/packet, the PDM module will map the number into different dialect indexes, resulting in different dialect selections and different input parameters of the MTC module for customizing packets. On the server-side, after performing the inverse process, the original packets are recovered, and the server will send the requested file to the client, as shown in Figure~\ref{fig:cmp} (a). To demonstrate the effectiveness of our moving target defense, we create an attacker FTP client that implements a spoofed (and fixed) dialect of the FTP. The attacker client connects to the FTP server equipped with our moving target defense and starts fabricating and sending the malicious command to the server to launch the command injection attack. The result are presented in Figure~\ref{fig:cmp} (b). It is obvious that the command injection attack fails to work. Since the attacker's client protocol dialect pattern is fixed, it is unable to understand the dialect evolution pattern generated in our moving target defense applied on the GET command. We prove that our method can effectively defend the system from such attacks. \begin{figure}[t] \centering \includegraphics[width=0.56\textwidth]{media/missing.png} \caption{Packets captured by Wireshark showing error propagation equal to the total number of previously sent packets cached in the buffer.} \label{fig:missing} \end{figure} \begin{table}[t] \caption{Self-synchronization mechanism demonstration with missing Pkt \#1.} \label{tab:syn_missing} \begin{threeparttable} \begin{tabular}{c|cc|cc|c} \Xhline{1pt} Iteration&Client&Cached Pkt&Server&Cached Pkt&Status\\[1pt] \Xhline{.5pt} 1&Pkt \#1&[Pkt \#p]&|&[Pkt \#p]&Synchronized, Nothing received\\[1pt] 2&Pkt \#2&[Pkt \#1]&Pkt \#2&[Pkt \#p]&Not synchronized\\[1pt] 3&Pkt \#3&[Pkt \#2]&Pkt \#3&[Pkt \#2]&Synchronized\\[1pt] 4&Pkt \#4&[Pkt \#3]&Pkt \#4&[Pkt \#3]&Synchronized\\[1pt] \Xhline{1pt} \end{tabular} \begin{tablenotes} \footnotesize \item[1] We assume \textit{Pkt \#1} is erased/missing during communication. \item[2] \textit{Pkt \#p} is previously sent packet that has already been cached in local buffer. \end{tablenotes} \end{threeparttable} \end{table} Next, we evaluate the effectiveness of our self-synchronization mechanism under erased/missing packets. In particular, we choose to store only one past packet in the buffer for self-synchronization. As is shown in Figure~\ref{fig:missing}, suppose that, due to some network error or intentional attack, one GET command packet before frame \#33 gets lost during communication (Thus, this missing packet cannot be captured by Wireshark in the screenshot). As we can observe, for the second GET command packets that the client sent, which in Figure \ref{fig:missing} corresponds to frame \#33, the server generates no response. Till now, the dialect selection at the client and the server are not synchronized because the previously sent packet cached in the local buffer is different, resulting in selecting different dialect for \textit{Pkt \#2} as shown in Table~\ref{tab:syn_missing} Row 3. However, starting from the third packet, which is frame \#39 in Figure \ref{fig:missing} and Row 4 in Table \ref{tab:syn_missing}, both sides are successfully re-synchronized, as the different packets move out of their local buffers after one (network) cycles, and as a result, the same previously sent packets are again shared between them. It demonstrates MPD 's ability to self-synchronize under packet erasures or in an unstable communication channel. Finally, we evaluate the overhead of MPD , by comparing it with the execution overhead of a standard FTP implementation on the server-side, concerning the network, CPU, and memory overhead. To this end, we use the \textit{time} command in the terminal to monitor several performance indexes such as running time and maximum resident set size. To avoid potential statistical bias, we run each experiment four times and record the average overhead in Table~\ref{tab:overhead}. For each test, we write a script to execute one billion FTP commands randomly (e.g., using GET to continuously retrieve a small text file from the server to the client). \begin{table}[t] \caption{Evaluating the overhead of designed moving target defense in FTP.} \label{tab:overhead} \centering \begin{threeparttable} \begin{tabular}{c|cc} \Xhline{1pt} Performance Index&Original FTP&Modified FTP\\[1pt] \Xhline{.5pt} System time/sec&0.57&0.60\\[1pt] Elapsed time/sec&42.64&44.53\\[1pt] Percent of CPU this job got&1\%&1\%\\[1pt] Maximum resident set size/KB&6402&7683\\[1pt] \Xhline{1pt} \end{tabular} \begin{tablenotes} \footnotesize \item[1] System time: time spent in kernel mode while running the program. \item[2] Maximum resident set size: Maximum memory space occupied while running the program. \end{tablenotes} \end{threeparttable} \end{table} As it is shown in the Table~\ref{tab:overhead}, while a number of functions (including dialect generating function, pseudo-random function, and consistent hashing mapping function) are added in FTP to support our moving target defense, the overhead is almost negligible as compared to a standard FTP program. More precisely, the execution overhead (as measured by elapsed time) increases by 4.43\%, while the maximum resident set size we need compared to standard FTP implementation increase by 1281 KB. \textbf{Target Protocol: MQTT } MQTT is a standard lightweight IoT protocol for transporting messages among IoT devices. Due to its small size, it is designed to provide efficient message delivery for the network where the bandwidth is limited. Apart from the similar reasons we mentioned for choosing FTP, we select MQTT as another target protocol for analysis because it is the typical IoT protocol used by many IoT devices, and shares many common characters compared with other IoT protocols. \begin{figure}[t] \centering \includegraphics[width=0.5\textwidth]{media/cmp_mqtt.png} \caption{Results of establishing connection with server (broker) by genuine MQTT client (a) and attacker MQTT client (b).} \label{fig:cmp_mqtt} \end{figure} MQTT usually runs over TCP/IP. Besides IP header and TCP header, an MQTT packet also contains a fixed header (including control field, e.g., \textit{CONNECT}/\textit{CONNACK}, and packet length field), variable length header, and payload on the application layer. We programmed a standard MQTT client and server (broker) and applied MPD on them. The timing diagram of establishing the MQTT connection is shown in Figure~\ref{fig:timing} (b). The client will send the \textit{CONNECT} to the server to request MQTT communication. After receiving \textit{CONNECT} packet, server will response with \textit{CONNACK} to confirm the connection. We add the MTC module using the byte-shuffling function on both sides, and thus the \textit{CONNECT}/\textit{CONNACK} packet will be cast into an appropriate dialect during each handshake. To demonstrate the capability of the modified MQTT system in defending the DoS attack, we implement an attacker MQTT client that tries to flush the modified server by continuously sending \textit{CONNECT} packets with different identities. The MTC module uses the byte-shuffling function, which will shuffle the control field with other fields in the \textit{CONNECT} packet. As shown in Figure~\ref{fig:cmp_mqtt} (b), the attacker fails to use multiple malicious \textit{CONNECT} packets to exhaust the server's buffer space because these packets will fail the dialect index check before getting accepted. In contrast, as shown in Figure~\ref{fig:cmp_mqtt} (a), the genuine client is able to successfully start communication and publish information on the server due to their having the synchronized dialect evolution pattern. Therefore, we can apply MPD on IoT protocol MQTT to prevent the system from some types of DoS attacks. \section{Implementation} We implement a prototype of MPD on FTP and MQTT, which includes the following main components. {\em Dialect Customization and Management:} We implement our dialect customization functions (e.g., shuffling-byte) in FTP and MQTT. Its input parameters are predefined and bundled together. We use indexes to indicate those parameter combinations. The index is decided by the consistent hashing mapping function in the PDM module. It determines which dialect is used for each communication packet. {\em Pseudo-Random Number Generator:} We implement a keyed-hash function as the pseudo-random number generator. The output pseudo-random number will be fed into the consistent hashing mapping function to compute the corresponding dialect index. {\em Self-Synchronization Module:} We implement our SSM module and cache one previous packet, which implies that any network errors will propagate for only one cycle during communication. The buffer will be filled with one predefined initial packet at the beginning of the program execution. \section{Introduction} \label{sec:intro} The Internet of Things (IoT) refers to the concept of a large number of smart objects and devices connected to the Internet, offering diverse capabilities, such as sensing, actuating, processing, and communication. It integrates and relies on various enabling components, e.g., software, application libraries, middleware, embedded systems, and network artifacts. Any vulnerability in these components would lead to exploitations to create serious threats - such as the denial of service attack and reconnaissance attack- to the IoT system. The Moving Target Defense (MTD) has been developed as an effective defense strategy to dynamically (and randomly) change the properties of configurations of a target system while maintaining its essential functionalities to diversify its defense mechanism and obfuscate the resulting attack surfaces. It significantly increases the work factor of an adversary to launch an effective attack towards a constantly evolving defense. System attributes (and thus the potential attack surfaces) that can be dynamically changed to confuse attackers include instruction sets, address space layouts, IP addresses, port numbers, proxies, virtual machines, and operating systems~\cite{cho2020toward}. Existing work on MTD has primarily focused on low-level attributes, such as instruction set randomization~\cite{barrantes2003randomized,kc2003countering} and address space layout randomization~\cite{hund2013practical,seibert2014information}. Some other MTD methods target network-level features, such as IP address randomization~\cite{al2011toward,jafarian2012openflow}, virtualization-based MTD~\cite{okhravi2012creating} and software-defined networking based MTD~\cite{macfarland2015sdn,wang2017random}. However, considering the security of communication protocols in IoT, these MTD methods cannot achieve desired defense diversity against potential attacks, as the low-level protocol properties to be mutated (e.g., IP addresses or port numbers) are minimal. In this paper, we propose a new approach for communication protocol MTD through protocol dialects, denoted by MPD. Our key idea of MPD is to automatically create many protocol dialects, which are variations of the target protocol created by mutating its handshake and message formats while keeping the communication functionalities unchanged. By selecting different protocol dialects and switching between them on the fly, we craft an MTD solution with substantially boosted diversity in communication. In order to enable easy management of dialects, our proposed solution also leverages distributed hash and keyspace partitioning to allow adding, removing, modifying any protocol dialect independent of others in MPD. We show that MPD requires very low overhead and is suitable for lightweight communication protocols in IoT using client-server architecture, e.g., File Transfer Protocol (FTP) and Message Queuing Telemetry Transport (MQTT). These protocols are usually lightweight and focus on data exchange efficiency while offering very limited security mechanism~\cite{andy2017attack, nebbione2020security}. We argue that vigilantly customizing protocol dialects for higher-level features such as handshake and message format is crucial for directly protecting IoT security from potential attacks, especially for insecure communication channels in IoT that adversaries frequently target. At the core of creating customized packet dialects, the critical problem is to design appropriate dialect generating functions that are easy to implement and will not introduce a high computational cost. Manually creating dialect templates is costly and also unrealistic for a huge program. Apart from that, we need to consider how to make the automatically created dialect applied on each packet vary following a random and unpredictable pattern. Only by increasing the uncertainty of the moving target can our approach help improve communication security. We propose two types of dialect generating functions - targeting at handshake and message format, respectively - to automatically create a number of protocol dialects as candidates for MTD. During communication, different dialects will be selected by a mapping function and applied to every handshake to dynamically change the defense properties, thus varying the attack surface on the fly. Furthermore, to deal with potential disruptions and attacks on the communication channel, we design the self-synchronization mechanism motivated by the self-synchronizing stream cipher~\cite{millerioux2010self} to ensure that the server and clients in an IoT system always return to a synchronized state after a limited error propagation. \vspace{0.1in} The main contributions of our work are as follows: \begin{itemize} \item We propose MPD , an automated and self-synchronous framework for creating and leveraging protocol dialects for effective MTD in IoT communications. Given a specific protocol, MPD can automatically generate enough dialects per user needs and apply them to certain packets/handshakes during communication to improve security. \item MPD leverages distributed hash and keyspace partition for easy dialect management. It also implements a self-synchronization mechanism. Packets are cached and used in conjunction with a pseudo-random generator to ensure the randomness and unpredictability of dialects. \item Evaluation using two communication protocols, namely FTP and IoT protocol MQTT, shows that MPD can efficiently generate a large number of dialects and dynamically select the dialects for a round of communication for improved security in IoT systems. \end{itemize} \section*{References}} \bibliographystyle{splncsnat} \section{Motivation} \label{overview} In this paper, we focus on IoT communication protocols that adopt a server-client architecture and employ packets for a communication, e.g., FTP and MQTT. Such protocols often emphasize communication efficiency and offer limited protection and security. We consider a threat model in which adversaries in an IoT system can dominate all communications, erase, replay, and replace arbitrary control packets between the client and server. In particular, an attacker can launch the Denial of Service (DoS) attack to exhaust the resources available at a victim network entity (e.g., a server or a client). This type of attack can be easily implemented by replaying the system with numerous connection requests~\cite{haripriya2019secure} or intentionally keeping all the connections alive on the network busy~\cite{vaccari2020slowite}. Besides, in our threat model, we also consider malicious modification of communication. Given a pre-owned privilege, an attacker can modify genuine control packets or inject malformed control packets to launch the command injection attack~\cite{liu2017vehicle,vuong2015decision}, and this could undermine the system's integrity by maliciously disrupting handshakes or guiding a victim to execute unauthorized code. These attacks can also be indicated by the documented real-world vulnerabilities such as CVE-2019-9760 on FTP and CVE-2016-10523 on MQTT. We note that the above-mentioned threat models support a proof-of-concept for MPD as a promising new direction of MTD in communication security without being over-aiming for an all-encompassing solution. Therefore, this work serves as an initial step toward a comprehensive solution based on MTD through protocol dialects. Moving target defense (MTD) \cite{nitrd2013iwg} enables us to create, analyze, evaluate, and deploy mechanisms and strategies that are diverse and that continually shift and change over time to increase complexity and cost for attackers, limit the exposure of vulnerabilities and opportunities for attacks, and increase system resiliency. Current MTD methods are limited in increasing complexity against potential attacks because low-level system attributes (e.g., IP addresses and port numbers) only offer a limited degree of freedom for mutation. Due to the above reasons, we design MPD , an MTD technique combined with protocol dialect that leverages more application-layer properties of communication protocols and efficiently increases the number of variations that we can create. The goal of MPD is to customize a standard protocol and create a moving target defense by (i) fabricating desired protocol dialects and (ii) allowing servers/clients to dynamically select dialects for communication in a synchronous fashion. We first introduce our definition of protocol dialects as follows: \begin{definition}{\textbf{Protocol Dialect.}}\label{def:dialect} Given a standard communication protocol, a dialect in this paper is a variation created by mutating its packets and handshakes while keeping the communication functionalities unchanged. More precisely, let $\mathcal{D}$ be a set of possible packet mutation functions, such as byte swapping and obfuscation. If the standard protocol employs packets $\{m_1, m_2, \ldots\}$, then the protocol dialect corresponding to $d_n\in \mathcal{D}$ (denoted as a dialect generating function) uses packets $\{d_n(m_1), d_n(m_2), \ldots\}$ for communication under the same communication rules. Thus, each protocol dialect is uniquely defined by a distinct dialect generating function $d_n\in \mathcal{D}$. \end{definition} From the given definition, protocol dialect can increase the variation of system attributes to improve communication security. We design a protocol dialect customization scheme to ensure the evolving of dialects used for every handshake. MPD allows the client and server, who are aware of the same protocol dialect varying pattern to communicate with each other correctly. Any infiltrated malicious packets sent by an attacker will be detected and discarded by the client/server without noticing the mutation pattern. In order to develop a moving target defense using protocol dialects, the critical challenge is keeping both client and server synchronized during communication. Under the ideal circumstance, both sides will follow the same protocol dialect variation pattern. Thus, the client/server can send and receive packets using the same dialect during one handshake. However, supposing there are disruptions in the channel (e.g., losing packets in transmission), the client and server will lose synchronization without an appropriate synchronization mechanism, leading to the disorder of selecting dialects within coming handshakes. This paper borrows the idea from the self-synchronizing stream cipher and redesigns its key mechanism in MPD . We use previously sent packets as the reference to guarantee that the dialect selected by the client and server will rematch after a limited time of error propagation when disruptions or attacks happen. \section{Introduction}\label{sec:intro} \lipsum[1-3]\todo{Refine me} The remainder of the paper starts with a presentation of related work (\cref{sec:relatedwork}). It is followed by a presentation of hints on \LaTeX{} (\cref{sec:hints}). Finally, a conclusion is drawn and outlook on future work is made (\cref{sec:outlook}). \section{Related Work} \label{sec:relatedwork} Winery~\cite{Winery} is a graphical \commentontext{modeling}{modeling with one \qq{l}, because of AE} tool. The whole idea of TOSCA is explained by \citet{Binz2009}. \section{LaTeX Hints} \label{sec:hints} \begin{figure} \centering \includegraphics[width=.8\textwidth]{example-image-golden} \caption{Simple Figure. \cite[based on][]{mwe}} \label{fig:simple} \end{figure} \begin{table} \caption{Simple Table} \label{tab:simple} \centering \begin{tabular}{ll} \toprule Heading1 & Heading2 \\ \midrule One & Two \\ Thee & Four \\ \bottomrule \end{tabular} \end{table} \begin{lstlisting}[ caption={Example Java Listing}, label=L1, language=Java, float] public class Hello { public static void main (String[] args) { System.out.println("Hello World!"); } } \end{lstlisting} \begin{lstlisting}[ caption={Example XML Listing}, label=L2, language=XML, float] <example attr="demo"> text content </example> \end{lstlisting} \begin{listing}[htbp] \begin{minted}[linenos=true,escapeinside=||]{java} public class Hello { public static void main (String[] args) { System.out.println("Hello World!"); } } \end{minted} \caption{Java code rendered using minted} \label{lst:java} \end{listing} \begin{listing}[htbp] \begin{minted}[linenos=true,escapeinside=||]{xml} <demo> <node> <!-- comment --> |\Vlabel{commentline}| </node> </demo> \end{minted} \caption{XML-Dokument rendered using minted} \label{lst:xml} \end{listing} \Cref{L1,L2} show listings typeset using the \texttt{lstlisting} environment. \href{https://github.com/gpoore/minted}{minted} is an alternative package, which enables syntax highlighting using \href{http://pygments.org/}{pygments}. This, in turn, requires Python, so it is disabled by default. In case you load it above, be sure to run pdflatex with \texttt{-shell-escape} option. \Cref{lst:java,lst:xml} show listings rendered using minted. You can point to a single line: \lref{commentline}. cref Demonstration: Cref at beginning of sentence, cref in all other cases. \Cref{fig:simple} shows a simple fact, although \cref{fig:simple} could also show something else. \Cref{tab:simple} shows a simple fact, although \cref{tab:simple} could also show something else. \Cref{sec:intro} shows a simple fact, although \cref{sec:intro} could also show something else. Brackets work as designed: <test> One can also input backquotes in verbatim text: \verb|`test`|. The symbol for powerset is now correct: $\powerset$ and not a Weierstrass p ($\wp$). \begin{inparaenum} \item All these items... \item ...appear in one line \item This is enabled by the paralist package. \end{inparaenum} Please use the \qq{qq command} or the \enquote{enquote command} to quote something. ``something in quotes'' using plain tex syntax also works. You can now write words containing hyphens which are hyphenated (application"=specific) at other places. This is enabled by an additional configuration of the babel package. In case you write \qq{application-specific}, then the word will only be hyphenated at the dash. You can also write applica\allowbreak{}tion-specific, but this is much more effort. The words \qq{workflow} and \qq{dwarflike} can be copied from the PDF and pasted to a text file. Numbers can written plain text (such as 100), by using the siunitx package like that: \SI{100}{\km\per\hour}, or by using plain \LaTeX{} (and math mode): $100 \frac{\mathit{km}}{h}$. \section{Conclusion and Outlook} \label{sec:outlook} \lipsum[1-2] \subsubsection*{Acknowledgments} \ldots In the bibliography, use \texttt{\textbackslash textsuperscript} for \qq{st}, \qq{nd}, \ldots: E.g., \qq{The 2\textsuperscript{nd} conference on examples}. When you use \href{https://www.jabref.org}{JabRef}, you can use the clean up command to achieve that. See \url{https://help.jabref.org/en/CleanupEntries} for an overview of the cleanup functionality. \renewcommand{\bibsection}{\section*{References}} \bibliographystyle{splncsnat} \begingroup \ifluatex \else \microtypecontext{expansion=sloppy} \fi \small \section{Related Work} \textbf{Pseudo-random number generator:} Pseudo-random number generator (PRNG) is an algorithm for generating a sequence of numbers that have properties approximating the properties of random-number sequences. Since we need to implement MPD with randomly varying moving targets, the basic idea is using PRNG to output a random number for self-synchronization and dialect selection. Many previous works~\cite{salmon2011parallel,akhshani2014pseudo,wang2019pseudo,lee2001elliptic} provide us with insight of choosing PRNG, such as PRNG based on logistic chaotic system~\cite{wang2019pseudo} and elliptic curve PRNG~\cite{lee2001elliptic}. \textbf{Self-synchronizing stream cipher:} Stream cipher~\cite{hell2007grain,paul2011rc4,mannai2018new} is a symmetric key cipher where the digits of plain text are combined with a pseudo-random cipher digit stream (which is the keystream), including two types: (i) synchronous stream cipher (SSC), and (ii) self-synchronizing stream cipher (SSSC). The latter uses previous ciphertext digits to compute the keystream. In paper~\cite{maurer1991new}, several alternative design approaches for SSSCs are proposed that are superior to the design based on the block cipher with respect to encryption speed and security. Joan Daemen et al. introduce another design approach for hardware-oriented SSSCs named Moustique~\cite{daemen2008self}. Besides, many other discussions are about SSSCs, which can be found in papers~\cite{khazaei2008new,millerioux2010self}. \textbf{Moving target defense (MTD):} Plenty of previous works about MTD have different areas of focus respectively. Some of them are about leveraging lower-level system configurations, which provide insights into our design. In OpenFlow random host mutation~\cite{jafarian2012openflow}, Jafarian et al. provided an MTD architecture that transparently mutates IP addresses with randomness. RPAH~\cite{luo2015rpah} achieved MTD by constantly changing IP addresses and ports to realize random port and address hoping. On protocol level, Ghost-MTD~\cite{park2020ghost} applied mutation on protocols to achieve MTD, while the protocol mutation pattern should be pre-defined and pre-shared between client and server. \textbf{Security risks of IoT protocols:} Internet of things is growing rapidly and reaches a multitude of different domains such as environmental monitoring, smart home, and automatic driving. However, many low-end IoT products do not usually have strong security mechanisms embedded~\cite{firdous2017modelling,hartzell2017automobile}. Hence, threats such as leakage of sensible information, DoS attacks, and unauthorized network access attacks bring a severe safety issue to the IoT system~\cite{meneghello2019iot}. Some researches are conducted previously to mitigate the threat of attacks targeting IoT protocols and devices. Besides, in the survey~\cite{nebbione2020security}, Nebbione et al. provided some basic ideas for solving the security issues in many IoT protocols. \section{First Section} \subsection{A Subsection Sample} Please note that the first paragraph of a section or subsection is not indented. The first paragraph that follows a table, figure, equation etc. does not need an indent, either. Subsequent paragraphs, however, are indented. \subsubsection{Sample Heading (Third Level)} Only two levels of headings should be numbered. Lower level headings remain unnumbered; they are formatted as run-in headings. \paragraph{Sample Heading (Fourth Level)} The contribution should contain no more than four levels of headings. Table~\ref{tab1} gives a summary of all heading levels. \begin{table} \caption{Table captions should be placed above the tables.}\label{tab1} \begin{tabular}{|l|l|l|} \hline Heading level & Example & Font size and style\\ \hline Title (centered) & {\Large\bfseries Lecture Notes} & 14 point, bold\\ 1st-level heading & {\large\bfseries 1 Introduction} & 12 point, bold\\ 2nd-level heading & {\bfseries 2.1 Printing Area} & 10 point, bold\\ 3rd-level heading & {\bfseries Run-in Heading in Bold.} Text follows & 10 point, bold\\ 4th-level heading & {\itshape Lowest Level Heading.} Text follows & 10 point, italic\\ \hline \end{tabular} \end{table} \noindent Displayed equations are centered and set on a separate line. \begin{equation} x + y = z \end{equation} Please try to avoid rasterized images for line-art diagrams and schemas. Whenever possible, use vector graphics instead (see Fig.~\ref{fig1}). \begin{figure} \includegraphics[width=\textwidth]{fig1.eps} \caption{A figure caption is always placed below the illustration. Please note that short captions are centered, while long ones are justified by the macro package automatically.} \label{fig1} \end{figure} \begin{theorem} This is a sample theorem. The run-in heading is set in bold, while the following text appears in italics. Definitions, lemmas, propositions, and corollaries are styled the same way. \end{theorem} \begin{proof} Proofs, examples, and remarks have the initial word in italics, while the following text appears in normal font. \end{proof} For citations of references, we prefer the use of square brackets and consecutive numbers. Citations using labels or the author/year convention are also acceptable. The following bibliography provides a sample reference list with entries for journal articles~\cite{ref_article1}, an LNCS chapter~\cite{ref_lncs1}, a book~\cite{ref_book1}, proceedings without editors~\cite{ref_proc1}, and a homepage~\cite{ref_url1}. Multiple citations are grouped \cite{ref_article1,ref_lncs1,ref_book1}, \cite{ref_article1,ref_book1,ref_proc1,ref_url1}. \subsection{Security Analysis} \label{sec:sys-anal} IoT protocol MQTT is widely applied in many popular IoT applications, such as IBM Watson IoT Platform and AWS IoT services. Four phases are in an MQTT session: (i) connection, (ii) authentication, (iii) communication, and (iv) termination. After establishing the connection by creating a TCP/IP connection with the server (broker) on a predefined port, the client will send the \textit{CONNECT} packet that includes user identification to start the MQTT communication. In this step, an attacker can launch a Denial of Service (DoS) attack by sending many \textit{CONNECT} requests continuously and thereby maliciously making the server busy as in requests flooding, which impairs the availability of the system~\cite{firdous2017modelling}. As the server is not able to differentiate the normal \textit{CONNECT} and the malicious \textit{CONNECT} packets, on receiving the flooding request messages, the broker starts to acknowledge all with \textit{CONNACK} message. Assuming many connection requests that arrive simultaneously, the server's buffer will be drained out, and the server will not be capable of processing new incoming requests. In our design, we will apply the MPD on \textit{CONNECT} packets. Before the server receiving incoming \textit{CONNECT} packets from the client, the system will check whether the dialect index of the current packet matches that of the server. For malicious \textit{CONNECT} packets, no dialect will be applied, and therefore they will fail the dialect index check. In this case, the server will refuse to start MQTT communication when receiving malicious request messages, saving its resources for processing valid requests. Attacks such as command injection attacks and cipher suite downgrade attacks can intentionally erase or alter the communication packets on a public channel, damaging the integrity of communication. In the light-weight controller area network (CAN) protocol, a message-based IoT protocol designed to allow microcontrollers and devices to communicate with each other's applications, no encryption or authentication is initially applied due to their high cost. Thus, it is vulnerable and threatens by safety issues such as command injection attack~\cite{liu2017vehicle}. In this type of attack, attackers will use reprogrammed electronic control units (ECU) to send malicious packets to the CAN bus. These malicious packets are fabricated and injected with forged ID and data to distract victim ECUs or make them execute malformed actions. For some other protocols, a standard mechanism such as SSL/TLS can be used to protect communication security. However, aiming to vulnerability existed in SSL/TLS, man in the middle (MITM) attackers can launch cipher suite downgrade attacks by abandoning the \textit{ClientHello} packet sent by the genuine client and replacing it with a malformed packet containing lower-version TLS~\cite{lee2020return,sjoholmsierchio2020strengthening}. In consequence, the server will start the communication with the client using insecure lower-version TLS. We propose the MPD that will guarantee the evolution of packet dialect patterns during communication. Even though an attacker leverage the fetched information to launch an attack, malicious packets such as command injection packet or lower-version \textit{ClientHello} packet will not be processed by the target because they cannot match the dialect evolution pattern of the receiver without acquiring knowledge of MPD . In the above threat models, if packets are missing in an unstable communication channel or maliciously erased/altered by an attacker, it will affect the current packet in communication and drive the moving target defense of the client and server into an asynchronous state. The self-synchronization mechanism in our proposed solution guarantees the resilience of our system under packet erasures and modifications. Supposing there is a packet missing or modified, according to Equation (\ref{eq:newHMAC}), it will only affect the dialect selection of the next packet (i.e., causing the server and the client to select different dialects for the next packet). Thus, such an error only propagates once. The server and the client will resume synchronization after a short, finite, and predictable transient time (when the erased or corrupted packet moves out of the local buffer). As shown in Figure \ref{fig:syn}, supposing the packet \textit{PKT \#2} is missing/erased, when transferring the \textit{PKT \#3}, previously cached packets stored in the local buffer of both sides are different. The sender assumes \textit{PKT \#2} was successfully sent and updates it into its local buffer, while the cached packets in the receiver's local buffer remain unchanged. The total number of past packets we cached is one, and therefore the error will propagate for only one handshake from \textit{PKT \#2} to \textit{PKT \#3}. After that, the sender and receiver return to a synchronous state again as the cached error packets move out of their local buffers. We note that if expanding the total number of cached packets in the local buffer to $h$, the error will propagate for $h$ handshakes accordingly. \subsection{Protocol Dialect Management} \label{sec:sys-manage} Finally, we need to map the pseudo-random number got from equation \ref{eq:newHMAC} to dialect index $n$. We implement the idea borrowed from consistent hashing in keyspace partitioning. Keyspace partitioning originally aims at mapping keys to nodes in a distributed hash table. We keep the concept but replace the keys with the pseudo-random number we had and nodes with protocol dialect indexes. Therefore, we design consistent hashing mapping that treats protocol dialect indexes as points on a circle, and $\delta(n, n+1)$ is the distance between two points, which is traveling clockwise around the circle from index $n$ to $n+1$, as shown in Figure \ref{fig:map}. Assuming the pseudo-random number value is $S$ and distance $\delta$ between every two indexes are equally distributed, in this case, consistent hashing mapping is presented as following: \begin{equation} n=S//(S_{max}//n_{max}+1)+1,\label{eq:mapping} \end{equation} where $S_{max}$ is the maximum pseudo-random number value we can generate, and $//$ denotes floor division. Considering that the total number of protocol dialects is $n_{max}$, we can decide the certain dialect with index $n$ that should be applied to the current packet by computing equation \ref{eq:mapping}. Due to the different protocol dialects that we created, consistent hashing mapping gives us the flexibility to add new dialects or drop existing dialects. Without making significant changes in the mapping function, we can minimize the update of equation \ref{eq:mapping} and thereby increase the efficiency of the design. \textit{Example 3:} Given the example illustrated in Figure \ref{fig:map}, the circle in this figure represents the range of pseudo-random numbers ranging from 0 to $S_{max}$. We have four existed dialects in use shown in Figure \ref{fig:map} (a), and each of them takes a quarter of the circle. Assuming that we add a new dialect 5 as a candidate, in equation \ref{eq:mapping}, the only changed parameter is the total number of protocol dialects $n_{max}$, which should increase from 4 to 5. Representing in Figure \ref{fig:map} (b), the distance $\delta$ will decrease to $\delta'$ which is one-fifth of the circle. Therefore, we can update $n_{max}$ value in the equation without modifying any parameters else and thereby mitigate the impact made by adding/dropping of dialect in our system. \begin{figure}[t] \centering \includegraphics[width=0.5\textwidth]{media/map.png} \caption{Consistent hashing mapping before (a) and after (b) adding new dialect.} \label{fig:map} \end{figure} \subsection{Moving Target Customization} \label{sec:sys-mtd} In our system, the MTC module can utilize any function $d_n$ to create protocol dialects, as long as (i) $d_n$ can sufficiently (i) mutate any communication packets $m$ and (ii) the inverse $d_n^{-1}$ exists and can be used to recover the original packets $m$, i.e., \begin{equation} \begin{array}{lr} m'=d_n(m) \ {\rm and} \ m=(d_n)^{-1}(m'), \ \forall m\label{eq:customize} \end{array}, \end{equation} where $m$ and $m'$ are the original and customized packets containing application layer information. To demonstrate the key ideas, we design two groups of dialects which are (i) shuffling dialect and (ii) packet-splitting dialect. \textbf{Shuffling:} The shuffling function aims to generate various protocol dialects by switching randomly selected segments from the original packet. Assuming a total number of $s$ switching-available segments within one packet, the mutations that we can create are the permutation of the $s$ segments. Each of these permutations is considered as one specific dialect. In practice, we often carefully select the subset of mentioned permutations to achieve better performance, as not the shuffling of every available segment will be valid given the possibility that some segments contain the same information or vital information that cannot be arbitrarily moved. \textit{Example 1:} In this example, we illustrate byte-shuffling by defining three parameters, i.e., position, length and offset. \textit{Position} decides starting byte of first segment. \textit{Offset} is the distance between starting bytes of two segments. \textit{Length} determines the length of two segments. We used $p$, $l$ and $o$ to denote those three parameters accordingly. The dialect index $n$ for bytes-shuffling maps to the available combination of three mentioned parameters, noted as $n(p,l,o)$, which is predefined. Meanwhile, letting $m$ be a packet of $k$ bytes, we denote its $i$ byte by $m(i)$ for $i=1,2,\ldots,k$. A segment of the packet $m$ from $i^{th}$ to $j^{th}$ is denoted as $m(i,j)$. Therefore, we describe the bytes-shuffling function as following: \begin{equation} \begin{split} d_{n(p,l,o)}(m)&=[m(1,p)||m(p+o,p+o+l)||m(p+l,p+o)||\\ &\quad\ \ m(p,p+l)||m(p+o+l,k)]\\ &=m', \label{eq:shuffle} \end{split} \end{equation} where $||$ denotes the separation of two segments within a packet. In equation, $m(p,p+l)$ and $m(p+o,p+o+l)$ are the two segments shuffled by function $d_{n(p,l,o)}$. The final result is the customized packets $m'$. In order to destruct the dialect and retrieve original packet $m$ on receiver side, we need to employ an inverse function of $(d_{(p,l,o)})^{-1}$, which is: \begin{equation} \begin{split} (d_{n(p,l,o)})^{-1}(m')&=[m(1,p)||m(p,p+l)||m(p+l,p+o)||\\ &\quad\ \ m(p+o,p+o+l)||m(p+o+l,k)]\\ &=m. \label{eq:inverse_shuffle} \end{split} \end{equation} Since three parameters will be synchronous for both sides during communication, we can locate two segments we need to shuffle back and then recover the original packets. By using the bytes-shuffling function and its inverse function, we can automatically generate the desired number of dialects for packets. \textbf{Packet-splitting:} The second type of dialect is packet-splitting dialect. This will generate protocol dialects by splitting a single packet into several sub-packets of any length (smaller than the original packet). In each sub-packet, the lower-layer header remains, but it only carries part of the application layer information of the original packet. Considering that each sub-packet will receive its corresponding response from the receiver, it breaks one handshake into several handshakes. Therefore, the packet-splitting dialect will change the original handshake pattern into a customized one with multiple handshakes. \textit{Example 2:} In this example, we split one single packet into four sub-packets. We introduce three predefined parameters $t_1$, $t_2$ and $t_3$ to denote the length of first three sub-packets correspondingly. Since the original packet has the fixed message length $k$, the length of the last sub-packets is equal to $k-t_1-t_2-t_3$. The combination of every four sub-packets is a new dialect. Therefore, we describe the packet-splitting function as following: \begin{equation} \begin{split} d_{n(t_1,t_2,t_3)}(m)&=d_{n(t_1,t_2,t_3)}([m(1,t_1) || m(t_1, t_1+t_2) ||\\ &\quad\ m(t_1+t_2,t_1+t_2+t_3) || m(t_1+t_2+t_3,k)])\\ &=[m'_1 || m'_2 || ... || m'_4], \label{eq:split} \end{split} \end{equation} where $m'_i$ denotes each sub-packet created by function $d_{n(t_1,t_2,t_3)}$. Each sub-packet accordingly contains a segment of the information from original packet. In order to retrieve the original packet $m$ from each sub-packet, receiver need to leverage inverse function $(d_{n(t_1,t_2,t_3)})^{-1}$ to merge all received sub-packets, which is: \begin{equation} \begin{split} (d_{n(t_1,t_2,t_3)})^{-1}([m'_1 || m'_2 || ... || m'_4])&=(d_{n(t_1,t_2,t_3)})^{-1}([m(1,t_1)||m(t_1, t_1+t_2)||\\ &\quad\ m(t_1+t_2,t_1+t_2+t_3)||m(t_1+t_2+t_3,k)])\\ &=m. \label{eq:inverse_split} \end{split} \end{equation} Each index indicates one specific combination of three defined parameters $t_1$, $t_2$ and $t_3$. With different dialect index $n$, the length of the sub-packets will be different. Due to the synchronization mechanism of protocol dialect, both sides will synchronize and generate the same dialect index $n$ during the communication. Hence, we can collect all the sub-packets and merge them to recover original packets. \section{System Design} \begin{figure}[t] \centering \includegraphics[width=0.9\textwidth]{media/sys.png} \caption{MPD System Diagram.} \label{fig:sys} \end{figure} MPD consists of three major modules: (i) Moving Target Customization (MTC), (ii) Self-Synchronization Mechanism (SSM), and (iii) Protocol Dialect Management (PDM). Its architecture is illustrated in Figure \ref{fig:sys}. When commands (e.g., GET command to retrieve a file or LS command to display a local directory in FTP) are received by a client, PDM automatically selects a protocol dialect (with a dialect index $n$) for the corresponding packets, and SSM ensures that the server and the client remain synchronized with respect to the dialect they decide to use. We note the dialect selections must be unpredictable to eves-droppers in order to prevent attacks on the protocol dialect, such as the denial of service attacks and command injection attacks. To this end, a pair of pseudo-random number generators that are self-synchronous are employed by SSM at the server and the client, respectively. Then, the consistent hashing mapping function is used in PDM to map the output of such pseudo-random number generators to proper dialect index $n$. Next, PDM instructs MTC to generate customized communication packets using the selected dialect generating function $d_n$ -- which are pre-installed on the server and the client -- and send those packets to the receiver. In particular, the sender's MTC module applies $d_n$ on each out-going packet while the receiver's MTC module employs its inverse function $d_n^{-1}$ to destruct the dialect and recover the standard packets for processing at the receiver. \subsection{Self-Synchronization Mechanism} \label{sec:sys-syn} Dialect selection by the server and client must be unpredictable to any eves-droppers and yet remain synchronized to ensure the proper functioning of the communication protocol. To this end, we propose a mechanism that leverages a pseudo-random number generator to create randomness for selecting random dialects. Similar to the idea of self-synchronizing stream ciphers, we make future dialect indexes depend on past packet values to allow the server and client to self-synchronize even under packet erasures/modifications. We cache one past packet (denoted by $m_{i-1}$) in a local buffer at the server and the client, i.e., \begin{equation} M_i=m_{i-1}.\label{eq:cat} \end{equation} The Keyed-hash function is widely used in the keyed-hash message authentication code (HMAC). However, we note that rather than using the keyed-hash value for message authentication, we consider it as a pseudo-random number that can be securely generated by the client and server and compute the current dialect index from the keyed-hash value. Therefore, next, we leverage the keyed-hash function to produce a pseudo-random number based on a shared secret key $K$ known to both the server and client. For a current cached message $M_i$ and shared secret $K$, the keyed-hash function will generate a pseudo-random number $S$ from $M_i$. More precisely, we present the keyed-hash function we used in our implementation as following: \begin{equation} \begin{split} S&=H((K \oplus opad) || H(K \oplus ipad) || M_i)\\ &=H((K \oplus opad) || H(K \oplus ipad) || m_{i-1}),\label{eq:newHMAC} \end{split} \end{equation} where $H$ is cryptographic hash function such as MD5, and $\oplus$ denotes bitwise exclusive or (XOR). According to RFC 2104, outer padding $opad$ consists of repeated bytes valued 0x5c, and inner padding $ipad$ consists of repeated bytes valued 0x36. As shown in equation \ref{eq:newHMAC}, the input becomes shared secret key $K$ and cached past packet $M_i$, and those work as random seed for each iteration. \begin{figure}[t] \centering \includegraphics[width=0.75\textwidth]{media/syn.png} \caption{An illustration of the proposed self-synchronization mechanism.} \label{fig:syn} \end{figure} The modified keyed-hash function used in this paper is just one among many possible candidates from the available pseudo-random functions family. Due to the security properties provided by keyed-hash functions, the resulting pseudo-random number $S$ is unpredictable to any eves-dropper who does not have access to the shared secret key $K$. It is also apparent that the server and the client will eventually resume synchronization even under packet erasures and modification. As illustrated in Figure~\ref{fig:syn}, as long as the past packets stored in local buffers are the same for both client and server, their pseudo-random number generator would compute the same pseudo-random number. Now the self-synchronous property is realized between client and server and can always be achieved if making the final dialect index depend on past packet values. We note that similar to self-synchronizing stream ciphers, caching more than one previous packet and concatenate them as new $M_i$ will leverage the contents of previously sent packets, which increases the complexity of modified keyed-hash function (i.e., equation \ref{eq:newHMAC}) and efficiency against brute force attacks aiming at the self-synchronizing mechanism. In contrast, it takes longer to re-synchronize the server and the client under network errors.

BkiUbGg5qsMAIvucb6Mc

\section{Introduction}\label{sec-intro} \IEEEPARstart{T}{his} paper considers a \emph{common-receiver reconstructions (CRR)} variant of the \emph{successive refinement} problem with receiver side information where the source reconstructions at the receivers are required to be identical (almost always). An encoder is required to compress the output of a discrete memoryless source (DMS) into two messages: \begin{itemize} \item a \emph{common} message that is reliably delivered to both receivers, and \item a \emph{private} message that is reliably delivered to one receiver. \end{itemize} Each receiver has some side information jointly correlated with the source (and the other receiver's side information), and is required to output source reconstruction that meets a certain fidelity requirement. The CRR condition requires that these reconstructions be identical to one another. The CRR problem described above can be viewed as an abstraction of a communication scenario that could arise when conveying data (e.g., meteorological or geological survey data, or an MRI scan) over a network for storage in separate data clusters storing (past) records of the data. The records, which serve as side information, could be an earlier survey data or a previous scan, depending on the specific application. The framework considered here is the source coding problem that arises when data is to be communicated over a degraded broadcast channel to two receivers that have prior side information, and the three terminals (the transmitter and the two receivers) use a separate source-channel coding paradigm~\cite{Vellambi-Timo-AusCTW2014}. The problem of characterizing the \emph{achievable rate-distortion region} of the general successive refinement problem with receiver side information is open~\cite{Steinberg-Merhav-SRProblem,Tian-Diggavi-SRProblem,Timo-Chan-Grant}. The version of the successive refinement problem where the private message is absent, known as the Heegard-Berger problem, is also open~\cite{Timo-Chan-Grant,HeegardBerger-BHProblem, Kaspi-HBProblem}. However, complete characterization exists for specific settings of both successive refinement and Heegard-Berger problems. For example, the rate region of the successive refinement problem is known when the side information of the receiver that receives one message is a degraded version of side information of the other receiver~\cite{Steinberg-Merhav-SRProblem}. Similarly, the Heegard-Berger problem has been solved when the side information is degraded~\cite{HeegardBerger-BHProblem}, mismatched degraded~\cite{Watanabe-HB}, or conditionally less noisy~\cite{Timo-Oechtering-Wigger-HB}. Additionally, the HB problem has also been solved under list decoding constraints (closely related to logarithmic-loss distortion functions)~\cite{Chai-HB}, degraded message sets~\cite{Benammar-Zaidi-HB}, and many vector Gaussian formulations~\cite{Unal-Wagner-HB-Vector-Gaussian1,Unal-Wagner-HB-Vector-Gaussian2}. The common reconstruction variant of the Wyner-Ziv problem was first motivated and solved by Steinberg~\cite{SteinbergCR}. Common reconstructions in other problems were then considered in~\cite {HB-CR2013}. Benammar and Zaidi recently considered the HB problem under a three-way common reconstructions condition with degraded message sets~\cite{Benammar-Zaidi-HB,Benammar-Zaidi-HB-IZS}. In our previous work~\cite{Vellambi-Timo-ITW2013}, we characterized the rate region for several cases of the HB problem with the CRR requirement. In this work, we present single-letter inner and outer bounds for the rate region of the successive refinement problem with receiver side information and the CRR requirement (termed as the SR-CRR problem). For several specific cases of the underlying joint distribution between the source and the side information random variables (including those in our previous work~\cite{Vellambi-Timo-ISIT2014}), we prove that the inner and outer bounds match, and therefore yield a characterization of the rate region. The characterization indicates that while the receiver side information can be fully utilized for reducing the communication rate by means of binning, only the G\'{a}cs-K\"{o}rner common randomness between the random variables (i.e., both auxiliary and side information) available to the two receivers can be used for generating the reconstructions. This feature is also seen in our characterization for the HB problem with the CRR requirement in~\cite{Vellambi-Timo-ITW2013}. This single-letter characterization for the rate region of the SR-CRR problem derived in this work is \emph{unique} in the sense that it is the first rate region formulation where the G\'{a}cs-K\"{o}rner common randomness explicitly appears in the single-letter constraint corresponding to the receiver source reconstructions. Unlike the best-known bounds for the successive refinement problem, the characterization of the SR-CRR rate region (when the source satisfies a certain support condition) requires only one auxiliary random variable that is decoded by both receivers. Thus, the CRR requirement obviates the need for a second auxiliary random variable to absorb the private message. The paper is organized as follows. Section~\ref{sec-N} introduces some basic notation; Section~\ref{Sec:Gacs-Korner} reviews the concept of G\'acs-K\"orner common randomness; and Section~\ref{sec-PD} formally defines the successive refinement problem with the common receiver reconstruction constraint. The characterization of the paper's main contributions are summarized in Section~\ref{sec-NewResults}, including the single-letter characterization of the rate region, and the proof of the discontinuity of the characterization with the source distribution. The reader will be directed to the respective appendices for the proofs of the results contained in Section~\ref{sec-NewResults}. Finally, Section~\ref{Sec:Conclu} concludes this work. \section{Notation, G\'acs-K\"orner Common Randomness, and Problem Setup} \subsection{Notation}\label{sec-N} Let $\mathbb{N} = \{1,2,\ldots\}$ denote the natural numbers. Uppercase letters (e.g., $S$, $U$, $V$) represent random variables (RVs), and the script versions (e.g., $\eS$, $\U$, $\V$) denote the corresponding alphabets. All alphabets in this work are assumed to be finite. Realizations of RVs are given by lowercase letters (e.g., $s$, $u$, $v$). For RVs $A,B,C$, we denote $A\mkv B\mkv C$ if they form a Markov chain. Given RVs $A$ and $B$, we let $A\equiv B $ if and only if \begin{equation} H(A|B)=H(B|A)=0. \end{equation} Given $k$ jointly correlated random variables $A_1,\ldots, A_l$, their support set is defined by \begin{align} \notag \supp(A_1,\ldots,A_k) &= \big\{(a_1,\ldots,a_k) \in \A_1 \times \cdots \times \A_k: p_{A_1\cdots A_k}(a_1,\ldots,a_k)>0\big\}. \end{align} For a set $T$, and $a,b\in T$, we let \begin{align} {\mathds{1}}\{a=b\} &:= \left\{\begin{array}{ll} 1, & a=b \\ 0, & a\neq b \end{array}\right., \end{align} and let $\bar{\mathds{1}}\{a= b\} := 1- \mathds{1}\{a= b\}$. Vectors are indicated by superscripts, and their components by subscripts; for example, $x^n:= (x_1,\ldots,x_n)$ and $x^{n\setminus i} := (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)$. We will use $\mpv$ to denote the least positive probability mass over the support of random variables; for example, given a joint probability mass function (henceforth, pmf) $p_{A_1\cdots A_k}$, \begin{align} \mpv_{A_1} &= \min\big\{p_{A_1}(a) : a_1 \in \supp(A_1) \big\},\\ \mpv_{A_1\cdots A_k} &= \min\big\{p_{A_1\cdots A_k}(a_1,\ldots,a_k): (a_1,\ldots,a_k)\in\supp(A_1,\ldots, A_k)\big\},\\ \mpv_{A_1|A_2} &= \min\big\{p_{A_1|A_2}(a_1|a_2): (a_1,a_2)\in\supp(A_1, A_2)\big\}. \end{align} The probability of an event $E$ is denoted by $\Pr(E)$, and $\Exp$ denotes the expectation operator. Lastly, for $\ensuremath{\varepsilon}>0$. the set of $\ensuremath{\varepsilon}$-letter-typical sequences of length $n$ according to pmf $p_X$ is denoted by $T^n_{\ensuremath{\varepsilon}}[p_X]$~\cite{Kramer-MUIF}. \subsection{G\'acs-K\"orner Common Randomness}\label{Sec:Gacs-Korner} Given two jointly correlated random variables $X$ and $Y$, the G\'{a}cs and K\"{o}rner's \emph{common randomness} between $X$ and $Y$~\cite{Gacs-KornerCI} is the random variable $Z$ with the largest entropy such that $H(Z|X)=H(Z|Y)=0$. This notion of common randomness will play a key role in this paper. To define this notion of common randomness we introduce the following terminology. Given $(X,Y) \sim p_{XY}$ on $\X\times \Y$, let $\mathbb{G}^{X,Y}[p_{XY}]$ denote the bipartite graph with left nodes $\X$, right nodes $\Y$, and edges between $x\in\X$ and $y\in\Y$ if and only if $p_{XY}(x,y)>0$. Now define an equivalence relation on $\Y$ by $y_1\xrightleftharpoons{} y_2$ if and only if they are in the same connected component of $\mathbb{G}^{X,Y}[p_{XY}]$. Finally, let \begin{equation}\label{Eqn:GKMapping1} \GK^{X,Y}: \Y \rightarrow \Y \end{equation} be any mapping satisfying \begin{equation}\label{Eqn:GKMapping} \GK^{X,Y}(y_1)=\GK^{X,Y}(y_2) \quad \text{iff} \quad y_1\xrightleftharpoons{} y_2. \end{equation} \begin{figure}[t!] \begin{center} \includegraphics[width=0.45\textwidth]{Fig1} \caption{Ilustration of $\mathbb{G}^{X,Y}[p_{XY}]$ and $\GK^{X,Y}$.}\label{FIGURE:1} \end{center} \end{figure} Of course, there are multiple choices for the \emph{G\'acs-K\"orner} mapping in~\eqref{Eqn:GKMapping1}. However, all such choices are equivalent in the sense that if $\GK^{X,Y}_1$ and $\GK^{X,Y}_2$ satisfy~\eqref{Eqn:GKMapping} then \begin{equation} \GK^{X,Y}_1(Y)\equiv \GK^{X,Y}_2(Y). \end{equation} As an illustration, let $\X=\{a,b,c,d,e,f,g,h\}$ and let $\Y=\{\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta\}$. Consider the following pmf $p_{XY}$. \begin{align} p_{XY}=\frac{1}{20} \begin{array}{cc} \color{red} a\,\,\,\,\,\, b\,\,\,\,\,\, c\,\,\,\,\, d\,\,\,\,\,\,\,e\,\,\,\,\, f\,\,\,\,\, g\,\,\,\,\, h \color{black}& \\ \left[\begin{array}{ccccccccc} 0&0&1& 2&\multicolumn{1}{|}{}\hspace{2.5mm}0&0&0&0\\ 1&2&1& 0&\multicolumn{1}{|}{}\hspace{2.5mm}0&0&0&0\\\cline{1-6} 0&0&0& 0&\multicolumn{1}{|}{}\hspace{2.25mm}1&2&\multicolumn{1}{|}{}\hspace{2.5mm}0 & 0\\ 0&0&0& 0&\multicolumn{1}{|}{}\hspace{2.25mm}2&1&\multicolumn{1}{|}{}\hspace{2.5mm}0 & 0\\\cline{5-8} 0&0&0& 0&\hspace{0.45mm}0&0&\multicolumn{1}{|}{}\hspace{2.5mm}2&1\\ 0&0&0&0&\hspace{0.45mm}0&0&\multicolumn{1}{|}{}\hspace{2.5mm}2&0\\ 0&0&0&0&\hspace{0.45mm}0&0&\multicolumn{1}{|}{}\hspace{2.5mm}0&2\\ \end{array}\right] & \color{red} \hspace{-2mm}\begin{array}{c} \alpha\\ \beta\\ \gamma\\ \delta\\ \epsilon\\ \zeta\\ \eta\end{array} \color{black} \end{array}\label{eqn-Fig1-example} \end{align} Figure~\ref{FIGURE:1} illustrates the bipartite graph representation of $p_{XY}$, and depicts one possible choice for the G\'acs-K\"orner mapping $\GK^{X,Y}$ satisfying the requirement in \eqref{Eqn:GKMapping}. Notice that that $\mathbb{G}^{X,Y}[p_{XY}]$ contains three connected components, and hence $\GK^{X,Y}$ is a ternary RV taking values in $\{b,d,g\}$ if the chosen mapping is the one illustrated in the figure. Note that for this pmf, there are $4\times 2\times 2=16$ equivalent choices for the mapping. From the definition above, two properties of the G\'{a}cs and K\"{o}rner's common randomness are evident. \begin{itemize} \item The G\'{a}cs and K\"{o}rner's common randomness between two random variables is \emph{symmetric}, i.e., \begin{equation} \GK^{X,Y}(Y) \equiv \GK^{Y,X}(X). \end{equation} Note however that the above two G\'{a}cs and K\"{o}rner's common randomness variables take values over different alphabets even though each is a function of the other, i.e., $\GK^{X,Y}(Y)$ and $\GK^{Y,X}(X)$ are random variables over $\Y$ and $\X$, respectively. \item Since the G\'{a}cs and K\"{o}rner's common randomness depends on the pmf $p_{XY}$ only through the bipartite graph, the G\'{a}cs and K\"{o}rner's common randomness between RVs computed using two pmfs $p_{XY}$ and $q_{XY}$ are identical if: (a) $\mathbb{G}^{X,Y}[p_{XY}]$ and $\mathbb{G}^{X,Y}[q_{XY}]$ are graph-isomorphic; and (b) the probabilities of the components of $p_{XY}$ and that of $q_{XY}$ are permutations of one another. To illustrate this, consider $p_{XY}$ of \eqref{eqn-Fig1-example} and the following pmf $q_{XY}$. \begin{align} q_{XY}=\frac{1}{20} \begin{array}{cc} \color{red} \,a\,\,\,\,\,\, b\,\,\,\,\,\, c\,\,\,\,\,\, d\,\,\,\,\,e\,\,\,\,\, f\,\,\,\,\, g\,\,\,\,\, h \color{black}& \\ \left[\begin{array}{ccccccccc} 0&1&1& 1&\multicolumn{1}{|}{}\hspace{2.5mm}0&0&0&0\\ 1&0&1& 2&\multicolumn{1}{|}{}\hspace{2.5mm}0&0&0&0\\\cline{1-6} 0&0&0& 0&\multicolumn{1}{|}{}\hspace{2.25mm}2&3&\multicolumn{1}{|}{}\hspace{2.5mm}0 & 0\\ 0&0&0& 0&\multicolumn{1}{|}{}\hspace{2.25mm}1&0&\multicolumn{1}{|}{}\hspace{2.5mm}0 & 0\\\cline{5-8} 0&0&0& 0&\hspace{0.45mm}0&0&\multicolumn{1}{|}{}\hspace{2.5mm}1&2\\ 0&0&0&0&\hspace{0.45mm}0&0&\multicolumn{1}{|}{}\hspace{2.5mm}3&0\\ 0&0&0&0&\hspace{0.45mm}0&0&\multicolumn{1}{|}{}\hspace{2.5mm}0&1\\ \end{array}\right] & \color{red} \hspace{-2mm}\begin{array}{c} \alpha\\ \beta\\ \gamma\\ \delta\\ \epsilon\\ \zeta\\ \eta\end{array} \color{black} \end{array} \end{align} The G\'{a}cs and K\"{o}rner's common randomness depends between $X$ and $Y$ computed using either $p_{XY}$ or $q_{XY}$ yields a ternary random variable with the pmf $[ \frac{7}{20}\,\, \frac{3}{10}\,\, \frac{7}{20}]$. \end{itemize} In the remainder of this work, for a given pmf $p_{XY}$ over $\X \times \Y$, we will assume an arbitrary but fixed choice for the G\'{a}cs and K\"{o}rner's common randomness mapping that satisfies~\eqref{Eqn:GKMapping} without explicitly specifying this mapping. On account of notational ease, we will also drop the argument, and abbreviate it by \begin{align} \GK^{X,Y} := \GK^{X,Y}(Y). \end{align} It is to be assumed that the argument is always the second random variable in the superscript, and consequently, the G\'{a}cs and K\"{o}rner's common randomness $\GK^{X,Y}$ is a random variable over the alphabet $\Y$ of the second variable. This is, quite simply, a only a \emph{notational bias}, since G\'{a}cs and K\"{o}rner's common randomness is indeed symmetric. Before we proceed to formally state the problem investigated and the main results, we present two results that pertain solely to G\'acs-K\"orner common randomness that we will need in the main section of this work. Both results present decompositions of the G\'{a}cs-K\"{o}rner common randomness between two random variables when additional information about the support of the joint pmf of the two variables is known. \begin{lemma}\label{Lem:Ancillary2} Suppose that the support set of $(A_1,A_2,U,V)\sim q_{A_1A_2UV}$ satisfies \begin{align} \supp(A_1,A_2,U,V) = \supp(A_1,A_2)\times \supp(U,V). \end{align} Then, \begin{equation} \GK^{A_1U,A_2V}\equiv (\GK^{A_1,A_2},\GK^{U,V}). \end{equation} \end{lemma} \begin{IEEEproof} The proof can be found in Appendix~\ref{APP-A}. \end{IEEEproof} \begin{lemma}\label{lem-drop1var} If $(X,Y,Z) \sim q_{XYZ}$ satisfies \begin{equation} \supp(X,Y,Z) = \supp(X,Y)\times \supp(Z), \end{equation} then \begin{equation} \GK^{X,YZ} \equiv \GK^{X,Y}. \end{equation} \end{lemma} \begin{IEEEproof} Define a constant random variable $W$ over a singleton alphabet, say $\{w\}$ and let $q_W$ denote the degenerate pmf of $W$. Define $q_{XYZ} = q_W q_{XYZ}$. Then, one can see that \begin{align} \supp(W,X,Y,Z) &= \supp(X,Y,Z) \times \supp(W) = \supp(X,Y)\times \supp(Z)\times\supp(W) \\ &=\supp(X,Y)\times \supp(Z,W). \end{align} Then, an application of Lemma~\ref{Lem:Ancillary2} yields \begin{align} \GK^{X,YZ} \equiv (\GK^{X,Y}, \GK(Z,W)) \step{(a)}\equiv\GK^{X,Y}, \end{align} where (a) follows since $W$ is a constant RV. \end{IEEEproof} \subsection{Problem Setup --- Successive Refinement with the CRR Constraint}\label{sec-PD} Let pmf $p_{SUV}$ on $\eS \times \U \times \V$, a reconstruction alphabet $\hat{\eS}$, and a bounded distortion function $d:\eS \times \hat{\eS} \rightarrow [0,\bar D]$ be given. We assume that $\bar D \in (0,\infty)$. A DMS $(S,U,V) \sim p_{SUV}$ emits an i.i.d. sequence \begin{equation} (S^n,U^n,V^n) = (S_1,U_1,V_1), (S_2,U_2,V_2), \ldots, (S_n,U_n,V_n). \end{equation} As illustrated in Fig.~\ref{FIGURE:2}, the engineering problem is to encode the \emph{source} $S^n$ into a common message $M_{uv}$ communicated to both receivers, and a private message $M_v$ communicated only the receiver having the side information $V^n$ so that the following three conditions are satisfied: \begin{enumerate} \item A receiver with access to the \emph{side information} $U^n$ and the common message $M_{uv}$ can output an estimate $\hat{S}^n$ of $S^n$ to within a prescribed average (per-letter) distortion $D$. \item A receiver with access to the \emph{side information} $V^n$ and both (common and private) messages $M_{uv}$ and $M_v$ can output an estimate $\tilde{S}^n$ of $S^n$ to within a prescribed average (per-letter) distortion $D$. \item The estimates $\hat{S}^n$ and $\tilde{S}^n$ (both defined on the set $\hat{\eS}^n$) are identical to one another almost always. \end{enumerate} The aim then is to characterize the rates of the common and private messages that need to be communicated to achieve the above requirements. The following definition formally defines the problem. \begin{figure}[h!] \begin{center} \includegraphics[width=0.5\textwidth]{Fig2} \caption{Successive refinement with receiver side information and common receiver reconstructions (CRR).}\label{FIGURE:2} \end{center} \end{figure} \begin{definition}\label{defn-CRRRateregion} Fix $D \geq 0$. We say that a rate pair $(r_{uv}, r_v) \in [0,\infty)^2$ is $D$-\emph{admissible} if for each $\ensuremath{\varepsilon}>0$ there exist a sufficiently large blocklength $n\in\mathbb N$, and: \begin{subequations} \begin{enumerate}[(a)] \item two encoders \begin{align} \mathtt E_{uv}:\eS^n&\rightarrow \big\{1,2,\ldots,\big\lceil 2^{n(r_{uv}+\ensuremath{\varepsilon})}\big\rceil\big\},\\ \mathtt E_{v}:\eS^n&\rightarrow \big\{1,2,\ldots,\big\lceil 2^{n(r_{v}+\ensuremath{\varepsilon})}\big\rceil\big\}; \end{align} \item and two decoders \begin{align} \mathtt D_u&:\big\{1,2,\ldots,\big\lceil 2^{n(r_{uv}+\ensuremath{\varepsilon})}\big\rceil\big\}\times \U^n \rightarrow \hat{\eS}^n,\\ \mathtt D_v&:\big\{1,2,\ldots,\big\lceil 2^{n(r_{uv}+\ensuremath{\varepsilon})}\big\rceil\big\} \times\big\{1,\ldots,\big\lceil 2^{n(r_{v}+\ensuremath{\varepsilon})}\big\rceil \big\}\times \V^n \rightarrow \hat{\eS}^n, \end{align} \end{enumerate} \end{subequations} such that the reconstructions \begin{subequations} \begin{align} \hat{S}^n &:=\mathtt D_u\big(\mathtt E_{uv}(S^n),U^n\big),\\ \tilde{S}^n &:= \mathtt D_v\big(\mathtt E_{uv}(S^n), \mathtt E_v(S^n),V^n\big), \end{align} \end{subequations} satisfy: \begin{subequations}\label{eqn-RRDefn1} \begin{align} \frac{1}{n} \sum_{i=1}^n \Exp\big[d(S_i,\hat{S}_i)\big] &\leq D+\ensuremath{\varepsilon},\\ \frac{1}{n} \sum_{i=1}^n \Exp\big[d(S_i,\tilde{S}_i)\big]&\leq D+\ensuremath{\varepsilon}, \\ \Pr\big[\hat S^n\neq \tilde S^n\big] &\leq \ensuremath{\varepsilon}. \end{align} \end{subequations} \end{definition} \begin{definition}\label{Def:RateRegion} The \emph{$D$-admissible rate region} $\mathsf{\R}(D)$ of the successive refinement problem with CRR is the set of all $D$-admissible rate pairs. \end{definition} The main problem of interest in this paper is to characterize the $D$-admissible rate region $\R(D)$. Define $\underline{D}$ by \begin{equation} \underline{D} :=\mathop{\min}_{\phi:\ \eS\rightarrow\hat\eS}\Exp d(S,\phi(S)). \end{equation} On one hand, if $0 \leq D < \underline{D}$, then $\mathsf{\R}(D) = \emptyset$. On the other, if $D > \overline{D}$, then $\mathsf{\R}(D) = [0,\infty)^2$. For the interesting interval of non-trivial values of $D \in [\underline{D},\overline{D}]$, the rate region $\R(D)$ is a closed and convex subset of $[0,\infty)^2$. This nontrivial interval of values of $D$ will be the subject of our investigation for the remainder of the paper. \section{Main Results}\label{sec-NewResults} In this section, we will present inner (achievability) and outer (converse) bounds for the $D$-admissible rate region $\R(D)$, and we show that these bounds are tight in a variety of nontrivial settings. We will characterize these bounds through the following three rate regions defined over three corresponding spaces of auxiliary random variable pmfs. \subsection{Three single-letter rate regions and their properties}\label{Mainresults-rrdefns} \begin{definition}\label{Def:PstarRegionk} For $k \in \mathbb{N}$, let $\mc{P}^*_{D,k}$ denote the set of all pmfs $ q_{ASUV}$ defined on $\A \times \eS \times \U \times \V$ such that $(A,S,U,V)\sim q_{ASUV}$ satisfies the following conditions: \begin{enumerate}[(i)] \item $q_{SUV}=p_{SUV}$; \item $|\A| \leq k$; \item $A\mkv S \mkv (U,V)$; and \item there exists a function $f:\A\times\V\rightarrow\hat \eS$ for which \begin{align}\label{eqn-Defn1eq3} \Exp \big[d \big(S, f(\GK^{ AU,AV})\big)\big] \leq D. \end{align} \end{enumerate} \end{definition} \begin{definition}\label{Def:RstarRegionk} Let $\R_{k}^*(D)$ denote the set of all rate pairs $(r_{uv},r_v) \in [0,\infty)^2$ satisfying \begin{subequations}\label{Eqn:Rsk:Rates} \begin{align} r_{uv} & \geq I(S;A|U)\label{eqn-*defnrate1}\\ r_v+r_{uv} & \geq I(S;A|V)\label{eqn-*defnrate2} \end{align} \end{subequations} for some $(A,S,U,V)\sim q_{ASUV} \in \mc{P}^*_{D,k}$. \end{definition} \begin{definition}\label{Def:PddagRegionk} For $k \in \mathbb{N}$, let $\mc{P}_{D,k}^\ddagger$ denote the set of all pmfs $q_{ABCSUV}$ defined on $\A \times \B \times \C \times \eS \times \U \times \V$ such that $(A,B,C,S,U,V) \sim q_{ABCSUV}$ satisfies the following constraints: \begin{enumerate}[(i)] \item $q_{SUV}=p_{SUV}$; \item $\max\{|\A|,|\B|,|\C|\}\leq k$; \item $(A,B,C)\mkv S \mkv (U,V)$; and \item there exists a function $f:\A\times\C\times\V\rightarrow\hat \eS$ for which \begin{align} {\textstyle\Exp \big[d (S, f\big(\GK^{ ABU,ACV}\big)\big]\leq D}.\label{eqn-Defn1eq1} \end{align} \end{enumerate} \end{definition} \begin{definition}\label{Def:RddagRegionk} Let $\R_k^\ddagger(D)$ denote the set of all rate pairs $(r_{uv},r_v) \in [0,\infty)^2$ satisfying \begin{subequations} \begin{align} r_{uv} &\geq I(S;A,B|U),\label{eqn-ddagdefnrate1}\\ r_v+r_{uv} &\geq I(S;B|A,C,U,V) +I(S;A,C|V) \label{eqn-ddagdefnrate2} \end{align} \end{subequations} for some $(A,B,C,S,U,V)\sim q_{ABCSUV} \in \mc{P}^\ddag_{D,k}$. \end{definition} \begin{definition}\label{Def:PdagRegionk} For $k \in \mathbb{N}$, let \begin{equation} \mc{P}_{D,k}^\dagger = \Big\{q_{ABCSUV} \in\mc{P}_{D,k}^\ddagger: B\mkv (A,S) \mkv C\Big\}. \end{equation} \end{definition} \begin{definition}\label{Def:RdagRegionk} Let $\R_{k}^{ \dagger}(D)$ denote the set of all rate pairs $(r_{uv},r_v)\in [0,\infty)^2$ satisfying \begin{subequations}\label{Eqn:DefRdagger} \begin{align} r_{uv} &\geq I(S;A,B|U)\label{eqn-dagdefnrate1}\\ r_v+r_{uv} &\geq \max\{I(S;A|U), I(S;A|V)\} + I(S;B|A,U) + I(S;C|A,V)\label{eqn-dagdefnrate2} \end{align} \end{subequations} for some $(A,B,C,S,U,V)\sim q_{ABCSUV} \in \mc{P}^\dag_{D,k}$. \end{definition} We can establish the following preliminary inclusions between the three rate regions defined above. \begin{lemma}\label{lem-*kddagkinclusion} For any $k\in\mathbb{N}$, $\R^*_{k}(D) \subseteq \R^\ddagger_k(D)$. \end{lemma} \begin{IEEEproof} By simply choosing $B=C=A$ with $|\mc A|\leq k$, we cover all rate pairs that line in $\R^*_k(D)$, and hence, $\R^*_{k}(D)$ is a subset of $\R^\ddagger_k(D)$. \end{IEEEproof} \begin{lemma}\label{lem-dagkddagkinclusion} For any $k\in\mathbb{N}$, $\R^\dag_{k}(D)\subseteq \R^\ddagger_k(D)$. \end{lemma} \begin{IEEEproof} First, note that $\mathcal P_{D,k}^\dagger \subseteq \mathcal P_{D,k}^\ddagger$. So we are done if we show that the RHS of \eqref{eqn-ddagdefnrate2} is numerically smaller than that of \eqref{eqn-dagdefnrate2} for any pmf in $\mathcal P_{D,k}^\dagger$. To do that, pick $p_{ABCSUV}\in \mathcal P_{D,k}^\dagger$ and consider the following argument. \begin{align} I(S;B|A,C,U,V) +I(S;A,C|V) &= H(B|A,C,U,V)-H(B|A,C,S,U,V) + I(S;A,C|V)\notag\\ &\step{(a)}=H(B|A,C,U,V)-H(B|A,C,S) + I(S;A,C|V)\notag\\ &\step{(b)}=H(B|A,C,U,V)-H(B|A,S) + I(S;A,C|V)\notag\\ &\step{(c)}\leq H(B|A,U)-H(B|A,S) + I(S;A,C|V)\label{eqn-Uintro}\\ & \step{(d)}= I(S; B|A,U) + I(S;A,C|U),\label{eqn-Uintro1} \end{align} where (a) follows from the chain $(A,B,C)\mkv S \mkv (U,V)$; (b) follows from the chain $B\mkv (A,S) \mkv C$; (c) follows by dropping variables in the conditioning; and finally (d) follows by reintroducing $U$ in the second term of \eqref{eqn-Uintro} without affecting the numerically affecting the terms. Finally, the claim follows by noting that \eqref{eqn-Uintro1} is bounded below by the RHS of \eqref{eqn-dagdefnrate2} thereby completing the proof of this claim. \end{IEEEproof} While the above two inclusions hold true for all DMSs $p_{SUV}$, we can establish stronger results if we know something more about $p_{SUV}$. In specific, if we know that the pmf $p_{SUV}$ satisfies the full-support condition of \eqref{eqn:Condition1}, then the following \emph{reverse} inclusion also holds albeit with some alphabet size readjustment. In other words, when the full-support condition is met, any rate pair that meets \eqref{eqn-ddagdefnrate1} and \eqref{eqn-ddagdefnrate2} (with auxiliary RVs $A,B$, and $C$) also meets \eqref{eqn-*defnrate1} and \eqref{eqn-*defnrate2} for a different auxiliary RV $A$ with an appropriately larger alphabet. \begin{lemma}\label{thm-RegionEquivalence} If the support of $(S,U,V) \sim p_{SUV}$ satisfies \begin{equation}\label{eqn:Condition1} \supp(S,U,V)=\supp(S)\times\supp(U,V), \end{equation} then \begin{equation} \R^{\ddagger}_{k}(D) \subseteq \R^{*}_{k^2}(D), \quad \forall\ k \in \mathbb{N}. \end{equation} \end{lemma} \begin{IEEEproof} Lemma~\ref{thm-RegionEquivalence} is proved in Appendix~\ref{APP-B}. \end{IEEEproof} Observe that each of the three rate regions defined above (Definitions~\ref{Def:RstarRegionk}, \ref{Def:RddagRegionk}, and \ref{Def:RdagRegionk}) can potentially be enlarged by merely increasing $k$. In other words, we are guaranteed to have \begin{align} \R_1^\lambda(D)\subseteq \R_2^\lambda(D)\subseteq\R_3^\lambda(D)\subseteq\cdots, \qquad \lambda\in\{\star,\ddagger,\dagger\}.\label{eqn-sequenceofsets} \end{align} Since we do not impose restrictions on how we encode the source, we can allow the alphabet sizes of the auxiliary RVs to be finite but arbitrarily large. Hence, it makes sense to introduce the following notation for the limiting rate regions allowing the alphabets of auxiliary random variables to be any finite set. \begin{definition} \begin{align} \R^\lambda(D) := \bigcup_{k\in\mathbb{N}} \R^{\lambda}_{k}(D), \qquad \lambda\in\{\star,\ddagger,\dagger\}. \end{align} \end{definition} However, from a computational point of view, it is preferable that the sequence of sets in \eqref{eqn-sequenceofsets} not grow indefinitely with $k$. The following result ensures that this, indeed, does not happen. It quantifies the bounds on the alphabet size of the auxiliary random variables beyond which there is no strict enlargement of these regions. \begin{lemma}\label{thm-RegionEquivalence2} For all integers $k \in \mathbb{N}$, we have the following. \begin{subequations} \begin{alignat}{3} \R^*_{k}(D) &\subseteq \R^*(D)=\R^*_{|\eS|+2}(D) ,\label{eqn-rateregionalphabetbnd1}\\ \R^\ddagger_{k}(D)&\subseteq \R^\ddagger(D)=\R^\ddagger_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}(D),\label{eqn-rateregionalphabetbnd2}\\ \R^\dagger_{k}(D)&\subseteq \R^\dagger(D)=\R^\dagger_{ |\eS|(|\eS|+4)|\hat\eS|^{|\U|}+1}(D).\label{eqn-rateregionalphabetbnd3} \end{alignat} \end{subequations} \end{lemma} \begin{IEEEproof} The proof for claim for $\R^\ddagger(D)$ is presented in detail in Appendix~\ref{APP-C}. The proof for \eqref{eqn-rateregionalphabetbnd1} and \eqref{eqn-rateregionalphabetbnd3} is almost identical to that of \eqref{eqn-rateregionalphabetbnd2}, and the difference are highlighted in Remarks~\ref{rem-regionequiv2} and \ref{rem-regionequiv2b} in Appendix~\ref{APP-C}. \end{IEEEproof} We conclude this section with two properties of the above three regions. \begin{lemma}\label{Lem:RstarConvex} The regions $\R^*(D)$, $\R^\ddagger(D)$, and $\R^\dagger(D)$ are convex. \end{lemma} \begin{IEEEproof} A proof of the claim for $\R^*(D)$ can be found in Appendix~\ref{APP-D}. The proofs of the convexity of the other two regions are identical, and are omitted. \end{IEEEproof} \begin{lemma}\label{Lem:RstarCont} Let for each $i\in\mathbb{N}$, $D_i\in [\underline{D},\overline{D}]$ and $(r_{uv}^{(i)},r_{v}^{(i)})\in \R^*_{D_i,k}$ be given. Suppose that $\lim\limits_{i\rightarrow \infty} D_i = \mathsf D $, and $\lim\limits_{i\rightarrow \infty} (r_{uv}^{(i)},r_{v}^{(i)}) = (\mathsf{r_{uv},r_v})$. Then, $(\mathsf{r_{uv},r_v}) \in \R^*_{\mathsf D,k}$. \end{lemma} \begin{IEEEproof} A proof of the claim for $\R^*(D)$ can be found in Appendix~\ref{APP-E}. The proofs for the other two regions are identical, and are omitted. \end{IEEEproof} By simply choosing $D_i = D\in[\underline D, \overline D]$, $i\in \mathbb N$, we obtain the following result. \begin{remark} For any $D\in[\underline D, \overline D]$, the regions $\R^*(D)$, $\R^\ddagger(D)$, and $\R^\dagger(D)$ are topologically closed. \end{remark} \subsection{A single-letter characterization for $\R(D)$}\label{Sec-SLChars} In this section, we present our main results on the single-letter characterization of the $D$-admissible rate region. The first two present inner and outer bounds sandwiching the $D$-admissible rate region using the three limiting rate regions given in Definitions~\ref{Def:RstarRegionk}, \ref{Def:RddagRegionk}, and \ref{Def:RdagRegionk}. \begin{theorem}\label{thm-achieve-thm-2} For any $D\in[\underline D, \overline D]$, the regions $\R^{*}(D)$ and $\R^{\dagger}(D)$ are inner bounds to the $D$-admissible rate region $\R(D)$ of the successive refinement problem with the CRR constraint, i.e., \begin{equation} \R^{*}(D) \step{(a)}{\subseteq} \R^\dagger(D)\step{(b)}\subseteq \mathsf{\R}(D). \end{equation} \end{theorem} \begin{IEEEproof} The inclusion (a) follows from Lemmas~\ref{lem-*kddagkinclusion} and~\ref{thm-RegionEquivalence2} above, and a proof of the inclusion (b) can be found in Appendix~\ref{APP-F}. \end{IEEEproof} \begin{theorem}\label{thm-achieve-thm-2a} For any $D\in[\underline D, \overline D]$, the rate region $\R^{\ddagger}(D)$ is an outer bound to the $D$-admissible rate region $\R(D)$ of the successive refinement problem with the CRR constraint, i.e., \begin{equation*} \mathsf{\R}(D) \step{(b)}\subseteq \R^\ddagger(D). \end{equation*} \end{theorem} \begin{IEEEproof} The proof of the inclusion in (a) can be found in Appendix~\ref{APP-G}. \end{IEEEproof} In the absence of the CRR constraint, Steinberg and Merhav's original solution to the \emph{physically-degraded side information} version of the successive refinement problem required three auxiliary random variables (later simplified to two by Tian and Diggavi~\cite{Tian-Diggavi-Aug-2007-A}) and two reconstruction functions. Benammar and Zaidi's solution to their formulation of the successive refinement problem with a common source reconstruction required two auxiliary random variables and a reconstruction function~\cite{Benammar-Zaidi-HB,Benammar-Zaidi-HB-IZS}. The following result, which is the main result in this work, establishes a single-letter characterization of the $D$-admissible rate region $\R(D)$ for several cases of side information. Unlike other characterizations, the rate region $\R(D)$ is completely described by a single auxiliary random variable, and a single reconstruction function whose argument is not the side information and the auxiliary random variable, but the G\'{a}cs-K\"{o}rner common randomness shared by the two receivers. \begin{theorem}\label{thm-3} If the DMS $(S,U,V) \sim p_{SUV}$ falls into one of the following cases, \begin{itemize} \item[A.] $\supp(S,U,V)=\supp(S)\times\supp(U,V)$, \item[B.] $S\mkv V \mkv U$, \item[C.] $S\mkv U\mkv V$ and $\supp(S,U)=\supp(S)\times\supp(U)$, \item[D.] $H(S|U)=0$, \item[E.] $S\mkv U\mkv V$ and $\supp(U,V)=\supp(U)\times\supp(V)$, or \item[F.] $\min\{H(U|S),H(V|S)\}=0$, \end{itemize} then for any $D\in[\underline D, \overline D]$, the inner bound $\R^{*}(D)$ and the outer bound $\R^{\ddagger}(D)$ match, and \begin{equation*} \mathsf{\R}(D) = {\R}^*(D) = \R^\ddagger (D). \end{equation*} \end{theorem} \begin{IEEEproof} Theorem~\ref{thm-3} is proved in Appendix~\ref{APP-I}. \end{IEEEproof} \begin{remark} $\mathsf{\R}(D) = {\R}^*(D)$ when $H(S|V)=0$ or $H(U|V)=0$, but these are subsumed by Case B above. \end{remark} Since in each of the cases in Theorem~\ref{thm-3}, the characterization is given by precisely one auxiliary random variable, it is only natural to wonder as to when a \emph{quantize-and-bin} strategy is optimal. In this strategy, the auxiliary random variable $A$ that the encoder encodes the source into is simply the reconstruction that the receivers require. The encoder upon identifying a suitable sequence of reconstruction symbols simply uses a binning strategy to reduce the rate of communication prior to forwarding the bin index to the receivers. Thus in this strategy, all three terminals (i.e., the transmitter included) are aware of the common reconstruction. To analyze cases in which the quantize-and-bin approach is optimal, we define the corresponding rate region. \begin{definition} The \emph{quantize-and-bin} rate region $\RQB(D)$ is the union of all pairs $(r_{uv},r_v)\in[0,\infty)^2$ such that \begin{subequations}\label{Eqn:QandBin} \begin{align} r_{uv}\geq & I(S;\hat S|U),\\ r_v+r_{uv}\geq & I(S;\hat S | V), \end{align} \end{subequations} where the union is taken over all reconstructions $\hat{S}$ on $\hat{\eS}$ with $\hat{S} \mkv S \mkv (U,V)$ and $\Exp[d(S,\hat S)] \leq D$. \end{definition} Clearly, by setting $A=\hat S$ in the proof of Theorem~\ref{thm-achieve-thm-2} detailed in Appendix~\ref{APP-F}, we infer that $\RQB(D)\subseteq \R^*(D)$. Consequently, we can see that the quantize-and-bin region is always achievable. The following result establishes three conditions under which the quantize-and-bin strategy is not merely achievable, but optimal, i.e., $\RQB(D)=\R^*(D)$. \begin{theorem}\label{thm-3-cor} If the DMS $(S,U,V) \sim p_{SUV}$ falls into one of the following cases, \begin{itemize} \item[ A.] $\supp(S,U,V)=\supp(S)\times\supp(U,V)$ and $H(\GK^{U,V}(V)) = 0$, \item[ B.] $S\mkv U\mkv V$ and $\supp(U,V)=\supp(U)\times\supp(V)$, or \item[ C.] $\min\{H(U|S),\ H(V|S)\} = 0$, \end{itemize} then, the quantize-and-bin strategy is optimal, i.e., \begin{equation} \mathsf{\R}(D) = {\R}^*(D) = \R^\ddagger (D) = \RQB(D). \end{equation} \end{theorem} \begin{IEEEproof} The proof can be found in Appendix~\ref{APP-J}. \end{IEEEproof} \subsection{A Binary Example}\label{Sec:BinaryExample} \begin{figure}[h!] \begin{center} \includegraphics[width=0.4\textwidth]{Fig3} \caption{A family of DMSs based on the binary symmetric channel.} \label{FIGURE:3} \end{center} \end{figure} In this section, we will present a binary example with $\eS = \hat{\eS} = \{0,1\}$ and with the reconstruction distortion measure $d$ being the binary Hamming distortion measure \begin{equation} d(s,\hat{s}) = \left\{ \begin{array}{ll} 1 & \text{ if } \hat{s} \neq s\\ 0 & \text{ otherwise} \end{array}. \right. \end{equation} As illustrated in Figure~\ref{FIGURE:3}, let $\big\{ (S,U,V) \sim p^{\rho,\delta}_{SUV}\big\}_{\rho,\delta \in [0,1]}$ denote a family of DMSs such that (a) $S$ is an equiprobable binary source; (b) $S\mkv U \mkv V$ form a Markov chain; and (c) the channels $p_{U|S}$, and $p_{V|U}$ are binary symmetric channels with crossover probabilities $\rho$ and $\delta$, respectively. Note that for any $0<\rho,\delta<1$, the pmf $p^{\rho,\delta}_{SUV}$ satisfies the conditions of both Case A and Case B of Theorem~\ref{thm-3-cor} above. Hence, we see that the quantize-and-bin strategy is optimal, and the optimal tradeoff between communication rates on the common and private links can be obtained without having to time-share between various operating points (corresponding to different average distortions). The following result presents an explicit characterization of the $D$-admissible rate region $\mathcal{R}(D)[p^{\rho,\delta}_{SUV}]$ for this class of sources. \begin{lemma}\label{Lem:BinaryExample} If\footnote{The case where $\delta = 0$ or $1$ will treated in the next section.} $0 < \rho,\delta < 1$ and $D\in[0,\frac{1}{2}]$, then \begin{equation}\label{Eqn:Lem:BinaryExample1} \R(D)[p^{\rho,\delta}_{SUV}] = \left\{ (r_{uv},r_v) \in [0,\infty)^2: \begin{array}{c} r_{uv} \geq h(\rho \ast D) - h(D)\\[2pt] r_v + r_{uv} \geq h\big((\rho \ast \delta) \ast D\big) - h(D)\\ \end{array} \right\}, \end{equation} where $x \ast y := x(1-y)+y(1-x)$ denotes the \emph{binary convolution} operation, and $h$ is the \emph{binary entropy function}. Otherwise if $D \geq 1/2$, then $R(D)[p^{\rho,\delta}_{SUV}] = [0,\infty)^2$. \end{lemma} \begin{IEEEproof} If the distortion $D \geq 1/2$, then we can trivially meet the distortion requirement by setting $\hat{S}^n = \tilde{S}^n = 0^n$. Consider the non-trivial range of distortions $0 \leq D < 1/2$. For $\delta \in (0,1)$, the corresponding joint pmf $p^{\rho,\delta}_{SUV}$ satisfies $S \mkv U \mkv V$ and $\supp(U,V)=\supp(U)\times\supp(V)$. Therefore, Case B of Theorem~\ref{thm-3-cor} applies, and we have \begin{equation}\label{Eqn:Lem:BinaryExample:1} \R(D)[p^{\rho,\delta}_{SUV}] = \RQB(D)[p^{\rho,\delta}_{SUV}]. \end{equation} \begin{figure}[h!] \begin{center} \includegraphics[width=0.5\textwidth]{Fig4} \caption{Illustration of the quantization $\hat{S}$ used in the proof of Lemma~\ref{Lem:BinaryExample}.} \label{FIGURE:4} \end{center} \end{figure} We first show that the right hand side of~\eqref{Eqn:Lem:BinaryExample1} is an inner bound for $\RQB(D)[p^{\rho,\delta}_{SUV}]$. Choose $\hat{S}$ as illustrated in Figure~\ref{FIGURE:4} with $\Pr[\hat{S} = 1] = \Pr[\hat{S} = 0] = 1/2$ and $\hat{S} \mkv S \mkv U \mkv V$. Then, $\Exp[d(S,\hat{S})] = D$ and \begin{subequations}\label{Eqn:Lem:BinaryExample:InnerRate} \begin{align} I(S;\hat{S}|U) &= h(D \ast \rho) - h(D),\\ I(S;\hat{S}|V) &= h( (\rho \ast \delta)\ast D) - h(D). \end{align} \end{subequations} Thus, from the above, we conclude that \begin{align} \left\{(r_{uv}, r_v) \in [0,\infty)^2\,\,\bigg | \begin{array}{rl} r_{uv}\geq & h\big(\rho \ast D \big) - h(D)\\[3pt] r_v+r_{uv}\geq &h\big((\rho \ast \delta) \ast D \big) - h(D) \end{array} \right\}\subseteq \RQB(D)[p^{\rho,\delta}_{SUV}]\step{\eqref{Eqn:Lem:BinaryExample:1}}{=} \R(D)[p^{\rho,\delta}_{SUV}]. \end{align} To establish that choosing $\hat S$ according to Fig.~\ref{FIGURE:4} suffices to cover the entire rate region, we need to show that the LHS of the above equation is an outer bound for $\RQB(D)[p^{\rho,\delta}_{SUV}]$. To that end, we proceed as follows. \begin{align} \R(D)[p^{\rho,\delta}_{SUV}] &\step{(a)}{=} \bigcup_{ \substack{\hat{S} \mkv S \mkv (U,V)\\\Exp[d(S,\hat{S})] \leq D}} \left\{(r_{uv}, r_v) \in [0,\infty)^2\,\,\bigg | \begin{array}{rl} r_{uv}\geq & I(S;\hat{S}|U)\\[5pt] r_v+r_{uv}\geq & I(S;\hat{S}|V) \end{array} \right\}\notag\\ &\step{(b)}{\subseteq} \left\{(r_{uv}, r_v) \in [0,\infty)^2\,\,\Bigg | \begin{array}{rl} r_{uv}\geq & \min\limits_{\substack{\hat{S} \mkv S \mkv U\\\Exp[d(S,\hat{S})] \leq D}} I(S;\hat{S}|U)\\[5pt] r_v+r_{uv}\geq & \min\limits_{\substack{\hat{S} \mkv S \mkv V\\\Exp[d(S,\hat{S})] \leq D}} I(S;\hat{S}|V) \end{array} \right\}\label{eqn-bimini}\\ &\step{(c)}{\subseteq} \left\{(r_{uv}, r_v) \in [0,\infty)^2\,\,\bigg | \begin{array}{rl} r_{uv}\geq & h\big(\rho \ast D \big) - h(D)\\[3pt] r_v+r_{uv}\geq &h\big((\rho \ast \delta) \ast D \big) - h(D) \end{array} \right\}, \end{align} where (a) follows from Case B of Theorem~\ref{thm-3-cor} of Sec.~\ref{Mainresults-rrdefns} and the definition of $\RQB(D)$; (b) creates an outer bound by relaxing the optimizing problem; and (c) follows from using Steinberg's \emph{common-reconstruction function for the binary symmetric source} (see (14) of \cite{SteinbergCR}) to obtain the solutions to the two minimizations in \eqref{eqn-bimini}. The channel $p_{S|\hat S}$ that \emph{simultaneously} minimizes both optimization problems in \eqref{eqn-bimini} is precisely the choice in Fig.~\ref{FIGURE:4}. \end{IEEEproof} Notice carefully that the proof of the above result uses one important aspect of Steinberg's characterization of the quantize-and-bin rate region for the point-to-point rate-distortion problem. \begin{itemize} \item When the source and the receiver side-information are related by a binary symmetric channel , the optimal \emph{reverse test channel} $p_{S|\hat S}$ to be a binary symmetric channel with crossover probability $D$ \emph{independent} of the crossover probability of the channel relating the source and the receiver side information (see (14) of \cite{SteinbergCR}). \end{itemize} Consequently, the following general observation holds. \begin{remark}\label{rem:3} The $D$-admissible rate region for any source $q_{SUV}$ that meets the conditions of Case A or Case B of Theorem~\ref{thm-3-cor} and for which $q_{U|S}$ and $q_{V|S}$ are binary symmetric channels with crossover probabilities $\rho, \rho\ast \delta$, respectively, is given by \eqref{Eqn:Lem:BinaryExample1}. \end{remark} Consider the two pmfs given in Fig.~\ref{FIGURE:5} in support of the remark. A simple computation will yield that when $(S,U,V)\sim q_{SUV}$ or $(S,U,V)\sim p_{SUV}^{0.2,0.2}$: (a) $\supp(S,U,V) = \eS\times\U\times \V$; (b) $q_{U|S}$ is a binary symmetric channel with crossover probability $0.2$; and (c) $q_{V|S}$ is a binary symmetric channel with crossover probability $0.32$. Remark~\ref{rem:3} assures that the quantize-and-bin strategy is optimal for both these sources, and that $\R(D)[p^{0.2,0.2}_{SUV}] =\R(D)[q_{SUV}] $. \begin{figure}[t!] \begin{center} \includegraphics[width=0.6\textwidth]{Fig5} \caption{Two pmfs that have the same $D$-admissible rate regions. } \label{FIGURE:5} \end{center} \end{figure} \begin{figure}[h!] \begin{center} \includegraphics[width=0.5\textwidth]{Fig6} \caption{The $D$-admissible rate region for $p^{\rho,\delta}_{SUV}$ for $\rho = 0.05$ and $\rho = 0.2$. } \label{FIGURE:6} \end{center} \end{figure} Figure~\ref{FIGURE:6} presents a simulation of the $D$-admissible rate region for $p^{\rho,\delta}_{SUV}$ for $\rho = 0.05$ and $\rho = 0.2$ as an illustration of the result in Lemma~\ref{Lem:BinaryExample}. The figure presents the rate region for four values of of $D$, namely $D=0.05, 0.1, 0.2$ and $0.3$. The rate region for each of these values for $D$ is bounded by two lines -- one with slope $-1$ corresponding to the message sum rate $r_{uv}+r_v$, and one with infinite slope corresponding to the common message rate $r_{uv}$. When $D=0.5$, no communication is required, and the rate region is the entire non-negative quadrant. As $D$ is made smaller, the $D$-admissible rate region shrinks, and the minimum required communication rate for the common message increases. The admissible rate region shrinks until eventually $D=0$, at which point, the corresponding admissible rate region is given by \begin{subequations} \begin{align} r_{uv} &\geq H(S|U) = h(\rho) = h(0.05)=0.2864,\\ r_{uv}+r_v& \geq H(S|V) = h(\rho\ast \delta) = h(0.23)= 0.7780. \end{align} \end{subequations} This region is entirely outside this figure except for the vertex, which is located at the top-right corner of the figure. \subsection{On the Discontinuity of $\R(D)$} In source coding problems, the continuity of rate regions with the underlying source distributions allows for small changes in source distributions to translate to small changes in boundary of the rate region. Continuity is therefore essential in practice to allow the communications system engineer to estimate the source distribution and use the estimate to choose a suitable system operating point. When a single-letter characterization of a source coding rate region is known, it is possible to establish its continuity w.r.t the underlying rate region using the continuity of Shannon's information measures on finite alphabets~\cite[Chap.~2.3]{Yeung-2008-B}. For example, \cite[Lem.~7.2]{Csiszar-2011-B} considers the continuity of the standard rate-distortion function, and~\cite{Gu-Sep-2008-C} and~\cite{Chen-Jul-2008-C} study the semicontinuity of various source networks. However, the rate regions of certain source-coding problems are known to be discontinuous in the source distribution especially when they involve zero-error or functional reconstruction constraints~\cite[Ch.~11]{Csiszar-2011-B},~\cite{Gu-Sep-2008-C, Han-Kobayashi-Functional1987}. Despite the absence of any such reconstruction constraints, it turns out that the $D$-admissible rate region studied here is \emph{discontinuous} in the pmf $p_{SUV}$. The discontinuity arises rather due to the fact that we require the two reconstructions that are generated at two different locations in the network to agree (albeit with vanishing block error probability). Intuitively, in each of the cases where a single-letter characterization of the $D$-admissible rate region is known, the discontinuity can be attributed to the G\'acs-K\"orner common randomness in the argument of the single-letter reconstruction function; the G\'acs-K\"orner common randomness, and more precisely, its entropy can easily be seen to be discontinuous in the pmf $p_{SUV}$. We illustrate this phenomenon by a simple example. Recall the $D$-admissible rate region $\R(D)[p^{\rho,\delta}_{SUV}]$ of the binary example given above. We now establish the discontinuity of this problem by showing that \begin{equation} \lim_{\delta \downarrow 0}\ \mathcal{R}(D)[p^{\rho,\delta}_{SUV}] \neq \mathcal{R}(D)[p^{\rho,0}_{SUV}], \qquad 0 < \rho < D < 1/2. \end{equation} Suppose that $(S,U,V) \sim p^{\rho,0}_{SUV}$ and $1>D>\rho>0$. Since $\delta=0$, we have $U=V$, and consequently, neither Case A nor Case B of Theorem~\ref{thm-3-cor} is no longer applicable in identifying the $D$-admissible rate region. However, since $D > \rho$ and $U = V$, we can obviously achieve the distortion $D$ at $(r_{uv},r_v) = (0, 0)$ by simply choosing $\hat{S}^n = U^n=V^n=\tilde{S}^n$. This yields an average distortion of $\rho<D$, since \begin{equation} \frac{1}{n} \sum_{i=1}^n \Exp[d(S_i,\hat{S}_i)] = \rho. \end{equation} Thus, \begin{equation}\label{Eqn:Lem:BinaryExample:q} \R(D)[p^{\rho,0}_{SUV}] = [0,\infty)^2. \end{equation} Lemma~\ref{Lem:BinaryExample} determines $\R(D)[p^{\rho,\delta}_{SUV}]$ for $\delta \in (0,1)$. The mapping $\delta \mapsto h( (\rho \ast \delta) \ast D)$ is continuous on $[0,1]$, so \begin{equation}\label{Eqn:Lem:BinaryExample2} \lim_{\delta \downarrow 0}\ \R(D)[p^{\rho,\delta}_{SUV}] = \left[\lim_{\delta\downarrow 0} h((\rho \ast \delta) \ast D) - h(D), \infty\right)\times \big[0,\infty\big) =\big[ h(\rho \ast D) - h(D), \infty\big)\times \big[0,\infty\big). \end{equation} From~\eqref{Eqn:Lem:BinaryExample:q} and~\eqref{Eqn:Lem:BinaryExample2}, we see that the $D$-admissible rate region of $p^{\rho,\delta}_{SUV}$ does not approach that of $p^{\rho,0}_{SUV}$ as $\delta \to 0$. \begin{remark} The above argument can also be used to show the discontinuity at $\delta = 1$, i.e., \begin{equation} \lim_{\delta \uparrow 1}\ \R(D)[p^{\rho,\delta}_{SUV}] \neq \mathcal{R}(D)[p^{\rho,1}_{SUV}]. \end{equation} \end{remark} \section{Conclusions}\label{Sec:Conclu} In this work, we look at a variant of the two-receiver successive refinement problem with the common receiver reconstructions requirement. We present general inner and outer bound for this variant. The outer bound is unique in the sense it is the first information-theoretic single-letter characterization where the source reconstruction at the receivers is explicitly achieved via a function of the G\'acs-K\"orner common randomness between the random variables (both auxiliary and side information) available to the two receivers. Using these bounds, we derive a single-letter characterization of the admissible rate region and the optimal coding strategy for several settings of the joint distribution between the source and the receiver side information variables. Using this characterization, we then establish the discontinuity of the admissible rate region with respect to the underlying source source distribution even though the problem formulation does not involve zero-error or functional reconstruction constraints. \appendices \section{Proof of Lemma~\ref{Lem:Ancillary2}}\label{APP-A} Suppose that we have $(a_1,u) \xrightleftharpoons{} (a_1',u')$ in $\mathbb{G}^{A_1U,A_2V}[p_{A_1UA_2V}]$. Then, there must exist a positive integer $k\in \mathbb N$ and sets $\big\{(a_2^{(j)},v^{(j)}): 1\leq j \leq k\big\} \textrm{ and } \big\{(a_1^{(j)},u^{(j)}): 1\leq j < k\big\}$ such that \begin{subequations} \begin{align} p_{A_1UA_2V} (a_1,u,a_2^{(1)},v^{(1)})&>0,\\ p_{A_1UA_2V} (a_1^{(j)},u^{(j)},a_2^{(j)},v^{(j)})&>0, \quad 1\leq j<k,\\ p_{A_1UA_2V} (a_1^{(j)},u^{(j)},a_2^{(j+1)},v^{(j+1)})&>0, \quad 1\leq j<k,\\ p_{A_1UA_2V} (a_1',u',a_2^{(k)},v^{(k)})&>0. \end{align} \end{subequations} Since $\supp(A_1,A_2,U,V) = \supp(A_1,A_2) \times \supp(U,V)$, it follows that the above equations hold if and only if \begin{subequations} \begin{alignat}{3} p_{A_1A_2} (a_1,a_2^{(1)})&>0\textrm{ and } p_{UV} (u,v^{(1)})&>0,&\quad \\ p_{A_1A_2} (a_1^{(j)},a_2^{(j)})&>0\textrm{ and } p_{UV} (u^{(j)},v^{(j)})&>0,&\quad 1\leq j<k,\\ p_{A_1A_2} (a_1^{(j)},a_2^{(j+1)})&>0\textrm{ and } p_{UV} (u^{(j)},v^{(j+1)})&>0, &\quad 1\leq j<k,\\ p_{A_1A_2} (a_1',a_2^{(k)})&>0\textrm{ and } p_{UV} (u',v^{(k)})&>0. &\quad \end{alignat} \end{subequations} Thus, it follows that \begin{align} (a_1,u) \xrightleftharpoons{} (a_1',u')\textrm{ in }\mathbb{G}^{A_1V,A_2U}[p_{A_1VA_2U}] \Leftrightarrow {\setstretch{1}\left\{\hspace{-3mm}\begin{array}{c} \,\,\,\,\,a_1\xrightleftharpoons{} a_1'\textrm{ in }\mathbb{G}^{A_1,A_2}[p_{A_1A_2}]\\ u\xrightleftharpoons{} u'\textrm{ in }\mathbb{G}^{U,V}[p_{UV}]\end{array}\right.}. \label{eqn-a1a2uvequiv2} \end{align} Similarly, it can be shown that \begin{align} (a_2,v) \xrightleftharpoons{} (a_2',v')\textrm{ in }\mathbb{G}^{A_1V,A_2U}[p_{A_1VA_2U}] \Leftrightarrow {\setstretch{1}\left\{\hspace{-3mm}\begin{array}{c} \,\,\,\,\,a_2\xrightleftharpoons{} a_2'\textrm{ in }\mathbb{G}^{A_1,A_2}[p_{A_1A_2}]\\ v\xrightleftharpoons{} v'\textrm{ in }\mathbb{G}^{U,V}[p_{UV}]\end{array}\right.}. \label{eqn-a1a2uvequiv2} \end{align} Consequently, from \eqref{Eqn:GKMapping}, it follows that $\GK^{A_1U,A_2V} \equiv(\GK^{A_1,A_2},\GK^{U,V})$.\hfill\qed \section{Proof of Lemma~\ref{thm-RegionEquivalence}}\label{APP-B} Let $(r_{uv},r_v)\in \R^\ddagger_{D,k}$. Then, there must exist pmf $p_{ABCSUV}\in\mc P^{ \ddagger}_{D,k}$ and function $f:\A\times\C\times\V\rightarrow \hat\eS$ such that \begin{align} {\Exp [d (S, f(\GK^{ ABU,ACV}))]\leq D},\label{eqn:reconfn} \end{align} and \begin{subequations} \begin{align} r_{uv} &\geq I(S;A,B|U),\label{eqn-appb1}\\ r_v+r_{uv} &\geq I(S;B|A,C,U,V) +I(S;A,C|V). \label{eqn-appb2} \end{align} \end{subequations} For this choice of $p_{ABCSUV}$, let us define \begin{align} \eta^* := \mathop{\min}\limits_{\substack{(u,v)\in\supp(U,V)\\ s\in \supp(S)}} \frac{p_{UV|S}(u,v|s)}{p_{UV}(u,v)}. \end{align} Since \eqref{eqn:Condition1} holds, we are guaranteed that $\eta^*>0$. Then, for any $(a,b,c)\in\A\times\B\times \C$ and $(u,v)\in\mathcal U\times \V$, \begin{align} p_{ABCUV}(a,b,c,u,v)&=\sum_{s\in\supp(S)}p_{ABCS}(a,b,c,s) p_{UV|S}(u,v|s)\notag\\ &\stackrel{}{\geq}\eta^*\sum_{s\in\supp(S)}p_{ABCS}(a,b,c,s) p_{UV}(u,v)\notag\\ &=\eta^*p_{ABC}(a,b,c)p_{UV}(u,v).\\ p_{ABCUV}(a,b,c,u,v)&=\sum_{s\in\supp(S)}p_{ABC|S}(a,b,c,s) p_{UV|S}(u,v|s)\notag\\ &\stackrel{}{\leq}\frac{1}{\mpv_S}\sum_{s\in\supp(S)}p_{ABCS}(a,b,c,s) p_{UVS}(u,v,s)\notag\\ &\leq \frac{p_{UV}(u,v)}{\mpv_S} \sum_{s\in\supp(S)}p_{ABCS}(a,b,c,s)\notag\\ &=\frac{p_{ABC}(a,b,c)p_{UV}(u,v)}{\mpv_S} . \end{align} Hence, we also have $ \supp(A,B,C,U,V) = \supp(A,B,C)\times \supp(U,V).$ An application of Lemma~\ref{Lem:Ancillary2} of Sec.~\ref{Sec:Gacs-Korner} with $A_1=(A,C)$ and $A_2=(A,B)$ then yields the following conclusion. \begin{align} \GK^{ABU,ACV}\equiv (\GK^{AB,AC}, \GK^{U,V}) \label{eqn:GK-decorr1}. \end{align} Now, define random variable $\tilde A:= \GK^{AB,AC}$ over $\tilde A \subseteq \A\times\C$, and let $\tilde p_{\tilde ASUV}$ denote the {pmf} of $(\tilde A,S,U,V)$. Then, by construction, $\tilde A\mkv S \mkv (U,V)$. Note that both $\tilde A$ and $\GK^{U,V}$ are functions of $\GK^{\tilde AV,\tilde AU}$, i.e., \begin{align} H\big(\tilde A, \GK^{U,V}\,\big|\,\GK^{\tilde AV,\tilde AU}\big)& = 0. \end{align} Combining the above with \eqref{eqn:GK-decorr1}, yields $H(\GK^{ABU,ACV}|\GK^{\tilde AU,\tilde AV})=0$. Let $\tilde{f}:\tilde \A \times \V\rightarrow \A\times\C\times\V$ be such that \begin{align} \tilde{f}(\GK^{\tilde AU,\tilde AV})=\GK^{ABU,ACV}. \end{align} Then, from \eqref{eqn:reconfn}, ${\Exp [d (S, f(\tilde{f}(\GK^{\tilde AV,\tilde AU})))]\leq D}$. Hence, since $\tilde \A \subseteq \A\times \C$, we definitely have $\tilde p_{\tilde ASUV}\in \mc{P}^*_{D,k^2}$. Lastly, note that \begin{align} r_{uv}\step{\eqref{eqn-appb1}}\geq I(S;A,B|U)\geq I(S;\GK^{AB,AC}|U)=I(S;\tilde A|U),\\ r_{v}+r_{uv}\step{\eqref{eqn-appb2}}\geq I(S;A,C|V)\geq I(S;\GK^{AB,AC}|V)=I(S;\tilde A|V). \end{align} Hence, it follows that $(r_{uv}, r_v) \in\R^*_{D,k^2}$.\hfill\qed \section{Proof of Lemma~\ref{thm-RegionEquivalence2}}\label{APP-C} We will begin by limiting the size for the auxiliary RV $A$, and then present bounds for the alphabet sizes for $B$ and $C$, respectively. To bound the size of $\A$, we need to preserve: (a) $\{p_S(s):s\in\eS\}$; (b) six information functionals $H(S|A,U)$, $H(S|A,V)$, $H(S|A,B,U)$, $H(S|A,C,V)$, $H(S|A,C,U,V)$, and $H(S|A,B,C,U,V)$; and lastly, (c) the reconstruction constraint \eqref{eqn-Defn1eq3}. We begin by fixing $\check{s}\in\eS$. Define the following $|\eS|+6$ continuous, real-valued functions on $\mathscr{P}(\B\times\C\times\eS)$, the set of all {pmf}s on $\B\times\C\times\eS$. Let $\pi\in\mathscr{P}(\B\times\C\times\eS)$. \begin{subequations} \begin{align} \psi_{s}(\pi)&=\sum_{(b,c)\in\B\times \C} \pi(b,c,s),\quad s\in\eS\setminus\{\check{s}\}\label{psifns-1},\\ \psi_{1}(\pi)&=-\sum\limits_{s,u} \left(\sum_{b,c} \pi(b,c,s)p_{U|S}(u|s)\right) \log_2 \left(\frac{\sum\limits_{b,c} \pi(b,c,s)p_{U|S}(u|s)}{\sum\limits_{b,c,s'} \pi(b,c,s')p_{U|S}(u|s')}\right),\\ \psi_{2}(\pi)&=-\sum\limits_{s,v} \left(\sum_{b,c} \pi(b,c,s)p_{V|S}(v|s)\right) \log_2 \left(\frac{\sum\limits_{b,c} \pi(b,c,s)p_{V|S}(v|s)}{\sum\limits_{b,c,s'} \pi(b,c,s')p_{V|S}(v|s')}\right),\\ \psi_{3}(\pi)&=-\sum\limits_{b,s,u} \left(\sum_{c} \pi(b,c,s)p_{U|S}(u|s)\right) \log_2 \left(\frac{\sum\limits_c \pi(b,c,s)p_{U|S}(u|s)}{\sum\limits_{c,s'} \pi(b,c,s')p_{U|S}(u|s')}\right),\\ \psi_{4}(\pi)&=-\sum\limits_{c,s,v} \left(\sum_{b} \pi(b,c,s)p_{V|S}(v|s)\right) \log_2 \left(\frac{\sum\limits_b \pi(b,c,s)p_{V|S}(v|s)}{\sum\limits_{b,s'} \pi(b,c,s')p_{V|S}(v|s')}\right),\\ \psi_{5}(\pi)&=-\sum\limits_{b,s,u,v} \left(\sum_{c} \pi(b,c,s)p_{UV|S}(u,v|s)\right) \log_2 \left(\frac{\sum\limits_c \pi(b,c,s)p_{UV|S}(u,v|s)}{\sum\limits_{c,s'} \pi(b,c,s')p_{UV|S}(u,v|s')}\right),\\ \psi_{6}(\pi)&=-\sum\limits_{b,c,s,u,v} \pi(b,c,s)p_{UV|S}(u,v|s) \log_2 \left(\frac{\pi(b,c,s)p_{UV|S}(u,v|s)}{\sum\limits_{s'} \pi(b,c,s)p_{UV|S}(u,v|s)}\right)\label{psifns-7}. \end{align} \end{subequations} Note that preserving condition \eqref{eqn-Defn1eq3} is not straightforward because of the presence of G\'{a}cs-K\"{o}rner common randomness function. Consequently, this condition has to be split into two parts, which have to be combined together non-trivially. However, this approach requires the application of the Support Lemma~\cite[p.~631]{ElGamalKimBook} infinitely many number of times, along with a suitable limiting argument. To preserve \eqref{eqn-Defn1eq3}, define for each $m\in\mathbb{N}$, a continuous function $\psi_{7,m}: \mathscr{P}(\B\times\C\times\eS)\rightarrow \mathbb{R}$ by \begin{equation} \psi_{7,m}(\pi)= \min_{\substack{g:\B\times\U\rightarrow \hat{\eS}\\f:\C\times\V\rightarrow \hat{\eS}}} \left[\sum\limits_{b,c,s,u,v}\pi(b,c,s)p_{UV|S}(u,v|s)\Big(m\bar{D}\bar{\mathds{1}}\{f (c,v)=g(b,u)\} +d(s,f(c,v))\Big)\right] \label{eqn:Cara1-1}. \end{equation} Note that $\psi_{7,m}$ links together the distortion requirement with the probability that the reconstructions are different. Pick any pmf $\mathsf p_{ABCSUV}\in \mathcal{P}^{\ddagger}_{D,k}$. Then, by definition, it follows that there exist functions $f:\A\times\C\times \V\rightarrow \hat \eS$ and $g:\A\times\B\times \V\rightarrow \hat \eS$ \begin{subequations} \begin{align} \Pr[f(A,C,V)\neq g(A,B,U)] &= 0,\\ \Exp[d(S,f(A,C,V))]\leq D. \end{align} \end{subequations} Consequently, \begin{align} \sum_{a\in\A}p_A(a)\psi_{7,m}(\mathsf p_{BCS|A=a})&\leq D, \quad \forall\, m\in\mathbb{N}. \end{align} Combining the above with \eqref{psifns-1}-\eqref{psifns-7}, we see that \begin{subequations} \begin{align} \sum_{a\in\A}p_A(a)\psi_s(\mathsf p_{BCS|A=a})&=\mathsf p_S(s)=p_S(s),\quad s\in\eS\setminus\{\check{s}\},\\ \sum_{a\in\A}p_A(a)\psi_1(\mathsf p_{BCS|A=a})&=H(S|A,U)[\mathsf p],\\ \sum_{a\in\A}p_A(a)\psi_2(\mathsf p_{BCS|A=a})&=H(S|A,V)[\mathsf p],\\ \sum_{a\in\A}p_A(a)\psi_3(\mathsf p_{BCS|A=a})&=H(S|A,B,U)[\mathsf p],\\ \sum_{a\in\A}p_A(a)\psi_4(\mathsf p_{BCS|A=a})&=H(S|A,C,V)[\mathsf p],\\ \sum_{a\in\A}p_A(a)\psi_5(\mathsf p_{BCS|A=a})&=H(S|A,B,U,V)[\mathsf p],\\ \sum_{a\in\A}p_A(a)\psi_6(\mathsf p_{BCS|A=a})&=H(S|A,B,C,U,V)[\mathsf p],\\ \sum_{a\in\A}p_A(a)\psi_{7,m}(\mathsf p_{BCS|A=a})&\leq D, \quad \forall\, m\in\mathbb{N}. \end{align} \end{subequations} For each $m\in\mathbb{N}$, apply the Support Lemma with $|\eS|+6$ functions $(\{\psi_s\}_{s\in\eS\setminus\{\check s\}}, \psi_1,\psi_2 , \psi_3,\psi_4, \psi_5, \psi_6, \psi_{7,m})$ to identify a {pmf} $q_{A_mBCSUV}$ with $|{\A}_m|\leq |\eS|+6$ such that $(A_m,B,C)\mkv S \mkv (U,V)$ and \begin{subequations} \begin{align} \sum_{a\in\A_m}q_{A_m}(a)\psi_s(q_{BCS|A_m=a})&=\mathsf p_S(s)=p_S(s), \quad s\in\eS\setminus\{\check{s}\} \label{eqn-Qm1},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_1(q_{BCS|A_m=a})&=H(S|A,U)[\mathsf p]\label{eqn-Qm2},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_2(q_{BCS|A_m=a})&=H(S|A,V)[\mathsf p]\label{eqn-Qm3},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_3(q_{BCS|A_m=a})&=H(S|A,B,U)[\mathsf p]\label{eqn-Qm4},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_4(q_{BCS|A_m=a})&=H(S|A,C,V)[\mathsf p]\label{eqn-Qm5},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_5(q_{BCS|A_m=a})&=H(S|A,B,U,V)[\mathsf p]\label{eqn-Qm6},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_6(q_{BCS|A_m=a})&=H(S|A,B,C,U,V)[\mathsf p]\label{eqn-Qm7},\\ \sum_{a\in\A_m}q_{A_m}(a)\psi_{7,m}(q_{BCS|A_m=a})&\leq D.\label{eqn-supportbound} \end{align} \end{subequations} After possibly renaming of elements, we may assume that the alphabet of each of the auxiliary RVs $A_m$ is the same, say $\A^*$. Note that the optimal reconstruction functions $f_m,\,g_m$ (see \eqref{eqn:Cara1-1}) for the choice $q_{A_mBCSUV}$ meeting \eqref{eqn-supportbound} satisfy \begin{subequations} \begin{align} \Pr\big[f_m(A_m,C,V)\neq g_m(A_m,B,U)\big]&\leq \textstyle \dfrac{D}{\bar{D}m}\leq \textstyle\frac{1}{m}\label{eqn-decoder1}, \\ \Exp\big[d(S,f_m(A_m,C,V)\big]&\leq D\label{eqn-decoder2}. \end{align} \end{subequations} Since the number of functions from the set $\A^*\times\B\times \U$ (or $\A^*\times\C\times\V$) to $\hat{\eS}$ is finite, the sequence $\{(f_m,g_m)\}_{m\in\mathbb N}$ must contain infinitely many repetitions of at least one pair of reconstruction functions. Therefore, let \begin{align} \omega_0=\min\big\{ i\in\mathbb{N}:\big|\{ j\in\mathbb{N}:(f_j,g_j)=(f_i,g_i)\}\big|=\infty\big\}.\nonumber \end{align} Let $\{q_{A_{m_j}BCSUV}\}_{j\in\mathbb{N}}$ be the subsequence of $\{q_{A_{m}BCSUV}\}_{m\in\mathbb{N}}$ with $(f_{m_j},g_{m_j})=(f_{\omega_0},g_{\omega_0})$. By the Bolzano-Weierstrass theorem~\cite{MathAnalysisApostol}, we can find a subsequence $\{q_{A_{m_{j_l}}SUV}\}_{l\in\mathbb{N}}\subseteq \{q_{A_{m_j}SUV}\}_{j\in\mathbb{N}}$ that converges to, say, $q^*_{A^*BCSUV}$. Since $\{\psi_s\}_{s\in\eS\setminus\{\check{s}\}}, \psi_1$, $\psi_2$, $\psi_3$, $\psi_4$, $\psi_5$, $\psi_6$ and $\{\psi_{7,m}\}_{m\in\mathbb N}$ are continuous in their arguments, by taking appropriate limits of \eqref{eqn-Qm1}-\eqref{eqn-Qm5}, we see that \begin{subequations} \begin{align} \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_s(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm1}}{=} \mathsf p_S(s),\,\,\, s\in\eS\setminus\{\check s\},\\ \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_1(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm2}}{=} H(S|A,U)[\mathsf p],\\ \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_2(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm3}}{=}H(S|A,V)[\mathsf p],\\ \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_3(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm4}}{=} H(S|A,B,U)[\mathsf p],\\ \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_4(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm5}}{=}H(S|A,C,V)[\mathsf p],\\ \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_5(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm6}}{=}H(S|A,B,U,V)[\mathsf p],\\ \sum_{a\in\A^*}\,\,q_{A^*}(a)\psi_6(q_{BCS|A^*=a})&\stackrel{\eqref{eqn-Qm7}}{=}H(S|A,B,C,U,V)[\mathsf p]. \end{align} \end{subequations} Let $\mc{S}:= \{(a,b,c,u,v):f_{\omega_0}(a,c,v)\neq g_{\omega_0}(a,b,u)\}$. Then, we have \begin{align} \Pr[f_{\omega_0}(A^*,C,V)\neq g_{\omega_0}(A,B,U)] &=\Pr[(A^*,B,C,U,V)\in\mc{S}]\notag\\ &= \lim_{l\rightarrow\infty} \Pr[(A_{m_{j_l}},B,C,U,V)\in\mc{S}]\notag \\ &\stackrel{}{=} \lim\limits_{l\rightarrow\infty} \Pr[f_{m_{j_l}}(A_{m_{j_l}},C,V)\neq g_{m_{j_l}}(A_{m_{j_l}},B,U)] \stackrel{\eqref{eqn-decoder1}}{=}0. \end{align} Note that the above equality holds because $(f_{\omega_0},g_{\omega_0})=(f_{m_{j_l}},g_{m_{j_l}})$ for all $l\in\mathbb{N}$. Similarly, \begin{align} \Exp[d(S,f_{\omega_0}(A^*,C,V)]=\lim_{j\rightarrow\infty} \Exp[d(S,f_{\omega_0}(A_{m_j},C,V)]\leq D\label{eqn-decoder3}. \end{align} Thus, we may, without loss of generality, restrict the size of the alphabet of $A$ to be $|\eS|+6$. Next, to restrict the size of the alphabets of $B$ and $C$, we proceed by first picking $(A,B,C,S,U,V)\in\mathcal P^{\ddagger}_{D,k}$ with $|\A|\leq |\eS|+6$. By definition, it follows that there exist functions $f:\A\times\C\times \V\rightarrow \hat \eS$ and $g:\A\times\B\times \V\rightarrow \hat \eS$ \begin{subequations} \begin{align} \Pr[f(A,C,V)\neq g(A,B,U)] &= 0,\label{eqn-reconconst-alphbnd1}\\ \Exp[d(S,f(A,C,V))]\leq D.\label{eqn-reconconst-alphbnd2} \end{align} \end{subequations} Unfortunately, we cannot limit the the sizes of $\B$ and ${\C}$ by invoking Carath\'{e}odory's Theorem because of the inability to preserve $I(S; {AB}|U)$, $(S; {AC}|V)$ and the constraint $\Pr[ f( A, C, V) \neq g( A, B, U) ]=0$ simultaneously. However, without loss of generality we can assume that $|\supp(\mathsf{B})| \leq |\hat \eS|^{|\U|}$. To see why that is the case, let $\{u_1,\ldots, u_k\}$ be an enumeration of $\U$. Define auxiliary random variable $\bar{B}$ by \begin{align} \bar{ B} &:= \big({g}( A, B, u_1),\ldots,{g}( A, B, u_k)\big) \in \hat\eS^{|\U|}. \end{align} By construction, $\bar B$ is a function of $A$ and $B$, and hence, we are guaranteed that $(A,\bar B, C) \mkv S \mkv (U,V)$, and \begin{align} H({g}(A,B,U) \mid A,\bar{ B}, U ) & = 0. \end{align} Combining the above with \eqref{eqn-reconconst-alphbnd1} and \eqref{eqn-reconconst-alphbnd2}, we conclude the existence of functions $\bar g: \A\times \bar{\B} \times \U \rightarrow \hat \eS$ such that \begin{align} \Pr[ f(A, C,V)\neq \bar g(A,\bar B,U)] &= 0,\\ \Exp[d(S, f(A, C,V))]\leq D. \end{align} Further, since $H(\bar{ B} \mid A,{ B}) = 0$, we are guaranteed that \begin{subequations} \begin{align} I(S; A,B|U) &\geq I(S; A,\bar B|U), \\ I(S; B|A,C,U,V) &\geq I(S; \bar{B}|A,C,U,V). \end{align} \end{subequations} It then follows from Definitions~ \ref{Def:PddagRegionk} and \ref{Def:RddagRegionk} of Sec.~\ref{Mainresults-rrdefns} that considering random variables $A$ and $B$ with $|\A|> |\eS|+6$ and $|\B|> |\hat \eS|^{|\U|}$ does not enlarge the region, i.e., we can identify different $A^*$ and $\bar B$ using the above argument that operate at the same rate pair. We are now only left with bounding the size of $\C$, for which, we can repeat now repeat steps similar to that of $A$. This time, we preserve: (a) the distribution $\mathsf p_{ABS}$; (b) three information functionals $H(S|A,C,V)$, $H(S|A,C,U,V)$ and $H(S|A,B,C,U,V)$; and (c) the reconstruction constraint. Proceeding similarly as in the case of for the random variable $A$, we conclude that $|\C| \leq |\eS|(|\eS|+6) |\hat \eS|^{|\U|}+3$ suffices. Thus, it follows that \begin{align} \R_{k}^\ddagger (D) \subseteq \R_{|\eS|(|\eS|+6) |\hat \eS|^{|\U|}+3}^\ddagger (D)=\R^\ddagger (D), \quad k\in\mathbb{N}. \end{align}\hfill\qed \begin{remark}\label{rem-regionequiv2} Since $\R^*_k(D)$ has only one auxiliary random variable, the proof of \eqref{eqn-rateregionalphabetbnd1} of Lemma~\ref{thm-RegionEquivalence2} follows closely the portion of the above proof corresponding to the reduction of the size of $A$ alone. For the purposes of Lemma~\ref{thm-RegionEquivalence2}, we only need to preserve two information functionals, namely $H(S|A,U)$, and $H(S|A,V)$. Hence, the proof will only use $\{\psi_s:s\in\eS\setminus\{\check s\}\}$, $\psi_1$, $\psi_2$, and $\psi_{7,m}$ to conclude that $|\A|\leq |\eS|+2$ suffices, and hence \begin{align} \R_{k}^* (D) \subseteq \R_{|\eS|+2}^* (D)=\R^*(D), \quad k\in\mathbb{N}. \end{align} \end{remark} \begin{remark}\label{rem-regionequiv2b} For the proof of \eqref{eqn-rateregionalphabetbnd3} of Lemma~\ref{thm-RegionEquivalence2}, we only need to preserve four information functionals, namely $H(S|A,U)$, $H(S|A,V)$, $H(S|A,B,U)$ and $H(S|A,C,V)$. Hence, the proof will only use $\{\psi_s:s\in\eS\setminus\{\check s\}\}$, $\psi_1$, $\psi_2$, $\psi_3$, $\psi_4$ and $\psi_{7,m}$ to conclude that $|\A|\leq |\eS|+4$ suffices. The bound for $\B$ is the same as in the above proof. The final argument for bounding $\C$ requires the preservation of (a) the distribution $\mathsf p_{ABS}$; (b) the information functionals $H(S|A,C,V)$; and (c) the reconstruction constraint. \begin{align} \R_{k}^\dagger (D) \subseteq \R_{|\eS|(|\eS|+4) |\hat \eS|^{|\U|}+1}^\dagger (D)=\R^\dagger (D), \quad k\in\mathbb{N}. \end{align} \end{remark} \section{Proof of Lemma~\ref{Lem:RstarConvex}}\label{APP-D} Since $\R^{*}(D)\subseteq \mathbb{R}^2$, we only need to show that the line segment between any two points in $\R^{*}(D)$ lies completely within $\R^{*}(D)$. To do so, pick $(r_u',r_{uv}'), (r_u',r_{uv}'') \in \R^{*}(D)$ and $ \lambda\in [0,1]$. Then, by definition, we can find pmfs $q_{A'SUV}, q_{A''SUV}\in \mc P_{D, |\eS|+2}^*$ such that \begin{subequations} \begin{align} r_{uv}' &\geq I(S; A'|U),\\ r_v'+r_{uv}' &\geq I(S; A'|V),\\ r_{uv}'' &\geq I(S; A''|U),\\ r_v''+r_{uv}'' &\geq I(S; A''|V). \end{align} \end{subequations} Further, there must also exist functions $f':\A' \times \U \rightarrow \hat \eS$, $g':\A' \times \V \rightarrow \hat \eS$, $f'':\A'' \times \U \rightarrow \hat \eS$, and $g'':\A'' \times \V \rightarrow \hat \eS$ such that \begin{subequations}\label{eqn:convexR*1} \begin{alignat}{3} \Exp(S, f'(A',U) ] &=\sum_{a',s,u,v} q_{A' SUV}(a',s,u,v) d(s, f'(a',u)) &\leq D,\\ \Exp(S,f''(A'',U) ] &=\sum_{a'',s,u,v} q_{A'' SUV}(a'',s,u,v) d(s, f''(a'',u))\, &\leq D, \end{alignat} \end{subequations} and \begin{subequations}\label{eqn:convexR*4} \begin{align} \Pr[f'(A',U) \neq g'(A',V)] &= 0,\\ \Pr[f''(A'',U) \neq g''(A'',V)] &= 0, \end{align} \end{subequations} Without loss of generality, we may assume that the alphabets of $A'$ and $A''$ are disjoint, i.e., $\A' \cap \A'' =\emptyset$. Let us define $\tilde \A \triangleq \A'\cup\A''$. Now, define a joint pmf $q_{T\tilde ASUV}^*$ over $\{0,1\}\times \tilde\A \times \eS \times \U \times \V$ as follows: \begin{itemize} \item[1.] Let $T\in \{0,1\}$ be an RV such that $\Pr[T=0] = \lambda$. \item[2.] Let \begin{align} q^*_{\tilde ASUV | T=\alpha} = \left\{ \begin{array}{cc} q_{A'SUV} ,& \alpha = 0\\ q_{A''SUV}, & \alpha = 1 \end{array}\right..\label{eqn:convexR*5} \end{align} \end{itemize} Then by definition $T$ is a function of $\tilde A$ (since $T=0$ if and only if $\tilde A\in \A'$) and $T$ is independent of $(S,U,V)$. Let $q_{\tilde A,S,U,V}^*$ be the marginal of $(\tilde A, S,U,V)$ obtained from $q_{T,\tilde A,S,U,V}^*$. Then, the following hold: \begin{itemize} \item[(i)] $q_{\tilde A,S,U,V}^* \in \mc P_{D, 2|\eS|+4}^*$. This follows by defining $\tilde f: \tilde \A \times \U \rightarrow \hat \eS$ and $\tilde f: \tilde \A \times \V \rightarrow \hat \eS$ by \begin{subequations} \label{eqn-convfuncdefn} \begin{align} \tilde f(\tilde A, U) = \left\{ \begin{array}{cc} f'(A',U), & \tilde A\in\A' \\f''(A'',U), & \tilde A\in\A'' \end{array}\right.,\\ \tilde g(\tilde A, V) = \left\{ \begin{array}{cc} g'(A',V) ,& \tilde A\in\A' \\g''(A'',V), & \tilde A\in\A'' \end{array}\right., \end{align} \end{subequations} and verifying that \begin{align} \Pr[\tilde f(\tilde A,U) \neq \tilde g(\tilde A,V)] &= \sum_{t\in\{0,1\}} q^*_T(t)\Pr[\tilde f(\tilde A,U) \neq \tilde g(\tilde A,V)| T=t] \notag \\ &\step{\eqref{eqn:convexR*5}, \eqref{eqn-convfuncdefn}}{=} \lambda \Pr[f'( A',U) \neq g'( A',V)]+(1-\lambda)\Pr[f''( A'',U) \neq g''( A'',V)] \step{\eqref{eqn:convexR*4}}{=} 0.\\ \Exp(S,\tilde f(\tilde A,U) ] &=\sum_{\tilde a,s,u,v} q_{\tilde ASUV}^*(\tilde a,s,u,v) d(s, \tilde f(\tilde a,u))\\ &=\sum_{\tilde a,s,u,v} q_T^*(t) q_{\tilde ASUV|T}^*(\tilde a,s,u,v|t) d(s, \tilde f(\tilde a,u))\\ &\step{\eqref{eqn:convexR*5}, \eqref{eqn-convfuncdefn}}{=} \lambda\Exp(S,f'(A',U) ]+ (1-\lambda)\Exp(S,f''(A'',U) ] \step{\eqref{eqn:convexR*1}}{\leq} D. \end{align} \item[(ii)] Further, we also have that \begin{align} I(S;\tilde A | U ) &\stackrel{(a)}{=} I(S;\tilde A, T | U ) \stackrel{(b)}{=} I(S; \tilde A | U, T) = \sum_{t\in\{0,1\}} \Pr[T=t] I (S; \tilde A| U, T=t) \notag\\ &= \lambda I(S; A'|U) + (1-\lambda) I(S; A''|U), \end{align} where (a) follows since $T$ is a function of $\tilde A$, and (b) follows by the independence of $T$ and $(A,U)$. Similarly, we can also show that $I(S;\tilde A|V) = \lambda I(S; A'|V) + (1-\lambda) I(S; A''|V)$. \end{itemize} It then follows that \begin{subequations} \begin{align} \lambda r_{uv}' + (1-\lambda) r_{uv}'' &\geq \lambda I(S; A'|U) + (1-\lambda) I(S; A''|U) = I (S; \tilde A | U),\\ \lambda (r_v+r_{uv}') + (1-\lambda) (r_v''+r_{uv}'') &\geq \lambda I(S; A'|V) + (1-\lambda) I(S; A''|V) = I (S; \tilde A | V). \end{align} \end{subequations} Hence, it follows that any point that is a linear combination of $(r_u',r_{uv}'), (r_u',r_{uv}'') \in \R^{*}(D)$ is a point in $ \R^{*}_{2|\eS|+6}(D)$, which by Lemma~\ref{thm-RegionEquivalence2}, is identical to $\R^{*}(D)$. Hence, the claim of convexity follows. \section{Proof of Lemma~\ref{Lem:RstarCont}}\label{APP-E} From Definition~\ref{Def:RstarRegionk}, we can find $q_{A_iSUV} \in \mc P_{D_i,k}^*$, and functions $f_i:\A_i\times \V \rightarrow \hat \eS$ and $g_i:\A_i\times \U \rightarrow \hat \eS$ such that \begin{subequations} \begin{align} r_{uv}^{(i)} & \geq I(S;A_i|U),\\ r_v^{(i)}+r_{uv}^{(i)} & \geq I(S;A_i|V), \end{align} \end{subequations} and \begin{subequations}\label{eqn-APPE3} \begin{align} \Pr[ g_i(A_i,U) \neq f_i(A_i,V) ] &=0,\\ \Exp \big[d \big(S, f_i(A_i,V)\big)\big]&\leq D_i. \end{align} \end{subequations} Perhaps after a round of renaming, we may assume that the alphabets of $\A_i$s are identical, i.e., $\A_i = \A$. Since there are only a finite number of functions from $\A\times \V$ or $\A\times \U$ to $\hat \eS$, the sequence $(f_1,g_1),(f_2,g_2),\ldots$ must contain infinitely many copies of some pair of functions. Let \begin{align} \omega = \min\big\{j\in\mathbb {N}: |\{i\in\mathbb {N}: (f_i,g_i) = (f_j,g_j)\}| =\infty\big\}. \end{align} Let $\{i_j\}_{j\in\mathbb{N}}$ be a subsequence of indices such that $(f_{i_j},g_{i_j})=(f_{\omega},g_\omega)$ for all $j\in\mathbb{N}$. Consider the sequence of pmfs $\{ q_{A_{i_j} SUV}\}_{j\in\mathbb{N}} $. Since $\A\times\eS\times\U\times\V$ is a finite set, by Bolzano-Weierstrass theorem~\cite{MathAnalysisApostol}, a subsequence of pmfs must be convergent. Let $\{{i_j}_k\}_{k\in\mathbb N}$ be one such subsequence, and let $q_{A_{{i_j}_k} SUV } \rightarrow q_{\mathring ASUV}$. By the continuity of the information functional~\cite{Yeung-2008-B}, we see that \begin{align} I(\mathring A; U,V | S) = \lim_{k\rightarrow \infty} I(A_{{i_j}_k}; U,V | S) = 0. \label{eqn-APPE5} \end{align} Now, let $\mc F = \{(a,u,v)\in\A\times \U\times \V: f_\omega(a,v) \neq g_\omega(a,u)\}$. Then, \begin{align} \Pr[f_\omega(\mathring A,V) \neq g_\omega (\mathring A,U) ] &= \Pr[(\mathring A,U,V)\in \mc F]\notag\\ &=\sum_{(a,u,v)\in\mc F} q_{\mathring AUV}(a,u,v) \notag\\ & =\sum_{(a,u,v)\in\mc F} \left(\lim_{k\rightarrow \infty} q_{A_{{i_j}_k}UV}(a,u,v)\right)\notag \\ & = \lim_{k\rightarrow \infty} \sum_{(a,u,v)\in\mc F}q_{A_{{i_j}_k}UV}(a,u,v)\notag\\ & = \lim_{k\rightarrow \infty} \Pr[f_\omega(A_{{i_j}_k},V) \neq g_\omega (A_{{i_j}_k},U) ]\step{\eqref{eqn-APPE3}}= 0.\label{eqn-APPE6} \end{align} Further, we see that \begin{align} \Exp \big[d \big(S, f_\omega(\mathring A,V)\big)\big] &= \sum_{a,s,v} q_{\mathring ASV} (a,s,v) d(s, f_\omega(a,v))\notag\\ & = \sum_{a,s,v} \left(\lim_{k\rightarrow \infty} q_{A_{{i_j}_k}UV}(a,u,v)d(s, f_\omega(a,v))\right)\notag\\ & = \lim_{k\rightarrow \infty} \sum_{a,s,v} q_{A_{{i_j}_k}UV}(a,u,v)d(s, f_\omega(a,v))\notag\\ &= \lim_{k\rightarrow \infty} \Exp \big[d \big(S, f_\omega(A_{{i_j}_k},V)\big)\big] \step{\eqref{eqn-APPE3}}\leq \lim_{k\rightarrow \infty} D_i = \mathsf D.\label{eqn-APPE7} \end{align} Note that in the above two arguments, we have used the fact that $(f_{{i_j}_k},g_{{i_j}_k})=(f_{\omega},g_\omega)$ for $k\in\mathbb N$. Combining \eqref{eqn-APPE5}, \eqref{eqn-APPE6}, and \eqref{eqn-APPE7}, we see that $q_{\mathring ASUV}\in \mc P^*_{D,k}$. Lastly, using the continuity of the information functional~\cite{Yeung-2008-B}, we see that \begin{subequations} \begin{align} \mathsf{r_{uv}} &= \lim_{k \rightarrow \infty} r_{uv}^{({i_j}_k)} \geq \lim_{k \rightarrow \infty} I(S; A_{{i_j}_k} | U ) = I(S;\mathring A|U), \\ \mathsf{r_{uv}}+ \mathsf{r_{v}}&= \lim_{k \rightarrow \infty} r_{uv}^{({i_j}_k)}+r_{v}^{({i_j}_k)} \geq \lim_{k \rightarrow \infty} I(S; A_{{i_j}_k} | V ) = I(S;\mathring A|U). \end{align} \end{subequations} Hence, it follows that $(\mathsf{r_{uv}}, \mathsf{r_{v}}) \in \R^*_{k}(\mathsf D)$. \hfill\qed \section{Proof of Theorem~\ref{thm-achieve-thm-2} (Inner Bound)}\label{APP-F} Pick $p_{ABCSUV}\in \mc{P}_{D,k}^{ \ddagger}$ and $\ensuremath{\varepsilon}\in(0,\frac{1}{6})$. Let $f:\A\times \B \times \U \rightarrow \A\times\C\times \V$ and $g:\A\times \C \times \V \rightarrow \A\times\C\times \V$ be such that $$ f_U(A,B,U) = f_V(A,C,V) = \GK^{ABU,ACV}(A,C,V).$$ Let us denote \begin{subequations} \begin{align} R_a &= I(S;A|U)+2\ensuremath{\varepsilon},\\ R_a' &=\max\{I(S;A|V)-I(S;A|U), 0\},\\ R_a''&=\max\{\min\{I(A;U), I(A;V)\}-\ensuremath{\varepsilon}, 0\},\\ R_b&=I(S;B|A,U)+2\ensuremath{\varepsilon},\\ R_b'&=\max\{I(U;B|A) -\ensuremath{\varepsilon}, 0\},\\ R_c&=I(S;C|A,V)+2\ensuremath{\varepsilon},\\ R_c'&=\max\{I(V;C|A) -\ensuremath{\varepsilon}, 0\}. \end{align}\label{eqn-InnerBndRates} \end{subequations} We build a codebook using the marginals $p_A$, $p_{B|A}$ and $p_{C|A}$ obtained from the chosen joint pmf. The codebooks for the three auxiliary RVs are constructed as follows using the structure illustrated in Fig.~\ref{FIGURE:7}. \begin{itemize} \item For each triple $(i,i',i'')\in\llbracket 1, 2^{nR_a}\rrbracket \times \llbracket 1, 2^{nR_a'}\rrbracket \times \llbracket 1, 2^{nR_a''}\rrbracket$, generate a random codeword $A^n(i,i',i'')\sim p_A^n$ independent of all other codewords. Note that by the choice of rates, the total rate of the $A$-codebook is \begin{align} R_a+R_a'+R_a'' \step{\eqref{eqn-InnerBndRates}}{>} I(S;A). \label{eqn-InnerBndArate} \end{align} \item For each triple $(i,i',i'')\in\llbracket 1, 2^{nR_a}\rrbracket \times \llbracket 1, 2^{nR_a'}\rrbracket \times \llbracket 1, 2^{nR_a''}\rrbracket$, and pair $(\ell,\ell')\in\llbracket 1, 2^{nR_b}\rrbracket \times \llbracket 1, 2^{nR_b'}\rrbracket $, generate a random codeword $B^n(i,i',i'',\ell,\ell')\sim \prod_{k=1}^n p_{B|A_k(i,i',i'')}$ independent of all other codewords. Note that by the choice of rates, the total rate of the $B$-codebook is \begin{align} R_b+R_b'' \step{\eqref{eqn-InnerBndRates}}{>} I(S;B|A).\label{eqn-InnerBndBrate} \end{align} \item Similarly, for each triple $(i,i',i'')\in\llbracket 1, 2^{nR_a}\rrbracket \times \llbracket 1, 2^{nR_a'}\rrbracket \times \llbracket 1, 2^{nR_a''}\rrbracket$, and pair $(l,l')\in\llbracket 1, 2^{nR_c}\rrbracket \times \llbracket 1, 2^{nR_c'}\rrbracket $, generate a random codeword $C^n(i,i',i'',l,l')$ randomly using $\prod_{k=1}^n p_{C|A_k(i,i',i'')}$ independent of all other codewords. Note that by the choice of rates, the total rate of the $B$-codebook is \begin{align} R_c+R_c'' \step{\eqref{eqn-InnerBndRates}}{>} I(S;C|A). \label{eqn-InnerBndCrate} \end{align} \end{itemize} \begin{figure}[h!] \begin{center} \includegraphics[width=0.6\textwidth]{Fig7} \caption{A random-coding achievable scheme.}\label{FIGURE:7} \end{center} \end{figure} Upon receiving a realization $s^n$ of $S^n$, the encoder does the following: \begin{itemize} \item[1.] It searches for a triple $(i,i',i'')$ such that $(A^n(i,i',i''),s^n)\in T_\ensuremath{\varepsilon}^n[p_{AS}]$. \item[2.] It then searches for a pair $(\ell,\ell')$ such that $(A^n(i,i',i''),B^n(i,i',i'',\ell,\ell',s^n)\in T_\ensuremath{\varepsilon}^n[p_{ABS}]$, and a pair $(l,l')$ such that $(A^n(i,i',i''),C^n(i,i',i'',l,l',s^n)\in T_\ensuremath{\varepsilon}^n[p_{ABS}]$. Using \eqref{eqn-InnerBndArate}-\eqref{eqn-InnerBndCrate} and invoking the lossy source coding theorem~\cite[p.~57]{ElGamalKimBook}, and the Covering Lemma~\cite[p.~62]{ElGamalKimBook}, we see that \begin{align} \Pr\left[ \exists\, (i,i',i'',\ell, \ell'): (A^n(i,i',i''), B^n(i,i',i'',\ell,\ell'), C^n(i,i',i'',l,l'),S^n)\in T_\ensuremath{\varepsilon}^n[p_{ABCS}] \right] \stackrel{n\rightarrow\infty}{\longrightarrow} 0.\label{eqn-codebook1} \end{align} \item[3.] The encoder conveys $(i,\ell)$ to both receivers, and $(i',l)$ to the receiver with side information $V$. Note that this strategy corresponds to the following rates \begin{subequations} \begin{align} r_{uv}&= R_a+R_b=I(S;A,B|U)+4\ensuremath{\varepsilon}, \\ r_{uv}+r_v&= R_a+R_a'+R_b+R_c\notag\\ &=\max\{I(S;A|U), I(S;A|V)\}+ I(S;B|A,U) + I(S;C|A,V)+6\ensuremath{\varepsilon}.\end{align} \end{subequations} \end{itemize} Further, for any $(i,i',i'',\ell,\ell',l,l')$, by the Markov Lemma~\cite[p.~296] {ElGamalKimBook}, we are guaranteed that \begin{align} \Pr\left[ \begin{array}{c} (A^n(i,i',i''), B^n(i,i',i'',\ell,\ell'), C^n(i,i',i'',l,l'), S^n,U^n,V^n)\notin T_{2\ensuremath{\varepsilon}}^n[p_{ABCSUV}]\\\Big |\, (A^n(i,i',i''), B^n(i,i',i'',\ell,\ell'), C^n(i,i',i'',l,l'), S^n)\in T_\ensuremath{\varepsilon}^n[p_{ABCS}]\end{array}\right]\stackrel{n\rightarrow\infty}{\longrightarrow} 0.\label{eqn-codebook2} \end{align} Thus for sufficiently large $n$, the probability with which we will find a tuple $(i,i',i'',\ell,\ell',l,l')$ such that the corresponding codewords and the source and side information realizations is $\ensuremath{\varepsilon}$-typical can be made arbitrarily small. Moreover, as a consequence of the Packing Lemma~\cite[p.~46]{ElGamalKimBook}, we are also guaranteed that \begin{subequations} \begin{align} \Pr\left[ \begin{array}{c} \exists\, (\tilde i',\tilde i'')\neq (i',i'') \textrm{ s.t. } (A^n(i,i\tilde ',\tilde i''), U^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{AU}] \\\qquad \Big |\,(A^n(i,i',i''),U^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{AU}] \end{array}\right]&\stackrel{n\rightarrow\infty}{\longrightarrow} 0,\\ \Pr\left[ \begin{array}{c} \exists\, (\tilde i',\tilde i'')\neq (i',i'') \textrm{ s.t. } (A^n(i,i\tilde ',\tilde i''), V^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{AV}] \\\qquad \Big |\,(A^n(i,i',i''),V^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{AV}] \end{array}\right]&\stackrel{n\rightarrow\infty}{\longrightarrow} 0,\\ \Pr\left[ \begin{array}{c} \exists\, \tilde \ell' \neq \ell \textrm{ s.t. } (A^n(i,i',i''), B^n(i,i',i'',\ell,\tilde \ell'), U^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{ABU}] \\\qquad \Big |\,(A^n(i,i',i''), B^n(i,i',i'',\ell,\ell'), U^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{ABU}] \end{array}\right]&\stackrel{n\rightarrow\infty}{\longrightarrow} 0,\\ \Pr\left[ \begin{array}{c} \exists\, \tilde l' \neq l \textrm{ s.t. } (A^n(i,i',i''), C^n(i,i',i'',l,l'), V^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{ACV}] \\\qquad \Big |\,(A^n(i,i',i''), C^n(i,i',i'',l,l'), V^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{ACV}] \end{array}\right]&\stackrel{n\rightarrow\infty}{\longrightarrow} 0. \end{align}\label{eqn-codebook4} \end{subequations} From~\eqref{eqn-codebook1}, \eqref{eqn-codebook2}, \eqref{eqn-codebook4}, we can choose $n$ large enough such that when averaging over all realizations of the random codebooks, the probability with which all of the following events occur is at least $\ensuremath{\varepsilon}$. \begin{itemize} \item[(a)] the encoder will be able to identify indices $(i,i',i'',l,l',\ell,\ell')$ such that the corresponding codewords and the realization of $S^n$ are jointly $\ensuremath{\varepsilon}$-letter typical; \item[(b)] the identified codewords and the realizations of $(S^n, U^n, V^n)$ are jointly $2\ensuremath{\varepsilon}$-letter typical; \item[(c)] the receiver with side information $V$ will identify the indices $(i'',l')$ determined by the encoder; and \item[(d)] the receiver with side information $U$ will identify the indices $i',i'',\ell'$ determined by the encoder. \end{itemize} Then, there must exist a realization of the three codebooks $\{\mathsf{a}^n(i,i',i'')\}$, $\{\mathsf{b}^n(i,i',i'',\ell,\ell')\}$, and $\{\mathsf{c}^n(i,i',i'',l,l')\}$ such that the above four events occur simultaneously with a probability of at least $1-\ensuremath{\varepsilon}$. For this realization of the codebooks, with probability of at least $1-\ensuremath{\varepsilon}$, the realizations of the source and side informations $(s^n,u^n,v^n)$, and the selected codewords will be jointly $2\ensuremath{\varepsilon}$-letter typical, i.e., \begin{align} (\mathsf{a}^n(i,i',i''), \mathsf{b}^n(i,i',i'',\ell,\ell'), \mathsf{c}^n(i,i',i'',l,l'), s^n,u^n,v^n)\in T_{2\ensuremath{\varepsilon}}^n[p_{ABCSUV}].\label{eqn-codebook5}\end{align} Note that letter typicality ensures that the support of the empirical distribution $\tilde{q}_{ABCSUV}$ induced by the tuple $ (\mathsf{a}^n(i,i',i''), \mathsf{b}^n(i,i',i'',\ell,\ell'), \mathsf{c}^n(i,i',i'',l,l'), s^n,u^n,v^n)$ matches that of $q_{ABCSUV}$. Thus, it follows that whenever \eqref{eqn-codebook5} holds, we have for each $k=1,\ldots, n$, \begin{align} \hat{s}_k &= f_U( \mathsf a_k(i,i',i''), \mathsf b_k(i,i',i'',\ell,\ell'),u_k)= f_V( \mathsf a_k(i,i',i''), \mathsf c_k(i,i',i'',l,l'),v_k) = \tilde{s}_k. \end{align} Lastly, since \eqref{eqn-codebook5} holds with a probability of at least $1-\ensuremath{\varepsilon}$, it is also true that the reconstructions at either receiver will offer an expected distortion of no more than $D(1+2\ensuremath{\varepsilon})$ with a probability of at least $1-\ensuremath{\varepsilon}$. The proof is complete by noting that $\ensuremath{\varepsilon}$ can be chosen to be arbitrarily small. \hfill\qed \section{{Proof of Theorem~\ref{thm-achieve-thm-2a} (Outer Bound)}}\label{APP-G} Let $(r_{uv},r_v)^T\in {\mathsf{\R}(D)}$. Let $0<\iota<1$. Choose $0<\eta<\log 2$ such that \begin{align} \max\left\{2|\eS|(e^\eta-1), \sqrt[4]{{2\eta}} \log \frac{|\eS|^2}{ \sqrt[4]{{2}\eta}} +2|\eS|(e^\eta-1)\log |\eS| \right\} <\iota. \end{align} Let $\ensuremath{\varepsilon}=\min\left\{\iota, \left(\frac{1}{4}\mpv_{SUV}\mpv_S(1-e^{-\eta})\right)^2\right\}$. By Definition~\ref{defn-CRRRateregion}, there exist functions $ \mathtt E_{uv}$, $ \mathtt E_v$, $ \mathtt D_u$, $ \mathtt D_v$ satisfying \eqref{eqn-RRDefn1}. Let $M_{uv}:= \mathtt E_{uv}(S^n)$, $M_v := \mathtt E_v(S^n)$, $\hat S^n:= \mathtt D_u(M_{uv},U^n)$, and $\tilde{S}^n := \mathtt D_v(M_{uv},M_v,V^n)$. Then, \begin{align} r_{uv}+\ensuremath{\varepsilon}&\geq \frac{1}{n}H(M_{uv})\notag \\ &\geq \frac{1}{n}H(M_{uv}|U^n) \notag\\ &\geq \frac{1}{n}I(S^n; M_{uv}|U^n)\notag\\ &=\frac{1}{n} \sum\limits_{i=1}^nI(S_i; M_{uv}|U^n ,S^{i-1})\notag\\ &=\frac{1}{n} \sum\limits_{i=1}^nI(S_i; M_{uv},U^{n\setminus i},S^{i-1}|U_i) \notag\\ &\step{(a)}{=} \frac{1}{n}\sum\limits_{i=1}^n I(S_i;A_i,B_i|U_i)\notag\\ & \step{(b)}{=} I (S_Q; A_Q,B_Q | U_Q,Q)\notag\\ & \step{(c)}{=} I (S_Q; A_Q,B_Q,Q | U_Q)\notag\\ & \step{(d)}{=} I(S;\bar{A},\bar{B}|U),\label{eqn-ruvexpn} \end{align} where in (a), we let $A_i:= (M_{uv}, U^{i-1})$ and $B_i:= (U_{i+1}^n,S^{i-1})$; in (b), we introduce the time-sharing random variable $Q$ that is uniform over $\{1,\ldots,n\}$; in (c) we use the fact that $Q$ is independent of $(S_Q,U_Q)$; and in (d), we denote $\bar{A}:= (A_Q,Q)$, and $\bar{B} = (B_Q,Q)$. Similarly, \begin{align} r_{uv}+r_v+2\ensuremath{\varepsilon}&\geq \frac{1}{n}H(M_{uv},M_v)\notag\\ &\geq \frac{1}{n}I(S^n; M_{uv},M_v|V^n)\notag\\ &=\frac{1}{n}\Big[I(S^n; M_{uv},M_v,U^n|V^n)-I(S^n;U^n|M_{uv},M_v,V^n)\Big]\notag\\ &=\frac{1}{n}\sum\limits_{i=1}^n \Big[ I(S_i; M_{uv},M_v, U^n|V^n ,S^{i-1})-I(S^n; U_i|M_{uv},M_v,V^n,U^{i-1})\Big]\notag\\ &=\frac{1}{n}\sum\limits_{i=1}^n \Big[ I(S_i; M_{uv},M_v,U^n,V^{n\setminus i},S^{i-1}|V_i)- I(S^n; U_i|M_{uv},M_v,V^n,U^{i-1})\Big]\notag\\ &=\frac{1}{n}\sum\limits_{i=1}^n \Big[ I(S_i; A_i,B_i,M_v,V^{n\setminus i},U_i|V_i)- H(U_i|A_i,M_v,V^n) +H(U_i|A_i,M_v,V^n,S^n)\Big]\notag\\ &\step{(a)}{=}\frac{1}{n}\sum\limits_{i=1}^n \Big[ I(S_i; A_i,B_i,C_i,U_i|V_i)- H(U_i|A_i,C_i,V_i)+ H(U_i|A_i,C_i,V_i,S_i)\Big]\notag\\ &=\frac{1}{n} \sum\limits_{i=1}^n \Big[ I(S_i; A_i,B_i,C_i,U_i|V_i) -I(S_i;U_i|A_i,C_i,V_i)\Big]\notag\\ &= \frac{1}{n}\sum\limits_{i=1}^n \Big[ I(S_i; A_i,C_i|V_i)+I(S_i; B_i|A_i,C_i,U_i,V_i)\Big]\notag\\ &\step{(b)}{=} I(S_Q; A_Q, C_Q|V_Q, Q) + I (S_Q; B_Q | A_Q, C_Q ,U_Q ,V_Q, Q) \notag\\ &\step{(c)}{=} I(S; \bar{A},\bar{C}|V)+I(S; \bar{B}|\bar A,\bar{C},U,V),\label{eqn-ruv+rvexpn} \end{align} where in (a), we have denoted $C_i:= (M_v,V^{n\setminus i})$ and used the chain \begin{align} U_i\mkv (M_{uv},M_v,V^n,U^{i-1},S_i) \mkv (M_{uv},M_v,V^n,U^{i-1},S^n)\label{eqn-OB-MkvChn}; \end{align} in (b) we make use of the uniform time-sharing random variable $Q$; and in (c), we use the independence of $Q$ and $(S_Q,U_Q,V_Q)$ and define $\bar{A}:= (A_Q,Q)$, $\bar{B} = (B_Q,Q)$, and $\bar{C} = (C_Q,Q)$. Note that the following holds. \begin{align} I (U,V ; \bar {A}, \bar B, \bar C | S) &= I( U_Q, V_Q; A_Q ,B_Q ,C_Q | S_Q, Q) = \sum_{i=1}^n \frac{1}{n }I( U_i,V_i; A_i ,B_i, C_i | S_i) \stackrel{\eqref{eqn-OB-MkvChn}}{=} 0. \end{align} Now, note that $H(M_{uv},M_v,V^n|A_i,C_i,V_i)=0$ and $H(M_{uv},U^n|A_i,B_i,U_i)=0$. Hence, $\hat S_i$ and $\tilde S_i$ are functions of $(A_i,B_i, U_i)$ and $(A_i,C_i, V_i)$, resp. Let $\hat{S}_i:= g_i(A_i,B_i,U_i)$ and $\tilde S_i := f_i(A_i,C_i,V_i)$. Let $\bar{f}: \bar{\mathcal{A}} \times \bar{\mathcal{C}} \times \V \rightarrow \hat{\eS}$ and $\bar{g}: \bar{\mathcal{A}} \times \bar{\mathcal{B}} \times \U \rightarrow \hat{\eS}$ be defined as follows. For each $i\in\{1,\ldots,n\}$ and $(a,b,c,u,v)\in\A_i\times\B_i\times \C_i\times \U\times\V$, \begin{subequations} \begin{align} \bar{f}((a,i),(c,i),v) &:= f_i(a,c,v),\\ \bar{g}((a,i),(b,i),u) &:= g_i(a,b,u). \end{align} \end{subequations} In other words, \begin{subequations} \begin{align} \tilde S_Q & =\bar{f}((A_Q,Q),(C_Q,Q), V_Q),\\ \hat S_Q & = \bar{g}((A_Q,Q),(B_Q,Q), U_Q). \end{align} \end{subequations} Using the above notation, we can then verify that \begin{align} \Pr[ \bar{f}(\bar A, \bar C, V) \neq \bar{g}(\bar A, \bar B, U) ] &=\sum_{i=1}^n \frac{1}{n}\Pr[ \bar{f}(\bar A, \bar C, V ) \neq \bar{g}(\bar A, \bar B, U) | Q=i ]\notag\\&= \sum_{i=1}^n \frac{\Pr[ \tilde{S}_i \neq \hat S_ i ]}{n} \leq \ensuremath{\varepsilon}\leq \iota, \label{eqn:InterAux1}\\ \Exp[d(S, \bar f(\bar A,\bar C, V)]& = \sum_{i=1}^n\frac{1}{n}\Exp[d({S}_i, f_i(A_i,C_i,V_i) | Q=i ] \notag\\&= \sum_{i=1}^n\frac{1}{n}\Exp[d({S}_i, \tilde{S}_i] \stackrel{(c)}{\leq} D+\ensuremath{\varepsilon}\leq D+\iota,\label{eqn:InterAux2}\\ \Exp[d(S, \bar g(\bar A,\bar B, U)]& = \sum_{i=1}^n\frac{1}{n}\Exp[d({S}_i, g_i(A_i,B_i,U_i) | Q=i ] \notag\\&= \sum_{i=1}^n\frac{1}{n}\Exp[d({S}_i, \hat{S}_i] \stackrel{(c)}{\leq} D+\ensuremath{\varepsilon}\leq D+\iota.\label{eqn:InterAux3} \end{align} Now, we have to establish the existence of auxiliary RVs such that the RHS of \eqref{eqn:InterAux1} is, in fact, zero. To do so, we make use of the two \emph{pruning theorems} in Appendix~\ref{APP-H}. The first step is to only allow realizations of auxiliary random variables for which the reconstructions agree most of the time, and prune out the rest. To this end, define \begin{align} \mc{E} := \left\{(\bar a,\bar b, \bar c)\in \supp(\bar A, \bar B, \bar C): \Pr\Big[ \bar{f}(\bar A, \bar C, V) \neq \bar{g}(\bar A, \bar B, U)\,\big|\, (\bar A,\bar B, \bar C)=(\bar a,\bar b, \bar c) \Big]\leq \sqrt{\ensuremath{\varepsilon}} \right\}. \label{eqn:mcEDefn} \end{align} By a simple application of Markov's inequality, one can argue that \begin{align} \Pr[ (\bar A, \bar B, \bar U)\in\mc E] \geq 1-\sqrt{\ensuremath{\varepsilon}}. \end{align} Define auxiliary RVs $\mathring{A}\in\bar\A$, $\mathring{B}\in\bar\B$ and $\mathring{C}\in\bar\C$ with $(\mathring A, \mathring B, \mathring C) \mkv \bar S \mkv (\bar U,\bar V)$ by \begin{align} p_{\mathring A \mathring B \mathring C S}(a,b,c,s) =\left\{\begin{array}{ll} \dfrac{p_{ABCS}(a,b,c, s)}{\Pr[(A,B,C)\in \E|S=s]},& (a,b,c)\in\E \textrm{ and } s\in\supp(S)\\ 0, &\textrm{otherwise}\end{array}\right.. \end{align} The above pmf $p_{\mathring A \mathring B \mathring C S}$ is precisely the pmf $p_{ABCS}$ obtained by an application of Pruning Method A defined in \eqref{eqn-Prune1Def} of Appendix~\ref{APP-H} applied with $\delta =\sqrt{\ensuremath{\varepsilon}}< m_S(1-e^{-\eta})$. Hence, the properties of Theorem~\ref{Thm:Ancillary3} of Appendix~\ref{APP-H} can be applied to $p_{\mathring A \mathring B \mathring C S}$. First, by an application of Property (d) of Theorem~\ref{Thm:Ancillary3} of Appendix~\ref{APP-H}, we see that \begin{align} | I(S; AB | U) - I(S; \mathring A \mathring B | U)| & = | H(S| ABU) - H(S| \mathring A \mathring B U)|\leq \iota.\label{eqn:AfterPrune1}\\ | I(S; AC | V) - I(S; \mathring A \mathring C | V)| & = | H(S| ACV) - H(S| \mathring A \mathring C V)|\leq \iota. \end{align} Similarly, using Property (e) of Theorem~\ref{Thm:Ancillary3} of Appendix~\ref{APP-H}, we see that \begin{align} | I(S; B | ACUV) - I(S; \mathring B | \mathring A \mathring C UV)| & = \left(\begin{array}{l} | H(S| ACUV) - H(S| \mathring A \mathring C UV)|\\ \quad + | H(S| ABCUV) - H(S| \mathring A \mathring B \mathring C UV)| \end{array}\right)\leq 2\iota. \end{align} From \eqref{eqn:InterAux2}, \eqref{eqn:InterAux3} and Property (a) of Theorem~\ref{Thm:Ancillary3} of Appendix~\ref{APP-H}, we see that \begin{subequations} \begin{align} \Exp[d({S}, \bar f(\mathring A,\mathring C, V)]&\leq \Exp[d({S}, \bar f(\bar A,\bar C, V)]+ \lVert p_{ACSV}-p_{\mathring A \mathring C SV}\rVert\overline{D}\\ & \leq D + \iota + \overline D \iota\\ \Exp[d({S}, \bar g(\mathring A,\mathring B, U)]& \leq \Exp[d({S}, \bar g(\bar A,\bar B, U)]+ \lVert p_{ABSU}-p_{\mathring A \mathring B SU}\rVert\overline{D}\\ &\leq D+\iota+\overline D \iota. \end{align}\label{eqn:AfterPrune2} \end{subequations} Since $\supp(\mathring A, \mathring B, \mathring C) = \mc E$, for any $(\bar a,\bar b, \bar c)\in\mc E$, we can invoke Property (b) of Theorem~\ref{Thm:Ancillary3} of Appendix~\ref{APP-H} with $\mathcal{F}:= \{(u,v): \bar{f}(\bar a, \bar c, v) \neq \bar g(\bar a, \bar b, u)\}$ to infer that \begin{align} \Pr\Big[ \bar{f}(\mathring A, \mathring C, V) \neq \bar{g}(\mathring A, \mathring B, U)\,\big|\, (\mathring A,\mathring B, \mathring C)=(\bar a,\bar b, \bar c) \Big]\leq e^\eta\sqrt{\ensuremath{\varepsilon}} = 2\sqrt{\ensuremath{\varepsilon}}, \label{eqn:mce'estim} \end{align} Now, let's proceed by pruning $p_{\mathring A\mathring B \mathring C S}$ further by using Pruning Method B defined in \eqref{eqn-Anc4-1} of Appendix~\ref{APP-H} with $\delta =\frac{2\sqrt{\ensuremath{\varepsilon}}}{\mpv_{UVS}}$. Let us define \begin{align} \mc{E}' = \left\{ (a,b,c,s): p_{S|\mathring A \mathring B \mathring C}(s|\bar a,\bar b, \bar c) > \frac{2\sqrt{\ensuremath{\varepsilon}}}{\mpv_{UVS}}\right\}, \end{align} and auxiliary RVs $\mathsf{A}\in\bar\A$, $\mathsf{B}\in\bar\B$, and $\mathsf{C}\in\bar\C$ with $(\mathsf A, \mathsf B, \mathsf C) \mkv S \mkv (U,V)$ by \begin{align} p_{\mathsf{ABC}S}(a,b,c,s) =\left\{\begin{array}{ll} \dfrac{p_{\mathring{A}\mathring B \mathring CS}(a,b,c,s)}{\Pr[(\mathring A,\mathring B, \mathring C, S)\in \mc E' |S=s]},& (a,b,c,s)\in\mc E' \\ 0, &\textrm{otherwise}\end{array}\right.. \end{align} Since $p_{\mathsf{ABC}S}$ is obtained from $p_{\mathring A \mathring B \mathring C S}$ by Pruning Method B, the properties of Theorem~\ref{Thm:Ancillary4} of Appendix~\ref{APP-H} can be applied to $p_{\mathring A \mathring B \mathring C S}$. Combining \eqref{eqn:mce'estim} with Property (b) of Theorem~\ref{Thm:Ancillary4} of Appendix~\ref{APP-H}, we see that \begin{align} \Pr\big[ \bar{f}(\mathsf A, \mathsf C, V) \neq \bar{g}(\mathsf A, \mathsf B, U)\,\big|\, (\mathsf A,\mathsf B, \mathsf C)=(\bar a,\bar b, \bar c) \big]=0, \quad (\bar a,\bar b, \bar c)\in\mc E. \end{align} Since by construction, $\supp(\mathsf{A},\mathsf{B},\mathsf{C}) \subseteq \supp(\mathring A,\mathring B, \mathring C) \subseteq \mc E$, it follows that \begin{align} \Pr[ \bar{f}(\mathsf A, \mathsf C, V) \neq \bar{g}(\mathsf A, \mathsf B, U) ] = 0. \end{align} Invoking Property (a) of Theorem~\ref{Thm:Ancillary4} of Appendix~\ref{APP-H}, it follows that \begin{subequations} \begin{align} \Exp[d({S}, \bar f(\mathsf A,\mathsf C, V)]&\leq \Exp[d({S}, \bar f(\mathring A,\mathring C, V)]+ \overline{D} \lVert p_{\mathsf{AC}SV}-p_{\mathring A \mathring C SV}\rVert\\ & \leq D + \iota + 2\overline D \iota\\ \Exp[d({S}, \bar g(\mathsf A,\mathsf B, U)]& \leq\Exp[d({S}, \bar f(\mathring A,\mathring B, U)]+ \overline{D} \lVert p_{\mathsf{AB}SU}-p_{\mathring A \mathring B SU}\rVert\\ & \leq D + \iota + 2\overline D \iota. \end{align}\label{eqn:AfterPrune3} \end{subequations} By an application of Property (e) of Theorem~\ref{Thm:Ancillary4} of Appendix~\ref{APP-H}, we see that \begin{subequations} \begin{align} | I(S; \mathsf{AB} | U) - I(S; \mathring A \mathring B | U)| & = | H(S| \mathsf{AB}U) - H(S| \mathring A \mathring B U)|\leq \iota,\\ | I(S; \mathsf{AC} | V) - I(S; \mathring A \mathring C | V)| & = | H(S| \mathsf{AC}V) - H(S| \mathring A \mathring C V)|\leq \iota. \end{align} \end{subequations} Similarly, an application of Property (f) of Theorem~\ref{Thm:Ancillary4} of Appendix~\ref{APP-H} yields \begin{align} | I(S; \mathsf B | \mathsf{AC}UV) - I(S; \mathring B | \mathring A \mathring C UV)| & = \left(\begin{array}{l} | H(S| \mathsf{AC}UV) - H(S| \mathring A \mathring C UV)|\\ \quad + | H(S| \mathsf{ABC}UV) - H(S| \mathring A \mathring B \mathring C UV)| \end{array}\right)\leq 2\iota\label{eqn:AfterPrune4}. \end{align} Combining \eqref{eqn:AfterPrune1}-\eqref{eqn:AfterPrune2} with \eqref{eqn:AfterPrune3}-\eqref{eqn:AfterPrune4}, we have auxiliary RVs $(\mathsf A, \mathsf B, \mathsf C) \mkv S \mkv (U,V)$ such that \begin{subequations} \begin{align} r_{uv} + 2\iota &\geq I(S; \mathsf{AB}|U),\\ r_v+ r_{uv} + 2\iota &\geq I(S; \mathsf{AC}|V)+I(S; \mathsf{B} | \mathsf{AC}UV),\\ \Pr[ \bar{f}(\mathsf A, \mathsf C, V) \neq \bar{g}(\mathsf A, \mathsf B, U) ] &= 0,\\ \Exp[d({S}, \bar f(\mathsf A,\mathsf C, V)]&\leq D+\iota+ 2\overline{D}\iota,\\ \Exp[d({S}, \bar g(\mathsf A,\mathsf B, U)]& \leq D+\iota+2\overline{D}\iota. \end{align} \end{subequations} Thus, from \eqref{eqn-rateregionalphabetbnd2} of Lemma~\ref{thm-RegionEquivalence2} of Sec.~\ref{Mainresults-rrdefns}, it follows that $(r_{uv} + \iota, r_{v} + \iota) \in \R^\ddagger (D+\iota+2\overline{D}\iota)$. Finally, by constructing an appropriate sequence of infinitesimals and invoking Lemma~\ref{Lem:RstarCont} of Sec.~\ref{Mainresults-rrdefns}, we conclude that $(r_{uv} , r_{v}) \in \R^\ddagger (D)$. \hfill\qed \section{Pruning Theorems}\label{APP-H} We now present two \emph{pruning theorems} that concern any five random variables $(A_1,A_2,S,B_1,B_2)$ where $(A_1,A_2)\mkv S \mkv (B_1,B_2)$ forms a Markov chain. The theorems will be applied to the CRR successive refinement problem, where $S$ will be identified as the source, $A_1, A_2$ will be associated with auxiliary random variables, and $B_1$ and $B_2$ are the side information random variables available at each of the receivers. The pruning theorems will help us to understand how small alterations to the marginal pmf of $(A_1,A_2,S)$ will change \begin{itemize} \item the joint pmf of $(A_1, A_2, S, B_1, B_2)$ with respect to variational distance, and \item information functionals such as $I(S; A_1 | B_1)$, $I(S; A_2| B_2)$ and $I(S; A_1 A_2 | B_1 B_2)$. \end{itemize} \begin{figure*}[t!] \begin{center} \includegraphics[width=0.7\textwidth]{Fig8} \caption{Illustration of the two kinds of pruning.}\label{FIGURE:8} \end{center} \end{figure*} Figure~\ref{FIGURE:8} illustrates the two kinds of pruning. In the Pruning Method A, we have $(A_1,A_2,S) \sim p_{A_1A_2S}$ as shown in Figure~\ref{FIGURE:8}(a). We select an appropriate threshold $0 < \delta < 1$, and consider any subset $\mathcal{E} \subseteq \mathcal{A}_1 \times \mathcal{A}_2$ with $\Pr[(A_1,A_2) \in \mathcal{E}] \geq 1-\delta$. We then take the joint pmf $p_{A_1A_2S}$ and construct a new joint pmf $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S}$ whose support set satisfies $\supp(\tilde A_1,\tilde A_2, S) \subseteq \mc E \times \eS$. In other words, the edges belonging to $\mathcal{E}^c \times \eS$ in the bipartite graph of $p_{A_1 A_2 S}$ indicated by red dashed lines in Figure~\ref{FIGURE:8}(a) are removed (and the remaining edges are scaled appropriately) to define $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S}$. In the Pruning Method B that is illustrated in Figure.~\ref{FIGURE:8}(b), we first select an appropriate threshold $0 < \delta < 1$, and consider any subset of edges $\mathcal{E} \subseteq \mathcal{A}_1 \times \mathcal{A}_2 \times \eS$ with \begin{equation} p_{S|A_1,A_2}(s| a_1,a_2)> \delta, \quad \forall\ (a_1,a_2,s) \in \mc{E} \end{equation} Edges that are not in $\mc E$ (shown by red dashed lines in Fig.~\ref{FIGURE:8}(b)) are removed, and the probability mass of the rest of the edges are scaled appropriately to construct a new $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S}$ such that $\supp(\tilde A_1,\tilde A_2, S) \subseteq \mc E$. The precise details of the two kinds of pruning, and the required results pertaining to them are elaborated below. \subsection{Pruning Method A} Suppose that we have a random tuple $(A_1,A_2,S,B_1,B_2)$ over $\mathcal{A}_1\times\mathcal{A}_2\times \mathcal{S} \times \mathcal{B}_1 \times \mathcal{B}_2$ distributed according to $p_{A_1A_2S B_1B_2}$ such that $(A_1,A_2) \mkv S \mkv (B_1,B_2)$ and $\supp(S)=\eS$. Pick any $\delta,\eta > 0$ satisfying \begin{equation}\label{Eqn:Proof:PruningThm:Delta1} \delta\leq\mpv_{S}(1-e^{-\eta}), \end{equation} and consider any subset $\E\subseteq \A_1\times\A_2$ satisfying $\Pr[(A_1,A_2)\in \E]\geq 1-\delta$. Let \begin{equation} \tilde{p}_{\tilde{A}_1\tilde{A}_2 S}(a_1,a_2,s) := \left\{ \begin{array}{ll} \dfrac{p_{A_1A_2S}(a_1,a_2,s)}{\Pr[(A_1,A_2)\in \E|S=s]} & \text{ if } (a_1,a_2)\in\E\\ 0 & \text{ if } (a_1,a_2)\in\E^c \end{array}, \right. \label{eqn-Prune1Def} \end{equation} and $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S B_1 B_2} :=\tilde{p}_{\tilde{A}_1\tilde{A}_2S}p_{B_1B_2|S}$. Define $\Xi: [0, \frac{\mpv_{SUV}}{|\eS|}) \rightarrow \mathbb R$ by \begin{equation} \Xi(x):= \frac{|\eS|x}{\mpv_{SUV}-|\eS|x}\log \frac{|\eS| ^2(\mpv_{SUV}-|\eS|x)}{x}. \label{eqn-XiDefn} \end{equation} \begin{theorem}\label{Thm:Ancillary3} $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S B_1 B_2}$ defines a valid joint pmf with $(\tilde{A}_1,\tilde{A}_2) \mkv S \mkv (B_1,B_2)$ for which the following holds: \begin{enumerate}[(a)] \item $ \big\lVert\tilde{p}_{\tilde A_1\tilde A_2 S B_1B_2} -p_{A_1A_2S B_1B_2}\big\rVert_1 \leq \frac{2\delta}{\mpv_S -\delta} \leq 2(e^\eta-1).\vspace{1mm} $ \item For any event $\mathcal{F}\subseteq \B_1\times \B_2$ and $(a_1,a_2) \in\mathcal{E}$, \begin{equation} \Pr[(B_1,B_2)\in \mathcal{F}| (\tilde A_1,\tilde A_2)=(a_1,a_2)]\\ \leq e^\eta\ \Pr[(B_1,B_2)\in \mathcal{F}| (A_1, A_2)=(a_1,a_2)]. \end{equation} \item $D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} ) \leq \eta. $ \item $ \big| H(S|\tilde{A}_1 B_1) - H(S|A_1 B_1) \big | \leq \sqrt[4]{{2\eta}} \log \dfrac{|\eS|^2}{ \sqrt[4]{{2}\eta}} +2(e^\eta-1)\log |\eS|. $ \item $ \big|H(S|\tilde A_1 \tilde A_2B_1B_2)-H(S|A_1A_2B_1B_2)\big| \leq \sqrt[4]{{2\eta}} \log \dfrac{|\eS|^2}{ \sqrt[4]{{2}\eta}} +2(e^\eta-1)\log |\eS|. $ \end{enumerate} \end{theorem} \begin{IEEEproof} For brevity, we will omit the subscripts on the joint distributions $p_{A_1 A_2 S B_1 B_2}$ and $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S B_1 B_2}$ (and their marginals) throughout the proof; for example, $p(a_1,s) = p_{A_1 S}(a_1,s)$ and $\tilde{p}(a_1,s) = \tilde{p}_{A_1 S}(a_1,s)$. \subsubsection*{Assertion (a)} For any $(a_1,a_2,s,b_1,b_2)$ with $(a_1,a_2) \in \E$ and $p(a_1,a_2,s,b_1,b_2)>0$, we have \begin{equation}\label{eqn:P1-1} 1 \leq \frac{\tilde{p}(a_1,a_2,s,b_1,b_2)}{p(a_1,a_2,s,b_1,b_2)} =\frac{\tilde{p}(a_1,a_2,s)}{p(a_1,a_2,s)} = \frac{p(s)}{\Pr[(A_1,A_2)\in\E, S=s]} \step{(a)}{\leq } \frac{\mpv_S}{\mpv_S-{\delta}}, \end{equation} which (a) follows because \begin{equation} 1 \geq \frac{\Pr[(A_1,A_2)\in\E, S=s]}{p(s)} \geq \frac{p(s) - \Pr[(A_1,A_2)\in\E]}{p(s)}\\ \geq 1- \frac{\Pr[(A_1,A_2)\in\E]}{p(s)} \geq \frac{\mpv_S - \delta}{\mpv_S}. \end{equation} Thus, from~\eqref{eqn:P1-1} we now have \begin{align} \notag \sum_{(a_1,a_2,s,b_1,b_2) \in \E \times \eS \times \B_1 \times \B_2} &\Big| p(a_1,a_2,s,b_1,b_2) - \tilde{p}(a_1,a_2,s,b_1,b_2) \Big|\\ \notag &\leq \sum_{(a_1,a_2,s,b_1,b_2) \in \E \times \eS \times \B_1 \times \B_2} \left(\frac{\mpv_S}{\mpv_S - \delta} - 1\right) p(a_1,a_2,s,b_1,b_2) \\ \label{Eqn:Proof:Thm:3:AssA:1} &= \left(\frac{\delta}{\mpv_S-{\delta}}\right)\Pr\big[(A_1,A_2)\in\mathcal{E}\big]. \end{align} Next, we can bound the contribution for $\mathcal{E}^c \times \mathcal{S}\times\mathcal{B}_1\times\mathcal{B}_2$ by \begin{equation}\label{Eqn:Proof:Thm:3:AssA:2} \sum_{(a_1,a_2,s,b_1,b_2) \in \E^c \times \eS \times \B_1 \times \B_2} \big| p(a_1,a_2,s,b_1,b_2) - \tilde{p}(a_1,a_2,s,b_1,b_2) \big| = \Pr\big[(A_1,A_2) \in \E^c\big] \leq \delta. \end{equation} Combining~\eqref{Eqn:Proof:Thm:3:AssA:1} and~\eqref{Eqn:Proof:Thm:3:AssA:2} yields Assertion (b), since $\delta+ \frac{\delta}{\mpv_S -\delta}<\frac{2\delta}{\mpv_S -\delta}=2(e^\eta-1)$. \subsubsection*{Assertion (b)} It is sufficient to prove the assertion for any singleton $\mathcal{F} = \{(b_1,b_2)\}$ with $(b_1,b_2)\in\B_1\times \B_2$. Let $(a_1,a_2)\in \mc E$. Without loss of generality, we may assume that $p(b_1,b_2|a_1,a_2) > 0$, since if not, the claim is trivially true. Thus, \begin{align} \frac{\tilde{p}(b_1,b_2|a_1,a_2)}{p(b_1,b_2|a_1,a_2)} &= \frac{\sum\limits_{s \in \supp(S)} \tilde{p}(a_1,a_2,s)\ p(b_1, b_2|s)} {\tilde{p}(a_1,a_1)\ p(b_1,b_2|a_1,a_2)} \step{{\eqref{eqn:P1-1}}}{\leq} \left({\frac{\mpv_S}{\mpv_S-{\delta}}}\right) \frac{\sum\limits_{s \in \supp(S)} p(a_1,a_2,s)\ p(b_1, b_2|s)} {\tilde{p}(a_1,a_2)\ p(b_1,b_2|a_1,a_2)}\notag\\ &= \left({\frac{\mpv_S}{\mpv_S-{\delta}}}\right) \frac{p(a_1,a_2,b_1, b_2)} {\tilde{p}(a_1,a_2)\ p(b_1,b_2|a_1,a_2)}\notag\\ &= \left({\frac{\mpv_S}{\mpv_S-{\delta}}}\right) \frac{ p(a_1,a_2)}{\tilde{p}(a_1,a_2)} \step{(a)} {\leq} \frac{\mpv_S}{\mpv_S-{\delta}} \step{\eqref{Eqn:Proof:PruningThm:Delta1}}{\leq} e^\eta, \end{align} where (a) follows by rearranging \eqref{eqn:P1-1}, and summing over elements of $\eS \times \B_1\times \B_2$. \subsubsection*{Assertion (c)} \begin{align} D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} ) &=D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S} \lVert\, p_{A_1 A_2 S} )\notag\\ &= \sum_{(a_1,a_2,s)\in\supp(\tilde{A}_1\tilde{A}_2 S)} \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) \log \frac{ \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) }{ {p}_{{A}_1{A}_2 S} (a_1,a_2,s) }\notag\\ &\stackrel{\eqref{eqn:P1-1}}{\leq} \sum_{(a_1,a_2,s)\in\supp(\tilde{A}_1\tilde{A}_2 S)} \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) \log \frac{\mpv_S}{\mpv_S-\delta}\leq \eta. \label{eqn:P3-1} \end{align} \subsubsection*{Assertion (d)} \begin{align} D_{KL}(\tilde{p}_{S|\tilde{A_1}B_1} \lVert\, p_{S|A_1 B_1}) &:= \sum_{a_1 b_1} \tilde{p}_{\tilde{A}_1B_1}(a_1,b_1) D_{KL} \big( \tilde{p}_{S\mid \tilde A_1 B_1} ( \cdot | a_1, b_1) | {p}_{S\mid A_1 B_1} ( \cdot | a_1, b_1)\big)\notag \\ &\leq D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} )\stackrel{\eqref{eqn:P3-1}}{\leq} \log \frac{\mpv_S}{\delta -\mpv_S}=\eta. \end{align} An application of Pinsker's followed by Jensen's inequality yields the following. \begin{align} \sum_{a_1 b_1} \tilde{p}_{\tilde{A}_1B_1}(a_1,b_1) \big\lVert\tilde{p}_{S\mid \tilde A_1 B_1} ( \cdot | a_1, b_1) - {p}_{S\mid A_1 B_1} ( \cdot | a_1, b_1)\big\rVert_1 \leq \sqrt{{2}\eta}. \end{align} Let $\mathcal{D} := \left\{(a_1,b_1): \big\lVert\tilde{p}_{S\mid \tilde A_1 B_1} ( \cdot | a_1, b_1) - {p}_{S\mid A_1 B_1} ( \cdot | a_1, b_1)\big\rVert_1 \leq \sqrt[4]{{2}\eta}.\right\}$. Then, by Markov's inequality, \begin{align} \Pr[ (\tilde A_1, B_1) \in \mathcal{D} ] &\geq 1- \sqrt[4]{{2}\eta}. \end{align} Further for every $(a_1,b_1)\in \mc D$, Lemma~2.5 of \cite{Csiszar-2011-B} guarantees that \begin{align} \Delta_{a_1,b_1} &:= \Big| H(S\,|\,(\tilde A_1 ,B_1)=(a_1,b_1)) - H(S\,|\,(A_1 ,B_1)=(a_1,b_1))\Big|\leq \sqrt[4]{{2}\eta} \log \frac{|\eS|}{ \sqrt[4]{{2}\eta}}. \end{align} Finally, \begin{align} \big|H(S|\tilde A_1 B_1) -H(S| A_1 B_1) \big| &\leq \sum_{a_1,b_1} \tilde{p}_{\tilde A_1,B_1} (a_1,b_1) \Delta_{a_1,b_1} + \big\lVert \tilde{p}_{\tilde A_1,B_1}- {p}_{A_1,B_1}\big\rVert_1 \cdot \log |\eS|\notag\\ &\leq \sum_{(a_1,b_1)\in \mc D} \tilde{p}_{\tilde A_1,B_1} (a_1,b_1) \Delta_{a_1,b_1}+\Pr[(\tilde A_1,B_1)\notin \mc D] \log |\eS|+ \frac{2\delta\log |\eS|}{\mpv_S -\delta}\notag\\ &\leq \sqrt[4]{{2}\eta} \log \frac{|\eS|^2}{ \sqrt[4]{{2}\eta}} +2(e^\eta-1) \log |\eS|. \end{align} \subsubsection*{Assertion (e)} The proof of (e) is identical to that of (d), with the only difference being that the commencement of the argument is as follows. \begin{align} D_{KL}(\tilde{p}_{S|\tilde{A_1}\tilde{A}_2 B_1 B_2} \lVert\, p_{S|A_1 A_2 B_1 B_1})\leq D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} )\stackrel{\eqref{eqn:P3-1}}{\leq} \log \frac{\mpv_S}{\delta -\mpv_S}=\eta. \end{align} The remaining steps are identical to those in (d), with the exception that all four variables (either $(\tilde A_1, \tilde A_2, B_1, B_2)$ or $(A_1, A_2, B_1, B_2) $) appear in the conditioning. \end{IEEEproof} \subsection{Pruning Method B} Let {pmf} $p_{A_1A_2S B_1B_2}$ over $\mathcal{A}_1\times\mathcal{A}_2\times \mathcal{S} \times ,\mathcal{B}_1\times\mathcal{B}_2$ be given such that $(A_1,A_2)\mkv S \mkv (B_1,B_2)$ and $\supp(S)=\eS$. Let $\delta\leq\mpv_{S}(1-e^{-\eta})$ for some $0<\eta<1$, and let \begin{subequations} \begin{align} \E&:= \left\{ (a_1,a_2,s)\in\supp(A_1,A_2,S): p_{S|A_1A_2}(s|a_1,a_2)> \delta\right\},\\ \mathcal{E}_s &:= \big\{(a_1,a_2): (a_1,a_2,s)\in \mc E\big\},\,\,\quad s\in\eS,\\ \mathcal{E}_{a_1,a_2} &:= \big\{s: (a_1,a_2,s)\in \mc E\big\},\qquad\qquad s\in\eS. \end{align} \end{subequations} Define {pmf} $\tilde{p}_{\tilde{A}_1\tilde{A}_2 S }$ by \begin{align} \tilde p_{A_1A_2S}(a_1,a_2,s) =\left\{\begin{array}{ll} {\displaystyle \frac{p_{A_1A_2S}(a_1,a_2,s)}{\Pr[(A_1,A_2)\in \E_s|S=s]}},& (a_1,a_2,s)\in\E\\ 0, &\textrm{otherwise}\end{array}\right., \label{eqn-Anc4-1} \end{align} and extend it to {pmf} $\tilde p_{\tilde A_1 \tilde A_2 S B_1 B_2}$ using the Markov chain $(\tilde A_1,\tilde A_2)\mkv S \mkv (B_1,B_2)$ with $\tilde{p}_{SB_1 B_2}:=p_{SB_1B_2}$. Let $ \Xi(\cdot)$ be as defined in \eqref{eqn-XiDefn}. The following properties hold for the pmf $\tilde{p}_{\tilde{A}_1 \tilde{A}_2 S B_1 B_2}$ defined above. \begin{theorem}\label{Thm:Ancillary4} \begin{itemize} \item[(a)]$ \lVert{\tilde{p}_{\tilde A_1\tilde A_2 S B_1B_2} -p_{A_1A_2S B_1B_2}}\rVert_1 \leq \frac{2\delta|\eS|}{\mpv_S -\delta}=2|\eS|(e^\eta-1).$\vspace{1mm} \item[(b)] Given $\mathcal{F}\subseteq \B_1\times \B_2$ and $(a_1,a_2)\in\supp(A_1,A_2)$ with $\Pr\big[(B_1,B_2)\in\mc F \big|(A_1,A_2)=(a_1,a_2)\big]\leq \delta\: \mpv_{B_1B_2|S}$, \begin{align}\Pr\big[(B_1,B_2)\in\mc F \big|(\tilde A_1,\tilde A_2)=(a_1,a_2)\big]\leq \delta \mpv_{B_1B_2|S} &= 0.\end{align} \item[(c)] $D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} ) \leq \log\dfrac{\mpv_S}{\mpv_S-\delta}=\eta$.\vspace{1mm} \item[(d)] $\big| H(S|\tilde{A}_1 B_1) - H(S|A_1 B_1) \big | \leq \sqrt[4]{{2}\eta} \log \dfrac{|\eS|^2}{ \sqrt[4]{{2}\eta}} +2|\eS|(e^\eta-1)\log |\eS|$.\vspace{0mm} \item[(e)] $\big|H(S|\tilde A_1 \tilde A_2B_1B_2)-H(S|A_1A_2B_1B_2)\big|\leq \sqrt[4]{{2}\eta} \log \dfrac{|\eS|^2}{ \sqrt[4]{{2}\eta}} +2|\eS|(e^\eta-1)\log |\eS|.$ \end{itemize} \end{theorem} \begin{IEEEproof} The proof follows on the same steps as Theorem~\ref{Thm:Ancillary3} and the difference is in the evaluation of the probabilities of the normalization term in \eqref{eqn-Anc4-1}. \subsubsection*{Assertion (a)} Since for $(a_1,a_2,s)\in \mc E$, we have $p(s|a_1,a_2) \leq \delta$, we can argue that \begin{align} \Pr[(A_1,A_2,S)\in\mathcal{E}^c] = \sum_{(a_1,a_2,s)\in\E^c} p_{A_1A_2 S}(a_1,a_2,s) \leq \delta \sum_{a_1,a_2,s} p_{A_1A_2 }(a_1,a_2) \leq \delta|\eS|. \label{eqn-Anc4-3} \end{align} Further, for any $s\in \supp(S)$, \begin{align} \Pr[(A_1,A_2)\in\E_s|\, S=s] &= \sum_{(a_1,a_2)\in \E_s} \frac{p_{A_1A_2S}(a_1,a_2,s) }{p_S(s)} = 1- \sum_{(a_1,a_2'\notin \E_s} \frac{p_{A_1A_2S}(a_1,a_2,s) }{p_S(s)} \notag\\ &\geq 1-\delta\sum_{a_1,a_2} \frac{ p_{A_1A_2}(a_1,a_2)}{p_S(s)} \geq 1-\frac{\delta}{\mpv_S}. \label{eqn-Anc4-4} \end{align} To prove (a), first note that for any $(a_1,a_2,b_1,b_2)$ and $s\in \mathcal{S}$, \begin{align} p_{A_1A_2SB_1B_2}(a_1,a_2,s,b_1,b_2)=0 \Rightarrow \tilde p_{\tilde A_1\tilde A_2S B_1 B_2}(a_1,a_2,s,b_1,b_2)=0, \end{align} and if $(a_1,a_2,s)\in\E$ and $p_{A_1A_2SB_1B_2}(a_1,a_2,s,b_1,b_2)>0$, then \begin{align} 1\leq \frac{\tilde p_{\tilde A_1\tilde A_2SB_1B_2}(a_1,a_2,s,b_1,b_2)}{p_{A_1A_2SB_1B_2}(a_1,a_2,s,b_1,b_2)} &= \frac{1}{\Pr[(A_1,A_2)\in\E_s\mid S=s]} \stackrel{\eqref{eqn-Anc4-4}}{\leq} \frac{\mpv_S}{\mpv_S-\delta} \label{eqn-Anc4-5}. \end{align} Thus, \begin{align} \sum_{\substack{b_1,b_2\\(a_1,a_2,s)\in\mathcal{E}}} \big| p_{A_1A_2SB_1B_2}(a_1,a_2,s,b_1,b_2) - \tilde p_{\tilde A_1\tilde A_2SB_1B_2}(a_1,a_2,s,b_1,b_2) \big| \leq \frac{\delta\,\Pr[(A_1,A_2,S)\in\mathcal{E}]}{\mpv_S-{\delta}}.\end{align} Next, we can bound the contribution for $\mathcal{E}^c \times\mathcal{B}_1\times\mathcal{B}_2$ by \begin{align} \sum_{\substack{b_1,b_2\\(a_1,a_2,s)\notin\mathcal{E}}} \big| p_{A_1A_2SB_1B_2}(a_1,a_2,s,b_1,b_2) - \tilde p_{\tilde A_1\tilde A_2SB_1B_2}(a_1,a_2,s,b_1,b_2) \big|& = \Pr[(A_1,A_2,S)\notin\mathcal{E}] \stackrel{\eqref{eqn-Anc4-3}}{\leq} \delta|\eS|. \end{align} Combining the above two equations together with the fact that $\delta|\eS|+ \frac{\delta}{\mpv_S -\delta}<\frac{2\delta|\eS|}{\mpv_S -\delta}$ establishes (a). \subsubsection*{Assertion (b)} We may assume that $\mc F\neq \emptyset$, or else the claim is trivial. Now, let $(a_1,a_2)\in\supp(A_1,A_2)$. Then, $(a_1,a_2)\in\supp(\tilde A_1,\tilde A_2)$ because \begin{align} \tilde{p}_{\tilde A_1, \tilde A_2}(a_1,a_2)\cdot \Pr[(A_1,A_2)\in \E_s|S=s] &= {\sum\limits_{s\in \mc{E}_{a_1,a_2}} p_{A_1A_2S}(a_1,a_2,s)}{}\notag\\ &= {p_{A_1,A_2}(a_1,a_2) -\sum\limits_{s\notin \mc{E}_{a_1,a_2}} p_{A_1A_2S}(a_1,a_2,s)}\notag\\ &\geq {p_{A_1,A_2}(a_1,a_2) - \sum\limits_{s\notin \mc{E}_{a_1,a_2}} \delta\, p_{A_1A_2}(a_1,a_2)}\notag\\ &= (1-\delta|\eS|)p_{A_1,A_2}(a_1,a_2) >0. \end{align} Now, by hypothesis, \begin{align} \Pr[(B_1,B_2)\in \mathcal{F} \mid (A_1,A_2)=(a_1, a_2)] = \sum_{(s',b_1,b_2)\in\,\supp(S)\times \mathcal F} p_{S|A_1A_2}(s'|a_1,a_2) p_{B_1B_2|S}(b_1,b_2|s')\leq \delta m_{B_1B_2|S}. \label{eqn-AssBFcond} \end{align} Thus, for any $(b_1,b_2)\in \mc F$ and $s\in \eS$ such that $p_{B_1B_2|S}(b_1,b_2|s)>0$, it must be true that \begin{align} p_{S|A_1A_2}(s|a_1,a_2) &\step{}{\leq} \sum\limits_{(s,b_1,b_2)\in\,\supp(S)\times \mathcal F} \frac{p_{S|A_1A_2}(s'|a_1,a_2) p_{B_1B_2|S}(b_1,b_2|s')}{p_{B_1B_2|S}(b_1,b_2|s)}\step{\eqref{eqn-AssBFcond}}{\leq} \frac{\delta m_{B_1B_2|S}}{p_{B_1B_2|S}(b_1,b_2|s)} \leq\delta, \end{align} Hence, $(a_1,a_2,s)\notin \mc{E}$. Thus, for any $(b_1,b_2)\in \mc F$ and $s\in \eS$ such that $p_{B_1B_2|S}(b_1,b_2|s)>0$, $\tilde{p}_{S|\tilde{A}_1\tilde A_2} (s|a_1,a_2)=0$, and therefore, (b) follows. \subsubsection*{Assertion (c)} To prove (c), we proceed as follows. \begin{align} D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} ) &=D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S} \lVert\, p_{A_1 A_2 S} )\notag\\ &= \sum_{a_1,a_2,s} \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) \log \frac{ \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) }{ {p}_{{A}_1{A}_2 S} (a_1,a_2,s) }\notag\\ &= \sum_{s\in\supp(S), (a_1,a_2)\in\mc E_s} \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) \log \frac{1}{ \Pr[ (A_1,A_2)\in \mc E_s|S=s]}\notag\\ &\stackrel{\eqref{eqn-Anc4-4}}{\leq} \sum_{\substack{s\in\supp(S)\\ (a_1,a_2)\in\mc E_s}} \tilde{p}_{\tilde{A}_1\tilde{A}_2 S} (a_1,a_2,s) \log \frac{\mpv_S}{\mpv_S-\delta}=\eta. \label{eqn-Anc4-12} \end{align} \subsubsection*{Assertion (d)} Consider the following. \begin{align} \sum_{a_1 b_1} \tilde{p}_{\tilde{A}_1B_1}(a_1,b_1) D_{KL} ( \tilde{p}_{S\mid \tilde A_1 B_1} ( \cdot | a_1, b_1) | {p}_{S\mid A_1 B_1} ( \cdot | a_1, b_1) &\leq D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} )\stackrel{\eqref{eqn-Anc4-12}}{\leq} \eta. \end{align} An application of Pinsker's followed by Jensen's inequality yields the following. \begin{align} \sum_{a_1 b_1} \tilde{p}_{\tilde{A}_1B_1}(a_1,b_1) \big\lVert\tilde{p}_{S\mid \tilde A_1 B_1} ( \cdot | a_1, b_1) - {p}_{S\mid A_1 B_1} ( \cdot | a_1, b_1)\big\rVert_1 \leq \sqrt{{2}\eta}. \end{align} Let $\mathcal{D} := \left\{(a_1,b_1): \big\lVert\tilde{p}_{S\mid \tilde A_1 B_1} ( \cdot | a_1, b_1) - {p}_{S\mid A_1 B_1} ( \cdot | a_1, b_1)\big\rVert_1 \leq \sqrt[4]{{2}\eta}.\right\}$. Then, by Markov's inequality, \begin{align} \Pr[ (\tilde A_1, B_1) \in \mathcal{D} ] &\geq 1- \sqrt[4]{{2}\eta}. \end{align} Further for every $(a_1,b_1)\in \mc D$, by Lemma~2.5 of \cite{Csiszar-2011-B}, we have \begin{align} \Delta_{a_1,b_1} &:= \Big| H(S| (\tilde A_1 ,B_1)=(a_1,b_1)) - H(S|(A_1 ,B_1)=(a_1,b_1))\Big| \leq \sqrt[4]{2\eta} \log \frac{|\eS|}{ \sqrt[4]{2\eta}}. \end{align} Finally, \begin{align} \big|H(S|\tilde A_1 B_1) -H(S| A_1 B_1) \big| &\leq \sum_{a_1,b_1} \tilde{p}_{\tilde A_1,B_1} (a_1,b_1) \Delta_{a_1,b_1} + \big\lVert \tilde{p}_{\tilde A_1,B_1}- {p}_{A_1,B_1}\big\rVert_1\log |\eS|\notag\\ &\leq \sum_{(a_1,b_1)\in \mc D} \tilde{p}_{\tilde A_1,B_1} (a_1,b_1) \Delta_{a_1,b_1}+ \Pr[(\tilde A_1,B_1)\notin \mc D]\log |\eS| + \frac{2\delta|\eS|\log |\eS|}{\mpv_S -\delta}\notag\\ &\leq \sqrt[4]{2\eta} \log \frac{|\eS|^2}{ \sqrt[4]{2\eta}}+2|\eS|(e^{\eta}-1)\log |\eS|. \end{align} \subsubsection*{Assertion (e)} The proof of (e) is identical to that of (d), with the only difference being that the commencement of the argument is as follows. \begin{align} \sum_{a_1 a_2,b_1,b_2} \tilde{p}_{\tilde{A}_1\tilde{A}_2,B_1,B_2}(a_1 a_2,b_1,b_2)& D_{KL} ( \tilde{p}_{S\mid \tilde A_1 \tilde A_2 B_1 B_2} ( \cdot | a_1 a_2,b_1,b_2) | {p}_{S\mid A_1 A_2 B_1 B_2} ( \cdot | a_1 a_2,b_1,b_2)\notag\\ &\leq D_{KL}(\tilde{p}_{\tilde{A}_1\tilde{A}_2 S B_1 B_2} \lVert\, p_{A_1 A_2 S B_1 B_2} )\stackrel{\eqref{eqn-Anc4-12}}{\leq} \eta. \end{align} The remaining steps are identical to those in (d), with the exception that all four variables, i.e., either $(\tilde A_1, \tilde A_2, B_1, B_2)$ or $(A_1, A_2, B_1, B_2) $, appear in the conditioning. \end{IEEEproof} \section{{Proof of Theorem~\ref{thm-3}}}\label{APP-I} In each case, the overall approach is to show that a rate pair $(r_{uv},r_v)$ that is included in the outer bound rate region $\R^\ddagger (D)$ is also included in $\R^D$. To establish that, in each case, we pick a rate pair $(r_{uv},r_v) \in {\R}_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D)$ and establish explicitly that the rate pair is also an element of ${\R}_{k}^{ *}(D)$ for some $k$. The proof is then complete by invoking the achievability of $\R_k^*(D)$ proved in Theorem~\ref{thm-achieve-thm-2} of Sec.~\ref{Sec-SLChars}. \subsubsection*{{Case A}}\label{proof-thm3-A} The proof follows from the following series of arguments: \begin{align} {\R}^{*}(D)\step{(a)}{\subseteq} {\mathsf{\R}(D)} &\step{(b)}{\subseteq } \R^\ddagger(D)\step{(c)}= {\R}^\ddagger_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}(D)\stackrel{(d)}{\subseteq } \R^{*}_{( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}(D), \end{align} where (a) and (b) follow from Theorems~\ref{thm-achieve-thm-2} and \ref{thm-achieve-thm-2a} of Sec.~\ref{Sec-SLChars}; (c) from Lemma~\ref{thm-RegionEquivalence2} of Sec.~\ref{Mainresults-rrdefns}; and (d) from Lemma~\ref{thm-RegionEquivalence} of Sec.~\ref{Mainresults-rrdefns}, where the full-support condition specific to this case is incorporated. \subsubsection*{{Case B}}\label{proof-thm3-B} Since $S\mkv V \mkv U$, from the outer bound in Theorem~\ref{thm-achieve-thm-2a} can be rewritten as follows. \begin{align} \R(D)\subseteq {{\R}_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D):= \bigcup_{q\in\mc{P}^{\ddagger}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}} \left\{(r_{uv}, r_v):{{\setstretch{1.25}\begin{array}{rl} r_{uv}\geq &\textstyle I(S;A,B|U)\\ r_v+r_{uv}\geq &I(S;A,B,C|V) \end{array}}}\right\}}. \label{eqn-CaseBRR} \end{align} Note that the sum rate constraint incorporates the chain $S\mkv V \mkv U$ specific to this case. Now, {let us} fix a pmf $q_{ABCSUV} \in \mc{P}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}$. Let $ f_U:\A\times\B\times \U \rightarrow \A\times\C\times \V$ and $f_V:\A\times\C\times \U \rightarrow \A\times\C\times \V$ be such that \begin{align} f_U(A,B,U) = f_V(A,C,V)= \GK^{ABU, ACV}. \end{align} Now, $\Pr[ f_U(A,B,U)\neq f_V(A,C,V)]=0$ can be expanded using $ C \mkv (A,B,V) \mkv U$ in the following manner. \begin{align} \sum_{a,b,v\in \supp(A,B,V)} q_{ABV}(a,b,v)\sum_{u,c} q_{C|ABV}(c|a,b,v) q_{U|V}(u|v)\bar{ \mathds 1}\left\{f_U(a,b,u) = f_V(a,c,v) \right\} = 0. \end{align} Hence, for any $(a,b,v)\in \supp(A,B,V)$, \begin{align} \sum_{u,c} q_{C|ABV}(c|a,b,v) q_{U|V}(u|v) \bar{\mathds 1}\left\{f_U(a,b,u)= f_V(a,c,v)\right\} = 0. \end{align} Then, by Lemma~\ref{lem-identical} of Appendix~\ref{APP-L}, it follows that $(a,b,v)\in \supp(A,B,V)$, there exists $\eta(a,b,v)$ such that \begin{subequations} \begin{align} \Pr[f_U(A,B,U)= \eta(a,b,v)\mid (A,B,V)=(a,b,v) ] &=1,\\ \Pr[f_V(A,C,V)= \eta(a,b,v)\mid (A,B,V)=(a,b,v) ] &=1. \end{align} \end{subequations} Hence, we are guaranteed of the existence of a function $g_V:\A\times\B\times \V\rightarrow\A\times\C\times \V$ such that \begin{align} g_V(A,B,V) = f_U(A,B,U) = f_V(A,C,V). \label{eqn-CaseBCondition1Met} \end{align} Thus, $\tilde{A} := g_V(A,B,V)$ is a function of $(A,B,U)$ as well as a function of $(A,B,V)$. Then, notice that \begin{subequations} \begin{alignat}{3} r_{uv}&\step{\eqref{eqn-CaseBRR}}{\geq} I(S;A,B|U)&\geq I(S; \tilde A|U),\\ r_v+r_{uv}&\step{\eqref{eqn-CaseBRR}}{\geq} I(S;A,B,C|V)\,\,&\geq I(S;\tilde A| V), \end{alignat} \end{subequations} and further $\tilde A \mkv S \mkv (U,V)$. Lastly, we see that the alphabet of $\tilde{A} \subseteq \A\times\C\times \V$, and hence \begin{align} |\tilde A| \leq \big( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\big)^2 |\U|. \label{eqn-CaseBCondition4Met} \end{align} From \eqref{eqn-CaseBCondition1Met}-\eqref{eqn-CaseBCondition4Met}, we conclude that $q_{\tilde ASUV} \in \mc P^*_{D,\big( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\big)^2 |\U|}$ and $(r_{uv}, r_v)\in \R^*_{\big( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\big)^2 |\U|}(D)$. Since $q_{ABCSUV}$ is any arbitrary pmf in $ \mc{P}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}$, it follows that \begin{align} {\textstyle {\R}_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D)} \subseteq\R^*_{( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2 |\U|}(D). \end{align} \subsubsection*{{Case C}}\label{proof-thm3-C} Let $(r_{uv}, r_v)\in\R_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D)$. Then, there must exist $q_{ABCSUV}\in \mc{P}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}$, and a function $f:\A\times\C\times\V \rightarrow \hat\eS$ such that $\Exp[d(S, f(\GK^{ABU,ACV}))] \leq D$, and \begin{subequations} \label{eqn-conditions-Case-C} \begin{align} r_{uv}&\geq \textstyle I(S;A,B|U),\\ r_v+r_{uv}&\geq I(S;A,C|V)+I(S;B|A,C,U). \end{align} \end{subequations} Since $(A,B,C) \mkv S \mkv (U,V)$, we have for any $(a,b,c,u,v)\in \supp(A,B,C,U,V)$, \begin{align} q_{ABCUV} (a,b,c,u,v) &= \sum_{s\in\supp(S)} q_{ABCSUV} (a,b,c,s,u,v) \notag\\ & = \sum_{s\in\supp(S)} q_{ABCS} (a,b,c,s) q_{UV|S} (u,v|s)\notag\\ & \leq \frac{1}{\mpv_S} \sum_{s\in\supp(S)} q_{ABCS} (a,b,c,s) q_{SUV}(s,u,v)\notag\\ & \leq \frac{1}{\mpv_S} q_{UV}(u,v)\sum_{s\in\supp(S)} q_{ABCS} (a,b,c,s)\notag\\ & \leq \frac{1}{\mpv_S} q_{ABC} (a,b,c) q_{UV}(u,v). \end{align} Further, since $\supp(S,U)=\supp(S)\times\supp(U)$, it follows that for any $(a,b,c,u,v)\in \supp(A,B,C,U,V)$, \begin{align} q_{ABCUV} (a,b,c,u,v) &= \sum_{s\in\eS} q_{ABCS}(a,b,c,s) q_{U|S}(u|s) q_{V|U}(v|u) \notag\\ &\geq \mpv_{U|S} \sum_{s\in\eS} q_{ABCS}(a,b,c,s) q_{V|U}(v|u)\notag \\ &= \mpv_{U|S} q_{ABC}(a,b,c) q_{V|U}(v|u)\notag\\ &\geq \mpv_{U|S} q_{ABC}(a,b,c) q_{UV}(u,v). \end{align} Hence, \begin{align} \supp(A,B,C,U,V) = \supp (A,B,C)\times \supp(U,V). \end{align} Then, from Lemma~\ref{Lem:Ancillary2} of Sec.~\ref{Sec:Gacs-Korner}, it follows that \begin{align} \GK^{ABU, ACV} \equiv (\GK^{AB, AC} , \GK^{U, V}).\label{eqn:GK-decorr} \end{align} Now, define $\tilde A:= \GK^{AB,AC}$, and let $ p_{\tilde ASUV}$ denote the {pmf} of $(\tilde A,S,U,V)$. Then, by construction the Markov chain $\tilde A\mkv S \mkv (U,V)$ holds. Further, \begin{align} H\big(\tilde A, \GK^{U,V}\,\big|\,\GK^{\tilde AV,\tilde AU}\big)& = 0. \end{align} Then, by \eqref{eqn:GK-decorr}, there must exist a function $\tilde{f}:\tilde \A \times \V\rightarrow \A\times\C\times\V$ such that \begin{align} \tilde{f}(\GK^{\tilde AU,\tilde AV})\equiv \GK^{ABU,ACV}. \end{align} Then, {from~\eqref{eqn-conditions-Case-C}}, we see that \begin{equation} {\Exp \big[d (S, f(\tilde{f}(\GK^{\tilde AV,\tilde AU})))\big]\leq D}. \end{equation} Since $\tilde \A\subseteq \A\times \C$, we are guaranteed that $|\tilde \A|\leq \big( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\big)^2 $, and hence, $ p_{\tilde ASUV}\in \mc{P}^*_{D,( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}$. Further, \begin{subequations} \begin{align} r_{uv} &\geq I(S;A,B|U)\geq I(S;\GK^{AC,AB}|U)=I(S;\tilde A|U),\\ r_{v}+r_{uv} &\geq I(S;A,C|V)\geq I(S;\GK^{AC,AB}|V)=I(S;\tilde A|V). \end{align} \end{subequations} Consequently, $(r_{uv}, r_v) \in \mathscr R^*_{( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}(D)$. Since the rate pair was chosen arbitrarily, it follows that \begin{equation} \R_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D) \subseteq \R_{ (|\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}^{*}(D). \end{equation} \subsubsection*{{Case D}}\label{proof-thm3-D} In this case, since $H(S|U)=0$, the receiver with side information $U$ does not require the encoder to communicate any message. Hence, we see that if $(r_{uv}, r_v)\in\R(D)$, then any $(r_{uv}', r_v')\in \R(D)$ provided $r_{uv}'+r_v' \geq r_{uv}+ r_v$, i.e., only the sum-rate constraint is relevant. Let $(r_{uv}, r_v)\in {\R}_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D)$. Then, there must exist {a joint pmf} $q_{ABCSUV}\in \mc{P}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}$ and a function $f:\A\times\C\times\V \rightarrow \hat\eS$ such that $\Exp[d(S, f(\GK^{ABU,ACV}))] \leq D$, and \begin{subequations} \begin{align} r_{uv}\geq &\textstyle I(S;A,B|U)= 0, \label{eqn-caseDrates1}\\ r_v+r_{uv}\geq &I(S;A,C|V)+I(S;B|A,C,U,V)= I(S;A,C|V). \label{eqn-caseDrates2} \end{align} \end{subequations} Let functions $f_U, g_V$ be defined such that $f_U(A,B,U) =f_V(A,C,V)=f(\GK^{ABU,ACV})$. Now, since $H(S|U)=0$, it follows that $B\mkv (A,C,U) \mkv V$. Then, we have \begin{align} \sum_{a,c,u} q_{ACU}(a,c,u) \sum_{b,v} q_{B|ACU}(b|a,c,u) p_{V|U}(v|u) \bar{\mathds 1} \left\{f_U(a,b,u) = f_V(a,c,v)\right\}= 0. \end{align} Then, by Lemma~\ref{lem-identical} of Appendix~\ref{APP-L}, for each $(a,c,u)\in \supp(A,C,U)$, there must exist an $\eta(a,c,u)$ such that \begin{subequations} \begin{align} \Pr\big[f_U(A,B,U) = \eta(a,c,u) \mid (A,C,U)=(a,c,u)\big] &= 1,\\ \Pr\big[f_V(A,C,V) = \eta(a,c,u) \mid(A,C,U)=(a,c,u)\big] &= 1. \end{align} \end{subequations} Hence, there must exist a function $\tilde f_U: \A\times\C\times \U \rightarrow \A\times\C\times \V$ such that \begin{align} \Pr\big[\tilde f_U (A,C,U) \neq f_V(A,C,V)\big] = 0. \label{eqn-existencefacu1} \end{align} Further, we also have \begin{align} \Exp[d(S, \tilde f_U (A,C,U) )]=\Exp[d(S, f_V(A,C,V))]=\Exp[d(S, f(\GK^{ABU,ACV}))] \leq D\label{eqn-existencefacu2} \end{align} Now, set $\tilde{A} = (A,C)$, and let $q_{\tilde A, S,U,V}$ be the joint {pmf} of $(\tilde A, S,U,V)$. From \eqref{eqn-existencefacu1}, \eqref{eqn-existencefacu2}, and from the fact that $|\tilde \A| \leq |\A||\C|$, it follows that $q_{\tilde A, S,U,V}\in \mc{P}^*_{D,\left( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\right)^2}$. Further, from \eqref{eqn-caseDrates1} and \eqref{eqn-caseDrates2}, it follows that \begin{subequations} \begin{align} r_{uv}&\geq I(S;A,B|U)=I(S;\tilde A|U)=0,\\ r_v+r_{uv}&\geq I(S;A,C|V) = I (S; \tilde A|V). \end{align} \end{subequations} Hence, $(r_{uv}, r_v)\in\R_{( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}^{*}(D)$. Since the rate pair was chosen arbitrarily, it follows that \begin{equation} \R_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D) \subseteq \R_{ (|\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}^{*}(D). \end{equation} \subsubsection*{Case E}\label{proof-thm3-E} Pick $(r_{uv}, r_v)\in {\R}_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D)$. Then, there must exist $q_{ABCSUV}\in \mc{P}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}$ and a function $f:\A\times\C\times\V \rightarrow \hat\eS$ such that $\Exp[d(S, f(\GK^{ABU,ACV}))] \leq D$, and \begin{subequations} \begin{align} r_{uv}&\geq I(S;A,B|U),\\ r_v+r_{uv}&\geq I(S;A,C|V)+I(S;B|A,C,U), \end{align} \end{subequations} where in the sum rate we have incorporated side information degradedness. Since $(A,B,C) \mkv S \mkv U \mkv V$, and $\supp(U,V)=\supp(U)\times\supp(V)$, it follows that \begin{align} \supp(A,B,C,U,V) = \supp(A,B,C,U) \times \supp(V). \end{align} Then, an invocation of Lemma~\ref{lem-drop1var} of Sec.~\ref{Sec:Gacs-Korner} yields \begin{align} \GK^{ABU,ACV} \equiv \GK^{ABU,AC}. \label{eqn-CaseEequivGK} \end{align} Hence, there must exist a function $\tilde f: \A\times \C \rightarrow \A \times \C \times \V$ such that $\tilde f (\GK^{ABU,AC}) = \GK^{ABU,ACV}$. Using this function, let us now define \begin{align} \tilde{A} := \tilde f(\GK^{ABU,AC}). \end{align} Then, $\hat S = f(\tilde A)$, and \begin{align} \Exp[d(S,\hat S )]=\Exp[d(S,f(\tilde A))]= \Exp[d(S, f(\tilde f (\GK^{ABU,AC}))] = \Exp[d(S, f(\GK^{ABU,ACV}))] \leq D. \label{eqn-CaseERes1} \end{align} Since $\tilde A$ is both a function of $(A,B,U)$ and $(A,C)$, we see that $|\tilde \A| \leq |\A||\C|= ( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2$, and \begin{subequations} \begin{alignat}{3} r_{uv}&\geq I(S;A,B|U)=I(S;A,B,\tilde{A}|U)&\geq I(S;\tilde A|U),\\ r_v+r_{uv}&\geq I(S;A,C,\tilde{A}|V)+I(S;B|A,C,U)\,&\geq I(S;\tilde A|V). \end{alignat}\label{eqn-CaseERes2} \end{subequations} Further, $\tilde A \mkv S \mkv U \mkv V$. Then, from \eqref{eqn-CaseERes1} and \eqref{eqn-CaseERes2}, we conclude that $q_{\tilde A SUV} \in \mc P^*_{D,( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}$ and $(r_{uv}, r_v)\in\R_{ (|\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^2}^{*}(D)$. Since $(r_{uv}, r_v)$ was chosen arbitrarily, it follows that \begin{equation} \R_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D) \subseteq \R_{ \left(|\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\right)^2}^{*}(D). \end{equation} \subsubsection*{{Case F}}\label{proof-thm3-F} Pick $(r_{uv}, r_v)\in {\R}_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D)$. Then, there must exist pmf $q_{ABCSUV}\in \mc{P}_{D, |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}$ and a reconstruction function $f:\A\times\V \rightarrow \hat \eS$ such that $\Exp[d(S, f(\GK^{ABU,ACV}))] \leq D$, and \begin{subequations} \begin{align} r_{uv}&\geq \textstyle I(S;A,B|U),\\ r_v+r_{uv}&\geq I(S;A,C|V)+I(S;B|A,C,U,V). \end{align} \label{eqn-conditions} \end{subequations} Suppose that $H(V|S) = 0$. Since $H(A|\GK^{ABU,ACV})=0$, we have \begin{align} I\big(\GK^{ABU,ACV};U,V|S\big)&\leq I\big(\GK^{ABU,ACV},A,C;U,V|S\big)\notag\\ &= I\big(A,C;U,V|S\big) + I\big(\GK^{ABU,ACV};U,V|A,C,S\big)\notag\\ &\step{(a)}= I(\GK^{ABU,ACV};U|A,C,S,V\big) \step{(b)} = 0, \end{align} where in (a), we use $(A,C)\mkv S\mkv (U,V)$ and that $V$ is a function of $S$; and in (b), we use the fact that $\GK^{ABU,ACV}$ is a function of $(A,V)$. Hence, \begin{align} \GK^{ABU,ACV} \mkv S \mkv (U,V).\end{align} Similarly, when $H(U|S)=0$, we arrive at the same conclusion by reversing the roles of $U$ and $V$. By defining $\tilde A := \GK^{ABU,ACV}$, we see that $\tilde A \mkv S \mkv (U,V)$ with $|\A| \leq \big( |\eS|(|\eS|+3)|\hat\eS|^{|\U|}+3\big)^3$, and that $\Exp[d(S, f(\tilde A)] \leq D$. Further, \begin{subequations} \begin{align} r_{uv}&\geq \textstyle I(S;A,B|U) \geq I(S;\tilde A|U)\\ r_v+r_{uv}&\geq I(S;A,C|V)+I(S;B|A,C,U,V)\geq I(S;\tilde A|V) \end{align} \end{subequations} Hence, $(r_{uv}, r_v)\in\R_{( |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4)^3}^{*}(D)$. Since the rate pair was chosen arbitrarily, it follows that \begin{equation} \R_{ |\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4}^{ \ddagger}(D) \subseteq \R_{ \left(|\eS|(|\eS|+6)|\hat\eS|^{|\U|}+4\right)^2}^{*}(D). \end{equation} \section{Proof of Theorem~\ref{thm-3-cor}}\label{APP-J} Clearly we have $\RQB(D) \subseteq \R^*(D)$. Since Theorem~\ref{thm-3} applies to Cases A, B and C, we need only show that $\RQB(D) \supseteq \R^*(D)$ in each case. Let $(r_{uv}, r_v)\in\R^*(D)$. Then, there must exist $q_{ASUV}\in \mc{P}_{D, |\eS|+2}^{ \ddagger}$, and a function $f:\A\times\V \rightarrow \hat \eS$ such that $\Exp[d(S, f(\GK^{AU,AV}))] \leq D$, and \begin{subequations} \begin{align} r_{uv}&\geq \textstyle I(S;A|U),\\ r_v+r_{uv}&\geq I(S;A|V). \end{align} \end{subequations} The rest of the proof for each case is as follows. \subsubsection*{Case A} Since $\supp(S,U,V) = \supp(S) \times \supp(U,V)$, it follows from that Lemma~\ref{Lem:Ancillary2} of Sec.~\ref{Sec:Gacs-Korner} that \begin{equation} \GK^{AU,AV} = \big(A,\GK^{U,V}\big). \end{equation} Since $H(\GK^{U,V}(V)) = 0$, we conclude from the above equation that $\GK^{AU,AV}(A,V) \equiv A$. Consequently, there must exist a function $\tilde f: \A\rightarrow \hat \eS$ such that reconstruction $ \tilde{f}(A)=f(\GK^{AU,AV})$. Define random variable $\hat S =\tilde{f}(A) =f(\GK^{AU,AV})$. Then, $(U,V) \mkv S \mkv A \mkv \hat{S}$ and \begin{subequations} \begin{align} r_{uv}&\geq \textstyle I(S;A|U)\geq I(S;\hat{S}|U),\\ r_v+r_{uv}&\geq I(S;A|V)\geq I(S;\hat{S}|V). \end{align}\label{eqn-APPI1} \end{subequations} Further, \begin{align} D\geq \Exp\big[d\big(S,f(\GK^{AU,AV})\big)\big] = \Exp\big[d\big(S,\hat{S}\big)\big], \label{eqn-APPI2} \end{align} Hence, it follows that $(r_{uv}, r_v)\in\RQB(D)$. \subsubsection*{Case B} In this setting, the Markov chain $A \mkv S \mkv U \mkv V$ and the support condition $\supp(U,V) = \supp(U) \times \supp(V)$ together imply $\supp(A,U,V) = \supp(A,U) \times \supp(V)$. Thus, from Lemma~\ref{lem-drop1var} of Sec.~\ref{Sec:Gacs-Korner}, we see that \begin{equation} \GK^{AU,AV} \equiv \GK^{AU,A} \equiv A. \end{equation} The rest of the proof then follows by setting $\hat S = f(A)=f(\GK^{AU,AV})$ and is identical to that of Case A above. \subsubsection*{Case C} Repeating the steps of Case F of Theorem~\ref{thm-3}, we see in this case that $\GK^{AU,AV} \mkv S \mkv (U,V)$. The proof is then complete by choosing $\hat{S} = f\big(\GK^{AU,AV}\big)$, and verifying that $\hat S \mkv S \mkv (U,V)$, \eqref{eqn-APPI1} and \eqref{eqn-APPI2} hold. \section{A Lemma on Functions of Independent Random Variables}\label{APP-L} \begin{lemma}\label{lem-identical} Let $X \sim p$ and $Y \sim q$ be independent random variables. Suppose that we are given a finite set $\Z$, and functions $f:\X \rightarrow \Z$ and $g:\Y\rightarrow \Z$ satisfying \begin{align} \Pr[f(X)\neq g(Y)] \leq \delta <\frac{1}{25}. \end{align} There exists an $z^*\in \Z$ such that \begin{subequations} \begin{align} \label{eqn:techlemma2-1} \Pr[f(X)=z^*] &\geq \frac{1+\sqrt{1-\delta-\sqrt{\delta}}}{2},\\ \label{eqn:techlemma2-2} \Pr[g(Y)=z^*] &\geq \frac{1+\sqrt{1-\delta-\sqrt{\delta}}}{2}. \end{align} \end{subequations} \end{lemma} \begin{IEEEproof} Let for $z\in \Z$, $p_z := \Pr[f(X)=z]$ and $q_z := \Pr[g(Y)=z]$. Then for any $z\in \Z$, \begin{align} |p_z-q_z| &= \sqrt{ (p_z-q_z)^2} \leq \sqrt{p_z(1-q_z)+(1-p_z)q_z}\leq \sqrt{\Pr[f(X)\neq g(Y)]} \leq\sqrt{\delta}.\label{eqn-SimpLem1} \end{align} Then, the following holds \begin{align} \sum_{z} p_z^2 & \geq \sum_{z} p_z( p_z+q_z-q_z) \geq \sum_z p_z q_z - \sum_z p_z |q_z-p_z| \notag\\& \geq \Pr[f(X)=g(Y)] -\sqrt{\delta} \geq 1-\delta-\sqrt \delta > \frac{19}{25}. \label{eqn-SimpLem2} \end{align} Now, let $M_p = \max\limits_z \,p_z$, $M_q= \max\limits_z \,q_z$, $z_p=\arg\max\limits_z\, p_z$ and $z_q=\arg\max\limits_z \,q_z$. Then, $M_p>\frac{19}{25}$, since \begin{align} M_p = \sum_z M_p \,p_z \geq \sum_z p_z^2 \geq 1-\delta-\sqrt \delta > \frac{19}{25}. \label{eqn-SimpLem3} \end{align} Also, $z_p$ and $z_q$, have to be identical, because $q_z$ can strictly exceed $\frac{1}{2}$ for only one $z\in\Z$, and \begin{align} q_{z_p} \geq p_{z_p}-|p_{z_p}-q_{z_p}| \stackrel{\eqref{eqn-SimpLem1}}{\geq} M_p -\sqrt{\delta} > \frac{19}{25}-\frac{1}{5}>\frac{1}{2}. \end{align} Since the problem is symmetric, we see that $M_q = q_{z_q} > 1-\delta -\sqrt{\delta}$ as well. At this point we are done if the RHS of \eqref{eqn:techlemma2-1} and \eqref{eqn:techlemma2-2} were $1-\delta-\sqrt\delta$. To improve this estimate, consider for $\gamma\in(0,1)$, the following optimization problem and its solution. \begin{align} \max_{\substack{ \forall z\in \Z, r_z \in [0,\gamma]\\ \sum_z r_z =1}} \sum_{z} r_z^2 = \left\lfloor\frac{1}{\gamma} \right\rfloor \gamma^2 + \left(1 - \left\lfloor\frac{1}{\gamma} \right\rfloor \gamma\right)^2 \leq \gamma^2 + (1-\gamma)^2.\label{eqn-SimpLem4} \end{align} Since $M_p$ is the maximum positive value taken by $\{p_z:z\in \Z\}$, it follows that \begin{align} 1-\delta-\sqrt{\delta} \leq \sum_z p_z^2 \stackrel{\eqref{eqn-SimpLem1}}{\leq} M_p^2 +(1-M_p)^2, \end{align} which necessitates that \begin{align} \Pr[f(X)=z_p] =M_p \in \left[ 0, \frac{1-\sqrt{1-\delta-\sqrt{\delta}}}{2}\right] \bigcup \left[ \frac{1+\sqrt{1-\delta-\sqrt{\delta}}}{2} ,1\right] .\end{align} Selecting the choice meeting \eqref{eqn-SimpLem3} eliminates the first interval. Finally, reversing the roles of $p$ and $q$ along with the fact that $z_p=z_q$ concludes the proof. \end{IEEEproof}

BkiUbGvxaJJQnKM6sK0P

\section{Introduction} Nearly three years have elapsed since the first kilohertz quasi-periodic oscillations (kHz QPOs) were discovered with the {\em Rossi X-ray Timing Explorer} (RXTE) in the X-ray flux of Scorpius X--1 (van der Klis et al. \cite{vanderklis96}) and 4U\,1728--34 (Strohmayer, Zhang, \& Swank \cite{strohmayer96circ}). In the meantime, kHz QPOs have been observed in the persistent flux of 18 low mass X-ray binaries (LMXBs; see van der Klis \cite{vanderklis98} for a review), both in the so called Z sources and in the atoll sources (Hasinger \& van der Klis \cite{hasinger89}, hereafter HK89). Except in Aql X--1, which showed a single kHz peak in its power spectrum (Zhang et al. \cite{zhangAqlx1}), in the other 17 sources two simultaneous kHz QPO peaks have been observed. In some sources a third, nearly-coherent, QPO peak has been detected during type-I X-ray bursts, with a frequency which was consistent with being equal to one or two times the frequency separation of the kHz QPOs observed in the persistent flux (Strohmayer, Swank, \& Zhang \cite{ssz98}). It was suggested that a beat frequency mechanism is responsible for this commensurability in the QPO frequencies (Strohmayer et al. \cite{strohmayer96circ}; Miller, Lamb \& Psaltis \cite{miller98}), but this interpretation is not without problems (van der Klis et al. \cite{vanderklisetal97a}; M\'endez et al. \cite{mendez98c}; M\'endez, van der Klis, \& van Paradijs \cite{mendez98b}). The dependence of the kHz QPO frequencies on X-ray luminosity, which is usually assumed to be a measure of the mass accretion rate, is complex. While in a given source, on a time scale of hours, there is a good correlation between frequency and luminosity, sources that span nearly three orders of magnitude in luminosity, such as Sco\,X--1 and 4U 0614+09, show kHz QPOs that cover the same range of frequencies (van der Klis \cite{vanderklis97}). It is as if the frequency of the kHz QPO depends on the difference between instantaneous and average luminosity in each source rather than on the luminosity itself. A similar effect is seen between observations of the same source at different epochs. On time scales of hours, frequency and X-ray flux are well correlated, but between different epochs the source covers the same range of frequencies even if the average flux is different by 40\,\% or more (e.g., Aql X--1; Zhang et al. \cite{zhangAqlx1}). In this paper we present new results that appear to resolve the latter of these two issues, at least in 4U\,1608--52. We show that while on time scales longer than $\sim 1$ day the frequency of the kHz QPOs is not well correlated to the X-ray flux, it is very well correlated to the position of the source in the color-color diagram. From this result we conclude that the observed changes in the frequency of the kHz QPOs in 4U\,1608--52 are driven by changes in the mass accretion rate, and that the lack of correlation between QPO frequency and X-ray count rate occurs because, as in the case of the Z sources, in atoll sources there is no one-to-one relationship between the observed X-ray flux and the mass accretion rate. \section{Observations and Results} All the observations presented here were obtained using the proportional counter array (PCA) onboard RXTE. We include the data of the decay of the 1996 outburst (Berger et al. \cite{berger96}; M\'endez et al. \cite{mendez98a}), and of the 1998 outburst (M\'endez et al. \cite{mendez98c}; Fig. 1 there shows a light curve of the 1998 outburst.) The observations as well as the modes used to record the data are described in M\'endez et al. (\cite{mendez98c}). We also include here a recent Public Target of Opportunity RXTE/PCA observation of $\sim 7.3$ ks performed on August 6 1998, 03:13 UTC. The observing modes for this last observation were similar to those used by M\'endez et al. (\cite{mendez98c}) after 1998 March 27. We calculated count rates in 5 energy bands, $2.0 - 3.5 - 6.4 - 9.7 - 16.0$ keV, and $2.0 - 16.0$ keV, taking into account the gain changes applied to the PCA in March and April 1996. In a few of the observations one or two of the five detectors of the PCA were switched off; we only used the three detectors which were always on to calculate these count rates. We subtracted the background contribution in each band using the standard PCA background model version 2.0c\footnote{The PCA Background Estimator is available at\\ http://lheawww.gsfc.nasa.gov/users/stark/pca/pcabackest.html, which is maintained by NASA/GSFC.}, and normalized the count rates to 5 detectors. In Figure \ref{figcolor} we show a color-color diagram of 4U\,1608--52. This is the first time that 4U\,1608--52 is observed to move across all the branches of the atoll, and constitutes one of the best examples of the color-color diagram of an atoll source. Based on this diagram we conclude that in 1996, as the source count rate decreased, 4U\,1608--52 gradually moved from the lower part of the banana to the island state. In 1998, at the peak of the outburst, 4U\,1608--52 was in the upper part of the banana, and gradually moved down to the lower part of the banana and the island state as the count rate decreased. In general, the count rate is observed to increase along the track from the island state to the banana branch, but the relation between count rate and either of the two colors is much less clean than that between colors. The high-time resolution data confirm this preliminary state classification. We divided the $2 - 60$ keV data into segments of 256 s and 512 s, and calculated power spectra up to a Nyquist frequency of $2048$ Hz, normalized to fractional rms squared per Hertz. The characteristics of the $\simless 100$ Hz part of these power spectra changed in correlation with the position of the source in the color-color diagram. When 4U 1608--52 was in the upper and lower parts of the banana, the power spectra fitted a power law below $\sim 1$ Hz (the Very Low Frequency Noise, VLFN), and an exponentially cut-off power law above $\sim 1$ Hz (the High Frequency Noise, HFN; see HK89). As 4U\,1608--52 moved from the upper to the lower parts of the banana, the fractional amplitude of the VLFN ($0.001 - 1$ Hz) decreased from $\sim 6$\,\% rms to $\sim 2$\,\% rms, while the fractional amplitude of the HFN ($1 - 100$ Hz) increased from $\sim 1$\,\% rms to $\sim 5$\,\% rms. In the island state the VLFN disappeared completely (the 95\,\% confidence upper limits were $\sim 1$\,\% rms), and the amplitude of the HFN increased further to $\sim 10 - 17$\,\% rms. (A more detailed analysis of the low frequency part of the power spectra will be presented elsewhere.) The $\simmore 100$ Hz part of the power spectra also changed in correlation with the position of the source in the color-color diagram: we only observed kHz QPOs when 4U\,1608--52 was at certain positions of the color-color diagram (see Fig. \ref{figcolor}). For those segments where we observed QPOs, the $2 - 60$ keV fractional amplitudes of the lower-frequency and higher-frequency, hereafter the lower and upper QPO, varied from 5.3\,\% to 9.1\,\% rms, and from 3.3\,\% to 8.8\,\% rms, respectively. We did not detect kHz QPOs in the upper part of the banana, with 95\,\% confidence upper limits of 0.8\,\% to 4.6\,\% rms depending on the source count rate and the assumed width of the QPO, nor in the extreme island state (upper right corner of the color-color diagram), with 95\,\% confidence upper limits of 3.5\,\% to 10\,\% rms. While these upper limits strongly suggest the absence in the upper banana of kHz QPOs as strong as those observed in the lower banana and the island states, we cannot rule out the presence of such kHz QPOs in the extreme island state, when the count rates were lowest. To further characterize the dependence of the kHz QPOs on other source parameters, we selected only those data where we detected two simultaneous kHz QPO peaks in the power spectrum (see M\'endez et al. \cite{mendez98c}), so that the identification of the observed kHz peaks is unambiguous. We divided the data in segments of 64 s and produced a power spectrum for each segment extending from 1/64 Hz to 2048 Hz. In Figure \ref{figrate} we show the dependence of the frequency of the lower QPO, $\nu_{\rm low}$, as a function of count rate for 4U\,1608--52. This figure shows several branches, which reflect the relation between $\nu_{\rm low}$ and count rate during individual observations that span from $\sim$ half an hour to $\sim 8$ hours. The only exception is the branch at the lowest count rate where two different observations taken 8 days apart overlap. This figure clearly shows that while on time scales of a few hours or less $\nu_{\rm low}$ is well correlated to count rate, on time scales greater than $\sim$ 1 day the relation is complex and $\nu_{\rm low}$ is not uniquely determined by the count rate. We obtained the same result using the $2 - 60$ keV source flux instead of the count rate. There is a much better correlation between $\nu_{\rm low}$ and the position of the source on the color-color diagram. In Figure \ref{fig_hc} we show $\nu_{\rm low}$ as a function of hard color (see Fig. \ref{figcolor}) for the same intervals shown in Figure \ref{figrate}. The complexity seen in the frequency vs. count rate diagram (Fig. \ref{figrate}) is reduced to a single track in the frequency vs. hard color diagram. The frequency of the lower kHz QPO increases as the hard color decreases, i.e., as 4U\,1608--52 moves from the island state to the lower banana, and keeps increasing at the turn of the lower banana, where the hard color reaches its lowest value of $\sim 0.40$. The shape of the track in Figure \ref{fig_hc} suggests that the hard color may not be sensitive to changes of state when the the source moves into the banana in the color-color diagram. To further investigate this, we applied the $S_{\rm Z}$ parameterization (e.g., Wijnands et al. \cite{wijnandsa&a97}), which we call $S_{\rm a}$ for atoll sources, to the color-color diagram. In this phenomenological approach we approximated the shape of the color-color diagram with a spline (Fig. \ref{figcolor}), and we used the parameter $S_{\rm a}$ to measure positions along this spline. We set $S_{\rm a}$ to 1 at (2.67,0.77), and to 2 at (2.19,0.42), as indicated in Figure \ref{figcolor}. We measured the position on the color-color diagram of each of the 64-s segments. We then grouped these segments into 36 sets, such that within each set $\nu_{\rm low}$ did not vary by more than $\sim 25$ Hz, and the source count rate did not vary by more than 10\,\%. (This is to avoid mixing segments that come from different branches in Figure \ref{figrate}.) For each set we calculated the average frequency of the lower and upper kHz QPO peak, and of $S_{\rm a}$. In Figure \ref{fig_sa} we plot the frequencies of both kHz QPOs as a function of $S_{\rm a}$. The error bars are the standard deviation of each selection. We also include in this figure several extra sets (open squares) for which we only detected one of the kHz QPOs, which therefore we could not a priori identify as the upper or lower peak. As expected, $S_{\rm a}$ is more sensitive than the hard color to changes of state when the the source moves from the island state into the turn of the banana: $S_{\rm a}$ increases from $\sim 2.04$ to $\sim 2.15$ there, while the hard color saturates at $\sim 0.40$. Figure \ref{fig_sa} confirms that the frequency of the kHz QPOs is strongly correlated to the position of the source in the color-color diagram. It also shows that the steep rise of kHz QPO frequency at the turn of the lower banana is not just due to a lack of sensitivity of the hard color to state changes in this part of the color-color diagram, but that it occurs as a function of the position in the color-color diagram as well. \section{Discussion} Previous observations of kHz QPOs in atoll sources have resulted in a confusing picture concerning the dependence of QPO frequency on mass accretion rate. Cases have been reported where, in a given source, QPO frequency showed a good correlation to count rate or spectral hardness (Ford, van der Klis, \& Kaaret \cite{fvdkk98}; Wijnands et al. \cite{wijnands1636}; Wijnands et al. \cite{wijnands1735}; Ford et al. \cite{ford1735}; Zhang et al. \cite{zhang1820}; Wijnands \& van der Klis \cite{wijnands1731}). In other cases, a correlation with count rate or flux was conspicuously lacking (Ford et al. \cite{ford97a}; M\'endez et al. \cite{mendez98a}; Zhang et al. \cite{zhangAqlx1}). Our observations show for the first time that a total lack of correlation between frequency and count rate on time scales longer than a day (Fig. \ref{figrate}) can coexist with a very good correlation between frequency and position in the X-ray color-color diagram (Fig. \ref{fig_hc}). The frequency increases with $S_{\rm a}$, as the source moves from the island to the banana. Only on time scales of hours does the QPO frequency appear to also correlate well with count rate. The presence of the QPOs also correlates well with the position in the color-color diagram: the QPOs are only detected in the lower banana and the moderate island states, and disappear both in the upper banana and in the extreme island states (Fig. \ref{figcolor}). In atoll sources $\dot M$ is thought to increase monotonically with $S_{\rm a}$ along the track in the color-color diagram, from the island to the upper banana (HK89), whereas X-ray count rate tracks $\dot M$ much less well (van der Klis et al. \cite{vdk90}; van der Klis \cite{vdk94apj}; Prins \& van der Klis \cite{prins97}). The properties of the $< 100 $ Hz power spectra depend monotonically on inferred $\dot M$ (HK89). Our result that the frequency of the kHz QPO is well correlated to $S_{\rm a}$, but not to X-ray count rate, implies that in 4U\,1608--52 the kHz QPO frequency {\em also} depends monotonically on inferred $\dot M$. By extension this conclusion applies to each atoll source; in Z sources similar conclusions were previously also reached (e.g., Wijnands et al. \cite{wijnands17+2}). Our analysis, however, sheds no light on the question why sources with very different inferred $\dot M$ (e.g., 4U 0614+09 and Sco\,X--1) can have kHz QPO in the same range of frequencies. Further indications for this interpretation come from the simultaneous analysis of the low and high frequency parts of the power spectra of other atoll sources. In 4U\,1728--34 the kHz QPO frequencies were recently found to be very well correlated to several $< 100$ Hz power spectral properties (Ford \& van der Klis \cite{ford&vdk98}), and Psaltis, Belloni and van der Klis (\cite{psaltisetal98}) obtained a similar result for a number of other atoll (and Z) sources. In all these sources, not only the position in the color-color diagram and the various low frequency power spectral parameters, but also the frequencies of the kHz QPOs are all well correlated with each other. This indicates that the single parameter, inferred to be $\dot M$, which governs all the former properties also governs the frequency of the kHz QPO. X-ray intensity is the exception: it can vary more or less independently from the other parameters. In 4U\,1608--52, it can change by a factor of $\sim 4$ (see Fig. \ref{figrate}) while the other parameters do not vary significantly. If as inferred, this constancy of the other parameters means that $\dot M$ does not change, then this indicates that strongly variable beaming of the X-ray flux, or large-scale redistribution of some of the radiation over unobserved energy ranges is occurring in order to change the flux by the observed factors, {\em without} any appreciable changes in the X-ray spectrum. We may need to scrutinize more closely the concept of $\dot M$ in order to solve this dilemma. For example, perhaps the $\dot M$ governing all the other parameters is the $\dot M$ through the inner accretion disk, whereas matter also flows onto the neutron star in a more radial inflow, or away from it in a jet. In 4U\,1608--52 we observe no evidence for a saturation of the frequency of the kHz QPOs at a constant maximum value as $\dot M$ increases, different from what Zhang et al. (\cite{zhang1820}) inferred for 4U 1820--30. They presented data in which the kHz QPO frequencies increase with count rate up to a threshold level, above which the frequencies remain approximately constant while the count rate keeps increasing. Interpreting count rate as a measure for $\dot M$ they argue that this is evidence for the inner edge of the disk reaching the general-relativistic innermost stable orbit. However, we have shown here that, at least in 4U\,1608--52, count rate is not a good measure for $\dot M$. Inspection of Figure \ref{figrate} suggests that with sparser sampling our plot could easily have looked similar to the one presented by Zhang et al. (\cite{zhang1820}) for 4U\,1820--30. It will therefore be of great interest to see if in 4U\,1820--30 the saturation of QPO frequency as a function of count rate is still there when this parameter is plotted as a function of position in the X-ray color-color diagram. \acknowledgements This work was supported in part by the Netherlands Organization for Scientific Research (NWO) under grant PGS 78-277 and by the Netherlands Foundation for research in astronomy (ASTRON) under grant 781-76-017. MM is a fellow of the Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\'ecnicas de la Rep\'ublica Argentina. JVP acknowledges support from the National Aeronautics and Space Administration through contract NAG5-3269, 5-4482 and 5-7382. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. {\footnotesize

BkiUbIw5qhLACD0Xwn79

\section{Introduction and Preliminaries} The optimization problem of approximate integration formulas in the modern sense appears as the problem of finding the minimum of the norm of a error functional $\ell$ given on some set of functions. The minimization problem of the norm of the error functional by coefficients was reduced in \cite{Sob74,SobVas} to the system of difference equations of Wiener-Hopf type in the space $L_2^{(m)}$, where $L_2^{(m)}$ is the space of functions with square integrable $m-$th generalized derivative. Existence and unique\-ness of the solution of this system was proved by S.L. Sobolev. In the works \cite{Sob74,SobVas} the description of some analytic algorithm for finding the coefficients of optimal formulas is given. For this S.L. Sobolev defined and investigated the discrete analogue $D_{hH}^{(m)}(h\beta)$ of the poly\-har\-monic operator $\Delta^m$. The problem of construction of the discrete opera\-tor $D_{hH}^{(m)}(h\beta)$ for $n-$ dimensional case was very hard. In one dimensional case the discrete analogue $D_{h}^{(m)}(h\beta)$ of the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}$ was constructed by Z.Zh. Zhamalov \cite{Zham78} and Kh.M. Shadimetov \cite{Shad85}. Further, in the work \cite{ShadHay04_1} the discrete analogue of the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}-\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}$ was constructed. The constructed discrete analogue of the operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}-\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}$ was applied for finding the coefficients of the optimal quadrature formulas (see \cite{ShadHay_04_2,ShadHay13}) and for construction of interpolation splines minimizing the semi-norm (see \cite{ShadHay12}) in the space $W_2^{(m,m-1)}(0,1)$, where $W_2^{(m,m-1)}(0,1)$ is the Hilbert space of functions $\varphi$ which $\varphi^{(m-1)}$ is absolutely continuous, $\varphi^{(m)}$ belongs to $L_2(0,1)$ and $\int_0^1(\varphi^{(m)}(x)+\varphi^{(m-1)}(x))^2\,{\rm d} x<\infty$. Here, we mainly use a concept of functions of a discrete argument and the corresponding operations \cite[Chapter VII]{Sob74}. For completeness we give some of definitions. Let $\beta\in \mathbb{Z}$, $h=\frac{1}{N}$, $N=1,2,...$ . Assume that $\varphi$ and $\psi$ are real-valued functions defined on the real line $\mathbb{R}$. \smallskip {\sc Definition 1.} The function $\varphi(h\beta)$ is a \emph{function of discrete argument} if it is defined on some set of integer values of $\beta $. \smallskip {\sc Definition 2.} The \emph{inner product} of two discrete argument functions $\varphi(h\beta)$ and $\psi(h\beta)$ is defined as \[ \left[ {\varphi,\psi} \right] = \sum_{\beta = - \infty }^\infty \varphi(h\beta) \cdot \psi(h\beta).\] Here we assume that the series on the right hand side converges absolutely. \smallskip {\sc Definition 3.} The \textit{convolution} of two functions $\varphi(h\beta)$ and $\psi(h\beta)$ is defined as the inner product \[ \varphi(h\beta)*\psi(h\beta) = \left[ {\varphi(h\gamma),\psi(h\beta-h\gamma)} \right] = \sum_{\gamma = - \infty }^\infty {\varphi (h\gamma)\cdot \psi(h\beta - h\gamma)}. \] {\sc Definition 4.} The function $\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{\varphi}\!\!(x)=\sum\limits_{\beta=-\infty}^{\infty}\varphi(h\beta)\delta(x-h\beta)$ is called \emph{the harrow-shaped function} corresponding to the function of discrete argument $\varphi(h\beta)$, where $\delta$ is Dirac's delta function. We are interested in to find a discrete function $D_m(h\beta)$ that satisfies the following equation \begin{equation}\label{eq.(1.1)} D_m(h\beta)*G_m(h\beta)=\delta_{\,{\rm d}}(h\beta), \end{equation} where \begin{eqnarray} G_m(h\beta)&=&\frac{(-1)^m\mathrm{sign}(h\beta)}{4\omega^{2m-1}}\Bigg[(2m-3)\sin(h\omega\beta)- h\omega\beta\cos(h\omega\beta)\nonumber \\ &&\qquad \qquad \qquad +2\sum\limits_{k=1}^{m-2}\frac{(-1)^{k}(m-k-1)(h\omega\beta)^{2k-1}}{(2k-1)!}\Bigg],\label{eq.(1.2)} \end{eqnarray} $m\geq 2$, $\omega>0$, $\delta_{\,{\rm d}}(h\beta)$ is equal to 1 when $\beta=0$ and is equal to 0 when $\beta\neq 0$, i.e. $\delta_{\,{\rm d}}(h\beta)$ is the discrete delta function. The discrete function $D_m(h\beta)$ plays an important role in the calculation of the coefficients of the optimal quadrature formulas and interpolation splines minimizing a semi-norm in the Hilbert space \[ K_2(P_m)=\Bigl\{\varphi:[0,1]\to \mathbb{R}\ \Bigm|\ \varphi^{(m-1)} \mbox{ is abs. cont. and } \varphi^{(m)}\in L_2(0,1)\Bigr\}, \] equipped with the norm \begin{equation}\label{eq.(1.3)} \|\varphi\|=\left\{\int\limits_0^1\left(P_m\left(\frac{\,{\rm d}}{\,{\rm d} x}\right) \varphi(x)\right)^2\,{\rm d} x\right\}^{\frac12}, \end{equation} where $P_m(\frac{\,{\rm d}}{\,{\rm d} x})=\frac{\,{\rm d}^m}{\,{\rm d} x^m}+\omega^2\frac{\,{\rm d}^{m-2}}{\,{\rm d} x^{m-2}}$, $\omega>0$ and $ {\int_0^1 {\left( {P_m \left( {{\,{\rm d} \over {\,{\rm d} x}}} \right)\varphi (x)} \right)} ^2 \,{\rm d} x}<\infty.$ Note that the equality (\ref{eq.(1.3)}) is semi-norm, moreover $\|\varphi\|=0$ if and only if $\varphi(x)=c_1\sin \omega x+c_2\cos \omega x+Q_{m-3}(x)$, with $Q_{m-3}(x)$ a polynomial of degree $m-3$. In particular, the optimal quadrature formulas and interpolation splines in the $K_2(P_m)$ space are exact for the trigonometric functions $\sin\omega x$, $\cos \omega x$ and polynomials of degree $m-3$. It should be noted that for a linear differential operator of order $m$,\linebreak $$ L:= P_m(\,{\rm d}/\,{\rm d} x)=\frac{\,{\rm d}^m}{\,{\rm d} x^m}+a_{m-1}(x)\frac{\,{\rm d}^{m-1}}{\,{\rm d} x^{m-1}}+...+a_1(x)\frac{\,{\rm d}}{\,{\rm d} x}+a_0(x),$$ Ahlberg, Nilson, and Walsh in the book \cite[Chapter 6]{Ahlb67} investigated the Hilbert spaces in the context of generalized splines. Namely, with the inner product \[\langle\varphi,\psi\rangle=\int_0^1 L\varphi (x)\cdot L\psi (x) \,{\rm d} x ,\] $K_2(P_m)$ is a Hilbert space if we identify functions that differ by a solution of $L\varphi=0$. Note that the equation (\ref{eq.(1.1)}) is the discrete analogue to the following equation \begin{equation}\label{eq.(1.4)} \left(\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}\right)G_m(x)=\delta(x), \end{equation} where \begin{eqnarray*} G_m(x)&=&\frac{(-1)^m\mathrm{sign}(x)}{4\omega^{2m-1}}\Bigg[(2m-3)\sin (\omega x)-\omega x\cdot \cos(\omega x)\\ && \qquad\qquad\qquad +2\sum_{k=1}^{m-2}\frac{(-1)^{k}(m-k-1)(\omega x)^{2k-1}}{(2k-1)!}\Bigg], \end{eqnarray*} and $\delta$ is the Dirac's delta function. Moreover the discrete function $D_m(h\beta)$ has analogous properties the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$: the zeros of the discrete operator $D_m(h\beta)$ coincides with the discrete functions corresponding to the zeros of the operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$. The discrete function $D_m(h\beta)$ is called \emph{the discrete analogue} of the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$. The rest of the paper is organized as follows: in Section 2 we give some known formulas and auxiliary results which will be used in the construction of the discrete function $D_m(h\beta)$. Section 3 is devoted to the construction of the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$. \section{Auxiliary results} \setcounter{equation}{0} In this section we explain some known formulas (see, for instance, \cite{Sob74,Vlad79}) and auxiliary results which we use in the construction of the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$. For the Fourier transformations and their properties: \begin{eqnarray} &&F[\varphi(x)]=\int\limits_{-\infty}^{\infty}\varphi(x)e^{2\pi ipx}\,{\rm d} x,\ \ \ F^{-1}[\varphi(p)]=\int\limits_{-\infty}^{\infty}\varphi(p)e^{-2\pi ipx}\,{\rm d} p,\label{eq.(2.1)}\\ &&F[\varphi*\psi]=F[\varphi]\cdot F[\psi], \label{eq.(2.2)}\\ && F[\varphi\cdot \psi]=F[\varphi]* F[\psi], \label{eq.(2.3)}\\ && F[\delta^{(\alpha)}(x)]=(-2\pi ip)^{\alpha},\ \ F[\delta(x)]=1, \label{eq.(2.4)} \end{eqnarray} where $*$ is the convolution and for two functions $\varphi$ and $\psi$ is defined as follows $$ (\varphi*\psi)(x)=\int\limits_{-\infty}^{\infty}\varphi(x-y)\psi(y)\,{\rm d} y=\int\limits_{-\infty}^{\infty}\varphi(y)\psi(x-y)\,{\rm d} y. $$ For the Dirac's delta function: \begin{eqnarray} &&\delta(hx)=h^{-1}\delta(x),\label{eq.(2.5)}\\ &&\delta(x-a)\cdot f(x)=\delta(x-a)\cdot f(a), \label{eq.(2.6)}\\ && \delta^{(\alpha)}(x)*f(x)=f^{(\alpha)}(x), \label{eq.(2.7)} \\ &&\phi_0(x)=\sum\limits_{\beta=-\infty}^{\infty}\delta(x-\beta)=\sum\limits_{\beta=-\infty}^{\infty}e^{2\pi i x\beta}. \label{eq.(2.8)} \end{eqnarray} It is known \cite{Froben,Sob06_1,SobVas} that {\it the Euler-Frobenius polynomials} $E_k (x)$ are defined by the following formula \begin{equation}\label{eq.(2.9)} E_k (x) = \frac{{(1 - x)^{k + 2} }}{x}\left( {x\frac{d}{{dx}}} \right)^k \frac{x}{{(1 - x)^2 }},\ \ k=1,2,... \end{equation} $E_0 (x) = 1$, moreover all roots $x_j^{(k)}$ of the Euler-Frobenius polynomial $E_k(x)$ are real, negative and simple, i.e. \begin{equation}\label{eq.(2.10)} x_1^{(k)}<x_2^{(k)}<...<x_k^{(k)}<0. \end{equation} Furthermore, the roots equally spaced from the ends of the chain (\ref{eq.(2.10)}) are reciprocal, i.e. $$ x_j^{(k)}\cdot x_{k+1-j}^{(k)}=1. $$ Euler obtained the following formula for the coefficients $a_s^{(k)}$, $s=0,1,...,k$ of the polynomial $E_k(x)=\sum\limits_{s=0}^ka_s^{(k)}x^s$: \begin{equation}\label{eq.(2.11)} a_s^{(k)}=\sum\limits_{j=0}^s(-1)^j{k+2\choose j} (s+1-j)^{k+1}. \end{equation} The polynomial $E_k(x)$ satisfies the following identity \begin{equation}\label{eq.(2.12)} E_k(x)=x^kE_k\left(\frac{1}{x}\right),\ \ x\neq 0, \end{equation} i.e. $a_s^{(k)}=a_{k-s}^{(k)},\ \ s=0,1,2,...,k$. Now we consider the following polynomial of degree $2m-2$ $$ {\cal P}_{2m-2}(x)=\sum_{s=0}^{2m-2}p_s^{(2m-2)}x^s =(1-x)^{2m-4}\bigg[[(2m-3)\sin h\omega-h\omega \cos h\omega]x^2 $$ $$ \qquad\qquad\qquad +[2h\omega-(2m-3)\sin(2h\omega)]x+[(2m-3)\sin h\omega-h\omega \cos h\omega]\bigg] $$ \begin{equation}\label{eq.(2.13)} +2(x^2-2x\cos h\omega+1)^2\sum_{k=1}^{m-2}\frac{(-1)^k(m-k-1)(h\omega)^{2k-1}(1-x)^{2m-2k-4}E_{2k-2}(x)}{(2k-1)!}, \end{equation} where $E_{2k-2}(x)$ is the Euler-Frobenius polynomial of degree $2k-2$, $\omega>0$, $h\omega\leq 1$, $h=1/N$, $N\geq m-1$, $m\geq 2$. The polynomial ${\cal P}_{2m-2}(x)$ has the following properties. \begin{lemma}\label{L2.1} The Polynomial ${\cal P}_{2m-2}(x)$ satisfies the identity $$ {\cal P}_{2m-2}(x)=x^{2m-2}{\cal P}_{2m-2}\left(\frac{1}{x}\right). $$ \end{lemma} \begin{lemma}\label{L2.3} For the derivative of the polynomial ${\cal P}_{2m-2}(x)$ the following equality holds $$ {\cal P}_{2m-2}'\left(\frac{1}{x_k}\right)=-\frac{1}{x_k^{2m-4}}{\cal P}_{2m-2}'(x_k), $$ where $x_k$ is a root of the polynomial ${\cal P}_{2m-2}(x)$. \end{lemma} Further we prove Lemmas \ref{L2.1}-\ref{L2.3}. \emph{Proof of Lemma \ref{L2.1}.} If we replace $x$ by $\frac1x$ in (\ref{eq.(2.13)}), we obtain $$ {\cal P}_{2m-2}\left(\frac1x\right)=\left(1-\frac1x\right)^{2m-4}\bigg[[(2m-3)\sin h\omega-h\omega \cos h\omega]\frac{1}{x^2} $$ $$ \qquad\qquad\qquad +[2h\omega-(2m-3)\sin(2h\omega)]\frac1x+[(2m-3)\sin h\omega-h\omega \cos h\omega]\bigg] $$ $$ +2\left(\frac{1}{x^2}-\frac{2}{x}\cos h\omega+1\right)^2\sum_{k=1}^{m-2}\frac{(-1)^k(m-k-1)(h\omega)^{2k-1}\left(1-\frac1x\right)^{2m-2k-4}E_{2k-2}\left(\frac{1}{x}\right)}{(2k-1)!}. $$ Hence, taking into account (\ref{eq.(2.12)}) and the fact that $\left(1-\frac{1}{x}\right)^{2k}=\frac{1}{x^{2k}}(1-x)^{2k}$, we get the statement of the lemma. Lemma \ref{L2.1} is proved.\hfill $\Box$ \emph{Proof of Lemma \ref{L2.3}.} Let $x_1,x_2,...,x_{2m-2}$ be the roots of the polynomial ${\cal P}_{2m-2}(x)$. Then from Lemma \ref{L2.1} we immediately get that if $x_k$ is the root then $\frac{1}{x_k}$ is also the root of the polynomial ${\cal P}_{2m-2}(x)$ and if $|x_k|<1$ then $|\frac{1}{x_k}|>1$. Suppose the roots $x_1,x_2,...,x_{2m-2}$ are situated such that \begin{equation}\label{eq.(2.14)} x_k\cdot x_{2m-1-k}=1. \end{equation} Therefore we have $$ {\cal P}_{2m-2}(x)=p_{2m-2}^{(2m-2)}(x-x_1)(x-x_2)...(x-x_{2m-2}), $$ where $p_{2m-2}^{(2m-2)}$ is the leading coefficient of the polynomial ${\cal P}_{2m-2}(x)$.\\ Hence $$ {\cal P}_{2m-2}'(x)=p_{2m-2}^{(2m-2)}\sum\limits_{j=1}^{2m-2}\frac{\prod\limits_{i=1}^{2m-2}(x-x_i)}{x-x_j}. $$ Then for $x=x_k$ we get \begin{equation}\label{eq.(2.15)} {\cal P}_{2m-2}'(x_k)=p_{2m-2}^{(2m-2)}\prod\limits_{i=1,i\neq k}^{2m-2}(x_k-x_i) \end{equation} and for $x=\frac{1}{x_k}$, taking into account (\ref{eq.(2.14)}), using the equality $$ \prod\limits_{i=1,i\neq 2m-1-k}^{2m-2}\frac{1}{x_{2m-1-i}}=\frac{1}{x_1x_2...x_{k-1}x_{k+1}x_{k+2}...x_{2m-2}}=\frac{1}{x_{2m-1-k}}=x_k $$ we have \begin{eqnarray} {\cal P}_{2m-2}'\left(\frac{1}{x_k}\right)&=&p_{2m-2}^{(2m-2)}\sum\limits_{j=1}^{2m-2}\frac{\prod\limits_{i=1}^{2m-2}(\frac{1}{x_k}-x_i)}{\frac{1}{x_k}-x_j}\nonumber\\ &=&p_{2m-2}^{(2m-2)}\frac{1}{x_k^{2m-3}}\prod\limits_{i=1,i\neq 2m-1-k}^{2m-2}(1-x_kx_i)\nonumber\\ &=&-p_{2m-2}^{(2m-2)}\frac{1}{x_k^{2m-3}}\prod\limits_{i=1,i\neq 2m-1-k}^{2m-2}\frac{x_k-x_{2m-1-i}}{x_{2m-1-i}}\nonumber\\ &=&-\frac{1}{x_k^{2m-4}}\ p_{2m-2}^{(2m-2)}\prod\limits_{i=1,i\neq k}^{2m-2}(x_k-x_i).\label{eq.(2.16)} \end{eqnarray} From (\ref{eq.(2.15)}) and (\ref{eq.(2.16)}) we get the statement of the lemma. Lemma \ref{L2.3} is proved.\hfill $\Box$ In the proof of the main result we have to calculate the following series \begin{eqnarray} S_1&=&\sum\limits_{\beta=-\infty}^{\infty}\frac{1}{[\beta-h(p+\frac{\omega}{2\pi})][\beta-h(p-\frac{\omega}{2\pi})]},\label{eq.(2.17)}\\ S_2&=&\sum\limits_{\beta=-\infty}^{\infty}\frac{1}{[\beta-h(p+\frac{\omega}{2\pi})]^2},\label{eq.(2.18)}\\ S_3&=&\sum\limits_{\beta=-\infty}^{\infty}\frac{1}{[\beta-h(p-\frac{\omega}{2\pi})]^2},\label{eq.(2.19)}\\ F[h\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_{k,1}]&=&\frac{(-1)^k}{(2\pi)^{2k}}\sum\limits_{\beta=-\infty}^{\infty}\frac{1}{(p-h^{-1}\beta)^{2k}}.\label{eq.(2.20)} \end{eqnarray} We denote $\lambda=e^{2\pi iph}$ then the following holds \begin{lemma}\label{L2.4} For the series \emph{(\ref{eq.(2.17)})-(\ref{eq.(2.20)})} the following are taken place $$ \begin{array}{rclrcl} S_1&=&\frac{-(2\pi)^2\lambda\sin h\omega}{h\omega (\lambda^2-2\lambda\cos h\omega)}, \ \ \ & S_2&=&\frac{-(2\pi)^2\lambda}{(\lambda^2+1)\cos h\omega -2\lambda+i(\lambda^2-1)\sin h\omega},\\ F[h\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_{k,1}]&=&\frac{h^{2k}\lambda\ E_{2k-2}(\lambda)}{(2k-1)!(1-\lambda)^{2k}},& S_3&=&\frac{-(2\pi)^2\lambda}{(\lambda^2+1)\cos h\omega -2\lambda+i(\lambda^2-1)\sin h\omega}, \end{array} $$ where $E_{2k-2}(\lambda)$ is the Euler-Frobenius polynomial of degree $2m-2$. \end{lemma} \emph{Proof.} To calculate the series (\ref{eq.(2.17)})-(\ref{eq.(2.20)}) we use the following well-known formula from the residual theory (see [6], p.296) \begin{equation}\label{eq.(2.21)} \sum\limits_{\beta=-\infty}^{\infty}f(\beta)=-\sum\limits_{z_1,z_2,...,z_n} \mathrm{res}(\pi \cot(\pi z)\cdot f(z)), \end{equation} where $z_1,z_2,...,z_n$ are poles of the function $f(z)$. At first we consider $S_1$. We denote $f_1(z)=\frac{1}{(z-h(p+\frac{\omega}{2\pi}))(z-h(p-\frac{\omega}{2\pi}))}$. It is clear that $z_1=h(p+\frac{\omega}{2\pi})$ and $z_2=h(p-\frac{\omega}{2\pi})$ are the poles of order 1 of the function $f_1(z)$. Then taking into account the formula (\ref{eq.(2.21)}) we have \begin{equation}\label{eq.(2.22)} S_1=\sum\limits_{\beta=-\infty}^{\infty}f_1(\beta)=-\sum\limits_{z_1,z_2}\mathrm{res}(\pi \cot(\pi z)\cdot f_1(z)). \end{equation} Since $$ \mathop{\mbox{res}}\limits_{z=z_1}(\pi \cot(\pi z)\cdot f_1(z))=\lim\limits_{z\to z_1}\frac{\pi\cot(\pi z)}{z-h(p-\frac{\omega}{2\pi})}=\frac{\pi^2}{h\omega}\cot(\pi hp+\frac{h\omega}{2}), $$ and $$ \mathop{\mbox{res}}\limits_{z=z_2}(\pi \cot(\pi z)\cdot f_1(z))=\lim\limits_{z\to z_2}\frac{\pi\cot(\pi z)}{z-h(p+\frac{\omega}{2\pi})}=-\frac{\pi^2}{h\omega}\cot(\pi hp-\frac{h\omega}{2}). $$ Using the last two equalities, from (\ref{eq.(2.22)}) we get $$ S_1=\frac{\pi^2}{h\omega}\left(\cot(\pi hp-\frac{h\omega}{2})-\cot(\pi hp+\frac{h\omega}{2})\right) =\frac{\pi^2\sin(h\omega)}{h\omega\sin(\pi hp-\frac{h\omega}{2})\sin(\pi hp+\frac{h\omega}{2})}. $$ Taking into account that $\lambda=e^{2\pi iph}$ and using the well known formulas $$ \cos z=\frac{e^{zi}+e^{-zi}}{2},\ \ \sin z=\frac{e^{zi}-e^{-zi}}{2i}, $$ after some simplifications for $S_1$ we have $$ S_1=\frac{-(2\pi)^2\lambda\sin h\omega}{h\omega(\lambda^2-2\lambda \cos h\omega+1)}. $$ Similarly for $S_2$ and $S_3$ we arrive the following equalities: $$ S_2=\frac{-(2\pi)^2\lambda}{(\lambda^2+1)\cos h\omega-2\lambda+i(\lambda^2-1)\sin h\omega} $$ and $$ S_3=\frac{-(2\pi)^2\lambda}{(\lambda^2+1)\cos h\omega-2\lambda-i(\lambda^2-1)\sin h\omega}. $$ The series (\ref{eq.(2.20)}) was calculated in \cite{Shad85}, where the following expression has been obtained $$ F[h\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_{k,1}]=\frac{h^{2k}\lambda E_{2k-2}(\lambda)}{(2k-1)!\ (1-\lambda)^{2k}}. $$ So, Lemma \ref{L2.4} is proved. \hfill $\Box$ \section{Construction of the discrete analogue of the operator} \setcounter{equation}{0} In this section we construct the discrete analogue $D_m(h\beta)$ of the operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$ and obtain some properties of the constructed discrete function $D_m(h\beta)$. The result is the following one: \begin{theorem}\label{THM3.1} The discrete analogue to the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$ satisfying equation \emph{(\ref{eq.(1.1)})} has the form \begin{equation}\label{eq.(3.1)} D_m(h\beta)=\frac{2\omega^{2m-1}}{(-1)^mp_{2m-2}^{(2m-2)}}\left\{ \begin{array}{ll} \sum\limits_{k=1}^{m-1}A_k \lambda_k^{|\beta|-1},& |\beta|\geq 2,\\ 1+\sum\limits_{k=1}^{m-1}A_k,& |\beta|=1,\\ C+\sum\limits_{k=1}^{m-1}\frac{A_k}{\lambda_k},& \beta=0, \end{array} \right. \end{equation} where \begin{eqnarray} A_k&=&\frac{(1-\lambda_k)^{2m-4}(\lambda_k^2-2\lambda_k\cos h\omega+1)^2p_{2m-2}^{(2m-2)}}{\lambda_k{\cal P}_{2m-2}'(\lambda_k)},\label{eq.(3.2)}\\ C&=&4-4\cos h\omega-2m-\frac{p_{2m-3}^{(2m-2)}}{p_{2m-2}^{(2m-2)}},\label{eq.(3.3)}\\ p_{2m-2}^{(2m-2)}&=&(2m-3)\sin h\omega-h\omega\cos h\omega\nonumber \\ &&\qquad\qquad\qquad +2\sum\limits_{k=1}^{m-2}\frac{(-1)^k(m-k-1)(h\omega)^{2k-1}}{(2k-1)!},\label{eq.(3.4)} \end{eqnarray} ${\cal P}_{2m-2}(\lambda)$ is the polynomial of degree $2m-2$ defined by \emph{(\ref{eq.(2.13)})}, $p_{2m-2}^{(2m-2)},\ p_{2m-3}^{(2m-2)}$ are the coefficients and $\lambda_k$ are the roots of the polynomial ${\cal P}_{2m-2}(\lambda)$, $|\lambda_k|<1$. \end{theorem} Before proving this result we present the following properties that are not difficult to verify. \begin{lemma} The discrete analogue $D_m(h\beta)$ of the differential operator $\frac{\,{\rm d}^{2m}}{\,{\rm d} x^{2m}}+2\omega^2\frac{\,{\rm d}^{2m-2}}{\,{\rm d} x^{2m-2}}+\omega^4\frac{\,{\rm d}^{2m-4}}{\,{\rm d} x^{2m-4}}$ satisfies the following equalities \emph{1)} $D_m(h\beta)*\sin(h\omega\beta)=0,$ \emph{2)} $D_m(h\beta)*\cos(h\omega\beta)=0,$ \emph{3)} $D_m(h\beta)*(h\omega\beta)\sin(h\omega\beta)=0,$ \emph{4)} $D_m(h\beta)*(h\omega\beta)\cos(h\omega\beta)=0,$ \emph{5)} $D_m(h\beta)*(h\beta)^\alpha=0,$ $\alpha=0,1,...,2m-5$. \end{lemma} \emph{Proof of Theorem \ref{THM3.1}.} According to the theory of periodic distribution (generalized) functions and Fourier transformations, instead of the function $D_m(h\beta)$ it is convenient to search the harrow-shaped function (see \cite{Sob74,SobVas}) $$ \stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m\!\!(x)=\sum\limits_{\beta=-\infty}^{\infty} D_m(h\beta)\delta(x-h\beta). $$ In the class of harrow-shaped functions, equation (\ref{eq.(1.1)}) takes the following form \begin{equation}\label{eq.(3.6)} \stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m\!\!(x)*\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)=\delta(x), \end{equation} where $\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)=\sum\limits_{\beta=-\infty}^{\infty} G_m(h\beta)\delta(x-h\beta)$ is the harrow-shaped function corresponding to the discrete function $G_m(h\beta)$. Applying the Fourier transformation to both sides of equation (\ref{eq.(3.6)}) and taking into account (\ref{eq.(2.2)}) and (\ref{eq.(2.4)}) we have \begin{equation}\label{eq.(3.7)} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m\!\!(x)]=\frac{1}{F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)]}. \end{equation} First, we calculate the Fourier transformation $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)]$ of the harrow-shaped function $\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)$. Using equalities (\ref{eq.(2.5)}), (\ref{eq.(2.6)}) and (\ref{eq.(2.8)}), we get \begin{eqnarray*} \stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)&=&\sum_{\beta=-\infty}^{\infty}G_m(h\beta)\delta(x-h\beta)\nonumber\\ &=&h^{-1}G_m(x)\sum_{\beta=-\infty}^{\infty}\delta(h^{-1}x-\beta).\nonumber \end{eqnarray*} Hence \begin{equation} \stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)=h^{-1}G_m(x)\cdot \phi_0(h^{-1}x).\label{eq.(3.8)} \end{equation} Taking into account (\ref{eq.(2.1)}) and (\ref{eq.(2.8)}), for $F[\phi_0(h^{-1}x)]$ we obtain \begin{eqnarray} F[\phi_0(h^{-1}x)]&=&\sum\limits_{\beta=-\infty}^{\infty}\int_{-\infty}^{\infty}\delta(h^{-1}x-\beta)e^{2\pi ipx}\,{\rm d} x\nonumber\\ &=&h\sum\limits_{\beta=-\infty}^{\infty}\int_{-\infty}^{\infty}\delta(x-h\beta)e^{2\pi ipx}\,{\rm d} x\nonumber\\ &=&h\sum\limits_{\beta=-\infty}^{\infty}e^{2\pi iph\beta}=h\sum\limits_{\beta=-\infty}^{\infty}\delta(hp-\beta)=h\phi_0(hp).\label{eq.(3.9)} \end{eqnarray} Applying the Fourier transformation to both sides of (\ref{eq.(3.8)}) and using (\ref{eq.(2.3)}), (\ref{eq.(3.9)}), we get \begin{equation} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)]=F[G_m(x)]*\phi_0(hp). \label{eq.(3.10)} \end{equation} To calculate the Fourier transformation $F[G_m(x)]$, we use equation (\ref{eq.(1.4)}). Taking into account (\ref{eq.(2.7)}), we rewrite equation (\ref{eq.(1.4)}) in the following form $$ (\delta^{(4)}(x)+2\omega^2\delta^{(2m-2)}(x)+\omega^4\delta^{(2m-4)}(x))*G_m(x)=\delta(x). $$ Hence, keeping in mind (\ref{eq.(2.2)}) and (\ref{eq.(2.4)}), we have $$ F[G_m(x)]=\frac{1}{(2\pi ip)^{2m}+2\omega^2(2\pi ip)^{2m-2}+\omega^4 (2\pi ip)^{2m-4}}. $$ Using the last equality, from (\ref{eq.(3.10)}) we arrive to \begin{eqnarray*} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)]&=&\frac{1}{(2\pi ip)^{2m}+2\omega^2(2\pi ip)^{2m-2}+\omega^4 (2\pi ip)^{2m-4}}*\phi_0(hp)\\ &=& \frac{1}{h}\sum\limits_{\beta=-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{\delta(y-h^{-1}\beta)\,{\rm d} y}{(2\pi i(p-y))^{2m}+2\omega^2(2\pi i(p-y))^{2m-2}+\omega^4 (2\pi i(p-y))^{2m-4}}. \end{eqnarray*} Hence, after some calculations we get \begin{equation}\label{eq.(3.11)} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_m\!\!(x)]= \sum\limits_{\beta=-\infty}^{\infty}\frac{h^{-1}}{(2\pi i(p-\frac{\beta}{h}))^{2m}+2\omega^2(2\pi i(p-\frac{\beta}{h}))^{2m-2}+\omega^4 (2\pi i(p-\frac{\beta}{h}))^{2m-4}}. \end{equation} Now, expanding to partial fractions of the right hand side of (\ref{eq.(3.11)}) and taking into account (\ref{eq.(3.7)}), we conclude \begin{eqnarray} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)&=&h\Bigg[\frac{(-1)^{m-1}(2m-3)h^2}{(2\pi)^2\cdot 2\omega^{2m-2}}S_1+\frac{(-1)^{m-2}h^2}{(2\pi)^2\cdot 4\omega^{2m-2}}(S_2+S_3) \nonumber\\ &&\qquad+\frac{1}{\omega^{2m}}\sum_{k=1}^{m-2}(-1)^{m-k}(m-k-1)\omega^{2k}F[h\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_{k,1}]\Bigg]^{-1},\label{eq.(3.12)} \end{eqnarray} where $S_1$, $S_2$, $S_3$ and $F[h\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_{k,1}]$ are defined by (\ref{eq.(2.17)})-(\ref{eq.(2.20)}). It is clear that the function $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)$ is a periodic function with respect to the variable $p$, and the period is $h^{-1}$. Moreover it is real and analytic for all real $p$. The zeros of the function $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)$ are $p=h^{-1}\beta+\frac{\omega}{2\pi}$, $p=h^{-1}\beta-\frac{\omega}{2\pi}$ and $p=h^{-1}\beta$. The function $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m ](p)$ can be expanded to the Fourier series as follows \begin{equation} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)=\sum\limits_{\beta=-\infty}^{\infty}\hat D_m(h\beta)e^{2\pi i ph\beta},\label{eq.(3.12)} \end{equation} where $\hat D_m(h\beta)$ are the Fourier coefficients of the function $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)$, i.e. \begin{equation}\label{eq.(3.13)} \hat D_m(h\beta)=\int\limits_0^{h^{-1}}F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)\ e^{-2\pi i ph\beta}\,{\rm d} p. \end{equation} Applying the inverse Fourier transformation to both sides of (\ref{eq.(3.12)}), we obtain the following harrow-shaped function $$ \stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m\!\!(x)=\sum\limits_{\beta=-\infty}^{\infty}\hat D_m(h\beta)\delta(x-h\beta). $$ Then, according to the definition of harrow-shaped functions, the discrete function $\hat D_m(h\beta)$ is the discrete argument function $D_m(h\beta)$, which we are searching. So, we have to find the discrete argument function $\hat D_m(h\beta)$. But here, to find the function $\hat D_m(h\beta)$ we will not use the formula (\ref{eq.(3.13)}). In this case to obtain $\hat D_m(h\beta)$ we have to calculate the series $S_1$, $S_2$, $S_3$ and $F[h\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{G}_{k,1}]$ defined by (\ref{eq.(2.17)})-(\ref{eq.(2.20)}). We have calculated these series in Lemma \ref{L2.4}. Therefore, using Lemma \ref{L2.4}, from (\ref{eq.(3.13)}) we get \begin{equation} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)=\frac{2\omega^{2m-1}}{(-1)^m}\ \frac{(1-\lambda)^{2m-4}(\lambda^2+1-2\lambda\cos h\omega)^2}{\lambda\ {\cal P}_{2m-2}(\lambda)}, \end{equation} where $\lambda=e^{2\pi iph}$ and ${\cal P}_{2m-2}(\lambda)$ is the polynomial of degree $2m-2$ defined by (\ref{eq.(2.13)}). Now we will get the Fourier series of the function $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)$. Dividing the polynomial ${(1-\lambda)^{2m-4}(\lambda^2+1-2\lambda\cos h\omega)^2}$ by the polynomial $\lambda {\cal P}_{2m-2}(\lambda)$ we obtain \begin{eqnarray}\label{eq.(3.15)} &&F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)=\frac{2\omega^{2m-1}}{(-1)^m}\ \frac{(1-\lambda)^{2m-4}(\lambda^2+1-2\lambda\cos h\omega)^2}{\lambda\ {\cal P}_{2m-2}(\lambda)}\nonumber\\ && =\frac{2\omega^{2m-1}}{(-1)^mp_{2m-2}^{(2m-2)}}\left[\lambda+4-4\cos h\omega-2m-\frac{p_{2m-3}^{(2m-2)}}{p_{2m-2}^{(2m-2)}}\right]+\frac{S_{2m-2}(\lambda)}{\lambda{\cal P}_{2m-2}(\lambda)}, \end{eqnarray} where $S_{2m-2}(\lambda)$ is a polynomial of degree $2m-2$. It is clear that $\frac{S_{2m-2}(\lambda)}{\lambda{\cal P}_{2m-2}(\lambda)}$ is the proper fraction. Since the roots of the polynomial ${\lambda{\cal P}_{2m-2}(\lambda)}$ are real and simple (see Appendix), then the rational fraction $\frac{S_{2m-2}(\lambda)}{\lambda{\cal P}_{2m-2}(\lambda)}$ is expanded to the sum of partial fractions, i.e. \begin{equation}\label{eq.(3.16)} \frac{S_{2m-2}(\lambda)}{\lambda {\cal P}_{2m-2}(\lambda)}=\frac{1}{p_{2m-2}^{(2m-2)}}\left[\frac{A_0}{\lambda}+\sum\limits_{k=1}^{m-1}\left(\frac{A_{1,k}}{\lambda-\lambda_{1,k}}+ \frac{A_{2,k}}{\lambda-\lambda_{2,k}}\right)\right], \end{equation} where $A_0,$ $A_{1,k}$ and $A_{2,k}$ are unknown coefficients, $\lambda_{1,k}$, $\lambda_{2,k}$ are the roots of the polynomial ${\cal P}_{2m-2}(\lambda)$ and $|\lambda_{1,k}|<1$, $|\lambda_{2,k}|>1$ and \begin{equation}\label{eq.(3.17)} \lambda_{1,k}\lambda_{2,k}=1. \end{equation} Keeping in mind equality (\ref{eq.(3.16)}) from (\ref{eq.(3.15)}), we obtain \begin{eqnarray} &&F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)=\frac{2\omega^{2m-1}}{(-1)^m}\ \frac{(1-\lambda)^{2m-4}(\lambda^2+1-2\lambda\cos h\omega)^2}{\lambda\ {\cal P}_{2m-2}(\lambda)}\label{eq.(3.18)}\\ && \qquad\qquad\qquad =\frac{2\omega^{2m-1}}{(-1)^mp_{2m-2}^{(2m-2)}}\left[\lambda+4-4\cos h\omega-2m-\frac{p_{2m-3}^{(2m-2)}}{p_{2m-2}^{(2m-2)}}\right]\nonumber\\ &&\qquad\qquad\qquad\ \ \ +\frac{1}{p_{2m-2}^{(2m-2)}}\left[\frac{A_0}{\lambda}+\sum\limits_{k=1}^{m-1}\left(\frac{A_{1,k}}{\lambda-\lambda_{1,k}}+ \frac{A_{2,k}}{\lambda-\lambda_{2,k}}\right)\right].\nonumber \end{eqnarray} Multiplying both sides of the last equality by $\lambda{\cal P}_{2m-2}(\lambda)$ we have \begin{eqnarray} &&\frac{2\omega^{2m-1}}{(-1)^m}\ {(1-\lambda)^{2m-4}(\lambda^2+1-2\lambda\cos h\omega)^2}\nonumber\\ && \qquad =\lambda\ {\cal P}_{2m-2}(\lambda)\Bigg[\frac{2\omega^{2m-1}}{(-1)^mp_{2m-2}^{(2m-2)}}\left[\lambda+4-4\cos h\omega-2m-\frac{p_{2m-3}^{(2m-2)}}{p_{2m-2}^{(2m-2)}}\right]\nonumber\\ &&\qquad\qquad\qquad\ \ \ +\frac{1}{p_{2m-2}^{(2m-2)}}\left[\frac{A_0}{\lambda}+\sum\limits_{k=1}^{m-1}\left(\frac{A_{1,k}}{\lambda-\lambda_{1,k}}+ \frac{A_{2,k}}{\lambda-\lambda_{2,k}}\right)\right]\Bigg].\label{eq.(3.19)} \end{eqnarray} To find unknown coefficients $A_0$, $A_{1,k}$ and $A_{2,k}$ in (\ref{eq.(3.19)}) we put $\lambda=0$, $\lambda=\lambda_{1,k}$ and $\lambda=\lambda_{2,k}$. Then we get \begin{eqnarray} A_0&=&\frac{2\omega^{2m-1}}{(-1)^m},\nonumber\\ A_{1,k}&=&\frac{2\omega^{2m-1}(1-\lambda_{1,k})^{2m-4}(\lambda_{1,k}^2+1-2\lambda_{1,k}\cos h\omega)^2p_{2m-2}^{(2m-2)}}{(-1)^m\lambda_{1,k}{\cal P}_{2m-2}'(\lambda_{1,k})},\label{eq.(3.20)}\\ A_{2,k}&=&\frac{2\omega^{2m-1}(1-\lambda_{2,k})^{2m-4}(\lambda_{2,k}^2+1-2\lambda_{2,k}\cos h\omega)^2p_{2m-2}^{(2m-2)}}{(-1)^m\lambda_{2,k}{\cal P}_{2m-2}'(\lambda_{2,k})}.\nonumber \end{eqnarray} Hence, taking into account (\ref{eq.(3.17)}) and Lemma \ref{L2.3}, we get \begin{equation}\label{eq.(3.21)} A_{2,k}=-\frac{A_{1,k}}{\lambda_{1,k}^2}. \end{equation} Since $|\lambda_{1,k}|<1$ and $|\lambda_{2,k}|>1$ the expressions $$ \sum\limits_{k=1}^{m-1}\frac{A_{1,k}}{\lambda-\lambda_{1,k}}\mbox{ and } \sum\limits_{k=1}^{m-1}\frac{A_{2,k}}{\lambda-\lambda_{2,k}} $$ can be expanded to the Laurent series on the circle $|\lambda| =1$, i.e. $$ \sum\limits_{k=1}^{m-1}\frac{A_{1,k}}{\lambda-\lambda_{1,k}}=\frac{1}{\lambda} \sum\limits_{k=1}^{m-1}\frac{A_{1,k}}{1-\frac{\lambda_{1,k}}{\lambda}}=\frac{1}{\lambda} \sum\limits_{k=1}^{m-1}A_{1,k}\sum\limits_{\beta=0}^{\infty}\left(\frac{\lambda_{1,k}}{\lambda}\right)^{\beta} $$ and taking into account (\ref{eq.(3.17)}), we deduce $$ \sum\limits_{k=1}^{m-1}\frac{A_{2,k}}{\lambda-\lambda_{2,k}}= \sum\limits_{k=1}^{m-1}\frac{A_{2,k}}{\lambda-\frac{1}{\lambda_{1,k}}}= -\sum\limits_{k=1}^{m-1}\frac{A_{2,k}\lambda_{1,k}}{1-\lambda\ \lambda_{1,k}}=-\sum_{k=1}^{m-1}A_{2,k}\lambda_{1,k}\sum\limits_{\beta=0}^{\infty}(\lambda\ \lambda_{1,k})^\beta. $$ Using the last two equalities and taking into account (\ref{eq.(3.3)}) and $\lambda=e^{2\pi iph}$ from (\ref{eq.(3.18)}) we get \begin{eqnarray*} F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)&=&\frac{1}{p_{2m-2}^{(2m-2)}}\Bigg\{\frac{2\omega^{2m-1}}{(-1)^m}[e^{2\pi iph}+C]+A_0e^{-2\pi iph}\\ &&+\sum\limits_{k=1}^{m-1}A_{1,k}\sum_{\beta=0}^{\infty}\lambda_{1,k}^\beta e^{-2\pi iph(\gamma+1)}-\sum_{k=1}^{m-1}A_{2,k}\sum_{\beta=0}^{\infty}\lambda_{1,k}^{\beta+1}e^{2\pi iph\gamma}\Bigg\}. \end{eqnarray*} Thus the Fourier series for $F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)$ has the following form $$ F[\stackrel{\mathop{\urcorner\!\!\!\!\!\ulcorner}}{D}_m](p)=\sum\limits_{\beta=-\infty}^{\infty}D_m(h\beta)e^{2\pi i hp\beta}, $$ where $$ D_m(h\beta)=\frac{1}{p_{2m-2}^{(2m-2)}}\left\{ \begin{array}{ll} \sum\limits_{k=1}^{m-1}A_{1,k}\lambda_{1,k}^{-\beta-1},&\beta\leq -2,\\ A_0+\sum\limits_{k=1}^{m-1}A_{1,k},& \beta=-1,\\ \frac{2\omega^{2m-1}}{(-1)^m}C-\sum\limits_{k=1}^{m-1}A_{2,k}\lambda_{1,k},& \beta=0,\\ \frac{2\omega^{2m-1}}{(-1)^m}-\sum\limits_{k=1}^{m-1}A_{2,k}\lambda_{1,k}^2,& \beta=1,\\ -\sum\limits_{k=1}^{m-1}A_{2,k}\lambda_{1,k}^{\beta+1},&\beta\geq 2. \end{array} \right. $$ Thus, using (\ref{eq.(3.21)}), we rewrite the discrete function $D_m(h\beta)$ in the following form \begin{equation}\label{eq.(3.22)} D_m(h\beta)=\frac{1}{p_{2m-2}^{(2m-2)}}\left\{ \begin{array}{ll} \sum\limits_{k=1}^{m-1}A_{1,k}\lambda_{1,k}^{|\beta|-1},&|\beta|\geq 2,\\ \frac{2\omega^{2m-1}}{(-1)^m}+\sum\limits_{k=1}^{m-1}A_{1,k},& |\beta|=1,\\ \frac{2\omega^{2m-1}}{(-1)^m}C+\sum\limits_{k=1}^{m-1}\frac{A_{1,k}}{\lambda_{1,k}},& \beta=0.\\ \end{array} \right. \end{equation} Combining (\ref{eq.(3.2)}) and (\ref{eq.(3.20)}) we get \begin{equation}\label{eq.(3.23)} A_{1,k}=\frac{2\omega^{2m-1}}{(-1)^m}\ A_k. \end{equation} Finally, denoting $\lambda_k=\lambda_{1,k}$ and taking into account (\ref{eq.(3.23)}) from (\ref{eq.(3.22)}) we get the statement of the Theorem. Theorem \ref{THM3.1} is proved.\hfill $\Box$ \textit{Remark 1.} {From (\ref{eq.(3.1)}) we note that $D_m(h\beta)$ is an even function, i.e. $$D_m(h\beta)=D_m(-h\beta)$$ and since $|\lambda_k|<1$, then the function $D_m(h\beta)$ decreases exponentially as $|\beta|\to \infty$.} \textit{Remark 2.} It should be noted that as consequence Theorem \ref{THM3.1} generalizes previous results given in \cite{Hay09,Hay12} for the particular cases $m=2,3$. \section{Appendix} \setcounter{equation}{0} Here we obtain explicate formulas for the coefficients of the polynomials ${\cal P}_{2m-2}(x)$ defined by (\ref{eq.(2.13)}), for the cases $m=2,3,...,6$. Furthermore, for these cases, we show connection of the polynomials ${\cal P}_{2m-2}(x)$ with the Euler-Frobenius polynomials $E_{2m-2}(x)$ defined by (\ref{eq.(2.9)}). Using equality (\ref{eq.(2.11)}), when $m=2,3,...,6$, for the Euler-Frobenius polynomials $E_{2m-2}(x)$ we have \begin{eqnarray}\label{eq.(4.1)} E_2(x)&=&x^2+4x+1,\nonumber\\ E_4(x)&=&x^4+26x^3+66x^2+26x+1,\nonumber\\ E_6(x)&=&x^6+120x^5+1191x^4+2416x^3+1191x^2+120x+1, \nonumber\\ E_8(x)&=&x^8+502x^7+14608x^6+88234x^5+156190x^4+88234x^3\\ &&+14608x^2+502x+1, \nonumber\\ E_{10}(x)&=&x^{10}+2036x^9+152637x^8+2203488x^7+9738114x^6\nonumber\\ &&+15724248x^5+9738114x^4+2203488x^3+152637x^2+2036x+1. \nonumber \end{eqnarray} Now, by direct computation from (\ref{eq.(2.13)}) when $m=2,3,...,6$, for the coefficients $p_s^{(2m-2)}$, $s=0,1,...,2m-2$, of the polynomials ${\cal P}_{2m-2}(x)$ we consequently get the following results. 1) For $m=2$: ${\cal P}_2(x)=\sum\limits_{s=0}^2p_s^{(2)}$ and \begin{eqnarray}\label{eq.(4.2)} p_2^{(2)}&=&p_0^{(2)}=\sin(h\omega)-h\omega\cos(h\omega),\nonumber\\ p_1^{(2)}&=&2h\omega-\sin(2h\omega),\nonumber\\ \frac{p_2^{(2)}}{p_2^{(2)}}&=&\frac{p_0^{(2)}}{p_2^{(2)}}=1,\\ \frac{p_1^{(2)}}{p_2^{(2)}}&=&4-\frac25(h\omega)^2+O(h^4).\nonumber \end{eqnarray} 2) For $m=3$: ${\cal P}_4(x)=\sum\limits_{s=0}^4p_s^{(4)}$ and \begin{eqnarray}\label{eq.(4.3)} p_4^{(4)}&=&p_0^{(4)}=3\sin(h\omega)-h\omega\cos(h\omega)-2h\omega,\nonumber\\ p_3^{(4)}&=&p_1^{(4)}=10h\omega\cos(h\omega)+2h\omega-3\sin(2h\omega)-6\sin(h\omega),\nonumber\\ p_2^{(4)}&=&-8h\omega(1+\cos^2(h\omega))+6\sin(h\omega)-2h\omega\cos(h\omega)+6\sin(2h\omega),\nonumber\\ \frac{p_4^{(4)}}{p_4^{(4)}}&=&\frac{p_0^{(4)}}{p_4^{(4)}}=1,\\ \frac{p_3^{(4)}}{p_4^{(4)}}&=&\frac{p_1^{(4)}}{p_4^{(4)}}=26-\frac{18}{7}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_2^{(4)}}{p_4^{(4)}}&=&66-\frac{64}{7}(h\omega)^2+O(h^4).\nonumber \end{eqnarray} 3) For $m=4$: ${\cal P}_6(x)=\sum\limits_{s=0}^6p_s^{(6)}$ and \begin{eqnarray}\label{eq.(4.4)} p_6^{(6)}&=&p_0^{(6)}=5\sin(h\omega)-h\omega\cos(h\omega)-4h\omega+\frac13(h\omega)^3,\nonumber\\ p_5^{(6)}&=&p_1^{(6)}=10h\omega-5\sin(2h\omega)-20\sin(h\omega)+4h\omega\cos(h\omega)+\frac43(h\omega)^3\nonumber\\ &&\ \ \ \ \ \ \ -8\cos(h\omega)\left(-2h\omega+\frac16(h\omega)^3\right),\nonumber\\ p_4^{(6)}&=&p_2^{(6)}=35\sin(h\omega)-7h\omega\cos(h\omega)-12h\omega+20\sin(2h\omega)+\frac13(h\omega)^3\nonumber\\ &&\ \ \ \ \ \ \ -8\cos(h\omega)\left(4h\omega+\frac23(h\omega)^3\right)+4(1+2\cos^2(h\omega))\left(-2h\omega+\frac16(h\omega)^3\right),\nonumber\\ p_3^{(6)}&=&-16\cos(h\omega)\left(-2h\omega+\frac16(h\omega)^3\right)+4(1+2\cos^2(h\omega))\left(4h\omega+\frac23(h\omega)^3\right)\nonumber\\ &&-40\sin(h\omega)+8h\omega\cos(h\omega)+12h\omega-30\sin(2h\omega),\nonumber\\ \frac{p_6^{(6)}}{p_6^{(6)}}&=&\frac{p_0^{(6)}}{p_6^{(6)}}=1,\\ \frac{p_5^{(6)}}{p_6^{(6)}}&=&\frac{p_1^{(6)}}{p_6^{(6)}}=120-\frac{26}{3}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_4^{(6)}}{p_6^{(6)}}&=&\frac{p_2^{(6)}}{p_6^{(6)}}=1191-\frac{470}{3}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_3^{(6)}}{p_6^{(6)}}&=&2416-\frac{1108}{3}(h\omega)^2+O(h^4).\nonumber \end{eqnarray} 4) For $m=5$: ${\cal P}_8(x)=\sum\limits_{s=0}^8p_s^{(8)}$ and \begin{eqnarray}\label{eq.(4.5)} p_8^{(8)}&=&p_0^{(8)}=7\sin(h\omega)-h\omega\cos(h\omega)-6h\omega+\frac{2}{3}(h\omega)^3-\frac{1}{60}(h\omega)^5,\nonumber\\ p_7^{(8)}&=&p_1^{(8)}=-\frac{13}{30}(h\omega)^5+26h\omega+\frac{4}{3}(hw)^3-8\cos(h\omega)\left(-3h\omega+\frac{1}{3}(h\omega)^3-\frac{1}{120}(h\omega)^5\right)\nonumber\\ &&\ \ \ \ \ \ \ -7\sin(2h\omega)-42\sin(h\omega)+6h\omega\cos(h\omega),\nonumber\\ p_6^{(8)}&=&p_2^{(8)}=-\frac{11}{10}(h\omega)^5-4(h\omega)^3-48h\omega-8\cos(h\omega)\left(-\frac{13}{60}(h\omega)^5+12h\omega+\frac{2}{3}(h\omega)^3\right)\nonumber\\ & &\ \ \ \ \ \ \ +4(1+2\cos^2(h\omega))\left(-3h\omega+\frac{1}{3}(h\omega)^3-\frac{1}{120}(h\omega)^5\right)+112\sin(h\omega)\nonumber\\ & &\ \ \ \ \ \ \ -16h\omega\cos(h\omega)+42\sin(2h\omega),\nonumber\\ p_5^{(8)}&=&p_3^{(8)}=-\frac{13}{30}(h\omega)^5+54h\omega+\frac{4}{3}(h\omega)^3+8\cos(h\omega)\left(\frac{67}{120}(h\omega)^5+\frac{5}{3}(h\omega)^3+21h\omega\right)\nonumber\\ &&\ \ \ \ \ \ \ +4(1+2\cos^2(h\omega))\left(-\frac{13}{60}(h\omega)^5+12h\omega+\frac{2}{3}(h\omega)^3\right)-182\sin(h\omega)\nonumber\\ &&\ \ \ \ \ \ \ +26h\omega\cos(h\omega)-105\sin(2h\omega),\nonumber\\ p_4^{(8)}&=&210\sin(h\omega)-30h\omega\cos(h\omega)-52h\omega+140\sin(2h\omega)+\frac{4}{3}(h\omega)^3-\frac{1}{30}(h\omega)^5\nonumber\\ &&-16\cos(h\omega) \left(-\frac{13}{60}(h\omega)^5+12h\omega+\frac{2}{3}(h\omega)^3\right)\nonumber\\ &&+4(1+2\cos^2(h\omega))\left(-\frac{11}{20}(h\omega)^5-2(h\omega)^3-18h\omega\right),\nonumber\\ \frac{p_8^{(8)}}{p_8^{(8)}}&=&\frac{p_0^{(8)}}{p_8^{(8)}}=1,\\ \frac{p_7^{(8)}}{p_8^{(8)}}&=&\frac{p_1^{(8)}}{p_8^{(8)}}=502-\frac{1426}{55}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_6^{(8)}}{p_8^{(8)}}&=&\frac{p_2^{(8)}}{p_8^{(8)}}=14608-\frac{86664}{55}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_5^{(8)}}{p_8^{(8)}}&=&\frac{p_3^{(8)}}{p_8^{(8)}}=88234-\frac{141798}{11}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_4^{(8)}}{p_8^{(8)}}&=&156190-\frac{273872}{11}(h\omega)^2+O(h^4).\nonumber \end{eqnarray} 5) For $m=6$: ${\cal P}_{10}(x)=\sum\limits_{s=0}^{10}p_s^{(10)}$ and \begin{eqnarray}\label{eq.(4.6)} p_{10}^{(10)}&=&p_0^{(10)}= 9\sin(h\omega)-h\omega\cos(h\omega)-8h\omega+\frac{1}{2520}(h\omega)^7+(hw)^3-\frac{1}{30}(h\omega)^5,\nonumber\\ p_9^{(10)}&=&p_1^{(10)}=50h\omega-9\sin(2h\omega)-72\sin(h\omega)+8h\omega\cos(h\omega)-\frac{4}{5}(h\omega)^5+\frac{1}{21}(h\omega)^7\nonumber\\ &&\ \ \ \ \ \ \ \ -8\cos(h\omega)\left(-4h\omega+\frac{1}{5040}(h\omega)^7+\frac{1}{2}(h\omega)^3-\frac{1}{60}(h\omega)^5\right),\nonumber\\ p_8^{(10)}&=&p_2^{(10)}=261\sin(h\omega)-29h\omega\cos(h\omega)-136h\omega+72\sin(2h\omega)+\frac{397}{840}(h\omega)^7-\frac{1}{2}(h\omega)^5\nonumber\\ &&\ \ \ \ \ \ \ \ -9(h\omega)^3-8\cos(h\omega)\left(24h\omega-\frac{2}{5}(h\omega)^5+\frac{1}{42}(h\omega)^7\right)\nonumber\\ &&\ \ \ \ \ \ \ \ +4(1+2\cos^2(h\omega))\left(-4h\omega+\frac{1}{5040}(h\omega)^7+\frac{1}{2}(h\omega)^3-\frac{1}{60}(h\omega)^5\right), \nonumber\\ p_7^{(10)}&=&p_3^{(10)}=8\cos(h\omega)\left(64h\omega-\frac{149}{630}(h\omega)^7-\frac{4}{15}(h\omega)^5-4(h\omega)^3\right)+216h\omega+\frac{8}{3}(h\omega)^5\nonumber\\ &&\ \ \ \ \ \ \ \ +16(h\omega)^3+\frac{302}{315}(h\omega)^7+4(1+2\cos^2(h\omega))\left(24h\omega-\frac{2}{5}(h\omega)^5+\frac{1}{42}(h\omega)^7\right)\nonumber\\ &&\ \ \ \ \ \ \ \ -576\sin(h\omega)+64h\omega\cos(h\omega)-252\sin(2h\omega),\nonumber\\ p_6^{(10)}&=&p_4^{(10)}=882\sin(h\omega)-98h\omega\cos(h\omega)-240h\omega+504\sin(2h\omega)+\frac{149}{315}(h\omega)^7-\frac{8}{15}(h\omega)^5\nonumber\\ &&\ \ \ \ \ \ \ \ -8(h\omega)^3-8\cos(h\omega)\left(104h\omega+\frac{14}{15}(h\omega)^5+8(h\omega)^3+\frac{151}{315}(h\omega)^7\right)\nonumber\\ &&\ \ \ \ \ \ \ \ +4(1+2\cos^2(h\omega))\left(-60h\omega+\frac{397}{1680}(h\omega)^7-\frac{1}{4}(h\omega)^5-\frac{9}{2}(h\omega)^3\right),\nonumber\\ p_5^{(10)}&=&-1008\sin(h\omega)+112h\omega\cos(h\omega)+236h\omega-630\sin(2h\omega)-\frac{8}{5}(h\omega)^5+\frac{2}{21}(h\omega)^7\nonumber \\ &&\ \ \ \ \ \ \ \ -16\cos(h\omega)\left(-60h\omega+\frac{397}{1680}(h\omega)^7-\frac{1}{4}(h\omega)^5-\frac{9}{2}(h\omega)^3\right)\nonumber\\ &&\ \ \ \ \ \ \ \ +4(1+2\cos^2(h\omega))\left(80h\omega+\frac{4}{3}(h\omega)^5+8(h\omega)^3+\frac{151}{315}(h\omega)^7\right),\nonumber\\ \frac{p_{10}^{(10)}}{p_{10}^{(10)}}&=&\frac{p_0^{(10)}}{p_{10}^{(10)}}=1,\\ \frac{p_9^{(10)}}{p_{10}^{(10)}}&=&\frac{p_1^{(10)}}{p_{10}^{(10)}}=2036-\frac{998}{13}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_8^{(10)}}{p_{10}^{(10)}}&=&\frac{p_2^{(10)}}{p_{10}^{(10)}}=152637-\frac{170920}{13}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_7^{(10)}}{p_{10}^{(10)}}&=&\frac{p_3^{(10)}}{p_{10}^{(10)}}=2203488-\frac{3644480}{13}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_6^{(10)}}{p_{10}^{(10)}}&=&\frac{p_4^{(10)}}{p_{10}^{(10)}}=9738114-\frac{19460312}{13}(h\omega)^2+O(h^4),\nonumber\\ \frac{p_5^{(10)}}{p_{10}^{(10)}}&=&15724248-\frac{33280180}{13}(h\omega)^2+O(h^4).\nonumber \end{eqnarray} From (\ref{eq.(4.1)})-(\ref{eq.(4.6)}) we state the following. \begin{conjecture}\label{L2.2} For the coefficients $p_s^{(2m-2)}$, $s=0,1,...,2m-2$ of the polynomial ${\cal P}_{2m-2}(x)$ the following holds $$ \frac{p_s^{(2m-2)}}{p_{2m-2}^{(2m-2)}}=a_s^{(2m-2)}+O(h^2), \ \ \ s=0,1,...,2m-2, $$ where $a_s^{(2m-2)}$ are the coefficients of the Euler-Frobenius polynomial $E_{2m-2}(x)$ of degree $2m-2$. \end{conjecture} \section*{Acknowledgements} The author thanks professor A.Cabada for discussion of the results. The present work was done in the University of Santiago de Compostela, Spain. A.R. Hayotov thanks the program Erasmus Mundus Action 2, Lot 10, Marco XXI for financial support (project number: Lot 10 - 20112572).

BkiUbJ7xK6-gDz87OAC1

\section{Introduction} We study local uniqueness properties of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation in the pair of sets consisting of a perforated domain and a small inclusion. We begin by presenting the geometric framework of our problem. We fix once for all a natural number \[ n\ge 3 \] that will be the dimension of the space $\mathbb{R}^n$ we are going to work in and a parameter \[ \alpha \in ]0,1[ \] which we use to define the regularity of our sets and functions. We remark that the case of dimension $n=2$ requires specific techniques and it is not treated in this paper (the analysis for $n=3$ and for $n\ge 3$ is instead very similar). Then, we introduce two sets $\Omega^o$ and $\Omega^i$ that satisfy the following conditions: \[ \begin{split} &\mbox{$\Omega^o$, $\Omega^i$ are bounded open connected subsets of $\mathbb{R}^n$ of class $C^{1,\alpha}$,} \\ &\mbox{their exteriors $\mathbb{R}^n\setminus \overline{\Omega^o}$ and $\mathbb{R}^n\setminus \overline{\Omega^i}$ are connected,} \\ &\mbox{and the origin $0$ of $\mathbb{R}^n$ belongs both to $\Omega^o$ and to $\Omega^i$.} \end{split} \] Here the superscript ``$o$" stands for ``outer domain" whereas the superscript ``$i$" stands for ``inner domain". We take \[ \epsilon_0 \equiv \mbox{sup}\{\theta \in ]0, +\infty[: \epsilon \overline{\Omega^i} \subseteq \Omega^o, \ \forall \epsilon \in ]- \theta, \theta[ \}, \] and we define the perforated domain $\Omega(\epsilon)$ by setting \[ \Omega(\epsilon) \equiv \Omega^o \setminus \epsilon \overline{\Omega^i} \] for all $\epsilon\in]-\epsilon_0,\epsilon_0[$. Then we fix three functions \begin{equation}\label{FGf^o} \begin{split} &F \colon ]-\epsilon_0,\epsilon_0[ \times \partial \Omega ^i \times \mathbb{R} \to \mathbb{R}\,,\\ &G \colon ]-\epsilon_0,\epsilon_0[ \times \partial \Omega ^i \times \mathbb{R} \to \mathbb{R}\,,\\ &f^o \in C^{1,\alpha}(\partial\Omega^o) \end{split} \end{equation} and, for $\epsilon \in ]0,\epsilon_0[$, we consider the following nonlinear nonautonomous transmission problem in the perforated domain $\Omega(\epsilon)$ for a pair of functions $(u^o,u^i)\in C^{1,\alpha}(\overline{\Omega(\epsilon)})\times C^{1,\alpha}(\overline{\epsilon\Omega^i})$: \begin{equation}\label{princeq} \begin{cases} \Delta u^o = 0 & \mbox{in } \Omega(\epsilon), \\ \Delta u^i = 0 & \mbox{in } \epsilon \Omega^i, \\ u^o(x)=f^o(x) & \forall x \in \partial \Omega^o, \\ u^o(x) = F\left(\epsilon,\frac{x}{\epsilon},u^i(x)\right) & \forall x \in \epsilon \partial \Omega^i, \\ \nu_{\epsilon \Omega^i} \cdot \nabla u^o (x) - \nu_{\epsilon \Omega^i} \cdot \nabla u^i (x) = G\left(\epsilon, \frac{x}{\epsilon},u^i(x)\right) & \forall x \in \epsilon \partial \Omega^i. \end{cases} \end{equation} Here $\nu_{\epsilon\Omega^i}$ denotes the outer exterior normal to $\epsilon\Omega^i$. Boundary value problems like \eqref{princeq} arise in the mathematical models for the heat conduction in (nonlinear) composite materials, as, for example, in Mishuris, Miszuris and \"Ochsner \cite{MiMiOc07,MiMiOc08}. In such cases, the functions $u^o$ and $u^i$ represent the temperature distribution in $\Omega(\epsilon)$ and in the inclusion $\epsilon \Omega^i$, respectively. The third condition in \eqref{princeq} means that we are prescribing the temperature distribution on the exterior boundary $\partial \Omega^o$. The fourth condition says that on the interface $\epsilon \partial \Omega^i$ the temperature distribution $u^o$ depends nonlinearly on the size of the inclusion, on the position on the interface, and on the temperature distribution $u^i$. The fifth conditions, instead, says that the jump of the heat flux on the interface depends nonlinearly on the size of the inclusion, on the position on the interface, and on the temperature distribution $u^i$. In literature, existence and uniqueness of solutions of nonlinear boundary value problems have been largely investigated by means of variational techniques (see, e.g., the monographs of Ne\v{c}as \cite{Ne83} and of Roub\'i\v{c}ek \cite{Ro13} and the references therein). On the other hand, boundary value problems in singularly perturbed domains are usually studied by expansion methods of asymptotic analysis, such as the methods of matching inner and outer expansions (cf., e.g., Il'in \cite{Il78, Il92}) and the multiscale expansion method (as in Maz'ya, Nazarov, and Plamenenvskii \cite{MaNaPl00}, see also Iguernane et al.~\cite{IgNaRoSoSz09} in connection with nonlinear problems). In this paper we do not use variational techniques and neither we resort to asymptotic expansion methods. Instead, we adopt the functional analytic approach proposed by Lanza de Cristoforis (cf., e.g., \cite{La02,La08}). The key strategy of such approach is the transformation of the perturbed boundary value problem into a functional equation that can be studied by means of certain theorems of functional analysis, for example by the implicit function theorem or by the Schauder fixed-point theorem. Typically, this transformation is achieved with an integral representation of the solution, a suitable rescaling of the corresponding integral equations, and an analysis based on results of potential theory. This approach has proven to be effective when dealing with nonlinear conditions on the boundary of small holes or inclusions. For example, it has been used in the papers \cite{La07} and \cite{La10} of Lanza de Cristoforis to study a nonlinear Robin and a nonlinear transmission problem, respectively, in the paper \cite{DaLa10} with Lanza de Cristoforis to analyze a nonlinear traction problem for an elastic body, in \cite{DaMi15} with Mishuris to prove the existence of solution for \eqref{princeq} in the case of a ``big" inclusion (that is, for $\epsilon>0$ fixed), and in \cite{Mo18} to show the existence of solution of \eqref{princeq} in the case of ``small" inclusion (that is, for $\epsilon>0$ that tends to $0$). In particular, in \cite{Mo18} we have proven that, under suitable assumptions on the functions $F$ and $G$, there exists a family of functions $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon\in]0,\epsilon'[}$, with $\epsilon'\in]0,\epsilon_0[$, such that each pair $(u^o_\epsilon,u^i_\epsilon)$ is a solution of \eqref{princeq} for the corresponding value of $\epsilon$. Moreover, the dependence of the functions $u^o_\epsilon$ and $u^i_\epsilon$ upon the parameter $\epsilon$ can be described in terms of real analytic maps of $\epsilon$. The aim of this paper is to show that each of such solutions $(u^o_\epsilon,u^i_\epsilon)$ is locally unique. In other words, we shall verify that, for $\epsilon>0$ smaller than a certain $\epsilon^\ast \in ]0,\epsilon_0[$, any solution $(v^o,v^i)$ of problem \eqref{princeq} that is ``close enough'' to the pair $(u^o_\epsilon,u^i_\epsilon)$ has to coincide with $(u^o_\epsilon,u^i_\epsilon)$. We will see that the ``distance'' from the solution $(u^o_\epsilon,u^i_\epsilon)$ can be measured solely in terms of the $C^{1,\alpha}$ norm of the trace of the rescaled function $v^i(\epsilon\cdot)$ on $\partial\Omega^i$. More precisely, we will prove that there is $\delta^*>0$ such that, if $\epsilon\in]0,\epsilon^*]$, $(v^o ,v^i )$ is a solution of \eqref{princeq}, and \begin{equation}\label{cond} \left\| v^i(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot ) \right\| _{C^{1,\alpha}(\partial \Omega^i)} < \epsilon\delta^\ast, \end{equation} then \[ (v^o,v^i)=(u^o_\epsilon,u^i_\epsilon) \] (cf.~Theorem \ref{Thmunisol2} below). We note that in \cite{DaMi15} it has been shown that for a ``big'' inclusion (that is, with $\epsilon>0$ fixed) problem \eqref{princeq} may have solutions that are not locally unique. Such a different result reflects the fact that the solutions of \cite{DaMi15} are obtained by the Schauder fixed-point theorem, whereas the solutions of the present paper are obtained by means of the implicit function theorem. We will not provide in this paper an estimate for the values of $\epsilon^*$ and $\delta^*$. In principle, they could be obtained studying the norm of certain integral operators that we employ in our proofs. We observe that, in specific applications, for example to the heat conduction in composite materials, it might be important to understand if $\epsilon^*$ and $\delta^*$ are big enough for the model that one adopts. In particular, for very small $\epsilon^*$, that correspond to very small inclusions, and very small $\delta^*$, that corresponds to very small differences of temperature, one may have to deal with different physical models. We also observe that uniqueness results are not new in the applications of the functional analytic approach to nonlinear boundary value problems (see, e.g., the above mentioned papers \cite{DaLa10,La07,La10}). However, the results so far presented concern the uniqueness of the entire family of solutions rather than the uniqueness of a single solution for $\epsilon>0$ fixed. For our specific problem \eqref{princeq}, a uniqueness result for the family $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon\in]0,\epsilon'[}$ would consist in proving that if $\{(v^o_\epsilon,v^i_\epsilon)\}_{\epsilon\in]0,\epsilon'[}$ is another family of solutions which satisfies a certain limiting condition, for example that \[ \lim_{\epsilon\to 0}\epsilon^{-1}\left\| v^i_\epsilon(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot ) \right\| _{C^{1,\alpha}(\partial \Omega^i)}=0, \] then \[ (v^o_\epsilon,v^i_\epsilon)=(u^o_\epsilon,u^i_\epsilon) \] for $\epsilon$ small enough. One can verify that the local uniqueness of a single solution under condition \eqref{cond} implies the uniqueness of the family of solutions $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon\in]0,\epsilon_0[}$ in the sense described here above (cf. Corollary \ref{cor} below). From this point of view, we can say that the uniqueness result presented in this paper strengthen the uniqueness result for families which is typically obtained in the application of the functional analytic approach. The paper is organized as follows. In Section \ref{notation} we define some of the symbols used later on. Section \ref{classresult} is a section of preliminaries, where we introduce some classical results of potential theory that we need. In Section \ref{existenceresult} we recall some results of \cite{Mo18} concerning the existence of a family $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon\in]0,\epsilon'[}$ of solutions of problem \eqref{princeq}. In Section \ref{localresult} we state and prove our main Theorem \ref{Thmunisol2}. The section consists of three subsections. In the first one we prove Theorem \ref{Thmunisol1}, which is a weaker version of Theorem \ref{Thmunisol2} and follows directly from the Implicit Function Theorem argument used to obtain the family $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon\in]0,\epsilon'[}$. The statement of Theorem \ref{Thmunisol1} is similar to that of Theorem \ref{Thmunisol2}, but the assumptions are much stronger. In particular, together with condition \eqref{cond}, we have to require other two conditions, namely that \begin{equation}\label{cond2} \begin{split} \left\| v^o - u^o_\epsilon \right\| _{C^{1,\alpha}(\partial \Omega^o)} < \epsilon\delta^\ast\quad\text{and}\quad \left\| v^o(\epsilon \cdot ) - u^o_\epsilon(\epsilon \cdot ) \right\| _{C^{1,\alpha}(\partial \Omega^i)} < \epsilon\delta^\ast, \end{split} \end{equation} in order to prove that $(v^o,v^i)=(u^o_\epsilon,u^i_\epsilon)$. In our main Theorem \ref{Thmunisol2} we will see that the two conditions in \eqref{cond2} can be dropped and \eqref{cond} is sufficient. The proof of Theorem \ref{Thmunisol2} is presented in Subsection \ref{second} where we first show some results on real analytic composition operators in Schauder spaces, then we turn to introduce certain auxiliary maps $N$ and $S$, and finally we will be ready to state and prove our main theorem. In the last Subsection \ref{family}, we see that the uniqueness of the family $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon\in]0,\epsilon'[}$ in the sense described here above can be obtained as a corollary of Theorem \ref{Thmunisol2}. At the end of the paper we have included an Appendix where we present some (classical) results on the product and composition in Schauder spaces. \section{Notation}\label{notation} We denote the norm of a real normed space $X$ by $\| \cdot \| _X$. We denote by $I_X$ the identity operator from $X$ to itself and we omit the subscript $X$ where no ambiguity can occur. For $x \in X$ and $R>0$, we denote by $B_X(x,R) \equiv \{y\in X : \|y-x\|_X < R \}$ the ball of radius $R$ centered at the point $x$. If $X=\mathbb{R}^d$, $d \in \mathbb{N} \setminus \{0,1\} $, we simply write $B_d(x,R)$. If $X$ and $Y$ are normed spaces we endow the product space $X \times Y$ with the norm defined by $\| (x,y) \|_{X \times Y} = \|x\|_X + \|y\|_Y $ for all $(x,y) \in X \times Y$, while we use the Euclidean norm for $\mathbb{R}^d$, $d\in\mathbb{N}\setminus\{0,1\}$. For $x \in \mathbb{R}^d$, $x_j$ denotes the $j$-th coordinate of $x$, $|x|$ denotes the Euclidean modulus of $x$ in $\mathbb{R}^d$. We denote by $\mathcal{L}(X,Y)$ the Banach space of linear and continuous map of $X$ to $Y$, equipped with its usual norm of the uniform convergence on the unit sphere of $X$. If $U$ is an open subset of $X$, and $F:U \to Y$ is a Fr\'echet-differentiable map in $U$, we denote the differential of $F$ by $dF$. The inverse function of an invertible function $f$ is denoted by $f^{(-1)}$, while the reciprocal of a function $g$ is denoted by $g^{-1}$. Let $\Omega \subseteq \mathbb{R}^n$. Then $\overline{\Omega}$ denotes the closure of $\Omega$ in $\mathbb{R}^n$, $\partial \Omega$ denotes the boundary of $\Omega$, and $\nu_\Omega$ denotes the outward unit normal to $\partial \Omega$. Let $\Omega$ be an open subset of $\mathbb{R}^n$ and $m \in \mathbb{N} \setminus \{0\}$. The space of $m$ times continuously differentiable real-valued function on $\Omega$ is denoted by $C^m(\Omega)$. Let $r \in \mathbb{N} \setminus \{0\}$, $f \in (C^m(\Omega))^r$. The $s$-th component of $f$ is denoted by $f_s$ and the gradient matrix of $f$ is denoted by $\nabla f$. Let $\eta=(\eta_1, \dots ,\eta_n) \in \mathbb{N}^n$ and $|\eta|=\eta_1+ \dots+\eta_n$. Then $D^\eta f \equiv \frac{\partial^{|\eta|}f}{\partial x^{\eta_1}_1, \dots , \partial x^{\eta_n}_n}$. If $r=1$, the Hessian matrix of the second-order partial derivatives of $f$ is denoted by $D^2 f$. The subspace of $C^m(\Omega)$ of those functions $f$ such that $f$ and its derivatives $D^\eta f$ of order $|\eta|\le m$ can be extended with continuity to $\overline{\Omega}$ is denoted $C^m(\overline{\Omega})$. The subspace of $C^m(\overline{\Omega})$ whose functions have $m$-the order derivatives that are H\"{o}lder continuous with exponent $\alpha \in ]0,1[$ is denoted $C^{m,\alpha}(\overline{\Omega})$. If $f \in C^{0,\alpha}(\overline{\Omega})$, then its H\"{o}lder constant is defined as $|f : \Omega|_\alpha\equiv \mbox{sup} \left\{\frac{|f(x)-f(y)|}{|x-y|^\alpha} : x,y \in \overline{\Omega}, x \neq y \right\}$. If $\Omega$ is open and bounded, then the space $C^{m,\alpha}(\overline{\Omega})$, equipped with its usual norm $\|f\|_{C^{m,\alpha}(\overline{\Omega})} \equiv \|f\|_{C^{m}(\overline{\Omega})} + \sum_{|\eta|=m}{|D^\eta f : \Omega|_\alpha}$, is a Banach space. We denote by $C^{m,\alpha}_{\mathrm{loc}}(\mathbb{R}^n \setminus \Omega)$ the space of functions on $\mathbb{R}^n \setminus \Omega$ whose restriction to $\overline{U}$ belongs to $C^{m,\alpha}(\overline{U})$ for all open bounded subsets $U$ of $\mathbb{R}^n \setminus \Omega$. On $C^{m,\alpha}_{\mathrm{loc}}(\mathbb{R}^n \setminus \Omega)$ we consider the natural structure of Fr\'echet space. Finally we set \begin{equation*} C^{m,\alpha}_{\mathrm{harm}}(\overline{\Omega}) \equiv \{ u \in C^{m,\alpha}(\overline{\Omega}) \cap C^2(\Omega): \Delta u = 0 \text{ in } \Omega \}. \end{equation*} We say that a bounded open subset of $\mathbb{R}^n$ is of class $C^{m,\alpha}$ if it is a manifold with boundary imbedded in $\mathbb{R}^n$ of class $C^{m,\alpha}$. In particular, if $\Omega$ is a $C^{1,\alpha}$ subset of $\mathbb{R}^n$, then $\partial\Omega$ is a $C^{1,\alpha}$ sub-manifold of $\mathbb{R}^n$ of co-dimension $1$. If $M$ is a $C^{m,\alpha}$ sub-manifold of $\mathbb{R}^n$ of dimension $d\ge 1$, we define the space $C^{m,\alpha}(M)$ by exploiting a finite local parametrization. Namely, we take a finite open covering $\mathcal{U}_1, \dots, \mathcal{U}_k$ of $M$ and $C^{m,\alpha}$ local parametrization maps $\gamma_l : \overline{B_{d}(0,1)} \to \overline{\mathcal{U}_l}$ with $l=1,\dots, k$ and we say that $\phi\in C^{m,\alpha}(M)$ if and only if $\phi\circ\gamma_l\in C^{m,\alpha}(\overline{B_{d}(0,1)})$ for all $l=1,\dots, k$. Then for all $\phi\in C^{m,\alpha}(M)$ we define \[ \|\phi\|_{C^{m,\alpha}(M)} \equiv \sum_{l=1}^k \|\phi\circ\gamma_l \|_{C^{m,\alpha}(\overline{B_{d}(0,1)})}\,. \] One verifies that different $C^{m,\alpha}$ finite atlases define the same space $C^{m,\alpha}(M)$ and equivalent norms on it. We retain the standard notion for the Lebesgue spaces $L^p$, $p\ge 1$. If $\Omega$ is of class $C^{1,\alpha}$, we denote by $d\sigma$ the area element on $\partial\Omega$. If $Z$ is a subspace of $L^1(\partial \Omega)$, then we set \begin{equation}\label{Z0} Z_0 \equiv \left\{ f \in Z : \int_{\partial\Omega} f \,d\sigma = 0 \right\}. \end{equation} \section{Classical results of potential theory}\label{classresult} In this section we present some classical results of potential theory. For the proofs we refer to Folland \cite{Fo95}, Gilbarg and Trudinger \cite{GiTr83}, Schauder \cite{Sc31}, and to the references therein. \begin{defin} We denote by $S_n$ the function from $\mathbb{R}^n \setminus \{0\}$ to $\mathbb{R}$ defined by \[ S_n(x) \equiv \frac{|x|^{2-n}}{(2-n) s_n} \qquad \forall x \in \mathbb{R}^n \setminus \{0\}, \] where $s_n$ denotes the $(n-1)$-dimensional measure of $\partial B_n(0,1)$. \end{defin} $S_n$ is well known to be a fundamental solution of the Laplace operator. Now let $\Omega$ be an open bounded connected subset of $\mathbb{R}^n$ of class $C^{1,\alpha}$. \begin{defin} We denote by $w_{\Omega}[\mu]$ the double layer potential with density $\mu$ given by \begin{equation*} w_{\Omega}[\mu](x) \equiv - \int_{\partial \Omega}{\nu_\Omega(y) \cdot \nabla S_n(x-y) \mu(y) \,d\sigma_y} \qquad \forall x \in \mathbb{R}^n \end{equation*} for all $\mu \in C^{1,\alpha}(\partial\Omega)$. \\ We denote by $W_{\partial\Omega}$ the boundary integral operator which takes $\mu \in C^{1,\alpha}(\partial\Omega)$ to the function $W_{\partial\Omega}[\mu]$ defined by \[ W_{\partial\Omega}[\mu](x) \equiv - \int_{\partial \Omega}{\nu_\Omega(y) \cdot \nabla S_n(x-y) \mu(y) \,d\sigma_y} \qquad \forall x \in \partial \Omega. \] \end{defin} It is well known that, if $\mu \in C^{1,\alpha}(\partial\Omega)$, then $w_{\Omega}[\mu]_{| \Omega}$ admits a unique continuous extension to $\overline{\Omega}$, which we denote by $w^+_{\Omega}[\mu]$, and $w_{\Omega}[\mu]_{| \mathbb{R} \setminus \overline{\Omega}}$ admits a unique continuous extension to $\mathbb{R} \setminus \Omega$, which we denote by $w^-_{\Omega}[\mu]$. In the following Theorem \ref{doublepot} we summarize classical results in potential theory. \begin{teo}\label{doublepot} The following statements holds. \begin{enumerate} \item[(i)] Let $\mu \in C^{1,\alpha}(\partial\Omega)$. Then the function $w_{\Omega}[\mu]$ is harmonic in $\mathbb{R}^n\setminus \partial\Omega$ and at infinity. Moreover, we have the following jump relations \begin{equation*} w^\pm_{\Omega}[\mu](x) = \left( \pm \frac{1}{2} I + W_{\partial\Omega} \right)[\mu](x) \qquad \forall x \in \partial \Omega; \end{equation*} \item[(ii)] The map from $C^{1,\alpha}(\partial\Omega)$ to $C^{1,\alpha}(\overline{\Omega})$ which takes $\mu$ to $w^+_{\Omega}[\mu]$ is linear and continuous and the map from $C^{1,\alpha}(\partial\Omega)$ to $C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}\setminus\Omega)$ which takes $\mu$ to $w^-_{\Omega}[\mu]$ is linear and continuous; \item[(iii)] The map which takes $\mu$ to $W_{\partial\Omega}[\mu]$ is continuous from $C^{1,\alpha}(\partial\Omega)$ to itself; \item[(iv)] If $\mathbb{R}\setminus\overline{\Omega}$ is connected, then the operator $\frac{1}{2} I + W_{\partial\Omega}$ is an isomorphism from $C^{1,\alpha}(\partial\Omega)$ to itself. \end{enumerate} \end{teo} \section{An existence result for the solutions of problem $(\ref{princeq})$}\label{existenceresult} In this section we recall some results of \cite{Mo18} on the existence of a family of solutions for problem \eqref{princeq}. In what follows $\mathfrak{u}^o$ denotes the unique solution in $C^{1,\alpha}(\overline{\Omega^o})$ of the interior Dirichlet problem in $\Omega^o$ with boundary datum $f^o$, namely \[ \begin{cases} \Delta \mathfrak{u}^o=0&\text{in }\Omega^o\,,\\ \mathfrak{u}^o=f^o&\text{on }\partial\Omega^o\,. \end{cases} \] Then we have the following Proposition \ref{rappresentsol}, where harmonic functions in $\overline{\Omega(\epsilon)}$ and $\overline{\epsilon \Omega^i}$ are represented in terms of $\mathfrak{u}^o$, double layer potentials with appropriate densities, and a suitable restriction of the fundamental solution $S_n$ (cf. \cite[Prop. 5.1]{Mo18}). \begin{prop}\label{rappresentsol} Let $\epsilon \in ]0,\epsilon_0[$. The map \[ (U^o_\epsilon[\cdot,\cdot,\cdot,\cdot], U^i_\epsilon[\cdot,\cdot,\cdot,\cdot]) \] from $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}_{\mathrm{harm}}(\overline{\Omega(\epsilon)}) \times C^{1,\alpha}_{\mathrm{harm}}(\epsilon \overline{\Omega^i})$ which takes $(\phi^o,\phi^i,\zeta,\psi^i)$ to the pair of functions \begin{equation*} (U^o_\epsilon[\phi^o,\phi^i,\zeta,\psi^i], U^i_\epsilon[\phi^o,\phi^i,\zeta,\psi^i]) \end{equation*} defined by \begin{equation}\label{rappsol} \begin{aligned} & U^o_\epsilon[\phi^o,\phi^i,\zeta,\psi^i](x)\\ &\ \equiv \mathfrak{u}^o(x) + \epsilon w^+_{\Omega^o}[\phi^o](x) + \epsilon w^-_{\epsilon\Omega^i}\left[\phi^i\left(\frac{\cdot}{\epsilon}\right)\right](x) + \epsilon^{n-1}\zeta\, S_n(x) && \qquad \forall x \in \overline{\Omega(\epsilon)}, \\ & U^i_\epsilon[\phi^o,\phi^i,\zeta,\psi^i](x) \equiv \epsilon w^+_{\epsilon\Omega^i}\left[\psi^i\left(\frac{\cdot}{\epsilon}\right)\right](x) + \zeta^i && \qquad \forall x \in \epsilon \overline{\Omega^i}, \end{aligned} \end{equation} is bijective. \end{prop} We recall that $C^{1,\alpha}(\partial\Omega^i)_0$ is the subspace of $C^{1,\alpha}(\partial\Omega^i)$ consisting of the functions with zero integral mean on $\partial\Omega^i$ (cf.~\eqref{Z0}). The following Lemma \ref{isolem} provides an isomorphism between $C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ and $C^{1,\alpha}(\partial\Omega^i)$ (cf. \cite[Lemma 4.2]{Mo18}). \begin{lemma}\label{isolem} The map from $C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^i)$ which takes $(\mu,\xi)$ to the function \begin{equation*} J[\mu,\xi] \equiv \left(-\frac{1}{2}I + W_{\partial\Omega^i}\right) [\mu] +\xi\,{S_n}_{|\partial \Omega^i} \end{equation*} is an isomorphism. \end{lemma} Let $F$ be as in \eqref{FGf^o}. We indicate by $\partial_\epsilon F$ and $\partial_\zeta F$ the partial derivative of $F$ with respect to the first the last argument, respectively. We shall exploit the following assumptions: \begin{equation}\label{zetaicond} \begin{split} &\text{There exist $\zeta^i \in \mathbb{R}$ such that $F(0,\cdot,\zeta^i) = \mathfrak{u}^o(0)$}\\ &\text{and $(\partial_\zeta F)(0,\cdot,\zeta^i)$ is constant and positive.} \end{split} \end{equation} and \begin{equation}\label{F1} \begin{split} &\mbox{For each $t\in\partial\Omega^i$ fixed, the map from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $\mathbb{R}$} \\ &\mbox{which takes $(\epsilon,\zeta)$ to $F(\epsilon,t,\zeta)$ is of class $C^2$.} \end{split} \end{equation} Then we have the following Lemma \ref{-Taylem}, concerning the Taylor expansion of the functions $\mathfrak{u}^o$ and $F$ (cf. \cite[Lemmas 5.2, 5.3]{Mo18}). \begin{lemma}\label{-Taylem} Let \eqref{zetaicond} and \eqref{F1} hold true. Let $a,b \in \mathbb{R}$. Then \begin{equation*} \begin{split} &F(\epsilon,t,a+\epsilon b)\\ &\quad = F(0,t,a) + \epsilon (\partial_\epsilon F) (0,t,a) + \epsilon b (\partial_\zeta F) (0,t,a) + \epsilon^2 \tilde{F}(\epsilon,t,a,b), \end{split} \end{equation*} for all $(\epsilon,t) \in ]-\epsilon_0,\epsilon_0[ \times \partial\Omega^i$, where \begin{equation*} \begin{split} &\tilde{F}(\epsilon,t,a,b) \equiv \int_{0}^{1} (1-\tau) \{ (\partial^2_\epsilon F)(\tau\epsilon,t,a + \tau\epsilon b)\\ &\qquad + 2b(\partial_\epsilon \partial_\zeta F)(\tau\epsilon,t,a + \tau\epsilon b) + b^2(\partial^2_\zeta F)(\tau\epsilon,t,a + \tau\epsilon b) \} \,d\tau . \end{split} \end{equation*} Moreover \begin{equation*} \mathfrak{u}^o(\epsilon t)-F(0,t,\zeta^i)=\epsilon\, t\cdot\nabla \mathfrak{u}^o(0)+\epsilon^2\,\tilde{\mathfrak{u}}^o(\epsilon,t) \end{equation*} for all $(\epsilon,t) \in ]-\epsilon_0,\epsilon_0[ \times \partial\Omega^i$, where \begin{equation*} \tilde{\mathfrak{u}}^o(\epsilon,t) \equiv\int_0^1(1-\tau)\sum_{i,j=1}^nt_i\,t_j\, (\partial_{x_i}\partial_{x_j} \mathfrak{u}^o)(\tau\epsilon t)\,d\tau\,. \end{equation*} \end{lemma} Then we introduce a notation for the superposition operators. \begin{defin}\label{Nemytskii} If $H$ is a function from $]-\epsilon_0,\epsilon_0[ \times \partial \Omega ^i \times \mathbb{R}$ to $\mathbb{R}$, then we denote by $\mathcal{N}_H$ the (nonlinear nonautonomous) superposition operator which takes a pair $(\epsilon,v)$ consisting of a real number $\epsilon\in ]-\epsilon_0,\epsilon_0[$ and of a function $v$ from $\partial\Omega^i$ to $\mathbb{R}$ to the function $\mathcal{N}_H(\epsilon,v)$ defined by \[ \mathcal{N}_H(\epsilon,v) (t) \equiv H(\epsilon,t,v(t))\qquad\forall t\in\partial\Omega^i\,. \] \end{defin} Here the letter ``$\mathcal{N}$" stands for ``Nemytskii operator". Having introduced Definition \ref{Nemytskii}, we can now formulate the following assumption on the funtions $F$ and $G$ of \eqref{FGf^o}: \begin{equation}\label{realanalhp} \begin{split} &\text{For all $(\epsilon,v)\in]-\epsilon_0,\epsilon_0[\times C^{1,\alpha}(\partial\Omega^i)$ we have $\mathcal{N}_F(\epsilon,v)\inC^{1,\alpha}(\partial\Omega^i)$}\\ &\text{and $\mathcal{N}_G(\epsilon,v)\in C^{0,\alpha}(\partial\Omega^i)$. Moreover, the superposition operator $\mathcal{N}_F$ is real}\\ &\text{analytic from $ ]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^i)$ and the superposition}\\ &\text{operator $\mathcal{N}_G$ is real analytic from $ ]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{0,\alpha}(\partial\Omega^i)$.} \end{split} \end{equation} Then, for real analytic superposition operators we have the following Proposition \ref{differenzialeN_H} (cf.~ Lanza de Cristoforis \cite[Prop 5.3]{La07}). \begin{prop}\label{differenzialeN_H} If $H$ is a function from $]-\epsilon_0,\epsilon_0[ \times \partial \Omega ^i \times \mathbb{R}$ to $\mathbb{R}$ such that the superposition operator $\mathcal{N}_H$ is real analytic from $ ]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^i)$, then the partial differential of $\mathcal{N}_H$ with respect to the second variable $v$, computed at the point $(\epsilon,\overline{v}) \in ]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^i)$, is the linear operator $d_v \mathcal{N}_H (\epsilon, \overline{v})$ defined by \begin{equation}\label{d_vN_Hformula} d_v \mathcal{N}_H (\epsilon, \overline{v}). \tilde{v} = \mathcal{N}_{(\partial_\zeta H)} (\epsilon,\overline{v}) \tilde{v} \qquad \forall \tilde{v} \in C^{1,\alpha}(\partial\Omega^i). \end{equation} The same result holds replacing the domain and the target space of the operator $\mathcal{N}_H$ with $]-\epsilon_0,\epsilon_0[ \times C^{0,\alpha}(\partial\Omega^i)$ and $C^{0,\alpha}(\partial\Omega^i)$, respectively, and using functions $\overline{v},\tilde{v} \in C^{0,\alpha}(\partial\Omega^i)$ in \eqref{d_vN_Hformula}. \end{prop} In what follows we will exploit an auxiliary map $M=(M_1,M_2,M_3)$ from $]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)$ defined by \begin{align*} &M_1[\epsilon, \phi^o,\phi^i, \zeta, \psi^i](x) \equiv \left(\frac{1}{2}I + W_{\partial\Omega^o}\right)[\phi^o](x) \nonumber \\ &\quad - \epsilon^{n-1} \int_{\partial\Omega^i}{\nu_{\Omega^i}(y) \cdot \nabla S_n(x-\epsilon y) \phi^i(y) \,d\sigma_y} + \epsilon^{n-2} S_n(x) \zeta\qquad\quad \forall x \in \partial\Omega^o\,, \\ &M_2[\epsilon, \phi^o, \phi^i, \zeta, \psi^i](t) \equiv \, t \cdot \nabla \mathfrak{u}^o(0) + \epsilon \tilde{\mathfrak{u}}^o(\epsilon, t) + \left(-\frac{1}{2}I + W_{\partial\Omega^i}\right)[\phi^i](t) \\ &\quad + \zeta\, S_n(t) + w^+_{\Omega^o}[\phi^o](\epsilon t) - (\partial_\epsilon F) (0,t,\zeta^i) - (\partial_\zeta F) (0,t,\zeta^i) \nonumber \\ &\quad \times \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) - \epsilon \tilde{F}\left(\epsilon,t,\zeta^i,\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t)\right)\ \forall t \in \partial\Omega^i, \nonumber \\ &M_3[\epsilon, \phi^o,\phi^i, \zeta, \psi^i](t) \equiv \, \nu_{\Omega^i}(t) \cdot \left( \nabla \mathfrak{u}^o(\epsilon t) + \epsilon \nabla w^+_{\Omega^o}[\phi^o](\epsilon t) + \nabla w^-_{\Omega^i}[\phi^i](t) \right. \\ &\quad \left. + \nabla S_n(t) \zeta - \nabla w^+_{\Omega^i}[\psi^i](t) \right) - G \left(\epsilon,t,\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) + \zeta^i \right) \nonumber \\ &\quad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad \forall t \in \partial\Omega^i\,, \end{align*} for all $(\epsilon, \phi^o, \phi^i, \zeta, \psi^i) \in ]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$. In Theorem \ref{existenceThm} here below we summarize some results of \cite{Mo18} on the operator $M$ (cf. \cite[Prop. 7.1, 7.5, 7.6, Lem. 7.7, Thm. 7.8]{Mo18}). \begin{teo}\label{existenceThm} Let assumptions \eqref{zetaicond}, \eqref{F1} and \eqref{realanalhp} hold. Then the following statement holds. \begin{enumerate} \item[(i)] The map $M$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)$. \item[(ii)] Let $\epsilon \in ]0,\epsilon_0[$ and $(\phi^o,\phi^i,\zeta,\psi^i) \in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$. Then the pair \[ (u^o_\epsilon[\phi^o,\phi^i,\zeta,\psi^i],u^i_\epsilon[\phi^o,\phi^i,\zeta,\psi^i]) \] defined by \eqref{rappsol} is a solution of \eqref{princeq} if and only if \begin{equation}\label{Me=0.e1} M[\epsilon, \phi^o, \phi^i, \zeta, \psi^i]=(0,0,0)\,. \end{equation} \item[(iii)] The equation \begin{equation*} M[0, \phi^o, \phi^i, \zeta, \psi^i]=(0,0,0) \end{equation*} has a unique solution $(\phi^o_0,\phi^i_0,\zeta_0,\psi^i_0) \in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$. \item[(iv)] The partial differential of $M$ with respect to $(\phi^o, \phi^i, \zeta, \psi^i)$ evaluated at $(0,\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)$, which we denote by \begin{equation*}\label{dMiso.eq1} \partial_{(\phi^o,\phi^i,\zeta,\psi^i)} M[0,\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0]\,, \end{equation*} is an isomorphism from $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)$. \item[(v)] There exist $\epsilon' \in ]0,\epsilon_0[$, an open neighborhood $U_0$ of $(\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)$ in $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$, and a real analytic map \begin{equation*} (\Phi^o,\Phi^i,Z,\Psi^i):\;]-\epsilon',\epsilon'[\to U_0 \end{equation*} such that the set of zeros of $M$ in $]-\epsilon',\epsilon'[ \times U_0$ coincides with the graph of $(\Phi^o[\cdot],\Phi^i[\cdot],Z[\cdot],\Psi^i[\cdot])$. In particular, \begin{equation*} (\Phi^o[0],\Phi^i[0],Z[0],\Psi^i[0]) = (\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0). \end{equation*} \end{enumerate} \end{teo} Then, in view of Theorem \ref{existenceThm} (ii) and Theorem \ref{existenceThm} (v), we can introduce a family of solutions $\{(u^o_\epsilon, u^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$ for problem \eqref{princeq}. \begin{teo}\label{uesol} Let assumptions \eqref{zetaicond}, \eqref{F1}, and \eqref{realanalhp} hold true. Let $\epsilon'$ and $(\Phi^o[\cdot],\Phi^i[\cdot],Z[\cdot],\Psi^i[\cdot])$ be as in Theorem \ref{existenceThm} (v). For all $\epsilon\in]0,\epsilon'[$, let \[ \begin{aligned} & u^o_\epsilon(x) \equiv U^o_\epsilon[\Phi^o[\epsilon] , \Phi^i[\epsilon], Z[\epsilon] , \Psi^i[\epsilon]](x)&&\forall x \in \overline{\Omega(\epsilon)}\,,\\ & u^i_\epsilon(x) \equiv U^i_\epsilon[\Phi^o[\epsilon] , \Phi^i[\epsilon], Z[\epsilon], \Psi^i[\epsilon]](x) &&\forall x \in \epsilon\overline{\Omega^i}\,, \end{aligned} \] with $U^o_\epsilon[\cdot,\cdot,\cdot,\cdot]$ and $U^i_\epsilon[\cdot,\cdot,\cdot,\cdot]$ defined as in \eqref{rappsol}. Then the pair of functions $(u^o_\epsilon,u^i_\epsilon) \in C^{1,\alpha}_{\mathrm{harm}}(\overline{\Omega(\epsilon)}) \times C^{1,\alpha}_{\mathrm{harm}}(\epsilon \overline{\Omega^i})$ is a solution of \eqref{princeq} for all $\epsilon \in ]0,\epsilon'[$. \end{teo} \section{Local uniqueness of the solution $(u^o_\epsilon,u^i_\epsilon)$}\label{localresult} In this section we prove local uniqueness results for the family of solutions $\{(u^o_\epsilon, u^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$ of Theorem \ref{uesol}. We will denote by $\mathcal{B}_{0,r}$ the ball in the product space $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ of radius $r>0$ and centered in the $4$-tuple $(\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)$ of Theorem \ref{existenceThm} (iii). Namely, we set \begin{equation}\label{B0r} \begin{split} &\mathcal{B}_{0,r} \equiv \bigg\{(\phi^o, \phi^i, \zeta, \psi^i)\in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)\,:\\ &\ \|\phi^o-\phi^o_0\|_{C^{1,\alpha}(\partial\Omega^o)}+ \|\phi^i-\phi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)}+ |\zeta-\zeta_0|+ \|\psi^i-\psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)}<r\bigg\}\end{split} \end{equation} for all $r>0$. Then for $\epsilon'$ as in Theorem \ref{existenceThm} (v), we denote by $\Lambda=(\Lambda_1,\Lambda_2)$ the map from $]-\epsilon',\epsilon'[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)$ defined by \begin{equation}\label{Aoperator} \begin{aligned} \Lambda_1[\epsilon,\phi^o,\phi^i,\zeta](x) &\equiv \left( \frac{1}{2}I + W_{\partial\Omega^o}\right) [\phi^o](x)\\ & - \epsilon^{n-1} \int_{\partial\Omega^i}{\nu_{\Omega^i}(y) \cdot \nabla S_n(x -\epsilon y) \phi^i(y) \,d\sigma_y} \\ & + \epsilon^{n-2} S_n (x) \zeta && \qquad \forall x \in \partial \Omega^o, \\ \Lambda_2[\epsilon,\phi^o,\phi^i,\zeta](t)& \equiv \left(-\frac{1}{2}I + W_{\partial\Omega^i}\right) [\phi^i](t) + w^+_{\Omega^o}[\phi^o](\epsilon t)\\ & + S_n(t) \zeta && \qquad\forall t \in \partial \Omega^i, \end{aligned} \end{equation} for all $(\epsilon,\phi^o,\phi^i,\zeta) \in ]-\epsilon',\epsilon'[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$. We now prove the following. \begin{prop}\label{propAoperator} There exist $\epsilon'' \in ]0,\epsilon'[$ and $C \in ]0,+\infty[$ such that the operator $\Lambda[\epsilon,\cdot,\cdot,\cdot]$ from $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)$ is linear continuous and invertible for all $\epsilon \in ]-\epsilon'',\epsilon''[$ fixed and such that \begin{equation*} \|\Lambda[\epsilon,\cdot,\cdot,\cdot] ^ {(-1)}\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R})} \leq C \end{equation*} uniformly for $\epsilon \in ]-\epsilon'',\epsilon''[$. \end{prop} \begin{proof} By the mapping properties of the double layer potential (cf. Theorem \ref{doublepot} (ii) and Theorem \ref{doublepot} (iii)) and of integral operators with real analytic kernels (cf. Lanza de Cristoforis and Musolino \cite{LaMu13}) one verifies that the map from $]-\epsilon',\epsilon'[$ to $\mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R},C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i))$ which takes $\epsilon$ to $\Lambda[\epsilon,\cdot,\cdot,\cdot]$ is continuous. Since the set of invertible operators is open in the space $\mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R},C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i))$, to complete the proof it suffices to show that for $\epsilon=0$ the map which takes $(\phi^o,\phi^i,\zeta) \in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ to \begin{equation*} \begin{split} &\Lambda[0,\phi^o,\phi^i,\zeta]\\ &\qquad = \left( \left( \frac{1}{2}I + W_{\partial\Omega^o}\right)[\phi^o] , \left(-\frac{1}{2}I + W_{\partial\Omega^i}\right) [\phi^i] + w^+_{\Omega^o}[\phi^o](0) + {S_n}_{|\partial\Omega^i} \zeta\right)\\ &\qquad\in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \end{split} \end{equation*} is invertible. To prove it, we verify that it is a bijection and then we exploit the Open Mapping Theorem. So let $(h^o,h^i) \in C^{1,\alpha}(\partial\Omega^o)\timesC^{1,\alpha}(\partial\Omega^i)$. We claim that there exists a unique $(\phi^o,\phi^i,\zeta) \in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ such that \begin{equation}\label{propAoperator.eq1} \Lambda[0,\phi^o,\phi^i,\zeta] = (h^o,h^i). \end{equation} Indeed, by Theorem \ref{doublepot} (iv), $ \frac{1}{2}I + W_{\partial\Omega^o}$ is an isomorphism from $C^{1,\alpha}(\partial\Omega^o)$ into itself and there exists a unique $\phi^o\in C^{1,\alpha}(\partial\Omega^o)$ that satisfies the first equation of \eqref{propAoperator.eq1}. Moreover, by Lemma \ref{isolem}, the map from $C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^i)$ that takes $(\phi^i,\zeta)$ to $\left(-\frac{1}{2}I + W_{\partial\Omega^i}\right) [\phi^i] + {S_n}_{|\partial\Omega^i} \zeta,$ is an isomorphism. Hence, there exists a unique $(\phi^i,\zeta) \in C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ such that \begin{equation*} \left(-\frac{1}{2}I + W_{\partial\Omega^i}\right) [\phi^i] + {S_n}_{|\partial\Omega^i} \zeta = h^i - w^+_{\Omega^o}[\phi^o](0). \end{equation*} Accordingly, there exists a unique $(\phi^i,\zeta) \in C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ that satisfies the second equation of \eqref{propAoperator.eq1}. Thus $\Lambda[0,\cdot,\cdot,\cdot]$ is an isomorphism from $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)$ and the proof is complete. \end{proof} \subsection{A first local uniqueness result} We are now ready to state our first local uniqueness result for the solution $(u^o_\epsilon,u^i_\epsilon)$. Theorem \ref{Thmunisol1} here below is, in a sense, a consequence of an argument based on the Implicit Function Theorem for real analytic maps (see, for example, Deimling \cite[Thm. 15.3]{De85}) that has been used in \cite{Mo18} to prove the existence of such solution. We shall see in the following Subsection \ref{second} that the statement of Theorem \ref{Thmunisol1} holds under much weaker assumptions. \begin{teo}\label{Thmunisol1} Let assumptions \eqref{zetaicond}, \eqref{F1}, and \eqref{realanalhp} hold true. Let $\epsilon' \in ]0,\epsilon_0[$ be as in Theorem \ref{existenceThm} (v). Let $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$ be as in Theorem \ref{uesol}. Then there exist $\epsilon^*\in]0,\epsilon'[$ and $\delta^\ast \in ]0,+\infty[$ such that the following property holds: If $\epsilon \in ]0,\epsilon^*[$ and $(v^o,v^i) \in C^{1,\alpha}(\Omega(\epsilon)) \times C^{1,\alpha}(\epsilon\Omega^i)$ is a solution of problem \eqref{princeq} with \begin{align} \left\|v^o - u^o_\epsilon \right\| _{C^{1,\alpha}(\partial \Omega^o)} &\leq \epsilon\delta^\ast, \label{uo} \\ \left\| v^o(\epsilon \cdot ) - u^o_\epsilon(\epsilon \cdot ) \right\| _{C^{1,\alpha}(\partial \Omega^i)} &\leq \epsilon\delta^\ast, \label{uoe} \\ \left\| v^i(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot ) \right\| _{C^{1,\alpha}(\partial \Omega^i)} &\leq \epsilon\delta^\ast, \label{uie} \end{align} then \begin{equation*} (v^o,v^i)=(u^o_\epsilon,u^i_\epsilon)\,. \end{equation*} \end{teo} \begin{proof} Let $U_0$ be the open neighborhood of $(\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)$ in $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ introduced in Theorem \ref{existenceThm} (v). We take $K>0$ such that \[ \overline{\mathcal{B}_{0,K}}\subseteq U_0\,. \] Since $(\Phi^o[\cdot],\Phi^i[\cdot],Z[\cdot],\Psi^i[\cdot])$ is continuous (indeed real analytic) from $]-\epsilon',\epsilon'[$ to $U_0$, then there exists ${\epsilon'_*} \in ]0,\epsilon'[$ such that \begin{equation}\label{(Phi^o,Phi^i,Z,Psi^i)K/2} (\Phi^o[\eta] , \Phi^i[\eta],Z[\eta] , \Psi^i[\eta])\in \mathcal{B}_{0,K/2} \qquad\forall \eta \in ]0,\epsilon'_*[\,. \end{equation} Let $\epsilon''$ be as in Proposition \ref{propAoperator} and let \begin{equation*} \epsilon^\ast \equiv \min\{\epsilon'_\ast,\epsilon''\}. \end{equation*} Let $\epsilon \in ]0,\epsilon^\ast[$ be fixed and let $(v^o,v^i) \in C^{1,\alpha}(\Omega(\epsilon)) \times C^{1,\alpha}(\epsilon\Omega^i)$ be a solution of problem \eqref{princeq} that satisfies \eqref{uo}, \eqref{uoe}, and \eqref{uie} for a certain $\delta^\ast \in ]0,+\infty[$. We show that for $\delta^\ast$ sufficiently small $(v^o,v^i)=(u^o_\epsilon,u^i_\epsilon)$. By Proposition \ref{rappresentsol}, there exists a unique quadruple $(\phi^o,\phi^i,\zeta,\psi^i) \in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ such that \begin{align}\label{uo[]} v^o &= U^o_{\epsilon}[\phi^o,\phi^i,\zeta,\psi^i] \qquad \text{in } \overline{\Omega(\epsilon)}, \\ \label{ui[]} v^i &= U^i_{\epsilon}[\phi^o,\phi^i,\zeta,\psi^i] \qquad \text{in } \epsilon \overline{\Omega^i}. \end{align} By \eqref{uie} and by \eqref{ui[]}, we have \begin{equation}\label{(u^i-u_e^i)} \begin{split} \delta^\ast &\geq \left\| \frac{v^i(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot )}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^i)}\\ & = \left\| \frac{U^i_\epsilon[\phi^o,\phi^i,\zeta,\psi^i](\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot )}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^i)} \\ & = \left\| \frac{\epsilon w^+_{\epsilon\Omega^i}\left[\psi^i\left(\frac{\cdot}{\epsilon}\right)\right](\epsilon \cdot) + \zeta^i -\epsilon w^+_{\epsilon\Omega^i}\left[\Psi^i[\epsilon]\left(\frac{\cdot}{\epsilon}\right)\right](\epsilon \cdot) - \zeta^i}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^i)} \\ & = \left\| w^+_{\Omega^i}[\psi^i] - w^+_{\Omega^i}[\Psi^i[\epsilon]] \right\| _{C^{1,\alpha}(\partial \Omega^i)} \,. \end{split} \end{equation} By the jump relations in Theorem \ref{doublepot} (i), we obtain \begin{equation}\label{Thmunisol1.eq1} \left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)} \leq \, \delta^\ast. \end{equation} By Theorem \ref{doublepot} (iv), the operator $\frac{1}{2}I + W_{\partial\Omega^i}$ is a linear isomorphism from $C^{1,\alpha}(\partial\Omega^i)$ to itself. Then, if we denote by $D$ the norm of its inverse, namely we set \[ D \equiv \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)^{(-1)} \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))}\,, \] we obtain, by \eqref{uie} and by \eqref{Thmunisol1.eq1}, that \begin{equation}\label{psi^i} \begin{split} &\| \psi^i - \Psi^i[\epsilon] \|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\quad\leq \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)^{(-1)} \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} \\ &\qquad\times\left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\quad \leq D \delta^\ast. \end{split} \end{equation} By \eqref{uo} and \eqref{uo[]} we have {\small \begin{equation}\label{(u^o - u_e^o)1} \begin{split} &\delta^\ast \geq \left\| \frac{v^o - u^o_\epsilon}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^o)} =\left\| \frac{U^o_{\epsilon}[\phi^o,\phi^i,\zeta,\psi^i] - u^o_\epsilon)}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^o)} \\ &= \left\| \frac{\epsilon w^+_{\Omega^o}[\phi^o -\Phi^o[\epsilon] ] + \epsilon w^-_{\epsilon\Omega^i}\left[\phi^i\left(\frac{\cdot}{\epsilon}\right) -\Phi^i[\epsilon]\left(\frac{\cdot}{\epsilon}\right)\right] + \epsilon^{n-1}\left(\zeta - Z[\epsilon] \right)\, S_n}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^o)} \\ &= \left\| w^+_{\Omega^o}[\phi^o -\Phi^o[\epsilon] ] + w^-_{\epsilon\Omega^i}\left[\phi^i\left(\frac{\cdot}{\epsilon}\right) -\Phi^i[\epsilon]\left(\frac{\cdot}{\epsilon}\right)\right] + \epsilon^{n-2}\left(\zeta - Z[\epsilon] \right)\, S_n \right\| _{C^{1,\alpha}(\partial \Omega^o)}. \end{split} \end{equation}} Similarly, \eqref{uoe} and \eqref{uo[]} yield {\small \begin{equation}\label{(u^o - u_e^o)2} \begin{split} &\delta^\ast \geq \left\| \frac{v^o(\epsilon \cdot) - u^o_\epsilon( \epsilon\cdot)}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^i)} =\left\| \frac{U^o_\epsilon[\phi^o,\phi^i,\zeta,\psi^i](\epsilon \cdot) - u^o_\epsilon(\epsilon\cdot )}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^i)} \\ &= \left\| w^+_{\Omega^o}[\phi^o -\Phi^o[\epsilon] ](\epsilon \cdot) + w^-_{\epsilon\Omega^i}\left[\phi\left(\frac{\cdot}{\epsilon}\right) -\Phi^i[\epsilon]\left(\frac{\cdot}{\epsilon}\right)\right](\epsilon \cdot)\right.\\ &\left.\qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad + \epsilon^{n-2}\left(\zeta - Z[\epsilon] \right)\, S_n(\epsilon \cdot) \right\| _{C^{1,\alpha}(\partial \Omega^i)} \\ &= \left\| w^-_{\Omega^i}\left[\phi^i -\Phi^i[\epsilon]\right] + w^+_{\Omega^o}[\phi^o -\Phi^o[\epsilon] ](\epsilon \cdot) + \left(\zeta - Z[\epsilon] \right)\, S_n \right\| _{C^{1,\alpha}(\partial \Omega^i)}. \end{split} \end{equation}} Then, by \eqref{(u^o - u_e^o)1} and \eqref{(u^o - u_e^o)2} and by the definition of the operator $\Lambda$ in \eqref{Aoperator}, we deduce that \begin{equation}\label{A[phi^o,phi^i,zeta]} \left\| \Lambda\left[\epsilon,\phi^o - \Phi^o[\epsilon],\phi^i - \Psi^i[\epsilon],\zeta - Z[\epsilon]\right] \right\| _{C^{1,\alpha}(\partial\Omega^o)\timesC^{1,\alpha}(\partial\Omega^i)} \leq 2\delta^\ast \end{equation} (see also the jump relations for the double layer potential in Theorem \ref{doublepot} (i)). Now let $C>0$ as in the statement of Proposition \ref{propAoperator}. Then, by the membership of $\epsilon$ in $]0,\epsilon^\ast[$, we have \begin{equation}\label{phi^o,phi^i,zeta} \begin{split} &\left(\phi^o - \Phi^o[\epsilon],\phi^i - \Psi^i[\epsilon],\zeta - Z[\epsilon]\right)\\ &\qquad = \Lambda[\epsilon,\cdot,\cdot,\cdot]^{(-1)} \Lambda\left[\epsilon,\phi^o - \Phi^o[\epsilon],\phi^i - \Psi^i[\epsilon],\zeta - Z[\epsilon]\right] \end{split} \end{equation} and, by \eqref{A[phi^o,phi^i,zeta]} and \eqref{phi^o,phi^i,zeta}, we obtain \begin{equation*} \begin{split} &\|\left(\phi^o - \Phi^o[\epsilon], \phi^i - \Psi^i[\epsilon],\zeta - Z[\epsilon]\right) \|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}} \\ & \quad\leq \|\Lambda[\epsilon,\cdot,\cdot,\cdot]^{(-1)} \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R})} \\ & \quad\quad\times \left\| \Lambda\left[\epsilon,\phi^o - \Phi^o[\epsilon],\phi^i - \Psi^i[\epsilon],\zeta - Z[\epsilon]\right] \right\|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R}} \\ &\quad\leq 2 C \delta^\ast\,. \end{split} \end{equation*} The latter inequality, combined with \eqref{psi^i}, yields \begin{equation}\label{phi^o,phi^i,zeta,psi^i} \begin{split} &\|\left(\phi^o - \Phi^o[\epsilon], \phi^i - \Psi^i[\epsilon],\zeta - Z[\epsilon], \psi^i - \Psi[\epsilon]\right) \|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq (2C + D) \delta^* \,. \end{split} \end{equation} Hence, by \eqref{(Phi^o,Phi^i,Z,Psi^i)K/2} and \eqref{phi^o,phi^i,zeta,psi^i} and by a standard computation based on the triangular inequality one sees that \begin{equation*} \begin{split} &\|(\phi^o, \phi^i, \zeta, \psi^i) - (\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)\|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \leq (2C+D)\delta^* + \frac{K}{2}. \end{split} \end{equation*} Accordingly, in order to have $(\phi^o, \phi^i, \zeta, \psi^i) \in \mathcal{B}_{0,K}$, it suffices to take \begin{equation*} \delta^\ast < \frac{K}{2(2C+D)} \end{equation*} in inequalities \eqref{uo}, \eqref{uoe}, and \eqref{uie}. Then, by the inclusion $\overline{\mathcal{B}_{0,K}}\subseteq U_0$ and by Theorem \ref{existenceThm} (v), we deduce that for such choice of $\delta^\ast$ we have \begin{equation*} (\phi^o, \phi^i, \zeta, \psi^i) = \left(\Phi^o[\epsilon], \Psi^i[\epsilon],Z[\epsilon],\Psi[\epsilon]\right) \end{equation*} and thus $(v^o,v^i)=(u^o_\epsilon,u^i_\epsilon)$ (cf. Theorem \ref{uesol}). \end{proof} \subsection{A stronger local uniqueness result}\label{second} In this Subsection we will see that we can weaken the assumptions of Theorem \ref{Thmunisol1}. In particular, we will prove in Theorem \ref{Thmunisol2} that the local uniqueness of the solution can be achieved with only one condition on the trace of the function $v^i(\epsilon\cdot)$ on $\partial\Omega^i$, instead of the three conditions used in Theorem \ref{Thmunisol1}. To prove Theorem \ref{Thmunisol2} we shall need some preliminary technical results on composition operators. \subsubsection{Some preliminary results on composition operators} We begin with the following Lemma \ref{lemmaboundA(ep,dot,dot)}. \begin{lemma}\label{lemmaboundA(ep,dot,dot)} Let $A$ be a function from $]-\epsilon_0,\epsilon_0[ \times \overline{B_{n-1}(0,1)} \times \mathbb{R} $ to $\mathbb{R}$. Let $\mathcal{M}_A$ be the map which takes a pair $(\epsilon,\zeta)\in]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to the function $\mathcal{M}_A (\epsilon,\zeta)$ defined by \begin{equation}\label{M_A} \mathcal{M}_A (\epsilon,\zeta)(z) \equiv A(\epsilon,z,\zeta)\qquad\forall z\in \overline{B_{n-1}(0,1)}. \end{equation} Let $m\in\{0,1\}$. If $\mathcal{M}_A (\epsilon,\zeta)\in C^{m,\alpha}( \overline{B_{n-1}(0,1)} )$ for all $(\epsilon,\zeta)\in]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ and if the map $\mathcal{M}_A$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{m,\alpha}( \overline{B_{n-1}(0,1)} )$, then for every open bounded interval $\mathcal{J}$ of $\mathbb{R}$ and every compact subset $\mathcal{E}$ of $]-\epsilon_0,\epsilon_0[$ there exists $C>0$ such that \begin{equation}\label{A<C} \sup_{\epsilon \in \mathcal{E}} \|A(\epsilon,\cdot,\cdot)\|_{C^{m,\alpha}( \overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})} \leq C. \end{equation} \end{lemma} \begin{proof} We first prove the statement of Lemma \ref{lemmaboundA(ep,dot,dot)} for $m=0$. If $\mathcal{M}_A$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{0,\alpha}( \overline{B_{n-1}(0,1)} )$, then for every $(\tilde{\epsilon},\tilde{\zeta}) \in ]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ there exist $M \in ]0,+\infty[$, $\rho \in ]0,1[$, and a family of coefficients $\{a_{jk}\}_{j,k \in \mathbb{N}} \subset C^{0,\alpha}( \overline{B_{n-1}(0,1)} )$ such that \begin{equation}\label{realanalcoeffajk} \|a_{jk}\|_{C^{0,\alpha}( \overline{B_{n-1}(0,1)} ) } \leq M \left( \frac{1}{\rho} \right)^{k+j}\qquad\forall j,k\in\mathbb{N}, \end{equation} and \begin{equation}\label{realanalexpA} \mathcal{M}_A (\epsilon,\zeta)(\cdot) = \sum_{j,k = 0}^{\infty} a_{jk}(\cdot) (\epsilon-\tilde{\epsilon})^k (\zeta-\tilde{\zeta})^j\qquad\forall(\epsilon,\zeta) \in ]\tilde{\epsilon}-\rho, \tilde{\epsilon}+\rho[ \times ]\tilde{\zeta}-\rho,\tilde{\zeta}+\rho[\,, \end{equation} where $\rho$ is less than or equal to the radius of convergence of the series in \eqref{realanalexpA}. Now let $\mathcal{J}\subset\mathbb{R}$ be open and bounded and $\mathcal{E} \subset \, ]-\epsilon_0,\epsilon_0[$ be compact. Since the product $\overline{\mathcal{J}} \times \mathcal{E}$ is compact, a standard finite covering argument shows that in order to prove \eqref{A<C} for $m=0$ it suffices to find a uniform upper bound (independent of $\tilde\epsilon$ and $\tilde\zeta$) for the quantity \begin{equation*} \sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \|A(\epsilon,\cdot,\cdot)\|_{C^{0,\alpha}( \overline{B_{n-1}(0,1)} \times [\tilde{\zeta} - \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}])}. \end{equation*} By \eqref{M_A}, \eqref{realanalcoeffajk}, and \eqref{realanalexpA} we have \begin{equation}\label{C0normA} \begin{split} \sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]}& \|A(\epsilon,\cdot,\cdot)\|_{C^0(\overline{B_{n-1}(0,1)}\times [\tilde{\zeta} - \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}])} \\ & \leq \sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \sum_{j,k=0}^{\infty} \|a_{jk}(\cdot) (\epsilon-\tilde{\epsilon})^k (\cdot-\tilde{\zeta})^j \|_{C^0(\overline{B_{n-1}(0,1)}\times [\tilde{\zeta} - \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}])} \\ &\leq \sum_{j,k=0}^{\infty} M \left(\frac{1}{\rho}\right)^{j+k} \left(\frac{\rho}{2}\right)^k \left(\frac{\rho}{2}\right)^j = \sum_{j=0}^{\infty} M \left(\frac{1}{2}\right)^{j+k} = 4M \end{split} \end{equation} for all $l=1,\dots, m$. Then inequality \eqref{C0normA} yields an estimate of the $C^0$ norm of $A$. To complete the proof of \eqref{A<C} for $m=0$ we have now to study the H\"older constant of $A(\epsilon,\cdot,\cdot)$ on $\overline{B_{n-1}(0,1)}\times [\tilde{\zeta} - \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}]$. To do so, we take $z',z'' \in \overline{B_{n-1}(0,1)}$, $\zeta',\zeta'' \in [\tilde{\zeta}-\frac{\rho}{4},\tilde{\zeta}+\frac{\rho}{4}]$, and $\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2}, \tilde{\epsilon} + \frac{\rho}{2}]$, and we consider the difference \begin{equation}\label{a'-a''} |a_{jk}(z') (\epsilon-\tilde{\epsilon})^k (\zeta'-\tilde{\zeta})^j - a_{jk}(z'') (\epsilon-\tilde{\epsilon})^k (\zeta''-\tilde{\zeta})^j | . \end{equation} For $j \geq 1$ and $k \geq 0$ we argue as follow: we add and subtract the term $a_{jk}(z'') (\epsilon-\tilde{\epsilon})^k (\zeta'-\tilde{\zeta})^j $ inside the absolute value in \eqref{a'-a''}, we use the triangular inequality to split the difference in two terms and then we exploit the membership of $a_{jk}$ in $C^{0,\alpha}( \overline{B_{n-1}(0,1)} )$ and an argument based on the Taylor expansion at the first order for the function from $[\tilde{\zeta}-\frac{\rho}{4},\tilde{\zeta}+\frac{\rho}{4}]$ to $\mathbb{R}$ that takes $\zeta$ to $(\zeta-\tilde{\zeta})^j$. Doing so we show that \eqref{a'-a''} is less than or equal to \begin{equation}\label{a'-a''.1} \begin{split} &|a_{jk}(z')-a_{jk}(z'')| |\epsilon-\tilde{\epsilon}|^k |\zeta'-\tilde{\zeta}|^j + |a_{jk}(z'')| |\epsilon-\tilde{\epsilon}|^k |(\zeta'-\tilde{\zeta})^j - (\zeta''-\tilde{\zeta})^j| \\ &\quad\leq \|a_{jk}\|_{C^{0,\alpha}( \overline{B_{n-1}(0,1)} )} |z'-z''|^\alpha |\epsilon-\tilde{\epsilon}|^k |\zeta'-\tilde{\zeta}|^j\\ &\qquad + \|a_{jk}\|_{C^{0,\alpha}( \overline{B_{n-1}(0,1)} )} |\epsilon-\tilde{\epsilon}|^k \left(j |\overline{\zeta}-\tilde{\zeta}|^{j-1} |\zeta'-\zeta''|\right) \end{split} \end{equation} for a suitable $\overline{\zeta} \in [\tilde{\zeta}-\frac{\rho}{4},\tilde{\zeta}+\frac{\rho}{4}]$. Then by \eqref{realanalcoeffajk}, by inequalities $|\epsilon-\tilde{\epsilon}|\le \frac{\rho}{2}$, $|\zeta'-\tilde\zeta|\le \frac{\rho}{4}$, and $|\overline\zeta-\tilde\zeta|\le \frac{\rho}{4}$, and by a straightforward computation we see that the right hand side of \eqref{a'-a''.1} is less than or equal to \begin{equation}\label{a'-a''.2} \begin{split} &M \left(\frac{1}{\rho}\right)^{j+k} |z'-z''|^\alpha \left(\frac{\rho}{2}\right)^k \left(\frac{\rho}{4}\right)^j + M \left(\frac{1}{\rho}\right)^{j+k} \left(\frac{\rho}{2}\right)^k j \left(\frac{\rho}{4}\right)^{j-1} |\zeta'-\zeta''| \\ &\quad = M \left(\frac{1}{2}\right)^j\left(\frac{1}{2}\right)^{j+k} |z'-z''|^\alpha + 4 M \rho^{-1}\frac{j}{2^j}\left(\frac{1}{2}\right)^{j+k} |\zeta'-\zeta''|^{1-\alpha} |\zeta'-\zeta''|^\alpha. \end{split} \end{equation} Now, since $\zeta'$ and $\zeta''$ are taken in the interval $[\tilde{\zeta}-\frac{\rho}{4},\tilde{\zeta}+\frac{\rho}{4}]$ we have $|\zeta'-\zeta''|^{1-\alpha}\le \left({\rho}/{2}\right)^{1-\alpha}$ and since $\rho \in ]0,1[$ and $\alpha \in ]0,1[$, we deduce that $|\zeta'-\zeta''|^{1-\alpha}\le 1$. Moreover, since $j\ge 1$, we have ${j}/{2^j} \leq 1$ and $(1/2)^j<1$. It follows that the right hand side of \eqref{a'-a''.2} is less than or equal to \begin{equation}\label{a'-a''.3} \begin{split} & M \left(\frac{1}{2}\right)^{j+k} |z'-z''|^\alpha + 4 M \rho^{-1} \left(\frac{1}{2}\right)^{j+k} |\zeta'-\zeta''|^\alpha\\ &\qquad\qquad \leq 4 M \rho^{-1} \left(\frac{1}{2}\right)^{j+k} \left(|z'-z''|^\alpha + |\zeta'-\zeta''|^\alpha\right) \end{split} \end{equation} (also note that $\rho^{-1}>1$). Finally, by inequality \[ a^\alpha+b^\alpha\le 2^{1-\frac{\alpha}{2}}(a^2+b^2)^{\frac{\alpha}{2}}\,, \] which holds for all $a,b>0$, we deduce that the right hand side of \eqref{a'-a''.3} is less than or equal to \begin{equation}\label{a'-a''.4} 2^{3-\frac{\alpha}{2}} M \rho^{-1} \left(\frac{1}{2}\right)^{j+k} |(z',\zeta')-(z'',\zeta'')|^\alpha, \end{equation} where $|(z',\zeta')-(z'',\zeta'')|$ denotes the Euclidean norm of the vector $(z',\zeta')-(z'',\zeta'')$ in $\mathbb{R}^{n-1}\times\mathbb{R}=\mathbb{R}^n$. Then, by \eqref{a'-a''.1}--\eqref{a'-a''.4}, we obtain that \begin{equation}\label{a'-a''.5} \begin{split} &|a_{jk}(z') (\epsilon-\tilde{\epsilon})^k (\zeta'-\tilde{\zeta})^j - a_{jk}(z'') (\epsilon-\tilde{\epsilon})^k (\zeta''-\tilde{\zeta})^j | \\ &\qquad\qquad \le 2^{3-\frac{\alpha}{2}} M \rho^{-1} \left(\frac{1}{2}\right)^{j+k} |(z',\zeta')-(z'',\zeta'')|^\alpha \end{split} \end{equation} for all $j \geq 1$, $k \geq 0$, and $\epsilon\in [\tilde{\epsilon} - \frac{\rho}{2}, \tilde{\epsilon} + \frac{\rho}{2}]$. Now, for every $\epsilon\in [\tilde{\epsilon} - \frac{\rho}{2}, \tilde{\epsilon} + \frac{\rho}{2}]$ we denote by $\tilde{a}_{jk, \epsilon}$ the function \begin{equation}\label{tildeajk} \begin{aligned} \tilde{a}_{jk,\epsilon}\,:\; \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right]&\to\mathbb{R} \\ (z,\zeta)&\mapsto \tilde{a}_{jk,\epsilon}(z,\zeta) \equiv a_{jk}(z) (\epsilon-\tilde{\epsilon})^k (\zeta-\tilde{\zeta})^j\,. \end{aligned} \end{equation} Then inequality \eqref{a'-a''.5} readily implies that \[ \begin{split} &\left|\tilde{a}_{jk,\epsilon}: \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \leq 2^{3-\frac{\alpha}{2}} M \rho^{-1} \left(\frac{1}{2}\right)^{j+k}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad \forall j \geq 1\,,\; k \geq 0\,,\; \epsilon\in \left[\tilde{\epsilon} - \frac{\rho}{2}, \tilde{\epsilon} + \frac{\rho}{2}\right]\,, \end{split} \] which in turn implies that \begin{equation}\label{lemmaboundA(ep,dot,dot)inequ4} \begin{split} & \sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \sum_{j=1,k=0}^{\infty} \left|\tilde{a}_{jk,\epsilon} : \overline{B_{n-1}(0,1)}\times \left[\tilde{\zeta} - \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \\ &\qquad\qquad\qquad \leq \sum_{j=1,k=0}^{\infty} 2^{3-\frac{\alpha}{2}} M \rho^{-1} \left(\frac{1}{2}\right)^{j+k} \leq 2^{4-\frac{\alpha}{2}} M \rho^{-1}. \end{split} \end{equation} We now turn to consider \eqref{a'-a''} in the case where $j=0$ and $k \geq 0$. In such case, one verifies that the quantity in \eqref{a'-a''} is less than or equal to \[ \|a_{0k}\|_{C^{0,\alpha}( \overline{B_{n-1}(0,1)} )} |\epsilon-\tilde{\epsilon}|^k |z'-z''|^\alpha\,, \] which, by \eqref{realanalcoeffajk} and by inequality $|\epsilon-\tilde{\epsilon}|\le \frac{\rho}{2}$, is less than or equal to \[ M \left( \frac{1}{\rho} \right)^{k} \left(\frac{\rho}{2}\right)^k |z'-z''|^\alpha=M\left(\frac{1}{2}\right)^k|z'-z''|^\alpha\,. \] Hence, for $\tilde{a}_{0k,\epsilon}$ defined as in \eqref{tildeajk} (with $j=0$) we have \begin{equation*} \begin{split} &\left|\tilde{a}_{0k,\epsilon} : \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \leq M\left(\frac{1}{2}\right)^k\\ &\qquad\qquad\qquad \qquad\qquad\qquad \qquad \forall k \geq 0\,,\; \epsilon\in \left[\tilde{\epsilon} - \frac{\rho}{2}, \tilde{\epsilon} + \frac{\rho}{2}\right]\,, \end{split} \end{equation*} which implies that \begin{equation}\label{lemmaboundA(ep,dot,dot)inequ3} \begin{split} &\sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \sum_{k=0}^{\infty} \left|\tilde{a}_{0k,\epsilon} : \overline{B_{n-1}(0,1)}\times \left[\tilde{\zeta} + \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \leq \sum_{k=0}^{\infty} M \left(\frac{1}{2}\right)^k = 2M\,. \end{split} \end{equation} Finally, by \eqref{realanalexpA}, \eqref{C0normA}, \eqref{lemmaboundA(ep,dot,dot)inequ4}, and \eqref{lemmaboundA(ep,dot,dot)inequ3} we obtain \[ \begin{split} &\sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \|A(\epsilon,\cdot,\cdot)\|_{C^{0,\alpha}( \overline{B_{n-1}(0,1)} \times [\tilde{\zeta} + \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}])} \\ &\qquad = \sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \|A(\epsilon,\cdot,\cdot)\|_{C^0(\overline{B_{n-1}(0,1)}\times [\tilde{\zeta} + \frac{\rho}{4}, \tilde{\zeta} + \frac{\rho}{4}])} \\ &\qquad\quad + \sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]} \left|A(\epsilon,\cdot,\cdot) : \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \\ & \qquad \le 4M +\sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]}\sum_{j,k=0}^\infty\left|\tilde{a}_{jk,\epsilon} : \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \\ & \qquad =4M +\sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]}\sum_{j=1,k=0}^\infty\left|\tilde{a}_{jk,\epsilon} : \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha \\ &\qquad\quad +\sup_{\epsilon \in [\tilde{\epsilon} - \frac{\rho}{2},\tilde{\epsilon} + \frac{\rho}{2}]}\sum_{k=0}^\infty\left|\tilde{a}_{0k,\epsilon} : \overline{B_{n-1}(0,1)} \times \left[\tilde{\zeta} - \frac{\rho}{4},\tilde{\zeta} + \frac{\rho}{4}\right] \right|_\alpha\\ &\qquad \leq 4M +2^{4-\frac{\alpha}{2}} M \rho^{-1} + 2M\,. \end{split} \] We deduce that \eqref{A<C} for $m=0$ holds with $C=6M+2^{4-\frac{\alpha}{2}} M \rho^{-1}$. We now assume that $\mathcal{M}_A$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to the space $C^{1,\alpha}(\overline{B_{n-1}(0,1)})$ and we prove \eqref{A<C} for $m=1$. To do so we will exploit the (just proved) statement of Lemma \ref{lemmaboundA(ep,dot,dot)} for $m=0$. We begin by observing that, since the imbedding of $C^{1,\alpha}(\overline{B_{n-1}(0,1)})$ into $C^{0,\alpha}(\overline{B_{n-1}(0,1)})$ is linear and continuous, the map $\mathcal{M}_A$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{0,\alpha}(\overline{B_{n-1}(0,1)})$. Hence, by Lemma \ref{lemmaboundA(ep,dot,dot)} for $m=0$ and by the continuity of the imbedding of $C^{0,\alpha}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})$ into $C^{0}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})$ we deduce that \begin{equation}\label{A<C1} \sup_{\epsilon \in \mathcal{E}} \|A(\epsilon,\cdot,\cdot)\|_{C^{0}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})} \leq C_1\,. \end{equation} Moreover, since differentials of real analytic maps are real analytic, we have that the map $\mathcal{M}_{\partial_\zeta A}= \partial_\zeta \mathcal{M}_{A}$ which takes $(\epsilon,\zeta)$ to $ \partial_\zeta A(\epsilon,\cdot,\zeta)$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{1,\alpha}(\overline{B_{n-1}(0,1)})$, and thus from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{0,\alpha}(\overline{B_{n-1}(0,1)})$. By Lemma \ref{lemmaboundA(ep,dot,dot)} for $m=0$ it follows that \begin{equation}\label{dzetaA<C2} \sup_{\epsilon \in \mathcal{E}} \| \partial_\zeta A(\epsilon,\cdot,\cdot)\|_{C^{0,\alpha}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})} \leq C_2, \end{equation} for some $C_2>0$. Finally, we observe that the map $\partial_z$ from $C^{1,\alpha}(\overline{B_{n-1}(0,1)}) $ to $C^{0,\alpha}(\overline{B_{n-1}(0,1)}) $ that takes a function $f$ to $\partial_z f$ is linear and continuous. Then, the map $\mathcal{M}_{\partial_z A}$ which takes $(\epsilon,\zeta)$ to the function \[ \partial_z A(\epsilon,z,\zeta)\qquad\forall z\in\overline{B_{n-1}(0,1)} \] is the composition of $\mathcal{M}_A$ and $\partial_z$. Namely, we can write \[ \mathcal{M}_{\partial_z A}= \partial_z\circ \mathcal{M}_{A}\,. \] Since $\mathcal{M}_A$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{1,\alpha}(\overline{B_{n-1}(0,1)})$, it follows that $\mathcal{M}_{\partial_z A}$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{0,\alpha}(\overline{B_{n-1}(0,1)})$. Hence Lemma \ref{lemmaboundA(ep,dot,dot)} for $m=0$ implies that there exists $C_3>0$ such that \begin{equation}\label{dzA<C3} \sup_{\epsilon \in \mathcal{E}} \| \partial_z A(\epsilon,\cdot,\cdot)\|_{C^{0,\alpha}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})} \leq C_3\,. \end{equation} Now, the validity of \eqref{A<C} for $m=1$ is a consequence of \eqref{A<C1}, \eqref{dzetaA<C2}, and \eqref{dzA<C3}. \end{proof} In the sequel we will exploit Schauder spaces over suitable subsets of $\partial\Omega^i \times\mathbb{R}$. We observe indeed that for all open bounded intervals $\mathcal{J}$ of $\mathbb{R}$, the product $\partial\Omega^i \times\overline{\mathcal{J}}$ is a compact sub-manifold (with boundary) of co-dimension $1$ in $\mathbb{R}^n\times\mathbb{R}=\mathbb{R}^{n+1}$ and accordingly, we can define the spaces $C^{0,\alpha}(\partial\Omega^i\times\overline{\mathcal{J}})$ and $C^{1,\alpha}(\partial\Omega^i\times\overline{\mathcal{J}})$ by exploiting a finite atlas (see Section \ref{notation}). \begin{lemma}\label{lemmaboundB(ep,dot,dot)} Let $B$ be a function from $]-\epsilon_0,\epsilon_0[ \times \partial\Omega^i \times \mathbb{R}$ to $\mathbb{R}$. Let $\tilde{\mathcal{N}}_B$ be the map which takes a pair $(\epsilon,\zeta)\in]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to the function $\tilde{\mathcal{N}}_B (\epsilon,\zeta)$ defined by \[ \tilde{\mathcal{N}}_B(\epsilon,\zeta)(t) \equiv B(\epsilon,t,\zeta)\qquad\forall t\in \partial\Omega^i\,. \] Let $m\in\{0,1\}$. If $\tilde{\mathcal{N}}_B (\epsilon,\zeta)\in C^{m,\alpha}(\partial\Omega^i)$ for all $(\epsilon,\zeta)\in]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ and the map $\tilde{\mathcal{N}}_B$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{m,\alpha}(\partial\Omega^i)$, then for every open bounded interval $\mathcal{J}$ of $\mathbb{R}$ and every compact subset $\mathcal{E}$ of $]-\epsilon_0,\epsilon_0[$ there exists $C>0$ such that \begin{equation}\label{lemmaboundB(ep,dot,dot).eq1} \sup_{\epsilon \in \mathcal{E}} \|B(\epsilon,\cdot,\cdot)\|_{C^{m,\alpha}( \partial\Omega^i \times \overline{\mathcal{J}})} \leq C\,. \end{equation} \end{lemma} \begin{proof} Since $\partial\Omega^i$ is a compact sub-manifold of class $C^{1,\alpha}$ in $\mathbb{R}^n$, there exist a finite open covering $\mathcal{U}_1$, \dots, $\mathcal{U}_k$ of $\partial\Omega^i$ and $C^{1,\alpha}$ local parametrization maps $\gamma_l \colon \overline{B_{n-1}(0,1)} \to \overline{\mathcal{U}_l}$ with $l=1,\dots, k$. Moreover, we can assume without loss of generality that the norm of $C^{m,\alpha}( \partial\Omega^i)$ is defined on the atlas $\{(\overline{\mathcal{U}_l},\gamma^{(-1)}_l)\}_{l = 1, \dots ,k}$ and the norm of $C^{m,\alpha}( \partial\Omega^i\times \overline{\mathcal{J}})$ is defined on the atlas $\{(\overline{\mathcal{U}_l}\times\overline{\mathcal{J}},(\gamma^{(-1)}_l,\mathrm{id}_{\overline{\mathcal{J}}}))\}_{l = 1, \dots ,k}$, where $\mathrm{id}_{\overline{\mathcal{J}}}$ is the identity map from $\overline{\mathcal{J}}$ to itself. Then, in order to prove \eqref{lemmaboundB(ep,dot,dot).eq1} it suffices to show that \begin{equation}\label{lemmaboundB(ep,dot,dot).eq2} \sup_{\epsilon \in \mathcal{E}} \|B(\epsilon,\gamma_l(\cdot),\cdot)\|_{C^{m,\alpha}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})} \leq C\qquad\forall l\in\{1,\dots,k\} \end{equation} for some $C>0$. Let $l\in\{1,\dots,k\}$ and let $A$ be the map from $]-\epsilon_0,\epsilon_0[ \times \overline{B_{n-1}(0,1)} \times \mathbb{R} $ to $\mathbb{R}$ defined by \begin{equation}\label{lemmaboundB(ep,dot,dot).eq3} A(\epsilon,z,\zeta)=B(\epsilon,\gamma_l(z),\zeta)\qquad\forall (\epsilon,z,\zeta)\in]-\epsilon_0,\epsilon_0[ \times \overline{B_{n-1}(0,1)} \times \mathbb{R}\,. \end{equation} Then, with the notation of Lemma \ref{lemmaboundA(ep,dot,dot)}, we have \[ \mathcal{M}_A(\epsilon,\zeta)=\gamma_l^*\left(\tilde{\mathcal{N}}_B(\epsilon,\zeta)_{|\overline{\mathcal{U}_l}}\right), \] where $\gamma_l^*\left(\tilde{\mathcal{N}}_B(\epsilon,\zeta)_{|\overline{\mathcal{U}_l}}\right)$ is the pull back of the restriction $\tilde{\mathcal{N}}_B(\epsilon,\zeta)_{|\overline{\mathcal{U}_l}}$ by the parametrization $\gamma_l$. Since the restriction map from $C^{m,\alpha}(\partial\Omega^i)$ to $C^{m,\alpha}(\overline{\mathcal{U}_l})$ and the pullback map $\gamma_l^*$ from $C^{m,\alpha}(\overline{\mathcal{U}_l})$ to $C^{m,\alpha}(\overline{B_{n-1}(0,1)})$ are linear and continuous and since $\mathcal{N}_B$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{m,\alpha}( \partial\Omega^i)$, it follows that the map $\mathcal{M}_A$ is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{m,\alpha}( \overline{B_{n-1}(0,1)})$. Then Lemma \ref{lemmaboundA(ep,dot,dot)} implies that \begin{equation}\label{lemmaboundB(ep,dot,dot).eq4} \sup_{\epsilon \in \mathcal{E}} \|A(\epsilon,\cdot,\cdot)\|_{C^{m,\alpha}(\overline{B_{n-1}(0,1)} \times \overline{\mathcal{J}})} \leq C \end{equation} for some $C>0$. Now the validity of \eqref{lemmaboundB(ep,dot,dot).eq2} follows by \eqref{lemmaboundB(ep,dot,dot).eq3} and \eqref{lemmaboundB(ep,dot,dot).eq4}. The proof is complete. \end{proof} \subsubsection{The auxiliary maps $N$ and $S$} In the proof of our main Theorem \ref{Thmunisol2} we will exploit two auxiliary maps, which we denote by $N$ and $S$ and are defined as follows. Let $\epsilon'$ be as in Theorem \ref{existenceThm} (v). We denote by $N=(N_1,N_2,N_3)$ the map from $]-\epsilon',\epsilon'[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)$ defined by \begin{align} \label{N1} \nonumber & N_1[\epsilon,\phi^o, \phi^i, \zeta,\psi^i](x) \equiv \left(\frac{1}{2}I + W_{\partial\Omega^o}\right)[\phi^o](x) \\ & \quad - \epsilon^{n-1} \int_{\partial\Omega^i}{\nu_{\Omega^i}(y) \cdot \nabla S_n(x-\epsilon y) \phi^i(y) \,d\sigma_y} + \epsilon^{n-2}\zeta S_n(x) && \forall x \in \partial\Omega^o, \\ \label{N2} \nonumber &N_2[\epsilon,\phi^o, \phi^i, \zeta,\psi^i](t) \equiv \left(-\frac{1}{2}I + W_{\partial\Omega^i}\right)[\phi^i](t) +\zeta\,S_n(t) \\ &\quad + w^+_{\Omega^o}[\phi^o](\epsilon t) - (\partial_\zeta F) (0,t,\zeta^i) \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) && \forall t \in \partial\Omega^i, \\ \label{N3}\nonumber &N_3[\epsilon,\phi^o, \phi^i, \zeta,\psi^i](t)\equiv \nu_{\Omega^i}(t) \cdot \left( \epsilon \nabla w^+_{\Omega^o}[\phi^o](\epsilon t)\right.\\ &\left. \qquad\qquad\qquad\quad\qquad + \nabla w^-_{\Omega^i}[\phi^i](t) + \zeta\nabla S_n(t) - \nabla w^+_{\Omega^i}[\psi^i](t) \right) && \forall t \in \partial\Omega^i, \end{align} for all $(\epsilon, \phi^o, \phi^i, \zeta, \psi^i) \in ]-\epsilon',\epsilon'[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ and we denote by $S=(S_1,S_2,S_3)$ the map from $ ]-\epsilon',\epsilon'[ \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)$ defined by \begin{align} \label{S1} &S_1[\epsilon,\psi^i] (x) \equiv 0 && \forall x \in \partial \Omega^o, \\ \label{S2} \nonumber &S_2[\epsilon,\psi^i] (t) \equiv - t \cdot \nabla \mathfrak{u}^o(0) - \epsilon \mathfrak{u}^o(\epsilon t) + (\partial_\epsilon F) (0,t,\zeta^i) \\ & \quad+ \epsilon \tilde{F}\left(\epsilon,t,\zeta^i,\left(\frac{1}{2}I + W_{\partial\Omega^i}\right) [\psi^i](t)\right) && \forall t \in \partial\Omega^i, \\ \label{S3}\nonumber &S_3[\epsilon,\psi^i] (t)\\ &\equiv - \nu_{\Omega^i}(t) \cdot \nabla \mathfrak{u}^o(\epsilon t) + G \left(\epsilon,t,\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) + \zeta^i \right) && \forall t \in \partial\Omega^i, \end{align} for all $(\epsilon,\psi^i) \in ]-\epsilon',\epsilon'[ \times C^{1,\alpha}(\partial\Omega^i)$. For the maps $N$ and $S$ we have the following result. \begin{prop}\label{N^-1} Let assumptions \eqref{zetaicond}, \eqref{F1}, and \eqref{realanalhp} hold true. Then there exists $\epsilon'' \in \, ]0,\epsilon'[$ such that the following statements hold: \begin{enumerate} \item[(i)] For all fixed $\epsilon\in]-\epsilon'',\epsilon''[$ the operator $N[\epsilon,\cdot,\cdot,\cdot,\cdot]$ is a linear homeomorphism from $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)$; \item[(ii)] The map from $]-\epsilon'',\epsilon''[$ to $\mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i),\\ C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i))$ which takes $\epsilon$ to $N[\epsilon,\cdot,\cdot,\cdot,\cdot]^{(-1)}$ is real analytic; \item[(iii)] Equation \eqref{Me=0.e1} is equivalent to \begin{equation}\label{N^-1S} (\phi^o, \phi^i, \zeta,\psi^i) = N[\epsilon,\cdot,\cdot,\cdot,\cdot]^{(-1)} [ S[\epsilon,\psi^i] ] \end{equation} for all $(\epsilon, \phi^o, \phi^i, \zeta, \psi^i) \in ]-\epsilon'',\epsilon''[ \times C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$. \end{enumerate} \end{prop} \begin{proof} By the definition of $N$ (cf. \eqref{N1}--\eqref{N3}), by the mapping properties of the double layer potential (cf. Theorem \ref{doublepot} (ii)-(iii)) and of integral operators with real analytic kernels and no singularity (see, e.g., Lanza de Cristoforis and Musolino \cite{LaMu13}), by assumption \eqref{realanalhp} (which implies that $(\partial_\zeta F) (0,\cdot,\zeta^i)$ belongs to $C^{1,\alpha}(\partial\Omega^i)$), and by standard calculus in Banach spaces, one verifies that the map from $]-\epsilon',\epsilon'[$ to \[ \mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)\,,\;C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)) \] which takes $\epsilon$ to $N[\epsilon,\cdot,\cdot,\cdot,\cdot]$ is real analytic. Then one observes that \begin{equation*} N[0,\phi^o, \phi^i, \zeta,\psi^i] = \partial_{(\phi^o,\phi^i,\zeta,\psi^i)} M[0,\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0].(\phi^o, \phi^i, \zeta,\psi^i) \end{equation*} and thus Theorem \ref{existenceThm} (iv) implies that $N[0,\cdot,\cdot,\cdot,\cdot]$ is an isomorphism from $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ to $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{1,\alpha}(\partial\Omega^i)$. Since the set of invertible operators is open in $\mathcal{L}(C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i), C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i))$ and since the map which takes a linear invertible operator to its inverse is real analytic (cf.~Hille and Phillips \cite{HiPi57}), we deduce the validity of $(i)$ and $(ii)$. To prove $(iii)$ we observe that, by the definition of $N$ and $S$ in \eqref{N1}--\eqref{S3} it readily follows that \eqref{Me=0.e1} is equivalent to \begin{equation*} N[\epsilon,\phi^o, \phi^i, \zeta,\psi^i] = S[\epsilon,\psi^i]\,. \end{equation*} Then the validity of $(iii)$ is a consequence of $(i)$. \end{proof} \subsubsection{The main theorem} We are now ready to prove our main Theorem \ref{Thmunisol2} on the local uniqueness of the solution $(u^o_\epsilon,u^i_\epsilon)$ provided by Theorem \ref{uesol}. \begin{teo}\label{Thmunisol2} Let assumptions \eqref{zetaicond}, \eqref{F1}, and \eqref{realanalhp} hold true. Let $\epsilon' \in ]0,\epsilon_0[$ be as in Theorem \ref{existenceThm} (v). Let $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$ be as in Theorem \ref{uesol}. Then there exist $\epsilon^\ast \in ]0,\epsilon'[$ and $\delta^\ast \in ]0,+\infty[$ such that the following property holds: If $\epsilon \in ]0,\epsilon^\ast[$ and $(v^o,v^i) \in C^{1,\alpha}(\Omega(\epsilon)) \times C^{1,\alpha}(\epsilon\Omega^i)$ is a solution of problem \eqref{princeq} with \[ \left\| {v^i(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot )} \right\| _{C^{1,\alpha}(\partial \Omega^i)} < \epsilon\delta^\ast, \] then \[ (v^o,v^i)=(u^o_\epsilon,u^i_\epsilon)\,. \] \end{teo} \begin{proof} {\it $\bullet$ Step 1: Fixing $\epsilon^\ast$.} \\ Let $\epsilon'' \in \, ]0,\epsilon'[$ be as in Proposition \ref{N^-1} and let $\epsilon'''\in]0,\epsilon''[$ be fixed. By the compactness of $[-\epsilon''',\epsilon''']$ and by the continuity of the norm in $\mathcal{L}(C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i) \times C^{1,\alpha}(\partial\Omega^o), C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i))$, there exists a real number $C_1>0$ such that \begin{equation}\label{unicconstantN-1} \begin{split} \| N[\epsilon,\cdot,\cdot,\cdot,\cdot]^{(-1)} \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i) \times C^{1,\alpha}(\partial\Omega^o), C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i))}&\\ \le C_1& \end{split} \end{equation} for all $\epsilon \in [-\epsilon''',\epsilon''']$ (see also Proposition \ref{N^-1} (ii)). Let $U_0$ be the open neighborhood of $(\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)$ in $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ introduced in Theorem \ref{existenceThm} (v). Then we take $K>0$ such that \[ \overline{\mathcal{B}_{0,K}}\subseteq U_0 \] (see \eqref{B0r} for the definition of $\mathcal{B}_{0,K}$). Since $(\Phi^o[\cdot],\Phi^i[\cdot],Z[\cdot],\Psi^i[\cdot])$ is continuous (indeed real analytic) from $]-\epsilon',\epsilon'[$ to $U_0$, there exists $\epsilon^* \in ]0,\epsilon'''[$ such that \begin{equation}\label{PhiBK/2} (\Phi^o[\eta] , \Phi^i[\eta],Z[\eta] , \Psi^i[\eta])\in \mathcal{B}_{0,K/2}\subset U_0\qquad\forall \eta \in ]0,\epsilon^*[\,. \end{equation} Moreover, we assume that \begin{equation*} \epsilon^\ast < 1. \end{equation*} We will prove that the theorem holds for such choice of $\epsilon^\ast$. We observe that the condition $\epsilon^\ast <1$ is not really needed in the proof but simplifies many computations. \\ {\it $\bullet$ Step 2: Planning our strategy.} \\ We suppose that there exists a pair of functions $(v^o,v^i) \in C^{1,\alpha}(\overline{\Omega(\epsilon)}) \times C^{1,\alpha}(\epsilon \overline{\Omega^i})$ that is a solution of problem \eqref{princeq} for a certain $\epsilon \in ]0,\epsilon^\ast[$ (fixed) and such that \begin{equation}\label{unidisuguiesp} \left\| \frac{v^i(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot )}{\epsilon} \right\| _{C^{1,\alpha}(\partial \Omega^i)} \leq \delta^\ast\,, \end{equation} for some $\delta^\ast \in ]0,+\infty[$. Then, by Proposition \ref{rappresentsol}, there exists a unique quadruple $(\phi^o,\phi^i,\zeta,\psi^i) \in C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ such that \begin{equation}\label{unicitauouirappr} \begin{aligned} & v^o = U^o_{\epsilon}[\phi^o,\phi^i,\zeta,\psi^i] && \text{in } \overline{\Omega(\epsilon)}, \\ & v^i = U^i_{\epsilon}[\phi^o,\phi^i,\zeta,\psi^i] && \text{in } \epsilon \overline{\Omega^i}. \end{aligned} \end{equation} We shall show that for $\delta^\ast$ small enough we have \begin{equation}\label{uniequalpsiPsi} (\phi^o,\phi^i,\zeta,\psi^i) = (\Phi^o[\epsilon] , \Phi^i[\epsilon], Z[\epsilon] , \Psi^i[\epsilon])\,. \end{equation} Indeed, if we have \eqref{uniequalpsiPsi}, then Theorem \ref{uesol} would imply that \begin{equation*} (v^o,v^i) = (u^o_\epsilon,v^i_\epsilon), \end{equation*} and our proof would be completed. Moreover, to prove \eqref{uniequalpsiPsi} it suffices to show that \begin{equation}\label{unidensitiesinB0K} (\phi^o,\phi^i,\zeta,\psi^i) \in \mathcal{B}_{0,K} \subset U_0\,. \end{equation} In fact, in that case, both $(\epsilon,\phi^o,\phi^i,\zeta,\psi^i)$ and $(\epsilon,\Phi^o[\epsilon] , \Phi^i[\epsilon], Z[\epsilon] , \Psi^i[\epsilon])$ would stay in the zero set of $M$ (cf. Theorem \ref{existenceThm} (ii)) and thus \eqref{unidensitiesinB0K} together with \eqref{PhiBK/2} and Theorem \ref{existenceThm} (v) would imply \eqref{uniequalpsiPsi}. So, our aim is now to prove that \eqref{unidensitiesinB0K} holds true for a suitable choice of $\delta^*>0$. It will be also convenient to restrict our search to \[ 0<\delta^*<1. \] As for the condition $\epsilon^*<1$, this condition on $\delta^*$ is not really needed, but simplifies our computations. Then to find $\delta^*$ and prove \eqref{unidensitiesinB0K} we will proceed as follows. First we obtain an estimate for $\psi^i$ and $\Psi^i[\epsilon]$ with a bound that does not depend on $\epsilon$ and $\delta^*$. Then we use such estimate to show that \begin{equation*} \| S[\epsilon,\psi^i] - S[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)} \end{equation*} is smaller than a constant times $\delta^*$, with a constant that does not depend on $\epsilon$ and $\delta^*$. We will split the analysis for $S_1$, $S_2$, and $S_3$ and we find convenient to study $S_3$ before $S_2$. Indeed, the computations for $S_2$ and $S_3$ are very similar but those for $S_3$ are much shorter and can serve better to illustrate the techniques employed. We also observe that the analysis for $S_2$ requires the study of other auxiliary functions $T_1$, $T_2$, and $T_3$ that we will introduce. Finally, we will exploit the estimate for $\| S[\epsilon,\psi^i] - S[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)}$ to determine $\delta^*$ and prove \eqref{unidensitiesinB0K}. \\ {\it $\bullet$ Step 3: Estimate for $\psi$ and $\Psi[\epsilon]$.} \\ By condition \eqref{unidisuguiesp}, by the second equality in \eqref{unicitauouirappr}, by Theorem \ref{uesol}, and by arguing as in \eqref{(u^i-u_e^i)} and \eqref{psi^i} in Theorem \ref{Thmunisol1}, we obtain \begin{equation}\label{inequality(1/2I + W)psiPsi} \left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)} \leq \, \delta^\ast \end{equation} and \begin{equation}\label{inequalitypsiPsi} \begin{split} \|& \psi^i - \Psi^i[\epsilon] \|_{C^{1,\alpha}(\partial\Omega^i)} \leq \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)^{(-1)} \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} \\ &\times\left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)} \leq C_2 \delta^\ast, \end{split} \end{equation} where \begin{equation*} C_2 \equiv \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)^{(-1)} \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))}. \end{equation*} By \eqref{PhiBK/2} we have \begin{equation}\label{psi-Psi} \|\psi^i_0 - \Psi^i[\eta]\|_{C^{1,\alpha}(\partial\Omega^i)} \leq \frac{K}{2} \qquad \forall \eta \in ]0,\epsilon^*[. \end{equation} Then, by \eqref{inequalitypsiPsi} and \eqref{psi-Psi}, and by the triangular inequality, we see that \begin{equation*} \begin{aligned} \|\psi^i \|_{C^{1,\alpha}(\partial\Omega^i)} & \leq \|\psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)} + \|\psi^i - \Psi^i[\epsilon] \|_{C^{1,\alpha}(\partial\Omega^i)} + \|\Psi^i[\epsilon] - \psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq \|\psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)} + C_2\, \delta^\ast + \frac{K}{2}, \\ \|\Psi^i[\epsilon]\|_{C^{1,\alpha}(\partial\Omega^i)}& \leq \|\psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)}+ \frac{K}{2}. \end{aligned} \end{equation*} Then, by taking $R_1 \equiv \|\psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)} + C_2+\frac{K}{2}$ and $R_2 \equiv \|\psi^i_0\|_{C^{1,\alpha}(\partial\Omega^i)} + \frac{K}{2}$ (and recalling that $\delta^\ast \in ]0,1[$), one verifies that \begin{equation}\label{uniccostantR_1R_2} \|\psi^i \|_{C^{1,\alpha}(\partial\Omega^i)} \leq R_1\qquad\text{and}\qquad \|\Psi^i[\epsilon]\|_{C^{1,\alpha}(\partial\Omega^i)} \leq R_2\,. \end{equation} We note here that both $R_1$ and $R_2$ do not depend on $\epsilon$ and $\delta^\ast$ as long they belong to $]0,\epsilon^\ast[$ and $]0,1[$, respectively. \smallskip {\it $\bullet$ Step 4: Estimate for $S_1$.} \\ We now pass to estimate the norm \begin{equation*} \| S[\epsilon,\psi^i] - S[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i)}. \end{equation*} To do so we consider separately $S_1$, $S_2$, and $S_3$. Since $S_1=0$ (cf. definition \eqref{S1}), we readily obtain that \begin{equation}\label{unicS1inqual} \| S_1[\epsilon, \psi^i] - S_1[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^o)} = 0. \end{equation} \smallskip {\it $\bullet$ Step 5: Estimate for $S_3$.} \\ We consider $S_3$ before $S_2$ because its treatment is simpler and more illustrative of the techniques used. By \eqref{S3} and by the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi \cite[Thm. 1.8]{AmPr95}), we compute that \begin{equation}\label{|S3|} \begin{split} &\| S_3[\epsilon, \psi^i] - S_3[\epsilon,\Psi^i[\epsilon]] \|_{C^{0,\alpha}(\partial\Omega^i)} \\ &\quad = \left\| G \left(\epsilon,\cdot,\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i \right)\right.\\ &\left.\qquad - G \left(\epsilon,\cdot,\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] + \zeta^i \right) \right\|_{C^{0,\alpha}(\partial\Omega^i)} \\ &\quad = \left\| \mathcal{N}_G\left(\epsilon, \epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right)\right.\\ &\left.\qquad - \mathcal{N}_G\left(\epsilon, \epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] + \zeta^i\right)\right\|_{C^{0,\alpha}(\partial\Omega^i)} \\ &\quad \leq \left\|d_v \mathcal{N}_G (\epsilon,\tilde{\psi}^i) \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{0,\alpha}(\partial\Omega^i))} \\ &\qquad \times\epsilon\left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]\right\|_{C^{0,\alpha}(\partial\Omega^i)}, \end{split} \end{equation} where \begin{equation*} \tilde{\psi}^i = \theta \left(\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i \right) + (1-\theta) \left(\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] +\zeta^i \right), \end{equation*} for some $\theta \in ]0,1[$. Then, by the membership of $\epsilon$ and $\theta$ in $]0,1[$ we have \[ \begin{split} \|\tilde{\psi}^i \|_{C^{1,\alpha}(\partial\Omega^i)} \leq &\left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i]+ \zeta^i\right\|_{C^{1,\alpha}(\partial\Omega^i)}\\ & + \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] +\zeta^i\right\|_{C^{1,\alpha}(\partial\Omega^i)} \end{split} \] and, by setting \[ C_3 \equiv \left\|\frac{1}{2}I + W_{\partial\Omega^i} \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))}\,, \] we obtain \begin{equation}\label{boundtildepsiiesp} \|\tilde{\psi}^i \|_{C^{1,\alpha}(\partial\Omega^i)} \leq C_3\|\psi^i\|_{C^{1,\alpha}(\partial\Omega^i)} + C_3\|\Psi^i[\epsilon]\|_{C^{1,\alpha}(\partial\Omega^i)} + 2 |\zeta^i| \leq R, \end{equation} with \begin{equation}\label{uniccostantR} R \equiv C_3(R_1 + R_2) + 2|\zeta^i| \end{equation} which does not depend on $\epsilon$. We wish now to estimate the operator norm \[ \left\|d_v \mathcal{N}_G (\eta,\tilde{\psi}^i) \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{0,\alpha}(\partial\Omega^i))} \] uniformly for $\eta \in ]0,\epsilon^\ast[$. However, we cannot exploit a compactness argument on $[0,\epsilon^*]\times \overline{B_{C^{1,\alpha}(\partial\Omega^i)}(0,R)}$, because $\overline{B_{C^{1,\alpha}(\partial\Omega^i)}(0,R)}$ is not compact in the infinite dimensional space $C^{1,\alpha}(\partial\Omega^i)$. Then we argue as follows. We observe that, by assumption \eqref{realanalhp}, the partial derivative $\partial_\zeta G(\eta,t,\zeta)$ exists for all $(\eta,t,\zeta) \in ]-\epsilon_0,\epsilon_0[ \times \partial\Omega^i \times \mathbb{R}$ and, by Proposition \ref{differenzialeN_H} and Lemma \ref{C0C1alphaalgebra} (i) in the Appendix, we obtain that \begin{equation}\label{norm.1} \begin{split} \|d_v \mathcal{N}_G (\eta,\tilde{\psi}^i) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{0,\alpha}(\partial\Omega^i))}& \leq \|\mathcal{N}_{\partial_\zeta G} (\eta,\tilde{\psi}^i) \|_{C^{0,\alpha}(\partial\Omega^i)} \\ & \leq \|\partial_\zeta G (\eta, \cdot, \tilde{\psi}^i (\cdot))\|_{C^{0,\alpha}(\partial\Omega^i)} \end{split} \end{equation} for all $\eta \in ]0,\epsilon^\ast[$. By Proposition \ref{proponcompC1alpha} (ii) in the Appendix, there exists $C_4 >0$ such that \begin{equation}\label{norm.1.1} \begin{split} &\|\partial_\zeta G (\eta, \cdot, \tilde{\psi}^i (\cdot))\|_{C^{0,\alpha}(\partial\Omega^i)}\\ &\quad\le C_4 \|\partial_\zeta G (\eta, \cdot,\cdot)\|_{C^{0,\alpha}(\partial\Omega \times [-R,R])} \left(1 + \|\tilde{\psi}^i\|^\alpha_{C^{1,\alpha}(\partial\Omega^i)} \right) \quad \forall \eta \in ]0,\epsilon^\ast[. \end{split} \end{equation} Moreover, by assumption \eqref{realanalhp} one deduces that the map $\tilde{\mathcal{N}}_G$ defined as in Lemma \ref{lemmaboundB(ep,dot,dot)} (with $B=G$) is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{0,\alpha}(\partial\Omega^i)$ and, by Proposition \ref{differenzialeN_H}, one has that $\partial_\zeta \tilde{\mathcal{N}}_G = \tilde{\mathcal{N}}_{\partial_\zeta G}$. Hence, by Lemma \ref{lemmaboundB(ep,dot,dot)} (with $m=0$), there exists $C_5>0$ (which does not depend on $\epsilon \in ]0,\epsilon^\ast[$ and $\delta^\ast \in ]0,1[$) such that \begin{equation}\label{norm.2} \sup_{\eta \in [-\epsilon^\ast,\epsilon^\ast]} \|\partial_\zeta G (\eta, \cdot,\cdot)\|_{C^{0,\alpha}(\partial\Omega^i \times [-R,R])} \leq C_5. \end{equation} Hence, by \eqref{boundtildepsiiesp}, \eqref{norm.1}, \eqref{norm.1.1} and \eqref{norm.2}, we deduce that \begin{equation}\label{norm.3} \|d_v \mathcal{N}_G (\epsilon,\tilde{\psi}^i) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{0,\alpha}(\partial\Omega^i))} \le C_4\, C_5\,(1+ R^\alpha). \end{equation} By \eqref{inequality(1/2I + W)psiPsi}, \eqref{|S3|}, and \eqref{norm.3}, and by the membership of $\epsilon$ in $]0,\epsilon^\ast[ \subset \,]0,1[$, we obtain that \begin{equation}\label{unicS2inequal} \| S_3[\epsilon, \psi^i] - S_3[\epsilon,\Psi^i[\epsilon]] \|_{C^{0,\alpha}(\partial\Omega^i)} \leq C_4\, C_5\, (1+ R^\alpha)\; \delta^\ast. \end{equation} \smallskip {\it $\bullet$ Step 6: Estimate for $S_2$.} \\ Finally, we consider $S_2$. By \eqref{S2} and by the fact that $\epsilon \in]0,1[$, we have {\small\begin{equation}\label{s1-S1<A123} \begin{split} &\| S_2[\epsilon, \psi^i] - S_2[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq \epsilon \left\|\tilde{F}\left(\epsilon, \cdot,\zeta^i,\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] \right) - \tilde{F}\left(\epsilon,\cdot,\zeta^i,\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq \left\| \int_{0}^{1} (1-\tau) \left\{ T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) + 2 \, T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) \right.\right.\\ &\left.\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad + T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) \right\} \,d\tau \right\|_{C^{1,\alpha}(\partial\Omega^i)}, \end{split} \end{equation}} where $T_1[\epsilon, \psi^i, \Psi^i[\epsilon]]$, $T_2[\epsilon, \psi^i, \Psi^i[\epsilon]]$, and $T_3[\epsilon, \psi^i, \Psi^i[\epsilon]]$ are the functions from $]0,1[ \times \partial\Omega^i$ to $\mathbb{R}$ defined by \begin{align} \label{A1} \nonumber &T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,t)\\ \nonumber &\equiv (\partial^2_\epsilon F)\left(\tau\epsilon,t, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) + \zeta^i \right) \\ & \quad- (\partial^2_\epsilon F)\left(\tau\epsilon,t, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]](t) + \zeta^i \right) , \\ \label{A2} \nonumber &T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,t)\\ \nonumber &\equiv \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) \, (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,t, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) + \zeta^i \right) \\& \quad- \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]](t) (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,t,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]](t) + \zeta^i \right), \\ \label{A3} \nonumber &T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,t)\\ \nonumber &\equiv \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i]^2(t) \, (\partial^2_\zeta F)\left(\tau\epsilon,t, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] (t) +\zeta^i \right) \\ & \quad- \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2(t) (\partial^2_\zeta F)\left(\tau\epsilon,t,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right) [\Psi^i[\epsilon] ] (t) + \zeta^i \right) \end{align} for every $(\tau,t) \in ]0,1[ \times \partial \Omega^i$. We now want to bound the $C^{1,\alpha}$ norm with respect to the variable $t \in \partial\Omega^i$ of \eqref{A1}, \eqref{A2}, and \eqref{A3} uniformly with respect to $\tau \in ]0,1[$. \smallskip {\it $\bullet$ Step 6.1: Estimate for $T_1$.} \\ First we consider $T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$. By the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi \cite[Thm. 1.8]{AmPr95}), we can estimate the $C^{1,\alpha}(\partial\Omega^i)$ norm of $T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$ (cf.~\eqref{A1}) as follows: \begin{equation}\label{unicA1equationnorm} \begin{split} &\|T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \qquad= \left\| \mathcal{N}_{\partial^2_\epsilon F}\left(\tau\epsilon, \tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right)\right.\\ &\left.\qquad\qquad - \mathcal{N}_{\partial^2_\epsilon F}\left(\tau\epsilon, \tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] + \zeta^i\right)\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \qquad\leq \left\|d_v \mathcal{N}_{\partial^2_\epsilon F} (\tau\epsilon,\tilde{\psi}^i_1)\right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} \\ &\qquad\qquad\times\tau\epsilon \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]\right\|_{C^{1,\alpha}(\partial\Omega^i)} \end{split} \end{equation} where \begin{equation*} \begin{split} &\tilde{\psi}^i_{1}\\ & = \theta_1 \left(\tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) + (1-\theta_1) \left(\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] +\zeta^i\right), \end{split} \end{equation*} for some $\theta_1 \in ]0,1[$. \smallskip {\it $\bullet$ Step 6.2: Estimate for $T_2$.} \\ We now consider $T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$. Adding and subtracting \begin{equation*} \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \, (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,t,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) + \zeta^i \right) \end{equation*} in the right hand side of \eqref{A2} and using the triangular inequality, we obtain {\small \begin{equation}\label{unicA2equationnorm1} \begin{split} \|&T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)}\\ & \leq \left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] \, (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) \right. \\ & \left. - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \, (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] +\zeta^i \right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & + \left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \, (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i \right) \right. \\ & \left. -\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] +\zeta^i \right) \right\|_{C^{1,\alpha}(\partial\Omega^i)}. \end{split} \end{equation}} By Lemma \ref{C0C1alphaalgebra} in the Appendix and by the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi \cite[Thm. 1.8]{AmPr95}), we can estimate the $C^{1,\alpha}(\partial\Omega^i)$ norm of $T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$ (cf.~\eqref{A2} and \eqref{unicA2equationnorm1}) as follows: \begin{equation}\label{unicA2equationnorm} \begin{split} &\|T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq 2 \left\| (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad\times \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad+ 2 \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad\times\left\| \mathcal{N}_{\partial_\epsilon\partial_\zeta F}\left(\tau\epsilon, \tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right)\right.\\ &\left.\qquad\qquad - \mathcal{N}_{\partial_\epsilon\partial_\zeta F}\left(\tau\epsilon, \tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] + \zeta^i \right)\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq 2 \left\| (\partial_\epsilon \partial_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i \right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\& \quad\times \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad+ 2\, C_3\, \| \Psi^i[\epsilon] \|_{C^{1,\alpha}(\partial\Omega^i)} \left\|d_v \mathcal{N}_{\partial_\epsilon\partial_\zeta F} (\tau\epsilon,\tilde{\psi}^i_{2}) \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} \\ & \quad\times \tau\epsilon \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]\right\|_{C^{1,\alpha}(\partial\Omega^i)}, \end{split} \end{equation} where \begin{equation*} \begin{split} \tilde{\psi}^i_{2}=& \theta_2 \left(\tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right)\\ & + (1-\theta_2) \left(\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] +\zeta^i\right), \end{split} \end{equation*} for some $\theta_2 \in ]0,1[$. \smallskip {\it $\bullet$ Step 6.3: Estimate for $T_3$.} \\ Finally we consider $T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$. Adding and subtracting the term \begin{equation*} \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2 \, (\partial^2_\zeta F)\left(\tau\epsilon,t, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i](t) + \zeta^i \right) \end{equation*} in the right hand side of \eqref{A3} and using the triangular inequality, we obtain \begin{equation}\label{unicA3equationnorm1} \begin{split} \|&T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)}\\ & \leq \left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i]^2 \, (\partial^2_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i ] + \zeta^i\right) \right. \\ & \left. - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2 \, (\partial^2_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & + \left\| \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2 \, (\partial^2_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i ]+\zeta^i \right) \right. \\ & \left.- \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2 (\partial^2_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right) [\Psi^i[\epsilon] ] +\zeta^i\right)\right\|_{C^{1,\alpha}(\partial\Omega^i)}\,. \end{split} \end{equation} By Lemma \ref{C0C1alphaalgebra} in the Appendix and by the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi \cite[Thm. 1.8]{AmPr95}), we can estimate the $C^{1,\alpha}(\partial\Omega^i)$ norm of $T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$ (cf. \eqref{A3} and \eqref{unicA3equationnorm1}) as follows: \begin{equation}\label{unicA3equationnorm} \begin{split} \|T_3&[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq 2 \left\| (\partial^2_\zeta F)\left(\tau\epsilon,\cdot, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad\times\left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i]^2 - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2 \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad+ 2 \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]]^2\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad\times\left\| \mathcal{N}_{\partial^2_\zeta F}\left(\tau\epsilon, \tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] +\zeta^i \right)\right.\\ & \left.\qquad - \mathcal{N}_{\partial^2_\zeta F}\left(\tau\epsilon, \tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] + \zeta^i \right)\right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq 4 \left\| (\partial^2_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i \right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad\times\left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad\times\left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \quad+ 4 C_3^2 \, \|\Psi^i[\epsilon]\|^2_{C^{1,\alpha}(\partial\Omega^i)} \left\|d_v \mathcal{N}_{\partial^2_\zeta F} (\tau\epsilon,\tilde{\psi}^i_{3}) \right\|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} \\ & \quad\times \tau \epsilon \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] - \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)}, \end{split} \end{equation} where \begin{equation*} \begin{split} \tilde{\psi}^i_{3} =& \theta_3 \left(\tau\epsilon\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right)\\ & + (1-\theta_3) \left(\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] +\zeta^i\right), \end{split} \end{equation*} for some $\theta_3 \in ]0,1[$. Let $R$ be as in \eqref{uniccostantR}. By the same argument used to prove \eqref{boundtildepsiiesp}, one verifies the inequalities \begin{equation}\label{boundoverpsiiesp} \|\tilde{\psi}^i_{1}\|_{C^{1,\alpha}(\partial\Omega^i)} \leq R, \quad \|\tilde{\psi}^i_{2}\|_{C^{1,\alpha}(\partial\Omega^i)} \leq R, \quad \|\tilde{\psi}^i_{3}\|_{C^{1,\alpha}(\partial\Omega^i)} \leq R. \end{equation} By assumption \eqref{realanalhp}, the partial derivatives $\partial_\zeta\partial^2_\epsilon F (\eta,t,\zeta)$, $\partial_\zeta\partial_\epsilon\partial_\zeta F (\eta,t,\zeta)$ and $\partial_\zeta\partial^2_\zeta F (\eta,t,\zeta)$ exist for all $(\eta,t,\zeta) \in ]-\epsilon_0,\epsilon_0[ \times \partial\Omega^i \times \mathbb{R}$ and by Proposition \ref{differenzialeN_H} and Lemma \ref{C0C1alphaalgebra} (ii) in the Appendix, we obtain \begin{equation}\label{norms.1} \begin{aligned} \| d_v \mathcal{N}_{\partial^2_\epsilon F } (\tau\eta, \tilde{\psi}^i_{1}) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} &\leq 2 \| \mathcal{N}_{\partial_\zeta\partial^2_\epsilon F } (\tau\eta, \tilde{\psi}^i_{1}) \|_{C^{1,\alpha}(\partial\Omega^i)} \\ & = 2\, \|\partial_\zeta \partial^2_\epsilon F(\tau\eta, \cdot, \tilde{\psi}^i_{1}(\cdot) ) \|_{C^{1,\alpha}(\partial\Omega^i)}\,, \\ \| d_v \mathcal{N}_{\partial_\epsilon \partial_\zeta F } (\tau\eta, \tilde{\psi}^i_{2}) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} & \leq 2\| \mathcal{N}_{\partial_\zeta\partial_\epsilon \partial_\zeta F } (\tau\eta, \tilde{\psi}^i_{2}) \|_{C^{1,\alpha}(\partial\Omega^i)} \\ & = 2\, \|\partial_\zeta \partial_\epsilon \partial_\zeta F(\tau\eta, \cdot, \tilde{\psi}^i_{2}(\cdot)) \|_{C^{1,\alpha}(\partial\Omega^i)}\,, \\ \| d_v \mathcal{N}_{\partial^2_\zeta F } (\tau\eta, \tilde{\psi}^i_{3}) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} &\leq \| \mathcal{N}_{\partial_\zeta\partial^2_\zeta F } (\tau\eta, \tilde{\psi}^i_{3}) \|_{C^{1,\alpha}(\partial\Omega^i)} \\ & = 2\|\partial_\zeta \partial^2_\zeta F(\tau\eta, \cdot, \tilde{\psi}^i_{3}(\cdot)) \|_{C^{1,\alpha}(\partial\Omega^i)}\,, \end{aligned} \end{equation} for all $ \eta \in ]0,\epsilon^\ast[$. By Proposition \ref{proponcompC1alpha} (ii) in the Appendix, there exists $C_6 > 0$ such that \begin{equation}\label{norms.1.1} \begin{aligned} &\|\partial_\zeta \partial^2_\epsilon F(\tau\eta, \cdot, \tilde{\psi}^i_{1}(\cdot) ) \|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\qquad\qquad\leq C_6 \|\partial_\zeta \partial^2_\epsilon F(\tau\eta, \cdot, \cdot) \|_{C^{1,\alpha}(\partial\Omega \times [-R,R])} \left(1+ \|\tilde{\psi}^i_{1}\|_{C^{1,\alpha}(\partial\Omega^i)}\right)^2, \\ &\|\partial_\zeta \partial_\epsilon \partial_\zeta F(\tau\eta, \cdot, \tilde{\psi}^i_{2}(\cdot) ) \|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\qquad\qquad\leq C_6 \|\partial_\zeta \partial_\epsilon \partial_\zeta F(\tau\eta, \cdot, \cdot ) \|_{C^{1,\alpha}(\partial\Omega^i \times [-R,R])} \left(1+ \|\tilde{\psi}^i_{2}\|_{C^{1,\alpha}(\partial\Omega^i)}\right)^2, \\ &\|\partial_\zeta \partial^2_\zeta F(\tau\eta, \cdot, \tilde{\psi}^i_{3}(\cdot) ) \|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\qquad\qquad\leq C_6 \|\partial_\zeta \partial^2_\zeta F(\tau\eta, \cdot, \cdot ) \|_{C^{1,\alpha}(\partial\Omega^i \times [-R,R])} \left(1+ \|\tilde{\psi}^i_{3}\|_{C^{1,\alpha}(\partial\Omega^i)}\right)^2, \end{aligned} \end{equation} for all $ \eta \in ]0,\epsilon^\ast[$. Now, by assumption \eqref{realanalhp} one deduces that the map $\tilde{\mathcal{N}}_F$ defined as in Lemma \ref{lemmaboundB(ep,dot,dot)} (with $B=F$) is real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^i)$. Then one verifies that also the maps \begin{equation*} \begin{split} &\partial^2_\epsilon \partial_\zeta \tilde{\mathcal{N}}_F = \tilde{\mathcal{N}}_{\partial^2_\epsilon \partial_\zeta F}, \quad \partial_\epsilon \partial^2_\zeta \tilde{\mathcal{N}}_F = \tilde{\mathcal{N}}_{\partial_\epsilon \partial^2_\zeta F},\quad \partial^3_\zeta \tilde{\mathcal{N}}_F = \tilde{\mathcal{N}}_{\partial^3 F}\,,\\ &\partial_\epsilon \partial_\zeta \tilde{\mathcal{N}}_F = \tilde{\mathcal{N}}_{\partial_\epsilon \partial_\zeta F}, \quad \partial^2_\zeta \tilde{\mathcal{N}}_F = \tilde{\mathcal{N}}_{\partial^2_\zeta F} \end{split} \end{equation*} are real analytic from $]-\epsilon_0,\epsilon_0[ \times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega^i)$. Hence, Lemma \ref{lemmaboundB(ep,dot,dot)} (with $m=1$) implies that there exists $C_7>0$ such that \begin{equation}\label{norms.2} \begin{aligned} &\sup_{\eta \in [-\epsilon^\ast,\epsilon^\ast]} \|\partial_\zeta \partial^2_\epsilon F(\tau\eta, \cdot, \cdot) \|_{C^{1,\alpha}(\partial\Omega \times [-R,R])} \leq C_7, \\ &\sup_{\eta \in [-\epsilon^\ast,\epsilon^\ast]} \|\partial_\zeta \partial_\epsilon \partial_\zeta F(\tau\eta, \cdot, \cdot) \|_{C^{1,\alpha}(\partial\Omega \times [-R,R])} \leq C_7, \\ &\sup_{\eta \in [-\epsilon^\ast,\epsilon^\ast]} \|\partial_\zeta \partial^2_\zeta F(\tau\eta, \cdot, \cdot) \|_{C^{1,\alpha}(\partial\Omega \times [-R,R])} \leq C_7, \\ &\sup_{\eta \in [-\epsilon^\ast,\epsilon^\ast]} \|\partial_\epsilon \partial_\zeta F(\tau\eta, \cdot, \cdot) \|_{C^{1,\alpha}(\partial\Omega \times [\zeta^i - C_3R,\zeta^i + C_3R])} \leq C_7, \\ &\sup_{\eta \in [-\epsilon^\ast,\epsilon^\ast]} \|\partial^2_\zeta F(\tau\eta, \cdot, \cdot) \|_{C^{1,\alpha}(\partial\Omega \times [\zeta^i - C_3R,\zeta^i + C_3R])} \leq C_7. \end{aligned} \end{equation} Thus, by \eqref{boundoverpsiiesp}, \eqref{norms.1}, \eqref{norms.1.1}, and \eqref{norms.2}, and by the membership of $\epsilon \in ]0,\epsilon^\ast[$ and $\delta^\ast \in ]0,1[$, we have \begin{equation}\label{unicboundNe^2FezFz^2F} \begin{aligned} \| d_v \mathcal{N}_{\partial^2_\epsilon F } (\tau\epsilon, \tilde{\psi}^i_{1}) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))} &\leq 2 C_6\, C_7\,(1+R)^2\,, \\ \| d_v \mathcal{N}_{\partial_\epsilon \partial_\zeta F } (\tau\epsilon, \tilde{\psi}^i_{2}) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))}& \leq 2 C_6\, C_7\,(1+R)^2\,, \\ \| d_v \mathcal{N}_{\partial^2_\zeta F } (\tau\epsilon, \tilde{\psi}^i_{3}) \|_{\mathcal{L}(C^{1,\alpha}(\partial\Omega^i),C^{1,\alpha}(\partial\Omega^i))}& \leq 2 C_6\, C_7\,(1+R)^2\,, \end{aligned} \end{equation} uniformly with respect to $\tau \in ]0,1[$. We can now bound the $C^{1,\alpha}$ norms with respect to the variable $t \in \partial\Omega^i$ of \eqref{A1}, \eqref{A2}, and \eqref{A3} uniformly with respect to $\tau \in ]0,1[$. Indeed, by \eqref{inequality(1/2I + W)psiPsi}, \eqref{unicA1equationnorm} and \eqref{unicboundNe^2FezFz^2F}, we obtain \begin{equation}\label{unicboundA1} \|T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)} \leq 2 C_6\, C_7\,(1+R)^2\delta^\ast \end{equation} for all $\tau\in]0,1[$. By Proposition \ref{proponcompC1alpha} (ii) in the Appendix, by \eqref{uniccostantR_1R_2} and \eqref{norms.2}, we obtain \begin{equation}\label{unicboundezFz^2F} \begin{aligned} &\left\|(\partial_\epsilon \partial_\zeta F) \left(\tau\epsilon,\cdot, \tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ &\qquad\leq C_6 \, C_7 \left(1 + \left\|\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right\|_{C^{1,\alpha}(\partial\Omega^i)}\right)^2\\ &\qquad \leq C_6\, C_7 \,\left(1 + C_3R_1 +|\zeta^i| \right)^2, \\ &\left\|(\partial^2_\zeta F)\left(\tau\epsilon,\cdot,\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \zeta^i\right) \right\|_{C^{1,\alpha}(\partial\Omega^i)} \\ &\qquad\leq C_6\, C_7 \left(1 + \left\|\tau\epsilon \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i]+\zeta^i\right\|_{C^{1,\alpha}(\partial\Omega^i)}\right)^2\\ &\qquad \leq C_6\, C_7 \left(1 + C_3R_1 +|\zeta^i| \right)^2, \end{aligned} \end{equation} for all $\tau\in]0,1[$. Hence, in view of \eqref{uniccostantR_1R_2}, \eqref{unicboundNe^2FezFz^2F} and \eqref{unicboundezFz^2F} and by \eqref{unicA2equationnorm} and \eqref{unicA3equationnorm} we have \begin{equation}\label{unicboundA2A3} \begin{aligned} &\|T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\quad\leq \left\{2 C_6 C_7 \left(1 + C_3R_1 +|\zeta^i| \right)^2 + 4 C_3 C_6 C_7 R_2 (1+R)^2 \right\} \delta^\ast\,, \\ &\|T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\quad \leq \left\{ 4 C_6 C_7 \left(1 + C_3 R_1 +|\zeta^i| \right)^2 R + 8 C_3^2 C_6 C_7 R_2^2 (1+R)^2 \right\} \delta^\ast\,, \end{aligned} \end{equation} for all $\tau\in]0,1[$, where to obtain the inequality for $T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)$ we have also used that \begin{equation*} \left\|\left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\psi^i] + \left(\frac{1}{2}I + W_{\partial\Omega^i}\right)[\Psi^i[\epsilon]] \right\|_{C^{1,\alpha}(\partial\Omega^i)} \leq C_3(R_1 + R_2) \leq R \end{equation*} (cf. \eqref{uniccostantR}). Moreover, since the boundedness provided in \eqref{unicboundA1} and \eqref{unicboundA2A3} is uniform with respect to $\tau \in ]0,1[$, one verifies that the following inequality holds: \begin{equation}\label{intA123} \begin{split} & \bigg\| \int_{0}^{1} (1-\tau) \left\{ T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) + 2 \, T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\right.\\ &\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad + T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) \right\} \,d\tau \bigg\|_{C^{1,\alpha}(\partial\Omega^i)} \\ & \leq \int_{0}^{1} (1-\tau) \| T_1[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) + 2 \, T_2[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad + T_3[\epsilon, \psi^i, \Psi^i[\epsilon]](\tau,\cdot) \|_{C^{1,\alpha}(\partial\Omega^i)} \,d\tau \\ & \le \bigg\{ 2 C_6\, C_7\,(1+R)^2 + 4 C_6 C_7 \left(1 + C_3R_1 +|\zeta^i| \right)^2 + 8 C_3 C_6 C_7 R_2 (1+R)^2 \\ &\quad +4 C_6 C_7 \left(1 + C_3 R_1 +|\zeta^i| \right)^2 R + 8 C_3^2 C_6 C_7 R_2^2 (1+R)^2 \bigg\}\;\delta^\ast\,. \end{split} \end{equation} Then, by \eqref{s1-S1<A123} and \eqref{intA123} we obtain \begin{equation}\label{s1-S1} \begin{split} &\| S_2[\epsilon, \psi^i] - S_2[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^i)}\\ &\qquad \le \bigg\{ 2 C_6\, C_7\,(1+R)^2 + 4 C_6 C_7 \left(1 + C_3R_1 +|\zeta^i| \right)^2 + 8 C_3 C_6 C_7 R_2 (1+R)^2 \\ &\qquad \quad +4 C_6 C_7 \left(1 + C_3 R_1 +|\zeta^i| \right)^2 R + 8 C_3^2 C_6 C_7 R_2^2 (1+R)^2 \bigg\}\;\delta^\ast \end{split} \end{equation} (also recall that $\epsilon\in]0,1[$). \smallskip {\it $\bullet$ Step 7: Conclusion for $S$.} \\ Finally, by \eqref{unicS1inqual}, \eqref{unicS2inequal} and \eqref{s1-S1}, we have \begin{equation*} \| S[\epsilon, \psi^i] - S[\epsilon,\Psi^i[\epsilon]] \|_{C^{1,\alpha}(\partial\Omega^i) \times C^{0,\alpha}(\partial\Omega^i) \times C^{1,\alpha}(\partial\Omega^o)} \leq C_{8}\; \delta^\ast, \end{equation*} with \[ \begin{split} C_8 \equiv& \,C_4\, C_5\, (1+ R^\alpha) + 2 C_6\, C_7\,(1+R)^2 + 4 C_6 C_7 \left(1 + C_3R_1 +|\zeta^i| \right)^2 \\ & + 8 C_3 C_6 C_7 R_2 (1+R)^2+4 C_6 C_7 \left(1 + C_3 R_1 +|\zeta^i| \right)^2 R \\ &+ 8 C_3^2 C_6 C_7 R_2^2 (1+R)^2. \end{split} \] \smallskip {\it $\bullet$ Step 8: Estimate for \eqref{unidensitiesinB0K} and determination of $\delta^\ast$.} \\ By \eqref{N^-1S} and \eqref{unicconstantN-1} we conclude that the norm of the difference between $(\phi^o, \phi^i, \zeta, \psi^i)$ and \begin{equation*} (\Phi^o[\epsilon] , \Phi^i[\epsilon], Z[\epsilon] , \Psi^i[\epsilon]) \end{equation*} in the space $C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)$ is less than $C_1C_8\, \delta^\ast$. Then, by \eqref{PhiBK/2} and by the triangular inequality we obtain \[ \|(\phi^o, \phi^i, \zeta, \psi^i) - (\phi^o_0, \phi^i_0, \zeta_0, \psi^i_0)\|_{C^{1,\alpha}(\partial\Omega^o) \times C^{1,\alpha}(\partial\Omega^i)_0 \times \mathbb{R} \times C^{1,\alpha}(\partial\Omega^i)} \leq C_1C_{8}\, \delta^\ast + \frac{K}{2} \,. \] Thus, in order to have $(\phi^o, \phi^i, \zeta, \psi^i) \in \mathcal{B}_{0,K}$, it suffices to take \[ \delta^\ast<\frac{K}{2C_1C_{8}} \] in inequality \eqref{unidisuguiesp}. Then, for such choice of $\delta^\ast$, \eqref{unidensitiesinB0K} holds and the theorem is proved. \end{proof} \subsection{Local uniqueness for the family of solutions.}\label{family} As a consequence of Theorem \ref{Thmunisol2}, we can derive the following local uniqueness result for the family $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$. \begin{cor}\label{cor} Let assumptions \eqref{zetaicond}, \eqref{F1}, and \eqref{realanalhp} hold true. Let $\epsilon' \in ]0,\epsilon_0[$ be as in Theorem \ref{existenceThm} (v). Let $\{(u^o_\epsilon,u^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$ be as in Theorem \ref{uesol}. Let $\{(v^o_\epsilon,v^i_\epsilon)\}_{\epsilon \in ]0,\epsilon'[}$ be a family of functions such that $(v^o_\epsilon,v^i_\epsilon) \in C^{1,\alpha}(\Omega(\epsilon)) \times C^{1,\alpha}(\epsilon\Omega^i)$ is a solution of problem \eqref{princeq} for all $\epsilon\in]0,\epsilon'[$. If \begin{equation}\label{cor.eq1} \lim_{\epsilon\to 0^+}\epsilon^{-1}\left\| {v^i_\epsilon(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot )} \right\| _{C^{1,\alpha}(\partial \Omega^i)}=0, \end{equation} then there exists $\epsilon_\ast\in]0,\epsilon'[$ such that \[ (v^o_\epsilon,v^i_\epsilon)=(u^o_\epsilon,u^i_\epsilon)\quad\forall \epsilon\in]0,\epsilon_\ast[\,. \] \end{cor} \begin{proof} Let $\epsilon^*$ and $\delta^*$ be as in Theorem \ref{Thmunisol2}. By \eqref{cor.eq1} there is $\epsilon_*\in]0,\epsilon^*[$ such that \[ \left\| {v^i_\epsilon(\epsilon \cdot ) - u^i_\epsilon(\epsilon \cdot )} \right\| _{C^{1,\alpha}(\partial \Omega^i)}\le \epsilon\delta^*\quad\forall\epsilon\in]0,\epsilon_*[. \] Then the statement follows by Theorem \ref{Thmunisol2}. \end{proof} \section{Appendix} In this Appendix, we present some technical facts, which have been exploited in this paper, on product and composition of functions of $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity. We start by introducing the following elementary result (cf. Lanza de Cristoforis \cite{La91}). \begin{lemma}\label{C0C1alphaalgebra} Let $n \in \mathbb{N}\setminus\{0\}$. Let $\Omega$ be a bounded connected open subset of $\mathbb{R}^n$ of class $C^{1,\alpha}$. Then \begin{equation*} \|uv\|_{C^{0,\alpha}(\partial\Omega)} \leq \|u\|_{C^{0,\alpha}(\partial\Omega)} \, \|v\|_{C^{0,\alpha}(\partial\Omega)} \qquad \forall u,v \in C^{0,\alpha}(\partial\Omega), \end{equation*} and \begin{equation*} \|uv\|_{C^{1,\alpha}(\partial\Omega)} \leq 2 \, \|u\|_{C^{1,\alpha}(\partial\Omega)} \, \|v\|_{C^{1,\alpha}(\partial\Omega)} \qquad \forall u,v \in C^{1,\alpha}(\partial\Omega). \end{equation*} \end{lemma} We now present a result on composition of a $C^{m,\alpha}$ function, with $m\in \{0,1\}$, with a $C^{1,\alpha}$ function. \begin{lemma}\label{lemmacompC1alpha} Let $n,d \in \mathbb{N} \setminus \{0\}$ and $\alpha \in \, ]0,1]$. Let $\Omega_1$ be an open bounded convex subset of $\mathbb{R}^n$ and $\Omega_2$ be an open bounded convex subset of $\mathbb{R}^d$. Let $v = (v_1,\dots,v_n) \in (C^{1,\alpha}(\overline{\Omega_2}))^n$ such that $v(\overline{\Omega_2}) \subset \overline{\Omega_1}$. Then the following statement holds. \begin{enumerate} \item[(i)] If $u \in C^{0,\alpha}(\overline{\Omega_1})$, then \begin{equation*} \|u(v(\cdot))\|_{C^{0,\alpha}(\overline{\Omega_2})} \leq \|u\|_{C^{0,\alpha}(\overline{\Omega_1})} \left(1 + \|v\|^\alpha_{(C^{1,\alpha}(\overline{\Omega_2}))^n} \right). \end{equation*} \item[(ii)] If $u \in C^{1,\alpha}(\overline{\Omega_1})$, then \begin{equation*} \|u(v(\cdot))\|_{C^{1,\alpha}(\overline{\Omega_2})} \leq (1 + nd)^2 \|u\|_{C^{1,\alpha}(\overline{\Omega_1})} \left(1 + \, \|v\|_{(C^{1,\alpha}(\overline{\Omega_2}))^n} \right)^2. \end{equation*} \end{enumerate} \end{lemma} Then, by Lemma \ref{lemmacompC1alpha}, we deduce the following Proposition \ref{proponcompC1alpha}. \begin{prop}\label{proponcompC1alpha} Let $n \in \mathbb{N} \setminus \{0\}$. Let $\alpha \in ]0,1]$. Let $\Omega$ be a bounded connected open subset of $\mathbb{R}^n$ of class $C^{1,\alpha}$. Let $R>0$. Then the following holds. \begin{enumerate} \item[(i)] There exists $c_0>0$ such that \begin{equation*} \|u(\cdot,v(\cdot))\|_{C^{0,\alpha}(\partial\Omega)} \leq c_0 \|u\|_{C^{0,\alpha}(\partial\Omega \times [-R,R])} \left(1 + \|v\|^\alpha_{C^{1,\alpha}(\partial\Omega)} \right). \end{equation*} for all $u \in C^{0,\alpha}(\partial\Omega \times \mathbb{R})$ and for all $v \in C^{1,\alpha}(\partial \Omega)$ such that $v(\partial \Omega) \subset [-R,R]$. \item[(ii)] There exists $c_1>0$ such that \begin{equation*} \|u(\cdot,v(\cdot))\|_{C^{1,\alpha}(\partial\Omega)} \leq c_1 \|u\|_{C^{1,\alpha}(\partial\Omega \times [-R,R])} \left(1 + \|v\|_{C^{1,\alpha}(\partial\Omega)} \right)^2. \end{equation*} for all $u \in C^{1,\alpha}(\partial\Omega \times \mathbb{R})$ and for all $v \in C^{1,\alpha}(\partial \Omega)$ such that $v(\partial \Omega) \subset [-R,R]$. \end{enumerate} \end{prop} \begin{proof} We prove only statement (ii). The proof of statement (i) can be obtained adapting the one of point (ii) and using Lemma \ref{lemmacompC1alpha} (i) instead of Lemma \ref{lemmacompC1alpha} (ii). Since $\partial\Omega$ is compact and of class $C^{1,\alpha}$, a standard argument shows that there are a finite cover of $\partial \Omega$ consisting of open subsets $\mathcal{U}_1,\dots,\mathcal{U}_k$ of $\partial\Omega$ and for each $j\in\{1,\dots,k\}$ a $C^{1,\alpha}$ diffeomorphism $\gamma_j$ from $\overline{B_{n-1}(0,1)}$ to the closure of $\mathcal{U}_j$ in $\partial\Omega$. Then, for a fixed $j \in \{1,\dots,k\}$ we define the functions $\tilde{u}^j: \overline{B_{n-1}(0,1)} \times \mathbb{R} \to \mathbb{R}$, $\tilde{v}^j: \overline{B_{n-1}(0,1)} \to \mathbb{R}$, and $\tilde w^j: \overline{B_{n-1}(0,1)} \to \overline{B_{n-1}(0,1)} \times \mathbb{R}$ by setting \begin{equation*} \begin{aligned} &\tilde{u}^j(t',s) \equiv u(\gamma_j(t'),s) \qquad &&\forall(t',s) \in \overline{B_{n-1}(0,1)} \times \mathbb{R}, \\ &\tilde{v}^j(t') \equiv v(\gamma_j(t')) \qquad &&\forall t' \in \overline{B_{n-1}(0,1)}, \\ &\tilde{w}^j(t') \equiv (t',\tilde{v}^j(t')) \qquad &&\forall t' \in \overline{B_{n-1}(0,1)}. \end{aligned} \end{equation*} Since $v(\partial\Omega) \subset [-R,R]$, it follows that $\tilde{w}^j(\overline{B_{n-1}(0,1)}) \subset \overline{B_{n-1}(0,1)} \times [-R,R]$. Moreover, we can see that there exists $d_j>0$ such that \begin{equation*} \|\tilde{w}^j\|_{(C^{1,\alpha}(\overline{B_{n-1}(0,1)}))^{n}} \leq d_j \|\tilde{v}^j\|_{\left(C^{1,\alpha}(\overline{B_{n-1}(0,1)})\right)}. \end{equation*} Then Lemma \ref{lemmacompC1alpha} implies that there exists $c_j>0$ such that \begin{equation}\label{proponcompC1alpha.eq1} \begin{split} \|\tilde{u}^j(\cdot,\tilde{v}^j(\cdot))\|&_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})} = \|\tilde{u}^j(\tilde{w}^j(\cdot))\|_{(C^{1,\alpha}(\overline{B_{n-1}(0,1)}))^n} \\ &\leq c_j \|\tilde{u}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)}\times [-R,R])} \left(1 + \|\tilde{w}^j\|_{(C^{1,\alpha}(\overline{B_{n-1}(0,1)}))^n}\right)^2 \\ &\leq c_j \|\tilde{u}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)}\times [-R,R])} \left(1 + d_j \|\tilde{v}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})}\right)^2. \end{split} \end{equation} Without loss of generality, we can now assume that the norm of $C^{1,\alpha}(\partial\Omega)$ is defined on the atlas $\{(\mathcal{U}_j,\gamma_j)\}_{j\in\{1,\dots,k\}}$ (cf.~Section \ref{notation}). Then by \eqref{proponcompC1alpha.eq1} we have \begin{equation}\label{proponcompC1alpha.eq2} \begin{split} \|u(\cdot,v(\cdot))\|_{C^{1,\alpha}(\partial\Omega)}&=\sum_{j=1}^k\|u(\gamma_j(\cdot),v(\gamma_j(\cdot)))\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})}\\ &=\sum_{j=1}^k\|\tilde{u}^j(\cdot,\tilde{v}^j(\cdot))\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})}\\ &\le \sum_{j=1}^kc_j \|\tilde{u}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)}\times [-R,R])} \left(1 + d_j \|\tilde{v}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})}\right)^2\,. \end{split} \end{equation} Moreover, \[ \|\tilde{u}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)}\times [-R,R])}\leq \|u\|_{C^{1,\alpha}(\partial\Omega \times [-R,R] )} \] and \[ \begin{split} &\left(1 + d_j \|\tilde{v}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})}\right)^2\\ &\quad\le(1+d_j)^2\left(1 + \|\tilde{v}^j\|_{C^{1,\alpha}(\overline{B_{n-1}(0,1)})}\right)^2\le (1+d_j)^2\left(1 + \|v\|_{C^{1,\alpha}(\partial\Omega)} \right)^2. \end{split} \] Hence, \eqref{proponcompC1alpha.eq2} implies that \[ \begin{split} &\|u(\cdot,v(\cdot))\|_{C^{1,\alpha}(\partial\Omega)}\\ &\quad\le k\, \max \{ c_1(1+d_1)^2,\dots,c_k(1+d_k)^2 \} \|u\|_{C^{1,\alpha}(\partial\Omega \times [-R,R] )} \left(1 + \|v\|_{C^{1,\alpha}(\partial\Omega)} \right)^2 \end{split} \] and the proposition is proved. \end{proof} \section*{Acknowledgement} The research of the R.~Molinarolo was supported by HORIZON 2020 RISE project ``MATRIXASSAY'' under project number 644175, during the author's secondment at the University of Texas at Dallas. The author gratefully acknowledges the University of Texas at Dallas and the University of Tulsa for the great research environment and the friendly atmosphere provided. P.~Musolino is member of the `Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni' (GNAMPA) of the `Istituto Nazionale di Alta Matematica' (INdAM).

BkiUbJk4uzqh_Fxeksub

\section{Introduction} A dynamical mechanism for the process $\gamma^* P \rightarrow \pi^0 P^\prime$ is proposed that allows for the extraction of the tensor charge from experiment. The basis for this approach is in the relation between a hadronic description of the process, in terms of Regge poles and cuts, and the partonic description, in terms of Generalized Parton Distributions (GPDs) \cite{DMul1,Ji1,Rad1}. The latter provides a formal connection to the transversity distribution of the nucleon, $h_1$, and the helicity amplitudes that are key to parameterizing the hadronic description. In the following we will use this connection as a guide in exploring the observables that isolate the transversity. The two models, GPD (Fig.\ref{fig1}a) and Regge (Fig.\ref{fig1}b), will illustrate how the extraction of transversity and the tensor charge can proceed experimentally. A key point in our approach is that deeply virtual $\pi^o$ (as well as $\eta, \eta^\prime$) production off a proton target is clearly distinct from the other types of meson production processes in that it involves the transition of a (virtual) photon with $J^{PC}=1^{--}$ to a $J^{PC}=0^{-+}$ state ({\it i.e.} the final $\pi^o$ or $\eta$, $\eta^\prime$) requiring odd $C-$parity and chiral odd t-channel quantum numbers. As a consequence, in a partonic description such as the one depicted in Fig.{\ref{fig1}a}, the "outgoing" and "returning" quark helicities need to be opposite to one another. A similar picture can be obtained in the Regge model as dictated by duality. Therefore, $\pi^o$ and $\eta$, $\eta^\prime$ electroproduction off a proton single out chiral-odd structures of the target. Another important consequence is that the collinear, leading twist $\gamma^\mu \gamma^5$ type contribution to the $\pi^o$ wave function does not have the correct chirality for the electroproduction process. Consideration of Orbital Angular Momentum (OAM) in the wave function allowed us however to overcome this problem, as we will explain below. GPDs are ``off-forward'' contributions, that allow access to partonic configurations with a given longitudinal momentum fraction, similarly to Deep Inelastic Scattering (DIS) processes, but also at a specific (transverse) location inside the hadron \cite{Bur}. They parameterize the nucleon vertex in the process depicted in Fig.~\ref{fig1}a in terms of three kinematical invariants, besides the initial photon's virtuality, $Q^2$: the longitudinal momentum transfer, $\zeta = Q^2/2(Pq)$, the four-momentum transfer squared, $\Delta^2=-t$, and the variable $X=(kq)/(Pq)$, representing the Light Cone (LC) momentum fraction carried by the struck parton with momentum $k$ (see \cite{Die_rev,BelRad,BofPas}) for reviews). \footnote{ The relations between the variables used in this paper and the analogous set of kinematical variables in the ``symmetric'' system, frequently used in the literature are given along with the definitions of the hadronic tensors components in Refs.~\cite{Die_rev,BelRad}.} As initially pointed out in Ref.\cite{Diehl_01} one can construct four quark helicity flip distributions: $H_T^q$, $E_T^q$, $\tilde{H}_T^q$, $\tilde{E}_T^q$, representing a complete set. \footnote{A similar argument as in Ref.\cite{Diehl_01} can be extended to define a corresponding number of gluon helicity flip distributions that, however, will not enter our discussion of transversity.} In particular $H_T^q$, which need not vanish in the forward limit, is related to transversity through the following properties: \begin{eqnarray} \label{transversity_def} \int\limits_{\zeta-1}^1 H_T^q(X,\zeta,t) \, dX = A_{T10}^q(t) \\ \label{transv_forward} H_T^q(X,0,0) = h_1^q(X) \end{eqnarray} The form factor $A_{T10}(t)$ gives the tensor charge for $t \rightarrow 0$, $A_{T10}(0) \equiv \delta q$. $h_1(X)$ is the transversity structure function~\cite{JafJi}. Eqs.(\ref{transversity_def}) and (\ref{transv_forward}) are analogous relations to the ones for the twist two, chiral-even, unpolarized and longitudinally polarized distributions cases. A substantial amount of literature exists on the connection between Transverse Momentum Distributions (TMDs) and GPDs \cite{Bur,Bur2,Bur3,DieHag,Metz}. TMDs are the soft matrix elements in deep inelastic semi-inclusive processes and therefore they are by definition forward quantities. The non-trivial role played by Initial State Interactions (ISI) and Final State Interactions (FSI) allows one however to access features of the motion of partons in the transverse direction. In particular, in Ref.\cite{Bur2} it was proposed that the transverse momentum asymmetry of the final quarks -- generating the Sivers function -- can be related to the transverse spatial asymmetry, through a chromodynamic lensing effect. Notwithstanding the appeal of the physical ideas connecting transversity and transverse spatial structure of hadrons, one should notice that the transverse spatial dependence appearing in \cite{Bur2} is, however, necessarily buried in the correlators expressions defining TMDs, and it can only be probed explicitly through exclusive measurements. The concrete possibility of testing the ideas on the role of transversity other than in semi-inclusive measurements, that are not directly sensitive to (transverse) spatial degrees of freedom, has been so far elusive. Suggestions were in fact made to obtain such information mainly from lattice calculations \cite{Bur2,Metz}. The main thrust of this paper is on the contrary to propose a new avenue to experimentally determine transversity and its connection to spatial degrees of freedom, using exclusive processes such as $\pi^o$ and $\eta$ production off nucleons and nuclei. In this context two other functions, namely the combination of GPDs $2 \tilde{H}_T^q + E_T \equiv \overline{E}_T$, and the GPD $E$ defining the spin flip component in scattering from an unpolarized nucleon, are essential. $\overline{E}_T$ is expected, in a class of models (\cite{Metz} and references therein), to be related to the Boer-Mulders function, $h_1^{\perp \, q}$ through: \begin{subequations} \begin{eqnarray} \label{E2} \int\limits_{\zeta-1}^1 dX \, \left[ 2 \tilde{H}_T^q(X,\zeta,t) + E_T^q(X,\zeta,t) \right]_{\zeta=t=0} = \kappa_T^q \\ \int d^2 k_T \, dX h_{1}^{\perp \, q}(X,k_T) \approx - \kappa_T^q \end{eqnarray} \end{subequations} Similarly, $E$, can be related to the Sivers function, $f_{1T}^\perp$: \begin{subequations} \begin{eqnarray} \label{E} \int\limits_{\zeta-1}^1 dX \, \left[ E(X,\zeta,t) \right]_{\zeta=t=0} = \kappa^q \\ \int d^2 k_T \, dX f_{1T}^{\perp \, q}(X,k_T) \approx - \kappa^q \end{eqnarray} \end{subequations} Here, $\kappa^q$ is the contribution to the proton anomalous magnetic moment of the quark $q$. In the transverse plane in coordinate space, $\kappa^q$ is a measure of an unpolarized quark's displacement along the $y$-axis in a proton polarized along the $x$-axis \cite{Bur}. $\kappa_T^q$, the transverse anomalous moment, measures the $y$-transverse displacement for a transversely polarized quark along the $x$-axis in an unpolarized proton \cite{Bur2,BofPas}. The relationships above express at a deeper level the fact that both pairs: $[ (k_1+ik_2) f_{1T}^\perp(X,k_T), (\Delta_1+i \Delta_2)E(X,k_T;\zeta,\Delta) ]$ and $[(k_1+ik_2) h_{1}^\perp(X,k_T), (\Delta_1+i \Delta_2) \overline{E}_T(X,k_T;\zeta,\Delta) ]$, have the same helicity structure. \footnote{$E(X,k_T;\zeta,\Delta)$ and $\overline{E}_T(X,k_T;\zeta,\Delta)$ are the unintegrated over $k_T$ GPDs.} They can therefore be obtained from the overlap of similar wave functions at the handbag level. Nevertheless, upon integration in $k_T$, $E$ does not necessarily vanish, while $f_{1T}^\perp$, because of it is $T$-odd, is non zero only if FSI is taken into account. This is the reason why in the literature \cite{Bur2,Metz} no direct relationship was written explicitly. In order to make a connection with observables for given processes one needs to transform the Dirac matrix elements in terms of which the defining quark correlation functions are written, to the quark chirality basis \cite{JafJi,BogMul}. One can see that $H_T(X,\zeta,t)$ appears in the off-forward helicity amplitude $A_{++,--}$ \cite{Diehl_01} while $\overline{E}_T(X,\zeta,t)$ is defined in $A_{++,+-}, A_{-+,--}$ (more details are given below). \footnote{We use the notation: $A_{\Lambda^\prime \lambda^\prime, \Lambda \lambda}$, where $\Lambda (\Lambda^\prime)$ are the initial (final) proton helicities, and $\lambda (\lambda^\prime)$ are the initial (final) quark helicities} As known from the helicity amplitude decomposition of observables in $\pi^0$ photoproduction~\cite{GolOwe}, measurements of either the polarized target asymmetry, $A$, or the recoil nucleon polarization, $T$, or polarized photon asymmetries, $P$ on nucleons can be used to determine the desired helicity structure which we will see is connected to the off-forward quark helicity amplitudes (and related GPDs). A particular model was developed in Ref.\cite{GolOwe} that, by accounting for the Regge cut corrections to the pole dominated amplitude, predicted non-negligible values of $A$, $T$ and $P$ in the range $0 \lesssim -t \lesssim 1.$ GeV$^2$. The model contains the nucleons' tensor charges as parameters -- or a combination of parameters. By extending this model to the case of virtual photon scattering (Fig.\ref{fig1}b), one can, therefore, extract the value of the tensor charge directly from the data using both the ideal intermediate energy kinematical range accessible at Compass, Hermes and Jefferson Lab, and the high precision provided by the latter. From the parton perspective, recent developments \cite{AHLT1,AHLT2} enable us to propose a detailed model where the quark degrees of freedom are described in terms of chiral-odd GPDs. The connections being explored here, among transversity distributions, Regge exchange models and GPDs had some early hints from an exchange model that was developed for the tensor charge~\cite{GG1}. There the isovector and isoscalar components of the tensor charge were shown to be proportional to the product of the matrix elements for the decay of the lightest axial vector mesons ($J^{PC}=1^{+-}$), $b_1$ and $h_1$ respectively, and for the coupling of these mesons to the nucleon (Fig.\ref{fig1}b). Both of them are needed in order to determine the isovector and isoscalar components of the tensor charge. Their coupling constants that entered the calculation were determined through the better known decay constant of the $a_1$ ($J^{PC}=1^{++}$) meson, by exploiting the fact that $a_1$ belongs to the same $SU(6)\otimes O(3)$ multiplet as $b_1, h_1$ \cite{GG1}. It is important to notice that the $a_1$ ($J^{PC}=1^{++}$) type exchanges do not enter directly the $\gamma^*p \rightarrow \pi^o p$ scattering amplitude. This will bear important consequences also in the description in terms of GPDs, namely in the identification of the correct structures entering the different helicity amplitudes. In the single meson or axial vector dominance approximation of Ref.\cite{GG1}, angular momentum conservation at point-like interaction vertices required that the coupling vanish in the forward limit. In order for the tensor charge not to vanish, a duality picture was envisaged where the struck quark undergoes Final State Interactions (FSI) whose effect is parametrized in \cite{GG1} in terms of the meson's constituent quarks' $\langle k_T^2 \rangle$. This interesting point of departure from other treatments is the appearance of the factor $\left< k_T^2 \right>$ in the expression for the tensor charge $\delta q$. This arises because of the kinematic structure of the exchange picture that was adopted. The approach to the tensor charge just summarized leaves several questions - two in particular. Is the extrapolation from the $b_1$ mass ($t = m_{b_1}^2$) to $t=0$ a reasonable one? Can FSI's enter the picture in a natural way to parallel the $\left<k_T^2\right>$ dependence that was needed? Both of these questions are answered affirmatively in a Regge exchange model for $\pi^0$ electroproduction. The $b_1$ Regge trajectory enables the extrapolation to the $t \leq 0$ region. Regge cuts restore the exchange amplitude in the forward limit, as shown in photoproduction analyses in the past~\cite{GolOwe}. The GPD, or off-forward amplitude approach also provides answers to these questions. GPDs are functions of off-forward two body kinematic invariants, including $t$, and allow for smooth extrapolation to $t=0$. Furthermore, the relevant off-forward helicity amplitude need not vanish in the forward limit, as will be seen later. A central question of factorization arises in relating the $\pi^0$ electroproduction single spin asymmetries to the relevant GPDs, because the important helicity amplitudes involve transverse virtual photons. Factorization in meson production was explained in Ref.\cite{CFS} for longitudinally polarized photons only, based on the fact that in this case the end-point effects from the wave function of the produced meson are suppressed with respect to the transverse case \cite{BroFraGun}. This point of view was reiterated in several papers ~\cite{CF,Mank,AniTer,Pire,DieVin,HuaKro}. On the other hand, GPD factorization was criticized recently by the authors of Ref.~\cite{SzcLlaLon} where it was claimed that the dominance of Regge type exchanges produces non-analytic terms that destroy factorization. This is an ongoing and important debate now underway. Since our starting point is a ``duality'' type of picture, we think that there is theoretical merit in both perspectives. On the experimental front, however, there seems to be little evidence in the data that one can rule out transverse vs. longitudinal factorization hypothesis. This appears in the $Q^2$ dependence of the recent HERMES $\rho$ production data showing a plateau in $R=\sigma_L/\sigma_T$ for vector meson production \cite{Hermes_rho} (intermediate energy), \cite{Hera_rho} (high energy). We propose a new mechanism to describe the $Q^2$ dependence at the meson vertex that distinguishes the longitudinal and transverse photon polarization contributions. This mechanism describes both the vector (natural parity) and axial vector (unnatural parity) channels taking into account the orbital angular momentum in the evaluation of the different quark helicity contributions to the pion wave function. In particular, the axial vector and vector channel differ by one unit of orbital angular momentum. Hence, as explained in detail later on, the longitudinal photon amplitude is dominated by the axial vector contributions. For the transverse photon both vector and axial vector channels will contribute. Our results, differing from Perturbative QCD (PQCD) type behavior, provide a less steep dependence on $Q^2$ of the longitudinal to transverse ratios. We would like to remark that independent of the way corrections to the standard PQCD approach are carried out, and of the interest in the process $per se$ as a probe of transversity, the issue of carefuly monitoring $\pi^0$ electroproduction at intermediate energies will be a prominent one in the analysis of many planned experiments at both Compass and Jlab. $\pi^0$'s constitute, in fact, a large background in $\gamma$ production from both protons and nuclei. Finally, we suggest a practical method to extract both the tensor charge, $\delta q$, and the transverse anomalous moment, $\kappa^q_T$, from experiment that makes use of the fact that such quantities enter as free parameters in both our Regge and partonic descriptions. We suggest a number of observables, {\it e.g.} the longitudinal/transverse interference term, $\sigma_{LT}$, the transverse spin asymmetry, $A_{UT}$, the beam spin polarization, that are sensitive to the values of $\delta q$ and $\kappa^q_T$, $q=u,d$. The paper is organized as follows: in Section \ref{sec-2} we present definitions and kinematics; in Section \ref{sec-3} we present our Regge approach; in Section \ref{sec-4} we introduce the connection to GPDs and we develop a parametrization of the chiral-odd ones, based on available experimental and phenomenological information; in Section \ref{sec-5} we propose a model of the $Q^2$ dependence of the various helicity amplitudes; in Section \ref{sec-6} we discuss results and propose an extraction method for the tensor charge; finally in Section \ref{sec-7} we draw our conclusions and present an outlook. The spirit of the paper is to suggest a method to obtain $\delta q$ and $\kappa_T^q$ from experiment, while exploring a number of questions: from the dominance of chiral-odd contributions in $\pi^o$ electroproduction, to the duality picture, and the transition from Regge to partonic contributions. \section{$t$-channel dominance picture} \label{sec-2} In the following discussion of exclusive $\pi^0$ electroproduction, the crucial nexus connecting observable quantities, transversity GPDs and the Regge description is provided by helicity amplitudes. The relation between the electroproduction, virtual photon helicity amplitudes, $f_{\Lambda_\gamma,\Lambda_N; 0, \Lambda_N\prime}$ and the relevant GPDs is expressed in the ``handbag'' picture, with the assumption that the ``hard part factorizes from the ``soft part'', as shown in Fig.~\ref{fig1}a. While this has been shown for the case of a high $Q^2$ longitudinal photon in exclusive vector meson electroproduction, it has not been demonstrated for transverse photons in $\pi^0$ production. The transverse photon contribution is kinematically suppressed by $1/Q$ relative to the longitudinal photon from the lepton tensor and the Hand convention for the virtual longitudinal photon flux. It has been presumed by many authors, without proof, that the transverse case does not factorize. Several papers have been written to carry the longitudinal case to higher twist, speculating that factorization still holds, while ignoring the transverse case, which should enter at the same order of $1/Q$ as twist 3 for longitudinal photons. Furthermore, the transverse virtual photon cross section does not show evidence of a decreasing ratio of transverse to longitudinal, as seen at HERMES~\cite{Hermes_rho} and in preliminary data from JLab~\cite{kubarovsky}, which leads us to reconsider the transverse case and to assume that factorization does hold. We therefore start out by defining the helicity amplitudes and the main observables related to transversity within the assumption of factorization for the partonic description. It is important to recall that in the Regge pole or single particle exchange picture there is factorization of the upper vertex ($\gamma^*\rightarrow \pi^0$) from the lower vertex ($p\rightarrow p\prime$). This factorization would correspond to the $t$-channel picture (as in Fig.~\ref{fig1}) in which the quark and antiquark exchanges are accompanied by ladder-like gluon links. The Regge pole coupling at the vertex is independent of the other vertex and satisfies parity conservation. \subsection{Kinematics and Definitions} \label{subsec-2.1} Exclusive $\pi^0$ electroproduction is shown in Fig.\ref{fig1}. The relevant four-momenta written in the laboratory frame are the initial (final) electrons: $k_{1(2)} \equiv (\epsilon_1,{\bf k_{1(2)}})$, the exchanged photon's: $q \equiv (\nu,\bf{q})$, with $\nu=\epsilon_1-\epsilon_2$, and ${\bf q} = {\bf k_{1}} -{\bf k_{2)}}$; the initial proton, $P\equiv (M,0))$. In addition, one has the final proton $P^\prime$, and the final pion $p_\pi$. We define the usual invariants for a Deeply Inelastic Scattering (DIS) process: the virtual photon's four-momentum squared, $Q^2= -q^2 = 4 \epsilon_1 \epsilon_2 \sin^2 \theta/2$, $(Pq)/M = \nu$, $x_{Bj} = Q^2/2M\nu$, $y=(Pq)/(P k_1)$. The Mandelstam invariants are defined with respect to the $\gamma^* N \rightarrow \pi^0 N^\prime $ process, namely: $s = W^2 =(P+q)^2$, $t=(P-P^\prime)^2$, and $u=(P-p_\pi)^2$. The amplitudes are decomposed into a purely leptonic part and the $\gamma^*+ N \rightarrow \pi^0 + N^\prime $ process for which there are six independent helicity amplitudes chosen as \begin{equation} f_1 = f_{1+,0+} \propto\Delta^1 ,\ \ f_2 = f_{1+,0-} \propto\Delta^0 ,\ \ f_3 = f_{1-,0+} \propto\Delta^2 ,\ \ f_4 = f_{1-,0-} \propto\Delta^1, \label{f1} \end{equation} for transverse photons and \begin{equation} f_5 = f_{0+,0-} \propto\Delta^1, \ \ f_6 = f_{0+,0+} \propto\Delta^0. \label{f0} \end{equation} for longitudinal photons. We use the notation: $f_{\Lambda_\gamma,\Lambda_N; 0, \Lambda_{N\prime}}$, $\Lambda_\gamma = \pm 1,0$ being the virtual photon spin, and $\Lambda_N (\Lambda_{N\prime}) = +,- \equiv +1/2,-1/2$ being the initial (final) nucleon spins; here $\Delta \equiv \mid {\bf \Delta} = {\bf P} - {\bf P}^\prime \mid$ is the magnitude of the three-momentum transfer, and the minimum kinematically allowed power is indicated for each helicity amplitude. In a single hadron exchange (or Regge pole exchange) factorization and parity conservation require \begin{equation} f_1 =\pm f_4 \ {\rm and} \ f_2 =\mp f_3 \end{equation} for even or odd parity exchanges, to leading order in $s$. These pair relations, along with a single hadron exchange model, force $f_2$ to behave like $f_3$ for small $\Delta$ or small transverse momentum for the outgoing particles. This introduces the $k_T^2$ factor into the $f_2$ amplitude, which is related to the transversity transfer, as we will see below from its connection with the GPD $H_T$. Observable quantities are bilinear combinations of these helicity amplitudes. The differential cross section for pion electroproduction off an unpolarized target is \cite{RasDon} \begin{equation} \label{unp-xs} \frac{d^4\sigma}{d\Omega d\epsilon_2 d\phi dt} = \Gamma \left\{ \frac{d\sigma_T}{dt} + \epsilon_L \frac{d\sigma_L}{dt} + \epsilon \cos 2\phi \frac{d\sigma_{TT}}{dt} + \sqrt{2\epsilon_L(\epsilon+1)} \cos \phi \frac{d\sigma_{LT}}{dt} \right\}. \end{equation} If the initial electron is polarized, with $h=\pm 1$, one has the additional contribution \begin{equation} \label{pol-xs} h \, \sqrt{2\epsilon_L(\epsilon-1)} \, \frac{d\sigma_{L^\prime T}}{dt} \sin \phi , \end{equation} The photon polarization parameter $\epsilon$ can be written in terms of invariants as \[ \epsilon^{-1} = 1 + 2\left( 1+\frac{\nu^2}{Q^2} \right)\left(4 \dfrac{\nu^2}{Q^2} \dfrac{1-y}{y^2}-1\right)^{-1}, \] \noindent and for longitudinal polarization alone, \[ \epsilon_L= \frac{Q^2}{\nu^2} \epsilon. \] \noindent The factor $\Gamma$ is given by \begin{equation} \label{Gamma} \Gamma = \sigma_{Mott} \, f_{rec} \, m_\pi \mid {\bf p}_\pi \mid J(Q^2,\nu,s) \end{equation} \noindent where the Mott cross section is \[ \sigma_{Mott}= \frac{4 \alpha^2 \epsilon_2^2 \cos^2 \theta/2}{Q^4}, \] \noindent and the hadronic recoil factor, $f_{rec}$, \[ f_{rec} = \lvert 1 + \frac{\nu - \mid {\bf q} \mid \cos \theta_\pi^{LAB}}{M} \rvert^{-1}. \] \noindent In Eq.(\ref{Gamma}), $J(Q^2,\nu,s)$ is the jacobian for the transformation from $\cos \theta_\pi^{LAB}$ to $t$, whose expression is given in Appendix \ref{app-1}. The different contributions in Eq.(\ref{unp-xs}) are written in terms of helicity amplitudes as \begin{eqnarray} \label{dsigT} \frac{d\sigma_T}{dt} & = & \mathcal{N} \, \left( \mid f_{1,+;0,+} \mid^2 + \mid f_{1,+;0,-} \mid^2 + \mid f_{1,-;0,+} \mid^2 + \mid f_{1,-;0,-} \mid^2 \right) \nonumber \\ & = & \mathcal{N} \, \left( \mid f_1 \mid^2 + \mid f_2 \mid^2 + \mid f_3 \mid^2 + \mid f_4 \mid^2 \right) \\ \label{dsigL} \frac{d\sigma_L}{dt} & = & \mathcal{N} \left(\mid f_{0,+;0,+} \mid^2 + \mid f_{0,+;0,-} \mid^2 \right) \nonumber \\ & = & \mathcal{N} \, \left( \mid f_5 \mid^2 + \mid f_6 \mid^2 \right), \end{eqnarray} for transverse and longitudinal virtual photon polarizations, respectively, and \begin{equation} \label{Hand} \mathcal{N} = \left[ M (s-M^2)^2 \right]^{-1} \, G \end{equation} where we used the Hand convention, multiplied by a geometrical factor $G$ that is given by $G=\pi/2$, and $G=1/8\pi$, in the Regge and GPD approaches considered later on. The cross section for the virtual photon linearly polarized out of the scattering plane minus that for the scattering plane is \begin{eqnarray} \label{dsigTT} \frac{d\sigma_{TT}}{dt} & = & 2 \, \mathcal{N} \Re e \left( f_{1,+;0,+}^*f_{1,-;0,-} - f_{1,+;0,-}^* f_{1,-;0,+} \right) \nonumber \\ & = & 2 \, \mathcal{N} \, \Re e \left( f_1^*f_4 - f_2^* f_3 \right). \end{eqnarray} The interference term for the transversely and longitudinally polarized virtual photons is \begin{eqnarray} \frac{d\sigma_{LT}}{dt} & = & 2 \, \mathcal{N} \, \Re e \left[ f_{0,+;0,+}^* (f_{1,+;0,-} + f_{1,-;0,+}) + f_{0,+;0,-}^* (f_{1,+;0,+} - f_{1,-;0,-}) \right] \nonumber \\ & = & 2 \, \mathcal{N} \, \Re e \left[ f_5^* (f_2 + f_3) + f_6^* (f_1 - f_4) \right]. \label{dsigLT} \end{eqnarray} Finally the beam polarization term is given by \begin{eqnarray} \frac{d\sigma_{LT^\prime}}{dt} & = & 2 \, \mathcal{N} \, \Im m \left[ f_{0,+;0,+}^* (f_{1,+;0,-} + f_{1,-;0,+}) + f_{0,+;0,-}^* (f_{1,+;0,+} - f_{1,-;0,-}) \right] \nonumber \\ & = & 2 \, \mathcal{N} \, \Im m \left[ f_5^* (f_2 + f_3) + f_6^* (f_1 - f_4) \right] \label{dsigLTp} \end{eqnarray} In addition to the unpolarized observables listed above, a number of observables directly connected to transversity can be written (see {\it e.g.} \cite{GolMor}). Here we give the transversely polarized target asymmetry, \begin{equation} \label{AUT} A_{UT} = \frac{2 \Im m (f_1^*f_3 - f_4^*f_2)}{{\displaystyle \frac{d\sigma_T}{dt}}}, \end{equation} and the beam spin asymmetry, \begin{equation} \label{BSA} A \approx \alpha \sin \phi, \end{equation} where \begin{equation} \label{alpha} \alpha = \frac{\sqrt{2 \epsilon_L (1-\epsilon)} \,\, {\displaystyle \frac{d \sigma_{LT^\prime}}{dt} }}% {{\displaystyle \frac{d\sigma_T}{dt} + \epsilon_L \frac{d\sigma_L}{dt} } } \end{equation} (note that the recoil nucleon polarization asymmetry, $\mathcal{T}$, defined analogously to $A_{UT}$ simply involves the switching of $f_1$ and $f_4$). It is important to realize that the relations between observables and helicity amplitudes are general, independent of any particular model. We will see that the Regge model, as well as the parameterization through GPDs, populate those helicity amplitudes related to transversity and thereby effect observables in important ways. \subsection{Connection to Generalized Parton Distributions} \label{subsec-2.2} The connection with the parton model and the transversity distribution is uncovered through the GPD decomposition of the helicity amplitudes. In the factorization scenario the amplitudes for exclusive $\pi^0$ electroproduction, $f_{\Lambda_\gamma,\Lambda_N; 0, \Lambda_{N\prime}}$ can be decomposed into a ``hard part'', $g_{\Lambda_\gamma,\lambda; 0, \lambda^\prime}$ describing the partonic subprocess $\gamma^* + q \rightarrow \pi^0 + q$ (top part of diagram in Fig\ref{fig1}a), and a ``soft part'', $A_{\Lambda^\prime,\lambda^\prime;\Lambda,\lambda}$ that, in turn, contains the GPDs, \begin{equation} f_{\Lambda_\gamma,\Lambda;0,\Lambda^\prime} = \sum_{\lambda,\lambda^\prime} g_{\Lambda_\gamma,\lambda;0,\lambda^\prime} (X,\zeta,t,Q^2) \otimes A_{\Lambda^\prime,\lambda^\prime;\Lambda,\lambda}(X,\zeta,t), \label{facto} \end{equation} where a sum over the different quark components is omitted for simplicity. The amplitudes in Eq.(\ref{facto}) implicitly contain an integration over the unobserved quark momenta, and are functions of $x_{Bj} \approx \zeta, t$ and $Q^2$; they are analogous to the Compton Form Factors in DVCS. In fact, in the more familiar case of DVCS, the upper part involves the matrix elements of $j_\mu^{EM}(x) j_\nu^{EM}(0)$, the tree level diagram for $\gamma^*+q \rightarrow \gamma + q$ with the high momentum struck quark intermediate state. In the Bjorken limit, in terms of light cone variables, this diagram has a simple Dirac structure of $\gamma^+$ with a denominator that becomes $1/(X-i\epsilon)$ (the corresponding crossed diagram, required for gauge invariance, yields $1/(X -\zeta+i\epsilon)$). The same structure would obtain for the production of vector mesons -- matrix elements of $j_\mu^{EM}(x) j_\nu^V(0)$), along with a hadronic wavefunction. One model for this, that is often used, has the hadronic wave function nicely factored from the partonic components of the PQCD diagrams. With this latter approach there is a depression of longitudinal to/from transverse transitions~\cite{CFS,Pire}. For pion production the upper, hard part of the diagram involves the matrix element between quark states of $j_\mu^{EM}(x) j^P(0)$, the latter being the pseudoscalar hadronic current operator. As we have emphasized, for $\pi^0$ the diagram is C-Parity odd and chiral odd in the $t-$channel. Because only one of these transverse photon functions survives the limits, the relation to the tensor charge is quite simple. Note that because of the pion chirality (0$^-$), the quark must flip helicity at the pion vertex where we take the coupling to be $\gamma^5$. Therefore, the corresponding Dirac structure for the hard subprocess diagram involves $\sigma^{+T}\gamma^5$, at variance with Refs.\cite{Mank,VGG} where the C-parity even axial vector structure $\gamma^\mu\gamma^5$ was considered. This is very significant for our reaction. Our observation also implies important changes in the $Q^2$ dependence of the process that will be discussed within a specific model in Section V. As we displayed for the t-channel picture of electroproduction, there are six independent helicity amplitudes for $\gamma^*+N\rightarrow\pi^0+N^\prime$, given parity conservation, four with $\lambda_\gamma=1$, Eq.~(\ref{f1}) and two with $\lambda_\gamma=0$, Eq.~(\ref{f0}). It will be important in the following to observe that each helicity amplitude $f_{\Lambda_\gamma,\Lambda;0,\Lambda^\prime}$ will have an angular momentum conserving factor of $sin^n(\theta_{CM}/2)$, where the minimum value of $n=\Lambda_\gamma-\Lambda+\Lambda^\prime$. This was written in terms of powers of $\mid {\bf \Delta} \mid $ $\propto sin(\theta_{CM}/2)$. in Eqs.~(\ref{f1}) and ~(\ref{f0}). In terms of the invariant variables used here for the GPDs, \begin{equation} sin^2(\theta_{CM}/2) = -\frac{\zeta}{Q^2} (t - t_{min}) \end{equation} where the limit of $Q^2 >> M^2$ is taken. Corresponding factors of $sin^{n^\prime}(\theta_{CM}/2)$ will occur in the $g$ and $A$ amplitudes. The helicity structure of the $g$ amplitudes is straightforward. It is the same as the $f$ amplitudes, so we can take the same labeling for $g_1,...,g_6$ as in Eqs.(\ref{f1}) and (\ref{f0}). Using parity conservation, $g_1= g_{1+,0+}$, $g_2= g_{1+,0-}$, $g_3= g_{-1+,0-} = g_{1-,0+}$, $g_4= g_{1-,0-}$, $g_5= g_{0+,0-}$, and $g_6=g_{0+,0+}$. If we have $${\rm \gamma^*(q) + q(k)\rightarrow \pi^0(p_\pi) + q(k^\prime)},$$ then $\hat{s}=(q+k)^2, t=(k^\prime-k)^2 = (q-p_\pi)^2, \mid {\bf \Delta} \mid^2=t_{min}-t$, with $t_{min}$ fixed in the center of mass (CM) of this reaction, and $\hat{u}=(q-k^\prime)^2$. With these variables, and taking the quarks and pion masses to zero, the $g$ amplitudes are quite simple. Only $g_2$ and $g_5$ survive in the $\hat{s}>>|t|$ limit. One has, \begin{subequations} \label{g_functions} \begin{eqnarray} g_1 & = & g_4= g_\pi \, \frac{\mathcal{C}_q }{\hat{s}} \frac{1}{4} N \, N^\prime \, {\rm Tr} \left \{\gamma^\lambda \gamma^o(1 + \gamma^3\gamma^5) \gamma^\mu \gamma^+ \gamma^\nu \right\} k_\lambda k^\prime_\mu \epsilon_\nu^T \nonumber \\ & = & g_\pi \, \frac{\mathcal{C}_q }{\hat{s}} N \, N^\prime \left[ \epsilon^{\lambda 3 \mu \nu} + \epsilon^{\lambda o \mu \nu} + (g^{\mu o} - g^{\mu 3} ) \epsilon^{\lambda o 3 \nu} \right] k_\lambda k^\prime_\mu \epsilon_\nu^T = 0 \\ g_2 & = & g_\pi \, \frac{\mathcal{C}_q }{\hat{s}} \frac{1}{4} N \, N^\prime \, {\rm Tr} \left \{\gamma^\lambda \gamma^o(-\gamma^1 + i\gamma^2)\gamma^\mu \gamma^+ \gamma^\nu \right\} k_\lambda k^\prime_\mu \epsilon_\nu^{(\lambda=+1)} \nonumber \\ & = & g_\pi \, \mathcal{C}_q \cos \theta/2 \simeq g_\pi \, \mathcal{C}_q \sqrt{\frac{-\hat{u}}{\hat{s}}} \\ g_3 & = & g_\pi \, \frac{\mathcal{C}_q }{\hat{s}} \frac{1}{4} N \, N^\prime \, {\rm Tr} \left \{\gamma^\lambda \gamma^o(-\gamma^1 + i\gamma^2)\gamma^\mu \gamma^+ \gamma^\nu \right\} k_\lambda k^\prime_\mu \epsilon_\nu^{(\lambda=-1)} =0 \\ g_5 & = & g_\pi \, \frac{\mathcal{C}_q }{\hat{s}} \frac{1}{4} N \, N^\prime \, {\rm Tr} \left \{\gamma^\lambda \gamma^o(-\gamma^1 + i\gamma^2)\gamma^\mu \gamma^+ \gamma^\nu \right\} k_\lambda k^\prime_\mu \epsilon_\nu^L \nonumber \\ & = & g_\pi \, \mathcal{C}_q \frac{\sqrt{2\hat{s}}}{Q} \sin \theta/2\simeq g_\pi \, \mathcal{C}_q \sqrt{\frac{-2t}{Q^2}}, \\ g_6 & = & g_\pi \, \frac{\mathcal{C}_q }{\hat{s}} \frac{1}{4} N \, N^\prime \, {\rm Tr} \left \{\gamma^\lambda \gamma^o(1 + \gamma^3\gamma^5) \gamma^\mu \gamma^+ \gamma^\nu \right\} k_\lambda k^\prime_\mu \epsilon_\nu^L = 0 \end{eqnarray} \end{subequations} where the photon polarization vectors are $\epsilon_\nu^T = \epsilon_\nu^{(\lambda=\pm 1)} \equiv 1/\sqrt{2} (0;\mp 1,- i,0)$, $\epsilon_\nu^L=\epsilon_\nu^{(\lambda=0)} \equiv 1/\sqrt{Q^2} (\mid {\bf q} \mid; 0_\perp,\nu)$, and \[ \mathcal{C}_q = \frac{1}{X- i \epsilon} + \frac{1}{X-\zeta + i \epsilon} \] \noindent $g_\pi \simeq \sqrt 15 (2/3)$ is the quark-pion coupling obtained from the nucleon-pion coupling in the additive quark model: we take this limiting value to show the structure of the upper part of the handbag. The composite structure of the pion production vertex generates an additional $Q^2$ dependence that will be described in detail in Section V. Our goal is to provide an alternative to the standard PQCD based meson production models that are well known to largely miss the behavior of current experiments in the Multi-GeV kinematical region (see however discussion in \cite{GolKro}). We obtain different $Q^2$ behaviors for the subprocess amplitudes, corresponding to either axial vector (A), or vector (V) $t$-channel exchanges, or depending, in other words, on the $C$ and $P$ quantum numbers. In our model this translates into a different dependence for each one of the helicity amplitudes entering Eq.(\ref{facto}), \begin{subequations} \label{f15} \begin{eqnarray} f_1 = f_4 & = & \int \limits_{-1+\zeta}^{1} dX \, g_2 (X,\zeta,t,Q^2) \, F_V(Q^2) \, A_{++,+-}(X,\zeta,t) \\ f_2 & = & \int \limits_{-1+\zeta}^{1} dX \, g_2 (X,\zeta,t,Q^2) \, [F_V(Q^2)+F_A(Q^2)] \, A_{--,++}(X,\zeta,t) \\ f_3 & = & \int \limits_{-1+\zeta}^{1} dX \, g_2 (X,\zeta,t,Q^2) \, [F_A(Q^2)-F_V(Q^2)] \, A_{+-,-+}(X,\zeta,t) \\ f_5 & = & \int \limits_{-1+\zeta}^{1} dX \, g_5 (X,\zeta,t,Q^2) \, F_A(Q^2) \, A_{--,++}(X,\zeta,t) \end{eqnarray} \end{subequations} Note that $A_{--, ++}$ involves no overall helicity change, and hence no required factor of a non-zero power of $ t - t_0$. Nevertheless, many models will lead to non-zero powers that kill the forward limit, and hence, do not contribute to the tensor charge. This point is explained in detail below for the Regge model adopted in this paper. \section{Weak-Cut Regge Model} \label{sec-3} To have a non-zero single spin asymmetry requires interference between single helicity flip and non-flip and/or double flip amplitudes. Asymmetry arises from rescattering corrections (or Regge cuts or eikonalization or loop corrections) to single hadron exchanges. That is, one of the amplitudes in the product must acquire a different phase, a relative imaginary part. We will construct a Regge pole model with cuts to account for measured photoproduction observables at moderate s and small t, and extend the model into electroproduction for moderate $Q^2$. We now employ a Regge pole description of the $\pi^0$ electroproduction, following the ``weak cut'' approach used by Goldstein and Owens \cite{GolOwe} where all the observable quantities are defined in terms of the $f$ amplitudes, as related above. From \cite{GolOwe} one can immediately see that the specific combination that contains $H_T$ is the one given above because this implies axial-vector t-channel exchanges. The leading Regge trajectories that are exchanged in this or any two-body diffractive process can be categorized by the signature and parity. The signature determines whether the poles in the exchange amplitudes will occur for even or odd positive integer values of the spin trajectory $\alpha(t) = J$. The leading axial vectors are $b_1$ and $h_1$. These are crucial for determining the tensor charge and the transversity distribution. The parameterization of these trajectories for each helicity amplitude takes the form of product of (1) a t-dependent coupling for both vertices, or the residue function, (2) a signature factor, dependent on the Regge trajectory, that determines the positions of the poles in the t-channel, and (3) the energy dependence, a power fixed by the trajectory. For the even signature exchanges, the natural parity $\rho$ and $\omega$, the amplitudes are \begin{eqnarray} f_1=f_4=\frac{\beta_1^V}{\Gamma(\alpha_V(t))} \Delta \frac{1-e^{-i\pi\alpha_V(t)}}{sin(\pi\alpha_V(t))}\nu^{\alpha_V(t)}e^{-\Delta^2c_V/2} \\ f_2=-f_3=\frac{-\beta_2^V \Delta^2}{2 M\Gamma(\alpha_V(t))} \frac{1-e^{-i\pi\alpha_V(t)}}{sin(\pi\alpha_V(t))}\nu^{\alpha_V(t)}e^{-\Delta^2c_V/2} \end{eqnarray} For the odd signature, unnatural parity $b_1$ and $h_1$, $f_1=f_4=0$, \begin{eqnarray} f_2=+f_3=\frac{\beta_1^A \Delta^2}{2 M\Gamma(\alpha_A(t)+1)} \frac{1-e^{-i\pi\alpha_A(t)}}{sin(\pi\alpha_A(t))}\nu^{\alpha_A(t)}e^{-\Delta^2c_A/2}. \end{eqnarray} The additional contributions to the longitudinal photon amplitudes are limited by parity, charge conjugation invariance and helicity conservation to the $b_1$ and $h_1$ only. To leading order in $s$ they contribute only to $f_5$, which has the minimal $\Delta^1$ dependence and is proportional to the same trajectories' contributions to $f_2$. The precise relation between these two amplitude contributions depends on $Q^2$ and can be related to the decay widths for the axial vectors using vector dominance of the photon. The vector meson trajectories do not couple to the helicity zero photon and pion vertex. Clearly all of these amplitudes vanish in the forward direction, as a result of the factorization of the Regge pole exchanges into two vertices and the fact that parity conservation holds separately for both vertices. This implies that $f_2$, which, in general has the minimal angular dependence of $\Delta^0$, for the Regge pole contributions alone, acquires the same angular dependence as $f_3$. Since the $\Delta\rightarrow 0$ limit is quite important for the identification with transversity through $A_{\Lambda,\lambda;\Lambda^\prime,\lambda^\prime}$ and thus through the GPD $H_T$, it behooves us to consider the rescattering or FSI in this picture. This leads to the Regge cut scheme for rescattering corrections, which was quite important for making contact with spin dependent data. The Regge cut scheme is implemented by first taking the impact parameter representation of the pole terms as an eikonal, \begin{equation} \chi_{Regge}(s,b)=\frac{1}{k\sqrt{s}}\int \Delta d\Delta J_n(b,\Delta)f(s,\Delta^2), \label{eikonal} \end{equation} with $n$ being the helicity change. Next convoluting with the eikonal obtained from helicity conserving Pomeron exchange, $\chi_P(s,b)$ and then transforming back to the momentum space representation \begin{equation} f(s,\Delta^2)=ik\sqrt{s}\int b db J_n(b,\Delta)\chi_{Regge}(s,b)\chi_P(s,b). \label{convolution} \end{equation} This restores the $\Delta^0$ behavior of $f_2$, thereby providing a non-zero tensor charge through the Regge couplings. The tensor charge is embedded in the residues of these $1^{+-}$ axial vector Regge trajectories, $A$ generically. The residue for $b_1$ or $h_1$ is expressed in terms of coupling constants and other dynamical factors that arise from evaluating the residue at the pole position, $t=m_{A}^2$. There \begin{equation} \beta_1^A=\frac{g_{\gamma A \pi} g_{AN\bar{N}}}{4\pi m_A}\frac{\pi \alpha_A^\prime}{4\sqrt{2}}e^{-m_A^2c_A/2}, \label{couplings} \end{equation} and the critical coupling is the $g_AN\bar{N}$ with factors from the Regge parameterization. \section{Generalized Parton Distributions} \label{sec-4} In order to explore the connection between the previous formalism with a partonic picture one needs to rewrite the various observables listed in Eqs.(\ref{dsigT},\ref{dsigL},\ref{dsigTT},\ref{dsigLT},\ref{AUT}) in terms of the correlation functions for the handbag diagram in Fig.\ref{fig1}a. The electroproduction amplitude for hard exclusive pseudoscalar meson production can be written in a factorized form where the soft, process independent, part involves both the description of the meson vertex and linear combinations of GPDs at the nucleon vertex, as in the expression presented in Eq.(\ref{facto}). A formal proof of factorization was given in the case of longitudinally polarized virtual photons producing longitudinally polarized vector mesons \cite{CFS}. The proof hinges on the hypothesis that the initial quark-antiquark pair produced in the hard interaction is in a pointlike configuration, thus granting the cancellation of soft gluons contributions, and only subsequently evolving into the observed meson. Endpoint contributions are surmised to be larger in electroproduction of transversely polarized vector mesons, and to therefore prevent factorization. Standard pQCD calculations \cite{BroFraGun} predict a ratio of $\sigma_L/\sigma_T \propto Q^2$. In \cite{Martinetal} it was observed that when this ratio is calculated in non-perturbative models, an even larger relative suppression of $\sigma_T$ might arise. Notwithstanding current theoretical approaches, many measurements conducted through the years, display larger transverse contributions than expected \cite{Hermes_rho,Hera_rho}. In order to explain the $Q^2$ dependence of the large $W^2$ data in \cite{Martinetal} the hypothesis of duality was used, whereby factorization was assumed, but the meson distribution amplitude was omitted. On the other side, with analogous arguments as for other reactions measured in the multi-GeV region, limitations to the factorization scenario challenging the ``point-like nature'' of the produced $q \overline{q}$ pair even in longitudinal polarization scattering, were suggested in \cite{HoyLen}. In the case of pseudoscalar ($\pi$ and $\eta$) production, preliminary data seem also to indicate transverse contributions larger than predicted within pQCD \cite{kubarovsky}. Lacking a complete formal proof of factorization (see however \cite{Ivanov}), it is therefore important to explore alternative avenues for the meson production mechanism. In this paper we suggest a QCD based model, described in more detail in Section \ref{sec-5}, that predicts different $Q^2$ behaviors for meson production via natural and unnatural parity channels.These are defined in the upper part of the diagram in Fig.\ref{fig2}, as $F_V(Q^2)$, and $F_A(Q^2)$, for the natural and unnatural parity exchanges, respectively. In what follows we give the expressions for the various terms in the $\pi^0$ electroproduction cross section in terms of GPDs and of the $Q^2$-dependent factors. The explicit form of the factors is explained in Section \ref{sec-5}. \subsection{Kinematics and Definitions} \label{sec-3_kin} The off-forward correlation matrix is defined in the collinear approximation as \footnote{Notice that we adopt the axial gauge, although results can be cast in a form highlighting gauge invariance \cite{Ji1}.} \begin{eqnarray} \label{corr1} \Phi_{ab} = \int \, \frac{ d y^-}{2 \pi} \, e^{i y^- X} \, \langle P^\prime S^\prime \mid \overline{\psi}_b(0) \psi_a(y^-) \mid P S \rangle \end{eqnarray} where $P,S$ ($P^\prime,S^\prime$) are the initial (final) nucleon momentum and spin, and we wrote explicitly the Dirac indices $a,b$. The C-parity odd and chiral odd quark density matrix we are interested in~\cite{RalSop,JafJi} involves the struck quark helicity flip via the contraction of $\Phi_{ab}$ with the Dirac matrix $(i \sigma^{+ \, i})_{ba}$. \noindent This contraction, as shown in Ref.\cite{Diehl_01}, gives rise to four chiral odd GPDs, \begin{eqnarray} \int dk^- \, d^2{\bf k} \, \, \rm{Tr} \left[ i\sigma^{+i} \Phi \right]_{X P^+=k^+} & & \nonumber \\ = \frac{1}{2 P^+} \, \overline{U}(P^\prime, S^\prime) \, [ H_T^q \, i \sigma^{+ \, i} + & \widetilde{H}_T^q \, \frac{P^+ \Delta^i - \Delta^+ P^i}{M^2} & \, +E_T^q \, \frac{\gamma^+ \Delta^i - \Delta^+ \gamma^i}{2M} \, \nonumber \\ & + \widetilde{E}_T^q \, \frac{\gamma^+P^i - P^+ \gamma^i}{M}] & U(P,S) \label{oddgpd} \end{eqnarray} where $q=u,d,s$. The nucleon spinors can be explicitly chosen in various ways. Diehl in Ref.~\cite{Diehl_01} chooses light-front helicity spinors to obtain 4 independent chiral odd amplitudes, the $A_{\Lambda^\prime,\lambda^\prime;\Lambda,\lambda}$'s in Eqn.~\ref{facto} and~\ref{f15}. These are linear combinations of the same 4 GPDs that appear in Eqn.~\ref{oddgpd}, namely \begin{subequations} \label{helamps_gpd} \begin{eqnarray} A_{+-,++} & = & - \frac{\sqrt{t_0-t}}{2M} \left[ \widetilde{H}_ T + \frac{1+\xi}{2}E_T - \frac{1+\xi}{2}\widetilde{E}_T \right] \\ A_{++,--} & = & \sqrt{1-\xi^2} \left[ H_ T + \frac{t_0-t}{4M^2} \widetilde{H}_T% - \frac{\xi^2}{1-\xi^2}E_T + \frac{\xi}{1-\xi^2}\widetilde{E}_T \right] \\ A_{+-,-+} & = & - \sqrt{1-\xi^2} \, \, \frac{t_0-t}{4M^2} \, \widetilde{H}_T \\ A_{++,+-} & = & \frac{\sqrt{t_0-t}}{2M} \left[ \widetilde{H}_T + \frac{1-\xi}{2}E_T + \frac{1-\xi}{2}\widetilde{E}_T \right], \end{eqnarray} \end{subequations} where for consistency with previous literature we have used $\xi=\zeta/(2-\zeta)$; $t_0 = - M^2 \zeta^2/(1-\zeta)$, $M$ being the proton mass. The exclusive process observables are defined in terms of the helicity amplitudes of Eqn.~\ref{facto}, which involves the integration over $X$ of the $A$'s with the $g$'s. In the Bjorken limit $g_2$ and $g_5$ are independent of $X$, except for the propagator denominators contained in the $\mathcal{C}_q$. Hence the overall helicity amplitudes $f_1$ to $f_6$ involve the analog of the Compton Form Factors, the Meson Production Form Factors (MPFFs) which can be written generically as~\cite{Die_rev,BelRad}: \begin{eqnarray} \mathcal{F}^q(\zeta,t) = i \pi \left[ F^q(\zeta,\zeta,t) - F^{\bar{q}}(\zeta,\zeta,t) \right] + \mathcal{P} \int\limits_{-1+\zeta}^1 dX \left(\frac{1}{X-\zeta} + \frac{1}{X} \right) F^q(X,\zeta,t). \label{cal_F} \end{eqnarray} where $\mathcal{P}$ indicates a principal value integration and $\mathcal{F}^q = \mathcal{H}_T^q, \mathcal{E}_T^q, \widetilde{\mathcal{H}}_T^q, \widetilde{\mathcal{E}}_T^q$. Notice that the $\Delta_T \equiv \sqrt{t_0-t}$ dependence in Eqs.(\ref{helamps_gpd}) is the same as the minimal $\Delta_T$ dependence predicted within the Regge model (see \cite{GolOwe} and Section \ref{sec-2}). The MPFFs appearing in Eq.(\ref{helamps_gpd}) are: \begin{equation} \mathcal{F} \equiv \mathcal{F}^{p\rightarrow \pi^o} = \frac{1}{\sqrt{2}} \left[ \frac{2}{3} \mathcal{F}^u + \frac{1}{3} \mathcal{F}^d \right]. \end{equation} To form the complete helicity amplitudes the $A$'s of Eqn.~\ref{helamps_gpd} are inserted into integrals over $X$ with the same form as Eqn.~\ref{cal_F}, \begin{eqnarray} \mathcal{A}^q(\zeta,t) = i \pi \left[ A^q(\zeta,\zeta,t) - A^{\bar{q}}(\zeta,\zeta,t) \right] + \mathcal{P} \int\limits_{-1+\zeta}^1 dX \left(\frac{1}{X-\zeta} + \frac{1}{X} \right) A^q(X,\zeta,t). \label{cal_A} \end{eqnarray} Being linear in the GPDs, the relations in Eqn.~\ref{helamps_gpd} are preserved for the integrated $A$'s and $\mathcal{F}$'s. This is due to the fact that the hard process factors, the $g_2$ and $g_5$, have only the $X$ dependence of their propagators, $\mathcal{C}_q$, which provide the denominators in the above integration over $X$. The complete helicity amplitudes, $f_1$ to $f_6$ are then given by the $\mathcal{A}$ multiplied by the $g_2/\mathcal{C}_q$ or $g_5/\mathcal{C}_q$ and the corresponding $Q^2$ dependent factors that are shown in Eqn.~\ref{f15}. The $Q^2$ dependent factors depend on the quantum numbers in the $t$-channel, and will be discussed in Section~\ref{sec-5}. The observables defined in Section \ref{sec-2} can be written in terms of the $p \rightarrow \pi^0$ MPFFs through the helicity amplitudes $f_1$ to $f_6$. \begin{subequations} \label{helamps_gpd2} \begin{eqnarray} f_1 = f_4 & = & \frac{g_2}{\mathcal{C}_q}F_V(Q^2) \frac{\sqrt{t_0-t}}{2M} \left[ \widetilde{\mathcal{H}}_ T + \frac{1-\xi}{2}\mathcal{E}_T + \frac{1-\xi}{2}\widetilde{\mathcal{E}}_T \right] \\ f_2 & = & \frac{g_2}{\mathcal{C}_q}[F_V(Q^2)+F_A(Q^2)]\sqrt{1-\xi^2} \left[ \mathcal{H}_ T + \frac{t_0-t}{4M^2} \widetilde{\mathcal{H}}_T% - \frac{\xi^2}{1-\xi^2}\mathcal{E}_T + \frac{\xi}{1-\xi^2}\widetilde{\mathcal{E}}_T \right] \\ f_3 & = & \frac{g_2}{\mathcal{C}_q}[F_V(Q^2)-F_A(Q^2)] \sqrt{1-\xi^2} \, \, \frac{t_0-t}{4M^2} \, \widetilde{\mathcal{H}}_T \\ f_5 & = & \frac{g_5}{\mathcal{C}_q}F_A(Q^2) \sqrt{1-\xi^2} \left[ \mathcal{H}_ T + \frac{t_0-t}{4M^2} \widetilde{\mathcal{H}}_T% - \frac{\xi^2}{1-\xi^2}\mathcal{E}_T + \frac{\xi}{1-\xi^2}\widetilde{\mathcal{E}}_T \right] , \end{eqnarray} \end{subequations} The $f_6$ amplitude is 0 in this model, since the corresponding $g_6$ is zero, as seen in Eqn.~\ref{g_functions}. With these amplitudes, all of the observables of Section IIA will contain bilinears in the MPFFs. The asymmetries will involve the interference between real and imaginary parts of the bilinear products. \subsection{Model for Transverse GPDs} \label{sec-3-model} We performed calculations using a model for the chiral-odd GPDs derived from the parametrization of Refs.\cite{AHLT1,AHLT2} (AHLT). The parameterization's form for the unpolarized GPD $H$ is \[ H(X,\zeta,t) = G(X,\zeta,t) R(X,\zeta,t), \] \noindent where $R(X,\zeta,t)$ is a Regge motivated term that describes the low $X$ and $t$ behaviors, while the contribution of $G(X,\zeta,t)$, obtained using a spectator model, is centered at intermediate/large values of $X$: \begin{equation} \label{diq_zeta} G(X,\zeta,t) = {\cal N} \frac{X}{1-X} \int d^2{\bf k}_\perp \frac{\phi(k^2,\lambda)}{D(X,\zeta,{\bf k}_\perp)} \frac{\phi({k^{\prime \, 2},\lambda)}}{D(X,\zeta,{ \bf k}_\perp^\prime)}. \end{equation} Here $k$ and $k^\prime$ are the initial and final quark momenta respectively (Fig.\ref{fig1}), $D(X,{\bf k}_\perp) \equiv k^2 - m^2$, $D(X-\zeta/(1-\zeta),{\bf k}_\perp^\prime) \equiv k^{\prime \, 2} - m^2$ ${\bf k}_\perp^\prime = {\bf k}_\perp - (1-X)/(1-\zeta)\Delta_\perp$, $m$ being the struck quark mass, $\Delta=P-P^\prime$ being the four-momentum transfer, and: \begin{eqnarray} k^2 & = & X M^2 - \frac{X}{1-X} M_X^{2} - \frac{{\bf k}_\perp^2}{1-X} \\ k^{\prime \, 2} & = & \frac{X-\zeta}{1-\zeta} M^2 - \frac{X-\zeta}{1-X} M_X^{2} - \left({\bf k}_\perp - \frac{1-X}{1-\zeta} \Delta_\perp \right)^2 \frac{1-\zeta}{1-X}, \end{eqnarray} with $M$, the proton mass, and $M_X$ the (flavor-dependent) diquark mass (we suppress the flavor indices for simplicity), $\phi(k^2,\lambda)$ defines the vertex functions in both the scalar and axial-vector cases \cite{AHLT1}. The normalization factor includes the nucleon-quark-diquark coupling, and it is set to ${\cal N} = 1$ GeV$^6$. $G(X,\zeta,t)$ reduces to the form given in Ref.\cite{AHLT1} in the $\zeta \rightarrow 0$ case. Similar equations were obtained for the spin flip GPD $E$. Similarly to \cite{VandH,DieKro} the $\zeta=0$ behavior is constrained by enforcing both the forward limit: \begin{equation} H^q(X,0,0) = q_{val}(X), \label{qpdf} \end{equation} where $q(X)$ is the valence quarks distribution, and the following relations: \begin{subequations} \begin{eqnarray} \int_0^1 dX H^q(X,\zeta,t) & = & F_1^q(t) \\ \int_0^1 dX E^q(X,\zeta,t) & = & F_2^q(t), \end{eqnarray} \label{FF} \end{subequations} \noindent which defines the connection with the quark's contribution to the Dirac and Pauli form factors. The proton and neutron form factors are obtained as: \begin{subequations} \begin{eqnarray} F_{1(2)}^p(t) & = & \frac{2}{3}F_{1(2)}^u(t) - \frac{1}{3}F_{1(2)}^d(t) + \frac{1}{3}F_{1(2)}^s(t) \\ F_{1(2)}^n(t) & = & -\frac{1}{3}F_{1(2)}^u(t)+ \frac{2}{3}F_{1(2)}^d(t) + \frac{1}{3}F_{1(2)}^s(t), \end{eqnarray} \label{FF2} \end{subequations} Notice that differently from \cite{VandH,DieKro} the AHLT parametrization does not make use of a ``profile function'' for the parton distributions, defined as: \[ H(X,0,t) = q(X) \exp-[t f(X)], \] \noindent whereas in our case the forward limit, $H(X,0,0) \equiv q(X)$, is enforced non trivially. In other words, with the effort of simultaneously having to provide a new parametrization of the PDFs at low initial scale, we gain both the flexibility and insight that are necessary to model the behavior at $\zeta, \, t \neq 0$. The $\zeta$-dependent constraints are given by the higher moments of GPDs. The $n=1,2,3$ moments of the NS combinations: $H^{u-d} = H^u-H^d$, and $E^{u-d} = E^u-E^d$ are available from lattice QCD \cite{zan_talk,haeg}, $n=1$ corresponding to the nucleon form factors. In a recent analysis a parametrization was devised that takes into account all of the above constraints. The parametrization gives an excellent description of recent Jefferson Lab data in the valence region, namely at $\zeta=0.36$. The connection to the chiral-odd GPDs is carried out by considering the following ansatz, similarly to what was adopted in Ref.\cite{Anselmino} for the transversity distribution, $h_1(X) \equiv H_T(X,0,0)$ \begin{eqnarray} H_T^q(X,\zeta,t) & = & \delta q H^{q, val}(X,\zeta,t) \\ \overline{E}_T^q & \equiv & 2 \widetilde{H}_T + E_T = \kappa_T^q H_T(X,\zeta,t) \end{eqnarray} where $\delta q$ is the tensor charge, and $\kappa_T^q$ is the transverse anomalous moment introduced, and connected to the transverse component of the total angular momentum in \cite{Bur2,Bur3}. \section{$Q^2$ dependence} \label{sec-5} The $Q^2$ dependence for $\pi^o$ electroproduction off a proton target, according to the factorization hypothesis, resides in the hard subprocess $\gamma^* q \rightarrow \pi^o q^\prime$, both in kinematical factors (see Eqs.(\ref{g_functions})), and in the description of the pion vertex. In determining the latter, particular care needs to be taken of the chiral odd nature of the reaction outlined in Section \ref{subsec-2.2}. This requires the pion wave function to be proportional to $\gamma_5$. In this way one obtains a chiral odd structure for the hard scattering amplitude as follows \begin{eqnarray} \label{sub_cs} \gamma_5 (\!\not{k} + \!\not{q}) \gamma^\mu & = & (k_\nu+q_\nu)\frac{\gamma_5}{2} \left( \ \left[ \gamma^\nu,\gamma^\mu \right] + \left\{ \gamma^\nu,\gamma^\mu \right\} \right) = (k_\nu+q_\nu)\gamma_5 \left( i \sigma^{\mu\nu} + g^{\mu\nu} \right) \nonumber \\ & \propto & i \gamma_5 \sigma^{\mu\nu} \end{eqnarray} $\pi^o$ production therefore singles out the proton's chiral odd structure. An important point made here is that the Lorentz structure of the process has to be taken into account in addition to the structure of the pion vertex. % By considering both parity and C-parity conservation, and by making use of duality one can view the $\gamma^* q \rightarrow \pi^o q^\prime$ reaction as a transition between a vector particle ($\gamma^*$), and either a vector or an axial vector particle (the two quark legs in Fig.\ref{fig1}a), with the emission of a $\pi^o$. Notice that had one used a $\gamma^\mu \gamma_5$ term at the pion vertex, based on the observation that this is the only collinear, leading twist contribution dictated by the Operator Product Expansion (OPE), one would have obtained $J^{PC}=1^{++}$, t-channel quantum numbers, and a clear violation of $C$-parity in the $\gamma^* q \rightarrow \pi^o q^\prime$ reaction would ensue. Our procedure is to model the pion wave function in this process after imposing $C$-parity conservation. This poses the problem of going beyond the collinear OPE-motivated description, an issue discussed also in \cite{JP}. Here we propose a new model, using crossing symmetry and duality, in which we replace the calculation of the hard subprocess amplitudes (given in Eqs.(\ref{g_functions})), that exhibit the structure in Eq.(\ref{sub_cs}), with the $\gamma^*$-axial/vector meson-$\pi^0 \approx \gamma^* (q \bar{q}) \rightarrow \pi^0$ vertex. This allows us to introduce OAM in our model. A similar structure can be considered both for the GPD and Regge based descriptions. For the latter, at non-zero $Q^2$ the kinematics shifts, so that $s$ depends on $Q^2$ and $x_{Bj}$, the preferred variables for the exclusive process. This kinematic generalization alters the Regge behavior vs. $Q^2$ and $t$ from the real photon limit. But furthermore there is a very strong suppression of the amplitudes and cross sections for increasing photon virtuality. This is indicated by data from DESY \cite{Hermes_rho,Hera_rho} and by theoretical expectations~\cite{BrodskyLepage} that for large $Q^2$ the amplitudes approach dimensional counting requirements from QCD, which predict $1/Q^4$. However, the transition from low to high $Q^2$ is subject to interpretation. Furthermore the data of interest for $\pi^0$ production at Jlab and Compass kinematics are at relatively low values of $Q^2$. Details of our model will be given in a forthcoming paper~\cite{GolLiu}. Here we notice that the quantum numbers of the desired transition form factors can be identified with the ones for the following $J=1$ mesons (Fig.\ref{fig2}), \[ \gamma^* \rho (\omega) \pi^o \] \[ \gamma^* b_1 (h_1) \pi^o, \] or with isovector (isoscalar) vector and axial vector exchanges, respectively. The quark contents of both vertices are: \footnote{In principle a strange quark component appears in $h_1$. This can be, however, disregarded due to the small contribution to the nucleon strange structure function.} \[ \rho \; (b_1) \rightarrow u \bar{u} - d \bar{d} \] \[ \omega \; (h_1) \rightarrow u \bar{u} + d \bar{d} \] In the transition between the vector mesons $J^{PC} = 1^{--}$, and the $\pi^o$, $J^{PC} = 0^{-+}$, the quark-antiquark pair carries an Orbital Angular Momentum (OAM) of $L=0$, both in the initial and final state. The transition between the axial-vector mesons $J^{PC} = 1^{+-}$, and the $\pi^o$, is instead characterized by a change of OAM ($L=1 \rightarrow L=0$). Both the vector (V) and axial-vector (A) vertices have the following Lorentz structure: \begin{equation} \Gamma_\mu=-ie^2F_{\pi\gamma}(Q^2) K_\mu, \end{equation} where the covariant kinematic factor for the vector case is \begin{equation} K_\mu=\epsilon_{\mu\nu\rho\sigma}p^\nu q^\rho \epsilon^\sigma(q^\prime), \label{Kmu} \end{equation} and the index $\mu$ refers to the virtual photon ($q^2=-Q^2$), while the $\epsilon^\sigma$ refers to the real photon ($q^{\prime 2}=0$) or the vector meson. For an axial vector the general form has two form factors, \begin{equation} \epsilon_\mu(q)K^\mu=-ie^2 [F_{\pi A}^{(1)}(Q^2) \epsilon(q)\cdot\epsilon^\prime(q^\prime)+F_{\pi A}^{(2)}(Q^2) (\epsilon^{\prime\mu}(q^\prime) \epsilon^\nu(q) + \epsilon^{\prime\nu}(q^\prime) \epsilon^\mu(q) + (\epsilon^{\prime}(q^\prime) \cdot \epsilon(q)) g^{\mu\nu})q_\mu q^\prime_\nu]. \label{bvertex} \end{equation} So, with no assumptions about the form factors, the longitudinal photon going to a transverse axial vector meson dominates the first, S-wave part (with a factor of $q^{\prime \perp}\nu$ in the Lab frame). For the second part the transverse to transverse transition carries a similar factor. Other transitions are suppressed. For the vector exchange we can see that the transverse to transverse dominates (with a factor of $\nu p_{long}$). The inclusion of the form factors into this mix leads to more complicated conclusions. The structures given above are equivalent to the ones obtained in Eqs.(\ref{g_functions}), where the coupling $g_\pi$ is now replaced by combinations of vector and axial vector form factors. While this will be explicitly shown in \cite{GolLiu}, here we state the essential result for the parameterization of the $Q^2$ dependence that axial vector exchange dominates the longitudinal photon amplitude. For the transverse photon both vector and axial vector will contribute. This is valid in a partonic picture as well as in the Regge approach since it is based only on the $J^{PC}$ quantum numbers for the different processes. We summarize our formulation of the $Q^2$ dependence in Table I. \begin{table}[h] \protect\label{table-1} \center \begin{tabular}{|c|c|c|c|} \hline \hline Photon polarization & t-channel parity & t-channel C-parity & t-channel polarization \\ \hline \hline L & A & -1 & L \\ \hline L & V & -1 & Not allowed \\ \hline T & A & -1 & L, T \\ \hline T & V & -1 & T \\ \hline \hline \end{tabular} \caption{Dominating transitions for the different t-channel exchanges in the reaction $\gamma^* p \rightarrow \pi^o p$.} \end{table} From the Table I it is clear that the following contributions from the $t$-channel spin/parity components will go into the helicity amplitudes $f_i$ ($i=1,6$) of interest, \[ f_1 = f_4 \propto V, \; f_2 \propto A+V, \; f_3 \propto A-V , \; f_5 \propto A. \] The transition form factors can be expressed in a PQCD model with transverse configuration space variables as \cite{StermanLi,KroJak} \begin{eqnarray} F_{\gamma^* V \pi^o} & = & \int dx_1 dy_1 \int d^2{\bf b} \, \psi_{V}(y_1,b) \, \mathcal{C} K_0(\sqrt{x_1(1-x_1)Q^2}b) \, \psi_{\pi^o}(x_1,b) exp(-S) \end{eqnarray} where $x_1(y_1)$ is the longitudinal momentum fraction carried by the quark, ${\bf b}$ is the Fourier transform of the transverse intrinsic momentum, ${\bf k_T}$, $\psi_{V}(y_1,b)$ and $\psi_{\pi^o}(x_1,b)$ are the vector meson and pion wave functions in configuration space, respectively, $\mathcal{C} K_0$ is the Fourier transform of the hard scattering amplitude, where $K_0$ is the modified Bessel function of order zero, $\mathcal{C} = 8 \alpha_S(x_1 y_1Q^2) C_F$, and $exp(-S)$ is the Sudakov exponential. It is important to observe that it is sufficient for our purpose to use the leading twist pion wave function since the power suppression due to the $\gamma^5$ coupling is already accounted for in $g_2$ and $g_5$ (Eqs.(\ref{g_functions})). Furthermore, in the vector case the OAM is the same in the initial and final states ($L=0$) whereas for an axial-vector meson in the initial state, the OAM changes from $L=1$ to $L=0$, leading to \begin{eqnarray} F_{\gamma^* A \pi^o} & = & \int dx_1 dy_1 \int d^2{\bf b} \, \psi_{A}^{(1)}(y_1,b) \, \mathcal{C} K_o(\sqrt{x_1(1-x_1)Q^2}b) \, \psi_{\pi^o}(x_1,b) exp(-S) \end{eqnarray} where now \begin{equation} \psi_{A}^{(1)}(y_1,b) = \int d^2 k_T J_1(y_1 b) \psi(y_1,k_T), \end{equation} a higher order Bessel function appears as a consequence of having $L=1$ in the initial state \cite{RCARnold}. In impact parameter space this yields configurations of larger radius. In terms of meson distribution amplitudes this is described by functions of higher twist originating from the ``bad'' components of the quark spinors \cite{BalBra}. We evaluated the form factors by using the asymptotic twist-two, $\phi_L(T) = 6x(1-x)$, and twist-three, $g_T = (3/4)[1+(2x-1)]^2$, amplitudes, defined in Ref.~\cite{BalBraTan}, corresponding to the same isospin but different spin configurations for the two mesons. We conclude by noting that while our approach might shed some light on the presence of large transverse polarization components in a number of recent exclusive measurements, we cannot straightforwardly apply it to vector meson production, since this is dominated by t-channel exchanges other than the axial and vector types governing $\pi^o$ production. In summary, we introduced a model for the $Q^2$ dependence for vector and axial vector exchanges. They differ because in the axial vector cases there is a change of one unit of OAM producing a suppression with respect to the vector. More details and more comparisons with $\pi^o$ electroproduction data will be given in a forthcoming manuscript \cite{GolLiu}. \section{Results} \label{sec-6} We now present our quantitative results for $\pi^o$ electroproduction cross sections and asymmetries both in the kinematical regime of currently analyzed experimental data obtained at $\gamma^* p$ CM energy, $4$ GeV$^2$ $ \lesssim W \lesssim$ $9$ GeV$^2$, and $Q^2$ in the multi-GeV region, and in a larger energy and momentum transfer regime. Approximate scaling was found to hold in the case of DVCS \cite{DVCSA,DVCSB}. We therefore expect our picture based on chiral odd GPDs to be valid in this regime. A Regge type description can also be reliably applied in this region for $-t << s$. The interplay between the Regge and partonic descriptions is key to the physical interpretation of GPDs and TMDs, and it constitutes the main motivation of our study. Measurements from Jefferson Lab on $\pi^0$ production \cite{HallA_proposal} show non-negligible, larger than theoretically surmised, contributions from transversely polarized photons. Important aspects of our approach that guided us towards an interpretation of $\pi^o$ electroproduction data are that: {\it i)} a multi-variable analysis needs to be performed that is sensitive to the values of the tensor charges, $\delta u$ and $\delta d$, and of the transverse moments, $\kappa_T^u$ and $\kappa_T^d$; {\it ii)} we consider a different $Q^2$ dependence of natural and unnatural t-channel exchanges governing both the Regge and GPD approaches. In what follows we provide a survey of the effects of the variations of the transversity parameters on different observables. \subsection{Cross Sections} In Figures \ref{fig3}, \ref{fig4} and \ref{fig5}, we show our predictions for the different contributions to the $\gamma^* p \rightarrow \pi^o p$ cross section, obtained both in the Regge model, and in the GPD-based calculations. Both models predict similar trends in the measured experimental regime despite their seemingly different physical nature. The different contributions to the cross section are in fact sensitive to the values of the tensor charge residing in the helicity amplitudes $f_2$ and $f_5$, as explained in the previous Sections, and defining their normalization as $t \rightarrow t_{min}$. As for the helicity amplitudes $f_1$ and $f_4$, a connection can be established between GPDs and the (normalization of the) Boer-Mulders function through the concept of ``transverse spin anomalous magnetic moment'', $\kappa_T$. We expect a similar connection to be established for the Regge amplitudes as well. In the GPD model we used $\delta u =0.48$, $\delta d =-0.62$, namely the values of the tensor charges extracted from the global analysis of semi-inclusive data in Ref.\cite{Anselmino}, and the values $\kappa_T^u =0.6$, $\kappa_T^d=0.3$. The latter are smaller than currently available lattice Ref.\cite{haeg}, and model Ref.\cite{BofPas} calculations. It should be remarked that the $t$-dependence in the GPD model follows closely what was found for the unpolarized case, {\it i.e.} $\mathcal{H}$, in DVCS data in a similar kinematical regime. The somewhat flatter $t$-dependence at large $x_{Bj}$ (lower panels in Fig.\ref{fig3}) is due to the interplay of the imaginary and real parts of the helicity amplitudes. In our approach, in fact, $\mathcal{H}_T$ has a similar trend to $\mathcal{H}$ in the unpolarized case. This is in turn determined by a parametrization constrained by the DVCS data. Both its real and imaginary parts are therefore decreasing with $-t$, but the real part being negative produces a less steep dependence of the cross sections with $-t$. \subsection{Asymmetries} More marked differences between the two approaches appear both in the transverse target spin asymmetry, $A_{UT}$ and in the Beam Spin Asymmetry (BSA), proportional to $d \sigma_{LT^\prime}/dt$. In the GPD model used in this paper the size of $A_{UT}$ in fact depends almost solely on the value of the tensor charge, due to the almost exact cancellation of the $\Im m (f_1^* f_3)$ term in Eq.(\ref{AUT}). $A_{UT}$ is therefore approximately proportional to $\mathcal{H}_T$. Such a cancellation does not occur in the Regge model, as it can be clearly seen at larger values of $-t$. This is a manifestation of the natural parity exchanges which become dominant at larger $-t$. However, because, of the proportionality of the helicity amplitudes of $f_1$ and $f_3$ to $\sqrt{t_0-t}$ and $t_0-t$, respectively, as $t$ approaches $t_{min}$, the amplitude $f_2$, measuring the tensor charge is the main contribution at low $t$. This is consistent with the approximations used in our GPD model, and our proposed extractions are indeed valid at $t < Q^2$ where the GPD-based description of the electroproduction cross sections applies. On the other hand, in $d \sigma_{LT^\prime}/dt$, a cancellation occurs of the $\Im m (f_5^* f_2)$ term, Eq.(\ref{dsigLTp}). Therefore $d \sigma_{LT^\prime}/dt$ is only the dependent on the GPD, $\bar{E}_2$, allowing one in principle to measure the sensitivity to $\kappa_T$. Again, in the Regge model the above cancellation is only partial because of a more complicated interplay between the natural and unnatural parity exchanges. However we observe a similar trend showing the suppression of the tensor charge dependent term. We are therefore able to single out the observables $A_{UT}$ and $d \sigma_{LT^\prime}/dt$ as probes of the tensor charge and of $\kappa_T$. Results for the asymmetries are shown in Figures \ref{fig6}, \ref{fig7},\ref{fig8},\ref{fig9}. $A_{UT}$, Eq.(\ref{AUT}), and $\alpha$, Eq.(\ref{alpha}), are given as a function of $t$ in Figs.\ref{fig6} and \ref{fig7}, respectively. In Fig.\ref{fig6} we compare the Regge and GPD models. For the Regge we show separately the contributions of the combination of helicity amplitudes including the tensor charge, and the ones sensitive to $\kappa_T$, together with the total contribution. More specifically we separate out the terms proportional to $\Im m (f_1^* f_3)$ and $\Im m (f_1^* f_2)$, respectively. The latter, given by the dot-dashed curve, clearly dominates the asymmetry at low $t$. The GPD model is instead governed entirely by the tensor charge term. A similar picture is obtained for $\alpha$, shown in Fig.\ref{fig7}, by inverting the role of the tensor charge and $\kappa_T^{u,d}$ terms. However, here the only surviving term in our GPD model is $\Im m (f_5^* f_3)$, which is very small due to the smallness of the helicity amplitude $f_3$. The Regge model gives larger contributions for $f_3$. These are shown at different recently measured kinematics. The sensitivity of $A_{UT}$ in the GPD model to the values of the u-quark and d-quark tensor charges, is shown in Fig.\ref{fig8}. The values in the figure were taken by varying up to $20 \%$ the values of the tensor charge extracted from the global analysis of Ref., {\it i.e.} $\delta u=0.48$ and $\delta d = -0.62$, keeping the transverse anomalous magnetic moment values, $\kappa_T^u = 0.6$ and $\kappa_T^d = 0.3$. Fig.\ref{fig8} is one of the main results of this paper: it summarizes our proposed method for extracting the tensor charge from $\pi^o$ electroproduction experiments. A practical extraction of the tensor charge can be obtained by noticing that for the asymmetry, as well as for other quantities evaluated in this paper such as $d \sigma_{TT}/dt$, and $d \sigma_{LT}/dt$ the tensor charges for the different isospin components might be treated as parameters related to the normalization of $H_T$ ($d \sigma_{LT}/dt$ is plotted in Fig.\ref{fig9}). Therefore our model can be used to constrain the range of values allowed by the data. In Fig.\ref{fig10} we show the sensitivity of $A_{UT}$ to the tensor charge values at fixed $t=-0.3$ GeV$^2$, and as a function of $x_{Bj}$. As in the previous figures we took $\kappa_T^u=0.6$ and $\kappa_T^d=0.3$. We performed calculations for a range of values of $Q^2$. We find that the $Q^2$ dependence of $A_{UT}$ is rather small due to the cancellations of the form factors in the ratio (Eq.(\ref{AUT})). On the contrary, as can be seen from Fig.\ref{fig11} for {\it e.g.} $d\sigma_{LT}/dt$, the single contributions to the cross section expectedly display a steep $Q^2$ dependence. Notice that the electroproduction data are essential in determining the tensor charge and other transversity related quantities. This is illustrated in Fig.\ref{fig12} where we show the photoproduction cross section calculated following the model in \cite{GolOwe}. The value of the tensor charge is extracted in this case from the the Regge residue of the axial vector contribution to the helicity amplitude $f_2$ (see Section \ref{sec-3}) at $t \rightarrow 0$. This is plotted in the lower panel of Fig.\ref{fig12} where the central value extracted by fitting the model parameters to the photoproduction data is shown along with curves corresponding to a $\pm 30 \%$ variation of the tensor charge (labeled correspondingly in the figure). From the figure it is clear that the photoproduction cross section is very little affected by variations in the values of the tensor charge, except very near forward. \subsection{$Q^2$ dependence} The $Q^2$ dependence of our model is illustrated in Figs. \ref{fig13} and \ref{fig14}. Fig.\ref{fig13} shows the different form factors describing the upper vertex in Fig.\ref{fig1}b, for the different helicity amplitudes, namely $F_1(Q^2) = F_V(Q^2)/2$, $F_2(Q^2) = (F_V(Q^2) + F_A(Q^2))/2$, $F_3(Q^2)= - (F_V(Q^2)-F_A(Q^2))/2$, and $F_5(Q^2) = F_A(Q^2)/2$, where $F_V$ and $F_A$ are a short notation for the vector and axial transition form factors,$F_{\gamma^*V\pi}$, and $F_{\gamma^*A\pi}$ introduced in Section \ref{sec-5}. All form factors were calculated using the approach described in Section \ref{sec-5}. The axial form factor displays a steeper $Q^2$ dependence at large $Q^2$, due to the difference in orbital angular momentum between the initial and final hadronic states. In Fig.~\ref{fig14} we show the impact of multiplying the different helicity amplitudes by different form factors on some of the observables which are governed by either longitudinal or transverse photon polarization. Results are shown at $x_{Bj}=0.36$ for two different values of $t$, $t=-0.3$ GeV$^2$ and $t=-0.7$ GeV$^2$, in the GPD model. Despite the fact that the $t$ dependence plays an important role, as can be seen from Figs. \ref{fig3},\ref{fig4}, \ref{fig5}, we expect the longitudinal to transverse ratios, $\sigma_L/\sigma_T \propto \left( F_A/(F_A+F_V) \right)^2$, and $ \sigma_{LT}/\sigma_{TT} $, to have a less steep $Q^2$ dependence than the one based on simple PQCD predictions. The shape of the curves is a consequence of the difference in the $Q^2$ behavior for the axial and vector form factors, whose ratio displays a $1/Q \log^a(Q^2/\Lambda)$ dependence. These however enter the cross section in different linear combinations, and with different weights depending on the values of $t$, giving rise to the curves shown in the figure. It should be noticed that this is qualitatively different from taking different monopole masses for axial vector and vector meson form factors, and assuming the same $Q^2$ behavior \cite{VdHLag}. \section{Conclusions and Outlook} \label{sec-7} In conclusion, we presented a framework for analyzing $\pi^o$ exclusive electroproduction where, by observing that this reaction proceeds through C-parity odd and chiral odd combinations of t-channel exchange quantum numbers, it can be selected to obtain direct measurements of the meson production form factors for the chiral odd generalized parton distributions. This is at variance with deeply virtual Compton scattering, and with both vector meson and charged $\pi$ electroproduction, where the axial charge corresponding to C-parity even exchanges can enter the amplitudes. We then studied the different terms appearing in the cross section for scattering from an unpolarized proton, including the beam polarization asymmetry using the helicity amplitudes formalism. A Regge based description based on the ``weak cut'' approach was adopted where the leading axial vectors exchanges, $b_1$ and $h_1$, determine the tensor charge and the transversity distribution, while the leading vector exchanges, $\rho$ and $\omega$ can be related to the transverse anomalous moment. The partonic description, singling out the chiral odd GPDs, was implemented to show the sensitivity of some of the observables, in particular the interference terms in the unpolarized cross section to the values of the $u$ and $d$ quark tensor charges, as well as to the values of the $u$ and $d$ quark transverse anomalous moments. Predictions were also given for the transverse target spin asymmetry, $A_{UT}$. The various correspondences between the Regge approach and the GPD models were highlighted. This aspect of the of analysis represents an avenue that we will continue to pursue in the near future. Finally, we expect a variety of new flavor sensitive observables to be extracted from the data in the near future using both unpolarized data and asymmetries from transversely polarized proton and deuteron data on $\pi^o$ and $\eta$ production at the higher $s$ values attainable at Jefferson Lab at 12 GeV. The extension of our analysis to these types of reactions promises to be a rich area for both theoretical and experimental exploration in the near future. \acknowledgments We thank Leonard Gamberg for participating in initial discussions. We are also thankful to John Ralston, Paul Stoler, and Christian Weiss for useful comments. This work is supported by the U.S. Department of Energy grants no. DE-FG02-01ER4120 (S.A. and S.L), and no. DE-FG02-92ER40702 (G.R.G.).

BkiUbJrxaJiQoRGw4hTL

\section{Introduction} The $\Lambda(1405)$ resonance is getting renewed experimental attention and different reactions have been used to produce it. One of the recent reactions is the $K^- p \to \pi^0 \pi^0 \Sigma^0$ measured at $p_{K^-}$ = 514 MeV/c to 750 MeV/c in \cite{Prakhov:2004an}. COSY at Juelich has also produced it in the $p p \to K^+ p \pi^0 \Sigma^0$ reaction \cite{Zychor:2007gf}. The photoproduction of the resonance has been investigated at LEPS \cite{Niiyama:2008rt} and CLAS \cite{Moriya:2009mx} and plans are made to continue its study in DAFNE and J-PARC among other facilities. One of the motivations to search for it in different reactions was the theoretical observation in \cite{Jido:2003cb} that the use of chiral dynamics led to two nearby poles in the region of the $\Lambda(1405)$ resonance, and that different reactions gave more weight to one or the other poles, such that the shape of the cross section would change from one reaction to another. The $\Lambda(1405)$ has been a historical example of a dynamically generated resonance in meson-baryon coupled-channels dynamics with $S=-1$~\cite{Dalitz:1967fp}. Modern investigations based on chiral dynamics with a unitary framework also reproduce well the observed spectrum of the $\Lambda(1405)$ together with cross sections of $K^{-}p$ to various channels~\cite{Kaiser:1995eg,Oset:1998it,Oller:2000fj,Oset:2001cn,GarciaRecio:2002td,Hyodo:2002pk,Hyodo:2003qa,Hyodo:2008xr}. The work of \cite{Jido:2003cb} also motivated further theoretical studies introducing terms of higher order in the kernel of the $K^-p$ interaction with its coupled channels \cite{Borasoy:2005ie,Oller:2006jw,Borasoy:2006sr}. The interesting thing observed in these works is that the two pole structure is always there independent of variations in the parameters allowed by the data. One of the states, located around 1420 MeV, couples dominantly to the $\bar KN$ channel and is very stable, while the other one, sitting around 1390 MeV and with a width of 130 MeV or larger, couples strongly to the $\pi \Sigma$ channel. This latter one is more model dependent but the mass is smaller than that of the other state and the width is considerably larger. Another interesting output of these works is that the results including the higher order kernel are compatible with those with the lowest order Lagrangian (the Weinberg Tomozawa term) within theoretical uncertainties \cite{Borasoy:2006sr}. Depending on the reaction mechanism and which is the channel predominantly chosen by the reaction to create the $\Lambda(1405)$, one obtains one shape or another from the superposition of the two resonances with different weights. Certainly, if one finds a reaction in which the $\Lambda(1405)$ is basically created through the entrance channel $\bar KN$, this will give more weight to the narrow, higher energy resonance, and one should see a spectrum peaking around 1420 MeV. The problem is that the $\bar KN$ threshold is already above the resonance mass. An ideal method to produce the narrow resonance is the $K^- p \to \gamma \Lambda(1405)$ reaction, since the mechanism where the photon is radiated from the initial $K^-$ is dominant. Then the emission of the photon reduces the $K^-$ energy, such as to be able to produce the $\Lambda(1405)$, and one ensures the creation of the resonance from $K^- p$. The theoretical study for this reaction was done in \cite{Nacher:1999ni}, but the experiment is not yet done. Conversely, the study done in \cite{Hyodo:2003jw} for the $\pi^-p\to K^0\pi\Sigma$ reaction showed that this was dominated by the $\pi \Sigma$ entrance channel and this justified the peak seen in the $\pi \Sigma$ invariant mass at around 1400 MeV in the experiment \cite{Thomas:1973uh}. Other reactions like the $p p \to K^+ p \pi^0 \Sigma^0$ \cite{Zychor:2007gf}, require a mixture of both resonances as seen in \cite{Geng:2007vm}. One of the strongest support for the idea of the two $\Lambda(1405)$ states comes from the experiment of \cite{Prakhov:2004an}, $K^- p \to \pi^0 \pi^0 \Sigma^0$, and the subsequent theoretical analysis of the reaction done in \cite{magas}. In this case one of the $\pi^0$ is ``radiated" from the initial proton, the $K^- p$ system loses energy, and as a consequence the $\Lambda(1405)$ production is initiated by the $K^- p$ system, exciting the higher mass state that couples mostly to the the $K^- p$ system and producing a peak in the $\pi \Sigma$ spectrum around 1420 MeV. Another great support for the idea of the two $\Lambda(1405)$ states comes from the $K^{-} d \to \pi\Sigma n$ reaction \cite{Braun:1977wd} with kaons in flight, where a clear peak is seen in the $\pi \Sigma$ spectrum around 1420 MeV. This reaction was studied recently in \cite{Jido:2009jf} and it was found that the rescattering mechanism of the $K^-$, in which the $K^-$ scatters on the neutron, loses some energy, propagates to the proton and produces the $\Lambda(1405)$ on the proton target, dominates the reaction in this invariant mass region. Since the production of the $\Lambda(1405)$ is done from the $K^- p$ channel, it also excites predominantly the higher mass $\Lambda(1405)$ state, which shows up in the $\pi \Sigma$ spectrum with a peak around 1420 MeV. The detail of kaons in flight is important. Indeed, in \cite{Jido:2009jf} it was shown that, together with the double scattering, where the $\Lambda(1405)$ shows up cleanly, there is the contribution of the single scattering, the impulse approximation. This contribution peaks around a value of the $\pi\Sigma$ invariant mass, corresponding to the invariant mass of the kaon in flight and one nucleon at rest. If one has sufficient kaon kinetic energy this peak and the one of the $\Lambda(1405)$ are quite separated. If one goes at threshold with kaons at rest, the single scattering peak is around $m_K +M_N$, does not have resonant structure and by interference with the double scattering distorts the shape of the $\Lambda(1405)$ contribution, and the reaction is not suited to investigate the properties of this resonance. In \cite{Esmaili:2009rf} a different opinion is expressed and claims are made that the impulse approximation can provide the shape of the resonance. We argue below that this is not the case. Indeed, one can get $K^- p $ invariant masses below threshold for kaons at rest, only through the Fermi motion of the deuteron. Yet, to reach invariant masses of the order of 1400 MeV, down from the threshold of 1432 MeV, one needs momentum components of the order of 300 MeV/c. Obviously, our knowledge of the deuteron wave function for such large momentum components is extremely poor. In fact, standard wave functions of the deuteron have dropped by two orders of magnitude at 300 MeV/c, with appreciable differences between different models. This is four orders of magnitude in the cross section, where the uncertainties are certainly not smaller than one order of magnitude. Even the concept of wave function, when one has a transfer of energy of 32 MeV to a second nucleon, is ambiguous. Trying to reconstruct a shape of the resonance, by dividing an experimental cross section by the theoretical deuteron wave function squared has very large uncertainties and should not be used. The possible use of an ``experimental" wave function for these high momentum components should be equally discouraged, since processes involving such momentum transfers have contributions from two body mechanisms difficult to quantify. DAFNE produces intense kaon beams from $\phi$ decay. Recently the $\phi$ mesons can even be produced with a small momentum of 12 MeV/c, by means of which the kaons have a range of 80 MeV/c to 130 MeV/c when they reach the target. We shall take advantage of this and investigate the production process for the most energetic kaons, looking at different kinematical conditions that maximize the clean production of the resonance. \section{Formulation} \label{sec:form} We briefly sketch here the formalism of \cite{Jido:2009jf} for this reaction. Further details can be seen in that paper. \begin{figure} \centerline{\includegraphics[width=6.5cm]{fig1.eps}} \caption{Kinematics of the $K^{-} d \to \pi \Sigma n$. \label{fig1}} \end{figure} We consider $\Lambda(1405)$ production induced by $K^{-}$ with a deuteron target, $K^{-} d \rightarrow \Lambda(1405) n$. The $\Lambda(1405)$ produced in this reaction decays into $\pi \Sigma$ with $I=0$ as shown in fig.~\ref{fig1}. The $\Lambda(1405)$ is identified by the $\pi \Sigma$ invariant mass spectra of this reaction. Figure~\ref{fig1} also gives the kinematical variables of the initial and final particles. The kinematics of the three-body final state is completely fixed by five variables, the $\pi\Sigma$ invariant mass $M_{\pi\Sigma}$, the neutron solid angle $\Omega_{n}$ in the c.m.\ frame and the pion solid angle $\Omega_{\pi}^{*}$ in the rest frame of $\pi$ and $\Sigma$~\cite{Amsler:2008zz}. Thus the differential cross section of this reaction can be written as \begin{equation} d \sigma = \frac{1}{ (2\pi)^{5}} \frac{M_{d}M_{\Sigma} M_{n}}{4k_{\rm c.m.}E_{\rm c.m.}^{2}} \, |{\cal T}|^{2} |\vec p_{\pi}^{\, *}|\, |\vec p_{n}|\, dM_{\pi\Sigma} d\Omega_{\pi}^{\,*} d\Omega_{n} \label{eq:difcross} \end{equation} where $\cal T$ is the $T$-matrix of this reaction, $E_{\rm c.m.}$ is the center of mass energy, $k_{c.m.}$ is the kaon CM momentum and $\vec p_{\pi}^{\, *} $ is the pion momentum in the rest frame of $\pi$ and $\Sigma$. The pion momentum $|\vec p_{\pi}^{\, *}|$ in the $\pi \Sigma$ rest frame can be fixed by the invariant mass $M_{\pi\Sigma}$. \begin{figure} \begin{center} \centerline{\includegraphics[width=8.5cm]{fig2.eps}} \caption{Diagrams for the calculation of the $K^{-}d \to \pi\Sigma n$ reaction. $T_{1}$ and $T_{2}$ denote the scattering amplitudes for $\bar KN \to \bar KN$ and $\bar K N \to \pi \Sigma$, respectively. \label{fig2}} \end{center} \end{figure} The $\Lambda(1405)$ production is investigated by limiting the kinematics with the invariant mass of the final $\pi \Sigma$ state around 1350 to 1450 MeV, in which the resonating $\pi\Sigma$ forms the $\Lambda(1405)$ and the resonance contribution may dominate the cross section. In this $\Lambda(1405)$ dominance approximation, we have three diagrams for this reaction as shown in fig.~\ref{fig2}. The left diagram of fig.~\ref{fig2} expresses the $\Lambda(1405)$ production in the impulse approximation. We refer to this diagram as direct production process. The middle and right diagrams are for two-step processes with $\bar K$ exchange. We refer to these diagrams as double scattering diagrams. In fig.~\ref{fig2}, $T_{1}$ and $T_{2}$ denote $s$-wave scattering amplitudes of $\bar K N \to \bar KN$ and $\bar K N \to \pi \Sigma$, respectively. These amplitudes are calculated in a coupled-channel approach based on chiral dynamics. The ${\cal T}$ matrix appearing in Eq. (\ref{eq:difcross}) is given by the sum of the contribution of the three diagrams of fig.~\ref{fig2} \begin{equation} {\cal T} = {\cal T}_{1} + {\cal T}_{2} + {\cal T}_{3} \end{equation} where the different amplitudes are given by \begin{equation} {\cal T}_{1} = T_{K^{-}p \rightarrow \pi \Sigma}(M_{\pi\Sigma}) \, \varphi(\vec p_{n} - \frac{\vec p_{d}}{2}). \label{eq:T1} \end{equation} \begin{eqnarray} {\cal T}_{2}& =& T_{K^{-}p \rightarrow \pi \Sigma}(M_{\pi\Sigma}) \int \frac{d^{3}q}{(2\pi)^{3}} \frac{\tilde \varphi (\vec q+\vec p_{n}-\vec k - \frac{\vec p_{d}}{2})}{q^{2}-m_{K}^{2} + i\epsilon} \nonumber \\ && \times T_{K^{-}n \rightarrow K^{-}n}(W_{1}) \ . \label{eq:T2} \end{eqnarray} \begin{eqnarray} {\cal T}_{3}& =& - T_{\bar K^{0}n \rightarrow \pi \Sigma}(M_{\pi\Sigma}) \int \frac{d^{3}q}{(2\pi)^{3}} \frac{\tilde \varphi (\vec q+\vec p_{n}-\vec k - \frac{\vec p_{d}}{2})}{q^{2}-m_{K}^{2} + i\epsilon} \nonumber \\ && \times T_{K^{-}p \rightarrow \bar K^{0}n}(W_{1}) \ . \label{eq:T3} \end{eqnarray} with $\varphi(\vec p_{n} - \frac{\vec p_{d}}{2})$ the deuteron wave function in momentum space and \begin{eqnarray} q^{0} &=& M_{N} + k^{0} - p_{n}^{0}\ , \\ W_{1} &=& \sqrt{(q^{0}+p_{n}^{0})^{2}-(\vec q + \vec p_{n})^{2}} \ . \end{eqnarray} For $q^0$ we have assumed that the deuteron at rest has energy $2M_N -B$, and we have taken half of it for one nucleon, neglecting the small binding energy. The variable $W_1$ depends, however, on the running $\vec{q}$ variable. It should be noted, as it was also the case in \cite{Jido:2009jf}, that taking an average value of $\vec q$, we can fix $W_{1}$ by the external variables as \begin{equation} W_{1} = \sqrt{(M_{N}+k^{0})^{2} - \vec k^{\, 2}}, \label{eq:W} \end{equation} which allows us to take the $T_{\bar KN \to \bar KN}$ amplitude out of the loop integral and produces results practically equal to those obtained with the values of these variables depending on the loop variable. Note also that $T_{\bar KN \to \pi\Sigma}$, which is the amplitude producing the resonance shape, has its argument fixed in all cases by the external $\pi \Sigma$. The $T$-matrix for the diagram 1 given in fig.~\ref{fig2} has been calculated in the impulse approximation in which the incident $K^{-}$ and the proton in the deuteron produce the $\pi \Sigma$ and the neutron behaves as a spectator of the reaction. As for the external particles, we have taken the wave functions of the incident kaon and the particles in the final state as plane waves. The double scattering requires a loop integral over the $\vec q$ variable in Eqs.~(\ref{eq:T2}) and (\ref{eq:T3}) with the deuteron wave function. Although the procedure is standard in multiple collision in nuclei, we find it appropriate to sketch the connection to a conventional loop evaluation with Feynman diagrams. The loop integral would be \begin{equation} \int \frac{d^{3} q}{(2\pi)^{3}} \frac{1}{E_{\rm tot} - k^{0} -H_{0NN}} t_{NN} \phi_{NN} \frac{T_{\bar KN \to \bar KN} T_{\bar KN \to \pi\Sigma}}{q^{2} - m_{K}^{2} + i \epsilon} \ , \end{equation} where $H_{0}$ is the free $NN$ Hamiltonian, $t_{NN}$ the appropriate $NN$ scattering matrix and $\phi_{NN}$ the free $NN$ wave function. By using the definition of the scattering matrix \begin{equation} t_{NN} \phi_{NN} = V_{NN} \Psi_{NN} \end{equation} where $V_{NN}$ is the $NN$ interaction potential and $\Psi_{NN}$ is the actual $NN$ wave function, the integral becomes \begin{equation} \int \frac{d^{3} q}{(2\pi)^{3}} \frac{1}{E_{NN} -H_{0NN}} V_{NN} \Psi_{NN} \frac{T_{\bar KN \to \bar KN} T_{\bar KN \to \pi\Sigma}}{q^{2} - m_{K}^{2} + i \epsilon} \ . \end{equation} Now taking into account the Schr\"odinger equation \begin{equation} \Psi_{NN} = \frac{1}{E_{NN}-H_{0NN}} V_{NN} \Psi_{NN} \ , \end{equation} one rewrites the integral as \begin{equation} \int \frac{d^{3} q}{(2\pi)^{3}} \Psi_{NN} \frac{T_{\bar KN \to \bar KN} T_{\bar KN \to \pi\Sigma}}{q^{2} - m_{K}^{2} + i \epsilon} \end{equation} which is the integral that appears in Eqs.~(\ref{eq:T2}) and (\ref{eq:T3}). \section{Results} In fig.~\ref{fig3} we show $d \sigma/dM_{\pi \Sigma}$ for $\pi^+ \Sigma^-$, $\pi^- \Sigma^+$ and $\pi^0 \Sigma^0$ production for 130 MeV/c $K^{-}$ momenta in the lab.~frame. What we see is a clear dominance of the single scattering. However, we observe a certain structure, more visible in the $\pi^+ \Sigma^-$ spectrum, with a peak at the invariant mass squared, $M_{\pi\Sigma}^{2}=(k+p_{N})^{2}$, with $p_{N}$ at rest in the deuteron rest frame, where the deuteron wave function has its largest weight. This peak has to do with the deuteron wave function and not with the amplitude $T_{K^{-}p\to \pi\Sigma}$ of Eq.~(\ref{eq:T1}). Then we see another peak in the region of 1420~MeV. This second peak comes from the shape of the $T_{K^{-}p\to \pi\Sigma}$ amplitude of Eq.~(\ref{eq:T1}), which reflects the excitation of the higher energy $\Lambda(1405)$ resonance appearing around 1420~MeV. We also see that the mechanism of double scattering also peaks around 1420 MeV and there is interference with the single scattering amplitude which, nevertheless, leaves the peak at about the same position. Yet, since the shape of the double scattering is cleaner and does not have the double shoulder, it would be interesting to find some situation in which it becomes dominant. It is easy to find such kinematical set up, since in the impulse approximation the neutron is a spectator and goes backwards in the $K^- d$ CM frame. If we demand that the neutron goes forward we shall be reducing drastically the single scattering contribution, giving room to the double scattering one. The results of imposing angular cuts can be seen in fig.~\ref{fig4}, in which $\theta_{\rm c.m.}^{n}$ is the angle between the outgoing neutron and the incident $K^{-}$ in the CM frame. We can see that for large angles of the neutron the impulse approximation is largely dominant, yet, as soon as we make the angles of the neutron smaller, the contribution of the single scattering becomes gradually smaller and at angles smaller than 30 degrees the double scattering has become dominant, with the total cross section showing a clean peak around 1420~MeV. To make it more precise, in fig.~\ref{fig5} we show the quantity $d \sigma/dM_{\pi \Sigma}\,d\cos(\theta_{\rm c.m.}^{n})$ for different angles and we find that for angles below 32 degrees the double scattering contribution is dominant. Hence, for practical purposes, and in order to gain statistics, gathering all events with angles smaller than 32 degrees in the CM is ideal in order to show a clean contribution of the $\Lambda(1405)$. The cross sections of 0.1 mb/MeV, from Fig. \ref{fig4}, are large enough to be observed in the DAFNE set up. A larger span of angles would also show a clear peak, with gain in statistics, since even at 60 degrees the peak is visible, as one can appreciate in figs.~\ref{fig4} and~\ref{fig5}. \begin{figure} \begin{center} \centerline{\includegraphics[width=7cm]{ChUM_PiSig_dia_130.eps}} \caption{$\pi\Sigma$ invariant mass spectra of $K^{-}d \to \pi \Sigma n$ with 130 MeV/c of $K^{-}$ incident momentum. The three panels correspond to $\pi\Sigma$ spectra for different $\pi\Sigma$ charge combinations, $\pi^{+}\Sigma^{-}$ (upper panel), $\pi^{-}\Sigma^{+}$ (middle panel) and $\pi^{0} \Sigma^{0}$ (lower panel). In each panel, the solid line denotes the total contributions of the three diagrams, while the dotted and dash-dotted lines show the calculations from the single and double scatterings, respectively. } \label{fig3} \end{center} \end{figure} \begin{figure} \begin{center} \centerline{\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_int_130.eps}} \caption{$\pi\Sigma$ invariant mass spectra of $K^{-}d \to \pi^{0}\Sigma^{0}n$ with 130 MeV/c of incident $K^{-}$ momentum imposing angular cuts for the emitted neutron with respect to the incident $K^{-}$ in the CM frame, $90^{\circ}<\theta^{n}_{\rm c.m.}<180^{\circ}$ (up-left), $60^{\circ}<\theta^{n}_{\rm c.m.}<90^{\circ}$ (up-right), $30^{\circ}<\theta^{n}_{\rm c.m.}<60^{\circ}$ (down-left) and $0^{\circ}<\theta^{n}_{\rm c.m.}<30^{\circ}$ (down-right). In each panel, the solid line denotes the total contributions of the three diagrams, while the dotted and dash-dotted lines show the calculations from the single and double scatterings, respectively. } \label{fig4} \end{center} \end{figure} \begin{figure} \begin{center} \centerline{\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_130.eps}} \caption{Differential $\pi\Sigma$ invariant mass spectra of $K^{-}d \to \pi^{0}\Sigma^{0}n$ for 130 MeV/c of incident $K^{-}$ momentum fixing the angle between the emitted neutron and the incident $K^{-}$ in the CM frame. In each panel, the solid line denotes the total contributions of the three diagrams, while the dotted and dash-dotted lines show the calculations from the single and double scatterings, respectively. } \label{fig5} \end{center} \end{figure} By producing the $\phi$ with a finite small momentum in the new DAFNE set up, one obtains kaons with larger momentum than before, when the $\phi$ was produced at rest. In what follows, we show the results for two other kaon momenta, within the present kaon momentum range, to emphasize the value of taking the kaons with larger momentum. In fig.~\ref{fig6}, we show the results equivalent to fig.~\ref{fig5} for momenta of the kaons 100 MeV/c and 80 MeV/c. At $k= 100$ MeV/c we can see in fig.~\ref{fig6} that the contribution of the single scattering dominates the cross section at neutron backward angles, and one finds two peaks, one coming from the resonance shape and the other one from the threshold effect. The two peaks are closer among themselves than in the case of $k= 130$ MeV/c. The final shape should then not be confused with the shape of the resonance. If one goes to neutron forward angles one can see that the single scattering is still as large as the double scattering one, and it shows its two peak structure, such that the shape of the global distribution is not the shape of the resonance. This momentum is not good to extract the properties of the $\Lambda(1405)$ resonance. The situation becomes worse for $k= 80$ MeV/c, as one can see in fig.~\ref{fig7}. Here, even at forward angles, the single scattering dominates the cross section and the shape that comes out does not reflect the shape of the resonance. Fortunately, the situation for $k=$ 130 MeV/c is satisfactory, with a clear dominance of the double scattering process and still a visible separation of the resonance and threshold peak contributions of the single scattering, such that the resulting peak around 1420 MeV clearly reflects the shape of the upper energy $\Lambda(1405)$ state peaking at 1420 MeV. Hence, the small increase in the kaon momentum from the former situation, where the kaons came from $\phi$ decay at rest, is most welcome and helps to get a cleaner signal of the $\Lambda(1405)$ resonance. One may wonder that, since double scattering becomes more important than single scattering in the kinematic conditions chosen, the triple scattering could also be important. However, as a general rule for inclusive processes, the single scattering is more important than the double and this one more important than the triple. We have observed this here, since the strength of the peak of single scattering is much bigger than that of double scattering. Yet, the single scattering has a very limited kinematics, a very restricted phase space, but this is not longer the case for double and triple scattering. Then the selection of forward neutron angles forces the single scattering out of its phase space, except for the finite momentum components of the deuteron, but does not put constraints on double scattering nor triple scattering, such that the triple scattering will remain small compared to the double. Note that there is an important difference between the case of the break up process that we have here and the coherent scattering of $K^-$ with deuteron without deuteron breakup. In the latter case the two nucleons in the deuteron stick together and it is relatively easy for the kaons to rescatter. This is indeed the case as shown in \cite{Toker:1981zh,Barrett:1999cw,kamalov}. But in the inclusive process, after the first scattering the nucleons move apart making more difficult the rescattering of the kaons with this pair of nucleons. We also do not consider double scattering diagrams with pion exchange in which the $\pi$ and the $\Sigma$ are emitted separately in the $T_{1}$ and $T_{2}$ amplitudes given in fig.~\ref{fig2}, respectively. It was discussed in \cite{Jido:2009jf} that such diagrams would give a smooth background in the $\pi \Sigma$ invariant mass spectra. \begin{figure} \begin{center} \centerline{\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_100.eps}} \caption{Differential $\pi\Sigma$ invariant mass spectra of $K^{-}d \to \pi^{0}\Sigma^{0}n$ for 100 MeV/c of incident $K^{-}$ momentum fixing the angle between the emitted neutron and the incident $K^{-}$ in the CM frame. In each panel, the solid line denotes the total contributions of the three diagrams, while the dotted and dash-dotted lines show the calculations from the single and double scatterings, respectively. } \label{fig6} \end{center} \end{figure} \begin{figure} \begin{center} \centerline{\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_80.eps}} \caption{Differential $\pi\Sigma$ invariant mass spectra of $K^{-}d \to \pi^{0}\Sigma^{0}n$ for 80 MeV/c of incident $K^{-}$ momentum fixing the angle between the emitted neutron and the incident $K^{-}$ in the CM frame. In each panel, the solid line denotes the total contributions of the three diagrams, while the dotted and dash-dotted lines show the calculations from the single and double scatterings, respectively. } \label{fig7} \end{center} \end{figure} \section{Summary} \label{sec:summary} We have studied the $K^{-}$ induced production of $\Lambda(1405)$ with a deuteron target in the reaction $K^{-} d \to \pi \Sigma n$, with the conditions of the DAFNE set up, where kaons are produced from the decay of slow moving $\phi$ mesons. The process proved well suited to investigate the predicted $\Lambda(1405)$ state around 1420 MeV, which couples mostly to $\bar{K}N$, since in the $K^{-} d \to \pi \Sigma n$ process, the $\Lambda(1405)$ resonance is produced by the $\bar KN$ channel. We found that for the most energetic kaons coming from the decay of the moving $\phi$ mesons, both the single scattering and the double scattering gave rise to a peak in the region of 1420 MeV for the spectrum of $\pi \Sigma $ invariant masses. Yet, it was also found that the contribution of single scattering can be easily distorted by a peak around threshold of the $K^-p$ channel. In order to obtain a cleaner signal, we evaluated the cross section at forward angles of the neutron and we found that the contribution of single scattering is drastically reduced and that of the double scattering takes over, leading to a cleaner signal of the $\Lambda(1405)$ resonance. The general rule is that having kaons in flight, at moderate energies of the kaons, becomes advantageous to distinguish the $\Lambda(1405)$ signal, since this automatically reduces the contribution of the single scattering versus the double scattering one. This is not so because the contribution of the single scattering is smaller, but because it peaks at a different region of invariant masses. Yet, we could see that, given the energy constraints at DAFNE, it was still possible to drastically reduce the contribution of single scattering at the expense of looking at a kinematical region in the phase space, with neutrons forward, where the single scattering contribution is reduced but the double scattering, which has more flexibility on momentum sharing, is not so much affected. In view of the results obtained here we can encourage to do this experiment at DAFNE where the fluxes of kaons are large enough to make the experiment feasible. Indeed, a first estimate \cite{nevio} indicates that with the energy resolution for charged particles of 2-4 \% of the kinetic energy and 4.2 MeV for neutrons for the energies of the experiment, measured with 20-35 \% efficiency in the KLOE calorimeter (not necessarily needed if the $\pi$ and $\Sigma$ are detected), the shape of the distribution can be obtained with the needed resolution. Taking into account present fluxes and the calculated cross sections one would need about eight months of beam time \cite{nevio}. \section*{Acknowledgements} One of us, E.~O.~wishes to acknowledge useful discussions with Nevio Grion and Alessandra Filippi. This work was partly supported by the Grant-in-Aid for Scientific Research from MEXT and JSPS (Nos. 22105507, 22740161, 20540273 and 22-3389), the collaboration agreement between the JSPS of Japan and the CSIC of Spain, and the Grant-in-Aid for the Global COE Program ``The Next Generation of Physics, Spun from Universality and Emergence" from MEXT of Japan. This work was also supported in part by DGICYT contract number FIS2006-03438. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity ``Study of Strongly Interacting Matter" (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU. One of us, T.~S.~acknowledges the support by the Grand-in-Aid for JSPS fellows. This work was supported in part by DFG (SFB/TR 16, ``Subnuclear Structure of Matter''). This work was done under the Yukawa International Program for Quark-hadron Sciences (YIPQS).

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\section{Introduction} Fusion of massive nuclei has recently drawn a lot of interest, especially at energies around the fusion barrier generated by the competition between the Coulomb and nuclear interactions. In this energy domain, the reaction mechanisms may depend strongly on the structure of the collision partners. The proper description of near-barrier fusion is thus a challenging N-body quantum dynamical problem involving the competition between various reaction channels. For example, the coupling between the internal degrees of freedom and the relative motion may generate a fusion barrier distribution \cite{das83}. Such couplings are needed to reproduce the sub-barrier fusion \cite{das98}. One of the internal degrees of freedom which can strongly affect the fusion is the static deformation \cite{rie70,jen70}. First, the fusion probability depends on the orientation of the deformed nucleus at the touching point. Second, a reorientation can occur under the torque produced by the long-range Coulomb force \cite{hol69,wil67,sim04,uma06b}. Such a reorientation is a consequence of the excitation of rotational states. It induces an anisotropy in the orientation distribution, thus modifying the near-barrier fusion \cite{bab00}. In this work we study the fusion of a spherical and a prolate deformed nucleus within the Time-Dependent Hartree-Fock (TDHF) theory. We first show the effect of the orientation at the touching point on the fusion probability. Then we include the long range Coulomb excitation of rotational states and study its effect on fusion. The results give a useful interpretation of full coupling channels calculations. \section{Time-Dependent Hartree-Fock theory} Let us first recall briefly some aspects of TDHF theory and of its numerical applications to nuclear collisions. TDHF is a mean field quantum dynamical theory \cite{har28,foc30,dir30} . It describes the evolution of occupied single particle wave functions in the mean field generated by all the particles. The total wave function of the system is constrained to be a Slater determinant at any time which assures an exact treatment of the Pauli principle during the dynamics. All standard applications of TDHF neglect pairing correlations so far. Like all mean-field methods, TDHF is best suited to desctibe average values of one-body operators. Such quantities are determined from the one-body density matrix $ \hat{\rho}=\sum_{n=1}^{N}\ll \varphi_{n}\> \<\varphi_{n}\right| $ where $\k \varphi_n \>$ denotes an occupied single particle state. In TDHF, its evolution is determined by a Liouville-von Neumann equation, $ i\hbar \partial_t \hat{\rho} =[\hat{h}(\rho ),\hat{\rho} ] $ where $\hat{h}(\rho )$ is the mean-field Hamiltonian. The great advantage of TDHF is that it treats the static properties {\it and} the dynamics of nuclei within the same formalism, i.e. using the same effective interaction (usually of the Skyrme type \cite{sky56}). The initial state is obtained through static Hartree-Fock (HF) calculations which are known to reproduce rather well nuclear binding energies and deformations. TDHF can be used in two ways to describe nuclear reactions: \begin{itemize} \item A single nucleus is evolved in an external field \cite{vau72}, simulating for instance the Coulomb field of the collision partner \cite{sim04}. \item The evolution of two nuclei, initially with a zero overlap, is treated in the same box with a single Slater determinant \cite{bon76,neg82}. \end{itemize} The first case is well suited for the description of inelastic scattering, like Coulomb excitation of vibrational and rotational states. The second case is used for more violent collisions like deep-inelastic and fusion reactions. In such cases, the lack of a collision term in TDHF might be a drawback. At low energy, however, the fusion is mainly driven by the one-body dissipation because the Pauli blocking prevents nucleon-nucleon collisions. The system fuses mainly by transfering relative motion into internal excitation via one-body mechanisms well treated by TDHF. Another important advantage of TDHF concerning its application to near-barrier reaction studies is that it contains implicitely all types of couplings between the relative motion and internal degrees of freedom whereas in coupling channels calculations one has to include them explicitely according to physical intuition which is not always straightforward for complex mechanisms. The only condition in TDHF is that the symmetries corresponding to the internal degrees of freedom of interest are relaxed. This is now the case with the latest TDHF codes in 3 dimensions (3D) which use a full Skyrme force \cite{kim97,uma06a}. However, TDHF gives only classical trajectories for the time-evolution and expectation values of one-body observables. In particular, TDHF does not include tunneling of the global wave function. We use the TDHF code built by P. Bonche and coworkers \cite{kim97} using a Skyrme functional \cite{sky56}. This code computes the evolution of each occupied single-particle wave function in a 3D box assuming one symmetry plane. The step size of the network is 0.8~fm and the step time 0.45~fm/c. We use the SLy4$d$ parametrization \cite{kim97} of the Skyrme force which is a variant of the SLy4 one specifically designed for TDHF calculations. \section{Fusion with a deformed nucleus} \subsection{Effect of the static deformation} Many nuclei exhibit static deformation, that is well described by mean-field calculations. Static deformation breaks the rotational invariance of the Slater determinant, which introduces an intrinsic frame of the nucleus. TDHF calculations of nuclear collisions, however, are performed in the laboratory frame, and one is left with an ambiguity concertning the relative orientation of the deformed nuclei. This is a critical point, because different orientations might ultimately lead to different reaction paths. To illustrate this point we consider central collisions of a prolate deformed $^{24}$Mg ($\beta_2=0.4$) with a spherical $^{208}$Pb. For symmetry reasons, the reaction mechanism will depend only on the energy and the angle between the deformation axis and the collision axis noted $\varphi$. Fig. \ref{fig:Tall} shows the time evolution of the density for two different initial orientations. We see that with an initial orientation $\varphi = 0 \deg$ the nuclei fuse whereas with $\varphi = 37.5 \deg$ the two fragments separate after a deep-inelastic collision. \begin{figure}[htbp] \includegraphics[height=.62\textheight]{./Tall.eps} \caption{Density plots of head-on $^{208}$Pb+$^{24}$Mg collisions at $E_{CM}=95$~MeV with an initial orientation at 20~fm of 0$^\circ$ (left) and 37.5$^\circ$ (right). The time step between each figure is 135~fm/c.} \label{fig:Tall} \end{figure} The technic used to overcome the ambiguity of the initial orientation is based on two prescriptions \cite{sim04,uma06b}: \begin{enumerate} \item It is necessary to assume an initial distribution of orientations. \item Interferences between different orientations are neglected. Then each Slater determinant evolves in its own mean field. \end{enumerate} Let us first assume an isotropic distribution of the orientations at the initial time, corresponding to a distance $D=20$~fm between the two centers of mass. This means that the $^{24}$Mg is supposed to be initially in its $0^+$ ground state and that all kind of long range Coulomb excitations are neglected up to this distance. Then, using the above prescriptions we get the fusion probability $$ P_{fus}(E)=\frac{1}{2}\int_0^\pi \!\!\! \d \varphi \,\,\, \sin \varphi \,\,\, P_{fus}(E,\varphi) $$ where $P_{fus}(E,\varphi)=0$ or 1. The solid line in Fig. \ref{fig:pfus_e}-a shows the resulting fusion probability as function of the center of mass energy. Below 93~MeV no orientation leads to fusion and above 99~MeV, all of them fuse. Between these two values, the higher the energy, the more orientations lead to fusion. As shown in Fig. \ref{fig:Tall}, configurations with small $\varphi$ are the first to fuse, even below the nominal barrier which would correspond to a spherical $^{24}$Mg case (dotted line). To conclude, sub-barrier fusion is described in TDHF through couplings between static deformation and relative motion. \begin{figure}[htbp] \includegraphics[height=.2\textheight]{./pfus_e} \caption{Fusion probability for a head-on collision or "penetrability" of $^{208}$Pb+$^{24}$Mg as a function of the c.m. energy. $a)$ TDHF results. Isotropic distribution of the initial orientations is supposed at 20~fm (solid line) and at 220~fm (dashed line). Step function expected in case of a spherical $^{24}$Mg (dotted line). $b)$ CCFULL results without (solid line) and with (dashed line) Coulomb excitation.} \label{fig:pfus_e} \end{figure} \subsection{Long range Coulomb reorientation} As a consequence of numerical limitations, actual TDHF calculations for collisions are performed in relatively small boxes and are started with internuclear distances of a few Fermi. However, the Coulomb interaction starts playing a role much earlier in the reaction process \cite{alder}. It is weaker than the nuclear interaction, but integrated over a long time it may induce a polarization, and then modify the reaction mechanism \cite{sim04,row06}. Long range Coulomb reorientation has been studied in Ref. \cite{sim04} with TDHF. The results have been interpreted using the classical formalism \cite{alder,broglia} where the motion of a deformed rigid projectile is considered in the Coulomb field of the target. An important conclusion of this work is that the reorientation, although being the result of a Coulomb effect, depends neither on the charges nor on the energy. Let us illustrate this phenomenon with a simple example. Consider a system at time $t$ formed by a deformed projectile at the distance $D(t)$ from the target. Increasing artificially the charge of one of the nuclei at this time has two effects. First, the Coulomb interaction increases and then the torque applied on the deformed nucleus should {\it increase} too. On the other hand, the distance $D(t)$ between the projectile and the target is larger because of the stronger Coulomb repulsion between the centers of mass. The latter effect leads to a {\it decrease} of the effective torque at time $t$ and both effects overall cancel exactly. One is left with a charge independent reorientation. The same argument applies for the energy. To study the effect of reorientation on fusion we calculate the reorientation in the approach phase between $D=220$~fm and 20~fm with TDHF using the technic described in Ref. \cite{sim04}. Assuming an isotropic distribution of orientations at 220~fm we get a new distribution at 20~fm which includes the reorientation coming from long range Coulomb excitation. The new fusion probability distribution (dashed line in Fig. \ref{fig:pfus_e}-a) is obtained with two additional assumptions: \begin{itemize} \item The rotational speed of the $^{24}$Mg is neglected at the initial time of the TDHF calculation (corresponding to $D=20$~fm), i.e. only a static reorientation is considered. \item The effect of the excitation energy on the relative motion is neglected, i.e. we assume a Rutherford trajectory before $D=20$~fm. \end{itemize} We observe in Fig. \ref{fig:pfus_e}-a a fusion hindrance up to 20$\%$ which is due to higher weights on orientations leading to compact configurations ($\varphi \sim 90\deg$ at the touching point) because of the reorientation \cite{sim04,row06}. The previous study is helpful to interpret coupling channels results. Calculations on the same system have been performed with the code CCFULL \cite{hag99} including coupling to the five first excited states of $^{24}$Mg rotational band. The fusion probability, or "penetrability" of the fusion barrier is given by the relation $ P_{fus}=\frac{\d\(\sigma E\)}{\d E}\pi R_B^2 $ where $R_B=11.49$~fm is the barrier position. Fig. \ref{fig:pfus_e}-b shows the fusion probability obtained from CCFULL including nuclear (solid line) and nuclear+Coulomb (dashed line) couplings. As with TDHF, an hindrance of the fusion due to Coulomb couplings is observed. However the shape of TDHF and CCFULL distributions are quite different. This is due to the fact that quantum mechanical effects are missing in TDHF. This point out the importance of improving the theory. It is also striking to see that TDHF "misses" the nominal barrier by about 15$\%$. TDHF is known to overestimate the fusion cross sections. One possible issue might be the time odd terms in the Skyrme energy functional. Their importance on fusion have been stressed recently \cite{uma06a,mar06}. \section{conclusion} To summarize, we performed a TDHF study of near-barrier fusion between a spherical and a deformed nucleus. The calculations show that, around the barrier, different orientations lead to different reaction path. Considering all possible orientations leads to a distribution of fusion probabilities interpreted as an effect of the coupling between the static deformation and the relative motion. We then included the long range Coulomb coupling which induces a {\it charge} and {\it energy} independent reorientation of the deformed nucleus. The effect of the reorientation is to hinder the near-barrier fusion. Finally the TDHF study have been used to interpret coupling channels calculations with the code CCFULL which show also an hindrance of near-barrier fusion due to Coulomb couplings. We also note some drawbacks of TDHF which, in one hand, underestimates the fusion barrier, and, in the other hand, miss important quantum effects. \begin{theacknowledgments} We warmly thank Paul Bonche for providing his TDHF code. This work has been partially supported by NSCL, Michigan State University and the National Science Foundation under the grant PHY-0456903. \end{theacknowledgments} \bibliographystyle{aipproc}

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\section{Introduction}\label{s:intro} For two variables holomorphic function germs, the first bi-Lipschitz invariants, different from the topological invariants, were found around 2003 by Henry and Parusi\' nski \cite{HP, HP2} who showed that there are moduli. In a different stream, Garcia Barroso and Teissier \cite{GT} had shown that the total curvature of the Milnor fibre concentrates in a multi-scale manner along a certain truncation of the generic polar curve of the two variable holomorphic function. More recently, Kuo, Koike and P\u aunescu \cite{kkp, KKP} studied the bumps of curvature on the Milnor fibre by using the gradient canyons as key devices. By using these three complementary viewpoints, all of which gravitate around the geometric and analytic properties of the polar curves, we show here that Lipschitz homeomorphisms preserve the gradient canyon structure of holomorphic functions $f : ({\mathbb C}^{2}, 0)\to ({\mathbb C}, 0)$, cf Theorem \ref{t:main2}. More precisely, the gradient canyons, together with their clustering and contact orders, are Lipschitz invariants of holomorphic functions of two variables. They complement the Henry-Parusi\' nski continuous invariants \cite{HP, HP2}, as demonstrated by Example \ref{e:2}. In order to state our main result, we give an account of the canyon data and send to the next sections for the details. Let $\gamma_{*}$ denote some \emph{polar} of $f = f(x,y) : ({\mathbb C}^{2}, 0) \to ({\mathbb C}, 0)$, i.e. an irreducible curve which is a solution of the equation $f_{x}=0$. We consider some Newton-Puiseux parametrization of it, i.e. of the form $\alpha(y) = (\gamma(y) , y)$, which can be obtained by starting from a holomorphic parametrization $\alpha : ({\mathbb C},0) \to ({\mathbb C}^{2}, 0)$, $\alpha(t) = (\alpha_{1}(t), \alpha_{2}(t))$ with ${\mathrm{ord}}_{t}\alpha_{2} \le {\mathrm{ord}}_{t}\alpha_{1}$, and then making the change of parameter $\alpha(y) = (\gamma(y) , y)$ with $y = \alpha_{2}(t)$. Then $m:= {\rm{mult}}(\gamma_{*}) := \min {\mathrm{ord}}_{t}\alpha_{2}$ (minimum over all parametrisations) is the \emph{multiplicity} of the polar $\gamma_{*}$; it is also equal to the total number of parametrizations of $\gamma_{*}$ of order $m$ which are conjugate. Let $d_{{\mathrm{gr}}}(\gamma)$ be the degree for which the order ${\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\gamma(y),y)\|)$ of the gradient is stabilized, see \eqref{GG}, and let $\mathcal{GC}(\gamma_{*})$ be \emph{the gradient canyon of $\gamma$} (Definition \ref{d:canyon}). Such a canyon contains one or more polars with the same canyon degree $d_{{\mathrm{gr}}}(\gamma)$. The \emph{multiplicity of the canyon} ${\rm{mult}}(\mathcal{GC}(\gamma_{*}))$ is the sum of the multiplicities of its polars. \smallskip Let us point out that the gradient canyons and their degrees are not topological invariants, see Example \ref{e:2}. \medskip While analytic maps do not preserve polars, we prove the analytic invariance of the canyons as a preamble for the definition of our new bi-Lipschitz invariants, Theorem \ref{t:main1}: \emph{If $f = g\circ \varphi$ with $\varphi$ analytic bi-Lipschitz, then $\varphi$ transforms canyons into canyons by preserving their degrees and multiplicities.} It follows that the map $\varphi$ establishes a bijection between the canyons of $f$ and those of $g$ such that the degrees $d_{{\mathrm{gr}}}(\gamma_{*})$ and the multiplicities ${\rm{mult}}(\mathcal{GC}(\gamma_{*}))$ are the same. \smallskip When we drop the analyticity assumption of the bi-Lipschitz map $\varphi$, the perspectives are challenging since not only that polar curves are not sent to polar curves, but we cannot prove anymore that gradient canyons are sent to gradient canyons. Up to now, the only result in full generality has been obtained by Henry and Parusi\'nski \cite{HP, HP2}, namely the authors have found that the leading coefficient in the expansion \eqref{eq:h-coefficient}, modulo an equivalence relation, is a bi-Lipschitz invariant. More than that, Henry and Parusi\'nski showed in \cite{HP, HP2} that a certain zone in the Milnor fibre, which is characterised by the higher order of the change of the gradient, is preserved by bi-Lipschitz homeomorphisms. Our new bi-Lipschitz invariants extend in a certain sense the discrete set of topological invariants of plane curves, but they refer to the branches of the polar curve instead of the branches of the curve $\{f=0\}$. Our clustering description of the polar curves and their associated zones refines in a multi-scale manner the Henry-Parusi\'nski zone. As of comparing our invariants to the Henry-Parusi\'nski Lipschitz continuous invariants, Example \ref{e:2} shows that they are complementary. We establish in \S \ref{s:genpolars} a faithful correspondence between the concentration of curvature invariants coming from Garcia Barroso and Teissier's geometric study \cite{GT} and those coming from the Koike, Kuo and P\u aunescu analytic study \cite{KKP}, in particular we prove (Theorem \ref{t:tau}): \emph{the contact degree $d_{\gamma(\tau)}$ and the gradient canyon $\mathcal{GC}(\gamma(\tau))$ do not depend of the direction $\tau$ of the polar $\gamma(\tau)$, for generic $\tau$.} This result also contributes to the proof of our main results Theorems \ref{t:main0} and \ref{t:main2} in Section \ref{s:disks}, of which we give a brief account in the following. \smallskip Let $f = g \circ \varphi$ with $\varphi$ a bi-Lipschitz homeomorphism. Even if the image by $\varphi$ of a gradient canyon is not anymore a gradient canyon like in the analytic category, our key result Theorem \ref{t:main0} says that: \emph{the bi-Lipschitz map $\varphi$ establishes a bijection between the canyon disks of $f$ and the canyon disks of $g$ by preserving the canyon degree}, where the \emph{canyon disks} are defined as the intersections of the \emph{horn domains} \eqref{eq:hd} with the Milnor fibre. We are therefore in position to prove that $\varphi$ induces a bijection between the gradient canyons of $f$ and those of $g$, and moreover, that there are \emph{clusters of canyons} of $f$ which correspond by $\varphi$ to similar clusters of $g$. Such clusters are defined in terms of orders of contact (i.e. certain rational integers) which are themselves bi-Lipschitz invariants. Our main result, Theorem \ref{t:main2}, states minutiously this correspondence, and we send to Section \ref{s:disks} for its formulation and the preparatory definitions. \begin{example} \label{e:1} The function germ $f :=z^{4}+ z^{2}w^{2}+ w^{4}$ has 3 polars with canyon degrees $d=1$ belonging to a single canyon of multiplicity 3. One can show\footnote{by using Kuo's trivialising vector field in the family $f_{t}=z^{4}+ tz^{2}w^{2}+ w^{4}$ which is homogeneous of degree 4.} that $f$ is bi-Lipschitz equivalent to the function germ $g :=z^{4}+ w^{4}$ which has a single polar, its canyon has degree $d=1$ and multiplicity 3. The canyon degree is indeed a bi-Lipschitz invariant, according to our Theorem \ref{t:main0}, but the number of polars in the canyon is not invariant. \end{example} \begin{example} \label{e:2} For instance, consider the function germs $f(x,y)=x^3+y^{12}$ and $g(x,y)=x^3+y^{12}+ x^2y^5$ which are topologically equivalent\footnote{since $f_{t}(x,y)=x^3+y^{12}+ tx^2y^5$ is a topologically trivial family.}. Then $g$ has two disjoint canyons corresponding to the two distinct polars, both having degree $d=6$, whereas $f$ has only one double polar with canyon degree $d=\frac{11}{2}$, hence only one canyon. According to our Theorem \ref{t:main0}, these two function germs are not Lipschitz equivalent. Nevertheless they have the same Henry-Parusinski invariants \cite{HP, HP2}. \end{example} \section{Gradient canyons}\label{structures} We recall from \cite{kuo-pau2} and \cite{KKP} some of the definitions and results that we shall use. One calls \textit{holomorphic arc} the image $\alpha_* :=\mathop{\rm{Im}}\nolimits(\tilde\alpha)$ of an irreducible plane curve germ: \[ \tilde\alpha: ({\mathbb C},0)\longrightarrow ({\mathbb C}^2,0), \quad \tilde\alpha(t)=(z(t),w(t)). \] It has a unique complex tangent line $T(\alpha_*)$ at $0$, considered as a point in the projective line, i.e. $T(\alpha_*)\in{\mathbb C} P^1$. The total space of holomorphic arcs was called ``enriched Riemann sphere'' in \cite{kuo-pau2, KKP}. The classical Newton-Puiseux Theorem asserts that the field $\mathbb{F}$ of convergent fractional power series in $y$ is algebraically closed, see e.g. \cite{walker}, \cite{wall}. A non-zero element of $\mathbb{F}$ is a (finite or infinite) convergent series with positive rational exponents: \[ \alpha(y)=a_0y^{n_0/N}+\cdots +a_iy^{n_i/N}+\cdots ,\quad n_0<n_1<\cdots, \] where $0\ne a_i\in {\mathbb C}$, $N, n_i \in {\mathbb N}$, $N>0$, with $\mathrm{gcd}(N, n_0, n_1, ...)=1$, $\limsup_{i}|a_i|^{\frac{1}{n_i}}<\infty$. The \textit{conjugates} of $\alpha$ are \[ \alpha_{conj}^{(k)}(y) :=\sum a_i \theta^{kn_i}y^{n_i/N},\mbox{ where } 0\leq k\leq N-1 \mbox{ and } \theta :=e^\frac{2\pi \sqrt{-1}}{N}. \] The \textit{order} of $\alpha$ is \begin{eqnarray*}& {\mathrm{ord}}(\alpha) :={\mathrm{ord}}_y(\alpha) =\frac{n_0}{N} \text{ if }\;\, \alpha\ne 0 \mbox{ and } {\mathrm{ord}}(\alpha) :=\infty \text{ if } \alpha=0,& \end{eqnarray*} and $m_{puiseux}(\alpha) :=N$ is the \textit{Puiseux multiplicity} of $\alpha$. For any $\alpha\in \mathbb{F}_1:=\{\alpha|{\mathrm{ord}}_y(\alpha)\geq 1\}$, the map germ \[ \label{para} \tilde\alpha: ({\mathbb C},0)\longrightarrow ({\mathbb C}^2,0), \quad t\mapsto (\alpha(t^N),t^N),\quad N :=m_{puiseux}(\alpha), \] is holomorphic and the holomorphic arc $\alpha_*$ is then well defined. \smallskip \noindent One defines several subspaces of holomorphic arcs, as follows. For some fixed $\alpha_* =\mathop{\rm{Im}}\nolimits(\tilde\alpha)$, one defines: \begin{equation}\label{ID} \mathcal{D}^{(e)}(\alpha_*;\rho):=\{\beta_*\,\mid \beta(y)=[J^{(e)}(\alpha)(y)+cy^e]+\mathrm{h.o.t.} ,\,\;|c|\leq \rho\}, \end{equation} where $1\leq e<\infty$, $\rho\geq 0$, and where $J^{(e)}(\alpha)(y)$ is the $e$-\textit{jet} of $\alpha$ and ``$\mathrm{h.o.t.}$'' means as usual ``higher order terms''. Moreover, one defines: \begin{equation}\label{IL} \begin{split}\mathcal{L}^{(e)}(\alpha_*)&:=\mathcal{D}^{(e)}(\alpha_*;\infty) :=\cup_{0<\rho<\infty}\mathcal{D}^{(e)}(\alpha_*;\rho)\\&= \{\beta_* \mid \beta(y)=[J^{(e)}(\alpha)(y)+cy^e]+\mathrm{h.o.t.},\; |c| \in {\mathbb R}\}. \end{split} \end{equation} Note that in the above definitions \eqref{ID} and \eqref{IL}, the parameter $\alpha\in \mathbb{F}_{1}$ runs over all its conjugates. \medskip Consider the Newton-Puiseux factorizations: \begin{eqnarray}\label{ff} f(x,y)=u\cdot \prod _{i=1}^m(x-\zeta_i(y)), \quad f_x(x,y)=v\cdot \prod_{j=1}^{m-1}(x-\gamma_j(y)), \end{eqnarray} where $\zeta_i$, $\gamma_j\in \mathbb{F}_1$ and $u,v$ are units. Note that all conjugates of roots are also roots. We call \textit{polar} any such root $\gamma_j$, as well as its geometric representation $\gamma_{j*}$. If a polar $\gamma$ is also a root of $f$, i.e. $f(\gamma(y),y)\equiv 0$, then it is a multiple root of $f$. \medskip From the Chain Rule it follows: \begin{equation}\label{EMF} f_x(\alpha(y),y)\equiv f_y(\alpha(y),y)\equiv 0 \implies f(\alpha(y),y)\equiv 0 \end{equation} for any $\alpha\in \mathbb{F}_1$. Let us fix a polar $\gamma$ with $f(\gamma(y),y)\not \equiv 0$. By (\ref{EMF}), $\gamma$ is not a common Newton-Puiseux root of $f_z$ and $f_y$. If $q$ is sufficiently large, then one has the equality: \begin{equation}\label{GG} {\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\gamma(y),y)\|)={\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\gamma(y)+uy^q,y)\|), \; \forall\, u\in {\mathbb C}. \end{equation} \begin{definition}\label{d:canyon} The \textit{gradient degree} $d_{{\mathrm{gr}}}(\gamma)$ is the smallest number $q$ such that (\ref{GG}) holds for generic $u\in {\mathbb C}$. In the case $f(\gamma(y),y)\equiv 0$, one sets $d_{{\mathrm{gr}}}(\gamma) :=\infty$. \end{definition} It turns out that the gradient degree $d_{{\mathrm{gr}}}(\gamma)$ is rational since it is a co-slope in a Newton polygon, see Lemma \ref{LH}(i) for $\alpha := \gamma$. It can also be interpreted as a \L ojasiewicz exponent. \begin{definition} Let $\gamma$ be a polar of gradient degree $d :=d_{{\mathrm{gr}}}(\gamma)$, $1\leq d \leq \infty$. The \textit{gradient canyon of $\gamma_*$} is by definition \[ \mathcal{GC}(\gamma_*) :=\mathcal{L}^{(d)}(\gamma_*). \] One calls $d_{{\mathrm{gr}}}(\gamma_*) :=d_{{\mathrm{gr}}}(\gamma)$ the \textit{gradient degree of $\gamma_{*}$}, or the \textit{degree of $\mathcal{GC}(\gamma_*)$}. One says that $\mathcal{GC}(\gamma_*)$ is \textit{minimal} if $d_{{\mathrm{gr}}}(\gamma_*)<\infty$ and if, for any polar $\gamma_i$ of finite degree, the inclusion $ \mathcal{GC}(\gamma_{i*}) \subseteq\mathcal{GC}(\gamma_*)$ implies the equality $\mathcal{GC}(\gamma_{i*})= \mathcal{GC}(\gamma_*)$. \end{definition} \begin{definition} The \textit{multiplicity} of the gradient canyon $\mathcal{GC}(\gamma_*)$ is defined as: \begin{equation}\label{multi} {\rm{mult}}(\mathcal{GC}(\gamma_*)) :=\sharp \{j\,\mid 1\leq j \leq m-1,\; \gamma_{j*}\in\mathcal{GC}(\gamma_*)\}, \end{equation} where $m$ is as in \eqref{ff}. \end{definition} \medskip Up to some generic unitary change of coordinates, one has the following presentation: \begin{equation}\label{mini} f(x,y) :=f_m(x,y)+f_{m+1}(x,y)+\mathrm{h.o.t.},\end{equation} where $f_k$ denotes a homogeneous $k$-form, with $f_m(1,0)\ne 0$ and $m={\mathrm{ord}}(f)$. The initial form $f_m(x,y)$ factors as: \begin{equation}\label{HG}f_m(x,y)=c(x-x_1y)^{m_1}\cdots (x-x_ry)^{m_r},\;\, m_i\geq 1, \;\, x_i\ne x_j \;\,\text{if}\;\,i\ne j, \end{equation} and $1\leq r\leq m$, $m=m_1+\cdots +m_r$, $c\ne 0$. \medskip We have that $f_m(x,y)$ is degenerate if and only if $r<m$. The following useful result sheds more light over the landscape of gradient canyons: \begin{theorem}\label{thmB} \cite[Theorem B]{KKP} Any gradient canyon of degree $1<d_{{\mathrm{gr}}}<\infty$ is a minimal canyon. The canyons of degrees $1<d_{{\mathrm{gr}}}\leq \infty$ are mutually disjoint. There are exactly $r-1$ polars of gradient degree $1$, counting multiplicities, and they belong to the unique gradient canyon of degree 1, denoted by ${\mathbb C}_{{\rm{enr}}}$. If $1<r\leq m$, then ${\mathbb C}_{{\rm{enr}}}$ is minimal if and only if $f(z,w)$ has precisely $r$ distinct roots $\zeta_i$ in (\ref{ff}). In particular, if $f_m(x,y)$ is non-degenerate then ${\mathbb C}_{{\rm{enr}}}$ is minimal. \hspace*{\fill}$\Box$ \end{theorem} \medskip \noindent \textbf{The horn, the partial Milnor number, and the total curvature of a gradient canyon.}\ A well-known formula to compute the Milnor number $\mu_f$ is the following: \begin{equation}\label{MWK} \mu_f=\sum_{j=1}^{m-1}\left[ {\mathrm{ord}}_y(f(\gamma_j(y),y))-1\right] , \end{equation} where the sum runs over all $\gamma_j$, i.e. over all polars and their conjugates. \medskip One defines the \emph{Milnor number of $f$ on a gradient canyon} $\mathcal{GC}(\gamma_*)$ with $d_{{\mathrm{gr}}}(\gamma_*)<\infty$, as: \begin{eqnarray}\label{M} &\mu_f(\mathcal{GC}(\gamma_*))\!:=\sum_j[{\mathrm{ord}}_y(f(\gamma_j(y),y))-1],& \end{eqnarray} where the sum is taken over all $j$, $1\leq j\leq m-1$, such that $\gamma_{j*}\in\mathcal{GC}(\gamma_*)$. From (\ref{M}) and (\ref{multi}) one has: \begin{eqnarray*}& \mu_f(\mathcal{GC}(\gamma_*))+{\rm{mult}}(\mathcal{GC}(\gamma_*))=\sum_j{\mathrm{ord}}_y(f(\gamma_j(y),y)),& \end{eqnarray*} where $j$ runs like in the sum of (\ref{M}). Consider $\mathcal{D}^{(e)}(\alpha_*;\rho)$ as in \eqref{ID}, of finite order $ e \geq1$ and finite radius $\rho >0 $, and a compact ball $B(0;\eta)\!:=\{(x,y)\in {\mathbb C}^2\,\mid \sqrt{|x|^2+|y|^2}\leq\eta\}$ with small enough $\eta>0$ (usually we consider a Milnor ball of $f$). Let then: \begin{equation}\label{eq:hd} {\rm{Horn}}^{(e)}(\alpha_*;\rho;\eta) : =\{(x,y)\in B(0;\eta) \mid x= \beta(y)=J^{(e)}(\alpha)(y)+cy^{e}, |c|\leq \rho\} \end{equation} be the \textit{horn domain} associated to $\mathcal{D}^{(e)}(\alpha_*;\rho)$; it is a compact subset of ${\mathbb C}^2$. The \textit{total asymptotic Gaussian curvature over} $\mathcal{D}^{(e)}(\alpha_*;\rho)$ is then by definition: \begin{equation}\label{eq:measure} \mathscr M_f(\mathcal{D}^{(e)}(\alpha_*;\rho)) :=\lim_{\eta\rightarrow 0}\left[ \lim_{\lambda\rightarrow 0}\int_{\{f=\lambda\}\cap {\rm{Horn}}^{(e)}(\alpha_*; \rho;\eta)}KdS\right] , \end{equation} where $S$ is the surface area and $K$ is the Gaussian curvature. The total asymptotic Gaussian curvature over $\mathcal{L}^{(e)}(\alpha_*)$ as in \eqref{IL} is by definition: \begin{equation}\label{eq:measure2} \mathscr M_f(\mathcal{L}^{(e)}(\alpha_*))\!:=\lim_{\rho\rightarrow \infty}\mathscr M_f(\mathcal{D}^{(e)}(\alpha_*;\rho)). \end{equation} The above definitions are easily extended to the case $e=\infty$, so that \[ \mathscr M_f(\mathcal{D}^{(\infty)}(\alpha_*))=\mathscr M_f(\mathcal{L}^{(\infty)}(\alpha_*))=\mathscr M_f(\{\alpha_*\})=0. \] Then, for the gradient canyon $\mathcal{GC}(\gamma_*) :=\mathcal{L}^{(d)}(\gamma_*)$ one has: \begin{theorem} \cite[Theorem C]{KKP} Let $\gamma_*$ be a polar, $1< d_{{\mathrm{gr}}}(\gamma_*)\leq\infty$. Then \begin{equation}\label{ttl}\mathscr M_f(\mathcal{GC}(\gamma_*))=\begin{cases}2\pi[\mu_f(\mathcal{GC}(\gamma_*))+{\rm{mult}}(\mathcal{GC}(\gamma_*)],&\quad 1<d_{{\mathrm{gr}}}(\gamma_*)<\infty,\\0,&\quad d_{{\mathrm{gr}}}(\gamma_*)=\infty.\end{cases} \end{equation} \hspace*{\fill}$\Box$ \end{theorem} \section{The arc valleys}\label{NParc} \subsection{Arc valleys and gradient canyons}\ We consider a Puiseux arc $\alpha : ({\mathbb C}, 0) \to ({\mathbb C}^{2}, 0)$ of order ${\mathrm{ord}}_y(\alpha)\geq 1$. We may assume, modulo transposition and rescaling, that $\alpha(y)=(\bar\alpha(y),y)$ where $\bar\alpha(y) $ is a fractional power series of order $\bar{\mathrm{ord}}_y(\alpha) \geq 1$; in the following we identify $\bar\alpha$ with $\alpha$. We define the contact degrees: $$d_{\mathrm{cont}}(\alpha) :=\inf \{d \mid ||\mathop{\rm{grad}}\nolimits f(\alpha+ cy^{d}+\mathrm{h.o.t.})||\sim ||\mathop{\rm{grad}}\nolimits f(\alpha)||, \mbox{ for generic } c\in {\mathbb C}^2 \}$$ and: $$ \tilde d_{\mathrm{cont}}(\alpha) :=\inf \{d\mid ||\mathop{\rm{grad}}\nolimits f(\alpha+cy^{d})||\sim ||\mathop{\rm{grad}}\nolimits f(\alpha)||, \, \text{ for generic } \, c\in {\mathbb C}^2 \}.$$ It follows from the definitions that $\tilde d_{\mathrm{cont}}(\alpha)\leq d_{\mathrm{cont}}(\alpha)$. Let us show that we actually have equality. If $\|\mathop{\rm{grad}}\nolimits f(\alpha(y)+u y^e)\|^2 :=D_{(\alpha,e)}(u)y^{2L_{gr}(\alpha,e)}+\mathrm{h.o.t.},\, D_{(\alpha,e)}(u)\not \equiv 0,$ so for generic $u\in {\mathbb C}^2$ it has the same order $L_{gr}(\alpha,e)$ which is increasing in $e,$ i.e. $\|\mathop{\rm{grad}}\nolimits f(\alpha(y)+u y^e)\|\sim \|\mathop{\rm{grad}}\nolimits f(\alpha(y)+vy^r)\|, \forall r>e, v\in {\mathbb C}^2 \,.$ If we set $\beta (y)=\alpha (y) +uy^e$ then $\|\mathop{\rm{grad}}\nolimits f(\alpha)\| \sim \|\mathop{\rm{grad}}\nolimits f(\alpha(y)+u y^e)\|=\|\mathop{\rm{grad}}\nolimits f(\beta)\| \sim \|\mathop{\rm{grad}}\nolimits f(\beta(y)+v y^r)\|=\|\mathop{\rm{grad}}\nolimits f(\alpha+uy^d+vy^r)\|$ and so on i.e. the two definitions give the same number (the order of the gradient stabilises, for instance if $r\geq {\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\alpha(y))\|^2$). \medskip Let $\alpha_*\in {\mathbb C}_{{\rm{enr}}}$ be given, $f(\alpha(y),y)\not \equiv 0$, $\alpha$ is not a common Newton-Puiseux root of $f_x$ and $f_y$. Hence, if $q$ is sufficiently large, then \begin{equation}\label{GGa} {\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\alpha(y),y)\|)={\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\alpha(y)+uy^q,y)\|), \; \forall\, u\in {\mathbb C}. \end{equation} Let $d_{\alpha}$ denote the \textit{smallest number} $q$ such that (\ref{GGa}) holds for \textit{generic} $u\in {\mathbb C}$. This definition gives the same degree as the previous definition and from now on we will use the later notation. In case $f(\alpha(y),y)\equiv 0$, we set $d_{\alpha} :=\infty$ if $\alpha$ is a multiple root of $f$. \begin{definition} The \textit{ valley } of $\alpha_*$ is, by definition: \[ \mathcal{V}(\alpha_*) :=\mathcal{L}^{(d_{\alpha})}(\alpha_*). \] We then call $d_{\alpha}$ the \textit{degree} of $\mathcal{V}(\alpha_*)$, or \textit{ the valley degree of $\alpha$} (since it does not depend of the representative $\alpha$ of $\alpha_*$). We say that $\mathcal{V}(\alpha_*)$ is \textit{minimal} if $d_{\alpha_*}<\infty$ and if for every arc $\beta$ with $d_{\beta}<\infty$, we have: \[ \mathcal{V}(\beta_{*}) \subseteq\mathcal{V}(\alpha_*) \Longrightarrow \mathcal{V}(\beta_{*})= \mathcal{V}(\alpha_*). \] \end{definition} \begin{remark} In the case $\gamma$ is a polar the construction above gives the notion of gradient canyon and canyon degree ($\mathcal{V}(\gamma_{*})=\mathcal{GC}(\gamma_{*}), d_{{\mathrm{gr}}}(\gamma)=d_{\gamma}$) as introduced in \cite{KKP} and mentioned earlier. If $\alpha$ is in a canyon then its valley coincides with the canyon. \end{remark} \subsection{Newton polygon} Let $\alpha$ be a given arc, $d_{\alpha}<\infty$. We can apply a unitary transformation, if necessary, so that $\alpha\in \mathbb{F}_{\ge 1}$, $T(\alpha_*)=[0:1]$. We then change variables (formally): \begin{equation}\label{3b} Z :=z-\alpha(w),\quad W :=w,\ \ F(Z,W) :=f(Z+\alpha(W),W). \end{equation} Since $\alpha\in \mathbb{F}_{\ge 1}$, i.e., ${\mathrm{ord}}(\alpha)>1$, it is easy to see that \begin{equation}\label{D} \|\mathop{\rm{grad}}\nolimits_{z,w} f\|\sim\|\mathop{\rm{grad}}\nolimits_{Z,W} F\|,\ \ \Delta_f(z,w)=\Delta_F(Z,W)+\alpha''(W)F_X^3. \end{equation} \medskip The Newton polygon $\mathcal{NP}(F)$ is defined in the usual way, as follows. Let us write $$F(Z,W)=\sum c_{iq}Z^iW^q,\quad c_{iq}\ne 0,\quad (i,q)\in \mathbb{Z}\times\mathbb{Q}.$$ A monomial term with $c_{iq}\ne 0$ is represented by a ``\textit{Newton dot}" at $(i,q)$. We shall simply call it a \textit{dot} of $F$. The \textit{boundary} of the convex hull generated by $\{(i+u,q+v)|u,\,v\geq 0\}$, for all dots $(i,q)$, is the Newton Polygon $\mathcal{NP}(F)$, having edges $E_i$ and angles $\theta_i$, as shown in Fig.\ref{fig:npF}. In particular, $E_0$ is the half-line $[m,\infty)$ on the $Z$-axis. \footnote{In\cite{kuo-lu}, \cite{kuo-par}, this is called the Newton Polygon of $f$ \textit{relative to} $\alpha$, denoted by $\mathcal{NP}(f,\alpha)$.} For a line in $\mathbb{R}^2$ joining $(u,0)$ and $(0,v)$, let us call $v/u$ its \textit{co-slope}. Thus $$\textit{co-slope of}\;\;E_s=\tan\theta_s.$$ Some elementary, but useful, facts are: \begin{itemize} \item If $i\geq 1$, then $(i,q)$ is a dot of $F$ \textit{if and only if} $(i-1,q)$ is one of $F_Z$. \item When $f(\alpha(w),w)\not\equiv 0$, we know $F(0,W)\not\equiv 0$. Let us write \begin{equation} \label{3k}F(0,W)=aW^h+\mathrm{h.o.t.}, \quad a\ne 0, \;\, h :={\mathrm{ord}}_W(F(0,W))\in \mathbb{Q}.\end{equation} Then $(0,h)$ is a vertex of $\mathcal{NP}(F)$, $(0,h-1)$ is one of $\mathcal{NP}(F_W)$. (See Fig.\,\ref{fig:npZ}.) \end{itemize} \begin{figure}[htt] \begin{minipage}[b]{.4\linewidth} \centering \begin{tikzpicture}[scale=0.44] \draw(0,0) -- (0,11.5); \path (0,11.5) node[above] {\tiny{W}}; \path (7.5,0) node[below] {\tiny{$m$}}; \draw(0,0) -- (8.1,0); \path (8,0) node[right] {\tiny{Z}}; \draw[dotted] (0.5,8)--(2,5); \draw plot[mark=*] coordinates {(2,5) (2.7,4)}; \draw plot[mark=*] coordinates {(0.5,8) (0,11)}; \draw plot[mark=*] coordinates {(5.6,1.4) (7.6,0)}; \draw[dotted] (3.6,3) -- (4.8,2); \draw[dotted] (1.5,4) -- (2.8,4); \path (2,3.7) node[above] {\tiny{$\theta_s$}}; \path (6.3,-0.3) node[above] {\tiny{$\theta_1$}}; \path (7.1,0.35) node[above] {\tiny{$E_1$}}; \path (3.7,4.8) node[left] {\tiny{$E_s$}}; \path (0.6,8) node[right] {\tiny{$(m_{\top},q_{\top})$}}; \path (2.2,9.2) node[left] {\tiny{$E_{\top}$}}; \draw (6.8,0) arc (180:150:1); \end{tikzpicture} \caption{$\mathcal{NP}(F)$}\label{fig:npF} \end{minipage} \begin{minipage}[b]{.4\linewidth} \centering \begin{tikzpicture}[scale=0.4] \draw(0,0) -- (0,13); \path (0,13) node[above] {\tiny{$W$}}; \draw[dotted](7,0.5) -- (10.5,0.5); \path (8.8,0.45) node[above] {$\theta_{\top}$}; \draw(0,0) -- (12,0); \path (12,0) node[below] {\tiny{$Z$}}; \draw[dotted] (5,0) -- (5,11); \draw plot[mark=*] coordinates {(0,12) (9,2) (10.5,0.5)}; \draw plot[mark=*] coordinates {(9,2) (7,5) (5.5,8)}; \draw[dotted] (5,11) -- (9,2); \draw[dashed] (5.5,8) -- (5,11); \draw[dashed] (0.5,8) -- (0,11); \path (5.1,9.5) node[right] {\tiny{$L$}}; \path (0.1,9) node[right] {\tiny{$L^*$}}; \path (4.4,11) node[right] {$\circ$}; \path (0,11) node {$\circ$}; \path (0,12) node[right] {\tiny{$(0,h)$}}; \path (0,10.9) node[left] {\tiny{$(0,h_{\alpha})$}}; \path (9,2) node[right] {\tiny{${(\widehat{m}_{\top},\widehat{q}_{\top})}$}}; \path (10.5,0.5) node[right] {\tiny{$(m_{\top},q_{\top})$}}; \path (5.5,8) node[right] {\tiny{$(m^*+1,q^*)$}}; \path (5.3,11) node[right] {\tiny{$(1,h_{\alpha})$}}; \path (5,0) node[below] {$1$}; \path (2.3,7) node[right] {\tiny{$E_{\top}$}}; \draw[dotted] (9.5,0.5) arc (180:145:1.3); \end{tikzpicture} \caption{$\mathcal{NP}(F)$~vs~$\mathcal{NP}(F_Z)$.} \label{fig:npZ} \end{minipage} \end{figure} \begin{notations}\label{top} In the case $d_{\alpha}<\infty$ , i.e. either $f(\alpha (w),w) \not\equiv 0$ or $f(\alpha (w),w)=0, f_z(\alpha(y),y)\not\equiv 0,$ let $E_{\top}$ denote the edge whose left vertex is, in the first case $(0,h)$, $h$ as in (\ref{3k}) or $(1,h')$ in the second case, and right vertex is $(m_{\top}, q_{\top})$, as shown in the figures. We call it the \textit{top} edge; the angle is $\theta_{\top}$. In the case $f(\alpha(w),w)\equiv 0$ the top edge $E_{\top}$ is not ending on $z=0$ but on $z=1$, except $\alpha$ is a multiple root of $f$ in which case we precisely have $d_{\alpha}=\infty$. However, except in the latter case, we always extend informally the top edge to virtually cut $z=0$ at $(0,h)$. \end{notations} Let $(\widehat{m}_{\top},\widehat{q}_{\top})\ne (0,h)$ be the dot of $F$ on $E_{\top}$ which is \textit{closest} to the left end of $E_{\top}$ ( which is $(0,h)$ if $\alpha$ not a multiple root of $f$) . (Of course, $(\widehat{m}_{\top},\widehat{q}_{\top})$ may coincide with $(m_{\top}, q_{\top})$.) Then, clearly, \[ \label{3d}2\leq \widehat{m}_{\top}\leq m_{\top},\quad \frac{h-\widehat{q}_{\top}}{\widehat{m}_{\top}}=\frac{h-q_{\top}}{m_{\top}}=\tan\theta_{\top}.\] Now we draw a line $L$ through $(1,h_\alpha), h_\alpha \leq h-1,$ with the following two properties (in particular this defines $h_\alpha$): \begin{itemize} \item[(a)] If $(m',q')$ is a dot of $F_Z$, then $(m'+1,q')$ lies on or above $L$; \item[(b)] There exists a dot $(m^*,q^*)$ of $F_Z$ such that $(m^*+1,q^*)\in L$. (Of course, $(m^*+1,q^*)$ may coincide with $(\widehat{m}_{\top},\widehat{q}_{\top})$.) \item[(c)] $h_\alpha$ is the largest with these properties. \end{itemize} \begin{lemma}\label{LH} Let $\sigma^*$ denote the co-slope of $L$. Then \begin{equation}\label{tan} (i)\;d_{\alpha}=\sigma^*;\,(ii)\,\;\sigma^*\geq \tan\theta_{\top};\;(iii)\; \sigma^*=\tan\theta_{\top}\Leftrightarrow (1,h_\alpha)\in E_{\top}. \end{equation} All dots of $F_W$ lie on or above $L^*$, $L^*$ being the line through $(0,h_\alpha)$ parallel to $L$. In the case $\tan\theta_{\top}>1$, $(0,h-1)$ may be the only dot of $F_W$ on $L^*$ (exactly when $h_\alpha=h-1$ and $\alpha$ not a root of $f$). \end{lemma} \begin{notations}\label{no} Take $e\geq 1$. Let $\omega(e)$ denote the weight system: $\omega(Z)=e$, $\omega(W)=1$. Let $G(Z,W^{1/N})\in {\mathbb C}\{Z,W^{1/N}\}$ be given. Consider its weighted Taylor expansion relative to this weight. We shall denote the \textit{weighted initial form} by $\mathcal{I}_{\omega(e)}(G)(Z,W)$, or simply $\mathcal{I}_\omega(G)$ when there is no confusion. If $\mathcal{I}_\omega(G)=\sum a_{ij}Z^iW^{j/N}$, the \textit{weighted order} of $G$ is ${\mathrm{ord}}_\omega(G) :=ie+\frac{j}{N}$. \end{notations} \begin{proof}Note that ($ii$) and ($iii$) are clearly true, since $(1,h_\alpha)$ lies on or above $E_{\top}$. Next, if $(i,q)$ is a dot of $F_W$, then $(i,q+1)$ is one of $F$, lying on or above $E_{\top}$. Hence, by ($ii$), all dots of $F_W$ lie on or above $L^*$. It also follows that if $\tan\theta_{\top}>1$, then $(0,h-1)$ may be the only dot of $F_W$ on $L^*$. \medskip Let us show $(i)$. It is easy to see that if $\tan\theta_{\top}=1$, then $d_{\alpha}=1$. It remains to consider the case $\sigma^*>1$. By construction, \begin{equation}\label{LI}{\mathrm{ord}}_y(\|\mathop{\rm{grad}}\nolimits f(\alpha(y),y)\|)=h_\alpha. \end{equation} Let us first take weight $\omega :=\omega(e)$ where $e\geq \sigma^*$. In this case, since $\sigma^*>1$: \[\mathcal{I}_{\omega}(F_W)(Z,W)=ahW^{h-1},\ {\mathrm{ord}}_W(F_Z)= h_\alpha, \mbox{ if } h_\alpha \leq h-1, \mbox{ and } > h-1 \mbox{ otherwise,}\] where $a$, $h$ are as in (\ref{3k}), $ah\ne 0$. Hence for generic $u\in {\mathbb C}$, $$ {\mathrm{ord}}_W(F_W(uW^e,W))=h-1,\quad {\mathrm{ord}}_W(F_Z(uW^e, W))= h_\alpha.$$ It follows that $d_{\alpha}\leq \sigma^*$. It remains to show that $\sigma^*>d_{\alpha}$ is impossible. \medskip Let us take $\omega(e)$ with $e<\sigma^*$. Note that $(m^*,q^*)$ is a dot of $F_Z$ on $L^*$, where $(m^*+1,q^*)$ is shown in Figure \ref{fig:npZ}. Hence, for generic $u$, $${\mathrm{ord}}_W(F_Z(uW^e, W))<h_\alpha,\quad {\mathrm{ord}}_W(\|\mathop{\rm{grad}}\nolimits F(uW^e,W)\|)<h_\alpha.$$ Thus, by (\ref{LI}), we must have $d(\alpha)>e$. This completes the proof of Lemma \ref{LH}. \end{proof} \begin{example}\label{ExampleGC} For $F(Z,W)=Z^4+Z^3W^{27}+Z^2W^{63}-W^{100}$ and $\gamma=0$, $\mathcal{NP}(F)$ has only two vertices $(4,0)$, $(0, 100)$, while $\mathcal{NP}(F_Z)$ has three: $(3,0)$, $(2,27)$, $(1,63)$. The latter two and $(0,99)$ are collinear, spanning $L^*$; $h=100$, $\sigma^*=(99-27)/2=36$. \end{example} In the following when we say ``arc'' we mean a complex Puiseux arc. \begin{proposition}\label{p:alpha} \begin{enumerate} \rm \item \it For any arc $\alpha$, there is some polar $\gamma$ of $f$ such that $d_{\gamma}\ge d_{\alpha}$. \rm \item \it For all $\beta \in \mathcal{GC}(\gamma_{*})$ one has $d_{\beta}=d_{{\mathrm{gr}}}(\gamma)$. \rm \item \it For any $\alpha \in \mathcal{GC} (\gamma_{*})$ one has $f(\alpha(y))=ay^h+\mathrm{h.o.t.}$, where $a$ and $h$ depend only on the canyon. \end{enumerate} \end{proposition} \begin{proof} (a). By using the Newton polygon relative to $\alpha$, $\mathcal{NP}(f,\alpha)$, see Figure \ref{fig:npF}, we observe that a polar can be obtained by pushing forward along $L^*$. Namely we construct a root of $f_x$ starting from $\mathcal{NP}(f_x,\alpha)$) by the Newton-Puiseux algorithm. This procedure adds up terms of degree at least $d_{\alpha}$, so we end up with at least one polar of the form $\gamma=\alpha +cy^{d_{\alpha}}+ \mathrm{h.o.t.}$, where $c$ is a root of the associated polynomial in $x$ (i.e. the derivative of the de-homogenisation of the polynomial associated to $L^*$). We then get ${\mathrm{ord}}_y(\alpha(y)-\gamma(y))\geq d_\alpha$, hence $d_{\alpha} \leq d_{{\mathrm{gr}}}(\gamma)$ for any such polar. Hence, starting with $\alpha$ one constructs polars by the diagram method and the process is not necessarily unique. Nevertheless, all such polars are clearly in the valley of $\alpha$. \noindent (b). Our assumption implies that $\beta=\gamma +cy^{d_{{\mathrm{gr}}}(\gamma)}+ \mathrm{h.o.t.}$ for some $c\in {\mathbb C}$, hence the Newton polygons $\mathcal{NP}(f,\beta)$ and $\mathcal{NP}(f,\gamma)$ will have the corresponding $L$ parallel, see Figure \ref{fig:npF}, and thus the same co-slope, which is $d_{{\mathrm{gr}}}(\gamma)$. \noindent (c). We have by definition $f(\gamma(y), y) = ay^{h}+\mathrm{h.o.t.}$ and by our assumption $\alpha(y) = \gamma(y) + cy^{d} + \mathrm{h.o.t.}$. Thus \begin{equation}\label{eq:h-coefficient} f(\gamma(y) + cy^{d} + \mathrm{h.o.t.}, y) = ay^{h}+ \cdots + \alpha(c)y^{d+h-1}+\mathrm{h.o.t.} , \end{equation} where the first terms depend only on the canyon (and not on the perturbation of $\gamma$), in particular the dependence of $c$ starts at the degree $d+h-1$. \end{proof} \begin{remark} \label{r:generic} Point (b) above holds for the gradient canyons but it is not necessarily true for arbitrary valleys. More precisely, in case of a valley $\mathcal{V}(\gamma_{*})$, the claim (b) holds only for arcs $\beta=\gamma +cy^{d_\gamma}+ \mathrm{h.o.t.}$ where the coefficient $c\in {\mathbb C}$ is generic. \end{remark} \begin{remark} In general, given $\alpha\in \mathcal{GC} (\gamma_{*})$ with ${\mathrm{ord}}_y(\alpha)=1$, to put it in the form $(\tilde\alpha(y),y)$ requires a rescaling of $y$ (i.e. replacing $y$ by $cy$ for some $c\neq 0$) and this yields $f(\alpha(y))=ac^hy^h+\mathrm{h.o.t.}$ \end{remark} \begin{corollary}\label{c:alpha} The function $\alpha \mapsto d_{\alpha}$ has its local maxima at the polars $\gamma$ of $f$ with $d_{\gamma} >1$. \hspace*{\fill}$\Box$ \end{corollary} \subsection{Analytic invariants} Let us show what are the canyon type invariants up to analytic equivalence, before entering the more involved study of the bi-Lipschitz invariants in \S \ref{s:disks}. So let $f= g \circ \varphi$, for some bi-holomorphic map $\varphi : ({\mathbb C}^{2}, 0) \to ({\mathbb C}^{2}, 0)$. For some arc $(\alpha(y), y)$, we then have $\varphi(\alpha(y), y) = (\varphi_{1}(\alpha(y), y), \varphi_{2}(\alpha(y), y))$ with \\ ${\mathrm{ord}}_{y}\varphi_{2}(\alpha(y), y) =1$, hence we may write $\varphi(\alpha(y), y) = (\beta(\bar y), \bar y)$ for some arc $\beta$, where $\bar y := \varphi_{2}(\alpha(y), y)$. Then we have: \begin{theorem}\label{t:degree} For any polar $\gamma_{f}$ of $f$ there exists a polar $\gamma_{g}$ of $g$ such that \[\varphi(\mathcal{GC}(\gamma_{f*})) = \mathcal{GC} (\gamma_{g*}) \] and the canyon degrees are the same. \end{theorem} \begin{proof} Let us prove first: \begin{lemma} \label{l:d}\label{p:varphi1} $d_{\varphi(\alpha)}= d_{\alpha}$ and $\varphi(\mathcal{V}(\alpha_{*})) = \mathcal{V} (\varphi(\alpha(y), y)_{*})$. \end{lemma} \begin{proof} We have $\varphi (\alpha + cy^{d_{\alpha}}, y)) = \varphi (\alpha) + a(c)\bar y^{d_{\alpha}} + \mathrm{h.o.t.}$ and the following equivalence: \[ \mathop{\rm{grad}}\nolimits f(\alpha + cy^{d_{\alpha}}, y) \simeq_{ord} \mathop{\rm{grad}}\nolimits g(\varphi (\alpha + cy^{d_{\alpha}}, y)) \] \[ = \mathop{\rm{grad}}\nolimits g(\varphi (\alpha) + a\bar y^{d_{\alpha}} + \mathrm{h.o.t.}, \bar y). \] By the definition \eqref{GGa} of the degree, one may consider some generic coefficient $c\in {\mathbb C}$, and its transform $a(c)$ which is also generic. Then by Remark \ref{r:generic} we may apply Proposition \ref{p:alpha}(b) for valleys, and get the inequality $d_{\alpha} \ge d_{\varphi(\alpha)}$. We apply the same to $\varphi^{-1}$ instead of $\varphi$ and obtain the converse inequality $d_{\varphi(\alpha)}\ge d_{\varphi^{-1}(\varphi(\alpha)) }= d_{\alpha}$, thus our first claim is proved. Next, we have: \[ {\mathrm{ord}}_{y}\| \varphi ((\alpha(y), y) - \varphi ((\beta(y), y)\| = {\mathrm{ord}}_{y}\| (\alpha(y), y) - (\beta(y), y)\| . \] By using the just proven equality of degrees we get the second claimed equality. \end{proof} We have $\varphi(\mathcal{GC}(\gamma)) = \mathcal{V} (\varphi(\gamma(y), y))$ by Lemma \ref{p:varphi1}. After Proposition \ref{p:alpha} we may associate to $\varphi(\gamma(y), y)$ some polar $\gamma_{g}$ in the valley of $\varphi(\gamma(y), y)$ with $d_{\gamma_{g}}\ge d_{\varphi(\gamma)}$, and therefore $\mathcal{V} (\varphi(\gamma(y), y)) \supset \mathcal{V} (\gamma_{g}) = \mathcal{GC} (\gamma_{g})$. We apply $\varphi^{-1}$ and get similarly: $\mathcal{GC}(\gamma_{*})\supset \varphi^{-1}(\mathcal{GC}(\gamma_{g*})) \supset \mathcal{GC}(\gamma_{f*})$ for some $\gamma_{f}$ constructed like in Proposition \ref{p:alpha}, with $d_{\gamma_{f}} \ge d_{\varphi^{-1}(\gamma_{g})}$. According to the minimality principle of polar canyons Theorem \ref{thmB}, we must have equality: $\mathcal{GC}(\gamma_{*})= \mathcal{GC}(\gamma_{f*})$ and $d_{\gamma_{f}} = d_{\gamma}$. Consequently we get that $\varphi(\mathcal{GC}(\gamma_{*})) = \mathcal{GC} (\gamma_{g*})$ and the degrees are equal. \end{proof} While analytic maps do not preserve polars, we may now prove the analytic invariance of the canyons: \begin{theorem}\label{t:main1} If $f = g\circ \varphi$ with $\varphi$ bi-holomorphic, then $\varphi$ transforms canyons into canyons by preserving their degrees and multiplicities. \end{theorem} \begin{proof} Theorem \ref{t:degree} shows that $\varphi $ sends a gradient canyon to a gradient canyon by preserving the degree. The preservation of the multiplicity follows from Proposition \ref{p:alpha}(c). \end{proof} \section{Generic polars}\label{s:genpolars} Based on Langevin's approach \cite{La-cmh} to the integral of the curvature of the Milnor fibre of a function of $n$ complex variables, Garcia Barroso and Teissier \cite{GT} gave a method to detect the concentration of curvature on the Milnor fibre of a function germ in 2 variables. Using Langevin's exchange formula which interprets the curvature in terms of polar curves, they showed that the intersections of the Milnor fibre with all generic polar curves is concentrated in certain small balls, and hence the curvature too. More recently, Koike, Kuo and P\u aunescu \cite{KKP} adopted a new viewpoint by looking into the curvature formula itself and studying its variation over the space of arcs. Their method uses the gradient canyons and provides sharper localization of the ``A'Campo bumps'' i.e. maxima of curvature. We shall find here the relations between the results obtained in \cite{GT} and in \cite{KKP}. Let $\gamma_{0}$ denote a solution of the equation $f_{x}(\gamma_{0}(y), y)=0$. Let $l_{\tau} \subset {\mathbb C}^{2}$ of coordinates $(x,y)$ denote the line $\{y-\tau x=0\}$, and call it \emph{the line of co-direction $\tau$}. The polars $\gamma_{\tau}$ are the solutions of the equation: \begin{equation}\label{eq:tau} (f_{x}+\tau f_{y})(\gamma_{\tau}(y), y)=0. \end{equation} \begin{theorem}\label{t:tau} The gradient canyon $\mathcal{GC}(\gamma_{\tau *})$ does not depend on the direction $\tau \in {\mathbb C}$, i.e. $\mathcal{GC}(\gamma_{0*}) =\mathcal{GC}(\gamma_{\tau *})$, $\forall \tau \in {\mathbb C}$. The canyon degree $d_{\gamma_{0}}$ is the lowest exponent from which the polar expansions $\gamma_{\tau}$ start to depend of $\tau$. The multiplicities $m_{\gamma_{\tau}}$ do not depend of $\tau$. \end{theorem} \begin{proof} In case $d_{\gamma_{0}}>1$ we consider the function $(f_{x} +\tau f_{y})(x + \gamma_{0}(y), y)$ from which we want to construct a solution of \eqref{eq:tau} by the method of ``pushing forward'' in the Newton diagram, as explained in the proof of Proposition \ref{p:alpha}. The polars associated to the direction $\tau$ are the Newton-Puiseux zeroes of the function $g(x,y) = (f_{x} +\tau f_{y})(x + \gamma_{0}(y), y)$ translated by $\gamma_{0}$, namely $\gamma_{\tau}(y) := x(y)+ \gamma_{0}(y)$. The top edge $E_{\top}$ of the Newton polygon of $g$ is parallel to the segment $L$ defined as in Figure \ref{fig:npZ}, taking $\alpha = \gamma_{0}$. Whenever $d_{\gamma_{0}}>1$, the segment $L$ has only one dot which depends on $\tau\not=0$, namely the one corresponding to the monomial $a(\tau)y^{h-1}$ (which comes from the contribution $\tau f_{y}$). Therefore the edging forward process will start with the initial term of the form $c(\tau)y^{d_{\gamma_{0}}}$ in order to annihilate $a(\tau)y^{h-1}$. Thus the Newton-Puiseux zero of $g$ will be of the form $x(y) = c(\tau)y^{d_{\gamma_{0}}} +\mathrm{h.o.t.}$, hence $\gamma_{\tau}(y) = \gamma_{0}(y) + c(\tau)y^{d_{\tau_{0}}} +\mathrm{h.o.t.}$ is a polar associated to $\tau$. This shows in particular that the constructed solution $\gamma_{\tau}$ is in the same gradient canyon as the polar $\gamma_{0}$. Note that the generic polars that we have constructed $\gamma_{\tau}(y) = \gamma_{0}(y) + c(\tau)y^{d_{\tau_{0}}} +\mathrm{h.o.t.}$ are in the canyon of $\gamma_0$ and therefore $f(\gamma_{\tau},y)=ay^h+\mathrm{h.o.t.}$, thus the initial term is constant in the canyon, and in particular the exponent $h$ does not depend of $\tau$. By construction the number of roots $x=x(y)$ of $g(x,y)=0$ is $m_{\gamma_{0}}$, where $(m_{\gamma_{0}} + 1, r)$ is the initial (lowest) dot of $L$. Consequently $(m_{\gamma_{0}},r)$ is the initial dot of $E_{\top}$, hence \[ \frac{h-1-r}{m_{\gamma_{0}}} = d_{\gamma_{0}} \] and $m_{\gamma_{0}}$ is the multiplicity of the canyon, i.e. the total number of polars in the canyon $\mathcal{GC}(\gamma_{0})$, for any $\tau \in {\mathbb C}$. \end{proof} \subsection{Garcia-Barroso and Teissier's approach \cite{GT}}\label{ss:GT} Let us recall some of the results obtained in \cite{GT} by following their original notations. \medskip \noindent (1). Let $P_{q}(\tau) = (x(t), y(t))$ where $x(t) = t^{m}$, $y(t)= at^{m} + \mathrm{h.o.t.}$, be a minimal parametrisation of an irreducible branch of the polar curve with respect to a direction $\tau \in {\mathbb P}^{1}$. Here $m_{q}= m_{q}(\tau)$ is the multiplicity at 0 of $P_{q}(\tau)$. Teissier had proved that the family $P_{q}(\tau)$ depending of $\tau$ is equisingular for generic $\tau$, thus the multiplicity $m_{q}(\tau)$ is constant for generic $\tau$. In the following we consider $\tau$ in such a generic set. Barroso and Teissier show in \cite{GT} that the polars fall into subsets called ``packets'' indexed by the black vertices of the Eggers diagram of $f$, such that they have the same contact with all branches of the curve $C:= \{ f=0\}$. Such a ``packet'' of polars is the set of polars from a certain union of canyons. \medskip \noindent (2). By \cite[Theorem 2.1]{GT}, the coefficients of $P_{q}(\tau)$ depend on $\tau$ only from a certain well-defined exponent of $t$. Let $g_{q}$ denote the first exponent of $y(t)$ the coefficient of which depends of $\tau$. It is shown that all the polars in the same packet have the same exponent $g_{q}$ and this is denoted by $\gamma_{q}$, cf \cite[pag. 406]{GT}. \medskip \noindent (3). Moreover, in the development of $f(t^{m_{q}}, y_{q}(t, \tau))$, the first exponent the coefficient of which depends of $\tau$ is $e_{q}+ g_{q}$. The geometric significance of $e_{q}$ is given by the identity \[ {\rm{mult}}_{0}(C, P_{q}(\tau))= e_{q} + m_{q},\] where $e_{q}= \mu_{q}(f)$ is a partial Milnor number in the sense that, by Teissier's formula for the polar multiplicity, the sum of $e_{q}$'s over all polar in the packet and over all packets is equal to the Milnor number $\mu_{f}$. \medskip \noindent (4). The concentration of points of intersection $P_{q}(\tau) \cap \{ f= \lambda\}$ on the Milnor fibre, for all generic $\tau$ and as $\lambda$ approaches 0, is equivalent to the concentration of curvature, according to Langevin's approach \cite{La-cmh}. In order to locate the zones of concentration on the Milnor fibre, i.e. the centers of the balls and their radii, Barroso and Teissier invert the convergent series $\lambda = \lambda(t)$ and expresses the coordinates $(x(t),y(t))$ as functions of $\lambda$ (see \cite[(5), page 408]{GT}). Let us now see what are the relations between these invariants and those defined in \cite{KKP} and in our previous sections. \subsection{A dictionary}\ We have shown that the value $h= {\mathrm{ord}}_{y}f(\gamma(y), y)$ is the same for all polars in some canyon (Proposition \ref{p:alpha}). Therefore $m_{\gamma}h := {\mathrm{ord}}_{y}f(\gamma(y^{m_{\gamma}}), y^{m_{\gamma}}) = {\rm{mult}}_{0}(C, \gamma_{*})$, where $m_{\gamma}$ is the multiplicity of the polar considered with its multiple structure. Note the difference to \cite{KKP} and the preceding sections, where by ``polar'' we mean with multiplicity 1, and precisely $m_{\gamma}$ such polars have the same image $\gamma_{*}$. On the other hand, by \S \ref{ss:GT}(3), from the \cite{GT} viewpoint we have ${\rm{mult}}_{0}(C, \gamma) = e_{\gamma} + m_{\gamma}$. We therefore conclude: \begin{equation} \label{eq:mu} e_{\gamma} = m_{\gamma}(h-1) \end{equation} which can be identified with a partial sum of \eqref{M}. By Theorem \ref{t:tau}: \[ \gamma_{\tau}(y) = \gamma_{0}(y)+ c(\tau) y^{d_{\gamma_{0}}} + \mathrm{h.o.t.} \] which implies \[ f(\gamma_{\tau}(y), y) = a y^{h} + \cdots + u(\tau)y^{d_{\gamma_{0}} +h-1} + \mathrm{h.o.t.} \] where $u(\tau)$ is the first coefficient which depends of $\tau$; thus: \[ f(\gamma_{\tau}(y^{m}), y^{m}) = a y^{mh} + \cdots + u(\tau)y^{md_{\gamma_{0}} +m(h-1)} + \mathrm{h.o.t.} \] By \cite[Lemma 2.2]{GT}: \[ f(\gamma_{\tau}(y^{m}), y^{m}) = a y^{mh} + \cdots + u(\tau)y^{e_{q} + g_{q}} + \mathrm{h.o.t.} \] using the notations $e_{q}$ and $ g_{q}$ from \S \ref{ss:GT}(3). We obtain: \[e_{q} + g_{q} =m_{q}d_{\gamma} +m_{q}(h-1)\] hence \[ g_{q}= m_{q} d_{\gamma} \] which shows that the exponent $g_{q}$ of \cite{GT} reminded at \S \ref{ss:GT}(2) is essentially the same as the degree $d_{\gamma}$ of the canyon, i.e. modulo multiplication by the multiplicity $m_{\gamma}$. \section{The correspondence of canyon disks}\label{s:disks} We consider in this section a gradient canyon $\mathcal{GC}(\gamma_*)$ of degree $d_{\gamma} >1$. Let $D^{(e')}_{\gamma_{*}, \varepsilon}(\lambda;\eta)$ be the union of disks in the Milnor fibre $\{ f=\lambda\}\cap B(0;\eta)$ of $f$ defined as follows (see \eqref{eq:hd} for the definition of the ${\rm{Horn}}$): \[ D^{(e')}_{\gamma_{*}, \varepsilon}(\lambda;\eta) := \{ f=\lambda\} \cap {\rm{Horn}}^{(e')}(\gamma_*;\varepsilon;\eta), \] for some rational $e'$ close enough to $d_{\gamma}$, with $1< e' <d_{\gamma}$, for some small enough $\varepsilon>0$, and where by ``disk'' we mean ``homeomorphic to an open disk''. In addition, we ask that $d< e' <d_{\gamma}$ for any other canyon degree $d < d_{\gamma}$. We have: \begin{equation}\label{eq:horn} \bigcap_{e' \in {\mathbb Q}, e' \to d_{\gamma}}D^{(e')}_{\gamma_{*}, \varepsilon}(\lambda;\eta) = \{ f=\lambda\} \cap {\rm{Horn}}^{(d_{\gamma})}(\gamma_*;\varepsilon;\eta) \end{equation} and we shall write $D_{\gamma_{*}}(\lambda)$ in the following as shorthand for $D^{(e')}_{\gamma_{*}, \varepsilon}(\lambda;\eta)$, keeping in mind the parameters $e', \varepsilon, \eta$. \smallskip By \cite[Lemma 6.6]{KKP} we have--see also \eqref{eq:measure} and \eqref{eq:measure2}: \begin{equation}\label{eq:curvintegralcanyon} \mathscr M_f({\mathcal D}^{(e')}(\gamma_{*}, \varepsilon)) = \mathscr M_f({\mathcal L}^{(e')}(\gamma_*)) = \mathscr M_f(\mathcal{GC}(\gamma_*)), \end{equation} and moreover ${\mathcal L}^{(e')}(\gamma_*)$ does not contain any other polar canyon besides $\mathcal{GC}(\gamma_*)$. This result means that a certain part of the curvature of the Milnor fibre is concentrated in the union $D_{\gamma_{*}}(\lambda) = \cup_{i}D_{\gamma_{*, i}}(\lambda)$, of connected disks $D_{\gamma_{*, i}}(\lambda)$. The number of disks is the intersection number ${\rm{mult}}_{0} (\{ f=0\}, \overline{\gamma_{*}})$, where $\overline{\gamma_{*}}$ is the truncation of the polar at the order $d_{\gamma}$. Here we have to understand $\gamma_{*}$ as image of $\gamma_{*}$, which is thus the same image for all conjugates of $\gamma$, and similarly for the truncations. There might be non-conjugate polars in the same canyon, and then the (centers of) the disks are the same. The centers and the radii of the disks $D_{\gamma_{*, i}}(\lambda)$ are given more explicitly in \cite[\S 3.1]{GT}, as we shall briefly describe in the following. First, one has to express the coordinates $x=\gamma(y)$ and $y$ in terms of $\lambda$. One obtains an expansion $(x(\lambda), y(\lambda))$ with complex coefficients: \begin{equation}\label{eq:centers} \left( \sum_{i=m}^{\infty} \alpha_{i}\lambda^{\frac{i}{mh}}, \sum_{i=m}^{\infty} \beta_{i}\lambda^{\frac{i}{mh}} \right). \end{equation} For polars $\gamma_{\tau}$ depending of the generic direction $\tau$, as we have discussed in \S \ref{s:genpolars}, the first coefficients of \eqref{eq:centers} which depend of $\tau$ are $\alpha_{md}$ and $\beta_{md}$, where $m$ is the multiplicity of $\gamma_{\tau}$ and $d$ is its polar degree, both of which are independent of the generic $\tau$, by Theorem \ref{t:tau}. Note that $h$ is also independent of $\tau$. The centers of the disks\footnote{The number of the disks is $mh$.} $D_{\gamma_{*, i}}(\lambda)$ are then the truncations of \eqref{eq:centers} up to the order $(md-1)/mh$. They are conjugated by $\lambda\to \omega^{i}\lambda$, where $\omega$ is a primitive root of order $mh$. The radii of the disks are of the form $r | \lambda|^{d/h} \sim | y |^{d}$, where $r\in {\mathbb R}_{+}$ depends on the compact subset of ${\mathbb P}^{1}$ in which $\tau$ varies. The distance between two (centers of) such disks is of order $| \lambda|^{1/h}\sim | y |$. \begin{theorem}\label{t:localcurv} Let $f= g\circ \varphi$, where $\varphi$ is bi-Lipschitz. Then: \[ \lim_{\lambda\to 0} \frac{1}{2\pi} \int_{\varphi(D_{\gamma_{*, i}}(\lambda))} K_{g} \ dS \ \ge 1\] for any $i$. \end{theorem} The proof consists of several steps. \begin{lemma}\label{l:totalcurvconcentr} $ \displaystyle \lim_{\lambda\to 0} \frac{1}{2\pi} \int_{D_{\gamma_{*, i}}(\lambda)} K_{f} \ dS \ $ is a positive integer, for any $i$. \end{lemma} \begin{proof} By \eqref{eq:curvintegralcanyon} the canyon $\mathcal{GC}(\gamma_*)$ concentrates the total curvature: \[ \frac{1}{2\pi} \mathscr M_{f}\mathcal{GC}(\gamma_*) = \sum {\rm{mult}}_{0}(\{ f=0\}, \gamma'_{*}), \] where the sum is taken over all polars $\gamma'_{*}$ in the canyon $\mathcal{GC}(\gamma_*)$. This is a multiple of the number ${\rm{mult}}_{0} (\{ f=0\}, \overline{\gamma_{*}})$ of the disks of the canyon $\mathcal{GC}(\gamma_*)$. These disks contain all the intersections of the Milnor fibre with the polars $\gamma_{\tau *}$, for $\tau$ in some dense subset of a compact $K(\lambda) \subset {\mathbb P}^{1}$ which tends to ${\mathbb P}^{1}$ when $\lambda \to 0$. On the other hand, as we have seen just above, these disks are conjugate. Therefore, when $\lambda \to 0$, each such disk concentrates the same total curvature, which must be a positive integer (modulo $2\pi$). \end{proof} We need the interpretation of Lemma \ref{l:totalcurvconcentr} in terms of the directions $\tau\in {\mathbb P}^{1}$. By applying Milnor's exchange formula (see Langevin's paper \cite{La-cmh}) we have the equalities: \begin{equation}\label{eq:exchange} \frac{1}{2\pi} \int_{D_{\gamma_{*, i}}(\lambda)} K_{f} \ dS = \frac{1}{2\pi} \int_{D_{\gamma_{*, i}}(\lambda)} |{\mathrm{Jac} \ } \nu_{{\mathbb C}}|^{2} \ dS \end{equation} where ${\mathrm{Jac} \ } \nu_{{\mathbb C}}$ denotes the Jacobian determinant of the complex Gauss map. In turn, this is equal, cf \cite{La-cmh}, to: \begin{equation}\label{eq:degree} \frac{1}{2\pi} \int_{ D_{\gamma_{*, i}}(\lambda)} \nu_{{\mathbb C}}^{*} dp = \frac{1}{2\pi} \int_{\nu_{{\mathbb C}}( D_{\gamma_{*, i}}(\lambda))} \deg ({\nu_{{\mathbb C}}}_{| D_{\gamma_{*, i}}(\lambda)}) \ dp \end{equation} where the last equality follows from the constancy of the degree $\deg ({\nu_{{\mathbb C}}}_{| D_{\gamma_{*, i}}(\lambda)})$ by Theorem \ref{t:tau}. Since $2\pi$ represents the volume of ${\mathbb P}^{1}$, we have proved: \begin{lemma}\label{l:dense} The image of the disk $D_{\gamma_{*, i}}(\lambda)$ by the Gauss map $\nu_{{\mathbb C}}$, as $\lambda$ tends to 0, is a dense subset of ${\mathbb P}^{1}$, the complementary of which has measure zero. \hspace*{\fill}$\Box$ \end{lemma} \medskip We continue the proof of Theorem \ref{t:localcurv}. From $f=g\circ \varphi$ we get the relation: \[ \mathop{\rm{grad}}\nolimits f (x,y)= \mathop{\rm{grad}}\nolimits g(\varphi(x,y))\circ M_{\varphi}(x,y) \] \noindent where $M_{\varphi}$ is a certain matrix, which plays the role of the Jacobian matrix, not everywhere defined but only in almost all points. Let us introduce it. The idea is that even if the partial derivatives of $\varphi$ do not exist at all points, the limits used to define them are bounded away from 0 in absolute value. Let us notice that the components $\varphi_{1}$ and $\varphi_{2}$ of the map $\varphi$ are real maps depending on coordinates $x, \bar x, y, \bar y$ but that only the derivatives with respect to $x$ and $y$ will play a role in the following. By the bi-Lipschitz property of $\varphi = (\varphi_{1}, \varphi_{2})$ we have, in some ball neighbourhood $B(0, \eta)$ of the origin $(0,0)$, for some $0<m <M$: \[ m \le \frac{\left\| (\varphi_{1}, \varphi_{2})(x,y_{0}) - (\varphi_{1}, \varphi_{2})(x_{0},y_{0})\right\| }{\lvert x-x_{0} \rvert }\le M \] and by taking the limit as $x\to x_{0}$ we get: \begin{equation}\label{eq:gradvarphi} m \le \left\| (\varphi_{1, x}, \varphi_{2, x})(x_{0}, y_{0}) \right\| \le M \end{equation} where the notation $\varphi_{1, x}$ suggests partial derivative with respect to $x$; it has a well-defined value at points where this derivative exists. This limit is not defined elsewhere, but it is however bounded by the values $m$ and $M$. We shall call \emph{pseudo-derivatives} such bounded quantities $\varphi_{1, x}$ and $\varphi_{2, x}$. Similarly we get, by taking the limit $y\to y_{0}$: \begin{equation}\label{eq:gradvarphi-sim} m \le \left\| (\varphi_{1, y}, \varphi_{2, y})(x_{0}, y_{0}) \right\| \le M. \end{equation} We shall also use the notations $\mathop{\rm{grad}}\nolimits \varphi_{i} := (\varphi_{i,x}, \varphi_{i,y}) $ for $i=1,2$. With these notations we shall prove that the matrix $M_{\varphi} = \begin{pmatrix} \varphi_{1,x} & \varphi_{1,y} \\ \varphi_{2,x} & \varphi_{2,y} \end{pmatrix}$ is bounded in some neighbourhood of the origin, in a strong sense that we shall define below. \begin{lemma}\label{l:bounds1} There exist $r_{1}, r_{2}>0$ such that: \[ \| \mathop{\rm{grad}}\nolimits \varphi_{1} \| \ge r_{1} \mbox{ and } \| \mathop{\rm{grad}}\nolimits \varphi_{2} \| \ge r_{2} \] in some neighbourhood of the origin. \end{lemma} \begin{proof} \noindent With the above notations, from $\varphi^{-1} \circ \varphi = {\mathrm{id}}$ on $B(0, \eta)$ we get: \[ \begin{pmatrix} {\varphi^{-1}}_{1,x} & {\varphi^{-1}}_{1,y} \\ {\varphi^{-1}}_{2,x} & {\varphi^{-1}}_{2,y} \end{pmatrix} \begin{pmatrix} \varphi_{1,x} & \varphi_{1,y} \\ \varphi_{2,x} & \varphi_{2,y} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \] and so: \[ \left\{ \begin{array}{l} {\varphi^{-1}}_{1,x} \varphi_{1,x} + {\varphi^{-1}}_{1,y}\varphi_{2,x} =1 \\ {\varphi^{-1}}_{2,x} \varphi_{1,y} + {\varphi^{-1}}_{2,y}\varphi_{2,y} =1. \end{array} \right. \] From this and from \eqref{eq:gradvarphi} we get that $ \| \mathop{\rm{grad}}\nolimits {\varphi^{-1}}_{1}\|$ and $ \| {\mathop{\rm{grad}}\nolimits \varphi^{-1}}_{2}\|$ are bounded away from $0$ in some neighbourhood of the origin. By symmetry we get the same conclusion for $ \varphi_{1}$ and $\varphi_{2}$, hence our claim is proved. \end{proof} \begin{lemma}\label{l:bounds2} There exists some $m_{1} >0$ such that: \[ \| \varphi_{2,y} (x,y) \| \ge m_{1} \] for any $(x,y)$ belonging to the canyon $\mathcal{GC}(\gamma_{*})$. \end{lemma} \begin{proof} Let $(x,y)\in \mathcal{GC}(\gamma_{*})$ in the following. By the definition of the canyon, and denoting $d:= d_{\gamma}$, we have: \[ \| (x,y) - (\gamma(y), y) \| \sim | y|^{d}. \] By the bi-Lipschitz property we then have the equivalence: \[\| \varphi(x,y) - \varphi(\gamma(y), y) \| \sim | y|^{d}. \] Since $d>1$ we get that $\| (x,y)\| \sim | y|$ and on the other hand, by dividing with $|y|$, the limit \[\left\| \frac{\varphi(x,y)}{| y|} - \frac{ \varphi(\gamma(y), y)}{ | y|} \right\| \to 0 \mbox{ as } y\to 0. \] These imply in particular: \begin{equation}\label{eq:varphi2} \left\| \frac{\varphi_{2}(x,y)}{| y|} - \frac{ \varphi_{2}(\gamma(y), y)}{ |y|} \right\| \to 0 \mbox{ as } y\to 0. \end{equation} We claim that $\|\varphi_{2}(\gamma(y), y) \| \sim |y| $. From this and from \eqref{eq:varphi2} we then get that \\ $\| \varphi_{2}(\gamma(y), y)/ |y| \|$ is bounded away from 0 as $y\to 0$, which means that the pseudo-derivative norm $\| \varphi_{2, y}\|$ is bounded away from 0 in the canyon; this proves our lemma. \medskip Let us now prove the above claim. From the very beginning we may choose the coordinates in ${\mathbb C}^{2}$ such that both $f$ and $g$ are \emph{miniregular}\footnote{in the terminology of \cite{HP, KKP}}, i.e. that the tangent cones of $f$ and $g$ do not contain the direction $[1;0]$. By our assumptions, the polar $\gamma$ is \emph{tangential}, i.e. its tangent cone is included in the one of $\{ f=0\}$. Let us assume without loss of generality that this is the $y$-axis. This means that $\gamma$ has contact $k>1$ with some root $(\xi(y),y)$ of $\{ f=0\}$. By the bi-Lipschitz property: \begin{equation}\label{eq:bilip} m \| (\xi(y),y) - (\gamma(y), y) \| \le \| \varphi(\xi(y),y) - \varphi(\gamma(y), y) \| \le M \| (\xi(y),y) - (\gamma(y), y) \| \end{equation} and we have the equivalence $\| (\xi(y),y) - (\gamma(y), y) \| \sim | y|^{k}$. Since by bi-Lipschitz we have $ \| \varphi(\gamma(y), y) \| \sim | y|$ then by the above facts we get: \begin{equation}\label{eq:varphi12} \| \varphi(\xi(y), y) \| \sim | y|. \end{equation} Next: \[ \| (\varphi_{1}(\xi(y),y), \varphi_{2}(\xi(y),y) \| = | \varphi_{2}(\xi(y),y)| \left\| \left( \frac{\varphi_{1}(\xi(y),y)}{\varphi_{2}(\xi(y),y)}, 1\right) \right\| \] Since $f = g\circ \varphi$, the root $\xi$ is sent by $\varphi$ to some root $\eta =(\eta_{1}, \eta_{2})$ of $g$, which means that the direction $ \left[ \frac{\varphi_{1}(\xi(y),y)}{\varphi_{2}(\xi(y),y)}, 1\right]$ is the same as the direction $[\frac{\eta_{1}}{ \eta_{2}}, 1]$. The later tends to the direction of the tangent line to $\eta$, which is different from $[1,0]$ by our assumption. Hence this is of the form $[a,1]$, where $a\in {\mathbb C}$. Consequently: \[ {\mathrm{ord}}_{y}\varphi_{1}(\xi(y),y) \ge {\mathrm{ord}}_{y}\varphi_{2}(\xi(y),y). \] Thus, with help of \eqref{eq:varphi12}, we get: \[ {\mathrm{ord}}_{y}\varphi_{2}(\xi(y),y) = {\mathrm{ord}}_{y}\varphi(\xi(y),y) = 1, \] which implies $\|\varphi_{2}(\xi(y), y) \| \sim |y|$ and which, in turn, implies our claim by using again \eqref{eq:bilip} and since $\gamma(y)$ has contact $>1$ with $\xi(y)$. \end{proof} \subsection{Continuation of the proof of Theorem \ref{t:localcurv}}\ \\ \noindent \textbf{Step 1.} We claim that $\varphi(D_{\gamma_{*, i}}(\lambda))$ intersects a disk cut out by some horn ${\rm{Horn}}^{(d)}(\gamma_{g,*};\varepsilon;\eta)$ of degree $d>1$ into the fibre $g=\lambda$ Let $G(\lambda)$ denote what remains from the Milnor fibre $g=\lambda$ after taking out all the horns of degree $d>1$. By \emph{reductio ad absurdum}, let us suppose that $\varphi(D_{\gamma_{*, i}}(\lambda))\subset G(\lambda)$ asymptotically, that is for any $\lambda$ close enough to $0$. We have seen (after \eqref{eq:centers}) that the distribution of curvature in $G(\lambda)$ is of order equal to ${\mathrm{ord}} |\lambda |^{1/h}$. The integral of curvature over $G(\lambda)$ equals $2\pi (r-1)$, which means that the image of the Gauss map on $G(\lambda)$ has dense image in ${\mathbb P}^{1}$ (see also Lemma \ref{l:dense}) and the degree of this map is $r-1$, thus it is at least 1 if $r\ge 2$. Moreover, by the results of Henry and Parusi\' nski \cite{HP, HP2}, the variation of the gradient itself on $G(\lambda)$ is of order equal to ${\mathrm{ord}} |\lambda |^{1/h}$, which means that there is no concentration of curvature on $G(\lambda)$ of higher order. The diameter of the disk $D_{\gamma_{*, i}}(\lambda)$ is of order ${\mathrm{ord}} |\lambda |^{d/h}$ with $d>1$, thus its bi-Lipschitz transform $\varphi(D_{\gamma_{*, i}}(\lambda))$ is of the same order. Therefore the integral of curvature over $\varphi(D_{\gamma_{*, i}}(\lambda))$ is asymptotically zero\footnote{this can be compared with the more particular case treated in \cite[Lemma 6.7]{KKP}}, namely: \begin{equation}\label{eq:lim} \lim_{\lambda\to 0}\int_{\varphi(D_{\gamma_{*, i}}(\lambda))} K_{g} \ dS =0. \end{equation} Then, by using the ``exchange formula'' \eqref{eq:degree} and \eqref{eq:exchange} for $\varphi(D_{\gamma_{*, i}}(\lambda))$, it follows that its image in ${\mathbb P}^{1}$ by the Gauss map $\nu_{{\mathbb C}, g} =\frac{\mathop{\rm{grad}}\nolimits g}{|\mathop{\rm{grad}}\nolimits g|} : B(0;\eta) \to {\mathbb P}^{1}$ is a contractible set which tends to a measure zero subset $A\subset {\mathbb P}^{1}$ as $\lambda \to 0$. But we claim more: \smallskip \noindent (*) \emph{If $\varphi(D_{\gamma_{*, i}}(\lambda))\subset G(\lambda)$ then the image $\nu_{{\mathbb C}, g}(\varphi(D_{\gamma_{*, i}}(\lambda)))$ tends to a constant when $\lambda \to 0$.} \smallskip The variation of the direction of the gradient of $g$ (i.e. of the Gauss map $\nu_{{\mathbb C}, g}$) on $\varphi(D_{\gamma_{*, i}}(\lambda))$ is of order $d/h$, with $d>1$. This means that it is asymptotically zero with respect to the variation on $G(\lambda)$, which is of order $1/h$ by the result of Henry and Parusi\' nski \cite{HP, HP2}. Thus the direction of the gradient of $g$ on $\varphi(D_{\gamma_{*, i}}(\lambda))$ tends to a constant and our claim is proved. We are now finishing the proof of Step 1. We have: \[ \mathop{\rm{grad}}\nolimits f (x,y)= \mathop{\rm{grad}}\nolimits g(\varphi(x,y))\circ M_{\varphi}(x,y). \] Let $(a,1)$ denote the limit direction of the gradient of $g$ that we obtain by the above condition (*). We then have: \begin{equation}\label{eq:bounded} (a,1) \begin{pmatrix} \varphi_{1,x} & \varphi_{1,y} \\ \varphi_{2,x} & \varphi_{2,y} \end{pmatrix} = (a \varphi_{1,x} + \varphi_{2,x}, a\varphi_{1,y} + \varphi_{2,y}) \end{equation} and, by using \eqref{eq:gradvarphi-sim} and Lemma \ref{l:bounds2}, we get \[ | a\varphi_{1,y} + \varphi_{2,y} | \ge | \varphi_{2,y} | - | a\varphi_{1,y}| \ge m_{1} - | a\varphi_{1,y}| >0. \] It then follows from the relation \eqref{eq:bounded} that the modulus of the direction of the gradient vector $\mathop{\rm{grad}}\nolimits f (x,y)$ on the disk $D_{\gamma_{*, i}}(\lambda)$, namely: \[ \frac{\|a \varphi_{1,x} + \varphi_{2,x}\|}{\|a\varphi_{1,y} + \varphi_{2,y}\|} \] is bounded, since the denominator is bounded away from 0 and the numerator is less or equal to $\max(m_{1}, M)$. This contradicts Lemma \ref{l:dense}. Step 1 is thus proved. \bigskip \noindent \textbf{Step 2.} We still refer to canyon disks of canyon degree $>1$. Let $D_{f}$ be some disk cut out on the Milnor fibre $f^{-1}(\lambda)$ by some horn ${\rm{Horn}}^{(d)}(\gamma_{g,*};\varepsilon;\eta)$ of a canyon $\mathcal{GC}(\gamma_{*})$ of degree $d =\deg D_{f}$. We recall\footnote{cf the discussion about radius before the statement of Theorem \ref{t:localcurv}} that the radius of $D_{f}$ is $k |y|^{d}\sim_{{\mathrm{ord}}}|\lambda|^{d/h}$, for some $k>0$ and that the distance between two conjugated disks is of order ${\mathrm{ord}} | y|$. If two polars are in the same canyon, then their associated disks coincide (by definition). By ``canyon disk'' we shall mean in the following such a disk of radius order $d/h$ with respect to $|\lambda|$, modulo some multiplicative constant $>0$ which is not specified. By Step 1, there is some canyon disk $D_{g}$ of $g$, of canyon order $>1$, such that: \[ \varphi(D_{f})\cap D_{g} \not= \emptyset. \] \begin{lemma}\label{l:incl} If $\varphi(D_{f})\cap D_{g} \not= \emptyset$ then: \ \deg D_{g} \ge \deg D_{f} \] and moreover $\varphi(D_{f})$ includes $D_{g}$. \end{lemma} \begin{proof} The diameter of $\varphi(D_{f})$ is asymptotically of order equal to $\frac{1}{h}\deg D_{f}$, since $\varphi$ is bi-Lipschitz. So if $\deg D_{g} < \deg D_{f}$ it follows as in the above proof of \eqref{eq:lim} that the total curvature over $\varphi(D_{f})$ must be zero asymptotically. This yields a similar contradiction as we have proved in Step 1 for \eqref{eq:lim}. Now if $\deg D_{g} \ge \deg D_{f}$ then, by the definition of the disks (i.e. with fixed order and arbitrary radius) and since $\varphi(D_{f})\cap D_{g} \not= \emptyset$, it follows that $\varphi(D_{f})$ includes $D_{g}$ for appropriate diameters. \end{proof} Applying now Lemma \ref{l:incl} and Lemma \ref{l:totalcurvconcentr} ends the proof of Theorem \ref{t:localcurv}. \hspace*{\fill}$\Box$ \subsection{The correspondence of canyon disks} Let us continue the above reasoning with: \begin{lemma}\label{l:single} $\varphi(D_{f})$ intersects a single disk $D_{g}$. \end{lemma} \begin{proof} If $D_{g}$ and $D'_{g}$ are two disjoint disks of $g$ which intersect $\varphi(D_{f})$, then they are of degree strictly greater than $\deg D_{f}$, otherwise they must be included one into the other up to rescaling their radii. Hence they are included in $\varphi(D_{f})$, by Lemma \ref{l:incl}. Next, by applying $\varphi^{-1}$ we get $\varphi^{-1}(D_{g}) \subset D_{f}$ with $\deg D_{g} > \deg D_{f}$, hence, by Step 1 and Lemma \ref{l:incl}, there must exist another disk $D'_{f} \subset \varphi^{-1}(D_{g})$ with $ \deg D'_{f}\ge \deg D_{g}$. But this means that we have the inclusion $D'_{f} \subset D_{f}$ with the inequality $\deg D'_{f} > \deg D_{f}$ and this contradicts one of the fundamental results of \cite{KKP} that canyons of degree $>1$ are disjoint. \end{proof} We therefore have a graduate bijection between canyon disks of $f$ and canyon disks of $g$, respecting the degrees. More precisely, we have shown the following: \begin{theorem}\label{t:main0} The bi-Lipschitz map $\varphi$ establishes a bijection between the canyon disks of $f$ and the canyon disks of $g$ by preserving the canyon degree. \hspace*{\fill}$\Box$ \end{theorem} We will show that this key theorem further yields bi-Lipschitz invariants. \noindent \subsection{The multi-layer cluster decomposition.}\label{ss:layer} Let $f = g \circ \varphi$, for a bi-Lipschitz homeomorphism $\varphi$. Even if one cannot prove anymore that the image by $\varphi$ of a canyon is a canyon, as we did in Theorem \ref{t:main1} for a bi-holomorphic $\varphi$, we will derive from Theorem \ref{t:main0} that $\varphi$ induces a bijection between the gradient canyons of $f$ and those of $g$. Moreover, we show here that there are well-defined ``clusters'' of canyons of $f$ which are sent by $\varphi$ into similar clusters of $g$, and that such clusters are determined by certain rational integers which are bi-Lipschitz invariants. We consider the tangential canyons only, i.e. those of degree more than 1. The canyon of degree 1 is preserved, since it covers the Milnor fibre entirely, together with its multiplicity, or, equivalently, its partial Milnor number, or its total curvature, as follows directly from \cite{GT}, \cite{KKP}. Note that the exponent $h$ from \eqref{eq:h-coefficient} is a topological invariant, see e.g. \cite{GT}, \cite{KKP}. We may group the canyons in terms of the essentials bars of the tree of $f$, namely those canyons departing from an essential bar $B(h)$ corresponding to $h$, i.e. associated to the polars leaving the tree of $f$ on that bar $B(h)$. Their contact, for distinct canyons, can be greater or equal to the co-slope of the corresponding bar, say $\theta_{B(h)}$, but less than their canyon degrees. The \emph{order of contact}, see \cite{KKP}, between two different holomorphic arcs $\alpha$ and $\beta$ is well defined as: \begin{equation}\label{eq:order} \max {\mathrm{ord}}_{y} (\alpha(y) - \beta(y)) \end{equation} where the maximum is taken over all conjugates of $\alpha$ and of $\beta$. Whenever the canyons $\mathcal{GC}(\gamma_{1*})\ni \alpha_{*}$ and $\mathcal{GC}(\gamma_{2*})\ni \beta_{*}$ are different and both of degree $d>1$, this order is lower than $d$ and therefore does not depend on the choice of $\alpha_{*}$ in the first canyon, and of $\beta_{*}$ in the second canyon. This yields a well-defined \emph{order of contact between two canyons of degree $d$}. In a similar way we can define the contact of any two canyons as the contact of the corresponding polars in the canyons. The contacts between the Puiseux roots of $f$ are automatically preserved by $\varphi$, because we have similar trees for $f$ and $g$ (topological equivalence). The more interesting situations appear after the polars leave the tree, namely at a higher level than the co-slopes $\theta_{B(h)}$. \medskip Let $G_{d}(f)$ be the union of gradient canyons of a fixed degree $d>1$. Let $G_{d, B(h)}(f)$ be the union of canyons the polars of which grow on the same bar $B(h)$, for $d>\theta_{B(h)} >1$, more precisely those canyons of degree $d$ with the same top edge relative to the Newton polygon relative to polar. One then has the disjoint union decomposition: \begin{equation}\label{eq:first} G_{d}(f) = \bigsqcup_{h}G_{d, B(h)}(f). \end{equation} Note that each canyon from $G_{d, B(h)}(f)$ has the same contact, higher than 1, with a fixed irreducible component $\{f_{i} =0\}$. Next, each cluster union of canyons $G_{d, B(h)}(f)$ has a partition into unions of canyons according to the mutual order of contact between canyons. More precisely, a fixed gradient canyon $\mathcal{GC}_i(f) \subset G_{d, B(h)}(f)$ has a well defined order of contact $k(i,j)$ with some other gradient canyon $\mathcal{GC}_j(f) \subset G_{d, B(h)}(f)$ from the same cluster; we count also the \emph{multiplicity} of each such contact, i.e. the number of canyons $\mathcal{GC}_j(f)$ from the cluster $G_{d, B(h)}(f)$ which have exactly the same contact with $\mathcal{GC}_i(f)$. Let then $K_{d,B(h), i}(f)$ be the (un-ordered) set of those contact orders $k(i,j)$ of the fixed canyon $\mathcal{GC}_i(f)$, counted with multiplicity. Let now $G_{d, B(h), \omega}(f)$ be the union of canyons from $G_{d,B(h)}(f)$ which have exactly the same set $\omega = K_{d,B(h), i}(f)$ of orders of contact with the other canyons from $G_{d,B(h)}(f)$. This defines a partition: \begin{equation}\label{eq:second} G_{d,B(h), \omega}(f) = \bigsqcup_{\omega} G_{d, B(h)}(f). \end{equation} In this way each canyon has its ``identity card'' composed of these orders of contact (which are rational numbers), and it belongs to a certain cluster $G_{d, B(h), \omega}(f)$ in the partition of $G_{d}(f)$. It is possible that two canyons have the same ``identity card''. We clearly have, by definition, the inclusions: \[ G_{d}(f) \supset G_{d, B(h)}(f) \supset G_{d, B(h), \omega}(f) \] for any defined indices. With these notations, we have the following far reaching extension of Theorem \ref{t:main0}: \begin{theorem}\label{t:main2} The bi-Lipschitz map $\varphi$ induces a bijection between the gradient canyons of $f$ and those of $g$. The following are bi-Lipschitz invariants: \begin{enumerate} \rm \item \it the set $G_{d}(f)$ of canyon degrees $d>1$, and for each fixed degree $d>1$, each bar $B$ and rational $h$, the cluster of canyons $G_{d, B(h)}(f)$. \rm \item \it the set of contact orders $K_{d,B(h), i}(f)$, and for each such set, the sub-cluster of canyons $G_{d,B(h), K_{d,B(h), i}}(f)$. \end{enumerate} Moreover, $\varphi$ preserves the contact orders between any two clusters of type $G_{d,B(h), K_{d,B(h), i}}(f)$. \end{theorem} \begin{proof} We know from Theorem \ref{t:main0}, Lemma \ref{l:incl} and Lemma \ref{l:single} that $\varphi$ induces a bijection between canyon disks since every canyon disk of $f$ is sent by $\varphi$ to a unique canyon disk of $g$. The contact between two canyons of degree $d>1$ translates to an asymptotic order of the distances between the canyon disks in the Milnor fibre. The map $\varphi$ transforms the Milnor fibre $\{f=\lambda\}$ into the Milnor fibre $\{g=\lambda\}$ and we know that canyon disks are sent to canyon disks of the same degree (Theorem \ref{t:main0}). In addition, the order of the distance between any two disks is preserved by $\varphi$ since it is bi-Lipschitz. \smallskip We then check the order of the distance between disks corresponding to two different canyons and translate it to the order of contact \eqref{eq:order} between these canyons, starting with the lowest orders which are higher than 1. Doing this on the set $G_{d}(f)$ will have as result the partition \eqref{eq:first}. Continuing to do this with each cluster of canyons $G_{d, B(h)}(f)$ will have as result the partition \eqref{eq:second}. This proves (a) and (b). \smallskip Our first assertion follows now from the bijective correspondence between the smallest clusters, as follows. In case if one small cluster of type $G_{d,h, K_{d,h, i}}(f)$ contains more than one gradient canyon, the number of canyons is detected by the multiplicity of the contact order, and this multiplicity is obviously preserved by the bi-Lipschitz map $\varphi$. By the same reasons as above, we get our last claim, that $\varphi$ preserves the contact orders between any cluster of canyons. \end{proof}

BkiUbMk4eILhQAZVCpbd

\section{Introduction} The present paper contains a probabilistic analysis of some mathematical model of asynchronous algorithm for parallel simulation. For the detailed discussion of synchronization issues in parallel and distributed algorithms we refer to~\cite{JeWI,BeTsi}. Here we give only a brief description of the problem. In large-scale parallel computation it is necessary to coordinate the activities of different processor which are working together on some common task. Usually such coordination is implemented by using a so-called message-passing system. This means that a processor shares data with other processors by sending timestamped messages. Between sending or receiving the messages the processors work independently. It can be happened that till the moment of receiving of a message some processor can proceed farther in performing its program than the value of timestamp indicated in this newly received message; in this case the state of the processor should be \emph{rolled back} to the indicated value. It is clear that due to possible rollbacks the mean speed of a given processor in the computing network will be lower than its proper speed. One of the most important performance characteristics of the system is the progress of the computing network on large time intervals. Probabilistic models for such system are studied already for twenty years. From the probabilistic point of view these models consists of many relatively independent components which synchronize from time to time their states according to some special algorithm. The detailed review of all existing publications is out of range of this paper. We would like to mention only that the bibliography on this subject consists mostly of two group of papers. The first group of publication \cite{MitMit,KuSh,GrShSt,GuKuSh,ShKuRe,GuKu} are devoted to the case of two processors. The paper~\cite{MitMit} is of special interest since it contains an exhaustive basic analysis of the two-dimensional model and had a big influence on further research. The case of many processors was studied in \cite{MadWalMes,GuAkFu,AkChFuSer,PGM,GrShSt,T1,T3}. An important difference of such models from two-dimensional case is that in a realistic model with more than two processors one message can provoke a multiple rollback of a chain of processors. Since the multi-dimensional model is much more complicated for a rigorous study, in the above papers the authors deal with the set of identical processors and their mathematical results are contained in preparatory sections before large numerical simulations. It should be noted also that probabilistic models with synchronization mechanism are interesting also for modelling database systems~(see for example, \cite{JeWI}). Moreover, now synchronization-like interactions are considered as well in the framework of interaction particle systems~\cite{MM-RR,MShCh,MM-TVP}. The model considered in the present paper is of special interest for the following reasons. We deals with a nonhomogeneus model consisting of several different processors. We consider case of message-passing topology other from the topology of complete graph which was considered in all previous papers. Our main interest is the cascade model which pressupose a subordination between processors. We put forward a conjecture on the cluster behavior of the system: processors can be divided into separated groups which are asymptotically independent and have their own proper performance characteristics. Our main goal is to justify this conjecture. One should point out that in the case of complete graph topology the cluster decomposition into groups is degenerated and, thus, not interesting. We describe our model in terms of multi-dimensional continuous time Markov process. To get asymptotical performance characteristics of the model we combine two probabilistic methods (stochastic comparison and Foster--Lyapunov functions). The paper is organized as follows. In Section~\ref{sec:Description-of-continuous} we introduce a general continuous time Markov model and define a cascade model as a special subclass of the general model. In~Section~\ref{sec:Definition-of-discrete} we pass to the embedded Markov chain. Main problem is now to study a long-time behavior of Markov chain with highly nonhomogeneous transition probabilities. To do this we consider relative coordinates and find groups of processors whose evolution is ergodic (convergences to a steady state) in these relative coordinates. To our opinion the method of Foster-Lyapunov functions seems to be the only one to prove the stability in relative coordinates for the Markov chain under consideration. First of all in Section~\ref{sec:Case-N-2} we start from the case of two processors ($N=2$) and the analysis here is rather simple and similar to~\cite{MitMit}. In the study of the three-dimensional case (Section~\ref{sec:N-3-Lyap}) the main point is the proof of ergodicity. We propose an explicit construction of some nonsmooth Foster-Lyapunov function. Our construction is rather nontrivial as it can be seen by comparing with already existing explicit examples of Lyapunov functions (see~\cite{FMM}). All this analysis bring us to some conclusions presented in Section~\ref{sec:Conclusions}. This section contains decomposition into groups (clusters) in the case of cascade model with any number of processors~$N$ and our main Conjecture~\ref{con-N-main}. We show that the proof of this conjecture could be related with progress in explicit construction of multi-dimensional Foster-Lyapunov functions. Analysis of random walks in~$\bZ _{+}^{n}$ (which was done in~\cite{FMM}) shows that, in general, this technical problem may be very difficult. In the next papers we hope to overcome these difficulties by using specific features of our concrete Markov processes. \paragraph*{Acknowledgements. } The first author is very grateful to the team TRIO (INRIA--Lorraine) and to l'Ecole des Mines de Nancy for their hospitality during his stay at Nancy in summer 2004 when the main results of this paper were obtained. \section{Description of continuous time model} \label{sec:Description-of-continuous} \subsection{General model} We present here some mathematical model for parallel computations. There are $N$ computing units (processors) working together on some common task. The state of a processor $k$ is described by an integer variable $x_{k}\in \mathbf{Z} $ which is called a local (or inner) time of the processor~$k$ and has a meaning of amount of job done by the processor $k$ up to the given time moment. Assume that the state $\left(x_{1},x_{2},\ldots ,x_{N}\right)$ of the system evolves in continuous time~$t\in \mathbf{R} _{+}$. Any change of a state is possible only at some special random time instants. Namely, with any processor~$k$ we associate a Poissonian flow $\Pi ^{k}=\left\{ 0=\sigma _{0}^{k}<\sigma _{1}^{k}<\cdots <\sigma _{n}^{k}<\cdots \right\} $ with intensity~$\lambda _{k}$ and with a pair $(k,l)$ of processors we associate a Poissonian flow $\Pi ^{kl}=\left\{ 0=\sigma _{0}^{kl}<\sigma _{1}^{kl}<\cdots <\sigma _{n}^{kl}<\cdots \right\} $ with intensity~$\beta _{kl}$. This means, for example, that $\left\{ \sigma _{n}^{k}-\sigma _{n-1}^{k}\right\} _{n=1}^{\infty }$ is a sequence of independent exponentially distributed random variables with mean~$\lambda _{k}^{-1}$: $\forall n=1,2,\ldots \quad \mathsf{P} \left\{ \sigma _{n}^{k}-\sigma _{n-1}^{k}>s\right\} =\exp \left(-\lambda _{k}s\right)$, and similarly for the flows $\Pi ^{kl}$. We also assume that all these flows $\Pi ^{k}$ and $\Pi ^{kl}$ are mutually independent. Let us now define a stochastic process $\left(X(t)=\left(x_{1}(t),\ldots ,x_{N}(t)\right),t\in \mathbf{R} _{+}\right)$ on the state space $\bZ ^{N}$ according to the following rules. 1) At time instants $\sigma _{n}^{k}$ the processor $k$ increases its local time~$x_{k}$ by $1$:~ $x_{k}(\sigma _{n}^{k}+0)=x_{k}(\sigma _{n}^{k})+1$. 2) There is an exchange of information between different processors. At time instant~$\sigma _{i}^{kl}$ the processor~$k$ sends a message $m_{kl}^{(x_{k})}$ to the processor~$l$. We assume that messages reach their destination immediately. A message $m_{kl}^{(x_{k})}$ coming to node $l$ from node~$k$ contains an information about local time $x_{k}(\sigma _{i}^{kl})=x_{k}$ of the sender~$k$. \emph{If} at the time instant~$\sigma _{i}^{kl}$ (when the message $m_{kl}^{(x_{k})}$ arrives to the node~$l$) we have $x_{l}(\sigma _{i}^{kl})>x_{k}(\sigma _{i}^{kl})$ \emph{then} the local time~$x_{l}$ rolls back to the value $x_{k}$: $x_{l}(\sigma _{i}^{kl}+0)=x_{k}(\sigma _{i}^{kl})$. Moreover, if the processor~$l$ rolls back, then all messages sent by the processor $l$ during the time interval~$\mathcal{I}=(\theta _{l}(x_{k},\sigma _{i}^{kl}),\sigma _{i}^{kl}),$ where $\theta _{l}(x,u):=\max \left\{ s\leq u:\, x_{l}(s)=x,\, x_{l}(s+0)=x+1\right\} ,\sigma _{i}^{kl}),$ should be eliminated. This may generate a cascading rollback of local times for some subset of processors. For example, assume that there is a processor $q$ which received a message $m_{lq}^{(x'_{l})}$ at some time instant $s'\in \mathcal{I}$ and $x_{q}(\sigma _{i}^{kl})>x_{l}(s')=x'_{l}$. Then the local clock~of~$q$ should be rolled back to the value $x_{l}(s')$: $x_{q}(\sigma _{i}^{kl}+0)=x_{l}(s')$ and, moreover, all messages sent by $q$ during the interval~$\mathcal{I}=(\theta _{q}(x_{l}(s'),\sigma _{i}^{kl}),\sigma _{i}^{kl})$ should be deleted, and so on. Hence, at time instant $\sigma _{i}^{kl}$ a message from~$k$ to $l$ can provoke a multiple rollback of processor~$l,q,\ldots $ in the system. \subsection{Cascade model} \label{sub:Cascade-model} From now we shall consider the following special subclass of the above general model. A chain of processors $1,2,\ldots ,N$ is called a \emph{cascade} if any processor $j$ can send a message only to its right neighbour $j+1$. Hence, the processor~$N$ does not send any message and the processor~$1$ does not receive any message. In other words, $\beta _{ij}\not =0\, \Leftrightarrow \, (j=i+1)$. A message sent from $j$ to $j+1$ can provoke a cascading roll-back of processors $j+2,\ldots $~. Recall that all above time intervals are exponentially distributed and assumed to be independent. Obviously, the stochastic process $X_{c}^{(N)}(t)=\left(\, x_{1}(t),\ldots ,x_{N}(t)\, \right)$ is Markovian. A very important property is that any {}``truncated'' marginal process $X_{c}^{(N_{1})}(t)=\left(\, x_{1}(t),\ldots ,x_{N_{1}}(t)\, \right)$, $N_{1}\leq N$, is also Markovian. Assume that for any $j$ the following limit \begin{equation} v^{*} _{j}=\lim _{t\rightarrow +\infty }\frac{x_{j}(t)}{t}\qquad (\textrm{in probability})\label{eq:v*j-lim}\end{equation} exists. Then the numbers $v^{*} _{j}$, $j=1,\ldots ,N$, characterize \emph{performance} of the model. The main goal of the present paper is to prove the existence of these limits and to calculate them. Note that if we uniformly transform the absolute time scale $t=cs$, where $c>0$ is a constant and $s$ is a new absolute time scale, the performance characteristics~(\ref{eq:v*j-lim}) will not change. \section{Definition of the discrete time cascade model} \label{sec:Definition-of-discrete} Consider a sequence \[ 0=\tau _{0}<\tau _{1}<\tau _{2}<\cdots <\cdots \] of time moments when changes of local time at nodes may happen (we mean local time updates and moments of sending of messages). It is clear that $\left\{ \tau _{r+1}-\tau _{r}\right\} _{r=0}^{\infty }$ is a sequence of independent identically distributed r.v. having exponential distribution with parameter\[ Z=\sum _{i=1}^{N}\lambda _{i}+\sum _{i=1}^{N-1}\beta _{i,i+1}.\] Observing the continuous time Markov process $(x_{1}(t),\ldots ,x_{N}(t))$ at epochs $\tau _{n}$ we get the so-called embedded discrete time Markov chain $\left\{ \Xd{n},n=0,1,\ldots \right\} $ with state space $\mathbf{Z} _{+}^{N}$. In the sequel we will be interested in the long-time behaviour of the chain $\left\{ \Xd{n},n=0,1,\ldots \right\} $. \paragraph*{Transition probabilities.} In the MC $\left\{ \Xd{n},n=0,1,\ldots \right\} $ there are transitions produced by the free dynamics and transitions generated by rollbacks. By the free dynamics we mean updating of local times\[ P\left\{ \Xd{n+1}=x+\mathbf{e} _{j}\, |\, \Xd{n}=x\right\} =\lambda _{j}Z^{-1},\quad j=1,\ldots ,N\, ,\] where $\mathbf{e} _{j}=(0,\ldots ,0,\begin{array}[t]{c} 1\\ ^{j}\end{array},0,\ldots ,0)$. It is easy to see that if a state $x=(x_{1},\ldots ,x_{N})$ is such that for some $j$ ~ $x_{j}<x_{j+1}$ then a message sent from $j$ to $j+1$ produces a transition of the following form \begin{equation} (x_{1},\ldots ,x_{j},x_{j+1},\ldots ,x_{l},x_{l+1},\ldots ,x_{N})\rightarrow (x_{1},\ldots ,x_{j},w_{j+1},\ldots ,w_{l},x_{l+1},\ldots ,x_{N})\label{eq:x-tran}\end{equation} with probability\begin{equation} Z^{-1}\beta _{j,j+1}\, \prod _{q=j+1}^{l-1}p(w_{q},x_{q};w_{q+1},x_{q+1})\, \times \, \left(1-b_{l}\right)^{\min \left(x_{l},x_{l+1}-1\right)-w_{l}+1},\label{eq:pr-x-tran}\end{equation} where \begin{itemize} \item sequence $\left(w_{j+1},\ldots ,w_{l}\right)$ is admissible in the following sense:\[ j<l\leq N,\quad \quad w_{j+1}=x_{j}\qquad w_{q}\leq w_{q+1}\leq \min \left(x_{q},x_{q+1}-1\right),\quad (j<q<l)\] \item $p(w_{q},x_{q};w_{q+1},x_{q+1})=b_{q}\left(1-b_{q}\right)^{w_{q+1}-w_{q}}$ \item $\displaystyle b_{q}=\frac{\lambda _{q}}{\lambda _{q}+\beta _{q,q+1}}$, $q<N$. \end{itemize} Here $b_{q}$ is the probability of an event that processor~$q$ in state~$x_{q}$ sends at least one message to~$q+1$ before updating its state $x_{q}\rightarrow x_{q}+1$. For $q=N$ we put $b_{N}=0$. So in the case $l=N$ the probability~(\ref{eq:pr-x-tran}) takes the form \[ Z^{-1}\beta _{j,j+1}\, \prod _{q=j+1}^{N-1}p(w_{q},x_{q};w_{q+1},x_{q+1})\, .\] \paragraph*{Relative coordinates.} Note that the first processor $x_{1}(t)$ evolves independently of other processors. It is useful to introduce new process $Y_{c}(t)=(y_{2}(t),\ldots ,y_{N}(t))\in \mathbf{Z} ^{N-1}$ in relative coordinates as viewing by an observer sitting at the point $x_{1}(t)$:\[ y_{j}(t):=x_{j}(t)-x_{1}(t),\quad j=2,\ldots ,N\, .\] In a similar way we define $\Yd{n}=Y_{c}(\tau _{n})$, $n=0,1,\ldots $~. The free dynamics produce the following transitions of $\Yd{n}$: \begin{eqnarray} P\left\{ \Yd{n+1}=y+\mathbf{e} _{j}\, |\, \Yd{n}=y\right\} & = & \lambda _{j}Z^{-1},\quad j=2,\ldots ,N\, ,\label{eq:trYo}\\ P\left\{ \Yd{n+1}=y-{\textstyle \sum _{j=2}^{N}}\mathbf{e} _{j}\, |\, \Yd{n}=y\right\} & = & \lambda _{1}Z^{-1}.\label{eq:trYd} \end{eqnarray} Since rollback does not affect on the first processor the corresponding transitions have the same form and the same probabilities as~(\ref{eq:x-tran}) and~(\ref{eq:pr-x-tran}). \section{Stochastic monotonicity} All statements of this section are valid for the both Markov processes $X_{c}^{(N)}(t)$, $t\in \mathbf{R} _{+}$, and $X(n)$, $n\in \mathbf{Z} _{+}$. For the sake of breavity we give here results only for the contionuous time model $X_{c}^{(N)}(t)$. The following results will play a significant part in the proof of the Theorem~\ref{t:N-3} in Section~\ref{sec:N-3}. \begin{theorem}\label{t-st-mon} \noindent Let us consider two cascade models (say $X_{c,1}^{(n)}(t)$ and $X_{c,2}^{(n)}(t)$~) with processors $1,2,\ldots ,n$ and parameters $\lambda _{1},\ldots ,\lambda _{n}$ and $\beta _{12}^{(1)},\beta _{23}^{(1)},\ldots ,\beta _{n-1,n}^{(1)}$ for the first model $X_{c,1}^{(n)}(t)$ and parameters $\lambda _{1},\ldots ,\lambda _{n}$ and $\beta _{12}^{(2)},\beta _{23}^{(2)},\ldots ,\beta _{n-1,n}^{(2)}$ for the second model $X_{c,2}^{(n)}(t)$. Assume that \[ \beta _{i,i+1}^{(1)}\leq \beta _{i,i+1}^{(2)}\qquad \forall i\, .\] Then $X_{c,1}^{(n)}$ is stochastically larger than $X_{c,2}^{(n)}$~, that is: if $X_{c,1}^{(n)}(0)=X_{c,2}^{(n)}(0)$ then $X_{c,1}^{(n)}(t)\geq _{{st}} X_{c,2}^{(n)}(t)$ for any~$t$.~% \footnote{It means that there exists a \emph{coupling} $\left(\widetilde{X}_{1}^{(n)}(t,\omega ),\widetilde{X}_{2}^{(n)}(t,\omega )\right)$of stochastic processes $X_{1}^{(n)}(t)$ and $X_{2}^{(n)}(t)$ such that $P\left\{ \omega :\, \widetilde{X}_{1}^{(n)}(t,\omega )\geq \widetilde{X}_{2}^{(n)}(t,\omega )\, \, \forall t\right\} =1$. If $w,z\in \mathbf{R} ^{n}$ we say $w\geq z$ if $w_{i}\geq z_{i}$ for all $i=1,\ldots ,n$~(partial order).% } \noindent \end{theorem} \smallskip{} \noindent \textbf{Proof} may be given by an explicit coupling construction of the processes $X_{c,1}^{(n)}(t)$ and $X_{c,2}^{(n)}(t)$ on the same probability space. The following fact should be used: a~Poisson flow with intensity $\beta _{12}^{(1)}$ can be obtained from a Poisson flow with intensity $\beta _{12}^{(2)}$ in which any point (independently from other) is killed with probability $1-\beta _{12}^{(1)}/\beta _{12}^{(2)}$. \medskip{} \begin{cor}[Solid barriers]\label{cor-barrier} \noindent Fix some $1\leq r_{1}<r_{2}<\cdots <r_{b}<n$ and consider two cascade models: $X_{c,1}^{(n)}(t)$ with parameters $\left(\, \lambda _{1},\ldots ,\lambda _{n}\, ;\, \beta _{12}^{(1)},\beta _{23}^{(1)},\ldots ,\beta _{n-1,n}^{(1)}\, \right)$ and $X_{c,2}^{(n)}(t)$ with parameters $\left(\, \lambda _{1},\ldots ,\lambda _{n}\, ;\, \beta _{12}^{(2)},\beta _{23}^{(2)},\ldots ,\beta _{n-1,n}^{(2)}\, \right)$, where\[ \beta _{i,i+1}^{(2)}=\beta _{i,i+1}^{(1)}\quad \forall i\not \in \left\{ r_{1},\ldots ,r_{b}\right\} ,\qquad \beta _{i,i+1}^{(2)}=0\quad \forall i\in \left\{ r_{1},\ldots ,r_{b}\right\} .\] We can say that the model $X_{c,2}^{(n)}(t)$ differs from the model $X_{c,1}^{(n)}(t)$ by the presence of $b$ solid barriers between processors $r_{1}$ and $r_{1}+1$, \ldots, $r_{b}$ and $r_{b}+1$. Then by Theorem~\ref{t-st-mon} we have that \[ X_{c,1}^{(n)}(t)\leq _{\mathrm{s}t} X_{c,2}^{(n)}(t)\, .\] \end{cor} \section{Case $N=2$} \label{sec:Case-N-2} We start with the Markov chain $X_{c}^{(2)}(t)$. Since processor $1$ works independently, it is enough to consider the Markov chain $Y_{c}^{(2)}(t)=x_{2}(t)-x_{1}(t)$. Bearing in mind the remark at the end of Subsection~\ref{sub:Cascade-model}, for brevity of notation let us rescale absolute time in such a way that $Z=1$. Then the Markov chain $Y(n)$ has the following transition probabilities\[ p_{i,i+1}=\lambda _{2},\quad p_{i,i-1}=\lambda _{1},\quad p_{i,0}=\beta _{12}\, \, (i\geq 0),\quad p_{i,i}=\beta _{12}\, \, (i<0)\] and $p_{i,j}=0$ for any another pair $i,j$~. \begin{center}\includegraphics [width=6in]{n-2.eps}\end{center} \begin{theorem}\label{t:n-2} If $\lambda _{1}<\lambda _{2}$ then the Markov chain $\Yd{n}$ is ergodic and we have $v^{*} _{1}=v^{*} _{2}=\lambda _{1}$. If $\lambda _{1}>\lambda _{2}$ then the Markov chain $\Yd{n}$ is transient and we have $v^{*} _{1}=\lambda _{1}$, $v^{*} _{2}=\lambda _{2}$. \end{theorem} \begin{proof}The Markov chain $Y(n)$ is one-dimensional and its analysis is quite easy. To~establish ergodicity under assumption $\lambda _{1}<\lambda _{2}$ we use the Foster-Lyapunov criterion (Theorem~\ref{t:Foster}, see~Appendix) with test function $f(y)=|y|$, $y\in \mathbf{Z} $. This implies that $x_{2}(t)-x_{1}(t)$ has a limit in distribution as $t\rightarrow \infty $. Recall that $x_{1}(t)$ is a Poissonian process hence the limit $\displaystyle v^{*} _{1}=t^{-1}\lim _{t}x_{1}(t)=\lambda _{1}$ exists (in probability). It follows from this that $\displaystyle v^{*} _{2}=t^{-1}\lim _{t}x_{2}(t)=\lambda _{1}$. Under assumption $\lambda _{1}>\lambda _{2}$ we get transience by choosing the function $f(y)=\min (e^{\delta y},1)$, $y\in \mathbf{Z} $, where we fix sufficiently small~$\delta >0$, and applying Theorem~\ref{t:kr-trans} from Appendix. Therefore any trajectory of $Y(n)$ spends a finite time in any prefixed domain $\{y\geq C\}$ entailing $\lim _{t\rightarrow \infty }x_{2}(t)-x_{1}(t)=-\infty $ (a.s.). It means that after some time, the messages from $1$ to~$2$ can not produce a rollback anymore, so $x_{1}(t)$ and $x_{2}(t)$ become asymptotically independent and hence $\displaystyle v^{*} _{2}=t^{-1}\lim _{t}x_{2}(t)=\lambda _{2}$. \end{proof} \section{Case $N=3$} \label{sec:N-3} \begin{theorem}\label{t:N-3} Four situations are possible. \begin{enumerate} \item If $\lambda _{1}<\min \left(\lambda _{2},\lambda _{3}\right)$ then $v^{*} _{1}=v^{*} _{2}=v^{*} _{3}=\lambda _{1}$. \item If $\lambda _{2}>\lambda _{1}>\lambda _{3}$ then $v^{*} _{1}=v^{*} _{2}=\lambda _{1}$, $v^{*} _{3}=\lambda _{3}$. \item If $\lambda _{2}<\min \left(\lambda _{1},\lambda _{3}\right)$ then $v^{*} _{1}=\lambda _{1}$, $v^{*} _{2}=v^{*} _{3}=\lambda _{2}$. \item If $\lambda _{1}>\lambda _{2}>\lambda _{3}$ then $v^{*} _{1}=\lambda _{1}$, $v^{*} _{2}=\lambda _{2}$, $v^{*} _{3}=\lambda _{3}$. \end{enumerate} \end{theorem} Items 2, 3 and 4 can be reduced in some sense to the results of the case $N=2$ (see Theorem~\ref{t:n-2}). We prove them in the current section. Proof of the item~1 is much more intricate and relies heavily on the construction of an adequate Lyapunov function needing lengthy developments deferred to the following section~\ref{sec:N-3-Lyap}. \paragraph{Proof of Theorem~\ref{t:N-3} (items 2--4).} We start from the item~2: $\lambda _{2}>\lambda _{1}>\lambda _{3}$. Since the first two processors are governed by the Markov chain $X_{c}^{(2)}(t)$ and do not depend on the state of processor~3 we apply Theorem~\ref{t:n-2} and conclude that $X_{c}^{(2)}(t)$ is ergodic and $v^{*} _{1}=v^{*} _{2}=\lambda _{1}$. Let us compare the following two cascade models \[ X_{c}^{(3)}(t):\qquad 1\stackrel{\beta _{1,2}}{\, \longrightarrow }\, 2\stackrel{\beta _{2,3}}{\, \longrightarrow }\, 3\] \[ X_{c,2}^{(3)}(t):\qquad 1\stackrel{\beta _{1,2}}{\, \longrightarrow }\, 2\stackrel{0}{\, \longrightarrow }\, 3\] (parameters $\lambda _{1}$, $\lambda _{2}$ and $\lambda _{3}$ are the same for the both models $X_{c}^{(3)}(t)$ and $X_{c,2}^{(3)}(t)$~). In the model $X_{c,2}^{(3)}$ the groups of processors $\{1,2\}$ and $\{3\}$ evolve independently. Evidently, an asymptotic speed of processor~$3$ in the model $X_{c,2}^{(3)}$ exists and is equal to~$\lambda _{3}$. By Corollary~\ref{cor-barrier} $X_{c}^{(3)}(t)\leq _{\mathrm{s}t} X_{c,2}^{(3)}(t)$. Hence in the model $X_{c}^{(3)}$ an asymptotic speed of the processor~$3$ is \emph{not greater} than $\lambda _{3}$. Since $\lambda _{3}<\lambda _{1}$ we conclude that there exists some time moment $T_{0}$ such that for $t\geq T_{0}$ in the model~$X_{c}^{(3)}$ messages from $2$ to $3$ that roll back the processor~3 will be very {}``rare''. So these rare rollbacks will be not essential for an asymptotical speed of the processor~3. In other words, as $t\rightarrow \infty $ the groups of processors $\{1,2\}$ and $\{3\}$ of the model $X_{c}^{(3)}$ become asymptotically independent, so the processor~3 will move with the average speed $\lambda _{3}$. Items 3 and 4 can be considered in a similar way. Note the item~3 consists of two subcases: $\lambda _{1}>\lambda _{3}>\lambda _{2}$ and $\lambda _{3}>\lambda _{1}>\lambda _{2}$. We omit details. \section{Explicit construction of Lyapunov function} \label{sec:N-3-Lyap} In this section we prove the item~1 of Theorem~\ref{t:N-3}. Recall that our key assumption here is \begin{equation} \lambda _{1}<\lambda _{2},\quad \lambda _{1}<\lambda _{3}.\label{eq:osn-predp}\end{equation} The main idea is to prove that the Markov chain~$Y(n)$ is ergodic. To do this we apply the Foster-Lyapunov criterion (see Theorem~\ref{t:Foster} in Appendix). As in the case of Theorem~\ref{t:n-2} ergodicity of $Y(n)$ implies that $v^{*} _{j}=\lambda _{1}$, $j=1,2,3$~. \subsection{Transition probabilities} \label{sub:Transition-probabilities} Consider the embedded Markov chain $Y(n)$. A stochastic dynamics produced by this Markov chain consists of two components: transitions generated by the free dynamics and transitions generated by roll-backs. For each transition probability $p_{\alpha \beta }$, $\alpha \not =\beta $, we have the following representation:\begin{equation} p_{\alpha \beta }=s_{\alpha \beta }+r_{\alpha \beta }\, ,\label{eq:p-s-r}\end{equation} where $s_{\alpha \beta }\geq 0$ corresponds to a transition $\alpha \rightarrow \beta $ which occurs due to the free dynamics and $r_{\alpha \beta }$ corresponds to a roll-back transition $\alpha \rightarrow \beta $. Taking into account the remark at the end of Subsection~\ref{sub:Cascade-model}, without loss of generality we assume that the time is rescaled in such way that $Z=1$. This slightly simplifies notation for transition probabilities. For example, free dynamics transitions~(\ref{eq:trYo})--(\ref{eq:trYd}) are equal to $\lambda _{2}$, $\lambda _{3}$ and $\lambda _{1}$ correspondingly. On the next figure we show all non-zero transitions $\alpha \rightarrow \beta $, $(\alpha \not =\beta )$. It is true, of course, that $p_{\alpha \alpha }=1-\sum _{\beta \not =\alpha }p_{\alpha \beta }$, but it is useless to put this information on the picture. Below we give the explicit form of rollback transition probabilities: \begin{eqnarray*} 1\rightarrow 2: & \qquad & r_{yz}=\beta _{12}\quad \quad \textrm{for }0<y_{2}\\ 2\rightarrow 3: & & r_{yz}=\beta _{23}\quad \quad \textrm{for }y_{2}<y_{3}\\ 1\rightarrow 2\rightarrow 3: & & r_{yz}=\left\{ \begin{array}{rl} \beta _{12}\left(1-b_{2}\right)^{z_{3}}b_{2},\quad & z_{3}<y_{3}\\ \beta _{12}\left(1-b_{2}\right)^{y_{3}},\quad & z_{3}=y_{3}\end{array}\right.\quad \textrm{for }0<y_{3}\leq y_{2}\\ 1\rightarrow 2\rightarrow 3: & & r_{yz}=\left\{ \begin{array}{rl} \beta _{12}\left(1-b_{2}\right)^{z_{3}}b_{2},\quad & z_{3}\leq y_{2}\\ \beta _{12}\left(1-b_{2}\right)^{y_{2}+1},\quad & z_{3}=y_{3}\end{array}\right.\quad \textrm{for }0<y_{2}<y_{3} \end{eqnarray*} where $y=(y_{2},y_{3})$, $z=(z_{2},z_{3})$. \includegraphics [width=5in]{n-3.eps} \subsection{Contour of level 1} In the plane $Oy_{2}y_{3}$ consider the ellipse\[ e(y_{2},y_{3})=ay_{2}^{2}+b\left(y_{2}-y_{3}\right)^{2}=1,\qquad a>0,b>0,\] and draw a tangent line to it with normal vector $\left(-\Delta ,1\right)$. Evidently, there exist two tangent lines with the same normal vector $\left(-\Delta ,1\right)$. If $\Delta >0$ is sufficiently large then one of this tangent line touches the ellipse at some point $T_{3}$ of the domain $y_{2}<0$, $y_{3}<0$. Take a segment on this line from the point $T_{3}$ to a point $K_{3}=(0,u_{3})$ of intersection with coordinate axis $Oy_{3}$. Now let us draw tangent lines to the ellipse corresponding to a normal vector $\left(1,-\Delta \right)$. If $\Delta >0$ is sufficiently large, then one of these lines touches the ellipse at some point $T_{2}$ of the domain $y_{3}<0$. Let us take this tangent line and fix a segment on it from the point $T_{2}$ to a point $K_{2}=(u_{2},0)$ of intersection with coordinate axis~$Oy_{2}$. It is evident that $[K_{2}K_{3}]=\mathbf{R} _{+}^{2}\cap \left\{ (y_{2},y_{3}):\, y_{2}/u_{2}+y_{3}/u_{3}=1\right\} $. Let us consider now a closed contour $L$, consisting of subsequently joined segment $K_{3}K_{2}$, segment $K_{2}T_{2}$, arc $T_{2}T_{3}$ of the ellipse and segment $T_{3}K_{3}$. This contour has the following property: any ray of the form$\{cv,\, c>0\}$, where $v\in \mathbf{R} ^{2}$, $v\not =0$, has exactly one common point with the contour $L$. \begin{center}\includegraphics [width=5in]{elli3.eps}\end{center} We denote by $n(y)$ the outer normal unitary vector of the contour $L$ corresponding to the point $y\in L$, $n(y)$ is well defined at all points of~$L$ except the points $K_{2}$ and $K_{3}$ and, moreover, this function is continuous on~$L$ except the points~$K_{2}$ and $K_{3}$. The~behaviour of $n(y)$ on the arc $T_{2}T_{3}$ is of prime interest: \[ n(y)=\frac{\nabla e(y)}{\left\Vert \nabla e(y)\right\Vert },\qquad \nabla e(y)=2(\, ay_{2}+b(y_{2}-y_{3}),-b(y_{2}-y_{3})\, ),\quad y\in T_{2}T_{3}\subset L\, .\] It is easy to see that $n(y)=n(T_{2})$ for $y\in (K_{2}T_{2}]$, $n(y)=n(T_{3})$ for $y\in [T_{3}K_{3})$ and\[ n(y)=\left(u_{2}^{-1},u_{3}^{-1}\right)\qquad y\in (K_{3}K_{2}).\] For the sequel it is important to point out the following points of the arc $T_{2}T_{3}$: $y^{(3)}=(-a^{-1/2},-a^{-1/2})$ and $y^{(2)}$, $\left\{ y^{(2)}\right\} =T_{2}T_{3}\cap \left\{ y_{3}^{(2)}=\frac{a+b}{b}y_{2}^{(2)}\right\} $. It is easy to check that \[ n(y^{(2)})\, \Vert \, Oy_{3},\quad n(y^{(3)})\, \Vert \, Oy_{2}\, .\] Obviously, both points belong to the domain $\left\{ y_{2}<0,\, y_{3}<0\right\} $. \begin{lemma} ~\label{l-normal} The function $n(y)$ has the following properties: \begin{itemize} \item $\skp{n(y)}{y}\not =0$ $\forall y\in L\backslash \{K_{2},K_{3}\}$ \item If $\Delta >0$ is sufficiently large then there exist continuous functions $c_{2}(y)$ and $c_{3}(y)$ such that \[ c_{2}(T_{2})=c_{3}(T_{3})=1,\qquad c_{2}(T_{3})=c_{3}(T_{2})=0,\qquad c_{2}(y)>0,\, c_{3}(y)>0\quad y\in (T_{2},T_{3})\] \[ n(y)=c_{2}(y)n(T_{2})+c_{3}(y)n(T_{3}),\qquad y\in (T_{2},T_{3}).\] \item $\skp{n(y)}{(0,-1)}<0$ if $y_{2}<0$, $y_{3}>y_{2}$, and $\skp{n(y)}{(-1,0)}<0$ if $y_{2}>0$, $y_{3}<0$. \end{itemize} \end{lemma} \subsection{Definition of function $\varphi $} For any point $(y_{2},y_{3})\in \mathbf{R} ^{2}\backslash \{0\}$ define $\varphi (y_{2},y_{3})>0$ such that \[ \frac{(y_{2},y_{3})}{\varphi (y_{2},y_{3})}\, \in \, L\, .\] For $(y_{2},y_{3})=0$ we put $\varphi (0,0)=0$. The function $\varphi (y_{2},y_{3})$ is well-defined and has the following properties: \begin{itemize} \item $\varphi :\, \mathbf{R} ^{2}\rightarrow \mathbf{R} _{+}$ (positivity) \item $\varphi (ry_{2},ry_{3})=r\varphi (y_{2},y_{3})$, $r>0$, (homogeneity) \item $L=\{y:\, \varphi (y)=1\}$. \end{itemize} To any point $y=(y_{2},y_{3})$ we put in correspondance a point $y^{*}:=\frac{y}{\varphi (y)}\in L$. Therefore, $\varphi (y^{*})=1$. \begin{center}\includegraphics [width=5in]{elli4.eps}\end{center} \begin{lemma} \label{l-grad-fi}~ \begin{itemize} \item The gradient $\nabla \varphi (y)$ exists at all points except that $y$ for which $y^{*}=K_{2}$ or $K_{3}$ and, moreover, the gradient is constant on rays of the form $\{cv,\, c>0\}$, $v\in \mathbf{R} ^{2}$:\[ \nabla \varphi (y)=\frac{n(y^{*})}{\skp{y^{*}}{n(y^{*})}}\, .\] \item Let $y=(y_{2},y_{3})$ be such that $y^{*}\in T_{2}T_{3}$. Then\begin{equation} \left|\varphi (w)-\skp{\nabla \varphi (y^{*})}{w}\right|\leq \frac{const}{\varphi (y)}\, \left\Vert w-y\right\Vert ^{2}.\label{eq:lagrange-ineq}\end{equation} In other words, in a neighbourhood of the point $y$ the function $\varphi $ can be approximated by the linear function $\skp{\nabla \varphi (y^{*})}{\cdot }$. \end{itemize} \end{lemma} In particular, $\varphi (y)=\skp{\nabla \varphi (y^{*})}{y}$. Proof of Lemma~\ref{l-grad-fi} is a straightforward computation. \subsection{Modification of the principle of local linearity} For any state $\alpha $ define a set $T_{\alpha }=\{\beta :\, p_{\alpha \beta }>0\}$. Recall decomposition~(\ref{eq:p-s-r}) and define $F_{\alpha }=\{\beta :\, s_{\alpha \beta }>0\}$ and $R_{\alpha }=\{\beta :\, r_{\alpha \beta }>0\}$. It is evident that $T_{\alpha }=F_{\alpha }\cup R_{\alpha }$. The most simple case is $F_{\alpha }\cup R_{\alpha }=\varnothing $. The case $F_{\alpha }\cup R_{\alpha }\not =\varnothing $ can be reduced to the previous one by a dilatation of the state space. Thus we assume that $F_{\alpha }\cup R_{\alpha }=\varnothing $ and consider the events $\left\{ Y (n+1)\in F_{\alpha }\right\} $ and $\left\{ Y (n+1)\in R_{\alpha }\right\} $. On the set $\left\{ \omega \in \Omega :\, Y (n,\omega )=\alpha \right\} $ we have $\IF{\alpha }(\omega )+\IR{\alpha }(\omega )\equiv 1$. Hence, \begin{eqnarray*} \mathsf{E}\, \left(f(Y (n+1))\, |\, Y (n)=\alpha \right)-f(\alpha ) & = & \mathsf{E}\, \left(\left(f(Y (n+1))-f(y)\right)\IF{\alpha }\, |\, Y (n)=\alpha \right)+\\ & & +\mathsf{E}\, \left(\left(f(Y (n+1))-f(y)\right)\IR{\alpha }\, |\, Y (n)=\alpha \right) \end{eqnarray*} It follows from definition of the Markov chain~$Y(n)$ (see Subsect.~\ref{sub:Transition-probabilities}) that the diameters $d_{\alpha }:=diam\, F_{\alpha }$ are uniformly bounded in~$\alpha $: $d=\max _{\alpha }d_{\alpha }<+\infty $. Define a vector\[ M_{F}(\alpha )=\mathsf{E}\, \left(\left(Y (n+1)-\alpha \right)\IF{\alpha }\, |\, Y (n)=\alpha \right)=\sum _{\beta \in F_{\alpha }}(\beta -\alpha )p_{\alpha \beta }\, .\] This is an analogue of a notion of mean jump (see~(\ref{eq:M-alpha}) in Appendix). In the next subsection we shall need the following modification of the principle of local linearity from~\cite{FMM} (see also Subsect.~\ref{sub:Principle-of-local} in Appendix). \begin{lemma}\label{l-mod-loclin}Assume that the following condition holds \[ \inf _{l}\, \sup _{\tilde{\alpha }\in \mathbf{R} ^{n},\| \tilde{\alpha }-\alpha \Vert \leq d_{\alpha }}\, \left|f(\tilde{\alpha })-l(\tilde{\alpha })\right|\, <\, \varepsilon \, ,\] where $\inf $ is taken over all linear functions~$l$. If \[ f\left(\alpha +M_{F}(\alpha )\right)-f(\alpha )<-5\varepsilon \, ,\] then the following inequality\[ \mathsf{E}\, \left(\, \left(\, f(Y (n+1))-f(\alpha )\, \right)\IF{\alpha }\, |\, Y (n)=\alpha \, \right)<-\varepsilon \, \] holds. \end{lemma} The proof of this statement repeats the proof of principle of local linearity presented in~\cite{FMM} and is omitted. \subsection{Proof of the Foster condition} The validity of the Foster condition will follow from several ancillary lemmas dealing with the following different domains of the state space: \begin{eqnarray*} E_{-} & := & \left\{ y=(y_{2},y_{3}):\, \min (y_{2},y_{3})<0\right\} ,\\ E_{1} & := & \left\{ y=(y_{2},y_{3}):\, y_{2}>0,y_{3}>0\right\} ,\\ E_{1,2} & := & \{y_{2}>0,\, y_{3}=0\},\\ E_{1,3} & := & \{y_{2}=0,\, y_{3}>0\}. \end{eqnarray*} \begin{lemma}\label{l-leftlower} Consider the domain $E_{-}=\left\{ y=(y_{2},y_{3}):\, \min (y_{2},y_{3})<0\right\} $. There exists $\CC{l-leftlower}>0$, such that if $\varphi (y)>\CC{l-leftlower}$, then ~\quad ~a) ~\[ \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IR{y}\, |\, Y (n)=y\right)\leq 0\, ,\] ~\quad ~b) there exists $\varepsilon >0$ such that\[ \varphi \left(y+M_{F}(y)\right)-\varphi (y)<-5\varepsilon \, .\] \end{lemma} \begin{proof} It is evident that the vector\[ M_{F}(y)=\left(\lambda _{2}-\lambda _{1},\lambda _{3}-\lambda _{1}\right)\] is constant (does not depend on $y$). Since the vector $n(T_{2})$ is co-directed with the vector $\left(1,-\Delta \right)$ and the vector $n(T_{3})$ is co-directed with the vector $\left(-\Delta ,1\right)$ and the conditions $\lambda _{2}>\lambda _{1}$, $\lambda _{3}>\lambda _{1}$ hold, we can find a large $\Delta _{1}>0$ such that \[ \skp{M_{F}(y)}{n(T_{2})}<0,\quad \skp{M_{F}(y)}{n(T_{3})}<0\, ,\qquad \forall \Delta >\Delta _{1}\, .\] Fix this $\Delta _{1}$. Hence, by Lemma~\ref{l-normal} there exists $\varepsilon >0$ such that\[ \skp{M_{F}(y)}{n(y)}<-6\varepsilon \quad \textrm{if }\, \min (y_{2},y_{3})<0.\] Put $w=y+M_{F}(y)$ and consider\begin{eqnarray*} \varphi (w)-\varphi (y) & = & \varphi (w)-\skp{\nabla \varphi (y^{*})}{y}\\ & = & \varphi (w)-\skp{\nabla \varphi (y^{*})}{w}+\skp{\nabla \varphi (y^{*})}{w-y}\\ & = & \varphi (w)-\skp{\nabla \varphi (y^{*})}{w}+\skp{\nabla \varphi (y^{*})}{M_{F}(y)}\, . \end{eqnarray*} By~(\ref{eq:lagrange-ineq}) for any given $\varepsilon >0$ we can choose $C_{0}>0$ such that \[ \left|\varphi (y+M_{F}(y))-\skp{\nabla \varphi (y^{*})}{M_{F}(y)}\right|\leq \frac{const}{\varphi (y)}\, d^{2}\leq \varepsilon \, \qquad \textrm{if }\varphi (y)\geq C_{0}\, .\] Now the item b) of the lemma easily follows. Let us prove the item a) of the lemma. Note that in the domain $y_{3}>y_{2},\, y_{2}<0$ a rollback decreases coordinate $y_{3}$: $(y_{2},y_{3})\rightarrow \left(y'_{2},y'_{3}\right)=(y_{2},y_{2})$. From geometrical properties of level sets of function $\varphi $ and item~3 of Lemma~\ref{l-normal} it follows that any transition generated by a rollback decreases a value of the function~$\varphi $: $\varphi (\, (y_{2},y_{3})\, )<\varphi (\, (y'_{2},y'_{3})\, )$. In the domain $y_{3}<0,\, y_{2}>0$ a rollback has the following form: $(y_{2},y_{3})\rightarrow \left(y'_{2},y'_{3}\right)=(0,y_{3})$. For similar reasons we again have $\varphi (\, (y_{2},y_{3})\, )<\varphi (\, (y'_{2},y'_{3})\, )$. In the domain $y_{3}\leq y_{2}<0$ there is no rollback. Now the item a) easily follows.\end{proof} \begin{lemma}\label{l-posoct} Consider the domain: $E_{1}=\left\{ y=(y_{2},y_{3}):\, y_{2}>0,y_{3}>0\right\} $. \begin{enumerate} \item The conditional expectation\[ \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IF{y}\, |\, Y (n)=y\right)=\skp{\left(u_{2}^{-1},u_{3}^{-1}\right)}{M_{F}(y)}\] does not depend on $y$. \item There exist constants $\CC{l-posoct},\ga{l-posoct}>0$ such that \begin{equation} \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IR{y}\, |\, Y (n)=y\right)\leq -\ga{l-posoct}\varphi (y)\, \quad \textrm{if}\, \textrm{ }\varphi (y)>\CC{l-posoct}\label{eq:pos-quat-ineq}\end{equation} \end{enumerate} \end{lemma} \begin{proof} The first statement follows from the fact that in this domain $\varphi (y)=\skp{\left(u_{2}^{-1},u_{3}^{-1}\right)}{y},$ and the vector $M_{F}(y)$ does not depend on $y$. Let us prove~(\ref{eq:pos-quat-ineq}). Fix some level set \[ L_{C}^{+}:=\left\{ y:\, \varphi (y)=C\right\} \cap E_{1}\equiv \left\{ y_{2}/u_{2}+y_{3}/u_{3}=C,\, y_{2}>0,y_{3}>0\right\} \] and consider an action of rollbacks for $y\in L_{C}^{+}$. We have three different situations. a) Let $y$ be such that $y_{2}\geq y_{3}>0$. It follows that $\displaystyle y_{2}\geq \frac{C}{\frac{1}{u_{2}}+\frac{1}{u_{3}}}\, .$ As it can be easily concluded from Subsection~\ref{sub:Transition-probabilities}, with probability $\beta _{12}$ we have a rollback of the following form $(y_{2},y_{3})\rightarrow (0,y_{3}')$ where $0\leq y_{3}'\leq y_{3}$. Then we obtain\begin{eqnarray*} \varphi \left((0,y_{3}')\right)-\varphi \left((y_{2},y_{3})\right) & = & \left(\frac{0}{u_{2}}+\frac{y_{3}'}{u_{3}}\right)-\left(\frac{y_{2}}{u_{2}}+\frac{y_{3}}{u_{3}}\right)\\ & \leq & -\frac{y_{2}}{u_{2}}\, \leq \, -\frac{C}{\frac{1}{u_{2}}+\frac{1}{u_{3}}}\, \, \left(u_{2}\right)^{-1}=-\frac{C}{1+u_{2}/u_{3}}\, \end{eqnarray*} uniformly in $y_{3}'$ such that $y_{3}'\leq y_{3}$. To phrase it, we will say that with probability~$\beta _{12}$ the increment of $\varphi (y)$ is less or equal to $\, -\, \displaystyle \frac{C}{1+u_{2}/u_{3}}$~. Hence the conditional mean \begin{equation} \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IR{y}\, |\, Y (n)=y\right)\label{eq:c-mean}\end{equation} does not exceed the value $\, -\displaystyle \frac{\beta _{12}C}{1+u_{2}/u_{3}}$ if $y\in L_{C}^{+}$, $y_{2}\geq y_{3}>0$. b) Let $y\in L_{C}^{+}$ be such that $0<\frac{1}{2}y_{3}\leq y_{2}<y_{3}$. It follows that $y_{2}\geq \frac{C}{2\left(\frac{1}{u_{2}}+\frac{1}{u_{3}}\right)}$. With probability $\beta _{12}$ we have a rollback $(y_{2},y_{3})\rightarrow (0,y_{3}')$ where $0\leq y_{3}'\leq y_{3}$ and with probability~$\beta _{23}$ we have a rollback $(y_{2},y_{3})\rightarrow (y_{2},y_{2})$. Both of them give negative increments of the function $\varphi $. But the first rollback gives the increment $\varphi \left((0,y_{3}')\right)-\varphi \left((y_{2},y_{3})\right)$ which is less or equal to~$\, -\, \displaystyle \frac{C}{2(1+u_{2}/u_{3})}$. So we conclude that the above conditional mean~(\ref{eq:c-mean}) will not exceed the value $\, -\frac{1}{2}\beta _{12}\left(1+u_{2}/u_{3}\right)^{-1}C$. c) Now let $y\in L_{C}^{+}$ be such that $0<y_{2}\leq \frac{1}{2}y_{3}$. It follows that $y_{3}-y_{2}\geq K(C)$ where\[ K(C):=\frac{1}{2}\cdot \frac{C}{\left(\frac{1}{2u_{2}}+\frac{1}{u_{3}}\right)}=\frac{Cu_{3}}{u_{3}/u_{2}+2}\, .\] With probability $\beta _{12}$ we have a rollback $(y_{2},y_{3})\rightarrow (0,y_{3}')$, $0\leq y_{3}'\leq y_{3}$, and with probability~$\beta _{23}$ we have a rollback $(y_{2},y_{3})\rightarrow (y_{2},y_{2})$. The first rollback gives a negative increment of the function~$\varphi $, and the second rollback gives the increment $\varphi \left((y_{2},y_{2})\right)-\varphi \left((y_{2},y_{3})\right)$ which is less or equal to $\, -\, K(C)/u_{3}$. Hence the conditional expectation~(\ref{eq:c-mean}) does not exceed the value $\, -\beta _{23}K(C)/u_{3}=-\beta _{23}\left(u_{3}/u_{2}+2\right)^{-1}C$. The proof of the lemma is completed. \end{proof} \begin{lemma}\label{l-axes} Consider the cases when $y$ belongs to the axes: $y\in E_{1,3}$, $y\in E_{2,3}$. Here\[ \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IR{y}\, |\, Y (n)=y\right)\leq -\ga{l-axes}\varphi (y)\, ,\] and\[ \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IF{y}\, |\, Y (n)=y\right)\] does not depend on $y$, where $y\in \{y:\, \varphi (y)>\CC{l-axes}\}$. \end{lemma} \begin{proof} We consider in details the case $E_{1,3}=\{y_{2}=0,\, y_{3}>0\}$. We start with the free dynamics. The following transition \[ (0,y_{3})\rightarrow (y_{2}',y_{3}')=(-1,y_{3}-1)\in E_{3}=\{y_{2}<0,y_{3}>y_{2}\}.\] occurs with probability $\relax \lambda _{1}$. It is easy to see that for $y_{3}>\CC{l-axes}$ values of the function $\varphi (\cdot )$ in both points $(0,y_{3})$ and $(-1,y_{3}-1)$ concide with the values of linear function $\skp{n(T_{3})}{\cdot }$. With probability $\relax \lambda _{2}$ we have a transition\[ (0,y_{3})\rightarrow (y_{2}',y_{3}')=(1,y_{3})\in E_{1}=\{y_{2}>0,y_{3}>0\},\] and with probability $\relax \lambda _{3}$ we have a transition\[ (0,y_{3})\rightarrow (y_{2}',y_{3}')=(0,y_{3}+1)\in E_{3}.\] Evidently, that in $(0,y_{3})$, $(1,y_{3})$ and $(0,y_{3}+1)$ the values of $\varphi (\cdot )$ coincide with the values of a linear function $\skp{n_{1}}{\cdot }$. Hence, \begin{eqnarray*} \mathsf{E}\, \left(\left(\varphi (Y (n+1))-\varphi (y)\right)\IF{y}\, |\, Y (n)=y\right) & = & \lambda _{1}\skp{n(T_{3})}{(-1,-1)}\\ & & \null +\lambda _{2}\skp{n_{1}}{(1,0)}+\lambda _{3}\skp{n_{1}}{(0,1)}.\\ & & \end{eqnarray*} Since the r.h.s.\ does not depend on~$y$ we get the second statement of the lemma. Due to a rollback the Markov chain $Y(n)$ goes from the state $(0,y_{3})\in E_{1,3}$ to a state $(0,0)$ with probability $\relax \beta _{23}$. Note that values of $\varphi (\cdot )$ at these two points can be calculated by using the linear function $\skp{n_{1}}{\cdot }$. Obviously, that the increment of~$\varphi $ corresponding to this rollback is equal to $\varphi \left(\, (0,0)\, \right)-\varphi \left(\, (0,y_{3})\, \right)=-C$, where $C=\varphi (\, (0,y_{3})\, )$. The first statement of the lemma is proved. The case of the domain $E_{1,3}=\{y_{2}>0,\, y_{3}=0\}$ is similar. \end{proof} \begin{lemma}\label{l-origin} For any $C_{0}>0$\[ \sup _{y:\, \varphi (y)\leq C_{0}}\mathsf{E}\, \left(\varphi (Y (n+1))\, |\, Y (n)=y\right)<+\infty \, .\] \end{lemma} \begin{proof} This statement follows from the fact that the jumps of any fixed neighbourhood of $(0,0)$ are bounded and the fact that the function $\varphi $ is continuous. \end{proof} In view of the Lemmas \ref{l-mod-loclin}--\ref{l-origin}, the Foster-Lyapunov criterion (Theorem~\ref{t:Foster}, Appendix) is fulfilled with $f(y)=\varphi (y)$ therefore the Markov chain~$Y(n)$ is ergodic and hence the proof of the item~1 of Theorem~\ref{t:N-3} is completed. \section{Conclusions, conjectures and perspectives} \label{sec:Conclusions} \subsection{Decomposition into groups} We shall always assume that all $\lambda _{1},\ldots ,\lambda _{N}$ are different. Define a function \[ \ell (m):=\min _{i\leq m}\lambda _{i}\, .\] Evidently, this function has the following property:\[ \ell (1)=\lambda _{1}\geq \cdots \geq \ell (m)\geq \ell (m+1)\geq \cdots \geq \min (\lambda _{1},\ldots ,\lambda _{N})=\ell (N)\, .\] Level sets of function~$\ell $ generate a partition of the set $\left\{ 1,\ldots ,N\right\} $. Namely, there exists a sequence $j_{1}=1<j_{2}<\cdots <j_{K}<j_{K+1}=N+1$ such that the set of all processors can be divided into several nonintersecting groups \begin{equation} \left\{ 1,\ldots ,N\right\} =\bigcup _{k=\overline{1,K}}G_{k}\, ,\label{eq:part-Gk}\end{equation} \[ G_{k}:=\, \left\{ j_{k},j_{k}+1,\ldots ,j_{k+1}-1\right\} ,\, \quad \, \ell (j_{k}-1)>\ell (j_{k})=\cdots =\ell (j_{k+1}-1)>\lambda _{j_{k+1}}\, .\] \begin{rem} \noindent \emph{An equivalent description of the group is possible. We say, for example, that $1,2,\ldots ,k$ is a group if \begin{equation} \lambda _{1}\leq \min \left(\lambda _{2},\ldots ,\lambda _{k}\right),\quad \lambda _{1}>\lambda _{k+1}.\label{eq:char-group}\end{equation} } \end{rem} \subsection{Long-time behaviour of the groups} Taking into account Theorems~\ref{t:n-2} and~\ref{t:N-3} and the above notion of groups of processors we put forward the following Conjecture. \noindent \begin{conj}\label{con-N-main} \emph{}Assume that all $\lambda _{1},\ldots ,\lambda _{N}$ are different. For any $j$ the following limit $\displaystyle v^{*} _{j}=\lim _{t\rightarrow +\infty }\frac{x_{j}(t)}{t}$ exists and $v^{*} _{j}=\ell (j)$ . \noindent \end{conj} Therefore this conjecture entails $v^{*} _{j}=\ell (j_{k})$ for $j\in G_{k}$. If for some $k$ the group $G_{k}$ consists of \emph{more than one} processor we may say that the processors of the group $G_{k}$ are \emph{synchronized}. \begin{rem}[On monotone cases] \noindent \emph{If $\lambda _{1}<\cdots <\lambda _{N}$ then $v^{*} _{j}=\lambda _{1}$ for any~ $j$. \\ If $\lambda _{1}>\cdots >\lambda _{N}$ then for all~ $j$ we have $v^{*} _{j}=\lambda _{j}$. } \smallskip{} \end{rem} Let us discuss briefly perspectives of rigorous proof of the above Conjecture for large values of~$N$. In fact, we have already proved this conjectures for a wide class of cascade models. \begin{theorem} Assume that all $\lambda _{1},\ldots ,\lambda _{N}$ are different and a partition~(\ref{eq:part-Gk}) of the set of processors $\left\{ 1,\ldots ,N\right\} $ is such that $\left|G_{k}\right|\leq 3$ for all~$k$. Then the limits $\displaystyle v^{*} _{j}=\lim _{t\rightarrow +\infty }\frac{x_{j}(t)}{t}$ exist and $v^{*} _{j}=\ell (j)$ . \end{theorem} The proof of this statement is just a combination of the result of Theorem~\ref{t:N-3} (item~1) and arguments of the proof of items 2-4 of Theorem~\ref{t:N-3}. We will not pursue further. So the key to the proof of Conjecture consists in generalization of item~1 of Theorem~\ref{t:N-3}. As it was seen in Section~\ref{sec:N-3-Lyap}, a possible way of such generalization is an explicit construction of Foster-Lyapunov function in high dimensions. This seems to be a difficult technical problem which is out of scope of this paper.

BkiUbQA25YjgKJ9_cPAd

\section{Introduction}\label{sec:intro} Heavy quarkonia, or mesons formed as heavy quark-antiquark bound states, constitute a valuable ground to study strong interactions, because they are related to both the perturbative and non-perturbative regimes of QCD. In heavy quarkonia energies are marginally high enough for perturbative methods to become useful, while the confining, non perturbative aspect of the interaction, is also of crucial importance. Recently a new state of the $b\bar{b}$ system has been observed by the ATLAS Collaboration, with a spin-weighted average (``barycenter") mass of $10530 \pm 5 \pm 9\mbox{ MeV}$ \cite{Aad:2011ih}. This state has been identified with the $n=3$ radial excitation of the $b\bar{b}$ system with angular momentum numbers $s=1$, $\ell=1$ (hence $J=0,1,2$), called $\chi_b(3P_J)$. The three states $J=0,1,2$ are closely spaced in mass, so the measured value of the mass corresponds to their average, while the splittings of these states are not yet resolved. In this work we present predictions from different potentials models for the values of these splittings. Since the discovery of the $J/\psi$ meson in 1974, understood as a $(c\bar c)$ system, many potential models have been proposed to describe the interaction between the quark and antiquark in a bound state \cite{ Eichten:1974af,Pumplin:1975cr,Quigg:1977dd,Eichten:1978tg,Richardson:1978bt,Eichten:1979ms,Buchmuller:1980su,Martin:1980rm,Buchmuller:1981aj,Gupta:1981pd, Gupta:1982kp,Gupta:1982qc,Gupta:1984um,Gupta:1986xt,Born:1989iv,Gupta:1993pd,Ding:1995he,Ebert:2002pp,Barnes:2003vb,Gonzalez:2003gx, Eichten:2004uh,Barnes:2005pb,Eichten:2005ga,Lakhina:2006vg}. One of the first proposed models was the so called \emph{Cornell potential} \cite{Eichten:1974af,Eichten:1978tg,Eichten:1979ms}, which is a Coulomb-plus-linear term combination that takes into account general properties expected from the interquark interaction: a Coulombic behavior at short distances and a linear confining term at long distances, representing the perturbative one-gluon exchange and the non-perturbative chromoelectric flux tube of confinement, respectively. Other models based on phenomenological grounds include a logarithmic potential \cite{Quigg:1977dd} and a non-integer power law potential \cite{Martin:1980rm}. Subsequently, elements from QCD were included in different ways in the potential formulation, from the inclusion of the running of the QCD coupling constant in the Coulombic interaction \cite{Richardson:1978bt}, to a derivation of the short distance quark-antiquark potential from perturbative QCD, where spin-dependent interactions naturally appear \cite{Buchmuller:1980su,Gupta:1981pd,Buchmuller:1981aj,Gupta:1982qc,Gupta:1982kp,Gupta:1984um,Gupta:1986xt}. A modification of the Cornell potential to take into account a saturation effect in the linearly growing confining part has also been proposed, inspired in lattice data \cite{Born:1989iv,Gonzalez:2003gx,Ding:1995he}. In the context of these potential models, a successful description of many properties of quarkonia has been done, both for mass spectrum and decay widths. A comprehensive review of the status of heavy quarkonium can be found in Ref. \cite{Brambilla:2010cs}. For specific studies of P-wave splittings in $b\bar b$, see also Ref.~\cite{chisplit}. Even though the $b$ quarks are heavy compared to the scale of confinement in QCD, the $b$-$\bar b$ interaction is sufficiently strong to make the bound state somewhat relativistic. Quark model calculations give typical velocities $v_q\sim 0.25-0.4$. Consequently, one expects relativistic effects to be important, and at a fundamental level the very use of a potential, which represents an instantaneous static interaction, is questionable. Nevertheless, potential models have been successful in predicting mass spectra and decays in quarkonia, and are particularly useful in describing the mass splitting among different states of orbital angular momentum and spin. Here we study the prediction on the mass splitting in $\chi_b(3P)$ from a set of potential models that have been able to predict other quarkonium spectra with relative success. We compare the average masses of known parts of the spectra in P-wave states with the results given by each of the potential models. An important issue to consider in the model is the separation between scalar vs. vector nature of the quark-antiquark interaction, as the spin-dependent interaction is sensitive to this separation. Since this separation is purely phenomenological, we must first use as input a known part of the spectrum in order to fit the parameters that model this separation; we then obtain results for other parts of the known spectrum, which we use to check the validity of the model. Then, taking into consideration the degree of success of these results, we adventure in predicting the yet unknown mass splitting in the $\chi_b(3P)$ state within each of these models. In Section \ref{models} we briefly describe the potential models that we consider in our analysis. In Section \ref{results} we present the results obtained from the different models including the predictions for the splittings of the $\chi_b(3P)$ state. Finally in Section \ref{conclusions} we state our conclusions. \section{Potential models}\label{models} Rigorously speaking, a potential in a bound state is a static instantaneous interaction. Such a description is good only in the non relativistic limit. A systematic relativistic expansion of the interaction provides additional terms that are corrections that depend on the orbital angular momentum and spin of the bound particles. The expansion should represent charmonium and bottomonium rather well since the motion of their constituents is not highly relativistic. In what follows we consider four representative models given in terms of a spin-independent, static central potential, where relativistic corrections are incorporated as the aforementioned expansion. We call these models a) the Cornell model b) the Screened Cornell model c) the Richardson model, and d) the p-QCD (perturbative QCD) model, which we describe in what follows. In addition, for the first two models we consider two variations, henceforth called Cornell-I and Cornell-II, and Screened Cornell-I and Screened Cornell-II, respectively, depending on the separation of the potential into a vector exchange and a scalar exchange, as explained below. \subsection{The Cornell model} The \emph{Cornell potential} \cite{Eichten:1974af} was originally proposed to describe masses and decay widths of charmonium states. It takes into account the following features of the interaction between the constituents: (i) a Coulomb-type interaction which should describe the short distance regime mediated by a single gluon exchange in perturbation theory, and (ii) a quark confining interaction that dominates at long distances. Explicitly, the potential is given by \begin{eqnarray} \label{e:cornell-a} V(r) = -\frac{\beta}{r} + \frac{r}{a^2} + V_0, \end{eqnarray} where the first term corresponds to the Coulombic interaction and the second is the interaction responsible for confinement that arises from the color field flux tube between the quarks. The coefficients $a$ and $\beta$ are adjusted to fit the charmonium spectrum, but with the assumption that, roughly speaking, it should be valid for all other heavy quarkonia. As such, the flavor dependence should arise solely from the mass of the bound quarks. Many fits have been done to adjust these parameters \cite{Eichten:1974af,Quigg:1979vr,Eichten:1979ms}. In this work we will consider the most recent values: $\beta=0.52$, $a=2.34\mbox{ GeV}^{-1}$, $m_c=1.84\mbox{ GeV}$ and $m_b = 5.17\mbox{ GeV}$ \cite{Eichten:1979ms}, where $\beta$, $a$ and $m_c$ were fitted using as inputs the mass differences of the charmonium states $J/\psi(1S)$, $\psi(2S)$ and the $\chi_c(1P)$ spin-weighted barycenter, as well as the requirement that the charm quark mass should be close to the lightest charmed-meson mass. The value of $m_b$ was chosen taking into account limits based on the $\Upsilon$ masses. The potential in Eq.\ (\ref{e:cornell-a}) represents the static limit of the interaction. A relativistic expansion will include additional terms, that will constitute smaller corrections in the case of heavy quarkonia, and some of them will depend on the orbital angular momentum and spin of the constituents. These terms are essential in order to reproduce or predict the mass splittings of quarkonium states that differ in angular momentum, such as the $\chi_b(3P_J)$, for $J=0, 1, 2$. A general expression for the spin-dependent part of the potential of a $q\bar q$ bound state in QCD using a Wilson loop approach was calculated to order $1/m^2$: \cite{Eichten:1980mw, Pineda:2000sz} \begin{eqnarray} \label{e:vsd1} V_{SD}(r) & = & \left[\frac{\mathbf{S_1}\cdot \mathbf{L_1}}{2m_1^2} - \frac{\mathbf{S_2}\cdot \mathbf{L_2}}{2m_2^2} \right] \left[\frac{dV(r)}{rdr}+2\frac{dV_1(r)}{rdr}\right] + \frac{(\mathbf{S_2}\cdot\mathbf{L_1}-\mathbf{S_1}\cdot\mathbf{L_2})}{2 m_1 m_2} \frac{dV_2(r)}{rdr}\nonumber\\ & & + \frac{1}{6 m_1 m_2}(6\mathbf{S_1}\cdot \mathbf{\hat{r}} \mathbf{S_2}\cdot \mathbf{\hat{r}} -2\mathbf{S_1}\cdot\mathbf{S_2} )V_3(r) + \frac{2}{3 m_1 m_2} \mathbf{S_1}\cdot\mathbf{S_2} \nabla^2 V_2(r), \end{eqnarray} where $V(r)$ is the spin-independent part of the potential, while $V_1(r)$, $V_2(r)$ and $V_3(r)$ are the spin-dependent parts, that can be expressed as expectation values of color electric and magnetic fields. These potentials are not all independent, as pointed out by Gromes \cite{Gromes:1984ma}. Lorentz invariance imposes the relation $V(r)+V_1(r)-V_2(r)= {\cal C}$, where $\cal C$ is an irrelevant integration constant. We can thus eliminate $dV_1(r)/dr$ from Eq.\ (\ref{e:vsd1}) and, in the equal-mass case, $m_1=m_2\equiv m$, we can rewrite it as: \begin{equation} \label{e:vsd2} V_{SD}(r) = \frac{\mathbf{S}\cdot \mathbf{L}}{2m^2} \left[-\frac{dV(r)}{rdr}+4\frac{dV_2(r)}{rdr}\right] + \frac{1}{12 m^2}(6\mathbf{S}\cdot \mathbf{\hat{r}} \mathbf{S}\cdot \mathbf{\hat{r}} -2\mathbf{S}\cdot\mathbf{S} )V_3(r) + \frac{1}{6 m^2} \left(2\mathbf{S}\cdot\mathbf{S}-3\right) \nabla^2 V_2(r), \end{equation} where $\mathbf{S}\equiv \mathbf{S_1}+\mathbf{S_2}$ and $\mathbf{L}\equiv \mathbf{L_1}=-\mathbf{L_2}$. The terms in Eq.\ (\ref{e:vsd2}) are referred to as spin-orbit, tensor and hyperfine interaction, respectively. Notice that only the spin-orbit and tensor interactions are responsible for the mass splittings of the $\chi_b(nP_J)$. Now, the otherwise independent potentials $V(r)$, $V_2(r)$ and $V_3(r)$ are further related if one assumes that the quark-antiquark potential due to the strong interaction arises from the effective exchange of a scalar and a vector particle. This assumption is clearly part of the modeling, as there could be further effective exchanges, but if we stick to it, the quark-antiquark lagrangian has the form: \begin{eqnarray} L_{int} = \tilde{s}(q^2)\bar{u}u\bar{v}v + \tilde{v}(q^2)\bar{u}\gamma_\mu u\bar{v}\gamma^\mu v. \end{eqnarray} The expansion of this expression in powers of $v^2/c^2$ gives a Breit-Fermi form of the spin-dependent potential \cite{Schnitzer:1975ux,Dib:1987cj}: \begin{eqnarray} \label{e:vsd3} V^{(eff)}_{SD}(r) & =& \frac{\mathbf{S}\cdot \mathbf{L}}{2m^2}\left[ -\frac{dv(r)+ds(r)}{rdr} + 4\frac{dv(r)}{rdr} \right] + \frac{1}{12m^2}(6\mathbf{S}\cdot \mathbf{\hat{r}} \mathbf{S}\cdot \mathbf{\hat{r}} -2\mathbf{S}\cdot\mathbf{S} ) \left[ \frac{dv(r)}{rdr} - \frac{d^2v(r)}{dr^2}\right]\nonumber\\ & & + \frac{1}{6m^2}(2\mathbf{S}\cdot \mathbf{S}-3)\nabla^2v(r). \end{eqnarray} Comparing \ Eqs.\ (\ref{e:vsd2}) and (\ref{e:vsd3}), one can make the following identifications, which relate $V(r)$, $V_2(r)$ and $V_3(r)$: \begin{eqnarray} V(r) = v(r)+s(r),\ \ \ \ \ V_2(r)=v(r),\ \ \ \ \ V_3(r) = \frac{dv(r)}{rdr} - \frac{d^2v(r)}{dr^2}.\nonumber \end{eqnarray} We then see that the spin-independent potential $V(r)$ is composed by the vector and scalar parts, while the spin-dependent parts, $V_2(r)$ and $V_3(r)$, are given in terms of the vector part only. To leading order in the wave function obtained from the spin-independent potential $V(r)$ (e.g. Eq.\ \ref{e:cornell-a}), one can express the spectrum of the $\chi_b(nP_J)$, for a fixed radial excitation $n$, as: \begin{eqnarray} \label{e:splittings} M(P_2) & = & \bar{M} + a - 2b/5,\nonumber\\ M(P_1) & = & \bar{M} - a + 2b,\\ M(P_0) & = & \bar{M} - 2a - 4b,\nonumber \end{eqnarray} where $a$ and $b$ are the expectation values of the radial functions in the spin-orbit and tensor terms, respectively: \begin{eqnarray} a & = & \frac{1}{2m^2} \left<-\frac{ds}{rdr} + 3 \frac{dv}{rdr}\right>,\nonumber\\ b & = & \frac{1}{12m^2} \left< \frac{dv}{rdr} - \frac{d^2v}{dr^2}\right>.\nonumber \end{eqnarray} As such, the potential $V(r)$ provides an explicit form for the sum $v(r)+s(r)$, and for the bound state wave function to leading order. However, to obtain the explicit form of the spin-dependent potential, Eq.\ (\ref{e:vsd3}), it is necessary to know, or to define within the model, the separation of $V(r)$ into the scalar and vector parts $s(r)$ and $v(r)$. For the Cornell potential, Eq.\ (\ref{e:cornell-a}), one can argue that a reasonable separation is to assign the Coulombic term to the vector part, coming from a single gluon exchange at short distances, and the linear term to the scalar part, coming from the flux tube at long distances: \begin{eqnarray} \label{e:sepcor1} v(r) = -\frac{\beta}{r},\ \ \ \ \ s(r) = \frac{r}{a^2}. \end{eqnarray} Here we call this prescription the \emph{Cornell-I} model. Alternatively one can relax this assumption by introducing two phenomenological parameters, $\eta_s$ and $\eta_v$, that define the combinations of Coulomb and linear terms of the scalar and vector potentials (where the case $\eta_s=\eta_v=1$ corresponds to Eq.\ \ref{e:sepcor1}): \begin{eqnarray} \label{e:cornell-b} s(r) & = & -(1-\eta_v) \frac{\beta}{r} + \eta_s\frac{r}{a^2},\nonumber\\ v(r) & = & -\eta_v \frac{\beta}{r} + (1-\eta_s)\frac{r}{a^2}. \end{eqnarray} One can then find the values of $\eta_s$ and $\eta_v$ that provide the best fit to a part of the spectrum. In this study we look for the values that minimize the following $\chi^2$ function, using the mass splittings in the experimentally known $\chi_b(1P_J)$ and $\chi_b(2P_J)$: \begin{eqnarray} \label{e:chi2} \chi^2 = \sum_{\substack{n=1,2 \\ij=21,10}} \left(\frac{\Delta M_{ij} (nP)^{(\mbox{model})} - \Delta M_{ij}(nP)^{(\mbox{exp})}}{\sigma_{ij}(nP)^{(\mbox{exp})}}\right)^2, \end{eqnarray} where \begin{equation}\label{SplitD} \Delta M_{21}{(1P)} \equiv M(1P_2) - M(1P_1), \quad \Delta M_{10}{(1P)} \equiv M(1P_1) - M(1P_0) \end{equation} are the mass splittings in the $\chi_b(1P_J)$, and similarly are the $\Delta M_{ij} (2P)$ defined for the $\chi_b(2P_J)$. Following this procedure, we find the optimal values $\eta_s =1$ and $\eta_v = 0.923$. We call this prescription the \emph{Cornell-II} model. It is interesting to notice that the optimal values of $\eta_s$ and $\eta_v$ are remarkably close to the Cornell-I prescription, which is the one commonly assumed in the literature. \subsection{The Screened Cornell model} A variation of the Cornell potential, which we call the \emph{Screened Cornell potential}, has been used to include the effect of saturation of the strong interaction at long distances that appears in lattice data \cite{Born:1989iv}: \begin{eqnarray} \label{e:screened1a} V(r) = \left(\frac{-\beta}{r}+\frac{r}{a^2}\right)\left(\frac{1-e^{-\mu r}}{\mu r}\right), \end{eqnarray} where $\mu$ is the \emph{screening parameter}. This potential behaves like a Coulomb potential at short distances but, unlike in the previous model, it tends to a constant value for large $r$ (namely, for $r\gg \mu^{-1})$. In other words, the linearly growing confining potential flattens to a finite value at large distances, corresponding to the saturation of $\alpha_S$ to a finite value for decreasing $Q^2$ \cite{Cornwall:1989gv,Papavassiliou:1991hx,Aguilar:2001zy}. This effect should be due to the creation of virtual light quark pairs that screen the interaction between the bound quarks at long distances. The values of $\beta$, $a$ and $\mu$ we use here are those of Ref.\ \cite{Gonzalez:2003gx} and are shown in Table \ref{t:parameters1}. Of these values, $\beta$ and $\mu$ are intrinsic to the model, while $a$, $m_c$ and $m_b$ were fixed by the authors in order to reproduce the $J/\psi$ mass in $c\bar c$ and the $\Upsilon(1S)$ and $\Upsilon(2S)$ in $b \bar b$. For the potential in Eq.\ (\ref{e:screened1a}), we can separate the scalar and vector parts in the same common way as in the Cornell potential (that is, the Coulombic term as vector and the linear term as scalar), a prescription we call here the \emph{Screened Cornell-I} model. Alternatively, just as before, we can introduce two phenomenological separation parameters, $\eta_s$ and $\eta_v$, which are fitted by the same optimization procedure: \begin{eqnarray} \label{e:screened1b} s(r) & = & \left(-(1-\eta_v) \frac{\beta}{r} + \eta_s\frac{r}{a^2}\right) \left(\frac{1-e^{-\mu r}}{\mu r}\right),\nonumber\\ v(r) & = & \left(-\eta_v \frac{\beta}{r} + (1-\eta_s)\frac{r}{a^2}\right) \left(\frac{1-e^{-\mu r}}{\mu r}\right). \end{eqnarray} The optimal values we find in this prescription are $\eta_s = 0.810$ and $\eta_v = 1$. We call this prescription the \emph{Screened Cornell-II} model. \subsection{The Richardson model} The \emph{ Richardson potential} is another well known model that incorporates the features of asymptotic freedom at short distances and linear confinement at long distances \cite{Richardson:1978bt}. With a minimal interpolation between these two asymptotic behaviors, the Richardson potential is obtained: \begin{eqnarray} \label{e:richardson} V(r) = \frac{8\pi}{33-2n_f} \Lambda \left[\Lambda r - \frac{f(\Lambda r)}{\Lambda r} \right], \end{eqnarray} with \begin{eqnarray} f(t) = 1-4\int_1^\infty \frac{dq}{q} \frac{e^{-qt}}{\ln^2(q^2-1)+\pi^2}.\nonumber \end{eqnarray} Here $n_f$ is the number of light quarks relevant to the renormalization scale, taken equal to three, while $\Lambda=0.398\mbox{ GeV}$ is the scale of interpolation between the two asymptotic regimes and $m_c = 1.49\mbox{ GeV}$ is the constituent mass of the charm quark in the model. These values have been fitted to reproduce the mass of the charmonium states $J/\psi(1S)$ and $\psi(2S)$. For bottomonium the same value of $n_f$ and $\Lambda$ should be used, setting the bottom quark mass at $m_b=4.8877\mbox{ GeV}$ in order to reproduce the current $\Upsilon(1S)$ mass. According to Richardson, the value of $n_f$ is kept equal to three (i.e. the number of light quarks), since the Appelquist Carazzone theorem \cite{Appelquist:1974tg} implies that the effect of quarks heavier than the energy scale that determines the dynamics should be small (the latter is related to the binding energy or the inverse of the radius). With these values for $\Lambda$, $n_f$ and $m_b$, the $\Upsilon(2S)$ mass is predicted with good agreement with experiment. In contrast to the Cornell potential, the Richardson potential has no obvious separation into scalar and vector parts. In order to be able to determine the splittings with this potential, one needs this separation. The following phenomenological separation has been used in previous works \cite{Dib:1987cj}: \begin{eqnarray} \label{e:richardsonsep} v(r) & = & V(r) e^{-r^2/a_r^2},\nonumber\\ s(r) & = & V(r) (1-e^{-r^2/a_r^2}). \end{eqnarray} Here $V(r)$ is the Richardson potential of Eq.\ (\ref{e:richardson}) and $a_r$ is a phenomenological length scale that separates the vector character at short distances and scalar character at large distances. Following the same $\chi^2$ optimization procedure as before, the value $a_r = 0.1782\mbox{ fm}$ is found. \begin{table}[t] \caption{Best values of the parameters for each of the potential models, according to previous authors, except $m_b$ in the Richardson model, which was updated with the current experimental data, and the separation parameters $\eta_s$, $\eta_v$ and $a_r$ obtained in our own fits.} \label{t:parameters1} \centering \begin{tabular}{l c c c c c c c c c c } \hline \hline & Equation \ \ \ & $\beta$ & $a\mbox{ GeV}^{-1}$ & $m_b\mbox{ GeV}$ & $V_0\mbox{ GeV}$ & $\mu\mbox{ fm}^{-1}$ & $\eta_s$ & $\eta_v$ & $\Lambda\mbox{ GeV}$ & $a_r \mbox{ fm}$ \\ \hline Cornell - I & (\ref{e:cornell-a}) & $0.52$ & $2.34$ & $5.17$ & $-0.50805$ & - & - & - & - & - \\ Cornell - II & (\ref{e:cornell-b}) & $0.52$ & $2.34$ & $5.17$ & $-0.50805$ & - & $1$ & $0.923$ & - & - \\ Screened Cornell -I & (\ref{e:screened1a}) & $0.423$ & $1.858$ & $4.6645$ & - & $0.71$ & - & - & - & - \\ Screened Cornell -II & (\ref{e:screened1b}) & $0.423$ & $1.858$ & $4.6645$ & - & $0.71$ & $0.810$ & $1$ & - & - \\ Richardson & (\ref{e:richardson}) & - & - & $4.8877$ & - & - & - & - & $0.398$ & $0.1782$ \\ \hline\hline \end{tabular} \end{table} \subsection{The p-QCD model} One last potential model we consider in our study \cite{Gupta:1982kp,Radford:2007vd} is based on a semi-relativistic treatment of perturbative QCD interactions to one loop, which we call the \emph{p-QCD model}, where the spin-dependent potentials, which appear in the perturbative treatment, are somewhat different than those in the previous models. The Hamiltonian in this model considers a relativistic expression for the quark kinetic energy, $H_0 = \sqrt{\vec{p}^2+m^2}$, unlike the previous models where a purely non-relativistic approximation, $H_0 = m + p^2/2m$, is used. Additionally, the potential is composed of a short distance part obtained from perturbative QCD to one loop, and a phenomenological long range confining part. The complete potential has the form \begin{eqnarray} V(r) = - \frac{4\alpha_S}{3r} \left[1-\frac{3\alpha_S}{2\pi}+\frac{\alpha_S}{6\pi}(33-2n_f) \left(\ln \mu r+\gamma_E\right)\right] + A \, r + V_S + V_L .\label{RR} \end{eqnarray} The first term is the spin independent Coulomb-like part, corrected to one loop in QCD. The term $A \, r$, where $A$ is a phenomenological constant, is the spin-independent, long range confining potential, assumed to be linear in $r$. $V_S$ is a short-range part of the potential that appears in the perturbative calculation to one loop, and includes most spin-dependent parts in the form of spin-orbit (LS), tensor (T) and hyperfine (HF) terms, and a short-interaction spin independent term (SI): \begin{eqnarray} V_{LS}&=&\frac{2\alpha_S\mathbf{L}\!\cdot\!\mathbf{S}}{m^2r^3}\!\left\{1- \frac{\alpha_S}{6\pi}\left[\frac{11}{3}-(33-2n_f)\left(\ln\mu r+\gamma_E-1\right)+12\left(\ln mr+\gamma_E-1\right)\right]\right\}\label{potb}, \nonumber\\ [4pt] V_{T\;} &=&\frac{4\alpha_S(3\mathbf{S_1}\!\cdot\!\mathbf{\hat{r}}\mathbf{S_2}\!\cdot\!\mathbf{\hat{r}}-\mathbf{S_1}\!\cdot\!\mathbf{S_2})} {3m^2r^3}\left\{1+\frac{\alpha_S}{6\pi}\left[8+(33-2n_f)\left(\ln\mu r+\gamma_E-\frac{4}{3}\right) -18\left(\ln mr+\gamma_E-\frac{4}{3}\right)\right]\right\},\label{potc} \nonumber\\ V_{HF}&=&\frac{32\pi\alpha_S\mathbf{S_1}\!\cdot\!\mathbf{S_2}}{9m^2}\left\{\left[1-\frac{\alpha_S}{12\pi} (26+9\ln\,2)\right]\delta(\mathbf{r})\right. \nonumber\\ & & \left.-\frac{\alpha_S}{24\pi^2}(33-2n_f)\nabla^2\left[\frac{\ln\,\mu r+\gamma_E}{r}\right]+\frac{21\alpha_S}{16\pi^2}\nabla^2\left[\frac{\ln\, mr+\gamma_E}{r}\right]\right\}\label{pota}, \nonumber \\ V_{SI}&=&\frac{4\pi\alpha_S}{3m^2}\left\{\left[1-\frac{\alpha_S}{2\pi}(1+\ln2)\right] \delta(\mathbf{r})-\frac{\alpha_S}{24\pi^2}(33-2n_f)\nabla^2\left[\frac{\ln\,\mu r+\gamma_E}{r}\right]-\frac{7\alpha_Sm}{6\pi r^2}\right\}. \label{potd}\nonumber \end{eqnarray} Finally $V_L$ is the relativistic correction arising from the confining potential (thus proportional to $A$), also composed of spin-dependent terms: \begin{equation} V_L = -(1-f_V)\frac{A}{2m^2 r}\mathbf{L}\cdot\mathbf{S} + f_V\left[\frac{A}{2m^2 r}\left(1+\frac{8}{3}\mathbf{S_1}\cdot \mathbf{S_2}\right) + \frac{3A}{2m^2 r}\mathbf{L}\cdot \mathbf{S} + \frac{A}{3m^2 r}(3\mathbf{S_1}\cdot \mathbf{\hat{r}}\mathbf{S_2}\cdot \mathbf{\hat{r}} - \mathbf{S_1}\cdot \mathbf{S_2}) \right]. \nonumber \end{equation} The phenomenological parameter $f_V$ was introduced by the authors to represent the fraction of vector vs. scalar character of the confining potential, which they fit with the available data. For bottomonium the parameters $A$, $m$, $\alpha_S$, $\mu$ and $f_V$ were fitted using as input eight masses of the $b\bar{b}$ spectrum. The authors obtained \cite{Radford:2009qi} $A=0.175\mbox{ GeV}^2$, $m=5.33\mbox{ GeV}$, $\alpha_S=0.295$, $\mu=4.82\mbox{ GeV}$ and $f_V=0$. The values of $\alpha_S$ and $m$ are given in the renormalization scheme of Ref.~\cite{Gupta:1982im} for the value of $\mu$ given above. Remarkably enough, $f_V=0$ implies, just like in the other models, that the long distance part of the potential corresponds to purely scalar exchange. \section{Calculations and Results} \label{results} \subsection{The spin-average masses of $\chi_b(1P)$, $\chi_b(2P)$ and $\chi_b(3P)$} As seen in Eq.\ (\ref{e:splittings}), the masses of the states $\chi_b(nP_0)$, $\chi_b(nP_1)$ and $\chi_b(nP_2)$ differ by small amounts. These differences are determined by spin-dependent interactions. On the other hand, the barycenter (or spin-averaged) mass of these states is defined as \begin{eqnarray}\label{bary} \bar{M} = \frac{5 M(P_2) + 3 M(P_1) + M(P_0)}{9}, \end{eqnarray} and is determined in each model by the leading, spin-independent part of the potential. Consequently, we expect the spin-independent part of the models we previously described to correctly reproduce the experimental barycenters. We have calculated $\bar{M}$ for the $\chi_b(1P)$, $\chi_b(2P)$ and $\chi_b(3P)$, numerically solving the Schr\"odinger equation \cite{Lucha:1998xc} for the potential models described in the previous section, using the parameters listed in Table \ref{t:parameters1}, for the first three models, respectively. The results we obtain are shown in Table \ref{t:avgmasses}. We include the results obtained in Ref.\ \cite{Radford:2009qi} by the authors of the p-QCD model. \begin{table}[h] \caption{The mass barycenters of the states $\chi_{b}(1P)$, $\chi_{b}(2P)$ and $\chi_{b}(3P)$ reproduced by the potential models described in Section \ref{models}, and compared to the corresponding experimental values. On the right of each reproduced mass is the discrepancy between this value and the corresponding experimental value, $\delta_{nP}\equiv \bar M(nP)_{(\mbox{model})}-\bar M(nP)_{(\mbox{exp})}$. The experimental values for $1P$ and $2P$ are from Ref.\ \cite{pdg:2010-11}, while those for $3P$ are from Ref.\ \cite{Aad:2011ih}. All values are in MeV. } \label{t:avgmasses} \centering \begin{tabular}{l l r l r l r c } \hline \hline Model & $\bar{M}(1P)$ & $\delta_{1P}$ \ \ \ & $\bar{M}(2P)$ & $\delta_{2P}$ \ \ \ & $\bar{M}(3P)$\ \ \ & $\delta_{3P}$ & $[\mbox{MeV}]$ \\ \hline Cornell & $9958.3$ & $58.4$ \ \ \ & $10312.6$ & $52.5$ \ \ \ & $10595.3$ & $65.3$ & \\ Screened Cornell & $9907.9$ & $8.0$ \ \ \ & $10261.2$ & $1.0$ \ \ \ & $10516.4$ & $-13.6$ & \\ Richardson & $9895.7$ & $-4.2$ \ \ \ & $10248.8$ & $-11.4$ \ \ \ & $10520.1$ & $-9.9$ & \\ p-QCD \cite{Radford:2009qi} & $9898.7$ & $-1.2$ \ \ \ & $10261.2$ & $1.0$ \ \ \ & $10543.9$ & $13.9$ & \\ Experiment & $9899.87 \pm 0.27$ & - \ \ \ & $10260.20 \pm 0.36$ & - \ \ \ & $10530 \pm 10 $ & - & \\ \hline\hline \end{tabular} \end{table} The value of the constant $V_0$ for the Cornell potential [Eq.\ (\ref{e:cornell-a})] was fitted in order to obtain the experimental value of the $\Upsilon(1S)$. We should recall that the parameters in the Cornell and Richardson potentials, which are shown in Table \ref{t:parameters1}, were fitted using charmonium states (except for the bottom constituent mass $m_b$), and those of the Screened Cornell model were fixed using both $c\bar c$ and $b\bar b$ masses. In contrast, in the p-QCD model the free parameters were fitted using $b\bar b$ states only. Consequently, one should naturally expect the latter model to give a closer prediction for the $\chi_b$, in comparison with the other models, where an accurate prediction for $\chi_b$ masses is clearly a more demanding requirement. The bottom line of Table \ref{t:avgmasses} shows the experimental values of the barycenter masses, in order to compare them with the model results. To ease the comparison we also include, for each $\chi_b$, the mass discrepancy $\delta_{nP}$ between the model result and the corresponding experimental value. As a first observation from Table \ref{t:avgmasses}, one can see that in most cases the model discrepancies are larger than the experimental uncertainties, indicating that in general there is still need for improvement in the models. Notwithstanding, the errors are quite small in all models except \emph{Cornell}. This tendency indicates that the inclusion of saturation (i.e. going from Cornell to Screened Cornell) clearly improves the predictions for the barycenter masses, but there is no clear distinction in the reliability between the last three models at this stage. We must also comment that the discrepancy in the Cornell model can be considerably improved in a very simple way: the offset $V_0$ was fitted using the $\Upsilon(1S)$ mass. If we had fitted $V_0$ using $\chi_b(1P)$ instead, the discrepancy in the barycenter mass of $\chi_b(2P)$ and $\chi_b(3P)$ in Table \ref{t:avgmasses} would have been $\delta_{2P} = - 6.1$ MeV and $\delta_{3P} = 6.8$ MeV, instead of $52.5$ and $65.3$ MeV, respectively. In this sense, the Cornell potential does not fit very well the whole $b\bar b$ spectrum, but could still be good in predicting mass \emph{differences} within $\chi_b(nP)$ states, which are all of the same spin and orbital angular momentum. \subsection{The splittings of $\chi_b(1P_J)$, $\chi_b(2P_J)$ and $\chi_b(3P_J)$}\label{s:splittings} To date, the splittings of the $J=0,1,2$ states in the $\chi_b(1P_J)$ and $\chi_b(2P_J)$ have been experimentally measured with reasonable precision, while there is still no data for the splitting in the $\chi_b(3P_J)$. Our main goal here is to predict the latter splitting according to each of the aforementioned models. We characterize the splittings of the $J=0, 1, 2$ states within a given $\chi_b(nP)$ by the mass differences $\Delta M_{21}(nP)$ and $\Delta M_{10}(nP)$, as defined in Eq.\ (\ref{SplitD}). An additional parameter which is used in the literature to characterize the splittings is the ratio of the mass differences within a given $\chi_b(nP)$, defined as \begin{eqnarray} R_{\chi}(nP) = \frac{\Delta M_{21}(nP)}{\Delta M_{10}(nP)}.\label{Splitx} \end{eqnarray} In our calculations, we first reproduce within each model the splittings in the $\chi_b(1P_J)$ and $\chi_b(2P_J)$ and compare these results with their corresponding experimental values, as a way to check the reliability of the models. These results are shown in Tables \ref{t:1psplit} and \ref{t:2psplit}, respectively. In these tables one can make several observations. First, from Table \ref{t:1psplit} for the splitting in $\chi_b(1P)$, the best results for the $\Delta M_{ij}$ are obtained with the Richardson and p-QCD models, followed by the Cornell-II model, while the largest discrepancies appear with the Screened Cornell-I model. On the other hand, concerning the splitting ratios $R_\chi$, the pattern of performance is different: the Screened Cornell-I joins the Richardson and p-QCD models in giving the best results, while the largest discrepancy occurs with the Screened Cornell-II model. Alternatively, we can see whether the results for the splitting in $\chi_b(1P)$ improve in going from the Cornell to the Screened Cornell model (i.e. by including saturation at long distances), as it happened in the results for the barycenters. Clearly this is not the case: there is no such an improvement on the splittings: the Screened Cornell models actually perform worse than the simple Cornell models. Finally, we can see whether the best fitted vector vs. scalar separation causes an improvement in the results (i.e. going from models of type -I to type -II). Indeed, a moderate improvement can be seen in Table \ref{t:1psplit} going from Cornell-I to -II or from Screened Cornell-I to -II, which in any case is expected, because the fitting was done with that purpose. However, the improvement occurs only in the $\Delta M_{ij}$, but not so in the ratio $R_\chi$. In turn, for the splittings in $\chi_b(2P)$, the best results for $\Delta M_{ij}$ are given by the p-QCD model, and Screened Cornell-I and -II models, while the largest discrepancy is found in the results given by the Cornell-I model. In contrast, for the splitting ratio $R_\chi$ the best results are given by the Screened Cornell I and p-QCD models, while the largest discrepancy occurs in the Screened Cornell-II model. From the above one sees that the results are rather disperse: models that reproduce well some of the splitting features do not do so well on other features. The inclusion of saturation in the Cornell model does not seem to make it much better, and the optimal adjustment of the vector vs. scalar separation does not cause a significant improvement either. The sole exception, to some degree, is the p-QCD model, which tends to be more often within the best results. However, this tendency was expected, since this model contains a larger number of parameters and they were fitted purely with $b\bar b$. In contrast, all the other models were built using the charmonium spectrum and are now required to fit the $b\bar b$ spectrum as well. One clear pattern in the results for $\chi_b(1P)$ and $\chi_b(2P)$ is that all models give values for the ratio $R_\chi$ which are consistently larger than experiment. This feature could be an indication that the separation of the potentials purely in terms of scalar and vector exchanges may not be sufficient, and an expansion into exchanges of additional spin and parity may be required \cite{Franzini:1992nk}. Now, concerning the prediction of the models for the so far unknown splitting in the $\chi_b(3P_J)$, Table \ref{t:3psplit} shows the results of our calculations, together with the prediction of the p-QCD model according to its authors \cite{Radford:2012ue}. Looking at the values for the ratio $R_\chi(3P)$, if we extrapolate from the results on $\chi_b(1P)$ and $\chi_b(2P)$ we may suspect that these are also overestimations, i.e. all models tend to give too large values for $R_\chi$. Within that assumption, we may consider more successful those models that give the smallest prediction for $R_\chi$, which correspond to the p-QCD and the Screened Cornell-I model. However, if we look at the $\Delta M_{ij}$ results in the 3P, the predictions of the Screened Cornell-I model are much lower than all the others. Then, going back to 1P and 2P, where there is experimental evidence, this model also gives too small splittings in those states. In that sense, concerning the splittings the reliability of this model is questionable. Besides that observation, among the other models there is a tendency in $\Delta M_{10}$ to be near $22\mbox{ MeV}$. For $\Delta M_{21}$ the predictions are more dispersed, but centered around a value of $15\mbox{ MeV}$. If we consider these two values as the best predictors for the splitting, one then predicts a ratio $R_\chi = 0.68$ for the $\chi_b(3P)$. Using the definition of the barycenter given in Eq.\ (\ref{bary}) we can deduce the masses of the states $J=0$, 1 and 2 of the $\chi_(3P_J)$ in terms of the barycenter and splittings: \begin{eqnarray} M(P_0) &=& \bar M - \frac{5\Delta M_{21} + 8 \Delta M_{10}}{9},\nonumber \\ M(P_1) &=& \bar M - \frac{5\Delta M_{21}-\Delta M_{10}}{9}, \\ M(P_2) &=& \bar M + \frac{4\Delta M_{21}+\Delta M_{10}}{9}.\nonumber \end{eqnarray} Using the experimental value and its uncertainty for the barycenter and the estimated splittings, the masses of the three $\chi_b(3P)$ states ($J=0,1,2$) would then be $10502\pm 10$ MeV, $10524\pm 10$ MeV and $10539\pm 10$ MeV, respectively. On the other hand, the threshold for $B^+ B^-$ decay is $10558.5\pm0.3$ MeV. Consequently our estimate of the $J=2$ state is just $20\pm 10$ MeV below the threshold. \begin{table}[t] \caption{The mass splittings in the $\chi_b(1P)$ mesons calculated for each of the models described in Section \ref{models}, and their experimental values: the mass differences $\Delta M_{21}$ and $\Delta M_{10}$ are defined as in Eq.\ (\ref{SplitD}), and the splitting ratio $R_\chi$ is defined as in Eq.\ (\ref{Splitx}). } \label{t:1psplit} \centering \begin{tabular}{l c c c} \hline \hline Model & $\Delta M_{21}(1P)\mbox{ [MeV]} \quad $ & $\Delta M_{10}(1P)\mbox{ [MeV]}$ & $R_{\chi}(1P)$\\ \hline Cornell-I & $23.13$ & $32.24$ & $0.717$\\ Cornell-II & $20.12$ & $29.14$ & $0.690$\\ Screened Cornell-I & $16.19$ & $25.21$ & $0.642$\\ Screened Cornell-II & $20.87$ & $28.46$ & $0.733$\\ Richardson & $18.58$ & $30.87$ & $0.602$\\ p-QCD \cite{Radford:2009qi} & $19.3$ & $29.94$ & $0.645$\\ Experiment & $19.43\pm 0.57$ & $33.34\pm 0.66$ & $0.583\pm 0.021$ \\ \hline\hline \end{tabular} \end{table} \begin{table}[t] \caption{The mass splittings in the $\chi_b(2P)$ mesons calculated for each of the models described in Section \ref{models}, and their experimental values: the mass differences $\Delta M_{21}$ and $\Delta M_{10}$ are defined as in Eq.\ (\ref{SplitD}), and the splitting ratio $R_\chi$ is defined as in Eq.\ (\ref{Splitx}). } \label{t:2psplit} \centering \begin{tabular}{l c c c c} \hline \hline Model & $\Delta M_{21}(2P)\mbox{ [MeV]} \quad $ & $\Delta M_{10}(2P)\mbox{ [MeV]}$ & $R_{\chi}(2P)$\\ \hline Cornell-I & $19.22$ & $26.43$ & $0.727$ \\ Cornell-II & $16.77$ & $23.91$ & $0.701$ \\ Screened Cornell-I & $12.91$ & $19.34$ & $0.668$ \\ Screened Cornell-II & $15.91$ & $21.45$ & $0.742$ \\ Richardson & $18.27$ & $25.36$ & $0.720$ \\ p-QCD \cite{Radford:2009qi} & $16.40$ & $24.30$ & $0.675$ \\ Experiment & $13.5\pm 0.6$ & $23.5\pm 1.0$ & $0.574\pm 0.035$ \\ \hline\hline \end{tabular} \end{table} \begin{table}[t] \caption{The predictions for the mass splittings in the $\chi_b(3P)$ mesons according to each of the models described in Section \ref{models}: the mass differences $\Delta M_{21}$ and $\Delta M_{10}$ are defined as in Eq.\ (\ref{SplitD}), and the splitting ratio $R_\chi$ is defined as in Eq.\ (\ref{Splitx}). } \label{t:3psplit} \centering \begin{tabular}{l c c c} \hline \hline Model & $\Delta M_{21}(3P)\mbox{ [MeV]} \quad $ & $\Delta M_{10}(3P)\mbox{ [MeV]}$ & $R_{\chi}(3P)$\\ \hline Cornell-I & $17.7$ & $24.0$ & $0.734$ \\ Cornell-II & $15.4$ & $21.8$ & $0.710$ \\ Screened Cornell-I & $10.8$ & $15.8$ & $0.683$ \\ Screened Cornell-II & $12.9$ & $17.3$ & $0.747$ \\ Richardson & $17.1$ & $22.1$ & $0.773$ \\ p-QCD \cite{Radford:2012ue} & $14.8$ & $22.1$ & $0.670$ \\ \hline\hline \end{tabular} \end{table} \section{Summary and Conclusions}\label{conclusions} Recently, the $\chi_b(3P)$ state was observed and its barycenter was determined, while there is still no available data regarding its mass splittings. Here we have used four well known potential models to predict the so far unresolved splitting in the $\chi_b(3P)$ states. The models were described in Section \ref{models} and are called here the \emph{Cornell}, \emph{Screened Cornell}, \emph{Richardson} and \emph{p-QCD} models, respectively. The first three models were originally adjusted to reproduce the charmonium spectrum, and are required to work for bottomonium as well. In contrast, the latter model was fitted using bottomonium states only. As a first attempt to test the reliability of these models, we use them to reproduce the known barycenter masses of the $\chi_b(1P)$, $\chi_b(2P)$ and $\chi_b(3P)$ states. The Cornell model, which is the simplest of all of them, shows the largest deviations in its results for the barycenter masses. The Screened Cornell model, which is similar to the previous one, but with the additional feature of saturation at long distances, shows a clear improvement in its results for the barycenters. The other two models, Richardson and p-QCD, give similar values. Up to here, it seems that the latter three models perform similar in terms of reliability. In any case, one should notice that the discrepancies in these model results are in general larger than their current experimental uncertainties. This is an indication that there is still need for model improvement. Now, concerning the splittings, we recall that the barycenters of the $\chi_b$ states are determined by the spin-independent part of the potential, while the splittings are determined by the spin-dependent part. The latter, which is a relativistic correction, should be sub-dominant for heavy quarkonia like the $\chi_b$. The spin-dependent part, however, is not unambiguously determined as an expansion of the potential, as it also depends on the Lorentz character (vector vs. scalar) of the effective interaction. In the models we study here, we have used a common phenomenological treatment of this feature, which is a separation of the potential into a vector and a scalar exchange, where it is usually assumed that the vector part should dominate the short distance regime, while the scalar part should represent the confining regime at longer distances. Accordingly, in the cases of the Cornell and the Screened Cornell models, we considered two variants for the scalar vs. vector separation. Our variant I considers the commonly assumed separation where the vector part corresponds to the short distance Coulombic term and the scalar part corresponds to the long distance linear (or linear-saturated) term. Our variant II uses an optimal scalar-vector separation that minimizes the error in reproducing the known splittings in the $\chi_b(1P_J)$ and $\chi_b(2P_J)$. Interesting enough, we found that this best fitted separation is very close to the usually assumed separation in variant I, in which the short distance part is purely vector exchange and the confining long distance part is purely scalar. This result also coincides with a similar analysis done in the p-QCD model by its authors, where they found that their best fit corresponds to a long distance part being purely scalar. For the Richardson potential, unlike the previous models, there is no obvious separation into scalar and vector parts, so we introduced a purely phenomenological function with one parameter to carry out a short distance vs. long distance separation. This parameter, also fitted with the $\chi_b(1P)$ and $\chi_b(2P)$ states, is a length scale that separates the two regions; the short distance part is then assumed to be mediated by vector exchange and the long distance part by scalar exchange. Of all the models studied, those that give more consistent results for the splittings are the p-QCD model and, to some degree, the Richardson model. One could have expected the p-QCD model to give the best results because of its more detailed short-distance treatment from perturbative QCD, and because it contains a larger number of parameters, all fitted to the bottomonium spectrum. Concerning the comparison between the Cornell and Screened Cornell models, we would have expected an improvement in the reproduction of the splittings in the latter, as was the case for the reproduction of the barycenters. However it was not the case: we found the Screened Cornell model to give results that are no better than the simpler Cornell model. On the other hand, when comparing the variants I and II of these models, we had expected better results in the variants II, because they were fitted to do so. This was indeed the case in the results for the $\Delta M_{ij}$, but not so for the ratios $R_\chi$. The one consistent pattern in all results for the splittings is that all the models give ratios $R_\chi$ in $\chi_b(1P)$ and $\chi_b(2P)$ that are larger than the corresponding experimental values. Concerning the reliability of the models, this tendency may be an indication that the composition of the potential in a scalar and a vector part may not be enough, and exchanges of further spins and parities may be present. From the point of view of predictions, on the other hand, we may then expect that the predictions for the $\chi_b(3P)$ splittings will follow that tendency, i.e. the model with best predictions should give the smallest value for $R_\chi$. This tendency again falls on the p-QCD model. Now, looking at the predictions for the $\Delta M_{10}$ and $\Delta M_{21}$ in $\chi_b(3P)$, one can see that the largest deviations around a central tendency are found in the Cornell-I and Screened Cornell-I models. Besides those predictions the tendency in $\Delta M_{10}$ is to be near $22\mbox{ MeV}$, and for $\Delta M_{21}$ the predictions, although more dispersed, are centered around a value of $15\mbox{ MeV}$. With these estimates and the experimental value for the barycenter, the masses of the three $\chi_b(3P)$ states ($J=0,1,2$) would be $10502\pm 10$ MeV, $10524\pm 10$ MeV and $10539\pm 10$ MeV, respectively (the model uncertainty is not included). This means that the $J=2$ state would be just $20\pm 10$ MeV below the $B^+B^-$ threshold. Finally, if we consider these values for $\Delta M_{21}$ and $\Delta M_{10}$ as the best predictions for the splitting, one then predicts a ratio $R_\chi = 0.68$ for the $\chi_b(3P)$, which is again quite close to the result of p-QCD. \section*{Acknowledgments} We are grateful to Will Brooks, Ryan White and Franz Sch\"{o}berl for useful discussions. This work was supported in part by Conicyt, Chile grant \emph{Institute for Advanced Studies in Science and Technology} ACT-119.

BkiUbQk5i7PA9Fz9mbgJ

\section{Lemma \ref{lemx} and its Proof} \label{sec:app_a} Here we state that we can condition on high-probability events. \begin{lem} \label{lemx} Let $p \in (0,1)$, and $X \sim Bernoulli (p)$ be defined on a probability space $(\Omega, \mathcal{F}, P)$. Consider $B_1, B_2, \cdots $ be a sequence of events defined on the same probability space such that $P(B_n) \rightarrow 1$ as $n$ goes to infinity. Also, let $\textbf{Y}$ be a random vector (matrix) in the same probability space, then: \[I(X; \textbf{Y}) \rightarrow 0\ \ \text{iff}\ \ I(X; \textbf{Y} {|} B_n) \rightarrow 0. \] \end{lem} \begin{proof} First, we prove that as n becomes large, \begin{align}\label{eq:H1} H(X {|} B_n)- H(X) \rightarrow 0. \end{align} Note that as $n$ goes to infinity, \begin{align} \nonumber P\left(X=1\right) &=P\left(X=1 \bigg{|} B_n\right) P\left(B_n\right) + P\left(X=1 \bigg{|} \overline{B_n}\right) P\left(\overline{B_n}\right)\\ \nonumber &=P\left(X=1 \bigg{|} B_n\right),\ \ \end{align} thus, $\left(X \bigg{|} B_n\right) \xrightarrow{d} X$, and as $n$ goes to infinity, \[H\left(X {|} B_n\right)- H(X) \rightarrow 0.\] Similarly, as $n$ becomes large, \[ P\left(X=1 \bigg{|} \textbf{Y}=\textbf{y}\right) \rightarrow P\left(X=1 \bigg{|} \textbf{Y}=\textbf{y}, B_n\right),\ \ \] and \begin{align}\label{eq:H2} H\left(X {|} \textbf{Y}=\textbf{y},B_n\right)- H\left(X {|} \textbf{Y}=\textbf{y}\right) \rightarrow 0. \end{align} Remembering that \begin{align}\label{eq:H3} I\left(X; \textbf{Y}\right)=H(X)-H(X {|} \textbf{Y}), \end{align} and using (\ref{eq:H1}), (\ref{eq:H2}), and (\ref{eq:H3}), we can conclude that as $n$ goes to infinity, \[I\left(X;\textbf{Y} {|} B_n\right) - I\left(X,\textbf{Y}\right) \rightarrow 0.\] As a result, for large enough $n$, \[I\left(X; \textbf{Y}\right) \rightarrow 0 \Longleftrightarrow I\left(X; \textbf{Y} {|} B_n\right) \rightarrow 0. \] \end{proof} \section{Proof of Lemma \ref{lem1}} \label{sec:app_c} Here, we provide a formal proof for Lemma \ref{lem1} which we restate as follows. The following lemma confirms that the number of elements in $J^{(n)}$ goes to infinity as $n$ becomes large. If $N^{(n)} \triangleq |J^{(n)}| $, then $N^{(n)} \rightarrow \infty$ with high probability as $n \rightarrow \infty$. More specifically, there exists $\lambda>0$ such that \[ P\left(N^{(n)} > \frac{\lambda}{2}n^{\frac{\beta}{2}}\right) \rightarrow 1. \] \begin{proof} Define the events $A$, $B$ as \[A \equiv p_1\leq P_u\leq p_1+\epsilon_n\] \[B \equiv p_1+\epsilon_n\leq Q_u\leq p_1+(1-2p_1)a_n.\] Then, for $u \in \{1, 2, \dots, n\}$ and $0\leq p_1<\frac{1}{2}$: \begin{align} \nonumber P\left(u\in J^{(n)}\right) &= P\left(A\ \cap \ B\right)\\ \nonumber &= P\left(A\right) P\left(B \big{|}A \right). \ \ \end{align} So, given $p_1 \in (0,1)$ and the assumption $0<\delta_1<f_p<\delta_2$, for $n$ large enough, we have \[ P(A) = \int_{p_1}^{ p_1+\epsilon_n}f_P(p) dp, \] so, we can conclude that \[ \epsilon_n\delta_1<P(A) <\epsilon_n\delta_2. \] We can find a $\delta$ such that $\delta_1<\delta<\delta_2$ and \begin{equation}\label{eq:5} P( A) = \epsilon_n\delta. \end{equation} We know \[Q_u\bigg{|}P_u=p_u \sim Uniform \left[p_u,p_u+(1-2p_u)a_n\right],\] so, according to Figure \ref{fig:piqi_b}, for $p_1\leq p_u\leq p_1+\epsilon_n$, \begin{align} \nonumber P\left(B | P_u=p_u\right ) &= \frac{p_1+(1-2p_1)a_n-p_1-\epsilon_n}{p_u+(1-2p_u)a_n-p_u} \\ \nonumber &= \frac{(1-2p_1)a_n-\epsilon_n}{(1-2p_u)a_n}\\ \nonumber &\geq \frac{(1-2p_1)a_n-\epsilon_n}{(1-2p_1)a_n} \\ \nonumber &= 1- \frac{\epsilon_n}{(1-2p_1)a_n}, \ \ \end{align} which implies \begin{align} P\left(B | A\right ) \geq 1- \frac{\epsilon_n}{(1-2p_1)a_n}. \label{eq:6} \end{align} Using (\ref{eq:5}) and (\ref{eq:6}), we can conclude \[P\left(u\in J^{(n)}\right)\geq \epsilon_n\delta \left(1- \frac{\epsilon_n}{(1-2p_1)a_n}\right).\] Then, we can say that $N^{(n)}$ has a binomial distribution with expected value of $N^{(n)}$ greater than $n\epsilon_n\delta \left(1- \frac{\epsilon_n}{(1-2p_1)a_n}\right)$, and by substituting $\epsilon_n$ and $a_n$, for any $c'>0$, we get \[E\left[N^{(n)}\right] \geq \delta\left(n^{\frac{\beta}{2}}- \frac{1}{{c'(1-2p_1)}}\right) \geq \lambda n^{\frac{\beta}{2}}.\] Now by using Chernoff bound, we have \[P\left(N^{(n)} \leq (1- \theta) E\left[N^{(n)}\right]\right) \leq e^{-\frac{\theta^2}{2}E\left[N^{(n)}\right]},\] so, if we assume $\theta=\frac{1}{2}$, we can conclude for large enough $n$, \begin{align} \nonumber P\left(N^{(n)} \leq \frac{\lambda}{2}n^{\frac{\beta}{2}}\right) &\leq P\left(N^{(n)} \leq \frac{E\left[N^{(n)}\right]}{2}\right)\\ \nonumber &\leq e^{-\frac{E[N^{(n)}]}{8}}\\ \nonumber &\leq e^{-\frac{\lambda n^{\frac{\beta}{2}}}{8}} \rightarrow 0. \end{align} As a result, $N^{(n)} \rightarrow \infty$ with high probability for large enough $n.$ \end{proof} \section{Proof of Lemma \ref{lemOnePointFive}} \label{sec:app_b} Here we provide a formal proof for Lemma \ref{lemOnePointFive} which we restate as follows. Let $N$ be a positive integer, and let $a_1, a_2, \cdots, a_N$ and $b_1, b_2, \cdots, b_N$ be real numbers such that $a_u \leq b_u$ for all $u$. Assume that $X_1, X_2, \cdots, X_N$ are $N$ independent random variables such that \[X_u \sim Uniform[a_u,b_u]. \] Let also $\gamma_1, \gamma_2, \cdots, \gamma_N$ be real numbers such that \[ \gamma_j \in \bigcap_{u=1}^{N} [a_u, b_u] \ \ \textrm{for all }j \in \{1,2,\cdots,N\}. \] Suppose that we know the event $E$ has occurred, meaning that the observed values of $X_u$'s is equal to the set of $\gamma_j$'s (but with unknown ordering), i.e., \[E \ \ \equiv \ \ \{X_1,X_2,\cdots,X_N\}= \{ \gamma_1, \gamma_2, \cdots, \gamma_N \}, \] then \[P\left(X_1=\gamma_j |E\right)=\frac{1}{N}. \] \begin{proof} Define sets $\mathfrak{P}$ and $\mathfrak{P}_j$ as follows: \[\mathfrak{P}= \textrm{The set of all permutations $\Pi$ on }\{1,2,\cdots,N\}. \] \[\mathfrak{P}_j= \textrm{The set of all permutations $\Pi$ on }\{1,2,\cdots,N\} \textrm{ such that } \Pi(1)=j. \] We have $|\mathfrak{P}|=N!$ and $|\mathfrak{P}|=(N-1)!$. Then \begin{align*} P(X_1=\alpha_j |E)&=\frac{\sum_{\pi \in \mathfrak{P}_j} f_{X_1,X_2, \cdots,X_N} (\gamma_{\pi(1)}, \gamma_{\pi(2)}, \cdots, \gamma_{\pi(N)})} {\sum_{\pi \in \mathfrak{P}} f_{X_1,X_2, \cdots,X_N} (\gamma_{\pi(1)}, \gamma_{\pi(2)}, \cdots, \gamma_{\pi(N)})}\\ &=\frac{(N-1)! \prod\limits_{u=1}^{N} \frac{1}{b_u-a_u}}{N! \prod\limits_{u=1}^{N} \frac{1}{b_u-a_u}}\\ &=\frac{1}{N}. \end{align*} \end{proof} \section{Completion of the Proof of Lemma \ref{lem4}}\label{sec:app_d} Let $p_1 \in (0,1)$, and let $N^{(n)}$ be a random variable as above, i.e., $N^{(n)} \rightarrow \infty$ as $n \rightarrow \infty$. Consider the sequence of independent random variables $Y_u \sim Bernoulli (p_u)$ for $u=1, 2, \cdots, N^{(n)}$ such that \begin{enumerate} \item For all $n$ and all $u \in \left\{1, 2, \cdots, N^{(n)}\right\}$, $\abs*{p_u-p_1} \leq \zeta_n.$ \item $\lim\limits_{n\to\infty} \zeta_n =0.$ \end{enumerate} Define \[\overline{Y}\triangleq \frac{1}{N^{(n)}}\sum_{u=1}^{N^{(n)}}Y_u,\] then $\overline{Y} \xrightarrow{d} p_1.$ \begin{proof} Note \begin{align} \nonumber E[\overline{Y}] &= \frac{1}{N^{(n)}}\sum_{u=1}^{N^{(n)}}p_u\\ \nonumber &\leq \frac{1}{N^{(n)}} \sum_{u=1}^{N^{(n)}}\left(p_1+\zeta_n\right)\\ \nonumber &= \frac{1}{N^{(n)}}\cdot N^{(n)}(p_1+\zeta_n) \\ \nonumber &= p_1+\zeta_n. \ \ \end{align} Similarly we can prove $E\left[\overline{Y}\right]\geq p_1-\zeta_n$. Since as $n$ becomes large, $\zeta_n\rightarrow 0$ and $p_1 \in (0,1)$, we can conclude \begin{align}\label{eq:H4} \lim\limits_{n\to\infty} E\left[\overline{Y}\right]=p_1. \end{align} Also, \begin{align} \nonumber Var\left(\overline{Y}\right) &= \frac{1}{\left(N^{(n)}\right)^2}\sum_{u=1}^{N^{(n)}}p_u\left(1-p_u\right) \\ \nonumber &\leq \frac{1}{(N^{(n)})^2} \sum_{u=1}^{N^{(n)}}\left(p_1+\zeta_n\right)\left(1-p_1+\zeta_n\right)\\ \nonumber &= \frac{1}{(N^{(n)})^2}\cdot N^{(n)}\left(p_1+\zeta_n\right)\left(1-p_1+\zeta_n\right) \\ \nonumber &= \frac{1}{N^{(n)}} \left(p_1+\zeta_n\right)\left(1-p_1+\zeta_n\right). \ \ \end{align} Thus, \begin{align}\label{eq:H5} \lim_{n\to\infty} Var\left(\overline{Y}\right)=0. \end{align} By using (\ref{eq:H4}), (\ref{eq:H5}), and Chebyshev's inequality, we can conclude \[\overline{Y}\xrightarrow{d} p_1.\] \end{proof} \section{Discussion} \label{discussion} \subsection{Markov Chain Model} \label{markov} As opposed to the i.i.d.\ case, we see from Section \ref{sec:perfect-MC} that if we do not limit $m=m(n)$, the assumed obfuscation method will not be sufficient to achieve perfect privacy. There are a few natural questions here. First, for a given noise level, what would be the maximum $m(n)$ that could guarantee perfect privacy in this model? The more interesting question is, how can we possibly modify the obfuscation technique to make it more suitable for the Markov chain model? A natural solution seems to be re-generating the obfuscation random variables $R_u$ periodically. This will keep the adversary from easily estimating them by observing a long sequence of data at a small increase in complexity. In fact, this will make the obfuscation much more \emph{robust} to modeling uncertainties and errors. It is worth noting, however, that this change would not affect the other results in the paper. That is, even if the obfuscation random variables are re-generated frequently, it is relatively easy to check that all the previous theorems in the paper remain valid. However, the increase in robustness to modeling errors will definitely be a significant advantage. Thus, the question is how often should the random variable $R_u$ be re-generated to strike a good balance between complexity and privacy? These are all interesting questions for future research. \subsection{Obfuscating the Samples of Users' Data Using Continuous Noise} Here we argue that for the setting of this paper, continuous noise such as that drawn from a Gaussian distribution is not a good option to obfuscate the sample of users' data drawn from a finite alphabet when we want to achieve perfect privacy. For a better understanding, let's consider a simple example. \begin{example} Consider the scenario where the users' datasets are governed by an i.i.d.\ model and the number of possible values for each sample of the users' data ($r$) is equal to 2 (two-states model). Note that the data sequence for user $u$ is a Bernoulli random variable with parameter $p_u$. Assume that the actual sample of the data of user $u$ at time $k$ ($X_u(k)$) is obfuscated using noise drawn from a Gaussian distribution ($S_u(k)$), and $Z_u(k)$ is the obfuscated version of $X_u(k)$. That is, we can write \[Z_u(k)=X_u(k)+S_u(k); \ \ \ \ \ S_u(k) \sim N\left(0, R_u\right), \] where $R_u$ is chosen from some distribution. For simplicity, we can consider $R_u\sim N\left(0, a^2_n\right)$ where $a_n$ is the noise level. We also apply anonymization to $Z_u(k)$, and, as before, $Y_u(k)$ is the reported sample of the data of user $u$ at time $k$ after applying anonymization. Per Section \ref{sec:framework}, anonymization is modeled by a random permutation $\Pi(u)$ on the set of $n$ users. Now, the question is as follows: Is it possible to achieve perfect privacy independent of the number of adversary's observation ($m$) while using this continuous noise ($S_u(k)$) to obfuscate the sample of users' data? Note that the density function of the reported sample of the data of user $u$ after applying obfuscation is \begin{align} \nonumber f_{Z_u}(z)&= p_u f_{S_u(k)}(z-1)+(1-p_u) f_{S_u(k)}(z) \\ \nonumber &= p_u \frac{1}{\sqrt{2\pi}R_u}e^{-\frac{(z-1)^2}{2R_u}}+(1-p_u)\frac{1}{\sqrt{2\pi}R_u}e^{-\frac{z^2}{2R_u}}.\ \ \end{align} In this case, when the adversary's number of observations is large, the adversary can estimate the values of $P_u$ and $R_u$ for each user with an arbitrarily small error probability. As a result, the adversary can de-anonymize the data and then recover $X_u(k)$. The conclusion here is that a continuous noise distribution gives too much information to the adversary when used for obfuscation of finite alphabet data. A method to remedy this issue is to regenerate the random variables $R_u$ frequently (similar to our previous discussion for Markov chains). Understanding the optimal frequency of such a regeneration and detailed analysis in this case is an interesting future research direction. \end{example} \subsection{Relation to Differential Privacy} Differential privacy is mainly used when there is a statistical database of users' sensitive information, and the goal is to protect an individual's data while publishing aggregate information about the database \cite{ lee2012differential, bordenabe2014optimal, chatzikokolakis2015geo, nguyen2013differential, machanavajjhala2008privacy, kousha2}. The goal of differential privacy is publishing aggregate queries with low sensitivity, which means the effect of changes in a single individual on the outcome of the aggregated information is negligible. In \cite{geo2013} three different approaches for differential privacy are presented. The one that best matches our setting is stated as \begin{align*} \frac{P(X_1(k)=x_1|\textbf{Y})}{P(X_1(k)=x_2|\textbf{Y})} \leq e^{\epsilon_r}\frac{P(X_1(k)=x_1)}{P(X_1(k)=x_2)}, \end{align*} where $\textbf{Y}$ is the set of reported datasets. It means $\textbf{Y}$ has a limited effect on the probabilities assigned by the attacker. In differential privacy, user $1$ has strongest differential privacy when $\epsilon_r=0$. In Lemma \ref{lem4}, we proved that if user $1$ has perfect privacy, this implies that asymptotically (for large enough $n$) \begin{align}\label{eq1} P\left(X_1(k)=x_1 \big{|} \textbf{Y}\right)\rightarrow P\left(X_1(k)=x_1\right). \end{align} \begin{align}\label{eq2} P\left(X_1(k)=x_2 \big{|} \textbf{Y}\right)\rightarrow P\left(X_1(k)=x_2\right). \end{align} As a result, by using (\ref{eq1}) and (\ref{eq2}), we can conclude that if we satisfy the perfect privacy condition given in this paper, we also satisfy differential privacy with $\epsilon_r=0$, i.e., the strongest case of differential privacy. \section{Conclusions} In this paper, we have considered both obfuscation and anonymization techniques to achieve privacy. The privacy level of the users depends on both $m(n)$ (number of observations per user by the adversary for a fixed anonymization mapping) and $a_n$ (noise level). That is, larger $m(n)$ and smaller $a_n$ indicate weaker privacy. We characterized the limits of privacy in the entire $m(n)-a_n$ plane for the i.i.d.\ case; that is, we obtained the exact values of the thresholds for $m(n)$ and $a_n$ required for privacy to be maintained. We showed that if $m(n)$ is fewer than $O\left(n^{\frac{2}{r-1}}\right)$, or $a_n$ is larger than $\Omega\left(n^{-\frac{1}{r-1}}\right)$, users have perfect privacy. On the other hand, if neither of these two conditions is satisfied, users have no privacy. For the case where the users' patterns are modeled by Markov chains, we obtained a no-privacy region in the $m(n)-a_n$ plane. Future research in this area needs to characterize the exact privacy/no-privacy regions when user data sequences obey Markov models. It is also important to consider different ways to obfuscate users' data sets and study the utility-privacy trade-offs for different types of obfuscation techniques. \section{Framework} \label{sec:framework} In this paper, we adopt a similar framework to that employed in~\cite{tifs2016,ciss2017}. The general set up is provided here, and the refinement to the precise models for this paper will be presented in the following sections. We assume a system with $n$ users with $X_u(k)$ denoting a sample of the data of user $u$ at time $k$, which we would like to protect from an interested adversary $\mathcal{A}$. We consider a strong adversary $\mathcal{A}$ that has complete statistical knowledge of the users' data patterns based on the previous observations or other resources. In order to secure data privacy of users, both obfuscation and anonymization techniques are used as shown in Figure \ref{fig:xyz}. In Figure \ref{fig:xyz}, $Z_u(k)$ shows the (reported) sample of the data of user $u$ at time $k$ after applying obfuscation, and $Y_u(k)$ shows the (reported) sample of the data of user $u$ at time $k$ after applying anonymization. The adversary observes only $Y_u(k)$, $k=1,2,\cdots, m(n)$, where $m(n)$ is the number of observations of each user before the identities are permuted. The adversary then tries to estimate $X_u(k)$ by using those observations. \begin{figure}[h] \centering \includegraphics[width = 0.75\linewidth]{fig/xyz} \caption{Applying obfuscation and anonymization techniques to users' data samples.} \label{fig:xyz} \end{figure} Let $\textbf{X}_u$ be the $m(n) \times 1$ vector containing the sample of the data of user $u$, and $\textbf{X}$ be the $m(n) \times n$ matrix with $u^{th}$ column equal to $\textbf{X}_u$; \[\textbf{X}_u = \begin{bmatrix} X_u(1) \\ X_u(2) \\ \vdots \\X_u(m) \end{bmatrix} , \ \ \ \textbf{X} =\left[\textbf{X}_{1}, \textbf{X}_{2}, \cdots, \textbf{X}_{n}\right]. \] \textit{Data Samples Model:} We assume there are $r \geq 2$ possible values ($0,1, \cdots, r-1$) for each sample of the users' data. In the first part of the paper (perfect privacy analysis), we assume an i.i.d.\ model as motivated in Section \ref{intro}. In the second part of the paper (converse results: no privacy region), the users' datasets are governed by irreducible and aperiodic Markov chains. At any time, $X_u(k)$ is equal to a value in $\left\{0,1, \cdots, r-1 \right\}$ according to a user-specific probability distribution. The collection of user distributions, which satisfy some mild regularity conditions discussed below, is known to the adversary $\mathcal{A}$, and he/she employs such to distinguish different users based on statistical matching of those user distributions to traces of user activity of length $m(n)$ \textit{Obfuscation Model:} The first step in obtaining privacy is to apply the obfuscation operation in order to perturb the users' data samples. In this paper, we assume that each user has only limited knowledge of the characteristics of the overall population and thus we employ a simple distributed method in which the samples of the data of each user are reported with error with a certain probability, where that probability itself is generated randomly for each user. In other words, the obfuscated data is obtained by passing the users' data through an $r$-ary symmetric channel with a random error probability. More precisely, let $\textbf{Z}_u$ be the vector which contains the obfuscated versions of user $u$'s data samples, and $\textbf{Z}$ is the collection of $\textbf{Z}_u$ for all users, \[\textbf{Z}_u = \begin{bmatrix} Z_u(1) \\ Z_u(2) \\ \vdots \\Z_u(m) \end{bmatrix} , \ \ \ \textbf{Z} =\left[ \textbf{Z}_{1}, \textbf{Z}_{2}, \cdots, \textbf{Z}_{n}\right]. \] To create a noisy version of data samples, for each user $u$, we independently generate a random variable $R_u$ that is uniformly distributed between $0$ and $a_n$, where $a_n \in (0,1]$. The value of $R_u$ gives the probability that a user's data sample is changed to a different data sample by obfuscation, and $a_n$ is termed the ``noise level'' of the system. For the case of $r=2$ where there are two states for users' data (state $0$ and state $1$), the obfuscated data is obtained by passing users' data through a Binary Symmetric Channel (BSC) with a small error probability~\cite{randomizedresponse}. Thus, we can write \[ {Z}_{u}(k)=\begin{cases} {X}_{u}(k), & \textrm{with probability } 1-R_u.\\ 1-{X}_{u}(k),& \textrm{with probability } R_u. \end{cases} \] When $r>2$, for $l \in \{0,1,\cdots, r-1\}$: \[ P({Z}_{u}(k)=l| X_{u}(k)=i) =\begin{cases} 1-R_u, & \textrm{for } l=i.\\ \frac{R_u}{r-1}, & \textrm{for } l \neq i. \end{cases} \] Note that the effect of the obfuscation is to alter the probability distribution function of each user across the $r$ possibilities in a way that is unknown to the adversary, since it is independent of all past activity of the user, and hence the obfuscation inhibits user identification. For each user, $R_u$ is generated once and is kept constant for the collection of samples of length $m(n)$, thus, providing a very low-weight obfuscation algorithm. We will discuss the extension to the case where $R_u$ is regenerated independently over time in Section \ref{sec:perfect-MC}. There, we will also provide a discussion about obfuscation using continuous noise distributions (e.g., Gaussian noise). \textit{Anonymization Model:} Anonymization is modeled by a random permutation $\Pi$ on the set of $n$ users. The user $u$ is assigned the pseudonym $\Pi(u)$. $\textbf{Y}$ is the anonymized version of $\textbf{Z}$; thus, \begin{align} \nonumber \textbf{Y} &=\textrm{Perm}\left(\textbf{Z}_{1}, \textbf{Z}_{2}, \cdots, \textbf{Z}_{n}; \Pi \right) \\ \nonumber &=\left[ \textbf{Z}_{\Pi^{-1}(1)}, \textbf{Z}_{\Pi^{-1}(2)}, \cdots, \textbf{Z}_{\Pi^{-1}(n)}\right ] \\ \nonumber &=\left[ \textbf{Y}_{1}, \textbf{Y}_{2}, \cdots, \textbf{Y}_{n}\right], \ \ \end{align} where $\textrm{Perm}( \ . \ , \Pi)$ is permutation operation with permutation function $\Pi$. As a result, $\textbf{Y}_{u} = \textbf{Z}_{\Pi^{-1}(u)}$ and $\textbf{Y}_{\Pi(u)} = \textbf{Z}_{u}$. \textit{Adversary Model:} We protect against the strongest reasonable adversary. Through past observations or some other sources, the adversary is assumed to have complete statistical knowledge of the users' patterns; in other words, he/she knows the probability distribution for each user on the set of data samples $\{0,1,\ldots,r-1\}$. As discussed in the model for the data samples, the parameters $\textbf{p}_u$, $u=1, 2, \cdots, n$ are drawn independently from a continuous density function, $f_\textbf{P}(\textbf{p}_u)$, which has support on a subset of a defined hypercube. The density $f_\textbf{P}(\textbf{p}_u)$ might be unknown to the adversary, as all that is assumed here is that such a density exists, and it will be evident from our results that knowing or not knowing $f_\textbf{P}(\textbf{p}_u)$ does not change the results asymptotically. Specifically, from the results of Section \ref{perfectsec}, we conclude that user $u$ has perfect privacy even if the adversary knows $f_\textbf{P}(\textbf{p}_u)$. In addition, in Section \ref{converse}, it is shown that the adversary can recover the true data of user $u$ at time $k$ without using the specific density function of $f_\textbf{P}(\textbf{p}_u)$, and as result, users have no privacy even if the adversary does not know $f_\textbf{P}(\textbf{p}_u)$. The adversary also knows the value of $a_n$ as it is a design parameter. However, the adversary does not know the realization of the random permutation $\Pi$ or the realizations of the random variables $R_u$, as these are independent of the past behavior of the users. It is critical to note that we assume the adversary does not have any auxiliary information or side information about users' data. In \cite{tifs2016}, perfect privacy is defined as follows: \begin{define} User $u$ has \emph{perfect privacy} at time $k$, if and only if \begin{align \nonumber \forall k\in \mathbb{N}, \ \ \ \lim\limits_{n\rightarrow \infty} I \left(X_u(k);{\textbf{Y}}\right) =0, \end{align} where $I(X;Y)$ denotes the mutual information between random variables (vectors) $X$ and $Y$. \end{define} \noindent In this paper, we also consider the situation in which there is no privacy. \begin{define} For an algorithm for the adversary that tries to estimate the actual sample of data of user $u$ at time $k$, define \[P_e(u,k)\triangleq P\left(\widetilde{X_u(k)} \neq X_u(k)\right),\] where $X_u(k)$ is the actual sample of the data of user $u$ at time $k$, $\widetilde{X_u(k)}$ is the adversary's estimated sample of the data of user $u$ at time $k$, and $P_e(u,k)$ is the error probability. Now, define ${\cal E}$ as the set of all possible adversary's estimators; then, user $u$ has \emph{no privacy} at time $k$, if and only if for large enough $n$, \[ \forall k\in \mathbb{N}, \ \ \ P^{*}_e(u,k)\triangleq \inf_{\cal E} {P\left(\widetilde{X_u(k)} \neq X_u(k)\right)} \rightarrow 0. \] Hence, a user has no privacy if there exists an algorithm for the adversary to estimate $X_u(k)$ with diminishing error probability as $n$ goes to infinity. \end{define} \textbf{\textit{Discussion:}} Both of the privacy definitions given above (perfect privacy and no privacy) are asymptotic in the number of users $(n \to \infty)$, which allows us to find clean analytical results for the fundamental limits. Moreover, in many IoT applications, such as ride sharing and dining recommendation applications, the number of users is large. \textbf{\textit{Notation:}} Note that the sample of data of user $u$ at time $k$ after applying obfuscation $\left(Z_u(k)\right)$ and the sample of data of user $u$ at time $k$ after applying anonymization $\left(Y_u(k)\right)$ depend on the number of users in the network $(n)$, while the actual sample of data of user $u$ at time $k$ is independent of the number of users $(n)$. Despite the dependency in the former cases, we omit this subscript $(n)$ on $\left(Z_u^{(n)}(k), Y_u^{(n)}(k) \right)$ to avoid confusion and make the notation consistent. \textbf{\textit{Notation:}} Throughout the paper, $X_n \xrightarrow{d} X$ denotes convergence in distribution. Also, We use $P\left(X=x\bigg{|} Y=y\right)$ for the conditional probability of $X=x$ given $Y=y$. When we write $P\left(X=x \bigg{|}Y\right)$, we are referring to a random variable that is defined as a function of $Y$. \section{Perfect Privacy Analysis: I.I.D.\ Case} \label{perfectsec} \subsection{Two-States Model} We first consider the two-states case $(r=2)$ which captures the salient aspects of the problem. For the two-states case, the sample of the data of user $u$ at any time is a Bernoulli random variable with parameter $p_u$, which is the probability of user $u$ having data sample $1$. Thus, \[X_u(k) \sim Bernoulli \left(p_u\right).\] Per Section \ref{sec:framework}, the parameters $p_u$, $u=1, 2, \cdots, n$ are drawn independently from a continuous density function, $f_P(p_u)$, on the $(0,1)$ interval. We assume there are $\delta_1, \delta_2>0$ such that:\footnote{The condition $\delta_1<f_P(p_u) <\delta_2$ is not actually necessary for the results and can be relaxed; however, we keep it here to avoid unnecessary technicalities.} \begin{equation} \nonumber\begin{cases} \delta_1<f_P(p_u) <\delta_2, & p_u \in (0,1).\\ f_P(p_u)=0, & p _u\notin (0,1). \end{cases} \end{equation} The adversary knows the values of $p_u$, $u=1, 2, \cdots, n$ and uses this knowledge to identify users. We will use capital letters (i.e., $P_u$) when we are referring to the random variable, and use lower case (i.e., $p_u$) to refer to the realization of $P_u$. In addition, since the user data $\left(X_u(k)\right)$ are i.i.d.\ and have a Bernoulli distribution, the obfuscated data $\left(Z_u(k)\right)$ are also i.i.d.\ with a Bernoulli distribution. Specifically, \[Z_u(k) \sim Bernoulli\left(Q_u\right),\] where \begin{align} \nonumber {Q}_u &=P_u(1-R_u)+(1-P_u)R_u \\ \nonumber &= P_u+\left(1-2P_u\right)R_u,\ \ \end{align} and recall that $R_u$ is the probability that user $u$'s data sample is altered at any time. For convenience, define a vector where element $Q_u$ is the probability that an obfuscated data sample of user $u$ is equal to one, and \[\textbf{Q} =\left[{Q}_{1},{Q}_{2}, \cdots,{Q}_{n}\right].\] Thus, a vector containing the permutation of those probabilities after anonymization is given by: \begin{align} \nonumber \textbf{V} &=\textrm{Perm}\left({Q}_{1}, {Q}_{2}, \cdots, {Q}_{n}; \Pi \right) \\ \nonumber &=\left[ {Q}_{\Pi^{-1}(1)}, {Q}_{\Pi^{-1}(2)}, \cdots, {Q}_{\Pi^{-1}(n)}\right ] \\ \nonumber &=\left[{V}_{1}, {V}_{2}, \cdots, {V}_{n}\right ] ,\ \ \end{align} where ${V}_{u} = {Q}_{\Pi^{-1}(u)}$ and ${V}_{\Pi(u)} = {Q}_{u}$. As a result, for $u=1,2,..., n$, the distribution of the data symbols for the user with pseudonym $u$ is given by: \[ Y_u(k) \sim Bernoulli \left(V_u\right) \sim Bernoulli\left(Q_{\Pi^{-1}(u)}\right) .\] The following theorem states that if $a_n$ is significantly larger than $\frac{1}{n}$ in this two-states model, then all users have perfect privacy independent of the value of $m(n)$. \begin{thm}\label{two_state_thm} For the above two-states model, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined above, and \begin{itemize} \item $m=m(n)$ is arbitrary; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(1-\beta\right)}$ for any $c'>0$ and $0<\beta<1$; \end{itemize} then, user 1 has perfect privacy. That is, \begin{align \nonumber \forall k\in \mathbb{N}, \ \ \ \lim\limits_{n\rightarrow \infty} I \left(X_1(k);{\textbf{Y}}\right) =0. \end{align} \end{thm} The proof of Theorem \ref{two_state_thm} will be provided for the case $0\leq p_1<\frac{1}{2}$, as the proof for the case $\frac{1}{2}\leq p_1\leq1$ is analogous and is thus omitted. \\ \noindent \textbf{Intuition behind the Proof of Theorem \ref{two_state_thm}:} Since $m(n)$ is arbitrary, the adversary is able to estimate very accurately (in the limit, perfectly) the distribution from which each data sequence $\textbf{Y}_u$, $u= 1, 2, \cdots, n$ is drawn; that is, the adversary is able to accurately estimate the probability $V_u$, $u= 1, 2, \cdots, n$. Clearly, if there were no obfuscation for each user $u$, the adversary would then simply look for the $j$ such that $p_j$ is very close to $V_u$ and set $\widetilde{X_j(k)}=Y_u(k)$, resulting in no privacy for any user. We want to make certain that the adversary obtains no information about $X_1(k)$, the sample of data of user $1$ at time $k$. To do such, we will establish that there are a large number of users whom have a probability $p_u$ that when obfuscated could have resulted in a probability consistent with $p_1$. Consider asking whether another probability $p_2$ is sufficiently close enough to be confused with $p_1$ after obfuscation; in particular, we will look for $p_2$ such that, even if the adversary is given the obfuscated probabilities $V_{\Pi(1)}$ and $V_{\Pi(2)}$, he/she cannot associate these probabilities with $p_1$ and $p_2$. This requires that the distributions $Q_{1}$ and $Q_{2}$ of the obfuscated data of user $1$ and user $2$ have significant overlap; we explore this next. Recall that $Q_u=P_u+ (1-2P_u)R_u$, and $R_u\sim Uniform [0, a_n]$. Thus, we know $Q_u {|} P_u=p_u$ has a uniform distribution with length $(1-2p_u)a_n$. Specifically, \[Q_u\bigg{|}P_u=p_u \sim Uniform \left[p_u,p_u+(1-2p_u)a_n\right].\] Figure \ref{fig:piqi_a} shows the distribution of $Q_u$ given $P_u=p_u$. \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/piqi_a} \caption{Distribution of $Q_u$ given $P_u=p_u$.} \label{fig:piqi_a} \end{figure} Consider two cases: In the first case, the support of the distributions $Q_1\big{|} P_1=p_1$ and $Q_2\big{|} P_2=p_2$ are small relative to the difference between $p_1$ and $p_2$ (Figure \ref{fig:case1}); in this case, given the probabilities $V_{\Pi(1)}$ and $V_{\Pi(2)}$ of the anonymized data sequences, the adversary can associate those with $p_1$ and $p_2$ without error. In the second case, the support of the distributions $Q_1\big{|} P_1=p_1$ and $Q_2\big{|} P_2=p_2$ is large relative to the difference between $p_1$ and $p_2$ (Figure \ref{fig:case2}), so it is difficult for the adversary to associate the probabilities $V_{\Pi(1)}$ and $V_{\Pi(2)}$ of the anonymized data sequences with $p_1$ and $p_2$. In particular, if $V_{\Pi(1)}$ and $V_{\Pi(2)}$ fall into the overlap of the support of $Q_1$ and $Q_2$, we will show the adversary can only guess randomly how to de-anonymize the data. Thus, if the ratio of the support of the distributions to $\big{|}p_1-p_2\big{|}$ goes to infinity, the adversary's posterior probability for each user converges to $\frac{1}{2}$, thus, implying no information leakage on the user identities. More generally, if we can guarantee that there will be a large set of users with $p_u$'s very close to $p_1$ compared to the support of $Q_1\big{|} P_1=p_1$, we will be able to obtain perfect privacy as demonstrated rigorously below. \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/case1} \caption{Case 1: The support of the distributions is small relative to the difference between $p_1$ and $p_2$.} \label{fig:case1} \end{figure} \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/case2} \caption{Case 2: The support of the distributions is large relative to the difference between $p_1$ and $p_2$.} \label{fig:case2} \end{figure} Given this intuition, the formal proof proceeds as follows. Given $p_1$, we define a set $J^{(n)}$ of users whose parameter $p_u$ of their data distributions is sufficiently close to $p_1$ (Figure \ref{fig:case2}; case 2), so that it is likely that $Q_1$ and $Q_u$ cannot be readily associated with $p_1$ and $p_u$. The purpose of Lemmas \ref{lemOnePointFive}, \ref{lem2}, and \ref{lem3} is to show that, from the adversary's perspective, the users in set $J^{(n)}$ are indistinguishable. More specifically, the goal is to show that the obfuscated data corresponding to each of these users could have been generated by any other users in $J^{(n)}$ in an equally likely manner. To show this, Lemma \ref{lemOnePointFive} employs the fact that, if the observed values of $N$ uniformly distributed random variables ($N$ is size of set $J^{(n)}$) are within the intersection of their ranges, it is impossible to infer any information about the matching between the observed values and the distributions. That is, all possible $N!$ matchings are equally likely. Lemmas \ref{lem2} and \ref{lem3} leverage Lemma \ref{lemOnePointFive} to show that even if the adversary is given a set that includes all of the pseudonyms of the users in set $J^{(n)}$ (i.e., $\Pi(J^{(n)})\overset{\Delta}{=} \left\{\Pi^{-1}(u) \in J^{(n)}\right\}$) he/she still will not be able to infer any information about the matching of each specific user in set $J^{(n)}$ and his pseudonym. Then Lemma \ref{lem4} uses the above fact to show that the mutual information between the data set of user $1$ at time $k$ and the observed data sets of the adversary converges to zero for large enough $n$. \noindent \textbf{Proof of Theorem \ref{two_state_thm}:} \begin{proof} Note, per Lemma~\ref{lemx} of Appendix \ref{sec:app_a}, it is sufficient to establish the results on a sequence of sets with high probability. That is, we can condition on high-probability events. Now, define the critical set $J^{(n)}$ with size $N^{(n)}=\big{|}J^{(n)}\big{|}$ for $0\leq p_1<\frac{1}{2}$ as follows: \[J^{(n)}= \left\{ u \in \{1, 2, \dots, n\}: p_1 \leq P_u\leq p_1+\epsilon_n; p_1+\epsilon_n\leq Q_u\leq p_1+(1-2p_1)a_n\right\}, \] where $\epsilon_n \triangleq \frac{1}{n^{1-\frac{\beta}{2}}}$, $a_n= c'n^{-\left(1-\beta\right)}$ ,and $\beta$ is defined in the statement of Theorem \ref{two_state_thm}. Note for large enough $n$, if $0\leq p_1<\frac{1}{2}$, we have $0\leq p_u<\frac{1}{2}$. As a result, \[Q_u\bigg{|}P_u=p_u \sim Uniform \left(p_u,p_u+(1-2p_u)a_n\right).\] We can prove that with high probability, $1 \in J^{(n)}$ for large enough $n$, as follows. First, Note that \[Q_1\bigg{|}P_1=p_1 \sim Uniform \left(p_1,p_1+(1-2p_1)a_n\right).\] Now, according to Figure \ref{fig:piqi_c}, \begin{align} \nonumber P\left(1 \in J^{(n)} \right) &= 1- \frac{\epsilon_n}{ \left(1-2p_1 \right)a_n}\\ \nonumber &= 1- \frac{1}{ \left(1-2p_1 \right)c'n^{\frac{\beta}{2}}}, \ \ \end{align} thus, for any $c'>0$ and large enough $n$, \begin{align} \nonumber P\left(1 \in J^{(n)} \right) \to 1. \end{align} \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/piqi_b} \caption{Range of $P_u$ and $Q_u$ for elements of set $J^{(n)}$ and probability density function of $Q_u\bigg{|}P_u=p_u$.} \label{fig:piqi_b} \end{figure} \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/piqi_c} \caption{Range of $P_u$ and $Q_u$ for elements of set $J^{(n)}$ and probability density function of $Q_1\bigg{|}P_1=p_1$.} \label{fig:piqi_c} \end{figure} Now in the second step, we define the probability $W_j^{(n)}$ for any $j \in \Pi(J^{(n)})=\{\Pi(u): u \in J^{(n)} \}$ as \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}, \Pi (J^{(n)})\right).\] $W_j^{(n)}$ is the conditional probability that $\Pi(1)=j$ after perfectly observing the values of the permuted version of obfuscated probabilities ($\textbf{V}$) and set including all of the pseudonyms of the users in set $J^{(n)}$ $\left(\Pi(J^{(n)})\right)$. Since $\textbf{V}$ and $\Pi (J^{(n)})$ are random, $W_j^{(n)}$ is a random variable. However, we will prove shortly that in fact $W_j^{(n)}=\frac{1}{N^{(n)}}$, for all $j \in \Pi (J^{(n)})$. Note: Since we are looking from the adversary's point of view, the assumption is that all the values of $P_u$, $u \in \{1,2,\cdots,n\}$ are known, so all of the probabilities are conditioned on the values of $P_1=p_1, P_2=p_2, \cdots, P_n=p_n$. Thus, to be accurate, we should write \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}, \Pi (J^{(n)}), P_1, P_2, \cdots, P_n\right).\] Nevertheless, for simplicity of notation, we often omit the conditioning on $P_1, P_2, \cdots, P_n$. First, we need a lemma from elementary probability. \begin{lem} \label{lemOnePointFive} Let $N$ be a positive integer, and let $a_1, a_2, \cdots, a_N$ and $b_1, b_2, \cdots, b_N$ be real numbers such that $a_u \leq b_u$ for all $u$. Assume that $X_1, X_2, \cdots, X_N$ are independent random variables such that \[X_u \sim Uniform [a_u,b_u]. \] Let also $\gamma_1, \gamma_2, \cdots, \gamma_N$ be distinct real numbers such that \[ \gamma_j \in \bigcap_{u=1}^{N} [a_u, b_u] \ \ \textrm{for all }j \in \{1,2,..,N\}.\] Suppose that we know the event $E$ has occurred, meaning that the observed values of $X_u$'s are equal to the set of $\gamma_j$'s (but with unknown ordering), i.e., \[E \ \ \equiv \ \ \{X_1, X_2, \cdots, X_N\}= \{ \gamma_1, \gamma_2, \cdots, \gamma_N \},\] then \[P\left(X_1=\gamma_j |E\right)=\frac{1}{N}. \] \end{lem} \begin{proof} Lemma \ref{lemOnePointFive} is proved in Appendix \ref{sec:app_b}. \end{proof} Using the above lemma, we can state our desired result for $W_j^{(n)}$. \begin{lem} \label{lem2} For all $j \in \Pi (J^{(n)})$, $W_j^{(n)}=\frac{1}{N^{(n)}}.$ \end{lem} \begin{proof} We argue that the setting of this lemma is essentially equivalent to the assumptions in Lemma \ref{lemOnePointFive}. First, remember that \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}, \Pi (J^{(n)})\right).\] Note that ${Q}_u= P_u+(1-2P_u)R_u$, and since $R_u$ is uniformly distributed, ${Q}_u$ conditioned on $P_u$ is also uniformly distributed in the appropriate intervals. Moreover, since ${V}_{u} = {Q}_{\Pi^{-1}(u)}$, we conclude ${V}_{u}$ is also uniformly distributed. So, looking at the definition of $W_j^{(n)}$, we can say the following: given the values of the uniformly distributed random variables ${Q}_u$, we would like to know which one of the values in $\textbf{V}$ is the actual value of ${Q}_1={V}_{\Pi(1)}$, i.e., is $\Pi(1)=j$? This is equivalent to the setting of Lemma \ref{lemOnePointFive} as described further below. Note that since $1 \in J^{(n)}$, $\Pi(1) \in \Pi (J^{(n)})$. Therefore, when searching for the value of $\Pi(1)$, it is sufficient to look inside set $\Pi (J^{(n)})$. Therefore, instead of looking among all the values of ${V}_{j}$, it is sufficient to look at ${V}_{j}$ for $j \in \Pi (J^{(n)})$. Let's show these values by $\textbf{V}_{\Pi} =\{v_1, v_2, \cdots, v_{N^{(n)}} \}$, so, \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}_{\Pi}, \Pi (J^{(n)})\right).\] Thus, we have the following scenario: $Q_u, u \in J^{(n)}$ are independent random variables, and \[Q_u\big{|} P_u=p_u \sim Uniform [p_u, p_u+(1-2p_u)a_n]. \] Also, $v_1, v_2, \cdots, v_{N^{(n)}}$ are the observed values of $Q_u$ with unknown ordering (unknown mapping $\Pi$). We also know from the definition of set $J^{(n)}$ that \[P_u \leq p_1+\epsilon_n \leq Q_u,\] \[Q_u \leq p_1(1-2a_n)+an \leq P_u(1-2a_n)+a_n,\] so, we can conclude \[ v_j \in \bigcap_{u=1}^{N^{(n)}} [p_u, p_u+(1-2p_u)a_n] \ \ \textrm{for all }j \in \{1,2,..,N^{(n)}\}. \] We know the event $E$ has occurred, meaning that the observed values of $Q_u$'s are equal to set of $v_j$'s (but with unknown ordering), i.e., \[E \ \ \equiv \ \ \{Q_u, u \in J^{(n)}\}= \{ v_1, v_2, \cdots, v_{N^{(n)}} \}. \] Then, according to Lemma \ref{lemOnePointFive}, \[P\left(Q_1=v_j |E, P_1, P_2, \cdots, P_n \right)=\frac{1}{N^{(n)}}. \] Note that there is a subtle difference between this lemma and Lemma \ref{lemOnePointFive}. Here $N^{(n)}$ is a random variable while $N$ is a fixed number in Lemma \ref{lemOnePointFive}. Nevertheless, since the assertion holds for every fixed $N$, it also holds for the case where $N$ is a random variable. Now, note that \begin{align*} P\left(Q_1=v_j |E, P_1, P_2, \cdots, P_n \right) &= P\left(\Pi(1)=j \bigg{|} E, P_1, P_2, \cdots, P_n \right)\\ &=P\left(\Pi(1)=j \bigg{|} \textbf{V}_{\Pi}, \Pi (J^{(n)}), P_1, P_2, \cdots, P_n \right)\\ &=W_j^{(n)}. \end{align*} Thus, we can conclude \[W_j^{(n)}=\frac{1}{N^{(n)}}.\] \end{proof} In the third step, we define $\widetilde{W_j^{(n)}}$ for any $j \in \Pi (J^{(n)})$ as \[\widetilde{W_j^{(n)}} = P\left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right).\] $\widetilde{W_j^{(n)}}$ is the conditional probability that $\Pi(1)=j$ after observing the values of the anonymized version of the obfuscated samples of the users' data ($\textbf{Y}$) and the aggregate set including all the pseudonyms of the users in set $J^{(n)}$ (i.e., $\Pi(J^{(n)})\overset{\Delta}{=} \left\{\Pi^{-1}(j) \in J^{(n)}\right\}$). Since $\textbf{Y}$ and $\Pi (J^{(n)})$ are random, $\widetilde{W_j^{(n)}} $ is a random variable. Now, in the following lemma, we will prove $\widetilde{W_j^{(n)}} =\frac{1}{N^{(n)}}$, for all $j \in \Pi (J^{(n)})$ by using Lemma \ref{lem3}. Note in the following lemma, we want to show that even if the adversary is given a set including all of the pseudonyms of the users in set $J^{(n)}$, he/she cannot match each specific user in set $J^{(n)}$ and his pseudonym. \begin{lem} \label{lem3} For all $j \in \Pi (J^{(n)})$, $\widetilde{W_j^{(n)}}=\frac{1}{N^{(n)}}.$ \end{lem} \begin{proof} First, note that \begin{align*} \widetilde{W_j^{(n)}} = \sum_{\text{for all v}} P\left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right), \textbf{V}=\textbf{v}\right) P\left(\textbf{V}=\textbf{v} \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right)\right). \end{align*} Also, we note that given $\textbf{V}$, $\Pi(J^{(n)})$, and $\textbf{Y}$ are independent. Intuitively, this is because when observing $\textbf{Y}$, any information regarding $\Pi(J^{(n)})$ is leaked through estimating $\textbf{V}$. This can be rigorously proved similar to the proof of Lemma 1 in \cite{tifs2016}. We can state this fact as \[ P\left(Y_u(k)\ \bigg{ | } \ {V}_u=v_u, \Pi(J^{(n)}) \right) = P\left(Y_u(k)\ \bigg{ | } \ {V}_u=v_u\right)=v_u. \] The right and left hand side are given by $Bernoulli (v_u)$ distributions. As a result, \[ \widetilde{W_j^{(n)}} = \sum_{\text{for all \textbf{v}}} P\left(\Pi(1)=j \bigg{|} \Pi (J^{(n)}), \textbf{V}=\textbf{v}\right) P\left(\textbf{V}=\textbf{v} \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right)\right). \] Note $W_j^{(n)}=P\left(\Pi(1)=j \bigg{|} \Pi (J^{(n)}), \textbf{V}\right)$, so \begin{align} \nonumber \widetilde{W_j^{(n)}} &= \sum_{\text{for all \textbf{v}}}W_j^{(n)} P\left(\textbf{V}=\textbf{v} \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right) \right) \\ \nonumber &= \frac{1}{N^{(n)}}\sum_{\text{for all \textbf{v}}} P\left(\textbf{V}=\textbf{v}\bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right)\right) \\ \nonumber &= \frac {1}{N^{(n)}}.\ \ \end{align} \end{proof} To show that no information is leaked, we need to show that the size of set $J^{(n)}$ goes to infinity. This is established in Lemma \ref{lem1}. \begin{lem} \label{lem1} If $N^{(n)} \triangleq |J^{(n)}| $, then $N^{(n)} \rightarrow \infty$ with high probability as $n \rightarrow \infty$. More specifically, there exists $\lambda>0$ such that \[ P\left(N^{(n)} > \frac{\lambda}{2}n^{\frac{\beta}{2}}\right) \rightarrow 1. \] \end{lem} \begin{proof} Lemma \ref{lem1} is proved in Appendix \ref{sec:app_c}. \end{proof} In the final step, we define $\widehat{W_j^{(n)}}$ for any $j \in \Pi (J^{(n)})$ as \[\widehat{W_j^{(n)}}=P\left(X_1(k)=1 \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right).\] $\widehat{W_j^{(n)}}$ is the conditional probability that $X_1(k)=1$ after observing the values of the anonymized version of the obfuscated samples of the users' data ($\textbf{Y}$) and the aggregate set including all of the pseudonyms of the users in set $J^{(n)}$ ($\Pi (J^{(n)})$). $\widehat{W_j^{(n)}} $ is a random variable because $\textbf{Y}$ and $\Pi (J^{(n)})$ are random. Now, in the following lemma, we will prove $\widehat{W_j^{(n)}}$ converges in distribution to $p_1$. Note that this is the probability from the adversary's point of view. That is, given that the adversary has observed $\textbf{Y}$ as well as the extra information $ \Pi (J^{(n)})$, what can he/she infer about $X_1(k)$? \begin{lem} \label{lem4} For all $j \in \Pi (J^{(n)})$, $\widehat{W_j^{(n)}} \xrightarrow{d} p_1.$ \end{lem} \begin{proof} We know \begin{align*} \widehat{W_j^{(n)}}= \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right) P\left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right), \end{align*} and according to the definition $\widetilde{W_j^{(n)}}=P \left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right)$, we have \begin{align} \nonumber \widehat{W_j^{(n)}} &= \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right) \widetilde{W_j^{(n)}}\\ \nonumber &= \frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right). \end{align} We now claim that \[ P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)=p_1+o(1). \] The reasoning goes as follows. Given $\Pi(1)=j$ and knowing $\textbf{Y}$, we know that \[ Y_{\Pi(1)}(k)={Z}_{1}(k)=\begin{cases} {X}_{1}(k), & \textrm{with probability } 1-R_1.\\ 1-{X}_{1}(k), & \textrm{with probability } R_1. \end{cases} \] Thus, given $Y_{j}(k)=1$, Bayes' rule yields: \begin{align*} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)&= \left(1- R_1 \right) \frac{P(X_1(k)=1)}{P(Y_{\Pi(1)}(k)=1)}\\ &=\left(1-R_1 \right) \frac{p_1}{p_1 (1- R_1)+(1-p_1)R_1}\\ &=1-o(1), \end{align*} and similarly, given $Y_{j}(k)=0$, \begin{align*} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)&= R_1 \frac{P(X_1(k)=1)}{P(Y_{\Pi(1)}(k)=0)}\\ &=R_1 \frac{p_1}{p_1 (1- R_1)+(1-p_1)R_1}\\ &=o(1). \end{align*} Note that by the independence assumption, the above probabilities do not depend on the other values of $Y_{u}(k)$ (as we are conditioning on $\Pi(1)=j$ ). Thus, we can write \begin{align} \nonumber \widehat{W_j^{(n)}} &= \frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)\\ \nonumber &=\frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)}), Y_{j}(k)=1} (1-o(1)) + \frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)}), Y_{j}(k)=0} o(1). \end{align} First, note that since $\abs*{\left\{j \in \Pi(J^{(n)}), Y_{j}(k)=0\right\}} \leq N^{(n)}$, the second term above converges to zero, thus, \begin{align} \nonumber \widehat{W_j^{(n)}} \rightarrow \frac{\abs*{\left\{ j \in \Pi(J^{(n)}), Y_{\Pi(1)}(k)=1\right\} }}{N^{(n)}}. \end{align} Since for all $j \in \Pi(J^{(n)})$, $ Y_{j}(k) \sim Bernoulli \left(p_1+o(1)\right)$, by a simple application of Chebyshev's inequality, we can conclude $\widehat{W_j^{(n)}}\rightarrow p_1$. Appendix \ref{sec:app_d} provides the detail. \end{proof} As a result, \begin{align*} X_1(k) {|} \textbf{Y}, \Pi (J^{(n)})\rightarrow \textit{Bernoulli} (p_1), \end{align*} thus, \[H\left(X_1(k) \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right)\rightarrow H\left(X_1(k)\right).\] Since conditioning reduces entropy, \[H\left(X_1(k) \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right)\leq H\left(X_1(k) \bigg{|} \textbf{Y}\right),\] and as a result, \[\lim_{n \rightarrow \infty} H\left(X_1(k)\right)-H\left(X_1(k) \bigg{|} \textbf{Y}\right) \leq 0,\] and \[\lim_{n \rightarrow \infty} I\left(X_1(k);\textbf{Y}\right)\leq 0.\] By knowing that $I\left(X_1(k);\textbf{Y}\right)$ cannot take any negative value, we can conclude that \[I\left(X_1(k);\textbf{Y}\right)\rightarrow 0.\] \end{proof} \subsection{Extension to $r$-States} Now, assume users' data samples can have $r$ possibilities $\left(0, 1, \cdots, r-1\right)$, and $p_u(i)$ shows the probability of user $u$ having data sample $i$. We define the vector $\textbf{p}_u$ and the matrix $\textbf{p}$ as \[\textbf{p}_u= \begin{bmatrix} p_u(1) \\ p_u(2) \\ \vdots \\p_u(r-1) \end{bmatrix} , \ \ \ \textbf{p} =\left[ \textbf{p}_{1}, \textbf{p}_{2}, \cdots, \textbf{p}_{n}\right]. \] We assume $p_u(i)$'s are drawn independently from some continuous density function, $f_\textbf{P}(\textbf{p}_u)$, which has support on a subset of the $(0,1)^{r-1}$ hypercube (Note that the $p_u(i)$'s sum to one, so one of them can be considered as the dependent value and the dimension is $r-1$). In particular, define the range of the distribution as \begin{align} \nonumber \mathcal{R}_{\textbf{p}} &= \{ (x_1,x_2, \cdots, x_{r-1}) \in (0,1)^{r-1}: x_i > 0 , x_1+x_2+\cdots+x_{r-1} < 1,\ i=1, 2, \cdots, r-1\}. \end{align} Figure~\ref{fig:rp} shows the range $\mathcal{R}_{\textbf{p}}$ for the case where $r=3$. \begin{figure}[h] \centering \includegraphics[width = 0.5\linewidth]{fig/rp} \caption{$\mathcal{R}_{\textbf{p}}$ for case $r=3$.} \label{fig:rp} \end{figure} Then, we assume there are $\delta_1, \delta_2>0$ such that: \begin{equation} \begin{cases} \nonumber \delta_1<f_{\textbf{P}}(\mathbf{p}_u) <\delta_2, & \textbf{p}_u \in \mathcal{R}_{\textbf{p}}.\\ f_{\textbf{P}}(\mathbf{p}_u)=0, & \textbf{p}_u \notin \mathcal{R}_{\textbf{p}}. \end{cases} \end{equation} The obfuscation is similar to the two-states case. Specifically, for $l \in \{0,1,\cdots, r-1\}$, we can write \[ P({Z}_{u}(k)=l| X_{u}(k)=i) =\begin{cases} 1-R_u, & \textrm{for } l=i.\\ \frac{R_u}{r-1}, & \textrm{for } l \neq i. \end{cases} \] \begin{thm}\label{r_state_thm} For the above $r$-states model, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined previously, and \begin{itemize} \item $m=m(n)$ is arbitrary; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(\frac{1}{r-1}-\beta\right)}$ for any $c'>0$ and $0<\beta<\frac{1}{r-1}$; \end{itemize} then, user 1 has perfect privacy. That is, \begin{align \nonumber \forall k\in \mathbb{N}, \ \ \ \lim\limits_{n\rightarrow \infty} I \left(X_1(k);{\textbf{Y}}\right) =0. \end{align}\end{thm} The proof of Theorem \ref{r_state_thm} is similar to the proof of Theorem \ref{two_state_thm}. The major difference is that instead of the random variables $P_u, Q_u, V_u$, we need to consider the random vectors $\textbf{P}_u, \textbf{Q}_u, \textbf{V}_u$. Similarly, for user $u$, we define the vector $\textbf{Q}_u$ as \[\textbf{Q}_u= \begin{bmatrix} Q_u(1) \\ Q_u(2) \\ \vdots \\Q_u(r-1) \end{bmatrix}. \] In the $r$-states case, \begin{align} \nonumber {Q}_u(i) &=P_u(i)\bigg(1-R_u(i) \bigg)+\bigg(1-P_u(i)\bigg)\frac{R_u}{r-1} \\ \nonumber &= P_u+\bigg(1-r P_u\bigg)\frac{R_u}{r-1}.\ \ \end{align} We also need to define the critical set $J^{(n)}$. First, for $i=0,1, \cdots, r-1$, define set $J_i^{(n)}$ as follows. If $0\leq p_1(i)<\frac{1}{r}$, then, \begin{align*} &J_i^{(n)}= \\ &\left\{u \in \{1, 2, \dots, n\}: p_1(i) \leq P_u(i)\leq p_1(i)+\epsilon_n; p_1(i)+\epsilon_n\leq Q_u(i)\leq p_1(i)+(1-r p_1(i))\frac{a_n}{r-1}\right\}, \end{align*} where $\epsilon_n \triangleq \frac{1}{n^{\frac{1}{r-1}-\frac{\beta}{2}}}$, $a_n = c'n^{-\left(\frac{1}{r-1}-\beta\right)}$, and $\beta$ is defined in the statement of Theorem \ref{r_state_thm}. We then define the critical set $J^{(n)}$ as: \[ J^{(n)}=\bigcap_{l=0}^{r-1} J_i^{(n)}. \] We can then repeat the same arguments in the proof of Theorem \ref{two_state_thm} to complete the proof. \section{Converse Results: No Privacy Region} \label{converse} In this section, we prove that if the number of observations by the adversary is larger than its critical value and the noise level is less than its critical value, then the adversary can find an algorithm to successfully estimate users' data samples with arbitrarily small error probability. Combined with the results of the previous section, this implies that asymptotically (as $n \rightarrow \infty$), privacy can be achieved \emph{if and only if} at least one of the two techniques (obfuscation or anonymization) are used above their thresholds. This statement needs a clarification as follows: Looking at the results of \cite{tifs2016}, we notice that anonymization alone can provide perfect privacy if $m(n)$ is below its threshold. On the other hand, the threshold for obfuscation requires some anonymization: In particular, the identities of the users must be permuted once to prevent the adversary from readily identifying the users. \subsection{Two-States Model} Again, we start with the i.i.d.\ two-states model. The data sample of user $u$ at any time is a Bernoulli random variable with parameter $p_u$. As before, we assume that $p_u$'s are drawn independently from some continuous density function, $f_P(p_u)$, on the $(0,1)$ interval. Specifically, there are $\delta_1, \delta_2>0$ such that \begin{equation} \nonumber\begin{cases} \delta_1<f_P(p_u) <\delta_2, & p_u \in (0,1).\\ f_P(p_u)=0, & p _u\notin (0,1). \end{cases} \end{equation} \begin{thm}\label{two_state_thm_converse} For the above two-states mode, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined, and \begin{itemize} \item $m =cn^{2 + \alpha}$ for any $c>0$ and $\alpha>0$; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(1+\beta\right)}$ for any $c'>0$ and $\beta>\frac{\alpha}{4}$; \end{itemize} then, user $1$ has no privacy as $n$ goes to infinity. \end{thm} Since this is a converse result, we give an explicit detector at the adversary and show that it can be used by the adversary to recover the true data of user $1$. \begin{proof} The adversary first inverts the anonymization mapping $\Pi$ to obtain $Z_1(k)$, and then estimates the value of $X_1(k)$ from that. To invert the anonymization, the adversary calculates the empirical probability that each string is in state $1$ and then assigns the string with the empirical probability closest to $p_1$ to user 1. \begin{figure} \centering \includegraphics[width=.8\linewidth]{fig/converse.jpg} \centering \caption{$p_1$, sets $B^{(n)}$ and $C^{(n)}$ for case $r=2$.} \label{fig:converse} \end{figure} Formally, for $u=1, 2, \cdots, n$, the adversary computes $\overline{Y_u}$, the empirical probability of user $u$ being in state $1$, as follows: \[ \overline{Y_u}=\frac{Y_u(1)+Y_u(2)+ \cdots +Y_u(m)}{m}, \] thus, \[ \overline{Y_{\Pi(u)}}=\frac{Z_u(1)+Z_u(2)+ \cdots +Z_u(m)}{m}. \] As shown in Figure \ref{fig:converse}, define \[B^{(n)}\triangleq \left\{x \in (0,1); p_1-\Delta_n \leq x \leq p_1+\Delta_n\right\},\] where $\Delta_n = \frac{1}{n^{1+\frac{\alpha}{4}}}$ and $\alpha $ is defined in the statement of Theorem \ref{two_state_thm_converse}. We claim that for $m =cn^{2 + \alpha}$, $a_n=c'n^{-(1 + \beta)}$, and large enough $n$, \begin{enumerate} \item $P\left( \overline{Y_{\Pi(1) }}\in B^{(n)}\right) \rightarrow 1.$ \item $P\left( \bigcup\limits_{u=2}^{n} \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right) \rightarrow 0.$ \end{enumerate} As a result, the adversary can identify $\Pi(1)$ by examining $\overline{Y_u}$'s and assigning the one in $B^{(n)}$ to user $1$. Note that $\overline{Y_{\Pi(u) }} \in B^{(n)}$ is a set (event) in the underlying probability space and can be written as $\left\{\omega \in \Omega: \overline{Y_{\Pi(u) }}(\omega) \in B^{(n)}\right\}$. First, we show that as $n$ goes to infinity, \[P\left( \overline{Y_{\Pi(1) }}\in B^{(n)}\right) \rightarrow 1.\] We can write \begin{align} \nonumber P\left(\overline{Y_{\Pi(1)}} \in B^{(n)}\right) &= P\left(\frac{\sum\limits_{k=1}^{m}Z_1(k)}{m} \in B^{(n)} \right)\\ \nonumber &= P\left(p_1-\Delta_n \leq\frac{\sum\limits_{k=1}^{m}Z_1(k)}{m}\leq p_1+\Delta_n \right)\\ \nonumber &= P\left(mp_1-m\Delta_n-mQ_1\leq \sum\limits_{k=1}^{m}Z_1(k)-mQ_1\leq mp_1+m\Delta_n-mQ_1 \right).\ \ \end{align} Note that for any $u \in \{1,2,\cdots, n \}$, we have \begin{align} \nonumber |p_u-{Q}_u| &=|1-2p_u|R_u \\ \nonumber & \leq R_u \leq a_n, \end{align} so, we can conclude \begin{align} \nonumber P\left(\overline{Y_{\Pi(1)}} \in B^{(n)}\right) &= P\left(mp_1-m\Delta_n-mQ_1\leq \sum\limits_{k=1}^{m}Z_1(k)-mQ_1 \leq mp_1+m\Delta_n-mQ_1 \right)\\ \nonumber &\geq P\left(-m\Delta_n+m a_n\leq \sum\limits_{k=1}^{m}Z_1(k)-mQ_1\leq -ma_n+m\Delta_n \right)\\ \nonumber &= P\left(\abs*{\sum\limits_{k=1}^{m}Z_1(k)-mQ_1}\leq m (\Delta_n-a_n) \right).\ \ \end{align} Since $a_n \rightarrow 0$, for $p_1 \in (0,1)$ and large enough $n$, we can say $p_1+a_n < 2 p_1$. From Chernoff bound, for any $c,c',\alpha>0$ and $\beta>\frac{\alpha}{4}$, \begin{align} \nonumber P\left(\abs*{\sum\limits_{k=1}^{m}Z_1(k)-mQ_1}\leq m (\Delta_n-a_n) \right) &\geq 1-2e^{-\frac{m(\Delta_n-a_n)^2}{3Q_1}} \\ \nonumber &\geq 1-2e^{-\frac{1}{3(p_1+a_n)}cn^{2+\alpha}\left(\frac{1}{n^{1+\frac{\alpha}{4}}}- \frac{c'}{n^{1 + \beta}}\right)^2}\\ \nonumber &\geq 1-2e^{-\frac{c''}{6 p_1}n^{\frac{\alpha}{2}}}\rightarrow 1. \ \ \end{align} As a result, as $n$ becomes large, \[P\left(\overline{Y_{\Pi(1)}} \in B^{(n)}\right) \rightarrow 1.\] Now, we need to show that as $n$ goes to infinity, \[P\left( \bigcup\limits_{u=2}^{n} \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right) \rightarrow 0.\] First, we define \[ C^{(n)}=\left\{x\in (0,1); p_1-2\Delta_n \leq x \leq p_1+2\Delta_n\right\} ,\] and claim as $n$ goes to infinity, \[P\left(\bigcup\limits_{u=2}^n \left(P_u \in C^{(n)}\right) \right)\rightarrow 0. \] Note \[4\Delta_n\delta_1< P\left( P_u\in C^{(n)}\right) < 4 \Delta_n\delta_2,\] and according to the union bound, for large enough $n$, \begin{align} \nonumber P\left( \bigcup\limits_{u=2}^n \left(P_u \in C^{(n)}\right) \right) &\leq \sum\limits_{u=2}^n P\left( P_u \in C^{(n)}\right) \\ \nonumber &\leq 4n \Delta_n \delta_2\\ \nonumber &= 4n \frac{1}{n^{1+{\frac{\alpha}{4}}}} \delta_2\\ \nonumber &= 4n^{-\frac{\alpha}{4}}\delta_2 \rightarrow 0. \ \ \end{align} As a result, we can conclude that all $p_u$'s are outside of $C^{(n)}$ for $u \in \left\{2,3, \cdots, n\right\}$ with high probability. Now, we claim that given all $p_u$'s are outside of $C^{(n)}$, $P\left(\overline{Y_{\Pi (u)}} \in B^{(n)}\right)$ is small. Remember that for any $u \in \{1,2,\cdots, n \}$, we have \begin{align} \nonumber |p_u-{Q}_u| \leq a_n. \end{align} Now, noting the definitions of sets $B^{(n)}$ and $C^{(n)}$, we can write for $u \in \left\{2,3, \cdots , n\right\}$, \begin{align} \nonumber \nonumber P\left(\overline{Y_{\Pi(u)}} \in B^{(n)}\right) &\leq P\left(\abs*{\overline{Y_{\Pi(u)}}-Q_u}\geq (\Delta_n-a_n) \right)\\ \nonumber &= P\left(\abs*{\sum\limits_{k=1}^{m}Z_u(k)-mQ_u}> m(\Delta_n-a_n) \right).\ \ \end{align} According to the Chernoff bound, for any $c,c',\alpha>0$ and $\beta>\frac{\alpha}{4}$, \begin{align} \nonumber P\left(\abs*{\sum\limits_{k=1}^{m}Z_u(k)-mQ_u}> m(\Delta_n-a_n) \right) &\leq 2e^{-\frac{m(\Delta_n-a_n)^2}{3Q_1}} \\ \nonumber &\leq 2e^{-\frac{1}{3(p_1+a_n)}cn^{2+\alpha}\left(\frac{1}{n^{1+\frac{\alpha}{4}}}- \frac{c'}{n^{1 + \beta}}\right)^2}\\ \nonumber &\leq 2e^{-\frac{c''}{6 p_1}n^{\frac{\alpha}{2}}}. \ \ \end{align} Now, by using a union bound, we have \begin{align} \nonumber P\left( \bigcup\limits_{u=2}^n \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right)&\leq \sum\limits_{u=2}^{n}P\left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\\ \nonumber &\leq n\left(2e^{-\frac{c''}{6 p_1}n^{\frac{\alpha}{2}}}\right),\ \ \end{align} and thus, as $n$ goes to infinity, \[P\left( \bigcup\limits_{u=2}^n \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right) \rightarrow 0.\] So, the adversary can successfully recover $Z_1(k)$. Since $Z_{1}(k)=X_1(k)$ with probability $1-R_1=1-o(1)$, the adversary can recover $X_{1}(k)$ with vanishing error probability for large enough $n$. \end{proof} \subsection{Extension to $r$-States} Now, assume users' data samples can have $r$ possibilities $\left(0, 1, \cdots, r-1\right)$, and $p_u(i)$ shows the probability of user $u$ having data sample $i$. We define the vector $\textbf{p}_u$ and the matrix $\textbf{p}$ as \[\textbf{p}_u= \begin{bmatrix} p_u(1) \\ p_u(2) \\ \vdots \\p_u(r-1) \end{bmatrix} , \ \ \ \textbf{p} =\left[ \textbf{p}_{1}, \textbf{p}_{2}, \cdots, \textbf{p}_{n}\right]. \] We also assume $\textbf{p}_u$'s are drawn independently from some continuous density function, $f_P(\textbf{p}_u)$, which has support on a subset of the $(0,1)^{r-1}$ hypercube. In particular, define the range of distribution as \begin{align} \nonumber \mathcal{R}_{\textbf{p}} &= \left\{ (x_1, x_2, \cdots, x_{r-1}) \in (0,1)^{r-1}: x_i > 0 , x_1+ x_2+\cdots+ x_{r-1} < 1,\ \ i=1, 2,\cdots, r-1\right\}. \end{align} Then, we assume there are $\delta_1, \delta_2>0$ such that: \begin{equation} \begin{cases} \nonumber \delta_1<f_{\textbf{P}}(\mathbf{p}_u) <\delta_2, & \textbf{p}_u \in \mathcal{R}_{\textbf{p}}.\\ f_{\textbf{P}}(\mathbf{p}_u)=0, & \textbf{p}_u \notin \mathcal{R}_{\textbf{p}}. \end{cases} \end{equation} \begin{thm}\label{r_state_thm_converse} For the above $r$-states mode, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined, and \begin{itemize} \item $m =cn^{\frac{2}{r-1} + \alpha}$ for any $c>0$ and $0<\alpha<1$; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(\frac{1}{r-1}+\beta\right)}$ for any $c'>0$ and $\beta>\frac{\alpha}{4}$; \end{itemize} then, user $1$ has no privacy as $n$ goes to infinity. \end{thm} The proof of Theorem \ref{r_state_thm_converse} is similar to the proof of Theorem \ref{two_state_thm_converse}, so we just provide the general idea. We similarly define the empirical probability that the user with pseudonym $u$ has data sample $i$ $\left(\overline{{Y}_{u}}(i)\right)$ as follows: \[ \overline{Y_u}(i)=\frac{\abs {\left\{k \in \{1, 2, \cdots, m\}:Y_u(k)=i\right\}}}{m}, \] thus, \[ \overline{Y_{\Pi(u)}}(i)=\frac{\abs {\left\{k \in \{1, 2, \cdots, m\}:Y_u(k)=i\right\}}}{m}. \] The difference is that now for each $u \in \{1,2,\cdots, n \}$, $\overline{\textbf{Y}_{u}}$ is a vector of size $r-1$. In other words, \[\overline{\textbf{Y}_{u}}=\begin{bmatrix} \overline{Y_u}(1) \\ \overline{Y_u}(2) \\ \vdots \\\overline{Y_u}(r-1) \end{bmatrix}.\] \begin{figure} \centering \includegraphics[width=.5\linewidth, height=0.5 \linewidth]{fig/R.jpg} \centering \caption{$\textbf{p}_1$, sets $B'^{(n)}$ and $C'^{(n)}$ in $\mathcal{R}_\textbf{p}$ for case $r=3$.} \label{fig:rpp} \end{figure} Define sets $B'^{(n)}$ and $C'^{(n)}$ as \begin{align} \nonumber B'^{(n)}\triangleq & \left\{(x_1,x_2, \cdots ,x_{r-1}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-\Delta'_n \leq x_i \leq p_1(i)+\Delta'_n,\ i=1,2, \cdots, r-1\right\}, \end{align} \begin{align} \nonumber C'^{(n)}\triangleq &\left\{(x_1,x_2, \cdots ,x_{r-1}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-2 \Delta'_n \leq x_i \leq p_1(i)+2 \Delta'_n,\ i=1,2, \cdots ,r-1\right\}, \end{align} where $\Delta'_n = \frac{1}{n^{\frac{1}{r-1}+\frac{\alpha}{4}}}.$ Figure \ref{fig:rpp} shows $\textbf{p}_1$ and sets $B'^{(n)}$ and $C'^{(n)}$ for the case $r=3.$ We claim for $m =cn^{\frac{2}{r-1} + \alpha}$ and large enough $n$, \begin{enumerate} \item $P\left( \overline{\textbf{Y}_{\Pi(1) }}\in B'^{(n)}\right) \rightarrow 1$. \item $P\left( \bigcup\limits_{u=2}^{n} \left(\overline{\textbf{Y}_{\Pi(u)}}\in B'^{(n)}\right)\right) \rightarrow 0.$ \end{enumerate} The proof follows that for the two-states case. Thus, the adversary can de-anonymize the data and then recover $X_1(k)$ with vanishing error probability in the $r$-states model. \subsection{Markov Chain Model} \label{subsec:markov} So far, we have assumed users' data samples can have $r$ possibilities $\left(0, 1, \cdots, r-1\right)$ and users' pattern are i.i.d.\ . Here we model users' pattern using Markov chains to capture the dependency of the users' pattern over time. Again, we assume there are $r$ possibilities (the number of states in the Markov chains). Let $E$ be the set of edges. More specifically, $(i, l) \in E$ if there exists an edge from $i$ to $l$ with probability $ p(i,l)>0 $. What distinguishes different users is their transition probabilities $p_u(i,l)$ (the probability that user $u$ jumps from state $i$ to state $l$). The adversary knows the transition probabilities of all users. The model for obfuscation and anonymization is exactly the same as before. We show that the adversary will be able to estimate the data samples of the users with low error probability if $m(n)$ and $a_n$ are in the appropriate range. The key idea is that the adversary can focus on a subset of the transition probabilities that are sufficient for recovering the entire transition probability matrix. By estimating those transition probabilities from the observed data and matching with the known transition probabilities of the users, the adversary will be able to first de-anonymize the data, and then estimate the actual samples of users' data. In particular, note that for each state $i$, we must have \[\sum\limits_{l=1}^{r} p_u(i,l)=1, \ \ \textrm{ for each }u \in \{1,2,\cdots, n \}, \] so, the Markov chain of user $u$ is completely determined by a subset of size $d=|E|-r$ of transition probabilities. We define the vector $\textbf{p}_u$ and the matrix $\textbf{p}$ as \[\textbf{p}_u= \begin{bmatrix} p_u(1) \\ p_u(2) \\ \vdots \\p_u(|E|-r) \end{bmatrix} , \ \ \ \textbf{p} =\left[ \textbf{p}_{1}, \textbf{p}_{2}, \cdots, \textbf{p}_{n}\right]. \] We also consider $\textbf{p}_u$'s are drawn independently from some continuous density function, $f_P(\textbf{p}_u)$, which has support on a subset of the $(0,1)^{|E|-r}$ hypercube. Let $\mathcal{R}_{\textbf{p}} \subset \mathbb{R}^{d}$ be the range of acceptable values for $\textbf{p}_{u}$, so we have \begin{align} \nonumber \mathcal{R}_{\textbf{P}} &= \left\{ (x_1,x_2 \cdots, x_{d}) \in (0,1)^{d}: x_i > 0 , x_1+x_2+\cdots+x_{d} < 1,\ \ i=1,2,\cdots, d\right\}. \end{align} As before, we assume there are $ \delta_1, \delta_2 >0$, such that: \begin{equation} \begin{cases} \nonumber \delta_1<f_{\textbf{P}}(\textbf{p}_u) <\delta_2, & \textbf{p}_u \in \mathcal{R}_{\textbf{p}}.\\ f_{\textbf{P}}(\textbf{p}_u)=0, & \textbf{p}_u \notin \mathcal{R}_{\textbf{p}}. \end{cases} \end{equation} Using the above observations, we can establish the following theorem. \begin{thm}\label{markov_thm} For an irreducible, aperiodic Markov chain with $r$ states and $|E|$ edges as defined above, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$, and \begin{itemize} \item $m =cn^{\frac{2}{|E|-r} + \alpha}$ for any $c>0$ and $\alpha>0$; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(\frac{1}{|E|-r}+\beta \right)}$ for any $c'>0$ and $\beta>\frac{\alpha}{4}$; \end{itemize} then, the adversary can successfully identify the data of user $1$ as $n$ goes to infinity. \end{thm} The proof has a lot of similarity to the i.i.d.\ case, so we provide a sketch, mainly focusing on the differences. We argue as follows. If the total number of observations per user is $m=m(n)$, then define $M_i(u)$ to be the total number of visits by user $u$ to state $i$, for $i=0, 1, \cdots, r-1$. Since the Markov chain is irreducible and aperiodic, and $m(n) \rightarrow \infty$, all $\frac{M_i(u)}{m(n)}$ converge to their stationary values. Now conditioned on $M_i(u)=m_i(u)$, the transitions from state $i$ to state $l$ for user $u$ follow a multinomial distribution with probabilities $p_u(i,l)$. Given the above, the setting is now very similar to the i.i.d.\ case. Each user is uniquely characterized by a vector $\textbf{p}_u$ of size $|E|-r$. We define sets $B^{''(n)}$ and $C^{''(n)}$ as \[ B^{''(n)}\triangleq \{(x_1, x_2, \cdots ,x_{d}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-\Delta''_n \leq x_i \leq p_1(i)+\Delta''_n, i=1,2 , \cdots ,d\}, \] \[ C^{''(n)}\triangleq \{(x_1, x_2, \cdots ,x_{d}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-2 \Delta''_n < x_i < p_1(i)+2 \Delta''_n, i=1,2, \cdots, d\}, \] where $\Delta''_n= \frac{1}{n^{\frac{1}{|E|-r}+\frac{\alpha}{4}}}$, and $d= |E|-r$. Then, we can show that for the stated values of $m(n)$ and $a_n$, as $n$ becomes large: \begin{enumerate} \item $P\left( \overline{\textbf{Y}_{\Pi(1) }}\in B^{''(n)}\right) \rightarrow 1$, \item $P\left( \bigcup\limits_{u=2}^{n} \left(\overline{\textbf{Y}_{\Pi(u)}}\in B{''}^{(n)}\right)\right) \rightarrow 0$, \end{enumerate} which means that the adversary can estimate the data of user $1$ with vanishing error probability. The proof is very similar to the proof of the i.i.d.\ case; however, there are two differences that need to be addressed: First, the probability of observing an erroneous observation is not exactly given by $R_u$. In fact, a transition is distorted if at least one of its nodes is distorted. So, if the actual transition is from state $i$ to state $l$, then the probability of an erroneous observation is equal to \begin{align} \nonumber R'_u&=R_u+R_u-R_uR_u =R_u(2-R_u). \end{align} Nevertheless, here the order only matters, and the above expression is still in the order of $a_n =O \left( n^{-\left(\frac{1}{|E|-r}+\beta \right)} \right)$. The second difference is more subtle. As opposed to the i.i.d.\ case, the error probabilities are not completely independent. In particular, if $X_u(k)$ is reported in error, then both the transition to that state and from that state are reported in error. This means that there is a dependency between errors of adjacent transitions. We can address this issue in the following way: The adversary makes his decision only based on a subset of the observations. More specifically, the adversary looks at only odd-numbered transitions: First, third, fifth, etc., and ignores the even-numbered transitions. In this way, the number of observations is effectively reduced from $m$ to $\frac{m}{2}$ which again does not impact the order of the result (recall that the Markov chain is aperiodic). However, the adversary now has access to observations with independent errors. \section{Perfect Privacy Analysis: Markov Chain Model}\label{sec:perfect-MC} So far, we have provided both achievability and converse results for the i.i.d.\ case. However, we have only provided the converse results for the Markov chain case. Here, we investigate achievability for Markov chain models. It turns out that for this case, the assumed obfuscation technique is not sufficient to achieve a reasonable level of privacy. Loosely speaking, we can state that if the adversary can make enough observations, then he can break the anonymity. The culprit is the fact that the sequence observed by the adversary is no longer modeled by a Markov chain; rather, it can be modeled by a hidden Markov chain. This allows the adversary to successfully estimate the obfuscation random variable $R_u$ as well as the $ p_u(i,l)$ values for each sequence, and hence successfully de-anonymize the sequences. More specifically, as we will see below, there is a fundamental difference between the i.i.d.\ case and the Markov chain case. In the i.i.d.\ case, if the noise level is beyond a relatively small threshold, the adversary will be unable to de-anonymize the data and unable to recover the actual values of the data sets for users, \emph{regardless of the (large) size of $m=m(n)$}. On the other hand, in the Markov chain case, if $m=m(n)$ is large enough, then the adversary can easily de-anonymize the data. To better illustrate this, let's consider a simple example. \begin{example} Consider the scenario where there are only two states and the users' data samples change between the two states according to the Markov chain shown in Figure \ref{fig:MC-diagram}. What distinguishes the users is their different values of $p$. Now, suppose we use the same obfuscation method as before. That is, to create a noisy version of the sequences of data samples, for each user $u$, we generate the random variable $R_u$ that is the probability that the data sample of the user is changed to a different data sample by obfuscation. Specifically, \[ {Z}_{u}(k)=\begin{cases} {X}_{u}(k), & \textrm{with probability } 1-R_u.\\ 1-{X}_{u}(k),& \textrm{with probability } R_u. \end{cases} \] \begin{figure}[H] \begin{center} \[ \SelectTips {lu}{12 scaled 2500} \xymatrixcolsep{6pc}\xymatrixrowsep{5pc}\xymatrix{ *++[o][F]{0} \ar@/^1pc/[r]^{1} & *++[o][F]{1} \ar@(dr,ur)[]_{1-p} \ar@/^1pc/[l]^{p} } \] \caption{A state transition diagram.}\label{fig:MC-diagram} \end{center} \end{figure} To analyze this problem, we can construct the underlying Markov chain as follows. Each state in this Markov chain is identified by two values: the real state of the user, and the observed value by the adversary. In particular, we can write \[\left(\text{Real value}, \text{Observed value}\right) \in \left\{\right(0,0), (0,1), (1,0), (1,1)\}.\] Figure \ref{fig:MC-diagram2} shows the state transition diagram of this new Markov chain. \begin{figure}[H] \begin{center} \[ \SelectTips {lu}{12 scaled 2000} \xymatrixcolsep{10pc}\xymatrixrowsep{8pc}\xymatrix{ *++[o][F]{00} \ar@//[d]^{R} \ar@/_1pc/[dr]_>>>>>{1-R} & *++[o][F]{01} \ar@//[d]^{1-R} \ar@/^1pc/[dl]^>>>>>{R} \\ *++[o][F]{10} \ar@(d,l)[]^{(1-p)R} \ar@/_2pc/[r]_{(1-p)(1-R)} \ar@/^2pc/[u]^{p(1-R)} \ar@/^1pc/[ur]^>>>>>{pR} & *++[o][F]{11} \ar@(d,r)[]_{(1-p)(1-R)} \ar@//[l]^{(1-p)R} \ar@/_2pc/[u]_{pR} \ar@/_1pc/[ul]_>>>>>{p(1-R)} } \] \caption{The state transition diagram of the new Markov chain.}\label{fig:MC-diagram2} \end{center} \end{figure} We know \[ \pi_{00}=\pi_0(1-R)=\frac{p}{1+p}(1-R).\] \[ \pi_{01}=\pi_0R= \frac{p}{1+p}R.\] \[ \pi_{10}=\pi_1R=\frac{1}{1+p}R.\] \[ \pi_{11}=\pi_1(1-R)=\frac{1}{1+p}(1-R).\] The observed process by the adversary is not a Markov chain; nevertheless, we can define limiting probabilities. In particular, let $\theta_0$ be the limiting probability of observing a zero. That is, we have \[ \frac{M_0}{m} \xrightarrow{d} \theta_0, \ \ \textrm{ as }n \rightarrow \infty, \] where $m$ is the total number of observations by the adversary, and $M_0$ is the number of $0$'s observed. Then, \[\theta_0= \pi_{00}+\pi_{10} =\frac{(1-R)p+R}{1+p}.\] Also, let $\theta_1$ be the limiting probability of observing a one, so \[\theta_1= \pi_{01}+\pi_{11} =\frac{pR+(1-R)}{1+p}=1-\theta_0.\] Now the adversary's estimate of $\theta_0$ is given by: \begin{align}\label{eq1} \hat{\theta}_0= \frac{(1-R)p+R}{1+p}. \end{align} Note that if the number of observations by the adversary can be arbitrarily large, the adversary can obtain an arbitrarily accurate estimate of $\theta_0$. The adversary can obtain another equation easily, as follows. Let $\theta_{01}$ be the limiting value of the portion of transitions from state $0$ to $1$ in the chain observed by the adversary. We can write \begin{align} \nonumber \theta_{01} &=P \left\{(00\rightarrow 01), (00\rightarrow 11), (10 \rightarrow 01), (10 \rightarrow 11) \right\}\\\ \nonumber &= \pi_{00}(1-R)+\pi_{10}PR+ \pi_{10}(1-p)(1-R).\ \ \end{align} As a result, \begin{align}\label{eq2} \hat{\theta}_{01}= \frac{p(1-R)^2+R\left(PR(1-R)(1-p)\right)}{1+p}. \end{align} Again, if the number of observations can be arbitrarily large, the adversary can obtain an arbitrarily accurate estimate of $\theta_{01}$. By solving the Equations \ref{eq1} and \ref{eq2}, the adversary can successfully recover $R$ and $p$; thus, he/she can successfully determine the users' data values. \end{example} \section{Introduction} \label{intro} Various emerging systems and applications work by analyzing the data submitted by their users in order to serve them; we call such systems \emph{user-data driven} (UDD) \amir{we need a name. how's this?} services. Examples of UDD services include smart cities, connected vehicles, smart homes, and connected healthcare devices, which are revolutionizing our lives in different ways. Unfortunately, the sheer volume of user data collected by these systems compromises users' privacy ostensibly~\cite{FTC2015}. Even the employment of standard privacy-protection mechanisms (e.g., anonymization of user identities and obfuscation of submitted data) does not mitigate the privacy leakage as the adversaries are able to use powerful statistical inference techniques to infer sensitive private information~\cite{FTC2015,0Quest2016, 2nia2016comprehensive, 3ukil2014iot, 4Hosseinzadeh2014,iotCastle,matching}. To illustrate their privacy leakage, consider the following three popular UDD services. (1) {\em Health care:} Wearable monitors that constantly track user health variables can be invaluable in assessing individual health trends and responding to emergencies. However, such monitors produce long time-series of user data uniquely matched to the health characteristics of each user; (2) {\em Smart homes:} Emerging smart-home technologies such as fine-grained power measurement systems can help users and utility providers to address one of the key challenges of the twenty-first century: energy conservation. But the measurements of power by such devices can be mapped to users and reveal their lifestyle habits; and, (3) {\em Connected vehicles:} The location data provided by connected vehicles promises to greatly improve everyday lives by reducing congestion and traffic accidents. However, the matching of such location traces to prior behavior not only allows for user tracking, but also reveals a user's habits. In summary, despite their critical importance to the society and their emerging popularity, these services have one thing in common: they collect long time-series of user data, for them to operate, that puts users' privacy at significant risk. There are two main approaches to augment privacy in UDD services: \emph{identity perturbation (anonymization)}, and \emph{location perturbation (obfuscation)}. In anonymization techniques, privacy is obtained by concealing the mapping between users and data, and the mapping is changed periodically to thwart statistical inference attacks that try to deanonymize the anonymized data traces by matching user data to known user profiles. On the other hands, obfuscation mechanisms aim at protecting privacy by perturbing user data, e.g., by adding noise to user locations. Anonymization and obfuscation improve user privacy at the cost of user utility. In anonymization, we need to change these pseudonym mappings frequently to achieve high privacy by reducing the length of time series exploited by statistical analysis. However, this frequent change could decrease usability and functionality by concealing the temporal relation between a user's locations, which may be critical in the utility of some systems, e.g., a dining recommendation system that makes suggestions based on the dining places visited by a user in the past. For obfuscation-based mechanisms, the added noise to the reported values of user locations will degrade the application utility; for example, user utility would be degraded by location obfuscation in ride-sharing systems. Thus, choosing the right level of privacy-protection mechanism is an important question, and understanding what levels of anonymization and obfuscation can provide theoretical guarantees of privacy is of interest. In this paper, we derive the fundamental limits of user privacy in UDD services in the presence of both anonymization and obfuscation protection mechanisms. This work is built on our previous work on formalizing privacy in location-based services~\cite{montazeri2016defining, Mont1610Achieving,tifs2016, ciss2017}, by extensively expanding not just the application area but also user models and settings. Particularly, our previous work introduced the notion of \emph{perfect privacy} for location-based services, and derives the rate at which an anonymization mechanism should change the pseudonyms in order to achieve the defined perfect privacy. In this work, we expand the notion of perfect privacy to UDD services in general, and derive the conditions for it to hold when \emph{both} anonymization and obfuscation-based protection mechanisms are employed. We study both achievability and converse results. \subsection{Summary of the Results} Given $n$, the total number of the users in a network, their degree of privacy depends on two parameters: (1) The number of observations $m=m(n)$ by the adversary per user for a fixed anonymization mapping (i.e., the number of observations before the pseudonyms are changed); and (2) the value of the noise added by the obfuscation technique (as defined in Section~\ref{sec:framework}, we quantify the obfuscation noise with a parameter $a_n$). Intuitively, smaller $m(n)$ and larger $a_n$ result in stronger privacy, at the expense of lower utility for the users. Our goal is to identify values of $a_n$ and $m(n)$ that satisfy perfect privacy as the number of users grows ($n \rightarrow \infty$). We show that when the users' data sets are governed by an i.i.d.\ process, the $m(n)- a_n$ plane can be divided into two areas. In the first area, all users have perfect privacy~\cite{tifs2016} (as defined in Section~\ref{sec:framework}), and, in the second area, users have no privacy. Figure~\ref{fig:region} shows the limits of privacy in the entire $m(n)- a_n$. As the figure shows, in regions $1$, $2$, and $3$, users have perfect privacy, while in region $4$ users have no privacy. \begin{figure}[h] \centering \includegraphics[width=0.8\linewidth, height=0.8 \linewidth]{fig/region} \caption{Limits of privacy in the entire $m(n)-a_n$ plane: in regions $1$, $2$, and $3$, users have perfect privacy, and in region $4$ users have no privacy.} \label{fig:region} \end{figure} For the case where the users' data sets are governed by irreducible and aperiodic Markov chains with $r$ states and $|E|$ edges, we show that users will have no privacy if $m =cn^{\frac{2}{|E|-r} + \alpha}$ and $a_n =c'n^{-\left(\frac{1}{|E|-r}+\beta \right)}$, for some constants $c>0$, $c'>0$, $\alpha>0$, and $\beta>\frac{\alpha}{4}$. We also provide some insights for the opposite direction (under which conditions users have perfect privacy) for the case of Markov chains. \section{Introduction} \label{intro} A number of emerging systems and applications work by analyzing the data submitted by their users in order to serve them; we call such systems \emph{User-Data Driven} (UDD) services. Examples of UDD services include smart cities, connected vehicles, smart homes, and connected healthcare devices, which have the promise of greatly improving users' lives. Unfortunately, the sheer volume of user data collected by these systems can compromise users' privacy~\cite{FTC2015}. Even the use of standard Privacy-Protection Mechanisms (PPMs), specifically anonymization of user identities and obfuscation of submitted data, does not guarantee users' privacy, as adversaries are able to use powerful statistical inference techniques to learn sensitive private information of the users~\cite{0Quest2016, 3ukil2014iot, 4Hosseinzadeh2014,iotCastle,matching}. To illustrate the threat of privacy leakage, consider three popular UDD services: (1) {\em Health care:} Wearable monitors that constantly track user health variables can be invaluable in assessing individual health trends and responding to emergencies. However, such monitors produce long time-series of user data uniquely matched to the health characteristics of each user; (2) {\em Smart homes:} Emerging smart-home technologies such as fine-grained power measurement systems can help users and utility providers to address one of the key challenges of the twenty-first century: energy conservation. But the measurements of power by such devices can be mapped to users and reveal their lifestyle habits; and, (3) {\em Connected vehicles:} The location data provided by connected vehicles promises to greatly improve everyday life by reducing congestion and traffic accidents. However, the matching of such location traces to prior behavior not only allows for user tracking, but also reveals a user's habits. In summary, despite their potential impact on society and their emerging popularity, these UDD services have one thing in common: their utility critically depends on their collection of user data, which puts users' privacy at significant risk. There are two main approaches to augment privacy in UDD services: \emph{identity perturbation (anonymization)}~\cite{1corser2016evaluating,hoh2005protecting,freudiger2007mix, ma2009location, shokri2011quantifying2, Naini2016,soltani2017towards, soltani2018invisible}, and \emph{data perturbation (obfuscation)}~\cite{shokri2012protecting, gruteser2003anonymous, bordenabe2014optimal}. In anonymization techniques, privacy is obtained by concealing the mapping between users and data, and the mapping is changed periodically to thwart statistical inference attacks that try to de-anonymize the anonymized data traces by matching user data to known user profiles. Some approaches employ $k$-anonymity to keep each user's identity indistinguishable within a group of $k-1$ other users ~\cite{2zhang2016designing,11dewri2014exploiting, gedik2005location, zhong2009distributed, sweeney2002k, kalnis2007preventing,liu2013game}. Other approaches employ users' pseudonyms within areas called mix-zones~\cite{beresford2003location, freudiger2009optimal, palanisamy2011mobimix}. Obfuscation mechanisms aim at protecting privacy by perturbing user data, e.g., by adding noise to users' samples of data. For instance, cloaking replaces each user's sample of data with a larger region~\cite{18shokri2014hiding,8zurbaran2015near,hoh2007preserving, wernke2014classification, chow2011spatial, um2010advanced}, while an alternative approach is to use dummy data in the set of possible data of the users~\cite{kido2005protection, shankar2009privately, chow2009faking, kido2005anonymous, lu2008pad}. In~\cite{randomizedresponse}, a mechanism of obfuscation was introduced where the answer was changed randomly with some small probability. Here we consider the fundamental limits of a similar obfuscation technique for providing privacy in the long time series of emerging applications. The anonymization and obfuscation mechanisms improve user privacy at the cost of user utility. The anonymization mechanism works by frequently changing the pseudonym mappings of users to reduce the length of time series that can be exploited by statistical analysis. However, this frequent change may also decrease the usability by concealing the temporal relation between a user's sample of data, which may be critical in the utility of some systems, e.g., a dining recommendation system that makes suggestions based on the dining history of its users. On the other hand, obfuscation mechanisms work by adding noise to users' collected data, e.g., location information. The added noise may degrade the utility of UDD applications. Thus, choosing the right level of the privacy-protection mechanism is an important question, and understanding what levels of anonymization and obfuscation can provide theoretical guarantees of privacy is of interest. In this paper, we will consider the ability of an adversary to perform statistical analyses on time series and match the series to descriptions of user behavior. In related work, Unnikrishnan~\cite{matching} provides a comprehensive analysis of the asymptotic (in the length of the time series) optimal matching of time series to source distributions. However, there are several key differences between that analysis and the work here. First, Unnikrishnan~\cite{matching} looks at the optimal matching tests, but does not consider any privacy metrics as considered in this paper, and a significant component of our study is demonstrating that mutual information converges to zero so that we can conclude there is no privacy leakage (hence, ``perfect privacy''). Second, the setting of~\cite{matching} is different, as it does not consider: (a) obfuscation, which is one of the two major protection mechanisms; and (b) sources that are not independent and identically distributed (i.i.d.). Third, the setting of Unnikrishnan~\cite{matching} assumes a fixed distribution on sources (i.e., classical inference), whereas we assume the existence of general (but possibly unknown) prior distributions for the sources (i.e., a Bayesian setting). Finally, we study the fundamental limits in terms of both the number of users and the number of observations, while Unnikrishnan~\cite{matching} focuses on the case where the number of users is a fixed, finite value. Numerous researchers have put forward ideas for quantifying privacy-protection. Shokri et al.~\cite{shokri2011quantifying, shokri2011quantifying2} define the expected estimation error of the adversary as a metric to evaluate PPMs. Ma et al.~\cite{ma2009location} use uncertainty about users' information to quantify user privacy in vehicular networks. To defeat localization attacks and achieve privacy at the same time, Shokri et al.~\cite{shokri2012protecting} proposed a method which finds optimal PPM for an LBS given service quality constraints. In~\cite{6li2016privacy} and~\cite{4olteanu2016quantifying}, privacy leakage of data sharing and interdependent privacy risks are quantified, respectively. A similar idea is proposed in \cite{14zhang2014privacy} where the quantification model is based on the Bayes conditional risk. Previously, mutual information has been used as a privacy metric in a number of settings,~\cite{kousha3,salamatian2013hide, csiszar1996almost, calmon2015fundamental, sankar2013utility, sankarISIT,sankar, yamamoto1983source, hyposankar}. However, the framework and problem formulation for our setting (Internet of Things (IoT) privacy) are quite different from those encountered in previous works. More specifically, the IoT privacy problem we consider here is based on a large set of time-series data that belongs to different users with different statistical patterns that has gone through a privacy-preserving mechanism, and the adversary is aiming at de-anonymizing and de-obfuscating the data. The discussed studies demonstrate the growing importance of privacy. What is missing from the current literature is a solid theoretical framework for privacy that is general enough to encompass various privacy-preserving methods in the literature. Such a framework will allow us to achieve provable privacy guarantees, obtain fundamental trade-offs between privacy and performance, and provide analytical tools to optimally achieve provable privacy. We derive the fundamental limits of user privacy in UDD services in the presence of both anonymization and obfuscation protection mechanisms. We build on our previous works on formalizing privacy in location-based services~\cite{tifs2016, ciss2017}, but we significantly expand those works here not just in application area but also user models and settings. In particular, our previous works introduced the notion of \emph{perfect privacy} for location-based services, and we derived the rate at which an anonymization mechanism should change the pseudonyms in order to achieve the defined perfect privacy. In this work, we expand the notion of perfect privacy to UDD services in general and derive the conditions for it to hold when \emph{both} anonymization and obfuscation-based protection mechanisms are employed. In this paper, we consider two models for users' data: i.i.d.\ and Markov chains. After introducing the general framework in Section \ref{sec:framework}, we consider an i.i.d.\ model extensively in Section \ref{perfectsec} and the first half of Section \ref{converse}. We obtain achievability and converse results for the i.i.d.\ model. The i.i.d.\ model would apply directly to data that is sampled at a low rate. In addition, understanding the i.i.d.\ case can also be considered the first step toward understanding the more complicated case where there is dependency, as was done for anonymization-only Location Privacy-Preserving Mechanisms (LPPMs) in \cite{tifs2016}, and will be done in Section \ref{subsec:markov}. In particular, in Section \ref{subsec:markov}, a general Markov chain model is used to model users' data pattern to capture the dependency of the user' data pattern over time. There, we obtain converse results for privacy for this model. In Section \ref{sec:perfect-MC}, we provide some discussion about the achievability for the Markov chain case. \subsection{Summary of the Results} Given $n$, the total number of the users in a network, their degree of privacy depends on two parameters: (1) The number of observations $m=m(n)$ by the adversary per user for a fixed anonymization mapping (i.e., the number of observations before the pseudonyms are changed); and (2) the value of the noise added by the obfuscation technique (as defined in Section~\ref{sec:framework}, we quantify the obfuscation noise with a parameter $a_n$, where larger $a_n$ means a higher level of obfuscation). Intuitively, smaller $m(n)$ and larger $a_n$ result in stronger privacy, at the expense of lower utility for the users. Our goal is to identify values of $a_n$ and $m(n)$ that satisfy perfect privacy in the asymptote of a large number of users ($n \rightarrow \infty$). When the users' datasets are governed by an i.i.d.\ process, we show that the $m(n)- a_n$ plane can be divided into two areas. In the first area, all users have perfect privacy (as defined in Section~\ref{sec:framework}), and, in the second area, users have no privacy. Figure~\ref{fig:region} shows the limits of privacy in the entire $m(n)- a_n$ plane. As the figure shows, in regions $1$, $2$, and $3$, users have perfect privacy, while in region $4$ users have no privacy. \begin{figure}[h] \centering \includegraphics[width=0.7\linewidth, height=0.6 \linewidth]{fig/region} \caption{Limits of privacy in the entire $m(n)-a_n$ plane: in regions $1$, $2$, and $3$, users have perfect privacy, and in region $4$ users have no privacy.} \label{fig:region} \end{figure} For the case where the users' datasets are governed by irreducible and aperiodic Markov chains with $r$ states and $|E|$ edges, we show that users will have no privacy if $m =cn^{\frac{2}{|E|-r} + \alpha}$ and $a_n =c'n^{-\left(\frac{1}{|E|-r}+\beta \right)}$, for any constants $c>0$, $c'>0$, $\alpha>0$, and $\beta>\frac{\alpha}{4}$. We also provide some insights for the opposite direction (under which conditions users have perfect privacy) for the case of Markov chains. \section{Related Work} \label{relatedwork} Previously, mutual information has been used as a privacy metric in a number of settings,~\cite{salamatian2013hide, csiszar1996almost, calmon2015fundamental, sankar2013utility, sankarISIT,sankar, yamamoto1983source, hyposankar}. However, the framework and problem formulation for our setting (Internet of Things (IoT) privacy) is quite different from those encountered in previous work. More specifically, the IoT privacy problem we consider here is based on a large set of time-series data that belong to different users with different statistical patterns that has gone through a privacy-preserving mechanism, and the adversary is aiming at de-anonymizing and de-obfuscating the data. Next, consider related works in smart-home/public environment Internet of Things (IoT) systems. The authors in \cite{8ukil2015privacy} investigate privacy for IoT smart energy metering where the concern results from the fact that any privacy breach can reveal in-house activity. A `Dynamic Privacy Analyzer' scheme is proposed as an attempt towards achieving a unique privacy metric that is derived from fundamental principles like robust statistics and information theory. To deal with concerns in different smart-home IoT devices such as lights, smoke-alarms, power switches, baby monitors, and weighing scales, which again makes it possible for the entities to snoop and intrude into the family's activities, the authors in \cite{12sivaraman2015network} suggest that device-level protections be augmented with network-level security solutions. This can be achieved through software defined networking technology when used to dynamically block/quarantine suspicious devices. In \cite{10harris2016security}, the authors consider the privacy issues in public environments such as an IoT-enabled retail store. An IoT-enable retail store has the potential to provide rich, targeted information and services to users within the environment. However, to fully realize such potential, users must be willing to share certain data regarding their habits and preferences with the public IoT facility. To encourage users to share such information, the paper proposes a protocol to ensure the customers that their data will not be leaked to third parties. As far as other applications of IoT is concerned, authors in \cite{9dalipi2016security} investigate privacy considerations for IoT applications on Smart Grids. In particular, the authors address three types of challenge domains: customer domain, information and communication domain, and the grid domain. In \cite{battery15}, the author tried to provide privacy for smart-metering systems with rechargeable batteries by obscuring the user's power demand. In \cite{11al2015security}, the authors consider the privacy issue of wearable devices. These devices make it possible for health care services start a new phase in serving patients' needs and monitor their health remotely. Finally, authors in \cite{14sadeghi2015security}, overview the privacy challenges in industrial IoT systems. Among these applications, location privacy constitutes one of the most important aspects of IoT privacy. Location privacy preserving mechanisms (LPPMs) can be categorized into two main classes: identity perturbation LPPMs (anonymization) ~\cite{1corser2016evaluating,hoh2005protecting,freudiger2007mix, ma2009location, shokri2011quantifying, Naini2016} and location perturbation LPPMs (obfuscation)~\cite{shokri2012protecting, gruteser2003anonymous, bordenabe2014optimal}. There are various approaches suggested for identity perturbation LPPMs. Particularly, k-anonymity approaches help to keep each user's identity indistinguishable within a group of $k-1$ other users ~\cite{1corser2016evaluating,2zhang2016designing,11dewri2014exploiting, gedik2005location, zhong2009distributed, mokbel2006new, sweeney2002k, bordenabe2014optimal, kalnis2007preventing,liu2013game}. Another proposal changes users' pseudonyms within areas called mix-zones~\cite{hoh2005protecting, beresford2003location, freudiger2009optimal, beresford2003location, palanisamy2011mobimix}. There are also various approaches used by location perturbation LPPMs. For instance, cloaking replaces each user's location information with a larger region~\cite{18shokri2014hiding,8zurbaran2015near,hoh2007preserving, gruteser2003anonymous, gedik2005location, gedik2008protecting, bordenabe2014optimal, duckham2005formal, duckham2006spatiotemporal, xue2009location, wernke2014classification, cai2015cloaking, chow2011spatial, cheng2006preserving, mokbel2006new, kalnis2006preserving, khoshgozaran2011location, bamba2008supporting, um2010advanced, zhangwei2010distributed}, while an alternative direction is to use dummy locations in the set of possible locations of users~\cite{kido2005protection, krumm2009survey, shankar2009privately, chow2009faking, kido2005anonymous, lu2008pad}. Some prior works combine techniques to achieve stronger location privacy. For instance, Freudiger et al.\ combine techniques from cryptography with mix-zones to improve location privacy~\cite{freudiger2007mix}. Differential privacy, which aims at protecting queries on aggregated data, has also been applied to the problem of location privacy. Particularly, several identity perturbation~\cite{lee2012differential, bordenabe2014optimal, chatzikokolakis2015geo, nguyen2013differential, machanavajjhala2008privacy} and location perturbation ~\cite{chatzikokolakis2013broadening,shokri2014optimal, chatzikokolakis2015location,andres2013geo,bordenabe2014optimal} LPPMs are based on differential privacy techniques. Dewri~\cite{dewri2013local} combines k-anonymity and differential privacy to achieve higher location privacy. Alternatively, Andres et al.\ hide the exact location of a user in a region by adding Laplacian distributed noise to achieve a desired level of geo-indistinguishability~\cite{andres2013geo}. In ~\cite{info2012}, it is proved that differential privacy arises out of maximizing entropy principle which is equal to minimizing information leakage and is measured using the mutual information notion. ~\cite{yeb17} tries to reduce the expected estonimation loss by using differential privacy and constraining the privacy level. Several studies aim at quantifying location privacy protection. Shokri et al.~\cite{shokri2011quantifying, shokri2011quantifying2} define the expected estimation error of the adversary as a metric to evaluate LPPM mechanisms. Ma et al.~\cite{ma2009location} use uncertainty about users' location information to quantify user location privacy in vehicular networks. To defeat localization attacks and achieve privacy at the same time, Shokri et al.~\cite{shokri2012protecting} proposed a method which finds optimal LPPM for an LBS given service quality constraints. Shokri et al.~\cite{shokri2012protecting} design LPPM mechanisms that will defeat localization attacks. In \cite{6li2016privacy} and \cite{4olteanu2016quantifying}, privacy leakage of location sharing and interdependent location privacy risks are quantified, respectively. A similar idea is proposed in \cite{14zhang2014privacy} where the quantification model is based on the Bayes conditional risk. Yu et al. \cite{diff2017} combine two complementary notations, geo-indstingushability and expected inference error, to achieve location privacy. The geo-indstingushability is derived from differential privacy but cannot adequately protect users' location against inference attacks of adversary by using prior information, and the expected inference error as privacy metric doesn't consider posterior information obtained from pseudo-location. As a result, combining both notations gives users the opportunity to have location privacy. In addition, users are allowed to personalized error bound for different locations. In \cite{5ullah2016novel}, the authors propose a novel model for preserving location privacy in IoT in which they propose the so-called Enhanced Semantic Obfuscation Technique (ESOT). In \cite{7zhou2012preserving}, in the context of IoT based wireless sensor networks, the authors propose a flexible routing strategy, called Multi-routing Random Walk, which protects the sensor's location. Another approach has been proposed in \cite{6sathishkumar2016enhanced}. The discussed studies demonstrate the growing importance of privacy. What is missing from the current literature is a solid theoretical framework for privacy that is general enough to encompass various privacy preserving methods in the literature. Such a framework will allow us to achieve provable privacy guarantees, obtain fundamental trade-offs between privacy and performance, and provide analytical tools to optimally achieve provable privacy. The closest works to this paper are ~\cite{montazeri2016defining, Mont1610Achieving,tifs2016,sit2017,ciss2017}. As mentioned, these works only consider anonymization and only achiveability. In this paper, by considering obfuscation and anonymization, we characterize the limits of privacy in the entire $m(n)-a_n$ plane by considering both achievability and converse results. \section{Lemma \ref{lemx} and its Proof} \label{sec:app_a} Here we state that we can condition on high-probability events. \begin{lem} \label{lemx} Let $p \in (0,1)$, and $X \sim Bernoulli (p)$ be defined on a probability space $(\Omega, \mathcal{F}, P)$. Consider $B_1, B_2, \cdots $ be a sequence of events defined on the same probability space such that $P(B_n) \rightarrow 1$ as $n$ goes to infinity. Also, let $\textbf{Y}$ be a random vector (matrix) in the same probability space, then: \[I(X; \textbf{Y}) \rightarrow 0\ \ \text{iff}\ \ I(X; \textbf{Y} {|} B_n) \rightarrow 0. \] \end{lem} \begin{proof} First, we prove that as n becomes large, \begin{align}\label{eq:H1} H(X {|} B_n)- H(X) \rightarrow 0. \end{align} Note that as $n$ goes to infinity, \begin{align} \nonumber P\left(X=1\right) &=P\left(X=1 \bigg{|} B_n\right) P\left(B_n\right) + P\left(X=1 \bigg{|} \overline{B_n}\right) P\left(\overline{B_n}\right)\\ \nonumber &=P\left(X=1 \bigg{|} B_n\right),\ \ \end{align} thus, $\left(X \bigg{|} B_n\right) \xrightarrow{d} X$, and as $n$ goes to infinity, \[H\left(X {|} B_n\right)- H(X) \rightarrow 0.\] Similarly, as $n$ becomes large, \[ P\left(X=1 \bigg{|} \textbf{Y}=\textbf{y}\right) \rightarrow P\left(X=1 \bigg{|} \textbf{Y}=\textbf{y}, B_n\right),\ \ \] and \begin{align}\label{eq:H2} H\left(X {|} \textbf{Y}=\textbf{y},B_n\right)- H\left(X {|} \textbf{Y}=\textbf{y}\right) \rightarrow 0. \end{align} Remembering that \begin{align}\label{eq:H3} I\left(X; \textbf{Y}\right)=H(X)-H(X {|} \textbf{Y}), \end{align} and using (\ref{eq:H1}), (\ref{eq:H2}), and (\ref{eq:H3}), we can conclude that as $n$ goes to infinity, \[I\left(X;\textbf{Y} {|} B_n\right) - I\left(X,\textbf{Y}\right) \rightarrow 0.\] As a result, for large enough $n$, \[I\left(X; \textbf{Y}\right) \rightarrow 0 \Longleftrightarrow I\left(X; \textbf{Y} {|} B_n\right) \rightarrow 0. \] \end{proof} \section{Proof of Lemma \ref{lem1}} \label{sec:app_c} Here, we provide a formal proof for Lemma \ref{lem1} which we restate as follows. The following lemma confirms that the number of elements in $J^{(n)}$ goes to infinity as $n$ becomes large. If $N^{(n)} \triangleq |J^{(n)}| $, then $N^{(n)} \rightarrow \infty$ with high probability as $n \rightarrow \infty$. More specifically, there exists $\lambda>0$ such that \[ P\left(N^{(n)} > \frac{\lambda}{2}n^{\frac{\beta}{2}}\right) \rightarrow 1. \] \begin{proof} Define the events $A$, $B$ as \[A \equiv p_1\leq P_u\leq p_1+\epsilon_n\] \[B \equiv p_1+\epsilon_n\leq Q_u\leq p_1+(1-2p_1)a_n.\] Then, for $u \in \{1, 2, \dots, n\}$ and $0\leq p_1<\frac{1}{2}$: \begin{align} \nonumber P\left(u\in J^{(n)}\right) &= P\left(A\ \cap \ B\right)\\ \nonumber &= P\left(A\right) P\left(B \big{|}A \right). \ \ \end{align} So, given $p_1 \in (0,1)$ and the assumption $0<\delta_1<f_p<\delta_2$, for $n$ large enough, we have \[ P(A) = \int_{p_1}^{ p_1+\epsilon_n}f_P(p) dp, \] so, we can conclude that \[ \epsilon_n\delta_1<P(A) <\epsilon_n\delta_2. \] We can find a $\delta$ such that $\delta_1<\delta<\delta_2$ and \begin{equation}\label{eq:5} P( A) = \epsilon_n\delta. \end{equation} We know \[Q_u\bigg{|}P_u=p_u \sim Uniform \left[p_u,p_u+(1-2p_u)a_n\right],\] so, according to Figure \ref{fig:piqi_b}, for $p_1\leq p_u\leq p_1+\epsilon_n$, \begin{align} \nonumber P\left(B | P_u=p_u\right ) &= \frac{p_1+(1-2p_1)a_n-p_1-\epsilon_n}{p_u+(1-2p_u)a_n-p_u} \\ \nonumber &= \frac{(1-2p_1)a_n-\epsilon_n}{(1-2p_u)a_n}\\ \nonumber &\geq \frac{(1-2p_1)a_n-\epsilon_n}{(1-2p_1)a_n} \\ \nonumber &= 1- \frac{\epsilon_n}{(1-2p_1)a_n}, \ \ \end{align} which implies \begin{align} P\left(B | A\right ) \geq 1- \frac{\epsilon_n}{(1-2p_1)a_n}. \label{eq:6} \end{align} Using (\ref{eq:5}) and (\ref{eq:6}), we can conclude \[P\left(u\in J^{(n)}\right)\geq \epsilon_n\delta \left(1- \frac{\epsilon_n}{(1-2p_1)a_n}\right).\] Then, we can say that $N^{(n)}$ has a binomial distribution with expected value of $N^{(n)}$ greater than $n\epsilon_n\delta \left(1- \frac{\epsilon_n}{(1-2p_1)a_n}\right)$, and by substituting $\epsilon_n$ and $a_n$, for any $c'>0$, we get \[E\left[N^{(n)}\right] \geq \delta\left(n^{\frac{\beta}{2}}- \frac{1}{{c'(1-2p_1)}}\right) \geq \lambda n^{\frac{\beta}{2}}.\] Now by using Chernoff bound, we have \[P\left(N^{(n)} \leq (1- \theta) E\left[N^{(n)}\right]\right) \leq e^{-\frac{\theta^2}{2}E\left[N^{(n)}\right]},\] so, if we assume $\theta=\frac{1}{2}$, we can conclude for large enough $n$, \begin{align} \nonumber P\left(N^{(n)} \leq \frac{\lambda}{2}n^{\frac{\beta}{2}}\right) &\leq P\left(N^{(n)} \leq \frac{E\left[N^{(n)}\right]}{2}\right)\\ \nonumber &\leq e^{-\frac{E[N^{(n)}]}{8}}\\ \nonumber &\leq e^{-\frac{\lambda n^{\frac{\beta}{2}}}{8}} \rightarrow 0. \end{align} As a result, $N^{(n)} \rightarrow \infty$ with high probability for large enough $n.$ \end{proof} \section{Proof of Lemma \ref{lemOnePointFive}} \label{sec:app_b} Here we provide a formal proof for Lemma \ref{lemOnePointFive} which we restate as follows. Let $N$ be a positive integer, and let $a_1, a_2, \cdots, a_N$ and $b_1, b_2, \cdots, b_N$ be real numbers such that $a_u \leq b_u$ for all $u$. Assume that $X_1, X_2, \cdots, X_N$ are $N$ independent random variables such that \[X_u \sim Uniform[a_u,b_u]. \] Let also $\gamma_1, \gamma_2, \cdots, \gamma_N$ be real numbers such that \[ \gamma_j \in \bigcap_{u=1}^{N} [a_u, b_u] \ \ \textrm{for all }j \in \{1,2,\cdots,N\}. \] Suppose that we know the event $E$ has occurred, meaning that the observed values of $X_u$'s is equal to the set of $\gamma_j$'s (but with unknown ordering), i.e., \[E \ \ \equiv \ \ \{X_1,X_2,\cdots,X_N\}= \{ \gamma_1, \gamma_2, \cdots, \gamma_N \}, \] then \[P\left(X_1=\gamma_j |E\right)=\frac{1}{N}. \] \begin{proof} Define sets $\mathfrak{P}$ and $\mathfrak{P}_j$ as follows: \[\mathfrak{P}= \textrm{The set of all permutations $\Pi$ on }\{1,2,\cdots,N\}. \] \[\mathfrak{P}_j= \textrm{The set of all permutations $\Pi$ on }\{1,2,\cdots,N\} \textrm{ such that } \Pi(1)=j. \] We have $|\mathfrak{P}|=N!$ and $|\mathfrak{P}|=(N-1)!$. Then \begin{align*} P(X_1=\alpha_j |E)&=\frac{\sum_{\pi \in \mathfrak{P}_j} f_{X_1,X_2, \cdots,X_N} (\gamma_{\pi(1)}, \gamma_{\pi(2)}, \cdots, \gamma_{\pi(N)})} {\sum_{\pi \in \mathfrak{P}} f_{X_1,X_2, \cdots,X_N} (\gamma_{\pi(1)}, \gamma_{\pi(2)}, \cdots, \gamma_{\pi(N)})}\\ &=\frac{(N-1)! \prod\limits_{u=1}^{N} \frac{1}{b_u-a_u}}{N! \prod\limits_{u=1}^{N} \frac{1}{b_u-a_u}}\\ &=\frac{1}{N}. \end{align*} \end{proof} \section{Completion of the Proof of Lemma \ref{lem4}}\label{sec:app_d} Let $p_1 \in (0,1)$, and let $N^{(n)}$ be a random variable as above, i.e., $N^{(n)} \rightarrow \infty$ as $n \rightarrow \infty$. Consider the sequence of independent random variables $Y_u \sim Bernoulli (p_u)$ for $u=1, 2, \cdots, N^{(n)}$ such that \begin{enumerate} \item For all $n$ and all $u \in \left\{1, 2, \cdots, N^{(n)}\right\}$, $\abs*{p_u-p_1} \leq \zeta_n.$ \item $\lim\limits_{n\to\infty} \zeta_n =0.$ \end{enumerate} Define \[\overline{Y}\triangleq \frac{1}{N^{(n)}}\sum_{u=1}^{N^{(n)}}Y_u,\] then $\overline{Y} \xrightarrow{d} p_1.$ \begin{proof} Note \begin{align} \nonumber E[\overline{Y}] &= \frac{1}{N^{(n)}}\sum_{u=1}^{N^{(n)}}p_u\\ \nonumber &\leq \frac{1}{N^{(n)}} \sum_{u=1}^{N^{(n)}}\left(p_1+\zeta_n\right)\\ \nonumber &= \frac{1}{N^{(n)}}\cdot N^{(n)}(p_1+\zeta_n) \\ \nonumber &= p_1+\zeta_n. \ \ \end{align} Similarly we can prove $E\left[\overline{Y}\right]\geq p_1-\zeta_n$. Since as $n$ becomes large, $\zeta_n\rightarrow 0$ and $p_1 \in (0,1)$, we can conclude \begin{align}\label{eq:H4} \lim\limits_{n\to\infty} E\left[\overline{Y}\right]=p_1. \end{align} Also, \begin{align} \nonumber Var\left(\overline{Y}\right) &= \frac{1}{\left(N^{(n)}\right)^2}\sum_{u=1}^{N^{(n)}}p_u\left(1-p_u\right) \\ \nonumber &\leq \frac{1}{(N^{(n)})^2} \sum_{u=1}^{N^{(n)}}\left(p_1+\zeta_n\right)\left(1-p_1+\zeta_n\right)\\ \nonumber &= \frac{1}{(N^{(n)})^2}\cdot N^{(n)}\left(p_1+\zeta_n\right)\left(1-p_1+\zeta_n\right) \\ \nonumber &= \frac{1}{N^{(n)}} \left(p_1+\zeta_n\right)\left(1-p_1+\zeta_n\right). \ \ \end{align} Thus, \begin{align}\label{eq:H5} \lim_{n\to\infty} Var\left(\overline{Y}\right)=0. \end{align} By using (\ref{eq:H4}), (\ref{eq:H5}), and Chebyshev's inequality, we can conclude \[\overline{Y}\xrightarrow{d} p_1.\] \end{proof} \section{Discussion} \label{discussion} \subsection{Markov Chain Model} \label{markov} As opposed to the i.i.d.\ case, we see from Section \ref{sec:perfect-MC} that if we do not limit $m=m(n)$, the assumed obfuscation method will not be sufficient to achieve perfect privacy. There are a few natural questions here. First, for a given noise level, what would be the maximum $m(n)$ that could guarantee perfect privacy in this model? The more interesting question is, how can we possibly modify the obfuscation technique to make it more suitable for the Markov chain model? A natural solution seems to be re-generating the obfuscation random variables $R_u$ periodically. This will keep the adversary from easily estimating them by observing a long sequence of data at a small increase in complexity. In fact, this will make the obfuscation much more \emph{robust} to modeling uncertainties and errors. It is worth noting, however, that this change would not affect the other results in the paper. That is, even if the obfuscation random variables are re-generated frequently, it is relatively easy to check that all the previous theorems in the paper remain valid. However, the increase in robustness to modeling errors will definitely be a significant advantage. Thus, the question is how often should the random variable $R_u$ be re-generated to strike a good balance between complexity and privacy? These are all interesting questions for future research. \subsection{Obfuscating the Samples of Users' Data Using Continuous Noise} Here we argue that for the setting of this paper, continuous noise such as that drawn from a Gaussian distribution is not a good option to obfuscate the sample of users' data drawn from a finite alphabet when we want to achieve perfect privacy. For a better understanding, let's consider a simple example. \begin{example} Consider the scenario where the users' datasets are governed by an i.i.d.\ model and the number of possible values for each sample of the users' data ($r$) is equal to 2 (two-states model). Note that the data sequence for user $u$ is a Bernoulli random variable with parameter $p_u$. Assume that the actual sample of the data of user $u$ at time $k$ ($X_u(k)$) is obfuscated using noise drawn from a Gaussian distribution ($S_u(k)$), and $Z_u(k)$ is the obfuscated version of $X_u(k)$. That is, we can write \[Z_u(k)=X_u(k)+S_u(k); \ \ \ \ \ S_u(k) \sim N\left(0, R_u\right), \] where $R_u$ is chosen from some distribution. For simplicity, we can consider $R_u\sim N\left(0, a^2_n\right)$ where $a_n$ is the noise level. We also apply anonymization to $Z_u(k)$, and, as before, $Y_u(k)$ is the reported sample of the data of user $u$ at time $k$ after applying anonymization. Per Section \ref{sec:framework}, anonymization is modeled by a random permutation $\Pi(u)$ on the set of $n$ users. Now, the question is as follows: Is it possible to achieve perfect privacy independent of the number of adversary's observation ($m$) while using this continuous noise ($S_u(k)$) to obfuscate the sample of users' data? Note that the density function of the reported sample of the data of user $u$ after applying obfuscation is \begin{align} \nonumber f_{Z_u}(z)&= p_u f_{S_u(k)}(z-1)+(1-p_u) f_{S_u(k)}(z) \\ \nonumber &= p_u \frac{1}{\sqrt{2\pi}R_u}e^{-\frac{(z-1)^2}{2R_u}}+(1-p_u)\frac{1}{\sqrt{2\pi}R_u}e^{-\frac{z^2}{2R_u}}.\ \ \end{align} In this case, when the adversary's number of observations is large, the adversary can estimate the values of $P_u$ and $R_u$ for each user with an arbitrarily small error probability. As a result, the adversary can de-anonymize the data and then recover $X_u(k)$. The conclusion here is that a continuous noise distribution gives too much information to the adversary when used for obfuscation of finite alphabet data. A method to remedy this issue is to regenerate the random variables $R_u$ frequently (similar to our previous discussion for Markov chains). Understanding the optimal frequency of such a regeneration and detailed analysis in this case is an interesting future research direction. \end{example} \subsection{Relation to Differential Privacy} Differential privacy is mainly used when there is a statistical database of users' sensitive information, and the goal is to protect an individual's data while publishing aggregate information about the database \cite{ lee2012differential, bordenabe2014optimal, chatzikokolakis2015geo, nguyen2013differential, machanavajjhala2008privacy, kousha2}. The goal of differential privacy is publishing aggregate queries with low sensitivity, which means the effect of changes in a single individual on the outcome of the aggregated information is negligible. In \cite{geo2013} three different approaches for differential privacy are presented. The one that best matches our setting is stated as \begin{align*} \frac{P(X_1(k)=x_1|\textbf{Y})}{P(X_1(k)=x_2|\textbf{Y})} \leq e^{\epsilon_r}\frac{P(X_1(k)=x_1)}{P(X_1(k)=x_2)}, \end{align*} where $\textbf{Y}$ is the set of reported datasets. It means $\textbf{Y}$ has a limited effect on the probabilities assigned by the attacker. In differential privacy, user $1$ has strongest differential privacy when $\epsilon_r=0$. In Lemma \ref{lem4}, we proved that if user $1$ has perfect privacy, this implies that asymptotically (for large enough $n$) \begin{align}\label{eq1} P\left(X_1(k)=x_1 \big{|} \textbf{Y}\right)\rightarrow P\left(X_1(k)=x_1\right). \end{align} \begin{align}\label{eq2} P\left(X_1(k)=x_2 \big{|} \textbf{Y}\right)\rightarrow P\left(X_1(k)=x_2\right). \end{align} As a result, by using (\ref{eq1}) and (\ref{eq2}), we can conclude that if we satisfy the perfect privacy condition given in this paper, we also satisfy differential privacy with $\epsilon_r=0$, i.e., the strongest case of differential privacy. \section{Conclusions} In this paper, we have considered both obfuscation and anonymization techniques to achieve privacy. The privacy level of the users depends on both $m(n)$ (number of observations per user by the adversary for a fixed anonymization mapping) and $a_n$ (noise level). That is, larger $m(n)$ and smaller $a_n$ indicate weaker privacy. We characterized the limits of privacy in the entire $m(n)-a_n$ plane for the i.i.d.\ case; that is, we obtained the exact values of the thresholds for $m(n)$ and $a_n$ required for privacy to be maintained. We showed that if $m(n)$ is fewer than $O\left(n^{\frac{2}{r-1}}\right)$, or $a_n$ is larger than $\Omega\left(n^{-\frac{1}{r-1}}\right)$, users have perfect privacy. On the other hand, if neither of these two conditions is satisfied, users have no privacy. For the case where the users' patterns are modeled by Markov chains, we obtained a no-privacy region in the $m(n)-a_n$ plane. Future research in this area needs to characterize the exact privacy/no-privacy regions when user data sequences obey Markov models. It is also important to consider different ways to obfuscate users' data sets and study the utility-privacy trade-offs for different types of obfuscation techniques. \section{Framework} \label{sec:framework} In this paper, we adopt a similar framework to that employed in~\cite{tifs2016,ciss2017}. The general set up is provided here, and the refinement to the precise models for this paper will be presented in the following sections. We assume a system with $n$ users with $X_u(k)$ denoting a sample of the data of user $u$ at time $k$, which we would like to protect from an interested adversary $\mathcal{A}$. We consider a strong adversary $\mathcal{A}$ that has complete statistical knowledge of the users' data patterns based on the previous observations or other resources. In order to secure data privacy of users, both obfuscation and anonymization techniques are used as shown in Figure \ref{fig:xyz}. In Figure \ref{fig:xyz}, $Z_u(k)$ shows the (reported) sample of the data of user $u$ at time $k$ after applying obfuscation, and $Y_u(k)$ shows the (reported) sample of the data of user $u$ at time $k$ after applying anonymization. The adversary observes only $Y_u(k)$, $k=1,2,\cdots, m(n)$, where $m(n)$ is the number of observations of each user before the identities are permuted. The adversary then tries to estimate $X_u(k)$ by using those observations. \begin{figure}[h] \centering \includegraphics[width = 0.75\linewidth]{fig/xyz} \caption{Applying obfuscation and anonymization techniques to users' data samples.} \label{fig:xyz} \end{figure} Let $\textbf{X}_u$ be the $m(n) \times 1$ vector containing the sample of the data of user $u$, and $\textbf{X}$ be the $m(n) \times n$ matrix with $u^{th}$ column equal to $\textbf{X}_u$; \[\textbf{X}_u = \begin{bmatrix} X_u(1) \\ X_u(2) \\ \vdots \\X_u(m) \end{bmatrix} , \ \ \ \textbf{X} =\left[\textbf{X}_{1}, \textbf{X}_{2}, \cdots, \textbf{X}_{n}\right]. \] \textit{Data Samples Model:} We assume there are $r \geq 2$ possible values ($0,1, \cdots, r-1$) for each sample of the users' data. In the first part of the paper (perfect privacy analysis), we assume an i.i.d.\ model as motivated in Section \ref{intro}. In the second part of the paper (converse results: no privacy region), the users' datasets are governed by irreducible and aperiodic Markov chains. At any time, $X_u(k)$ is equal to a value in $\left\{0,1, \cdots, r-1 \right\}$ according to a user-specific probability distribution. The collection of user distributions, which satisfy some mild regularity conditions discussed below, is known to the adversary $\mathcal{A}$, and he/she employs such to distinguish different users based on statistical matching of those user distributions to traces of user activity of length $m(n)$ \textit{Obfuscation Model:} The first step in obtaining privacy is to apply the obfuscation operation in order to perturb the users' data samples. In this paper, we assume that each user has only limited knowledge of the characteristics of the overall population and thus we employ a simple distributed method in which the samples of the data of each user are reported with error with a certain probability, where that probability itself is generated randomly for each user. In other words, the obfuscated data is obtained by passing the users' data through an $r$-ary symmetric channel with a random error probability. More precisely, let $\textbf{Z}_u$ be the vector which contains the obfuscated versions of user $u$'s data samples, and $\textbf{Z}$ is the collection of $\textbf{Z}_u$ for all users, \[\textbf{Z}_u = \begin{bmatrix} Z_u(1) \\ Z_u(2) \\ \vdots \\Z_u(m) \end{bmatrix} , \ \ \ \textbf{Z} =\left[ \textbf{Z}_{1}, \textbf{Z}_{2}, \cdots, \textbf{Z}_{n}\right]. \] To create a noisy version of data samples, for each user $u$, we independently generate a random variable $R_u$ that is uniformly distributed between $0$ and $a_n$, where $a_n \in (0,1]$. The value of $R_u$ gives the probability that a user's data sample is changed to a different data sample by obfuscation, and $a_n$ is termed the ``noise level'' of the system. For the case of $r=2$ where there are two states for users' data (state $0$ and state $1$), the obfuscated data is obtained by passing users' data through a Binary Symmetric Channel (BSC) with a small error probability~\cite{randomizedresponse}. Thus, we can write \[ {Z}_{u}(k)=\begin{cases} {X}_{u}(k), & \textrm{with probability } 1-R_u.\\ 1-{X}_{u}(k),& \textrm{with probability } R_u. \end{cases} \] When $r>2$, for $l \in \{0,1,\cdots, r-1\}$: \[ P({Z}_{u}(k)=l| X_{u}(k)=i) =\begin{cases} 1-R_u, & \textrm{for } l=i.\\ \frac{R_u}{r-1}, & \textrm{for } l \neq i. \end{cases} \] Note that the effect of the obfuscation is to alter the probability distribution function of each user across the $r$ possibilities in a way that is unknown to the adversary, since it is independent of all past activity of the user, and hence the obfuscation inhibits user identification. For each user, $R_u$ is generated once and is kept constant for the collection of samples of length $m(n)$, thus, providing a very low-weight obfuscation algorithm. We will discuss the extension to the case where $R_u$ is regenerated independently over time in Section \ref{sec:perfect-MC}. There, we will also provide a discussion about obfuscation using continuous noise distributions (e.g., Gaussian noise). \textit{Anonymization Model:} Anonymization is modeled by a random permutation $\Pi$ on the set of $n$ users. The user $u$ is assigned the pseudonym $\Pi(u)$. $\textbf{Y}$ is the anonymized version of $\textbf{Z}$; thus, \begin{align} \nonumber \textbf{Y} &=\textrm{Perm}\left(\textbf{Z}_{1}, \textbf{Z}_{2}, \cdots, \textbf{Z}_{n}; \Pi \right) \\ \nonumber &=\left[ \textbf{Z}_{\Pi^{-1}(1)}, \textbf{Z}_{\Pi^{-1}(2)}, \cdots, \textbf{Z}_{\Pi^{-1}(n)}\right ] \\ \nonumber &=\left[ \textbf{Y}_{1}, \textbf{Y}_{2}, \cdots, \textbf{Y}_{n}\right], \ \ \end{align} where $\textrm{Perm}( \ . \ , \Pi)$ is permutation operation with permutation function $\Pi$. As a result, $\textbf{Y}_{u} = \textbf{Z}_{\Pi^{-1}(u)}$ and $\textbf{Y}_{\Pi(u)} = \textbf{Z}_{u}$. \textit{Adversary Model:} We protect against the strongest reasonable adversary. Through past observations or some other sources, the adversary is assumed to have complete statistical knowledge of the users' patterns; in other words, he/she knows the probability distribution for each user on the set of data samples $\{0,1,\ldots,r-1\}$. As discussed in the model for the data samples, the parameters $\textbf{p}_u$, $u=1, 2, \cdots, n$ are drawn independently from a continuous density function, $f_\textbf{P}(\textbf{p}_u)$, which has support on a subset of a defined hypercube. The density $f_\textbf{P}(\textbf{p}_u)$ might be unknown to the adversary, as all that is assumed here is that such a density exists, and it will be evident from our results that knowing or not knowing $f_\textbf{P}(\textbf{p}_u)$ does not change the results asymptotically. Specifically, from the results of Section \ref{perfectsec}, we conclude that user $u$ has perfect privacy even if the adversary knows $f_\textbf{P}(\textbf{p}_u)$. In addition, in Section \ref{converse}, it is shown that the adversary can recover the true data of user $u$ at time $k$ without using the specific density function of $f_\textbf{P}(\textbf{p}_u)$, and as result, users have no privacy even if the adversary does not know $f_\textbf{P}(\textbf{p}_u)$. The adversary also knows the value of $a_n$ as it is a design parameter. However, the adversary does not know the realization of the random permutation $\Pi$ or the realizations of the random variables $R_u$, as these are independent of the past behavior of the users. It is critical to note that we assume the adversary does not have any auxiliary information or side information about users' data. In \cite{tifs2016}, perfect privacy is defined as follows: \begin{define} User $u$ has \emph{perfect privacy} at time $k$, if and only if \begin{align \nonumber \forall k\in \mathbb{N}, \ \ \ \lim\limits_{n\rightarrow \infty} I \left(X_u(k);{\textbf{Y}}\right) =0, \end{align} where $I(X;Y)$ denotes the mutual information between random variables (vectors) $X$ and $Y$. \end{define} \noindent In this paper, we also consider the situation in which there is no privacy. \begin{define} For an algorithm for the adversary that tries to estimate the actual sample of data of user $u$ at time $k$, define \[P_e(u,k)\triangleq P\left(\widetilde{X_u(k)} \neq X_u(k)\right),\] where $X_u(k)$ is the actual sample of the data of user $u$ at time $k$, $\widetilde{X_u(k)}$ is the adversary's estimated sample of the data of user $u$ at time $k$, and $P_e(u,k)$ is the error probability. Now, define ${\cal E}$ as the set of all possible adversary's estimators; then, user $u$ has \emph{no privacy} at time $k$, if and only if for large enough $n$, \[ \forall k\in \mathbb{N}, \ \ \ P^{*}_e(u,k)\triangleq \inf_{\cal E} {P\left(\widetilde{X_u(k)} \neq X_u(k)\right)} \rightarrow 0. \] Hence, a user has no privacy if there exists an algorithm for the adversary to estimate $X_u(k)$ with diminishing error probability as $n$ goes to infinity. \end{define} \textbf{\textit{Discussion:}} Both of the privacy definitions given above (perfect privacy and no privacy) are asymptotic in the number of users $(n \to \infty)$, which allows us to find clean analytical results for the fundamental limits. Moreover, in many IoT applications, such as ride sharing and dining recommendation applications, the number of users is large. \textbf{\textit{Notation:}} Note that the sample of data of user $u$ at time $k$ after applying obfuscation $\left(Z_u(k)\right)$ and the sample of data of user $u$ at time $k$ after applying anonymization $\left(Y_u(k)\right)$ depend on the number of users in the network $(n)$, while the actual sample of data of user $u$ at time $k$ is independent of the number of users $(n)$. Despite the dependency in the former cases, we omit this subscript $(n)$ on $\left(Z_u^{(n)}(k), Y_u^{(n)}(k) \right)$ to avoid confusion and make the notation consistent. \textbf{\textit{Notation:}} Throughout the paper, $X_n \xrightarrow{d} X$ denotes convergence in distribution. Also, We use $P\left(X=x\bigg{|} Y=y\right)$ for the conditional probability of $X=x$ given $Y=y$. When we write $P\left(X=x \bigg{|}Y\right)$, we are referring to a random variable that is defined as a function of $Y$. \section{Perfect Privacy Analysis: I.I.D.\ Case} \label{perfectsec} \subsection{Two-States Model} We first consider the two-states case $(r=2)$ which captures the salient aspects of the problem. For the two-states case, the sample of the data of user $u$ at any time is a Bernoulli random variable with parameter $p_u$, which is the probability of user $u$ having data sample $1$. Thus, \[X_u(k) \sim Bernoulli \left(p_u\right).\] Per Section \ref{sec:framework}, the parameters $p_u$, $u=1, 2, \cdots, n$ are drawn independently from a continuous density function, $f_P(p_u)$, on the $(0,1)$ interval. We assume there are $\delta_1, \delta_2>0$ such that:\footnote{The condition $\delta_1<f_P(p_u) <\delta_2$ is not actually necessary for the results and can be relaxed; however, we keep it here to avoid unnecessary technicalities.} \begin{equation} \nonumber\begin{cases} \delta_1<f_P(p_u) <\delta_2, & p_u \in (0,1).\\ f_P(p_u)=0, & p _u\notin (0,1). \end{cases} \end{equation} The adversary knows the values of $p_u$, $u=1, 2, \cdots, n$ and uses this knowledge to identify users. We will use capital letters (i.e., $P_u$) when we are referring to the random variable, and use lower case (i.e., $p_u$) to refer to the realization of $P_u$. In addition, since the user data $\left(X_u(k)\right)$ are i.i.d.\ and have a Bernoulli distribution, the obfuscated data $\left(Z_u(k)\right)$ are also i.i.d.\ with a Bernoulli distribution. Specifically, \[Z_u(k) \sim Bernoulli\left(Q_u\right),\] where \begin{align} \nonumber {Q}_u &=P_u(1-R_u)+(1-P_u)R_u \\ \nonumber &= P_u+\left(1-2P_u\right)R_u,\ \ \end{align} and recall that $R_u$ is the probability that user $u$'s data sample is altered at any time. For convenience, define a vector where element $Q_u$ is the probability that an obfuscated data sample of user $u$ is equal to one, and \[\textbf{Q} =\left[{Q}_{1},{Q}_{2}, \cdots,{Q}_{n}\right].\] Thus, a vector containing the permutation of those probabilities after anonymization is given by: \begin{align} \nonumber \textbf{V} &=\textrm{Perm}\left({Q}_{1}, {Q}_{2}, \cdots, {Q}_{n}; \Pi \right) \\ \nonumber &=\left[ {Q}_{\Pi^{-1}(1)}, {Q}_{\Pi^{-1}(2)}, \cdots, {Q}_{\Pi^{-1}(n)}\right ] \\ \nonumber &=\left[{V}_{1}, {V}_{2}, \cdots, {V}_{n}\right ] ,\ \ \end{align} where ${V}_{u} = {Q}_{\Pi^{-1}(u)}$ and ${V}_{\Pi(u)} = {Q}_{u}$. As a result, for $u=1,2,..., n$, the distribution of the data symbols for the user with pseudonym $u$ is given by: \[ Y_u(k) \sim Bernoulli \left(V_u\right) \sim Bernoulli\left(Q_{\Pi^{-1}(u)}\right) .\] The following theorem states that if $a_n$ is significantly larger than $\frac{1}{n}$ in this two-states model, then all users have perfect privacy independent of the value of $m(n)$. \begin{thm}\label{two_state_thm} For the above two-states model, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined above, and \begin{itemize} \item $m=m(n)$ is arbitrary; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(1-\beta\right)}$ for any $c'>0$ and $0<\beta<1$; \end{itemize} then, user 1 has perfect privacy. That is, \begin{align \nonumber \forall k\in \mathbb{N}, \ \ \ \lim\limits_{n\rightarrow \infty} I \left(X_1(k);{\textbf{Y}}\right) =0. \end{align} \end{thm} The proof of Theorem \ref{two_state_thm} will be provided for the case $0\leq p_1<\frac{1}{2}$, as the proof for the case $\frac{1}{2}\leq p_1\leq1$ is analogous and is thus omitted. \\ \noindent \textbf{Intuition behind the Proof of Theorem \ref{two_state_thm}:} Since $m(n)$ is arbitrary, the adversary is able to estimate very accurately (in the limit, perfectly) the distribution from which each data sequence $\textbf{Y}_u$, $u= 1, 2, \cdots, n$ is drawn; that is, the adversary is able to accurately estimate the probability $V_u$, $u= 1, 2, \cdots, n$. Clearly, if there were no obfuscation for each user $u$, the adversary would then simply look for the $j$ such that $p_j$ is very close to $V_u$ and set $\widetilde{X_j(k)}=Y_u(k)$, resulting in no privacy for any user. We want to make certain that the adversary obtains no information about $X_1(k)$, the sample of data of user $1$ at time $k$. To do such, we will establish that there are a large number of users whom have a probability $p_u$ that when obfuscated could have resulted in a probability consistent with $p_1$. Consider asking whether another probability $p_2$ is sufficiently close enough to be confused with $p_1$ after obfuscation; in particular, we will look for $p_2$ such that, even if the adversary is given the obfuscated probabilities $V_{\Pi(1)}$ and $V_{\Pi(2)}$, he/she cannot associate these probabilities with $p_1$ and $p_2$. This requires that the distributions $Q_{1}$ and $Q_{2}$ of the obfuscated data of user $1$ and user $2$ have significant overlap; we explore this next. Recall that $Q_u=P_u+ (1-2P_u)R_u$, and $R_u\sim Uniform [0, a_n]$. Thus, we know $Q_u {|} P_u=p_u$ has a uniform distribution with length $(1-2p_u)a_n$. Specifically, \[Q_u\bigg{|}P_u=p_u \sim Uniform \left[p_u,p_u+(1-2p_u)a_n\right].\] Figure \ref{fig:piqi_a} shows the distribution of $Q_u$ given $P_u=p_u$. \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/piqi_a} \caption{Distribution of $Q_u$ given $P_u=p_u$.} \label{fig:piqi_a} \end{figure} Consider two cases: In the first case, the support of the distributions $Q_1\big{|} P_1=p_1$ and $Q_2\big{|} P_2=p_2$ are small relative to the difference between $p_1$ and $p_2$ (Figure \ref{fig:case1}); in this case, given the probabilities $V_{\Pi(1)}$ and $V_{\Pi(2)}$ of the anonymized data sequences, the adversary can associate those with $p_1$ and $p_2$ without error. In the second case, the support of the distributions $Q_1\big{|} P_1=p_1$ and $Q_2\big{|} P_2=p_2$ is large relative to the difference between $p_1$ and $p_2$ (Figure \ref{fig:case2}), so it is difficult for the adversary to associate the probabilities $V_{\Pi(1)}$ and $V_{\Pi(2)}$ of the anonymized data sequences with $p_1$ and $p_2$. In particular, if $V_{\Pi(1)}$ and $V_{\Pi(2)}$ fall into the overlap of the support of $Q_1$ and $Q_2$, we will show the adversary can only guess randomly how to de-anonymize the data. Thus, if the ratio of the support of the distributions to $\big{|}p_1-p_2\big{|}$ goes to infinity, the adversary's posterior probability for each user converges to $\frac{1}{2}$, thus, implying no information leakage on the user identities. More generally, if we can guarantee that there will be a large set of users with $p_u$'s very close to $p_1$ compared to the support of $Q_1\big{|} P_1=p_1$, we will be able to obtain perfect privacy as demonstrated rigorously below. \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/case1} \caption{Case 1: The support of the distributions is small relative to the difference between $p_1$ and $p_2$.} \label{fig:case1} \end{figure} \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/case2} \caption{Case 2: The support of the distributions is large relative to the difference between $p_1$ and $p_2$.} \label{fig:case2} \end{figure} Given this intuition, the formal proof proceeds as follows. Given $p_1$, we define a set $J^{(n)}$ of users whose parameter $p_u$ of their data distributions is sufficiently close to $p_1$ (Figure \ref{fig:case2}; case 2), so that it is likely that $Q_1$ and $Q_u$ cannot be readily associated with $p_1$ and $p_u$. The purpose of Lemmas \ref{lemOnePointFive}, \ref{lem2}, and \ref{lem3} is to show that, from the adversary's perspective, the users in set $J^{(n)}$ are indistinguishable. More specifically, the goal is to show that the obfuscated data corresponding to each of these users could have been generated by any other users in $J^{(n)}$ in an equally likely manner. To show this, Lemma \ref{lemOnePointFive} employs the fact that, if the observed values of $N$ uniformly distributed random variables ($N$ is size of set $J^{(n)}$) are within the intersection of their ranges, it is impossible to infer any information about the matching between the observed values and the distributions. That is, all possible $N!$ matchings are equally likely. Lemmas \ref{lem2} and \ref{lem3} leverage Lemma \ref{lemOnePointFive} to show that even if the adversary is given a set that includes all of the pseudonyms of the users in set $J^{(n)}$ (i.e., $\Pi(J^{(n)})\overset{\Delta}{=} \left\{\Pi^{-1}(u) \in J^{(n)}\right\}$) he/she still will not be able to infer any information about the matching of each specific user in set $J^{(n)}$ and his pseudonym. Then Lemma \ref{lem4} uses the above fact to show that the mutual information between the data set of user $1$ at time $k$ and the observed data sets of the adversary converges to zero for large enough $n$. \noindent \textbf{Proof of Theorem \ref{two_state_thm}:} \begin{proof} Note, per Lemma~\ref{lemx} of Appendix \ref{sec:app_a}, it is sufficient to establish the results on a sequence of sets with high probability. That is, we can condition on high-probability events. Now, define the critical set $J^{(n)}$ with size $N^{(n)}=\big{|}J^{(n)}\big{|}$ for $0\leq p_1<\frac{1}{2}$ as follows: \[J^{(n)}= \left\{ u \in \{1, 2, \dots, n\}: p_1 \leq P_u\leq p_1+\epsilon_n; p_1+\epsilon_n\leq Q_u\leq p_1+(1-2p_1)a_n\right\}, \] where $\epsilon_n \triangleq \frac{1}{n^{1-\frac{\beta}{2}}}$, $a_n= c'n^{-\left(1-\beta\right)}$ ,and $\beta$ is defined in the statement of Theorem \ref{two_state_thm}. Note for large enough $n$, if $0\leq p_1<\frac{1}{2}$, we have $0\leq p_u<\frac{1}{2}$. As a result, \[Q_u\bigg{|}P_u=p_u \sim Uniform \left(p_u,p_u+(1-2p_u)a_n\right).\] We can prove that with high probability, $1 \in J^{(n)}$ for large enough $n$, as follows. First, Note that \[Q_1\bigg{|}P_1=p_1 \sim Uniform \left(p_1,p_1+(1-2p_1)a_n\right).\] Now, according to Figure \ref{fig:piqi_c}, \begin{align} \nonumber P\left(1 \in J^{(n)} \right) &= 1- \frac{\epsilon_n}{ \left(1-2p_1 \right)a_n}\\ \nonumber &= 1- \frac{1}{ \left(1-2p_1 \right)c'n^{\frac{\beta}{2}}}, \ \ \end{align} thus, for any $c'>0$ and large enough $n$, \begin{align} \nonumber P\left(1 \in J^{(n)} \right) \to 1. \end{align} \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/piqi_b} \caption{Range of $P_u$ and $Q_u$ for elements of set $J^{(n)}$ and probability density function of $Q_u\bigg{|}P_u=p_u$.} \label{fig:piqi_b} \end{figure} \begin{figure}[h] \centering \includegraphics[width = 0.8\linewidth]{fig/piqi_c} \caption{Range of $P_u$ and $Q_u$ for elements of set $J^{(n)}$ and probability density function of $Q_1\bigg{|}P_1=p_1$.} \label{fig:piqi_c} \end{figure} Now in the second step, we define the probability $W_j^{(n)}$ for any $j \in \Pi(J^{(n)})=\{\Pi(u): u \in J^{(n)} \}$ as \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}, \Pi (J^{(n)})\right).\] $W_j^{(n)}$ is the conditional probability that $\Pi(1)=j$ after perfectly observing the values of the permuted version of obfuscated probabilities ($\textbf{V}$) and set including all of the pseudonyms of the users in set $J^{(n)}$ $\left(\Pi(J^{(n)})\right)$. Since $\textbf{V}$ and $\Pi (J^{(n)})$ are random, $W_j^{(n)}$ is a random variable. However, we will prove shortly that in fact $W_j^{(n)}=\frac{1}{N^{(n)}}$, for all $j \in \Pi (J^{(n)})$. Note: Since we are looking from the adversary's point of view, the assumption is that all the values of $P_u$, $u \in \{1,2,\cdots,n\}$ are known, so all of the probabilities are conditioned on the values of $P_1=p_1, P_2=p_2, \cdots, P_n=p_n$. Thus, to be accurate, we should write \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}, \Pi (J^{(n)}), P_1, P_2, \cdots, P_n\right).\] Nevertheless, for simplicity of notation, we often omit the conditioning on $P_1, P_2, \cdots, P_n$. First, we need a lemma from elementary probability. \begin{lem} \label{lemOnePointFive} Let $N$ be a positive integer, and let $a_1, a_2, \cdots, a_N$ and $b_1, b_2, \cdots, b_N$ be real numbers such that $a_u \leq b_u$ for all $u$. Assume that $X_1, X_2, \cdots, X_N$ are independent random variables such that \[X_u \sim Uniform [a_u,b_u]. \] Let also $\gamma_1, \gamma_2, \cdots, \gamma_N$ be distinct real numbers such that \[ \gamma_j \in \bigcap_{u=1}^{N} [a_u, b_u] \ \ \textrm{for all }j \in \{1,2,..,N\}.\] Suppose that we know the event $E$ has occurred, meaning that the observed values of $X_u$'s are equal to the set of $\gamma_j$'s (but with unknown ordering), i.e., \[E \ \ \equiv \ \ \{X_1, X_2, \cdots, X_N\}= \{ \gamma_1, \gamma_2, \cdots, \gamma_N \},\] then \[P\left(X_1=\gamma_j |E\right)=\frac{1}{N}. \] \end{lem} \begin{proof} Lemma \ref{lemOnePointFive} is proved in Appendix \ref{sec:app_b}. \end{proof} Using the above lemma, we can state our desired result for $W_j^{(n)}$. \begin{lem} \label{lem2} For all $j \in \Pi (J^{(n)})$, $W_j^{(n)}=\frac{1}{N^{(n)}}.$ \end{lem} \begin{proof} We argue that the setting of this lemma is essentially equivalent to the assumptions in Lemma \ref{lemOnePointFive}. First, remember that \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}, \Pi (J^{(n)})\right).\] Note that ${Q}_u= P_u+(1-2P_u)R_u$, and since $R_u$ is uniformly distributed, ${Q}_u$ conditioned on $P_u$ is also uniformly distributed in the appropriate intervals. Moreover, since ${V}_{u} = {Q}_{\Pi^{-1}(u)}$, we conclude ${V}_{u}$ is also uniformly distributed. So, looking at the definition of $W_j^{(n)}$, we can say the following: given the values of the uniformly distributed random variables ${Q}_u$, we would like to know which one of the values in $\textbf{V}$ is the actual value of ${Q}_1={V}_{\Pi(1)}$, i.e., is $\Pi(1)=j$? This is equivalent to the setting of Lemma \ref{lemOnePointFive} as described further below. Note that since $1 \in J^{(n)}$, $\Pi(1) \in \Pi (J^{(n)})$. Therefore, when searching for the value of $\Pi(1)$, it is sufficient to look inside set $\Pi (J^{(n)})$. Therefore, instead of looking among all the values of ${V}_{j}$, it is sufficient to look at ${V}_{j}$ for $j \in \Pi (J^{(n)})$. Let's show these values by $\textbf{V}_{\Pi} =\{v_1, v_2, \cdots, v_{N^{(n)}} \}$, so, \[W_j^{(n)}= P\left(\Pi(1)=j \bigg{|} \textbf{V}_{\Pi}, \Pi (J^{(n)})\right).\] Thus, we have the following scenario: $Q_u, u \in J^{(n)}$ are independent random variables, and \[Q_u\big{|} P_u=p_u \sim Uniform [p_u, p_u+(1-2p_u)a_n]. \] Also, $v_1, v_2, \cdots, v_{N^{(n)}}$ are the observed values of $Q_u$ with unknown ordering (unknown mapping $\Pi$). We also know from the definition of set $J^{(n)}$ that \[P_u \leq p_1+\epsilon_n \leq Q_u,\] \[Q_u \leq p_1(1-2a_n)+an \leq P_u(1-2a_n)+a_n,\] so, we can conclude \[ v_j \in \bigcap_{u=1}^{N^{(n)}} [p_u, p_u+(1-2p_u)a_n] \ \ \textrm{for all }j \in \{1,2,..,N^{(n)}\}. \] We know the event $E$ has occurred, meaning that the observed values of $Q_u$'s are equal to set of $v_j$'s (but with unknown ordering), i.e., \[E \ \ \equiv \ \ \{Q_u, u \in J^{(n)}\}= \{ v_1, v_2, \cdots, v_{N^{(n)}} \}. \] Then, according to Lemma \ref{lemOnePointFive}, \[P\left(Q_1=v_j |E, P_1, P_2, \cdots, P_n \right)=\frac{1}{N^{(n)}}. \] Note that there is a subtle difference between this lemma and Lemma \ref{lemOnePointFive}. Here $N^{(n)}$ is a random variable while $N$ is a fixed number in Lemma \ref{lemOnePointFive}. Nevertheless, since the assertion holds for every fixed $N$, it also holds for the case where $N$ is a random variable. Now, note that \begin{align*} P\left(Q_1=v_j |E, P_1, P_2, \cdots, P_n \right) &= P\left(\Pi(1)=j \bigg{|} E, P_1, P_2, \cdots, P_n \right)\\ &=P\left(\Pi(1)=j \bigg{|} \textbf{V}_{\Pi}, \Pi (J^{(n)}), P_1, P_2, \cdots, P_n \right)\\ &=W_j^{(n)}. \end{align*} Thus, we can conclude \[W_j^{(n)}=\frac{1}{N^{(n)}}.\] \end{proof} In the third step, we define $\widetilde{W_j^{(n)}}$ for any $j \in \Pi (J^{(n)})$ as \[\widetilde{W_j^{(n)}} = P\left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right).\] $\widetilde{W_j^{(n)}}$ is the conditional probability that $\Pi(1)=j$ after observing the values of the anonymized version of the obfuscated samples of the users' data ($\textbf{Y}$) and the aggregate set including all the pseudonyms of the users in set $J^{(n)}$ (i.e., $\Pi(J^{(n)})\overset{\Delta}{=} \left\{\Pi^{-1}(j) \in J^{(n)}\right\}$). Since $\textbf{Y}$ and $\Pi (J^{(n)})$ are random, $\widetilde{W_j^{(n)}} $ is a random variable. Now, in the following lemma, we will prove $\widetilde{W_j^{(n)}} =\frac{1}{N^{(n)}}$, for all $j \in \Pi (J^{(n)})$ by using Lemma \ref{lem3}. Note in the following lemma, we want to show that even if the adversary is given a set including all of the pseudonyms of the users in set $J^{(n)}$, he/she cannot match each specific user in set $J^{(n)}$ and his pseudonym. \begin{lem} \label{lem3} For all $j \in \Pi (J^{(n)})$, $\widetilde{W_j^{(n)}}=\frac{1}{N^{(n)}}.$ \end{lem} \begin{proof} First, note that \begin{align*} \widetilde{W_j^{(n)}} = \sum_{\text{for all v}} P\left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right), \textbf{V}=\textbf{v}\right) P\left(\textbf{V}=\textbf{v} \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right)\right). \end{align*} Also, we note that given $\textbf{V}$, $\Pi(J^{(n)})$, and $\textbf{Y}$ are independent. Intuitively, this is because when observing $\textbf{Y}$, any information regarding $\Pi(J^{(n)})$ is leaked through estimating $\textbf{V}$. This can be rigorously proved similar to the proof of Lemma 1 in \cite{tifs2016}. We can state this fact as \[ P\left(Y_u(k)\ \bigg{ | } \ {V}_u=v_u, \Pi(J^{(n)}) \right) = P\left(Y_u(k)\ \bigg{ | } \ {V}_u=v_u\right)=v_u. \] The right and left hand side are given by $Bernoulli (v_u)$ distributions. As a result, \[ \widetilde{W_j^{(n)}} = \sum_{\text{for all \textbf{v}}} P\left(\Pi(1)=j \bigg{|} \Pi (J^{(n)}), \textbf{V}=\textbf{v}\right) P\left(\textbf{V}=\textbf{v} \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right)\right). \] Note $W_j^{(n)}=P\left(\Pi(1)=j \bigg{|} \Pi (J^{(n)}), \textbf{V}\right)$, so \begin{align} \nonumber \widetilde{W_j^{(n)}} &= \sum_{\text{for all \textbf{v}}}W_j^{(n)} P\left(\textbf{V}=\textbf{v} \bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right) \right) \\ \nonumber &= \frac{1}{N^{(n)}}\sum_{\text{for all \textbf{v}}} P\left(\textbf{V}=\textbf{v}\bigg{|} \textbf{Y}, \Pi \left(J^{(n)}\right)\right) \\ \nonumber &= \frac {1}{N^{(n)}}.\ \ \end{align} \end{proof} To show that no information is leaked, we need to show that the size of set $J^{(n)}$ goes to infinity. This is established in Lemma \ref{lem1}. \begin{lem} \label{lem1} If $N^{(n)} \triangleq |J^{(n)}| $, then $N^{(n)} \rightarrow \infty$ with high probability as $n \rightarrow \infty$. More specifically, there exists $\lambda>0$ such that \[ P\left(N^{(n)} > \frac{\lambda}{2}n^{\frac{\beta}{2}}\right) \rightarrow 1. \] \end{lem} \begin{proof} Lemma \ref{lem1} is proved in Appendix \ref{sec:app_c}. \end{proof} In the final step, we define $\widehat{W_j^{(n)}}$ for any $j \in \Pi (J^{(n)})$ as \[\widehat{W_j^{(n)}}=P\left(X_1(k)=1 \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right).\] $\widehat{W_j^{(n)}}$ is the conditional probability that $X_1(k)=1$ after observing the values of the anonymized version of the obfuscated samples of the users' data ($\textbf{Y}$) and the aggregate set including all of the pseudonyms of the users in set $J^{(n)}$ ($\Pi (J^{(n)})$). $\widehat{W_j^{(n)}} $ is a random variable because $\textbf{Y}$ and $\Pi (J^{(n)})$ are random. Now, in the following lemma, we will prove $\widehat{W_j^{(n)}}$ converges in distribution to $p_1$. Note that this is the probability from the adversary's point of view. That is, given that the adversary has observed $\textbf{Y}$ as well as the extra information $ \Pi (J^{(n)})$, what can he/she infer about $X_1(k)$? \begin{lem} \label{lem4} For all $j \in \Pi (J^{(n)})$, $\widehat{W_j^{(n)}} \xrightarrow{d} p_1.$ \end{lem} \begin{proof} We know \begin{align*} \widehat{W_j^{(n)}}= \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right) P\left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right), \end{align*} and according to the definition $\widetilde{W_j^{(n)}}=P \left(\Pi(1)=j \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right)$, we have \begin{align} \nonumber \widehat{W_j^{(n)}} &= \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right) \widetilde{W_j^{(n)}}\\ \nonumber &= \frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right). \end{align} We now claim that \[ P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)=p_1+o(1). \] The reasoning goes as follows. Given $\Pi(1)=j$ and knowing $\textbf{Y}$, we know that \[ Y_{\Pi(1)}(k)={Z}_{1}(k)=\begin{cases} {X}_{1}(k), & \textrm{with probability } 1-R_1.\\ 1-{X}_{1}(k), & \textrm{with probability } R_1. \end{cases} \] Thus, given $Y_{j}(k)=1$, Bayes' rule yields: \begin{align*} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)&= \left(1- R_1 \right) \frac{P(X_1(k)=1)}{P(Y_{\Pi(1)}(k)=1)}\\ &=\left(1-R_1 \right) \frac{p_1}{p_1 (1- R_1)+(1-p_1)R_1}\\ &=1-o(1), \end{align*} and similarly, given $Y_{j}(k)=0$, \begin{align*} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)&= R_1 \frac{P(X_1(k)=1)}{P(Y_{\Pi(1)}(k)=0)}\\ &=R_1 \frac{p_1}{p_1 (1- R_1)+(1-p_1)R_1}\\ &=o(1). \end{align*} Note that by the independence assumption, the above probabilities do not depend on the other values of $Y_{u}(k)$ (as we are conditioning on $\Pi(1)=j$ ). Thus, we can write \begin{align} \nonumber \widehat{W_j^{(n)}} &= \frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)})} P\left(X_1(k)=1 \bigg{|} \Pi(1)=j, \textbf{Y}, \Pi (J^{(n)})\right)\\ \nonumber &=\frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)}), Y_{j}(k)=1} (1-o(1)) + \frac{1}{N^{(n)}} \sum_{j \in \Pi(J^{(n)}), Y_{j}(k)=0} o(1). \end{align} First, note that since $\abs*{\left\{j \in \Pi(J^{(n)}), Y_{j}(k)=0\right\}} \leq N^{(n)}$, the second term above converges to zero, thus, \begin{align} \nonumber \widehat{W_j^{(n)}} \rightarrow \frac{\abs*{\left\{ j \in \Pi(J^{(n)}), Y_{\Pi(1)}(k)=1\right\} }}{N^{(n)}}. \end{align} Since for all $j \in \Pi(J^{(n)})$, $ Y_{j}(k) \sim Bernoulli \left(p_1+o(1)\right)$, by a simple application of Chebyshev's inequality, we can conclude $\widehat{W_j^{(n)}}\rightarrow p_1$. Appendix \ref{sec:app_d} provides the detail. \end{proof} As a result, \begin{align*} X_1(k) {|} \textbf{Y}, \Pi (J^{(n)})\rightarrow \textit{Bernoulli} (p_1), \end{align*} thus, \[H\left(X_1(k) \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right)\rightarrow H\left(X_1(k)\right).\] Since conditioning reduces entropy, \[H\left(X_1(k) \bigg{|} \textbf{Y}, \Pi (J^{(n)})\right)\leq H\left(X_1(k) \bigg{|} \textbf{Y}\right),\] and as a result, \[\lim_{n \rightarrow \infty} H\left(X_1(k)\right)-H\left(X_1(k) \bigg{|} \textbf{Y}\right) \leq 0,\] and \[\lim_{n \rightarrow \infty} I\left(X_1(k);\textbf{Y}\right)\leq 0.\] By knowing that $I\left(X_1(k);\textbf{Y}\right)$ cannot take any negative value, we can conclude that \[I\left(X_1(k);\textbf{Y}\right)\rightarrow 0.\] \end{proof} \subsection{Extension to $r$-States} Now, assume users' data samples can have $r$ possibilities $\left(0, 1, \cdots, r-1\right)$, and $p_u(i)$ shows the probability of user $u$ having data sample $i$. We define the vector $\textbf{p}_u$ and the matrix $\textbf{p}$ as \[\textbf{p}_u= \begin{bmatrix} p_u(1) \\ p_u(2) \\ \vdots \\p_u(r-1) \end{bmatrix} , \ \ \ \textbf{p} =\left[ \textbf{p}_{1}, \textbf{p}_{2}, \cdots, \textbf{p}_{n}\right]. \] We assume $p_u(i)$'s are drawn independently from some continuous density function, $f_\textbf{P}(\textbf{p}_u)$, which has support on a subset of the $(0,1)^{r-1}$ hypercube (Note that the $p_u(i)$'s sum to one, so one of them can be considered as the dependent value and the dimension is $r-1$). In particular, define the range of the distribution as \begin{align} \nonumber \mathcal{R}_{\textbf{p}} &= \{ (x_1,x_2, \cdots, x_{r-1}) \in (0,1)^{r-1}: x_i > 0 , x_1+x_2+\cdots+x_{r-1} < 1,\ i=1, 2, \cdots, r-1\}. \end{align} Figure~\ref{fig:rp} shows the range $\mathcal{R}_{\textbf{p}}$ for the case where $r=3$. \begin{figure}[h] \centering \includegraphics[width = 0.5\linewidth]{fig/rp} \caption{$\mathcal{R}_{\textbf{p}}$ for case $r=3$.} \label{fig:rp} \end{figure} Then, we assume there are $\delta_1, \delta_2>0$ such that: \begin{equation} \begin{cases} \nonumber \delta_1<f_{\textbf{P}}(\mathbf{p}_u) <\delta_2, & \textbf{p}_u \in \mathcal{R}_{\textbf{p}}.\\ f_{\textbf{P}}(\mathbf{p}_u)=0, & \textbf{p}_u \notin \mathcal{R}_{\textbf{p}}. \end{cases} \end{equation} The obfuscation is similar to the two-states case. Specifically, for $l \in \{0,1,\cdots, r-1\}$, we can write \[ P({Z}_{u}(k)=l| X_{u}(k)=i) =\begin{cases} 1-R_u, & \textrm{for } l=i.\\ \frac{R_u}{r-1}, & \textrm{for } l \neq i. \end{cases} \] \begin{thm}\label{r_state_thm} For the above $r$-states model, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined previously, and \begin{itemize} \item $m=m(n)$ is arbitrary; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(\frac{1}{r-1}-\beta\right)}$ for any $c'>0$ and $0<\beta<\frac{1}{r-1}$; \end{itemize} then, user 1 has perfect privacy. That is, \begin{align \nonumber \forall k\in \mathbb{N}, \ \ \ \lim\limits_{n\rightarrow \infty} I \left(X_1(k);{\textbf{Y}}\right) =0. \end{align}\end{thm} The proof of Theorem \ref{r_state_thm} is similar to the proof of Theorem \ref{two_state_thm}. The major difference is that instead of the random variables $P_u, Q_u, V_u$, we need to consider the random vectors $\textbf{P}_u, \textbf{Q}_u, \textbf{V}_u$. Similarly, for user $u$, we define the vector $\textbf{Q}_u$ as \[\textbf{Q}_u= \begin{bmatrix} Q_u(1) \\ Q_u(2) \\ \vdots \\Q_u(r-1) \end{bmatrix}. \] In the $r$-states case, \begin{align} \nonumber {Q}_u(i) &=P_u(i)\bigg(1-R_u(i) \bigg)+\bigg(1-P_u(i)\bigg)\frac{R_u}{r-1} \\ \nonumber &= P_u+\bigg(1-r P_u\bigg)\frac{R_u}{r-1}.\ \ \end{align} We also need to define the critical set $J^{(n)}$. First, for $i=0,1, \cdots, r-1$, define set $J_i^{(n)}$ as follows. If $0\leq p_1(i)<\frac{1}{r}$, then, \begin{align*} &J_i^{(n)}= \\ &\left\{u \in \{1, 2, \dots, n\}: p_1(i) \leq P_u(i)\leq p_1(i)+\epsilon_n; p_1(i)+\epsilon_n\leq Q_u(i)\leq p_1(i)+(1-r p_1(i))\frac{a_n}{r-1}\right\}, \end{align*} where $\epsilon_n \triangleq \frac{1}{n^{\frac{1}{r-1}-\frac{\beta}{2}}}$, $a_n = c'n^{-\left(\frac{1}{r-1}-\beta\right)}$, and $\beta$ is defined in the statement of Theorem \ref{r_state_thm}. We then define the critical set $J^{(n)}$ as: \[ J^{(n)}=\bigcap_{l=0}^{r-1} J_i^{(n)}. \] We can then repeat the same arguments in the proof of Theorem \ref{two_state_thm} to complete the proof. \section{Converse Results: No Privacy Region} \label{converse} In this section, we prove that if the number of observations by the adversary is larger than its critical value and the noise level is less than its critical value, then the adversary can find an algorithm to successfully estimate users' data samples with arbitrarily small error probability. Combined with the results of the previous section, this implies that asymptotically (as $n \rightarrow \infty$), privacy can be achieved \emph{if and only if} at least one of the two techniques (obfuscation or anonymization) are used above their thresholds. This statement needs a clarification as follows: Looking at the results of \cite{tifs2016}, we notice that anonymization alone can provide perfect privacy if $m(n)$ is below its threshold. On the other hand, the threshold for obfuscation requires some anonymization: In particular, the identities of the users must be permuted once to prevent the adversary from readily identifying the users. \subsection{Two-States Model} Again, we start with the i.i.d.\ two-states model. The data sample of user $u$ at any time is a Bernoulli random variable with parameter $p_u$. As before, we assume that $p_u$'s are drawn independently from some continuous density function, $f_P(p_u)$, on the $(0,1)$ interval. Specifically, there are $\delta_1, \delta_2>0$ such that \begin{equation} \nonumber\begin{cases} \delta_1<f_P(p_u) <\delta_2, & p_u \in (0,1).\\ f_P(p_u)=0, & p _u\notin (0,1). \end{cases} \end{equation} \begin{thm}\label{two_state_thm_converse} For the above two-states mode, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined, and \begin{itemize} \item $m =cn^{2 + \alpha}$ for any $c>0$ and $\alpha>0$; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(1+\beta\right)}$ for any $c'>0$ and $\beta>\frac{\alpha}{4}$; \end{itemize} then, user $1$ has no privacy as $n$ goes to infinity. \end{thm} Since this is a converse result, we give an explicit detector at the adversary and show that it can be used by the adversary to recover the true data of user $1$. \begin{proof} The adversary first inverts the anonymization mapping $\Pi$ to obtain $Z_1(k)$, and then estimates the value of $X_1(k)$ from that. To invert the anonymization, the adversary calculates the empirical probability that each string is in state $1$ and then assigns the string with the empirical probability closest to $p_1$ to user 1. \begin{figure} \centering \includegraphics[width=.8\linewidth]{fig/converse.jpg} \centering \caption{$p_1$, sets $B^{(n)}$ and $C^{(n)}$ for case $r=2$.} \label{fig:converse} \end{figure} Formally, for $u=1, 2, \cdots, n$, the adversary computes $\overline{Y_u}$, the empirical probability of user $u$ being in state $1$, as follows: \[ \overline{Y_u}=\frac{Y_u(1)+Y_u(2)+ \cdots +Y_u(m)}{m}, \] thus, \[ \overline{Y_{\Pi(u)}}=\frac{Z_u(1)+Z_u(2)+ \cdots +Z_u(m)}{m}. \] As shown in Figure \ref{fig:converse}, define \[B^{(n)}\triangleq \left\{x \in (0,1); p_1-\Delta_n \leq x \leq p_1+\Delta_n\right\},\] where $\Delta_n = \frac{1}{n^{1+\frac{\alpha}{4}}}$ and $\alpha $ is defined in the statement of Theorem \ref{two_state_thm_converse}. We claim that for $m =cn^{2 + \alpha}$, $a_n=c'n^{-(1 + \beta)}$, and large enough $n$, \begin{enumerate} \item $P\left( \overline{Y_{\Pi(1) }}\in B^{(n)}\right) \rightarrow 1.$ \item $P\left( \bigcup\limits_{u=2}^{n} \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right) \rightarrow 0.$ \end{enumerate} As a result, the adversary can identify $\Pi(1)$ by examining $\overline{Y_u}$'s and assigning the one in $B^{(n)}$ to user $1$. Note that $\overline{Y_{\Pi(u) }} \in B^{(n)}$ is a set (event) in the underlying probability space and can be written as $\left\{\omega \in \Omega: \overline{Y_{\Pi(u) }}(\omega) \in B^{(n)}\right\}$. First, we show that as $n$ goes to infinity, \[P\left( \overline{Y_{\Pi(1) }}\in B^{(n)}\right) \rightarrow 1.\] We can write \begin{align} \nonumber P\left(\overline{Y_{\Pi(1)}} \in B^{(n)}\right) &= P\left(\frac{\sum\limits_{k=1}^{m}Z_1(k)}{m} \in B^{(n)} \right)\\ \nonumber &= P\left(p_1-\Delta_n \leq\frac{\sum\limits_{k=1}^{m}Z_1(k)}{m}\leq p_1+\Delta_n \right)\\ \nonumber &= P\left(mp_1-m\Delta_n-mQ_1\leq \sum\limits_{k=1}^{m}Z_1(k)-mQ_1\leq mp_1+m\Delta_n-mQ_1 \right).\ \ \end{align} Note that for any $u \in \{1,2,\cdots, n \}$, we have \begin{align} \nonumber |p_u-{Q}_u| &=|1-2p_u|R_u \\ \nonumber & \leq R_u \leq a_n, \end{align} so, we can conclude \begin{align} \nonumber P\left(\overline{Y_{\Pi(1)}} \in B^{(n)}\right) &= P\left(mp_1-m\Delta_n-mQ_1\leq \sum\limits_{k=1}^{m}Z_1(k)-mQ_1 \leq mp_1+m\Delta_n-mQ_1 \right)\\ \nonumber &\geq P\left(-m\Delta_n+m a_n\leq \sum\limits_{k=1}^{m}Z_1(k)-mQ_1\leq -ma_n+m\Delta_n \right)\\ \nonumber &= P\left(\abs*{\sum\limits_{k=1}^{m}Z_1(k)-mQ_1}\leq m (\Delta_n-a_n) \right).\ \ \end{align} Since $a_n \rightarrow 0$, for $p_1 \in (0,1)$ and large enough $n$, we can say $p_1+a_n < 2 p_1$. From Chernoff bound, for any $c,c',\alpha>0$ and $\beta>\frac{\alpha}{4}$, \begin{align} \nonumber P\left(\abs*{\sum\limits_{k=1}^{m}Z_1(k)-mQ_1}\leq m (\Delta_n-a_n) \right) &\geq 1-2e^{-\frac{m(\Delta_n-a_n)^2}{3Q_1}} \\ \nonumber &\geq 1-2e^{-\frac{1}{3(p_1+a_n)}cn^{2+\alpha}\left(\frac{1}{n^{1+\frac{\alpha}{4}}}- \frac{c'}{n^{1 + \beta}}\right)^2}\\ \nonumber &\geq 1-2e^{-\frac{c''}{6 p_1}n^{\frac{\alpha}{2}}}\rightarrow 1. \ \ \end{align} As a result, as $n$ becomes large, \[P\left(\overline{Y_{\Pi(1)}} \in B^{(n)}\right) \rightarrow 1.\] Now, we need to show that as $n$ goes to infinity, \[P\left( \bigcup\limits_{u=2}^{n} \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right) \rightarrow 0.\] First, we define \[ C^{(n)}=\left\{x\in (0,1); p_1-2\Delta_n \leq x \leq p_1+2\Delta_n\right\} ,\] and claim as $n$ goes to infinity, \[P\left(\bigcup\limits_{u=2}^n \left(P_u \in C^{(n)}\right) \right)\rightarrow 0. \] Note \[4\Delta_n\delta_1< P\left( P_u\in C^{(n)}\right) < 4 \Delta_n\delta_2,\] and according to the union bound, for large enough $n$, \begin{align} \nonumber P\left( \bigcup\limits_{u=2}^n \left(P_u \in C^{(n)}\right) \right) &\leq \sum\limits_{u=2}^n P\left( P_u \in C^{(n)}\right) \\ \nonumber &\leq 4n \Delta_n \delta_2\\ \nonumber &= 4n \frac{1}{n^{1+{\frac{\alpha}{4}}}} \delta_2\\ \nonumber &= 4n^{-\frac{\alpha}{4}}\delta_2 \rightarrow 0. \ \ \end{align} As a result, we can conclude that all $p_u$'s are outside of $C^{(n)}$ for $u \in \left\{2,3, \cdots, n\right\}$ with high probability. Now, we claim that given all $p_u$'s are outside of $C^{(n)}$, $P\left(\overline{Y_{\Pi (u)}} \in B^{(n)}\right)$ is small. Remember that for any $u \in \{1,2,\cdots, n \}$, we have \begin{align} \nonumber |p_u-{Q}_u| \leq a_n. \end{align} Now, noting the definitions of sets $B^{(n)}$ and $C^{(n)}$, we can write for $u \in \left\{2,3, \cdots , n\right\}$, \begin{align} \nonumber \nonumber P\left(\overline{Y_{\Pi(u)}} \in B^{(n)}\right) &\leq P\left(\abs*{\overline{Y_{\Pi(u)}}-Q_u}\geq (\Delta_n-a_n) \right)\\ \nonumber &= P\left(\abs*{\sum\limits_{k=1}^{m}Z_u(k)-mQ_u}> m(\Delta_n-a_n) \right).\ \ \end{align} According to the Chernoff bound, for any $c,c',\alpha>0$ and $\beta>\frac{\alpha}{4}$, \begin{align} \nonumber P\left(\abs*{\sum\limits_{k=1}^{m}Z_u(k)-mQ_u}> m(\Delta_n-a_n) \right) &\leq 2e^{-\frac{m(\Delta_n-a_n)^2}{3Q_1}} \\ \nonumber &\leq 2e^{-\frac{1}{3(p_1+a_n)}cn^{2+\alpha}\left(\frac{1}{n^{1+\frac{\alpha}{4}}}- \frac{c'}{n^{1 + \beta}}\right)^2}\\ \nonumber &\leq 2e^{-\frac{c''}{6 p_1}n^{\frac{\alpha}{2}}}. \ \ \end{align} Now, by using a union bound, we have \begin{align} \nonumber P\left( \bigcup\limits_{u=2}^n \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right)&\leq \sum\limits_{u=2}^{n}P\left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\\ \nonumber &\leq n\left(2e^{-\frac{c''}{6 p_1}n^{\frac{\alpha}{2}}}\right),\ \ \end{align} and thus, as $n$ goes to infinity, \[P\left( \bigcup\limits_{u=2}^n \left(\overline{Y_{\Pi(u)}}\in B^{(n)}\right)\right) \rightarrow 0.\] So, the adversary can successfully recover $Z_1(k)$. Since $Z_{1}(k)=X_1(k)$ with probability $1-R_1=1-o(1)$, the adversary can recover $X_{1}(k)$ with vanishing error probability for large enough $n$. \end{proof} \subsection{Extension to $r$-States} Now, assume users' data samples can have $r$ possibilities $\left(0, 1, \cdots, r-1\right)$, and $p_u(i)$ shows the probability of user $u$ having data sample $i$. We define the vector $\textbf{p}_u$ and the matrix $\textbf{p}$ as \[\textbf{p}_u= \begin{bmatrix} p_u(1) \\ p_u(2) \\ \vdots \\p_u(r-1) \end{bmatrix} , \ \ \ \textbf{p} =\left[ \textbf{p}_{1}, \textbf{p}_{2}, \cdots, \textbf{p}_{n}\right]. \] We also assume $\textbf{p}_u$'s are drawn independently from some continuous density function, $f_P(\textbf{p}_u)$, which has support on a subset of the $(0,1)^{r-1}$ hypercube. In particular, define the range of distribution as \begin{align} \nonumber \mathcal{R}_{\textbf{p}} &= \left\{ (x_1, x_2, \cdots, x_{r-1}) \in (0,1)^{r-1}: x_i > 0 , x_1+ x_2+\cdots+ x_{r-1} < 1,\ \ i=1, 2,\cdots, r-1\right\}. \end{align} Then, we assume there are $\delta_1, \delta_2>0$ such that: \begin{equation} \begin{cases} \nonumber \delta_1<f_{\textbf{P}}(\mathbf{p}_u) <\delta_2, & \textbf{p}_u \in \mathcal{R}_{\textbf{p}}.\\ f_{\textbf{P}}(\mathbf{p}_u)=0, & \textbf{p}_u \notin \mathcal{R}_{\textbf{p}}. \end{cases} \end{equation} \begin{thm}\label{r_state_thm_converse} For the above $r$-states mode, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$ as defined, and \begin{itemize} \item $m =cn^{\frac{2}{r-1} + \alpha}$ for any $c>0$ and $0<\alpha<1$; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(\frac{1}{r-1}+\beta\right)}$ for any $c'>0$ and $\beta>\frac{\alpha}{4}$; \end{itemize} then, user $1$ has no privacy as $n$ goes to infinity. \end{thm} The proof of Theorem \ref{r_state_thm_converse} is similar to the proof of Theorem \ref{two_state_thm_converse}, so we just provide the general idea. We similarly define the empirical probability that the user with pseudonym $u$ has data sample $i$ $\left(\overline{{Y}_{u}}(i)\right)$ as follows: \[ \overline{Y_u}(i)=\frac{\abs {\left\{k \in \{1, 2, \cdots, m\}:Y_u(k)=i\right\}}}{m}, \] thus, \[ \overline{Y_{\Pi(u)}}(i)=\frac{\abs {\left\{k \in \{1, 2, \cdots, m\}:Y_u(k)=i\right\}}}{m}. \] The difference is that now for each $u \in \{1,2,\cdots, n \}$, $\overline{\textbf{Y}_{u}}$ is a vector of size $r-1$. In other words, \[\overline{\textbf{Y}_{u}}=\begin{bmatrix} \overline{Y_u}(1) \\ \overline{Y_u}(2) \\ \vdots \\\overline{Y_u}(r-1) \end{bmatrix}.\] \begin{figure} \centering \includegraphics[width=.5\linewidth, height=0.5 \linewidth]{fig/R.jpg} \centering \caption{$\textbf{p}_1$, sets $B'^{(n)}$ and $C'^{(n)}$ in $\mathcal{R}_\textbf{p}$ for case $r=3$.} \label{fig:rpp} \end{figure} Define sets $B'^{(n)}$ and $C'^{(n)}$ as \begin{align} \nonumber B'^{(n)}\triangleq & \left\{(x_1,x_2, \cdots ,x_{r-1}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-\Delta'_n \leq x_i \leq p_1(i)+\Delta'_n,\ i=1,2, \cdots, r-1\right\}, \end{align} \begin{align} \nonumber C'^{(n)}\triangleq &\left\{(x_1,x_2, \cdots ,x_{r-1}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-2 \Delta'_n \leq x_i \leq p_1(i)+2 \Delta'_n,\ i=1,2, \cdots ,r-1\right\}, \end{align} where $\Delta'_n = \frac{1}{n^{\frac{1}{r-1}+\frac{\alpha}{4}}}.$ Figure \ref{fig:rpp} shows $\textbf{p}_1$ and sets $B'^{(n)}$ and $C'^{(n)}$ for the case $r=3.$ We claim for $m =cn^{\frac{2}{r-1} + \alpha}$ and large enough $n$, \begin{enumerate} \item $P\left( \overline{\textbf{Y}_{\Pi(1) }}\in B'^{(n)}\right) \rightarrow 1$. \item $P\left( \bigcup\limits_{u=2}^{n} \left(\overline{\textbf{Y}_{\Pi(u)}}\in B'^{(n)}\right)\right) \rightarrow 0.$ \end{enumerate} The proof follows that for the two-states case. Thus, the adversary can de-anonymize the data and then recover $X_1(k)$ with vanishing error probability in the $r$-states model. \subsection{Markov Chain Model} \label{subsec:markov} So far, we have assumed users' data samples can have $r$ possibilities $\left(0, 1, \cdots, r-1\right)$ and users' pattern are i.i.d.\ . Here we model users' pattern using Markov chains to capture the dependency of the users' pattern over time. Again, we assume there are $r$ possibilities (the number of states in the Markov chains). Let $E$ be the set of edges. More specifically, $(i, l) \in E$ if there exists an edge from $i$ to $l$ with probability $ p(i,l)>0 $. What distinguishes different users is their transition probabilities $p_u(i,l)$ (the probability that user $u$ jumps from state $i$ to state $l$). The adversary knows the transition probabilities of all users. The model for obfuscation and anonymization is exactly the same as before. We show that the adversary will be able to estimate the data samples of the users with low error probability if $m(n)$ and $a_n$ are in the appropriate range. The key idea is that the adversary can focus on a subset of the transition probabilities that are sufficient for recovering the entire transition probability matrix. By estimating those transition probabilities from the observed data and matching with the known transition probabilities of the users, the adversary will be able to first de-anonymize the data, and then estimate the actual samples of users' data. In particular, note that for each state $i$, we must have \[\sum\limits_{l=1}^{r} p_u(i,l)=1, \ \ \textrm{ for each }u \in \{1,2,\cdots, n \}, \] so, the Markov chain of user $u$ is completely determined by a subset of size $d=|E|-r$ of transition probabilities. We define the vector $\textbf{p}_u$ and the matrix $\textbf{p}$ as \[\textbf{p}_u= \begin{bmatrix} p_u(1) \\ p_u(2) \\ \vdots \\p_u(|E|-r) \end{bmatrix} , \ \ \ \textbf{p} =\left[ \textbf{p}_{1}, \textbf{p}_{2}, \cdots, \textbf{p}_{n}\right]. \] We also consider $\textbf{p}_u$'s are drawn independently from some continuous density function, $f_P(\textbf{p}_u)$, which has support on a subset of the $(0,1)^{|E|-r}$ hypercube. Let $\mathcal{R}_{\textbf{p}} \subset \mathbb{R}^{d}$ be the range of acceptable values for $\textbf{p}_{u}$, so we have \begin{align} \nonumber \mathcal{R}_{\textbf{P}} &= \left\{ (x_1,x_2 \cdots, x_{d}) \in (0,1)^{d}: x_i > 0 , x_1+x_2+\cdots+x_{d} < 1,\ \ i=1,2,\cdots, d\right\}. \end{align} As before, we assume there are $ \delta_1, \delta_2 >0$, such that: \begin{equation} \begin{cases} \nonumber \delta_1<f_{\textbf{P}}(\textbf{p}_u) <\delta_2, & \textbf{p}_u \in \mathcal{R}_{\textbf{p}}.\\ f_{\textbf{P}}(\textbf{p}_u)=0, & \textbf{p}_u \notin \mathcal{R}_{\textbf{p}}. \end{cases} \end{equation} Using the above observations, we can establish the following theorem. \begin{thm}\label{markov_thm} For an irreducible, aperiodic Markov chain with $r$ states and $|E|$ edges as defined above, if $\textbf{Z}$ is the obfuscated version of $\textbf{X}$, and $\textbf{Y}$ is the anonymized version of $\textbf{Z}$, and \begin{itemize} \item $m =cn^{\frac{2}{|E|-r} + \alpha}$ for any $c>0$ and $\alpha>0$; \item $R_u \sim Uniform [0, a_n]$, where $a_n \triangleq c'n^{-\left(\frac{1}{|E|-r}+\beta \right)}$ for any $c'>0$ and $\beta>\frac{\alpha}{4}$; \end{itemize} then, the adversary can successfully identify the data of user $1$ as $n$ goes to infinity. \end{thm} The proof has a lot of similarity to the i.i.d.\ case, so we provide a sketch, mainly focusing on the differences. We argue as follows. If the total number of observations per user is $m=m(n)$, then define $M_i(u)$ to be the total number of visits by user $u$ to state $i$, for $i=0, 1, \cdots, r-1$. Since the Markov chain is irreducible and aperiodic, and $m(n) \rightarrow \infty$, all $\frac{M_i(u)}{m(n)}$ converge to their stationary values. Now conditioned on $M_i(u)=m_i(u)$, the transitions from state $i$ to state $l$ for user $u$ follow a multinomial distribution with probabilities $p_u(i,l)$. Given the above, the setting is now very similar to the i.i.d.\ case. Each user is uniquely characterized by a vector $\textbf{p}_u$ of size $|E|-r$. We define sets $B^{''(n)}$ and $C^{''(n)}$ as \[ B^{''(n)}\triangleq \{(x_1, x_2, \cdots ,x_{d}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-\Delta''_n \leq x_i \leq p_1(i)+\Delta''_n, i=1,2 , \cdots ,d\}, \] \[ C^{''(n)}\triangleq \{(x_1, x_2, \cdots ,x_{d}) \in \mathcal{R}_{\textbf{p}}: p_1(i)-2 \Delta''_n < x_i < p_1(i)+2 \Delta''_n, i=1,2, \cdots, d\}, \] where $\Delta''_n= \frac{1}{n^{\frac{1}{|E|-r}+\frac{\alpha}{4}}}$, and $d= |E|-r$. Then, we can show that for the stated values of $m(n)$ and $a_n$, as $n$ becomes large: \begin{enumerate} \item $P\left( \overline{\textbf{Y}_{\Pi(1) }}\in B^{''(n)}\right) \rightarrow 1$, \item $P\left( \bigcup\limits_{u=2}^{n} \left(\overline{\textbf{Y}_{\Pi(u)}}\in B{''}^{(n)}\right)\right) \rightarrow 0$, \end{enumerate} which means that the adversary can estimate the data of user $1$ with vanishing error probability. The proof is very similar to the proof of the i.i.d.\ case; however, there are two differences that need to be addressed: First, the probability of observing an erroneous observation is not exactly given by $R_u$. In fact, a transition is distorted if at least one of its nodes is distorted. So, if the actual transition is from state $i$ to state $l$, then the probability of an erroneous observation is equal to \begin{align} \nonumber R'_u&=R_u+R_u-R_uR_u =R_u(2-R_u). \end{align} Nevertheless, here the order only matters, and the above expression is still in the order of $a_n =O \left( n^{-\left(\frac{1}{|E|-r}+\beta \right)} \right)$. The second difference is more subtle. As opposed to the i.i.d.\ case, the error probabilities are not completely independent. In particular, if $X_u(k)$ is reported in error, then both the transition to that state and from that state are reported in error. This means that there is a dependency between errors of adjacent transitions. We can address this issue in the following way: The adversary makes his decision only based on a subset of the observations. More specifically, the adversary looks at only odd-numbered transitions: First, third, fifth, etc., and ignores the even-numbered transitions. In this way, the number of observations is effectively reduced from $m$ to $\frac{m}{2}$ which again does not impact the order of the result (recall that the Markov chain is aperiodic). However, the adversary now has access to observations with independent errors. \section{Perfect Privacy Analysis: Markov Chain Model}\label{sec:perfect-MC} So far, we have provided both achievability and converse results for the i.i.d.\ case. However, we have only provided the converse results for the Markov chain case. Here, we investigate achievability for Markov chain models. It turns out that for this case, the assumed obfuscation technique is not sufficient to achieve a reasonable level of privacy. Loosely speaking, we can state that if the adversary can make enough observations, then he can break the anonymity. The culprit is the fact that the sequence observed by the adversary is no longer modeled by a Markov chain; rather, it can be modeled by a hidden Markov chain. This allows the adversary to successfully estimate the obfuscation random variable $R_u$ as well as the $ p_u(i,l)$ values for each sequence, and hence successfully de-anonymize the sequences. More specifically, as we will see below, there is a fundamental difference between the i.i.d.\ case and the Markov chain case. In the i.i.d.\ case, if the noise level is beyond a relatively small threshold, the adversary will be unable to de-anonymize the data and unable to recover the actual values of the data sets for users, \emph{regardless of the (large) size of $m=m(n)$}. On the other hand, in the Markov chain case, if $m=m(n)$ is large enough, then the adversary can easily de-anonymize the data. To better illustrate this, let's consider a simple example. \begin{example} Consider the scenario where there are only two states and the users' data samples change between the two states according to the Markov chain shown in Figure \ref{fig:MC-diagram}. What distinguishes the users is their different values of $p$. Now, suppose we use the same obfuscation method as before. That is, to create a noisy version of the sequences of data samples, for each user $u$, we generate the random variable $R_u$ that is the probability that the data sample of the user is changed to a different data sample by obfuscation. Specifically, \[ {Z}_{u}(k)=\begin{cases} {X}_{u}(k), & \textrm{with probability } 1-R_u.\\ 1-{X}_{u}(k),& \textrm{with probability } R_u. \end{cases} \] \begin{figure}[H] \begin{center} \[ \SelectTips {lu}{12 scaled 2500} \xymatrixcolsep{6pc}\xymatrixrowsep{5pc}\xymatrix{ *++[o][F]{0} \ar@/^1pc/[r]^{1} & *++[o][F]{1} \ar@(dr,ur)[]_{1-p} \ar@/^1pc/[l]^{p} } \] \caption{A state transition diagram.}\label{fig:MC-diagram} \end{center} \end{figure} To analyze this problem, we can construct the underlying Markov chain as follows. Each state in this Markov chain is identified by two values: the real state of the user, and the observed value by the adversary. In particular, we can write \[\left(\text{Real value}, \text{Observed value}\right) \in \left\{\right(0,0), (0,1), (1,0), (1,1)\}.\] Figure \ref{fig:MC-diagram2} shows the state transition diagram of this new Markov chain. \begin{figure}[H] \begin{center} \[ \SelectTips {lu}{12 scaled 2000} \xymatrixcolsep{10pc}\xymatrixrowsep{8pc}\xymatrix{ *++[o][F]{00} \ar@//[d]^{R} \ar@/_1pc/[dr]_>>>>>{1-R} & *++[o][F]{01} \ar@//[d]^{1-R} \ar@/^1pc/[dl]^>>>>>{R} \\ *++[o][F]{10} \ar@(d,l)[]^{(1-p)R} \ar@/_2pc/[r]_{(1-p)(1-R)} \ar@/^2pc/[u]^{p(1-R)} \ar@/^1pc/[ur]^>>>>>{pR} & *++[o][F]{11} \ar@(d,r)[]_{(1-p)(1-R)} \ar@//[l]^{(1-p)R} \ar@/_2pc/[u]_{pR} \ar@/_1pc/[ul]_>>>>>{p(1-R)} } \] \caption{The state transition diagram of the new Markov chain.}\label{fig:MC-diagram2} \end{center} \end{figure} We know \[ \pi_{00}=\pi_0(1-R)=\frac{p}{1+p}(1-R).\] \[ \pi_{01}=\pi_0R= \frac{p}{1+p}R.\] \[ \pi_{10}=\pi_1R=\frac{1}{1+p}R.\] \[ \pi_{11}=\pi_1(1-R)=\frac{1}{1+p}(1-R).\] The observed process by the adversary is not a Markov chain; nevertheless, we can define limiting probabilities. In particular, let $\theta_0$ be the limiting probability of observing a zero. That is, we have \[ \frac{M_0}{m} \xrightarrow{d} \theta_0, \ \ \textrm{ as }n \rightarrow \infty, \] where $m$ is the total number of observations by the adversary, and $M_0$ is the number of $0$'s observed. Then, \[\theta_0= \pi_{00}+\pi_{10} =\frac{(1-R)p+R}{1+p}.\] Also, let $\theta_1$ be the limiting probability of observing a one, so \[\theta_1= \pi_{01}+\pi_{11} =\frac{pR+(1-R)}{1+p}=1-\theta_0.\] Now the adversary's estimate of $\theta_0$ is given by: \begin{align}\label{eq1} \hat{\theta}_0= \frac{(1-R)p+R}{1+p}. \end{align} Note that if the number of observations by the adversary can be arbitrarily large, the adversary can obtain an arbitrarily accurate estimate of $\theta_0$. The adversary can obtain another equation easily, as follows. Let $\theta_{01}$ be the limiting value of the portion of transitions from state $0$ to $1$ in the chain observed by the adversary. We can write \begin{align} \nonumber \theta_{01} &=P \left\{(00\rightarrow 01), (00\rightarrow 11), (10 \rightarrow 01), (10 \rightarrow 11) \right\}\\\ \nonumber &= \pi_{00}(1-R)+\pi_{10}PR+ \pi_{10}(1-p)(1-R).\ \ \end{align} As a result, \begin{align}\label{eq2} \hat{\theta}_{01}= \frac{p(1-R)^2+R\left(PR(1-R)(1-p)\right)}{1+p}. \end{align} Again, if the number of observations can be arbitrarily large, the adversary can obtain an arbitrarily accurate estimate of $\theta_{01}$. By solving the Equations \ref{eq1} and \ref{eq2}, the adversary can successfully recover $R$ and $p$; thus, he/she can successfully determine the users' data values. \end{example} \section{Introduction} \label{intro} A number of emerging systems and applications work by analyzing the data submitted by their users in order to serve them; we call such systems \emph{User-Data Driven} (UDD) services. Examples of UDD services include smart cities, connected vehicles, smart homes, and connected healthcare devices, which have the promise of greatly improving users' lives. Unfortunately, the sheer volume of user data collected by these systems can compromise users' privacy~\cite{FTC2015}. Even the use of standard Privacy-Protection Mechanisms (PPMs), specifically anonymization of user identities and obfuscation of submitted data, does not guarantee users' privacy, as adversaries are able to use powerful statistical inference techniques to learn sensitive private information of the users~\cite{0Quest2016, 3ukil2014iot, 4Hosseinzadeh2014,iotCastle,matching}. To illustrate the threat of privacy leakage, consider three popular UDD services: (1) {\em Health care:} Wearable monitors that constantly track user health variables can be invaluable in assessing individual health trends and responding to emergencies. However, such monitors produce long time-series of user data uniquely matched to the health characteristics of each user; (2) {\em Smart homes:} Emerging smart-home technologies such as fine-grained power measurement systems can help users and utility providers to address one of the key challenges of the twenty-first century: energy conservation. But the measurements of power by such devices can be mapped to users and reveal their lifestyle habits; and, (3) {\em Connected vehicles:} The location data provided by connected vehicles promises to greatly improve everyday life by reducing congestion and traffic accidents. However, the matching of such location traces to prior behavior not only allows for user tracking, but also reveals a user's habits. In summary, despite their potential impact on society and their emerging popularity, these UDD services have one thing in common: their utility critically depends on their collection of user data, which puts users' privacy at significant risk. There are two main approaches to augment privacy in UDD services: \emph{identity perturbation (anonymization)}~\cite{1corser2016evaluating,hoh2005protecting,freudiger2007mix, ma2009location, shokri2011quantifying2, Naini2016,soltani2017towards, soltani2018invisible}, and \emph{data perturbation (obfuscation)}~\cite{shokri2012protecting, gruteser2003anonymous, bordenabe2014optimal}. In anonymization techniques, privacy is obtained by concealing the mapping between users and data, and the mapping is changed periodically to thwart statistical inference attacks that try to de-anonymize the anonymized data traces by matching user data to known user profiles. Some approaches employ $k$-anonymity to keep each user's identity indistinguishable within a group of $k-1$ other users ~\cite{2zhang2016designing,11dewri2014exploiting, gedik2005location, zhong2009distributed, sweeney2002k, kalnis2007preventing,liu2013game}. Other approaches employ users' pseudonyms within areas called mix-zones~\cite{beresford2003location, freudiger2009optimal, palanisamy2011mobimix}. Obfuscation mechanisms aim at protecting privacy by perturbing user data, e.g., by adding noise to users' samples of data. For instance, cloaking replaces each user's sample of data with a larger region~\cite{18shokri2014hiding,8zurbaran2015near,hoh2007preserving, wernke2014classification, chow2011spatial, um2010advanced}, while an alternative approach is to use dummy data in the set of possible data of the users~\cite{kido2005protection, shankar2009privately, chow2009faking, kido2005anonymous, lu2008pad}. In~\cite{randomizedresponse}, a mechanism of obfuscation was introduced where the answer was changed randomly with some small probability. Here we consider the fundamental limits of a similar obfuscation technique for providing privacy in the long time series of emerging applications. The anonymization and obfuscation mechanisms improve user privacy at the cost of user utility. The anonymization mechanism works by frequently changing the pseudonym mappings of users to reduce the length of time series that can be exploited by statistical analysis. However, this frequent change may also decrease the usability by concealing the temporal relation between a user's sample of data, which may be critical in the utility of some systems, e.g., a dining recommendation system that makes suggestions based on the dining history of its users. On the other hand, obfuscation mechanisms work by adding noise to users' collected data, e.g., location information. The added noise may degrade the utility of UDD applications. Thus, choosing the right level of the privacy-protection mechanism is an important question, and understanding what levels of anonymization and obfuscation can provide theoretical guarantees of privacy is of interest. In this paper, we will consider the ability of an adversary to perform statistical analyses on time series and match the series to descriptions of user behavior. In related work, Unnikrishnan~\cite{matching} provides a comprehensive analysis of the asymptotic (in the length of the time series) optimal matching of time series to source distributions. However, there are several key differences between that analysis and the work here. First, Unnikrishnan~\cite{matching} looks at the optimal matching tests, but does not consider any privacy metrics as considered in this paper, and a significant component of our study is demonstrating that mutual information converges to zero so that we can conclude there is no privacy leakage (hence, ``perfect privacy''). Second, the setting of~\cite{matching} is different, as it does not consider: (a) obfuscation, which is one of the two major protection mechanisms; and (b) sources that are not independent and identically distributed (i.i.d.). Third, the setting of Unnikrishnan~\cite{matching} assumes a fixed distribution on sources (i.e., classical inference), whereas we assume the existence of general (but possibly unknown) prior distributions for the sources (i.e., a Bayesian setting). Finally, we study the fundamental limits in terms of both the number of users and the number of observations, while Unnikrishnan~\cite{matching} focuses on the case where the number of users is a fixed, finite value. Numerous researchers have put forward ideas for quantifying privacy-protection. Shokri et al.~\cite{shokri2011quantifying, shokri2011quantifying2} define the expected estimation error of the adversary as a metric to evaluate PPMs. Ma et al.~\cite{ma2009location} use uncertainty about users' information to quantify user privacy in vehicular networks. To defeat localization attacks and achieve privacy at the same time, Shokri et al.~\cite{shokri2012protecting} proposed a method which finds optimal PPM for an LBS given service quality constraints. In~\cite{6li2016privacy} and~\cite{4olteanu2016quantifying}, privacy leakage of data sharing and interdependent privacy risks are quantified, respectively. A similar idea is proposed in \cite{14zhang2014privacy} where the quantification model is based on the Bayes conditional risk. Previously, mutual information has been used as a privacy metric in a number of settings,~\cite{kousha3,salamatian2013hide, csiszar1996almost, calmon2015fundamental, sankar2013utility, sankarISIT,sankar, yamamoto1983source, hyposankar}. However, the framework and problem formulation for our setting (Internet of Things (IoT) privacy) are quite different from those encountered in previous works. More specifically, the IoT privacy problem we consider here is based on a large set of time-series data that belongs to different users with different statistical patterns that has gone through a privacy-preserving mechanism, and the adversary is aiming at de-anonymizing and de-obfuscating the data. The discussed studies demonstrate the growing importance of privacy. What is missing from the current literature is a solid theoretical framework for privacy that is general enough to encompass various privacy-preserving methods in the literature. Such a framework will allow us to achieve provable privacy guarantees, obtain fundamental trade-offs between privacy and performance, and provide analytical tools to optimally achieve provable privacy. We derive the fundamental limits of user privacy in UDD services in the presence of both anonymization and obfuscation protection mechanisms. We build on our previous works on formalizing privacy in location-based services~\cite{tifs2016, ciss2017}, but we significantly expand those works here not just in application area but also user models and settings. In particular, our previous works introduced the notion of \emph{perfect privacy} for location-based services, and we derived the rate at which an anonymization mechanism should change the pseudonyms in order to achieve the defined perfect privacy. In this work, we expand the notion of perfect privacy to UDD services in general and derive the conditions for it to hold when \emph{both} anonymization and obfuscation-based protection mechanisms are employed. In this paper, we consider two models for users' data: i.i.d.\ and Markov chains. After introducing the general framework in Section \ref{sec:framework}, we consider an i.i.d.\ model extensively in Section \ref{perfectsec} and the first half of Section \ref{converse}. We obtain achievability and converse results for the i.i.d.\ model. The i.i.d.\ model would apply directly to data that is sampled at a low rate. In addition, understanding the i.i.d.\ case can also be considered the first step toward understanding the more complicated case where there is dependency, as was done for anonymization-only Location Privacy-Preserving Mechanisms (LPPMs) in \cite{tifs2016}, and will be done in Section \ref{subsec:markov}. In particular, in Section \ref{subsec:markov}, a general Markov chain model is used to model users' data pattern to capture the dependency of the user' data pattern over time. There, we obtain converse results for privacy for this model. In Section \ref{sec:perfect-MC}, we provide some discussion about the achievability for the Markov chain case. \subsection{Summary of the Results} Given $n$, the total number of the users in a network, their degree of privacy depends on two parameters: (1) The number of observations $m=m(n)$ by the adversary per user for a fixed anonymization mapping (i.e., the number of observations before the pseudonyms are changed); and (2) the value of the noise added by the obfuscation technique (as defined in Section~\ref{sec:framework}, we quantify the obfuscation noise with a parameter $a_n$, where larger $a_n$ means a higher level of obfuscation). Intuitively, smaller $m(n)$ and larger $a_n$ result in stronger privacy, at the expense of lower utility for the users. Our goal is to identify values of $a_n$ and $m(n)$ that satisfy perfect privacy in the asymptote of a large number of users ($n \rightarrow \infty$). When the users' datasets are governed by an i.i.d.\ process, we show that the $m(n)- a_n$ plane can be divided into two areas. In the first area, all users have perfect privacy (as defined in Section~\ref{sec:framework}), and, in the second area, users have no privacy. Figure~\ref{fig:region} shows the limits of privacy in the entire $m(n)- a_n$ plane. As the figure shows, in regions $1$, $2$, and $3$, users have perfect privacy, while in region $4$ users have no privacy. \begin{figure}[h] \centering \includegraphics[width=0.7\linewidth, height=0.6 \linewidth]{fig/region} \caption{Limits of privacy in the entire $m(n)-a_n$ plane: in regions $1$, $2$, and $3$, users have perfect privacy, and in region $4$ users have no privacy.} \label{fig:region} \end{figure} For the case where the users' datasets are governed by irreducible and aperiodic Markov chains with $r$ states and $|E|$ edges, we show that users will have no privacy if $m =cn^{\frac{2}{|E|-r} + \alpha}$ and $a_n =c'n^{-\left(\frac{1}{|E|-r}+\beta \right)}$, for any constants $c>0$, $c'>0$, $\alpha>0$, and $\beta>\frac{\alpha}{4}$. We also provide some insights for the opposite direction (under which conditions users have perfect privacy) for the case of Markov chains. \section{Introduction} \label{intro} Various emerging systems and applications work by analyzing the data submitted by their users in order to serve them; we call such systems \emph{user-data driven} (UDD) \amir{we need a name. how's this?} services. Examples of UDD services include smart cities, connected vehicles, smart homes, and connected healthcare devices, which are revolutionizing our lives in different ways. Unfortunately, the sheer volume of user data collected by these systems compromises users' privacy ostensibly~\cite{FTC2015}. Even the employment of standard privacy-protection mechanisms (e.g., anonymization of user identities and obfuscation of submitted data) does not mitigate the privacy leakage as the adversaries are able to use powerful statistical inference techniques to infer sensitive private information~\cite{FTC2015,0Quest2016, 2nia2016comprehensive, 3ukil2014iot, 4Hosseinzadeh2014,iotCastle,matching}. To illustrate their privacy leakage, consider the following three popular UDD services. (1) {\em Health care:} Wearable monitors that constantly track user health variables can be invaluable in assessing individual health trends and responding to emergencies. However, such monitors produce long time-series of user data uniquely matched to the health characteristics of each user; (2) {\em Smart homes:} Emerging smart-home technologies such as fine-grained power measurement systems can help users and utility providers to address one of the key challenges of the twenty-first century: energy conservation. But the measurements of power by such devices can be mapped to users and reveal their lifestyle habits; and, (3) {\em Connected vehicles:} The location data provided by connected vehicles promises to greatly improve everyday lives by reducing congestion and traffic accidents. However, the matching of such location traces to prior behavior not only allows for user tracking, but also reveals a user's habits. In summary, despite their critical importance to the society and their emerging popularity, these services have one thing in common: they collect long time-series of user data, for them to operate, that puts users' privacy at significant risk. There are two main approaches to augment privacy in UDD services: \emph{identity perturbation (anonymization)}, and \emph{location perturbation (obfuscation)}. In anonymization techniques, privacy is obtained by concealing the mapping between users and data, and the mapping is changed periodically to thwart statistical inference attacks that try to deanonymize the anonymized data traces by matching user data to known user profiles. On the other hands, obfuscation mechanisms aim at protecting privacy by perturbing user data, e.g., by adding noise to user locations. Anonymization and obfuscation improve user privacy at the cost of user utility. In anonymization, we need to change these pseudonym mappings frequently to achieve high privacy by reducing the length of time series exploited by statistical analysis. However, this frequent change could decrease usability and functionality by concealing the temporal relation between a user's locations, which may be critical in the utility of some systems, e.g., a dining recommendation system that makes suggestions based on the dining places visited by a user in the past. For obfuscation-based mechanisms, the added noise to the reported values of user locations will degrade the application utility; for example, user utility would be degraded by location obfuscation in ride-sharing systems. Thus, choosing the right level of privacy-protection mechanism is an important question, and understanding what levels of anonymization and obfuscation can provide theoretical guarantees of privacy is of interest. In this paper, we derive the fundamental limits of user privacy in UDD services in the presence of both anonymization and obfuscation protection mechanisms. This work is built on our previous work on formalizing privacy in location-based services~\cite{montazeri2016defining, Mont1610Achieving,tifs2016, ciss2017}, by extensively expanding not just the application area but also user models and settings. Particularly, our previous work introduced the notion of \emph{perfect privacy} for location-based services, and derives the rate at which an anonymization mechanism should change the pseudonyms in order to achieve the defined perfect privacy. In this work, we expand the notion of perfect privacy to UDD services in general, and derive the conditions for it to hold when \emph{both} anonymization and obfuscation-based protection mechanisms are employed. We study both achievability and converse results. \subsection{Summary of the Results} Given $n$, the total number of the users in a network, their degree of privacy depends on two parameters: (1) The number of observations $m=m(n)$ by the adversary per user for a fixed anonymization mapping (i.e., the number of observations before the pseudonyms are changed); and (2) the value of the noise added by the obfuscation technique (as defined in Section~\ref{sec:framework}, we quantify the obfuscation noise with a parameter $a_n$). Intuitively, smaller $m(n)$ and larger $a_n$ result in stronger privacy, at the expense of lower utility for the users. Our goal is to identify values of $a_n$ and $m(n)$ that satisfy perfect privacy as the number of users grows ($n \rightarrow \infty$). We show that when the users' data sets are governed by an i.i.d.\ process, the $m(n)- a_n$ plane can be divided into two areas. In the first area, all users have perfect privacy~\cite{tifs2016} (as defined in Section~\ref{sec:framework}), and, in the second area, users have no privacy. Figure~\ref{fig:region} shows the limits of privacy in the entire $m(n)- a_n$. As the figure shows, in regions $1$, $2$, and $3$, users have perfect privacy, while in region $4$ users have no privacy. \begin{figure}[h] \centering \includegraphics[width=0.8\linewidth, height=0.8 \linewidth]{fig/region} \caption{Limits of privacy in the entire $m(n)-a_n$ plane: in regions $1$, $2$, and $3$, users have perfect privacy, and in region $4$ users have no privacy.} \label{fig:region} \end{figure} For the case where the users' data sets are governed by irreducible and aperiodic Markov chains with $r$ states and $|E|$ edges, we show that users will have no privacy if $m =cn^{\frac{2}{|E|-r} + \alpha}$ and $a_n =c'n^{-\left(\frac{1}{|E|-r}+\beta \right)}$, for some constants $c>0$, $c'>0$, $\alpha>0$, and $\beta>\frac{\alpha}{4}$. We also provide some insights for the opposite direction (under which conditions users have perfect privacy) for the case of Markov chains. \section{Related Work} \label{relatedwork} Previously, mutual information has been used as a privacy metric in a number of settings,~\cite{salamatian2013hide, csiszar1996almost, calmon2015fundamental, sankar2013utility, sankarISIT,sankar, yamamoto1983source, hyposankar}. However, the framework and problem formulation for our setting (Internet of Things (IoT) privacy) is quite different from those encountered in previous work. More specifically, the IoT privacy problem we consider here is based on a large set of time-series data that belong to different users with different statistical patterns that has gone through a privacy-preserving mechanism, and the adversary is aiming at de-anonymizing and de-obfuscating the data. Next, consider related works in smart-home/public environment Internet of Things (IoT) systems. The authors in \cite{8ukil2015privacy} investigate privacy for IoT smart energy metering where the concern results from the fact that any privacy breach can reveal in-house activity. A `Dynamic Privacy Analyzer' scheme is proposed as an attempt towards achieving a unique privacy metric that is derived from fundamental principles like robust statistics and information theory. To deal with concerns in different smart-home IoT devices such as lights, smoke-alarms, power switches, baby monitors, and weighing scales, which again makes it possible for the entities to snoop and intrude into the family's activities, the authors in \cite{12sivaraman2015network} suggest that device-level protections be augmented with network-level security solutions. This can be achieved through software defined networking technology when used to dynamically block/quarantine suspicious devices. In \cite{10harris2016security}, the authors consider the privacy issues in public environments such as an IoT-enabled retail store. An IoT-enable retail store has the potential to provide rich, targeted information and services to users within the environment. However, to fully realize such potential, users must be willing to share certain data regarding their habits and preferences with the public IoT facility. To encourage users to share such information, the paper proposes a protocol to ensure the customers that their data will not be leaked to third parties. As far as other applications of IoT is concerned, authors in \cite{9dalipi2016security} investigate privacy considerations for IoT applications on Smart Grids. In particular, the authors address three types of challenge domains: customer domain, information and communication domain, and the grid domain. In \cite{battery15}, the author tried to provide privacy for smart-metering systems with rechargeable batteries by obscuring the user's power demand. In \cite{11al2015security}, the authors consider the privacy issue of wearable devices. These devices make it possible for health care services start a new phase in serving patients' needs and monitor their health remotely. Finally, authors in \cite{14sadeghi2015security}, overview the privacy challenges in industrial IoT systems. Among these applications, location privacy constitutes one of the most important aspects of IoT privacy. Location privacy preserving mechanisms (LPPMs) can be categorized into two main classes: identity perturbation LPPMs (anonymization) ~\cite{1corser2016evaluating,hoh2005protecting,freudiger2007mix, ma2009location, shokri2011quantifying, Naini2016} and location perturbation LPPMs (obfuscation)~\cite{shokri2012protecting, gruteser2003anonymous, bordenabe2014optimal}. There are various approaches suggested for identity perturbation LPPMs. Particularly, k-anonymity approaches help to keep each user's identity indistinguishable within a group of $k-1$ other users ~\cite{1corser2016evaluating,2zhang2016designing,11dewri2014exploiting, gedik2005location, zhong2009distributed, mokbel2006new, sweeney2002k, bordenabe2014optimal, kalnis2007preventing,liu2013game}. Another proposal changes users' pseudonyms within areas called mix-zones~\cite{hoh2005protecting, beresford2003location, freudiger2009optimal, beresford2003location, palanisamy2011mobimix}. There are also various approaches used by location perturbation LPPMs. For instance, cloaking replaces each user's location information with a larger region~\cite{18shokri2014hiding,8zurbaran2015near,hoh2007preserving, gruteser2003anonymous, gedik2005location, gedik2008protecting, bordenabe2014optimal, duckham2005formal, duckham2006spatiotemporal, xue2009location, wernke2014classification, cai2015cloaking, chow2011spatial, cheng2006preserving, mokbel2006new, kalnis2006preserving, khoshgozaran2011location, bamba2008supporting, um2010advanced, zhangwei2010distributed}, while an alternative direction is to use dummy locations in the set of possible locations of users~\cite{kido2005protection, krumm2009survey, shankar2009privately, chow2009faking, kido2005anonymous, lu2008pad}. Some prior works combine techniques to achieve stronger location privacy. For instance, Freudiger et al.\ combine techniques from cryptography with mix-zones to improve location privacy~\cite{freudiger2007mix}. Differential privacy, which aims at protecting queries on aggregated data, has also been applied to the problem of location privacy. Particularly, several identity perturbation~\cite{lee2012differential, bordenabe2014optimal, chatzikokolakis2015geo, nguyen2013differential, machanavajjhala2008privacy} and location perturbation ~\cite{chatzikokolakis2013broadening,shokri2014optimal, chatzikokolakis2015location,andres2013geo,bordenabe2014optimal} LPPMs are based on differential privacy techniques. Dewri~\cite{dewri2013local} combines k-anonymity and differential privacy to achieve higher location privacy. Alternatively, Andres et al.\ hide the exact location of a user in a region by adding Laplacian distributed noise to achieve a desired level of geo-indistinguishability~\cite{andres2013geo}. In ~\cite{info2012}, it is proved that differential privacy arises out of maximizing entropy principle which is equal to minimizing information leakage and is measured using the mutual information notion. ~\cite{yeb17} tries to reduce the expected estonimation loss by using differential privacy and constraining the privacy level. Several studies aim at quantifying location privacy protection. Shokri et al.~\cite{shokri2011quantifying, shokri2011quantifying2} define the expected estimation error of the adversary as a metric to evaluate LPPM mechanisms. Ma et al.~\cite{ma2009location} use uncertainty about users' location information to quantify user location privacy in vehicular networks. To defeat localization attacks and achieve privacy at the same time, Shokri et al.~\cite{shokri2012protecting} proposed a method which finds optimal LPPM for an LBS given service quality constraints. Shokri et al.~\cite{shokri2012protecting} design LPPM mechanisms that will defeat localization attacks. In \cite{6li2016privacy} and \cite{4olteanu2016quantifying}, privacy leakage of location sharing and interdependent location privacy risks are quantified, respectively. A similar idea is proposed in \cite{14zhang2014privacy} where the quantification model is based on the Bayes conditional risk. Yu et al. \cite{diff2017} combine two complementary notations, geo-indstingushability and expected inference error, to achieve location privacy. The geo-indstingushability is derived from differential privacy but cannot adequately protect users' location against inference attacks of adversary by using prior information, and the expected inference error as privacy metric doesn't consider posterior information obtained from pseudo-location. As a result, combining both notations gives users the opportunity to have location privacy. In addition, users are allowed to personalized error bound for different locations. In \cite{5ullah2016novel}, the authors propose a novel model for preserving location privacy in IoT in which they propose the so-called Enhanced Semantic Obfuscation Technique (ESOT). In \cite{7zhou2012preserving}, in the context of IoT based wireless sensor networks, the authors propose a flexible routing strategy, called Multi-routing Random Walk, which protects the sensor's location. Another approach has been proposed in \cite{6sathishkumar2016enhanced}. The discussed studies demonstrate the growing importance of privacy. What is missing from the current literature is a solid theoretical framework for privacy that is general enough to encompass various privacy preserving methods in the literature. Such a framework will allow us to achieve provable privacy guarantees, obtain fundamental trade-offs between privacy and performance, and provide analytical tools to optimally achieve provable privacy. The closest works to this paper are ~\cite{montazeri2016defining, Mont1610Achieving,tifs2016,sit2017,ciss2017}. As mentioned, these works only consider anonymization and only achiveability. In this paper, by considering obfuscation and anonymization, we characterize the limits of privacy in the entire $m(n)-a_n$ plane by considering both achievability and converse results.

BkiUbSTxK1ThhAqzUzK9

\section{Experiments} To illustrate the efficiency and capacity of \modelm, we conduct two sets of experiments regarding to the architecture design, filter pruning, and integer-only inference. The models are trained with PyTorch~\cite{paszke2019pytorch} implementation and latency results are measured by deploying on a Tesla T4 GPU with the TensorRT library. Density estimation performance is reported in bits per dimension (bpd). We compare \modelm~with IDF on ImageNet32 and ImageNet64~\cite{deng2009imagenet} dataset\footnote{There are two different versions of ImageNet32 and ImageNet64 datasets. We use the down-sampled ImageNet datasets from \url{https://image-net.org/data/downsample/Imagenet32_train.zip}, following~\citet{grcic2021densely, hazami2022efficient}. \citet{hoogeboom2019integer} use the datasets downloaded from \url{http://image-net.org/small/train_32x32.tar}. }. The flow architecture is taken from IDF, which has 3 levels of flow steps at the resolution $16\times 16$, $8\times 8$, and $4\times 4$ on ImageNet32, and 4 levels of flow steps at the resolution $32\times 32$, $16\times 16$, $8\times 8$, and $4\times 4$. Each resolution level has 8 additive coupling layers. Batch normalization is not used in both models. Following IDF, we adopt the rezero trick \cite{kingma2018glow} to realize identity mapping initialization which is helpful to improving training stability. The models are trained 100 epochs for ImageNet32 and 50 epochs for ImageNet64. See Appendix~\ref{app:arch} for architecture and training details. Open-source code is available at \url{https://github.com/thu-ml/IODF}. \begin{table}[t] \caption{Overall evaluation results on test datasets (measured in bits per dimension) of IDF-DenseNets, IDF-ResNets, and pruned models of different FLOPs pruning ratio on ImageNet32 and ImageNet64. FLOPs are measured in floating-point operations.} \label{tab:bpd_pruned} \vskip 0.15in \begin{center} \begin{small} \begin{sc} \begin{tabular}{lccc} \toprule Model & \#FLOPs & \#Parameters & bpd \\ \midrule ImageNet32 \\ \midrule IDF-Dense & 6.43G & 58.4M & 3.890 \\ IDF-Res & 7.15G & 62.2M & 3.916 \\ IDF-Res-Pruned1 & 5.67G & 38.2M & 3.916 \\ IDF-Res-Pruned2 & 4.25G & 23.5M & 3.920 \\ IDF-Res-Pruned3 & 3.28G & 17.0M & 3.930 \\ IDF-Res-Pruned4 & 1.60G & 7.5M & 4.048 \\ \midrule ImageNet64 \\ \midrule IDF-Dense & 26.27G & 84.3M & 3.629 \\ IDF-Res & 29.09G & 84.5M & 3.630\\ IDF-Res-Pruned1 & 23.18G & 39.1M & 3.642 \\ IDF-Res-Pruned2 & 17.27G & 26.3M & 3.665 \\ IDF-Res-Pruned3 & 12.35G & 18.6M & 3.700 \\ \bottomrule \end{tabular} \end{sc} \end{small} \end{center} \vskip -0.2in \end{table} \subsection{Network Architecture and Filter Pruning} \label{exp:1} \paragraph{Network Architecture} IDF and IODF differ by the network architecture adopted in the additive coupling layer. We first compare the DenseNet architecture used in IDF (denoted as IDF-Dense) and the more hardware-friendly ResNet architecture discussed in Sec.~\ref{sec:arch} (denoted as IDF-Res). The models are used in full precision. Table~\ref{tab:bpd_pruned} compares these two architectures. IDF-Res and IDF-Dense achieve similar bpd under comparable FLOPs and number of parameters, indicating that replacing DenseNet with ResNet will not sacrifice much modeling capacity. However, IDF-Res is much more efficient than IDF-Dense with integer arithmetic, as we shall see in Sec.~\ref{exp:3}. \paragraph{Filter Pruning} Next, we study the effectiveness of the learned binary gates proposed in Sec.~\ref{sec:pruning} on pruning redundant filters. We insert learnable binary gates into a well-trained IDF-Res model and perform training stage 2 and stage 3 depicted in Alg.~\ref{alg:binary_gate} until it satisfies targeted FLOPs and prune out $0$-gated filters as illustrated in Sec.~\ref{sec:pruning}. See Appendix \ref{app:optim} for more details about optimization parameters. We set different targeted FLOPs reduction (20\%, 40\%, 60\%, 80\% of original IDF-Res) and obtain several pruned models IDF-Res-Pruned\{1,2,3,4\}. Table~\ref{tab:bpd_pruned} compares the FLOPs and density estimation performance of the pruned models. Binary gated convolution can effectively reduce the number of parameters and FLOPs of IDF-Res with little harm to modeling capacity. 60\% computations can be cut with only a 0.015 bpd drop on ImageNet32. For ImageNet64, pruning is relatively harder, $57.6\%$ reduced FLOPs can cause a $0.07$ bpd increase from $3.630$ of IDF-Res to $3.700$ of IDF-Res-Pruned3. \input{table_whole} \begin{figure}[t] \begin{center} \centerline{\includegraphics[width=0.9\columnwidth]{figs/pruning_layers.pdf}} \caption{Number of remaining filters in different coupling layers after pruning. Layers in different flow levels are distinguished by their colors.} \label{fig:exp_pruning} \end{center} \vskip -0.5in \end{figure} Figure~\ref{fig:exp_pruning} displays the distribution of the percentage of remained filters among different coupling layers in a pruned model trained on ImageNet32. Most filters that are pruned out concentrate in the third flow level of IDF-Res, even we set the regularization strength $\lambda$ according to the size of feature maps, i.e., $\lambda$ is larger for convolutions in the first flow level whose feature map size is $16\times 16$ and smaller for those in the third flow level whose feature map size is $4\times 4$. We observe this phenomenon in all the pruned models. One possible explanation is that these filters act on feature maps of very small size, which have minor functionalities in density estimation. Another possible answer is that, due to the multi-scale architecture of the IDF model, part of the objective only relies on the first and second flow levels. Hence, parameters in these levels play more critical roles in density estimation. \subsection{Latency Evaluation of IODF} \label{exp:3} Now we consider the full IODF with integer-only arithmetic. We conduct a set of experiments to display density estimation performance and inference speedup of quantized models. We use unsigned per-tensor quantization for activations and signed per-channel quantization for weight tensors\footnote{Per-tensor quantization uses one scale and offset parameters for the activation tensor. Per-channel quantization has a different scale and offset for each channel of weight tensor.}. IODF is trained with the complete 5-stage procedure depicted in Alg.~\ref{alg:binary_gate}. The first convolution layer within each coupling transformation step is not quantized. Afterwards, we deploy these low-precision models on a Tesla T4 GPU and evaluate their inference latency. See Appendix for the detailed environment setup. For rigorous comparison, INT8 models and FP32 models are built into inference engines with the TensorRT library. We consider the following models: (1) pure FP32 IDF-Dense and IDF-Res; (2) INT8 quantized IDF-Dense and IDF-Res; (3) FP32 IDF-Res with half of the FLOPs pruned; and (4) \modelm, which is quantized INT8 IDF-Res with half of the FLOPs pruned. We evaluate latency for different batch sizes. Table~\ref{tab:latency} shows the overall results. We see that the INT8 inference of \model is $5.9\times$ faster on ImageNet32 and $8.7\times$ faster on ImageNet64 than the baseline IDF-Dense on average. \model achieves up to $10.4\times$ speedup with a batch size 16 on ImageNet64. Comparing IDF-Res and IDF-Dense and their INT8 versions, we see that the model architecture improvement in Sec.~\ref{sec:arch} is necessary for efficient inference. Pure FP32 inference of IDF-Res is faster than IDF-Dense even the former has more parameters and FLOPs. Additionally, INT8 inference of IDF-Res is on average $5.3\times$ faster than FP32 inference while only $2.3\times$ faster for IDF-Dense. For larger batch sizes, inference latency per sample is lower, and the speedup effect of \model is more remarkable. This is promising in commercial applications since real-world images are mainly high resolution. Limited by hardware, training generative models is impossible on such large images directly. Thus we train the model on smaller patches and perform encoding in a patch-based manner, which naturally gives rise to a large batch size scenario. \begin{figure}[t] \begin{center} \centerline{\includegraphics[width=\columnwidth]{figs/bpd-latency.pdf}} \caption{Bpd-latency trade-off of IDF models with a varying number of filters (IDF-Res) and their corresponding quantized models (8bit IDF-Res), different-sized models pruned from a single large IDF model (Pruned IDF-Res), and \modelm. Models are trained on ImageNet32 and evaluated with batch size=32.} \label{fig:bpd-latency} \end{center} \vskip -0.3in \end{figure} We also implement rANS\cite{duda2009asymmetric, duda2013asymmetric} on the CPU to do actual encoding and report the actual size of compressed images as coding bpd. The coding BPD aligns well with the analytic bpd. Furthermore, the compression bandwidth is determined by both the inference latency and the cost of running rANS. IODF achieves 5.6$\times$ higher bandwidth than IDF-Dense. The speedup is lower than that for inference latency due to our suboptimal CPU implementation of rANS. An optimized GPU implementation of rANS should fill the gap, which we leave as future work. Additionally, we train IDF models with a varying number of filters (IDF-Res), prune a single large model to different extents (Pruned IDF-Res), and quantize the unpruned, different-sized IDF model (8bit IDF-Res). Figure~\ref{fig:bpd-latency} presents a bpd-latency trade-off curve of these models and shows that pruned IDF-Res and 8bit IDF-Res achieve better Pareto frontier than IDF-Res, and \model is better than both. We also conduct experiments to evaluate memory usage, generalization ability, and practical applicability of \modelm. We evaluate the models' compression performance on the high-resolution image dataset CLIC \footnote{\url{http://compression.cc/tasks/\#image}} ($\sim 10^6$ pixels per image) \cite{agustsson2017ntire} and compare the performance with non-neural compression algorithms. As shown in Table~\ref{tab:high_reso}, the models can generalize to realistic images well. Compared to IDF, \model improves the bandwidth by ten times and reduces memory usage by 47\%, with little compression rate drop. \model can attain better compression ratio over the traditional PNG codec~\cite{boutell1997png}, while being about 3 times slower than the CPU-based PNG compressor implemented in the \texttt{Pillow-SIMD} package. We also compare the speed with GPU implementations of JPEG2000 codecs, where the bandwidth of CUJ2K and Fastvideo JPEG2000 encoders are 60.5MB/s and 985MB/s, respectively\footnote{\url{https://www.fastcompression.com/benchmarks/benchmarks-j2k.htm}}. \begin{table} \caption{Compression performance on high resolution image dataset CLIC. (Models are trained on ImageNet64 and evaluated with batch size=32). Compression rate is measured in bpd, bandwidth in MB/s, and GPU memory usage in GB.} \label{tab:high_reso} \begin{center} \begin{small} \resizebox{\linewidth}{!}{ \begin{tabular}{l|ccccc|c} \toprule Model & \makecell[c]{IDF-\\Dense} & \makecell[c]{IDF-\\Res} & \makecell[c]{8bit\\IDF-Res} & \makecell[c]{Pruned\\IDF-Res} & IODF & PNG \\ \midrule BPD & \textbf{2.438} & 2.430 & 2.499 & 2.451 & 2.505 & 3.62 \\ Bandwidth & 0.84 & 1.28 & 7.57 & 1.95 & \textbf{9.17} & {29.8} \\ Memory & 3.2 & 2.8 & 1.7 & 2.4 & \textbf{1.7} & * \\ \bottomrule \end{tabular} } \vskip -0.4in \end{small} \end{center} \end{table} \section{Asymmetric Numeral System} Asymmetric Numeral Systems (ANS) \cite{duda2009asymmetric, duda2013asymmetric} is an approach to encoding a string of discrete symbols with a known distribution into a bit stream and decoding symbols from the bit stream. ANS is a kind of arithmetic coding algorithms, achieving approximate optimal code length, i.e. entropy of the distribution. Range-base ANS (rANS) is a variant of ANS with fast coding speed. Let $S=(s_1, \dots, s_n)$ be the input string of symbols with each symbol taken from the alphabet set $\mathcal{A} = \{ a_1, \dots, a_k \}$. Assume the distribution over alphabet is given by $\mbf p = \{p_1, \dots, p_k\}$ with $\sum_{i=1}^k p_i = 1$. Then a large integer $M$ a chosen as total mass and integers $\{ F_{a_1}, \dots, F_{a_k}\}$ represent mass of each symbol in the alphabet, with $p_i\approx F_{a_i} / M$. Then define a cumulative mass $C_{a_i} = \sum_{j=1}^{i-1} F_{a_j}$. rANS keeps track of input symbols with a single integer state. Let $X_t$ represent the state after rANS encodes $t$ symbols in string $S$. $X_0$ is initialized to $0$. When $X_{t}$ comes, rANS update the state $X_t$ based on $X_{t-1}$ and $s_t$ in the form \begin{equation} X_{t} = \lfloor {\frac{X_{t-1}}{F_{s_{t}}}} \rfloor * M + C_{s_{t}} + X_{t-1}\smod F_{s_{t}}. \end{equation} rANS decoder takes in a state $X_{t}$ and retrieves previous state $X_{t-1}$ and encoded symbol $s_t$. Consider that \begin{equation} X_{t}\smod M = C_{s_{t}} + X_{t-1}\smod F_{s_{t}} \end{equation} must lies in $[C_{s_t}, C_{s_t} + F_{s_t} )$. Thus the symbol $s_{t+1}$ can be retrieved by \begin{equation} s_t = a_{l} \quad C_{a_l} \leq X_t\smod M < C_{a_{l+1}}. \end{equation} Then \begin{equation} X_{t-1} = \lfloor \frac{X_{t-1}}{F_{s_{t}}} * F_{s_t} \rfloor + {X_{t-1}}\smod F_{s_t} = \lfloor \frac{X_t}{M} \rfloor * F_{s_t}+ (X_t\smod M - C_{s_t}) \end{equation} \section{Experimental Details} \subsection{Network Architecture} \label{app:arch} The overall architecture of \model is the same as IDF introduced in Sec.~\ref{sec:idf}. The entire invertible transformation from $x$ to $z$ has $L$ levels and each level is composed of $D$ coupling layers. The neural network $t_\theta(\cdot)$ in each coupling layer consists of $8$ residual blocks as Figure.~\ref{fig:network_arch} shows. \begin{figure}[h] \label{fig:resnets} \vskip 0.2in \begin{center} \centerline{\includegraphics[width=\columnwidth]{figs/network_arch.pdf}} \caption{$t_\theta(\cdot)$ in each coupling transformation consist of $8$ residual blocks and two convolutions. All convolutions use $3\times 3$ kernel and 128 hidden channels. We set \emph{padding=1} and \emph{stride=1} in convolution for not changing the shape of feature maps.} \label{fig:network_arch} \end{center} \vskip -0.2in \end{figure} The architecture for IDF-ResNets and optimization parameters are shown in Tab.~\ref{tab:exp}. \begin{table}[t] \caption{IDF-ResNets architecture and optimization parameters for each experiment.} \label{tab:exp} \vskip 0.15in \begin{center} \begin{small} \begin{sc} \begin{tabular}{lcccccccr} \toprule Dataset & L & D & batchsize & train samples & optimizer & lr & lr decay & epochs \\ \midrule ImageNet32 & 3 & 128 & 512 & 1230000 & Adamax & 0.001 & 0.99 & 100 \\ ImageNet64 & 4 & 128 & 256 & 1230000 & Adamax & 0.0001 & 0.99 & 50 \\ \bottomrule \end{tabular} \end{sc} \end{small} \end{center} \vskip -0.2in \end{table} \subsection{Optimization Parameters for Binary Gates and LSQ} \label{app:optim} In training model with gated convolutions, we initialize gates with $\alpha=0.8$ and set $lr=0.00005,\ lr\_decay=0.99$. We set strength parameter $\lambda$ in Eqn.~\ref{eq:obj_gated} according to which layer the convolution locates in, as shown in Tab.~\ref{tab:lambda}. In fine-tuning pruned model, we set $lr = 0.00005,\ lr\_decay=0.99$. Model with gated convolutions is trained for 50 epochs and pruned models are fine-tuned for 5 epochs. \begin{table}[t] \caption{We choose larger strength parameters for gated convolutions in shallower level.} \label{tab:lambda} \vskip 0.15in \begin{center} \begin{small} \begin{sc} \begin{tabular}{lcccc} \toprule Dataset & level 1 & level 2 & level 3 & level 4 \\ \midrule ImageNet32 & 1 & 2 & 4 & - \\ ImageNet64 & 1 & 2 & 4 & 8 \\ \bottomrule \end{tabular} \end{sc} \end{small} \end{center} \vskip -0.2in \end{table} For quantized model, we initialize scale parameters in quantizers data dependently in the form \begin{equation} s_\rr = \frac{2\frac{1}{n_{\rr}}\sum |r|}{\sqrt {2^b-1}},~~\rr=\WW~or~\xx, \end{equation} where $b$ is the bit width and $n_\xx$ denotes for the number of elements in tensor $\xx$. We set $lr=1e-4,\ lr\_decay=0.99$ in simulated quantization training and quantized models are fine-tuned for 10 epochs. \subsection{Hardware and Software} \label{app:hardware} The codes for our experiments are implemented with PyTorch \cite{paszke2019pytorch}. The model implementation is based on IDF codes released by \cite{hoogeboom2019integer}. rANS implementation is based on local bits back code released by \cite{ho2019compression} in C language. We train \model using 8 Nvidia RTX 2080Ti GPUs. We build inference engine and evaluate the latency on a Tesla T4 GPU and Intel(R) Xeon(R) Platinum 8259CL CPU @ 2.50GHz with TensorRT8.2.0.6, CUDA10.2. \section{Introduction} As a growing amount of data is produced every day, efficient lossless compression is of significance in storing and transmitting them. Shannon's source coding theorem \cite{shannon1948mathematical} states that the average code length needed to encode data is lower bounded by entropy of its distribution: \begin{equation} \EXP_{\xx\sim p_\cD}[|c(\xx)|] \geq \EXP_{\xx \sim p_\cD}[-\log p_\cD(\xx)], \end{equation} where $|c(\xx)|$ is the code length and $p_\cD$ is the data distribution. Based on the insight that the optimal code length for a single symbol $\xx$ is $-\log p_\cD(\xx)$, many efficient entropy codecs \cite{huffman1952method, duda2009asymmetric, duda2013asymmetric} have been developed, and they have achieved nearly optimal code length given known data distribution. However, the data distribution is generally unknown in practice. \begin{figure}[ht] \vskip 0.2in \begin{center} \centerline{\includegraphics[width=\columnwidth]{figs/intro.pdf}} \caption{Encoding latency of different neural compressors measured by milliseconds per sample. Statistics of LBB, iVPF, and iFlow are directly picked from \cite{zhang2021ivpf, zhang2021iflow}. The inner figure shows inference and entropy coding latency of IDF and \model separately. For IDF, inference is the time bottleneck in the encoding process. \model significantly improves the inference speed, making it comparable to entropy coding.} \label{fig:dgm_comp} \end{center} \vskip -0.2in \end{figure} In the machine learning community, various likelihood-based generative models have been developed, including normalizing flows (\citet{dinh2016density, ho2019flow++, chen2020vflow, lu2021implicit}, NFs), variational auto-encoders (\citet{Kingma2014AutoEncodingVB, rezende2014stochastic}, VAEs), auto-regressive models (\citet{4587817, larochelle2011neural, salimans2017pixelcnn++}, ARMs), and diffusion models (\citet{sohl2015deep, ho2020denoising, song2020denoising}, DPMs). These deep generative models (DGMs) are powerful density estimators. With a model distribution close enough to the data distribution, many efficient lossless \emph{neural compressors} have been explored \cite{hoogeboom2019integer, van2020idf++, ho2019compression, townsend2019practical, kingma2021variational}. Neural compressors have achieved superior compression ratios on standard datasets than traditional lossless compression methods, such as JPEG2000 \cite{joint2000jpeg2000}, PNG \cite{boutell1997png} and ELIF \cite{sneyers2016flif}. Despite the high compression rates, existing neural compressors still suffer from a low coding bandwidth, which has hindered practical applications. Figure~\ref{fig:dgm_comp} shows the encoding latency of several NFs-based neural compressors on the ImageNet32 dataset, of which Integer Discrete Flows (IDF) \cite{hoogeboom2019integer} is the fastest. However, the coding bandwidth of IDF is still more than an order of magnitude slower than traditional image codecs such as JPEG2000. Technically, by viewing both image data and latent variables in a discrete integer space, IDF designs a bijective mapping between them for exact likelihood inference. Then it utilizes the rANS \cite{duda2009asymmetric, duda2013asymmetric} coding algorithm to encode images into bits stream. Inference and entropy encoding are two separate steps of compression with IDF. Figure~\ref{fig:dgm_comp} illustrates that inference is over ten times slower than entropy coding for IDF. Thus, we seek to improve the coding bandwidth by accelerating the inference of flow-based models. In this work, we present integer-only discrete flow (\modelm), an efficient discrete flow for neural compression. We leverage quantization methods~\citep{jacob2018quantization,esser2019learned} to accelerate the inference. Unlike IDF, which defines discrete bijections with expensive continuous neural networks, \model performs its basic operations with the efficient \emph{integer} arithmetic. Furthermore, \model is equipped with learnable binary gates to identify and prune out redundant computations during inference. Our experiments demonstrate that \model achieves up to 10$\times$ inference speedup compared to IDF. In summary, our contributions include: \begin{itemize} \item We propose an efficient integer-only neural architecture for discrete flows. The architecture is carefully designed to be hardware-friendly to allow fast inference. We propose an algorithm to train such integer-only architectures, where \model achieves comparable density estimation performance with the full precision IDF. \item We propose to prune IDF with learnable integer (more specifically, binary) gates. By removing redundant filters, we reduce FLOPs of IDF from $7.2$G to $3.2$G with only a tiny increase in bits per dimension (bpd). \item We deploy \model with integer-only computational kernels on a Tesla T4 GPU using the TensorRT library~\cite{tensorrt}. We show that with integer arithmetic and pruning, \model can achieve up to $10\times$ speedup compared to IDF during inference. \model makes a step forward towards practical application of deep generative models in data compression. \end{itemize} \section{Related Work} {\textbf {Coding with DGMs}} Based on the \emph{change-of-variable formula}, NFs perform exact data distribution inference by designing bijective maps between data $\xx$ and latent representation $\zz$. Combining with entropy coding algorithms, NFs are applied to data lossless compression. Different from IDF \cite{hoogeboom2019integer, van2020idf++} which model $\xx$, $\zz$ both with discrete integers, \citet{ho2019compression} dicretize continuous variables and propose a \emph{local bits back} (LBB) scheme for compression with a general class of NFs which model $\xx$, $\zz$ as continuous. \citet{zhang2021ivpf, zhang2021iflow} aim to handle the problem that continuous NFs are numerically non-invertible in data compression, and they propose novel NFs with numerically invertible transformations. Although these continuous flow-based models have achieved higher compression rates than IDF, their inference and coding procedures are generally more complex and slower. Currently, hierarchical VAEs attain theoretical code lengths comparable to NFs \cite{maaloe2019biva, ho2019flow++} and DPMs achieve the best code lengths \cite{kingma2021variational} (2.49 bits/dim on CIFAR10, 3.72 bits/dim on ImageNet32 and 3.40 bits/dim on ImageNet64). However, the bandwidth of the corresponding compressors is much lower than IDF ($\sim$1MB/s) and far from practical demand, e.g., \citet{townsend2019hilloc} compresses at $\sim$ 0.02MB/s and \citet{kingma2021variational} requires expensive computation due to a large number of timesteps with a network forward in each timestep. Moreover, continuous NFs, VAEs and DPMs based compression algorithms rely on the bits back coding scheme and thus suffer from non-negligible auxiliary bits when there are only a small amount of data to compress. Meanwhile, ARMs attain theoretical code lengths similar to DPMs, but decoding is extremely slow due to their serial sampling procedure. Overall, IDF is the most efficient neural compressor at the cost of a weak compression rate loss. \textbf{Pruning and Quantization} Pruning and quantization are popular methods for reducing the memory and latency of deep neural networks (DNNs). Pruning strategies generally follow a procedure that first trains a network to convergence, scores the weight parameters (usually based on $l_1 / l_2$-norms), prunes out low-scored parameters, and fine-tunes the pruned networks \cite{lecun1990optimal, han2015learning, li2016pruning, lebedev2016fast, wen2016learning, he2017channel, frankle2018lottery}. Although various pruning methods have been developed for classification models, there is little attention on pruning NFs. Quantization represents parameters and activations in low precision format instead of 32-bit floating-point numbers~\cite{courbariaux2015binaryconnect, courbariaux2016binarized, rastegari2016xnor,zhou2016dorefa, jacob2018quantization, choi2018pact, dong2019hawq, esser2019learned, chen2020statistical, van2020bayesian}. While researches in low-bit neural networks mainly focus on the quantization of discriminative models, there have been recent attempts to introduce quantization techniques into generative models. \citet{bird2020reducing} binarize the majority of weights and activations in deep hierarchical VAE and NFs while retaining a valid probabilistic model. However, binarizing weights and activations leads to significant performance degradation Moreover, they use \emph{simulated quantization} and all operations are still performed in floating points, which can not improve inference speed in reality. \citet{balle2018integer} explores integer networks and quantization in generative models for lossy compression, aiming to address the problem that floating-point arithmetic are not deterministic across different platforms. Briefly, current DGMs based neural compressors still suffer from high inference latency, among which IDF is the most time efficient. On the other hand, pruning and quantization techniques for NFs need further exploration. Our work aims to speed up inference of IDF with integer computations and pruning out redundant computations. \section{Background: Integer Discrete Flows} \label{sec:idf} Normalizing flows (NFs) provide a general framework for learning probability distributions over continuous and discrete random variables. In the discrete case, NFs consider $\xx$ to be a discrete random variable with unknown distribution $p_X(\xx)$. Then NFs construct an invertible transformation $f: \mathcal{X} \mapsto \mathcal{Z}$, mapping $\xx$ to latent representation $\zz = f(\xx)$ on which we impose a tractable density $p_Z(\zz)$. Then the density of $\xx$ can be obtained by the change-of-variables formula: \begin{equation*} p_X(\xx) = \sum_{z\in \{f\rev(z)=x\}} p_Z(\zz). \end{equation*} Consider that $f:\cX \mapsto \mathcal{Z}$ is invertible, the summation set contains only $\zz=f(\xx)$, so the probability mass of $\xx$ is given by \begin{equation*} p_X(\xx) = p_Z(\zz). \end{equation*} IDF \cite{hoogeboom2019integer} assumes that both $\xx$ and $\zz$ lie in the $d-$dimensional integer space so $\cX = \mathcal{Z} = \ints^d$, and the prior distribution $p_Z(\zz)$ is chosen as factorized discrete logistic distribution in the form \begin{equation*} p_Z(\zz|\bm{\mu},\bm{s}) = \prod_{i=1}^d \left(\sigma(\frac{z_i+\frac12-\mu_i}{s_i}) - \sigma(\frac{z_i-\frac12-\mu_i}{s_i})\right), \end{equation*} where $\sigma(\cdot)$ denotes the sigmoid function. To obtain an invertible function, IDF designs its basic building block as an additive coupling layer \cite{dinh2016density}: \begin{equation} \label{eq:coupling} \left[\begin{array}{l}\zz_{a} \\ \zz_{b}\end{array}\right]=\left[\begin{array}{c}\xx_{a} \\ \xx_{b}+ \round{t_{\theta}\left(\xx_{a}\right)}\end{array}\right]. \end{equation} Here $\xx$ is split into two parts $\xx_a \in \ints^m, \xx_b\in\ints^n$ with $m+n=d$; likewise for $\zz_a, \zz_b$. $t_\theta(\cdot)$ is a neural network parameterized by $\theta$. The network $t_\theta(\cdot): \real^m \rightarrow \real^n$ defines a continuous mapping, which is projected to the discrete domain by the rounding operator $\round{\cdot}$. We defer the optimization of neural networks with the rounding operator to Sec.~\ref{sec:optim_quant}. IDF defines a discrete invertible mapping with a continuous function $t_\theta(\cdot)$, but the network $t_\theta(\cdot)$ still operates in the continuous domain internally. This makes IDF slow since expensive float-point operations are performed within the network. \section{Methodology} To make neural compression algorithms more efficient, we propose integer-only discrete flows (\modelm). \model consists of a novel network architecture for $t_\theta(\cdot)$, where most of the computations are achieved by efficient integer operations. \model also prunes redundant convolution filters with learnable binary gates, which are again implemented with integer operations. We discuss how to train such integer-only networks. Finally, we propose hardware-friendly optimizations to maximize the efficiency of the integer arithmetic in \modelm, including a reconsideration of the backbone architecture and a carefully implemented shortcut path. \subsection{Integer-Only Residual Block} \label{sec:int_resb} We first present the basic integer-only building block of IODF. The methodology is mostly based on existing works on neural network quantization~\citep{jacob2018quantization,esser2019learned}, but we present the details below in the normalizing flow context. In \modelm, each network $t_\theta(\cdot)$ is made of a sequence of $L$ \emph{integer-only residual blocks} $t_\theta(\xx) = t^{(1)}_\theta\circ \dots \circ t^{(L)}_\theta(\xx)$, where each block is defined as \begin{equation} \label{eq:resb} t_\theta^{(l)}(\xx) = \relu(Q(\xx) + \conv(\relu(\conv(Q(\xx))))). \end{equation} Note that this ResNet-like architecture~\cite{he2016deep} differs from the DenseNet architecture~\cite{huang2017densely} used in IDF, which we will explain in Sec.~\ref{sec:arch}. In the integer-only residual block, all the tensors are represented with a hybrid numerical format, where a \emph{quantizer} $Q$ is used to convert floating-point tensors to the hybrid format. For a real-valued tensor $\rr$, the quantizer outputs \begin{equation} \label{eq:q} \tilde{\rr} := Q(\rr) = s_\rr\hat{\rr} \approx \rr, \end{equation} where $s_\rr$ is a real-valued \emph{scale} scalar and $\hat{\rr}$ is an integer tensor. The scale captures the wide common range of the numerical values, and $\hat{\rr}$ encodes the actual value. In \modelm, $\hat{\rr}$ consists of 8-bit signed integers in $\{-128, \dots, +127\}$ or $\{0,\dots,+255\}$, depending on whether the tensor is non-negative. The Conv, ReLU, and addition operations are defined with this hybrid format, and they can be implemented efficiently with integer-only arithmetic. With a little abuse of notation, we still call numbers in this hybrid format 8-bit integers, though it can represent non-integer values with the scale scalar. We defer the discussion of the implementation of quantizer to Sec.~\ref{sec:optim_quant}. \paragraph{Integer-Only Convolution} The integer-only convolution is defined as $\yy = \conv(\xx;\WW,\bb)$, where $\WW$ is a $C\times D\times k\times k$ convolution kernel tensor with $C$, $D$ denoting the number of output / input channels, $\bb$ is a $C$-dimensional bias vector, $\xx$ is a $D\times h \times w$ input tensor, and $\yy$ is a $C\times h'\times w'$ output tensor. $(\yy, \xx, \WW)$ are all integer tensors with a floating-point scale scalar, while $\bb$ is kept in the floating-point format. The convolution is performed in the form \begin{equation} \label{eq:conv} y_c = \sum_{c'=1}^D W_{c,c'}\circledast x_{c'}+b_c, \quad c\in \{1,\dots, C\} \end{equation} Here, $W_{c,c'}$ is a $k\times k$ $2$-D kernel, $b_c$ is a scalar, $x_{c'}$ is a $h\times w$ $2$-D input feature map, $y_c$ is a $h'\times w'$ $2$-D output feature map, and $\circledast$ denotes for 2D-convolution. In our architecture, we fix $C=D, h'=h, w'=w$ within a residual block. Using the hybrid format defined as Eqn.~(\ref{eq:q}), we have $\yy\approx s_{\yy}\hat \yy$, $\xx\approx s_{\xx}\hat \xx$, $\WW\approx s_{\WW}\hat \WW$. Plugging them into Eqn.~(\ref{eq:conv}) yields \begin{align*} s_{\yy}\hat y_c \approx y_c & = \sum_{c'=1}^D W_{c,c'}\circledast x_{c'} + b_c \approx \sum_{c'=1}^D \tilde W_{c,c'}\circledast\tilde x_{c'} + b_c \\ & = \sum_{c'=1}^D \left(s_\WW \hat W_{c,c'}\right) \circledast \left(s_\xx \hat x_{c'}\right) + b_c \\ & = s_\WW s_\xx \sum_{c'=1}^D \hat W_{c,c'}\circledast\hat x_{c'} + b_c. \end{align*} \begin{figure}[t] \vskip 0.2in \subfigure[Integer-arithmetic inference.]{ \begin{minipage}{0.95\columnwidth} \centering \includegraphics[width=\columnwidth]{figs/true_quant_v2.pdf} \end{minipage} \label{fig:true_q} } \subfigure[Fake quantization in training.]{ \begin{minipage}{0.95\columnwidth} \centering \includegraphics[width=\columnwidth]{figs/simu_quant_v2.pdf} \end{minipage} \label{fig:fake_q} } \vskip -0.1in \label{fig:quant} \caption{True quantization and fake quantization. Best viewed in color. } \end{figure} Reorganizing the terms, we have \begin{equation} \label{eq:quant_conv} \hat y_c \approx \frac{s_\WW s_\xx}{s_\yy} \sum_{c'=1}^D \hat W_{c,c'}\circledast\hat x_{c'} + \frac{b_c}{s_\yy}. \end{equation} Eqn.~(\ref{eq:quant_conv}) can be implemented as the convolution of two signed 8-bit integer tensors $\hat \WW$ and $\hat \xx$, followed by element-wise floating-point multiplication and additions. This can be realized efficiently on GPUs as a single operation in the TensorRT library, as illustrated in Figure.~\ref{fig:true_q}. Matrix multiplications in the convolution are performed with the INT8 kernel, and the summation is performed by a 32-bit integer accumulator. Bias addition is also performed with 32-bit integers. The outputs are again quantized to an 8-bit integer using Eqn.~\ref{eq:quant_conv}. Because computations of a convolution layer are dominated by the multiplications in the summation, substituting floating-point operations with integer-arithmetic leads to a significant computation reduction when deploying on hardware. \paragraph{Integer-Only ReLU} Given an input tensor $\xx \approx s_\xx \hat \xx$, the ReLU is directly applied to the signed 8-bit integer part as $\hat \yy = \mbox{ReLU}(\hat \xx) = \max\{0,\hat\xx\}$. The scale scalar is not affected $s_\yy=s_\xx$. Hence, we have $\yy \approx s_\yy\hat \yy = s_\xx \max\{0, \hat \xx\} \approx \max\{0,\xx\} = \mbox{ReLU}(\xx)$. \subsection{Training Integer-Only Residual Blocks} \label{sec:optim_quant} So far, we have defined the integer-only residual blocks with a general definition of the quantizer. We have not yet discussed how to train such blocks or implement the quantizer, which we shall do now. As mentioned in the last subsection, integer-only residual blocks rely on quantizers to convert real-valued tensors to integers. The conversion is lossy, which usually causes inaccurate outputs of the model. Additionally, the scale parameter in Eqn.~\ref{eq:q} is crucial to the performance of quantized networks. This section focuses on fine-tuning the integer-only network with \emph{simulated quantization training} (a.k.a. fake quantization)~\cite{jacob2018quantization}. The true integer-only inference must be deployed on hardware with special tools, which is not convenient for our training. So we implement fake quantization in PyTorch, which still uses floating-point operations but simulates the integer arithmetic by the quantizers as shown in Figure.~\ref{fig:fake_q}. This corresponds to using the convolution defined in Eqn.~(\ref{eq:conv}). Given a known scale $s$, we define the quantizer in Eqn.~(\ref{eq:q}) in the specific form: \begin{equation} \label{eq:q2} \begin{aligned} \tilde{\rr} & = Q(\rr) = s\hat{\rr} = s\cdot\round{\mrm{clip}\left(\frac{\rr}{s}, -Q_N, Q_P\right)}. \end{aligned} \end{equation} The clipping operation acts element-wise on the tensor $\rr/s$: \begin{equation} \mrm{clip}\left(\frac \rr s, -Q_N, Q_P\right) = \max\left(\min\left(\frac \rr s, Q_P\right), Q_N\right). \end{equation} We set $Q_N=-128, Q_P=127$ for weight tensors and $Q_N=0, Q_P=255$ for non-negative activation tensors. To optimize the network parameters with the quantizer, we leverage learned step-size quantization (LSQ) \cite{esser2019learned}. LSQ treats the scalar $s$ as a learnable parameter, which is updated by a gradient-based optimization algorithm. Back-propagation through quantizer is performed with the Straight Through Estimator (STE)\cite{bengio2013estimating} in the form $ \partial \round{\uu} / \partial \uu = \II$ for real-valued vectors $\uu$. Thus we have \begin{equation} \label{eq:grad_flow} \frac{\partial \mathcal{L}}{\partial \rr} = \frac{\partial \mathcal{L}}{\partial \tilde{\rr}}\frac{\partial \tilde \rr }{\partial \rr} = \frac{\partial \mathcal{L}}{\partial \tilde \rr},~ \rr = \WW~or~\xx, \end{equation} where $\mathcal{L}$ denotes for the objective function and $\rr$ applies to $\WW$ or $\xx$. With Eqn.~(\ref{eq:grad_flow}) we can perform gradients back-propagation in \model normally. For scale parameters, the gradient of $s$ can be calculated as follows, \begin{equation*} \frac{\partial \tilde{\rr}}{\partial s} = \left \{ \begin{array}{ll} \left ( -\rr / s + \round{ \rr / s } \right ) \odot \mathbb{I}(-Q_N < \rr / s < Q_P) \\ [0.2cm] -Q_N \cdot \mathbb{I}(\rr / s < -Q_N) \\ [0.2cm] Q_P \cdot \mathbb{I}(\rr / s > Q_P) \end{array} \right . \end{equation*} where $\mathbb{I}(\cdot)$ is an indicator function that returns a tensor of the same shape as $\rr$, and $\odot$ is the element-wise multiplication. Applying the chain rule, we have \begin{equation*} \frac{\partial \mathcal{L}}{\partial s} = \frac{\partial \mathcal{L}}{\partial \tilde \rr} \frac{\partial \tilde \rr}{\partial s} \end{equation*} We adopt the gradient re-scaling trick introduced in \cite{esser2019learned}, multiplying the gradient of $s$ by a scale factor $g=1/\sqrt{C Q_P}$, where $C$ is the number of channels. This gradient re-scaling trick helps to stabilize the learning of the scale parameter $s$. \begin{table}[t] \caption{Latency of floating-point and integer-only inference for convolutions with varying number of input and output channels. Obtained by averaging over 1000 runs (milliseconds).} \label{tab:conv_speedup} \vskip 0.15in \begin{center} \begin{small} \begin{sc} \begin{tabular}{cc|ccc} \toprule In Chn & Out Chn & Fp32 & Int8 & Speedup \\ \midrule 128 & 128 & 0.040 & 0.0039 & 10.2$\times$\\ 64 & 256 & 0.035 & 0.0098 & 3.6$\times$\\ 32 & 512 & 0.037 & 0.0131 & 2.8$\times$\\ \bottomrule \end{tabular} \end{sc} \end{small} \end{center} \vskip -0.1in \end{table} \subsection{A More Efficient Architecture for Quantization} \label{sec:arch} To make the inference of the integer-only model more efficient on hardware, we improve the network architecture. We first replace dense blocks in IDF with residual blocks for their more regular architecture and fewer connections across layers. DenseNets are more memory-intensive yet less computation-intensive~\cite{zhang2021resnet} since it has many concatenation operations, which lead to many expensive quantization and dequantization operations during inference. Furthermore, DenseNets have many convolutions with a small number of input / output channels, which have unsatisfactory INT8 speedup. Tab.~\ref{tab:conv_speedup} shows the inference latency of floating-point and INT8 convolution layers for a varying number of input / output channels. All convolutions use $3\times 3$ kernels and $16\times 16$ input / output feature maps, so their FLOPs are kept the same. The floating-point inference latency of these three convolutions is similar, but the integer inference latency differs significantly. When the channels of input and output feature maps are identical, integer arithmetic can bring better speedup than when there is a big difference between input and output channels. Convolutions in ResNets are mainly of the former shapes, while those in DenseNets are the opposite. Therefore, residual blocks are better building blocks than dense blocks for \modelm. \subsection{Learnable Binary-Gated Convolution} \label{sec:pruning} \begin{figure}[t] \vskip 0.2in \begin{center} \centerline{\includegraphics[width=\columnwidth]{figs/pruning_v2.pdf}} \caption{A gated convolution layer with $C\times D \times k \times k$ kernel weights $\WW$. The weights are represented by a $C$-by-$D$ chessboard with each element as a $2$-D $k$-by-$k$ kernel. Each gate entry $\tilde \vg$ determines whether to disable a filter (a row of kernels) in the kernel weights. Disabling a filter leads to removing the output feature map and corresponding kernels of the successive convolution that act on this feature map.} \label{fig:prune_filters} \end{center} \vskip -0.2in \end{figure} Neural networks have many redundant computations. Many channels die during training, and they are rarely used for inference. This problem is particularly severe for flow-based models since each transformation step has a separate network, and the required network width may vary for each transformation step. \model addresses this problem by adding learnable binary gates to integer-only convolutions, where the masked gates can be removed at the inference time. Formally, we denote a learnable binary gate by $\tilde{\vg} = b(\vg) \coloneqq \mathbb{I}(\vg > 0.5)$, where $\vg \in [0,1]^C$. Then a gated convolution is defined as (omitting the bias): \begin{equation} \begin{aligned} \yy & = \mrm{GConv}(\xx;\WW, \vg)\coloneqq B(\tilde \vg) \odot \mrm{Conv}(\xx;\WW) \\ & = \sum_{c'=1}^D \left(\tilde g_{c} W_{c,c'}\right) \circledast x_{c'}, \end{aligned} \end{equation} where $B(\tilde \vg)$ is a broadcast operation to a $C\times h' \times w'$ tensor with entries $B(\tilde \vg)_{c,i,j} = \tilde g_c, \forall i=1,\dots,h', j=1,\dots,w'$. Figure~\ref{fig:prune_filters} illustrates the process of pruning out a filter with a binary gate. $\tilde g_c=0$ relates to disabling the filter $W_{c,:}$ and zeroing a output feature map $y_c$. Then the corresponding entries in the next convolution layer's weight that apply on this feature map are also removed. Therefore, pruning out $m$ out of $C$ filters of a convolution will reduce $m/C$ computations for both current and the next layer. \paragraph{Training Binary-Gated Convolution} In the direction of obtaining good gates which contain as many zeros as possible while doing little harm to the performance of the whole model, we optimize $\vg$ based on both the original training objective and the $l_1$-norm of $\tilde{\vg}$, i.e., the number of ones in the vector $\tilde{\vg}$. Let $\mathcal{L}_{IDF}$ be the original IDF objective function, the objective of \modelm~is formalized as follows, \begin{equation} \label{eq:obj_gated} \begin{aligned} \mathcal{L}(\XX; & \{ \WW \}, \{ \vg \}) = \mathcal{L}_{IDF} (\XX; \{\WW\}, \{\vg\}) + \sum \lambda \norm{\tilde \vg}_1, \end{aligned} \end{equation} Here $\{\WW\}, \{\vg\}$ denote for the sets of all convolution kernel matrices and all gates vectors respectively. All gate vectors $\vg$ are initialized as $\alpha \bm{1}$ with $0.5 \leq \alpha < 1$ so original gated convolution retains all filters. The $\mathcal{L}_{IDF}$ term tends to keep all entries of $\tilde \vg$ close to $1$ while $\norm{\tilde \vg}_1$ term pushes the gates to be sparse. $\lambda$ acts as a strength hyper-parameter balancing them. See Appendix~\ref{app:optim} for settings of $\lambda$ in different parts of our model. Taking derivative of $\mathcal{L}$ w.r.t. $\vg$, we have \begin{equation*} \label{ea:grad_gates} \frac{\partial \mathcal{L}}{\partial \vg } = \frac{\partial \tilde{\vg}}{\partial \vg} \left ( \frac{\partial \mathcal{L}_{IDF}}{\partial \tilde \vg} + \lambda \frac{\partial \norm{\tilde \vg}_1}{\partial \tilde \vg} \right ) \end{equation*} ${\partial \mathcal{L}_{IDF}} / {\partial \tilde \vg}$ can be obtained by backward-propagating through the neural network. $\partial {\norm{\tilde\vg}_1} / {\partial \tilde \vg}= \bm{1}$ since $\tilde \vg$ is binary. We adopt STE to make gradients flow through the binarize operation by taking $\partial \tilde \vg / \partial \vg = \II$. Based on the gradients $\partial \mathcal{L} / \partial \vg$, gates can be optimized with gradient-based algorithms to achieve a good balance between efficiency and density modeling performance. \paragraph{Binary-Gated Residual Blocks} \begin{algorithm}[t] \caption{Training \modelm} \label{alg:binary_gate} \begin{algorithmic} \STATE {\bfseries Input:} $r_{target}$, remaining target proportion of FLOPs and $\XX$, the training dataset. \STATE \#Stage 1: \STATE $\WW \longleftarrow$ InitializeParameter() \STATE Train $\mathcal{L}_{IDF} (\XX; \{\WW\}, \{\mathbf{1}\})$ to convergence \STATE $F_0 \longleftarrow$ CalculateFLOPs($\WW$) \STATE \#Stage 2: \STATE $\vg \longleftarrow \alpha \bm{1},\ \lambda \longleftarrow$ InitializeLambda() \STATE Train $\mathcal{L} (\XX; \{\WW\}, \{\mathbf{\vg}\}) $ until CalculateFLOPs($\WW, \vg$)$<r_{target}F_0$ \STATE \#Stage 3: \STATE Fine-tune $\mathcal{L}_{IDF} (\XX; \{\WW\}, \vg)$ with fixed $\vg$ \STATE \#Stage 4: \STATE Fine-tune the model with fake quantization applied to activations \STATE \#Stage 5: \STATE Fine-tune the model with fake quantization applied to activations and weights \end{algorithmic} \end{algorithm} To prune out filters, we need to carefully consider related layers which can be influenced by the filter removal. For simple network architectures, pruning across consecutive layers is relatively straightforward, as shown in Figure.~\ref{fig:prune_filters}. However, pruning residual blocks is a little more complicated since the addition operation in Eqn.~\ref{eq:resb} requires the two operands, i.e., the shortcut and the convolution output, to have the same number of feature maps. This is not guaranteed in general since the convolution has a trainable gate. We solve this problem by performing the addition in the original unpruned feature map with a \emph{scatter add ($\mrm{SAdd}$)} operation: \begin{align*} t_\theta^{(l)}(\xx) = \relu(\mrm{SAdd}(&Q(\xx), \\ &\mrm{GConv}(\relu(\mrm{GConv}(Q(\xx))). \end{align*} The scatter add operation directly sums up the unpruned $Q(\xx)$ and the sparse pruned output of the gated convolution by maintaining indices representing which feature maps are removed. As a result, convolution layers within the residual blocks can be pruned arbitrarily without considering the alignment with the shortcut path. \subsection{Training Workflow} We propose a 5-stage training algorithm for \modelm, as shown in Alg.~\ref{alg:binary_gate}. We first train a full-precision, non-gated model to convergence by optimizing $\mathcal{L}_{IDF} (\XX; \{\WW\}, \{\mathbf{1}\}) $. Next, we insert learnable binary gates into convolutions, set the strength hyper-parameter $\lambda$, and train the gated model by optimizing $\mathcal{L} (\XX; \{\WW\}, \{\mathbf{\vg}\})$. After removing all zeroed-out input and output filters in $\WW$, we obtain a \emph{pruned} model. We train the gated model until the FLOPs of the pruned model reaches our target. Then, we fine-tune the pruned model by optimizing $\mathcal{L}_{IDF} (\XX; \{\WW\}, \{\mathbf{\vg}\}) $, keeping the gates fixed. The final two stages fine-tune the model to use integer-only arithmetic by incorporating fake quantization. \subsection{Deployment on Hardware} So far, \model is still at the simulation level with quantizers mimicking the behaviors of integer arithmetic. To earn true inference speedup effects, we must deploy \model on specific hardware that support accelerated integer operations. As an example, NVIDIA T4 GPUs attain 16 times as many integer operations per second as floating point numbers\footnote{\url{https://www.nvidia.com/en-us/data-center/tesla-t4/}}. In this work, the T4 GPU is selected due to more convenient software support, specifically, the TensorRT library \cite{tensorrt}. Notably, we do not rely upon any unique features of T4, so \model can be successfully deployed to other hardware and achieve the corresponding acceleration effect. \input{5-experiments} \section{Conclusion} We propose Integer-Only Discrete Flows (\modelm), an efficient flow-based neural compressor. We propose a hardware-friendly backbone architecture with integer-only residual blocks. By equipping integer-arithmetic convolutions with learnable binary gates, we prune out redundant filters, significantly reducing the number of parameters and the amount of computation. Furthermore, we directly deploy \model on a Tesla T4 GPU and measure the encoding latency, showing that \model achieves up to $10\times$ speedup compared to IDF. \section*{Acknowledgements} This work was supported by National Key Research and Development Project of China (No. 2021ZD0110502); NSF of China Projects (Nos. 62061136001, 61620106010, 62076145, U19B2034, U1811461, U19A2081, 6197222, 62106120); Beijing NSF Project (No. JQ19016); Beijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098; a grant from Tsinghua Institute for Guo Qiang; the NVIDIA NVAIL Program with GPU/DGX Acceleration; the High Performance Computing Center, Tsinghua University; and Major Innovation \& Planning Interdisciplinary Platform for the ``Double-First Class" Initiative, Renmin University of China.

BkiUbTA4uzki04AfVOYx

\section*{Draft} \begin{abstract} Radio frequency sources are observed at a fusion center via sensor measurements made over slow flat-fading channels. The number of sources may be larger than the number of sensors, but their activity is sparse and intermittent with bursty transmission patterns. To account for this, sources are modeled as hidden Markov models with known or unknown parameters. The problem of blind source estimation in the absence of channel state information is tackled via a novel algorithm, consisting of a dictionary learning (DL) stage and a per-source stochastic filtering (PSF) stage. The two stages work in tandem, with the latter operating on the output produced by the former. Both stages are designed so as to account for the sparsity and memory of the sources. To this end, smooth LASSO is integrated with DL, while the forward-backward algorithm and Expectation Maximization (EM) algorithm are leveraged for PSF. It is shown that the proposed algorithm can enhance the detection and the estimation performance of the sources, and that it is robust to the sparsity level. \end{abstract} \begin{IEEEkeywords} Blind source separation, Wireless networks, Dictionary learning, Intermittent and sparse sources, Hidden Markov model \end{IEEEkeywords} \section{Introduction} Blind source separation (BSS) refers to the separation of a set of source signals from a set of mixed signals, without resorting to any a priori information about the source signals or the mixing process \cite{BSShandbook}. BSS exploits only the information carried by the received signals themselves, hence the term \textit{blind}. BSS has numerous applications in speech recognition \cite{Speech03,BD11}, image extraction \cite{Image02,Image97}, and surveillance \cite{Surv00,Surv04}. Different metrics are used to evaluate the performance of BSS methods depending on the applications. For example, signal-to-interference ratio is used in \cite{Speech03} for speech separation, and a performance index is introduced in \cite{Image02} for image feature extraction. Based on these metrics, many approaches have been proposed to solve BSS problems, such as independent component analysis (ICA) \cite{ICA00}, principal component analysis (PCA) \cite{PCA87}, and singular value decomposition (SVD) \cite{SVD70}. This paper addresses BSS in wireless networks. We are specifically interested in the set-up illustrated in Fig.~\ref{fig:SM}, in which a fusion center observes a number of radio sources via noisy sensor measurements over unknown channels. The system may model an Internet-of-Things (IoT) system, such as LoRa, Sigfox, or Narrow Band-IoT (NB-IoT) \cite{IoT1,IoT2}. In wireless networks involving multiple terminals operating over flat fading channels, the signals received at a terminal are linear mixtures of the signals emitted by the transmitting terminals. The need for BSS arises in non-collaborative applications in which the signals and the channels through which they are received at a terminal are both unknown. ICA has been widely applied to solve BSS problems in wireless networks \cite{Alavi16,ICACRN11,ICAMIMO04,ICAOFDM07}, since it yields a useful decomposition with only scaling, and permutation ambiguities \cite{ICA}. To achieve signal separation, ICA relies on the statistical independence and on the non-Gaussian distribution of the components of the mixture. Key assumptions made in the implementation of the various forms of ICA are that the underlying mixing process has the same number of inputs and outputs, and that all sources are active throughout the observation interval. These assumptions are limiting and not suitable for the applications under study in this work, as discussed next. With the aim of capturing key aspects of IoT systems, this paper focuses on practical wireless scenarios in which the number of latent sources is generally larger than the number of sensors, but the sources are active intermittently with bursty transmission patterns. The cumulative time a source is active is a small fraction of the overall observation time, and the sources' on-off patterns vary slowly. In order to capture these properties, the sources are modeled as hidden Markov processes with known or unknown parameters. Source memory will be seen to be instrumental in enabling source separation. The wireless network operates over slow flat-fading channels; sensors communicate with a fusion center over ideal channels; and all nodes are time-synchronized to the same clock by the fusion center. The general BSS problem with more sources than sensors can be formulated as an underdetermined linear system $\mathbf{A}\mathbf{x}=\mathbf{y}$ in the absence of noise. The columns of the matrix $\mathbf{A}$ serve as a basis for expressing the observations $\mathbf{y}$. The set of basis signals that form the matrix $\mathbf{A}$ is called a dictionary. Underdetermined linear systems of equations of the form $\mathbf{A}\mathbf{x}=\mathbf{y}$ have infinitely many solutions when the matrix $\mathbf{A}$ is full rank. Regularization may introduce additional conditions on the solutions, for example favoring smaller values of $\mathbf{x}$, leading to unique solutions of the underdetermined linear system. Sparse representations for which the solution $\mathbf{x}$ is unique have been the subject of intensive research resulting in a large body of literature \cite{OMPbook,FOCUSS97,OOMP02}. Pre-defined dictionaries, such as based on Fourier transforms, are convenient and computationally fast, but in the BSS problem, both the dictionary $\mathbf{A}$ and the signals $\mathbf{x}$ are unknown. Sparse representation problems for which the dictionary is unknown require \textit{dictionary learning} (DL) in addition to signal recovery \cite{DL10,DL11}. The advantage of DL is to enable a system to learn a dictionary adaptively from a set of observations rather than assume a prescribed rigid dictionary. Among other problems, DL methods have been applied to joint direction of arrival (DOA) estimation and array calibration \cite{DLWC17}, linear transceiver design \cite{DLWC13}, cloud K-singular value decomposition (K-SVD) for big, distributed data \cite{DLWC15}, and channel representation for frequency-division duplexing (FDD) massive MIMO system \cite{DLWC16}. In these examples, the DL algorithms solve BSS problems in which the number of sources is larger than the number of sensors, but the methods are agnostic to time variability properties of sources with memory. In \cite{HMM09} and \cite{HMM10}, the authors set up a hidden Markov model (HMM) to solve a BSS problem. However, in \cite{HMM09} the memory of the sources is not accounted for, while in \cite{HMM10} a simplified model is assumed whereby only one source can appear or disappear at any given time. In this context, we propose a two-stage algorithm for solving the BSS problem for sources with memory modeled by an HMM and observed over slow flat-fading channels. The proposed algorithm comprises a DL stage and a Per-source Stochastic Filtering (PSF) stage. The DL stage of the proposed algorithm exploits knowledge about the source sparsity and memory to aid with the source separation. The effect of source memory is introduced by a penalty term that discourages solutions with short-duration transmissions by means of a smooth LASSO algorithm. The input to the DL stage are observations from the sensors. The output from the DL stage are channel estimates, source signal estimates and source states (active or inactive). The source state estimates produced by the DL algorithm may be viewed as the output of a binary asymmetric channel in which some of the states are "flipped" with respect to the true states. The error probabilities associated with the flipped states are referred to as flipping probabilities. The PSF stage consists of a forward-backward step along with an Expectation Maximization (EM) step that estimate the unknown HMM transition probabilities and flipping probabilities. The main contributions of this paper are summarized as follows: \begin{itemize} \item A two-stage architecture is introduced for solving the problem of blind source estimation of HMM sources over slow flat-fading channel. The sources feature intermittent activity, and the number of latent source may be larger than the number of sensors; \item A smooth DL algorithm is proposed that is capable of exploiting source memory to support channel estimation, signal estimation and source detection. Two simplified versions of the smooth DL algorithms are introduced to reduce the computational complexity; \item An PSF algorithm is introduced that is capable of operating in the absence of a priori information about the HMM parameters and the state estimation flipping probabilities. This algorithm integrates an forward-backward step with an EM step. \end{itemize} The rest of the paper is organized as follows. The system model and a hidden Markov source model are presented in Sec.~\ref{sec:SM}. Background on DL algorithms is provided in Sec.~\ref{sec:pre}. In Sec.~\ref{sec:SDL}, we propose a two-stage DL-based algorithm to solve the BSS problem for sources with memory in wireless networks. The PSF is described in detail in Sec.~\ref{sec:PSF} for the two cases with known and unknown source parameters. Finally in Sec.~\ref{sec:NR}, simulation numerical results are shown to support that our proposed algorithm can separate sources and recover source signals with higher accuracy than existing DL algorithms. \section{System Model} \label{sec:SM} Consider a system that includes $M$ receiving antennas, or radio sensors, and $N$ sources, as illustrated in Fig.~\ref{fig:SM}. The number of sources $N$ is generally larger than the number of receive antennas $M$. All sensors are connected to a fusion center, which may be implemented in a cloud processor, via backhaul links. Equivalently, the fusion center has access to $N$ receive antennas. Models with a fusion center reflect the architecture of IoT networks, such as LoRa, Sigfox, and NB-IoT \cite{IoT1,IoT2}. In our model, each source transmits intermittently, and hence is generally active only for a subset of the $T$ symbol periods over which the sensors collect data. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \includegraphics[width=\columnwidth]{./Figure1.pdf} \caption{Wireless Sources transmit sporadically over a flat fading channel. The fusion center in the ``cloud" wishes to estimate the activity and transmitted signals based on the signals received by the distributed radio sensors.} \label{fig:SM} \end{minipage} \end{figure} Assuming that all nodes are time-synchronous, the discrete-time signal received by the $M$ sensors over $T$ symbol periods is given in matrix form as \begin{equation} \label{eq:signal model} \mathbf{Y}=\mathbf{H}\mathbf{X} +\mathbf{Z}, \end{equation} where $\mathbf{Y}=[\mathbf{y}(1),\dots,\mathbf{y}(T)]$ is an $M \times T$ matrix collecting as columns the $M \times 1$ received signals $\mathbf{y}(t)$ across all $T$ symbols $t=1,\dots,T$; $\mathbf{X}=[\mathbf{x}(1),\dots,\mathbf{x}(T)]$ is an $N \times T$ matrix that gathers the $N \times 1$ signals $\mathbf{x}(t)$ transmitted from all $N$ sources over time; $\mathbf{Z}=[\mathbf{z}(1),\dots,\mathbf{z}(T)]$ contains independent zero-mean complex Gaussian noise entries with variance $\sigma^2$; and $\mathbf{H}$ is the $M \times N$ complex fading channel matrix. The channel matrix $\mathbf{H}$ is assumed to be constant for $T$ symbol periods. Given the intermittent nature of the traffic pattern of the sources, the $t$-th column vector $\mathbf{x}(t)$ that collects the $M$ symbols transmitted by the sources at time $t$, is generally sparse. In other words, only a subset of the entries of $\mathbf{x}(t)$ is non-zero. The signals $\mathbf{y}(t)$, for $t=1,\dots,T,$ are collected at the fusion center. Based on the received signals $\mathbf{Y}$, the goal of the fusion center is to detect the sources' activity and to recover the signals $\mathbf{x}(t),$ for $t=1,\dots,T$, or equivalently the matrix $\mathbf{X}$, in the absence of information about the channel matrix $\mathbf{H}$. For each source $n$, we define the activation pattern as a binary sequence $s_n(t)$, where the binary state $s_n(t)$ indicates whether a source is active or not. Specifically, when the state of source $n$ is $s_n(t)=1$, then the source is active, while it is inactive when $s_n(t)=0$. The binary state is assumed to be described by the following models. \begin{enumerate} \item \textit{Intermittent and smooth deterministic model}: Each source is active for a small fraction of time, and its on-off patterns tend to have few switches between on and off states. As a result, the sequence $s_n(t)$ is ``smooth", i.e., it has a small number of transitions between on and off states; \item \textit{Probabilistic hidden Markov model}: As illustrated in Fig.~\ref{fig:HMM}, the activity $s_n(t)$ of each source $n$ follows a two-state Markov chain. The transition probabilities of the two-state Markov chain are defined as $p_n=\text{Pr}(s_n(t)=1|s_{n-1}(t)=0)$ and $q_n=\text{Pr}(s_n(t)=0|s_{n-1}(t)=1)$. We will consider both the cases in which the probabilities $p_n$ and $q_n$ are known and unknown. \end{enumerate} We emphasize that the two models are not mutually exclusive and can be assumed to hold simultaneously. In particular, we will propose to leverage the deterministic model in order to improve the performance of source separation and the stochastic model to refine the estimates obtained by sources separation. When a source $n$ is active, i.e., when $s_n(t)=1$, it transmits an independent sample $x_n(t) \sim f_n(t)$ with a given distribution $f_n(t)$, e.g., Gaussian or binary. Instead, when the source $n$ is inactive, i.e., $s_n(t)=0$, it does not transmit and we have $x_n(t)=0$. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \includegraphics[width=\columnwidth]{./Figure2.pdf} \caption{Hidden Markov Model (HMM) for a source $n$.} \label{fig:HMM} \end{minipage} \end{figure} A final remark is in order concerning synchronization requirements. The described model \eqref{eq:signal model} applies even in the absence of time synchronization among the sources, since the vector $\mathbf{x}(t)$ can model a generic sample at discrete time $t$ of the transmitted signals. However, if time synchronization is assumed, and the sources transmit digitally modulated signals, the vector $\mathbf{x}(t)$ can be assumed to contain the constellation points transmitted at the $t$-th symbol period. We note that time and frequency synchronization in sensor networks is a topic of great interest, and it has been investigated in blind scenarios with no pilot symbols as well \cite{Bsync1,Bsync2}. However, the literature about BSS often assumes that sensors and sources are synchronized in order to focus on the technical challenges of BSS \cite{DLWC17,DLWC13,DLWC15,DLWC16}. \section{Preliminaries: Dictionary Learning} \label{sec:pre} The DL method proposed in this work leverages prior information about the memory of the sources. We start by reviewing DL methods that do not exploit such information. These methods use only the fact that the signal $\mathbf{x}(t)$ is sparse at any time $t$. Prior information about the memory of each source $x_n(t)$, to be considered in the next section, includes smoothness properties or statistical models. Assuming only information about the sparseness of $\mathbf{x}(t)$ at each time $t$, a standard approach is to utilize the channel matrix $\mathbf{H}$ as a \textit{dictionary} to be learned to recover $\mathbf{X}$. DL techniques approximate the solution of the maximum likelihood (ML) problem if the fusion center acts as a master clock synchronizing all other nodes. \begin{equation} \label{eq:DL} \underset{\mathbf{H},\mathbf{X} \in \mathcal{X}}{\text{minimize }} ||\mathbf{Y}-\mathbf{H}\mathbf{X}||^2, \end{equation} where $\mathcal{X}$ is the set of matrices with sparse columns, that is, with columns containing a limited number of non-zero entries. This problem is not convex with respect to the pair ($\mathbf{H}$, $\mathbf{X}$). DL methods use an iterative procedure, whereby the signal $\mathbf{X} \in \mathcal{X}$ and the channel $\mathbf{H}$ are optimized alternately \cite{DL15}. In the following, we first discuss solutions for the optimization over the signal $\mathbf{X}$ for a given channel matrix $\mathbf{H}$, and then over the channel $\mathbf{H}$ for a given signal matrix $\mathbf{X}$. DL methods are subject to inherent permutation and sign ambiguities \cite{Ambi10}. The scaling ambiguity can be instead resolved if one imposes that the channel matrix columns are normalized \cite{MOD99,MDU13,SGK13,Sadeghi13}. \subsection{Signal Estimation} \label{sec:SE} For any fixed iterate $\mathbf{H}^{(k)}$ at the $k$-th iteration, from \eqref{eq:DL}, the problem of estimating the signal $\mathbf{X}$ reduces to \begin{equation} \label{eq:SE} \mathbf{X}^{(k+1)}=\underset{\mathbf{X} \in \mathcal{X}}{\text{argmin}} ||\mathbf{Y}-\mathbf{H}^{(k)}\mathbf{X}||^2. \end{equation} Standard sparse optimization estimators, such as orthogonal matching pursuit (OMP) \cite[\S 3.1.2]{OMPbook} can be used to address problem \eqref{eq:SE}. Alternatively, one can use the LASSO algorithm \cite{lasso96} to solve the convex problem \begin{equation} \label{eq:LASSO} \underset{{\mathbf{x}(t)}}{\text{minimize }} || \mathbf{y}(t)-\mathbf{H}^{(k)}\mathbf{x}(t)||_2 + \lambda ||\mathbf{x}(t)||_1, \qquad t=1,\dots,T, \end{equation} separately for each $t$, where the weight $\lambda$ is a parameter to be determined as a function of the sparsity of vector $\mathbf{x}(t)$. \subsection{Channel Estimation} At the $k$-th iteration, for a fixed iterate $\mathbf{X}^{(k+1)}$, the channel estimation step can obtain the next channel iterate $\mathbf{H}^{(k+1)}$ by using different algorithms, such as the Method of Optimal Directions (MOD) \cite{MOD99}, the Multiple Dictionary Update (MDU) \cite{MDU13}, or their enhanced versions proposed in \cite{Sadeghi13}. Here we summarize the enhanced MDU approach, which was shown in \cite{Sadeghi13} to provide the best performance via simulation results. The MDU approach estimates the channel matrix $\mathbf{H}$ for a given $\mathbf{X}^{(k+1)}$ by following an iterative approach. To elaborate, denote $\mathcal{S}(\mathbf{x})$ the set of indices of the non-zero elements in vector $\mathbf{x}$. Also, index $(j,k)$ the $j$-th iteration of the MDU algorithm within the $k$-th step of the DL alternate optimization scheme. At the $(j,k)$ iteration, for $j=1,2,\dots$, MDU computes \begin{equation} \label{eq:MOD} \mathbf{H}^{(j,k)}=\mathbf{Y}\mathbf{X}^{(j,k)T}(\mathbf{X}^{(j,k)}\mathbf{X}^{(j,k)T})^{-1}, \end{equation} and \begin{equation} \label{eq:MDU} \mathbf{x}^{(j+1,k)}(t)=\mathbf{D}^{(k)}(\mathbf{H}^{(j,k)T}\mathbf{H}^{(j,k)})^{-1}\mathbf{H}^{(j,k)T}\mathbf{y}(t), \end{equation} where for all $t$, $\mathbf{D}^{(k)}$ is a diagonal matrix with elements having indices in $\mathcal{S}(\mathbf{x}^{(k+1)}(t))$ equal to $1$ and zero otherwise. The iteration is initialized with $\mathbf{X}^{(1,k)}=\mathbf{X}^{(k+1)}$. Therefore, for a fixed sparsity pattern $\mathcal{S}(\mathbf{x}^{(k+1)}(t))$, MDU alternately estimates channel and signals. The enhancement proposed in \cite{Sadeghi13} substitutes at iteration $k$ the received signal $\mathbf{Y}$ in \eqref{eq:MOD} and \eqref{eq:MDU} with $\mathbf{Y}^{(k)}=\mathbf{Y}+\mathbf{H}^{(k)}\mathbf{X}^{(k)}-\mathbf{H}^{(k)}\mathbf{X}^{(k+1)}$. A final remark is that the reviewed DL techniques are designated for flat-fading channels. Multipath scenarios would require convolutional dictionary learning \cite{CDL18}, and would be the topic of a future study. \section{Smooth Dictionary Learning} \label{sec:SDL} In this section, we propose an improved DL-based source separation algorithm that exploits knowledge about the memory of the sources $n=1,\dots,N$. In particular, we introduce a modified DL scheme that accounts for the intermittent and smooth deterministic model discussed in Sec.~\ref{sec:SM}. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \includegraphics[width=\columnwidth]{./Figure3.pdf} \caption{Block diagram of the proposed algorithm with Per-Source Filtering under the hidden Markov model for the sources' activity.} \label{fig:AF} \end{minipage} \end{figure} To account for the assumption of smoothness, the signal estimation step described in Sec.~\ref{sec:SE} is modified by substituting the LASSO algorithm \eqref{eq:LASSO} with a smooth LASSO algorithm \cite{SL11}. Smooth LASSO adds a penalty term for on/off switches within the transmitted signals. This penalty reflects prior knowledge that the transmitted signals do not switch on/off an excessive number of times. Accordingly, at each iteration $k$, given the channel iterate $\mathbf{H}^{(k)}$, instead of using \eqref{eq:LASSO}, the proposed method obtains the updated estimate $\mathbf{X}^{(k)}$ of the channel matrix by solving the problem \begin{equation} \label{eq:SL} \underset{\mathbf{X}}{\text{minimize }}|| \mathbf{Y}-\mathbf{H}^{(k)}\mathbf{X}||^2_2 + \lambda \sum_{t=1}^T ||\mathbf{x}(t)||_1 +\mu \sum_{t=2}^T||\mathbf{x}(t)-\mathbf{x}(t-1)||^2, \end{equation} where $\mu$ is a weight parameter that is set depending the level of smoothness expected in the transmitted signals. Larger values of $\mu$ indicate fewer expected changes in the transmitted signal. Solving \eqref{eq:SL} directly has a complexity in the order of $O(MN^2T^2)$ \cite{CC07}. To reduce the computational complexity, we introduce two approximations of \eqref{eq:SL}, namely Sequential Smooth LASSO (SL-SEQ) and Alternating Direction Method of Multipliers-based Smooth LASSO (SL-ADMM). SL-SEQ solves \eqref{eq:SL} column-wise sequentially, and can be found in \cite{SLSEQ05} as an approximation for \eqref{eq:SL}; while SL-ADMM is an ADMM-based \cite{ADMM11,ADMM14} iterative approach we propose to tackle \eqref{eq:SL} with lower computational complexity. \subsection{Sequential Smooth LASSO (SL-SEQ)} SL-SEQ solves the problem \begin{equation} \label{eq:SSL} \underset{\mathbf{x}(t)}{\text{minimize }}|| \mathbf{y}(t)-\mathbf{H}^{(k)}\mathbf{x}(t)||^2_2 + \lambda ||\mathbf{x}(t)||_1+\mu ||\mathbf{x}(t)-\mathbf{x}(t-1)||^2, \end{equation} sequentially for $t=1,\dots,T$, where $\mathbf{x}(t-1)$ is the solution obtained at the previous step \cite{SLSEQ05}. With this scheme, the quadratic penalty term imposes on the current solution a proximity constraint with respect to the previous one. SL-SEQ has a complexity that in the order of $O(MN^2T)$ as opposed to $O(MN^2T^2)$ since it solves the $T$ problems in \eqref{eq:SSL} sequentially. \subsection{Alternating Direction Method of Multipliers-Based Smooth LASSO (SL-ADMM)} We now propose an approximation to Smooth LASSO with ADMM approach \cite{ADMM11,ADMM14}. The novelty of SL-ADMM is to solve BSS problems with lower computational complexity than smooth LASSO for sources that have memory. To this end, we introduce a copy $\mathbf{x}'(t)$ of $\mathbf{x}(t)$ and rewrite problem \eqref{eq:SL} as \begin{equation} \label{eq:ADMM1} \begin{aligned} \underset{\mathbf{X},\mathbf{X}'}{\text{minimize}} \text{ } &|| \mathbf{Y}-\mathbf{H}^{(k)}\mathbf{X}||_2 + \lambda \sum_{t=1}^T||\mathbf{x}(t)||_1 + \mu \sum_{t=2}^T||\mathbf{x}(t)-\mathbf{x}'(t-1)||^2, \\ \text{subject to }& \mathbf{x}'(t)=\mathbf{x}(t), \qquad t=1,\dots,T-1, \end{aligned} \end{equation} where we have defined the matrix $\mathbf{X}'=[\mathbf{x}'(1),\dots,\mathbf{x}'(T-1)].$ Define the functions $l_1(\mathbf{x}(1))=|| \mathbf{y}(1)-\mathbf{H}^{(k)}\mathbf{x}(1)||_2 + \lambda ||\mathbf{x}(1)||_1$ and $l_t(\mathbf{x}(t),\mathbf{x}'(t-1))=|| \mathbf{y}(t)-\mathbf{H}^{(k)}\mathbf{x}(t)||_2 + \lambda ||\mathbf{x}(t)||_1 + \mu ||\mathbf{x}(t)-\mathbf{x}'(t-1)||^2$, for $t \ge 2$. Then, the augmented Lagrangian \cite{AL69} for problem \eqref{eq:ADMM1} can be written as: \begin{equation} \label{eq:ADMM2} \begin{aligned} &\mathcal{L}(\mathbf{x},\mathbf{x}',\boldsymbol{\alpha}) = l_1(\mathbf{x}(1)) + \sum_{t=2}^T l_t(\mathbf{x}(t),\mathbf{x}'(t-1)) + \sum_{t=1}^{T-1} \boldsymbol{\alpha}^T(t) (\mathbf{x}(t)-\mathbf{x}'(t)) + \rho \sum_{t=2}^T||\mathbf{x}(t) - \mathbf{x}'(t-1)||^2 \\ &= l_1(\mathbf{x}(1))+\boldsymbol{\alpha}^T(1) \mathbf{x}(1) + \sum_{t=2}^{T-1} (l_t(\mathbf{x}(t),\mathbf{x}'(t-1))+\boldsymbol{\alpha}^T(t) \mathbf{x}(t) - \boldsymbol{\alpha}^T(t-1) \mathbf{x}'(t-1)) \\ &+ l(\mathbf{x}(T),\mathbf{x}'(T-1)) - \boldsymbol{\alpha}^T(T-1) \mathbf{x}'(T-1) + \rho \sum_{t=2}^T||\mathbf{x}(t) - \mathbf{x}'(t-1)||^2, \end{aligned} \end{equation} where $\boldsymbol{\alpha}(t)$ is the $N \times 1$ Lagrange multipliers for the constraints in \eqref{eq:ADMM1} and $\rho \ge 0$ is a parameter. The proposed method tackles problem \eqref{eq:ADMM1} via a primal-dual subgradient method that carries out the following steps at each iteration $i$ of ADMM \begin{itemize} \item For the current iterates $\boldsymbol{\alpha}^{(i)}(t)$, solve in parallel the $T$ problems: \begin{subequations} \label{eq:ADMMpara} \begin{itemize} \item[$\cdot$] $\underset{\mathbf{x}(1)}{\text{minimize}} \text{ } l_1(\mathbf{x}(1))+\boldsymbol{\alpha}^{(i)T}(1)\mathbf{x}(1)$, \hfill\refstepcounter{equation}\textup{(\theequation)} \item[$\cdot$] $\underset{\mathbf{x}(t),\mathbf{x}'(t-1)}{\text{minimize}} \text{ } l_t(\mathbf{x}(t),\mathbf{x}'(t-1))+\boldsymbol{\alpha}^{(i)T}(t)\mathbf{x}(t)-\boldsymbol{\alpha}^{(i)T}(t-1)\mathbf{x}'(t-1) +\rho ||\mathbf{x}(t) - \mathbf{x}'(t-1)||^2, \\ \text{ for } t=2,\dots,T-1$, \hfill\refstepcounter{equation}\textup{(\theequation)} \item[$\cdot$] $\underset{\mathbf{x}(T),\mathbf{x}'(T-1)}{\text{minimize}} \text{ } l_T(\mathbf{x}(T),\mathbf{x}'(T-1))-\boldsymbol{\alpha}^{(i)T}(T-1)\mathbf{x}'(T-1)+\rho ||\mathbf{x}(T) - \mathbf{x}'(T-1)||^2$, \hfill\refstepcounter{equation}\textup{(\theequation)} \item[] obtaining the new iterates $\mathbf{x}^{(i)}(t)$ for $t=1,\dots,T$ and $\mathbf{x}'^{(i)}(t)$ for $t=2,\dots,T$; \end{itemize} \end{subequations} \item Update the Lagrange multipliers as \begin{equation} \boldsymbol{\alpha}^{(i+1)}(t) \leftarrow \boldsymbol{\alpha}^{(i)}(t)+\rho (\mathbf{x}^{(i)}(t) - \mathbf{x}'^{(i)}(t-1)), \end{equation} \end{itemize} where $(\cdot)^T$ represents the transpose of a vector or a matrix. We emphasize that SL-SEQ has a computational complexity in the order of $O(MN^2T)$, and SL-ADMM has a complexity in the order of $O(MN^2TI)$, where $I$ is the number of iterations in the ADMM approach. In general, $I$ is chosen to be a small value compared to $T$. It is seen that the computational complexity is greatly reduced with SL-SEQ and SL-ADMM compared to solving \eqref{eq:SL} directly. \section{Per-Source Stochastic Filtering} \label{sec:PSF} In this section, we propose a further improvement to the DL-based source separation schemes discussed thus far which as shown in Fig.~\ref{fig:AF}, consists of a per-source post-processing step. Specifically, once we have obtained an estimate $\tilde{x}_n(t),t=1,\dots,T$, of the signal of any source $n$ based on one of the DL-based methods discussed above, a perform per-source stochastic filtering (PSF) is applied that leverages the HMM of the sources to remove some of the "noise" in the estimates $\tilde{x}_n(t)$. The "noise" flips the state of the signal from active to inactive and vice-versa. Filtering of the noise leads to a more accurate activity map of the signals. PSF is based on the hidden Markov model of Fig.~\ref{fig:HMM}. We first consider the case in which the transition probabilities $(p_n, q_n)$ of the hidden Markov model are known, and then study the scenario in which this information is not available. \subsection{Known Model Parameters} Adopting a Bayesian formulation, a first approach would be to interpret the signal $\tilde{\mathbf{x}}_n(t)$ obtained from each source $n$ via DL-based source separation as the output of a memoryless channel $p(\tilde{\mathbf{x}}_n|\mathbf{x}_n)$ whose input is the current transmitted signal $\mathbf{x}_n$. Composing this channel with the hidden Markov model of Fig.~\ref{fig:HMM}, one could apply the forward-backward message passing algorithm on the resulting Bayesian network in order to compute the posterior probabilities $p(x_n(t)|\tilde{x}_n(1),\dots,\tilde{x}_n(T))$ for all $t=1,\dots,T$ \cite{BN}. Here we do not follow this approach since it is generally unclear how to define the channel $p(\tilde{\mathbf{x}}_n|\mathbf{x}_n)$, and the resulting message passing algorithm typically requires computationally expensive integrations. In contrast, we take a more pragmatic low-complexity approach, which is described next. First, we perform a binary quantization of the available estimates $\tilde{x}_n(t),t=1,\dots,T$, by thresholding the absolute value of $\tilde{x}_n(t)$. This provides an estimate $\tilde{s}_n(t)$ of the state $s_n(t)$, since small absolute values of $\tilde{\mathbf{x}}_n(t)$ suggest the absence of signal for source $n$ at time $t$. As a result, we obtain the binary sequence $\tilde{s}_n(t),t=1,\dots,T$, where $\tilde{s}_n(t)=1$ if $|\tilde{x}_n(t)|>\gamma$ for a threshold $\gamma$, and $\tilde{s}_n(t)=0$ otherwise. The selection of the threshold $\gamma$ will allow us to obtain different points on the trade-off between the probability of false alarm and missed detection, as discussed in Sec.~\ref{sec:NR}. To formulate the filtering problem, we model the observations $\tilde{s}_n(t),t=1,\dots,T$, as being received at the output of a binary asymmetric channel (BAC) with flip probabilities $p_n'$ and $q_n'$, as illustrated in Fig.~\ref{fig:BAC}. The BAC models the errors made by the DL-based source separation algorithm that produced $\tilde{x}_n(t),t=1,\dots,T$, in detecting the state variables $s_n(t)$. Parameters $p_n'$ and $q_n'$ can be used as additional degrees of freedom in exploring the trade-off points between the probability of false alarm and missed detection. As an alternative, we will discuss in the next subsection how to estimate them from the data. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=0.6\columnwidth]{./Figure4.pdf} \caption{Binary Asymmetric Channel (BAC) describing the relationship between the true state $s_n(t)$ and the observed state $\tilde{s}_n(t)$ obtained from the DL-based source separation step that is assumed by PSF.} \label{fig:BAC} \end{minipage} \end{figure} Given the observations $\tilde{s}_n(t),t=1,\dots,T$, and the parameters $p_n$, $q_n$, $p_n'$ and $q_n'$, we compute the posterior distribution $p(s_n(t)|\tilde{s}_n(1),\dots,\tilde{s}_n(T))$ by using the forward-backward algorithm \cite{ML}. Accordingly, the posterior probability can be written as \begin{equation} \label{eq:PP} \text{Pr}(s_n(t)=1|\tilde{s}_n(1),\dots,\tilde{s}_n(T))=\alpha_t \beta_t, \end{equation} where the probability $\alpha_t = \text{Pr}(s_n(t)=1|\tilde{s}_n(1),\dots,\tilde{s}_n(t))$ is obtained as a result of the forward pass, while the probability $\beta_t= \text{Pr}(\tilde{s}_n(t+1),\dots,\tilde{s}_n(T)|s_n(t)=1)$ is obtained from the backward pass as explained in \cite[\S 17.4.3]{ML}. We then estimate $s_n(t)$ using the maximum a posteriori (MAP) approach, i.e., we set $\tilde{s}_n(t)=1$ if the inequality $p(s_n(t)|\tilde{s}_n(1),\dots,\tilde{s}_n(T))>0.5$ holds, and $\tilde{s}_n(t)=0$ otherwise. To recover an estimate $\hat{x}_n(t)$ of the transmitted signal $x_n(t),t=1,\dots,T$, we finally null all entries of $\tilde{x}_n(t)$ corresponding to states $s_n(t)$ that are estimated to be zero, i.e., $\hat{s}_n(t)=0$, while leaving unaltered the values of $\tilde{x}_n(t)$ for times $t$ at which $s_n(t)$ is estimated to be $\hat{s}_n(t)=1$. This can be summarized as \begin{equation} \hat{x}_n(t) = \begin{cases} \tilde{x}_n(t) & \quad \text{if } \hat{s}_n(t)=1,\\ 0 & \quad \text{if } \hat{s}_n(t)=0.\\ \end{cases} \end{equation} \subsection{Unknown Model Parameters} \label{sec:UMP} We now study the case in which the parameters $p_n$ and $q_n$ of the HMM are unknown, and need to be estimated. We also jointly estimate the parameters $p_n'$ and $q_n'$ in the BAC of Fig.~\ref{fig:BAC} that is used to derive the forward-backward filtering algorithm. To this end, we apply the Expectation-Maximization (EM) algorithm \cite{ML}, as described next. Given the estimated states $\tilde{s}_n(t)$ of each source $n$ obtained from the source separation and quantization steps, we would like to estimate the state sequence $s_n(t), \text{ } t=1,\dots,T$, the probabilities $p_n$ and $q_n$ in the transition matrix of the Markov chain, as well as the probabilities $p_n'$ and $q_n'$ in the BAC. The EM algorithm can be detailed as follows. \begin{itemize} \item \textbf{Initialization:} Initialize $p^{(0)}_n$, $q^{(0)}_n$, $p'^{(0)}_n$ and $q'^{(0)}_n$. For each iteration $\nu=1,2,\dots$: \item \textbf{E Step}: Given the probabilities $p_n=p^{(\nu-1)}_n$, $q_n=q^{(\nu-1)}_n$, $p'_n=p'^{(\nu-1)}_n$ and $q'_n=q'^{(\nu-1)}_n$, apply the forward-backward algorithm to calculate the probabilities $\alpha_t$ and $\beta_t$ and hence the posterior probabilities \eqref{eq:PP}, for $n=1,\dots,N$. \item \textbf{M Step}: Update the probability parameters by averaging the sufficient statistics with respect to the posterior distributions identified during the E step \cite{Bishop}. This leads to the updates \begin{equation} \label{eq:pn} p^{(\nu +1)}_n \leftarrow \frac{\sum_{t=2}^T \text{Pr}(s_n(t)=1,s_n(t-1)=0|\tilde{\mathbf{s}}_n)}{\sum_{t=2}^T \text{Pr}(s_n(t-1)=0|\tilde{\mathbf{s}}_n)}, \end{equation} \begin{equation} \label{eq:qn} q^{(\nu +1)}_n \leftarrow \frac{\sum_{t=2}^T \text{Pr}(s_n(t)=0,s_n(t-1)=1|\tilde{\mathbf{s}}_n)}{\sum_{t=2}^T \text{Pr}(s_n(t-1)=1|\tilde{\mathbf{s}}_n)}, \end{equation} where $\tilde{\mathbf{s}}_n=[\tilde{s}_n(1),\dots,\tilde{s}_n(T))]$, $\text{Pr}(s_n(t-1)=0|\tilde{\mathbf{s}}_n)=1-\text{Pr}(s_n(t-1)=1|\tilde{\mathbf{s}}_n)$ is obtained from \eqref{eq:PP}, and the posterior joint probabilities in \eqref{eq:pn} and \eqref{eq:qn} are computed as detailed below; and \begin{equation} \label{eq:ppn} p'^{(\nu +1)}_n \leftarrow \frac{\sum_{t=1}^T \mathbbm{1}({\tilde{s}_n(t)=1}) \text{Pr}(s_n(t)=0|\tilde{\mathbf{s}}_n)}{\sum_{t=1}^T \text{Pr}(s_n(t)=0|\tilde{\mathbf{s}}_n)}, \end{equation} \begin{equation} \label{eq:qpn} q'^{(\nu +1)}_n \leftarrow \frac{\sum_{t=1}^T \mathbbm{1}({\tilde{s}_n(t)=0}) \text{Pr}(s_n(t)=1|\tilde{\mathbf{s}}_n)}{\sum_{t=1}^T \text{Pr}(s_n(t)=1|\tilde{\mathbf{s}}_n)}, \end{equation} where $\mathbbm{1}(a)$ is the indicator function, i.e. $\mathbbm{1}(a)=1$ if $a$ is true and 0 otherwise. \item \textbf{Stopping Criterion:} The iteration stops when the parameters converge, i.e. $|p^{(\nu)}_n-p^{(\nu-1)}_n|<\epsilon$, where $\epsilon$ is a small value. \end{itemize} The posterior joint probabilities in \eqref{eq:pn} and \eqref{eq:qn} are computed as \cite{EMHMM01} \begin{equation} \label{eq:M} \text{Pr}(s_n(t)=i,s_n(t-1)=j|\tilde{s}_n(1),\dots,\tilde{s}_n(T))=\alpha_{t-1,j+1} \Phi_{j+1,i+1}^{(\nu)} \text{Pr}(\tilde{s}_n(t)|s_n(t)=i) \beta_{t,i+1}, \qquad i,j=0,1, \end{equation} where $\alpha_{t,j}$ is the $j$-th element of vector $\boldsymbol{\alpha}_t=[1-\alpha_t,\alpha_t]^T$; $\beta_{t,i}$ is the $i$-th element of vector $\boldsymbol{\beta}_t=[1-\beta_t,\beta_t]^T$; $\Phi_{j,i}^{(\nu)}$ is the $(j,i)$-th element of the transition matrix $\boldsymbol{\Phi}^{(\nu)}_n$ in the $\nu$-th iteration; and matrix $\boldsymbol{\Phi}^{(\nu)}_n$ is defined as \begin{equation} \boldsymbol{\Phi}^{(\nu)}_n= \begin{bmatrix} 1-p^{(\nu)}_n & p^{(\nu)}_n \\ q^{(\nu)}_n & 1-q^{(\nu)}_n \end{bmatrix}. \end{equation} \section{Numerical Results} \label{sec:NR} In this section, we present numerical results to obtain insights into the performance of different DL-based source separation schemes and on the advantage of PSF. We consider the following source separation schemes, all implemented with and without the post-processing PSF step: OMP, LASSO, SL-SEQ and SL-ADMM. As performance criteria, we adopt the probability of false alarm $P_{fa}$ and probability of detection $P_d$. The detection probability $P_d$ is the ratio between the number of correctly detected active sources and the total number of active sources over the $T$ symbols, and the false alarm probability $P_{fa}$ is the ratio between the number of incorrectly detected active sources and the total number of inactive sources over the $T$ time samples. We also consider the performance of source estimation in terms of error vector magnitude (EVM) to synchronized digital transmission (see Sec.~\ref{sec:SM}). Unless stated otherwise, the numerical results were obtained for $N=30$ sources, $M=20$ sensors, signal to noise ratio (SNR) per source per sample 30 dB, $T=1000$ time samples. We assume an HMM for the state $s_n(t)$, which is defined by transition probabilities $p_n=0.0022$, $q_n=0.02$, for all $N$ sources, so that an average $Np_n/(p_n+q_n)=3$ sources are active at each time sample $t$, and the average duration of transmission is $1/q_n=50$ time samples. Also, unless stated otherwise, the algorithms with PSF are implemented with fixed values $p'_n=0.02$ and $q'_n=0.27$. We optimize numerically over the multipliers $\lambda$ and $\mu$ in order to satisfy given constants on the probability of detection $P_d$ or the probability of false alarm $P_{fa}$. For PSF with unknown model parameters, we initialize the transition probabilities of the Markov model as $p_n=0.5$, $q_n=0.5$, and the flip probabilities of the BAC as $p'_n=0.1$, $q'_n=0.2$. Throughout this section, the number of iterations for SL-ADMM was fixed at $K=30$; the Lagrange multipliers $\boldsymbol{\alpha}$ in \eqref{eq:ADMMpara} are initialized as the all-one vector; $\rho$ is set as 0.1 and $\gamma$ is fixed at 0.5. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure5.pdf} \caption{Probability of detection $P_d$ versus probability of false alarm $P_{fa}$ for the considered algorithms ($N=30$, $M=20$, $T=1000$, SNR$=30$ dB, $p_n=0.0022$, $q_n=0.02$).} \label{fig:Comp} \end{minipage} \end{figure} \subsection{Tuning Multipliers $\lambda$ and $\mu$} We start by discussing the choice of the multipliers $\lambda$ and $\mu$ in optimization problems \eqref{eq:SSL} and \eqref{eq:ADMM1} for SL-SEQ and SL-ADMM. These parameters penalize the number of non-zero elements in the solution vector and the number of state changes of each source, respectively. The choice of these two multipliers is scenario-dependent, and a discussion on their optimal selection can be found in \cite{GENR13}. In our extensive simulations, we found that the rule-of-thumb choices $\lambda = 1/$SNR and $\mu =0.1/q_n$, where $q_n$ is the transition probability from an active to an inactive state (see Fig.~\ref{fig:HMM}), works well in practice. The selection $\lambda = 1/$SNR reflects a decrease in the relevance of the sparsity prior as the quality of the observations increases \cite{GENR13}. In contrast, the multiplier $\mu \propto 1/q_n$ increases as the average transmission duration $1/q_n$ increases, implying fewer state changes. As an example, Fig.~\ref{fig:lambda} and Fig.~\ref{fig:mu} show the probability of detection $P_d$ for a fixed probability of false alarm $P_{fa}=0.1$ versus $\lambda$ and $\mu$, respectively. From Fig.~\ref{fig:lambda}, we observe that, when varying the SNR level, the optimal selection of $\lambda$ changes according to the mentioned inverse proportionality rule $1/$SNR. In a similar manner, Fig.~\ref{fig:mu} confirms the validity of our choice for multiplier $\mu$. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure6.pdf} \caption{Probability of detection $P_d$ when the probability of false alarm is $P_{fa}=0.1$ versus $\lambda$ for the SL-ADMM algorithm ($N=30$, $M=20$, $T=1000$, $p_n=0.0022$, $q_n=0.02$).} \label{fig:lambda} \end{minipage} \end{figure} \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure7.pdf} \caption{Probability of detection $P_d$ when the probability of false alarm is $P_{fa}=0.1$ versus $\mu$ for the SL-ADMM algorithm ($N=30$, $M=20$, $T=1000$, SNR$=30$ dB.} \label{fig:mu} \end{minipage} \end{figure} \subsection{Source Activity Detection} We then investigate the trade-off between the probability of detection $P_d$ versus the probability of false alarm $P_{fa}$ in Fig.~\ref{fig:Comp}. The transmitted signal $x_n(t)$ is assumed here to be distributed as $x_n(t) \sim \mathcal{N}(0,1)$ whenever $s_n(t)=1$. We also assume known source statistics $p_n$ and $q_n$. A first observation is that both SL-SEQ and SL-ADMM significantly outperform all other source separation schemes, with the latter obtaining some performance gain over the former. Furthermore, PSF provides a significant performance boost for all algorithms. For example, at $P_{fa}=0.07$, the probability of detection with SL-ADMM is increased from 0.87 to 0.98. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure8.pdf} \caption{Probability of detection $P_d$ when the probability of false alarm is $P_{fa}=0.1$ versus SNR for the considered algorithms ($N=30$, $M=20$, $T=1000$, $p_n=0.0022$, $q_n=0.02$).} \label{fig:SNR} \end{minipage} \end{figure} We now explore the impact of different operating regimes on the probability of detection when the probability of false alarm is constrained to be smaller than 0.1. In particular, Fig.~\ref{fig:SNR} shows performance with respect to different SNR values, and Fig.~\ref{fig:SP} investigates the impact of the sparsity level, which is defined as the ratio between the average number $Np_n/(p_n+q_n)$ of active sources and the total number $N$ of sources. The former is modified by changing $p_n$. Both figures confirm the main conclusions obtained above in terms of the relative performance of the considered schemes. Moreover, Fig.~\ref{fig:SNR} suggests that the performance undergoes a threshold phenomenon with respect to the SNR, particularly if implemented without PSF. It is also seen that SL with PSF is able to obtain a vanishing probability of missed detection as the SNR increases, unlike the other schemes whose probability of detection reaches a ceiling lower than one. Finally, Fig.~\ref{fig:SP} indicates that SL with PSF is robust to the sparsity level, while the other schemes are extremely sensitive to an increase in the average number of active sources. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure9.pdf} \caption{Probability of detection $P_d$ when the probability of false alarm is $P_{fa}=0.1$ versus average number of active sources ($N=30$, $M=20$, $T=1000$, SNR$=30$ dB, $q_n=0.02$).} \label{fig:SP} \end{minipage} \end{figure} \subsection{Signal Estimation} \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure10.pdf} \caption{EVM versus SNR ($N=30$, $M=20$, $T=1000$, $p_n=0.0022$, $q_n=0.02$).} \label{fig:EVM} \end{minipage} \end{figure} As seen above, the SL schemes with PSF have the best performance in terms of source activity detection. Here, we study the performance in terms of the quality of signal estimation. To this end, we adopt the criterion of the EVM. EVM is defined as \begin{equation} \text{EVM}(\%)=\sqrt{\frac{\sum_{n=1}^{N} \sum_{t \in \mathcal{D}_n} (x_n(t)-\tilde{x}_n(t))^2}{\sum_{n=1}^{N} \sum_{t \in \mathcal{D}_n} x_n^2(t)}} \times 100\%, \end{equation} where $\mathcal{D}_n$ is the set of time samples in which user $n$ is correctly detected as active. Here we assume that, when $s_n(t)=1$, the transmitted signal is binary, that is $x_n(t)=$1 or $-1$ with equal probability. Assuming again known sources' transition probabilities, Fig.~\ref{fig:EVM} shows the EVM for all the considered algorithms with respect to SNR where the probability of missed detection $P_{md}$ is constrained to be smaller than 0.2. The behavior of the EVM is in line with the discussion above regarding source activity detection. In particular, we observe a 40\% decrease at 30 dB SNR that are achievable with SL and PSF, as well as the threshold behavior as a function of the SNR. \subsection{Parameter Estimation} Here, we evaluate the performance loss incurred when the parameters $p_n$ and $q_n$ in the Markov model defining the sources' activities, are not known. Using the framework of Sec.~\ref{sec:UMP}, We jointly estimate the parameters $p_n$, $q_n$, $p'_n$ and $q'_n$ with the EM algorithm. Fig.~\ref{fig:HP} investigates the performance of the estimation of the hidden parameters $p'_n$ and $q'_n$ in the BAC. From the figure, the estimate is seen to be close to the real value. The parameters $p_n$ and $q_n$ in the HMM have similar estimation accuracy as $p'_n$ and $q'_n$. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure11.pdf} \caption{Estimation performance of $p'_n$ and $q'_n$ for BAC in the EM algorithm ($N=30$, $M=20$, $T=1000$, SNR$=30$ dB, $p_n=0.0022$, $q_n=0.02$).} \label{fig:HP} \end{minipage} \end{figure} Fig.~\ref{fig:EM} shows the probability of detection $P_d$ versus the probability of false alarm $P_{fa}$ under the same conditions as Fig.~\ref{fig:Comp}. A first observation is that, even with unknown source parameters, PSF can provide a significant performance boost. Nevertheless, the gain is somewhat reduced as compared to the case with perfect knowledge. For instance, for SL-ADMM, when $P_{fa}=0.07$, the probability of detection $P_d$ is reduced from 0.98 to 0.89. The loss is generally increases when $T$ is smaller. \begin{figure} \centering \begin{minipage}[c]{0.7\columnwidth} \centering \includegraphics[width=\columnwidth]{./Figure12.pdf} \caption{Probability of detection $P_d$ versus probability of false alarm $P_{fa}$ for the considered algorithms with EM algorithm ($N=30$, $M=20$, $T=1000$, SNR$=30$ dB, $p_n=0.0022$, $q_n=0.02$).} \label{fig:EM} \end{minipage} \end{figure} \section{Conclusions} In this paper, we have introduced a two-stage Dictionary Learning (DL)-based algorithm for solving the BSS problem as the presence of radio sources with memory that are observed over slow flat-fading wireless channels. The DL stage of the proposed algorithm exploits source memory information to aid with the source separation. The effect of source memory is accounted for by a penalty term that discourages short-duration transmissions. The PSF stage utilizes source model information to learn about unknown source model parameters to further enhance source estimation. Numerical results show that the proposed algorithm outperforms existing DL algorithms in terms of source activity detection as well as source signal estimation. As a representative numerical example, even with unknown source statistics, it is shown that with 30 potential sources, 20 sensors and average 3 sources active in each time sample, it is possible to increase the probability of detection from 0.9 without PSF to 0.96 with PSF at 0.1 probability of false alarm and 30 dB SNR. The proposed algorithm can also decrease the EVM from by 40\% at 0.8 probability of detection at 30 dB SNR. In addition, the proposed algorithm is robust to the source sparsity level, while existing DL algorithms are extremely sensitive to an increase in the number of active sources. \bibliographystyle{IEEEtran}

BkiUbTI241xg-GrQoAAD

\section{Introduction} The mapping class group $\Mod(M)$ of a smooth, closed oriented manifold $M$, is the group of isotopy classes of orientation-preserving diffeomorphisms of $M$. Denoting $\Diff^+(M)$ the group of orientation-preserving diffeomorphisms, there is a natural projection map \[\pi:\Diff^+(M)\to\Mod(M)\] sending a diffeomorphism to its isotopy class. We say that a subgroup $i: G\hookrightarrow \Mod(M)$ is \emph{realizable} if there is a homomorphism of $\rho: G\to \Diff^+(M)$ such that $\pi \circ \rho = i$. \[\begin{xy} (-30,0)*+{G}="A"; (0,15)*+{\Diff^+(M)}="B"; (0,0)*+{\Mod(M)}="C"; {\ar@{-->}"A";"B"}?*!/_3mm/{\rho}; {\ar "B";"C"}?*!/_3mm/{\pi}; {\ar "A";"C"}?*!/^3mm/{i}; \end{xy}\] The Nielsen realization problem asks which finite groups $G<\Mod(M)$ are realizable. When $M$ is a surface, every finite subgroup of $\Mod(M)$ is realizable by work of Kerkchoff \cite{kerckhoff}. For other manifolds, only sporadic results have been obtained; see \cite[\S2]{pardon} for a summary of results in dimension 3, \cite{farb-looijenga,Konno2, lee, Konno} for results in dimension 4, and \cite{farrell-jones,block-weinberger,bustamante-tshishiku} for higher dimensions. When $M$ is a 3-manifold, the \emph{twist group} $\Twist(M)<\Mod(M)$ is the subgroup generated \emph{sphere twists} (defined in \S\ref{sec:sphere-twists}). McCullough \cite{mccullough} proved that $\Twist(M)\cong(\Z/2\Z)^d$ for some $d\ge0$. We address the Nielsen realization problem for subgroups of $\Twist(M)$. This problem was studied by Zimmermann \cite{zimmermann} for $M=\#_k(S^1\times S^2)$, but there is an error in his argument; see \cite{zimmermann-erratum}. Nevertheless, using some of the same ideas, we correct the argument, and we generalize from $\#_k(S^1\times S^2)$ to all $3$-manifolds, giving a precise condition for which subgroups can be realized or not. To state the main result, recall that for any pair of coprime integers $p,q$, there is a lens space $L(p,q)$. Every lens space is covered by $S^3$ with the exception of $L(0,1)\cong S^1\times S^2$. \begin{mainthm} Fix a closed, oriented $3$-manifold $M$, and fix a nontrivial subgroup $1\neq G<\Twist(M)$. Then $G$ is realizable if and only if $G$ is cyclic and $M$ is diffeomorphic to a connected sum of lens spaces. \end{mainthm} \p{Application to the Burnside problem for diffeomorphism groups} Recall that a \emph{torsion group} is a group where every element has finite order. The existence of finitely-generated, infinite torsion groups, known as \emph{Burnside groups}, was proved by Golod--Shafarevich \cite{Golod} \cite{GS} and Adian-Novikov \cite{AN}. E. Ghys and B. Farb asked if the homeomorphism group of a compact manifold can contain a Burnside group; see \cite[Question 13.2]{Fisher2} and \cite[\S5]{Fisher1}. As an application of the tools used to prove the Main Theorem, we prove the following result on Burnside problem for $\Diff(M)$. \begin{theorem}\label{Burnside} Let $M$ be a compact oriented $3$-manifold. Assume that $M$ is reducible and not a connected sum of lens spaces. Then $\Diff(M)$ does not contain a Burnside group. \end{theorem} \p{Smooth vs.\ topological Nielsen realization} The topological mapping class group $\Mod_H(M)$ is defined as the group of isotopy classes of orientation-preserving homeomorphisms of $M$. There is a natural projection map \[\Homeo^+(M)\to \Mod_H(M),\] and the Nielsen realization problem can also be asked for subgroups of $\Mod_H(M)$. For 3-manifolds, the smooth and topological mapping class groups coincide $\Mod(M)\cong\Mod_H(M)$ by Cerf \cite{cerf}, who proved that $\Diff^+(M)$ and $\Homeo^+(M)$ are homotopy equivalent. Surprisingly, the realization problem for finite groups is also the same in the topological and smooth categories in dimension 3. For dimension $4$, even the connected components are different by work of Ruberman \cite{Ruberman}. The work of Baraglia--Konno \cite{Konno2} gives a mapping class that can be realized as homeomorphism but not as diffeomorphism and Konno \cite{Konno} generalizes the results to more manifolds. \begin{theorem}[Pardon, Kirby--Edwards]\label{thm:pardon} Let $M$ be a closed oriented 3-manifold. A finite subgroup $G<\Mod(M)$ is realizable by homeomorphisms if and only if it is realizable by diffeomorphisms. \end{theorem} In particular, this allows us to strengthen the conclusion of the Main Theorem. We emphasize that the group $G$ in Theorem \ref{thm:pardon} is finite; however, we do not know an example of a 3-manifold $M$ and an infinite group $G<\Mod(M)$ that is realizable by homeomorphisms but not diffeomorphisms. \begin{proof}[Proof of Theorem \ref{thm:pardon}] Let $\rho:G\to\Homeo^+(M)$ be a realization of $G<\Mod(M)$. By Pardon \cite{pardon}, $\rho$ can be approximated uniformly by a smooth action $\rho': G\to\Diff^+(M)$. Since $\Homeo^+(M)$ is locally path-connected by Kirby--Edwards \cite{kirby-edwards}, we know that $\rho'(g)$ and $\rho(g)$ are isotopic in $\Homeo^+(M)$ for each $g\in G$, and hence also isotopic in $\Diff^+(M)$ since $\Mod_H(M)=\Mod(M)$. \end{proof} \p{Related work} The following remarks connect the Main Theorem to other previous work. \begin{remark}[Twist group for $S^1\times S^2$] The group $\Twist(S^1\times S^2)$ is isomorphic to $\Z/2\Z$. We construct a realization of this group in \S\ref{sec:construction}. This example seems to be overlooked in some of the literature on finite group actions on geometric 3-manifolds. It is a folklore conjecture of Thurston that any finite group action on a geometric 3-manifold is geometric (i.e.\ acts isometrically on some geometric structure). It is easy to see that our realization of the twist group, which also appears in work of Tollefson \cite{tollefson_involutions}, does not preserve any geometric structure on $S^1\times S^2$, so it is a simple counterexample to Thurston's conjecture. According to Meeks--Scott \cite{meeks-scott}, Thurston proved some cases of his conjecture, but these results were not published. Meeks--Scott \cite{meeks-scott} proved Thurston's conjecture for manifolds modeled on $\mathbb H^2\times\R$, $\widetilde{\SL_2(\R)}$, $\Nil$, $\mathbb E^3$, and $\Sol$. In \cite[Thm.\ 8.4]{meeks-scott} it is asserted (incorrectly) that Thurston's conjecture also holds for 3-manifolds modeled on $S^2\times\R$ (in particular $S^2\times S^1$); they give an argument, but in the case when some $g\in G$ has positive-dimensional fixed set (as is the case for our realization of $\Twist(S^1\times S^2)$), they cite a preprint of Thurston that seems to have never appeared. \end{remark} \begin{remark}[Sphere twists in dimension 4] For a 4-manifold $W$, for each embedded 2-sphere $S\subset W$ with self-intersection $S\cdot S=-2$, there is a sphere twists $\tau_S\in\Mod(W)$, which has order 2. There are several results known about realizing the subgroup generated by a sphere twist, both positive and negative; see Farb--Looijenga \cite[Cor.\ 1.10]{farb-looijenga}, Konno \cite[Thm.\ 1.1]{Konno}, and Lee \cite[Rmk.\ 1.7]{lee}. It would be interesting to determine precisely when a sphere twist is realizable in dimension 4. \end{remark} \p{About the proof of the Main Theorem} The proof is divided into two parts: construction and obstruction (corresponding to the ``if" and ``only if" directions in the theorem statement). For the obstruction part of the argument, we prove the following constraint on group actions on reducible 3-manifolds. \begin{theorem}\label{thm:contraint} Let $M$ be a closed, oriented, reducible $3$-manifold. Let $G<\Diff^+(M)$ be a finite subgroup that acts trivially on $\pi_1(M)$. Then $G$ is cyclic. If $G$ is nontrivial, then $M$ is a connected sum of lens spaces. \end{theorem} \p{Section outline} In \S\ref{sec:sphere-twists}, we recall results about sphere twists and the twist group. In \S\ref{sec:invariant-spheres} we explain results from minimal surface theory that allow us to decompose a given action into actions on irreducible 3-manifolds. In \S\ref{sec:obstruction} and \ref{sec:construction} we prove the ``obstruction" and ``construction" parts of the Main Theorem, respectively. In \S\ref{Burn}, we prove Theorem \ref{Burnside}. \p{Acknowledgement} The authors are supported by NSF grants DMS-2203178, DMS-2104346 and DMS-2005409. This work is also supported by NSF Grant No. DMS-1439786 while the first author visited ICERM in Spring 2022. \section{Sphere twists and the twist subgroup} \label{sec:sphere-twists} In this section we collect some facts about sphere twists and the group they generate. In \S\ref{sec:individual-twist} we recall the definition of sphere twists and recall that they act trivially on $\pi_1(M)$ and $\pi_2(M)$. In \S\ref{sec:twist-group} we give a computation for the twist group of any closed, oriented 3-manifold (Theorem \ref{thm:twist-group}). The computation can be deduced by combining different results from the literature and gives a precise generating set. \subsection{Sphere twists and their action on homotopy groups} \label{sec:individual-twist} Fix a closed oriented $3$-manifold $M$. We recall the definition of a sphere twist. Fix an embedded $2$-sphere $S\subset M$ with a tubular neighborhood $U\cong S\times [0,1]\subset M$, and fix a closed path $\phi:[0,1]\to\SO(3)$ based at the identity that generates $\pi_1(\SO(3))$. Define a diffeomorphism of $U$ by \begin{equation}\label{eqn:twist} T_S(x,t)=\big(\phi(t)(x),t\big) \end{equation} and extend by the identity to obtain a diffeomorphism $T_S$ of $M$. The isotopy class $\tau_S\in\Mod(M)$ of $T_S$ is called a \emph{sphere twist}. The \emph{twist subgroup} of $\Mod(M)$, denoted $\Twist(M)$, is the subgroup generated by all sphere twists. \begin{lemma}[Action of sphere twists on homotopy groups]\label{lem:twist-action} Let $M$ be a closed, oriented $3$-manifold with a $2$-sided embedded sphere $S\subset M$. Then $\tau_S$ acts trivially on $\pi_1(M)$ and $\pi_2(M)$. \end{lemma} \begin{remark}[Action on $\pi_1(M)$ vs.\ $\pi_1(M,*)$] When we refer to the action of a diffeomorphism $f\in\Diff(M)$ on $\pi_1(M)$, we mean as an outer automorphism. Technically, this is not an action, but there is a well-defined homomorphism $\Diff(M)\to\Out(\pi_1(M))$. When $f$ has a fixed point $*\in M$, the action of $f$ on $\pi_1(M,*)$ refers to the induced automorphisms $f_*:\pi_1(M,*)\to\pi_1(M,*)$. This distinction will be important in later sections. If $f$ acts trivially on $\pi_1(M)$ it can be isotoped to $f'\in\Diff(M,*)$ that acts on $\pi_1(M,*)$ by conjugation (generally nontrivial). \end{remark} Lemma \ref{lem:twist-action} is well-known. That sphere twists act trivially on $\pi_1(M)$ is implicit in \cite{mccullough}. This fact may be proved as follows. Let $*\in M$ be a fixed point of the diffeomorphism $T_S$ defined in (\ref{eqn:twist}). After choosing a prime decomposition of $M$, one can show that each element of $\pi_1(M,*)$ is represented by a loop contained entirely in the fixed set of $T_S$. That sphere twists act trivially on $\pi_2(M)$ follows from \cite[Lem.\ 1.1]{mccullough}. It can also be deduced from a general result of Laudenbach that says that if $f\in\Diff(M,*)$ acts trivially on $\pi_1(M,*)$, then it also acts trivially on $\pi_2(M,*)$; see \cite[Thm.\ 2.4]{BBP}. \subsection{Generators and relations in the twist group} \label{sec:twist-group} In this section we compute $\Twist(M)$ for every closed, oriented 3-manifold. \begin{theorem}\label{thm:twist-group} Let $M$ be a closed, orientable 3-manifold with prime decomposition $M=\#_k(S^1\times S^2)\#P_1\#\cdots\#P_\ell$, where the $P_i$ are irreducible. Let $\ell'\le\ell$ be the number of the $P_i$ that are lens spaces. Then \[\Twist(M)\cong\begin{cases} (\Z/2\Z)^k&\text{ if } \ell=\ell'\\ (\Z/2\Z)^{k+\ell-\ell'-1}&\text{ if } \ell>\ell'. \end{cases}\] \end{theorem} We were not able to find Theorem \ref{thm:twist-group} in the literature, although it can be deduced by combining various old results. For the proof of Theorem \ref{thm:twist-group}, we will use the following explicit construction of $M$. Let $X$ be the complement of $\ell+2k$ open disks in $S^3$. For each $P_i$ choose a closed embedded disk $D_i\subset P_i$, and let $Y$ be the compact manifold \begin{equation}\label{eqn:prime-factors}Y:=\left[\coprod_{i=1}^\ell P_i\setminus \text{int}(D_i)\right]\sqcup\left[\coprod_{k}S^2\times[-1,1]\right].\end{equation} We form $M$ by gluing $X$ and $Y$ along their boundary $\partial X\cong\partial Y$. See Figure \ref{fig:sphere-twist}. \begin{figure}[h!] \labellist \pinlabel $X$ at 100 80 \pinlabel $P_1$ at 10 40 \pinlabel $P_2$ at 115 10 \pinlabel $P_3$ at 190 50 \pinlabel $[-1,1]\times S^2$ at -20 110 \pinlabel $[-1,1]\times S^2$ at 215 110 \endlabellist \centering \includegraphics[scale=.6]{sphere-twist} \caption{Construction of $M=\#_k(S^1\times S^2)\# P_1\#\cdots\# P_\ell$. The red set represents spheres that generate $\Twist(M)$.} \label{fig:sphere-twist} \end{figure} \begin{lemma}[Pants relation]\label{lem:pants} Let $Z$ be the complement in $S^3$ of three disjoint $3$-balls. Let $S_1,S_2,S_3$ be the three boundary components of $Z$. Then $\tau_{S_1} \tau_{S_2} \tau_{S_3}=1$ in $\Mod(Z)$. \end{lemma} Lemma \ref{lem:pants} can be proved in an elementary fashion by constructing an explicit isotopy. We explain the idea briefly. Fix a properly embedded arc $\alpha\subset Z$ joining the spheres $S_1$ and $S_2$. Let $N$ be a regular neighborhood of $S_1\cup\alpha\cup S_2$. Observe that $N$ is diffeomorphic to $Z$ and one of its boundary components $S_3'$ is parallel to $S_3$. Using $\alpha$ is a guide, one can isotope $T_{S_1}\circ T_{S_2}$ to a diffeomorphism supported on a collar neighborhood of $S_3'$ and that is a sphere twist on this collar neighborhood. Since $S_3'$ is parallel to $S_3$, this gives the desired relation. \begin{proof}[Proof of Theorem \ref{thm:twist-group}] We prove the theorem in 3 steps. \p{Step 1: a generating set for \boldmath $\Twist(M)$} For $i=1,\ldots,\ell$, fix an embedded sphere $S_i\subset Y$ that is parallel to the boundary component of $P_i\setminus\text{int}(D_i)$. For $j=1,\ldots,k$, let $S_j'\subset Y$ be the embedded sphere $S^2\times\{0\}$ in the $j$-th copy of $S^2\times[0,1]$. Let $\Twist(Y)$ be the subgroup of $\Mod(Y)$ generated by the sphere twists $\{\tau_{S_1},\ldots,\tau_{S_\ell}\}\cup\{\tau_{S_1'},\ldots,\tau_{S_k'}\}$. (Recall that the mapping class group $\Mod(Y)$ of a manifold with boundary is the group $\Diff_\partial(Y)$ of diffeomorphisms that restrict to the identity on $\partial Y$, modulo isotopies that are the identity on $\partial Y$.) Consider the composition \begin{equation}\label{eqn:twist-composition}(\Z/2\Z)^{\ell+k}\xrightarrow{\rho}\Twist(Y)\xrightarrow{\pi}\Twist(M),\end{equation} where \[\rho(a_1,\ldots,a_\ell,b_1,\ldots,b_k)=\tau_{S_1}^{a_1}\>\cdots\>\tau_{S_\ell}^{a_\ell}\>\tau_{S_1'}^{b_1}\>\cdots\>\tau_{S_k'}^{b_k},\] and $\pi$ is the restriction of the homomorphism $\Mod(Y)\to\Mod(M)$. The composition $\pi\circ\rho$ is surjective by \cite[Prop.\ 1.2]{mccullough}. In the rest of the proof we compute the kernels of $\pi$ and $\rho$. \p{Step 2: global relation among sphere twists} In this step we compute the kernel of $\pi:\Twist(Y)\to\Twist(M)$. Let $D\subset X\subset M$ be an embedded ball. Let $\Emb(D,M)$ be the space of embeddings that respect the orientation. Let $\Emb_D^e(X,M)$ be the space of embeddings $X\to M$ that (i) restrict to the inclusion on $D$ and (ii) that extend to a diffeomorphism of $M$. Consider the following diagram, which consists of two fiber sequences (c.f.\ \cite{palais}). \[ \begin{xy} (-30,0)*+{\Diff_\partial(Y)}="A"; (0,0)*+{\Diff(M,D)}="B"; (30,0)*+{\Emb_{D}^e(X,M)}="C"; (0,-15)*+{\Diff^+(M)}="E"; (0,-30)*+{\Emb(D,M)}="H"; {\ar"A";"B"}?*!/_3mm/{}; {\ar "B";"C"}?*!/_3mm/{}; {\ar "B";"E"}?*!/_3mm/{}; {\ar "E";"H"}?*!/_3mm/{}; \end{xy}\] The ``horizontal" fiber bundle in the diagram splits as a product by \cite[Thm.\ 1]{hendriks-mccullough}. Consequently, $\Mod(Y)\to\Mod(M,D)$ is injective. Then from the proceeding diagram, we obtain \[\begin{xy} (-30,-15)*+{\Mod(Y)}="A"; (0,0)*+{\pi_1\big(\Emb(D,M)\big)}="B"; (0,-15)*+{\Mod(M,D)}="E"; (0,-30)*+{\Mod(M)}="H"; {\ar@{^{(}->}"A";"E"}?*!/_3mm/{}; {\ar "B";"E"}?*!/_3mm/{\delta}; {\ar "E";"H"}?*!/_3mm/{}; \end{xy}\] As is well-known, the space $\Emb(D,M)$ is homotopy equivalent to the (oriented) frame bundle of $M$, which is diffeomorphic to $M\times\SO(3)$ since closed oriented 3-manifolds are parallelizable. Then $\pi_1\big(\Emb(D,M)\big)\cong\pi_1(M)\times\Z/2\Z$. A generator of the $\Z/2\Z$ factor maps under $\delta$ to a sphere twist about $S:=\partial D$. From Lemma \ref{lem:pants} we deduce that $\tau_S=\tau_{S_1}\cdots\tau_{S_\ell}$ in $\Mod(M,D)$, so $\tau_{S_1}\cdots\tau_{S_\ell}$ belongs to $\ker(\pi)$. We claim that $\ker(\pi)=\langle\tau_{S_1}\cdots\tau_{S_\ell}\rangle$. To see that $\ker(\pi)$ is not larger, it suffices to show that if $\gamma\in\pi_1(M)<\pi_1\big(\Emb(D,M)\big)$ is nontrivial, then $\delta(\gamma)$ is not in the image of $\Twist(Y)\to\Mod(M,D)$. This is easy to see because each element of $\Twist(Y)<\Mod(M,D)$ acts trivially on $\pi_1(M,*)$ (where the basepoint $*$ belongs to $D$), whereas $\delta(\gamma)$ acts by a nontrivial conjugation on $\pi_1(M,*)$ (note that the fundamental group of a reducible 3-manifold has trivial center). \p{Step 3: local triviality of sphere twists} Here we compute the kernel of the map $\rho:(\Z/2\Z)^{\ell+k}\to\Twist(Y)$ defined in (\ref{eqn:twist-composition}). Since the spheres $S_1,\ldots,S_\ell,S_1',\ldots,S_k'$ belong to distinct components of $Y$, it suffices to determine which of the given generators for $\Twist(Y)$ is trivial in $\Twist(Y)$. Sphere twists in $S^2\times[-1,1]$ components are nontrivial by \cite{laudenbach}. See also \cite{BBP} who prove this by considering the action on framings. It remains then to consider when the twists $\tau_{S_i}$ (about the boundary of $P_i\setminus\text{int}(D_i)$) are nontrivial. By Theorem \ref{thm:HFW} below, $\tau_{S_i}\in\Mod(P_i,D_i)<\Mod(Y)$ is trivial if and only if $P_i$ is a lens space. Combining this with the proceeding steps finishes the proof. \end{proof} \begin{theorem}[Hendriks, Friedman-Witt]\label{thm:HFW} Let $P$ be an irreducible $3$-manifold. Fix an embedded ball $D\subset P$, and let $\tau_S\in\Mod(P,D)$ be the sphere twist about a sphere $S$ parallel to $\partial D$. Then $\tau_S=1$ if and only if $P$ is a lens space. \end{theorem} \begin{proof} By work of Hendriks \cite{hendriks} (see \cite[Cor.\ 2.1]{friedman-witt}), the twist $T_S\in\Diff_\partial(P\setminus \text{int}(D))$ is not homotopic to the identity (rel boundary), unless $P$ is either a lens space or a prism manifold (the latter are manifolds covered by $S^3$ whose fundamental group is an extension of a dihedral group). First consider the case when $P$ is a lens space. Since we assume $P$ is irreducible, $P\neq S^1\times S^2$, so $P$ is covered by $S^3$. In this case $T_S$ is isotopic to the identity (rel boundary) by \cite[Lem.\ 3.5]{friedman-witt}. (Aside: $T_S$ is also isotopic to the identity when $P=S^1\times S^2$, which can be seen using Lemma \ref{lem:pants}.) When $P$ is a prism manifold, $T_S$ is not isotopic to the identity (rel boundary). See \cite[Thm.\ 2.2]{friedman-witt}. The statement there does not include one family of prism manifolds $S^3/D_{4m}^*$. This is because the argument uses the (generalized) Smale conjecture, which was not proved for $S^3/D_{4m}^*$ at the time the paper was written. See \cite[Remark after Corollary 2.2]{friedman-witt}. Fortunately, the generalized Smale conjecture has now been confirmed for all prism manifolds (in fact for all elliptic 3-manifold, with the exception of $\R P^3$). See \cite{smale-elliptic} and \cite{balmer-kleiner}. \end{proof} For later use, we record the following Corollary of Theorem \ref{thm:twist-group}. For the manifold $S^1\times S^2$, we call a sphere of the form $*\times S^2$ a \emph{belt sphere} (we use this terminology because this sphere can be viewed as the belt sphere of a handle attachment). \begin{corollary}\label{cor:twist-generator} Let $M=\#_k(S^1\times S^2)\#P_1\#\cdots\#P_\ell$, and assume each $P_i$ is a lens space. Then $\Twist(M)\cong(\Z/2\Z)^k$ is generated by twists about the belt spheres of the $S^1\times S^2$ summands. \end{corollary} \section{Decomposing finite group actions on 3-manifolds}\label{sec:invariant-spheres} In this section we explain some general structural results for certain finite group actions on $3$-manifolds, which will allow us to decompose a $G$-manifold $M^3$ into simpler $G$-invariant pieces. For our application to the Main Theorem we are particularly interested in actions that are trivial on $\pi_i(M)$ for $i=1,2$. \subsection{Equivariant sphere theorem} The main result of this section is Theorem \ref{thm:invariant-spheres}. In order to state it, we introduce some notation. Let $\mathbb S$ be a collection of disjoint embedded spheres in a 3-manifold $M$. Define $M_{\mathbb S}$ as the result of removing an open regular neighborhood of each $S\in\mathbb S$ and capping each boundary component with a 3-ball. The 3-manifold $M_{\mathbb S}$ is a closed, but usually not connected. This process is pictured in Figure \ref{fig:cut-cap}. \begin{figure}[h!] \labellist \pinlabel $M$ at 60 10 \pinlabel $M_{\mathbb S}$ at 580 10 \endlabellist \centering \includegraphics[scale=.6]{cut-cap} \caption{Cutting and capping along spheres $M\leadsto M_{\mathbb S}$.} \label{fig:cut-cap} \end{figure} \begin{theorem}\label{thm:invariant-spheres} Let $M$ be a closed oriented $3$-manifold and let $G$ be a finite subgroup of $\Diff^+(M)$. \begin{enumerate} \item There exists a $G$-invariant collection $\mathbb S$ of disjoint embedded spheres in $M$ such that the components of $M_{\mathbb S}$ are irreducible. \item If $M\neq S^1\times S^2$ and $G$ acts trivially on $\pi_1(M)$ and $\pi_2(M)$, then $G$ preserves every element in $\mathbb S$. \end{enumerate} \end{theorem} We call a collection of spheres as in the statement of Theorem \ref{thm:invariant-spheres} a \emph{sphere system} for $G$. \begin{remark}\label{rmk:essential} Without loss of generality, one can assume that no $S\in\mathbb S$ bounds a ball in $M$ by removing any sphere that bounds a ball from $\mathbb S$. Similarly, if $G$ preserves every element of $\mathbb S$, then we can also assume that no pair $S\neq S'\in\mathbb S$ bound an embedded $S^2\times[0,1]$ in $M$. \end{remark} Part (1) of Theorem \ref{thm:invariant-spheres} is due to Meeks--Yau; see \cite[c.f.\ Thm.\ 7]{MY}. An alternate approach was given by Dunwoody \cite[Thm.\ 4.1]{dunwoody}. The main tools used in these works are minimal surface theory (in the smooth and PL categories). For the proof of Theorem \ref{thm:invariant-spheres}(2), we use the following lemmas. \begin{lemma}\label{lem:annulus} Let $S_0,S_1\subset M$ be disjoint embedded spheres. If $S_0$ and $S_1$ are ambiently isotopic, then they bound an embedded $S^2\times[0,1]$ in $M$. \end{lemma} Lemma \ref{lem:annulus} follows from \cite[Lem.\ 1.2]{laudenbach} and the Poincar\'e conjecture (Laudenbach proves that homotopic spheres bound an $h$-cobordism, and every $h$-cobordism is trivial by Perelman's resolution of the Poincar\'e conjecture). \begin{lemma}\label{lem:annulus-homeo} Let $h$ be an orientation-preserving homeomorphism of $S^2\times [0,1]$. If $h$ that interchanges the two boundary components, then $h$ acts on $H_2(S^2\times[0,1])\cong\Z$ by $-1$. \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:annulus-homeo}] Set $A=S^2\times[0,1]$. Consider the arc $\alpha=*\times[0,1]$ and the sphere $\beta=S^2\times 0$. After orienting $\alpha$ and $\beta$, we view them as homology classes $\alpha\in H_1(A,\partial A)$ and $\beta\in H_2(A)$, which generate these groups. Since $h$ interchanges the components of $\partial A$, $h(\alpha)=-\alpha$. Since $h$ is orientation-preserving, \[\alpha\cdot\beta=h(\alpha)\cdot h(\beta)=-\alpha\cdot h(\beta).\] This implies $h(\beta)=-\beta$ because the intersection pairing $H_1(A,\partial A)\times H_2(A)\rightarrow \Z$ is a perfect pairing by Poincar\'e--Lefschetz duality. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:invariant-spheres}(2)] Let $\mathbb S$ be a $G$-invariant collection of embedded spheres as in Theorem \ref{thm:invariant-spheres}(1). Fix $S\in\mathbb S$ and $g\in G$. We want to show that $g(S)=S$. Suppose for a contradiction that $g(S)$ is disjoint from $S$. Fix an embedding $f:S^2\rightarrow M$ with $f(S^2)=S$. Since $g$ acts trivially on $\pi_2(M)$, the maps $f$ and $g\circ f$ are homotopic, hence isotopic by a result of Laudenbach and the Poincar\'e conjecture; c.f.\ \cite[Thm.\ 1]{laudenbach}. By Lemma \ref{lem:annulus}, the spheres $S$ and $g(S)$ bound a submanifold $A\cong S^2\times[0,1]$. Let $k\ge2$ be the smallest power of $g$ so that $g^k(S)=S$. First assume $k\ge3$. Since we can find a $G$-invariant metric on $M$, the interiors of $A$ and $g(A)$ are either equal or disjoint. Since $k\ge3$, $g^2(S)\neq S$, so $g(A)\neq A$. Consequently, $A\cup g(A)$ is diffeomorphic to $S^2\times[0,1]$. Similarly, we conclude that $A\cup g(A)\cup\cdots\cup g^{k-1}(A)$ is diffeomorphic to $S^2\times S^1$ (this is the only $S^2$-bundle over $S^1$ with orientable total space). This contradicts the assumption that $M\neq S^2\times S^1$. Now assume that $k=2$. As in the preceding paragraph, if $A$ and $g(A)$ have disjoint interiors, then $M=S^2\times S^1$. Therefore, $g(A)=A$. By Lemma \ref{lem:annulus-homeo}, $[g(S)]=-[S]$ in $H_2(M)$. By assumption that $G$ acts trivially on $\pi_2(M)$, we know $[g(S)]=[S]$. Therefore, $2[S]=0$ in $H_2(M)$. If $S\subset M$ is non-separating, then there is a closed curve $\gamma\subset M$ so that $[S]\cdot[\gamma]=1$; this implies that $[S]$ has infinite order in $H_2(M)$, which is a contradiction. If $S\subset M$ is separating, then $M\setminus A$ is a union of two components $M_1\sqcup M_2$ that are interchanged by $g$. This contradicts the fact that $G$ acts trivially on $\pi_1(M)\cong\pi_1(M_1)*\pi_1(M_2)$. \end{proof} \subsection{Decomposing an action along invariant spheres}\label{sec:decompose} Here we explain how we use Theorem \ref{thm:invariant-spheres} to decompose an action $G\curvearrowright M$ into smaller pieces. We also prove a result about the action on the fundamental group of the pieces under the assumption that $G$ acts trivially on $\pi_1(M)$. Fix a finite subgroup $G<\Diff^+(M)$ and assume $G$ acts trivially on $\pi_1(M)$. Let $\mathbb S$ be a sphere system for $G$ (Theorem \ref{thm:invariant-spheres}). We will always assume that $\mathbb S$ has the additional properties discussed in Remark \ref{rmk:essential}: no $S\in\mathbb S$ bounds a ball and no pair $S,S'\in\mathbb S$ bound an embedded $S^2\times[0,1]$. Observe that there is an induced action of $G$ on $M_{\mathbb S}$. To construct it, recall a classical result of Brouwer, Eilenberg, and de Ker\'ekj\'art\'o \cite{Brouwer,K,Eilenberg} that every finite subgroup of $\Homeo^+(S^2)$ is conjugate to a finite subgroup of $\SO(3)$, hence extends from the unit sphere $S^2\subset\R^3$ to the unit ball $D^3\subset\R^3$. In this way the action of $G$ on $M\setminus\bigcup_{S\in\mathbb S}S$ extends to an action on $M_{\mathbb S}$, which can be made smooth as well. \begin{remark}[global fixed points]\label{rmk:fixed-points} Since $G$ acts trivially on $\mathbb S$, then the center of each of the added 3-balls in $M_{\mathbb S}$ contains a global fixed point for the $G$-action; we call these \emph{canonical fixed points}. Each component of $M_{\mathbb S}$ contains at least one canonical fixed point. \end{remark} The following proposition will be important for our proof of the Main Theorem. \begin{proposition}\label{prop:fundamental-group} Fix $G<\Diff^+(M)$ acting trivially on $\pi_1(M)$ and fix a $G$-invariant collection $\mathbb S$ of disjoint, embedded spheres such that $M_{\mathbb S}$ has irreducible components. Let $N$ be a component of $M_{\mathbb S}$, and let $p\in N$ be a canonical fixed point, as defined in Remark \ref{rmk:fixed-points}. Then $G$ acts trivially on $\pi_1(N,p)$. \end{proposition} \begin{proof} Fix $g\in G$. We show that the action of $g$ on $\pi_1(N,p)$ is trivial. The statement is only interesting when $N$ is not simply connected, so we assume this. Let $k$ be the number of elements of $\mathbb S$ that meet $N$. We separate the argument into the cases $k=1$, $k=2$, and $k\ge3$. \p{Case: $k=1$} Let $S\in\mathbb S$ be the sphere that meets $N$. The sphere $S$ is separating and gives a description of $M$ as a connected sum $M\cong N\# N'$. Since $g\in G$ has finite order and preserves $S$, there is a fixed point $q\in S^g$. By van Kampen's theorem, $\pi_1(M,q)\cong\pi_1(N,q)*\pi_1(N',q)$. By Lemma \ref{lem:twist-action}, the action of $g$ on $\pi_1(M,q)$ is by conjugation by some element $\pi_1(M,q)$. Since $g$ preserves the decomposition $M=N\#N'$, it also preserves the factors in the splitting $\pi_1(N,q)*\pi_1(N',q)$. Note that $\pi_1(N',q)$ is nontrivial, since otherwise $S$ bounds a ball in $M$, contrary to our assumption. The only conjugation of $A*A'$ that preserves both $A\neq1$ and $A'\neq1$ is the trivial conjugation, so $g$ acts trivially on $\pi_1(M,q)$. Consequently, $g$ acts trivially on $\pi_1(N,q)$, and also on $\pi_1(N,p)$. (In general, changing the basepoint can change the automorphism to a nontrivial conjugation, but in this does not happen here since e.g.\ the points $p,q\in N$ are connected by an arc contained in the fixed set $N^g$.) \p{Case: $k=2$} Let $S,S'\in\mathbb S$ denote the spheres the meet $N$. Observe that these sphere are either both separating or both nonseparating. Let $N'$ be the closed 3-manifold such that $M_{\mathbb S}=N\sqcup N'$. If $S$ and $S'$ are both separating, then the argument is similar to the case $k=1$. Apply that argument to either sphere to see that the action is trivial at the corresponding canonical fixed point. Assume then that both $S$ and $S'$ are nonseparating; in particular, this implies that $N'$ is connected. Then $M$ is obtained from $N\sqcup N'$ by removing balls $B_1,B_2\subset N$ and $B_1',B_2'\subset N'$ and gluing $\partial B_i$ to $\partial B_i'$. Choose a fixed points $q\in S^g$ and $q'\in (S')^g$. There is an isomorphism $\pi_1(M)\cong \pi_1(N,q)*\pi_1(N',q)*\Z$. (The $\Z$ factor is not important for this part of the argument, but will play a role when $k\ge3$.) By assumption $g$ acts on $\pi_1(M,q)$ by conjugation and preserves the free factors $\pi_1(N,q)$ and $\pi_1(N',q)$. Both of these groups is nontrivial, by our assumption that no two spheres in $\mathbb S$ are parallel. Then as before, we conclude that $g$ acts trivially on $\pi_1(M,q)$, hence also on $\pi_1(N,q)$ and $\pi_1(N,p)$. \p{Case $k\ge3$} Let $S_0,\ldots,S_{k-1}\in\mathbb S$ denote the spheres that meet $N$. If some $S_i$ separates, then we can proceed similar to the case $k=1$, so we can assume each $S_i$ is nonseparating. Then $M_{\mathbb S}=N\sqcup N'$, where $N'$ is connected, and $M$ is obtained from $N\sqcup N'$ by removing balls $B_0,\ldots,B_{k-1}\subset N$ and $B_0',\ldots,B_{k-1}'\subset N'$ and gluing $B_i$ and $B_i'$ along their boundary (which is identified with $S_i\subset M$). Choose fixed points $q_i\in (S_i)^g$. There is an isomorphism \[\pi_1(M,q_0)\cong\pi_1(N,q_0)*\pi_1(N',q_0)*F_{k-1}.\] If $\pi_1(N',q_0)\neq1$, then we can argue similar to the case $k=2$. Therefore, we assume that $N'$ is simply connected, which means $\pi_1(M,q_0)\cong\pi_1(N,q_0)*F_{k-1}$. The free group $F_{k-1}$ is generated by loops $\gamma_i=\eta_i*\eta_i'$, where $\eta_i$ is a path in $N$ from $q_0$ to $q_i$ (and disjoint from the interiors of the balls $B_0,\ldots,B_{k-1}$), and $\eta_i'$ is a path in $N'$ from $q_i$ to $q_0$. On the one hand,\footnote{Here the symbol $\sim$ indicates homotopic loops based at $q_0$. For the first homotopy, note that the paths $g(\eta_i')$ and $\eta_i'$ are homotopic rel endpoints because $N'$ is simply connected.} \[g(\gamma_i)\sim g(\eta_i)*\eta_i'\sim g(\eta_i)*\overline{\eta_i}*\eta_i*\eta_i'=(g(\eta_i)*\overline{\eta_i})*\gamma_i,\] so $g$ acts on $\gamma_i$ by left multiplication by the element $\beta_i = g(\eta_i)*\overline{\eta_i}\in\pi_1(N,q_0)$. On the other hand, $g$ acts on $\gamma_i$ by conjugation by an element $\alpha\in \pi_1(M,q_0)$. The only way these actions are equal is if both are trivial. If we use the word length on $\pi_1(M,q_0)$ given by the generating set $\{s: s\in \pi_1(N,q_0) \text{ or } s\in F_{k-1}\}$, then the word length of $\alpha\gamma_i\alpha^{-1}$ is odd, but the word length for $\beta_i\gamma_i$ is 2 unless $\beta_i=1$. This implies that $\beta_i=1$. Then we know that $\gamma_i= \alpha\gamma_i\alpha^{-1}$ for every $i$, which implies that $\alpha=1$. In particular, this implies that $g$ acts trivially on $\pi_1(M,q_0)$ and hence also on $\pi_1(N,q_0)$ and $\pi_1(N,p)$. \end{proof} \section{Obstructing realizations}\label{sec:obstruction} In this section we prove the ``only if" direction of the Main Theorem. This can be deduced quickly from the following more general statements. \begin{theorem}\label{thm:holder-argument} Let $N$ be a closed, oriented, irreducible $3$-manifold with basepoint $p\in N$. Suppose there exists a nontrivial, finite-order element $f\in\Diff^+(N,p)$ that acts trivially on $\pi_1(N,p)$. Then $N$ is a lens space. \end{theorem} \begin{theorem}\label{thm:noncyclic} Let $M$ be a closed, oriented, reducible $3$-manifold. Let $G<\Diff^+(M)$ be a finite subgroup that acts trivially on $\pi_1(M)$. Then $G$ is cyclic. \end{theorem} \begin{proof}[Proof of Main Theorem: obstruction] Suppose $1\neq G<\Twist(M)$ is realizable. The fact that $\Twist(M)\neq1$ implies that either $M=S^2\times S^1$ or $M$ is reducible. In the former case, there is nothing to prove, so we assume $M$ is reducible. This assumption together with Lemma \ref{lem:twist-action} allow us to apply Theorem \ref{thm:noncyclic} and conclude that $G$ is cyclic. It remains to show $M$ is a connected sum of lens spaces, or, equivalently, that each component of $M_{\mathbb S}$ is a lens space. This is implied directly by Proposition \ref{prop:fundamental-group} and Theorem \ref{thm:holder-argument}. \end{proof} Next we use Theorem \ref{thm:holder-argument} to deduce Theorem \ref{thm:noncyclic}. Then we prove Theorem \ref{thm:holder-argument}. \begin{proof}[Proof of Theorem \ref{thm:noncyclic}] Let $\mathbb S$ be a sphere system for $G$ (Theorem \ref{thm:invariant-spheres}). We also assume that no $S\in\mathbb S$ bounds a ball and that no two spheres $S,S'\in\mathbb S$ bound an embedded $S^2\times[0,1]$ (Remark \ref{rmk:essential}). Since $G$ acts trivially on $\pi_1(M)$, Proposition \ref{prop:fundamental-group} and Theorem \ref{thm:holder-argument} combine to show that each component of $M_{\mathbb S}$ is a lens space. Suppose that there exists a component $N$ of $M_{\mathbb S}$ that is diffeomorphic to $S^3$. Let $k$ be the number of elements of $\mathbb S$ that meet $N$. If $k=1$, then $N$ is obtained from $M$ by cutting along a sphere that bounds a ball. Similarly, if $k=2$, then $N$ is obtained by cutting $M$ along two parallel spheres. Both of these are contrary to our assumption about $\mathbb S$. Therefore $k\ge3$, which implies that $N^G$ has at least 3 points. By the Smith conjecture \cite{smith-conjecture}, the action of $G$ on $N\cong S^3$ is conjugate into $\SO(4)$, and the fact that $|N^G|\ge3$ implies that $G$ is conjugate into $\SO(2)$. Therefore $G$ is cyclic. The remaining case is that every component of $M_{\mathbb S}$ is a lens space different from $S^3$ and $S^1\times S^2$. In this case $\Twist(M)$ is the trivial group by Corollary \ref{cor:twist-generator}. \end{proof} We proceed to the proof of Theorem \ref{thm:holder-argument}. Our argument is inspired by an argument of Borel \cite{borel-isometry-aspherical} that shows that a finite group $G$ acting faithfully on a closed aspherical manifold $N$ and $\pi_1(N)$ has trivial center, then $G$ also acts faithfully on $\pi_1(N)$ (by outer automorphisms). Before starting the proof, we recall some facts about lifting actions to universal covers. Let $N$ be a closed manifold. Recall that $\widetilde N$ can be defined as the set of paths $\alpha:[0,1]\to N$ with $\alpha(0)=*$, up to homotopy rel endpoints. Using this description, there is a left action $\pi_1(N,*)\times \widetilde N\to \widetilde N$ given by pre-concatenation of paths $[\gamma].[\alpha]=[\gamma*\alpha]$, and there is a left action \[\Diff(N,*)\times\widetilde N \to\widetilde N\] given by post-composition $f.[\alpha]=[f\circ \alpha]$. If $f\in\Diff(N,*)$ acts trivially on $\pi_1(N,*)$, then $F([\alpha])=[f\circ \alpha]$ is a lift of $f$ that commutes with the deck group action and fixes the homotopy class of the constant path (as well as every other homotopy class corresponding to an element of $\pi_1(N,*)$). \begin{proof}[Proof of Theorem \ref{thm:holder-argument}] As observed above, we can lift $f$ to a finite-order diffeomorphism $F$ that commutes with the deck group $\pi_1(N,*)$ and has a global fixed point. First we show that $\pi_1(N)$ is finite. Suppose for a contradiction that $\pi_1(N)$ is infinite. This implies $\widetilde N$ is contractible.\footnote{By Hurewicz, $\pi_3(\widetilde N)\cong H_3(\widetilde N)$. Since $\pi_1(N)$ is infinite, $\widetilde N$ is noncompact, so $H_3(\widetilde N)=0$. Similarly, all higher homotopy groups vanish by Hurewicz's theorem.} By Smith theory \cite{Smith}, the fixed set $(\widetilde{N})^F$ is connected, and simply connected. Since $F$ acts smoothly, $(\widetilde{N})^F$ is a smooth 1-dimensional manifold, hence it is homeomorphic to $\R$. Since $\pi_1(N)$ commutes with $F$, it acts on $(\widetilde N)^F\cong\R$, and this action is free and properly discontinuous since the action of $\pi_1(N,*)$ on $\widetilde{N}$ has these properties. This implies that $\pi_1(N,*)\cong\Z$, which contradicts the fact that $N$ is a closed, aspherical 3-manifold ($\Z$ is not a 3-dimensional Poincar\'e duality group). Since $\pi_1(N)$ is finite, its universal cover is diffeomorphic to $S^3$ by the Poincare-conjecture. As in the preceding paragraph, consider the action of $F$ on $\widetilde N\cong S^3$. By Smith theory and smoothness of the action, the fixed set is a smooth, connected 1-dimensional manifold with nontrivial fundamental group. Hence $(\widetilde N)^F\cong S^1$. Since $\pi_1(N)$ acts freely on $(\widetilde N)^F$ this implies $\pi_1(N)$ is cyclic, which implies that $N$ is a lens space. \end{proof} \section{Constructing realizations}\label{sec:construction} In this section we prove the ``if" direction of the Main Theorem. We state this as the following theorem. \begin{theorem}\label{thm:realization} Let $M$ be a connected sum of lens spaces. Then every cyclic subgroup of $\Twist(M)$ is realizable. \end{theorem} Fix $M$ as in Theorem \ref{thm:realization}, and write the prime decomposition \[M=\#_k(S^1\times S^2)\# P_1\#\cdots\# P_\ell,\] where each $P_i$ is a lens space different from $L(0,1)\cong S^1\times S^2$. To prove Theorem \ref{thm:realization}, given a nontrivial element $g\in\Twist(M)$ we define $\gamma\in\Diff(M)$ such that $\gamma^2=id$ and $[\gamma]=g$ in $\Mod(M)$. The basic approach is to define an order-2 diffeomorphism of \begin{equation}\label{eqn:disjoint-union}\sqcup_k(S^1\times S^2)\sqcup P_1\sqcup\cdots\sqcup P_\ell\end{equation} in such a way that the diffeomorphisms on the components can be glued to give an order-2 diffeomorphism of $M$. On each component of (\ref{eqn:disjoint-union}) we perform one of the following diffeomorphisms. \begin{itemize} \item (constant $\pi$ rotation) Define \[R_0:S^1\times S^2\to S^1\times S^2\] by $id\times r$, where $r:S^2\to S^2$ is any $\pi$ rotation (choose one -- the particular axis is not important). \item (nonconstant $\pi$ rotation) Let $c:[0,1]\to\R P^2$ be a closed path that generates $\pi_1(\R P^2)$, and let $\alpha:\R P^2\rightarrow\SO(3)$ be the map that sends $\ell\in\R P^2$ to the $\pi$-rotation whose axis is $\ell$. Now define \[R_1:S^1\times S^2\to S^1\times S^2\] by $(t,x)\mapsto(t,\alpha(c(t))(x))$. Since $\alpha\circ c:[0,1]\rightarrow\SO(3)$ defines a nontrivial element of $\pi_1(\SO(3))$, the diffeomorphism $R_1$ represents the generator of $\Twist(S^1\times S^2)\cong\Z/2\Z$. This shows that $\Twist(S^1\times S^2)$ is realized. This involution appears in \cite[\S1]{tollefson_involutions} in a slightly different form. \item (lens space rotation) Fix $p,q$ relatively prime and with $p\ge2$. View $L(p,q)$ as the quotient of $S^3\subset\mathbb C^2$ by the $\Z/p\Z$ action generated by $(z,w)\mapsto(e^{2\pi i/p}z, e^{2\pi iq/p}w)$. Define \[R_{p,q}:L(p,q)\to L(p,q)\] as the involution induced by $(z,w)\mapsto(z,-w)$ on $S^3$ (which descends to $L(p,q)$ since it commutes with the $\Z/p\Z$ action). \end{itemize} Each of the diffeomorphisms $R_0$, $R_1$, and $R_{p,q}$ has 1-dimensional fixed set. The representation in the normal direction at a fixed point is the antipodal map on $\R^2$ (there is no other option since these diffeomorphisms are involutions). Lemma \ref{lem:gluing} below allows us to glue these actions along their fixed sets. \begin{lemma}\label{lem:gluing} Suppose $M,M'$ are oriented manifolds, each with a smooth action of a finite group $G$. Assume that $x\in M$ and $x'\in M'$ are fixed points of $G$, and that the representations $T_xM$ and $T_{x'}M'$ are isomorphic by an orientation reversing map. Then $M$ and $M'$ can be glued along regular neighborhoods $B$ and $B'$ of $x$ and $x'$ so that there is a smooth action of $G$ on $M\#M'$ that restricts to the given action on $M\setminus B$ and $M'\setminus B'$. \qed \end{lemma} \begin{remark} The condition that the isomorphism $T_xM\cong T_{x'}M'$ be orientation-reversing appears because the connected sum of two oriented manifolds is defined by deleting an open ball from each and identifying the boundaries of these balls by an orientation-reversing diffeomorphism. This condition is always satisfied if each tangent space contains a copy of the trivial representation (choose an appropriate reflection). \end{remark} \begin{remark}[Useful isotopies]\label{rmk:isotopy} To prove that $\gamma\in\Diff(M)$ is in the isotopy class of $g\in\Twist(M)$, the following observation will be useful. The fixed set of $R_{p,q}$ acting on $L(p,q)$ contains\footnote{It's possible that the fixed set is larger (this is true for $L(2,1)\cong\R P^3$), but this is not important.} the image $C$ of the circle $\{(z,0): |z|=1\}\subset S^3$. The isotopy $h_t(z,w)=(z,e^{\pi i(1-t)}w)$, $0\le t\le 1$, descends to $L(p,q)$ to give an isotopy between $R_{p,q}$ and the identity, and $h_t$ fixes $C$ for each $t$. Similarly, it's possible to isotope $R_1$ to $R_1'$, which is a constant $\pi$-rotation (say about the $z$-axis) on a neighborhood of $*\times S^2$, for some fixed basepoint $*\in S^1$ (observe that $R_1'$ is still an involution). Furthermore, we can isotope $R_1'$ to a diffeomorphism that is the identity near $*\times S^2$ and in such a way that the isotopy at time $t\in[0,1]$ is a rotation by angle $\pi (1-t)$ (about the $z$-axis) on each sphere in a regular neighborhood of $*\times S^2$. The fixed set restricted to a neighborhood of $*\times S^2$ remains constant during this isotopy. Finally, we can isotope $R_0$ to the identity so that at time $t$ the diffeomorphism is a constant rotation by angle $\pi(1-t)$ (about the fixed axis). On a neighborhood of a fixed point, the local picture of the isotopies of $R_{p,q}$, $R_1'$, and $R_0$ looks the same. This will allow us to perform these isotopies equivariantly on connected sums. \end{remark} We proceed now to the proof of Theorem \ref{thm:realization}. First we warm up with the case $M=\#_k(S^1\times S^2)$ and then we do the general case. \subsection{Realizations for connected sums of \boldmath $S^1\times S^2$}\label{sec:torus-case} Fix $k\ge1$ and consider \[M_k:=\#_k(S^1\times S^2).\] Let $S_i$ be a belt sphere in the $i$-th connect summand, and denote the sphere twist about $S_i$ by $\tau_i$. The twists $\tau_1,\ldots,\tau_k$ form a basis for $\Twist(M_k)\cong(\Z/2\Z)^k$, c.f.\ Theorem \ref{thm:twist-group}. Fix a nonzero element \[g=a_1\tau_1+\cdots+a_k\tau_k\] in $\Twist(M_k)$. We start by defining an involution $\hat\gamma$ of $\sqcup_k(S^1\times S^2)$. For ease of exposition, let $W_i=S^1\times S^2$ denote the $i$-th component of $\sqcup_k(S^1\times S^2)$. Define $\hat\gamma$ on $W_i$ to be $R_0$ or $R_1$, depending on whether the coefficient $a_i$ is $0$ or $1$, respectively. Next we glue using Lemma \ref{lem:gluing} to obtain an involution $\gamma$ of $M_k=W_1\#\cdots\#W_k\cong\#_k(S^1\times S^2)$. There are multiple ways to describe the gluing; for example, choose $k-1$ distinct fixed points $x_1,\ldots,x_{k-1}\in W_k$, and for $1\le i\le k-1$, glue $W_i$ to $W_k$ along regular (equivariant) neighborhoods of $x_i$ and an arbitrary fixed point $y_i\in W_i$ (the neighborhoods of $x_1,\ldots,x_k$ should be chosen to be small enough so that they are disjoint). To see that $\gamma\in\Diff(M_k)$ is in the isotopy class $g$, recall the short exact sequence of Laudenbach \[1\to\Twist(M_k)\to\Mod(M_k)\to\Out(\pi_1(M_k))\to1.\] It's easy to check that $\gamma$ acts trivially on $\pi_1(M_k)$, so $\gamma$ represents a mapping class in $\Twist(M_k)$. The particular isotopy class is determined by the action on trivializations of the tangent bundle of $M_k$, and in this way one can check that $[\gamma]=g$ in $\Twist(M_k)$. We do not spell out the details of this because we give an alternate argument in the next section in the general case. \begin{remark} We cannot realize a non-cyclic subgroup of $\Twist(M_n)$ using this construction because it is not possible to choose the axis for $R_0$ so that (1) $R_0$ and $R_1$ have a common fixed point and (2) $R_0$ and $R_1$ commute. Indeed, \S\ref{sec:obstruction} proves no non-cyclic subgroup of $\Twist(M_n)$ is realized. \end{remark} \subsection{Realizations for connected sum of lens spaces} Now we treat the general case \[M=\#_k(S^1\times S^2)\# L(p_1,q_1)\#\cdots\# L(p_\ell,q_\ell),\] where each $L(p_j,q_j)$ is a lens space different from $L(0,1)\cong S^1\times S^2$. Our approach is similar to the preceding section. Recall from Corollary \ref{cor:twist-generator} that $\Twist(M)\cong(\Z/2\Z)^k$ is generated by twists $\tau_1,\ldots,\tau_k$ in the belt spheres of the $S^1\times S^2$ summands. Fix a nonzero element \[g=a_1\tau_1+\cdots+a_k\tau_k\] in $\Twist(M)$. We start by defining an involution $\hat\gamma$ of \[\sqcup_k(S^1\times S^2)\sqcup L(p_1,q_1)\sqcup\cdots\sqcup L(p_\ell,q_\ell).\] Let $W_i$ denote the $i$-th component diffeomorphic to $S^1\times S^2$. Define $\hat\gamma$ on $L(p_j,q_j)$ to be $R_{p_j,q_j}$, and on $W_i$ to be $R_0$ or $R_1'$, depending on whether the coefficient $a_i$ is $0$ or $1$, respectively. (Recall that $R_1'$ is similar to $R_1$, but it has a product region.) Next we glue using Lemma \ref{lem:gluing} to obtain an involution $\gamma$ of $M$. We glue by the following pattern. First we glue $W_1,\ldots,W_k$. Choose $k-1$ distinct fixed points $x_1,\ldots,x_{k-1}\in W_k$, and for $1\le i\le k-1$, glue $W_i$ to $W_k$ along regular (equivariant) neighborhoods of $x_i$ and an arbitrary fixed point $y_i\in W_i$ (as was done in the preceding section). Next glue $L(p_j,q_j)$ to $W_k$ in a region where $\hat\gamma$ acts as a product (we can choose $x_1,\ldots,x_{k-1}$ and the regular neighborhoods of these points to ensure that there is room to do this). In this way we obtain an involution $\gamma\in\Diff(M)$. We need to check that $\gamma$ is in the isotopy class of $g$. Using the isotopies defined in Remark \ref{rmk:isotopy}, we can isotope $\hat \gamma$ to a map that is the identity on each $L(p_j,q_j)$ component and each component $W_i$ such that $a_i=0$, and is a sphere twist on each component $W_i$ such that $a_i=1$. By construction these isotopies glue to give an isotopy of $\gamma$ to a product of sphere twists representing $g$. This completes the proof of Theorem \ref{thm:realization}.\qed \begin{question} Are any two realizations of $g\in\Twist(M)$ conjugate in $\Diff(M)$? \end{question} \section{Burnside Problem for $\Diff(M)$ for reducible 3-manifold $M$}\label{Burn} In this section, we prove Theorem \ref{Burnside}, which follows quickly from Lemma \ref{Burnside_lemma}. \begin{lemma}\label{Burnside_lemma} Fix a closed, oriented $3$-manifold $M$, and consider the group \[K:=\ker\big[\Diff(M)\xrightarrow{\Phi} \Out(\pi_1(M))\big].\] If $M$ is reducible and that $M$ is not a connected sum of lens spaces, then $K$ is torsion free. \end{lemma} \begin{remark}[A strong converse to Lemma \ref{Burnside_lemma}] If $M$ is a connected sum of lens spaces, then $M$ has a faithful $S^1$-action, so $K$ contains $S^1$ as a subgroup. To see this, observe that each lens space has an $S^1$ action with global fixed points, so by performing the connected sum equivariantly along fixed points (similar to the construction in Section \ref{sec:construction}) we obtain an $S^1$ action on $M$. \end{remark} \begin{proof}[Proof of Theorem \ref{Burnside}] If $\Diff(M)$ has an infinite torsion group $G$, then either the image of $\Phi$ is an infinite torsion group or the kernel $K$ of $\Phi$ has infinite torsion group. By \cite[Thm.\ 5.2]{Hong_McCullough}, $\Out(\pi_1(M))$ contains a torsion-free finite-index subgroup. This implies that $\Phi(G)$ is finite, since every element of $\Phi(G)$ has finite order. On the other hand, by Lemma \ref{Burnside_lemma} that $K$ is torsion free, so $G\cong \Phi(G)$ is finite. \end{proof} \begin{proof}[Proof of Lemma \ref{Burnside_lemma}] Fix a nontrivial subgroup $G=\Z/p\Z<K$, and fix a sphere system $\mathbb S$ for $G$ (Theorem \ref{thm:invariant-spheres}). By Proposition \ref{prop:fundamental-group}, the action of $G$ on each component $N$ of $M_\mathbb{S}$ is trivial on $\pi_1(N,p)$ as an automorphism. This implies that each component $N$ of $M_\mathbb{S}$ is a lens space by Theorem \ref{thm:holder-argument}. \end{proof} \bibliographystyle{alpha}

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\section{Introduction} In previous articles \cite{Paschos:2003qr,Lalakulich:2005cs} we described the formalism for the excitation of the $P_{33}(1232)$ resonance. In the meanwhile, we extended the analysis to the second resonance region, which includes three isospin $1/2$ states: $P_{11}(1440)$, $D_{13}(1520)$, $S_{11}(1535)$. The dominant $P_{33}(1232)$ has been observed in neutrino reactions and there are several theoretical articles which describe it with dynamical calculations based on unitarized amplitudes through dispersion relations \cite{Adler:1968tw}, phenomenological \cite{Schreiner:1973mj,Alvarez-Ruso:1998hi,Hagiwara:2003di,Amaro:2004bs} and quark models \cite{Rein:1980wg,Kuzmin:2003ji,Kuzmin:2004ya}, as well as models incorporating mesonic states \cite{Sato:2003rq} including a cloud of pions. Articles in the past five years have taken a closer look at the data by analysing the dependence on neutrino energy and $Q^2$ \cite{Alvarez-Ruso:1998hi,Paschos:2003qr,Lalakulich:2005cs}. So far a consistent picture emerged to be tested in the new accurate experiments. For the higher resonances there are several articles, that describe their excitation by electrons \cite{Ochi:1997ev,Armstrong:1998wg,Kamalov:2001yi,Yang:2005cq}, and only one \cite{Rein:1980wg} by neutrinos. Experimental data for neutrino excitation of these resonances are very scarce and come from old bubble--chamber experiments \cite{Barish:1978pj,Radecky:1981fn,Kitagaki:1986ct,Grabosch:1988gw,Allasia:1990uy}. In the new experiments, studying neutrino oscillations, there is a strong interest to go beyond the QE scattering \cite{Morfin:2005yt,Morfin:2005yp} and understand the excitation of these resonances. One reason comes from the long--baseline experiments where the two detectors (nearby and faraway) observe different regions of neutrino fluxes and kinematic regions of the produced particles. A basic problem with resonances deals with the determination of their form factors (coupling constants and $Q^2$ dependences). The problem was apparent in the $\Delta$ resonance where after many years several of the form factors and their $Q^2$ dependence became accurately known and were found to deviate from the dipoles. The situation is more serious for the higher resonances where the results of specific models are used. In this article we adopt the approach of determining the vector couplings from helicity amplitudes of electroproduction data, which became recently available from the Jefferson Laboratory \cite{Burkert:2002zz,Burkert:2004sk,Aznauryan:2004jd} and Mainz accelerator \cite{Tiator:2003uu}. This requires that we write amplitudes for electroproduction in terms of the electromagnetic form factors and then relate them to the vector form factors that we use in neutrino reactions. The above approach together with CVC uniquely specifies the couplings and the $Q^2$-dependences in the region of $Q^2$ where data is available, that is for $Q^2<3.5 \; {\mathrm{GeV}}^2$. The axial form factors are more difficult to determine. For the axial form factors we adopt an effective Lagrangian for the $R \rightarrow N \pi $ couplings and calculate the decay widths. For each resonance we assume PCAC which gives us one relation and a second coupling is determined using the decay width of each resonance. Having determined the couplings for the four resonances, we are able to calculate differential and integrated cross sections. This way we investigate several properties in the excitation of the resonances. We find that a second resonance peak with an invariant mass between $1.4\; {\mathrm{GeV}}$ and $1.6 \; {\mathrm{GeV}}$ should be observable provided that neutrino energy is larger than $2-3\; {\mathrm{GeV}}$. Calculating cross sections in terms of the resonances provides a benchmark for their contribution and allows investigators to decide, when more precise data becomes available, whether a smooth background contribution is required. Integrated cross section already suggest the presence of a background. In sections~\ref{electro-hel-amp}, \ref{isospin-relat} and \ref{resonances} we present the formalism and give expressions for the vector form factors. Estimates for the axial form factors are presented in the Appendix~\ref{gold-trei}. Section~\ref{tau} points out that the structure functions ${\cal W}_4$ and ${\cal W}_5$ are important for reactions with tau neutrinos. We analyze differential and integrated cross sections in section~\ref{xsec-2nd}. We discuss there the existing data and point out a discrepancy in the normalization to be resolved in the next generation of experiments. \section{Electroproduction via helicity amplitudes \label{electro-hel-amp}} One of the main contributions of this article is the determination of the vector form factors for weak processes. Our work relates form factors to electromagnetic helicity amplitudes, whose numerical values are available from the Jefferson Laboratory and the University of Mainz. Values for the form factors at $Q^2=0$ are presented in Table~\ref{res-couplings}. Later on we express the weak structure functions in terms of form factors. In applying this method we must still define the normalization of electromagnetic amplitudes, which is done in this section. Before we address this topic we discuss the kinematics, the polarizations and spinors entering the problem. We shall calculate Feynman amplitudes shown in Fig.\ref{Fey-dia}. We estimate the amplitudes in the laboratory frame with the initial nucleon at rest and with the intermediate photon moving along the $z-$axis. \begin{figure}[th] \includegraphics[angle=-90,width=0.7\columnwidth]{feynman.ps} \caption{Electroproduction of the resonance.} \label{Fey-dia} \end{figure} We define the four--momenta \[ p^{\mu}=(m_N; 0,0,0), \qquad q^{\mu}=(q^0; 0,0,q^z) \] \[ \mbox{and}\quad {p'}^{\mu}=p^\mu + q^\mu = (q^0+m_N; 0,0,q^z) . \] The intermediate photon or $W-$boson can have three polarizations $\varepsilon^{\mu}_{(i)}$ defined as \begin{equation} \begin{array}{c} \displaystyle \varepsilon^\mu_{(R,L)}=\mp \frac1{\sqrt{2}}\bigl(0; 1, \pm i, 0), \qquad \\[5mm] \displaystyle \varepsilon^\mu_{(S)}=\frac1{\sqrt{Q^2}}\bigl(q^z; 0, 0, q^0) \end{array} \end{equation} and $Q^2=-q^2$. The spinors for the target nucleon, normalized as $\bar u (p,s_z) u (p,s_z) = 2m_N$, are given by \begin{equation} \displaystyle u(p,s_z)=\sqrt{2m_N}\left( \begin{array}{c} u_{s_z} \\ 0 \end{array} \right) \label{spinor-1/2} \end{equation} where $u_{s_z}$ can be either \[ \displaystyle u_{+1/2}= \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \quad \mbox{or} \quad u_{-1/2}= \left( \begin{array}{c} 0 \\ 1 \end{array} \right). \] The final resonances will have spin $1/2$ or $3/2$. For spin-3/2 resonances we shall use Rarita--Schwinger spinors constructed as the product of a polarization vector $e^{\mu}_{(i)}$ with a spinor $U$. The states with various helicities are defined by \begin{equation} \begin{array}{l} \psi_\mu^{(3/2)}= e_\mu^{(R)} U({p'}, +1/2) \\[3mm] \psi_\mu^{(1/2)} =\sqrt{\frac23} e_\mu^{(S)} U({p'}, +1/2) + \sqrt{\frac13} e_\mu^{(R)} U({p'}, -1/2) \\[3mm] \psi_\mu^{(-1/2)}=\sqrt{\frac23} e_\mu^{(S)} U({p'}, -1/2) + \sqrt{\frac13} e_\mu^{(L)} U({p'}, +1/2) \\[3mm] \psi_\mu^{(-3/2)}=e_\mu^{(L)} U({p'}, -1/2) \end{array} \label{spinvector-3/2} \end{equation} with the spinor given as \[ \displaystyle U(p',s_z)=\sqrt{{p'}^0+M_R}\left( \begin{array}{c} u_{s_z} \\ \frac{ \vec{{p'}}\cdot \vec{\sigma} }{{p'}^0+M_R} u_{s_z} \end{array} \right), \] and the polarization vectors by \[ e^\mu_{(R,L)}=\mp \frac1{\sqrt{2}}\bigl(0; 1, \pm i, 0), \qquad e^\mu_{(S)}=\frac1{M_R}\bigl(p'{}^z; 0, 0, p'{}^0) . \] We emphasize that $\varepsilon^{\mu}_{(i)}$ refer to the intermediate photon and $e^{\mu}_{(i)}$ belong to the $J=3/2$ spin state. For spin-1/2 resonances the spinors are \begin{equation} \displaystyle u(p',s_z)=\sqrt{{p'}^0+M_R}\left( \begin{array}{c} u_{s_z} \\ \frac{ \vec{{p'}}\cdot \vec{\sigma} }{{p'}^0+M_R} u_{s_z} \end{array} \right) . \label{spinor-resonance} \end{equation} With this notation we can calculate three helicity amplitudes for the electromagnetic process. For instance, for the $D_{13}$ resonance the amplitude $\langle R,+1/2|J_{em}\cdot \varepsilon_{(R)}| N,-1/2\rangle$ in terms of form factors is presented in eqs.~(\ref{vertexD13V}) and (\ref{D1520-A12}). We now define the overall normalization. Analyses of electroproduction data give numerical values for cross sections at the peak of each resonance \cite{Burkert:2002zz,Gorchtein:2004jd,Aznauryan:2004jd,Tiator:2003uu} \begin{equation} \sigma_{T}(W \! = \! M_R)=\frac{2m_N}{M_{R}\Gamma_{R}}(A^2_{1/2}+A^2_{3/2}), \label{sigmaTviaA} \end{equation} \begin{equation} \sigma_{L}(W \! = \! M_R)=\frac{2m_N}{M_{R}\Gamma_{R}}\frac{Q^2}{q^2_z}S^2_{1/2} . \label{sigmaLviaA} \end{equation} These are helicity cross sections for the absorption of the "virtual" photon by the nucleon to produce the final resonance \cite{Hand:1963bb,Bjorken:1969ja}. They are defined as \begin{eqnarray}\nonumber \sigma_{(i)}(W)=\frac12 K \sum\limits_{\lambda, \, s}\left| \langle R,\lambda| \varepsilon^\nu_{(i)} J^{el}_{\nu}| N,s \rangle \right|^2 R(W,M_R) \label{sigmai} \end{eqnarray} \[ \mbox{with} \quad K=\frac{4\pi^{2}\alpha}{W^2-m_N^2} . \] The last factor in the cross section is the Breit--Wigner term of a resonance: \begin{equation} R(W,M_R)=\frac{M_R \Gamma_R}{\pi} \frac{1}{(W^2-M^2_R)^2+M_R^2 \Gamma_R^2} . \label{delta-function} \end{equation} For a very narrow resonance or a stable particle it reduces to a $\delta-$function. Numerical values have been reported for the amplitudes in (\ref{sigmaTviaA}), (\ref{sigmaLviaA}) which are related to the following matrix elements \begin{equation} \begin{array}{l} \displaystyle A_{1/2} = \sqrt{\frac{\pi\alpha}{m_N(W^2-m_N^2)}}\langle R,+\frac{1}{2}|J_{em}\cdot \varepsilon^{(R)}| N,-\frac{1}{2}\rangle , \\[4mm] \displaystyle A_{3/2} = \sqrt{\frac{\pi\alpha}{m_N(W^2-m_N^2)}}\langle R,+\frac{3}{2}|J_{em}\cdot \varepsilon^{(R)}| N,+\frac{1}{2}\rangle , \\[4mm] \displaystyle S_{1/2} = \sqrt{\frac{\pi\alpha}{m_N(W^2-m_N^2)}} \frac{q_z}{\sqrt{Q^2}} \langle R,+\frac{1}{2}|J_{em}\cdot \varepsilon^{(S)}| N,+\frac{1}{2}\rangle , \end{array} \label{helic-elem} \end{equation} which we calculate in this article. Following standard rules for the calculation of the expectation values the signs in these equations are determined. There is an ambiguity for the sign of the square root, which we select for all resonances to be positive. Later on we must also select the sign for axial form factors. We shall choose them in such a way that the structure functions $W_3$ for all resonances are positive, as indicated or suggested by the data. As a consequence the neutrino induced cross sections are larger than the corresponding antineutrino cross sections. \section{Isospin relations between electromagnetic and weak vertices \label{isospin-relat}} Our aim is to relate the electromagnetic to weak form factors using isotopic symmetry. The photon has two isospin components $|I,I_3\rangle=|1,0\rangle$ and $|0,0\rangle$. The isovector component belongs to the same isomultiplet with the vector part of the weak current. Each of the amplitudes $A_{3/2}$, $A_{1/2}$, $S_{1/2}$ can be further decomposed into three isospin amplitudes. Let us use a general notation and denote by $b$ the contribution from the isoscalar photon; similarly $a^1$ and $a^3$ denote contributions of isovector photon to resonances with isospin $1/2$ and $3/2$, respectively. A general helicity amplitude on a proton ($A_p$) and neutron ($A_n$) target has the decomposition \begin{eqnarray} \displaystyle A_{p}&=&A_{p}(\gamma p \to R^{+}) = b-\sqrt{\frac13}a^{1} + \sqrt{\frac23} a^3, \nonumber \\[4mm] \displaystyle A_{n}&=&A_{n}(\gamma n \to R^{0}) = b+\sqrt{\frac13}a^{1} + \sqrt{\frac23} a^3 . \label{p1} \end{eqnarray} For the weak current we have only an isovector component of the vector current, therefore the $b$ amplitude never occurs in weak interactions. A second peculiarity of the charged current is that $V_1\pm i V_2$ does not have the normalization for the Clebsch--Gordon coefficients, it must be normalized as $(V_1\pm i V_2)/\sqrt{2}$, which brings an additional factor of $\sqrt{2}$ to each of the charged current in comparison with the Clebsch--Gordon coefficients: \begin{equation} A(W^+ n \to R^{(1)+}) = \frac{2}{\sqrt{3}}a^1, \label{w1} \end{equation} \begin{equation} \begin{array}{c} \displaystyle A(W^+ p \to R^{(3)++}) = \sqrt{2}a^3, \\[4mm] \displaystyle A(W^+ n \to R^{(3)+}) = \sqrt{\frac23}a^3 . \end{array} \label{w3} \end{equation} Comparing (\ref{p1}) with (\ref{w1}), one easily sees, that, for the isospin-1/2 resonances, the weak amplitude satisfies the equality $A(W^+ n \to R^{(1)+}) = A_n^{1} - A_p^{1}$. Since the amplitudes are linear functions of the form factors, the weak vector form factors are related in the same way to electromagnetic form factors for neutrons $C_i^n$ and protons $C_i^p$: \begin{equation} I=1/2: \quad C_i^V=C_i^{n}-C_i^{p}, \label{iV-1/2} \end{equation} with index $i$ distinguishing the Lorenz structure of the form factors. For the isospin-3/2 resonances one gets $A_n^3(W^+ n \to R^{(3)+}) = A_p^3(W^- p \to R^{(3)0})=\sqrt{2/3}a^3$. The weak form factors, which are conventionally specified for these two processes, are \begin{equation} I=3/2: \quad C_i^V=C_i^{p}=C_i^{n}. \label{iV-3/2} \end{equation} For the process $W^+ p \to R^{(3)++}$ the amplitude is $\sqrt{3}$ times bigger: $A(W^+ p \to R^{(3)++})=\sqrt{3}A(W^+ n \to R^{(3)+})$. Some of the above relations were explicitly used in earlier articles \cite{Schreiner:1973mj,Paschos:2003qr,Lalakulich:2005cs}. \section{Matrix elements of the resonance production, form factors \label{resonances}} Following the notation of our earlier article \cite{Lalakulich:2005cs}, we write the cross section for resonance production in a notation close to that of DIS, that is we express it in terms of the hadronic structure functions. In the present notation the sign in front of ${\cal W}_3$ in the cross section (\ref{cross-sec}) is plus, \begin{eqnarray} \displaystyle \frac{d\sigma^{\nu N}}{dQ^2 dW}=\frac{G^2}{4\pi}\cos^2\theta_C\frac{W}{m_N E^2} \Biggl\{ {\cal W}_1(Q^2+m_\mu^2) \nonumber \\ \displaystyle +\frac{{\cal W}_2}{m_N^2}\left[ 2({k \cdot p})({k' \cdot p}) - \frac12 m_N^2 (Q^2+m_\mu^2)\right] \nonumber \\ \displaystyle \nonumber +\frac{{\cal W}_3}{m_N^2} \left[ Q^2{k \cdot p} - \frac12{q\cdot p} (Q^2+m_\mu^2) \right] \\ \displaystyle +\frac{{\cal W}_4}{m_N^2}m_\mu^2 \frac{(Q^2+m_\mu^2)}{2} -2\frac{{\cal W}_5}{m_N^2} m_\mu^2 ({k \cdot p}) \Biggr\} , \label{cross-sec} \end{eqnarray} which implies that cross section for a reaction with neutrino exceeds that with antineutrino if the structure function ${\cal W}_3$ is positive. The corresponding formula for electroproduction is obtained by replacing the overall factor $G^2\cos^2\theta_C/4\pi$ by $2\pi \alpha^2/Q^4$ and using the electromagnetic structure functions ${\cal W}_1^{em}$ and ${\cal W}_2^{em}$ instead of weak ones. In this case the contribution from ${\cal W}_4^{em}$ and ${\cal W}_5^{em}$ is negligible and ${\cal W}_3^{em}=0$. \begin{widetext} \begin{table}[hb] \arraycolsep=3mm \doublerulesep=1mm \caption{Vector and axial couplings for the excitation of resonances ($Q^2=0$)} \[ \begin{array}{ccccccccccc} \hline R & M_R, \; {\mathrm{GeV}} & \Gamma_{tot}, \; {\mathrm{GeV}} & \mbox{elast} & g_{\pi N R} & \mbox{} & C_5^A & \mbox{} & C_3^V & C_4^V & C_5^V \\ \hline P_{33}(1232) & 1.232 & 0.120 & 0.995 & 15.3 & \mbox{} & 1.2 & \mbox{} & 2.13 & -1.51 & 0.48 \\[1mm] D_{13}(1520) & 1.520& 0.125 & 0.5 & 19.0 & \mbox{} & -2.1 & \mbox{} & -4.08 & 1.51 & 0.31 \\[4mm] \hline R & M_R, \; {\mathrm{GeV}} & \Gamma_{tot}, \; {\mathrm{GeV}} & \mbox{elast} & g_{\pi N R} & \mbox{} & g_1^A & \mbox{} & g_1^V & g_2^V & \\ \hline P_{11}(1440) & 1.440 & 0.350 & 0.6 & 10.9 & \mbox{} & -0.51 & \mbox{} & -4.6 & 1.52 \\[1mm] \SS & 1.535 & 0.150 & 0.4 & 1.12 & \mbox{} & -0.21 & \mbox{} & -4.0 & -1.68 \\ \hline \end{array} \] \label{res-couplings} \end{table} \end{widetext} In the following subsections we specify the structure functions ${\cal W}_i$ for each resonance by relating them to form factors. To give the reader a quick overview of the results, we summarize the couplings (the values of the form factor at $Q^2=0$) in Table~\ref{res-couplings}. One should keep in mind, however, that all form factors have different $Q^2$--dependences, which are given explicitly in the following subsections. \subsection{Resonance $D_{13}(1520)$} We begin with a $D_{13}$ resonance, which has spin-3/2 and negative parity. The matrix element of the charged current for the resonance production is expressed as \begin{equation} \langle D_{13} | J^\nu | N \rangle = \bar \psi_\lambda^{(D)} ({p'}) d^{\lambda\nu}_{D_{13}} u(p) \label{d13} \end{equation} with $u(p)$ the spinor of the target and $\psi_\lambda^{(D)}$ the Rarita-Schwinger field for a $D_{13}$ resonance. The structure of $d^{\lambda\nu}_D$ is given in term of the weak form factors \begin{widetext} \begin{eqnarray} \displaystyle d^{\lambda\nu}_{D_{13}}&=& \displaystyle g^{\lambda\nu}\left[ \frac{C_3^V}{m_N} \slash{q} + \frac{C_4^V}{m_N^2}(p'q) + \frac{C_5^V}{m_N^2} (pq) + C_6^V \right] -q^\lambda \left[ \frac{C_3^V}{m_N} \gamma^\nu + \frac{C_4^V}{m_N^2} p'{}^\nu + \frac{C_5^V}{m_N^2} p^\nu \right] \nonumber \\[5mm] \hspace*{12mm} \displaystyle &+& \displaystyle g^{\lambda\nu}\left[ \frac{C_3^A}{m_N} \slash{q} + \frac{C_4^A}{m_N^2} (p'q) \right] \gamma_5 - q^\lambda \left[ \frac{C_3^A}{m_N} \gamma^\nu + \frac{C_4^A}{m_N^2} p'{}^\nu \right] \gamma_5 + \left[ g^{\lambda\nu} C_5^A + q^\lambda q^\nu \frac{C_6^A}{m_N^2} \right] \gamma_5. \label{dlanu-d13} \end{eqnarray} \end{widetext} The general form of the current for $D_{13}$ differs from that of $P_{33}$ in the location of the $\gamma_5$ matrix, which is due to the parity of the resonance. The form factors $C^V_i(Q^2)$ and $C^A_i(Q^2)$ refer now to any $D_{13}$ resonance. Later we'll specify them for the $D_{13}(1520)$. The vector form factors are extracted from the electroproduction data, in particular from the helicity amplitudes. We use recent data from \cite{Aznauryan:2004jd,Aznauryan:2005oral}, which were kindly provided to us by I. Aznauryan. Helicity amplitudes are expressed via the $J_{em}\cdot \varepsilon$, which can be obtained from (\ref{d13}) and (\ref{dlanu-d13}) by setting the axial couplings equal to zero and replacing the vector form factors by the electromagnetic form factors. This results in the following expression \begin{eqnarray} \displaystyle \langle D_{13}| J_{em}\cdot \varepsilon |N \rangle &=& \bar{\psi}_{\mu} \Gamma_\nu^{(em)} \, F^{\mu\nu} u(N), \nonumber \\[2mm] \mbox{with} \qquad \Gamma_\nu^{(em)}&=&\frac{C_3^{(em)}}{m_N}\gamma_{\nu} +\frac{C_4^{(em)}}{m_N^2} p^{\prime}_{\nu}+\frac{C_5^{(em)}}{m_N^2}p_{\nu} \nonumber \\[2mm] F^{\mu\nu}&=& q^\mu\varepsilon^\nu - q^\nu\varepsilon^\mu \label{vertexD13V} \end{eqnarray} As it was discussed in the previous section, the electromagnetic form factors of $D_{13}$ resonance are different for proton and neutron. Substituting the matrix element (\ref{vertexD13V}) into Eqs. (\ref{helic-elem}) and carrying out the products with the spinors and Rarita--Schwinger field we obtain the following helicity amplitudes for electroproduction \begin{eqnarray} \displaystyle A_{3/2}^{D_{13}}&=& \sqrt{N} \biggl[ \frac{C_3^{(em)}}{m_N}(M_R-m_N) \nonumber \\[1mm] \displaystyle &+&\frac{C_4^{(em)}}{m_N^2}{q \cdot p'} +\frac{C_5^{(em)}}{m_N^2}{q\cdot p} \biggr] \label{D1520-A32} \end{eqnarray} \begin{eqnarray} \displaystyle A_{1/2}^{D_{13}}&=&\sqrt{\frac{N}{3}} \biggl[ \frac{C_3^{(em)}}{m_N}(M_R-m_N -\frac{2m_N}{M_R}\frac{q_z^2}{{p'}^0+M_R}) \nonumber \\[1mm] \displaystyle &+&\frac{C_4^{(em)}}{m_N^2}{q \cdot p'} +\frac{C_5^{(em)}}{m_N^2}{q\cdot p} \biggr] \phantom{ccccccc} \label{D1520-A12} \end{eqnarray} \begin{eqnarray} \displaystyle S_{1/2}^{D_{13}}&=&\sqrt{\frac{2N}{3}}\frac{q^z}{M_R} \biggl[ \frac{C_3^{(em)}}{m_N}\left( -M_R \right) +\frac{C_4^{(em)}}{m_N^2}(Q^2 \nonumber \\[1mm] \displaystyle &-&2m_N q^0-m_N^2) -\frac{C_5^{(em)}}{m_N}(q^0+m_N) \biggr] \label{D1520-S12} \end{eqnarray} where $\displaystyle N= \frac{\pi\alpha_{em}}{m_N(W^2-m_N^2)}2m_N(p'{}^0+M_R)$. \begin{figure}[ht] \begin{center} \includegraphics[angle=-90,width=\columnwidth]{D1520-AS.ps} \end{center} \caption{Helicity amplitudes for the $D_{13}(1520)$ resonance, calculated with the form factors from Eq.(\ref{ff-D1520}).} \label{fig-D1520-AS} \end{figure} Comparing Eqs. (\ref{D1520-A32}), (\ref{D1520-A12}), (\ref{D1520-S12}) for each value of $Q^2$ with the recent data on helicity amplitudes \cite{Burkert:2002zz,Aznauryan:2004jd,Aznauryan:2005oral}, we extract the following form factors: \begin{equation} \begin{array}{cl} \displaystyle D_{13}(1520): \quad & \displaystyle C_3^{(p)}=\frac{2.95/D_V}{1+Q^2/8.9 M_V^2}, \\ & \displaystyle C_4^{(p)}=\frac{-1.05/D_V}{1+Q^2/8.9 M_V^2}, \\ & \displaystyle C_5^{(p)}=\frac{-0.48}{D_V} . \\[4mm] & \displaystyle C_3^{(n)}=\frac{-1.13/D_V}{1+Q^2/8.9 M_V^2}, \\ & \displaystyle C_4^{(n)}=\frac{0.46/D_V}{1+Q^2/8.9 M_V^2} \\ & \displaystyle C_5^{(n)}=\frac{-0.17}{D_V} , \end{array} \label{ff-D1520} \end{equation} This is a simple algebraic solution with the numerical values for the form factors being unique. The function $D_V=(1+Q^2/M_V^2)^2$ denotes the dipole function with the vector mass parameter $M_V=0.84\; {\mathrm{GeV}}$. To give an impression, how good this parametrisation is, we plot in Fig.\ref{fig-D1520-AS} the helicity amplitudes, obtained with these form factors. Vector form factors for the neutrino--nucleon interactions are calculated according to Eq. (\ref{iV-1/2}) For the axial form factors we derive in the Appendix~\ref{gold-trei} \begin{equation} C_6^A{}^{(D)}=m_N^2 \frac{C_5^A{}^{(D)}}{m_\pi^2+Q^2}, \quad C_5^A{}^{(D)}(0)=-2.1 \label{c6a-D13} \end{equation} Two other form factors and the $Q^2$ behaviour of the $C_5^A$ can be determined either experimentally or in a specific theoretical model. To check how big the contribution of the $C_3^A$ and $C_4^A$ could be, we set them $C_3^A=C_4^A=1$ and computed in Fig.\ref{fig-difrFF} the various contributions to the differential cross section for $E_\nu=2\; {\mathrm{GeV}}$. Motivated by the results on $P_{33}(1232)$ resonance \cite{Paschos:2003qr}, the $Q^2$ dependence in our calculations is taken as \begin{equation} C_i^A{}^{(D)} = \frac{C_i^A{}^{(D)}(0)/D_A}{1+Q^2/3M_A^2}, \end{equation} where $D_A=(1+Q^2/M_A^2)^2$ denotes the dipole function with the axial mass parameter $M_A=1.05 \; {\mathrm{GeV}}$. \begin{figure}[ht] \includegraphics[angle=-90,width=\columnwidth]{D13a.ps} \caption{Contribution of various form factors to the differential cross section.} \label{fig-difrFF} \end{figure} We conclude from Figure \ref{fig-difrFF}, that the contribution of $(C_4^A)^2$ and $C_4^AC_5^A$ are small, but the other terms could be sizable. Their importance depends on the relative signs. It is possible, that $C_3^A C_4^A$ and $C_3^A C_5^A$ are positive and together with $(C_3^A)^2$ give an additional $100\%$ to the $D_{13}$ cross section. In case that $C_3^A C_4^A$ and/or $C_3^A C_5^A$ are negative, there are cancellations. In the following calculations of this article we set for simplicity $C_3^A=C_4^A=0$. The hadronic structure functions for $D_{13}$ resonance are similar to those for $P_{33}$, presented in our earlier paper \cite{Lalakulich:2005cs} and can be obtained from them by replacing $m_N M_R$ by $-m_NM_R$. We repeat the corresponding formulas here including the terms with $C_5^V$ and $C_3^A$, which could be nonzero. The structure functions have a form \begin{equation} {\cal W}_i(Q^2,\nu)=\frac{2}{3m_N} V_i(Q^2,\nu) R(W,M_R) \end{equation} where $R(W,M_R)$ was defined in (\ref{delta-function}) and $V_i$ are given below. In the following equations the upper sign corresponds to $P_{33}$ resonance and the lower sign to the $D_{13}$. \begin{widetext} \begin{eqnarray} \displaystyle \nonumber V_1 &=& \frac{(C_3^V)^2}{m_N^2 M_R^2} \left[(Q^2-{q\cdot p})^2({q\cdot p}+m_N^2) + M_R^2( ({q\cdot p})^2 +Q^2 m_N^2 \pm Q^2 m_N M_R) \right] \\[3mm] \displaystyle \nonumber &+& \frac{(C_3^A)^2}{m_N^2 M_R^2} \left[(Q^2-{q\cdot p})^2({q\cdot p}+m_N^2) + M_R^2(({q\cdot p})^2+Q^2 m_N^2 \mp Q^2 m_N M_R) \right] \\[3mm] \displaystyle \nonumber &+& \frac{C_3^V C_4^V (Q^2-{q\cdot p}) - C_3^V C_5^V {q\cdot p}}{m_N^3 M_R} \left[ (Q^2-{q\cdot p})({q\cdot p} +m_N^2 \mp 2m_N M_R)- M_R^2 {q\cdot p} \right] \\[3mm] \displaystyle &+& \frac{C_3^A C_4^A (Q^2-{q\cdot p}) - C_3^A C_5^A m_N^2}{m_N^3 M_R} \left[ (Q^2-{q\cdot p}) \left({q\cdot p} + (M_R\pm m_N)^2 \right) - M_R^2 Q^2 \right] \\[3mm] \displaystyle &+& \frac{\left[ C_4^V (Q^2-{q\cdot p}) - C_5^V {q\cdot p} \right]^2 }{m_N^4} ({q\cdot p}+m_N^2 \mp m_N M_R) +\left[ C_5^A - \frac{ C_4^A (Q^2-{q\cdot p})}{m_N^2} \right]^2 \left[ {q\cdot p} +m_N^2 \pm m_N M_R \right] \nonumber \label{calW1} \end{eqnarray} \begin{eqnarray} \displaystyle V_2 &=& \frac{(C_3^V)^2 + (C_3^A)^2}{M_R^2} Q^2 \left[ {q\cdot p} +m_N^2 +M_R^2 \right] +\frac{C_3^V C_4^V}{m_N M_R} Q^2 \left[ {q\cdot p} + (M_R \mp m_N)^2 \right] +\frac{C_3^A C_4^A}{m_N M_R} Q^2 \left[ {q\cdot p} + (M_R \pm m_N)^2 \right] \nonumber \\[3mm] \displaystyle &+&\frac{C_3^V C_5^V}{m_N M_R } Q^2 \left[ {q\cdot p} +(M_R\mp m_N)^2 +Q^2\right] +\left( \frac{(C_4^V)^2}{m_N^2} + \frac{(C_5^V)^2(Q^2+M_R^2)}{m_N^2 M_R^2} +\frac{2 C_4^V C_5^V}{m_N^2} \right)Q^2 \left[ {q\cdot p} +m_N^2 \mp m_N M_R \right] \nonumber \\[3mm] \displaystyle &+& C_3^A C_5^A \frac{m_N}{M_R} Q^2 +\left[ ({C_5^A})^2 \frac{m_N^2}{M_R^2} + \frac{(C_4^A)^2}{m_N^2} Q^2 \right] \left[ {q\cdot p}+m_N^2 \pm m_N M_R \right] \label{calW2} \end{eqnarray} \begin{eqnarray} \displaystyle V_3 &=& \displaystyle 2\frac{C_3^V C_3^A}{M_R^2} \left[ 2(Q^2-{q\cdot p})^2 +M_R^2(3Q^2-4{q\cdot p}) \right] +2\left[ \frac{C_4^V C_4^A}{m_N^2}(Q^2 -{q\cdot p}) -C_4^V C_5^A \right] (Q^2-{q\cdot p}) \nonumber \\[3mm] \displaystyle &+& 2\frac{C_5^V C_3^A {q\cdot p} - C_4^V C_3^A(Q^2-{q\cdot p})}{M_R m_N} \left[2M_R^2 \mp 2 m_N M_R +Q^2 -{q\cdot p} \right] + 2\left[ C_5^V C_5^A -\frac{C_5^V C_4^A}{m_N^2}(Q^2-{q\cdot p}) \right] {q\cdot p} \nonumber \\[3mm] \displaystyle & + & 2 \left[C_3^V C_5^A \frac{m_N}{M_R} - \frac{C_3^V\, C_4^A}{M_R m_N} (Q^2-{q\cdot p}) \right] \left(2 M_R^2 \pm 2 m_N M_R + Q^2 - {q\cdot p} \right) \label{calW3} \end{eqnarray} These are the important structure functions for most of the kinematic region. There are two additional structure functions, whose contribution to the cross section is proportional to the square of the muon mass. \begin{eqnarray} \displaystyle \nonumber V_4 &=& \frac{(C_3^V)^2}{M_R^2} \left[ (2{q\cdot p}-Q^2)({q\cdot p}+m_N^2) -M_R^2(m_N^2 \pm m_N M_R) \right] \\[3mm] \nonumber \displaystyle &+& \frac{(C_3^A)^2}{M_R^2} \left[ (2{q\cdot p}-Q^2)({q\cdot p}+m_N^2) -M_R^2(m_N^2 \mp m_N M_R) \right] \\[3mm] \nonumber \displaystyle &+&\left[ \frac{(C_4^V)^2 (2{q\cdot p}-Q^2) }{m_N^2} + \frac{(C_5^V)^2 ({q\cdot p})^2 }{m_N^2 M_R^2} + 2\frac{C_4^V C_5^V}{m_N^2} {q\cdot p} \right] \left[ {q\cdot p} +m_N^2 \mp m_N M_R \right] \\[3mm] \nonumber \displaystyle &+& \frac{C_3^V C_4^V}{m_N M_R} \left[ (2{q\cdot p}-Q^2)({q\cdot p} +m_N^2 \mp 2 m_N M_R)+ {q\cdot p} M_R^2 \right] +\frac{C_3^V C_5^V}{m_N M_R} pq \bigl[2{q\cdot p} +m_N^2 \mp 2 m_N M_R +M_R^2 +Q^2\bigr] \\[3mm] \nonumber \displaystyle &+& \left[ (C_5^A)^2 \frac{m_N^2}{M_R^2} + \frac{(C_4^A)^2}{m_N^2}(2{q\cdot p}-Q^2) +\frac{(C_6^A)^2}{m_N^2 M_R^2}\biggl((Q^2-{q\cdot p})^2+Q^2M_R^2\biggr) \right. \\[3mm] \displaystyle \nonumber & & \hspace*{25mm} \left. +2 C_4^A C_5^A -2\frac{C_4^A C_6^A}{m_N^2}{q\cdot p} -2\frac{C_5^A C_6^A}{M_R^2}(M_R^2+Q^2-{q\cdot p}) \right] \left[{q\cdot p} +m_N^2 \pm m_N M_R \right] \\[3mm] \displaystyle \nonumber &+& \frac{C_3^A C_4^A}{m_N M_R} \left[ (2{q\cdot p} - Q^2)({q\cdot p}+m_N^2 \pm 2m_N M_R) + M_R^2 {q\cdot p} \right] \\[3mm] \displaystyle &+& C_3^A C_5^A\frac{m_N}{ M_R} (2{q\cdot p}+m_N^2 \pm 2m_N M_R) -\frac{C_3^A C_6^A}{m_N M_R} {q\cdot p} \biggl(Q^2 +(M_R\pm m_N)^2 \biggr) \end{eqnarray} \begin{eqnarray} \nonumber V_5 &=& \frac{(C_3^V)^2 + (C_3^A)^2}{M_R^2}{q\cdot p} \left[ {q\cdot p} +m_N^2+ M_R^2 \right] +\frac{C_3^V C_5^V}{m_N M_R} {q\cdot p} \left[ {q\cdot p} + (M_R \mp m_N)^2 +Q^2 \right] \\[3mm] \displaystyle \nonumber &+&\left[ \frac{(C_4^V)^2}{m_N^2} + \frac{(C_5^V)^2(Q^2+M_R^2)}{m_N^2 M_R^2} + 2\frac{C_4^V C_5^V}{m_N^2} \right] {q\cdot p} \left[ {q\cdot p} +m_N^2 \mp m_N M_R \right] +\frac{C_3^V C_4^V}{m_N M_R} {q\cdot p} \left[ {q\cdot p} + (M_R \mp m_N)^2 \right] \\[3mm] \displaystyle \nonumber &+&\frac{C_3^A C_4^A}{m_N M_R} {q\cdot p} \left[ {q\cdot p}+ (M_R\pm m_N)^2 \right] +C_3^A C_5^A\frac{m_N}{2M_R} \left[ 2{q\cdot p}+ Q^2+ (M_R\pm m_N)^2 \right] +\frac{C_3^A C_6^A}{2 m_N M_R} Q^2 \left[ Q^2+ (M_R\pm m_N)^2 \right] \\[3mm] \displaystyle &+&\left[ \frac{(C_4^A)^2}{m_N^2}{q\cdot p} + (C_5^A)^2 \frac{m_N^2}{M_R^2} +C_4^A C_5^A - \frac{C_4^A C_6^A}{m_N^2}Q^2 + \frac{C_5^A C_6^A}{M_R^2}({q\cdot p} - Q^2) \right] \left[ {q\cdot p} +m_N^2 \pm m_N M_R \right] \label{calW5} \end{eqnarray} \end{widetext} \subsection{Resonance $P_{33}(1232)$} The method of extracting the form factors from the helicity amplitudes, described in the previous section is applicable to any resonance. The helicity amplitudes for the $P_{33}(1232)$ resonance were calculated in a similar manner and we obtain the following results: \begin{eqnarray} \displaystyle A_{3/2}^{P_{33}}&=&-\sqrt{N}\frac{q^z}{{p'}^0+M_R}\left[ \frac{C_3^{(em)}}{m_N}(m_N+M_R) \right. \nonumber \\[4mm] \displaystyle &+&\left. \frac{C_4^{(em)}}{m_N^2}{q \cdot p'} +\frac{C_5^{(em)}}{m_N^2}{q\cdot p} \right] \label{P1232-A32} \end{eqnarray} \begin{eqnarray} \displaystyle A_{1/2}^{P_{33}}&=&\sqrt{\frac{N}{3}} \biggl[ \frac{C_3^{(em)}}{m_N}\bigl(m_N+M_R -2\frac{m_N}{M_R}({p'}^0+M_R)\bigr) \nonumber \\[4mm] \displaystyle &+&\frac{C_4^{(em)}}{m_N^2}{q \cdot p'} +\frac{C_5^{(em)}}{m_N^2}{q\cdot p} \biggr] \frac{q^z}{{p'}^0+M_R} \label{P1232-A12} \end{eqnarray} \begin{eqnarray} \displaystyle S_{1/2}^{P_{33}}&=&\sqrt{\frac{2N}{3}}\frac{q_z^2}{M_R({p'}^0+M_R)} \biggl[ \frac{C_3^{(em)}}{m_N}M_R \nonumber \\[4mm] \displaystyle &+&\frac{C_4^{(em)}}{m_N^2} W^2 +\frac{C_5^{(em)}}{m_N^2}m_N(m_N+q^0) \biggr] \phantom{cccc} \label{P1232-S12} \end{eqnarray} Comparing helicity amplitudes from Eqs. (\ref{P1232-A32}),(\ref{P1232-A12}),(\ref{P1232-S12}) with the available data \cite{Tiator:2003uu} allows us to extract the form factors \begin{eqnarray} C_3^{(p)}=\frac{2.13/D_V}{1+Q^2/4 M_V^2}, \nonumber \\ C_4^{(p)}=\frac{-1.51/D_V}{1+Q^2/4 M_V^2}, \label{ff-P1232} \\ C_5^{(p)}=\frac{0.48/D_V}{1+Q^2/0.776 M_V^2} . \nonumber \end{eqnarray} Form factors $C_3^{(p)}$ and $C_4^{(p)}$ are in agreement with those obtained in the magnetic dominance approximation (which was used in all the previous papers on neutrinoproduction). The agreement has $5\%$ accuracy and at the same time the nonzero scalar helicity amplitude is described correctly. The fit of the proton helicity amplitudes for the form factors from Eq.(\ref{ff-P1232}) is shown in Fig.\ref{fig-P1232-AS}. \begin{figure}[ht] \begin{center} \includegraphics[angle=-90,width=\columnwidth]{P1232-AS.ps} \end{center} \caption{Helicity amplitudes for the $P_{33}(1232)$ resonance, calculated with the form factors from Eq.(\ref{ff-P1232}). Data are from \cite{Tiator:2003uu}.} \label{fig-P1232-AS} \end{figure} Electromagnetic neutron form factors and vector form factors for the neutrino--nucleon interactions can be calculated according to Eq. (\ref{iV-3/2}). Axial form factors have already been discussed several times, the way to obtain them is illustrated in Appendix~\ref{gold-trei}, Eq. (\ref{P1232-C5A}). The result is \[ C_6^A{}^{\Delta}=m_N^2 \frac{C_5^A{}^{\Delta}}{m_\pi^2+Q^2}, \quad C_5^A{}^{\Delta}=\frac{1.2/D_A}{1+Q^2/3M_A^2} \] A practical aspect with this resonance concerns the cross section of the tau neutrino interactions, which is discussed in Section \ref{tau}. \subsection{Resonance $P_{11}(1440)$} For spin-1/2 resonances the parametrization of the weak vertex for the resonance production is simpler than for the spin-3/2 resonances and is similar to the parametrization for quasi--elastic scattering. The matrix elements of the $P_{11}$ resonance production can be written as follows: \begin{eqnarray} \displaystyle \langle P_{11} | J^\nu | N\rangle =\bar{u}(p') \left[ \frac{g_1^V}{\mu^2}(Q^2\gamma^\nu + \slash{q} q^\nu) \right. \nonumber \\ \left. \displaystyle + \frac{g_2^V}{\mu} i \sigma^{\nu\rho} q_\rho - g_1^A \gamma^\nu \gamma_5 - \frac{g_3^A}{m_N} q^\nu \gamma_5\right] u(p), \label{meP1440} \end{eqnarray} where we use the standard notation $\sigma^{\nu\rho} = \frac{i}{2}[\gamma^\nu, \gamma^\rho]$ and the kinematic factors are scaled by $\mu=m_N+M_R$ in order to make them dimensionless. To extract the vector form factors we use the same procedure as before and calculate the helicity amplitudes for the virtual photoproduction process. Since the resonance has spin $1/2$, only the $A_{1/2}$ and $S_{1/2}$ amplitudes occur: \begin{eqnarray} \displaystyle A^{P_{11}}_{1/2}=\sqrt{N} \frac{\sqrt{2} q^z}{p'{}^0+M_R} \left[ \frac{g_1^{(em)}}{\mu^2}Q^2 \right. \nonumber \\ \displaystyle \left. +\frac{g_2^{(em)}}{\mu}(M_R+m_N)\right] \label{P1440-A12} \end{eqnarray} \begin{eqnarray} \displaystyle S^{P_{11}}_{1/2}=\sqrt{N} \frac{q_z^2}{p'{}^0+M_R} \left[\frac{g_1^{(em)}}{\mu^2}(M_R+m_N) \right. \nonumber \\ \displaystyle \left. -\frac{g_{2}^{(em)}}{\mu} \right] \label{P1440-S12} \end{eqnarray} At nonzero $Q^2$ data on helicity amplitudes for the $P_{11}(1440)$ are available only for proton. Unlike the other resonances, the accuracy of data in this case is low and numerical values, provided by different groups differ significantly, as illustrated in Fig.\ref{fig-P1440-AS}. In this situation we use for our fit only the recent data \cite{Aznauryan:2004jd, Aznauryan:2005oral}. \begin{figure}[b!ht] \begin{center} \includegraphics[angle=-90,width=\columnwidth]{P1440-AS.ps} \end{center} \caption{Helicity amplitudes for the $P_{11}(1440)$ resonance, calculated with the form factors from Eq.(\ref{ff-P1440}). For $A_{1/2}$ the ata are from: \cite{Tiator:2003uu} (unshaded circles), \cite{Aznauryan:2004jd} (unshaded pentagons), \cite{Aznauryan:2005oral} (full circles); for $S_{1/2}$: \cite{Tiator:2003uu} (unshaded up triangles), \cite{Aznauryan:2004jd} (unshaded down triangles), \cite{Aznauryan:2005oral} (full triangles)} \label{fig-P1440-AS} \end{figure} Matching the data against Eqs. (\ref{P1440-A12}), (\ref{P1440-S12}) allows us to parametrize the proton electromagnetic form factors as follows: \begin{equation} \begin{array}{ll} \displaystyle P_{11}(1440): & \displaystyle g_1^{(p)} = \frac{2.3/D_V}{1+Q^2/4.3 M_V^2}, \\ & \displaystyle g_2^{(p)} = \frac{-0.76}{D_V} \left[1 - 2.8 \ln\left(1+\frac{Q^2}{1\; {\mathrm{GeV}}^2}\right) \right]\ . \end{array} \label{ff-P1440} \end{equation} The difference among the reported helicity amplitudes are larger than the estimated contribution of the isoscalar part of the electromagnetic current. For this reason we shall assume that the isoscalar contribution is negligibly small and use the relation $A^{(n)}_{1/2}=-A^{(p)}_{1/2}$. The isovector contribution in the neutrinoproduction is now given as $g_i^V=-2g_i^{(p)}$. The differential cross section is expressed again with the general formula (\ref{cross-sec}). The hadronic structure functions are calculated explicitly to be: \[ {\cal W}_i(Q^2,\nu)=\frac{1}{m_N} V_i(Q^2,\nu) R(W,M_R) \] \begin{eqnarray} \displaystyle V_1&=& \frac{(g_1^V)^2}{\mu^4} Q^4 \left[ (pq+m_N^2\mp m_N M_R) \right] \nonumber \\ \displaystyle &+&\frac{(g_2^V)^2}{\mu^2} \left[ 2(pq)^2+Q^2(m_N^2\pm m_N M_R-{q\cdot p}) \right] \nonumber \\ \displaystyle &+&\frac{g_1^V g_2^V}{\mu^3} 2 Q^2 \left[ (pq)(M_R\mp m_N) \pm m_N Q^2 \right] \nonumber \\ \displaystyle &+& (g_1^A)^2(m_N^2\pm m_N M_R+{q\cdot p}) \phantom{cccccc} \end{eqnarray} \begin{equation} V_2=2 m_N^2 \left[\frac{(g_1^V)^2}{\mu^4}Q^4 + \frac{(g_2^V)^2}{\mu^2}Q^2 + (g_1^A)^2 \right] \end{equation} \begin{equation} V_3= 4 m_N^2 \left[ \frac{g_1^V g_1^A}{\mu^2}Q^2 +\frac{g_2^V g_1^A}{\mu}(M_R\pm m_N) \right] \end{equation} \begin{eqnarray} \displaystyle V_4 & = & m_N^2\biggl[ \frac{(g_2^V)^2}{\mu^2} \left[ {q\cdot p} -m_N^2 \mp m_N M_R \right] \nonumber \\ \displaystyle &+&\frac{(g_1^V)^2}{\mu^4} \left[ 2(pq)^2 - Q^2(pq+m_N^2\mp m_N M_R) \right] \nonumber \\ \displaystyle &-&\frac{g_1^V g_2^V}{\mu^3} \left[ {q\cdot p}(M_R \mp m_N) \pm m_N Q^2 \right] \\ \displaystyle &\mp& 2 g_1^A g_3^A + \frac{(g_3^A)^2}{m_N^2} \left[ (pq) + m_N^2 \mp m_N M_R \right] \biggr] \nonumber \end{eqnarray} \begin{eqnarray} \displaystyle V_5&=& m_N^2 \biggl[ 2\frac{(g_1^V)^2}{\mu^4}Q^2 {q\cdot p} +2\frac{(g_2^V)^2}{\mu^2}{q\cdot p} + \nonumber \\ \displaystyle &+&(g_1^A)^2 + \frac{g_1^A g_3^A}{m_N}(M_R \mp m_N) \biggr] \end{eqnarray} where the upper sign corresponds to $P_{11}$ and the lower sign to $S_{11}$ resonance. As it is shown in Appendix~\ref{gold-trei}, PCAC allows us to relate the two axial form factors and fix their values at $Q^2=0$: \[ g_3^A{}^{(P)}=\frac{(M_R+m_N)m_N}{Q^2+m_\pi^2} g_1^A{}^{(P)}, \quad g_1^A{}^{(P)}(0)=-0.51 . \] The $Q^2$ dependence of the form factors cannot be determined by general theoretical consideration and will have to be extracted from the experimental data. We again suppose, that the form factors are modified dipoles \begin{equation} g_1^A{}^{(P)}= \frac{g_1^A{}^{(P)}(0)/D_A}{1+Q^2/3M_A^2} . \end{equation} \subsection{Resonance $S_{11}(1535)$} For the $S_{11}$ the amplitude of resonance production is similar to that for $P_{11}$ with the $\gamma_5$ matrix exchanged between the vector and the axial parts \begin{eqnarray} \displaystyle \langle S_{11} | J^\nu | N\rangle =\bar{u}(p') \left[ \frac{g_1^V}{\mu^2}(Q^2\gamma^\nu + \slash{q} q^\nu) \gamma_5 \right. \nonumber \\ \left. \displaystyle + \frac{g_2^V}{\mu} i \sigma^{\nu\rho} q_\rho \gamma_5 - g_1^A \gamma^\nu - \frac{g_3^A}{m_N} q^\nu \right] u(p) . \phantom{cccccc} \label{meS1535} \end{eqnarray} The helicity amplitudes \begin{eqnarray} A^{S_{11}}_{1/2}=\sqrt{2N} \left[ \frac{g_1^{(em)}}{\mu^2}Q^2 + \frac{g_{2}^{(em)}}{\mu}\left(M_R-m_N\right)\right] \label{S1535-A12} \end{eqnarray} \begin{eqnarray} \nonumber S^{S_{11}}_{1/2}=\sqrt{N} q_z \left[ -\frac{g_1^{(em)}}{\mu^2} \left(M_R-m_N\right) +\frac{g_{2}^{(em)}}{\mu}\right] \label{S1535-S12} \end{eqnarray} are used to extract the electromagnetic form factors. As in the case of $P_{11}(1440)$ resonance, we choose here to fit only proton data on helicity amplitudes and neglect the isoscalar contribution to the electromagnetic current \begin{figure}[ht] \begin{center} \includegraphics[angle=-90,width=\columnwidth]{S1535-AS.ps} \end{center} \caption{Helicity amplitudes for the $P_{11}(1440)$ resonance, calculated with the form factors from Eq.(\ref{ff-S1535}). For $A_{1/2}$ data are from: \cite{Tiator:2003uu} (unshaded circles), \cite{Aznauryan:2004jd} (unshaded pentagons), \cite{Aznauryan:2005oral} (full circles), \cite{Armstrong:1998wg} (unshaded diamond); for $S_{1/2}$: \cite{Tiator:2003uu} (unshaded up triangles), \cite{Aznauryan:2004jd} (unshaded down triangles), \cite{Aznauryan:2005oral} (full triangles) } \label{fig-S1535-AS} \end{figure} We obtain the form factors \begin{equation} \begin{array}{l} \displaystyle S_{11}(1535): \nonumber \\ \displaystyle g_1^{(p)}=\frac{2.0/D_V}{1+Q^2/1.2 M_V^2} \left[1+7.2\ln\left(1+ \frac{Q^2}{1\; {\mathrm{GeV}}^2}\right) \right]\ , \nonumber \\[4mm] \displaystyle g_2^{(p)}=\frac{0.84}{D_V} \left[1 + 0.11 \ln\left(1+\frac{Q^2}{1\; {\mathrm{GeV}}^2}\right) \right]\ . \label{ff-S1535} \end{array} \end{equation} Notice here, that $g_2^p$ falls down slower than dipole (at least for $Q^2<3.5\; {\mathrm{GeV}}^2$), supplying the most prominent contribution among the three isospin-1/2 resonances discussed in this paper. This means experimentally, that the relative role of the second resonance region increases with increasing of $Q^2$. Values of $Q^2=1-2\; {\mathrm{GeV}}^2$ are accessible (and are not suppressed kinematically) for $E_\nu\ge 1.5- 2 \; {\mathrm{GeV}}$. For these energies the $\SS$ and $D_{13}(1520)$ resonances are observable in the differential cross section. The axial form factors are determined from PCAC as described in the Appendix~\ref{gold-trei}: \[ g_3^A{}^{(S)}=\frac{(M_R-m_N)m_N}{Q^2+m_\pi^2} g_1^A{}^{(S)}, \qquad g_1^A{}^{(S)}(0)=-0.21 . \] The $Q^2$ dependence is again taken \begin{equation} g_1^A{}^{(S)}= \frac{g_1^A{}^{(S)}(0)/D_A}{1+Q^2/3M_A^2} . \end{equation} We adopt this functional form, but one must keep open the possibility that it may change when experimental results become available. \section{Cross section for the tau neutrinos \label{tau}} Before we describe numerical results for the cross sections in the second resonance region, we shortly discuss the cross section for $\tau$--neutrinos and the accuracy achieved in different calculations. Recently we calculated the cross section of the resonance production \cite{Lalakulich:2005cs} by taking into account the effects from the nonzero mass of the outgoing leptons. They generally decrease the cross section at small $Q^2$. For the muon the effect is noticeable in the $Q^2-$dependence of the differential cross section, but is very small in the integrated cross section. It was shown, that the cross section is decreased at small $Q^2$ via 1) reduction of the available phase space; 2) nonzero contributions of the ${\cal W}_4$ and ${\cal W}_5$ structure functions. Following \cite{Kuzmin:2004ya}, we'll refer to the latter effect as "dynamic correction". To date, several Monte Carlo Neutrino Simulators use the Rein-Sehgal model \cite{Rein:1980wg} of the resonance production as an input. In this model the lepton mass is not included. Thus, in Monte Carlo simulations the phase space is restricted simply by kinematics, but they do not take into account effects from ${\cal W}_4$ and ${\cal W}_5$ structure function. Some calculations are also available, where the partonic values for the structure functions \begin{equation} {\cal W}_4=0 \quad (a), \quad {\cal W}_5={\cal W}_2\cdot ({q\cdot p}) / Q^2 \quad(b) \label{calW-DIS} \end{equation} are included. We compare these two approximations with our full calculations for the integrated and differential cross sections. Fig.~\ref{fig-P1232-tau-integr} shows the integrated cross section for the reaction $\nu_\tau p \to \mu^- \Delta^{++}$. \begin{figure}[ht] \begin{center} \includegraphics[angle=-90,width=\columnwidth]{tau-tot.ps} \end{center} \caption{The integrated cross section $\sigma(E)$ as function of the neutrino energy. The dashed curve includes phase space corrections only; the dotted curve includes phase space corrections and structure functions from the parton model; the solid curve includes phase space and structure functions calculated in our model.} \label{fig-P1232-tau-integr} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[angle=-90,width=\columnwidth]{tau-Q2.ps} \end{center} \caption{The differential cross section $d\sigma/d Q^2$ for the reaction $\nu_\tau p \to \mu^- \Delta^{++}$ at $E_\nu=5\; {\mathrm{GeV}}$. The meaning of the curves is the same as in Fig.~\ref{fig-P1232-tau-integr}.} \label{fig-P1232-tau-Q2} \end{figure} One can easily see that taking the partonic limit for the structure functions is a good approximation in this case. Ignoring the structure functions, however, leads to a $100\%$ overestimate of the cross section which is inaccurate. In both cases the difference comes mainly from the "low" (close to the threshold) $Q^2$ region, as it is illustrated in Fig.~\ref{fig-P1232-tau-Q2}, where the differential cross section for the tau neutrino energy $E_\nu= 5\; {\mathrm{GeV}}$ is shown. One easily sees, that the discrepancy at low $Q^2$ reaches $30\%$ for the partonic structure functions and more than $100\%$ when ${\cal W}_4$ and ${\cal W}_5$ are ignored. \section{Cross sections in the second resonance region \label{xsec-2nd}} \begin{figure}[ht] \includegraphics[angle=-90,width=\columnwidth]{PPDS-W.ps} \caption{Differential cross section $d\sigma / dW$ for the one-pion neutrinoproduction on neutron} \label{dsidW-ppds} \end{figure} We finally return to the second resonance region and use the isovector form factors to calculate the cross section for neutrinoproduction of the resonances. We specialize to the final channels $\nu n \to R \to p \pi^0$ and $\nu n \to R \to n \pi^+$, where both $I=3/2$ and $I=1/2$ resonances contribute. The data that we use is from the ANL \cite{Barish:1978pj,Radecky:1981fn}, SKAT \cite{Grabosch:1988gw} and BNL \cite{Kitagaki:1986ct} experiments. The ANL and BNL experiments were carried on Hydrogen and Deuterium targets, while the SKAT experiment used freon ($CF_3Br$). The experiments use different neutrino spectra, there is, however, an overlap region for $E_\nu<2.0 \; {\mathrm{GeV}}$. The data show that the BNL points are consistently higher that those of the other two experiments (see figure 4a,b in ref. \cite{Grabosch:1988gw}). This is also evident in earlier compilations of the data. For instance, Sakuda \cite{Sakuda:2002rx} used the BNL data and his cross sections are larger that those of Paschos et al.\cite{Paschos:2002mb} where ANL and SKAT data were used. A recent article \cite{Sato:2003rq} uses data from a single experiment \cite{Barish:1978pj}, where the differences between the experimental results is not evident. The error bars in these early experiments are rather large and it should be the task of the next experiments to improve them and settle the issue. The differential cross section $d \sigma / d W$ was reported in several experiments (see figures 4 in \cite{Barish:1978pj}, 1 in \cite{Radecky:1981fn}, 4 in \cite{Kitagaki:1986ct}, 7 in \cite{Allasia:1990uy}). We plot the differential cross section $d\sigma / d W$ in figure~\ref{dsidW-ppds} for incoming neutrino energies $E_\nu=1,2$ and $3\; {\mathrm{GeV}}$. We note, that the second resonance peak grows faster than the first one with neutrino energy and becomes more pronounced at $E_\nu=3 \; {\mathrm{GeV}}$. \begin{figure}[ht] \includegraphics[angle=-90,width=\columnwidth]{PPDS-W-BEBC.ps} \caption{Differential cross section $d\sigma / dW$ for the one-pion neutrinoproduction for BEBC experiment} \label{fig-BEBC} \end{figure} In Fig.\ref{fig-BEBC} we plot our theoretical curves together with the experimental data from the BEBC experiment \cite{Allasia:1990uy} for $ \langle E_\nu \rangle =54\; {\mathrm{GeV}}$. The theoretical curve clearly shows two peaks with comparable areas under the peaks. The experimental points are of the same order of magnitude and follow general trends of our curves, but are not accurate enough to resolve two resonant peaks. The spectra of the invariant mass are also plotted in figure 4 in Ref.\cite{Kitagaki:1986ct} up to $W=2.0 \; {\mathrm{GeV}}$, but there is no evident peak at $1.4<W<2.0 \; {\mathrm{GeV}}$, in spite of the fact that the number of events is large. This result together with the fact that the integrated cross sections for $p\pi^0$ and $n\pi^+$ are within errors comparable suggest that the $I=1/2$ and $I=3/2$ amplitudes are comparable. We study next the integrated cross sections for the final states $\mu^- p \pi^0$ and $\mu^- n\pi^+$ as functions of the neutrino energy. The solid curves in Fig.~\ref{fig-si-integr} show the theoretically calculated cross sections with the cut $W<2.0 \; {\mathrm{GeV}}$ and the dashed curve with the cut $W<1.6\; {\mathrm{GeV}}$. For $p \pi^0$ the solid curve goes through most of the experimental points except for those of the BNL experiment, which are consistently higher than those of the other experiments. \begin{figure}[ht] \includegraphics[angle=-90,width=\columnwidth]{nun-tot-ppi0.ps} \includegraphics[angle=-90,width=\columnwidth]{nun-tot-npiplus.ps} \caption{Integrated cross section for the $\mu^- p \pi^0$ and $\mu^- n\pi^+$ final states} \label{fig-si-integr} \end{figure} For the $n\pi^+$ channel our curve is a little lower than the experimental points. This means that there are contributions from higher resonances or additional axial form factors. Another possibility is to add a smooth background which grows with energy. An incoherent isospin-1/2 background of approximately $5\cdot 10^{-40}(E_\nu/1\; {\mathrm{GeV}}-0.28)^{1/4} \; {\mathrm{cm}}^2$ would be sufficient to fit the data, as it is shown by a double--dashed curve. By isospin conservation, the background for the $p \pi^0$ channel is determined to be half as big. Including this background, which may originate from various sources, produces the double-dashed curves in Fig.~\ref{fig-si-integr}. Since experimental points are not consistent with each other, it is premature for us to speculate on the additional terms. For the $P_{33}(1232)$ the elasticity is high, but for the other resonances is $\approx 0.5$, which implies substantial decays to multipion final states. We computed in our formalism the integrated cross section for multipion production. The results are shown in Fig.~\ref{fig-pds-multipion} with two cuts $W<1.6 \; {\mathrm{GeV}}$ and $W<2 \; {\mathrm{GeV}}$. The experimental points are from Ref.~\cite{Barish:1978pj}. \begin{figure}[ht] \includegraphics[angle=-90,width=\columnwidth]{nun-tot-multi.ps} \caption{Integrated cross section for the multipion production by neutrinos} \label{fig-pds-multipion} \end{figure} \section{Conclusions} We described in this article a general formalism for analysing the excitation of resonances by neutrinos. We adopt a notation for the cross section very similar to that of DIS by introducing structure functions. We give explicit formulas for the structure functions in terms of form factors. The form factors describe the structure of the transition amplitudes from nucleons to resonances. The vector components appear in electro- and neutrino--production. We use recent data on helicity amplitudes from JLAB and the Mainz accelerator to determine the form factors including $Q^2$ dependences. We found out, that several of them fall slower than the dipole form factor, at least for $Q^2<2 - 3 \; {\mathrm{GeV}}^2$. The accuracy of these results is illustrated in Figures \ref{fig-D1520-AS}, \ref{fig-P1232-AS}, \ref{fig-P1440-AS}, \ref{fig-S1535-AS}. We obtain values for two axial form factors by applying PCAC (see Appendix~\ref{gold-trei}) whenever the decay width and elasticity is known. For the spin-3/2 resonances there is still freedom for two additional axial form factors whose contribution may be important. This should be tested in the experiments. We present differential and integrated cross sections in Sections \ref{tau} and \ref{xsec-2nd}. For the $P_{33}(1232)$ we point out, that the structure functions ${\cal W}_4$ and ${\cal W}_5$ are important for experiments with $\tau-$leptons because they modify the $Q^2$ dependence and influence the integrated cross section. The second resonance region has a noticeable peak in $d\sigma / dW$ (Fig.\ref{dsidW-ppds}), which grows as $E_\nu$ increases from $1$ to $3\; {\mathrm{GeV}}$. The integrated cross section for the $I=1/2$ channel also grows with energy of the beam and may require stronger contribution from the resonance region and a non-resonant background (Fig.\ref{fig-si-integr}). Our results are important for the new oscillation experiments. In addition to the production of the resonances and the decays to one pion and a nucleon, there are also decays to two and more pions. Multipion decays contribute to the integrated cross section with the cut $W<2\; {\mathrm{GeV}}$ at the level of $(2 - 3) \cdot 10^{-39} \; {\mathrm{cm}}^2$ for $E_\nu> 4 \; {\mathrm{GeV}}$. Thus our results are useful in understanding the second resonance region and may point the way how the resonances sum up to merge at higher $Q^2$ into the DIS region. \begin{acknowledgements} We thank Dr. I. Aznauryan for providing us recent electroproduction data. The financial support of BMBF, Bonn under contract 05HT 4 PEA/9 is gratefully acknowledged. GP wishes to thank the Graduiertenkolleg 841 of DFG for financial support. One of us (EAP) wishes to thank the theory group of JLAB for its hospitality where part of this work was done. \end{acknowledgements}

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\section{Introduction} Consider a queueing/scheduling system (as in Fig. \ref{fig:sys}), where a finite number $J$ jobs wait in a buffer, each to be processed by one of $N$ servers/processors. Time is slotted. The service/processing requirement, $\sigma_j$, of each job $j$ is random and its distribution, $f_j(\sigma_j)$, is known. All processors operate at service rate $1$; hence, the service time for each job is invariant to the processor which it assigned. Service is {\em non-preemptive} (job service cannot be interrupted mid-processing to be resumed later or discontinued). The completion of job $j$ in time slot $t$ garners a reward $w_j(t)\geq 0$, which decays with time (i.e. $w_j(t)$ is non-increasing in $t$). The goal is to schedule the jobs on the processors so as to maximize the aggregate reward accrued when all jobs complete execution. As will become clear below, a key complicating factor is that the job service is {\em non-preemptive}, inducing a `combinatorial twist' on the problem. Under preemptive processing, the latter would wash away and the problem would become much simpler. Another complicating factor is the fact that the rewards/values $w_j(t)$ decay over time in a general way; special cases might be significantly easier to handle (though still not necessarily easy). A third complicating factor is the general distributions of the stochastic job processing times $\sigma_j$ (even though these are independent across different jobs); for special distributions the problem can become significantly simpler (and the results tighter). We aim to address the problem in the most general setting arising in a variety of applications (see below), which may actually require online (real-time) schedule implementation. In that case, since the complexity of computing the optimal job schedule is prohibitive, one seeks simple and practical schedules (implementable online), which have performance within provable bounds from optimal. In this paper, we focus on a \emph{greedy/myopic} schedule defined below and study its efficiency. We discuss these factors below in conjunction with prior work and a variety of applications. \begin{figure \begin{center} \psfrag{jobs}{jobs $ = \{1,2,\dots,j,\dots,J-1,J\}$}\psfrag{m1j}{$\sigma_j$}\psfrag{m2j}{$\sigma_j$}\psfrag{mn1j}{$\sigma_j$}\psfrag{mnj}{$\sigma_j$} \includegraphics[scale=.8]{figs/sys} \caption{System Diagram: $J$ jobs wait to be processed on one of $n$ machines. The processing time of each job is independent of the processor and other jobs.}\label{fig:sys} \end{center} \end{figure} \subsection{Applications} There are diverse applications where job completion rewards decay over time. For example, such is the case with patient scheduling in health-care systems. Delays in treatment often lead to deterioration of patient health (see, for instance,\cite{mcquillan}) which may result in reduction of the eventual treatment impact; this is obviously the case with various medical procedures, operations, etc. Indeed, a number of studies have demonstrated that delayed treatment results in increased patient mortality \cite{chan,luca,buist,bellomo,sharek_jama}. Moreover, in a related study \cite{poon_arch_internmed}, over 60\% of physicians reported dissatisfaction with delays in viewing test results, which subsequently led to delays in treatment. It is likely that increased mortality is primarily induced via deterioration of patient health condition and resulting reduction of benefit from eventual treatment. This is how the effect of treatment delay is modeled in this paper. On the other hand, in information technology, reward decay occurs in various situations--for example, in multimedia packet scheduling for transmission over wireless links. Each packet corresponds to a job which is completed once the packet is successfully received at the receiver; until then, it is repeatedly transmitted (non-preemptive processing). Transmission time until successful reception is random, due both to random packet sizes and randomly varying wireless channel quality. In the simplest case, video packets have a single deadline and reward is only received if the packet is received prior to its deadline expiration. In more advanced schemes, multiple deadlines are considered (decreasing, piecewise-constant reward decay function), reflecting coding interdependencies across packets. Indeed, even if a packet misses its initial deadline, it could improve the quality of the received and reconstructed video because other packets which depend on it may still be able to meet their deadline \cite{wee_icme02}. As with multimedia packet scheduling above and similar situations of task scheduling in parallel computing systems, we can consider jobs that contain interdependencies within our model. The completion of a single job $j$ garners reward $r_j$. However, other jobs may rely on that one too, either because they cannot begin processing until that is completed (due to data-passing, precedence constraints, etc.) or their processing accuracy/quality depends on output from that job (e.g. decoding dependencies). Therefore, the `effective' reward generated is actually $w_j(t) = r_j - f(t)$, where the increasing function $f(t)$ reflects the detrimental effect that completing job $j$ after delay $t$ has on other jobs depending on it. In fact, our formulation allows for the case where even $r_j$ is a decaying function in time. A third application area where job completion rewards may decay over time is in the case of perishable items, like food, medicine, etc. For example, the quality of food items (milk, eggs, etc.) decays with time. The scheduling problem is when to release these items for sale given varying transportation times (from storage to shelf) and the decaying reward $R(t)$. It is also possible to have a cost $s$ for each time slot the item remains in storage so that the effective reward of an item once it is released for sale is $C(t) = R(t) - st$. \subsection{Literature Review} When rewards do not decay over time but stay constant, job scheduling problems may be cast in the framework of `multiarmed bandit' problems \cite{gittins, walrand}. Furthermore, optimal policies for certain `well-behaved' decaying reward functions (such as linear and exponential) have been developed (see \cite{gittins, walrand} and related works). Unfortunately, under general decaying rewards, solving for the optimal schedule becomes very difficult. There has been related work on delay-sensitive scheduling in networking. In the case of broadcast scheduling in computer networks, jobs correspond to requests for pages (files). Due to the broadcast nature of a wireless channel, multiple requests can be satisfied with the transmission of a single page. In \cite{kim04}, a greedy algorithm is shown to be a $2$-approximation for throughput maximization of broadcast scheduling in the case of {\em equal sized pages}. In a similar scenario, an online {\em preemptive} algorithm is shown to be $\Omega(\sqrt{n})$ competitive where $n$ is the number of pages that can be requested \cite{lipton94}. Our work differs from this prior work in that we allow for 1) arbitrary decaying rewards, rather restricting to step functions when the deadline expires, 2) jobs are non-preemptive and have varied lengths (and all jobs are available at time $0$). A substantial body of work has focused on scheduling for perishable products (see \cite{nahmias82} for a review). The focus is on finding an optimal ordering policy given the lifetime and demand of the perishable items. In \cite{naso07}, the authors study how to maximize utility garnered by delivering perishable goods, such as ready-mixed concrete, and minimize costs subject to stochasticity in transportation times. The authors formulate a mathematical program to solve the problem and propose heuristic algorithms for use in practice. Interestingly, the perishable items in this case have a fixed lifetime, after which they are rendered useless (deadline). Our formulation here allows for general decay. In \cite{dalal05}, the authors look at how to schedule an M/M/1 queue where rewards decay exponentially dependent on each job's sojourn time due to the `impatient' nature of the users. A greedy policy is shown to be optimal in the case of identical decay rates of these impatient users. Our scheduling problem is closely related to a number of instances of the Multiarmed Bandit Problem. When rewards exhibit `well-behaved' decay, (identical rates, constant rates, etc.) it is possible to find optimal, or near-optimal algorithms \cite{pinedo,schuurman99,walrand,gittins}. This is not always the case for arbitrary decay. In a problem similar to the one we study in this paper, a greedy algorithm is shown to be a $2$-approximation when job completions generate rewards according to general decaying reward functions \cite{chan08}. The main distinction between this work and ours is that the previous work allows for job preemption while we consider the case that once a job is scheduled it occupies the machine until it completes. This constraint adds an extra layer of complexity. Indeed, non-preemption makes the scheduling problem we study substantially more difficult. Non-preemptive interval scheduling is studied in \cite{goldman00,lipton94} among others. Jobs can either be scheduled during their specified interval or rejected. The end of the interval corresponds to the deadline of the corresponding job. If $\Delta$ is the ratio of the large job size to the smallest job size, then an online algorithm cannot be better than $O(\log \Delta)$. Our work differs from this prior work because we consider arbitrary decay of rewards and assume all jobs are available at time $0$. The decaying reward functions make this a more general and difficult scheduling problem. However, our result also relies on $\Delta$, the ratio between largest and smallest jobs. Still, there are instances where optimal schedules can be found for arbitrary decaying rewards. In a parallel scenario to ours, jobs can be scheduled, non-preemptively, multiple times. For this problem, the reward function for completing a particular job decays with the number of times that job has been completed. In this case, a greedy policy is optimal for arbitrary decaying rewards \cite{walrand}. This problem is parallel to ours in that it allows for arbitrary decaying rewards. However, the decay does not depend on the completion time of the job, but rather on the number of times that job has been completed. In our case, each job is only processed a single time. Relating back to our scenario where the rewards decay with time, it is again the case that for `well-behaved' decaying functions (linear and exponential), policies based on an index rule are optimal \cite{gittins,walrand}. The policy we propose in this paper is also an index rule. In fact, the proposed policy is very closely related to the `c-$\mu$'-type scheduling rules (see, for instance \cite{VanMieghem,walrand}) where the objective is to minimize cost (rather than maximize rewards) when costs are linearly or concavely increasing. One of the main distinctions between our work and this is that we consider multiple servers. Unfortunately, the optimality of the `c-$\mu$' rule does not extend to this case. Furthermore, linear/concave decaying rewards are just single instances of our more general formulation of decaying rewards. It is also important to recognize that many of the results of this prior work are in heavy-traffic regimes where a lot of the fine-grained optimization required in non-heavy-traffic is washed out. \subsection{Summary of Results} In this paper, we study the efficacy of a greedy scheduling algorithm for non-preemptive jobs whose rewards decay arbitrarily with time. There are a number of applications which exhibit such behavior such as patient scheduling in hospitals, packet scheduling in multimedia communication systems, and supply chain management for perishable goods. It is shown that finding an optimal scheduling policy for such systems is NP-hard. As such, finding simple heuristics is highly desirable. We show that a greedy algorithm is guaranteed to be within a factor of $\Delta+2$ of optimal where $\Delta$ is the ratio of the largest job completion time to the smallest. This bound is improved in some special cases. Via numerical studies, we see that, in practice, the greedy policy is likely to perform much closer to optimal which suggests it is a reasonable heuristic for practical deployment. To the best of our knowledge this is the first look at non-preemptive scheduling of jobs with arbitrary decaying rewards. The rest of the paper is structured as follows. In Section \ref{sec:model} we formally introduce the scheduling model we will study. In Section \ref{sec:greedy} we propose and study the performance of a greedy scheduling policy. The main result, which is a bound on the loss of efficiency due to greedy scheduling, is given in Section \ref{ssec:bound}. In Section \ref{sec:special}, we examine some special cases where this bound can be improved. In Section \ref{sec:perfeval}, we do some performance evaluation of the greedy policy via a simulation study. Finally, we conclude in Section \ref{sec:conc}. \section{Model Formulation}\label{sec:model} Consider a set of $J$ jobs, indexed by $j \in {\mathcal J} =\{1,2,\dots,J\}$, and $N$ processors/servers, indexed by $n \in {\mathcal N} = \{1,2,\dots, N\}$. Each job $j\in {\mathcal J}$ has a random processing requirement $\sigma_j$ and can be processed by any processor $n\in {\mathcal N}$. All processors have service rate 1 and each one can process a single job at a time. Service is {\em non-preemptive} in the sense that once a processor starts executing a job it cannot stop until completion. Time is slotted and indexed by $t\in\{1,2,3,...\}$. We denote the distribution of the service times by $f_j(\sigma_j)$. \begin{assumption} \label{as:sigma} The random job processing times $\sigma_j,j\in\cal J$ are 1) statistically independent with $P(\sigma_j < \infty) = 1$ and 2) their distributions, $f_j(\sigma_j)$, do not depend on time. \end{assumption} However, the jobs processing times are {\em not} necessarily identically distributed. Let $b_j(t)$ be the residual service time of job $j$ at time $t$. Initially, $b_j(0)=\sigma_j$, for each $j\in\cal J$. The backlog state of the system at time $t$ is the vector \begin{equation} {\bf b}(t)= \left(b_1(t), b_2(t),..., b_j(t), ..., b_J(t)\right). \end{equation} It evolves from initial state ${\bf b}(0)= (\sigma_1,\sigma_2,...,\sigma_j,...,\sigma_J)$ to final state $\vec \sigma(T)=(0,0,...,0,...,0)$ by assigning processors to process the jobs \emph{non-preemptively}, until all jobs have finished execution at some (random) time $T$. Note that for each job $j\in\cal J$, $b_j(t)=\sigma_j$ implies that $j$ has not started processing by $t$ (has not been scheduled before $t$), while $b_j(t)=0$ implies that the job finished execution before (or at) time $t$. Indeed, if job $j$ starts execution at time slot $t_j$ and finishes at the beginning of time slot $T_j$ then $\sigma_j=T_j-t_j$ and \begin{equation} b_j(t) = \left\{ \begin{array}{lll} \sigma_j, & t < t_j \\ \sigma_j-[t-t_j], & t = t_j, t_j+1, ..., T_j-1\\ 0, & t \geq T_j \end{array} \right. \end{equation} As discussed later, the start times $t_j$ are chosen by the scheduling policy, while the end times $T_j$ are then determined by the fact that scheduling is non-preemptive so that $T_j=t_j+ \sigma_j$. The job service times are random and their true values are not observable ex ante or known a priori; they can only be seen ex post, after a job has completed processing. However, the values $x_j(t)$ tracking which jobs are completed at each time $t$ \begin{equation} x_j(t) = \left\{ \begin{array}{ll} 1, & \mbox{if job $j$ has not completed processing at time slot $t$} \\ 0, & \mbox{if job $j$ has completed by time slot $t$} \end{array} \right. \end{equation} are directly observable for each job $j\in\cal J$. We work below with the {\em observable} `backlog state' \begin{equation} {\bf x}(t)=\left(x_1(t),x_2(t),...,x_j(t),...,x_J(t)\right) \end{equation} in $\{0,1\}^J$ which tracks which jobs are completed and which are still waiting to complete processing at time $t$. To fully specify the state of a job, we define $y_j(t)$ as the time slot $t' < t$ in which job $j$ begins processing. Specifically, \begin{equation} y_j(t) = \left\{ \begin{array}{ll} t', & \mbox{if job $j$ began procesing in time slot $t' < t$} \\ \emptyset, & \mbox{if job $j$ has not begun processing prior to time slot $t$ (necessarily, $x_j(t) = 1$)} \end{array} \right. \end{equation} Hence, any job with $y_j(t) = \emptyset$ (where $\emptyset$ is some null symbol) has not yet begun processing and is free to be scheduled. If $x_j(t) = 1$, then job $j$ has not completed and it is still being processed due to the non-preemptive nature of the service discipline. Once a job is scheduled in time slot $t_j$, then $y_j(t) = t_j$ for all $t > t_j$. The service state is then, \begin{equation} {\bf y}(t)=(y_1(t),y_2(t),...,y_j(t),...,y_J(t)) \end{equation} in $\{\{0,1,...,t-1\} \cup \emptyset \}^J$ and tracks when (and if) each job began processing. In time slot $t$, one can calculate the distribution for the remaining service time $b_j(t)$ given the distribution of $\sigma_j$ based on when (if) the job has started processing and whether it has completed. Only the distribution of $b_j(t)$ is known as the job service time is only observable once the job completes processing. Therefore, $x_j$ and $y_j$ can be jointly leveraged to compute the distribution of the residual service time of job $j$. We next define the state $z_n(t)$ of processor $n\in\cal N$ which tracks which job it is assigned to process in time slot $t$. Specifically, \begin{equation} z_n(t)= \left\{ \begin{array}{ll} j, & \mbox{if processor $n$ is still executing job $j\in\cal J$ at the beginning of time slot $t$} \\ 0, & \mbox{if processor $n$ is free at the beginning of time slot $t$, hence, available for allocation} \end{array} \right. \end{equation} and the processor state is \begin{equation} {\bf z}(t)=(z_1(t),z_2(t),...,z_n(t),...,z_N(t)) \end{equation} in $\{0,1,...,j,...J\}^N$ and tracks the free vs. allocated processors at the beginning of time slot $t$. At the beginning of each time slot $t$, each job $j$ with $y_j(t) = \emptyset$ (not yet started) can be scheduled on (matched with) a processor $n$ with $z_n(t) = 0$ (free) to start execution. The {\em observable} state of the system at the beginning of time slot $t$ is \begin{equation}\label{eq:state} s_t = \left( {\bf x}(t),{\bf y}(t),{\bf z}(t) \right) \end{equation} Recall that from ${\bf x}(t)$ and ${\bf y}(t)$ we can determine the distribution of the remaining service time ${\bf b}(t)$. So the global state (\ref{eq:state}) yields the distribution for the remaining backlog and also tracks the processor state. The state space ${\cal S}$ is the set of all states the system may attain throughout its evolution. We denote by $\mathbf{x}(s)$ the projection of the state onto the $\mathbf{x}$-coordinate. We similarly apply notation for $\mathbf{y}(s)$ and $\mathbf{z}(s)$. Given the free jobs and processors at state $s_t$, we denote by ${\mathcal A}(s_t)$ the set of job-processor matchings (schedules) that can be selected, i.e. they are feasible, at the beginning of time slot $t$. These matchings are in addition to those already in place for jobs which are in mid-processing due to the non-preemptive nature of execution. Note that at each time $t$, for any feasible job-processor matching $A \in {\mathcal A}(s_t)$ we have that $(j,n) \in A$ implies $x_j (t)= 1$, $y_j(t) = \emptyset$ and $z_n(t) = 0$, meaning processor $n$ is free and job $j$ has not started processing. Also, only one free job can be matched to each free processor and vice-versa (hence, $(j,n), (k,m) \in A$ with $(j,n) \not = (k,m)$ implies $j \not = k$ and $n \not = m$). Despite the fact that ${\mathcal A}(s_t)$ is clearly a function of $s_t$, we may occasionally suppress $s_t$ for notational simplicity. The completion of job $j$ by the end of time slot $t$ garners non-negative reward $w_j(t)$. We assume the reward decays over time, as follows. \begin{assumption} \label{as:reward} For each job $j\in\cal J$, the reward function $w_j(t) \geq 0$ decays over time; that is, it is non-increasing in $t$ (may be piece-wise constant). \end{assumption} This immediately accounts for raw deadlines by setting $w_j(t)={\bf 1}_{ \{t\leq d_j\} }$ when $d_j$ is the deadline of job $j$. Recall that if job $j$ is scheduled on processor $n$ at the beginning of time slot $t$, it will finish by the beginning of time slot $t+\sigma_j$. Therefore, the reward `locked' at the beginning of time slot $t$, given that a job-processor match $A\in{\mathcal A}(s_t)$ is chosen to be used in this slot, is simply \begin{equation} \begin{split} R_t(s,A) = \sum_{(j,n) \in A}w_j(t + \sigma_j) \end{split} \end{equation} It is desirable to design a control (scheduling, matching) policy choosing at each $t$ a job-processor matching in ${\mathcal A}(s_t)$ to maximize the total expected reward accrued until all jobs have been executed. Since at time $t$ the realization of $\sigma_j$ is unknown for each job $j$ that has not completed by $t$, \emph{any} control policy is a-priori unaware of the exact reward accrued from a particular action at $t$. Only the statistics of this reward are known. Specifically, let $\pi$ be a scheduling policy which chooses a job-processor matching $\pi_t(s_t)\in{\mathcal A}(s_t)$ at $t$, and let $\Pi$ be the set of all such policies. Define the expected total reward-to-go under a policy $\pi$ starting at state $s \in{\cal S}$ at time slot $t$, as \begin{equation} \begin{split} J^\pi_t(s) = E\left[\sum_{t'=t}^T R_{t'}(s_{t'},\pi_{t'}(s_{t'})) | s_t = s \right] \end{split} \end{equation} where $T$ is the (random) time where all jobs have completed execution. $T$ may depend on the policy $\pi$ used. Note that if we wanted to consider a finite, deterministic horizon $\tilde{T}$, we could appropriately generate a schedule based on the modified, truncated reward functions, $\tilde{w}_j(t)$, such that for all $t \leq \tilde{T}$, $\tilde{w}_j(t) = w_j(t)$, otherwise $\tilde{w}_j(t) = 0$. The expectation is taken over the random service times $\sigma_j$ of the jobs. We let \begin{equation} J_t^*(s) = \max_{\pi \in \Pi} J_t^{\pi}(s) \end{equation} denote the expected total reward-to-go under the optimal policy, $\pi^* = \argmax_{\pi \in \Pi} J_t^{\pi}(s)$. The optimal reward-to-go function (or \emph{value} function) $J^*$ and the optimal scheduling policy $\pi^*$ can in principle be computed via dynamic programming. Once all jobs have been completed, ${\bf x} = \mathbf{0}$ and no more reward can be earned. Therefore, $J_t^*(s) = 0$ for all $s=({\bf x}, {\bf y},{\bf z})$ such that ${\bf x} = \mathbf{0}$. Given the current state $s_t=s$ and the matching $A$ between free jobs and processors enabled at the beginning of time slot $t$, the system will transition to state $s_{t+1}=s'$ at the beginning of time slot $t+1$ with probabilities $P_A(s_{t+1}=s'|s_t=s)$. For example, if the service times $\sigma_j$ are geometrically distributed with probabilities $p_j$ correspondingly, and the system is in state $s_t=s=({\bf x}, {\bf y},{\bf z})$ and matching $A\in{\mathcal A}(s_t)$ is chosen, then the system transitions to state $s_t=s'=({\bf x}',{\bf y}',{\bf z}')$ with the following probabilities: \begin{eqnarray} P_A(x'_j = 0|s) &=& \left\{ \begin{array}{ll} 1, & \hbox{if $x(s)_j = 0$;} \\ p_j, & \hbox{if $y_j(s) < t$ or $(j,n) \in A$ for some $n$;} \\ 0, & \hbox{otherwise.} \end{array} \right. \nonumber \\ P_A(x'_j = 1|s) &=& \left\{ \begin{array}{ll} 1, & \hbox{if $x(s)_j = 1$ and $(j,n) \not \in A$ for all $n$;} \\ 1-p_j, & \hbox{if $y_j(s) < t$ or $(j,n) \in A$ for some $n$;} \\ 0, & \hbox{otherwise.} \end{array} \right.\nonumber \\ P_A(y'_j = t|s) &=& \left\{ \begin{array}{ll} 1, & \hbox{if $(j,n) \in A$ for some $n$;} \\ 0, & \hbox{otherwise.} \end{array} \right. \nonumber \\ P_A(y'_j = y_j|s) &=& \left\{ \begin{array}{ll} 1, & \hbox{$(j,n) \not \in A$ for all $n$;} \\ 0, & \hbox{otherwise.} \end{array} \right.\nonumber \\ P_A(z'_n = j|s) &=& \left\{ \begin{array}{ll} 1-p_j, & \hbox{if $z(s)_n = j$ or $(j,n) \in A$ for some $n$;} \\ 0, & \hbox{otherwise.} \end{array} \right. \nonumber \\ P_A(z'_n = 0|s) &=& \left\{ \begin{array}{ll} p_j, & \hbox{if $z(s)_n = j$ or $(j,n) \in A$ for some $n$;} \\ 1, & \hbox{if $z(s)_n = 0$ and $(j,n) \not \in A$ for all $j$;} \\ 0, & \hbox{otherwise.} \end{array} \right. \end{eqnarray} We can now recursively obtain $J^*$ using the Bellman recursion \begin{eqnarray}\label{eqn:DP} J_t^*(s) &= &\max_{A \in {\mathcal A}} \Big \{ E \big[\sum_{(j,n)\in A} w_j(t+\sigma_j) + \sum_{s'\in\cal S} P_A[s_{t+1}=s'|s_t=s] J_{t+1}^*(s') \Big \} \nonumber \\ &= &\max_{A \in {\mathcal A}}\Big \{E[R_t(s,A)] + J_{t+1}^*(\tilde{S}(s,A)) \big]\Big \} \end{eqnarray} where $\tilde{S}(s,A)$ is the random next state encountered given that we start in state $s$ and action $A$ is taken. The solution can be found using the \textit{value iteration} method. \begin{proposition} \label{prop:termination} There exists an optimal control solution to (\ref{eqn:DP}) which is obtainable via value iteration. \end{proposition} \begin{proof}Once the queue is emptied, Bellman's recursion terminates. When $\mathbf{x} = \mathbf{0}$, there are no more jobs left to be processed. No action can generate any reward and the optimal policy will never leave this state once it reaches it. There exists a policy which will complete all jobs and cause the Bellman's recursion to terminate in finite time. (i.e. we process \emph{all} jobs on a single server, $n$, in random order. Because $P(\sigma_j < \infty) = 1$, all jobs will be completed in finite time.) This guarantees the existence of a stationary optimal policy which is obtainable via value iteration \cite{bertsekas}. \end{proof} Of course, this approach is computationally intractable: the state space (the set of all $(\mathbf{x},\mathbf{y},\mathbf{z})$) is exponentially large. As such, this makes such problems pragmatically difficult. \subsection{A Hardness Result} We now show that a special case of the non-preemptive scheduling problem is NP-hard. Consider a deterministic version of the problem where the completion time of job $j$ is $\sigma_j$ with probability $1$. Let $w_j(t) = v_j$ for $t \leq K$ and $w_j(t) = 0$ otherwise. We can think of $v_j$ as the value of job $j$ and $K$ as the shared deadline amongst all jobs. This version of the non-preemptive scheduling problem with decaying rewards can be reduced to the 0/1 Multiple-Knapsack Problem which is known to be NP-complete. \begin{theorem}\label{th:NPhard} The non-preemptive scheduling problem with decaying rewards is NP-hard. \end{theorem} \begin{proof} In the case of the 0/1 Multiple Knapsack Problem, there are $J$ objects of size $\sigma_j$ and value $v_j$ to be placed in $N$ knapsacks of capacity $K$. Reward is only accrued if the entire object is placed in a knapsack--fractional objects are not possible. The optimal packing of objects is equal to the optimal scheduling policy for the non-preemptive scheduling with decaying rewards problem. This reduction takes constant time. This completes the proof. \end{proof} \section{A Greedy Heuristic for Non-preemptive Scheduling of Decaying Jobs}\label{sec:greedy} In light of Theorem \ref{th:NPhard}, finding an optimal policy for the scheduling problem at hand is computationally intractable. Therefore, it is highly desirable to find simple, but effective heuristics for practical deployment. In this section, we examine one such policy. A natural heuristic policy one may consider for the stochastic depletion problem is given by the greedy policy which in state $s$ with $F = \sum_n {\bf 1}_{\{z(s)_n = 0\}}$ free processors chooses the $F$ available jobs with maximum expected utility rate earned over the following time-step, $\frac{E[w_j(t(s)+\sigma_j)]}{E[\sigma_j]}$. That is \begin{equation} \begin{split} \pi_t^{g}(s) = \argmax_{A \in {\mathcal A}} \sum_{(j,n)\in A} \frac{E[w_j(t(s)+\sigma_j)]}{E[\sigma_j]} \end{split} \end{equation} Such a policy is \emph{adaptive} but ignores the evolution of the reward functions, $w_j(t)$, and its impact on rewards accrued in future states. We denote by $J_t^g(s)$ the reward garnered by the greedy policy starting in state $s$. \subsection{Sub-optimality of Greedy Policy} We start with an instructive example which demonstrates the nature (and degree) of sub-optimality of the greedy policy. \begin{example}(Greedy Sub-Optimality) \label{ex:suboptimality} Consider the case with $2$ jobs and $1$ machine. Time is initialized to $0$ so that $t=0$, $J = 2$ and $N = 1$. Assume that each job is waiting to begin processing so that $x_1 = x_2 = 1$ and $y_1 = y_2 = \emptyset$. The service times are Geometric and the expected service times for job $1$ and $2$ are $M$ and $1$, respectively, i.e. $p_1 = \frac{1}{M}$ and $p_2 = 1$. The reward functions are: \[ \begin{split} {\rm For} \ j=1:& \ w_j(t) = \begin{cases} M^2, & \text{$t = 1$}\\ 0, & \text{$t > 1$} \end{cases} \\ {\rm For} \ j=2:& \ w_j(t) = \begin{cases} 1+\epsilon, & \text{$\forall t$} \end{cases} \end{split} \] for $\epsilon > 0$. Hence, the completion of job $1$ generates rewards of $M^2$ if it is completed in the first time slot; otherwise, no revenue is received. On the other hand, job $2$ results generates reward of $1+\epsilon$, regardless of which time slot it is completed in. Therefore, the reward rates are: \begin{eqnarray} \frac{E[w_1(t+\sigma_1)]}{E[\sigma_1]} & = \begin{cases} 1, & \text{$t = 1$}\\ 0, & \text{$t > 1$} \end{cases} \nonumber\\ \frac{E[w_2(t+\sigma_2)]}{E[\sigma_2]} & = \begin{cases} 1+\epsilon, & \text{$\forall t$} \end{cases} \nonumber \end{eqnarray} In time slot $t = 0$, the greedy policy schedules job $2$ because its reward rate ($1+\epsilon$) is great than that of job $1$ ($1$). Job $2$ completes processing in one time slot and generates reward of $w_2(1) = 1 + \epsilon$. At $t=1$, only job $1$ remains to be processed. However, the service time for job $1$ is at least one time slot, so when job $1$ is completed at $t = 1+\sigma_2>1$, $0$ reward is generated. Hence the greedy policy generates a total expected reward of $1+\epsilon$. On the other hand, the optimal policy realizes the reward of job $1$ is degrading and schedules it first. With probability $1/M$, job $1$ will complete by time slot $t = 1$ and generate reward $M^2$. However, with probability $1-1/M$ it will take more than one time slot and generate no reward since $w_1(t) = 0$ for $t > 1$. Upon the completion of job $1$, job $2$ is scheduled and it completes processing in $1$ time slot. Since $w_2(t) = 1 + \epsilon$ for all $t$, this results in additional reward of $1+\epsilon$. Hence, the total expected reward generated by the optimal policy is $M+1+\epsilon$. Comparing the performance of the optimal and greedy policies gives $J^*(s)/J^g(s) = (M+1+\epsilon)/(1 + \epsilon)$. \end{example} Letting $\epsilon \rightarrow 0$, it is easy to see that the greedy policy results in an $M+1$ approximation, where $M = \frac{E[\max_j \sigma_j]}{\min_j E[\sigma_j]} = \Delta$. This suggests that the approximation of the greedy policy is dependent on the relationship between job service times. The following subsection specifies this relation. \subsection{The Greedy Heuristic is an online $(2+\Delta)$-Approximation}\label{ssec:bound} In this section we will show that the greedy heuristic is within a factor of $2+\Delta$ of optimal, where \begin{equation}\label{eq:delta} \Delta =\frac{E[\sigma_{\max}]}{\min_jE[\sigma_j]} \end{equation} Before we can prove this result, we need to first show a few properties of the system and the optimal value function, $J_t^*$. We begin with a monotonicity property based on the number of jobs remaining to be processed. Intuitively, if one were given an additional set of jobs to process, the reward that can be garnered by the completion of the original set of jobs in conjunction with the additional jobs will be more than if those extra jobs were not available. Consider two states: $s$ and $s'$ which are nearly identical except state $s$ has more jobs to process than state $s'$. In other words, all jobs that have been completed in the $s$-system have also been completed in the $s'$-system. Similarly, any job that has started processing in the $s$-system has also started processing in the $s'$-system at the exact same time on the same machine. Any additional jobs in state $s$ are jobs that have not started processing, but have already been completed in state $s'$. That is, the additional jobs are only available for processing in the $s$-system. Then the reward-to-go generated starting in state $s$ is larger than that starting in state $s'$. The following lemma formalizes this intuition. \begin{lemma}(Monotonicity in Jobs)\label{lemma:mono_job} Consider states $s$ and $s'$ such that state $s$ has more jobs than state $s'$ and any job that has started in state $s'$ started processing in the exact same time slot in state $s$ so that for each job $j$: $x(s)_j \geq x_j(s')$ and \begin{eqnarray} y(s)_j = \left\{ \begin{array}{ll} y(s')_j, & \hbox{if $x(s)_j = x(s')_j$;} \\ \emptyset, & \hbox{if $x(s)_j > x(s')_j$.} \end{array} \right. \nonumber \end{eqnarray} Also, in both states, each processor $n$ is either not busy or busy processing the same job: $z(s)_n = z(s')_n$. For all states $s$ and $s'$ which satisfy these conditions, the following holds: $$J_t^*(s) \geq J_t^*(s').$$ \end{lemma} \begin{proof} Consider a coupling of the systems starting at $s$ and $s'$ such that they see the same realizations of service times $\sigma_j$ (and residual service times for jobs that have already started processing). This is possible for all jobs $j \in {\mathcal J}_{s'} = \{j \in {\mathcal J}|x(s')_j = 1\} \subseteq {\mathcal J}_s = \{j\in {\mathcal J}|x(s)_j = 1\}$ because they have the same distributions. ${\mathcal J}_s$ and ${\mathcal J}_{s'}$ denote the jobs to be completed under the systems starting in states $s$ and $s'$, respectively. Let $\pi^*(s')$ denote the optimal scheduling policy starting from state $s'$. Consider a policy $\tilde{\pi}$ that starts in state $s$ and mimics $\pi^*(s')$ until all jobs $j \in {\mathcal J}_{s'}$ are completed and completes the rest of the jobs $j \in {\mathcal J}_s \setminus {\mathcal J}_{s'}$ in sequential order. That is, under $\tilde{\pi}$ the scheduler initially pretends that jobs $j\in {\mathcal J}_s \setminus {\mathcal J}_{s'}$ do not exist and uses the optimal policy under this assumption; once these jobs are completed, it processes the remaining jobs in an arbitrary order. Said another way, the $\tilde{\pi}$ policy blocks processing of the additional jobs in state $s$ ($j\in {\mathcal J}_s \setminus {\mathcal J}_{s'}$) and optimally processes the remaining jobs. Once these jobs ($j\in {\mathcal J}_{s'}$) are completed, the $\tilde{\pi}$ policy `unlocks' the remaining, additional jobs and processes them in an arbitrary manner. Fig. \ref{fig:monojobs} demonstrates the relationship between $\pi^*(s')$ and $\tilde{\pi}$ for a single server over a particular sample path for service times. \begin{figure}[htb] \begin{center} \psfrag{spsys}{$s'$-system} \psfrag{ssys}{$s$-system} \psfrag{s1}{$\sigma_1$} \psfrag{s2}{$\sigma_2$} \psfrag{s3}{$\sigma_3$} \psfrag{s4}{$\sigma_4$} \psfrag{s5}{$\sigma_5$} \psfrag{s6}{$\sigma_6$} \psfrag{t}{$t$}\psfrag{t1}{$T_{s'}$}\psfrag{t2}{$T_{s}$} \includegraphics[scale=.8]{figs/monojobs} \caption{Monotonicity in Jobs: A single server scenario. The $s$-system is given $2$ additional jobs, $j =5$ and $j = 6$. The $s'$-system uses policy $\pi^*(s')$ to optimally process all jobs $j = 1,2,3,4$. The $s$-system uses policy $\tilde{\pi}$ which mimics $\pi^*(s')$ until all jobs $j = 1,2,3,4$ are completed at time $T_{s'}$ and then processes the remaining additional jobs. }\label{fig:monojobs} \end{center} \end{figure} Let $T_j$ be the completion time of job $j$ for the $s$-system when using policy $\tilde{\pi}$. Similarly, let $T_j^*$ be the completion time of job $j$ for the $s'$-system under the optimal policy, $\pi^*(s')$. By our coupling, for all $j \in {\mathcal J}_{s'}$, $T_j = T^*_j$, i.e. the completion time of job $j$ is identical under the $s$-system which uses policy $\tilde{\pi}$ and under the $s'$-system which uses policy $\pi^*(s')$. (Notice in Fig. \ref{fig:monojobs}, jobs $1,2,3,4$ complete at the same time in the $s'$ and $s$-systems). We use the notation $J_t^*(s|\sigma)$ as the optimal reward-to-go given the filtration of the job service times, i.e. given a sample path of realizations of the $\sigma_j$. We employ similar notation for $J^{\tilde{\pi}}_t$. We have, \begin{eqnarray} J_t^*(s|\mathbf{\sigma}) & \geq & J_t^{\tilde{\pi}}(s|\mathbf{\sigma}) \nonumber\\ & = & \sum_{j \in {\mathcal J}}w_j(T_j) \nonumber \\ & = & \sum_{j \in {\mathcal J}_{s'}}w_j(T_j) + \sum_{j \in {\mathcal J}_s \setminus {\mathcal J}_{s'}}w_j(T_j) \nonumber \\ & = & J_t^*(s'|\mathbf{\sigma}) + \sum_{j \in {\mathcal J}_s \setminus {\mathcal J}_{s'}}w_j(T_j) \nonumber \\ & \geq & J_t^*(s'|\mathbf{\sigma})\nonumber \end{eqnarray} The first inequality comes from the optimality of $J_t^*(\cdot)$. The first equality comes from the definition of the reward function, $T_j$, and $\tilde{\pi}$ policy. The third equality comes from the coupling of the two systems so that $T_j = T_j^*$ for all $j\in {\mathcal J}_{s'}$. The last inequality comes from non-negative property of the rewards in Assumption \ref{as:reward}. Taking expectations over $\sigma_j$ yields the desired result. \end{proof} Next, we consider a property of the optimal policy. In every time slot, there will be a set (possibly empty) of free machines ($z_n = 0$). In each time slot, the optimal policy will assign a job to all \emph{free} machines, assuming there are enough available jobs. That is, while there are still jobs waiting to be processed, no machine will idle under the optimal policy. \begin{lemma}(Non-idling)\label{lemma:freemach} Suppose in state $s$, there are $F = |\{n \in {\mathcal N}|z(s)_n = 0\}|$ free machines, and the number of jobs remaining to be processed is $K = |\{j \in {\mathcal J}| x(s)_j = 1, y(s)_j = \emptyset\}|$. Then, under the optimal policy $\pi^*(s)$, the number of job-processor pairs executed in the next time slot will be: $$|A| = \min\{K,F\}.$$ i.e. the optimal policy is non-idling. \end{lemma} \begin{proof} The proof is by contradiction. What needs to be shown is that nothing can be gained by idling ($|A| < \min\{K,F\}$). Suppose that under the optimal policy, a processor remains free (idles), even though there is an available job to work on. Consider another policy $\tilde{\pi}$ which is identical to the $\pi^*$ policy except it begins processing all jobs on the idling machine one time slot earlier. Due to assumption \ref{as:reward}, by processing the jobs earlier, this will result in an increase in reward. This contradicts the optimality of the idling policy; hence, no optimal policy will idle. \end{proof} Now consider two systems which are identical, except one machine is tied up longer in the second system. The following lemma says that the maximum amount of additional revenue accrued by the first system for being able to start processing earlier is given by the reward rate of the greedy job; that is, the job of maximum reward rate amongst those in processing or waiting. \begin{lemma}(Greedy Revenue)\label{lemma:scaled_revenue} Consider a state $s_t = s$ in time slot $t$ and let $g$ denote the index of a greedy job, i.e. $g = \argmax_{j\in {\mathcal J}_s} \frac{E[w_j(t+\sigma_j)]}{E[\sigma_j]}$ for all jobs which are mid-processing or have not started (${\mathcal J}_s = \{k \in {\mathcal J}|x_k(s) = 1\}$). Denote by $s_g$ and $s_i$ two states which are related to state $s$ in the following manner. The two states are identical to state $s$, except on free machine $n_g$ ($z(s)_{n_g} = 0$). In state $s_g$, machine $n_g$ is occupied by a {\bf replica} of job $g$ meaning it has the same service time as job $g$, however, its completion does not generate any rewards nor does it effect the completion of the original job $g$. Similarly, in state $s_i$, machine $n_g$ is occupied by a replica of job $i$. Said in notation: $x(s_i)_j = x(s_g)_j = x(s)_j$ and $y(s_i)_j = y(s_g)_j = y(s)_j$ for all $j$, $z(s_i)_n = z(s_g)_n = z(s)_n$ for all $n \not = n_g$, and $z (s_g)_{n_g} = g$ while $z(s_i)_{n_g} = i$ for some arbitrary job index $i$ and machine $n_g$. Then, $$E[J_t^*(s_i)] \leq \Big(1-\frac{E[\sigma_i]}{E[\sigma_g]} + \frac{E[\max_{j\in{\mathcal J}_s}\sigma_j]}{E[\sigma_g]}\Big)E[w_g(t+\sigma_g)] + E[J_t^*(s_g)]$$ \end{lemma} \begin{proof} We begin by coupling the systems such that they see the same realizations for service times. Note that the replicated jobs which currently occupy machine $n_g$ need not have the same service time of their original jobs, $i$ or $g$--despite having the same distribution. Consider a policy $\tilde{\pi}$ for the $s_g$-system which mimics the $\pi^*(s_i)$ policy. While processor $n_g$ is occupied by replica job $g$, which blocks processing of other jobs, the $s_g$-system will simulate the service time of jobs on processor $n_g$. There are two possible cases, $\sigma_i \geq \sigma_g$ and $\sigma_i < \sigma_g$. \begin{description} \item[Case 1, $\sigma_i \geq \sigma_g$:] the $\tilde{\pi}$ policy idles on machine $n_g$ until $t+\sigma_i$ (time which machine $n_g$ is free in the $s_i$-system). At this point, the $s_g$-system is `synced' with the $s_i$-system and it proceeds with executing the optimal policy for the $s_i$ system, $\pi^*(s_i)$. See Fig. \ref{fig:case1} for a single processor example of such a scenario. \begin{figure}[htb] \begin{center} \psfrag{sisys}{$s_i$-system} \psfrag{sgsys}{$s_g$-system} \psfrag{si}{$\sigma_i$} \psfrag{sg}{$\sigma_g$} \psfrag{s2}{$\sigma_2$} \psfrag{s3}{$\sigma_3$} \psfrag{s4}{$\sigma_4$} \psfrag{s5}{$\sigma_5$} \psfrag{s6}{$\sigma_6$} \psfrag{t}{$t$}\psfrag{tg}{$t+\sigma_g$} \psfrag{t1}{$t+\sigma_i$} \psfrag{t2}{$T_{s_i}$} \includegraphics[scale=.8]{figs/case1} \caption{Case 1, $\sigma_i \geq \sigma_g$. The optimal policy is used for the $s_i$-system, which processes jobs in order $2,3,4,g$. The $s_g$-system uses policy $\tilde{\pi}$ which mimics $\pi^*(s_i)$. Because job $i$ completes after job $g$, the $\tilde{\pi}$ policy idles. Note that job $g$ is processed twice in the $s_g$ system because the first job is just a replica. Job $i$ is only processed once in the $s_i$ system because even though $i$ is a replica, the original had already completed processing. }\label{fig:case1} \end{center} \end{figure} If $T_j^*(s_i)$ is the completion time of job $j$ in the $s_i$-system under optimal policy $\pi^*(s_i)$,and $T_j$ is the completion time of job $j$ in the $s_g$-system under the $\tilde{\pi}$ policy, then $T_j = T_j^*(s_i)$. Employing similar notation as before, we consider the reward-to-go on a single realized sample path of service times, given by $\sigma$ and the event $\sigma_i \geq \sigma_g$: \begin{eqnarray}\label{eqn:gr1} J_t^*(s_i|\sigma, \sigma_i \geq \sigma_g) & = & \sum_{j\in {\mathcal J}_s} w_j(T_j^*(s_i))\\ & = &\sum_{j\in {\mathcal J}_s} w_j(T_j)\nonumber \\ & = & J_t^{\tilde{\pi}}(s_g|\sigma,\sigma_i \geq \sigma_g) \nonumber \\ & \leq & J_t^*(s_g|\sigma,\sigma_i \geq \sigma_g) \nonumber \\ & \leq & J_t^*(s_g|\sigma,\sigma_i \geq \sigma_g) + \frac{E[w_g(t+\sigma_g)|\sigma,\sigma_i < \sigma_g]}{E[\sigma_g|\sigma,\sigma_i < \sigma_g]} E[\sigma_g-\sigma_i+\sigma_{\max}|\sigma,\sigma_i < \sigma_g] \nonumber \end{eqnarray} \item[Case 2, $\sigma_i < \sigma_g$:] In this case, $\tilde{\pi}$ cannot exactly mimic $\pi^*(s_i)$ policy because machine $n_g$ will continue to be busy after $i$ completes in the $s_i$-system. The $\tilde{\pi}$ policy will simulate the processing of jobs on $n_g$, while the machine is still busy. Let ${\mathcal J}_{sim}$ denote the set of jobs whose processing is simulated. Despite the fact that these simulated jobs will not actually be completed, the $\tilde{\pi}$ policy assumes they are. The $\tilde{\pi}$ policy continues to follow the $\pi^*(s_i)$ policy until all jobs are `completed' in the sense that they are actually completed or their completion was simulated because processor $n_g$ was busy under the $s_g$-system when it was free under the $s_i$-system. The $\tilde{\pi}$ policy then finishes processing the simulated jobs ($j \in {\mathcal J}_{sim}$) in an arbitrary manner so that they are actually completed. That is, the actual completion of the simulated jobs is transferred to after the rest of the jobs have completed processing. Fig. \ref{fig:case2} shows an example sample path of this scenario. \begin{figure}[htb] \begin{center} \psfrag{sisys}{$s_i$-system} \psfrag{sgsys}{$s_g$-system} \psfrag{si}{$\sigma_i$} \psfrag{sg}{$\sigma_g$} \psfrag{s2}{$\sigma_3$} \psfrag{s3}{$\sigma_4$} \psfrag{s4}{$\sigma_5$} \psfrag{s5}{$\sigma_2$} \psfrag{t}{$t$}\psfrag{tg}{$t+\sigma_g$} \psfrag{t1}{$t+\sigma_i$} \psfrag{t3}{$\tau$} \psfrag{sim}{simulated jobs} \psfrag{t2}{$T_{s_i}$} \psfrag{t4}{$T_{s_g}$} \includegraphics[scale=.8]{figs/case2} \caption{Case 2, $\sigma_i < \sigma_g$. The optimal policy is used for the $s_i$-system, which processes jobs in order $2,3,4$. The $s_g$-system uses policy $\tilde{\pi}$ which mimics $\pi^*(s_i)$. Because job $i$ completes before job $g$, the $\tilde{\pi}$ policy is blocked until $t + \sigma_g$. At time $t+\sigma_i$, the $\tilde{\pi}$ policy simulates the processing of jobs $2$ and $3$ on machine $n_g$. The machine will idle once replica job $g$ completes and before job $3$ finishes its simulated processing. At time $\tau$, the $\tilde{\pi}$ policy is able to follow the $\pi^*(s_i)$ policy. Then the simulated jobs $2$ and $3$ are completed in an arbitrary order after the $\pi^*(s_i)$ policy completes at time $T_{s_i}$. Note that job $g$ and $i$ are processed once in each system because the original jobs have already completed processing (the replicas are processed by time $t$). }\label{fig:case2} \end{center} \end{figure} If $T_j^*(s_i)$ is the completion time of job $j$ in the $s_i$-system under optimal policy, $\pi^*(s_i)$, and $T_j$ is the completion time of job $j$ in the $s_g$-system under the $\tilde{\pi}$ policy, then $T_j = T_j^*(s_i)$ for all $j\not \in {\mathcal J}_{sim}$. Then (again employing the notation given the filtration of $\sigma_j$ and the case $\sigma_i < \sigma_g$): \begin{eqnarray}\label{eqn:gr_help} J_t^*(s_i|\sigma,\sigma_i < \sigma_g)& = & \sum_{j \in {\mathcal J}_s} w_j(T_j^*(s_i)) \nonumber \\ & = & \sum_{j \in {\mathcal J}_{sim}} w_j(T_j^*(s_i)) + \sum_{j \not \in {\mathcal J}_{sim}} w_j(T_j^*(s_i)) \nonumber \\ & \leq & \sum_{j \in {\mathcal J}_{sim}} w_j(T_j^*(s_i)) + \sum_{j \not \in {\mathcal J}_{sim}} w_j(T_j^*(s_i)) + \sum_{j \in {\mathcal J}_{sim}} w_j(T_j') \nonumber \\ & = &\sum_{j \in {\mathcal J}_{sim}} w_j(T_j^*(s_i)) + J_t^{\tilde{\pi}}(s_g|\sigma,\sigma_i < \sigma_g)\nonumber \\ & \leq &\sum_{j \in {\mathcal J}_{sim}} w_j(T_j^*(s_i)) + J_t^*(s_g|\sigma,\sigma_i < \sigma_g) \end{eqnarray} The first inequality comes from the non-negativity of rewards. The third equality comes from our coupling and the definition of the $\tilde{\pi}$ policy. The last inequality comes from the optimality of $J^*_t$. Taking expectations over the $\sigma_j$, or equivalently the $T^*_j(s_i)$, and using a little algebra for (\ref{eqn:gr_help}): \begin{eqnarray}\label{eqn:gr2} J_t^*(s_i|\sigma_i < \sigma_g) & \leq &\sum_{j \in {\mathcal J}_{sim}} E[w_j(T_j^*(s_i))|\sigma_i < \sigma_g] + J_t^*(s_g|\sigma_i < \sigma_g) \\ & \leq & \sum_{j \in {\mathcal J}_{sim}} E[\sigma_j|\sigma_i < \sigma_g]\frac{E[w_j(t + \sigma_j)|\sigma_i < \sigma_g]}{E[\sigma_j|\sigma_i < \sigma_g]} + J_t^*(s_g|\sigma_i < \sigma_g) \nonumber \\ & \leq & \max_k \frac{E[w_k(t+\sigma_k)|\sigma_i < \sigma_g]}{E[\sigma_k|\sigma_i < \sigma_g]}\sum_{j \in {\mathcal J}_{sim}} E[\sigma_j|\sigma_i < \sigma_g] + J_t^*(s_g|\sigma_i < \sigma_g)\nonumber \\ & \leq &\frac{E[w_g(t+\sigma_g)|\sigma_i < \sigma_g]}{E[\sigma_g|\sigma_i < \sigma_g]} E[\sigma_g + \sigma_l- \sigma_i|\sigma_i < \sigma_g]+ J_t^*(s_g|\sigma_i < \sigma_g)\nonumber \\ & \leq & \frac{E[w_g(t+\sigma_g)|\sigma_i < \sigma_g]}{E[\sigma_g|\sigma_i < \sigma_g]} E[\sigma_g-\sigma_i+\sigma_{\max}|\sigma_i < \sigma_g] + J_t^*(s_g|\sigma_i < \sigma_g)\nonumber \end{eqnarray} The second inequality comes from the fact that for all $j$, $T^*_j(s_i) \geq t + \sigma_j$ since the earliest time a job can begin processing is $t$ and all $w_j(t)$ are non-increasing in $t$ (Assumption \ref{as:reward}). The forth inequality comes from the definition of job $g$. Now, consider the total service time of simulated jobs. Simulated jobs begin at $t+\sigma_i$ and finish at $\tau > t+\sigma_g$. In particular, there exists some $l$ such that the first time machine $n_g$ is free under policy $\tilde{\pi}$ is $\tau < t + \sigma_g + \sigma_l$, i.e. $l$ is the last simulated job (job $2$ in Fig. \ref{fig:case2}). Hence the total service time of simulated jobs is bounded above by $(t + \sigma_g + \sigma_l) - (t + \sigma_i)$. This yields inequality four. \end{description} Combining (\ref{eqn:gr1}) and (\ref{eqn:gr2}), and taking expectations over $\sigma_i \geq \sigma_g$ and $\sigma_i < \sigma_g$ yields: \begin{eqnarray} J_t^*(s_i)& \leq &\frac{E[w_g(t+\sigma_g)]}{E[\sigma_g]}\Big( E[\sigma_g]-E[\sigma_i]+E[\sigma_{\max}]\Big ) + J_t^*(s_g)\nonumber \\ & = &\Big(1 - \frac{E[\sigma_i]}{E[\sigma_g]} + \frac{E[\sigma_{\max}]}{E[\sigma_i]}\Big)E[w_g(t+\sigma_g)] + J_t^*(s_g)\nonumber \end{eqnarray} which concludes the proof. \end{proof} Suppose we were able to process a job without using a machine. The total reward gained by the use of this `virtual machine' is greater than the reward gained without the use of it. Define $S': {\cal S} \times {\mathcal J} \rightarrow {\cal S}$ as the operation/function which reduces state $s$ to state $s_i' = S'(s,i)$ by removing job $i$ which has not yet begun processing in state $s$. That is, starting in state $s$, select a job $i$ that has not been completed. Complete job $i$ and generate its associated reward without tying up a processor. Said in notation, $\forall n: z(s_i')_n = z(s)_n$; and $\forall j \not = i$: $x(s_i')_j = x(s)_j$ and $y(s_i')_j = y(s)_j$, but $x (s_i')_i= (x(s)_i - 1)^+$ and $y(s_i')_i \not = \emptyset$ . \begin{lemma}(Virtual Machine Rewards)\label{lemma:vm_rewards} For all states $s$ and any job $i$, let state $S'(s,i)$ denote the resulting state if job $i$ were processed without occupying a processor. Also, reward $w_i$ is generated upon completion. Then: $$J_t^*(s) \leq E[w_i(t+\sigma_i)] + J_t^*(S'(s,i)),$$ \end{lemma} \begin{proof} Consider a coupling of the systems starting in state $s$ and $s'_i = S'(s,i)$ such that they see the same realizations of the service times for all jobs. Let $\pi^*(s)$ denote the optimal scheduling policy starting from state $s$. In the $s_i'$-system, we call job $i$ a `fictitious' job. It is fictitious because it does not actually exist (it has already completed) under the $s_i'$-system. Consider a policy $\tilde{\pi}$ which assumes that job $i$ is a `real' (available/not processed) job and executes the optimal policy under this assumption, i.e. it at time slot $t$, it assumes it is in state $s$ (rather than $s_i'$) and executes the optimal policy $\pi_t^*(s)$. When $\tilde{\pi}$ schedules job $i$, there is no job to actually process, so the processor will idle while it simulates the processing time for job $i$ which is identically distributed to $\sigma_i$ under the $s$-system. See Fig. \ref{fig:vmachine} for a single machine example of the $\tilde{\pi}$ and $\pi^*(s)$ policies given a sample path for service time realizations. \begin{figure}[htb] \begin{center} \psfrag{sisys}{$s$-system} \psfrag{vm}{virtual machine} \psfrag{sgsys}{$s'_i$-system} \psfrag{si}{$\sigma_1$} \psfrag{s2}{$\sigma_2$} \psfrag{s3}{$\sigma_3$} \psfrag{s4}{$\sigma_4$} \psfrag{s5}{$\sigma_5$} \psfrag{s6}{$\sigma_6$} \psfrag{t}{$t$}\psfrag{t2}{$T_{s'}$} \includegraphics[scale=.8]{figs/virtualmachine} \caption{Virtual machine: A single server scenario. Under the $s'_i$-system, job $2$ is processed on a virtual machine. The $s$-system uses policy $\pi(s')$ to optimally process all jobs $j = 1,2,3,4$. The $s_i'$-system uses policy $\tilde{\pi}$ which mimics $\pi^*(s)$. Because job $j = 2$ has already been processed on the `virtual machine', the $\tilde{\pi}$ policy idles. }\label{fig:vmachine} \end{center} \end{figure} Let $T_j$ be the completion time of job $j$ under the $\tilde{\pi}$ policy. Note that $T_i$ is the completion time of the fictitious job, $i$. Let $t_i$ denote the random time which job $i$ begins `processing' under this policy. Under our coupling, $T_j$ is precisely $t_j$ plus the processing time of job $j$ under $\pi^*(s)$ for the $s$-system. Hence, \begin{eqnarray} J_t^*(s|\sigma_j) & = & \sum_{j} w_j(T_j) \nonumber\\ & = & \sum_{j\not = i} w_j(T_j) + w_i(T_i) \nonumber \\ & = & J_t^{\tilde{\pi}}(s'_i|\sigma_j) + w_i(t_i + \sigma_i) \nonumber\\ & \leq & J_t^*(s'_i|\sigma_j) + w_i(t + \sigma_i) \nonumber \end{eqnarray} The inequality results from the non-increasing property of the reward functions in Assumption \ref{as:reward} and from the optimality of $J_t^*(\cdot)$. Taking expectations over $\sigma_j$ yields the desired result. \end{proof} We are now in position to prove the main result of this paper. Let $\Delta = \frac{E[\max_j \sigma_j]}{\min_j E[\sigma_j]}$ as in (\ref{eq:delta}). \begin{theorem}\label{theorem:bound} For all states $s \in {\cal S}$, the following performance guarantee for the greedy policy holds: $$J_t^*(s) \leq (2+\Delta)J_t^g(s).$$ \end{theorem} \begin{proof} The proof proceeds by induction on the number of jobs remaining to be processed, $\sum_{j\in {\mathcal J}} {\bf 1}_{\{y_j = \emptyset\}}$. The claim is trivially true if there is only one job remaining to be processed--the greedy and optimal policies will coincide. Now consider a state $s$ such that $\sum_j {\bf 1}_{\{y(s)_j = \emptyset\}} = K$, and assume that the claim is true for all states $s'$ with $K > \sum_j {\bf 1}_{\{y(s')_j = \emptyset\}}$. Now if $\pi_t^*(s) = \pi_t^g(s)$ the then the next state encountered and rewards generated in both systems are identically distributed so that the induction hypothesis immediately yields the result for state $s$. Consider the case where $\pi_t^*(s) \neq \pi_t^g(s)$. Denote by ${\mathcal J}_*$ and ${\mathcal J}_g$ the set of jobs processed by the optimal and greedy policies in state $s$. Note that these sets depend on the current time slot $t$ and the state $s$; however, we suppress them for notational compactness. Recall that, by Lemma \ref{lemma:freemach}, $|{\mathcal J}_*| = |{\mathcal J}_g|$. Let $A_*$ and $A_g$ denote the optimal and greedy scheduling policy, respectively, given state $s$ in time slot $t$. Taking definitions from before, we define $\tilde{S}(s,A)$ as the random next state encountered given that we start in state $s$ and action $A$ is taken. Also, define $S'(s,i)$ as state $s$ with the completion of job $i$, i.e. job $i$ is completed ($x_i = 0$) without using a processor. Define the operator $\hat{S} : {\cal S} \times {\mathcal A} \rightarrow {\cal S}$ which transforms state $s$ by tying up machines with replicated the jobs defined by $A$. That is, $\hat{s} = \hat{S}(s,A)$ is the state where jobs begin processing on the machines given by $A$, but no reward is generated for their completion and they remain to be processed at a later time (reward is generate upon this second completion). This second completion may occur prior or following the completion of the replicated job. $A$ defines which jobs are replicated and which machine they are processed on, and hence occupy--replicated jobs do not generated any reward. Put another way, $\hat{s}$ is a new state where machines are occupied for an amount of time defined by the service times of jobs in $A$. Said in notation, $x(\hat{S}(s,A))_j = x(s)_j$ and $y(\hat{S}(s,A))_j = y(s)_j$ for all $j$, while $z(\hat{S}(s,A))_n = j$ if $(j,n) \in A$ and $z(\hat{S}(s,A))_n = y(s)_n$ otherwise. We have: \begin{eqnarray} \label{eq:th1} J_t^*(s) & = & \sum_{j\in{\mathcal J}_*}E[w_j(t+\sigma_j)] + E[J_t^*(\tilde{S}(s,A_*))] \nonumber \\ & \leq & \sum_{(i,g)\in ({\mathcal J}_*,{\mathcal J}_g)}\frac{E[\sigma_i]}{E[\sigma_g]}E[w_g(t+\sigma_g)] + E[J_t^*(\tilde{S}(s,A_*))]\nonumber \\ & \leq & \sum_{(i,g)\in ({\mathcal J}_*,{\mathcal J}_g)}\frac{E[\sigma_i]}{E[\sigma_g]}E[w_g(t+\sigma_g)] + + E[J_t^*(\hat{S}(s,A_*))] \end{eqnarray} The first inequality comes from the definition of the greedy policy; the reward rate for greedy jobs is higher than for the optimal jobs. The second inequality comes from Lemma \ref{lemma:mono_job} by putting back the jobs in $A_*$. That is the machines are occupied by replicas of jobs defined in $A_*$, but the original jobs are placed back to be completed at a later date. These additional jobs generate more reward as shown in Lemma \ref{lemma:mono_job}. Continuing (\ref{eq:th1}), we switch $A_*$ with $A_g$. That is, instead of tying up the machines with replicas of the optimal jobs, they are typed up with replicas of the greedy jobs. Because $|{\mathcal J}_*| = |{\mathcal J}_g|$ and the processing times on each machines are identical, we can consider each machine individually and use Lemma \ref{lemma:scaled_revenue} so that, \begin{eqnarray}\label{eq:th2} \sum_{(i,g)\in ({\mathcal J}_*,{\mathcal J}_g)}\frac{E[\sigma_i]}{E[\sigma_g]}E[w_g(t+\sigma_g)] &+& E[J_t^*(\hat{S}(s,A_*))]\nonumber\\ & \leq & \sum_{(i,g)\in ({\mathcal J}_*,{\mathcal J}_g)}\frac{E[\sigma_i]}{E[\sigma_g]}E[w_g(t+\sigma_g)] + E[J_t^*(\hat{S}(s,A_g))] \nonumber\\ &&\sum_{(i,g)\in ({\mathcal J}_*,{\mathcal J}_g)} E[w_g(t+\sigma_g)]\Big(1 - \frac{E[\sigma_i]}{E[\sigma_g]} + \frac{E[\sigma_{\max}]}{E[\sigma_g]}\Big) + \nonumber \\ & = & \sum_{g\in{\mathcal J}_g} E[w_g(t+\sigma_g)]\Big(1 + \frac{E[\sigma_{\max}]}{E[\sigma_g]}\Big) + E[J_t^*(\hat{S}(s,A_g))] \end{eqnarray} Continuing (\ref{eq:th2}), we now complete the greedy jobs without occupying any machines: \begin{eqnarray}\label{eq:th3} \sum_{j\in{\mathcal J}_g} E[w_g(t+\sigma_g)]\Big(1 + \frac{E[\sigma_{\max}]}{E[\sigma_g]}\Big) &+ & E[J_t^*(\hat{S}(s,A_g))]\nonumber \\ & \leq & \sum_{g\in{\mathcal J}_g} E[w_g(t+\sigma_g)]\Big(2 +\frac{E[\sigma_{\max}]}{E[\sigma_g]}\Big) + E[J_t^*(\tilde{S}(s,A_g))] \nonumber \\ & \leq & \sum_{g\in{\mathcal J}_g}\Big(2 + \frac{E[\sigma_{\max}]}{\min_kE[\sigma_k]}\Big) E[w_g(t+\sigma_g)] + \nonumber \\ &&\Big(2 + \frac{E[\sigma_{\max}]}{\min_kE[\sigma_k]}\Big)E[J_t^g(\tilde{S}(s,A_g))] \nonumber \\ & =& \Big(2+ \Delta\Big)J_t^g(s)\nonumber \end{eqnarray} The first inequality comes from use of `virtual machines' for the greedy jobs under Lemma \ref{lemma:vm_rewards}. The second inequality comes from the induction hypothesis. This concludes the proof. \end{proof} \section{Special Cases}\label{sec:special} As shown in \cite{gittins,walrand} the greedy policy is optimal, for linear or exponential decaying reward functions. Under a few other special cases, the bound in Theorem \ref{theorem:bound} can be improved. \subsection{Identical Processing Times} Suppose that all job service times are independent and identically distributed, i.e. in the case of Geometric service times, $p_j = p$ for all $j$. In general, there is no closed form equation for $E[\sigma_{\max}]$; however, in this case, the bound can be improved to a factor of $2$. To do this, Lemma \ref{lemma:scaled_revenue} needs to be modified. \begin{lemma}(Greedy Revenue, I.I.D processing times)\label{lemma:scaled_revenue_identical} Consider a state $s_t = s$ in time slot $t$ and let $g$ denote the index of a greedy job, i.e. $g = \argmax_{j\in {\mathcal J}_s} \frac{E[w_j(t(s)+\sigma_j)]}{E[\sigma_j]}$ for all jobs which are mid-processing or have not started (${\mathcal J}_s = \{k \in {\mathcal J}|x_k(s) = 1\} $). Denote by $s_g$ and $s_i$ two states which are related to state $s$ as follows: $x(s_i)_j = x(s_g)_j = x(s)_j$ and $y(s_i)_j = y(s_g)_j = y(s)_j$ for all $j$, $z(s_i)_n = z(s_g)_n = z(s)_n$ for all $n \not = n_g$, and $z (s_g)_{n_g} = g$ while $z(s_i)_{n_g} = i$ for some arbitrary job index $i$ and machine $n_g$. That is in state $s_i$, machine $n_g$ is occupied by a replica of job $g$; and in state $s_i$, machine $n_g$ is occupied by a replica of job $i$. Then, $$E[J_t^*(s_i)] = E[J_t^*(s_g)]$$ \end{lemma} \begin{proof} Couple the systems such that they see the same realizations for service times of job $i$ and job $g$ which are currently occupying machine $n_g$. This coupling is possible since the jobs are i.i.d. Therefore, under this coupling there is no difference between state $s_i$ and $s_g$ since these `jobs' are only occupying the machine but are not generating any rewards. Hence, $E[J_t^*(s_i)] = E[J_t^*(s_g)]$. \end{proof} Now we are able to prove an improved bound on the performance of the greedy policy. \begin{theorem}\label{theorem:bound_up} Let the service time for job $j$ be distributed according to density function $f_j(\sigma)$. If all job service times are independent and identically distributed according to $f(\sigma)$, i.e. $f_j(\sigma) = f(\sigma)$ $\forall j$, then for all states $s \in {\cal S}$, the greedy policy is guaranteed to be within a factor of $2$ of optimal: $$J_t^*(s) \leq 2J_t^g(s).$$ \end{theorem} \begin{proof} Under this scenario, Lemma \ref{lemma:scaled_revenue} can be replaced by Lemma \ref{lemma:scaled_revenue_identical} in the proof of Theorem \ref{theorem:bound}. Hence, $E[J_t^*(\hat{S}(s,A_*))] = E[J_t^*(\hat{S}(s,A_g))]$ since the distribution of completion times is identical, the amount of time a processor is busy is independent of which job it is processing. Instead of replicating the entire proof here, we examine how (\ref{eq:th1}), (\ref{eq:th2}), and (\ref{eq:th3}) change. The only difference for (\ref{eq:th1}) is that $E[\sigma_j] = E[\sigma_i]$ for $i,j$ which allows for a slight simplification. \begin{eqnarray} J_t^*(s) & = & \sum_{j\in{\mathcal J}_*}E[w_j(t+\sigma_j)] + E[J_t^*(\tilde{S}(s,A_*))] \nonumber \\ & \leq & \sum_{(i,g)\in ({\mathcal J}_*,{\mathcal J}_g)}\frac{E[\sigma_i]}{E[\sigma_g]}E[w_g(t+\sigma_g)] + E[J_t^*(\tilde{S}(s,A_*))]\nonumber \\ & \leq & \sum_{g\in{\mathcal J}_g}E[w_g(t+\sigma_g)] + + E[J_t^*(\hat{S}(s,A_*))] \end{eqnarray} Now, with improvement to Lemma \ref{lemma:scaled_revenue} in Lemma \ref{lemma:scaled_revenue_identical}, (\ref{eq:th2}) is reduced significantly \begin{eqnarray} \sum_{g \in {\mathcal J}_g}E[w_g(t+\sigma_g)] + E[J_t^*(\hat{S}(s,A_*))] = \sum_{g\in{\mathcal J}_g}E[w_g(t+\sigma_g)] + E[J_t^*(\hat{S}(s,A_g))] \end{eqnarray} Finally, utilizing Lemma \ref{lemma:vm_rewards} and completing/generating rewards for the greedy jobs gives: \begin{eqnarray} \sum_{g\in{\mathcal J}_g}E[w_g(t+\sigma_g)] + E[J_t^*(\hat{S}(s,A_g))] & \leq & 2\sum_{g\in{\mathcal J}_g}E[w_g(t+\sigma_g)] + E[J_t^*(\tilde{S}(s,A_g))] \\ & \leq & 2\sum_{g\in{\mathcal J}_g}E[w_g(t+\sigma_g)] + 2E[J_g^*(\tilde{S}(s,A_g))] \nonumber \\ & =& 2J_t^g(s)\nonumber \end{eqnarray} \end{proof} In the case of i.i.d. service times, the greedy policy corresponds to scheduling the job with the highest expected rewards over their identical completion times. While this seems to be an intuitive policy, the following example shows what can go wrong. \begin{example}\label{ex:iid} Consider the case with $2$ jobs and $1$ machine ($J = 2$ and $N = 1$). We begin at $t=0$. Assume that neither job has begun processing so that $x_1 = x_2 = 1$ and $y_1 = y_2 = \emptyset$. The service times for job $1$ and $2$ are both deterministic and equal to $1$. The reward functions are: \[ \begin{split} {\rm For} \ j=1:& \ w_j(t) = \begin{cases} 1-\epsilon, & \text{$t = 1$}\\ 0, & \text{$t > 1$} \end{cases} \\ {\rm For} \ j=2:& \ w_j(t) = \begin{cases} 1, & \text{$\forall t$} \end{cases} \end{split} \] for $\epsilon > 0$. So that the completion of job $1$ only results in revenue if it is completed in the first time slot, but job $2$ results in the same revenue, regardless of which time slot it is completed in. Therefore, the reward rates are: \begin{eqnarray} \frac{E[w_1(t+\sigma_1)]}{E[\sigma_1]} & =& \begin{cases} 1-\epsilon, & \text{$t = 1$}\\ 0, & \text{$t > 1$} \end{cases} \nonumber\\ \frac{E[w_2(t+\sigma_2)]}{E[\sigma_2]} & =& \begin{cases} 1, & \text{$\forall t$} \end{cases} \nonumber \end{eqnarray} Clearly, the greedy policy is to schedule job $2$ and then job $1$ since the reward rate for job $2$ is greater than that for job $1$ ($1+\epsilon > 1$). However, when job $1$ completes at $t = 2$, it generates no reward since $w_1(2) = 0$. This results in reward $1$. On the other hand, the optimal policy realizes the reward of job $1$ is degrading and schedules it first and schedules job $2$ second. This results in reward $2-\epsilon$. We thus see that $J_t^*(s) = (2-{\epsilon}) J_t^{g}(s)$ here. \end{example} In light of the example just shown, the bound in Theorem \ref{theorem:bound_up} is \emph{tight}. \subsection{Slowly Decaying Rewards}\label{ssec:asym_opt} We have proven a worse case bound for arbitrary decaying rewards. If the time-scale of decay is very long compared to the time-scale of job completion times, then the rewards would be nearly constant during the processing time of a job. In particular, as the decay goes to zero over the time-scale of job completion times, the performance of the greedy heuristic approaches the performance of the optimal policy. We will now formally define the time-scale of decay. Consider a difference equation specification for the time-scale of decay. Let $$\delta = \max_{t,k,m} E[w_k(t) - w_k(t + \sigma_m)] \geq 0.$$ We will show that as $\delta \rightarrow 0$, $J_t^g(s) \rightarrow J_t^*(s)$. To do this, we must start with a few preliminary results. The first is, as $\delta \rightarrow 0$, rewards become invariant to the completion time. Rewards are generated upon the completion of each job. However, as $\delta \rightarrow 0$, the rewards generated at the completion time of a job is nearly the rewards that would have been generated had the job had $0$ processing time. \begin{lemma}(Time-Invariant Rewards)\label{lemma:tinvariant} For any jobs $i,j$ and time slot $t$, as the time-scale of decay, $\delta$, approaches $0$, the reward generated for completing job $j$ is invariant to shifts in time by the service time of job $i$, $\sigma_i$. In particular, $$\lim_{\delta \rightarrow 0} E[w_j(t + \sigma_i)] = w_j(t)$$ \end{lemma} \begin{proof} For any job indices $i,j$ and time slot $t$: \begin{eqnarray} \big | E[w_j(t + \sigma_i)] - w_j(t) \big | & \leq & \max_{\tau,k,m} \big | E[w_k(\tau + \sigma_m)] - w_k(\tau) \big | \nonumber \\ & = & \delta \end{eqnarray} which implies that $ \big | E[w_j(t + \sigma_i)] - w_j(t) \big | \rightarrow 0$ as $\delta \rightarrow 0$. \end{proof} Because rewards are nearly constant over the time-scale of job completions, starting a job $\sigma_j$ time slots later does not significantly reduce the aggregate reward accrued. The following lemma is similar to Lemma \ref{lemma:scaled_revenue} for slowly decaying reward functions. Define $\hat{S}(s,A)$ as in Section \ref{ssec:bound}, so that $\hat{S}(s,A)$ is the state where jobs are processed on the machines given by $A$, but they are not removed and no reward is generated for this initial processing. These replica jobs occupy the machines, making them unable to process other jobs in the meantime. However, they do not generate reward. In notation, $x(\hat{S}(s,A))_j = x(s)_j$ and $y(\hat{S}(s,A))_j = y(s)_j$ for all $j$, while $z(\hat{S}(s,A))_n = j$ for all $(j,n) \in A$ and $z(\hat{S}(s,A))_n = z(s)_n$ otherwise. \begin{lemma}(Delayed Machine)\label{lemma:delayedmach} Let $\hat{s} = \hat{S}(s,A)$ denote the resulting state if machines in $A$ are occupied, but all the jobs have the same (un)processed state as in state $s$. Then, starting in any state $s$ and given action $A$, the difference in optimal reward-to-go generated in states $s$ and $\hat{s}$ goes to $0$ as the time-scale of decay, $\delta$, goes to $0$, i.e. $$\big | J_t^*(s) - E[J_t^*(\hat{S}(s,A) ]\big | \rightarrow 0\textrm{ as }\delta \rightarrow 0$$ \end{lemma} \begin{proof} To begin, note that $J_t^*(s) \geq E[J_t^*(\hat{S}(s,A))]$. To see this, we couple the job completion times. Let $\tilde{\pi}$ denote a policy starting from state $s$, but mimicking the optimal policy starting from state $\hat{S}(s,A) = \hat{s}$, $\pi^*(\hat{s})$. Therefore, under the $\tilde{\pi}$ policy, machine $n_k$ will idle for $\sigma_k$ time slots before proceeding if $(k,n_k) \in A$. The $s$-system simply delays processing any new jobs until the replica jobs in the $\hat{s}$-system are completed. In this case, the completion time for jobs will be identical under the $\tilde{\pi}$ and $\pi^*(\hat{s})$ policies. Hence $J_t^*(\hat{s}) = J_t^{\tilde{\pi}}(s) \leq J_t^*(s)$, by the optimality of $J_t^*(s)$. Now to show the convergence result, couple the job completion times under the $s$ and $\hat{s}$-systems. Let $\sigma_k^* = \max_{(k,n) \in A} \sigma_k$ be the maximal service time for jobs in $A$. Consider a policy $\tilde{\pi}$ for the $\hat{s}$-system which idles for $\sigma^*_k$ time slots and begins processing new jobs at time $t' = t+\sigma_k^*$, but assumes that $t' = t$. Therefore, $\tilde{\pi}$ coincides precisely with $\pi^*(s)$ shifted in time by $\sigma_k^*$. In other words, the $\tilde{\pi}$ policy waits until $t'$ at which point all replica jobs are completed and then begins processing new jobs as if no time has passed and $t' = t$. For the $s$-system, let $T_j^*$ be the completion time of job $j$ under the optimal policy $\pi^*(s)$. Then $T_j^{\tilde{\pi}} = T^*_j + \sigma_k^*$ is the completion time for job $j$ under the $\tilde{\pi}$ policy. Now, given some $\epsilon > 0$ \begin{eqnarray} \big | J_t^*(s) - E[J_t^*(\hat{s})] \big | & \leq & \big |J_t^*(s) - E[J_t^{\hat{\pi}}(\hat{s}) ]\big | \nonumber \\ & = & \Big |E\big [\sum_{j}w_j(T^*_j)\big ] - E\big [\sum_j w_j(T^{\tilde{\pi}}_j)\big ] \Big | \nonumber \\ & = & \Big |\sum_{j}E\big [w_j(T^*_j) - w_j(T^*_j + \sigma_k^*)\big ] \Big | \nonumber \\ & = & \Big |\sum_{j}E\Big[w_j(T^*_j) - E\big[w_j(T^*_j + \sigma_k^*)\big ]\Big] \Big | \nonumber \\ & \leq & J\delta < \epsilon \end{eqnarray} The inequality comes from Lemma \ref{lemma:tinvariant} and because $\delta \rightarrow 0$ there exists $\delta < \epsilon/J$. \end{proof} Now, we are in position to prove that the performance of the greedy policy approaches the performance of the optimal policy when the decay of rewards is slow compared to the job completion time. \begin{theorem}(Slowly Decaying Rewards)\label{theorem:slowdecay} For any state $s\in {\cal S}$, as the time-scale of decay goes to $0$, i.e $\delta \rightarrow 0$, the performance of the greedy policy approaches that of the optimal policy. $$J_t^g(s) \rightarrow J_t^*(s).$$ \end{theorem} \begin{proof} The proof is by induction on the number of jobs remaining to begin processing. Clearly, when only one job remains the greedy and optimal policies coincide. Now we assume it is true for $K-1$ jobs remaining and show that it is true for $K$ jobs. Denote by ${\mathcal J}_*$ and ${\mathcal J}_g$ the set of jobs processed by the optimal and greedy policies in state $s$. Recall that, by Lemma \ref{lemma:freemach}, $|{\mathcal J}_*| = |{\mathcal J}_g|$. Let $A_*$ and $A_g$ denote the optimal and greedy scheduling policy, respectively. As before, we define $\tilde{S}(s,A)$ which is the next state given we start in state $s$ and take action $A$ and $\hat{S}(s,A)$ which is the state with machines in $A$ occupied by replica jobs which generate $0$ reward. Suppose we are given $\epsilon > 0$. Define $\delta_{\epsilon,1}$ such that for all $\delta < \delta_{\epsilon,1}$, $\big | J_t^*(s|\sigma_j) - J_t^*(\hat{S}(s,A)|\sigma_j) \big | < \epsilon/2$; this is possible due to Lemma \ref{lemma:delayedmach}. Define $\delta_{\epsilon,2}$ such that for all $\delta < \delta_{\epsilon,2}$, $\big | J_t^*(s') - J_t^g(s') \big | < \epsilon/2$ for any $s'$ with $K-1$ jobs remaining; this is possible due to our inductive hypothesis. Then let $\delta_{\epsilon} = \min\{\delta_{\epsilon,1},\delta_{\epsilon,2}\}$. For any $\delta < \delta_{\epsilon}$: \begin{eqnarray} J_t^*(s) &\leq& E[J_t^*(\hat{S}(s,A_g))] +\epsilon/2 \nonumber \\ &\leq&\sum_{j\in{\mathcal J}_g}E[w_j(t+\sigma_j)] + E[J_t^*(\tilde{S}(s,A_g))] +\epsilon/2 \nonumber \\ &\leq&\sum_{j\in{\mathcal J}_g}E[w_j(t+\sigma_j)] + E[J_t^g(\tilde{S}(s,A_g))] +\epsilon \nonumber \\ & = & J_t^g(s) + \epsilon \nonumber \end{eqnarray} The first inequality is due to Lemma \ref{lemma:delayedmach}, for state $s$ and action given by $A_g$. The second inequality is by Lemma \ref{lemma:vm_rewards} for removing the greedy jobs. The third inequality is by the inductive hypothesis. By the optimality of $J_t^*$, $\big | J_t^*(s) - J_t^g(s) \big | = J_t^*(s) - J_t^g(s)$. So for $\delta < \delta_{\epsilon}$, $\big | J_t^*(s) - J_t^g(s) \big | < \epsilon$, which proves our claim. \end{proof} This result is intuitive because as the time-scale of decay becomes negligible to the time-scale of job completion times, rewards can be viewed as essentially constant. As such, it does not matter which order jobs are completed, since all will be completed. Hence, any policy, and certainly the greedy policy, is nearly optimal. However, the convergence rate to optimality will vary across policies. \section{Performance Evaluation}\label{sec:perfeval} In the previous sections, we have shown performance guarantees for a greedy policy when scheduling jobs with decaying rewards. In light of Example \ref{ex:suboptimality} and Theorem \ref{theorem:bound}, the loss in performance due to use of the greedy policy can be at least $\Delta +1$ but can do no worse that $\Delta + 2$. In this section, we show that, in practice, the greedy performance is likely to be much better. \begin{figure}[hbt] \begin{center} \includegraphics[scale=.5]{figs/wdecay} \caption{Different types of decaying functions.}\label{fig:decayplots} \end{center} \end{figure} In order to enable computation of an optimal policy we assume that the number of jobs is finite and small ($2$-$10$). Even with a finite number of jobs, $|{\cal S}|$ grows exponentially fast in several problem parameters which forces us to limit the size of the problem instances we consider. In particular, we consider problems with a single machine, $M=1$, and jobs with finite deadlines less than $100$. That is no reward is accrued after $t = 100$. We assume job completion times are Geometric with $p_j$ evenly distributed between $p_{min}$ and $p_{max} = .9$. Since there is no closed form distribution for $\sigma_{\max}$, see Appendix \ref{ap:geo} for how to find an upper-bound to $E[\sigma_{\max}]$ and, subsequently, an upper-bound to $\Delta$. We consider a number of decaying reward functions depicted in Fig. \ref{fig:decayplots}. The constants defining each reward function are drawn uniformly; all experimental results are averaged over $100$ different realizations of these constants, with $1000$ experiments for each such set. In Table \ref{table:perf}, we summarize the performance of the greedy policy for the reward functions shown in Fig. \ref{fig:decayplots}. In this case, $p_{\min} = .1$ and $p_{\max} = .9$; $\min_j E[\sigma_j] = \frac{1}{.9} = 1.11$ and $E[\max_j \sigma_j] < 27.3$, therefore, $2+\Delta< 26.6$. We can see that while the optimal policy achieves larger reward than the greedy policy, the gains are within a factor of $1.20$--much better than the guarantee provided by Theorem \ref{theorem:bound}. Because we have finite deadlines for each job, there exists some $T_{max}$ such that for all $j$, $w_j(t) = 0$ for all $t > T_{max}$. Therefore, the reward function with exponential decay is slightly modified from the standard notion of exponential decay where $w_j(t) \rightarrow 0$, but $w_j(t) > 0$ for any $t < \infty$. Hence, the greedy policy is \emph{not} optimal for this exponential decay with finite deadline. \begin{table}[htbp] \begin{center} \begin{tabular}{|l|c|c|c|} \hline &\multicolumn{3}{c|}{$J_t^*/J_t^g$} \\ \cline{2-4} Type & $J = 2$ & $J = 5$ & $J = 8$ \\ \hline \hline Step & 1.0065 & 1.0931 & 1.1287 \\ \hline Linear & 1.0133 & 1.0576 & 1.1289 \\ \hline Exponential & 1.0609 & 1.0433 & 1.0590 \\ \hline Parabolic & 1.0265 & 1.0382 & 1.0667 \\ \hline 2-step & 1.0218 & 1.1007 & 1.1520 \\ \hline \end{tabular} \end{center} \caption{Performance of Greedy policy versus Optimal Policy for different types of decaying reward functions for $J$ jobs.} \label{table:perf} \end{table} \begin{figure}[htb] \begin{center} \psfrag{N}[][][1.1]{$J$} \psfrag{JoJg}[][][1.4][-90]{ $\frac{J_t^*}{J_t^g}$ } \includegraphics[scale=.5]{figs/runN} \caption{Performance loss ($\frac{J_t^*}{J_t^g}$) as the number of jobs ($J$) increases.}\label{fig:runN} \end{center} \end{figure} It is interesting to note that the performance of the greedy policy seems to degrade as the number of jobs increases. We examine this more closely in Fig. \ref{fig:runN} under a step function reward function where rewards are constant until a fixed deadline as in Fig. \ref{fig:decayplots}a. Clearly, the greedy and optimal policies coincide when there is only one job. As the number of jobs increases, the performance of the greedy policy degrades; however, the loss in performance is much less than the bound of $2+\Delta< 26.6$ guarantees. $2+\Delta$ is a worse-case bound and while there are degenerate cases whose performance approaches this bound; it seems that in practice, the performance of the greedy policy is likely to be much better. \begin{figure}[htb] \begin{center} \psfrag{pmaxpmin}[][][1.1]{$\Delta_{UB}$} \psfrag{JoJg}[][][1.4][-90]{$\frac{J_t^*}{J_t^g}$} \includegraphics[scale=.5]{figs/runpax} \caption{Performance loss ($\frac{J_t^*}{J_t^g}$) as $\Delta_{UB}$ increases.}\label{fig:runpmax} \end{center} \end{figure} From Theorem \ref{theorem:bound}, the performance of the greedy policy is dependent on $\Delta$, the ratio between the largest and smallest expected service times. In our previous experiments, we have seen that $\frac{J_t^*}{J_t^g} \ll 2+\Delta$. We now examine if the performance of the greedy policy will vary significantly as we change $\Delta$. We fix $p_{\max} = .9$ and vary $p_{\min}\in [.01,.8]$; this varies the upper-bound of $\Delta$ (as calculated in Appendix \ref{ap:geo}), $\Delta_{UB} \in [5.2,265.9]$. In Fig. \ref{fig:runpmax}, we see how the performance of the greedy policy ($\frac{J_t^*}{J_t^g}$) varies with $\Delta$. As expected, as $\Delta$ increases, so does the loss in performance. However, it is interesting to note that $\Delta$ must be very large before the degradation in performance is significant. In fact, for a large range of $\Delta_{UB} \in [1,100]$, $\frac{J_t^*}{J_t^g}$ is nearly constant and the greedy policy performs within $10\%$ of optimal. Even when $\Delta = 260$, $\frac{J_t^*}{J_t^g} < 1.3$. A loss of $30\%$ is \emph{much} better than the theory guarantees. Depending on the system parameters, $\Delta$ can be arbitrarily large which would lead to arbitrarily large degradation in performance of the greedy policy. While we have seen via Example \ref{ex:suboptimality} that the performance of the greedy policy can be highly dependent on $\Delta$, we suspect this to be a degenerate example. We expect that in practice, the performance of the greedy policy to be closer to performance of the optimal policy. \section{Conclusion}\label{sec:conc} In this paper, we have studied online stochastic non-preemptive scheduling of jobs with decaying rewards. Arbitrary decaying reward functions enables this model to capture various distastes for delay which are more general than the standard exponential or linear decay as well as fixed (random or deterministic) deadlines. Using stochastic Dynamic Programming techniques, we are able to show that a greedy heuristic is guaranteed to be within a factor of $\Delta + 2$ of optimal where $\Delta = \frac{E[\max_j \sigma_j]}{\min_j E[\sigma_j]}$ is the ratio of largest to shortest service times. While there exist degenerate scenarios where the performance loss of the proposed policy is near this worse-case bound, we expect that the performance loss to be much smaller for many practical scenarios of interest. This is a first look at non-preemptive scheduling with arbitrary decaying rewards. Some questions that remain are how to account for job arrivals and processor dependent service times. When there are job arrivals, due to the non-preemptive service discipline, it may be optimal for a machine to idle in order to allow the machine to be free upon arrival of the new job. However, doing so requires some estimate or knowledge of future jobs arrivals, which may not be available. Also with processor dependent service times, optimal policies may also call for idling. Consider a scenario where one machine is much faster than the rest. Then an optimal policy may process \emph{all} jobs on this fast machine, causing the other machines to idle. Allowing for idling policies significantly complicates the optimization problem at hand. One option is to only consider non-idling policies and maximize reward over this class of policies. It can be shown via a highly degenerate example that requiring non-idling service disciplines can significantly degrade performance. However, for many scenarios this constraint is very natural. For instance, in service applications, such as health-care facilities, making customers (patients) wait when there are available servers (doctors) is unlikely to be tolerated. These are just some extensions to this general model we have analyzed. In this paper, we have considered the performance of an online scheduling algorithm for jobs with arbitrary decaying rewards. We have shown a worse-case performance bound for this policy compared to the optimal \emph{off-line} algorithm. While there are some rare instances when the loss in performance of the proposed greedy policy is significant, in practice, we expect the performance loss to be small. This, along with the simplicity of this algorithm, makes it highly desirable for real world implementation. \bibliographystyle{IEEEtran}

BkiUbY_xaJiQnbugTf0s

\section{Introduction}\label{sec:intro} Models of quantum systems interacting with their environment are of primary importance in the field of open quantum systems \cite{breuer:open-quantum,Weiss:open-quantum2}. In the last decades, several experimental achievements, both in the field of quantum information and quantum simulation, have stimulated a renewed interest in these problems \cite{harris:d-wave}. In Adiabatic Quantum Computation (AQC) protocols, the interaction of a system of mutually coupled qubits with a thermal bath can lead to noticeable changes in the mechanism of defects creation, affecting the success probability of the computation \cite{Smelyanskiy:decoherence,Keck:dissipationadiab}.~However, decoherence effects are not necessarily detrimental to the success of AQC \cite{childs:robustness,albash:decoherence}; further, it has also been shown that the interaction with an external bath at finite temperature can have beneficial effects on the success of AQC \cite{amin:thermal-qa,dickson:thermal-qa,Gianluca}, though the underlying mechanism providing such an advantage over closed-system dynamics, as well as its dependence on the nature of quantum critical point remains rather unclear \cite{arceci:dissipative-lz,mishra:thermal-gap-qa}. Several prototypical models of driven open quantum systems \cite{Grifoni:drivenquantum} can prove useful in the study of out-of-equilibrium quantum thermodynamics \cite{SassettiWeiss:energyexchange, Esposito:Noneq1,Campisi:Noneq2}, a well-estabilished field that in the last decades has regained popularity since the discovery of its links with quantum information theory \cite{Goold:entropyandinfo}. In this context, entropy production in systems interacting with a heat bath as well as with finite, engineered baths has been the focus of recent experimental and theoretical work \cite{Brunelli:entropyexp,Manzano:entropy}. Moreover, the study of energy and heat exchanges mechanism between an externally driven quantum system and one or more heat baths is relevant for the theoretical understanding of quantum heat engines \cite{CampisiPekolaFazio:quantheng,Manzano:entropy2}. The prototypical model of a two level system (TLS) interacting with a thermal bath, also known as Spin Boson Model (SBM) has been considered in several works, aimed at describing its out-of-equilbrium dynamics and quantum phase transition arising from different kinds of dissipation \cite{caldeira:caldeira-leggett,Leggett,Sassetti:univ,LeHurNRG1,LeHurImpurity}. Energy exchange in dissipative driven TLS has also been considered \cite{SassettiWeiss:energyexchange} from weak to strong coupling regimes. Further, the effect of the environment on the ground state topology of SBM has been studied \cite{henriet:Topologysbm}, showing that only local geometric properties are noticeably affected, while global properties remain unchanged as long as the system is in the delocalized phase, i.\,e.~the coupling to the bath degrees of freedom does not exceed the critical value ($\alpha < \alpha\ped{c}=1$). In addition, the bath-induced non-adiabaticity has been addressed \cite{henriet:Topologysbm}, and at strong coupling regime the crossover from quasi-adiabatic to non-adiabatic dynamics due to the environment has been studied. While this picture holds true in the quasi-adiabatic regime, it is not a priori clear how the environment affects the dynamics of the TLS at not-so-low sweep velocities; furthermore, different forms of the coupling could lead to changes in this scenario, as stressed in several works \cite{thorwart1,thorwart4,arceci:dissipative-lz} addressing the dissipative dynamics of Landau-Majorana-St\"uckelberg-Zener (LMSZ) model \cite{landau:crossings,Majorana1932,zener:crossings}. In this paper, we study the dissipative dynamics of a two level system, i.\,e.~a qubit subject to external driving fields and interacting with its environment, from weak to strong coupling regimes. We first address the static field case, focusing on the biased SBM in contact with an Ohmic bath, a widely-studied model which describes the effects of dissipation and decoherence on a TLS. In addition, we consider a time-dependent protocol, which has been recently implemented in solid-state devices in order to realize dynamical measurements of topological phase transitions \cite{Gritsev,RoushanNature:topotrans}. We employ an exact numerical approach based on a truncation scheme of the bath Hilbert space and on the Short Iterative Lanczos (SIL) diagonalization \cite{DeFilippisOc,DeFilippisSpinLattice,nature:Giulio,letters:DeFilippisPumpProbe,letters:DeFilippisSharpTrans,Lanczos1,Lanczos2}, which allows us to follow the dynamics of the observables of both the qubit system and the environment, without the need of tracing out the bath degrees of freedom. This method has proved useful in reproducing the correct physical behavior of the SBM in the weak coupling regime \cite{Cangemi:SIL}, and we show how the inclusion of higher order excitation processes in the physical description can noticeably widen the range of coupling strengths to be investigated, allowing us to describe the physics from intermediate to strong coupling regime where no analytical scheme is known to hold. We show that in the case of unbiased SBM our approach is successful in describing the crossover from coherent to incoherent behavior of magnetization dynamics, occurring at the Toulouse point at $\alpha=1/2$. In addition, taking advantage of our technique we find the dynamical evolution of the mean population of the bath modes as a function of time, and we observe a change from non-resonant to resonant response at fixed coupling strengths. Furthermore, in the case of the driven qubit, we show a non-monotonic behavior of the fidelity at fixed final times as a function of the dissipation strength: this behavior, which is found to depend on the detailed form of interaction with the environment, signals the complex interplay between non-adiabatic effects due to the external time-dependent driving force and dissipation. The paper is organized as follows: in section \ref{sec:modelHam} we introduce the general Hamiltonian scheme we intend to study, focusing on the characteristic form of coupling with the bath. In section \ref{sec:Toulouse},\ref{sec:Magnetic} we discuss two different prototypical models of a TLS interacting with external environment, subject to static and time-dependent external fields, and we analyze numerically their open dynamics from weak to strong coupling regimes. We test our predictions by a comparison with well known theoretical approximations, and discuss their possible physical intepretations. Eventually, in \ref{sec:conclusions} we discuss viable extensions of this work along with future perspectives. \section{Model Hamiltonian}\label{sec:modelHam} We focus on a TLS system, i.\,e.~a qubit subject to time-dependent external fields, which is interacting with its environment. The qubit is described by a time-dependent Hamiltonian $H\ped{S}(t)$. The total Hamiltonian of the interacting system is written as \begin{equation}\label{eq:TotalHam} H(t)= H\ped{S}(t) + H\ped{B} + H\ped{I}, \end{equation} where $H\ped{B}$ is the free Hamiltonian of the bath and $H\ped{I}$ is the coupling energy between the qubit and the bath. Making use of the spin $1/2$ Pauli matrices $\boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$, the qubit Hamiltonian can be written in a very simple form, \begin{equation}\label{eq:QubitHam} H\ped{S}(t)=-\frac{1}{2}\boldsymbol{h}(t)\cdot \boldsymbol{\sigma}, \end{equation} where we take as $\boldsymbol{h}$ a time-dependent magnetic field vector which at fixed time $t$ points in a given direction of the three-dimensional coordinate space. We conventionally adopt as a basis for the qubit states, i.\,e.~the computational basis, the set of eigenstates of $\sigma_z$ operator, namely $\sigma_z\ket{\hat{z}; \pm}=\pm\ket{\hat{z}; \pm}$: as a consequence, the component of $\boldsymbol{h}$ along the $\hat{z}$ axis acts as a bias on the energy levels of the two states, while linear combinations of $\sigma_\pm$ operators give rise to tunnelling between these two states. As usual, we model our bath by means of a collection of bosonic oscillators of frequency $\omega_k$, and the Hamiltonian $H\ped{B}$ can be conveniently written in terms of creation (annihilation) operators $b^{\dagger}_k (b^{\phantom{\dagger}}_k) $ obeying to bosonic commutation relations $\comm{b^{\dagger}_k}{b^{\phantom{\dagger}}_l}=\delta_{kl}$, \begin{equation}\label{eq:BathHam} H\ped{B}=\sum_k\omega_k b^{\dagger}_k b_k, \end{equation} The time-independent interaction term couples the qubit with the external bath along a given direction $\hat{n}$ as follows \begin{equation}\label{eq:IntHam} H\ped{I}=\frac{1}{2}\boldsymbol{\sigma}\cdot\boldsymbol{\hat{n}}\sum_k \lambda_k (b^{\dagger}_k + b_k ), \end{equation} where $\lambda_k$ is the coupling strength with the k-th oscillator. The bath properties are described by its spectral density $J(\omega)$ \cite{Leggett}: it can be written as a sum over discrete frequencies of the bath modes, ranging from $0$ up to a cutoff frequency $\omega\ped{c}$, which is the greatest energy scale of the system. In the continuum limit, the spectral density takes the form \begin{equation}\label{eq:SpectDensity} J(\omega)= \sum_k \lambda_k^2 \delta(\omega-\omega_k)= 2 \alpha \frac{\omega^s}{\omega^{s-1}_c}\eu^{-\frac{\omega}{\omega\ped{c}}}, \end{equation} Here the adimensional parameter $\alpha$ measures the strength of the dissipation, while the parameter $s$ distinguishes among three different kinds of dissipation that have been studied in the recent literature \cite{Weiss:open-quantum2,Leggett,bulla:nrg2}: Ohmic ($s=1$), sub-Ohmic ($s<1$) and super-Ohmic case ($s>1$). The expression for the coupling term in Eq.~\eqref{eq:IntHam} is rather general: it has been proposed to study the effect of a thermal environment on qubits subject to different time-dependent protocols, including the widely studied LMSZ sweeps \cite{Vavilov:Decoherence,thorwart4,arceci:dissipative-lz}; in the latter case, it has been argued that the introduction of a "transverse" coupling direction, i.\,e.~orthogonal to the time-dependent bias field, could provide a simple theoretical explanation of the experimental findings regarding D-Wave Rainier's chip \cite{dickson:thermal-qa}. In the following we extend our analysis to a qubit coupled to a thermal bath along different spatial directions. As a first example, we consider the dissipative dynamics of a widely studied model in the spin-boson literature, which accounts for decoherence and dissipation effects on the qubit dynamics \cite{Leggett,LeHurImpurity,LeHurNRG1}; here the qubit is coupled with the heat bath along $\hat{z}$ axis, playing the role of an additional bias field on the computational basis states $\ket{\hat{z}; \pm}$; if the external fields are time-independent, in the case of Ohmic dissipation ($s=1$), as the coupling strength reaches its critical value $\alpha_c=1$ this model predicts the occurrence of a Quantum Phase Transition (QPT) of Kosterlitz-Thouless kind. As a second example, we analyze a time-dependent protocol where the qubit is subject to a rotating magnetic field $\boldsymbol{h}(t)$ performing a sweep in a fixed plane, and the dissipation can take place along two particular directions in the plane of rotation. \section{Spin Boson model}\label{sec:Toulouse} We study the dynamics of model in Eq.~\eqref{eq:TotalHam}, taking a static tunnelling element along $\hat{x}$ axis, i.\,e.~$h_x=\Delta$, a bias field along $\hat{z}$ axis, $h_z=h_0\mbox{, } h_y=0$, and restricting to the case of Ohmic dissipation ($s=1$), with $\boldsymbol{\hat{n}}=\hat{z}$ : hence our model reduces to the biased SBM. In the last decades, several works have been written in order to characterize its quantum phase diagram and describe the corresponding dynamical properties under different parameter regimes. A number of approximate analytical treatments have been devised in order to compute the behavior of one of the most important correlators as a function of time, i.\,e.~the magnetization along the $\hat{z}$ axis $\ev{\sigma_z(t)}$, which is directly related to experiments. In the unbiased case ($h_0=0$), theoretical approaches based on Conformal Field Theory (CFT) predict that $\ev{\sigma_z(t)}$ exhibits underdamped oscillations in time, for fixed values of the coupling strength $0\leq\alpha<1/2$; however, the detailed expressions of the oscillation frequency and the damping rate depend on the adopted approximation scheme \cite{orth-lehur}. One of the most popular approximate treatments is the Non-Interacting Blip Approximation (NIBA) \cite{Leggett,Grifoni:drivenquantum}: according to NIBA scheme, at $T=0$ the underdamped oscillation in time can be described in terms of analytical functions of the inverse time scale $\Delta\ped{eff}^{-1}$, with the effective tunnelling energy $\Delta\ped{eff}$ reading \begin{equation} \Delta\ped{eff}=[\Gamma(1-2\alpha) \cos\pi\alpha]^{\frac{1}{2(1-\alpha)} }\Delta\ped{r}\mbox{ , } \Delta\ped{r} = \Delta \left(\frac{\Delta}{\omega\ped{c}}\right)^{\frac{\alpha}{1-\alpha}} \end{equation} where $\Delta\ped{r}$ is the renormalized gap \cite{Leggett,orth-lehur}. While the resulting expression for the oscillation frequency is expected not to be valid \cite{EggerCrossover}, this analytical treatment provides the correct result for the quality factor of the damped oscillation (see App.~\ref{app:Toul} for details), which is a monotonic decreasing function of $\alpha$. Although the NIBA approach successfully describes the main features of the system in the unbiased case, in the presence of an external bias field $h_0$, it fails in describing the long-time limit of $\ev{\sigma_z(t)}$. Several exact analytical treatments based on perturbation theory are known which, in the weak coupling limit, give the correct results for the qubit observables, taking into account the fully-quantum correlations between the qubit and bath degrees of freedom. These theories have been employed to derive analytical results for the dynamics of heat exchanges of the qubit with the reservoir \cite{Carrega15:heatexchange}. The SBM problem can be analytically solved for the coupling strength $\alpha = 1/2$ (Toulouse limit). Here a crossover from coherent to incoherent dynamics takes place: at this point, at zero bias, the oscillation frequency tends to vanish, as well as the quality factor. The analytical solution can be found by mapping the SBM into a Resonant Level Model (RLM), describing a single localized impurity in contact with a bath of spinless fermions at the Fermi level; in addition, Coulomb interactions between the impurity and the fermionic bath are present \cite{Leggett,SassettiWeiss:SpinBoson,LeHurNRG1}, and the resulting coupling strength is a function of $\alpha$. For $\alpha = 1/2$, Coulomb interactions vanish and the model can be exactly solved: by preparing the qubit in the state $\ket{\hat{z}; +}$, i.\,e.~$\ev{\sigma_z(0)}=1$, in the limit of small $\Delta/\omega\ped{c}$, i.\,e.~$\omega\ped{c}\to\infty$ the magnetization dynamics takes the form \begin{equation}\label{eq:Toulouse} \ev{\sigma_z(t)}=e^{-\gamma t} + 2 \int_0^t d\tau\frac{\sin(h_0 \tau)}{\beta \sinh(\frac{\pi \tau}{\beta}) }\left( e^{-\gamma \frac{\tau}{2}}-e^{-\gamma t}e^{\gamma \frac{\tau}{2}}\right) \end{equation} where $\beta=1/T$ ($k\ped{B}=1$) and the damping rate is proportional to the renormalized gap at $\alpha=1/2$, i.\,e.~$\gamma=\pi \Delta^2/2 \omega_c$. In the absence of external bias, $\ev{\sigma_z(t)}$ takes the exponential form which is also recovered in the NIBA approximation. Different numerical approaches have been devised in order to compute the real time dynamics of this problem in the range of coupling strengths $0 <\alpha < 1/2$, where no exact analytical solution is known to exist \cite{PathInt1,PathInt2,quapi,bulla:nrg1,orth-lehur,Strathearn:SBMMPS}. Here we apply the numerical SIL technique (see App.~\ref{app:SIL}) in order to describe dynamics of the reduced density matrix of the qubit up to the Toulouse point. As stressed in \cite{Cangemi:SIL}, this technique allows us to simulate the exact dynamical evolution of the whole system density matrix $\rho(t)$ in a suitably truncated bath Hilbert space: hence, the observables of the whole qubit + bath system can be computed. We start by preparing the system and the bath at initial time $t\ped{0}$ in a factorized state: \begin{equation}\label{eq:startcond} \rho(t\ped{0})=\rho\ped{S}(t\ped{0})\otimes \frac{e^{-\beta H\ped{B}}}{Z\ped{B}} \end{equation} where $\rho\ped{S}(t)$ is the reduced density matrix of the qubit that can be computed by tracing out the bath degrees of freedom, i.\,e.~$\rho\ped{S}(t)=\tr\ped{B}{\rho(t)}$. We choose $\ket{\hat{z}; +}$ as the initial state of the qubit, while the bath state is taken as the equilibrium state at $T=0$. We model our bath with a collection of $M=50$ bosonic modes, choosing the absolute maximum number of excitations up to $N\ped{ph}=6$ (see App.~\ref{app:SIL} for details) and we fix the cutoff frequency of the bath to $\omega\ped{c}=5 \Delta$. In the following, we restrict to the unbiased case ($h_0=0$), while in App.~\ref{app:Biased}, \ref{app:Toul}, we discuss the biased case along with a comparison with analytical results. We simulate the dynamics of the system for different values of the coupling strength $\alpha$, ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1e-1}{5e-1}$. In Fig.~\ref{fig:sigmazt}, we plot the qubit magnetization $\ev{\sigma_z(t)}$ as a function of the rescaled time $\Delta\ped{r} t$: we show that the magnetization dynamics experiences a crossover from a regime of underdamped oscillations in time to an incoherent regime where the oscillation frequency tends to vanish, as long as the coupling strength approaches the expected crossover value $\alpha=0.50$. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_sigmaz_toul.pdf} \caption{Magnetization $\ev{\sigma_z(t)}$ as a function of the rescaled time $\Delta\ped{r}t$, in the case of Ohmic bath ($ s = 1 $), $T=0$, $h_0=0$, for different values of the coupling strengths $\alpha$ in the range $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1e-1}{5e-1}$. The number of bath modes is $M=50$, the cutoff frequency $\omega\ped{c}=5\Delta$ and the maximum number of excitations is $N\ped{ph}=6$.} \label{fig:sigmazt} \end{figure} The crossover from coherent to incoherent behavior can be interpreted in terms of the growth of the entanglement between the qubit and its bath \cite{LeHurNRG1}, a mechanism which can be found in several bipartite systems \cite{AmicoFazioFalci:Entanglement}. Starting from the initial condition in Eq.~\eqref{eq:startcond}, where the state of the system is factorized into a product of states of the two subsystem, the state of the qubit thermalizes towards the equilibrium state of the whole Hamiltonian in Eq.~\eqref{eq:TotalHam} at $T=0$, showing entanglement with the bath degrees of freedom. Each numerical curve reported in Fig.~\ref{fig:sigmazt} describes the correct dissipative behavior of the qubit, as it can be shown by a direct comparison with the theoretical result for the quality factor as a function of the coupling strength $\alpha$ (see App.~\ref{app:Toul}). Our results are also in good agreement with recent findings obtained through novel numerical approaches based on non-perturbative techniques \cite{orth-lehur}. A detailed comparison with the case $\alpha=1/2$ in Eq.~\eqref{eq:Toulouse}, reported in App.~\ref{app:Toul}, shows also good agreement with theory at long times, while at shorter times small deviations start to appear: it can be explained by the small cutoff value chosen $\omega\ped{c}=5\Delta$, which cannot meet the parameters conditions ensuring the validity of Eq.~\eqref{eq:Toulouse}; we argue that this limit is also responsible for the residual coherent behavior of the magnetization at $\alpha=1/2$ observed in Fig.~\ref{fig:sigmazt} for long times. A similar analysis of the magnetization dynamics can be performed in the biased case, considering both the well-known limits of weak coupling and Toulouse point (see respectively App.~\ref{app:Biased} and \ref{app:Toul}). As expected, in the weak coupling regime an excellent agreement can be found with the analytical curves, while in the strong coupling limit the observed deviations from the analytic results can be traced back to the same reason as in the unbiased case. Additional insights can be derived from the analysis of the expectation values of the difference of number operators from their initial equilibrium values, i.\,e.~$\ev{\Delta n_{k}(t)}=\ev{n_{k}(t)}-n\api{0}\ped{k}$, computed for each bosonic mode at fixed time intervals in the range $[t\ped{0}, t\ped{f}]$, as shown in Fig.~\ref{fig:phonons}; \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_fononi_riazzerati.pdf} \caption{Expectation values of $\Delta n_{k}(t)$ computed for each bath mode $k$ at different times $\set{t\ped{1},t\ped{2},t\ped{3},t\ped{f}}=\Set{0.03,0.06,1.14,6.00}$ (in units of $\Delta\ped{r}^{-1}$), for fixed coupling strength $\alpha=0.40$, $\omega\ped{c}=5\Delta$, $T=0$, $h_0=0$, $M=50$ and $N\ped{ph}=6$.} \label{fig:phonons} \end{figure} it can be inferred that at short times the bath response extends over the whole frequency spectrum, high-frequency modes showing slightly greater occupation than the slower ones, even if the occupation is quite small. At intermediate times, a set of peaks start to come into play, due to multiple scattering processes of the qubit with the bath modes. The bath response shows a first order peak which signals the onset of a resonant behavior, its position shifting towards lower frequencies as long as time increases. As expected, the behavior of each curve at intermediate times, as well as at longer times shows a clear dependence on the value of the coupling strength $\alpha$. In Fig.~\ref{fig:phonons2}, we plot $\ev{\Delta n_{k}(t)}$, computed at sufficiently long time $t\ped{sat}$ for different coupling strengths $\alpha$: the results show that \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_fononi_varialpha.pdf} \caption{Expectation values of $\ev{\Delta n_{k}}$ computed for each bath mode $k$ at rescaled time $t\ped{sat}=4.85$ (in units of $\Delta\ped{r}^{-1}$), for different coupling strengths $\alpha$ in the range $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1e-1}{4e-1}$, $\omega\ped{c}=5\Delta$, $T=0$, $h_0=0$, $M=50$ and $N\ped{ph}=6$.} \label{fig:phonons2} \end{figure} the position of the first-order peak shifts towards lower energies for increasing coupling strengths, and the characteristic energy of the system is proportional to the effective tunnelling energy $\Delta\ped{eff}$. Moreover, the curves of bosonic excitations exhibit oscillations in $\omega_k/\omega\ped{c}$ that tend to disappear as the coupling strength approaches the crossover value: this effect can be seen as a consequence of the increasingly incoherent behavior of the system. These features confirm that, for coupling strengths in the range $0\leq\alpha \leq 0.5$ the dynamical evolution of the whole system reaches an equilibrium state that can be interpreted in terms of a single qubit whose tunnelling energy is renormalized proportionally to $\Delta\ped{r}$, experiencing incoherent tunnelling between localized states. The exchanged energy with the bath can also be studied for different values of the coupling strengths $\alpha$: from Fig.~\ref{fig:bathenergy}, it can be shown that all the curves tend to a saturation value, which in our simulation is fixed by the energy conservation: as shown in \cite{Cangemi:SIL}, at every time $t$ the energy of the non-equilibrium initial state of the qubit is equal to the sum of the expectation values of the different operators in Eq.~\eqref{eq:TotalHam}. It can also be noticed that the saturation value of $\ev*{H\ped{B}(t)}$ strongly depends on the coupling strength $\alpha$. Moreover, for increasing coupling strengths, it can be observed that the bath energy exhibits an oscillatory behavior at short times, and a moderately prononunced peak which tends to be greater than its long-time value. This feature is due to the increasing importance of the qubit-bath correlations which, in the strong coupling regime, can modify the mechanism of energy exchange. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_heat_toul.pdf} \caption{Expectation values of $\ev*{H\ped{B}(t)}$ (in units of $\Delta\ped{r}$ ) computed as a function of the rescaled time $\Delta\ped{r} t$, for different coupling strengths $\alpha$ in the range $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1e-1}{5e-1}$, $\omega\ped{c}=5\Delta$, $T=0$, $h_0=0$, $M=50$ and $N\ped{ph}=6$.} \label{fig:bathenergy} \end{figure} \section{Time dependent protocol}\label{sec:Magnetic} In this section, we study the effect of decoherence and dissipation on a two level system subject to a rotating magnetic field. We take the qubit Hamiltonian as in Eq.~\eqref{eq:QubitHam}, where $h$ is the magnitude of the applied magnetic field; we adopt a system of polar coordinates $(\theta,\phi)$, i.\,e.~$\boldsymbol{h}=(h\sin\theta\cos\phi,h\sin\theta\sin\phi, h\cos\theta)$. We also introduce an additional static magnetic field along the positive $\hat{z}$ direction, i.\,e.~$h\ped{0}\hat{z}$. We restrict the rotating magnetic field $\boldsymbol{h}$ in the $\hat{x}\mbox{-}\hat{z}$ plane by fixing $\phi=0$. The qubit Hamiltonian thus reads \begin{equation}\label{eq:qubitperiodic} H\ped{S}(t)= -\frac{1}{2}(h\ped{0}+h\cos\theta(t))\sigma_z - \frac{h}{2}\sin\theta(t) \sigma_x \end{equation} The magnetic field $\boldsymbol{h}$ evolves performing a sweep in $\hat{x}\mbox{-}\hat{z}$ plane in a total time $t\ped{f}$, i.\,e.~the polar angle changes according to $\theta\left(t\right)=\pi(t-t\ped{0})/t\ped{f}$, from $\theta(t\ped{0})=0$ to $\theta(t\ped{f})=\pi(1-t\ped{0}/t\ped{f})$. This protocol, widely studied in the field of Nuclear Magnetic Resonance (NMR), has regained attention following recent theoretical and experimental works \cite{Gritsev,RoushanNature:topotrans}. It has been shown that physical implementations of Hamiltonians of the form of Eq.~\eqref{eq:qubitperiodic} can be achieved with high level of control employing superconducting circuits; moreover, a simple mapping exists from Eq.~\eqref{eq:qubitperiodic} to the Haldane model at half filling on a honeycomb lattice \cite{Haldane88}, which is a prototypical model of a Chern insulator. Following this mapping, every qubit state on the Bloch sphere at fixed coordinates $(\theta,\phi)$ can be mapped onto a single quasi-momentum state $(k\ped{x},k\ped{y})$ around the high-symmetry points of the first Brillouin zone of the honeycomb lattice. As a consequence, it allows for a dynamical measurement of the topological properties of the Haldane model by making use of a superconducting qubit, e.\,g.~the first Chern number can be probed. It follows that, by tuning the ratio of the field amplitudes $h\ped{0}/h$ and performing quantum state tomography at different times $t$ during the sweep, topological transitions can be measured with high level of accuracy. In the following, we analyze the dissipative dynamics of a qubit described by Eq.~\eqref{eq:qubitperiodic} at weak and strong coupling strengths, both for long and short sweep times $t\ped{f}$ as compared with the time scale $1/h$, i.\,e.~we consider both adiabatic and anti-adiabatic regimes. The qubit is coupled to the environment along a direction which lies in the plane of rotation of the magnetic field, and we focus on the two particular cases $\boldsymbol{\hat{n}}=\hat{z}\mbox{, }\hat{x}$. We compute the excess energy of the qubit at the end of the sweep, i.\,e.~the difference between the mean value of the reduced system energy and the ground state energy $\epsilon_{gs}(t\ped{f})$ of the non-interacting qubit Hamiltonian in Eq.~\eqref{eq:qubitperiodic}, computed at final time $t\ped{f}$ \begin{equation} \epsilon\ped{res}=\Tr[\rho(t\ped{f})H\ped{S}(t\ped{f})]-\epsilon_{gs}(t\ped{f}) \end{equation} Due to the simple form of Eq.~\eqref{eq:qubitperiodic}, the excess energy can also be linked to the fidelity $\mathcal{F}(t\ped{f})$ at the end of the sweep \begin{equation} \epsilon\ped{res}=\abs{h-h\ped{0}}(1-\mathcal{F}(t\ped{f})) \end{equation} where $\mathcal{F}(t\ped{f})=\bra*{\psi\ped{gs}(t\ped{f})}\rho\ped{S}(t\ped{f})\ket*{\psi\ped{gs}(t\ped{f})}$ and $\ket*{\psi\ped{gs}(t\ped{f})}$ is the ground state of qubit Hamiltonian in Eq.~\eqref{eq:qubitperiodic} at $t=t\ped{f}$. In addition, we compute the expectation values of qubit operators $\ev{\boldsymbol{\sigma}}=(\ev{\sigma_x(t)}, \ev*{\sigma_y(t)}, \ev{\sigma_z(t)})$ as functions of time, i.\,e.~the dynamical evolution of the Bloch vector, at fixed final times $t\ped{f}$ and for different values of the coupling strength. We first consider the qubit system in the absence of dissipation, taking the static bias field $h\ped{0}=0$: at initial time $t_0=0$, the magnetic field is aligned along the positive $\hat{z}$ direction and the qubit is prepared in its ground state, i.\,e.~$\ket{\psi(t\ped{0})}=\ket{\hat{z},+}$. For $t>0$, the field $\boldsymbol{h}$ rotates around the $\hat{y}$ axis. The qubit dynamics can be straightforwardly solved in the counter-rotating frame around the $\hat{y}$ axis (see App.~\ref{app:unitary}): due to its simple form, the Hamiltonian in Eq.~\eqref{eq:qubitperiodic} in the rotating frame is time-independent, and it follows that the excess energy of the closed system reads \begin{equation}\label{eq:eresclosed} \epsilon\ped{res} =\frac{h \dot{\theta}^2}{2} \frac{1-\cos(\pi\sqrt{h^2 + \dot{\theta}^2}/\dot{\theta})}{h^2+ \dot{\theta}^2} \end{equation} where we put for brevity $\dot{\theta}=\pi/t\ped{f}$. The qubit dynamics is described by a cycloid on the Bloch sphere, i.\,e.~the Bloch vector periodically points out of the $\hat{x}\mbox{-}\hat{z}$ plane. This trajectory is due to the oscillations in time of the magnetization along the $\hat{y}$ axis: therefore, $\ev*{\sigma_y(t)}$ can serve as a measure of deviation from the adiabatic path, which is a circle in the $\hat{x}\mbox{-}\hat{z}$ plane. The non-adiabatic response of the Bloch vector is thus proportional to $\ev*{\sigma_y(t)}$ \cite{Gritsev}; furthermore, using perturbation theory it has been shown that at first order in $\dot{\theta}/h$ the non-adiabatic response can be linked to the curvature of the ground state manifold of the Hamiltonian in Eq.~\eqref{eq:qubitperiodic}, i.\,e.~to the Berry phase of the qubit. As a consequence, the measure of $\ev*{\sigma_y(t)}$ at each time $t$ allows to achieve the fidelity at final time $t\ped{f}$ that, in the quasi-adiabatic limit, can be used to compute the first Chern number of the system \cite{RoushanNature:topotrans}. The deviation from the adiabatic path can also be seen from the excess energy in Eq.~\eqref{eq:eresclosed}, which is plotted in Fig.~\ref{fig:eressigmaz} (black curve): notice that it exhibits several maxima corresponding to different final times $t\ped{f}$ owing to the fact that, in the non-adiabatic regime, the qubit dynamics cannot follow the evolution of the externally driven magnetic field, and the state vector of the qubit at the end of the sweep differs from the corresponding ground state. However, the amplitude of these maxima is decreasing as long as the time $t\ped{f}$ is increased, i.\,e.~the dynamics can be considered truly adiabatic only in the limit $t\ped{f}\gg1/h$. This scenario undergoes several changes if the interaction with the external bath is considered. \subsection{Coupling along $\hat{z}$}\label{subsec:couplingz} We first analyze the case of interaction along $\hat{z}$ axis. In Fig.~\ref{fig:eressigmaz}, we plot the excess energy curve of the qubit interacting with a bath at $T=0$, for different values of the coupling strength $\alpha$ ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{0}{2e-1}$. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_excessenergy_lowcoup.pdf} \caption{Excess energy plotted as a function of the final time $t\ped{f}$ (in units of $h^{-1}$), for different coupling strengths ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1e-2}{2e-1}$, in the case $\boldsymbol{\hat{n}}=\hat{z}$. The number of modes has been fixed to $M=80$, the cutoff frequency $\omega\ped{c}=5 h$, $N\ped{ph}=3$ and $T=0$. Inset: semi-logarithmic plot of the same curves as in the main plot.} \label{fig:eressigmaz} \end{figure} As it can be noticed, the interaction with the external bath acts to reduce the coherence of the system dynamics. The effect of decoherence results in a smoothing of the excess energy curve with respect to the closed case. However, the difference between the closed and the open system curve depends on the final time $t\ped{f}$: at short final times $t\ped{f}$, the interaction with the environment generally leads to an increase of the residual energy, resulting in a non-adiabatic behavior of qubit dynamics; conversely, at intermediate times $t\ped{f}$, the effect of the bath can lead to a decrease of the local maxima of the excess energy as compared with the closed case, i.\,e.~the state of the qubit at $\theta(t\ped{f})$ is closer to the corresponding state on the adiabatic path. It follows that, at weak coupling regime the effect of friction counteracts the non-adiabaticity of the system induced by the fast external drive, thus resulting in a reduction of the excess energy. This scenario changes in the intermediate coupling regime: for coupling strengths $\alpha > 0.1$, it can be noticed that the excess energy starts to increase, and the system definitely misses the adiabatic path. The resulting non-monotonic behavior of the excess energy can be clearly observed for final times $t\ped{f}$ where the closed system curve shows secondary maxima of excitation, while for values of $t\ped{f}$ corresponding to minima the interaction with the bath leads to monotonic non-adiabaticity. It should also be noticed that, as depicted in the inset of Fig.~\ref{fig:eressigmaz}, for very slow sweeps the open system curves at weak coupling strengths tend to coincide, and they are consistent with the closed system result. Interestingly, the monotonic non-adiabaticity at strong coupling regime was recently observed in \cite{henriet:Topologysbm}, where the dynamical behavior of the Chern number in a dissipative environment was studied and a description of the bath-induced non-adiabaticity was achieved using non-perturbative Stochastic Schr\"odinger equation. Further information on the dynamics of the open system at intermediate final times $t\ped{f}$ can be derived from the analysis of the expectation values $\ev{\sigma_x(t)},\ev*{\sigma_y(t)},\ev{\sigma_z(t)}$: in Fig.~\ref{fig:evolutionsigmaz}, \ref{fig:evolutionsigmay}, \ref{fig:evolutionsigmax}, we plot the expectation values of the spin operators as a function of time $t$, from weak to strong coupling regime and for fixed final time $t\ped{f}=t\ped{fmax}=8.42/h$, corresponding to the first second-order maximum of Eq.~\eqref{eq:eresclosed}. It can be noticed that, following the Heisenberg equations which link the time derivative of $\ev{\sigma_z(t)}$ to $\ev*{\sigma_y(t)}$, the decrease in the excess energy occurring for $t\ped{f}=t\ped{fmax}=8.42/h$ observed at weak coupling can be traced back to the progressive change of $\ev*{\sigma_y(t)}$. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_sigmaz_strongcoupl.pdf} \caption{Plot of $\ev{\sigma_z(t)}$ as a function of time $t$ for the protocol in Eq.~\eqref{eq:qubitperiodic}, with fixed final time $t\ped{f}=t\ped{fmax}=8.42/h$. The qubit couples to an Ohmic bath ($s=1$) along $\boldsymbol{\hat{n}}=\hat{z}$ , for different coupling strengths ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{0}{4e-1}$. The number of modes has been fixed to $M=70$, the cutoff frequency $\omega\ped{c}=5 h$, $N\ped{ph}=5$ and $T=0$.} \label{fig:evolutionsigmaz} \end{figure} Hence, for increasing coupling strengths $\alpha$ it can be noticed that the magnetization along $\hat{y}$ loses the oscillatory behavior with frequency $\sqrt{\dot{\theta}^2 + h^2}$ (see App.~\ref{app:unitary}), which is a characteristic feature of dynamics in the absence of dissipation: actually, as it can be inferred from Fig.~\ref{fig:evolutionsigmay} the second local minimum turns into a local maximum, its position drifts towards higher times $t$, causing the inflection point of $\ev{\sigma_z(t)}$ in Fig.~\ref{fig:evolutionsigmaz} to change accordingly; eventually, at the end of the sweep the magnetization along $\hat{z}$ tends to the adiabatic value, and the state of the qubit in the open dynamics at final time $t\ped{f}$ is closer to the ground state $\ket*{\psi\ped{gs}(t\ped{f})}$. While the previous description holds true also when the closed-system excess energy in Eq.~\eqref{eq:eresclosed} shows local minimum values, e.\,g.~$t\ped{f}=t\ped{fmin}=12.7/h$, it can be observed that at weak coupling strengths the interaction with the bath cannot noticeably change the excess energy. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_sigmay_strongcoupling.pdf} \caption{Plot of $\ev*{\sigma_y(t)}$ as a function of time $t$ for the protocol in Eq.~\eqref{eq:qubitperiodic}, with fixed final time $t\ped{f}=t\ped{fmax}=8.42/h$. The qubit couples to an Ohmic bath ($s=1$) along $\boldsymbol{\hat{n}}=\hat{z}$, for different coupling strengths ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{0}{4e-1}$. The number of modes has been fixed to $M=70$, the cutoff frequency $\omega\ped{c}=5 h$, $N\ped{ph}=5$ and $T=0$.} \label{fig:evolutionsigmay} \end{figure} \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_sigmax_strongcoupling.pdf} \caption{Plot of $\ev{\sigma_x(t)}$ as a function of time $t$ for the protocol in Eq.~\eqref{eq:qubitperiodic}, with fixed final time $t\ped{f}=t\ped{fmax}=8.42/h$. The qubit couples to an Ohmic bath ($s=1$) along $\boldsymbol{\hat{n}}=\hat{z}$, for different coupling strengths ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{0}{4e-1}$. The number of modes has been fixed to $M=70$, the cutoff frequency $\omega\ped{c}=5 h$, $N\ped{ph}=5$ and $T=0$.} \label{fig:evolutionsigmax} \end{figure} More information on the physics at strong coupling regime can be drawn: as shown in Fig.~\ref{fig:evolutionsigmaz}, for $\alpha > 0.10$ $\ev{\sigma_z(t\ped{f})}$ starts to increase. This behavior clearly depends on the final time $t\ped{f}$, i.\,e.~on the slope of the external drive: for faster sweeps, the non-adiabatic behavior due to the interaction with the environment occurs at lower coupling strengths as compared to slower evolutions; \begin{figure}[thb] \centering \includegraphics[width=\linewidth]{Fidelities.pdf} \caption{Fidelity at final time $t\ped{f}$ $\mathcal{F}(t\ped{f})$, plotted against the coupling strength $\alpha$ in the range $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1.0e-2}{4.5e-1}$, for two fixed final times $t\ped{f}=\set{t\ped{fmax},t\ped{fmin}}=\set{8.42/h,12.17/h}$ corresponding to the first second order maximum and the second minimum of Eq.~\eqref{eq:eresclosed}. The number of modes has been fixed to $M=70$, the cutoff frequency $\omega\ped{c}=5 h$, $N\ped{ph}=5$ and $T=0$.} \label{fig:fidelity} \end{figure} as a consequence, the coupling strength directly influences the adiabaticity condition. This feature can be inferred from Fig.~\ref{fig:fidelity}, where we plot the behavior of fidelity $\mathcal{F}(t\ped{f})$ at the end of the sweep, computed for two different fixed final times $t\ped{f}=\set{t\ped{fmax},t\ped{fmin}}=\set{8.42/h,12.17/h}$, corresponding to the first second-order maximum and the second minimum of Eq.~\eqref{eq:eresclosed} (see Fig.~\ref{fig:eressigmaz}, black curve), for different coupling strengths $\alpha$ taken in the range $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1.0e-1}{4.5e-1}$. At final time $t\ped{f}=t\ped{fmax}$, where the closed system excess energy exhibits a local maximum, fidelity shows a small non-monotonic behavior, due to the previously described effect; conversely, at $t\ped{f}=t\ped{fmin}$ a flat behavior at weak coupling, followed by monotonic decrease occurring at higher values of $\alpha$ can be observed. As shown in \cite{henriet:Topologysbm}, an adiabaticity criterion for the protocol in Eq.~\eqref{eq:qubitperiodic} has been proposed which links the velocity of the sweep $\dot{\theta}$ to the renormalized field $\Delta\ped{r}$ along $\hat{x}$ direction (with $\Delta=h$), i.\,e.~$\dot{\theta}\ll \Delta\ped{r}$, provided that $\dot{\theta}\ll h$. Further, for fixed values of $\dot{\theta}$ well below $h$, at strong coupling a crossover from quasi-adiabatic to non-adiabatic behavior occurs at $\dot{\theta}\simeq \Delta\ped{r}$. We find that our numerical results at strong coupling generally agree with this scenario, while in the weak coupling regime several intervals of final times $t\ped{f}$ exist where the bath can act to improve the adiabaticity. It follows that, at weak coupling strengths the dynamical measure of the topological properties shows robustness to the external noise. \subsection{Coupling along $\hat{x}$}\label{subsec:couplingx} Qualitatively different results can be found if the qubit couples with the bath along $\hat{x}$ axis. Here we restrict to weak coupling regime and simulate the dissipative dynamics at $T=0$ of the time-dependent protocol in Eq.~\eqref{eq:qubitperiodic} with $\boldsymbol{\hat{n}}=\hat{x}$ and $h\ped{0}=0$. In Fig.~\ref{fig:eres-sigmax}, we plot the excess energy as a function of the final time $t\ped{f}$ for different coupling strengths, taken in the same range as in Fig.~\ref{fig:eressigmaz}. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_excessenergy_sigmax.pdf} \caption{Excess energy plotted as a function of the final time $t\ped{f}$ (in units of $h^{-1}$), for different coupling strengths $\alpha$ ranging from $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{1e-2}{2e-1}$, in the case of $\boldsymbol{\hat{n}}=\hat{x}$. The number of modes has been fixed to $M=80$, the cutoff frequency $\omega\ped{c}=5 h$, $N\ped{ph}=3$ and $T=0$.} \label{fig:eres-sigmax} \end{figure} It can be shown that, for very fast sweeps the excess energy can be lower than the closed system result: the actual numerical results at short final times $t\ped{f}$ depend on the coupling strength $\alpha$ at fixed cutoff frequency $\omega\ped{c}$. As shown in Fig.~\ref{fig:eres-sigmax-highomega}, by increasing the cutoff frequency $\omega\ped{c}$, the short final time limit of the excess energy curve decreases, as a result of the reduced reaction time of the bath. However, the choice of different cutoff frequencies $\omega\ped{c}$ does not qualitatively change the physics at long times $t\ped{f}$. The decrease in the excess energy at short $t\ped{f}$ is due to the peculiar form of the coupling to the external environment, which causes the qubit to flip at a fixed rate proportional to the coupling strength. This effect can provide a slight advantage to the success of the protocol, as long as the final time $t\ped{f}$ is sufficiently short. However, for longer final times $t\ped{f}$ the open system excess energy tends to be greater than the closed curve: this effect leads to an increasingly non-adiabatic dynamics, even at weak coupling strengths, as opposed to the case studied in \ref{subsec:couplingz} where, as long as the closed system dynamics is quasi-adiabatic, the dynamics is unaffected by the environment. This result shows several analogies with a recent study of the finite-time LMSZ protocol \cite{SunfinitetimeLandZen}, showing that the effect of a transverse coupling to the bath at long final times $t\ped{f}$ can lead to a fidelity $\mathcal{F}(t\ped{f})$ lower than $1$, as opposed to the exact result proposed in \cite{wubs:lz-zero-t}. In addition, here the effect of time-periodic driving can be clearly observed, noticing the persistence of a structure made of several secondary maxima in the excess energy. These findings point towards an increasingly non-adiabatic behavior due to the bath, as long as the coupling strengths increases, occurring at intermediate up to long final times $t\ped{f}$. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{fig_excessenergy_omegac.pdf} \caption{Excess energy plotted as a function of the final time $t\ped{f}$ chosen in the range $\numrange[range-phrase = \text{~to~},exponent-product=\cdot]{0.2}{1.5}$ (in units of $h^{-1}$), for fixed $\alpha=0.10$, $M=80$, $N\ped{ph}=3$, $T=0$ and different cutoff frequencies $\omega\ped{c}$.} \label{fig:eres-sigmax-highomega} \end{figure} \section{Conclusions}\label{sec:conclusions} In this work, we studied the dynamics of a qubit in contact with its environment, subject both to static and driven external fields, from weak to strong coupling strengths, using the SIL approach. We showed that our method can provide a good description of the physics of the SBM as a function of the coupling strength up to Toulouse point, where a crossover from coherent to incoherent behavior of the qubit magnetization takes place. We provided additional insights on the dynamics of the bath degrees of freedom, showing the changes in the bath response as a function of time. Moreover, we studied a protocol of a driven qubit subject to a time-periodic driving, with dissipation taking place along different directions. We showed that in the case of coupling along $\hat{z}$, if the dissipation strength is sufficiently weak, the influence of the bath can counteract the non-adiabaticity of the closed system evolution, leading to a non-monotonic behavior of the fidelity as a function of the coupling strength at fixed values of the final times. Conversely, at strong coupling bath-induced non adiabaticity \cite{henriet:Topologysbm} takes place, hindering the success of the protocol. This scenario changes if the coupling along $\hat{x}$ axis is considered: a measurable advantage over the closed system dynamics can be observed only for very fast sweeps, while for longer sweep durations we predict an increasingly non-adiabatic behavior, i.\,e.~the excess energy tends to increase at increasing coupling strength. In the near future, we plan to extend our analysis to recently proposed time-dependent protocols implementing counter-diabatic driving \cite{Polkovnikov:CD,SelsE3909}, in order to investigate the influence of the environment on the final success probability of these protocols in a broad range of coupling regimes. In addition, energy exchanges between systems of externally driven interacting qubits and the bath will also be analyzed, as well as prototypical models of quantum heat engines.

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\section{Introduction} The backbone of many-body perturbation theory (MBPT) is the interaction self-energy $\Sigma$ which appears in the Dyson equation for equilibrium or non-equilibrium Green's function (NEGF). At equilibrium, the structure of $\Sigma$ is guessed by systematically using Wick's theorem and by analysing the resulting expansion into Feynman diagrams \cite{FW}. Approximation schemes (e.g.\;mean-field approach or RPA) correspond to partial resummation of series of diagrams contributing to $\Sigma $. In the finite temperature non-equilibrium regime of interacting systems the initial state cannot be in general connected to a non-interacting state in the remote past \cite{Adiabatic} and writing down statistical averages of time-dependent observables becomes cumbersome. The remedy for these technical difficulties is to combine the chronological $T$ and anti-chronological $\overline T$ time-ordering operators into a single operator ${\cal T}_C$ which allows an unambiguous book-keeping of time arguments on the two-branch Schwinger-Keldysh contour \cite{Sc,Ke,Danielewicz1}. This construction comes with a price: the non-equilibrium GFs turn to contour-ordered quantities as well and the various identities among them are not easy to recover. At a formal level one assumes the existence of a well-defined self-energy and then the contour-ordered Dyson equation splits via the Langreth rules \cite{La} into the Keldysh equation for the lesser/greater GFs and the Dyson equation for the retarded/advanced GFs (see the textbook \cite{HJ}). The existence of a self-energy for the contour-ordered GF is argued by the {\it formal} analogy between equilibrium and non-equilibrium quantum averages. Then a complete interaction self-energy can be defined \cite{Eq3.16}. In more recent formulations \cite{SvL} one starts from the differential equations of motions relating higher $n$-particle Green-Keldysh functions and then {\it truncates} the so-called Martin-Schwinger hierarchy \cite{Bonitz1} to identify various approximate interaction self-energies. Nowadays, the NEGFs formalism has grown up as a remarkable machinery, being extensively used for modelling quantum transport in mesoscopic systems \cite{Myohanen}, molecules \cite{DvL} or even nuclear reactions \cite{Danielewicz2}. Nonetheless, some fundamental theoretical questions were only recently answered by fully exploiting the mathematical structure of the theory and {\it without} making any approximations. We refer here to: (i) the existence of non-equilibrium steady-state (NESS) in interacting open systems and (ii) the independence of the steady-state quantities from the initial state of the sample \cite{FMU,AJPP1,JOP3,CMP1,CMP2} both in the partitioning \cite{Ca1} and partition free \cite{Ci,HDP} settings. We recall here that in the partitioned case the system and the biased leads are initially decoupled. In our recent work \cite{AHP} the NEGF formalism for open systems in the partitioning transport setting was rigorously treated in great detail and generality. In particular, we derived the Jauho-Wingreen-Meir formula (JWM) \cite{JWM} for the time-dependent current through an interacting sample by using only real-time quantities. In this short note we are interested in the partition-free regime which was adapted for interacting systems by Stefanucci and Almbladh \cite{SA}. Recently, the long-time limit of the energy current in the partition-free setting was discussed in Ref.\cite{Eich1} and the transient heat currents due to a temperature gradient were calculated in \cite{Eich2}. We briefly outline a rigorous formulation of the non-equilibrium Dyson equation for the retarded Green's function. Mathematical details are kept to a minimum while focusing on the explicit construction of a complete interaction self-energy. The content of the paper goes as follows: the model and the notations are introduced in Section \ref{section2}, the main result and its proof are given in Section \ref{section3} while Section \ref{section4} is left for conclusions. \section{Setting and notation.}\label{section2} \subsection{Configuration space and Hamiltonians.} We assume that a small sample is coupled to $M$ leads. The one-particle Hilbert space is of tight-binding type and can be written as $\mathfrak{h}=\mathfrak{h}_{\cal S}\oplus \mathfrak{h}_{\cal R}$ where $\mathfrak{h}_{\cal S}$ is finite dimensional and $\mathfrak{h}_{\cal R}=\oplus_{\nu=1}^M\mathfrak{h}_\nu$ describes the (finite or not) leads. Particles can only interact in the sample. One-particle operators are denoted with lower-case letters and their second quantized versions will be labeled by capital letters. The one-particle Hamiltonian of the decoupled system acquires a block-diagonal structure $h_{\rm D}=h_{\cal S}\oplus h_{\cal R}$ where $h_{\cal R}=\oplus_{\nu=1}^Mh_\nu$ is supposed to be bounded. The lead-sample tunnelling Hamiltonian is defined as: \begin{equation}\label{april1} h_{\rm T}=\sum_{\nu=1}^M d_\nu\big (|f_\nu\rangle\langle g_\nu| +|g_\nu\rangle\langle f_\nu|\big ), \end{equation} where $\nu$ counts the particle reservoirs, $f_\nu\in{\mathfrak h}_\nu$ and $g_\nu\in{\mathfrak h}_{\cal S}$ are unit vectors and $d_\nu\in{\mathbb R}$ are coupling constants. The one-particle Hamiltonian of the fully coupled system is then $h=h_{\rm D}+h_{\rm T}$. We summarize below some useful identities from the second quantization machinery (see e.g. \cite{MR}). The total Fock space admits a factorization $\mathcal{F}=\mathcal{F}_\mathcal{S}\otimes \mathcal{F}_\mathcal{R}$. By $a^{\#}(f)$ we mean either the creation operator $a^*(f)$ or the annihilation operator $a(f)$. We have $a^*(\lambda f)=\lambda a^*(f)$ and $a(\lambda f)=\overline\lambda a(f)$. The general form of the canonical anticommutation relations is: \begin{equation} \{a(f),a^\ast(g)\}=\langle f|g\rangle,\qquad\{a(f),a(g)\}=0. \label{CAR} \end{equation} Here $\langle f|g\rangle$ denotes the scalar product in $\mathfrak{h}$. Also, $a^{\#}(f)$ is bounded on the Fock space and $\|a^{\#}(f)\|\leq \|f\|$. The interacting, coupled system, and with a potential bias $v_\nu$ on lead $\nu$ is described by: \begin{equation} K_v:=H+\sum_{\nu=1}^Mv_\nu N_\nu+\xi W, \label{H_int} \end{equation} where $N_\nu$ is the particle number operator on lead $\nu$ (i.e., the second quantization of the orthogonal projection onto $\mathfrak{h}_\nu$), $v:=(v_1,\dots,v_M)\in{\mathbb R}^M$ is the bias vector and $$W=\frac{1}{2}\sum_{x,y\in\mathcal{S}}w(x,y) a^*(|x\rangle)a^*(|y\rangle)a(|y\rangle)a(|x\rangle)$$ is the second quantization of a two-body potential satisfying $w(x,y)=w(y,x)$ and $w(x,x)=0$ for all $x,y\in{\cal S}$. Here $\xi\in{\mathbb R}$ stands for the interaction strength. Assume that the bias is turned on at time $t=0$. Then the Heisenberg evolution of an observable $A$ at $t>0$ is \begin{equation}\label{hc0} \tau_{K_v}^t(A):={\rm e}^{{\rm i} tK_v}A{\rm e}^{-{\rm i} tK_v},\quad t>0. \end{equation} If $h$ is a single-particle Hamiltonian, the associated Heisenberg evolution obeys: \begin{equation}\label{Q1} \tau_{H}^t(a^\#(f)):= {\rm e}^{{\rm i} t H}a^\#(f){\rm e}^{-{\rm i} t H}=a^\#({\rm e}^{{\rm i} th}f), \end{equation} and one has \begin{equation}\label{hc0'} [H,a^{*}(f)]=a^{*}(hf),\quad [H,a(f)]=-a(hf). \end{equation} Along the proof of the Dyson equation we shall encounter the operators: \begin{equation}\label{W-comm} b(f):={\rm i}\xi[W,a (f)],\quad b^\ast(f):={\rm i}\xi [W,a^\ast(f)]. \end{equation} These operators vanish if $f$ is supported in the leads. \subsection{The partition-free initial state.} The initial state in the partition-free case is a Gibbs state characterized by the inverse temperature $\beta>0$ and the chemical potential $\mu\in{\mathbb R}$. It is given by the thermodynamic (i.e., infinite leads) limit of the density operator $\rho_{\rm pf}=Z^{-1}{\rm e}^{-\beta(K_0-\mu N)}$ where $Z={\rm Tr}_\mathcal{F}\;{\rm e}^{-\beta(K_0-\mu N)}$. In what follows we briefly explain how it is constructed. The interacting but decoupled and unbiased Hamiltonian is denoted by: \begin{equation*} K_D:=H_{\mathcal{S}}+\xi W +H_{\mathcal{R}}=K_0-H_T. \end{equation*} The thermodynamic limit of $\rho_D=Z_D^{-1}{\rm e}^{-\beta(K_D-\mu N)}$ where $Z_D={\rm Tr}_\mathcal{F}\;{\rm e}^{-\beta(K_D-\mu N)}$ is a tensor product between a many-body Gibbs state $$\rho_S=\frac{1}{{\rm Tr}_{\mathcal{F}_{\mathcal{S}}}{\rm e}^{-\beta(H_\mathcal{S}+\xi W-\mu N_\mathcal{S})}}{\rm e}^{-\beta(H_\mathcal{S}+\xi W-\mu N_\mathcal{S})}$$ only acting on the finite dimensional Fock space $\mathcal{F}_{\mathcal{S}}$, and $M$ non-interacting $(\beta,\mu)$ Fermi-Dirac quasi-free states acting on each lead separately, where expectations can be computed with the usual Wick theorem. This special factorized initial state is denoted by $\big \langle \cdot \big\rangle _{\beta,\mu}$. For example, the expectation of a factorized observable of the type $\mathcal{O}=\mathcal{O}_\mathcal{S}\prod_{\nu=1}^M a^*(\tilde{f}_\nu)a(f_\nu)$ where $\tilde{f}_\nu,f_\nu\in \mathfrak{h}_\nu$ is: $$\big \langle \mathcal{O} \big\rangle_{\beta,\mu}={\rm Tr}_{\mathcal{F}_{\mathcal{S}}}(\rho_\mathcal{S} \mathcal{O}_\mathcal{S})\prod_{\nu=1}^M\langle f_\nu|({\rm Id}+{\rm e}^{\beta(h_\nu-\mu)})^{-1}\tilde{f}_\nu\rangle.$$ Its connection with the partition-free state is as follows. Consider the operator $B(\alpha):={\rm e}^{-i\alpha K_D}H_T{\rm e}^{i\alpha K_D}$, $\alpha\in{\mathbb R}$. From \eqref{april1} and using \eqref{Q1} we see that a generic term entering $B(\alpha)$ is $$\sum_\nu d_\nu \; a^*({\rm e}^{-{\rm i}\alpha h_\nu}f_\nu)\; \tau_{H_\mathcal{S}+\xi W}^{-\alpha}\big (a(g_\nu)\big ).$$ Since $h_\nu$ is bounded and $\mathcal{F}_\mathcal{S}$ is finite dimensional, this expression remains bounded for all complex values of $\alpha$. Then, the initial value problem $$\Gamma'(x)= B(ix)\Gamma(x),\qquad\Gamma(0)={\rm Id},$$ has a unique solution given by a norm convergent Picard/Dyson/Duhamel iteration, with terms containing products of operators either living in the sample or in the leads. Before the thermodynamic limit, the operators $\Gamma(\beta)$ and $e^{\beta K_D}e^{-\beta K_0}$ satisfy the same differential equation and obey the same initial condition at $\beta=0$, hence they must coincide. Consequently, writing ${\rm e}^{-\beta K_0}={\rm e}^{-\beta K_D} \Gamma(\beta)$ we obtain an appropriate expression for the thermodynamic limit: $\mathcal{O}$ being an arbitrary bounded physical observable, we have \begin{align}\label{hc1} \big\langle \mathcal{O}\big\rangle_{\rm pf }=\frac{\big\langle\Gamma(\beta)\mathcal{O}\big\rangle_{\beta,\mu}}{\big\langle\Gamma(\beta)\big\rangle_{\beta,\mu}}. \end{align} \subsection{Function spaces and Volterra operators.} Let $0<T<\infty$ be fixed and let $C_0^1([0,T];\mathfrak{h})$ be the space consisting of time dependent vectors $\phi(t)\in \mathfrak{h}$, $0\leq t\leq T$, which are continuously differentiable with respect to $t$, and $\phi(0)=0$. We also define $C([0,T];\mathfrak{h})$ to be the space of vectors which are only continuous in $t$, with no additional condition at $t=0$. We note that $C([0,T];\mathfrak{h})$ is a Banach space if we introduce the norm $$|||\psi|||:=\sup_{0\leq t\leq T}\|\psi(t)\|_{\mathfrak{h}}.$$ We say that an operator $A$ which maps $C([0,T];\mathfrak{h})$ into itself is a Volterra operator if there exists a constant $C_A<\infty$ such that $$ \|(A\psi)(t)\|_{\mathfrak{h}}\leq C_A\int_0^t \|\psi(t')\|_{\mathfrak{h}} dt',\quad 0\leq t\leq T.$$ By induction one can prove: $$ \|(A^n\psi)(t)\|_{\mathfrak{h}}\leq C_A\frac{(C_AT)^{n-1}}{(n-1)!}\int_0^t \|\psi(t')\|_{\mathfrak{h}} dt',\quad n\geq 1. $$ This implies: $$|||A^n\psi|||\leq \frac{(C_A T)^n}{(n-1)!}|||\psi|||$$ which leads to the conclusion that the operator norm of $A^n$ is bounded by $\frac{(C_A T)^n}{(n-1)!}$. In particular, the series $\sum_{n\geq 1}(-1)^nA^n$ converges in operator norm and defines a Volterra operator with a constant less than $C_A{\rm e}^{TC_A}$. Thus, $({\rm Id}+A)^{-1}={\rm Id}+\sum_{n\geq 1}(-1)^nA^n$ always exists and $A({\rm Id}+A)^{-1}$ is a Volterra operator. \subsection{Retarded NEGF's.} Let $\{e_j\}$ be an arbitrary orthonormal basis in $\mathfrak{h}$. Define the map $G_0: C([0,T];\mathfrak{h})\mapsto C_0^1([0,T];\mathfrak{h})$ given by: \begin{align}\label{hc2} \langle e_j|(G_0\psi)(t)\rangle:=-{\rm i} \int_0^t \langle e_j|{\rm e}^{-{\rm i}(t-t')h_v}\psi(t')\rangle dt', \end{align} where $h_v$ denotes the single-particle Hamiltonian of the non-interacting coupled and biased system. One can check that $G_0$ is invertible and if $\phi\in C_0^1([0,T];\mathfrak{h})$: \begin{align}\label{hc3} (G_0^{-1}\phi)(t)={\rm i} \partial_t\phi(t)-h_v\phi(t)\in C([0,T];\mathfrak{h}). \end{align} By definition, the retarded non-equilibrium Green operator in the partition-free setting $G_\xi:C([0,T];\mathfrak{h})\mapsto C_0^1([0,T];\mathfrak{h})$ is given by: \begin{align}\label{hc4} \langle e_j|(G_\xi\psi)(t)\rangle:=-{\rm i} \int_0^t \big \langle \{ \tau_{K_v}^{t'}(a^*(\psi(t'))), \tau_{K_v}^{t}(a(e_j))\}\big\rangle_{\rm pf}\; dt'. \end{align} Using \eqref{Q1} and \eqref{CAR} we see that $G_\xi$ coincides with $G_0$ when $\xi=0$. One can show that \begin{align}\label{hc5} \|(G_\xi\psi)(t)\|_{\mathfrak{h}}\leq 2\int_0^t \|\psi(t')\|_{\mathfrak{h}} dt', \end{align} so that $G_\xi$ is a Volterra operator. The integral kernel of $G_\xi$ is nothing but the more familiar retarded NEGF given by: \begin{align}\label{hc4'} G_\xi^R(e_j,t;e_m,t'):=-{\rm i} \theta(t-t') \big\langle \{ \tau_{K_v}^{t'}(a^*(e_m)), \tau_{K_v}^{t}(a(e_j))\}\big\rangle_{\rm pf}\;, \end{align} and \begin{align}\label{hc4''} \langle e_j|(G_\xi\psi)(t)\rangle=\sum_m\int_0^t G_\xi^R(e_j,t;e_m,t')\langle e_m|\psi(t')\rangle dt'. \end{align} The advanced NEGF can be defined as: \begin{align*} G_\xi^A(e_j,t;e_m,t'):=-G_\xi^R(e_j,t';e_m,t)={\rm i} \theta(t'-t) \big\langle \{ \tau_{K_v}^{t'}(a^*(e_m)), \tau_{K_v}^{t}(a(e_j))\}\big\rangle_{\rm pf}\;. \end{align*} All properties of the advanced NEGF can be immediately read off from those of the retarded one. \section{Irreducible self-energy and Dyson equation.}\label{section3} Here is the main result of our paper. \begin{theorem}\label{thm1} The bounded linear map $\widetilde{\Sigma}_\xi$ defined on $C([0,T];\mathfrak{h})$ by \begin{align}\label{hc9} \langle e_j|(\widetilde{\Sigma}_\xi\phi)(t)\rangle:=-{\rm i} \int_0^t \big\langle\{ \tau_{K_v}^{t'}(b^*(\phi(t'))), \tau_{K_v}^{t}(b(e_j))\}\big\rangle_{\rm pf}\; dt'+ {\rm i} \big\langle\tau_{K_v}^{t}(\{a^*(\phi(t)),b(e_j)\})\big\rangle_{\rm pf}\;, \end{align} obeys: \begin{align}\label{hc6} G_\xi=G_0+G_0\widetilde{\Sigma}_\xi G_0. \end{align} Moreover, the operator $G_0\widetilde{\Sigma}_\xi$ is a Volterra operator, the inverse $({\rm Id} +G_0\widetilde{\Sigma}_\xi)^{-1}$ exists, and by defining \begin{align}\label{hc20} \Sigma_\xi:=\widetilde{\Sigma}_\xi\left ({\rm Id}+ G_0 \widetilde{\Sigma}_\xi\right )^{-1} \end{align} we have: \begin{align}\label{hc21} G_\xi=G_0+G_0\Sigma_\xi G_\xi. \end{align} Finally, $G_0 \Sigma_\xi$ is also a Volterra operator and \begin{align}\label{hc30} G_\xi=\left ({\rm Id}- G_0 \Sigma_\xi\right )^{-1}G_0. \end{align} As in the physical literature Eq.(\ref{hc20}) defines the irreducible self-energy operator $\Sigma_\xi$ in terms of the reducible part $ \widetilde{\Sigma}_\xi$. \end{theorem} \vspace{0.5cm} \subsection{Proof: step 1.} First we will show that the identity: \begin{align}\label{hc7} G_0^{-1}G_\xi={\rm Id}+F_\xi \end{align} holds on $C([0,T];\mathfrak{h})$, where the map $F_\xi$ is given by \begin{align}\label{hc8} \langle e_j|(F_\xi\psi)(t)\rangle:=\int_0^t \big\langle\{ \tau_{K_v}^{t'}(a^*(\psi(t'))), \tau_{K_v}^{t}(b(e_j))\}\big\rangle_{\rm pf}\; dt'. \end{align} Using \eqref{hc3} and \eqref{hc4} we have: \begin{align}\nonumber &\langle e_m|(G_0^{-1}G_\xi\psi)(t)\rangle =\langle e_m|\psi(t)\rangle-\sum_j \langle e_m|h_ve_j\rangle \langle e_j|(G_\xi\psi)(t)\rangle \label{hc11}\\ &+\int_0^t \big\langle\{ \tau_{K_v}^{t'}(a^*(\psi(t'))), \partial_t\tau_{K_v}^{t}(a(e_m))\}\big\rangle_{\rm pf}\; dt'. \end{align} From the antilinearity of the annihilation operators we get \begin{align*} \sum_j \langle e_m|h_ve_j\rangle \langle e_j|(G_\xi\psi)(t)\rangle= -{\rm i} \int_0^t \big\langle\{ \tau_{K_v}^{t'}(a^*(\psi(t'))), \tau_{K_v}^{t}(a(h_ve_m))\}\big\rangle_{\rm pf}\; dt'. \end{align*} Also, using \eqref{hc0}, \eqref{hc0'} and \eqref{W-comm} we obtain the identity: $$\partial_t\tau_{K_v}^{t}(a(e_m))=-i\tau_{K_v}^{t}(a(h_ve_m))+\tau_{K_v}^{t}(b(e_m)).$$ After introducing the last two identities into \eqref{hc11} we see that two terms cancel each other and we obtain \eqref{hc8}. \subsection{Proof: step 2.} The second step consists of showing that $F_\xi$ can be written as $\widetilde{\Sigma}_\xi G_0$, with $\widetilde{\Sigma}_\xi$ as in \eqref{hc9}. In order to identify $\widetilde{\Sigma}_\xi$ we compute for every $\phi\in C_0^1([0,T];\mathfrak{h})$ the quantity (remember that $a^*$ is linear): \begin{align}\nonumber \langle e_j|(F_\xi G_0^{-1}\phi)(t)\rangle &={\rm i}\int_0^t \big\langle\{ \tau_{K_v}^{t'}(a^*(\partial_{t'}\phi(t'))), \tau_{K_v}^{t}(b(e_j))\}\big\rangle_{\rm pf}\; dt'\label{hc12}\\ &-\int_0^t \big\langle\{ \tau_{K_v}^{t'}(a^*(h_v\phi(t'))), \tau_{K_v}^{t}(b(e_j))\}\big\rangle_{\rm pf}\; dt'. \end{align} Another key identity is: \begin{align*} \tau_{K_v}^{t'}(a^*(\partial_{t'}\phi(t')))=\partial_{t'}\left (\tau_{K_v}^{t'}(a^*(\phi(t')))\right )-{\rm i} \tau_{K_v}^{t'}(a^*(h_v\phi(t')))-\tau_{K_v}^{t'}(b^*(h_v\phi(t'))). \end{align*} Inserting this identity in \eqref{hc12}, integrating by parts with respect to $t'$ and using that $\phi(0)=0$, we obtain \eqref{hc9}. \subsection{Proof: step 3.} From the first two steps we derive \eqref{hc6}. From \eqref{hc9} and \eqref{hc5} we see that $A=G_0 \widetilde{\Sigma}_\xi$ is a Volterra operator for which there exists a $T$-dependent constant $C<\infty$ such that \begin{align}\label{hc10} \|(A\psi)(t)\|_{\mathfrak{h}}\leq C\; \int_0^t \|\psi(t')\|_{\mathfrak{h}} dt',\quad 0\leq t\leq T. \end{align} Then $({\rm Id} +A)^{-1}$ exists and it is given by a norm convergent Neumann series $\sum_{n\geq 0}(-1)^nA^n$, as long as $T<\infty$. We write $$G_0=\left ({\rm Id}+ G_0 \widetilde{\Sigma}_\xi\right )^{-1}G_\xi$$ and we can choose $\Sigma_\xi$ as in \eqref{hc20}, which finishes the construction of the proper self-energy. \subsection{Consequences.} We list a few remarks concerning our main theorem. \noindent (i) The integral kernel of $\widetilde{\Sigma}_\xi$ (see \eqref{hc9}) is given by \begin{align}\label{hc22} \widetilde{\Sigma}_\xi^R(e_j,t;e_m,t'):=-{\rm i} \theta(t-t') \big\langle\{ \tau_{K_v}^{t'}(b^*(e_m)), \tau_{K_v}^{t}(b(e_j))\}\big\rangle_{\rm pf} + {\rm i} \delta(t-t')\big\langle\tau_{K_v}^{t}(\{a^*(e_m),b(e_j)\})\big\rangle_{\rm pf}\;.\nonumber \end{align} If either $e_j$ or $e_m$ belongs to the leads, then the above matrix element equals zero. The explanation for the first term is that at least one of the two operators $b(e_j)$ and $b^*(e_m)$ defined through \eqref{W-comm} would be zero in this case, because the self-interaction $W$ is only supported in the sample, hence it commutes with any observable supported on the leads. For the second term, assume that $e_j$ is from the sample while $e_m$ is from the leads. Then since $b(e_j)$ is a sum of products of three creation/annihilation operators from the sample, it anticommutes with $a^*(e_m)$. The proper self-energy $\Sigma_\xi$ has the same support property. One recognizes that $\widetilde{\Sigma}_\xi^R(e_j,t;e_m,t')$ is a reducible self-energy . In the diagrammatic language all terms contributing to $\widetilde{\Sigma}_\xi^R(e_j,t;e_m,t')$ connect to other diagrams by incoming and outgoing $G_0$-lines. \noindent (ii) If both $e_j=x$ and $e_m=y$ are located in the small sample, then from \eqref{hc21} we see that in order to compute $G_\xi^R(x,t;y,t')$ we only need to know the values of $G_0$ restricted to the small sample (besides $\Sigma_\xi$, of course). From \eqref{hc2} we have: \begin{align*} &G_0^R(x,t;y,t')=-{\rm i} \theta(t-t') \langle x|{\rm e}^{-{\rm i}(t-t')h_v}y\rangle , \end{align*} with $x,y\in\mathcal{S}$. Such matrix elements can be computed from the resolvent $(h_v-z)^{-1}$ restricted to the small sample; we note that via the Feshbach formula, the biased leads appear as a non-local ``dressing'' potential which perturbs $h_\mathcal{S}$, see \cite{CMP1} for details. At the level of integral kernels, the Dyson equation \eqref{hc21} reads as: \begin{align*} G_\xi^R(x,t;y,t')=G_0^R(x,t;y,t')+\sum_{u,v\in \mathcal{S}}\int_0^t ds\int_0^s ds' G_0^R(x,t;u,s)\Sigma_\xi^R (u,s;v,s')G_\xi^R(v,s';y,t'). \end{align*} \noindent (iii) Assume that we can write $\Sigma_\xi$ as $\Sigma_{\rm app}+\Sigma'$, where $\Sigma_{\rm app}$ is an approximating Volterra operator. If $G_{\rm app}=({\rm Id}-G_0\Sigma_{\rm app})^{-1}G_0$ is the solution of the approximate Dyson equation $G_{\rm app}=G_0+G_0\Sigma_{\rm app}G_{\rm app}$, then we have: $$G_\xi=G_{\rm app}+G_{\rm app}\Sigma'G_\xi$$ and $G_\xi=({\rm Id}-G_{\rm app}\Sigma')^{-1}G_{\rm app}$. \noindent (iv) The limit $T\to\infty$ is a difficult problem. To the best of our knowledge, the only rigorous mathematical results concerning the existence of a steady-state regime in partition free-systems are \cite{CMP1,CMP2}. Under certain non-resonant conditions and for $\xi$ small enough, one can prove that a quantity like $G_\xi^R(e_m,t'+s;e_n,t')$, where $s>0$ is fixed, will have a limit as $t'\to\infty$. This is definitely not guaranteed to happen in all cases, not even in non-interacting systems, due to bound states which may produce persistent oscillations. (v) One may generalize the present setting in order to allow a non-trivial time dependence of the bias, the only difference would appear in the evolution groups which now would have time-dependent generators. Also, the notation and formulas would be more involved, but no new mathematical issues would appear. \section{Conclusions}\label{section4} We presented a non-perturbative approach to the partition-free transport problem. Starting from the Volterra operator associated to the retarded Green's function we establish its Dyson equation, and we derive closed formulas for the reducible and irreducible self-energies. The proof is rigorous yet elementary in the sense that although the partition-free scenario is a genuine non-equilibrium regime we do not use contour-ordered operators. A Keldysh equation for the lesser Green's function should be established following the same lines of reasoning, with the extra difficulty induced by the fact that in the partition free setting, the small sample is not empty at $t=0$. Unravelling the connection between the closed formula \eqref{hc9} and the diagrammatic approach remains an open problem. Although the anti-commutator structure $\big\langle\tau_{K_v}^{t}(\{a^*(\phi(t)),b(e_j)\})\big\rangle_{\rm pf} $ in Eq. \eqref{hc9} looks less familiar one can speculate that the systematic application of the Wick theorem should eventually recover various classes of diagrams. A possible approximation in the self-energy would be to replace the interacting propagator $\tau_{K_v}^{t}(\cdot)$ with the non-interacting one $\tau_{H_v}^{t}(\cdot)$, where $K_v=H_v+\xi W$. Note however that the application of the Wick theorem is technically challenging due to the extra term $\Gamma(\beta)$ appearing in \eqref{hc1}. Given the fact that the partition-free setting is less studied in the literature, yet more intuitive on physical grounds than the partitioned approach, we hope that our investigation will trigger more efforts from both the physical and mathematical-physics communities. Our main message is that one can properly formulate some of the central equations of the many-body perturbation theory (MBPT) in a direct way, paying close attention to fundamental issues like convergence, existence, uniqueness, stability, and at the same time, trying to obtain precise error bounds for a given approximation of the self-energy. The Volterra theory guarantees that for relatively small $T$'s one can "keep doing what one has been doing"; however, the large time behavior like for example the existence of steady states and the speed of convergence seem to be very much dependent on the system and no general recipe can work out in all cases. \textbf{Acknowledgments.} V.M.\;acknowledges financial support by the CNCS-UEFISCDI Grant PN-III-P4-ID-PCE-2016-0084 and from the Romanian Core Research Programme PN16-480101. H.C.\;acknowledges financial support by Grant 4181-00042 of the Danish Council for Independent Research $|$ Natural Sciences. C.A.P.\;acknowledges financial support by the ANR, Grant NONSTOPS (ANR-17-CE40-0006),

BkiUbZA4eIXgu1N6gNbE

\section{Introduction} Inspite of the belief that the structure of baryons in the octet and decuplet representation is roughly understood (and exhausted), recent claims on the discovery of the manifestly exotic baryon $\Theta^+$ have opened a new chapter in hadron physics (see e.g. Refs.~\cite{Nakano,Barmin,Stepanyan,Barth,HERMES,Kubarovsky,Asratyan,Alt,Kubarovsky2,Aleev,Abdel-Bary}). Although the existence and properties of this exotic and long-lived baryonic state still need final experimental confirmation a variety of theoretical models have been set up with different assumptions about the internal structure of the $\Theta^+$. Here the quark-soliton model of Diakonov, Petrov and Polyakov \cite{Diakonov} was the first to claim the existence of an antidecuplet with rather narrow spectral width (actually prior to the experimental observations). However, the $\Theta^+$ might also be 'bound' due to strong diquark correlations (in a relative $p$-wave) as proposed in Ref. \cite{Jaffe} or due to a strong mixing with the octet~\cite{Hyodo}. Alternatively, it might even be explained on the basis of the constituent quark model involving clusters \cite{Oka}. Further models have been proposed in the last 2 years that all claim a different dynamical origin of the pentaquark $\Theta^+$ \cite{Beane,Ioffe} (cf. \cite{Jafferev} and Refs. cited therein). Additionally, the properties of the exotic state have been analysed within the framework of QCD sum rules \cite{Sugiyama} and even lattice QCD \cite{Fodor,Bali,LQCDlast,Csikor}. However, for a better understanding of this exotic state and its wave function (in terms of the elementary degrees of freedom) it is very important to study the dynamics of $\Theta^+$ production in comparison to the production of non-exotic strange baryons. In this respect exclusive reactions with strangeness ($s\bar{s}$) production are of interest, i.e. (starting with $\gamma$ induced reactions): \begin{equation} \gamma p \to \bar{K}^0 \Theta ^+ \ , \label{gpKTheta} \end{equation} \begin{equation} \gamma p \to \bar{K}^{*0} \Theta ^+ \label{gpK*Theta} \end{equation} and \begin{equation} \gamma d \to \Lambda \Theta ^+ \ . \label{gdLTheta} \end{equation} The first two reactions (\ref{gpKTheta}) and (\ref{gpK*Theta}) -- where the $s$ quark ends up in the mesonic final state -- can be compared with $\Lambda$ production in the binary reactions \begin{equation} \gamma p \to K^+ \Lambda, \label{gpKL} \end{equation} \begin{equation} \gamma p \to K^{*+} \Lambda \label{gpK*L} \end{equation} where the $s$ quark ends up in the hyperon. Note that very detailed measurements of the reaction (\ref{gpKL}) have been performed in the energy range from threshold up to a photon energy of 2.6 GeV \cite{Glander04}. The third reaction (\ref{gdLTheta}), furthermore, can be compared with the two-body deuteron photodisintegration reaction $\gamma d \to p n $ which has been studied recently at Jlab \cite{Rossi}. We recall that several studies of $\Theta^+$ photoproduction have been performed within the framework of isobar models using the Born approximation \cite{Oh,Nam,Li,Ko,Liu2,Yu}. Since these models involve a variety of uncertain parameters (coupling constants and cutoff's) the resulting cross sections differ from several nb to almost 1~$\mu$b. On the other hand the Regge model has a substantial advantage that the amount of uncertain parameters is much lower and that the latter can be fixed by other reactions in a more reliable fashion \cite{Mart}. Accordingly, in this work we will apply the Quark-Gluon Strings Model (QGSM) combined with Regge phenomenology to the analysis of the differential and total cross sections of the exclusive reactions $\gamma p \to \ K^+ \Lambda$, $\gamma p \to \ K^+ \Sigma^0$ and $\gamma p \to \bar{K}^0 \Theta^+$. Similar final channels will be investigated also for pion and proton induced reactions. We note that the QGSM was originally proposed by Kaidalov in Ref. \cite{Kaidalov} for the description of binary hadronic reactions; as demonstrated in Refs. \cite{Kaidalov,KaidalovSurveys} the QGSM describes rather well the experimental data on exclusive and inclusive hadronic reactions at high energy. More recently this model has been also successfully applied to the description of the nucleon and pion electromagnetic form factors \cite{Tchekin} as well as deuteron photodisintegration \cite{Rossi,Grishina,Grishina2}. We recall that the QGSM is based on two ingredients: i) a topological expansion in QCD and ii) the space-time picture of the interactions between hadrons, that takes into account the confinement of quarks. The $1/N$ expansion in QCD (where $N$ is the number of colors $N_c$ or flavors $N_f$) was proposed by 't Hooft \cite{Hooft}; the behavior of different quark-gluon graphs according to their topology, furthermore, was analyzed by Veneziano \cite{Veneziano} with the result that in the large $N$ limit the planar quark-gluon graphs become dominant. This approach -- based on the $1/N_f$ expansion \cite{Veneziano} with $N_c \sim N_f$ -- was used by Kaidalov \cite{Kaidalov,KaidalovSurveys} in formulating the QGSM. Again for sufficiently large $N_f$ the simplest planar quark-gluon graphs were found to give the dominant contribution to the amplitudes of binary hadronic reactions. Moreover, it can be shown that (in the space-time representation) the dynamics described by planar graphs corresponds to the formation and break-up of a quark-gluon string (or color tube) in the $s$-channel (see e.g. \cite{Casher,Artru,Casher2,Andersson,Gurvich}). On the other hand an exchange of the $u$ and $\bar s$ quarks in the $t$-channel implies that the energy behavior of the amplitudes -- described by quark diagrams in Fig.~\ref{fig:qdiagr} -- is given by the contribution of the $K^*$ Regge trajectory. In this sense the QGSM can be considered as a microscopic model of Regge phenomenology. This in turn allows to obtain many relations between amplitudes of different binary reactions and residues of Regge poles which determine these amplitudes \cite{Kaidalov,KaidalovSurveys,Boreskov}. \begin{figure} \begin{center} \leavevmode \psfig{file=gampklkt.eps,width=6.cm} \caption{Quark planar diagrams describing the binary reactions: $\gamma p \to \ K^+ \Lambda$ (a), $\gamma p \to \bar{K}^0 \Theta^+$ (b and c).} \label{fig:qdiagr} \end{center} \end{figure} Our investigation is organized as follows: In Section 2 we outline our approach and present the results for the differential cross sections $\gamma p \to K^+ \Lambda,\ K^+ \Sigma^0,\ \bar{K}^{0} \Theta^+$. In Section 3 we compare total cross sections for the reactions $\gamma p \to K^+ \Lambda(1520)$ and $\gamma p \to \bar{K^0} \Theta ^+$ while in Section 4 we step on with the pion induced reactions $\pi^- p \to K^0 \Lambda$ and $\pi^- p \to K^- \Theta ^+$. An analysis of $\Theta^+$ production in exclusive and inclusive $NN$ collisions is presented in Section 5 while a summary of our studies is given in Section 6. \section{The reactions $\gamma p \to \ K^+ \Lambda$, $\gamma p \to \ K^+ \Sigma^0$ and $\gamma p \to \bar{K}^{0} \Theta^+$} We first concentrate on $\gamma$ induced reactions and work out the Regge model in more detail. The reaction $\gamma p \to \ K^+ \Lambda$ can be described by the exchange of two valence ($u$ and $\bar s$) quarks in the $t$-channel with any number of gluon exchanges between them (Fig.~\ref{fig:qdiagr} a) ). Alternatively, in terms of the Regge phenomenology this diagram corresponds to the $K^*$- Reggeon exchange mechanism shown in Fig.~\ref{fig:Regdiagr}~a). \begin{figure} \begin{center} \leavevmode \psfig{file=gpklkt.eps,width=6.cm} \caption{$K^*$ Reggeon exchanges corresponding to the quark planar diagrams of Fig. \ref{fig:qdiagr}.} \label{fig:Regdiagr} \end{center} \end{figure} Employing the Regge model further on we can write the $\gamma p \to \ K^+ \Lambda$ amplitude in the form \begin{eqnarray} && T(\gamma p \to K^+ \Lambda) \simeq \frac{e}{2 \gamma_{\rho}} T(\rho^0 p \to K^+ \Lambda) = \nonumber \\ && \frac{e}{2 \gamma_{\rho}}\ g_{\rho K K^*} \ g_{pK^* \Lambda} \ F_1(t) \ (-s/s_0^{K \Lambda})^{\alpha_{K^*}(t)} \ . \label{gpKLf} \end{eqnarray} Here $e^2/4\pi$ is the fine structure constant, $\gamma_{\rho}^2 /4 \pi$ = 0.55, ${\alpha_{K^*}(t)}$ is the $K^*$ Regge trajectory, $s_0^{K\Lambda}= (M_{\Lambda} + m_{K})^2$, $g_{\rho K K^*}$ and $g_{pK^* \Lambda}$ are the coupling constants describing the interaction of the $K^*$ Reggeon with the $ \rho K$ and $p \Lambda$ systems. Within the Reggeized-Born-term model (see eg. Refs.~\cite{Irving,Levy,Guidal}) it is assumed that the coupling constants $g_i$ in the Regge amplitude of Eq.~(\ref{gpKLf}) can be identified with the coupling constants in an effective Lagrangian model. However it is difficult to justify this assumption and we do not address this model here. We follow another approach for a 'Reggeization of the amplitude' as proposed in Refs. \cite{Tchekin,Grishina,Grishina2}, i.e. by using the $s-$channel convolution representation in the QGSM. In this approach one can express the amplitude for the reaction $\gamma p \to \ K^+ \Lambda$ in terms of the $s$-channel convolution of two amplitudes: $ T(\gamma p \to q + qq)$ and $T(q+qq \to K^+ \Lambda)$ (see Fig.~1). Then -- using the Regge representation for the hadron-quark and quark-hadron transition amplitudes -- we can Reggeize the binary amplitude $\gamma p \to \ K^+ \Lambda$. Such a procedure was applied in Refs. \cite{Grishina,Grishina2} to define the spin structure of the deuteron photoproduction amplitude. We point out that this approach is more general and gives a vertex structure of the amplitude at negative $t$ different from the Reggeized-Born term. Assuming, furthermore, that the $\Theta^+$ is a pentaquark of structure ($uudd\bar{s}$) we can use a similar strategy for the $\gamma p \to \bar{K}^0 \Theta^+$ reaction. The relevant quark diagrams for this reaction are shown in Fig. \ref{fig:qdiagr} b), c). It is obvious that in terms of the Regge phenomenology we can also use the $K^*$- Reggeon exchange model to describe the reaction $\gamma p \to \bar{K}^0 \Theta^+$ (cf. Fig.~\ref{fig:Regdiagr}~b). The $\gamma p \to \bar{K}^0 \Theta^+$ amplitude reads accordingly \begin{eqnarray} && T(\gamma p \to \bar{K}^0 \Theta^+ ) \simeq \frac{e}{2 \gamma_{\rho}} T(\rho_0 p \to \bar{K}^0 \Theta^+ ) = \nonumber \\ &&\frac{e}{2 \gamma_{\rho}} \ g_{\rho K K^*} g_{pK^* \Theta} \ F_2(t) \ (-s/s_0^{K \Theta})^{\alpha_{K^*}(t)} \ \label{gpKThetaf} \end{eqnarray} with $s_0^{K\Theta}= (M_{\Theta} + m_{K})^2$. In the following calculations the form factor squared $|F_i|^ 2 $ in (\ref{gpKLf}), (\ref{gpKThetaf}) is chosen always in the form \begin{equation} |F_i|^2 = (1 - B_i t) \, \exp (2 R_i^2 t). \label{FF} \end{equation} For the further developments it is important to recall that the QGSM originally was formulated for small scattering angles (or small negative 4-momentum transfer (squared) $-t$). Thus the question arises about the extrapolation of the QGSM amplitudes to large angles (or large $-t$). Here we adopt the same concept as in our previous works \cite{Grishina,Grishina2}: following Coon et al.~\cite{Coon} we assume that only a single analytic Regge term with a logarithmic trajectory gives the dominant contribution to large momentum transfer processes. As shown in \cite{Coon} such a model (denoted as 'logarithmic dual model') can describe very well the differential cross section $d\sigma/dt$ for elastic $pp$ scattering in the energy range of $5-24$ GeV/c for $-t$ up to 18~GeV$^2$. The logarithmic Regge trajectory itself can be written in the form \begin{equation} \label{nonlin} \alpha(t)= \alpha(0)- (\gamma \nu) \ln (1 - {t}/{T_B}). \end{equation} with the parameters $\alpha (0)=0.32$ and $T_B = 6$~GeV$^2$ that have been fixed in Refs.\cite{Volkovitsky94,Burak}. To describe the energy dependence of the $\gamma p \rightarrow K^+ \Lambda$ differential cross section at fixed $t$ we have found $\gamma \nu =2.75$. We note in passing that logarithmic Regge trajectories have also been discussed in Refs. \cite{Bugrij,Ito,Chikovani}. The special limit $\gamma \nu \to 0$ at large $-t$ corresponds to 'saturated' trajectories, i.e. all trajectories approach a constant asymptotically. Such a case leads to the 'constituent-interchange model' that can be considered as a predecessor of the 'asymptotic quark counting rules' \cite{Brodsky,Hiller}. Moreover, the model with 'saturated' trajectories has also successfully been applied to the large-$t$ behavior of exclusive photon- and hadron-induced reactions in Refs. \cite{Guidal,Fiore,White,Battaglieri}. Formally, the amplitude (\ref{gpKLf}) does not contain spin variables. Nevertheless it can be used for a description of the differential cross section that is averaged over the spin states of the initial particles and summed up over the polarizations of the final particles, \begin{eqnarray} &&\displaystyle \frac{d\sigma_{\gamma p \to K^+ \Lambda }}{d t} = \frac{1}{64\,\pi s}\ \frac{1}{(p_{\pi}^{\mathrm{cm}})^2}\ \times \nonumber \\ &&\frac{1}{4} \sum_{\lambda_{\gamma}, \lambda_{p}, \lambda_{\Lambda}} \ \left| \langle \lambda_{\Lambda} | T_{\gamma p \to K^+ \Lambda}(s,t) | \lambda_{p}, \lambda_{\gamma} \rangle \right|^2 \ . \end{eqnarray} Here the amplitude squared can be written as \begin{eqnarray} &&\displaystyle \frac{1}{4} \sum_{\lambda_{\gamma}, \lambda_{p}, \lambda_{\Lambda}} \ \left| \langle \lambda_{\Lambda} | T_{\gamma p \to K^+ \Lambda}(s,t) | \lambda_{p}, \lambda_{\gamma} \rangle \right|^2 = \nonumber \\ && \frac{e^2}{4 \gamma_{\rho}^2}g_{\rho K K^*}^2 \: g_{pK^* \Lambda}^2 |F_1(t)|^2 \left|-s/s_0^{K \Lambda}\right|^{2 \alpha_{K^*}(t)} \ . \end{eqnarray} Let us now discuss constraints that have to be fulfilled for the residues and coupling constants. In line with Refs.~\cite{KaidalovSurveys,Volkovitsky94,Nogteva} we assume that for the planar quark diagrams with light quarks there is some kind of 'universality' of the secondary Reggeon couplings to $q \bar q$ mesons involved in a binary reaction, i.e. in particular \begin{equation} g_{\rho K K^{*}}\simeq g_{\pi K K^*} \simeq g_{\rho \pi \pi} = g_0 \label{univers} \end{equation} with $g_0 \simeq$ 5.8. Taking $g_{\rho K K^*}=5.8$ and normalising the differential cross section of the reaction $\gamma p \to K^+ \Lambda$ at $t=0$ we find $g_{p K^* \Lambda} \simeq$ 3.5. This result shows that the Reggeon couplings to mesons and baryons might be, in general, different by up to a factor of 2. We mention that the form factor $F_i$ -- determining the $t$-dependence of the residue -- was parametrized in Refs.~\cite{KaidalovSurveys,Volkovitsky94} as \begin{equation} \label{dual} F_i(t)= \Gamma(1-\alpha_i(t)). \end{equation} Indeed, such a choice of the form factor is convenient for an analytical continuation of the amplitude to positive $t$ where the $\Gamma$ function decreases exponentially with $t$. However, in the region of negative $t$ the parametrization (\ref{dual}) exhibits a factorial growth and is not acceptable (see e.g. the discussion in Ref. \cite{Cassing}). Accordingly we use the parametrization of the form factor (\ref{FF}), which decreases with $t$. To keep the same normalization of the amplitude at $t=0$ we have to change the coupling constant squared as \begin{equation} \label{g_M} g_0^2 \to g_M^2=g_0^2 \Gamma(1-\alpha_i(0)), \end{equation} where for the $K^*$ trajectory we have $\Gamma(1-\alpha_{K^*}(0)) \simeq 1.32$. \begin{figure}[t] \centerline{\psfig{file=dst20.eps,width=7.0cm}} \centerline{\psfig{file=dst45.eps,width=7.0cm}} \centerline{\psfig{file=dst165.eps,width=7.0cm}} \centerline{\psfig{file=dst320.eps,width=7.0cm}} \centerline{\psfig{file=dst450.eps,width=7.0cm}} \caption{Differential cross section of the reaction $\gamma p \to K^+ \Lambda$ at $t=-0.02,-0.045,-0.165,-0.32$ and --0.45 GeV$^2$ as a function of the laboratory photon energy. The experimental data are from Refs.\cite{Glander04} (full triangles), \cite{Boyarski69} (full circles),\cite{Boyarski71}(full stars), and \cite{Feller72}(empty triangles). The solid line is the result of the QGSM for the contribution of the $K^*$ logarithmic Regge trajectory defined by Eq. (\ref{nonlin}). The dashed line (for $t = -0.165$ GeV$^2$) describes the result for the $K$ Regge trajectory $\alpha_K(t)=0.7 (t - m_K^2)$ normalized to the data of Ref. \cite{Glander04} at $E_{\gamma}$ = 2 GeV. } \label{dstklam045} \end{figure} \begin{figure}[t] \centerline{\psfig{file=dst600.eps,width=7.0cm}} \centerline{\psfig{file=dst790.eps,width=7.0cm}} \centerline{\psfig{file=dst1000.eps,width=7.0cm}} \centerline{\psfig{file=dst1300.eps,width=7.0cm}} \centerline{\psfig{file=dst2000.eps,width=7.0cm}} \caption{Differential cross section of the reaction $\gamma p \to K^+ \Lambda$ at $t=-0.6,-0.79,-1.0,-1.3$ and --2.0 GeV$^2$ as a function of the laboratory photon energy. The experimental data are from Refs. \cite{Glander04} (full triangles), \cite{Boyarski69} (full circles), \cite{Boyarski71}(full stars), and \cite{Anderson76}(empty circle). The solid lines are the results of the QGSM for the contribution of the $K^*$ logarithmic Regge trajectory defined by Eq. (\ref{nonlin}). } \label{dstklam2} \end{figure} The differential cross section for the reaction $\gamma p \to K^+ \Lambda$ is presented in Figs. \ref{dstklam045}~--~\ref{dstklam2} as a function of the laboratory photon energy at fixed values of $t$. The solid lines are calculated using our model with the following coupling constants and parameters of the form factor: $g_{\rho K K^*}=5.8, \ g_{p K^* \Lambda} \simeq 3.5$, $B_1= 5 $~GeV$^{-2}$, $R^2_1=1.13$~GeV$^{-2}$. The experimental data are from Refs.\cite{Glander04} (full triangles), \cite{Boyarski69} (full circles), \cite{Boyarski71}(full stars), \cite{Feller72}(empty triangles) and \cite{Anderson76} (empty circles). The agreement between the solid curves and the experimental data clearly supports the dominant role of the $K^*$ Regge trajectory in the reaction $\gamma p \to K^+ \Lambda$. The dashed line in Fig.~\ref{dstklam045} -- calculated at $t=-0.165$ GeV$^2$ -- describes the result for a $K$ Regge trajectory normalized to the data at $E_{\gamma} = 2$~GeV. Definitely, it can not describe the energy dependence of the differential cross section and we may conclude that the $K$ Regge trajectory is subdominant. We have applied our model also to the description of the reaction $\gamma p \to K^+ \Sigma^0$ adopting the same coupling constants and form factor as for the reaction $\gamma p \to K^+ \Lambda$, however, modifying the scaling factor to $s_0^{K\Sigma}= (M_{\Sigma} + m_{K})^2$. The total cross sections of the reactions $\gamma p \to K^+ \Lambda$ and $\gamma p \to K^+ \Sigma^0$ as a function of the laboratory photon energy are shown in Fig. \ref{slam} (upper and lower parts describe the reactions $\gamma p \to K^+ \Lambda$ and $\gamma p \to K^+ \Sigma^0$, respectively) in comparison to the experimental data from Ref. \cite{Glander04}. In this context one has to note that the Regge model gives only the average cross section for a particular channel and misses resonant amplitudes at low energy. For example, according to recent data on the reaction $\gamma p \to K^+ \Sigma^0$~\cite{McNabb} the $s$-channel resonance contributions are found to be important for photon beam energies at least up to 1.5~GeV. Since the $K^+\Lambda$ and $K^+\Sigma^0$ systems show strong resonances in the 1.3 to 2 GeV invariant mass region the latter cannot be described in the Regge approach. Nevertheless, the results of our model calculations (presented as the solid lines) are in a good agreement with the data -- except for the resonance structures mentioned above -- and support the 'universality' of Reggeon couplings. \begin{figure}[t] \centerline{\psfig{file=gpkplamtot.eps,width=7.0cm}} \centerline{\psfig{file=gpkpsigtot.eps,width=7.0cm}} \caption{Total cross section of the reactions $\gamma p \to K^+ \Lambda$ and $\gamma p \to K^+ \Sigma^0$ as a function of the laboratory photon energy in comparison to the experimental data from \cite{Glander04}. The solid line is the result of the QGSM for the contribution of the $K^*$ logarithmic Regge trajectory defined by Eq. (\ref{nonlin}).} \label{slam} \end{figure} \section{Total cross sections for the reactions $\gamma p \to K^+ \Lambda(1520)$ and $\gamma p \to \bar{K}^0 \Theta ^+$} In this Section we explore if the universality of the $K^*$ trajectory coupling to baryons with constituent $qqs$ quarks also holds in the binary reactions \begin{equation} \gamma p \to K^+ Y_i \label{eq:KY} \end{equation} with $Y_1= \Lambda(1116), Y_2=\Lambda(1520),...$. In case of the universality to hold we have \begin{equation} g_{pK^* {\Lambda(1520)}} = g_{pK^* \Lambda} \simeq 3.5, ~F_{\Lambda(1520)}(t)=F_{1}(t) . \end{equation} The resulting total cross section of the reaction $\gamma p \to K^+ \Lambda(1520)$ is presented in Fig. \ref{slam1520} in comparison to the experimental data from \cite{Barber} (full squares) and \cite{Barth} (empty square). The solid curve is calculated for the coupling constant $g_{pK^* Y}=3.5$ which corresponds directly to the prediction from the universality principle. The dashed curve is calculated using $g_{pK^* Y}=4.14$ and is in a good agreement with the data; the deviation between the two curves does not exceed 40\%. Therefore, the data on the reactions $\gamma p \to K^+ \Lambda$ and $\gamma p \to K^+ \Lambda(1520)$ support the assumption on the universality of the $K^*$ trajectory coupling to $q \bar q$ mesons as well as to baryons with constituent $qqs$ quarks (at least within a factor of 2). We, accordingly, consider (or define) the universality principle to hold if a variety of cross sections is described (predicted) within a factor better than 2. We continue with the $\gamma p \rightarrow \bar{K}^0 \Theta ^+$ reaction and explore if the unversality principle (in the sense defined above) also holds in this case. The cross section of the reaction $\gamma p \to \bar{K}^0 \Theta ^+$ was estimated by the SAPHIR collaboration \cite{Barth} as \begin{equation} \sigma_{\gamma p \to \bar{K}^0 {\Theta}^+} \simeq 200 \ {\mbox{nb}} \label{Theta_cross_section} \end{equation} at an average photon energy of $\sim$2 GeV. Let's first adopt 200 nb as an upper limit for the total cross section of the reaction $\gamma p \to \bar{K}^0 \Theta ^+$ at 2 GeV. In this case equation (\ref{Theta_cross_section}) implies already a noticeable violation of the universality principle for $\Theta^+$ photoproduction since -- assuming the universality principle to hold for $\Theta^+$ -- we get \begin{equation} g_{pK^* \Theta} = g_{pK^* {\Lambda(1520)}} \simeq 4.14,~ \ F_2(t)=F_1(t). \end{equation} This leads to a total cross section of the reaction $\gamma p \to \bar{K}^0 \Theta ^+$ shown by the solid line in Fig.~\ref{stht}. The dash-dotted curve in Fig.~\ref{stht} is calculated assuming \begin{equation} g_{pK^* \Theta}^{\mbox{\tiny SAPHIR}} \simeq 0.4 \ g_{pK^*{\Lambda(1520)} } \simeq 1.8 \ , \ F_2(t)=F_1(t) \label{redu} \end{equation} in order to match the quoted cross section in (\ref{Theta_cross_section}). However, the new preliminary results from the CLAS collaboration \cite{DeVita} do not support the estimate (\ref{Theta_cross_section})and indicate that the upper limit on the total cross section of the reaction $\gamma p \to \bar{K}^0 \Theta ^+$ should be much lower: \begin{equation} \sigma_{\gamma p \to \bar{K}^0 \Theta ^+} \leq 1\div 4 \ \mbox{nb}\ . \label{CLAS_ limit} \end{equation} When taking 4~$nb$ as an upper limit we find \begin{equation} g_{pK^* \Theta}^{\mbox{\tiny CLAS}} \simeq 0.06 \ g_{pK^*{\Lambda(1520)} } \simeq 0.25 \ , \ F_2(t)=F_1(t) \ . \label{redu1} \end{equation} Therefore, using the preliminary result from the CLAS collaboration we find a very strong suppression of the $\gamma p \to \bar{K}^0 \Theta ^+$ cross section relative to the prediction from the universality principle for photoproduction of the lowest $qqs$ baryons. If the pentaquark exists, we may interpret this finding as a clear indication of a substantially different quark structure of the $\Theta^+$. \begin{figure} \begin{center} \leavevmode \psfig{file=gampklam1520.eps,width=8.cm} \caption{Total cross section of the reaction $\gamma p \to K^+ \Lambda(1520)$ as a function of the lab. photon energy in comparison to the experimental data from Ref. \cite{Barber} (full squares) and Ref. \cite{Barth} (open square). The solid line is the result of the QGSM for the contribution of the $K^*$ logarithmic Regge trajectory defined by Eq. (\ref{nonlin}) (see text); the dashed line results for a coupling $g_{pK^* Y}=4.14$.} \label{slam1520} \end{center} \end{figure} \begin{figure} \begin{center} \leavevmode \psfig{file=gampkthetcl.eps,width=8.cm} \caption{Total cross section of the reaction $\gamma p \to \bar{K}^0 \Theta ^+$. The solid line results when assuming the validity of the universality principle also for the $\Theta^+$ baryon. The dash-dotted curve is calculated for the reduced couplings~(\ref{redu}) assuming the validity of the quoted cross section ~(\ref{Theta_cross_section}). The dotted curve is calculated using the coupling constant $g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$~(\ref{redu1}) that corresponds to the preliminary upper limit from the CLAS collaboration (\ref{CLAS_ limit}) on the total cross section of the reaction $\gamma p \to \bar{K}^0 \Theta ^+$ \cite{DeVita}. In the latter case the 'universality principle' is found to be violated by almost 3 orders of magnitude. } \label{stht} \end{center} \end{figure} \section{Pion induced reactions: $\pi^- p \to K^0 \Lambda$ and $\pi^- p \to K^- \Theta ^+$ } We continue with $\pi^-$ induced reactions and assume that the amplitudes of the reactions $\pi^- p \to K^0 \Lambda$ and $\pi^- p \to {K^-} \Theta ^+$ are also dominated by the contribution of the $K^*$ Regge trajectory (see Fig.~\ref{fig:pipklkt} a) and b)) such that the cross sections are fully determined. We directly step on with the results for the differential cross section of the $\pi^- p \to K^0 \Lambda$ reaction as a function of $t$ at $p_{\mathrm{lab}}$=4.5, 6, 8, 10.7 and 15.7~GeV/c shown in Fig. \ref{dstpipkl} in comparison to the experimental data from~\cite{Foley73,Crennel}. The results of our model are displayed by the dashed lines calculated for the following parameters: \begin{equation} g_{\pi K^* K} =5.8, ~g_{pK^* \Lambda} \simeq 4.5, ~B=0,~R_1^2=2.13~ \mathrm{GeV^{-2}} \label{pipkl_g}. \end{equation} We see that the coupling constants $g_{\pi K^* K}$ and $g_{pK^* \Lambda}$ -- in the case of the reaction $\pi^- p \to K^0 \Lambda$ -- are also in agreement with the universality assumption for the coupling constants. However, to describe the $t$-dependence of the differential cross section we have to employ different parameters for the form factor $F(t)$ as compared to the reaction $\gamma p \to {K^+} \Lambda$. This can be explained, in particular, by the different relative contributions of the baryon spin-flip terms in the reactions $\gamma p \to K^+ \Lambda$ and $\pi^- p \to K^0 \Lambda$. \begin{figure} \begin{center} \leavevmode \psfig{file=pipklkt.eps,width=6.cm} \caption{$K^*$ Reggeon exchanges corresponding to the quark diagrams of Fig. \ref{fig:qdiagr}.} \label{fig:pipklkt} \end{center} \end{figure} \begin{figure}[t] \centerline{\psfig{file=pipkl45.eps,width=7.0cm}} \centerline{\psfig{file=pipkl6.eps,width=7.0cm}} \centerline{\psfig{file=pipkl8.eps,width=7.0cm}} \centerline{\psfig{file=pipkl107.eps,width=7.0cm}} \centerline{\psfig{file=pipkl157.eps,width=7.0cm}} \caption{The differential cross section of the $\pi^- p \to K^0 \Lambda$ reaction as a function of $t$ at different laboratory momenta p$_{\mathrm{lab}}$ in comparison to the experimental data from \cite{Foley73} (full circles) and~\cite{Crennel} (full triangles). The results of our model are shown by the dashed lines.} \label{dstpipkl} \end{figure} The total cross section of the reaction $\pi^- p \to K^0 \Lambda$ is presented in Fig.~\ref{pipkltot} where the dashed curve is the result of our calculations. As expected for a Regge model -- and discussed above -- we see some deviation of the QGSM from the data~\cite{Landoldt} in the resonance region. However, at higher energies, where the explicit resonance structure disappears, the theoretical calculations are in a good agreement with the data. Now using the coupling constants $g_{\pi K^* K} =5.8$, $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny SAPHIR}}$ and $g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$ and assuming $R_1^2$ and $B$ to be the same as for the reaction $\pi^- p \to K^0 \Lambda$ we can calculate the cross section for the reaction $\pi^- p \to K^- \Theta^+$ . The results are shown in Fig.~\ref{pipkltot} by the solid and dotted line, respectively; these cross sections reach about 10 (0.2)~$\mu$b in their maximum. \begin{figure} \begin{center} \leavevmode \psfig{file=pipkltotcl.eps,width=7.cm} \caption{Total cross section of the reaction $\pi^- p \to K^0 \Lambda$ as a function of the c.m. excess energy $Q= \sqrt{s} - m_K - M_{\Lambda}$ (dashed line) in comparison to the data from Ref.~\cite{Landoldt}. The solid and dotted line show the expected total cross section of the reaction $\pi^- p \to K^- \Theta^+$ as a function of the c.m. excess energy $Q= \sqrt{s} - m_K - M_{\Theta^+ (1540)}$ calculated with the coupling constants $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny SAPHIR}}$ (19) and $g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$ (21). } \label{pipkltot} \end{center} \end{figure} \section{$\Theta^+$ production in exclusive and inclusive $NN$ collisions} Further constraints on the universality of amplitudes and cross sections for $\Theta^+$ production are provided by $NN$ collisions. Here the amplitude of the reaction $pp \to \Theta^+ \Sigma^+ $ (cf. upper diagram in Fig.~\ref{fig:ppthetax}) has been calculated using the coupling constant $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny SAPHIR}}$ ($g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$). Of course, taking into account the results of Section~2 we can safely assume that $g_{pK^* \Sigma}\simeq g_{pK^* \Lambda}$. At the same time the coupling constant $g_{pK^* \Lambda}$ may vary from 3.5 (if we define it via the reaction $\gamma p \to K^+ \Lambda$) to 4.5 (if we define it via the $\pi^- p \to K^0 \Lambda$ reaction ). In this section we use $g_{pK^* \Lambda}$ = 4.5. The predictions for the total cross section of the $pp \to \Theta^+ \Sigma^+ $ reaction are shown by the solid (dotted) line in Fig.~\ref{ppthetsig} using $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny SAPHIR}}$ ($g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$) . The cross section described by the dotted line is $\sim$ 20--30 times smaller than the experimental value $\sigma=0.4\pm 0.1(\mbox{stat}) \pm 0.1 (\mbox{syst})~\mu$b~\cite{Abdel-Bary} measured at a beam momentum of $2.95$~GeV/c (open square) and clearly signals an incompatibility of the different measurements. \begin{figure} \begin{center} \leavevmode \psfig{file=ppthetsigcl.eps,width=7.cm,height=4.3cm} \caption{Total cross section for $\Theta^+$ production as a function of the excess energy $Q$ in the reaction $pp \to \Theta^+ \Sigma^+$. The solid (dotted) line is the result of a calculation with the coupling constants $g_{p K^* \Sigma}=g_{p K^* \Lambda}=4.5$, $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny SAPHIR}}$ ($g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$). The data point (open square) is from the COSY-TOF collaboration~\cite{Abdel-Bary}. } \label{ppthetsig} \end{center} \end{figure} \begin{figure}[t] \centerline{\psfig{file=ppsigt.eps,width=5.0cm}} \centerline{\psfig{file=ppktp.eps,width=5.0cm}} \centerline{\psfig{file=ppthetxf.eps,width=5.0cm}} \caption{The diagrams describing the $\Theta^+$ production in the reactions: $pp \to \Theta^+ \Sigma^+$ or $pn \to \Theta^+ \Sigma^0 (\Lambda)$ (upper diagram), $pp \to \Theta^+ \bar{K}^0 p$ with pion exchange (middle diagram) and $pp \to \Theta^+ X$ with kaon exchange (lower diagram).} \label{fig:ppthetax} \end{figure} We note, that the first analysis of the reaction $pp \to \Theta^+ \Sigma^+ $ was performed by Polyakov et al. in Ref. \cite{Polyakov} even before the $\Theta^+$ baryon was 'discovered' from the experimental side. Their estimation of the cross section -- within the kaon-exchange approximation -- was about 2 $\mu$b at the initial momentum $\sim$3 GeV/c. Approximately the same value of the total cross section of the reaction $pp \to \Theta^+ \Sigma^+ $ was found by Liu and Ko in Ref. \cite{LiuKo} later on. Our calculated cross section is about 0.8~$\mu$b on the basis of the coupling (19) and 16 $nb$ for the coupling (21) at the excess energy $Q=22$~MeV ($p_{\mathrm{lab}}=2.95$~GeV/c). The maxima of cross sections are $3.3 \ \mu$b and 66$~nb$, respectively, at the excess energy $Q=460$~MeV ($p_{\mathrm{lab}}=4.4$~GeV/c). These cross sections are smaller by factors of 2 and 90, respectively, compared to the early estimates. We continue with alternative production mechanisms for $\Theta^+$ production in $pp$ collisions as described by the middle and lower diagrams in Fig.~\ref{fig:ppthetax}. In the following we use the method of Yao~\cite{Yao} to calculate the cross sections of the reactions: \\ 1) $pp\to p \bar {K}^0 \Theta^+$ with the pion exchange (middle diagram in Fig.~\ref{fig:ppthetax})\\ 2) $pp\to p \bar {K}^0 \Theta^+$ with the kaon exchange (lower diagram in Fig.~\ref{fig:ppthetax} with $X=\bar {K}^0 p$), \\ 3) $pp\to \Theta^+ X$ with the kaon exchange (lower diagram in Fig.~\ref{fig:ppthetax}).\\ In the case of pion exchange the expression for the total cross section can be written in the form: \begin{eqnarray} && \sigma (pp \to p \bar{K}^0 \Theta^+)= \nonumber \\ &&\frac{G^2_{\pi NN}}{8 {\pi}^2 p_{1} s} \int_{W_{\mathrm {min}}}^{W_{\mathrm {max}}} k \, W^2 \ \sigma ({\pi}^0 p \to \bar{K}^0 {\Theta}^+, W)\ dW \nonumber \\ &&\times \int_{t_{\mathrm{min}}(W)}^{t_{\mathrm{max}}(W)} F_{\pi}^4(t)\ \frac{1}{(t-m_{\pi}^2)^2} \ t\, dt \ , \label{pppiex} \end{eqnarray} where $W$ is the invariant mass of the $\bar{K}^0 {\Theta}^+$ system, $k$ is defined as $$ k=\left((W^2-(m_p-m_{\pi})^2)(W^2-(m_p+m_{\pi})^2)\right)^{1/2}/2W \ , $$ $p_1$ is the initial proton momentum in the c.m. system, $t=(p_2-p_4)^2$, and $G_{\pi NN}=13.45$. Assuming that $J^P(\Theta^+)=\frac12^+$ we have the following expression for the kaon exchange contribution \begin{eqnarray} && \sigma (pp \to {\Theta}^+ X)= \nonumber \\ &&\frac{G^2_{\Theta KN}}{8 {\pi}^2 p_{1} s} \int_{W_{\mathrm {min}}}^{W_{\mathrm {max}}} k \, W^2 \ \sigma (\bar{K}^0 p \to X, W)\ dW \nonumber \\ &&\times \int_{t_{\mathrm{min}}(W)}^{t_{\mathrm{max}}(W)} F_{K}^4(t)\ \frac{1}{(t-m_{K}^2)^2} \ \left(t-\Delta_{M43}^2 \right)\, dt \ . \label{ppkex} \end{eqnarray} Here $W$ is the invariant mass of the system $X$ and $\Delta_{M43}^2=(m_{\Theta}-m_p)^2$. The form-factors for the virtual pion and kaon exchange have been chosen of the monopole type \begin{equation} F_j(t)=\frac{\Lambda_j^2-m_j^2}{\Lambda_j^2-t} \, \end{equation} with $\Lambda_{\pi}=1.3$~GeV and $\Lambda_K=1$~GeV. These parameters have been used in Ref.~\cite{Gasparyan} to describe the total cross section of the reaction $pp \to K^+ \Lambda p$. Obviously, the contribution of the kaon exchange to the reaction $pp \to \Theta^+ X$ depends on the coupling constant $G_{\Theta KN}$, which can only be estimated (or fixed by upper limits). If the ${\Theta}^+$ decay width is less than $1$~MeV~\cite{PDG} we have $G_{\Theta KN} \leq 1.4$. The solid and dashed lines in Fig.~\ref{ppthetax} present our results for the inclusive $pp\to \Theta^+ X$ and exclusive $pp\to \Theta^+ \bar{K}^0 p$ reactions as calculated within the kaon-exchange model for $G_{\Theta KN} = 1.4$ (as an upper limit). The inclusive cross section turns out to be about $1.5~\mu$b at high energies while the exclusive $pp\to \Theta^+ \bar{K}^0 p$ cross section is at least one order of magnitude smaller. \begin{figure} \begin{center} \leavevmode \psfig{file=ppthetaxcl.eps,width=8.cm} \caption{Total cross sections for $\Theta^+$ production as a function of the c.m. energy in the reactions: $pp \to \Theta^+ \Sigma^+$ (dotted curve) and $pp \to \Theta^+ \bar{K}^0 p$ with pion exchange (dash-dotted line) calculated with the coupling constant $g_{p K^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$ from Eq.~(\ref{redu1}), $pp \to \Theta^+ \bar{K}^0 p$ with kaon exchange (dashed line), and $pp \to \Theta^+ X$ with kaon exchange (solid line).} \label{ppthetax} \end{center} \end{figure} The reaction $pp\to p \bar {K}^0 \Theta^+$ may proceed also via $\pi$ exchange. The corresponding contribution to the total cross section calculated with the coupling constant $g_{p K^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$ from Eq.~(\ref{redu1}) is presented by the dash-dotted line in Fig.~\ref{ppthetax}. It reaches about $10$~nb in its maximum. Therefore, we can conclude that the $\Theta^+$ production cross section in $pp$-collisions is dominated by the $K$ exchange and should be about a few $\mu$b. Note that using the hadronic Lagrangian model Liu and Ko \cite{LiuKo} found a considerably larger cross section for $\Theta^+$ production, i.e. about 50 $\mu$b in pion-nucleon reactions and $\sim$20 $\mu$b in proton-proton reactions. From our point of view the latter results are essentially due to a large coupling constant $G_{\Theta KN} \simeq 4.4$ in \cite{LiuKo}, which corresponds to $\Gamma_{\Theta} \approx 20$ MeV. Such large couplings, however, should be excluded according to the more recent analysis in Ref.~\cite{Sibirtsev}. \section{Conclusions} In this study we have analyzed $\Lambda$, $\Sigma^0$, $\Lambda(1520)$ and $\Theta^+$ production in binary reactions induced by photon, pion and proton beams in the framework of the Quark-Gluon Strings Model combined with Regge phenomenology. Starting with the existing experimental data on the $\gamma p \to K^+ \Lambda$ reaction we have demonstrated that the differential and total cross sections at photon energies $1 - 16$~GeV and $-t < 2$~GeV${}^2$ can be described very well by the model with a dominant contribution of the $K^*$ Regge trajectory. We stress that the rather good description of the large $t$ region was possible only due to the logarithmic form of the $K^*$ Regge trajectory (\ref{nonlin}). It has been demonstrated, furthermore, that the data on the reactions $\gamma p \to K^+ \Lambda$, $\gamma p \to K^+ \Sigma^0$ and $\gamma p \to K^+ \Lambda(1520)$ -- at least within a factor of 2 -- support the assumption on the universality of the $K^*$ trajectory coupling to $q \bar q$ mesons as well as to baryons with $qqs$ constituent quarks. This implies that -- using the same parameters as for $\gamma p \to K^+ \Lambda$ -- we are able to reproduce the total $\gamma p \to K^+ \Sigma^0$ and $\gamma p \to K^+ \Lambda(1520)$ cross sections (within an accuracy of 40\%). On the other hand, as a consequence of the SAPHIR data \cite{Barth} and the preliminary data from CLAS \cite{DeVita}, there is an essential suppression of the $\gamma p \to \bar{K}^0 \Theta ^+$ cross section relative to the prediction within the universality principle that was shown to hold (with reasonable accuracy) for the photoproduction of the lowest $qqs$ baryons. We conclude that this suppression indicates a substantially different quark structure and wave function of the $\Theta^+$ (in case of its final experimental confirmation). Moreover, we have suggested that the amplitudes of the reactions $\pi^- p \to K^0 \Lambda$ and $\pi^- p \to {K^-} \Theta ^+$ are also dominated by the contribution of the $K^*$ Regge trajectory (cf. Fig.\ref{fig:pipklkt} a) and b)). Indeed, the differential and total cross sections of the $\pi^- p \to K^0 \Lambda$ reaction are found to be in a reasonable agreement with the universality principle. Using parameters defined by the analysis of the reactions $\gamma p \to K^+ \Lambda$, $\gamma p \to K^+ \Sigma^0$, $\gamma p \to K^+ \Lambda(1520)$, $\gamma p \to \bar{K}^0 \Theta^+$ and $\pi^- p \to K^0 \Lambda$ we have calculated the cross section for the reaction $\pi^- p \to {K^-} \Theta ^+$ . We predicted a maximum cross section of about 200~nb (cf. Fig. 10) for the coupling constant $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$ extracted from the new preliminary CLAS result on the reaction $\gamma p \to \bar{K}^0 \Theta^+$~\cite{DeVita}. We extended our model additionally to the analysis of the binary reaction $pp \to \Sigma^+ \Theta^+$. We found that the cross section of this reaction -- measured by the COSY-TOF collaboration \cite{Abdel-Bary}-- is 20--30 times larger than the value predicted by the model with the coupling constant $g_{pK^* \Theta}=g_{pK^* \Theta}^{\mbox{\tiny CLAS}}$. Furthermore, we have investigated the exclusive and inclusive $\Theta^+$ production in the reactions $pp \to p \bar{K}^0 \Theta^+$ and $pp \to \Theta^+ X$ and found that the inclusive $\Theta^+$ production in pp collisions at high energy should be on the level of 1 $\mu$b. The systematic and comparative Regge analysis -- provided by our study -- will also allow in future to relate different data sets from $\gamma$, $\pi$ and proton induced reactions on $s\bar{s}$ pair production and finally should yield a transparent picture of the dynamics as well as the properties of (possible) exotic states. \section*{Acknowledgments} The authors are grateful to A.B. Kaidalov and E.~De~Sanctis for useful discussions. This work was partially supported by DFG (Germany) and INFN (Italy). One of us (V. G.) acknowledges the financial support from For\-schungszentrum J\"ulich (FFE grant 41520739 (COSY - 071)).

BkiUbdA5qsNCPiNSnCq-

\section{Introduction} Sequential decision making tasks such as web-marketing and robot control can often be formalized as Markov decision processes and solved using reinforcement learning (RL). RL algorithms produce policies that map states of the world to actions in order to maximize the expected return, i.e., the sum of a scalar reward signal over time. The sub-problem of policy evaluation in RL is the problem of estimating the expected return of a particular \textit{evaluation policy} using a finite set of data collected by interacting with the task of interest. Policy evaluation allows practitioners to estimate the expected return of a particular policy before extensively deploying it. In many RL applications, data-efficient policy evaluation is of the upmost importance -- we desire the most accurate estimate with minimal collected data. Towards data-efficient policy evaluation, much research has gone into how to most efficiently use a set of already collected data~\citep{precup2000eligibility,thomas2016data-efficient}. Comparatively less research has considered the question of how to improve data collection for the data-efficiency of policy evaluation. This paper focuses on this latter question. In the policy evaluation literature, a widely held assumption is that the optimal data collection strategy is to simply run the evaluation policy~\citep[c.f.][Chapter 5]{sutton1998reinforcement}. This strategy, known as on-policy data collection, typically leads to lower variance policy evaluation than off-policy data collection, in which a different policy (called the \emph{behaviour policy}) is executed to collect data. One exception to this assumption is the work of \citet{hanna2017data-efficient} who show that off-policy evaluation can be more data-efficient than on-policy evaluation if importance sampling \citep{precup2000eligibility} is the policy evaluation method, and propose the behavior policy gradient (BPG) algorithm to search for a behavior policy that can increase the probability of rare, high-magnitude returns. However, both on-policy data collection and BPG fail to consider what additional data best supplements the previously collected data. Specifically, on-policy data collection samples each trajectory independently and BPG only attempts to minimize the error of policy evaluation on data collected in the future. In this paper, we study the problem of how to take previously collected data into account when collecting data for Monte Carlo policy evaluation. We design two data collection strategies\footnote[1]{ We provide an open-source implementation of this work here: \url{https://github.com/uoe-agents/robust_onpolicy_data_collection} } -- robust on-policy sampling (ROS) and robust on-policy acting (ROA) -- both of which consider previously collected data when collecting new data to move the empirical distribution of the entire dataset closer to the expected on-policy distribution (referred to as lower \emph{sampling error}). We then show across multiple RL domains that data from ROS and ROA generally leads to lower mean-squared error (MSE) policy evaluation than Monte Carlo evaluation with on-policy sampling data collection. We also show that ROS and ROA can be initialized with offline data and collect additional data to be able to use the entire dataset \textit{without off-policy estimation methods}. While both ROS and ROA lower MSE compared to on-policy data collection, ROA generally lowers MSE further while ROS is a simpler strategy to implement. Taken together, our contributed strategies and empirical study demonstrate the promise of taking past data into account when collecting data for policy evaluation in reinforcement learning. \section{Related Work} Data collection is a fundamental part of the RL problem. The most widely studied data collection problem is the question of how an agent should explore its environment to learn an optimal policy~\citep{ostrovski2017count,tang2017exploration}. Many approaches introduce intrinsic rewards~\citep{oudeyer2009intrinsic,bellemare2016unifying,pathak2017curiosity} or consider uncertainty in their value function~\citep{osband2016bootstrapped,Zintgraf2020VariBADAV,ciosek2019optimistic} to guide policy optimisation. \citet{schaefer2021decoupling} instead propose to learn a separate exploration strategy to collect off-policy data used to optimise a policy. In contrast to these approaches, our work focuses on the question of how an agent should collect data to evaluate a fixed policy. When given a choice of how to collect data for policy evaluation, on-policy data collection is generally preferable to off-policy data collection \cite{sutton1998reinforcement}. One exception is the adaptive importance sampling method introduced by \citet{hanna2017data-efficient} which leads to lower MSE policy evaluation than on-policy Monte Carlo sampling. Adaptive importance sampling has also been used in RL for policy gradient learning \cite{ciosek2017offer,bouchard2016online} and temporal difference learning \cite{frank2008reinforcement}. Adaptive importance sampling focuses on lowering variance in estimates from future samples while our work aims to lower variance in the estimate computed from both past and future samples. The strategies we introduce in this paper are motivated by the idea of decreasing sampling error in collected data. Previous work has considered how sampling error can be reduced \textit{after} data collection by re-weighting the obtained samples. For example, \citet{hanna2021importance} show how importance sampling with an estimated behavior policy can lower sampling error and lead to more accurate policy evaluation. Similar methods have also been studied for policy evaluation in multi-armed bandits \cite{li2015toward,narita2019efficient}, temporal-difference learning \cite{pavse2020reducing}, and policy gradient RL \cite{hanna2021importance}. All of these prior works assume that data is available a priori and ignore the question of how to collect data if it is unavailable. Finally, the idea of adapting the \textit{sampling distribution}, (i.e., behavior policy) has analogs outside of policy evaluation in Markov decision processes. \citet{ohagan1987monte} identifies flaws in Monte Carlo sampling that motivate taking past samples into account. \citet{rasmussen2003bayesian} use Gaussian processes to represent uncertainty in an expectation to be evaluated and use this uncertainty to guide future sample generation. However, we are unaware of any adaptations of these ideas for RL. \section{Preliminaries} \subsection{Notation} We assume the environment is an episodic \emph{Markov decision process} with state set $\mathcal{S}$, action set $\mathcal{A}$, transition function, $P: \mathcal{S} \times \mathcal{A} \times \mathcal{S} \rightarrow [0,1]$, reward function $R: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$, discount factor $\gamma$, and initial state distribution $d_0$ \cite{puterman2014markov}. For simplicity, we assume that $\mathcal{S}$ and $\mathcal{A}$ are finite though all methods discussed are applicable to infinite-sized state and action spaces and our empirical analysis considers both settings. We assume that the transition and reward functions are unknown. A policy, $\pi: \mathcal{S} \times \mathcal{A} \rightarrow [0,1]$, is a function mapping states and actions to probabilities. We use $\pi(a|s) \coloneqq \pi(s,a)$ to denote the conditional probability of action $a$ given state $s$ and $P(s^\prime | s,a) \coloneqq P(s, a, s^\prime)$ to denote the conditional probability of state $s^\prime$ given state $s$ and action $a$. Let $h \coloneqq (s_0,a_0,r_0,s_1, \dotsc, s_{{l} - 1},a_{{l} - 1},r_{{l} - 1})$ be a possible \textit{trajectory} and $g(h) \coloneqq \sum_{t=0}^{{l} - 1} \gamma^t r_t$ be the \textit{discounted return} of $h$. Any policy induces a distribution over trajectories, $\Pr(h | \pi)$. We define the \textit{value} of a policy, $v(\pi) \coloneqq \sum_{h} \Pr(h|\pi) g(h)$, as the expected discounted return when sampling a trajectory with policy $\pi$. \subsection{Policy Evaluation} In the policy evaluation problem, we are given an \textit{evaluation policy}, ${\pi_e}$, for which we would like to estimate $v({\pi_e})$. Conceptually, algorithms for policy evaluation involve two steps: collecting data (or receiving previously collected data) and producing an estimate. We assume that data is collected by running a policy which we call the \textit{behavior policy}. If the behavior policy is the same as the evaluation policy data collection is \textit{on-policy}; otherwise it is \textit{off-policy}. Whether on-policy or off-policy, we assume the data collection process produces a set of trajectories, $D \coloneqq \{H_i\}_{i=1}^n$. The final value estimate is then computed by a policy evaluation method ($\operatorname{PE}$) that maps the set of trajectories to a scalar-valued estimate of $v({\pi_e})$. Our goal is policy evaluation with low \textit{mean squared error} (MSE), which is formally defined as: \begin{equation} {\operatorname{MSE}} \biggl [\operatorname{PE} \biggr] \coloneqq \mathbf{E}\biggl [\biggl(\operatorname{PE}(D) - v(\pi_e)\biggr)^2 \biggr], \end{equation} where the expectation is taken with respect to the collected data, $D$. MSE is the most common metric in the policy evaluation literature~\citep[e.g.][]{thomas2016data-efficient,ris}. \subsection{Monte Carlo Policy Evaluation} Perhaps the most fundamental, model-free policy evaluation method is the \textit{on-policy Monte-Carlo} (MC) estimator. As an on-policy method, the Monte Carlo estimator collects data by sampling trajectories with ${\pi_e}$, which is known as on-policy sampling. The estimate of $v(\pi_e)$ is then the mean return: \begin{equation} \operatorname{MC}(D) \coloneqq \frac{1}{n}\sum_{i=1}^n g(H_i) = \sum_{h} \Pr(h|D) g(h), \end{equation} where $\Pr(h|D)$ denotes the empirical probability of $h$, i.e.\ how often $h$ appears in $D$. This estimator is unbiased and strongly consistent given mild assumptions \cite{sen1993large}. However, this method can have high variance as on-policy sampling may require many trajectories for the empirical trajectory distribution $\Pr(h|D)$ to accurately approximate $\Pr(h|{\pi_e})$. Since on-policy sampling collects each trajectory i.i.d., it relies on the law of large numbers for an accurate weighting on each possible return. We refer to the discrepancy between $\Pr(h|D)$ and $\Pr(h|{\pi_e})$ as \textit{sampling error}. \section{Data Completion Problem} Our starting point for this work is the question, ``given an available dataset of trajectories, how should additional trajectories be collected for minimal MSE policy evaluation of a fixed evaluation policy?" This question amounts to asking what behavior policy should be used to collect additional trajectories. The answer to this question depends on the policy evaluation estimator used to produce the value estimate. In this paper we focus on the sample-average Monte Carlo estimator without importance sampling which may be the simplest method for policy evaluation. Given our choice of Monte Carlo estimation, on-policy data collection may seem like the only generally viable choice for accurate policy evaluation. However, when including previously collected trajectories in the policy evaluation data, on-policy future data collection for the Monte Carlo estimator may produce an estimate that is biased in expectation -- even if the older trajectories were collected on-policy as well. Consider a one-step MDP with one state, two actions $a_0$ and $a_1$. The return following $a_0$ is $2$ and the return following $a_1$ is $4$. The evaluation policy puts equal probability on both actions and suppose that, after sampling 3 trajectories, we have observed $\{a_0,2\}$ twice and $\{a_1,4\}$ once. If we collect an additional trajectory with ${\pi_e}$ the expected value of the Monte Carlo estimate is: $\frac{1}{4} (2 + 2 + 4 + 2{\pi_e}(a_0) + 4{\pi_e}(a_1)) = \frac{11}{4} = 2.75$. The true value, $v({\pi_e}) = 3$ and thus, \textit{conditioned on prior data}, the Monte Carlo estimate is biased in expectation. If instead we choose the behavior policy such that ${\pi_b}(a_1)=1$ then the expected value of the Monte Carlo estimate becomes: $\frac{1}{4} (2 + 2 + 4 + 4) = \frac{12}{4} = 3$. By considering the previously collected data when choosing the behavior policy for collecting additional data the final estimate is less biased in expectation. This example highlights that adapting the behavior policy to consider previously collected data can lead to more accurate policy evaluation. We call this problem -- how to choose a behavior policy that depends on what data is already collected -- the dataset completion problem (DCP). DCP has a resemblance to the behavior policy search problem introduced by \citet{hanna2017data-efficient}. The crucial difference is that \citeauthor{hanna2017data-efficient} only considered how to minimize MSE using the data newly collected by the chosen behavior policy. We are interested in minimizing MSE using the newly collected data \textit{and} the data already collected. In the next section, we will introduce two behavior policy adaptation strategies that improve upon on-policy sampling for the Monte Carlo estimator. \section{Robust On-Policy Data Collection} In this section, we introduce two data collection strategies that adapt the data collecting behavior policy online so as to minimize sampling error in the Monte Carlo estimator. These strategies are also able to start with data collected from a different policy than ${\pi_e}$ and collect additional data to reduce sampling error in the entire dataset, allowing policy evaluation without using off-policy corrections. To reduce sampling error, our goal is to reduce the discrepancy between the evaluation policy ${\pi_e}$ and the \emph{empirical policy} ${\pi_{D}}$, which is the maximum-likelihood policy under collected data $D$. As the distribution of the collected data $D$ could be decomposed as $\Pr(h|D)\approx\Pr(h|{\pi_{D}})$, we are more likely estimating the policy value of ${\pi_{D}}$ when performing Monte Carlo estimation, and thus may get large error if there is a large difference between ${\pi_{D}}$ and ${\pi_e}$. In the next two sub-sections, we will introduce two concrete strategies for reducing this discrepancy (sampling error). In particular, we assume that the evaluation policy is from a class of policies parameterized by a vector ${\boldsymbol\theta}$ and we write ${\boldsymbol\theta}_e$ to denote the evaluation policy's parameter values, and ${\pi_\btheta}$ to denote a policy with parameters ${\boldsymbol\theta}$. \subsection{Robust On-policy Sampling} Our first strategy -- \textbf{R}obust \textbf{O}n-Policy \textbf{S}ampling (ROS) -- reduces sampling error by adapting the behavior policy ${\pi_b}$ for future data collection, which -- compared to ${\pi_e}$ -- has higher probabilities for under-sampled actions and lower probabilities for over-sampled actions, i.e., $({\pi_b} - {\pi_e})({\pi_{D}} - {\pi_e}) \leq 0$. If we perform one-step gradient ascent on the log-likelihood evaluated at ${\boldsymbol\theta}_e$, we obtain a new policy that is closer to ${\pi_{D}}$. ROS therefore updates the behavior policy in the \textit{opposite} direction with a step of gradient \textit{descent} on the log-likelihood evaluate at ${\boldsymbol\theta}_e$. Pseudocode for ROS is shown in Algorithm~\ref{alg:ros}. Importantly, we can use an incremental implementation to store and compute the gradient. In this way, we maintain linear per-time-step computation while considering all historical data in the behavior policy update \begin{algorithm}[tb] \begin{algorithmic}[1] \STATE {\bfseries Input:} evaluation policy $\pi_e$, step size $\alpha$ \STATE Initialize $\nabla \leftarrow 0$ \FOR{step $i$} \STATE Observe $s$ from the environment \STATE ${\boldsymbol\theta}_b \leftarrow {\boldsymbol\theta}_e - \alpha\nabla$ \STATE Choose $a \sim \pi_{{\boldsymbol\theta}_b}(s)$ \STATE $\nabla \leftarrow \frac{i}{i+1}\nabla + \frac{1}{i+1}\nabla_{\boldsymbol\theta} \log{\pi_\btheta}(a|s) \rvert_{{\boldsymbol\theta}={\boldsymbol\theta}_e}$ \ENDFOR \end{algorithmic} \caption{Robust On-policy Sampling.} \label{alg:ros} \end{algorithm} While we will show empirically that ROS reduces sampling error, we highlight two limitations that motivate the second strategy we introduce. Firstly, choosing under-sampled actions cannot always reduce sampling error. Suppose we have the evaluation policy with $\pi_e(a_1|s)=0.9$ and $\pi_e(a_2|s)=0.1$. If we have only collected one-step data of action $a_1$ in state $s$, the empirical policy can be denoted as $\pi_D(a_1|s)=1$ and $\pi_D(a_2|s)=0$. At this point, action $a_2$ is a under-sampled action for state $s$. However, if we choose action $a_2$, the empirical policy will become $\pi_D(a_1|s)=0.5$ and $\pi_D(a_2|s)=0.5$, which will significantly change the empirical policy and enlarge sampling error. This phenomenon (referred to as \emph{over-correction}) could easily occur when using ROS with large step sizes. Second, in continuous action domains, the parameterization of policy class ${\pi_\btheta}$ may be too inflexible to adequately reduce sampling error. For example, a common distribution used in these domains is the normal distribution where the policy is defined as $\pi(a|s) \coloneqq \mathcal{N}(a, \mu, \sigma^2)$, where $\mu$ and $\sigma^2$ are the mean and variance, respectively. In this case the ROS behavior policy $\pi_{{\boldsymbol\theta}_b}$ can only output a normal distribution as it shares the same structure as the evaluation policy. As an example, we may obtain the ROS behavior policy $\pi_{{\boldsymbol\theta}_b}$ with $\pi_e$ and $\pi_D$ as shown in Figure~\ref{fig:dist_pieDb_continuous}. This behavior policy fails to increase the probability of some under-sampled actions. \begin{figure}[tb] \centering \includegraphics[width=0.5\columnwidth]{figs/dist_pieDb_continuous} \caption{Distribution of evaluation policy $\pi_e$, empirical policy $\pi_D$ and ROS behavior policy $\pi_{{\boldsymbol\theta}_b}$. This figure gives an example where ROS fails to increase the probability of all under-sampled actions because it is constrained by the functional form of the policy class.} \label{fig:dist_pieDb_continuous} \end{figure} \subsection{Robust On-policy Acting} To mitigate both problems of ROS, we propose \textbf{R}obust \textbf{O}n-policy \textbf{A}cting (ROA) that aims to identify and choose the action that can minimize the magnitude of the gradient of the log-likelihood, as shown in Algorithm~\ref{alg:roa}. This strategy is based on the observation that when there is no sampling error, the gradient of the log-likelihood should be zero when evaluated at ${\boldsymbol\theta}_e$. As the empirical policy ${\pi_{D}}$ is the maximum likelihood policy of collected data $D$, its parameters can be obtained by ${\boldsymbol\theta}_D \coloneqq \argmax_{\boldsymbol\theta} \sum_{s,a\in D} \log{\pi_\btheta}(a|s)$. When there is no sampling error, ${\boldsymbol\theta}_D = {\boldsymbol\theta}_e$ and the log-likelihood has zero-gradient when evaluated at ${\boldsymbol\theta}_e$. This observation provides a goal for adapting the behavior policy to reduce sampling error: taking actions that reduce the magnitude of the gradient of the log-likelihood of actions in $D$. Deterministicly choosing the gradient minimizing action can fail to reduce sampling error when done in a state that has not been previously visited. To reduce sampling error, the algorithm should only consider the historical data in states similar to the current state. To mitigate this issue, ROA reserves a probability of $1-\rho$ to sample actions from ${\pi_e}$ (line $6$), so as to collect data for identifying the minimum-sampling-error action precisely. For computational tractability, ROA only considers an $m$-action subset for choosing the minimum-gradient action in continuous action domains. This finite subset is obtained by computing the inverse cumulative distribution function (CDF) of ${\pi_e}$ and choosing actions at uniform percentiles (line $10$). Such a subset considers a wide range of actions while focusing on actions more likely to be represented in the data. \begin{algorithm}[tb] \begin{algorithmic}[1] \STATE {\bfseries Input:} evaluation policy $\pi_e$, correction probability $\rho$, potential action number $m$ \STATE Initialize $\nabla \leftarrow 0$ \FOR{step $i$} \STATE Observe $s$ from the environment \STATE Sample $u$ from $\mathcal{U}(0,1)$ \IF{$u < \rho$} \IF{$\mathcal{A}$ is a finite set} \STATE $\widetilde{\mathcal{A}} \gets \mathcal{A}$ \ELSE \STATE $\widetilde{\mathcal{A}} \gets \left\{ F^{-1}(\frac{i}{m+1}) \right\}_{i=1}^m$ where $F^{-1}$ is the inverse CDF of evaluation distribution ${\pi_e}(s)$. \ENDIF \STATE $a = \underset{a' \in \widetilde{\mathcal{A}}}{\text{argmin}} \lVert \frac{i}{i+1}\nabla + \frac{1}{i+1}\nabla_{\boldsymbol\theta} \log{\pi_\btheta}(a'|s)\rvert_{{\boldsymbol\theta}={\boldsymbol\theta}_e} \rVert$ \ELSE \STATE Choose $a \sim {\pi_e}(s)$ \ENDIF \STATE $\nabla \leftarrow \frac{i}{i+1}\nabla + \frac{1}{i+1}\nabla_{\boldsymbol\theta} \log{\pi_\btheta}(a|s)\rvert_{{\boldsymbol\theta}={\boldsymbol\theta}_e}$ \ENDFOR \end{algorithmic} \caption{Robust On-policy Acting.} \label{alg:roa} \end{algorithm} \section{Empirical Study} We next conduct an empirical study of our proposed strategies for the dataset completion problem. In this empirical study, our goal is to answer the following questions: \begin{enumerate} \item Do ROS and ROA collect data with lower sampling error? \item Do ROS and ROA lower the MSE of policy evaluation when starting with or without off-policy data? \item How is the MSE of ROS and ROA affected by different evaluation policies, hyper-parameter and environment settings? \end{enumerate} In particular, we use KL-divergence $D_\text{KL}\left(\pidata \parallel \pieval \right)$ of the empirical policy ${\pi_{D}}$ and the evaluation policy ${\pi_e}$ to measure sampling error. Similar measurement has also been applied in offline RL~\citep{jaques2019way}, and the complete definition is presented in Appendix~\ref{sec:kl} We conduct policy evaluation experiments in four domains covering discrete and continuous state and action spaces: A multi-armed bandit problem~\citep{sutton1998reinforcement}, Gridworld~\citep{hanna2017data-efficient}, CartPole and Continuous CartPole~\citep{gym}. Since these domains are widely used, we omit their description here; full details are presented in Appendix~\ref{sec:env}. Our primary point of comparison is to on-policy sampling (OS) with Monte Carlo estimation. We also compare to Behavior Policy Gradient (BPG) which is designed to find a minimum variance behavior policy for ordinary importance sampling~\citep{hanna2017data-efficient}. We obtain the evaluation policy, ${\pi_e}$, in each domain by improving an initial random policy with REINFORCE~\citep{reinforce} and stopping once the policy has improved but is still far from convergence. We obtain the true value $v({\pi_e})$ and the average episode steps $\overline{T}$ in each domain with $10^6$ Monte Carlo roll-outs. Additional details are found in Appendix~\ref{sec:pi_e}. \subsection{Policy Evaluation without Initial Data} \begin{table*}[t] \centering \scriptsize \begin{tabular}{ccccc} \hline Policy Evaluation & MultiBandit & GridWorld & CartPole & CartPoleContinuous\\ \hline OS - MC & 4.01e-05 $\pm$ 3.89e-06 & 2.68e-04 $\pm$ 2.42e-05 & 2.61e-05 $\pm$ 2.40e-06 & 4.44e-05 $\pm$ 4.01e-06\\ BPG - OIS & 3.52e-05 $\pm$ 3.03e-06 & 2.81e-04 $\pm$ 2.63e-05 & 1.20e-05 $\pm$ 1.11e-06 & 3.62e-05 $\pm$ 3.54e-06\\ ROS - MC & 2.93e-05 $\pm$ 2.79e-06 & 1.36e-05 $\pm$ 1.33e-06 & 1.01e-05 $\pm$ 9.79e-07 & 3.29e-05 $\pm$ 3.01e-06\\ ROA - MC & 2.83e-05 $\pm$ 3.03e-06 & 2.10e-07 $\pm$ 2.39e-08 & 3.74e-06 $\pm$ 3.34e-07 & 1.02e-05 $\pm$ 1.06e-06\\ \hline \end{tabular} \caption{Final MSE of policy evaluation without initial data. These results give the MSE for policy evaluation at the end of data collection, averaged over 200 trials with one standard error.} \label{tab:mse} \end{table*} \begin{figure}[tb] \centering \begin{subfigure}{0.3\columnwidth} \centering \includegraphics[width=\textwidth]{figs/basic_kl_GridWorld.pdf} \caption{Without initial data} \label{fig:_kl_basic} \end{subfigure}\qquad \begin{subfigure}{0.3\columnwidth} \centering \includegraphics[width=\textwidth]{figs/comb_kl_GridWorld.pdf} \caption{With initial data} \label{fig:_kl_comb} \end{subfigure} \caption{Sampling error ($D_\text{KL}\left(\pidata \parallel \pieval \right)$) curves of data collection in the GridWorld domain. Each strategy is followed to collect data with $2^{13}\overline{T}$ steps, and all results are averaged over $200$ trials with shading indicating one standard error intervals. (\ref{fig:_kl_basic}) and (\ref{fig:_kl_comb}) show the sampling error curves of data collection without and with initial data, respectively. Axes in these figures are log-scaled. } \label{fig:_kl} \end{figure} We first start from the setting where we have no available initial data and collect data from scratch for policy evaluation. For each domain, we follow different data collection strategies to collect data with $2^{13}\overline{T}$ steps, which is equivalent to around $2^{13}$ trajectories. Note that we specify the number of steps for any evaluation, as it is a fairer control variable than the number of trajectories to compare different strategies for data-efficient policy evaluation. The hyper-parameter settings for these experiments are presented in Appendix~\ref{sec:hp}. We first verify that ROS and ROA reduce sampling error compared to on-policy sampling. Figure~\ref{fig:_kl_basic} shows sampling error in the GridWorld domain, and we observe that both ROS and ROA can collect data with less sampling error than other strategies. This pattern can also hold for other domains in our study, which can be found in Appendix~\ref{sec:kl}. We also measure sampling error using the magnitude of the log-likelihood gradient and present results in Appendix~\ref{sec:grad} and these gradient curves are generally consistent with the sampling error curves. This result supports ROA identifying the minimum-gradient action as the next action to take to reduce sampling error. Ultimately, we aim to reduce sampling error for lower MSE policy evaluation. In our next experiment, we compare different data collection strategies for policy evaluation. Figure~\ref{fig:mse} shows that both ROS and ROA can generally lead to lower MSE compared to both OS and BPG. However, the small gap between ROS and OS in Figure~\ref{fig:mse_cpc} shows the limited ability of ROS to generalize in continuous action domains. The numerical results of the final MSE are reported in Table~\ref{tab:mse}. In order to evaluate the stability of different strategies, we also present the median and inter-quartile range of the squared error (SE) of the policy evaluation for these experiments in Appendix~\ref{sec:mq}, and results show that ROS and ROA will not introduce extra error range and can also enable estimation with lower median of SE. Other policy evaluation methods such as weighted regression importance sampling~\citep{hanna2021importance} and fitted q-evaluation~\citep{le2019batch} are also conducted on this collected data, and the results can be found in Appendix~\ref{sec:mse_others}, showing that these off-policy evaluation methods could also benefit from this lower-sampling-error data. \begin{figure}[tb] \centering \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/basic_mse_MultiBandit.pdf} \caption{MultiBandit} \label{fig:mse_mb} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/basic_mse_GridWorld.pdf} \caption{GridWorld} \label{fig:mse_gw} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/basic_mse_CartPole.pdf} \caption{CartPole} \label{fig:mse_cp} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/basic_mse_CartPoleContinuous.pdf} \caption{CartPoleContinuous} \label{fig:mse_cpc} \end{subfigure} \caption{Mean squared error (MSE) of different policy evaluation without initial data. Policy evaluation is conducted on the data collection from each strategy, and these curves show the MSE of estimations and the true policy value. Lower MSE denotes more accurate policy evaluation. Steps, axes, trials and intervals are the same as Figure~\ref{fig:_kl}. } \label{fig:mse} \end{figure} \subsection{Policy Evaluation with Initial Data} \begin{figure}[tb] \centering \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/comb_mse_MultiBandit} \caption{MultiBandit} \label{fig:comb_mse_mb} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/comb_mse_GridWorld} \caption{GridWorld} \label{fig:comb_mse_gw} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/comb_mse_CartPole} \caption{CartPole} \label{fig:comb_mse_cp} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/comb_mse_CartPoleContinuous} \caption{CartPoleContinuous} \label{fig:comb_mse_cpc} \end{subfigure} \caption{Mean squared error (MSE) of different policy evaluation with initial data. In particular, the estimation at step 0 is based on only the off-policy data (OPD), while there is no data available for OS \emph{only} to make estimations and thus no MSE presented. Steps, axes, trials and intervals are the same as in Figure~\ref{fig:mse}.} \label{fig:comb_mse} \end{figure} We then conduct experiments in the setting where a small amount of off-policy data (OPD) is already available, and we follow a strategy to collect additional data for policy evaluation. To simulate this scenario, we first follow a behavior policy that is slightly different from the evaluation policy to collect $100$ trajectories as OPD (more details see Appendix~\ref{sec:opd}). We then follow different strategies respectively to continue collecting data with $2^{13}\overline{T}$ steps. All strategies except OS will consider this OPD as the historical data and adjust their behavior policies according to the corresponding algorithms. The hyper-parameter settings for these experiments are presented in Appendix~\ref{sec:hp}. This newly collected data will be combined with OPD for policy evaluation. Sampling error and log-likelihood gradient are computed for each combined data, and ROS and ROA can reduce both of them faster than OS, which can be found in Appendix~\ref{sec:kl} and~\ref{sec:grad}. The MSE curves of policy evaluation are shown in Figure~\ref{fig:comb_mse}. In particular, OS can be regarded as a baseline that only uses new data for policy evaluation. Additionally, we also consider using an off-policy estimator -- weighted importance sampling (WIS) -- for OPD and MC for new OS data, and make combined estimations with the sum weighted by their sizes of data. It is shown from these figures that although OIS and WIS can generally make more accurate estimation for OPD than MC, ROS and ROA can collect additional data that can quickly reduce sampling error and produce lower MSE MC estimation compared to other data collection strategies. It is worth noting that in Figure~\ref{fig:comb_mse_cpc}, it is hard for OS to correct the bias brought by OPD, while ROS and ROA are able to correct the off-policy distribution to on-policy distribution, and thus make estimation without any off-policy estimator. The numerical results of the final MSE can be found in Appendix~\ref{sec:stat}. \subsection{Hyper-parameter Sensitivity} In this sub-section, we explore the effects of hyper-parameter settings on the performance of ROS and ROA, based on the settings of policy evaluation without initial data. ROS only involves one hyper-parameter, step size $\alpha$, which controls how much the ROS behavior policy reduces its probabilities for over-sampled actions, and increases those of under-sampled actions. By taking GridWorld and CartPole as examples, we show the MSE curves of ROS with different step size $\alpha$ in Figures~\ref{fig:alpha}. In particular, OS can be seen as ROS with $\alpha=0$. It can be observed from Figure~\ref{fig:alpha_gw} that ROS with large step size may produce higher MSE when collecting small sizes of data, which reflects its problem of over-correction. As it collects more data, ROS with larger $\alpha$ can generally enable lower MSE because the norm of the gradient reduces dramatically, and thus requires large $\alpha$ to make significant updates. However, in the domain with a continuous state space (Figure~\ref{fig:alpha_cp}), ROS with extremely large $\alpha$ ($1000$) cannot reduce MSE with more data, because performing gradient descent with such a large step size on a neural network's parameters can make the behavior policy diverge. \begin{figure}[tb] \centering \begin{subfigure}{0.3\columnwidth} \centering \includegraphics[width=\textwidth]{figs/alpha_GridWorld} \caption{GridWorld} \label{fig:alpha_gw} \end{subfigure}\qquad \begin{subfigure}{0.3\columnwidth} \centering \includegraphics[width=\textwidth]{figs/alpha_CartPole} \caption{CartPole} \label{fig:alpha_cp} \end{subfigure} \caption{MSE curves of ROS with different step size $\alpha$. These figures covers the MSE curves of ROS with $\alpha \in \{1,10,100,1000\}$. Steps, axes, trials and intervals are the same as Figure~\ref{fig:mse}.} \label{fig:alpha} \end{figure} On the other hand, ROA involves two hyper-parameters, correction probability $\rho$ and potential action number $m$. By taking GridWorld, CartPole, CartPoleContinuous as examples, we show the MSE curves with different $\rho$ and $m$ in Figure~\ref{fig:eps_m}. It can be observed that ROA can further lower the MSE with increasing $\rho$ in the pure tabular domain (Figure~\ref{fig:eps_gw}), showing it mitigates the problem of over-correction. However, this pattern cannot hold in non-tabular domains (Figure~\ref{fig:eps_cp} and~\ref{fig:eps_cpc}), which is because ROA with large $\rho$ cannot collect enough on-policy data to identify the minimum-sampling-error actions precisely. Another hyper-parameter $m$ is the number of actions that ROA can choose from and thus also affects the precision of ROA identifying minimum-sampling-error actions. In Figure~\ref{fig:m_cpc}, ROA could slightly lower the MSE with $m$ increasing from $5$ to $9$, but could hardly further improve with $m$ further increasing. \begin{figure}[tb] \centering \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/eps_GridWorld} \caption{GridWorld} \label{fig:eps_gw} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/eps_CartPole} \caption{CartPole} \label{fig:eps_cp} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/eps_CartPoleContinuous} \caption{CartPoleContinuous} \label{fig:eps_cpc} \end{subfigure} \begin{subfigure}{0.245\columnwidth} \centering \includegraphics[width=\textwidth]{figs/m_CartPoleContinuous} \caption{CartPoleContinuous} \label{fig:m_cpc} \end{subfigure} \caption{MSE curves of ROA with different correction probability $\rho$ and potential action number $m$. (\ref{fig:eps_gw}) (\ref{fig:eps_cp}) (\ref{fig:eps_cpc}) consider ROA with $\rho$ in $\{0.6, 0.8, 1.0\}$ for tabular domains and $\{0.05, 0.1, 0.2\}$ for non-tabular domains, and (\ref{fig:m_cpc}) considers $m \in \{5,9,15,19\}$ for domains with continuous action space. Steps, axes, trials and intervals are the same as Figure~\ref{fig:mse}.} \label{fig:eps_m} \end{figure} \subsection{Environment and Policy Sensitivity} We then explore how different environment settings and evaluation policies can affect the improvement of ROS and ROA to OS. In this sub-section, we take the MultiBandit domain as the example for simplicity, while similar experimental results in GridWorld can be found in Appendix~\ref{sec:imp_gw}. In particular, we choose $\alpha=1000$ for ROS and $\rho=1.0$ for ROA for the following experiments. We first create MultiBandit environments with different randomness by multiplying the means and scales of the reward by different factors. OS, ROS and ROA are followed to collect data with $1000\overline{T}$ steps and MC estimation are performed on this data, respectively. The relative MSEs are shown in Figure~\ref{fig:imp_env_mb}, we can observe that 1) as the factors for means increase, MC estimation could suffer from larger error for the wrong action proportions and thus higher improvement can be brought by ROS and ROA (lower sampling error); 2) as the factors for scales increase, the environment contains more noise, which makes it harder to make accurate estimations even with low-sampling-error data. To evaluate the improvement for evaluation policies with different randomness, we create parameterized epsilon-greedy policies with epsilon from $0$ to $1$. In particular, the epsilon-greedy policy with $0$ epsilon represents the deterministic optimal policy, and that with $1$ epsilon represents the uniformly random policy. Complete details about the creation of these policies can be found in Appendix~\ref{sec:eps_policy}. We then perform the same policy evaluation as above and present the relative MSEs in Figure~\ref{fig:imp_pie_mb}. It can be observed that as the policies contain more stochasticity (with larger epsilon), ROS and ROA generally improve upon OS by a larger margin, except when the policies are close to uniformly random. \begin{figure}[tb] \centering \begin{subfigure}{0.3\columnwidth} \centering \includegraphics[width=\textwidth]{figs/imp_env_MultiBandit.pdf} \caption{Improvement with different environment settings.} \label{fig:imp_env_mb} \end{subfigure}\qquad \begin{subfigure}{0.3\columnwidth} \centering \includegraphics[width=\textwidth]{figs/imp_pie_MultiBanditM1S1.pdf} \caption{Improvement with different evaluation policies.} \label{fig:imp_pie_mb} \end{subfigure} \caption{Improvement of ROS and ROA to OS with different settings in MultiBandit. (\ref{fig:imp_env_mb}) shows improvement curves with changing reward mean factors and reward scale factors where higher factors represent greater randomness of the environment. (\ref{fig:imp_pie_mb}) shows the improvement curves with changing epsilon-greedy policies where higher epsilon represent greater randomness of ${\pi_e}$. Results in these figures are averaged over $500$ trials.} \label{fig:imp_mb} \end{figure} \section{Discussion and Future Work} This work has shown across multiple domains that ROS and ROA are more data-efficient than OS for accurate policy evaluation, which benefits from the consideration of previously collected data. To the best of our knowledge, these methods are the first data collection methods for policy evaluation that consider how to combine future data with the data that has already been collected. The strong performance of ROS and ROA suggests that this general approach for data collection is a direction that should be explored more in the policy evaluation literature. In our experiments, ROA generally led to more accurate policy evaluation compared to ROS. On the other hand, ROS is simpler to implement than ROA, has one fewer hyper-parameters, and avoids the need to find a minimum over the action space. These characteristics give a trade-off for practitioners interested in using these methods. Fortunately, choosing either method lowered MSE compared to on-policy sampling. In the future, we would like to investigate improved data collection for off-policy estimators. ROS and ROA reduce sampling error, producing data that matches the expected on-policy distribution and allowing them to use the mean return as the value estimate. An alternative could be to use data collection strategies that perform wider exploration and then rely on off-policy estimators to correct the discrepancy between the empirical distribution and the expected on-policy distribution. In preliminary results we have observed ROS and ROA reducing the MSE of weighted regression importance sampling \cite{hanna2021importance} and fitted q-evaluation \cite{le2019batch} compared to on-policy sampling. However, novel data collection strategies specifically designed for off-policy estimators may further increase data efficiency. We are also interested in increasing the data efficiency of policy improvement methods with ROS and ROA. While exploration for policy improvement is typically framed as uncovering new, high-rewarding action sequences, ROS and ROA take actions to boost data efficiency. Integrating these two types of exploration for policy improvement is an interesting direction for future research. \section{Conclusion} In this paper we have introduced two novel strategies for data collection for policy evaluation in reinforcement learning. Our strategies -- robust on-policy sampling (ROS) and robust on-policy acting (ROA) -- consider previously collected data when selecting actions to reduce sampling error in the entire collected dataset. We show empirically that ROS and ROA produce lower MSE policy value estimates than on-policy sampling without the need for off-policy corrections such as importance sampling. \bibliographystyle{unsrtnat}

BkiUbhI5qhDACqRGRQO2

\section{Introduction} \label{introduction} The ever-growing demand for high data rate and massive connectivity has necessitated developing advanced technologies that can more efficiently exploit a finite radio frequency spectrum. Among such, cognitive radio (CR) is regarded as a promising approach to improve spectrum utilization \cite{Haykin05}. Specifically, primary users (PUs) in underlay CR systems have prioritized access to the available radio spectrum, and secondary users (SUs) are allowed to transmit simultaneously with PUs as long as predefined interference power constraints are satisfied at the PUs \cite{Zhang08}. To improve the performance of a secondary system, the transmission strategies for SUs should be designed properly to meet a given interference power constraint. Notably, linear beamforming (BF) design has been considered as a powerful technique that can improve secondary throughput. Thus, BF approaches for CR have been investigated in multiple-input single-output (MISO) \cite{HeTWC14,MaCL13} and multiple-input multiple-output (MIMO) broadcast channels \cite{Zhang09,ZhangTSP08,NguyenTVT16,NguyenCL16}. In conjunction with BF designs, the sum rate maximization (SRM) problem of the CR network has been extensively studied recently. For instance, the SRM problem was investigated with the sum power constraint (SPC) \cite{Zhang09,ZhangTSP08} and the per-antenna power constraints (PAPCs) \cite{NguyenTVT16,NguyenCL16}. However, the quality-of-service (QoS) of SUs for the SRM problem was not addressed in \cite{Zhang08,HeTWC14,Zhang09,ZhangTSP08,NguyenTVT16,NguyenCL16}, although such additional constraints are crucial to resolving the so-called user fairness. To reduce the interference in underlay CR systems, antenna selection (AS) was proposed to select antennas at the SUs \cite{LiTVT16} and only the best antenna at the transmitters (e.g., the source and the relay) \cite{YeohTVT14}. In this letter, we study the SRM problem of a CR network with constraints for the secondary base station (BS) transmit power, SUs' minimum achievable rates, and interference power at the PUs. To mitigate the effects of the interference power constraints at the PUs which in turn improve the sum rate (SR) of the SUs, we consider joint BF and AS (JBFAS). In addition, the proposed design incorporates antenna selection into the power constraint to select proper antennas at the secondary BS, differently from \cite{LiTVT16} and \cite{YeohTVT14}. To the authors' best knowledge, existing works cannot address the present optimization problem in that it is difficult to even find a feasible point from a nonconvex set due to the mixed integer nature of the problem. To solve the JBFAS problem, we propose a new iterative algorithm with low complexity. The proposed design is based on an inner approximation method that invokes a simple convex quadratic program, which requires a lower computational effort than an exhaustive search. The obtained solutions are at least local optima since they satisfy the Karush-Kuhn-Tucker (KKT) conditions. Numerical results show fast convergence of the proposed algorithm and a significant performance improvement over existing approaches. \emph{Notation}: $\mathbf{H}^{H}$ and $\mathbf{H}^{T}$ are the Hermitian transpose and normal transpose of a matrix $\mathbf{H}$, respectively. $\|\cdot\|$ and $|\cdot|$ denote the Euclidean norm of a matrix or vector and the absolute value of a complex scalar, respectively. $\Re\{\cdot\}$ represents the real part of a complex number. $\mathbb{E}[\cdot]$ denotes a statistical expectation. $\nabla_{\mathbf{x}}f(\mathbf{x})$ represents the gradient of $f(\cdot)$ with respect to $\mathbf{x}$. \vspace*{-0.4cm} \section{System Model and Problem Formulation}\label{Systemmodel}\vspace*{-0.08cm} We consider the downlink transmissions in a CR network, where a secondary BS equipped with $N_t$ transmit antennas serves $K$ single-antenna SUs in the presence of $M$ single-antenna PUs. It is assumed that all SUs are allowed to share the same bandwidth with the PUs for transmission \cite{ZhangTSP08,NguyenTVT16}. The channel vectors from the secondary BS to the $k$-th SU and $m$-th PU are represented by $\mathbf{h}_{k}\in\mathbb{C}^{N_t\times 1},\, k\in\mathcal{K}\triangleq\{1,2,\cdots, K\}$ and $\mathbf{g}_{m}\in\mathbb{C}^{N_t\times 1},\, m\in\mathcal{M}\triangleq\{1,2,\cdots, M\}$, respectively. We assume that instantaneous channel state information (CSI) is available at the transceivers for all channels, which is consistent with several previous works on information theoretic analysis and optimization for similar kinds of problems \cite{Zhang08,HeTWC14,Zhang09,ZhangTSP08,NguyenTVT16}. Although this assumption is quite ideal, the assumption of perfect CSI is still of practical importance since the resulting performance serves as a benchmark for how the CR system will perform in more realistic conditions \cite{NguyenCL16}. The information signals are precoded at the secondary BS prior to being transmitted to the SUs. Specifically, the information intended for the $k$-th SU is $x_k\in\mathbb{C}$ with $\mathbb{E}\{|x_k|^2\} = 1$, which is precoded by beamforming vector $\mathbf{w}_k\in\mathbb{C}^{N_t\times 1}$. Then, the received signal at the $k$-th SU is given as \begin{equation} y_{k}=\mathbf{h}^H_{k}\mathbf{w}_{k}x_{k}+\sum\nolimits_{j\in\mathcal{K}\backslash \{k\}}\mathbf{h}^{H}_{k}\mathbf{w}_{j}x_{j}+n_{k},\label{eq:signalmodel} \end{equation} where $n_k\sim\mathcal{CN}(0,\sigma_k^2)$ is the additive white Gaussian noise.\footnote{Note that the background noise at the SUs also contains the interference from the primary BS, which is nonwhite in general. However, it can be assumed to be approximately white Gaussian by applying a noise-whitening filter at the SUs if the primary BS uses a Gaussian codebook \cite{ZhangTSP08}.} Correspondingly, the achievable rate for the $k$-th SU is computed as \begin{equation} R_k(\mathbf{w}) = \ln\Bigl(1+\frac{|\mathbf{h}^H_{k}\mathbf{w}_{k}|^2}{\sum_{j\in\mathcal{K}\backslash \{k\}}|\mathbf{h}^{H}_{k}\mathbf{w}_{j}|^2 + \sigma_k^2}\Bigr) \label{eq:Rateformulation} \end{equation} where $\mathbf{w}\triangleq[\mathbf{w}_1^T,\cdots,\mathbf{w}_K^T]^T$. For transmit antenna selection design, let $\alpha_{n}\in\{0,1\}$ be the binary variable indicating the association of the $n$-th transmit antenna: \begin{equation} \alpha_n = \left\{ \begin{array}{ll} 1, \mbox{ if the $n$-th antenna is selected}, \\ 0, \mbox{ otherwise}. \end{array} \right. \label{eq:proba} \end{equation} Let us define $\tilde{\mathbf{w}}_n\triangleq\bigl[[\mathbf{w}_1]_n,\cdots,[\mathbf{w}_K]_n\bigr]^T$ to be the beamforming weights of all SUs associated with the $n$-th antenna, where $[\mathbf{w}_k]_n$ is the $n$-th element of $\mathbf{w}_k$. We impose the following constraints: \begin{equation} \|\tilde{\mathbf{w}}_n\|^2 \leq \alpha_n\rho_n,\,\forall n\in\mathcal{N}\triangleq\{1,\cdots,N_t\} \end{equation} where $\rho_n$ is a newly introduced optimization variable representing as the soft power level for the $n$-th antenna. With the setting and explanation given above, the SRM problem based on JBFAS (JBFAS-SRM) for the CR system can be formulated as \begin{IEEEeqnarray}{rCl} \label{eq:Sumrate:JBDAS} & &\underset{\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho}}{\mathrm{maximize}}\quad \sum\nolimits_{k=1}^{K}R_k(\mathbf{w})\IEEEyessubnumber\label{eq:JBDAS:a}\\ & &\st\; R_k(\mathbf{w}) \geq \mathsf{\bar{r}}_k,\;\forall k\in\mathcal{K},\IEEEyessubnumber\label{eq:JBDAS:b}\\ & &\qquad\qquad\quad \sum\nolimits_{k=1}^K|\mathbf{g}_m^H\mathbf{w}_k|^2 \leq \mathcal{I}_m,\,\forall m\in\mathcal{M},\IEEEyessubnumber\label{eq:JBDAS:c}\\ & &\qquad\qquad\quad \|\tilde{\mathbf{w}}_n\|^2 \leq \alpha_n\rho_n,\,\forall n\in\mathcal{N},\IEEEyessubnumber\label{eq:JBDAS:d}\\ & &\qquad\qquad\quad \sum\nolimits_{n=1}^{N_t}\alpha_n\rho_n \leq P_{bs},\IEEEyessubnumber\label{eq:JBDAS:e}\\ & &\qquad\qquad\quad \alpha_n\in\{0,1\},\,\forall n\in\mathcal{N}\IEEEyessubnumber\label{eq:JBDAS:f} \end{IEEEeqnarray} where $\boldsymbol{\alpha}\triangleq[\alpha_1,\cdots,\alpha_{N_t}]^T$ and $\boldsymbol{\rho}\triangleq[\rho_1,\cdots,\rho_{N_t}]^T$. $\mathsf{\bar{r}}_k$ and $\mathcal{I}_m$ are the minimum required rate for the $k$-th SU and the predefined interference power at the $m$-th PU, respectively. $P_{bs}$ denotes the transmit power budget at the secondary BS. Note that the power constraint in \eqref{eq:JBDAS:d}-\eqref{eq:JBDAS:f} is different from SPC \cite{Zhang09,ZhangTSP08}: \begin{IEEEeqnarray}{rCl}\label{eq:PT:SPC} \sum\nolimits_{k=1}^{K}\|\mathbf{w}_k\|^2 \leq P_{bs}, \end{IEEEeqnarray} and PAPCs \cite{NguyenTVT16,NguyenCL16}: \begin{IEEEeqnarray}{rCl}\label{eq:PT:PAPCs} \sum\nolimits_{k=1}^{K}\|\mathbf{w}_k\|^2_{n} \leq P_{n},\,\forall n\in\mathcal{N} \end{IEEEeqnarray} where $P_n$ represents the power constraint for the $n$-th antenna at the secondary BS. The antenna selection is also incorporated into the power constraint. The effect of different types of power constraints on the system performance will be discussed in Section~\ref{Numericalresults}. \section{Proposed Iterative Optimization Based Solution}\label{Proposedalgorithm} \subsection{Relaxed Optimization Problem} We can see that the major difficulty of solving \eqref{eq:Sumrate:JBDAS} is in finding the optimal solution for $\alpha_n$ since it is a discrete variable. Note that once $\alpha_n,\,\forall n\in\mathcal{N}$ is set to 1 or 0, the selected antennas will be fixed and thus the obtained solution may not be optimal. To circumvent this issue, we relax the constraint \eqref{eq:JBDAS:f} to $0 \leq \alpha_n\leq 1$. Consequently, the relaxed JBFAS-SRM optimization problem of \eqref{eq:Sumrate:JBDAS} can be written as \begin{IEEEeqnarray}{rCl} \label{eq:JBDASrelaxed} & &\underset{\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho}}{\mathrm{maximize}}\quad \sum\nolimits_{k=1}^{K}R_k(\mathbf{w})\IEEEyessubnumber\label{eq:JBDASrelaxed:a}\\ & &\st\; R_k(\mathbf{w}) \geq \mathsf{\bar{r}}_k,\;\forall k\in\mathcal{K},\IEEEyessubnumber\label{eq:JBDASrelaxed:b}\\ & &\qquad\qquad\quad \sum\nolimits_{n=1}^{N_t}\alpha_n\rho_n \leq P_{bs},\IEEEyessubnumber\label{eq:JBDASrelaxed:c}\\ & &\qquad\qquad\quad 0 \leq\alpha_n \leq 1,\,\forall n\in\mathcal{N},\IEEEyessubnumber\label{eq:JBDASrelaxed:d}\\ & &\qquad\qquad\quad \eqref{eq:JBDAS:c}, \eqref{eq:JBDAS:d}.\IEEEyessubnumber\label{eq:JBDASrelaxed:e} \end{IEEEeqnarray} Even with the relaxation in \eqref{eq:JBDASrelaxed:d}, the optimization problem \eqref{eq:JBDASrelaxed} is still nonconvex. Thus, it is challenging to find an optimal solution of \eqref{eq:JBDASrelaxed} due to the nonconcavity of the objective function and nonconvexity of its feasible set. In what follows, we propose an iterative algorithm that can obtain a local optimum of \eqref{eq:JBDASrelaxed}. For the set of constraints, \eqref{eq:JBDASrelaxed:b} and \eqref{eq:JBDASrelaxed:c} are nonconvex constraints while \eqref{eq:JBDASrelaxed:d} and \eqref{eq:JBDASrelaxed:e} are convex and linear constraints. Let us treat the objective function \eqref{eq:JBDASrelaxed:a} first. As observed in \cite{WES06}, \eqref{eq:Rateformulation} can be equivalently replaced by \begin{equation} R_k(\mathbf{w}) = \ln\Bigl(1+\frac{(\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\})^2}{\sum_{j\in\mathcal{K}\backslash \{k\}}|\mathbf{h}^{H}_{k}\mathbf{w}_{j}|^2 + \sigma_k^2}\Bigr). \label{eq:RateRewrite} \end{equation} Let $\varphi_k(\mathbf{w})\triangleq\bigl(\sum_{j\in\mathcal{K}\backslash \{k\}}|\mathbf{h}^{H}_{k}\mathbf{w}_{j}|^2 + \sigma_k^2\bigl)/(\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\})^2$, then \eqref{eq:RateRewrite} becomes $R_k(\mathbf{w}) = \ln\bigl(1+1/\varphi_k(\mathbf{w})\bigl)$. Note that $R_k(\mathbf{w})$ is convex in the domain $\varphi_k(\mathbf{w}) > 0$. Thus, this is useful to develop an inner approximation of $R_k(\mathbf{w})$. At the feasible point $\mathbf{w}^{(\kappa)}$ of $\mathbf{w}$ at the $(\kappa + 1)$-th iteration of an iterative algorithm presented shortly, a global lower bound of $R_k(\mathbf{w})$ can be obtained as \cite{Tuybook} \begin{IEEEeqnarray}{rCl}\label{zf8:ineq} R_k(\mathbf{w})&\geq& R_k(\mathbf{w}^{(\kappa)}) + \nabla_{\varphi_k(\mathbf{w})}R_k(\mathbf{w}^{(\kappa)})\bigl(\varphi_k(\mathbf{w})-\varphi_k(\mathbf{w}^{(\kappa)})\bigl)\nonumber\\ &=& a_k - b_k\frac{\sum_{j\in\mathcal{K}\backslash \{k\}}|\mathbf{h}^{H}_{k}\mathbf{w}_{j}|^2 + \sigma_k^2}{(\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\})^2} := \mathsf{R}_k^{(\kappa)}(\mathbf{w}) \end{IEEEeqnarray} where $a_k$ and $b_k$ are defined as \begin{IEEEeqnarray}{rCl} a_k &\triangleq& R_k(\mathbf{w}^{(\kappa)}) + \frac{1}{1+\varphi_k(\mathbf{w}^{(\kappa)})} > 0,\nonumber\\ b_k &\triangleq& \frac{1}{\varphi^2_k(\mathbf{w}^{(\kappa)})+\varphi_k(\mathbf{w}^{(\kappa)})} > 0.\nonumber \end{IEEEeqnarray} Next, at the feasible point $\mathbf{w}^{(\kappa)}$, the first-order approximation of $(\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\})^2$ in \eqref{zf8:ineq} is $2\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\mathbf{h}^H_{k}\mathbf{w}_{k}\}-(\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\})^2$. Then, we have \begin{IEEEeqnarray}{rCl}\label{zf8:ineq2} \mathsf{R}_k^{(\kappa)}(\mathbf{w}) &\geq & a_k - b_k\frac{\sum_{j\in\mathcal{K}\backslash \{k\}}|\mathbf{h}^{H}_{k}\mathbf{w}_{j}|^2 + \sigma_k^2}{\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\}\bigl(2\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\}-\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\}\bigl)}\nonumber \\ &:=& \widetilde{\mathsf{R}}_k^{(\kappa)}(\mathbf{w}) \end{IEEEeqnarray} over the trust region \begin{equation}\label{eq:trustregion} 2\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\}-\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\} > 0, \forall k\in\mathcal{K}. \end{equation} It should be noted that $\widetilde{\mathsf{R}}_k^{(\kappa)}(\mathbf{w})$ is concave, and \eqref{zf8:ineq} and \eqref{zf8:ineq2} are active at optimum, i.e., \begin{equation}\label{eq:active} \widetilde{\mathsf{R}}_k^{(\kappa)}(\mathbf{w}^{(\kappa)}) = R_k(\mathbf{w}^{(\kappa)}). \end{equation} In order to solve \eqref{zf8:ineq2} using existing solvers such as MOSEK \cite{mosek}, we further transform \eqref{zf8:ineq2} to \begin{IEEEeqnarray}{rCl}\label{zf8:ineq3} \widetilde{\mathsf{R}}_k^{(\kappa)}(\mathbf{w}) \geq a_k - b_k\gamma_k :=\ddot{\mathsf{R}}_k^{(\kappa)}(\boldsymbol{\gamma}) \end{IEEEeqnarray} with additional convex constraint \begin{equation}\label{zf8:ineq4} \frac{\sum_{j\in\mathcal{K}\backslash \{k\}}|\mathbf{h}^{H}_{k}\mathbf{w}_{j}|^2 + \sigma_k^2}{\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\}\bigl(2\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}\}-\Re\{\mathbf{h}^H_{k}\mathbf{w}_{k}^{(\kappa)}\}\bigl)} \leq \gamma_k,\forall k\in\mathcal{K} \end{equation} where $\boldsymbol{\gamma}\triangleq[\gamma_1,\cdots,\gamma_K]$ is a newly introduced variable. Turning our attention to the constraints in \eqref{eq:JBDASrelaxed}, we see that \eqref{eq:JBDASrelaxed:b} is convex and admits the following form: \begin{equation}\label{eq:JBDASrelaxed:b1} \ddot{\mathsf{R}}_k^{(\kappa)}(\boldsymbol{\gamma}) \geq \mathsf{\bar{r}}_k,\;\forall k\in\mathcal{K}. \end{equation} Next, for the nonconvex constraint \eqref{eq:JBDASrelaxed:c}, a convex upper bound of $\chi_n(\alpha_n,\rho_n)\triangleq \alpha_n\rho_n$ can be found as \cite{BBA10}: \begin{equation}\label{eq:problem:31:d4} \alpha_n\rho_n\leq \frac{(\alpha_n)^2}{2r^{(\kappa)}(\alpha_n,\rho_n)}+ \frac{r^{(\kappa)}(\alpha_n,\rho_n)}{2}(\rho_n)^2 \ := \chi_n^{(\kappa)}(\alpha_n,\rho_n) \end{equation} where $r^{(\kappa)}(\alpha_n,\rho_n) \triangleq \alpha_n^{(\kappa)}/\rho_n^{(\kappa)} $. It is readily seen that \eqref{eq:problem:31:d4} holds with equality at optimum. Thus, \eqref{eq:JBDASrelaxed:c} is transformed to the following convex constraint: \begin{IEEEeqnarray}{rCl}\label{eq:problem:31:d5} \sum\nolimits_{n=1}^{N_t} \chi_n^{(\kappa)}(\alpha_n,\rho_n) \leq P_{bs}. \end{IEEEeqnarray} With the above results, we now find the solution of \eqref{eq:JBDASrelaxed} by successively solving the following convex quadratic program $(\mathsf{JBFAS-relaxed})$: \begin{IEEEeqnarray}{rCl} \label{eq:JBDASoptimal} & &\underset{\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho},\boldsymbol{\gamma}}{\mathrm{maximize}}\quad \sum\nolimits_{k=1}^{K}\ddot{\mathsf{R}}_k^{(\kappa)}(\boldsymbol{\gamma})\IEEEyessubnumber\label{eq:JBDASoptimal:a}\\ & &\st\quad \eqref{eq:JBDAS:c}, \eqref{eq:JBDAS:d}, \eqref{eq:JBDASrelaxed:d}, \eqref{eq:trustregion}, \eqref{zf8:ineq4}, \eqref{eq:JBDASrelaxed:b1}, \eqref{eq:problem:31:d5}.\IEEEyessubnumber\label{eq:JBDASoptimal:b} \end{IEEEeqnarray} Algorithm~\ref{algo:proposedalgorithm} outlines the proposed iterative algorithm to solve the JBFAS-SRM problem \eqref{eq:JBDASrelaxed}. To generate a feasible point $(\mathbf{w}^{(0)},\boldsymbol{\alpha}^{(0)},\boldsymbol{\rho}^{(0)})$ to \eqref{eq:JBDASoptimal}, we successively solve \begin{IEEEeqnarray}{rCl} \label{eq:feasiblepoint} & &\underset{\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho},\boldsymbol{\gamma}}{\mathrm{maximize}}\quad \underset{k\in\mathcal{K}}{\min}\quad \{\ddot{\mathsf{R}}_k^{(\kappa)}(\boldsymbol{\gamma}) - \mathsf{\bar{r}}_k\}\IEEEyessubnumber\label{eq:feasiblepoint:a}\\ & &\st\quad \eqref{eq:JBDAS:c}, \eqref{eq:JBDAS:d}, \eqref{eq:JBDASrelaxed:d}, \eqref{eq:trustregion}, \eqref{zf8:ineq4}, \eqref{eq:problem:31:d5}.\IEEEyessubnumber\label{eq:feasiblepoint:b} \end{IEEEeqnarray} until reaching $\ddot{\mathsf{R}}_k^{(\kappa)}(\boldsymbol{\gamma}) - \mathsf{\bar{r}}_k \geq 0,\;\forall k\in\mathcal{K}$. We see that all the constraints in \eqref{eq:JBDASoptimal} are linear and quadratic. In addition, the quadratic constraints \eqref{eq:JBDAS:c}, \eqref{eq:JBDAS:d}, \eqref{zf8:ineq4}, and \eqref{eq:problem:31:d5} admit second order cone (SOC) representable \cite[Sec. 3.3]{Ben:2001}. Thus, we are able to arrive at a SOC program, which helps reduce the overall run-time of Algorithm~\ref{algo:proposedalgorithm}. \begin{algorithm}[t] \begin{algorithmic}[1] \protect\caption{Proposed iterative algorithm for JBFAS-SRM } \label{algo:proposedalgorithm} \global\long\def\textbf{Initialization:}{\textbf{Initialization:}} \REQUIRE Set $\kappa:=0$ and solve \eqref{eq:feasiblepoint} to generate an initial feasible point $(\mathbf{w}^{(0)},\boldsymbol{\alpha}^{(0)},\boldsymbol{\rho}^{(0)})$. \REPEAT \STATE Solve \eqref{eq:JBDASoptimal} to obtain the optimal solution $(\mathbf{w}^{*},\boldsymbol{\alpha}^{*},\boldsymbol{\rho}^{*},\boldsymbol{\gamma}^{*})$. \STATE Update $\mathbf{w}^{(\kappa+1)}:=\mathbf{w}^{*},$ $\boldsymbol{\alpha}^{(\kappa+1)}:=\boldsymbol{\alpha}^{*},$ and $\boldsymbol{\rho}^{(\kappa+1)}:=\boldsymbol{\rho}^{*}.$ Set $\kappa:=\kappa+1.$ \UNTIL Convergence\\ \end{algorithmic} \end{algorithm} \textit{Convergence analysis:} Let $\mathcal{F}(\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho})$ and $\mathcal{F}^{(\kappa)}(\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho})$ denote the objective values of \eqref{eq:JBDASrelaxed} and \eqref{eq:JBDASoptimal}, respectively. It follows that $\mathcal{F}(\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho})\geq \mathcal{F}^{(\kappa)}(\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho})$ due to \eqref{zf8:ineq}, \eqref{zf8:ineq2}, and \eqref{zf8:ineq3}. Also, we have $\mathcal{F}(\mathbf{w}^{(\kappa)},\boldsymbol{\alpha}^{(\kappa)},\boldsymbol{\rho}^{(\kappa)})= \mathcal{F}^{(\kappa)}(\mathbf{w}^{(\kappa)},\boldsymbol{\alpha}^{(\kappa)},\boldsymbol{\rho}^{(\kappa)})$ since both \eqref{eq:active} and \eqref{zf8:ineq4} hold with equality at optimum. This implies that Algorithm~\ref{algo:proposedalgorithm} yields a non-decreasing sequence of the objective value, i.e., $\mathcal{F}^{(\kappa)}(\mathbf{w}^{(\kappa+1)},\boldsymbol{\alpha}^{(\kappa+1)},\boldsymbol{\rho}^{(\kappa+1)}) \geq \mathcal{F}^{(\kappa)}(\mathbf{w}^{(\kappa)},\boldsymbol{\alpha}^{(\kappa)},\boldsymbol{\rho}^{(\kappa)})$. In addition, the sequence of the objective is bounded above due to \eqref{eq:problem:31:d5}. By following the same arguments as those in \cite[Theorem 1]{Marks:78}, we can show that Algorithm~\ref{algo:proposedalgorithm} converges to a KKT point of \eqref{eq:JBDASrelaxed}. \textit{Complexity analysis:} The convex problem \eqref{eq:JBDASoptimal} involves $N_t(K+2)+K$ real scalar variables and $M+2N_t+3K+1$ quadratic and linear constraints. The computational complexity for solving \eqref{eq:JBDASoptimal} is thus $\mathcal{O}\bigl((M+2N_t+3K+1)^{2.5}\bigl((N_t(K+2)+K)^2+ M+2N_t+3K+1\bigr) \bigr)$ \cite{Ben:2001}. \vspace*{-0.4cm} \subsection{Improvement to Relaxed Problem} We have numerically observed that there exists a case where $\alpha_n$ is close to 1 and $\rho_n$ is negligibly smaller than $P_{bs}$ to indicate that the $n$-th antenna is not selected. However, this value of $\rho_n$ cannot be neglected and this will make the antenna selection procedure become inefficient. To manage the selection exactly to in turn improve the SR, we incorporate the following additional linear constraint: \begin{IEEEeqnarray}{rCl}\label{eq:assigningconstraint} \alpha_n \leq \Omega \rho_n,\;\forall n\in\mathcal{N} \end{IEEEeqnarray} where $ \Omega $ is a given constant and is large enough to force $ \alpha_n $ to reach 0 or 1 quickly. In fact, when $\rho_n$ is comparable to $P_{bs}$, the antenna selection satisfies $ 0 \leq \alpha_n \leq 1 \leq \Omega\rho_n,\;\forall n\in\mathcal{N}, $ to rapidly boost $\alpha_n$ up to 1. Otherwise, it is depressed to 0 by warranting $ 0 \leq \alpha_n \leq \Omega\rho_n \leq 1,\;\forall n\in\mathcal{N}, $ when $\rho_n$ is negligibly smaller than $P_{bs}$. Here, we chose $\Omega = 100$, and this choice will not affect the optimal solution. Summing up, the improved solution for the JBFAS-SRM problem in \eqref{eq:JBDASrelaxed} can be found by solving \begin{IEEEeqnarray}{rCl} \label{eq:JBDASimproved} & &\underset{\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\rho},\boldsymbol{\gamma}}{\mathrm{maximize}}\quad \sum\nolimits_{k=1}^{K}\ddot{\mathsf{R}}_k^{(\kappa)}(\boldsymbol{\gamma})\IEEEyessubnumber\label{eq:JBDASimprovedl:a}\\ & &\st\;\, \alpha_n \leq \Omega \rho_n,\;\forall n\in\mathcal{N}, \IEEEyessubnumber\label{eq:JBDASimproved:b}\\ & &\qquad\qquad\quad\eqref{eq:JBDAS:c}, \eqref{eq:JBDAS:d}, \eqref{eq:JBDASrelaxed:d}, \eqref{eq:trustregion}, \eqref{zf8:ineq4}, \eqref{eq:JBDASrelaxed:b1}, \eqref{eq:problem:31:d5}.\IEEEyessubnumber\label{eq:JBDASimproved:c} \end{IEEEeqnarray} \vspace*{-0.4cm} \section{Numerical Results} \label{Numericalresults} In this section, we use computer simulations to investigate the performance of the proposed solution. The entries for $\mathbf{h}_k,\forall k\in\mathcal{K}$ and $\mathbf{g}_m,\forall m\in\mathcal{M}$ are generated from independent circularly-symmetric complex Gaussian random variables with zero mean and unit variance. It is assumed that $\sigma_k^2 =1$ for all SUs, and $P_{bs}$ and $\mathcal{I}_m,\forall m$ are defined in dB scale relative to the noise power. The interference power constraints for all PUs and rate constraints for all SUs are set to be equal, i.e., $\mathcal{I}_m = \mathcal{I}$, $\forall m\in\mathcal{M}$ and $\mathsf{\bar{r}}_k = \mathsf{\bar{r}}$, $\forall k\in\mathcal{K}$. We compare the system performance of the proposed design with that of the BF design using SPC in \eqref{eq:PT:SPC} and PAPCs in \eqref{eq:PT:PAPCs}. In the PAPCs design, the power constraint for each antenna is $P_n = P_{bs}/N_t$, $\forall n\in\mathcal{N}$ \cite{NguyenTVT16,NguyenCL16}. We divide the achieved SR by $\ln(2)$ to arrive at the unit of the bps/channel-use. The simulation parameters are described in the caption for ease of reference. \begin{figure} \centering \includegraphics[width=0.26\textwidth,trim={1cm 0.3cm 1cm 0.5cm}]{ConvergenceRate.eps} \caption{Convergence behavior of Algorithm~\ref{algo:proposedalgorithm} ($K = M = 3$, $\mathcal{I} = 1$ dB, $\mathsf{\bar{r}} = 0.5$ bps/Hz, and $P_{bs} = 20$ dB).} \label{fig:convergencerate} \end{figure} In Fig.~\ref{fig:convergencerate}, we investigate the typical convergence behavior of Algorithm~\ref{algo:proposedalgorithm} and also compare it with that of the case of fixing $\alpha_n = 1,\forall n$. As seen, Algorithm~\ref{algo:proposedalgorithm} converges very fast to the optimal value of SR, and it is insensitive to the problem size. Notably, Algorithm~\ref{algo:proposedalgorithm} with \eqref{eq:JBDASimproved} achieves a better objective value and converges faster than the case of $\alpha_n = 1,\forall n$. As expected, with a larger number of transmit antennas, we obtain a higher SR. \begin{figure} \begin{center} \begin{subfigure}[Average SR of SUs versus $P_{bs}$ with $\mathcal{I} = 1$ dB.]{ \includegraphics[width=0.17\textwidth,trim={2cm -0.0cm 0.1cm 0.6cm}]{SumratevsPbs.eps}} \label{fig:SRvsP} \end{subfigure} \begin{subfigure}[Average SR of SUs versus $\mathcal{I}$ with $P_{bs} = 20 $ dB.]{ \includegraphics[width=0.17\textwidth,trim={0.0cm -0.0cm 2cm 0.6cm}]{SumratevsI.eps}} \label{fig:SRvsI} \end{subfigure} \caption{Average SR of SUs, (a) versus $P_{bs}$ and (b) versus $\mathcal{I}$ ($N_t = 6, K = M = 3$, and $\mathsf{\bar{r}} = 0.5$ bps/Hz).}\label{fig:SRvsPI} \end{center} \end{figure} Next, we illustrate the average SR of the SUs versus $P_{bs}$ in Fig.~\ref{fig:SRvsPI}(a) and versus $\mathcal{I}$ in Fig.~\ref{fig:SRvsPI}(b). We also compare the performance of the proposed design with zero-forcing (ZF) beamforming using PAPCs \cite{NguyenTVT16}. As can be seen, the SRs of the JBFAS designs outperform that of the others in all cases, and the gains of JBFAS designs are even higher than those of other designs for higher $P_{bs}$ (see Fig.~\ref{fig:SRvsPI}(a)) and for smaller $\mathcal{I}$ (see Fig.~\ref{fig:SRvsPI}(b)). These results are probably attributed to the fact that the secondary BS in the other designs needs to scale down its transmit power when $P_{bs}$ is high (or $\mathcal{I}$ is small) to satisfy \eqref{eq:JBDAS:c} which leads to a loss in the system performance. In contrast, only the best antennas with respect to \eqref{eq:JBDAS:c} are selected to transmit in the proposed JBFAS designs. As expected, the improved solution of \eqref{eq:JBDASimproved} achieves a larger SR compared to the relaxed solution of \eqref{eq:JBDASoptimal}. \begin{figure} \begin{center} \begin{subfigure}[Average SR of SUs versus $M$.]{ \includegraphics[width=0.17\textwidth,trim={2cm -0.0cm 0.1cm 0.6cm}]{SumratevsM.eps}} \label{fig:SRvsM} \end{subfigure} \begin{subfigure}[Average SR of SUs versus channel uncertainties with $M=2$.]{ \includegraphics[width=0.18\textwidth,trim={0.0cm -0.0cm 2cm 0.6cm}]{SRvsCSI.eps}} \label{fig:SRvsCSI} \end{subfigure} \caption{Average SR of SUs, (a) versus $M$ and (b) versus $(\epsilon_s,\epsilon_p)$ ($N_t = 8, K = 2$, $\mathcal{I} = 2$ dB, $\mathsf{\bar{r}} = 1$ bps/Hz, and $P_{bs} = 30$ dB).}\label{fig:SRvsMCSI} \end{center} \end{figure} Increasing the number of PUs drastically degrades the system performance for all designs due to a lack of the degree of freedom for leveraging multiuser diversity, as shown in Fig.~\ref{fig:SRvsMCSI}(a). Again, the proposed JBFAS designs achieve better performance in terms of the SR compared to the others thanks to the optimized transmission. Interestingly, the proposed JBFAS with improved solution is quite robust to the number of PUs. Fig.~\ref{fig:SRvsMCSI}(b) plots the average SR of the SUs versus different levels of channel uncertainty. The channel vectors are modeled as $\mathbf{f} = \hat{\mathbf{f}} + \Delta\mathbf{f}$ for $\mathbf{f}\in\{\mathbf{h}_k,\forall k,\mathbf{g}_m,\forall m\}$, where $\hat{\mathbf{f}}$ is the channel estimate available at the secondary BS and $\Delta\mathbf{f}$ represents the associated CSI error which is bounded by the uncertainty $\delta_{\mathbf{f}}$ as $\|\Delta\mathbf{f}\|^2 \leq \delta_{\mathbf{f}}$ \cite{NguyenCL16}. We define the normalized channel uncertainties as $\epsilon_s = \delta_{\mathbf{h}_k}/\|\mathbf{h}_k\|^2,\forall k$ and $\epsilon_p = \delta_{\mathbf{g}_m}/\|\mathbf{g}_m\|^2,\forall m$. From Fig.~\ref{fig:SRvsMCSI}(b), we can see that the SR is more sensitive to the CSI errors of the SUs' channels compared to those of the PUs' channels. \vspace*{-0.41cm} \section{Conclusions} \label{Conclusion} We presented joint beamforming and antenna selection to maximize the SR of the secondary system of a CR network. We developed a new iterative algorithm that quickly converges at least to a locally optimal solution. The relaxed version of the original problem was first presented using an inner approximation method to derive a solution. Then, we provided an improved solution by adding an additional constraint to the relaxed problem. Numerical results were also provided to demonstrate the effectiveness of the proposed design. \vspace*{-0.4cm} \bibliographystyle{IEEEtran}

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\section{An (Almost) 2/3-Approximation}\label{sec:23-algo} Throughout this section, we introduce an algorithm that achieves an almost 2/3 approximation in sublinear time in both the adjacency list and matrix models, thereby proving \cref{thm:2/3}. Specifically, we prove the following theorems, which further formalize \cref{thm:2/3} of the introduction. \begin{restatable}{theorem}{apprxtheoremA}\label{thm:apprx23theorem} For any constant $\epsilon > 0$, there exists an algorithm that estimates the size of the maximum matching in $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$ time up to a multiplicative-additive factor of $(\frac{2}{3} - \epsilon, o(n))$ with high probability in both adjacency matrix and adjacency list model. \end{restatable} \begin{restatable}{theorem}{apprxtheoremAlist}\label{thm:apprx23theoremlist} For any constant $\epsilon > 0$, there exists an algorithm that estimates the size of the maximum matching in $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$ time up to a multiplicative factor of $(\frac{2}{3} - \epsilon)$ with high probability in the adjacency list model. \end{restatable} One key component of our algorithm is the {\em edge-degree constrained subgraph} (EDCS) of Bernstein and Stein \cite{bernsteinstein2015}. An EDCS is a maximum matching sparsifier that has been used extensively in the literature on maximum matching in different settings such as dynamic and streaming. It is known that an EDCS contains an almost 2/3-approximation of the maximum matching of the original graph. Formally, we can define EDCS as follows: \begin{definition}[EDCS] \label{def:edcs} For any $\lambda < 1$ and $\beta \geq 2$, subgraph $H$ is a $(\beta, (1-\lambda)\beta)$-EDCS of $G$ if \begin{itemize} \item (Property P1:) for all edges $(u, v) \in H$, $\deg_H(u) + \deg_H(v) \leq \beta$, and \item (Property P2:) for all edges $(u, v) \in G \setminus H$, $\deg_H(u) + \deg_H(v) \geq (1-\lambda)\beta$. \end{itemize} \end{definition} We use a more relaxed version of EDCS in our algorithm that first appeared in the literature on random-order streaming matching \cite{bernsteinstreaming2020}. Let $H = (V, E_H)$ be a subgraph of $G$ such that it satisfies the first property of the EDCS (P1) in \Cref{def:edcs} for some $\beta = \widetilde{O}(1)$, and $E_U$ be the set of all edges of $G$ that violate the second property of EDCS (P2). It is possible to show that the union of $E_H$ and $E_U$ contains an almost 2/3-approximation of the maximum matching of the original graph. \begin{table}[] \begin{center} \caption{Subgraphs considered by algorithms in Sections~\ref{sec:23-algo} and~\ref{sec:beating}.} \begin{tabular}{|l|l|l|} \hline Variable & Definition & Intuition \\ \hline\hline $H$ & See \Cref{alg:matching23} & Relaxed EDCS \\ \hline $G^{Unsampled}$ & Edges not sampled by \Cref{alg:matching23} & $\mu(G^{Unsampled}) \geq (1-2\epsilon)\mu(G)$ \\ \hline $U$ & See \Cref{alg:matching23} & \begin{tabular}{@{}c@{}} Unsampled low $H$-degree edges \\(violating P1)\end{tabular} \\ \hline $G'$ & $G' = (V, E_H \cup E_U)$ & \begin{tabular}{@{}c@{}}We can estimate $\mu(G')$, and also \\ $\mu(G')$ is a $2/3$-approx of $\mu(G)$ \end{tabular}\\ \hline\hline $G''$ & $G'' = G'[V_{low}, V_{mid}]$ & \begin{tabular}{@{}c@{}} Subgraph that contains 2/3-approx\\matching and is far from being maximal \end{tabular}\\ \hline $G[A]$ & $A = V_{mid} \setminus V(M)$ & Edges in $G[A]$ can augment $\mu(G'')$\\ \hline $G[V_{mid} \setminus V(M^i_{AB})]$ & $M^i_{AB}$ defined in \Cref{alg:matchingbetterthan23} & $G[V_{mid} \setminus V(M^i_{AB})]$ is non-trivially sparse\\ \hline $G_{M_{AB}}$ & \begin{tabular}{@{}c@{}} $G_{M_{AB}} = (V_{mid}, \bigcup_i^k M_{AB}^i)$\\ defined in \Cref{alg:matchingbetterthan23} \end{tabular} & \begin{tabular}{@{}c@{}}Helps to augment $G''$ or remove \\vertices of $V_{mid} \setminus A$ in case 3 \\ \end{tabular}\\ \hline \end{tabular} \end{center} \end{table} \begin{definition}[Bounded edge-degree] A graph $H$ has a bounded edge-degree $\beta$ if for each edge $(u, v) \in H$, we have $\deg_H(u) + \deg_H(v) \leq \beta$. \end{definition} \begin{proposition}[Lemma 3.1 of \cite{bernsteinstreaming2020}, (Relaxed EDCS)]\label{pro:large-matching-union-edcs} Let $\epsilon \in [0, 1/2)$ and $\lambda, \beta$ be parameters such that $\lambda \leq \frac{\epsilon}{128}$, $\beta \geq \frac{16\log(1/\lambda)}{\lambda^2}$. Let $H = (V, E_H)$ be a subgraph of $G$ with bounded edge-degree $\beta$. Furthermore, let $\widetilde{H} = (V, E_U)$ be a subgraph that contains all edges $(u,v)$ in $G$ such that $\deg_H(u) + \deg_H(v) < (1-\lambda)\beta$. Then we have $\mu(H \cup \widetilde{H}) \geq (3/2-\epsilon) \cdot\mu(G)$. \end{proposition} \paragraph{An Informal Overview of the Algorithm:} In the next few paragraphs, we give an informal overview so the readers know what to expect. \Cref{alg:matching23} consists of two main parts. Let $H$ initially be an empty subgraph. First, in $n\beta^2 + 1$ rounds ($\beta$ is the parameter of EDCS), we sample $n^{1-\epsilon^3}$ pairs of vertices in the graph (in the adjacency list model, we sample $\delta$ edges). After sampling in each round, we update the bounded edge-degree subgraph $H$. For each edge $(u,v)$, if the $\deg_H(u) + \deg_H(v) < (1-\lambda) \beta$, we add it to $H$. This change can affect the bounded edge-degree property of $H$. In order to avoid this issue, we iterate over incident edges of $(u,v)$ to remove edges with degree more than $\beta$. Note that at most one neighbor of each $u$ and $v$ will be deleted after this iteration. Since $\beta = \widetilde{O}(1)$, this part can be done in $\widetilde{O}(n^{2-\epsilon^3})$ time. Furthermore, since we have chosen enough random edges, the number of {\em unsampled} edges in the graph that violate Property (P2) of the EDCS is relatively small. Note that when we remove edges to fix Property (P1), it is possible that an edge that was previously sampled but rejected due to the high $H$-degree, now has a lower $H$-degree and thus violates Property (P2). However, in the analysis, it will be sufficient to consider only unsampled edges that violate Property P2. More specifically, we show that the total number of unsampled violating edges is $\widetilde{O}(n^{1+\epsilon^3})$. Note that the union of approximate EDCS $H$ and unsampled violating edges preserves an almost 2/3-approximate maximum matching of the original graph since most of the edges of the graph are unsampled and the unsampled subgraph itself, contains a large matching. Hence, we can apply \Cref{pro:large-matching-union-edcs} to the union of $H$ and all unsampled violating edges. Also, graph $G' = (V, E_H, E_U)$ has a low average degree. Hence, it remains to estimate a $(1-\epsilon)$-approximate matching of graph $G'$ that has a low average degree. For this part, we use the $\widetilde{O}_{\epsilon}(\Delta ^ {1/\epsilon^2})$ time {\em local computation algorithms} (LCA) of \citet*{levironitt} which itself builds on the sublinear time algorithm of \citet*{YoshidaYISTOC09}. In local computation algorithms, the goal is to compute a queried part of the output in sublinear time. One challenge here is that we do not have access to the adjacency list of graph $G'$ to use the algorithm of \cite{levironitt} as a black box. We slightly modify this algorithm to run in $\widetilde{O}_\epsilon(n\Delta^{1/\epsilon^2})$ with access to the adjacency matrix or the adjacency list of the original graph $G$. Furthermore, since this algorithm works with the maximum degree, we eliminate some high-degree vertices by losing an additive error. In what follows, we formalize the intuition given in previous paragraphs. \begin{proposition}[\cite{levironitt}]\label{pre:lca-alg} There exists a randomized $(1-\epsilon)$-approximation local computation algorithm for maximum matching with running time $\widetilde{O}_\epsilon(\Delta^{1/\epsilon^2})$ using access to adjacency list. \end{proposition} \begin{lemma}\label{lem:lca-reduction} Let $G'$ be a subgraph of graph $G$. Also, let $\mathcal{A}$ be a local computation algorithm for maximum matching in $G'$ with running time $O(T)$ using access to the adjacency list of $G'$. There exists an algorithm with exactly the same approximation ratio that runs in $O(nT)$ time with access to the adjacency matrix or the adjacency list of $G$. \end{lemma} \begin{proof} Since algorithm $\mathcal{A}$ runs in $O(T)$ time, it visits at most $O(T)$ vertices in the graph. For each of these vertices, we can simply query all their edges in $G'$ using $O(n)$ time having access to the adjacency matrix or the adjacency list of $G$. Therefore, for each vertex that $\mathcal{A}$ visits, we can construct the adjacency list of the vertex by spending $O(n)$ time. \end{proof} \begin{corollary}\label{cor: fast-low-deg-alg} Let $G'$ be a subgraph with maximum degree $\Delta$ of graph $G$. There exists a local computation algorithm for that for a given vertex $v$, determines if it is in the output of a $(1-\epsilon)$-approximate maximum matching of $G'$ with a running time $\widetilde{O}_\epsilon(n\Delta^{1/\epsilon^2})$ with access to the adjacency matrix or the adjacency list of $G$. \end{corollary} \begin{proof} The proof follows by combining \Cref{pre:lca-alg} and \Cref{lem:lca-reduction}. \end{proof} \begin{algorithm}[H] \caption{A 2/3-Approximate Matching Algorithm in Adjacency List and Matrix} \label{alg:matching23} \textbf{Parameter:} $\epsilon$. $\delta \leftarrow n^{1-\epsilon^3}$, $H \leftarrow \emptyset$, $\lambda \leftarrow \frac{\epsilon}{128}$, $\beta \leftarrow \frac{16\log(1/\lambda)}{\lambda^2}$, $T \leftarrow n\beta^2 + 1$, $r \leftarrow 36 \log^3 n$. \For{$i$ in $1 \ldots T$}{ Sample $\delta$ pairs of vertices and query if there is an edge between them (in the adjacency list model, sample $\delta$ edges). Let $E_S$ be the set of sampled edges. Run the \Cref{alg:bernstein2020streaming} on $E_S$ for one epoch to update $H$. \lIf{subgraph $H$ did not change in this iteration}{\textbf{break}} } Let $E_U = \{(u, v) \lvert (u,v) \in E, \deg_H(u) + \deg_H(v) < (1-\lambda)\beta \}$. Sample $r$ vertices $v_1, \ldots v_r$ from $V$ with replacement. Let $X_i$ be the indicator if vertex $v_i$ is in the solution of maximum matching of graph $G' = (V, E_H \cup E_U)$ up to a multiplicative-additive factor of $(1-\epsilon)$ using the algorithm of \Cref{lem:1-eps-approx-subroutine-new}. Let $X \leftarrow \sum_{i=1}^{r} X_i$ and $\widetilde{\mu} \leftarrow \frac{nX}{2r} - \frac{n}{2 \log n}$. \Return $\widetilde{\mu}$. \end{algorithm} \begin{algorithm}[H] \caption{Algorithm of \cite{bernsteinstreaming2020} on $E_S$ for One Epoch to Update $H$.} \label{alg:bernstein2020streaming} \For{$(u, v)$ in $E_S$}{ \If{$\deg_H(u) + \deg_H(v) < (1-\lambda)\beta$}{ $H \leftarrow H \cup {(u,v)}$ \For{$(u, w)$ in $N_H(u)$}{ \If{$\deg_H(u) + \deg_H(w) > \beta$}{ Remove edge $(u, w)$ from $H$. \textbf{break.} } } \For{$(v, w)$ in $N_H(v)$}{ \If{$\deg_H(u) + \deg_H(w) > \beta$}{ Remove edge $(u, w)$ from $H$. \textbf{break.} } } } } \end{algorithm} We use the same argument as \cite{bernsteinstreaming2020} to show that the average degree of violating unsampled edges is low. The only difference here is that in each epoch, we have $\widetilde{O}(n^{1-\epsilon^3})$ sampled edges but the algorithm of \cite{bernsteinstreaming2020} use $\widetilde{O}(n)$ sampled edges which causes us to get weaker bound on the average degree. First, we rewrite a useful lemma from \cite{bernsteinstreaming2020} that also holds in our case which shows that in one of $T$ epochs of \Cref{alg:matching23} the subgraph $H$ does not change and the algorithm breaks the loop in Line 3. \begin{lemma}[Lemma 4.2 of \cite{bernsteinstreaming2020}]\label{lem:edcs-moves-bound} Let $\beta > 2$ and $H = (V_H, E_H)$ be a subgraph with no edges at the beginning. An adversary inserts and deletes edges from $H$ with the following rules: \begin{itemize} \item delete an edge $(u,v)$ if $\deg_H(u) + \deg_H(v) > \beta$, \item insert an edge $(u, v)$ if $\deg_H(u) + \deg_H(v) < (1-\lambda)\beta$. \end{itemize} Then after at most $n\beta^2$ insertions and deletions, no legal move remains. \end{lemma} Note that in the original version of this lemma in \cite{bernsteinstreaming2020}, the constraint for insertion rule is $\deg_H(u) + \deg_H(v) < \beta - 1$, but essentially the same proof carries over without any change. Therefore, the same $n\beta^2$ bound still holds for this weaker version. By \Cref{lem:edcs-moves-bound}, in at least one of $T$ rounds of \Cref{alg:matching23}, the subgraph $H$ remains unchanged and the algorithm exits the loop in Line 3. Next, we prove that after the algorithm exits in Line 3 because of no change in $H$, the number of remaining unsampled edges that violate the second property of EDCS (P2) is small. (The proof of this lemma is similar to Lemma 4.1 of \cite{bernsteinstreaming2020}.) Next, after the algorithm exits the loop in line 3 because of no change in $H$, the number of remaining unsampled edges that vaiolate the second property of EDCS (P2) is small. Assume that the algorithm sample $p$ fraction of edges of the original graph at random and constructs the bounded edge-degree subgraph $H$. Behnezhad and Khanna \cite{soheilsanjeev} (see Claim 4.16 of the paper) proved that the number of unsampled violating edges is at most $O(\mu(G)\cdot\beta^2 \cdot \log n/p)$. In our case, $p = O(\beta^2/n^{\epsilon^3})$ by the choice of $\delta$ and $T$. We restate the lemma with this specific $p$. \begin{lemma}\label{lem:sparsification} Let $\widetilde{H} = (V, E_U)$ be the subgraph consisting of violating unsampled edges in Line 7 of \Cref{alg:matching23}. Then we have $|E_U| = O(\mu(G)\cdot n^{\epsilon^3} \cdot \log n)$ with probability at least $1 - 1/n^5$. \end{lemma} \begin{claim}\label{clm:constructing-edcs} Let $E_S$ be the set of sampled edges in Line 4 of \Cref{alg:matching23}. Then running the \Cref{alg:bernstein2020streaming} for one epoch to update $H$ can be done in $O_\epsilon(n^{1-\epsilon^3})$ time. Moreover, the whole process of constructing $H$ takes $O_\epsilon(n^{2-\epsilon^3})$ time in both adjacency lists and matrix models. \end{claim} \begin{proof} Since \Cref{alg:matching23} samples at most $\delta = n^{1-\epsilon^3}$ pairs of vertices, we have $|E_S| = O(n^{1-\epsilon^3})$. \Cref{alg:bernstein2020streaming} iterates over the edges one by one in each epoch and adds edge $(u,v)$ to $H$ if $\deg_H(u)+\deg_H(v) < (1-\lambda)\beta$. Adding this edge can cause incident edges of $u$ and $v$ to have a degree higher than $\beta$ in $H$. In order to remove those edges, we iterate over edges of $u$ and $v$ in $H$ and remove the first edge with a degree higher than $\beta$. This can be done in $O(\beta)$ since each vertex has at most $\beta$ neighbors. Thus, the running time of each epoch is $O(n^{1-\epsilon^3}\beta) = O_\epsilon(n^{1-\epsilon^3})$. Since we have $O(n\beta^2)$ epochs, the total running time is $O(n^{2-\epsilon^3}\beta^3) = O_\epsilon(n^{2-\epsilon^3})$ by our choice of $\beta$. \end{proof} \begin{lemma}\label{lem:1-eps-approx-subroutine-new} Let $G' = (V, E_H \cup E_U)$ be as defined in \Cref{alg:matching23}. Then there exists a $(1-\epsilon)$-approximation LCA algorithm for maximum matching of $G'$ with $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$ preprocessing time and $\widetilde{O}_\epsilon(n^{1+\epsilon})$ additional time per query. \end{lemma} \begin{proof} Combining \Cref{lem:sparsification} and our choice of $\beta$, the average degree of graph $G'$ is $O((\mu(G) \cdot n^{\epsilon^3} \cdot \log n )/ n)$. Note that the algorithm of \cite{levironitt} works with maximum degree. In order to use this algorithm as a subroutine in Line 9 of our algorithm, we ignore vertices of high degree and find a $(1-\epsilon)$-approximate maximum matching in the remaining graph by losing an additive error which depends on $\mu(G)$. However, we do not have access to the degree of vertices in $G'$ since we do not have access to the adjacency list of $E_U$. For each vertex $v$, we sample $k = 100n^{1-\epsilon^3}\log^2 n$ vertices to estimate the degree of $v$. Let $X$ be the number of neighbors that are in $G'$ and $\widetilde{\deg}_{G'}(v) = nX/k$ be our estimate for the degree of $v$ in $G'$. Using Chernoff bound, we get \begin{align*} \Pr\left[|\widetilde{\deg}_{G'}(v) - \deg_{G'}(v) | > n^{\epsilon^3}\right] \leq 2\exp\left(-\frac{10000\log^4 n}{1500\log^3 n}\right) < \frac{2}{n^6}. \end{align*} A union bound over all $n$ vertices yields that with a probability of $1-1/n^4$, we have an additive error of at most $n^{\epsilon^3}$ for the degree of all vertices. Now we ignore the vertices with estimated degrees larger than $n^{\epsilon^3} \log^2 n$. Combining the additive error bound for degree estimation and the fact that the average degree of the graph is at most $O((\mu(G) \cdot n^{\epsilon^3} \cdot \log n )/ n)$, the total number of ignored vertices is at most $o(\mu(G))$. For the remaining vertices, the maximum degree is at most $\widetilde{O}(n^{\epsilon^3})$. Moreover, for an edge in $G$ we can check if the degree of the edge is smaller than $(1-\lambda)\beta$ or not (the edge belongs to $G'$ or not). Thus, we can run the algorithm of \Cref{cor: fast-low-deg-alg} in $\widetilde{O}_\epsilon(n^{1 + \epsilon})$ since the maximum degree is $\widetilde{O}(n^{\epsilon^3})$ and either we have access to adjacency matrix or adjacency list of $G$. Also, note that since the additive error is $o(\mu(G))$, the approximation ratio cannot be worse than $(1-\epsilon)\mu(G') - o(\mu(G)) \geq (1-2\epsilon)\mu(G')$. Proof of claim follows from using $\epsilon/2$ as the parameter. \end{proof} \begin{claim}\label{clm: large-matching-union} $ (\frac{2}{3} - \epsilon) \cdot \mu(G) \leq \mu(G') \leq \mu(G)$. \end{claim} \begin{proof} The fact that $G'$ is a subgraph of $G$ implies the second inequality. Let $G^{Unsampled}$ be the subgraph consisting of unsampled edges. By Lemma 2.2 of \cite{bernsteinstreaming2020}, we have $\mu(G^{Unsampled}) \geq (1-2\epsilon)\mu(G)$ with high probability. To see this, we sample $\Tilde{O}(m/n^{\epsilon^3})$ fraction of edges, however, in \cite{bernsteinstreaming2020}, they sample $\epsilon m$ edges. Hence, the graph $G^{Unsampled}$ has more unsampled edges and therefore a larger matching compared to unsampled edges of \cite{bernsteinstreaming2020}. Moreover, by \Cref{pro:large-matching-union-edcs}, we have $\mu(G') \geq (\frac{2}{3}-\epsilon) \mu(G^{Unsampled})$. Therefore, we obtain \begin{align*} \mu(G') \geq (\frac{2}{3}-\epsilon) \mu(G^{Unsampled}) \geq (\frac{2}{3}-\epsilon) (1-2\epsilon)\mu(G) \geq (\frac{2}{3} - 3\epsilon) \mu(G). \end{align*} Proof of claim follows from using $\epsilon/3$ as the parameter. \end{proof} \begin{lemma}\label{lem:approx-ratio-32} let $\widetilde{\mu}$ be the output of \Cref{alg:matching23} on graph $G$. With high probability, $$ (\frac{2}{3} - \epsilon) \mu(G) - o(n) \leq \widetilde{\mu} \leq \mu(G). $$ \end{lemma} \begin{proof} Let $\hat{M}$ be the $(1-\epsilon)$-approximate matching of \Cref{lem:1-eps-approx-subroutine-new} on graph $G'$ and $X_i$ be the indicator if vertex $i$ is matched in $\hat{M}$ or not. By \Cref{clm: large-matching-union}, we have \begin{align}\label{eq:bounds-on-M} (1-\epsilon)(\frac{2}{3} - \epsilon) \mu(G) \leq \E|\hat{M}| \leq \mu(G). \end{align} Because the number of matching edges is half of the matched vertices, \begin{align*} \E[X_i] = \Pr[X_i = 1] = \frac{2\E|\hat{M}|}{n}. \end{align*} Thus, \begin{align}\label{eq:expected-matching} \E[X] = \frac{2r\E|\hat{M}|}{n}. \end{align} Using Chernoff bound and the fact that $X$ is the sum of $r$ independent Bernoulli random variables, we have \begin{align*} \Pr[|X - \E[X]| \geq \sqrt{18\E[X]\log n}] \leq 2 \exp\left(-\frac{18\E[X]\log n}{3\E[X]}\right) = \frac{2}{n^6}. \end{align*} Hence, with probability $1-2/n^6$, \begin{align*} \widetilde{\mu} = \frac{nX}{2r} - \frac{n}{2\log n} & \in \frac{n(\E[X] \pm \sqrt{18\E[X]\log n})}{2r} - \frac{n}{2\log n} \\ & = \E|\hat{M}| \pm \sqrt{\frac{9n\E|\hat{M}|\log n}{r}}- \frac{n}{2\log n} & (\text{By } \Cref{eq:expected-matching})\\ & = \E|\hat{M}| \pm \sqrt{\frac{n\E|\hat{M}|}{4\log^2 n}}- \frac{n}{2\log n} & (\text{Since } r = 36 \log^3 n)\\ & = \E|\hat{M}| \pm \frac{n}{2\log n}- \frac{n}{2\log n} & (\text{Since } \E|\hat{M}| \leq n). \end{align*} Plugging \Cref{eq:bounds-on-M} in the above range implies \begin{align*} (1-\epsilon)(\frac{2}{3} - \epsilon) \mu(G) - \frac{n}{\log n} \leq \widetilde{\mu} \leq \mu(G). \end{align*} Since $\frac{n}{\log n} = o(n)$ and $(1-\epsilon)(\frac{2}{3} - \epsilon) > (\frac{2}{3} - 3\epsilon)$, we get \begin{align*} (\frac{2}{3} - 3\epsilon) \mu(G) - o(n) \leq \widetilde{\mu} \leq \mu(G). \end{align*} Proof of lemma follows from using $\epsilon/3$ as the parameter. \end{proof} Now we are ready to complete the analysis of the 2/3-approximate maximum matching algorithm. \apprxtheoremA* \begin{proof} We run \Cref{alg:matching23}. By \Cref{lem:approx-ratio-32}, we obtain the claimed approximation ratio. The proof of running time follows from combining \Cref{clm:constructing-edcs}, \Cref{lem:1-eps-approx-subroutine-new}, and $r = \widetilde{O}(1)$. \end{proof} \paragraph{Multiplicative Approximation for Adjacency List:} In the adjacency list model, we can assume that we are given the degree of each vertex. This assumption is without a loss of generality since we can use binary search for each vertex to find its exact degree. Thus, we know the total number of edges in the graph and if this number is not larger than $n^{1.99}$, then we can use linear time $1-\epsilon$ approximation for maximum matching. Equipped with this observation, we assume that $\mu(G) \geq n^{0.99} / 2$, otherwise, the number of edges cannot be more than $2n\cdot \mu(G)$. Moreover, the size $\mu(G')$ is at least $(2/3 - \epsilon)\mu(G)$, which implies that we can assume $\mu(G') = \Omega(n^{0.99})$. With this assumption and using standard Chernoff bound, it is not hard to see that we can estimate the size of the maximum matching of $G'$ with a multiplicative factor by sampling $r = \widetilde{O}(n^{0.01})$ vertices. By the running time of \Cref{clm:constructing-edcs}, preprocessing time of \Cref{lem:1-eps-approx-subroutine-new}, and query time of \Cref{lem:1-eps-approx-subroutine-new} combined with $r$ queries, results in multiplicative $(2/3 - \epsilon)$-approximation algorithm with $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$ running time. \apprxtheoremAlist* \section{Beating 2/3-Approximation in Bipartite Graphs}\label{sec:beating} In this section, we design a new algorithm that gets slightly better than $2/3$ approximation in sublinear time in both the adjacency list and matrix models for bipartite graphs. Our starting point is to use the tight case characterization of the instance that our algorithm in the previous part cannot obtain better than a $2/3$ approximation. Then we use that characterization to design a new algorithm to go beyond $2/3$. We prove the following theorem, which further formalizes the statement of \cref{thm:beating-2/3} in the introduction. \begin{restatable}{theorem}{apprxtheoremB}\label{thm:apprxbetter23theorem} For an absolute constant $\alpha$, there exists an algorithm that estimates the size of the maximum matching in $\widetilde{O}(n^{2-\Omega(1)})$ time up to a multiplicative-additive factor of $(\frac{2}{3} + \alpha, o(n))$ with high probability in both adjacency list and matrix models. \end{restatable} \begin{restatable}{theorem}{apprxtheoremBlist}\label{thm:apprxbetter23theoremlist} For an absolute constant $\alpha$, there exists an algorithm that estimates the size of the maximum matching in $\widetilde{O}(n^{2-\Omega(1)})$ time up to a multiplicative factor of $(\frac{2}{3} + \alpha)$ with high probability in the adjacency list model. \end{restatable} To break $2/3$-approximation, we build on a characterization of tight instances of EDCS due to a recent work of \citet*{behnezhad2022dynamic}. In our context, the characterization asserts that if the output of our \Cref{alg:matching23} is not already a $(2/3 + \alpha)$-approximate matching of $G$ for some constant $\alpha \ge 10^{-24}$ (we did not try to optimize the constant), then there exists a $2/3$-approximate matching in $G'$ (defined in \Cref{alg:matching23}) such that it is far from being maximal in the original graph $G$. \paragraph{Notation} Let $G$ be the input bipartite graph, and let $G' = (V, E_H \cup E_U)$ and $H = (V, E_H)$ be as defined in \Cref{alg:matching23}. Let $V_{low}$ and $V_{mid}$ be the set of vertices $v$ such that $\deg_H(v) \in [0, 0.2\beta]$ and $\deg_H(v) \in [0.4\beta, 0.6\beta]$, respectively. Let $G'' \coloneqq G'[V_{low}, V_{mid}]$. Also, fix $M$ to be a $(1-\epsilon)$-approximate matching of $G'' \coloneqq G'[V_{low}, V_{mid}]$. We let $A \coloneqq V_{mid} \setminus V(M)$ and $B \coloneqq V_{mid} \cap V(M)$ for the rest of this section. \begin{lemma}[\cite{behnezhad2022dynamic}]\label{lem:tight-instance-edcs} Let $\epsilon < 1/120$ and $\alpha$ be some constant. If $\mu(G') \leq (\frac{2}{3} + \alpha)\mu(G)$, then both of the following guarantees hold: \begin{itemize} \item {\bf (G1 - $G''$ contains a large matching $M$)} $\mu(G'') \geq (\frac{2}{3} - 120 \sqrt{\alpha}) \cdot\mu(G),$ \item {\bf (G2 - $M$ can be augmented in $G[A]$)} For any matching $M$ in $G''$, we have $\mu(G[V_{mid} \setminus V(M)]) \geq (\frac{1}{3} - 800 \alpha)\cdot \mu(G).$ \end{itemize} \end{lemma} \paragraph{An Informal Overview of the Algorithm:} Let $M$ be a $(1-\epsilon)$-approximate matching of $G'' \coloneqq G'[V_{low}, V_{mid}]$. According to this characterization, if \Cref{alg:matching23} does not obtain better than a $(2/3 + \alpha)$ approximation, then matching $M$ preserves the size of matching of $G'$ and also, it can be augmented using a matching with edges of $G[V_{mid} \setminus V(M)]$. By \Cref{lem:tight-instance-edcs}, if the matching that we estimate in \Cref{alg:matching23} does not obtain better than $2/3$ approximation, then there exists a large matching in $G''$ that can be augmented using a matching between vertices of $A$. Therefore, if we want to obtain a better than $2/3$ approximation, all we need is to estimate the size of a constant fraction of a matching in $G[A]$. The first challenge is that we do not know which vertices of $V_{mid}$ are in $A$ and which are in $B$. With a similar proof as proof of \Cref{lem:1-eps-approx-subroutine-new}, we can show that it is possible to query if a vertex is matched in $M$ in $\widetilde{O}_\epsilon(n^{1+\epsilon})$ time, since $G''$ is a subgraph of $G'$ and its average degree is smaller than $G'$. Moreover, for an edge in $G$, we can check if the $H$-degree of the edge is smaller than $(1-\lambda)\beta$ (the edge belongs to $G'$ or not), and one endpoint is in $V_{mid}$ and the other one in $V_{low}$ (the edge belongs to $G''$ or not) in $O(1)$. Thus, we can run the algorithm of \Cref{cor: fast-low-deg-alg}. We restate the lemma for subgraph $G''$. \begin{lemma}\label{lem:mid-low-lca} There exists a $(1-\epsilon)$-approximation LCA algorithm for maximum matching of $G'' \coloneqq G'[V_{low}, V_{mid}]$ with $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$ preprocessing time and $\widetilde{O}_\epsilon(n^{1+\epsilon})$ additional time per query. \end{lemma} Since $G'[V_{low}, V_{mid}]$ is a subgraph of $G'$, by \Cref{lem:1-eps-approx-subroutine-new}, we can query if a vertex is matched in $M$ in $\widetilde{O}_\epsilon(n^{1+\epsilon})$. So the time to classify if a vertex is in $A$ or $B$ is $\widetilde{O}_\epsilon(n^{1+\epsilon})$ since we can query if it is matched in matching $M$ or not. Unfortunately, the subgraph $G[A\cup B]$ can be a dense graph (possibly with an average degree of $\Theta(n)$), so it is not possible to run an adaptive sublinear algorithm that adaptively queries if a vertex is in $A$ or not while looking for a matching. So we first try to sparsify the graph by sampling edges and constructing several maximal matchings. More specifically, we query $\delta k$ pairs of vertices (for $\delta = n^{1+\gamma}$ and $k = n^{\gamma/2}$) and partition the queries to $k$ equal buckets. Then we build a greedy maximal matching $M_{AB}^i$ among those sampled edges of bucket $i$. By this construction and the sparsification property of greedy maximal matching, we are able to show that the maximum degree of subgraph $G[V_{mid} \setminus V(M^i_{AB})]$ is $O(n^{1-\gamma})$ with high probability for each $i$. We split the rest of the analysis into three possible cases. \paragraph{(Case 1) A constant fraction of vertices of $A$ are matched in at least one of $M_{AB}^i$:} in this case we immediately beat $2/3$ since we found a constant fraction of a maximum matching edges of $G[A]$. \paragraph{(Case 2) A constant fraction of maximum matching of $G[A]$ has both endpoints unmatched in at least one of $M_{AB}^i$:} Let $i^*$ be an index of matching $M_{AB}^{i^*}$ that a constant fraction of maximum matching of $G[A]$ has both endpoints unmatched. In this case, we prove that it is possible to estimate a constant approximation of the size of the matching between unmatched vertices of $A$ in sublinear time. Let $G_{AB}^{i^*}$ be the same as subgraph $G[V_{mid} \setminus V(M_{AB}^{i^*})]$, except that we connect each vertex in $B \setminus V(M_{AB}^{i^*})$ to $\kappa = \widetilde{O}(n^{1-\gamma})$ dummy singleton vertices. The reason to connect singleton vertices to $B$ is that if we run a random greedy maximal matching on $G_{AB}^{i^*}$, most of the vertices in $B$ will match to singleton vertices with some additive error. Thus, if there exists a large matching between unmatched $A$ vertices, random greedy maximal matching will find at least half its edges. We sample $\widetilde{O}(1)$ vertices and test if they are matched in the random greedy maximal matching of $G_{AB}^{i^*}$. To do this, we use the algorithm of \cite{behnezhad2021} which has a running time of $\widetilde{O}(n^{1-\gamma})$ for a random vertex. However, for each vertex that this algorithm recursively makes a query, we have to classify if it is in $A$ or $B$ since we do not explicitly construct $G_{AB}^{i^*}$. As we discussed at the beginning of this overview, this task can be done in $\widetilde{O}_\epsilon(n^{1+\epsilon})$ time. For each maximal matching, the running time for this case is $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma})$. Therefore, the total running time is $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma/2})$ for all $n^{\gamma/2}$ maximal matchings which is sublinear if we choose $\gamma$ sufficiently larger than $\epsilon$. Note that if we are not in one of the above cases, almost all edges of $n^{\gamma/2}$ maximal matchings have at most one endpoint in $A$ (as a result of not being in Case 1), and almost all edges of the maximum matching of $G[A]$ have at least one endpoint matched by all $n^{\gamma/2}$ maximal matchings (as a result of not being in Case 2). Thus, if we consider a vertex in $A$, almost all of its incident maximal matching edges go to set $B$. For this case, we now describe how it is possible to estimate the matching of $G[A]$. \paragraph{(Case 3) Almost every edge of maximum matching of $G[A]$ has at least one of its endpoints matched by almost all $M_{AB}^i$:} In this case we will show that we can efficiently remove many $B$-vertices, and then repeat the algorithm and and analysis of Cases 1 and 2 (using fresh samples of $M^i_{AB}$. Because we can't keep removing $B$-vertices indefinitely, eventually we have to end up at either Case 1 or 2. Our goal is henceforth to efficiently identify many $B$-vertices. Suppose that we sample a vertex and check if it is in $A$ or $B$. If the sampled vertex is in $A$ and is one of the endpoints of maximum matching of $A$, most of its incident edges in $M_{AB}^1, M_{AB}^2, \ldots, M_{AB}^k$ are connected to vertices of $B$. Therefore, we are able to find $k = \Theta(n^{\gamma/2})$ vertices of the set $B$ by spending $\widetilde{O}_\epsilon(n^{1+\epsilon})$ to find a vertex in $A$. Hence, if we sample $\widetilde{\Theta}(n^{1-\gamma/2})$ such vertices, we are expecting to see a constant fraction of vertices of $B$; we can remove these vertices and run the same algorithm on the remaining graph again. One challenge that arises here is that some of the neighbors of the sampled vertex might be in $A$. To resolve this issue, we choose a threshold $\eta$ and only remove vertices that have at least $\eta$ maximal matching edges that are connected to sampled $A$ vertices. This will help us to avoid removing many $A$ vertices in each round. Since initially the size of $B$ and $A$ is roughly the same (up to a constant factor), after a few rounds, we can estimate the size of the maximum matching of $A$. The total running time of this case is also $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma/2})$ which is the same as the previous case. In what follows, we present the formal algorithm (\Cref{alg:matchingbetterthan23}) and formalize the technical overview given in previous paragraphs. \begin{algorithm}[] \caption{Better Than 2/3-Approximate Matching Algorithm in Adjacency List and Matrix} \label{alg:matchingbetterthan23} \textbf{Parameter:} $\epsilon, \gamma$. $r_1 \leftarrow 72 \log^3 n, r_2 \leftarrow 288 \log^3 n, r_3 \leftarrow 10n^{1-\gamma/2}\log n$. $T \leftarrow 200, \alpha \leftarrow 10^{-10}, \kappa \leftarrow 48n^{1-\gamma}\log^2 n, \delta \leftarrow n^{1+\gamma}, k \leftarrow n^{\gamma/2}$, $c \leftarrow 10^{16}$. Let $\widetilde{\mu}$ be the output of \Cref{alg:matching23} on $G$ with parameter $\epsilon$. Also, let $\beta$, $H$, $E_U$, and $G' = (V, E_H \cup E_U)$ be as defined in \Cref{alg:matching23}. $\eta \leftarrow \frac{n\log n}{100\widetilde{\mu}_1}$. Let $V_{low} = \{v | \deg_H(v) \in [0, 0.2\beta] \}$, $V_{mid} = \{v | \deg_H(v) \in [0.4\beta, 0.6\beta] \}$, and $G'' \coloneqq G'[V_{low}, V_{mid}]$. Let $\widetilde{\mu}_1$ be an estimate with an additive error of $o(n)$ for the size of $(1-\epsilon)$-approximate matching of $G''$ by sampling $\widetilde{O}(1)$ vertices and running the algorithm of \Cref{lem:mid-low-lca}. \For{$j$ in $1 \ldots T$}{ Sample $k\delta$ pairs of vertices of $V_{mid}$ and partition them into $k$ buckets of equal size. Let $M_{AB}^i$ be the greedy maximal matching on the sampled edges of bucket $i$. \For{$i$ in $1 \ldots k$}{ \textbf{// Beginning of case 1} Sample $r_1$ random edges $e_1, \ldots, e_{r_1}$ of $M_{AB}^i$. Let $X_{\ell}$ be the indicator if neither endpoint of $e_{\ell}$ is matched in the $(1-\epsilon)$-approximate matching of $G''$ by running algorithm of \Cref{lem:mid-low-lca}. Let $X \leftarrow \sum^{r_1}_{\ell=1} X_{\ell}$ and $\widetilde{\mu}_2 = \frac{nX}{r_1} - \frac{n}{2\log n}$. \lIf{$\widetilde{\mu}_2 \geq c\alpha \widetilde{\mu}_1$}{ \Return $\widetilde{\mu}_1 + \widetilde{\mu}_2$ } \textbf{// End of case 1} \textbf{// Beginning of case 2} Let $G_{AB}^{i}$ be the same as subgraph $G[V_{mid} \setminus V(M_{AB}^{i})]$, except that we connect each vertex in $B \setminus V(M_{AB}^{i})$ to $\kappa$ singleton vertices. Sample $r_2$ random vertices $v_1, \ldots, v_{r_2}$ of vertex set $A \setminus V(M_{AB}^{i})$. Let $Y_{\ell}$ be the indicator if $v_{\ell}$ is matched in the random greedy maximal matching of $G_{AB}^i$ (algorithm of \cite{behnezhad2021}). Let $Y \leftarrow \sum^{r_2}_{\ell=1} Y_{\ell}$ and $\widetilde{\mu}_3 = \frac{nY}{2r_2} - \frac{n}{2\log n}$. \lIf{$\widetilde{\mu}_3 \geq c\alpha \widetilde{\mu}_1$}{ \Return $\widetilde{\mu}_1 + \widetilde{\mu}_3$ } \textbf{// End of case 2} } \textbf{// Beginning of case 3} Sample $r_3$ vertices $u_1, u_2, \ldots u_{r_3}$ of vertex set $A$. Define $G_{M_{AB}} = (V_{mid}, \bigcup_i^k M_{AB}^i)$. Let $C \subseteq V_{mid}$ be the set of vertices that has at least $\eta$ neighbors in $G_{M_{AB}}$ among $r_3$ sampled vertices of $A$. \algorithmiccomment{$C$ vertices likely in $B$} $V_{mid} \leftarrow V_{mid} \setminus C$. \algorithmiccomment{Remove those vertices} \label{Line:remove} \textbf{// End of case 3} } \Return $\widetilde{\mu}$ \end{algorithm} \subsection{Analysis of Running Time} We analyze each part of \Cref{alg:matchingbetterthan23} separately in this section and at the end, we put everything together. Before starting the analysis of the running time of \Cref{alg:matchingbetterthan23}, we first restate the following result on the time complexity of estimating the size of the maximal matching by Behnezhad \cite{behnezhad2021}. \begin{lemma}[Lemma 4.1 of \cite{behnezhad2021}]\label{lem:RGMM-query-complexity} There exists an algorithm that draws a random permutation over the edges of the graph and for an arbitrary vertex $v$ in graph $G$, it determines if $v$ is matched in the random greedy maximal matching of $G$ in $\widetilde{O}(\bar{d})$ time with high probability, using adjacency list of $G$, where $\bar{d}$ is the average degree of $G$. \end{lemma} \begin{claim}\label{clm:fast-maximal-matching} Let $M_{AB}^i$ be as defined in \Cref{alg:matchingbetterthan23}. The running time for constructing $M_{AB}^i$ for all $1 \leq i\leq k$, takes $O(n^{1+3\gamma/2})$ time for all $T$ iterations of the algorithm. \end{claim} \begin{proof} Fix an iteration in the algorithm. Since we sample $\delta k = n^{1+3\gamma/2}$ pairs of vertices and we have $k$ buckets of equal size, each bucket can have at most $\delta$ edges. For each of the buckets, in order to construct the maximal matching we need to iterate over the edges of the bucket one by one which takes $O(\delta k)$ time for all buckets. The proof follows from $T = O(1)$. \end{proof} \begin{claim}\label{clm:preprocessing-time} The total preprocessing time before all iterations of \Cref{alg:matchingbetterthan23} is $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$. \end{claim} \begin{proof} \Cref{alg:matchingbetterthan23} uses subroutine of \Cref{lem:mid-low-lca} to query if a vertex is matched in $(1-\epsilon)$-approximate matching of $G''$ during its execution. Thus, by \Cref{lem:mid-low-lca}, the total preprocessing time needed for the algorithm is $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$. \end{proof} \begin{claim}\label{clm:maximal-matching-aa-edges} Every time that \Cref{alg:matchingbetterthan23} calculates $\widetilde{\mu}_2$, it takes $\widetilde{O}_\epsilon(n^{1+\epsilon})$ time. Furthermore, the total running time for computing $\widetilde{\mu}_2$ in the whole process of algorithm is $\widetilde{O}_\epsilon(n^{1+\epsilon + \gamma/2})$ time. \end{claim} \begin{proof} By \Cref{lem:mid-low-lca}, each query to see if a vertex is matched in $(1-\epsilon)$-approximate matching of $G''$, takes $\widetilde{O}_\epsilon(n^{1+\epsilon})$ time. Since we sample $\widetilde{O}(1)$ vertices, we get the claimed time complexity. \end{proof} \begin{lemma}\label{lem: sparsification-property} Subgraph $G[V_{mid} \setminus V(M^i_{AB})]$ has a maximum degree of $6n^{1-\gamma}\log n$ with high probability. \end{lemma} \begin{proof} Consider a vertex $v$ that is not matched in $M^i_{AB}$ and has a degree larger than $6n^{1-\gamma}\log n$ in $G[V_{mid} \setminus V(M^i_{AB})]$. The only way that $v$ remains unmatched is that none of its incident edges in $G[V_{mid} \setminus V(M^i_{AB})]$ get sampled in $n^{1+\gamma}$ pairs of vertices we sampled. The probability that we sample one of these edges is at least $6n^{1-\gamma}\log n/n^2$, since there are at most $n^2$ possible pairs of vertices. Thus, the probability that none of these edges sampled in $n^{1+\gamma}$ samples when constructing $M_{AB}^i$ is at most \begin{align*} \left( 1 - \frac{6n^{1-\gamma}\log n}{n^2} \right)^{n^{1 + \gamma}} = \left( 1 - \frac{6\log n}{n^{1+\gamma}} \right)^{n^{1 + \gamma}} \leq n^{-6}. \end{align*} Taking a union bound over all vertices of the graph completes the proof. \end{proof} \begin{claim}\label{clm:estimating-aa-missing-matching} Every time that \Cref{alg:matchingbetterthan23} calculates $\widetilde{\mu}_3$, it takes $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma})$ time. Furthermore, the total running time for computing $\widetilde{\mu}_3$ in the whole process of algorithm is $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma/2})$ time. \end{claim} \begin{proof} By \Cref{lem: sparsification-property}, the maximum degree of subgraph $G_{AB}^i$ is $\widetilde{O}(n^{1-\gamma})$ with high probability. Thus, by \Cref{lem:RGMM-query-complexity}, indicating if a vertex is matched in the random greedy maximal matching of $G^i_{AB}$ takes $\widetilde{O}(n^{1-\gamma})$ time. Moreover, for each vertex in the process of running a random greedy maximal matching local query algorithm, we need to distinguish if it is a vertex in $A$ or $B$ since we build subgraph $G^i_{AB}$ on the fly (we cannot afford to build the whole graph at the beginning which takes $\Theta(n^2)$ time). By \Cref{lem:mid-low-lca}, each of these queries takes $\widetilde{O}_\epsilon(n^{1+\epsilon})$ time. Therefore, it takes $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma})$ time to calculate $\widetilde{\mu}_3$ each time. Combining with the fact that $r_2 = \widetilde{O}(1)$, we have $O(n^{\gamma/2})$ maximal matchings, the size of set $A$ is a constant fraction of $V_{mid}$, and $T = O(1)$ implies the claimed running time. \end{proof} \begin{claim}\label{clm:updating-vmid} Updating vertex set $V_{mid}$ at the end of case 3 in \Cref{alg:matchingbetterthan23} takes $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma})$ time. Furthermore, the total running time for computing $V_{mid}$ in the whole process of algorithm is $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma})$ time. \end{claim} \begin{proof} Each time that we sample a vertex, we need to test if it is a vertex in $A$. This takes $\widetilde{O}_\epsilon(n^{1+\epsilon})$ by \Cref{lem:mid-low-lca}. Since we need to sample $r_3 = \widetilde{O}(n^{1-\gamma/2})$ vertices of $A$ and the size of set $A$ is a constant fraction of $V_{mid}$ by \Cref{lem:tight-instance-edcs}, we need to sample at most $\widetilde{O}(n^{1-\gamma})$ vertices of $V_{mid}$. Hence, it takes $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma/2})$ time to find vertices $u_1, u_2, \ldots, u_k \in A$. Each of the vertices has at most $k$ neighbors in all maximal matchings $M_{AB}^1, M_{AB}^2 \ldots, M_{AB}^{k}$ which implies that we need to spend $\widetilde{O}(n)$ time to find and remove vertices of set $C$ which completes the proof since $T = O(1)$. \end{proof} Now we are ready to complete the analysis of the running time of \Cref{alg:matchingbetterthan23}. \begin{lemma}\label{lem:time-better-23} The total running time of \Cref{alg:matchingbetterthan23} is $\widetilde{O}_\epsilon(\max(n^{2+\epsilon-\gamma/2}, n^{2-\epsilon^3}, n^{1+3\gamma/2}, n^{1+\epsilon+\gamma/2}))$. \end{lemma} \begin{proof} The proof follows from combining \Cref{lem:mid-low-lca}, \Cref{clm:fast-maximal-matching}, \Cref{clm:preprocessing-time}, \Cref{clm:maximal-matching-aa-edges}, \Cref{clm:estimating-aa-missing-matching}, and \Cref{clm:updating-vmid}. \end{proof} \subsection{Analysis of Approximation Ratio} In this section, we assume that $\widetilde{\mu} \leq (\frac{2}{3} + \alpha)\cdot\mu(G)$ and we prove with this assumption, the algorithm outputs a better than $2/3$ approximation in one of $T$ iterations. The proof for the other case where $\widetilde{\mu} > (\frac{2}{3} + \alpha)\cdot\mu(G)$ is trivial since the algorithm outputs an estimation which is at least $\widetilde{\mu}$. \subsubsection{Approximation analysis of Case 1: Many $A$-vertices matched in some $M_{AB}^i$} \begin{lemma}\label{lem:case1-approx} If $\widetilde{\mu}_2 \geq c\alpha\widetilde{\mu}_1$, then $\widetilde{\mu}_1 + \widetilde{\mu}_2$ is an estimate for $\mu(G)$ up to a multiplcative-additive factor of $(\frac{2}{3} + \alpha, o(n))$ with high probability. \end{lemma} \begin{proof} First, we prove that $\widetilde{\mu}_1 + \widetilde{\mu}_2 \leq \mu(G)$. Let $X$ be as defined in \Cref{alg:matchingbetterthan23}. Note that $\frac{n\E[X]}{r_1} \leq \mu(G[A])$ since it counts the number of edges in $M_{AB}^i$ that have both of their endpoints in $A$. Since $X$ is the sum of $r_1$ independent Bernoulli random variables, using Chernoff bound we have \begin{align*} \Pr[|X - \E[X]| \geq \sqrt{18\E[X]\log n}] \leq 2 \exp\left(-\frac{18\E[X]\log n}{3\E[X]}\right) = \frac{2}{n^6}. \end{align*} Thus, with a probability of at least $1-2/n^6$, \begin{align*} \widetilde{\mu}_2 \leq \frac{nX}{r_1} - \frac{n}{2\log n} & \leq \frac{n\left(\E[X] + \sqrt{18\E[X]\log n}\right)}{r_1} - \frac{n}{2\log n} \\ & \leq \mu(G[A]) + \sqrt{\frac{18n\cdot\mu(G[A])\log n}{r_1}} - \frac{n}{2\log n} & (\text{Since } \frac{n\E[X]}{r_1}\leq \mu(G[A]))\\ & \leq \mu(G[A]) + \sqrt{\frac{n\cdot \mu(G[A])}{4\log^2 n}} - \frac{n}{2\log n} & (\text{Since } r_1 = 72 \log^3 n)\\ & \leq \mu(G[A]) & (\text{Since } \mu(G[A]) \leq n). \end{align*} Moreover, we have that $\widetilde{\mu}_1 \leq \mu(G'')$. Since $G[A]$ only contains vertices that are not matched in the matching that we found in $G''$, we get $\widetilde{\mu}_1 + \widetilde{\mu}_2 \leq \mu(G)$. On the other hand, by \Cref{lem:tight-instance-edcs}, we have $\widetilde{\mu}_1 \geq (1-\epsilon)\cdot (\frac{2}{3} - 120\sqrt{\alpha})\mu(G) - o(n)$. Thus, \begin{align*} \widetilde{\mu}_1 + \widetilde{\mu}_2 \geq (1+c\alpha)\widetilde{\mu}_1 & \geq (1+c\alpha)\left((1 - \epsilon)\left(\frac{2}{3}-120\sqrt{\alpha}\right)\cdot\mu(G) - o(n)\right) \\ & \geq (1+c\alpha)\cdot \left(\frac{2}{3} - 120\sqrt{\alpha} - \frac{2}{3}\epsilon\right)\cdot \mu(G) - o(n)\\ & \geq \left(\frac{2}{3} - 120\sqrt{\alpha} - \frac{2}{3}\epsilon +\frac{2}{3}c\alpha - 120c\alpha\sqrt{\alpha} - \frac{2}{3}c\alpha \epsilon \right)\cdot\mu(G) - o(n) \\ & \geq \left(\frac{2}{3} + \alpha \right)\mu(G) - o(n), \end{align*} where the last inequality holds by our choice of $c$, $\alpha$, and if we choose $\epsilon$ sufficiently small enough, which leads to our claimed approximation ratio. \end{proof} \subsubsection{Approximation analysis of Case 2: Large matching in $G[A \setminus M^i_{AB}]$ (for some $M^i_{AB}$)} \begin{lemma}\label{lem:case2-approx} If $\widetilde{\mu}_3 \geq c\alpha\widetilde{\mu}_1$, then $\widetilde{\mu}_1 + \widetilde{\mu}_3$ is an estimate for $\mu(G)$ up to a multiplcative-additive factor of $(\frac{2}{3} + \alpha, o(n))$ with high probability. \end{lemma} \begin{proof} Note that since we connect each vertex of $B \setminus V(M_{AB}^{i})$ to $\kappa = \widetilde{O}(n^{1-\gamma})$ singleton vertices, most of the vertices of $B \setminus V(M_{AB}^{i})$ will be matched to singleton vertices when running random greedy maximal matching. Let $v \in B \setminus V(M_{AB}^{i})$. The first incident edge of $v$ in the permutation of edges is not an edge to singleton vertices with a probability of at most $(6n^{1-\gamma}\log n) / (\kappa + 6n^{1-\gamma}\log n)$ since the maximum degree of subgraph $G[V_{mid} \setminus V(M^i_{AB})]$ is at most $6n^{1-\gamma}\log n$ by \Cref{lem: sparsification-property}. We denote the vertices of $B \setminus V(M^i_{AB})$ that are not matched with singleton vertices by $R$. Hence, we have the following bounds, \begin{align*} \E|R| \leq n\cdot\left(\frac{6n^{1-\gamma}\log n}{\kappa + 6n^{1-\gamma}\log n}\right) \leq n\cdot\left(\frac{6n^{1-\gamma}\log n}{\kappa}\right) = \frac{n}{8\log n},\\ \E|R| \geq n\cdot\left(\frac{6n^{1-\gamma}\log n}{\kappa + 6n^{1-\gamma}\log n}\right) \geq n\cdot\left(\frac{6n^{1-\gamma}\log n}{2\kappa}\right) = \frac{n}{16\log n}. \end{align*} Therefore, using a Chernoff bound, we can show that $|R| \leq \frac{n}{4\log n}$ with a high probability. Now, we prove that $\widetilde{\mu}_1 + \widetilde{\mu}_3 \leq \mu(G)$. Let $Y$ be as defined in \Cref{alg:matchingbetterthan23}, then we have $\frac{n\E[Y]}{2r_2} \leq \mu(G[A]) + |R|$, since it counts the number of edges in output of random greedy maximal matching with both endpoints in $A$ and there are at most $|R|$ vertices of $B$ can match to vertices of $A$. Also, each edge is counted at most twice since it has two endpoints in $A$. Thus, \begin{align*} \Pr[|Y - \E[Y]| \geq \sqrt{18\E[Y]\log n}] \leq 2 \exp\left(-\frac{18\E[Y]\log n}{3\E[Y]}\right) = \frac{2}{n^6}, \end{align*} since $Y$ is the sum of independent Bernoulli random variables. Therefore, with a probability of at least $1 - 2/n^6$, \begin{align*} \widetilde{\mu}_3 & \leq \frac{nY}{2r_2} - \frac{n}{2\log n} \\ & \leq \frac{n\left(\E[Y] + \sqrt{18\E[Y]\log n}\right)}{2r_2} - \frac{n}{2\log n} \\ & \leq \mu(G[A]) + |R| + \sqrt{\frac{9n\cdot(\mu(G[A]) + |R|)\log n}{r_2}} - \frac{n}{2\log n} & (\text{Since } \frac{n\E[Y]}{2r_2}\leq \mu(G[A]) + |R|)\\ & \leq \mu(G[A]) + |R| + \sqrt{\frac{n\cdot \left(\mu(G[A]) + |R| \right)}{32\log^2 n}} - \frac{n}{2\log n} & (\text{Since } r_2 = 288 \log^3 n)\\ & \leq \mu(G[A]), & (\text{Since } \mu(G[A])\leq n \text{ and }|R| \leq \frac{n}{4\log n}) \end{align*} which implies that $\widetilde{\mu}_1 + \widetilde{\mu}_3 \leq \mu(G)$ with the same argument as proof of \Cref{lem:case1-approx}. Now we are ready to finish the proof since \begin{align*} \widetilde{\mu}_1 + \widetilde{\mu}_3 \geq (1+c\alpha)\widetilde{\mu}_1 & \geq (1+c\alpha)\left((1 - \epsilon)\left(\frac{2}{3}-120\sqrt{\alpha}\right)\cdot\mu(G) - o(n)\right) \\ & \geq (1+c\alpha)\cdot \left(\frac{2}{3} - 120\sqrt{\alpha} - \frac{2}{3}\epsilon\right)\cdot \mu(G) - o(n)\\ & \geq \left(\frac{2}{3} - 120\sqrt{\alpha} - \frac{2}{3}\epsilon +\frac{2}{3}c\alpha - 120c\alpha\sqrt{\alpha} - \frac{2}{3}c\alpha \epsilon \right)\cdot\mu(G) - o(n) \\ & \geq \left(\frac{2}{3} + \alpha \right)\mu(G) - o(n), \end{align*} \end{proof} \subsubsection{Approximation analysis of Case 3: Almost every edge of maximum matching of $G[A]$ has at least one of its endpoints matched by almost all $M_{AB}^i$} \begin{lemma}\label{lem:bound-on-A} Before the start of the $i$-th iteration of \Cref{alg:matchingbetterthan23} for $i \in [1, T]$: if the algorithm has not terminated yet, then it holds that $\mu(G[A]) \geq \left(\frac{1}{4} - (i-1)\cdot(10^{20}\alpha)\right) \cdot \mu(G)$. \end{lemma} \begin{proof} In the beginning of the section we assumed that $\widetilde{\mu} \leq (\frac{2}{3} + \alpha)\cdot\mu(G)$. Note that $A$ contains vertices that are not matched by the matching of estimation $\widetilde{\mu}_1$. Therefore, before the first iteration of the algorithm, by \Cref{lem:tight-instance-edcs}, set $A$ must contain a matching of size larger than $(\frac{1}{3}-800\alpha)\cdot \mu(G) \geq \mu(G)/4$ where the inequality follows from our choice of $\alpha$. We use induction to prove the lemma. Suppose that the algorithm has not terminated before the $i$-th iteration for $i > 1$. Note that since the algorithm has not terminated in the previous iteration for any of the maximal matching $M_{AB}^i$, the total number of maximal matching edges between vertices of $A$ is at most $c\alpha \widetilde{\mu}_1 n^{\gamma/2} \leq c\alpha \mu(G)n^{\gamma/2}$, which implies that the average degree of a vertex in $G_{M_{AB}}[A]$ is at most \begin{align*} \frac{c\alpha\mu(G)n^{\gamma/2}}{|A|} & \leq \frac{c\alpha\mu(G)n^{\gamma/2}}{(\frac{1}{4} - (i-2)(10^5\alpha)) \cdot \mu(G)} & (\text{By induction hypothesis})\\ & \leq \frac{c\alpha\mu(G)n^{\gamma/2}}{1/6\cdot \mu(G)} & (i \leq T \text{ and } 10^{20} T \alpha \leq \frac{1}{12})\\ & = 6c\alpha n^{\gamma/2}. \end{align*} Let $Z$ be the number of edges of $G[A]$ that are in one of the maximal matchings. Since we sample $r_3 = 10n^{1 - \gamma/2}\log n$ vertices of $A$, we have \begin{align*} \E[Z] \leq 60 c \alpha n \log n. \end{align*} Since each vertex of $A$ have at most $n^{\gamma/2}$ neighbors in $A$ (because we have $k = n^{\gamma/2}$ maximal matchings), then by using Hoeffding's inequality, we obtain \begin{align*} \Pr[|Z-\E[Z]| \geq 30 c \alpha n \log n] \leq 2 \exp\left(- \frac{2 \cdot(10n^{1-\gamma/2}\log n) \cdot (12 \alpha n^{\gamma /2})}{n^\gamma} \right) \leq \frac{1}{n^{10}}. \end{align*} Therefore, with high probability, there are at most $90 c \alpha n \log n$ edges of maximal matchings of sampled vertices with both endpoints in $A$ (*). Now we show that at most $10^{20} \alpha \mu(G)$ vertices of $A$ are removed in Line~\ref{Line:remove} (of Algorithm~\ref{alg:matchingbetterthan23}) in the previous iteration. For the sake of contradiction, assume that more than $10^{20} \alpha \mu(G)$ are removed in the previous iteration. Then, we have at least \begin{align*} 10^{20} \alpha \mu(G) \cdot \eta & = 10^{20} \alpha \mu(G) \cdot \left( \frac{n \log n}{100 \widetilde{\mu}_1}\right)\\ & \geq 10^{20} \alpha \mu(G) \cdot \left( \frac{n \log n}{100 \mu(G)}\right) & (\text{Since } \widetilde{\mu}_1 \leq \mu(G))\\ & = 100 c \alpha n \log n & (\text{Since } c = 10^{16}) \end{align*} edges of maximal matching between vertices of $A$ since each removed vertex must have at least $\eta$ neighbors in the sampled vertices, which is a contradiction to (*). By induction, before iteration $i - 1$, we have $\mu(G[A]) \geq (\frac{1}{4} - (i-1)\cdot(10^{20}\alpha)) \cdot \mu(G)$. Also, at most $10^{20} \alpha \mu(G)$ vertices of $A$ can be removed in the previous step which implies the claimed bound for iteration $i$. \end{proof} \begin{corollary}\label{cor:size-of-a} Before the start of the $i$-th iteration of \Cref{alg:matchingbetterthan23} for $i \in [1, T]$: if the algorithm has not terminated yet, then it holds that $|A| \geq \frac{1}{5}\mu(G)$ \end{corollary} \begin{proof} By \Cref{lem:bound-on-A}, we have that \begin{align*} \mu(G[A]) \geq (\frac{1}{4} - (i-2)(10^{20}\alpha)) \cdot \mu(G) \geq \frac{1}{5}\mu(G), \end{align*} where the last inequality follows from our choice of $\alpha$ and $T$. We finish the proof by the fact that the number of vertices of a graph is larger than its matching size. \end{proof} \begin{observation} Suppose that \Cref{alg:matchingbetterthan23} does not terminate in iteration $i$. Then, it holds $|B| \geq \mu(G)/5$ before iteration $i$. \end{observation} \begin{proof} Consider maximal matching $M_{AB}^1$. Since the algorithm has not terminated in iteration $i$, at least $\mu(G[A]) - 2c\alpha \widetilde{\mu}_1$ edges of $\mu(G[A])$ are blocked by maximal matching edges since the algorithm does not find more than $2c\alpha \widetilde{\mu}_1$ edges of matching $\mu(G[A])$ in case 1 and 2. This means that at least one endpoint of these edges is matched with a vertex in $B$. Thus, we have \begin{align*} |B| \geq \mu(G[A]) - 2c\alpha \widetilde{\mu}_1 \geq (\frac{1}{4} - (i-1)(10^{20} \alpha) - 8\alpha)\cdot\mu(G) \geq \frac{1}{5} \cdot \mu(G), \end{align*} where the second inequality follows from \Cref{lem:bound-on-A} and $\widetilde{\mu}_1 \leq \mu(G)$, and the last inequality follows by the choice of $T$, $\alpha$, and $c$. \end{proof} \begin{lemma}\label{lem:removing-b} Suppose that the \Cref{alg:matchingbetterthan23} does not terminate in iteration $i$. Then, at the end of this iteration, the algorithm removes at least $0.01 \cdot |B|$ vertices of $B$. \end{lemma} \begin{proof} Since the algorithm does not terminate in iteration $i$ for any of $k = n^{\gamma/2}$ maximal matchings, at least $(\mu(G[A]) - 2c\alpha\mu(G))\cdot n^{\gamma/2}$ edges of $\mu(G[A])$ are blocked by each of maximal matchings. This means that at least one endpoint of these edges is matched with a vertex in $B$. Thus, there are at least \begin{align*} (\mu(G[A]) - 2c\alpha\cdot \mu(G))\cdot n^{\gamma/2} \geq (\frac{1}{4} - (i-1)(10^{20} \alpha) - 2c\alpha)\cdot\mu(G) n^{\gamma /2} \geq \frac{1}{5} \cdot \mu(G)n^{\gamma/2} \end{align*} edges with one endpoint in $A$ and one endpoint in $B$ in all maximal matchings. Hence, the expected average degree of a vertex in $B$ in the subgraph $G_{M_{AB}}$ is at least \begin{align*} \frac{\frac{1}{5} \cdot \mu(G)n^{\gamma/2}}{|B|} \leq \frac{1}{5} \cdot n^{\gamma/2}, \end{align*} where the inequality is because $|B| \leq \mu(G'') \leq \mu(G)$. Let $Z$ be the number of edges of $G_{M_{AB}}$ with one endpoint in $A$ that is among $r_3$ sampled vertices and one endpoint in $B$. Since we sample $10n^{1-\gamma /2} \log n$ vertices of $A$, we have \begin{align*} \E[Z] \geq 2n\log n. \end{align*} With the same argument as proof of \Cref{lem:bound-on-A}, each vertex of $A$ can have at most $n^{\gamma/2}$ neighbors in $B$, then by using Hoeffding's inequality, we get \begin{align*} \Pr[|Z-\E[Z]| \geq n \log n] \leq 2 \exp\left(- \frac{2 \cdot(10n^{1-\gamma/2}\log n) \cdot ( \frac{1}{10} n^{\gamma /2})}{n^\gamma} \right) \leq \frac{1}{n^{10}}, \end{align*} which implies that with high probability, there are at least $n\log n$ edges of maximal matchings with one endpoint in sampled $A$ vertices and one in $B$. Next, we prove for each vertex in $B$, the number of sampled $A$ vertices in $G_{M_{AB}}$ is small with high probability. Note that each vertex in $A$ is sampled with a probability of $\frac{10n^{1-\gamma/2}}{|A|}$. Moreover, each vertex in $B$ can have at most $n^{\gamma/2}$ neighbors in $G_{M_{AB}}$. Let $W_v$ be the expected number of sampled $A$ vertices that are connected to $v$ in $G_{M_{AB}}$ for $v \in B$. Therefore, \begin{align*} \E[W_v] \leq \frac{10n \log n}{|A|} \leq \frac{50 n \log n}{\mu(G)}. \end{align*} where the last inequality holds because of \Cref{cor:size-of-a}. Then, by Chernoff bound \begin{align*} \Pr\left[|W_v - \E[W_v]| \geq \frac{25n\log n}{\mu(G)}\right] \leq 2\exp\left(-\frac{625n\log n}{150\mu(G)}\right) \leq \frac{2}{n^8}. \end{align*} Using a union bound on all vertices of $B$ implies that with high probability, each of the vertices of $B$ has at most $75n\log n / \mu(G)$ neighbors in $G_{M_{AB}}$ among sampled $A$ vertices. For the sake of contradiction, suppose that the algorithm removes less than $0.01|B|$ vertices of $B$ at the end of the iteration. This implies that the maximum number of edges between vertices of $B$ and sampled $A$ vertices is \begin{align*} 0.01|B| \cdot \left(\frac{75n\log n}{\mu(G)}\right) + 0.99|B| \cdot \left(\frac{n\log n}{100\widetilde{\mu}_1}\right) & \leq 0.01|B| \cdot \left(\frac{75n\log n}{\widetilde{\mu}_1}\right) + 0.99|B| \cdot \left(\frac{n\log n}{100\widetilde{\mu}_1}\right) \\ &\leq \frac{0.76|B|\cdot n\log n}{\widetilde{\mu}_1}\\ & \leq 0.76 n \log n, \end{align*} where the first term is the total number of edges of removed vertices of $B$ and the second term is the total number of edges of vertices of $B$ that are not removed. Since we proved that with high probability, there are at least $n\log n$ edges, then the above upper bound on the number of edges is a contradiction that completes the proof. \end{proof} \subsection{Putting it all together: Beating two-thirds} \begin{lemma}\label{lem:termination} If $\widetilde{\mu} \leq (\frac{2}{3} + \alpha)\cdot\mu(G)$, then the \Cref{alg:matchingbetterthan23} terminates in one of $T$ iterations. \end{lemma} \begin{proof} Suppose that the algorithm does not terminate in any of $T$ rounds. By \Cref{lem:removing-b}, in each iteration, the algorithm will remove $0.01 |B|$ vertices of $B$. Thus, after $T = 200$ iterations, none of the vertices of $B$ will remain in the graph. However, plugging $T =200$ in \Cref{lem:bound-on-A}, we have $\mu(G[A]) \geq \frac{1}{5}\mu(G)$. Therefore, either $\widetilde{\mu}_2 \geq c\alpha \widetilde{\mu}_1$ or $\widetilde{\mu}_3 \geq c\alpha \widetilde{\mu}_1$ in iteration $200$, since there is no vertices of $B$ to block the matching of $G[A]$. \end{proof} \begin{lemma}\label{lem:approx-ratio-better-32} The output of \Cref{alg:matchingbetterthan23} is a $(\frac{2}{3} + \alpha, o(n))$ estimate for maximum matching of $G$. \end{lemma} \begin{proof} If $\widetilde{\mu}$ is not a $(\frac{2}{3} + \alpha, o(n))$ estimate for maximum matching of $G$, then by \Cref{lem:termination}, the algorithm terminates in one of the rounds. Therefore, by \Cref{lem:case1-approx} and \Cref{lem:case2-approx}, the output is a $(\frac{2}{3} + \alpha, o(n))$ estimate for maximum matching of $G$. \end{proof} \apprxtheoremB* \begin{proof} By \Cref{lem:approx-ratio-better-32}, we obtain the claimed approximation ratio. Also, since we choose $\epsilon$ to be smaller than $\gamma$ in \Cref{alg:matchingbetterthan23}, the running time is $\widetilde{O}_\epsilon(n^{2-\epsilon^3})$ by \Cref{lem:time-better-23}. \end{proof} \paragraph{Multiplicative Approximation for Adjacency List:} With the same argument as the multiplicative approximation for the adjacency list in the previous section, we can assume $\mu(G') = \Omega(n^{0.99})$. Hence, if $\widetilde{\mu} < (2/3 + \alpha)\mu(G)$, we have $\mu(G[A]) = \Omega(n^{0.99}) $ by \Cref{lem:tight-instance-edcs}. Therefore, with the exact same argument as previous section, using a standard Chernoff bound, for getting multiplicative approximation, we can use $r_1 = \widetilde{O}(n^{0.01})$ and $r_2 = \widetilde{O}(n^{0.01})$ samples for estimation of $\widetilde{\mu}_2$ and $\widetilde{\mu}_3$. Note that the running time of \Cref{clm:maximal-matching-aa-edges} and \Cref{clm:estimating-aa-missing-matching} will increase to $\widetilde{O}_\epsilon(n^{1+\epsilon + \gamma/2 + 0.01})$ and $\widetilde{O}_\epsilon(n^{2+\epsilon-\gamma/2 + 0.01})$, respectively because of the larger number of samples. However, if we choose $\gamma$ larger than $0.02 + 2\epsilon + 2\epsilon^3$, the running time of the algorithm will remain $\widetilde{O}(n^{2-\epsilon^3})$ by \Cref{lem:time-better-23}. \apprxtheoremBlist* \section{Introduction} We study algorithms for estimating the size of {\em maximum matching}. Recall that a {\em matching} is a set of edges no two of which share an endpoint, and a {\em maximum matching} is a matching of the largest size. This problem has been studied for decades. There are traditional algorithms that can obtain an arbitrary good approximation of the solution in time linear in the input size (see e.g. the celebrated algorithm of Hopcroft and Karp \cite{HopcroftK73}). A natural question, that has received significant attention over the past two decades, is whether it is possible to estimate the size of maximum matching in {\em sublinear time} in the input size \cite{ParnasRon07,NguyenOnakFOCS08,YoshidaYISTOC09,OnakSODA12,KapralovSODA20,ChenICALP20,behnezhad2021,behnezhadsublinear22}. Since any sublinear time algorithm can read only a small fraction of the input, it is important to specify how the input is represented. Our focus in this work is particularly on the standard {\em adjacency list} model. Here, the algorithm can specify a vertex $v$ of its choice and any integer $i$, and in response receives the $i$-th neighbor of $v$ stored in an arbitrarily ordered list, or receives NULL if $v$ has less than $i$ neighbors.\footnote{Another common model is the {\em adjacency matrix} model where the algorithm can specify a pair of vertices $(u, v)$ and in response gets whether they are adjacent or not.} Given a graph $G$ represented in this model, the goal is to provide an estimate $\widetilde{\mu}(G)$ of the size $\mu(G)$ of the maximum matching of $G$ such that $\widetilde{\mu}(G)$ is ``close'' to $\mu(G)$ while the total number of queries made to the graph is much smaller than the size of the graph. \subsection{Known Bounds} It is not hard to see that to provide an exact estimate satisfying $\widetilde{\mu}(G) = \mu(G)$, one has to make $\Omega(n^2)$ queries to the graph, even if the algorithm is randomized (with constant success probability). Thus, any sublinear time algorithm must resort to approximations. Indeed, multiple algorithms have been devised that run in sublinear time and approximate maximum matching size. Earlier results, pioneered in the works of \citet*{ParnasRon07,NguyenOnakFOCS08} and \citet*{YoshidaYISTOC09} focused mainly on bounded-degree graphs. The more recent works of \cite{ChenICALP20,KapralovSODA20,behnezhad2021,behnezhadsublinear22} run in sublinear time even on general $n$-vertex graph. For instance, a result of \citet*{behnezhad2021} shows an (almost) 1/2-approximation in the adjacency list model can be obtained in $\widetilde{O}(n)$ time. More recently, \citet*{behnezhadsublinear22} broke the half-approximation barrier, obtaining a (slightly) better $(\frac{1}{2}+\Omega(1))$-approximation in $O(n^{1+\epsilon})$-time. With the 1/2-approximation now broken, a natural next step, posed explicitly in \cite{behnezhadsublinear22} as an open problem, is determining the best approximation achievable in $n^{2-\Omega(1)}$ time. The situation on the lower bound side is much less understood. The only known lower bound is that $\Omega(n)$ time is needed for obtaining any constant approximation of maximum matching, which was proved 15 years ago by \citet*{ParnasRon07}. While this lower bound was (nearly) matched by \cite{behnezhad2021} for 1/2-approximations, it is not known whether it is optimal or can be improved for better approximations. In particular, the current state of affairs leave it possible to obtain even a $(1-\epsilon)$-approximation, for any fixed $\epsilon > 0$, in just $O(n)$ time. Not only such a result would be amazing on its own, but as we will later discuss, it will have deep consequences in the study of dynamic graphs. It is also worth noting that in their beautiful work, \citet*{YoshidaYISTOC09} showed existence of an $O(n) + \Delta^{O(1/\epsilon^2)}$ time algorithm that obtains a $(1-\epsilon)$-approximation\footnote{More precisely, \cite{YoshidaYISTOC09} give a multiplicative-additive $(1-\epsilon, o(n))$ approximation in $\Delta^{O_\epsilon(1)}$ time. The claimed bound follows by slightly tweaking their algorithm using techniques developed in \cite{behnezhad2021} for multiplicative approximations.}. While this is not a sublinear time algorithm for the full range of $\Delta$, it runs in $O(n)$ time for $\Delta = n^{O(\epsilon^2)}$. This shows that any potential $\omega(n)$ time lower bound must be proved on graphs of large degree. \subsection{Our Results} \paragraph{Improved Lower Bound:} In this work, we give the first super-linear in $n$ lower bound for the sublinear matching problem. We show that not only $(1-\epsilon)$-approximations are not achievable in $O(n)$ time, but in fact any better than $2/3$-approximation requires at least $n^{1.2-o(1)}$ time. \begin{graytbox} \begin{theorem}\label{thm:lb} For any fixed $\alpha > 0$, any (possibly randomized) algorithm obtaining a $(2/3+\alpha)$-approximation of the size of maximum needs to make at least $n^{1.2 - o(1)}$ adjacency list queries to the graph. This holds even if the graph is bipartite and has a matching of size $\Theta(n)$. \end{theorem} \end{graytbox} Our proof of \cref{thm:lb} relies crucially on {\em correlation decay}. To our knowledge, this is the first application of correlation decay in proving sublinear time lower bounds. We give an informal overview of our techniques in proving \cref{thm:lb} in \cref{sec:techniques}, and discuss why correlation decay is extremely helpful for us. The formal proof of \cref{thm:lb} is then presented in \cref{sec:lb}. We note that the essence of the $\Omega(n)$ lower bound of \citet*{ParnasRon07} is that $o(n)$ queries are not enough to even see all neighbors of a single vertex. Using this, \cite{ParnasRon07} constructs an input distribution where no $o(n)$ time algorithm can see any edge of any (approximately) optimal matching. Indeed one key challenge that our lower bound of \cref{thm:lb} overcomes is to show that even though there are, say, $O(n^{1.1})$ time algorithms that ``see'' as many as $n^{\Omega(1)}$ edges of an optimal matching, they are still unable to obtain a $(2/3+\Omega(1))$-approximation. \paragraph{Improved Upper Bounds:} We also present two algorithms that run in strictly sublinear time of $n^{2-\Omega(1)}$. Our first result is an algorithm that works on general (i.e., not necessarily bipartite) graphs and obtains an (almost) 2/3-approximation. This significantly improves prior close-to-1/2 approximations of \cite{behnezhad2021,behnezhadsublinear22}. \begin{graytbox} \begin{theorem}\label{thm:2/3} For any fixed $\epsilon > 0$, there are algorithms for approximating the maximum matching size of any (general) $n$-vertex graph that take $n^{2-\Omega_\epsilon(1)}$ time and obtain \begin{itemize}[itemsep=0pt, topsep=5pt] \item a multiplicative $(2/3-\epsilon)$-approximation in the adjacency list model, and \item a multiplicative-additive $(2/3-\epsilon, o(n))$-approximation in the adjacency matrix model. \end{itemize} \end{theorem} \end{graytbox} \begin{remark*} A subquadratic time multiplicative $O(1)$-approximation is impossible in the adjacency matrix model since even distinguishing an empty graph from one including only one random edge requires $\Omega(n^2)$ adjacency matrix queries. Therefore, a multiplicative-additive approximation (defined formally in \cref{sec:prelim}) is all we can hope for under adjacency matrix queries. \end{remark*} Although \cref{thm:2/3} significantly improves prior approximations and comes close to the 2/3 barrier, but it does not break it. As such, one may still wonder whether the $n^{1.2-o(1)}$ lower bound of \cref{thm:lb} for $(2/3+\Omega(1))$-approximations can be improved to $n^{2-o(1)}$. Our next result rules this possibility out, and gives a subquadratic time algorithm that indeed breaks $2/3$-approximation provided that the input graph is bipartite. \begin{graytbox} \begin{theorem}\label{thm:beating-2/3} There are algorithms for approximating the maximum matching size of any bipartite $n$-vertex graph that take $n^{2-\Omega(1)}$ time and obtain \begin{itemize}[itemsep=0pt, topsep=5pt] \item a multiplicative $(2/3+\Omega(1))$-approximation in the adjacency list model, and \item a multiplicative-additive $(2/3+\Omega(1), o(n))$-approximation in the adjacency matrix model. \end{itemize} \end{theorem} \end{graytbox} \cref{thm:lb,thm:beating-2/3} together imply a rather surprising bound on the complexity of $(2/3+\Omega(1))$-approximating maximum matching size for bipartite graphs: They show that the right time-complexity is $n^{1+c}$ for some $c$ that is strictly smaller than 1 but no smaller than $.2$. \paragraph{Independent Work:} In a concurrent and independent work \citet*{BKS22} gave $n^{2-\Omega(1)}$ time algorithms for obtaining (almost) 2/3-approximation in both adjacency list and adjacency matrix models similar to our \cref{thm:2/3}. We note that the lower and upper bounds of \cref{thm:lb,thm:beating-2/3} are unique to our paper. \subsection{Further Related Works and Implications} \paragraph{The 2/3-Barrier for Approximating Matchings:} Prior to our work, it was shown in a beautiful paper of \citet*{GoelKK12} that obtaining a better than 2/3-approximation of maximum matching in the one-way communication model requires $n^{1+\Omega(1/\log \log n)} \gg n\poly(\log n)$ communication from Alice to Bob. This was, to our knowledge, the first evidence that going beyond 2/3-approximation for maximum matching is ``hard'' in a certain model. Our \cref{thm:lb} shows that this barrier extends to the sublinear model. While the two models are completely disjoint and the constructions are different, we note that unlike \cref{thm:lb}, the $n^{1+\Omega(1/\log \log n)}$ lower bound of \cite{GoelKK12} is still $n^{1+o(1)}$. Obtaining an $n^{1+\Omega(1)}$ lower bound for approximating matchings in the one-way communication model remains open (and, in fact, impossible short of a breakthrough in combinatorics --- see \cite{GoelKK12}). \paragraph{Implications for Dynamic Algorithms:} In the fully dynamic maximum matching problem, we have a graph that is subject to both edge insertions and deletions. The goal is to maintain a good approximation of maximum matching by spending a small time per update. This is a very well studied problem for which several update-time/approximation trade-offs have been shown. Recent works of \citet*{behnezhad2022dynamic} and \citet*{bhattacharyaKSW-SODA23} established a new connection between this problem and (static) sublinear time algorithms. In particular, they showed that any $T$ time algorithm for $\alpha$-approximating maximum matching size leads to an algorithm that maintains an (almost) $\alpha$-approximation in $\widetilde{O}(T/n)$ time per update in the dynamic setting. This, in particular, motivates the study of sublinear time maximum matching algorithms that run in $\widetilde{O}(n)$ time as they would lead to $\widetilde{O}(1)$ update-time algorithms which are the holy grail of dynamic algorithms. Our \cref{thm:lb} shows that, at least in this framework, better than a $2/3$-approximation requires $n^{0.2-o(1)}$ update-time. We note that proving such a lower bound for all algorithms (i.e., not necessarily in this framework) remains an important open problem (see \cite{AbboudW14,HenzingerKNS15} and the references therein). \section{Our Techniques}\label{sec:techniques} \subsection{The Lower Bound of \cref{thm:lb}} \paragraph{An insightful, but broken, input distribution:} Let us start with the lower bound. Consider the following input distribution, illustrated in the figure below. While we emphasize that this is not the final input distribution that we prove the lower bound with, it provides the right intuition and also highlights some of the challenges that arise in proving the lower bound. There are four sets of vertices, $A, B, S, T$ with $|A| = |B| = |S| = N$ and $T = \epsilon N$. The $T$ vertices are ``dummy'' vertices, they are adjacent to every other vertex in $A \cup B \cup S$. While this increases the degree of every vertex in $A \cup B \cup S$ to at least $\epsilon N$, the $T$ vertices cannot contribute much to the output since $|T| = \epsilon N$. The important edges, that determine the output are among the rest of the vertices. In particular, there is always a perfect matching between the $B$ vertices and the $S$ vertices. Additionally, there is a random Erd\"os-Renyi graph between $A$ and $B$ for a suitable expected degree $d = N^{\Theta(1)}$. Now in the \yes{} distribution, we also add a perfect matching among the $A$ vertices (the red matching) but in the \no{} distribution, we do not add this matching. Observe that in the \yes{} case, we can match all of $B$ to $S$ and all of $A$ together, obtaining a matching of size at least $3N$. However, in the \no{} case, it can be verified that the maximum matching is only at most $(2+\epsilon)N$. Thus, any algorithm that beats 2/3-approximation by a margin more than $\epsilon$, should be able to distinguish whether we are in the \yes{} distribution or in the \no{} distribution. \begin{figure}[h] \centering \includegraphics[scale=0.5]{figs/input-dist-techniques.pdf} \end{figure} Suppose now that we give away which vertices belong to $S$ and $T$ for free with no queries, but keep it secret whether a vertex $v \in A \cup B$ belongs to $A$ or $B$. To examine whether a vertex $v$ belongs to $A$ or $B$, the naive approach is to go over all of its neighbors, and see whether we see an $S$ neighbor or not. But because the degree of each vertex is $\Omega(\epsilon N)$ due to the edges to $T$, this requires at least $\Omega(\epsilon N) = \Omega(N)$ time. But this is not enough. To separate the two cases, one has to actually determine if there are any edges among the $A$ vertices or not. Here, the naive approach is to first find a vertex $v \in A$ which can be done in $O(N)$ time, and then go over all neighbors $u$ of $v$ in $A \cup B$, and examine whether $u$ belongs to $A$. Since $v$ has $d = N^{\Omega(1)}$ neighbors in $A \cup B$, this naive approach takes $N^{1+\Omega(1)}$ time. We would like to argue that this naive approach is indeed the best possible, and that any algorithm distinguishing the two distributions must make $N^{1+\Omega(1)} = n^{1+\Omega(1)}$ queries to the graph. Unfortunately, this is not the case with the distribution above as we describe next. Consider the following algorithm. We first start by finding an $A$ vertex $v$ in $O(n)$ time. Then we go over the neighbors of $v$ in a random order until we find the first neighbor $u$ that belongs to $A \cup B$. Note that this takes $O(n/d)$ time. We then do the same for $u$, finding a random neighbor to another vertex in $A \cup B$. Since each step of this random walk takes $O(n/d)$ time, we can continue it for $2d$ steps in $O(n)$ total time. Let $w$ be the last vertex of the walk. We now examine whether $w$ belongs to $A$ or $B$ in $O(n)$ time. We argue that by doing so, there is a constant probability that we can distinguish the \yes{} case from the \no{} case. To see this, observe that in the \no{} distribution, every $A$ vertex goes to a $B$ vertex and every $B$ vertex goes to an $A$ vertex with probability one. As such, since the walk continues for an even number of steps, the last vertex $w$ must belong to $A$ with probability 1. But in the \yes{} distribution, there is a constant probability that we go through an $A$-$A$ edge exactly once. In this case, the last vertex $w$ of the random walk must belong to $B$. Repeating this process a constant number of times allows us to distinguish the two distributions with probability $0.99$ in merely $O(n)$ time! \paragraph{Correlation Decay to the Rescue:} To get around this challenge, we add $\epsilon d$ edges also among the $B$ vertices (while modifying the size of $A$ and $B$ slightly to ensure that the degrees in $A$ and $B$ do not reveal any information --- see \cref{fig:dist_common}). Observe that this completely destroys the algorithm above. In particular, it no longer holds that any $B$ vertex goes to an $A$ vertex with probability $1$. Rather, there is a probability $\epsilon$ that we go from $B$ to $B$. Hence, intuitively, whether the last vertex $w$ of the random walk belongs to $A$ or $B$ does not immediately reveal any information about whether an $A$-$A$ edge was seen or not. We show that indeed, this can be turned into a formal lower bound against all algorithms via \underline{correlation decay}. First, we show that for suitable $d$, the queried subgraph will only be a tree. We then show that conditioning on the queries conducted far away from a vertex $v$, the probability of $v$ being an $A$ or $B$ vertex will not change, and use this to prove that the algorithm cannot distinguish the \yes{} and \no{} distributions. \subsection{The Algorithms of \cref{thm:2/3,thm:beating-2/3}} We now turn to provide an overview of the techniques in obtaining the (almost) 2/3-approximation of \cref{thm:2/3}, and the $(2/3 + \Omega(1))$-approximation of \cref{thm:beating-2/3} for bipartite graphs. \paragraph{An (Almost) 2/3-Approximation via EDCS:} The {\em edge-degree constrained subgraph} (EDCS) introduced by \citet*{bernsteinstein2015,bersteinsteingeneral} has been a powerful tool in obtaining an (almost) $2/3$-approximation of maximum matching in various settings including dynamic algorithms \cite{bernsteinstein2015,bersteinsteingeneral}, communication complexity \cite{AssadiB19}, stochastic matchings \cite{AssadiB19}, and (random order) streaming \cite{bernsteinstreaming2020,AssadiB21}. In this work, we use it for the first time in the sublinear time model. In particular, both \cref{thm:2/3,thm:beating-2/3} build on EDCS. For a parameter $\beta$ (think of it as a large constant), a subgraph $H$ of a graph $G$ is a $\beta$-EDCS of $G$ if $(P1)$ all edges $(u, v)$ in $H$ satisfy $\deg_H(u) + \deg_H(v) \leq \beta$ and $(P2)$ all edges $(u, v) \in G \setminus H$ satisfy $\deg_H(v) + \deg_H(v) \geq (1-\epsilon)\beta$. The main property of EDCS, proved first by \cite{bernsteinstein2015,bersteinsteingeneral} and further strengethened in \cite{AssadiB19,BehnezhadEdmondsGallai}, is that for any $\beta = \Omega(1/\epsilon)$, any $\beta$-EDCS includes a $(1-O(\epsilon))2/3$ approximate maximum matching of its base graph $G$. It is not hard to see that {\em finding} any $\beta$-EDCS of the whole input graph $G$ requires $\Omega(n^2)$ queries.\footnote{In fact, finding the edges of any constant approximate matching requires $\Omega(n^2)$ time. Our focus throughout this paper is on estimating only the size of the maximum matching.} Therefore, instead, we first sub-sample some $p=1/n^{\delta}$ fraction of the edges (or pairs in the adjacency matrix model) of $G$ for some fixed $\delta > 0$ in $O(n^2 p) = n^{2-\Omega(1)}$ time and construct an EDCS $H$ over those edges. Building on an approach of \citet*{bernsteinstreaming2020} in the random-order streaming setting, this can be done in a way such that at most $\widetilde{O}(\mu(G) / p ) = n^{2-\Omega(1)}$ edges in the whole graph $G$ remain {\em underfull}, i.e., those that do not satisfy property $(P2)$. The union of the set $U$ of underfull edges and the EDCS $H$ can be shown to include an (almost) 2/3-approximation. The next challenge is that we do not have the set $U$. However, to estimate the maximum matching of $H \cup U$, it suffices to design an oracle that upon querying a vertex $v$, determines whether it belongs to an approximately optimal maximum matching of $H \cup U$ and run this oracle on a few randomly sampled vertices. To do so, we build on a local computation algorithm (LCA) of \citet*{levironitt} (which itself builds on a result of \citet*{YoshidaYISTOC09}) that takes $\poly_\epsilon(\Delta)$ time to return if a given vertex $v$ belongs to some $(1-\epsilon)$-approximate matching of its input graph, where here $\Delta$ is the maximum degree. Modifying $H \cup U$ by getting rid of its high-degree vertices, we show the algorithm of \cite{levironitt} can be used to $(1-\epsilon)$-approximate the size of the maximum matching of $H \cup U$ in $O(n^{1+\delta/\epsilon^2})$ time. Picking $\delta$ sufficiently small, we arrive at an algorithm that obtains a $(2/3-\epsilon)$-approximation in $n^{2-\Omega_\epsilon(1)}$ time. \paragraph{Going Beyond $2/3$-Approximations:} It is known that the (almost) $2/3$-approximation analysis for EDCS is tight. That is, there are instances on which the EDCS does not include a better than $2/3$-approximation. However, a characterization of such tight instances of EDCS was recently given in the work of \citet*{behnezhad2022dynamic} that we use in beating $2/3$-approximation. Consider a $\beta$-EDCS $H$, for sufficiently large constant $\beta$, that does not include a strictly better than 2/3 approximation of $(2/3+\delta)$-approximation for some small $\delta > 0$. The characterization divides the vertices into two subsets $V_{mid}$ and $V_{low}$ depending on their degrees in $H$. Then shows that there must be an (almost) 2/3-approximate matching $M$ in the induced bipartite subgraph $G[V_{mid}, V_{low}]$ that is far from being maximal. Namely, it leaves an (almost) 1/3-approximate matching in $G[V \setminus M]$ that can be directly added to $M$. Interestingly, this characterization coincides with our lower bound construction! The set $V_{mid}$, here, corresponds to the set $A \cup B$ in the lower bound and the set $V_{low}$ corresponds to $S$. This essentially reduces the problem to showing that our lower bound construction can indeed be solved in $n^{2-\Omega(1)}$ time for all choices of the degree $d$. Here we show why our lower bound construction fundamentally cannot lead to a better than $n\sqrt{n}$ time lower bound. Indeed the ideas behind our algorithm for \cref{thm:beating-2/3} are very close. First, suppose that $d < \sqrt{n}$. In this case, we can random sample a vertex in $A$ and examine the $A$/$B$ value of each of its $d$ neighbors in total $O(nd) = O(n\sqrt{n})$ time to see if it has any $A$ neighbors, thereby solving the instance. So let us suppose that $d \geq \sqrt{n}$. Now in this case, we can first sample a vertex $v$ in $A$ and list all of its $A \cup B$ neighbors in $O(n)$ time. Since the vast majority (i.e., all but one) of the neighbors of $v$ go to $B$, we have essentially found $d$ vertices in $B$ in $O(n)$ time. Repeating this $\Theta(n/d)$ times, we find all of the $B$ vertices in total $O(n^2/d) = O(n\sqrt{n})$ time. We note that the final running time of our algorithm in \cref{thm:beating-2/3} is only $n^{2-\Omega_\epsilon(1)}$ and not $O(n\sqrt{n})$ since the bottleneck is in finding the EDCS and estimating the maximum matching size of $H \cup U$ as discussed above. For a more detailed high level overview of the algorithm, see \cref{sec:beating}. \section{Preliminaries}\label{sec:prelim} \paragraph{Notation:} Throughout this paper, we let $G = (V, E)$ to denote the input graph. We use $n$ and $m$ to denote the number of vertices and number of edges of the input graph $G$. Also, we use $\Delta$ to show the maximum degree, $\bar{d}$ to show the average degree, and $\mu(G)$ to show the size of the maximum matching of graph $G$. Throughout this paper, for $\alpha \in (0, 1]$ and $\delta \in [0, n]$, we say $\widetilde{\mu}(G)$ is an approximation of $\mu(G)$ up to a multiplicative-additive error of ($\alpha, \delta$), if $\alpha \mu(G) - \delta \leq \widetilde{\mu}(G) \leq \mu(G)$. Also, we use $\widetilde{O}(\cdot)$ to hide $\poly \log n$ factors. \paragraph{Problem Definition:} Given a graph $G$, we are interested in estimating the size of the maximum matching of $G$. The graph is given in one of the following representations: \begin{itemize} \item \textit{Adjacency Matrix:} In this model, the algorithm can query a pair of vertices $(u,v)$. The answer is 1 if there is an edge between $u$ and $v$, and 0 otherwise. \item \textit{Adjacency List:} In this model, a list of neighbors of each vertex is stored in arbitrary order in a list. The algorithm can query the $i$'th element in the list of neighbors of a vertex. The answer is empty if the $i$'th element does not exist. \end{itemize} \paragraph{Random Greedy Maximal Matching:} Let $G = (V, E)$ be the input graph and $\pi$ be a random permutation over all possible permutation of edges $E$. A random greedy maximal matching can be obtained by iterating over $E$ with respect to the permutation $\pi$ and constructing a matching by adding edges one by one if they do not violate the matching constraint. \paragraph{Probabilistic Tools:} In this paper, we use the following standard concentration inequalities. \begin{proposition}[Chernoff Bound]\label{prop:chernoff} Let $X_1, X_2, \ldots, X_n$ be independent Bernoulli random variables, and let $X = \sum_{i=1}^{n} X_i$. For any $t > 0$, $\Pr[|X - \E[X]| \geq t] \leq 2\exp\left(-\frac{t^2}{3\E[X]}\right).$ \end{proposition} \begin{proposition}[Hoeffding's Inequality]\label{prop:hoeffding} Let $X_1, X_2, \ldots, X_n$ be independent random variables such that $a \leq X_i \leq b$. Let $\overline{X} = (\sum_{i=1}^n X_i)/n$. For any $t > 0$, $ \Pr[|\overline{X} - \E[\overline{X}]| \geq t] \leq 2\exp\left(-\frac{2nt}{(b-a)^2}\right). $ \end{proposition} \paragraph{Graph Theory:} We use the following classic theorem by K\"{o}nig's. \begin{proposition}[K\"{o}nig's Theorem] In any bipartite graph, the size of maximum matching is equal to the size of the minimum vertex cover. \end{proposition} \section{The Lower Bound}\label{sec:lb} In this section, we prove the lower bound of \cref{thm:lb}. To prove \cref{thm:lb}, we first give an input distribution in \cref{sec:input-dist}. We then prove that any {\em deterministic} algorithm which with probability at least .51 (taken over the randomization of the input distribution) returns a $(2/3+\alpha)$-approximation of the size of maximum matching of the input must make at least \thelb{} queries to the graph. By the `easy' direction of Yao's minimax theorem \cite{Yao77}, this also implies that any randomized algorithm with success probability .51 over worst-case inputs must make at least $\thelb$ queries to obtain a $(2/3+\alpha)$-approximation. \subsection{The Input Distribution}\label{sec:input-dist} We start by formalizing the input distribution. Let $\epsilon := \alpha/100$, where $\alpha$ is as in \cref{thm:lb}, let $N$ be a parameter which controls the number of vertices and let $d = N^{1/5}$. In our construction, we assume $N$, $\epsilon N$, $d$, and $\epsilon d$ are all integers; note that this holds for infinitely many choices of $N$. For $n := 6N$, we construct an $n$-vertex bipartite graph $G=(V, U, E)$. We first categorize the vertices in $G$, then specify its edges. \paragraph{The vertex set:} The vertex set $V$ is composed of four distinct subsets $A_V, B_V, S_V, T_V$ and similarly $U$ is composed of $A_U, B_U, S_U, T_U$. Throughout, we may write $A, B, S, T$ to respectively refer to sets $A_V \cup A_U$, $B_V \cup B_U$, $S_V \cup S_U$, and $T_V \cup T_U$. In our construction, we will have $$ |A_V| = |A_U| = (1-\epsilon) N, \qquad |B_V| = |B_U| = |S_V| = |S_U| = N, \qquad |T_U| = |T_V| = \epsilon N. $$ We randomly permute all the vertices in $A_V$ and use $v_i(A_V)$ to denote the $i$-th vertex of $A_V$ in this permutation. We do the same for $A_U, B_V, B_U, S_V, S_U$. \paragraph{The edge set:} For the edge-set $E$, we give two distributions: in distribution \yesdist{} the maximum matching of $G$ is sufficiently large, and in distribution \nodist{} the maximum matching of $G$ is sufficiently small. The final input distribution $\dist := (\frac{1}{2} \yesdist + \frac{1}{2} \nodist)$ draws its input from \yesdist{} with probability 1/2 and from \nodist{} with probability 1/2. The following edges are common in both disributions \yesdist{} and \nodist{}: \begin{itemize} \item All vertices in $T_U$ (resp. $T_V$) are adjacent to all of $A_V, B_V, S_V$ (resp. $A_U, B_U, S_U$). \item For any $i \in [N]$, we add edges $(v_i(B_U), v_i(S_V))$ and $(v_i(B_V), v_i(S_U))$ to the graph. In words, there are perfect matchings between $B_U$ and $S_V$ and between $B_V$ and $S_U$. \item Let $\mathcal{R}(B_V, B_U)$ be the set of all $(\epsilon d - 1)$-regular graphs $H$ between $B_V$ and $B_U$ such that for all $i \in [N]$, $(v_i(B_V), v_i(B_U)) \not\in H$. We draw one regular graph $R(B_V, B_U)$ from $\mathcal{D}(B_V, B_U)$ uniformly at random and add its edges to $G$. \item Let $\mathcal{R}(A_V, B_U)$ be the set of all graphs $H$ between $A_V$ and $B_U$ such that $\deg_H(v) = d$ for all $v \in A_V$, $\deg_H(u) = (1-\epsilon)d$ for all $u \in B_U$, and for all $i \in [(1-\epsilon)N]$, $(v_i(A_V), v_i(B_U)) \not\in H$. We draw one regular graph $R(A_V, B_U)$ from $\mathcal{R}(A_V, B_U)$ uniformly at random and add its edges to $G$. \item Let $\mathcal{R}(B_V, A_U)$ be the set of all graphs $H$ between $B_V$ and $A_U$ such that $\deg_H(v) = d$ for all $v \in A_U$, $\deg_H(u) = (1-\epsilon)d$ for all $u \in B_V$, and for all $i \in [(1-\epsilon)N]$, $(v_i(B_V), v_i(A_U)) \not\in H$. We draw one regular graph $R(B_V, A_U)$ from $\mathcal{R}(B_V, A_U)$ uniformly at random and add its edges to $G$. \item For all $i \in \{(1-\epsilon)N+1, \ldots, N\}$ we add edge $(v_i(B_V), v_i(B_U))$ to $G$. \end{itemize} The following edges are specific to \yesdist{} and \nodist{} respectively: \begin{itemize} \item In \yesdist{}, we additionally add edges $(v_i(A_V), v_i(A_U))$ and $(v_i(B_V), v_i(B_U))$ for all $i \in [(1-\epsilon)N]$. \item In \nodist{}, we additionally add edges $(v_i(A_V), v_i(B_U))$ and $(v_i(B_V), v_i(A_U))$ for all $i \in [(1-\epsilon)N]$. \end{itemize} This concludes the construction of graph $G$. We also emphasize that the adjacency list of each vertex includes its neighbors in a random order chosen uniformly and independently. Finally, we note that we give away the bipartition $V, U$ of the graph for free. Additionally, we also give away which of the sets $S, T, A \cup B$ any vertex belongs to. What is crucially hidden from the algorithm, however, is whether a vertex $v \in A \cup B$ belongs to $A$ or $B$. \begin{figure} \centering \includegraphics[scale=0.5]{figs/input-dist-common.pdf} \caption{The common edges in both \yesdist{} and \nodist{} distributions. For simplicity, we have not illustrated the edges of $T_V$ and $T_U$ (which are adjacent to all vertices on the opposite side). See \cref{fig:dist_yesno} for edges specific to the two distributions.} \label{fig:dist_common} \end{figure} \begin{figure} \centering \includegraphics[scale=0.4]{figs/input-dist-yesno.pdf} \caption{In addition to the common edges in distributions \yesdist{} and \nodist{} that were illustrated in \cref{fig:dist_common}, we have a special perfect matching between vertices $A_V \cup B_V$ and $A_U \cup B_U$. In distribution \yesdist{}, this perfect matching matches all of $A_V$ to $A_U$. But in distribution \nodist{}, none of the edges of this matching go from $A_V$ to $A_U$. This ensures that the maximum matching of $G$ in the \yes{} case is (almost) $1.5$ times that of $G$ in the \no{} case.} \label{fig:dist_yesno} \end{figure} \subsection{Basic Properties of the Input Distribution} The following bounds on vertex degrees follows immediately from the construction. \begin{observation}\label{obs:degrees} For any graph $G$ drawn from $\yesdist$ or $\nodist$ with probability 1 it holds that: \begin{enumerate} \item $\deg(v) = \epsilon N + d + 1$ for all $v \in A_V, A_U, B_V, B_U$. \item $\deg(v) = \epsilon N + 1$ for all $v \in S_V, S_U$. \item $\deg(v) = (3-\epsilon)N$ for all $v \in T_V, T_U$. \end{enumerate} \end{observation} Since for any vertex $v$, we know which of the sets $S_V, S_U, T_V, T_U, A_V \cup B_V, A_U \cup B_U$ it belongs to, \cref{obs:degrees} implies we know the degree of every vertex in the graph before making any queries. Thus, there is no point for the algorithm to make any degree queries. \begin{lemma} Let $G_\yes \sim \yesdist$ and $G_\no \sim \nodist$. Then it holds with probability 1 that $$ \mu(G_\yes) \geq (3-\epsilon) N \qquad \text{and} \qquad \mu(G_\no) \leq (2+2\epsilon) N.$$ This, in particular, implies that any algorithm satisfying the condition of \cref{thm:lb} must be able to distinguish whether a graph $G$ drawn from distribution $(\frac{1}{2}\yesdist + \frac{1}{2} \nodist{})$ belongs to the support of \yesdist{} or \nodist{} with probability at least $.51$. \end{lemma} \begin{proof} For $G_{\yes}$, we take edges $(v_i(A_V), v_i(A_U))$ for $1\leq i \leq (1-\epsilon)N$, $(v_i(B_V), v_i(S_V))$ for $1\leq i \leq N$, and $(v_i(B_U), v_i(S_U))$ for $1\leq i \leq N$. Since none of the edges have the same endpoint, the union creates a matching with size $(3-\epsilon)N$ which implies $\mu(G_{\yes}) \geq (3 - \epsilon) N$. For $G_{\no}$, we take $B_U \cup B_V \cup T_U \cup T_V$ as a vertex cover. Note that there is no edge in the induced subgraph between vertices of $A_U \cup A_V \cup S_U \cup S_V$. The proof follows by K\"{o}nig's Theorem since there exists a vertex cover with $(2 + 2\epsilon)N$ vertices. Furthermore, if $\alpha > 0$ be a constant, since $(2+2\epsilon) / (3-\epsilon) \leq 2/3 + \alpha$ (if we choose $\epsilon$ sufficiently smaller than $\alpha$), any algorithm satisfying the condition of \cref{thm:lb}, must be able to distinguish whether a graph $G$ drawn from distribution $(\frac{1}{2}\yesdist + \frac{1}{2} \nodist{})$ belongs to the support of \yesdist{} or \nodist{} with probability at least $.51$. \end{proof} \begin{remark} Note that our construction can be slightly modified to have a perfect matching in \yesdist{}. We can add a perfect matching between vertices of $T_U$ and $T_V$ to have a matching of size $3N$. Therefore, our lower bound also holds for the case that the \yesdist{} has a perfect matching. \end{remark} \subsection{Queried Edges Form a Rooted Forest} Here and throughout the rest of the paper, we use $G'$ to denote the induced subgraph of $G$ excluding all the dummy vertices in $T_U \cup T_V$. Our main result of this section is that any algorithm that makes at most $\K{}$ queries to graph $G$, only discovers a rooted forest in $G'$: \begin{lemma}\label{lem:forest} Let $\mc{A}$ be any algorithm making at most $K = \K{}$ queries to graph $G \sim \dist$. Let $F_0$ be the empty graph on the vertex set of $G'$, and for $t \geq 1$ let $F_t$ be the subgraph of $G'$ that $\mc{A}$ discovers after $t$ queries. The following property holds throughout the execution of $\mc{A}$ with probability $1-o(1)$: For any $t$, if the $t$'th query is to the adjacency list of a vertex $v$ and returns edge $(v, u)$ then either $(v, u) \not\in G'$ or if $(v, u) \in G'$ then $u$ is singleton in $F_{t-1}$. Equivalently, this implies that each $F_t$ can be thought of as a rooted forest where $F_t \setminus F_{t-1}$ may only include one edge $(v, u)$ and if $\mc{A}$ discovered $(v, u)$ by querying $v$, then $u$ becomes a leaf of $v$ in $F_t$. \end{lemma} \begin{remark} Our proof of \cref{lem:forest} crucially relies on the fact that the adjacency list of each vertex is randomly permuted in the input distribution. However, we emphasize here that our proof holds even if the internal permutation of the $G'$-neighbors of a vertex $v$ in the adjacency list of $v$ is adversarial. This implies, in particular, that if we condition on the high probability event of \cref{lem:forest}, the internal permutation of the $G'$-neighbors will still be uniform. \end{remark} To prove \cref{lem:forest}, we first bound the number of edges of $G'$ that we see with $K = \K{}$ queries. \begin{claim}\label{clm:few-edges} Any algorithm $\mc{A}$ that makes $K = \K{}$ queries to $G$, discovers at most $o(n^{2/5})$ edges of $G'$ with probability $1-1/\poly(n)$. \end{claim} \begin{proof} Since we assume $\mc{A}$ makes at most $K = o(n^{6/5})$ queries, there will be at most $o(n^{1/5})$ vertices for which the algorithm makes more than $\epsilon N/2 = \Omega(n)$ adjacency list queries. For each of these $o(n^{1/5})$ vertices, we assume that we discover all their neighbors in $G'$ that is at most $d+1 = O(n^{1/5})$, which in total is $o(n^{2/5})$ edges. Now let $V'$ be the set of vertices for which $\mc{A}$ makes at most $\epsilon N /2$ queries. For each new query to a vertex $v \in V'$, since there are $\epsilon N$ edges to $T_U \cup T_V$ in $G$ and that we have already made at most $\epsilon N/2$ queries to $v$, the new query goes to $G'$ with probability at most $\deg_{G'}(v)/(\deg_{G'}(v) + \epsilon N - \epsilon N/2) = O(d/n)$. Next, let $X_i$ be the indicator of the event that the $i$'th query to $V'$ discovers a $G'$ edge. From our earlier discussion, we get $\E[X_i] = O(d/n)$. Moreover, these $X_i$'s are negatively correlated. Hence, denoting $X = \sum X_i$, we get $\E[X] \leq O(Kd/n)$ and can apply the Chernoff bound to obtain that with probability at least $1-1/\poly(n)$, $$ X \leq \E[X] + O(\sqrt{\E[X] \log n}) = O(Kd/n) = o(n^{2/5}), $$ concluding the proof. \end{proof} Next, we show that conditioning on the fact that we can discover at most $o(n^{2/5})$ edges, the probability of having an edge between any pairs of vertices is at most $O(1/n^{4/5})$. To give an intuition of why this claim holds, assume that instead of the regular graphs between $B_U$ and $B_V$ (similarly between $A_U$ and $B_V$, and $A_V$ and $B_U$), we have an Erdos-Renyi with the same expected degree as the regular graphs. Since the degree of each regular graph is $O(d)$, then the probability of having an edge between a pair of vertices must be $O(d/n) = O(1/n^{4/5})$ since the existences of edges are independent. However, since we draw a random $O(d)$-regular graph, edge realizations are not independent the same argument does not work. But using careful coupling, we prove that the same claim holds for this construction. \begin{claim}\label{clm:no-edge-between-edge} Let us condition on the high probability event of \cref{clm:few-edges} that the algorithm has discovered at most $o(n^{2/5})$ edges in $G'$. Then for any pair of vertices $u, v$ such that $(u, v)$ is not among the discovered edges, the probability that $(u, v)$ is an edge in $G'$ is at most $O(1/n^{4/5})$. \end{claim} \begin{proof} Let $A = A_U \cup A_V$ and $B = B_U \cup B_V$. Also, assume that by index of a vertex we mean the index in the permutation of its corresponding subset in the construction. There are three possible cases for the type of subsets that $u$ and $v$ belong to: \paragraph{(Case 1) $u, v \in A$:} note that since the algorithm makes at most $o(n^{6/5})$ queries, using Chernoff bound, it is easy to see that at least a constant fraction of perfect matching edges in the induced subgraph of $A$ remains undiscovered. If $u$ and $v$ both belong to $A$, the probability of having an edge between the pair is at most $O(1/n)$ since we only have a perfect matching between vertices with the same indices in $A_U$ and $A_V$ for $\mc{D}_{\yes}$, and vertices are randomly permuted. For the other two cases, we bound the probability of having an edge between $u$ and $v$ by considering two possible scenarios. First, the probability that both $u$ and $v$ have the same indices is equal to $1/n$. Next, assume that $u$ and $v$ have different indices. Let $\mc{G}_e$ be the set of all graphs in $\mc{D}$ such that all of them have the edge $e = (u,v)$. Also, let $\overline{\mc{G}}_e$ be the set of all graphs in $\mc{D}$ with no edge between pair $(u,v)$. In the next two cases, we use coupling to show that conditioned on discovered edges, we have $|\mc{G}_e| / |\overline{\mc{G}}_e| \leq O(1/n^{4/5})$ which implies that the probability of having an edge for a pair $(u, v)$ is at most $O(1/n^{4/5})$. \paragraph{(Case 2) $u, v \in B$:} without loss of generality, assume that $u \in B_U$ and $v \in B_V$. First, note that with a probability of $O(1/n)$, the index of $u$ and $v$ is the same, which implies that there is an edge between $u$ and $v$. Let $i_x$ denote the index of vertex $x$. Now assume that $i_u \neq i_v$. We claim if there exists edge $(u,v)$, then there are at least $N\epsilon d /2$ edges $(w,z)$ such that $z \in B_U$, $w \in B_V$, $\{i_u, i_v, i_w, i_z \} = 4$, and the induced subgraph of these four vertices only has two edges $(u,v)$ and $(w, z)$. Since the number of incident edges of $u$ to $B_V$ is at most $\epsilon d$, then there are at least $N - \epsilon d$ vertices in $B_V$ that are not connected to $u$. Also, we exclude the vertex with index $i_u$ in $B_V$ from this set. Let $Q$ be the set of such vertices in $B_V$. Hence, $|Q| \geq N -\epsilon d - 1$. Furthermore, by the construction, each vertex $w \in Q$ has at least $\epsilon d - 3$ neighbors in $B_U$ with an index not in $\{i_u, i_v, i_w\}$. Hence, there are at least $(\epsilon d - 3)(N -\epsilon d - 1)$ edges between vertices of $Q$ and $B_U$ with endpoints having different labels. Moreover, at most $\epsilon^2 d^2$ of these edges are incident to an edge that one of its endpoints is $v$. Therefore, there are at least $(\epsilon d - 3)(N -\epsilon d - 1) - \epsilon^2 d^2 \geq N\epsilon d /2$ edges $(w,z)$ that satisfy the claimed properties where the inequality follows by the choice of $d$. Now assume that we remove all edges that we discovered so far. Since we discover at most $o(n^{2/5})$ edges, by removing these edges we still have $O(n\epsilon d)$ edges $(w,z)$ with mentioned properties. Let $(w,z)$ be such an edge where $w \in B_V$ and $z \in B_U$. We construct a graph by removing edge $(u, v)$ and $(z, w)$, and adding edges $(u, w)$ and $(v, z)$. If the original graph is in $\mc{D}_{\yes}$ ($\mc{D}_{\no}$), the new graph is in $\mc{D}_{\yes}$ ($\mc{D}_{\no}$) according to the construction since we only change the random regular bipartite graph between $B_U$ and $B_V$ without changing the degrees. We construct a bipartite graph where each vertex of the first part represents a graph in $\mc{G}_e$ and each vertex in the second part represents a graph in $\overline{\mc{G}}_e$. For each vertex in the first part, we connect it to the vertices of the second part that can be produced by the above operation. Thus, the degree of vertices in the first part is at least $O(n\epsilon d)$. On the other hand, each vertex in the second part is connected to at most $O(\epsilon^2 d^2)$ vertices of the first part since the degree of each $u$ and $v$ is at most $\epsilon d$ in the induced subgraph of $B$. Counting the edges from both sides of the constructed bipartite graph yields \begin{align*} \frac{|\mc{G}_e|}{|\overline{\mc{G}}_e|} \leq \frac{O( \epsilon^2 d^2)}{O(N\epsilon d)} = O\left(\frac{\epsilon d}{N}\right) = O_\epsilon\left(\frac{1}{n^{4/5}}\right). \end{align*} \paragraph{(Case 3) $u \in A, v \in B$ or $u \in B, v \in A$:} without loss of generality, assume that $u \in A_U$ and $v \in B_V$. With the same argument as the previous case, the probability that both $u$ and $v$ have the same index is $O(1/n)$. Moreover, since the degree of the induced subgraph of $A_U \cup B_V$ differs from the previous case by a constant factor, with the same reasoning as the previous part, the probability that there exists an edge between $u$ and $v$ is at most $O(1/n^{4/5})$. Therefore, conditioning on the discovered edges, the probability of having an edge $(u,v)$, is at most $O(1/n^{4/5})$ \end{proof} Now we are ready to complete the proof of \Cref{lem:forest}. \begin{proof}[Proof of \Cref{lem:forest}] First, we prove that for any $t \geq 1$, if two vertices $u, v$ are non-singleton in $F_t$ and $(u, v) \not\in F_t$, then there is no edge between $u$ and $v$ in graph $G'$. This will imply that any time that we discover a new edge of a non-singleton vertex in $F_t$, it must go to a vertex that is singleton in $F_t$. We prove this by induction on $t$. For $F_0$, the graph is empty and so the claim holds. Suppose that there are no undiscovered edges among the non-singleton vertices of $F_{t-1}$. We prove that this continues to hold for $F_t$. If $F_t \setminus F_{t-1} = \emptyset$, i.e., if we do not discover any new edge of $G'$ at step $t$, then the claim clearly holds. So suppose that we query some vertex $v$ and find an edge $(v, u)$ at step $t$. It suffices to show that in this case, $u$ will not have any non-singleton neighbors in $F_{t-1}$ other than $v$. To see this, recall by \Cref{clm:few-edges} that there are at most $o(n^{2/5})$ non-singleton vertices in $F_{t-1}$. Fixing any such vertex $w$ with $w \not= v$, we get by \cref{clm:no-edge-between-edge} that the conditional probability of $(u, w)$ being an edge in $G'$ is at most $O(1/n^{4/5})$. By a union bound over all $o(n^{2/5})$ choices of $w$, we get that the probability of $u$ having an edge to any of the non-singleton vertices of $F_{t-1}$ is $o(n^{2/5}) \cdot O(1/n^{4/5}) = o(1/n^{2/5})$. Finally, since by \cref{clm:few-edges} there are at most $O(n^{2/5})$ steps where we discover any edge of $G'$, the failure probability over all the steps of the induction is $O(n^{2/5}) \cdot o(1/n^{2/5}) = o(1)$. Hence, the claim is true throughout with probability at least $1-o(1)$. We proved above that by querying an already non-singleton vertex $v$, its discovered neighbor $u$ must be singleton w.h.p. We will now prove that this also holds if $v$ itself is singleton. As discussed, the number of non-singleton vertices in $F_t$ is $o(n^{2/5})$ by \cref{clm:few-edges}. For each of these vertices $u$, the conditional probability of $v$ having an edge to $u$ is at most $O(1/n^{4/5})$ by \cref{clm:no-edge-between-edge}. Hence, the expected number of edges of $v$ to non-singleton vertices is $o(1/n^{2/5})$. Since $v$ has $\Omega(n)$ edges in $G$ and its adjacency list is uniformly sorted, the probability of discovering one of such edges of $v$ is at most $o(1/n^{2/5}) / n = o(1/n^{7/5})$. A union bound over all $o(n^{6/5})$ queries of the algorithm implies that with probability $1-o(1)$, any time that we query a singleton vertices of $F_t$, its discovered edge will not be to another non-singleton vertex of $F_t$. \end{proof} \subsection{The Tree Model} In our proofs, we condition on the high probability event of \cref{lem:forest} that $F_t$ for any $t$ forms a rooted forest. Under this event, we prove that the algorithm cannot distinguish the \yes{} distribution from the \no{} distribution. Recall that from our definition in \cref{lem:forest}, every vertex in $F_t$ for any $t$ is a vertex in set $A \cup B \cup S$. Given the forest $F_t$, we do not necessarily know for a given a vertex $v$ in $F_t$ whether it belongs to $A$ or $B$ (but if it belongs to $S$ we know this since recall all the $S$ vertices are given for free by the distribution). Inspired by this, we can see each vertex $v$ in $F_t$ as a random variable taking one of the values $\{A, B, S\}$. Importantly, depending on the value of $v$ we have a different distribution on what values its children in the tree will take. This distribution is also different for the \yes{} and \no{} distributions. For example, in the \no{} distribution all children of any vertex $v \in A$ will be $B$ vertices whereas in the \yes{} distribution there is a small chance of seeing an $A$ child. We prove that although these distributions are different for the \yes{} and \no{} distributions, no algorithm can distinguish (with a sufficiently large probability) whether the observed forest $F_t$ was drawn from \yesdist{} or \nodist{}. The proof consists of multiple steps. First, in Section~\cref{sec:cor-decay}, we prove a correlation decay property on the tree. Then equipped with the correlation decay property, we are able to show the probability of seeing the same forest by the algorithm in a \yesdist{} and \nodist{} is almost the same which implies that the algorithm cannot decide if the graph is drawn from \yesdist{} or \nodist{}. \subsection{Correlation Decay}\label{sec:cor-decay} \cref{lem:cor-decay} below is our main result in this section. Before stating the lemma, let us start with some definitions. Consider a tree $T(v)$ of any arbitrary depth rooted at vertex $v$. We say $T(v)$ is $\ell$-$S$-free if no vertex in $T(v)$ with distance at most $\ell$ from the root belongs to $S$. We also use $Q(T(v), \ell)$ to denote the total number of vertices of distance at most $\ell$ from the root $v$ in $T(v)$. With a slight abuse of notation, we may also use $T(v)$ to denote the event that the sub-tree of $F_t$ rooted at vertex $v$ by the end of the algorithm (i.e., when $t = K$) is exactly the same as $T(v)$. \begin{lemma}\label{lem:cor-decay} Let $v$ be any vertex, $\ell \geq 20(\log^2 n)/\epsilon$, and let $T(v)$ be any $\ell$-$S$-free tree. Let $P(v)$ and $P'(v)$ each be an arbitrary outcome of $v$ from $\{A, B\}$ and the entire forest $F_t(v)$ excluding the sub-tree of $v$. Letting $Q := Q(T(v), \ell)$ and assuming that $Q = o(d)$, we have $$ \Pr_{\yesdist{}}[T(u_{\ell'}) \mid P(v)] \leq \Big(1+\frac{1}{d}\Big)^{O(Q)}\Pr_{\yesdist{}}[T(u_{\ell'}) \mid P'(v)], $$ and $$ \Pr_{\nodist{}}[T(u_{\ell'}) \mid P(v)] \leq \Big(1+\frac{1}{d}\Big)^{O(Q)} \Pr_{\nodist{}}[T(u_{\ell'}) \mid P'(v)]. $$ \end{lemma} \begin{proof} Call a vertex $u \in T(v)$ an {\em internal} vertex if it has distance at most $\ell$ from $v$. Note that $Q$ is the number of internal vertices in $T(v)$. We say a subset of internal vertices $u_1, \ldots, u_k$ form an {\em internal path} in a tree $T$ if for any $1 \leq i \leq k-1$, $u_{i}$ is the parent of $u_{i+1}$ in $T$, and additionally each $u_i$, for $i \in [k]$, only has one child in $T$. We use the following two auxiliary \cref{cl:internal-paths,cl:cor-decay-on-paths} to prove \cref{lem:cor-decay}. \begin{claim}\label{cl:internal-paths} Suppose that every vertex in $T(v)$ has at least one non-internal descendant. Then $T$ must have an internal path of length at least $10\log n/\epsilon$. \end{claim} \begin{myproof} Suppose for contradiction that $T(v)$ does not have any internal path of length $\ell'$. Construct a tree $T'(v)$ by contracting all maximal internal paths of $T(v)$ (of any length). Since every vertex in $T(v)$ has a non-internal descendant, which is of distance at least $\ell$ from the root by definition, and every contracted path has a length less than $\ell'$, every vertex in $T'(v)$ must have a descendant of distance at least $\ell/\ell'$ from the root. Furthermore, every vertex in $T'(v)$ must have at least two children. As such, we get that $T'(v)$ must have $2^{\ell/\ell'} > n$ vertices, a contradiction. \end{myproof} The proof of \cref{cl:cor-decay-on-paths} below is involved; we state it here and present the proof in \cref{sec:cor-decay-paths}. \begin{claim}\label{cl:cor-decay-on-paths} Let $(u_1, \ldots, u_{\ell'})$ be an internal path in $T(v)$ of length $\ell' \geq 10\log n / \epsilon$ such that each $u_i$ is the parent of $u_{i+1}$. Let $T(u)$ be the sub-tree of $T(v)$ rooted at vertex $u$ and let $P(u_1)$ and $P'(u_1)$ be any two events on the vertices outside $u_1$'s sub-tree. Then $$ \Pr_{\yesdist{}}[T(u_{\ell'}) \mid P(u_1)] \leq \Big(1+\frac{1}{d}\Big)^{O(\ell')}\Pr_{\yesdist{}}[T(u_{\ell'}) \mid P'(u_1)], $$ and $$ \Pr_{\nodist{}}[T(u_{\ell'}) \mid P(u_1)] \leq \Big(1+\frac{1}{d}\Big)^{O(\ell')} \Pr_{\nodist{}}[T(u_{\ell'}) \mid P'(u_1)]. $$ \end{claim} We are now ready to present the proof of \cref{lem:cor-decay}. Given the tree $T(v)$, to measure the probability of $T(v)$ actually happening we start by querying from $v$ the same tree topology. That is, if some vertex $u$ has $x$ children in $T(v)$, we reveal $x$ children of $u$ from the distribution and measure the probability that the resulting tree is exactly the same as $T(v)$. First, let us measure the probability that the resulting tree is indeed $\ell$-$S$-free. That is, no internal vertex in the queried subtree is an $S$ vertex. To do this, take an internal vertex $u$ in $T(v)$. Conditioned on either of $P(u_1)$ or $P'(u_1)$, the probability of $u$ being an $S$ vertex is at most $O(1/d)$ since every vertex in $G'$ has at most one $S$ neighbor, has degree $d+1$, and its adjacency list is uniformly sorted. On the other hand, since the total number of internal vertices in $T(v)$ is $Q$, by a union bound, the probability of seeing at least one $S$ vertex is at most $O(Q/d)$. Let us condition on the event that we see no internal $S$ vertex. This only multiplies the final probability of $T(v)$ occuring by some $1 \pm O(Q/d) \leq (1 \pm 1/d)^{O(Q)}$ factor. Conditioned on the observed tree being $\ell$-$S$-free, note that if some internal vertex $u$ in the tree has no non-internal descendants, then indeed its sub-tree exactly matches $T(v)$. So let us peel off all such sub-trees, arriving at a tree where every internal vertex has at least one non-internal descedntant. From \cref{cl:internal-paths}, we get that there must be a an internal path $u_1, \ldots, u_{\ell'}$ of length at least $\ell' \geq 10\log n / \epsilon$ in this tree. Applying \cref{cl:cor-decay-on-paths}, the probability of $T(u_{\ell'})$ happening remains the same up to a factor of $(1 + 1/d)^{O(\ell')}$, no matter what we condition on above $u_1$. Thus, we can peel off the whole sub-tree of $u_{1}$ and repeat. Assuming that we repeat this process $K$ times until we peel off all vertices, we get that the probability of $T(v)$ occuring under $P(v)$ and $P'(v)$ is the same up to a factor of $(1 \pm 1/d)^{O(Q)} \times (1\pm 1/d)^{\ell' K} = (1 \pm 1/d)^{O(Q + \ell'K)}$. Finally, note that $\ell' K = O(Q)$ since every time we peel off the sub-tree of an internal path, we remove $\ell'$ internal vertices in the path, and so $K \times \ell'$ is at most the number of internal vertices. This completes the proof of \cref{lem:cor-decay}. \end{proof} \subsubsection{Correlation Decay on Internal Paths: Proof of \cref{cl:cor-decay-on-paths}}\label{sec:cor-decay-paths} In this section, we present the proof of \cref{cl:cor-decay-on-paths}, which as discussed implies correctness of \cref{lem:cor-decay}. \begin{proof}[Proof of \cref{cl:cor-decay-on-paths}] For any $i \in [\ell']$ we define $$ b_i = \Pr[u_i \in B \mid P(u)], \qquad a_i = \Pr[u_i \in A \mid P(u)]. $$ We claim that for both distributions \yesdist{} and \nodist{}, and for any $i > 1$, \begin{flalign}\label{eq:recursive} b_i \in \left(1 \pm \frac{2}{\epsilon d}\right) \left( \epsilon b_{i-1} + a_{i-1} \right), \qquad a_i \in \left(1 \pm \frac{2}{\epsilon d}\right) (1-\epsilon) b_{i-1} \pm \frac{a_{i-1}}{d}. \end{flalign} To see this, let us first focus on $b_i$. We have \begin{flalign} \nonumber \Pr[u_i \in B \mid P(u)] &= \Pr[u_i \in B \mid u_{i-1} \in B, P(u)] \cdot \Pr[u_{i-1} \in B \mid P(u)]\\ \nonumber &+ \Pr[u_i \in B \mid u_{i-1} \in A, P(u)] \cdot \Pr[u_{i-1} \in A \mid P(u)]\\ &= \Pr[u_i \in B \mid u_{i-1} \in B, P(u)] b_{i-1} + \Pr[u_i \in B \mid u_{i-1} \in A, P(u)] a_{i-1}.\label{eq:hcjj-299283780} \end{flalign} Let us now examine $\Pr[u_{i} \in B \mid u_{i-1} \in B, P(u)]$. Given that $u_{i-1} \in B$, vertex $u_{i-1}$ has exactly $d+1$ neighbors in graph $G'$. Among them, either $\epsilon d$ or $\epsilon d - 1$ neighbors of $u_i$ are in $B$. Additionally, since $u_{i-1}$ has exactly one child (by definition of internal paths) then the conditional event $P(u)$ may only reveal the $A$/$B$-value of one neighbor of $u_{i-1}$, which would be its parent. Since the adjacency list of vertex $u_{i-1}$ is randomly sorted, its discovered child $u_{i}$ belongs to $B$ with probability at least $\frac{\epsilon d - 2}{d} \geq (1 - \frac{2}{\epsilon d}) \epsilon$ and at most $\frac{\epsilon d}{d} = \epsilon$. Thus, $$\Pr[u_{i} \in B \mid u_{i-1} \in B, P(u)] \in (1 \pm \frac{2}{\epsilon d}) \epsilon.$$ On the other hand, since every vertex in $A$ has at least $d$ and at most $d+1$ neighbors in $A$, $$\Pr[u_{i} \in B \mid u_{i-1} \in A, P(u)] \in (1 \pm \frac{1}{d}).$$ Replacing these two bounds in \cref{eq:hcjj-299283780}, we indeed arrive at the recursion of \cref{eq:recursive} for $b_i$. The calculation for $a_i$ is similar. In particular, as in \cref{eq:hcjj-299283780}, we have \begin{flalign} \nonumber \Pr[u_i \in A \mid P(u)] &= \Pr[u_{i} \in A \mid u_{i-1} \in B, P(u)] b_{i-1} + \Pr[u_{i} \in A \mid u_{i-1} \in A, P(u)] a_{i-1}. \end{flalign} We have $\Pr[u_{i} \in A \mid u_{i-1} \in B, P(u)] \in (1 \pm \frac{2}{\epsilon d}) (1-\epsilon)$ since almost $(1-\epsilon)$ fraction of neighbors of each $B$ vertex go to $A$, and we have $\Pr[u_{i} \in A \mid u_{i-1} \in A, P(u)] \leq 1/d$ since each $A$ vertex has at most one $A$ neighbor among its $d+1$ neighbors. Replacing these into the inequality above, implies the recursion of \cref{eq:recursive} for $a_i$. The following claim gives an explicit (i.e., non-recursive) bound for $b_i$ and $a_i$ as a function of just $a_1$ and $b_1$. \begin{claim}\label{cl:explicit} Let $$ a'_i := \frac{(1-\epsilon)^2(a_1+b_1) - (\epsilon - 1)^i (a_1 - (1-\epsilon)b_1) }{(2-\epsilon)(1-\epsilon)}, $$ and $$ b'_i := \frac{(1- \epsilon)(a_1 + b_1) + (\epsilon - 1)^i (a_1 - (1-\epsilon)b_1)}{(2-\epsilon)(1-\epsilon)}. $$ Then for any $i \geq 3$, we have \begin{flalign*} b_i \in \left( 1 \pm \frac{2}{\epsilon d} \right)^{2i} b'_i, \qquad a_i \in \left( 1 \pm \frac{2}{\epsilon d} \right)^{2i} a'_i. \end{flalign*} \end{claim} \begin{proof} First, we note a couple of useful properties of $b'_i$ and $a'_i$ that all can be verified from their definitions: \begin{equation}\label{eq:bpap} a'_1 = a_1, \qquad b'_1 = b_1, \qquad b'_i = \epsilon b'_{i-1} + a'_{i-1}, \qquad a'_i = (1-\epsilon)b'_{i-1}. \end{equation} To prove the stated bounds on $b_i$ and $a_i$ we use induction on $i$. For the base case $i=3$, directly applying \cref{eq:recursive} we get \begin{flalign*} b_3 &\in \left(1 \pm \frac{2}{\epsilon d} \right) (\epsilon b_{2} + a_{2})\\ &\in \left(1 \pm \frac{2}{\epsilon d} \right) \left[ \epsilon\left( \left(1 \pm \frac{2}{\epsilon d} \right) (\epsilon b_{1} + a_{1}) \right) + \left(\left(1 \pm \frac{2}{\epsilon d} \right)(1-\epsilon)b_1 \pm \frac{a_1}{d} \right) \right]\\ &\in \left(1 \pm \frac{2}{\epsilon d} \right)^{2} \left[ \epsilon^2 b_1 + \epsilon a_1 + (1-\epsilon) b_1 \pm \frac{a_1}{d}\right]\\ &\in \left(1 \pm \frac{2}{\epsilon d} \right)^{3} \left[ (1 - \epsilon + \epsilon^2) b_1 + \epsilon a_1 \right]\\ &= \left(1 \pm \frac{2}{\epsilon d} \right)^{3} b'_3 \in \left(1 \pm \frac{2}{\epsilon d} \right)^{6} b'_3. \end{flalign*} Similarly, we have \begin{flalign*} a_3 &\in \left( 1 \pm \frac{2}{\epsilon d} \right) (1-\epsilon) b_2 \pm \frac{a_2}{d}\\ &\in \left( 1 \pm \frac{2}{\epsilon d} \right) (1-\epsilon) \left( \left(1 \pm \frac{2}{\epsilon d} \right) (\epsilon b_{1} + a_{1}) \right) \pm \frac{1}{d} \left( \left( 1 \pm \frac{2}{\epsilon d} \right) (1-\epsilon) b_1 \pm \frac{a_1}{d} \right)\\ &\in \left( 1 \pm \frac{2}{\epsilon d} \right)^3 (1-\epsilon) \left( \epsilon b_1 + a_1 \right)\\ &= \left( 1 \pm \frac{2}{\epsilon d} \right)^3 a'_3 \in \left( 1 \pm \frac{2}{\epsilon d} \right)^6 a'_3. \end{flalign*} We now turn to prove the induction step for $i$, assuming that it holds for $i-1$. Let us start with $b_i$. We have \begin{flalign*} b_i &\in \left(1 \pm \frac{2}{\epsilon d} \right) (\epsilon b_{i-1} + a_{i-1}) \tag{From \cref{eq:recursive}.}\\ &\in \left(1 \pm \frac{2}{\epsilon d} \right) \left( \epsilon \left(1 \pm \frac{2}{\epsilon d} \right)^{2(i-1)} b'_{i-1} + \left(1 \pm \frac{2}{\epsilon d} \right)^{2(i-1)} a'_{i-1} \right) \tag{By the induction hypothesis.}\\ &\in \left(1 \pm \frac{2}{\epsilon d} \right)^{2i} \left( \epsilon b'_{i-1} + a'_{i-1} \right)\\ &= \left(1 \pm \frac{2}{\epsilon d} \right)^{2i} b'_i. \tag{By \cref{eq:bpap}.} \end{flalign*} Moreover, for $a_i$ we have \begin{flalign*} a_i &\in \left(1 \pm \frac{2}{\epsilon d}\right) (1-\epsilon) b_{i-1} \pm \frac{a_{i-1}}{d} \tag{From \cref{eq:recursive}.}\\ &\in \left(1 \pm \frac{2}{\epsilon d}\right)^{2(i-1)+1} (1-\epsilon) b'_{i-1} \pm \left(1 \pm \frac{2}{\epsilon d}\right)^{2(i-1)} \frac{a'_{i-1}}{d} \tag{By the induction hypothesis.}\\ &\in \left(1 \pm \frac{2}{\epsilon d}\right)^{2i} (1-\epsilon) b'_{i-1} \tag{Since $a'_{i-1} \leq b'_{i-1}/\epsilon$ for $i \geq 4$.}\\ &\in \left(1 \pm \frac{2}{\epsilon d}\right)^{2i} a'_i. \tag{By \cref{eq:bpap}.} \end{flalign*} This completes the proof of \cref{cl:explicit}. \end{proof} Observe that $a_1 + b_1 = 1$ since $u_1 \not\in S$ and so either $u_1 \in A$ or $u_1 \in B$. Combined with \cref{cl:explicit}, we get that \begin{flalign*} b_{\ell'} &\in \left( 1 \pm \frac{2}{\epsilon d} \right)^{2\ell'} \frac{(1- \epsilon)(a_1 + b_1) + (\epsilon - 1)^{\ell'} (a_1 - (1-\epsilon)b_1)}{(2-\epsilon)(1-\epsilon)}\\ &\in \left( 1 \pm \frac{2}{\epsilon d} \right)^{2\ell'} \frac{1-\epsilon + (\epsilon - 1)^{10\log n /\epsilon}(a_1 - (1-\epsilon)b_1)}{(2-\epsilon)(1-\epsilon)} \tag{Since $\ell' \geq 10\log n \epsilon$.}\\ &\in \left( 1 \pm \frac{2}{\epsilon d} \right)^{2\ell'+1} \frac{1-\epsilon}{(2-\epsilon)(1-\epsilon)}\\ &= \left( 1 \pm \frac{1}{d} \right)^{O(\ell')} \frac{1}{2-\epsilon}. \end{flalign*} Observe that the conditioning $P(u_1)$ determines the values of $a_1$ and $b_1$. However, for large enough $\ell'$, as we see above, the dependence of $b_{\ell'}$ on $a_1$ and $b_1$ vanishes. In other words, by changing the conditioning $P(u_1)$ the value of $b_{\ell'}$ for $\ell' \geq 10\log n / \epsilon$ only changes by a $(1 \pm 1/d)^{O(\ell')}$ factor. The same also holds for $a_{\ell'}$. As such, whether vertex $u_{\ell'}$ belongs to $A$ or $B$ is essentially independent of the events above the root vertex $u_1$. To see why this implies \cref{cl:cor-decay-on-paths}, take for example the \yesdist{} distribution and note that \begin{flalign*} \Pr_{\yesdist{}}[T(u_{\ell'}) \mid P(u_1)] &= \Pr_{\yesdist{}}[u_{\ell'} \in A \mid P(u_1)] \cdot \Pr_{\yesdist{}}[T(u_\ell') \mid u_{\ell'} \in A]\\ &+ \Pr_{\yesdist{}}[u_{\ell'} \in B \mid P(u_1)] \cdot \Pr_{\yesdist{}}[T(u_\ell') \mid u_{\ell'} \in B]\\ &= a_{\ell'} \Pr_{\yesdist{}}[T(u_\ell') \mid u_{\ell'} \in A] + b_{\ell'} \Pr_{\yesdist{}}[T(u_\ell') \mid u_{\ell'} \in B]. \end{flalign*} Since $a_{\ell'}$ and $b_{\ell'}$ remain the same up to a $(1 \pm 1/d)^{O(\ell')}$ factor by changing the conditioning $P(u_1)$ to $P'(u_1)$, we arrive at the desired inequality of \cref{cl:cor-decay-on-paths} for the \yesdist{} distribution. The proof for \nodist{} is exactly the same. \end{proof} \subsection{Limitation of the Algorithm} Let us define $E_A$ to be the set of all discovered edges of $G[A]$ by the algorithm. Also, let $V_A = \{ v \mid (u, v) \in E_A\}$, where here $(u, v)$ is directed and $u$ is the parent of $v$ in the forest of discovered edges. \begin{claim}\label{clm:few-bs-edges} With high probability, any algorithm $\mc{A}$ that makes at most $o(n^{6/5} / \log^2 n)$ queries, discovers at most $o(d / \log^2 n)$ of edges in subgraph $G'$ with one endpoint in $S$. \end{claim} \begin{proof} The proof is similar to the proof of \Cref{clm:few-edges}. Since we assume $\mc{A}$ makes at most $o(n^{6/5} / \log^2 n)$ queries, there will be at most $o(n^{1/5} /\log^2 n)$ vertices for which the algorithm makes more than $\epsilon N/2 = \Omega(n)$ adjacency list queries. For these vertices, we assume that $\mc{A}$ finds its edge in $G'$ with one endpoint in $S$ (note that the degree of $S$ vertices in $G'$ is one and each vertex of $G'$ is connected to at most one vertex of $S$). Now let $V'$ be the set of vertices for which $\mc{A}$ makes at most $\epsilon N /2$ queries. For each new query to a vertex $v \in V'$, since there are $\epsilon N$ edges to $T_U \cup T_V$ in $G$ and that we have already made at most $\epsilon N/2$ queries to $v$, the new query goes to a vertex in $S$ with probability at most $O(1/n)$. Hence, by applying the Chernoff bound, with probability at least $1-1/\poly(n)$, there are at most $o(d / \log^2 n)$ edges in subgraph $G'$ with one endpoint in $S$. \end{proof} \begin{lemma}\label{lem:always-few-in-subtree} Suppose that the algorithm $\mc{A}$ has made $t$ queries for some $t \leq O(n^{6/5} / \log^2)$. Then with probability of $1 - o(1)$, all the following hold: \begin{enumerate} \item[(i)] $\mc{A}$ has found at most $o(d)$ vertices in total for all subtrees $T(v)$ for $v \in V_A$ up to a distance $20(\log^2 n)/\epsilon$ from root $v$. \item[(ii)] $\mc{A}$ has not found any edge in $G'$ with one endpoint in $S$ in subtree $T(v)$ up to distance $20(\log^2 n)/\epsilon$ from root $v$, for all $v \in V_A$. \item[(iii)] Conditioning on what $\mc{A}$ has queried so far, the probability of each edge in $F_t$ belonging to $G[A]$ is $O(1/d)$. \end{enumerate} \end{lemma} \begin{proof} We prove the lemma using induction. For $t = 0$, trivially the statement is true. Now assume that the lemma holds for $t - 1$ and $\mc{A}$ makes a new query. If the new queried edge is an edge to $T_U \cup T_V$, we are done since none of the conditions in the lemma statement will change. So we assume that the newly queried edge is in $G'$. We prove each of the three claims separately. \paragraph{Induction step for (ii):} if the newly queried edge is between $B$ and $S$, the probability that one of its $20\log^2 n /\epsilon$ most recent predecessors is an edge in $G[A]$ is $O(\log^2 n)/ d)$ using the induction hypothesis (iii). Since $\mc{A}$ can discover at most $o(d / \log^2 n)$ edges in $G'$ with one endpoint in $S$ by \Cref{clm:few-bs-edges}, then with probability at least $1- o(\frac{\log^2 n}{d} \cdot \frac{d}{\log^2 n}) = 1 - o(1)$, the statement of (ii) remains true throughout all the steps of the induction. \paragraph{Induction step for (i):} the probability that at least one of the $20\log^2 n /\epsilon$ most recent predecessors of the newly queried edge is an edge in $G[A]$ is $O(\log^2 n / d)$ using the induction hypothesis (iii). Since $\mc{A}$ discovers at most $o(n^{2/5} / \log^2 n) = o(d^2/\log^2 n)$ edges of $G'$ by \Cref{clm:few-edges}, then with probability at least $1 - o(\frac{\log^2 n }{d^2} \cdot \frac{d^2}{\log^2 n})= 1 - o(1)$, the statement of (i) remains true throughout all the steps of the induction. \paragraph{Induction step for (iii):} let $(u, v)$ be a discovered edge in the forest ($u$ is parent of $v$). First, we condition on $u \in A$, otherwise, the probability of $(u,v)$ being an edge in $G[A]$ is zero. Note that $u$ has at least $d$ neighbors in $G'$ that are not the parent of $u$ in the forest $F_t$. By \Cref{clm:few-bs-edges}, \mc{A} discovers at most $o(d)$ edges in subgraph $G'$ with one endpoint in $S$. Hence, at most $o(d)$ of neighbors of $u$ in $G'$ have an edge with one endpoint in $S$ in their subtree. Furthermore, by \Cref{clm:few-edges}, at least $\Theta(d)$ neighbors of $u$ in $G'$, have at most $o(d)$ vertices in their subtree. Also, note that by proof of \Cref{lem:forest}, neighbors of $u$ in $G'$ that are not adjacent to $u$ in the forest are singleton vertices in the forest. Let $v_1, v_2, \ldots, v_r$ be the union of (1): children of $u$ in the forest such that each of them has no edge with one $S$ endpoint in their subtree up to distance $20 \log^2 n /\epsilon$, and each of them has $o(d)$ vertices in their subtree up to distance $20 \log^2 n /\epsilon$, (2): the neighbors of $u$ in $G'$ that are singleton in the forest. Hence, we have $r = \Theta(d)$. By (i) and (ii), if $v \in A$ then there are at most $o(d)$ vertices in subtree $T(v)$ up to distance $20 \log^2 n /\epsilon$ from root $v$, and there is no edge in subtree $T(v)$ with one endpoint in $S$ up to distance $20 \log^2 n /\epsilon$ from the root. Thus, if $v \notin \{v_1, v_2, \ldots v_r\}$, then $(u,v)$ is not an edge in $G[A]$. Now assume that $v \in \{v_1, v_2, \ldots v_r\}$. Note that in \nodist{}, there is no neighbor of $u$ with label $A$ and in \yesdist{}, there is exactly one neighbor in $A$. Therefore, for \nodist{}, the probability of $v \in A$ is zero. Now assume that the graph is drawn from \yesdist{}. Hence, exactly one of $v_1, v_2, \ldots, v_r$ is in $A$ (if parent of $u$ is $A$ then the probability of $(u, v)$ being in $G[A]$ is zero). We prove that each of $v_i$ can be in $A$ with almost the same probability using a coupling argument. Let $c_i$ be the subset that vertex $v_i$ belongs to ($c_i \in \{A, B\})$. We say $C = (c_1, c_2, \ldots, c_r)$ is a {\em profile} for labels of vertices $v_1, \ldots v_r$. Therefore, exactly one of $c_i$ is equal to $A$ and all others are equal to $B$ which implies that there are $r$ different possible profiles. Assume that $C$ and $C'$ are two different profiles. Let $v_i$ be a vertex with label $A$ in $C$ and $v_j$ be a vertex with label $A$ in $C'$. By \Cref{lem:cor-decay}, the probability of sampling subtree below $v_i$ with label $B$ is the same up to a factor of $\left(1 \pm \frac{1}{d}\right)^{O(Q(T(v_i), \ell))}$. Similarly, the probability of sampling subtree below $v_j$ with label $A$ is the same up to a factor of $\left(1 \pm \frac{1}{d}\right)^{O(Q(T(v_j), \ell))}$. Since $Q(T(v_j), \ell) = o(d)$, then the probability of having profile $C$ and $C'$ is the same up to a $(1 + o(1))$ factor. Therefore, the probability of having one specific $v_i$ to be in $A$ is $O(1/d)$ since $r = \Theta(d)$. \end{proof} \subsection{Indistinguishability of the \yes{} and \no{} distributions} We define a \emph{bad event} to be the event of $\mc{A}$ discovering more than $o(d)$ vertices in total for all subtrees $T(v)$ for $v \in V_A$ up to a distance $20 \log^2 n /\epsilon$ from root $t$, or $\mc{A}$ has found at least one edge in $G'$ with one endpoint in $S$ in subtree $T(v)$ up to distance $20 \log^2 n /\epsilon$ from root $v$ for at least one $v \in V_A$. By \Cref{lem:always-few-in-subtree}, the bad event happens with probability $o(1)$. For the next lemma, we condition on not having a bad event. \begin{lemma}\label{lem:same-distribution} Let us condition on not having the bad event defined above. Let $F$ be the final forest found by algorithm $\mc{A}$ on a graph drawn from \yesdist{} after at most $O(n^{6/5}/\log^2 n)$ queries. Then, the probability of querying the same forest in a graph that is drawn from \nodist{} is at least almost as large, up to $1 + o(1)$ multiplicative factor. \end{lemma} \begin{proof} First, we define $n$ hybrid distributions $\mc{D}_0, \mc{D}_1, \ldots, \mc{D}_{n-1}$ as follows. Distribution $\mc{D}_i$ can be obtained by sampling from \yesdist{} until the $i$-th level in any tree in the forest, and then sampling from \nodist{} below the $i$-th level. Hence, $\mc{D}_0$ is exactly the same as \nodist{} and $\mc{D}_{n-1}$ is the same as \yesdist{}. Our goal is, starting from a forest sampled according $\mc{D}_{n-1} = \yesdist{}$, to inductively show that we can switch from $\mc{D}_i$ to $\mc{D}_{i-1}$ with only negligble total decrease in the probability. To formalize our coupling argument, recall the notion of {\em special edges} from the input distribution: In \yesdist{}, edges in $G[A]$ are special; in \nodist{}, every vertex in $A$ has one special edge to $B$ all forming a matching. We also extend the definition of bad events to \nodist{} and hybrid distribuitons to include bad events in subtrees originating from special edges. Consider the forest $F$ found by algorithm $\mc{A}$ on a graph drawn from \yesdist{}, and let $F^{\le i}$ denote the forest as well as any choice of its special edges in all levels ${\le i}$ (arbitrarily, conditioning on not having a bad event). We compare the probability of seeing $F^{\le i}$ when making the same queries when the oracle samples its answers from $\mc{D}_{i}$ vs $\mc{D}_{i-1}$. We can couple the sampling for the two distributions so that it is identical for everything in levels ${<i}$ (including the choice of special edges), and also for levels $\ge i$ in all the sub-trees that are not descendants of special level-$i$ edges. For each special edge $(u \rightarrow v)$ in the $i$-th level, the label of vertex $v$ is $A$ when sampling from $\mc{D}_{i}$ and $B$ when sampling from $\mc{D}_{i-1}$. Below this vertex (aka levels $>i$), both distributions $\mc{D}_{i}$ and $\mc{D}_{i-1}$ sample according to \nodist{}. Thus, by~\cref{lem:cor-decay}, the probability of sampling the subtree below $v$ is the same regardless of the label of $v$, up to a factor of $\left(1 \pm \frac{1}{d}\right)^{O(Q(T(v_i), \ell))}$. Let $V_{A,i}$ denote the set of vertices pointed to by $i$-th level special edges. By the argument in the previous paragraphs, we have \begin{align}\label{eq:special-edges} \Pr[F^{\le i} | F^{\le i} \sim \mc{D}_{i-1} ] & \geq \Pr[F^{\le i} | F^{\le i} \sim \mc{D}_i] \left(1 - \frac{1}{d}\right)^{\sum_{v\in V_{A,i}} O(Q(T(v), \ell))}. \end{align} Let $\mc{G}^{\le i}$ denote the event of having no bad events corresponding to level-$(\le i)$ special edges. Summing over~\eqref{eq:special-edges} for all valid choices of special edges, we have the following: \begin{align*} \Pr[F \wedge \mc{G}^{\le i-1} | F \sim \mc{D}_{i-1} ] & \geq \Pr[F \wedge \mc{G}^{\le i} | F \sim \mc{D}_{i-1} ] \\ & \geq \Pr[F \wedge \mc{G}^{\le i} | F \sim \mc{D}_i] \cdot \left(1 - \frac{1}{d}\right)^{\sum_{v\in V_{A,i}} O(Q(T(v), \ell))}. \end{align*} We can now bound the total distribution shift across all hybrid steps: \begin{flalign*} \Pr[F | F \sim \mc{D}_{0}] & \geq \Pr[F \wedge \mc{G}^{\le n-1} | F \sim \mc{D}_{n-1}] \cdot \prod_{i=0}^{n-1} \left(1 - \frac{1}{d}\right)^{\sum_{v\in V_{A,i}} O(Q(T(v), \ell))}\\ & = \Pr[F \wedge \mc{G}^{\le n-1} | F \sim \mc{D}_{n-1}] \cdot \left(1 - \frac{1}{d}\right)^{\sum_{v\in V_{A}} O(Q(T(v), \ell))} \\ & \geq \Pr[F \wedge \mc{G}^{\le n-1} | F \sim \mc{D}_{n-1}] \cdot \left(1 - \frac{1}{d}\right)^{o(d)} \tag{By \Cref{lem:always-few-in-subtree}.} \\ & \geq \Pr[F \wedge \mc{G}^{\le n-1} | F \sim \mc{D}_{n-1}] \cdot (1-o(1)). \end{flalign*} This completes the proof of \cref{lem:same-distribution}. \end{proof} \begin{proof}[Proof of \Cref{thm:lb}] Note that the probability of having the bad event that we defined is $o(1)$. Conditioning on not having the bad event, by \Cref{lem:same-distribution}, the distribution of the outcome that the algorithm discovers is in $o(1)$ total variation distance for \yesdist{} and \nodist{}. Therefore, the algorithm is not able to distinguish between the support of two distributions with constant probability. According to our construction, the size of the maximum matching of both \yesdist{} and \nodist{} is $\Theta(n)$, and in both cases, the input graph is bipartite. \end{proof}

BkiUbi3xK0-nUh8iIO3H

\section{Introduction}\label{sec:intro} Reinforcement learning (RL) framework studies the problem of efficient agent-environment interaction, where the agent learns to maximize a given reward function in the long run~\citep{bertsekas1996neuro-dynamic,sutton1998introduction}. At the beginning of the interaction, the agent is uncertain about the environment's dynamics and must \textit{explore} different policies in order to gain information about it. Once the agent is fairly certain, the knowledge about the environment can be \textit{exploited} to compute a good policy attaining a large cumulative reward. Designing algorithms that achieve an effective trade-off between exploration and exploitation is the primary goal of reinforcement learning. The trade-off is commonly measured in terms of {\em cumulative regret}, that is the difference between the rewards accumulated by the optimal policy (which requires exact knowledge of the environment) and those obtained by the learning algorithm. In practice, we often deal with environments with large observation state spaces (e.g., robotics). In this case the regret of standard RL algorithms grows quickly with the size of the observation state space. (We use observation state and observation interchangeably.) Nonetheless, in many domains there is an underlying low dimensional latent space that summarizes the large observation space and its dynamics and rewards. For instance, in robot navigation, the high-dimensional visual and sensory input can be summarized into a 2D position map, but this map is typically unknown. This makes the problem challenging, since it is not immediately clear how to exploit the low-dimensional latent structure to achieve low regret. \textbf{Contributions.} In this paper we focus on rich-observation Markov decision processes (ROMDP\xspace), where a small number of $X$ hidden states are mapped to a large number of $Y$ observations through an injective mapping, so that an observation can be generated by only one hidden state and hidden states can be viewed as clusters. In this setting, we show that it is indeed possible to devise an algorithm that starting from observations can progressively cluster them in ``smaller'' states and eventually converge to the hidden MDP. We introduce \textsc{\small{SL-UCRL}}\xspace, where we integrate spectral decomposition methods into the upper-bound for RL algorithm (UCRL)~\citep{jaksch2010near}. The algorithm proceeds in epochs in which an estimated mapping between observations and hidden state is computed and an optimistic policy is computed on the MDP (called auxiliary MDP) constructed from the samples collected so far and the estimated mapping. The mapping is computed using spectral decomposition of the tensor associated to the observation process. We prove that this method is guaranteed to correctly ``cluster'' observations together with high probability. As a result, the dimensionality of the auxiliary MDP decreases as more observations are clustered, thus making the algorithm more efficient computationally and more effective in finding good policies. Under transparent and realistic assumptions, we derive a regret bound showing that the per-step regret decreases over epochs, and we prove a worst-case bound on the number of steps (and corresponding regret) before the full mapping between states and observations is computed. The regret accumulated over this period is actually constant as the time to correct clustering does not increase with the number of steps $N$. As a result, \textsc{\small{SL-UCRL}}\xspace asymptotically matches the regret of learning directly on the latent MDP. We also notice that the improvement in the regret comes with an equivalent reduction in time and space complexity. In fact, as more observations are clustered, the space to store the auxiliary MDP decreases and the complexity of the extended value iteration step in UCRL decreases from $O(Y^3)$ down to $O(X^3)$ \textbf{Related work.} The assumption of the existence of a latent space is often used to reduce the learning complexity. For multi-armed bandits, \citet{gheshlaghi-azar2013sequential} and \citet{maillard2014latent} assume that a bandit problem is generated from an unknown (latent) finite set and show how the regret can be significantly reduced by learning this set. \citet{gentile2014online} consider the more general scenario of latent contextual bandits, where the contexts belong to a few underlying hidden classes. They show that a uniform exploration strategy over the contexts, combined with an online clustering algorithm achieve a regret scaling only with the number of hidden clusters. An extension to recommender systems is considered in~\citet{gopalan2016low} where the contexts for the users and items are unknown a priori. Again, uniform exploration is used together with the spectral algorithm of~\citet{anandkumar2014tensor} to learn the latent classes. Bartók et al. ~\citep{bartok2014partial} tackles a general case of partial monitoring games and provides minimax regret guarantee which is polynomial in certain dimensions of the problem. The ROMDP\xspace model considered is a generalization of the latent contextual bandits, where actions influence the contexts (i.e., the states) and the objective is to maximize the long-term reward. ROMDPs\xspace have been studied in~\citet{krishnamurthy2016pac} in the PAC-MDP setting and episodic deterministic environments using an algorithm searching the best $Q$-function in a given function space. This result is extended to the general class of contextual decision processes in~\citet{jiang2016contextual}. While the resulting algorithm is proved to achieve a PAC-complexity scaling with the number of hidden states/factors $X$, it suffers from high computations complexity. \citet{ortner2013adaptive} proposes an algorithm integrating state aggregation with UCRL but, while the resulting algorithm may significantly reduce the computational complexity of UCRL, the analysis does not show any improvement in the regret. Learning in ROMDPs\xspace can be also seen as a state-aggregation problem, where observations are aggregated to form a small latent MDP. While the literature on state-aggregation in RL is vast, most of the results have been derived for the batch setting (see e.g.,~\citet{li2006towards}). Finally, we notice that ROMDPs\xspace are a special class of partially observable MDPs (POMDP). \citet{azizzadenesheli2016reinforcement} recently proposed an algorithm that leverages spectral methods to learn the hidden dynamic of POMDPs and derived a regret scaling as $\sqrt{Y}$ using fully stochastic policies (which are sub-optimal in ROMDPs\xspace). While the computation of the optimal memoryless policy relies on an optimization oracle, which in general is NP-hard~\citet{littman1994memoryless,vlassis2012computational,porta2006point,azizzadenesheli2016open,shani2013survey}, computing the optimal policy in ROMDPs\xspace amounts to solving a standard MDP. Moreover, \citet{guo2016pac} develops a PAC-MDP analysis for learning in episodic POMDPs and obtain a bound that depends on the size of the observations. The planning, in general, is computationally hard since it is a mapping from history to action. \section{Rich Observation MDPs} \begin{figure}[t!] \hspace{-0.2in} \centering \begin{psfrags} \psfrag{x1}[][1]{$x_{t}$} \psfrag{x0}[][1]{$x_t$} \psfrag{x2}[][1]{$x_{t+1}$} \psfrag{x3}[][1]{$x_{t+2}$} \psfrag{y0}[][1]{$y_t$} \psfrag{y1}[][1]{$y_{t}$} \psfrag{y2}[][1]{$y_{t+1}$} \psfrag{r0}[][1]{$r_t$} \psfrag{r1}[][1]{$r_{t}$} \psfrag{r2}[][1]{$r_{t+1}$} \psfrag{a0}[][1]{$a_{t}$} \psfrag{a1}[][1]{$a_{t}$} \psfrag{a2}[][1]{$a_{t+1}$} \includegraphics[scale=0.20]{Fig/Plot/temp/GModel.eps} \end{psfrags} \caption{Graphical model of a ROMDP\xspace.} \label{fig:Gmodel} \end{figure} A rich-observation MDP (ROMDP\xspace) (Fig.~\ref{fig:Gmodel}) is a tuple $M=\langle\mathcal{X},\mathcal{Y},\mathcal{A},R, f_T,f_O\rangle$, where $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{A}$ are the sets of hidden states, observations, and actions. We denote by $X$, $Y$, $A$ their cardinality and we enumerate their elements by $i\in[X]=\{1..X\}$, $j\in[Y]=\{1..Y\}$, $l\in[A]=\{1..A\}$. We assume that the hidden states are fewer than the observations, i.e., $X \leq Y$. We consider rewards bounded in $[0,1]$ that depend only on hidden states and actions with a reward matrix $R\in \mathbb{R}^{A\times X}$ such that $[R]_{i,l}=\mathbb{E}[r(x=i,a=l)]$. The dynamics of the MDP is defined on the hidden states as $T_{i',i,l}:=f_T(i'|i,l)=\mathbb P(x'\!\!=\!i'|x\!=\!i,a\!=\!l)$, where $T\in\mathbb{R}^{X\times X\times A}$ is the transition tensor. The observations are generated as $[O]_{j,i}=f_O(j|i)=\mathbb P(y\!=\!j|x\!=\!i)$, where the observation matrix $O\in \mathbb{R}^{Y\times X}$ has minimum \textit{non-zero} entry $O_{\min}$. This model is a strict subset of POMDPs since each observation $y$ can be generated by only one hidden state (see Fig.~\ref{fig:observation.policy}-\textit{left}) and thus $\mathcal X$ can be seen as a non-overlapping clustering of the observations. \begin{figure}[ht] \centering \includegraphics[height=4.cm]{Fig/O_matrix_orig.eps} \hspace{0.5in} \centering \includegraphics[height=4.cm]{Fig/O_matrix_policy.eps} \caption{\textit{(left)} Example of an observation matrix $O$. Since state and observation labeling is arbitrary, we arranged the non-zero values so as to display a diagonal structure. \textit{(right)} Example of clustering that can be achieved by policy $\pi$ (e.g., $\mathcal X_{\pi}^{(a_1)} = \{x_2, x_3\}$). Using each action we can recover \textit{partial} clusterings corresponding to 7 auxiliary states $\S =\{s_1 .. s_7\}$ with clusters $\mathcal Y_{s_1} = \{y_1,y_2\}$, $\mathcal Y_{s_2} = \{y_3,y_4,y_5\}$, $\mathcal Y_{s_3} = \{y_6\}$, and $\mathcal Y_{s_8} = \{y_{10}, y_{11}\}$, while the remaining elements are the singletons $y_6$, $y_7$, $y_8$, and $y_9$. Clusters coming from different actions cannot be merged together because of different labeling of the hidden state, where, e.g., $x_2$ may be labeled differently depending on whether action $a_1$ or $a_2$ is used. } \label{fig:observation.policy} \end{figure} We denote by $\mathcal{Y}_x = \mathcal{Y}_i = \{y=j\in\mathcal{Y}: [O]_{j,i} > 0\}$ the set of observations in cluster $x$, while $x_y = x_j$ is the cluster observation $y=j$ belongs to.\footnote{Throughout the paper we use the indices $i$, $j$, and $l$ and the ``symbolic'' values $x$, $y$, and $a$ interchangeably.} This structure implies the existence of an \textit{observable} MDP $M_{\mathcal Y} = \langle \mathcal {Y}, \mathcal{A}, R', f_T'\rangle$, where $R' = R$ as the reward of an observation-action pair $(y,a)$ is the same as in the hidden state-action pair $(x_y,a)$, and the dynamics can be obtained as $f_T'(j'|j,a) = \mathbb P(y'\!\!=\!j' | y\!=\!j,a\!=\!l) = \mathbb P(y'\!\!=\!j' | x'\!\!=\!x_{j'}) \mathbb P(x'\!\!=\!x_{j'} | x\!=\!x_j, a=l) = [O]_{j',x_{j'}} [T]_{x_{j'},x_j,l}$. We measure the performance of an observation-based policy $\pi_{\mathcal Y}:\mathcal Y\rightarrow \mathcal A$ starting from a hidden state $x$ by its asymptotic average reward $\rho(x; \pi_\mathcal Y) = \lim_{N\rightarrow \infty} \mathbb{E}\big[\sum_{t=1}^N r_t /N \big| x_0 = x, \pi_{\mathcal Y}\big]$. Given the mapping between the ROMDP\xspace to the hidden MDP, the optimal policy $\pi_{\mathcal Y}^*(y)$ is equal to the optimal hidden-state policy $\pi^*_{\mathcal X}: \mathcal X\rightarrow \mathcal A$ for all $y\in\mathcal Y_x$. The learning performance of an algorithm run over $N$ steps is measured by the regret \begin{align*} R_N = N\rho^* - \big[\sum_{t=1}^N r_t\big],~ \textit{where } \rho^* = \rho(\pi_{\mathcal X}^*). \end{align*} Finally we recall that the diameter of the observation MDP is defined as \begin{align*} D_{\mathcal Y} = \max_{y,y'\in\mathcal Y} \min_{\pi:\mathcal Y\rightarrow \mathcal A} \mathbb{E}\big[\tau_{\pi}(y,y')\big], \end{align*} where $\tau_{\pi}(y,y')$ is the (random) number of steps from $y$ to $y'$ by following the observation-based policy $\pi$ (similar for the diameter of the hidden MDP). \section{Experiments} \begin{figure} \centering \begin{psfrags} \psfrag{sqrt(N)}[][1]{$\sqrt{N}$} \psfrag{Regret}[][1]{Regret} \psfrag{SL-UC}[][1]{\scalebox{.4}\textsc{\small{SL-UCRL}}\xspace} \psfrag{DQN}[][1]{\scalebox{.4}{DQN}} \psfrag{UCRL}[][1]{\scalebox{.4}{UCRL}} \includegraphics[height=5.cm]{Fig/Plot/regretcompare.eps} \end{psfrags} \caption{Regret comparison for ROMDPs\xspace with $X=5,A=4$ and from top to bottom $Y=10,20,30$.} \label{fig:comparison} \vspace{-.2in} \end{figure} We validate our theoretical results by comparing the performance of \textsc{\small{SL-UCRL}}\xspace, UCRL2 (model based) and DQN (model free, function approximation)~\cite{mnih2013playing}, two well known RL algorithms. The goal of this experiment is to evaluate the dependency of regret to dimensionality of observation space. Generally, DQN is considered as model free RL method which extend the notion of Bellman residual \citep{antos2008learning} to deep RL. For DQN, we implement a three layers feed forward network (with no CNN block), equipped with RMSprop and replay buffer. We tune the hyper parameters of the network and report the best performance achieved by network of size $30\times 30\times 30$. We consider three randomly generated ROMDPs\xspace (Dirichlet transition and Uniform reward with different bias) with $X=5$, $A=4$ and observation spaces of sizes $Y=10, 20, 30$. Fig.~\ref{fig:comparison} reports the regret on a $\sqrt{N}$ scale where regret of UCRL and DQN grows much faster than \textsc{\small{SL-UCRL}}\xspace's. While all regrets tend to be linear (i.e., growing as $\sqrt{N}$), we observe that the regret slope of UCRL and DQN are negatively affected by the increasing number of observations, while the regret slope of \textsc{\small{SL-UCRL}}\xspace stays almost constant, confirming that the hidden space $\mathcal X$ is learned rapidly. These experiments are the first step towards more practical applications. Additional experiments App.~\ref{app:experiment}. \section{Learning in ROMDPs\xspace}\label{sec:stochastic} \section{Learning ROMDP\xspace}\label{sec:stochastic} In this section we introduce the spectral method used to learn the structure of the observation matrix $O$. In particular, we show that we do not need to estimate $O$ exactly as the clusters $\{\mathcal Y_x\}_{x\in\mathcal X}$ can be recovered by identifying the non-zero entries of $O$. We need a first assumption on the ROMDP\xspace. \begin{assumption}\label{asm:ergodicity} The Markov chain induced on the hidden MDP $M$ by any policy $\pi_{\mathcal Y}$ is ergodic. \end{assumption} Under this assumption for any policy $\pi$ there is a stationary distribution over hidden states $\omega_{\pi}$ and a stationary distribution conditional on an action $\omega_{\pi}^{(l)}(i) = \mathbb P_\pi(x\!=\!i | a\!=\!l)$. Let $\mathcal X_{\pi}^{(l)} = \{i\in[X]: \omega_{\pi}^{(l)}(i) > 0\}$ be the hidden states where action $l$ could be taken according to policy $\pi$. In other words, if $\mathcal Y_{\pi}^{(l)} = \{j\in[Y]: \pi(j)=l\}$ is the set of observations in which policy $\pi$ takes action $l$, then $\mathcal X_{\pi}^{(l)}$ is the set of hidden states $\{x_y\}$ with $y\in\mathcal Y_{\pi}^{(l)}$ (see Fig.~\ref{fig:observation.policy}-\textit{right}). We also define the set of all hidden states that can be reached starting from states in $\mathcal X_{\pi}^{(l)}$ and taking action $l$, that is \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} \begin{align*} \overline\mathcal X_{\pi}^{(l)} = \bigcup_{i\in\mathcal X_{\pi}^{(l)}}\!\!\! \Big\{i'\in[X]: \mathbb P\big(x'=i' | x=i, a=l\big)>0\Big\}. \end{align*} Similarly $\underline\mathcal X_{\pi}^{(l)}$ is the set of hidden states from which we can achieve the states $\mathcal X_{\pi}^{(l)}$ by policy $\pi$. We need the following assumption. \begin{assumption}[Full-Rank]\label{asm:expansive} Given any action $l$, the slice of transition tensor $[T]_{\cdot,\cdot,l}$ is full rank. \end{assumption} Asm.~\ref{asm:expansive} implies that for any action $l$ the dynamics of $M$ is ``expansive'', i.e., $|\mathcal X_{\pi}^{(l)}| \leq |\overline\mathcal X_{\pi}^{(l)}|$. In other words, the number of hidden states where policy $\pi$ can take an action $l$ (i.e., $\mathcal X_{\pi}^{(l)}$) is smaller than the number of states that can be reached when executing action $l$ itself (i.e., $\overline\mathcal X_{\pi}^{(l)}$). These two assumptions ensure that the underlying Markov process is stochastic. \textbf{Multi-view model and exact recovery.} We are now ready to introduce the multi-view model~\citep{anandkumar2014tensor} that allows us to reconstruct the clustering structure of the ROMDP\xspace Alg.~\ref{alg:spectral.learning}. We consider the trajectory of observations and actions generated by an arbitrary policy $\pi$ and we focus on three consecutive observations $y_{t-1}, y_t, y_{t+1}$ at any step $t$. As customary in multi-view models, we \textit{vectorize} the observations into three one-hot view vectors $\vec{v}_1$, $\vec{v}_2$, $\vec{v}_3$ in $\{0,1\}^Y$ such that $\vec{v}_1 = \vec{e}_j$ means that the observation in the first view is $j\in[Y]$ and where we remap time indices $t-1, t, t+1$ onto $1$, $2$, and $3$. We notice that these views are indeed independent random variables when conditioning on the state $x_2$ (i.e., the hidden state at time $t$) and the action $a_2$ (i.e., the action at time $t$), thus defining a multi-view model for the hidden state process. Let $k_1 = |\underline\mathcal X_{\pi}^{(l)}|$, $k_2 = |\mathcal X_{\pi}^{(l)}|$ and $k_3 = |\overline \mathcal X_{\pi}^{(l)}|$, then we define the factor matrices $V_1^{(l)}\!\in\mathbb{R}^{Y\times k_1},V_2^{(l)}\!\in\mathbb{R}^{Y\times k_2},V_3^{(l)}\!\in\mathbb{R}^{Y\times k_3}$ as follows \begin{align*} [V_p^{(l)}]_{j,i}=\mathbb P(\vec v_p=\vec{e}_j|x_2=i,a_2=l), \end{align*} where for $p\!=\!1,~i\in\underline\mathcal X_{\pi}^{(l)}$, for $p\!=\!2,~i\in\mathcal X_{\pi}^{(l)}$, and for $p\!=\!3,~ i\in\overline\mathcal X_{\pi}^{(l)}.$ We are interested in estimating $V_2^{(l)}$ since it directly relates to the observation matrix as \begin{align}\label{eq:v2toO} [V_2^{(l)}]_{j,i} = \frac{\mathbb P(a_2=l | y_2=j) \mathbb P(y_2=j | x_2=i)}{\mathbb P(a_2=l | x_2=i)} = \frac{\mathbb{I}\{\pi(j)=l\} f_O(j|i)}{\mathbb P(a_2=l | x_2=i)}, \end{align} where $\mathbb{I}$ is the indicator function. As it can be noticed, $V_2^{(l)}$ borrows the same structure as the observation matrix $O$ and since we want to recover only the clustering structure of $M$ (i.e., $\{\mathcal Y_i\}_{i\in[X]}$), it is sufficient to compute the columns of $V_2^{(l)}$ up to any multiplicative constant. In fact, any non-zero entry of $V_2^{(l)}$ corresponds to a non-zero element in the original observation matrix (i.e., $[V_2^{(l)}]_{j,i} > 0 \Rightarrow [O]_{j,i} > 0$) and for any hidden state $i$, we can construct a cluster $\mathcal Y_i^{(l)} = \{j\in[Y]: [V_2^{(l)}]_{j,i} > 0\}$, which is accurate up to a re-labelling of the states. More formally, there exists a mapping function $\sigma^{(l)}:\mathcal X\rightarrow \mathcal X$ such that any pair of observations $j,j'\in\mathcal Y_i^{(l)}$ is such that $j,j'\in\mathcal Y_{\sigma(i)}$. Nonetheless, as illustrated in Fig.~\ref{fig:observation.policy}-\textit{right}, the clustering may not be minimal. In fact, we have $[O]_{j,i} > 0 \not\Rightarrow [V_2^{(l)}]_{j,i} > 0$ since $[V_2^{(l)}]_{j,i}$ may be zero because of policy $\pi$, even if $[O]_{j,i} > 0$. Since the (unknown) mapping function $\sigma^{(l)}$ changes with actions, we are unable to correctly ``align'' the clusters and we may obtain more clusters than hidden states. We define $\S$ as the auxiliary state space obtained by the partial aggregation and we prove the following result. \begin{figure} \vspace*{-0.1in} \begin{algorithm}[H] \begin{small} \setstretch{1.3} \begin{algorithmic} \STATE \hspace{-0.1in}\textbf{Input:} Trajectory $(y_1,a_1,\ldots,y_N)$ \STATE \hspace{-0.1in}\textbf{For} Action $l\in[A]$ \textbf{do} \STATE \hspace{-0.05in}Estimate second moments $\widehat{K}_{2,3}^{(l)}$, $\widehat{K}_{1,3}^{(l)}$, $\widehat{K}_{2,1}^{(l)}$, and $\widehat{K}_{3,1}^{(l)}$ \STATE \hspace{-0.05in}Estimate the rank of matrix $\widehat{K}_{2,3}^{(l)}$ (see App.~\ref{sec:rank}) \STATE \hspace{-0.05in}Compute symmetrized views $\widetilde v_{1,t}$ and $\widetilde v_{3,t}$, for $t=2..N-2$ \STATE \hspace{-0.05in}Compute second and third moments $\widehat M_{2}^{(l)}$ and $\widehat M_{3}^{(l)}$ \STATE \hspace{-0.05in}Compute $\widehat V_2^{(l)}$ from the tensor decomposition of (an orthogonalized version of) $\widehat M_{3}^{(l)}$ \STATE \hspace{-0.05in} return clusters $$\widehat{\mathcal Y}_i^{(l)} = \{j\in[Y]: [\widetilde V_2^{(l)}]_{j,i} > 0\}$$ \end{algorithmic} \caption{Spectral learning algorithm.} \label{alg:spectral.learning} \end{small} \end{algorithm} \vspace*{-0.5cm} \end{figure} \begin{lemma}\label{lem:partial.cluster} Given a policy $\pi$, for any action $l$ and any hidden state $i\in\mathcal X_{\pi}^{(l)}$, let $\mathcal Y_i^{(l)}$ be the observations that can be clustered together according to $V_2^{(l)}$ and $\mathcal Y^{\mathsf{c}} = \mathcal Y \setminus \bigcup_{i,l}\mathcal Y_i^{(l)}$ be the observations not clustered, then the auxiliary state space $\S$ contains all the clusters $\{\bigcup_{i,l}\mathcal Y_i^{(l)}\}$ and the singletons in $\mathcal Y^{\mathsf{c}}$ for a total number of elements $S = |\S| \leq A X$. \end{lemma} \vspace*{-.1in} We now show how to recover the factor matrix $V_2^{(l)}$. We introduce mixed second and third order moments as $K^{(l)}_{p,q}=\mathbb{E}[\vec v_p\otimes \vec v_q], K^{(l)}_{p,q,r}=\mathbb{E}[\vec v_p\otimes \vec v_q\otimes\vec v_r]$ where $p,q,r$ is any permutation of $\lbrace 1,2,3\rbrace$. Exploiting the conditional independence of the views, the second moments can be written as \begin{align*} K^{(l)}_{p,q} = \sum_{i\in\mathcal X^{l}_{\pi}} \omega_\pi^{(l)}(i) [V_p^{(l)}]_{:,i} \otimes [V_q^{(l)}]_{:,i} \end{align*} where $[V_p^{(l)}]_{:,i}$ is the $i$-th column of $V_p^{(l)}$. In general the second moment matrices are rank deficient, with rank $X_{\pi}^{(l)}$. We can construct a symmetric second moment by introducing the symmetrized views \begin{align}\label{eq:symmetric.views} \widetilde v_1 = K^{(l)}_{2,3} (K^{(l)}_{1,3})^{\dagger}\vec v_1,\enspace\enspace\quad \widetilde v_3 = K^{(l)}_{2,1}(K^{(l)}_{3,1})^{\dagger}\vec v_3, \end{align} where $K^\dagger$ denotes the pseudoinverse. Then we can construct the second and third moments as \begin{align}\label{eq:second.moment} M_2^{(l)} \!=\! \mathbb E\big[\widetilde v_1 \otimes \widetilde v_3 \big] \!=\! \sum_{i\in\mathcal X^{(l)}_{\pi}} \!\!\omega_\pi^{(l)}(i) [V_2^{(l)}]_{:,i} \otimes [V_2^{(l)}]_{:,i}. \end{align} \begin{align}\label{eq:third.moment} M_3^{(l)} = \mathbb E\big[\widetilde v_1 \otimes \widetilde v_3 \otimes \vec v_2 \big] = \sum_{i\in\mathcal X^{l}_{\pi}}\omega_\pi^{(l)}(i) [V_2^{(l)}]_{:,i} \otimes [V_2^{(l)}]_{:,i}\otimes [V_2^{(l)}]_{:,i}. \end{align} We can now employ the standard machinery of tensor decomposition methods to orthogonalize the tensor $M_3^{(l)}$ using $M_2^{(l)}$ and recover $V_2^{(l)}$ (refer to~\citep{anandkumar2014tensor} for further details) and a suitable clustering. \begin{lemma}\label{lem:tensor.decomposition} For any action $l\in[A]$, let $M_3^{(l)}$ be the third moment constructed on the symmetrized views as in Eq.~\ref{eq:third.moment}, then we can orthogonalize it using the second moment $M_2^{(l)}$ and obtain a unique spectral decomposition from which we compute the exact factor matrix $[V_2^{(l)}]_{j,i}$. As a result, for any hidden state $i\in\mathcal X_\pi^{(l)}$ we define the cluster $\widetilde{\mathcal Y}_i^{(l)}$ as \begin{align}\label{eq:estimated.cluster2} \widetilde{\mathcal Y}_i^{(l)} = \{j\in[Y]: [V_2^{(l)}]_{j,i} > 0\} \end{align} and there exists a mapping $\sigma^{(l)}:X\rightarrow X$ such that if $j,j'\in \widetilde{\mathcal Y}_i^{(l)}$ then $j,j'\in \mathcal Y_{\sigma^{(l)}(i)}$ (i.e., observations that are clustered together in $\widetilde{\mathcal Y}_i^{(l)}$ are clustered in the original ROMDP\xspace). \end{lemma} \begin{figure} \vspace*{-0.5cm} \hspace*{.2in} \begin{algorithm}[H] \begin{small} \setstretch{1.2} \begin{algorithmic} \STATE \hspace{-0.1in}\textbf{Initialize:} $t=1$, initial state $x_1$, $k=1$, $\delta/N^6$ \STATE \hspace{-0.1in}\textbf{While} {$t < N$} \textbf{do} \STATE \hspace{-0.05in}Run Alg.~\ref{alg:spectral.learning} on samples from epoch $k-1$ and obtain $\widehat\S$ \STATE \hspace{-0.05in}Compute aux. space ${\widehat\S}^{(k)}$ by merging $\widehat\S$ and $\widehat\S^{(k-1)}$ \STATE \hspace{-0.05in}Compute the estimate reward $r^{(k)}$ and dynamics $p^{(k)}$ \STATE \hspace{-0.05in}Construct admissible AuxMDPs $\mathcal{M}^{(k)}$ \STATE \hspace{-0.05in}Compute the optimistic policy \begin{align}\label{eq:optimism} \widetilde{\pi}^{(k)} = \arg\max\limits_{\pi}\max\limits_{M\in\mathcal{M}^{(k)}} \rho(\pi; M) \end{align} \STATE \hspace{-0.05in}Set $v^{(k)}( s,l) = 0$ for all actions $l\in\mathcal{A}, s\in {\widehat\S}^{(k)}$ \STATE \hspace{-0.05in}\textbf{While} $\forall l, \forall s, v^{(k)}(s,l) < \max\lbrace 1,N^{(k)}(s,l)\rbrace$ \textbf{do} \STATE Execute $a_t = \widetilde\pi^{(k)}(s_t)$ \STATE Observe reward $r_t$ and observation $y_t$ \end{algorithmic} \caption{Spectral-Learning UCRL(\textsc{\small{SL-UCRL}}\xspace).} \label{alg:sm.ucrl} \end{small} \end{algorithm} \vspace{-0.3in} \end{figure} The computation complexity of Alg.~\ref{alg:spectral.learning} has been studied by~\citet{song2013nonparametric} and is polynomial in the rank of third order moment. \textbf{Spectral learning.} \footnote{We report the spectral learning algorithm for the tensor decomposition but a very similar algorithm and guarantees can be derived for the matrix decomposition approach when the eigenvalues of $\widehat M_2^{(l)}$ for all actions and all posible policy have multiplicity 1. This further condition is not required when the tensor decomposition is deployed.} While in practice we do not have the exact moments, we can only estimates them through samples. Let $N$ be the length of the trajectory generated by policy $\pi$, then we can construct $N-2$ triples $\{y_{t-1}, y_t, y_{t+1}\}$ that can be used to construct the corresponding views $\vec v_{1,t},\vec v_{2,t},\vec v_{3,t}$ and to estimate second mixed moments as \begin{align*} \widehat{K}^{(l)}_{p,q}=\frac{1}{N{(l)}} \sum_{t=1}^{N(l)-1}\mathbb{I}(a_t=l) ~\vec v_{p,t} \otimes \vec v_{q,t}, \end{align*} with $p,q\in\lbrace 1,2,3\rbrace$ and $N{(l)}=\sum_t^{N-1}\mathbb{I}(a_t=l)$. Furthermore, we require knowing $|\mathcal X_{\pi}^{(l)}|$, which is not known apriori. Under Asm.~\ref{asm:ergodicity} and~\ref{asm:expansive}, for any action $l$, the rank of ${K}^{(l)}_{2,3}$ is indeed $|\mathcal X_{\pi}^{(l)}|$ and thus $\widehat{K}^{(l)}_{2,3}$ can be used to recover the rank. The actual way to calculate the efficient rank of $\widehat{K}^{(l)}_{2,3}$ is quite intricate and we postpone the details to App.~\ref{sec:rank}. From $\widehat{K}^{(l)}_{p,q}$ we can construct the symmetric views $\widetilde v_{1,t}$ and $\widetilde v_{3,t}$ as in Eq.~\ref{eq:symmetric.views} and compute the estimates of second and third moments as \begin{align* \widehat{M}_2^{(l)} &= \frac{1}{N{(l)}} \sum_{t=1}^{N-1} \mathbb{I}(a_t=l)\widetilde v_{1,t} \otimes \widetilde v_{3,t},\\ \widehat{M}_3^{(l)} &= \frac{1}{N{(l)}} \sum_{t=1}^{N-1} \mathbb{I}(a_t=l)\widetilde v_{1,t} \otimes \widetilde v_{3,t} \otimes \vec v_{2,t}. \end{align*} Following the same procedure as in the exact case, we are then able to recover estimates of the factor matrix $\widehat V_2^{(l)}$, which enjoys the following error guarantee. \begin{lemma}\label{lem:decomp} Under Asm.~\ref{asm:ergodicity} and~\ref{asm:expansive}, let $\widehat{V}^{(l)}_2$ be the empirical estimate of ${V}^{(l)}_2$ obtained using $N$ samples generated by a policy $\pi$. There exists $N_0$ such that for any $N(l) > N_0$, $l\in\mathcal A$, $i\in \mathcal X_{\pi}^{(l)}$ w.p. $1-\delta$ \begin{align}\label{eq:BO} \|[V_2^{(l)}]_{\cdot,i}\!-\![\widehat{V}_2^{(l)}]_{\cdot,i}\|_2\!\leq\! C_2\sqrt{\frac{\log(2Y^{3/2}/\delta)}{N(l)}}:=\mathcal{B}_O^{(l)} \end{align} where $C_2$ is a problem-dependent constant independent from the number of observations $Y$. \end{lemma} While this estimate could be directly used to construct a clustering of observations, the noise in the empirical estimates might lead to $[\widehat V_2^{(l)}]_{j,i} > 0$ for any $(j,i)$ pair, which prevents us from generating any meaningful clustering. On the other hand, we can use the guarantee in Lem.~\ref{lem:decomp} to single-out the entries of $\widehat V_2^{(l)}$ that are non-zero w.h.p. We define the binary matrix $\widetilde V_2^{(l)}\in\{0,1\}^{Y\times X}$ as \begin{align* [\widetilde V_2^{(l)}]_{j,i} = \begin{cases} 1 &\mbox{if } [\widehat V_2^{(l)}]_{j,i} \geq \mathcal{B}_O^{(l)} \\ 0 & \mbox{otherwise} \end{cases}, \end{align*} which relies on the fact that $[\widehat V_2^{(l)}]_{j,i} -\mathcal{B}_O^{(l)} > 0$ implies $[V_2^{(l)}]_{j,i} > 0$. At this point, for any $l$ and any $i\in\mathcal X_\pi^{(l)}$, we can generate the cluster % \begin{align}\label{eq:estimated.cluster3} \widehat{\mathcal Y}_i^{(l)} = \{j\in[Y]: [\widetilde V_2^{(l)}]_{j,i} > 0\}, \end{align} which is guaranteed to aggregate observations correctly in high-probability. We denote be $\widehat\mathcal Y^{\mathsf{c}} = \mathcal Y \setminus \bigcup_{i,l}\widehat{\mathcal Y}_i^{(l)}$ the set of observations which are not clustered through this process. Then we define the auxiliary state space $\widehat\S$ obtained by enumerating all the elements of non-clustered observations together with clusters $\{\widehat{\mathcal Y}_i^{(l)}\}_{i,l}$, for which we have the following guarantee. \begin{corollary}\label{cor:partial.cluster.est} Let $\widehat\S$ be the auxiliary states composed of clusters $\{\widehat{\mathcal Y}_i^{(l)}\}$ and singletons in $\mathcal Y^{\mathsf{c}}$ obtained by clustering observations according to $\widetilde V_2^{(l)}$, then for any pair of observations $j,j'$ clustered together in $\widehat \S$, there exists a hidden state $i$ such that $j,j'\in \mathcal Y_i$. Finally, $\widehat\S \rightarrow \S$ as $N$ tends to infinity. \end{corollary} \section{RL in ROMDP\xspace} We now describe the spectral learning UCRL (\textsc{\small{SL-UCRL}}\xspace) (Alg.~\ref{alg:sm.ucrl}) obtained by integrating the spectral method above with the UCRL strategy. The learning process is split into epochs of increasing length. At the beginning of epoch $k$, we use the trajectory $(s_1,a_1,..,s_{N^{(k-1)}})$ generated at previous epoch using auxiliary states $s\in\widehat\S^{(k)}$ to construct the auxiliary state space ${\widehat\S}$ using Alg.~\ref{alg:spectral.learning}.\footnote{Since Alg.~\ref{alg:spectral.learning} receives as input a sequence of auxiliary states rather than observations as in Sect.~\ref{sec:stochastic} the spectral decomposition runs on a space of size $|\widehat S^{(k-1)}|$ instead of $Y$, thus reducing the computation complexity.} As discussed in the previous section, the limited number of samples and the specific policy executed at epoch $k-1$ may prevent from clustering many observations together, which means that despite ${\widehat\S}$ being a \textit{correct} clustering (see Cor.~\ref{cor:partial.cluster.est}), its size may still be large. While clusterings obtained at different epochs cannot be ``aligned'' because of different labelling, we can still effectively merge together any two clusterings $\widehat\S$ and $\widehat\S'$ generated by two different policies $\pi$ and $\pi'$. We illustrate this procedure through Fig.~\ref{fig:mappolicy}. Observations $y_3$, $y_4$, and $y_5$ are clustered together in the auxiliary space generated by $\pi$, while $y_5$ and $y_6$ are clustered together using $\pi'$. While the labeling of the auxiliary states is arbitrary, observations preserve their labels across epochs and thus we can safely conclude that observations $y_3$, $y_4$, $y_5$, and $y_6$ belong to the same hidden state. Similarly, we can construct a new cluster with $y_9$, $y_{10}$, and $y_{11}$, which, in this case, returns the exact hidden space $\mathcal X$. Following this procedure we generate $\widehat\S^{(k)}$ as the clustering obtain by merging $\widehat\S$ and $\widehat\S^{(k-1)}$ (where $\widehat\S^{1} = \mathcal Y$). At this point we can directly estimate the reward and transition model of the auxiliary MDP constructed on $\widehat\S^{(k)}$ by using empirical estimators. For a sequence of clustering $\widehat\S^{(0)},\ldots, \widehat\S^{(k)}$, since the clustering $\widehat\S^{(k)}$ is \textit{monotonic} (i.e., observations clustered at epoch $k$ stay clustered at any other epoch $k' > k$) any cluster $s^{(t^k)} \in \widehat\S^{(k)}$ can be represented as result of a \textit{monotonically} aggregating observations as a increasing series of $s^1\subseteq s^2\subseteq \ldots \subseteq s^{(t^k-1)}$ (not unique, and random. As it is has been shown in Fig.~\ref{fig:evolution} any branching can be considered as one of these series. Let's choose one of them. \begin{figure}[ht] \centering \includegraphics[height=4.0cm]{Fig/evolution_1_.eps} \caption{Monotonic evolution of clusters, each layer is the beginning of an epoch. The green and red paths are two examples for two different cluster aggregation.} \label{fig:evolution} \end{figure} Here, $s^t$ is a cluster at time point $t(\leq t_k)$ which evolves to the cluster $s^{(t_k)}$. For a clustering sequence $s^1\subseteq s^2\subseteq \ldots \subseteq s^{(t^k-1)}$, evolving to $s^{(t^k)}$, define $N^{(k)}(s,a)$, the number of samples in interest is: \begin{align*} N^{(k)}(s^{(t^k)},a) = \sum_t^{t^{(k)}}\mathbbm{1}(y_t\in s^t)\mathbbm{1}(a_t = a) \end{align*} with an abuse of notation, we write $y\in s^t$ to denote that the observation $y$ has been clustered into an auxiliary state $s^t$ at time step $t$. For any observation $y$, we use all the samples of $y$ to decide whether to merge this observation to any cluster. When we merge this observation to a cluster, we do not use the past sample of $y$ for the empirical estimates of reward and transition. For example, Fig.~\ref{fig:mappolicy}, we cluster together $\lbrace y_3, y_4,y_5\rbrace$. At the beginning of each epoch, we use all the samples to decide whether $y_6$ belongs to this cluster. For an epoch, when we decide that $y_6$ belongs to this cluster, we do not use the samples of $y_6$ up to this epoch to estimate the reward and transition estimates of cluster $\lbrace y_3, y_4,y_5,y_6\rbrace$. \begin{figure}[t] \centering \includegraphics[height=4.cm]{Fig/O_matrix_policy_merge.eps} \caption{ Examples of clusterings obtained from two policies that can be effectively merged.} \label{fig:mappolicy} \vspace{-.0in} \end{figure} Therefore, to estimate the empirical mean of reward and the transition kernel, we have \begin{align*} \widehat r^{(k)}(s^{(t^k)},a) = \sum_t^{t^{(k)}}r_t \mathbbm{1}(y_t\in s^t)\mathbbm{1}(a_t = a)/N^{(k)}(s,a) \end{align*} and for transitions, let's define the following count \begin{align*} N^{(k)}(s^{(t^k)},a,s') = \sum_t^{t^{(k)}}\mathbbm{1}(y_{t+1}\in s')\mathbbm{1}(y_t\in s^t)\mathbbm{1}(a_t = a) \end{align*} therefore \begin{align*} \widehat p^{(k)}(s' | s^{(t^k)},a) = N^{(k)}(s^{(t^k)},a,s')/N^{(k)}(s^{(t^k)},a) \end{align*} then we return the estimates.\footnote{Since the clustering $\widehat\S^{(k)}$ is \textit{monotonic}, $\widehat r^{(k)}$ and $\widehat p^{(k)}$ can be computed incrementally without storing the statistics $N^{(k)}(y,a,y')$, $N^{(k)}(y,a)$, and $R^{(k)}(y,a)$ at observation level, thus significantly reducing the space complexity of the algorithm.} For further use, we define the per-epoch samples of interest for $s\in\widehat\S^{(k)}$ as $\nu^{(k)}( s^{(t^k)},a) := \sum_{y\in\mathcal Y}\sum_{t^{(k-1)}}^{t^{(k)}}z_{t,t^{(k)}}(y)$ The corresponding confidence intervals are such that for any $s\in {\widehat\S}^{(k)}$ and $a\in\mathcal{A}$ \begin{align*} &\|p(\cdot|s,a)\text{--}\widehat{p}^{(k)}(\cdot|s,a)\|_1 \!\leq\! d_p(s,a)\!=\! \sqrt{\frac{28S^{(k)}\log(\frac{2AN^{(k)}}{\delta})}{N^{(k)}(s,a)}},\\ &|\bar{r}(s,a)-\widehat{r}^{(k)}(s,a)|\leq d_r(s,a)= \sqrt{\frac{28\log(\frac{2YAN^{(k)}}{\delta})}{N^{(k)}(s,a)}}, \end{align*} hold w.p. $1-\delta$, where $p(\cdot|s,a)$ and $\bar{r}$ are the transition probabilities and reward of the auxiliary MDP $M_{\widehat\S^{(k)}}$ App.~\ref{Apx:concentration}. Given the estimates and the confidence intervals, we construct a set of plausible auxiliary MDPs, $\mathcal M^{(k)}$, where the reward means and transition probabilities satisfy the confidence intervals. At this point we can simply apply the same steps as in standard UCRL, where an optimistic auxiliary MDP $\widetilde{M}^{(k)}$ is constructed using the confidence intervals above and extended value iteration (EVI)~\citep{jaksch2010near}. The resulting optimal optimistic policy $\widetilde\pi^{(k)}$ is then executed until the number of samples at least for one pair of auxiliary state and action is doubled. EVI has a per-iteration complexity which scales as $\mathcal{O}((\widehat S^{(k)})^2A)$ thus gradually reducing the complexity of \textsc{\small{UCRL}}\xspace on the observation space (i.e., $\mathcal{O}((Y)^2A)$) as soon as observations are clustered together. When the whole clustering is learnt, the computational complexity of EVI tends to $\mathcal{O}((X)^2A)$. Moreover, since we aggregate the samples of the elements in clusters, therefore more accurate estimates, the number of times we call EVI algorithm goes from $\mathcal{O}(Y\log(N))$ to $\mathcal{O}(X\log(N))$. \begin{theorem}\label{thm:regret} Consider a ROMDP\xspace $M=\langle\mathcal{X},\mathcal{Y},\mathcal{A},R,f_T,f_O\rangle$ with diameter $D_{\mathcal X}$. If \textsc{\small{SL-UCRL}}\xspace is run over $N$ time steps, under Asm.~\ref{asm:ergodicity} and \ref{asm:expansive}, with probability $1-\delta$ it suffers the total regret of \begin{align*} \textit{Reg}_{N}\leq \sum_{k= 1}^{K}\Big( D_{\widehat S^{(k)}}\sqrt{\widehat S^{(k)}\log\Big(\frac{N^{(k)}}{\delta}\Big)}\hspace*{-.4em}\sum_{s\in\mathcal{\widehat S}^{(k)},a}\hspace*{-.4em}\frac{\nu^{(k)}( s,a)}{\sqrt{N^{(k)}(s,a)}}\Big), \end{align*} where $(\S^{(k)})$ is the sequence of auxiliary state spaces generated over $K$ epochs. \end{theorem} \textbf{Remark.} This bound shows that the per-step regret decreases over epochs. First we notice that only the regret over the first few (and short) epochs actually depends on the number of observations $Y$ and the diameter $D_\mathcal Y$. As soon as a few observations start being clustered into auxiliary states, the regret depends on the number of auxiliary states $\widehat S^{(k)}$ and the diameter $D_{\S^{(k)}}$. Since $\widehat S^{(k)}$ decreases every time an observation is added to a cluster and $D_{\S^{(k)}}$ is monotonically decreasing with of $\widehat \S^{(k)}$, the per-step regret significantly decreases with epochs.\footnote{We refer to the per-step regret since an epoch may be longer, thus making the cumulative epoch regret larger.} Cor.~\ref{cor:partial.cluster.est} indeed guarantees that the number of auxiliary states in $\widehat\S$ reduces down to $|\S|$ ($XA$ in the worst case) as epochs get longer. Furthermore we recall that even if the clustering $\widehat \S$ returned by the spectral method is not minimal, merging clusters across epochs may rapidly result in very compact representations even after a few epochs. \textbf{Minimal clustering.} While Thm.~\ref{thm:regret} shows that the performance of \textsc{\small{SL-UCRL}}\xspace improves over epochs, it does not relate it to the (ideal) performance that could be achieved when the hidden space had been known. Unfortunately, even if the number of clusters in $\widehat{\S}^{(k)}$ is nearly-minimal, the MDP constructed on the auxiliary state space may have a large diameter. In fact, it is enough that an observation $j$ with very low probability $O_{j,i}$ is not clustered (it is a singleton in $\S^{(k)}$) to have a diameter that scales as $1/O_{\min}$ (although its \textit{actual} impact on the regret may be negligible, for instance when $j$ is not visited by the current policy). , in general the advantage obtained by clustering reduces the dependency on the number of states from $Y$ to $XA$ but it may not be effective in reducing the dependency on the diameter from $D_\mathcal Y$ to $D_\mathcal X$. In order to provide a minimal clustrting, we integrate Alg.~\ref{alg:sm.ucrl} with a clustering technique similar to the one used in~\cite{gentile2014online} and~\cite{ortner2013adaptive}. At any epoch $k$, we proceed by merging together all the auxiliary states in $\widehat \S^{(k)}$ whose reward and transition confidence intervals overlap (i.e., $s$ and $s'$ are merged if the confidence interval $[\widehat r(s,a) \pm d_r(s,a)]$ overlaps with $[\widehat r(s',a) \pm d_r(s',a)]$ and $[\widehat p(\cdot|s,a) \pm d_p(s,a)]$\footnote{Deviation $d_p(s,a)$ on a $\widehat S$ dimensional simplex} overlaps with $[\widehat p(\cdot|s',a) \pm d_p(s',a)]$. If the number of new clusters is equal to $X$, then we claim we learned the true clustering, if it is less than $X$ we ignore this temporary clustering and proceed to the next epoch. It is worth noting that this procedure requires the knowledge of $X$, while the spectral method, by its own, does not. While an explicit rate of clustering is very difficult to determine (the merging process depends on the spectral method, whose result depends on the policy, which in turn is determined according to the clustering at previous epochs), we derive worst-case bounds on the number of steps needed to start clustering at least one observation (i.e., steps before avoiding the dependency on $Y$ and $D_\mathcal Y$) and before the exact clustering is recovered. \begin{corollary}\label{cor:regret} Let $\tau_M=\max_{x,\pi}\mathbb{E}_{\pi}[\tau_\pi(x, x)]$ the maximum expected returning time in MDP $M$ (bounded due to ergodicity) and \begin{align}\label{eq:firstlast} \overline N_{\text{first}} &= \frac{AY \tau_M}{O_{\min}}\frac{C_2\log(1/\delta)}{\max_{i,j}f_{O}(y=j|x=i)^2}; \nonumber\\ \overline N_{\text{last}} &= \frac{AY \tau_M}{O_{\min}^3}C_2\log(1/\delta). \end{align} After $\overline N_{\text{first}}$ steps at least two observations are clustered and after $\overline N_{\text{last}}$ steps all observations are clustered (but not necessarily in the minimum hidden space configuration) with probability $1-\delta$. This implies that after $\overline N_{\text{last}}$ steps $|\widehat{S}^{(k)}| \leq XA$. Furthermore, let $\gamma_r = \min_{x,x',a} |r(x,a)-r(x',a)|$ and $\gamma_p = \min_{x,x',a} \|p(\cdot|x,a) - p(\cdot|x',a)\|_1$ be the smallest gaps between rewards and transition probabilities and let $\gamma = \max\{\gamma_r, \gamma_p\}$ the maximum between the two. In the worst case, using the additional clustering step together with \textsc{\small{SL-UCRL}}\xspace guarantees that after $\overline N_{\mathcal X}$ \begin{align*} \min \Big\{\frac{AY^2 \tau_M}{\gamma^2}\log(1/\delta), \max\big\lbrace \frac{AS^2 \tau_M}{\gamma^2}\log(1/\delta) , \overline N_{\text{last}}\big\rbrace\Big\} \end{align*} samples the hidden state $\mathcal X$ is correctly reconstructed (i.e., $\widehat\S^{(k)} = \mathcal X$), therefore \begin{align*} \textit{Reg}_{N}&\leq 34D_{\mathcal X}X\sqrt{A(N-\overline N_{\mathcal X})\log(N/\delta)}\mathbb{I}(N\geq \overline N_{\mathcal X})\nonumber\\ &+\min\lbrace \overline N_{\mathcal X},34D_{\mathcal Y}Y\sqrt{A(\overline N_{\mathcal X})\log(N_{\mathcal X}/\delta)} \rbrace \end{align*} \end{corollary} We first notice that this analysis is constructed over a series of worst-case steps (see proof in App.~\ref{app:clustering.rate}). Nonetheless, it first shows that the number of observations $Y$ does impact the regret only over the first $\overline N_{\text{first}}$ steps, after which $\widehat S^{(k)}$ is already smaller than $\mathcal Y$. Furthermore, after at most $\overline N_{\text{last}}$ the auxiliary space has size at most $XA$ (while the diameter may still be as large as $D_{\mathcal Y}$). Finally, after $\overline N_{\mathcal X}$ steps $\widehat{S}^{(k)}$ reduces to $\mathcal X$ and the performance of \textsc{\small{SL-UCRL}}\xspace tends to the same performance of UCRL in the hidden MDP. \section{Conclusion} We introduced \textsc{\small{SL-UCRL}}\xspace, a novel RL algorithm to learn in ROMDPs\xspace combining a spectral method for recovering the clustering structure of the problem and UCRL to effectively trade off exploration and exploitation. We proved theoretical guarantees showing that \textsc{\small{SL-UCRL}}\xspace progressively refines the clustering so that its regret tends to the regret that could be achieved when the hidden structure is known in advance (in higher order term). Despite this result almost matching the regret obtained by running UCRL directly on the latent MDP, the regret analysis requires ergodicity of the MDP. One of the main open questions is whether the spectral clustering method could still provide ``useful'' clusterings when the state space is not fully visited (i.e., in case of non-ergodic MDP), so that observations are properly clustered where it is actually needed to learn the optimal policy App.~ \ref{Aux:deter}. At the beginning of the learning, the algorithm deals with larger MDPs and gradually, while learning bigger cluster, it starts to deal with smaller MDPs. Reducing the state space of MDPs means lower cost in computing optimistic policy, having fewer number of epochs, and suffer from lower computation cost. Finally, this work opens several interesting directions to extend the results for variety of state aggregation topologies \citep{li2006towards}. Furthermore, one can aggregate the proposed method with other regret analyses, e.g. \citep{dann2017ubev}, \citep{azar2017minimax} to provide a new regret bound. \section{Proofs of Section~\ref{sec:stochastic}} \section{Concentration inequalities for transition kernel and reward process}\label{Apx:concentration} In this section we construct the concentration inequality mentioned in Eq.~\ref{eq:confidence1}. \textbf{Reward Process}. At first we derive the concentration inequality for the reward process by defining the following martingale sequence at time $\tau$, for a random and incremental sequence of $s^1\subseteq s^2\subseteq \ldots \subseteq s^{\tau}$\footnote{To simplify the notation we use $\S^{\tau'}~,~\tau'\in[1,\ldots]$ to note a random set of clustered which are derived by Alg.~\ref{alg:sm.ucrl} at time $\tau$(we omitted that hat notation for simplicity), and $s^{\tau'}$ as a element in set $\S^{\tau'}$ } (we construct the confidence bound for this cluster as a reference cluster and show the analysis holds for any cluster) \begin{align*} \varpi_{\tau} = \sum_{y\in\mathcal Y}\sum_{t=1}^{N}[r_t-\overline r(s^\tau,a)]z_{t,\tau}(y) \end{align*} where $z_{t,\tau}(y):=\mathbbm{1}(y_t = y)\mathbbm{1}(y\in s^t) \mathbbm{1}(a_t = a)\mathbbm{1}(t\leq\tau)$ is a function of set $\S^{\tau}$. Instead of $\overline r(s^\tau,a)$, we can directly use $\overline r(x,a)$ where $x$ is the corresponding latent state of sequence $s^1\subseteq s^2\subseteq \ldots \subseteq s^{\tau}$. \footnote{Since the regret due to incorrect clustering is considered in the final regret, we assume that the clusters $s^{\tau'}~\tau'\in[1,\ldots]$ are consistent and do not cluster together any two observation which are not from same hidden state.} We define $N(s,a)$ as a count for state $s$ and action $a$ at time $N$; \begin{align*} N(s,a) = \sum_{y\in\mathcal Y}\sum_{t=1}^{N}z_{t,\tau}(y) \end{align*} Let's define a Martingale difference for the sequence $\varpi_{\tau}$ as follows: \begin{align*} \delta\varpi_{\tau} = \varpi_{\tau} - \varpi_{\tau-1} = \sum_{y\in\mathcal Y}\sum_{t=1}^{N}[r_t-\overline r(s^\tau,a)][z_{t,\tau}(y)-z_{t,\tau-1}(y)] \end{align*} Therefore given a filtration $\mathcal F$ and Azuma-Hoeffding inequality \citep{hoeffding1963probability}, since \begin{align*} \sum_{\tau}^N|\delta\varpi|^2\leq N(s,a)~~~~ ,and ~~ \sum_{\tau}^N (Var\lbrace \varpi_{\tau}|\mathcal F_{\tau-1}\rbrace)^2\leq N(s,a)^2 \end{align*} we have the following inequality; \begin{align*} \mathbb P\lbrace \sum_{\tau}^N\delta\varpi_{\tau}\geq \epsilon\rbrace \leq \exp(-\frac{\epsilon^2}{4N(s,a)}) \end{align*} Therefore we have \begin{align*} \mathbb P\lbrace |\hat r(s,a)-\overline r(s,a)|\geq \sqrt{\frac{28\log(2 SAN/\delta)}{\max\lbrace 1,N(s,a)\rbrace}}\rbrace\leq \frac{\delta}{60N^7SA} \end{align*} Where $\hat r(s,a) := \sum_{y\in\mathcal Y}\sum_{t=1}^{N}r_t\mathbbm{1}(y=y_t)z_{t,N}(y)$. Since the $N(s,a)$ is a random variable by its own, we have union bound over time, therefore we have \begin{align*} \mathbb P\lbrace |\hat r(s,a)-\overline r(s,a)|\geq \sqrt{\frac{28\log(2 SAN/\delta)}{\max\lbrace 1,N(s,a)\rbrace}}\rbrace\leq \sum_n^{N}\frac{\delta}{60N^7SA}<\frac{\delta}{60N^6SA} \end{align*} \textbf{Transition Kernel}. Same as before, we need to define a Martingale sequence for transition process. For any possible and fixed clustering $\S$ and a cluster $s$, let's define \begin{align*} \psi_{\tau} = \sum_{y\in\mathcal Y}\sum_{t=1}^{N}[\phi_t-\overline \phi(s,a)]\mathbbm{1}(y\in s) z_{t,\tau}(y) \end{align*} where $\phi,\overline \phi\in\Re^S$. \begin{align*} [\phi_t]_i= \begin{cases} 1 &\mbox{if } y_{t+1} \in s_i \\ 0 & \mbox{otherwise} \end{cases}, \end{align*} $[\overline\phi(s,a)]_i= \mathbb P(s_i|s,a), \forall i\in S$, $s_i$ is $i$'th cluster in $\S$, and $s^\tau,\tau \in[N]$ is the sequnec of clustering under Alg.~\ref{alg:sm.ucrl}. Given the martingale sequence $\psi_{\tau} $ ( a vector) and the corresponding Martingale difference $\delta\psi_{\tau}:=\psi_{\tau}-\psi_{\tau-1} $ , we construct a new martingale sequnce using dilation technique \cite{tropp2012user}. \begin{align*} \Psi_{\tau} = \begin{bmatrix} \textbf{0} & \psi_{\tau} \\[0.3em] \psi_{\tau}^\top & \textbf{0} \end{bmatrix} \end{align*} and \begin{align*} \Delta\Psi_{\tau} = \begin{bmatrix} \textbf{0} & \delta\psi_{\tau} \\[0.3em] \delta\psi_{\tau}^\top & \textbf{0} \end{bmatrix} \end{align*} The corresponding matrices $\Psi_{\tau}$ (and $\Delta\Psi_{\tau}$) is self adjoint matrix which have the following properties; \begin{align*} \Psi_{\tau}^2 = \begin{bmatrix} \psi_{\tau} \psi_{\tau}^\top &\textbf{0} \\[0.3em] \textbf{0} &\psi_{\tau}^\top \psi_{\tau} \end{bmatrix} \end{align*} \begin{align*} \Delta\Psi_{\tau}^2 = \begin{bmatrix} \delta\psi_{\tau} \delta\psi_{\tau}^\top &\textbf{0} \\[0.3em] \textbf{0} &\delta\psi_{\tau}^\top \delta\psi_{\tau} \end{bmatrix} \end{align*} and $\lambda_{max}(\Psi_{\tau})=\|\Psi_{\tau}\|_2=\|\psi_{\tau}\|_2$. Therefore using Freedman inequality \citep{tropp2011freedman}, and the covariance matrix: \begin{align*} W_\tau := \sum_{\tau'=1}^{\tau}\mathbb E[\Delta\Psi^2_{\tau'}|\mathcal F_{\tau'-1}],~\textit{and}, \lambda_{\max}(\Delta\Psi_{\tau})\leq C ~,~\forall \tau \end{align*} for all $\epsilon\geq 0$ and $\sigma^2>0$ \begin{align*} \mathbb P[\exists \tau'\geq 0:\lambda_{\max}(\Psi_{\tau'})\geq \epsilon~ \textit{and}~ \|W_{\tau'}\|\leq \sigma^2]\leq 2S.\exp\lbrace-\frac{-\epsilon^2}{2\sigma^2+2C t/3}\rbrace \end{align*} in other word \begin{align*} \|\Psi_\tau\|_2\leq \frac{2C}{3}+\sqrt{2\sigma^2\log(S/\delta)} \end{align*} with probability at least $1-\delta$. Since $\sigma^2\leq \|W_\tau\|_1\leq N'(s,a)$ where $N'(s,a):=\sum_{y\in\mathcal Y}\sum_{t=1}^{N}\mathbbm{1}(y\in s) z_{t,\tau}(y)$, we have \begin{align*} \|\psi_\tau\|_2\leq \frac{2C}{3}+\sqrt{2N'(s,a)\log(S/\delta)} \end{align*} and for the average $\alpha_\tau := \psi_\tau/(N'(s,a))$ \begin{align*} \|\alpha_\tau\|_2\leq \frac{2C}{3N'(s,a)}+\sqrt{\frac{2\log(S/\delta)}{N'(s,a)}} \end{align*} Since $C\leq 2$ and $S\leq Y$, for $N(s,a)\geq 1$ \begin{align*} \|\alpha_\tau\|_2\leq \sqrt{\frac{4\log(Y/\delta)}{N'(s,a)}} \end{align*} Therefore, if we replace $\delta$ with $\frac{\delta}{20N^7YA}$ we get \begin{equation} \begin{split} &\|\mathbb P(\cdot|s,a)-\widehat{\mathbb P}(\cdot|s,a)\|_1\leq d(s,a) := \sqrt{\frac{28 S\log(2AN/\delta)}{\max\lbrace 1,N'(s,a)\rbrace}}\\ \end{split} \end{equation} Since this inequality holds for any clustering $\S$, it holds for $\S^N$, generated by Alg.~\ref{alg:sm.ucrl} as well. Therefore, at time step $N$, if we are interested in $s^N \in \S^N$, choose $s = s^N$ while $N'(s,a) = N(s^N,a)$, then we have \begin{equation} \begin{split} &\|\mathbb P(\cdot|s^N,a)-\widehat{\mathbb P}(\cdot|s^N,a)\|_1\leq d(s,a) := \sqrt{\frac{28 S^N\log(2AN/\delta)}{\max\lbrace 1,N(s^N,a)\rbrace}}\\ \end{split} \end{equation} \subsection{Minimal Clustering}\label{app:further.clustering} The spectral learning algorithm has been shown to efficiently cluster the observation set to an auxiliary state space of size $X\leq S\leq XA$. As long as different clusters are merged across epochs, we expect $\S^{(k)}$ to tends to $\mathcal X$, yet there is a chance that it converges to a number of auxiliary states $S\neq X$. To make sure that the algorithm eventually converges to the hidden space $\mathcal X$, we include a further clustering technique. We adapt the idea of~\citet{gentile2014online}, \citet{cesa2013gang} and the state aggregation analysis of~\citet{ortner2013adaptive} and perform an additional step of \textit{Reward and Transition Clustering}. In order to simplify the notation, in the following we remove the dependency on $k$, even if all the quantities should be intended as specifically computed at the beginning of epoch $k$. We first recall that given any hidden state $x$ and any action $a$ we have $r(y,a) = r(x,a)$ (\textit{reward similarity}) and $p(\cdot|y,a)=p(\cdot|x,a)$ for all observations $y\in\mathcal Y_x$ (\textit{transition similarity}). The same similarity measures work for auxiliary states via replacing observations with auxiliary states in the above definitions, i.e., given any hidden state $x$ and any action $a$ we have $r(s,a) = r(s,a)$ and $p(\cdot|s,a)=p(\cdot|x,a)$ for all auxiliary states $s\in\S$ that belong to hidden state $x$.\footnote{Notice that this holds since $\S$ is a ``valid'' clustering in high probability.} We also recall that high-probability confidence intervals can be computed for any $s\in \widehat\S$ any $a\in\mathcal{A}$ as \begin{equation}\label{eq:confidence1} \begin{split} &\|p(\cdot|s,a)-\widehat{p}(\cdot|s,a)\|_1\leq d(s,a) := \sqrt{\frac{28\widehat S\log(2AN/\delta)}{\max\lbrace 1,N(s,a)\rbrace}}\\ &|\bar{r}(s,a)-\widehat{r}(s,a)|\leq d'(s,a):=\sqrt{\frac{28\log(2\widehat SAN/\delta)}{\max\lbrace 1,N(s,a)\rbrace}}. \end{split} \end{equation} At any epoch, we proceed by merging together all the auxiliary states in $\widehat S$ whose reward and transition confidence intervals overlap (i.e., $s$ and $s'$ are merged if the confidence interval $[\widehat r(s,a) \pm d_r(s,a)]$ overlaps with $[\widehat r(s',a) \pm d_r(s',a)]$ and $[\widehat p(\cdot|s,a) \pm d_p(s,a)]$ overlaps with $[\widehat p(\cdot|s',a) \pm d_p(s',a)] $) and construct a new set $\widetilde\S$. In practice, the set $\widetilde S$ is constructed by building a fully connected graph on $s\in\widehat S$ where each state $s$ as a node. The algorithm deletes the edges between the nodes when $|\widehat{r}(s,a)-\widehat{r}(s',a)|>d_r(s,a)+d_r(s',a)$ or $|\widehat{p}(\cdot|s,a)-\widehat{p}(\cdot|s',a)|_1>d_p(s,a)+d_p(s',a)$. The algorithm temporarily aggregates the connected components of the graph and consider each disjoint component as a cluster. If the number of disjoint components is equal to $X$ then it returns $\widehat\S$ as the final hidden state $\mathcal X$, otherwise the original auxiliary state space $\widehat \S$ is preserved and the next epoch is started. Notice that if $s$ and $s'$ belong to the same hidden state $x$ then w.h.p.\ their confidence reward and transition intervals in Eq.~\ref{eq:confidence1} overlap at any epoch. Thus in general $\widetilde S\leq X$. Let's define the reward gaps as follows (similar for the transitions) $\forall s,s'\in\widehat \S, \forall a\in\mathcal A$ and the corresponding $x,x'$ \begin{align*} \gamma_r^a(s,s')=\gamma_r^a(x,x'):=\big|\bar{r}(x,a)-\bar{r}(x',a)\big|=\big|\bar{r}(s,a)-\bar{r}(s',a)\big|,\\ \gamma_p^a(s,s')=\gamma_p^a(x,x'):=\|p(\cdot|x,a)-p(\cdot|x',a)\|_1=\|p(\cdot|s,a)-p(\cdot|s',a)\|_1 . \end{align*} where $p(\cdot|x,a),p(\cdot|s,a)\in\widetilde\Delta_{\widehat S-1}$, where $\widetilde\Delta_{\widehat S-1}$ is $(\widehat S-1)$ dimensional simplex. To delete an edge between two states $s,s'$ belonging to two different hidden states, one of the followings needs to be satisfied for at least for one action \begin{align} |\widehat{r}(s,a)-\widehat{r}(s',a)|>d_r(s,a)+d_r(s',a)\Rightarrow \gamma_r^a(s,s')>\sqrt{\frac{28\log(2\widehat SAN/\delta)}{\max\lbrace 1,N(s,a)\rbrace}}+\sqrt{\frac{28\log(2\widehat SAN/\delta)}{\max\lbrace 1,N(s',a)\rbrace}} \end{align} \begin{align}\label{eq:rew_bound} |\widehat{p}(\cdot|s,a)-\widehat{p}(\cdot|s',a)|_1>d_p(s,a)+d_p(s',a)\Rightarrow \gamma_p^a(s,s') >\sqrt{\frac{28\widehat S\log(2AN/\delta)}{\max\lbrace 1,N(s,a)\rbrace}}+\sqrt{\frac{28\widehat S\log(2AN/\delta)}{\max\lbrace 1,N(s',a)\rbrace}} \end{align} For simplicity, we proceed the analysis with respect to reward, the same analysis holds for transition probabilities. The Eq.~\ref{eq:rew_bound} can be rewritten as follows; \begin{align*} \left(\frac{1}{\sqrt{\max\lbrace 1,N(s,a)\rbrace}}+\frac{1}{\sqrt{\max\lbrace 1,N(s',a)\rbrace}}\right)^{-1}\geq\sqrt{\frac{28\log(2\widehat SA^2N/\delta)}{\gamma_r^a(s,s')^2}} \end{align*} which hold when \begin{align}\label{eq:rcluss} \min\lbrace N(s,a),N(s',a) \rbrace > \frac{112\log(2\widehat S AN/\delta)}{\gamma_r^a(s,s')^2}. \end{align} This implies that after enough visits to the auxiliary states $s$ and $s'$, the two states would be split if they belong to different hidden states. We notice that as $\S$ becomes smaller, more and more samples from raw observations are clustered into the auxiliary states, thus making $N(s,a)$ larger and larger. Furthermore, we can expect that the transition gaps may become bigger and bigger as observations are clustered together. The way that spectral method clusters the observation is effected by separability of observations' probability. But the clustering due to reward analysis (or transition or both) is influenced by the separability in reward function (or transition function or both) and depends on gaps. These two methods look at the clustering problem from different point of view, as a consequence, their combination speeds up the clustering task. For simplicity we just again look at the reward function, same analysis applies to transition function as well. \textbf{Regret due to slowness of Reward Clustering (Transition Clustering)} Let ${N}^r_a(s,s')$ denote the required number of sample for each of $s$ and $s'$ to disjoint them. \begin{align*} {N}^r_a(s,s'):=\frac{56\log(2\widehat S AN/\delta)}{\gamma_a(s,s')^2} \end{align*} While the underlying Markov chain is ergodic, with high probability we can say at time step $N(s,s')$, at least for one action $\min\lbrace N(s,a),N(s',a) \rbrace \leq\overline{N}_a(s,s')$ where \begin{align*} N_r(s,s')=\displaystyle\min_a\lbrace\frac{4\tau_{M}}{\alpha_p} \left({N}^r_a(s,s')+\log(YA/\delta)\right)\rbrace \end{align*} In the worse case analysis we might need to have $N_r=\max_{s,s'}N_r(s,s')$ samples, which corresponds to at most $AY\log(N_r)$ episode. At this time, the reward clustering procedure can output the exact mapping. This bound can be enhance even further by considering the reward function together with the transition process. With the same procedure we can define $N_p=\max_{s,s'}N_p(s,s')$ where \begin{align} N_p(s,s')=\displaystyle\min_a\lbrace\frac{4\tau_{M}}{\alpha_p} \left({N}^p_a(s,s')+\log(YA/\delta)\right)\rbrace \end{align} with \begin{align*} {N}^p_a(s,s'):=\frac{112\widehat S\log(2\widehat S AN/\delta)}{\gamma_a(s,s')^2} \end{align*} Again, in the worse case analysis we might need to have $N_p=\max_{s,s'}N_p(s,s')$ samples, which corresponds to at most $AY\log(N_r)$ episode. Therefore the number of required episode for the agent to declare the true mapping w.h.p., is $AY\log (\min \lbrace N_r,N_p\rbrace)$. \section{Proof of Theorem~\ref{thm:regret}} The overall proof is mostly based on the original UCRL proof in~\cite{jaksch2010near}. In the following we refer to the exact steps in the original proof whenever we borrow results directly from it. The regret can be decomposed as follows; \begin{align*} \textit{Reg}_N=N\eta^*-\sum_{t=1}^Nr_t = \sum_{k=1}^K\underbrace{\sum_{t=t_k}^{t_{k+1}-1} \big(\eta^* - \overline r(x_t,a_t)\big)}_{\Delta_k} + \sum_{t=1}^N \big( \overline r(x_t,a_t) - r_t \big), \end{align*} where $K$ is the total (random) number of episodes, $x_t$ is the hidden state of the MDP at time $t$, same as $r_t$ is the reward at time $t$, and $\overline r(x_t,a_t)$ is the true mean of reward. Using Heffting inequallity, as in Eq.~8 in~\cite{jaksch2010near}, the last term can be bounded as $O(\sqrt{N \log(1/\delta)}$ with high probability. We then focus on the per-step regret $\Delta_k$. At any epoch $k$, from Corollary~\ref{cor:partial.cluster.est} we know that any auxiliary state $s\in\widehat\S^{(k)}$ is a cluster of observations with same same hidden state. As a result, the reward $\overline r(x_t,a_t)$ is equivalent to $\overline r(y_t,a_t)$ (recall that all observations have the same reward as their hidden state). Therefore; \begin{align*} \Delta_k = \sum_{a,s\in\widehat\S^{(k)}} \nu^{(k)}(s,a) \big(\eta^* - r(s,a)\big). \end{align*} The case when confidence intervals fail is bounded as in the original analysis. We now proceed with the same decomposition as done in~\cite{jaksch2010near} (Eqs.10,13,16) and obtain\footnote{Here we ignore the additive regret coming from approximate extended value iteration that accounts for an extra $O(\sqrt{N})$ regret at the end.} \begin{align*} \Delta_k = \bm{\nu}^{(k)} \big( \widetilde{P}^{(k)} - P^{(k)} \big)w^{(k)} + \sum_{s\in\widehat\S^{(k)},a} \nu^{(k)}(s,a)\big( \widetilde{r}^{(k)}(s,a) - \overline r(s,a) \big) + \bm{\nu}^{(k)} \big( I - P^{(k)} \big)w^{(k)}, \end{align*} where $\bm{\nu}^{(k)}$ is the vector of number of samples to auxiliary states in epoch $k$, $P^{(k)}$ (resp. $\widetilde{P}^{(k)}$) is the true (resp. optimistic) transition matrix over auxiliary states of policy $\pi^{(k)}$, $w^{(k)}$ is the centered version of the bias function returned by extended value iteration and $\widetilde{r}^{(k)}(s,a)$ is the optimistic reward. The first two terms account for the errors in estimating the dynamics and rewards of the (auxiliary) MDP and can be bounded as \begin{align*} \Delta_k \leq \square D_{\widehat\S^{(k)}} \sqrt{\widehat S^{(k)} \log(1/\delta)} \sum_{s\in\widehat\S^{(k)},a} \sqrt{N^{(k)}(s,a)} + \bm{\nu}^{(k)} \big( I - P^{(k)} \big)w^{(k)}, \end{align*} where $D_{\widehat\S^{(k)}}$ is the diameter of the auxiliary MDP at epoch $k$ and $\square$ denotes universal numerical constants. The remaining term can be cumulatively bounded following similar steps as in Eq.18 in~\cite{jaksch2010near} with the only difference that the range of $w^{(k)}$ changes at each epoch. Thus we have \begin{align*} \sum_{k=1}^K \bm{\nu}^{(k)} \big( I - P^{(k)} \big)w^{(k)} \leq \square \sqrt{\sum_{k=1}^K D_{\widehat\S^{(k)}} \nu^{(k)}}, \end{align*} where $\nu^{(k)} = \sum_{s,a} \nu^{(k)}(s,a)$ is the length of epoch $k$. Grouping all the terms lead to the first regret statement \begin{align*} \textit{Reg}_N \leq \square\bigg(\sum_{k=1}^K D_{\widehat\S^{(k)}} \sqrt{\widehat S^{(k)} \log(1/\delta)} \sum_{s\in\widehat\S^{(k)},a} \sqrt{N^{(k)}(s,a)} + \sqrt{\sum_{k=1}^K D_{\widehat\S^{(k)}} \nu^{(k)}}\bigg). \end{align*} \subsection{Clustering Rate}\label{app:clustering.rate} The first regret bound still contains random quantities in terms of the auxiliary MDPs generated over episodes. In this section we derive bounds on the number of steps needed to cluster observations. We notice that the analysis is extremely ``pessimistic'' and as we take worst-case values for all the quantities involved in the analysis. \textbf{Time to clustering.} We proceed as follows. We first compute the minimum number of samples $\overline N(y)$ to guarantee that an observation $y$ is correctly clustered. We then compute the length $\overline \nu(y)$ of an epoch so that $\overline N(y)$ samples are collected. Finally, we derive how many epochs $\overline K(y)$ are needed before an epoch of length $\overline \nu(y)$ is run. We start by defining the probability that a certain action is explored. Let $\pi$ be an arbitrary policy such that action $a$ is taken in at least one observation $y$ belonging to a hidden state $x=x_y$. Whenever an agent is in state $x$, there is a probability $\mathbb P(y|x)$ to observe $y$ and thus trigger action $a$. We define the probability of ``observing'' an action $a$ in state $x$ under policy $\pi$ as \begin{align}\label{eq:prob.action} \alpha_\pi(l) = \sum_{y\in x} \mathbb P(y|x) \mathbbm{1}(\pi(y)=a). \end{align} Since we assumed that $l$ is taken in at least one observation $\alpha_\pi(l)$ is always non-zero and it indeed lower-bounded by $O_{\min}$. We define $\alpha_p:=\min_{k\in[K],x\in\mathcal{X},a\in\mathcal{A}} \alpha_{\pi^{(k)}}(a)$ as the worst proportion across all epochs, states, and actions. Now we need need to know how fast the set $\mathcal{ S}^{(k)}$ converges to set $\mathcal{X}$, in other work how fast is the clustering process. From Eq.~\ref{eq:BO} and the clustering process, we know that an observation $y$ is clustered in $x_y$ if the number of samples $\nu^{(k)}(l)$ obtained from the action $l$ executed in $y$ within a given epoch $k$ is such that \begin{align}\label{eq:clustering.condition} f_O(y|x(y))\geq 2C^{(k)}_O(l)\sqrt{\frac{\log(1/\delta)}{\nu^{k}(l)}}. \end{align} By reverting the bound and taking the worse case over actions and epochs, we obtain that a sufficient condition is to collect at least $\overline N(y)$ samples, with \begin{align*} \overline N(y) := \max_{l,k} 4C^{(k)}_O(l)\frac{\log(1/\delta)}{(f_O(y|x(y)))^2}. \end{align*} We can now leverage on the ergodicity of the MDP and the probability of observation of an action $\alpha_p$ to find the minimum number of steps with an epoch to guarantee that with high probability the condition in Eq.~\ref{eq:clustering.condition} is satisfied. We define the worst-case mean returning time as $\tau_M=\max_\pi\max_x \mathbb{E}[\tau_\pi(x\rightarrow x)]$, where $\tau_\pi(x\rightarrow x)$ is the random time to go from $x$ back to $x$ through policy $\pi$. By Markov inequality, the probability that it takes more than $2\tau_M$ time step to from first visit of state $x$ to its second visit is at most $1/2$. Given the definition of $\alpha_p$, it is clear that if the action $l$ is taken in state $x$ then, this action will be taken at state $x$ for $\alpha_p$ portion of the time. If we divide the episode of length $\nu$ into $\nu\alpha_p/2\tau_M$ intervals of length $2\tau_M/\alpha_p$, we have that within each interval we have a probability of $1/2$ to observe a sample from state $x$ and take a particular action. Therefore, the lower bound on the average number of time that the agent takes any action $l$ (that has a non zero probability to be executed in a state $x$) is $\nu\alpha_p/4\tau_M$ samples. Thus from Chernoff-Hoeffding, we obtain that the number of samples of any feasible action in the epoch with length $\nu$ is as follows; \begin{align*} \forall x\in \mathcal{X},\forall l\in range\lbrace\widetilde{\pi}(\cdot|x)\rbrace;~\nu(l)\geq \frac{\nu\alpha_p}{4\tau_M}-\sqrt{\frac{\nu\alpha_p\log(XA/\delta)}{2\tau_M}} \end{align*} with probability at least $1-\delta$. At this point, we can derive a lower bound on the length of the episode that guarantee the desired number of samples to reveal the identity of any observation is reached. For observation $y$, we solve \begin{align*} \frac{\nu\alpha_p}{4\tau_M}-\sqrt{\frac{\nu\alpha_p\log(XA/\delta)}{2\tau_M}}\geq \overline{N}(y) \end{align*} and we obtain the condition \begin{align*} \sqrt{\nu} \geq \sqrt{\frac{2\tau_{M}}{\alpha_p} \log(XA/\delta)} + \sqrt{\frac{2\tau_{M}}{\alpha_p} \log(XA/\delta) + \frac{4\tau_{M}}{\alpha_p}\overline{N}(y)}, \end{align*} which can be simplified to \begin{align}\label{eq:v.condition} \nu \geq \overline{\nu}(y) := \frac{4\tau_{M}}{\alpha_p} \left(\overline{N}(y)+\log(XA/\delta)\right). \end{align} With the same argument in App.~D in~\cite{azizzadenesheli2016reinforcement} the number of required epochs to reveal observation $y$ is $\overline{K}(y)\leq AY\log_2(\overline{\nu}(y))+1$. \textbf{Time to clustering.} Let $y_{\text{first}} = \arg\min_{y\in\mathcal Y} \overline K(y)$ be the first observation that could be clustered\footnote{This should be intended as the first observation that is clustered in the worst case. In practice, depending on the policy and the randomness in the process, other observation may actually be clustered well before $y_1$.} then we define $K_{\text{first}} = \overline K(y_{\text{last}})$ as the number of episodes and $N_{\text{first}} = 4 AY \overline\nu(y_{\text{first}})$ the total number of steps needed before clustering $y_1$ correctly. Similarly, let $y_{\text{last}} = \arg\max_{y\in\mathcal Y}$ and $K_{\text{last}} = \overline K(y_{\text{last}})$ as the number of episodes and $N_{\text{last}} = 4 AY \overline\nu(y_{\text{last}})$ the total number of steps needed before clustering $y_{\text{last}}$ correctly. Since all the other observations will be clustered before $y_{\text{last}}$ we can say that by epoch $K_{\text{last}}$ all observations will be clustered. As discussed in Lem.~\ref{lem:partial.cluster} this does not necessarily correspond to the hidden state $\mathcal X$ but it could be an auxiliary space $\S$ with at most $AX$ states. \textbf{Validity of the bound in Lemma~\ref{lem:decomp}.} We also notice that Lemma~\ref{lem:decomp} requires a minimum number of samples $N_0$ before the concentration inequality on the estimate of $V_2$ holds. Applying a similar reasoning as for the time for clustering, we can derive a bound on the number of episodes $K_{\text{sm}}$ and number of samples $N_{\text{sm}}$ needed before the spectral method actually works (from a theoretical point of view). As a result, a more accurate definition of $K_{\text{first}}$ and $K_{\text{last}}$ (resp. for $N$) should take the maximum between the values derived above and $K_{\text{sm}}$. \section{Proofs of Section~\ref{sec:stochastic}} \section{Proof of Lemma~\ref{lem:decomp}} At the end of each epoch, e.g. $k$, we estimate the factor matrix $V^{(l)}_2$ (for all $l\in[A]$) using all the samples collected during that epoch according to policy $\widetilde\pi^{(k)}$. In order to simplify the notation, in the following we remove the dependency on $k$, even if all the quantities should be intended as specifically computed at the beginning of epoch $k$. In order to bound the empirical error of the moment estimators, we need to consider the properties of the Markov chain generated by policy $\widetilde\pi$ and the fact that a single continuous trajectory is observed. In particular, we have to carefully consider the mixing time of the underlying Markov chain (the amount of time it takes that the underlying Markov chain converges to its stationary distribution) and exploit the martingale property of the trajectory. This problem has been previously studied by~\citet{azizzadenesheli2016reinforcement} for the general case of partially observable MDPs. Since ROMDPs\xspace are a special case of POMDPs, we can directly rely on the following general concentration inequality. For any ergodic Markov chain with stationary distribution $\omega_{\widetilde\pi}$, let $f_{1\rightarrow t}(x_t|x_1)$ by the distribution over states reached by a policy $\widetilde\pi$ after $t$ steps starting from an initial state $x_1$. The inverse mixing time $\rho_{\text{mix},\pi}(t)$ of the chain is defined as \begin{align*} \rho_{\text{mix},\widetilde\pi}(t)=\sup_{x_1}\left\|f_{1\rightarrow t}(\cdot|x_1)-\omega_{\widetilde\pi}\right\|_{\text{TV}}, \end{align*} where $\| \cdot\|_{\text{TV}}$ is the total-variation metric. \citet{kontorovich2014uniform} show that for any ergodic Markov chain the mixing time can be bounded as \begin{align*} \rho_{\text{mix},\widetilde\pi}(t)\leq G(\widetilde\pi)\theta^{t-1}(\widetilde\pi), \end{align*} where $1\leq G(\widetilde\pi)<\infty$ is the \textit{geometric ergodicity} and $0\leq \theta(\widetilde\pi)<1$ is the \textit{contraction coefficient} of the Markov chain generated by policy $\widetilde\pi$. \begin{proposition}[Theorem 11 \cite{azizzadenesheli2016reinforcement} POMDP concentration bound]\label{lem:POMDPconstration} Consider a sequence of $\nu$ observations $\{y_1,\ldots,y_\nu\}$ obtained by executing a policy $\widetilde\pi$ in a POMDP starting from an arbitrary initial hidden state. For any action $l\in[A]$, $\nu^{(l)}$-length sequence $b^{(l)} = \{(y_{t-1},y_{t},y_{t+1});a_t=l\}$, and any $c$-Lipschitz\footnote{under the Hamming metric} matrix valued function $\Phi(\cdot):b^{(l)}\rightarrow \mathbb{R}^{Y\times Y}$, we have \begin{align*} \big\|\Phi(b^{(l)})-\mathbb{E}[\Phi(b^{(l)})]\big\|_2\leq \frac{G(\widetilde\pi)}{1-\theta(\widetilde\pi)} \Big(1+\frac{1}{\sqrt{2}c(\nu^{(l)})^\frac{3}{2}}\Big)\sqrt{8c^2\nu^{(l)}\log\Big(\frac{2Y}{\delta}\Big)} \end{align*} with probability at least $1-\delta$, where $G(\widetilde\pi)$ and $\theta(\widetilde\pi)$ are, respectively, the geometric ergodicity and the contraction coefficient of the underlying Markov chain on the hidden states (they define how fast the underlying Markov chain converges to its stationary distribution), and the expectation is with respect to the distribution of initial state equals to the stationary distribution. \end{proposition} The parameters $1\leq G(\pi^{(k)})<\infty$ and $0\leq\theta(\pi^{(k)})<1$ are well defined for Markov chain and shows the state distribution of the Markov chain convergence to its stationary distribution (if such distribution exists) with rate of $G(\pi)\theta(\pi)^t$. (the lower $G(\pi)$ and $\theta(\pi)$ give the lower mixing for corresponding Markov chain.) In this paper, we are interested in moments of our data, therefore, $\Phi(\cdot)$ is considered as a moment estimator. Given the ergodicity assumption (Asm.~\ref{asm:ergodicity}) under any policy, we can apply Proposition~\ref{lem:POMDPconstration} to bound the errors for both second and third order moments. For any $\lbrace p,q,r\rbrace$ a permutation of set $\lbrace 1,2,3 \rbrace$ \begin{align}\label{eq:perterbationM_2} \|\widehat K_{p,q}^{(l)}-K^{(l)}_{p,q}\|_2&\leq G(\pi)\frac{1+\frac{1}{\sqrt{2}c(\nu^{(k)}(l))^\frac{3}{2}}}{1-\theta }\sqrt{8c^2\nu^{(k)}(l)log(\frac{2Y}{\delta})}\\ \|\widehat M_{p,q,r}^{(l)}-M_{p,q,r}^{(l)}\|_2&\leq G(\pi)\frac{1+\frac{1}{\sqrt{2}c(\nu^{(k)}(l))^\frac{3}{2}}}{1-\theta }\sqrt{8c^2\nu^{(k)}(l)log(\frac{2Y^{1.5}}{\delta})} \end{align} with probability at least $1-\delta$. At this point we can proceed with applying the robust tensor power method proposed in~\citep{anandkumar2012method} to recover $V_2^{(l)}$ and obtain the guarantees of Lemma 5 of~\citet{azizzadenesheli2016reinforcement} through Proposition~\ref{lem:concentration}, where $c=\frac{1}{\nu^{(k)}(l)}$. We report a more detailed version of the statement of Lemma~\ref{lem:decomp}. \begin{lemma}[Concentration Bounds]\label{lem:concentration} The robust power method of~\citet{anandkumar2012method} applied to tensor $\widehat M_3^{(l)}$ returns the $X^{(l)}$ columns of matrix $V_2^{(l)}$ with the following confidence bounds \begin{align}\label{eq:factor} \Big\|[V_2^{(l)}]_{(\cdot|i)}-[\widehat{V}^{(l)}_2]_{(\cdot|i)}\Big\|_2\leq \epsilon_3^{(l)}=C^{l}_O\sqrt{\frac{\log(2Y^{3/2}/\delta)}{\nu^{(l)}}}:=\mathcal{B}_O^{(l)} \end{align} if \begin{align}\label{eq:nl.condition1} \nu^{(l)}&\geq\overline{N}:=\max_\pi\left(\frac{4}{{\omega^{(l)}_{\widetilde\pi_{\min}}\min\limits_{m\in\{1,2,3\}}\lbrace\sigma^2_{\min}(V^{(l)}_m)\rbrace}}\right)^2\log(2\frac{(Y^{1.5})}{\delta})\Theta^{(l)}\\ \Theta^{(l)}&:=\max\left\{\frac{16(X^{(l)})^{\frac{1}{3}}}{C_1^{\frac{2}{3}}(\omega^{(l)}_{\widetilde\pi_{\min}})^{\frac{1}{3}}} ,4,\frac{2\sqrt{2}X^{(l)}}{C_1^2\omega^{(l)}_{\min}\min\limits_{m\in\{1,2,3\}}\lbrace\sigma^2_{\min}(V^{(l)}_m)\rbrace}\right\}, \end{align} with probability at least $1-\delta$, where $C_1$ is a problem-independent constants and $\omega^{(l)}_{\widetilde\pi_{\min}}:=\min_{i\in\mathcal X^{(l)}}\mathbb P_{\widetilde\pi}(x=i|a=l)$ where the minimization is over non-zero probabilities. In addition, the $\sigma_{\min}(\cdot)$ operator returns the smallest non-zero singular value of its input matrix. The values of the error $\mathcal{B}_O^{(l)}$ under policy $\widetilde\pi$ is defined (see Eq.29 of~\citet{azizzadenesheli2016reinforcement}) \begin{equation} \label{eq:epst1} \mathcal{B}_O^{(l)}:=G(\widetilde\pi)\frac{4\sqrt{2}+2}{(\omega^{(l)}_{\widetilde\pi_{\min}})^{\frac{1}{2}}(1-\theta(\widetilde\pi))}\sqrt{\frac{\log(2\frac{(2Y)}{\delta})}{\nu^{(l)}}}+\frac{8\widetilde{\epsilon}^{(l)}}{\omega^{(l)}_{\widetilde\pi_{\min}}}, \end{equation} where \begin{align*} \widetilde{\epsilon}^{(l)} \leq \frac{2\sqrt{2} G(\widetilde\pi)\frac{2\sqrt{2}+1}{1-\theta(\widetilde\pi)}\sqrt{\frac{\log(\frac{2(Y^{\frac{3}{2}})}{\delta})}{\nu^{(l)}}}}{((\omega^{(l)}_{\widetilde\pi_{\min}})^{\frac{1}{2}}\min\limits_{m\in\{1,2,3\}}\lbrace\sigma_{\min}(V^{(l)}_m)\rbrace)^3}+\frac{\left(64 G(\widetilde\pi)\frac{2\sqrt{2}+1}{1-\theta(\widetilde\pi)}\right)}{{\min\limits_{m\in\{1,2,3\}}\lbrace\sigma^2_{\min}(V^{(l)}_m)\rbrace}(\omega^{(l)}_{\widetilde\pi_{\min}})^{1.5}}\sqrt{\frac{\log\Big(2\frac{Y^{\frac{3}{2}}}{\delta}\Big)}{\nu^{(l)}}}, \end{align*} \end{lemma} We notice that the columns of $V_2^{(l)}$ are all orthogonal (but not orthonormal) since the clusters are non-overlapping and an observation $j$ that can be obtained from a state $i$ cannot be generated by any other state $i'$ (i.e., for any $i\neq i'$, $[V_2^{(l)}]_{:,i}^\mathsf{T} [V_2^{(l)}]_{:,i'} = 0$). As a result, Eq.~\ref{eq:second.moment} can be seen as an eigendecomposition of $M_2^{(l)}$, where the columns $[V_2^{(l)}]_{:,i}$ are the eigenvectors and $\omega_\pi^{(l)}(i)$ are the eigenvalues. More formally, let $M_2^{(l)} = U\Sigma U^{\mathsf{T}}$ be the eigendecomposition of $M_2^{(l)}$, if all eigenvalues are distinct, the eigenvectors in $U$ can be used to recover $V_2^{(l)}$ up to a mapping function and multiplicative factors. Nonetheless, in general $V_2^{(l)}$ may have eigenvalues with multiplicity and the eigendecomposition of $M_2^{(l)}$ may return a wrong clustering since observations generated by distinct states (and thus with different rewards and dynamics) may be aggregated together. In this case, we have to move to the third order statistics to disambiguate between observations and cluster them properly. The computational complexity of this methods appeared at \citep{wang2015fast}, and \citep{song2013nonparametric}. \section{Rank recovery}\label{sec:rank} Lemma~\ref{lem:concentration} holds when the rank of matrix $V_2^{(l)}$ is known in advance. While this is not the case in practice, here we show how one can estimate the rank $r = \big|\mathcal X^{(l)}_{\pi^{(k)}}\big|$ of $V_2^{(l)}$. Given the expansiveness of latent MDP (Asm.~\ref{asm:expansive}), we have that for any policy $\pi$ and any action $l$, $\big|\mathcal X^{(l)}_{\pi}\big|\leq\big|\overline\mathcal X^{(l)}_{\pi}\big|$. The rank of the second moment matrix $K^{(l)}_{2,3}$ is then $\min\lbrace \big|\mathcal X^{(l)}_{\pi^{(k)}}\big|,\big|\overline\mathcal X^{(l)}_{\pi^{(k)}}\big| \rbrace = r$, which also corresponds to the number of non-zero columns in matrix $V_2^{(l)}$. We can then try to estimate $r$ through the estimate second moment $\widehat{K}^{(l)}_{2,3}$, which according to Eq.~\ref{eq:perterbationM_2}, estimates $K^{(l)}_{2,3}$ up to an additive error $\epsilon^{(l)}_{2,3}$ that decreases as $O(\sqrt{\frac{1}{\nu^{(l)}}})$. This means that the highest perturbation over its singular values is also at most $O(\sqrt{\frac{1}{\nu^{(l)}}})$. We introduce a threshold function $g^\epsilon(\nu^{(l)})$ that satisfies the condition \begin{align}\label{eq:rankcondition} \epsilon^{(l)}_{2,3} \leq g^\epsilon(\nu^{(l)})\leq 0.5\sigma_{r}, \end{align} where $\sigma_{r}$ is the smallest non-zero singular value of $K^{(l)}_{2,3}$. We then perform a SVD of $\widehat K^{(l)}_{2,3}$ and discard all singular values with value below the threshold $g^\epsilon(\nu^{(l)})$. Therefore, with probability at least $1-\delta$, the number of remaining singular values is equal to the true rank $r$. We are left with finding a suitable definition for the threshold function $g^\epsilon$. From the condition on Eq.~\ref{eq:rankcondition}, we notice that we need $g^{\epsilon}$ to be smaller than a fixed value (RHS) and, at the same time, greater than a decreasing function of order $\mathcal{O}(\sqrt{\frac{1}{\nu^{(l)}}})$ (LHS). Then it is natural to define \begin{align*} g^\epsilon(\nu^{(k)}(l))=\frac{g}{\nu^{(k)}(l)^{0.5-\epsilon}} \end{align*} for a suitable $g>0$ and with $0<\epsilon<0.5$. Therefore there is a number $N_0^{(l)}$ such that for all $\nu^{(l)}\geq N_0^{(l)}$ the condition on Eq.~\ref{eq:rankcondition} is satisfied and Lemma~\ref{lem:concentration} holds. Therefore we restate the sample complexity in Lemma~\ref{lem:concentration} by adding the extra term to \begin{align*} \overline{N}\leftarrow \overline{N} + N_0(l). \end{align*} Let $\overline{N}_{\max}$ denotes the maximum of this threshold for any action and policy. \section{Proof of Lemma \ref{lem:partial.cluster}} Under policy $\pi$, Fig.~\ref{fig:observation.policy}\textit{-right} shows the structure of $V_2^{(l)}$. Given action $l$, the matrix $V_2^{(l)}$ contains $X_{\pi}^{(l)}$ columns and each column corresponds to a column in emission matrix (up to permutation). We showed that the knowledge about a column of $V_2^{(l)}$ reveals part of the corresponding column in emission matrix, the entries with non-zero $\pi(y|l)$. The policy, in general, partitions the observation space to at most $A$ partitions, $\mathcal Y_l\forall l\in\mathcal A$ and maps each partition to an action. It means that when we condition on an action, e.g., $l$, we restrict ourselves to the part of observation space $\mathcal Y_l$ and the input to the spectral learning algorithm is set $\mathcal Y_l$. Therefore, the algorithm is able to partition this set to $X_{\pi}^{(l)}$ partition. Because of the unknown permutation over columns of $V_2^{(l)}$ for different actions, we are not able to combine the resulting clustering give different actions. If we enumerate over actions, we end up with $A$ partition $\mathcal Y_l$ and then we partition each set $\mathcal Y_l$ to at most $X$(upper bound on $X_{\pi}^{(l)}$), as a consequence, we might end up with at most $XA$ disjoint clusters. \section{Proof of Lemma \ref{lem:tensor.decomposition}} We first study the eigendecomposition of $M_2^{(l)}$ when its eigenvalues have multiplicity 1. \begin{lemma}\label{lem:matrix.decomposition} For any action $l\in[A]$, let the second moment in Eq.~\ref{eq:second.moment} have the eigendecomposition $M_2^{(l)} = U\Sigma U^{\mathsf{T}}$. If all eigenvalues of $M_2^{(l)}$ have multiplicity 1, there exists a mapping $\sigma^{(l)}:X\rightarrow X$ and multiplicative constants $\{C_i^{(l)}\}_{i\in[X]}$, such that for any $i\in\mathcal X_\pi^{(l)}$ and $j\in[Y]$, $[V_2^{(l)}]_{j,\sigma^{(l)}(i)} = C^{(l)}_i[U]_{j,i}$. As a result, for any hidden state $i\in\mathcal X_\pi^{(l)}$ we define the cluster $\widetilde{\mathcal Y}_i^{(l)}$ as \begin{align}\label{eq:estimated.cluster1} \widetilde{\mathcal Y}_i^{(l)} = \{j\in[Y]: [U]_{j,i} > 0\} \end{align} and we have that if $j,j'\in \widetilde{\mathcal Y}_i^{(l)}$ then $j,j'\in \mathcal Y^{(l)}_{\sigma(i)}$ (i.e., observations that are clustered together in $\widetilde{\mathcal Y}_i^{(l)}$ are clustered in the original ROMDP\xspace). \end{lemma} In Eq.~\ref{eq:second.moment} we show that matrix $M_2^{(l)}$ is a symmetric matrix and has the following representation; \begin{align*} M_2^{(l)}:= \sum_{i\in\mathcal X^{(l)}_{\pi}} \!\!\omega_\pi^{(l)}(i) [V_2^{(l)}]_{:,i} \otimes [V_2^{(l)}]_{:,i}. \end{align*} As long as $V_2^{(l)}]_{:,i}$ for $i\in\mathcal X^{(l)}_{\pi}$ are orthogonal vectors, this matrix has rank of $X^{(l)}_{\pi}$ with the following eigendecomposition; \begin{align*} M_2^{(l)} = U\Sigma U^{\mathsf{T}} \end{align*} where the matrix $U$, up to permutation, is the orthonormal version of $V_2^{(l)}$, and $\Sigma$ is a diagonal matrix of rank $X^{(l)}_{\pi}$ with diagonal entries equal to $\omega_\pi^{(l)}$ multiplied by the normalization factors. As a result we can use the decomposition to directly recover the non zero elements of $V_2^{(l)}$ and the corresponding partial clustering. Let's consider the $i$'th and $j$'th nonzero diagonal entries of matrix $\Sigma$, $\sigma_i$ and $\sigma_j$, with eigenvectors of $U_i, U_j$, i.e., $M_2^{(l)}U_i=\sigma_iU_i~,~M_2^{(l)}U_j=\sigma_jU_j$. In the case of no eigengap, i.e., $\sigma_i=\sigma_j$, for any $0\leq\lambda\leq 1$ we have $M_2^{(l)}(\lambda U_i+(1-\lambda)U_j)= \sigma_i (\lambda U_i+(1-\lambda)U_j)=\sigma_j(\lambda U_i+(1-\lambda)U_j)$. Therefore, any direction in the span of $span(U_i,U_j)$ is an eigenvector and the matrix decomposition is not unique, and we can not learn the true $V_2^{(l)}$. We relax this issue by deploying tensor decomposition of higher order moments. The proof of Lemma \ref{lem:tensor.decomposition} directly follows from the properties of tensor decomposition in~\cite{anandkumar2014tensor} and the use of $V_2^{(l)}$ to generate a partial clustering. \section{Extended Discussion}\label{Aux:deter} \begin{conjecture} We can extend this results to deterministic MDP. In this case we can first uniformly explore the latent space and collect sufficient number of sample to find the exact clustering and reduce the large MDP to the latent MDP and then apply \textsc{\small{UCRL}}\xspace on the latent MDP. Which can suffer a constant regret of pure exploration at the beginning and regret of $\mathcal{\widetilde O}(D_\mathcal X X\sqrt{AN})$ due to \textsc{\small{UCRL}}\xspace in the second phase. One of the main open questions is whether the spectral clustering method could still provide ``useful'' clusterings when the state space is not fully visited (i.e., in case of non-ergodic MDP), so that observations are properly clustered where it is actually needed to learn the optimal policy. We can provide a partial answer in the case of deterministic ROMDPs\xspace. In fact, despite not being ergodic, in this case we can first uniformly explore the latent space and collect sufficient number of sample to find the exact clustering and reduce the large MDP to the latent MDP and then apply \textsc{\small{UCRL}}\xspace on the latent MDP. These two-phase algorithm would suffer a constant regret of pure exploration at the beginning and regret of $\mathcal{\widetilde O}(D_\mathcal X X\sqrt{AN})$ due to \textsc{\small{UCRL}}\xspace in the second phase. \end{conjecture} In RL problems, the principle of Optimism-in-Face-of-Uncertainty contributes in designing a policy that locally improves the model uncertainty and average reward which has been shown to be an optimal strategy. It is an open question to analyze and modify this principle for the models with clustering where global improvement of the information in model uncertainty is required. While the $\textsc{\small{SL-UCRL}}\xspace$ for deterministic models reaches order optimal regret, it is not still clear how to modify the exploration to enhance the constant regret of the pure exploration phase. \textbf{Compared to POMDP}. We can compare this result with the regret bound of~\citet{azizzadenesheli2016reinforcement} for POMDPs, which are a more general class than ROMDPs\xspace. Recalling that POMDPs are characterized by a diameter \begin{align*} D_{pomdp}:=\max_{x,x',a,a'}\min_{\pi\in\mathcal{P}} \mathbb{E}[\tau\left(\left(x,a\right)\rightarrow\left(x',a'\right)\right)], \end{align*} the regret derived by~\citet{azizzadenesheli2016reinforcement} scales as $\widetilde{\mathcal{O}}(D_{pomdp}X^{3/2}\sqrt{AYN})$. The regret suffers from additional term $\sqrt{Y}$ because the RL algorithm in POMDP put much effort on accurate estimation of entries of $O$ matrix and does not exploit its specific structure. Moreover, there is an additional factor $X$ in regret bound due to learning of transition tensor through spectral methods. \section{Additional Experiments}\label{app:experiment} While the results reported in the main text are obtained on actual ROMDPs\xspace, here we test \textsc{\small{SL-UCRL}}\xspace on random MDPs with no explicit hidden space. The objective is to verify whether \textsc{\small{SL-UCRL}}\xspace can be used to identify (approximate) clusters. Since \textsc{\small{SL-UCRL}}\xspace in high probability only clusters observations that \textit{actually} belong to the same hidden state, in this case \textsc{\small{SL-UCRL}}\xspace would reduce to run simple UCRL, as there is no two observations that can be \textit{exactly} clustered. In order to encourage clustering, we half the (exact) confidence intervals in the attempt of trading off a small bias with a significant reduction in the variance. We compare the regret on three random MDPs with increasing number of states. As it is shown in Fig.~\ref{fig:comparisonCar}, \textsc{\small{SL-UCRL}}\xspace is effective even in this scenario compared to UCRL and DQN. In fact, we see from Fig.~\ref{fig:comparisonCar}-right that \textsc{\small{SL-UCRL}}\xspace is able to find clusters without compromising the overall regret. While the number of states now directly affects the performance of \textsc{\small{SL-UCRL}}\xspace, we see that it is more robust than the other algorithms and its regret is not severely affected by an increasing number of observations. \begin{figure}[ht] \centering \includegraphics[height=4.5cm]{Fig/Plot/regretcompareMDP.eps} \hspace{0.5in} \centering \includegraphics[height=4.5cm]{Fig/Plot/regretcompareMDP.eps} \caption{\textit{(left)}The regret comparison, $A=4$, from top to bottom, $Y=10,20,30$. The scale is $\sqrt{T}$. \textit{(right)} Learning rate of \textsc{\small{SL-UCRL}}\xspace compared to \textsc{\small{UCRL}}\xspace and DQN. After first few rounds, it learns the the true mapping matrix. The numbers in the bulbs are the cardinality of Aux-MDP. } \label{fig:comparisonCar} \end{figure}

BkiUbi_xaKgS2Gx0cWZR

\section{Introduction}\label{Sec:Intro} Plasmonics has experienced a drastic paradigm shift during the past decade, with rapid advances in nanofabrication and characterisation allowing metallic nanostructures to enter the quantum regime~\cite{tame_natphys9,zhu_natcom7,bozhevolnyi_nanophot6,Bozhevolnyi,tserkezis_ijmpb31_0}, defined by components of just a few nm in size~\cite{palpant_prb57,berciaud_nl5,scholl_nat483,raza_natcom6} and separations of only a few \AA~\cite{duan_nl12,kern_nl12}. In such situations, quantum effects like nonlocal screening~\cite{abajo_jpcc112,raza_jpcm27}, surface-enhanced Landau damping~\cite{shahbazyan_prb94,khurgin_acsphoton4}, electron spill-out~\cite{teperik_prl110,toscano_natcom6} and tunnelling~\cite{savage_nat491,scholl_nl13,tan_sci343,lerch_ijmpb31}, which cannot be captured by the common local response approximation (LRA) of classical electrodynamics, quickly become important and dominate the optical response. To deal with this issue, and overcome the limitations of the common free-electron-gas description in theoretical studies, a variety of quantum-informed models has been proposed. Benefiting from the enormous experience acquired over the decades through studies of metals in solid-state physics, models based on screening and the resulting nonlocal dielectric function~\cite{Ashcroft}, the $d$-parameter formalism of Feibelman~\cite{feibelman_progsurfsci12}, or (time-dependent) density functional theory (TD-DFT)~\cite{lang_prb1,Marques,varas_nanophot5,zhang_ijmpb31} have been developed. For instance, the hydrodynamic Drude model (HDM), already introduced in the 1930s~\cite{bloch_zphys81}, keeps reemerging in various forms due to its success in reproducing screening effects in noble metals~\cite{ruppin_prl31,fuchs_prb35,leung_prb42,mcmahon_jpcc114,raza_prb84,david_acsnano8,trugler_ijmpb31}, and approaches to extend its applicability are being proposed~\cite{toscano_natcom6,luo_prl111,ciraci_prb95}. One such extension, that also accounts in an efficient way for Landau damping, is provided by the generalised nonlocal optical response (GNOR) theory~\cite{mortensen_natcom5}. Other approaches to tackle Landau damping are typically based on a modified, quantum-mechanically obtained damping rate~\cite{shahbazyan_prb94,khurgin_acsphoton4}. On the other hand, the quantum-corrected model (QCM)~\cite{esteban_natcom3} has proven efficient in treating plasmonic dimers with gap distances in the quantum-tunnelling regime~\cite{savage_nat491,scholl_nl13} when the sizes involved render full TD-DFT calculations prohibitive. Finally, $d$-parameter approaches~\cite{yan_prl115,christensen_prl118} have attracted much interest recently, as they introduce the induced charges and currents of a quantum-mechanical calculation into an otherwise classical computation, accounting in principle for all aspects of quantum predictions relevant to plasmonics. To move through this diversity of complementing -- or sometimes contradicting -- approaches, a robust test bed for assessing their validity and pertinence is required. Here we show that a promising such template can be found in chiral metallic nanoparticle (NP) helices~\cite{fan_jpcc115,kuzyk_nat483}, whose optical response is dominated by a double-peak absorption resonance in the visible. By calculating absorption spectra of such helices for left- and right-circularly polarised (LCP and RCP) incident light, we show that different models produce fundamentally different circular dichroism (CD) spectra, characterised by a sign change that can be traced even when the two individual absorption peaks are hard to resolve. In particular, we compare HDM, GNOR and QCM to the standard LRA, and explain how gradually decreasing the distance between NPs in a helical arrangement allows to monitor the transition from one model's predominance to the other's, through distinct, experimentally resolvable spectral features that are significantly different in each case, in accordance with the particular physics governing each model. \section{Theoretical methods}\label{Sec:models} A fundamental factor to the rapid growth of plasmonics has been that, in the vast majority of situations, the theoretical description is extremely efficient within classical electrodynamics, with the metal adequately described by a \emph{local}, usually scalar dielectric function $\varepsilon$. The main assumption of LRA is that, in the linear regime, the displacement vector $\mathbf{D}$ in a metallic NP is simply proportional to the externally applied electric field $\mathbf{E}$~\cite{Bohren}, \begin{equation}\label{Eq:localD} \mathbf{D} (\mathbf{r}, \omega) = \varepsilon_{0} \varepsilon (\omega) \mathbf{E} (\mathbf{r}, \omega)~, \end{equation} where $\varepsilon_{0}$ is the vacuum permittivity, and $\varepsilon$ depends only on the angular frequency $\omega$ (dispersive medium), but not on the position $\mathbf{r}$. For simple free-electron metals $\varepsilon$ follows the Drude model~\cite{Ashcroft}, \begin{equation}\label{Eq:Drude} \varepsilon = \varepsilon_{\infty} - \frac{\omega_{\mathrm{p}}^{2}}{({\omega^{2} + \mathrm {i} \omega \gamma)}}~, \end{equation} where $\omega_{\mathrm{p}}$ is the plasmon frequency, $\gamma$ is the damping rate, and $\varepsilon_{\infty}$ the contribution of bound electrons. In noble metals, the contribution from interband transitions is typically included in $\varepsilon_{\infty}$, either through fitting Eq.~(\ref{Eq:Drude}) to measured data~\cite{johnson_prb6}, or by subtracting the Drude part from the experimental dielectric function~\cite{abajo_jpcc112}. With these assumptions, one just needs to solve the wave equation for the electric field, \begin{equation}\label{Eq:WaveLRA} \nabla \times \nabla \times \mathbf{E} (\mathbf{r}, \omega) = \left(\frac{\omega}{c}\right)^{2} \varepsilon (\omega) \mathbf{E} (\mathbf{r}, \omega)~, \end{equation} subject to the appropriate boundary conditions. In LRA, a hard wall boundary condition is usually adopted, implying that the induced charges are exactly localised at the metal-dielectric interface, explicitly excluding the possibility of spill-out, while the electron gas at the NP interior is assumed incompressible. The assumption of locality starts to become problematic when NP sizes, or their separations in plasmonic aggregates, are comparable to the electron mean free path. Going one step backwards, a more fundamental (still in the linear regime) constitutive relation is \begin{equation}\label{Eq:nonlocalD} \mathbf{D} (\mathbf{r}, \omega) = \varepsilon_{0} \int \mathrm{d} \mathbf{r}' \varepsilon (\mathbf{r}, \mathbf{r}', \omega) \mathbf{E} (\mathbf{r}', \omega)~, \end{equation} where the displacement at position $\mathbf{r}$ might depend on the electric field at positions $\mathbf{r}'$. In a homogeneous medium the spatial dependence becomes $\mathbf{r}-\mathbf{r}'$, simplifying the Fourier-transformed Eq.~(\ref{Eq:nonlocalD}) to $\mathbf{D} (\mathbf{k}, \omega) = \varepsilon_{0} \varepsilon(\mathbf{k}, \omega) \mathbf{E} (\mathbf{k}, \omega)$, where $\mathbf{k}$ is the wavevector. To proceed one needs an explicit form for $\varepsilon(\mathbf{k}, \omega)$, such as those obtained in the theory of screening~\cite{Ashcroft}. An efficient description is provided by HDM, where the equation of motion (at time $t$) for an electron moving with velocity $\mathbf{v} (\mathbf{r}, t)$ in a metal with electron density $n (\mathbf{r}, t)$ is given by \begin{equation}\label{Eq:HydroMotion} \left[\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \right] \mathbf{v} = - \frac{e}{m} \left(\mathbf{E} + \mathbf{v} \times \mathbf{B} \right) - \frac{1}{m} \nabla \frac{\delta G[n]}{\delta n} - \gamma \mathbf{v}~. \end{equation} In the above, $e$ is the (positive) electron charge and $m$ its mass; $\mathbf{B}$ is the magnetic field, so that the first term on the right-hand side corresponds to the Lorentz force, while the last term describes damping. Finally, the functional $G$ takes into account the internal kinetic energy of the electron gas, together with the exchange and correlation contributions. In its simplest form, the Thomas--Fermi approximation, only the kinetic energy is accounted for, and the second term becomes $(\beta^{2}/n) \nabla n$, where, in the high-frequency limit, $\beta^{2} = 3 v_{\mathrm{F}}^{2}/5$ ($v_{\mathrm{F}}$ being the Fermi velocity of the metal). This hydrodynamic pressure term accounts for the finite compressibility and the fact that electrons are fermions obeying the exclusion principle~\cite{Griffiths}. Eq.~(\ref{Eq:HydroMotion}), combined with the continuity equation, $\frac{\partial}{\partial t} n = - \nabla \cdot (n \mathbf{v})$, can be solved assuming a small deviation from the equilibrium electron density $n_{0}$ to produce the system of coupled electromagnetic (EM) equations~\cite{raza_jpcm27} \begin{eqnarray}\label{Eq:CoupledHydro} && \nabla \times \nabla \times \mathbf{E} (\mathbf{r}, \omega) = \left(\frac{\omega}{c}\right)^{2} \varepsilon_{\infty} \mathbf{E} (\mathbf{r}, \omega) + \mathrm{i} \omega \mu_{0} \mathbf{J} (\mathbf{r}, \omega)~, \nonumber\\ && \frac{\beta^{2}}{\omega \left(\omega + \mathrm{i} \gamma \right)} \nabla \left[\nabla \cdot \mathbf{J} (\mathbf{r}, \omega) \right] + \mathbf{J} (\mathbf{r}, \omega) = \sigma \mathbf{E} (\mathbf{r}, \omega)~, \end{eqnarray} where $\mu_{0}$ is the vacuum permeability, $\mathbf{J} (\mathbf{r}, \omega) = -e n_{0} \mathbf{v} (\mathbf{r}, \omega)$ is the induced current density, and $\sigma = \varepsilon_{0} \mathrm{i} \omega_{\mathrm{p}}^{2}/[\omega (\omega + \mathrm{i} \gamma)]$ is the Drude conductivity. While exact analytic solutions can be found for spherical and cylindrical NPs~\cite{ruppin_prl31,raza_prb84}, for arbitrary geometries one solves Eqs.~(\ref{Eq:CoupledHydro}) numerically. Here we use a commercial finite-element method (FEM) solver (Comsol Multiphysics 5.1)~\cite{toscano_oex20}, assuming the additional boundary condition $\mathbf{J} \cdot \hat{\mathbf{n}} = 0$ (where $\hat{\mathbf{n}}$ is the unit vector normal to the metal-dielectric interface), which implies that the electron density vanishes abruptly at the interface (no spill-out). This condition can be relaxed self-consistently, by adding exchange-correlation terms to the functional $G[n]$ of Eq.~(\ref{Eq:HydroMotion})~\cite{toscano_natcom6}. With HDM describing electron convection, one might ask how the optical response is affected by diffusive currents. In the presence of diffusion the continuity equation becomes \begin{equation}\label{Eq:ContinuityDiffusion} -\mathrm{i} \omega e n(\mathbf{r}, \omega) = D \nabla^{2}\left[e n(\mathbf{r}, \omega)\right] + \nabla \cdot \left[-e n_{0} \mathbf{v}(\mathbf{r}, \omega)\right]~, \end{equation} where $D$ is the diffusion constant. Combining this with the diffusive version of Fick's law, \begin{equation}\label{Eq:Fick} \mathbf{J}(\mathbf{r}, \omega) = -e n_{0} \mathbf{v}(\mathbf{r}, \omega) + e D \nabla n(\mathbf{r}, \omega)~, \end{equation} simply modifies the second of Eqs.~(\ref{Eq:CoupledHydro}) by adding the term $-\mathrm{i} D/\omega$ to the prefactor of $\nabla \left[\nabla \cdot \mathbf{J} (\mathbf{r}, \omega) \right]$; the solving methodologies developed for HDM apply thus immediately. Using the Boltzman equation of motion as the starting point, it can be shown that diffusion is the main \textit{low-frequency} contribution in the bulk, and becomes negligible at optical frequencies. Consequently, if a diffusion term is to be included, it must describe processes at the surface of the metal. Indeed, comparison with TD-DFT and modified-damping models shows that GNOR accounts efficiently for surface-enhanced Landau damping~\cite{tserkezis_ijmpb31}. For NP separations of just few {\AA}, a prominent feature left out of the hydrodynamic treatment (by assuming that $\mathbf{J} \cdot \hat{\mathbf{n}} = 0$ at the metal surface) is the probability of direct electron tunnelling between NPs~\cite{savage_nat491,scholl_nl13}. This is where QCM~\cite{esteban_natcom3,esteban_faraday178} comes into play, to account for the infeasibility of fully quantum mechanical calculations for realistic NP sizes, where millions of electrons are involved. The model starts by calculating (typically with TD-DFT) the electron tunnelling probability in a nanogap (e.g. two flat interfaces), $T(\Omega, l)$, as a function of energy $\Omega$ and position in the gap $l$. This in turn is translated into a distance-dependent dc conductivity through \cite{esteban_natcom3} \begin{equation}\label{Eq:QCMconductivity} \sigma_{0}(l) = \frac{l}{2\pi^{2}} \int_{0}^{\Omega_{\mathrm{F}}} \mathrm{d}\Omega \; T(\Omega, l)~, \end{equation} where $\Omega_{\mathrm{F}}$ is the Fermi energy of the metal. A tunnelling damping rate $\gamma_{\mathrm{g}}$ is then obtained from \begin{equation}\label{Eq:QCMGamma} \gamma_{\mathrm{g}} = \frac{\omega_{\mathrm{g}}^{2}}{4\pi \sigma_{0}(l)}~, \end{equation} where the gap plasma frequency $\omega_{\mathrm{g}}$ is assumed equal to that of the bulk metal, $\omega_{\mathrm{p}}$. These two parameters, $\omega_{\mathrm{g}}$ and $\gamma_{\mathrm{g}}$, are finally introduced into a classical EM calculation, to model the Drude permittivity of a bridge connecting the metallic components [see schematics in Fig.~\ref{fig5}(a)]. This approach has been employed to mimic the role of tunnelling-mediated dissipation in different geometries, such as NP aggregates~\cite{hohenester_prb91} or nanomatryoshkas of alternating metal and dielectric layers~\cite{zapata_oex23}. \section{Optical response of NP helices}\label{Sec:LRA} The quantum-informed models described in the previous section will be employed to explore the optical response of metallic NP helices. The helices considered in the largest part of the paper consist of 9 silver spheres with radius $R = 5$\;nm, described by the experimental dielectric function of Johnson and Christy~\cite{johnson_prb6}, in air. The spheres revolve around the $z$ axis by $\pi/2$ angle steps, so that 9 NPs produce two full revolutions, as shown schematically in Fig.~\ref{fig1}(b). While it was not modelled in our simulations, a supporting pillar of diameter $d$ and height 40\;nm is implied and shown in the schematics: such a pillar is usually produced with DNA-origami nanofabrication techniques~\cite{kuzyk_acsphoton5}. Our choice of NP size serves therefore a dual purpose: it is small enough to ensure that quantum effects will be relevant even for non-interacting NPs, while being the typical size supported by DNA-origami pillars~\cite{zhou_acr50}. The centre of each NP is vertically shifted by $R$, while $d$ can vary from 14 to 12.5\;nm, producing surface-to-surface NP distances of 3.5 to 0.3\;nm. The system is illuminated by circularly polarised light propagating either along or normally to the helix axis ($z$ axis). \begin{figure}[h] \centerline{\includegraphics*[width=1\columnwidth]{figure1.eps}} \caption{(a) Absorption cross section ($\sigma_{\mathrm{abs}}$, normalised to the geometrical cross section of a single NP), calculated within LRA for a linearly ($z$)- or circularly-polarised incident plane wave propagating along the $x$ axis, for a helix of 9 silver NPs ($R = 5$\;nm) revolving around a supporting pillar of diameter $d = 12.5$\;nm, as shown schematically in (b) (left-hand schematic). Black, red, and blue lines correspond to $z$-polarised, LCP, and RCP light, respectively. (c) Same as in (a), for propagation along the helix ($z$) axis [right-hand schematics in (b)]. (d) Zoom in the 440--480\;nm spectral window of (c), for silver NPs described by a dielectric function whose imaginary part is artificially reduced by 90\%. The spectra for linearly $x$- and $y$-polarised light are shown by a thick black and grey line respectively, while LCP (RCP) spectra are represented by a thin red (blue) line.} \label{fig1} \end{figure} In Fig.~\ref{fig1}(a) we plot the calculated absorption cross section, $\sigma_{\mathrm{abs}}$, normalised to the geometrical cross section of a single NP, as a function of wavelength $\lambda$, for a helix revolving around a pillar with $d = 12.5$\;nm, which reduces the interparticle gap to just 0.3\;nm. For the strong interaction resulting from such narrow gaps, the long-wavelength resonances in the spectra can be understood in view of the embedded-chain model as collective chain modes in the direction of the electric field of the incident plane wave~\cite{taylor_jpcc120}, while short-wavelength resonances are associated with single-NP and higher-order hybrid modes. In this respect, when the incident plane wave propagates normally to the helix axis (here along the $x$ axis), the main difference between a linearly, $z$-polarised wave with electric field $\mathbf{E} = E_{0}\;\widehat{\mathbf{z}}$ ($E_{0}$ being the amplitude of the electric field) and an L(R)CP wave [taken as $\mathbf{E} = (E_{0}/\sqrt{2}) (\widehat{\mathbf{z}} \pm \mathrm{i} \widehat{\mathbf{y}})$ here] regards the excitation of short, kinked chains along the $y$ axis in the latter case, which manifests itself as an intense absorption peak around 450\;nm in the spectra. Naturally, since the system lacks mirror symmetry, LCP and RCP illumination produces different intensities of the absorption peaks, and thus a strong CD signal, in agreement with the results of Ref.~\cite{fan_jpcc115} (emphasis will be placed on CD spectra in the next section, where the different quantum-informed models are compared). A far more interesting optical response is obtained for propagation along the helix axis, as can be seen in Fig.~\ref{fig1}(c). In this case, instead of a single long-wavelength peak as in Fig.~\ref{fig1}(a), a double peak appears in the spectral window of 440--480\;nm, centred at 452 and 464\;nm, with different intensity for each branch under LCP and RCP illumination [$\mathbf{E} = (E_{0}/\sqrt{2}) (\widehat{\mathbf{x}} \pm \mathrm{i} \widehat{\mathbf{y}})$ for L(R)CP light]. An additional, higher-order hybrid chain mode is excited at 448\;nm under $x$ polarisation, but as it always appears just as a shoulder at the high-energy end of the fundamental chain resonance~\cite{tserkezis_ppsc31}, its presence does not significantly affect the doublet of interest here. To understand the origin of this doublet, we plot in Fig.~\ref{fig1}(d) the spectra for linear polarisation along the $x$ or $y$ axis (thick black and grey line, respectively), with absorptive losses artificially reduced by 90\%, by manually modifying the imaginary part of the dielectric function of silver. It is clear that the 452\;nm mode is only excited by $y$ polarisation, while the 464\;nm only by $x$ polarisation. These resonances can therefore be understood as chain modes of embedded chains growing along the $x$ or $y$ axis, depending on the polarisation, appearing at different wavelengths due to the finite size of the helix. In an infinite helix, the number of embedded chains along both axes is the same, and the two peaks coincide. This can be verified by calculations for shorter (5 NPs, one revolution) and longer (13 NPs, three revolutions) helices, with the spectral split gradually closing, from 21 to 8\;nm, as shown in Fig.~\ref{fig2}(a). This interpretation of the modes sustained by such helices is further supported by the near-field profiles of Fig.~\ref{fig2}(b), where the highest field intensities are obtained at the gaps along the corresponding effective chain for each incident linear ($x$ or $y$) polarisation. \begin{figure}[h] \centerline{\includegraphics*[width=0.9\columnwidth]{figure2.eps}} \caption{(a) Normalised absorption spectra (vertically shifted for clarity) for the Ag NP helices schematically shown on the right. From bottom to top, LCP (red) and RCP (blue) absorption spectra are calculated for helices of 5 (one full revolution, solid lines), 9 (two revolutions, dashed lines), and 13 NPs (three revolutions, dotted lines). (b) Near-field enhancement ($|\mathbf{E}|/E_{0}$) plots for the 9-NP helices of Fig.~\ref{fig1}(c), at $\lambda = 464$\;nm. Left- and right-hand panels correspond to $x$- and $y$-linearly polarised incident light, respectively. In the top row the field is plotted in the $x-z~(y=0)$ plane, and in the bottom row in the $y-z~(x=0)$ plane, as shown in the schematics in the middle panel.}\label{fig2} \end{figure} \begin{table}[ht] \centering \caption{Character table of the $C_{2}$ point group.}\label{Table1} \begin{tabular}{|c|c c|} \hline ~ & $\quad \mathcal{E} \quad$ & $\quad \mathcal{C}_{2x} \quad$ \\ \hline $Q_{1}$ & 1 & 1 \\ $Q_{2}$ & 1 & -1 \\ \hline \end{tabular} \end{table} To further analyse the spectral doublet, we turn to group theory. Since the only symmetry operations $\widehat{P}$ leaving the finite helices unaltered are identity, $\mathcal{E}$, and a $\pi$ rotation about the $x$ axis, $\mathcal{C}_{2x}$, the appropriate point group is $C_{2}$, whose character ($\chi$) table is given in Table~\ref{Table1}~\cite{Cornwell}. All modes must have the symmetry of the irreducible representations of this group, namely $Q_{1}$ and $Q_{2}$, while a mode is excited only if the electric field of the incident light has a nonvanishing projection onto the appropriate subspace. For the $C_{2}$ group the projection operator is \begin{equation}\label{Eq:projection} \mathcal{P}^{(\Gamma)} = \frac{1}{2} \sum_{\Gamma} \chi^{\ast} (\Gamma) \widehat{P} \mathbf{E}~, \end{equation} where $\Gamma = \{\mathcal{E}, \mathcal{C}_{2x}\}$ is the set of group operations. Since a $\pi$ rotation about the $x$ axis leads to the following transformations: $x \rightarrow x$, $y \rightarrow -y$, $z \rightarrow -z$, $\mathbf{k} \rightarrow -\mathbf{k}$, it is straightforward to show that an $x$-polarised plane wave projects only onto $Q_{1}$, while a $y$-polarised plane wave only onto $Q_{2}$, in agreement with the absorption spectra of Fig.~\ref{fig1}(d). For circularly polarised light, the electric field has both $x$ and $y$ components and both modes are efficiently excited. The reported optical response is rather robust against manufacturing deviations, facilitating realisation of the proposed experiment. We have already discussed how the number of NPs forming the helix affects the measured spectra, and concluded that short chains are much more preferable than longer ones. The number of helical revolutions is in fact the factor that calls for most attention. In the Supporting Information we present spectra for intermediate NP numbers (incomplete revolutions), increasing from 5 to 9 (Fig.~S1). It is shown that in those cases the spectral doublet is always efficiently excited. We also explore the role of the revolution angle, by considering $\pi/3$ steps (Fig.~S2), which are often more feasible with DNA-origami methods~\cite{kuzyk_nat483}. Similarly, the spectra are little affected by variations in the NP size and interparticle gap (we modelled up to $\pm 10 \%$ size variations in our calculations) within the same helix (Fig.~S3). On the other hand, intrinsic losses in the chosen material can be important, as we show through the corresponding spectra for helices of 9 gold NPs (Fig.~S4). There, the resonances are not only shifted to longer wavelengths, but also broadened so much that the doublet is not distinguishable in the absorption spectra; it might survive as a sign change in CD for some chain lengths. Finally, it is important to notice that in the following analysis, and in any experimental realisation, the exact values of resonance shifts or CD measurements are not important, and only the qualitative features of the spectra matter. \section{Quantum corrections in the helix response}\label{Sec:Quantum} In the remaining of the paper we explore how the absorption doublet discussed in the previous section, and the resulting CD signal (calculated here as the difference of absorption cross sections for LCP and RCP incident light), behave within the three different quantum-informed models presented above. We start with the HDM calculation, whose screening mechanism is expected to dominate for larger NP separations. We use $v_{\mathrm{F}} = 1.39 \times 10^{6}$\;m\;s$^{-1}$ for the Fermi velocity of silver, while $\varepsilon_{\infty}$ is obtained by subtracting from the experimental dielectric function a Drude permittivity with $\hbar \omega_{\mathrm{p}} = 8.99$\;eV and $\hbar \gamma = 0.25$\;eV~\cite{tserkezis_scirep6}. In Fig.~\ref{fig3}(a) we compare the LRA absorption spectra of Fig.~\ref{fig1}(c) (dashed lines) with those obtained by HDM (solid lines). As expected, the main difference is a large blueshift of the modes due to screening, which is the characteristic of HDM: since the induced charges are ``pushed'' inwards, the NPs behave as if they were effectively smaller, and therefore their separations larger, decreasing the strength of their interaction. This can be observed more clearly in CD spectra, through the transition from negative (short-wavelength branch) to positive (long-wavelength branch) peaks, as shown in Fig.~\ref{fig3}(b). The fingerprint of the blueshifting doublet is clearly visible even for $d = 14$\;nm, corresponding to an interparticle gap as wide as 2.2\;nm, for which individual absorption peaks are hard to resolve, as they nearly coincide in wavelength with the strong single-NP resonance (chain modes are only very weakly excited). \begin{figure}[h] \centerline{\includegraphics*[width=0.9\columnwidth]{figure3.eps}} \caption{(a) Normalised absorption cross section for the helix of Fig.~\ref{fig1}(b) (right-hand schematic), for LCP (red) and RCP (blue) incident light, within the LRA and HDM description (dashed and solid lines respectively). (b) CD spectra for the two models, when increasing the interparticle gap through $d = 12.5, 13, 13.5, 14$\;nm (red, green, light-blue and blue lines respectively, corresponding to gaps of 0.31\;nm, 0.93\;nm, 1.55\;nm, and 2.18\;nm). The arrows in both panels trace the transition from the double-peak feature in absorption to the positive-negative signal in CD, highlighted here for the $d = 12.5$\;nm case (LRA calculation).}\label{fig3} \end{figure} It is now interesting to apply the same analysis for the GNOR model. To that end, we perform FEM simulations to solve the coupled Eqs.~(\ref{Eq:CoupledHydro}) (as modified for GNOR), with a diffusion constant $D = 2.684 \times 10^{-4}$\;m$^{2}$\;s$^{-1}$, a value which allows direct comparison with previous studies of silver NP dimers and chains~\cite{tserkezis_prb96}. Since GNOR is specifically designed to account for increased absorptive losses when NP sizes and distances are small, and CD is by definition an observable depending on absorption, one expects substantial differences from the response predicted by LRA. Indeed, as seen in Fig.~\ref{fig4}(a), GNOR introduces such a degree of plasmon damping and broadening, that the two separate peaks of interest merge into a single broad resonance (blueshifted with respect to LRA due to nonlocal screening, already accounted for with HDM). In this case, the only way to trace the double resonance of the helix chain modes is through its CD signal, which, even in this case, retains the characteristic sign change. This is shown in Fig.~\ref{fig4}(b), for the same four NP separations as in Fig.~\ref{fig3}(b). One should notice, however, that broadening of the modes could have many different origins, and is not necessarily an indication of Landau damping (see also the spectra for gold NP helices in the Supporting Information, Fig.~S4). Consequently, in an experiment, one should take care to choose materials with low intrinsic loss, and NPs as smooth and with as homogeneous size distributions as possible. \begin{figure}[h] \centerline{\includegraphics*[width=0.9\columnwidth]{figure4.eps}} \caption{(a) Normalised absorption cross section for the helix of Fig.~\ref{fig1}(b) (right-hand schematic), for LCP (red) and RCP (blue) incident light, within the LRA and GNOR description (dashed and solid lines respectively). (b) CD spectra for the two models, when increasing the interparticle gap through $d = 12.5, 13, 13.5, 14$\;nm (red, green, light-blue and blue lines respectively, corresponding to gaps of 0.31\;nm, 0.93\;nm, 1.55\;nm, and 2.18\;nm). The arrows indicate the wavelength at which the CD signal changes sign, in the region of the double resonance of interest.}\label{fig4} \end{figure} Finally, for gaps as narrow as a few \AA, it is important to consider the possibility of direct electron tunnelling from one NP to another~\cite{savage_nat491,yan_prl115}. As described in Sec.~\ref{Sec:models}, QCM seeks to mimic this regime through a (usually multilayered) conductive junction, introduced in a classical EM calculation to connect neighbouring NPs, as shown schematically in Fig.~\ref{fig5}(a). For a strict, quantitative description, the conductivity of such a bridge must be obtained from quantum mechanical calculations (for reduced NP sizes that make it feasible), typically with TD-DFT~\cite{esteban_natcom3}. Nevertheless, since we are only interested in a qualitative description here, we use a homogeneous cylinder described by a Drude model, with $\hbar \omega_{\mathrm{g}} = 8.99$\;eV and $\hbar \gamma_{\mathrm{g}} = 0.025$\;eV (but with $\varepsilon_{\infty} = 1$, as in the previous calculations. Omitting the multilayered structure of the bridge can be compensated by considering different bridge diameters, which, through facilitating or hindering charge transfer between NPs, tune the optical response of the entire structure. On the other hand, the tunnelling conductivity might in fact be smaller than that of the bulk metal, and therefore, the Drude damping rate might be larger, according to Eq.~(\ref{Eq:QCMGamma}), leading to broadening of the modes. It turns out that in our case this broadening is rather small, as shown by the shaded blue line ($w = 4$\;nm) in Fig.~\ref{fig5}(c) which corresponds to $\hbar \gamma_{\mathrm{g}} = 0.04$\;eV, and is practically indistinguishable from the $\hbar \gamma_{\mathrm{g}} = 0.025$\;eV case (solid line). In the Supporting Information we show how the optical response of silver NP dimers is affected by these two parameters (Fig.~S5). \begin{figure}[h] \centerline{\includegraphics*[width=0.9\columnwidth]{figure5.eps}} \caption{(a) Schematic of the bridged NP helices used for a qualitative implementation of QCM: cylindrical bridges of diameter $w$, described by a Drude dielectric function with values for $\omega_{\mathrm{g}}$ and $\gamma_{\mathrm{g}}$ appropriate for silver, connect the NPs. (b) Absorption spectra for LCP and RCP (red and blue lines) incident light, calculated for the helix of Fig.~\ref{fig1}(b) within LRA (dashed lines) and QCM (solid lines) ($w = 4$\;nm). (c) CD spectra in QCM for varying $w$, from $w = 1$ to 4\;nm (from top to bottom). The shaded line for $w = 4$\;nm represents the spectra for a higher value of $\hbar \gamma_{\mathrm{g}} = 0.04$\;eV, corresponding to lower tunnelling conductivities in the gap. The arrows indicate the wavelength at which the CD signal changes sign in the region of the double resonance of interest.}\label{fig5} \end{figure} Fig.~\ref{fig5}(b) shows the absorption spectra obtained with QCM when Drude bridges of diameter $w = 4$\;nm connect each NP in the helix with its nearest neighbours. Once such junctions are created, threaded chain plasmons (TCPs) are excited in the infrared~\cite{herrmann_natcom5,tserkezis_oex22}, at longer wavelengths for thinner bridges in which charge transfer is more hindered, as can also be seen in the CD spectra of Fig.~\ref{fig5}(c). It is important to notice that these modes are not of the same nature as the chain modes discussed above and just redshifted, but hybrid, chain/rod (or, in our case, chain/spiral) modes. The unthreaded chain modes become screened chain plasmons, getting rapidly damped and blushifting due to the screening from the field accumulated around the bridge~\cite{perez_njp13}. The two limiting cases in threaded chains are therefore i) very weak, deep-infrared modes for $w \rightarrow 0$ and ii) single spiral (threaded helix) modes for $w \rightarrow 2R$~\cite{tserkezis_oex22}. Nevertheless, if tunnelling prevails, the spectra are dominated by a strongly redshifted double peak, predicting yet another qualitatively different optical response, characteristic of QCM, the exact quantitative aspects of which (in terms of resonance shift and strength) depend on the precise TD-DFT input~\cite{esteban_natcom3}. \section{Discussion}\label{Sec:discussion} Having explored three different quantum-informed models, one might ask what the real optical response of the helix will be, and what kind of resonances one should expect in experiments. The truth however is that, before measuring, there is no real way to know which predictions are more accurate. It can be expected that for larger NP separations, screening will prevail and spectra closer to those discussed in relation to Fig.~\ref{fig3} will be obtained. On the other hand, for small separations, both damping and tunnelling could become important, leading to broad, merged absorption peaks (but probably still traceable in CD), both at shorter (screened chain plasmons) and at longer (TCPs) wavelengths. Such a combination of effects could be predicted by more elaborate models, e.g. $d$-parameter based ones. But the situation is actually reminiscent of the debate about nonlocality a few years ago. The classical HDM predicts invariably resonance blueshifts. It took \emph{ab initio} calculations~\cite{teperik_prl110}, which for small monomers and dimers were feasible, together with sensitive experiments~\cite{rainers_prl74}, to establish the fact that for good free-electron metals (e.g. Na; in noble metals spill-out is much less important due to d-electron screening) spill-out prevails over screening, thus leading to resonance redshifts. Similarly, only fine experiments (the number of electrons involved here make \emph{ab initio} calculations impossible) can tell what the actual optical response of the helices is, and in doing so they will most likely call for the extension of the existing, or the development of new quantum-informed models. Finally, let us briefly comment on the feasibility of the proposed architectures. One way of fabricating advanced plasmonic devices that exhibit novel optical properties is utilising DNA origami \cite{liu_chemrev118}. Due to its programmability and specificity, DNA origami enables NP organisation at the sub-nm scale \cite{bidault_jacs130}, that is still a challenge for top-down techniques. For instance, a single stranded DNA could be folded by smaller single DNA strands, called ssDNAs, to build a pillar with a diameter 14-12.5\;nm. It has been shown that 10-15 nucleotides, approximately 1.3\;nm in length, are enough to provide this connection \cite{heck_acsphoton4}. The pillar would be robust enough to support metallic NPs of diameter 10\;nm \cite{kuzyk_nat483}, and NPs would stick onto their predefined locations by the use of capture strands protruding from the pillar. Hence, it should be possible to obtain interparticle gaps of 0.3-3.5\;nm. Another possibility could be directly growing NPs on a DNA template \cite{shemer_jacs128}. In the case of silver in particular, oxidation can also be an issue, and experiments should be performed as soon as carefully synthesised NPs are prepared \cite{scholl_nature483}, to reduce its influence \cite{weller_acsphoton3}. \section{Conclusions}\label{Sec:conclusion} In summary, we have shown that CD measurements in chiral chains of metallic NPs, experimentally feasible nowadays with DNA-origami techniques, serve as an excellent test bed for assessing quantum-informed models in plasmonics. By comparing HDM, GNOR, and QCM, we obtained three fundamentally different optical responses, each one characteristic of the specific model. The spectral doublet calculated within LRA shifts to the blue in HDM due to screening, merges into a single resonance due to broadening in GNOR, and strongly redshifts once tunnelling becomes important in QCM. CD measurements allow to monitor these changes simply through a change in sign. Even though our calculations only intend to qualitatively describe each model, we anticipate that quantitative differences should be large enough to be experimentally resolved, thus allowing to understand the limitations and range of validity of, in principle, any existing or new quantum-informed model. \begin{acknowledgements} We thank C. Wolff for discussions. N.~A.~M. is a VILLUM Investigator supported by VILLUM FONDEN (grant No. 16498). The Center for Nano Optics is financially supported by the University of Southern Denmark (SDU 2020 funding). Simulations were supported by the DeIC National HPC Centre, SDU. \end{acknowledgements}

BkiUbjo4eIXguy2tOD_h

\section{Conclusion and Future Work} \label{sec:concl} In this paper, we have considered the problem of vulnerability analysis for nonlinear control systems with Gaussian noise, when attacker can compromise sensor measurements from any subset of sensors. Notions of strict stealthiness and $\epsilon$-stealthiness have been defined, and we have shown that these notions are independent of the deployed intrusion detector. Using the KL-divergence, we have presented conditions for the existence of stealthy yet effective attacks. Specifically, we have defined the $(\epsilon,\alpha)$-successful attacks where the goal of the attacker is to be $\epsilon$-stealthy while moving the system states into an unsafe region, determined by the parameter $\alpha$. We have then derived a condition for which there exists a sequence of such $(\epsilon,\alpha)$-successful false-data injection attacks. In particular, we showed that if the closed-loop system is incrementally exponentially stable and the open-loop system is incrementally unstable, then there exists a sequence of $(\epsilon,\alpha)$-successful attacks. We also provided the results for LTI systems, showing that they are compatible with the existing results for LTI systems and $\chi^2$-based detectors. Our results assume that the attacker has knowledge of the state evolution function $f$, as well as access to the values of the actual system states and the control inputs during the attack. Future work will be directed toward deriving conditions when the attacker has limited knowledge about the states, control input and the function $f$. We will also study the effects of specific previously reported attacks (e.g., replay attack) on general nonlinear control systems using the defined notions of strict and $\epsilon$-stealthiness. \section{Introduction} \label{sec:intro} Cyber-physical systems (CPS) are characterized by the tight integration of controllers and physical plants, potentially through communication networks. As such, they have been shown to be vulnerable to various types of cyber and physical attacks with disastrous impact (e.g.,~\cite{chen2011lessons}). Consequently, as part of the control design and analysis process, it is critical to identify early any vulnerability of the considered system to impactful attacks, especially the ones that are potentially stealthy to the deployed intrusion detection mechanisms. Depending on attacker capabilities, different types of stealthy attacks have been proposed. For instance, when only sensor measurements can be compromised by the attacker, it has been shown that false data injection attacks are capable of significantly impacting the system while remaining undetected (i.e., stealthy) by a particular type of residual-based anomaly detectors (e.g.,~\cite{mo2010false,jovanov_tac19,kwon2014analysis,khazraei_automatica21,khazraei_acc20,zhang2020false,shang2021optimal}). For example, for linear time invariant (LTI) systems, if measurements from all sensors can be compromised , the plant's (i.e., open-loop) instability is a necessary and sufficient condition for the existence of impactful stealthy attacks. Similarly, for LTI systems with strictly proper transfer functions, the attacker that compromises the control input can design effective stealthy attacks if the system has unstable zero invariant (e.g.,~\cite{teixeira2012revealing,pasqualetti2013attack}); however, when the transfer function is not strictly proper, the attacker needs to compromise both plant's inputs and outputs. When the attacker compromises both the plant's actuation and sensing, e.g., ~\cite{sui2020vulnerability} derives the conditions under which the system is vulnerable to stealthy~attacks. However, the common assumption for all these results is that the considered plant is an LTI system. Furthermore, the notion of stealthiness is only characterized for a \emph{specific type} of the employed intrusion detector (e.g., $\chi^2$-based detectors). In~\cite{bai2017kalman,bai2017data}, the notion of attack stealthiness is generalized, defining an attack as stealthy if it is stealthy from the best existing intrusion detector. In addition, the authors show that a sufficient condition for such notion of stealthiness is that the Kullback–Leibler (KL) divergence between the probability distribution of compromised system measurements and the attack-free measurements is close to zero, and consider stealthiness of such attacks on control systems with an LTI plant and an LQG controller. To the best of our knowledge, no existing work provides vulnerability analysis for systems with nonlinear dynamics, while considering general control and intrusion detector designs. In~\cite{smith2015covert}, covert attacks are introduced as stealthy attacks that can target a potentially nonlinear system. However, the attacker needs to have perfect knowledge of the system's dynamics and be able to compromise \emph{both} the plant's input and outputs. Even more importantly, as the attack design is based on attacks on LTI systems, no guarantees are provided for effectiveness and stealthiness of attacks on nonlinear systems. More recently, \cite{zhang2021stealthy} introduced stealthy attacks on a \emph{specific class} of nonlinear systems with residual-based intrusion detector, but provided effective attacks only when \emph{both} plant's inputs and outputs are compromised by the attacker. On the other hand, in this work, we assume the attacker can only compromise the plant's sensing data and consider systems with \emph{general} nonlinear dynamics. For systems with general nonlinear dynamics and residual-based intrusion detectors, machine learning-based methods to design the stealthy attacks have been introduced (e.g.,~\cite{khazraei2021learning}), but without any theoretical analysis and guarantees regarding the impact of the stealthy attacks. Consequently, in this work we provide conditions for existence of effective yet stealthy attacks on nonlinear systems without limiting the analysis on particular type of employed intrusion detectors. Our notion of attack stealthiness and system performance degradation is closely related to~\cite{khazraeReport21}. However, we extend these notions for systems with general nonlinear plants and controllers. To the best of our knowledge, this is the first work that considers the problem of stealthy impactful sensor attacks for systems with general nonlinear dynamics that is independent of the deployed intrusion detector. The main contributions of the paper are twofold. First, we introduce the notions of \emph{strict} and \emph{$\epsilon$-stealthiness}. Second, using the well-known results for incremental stability introduced in~\cite{angeli2002lyapunov}, we derive conditions for the existence of effective stealthy attacks that move the system into an unsafe operating region. We show that if the closed-loop system is incrementally stable while the open-loop plant is incrementally unstable, then the closed-loop system is strictly vulnerable to stealthy sensing attacks. The paper is organized as follows. In~\cref{sec:prelim}, we introduce preliminaries, whereas \cref{sec:motive} presents the system and attack model, before formalizing the notion of stealthiness in \cref{sec:stealthy}. \cref{sec:perfect} provides sufficient conditions for existence of the impactful yet stealthy attacks. Finally, in \cref{sec:simulation}, we illustrate our results on a case-study, before concluding remarks in \cref{sec:concl}. \paragraph*{Notation} We use $\mathbb{R, Z}, \mathbb{Z}_{t\geq 0}$ to denote the sets of reals, integers and non-negative integers, respectively, and $\mathbb{P}$ denotes the probability for a random variable. For a square matrix $A$, $\lambda_{max}(A)$ denotes the maximum eigenvalue. For a vector $x\in{\mathbb{R}^n}$, $||x||_p$ denotes the $p$-norm of $x$; when $p$ is not specified, the 2-norm is implied. For a vector sequence, $x_0:x_t$ denotes the set $\{x_0,x_1,...,x_t\}$. A function $f:\mathbb{R}^{n}\to \mathbb{R}^{p}$ is Lipschitz with constant $L$ if for any $x,y\in \mathbb{R}^{n}$ it holds that $||f(x)-f(y)||\leq L ||x-y||$. Finally, if $\mathbf{P}$ and $\mathbf{Q}$ are probability distributions relative to Lebesgue measure with densities $\mathbf{p}$ and $\mathbf{q}$, respectively, then the Kullback–Leibler (KL) divergence between $\mathbf{P}$ and $\mathbf{Q}$ is defined as $KL(\mathbf{P},\mathbf{Q})=\int \mathbf{p}(x)\log{\frac{\mathbf{p}(x)}{\mathbf{q}(x)}}dx$. \section{System Model} \label{sec:motive} In this section, we introduce the considered system and attack model, allowing us to formally capture the problem addressed in this work. \subsection{System and Attack Model} \label{sec:A} We consider the setup from \cref{fig:architecture} where each of the components is modeled as follows. \subsubsection{Plant} We assume that the states of the system evolve following a general nonlinear discrete-time dynamics that can be captured in the state-space form~as \begin{equation}\label{eq:plant} \begin{split} {x}_{t+1} &= f(x_t,u_t)+w_t,\\ y_t &= h(x_t)+v_t; \end{split} \end{equation} here, $x \in {\mathbb{R}^n}$, $u \in {\mathbb{R}^m}$, $y \in {\mathbb{R}^p}$ are the state, input and output vectors of the plant, respectively. In addition, $f$ is a nonlinear mapping from previous time state and control input to the current state, and $h$ is the mapping from the states to the sensor measurements; we assume here that $h$ is Lipschitz with a constant $L_h$. The plant output vector captures measurements from the set of plant sensors $\mathcal{S}$. Furthermore, $w \in {\mathbb{R}^{n}}$ and $v \in {\mathbb{R}^p}$ are the process and measurement noises that are assumed to be Gaussian with zero mean, and $\Sigma_w$ and $\Sigma_v$ covariance matrices, respectively. As we show later, it will be useful to consider the input to state relation of the dynamics~\eqref{eq:plant}; if we define $U=\begin{bmatrix}u^T&w^T\end{bmatrix}^T$, the first equation in~\eqref{eq:plant} becomes \begin{equation}\label{eq:input-state} x_{t+1}=f_u(x_t,U_t). \end{equation} \begin{figure}[!t] \centering \vspace{6pt} \includegraphics[width=0.468\textwidth]{./Pictures/Picture} \caption{Control system architecture considered in this work, in the presence of network-based attacks.} \label{fig:architecture} \end{figure} \subsubsection{Control Unit} The controller, illustrated in~\cref{fig:architecture}, is equipped with a feedback controller in the most general form, as well as an intrusion detector (ID). In what follows, we provide more details on the controller design. Intrusion detector will be discussed after introducing the attack model. \paragraph*{Controller} A large number of dynamical systems are intrinsically unstable or are designed to be unstable (e.g., if an aircraft is unstable, it is easier to change its altitude). Thus, it is critical to stabilize such systems using a proper controller. Due to their robustness to uncertainties, closed-loop controllers are utilized in most control systems. In the most general form, a feedback controller design can be captured in the state-space form~as \begin{equation} \label{eq:plant_withoutB} \begin{split} \mathpzc{X}_{t} &= f_c(\mathpzc{X}_{t-1},y_t^c),\\ u_t &= h_c(\mathpzc{X}_{t},y_t^c), \end{split} \end{equation} where $\mathpzc{X}$ is the internal state of the controller, and $y^c$ captures the sensor measurements received by the controller. Thus, without malicious activity, it holds that $y^c=y$.\footnote{Here we assume that the employed communication network is reliable (e.g., wired).} Note that the control model~\eqref{eq:plant_withoutB} is general, capturing for instance nonlinear filtering followed by a classic nonlinear controller (e.g., $f_c$ can model an extended Kalman filter and $h_c$ any full-state feedback controller). We define the full state of the closed-loop control system as $\mathbf{X}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} \begin{bmatrix} {x}\\{\mathpzc{X}} \end{bmatrix}$, and exogenous disturbances as $\mathbf{W}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} \begin{bmatrix} w\\{v} \end{bmatrix}$; then, the dynamics of the closed-loop system can be captured~as \begin{equation}\label{eq:closed-loop} \mathbf{X}_{t+1}=F(\mathbf{X}_t,\mathbf{W}_t). \end{equation} We assume that $\mathbf{X}=0$ is the operating point of the noiseless system (i.e., when $w=v=0$). {Moreover, we assume $f_c$ and $h_c$ are designed to keep the system within a safe region around the equilibrium point.} Here, without loss of generality, we define the safe region as $\mathbf{S}=\{x\in \mathbb{R}^n\,\,|\,\,\Vert x\Vert_2 \leq R_{\mathbf{S}}\}$, for some $R_{\mathbf{S}}>0$. \subsubsection{Attack Model} \label{sec:attack_model} We consider a sensor attack model where, for sensors from the set $\mathcal{K}\subseteq{\mathcal{S}}$, the information delivered to the controller differs from the non-compromised sensor measurements. The attacker can achieve this via e.g., noninvasive attacks such sensor spoofing (e.g.,~\cite{kerns2014unmanned}) or by compromising information-flow from the sensors in $\mathcal{K}$ to the controller (e.g., as in network-based attacks~\cite{lesi_rtss17}). In either cases, the attacker can launch false-date injection attacks, inserting a desired value instead of the current measurement of a compromised sensor.\footnote{We refer to sensors from $\mathcal{K}$ as compromised, even if a sensor itself is not directly compromised but its measurements may be altered due to e.g., network-based attacks.} Thus, assuming that the attack starts at time $t=0$, the sensor measurements delivered to the controller for $t\geq 0$ can be modeled~as~\cite{ncs_attack_model} \begin{equation}\label{att:model} y^{c,a}_t = y_t^a+a_t; \end{equation} here, $a_t\in {\mathbb{R}^p}$ denotes the attack signal injected by the attacker at time $t$ via the compromised sensors from $\mathcal{K}$, $y_t^a$ is the true sensing information (i.e., before the attack is injected at time $t$). In the rest of the paper we assume $\mathcal{K}=\mathcal{S}$; for some systems, we will discuss how the results can be generalized for the case when $\mathcal{K}\subset \mathcal{S}$. Note that since the controller uses the received sensing information to compute the input $u_t$, the compromised sensor values affect the evolution of the system and controller states. Hence, we add the superscript $a$ to denote any signal obtained from a compromised system -- e.g., thus, $y_t^a$ is used to denote before-attack sensor measurements when the system is under attack in~\eqref{att:model}, and we denote the closed-loop plant and controller state when the system is compromised as $\mathbf{X}^a\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} \begin{bmatrix} {x^a}\\{{\mathpzc{X}^a}} \end{bmatrix}$. In this work, we consider the commonly adopted threat model as in majority of existing stealthy attack designs, e.g.,~\cite{mo2009secure,mo2010false,smith2015covert, khazraei_automatica21, jovanov_tac19}, where the attacker has full knowledge of the system, its dynamics and employed architecture. In addition, the attacker has the required computational power to calculate suitable attack signals to be injected, while planning ahead as needed. Finally, the attacker's goal is to design an attack signal $a_t$, $t\geq 0$, such that it always remains \emph{stealthy} -- i.e., undetected by the intrusion detection system -- while \emph{maximizing control performance degradation}. The notions of \emph{stealthiness} and \emph{control performance degradation} depend on the employed control architecture, and thus will be formally defined after the controller and intrusion detection have been~introduced. \subsubsection{Intrusion Detector} To detect system attacks (and anomalies), we assume that an intrusion detector (ID) is employed, analyzing the received sensor measurements and internal state of the controller. Specifically, by defining $Y\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} \begin{bmatrix} {y^c}\\{\mathpzc{X}} \end{bmatrix}$, as well as $Y^a\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} \begin{bmatrix} {y^{c,a}}\\{\mathpzc{X}^a} \end{bmatrix}$ when the system is under attack, we assume that the intrusion detector has access to a sequence of values $Y_{-\infty}:Y_t$ until time $t$ and solves the binary hypothesis checking\\ $H_0$: normal condition (the ID receives $Y_{-\infty}:Y_t$);~~ $H_1$: abnormal behaviour (receives $Y_{-\infty}:Y_{-1},Y_{0}^a:Y_t^a$).\footnote{Since the attack starts at $t=0$, we do not use superscript $a$ for the system evolution for $t<0$, as the trajectories of the non-compromised and compromised systems do not differ before the attack~starts.} \\ Given a sequence of received data denoted by $\bar{Y}^t=\bar{Y}_{-\infty}:\bar{Y}_t$, it is either extracted from the distribution of the null hypothesis $H_0$, which we refer to as~$\mathbf{P}$, or from an \textbf{\emph{unknown}} distribution of the alternative hypothesis $H_1$, which we denote as~$\mathbf{Q}$. {Note here that, for known noise profiles, the distribution $\mathbf{Q}$ is controlled by the injected attack signal.} Defining the intrusion detector mapping as $D: \bar{Y}^t\to \{0,1\}$, two possible errors may occur. The error type ($\rom{1}$) known as \emph{false alarm}, occurs if $D(\bar{Y}^t)=1$ when $\bar{Y}^t \sim \mathbf{P}$ and error type ($\rom{2}$), also known as \emph{miss-detection}, occurs when $D(\bar{Y}^t)=0$ for $\bar{Y}^t \sim \mathbf{Q}$. Hence, we define the \underline{\emph{sum of conditional error probabilities}} of the intrusion detector for a given random sequence $\bar{Y}^t$, at time $t$~as \begin{equation} \label{eq:pe} p_t^e=\mathbb{P}(D(\bar{Y}^t)=0|\bar{Y}^t\sim \mathbf{Q})+\mathbb{P}(D(\bar{Y}^t)=1|\bar{Y}^t \sim \mathbf{P}). \end{equation} Note that $p_t^e$ is not a probability measure as it can take values larger than one. However, it will be useful when we define the notion of stealthy attacks in the following~section. \section{Formalizing Stealthy Attacks Requirements} \label{sec:stealthy} In this section, we capture the conditions for which an attack sequence is stealthy even from {{an optimal}} intrusion detector. Specifically, we define an attack to be strictly stealthy if \emph{there exists no detector that can perform better than random guess between the two hypothesis}; by better we mean the true attack detection probability is higher than the false alarm probability. However, reaching such stealthiness guarantees may not be possible in general. Therefore, we define the notion of $\epsilon$-stealthiness, which as we will show later, is attainable for a large class of nonlinear systems. Before formally defining the notion of attack stealthiness, we introduce the following lemma. \begin{lemma}\label{lemma:stealthy} Any intrusion detector $D$ cannot perform better than a random guess between the two hypothesis if and only if $p^e\geq 1$. Also, $p^e=1$ if and only if $D$ performs as well as a random guess detector. \end{lemma} \begin{proof} First, we consider the case $p^e> 1$. From~\eqref{eq:pe}, we have \begin{equation} \begin{split} 1&< p^e=\mathbb{P}(D(\bar{Y})=0|\bar{Y}\sim \mathbf{Q})+\mathbb{P}(D(\bar{Y})=1|\bar{Y} \sim \mathbf{P})\\ &=1-\mathbb{P}(D(\bar{Y})=1|\bar{Y}\sim \mathbf{Q})+\mathbb{P}(D(\bar{Y})=1|\bar{Y} \sim \mathbf{P})\\ \end{split} \end{equation} Thus, $\mathbb{P}(D(\bar{Y})=1|\bar{Y}\sim \mathbf{Q})< \mathbb{P}(D(\bar{Y})=1|\bar{Y} \sim \mathbf{P})$. This means the probability of attack detection is less than the false alarm rate; therefore, $D$ is performing worse than random guess as in random guess we have $\mathbb{P}(D(\bar{Y})=1|\bar{Y}\sim \mathbf{Q})= \mathbb{P}(D(\bar{Y})=1|\bar{Y} \sim \mathbf{P})=\mathbb{P}(D(\bar{Y})=1)$ because random guess is independent of the given distribution. When the equality holds (i.e., $p^e=1$), it holds that $\mathbb{P}(D(\bar{Y})=1|\bar{Y}\sim \mathbf{Q})= \mathbb{P}(D(\bar{Y})=1|\bar{Y} \sim \mathbf{P})$ where the decision of the detector $D$ is independent of the distribution of $\bar{Y}$ and therefore, the detector performs as the random guess detector. Since the reverse of all these implications hold, the other (i.e., necessary) conditions of the theorem also hold. \end{proof} Now, using Lemma~\ref{lemma:stealthy}, we can define the notions of strict stealthiness and $\epsilon$-stealthiness as follows. \begin{definition} \label{def:stealthiness} Consider the system from~\eqref{eq:plant}. An attack sequence is \textbf{strictly stealthy} if there exists no detector such that the total error probability $p_t^e$ satisfies $p_t^e<1$, for any $t\in \mathbb{Z}_{\geq 0}$. An attack is \textbf{$\epsilon$-stealthy} if for a given $\epsilon >0$, there exists no detector such that $p_t^e<1-\epsilon$, for any $t\in \mathbb{Z}_{\geq 0}$. \end{definition} The following theorem uses Neyman-Pearson lemma to capture the condition for which the received sensor measurements satisfy the stealthiness condition in Definition~\ref{def:stealthiness}. \begin{theorem}[\cite{khazraeReport21}] An attack sequence is \begin{itemize} \item strictly stealthy if and only if $KL\big(\mathbf{Q}(Y_{0}^a:Y_t^a)||\mathbf{P}(Y_{0}:Y_t)\big)=0$ for all $t\in \mathbb{Z}_{\geq 0}$, where $KL$ represents the Kullback–Leibler divergence operator. \item is $\epsilon$-stealthy if the corresponding observation sequence $Y_{0}^a:Y_t^a$ satisfies \begin{equation}\label{ineq:stealthiness} KL\big(\mathbf{Q}(Y_{0}^a:Y_t^a)||\mathbf{P}(Y_{0}:Y_t)\big)\leq \log(\frac{1}{1-\epsilon^2}). \end{equation} \end{itemize} \end{theorem} \begin{remark} The $\epsilon$-stealthiness condition defined in~\cite{bai2017data,bai2017kalman} requires $$\lim_{t\to \infty}\frac{KL\big(\mathbf{Q}(Y_{0}^a:Y_t^a)||\mathbf{P}(Y_{0}:Y_t)\big)}{t}\leq \epsilon.$$ This allows for the KL divergence to linearly increase over time for any $\epsilon>0$, and as a result, after large-enough time period the attack may be detected. On the other hand, our definition of $\epsilon$-stealthy only depends on $\epsilon$ and is fixed for any time $t$; thus, it introduces a stronger notion of stealthiness for the attack. \end{remark} \subsection{Formalizing Attack Goal}\label{sec:attack_goal} As previously discussed, the attacker intends to \emph{maximize} degradation of control performance. Specifically, as we consider the origin as the operating point, we formalize the attack objective as \emph{maximizing (the norm of) the states $x_t$}; i.e., moving the system's states into an unsafe region. Since there might be a zone between the safe and unsafe region, we define the the unsafe region as $\mathbf{U}=\{x\in \mathbb{R}^n\,\,|\,\,\Vert x\Vert_2 \geq \alpha\}$ for some $\alpha>R_{\mathbf{S}}$, where $R_{\mathbf{S}}$ is the radius of the safe region $\mathbf{S}$. Moreover, the attacker wants \emph{to remain stealthy (i.e., undetected by the intrusion detector)}, as formalized below. \begin{definition} \label{def:eps_alpha} The attack sequence, denoted by $\{a_{0}, a_{1},...\}$ is referred to as $(\epsilon,\alpha)$-successful attack if there exists $t'\in \mathbb{Z}_{\geq 0}$ such that $ \Vert x_{t'}^a \Vert \geq \alpha$ and the attack is $\epsilon$-stealthy for all $t\in \mathbb{Z}_{\geq 0}$. When such a sequence exists for a system, the system is called $(\epsilon,\alpha)$-attackable. When the system is $(\epsilon,\alpha)$-attackable for arbitrarily large $\alpha$, the system is referred to as a perfectly attackable system. \end{definition} Now, the problem considered in this work can be formalized as capturing the potential impact of stealthy attacks on a considered system; specifically, in the next section, we derive conditions for existence of a \emph{stealthy} yet \emph{effective} attack sequence $a_{0}, a_{1},...$ resulting in $\Vert x_t^a\Vert \geq \alpha$ for some $t\in \mathbb{Z}_{\geq 0}$ -- i.e., we find conditions for the system to be $(\epsilon,\alpha)$-attackable. Here, for an attack to be stealthy, we focus on the $\epsilon-$stealthy notion; i.e., that even the best intrusion detector could only improve the detection probability by $\epsilon$ compared to the random-guess baseline detector. \section{Vulnerability Analysis of Nonlinear Systems to Stealthy Attacks} \label{sec:perfect} In this section, we derive the conditions such that the nonlinear system~\eqref{eq:plant} with closed-loop dynamics~\eqref{eq:closed-loop} is vulnerable to effective stealthy attacks formally defined in Section~\ref{sec:stealthy}. The following theorem captures such condition. \begin{theorem} \label{thm:PAt} The system~\eqref{eq:plant} is ($\epsilon,\alpha$)-attackable for arbitrarily large $\alpha$ and arbitrarily small $\epsilon$, if the closed-loop system~\eqref{eq:closed-loop} is incrementally exponentially stable (IES) in the set $\mathbf{S}$ and the system~\eqref{eq:input-state} is incrementally unstable (IU) in the set $\mathbf{S}$. \end{theorem} \begin{proof} Assume that the trajectory of the system and controller states for $t\in \mathbb{Z}_{<0}$ is denoted by {$\mathbf{X}_{-\infty}:\mathbf{X}_{-1}$}. Following attack start at $t=0$, let us consider the evolutions of the system with and without attacks during $t\in \mathbb{Z}_{\geq 0}$. For the system under attack, starting at time zero, the trajectory {$\mathbf{X}_{0}^a:\mathbf{X}_{t}^a$} of the system and controller states is governed by \begin{equation}\label{eq:attack_trajec} \begin{split} x_{t+1}^a=&f(x_t^a,u_t^a)+ w_t,\quad y_t^{c,a}=h(x_t^a)+v_t+a_t\\ \mathpzc{X}_{t}^a=&f_c(\mathpzc{X}^a_{t-1},y_t^{c,a}),\quad u_t^a=h_c(\mathpzc{X}^a_{t},y_t^{c,a}).\\ \end{split} \end{equation} On the other hand, if the system were not under attack during $t\in \mathbb{Z}_{\geq 0}$, we denote the plant and controller state evolution by {$\mathbf{X}_{0}:\mathbf{X}_{t}$}. Hence, it is a continuation of the system trajectories {$\mathbf{X}_{-\infty}:\mathbf{X}_{-1}$} if hypothetically no data-injection attack occurs during $t\in \mathbb{Z}_{\geq 0}$. Since the system and measurement noises are independent of the state, we can assume that $w_t^a=w_t$ and $v_t^{a}=v_t$. In this case, the dynamics of the plant and controller state evolution satisfies \begin{equation}\label{eq:free_trajec} \begin{split} x_{t+1}=&f(x_t,u_t)+ w_t,\quad y_t^c=h(x_t)+v_t,\\ \mathpzc{X}_{t}=&f_c(\mathpzc{X}_{t-1},y_t^c),\quad u_t=h_c(\mathpzc{X}_{t},y_t^c),\\ \end{split} \end{equation} which can be captured in the compact form~\eqref{eq:closed-loop}, with $\mathbf{X}_{0}=\begin{bmatrix}x_{0}\\\mathpzc{X}_{0}\end{bmatrix}$. Now, consider the sequence of attack vectors injected in the system from~\eqref{eq:attack_trajec}, which are constructed using the following dynamical model \begin{equation}\label{eq:attack_seq} \begin{split} s_{t+1}&=f(x_t^a,u_t^a) - f(x_t^a-s_t,u_t^a) \\ a_t&=h(x_t^a-s_t)-h(x_t^a), \end{split} \end{equation} for $t\in \mathbb{Z}_{\geq 0}$, and with some arbitrarily chosen nonzero initial value of $s_0$. By injecting the above attack sequence into the sensor measurements, we can verify that $y_t^{c,a}=h(x_t^a)+v_t+a_t=h(x_t^a-s_t)+v_t$. After defining \begin{equation} e_t\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} x_t^a-s_t, \end{equation} and combining~\eqref{eq:attack_seq} with~\eqref{eq:attack_trajec}, the dynamics of $e_t$ and the controller, and the corresponding input and output satisfy \begin{equation}\label{eq:closed_loop_attack} \begin{split} e_{t+1}=&f(e_t,u_t^a)+ w_t,\quad y_t^{c,a}=h(e_t)+v_t,\\ \mathpzc{X}_{t}^a=&f_c(\mathpzc{X}^a_{t-1},y_t^{c,a}), \quad u_t^a=h_c(\mathpzc{X}^a_{t},y_t^{c,a}), \end{split} \end{equation} with the initial condition $e_0=x_0^a-s_0$. Now, if we define $\mathbf{X}^e_t=\begin{bmatrix}e_{t}\\\mathpzc{X}_{t}^a\end{bmatrix}$, it holds that \begin{equation}\label{eq:closed_attack} \mathbf{X}^e_{t+1}=F(\mathbf{X}^e_{t},\mathbf{W}_t). \end{equation} with $\mathbf{X}^e_{0}=\begin{bmatrix}e_{0}\\\mathpzc{X}_{0}^a\end{bmatrix}$. Since we have that $x_0^a=x_0$ and $\mathpzc{X}_{0}^a=\mathpzc{X}_{0}$, it holds that $\mathbf{X}_{0}-\mathbf{X}^e_{0}=\begin{bmatrix}s_{0}\\0\end{bmatrix}$. On the other hand, since both~\eqref{eq:closed_attack} and~\eqref{eq:closed-loop} share the same function and argument $\mathbf{W}_t$, the closed-loop system~\eqref{eq:closed_attack} is IES, and it also follows that \begin{equation}\label{eq:error_bound} \begin{split} \Vert \mathbf{X}(t,\mathbf{X}_{0},\mathbf{W})-\mathbf{X}^e(t,\mathbf{X}^e_{0},\mathbf{W})\Vert &\leq \kappa \Vert \mathbf{X}_{0}-\mathbf{X}^e_{0}\Vert \lambda^{-t}\\ &\leq \kappa \Vert s_0\Vert \lambda^{-t}; \end{split} \end{equation} therefore, the trajectories of $\mathbf{X}$ (i.e., the system without attack) and $\mathbf{X}^e$ converge to each other exponentially fast. We now use these results to show that the generated attack sequence satisfies the $\epsilon$-stealthiness condition. By defining $\mathbf{Z}_{t}=\begin{bmatrix}x_{t}\\y_t^{c}\end{bmatrix}$ and $\mathbf{Z}_{t}^e=\begin{bmatrix}e_{t}\\y_t^{c,a}\end{bmatrix}$, it holds that \begin{equation}\label{ineq:1} \begin{split} &KL\big(\mathbf{Q}(Y_{0}^a:Y_t^a)||\mathbf{P}(Y_{0}:Y_t)\big) \\ &\stackrel{(i)}\leq KL\big(\mathbf{Q}(\mathbf{X}_{0}^e:\mathbf{X}_{t}^e)||\mathbf{P}(\mathbf{X}_{0}:\mathbf{X}_{t})\big) \\ &\stackrel{(ii)}\leq KL\big(\mathbf{Q}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1},\mathbf{Z}_{0}^e:\mathbf{Z}_{t}^e)||\mathbf{P}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1},\mathbf{Z}_{0}:\mathbf{Z}_{t})\big), \end{split} \end{equation} where we applied the data-processing inequality property of KL-divergence for $t\in \mathbb{Z}_{\geq 0}$ to obtain $(i)$, and the monotonicity property of KL-divergence to obtain the inequality $(ii)$.\footnote{Due to the space limitation, we do not introduce data-processing, chain-rule, and monotonicity properties of KL-divergence. More information about these terms can be found in~\cite{thomas2006elements}.} Then, we apply the chain-rule property of KL-divergence on the right-hand side of \eqref{ineq:1} to obtain the following \begin{equation}\label{ineq:2} \begin{split} &KL\big(\mathbf{Q}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1},\mathbf{Z}_{0}^e:\mathbf{Z}_{t}^e)||\mathbf{P}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1},\mathbf{Z}_{0}:\mathbf{Z}_{t})\big)\\ &= KL\big(\mathbf{Q}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})||\mathbf{P}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})\big)+\\ &\,\,\,\,\,\,\,KL\big(\mathbf{Q}(\mathbf{Z}_{0}^e:\mathbf{Z}_{t}^e|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})||\mathbf{P}(\mathbf{Z}_{0}:\mathbf{Z}_{t}|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})\big)\\ &=KL\big(\mathbf{Q}(\mathbf{Z}_{0}^e:\mathbf{Z}_{t}^e|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})||\mathbf{P}(\mathbf{Z}_{0}:\mathbf{Z}_{t}|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})\big); \end{split} \end{equation} here, we used the fact that the KL-divergence of two identical distributions (i.e., $\mathbf{Q}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})$ and $\mathbf{P}(\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})$ since the system is not under attack for $t<0$) is zero. Applying the chain-rule property of KL-divergence to~\eqref{ineq:2} results in \begin{equation}\label{ineq:3} \begin{split} K&L\big(\mathbf{Q}(\mathbf{Z}_{0}^e:\mathbf{Z}_{t}^e|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})||\mathbf{P}(\mathbf{Z}_{0}:\mathbf{Z}_{t}|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})\big)\\ &\leq KL\big(\mathbf{Q}(e_0|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})||\mathbf{P}(x_{0}|\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})\big)\\ &+ KL\big(\mathbf{Q}(y_0^{c,a}|e_0,\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})||\mathbf{P}(y_{0}|x_0,\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1})\big)\\ &+...+ KL\big(\mathbf{Q}(e_t|\mathbf{Z}_{-\infty}:\mathbf{Z}_{t-1}^e)||\mathbf{P}(x_{t}|\mathbf{Z}_{-\infty}:\mathbf{Z}_{t-1})\big)\\ &+ KL\big(\mathbf{Q}(y_t^{c,a}|e_t,\mathbf{Z}_{-\infty}:\mathbf{Z}^e_{t-1})||\mathbf{P}(y_{t}|x_t,\mathbf{Z}_{-\infty}:\mathbf{Z}_{t-1})\big). \end{split} \end{equation} Given $\mathbf{Z}_{-\infty}:\mathbf{Z}_{t-1}$, the distribution of $x_t$ is a Gaussian with mean $f(x_{t-1},u_{t-1})$ and covariance $\Sigma_w$. Similarly given $\mathbf{Z}_{-\infty}:\mathbf{Z}_{-1},\mathbf{Z}_{0}^e:\mathbf{Z}_{t-1}^e$, the distribution of $e_t$ is a Gaussian with mean $f(e_{t-1},u_{t-1}^a)$ and covariance $\Sigma_w$. Since we have that $x_t=f(x_{t-1},u_{t-1})+w_t$ and $e_t=f(e_{t-1},u_{t-1}^a)+w_t$ according to~\eqref{eq:free_trajec} and~\eqref{eq:closed_loop_attack}, it holds that $f(x_{t-1},u_{t-1})-f(e_{t-1},u_{t-1}^a)=x_t-e_t$. On the other hand, in~\eqref{eq:error_bound} we showed that $\Vert x_t-e_t\Vert \leq \kappa \Vert s_0\Vert \lambda^{-t}$ holds for $t\in \mathbb{Z}_{\geq 0}$. Therefore, for all $t\in \mathbb{Z}_{\geq 0}$, it holds that \begin{equation}\label{ineq:4} \begin{split} KL\big(\mathbf{Q}(e_t|\mathbf{Z}_{-\infty}:&\mathbf{Z}_{t-1}^e)||\mathbf{P}(x_{t}|\mathbf{Z}_{-\infty}:\mathbf{Z}_{t-1})\big)= \\ =& (x_t-e_t)^T \Sigma_w^{-1} (x_t-e_t) \\\leq& \kappa^2 \Vert s_0\Vert^2 \lambda^{-2t} \lambda_{max}(\Sigma_w^{-1}), \end{split} \end{equation} where $\lambda_{max}(\Sigma_w^{-1})$ is the maximum eigenvalue of the matrix~$\Sigma_w^{-1}$. Now, using the Markov property it holds that $\mathbf{Q}(y_t^{c,a}|e_t,\mathbf{Z}_{-\infty}:\mathbf{Z}^e_{t-1})=\mathbf{Q}(y_t^{c,a}|e_t)$ and $\mathbf{P}(y_{t}|x_t,\mathbf{Z}_{-\infty}:\mathbf{Z}_{t-1})=\mathbf{P}(y_{t}|x_t)$; also, from~\eqref{eq:free_trajec} and~\eqref{eq:closed_loop_attack} it holds that given $x_t$ and $e_t$, $\mathbf{P}(y_{t}|x_t)$ and $\mathbf{Q}(y_t^{c,a}|e_t)$ are both Gaussian with mean $h(x_t)$ and $h(e_t)$, respectively and covariance $\Sigma_v$. Thus, it follows that \begin{equation}\label{ineq:5} \begin{split} KL\big(\mathbf{Q}&(y_t^{c,a}|e_t)||\mathbf{P}(y_{t}|x_t)\big)\\ &=\big(h(x_t)-h(e_t)\big)^T \Sigma_v^{-1} \big(h(x_t)-h(e_t)\big)\\ &\leq L_h^2 (x_t-e_t)^T \Sigma_v^{-1} (x_t-e_t)\\ &\leq L_h^2 \kappa^2 \Vert s_0\Vert^2 \lambda^{-2t} \lambda_{max}(\Sigma_v^{-1}). \end{split} \end{equation} Combining~\eqref{ineq:1}-\eqref{ineq:5} results in \begin{equation}\label{eq:epsilon} \begin{split} &KL\big(\mathbf{Q}(Y_{0}^a:Y_t^a)||\mathbf{P}(Y_{0}:Y_t)\big) \leq \\ &\sum_{i=0}^t \kappa^2 \Vert s_0\Vert^2 \lambda^{-2t} \lambda_{max}(\Sigma_w^{-1})+L_h^2 \kappa^2 \Vert s_0\Vert^2 \lambda^{-2t} \lambda_{max}(\Sigma_v^{-1})\\ &\leq \frac{\kappa^2 \Vert s_0\Vert^2}{1-\lambda^2} \big(\lambda_{max}(\Sigma_w^{-1})+L_h^2 \lambda_{max}(\Sigma_v^{-1})\big)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}} b_{\epsilon}. \end{split} \end{equation} Finally, with $b_{\epsilon}$ defined as in~\eqref{eq:epsilon}, the attack sequence defined in~\eqref{eq:attack_seq} satisfies the $\epsilon$-stealthiness condition with $\epsilon=\sqrt{1-e^{-b_{\epsilon}}}$. We now show that the proposed attack sequence is effective; i.e., there exists $t'\in \mathbb{Z}_{\geq 0}$ such that $\Vert x_{t'}^a\Vert \geq \alpha$ for arbitrarily large $\alpha$. To achieve this, consider the two dynamics from~\eqref{eq:attack_trajec} and~\eqref{eq:closed_loop_attack} for any $t\in \mathbb{Z}_{\geq 0}$ \begin{equation} \begin{split} x_{t+1}^a=&f(x_t^a,u_t^a)+ w_t=f_u(x_t^a,U_t^a)\\ e_{t+1}=&f(e_t,u_t^a)+ w_t=f_u(e_t,U_t^a) \end{split} \end{equation} with $U_t^a=\begin{bmatrix}{u_t^a}^T&w_t^T\end{bmatrix}^T$, for $t\in \mathbb{Z}_{\geq 0}$. Since we assumed that the open-loop system~\eqref{eq:input-state} is IU on the set $\mathbf{S}$, it holds that for all $x_0^a=x_0\in \mathbf{S}$, there exits a nonzero $s_0$ such that for any $M>0$ \begin{equation}\label{eq:alpha} \Vert x^a(t,x_0^a,U^a)-e(t,x_0^a-s_0,U^a)\Vert \geq M \end{equation} holds in $t\geq t'$, for some $t'\in \mathbb{Z}_{\geq 0}$. On the other hand, we showed in~\eqref{eq:error_bound} that $\Vert x(t,x_0,U)-e(t,x_0^a-s_0,U^a)\Vert \leq \kappa \Vert s_0\Vert \lambda^{-t}$. Combining this with~\eqref{eq:alpha} and using the fact that $\Vert x(t,x_0,U)\Vert \leq R_{\mathbf{S}}$ results in \begin{equation}\label{eq:alpha_x_a} \begin{split} &\Vert x^a(t,x_0^a,U^a)-x(t,x_0-s_0,U)\Vert = \\ &\Vert x^a(t,x_0^a,U^a)-e(t,x_0^a-s_0,U^a)+e(t,x_0^a-s_0,U^a)\\&-x(t,x_0-s_0,U)\Vert \geq \Vert x^a(t,x_0^a,U^a)-e(t,x_0^a-s_0,U^a)\Vert \\ & -\Vert e(t,x_0^a-s_0,U^a)-x(t,x_0-s_0,U)\Vert \geq M-\kappa \Vert s_0\Vert \lambda^{-t} \\ &\Rightarrow \Vert x^a(t,x_0^a,U^a)\Vert \geq M-\kappa \Vert s_0\Vert \lambda^{-t}-R_{\mathbf{S}}\\ &\qquad \qquad \qquad \qquad \geq M-\kappa \Vert s_0\Vert -R_{\mathbf{S}}. \end{split} \end{equation} Since $M$ is arbitrarily, we can choose it to satisfy $M>\alpha+R_s+\kappa \Vert s_0\Vert$, for arbitrarily large $\alpha$. Thus, the system is $(\epsilon,\alpha)$-attackable. \end{proof} From~\eqref{eq:closed_loop_attack}, we can see that the false sensor measurements are generated by the evolution of $e_t$. Therefore, intuitively, the attacker wants to fool the system into believing that $e_t$ is the actual state of the system instead of $x_t^a$. Since $e_t$ and $x_t$ (i.e., the system state if no attack occurs during $t\in \mathbb{Z}_{\geq 0}$) converge to each other exponentially fast, the idea is that the system almost believes that $x_t$ is the system state (under attack), while the actual state $x_t^a$ becomes arbitrarily large. Furthermore, all parameters $\kappa$, $ \lambda$, $L_h$, $\Sigma_w$, and $\Sigma_v$ in~\eqref{eq:epsilon} are some constants that depend either on system properties ($L_h$, $\Sigma_w$, and $\Sigma_v$) or are determined by the controller design ($\kappa$, $ \lambda$). However, $s_0$ is set by the attacker, and \emph{it can be chosen arbitrarily small to make $\epsilon$ arbitrarily close to zero}. Yet, $s_0$ can not be equal to zero; in that case~\eqref{eq:alpha} would not hold -- i.e., the attack would not not be impactful. Therefore, as opposed to attack methods targeting the prediction covariance in~\cite{bai2017kalman} where the attack impact linearly changes with $\epsilon$, here arbitrarily large $\alpha$ (high impact attacks) can be achieved even with an arbitrarily small $\epsilon$ -- it may only take more time to get to $\Vert x^a_{t'}\Vert \geq \alpha$. \begin{remark} Even though we assumed that the closed-loop dynamics is IES, slightly weaker results can still be obtained for closed-loop dynamics with incrementally asymptotic stability. We will consider this case as future work. \end{remark} \begin{remark} For constructing the attack sequence in~\eqref{eq:attack_seq} we assumed that the attacker has knowledge of the system's nonlinear functions $f$ and $h$, as well as has access to the values of the system state. In future work, we will show how these assumptions can be relaxed for systems with general nonlinear dynamics \end{remark} \begin{remark} In case that either $w=0$ or $v=0$ (i.e., when there is no process or measurement noise), one can still get a similar bound on the KL-divergence only as a function of the nonzero noise covariance by applying monotonicity and data-processing inequality. However, ensuring stealthiness requirement is not possible if both $w=0$ and $v=0$ (i.e., for the noiseless system), as the system would be completely deterministic, and thus theoretically any small perturbation to the sensor measurements could be detected. \end{remark} \iffalse \subsection{Discussion on Incremental Stability} As previously discussed, the notions of incremental stability and contraction theory are well-defined \cite{angeli2002lyapunov,tran2018convergence,tran2016incremental} and going over their satisfying conditions is beyond the scope of this paper. However, to provide some useful intuition, we start with the following example. \begin{example} Consider the system \begin{equation*} x_{t+1}=x_{t}+a x_{t}^3 +bu_t+w_t, \end{equation*} where $a>0$ and $w$ is a Gaussian noise with zero mean and variance $\sigma_w$. Also, consider the feedback controller $$u_t=-\frac{a}{b}x_t^3-\frac{1}{2b}x_t.$$ With such control input, the closed-loop system becomes \begin{equation}\label{eq:example1} x_{t+1}=.5x_{t}+w_t. \end{equation} The above system is GIES since the difference between any two trajectories of~\eqref{eq:example1} with the same noise $w_0:w_{t-1}$ and different initial conditions ($\xi_1$ and $\xi_2$), defined as $\Delta x_{t}=x(t,\xi_1,w)-x(t,\xi_2,w)$, has the dynamics $\Delta x_{t+1}=.5\Delta x_{t}$ with the nonzero initial condition $\Delta x_{0}=\xi_1-\xi_2$. Hence, $\Delta x_t$ converge exponentially to zero and the closed-loop control system is exponentially stable. Now, let us consider \begin{equation}\label{eq:example2} x_{t+1}=x_{t}+a x_{t}^3 +d_t, \end{equation} with $d_t=bu_t+w_t$. The difference between two trajectories with the same input $\{d_0,...,d_{t-1}\}$ and different initial condition ($\xi'_1$ and $\xi'_2$) is $\Delta x'_{t}=x(t,\xi'_1,d)-x(t,\xi'_2,d)$. Therefore, the difference dynamics becomes $\Delta x'_{t+1}=\Delta x'_{t}+a (x^3(t,\xi'_1,d)-x^3(t,\xi'_2,d))$. One can use Lyapunov function method by defining $V_t={\Delta x_{t}'}^2$ and verify that \begin{equation} \begin{split} V_{t+1}&-V_t= {\Delta {x'}^2_{t+1}}-{\Delta x'_{t}}^2 = 2a{\Delta x'_{t}}^2\Big(x^2(t,\xi'_1,d)\\ &+x^2(t,\xi'_2,d)+x(t,\xi'_1,d)x(t,\xi'_2,d)\Big)+a^2{\Delta x'_{t}}^2\times \\ &\Big(x^2(t,\xi'_1,d)+x^2(t,\xi'_2,d) +x(t,\xi'_1,d)x(t,\xi'_2,d)\Big). \end{split} \end{equation} Since $x^2(t,\xi'_1,d)+x^2(t,\xi'_2,d)+x(t,\xi'_1,d)x(t,\xi'_2,d)>0$ and $V_0>0$, it holds that $V_{t+1}-V_t>0$ for all $t\in \mathbb{Z}_{\geq 0}$; therefore, $\Delta x'_{t}$ will diverge to infinity, and \eqref{eq:example2} is globally incrementally unstable. \end{example} \todo{so what is the intuition? what did I learn from this example? especially in light of your earlier results} \fi \subsection{Vulnerability Analysis of LTI Systems} Theorem~\ref{thm:PAt} can also be applied to find the condition for the existence of ($\epsilon,\alpha$)-successful attacks on LTI systems. Specifically, the LTI formulation of~\eqref{eq:plant} and~\eqref{eq:plant_withoutB} is \begin{equation}\label{eq:lti} \begin{split} {x}_{t+1} &= Ax_t+Bu_t+w_t,\quad y_t^c = Cx_t+v_t,\\ \mathpzc{X}_{t} &= A_c\mathpzc{X}_{t-1}+B_cy_t^c,\quad u_t = C_c\mathpzc{X}_{t};\\ \end{split} \end{equation} LTI systems with any controller (e.g., LQG controllers) can be captured in the above form. The following lemma provides the conditions for IES and IU for the above LTI~system. \begin{lemma}\label{lemma:LTI} Consider the LTI dynamical system in the form of $x_{t+1}=Ax_t+Bd_t$. The system is IES if and only if all eigenvalues of the matrix $A$ are inside the unit circle. The system is IU if and only if $A$ has an unstable eigenvalue. \end{lemma} \begin{proof} The proof is straightforward and follows from the definition and the direct method of Lyapunov. \end{proof} This allows us to directly capture conditions for stealthy yet effective attacks on LTI systems. \begin{corollary} \label{cor:cor1} The LTI system~\eqref{eq:lti} is ($\epsilon,\alpha$)-attackable for arbitrarily large $\alpha$ if the matrix $A$ is unstable and the closed-loop control system is asymptotically stable. \end{corollary} \begin{proof} The proof is directly obtained\textbf{} by combining Theorem~\ref{thm:PAt} and Lemma~\ref{lemma:LTI}. \end{proof} Asymptotic stability of the closed-loop system is not a restrictive assumption as stability is commonly the weakest required performance guarantee for a control system. Matrix $A$ being unstable is a necessary and sufficient condition for satisfying ($\epsilon,\alpha$)-attackability when any set of sensors can be compromised. Note that the ($\epsilon,\alpha$)-attackability condition for LTI systems with an optimal detector complies with the results from~\cite{mo2010false,jovanov_tac19} where LQG controllers with residue based detectors (e.g., $\chi^2$ detectors) have been considered. \begin{remark} The false-date injection attack sequence design method from~\eqref{eq:attack_seq} will reduce into a simple dynamical model \begin{equation} \begin{split} s_{t+1}&=Ax_t^a+Bu_t^a - (A(x_t^a-s_t)+Bu_t^a)=As_t \\ a_t&=C(x_t^a-s_t)-C(x_t^a)=-Cs_t, \end{split} \end{equation} that only requires knowledge about the matrices $A$ and $C$. In addition, unlike the case for nonlinear systems, there is no need to have access to the actual states of the system. \end{remark} \begin{remark} In~\cref{sec:attack_model} we assumed that $\mathcal{K}=\mathcal{S}$; i.e., the attacker can compromise all sensors. However, when the system is LTI, the minimum subset of compromised sensors can be obtained as \begin{equation} \min_{v_i\in \{v_1,...,v_q\}} \Vert \text{supp} (Cv_i)\Vert_0, \end{equation} where $ \{v_1,...,v_q\}$ denotes the set of unstable eigenvectors of the matrix $A$, and $\text{supp}$ denotes the set of nonzero elements of the vector. \end{remark} \section{Preliminaries}\label{sec:prelim} Let $\mathbb{X}\subseteq \mathbb{R}^n$ and $\mathbb{D}\subseteq \mathbb{R}^m$, with $0\in \mathbb{X}, \mathbb{D}$. Consider a discrete-time nonlinear system with an exogenous input, modeled in the state-space form as \begin{equation}\label{eq:prilim} x_{t+1}=f(x_t,d_t),\quad x_t\in \mathbb{X},\,\,t\in \mathbb{Z}_{t\geq 0}, \end{equation} where $f:\mathbb{X}\times\mathbb{D}\to \mathbb{X}$ is continuous and $f(0,0)=0$. We denote by $x(t,\xi,d)$ the trajectory (i.e., the solution) of~\eqref{eq:prilim} at time $t$, when the system has the initial condition $\xi$ and is subject to the input sequence $\{d_0:d_{t-1}\}$.\footnote{To simplify our notation, we denote the sequence $\{d_0:d_{t-1}\}$ as $d$.} The following definitions are derived from~\cite{angeli2002lyapunov,tran2018convergence,tran2016incremental}. \begin{definition} The system~\eqref{eq:prilim} is incrementally exponentially stable (IES) in the set $\mathbb{X}\subseteq \mathbb{R}^n$ if there exist $\kappa>1$ and $\lambda>1$ such that \begin{equation} \Vert x(t,\xi_1,d)-x(t,\xi_2,d)\Vert \leq \kappa \Vert \xi_1-\xi_2\Vert \lambda^{-t}, \end{equation} holds for all $\xi_1,\xi_2\in \mathbb{X}$, any $d_t\in \mathbb{D}$, and $t\in \mathbb{Z}_{t\geq 0}$. When $\mathbb{X}=\mathbb{R}^n$, the system is referred to as globally incrementally exponentially stable (GIES). \end{definition} \begin{definition} The system~\eqref{eq:prilim} is incrementally unstable (IU) in the set $\mathbb{X}\subseteq \mathbb{R}^n$ if for all $\xi_1\in \mathbb{X}$ and any $d_t\in \mathbb{D}$, there exists a $\xi_2$ such that for any $M>0$, \begin{equation} \Vert x(t,\xi_1,d)-x(t,\xi_2,d)\Vert \geq M, \end{equation} holds for all $t\geq t'$, for some $t'\in \mathbb{Z}_{t\geq 0}$. \end{definition} \section{Simulation Results} \label{sec:simulation} We illustrate our results on a case-study. Specifically, we consider a fixed-base inverted pendulum equipped with an Extended Kalman Filter to estimate the states of the system followed by a feedback full state controller to keep the pendulum rod in the inverted position. Using $x_1=\theta$ and $x_2=\dot{\theta}$, the inverted pendulum dynamics can be modeled~a \begin{equation} \label{eq:state_Nonlin} \begin{split} \dot{x}_1&=x_2\\ \dot{x}_2&=\frac{g}{r}\sin{x_1} - \frac{b}{mr^2} x_2+ \frac{L}{mr^2}; \end{split} \end{equation} here, $\theta$ is the angle of pendulum rod from the vertical axis measured clockwise, $b$ is the Viscous friction coefficient, $r$ is the radius of inertia of the pendulum about the fixed point, $m$ is the mass of the pendulum, $g$ is the acceleration due to gravity, and $L$ is the external torque that is applied at the fixed base. We assumed that both the states are measured by sensors. Finally, we assumed $g=9.8$, $m=.2 Kg$, $b=.1$, $r=.3m$, $\Sigma_w=\Sigma_v=\begin{bmatrix}.01&0\\0&.01\end{bmatrix}$ and discretized the model with $T_s=10~ms$. We assume the safe region for angle around the equilibrium point $\theta=0$ is $\mathbf{S}=(-\frac{\pi}{3},\frac{\pi}{3})$. To detect the presence of attack, we designed a standard $\chi^2$-based anomaly detector that receives the sensor values and outputs the residue/anomaly alarm. We used the attack model considered in~\eqref{eq:attack_seq} to generate the sequence of false-data injection attacks over time. Fig.~\ref{fig:attack}(a) shows the angle of the pendulum pod over time. Before the attack starts at time zero, the pendulum pod is around the angle zero; however, after initiating the attack it can be observed that the absolute value of the angle increases over time until it leaves the safe set and even becomes more than $\pi$. Note that having values more than $\pi$ does not make a difference because we have a periodic system, {and $\pi$ corresponds to the pendulum falling down}. Meanwhile, the distribution of the norm of the residue signal (see Fig.~\ref{fig:attack}(b)) does not change before and after attack initiation -- i.e., the attack remains~stealthy. \begin{figure}[!t] \centering {\label{fig:theta} \includegraphics[clip,width=0.484\columnwidth]{./Pictures/theta.pdf } {\label{fig:residue} \includegraphics[clip,width=0.484\columnwidth]{./Pictures/residue.pdf}% } \caption{(a) Angle's ($\theta$) absolute value over time for the under-attack system, when the attack starts at time zero; (b) The residue norm over time for the under-attack system, when the attack starts at time zero. }\label{fig:attack} \end{figure}

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\section{Introduction} Ticket or Incident Management is an important ITIL~\cite{Ref1} process, where an incoming ticket logged by the users of the system is analyzed and appropriate measures are taken for remediation within pre-specified time constraints. In terms of an activity pipeline, the tickets are first triaged, starting with a check for duplication so that one does not start working on a ticket which has already been closed or remediated, the tickets are then clustered based on themes around business process or techno-functional areas~\cite{Ref2}; further, the tickets are assigned to service engineers who have already worked on similar incidents or technology; the service engineers may take recourse to knowledge management tools which may help them to resolve the problem in a guided way by providing contextual search facility on product knowledge base or literature~\cite{Ref3}. Volume of research has been conducted on each of the steps of the process. The problem of triaging and identifying duplicate incidents or tickets has been discussed in ~\cite{Ref4}, especially in the context of mining large-scale open source and other software repositories and developer comments. Mapping to business processes has been studied in~\cite{Ref6}. Recommendation of assignee has been discussed in~\cite{Ref7}. While there is no dearth in investigation on and application of advanced techniques like Machine Learning (classification and clustering), Natural Language Processing and Guided Search; in practice, the problem remains largely unsolved. There are several contributing factors that make ``in the trench'' application of sophisticated technologies challenging. We list a few of them – \begin{description} \item[The data conundrum] Industry data is either very noisy or very sparse. This is especially true when the systems have wide variety of applications and usage scenarios. This makes data sampling and preparation for machine learning fairly difficult. \item[SME support] Most classification techniques would require proper identification of features which are discriminating enough to render the data set classifiable. Often times, we find that these discriminating features are buried in the semantics of the description and it becomes very difficult for a data-scientist to conduct proper feature engineering without help from the subject matter experts (SMEs) to understand the semantics. However, the SMEs may often not have enough time to support. \item[Cost and Time to market] In practice, for the solution to be usable, the time to maturity would have to be short and implementable without much penalty in terms of cost and time. This is because such systems are often used to augment the support engineers, who may not have enough expertise and training to deal with intricacies. \end{description} In this report, we describe a ``shop usable'' for some aspects of the incident management problem and share experience from the implementation details. We believe that this paper would illustrate the contours and challenges of a practical research where unsupervised semantic similarity and ML techniques have been deployed to provide good enough solution. We also share some thoughts on the challenges for roll-out and feedback from the actual users of this incident management system. {\sf Notations:} In the rest of the paper, we will use `ticket', `incident', and `bug' interchangeably. The PoC prototype tool will be referred henceforth as \textsl{CogAM} (Cognitive Application Maintenance). \section{Problem Scenarios}\label{prob_scene} \begin{figure} \centering \includegraphics[scale=0.5]{2_new} \caption{Problem Scenarios}\label{pv} \end{figure} \subsection{Bug Deduplication: Identify Similar Incidents} \sf{Current Scenario:} When a new ticket is logged, users need to search existing ticket logs to find out if a duplicate bug has been already resolved/closed in past. Effectiveness of the search process not only depends upon nature of search query in relation to existing ticket log data but also on knowledge and expertize of the user. Even though existing application maintenance (AM) tools (e.g., Remedy~cite{remedy}, ServiceNow~\cite{servicenow}) offer many features to manage incidents and other associated processes, they fall short when it comes to offering plain text based semantic relevance search (often found in the realm of search engines). This makes it hard for users having relatively less experience to determine if semantically similar defects have already been raised in past and what actions were performed for those bugs. {\sf Solution Idea:} In order to keep barriers to adoption low, we adopted IR (information retrieval) based solution (similar to ~\cite{amoui2013search}) by aiming to enable search based bug deduplication and identification of semantically related incidents. IR based approaches are largely unsupervised in nature, hence saving initial efforts from SMEs towards training the tool and appear to solve deduplication problem in AM domain at right level of complexity. It's success is based upon implicit assumption that if two bugs are similar (or related), they would have overlapping surface form representations (ref. `distributional hypothesis'~\cite{}). Solution starts with generating a temporal informational model of incidents and their associations with assignees, business processes, and other semantically relevant factors. This model is used to estimate strengths of semantic associations to recommend related incidents (including duplicates), suggest assignees, identify potential business process associations, and pointers to potential resolutions. Ability to learn incrementally is designed to improve relevance of results from user interactions. Performance of CogAM could be improved further if it is coupled with domain knowledge sources, e.g., alias glossary consisting of synonymous terms. \subsection{Mapping Tickets to Business Processes} \sf{Current Scenario:} When a new ticket is logged in an AM tool, SME may map reported incident to specific business process as a starting point to provide right level of intervention. Correctness of mapping primarily depends upon knowledge and expertize of the SME in conjunction with the degree of comprehensiveness of business process documentation. {\sf Solution Idea:} Using existing log of tickets and associated business process mappings as recoded in the ticket log, CogAM would learn semantic associations between tickets and business processes by way of generating a ML model. When a new ticket arrives, CogAM would identify most probable business process, which current ticket should be mapped to and would recommend that to user. Based upon user acceptance or rejection of the recommendation, CogAM incrementally learns and improves its performance on the ticket mapping process. \subsection{Assignee Recommendation} \sf{Current Scenario:} When a new ticket is logged, based upon expertize, availability, and other factors, suitable team member is selected and ticket is assigned to him/her for resolution. If an AM tool has to be used for an automated assignment, it requires manual configuration of parameters and tickets in the AM tool. However, state-of-the-art AM tools ignore historical evidence from past experience of working on similar tickets and are generally designed as rule based systems. however, current practice is far from being optimal. From discussions with SMEs it turned out that when team sizes are not small (> 20) and ticket influx rates are high (> 50 per week), not always first time fixes of tickets were found to be correct and for 12-15\% occasions tickets were either reopened owing to incorrect fixes or got reassigned to different team member. {\sf Solution Idea:} When a new ticket arrives, CogAM parses its details and would identify potential relationship with currently available team members (having worked in past) to map the ticket to those team members who could potentially be assigned to it because they worked on similar tickets in past. CogAM would produce a ranked list of team members so that right decision could be made even when the most suitable team member is not available. Based upon user acceptance or rejection of the recommendation, CogAM would incrementally learn and improve its performance on the assignment process. Performance of CogAM could be improved further if it is coupled with domain heuristics (e.g., documented expertize of the team members). \subsection{Theme Mining: Identifying Central Problem Areas} \sf{Current Scenario:} When a ticket is raised, it may sometimes get tagged against high level functionality or component of the system or type of error potentially underlying the bug. Beyond this tagging, getting any further insights on the problem areas around which bugs are being raised remains limited to subjective experiences of the SMEs. Large scale collective analysis of ticket text is necessary to identify central problem areas around which bugs are being raised over a course of time and semantic relationships among these problem areas. Such insights enable better decision making towards allocating resources during design of newer versions of the applications and also during their maintenance. {\sf Solution Idea:} Using existing log of tickets CogAM applies combination of theme mining techniques including topic modelling, frequency based centrality, latent semantic analysis based centrality~\cite{raise2016} on the terms (noun phrases and verb phrases) extracted from the bug data. Based upon that it would identify central terms with at least 85\% bug coverage and their spread within user defined tagging (e.g., within modules). \section{Tool Design}\label{design} CogAM is implemented as web application in Java using Spring MVC architecture~\cite{mvc}. User experience is the most important thing in any application. We used AdminLTE template~\cite{lte} that is built on Bootstrap 3, JSP and JQuery to make user interface (UI) more user friendly, responsive and informative. We use Apache Lucene~\cite{lucene} for indexing and retrieving data and OpenNLP~\cite{8} for natural language processing. The high level architecture is shown in Figure~\ref{hla}. \begin{figure*}[ht!] \centering \includegraphics[width=\textwidth]{HLA_op} \caption{High level architecture}\label{hla} \end{figure*} The first step is to configure fields that are used to retrieve data and apply filters to refine search results. Tool provides some fields as pre-defined fields like Incident Id, Short description, Long Description etc. Beyond these, tool allows users to define custom fields as per business requirements. Figure~\ref{config} describes filed configuration. \begin{figure*} \centering \includegraphics[width=\textwidth]{config-fields_op} \caption{Field configurations}\label{config} \end{figure*} A field can be marked as information type where details of that particular field will be available in output screen or filter type with filter level where the field can be used as filter during search to refine results as per the requirements of the user. Filter level is used to define filter hierarchy e.g. Country, State, City is filter hierarchy while providing address details. The second step is to ingest data. The tool supports MS Excel format for input data. Table~\ref{t1} presents input data model, in which inputs to the designed tool need to be given as inputs. \begin{table} \centering \label{t1} \begin{tabular}{l|l} \hline \hline {\bf Field Name} & {\bf Field Description} \\ \hline \hline {ID} & Unique identifier of a ticket \\ \hline {Summary} & Brief description of error \\ \hline {Description} & Detailed description of error\\ \hline {Assignee} & Team member who fixed the bug \\ \hline {Business Process} & Associated business process step \\\hline {Created Date} & Date and time of creation of ticket \\\hline {Module} & Components or functionality affected \\\hline {Priority} & Criticality level assigned to the ticket \\ \hline {Status} & [Closed | Open | Assigned | \\ & Re-Opened | ...] \\ \hline \end{tabular} \caption{Input data model} \end{table} Fields with plain-text details are of primary interest in this paper: (i) short description of an incident, (ii) detailed description of the incident, and (iii) reproduction steps. The tool expects users to map fields to MS Excel columns. Also, users can configure name of the worksheet, date and time for data ingestion as shown below: \begin{figure*} \centering \includegraphics[width=\textwidth]{config-data_op} \caption{Data Ingestion Configuration}\label{settings} \end{figure*} Once the data is ingested then tool is ready for providing ranked list of similar tickets for a new bug, details of which are given to the tool as search query. Here user can apply filters if configured earlier and narrow down search by applying date range as shown below: \begin{figure} \centering \includegraphics[scale=0.6]{ingest_op} \caption{Search User Interface}\label{ingest_op} \end{figure} The results are categorized into four classes and users can get details of each output. The threshold of those four categories were decided based on empirical studies performed internally on various data sets. \begin{figure*} \centering \includegraphics[width=\textwidth]{results_op} \caption{Search results}\label{results} \end{figure*} \section{Pilot Studies and Observations}\label{exp} For experimental analysis, PoC tool was given to SMEs for evaluation in their actual practice from multiple client accounts spread across various industries including retail, leisure, energy and petrochemicals, and mining. In each case, we asked SMEs to consider multiple projects satisfying one or more of the following requirements: \begin{enumerate} \item Rate at which bugs are being raised are not very low (< 10 per week) \item Historical bug data-base contains at least 100 tickets \item Degree of redirection of bugs is at least 10\% i.e., bugs are not getting closed correctly for at least 10\% cases \item There exist sizable number of bugs affecting multiple functionality and/or modules \end{enumerate} Let us discuss for each use case eventual qualitative outcomes in light of the specific nature of the input data and other circumstantial factors which mattered in practice and challenges in adoption of current research in practice. \subsection{Identifying Semantically Similar Bugs} \sf{\bf Nature of Data:} Most of the time only single line details were available as bug description apart from information on other fields (see Table 1). Detailed descriptions were not available in all cases or were not very different from brief descriptions. Resolution notes or root cause analysis or steps to reproduce bugs were also often missing or were very brief or very extensive involving detailed communication which took place (often using emails) among team members while resolving bugs. Additionally, plain text descriptions contained specific symbols (e.g. '\#* ... *\#') to enclose terms (meaning of which known only within team) as visual clues about the nature of bugs. Sometime bugs having exactly same text but differing in these localized terms were considered semantically related but not duplicate by SMEs. Also, during experimental analysis of many other use cases it turned out that such code words contained critical semantic information, however, this was something not possible to extract without SME help. \sf{\bf Experimental Analysis:} Table~\ref{t_dedup} presents results from a pilot study with a retail account, wherein tool was injested with ticket log of nearly twenty six thousand incidents and SMEs designed $59$ sample bugs as queries based upon recent tickets which were not part of the ingested ticket log and evaluated relevance of the outputs from the tool from view point of resolving these bugs. Since it was not known to SMEs how many relevant bugs actually existed in the ticket log, only precision of the outputs at two different levels (top 5 and top 20) was measured. \begin{table*}[ht] \centering \begin{tabular}{l|l|l} \hline \hline & {\bf Relevant Results} & {\bf Relevant + Related Results} \\ \hline \hline \textbf{Top 5} & 67\% & 71\% \\ \hline \textbf{Top 20} & 61\% & 69\% \\ \hline \end{tabular} \caption{Precision of Search Results towards their relevance for users to identify duplicate bugs or identifying potential resolutions} \label{t_dedup} \end{table*} \sf{\bf Requirements for Minimal Solution:} In practice, domain familiarity and past experience of the users mattered a lot and often junior team members which were in sizable numbers, needed help from senior team members to figure out how to start understanding a bug. Therefore, when we asked them to use the tool, their primary requirement was that tool should be able to provide critical clues regarding a new bug, which otherwise would have come from senior SMEs. From operational perspective, users across levels appreciated that tool had small learning curve and one could start using it right away after installation with minimal configuration requirements. \sf{\bf Challenges in Adoption of Research:} When authors were designing the solution, it was not clear whether we need to adopt techniques which are claimed to be achieving best performance on open source data sets(~\cite{defectdatabse}) for the problem of duplicate defect identification. Interestingly most of the published works with high scores in detecting duplicate bugs apply supervised machine learning techniques (including deep learning), which though work well on large open source data sets, however, can't be easily adopted in practice without incurring significant cost to create desired training data in form of semantically equivalent pairs of bugs by the experienced SMEs before tool could be adopted in practice. Interestingly, parallel efforts were in place in some other client account towards adoption of such supervised ML based tools and in detailed comparative analysis, it turned out that our PoC tool was only 5\% below the other tool in performance (precision of top 20 relevant bugs for a new bug). On the other hand, simpler approaches based upon information retrieval methods have also been found to be reasonably effective in industrial settings~\cite{amoui2013search} and our experience confirms this. \subsection{Assignee Recommendation} \sf{\bf Nature of Data:} Most often ticket data only contains name or employee ID of the team member who raised and/or fixed a bug. However, when a team member is selected for an assignment, external factors like documented skills as recorded in HR databases, team-organization, current work load, etc. also play very important role in decision making process, which however are not recorded as a part of ticket details. \sf{\bf Requirements for Minimal Solution:} A solution which brings additional insights from the analysis of historical evidence received positive attention from the AM management. Time based filtering options were found to be of specific help since in many cases, at least some similar or related bugs had been fixed during recent past and older bugs might have been fixed by those who were no longer part of the team currently. \sf{\bf Experimental Analysis:} Table~\ref{t_assignee_1} presents results from pilot studies from three different industry accounts with varying ticket log sizes. In each case, we randomly identified more than 30 existing tickets from each log for validation purpose and estimated accuracy of predicting assignee when compared with the actual assignment as recorded in the log data. One challenge in this analysis is that our target set is relatively narrow (among many capable team members, only one is listed in the log-data set and we need to predict that!) Primary reason for adopting this intrinsic validation as against SME based validation was that SMEs expressed their lack of familiarity with the team members who worked long back and in some cases teams had just acquired the contract to maintain the application from another firm, hence SMEs did not have much knowledge to validate outputs. \begin{table}[ht] \centering \begin{tabular}{l|l|l|l} \hline \hline {\bf Data Set} & {\bf Size of} & {Number of} & {\bf Accuracy} \\ & {\bf Ticket Log} & {Assignees} & \\ \hline \hline \textbf{Data-Set 1} & 1850 & 31 & 0.50 \\ \hline \textbf{Data-Set 2} & 10670 & 13 & 0.47 \\ \hline \textbf{Data-Set 3} & 166902 & 26 & 0.47 \\ \hline \end{tabular} \caption{Average accuracy in predicting assignee using historical log as reference data} \label{t_assignee_1} \end{table} Realizing that on average results were relatively poor, we investigated this further to learn whether there existed any correlation between number of tickets which were assigned to a team member and accuracy in being able to predict that team member as recommended assignee during validation. Graph~\ref{fig:assignee} depicts that visually and as a numerical indicator, Pearson correlation coefficient between these factors was estimated as $0.34$, which clearly indicated that higher number of assigned tickets to a team member may not imply higher accuracy in predicting whether a team member should work on a ticket. This could be owing to the factors which were not recorded in the ticket log, however were playing key role during ticket assignment. \begin{figure} \centering \includegraphics[scale=0.6]{assignee} \caption{Graphical representation of association between tickets on which a team member has worked and accuracy in prediction of that team member during validation}\label{fig:assignee} \end{figure} \sf{\bf Challenges in Adoption of Research:} Despite the fact that there exist many published research works on this problem, in practice today, often ad-hoc approaches reflecting prevailing organizational processes and practices are to be found. One reason for this is that often there are some transient and circumstantial factors, which affect the decision making process. Hoever, such factors are not recorded in a way that these could be given as inputs to a tool. For example, consider the role of training programs, which play a role while assigning tickets to those team members who had undergone training on related functional areas recently. However, it would be hardly captured in an AM tool. This became further evident when we tried training a ML model on the ticket log (using features from vector space modeling and TF-IDf based information theoretic weighing) and Table~\ref{t_assignee_2} presents the results. It is clear from these results that ticket details as captured in the AM tool are hardly sufficient for achieving high accuracy in automated assignee recommendation. However, in practice, these outputs can still add value in assisting users in making informed decisions. \begin{table}[ht] \centering \begin{tabular}{l|l|l|l|l} \hline \hline {\bf Size of} & {\bf Number of} & {\bf P}& {\bf R} & {\bf F1-Score}\\ {\bf Ticket Log} & {\bf Assignees} & & & \\ \hline \hline 5600 & 28 & 0.35 & 0.31 & 0.33 \\ \hline \end{tabular} \caption{10 fold cross validation of SVM Classifier for Assignee Prediction} \label{t_assignee_2} \end{table} \subsection{Theme Mining} \sf{\bf Nature of Data:} Ticket log data often contains primary classification of the bugs in terms of modules or types of bugs. However, it misses giving any further insights, which can only be gleaned by detailed analysis of the bug descriptions, priority levels, number of re-directions, priority-escalations etc. \sf{\bf Requirements for Minimal Solution:} Themes or central problem areas must be extracted such that their correlations with user categorizations are explicit. Furthermore, tool must provide detailed navigation support so that users get explanatory support underlying extracted results by the tool. For example, if a tool depicts that a problem area {\it p = `page access issue'} is weakly connected with problem area {\it q = `password reset'} but is strongly connected with {\it r = `login failure'}, user should be given underlying justification in terms of displaying list of tickets where first pair of problem areas (p, q) appear together versus where second pair (p, r) of problem areas appear in the ticket log. {\sf Experimental Analysis:} Table~\ref{t_tm} shows results of various techniques, which we experimented with for mining central problem areas using examples ticket log consisting of twenty six thousand ticket details. Considered techniques included including latent semantic analysis based centrality; latent Dirichlet allocation; term frequency, and term frequency and inverse document frequency based centrality. Themes were extracted using textual fields as source of analysis. These themes were mined and analyzed for whole ticket-log and compared with project SMEs specified tagging. Precision of mined themes was not considered since SMEs were not available for analysis of the results and it would have been incorrect to assume that any theme not present in user specified tags must be considered wrong. \begin{table}[ht] \centering \begin{tabular}{l|l} \hline \hline {\bf Approach} & {\bf Recall}\\ \hline \hline LSA + TF & 48\% \\ \hline LSA + TF\_IDF & 48\% \\ \hline LSA + LDA (Mallet) & 48\% \\ \hline LDA + TF & 43\% \\ \hline LDA + TF\_IDF & 43\% \\ \hline TF & 38\% \\ \hline TF\_IDF & 38\% \\ \hline LSI & 38\% \\ \hline LDA & 33\% \\ \hline \end{tabular} \caption{Recall Levels for Theme Mining approaches (top 50 results compared to user specified 21 themes. LSA: latent semantic analysis; LDA: latent Dirichlet allocation; TF: Term Frequency; TF\_IDF: Term Frequency Inverse Document Frequency)} \label{t_tm} \end{table} \sf{\bf Challenges in Adoption of Research:} As can be seen from relatively poor results in Table~\ref{t_tm}, primary challenge in adoption of existing approaches (mostly unsupervised in nature) is that they do not appear to align well with users' expectations. Alternative approaches which aim at filling this gap are supervised ML based approaches, which however require significant amount of annotated data to start with. \subsection{Business Process Mapping} \sf{\bf Nature of Data:} In practice, it turned out that often mapping of bugs with standard business processes is not mentioned and only proxy mappings with modules or components are recorded in AM tools. However, in case of large scale applications, it can be assumed that component or module level architecture approximately aligns with underlying business process hierarchy and relationships. Furthermore, assuming that business process maps are generally represented as a tree (acyclic undirected graph), it is sufficient to consider leaf level business process nodes (i.e., business process steps) alone as representative for who business process flow-path. {\sf Experimental Analysis:} Table~\ref{t_bpm} shows results of ML based prediction approach for associating business processes for tickets based upon learning from historical log data. We employed 10-fold cross validation as evaluation measure and experimented with six different data sets from different industry accounts with sizes of ticket logs ranging from 448 to 22000. \begin{table}[ht] \centering \begin{tabular}{l|l|l|l|l|l} \hline \hline \textbf{Ticket-Log} & \textbf{\#BP} & \textbf{\#Tkts} & \textbf{P} & \textbf{R} & {\bf F1-score} \\ \hline \hline Data-Set\#1 & 14 & 448 & 0.67 & 0.66 & 0.66 \\ \hline Data-Set\#2 & 11 & 22000 & 0.78 & 0.76 & 0.77 \\ \hline Data-Set\#3 & 3 & 6420 & 0.86 & 0.85 & 0.85 \\ \hline Data-Set\#4 & 8 & 3600 & 0.67 & 0.67 & 0.67 \\ \hline Data-Set\#5 & 5 & 5630 & 0.54 & 0.53 & 0.53 \\ \hline Data-Set\#6 & 4 & 2000 & 0.90 & 0.90 & 0.90 \\ \hline \end{tabular} \caption{Precision, Recall, and F1-Scores for SVM (Gaussian kernel) classifier based ticket to business process mapping problem. \#BP: No. of Business Processes; \#Tkts: Number of tickets} \label{t_bpm} \end{table} Noticing that performance of the tool was comparatively poorer in Data-Set\#5, we investigated the cases where tool predicted incorrectly and it turned out that in most of the cases, bug description, which formed primary feature space for the ML technique, consisted of cryptic terms (often unique) like `Kfeil: JSP045ABCD \#ABCDD11759 [(231630 07/32/15), (0AAAAAAAAAAXXZW2M2U)]. ABCD1168 (\#J137891)'. We extended this analysis to other Data-Sets as well and realized that presence of such terms, which can't be parsed without SME intervention, was primary source of errors in outputs. \sf{\bf Challenges in Adoption of Research:} Application of supervised ML based approach resulted into comparatively better results and could be adopted in practice if designed as a separate tool which works in conjunction with existing AM tools. However, close integration with AM tools is not always possible since not all proprietary tools expose APIs for automated data export and in those cases, manual efforts in exporting data from AM tool and feeding it into tools like CogAM is necessary. \subsection{Threats to Validity} Even though we tried working with multiple industry accounts and had variety of data to analyze, it might still be limited by way of being influenced by circumstantial organizational factors which determined the nature of projects which we could work with currently. Furthermore, nature of software applications' data is a highly dynamic in nature and hence temporal reliability of the results would definitely demand ongoing reevaluations. \section{Conclusion and Future Work}\label{con} In this paper, we discussed experiences with a data-driven application maintenance tool called CogAM, which applies information retrieval based semantic search technology and unsupervised / supervised machine learning approaches for addressing spectrum of interrelated problems in AM domain. In specific, we tackle the critical problems of checking duplicate incidents, assignment of tickets to appropriate resources, mining the central themes for a cluster of incidents, and mapping incidents to the underlying business processes. Today, each of these tasks are carried out manually with considerable inefficiency and cost. While some of these problems have been researched extensively with sophisticated techniques on open source repositories and boards like Apache Foundation or Stack Overflow; we hardly find them to be shop usable as they impose challenges for adoption. Approach and tool has been piloted in real customer projects and we have received valuable lessons in terms of fitment, usability and adoption. Our experience suggests that in some cases search based solutions are good enough while in others detailed machine learned classifiers with annotated data would provide better results. Still in others, the problem becomes difficult as the ticket data as a single source of information is not enough to perform the necessary tasks which depend on external circumstantial inputs. In all cases, a richer domain knowledge becomes a key ingredient to success. In future, we propose to carry out research in two main directions. First, there is clear need to carry out extensive experimentation with mass deployments to study the problem patterns across different types of maintenance projects and gather further insights. Second, we realize that rather than using pre-trained models, it would be better to experiment with active learning approaches where the engineers can interact and provide iterative supervision for better contextual refinement. \bibliographystyle{abbrv}

BkiUblo5jDKDyFgbAFpl

\section{Introduction} Recently Bose condensed gases in an optical lattice are extensively studied both experimentally and theoretically. Much of attention is attracted to the breakdown of the superfluidity in a condensate. A lot of works have discussed the stability condition of the superfluidity by using the zero-temperature Gross-Pitaevskii equation.~\cite{wu,machholm2003,kramer2003,menotti2003} As emphasizing in our previous works,~\cite{konabe2006_2} it is crucial to explicitly include the degree of freedom of noncondensate atoms into the theory to understand the Landau instability from the microscopic point of view. In the present paper, by means of an effective action and an associated a coupled set of generalized hydrodynamic equations, we study the microscopic mechanism of the breakdown of the superfluidity having experiments such as reported in Ref.~\onlinecite{sarlo2005} in mind. \section{Generalized Hydrodynamic Equations in an Optical Lattice}\label{sec:generalized_hyd} We consider the Bose condensate at finite temperatures trapped in a one-dimensional optical lattice in addition to the harmonic potential \begin{eqnarray} V_{\rm ext}(\r)=\frac{m}{2}(\omega_x^2x^2+\omega_y^2y^2+\omega_z^2z^2)+sE_{R}\cos^2\left(\frac{\pi}{d}z\right),\label{generalized:potential} \end{eqnarray} where $\omega_x$, $\omega_y$, $\omega_z$ are the frequencies of the harmonic trap potential, $s$ is the strength of the lattice potential, $E_{R}$ is the recoil energy, and $d$ is a lattice constant. In the external potential represented by Eq.~(\ref{generalized:potential}), the Bose condensates are described by the following action \begin{eqnarray} S&=&\int d\r\int dt\ \phi^*(\r,t)\biggl[ i\hbar\frac{\partial}{\partial t}+\frac{\hbar^2}{2m}\nabla^2-V_{{\rm ext}}(\r) -\frac{g}{2}|\phi(\r,t)|^2\nonumber\\ &&{}-2g\tilde{n}(\r,t)\biggl]\phi(\r,t)\nonumber\\ &&{}+\frac{g^2}{2}\int d\r d\r'\int_{c} dtdt'\phi^*(\r,t)F(\r,t;\r',t')\phi(\r',t'),\label{dissipative:action} \end{eqnarray} where $\phi(\r,t)$ is a complex field that represents the degree of freedom of condensate atoms, $\tilde{n}(\r,t)\equiv\langle\tilde\psi^{\dag}(\r,t)\tilde\psi(\r,t)\rangle$ is the noncondensate density with $\tilde\psi(\r,t)$ being the noncondensate field operator, and $F(\r,t;\r',t')\equiv G(\r,t;\r',t')G(\r,t;\r',t')G(\r',t';\r,t)$ is defined in terms of nonequilibrium Green's functions of the noncondensate atoms $ G(\r,t;\r',t')\equiv-i\langle \tilde\psi(\r,t)\tilde\psi^{\dag}(\r',t')\rangle. $ In Eq.~(\ref{dissipative:action}), $g$ is a coupling constant which is related to the $s$-wave scattering length of atoms. We note that the index $c$ of integral in Eq.~(\ref{dissipative:action}) means that time integrations should be performed on Keldysh contour path.~\cite{danielewicz1984} The derivation and detailed discussion of the action~(\ref{dissipative:action}) will be presented elsewhere.~\cite{konabe2006_3} Considering a relatively deep lattice potential with large $s$, we introduce the ansatz (tight-bindeing approximation) \begin{eqnarray} \phi(\r,t)=\sum_{l}w_l(z)\phi_l(\r_{\bot},t)e^{i\theta_l(\r_{\bot},t)},\label{generalized:ansatz} \end{eqnarray} for the order parameter field, where $\theta_l({\bf r}_{\bot},t)$ is the phase of the order parameter at the $l$th lattice cite, and $w_l$ and $\phi_l$ are real functions. Here we assume the periodicity for $w_l(z)=w_0(z-ld)$, where $w_0$ is localized near $z=0$. Substituting the ansatz (\ref{generalized:ansatz}) into the action~(\ref{dissipative:action}) and performing a smoothing procedure from discrete variables to continuous ones following Ref.\onlinecite{kramer2002}, we obtain after some algebra \begin{eqnarray} S&=&\int dt\int d\r\biggl[ \sqrt{n_c(\r,t)}i\hbar\frac{\partial}{\partial t}\sqrt{n_c(\r,t)}-\hbar n_c(\r,t)\frac{\partial \theta(\r,t)}{\partial t}\biggl]\nonumber\\ &&{} +J\int dt\int d\r n_c(\r,t)\cos\left(d \frac{\partial \theta(\r,t)}{\partial z}\right)\nonumber\\ &&{}+\frac{\hbar^2}{2m}\int dt\int d\r\sqrt{n_c(\r,t)}\biggl\{ \nabla_{\bot}^2\sqrt{n_c(\r,t)}-\sqrt{n_c(\r,t)}\left(\nabla_{\bot}\theta(\r,t)\right)^2\nonumber\\ &&{}+i\biggl[ 2\nabla_{\bot}\sqrt{n_c(\r,t)}\cdot\nabla_{\bot}\theta(\r,t)+\sqrt{n_c(\r,t)}\nabla_{\bot}^2\theta(\r,t)\biggl]\biggl\}\nonumber\\ &&{}-\int dt\int d\r\ V_{\rm ho}(\r)n_c(\r,t) -\frac{\tilde g}{2}\int dt\int d\r\ n_c^2(\r,t)\nonumber\\ &&{} -\tilde J\int dt\int d\r\ n_c^2(\r,t)\cos\left(d\frac{\partial\theta(\r,t)}{\partial z}\right)\nonumber\\ &&{}-2\tilde g\int dt\int d\r\ n_c(\r,t)\tilde{n}(\r,t)\nonumber\\ &&{}+\frac{\tilde g^2}{2}\int dtdt'\int d\r d\r \sqrt{n_c(\r,t)}e^{-i\theta(\r,t)}F(\r,t;\r',t')\sqrt{n_c(\r',t')}e^{i\theta(\r,t)},\nonumber\\ \label{derivation_smoothed:smoothed_action} \end{eqnarray} where $\tilde g\equiv gd\int dz w_0^4(z)$ is the effective renormalized coupling constant, $V_{\rm ho}(\r)$ is the harmonic potential in Eq.~(\ref{generalized:potential}), and $n_c(\r,t)=\phi_l^2({\bf r}_{\bot},t)/d$ and $\theta(\r,t)=\theta_l({\bf r}_{\bot},t)$ represent the density and phase after the smoothing procedure. In Eq.~(\ref{derivation_smoothed:smoothed_action}), we defined the following quantities: \begin{eqnarray} J&\equiv&\int dz\ w_l(z)\biggl[\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2}-V_{\rm opt}(z)\biggl]w_{l+1}(z),\label{derivation:tunneling_rate}\\ \tilde J&\equiv& 4gd\int dz w_l^3(z)w_{l+1}(z), \end{eqnarray} where $V_{\rm opt}(z)$ is the optical lattice potential in Eq.~(\ref{generalized:potential}). The hydrodynamic equations are derived by imposing the stationary condition on the smoothed effective action~(\ref{derivation_smoothed:smoothed_action}) with respect to arbitrary variations of the density $n_c(\r,t)$ and of the phase $\theta(\r,t)$. After performing the Markovian approximation and the gradient expansion for the non-local term related to $F(\r,t;\r',t')$, we obtain \begin{eqnarray} &&m\frac{\partial\v_c(\r,t)}{\partial t}+\nabla\biggl[ \frac{m}{2}\frac{m}{m^*_{\mu}}v_z^2(\r,t)+\frac{m}{2}\v_{\bot}^2(\r,t)+V_{\rm ho}(\r)\nonumber\\ &&{}+\tilde g n_c(\r,t)+2\tilde g \tilde{n}(\r,t)\biggl]=0,\label{derivation_dissipative:Josephson_eq}\\ &&\frac{\partial n_c(\r,t)}{\partial t}+\frac{\partial}{\partial z}\left[ \frac{m}{m^*_{\mu}}n_c(\r,t)v_z(\r,t)\right] +\nabla_{\bot}\left(n_c(\r,t)\v_{\bot}(\r,t)\right)=-\Gamma_{12}(\r,t).\label{derivation_dissipative:continuity_eq}\nonumber\\ \end{eqnarray} where $\Gamma_{12}(\r,t)\equiv 2n_c(\r,t)R(\r,t)$ with \begin{eqnarray} R(\r,t)&\equiv& \frac{\pi \tilde g^2}{2\hbar}\int\frac{d\k_1}{(2\pi)^3}\frac{d\k_2}{(2\pi)^3}\frac{d\k_3}{(2\pi)^3}\nonumber\\ &&{}\times\delta(\omega_c+\tilde\omega_1-\tilde\omega_2-\tilde\omega_3) \delta(\k_c+\k_1-\k_2-\k_3) \nonumber\\ &&{}\times\biggl\{[1+f(\k_1,\r,t)][1+f(\k_2,\r,t)]f(\k_3,\r,t)\nonumber\\ &&{}\quad -f(\k_1,\r,t)f(\k_2,\r,t)[1+f(\k_3,\r,t)]\biggl\},\label{dissipative:R} \end{eqnarray} where ${\bf p}_c=\hbar\k_c$ and $\epsilon_c=\hbar\omega_c$ is a condensate momentum and energy, $f$ is the nonequilibrium distribution function of the noncondensate, and $\hbar\tilde\omega_i\equiv \hbar^2k_i^2/2m+2\tilde g(n_c(\r,t)+\tilde{n}(\r,t))$ is a noncondensate energy. The condensate chemical potential and energy are defined by \begin{eqnarray} \mu_c(\r,t)&=&\tilde gn_c(\r,t)+2\tilde g\tilde{n}(\r,t),\\ \epsilon_c(\r,t)&=&\mu_c(\r,t)+\frac{m}{2}v_x^2(\r,t)+\frac{m}{2}v_y^2(\r,t)+\frac{m}{2} \left(\frac{m}{m_{\mu}^*}\right)v_z^2(\r,t). \end{eqnarray} In deriving above hydrodynamic equations~(\ref{derivation_dissipative:Josephson_eq}) and (\ref{derivation_dissipative:continuity_eq}), we have expanded the gradient term of the phase regarding the term as small, which is the case in the study of small amplitude oscillations, and introduced the effective mass and the chemical potential effective mass, $m^*\equiv\hbar^2/d^2\left[J-\tilde J n_c(\r,t)\right]^{-1}$ and $m^*_{\mu}\equiv\hbar^2/d^2\left[J-2\tilde J n_c(\r,t)\right]^{-1}$, respectively. We neglected the $n_c$ dependence of $m^*_{\mu}$, quantum pressure term, and $\nabla R(\r,t)$, in deriving Eqs.~(\ref{derivation_dissipative:Josephson_eq}) and (\ref{derivation_dissipative:continuity_eq}). The coupled set of hydrodynamic equations~(\ref{derivation_dissipative:Josephson_eq}) and (\ref{derivation_dissipative:continuity_eq}) are useful starting point for discussing the collective oscillations of the condensate in an optical lattice at finite temperatures.~\cite{ferlaino2002,fertig2005,sarlo2005} \section{Instability due to the Collisional Damping Process} In this section, we demonstrate the useful application of the hydrodynamic equations~(\ref{derivation_dissipative:Josephson_eq}) and (\ref{derivation_dissipative:continuity_eq}) derived in Sec.~\ref{sec:generalized_hyd} by investigating the stability of the superfluidity. We consider the system in which the condensates are moving with a constant velocity~\cite{sarlo2005} and neglect the confining trapping potential. Nonequilibrium distribution functions in Eq.~(\ref{dissipative:R}) should be determined by solving a kind of kinetic equation. Instead of solving a kinetic equation, we approximate the nonequilibrium distribution function by the static Bose distribution function $f({\bf p},\r)=\frac{1}{e^{\beta(\tilde{\epsilon}_p-\tilde{\mu})}-1}$,~\cite{williams} where $\tilde\epsilon_p\equiv\hbar\tilde{\omega}_p\equiv\frac{p^2}{2m}+2\tilde g(n_c+\tilde{n})$ and $\tilde\mu$ is the chemical potential of the noncondensate atoms. Expanding $\epsilon_c(\r,t)$ around equilibrium, one can linearize $\Gamma_{12}(\r,t)$ as \begin{eqnarray} \delta\Gamma_{12}(\r,t)&=&\frac{\pi \tilde g^2}{2\hbar}\int\frac{d\k_1}{(2\pi)^3}\frac{d\k_2}{(2\pi)^3}\frac{d\k_3}{(2\pi)^3}\delta(\omega_{c,0}+\omega_1-\omega_2-\omega_3)\nonumber\\&&{}\times \delta(\k_{c,0}+\k_1-\k_2-\k_3) [1+f(\k_1,\r)]f(\k_2,\r)f(\k_3,\r)\nonumber\\ &&{}\times\beta\left[\tilde g\delta n_c+m\v_{\bot,0}\cdot\delta\v_{\bot} +m\left(\frac{m}{m_{\mu}^*}\right)v_{z,0}\delta v_z\right]\nonumber\\ &\equiv&\frac{1}{\tau'(\r,t)}\left[\tilde g\delta n_c+m\v_{\bot,0}\cdot\delta\v_{\bot} +m\left(\frac{m}{m_{\mu}^*}\right)v_{z,0}\delta v_z\right],\nonumber\\ \end{eqnarray} where we define the relaxation time $\tau'(\r,t)$ by \begin{eqnarray} \frac{1}{\tau'(\r,t)}&\equiv&\frac{\pi \tilde g^2\beta}{2\hbar}\int\frac{d\k_1}{(2\pi)^3}\frac{d\k_2}{(2\pi)^3}\frac{d\k_3}{(2\pi)^3}\nonumber\\ &&{}\times\delta(\omega_{c,0}+\omega_1-\omega_2-\omega_3) \delta(\k_{c,0}+\k_1-\k_2-\k_3). \end{eqnarray} Finally we obtain the linearized hydrodynamic equations \begin{eqnarray} &&\frac{\partial \delta n_c(\r,t)}{\partial t} +\nabla_{\bot}\cdot\left[n_{c,0}(\r)\delta{\bf v}_{c}(\r,t)\right] +\frac{\partial}{\partial z}\left[\frac{m}{m^*}n_{c,0}(\r)\delta v_{z}(\r,t)\right]\nonumber\\ &&{}+\nabla_{\bot}\cdot\left[\delta n_c(\r,t){\bf v}_{c,0}(\r)\right] +\frac{\partial}{\partial z}\left[\frac{m}{m^*_{\mu}}\delta n_c(\r,t)v_{z,0}(\r)\right] =-\delta\Gamma_{12}(\r,t),\label{collisional_linearized_continuity_eq}\nonumber\\ \\ &&m\frac{\partial}{\partial t}\delta\v_c(\r,t)=-\nabla \biggl[m\v_{\bot,0}(\r)\cdot\delta\v_{\bot}(\r,t)+m\left(\frac{m}{m_{\mu}^*}\right)v_{z,0}(\r)\delta v_z(\r,t)\nonumber\\ &&{}\qquad\qquad\qquad\qquad\quad+\tilde g\delta n_c(\r,t)\biggl].\label{collisional_linearized_josephson_eq} \end{eqnarray} Looking for a plane-wave solution such as ${\rm exp}[i({\bf k}\cdot\r-\omega t)]$ with $\k=(0,0,k)$, we obtain $\omega \equiv\v_{c,0}^{\mu}\cdot\k+c^*k\equiv\Omega$ in the limit $1/\tau'\to0$, where $c^*\equiv\sqrt{\tilde g n_{c,0}/m(m/m^*)}$ is the sound velocity modified due to the lattice potential and $v_{c,0}^{\mu}\equiv (m/m_{\mu}^*)v_{c,0}$ is the chemical potential group velocity.~\cite{kramer2003,menotti2003} When we include $1/\tau'$, the dispersion relation is modified as $\omega=\Omega-i\Gamma$. To first order in $1/\tau'$, we obtain the damping term \begin{eqnarray} \Gamma=\frac{1}{2\tau'}\left(1+\frac{m}{m_{\mu}^*}\frac{v_{c,0}}{c^*}\hat\v_{c,0}\cdot\hat\k\right),\label{collisional:damping_term} \end{eqnarray} where $\hat\v_{c,0}$ and $\hat\k$ are normal vectors for $\v_{c,0}$ and $\k$, respectively. From the expression (\ref{collisional:damping_term}) for $\Gamma$, it can be seen that the instability occurs for the mode with $\hat\v_{c,0}\cdot\hat\k=-1$ when $v_{c,0}^{\mu}>c^*$,~\cite{machholm2003,kramer2003,menotti2003} since the damping term $\Gamma$ changes its sign. \section{Summary} In summary, we have investigated the stability of the superfluidity of the condensates in an optical lattice by using the generalized hydrodynamic equations. We derived the Landau criterion microscopically and showed that the criterion is modified due to the lattice potential as expected. In the present paper, while we only consider the collisional damping process as a destabilizing process, one can also derive the Landau instability due to the Landau damping process, which is also important in the collisionless regime of interest, from the generalized hydrodynamic equations.~\cite{konabe2006_3} The generalized hydrodynamic equations derived in the present paper can be used for analyzing the dipole oscillations and its damping in the presence of the noncondensate atoms such as observed in the experiment by Florence group.~\cite{ferlaino2002} \section*{ACKNOWLEDGMENTS} S. K. is supported by JSPS (Japan Society for the Promotion of Science) Research Fellowship for Young Scientists.

BkiUboc5i7PA9KOTUaOk

\section{Introduction} Since the discovery of the Drinfel'd-Sokolov hierarchy \cite{ds84}, a number of integrable systems have been constructed in the same spirits cf. \cite{bdghm93, dghm92, fhm93, fgm96, fgm95}. Those integrable systems are not of finite dimension, which are formulated by Poisson algebras, but of infinite dimension, reflecting classical field theory as their origin. In \cite{bdk09}, Barakat, De Sole and Kac used \emph{Poisson vertex algebras}, which play a role of Poisson algebras for integrable systems as above, and introduced the notion of \emph{integrable Hamiltonian hierarchies} as a framework of integrability for Poisson vertex algebras. Since then De Sole, Kac, and Valeri e.g.\ \cite{dkv13, dkv15a, dkv16a, dkv16b, dkv16c} have studied systematically integrable Hamiltonian hierarchies associated with Poisson vertex algebras, called the {\em classical (affine) $\W$-algebras}, which are obtained as classical limit of vertex algebras, called the (affine) $\W$-algebras \cite{dk06,krw03}, see also \cite{dk13a, suh18}. The $\W$-algebras are parametrized by a triple $(\g,f,k)$ consisting of a finite dimensional simple Lie algebra $\g$ over $\C$, a nonzero nilpotent element $f\in\g$, and a complex number $k\in\C$, called the \emph{level} and are denoted by $\W^k(\g, f)$. The classical $\W$-algebras are also parametrized by the same data and thus we denote them also by $\W^k(\g, f)$. De Sole, Kac and Valeri recovered the earlier results on the integrable Hamiltonian hierarchies mentioned above in their study and obtained a culminating result in \cite{dkv16c} on the existence of integrable Hamiltonian hierarchies associated with $\W^k(\g, f)$, which states: \emph{the classical affine $\W$-algebra $\W^k(\g, f)$ admits an integrable Hamiltonian hierarchy for any finite dimensional simple Lie algebra $\g$ of classical type, nonzero nilpotent element $f$, and $k\neq0$.} Besides the algebraic theory of integrable Hamiltonian hierarchies mentioned above, there is a result of Feigin-Frenkel \cite{feiginfrenkel95,feiginfrenkel93} and Enriquez-Frenkel \cite{enriqfrenkel97} which constructs the Drinfel'd-Sokolov hierarchies by using a geometric realization of the classical affine $\W$-algebra $\W^1(\g,f_\mathrm{prin})$, where $f_\mathrm{prin}$ denotes the principal nilpotent element in $\mathfrak{g}$. More precisely, they proved that $\W^1(\g,f_{\mathrm{prin}})$ are isomorphic as differential algebras to the coordinate rings of certain double coset spaces of prounipotent proalgebraic groups and that the Drinfel'd-Sokolov hierarchies are induced from natural group actions on these spaces. The aim of this paper is to generalize the those results to the classical $\W$-algebras $\W^k(\g, f)$ when $f$ satisfies certain properties (see the condition (F) in the below) and the level $k\in\C$ is generic. We note that the choice of $f$ is a special case of the so-called Type I in \cite{bdghm93, dghm92,fhm93}. The integrable Hamiltonian hierarchies obtained in the paper for the case $k=1$ coincide with special cases of those considered in \cite{bf01} by construction, and are special cases of the result \cite{dkv13}. Let $\g$ be a finite dimensional Lie algebra over $\C$, and $f\in \g$ be a non-zero nilpotent element. We fix an $\mathfrak{sl}_2$-triple $\{e,h=2x,f\}$ containing $f$. We denote by $\Gamma: \g=\oplus_{j=-d}^d\g_j$ the $\ad_{x}$-grading and by $\g=\h\oplus (\oplus_{\al\in\Delta}\g_\al)$ a root space decomposition of $\g$ which is homogeneous with respect to $\Gamma$. Then the root system $\Delta$ admits an induced grading $\Delta=\sqcup_{j=-d}^d \Delta_j$. Let $\Pi$ denote the subset of $\Delta_{>0}=\sqcup_{j>0} \Delta_j$ consisting of the elements indecomposable in $\Delta_{>0}$. Let $V^k(\g_0)$ denote the universal affine Poisson vertex algebra associated with the reductive Lie algebra $\g_0$ at level $k$ and $F(\g_\half)$ the $\beta\gamma$-system Poisson vertex algebra associated with the symplectic vector space $\g_\half$. Then the $\W$-algebra $\W^k(\g,f)$ is defined as a 0-th cohomology of the BRST complex $C^k(\g,f)$, see Section \ref{affW} for details. For generic level $k$, the $\W$-algebras are realized as the joint kernel of certain screening operators \cite{gen17}. Our first result (Theorem \ref{main1}) is a Poisson vertex algebra analogue of this result. Namely, for generic $k\in\C$, we realize the classical affine $\W$-algebra $\W^k(\g,f)$ as the Poisson vertex subalgebra of $V^k(\g_0)\otimes F(\g_\half)$ invariant under the derivations $Q_\al^W$, $(\al\in \Pi)$: \begin{equation}\label{eq0}\W^k(\g,f) \cong \bigcap_{\al\in \Pi}\Ker\Bigl(Q_\al^W: V^k(\g_0)\otimes F(\g_\half)\rightarrow V^k(\g_0)\otimes F(\g_\half)\Bigr). \end{equation} \noindent See \eqref{eq;2} and \eqref{eq;1} for the definition of $Q_\al^W$. We now suppose that the pair $(\g,f)$ satisfies the following condition (F): \begin{enumerate} \item[(F1)] The grading $\Gamma$ is a $\Z$-grading. \item[(F2)] There exists an element $y\in\g_d$ such that $s=f +y t^{-1}\in\gpoly$ is semisimple. \item[(F3)] The Lie subalgebra $\Ker(\ad_s)\subset\g[t^{\pm1}]$ is abelian and $\Im(\ad_s)\cap \gpoly_0=\g_0$. \end{enumerate} Here $\gpoly=\g \otimes \C[t,t^{-1}]$ and $\gpoly=\oplus_{j\in\Z}\gpoly_j$ is the $\Z$-grading given by $\deg(X t^n)= j +(d+1)n$, $(X\in \g_j)$. The nilpotent elements $f$ satisfying (F1)-(F2) are called Type I in the literature e.g.\ \cite{fhm93, bdghm93, dghm92}. We consider the completion $\glau=\g\otimes \C((t))$. It has subalgebras $\lgp$, which is the completion of $\gpoly_{>0}$, and $\lgn=\gpoly_{\leq0}$. We have the corresponding closed subgroups $\lGp$, $\lGn$ of the loop group $\lG$ of $G$. Let $\abel$ denote the completion of $\Ker(\ad_s)$ in $\glau$, and $\abel_{\pm}=\abel\cap\glau_{\pm}$. Let $A$, (resp.\ $A_{\pm}$) be the closed subgroup of $\lG$ corresponding to $\abel$, (resp.\ $\abel_\pm$). The right $A$-action $\lGn\backslash \lG/\Ap\times A\rightarrow \lGn\backslash \lG/\Ap$, induces a Lie algebra homomorphism $\abel\rightarrow \Der(\C[\lGn\backslash\lG/\Ap])$, and so that $\abel\rightarrow \Der(\C[\lGp/\Ap])$ since $\lGp/A_+\subset \lGn\backslash\lG/\Ap$ is an open subset. We denote this action by $a\mapsto a^R$. In particular, we obtain a differential algebra $(\C[\lGp/\Ap],s^R)$. On the other hand, the natural left $\lGp$-action $\lGp\times\lGp/\Ap\rightarrow \lGp/A_+$ induces a Lie algebra homomorphism $\lgp\rightarrow \Der(\C[\lGp/\Ap])$, which we denote by $X\mapsto X^L$. Our second result (Theorem \ref{main2}) is the following geometric interpretation of \eqref{eq0} under the condition (F): there exists an isomorphism of differential algebras $$\Psi_k: V^k(\g_0)\rightarrow \C[\lGp/\Ap],$$ such that the derivation $Q_\al^W$ is identified with $X_\al^L$ for some element $X_\al\in \g_\al$, ($\al\in\Pi$). Since such root vectors $X_\al$ generate a Lie algebra $\g_+=\oplus_{j>0}\g_j$, it implies that $\W^k(\g,f)$ is isomorphic to $\C[\dcoset]$ as a differential algebra (Corollary \ref{doublecoset}). Here $G_+$ is the closed subgroup of $\lGp$ corresponding to $\g_+$. The action $\abel\rightarrow \Der(\C[\lGp/A_+])$ preserves the subalgebra $\C[\dcoset]$. Thus we obtain a space of mutually commutative derivations of $\W^\k(\g,f)$ as the image of $\abel_-$. We denote this space by $\mathcal{H}^\k(\g,f)$. Our third result (Theorem \ref{main3}) states that $\mathcal{H}^k(\g,f)$ is an integrable Hamiltonian hierarchy associated with $\W^k(\g,f)$. The double coset spaces $\dcoset$ considered here are embedded into the abelianized Grassmanians $G[t^{-1}]\backslash G((t))/A_+$ as a Zariski open subset. The abelianized Grassmanians have been used to construct Drinfeld-Sokolov hierarchies geometrically \cite{bf01}. As pointed in \emph{loc.cit.}, such a construction implies a strong compatibility of integrable Hamiltonian systems associated with classical affine $\W$-algebras and Hitchin systems. The author hope to investigate the relationship between classical affine $\W$-algebras and Hitchin systems in future works. \vspace{3mm} {\it Acknowledgements}\quad This paper is the master thesis of the author. He wishes to express his gratitude to Professor Atsushi Matsuo for encouragement throughout this work and numerous advices. \section{Poisson vertex superalgebras} \subsection{Poisson vertex superalgebra} We recall here some basics about Poisson vertex superalgebras and their relation to the theory of integrable systems, following \cite{bdk09,suh18}. We remove the prefix ``super'' whenever we consider the non-super cases. A \emph{differential $\C$-superalgebra} is a pair $(V,\pd)$ consisting of a supercommutative $\C$-superalgebra $V$ and an even derivation $\pd$ on it. We denote by $\bar{a}$ the parity of For $a\in V$ and by $\Der(V)$ the set of super derivations of $V$. A differential $\C$-superalgera of the form $$V=\C[u_i^{(n)}| i\in I_{\bar{0}}, n\geq 0]\otimes \bigwedge\ _{\oplus_ {i\in I_{\bar{1}}}\C u_i^{(n)}}$$ as $\C$-superalgebras for some index set $I=I_{\bar{0}}\sqcup I_{\bar{1}}$ where $u_i^{(n)}=\pd^n u_i$, is called the \emph{superalgebra of differential polynomials} in the variables $u_i,\ (i\in I)$. A \emph{Poisson vertex superalgebra} is a triple $(V,\pd,\{-_\lam-\})$ consisting of a differential $\C$-superalgebra $(V,\pd)$ and an even $\C$-bilinear map \begin{equation}\label{lambda bracket} \{-_\lam -\}:\ V\times V\rightarrow V[\lam],\quad (f,g)\mapsto \{f_\lam g\}=\sum_{n\geq0} \frac{\lam^n}{n!} f_{(n)}g, \end{equation} called the $\lambda$-bracket, satisfying \begin{eqnarray} &&\label{sesqui}\{ \pd f_\lam g\}=-\lam \{f_\lam g\},\quad \{f_\lam \pd g\}=(\lam+\pd)\{f_\lam g\},\\ &&\label{skew}\{g_\lam f\}=- (-1)^{\bar{f}\bar{g}}_\leftarrow\{f_{-\pd-\lam} g\},\\ &&\label{Jacobi}\{f_\lam \{g_\mu h\}\}-(-1)^{\bar{f}\bar{g}}\{g_\mu \{f_\lam h\}\}=\{\{f_\lam g\}_{\lam+\mu} h\},\\ &&\label{rightLeibniz}\{f_\lam gh\}=\{f_\lam g\} h+(-1)^{\bar{g}\bar{h}}\{f_\lam h\} g,\\ &&\label{leftLeibniz}\{fg_\lam h\}=(-1)^{\bar{g}\bar{h}}\{f_{\lam+\pd} h\}_\rightarrow g+(-1)^{\bar{f}(\bar{g}+\bar{h})}\{g_{\lam+\pd}h\}_\rightarrow f, \end{eqnarray} for $f,g,h\in V$. Here we denote \begin{equation*} \{f_\lam g\}_\rightarrow= \sum \frac{1}{n!} f_{(n)}g \lam^n,\quad _\leftarrow \{f_\lam g\}=\sum \lam^n\frac{1}{n!} f_{(n)}g. \end{equation*} \begin{remark} \label{1.1} Given an algebra of differential polynomials $V$ in the variables $u_i$, $(i\in I)$, a linear map $F: \operatorname{span}_\C \{u_i\}_{i\in I}\rightarrow V[\lam]$ uniquely extends to $F:V\rightarrow V[\lam]$ by \eqref{sesqui}, \eqref{leftLeibniz}. \end{remark} Let $V$ be a superalgebra of differential polynomials in the variables $u_i$, $(i\in I)$. We denote by $$\frac{\pd}{\pd u_i^{(n)}}, \frac{\pd_R}{\pd_R u_i^{(n)}}\in \mathrm{Der}(V)$$ denote the derivation with respect to $u_i^{(n)}$ from the left and the right respectively, (which coincide if $V$ is non-super). Then the $\lambda$-bracket on $V$ is determined by the values $\{u_{i\lambda}u_j\}$ in the following sense. \begin{theorem}[\cite{bdk09,suh18}] \label{1.2} Let $(V,\pd)$ be a superalgebra of differential polynomials in the variables $u_i$, $(i\in I)$, and $H_{ij}(\lam)$, $(i,j\in I)$, an element of $V[\lam]$ with the same parity as $u_iu_j$. Then there is a unique Poisson vertex superalgebra structure on $V$ satisfying $\{u_{i\lam}u_j\}=H_{ji}(\lam)$ if and only if the $\lam$-bracket satisfies \eqref{skew} and \eqref{Jacobi} for $u_i,\ (i \in I)$. Moreover, the $\lam$-bracket is given by $$\{f_\lam g\}= \sum_{\begin{subarray}{c} i,j\in I,\\ n,m\geq 0\end{subarray}} (-1)^{\bar{f}\bar{g}+\bar{i}\bar{j}}\frac{\pd_R g}{\pd_R u_j^{(n)}}(\lam+\pd)^nH_{ji}(\lam+\pd)_\rightarrow(-\lam-\pd)^m\frac{\pd f}{\pd u_i^{(m)}},\quad f,g\in V.$$ \end{theorem} \begin{ex}[Universal affine Poisson vertex algebra] \emph{ Given a finite dimensional Lie algebra $L$ over $\C$ and a nondegenerate symmetric invariant bilinear form $\k$ on $L$, let $V^\k(L)$ denote the algebra of differential polynomials in the variables given by a basis of $L$. Then a $\lam$-bracket $\{-_\lam-\}: L\times L\rightarrow V^\k(L)[\lam]$ given by $\{u_\lam v\}=[u,v]+\k(u,v)\lam$, $(u,v\in L)$, defines a Poisson vertex algebra structure on $V^\k(L)$. This is called the universal \emph{affine Poisson vertex algebra} associated with $L$ at level $\k$.} \end{ex} Given a Poisson vertex superalgebra $V$, a \emph{Poisson vertex module} over $V$ (cf. \cite {A}) is a vector superspace $M$ which is a $V$-module as a supercommutative algebra $V$ and endowed with an even $\C$-bilinear map $\{-_\lam-\}: V\times M\rightarrow M[\lam]$ satisfying \begin{equation} \{\pd f_\lam m\}=-\lam \{f_\lam m\}, \end{equation} \begin{equation} \{f_\lam\{g_\mu m\}\}-(-1)^{\bar{f}\bar{g}}\{g_\mu\{f_\lam m\}\}=\{\{f_\lam g \}_{\lam+\mu} m\}, \end{equation} \begin{equation} \{f_\lam g\cdot m\}=\{f_\lam g\}\cdot m+(-1)^{\bar{f}\bar{g}}g\cdot \{f_\lam m\}, \end{equation} \begin{equation} \{f\cdot g_\lam m\}=(-1)^{\bar{m}\bar{g}}\{f_{\lam+\pd}m\}_\rightarrow g+(-1)^{\bar{f}(\bar{g}+\bar{m})}\{g_{\lam+\pd}m\}_\rightarrow f \end{equation} for $f,g\in V$, $m\in M$. Here the right $V$-action $M\times V\rightarrow M$ is defined by the action of $V$ as a supercommutative algebra. In this case, we define the $\lam$-bracket $$\{-_\lam-\}:M\times V\rightarrow M[\lam],\quad (m,f)\mapsto \{m_\lam f\}=-(-1)^{\bar{m} \bar{f}}_\leftarrow \{f_{-\pd-\lam}m\}.$$ For $m\in M$, the linear map $\{m_\lam-\}:V\rightarrow M[\lam]$ is called an \emph{intertwining operator} and $m$ is called the Hamiltonian. For a Poisson vertex superalgebra $V$, the vector superspace $\mathrm{Lie}(V)=V/\pd V$ is called \emph{the space of local functionals}. We denote by \begin{center} $\int: V\rightarrow \mathrm{Lie}(V),\quad f\mapsto \int f$ \end{center} the canonical projection. \begin{proposition}[\cite{bdk09,suh18}] \label{1.3}\hspace{1mm}\\ \begin{enumerate} \item The bilinear map \begin{center} $\mathrm{Lie}(V)\times \mathrm{Lie}(V)\rightarrow \mathrm{Lie}(V),\quad (\int f,\int g)\mapsto \int \{f_\lam g\}_{\lam=0}$ \end{center} is well-defined and defines a Lie superalgebra structure. Moreover, if $V$ is even and an algebra of differential polynomials in the variables $\{u_i\}_{i\in I}$, then \begin{center} $\left[\int f,\int g\right]=\sum_{i.j \in I}\int\frac{\delta_R g}{\delta_R u_j} \{u_{i\pd} u_j\}_\rightarrow \frac{\delta f}{\delta u_i},$ \end{center} where $\frac{\delta f}{\delta u_i}=\sum_{n\geq 0}(-\pd)^n \frac{\pd f}{\pd u_i^{(n)}}$ and $\frac{\delta_R f}{\delta_R u_i}=\sum_{n\geq 0}(-\pd)^n \frac{\pd_R f}{\pd_R u_i^{(n)}}$ denote the left and right variational derivative of $f$ with respect to $u_i$. \item The Lie superalgebra $\Lie(V)$ acts on $V$ by \begin{center} $\eta:\Lie(V)\rightarrow \Der(V),\quad \int f \mapsto \{f_\lam -\}|_{\lam=0}.$ \end{center} \end{enumerate} \end{proposition} We use the following lemma in the below. \begin{lemma}[{cf. \cite[Proposition 1.33]{bdk09}}] \label{1.4} Let $L$ be a finite dimensional Lie algebra over $\C$ equipped with a nondegenerate symmetric invariant bilinear form $\k$. For the universal affine Poisson vertex algebra $V^\k(L)$, the kernel of $\eta: \Lie(V^\k(L)) \rightarrow \Der(V^\k(L))$ is \begin{center} $\Ker(\eta) =\mathrm{span}_\C \bigl\{\int 1, \int u \mid u\in Z(L)\bigr\},$ \end{center} where $Z(L)$ is the center of $L$. \end{lemma} \begin{proof} Let $\{u_i\}_{i\in I}$ be a basis of $L$. Suppose $\int F\in \Ker(\eta)\subset \Lie(V^\k(L))$. By Theorem \ref{1.2}, \begin{center} $\eta\bigl(\int F\bigr)= \sum\limits_{i,j,n} \pd^n \Bigl(([u_i, u_j]+ \k(u_i, u_j)\pd) \frac{\delta F}{\delta u_i}\Bigr) \frac{\pd}{\pd u_j^{(n)}}$. \end{center} It follows $M_j=\sum\limits_{i} ([u_i, u_j]+ \k(u_i, u_j)\pd) \frac{\delta F}{\delta u_i}=0$ for $j\in I$. Define a degree on $V^\k(L)$ by $\deg(u_i^{(n)})=n$ and $\deg(AB)=\deg(A)+\deg(B)$. Then $\delta/\delta u$ preserves the degree. Let $G_i$ be the top degree component of $\delta F/\delta u$. Then the top component $M_j^{\mathrm{top}}$ of $M_j$ is $M_j^{\mathrm{top}}=\sum_i \k(u_i,u_j)\pd G_i$. Since $\k$ is nondegenerate, we obtain $\pd G_i=0$, which implies $G_i\in \C$. Since $V^\k(L)$ is an algebra of differential polynomials, we conclude $F\in\C\oplus L$. (See the proof of \cite[Proposition 1.5]{bdk09}.) Set $F=a+ \sum_i b_i u_i$, $(a, b_i\in\C)$. Then $M_j=[\sum_i b_i u_i, u_j]=0$, ($j\in I$), which implies $\sum_i b_i u_i\in Z(L)$. \end{proof} Finally, given a Poisson vertex superalgebra $V$, an element $\int f\in\text{Lie}(V)$ is called \emph{integrable} if there exists an infinite dimensional abelian Lie subsuperalgebra $\mathcal{H}$ of $\text{Lie}(V)$ which contains $\int f$. In this case, $\mathcal{H}$ is called an \emph{integrable Hamiltonian hierarchy} associated with $V$. \subsection{Differential graded Poisson vertex superalgebra} A \emph{differential graded Poisson vertex superalgebra} (d.g.\ Poisson vertex superalgebra) is a pair $(V,d)$ consisting of a Poisson vertex superalgebra $V=\oplus_{n\in \Z}V_n$ and a linear map $d:V\rightarrow V$, called the differential, satisfying \begin{itemize} \item $V$ is a $\Z$-graded Poisson vertex superalgebra, i.e., $V=\oplus_{n\in\Z}V_n$ as a vector superspace satisfying $$V_n\cdot V_m\subset V_{n+m},\quad \{V_{n\lam}V_m\}\subset V_{n+m}[\lam],$$ \item the linear map $d$ is of homogeneous parity and satisfies $d^2=0$, $$d:V_n\rightarrow V_{n+1},$$ $$d(ab)=d(a)\cdot b+(-1)^{\bar{d}\bar{a}}a\cdot d(b),\quad d(\{a_\lam b\})=\{d(a)_\lam b\}+(-1)^{\bar{d}\bar{a}}\{a_\lam d(b)\}.$$ \end{itemize} The cohomology $H^*(V;d)=\oplus_{n\in \Z}H^n(V;d)$ inherits a $\Z$-graded Poisson vertex algebra structure. Moreover, $H^0(V;d)$ is a Poisson vertex subsuperalgebra and $H^n(V;d)$, $(n\in\Z)$, is a Poisson vertex module over $H^0(V;d)$. In the sequel, we also use the notion of a \emph{differential graded vertex superalgebra}. The definition is similar and therefore we omit the details. \subsection{Classical limit}\label{classical limit} Let $V$ be a vertex superalgebra over a polynomial ring $\C[\epsilon]$. Suppose that $V$ is free as a $\C[\epsilon]$-module and the $\lambda$-bracket satisfies \begin{equation}\label{condition} [V_\lam V]\subset \epsilon V. \end{equation} Define the vector superspace $V^{\mathrm{cl}}=V/\epsilon V$ and let $V\rightarrow V^{\mathrm{cl}}$, ($f\mapsto \bar{f}$) denote the canonical projection. Then $$V^{\mathrm{cl}}\times V^{\mathrm{cl}}\rightarrow V^{\mathrm{cl}},\quad (\bar{f},\bar{g})\mapsto \overline{f_{(-1)}g}$$ is well-defined and defines an associative supercommutative algebra structure on $V^{\mathrm{cl}}$. Since the translation operator $\pd$ of $V$ preserves $\epsilon V$, it induces a linear map $$\pd: V^{\mathrm{cl}}\rightarrow V^{\mathrm{cl}},\quad \pd\bar{f}\mapsto \overline{\pd(f)},$$ which is a derivation of $V^{\mathrm{cl}}$. Since the $\epsilon V$ is an ideal of $V$ by \eqref{condition}, the $\lam$-bracket of $V$ induces a bilinear map $$\{-_\lam -\}: V^{\mathrm{cl}}\times V^{\mathrm{cl}}\rightarrow V^{\mathrm{cl}}[\lam],\quad (\bar{f} \bar{g})\mapsto\overline{[f_\lam g]}.$$ The triple $(V^{\mathrm{cl}},\pd, \{-_\lam -\})$ defines a Poisson vertex superalgebra, called the \emph{classical limit} of $V$ (cf. \cite{dk06,fbz}). \section{Screening operators for classical affine $\W$-algebras} In this section, we describe the classical affine $\W$-algebras by using screening operators. They will be obtained as a classical limit of the screening operators for the affine $\W$-algebras obtained in \cite{gen17}. We will use the same notation for Poisson vertex algebras as vertex algebras since there will be no confusion. \subsection{Affine $\W$-algebras}\label{affW} Let $\g$ be a finite dimensional simple Lie algebra over $\C$ with the normalized symmetric invariant bilinear form $\k=\inv$. Let $f\in\g$ be a nonzero nilpotent element, fix an $\Sl_2$-triple $\{e,h=2 x,f\}$ containing $f$ and denote by $\Gamma: \g=\oplus_{-d\leq j\leq d} \g_j$ the $\frac{1}{2}\Z$-grading given by $\ad_x$, with $d$ the largest number such that $\g_d\neq0$. We fix a triangular decomposition $\g=\n_+\oplus\h\oplus \n_-$ so that $x\in \h$, $\g_{>0}=\oplus_{j>0}\g_j\subset \n_+$, and $\g_{<0}=\oplus_{j<0}\g_j\subset \n_-$. Let $\g=\h\bigoplus\oplus_{\al\in\Delta}\g_\al$ be a root space decomposition, $\Delta_j=\{\al\in\Delta\mid \g_\al\subset \g_j\}$, and $\Delta_{>0}=\sqcup_{j>0} \Delta_j$. Fix a nonzero root vector $e_\al$ in $\g_\al$ and a basis $e_i$, $(i\in I)$, of $\h$. Then $e_\al$, $(\al\in I\sqcup \Delta)$, form a basis of $\g$. We denote by $\{e_{\bar{\al}}\}_{\al\in I\sqcup \Delta}$ its dual basis of $\g$ with respect to $\k$. Let $c_{\al,\bt}^\gam$ denote the structure constants of $\g$, i.e., $[e_\al,e_\bt]=\sum_{\gam\in I \sqcup\Delta} c_{\al,\bt}^\gam e_\gam$. Let $V^k(\g)$ be the universal affine vertex algebra of $\g$ at level $k$, generated by the even elements $e_\al$, $(\al\in I\sqcup \Delta)$, with $\lam$-bracket $[e_{\al\lam}e_\bt]=[e_\al,e_\bt]+k(e_\al|e_\bt)\lam$. Let $F^{\operatorname{ch}}(\g_{>0})$ be the charged free fermion vertex superalgebra associated with the symplectic odd vector superspace $\g_{>0}\oplus \g_{>0}^*$, generated by the odd elements $\vp_\al$, $\vp^\al$, $(\al\in\Delta_{>0})$, with $\lam$-bracket $[\vp_{\al\lam}\vp^\bt]=\delta_{\al,\bt}$, $[\vp_{\al\lam}\vp_\bt]=[\vp^\al_\lam\vp^\bt]=0$. Let $F(\g_\half)$ be the $\beta\gamma$-system vertex algebra associated with the symplectic vector space $\g_\half$, generated by $\Phi_\al$, ($\al\in \Delta_\half$), with $\lam$-bracket $[\Phi_{\al\lam}\Phi_\beta]=\chi([e_\al,\beta])$, where $\chi(-)=(f\mid-)$. The affine $\W$-algebra $\W^k(\g,f)$ associated with the triple $(\g,f, k)$, ($k\in\C$), is the vertex algebra defined as the $0$-th cohomology of the differential graded vertex algebra $$C^k(\g,f)=V^k(\g)\otimes F^{\operatorname{ch}}(\g_{>0})\otimes F(\g_\half),$$ with differential $$d_{(0)}=\Bigl[\sum_{\al\in \Delta_{>0}}\big((e_\al+\Phi_\al+\chi(e_\al) \big)\vp^\al-\frac{1}{2} \sum_{\al,\bt,\gam\in\Delta_{>0}} c_{\al,\bt}^\gam \vp_\gam \vp^\al\vp^\bt_\lam-\Bigr]|_{\lam=0},$$ called the \emph{BRST complex}. The grading $C^k(\g,f)=\oplus_{n\in\Z}C_n^k(\g,f)$ is given by $\gr(e_\al)=\gr(\Phi_\bt)=0$, $\gr(\vp^\al)=-\gr(\vp_\al)=1$ with $\gr(AB)=\gr(A)+\gr(B)$ and $\gr(\pd A)=\gr(A)$. Then we have $d_{(0)}: C_n^k(\g,f)\rightarrow C_{n+1}^k(\g,f)$. We have $H^n(C^k(\g,f))=0$ for $n\neq 0$ (see \cite{kw04,kw05}). \subsection{Classical limit}\label{classical limit} By \cite{kw04,kw05}, we have vertex subsuperalgebras $C^k_\pm(\g,f)$, which gives a decomposition of a complex $C^k(\g,f)=C^k_-(\g,f)\otimes C^k_+(\g,f)$ and satisfies $H^n(C^k_+(\g,f))\cong \delta_{n,0} \C$. Thus $H^0(C^k_-(\g,f))\cong \W^k(\g,f)$. As a vertex superalgebra, $C^k_-(\g,f)$ is generated by \begin{eqnarray*} &&J^u=u+\sum_{\bt,\gam\in \Delta_{>0}} c_{u,\bt}^\gam \vp_\gam \vp^\bt,\ (u\in\g_{\leq0}),\\ &&\Phi_\al, (\al\in \Delta_\half),\quad \vp^\al, (\al\in \Delta_{>0}). \end{eqnarray*} Following \cite{gen17}, we introduce the \emph{classical affine $\W$-algebra} as the cohomology of the differential graded Poisson vertex algebra in the classical limit of $C^k_-(\g,f)$. Suppose $k+h^\vee\neq0$. Set $\ep=\frac{k'}{k+h^\vee}$, $(k'\in\C\backslash\{0\})$, $\bar{J}^u=\ep J^u$, $(u\in \g_{\leq0})$, and $\bar{\Phi}_\al=\ep \Phi_\al$, $(\al\in\Delta_\half$). Then we have \begin{equation}\label{ope1} [\bar{J}^u_\lam \bar{J}^v]=\ep\bigl(\bar{J}^{[u,v]}+\big(k'(u|v)+o(\ep))\lam\bigr),\quad [\vp^\al_\lam \bar{J}^u]=\ep\sum_{\bt\in \Delta_{>0}}c_{u,\bt}^\al \vp^\bt, \end{equation} \begin{equation}\label{ope2} [\bar{\Phi}_{\al\lam}\bar{\Phi}_\bt]=\ep\cdot\chi([e_\al,e_\bt]),\quad [\bar{J}^u_\lam\bar{\Phi}_\al]=[\vp^\al_\lam\vp^\bt]=[\vp^\al_\lam\bar{\Phi}_\bt]=0, \end{equation} (cf. \cite{gen17}). Viewing $\epsilon$ as an indeterminate in \eqref{ope1}, \eqref{ope2}, we obtain a vertex superalgebra $\tilde{C}^{k}_-$ over the polynomial ring $\C[\epsilon]$. By Section \ref{classical limit}, we obtain a Poisson vertex superalgebra $\tilde{C}^{k}_-/\epsilon \tilde{C}^{k}_-$, which we denote by $C_{k'}^\mathrm{cl}=C_{k'}^\mathrm{cl}(\g,f)$. We have an isomorphism $$C_{k'}^\mathrm{cl}\cong V^{k'}(\g_{\leq0})\otimes F(\g_\frac{1}{2})\otimes \mathrm{Sym}(\C[\pd]\g_{>0}^*)$$ of Poisson vertex superalgebra where $V^{k'}(\g_{\leq0})$ is the universal affine Poisson vertex algebra generated by $\g_{\leq0}$ with $\lam$-bracket $\{u_{\lam} v \}=[u,v]+k'(u|v)\lam$, $F(\g_\half)$ the $\beta\gamma$-system Poisson vertex algebra generated by $\Phi_\al$, $(\al\in\Delta_\half)$, with $\lam$-bracket $\{\Phi_{\al\lam} \Phi_\bt \}=( f|[e_\al,e_\bt])$, and $\mathrm{Sym}(\C[\pd]\g_{>0}^*)$ the differential $\C$-superalgebra generated by odd elements $\vp^\al$, $(\al\in \Delta_{>0})$, which satisfy $$\{\vp^\al_{\lam} u\}=\sum_{\bt\in\Delta_{>0}} c^\al_{u,\bt} \vp^\bt,\quad \{\vp^\al_\lam \vp^\bt\}=\{ \vp^\bt_\lam\Phi_\al \}=0.$$ Decompose the differential $d_{(0)}$ as $d_{(0)}=\dstz+\dnez+\dchz$ where $$\dst=\sum_{\al\in\Delta_{>0}}e_\al \vp^\al-\frac{1}{2}\sum_{\begin{subarray}{c}\al,\bt\in \Delta_{>0}\\ \gam\in \Delta_{>0}\end{subarray}}c_{\al,\bt}^\gam \vp_\gam \vp^\al \vp^\bt,\ \dne=\sum_{\al\in \Delta_\frac{1}{2}}\Phi_\al \vp^\al,\ \dch=\sum_{\al\in\Delta_{>0}} \chi(e_\al)\vp^\al.$$ Then we have $$[d_{\mathrm{st}\lam}\bar{J}^u]=-\sum_{\begin{subarray}{c}\al\in I\sqcup \Delta_{\leq 0}\\ \bt\in\Delta_{>0}\end{subarray}} c_{u,\bt}^\al \bar{J}^{e_\al}\vp^\bt+\sum_{\bt\in \Delta_{>0}}(k'(u|v)(\pd+\lam)+o(\ep))\vp^\bt,$$ $$[d_{\mathrm{ne}\lam}\bar{J}^u]=\sum_{\begin{subarray}{c}\al\in\Delta_\half\\ \bt\in\Delta_{>0}\end{subarray}} c_{\bt}^\al \bar{\Phi}_\al\vp^\bt,\quad [\frac{1}{\epsilon}d_{\chi\lam}\bar{J}^u]=\sum_{\bt\in\Delta_{>0}}\chi([u,e_\bt])\vp^\bt,$$ $$[d_{\mathrm{st}\lam}\vp^\al]=-\frac{1}{2} \sum_{\bt,\gam\in \Delta_{>0}}c_{\bt,\gam}^\al \vp^\bt \vp^\gam,\quad [d_{\mathrm{ne}\lam} \bar{\Phi}_\al]=\sum_{\bt\in \Delta_\half}\chi([e_\bt,e_\al])\vp^\bt,$$ $$[d_{\mathrm{ne}\lam}\vp^\al]=[d_{\chi\lam}\vp^\al]=[d_{\mathrm{st}\lam}\bar{\Phi}_\al]=[\frac{1}{\epsilon}d_{\chi\lam}\bar{\Phi}_\al]=0.$$ The differential $d_{(0)}=\{(\dst+\dne+\dch)_\lam-\}|_{\lam=0}$ is given by $$\{d_{\mathrm{st}\lam}-\}, \{d_{\mathrm{ne}\lam}-\}, \{d_{\chi\lam}-\}: C_k^\cl(\g,f) \rightarrow C_k^\cl(\g,f)[\lam],$$ which satisfy $$\{d_{\mathrm{st}\lam} u \}=-\sum_{\begin{subarray}{c}\al\in I \sqcup\Delta_{\leq0}\\ \bt\in\Delta_{>0} \end{subarray}} c^\al_{u,\bt} e_\al\vp^\bt +\sum_{\bt\in\Delta_{>0}}k(u|e_\bt)(\pd+\lam)\vp^\bt, $$ $$\{d_{\mathrm{st}\lam} e_\al \}=0,\quad \{d_{\mathrm{st}\lam} \vp^\al \}=-\frac{1}{2} \sum_{\bt,\gam\in\Delta_{>0}}c^\al_{\bt,\gam} \vp^\bt \vp^\gam, $$ $$ \{d_{\mathrm{ne}\lam} u \}=\sum_{\begin{subarray}{c}\al\in\Delta_\half\\ \bt\in\Delta_{>0}\end{subarray}}c^\al_{u,\bt} e_\al\vp^\bt,\quad \{d_{\mathrm{ne}\lam}e_\al \}=\sum_{\bt\in\Delta_\half}(f|[e_\bt,e_\al])\vp^\bt,\quad \{d_{\mathrm{ne}\lam}\vp^\al \}=0,$$ and $$\{d_{\chi\lam}u \}=\sum_{\bt\in\Delta_{>0}}(f|[u,e_\bt])\vp^\bt,\quad \{d_{\chi\lam}\vp^\al \}=\{d_{\chi\lam}e_\al \}=0,$$ and are extended to $C_k^\mathrm{cl}$ by \eqref{sesqui}, \eqref{rightLeibniz} (see Remark \ref{1.1}). The 0-th cohomology of $C_k^\cl(\g,f)$, which we denote by $\W^k(\g,f)=H^0( C_k^\cl(\g,f))$, is a Poisson vertex algebra called \emph{the classical affine $\W$-algebra associated with $(\g,f,k)$} (\cite{gen17}). Note that $H^n( C_k^\cl(\g,f))=0$ holds for all $n\neq0$. \subsection{Screening operators} Introduce another grading wt on $C_k^\cl$ by $$\wt(u)= -2j, (u\in\g_j),\quad \wt(\Phi_\al)=0,\ (\al\in\Delta_\half),\quad \wt(\vp^\al)=2j \ (\al\in\Delta_j),$$ $$\wt(\pd A)=\wt(A),\ \text{and}\ \wt(AB)=\wt(A)+\wt(B). $$ and a decreasing filtration $\{F_pC_k^\cl \}_{p\geq0}$ on $C_k^\cl$ by $$F_pC_k^\cl=\text{span}\{A\in C_k^\cl|\wt(A)\geq p \},$$ This filtration is exhaustive, separated, and compatible with the grading of $C_k^\cl$ as a complex. The associated spectral sequence $\{E_r, d_r\}_{r\geq0}$ has the differentials $$d_0=d_{\mathrm{st}(0)}, \quad d_1=d_{\mathrm{ne}(0)}, \quad d_2=d_{\chi(0)},\quad d_r=0,\ (r\geq3),$$ and thus converges at $r=3$. We will describe $\W^k(\g,f)=H^0(C_k^\cl(\g,f))$ by using it. Since the calculation is straightforward, we omit the details. (The analogous argument for vertex algebras can be found in \cite{gen17}.) To calculate $E_1=H^*(C_k^\cl;d_0)$, notice that $d_0$ acts by $0$ on $V^k(\g_0)\otimes F(\g_\half)$ and that $(\bigwedge(\oplus_{\al\in\Delta_{>0}}\C \vp^\al), d_0)$ is a subcomplex isomorphic to the Chevalley-Eilenberg complex of the Lie algebra $\g_{>0}$ with coefficients in the trivial representation $\C$. Thus $(V^k(\g_0)\otimes F(\g_\half)\otimes\bigwedge(\oplus_{\al\in\Delta_{>0}}\C \vp^\al), d_0)$ is a subcomplex, whose cohomology is $V^k(\g_0)\otimes F(\g_\half)\otimes H^*(\g_{>0};\C)$. \begin{lemma}{\quad} \begin{enumerate} \item The natural map $V^k(\g_0)\otimes F(\g_\half)\otimes H^*(\g_{>0};\C)\rightarrow E_1$ is an isomorphism of graded vector spaces for generic $k\in\C$. \item The isomorphism $E_1^{(0)}\cong V^k(\g_0)\otimes F(\g_\half)$ is an isomorphism of Poisson vertex algebras. \item Each cohomology $E_1^{(n)}$ is a Poisson vertex module over $E_1^{(0)}$. Moreover, $E_1^{(n)}$ is isomorphic to $V^k(\g_0)\otimes F(\g_\half)\otimes H^n(\g_{>0};\C)$ as vector spaces. \end{enumerate} \end{lemma} Let us describe the Poisson vertex modules $E_1^{(n)}$ more explicitly. Recall that the coadjoint representation on $\g_{>0}^*$ of $\g_0$ induces a representation of $\g_0$ on $H^n(\g_{>0},\C)$ as described as follows. (cf. \cite[Chapter 3]{kumar02}) Let $W$ denote the Weyl group of $\g$ and set $I_0=\{i\in I| \al_i\in \Delta_0\}$, $W'_0=\{w\in W| w \Delta_0^+\subset \Delta^+\}$. Let $*: W\times \h^*\rightarrow\h^*$ denote the shifted action of $W$ and $l:W\rightarrow \Z_{\geq0}$ the length function. Then there is an isomorphism of $\g_0$-modules $$H^n(\g_{>0};\C)\cong \bigoplus_{\begin{subarray}{c}w\in W'_0\\ l(w)=n\end{subarray}}L_0(w^{-1}*0),$$ where $L_0(w^{-1}*0)$ is the integrable highest weight $\g_0$-module with highest weight $w^{-1}*0$. For a $\g_0$-module $M$, set $\mathbb{M}^k=V^k(\g_0)\otimes_\C M$. The space $\mathbb{M}^k$ has a unique Poisson vertex module over $\mathbb{M}^k$ such that $V^k(\g_0)$ acts as a commutative algebra by multiplication on the first component and the $\lam$-bracket $\{-_\lam-\}:V^k(\g_0)\otimes \mathbb{M}^k\rightarrow \mathbb{M}^k[\lam]$ satisfies $\{a_\lam b\otimes m\}=\{a_\lam b\}\otimes m+b \otimes a\cdot m$ for $a\in \g_0$, $b\in V^k(\g_0)$, and $m\in M$. We denote by $\L(w^{-1}*0)$ the Poisson vertex module obtained from $L_0(w^{-1}*0)$. \begin{lemma} There is an isomorphism $$E_1^{(n)}\cong \bigoplus_{\begin{subarray}{c}w\in W'_0\\ l(w)=n\end{subarray}}\L(w^{-1}*0)\otimes F(\g_\half),$$ as Poisson vertex modules over $V^k(\g_0)\otimes F(\g_\half)$. \end{lemma} In particular, we have $E_1^{(1)}\cong \bigoplus_{i\in I\backslash I_0}\L(-\al_i)\otimes F(\g_\half)$ and the subspace $L_0(-\al_i)$ is identified as $$L_0(-\al_i)\cong \bigoplus_{\bt\in [\al_i]}\C \vp^\bt\subset E_1^{(1)},$$ where $[\al_i]=\Delta_{>0}\cap (\al_i+Q_0)$ and $Q_0$ denotes the root lattice of $\Delta_0$. For $\al\in [\al_i]$ with $i\in I\backslash I_0$, we have \begin{equation}\label{eq;3} \pd \vp^\al=\frac{1}{k}\sum_{\begin{subarray}{c}\bt\in [\al],\\ \gam\in I\sqcup \Delta_0 \end{subarray}}c_{\bt,\gam}^\al e_{\bar{\gam}} \vp^\bt. \end{equation} Let us describe the differentials on $E_1$ induced from $d_{\mathrm{ne}(0)}$ and $d_{\chi(0)}$. Consider the intertwining operators $Q_i^W: V^k(\g_0)\otimes F(\g_\half)\rightarrow \L(-\al_i)\otimes F(\g_\half)$ given by \begin{eqnarray*} Q_i^W=\begin{cases}{} \sum_{\bt\in[\al_i]} \{\Phi_\bt \vp^\bt_\lam- \}|_{\lam=0},&(i\in I_\half),\\ \sum_{\bt\in[\al_i]} \{(f|e_\bt) \vp^\bt_\lam- \}|_{\lam=0},& (i \in I_1). \end{cases} \end{eqnarray*} Then we have: \begin{lemma} The differentials on $E_1$ induced by $d_{\mathrm{ne}(0)}$ and $d_{\chi(0)}$ are given by $$d_{\mathrm{ne}(0)}=\sum_{i\in I_\half} Q_i^W\ \text{and}\ d_{\chi(0)}=\sum_{i\in I_1} Q_i^W.$$ \end{lemma} Recall that the complex $C_k^\mathrm{cl}=C_k^\mathrm{cl}(\g,f)$ is $\Z_{\geq0}$-graded and that $H^n(C_k^\mathrm{cl}(\g,f)\cong \delta_{n,0}\W^k(\g,f)$. Then we see that $\W^k(\g,f)$ is a subalgebra of the 0-th degree $C_{k,0}^\mathrm{cl}=V^k(\g_{\leq0})\otimes F(\g_\half)$. Since $\mathcal{I}=V^k(\g_{<0})\otimes F(\g_\half)$ is a Poisson vertex ideal of $C_{k,0}^\mathrm{cl}$, we obtain a homomorphism of Poisson vertex algebras $$\W^k(\g,f)\rightarrow C_{k,0}^\mathrm{cl}/\mathcal{I}\cong V^k(\g_0)\otimes F(\g_{\half}).$$ It is injective and, by using the differentials $d_{\mathrm{ne}(0)}$ and $d_{\chi(0)}$, the image is described as in the following theorem. \begin{theorem} \label{2.2} For generic $k\in\C$, there is an isomorphism $$j: \W^k(\g,f)\cong \bigcap\limits_{i\in I_\half\sqcup I_1} \Ker\Bigl(Q_i^W:V^k(\g_0)\otimes F(\g_\half)\rightarrow \L(-\al_i)\otimes F(\g_\half)\Bigr),$$ of Poisson vertex algebras. \end{theorem} \noindent The operators $\{ Q_i^W\}_ {i\in I_\half\sqcup I_1}$ are called the \emph{screening operators} for $\W^k(\g,f).$ We note that the level $k=1$ is generic \cite{gen17}. The inclusion $j:\ \W^k(\g,f) \rightarrow V^k(\g_0)\otimes F(\g_\half)$ in Theorem \ref{2.2} induces a Lie algebra homomorphism between their spaces of local functionals $$j_*: \Lie\bigl(\W^k(\g,f)\bigr) \rightarrow \Lie\bigl(V^k(\g_0)\otimes F(\g_\half)\bigr).$$ (See Proposition \ref{1.3}.) \begin{lemma} \label{2.3} The Lie algebra homomorphism $j_*$ is injective. \end{lemma} \begin{proof} It is easy to see $$\Ker\Bigl(\pd:\L(-\al_i)\otimes F(\g_\half) \rightarrow \L(-\al_i)\otimes F(\g_\half)\Bigr) =0,\quad (i\in I_\half\sqcup I_1).$$ Take an element $f\in \W^k(\g,f)$ such that $\int f\in\Ker j_*$. Then there exists an element $G\in V^k(\g_0)$ such that $j(f)=\pd g$. Let $H_i^W$ denote the Hamiltonian of $Q_i^W$. Then we have $$0=Q_i^W j(f)=\{H^W_{i\lam}j(f)\}|_{\lam=0}=\{H^W_{i\lam}\pd g\}|_{\lam=0}=\pd \{H^W_{i\lam}g\}|_{\lam=0},$$ and so that $\{H^W_{i\lam}g\}|_{\lam=0}\in\Ker(\pd: \L(\al_i)\otimes F(\g_\half)) \ \rightarrow \L(\al_i)\otimes F(\g_\half))=0$. Therefore, we obtain $g\in j(\W^k(\g,f))$ and so that $\int f=0$. \end{proof} Let $Q_\al^W: V^k(\g_0)\otimes F(\g_\half)\rightarrow V^k(\g_0)\otimes F(\g_\half)$, ($\alpha\in\Pi$), be the derivation determined by \begin{equation} \label{eq;2} \begin{cases}{}Q_\al^W e_\bt=\sum\limits_{\gam\in[\al]} c^\gam_{\bt\al} \Phi_\gam,\quad Q_\al^W \Phi_\bt=(f|[e_\al,e_\bt]), & (\al\in\Pi_\half), \\ Q_\al^W e_\bt=(f|[e_\bt,e_\al]),\quad Q_\al^W \Phi_\bt=0,& (\al\in\Pi_1), \end{cases} \end{equation} \begin{equation} \label{eq;1} [Q_\al^W,\pd]=\frac{1}{k}\sum_{\begin{subarray}{c}\bt\in I\sqcup \Delta_0\\ \gam \in [\al]\end{subarray}} c_{\al,\bt}^\gam e_{\bar{\bt}} Q_\gam^W. \end{equation} \begin{theorem} \label{main1} For generic $k\in\C$, the classical affine $\W$-algebra $\W^k(\g,f)$ is isomorphic to the Poisson vertex subalgebra \begin{equation}\label{isom} \W^k(\g,f) \cong \bigcap_{\al\in \Pi}\Ker\Bigl(Q_\al^W: V^k(\g_0)\otimes F(\g_\half)\rightarrow V^k(\g_0)\otimes F(\g_\half)\Bigr), \end{equation} of $V^k(\g_0)\otimes F(\g_\half)$ invariant under the derivations $Q_\al^W$, $(\al\in \Pi)$. \end{theorem} \noindent We call the level $k\in\C$ \emph{generic} when \eqref{isom} holds. \begin{proof} Since $Q_i^W:V^k(\g_0)\otimes F(\g_\half)\rightarrow \L(-\al_i)\otimes F(\g_\half)$ acts by derivation, it decomposes as $Q_i^W=\sum_{\al\in[\al_i]} \vp^\al Q_\al^W$, where $Q_\al^W \in \Der(V^k(\g_0)\otimes F(\g_\half))$ and $\vp^\al$ is the multiplication by $\vp^\al$. We check that $Q_\al^W$ satisfies \eqref{eq;1} and \eqref{eq;2}. By direct calculation, \eqref{eq;2} follows from the definition of $Q_i^W$. To show \eqref{eq;1}, recall that $V^k(\g_0)$ has a Virasoro element $L=\frac{1}{2k}\sum_{\al \in I\sqcup \Delta_0} e_\al e_{\bar{\al}}$, i.e., it satisfies $\{L_\lam L\}=(\lam+\pd)L+c_k L$ for some $c_k\in \C$ and $\pd=L_{(0)}$. Then we have $[Q_i^W,\pd]=[\{Q_{i,\lam}^W-\}|_{\lam=0},\{L_{\mu}-\}|_{\mu=0}]=\{\{Q_{i\lam}^W L\}_\mu-\}|_{\lam=\mu=0}=\{-\pd Q_{i\lam}^W-\}|_{\lam=0}=0$. Here, we have used \eqref{Jacobi} in the second, \eqref{skew}in the third, and \eqref{sesqui} in the last equality. Therefore, $0=\sum_{\al\in [\al_i]} \vp^\al [Q_\al^W,\pd]-(\pd \vp^\al) Q_\al^W$. Now \eqref{eq;1} follows from this by \eqref{eq;3}. \end{proof} \section{Geometric Realization of $\W^k(\g,f)$} \subsection{Double coset space} Consider the Lie algebra $\gpoly=\g\otimes \C[t^{\pm1}]$. By abuse of notation, we denote by $\k=\inv$ the invariant bilinear form on $\gpoly$ given by $(a t^n| b t^m)=(a|b) \delta_{n+m,0}$, which extends the one on $\g$. We extend the grading $\Gamma$ on $\g$ to $\gpoly$ by setting $\deg(a t^n)=\deg(a)+(d+1) n$ (see Section \ref{affW}) and fix a homogeneous basis $e_\al$, $(\al \in I \sqcup \afD)$, extending the basis $e_\al,$ $(\al\in I \sqcup \Delta)$, of $\g$. We denote $|\al|=\deg(e_\al)$ for simplicity and $c_{\al,\bt}^\gam$ the structure constants, i.e., $[e_\al,e_\bt]=\sum_{\gam}c_{\al,\bt}^\gam e_\gam$. Let $e_{\bar{\al}}$, $(\al \in I \sqcup \afD)$, denote the dual basis of $e_\al$, $(\al \in I \sqcup \afD)$, with respect to $\k$. Consider the completion $\glau=\varprojlim_{n>0} (\gpoly/\gpoly_{>n})$, which we call the \emph{loop algebra} of $\g$. The grading on $\gpoly$ induces a decomposition $\glau=\lgp\oplus \glau_-$ where $\lgp=\varprojlim_{n>0} (\gpoly_{>0}/\gpoly_{>n})$ and $\glau_-=\gpoly_{\leq0}$. We decompose $X\in\glau$ as $X=X_++X_-\in \lgp\oplus \glau_{-}$. The subspace $\lgp$ is an affine scheme of infinite type. By setting $z_\al$ the coordinates of the basis $e_\al$, $(\al\in\afDp\subset \afD)$, of $\lgp$, we have $\C[\lgp]=\C[z_\al\mid \al\in \afDp]$. Let $G$ be the connected simply-connected algebraic group whose Lie algebra is $\g$ and $\lG$ the loop group of $G$, whose Lie algebra is $\glau$. We have closed subgroups $\lG_{\pm}$ of $\lG$ whose Lie algebra is $\glau_{\pm}$. Let us consider the quotient space $\lGn\backslash \lG$, which is an ind-scheme. It has an open subscheme $\lGp$. The $\C$-points of $\lGp$ is identified with the $\lGp$-orbits of the image of the identity in $\lGn\backslash \lG$. Since $\lGp$ is a prounipotent proalgebraic group, the exponential map $\exp: \lgp\rightarrow \lGp$ is an isomorphism of schemes. Thus the coordinate ring $\C[\lGp]$ is identified with $\C[\lgp]=\C[z_\al \mid \al\in\afDp]$. The left multiplication $\lGp\times \lGp\rightarrow \lGp$, $((g_1,g_2)\mapsto g_1^{-1}g_2)$ induces a $\lgp$-action on $\C[\lGp]$ as derivations \begin{equation}\label{leftaction} \xi^L: \lgp\rightarrow \Der(\C[\lGp]),\quad \phi\mapsto \phi^L. \end{equation} Since $\lGp$ is embedded into $\lGn\backslash \lG$ as an open subscheme, the right multiplication $\lGn\backslash \lG\times \lGn\rightarrow \lGn\backslash \lG$, $(([g_1],g_2)\mapsto [g_1g_2])$, induces a $\lgn$-action on $\C[\lGp]$ as derivations \begin{equation}\label{rightaction} \xi^R: \glau\rightarrow \Der(\C[\lGp]),\quad \phi\mapsto \phi^R. \end{equation} The commutator of these actions $\xi^L$, $\xi^R$ can be expressed by using the distinguished element $K=\exp(\sum_{\al\in\afDp} z_\al e_\al)\in\lGp(\C[\lGp])$. Note that this element is coordinate independent since it is identified with the identity morphism under the correspondence $$\lGp(\C[\lGp])=\Hom(\C[\lGp],\C[\lGp]).$$ \begin{lemma} \label{3.1} For $u\in \lgp$ and $v\in \glau$, $[u^L,v^R]=[u,(K v K^{-1})_-]_+^L$ holds. \end{lemma} \begin{proof} We use the notation $\e^X=\exp(X)$ for brevity. Let $\epsilon_i$ be dual numbers, i.e., satisfies $\epsilon_i^2=0$. For $F\in\C[\lGp]$ and $g\in \lGp$, we have \begin{align*} (v^R F)(g)&= \text{the}\ \ep_2\text{-linear term of}\ F(g \e^{\ep_2 v})\\ &= \text{the}\ \ep_2\text{-linear term of}\ F((\e^{(g v g^{-1})_+})g), \end{align*} and \begin{align*} (u^L v^R F)(g)&= \text{the}\ \ep_1\text{-linear term of}\ (v^RF)(\e^{-\ep_1 u}g)\\ &= \text{the}\ \ep_1\ep_2\text{-linear term of}\ F((\e^{(\e^{-\ep_1 u}g v g^{-1}\e^{\ep_1 u})_+})\e^{-\ep_1 u}g)\\ &= \text{the}\ \ep_1\ep_2\text{-linear term of}\ F(g_1(\ep_1,\ep_2)), \end{align*} where $$g_1(\ep_1,\ep_2)=\e^{-u \ep_1+(g v g^{-1})_+ \ep_2-([u,g v g^{-1}]_++(gv g^{-1})_+u) \ep_1 \ep_2}g.$$ Similarly, we obtain $$(v^R u^L)(F)(g)=\text{the}\ \ep_1\ep_2\text{-linear term of}\ F(g_2(\ep_1,\ep_2))$$ where $$g_2(\ep_1,\ep_2)=\e^{-u \ep_1+(g vg^{-1})_+ \ep_2-u(g vg^{-1})_+ \ep_1\ep_2}g.$$ Since $$g_1(\ep_1,\ep_2)=\e^{-[u,(gvg^{-1})_-]_+\ep_1\ep_2}g_2(\ep_1,\ep_2),$$ we obtain $[u^L, v^R]F(g)=[u,(g v g^{-1})_-]_+^L F(g)$. \end{proof} In the sequel, we assume that $(\g,f)$ satisfies the condition (F) introduced in Section 1: \begin{enumerate} \item[(F1)] The grading $\Gamma$ is a $\Z$-grading. \item[(F2)] There exists an element $y\in\g_d$ such that $s=f +y t^{-1}\in\gpoly$ is semisimple. \item[(F3)] The Lie subalgebra $\Ker(\ad_s)\subset\g[t^{\pm1}]$ is abelian and $\Im(\ad_s)\cap \gpoly_0=\g_0$. \end{enumerate} \begin{ex} \emph{ The following pairs $(\g,f)$ satisfy (F). \begin{enumerate} \item $(\g,f)$ with $f$ principal nilpotent element. \item $(\g(C_n),f_{(2^n)})$ and $(\g(C_{2n}),f_{(4^n)})$, $(n \geq2)$. \item $(\g(X_n),f)$ with $X=E,F,G$ listed in Table \ref{tb:exceptional}. \end{enumerate}} \end{ex} Here we have used the classification of nilpotent elements of $\g(C_n)$ by symplectic partitions and that of $\g(X_n)$, $(X=E,F,G)$, by the weighted Dynkin diagram, (see \cite{collmcg93}.) \begin{table}[h]{Table 4.1} \label{tb:exceptional} \begin{tabular}{|c|c||c|c|} \hline Dynkin diagram&weights of vertices& Dynkin diagram&weights of vertices \\ \hline $G_2$ &\wtGa& $E_{6\ \text{(continued)}}$ & \wtEac \\ \cline{2-4} \dynkinGtwo&\wtGb&$E_7$ & \wtEba \\ \cline{1-2}\cline{4-4} $F_4$ &\wtFa&\dynkinEseven & \wtEbb \\ \cline{2-4} \dynkinFfour&\wtFb& $E_8$ & \wtEca \\ \cline{2-2}\cline{4-4} &\wtFc&\dynkinEeight & \wtEcb \\ \cline{2-2}\cline{4-4} &\wtFd& &\wtEcc \\ \cline{1-2}\cline{4-4} $E_6$ &\wtEaa& &\wtEcd \\ \cline{4-4} \dynkinEsix &\wtEab& &\\ \hline \end{tabular} \end{table} Let $\abel$ denote the completion of $\Ker(\ad_s)$ in $\glau$. It is abelian by (F2). Let $A\subset \lG$, (resp.\ $A_\pm\subset \lG$) be the closed subgroup of $\lG$ whose Lie algebra is $\abel$, (resp.\ $\abel_{\pm}=\abel\cap\glau_{\pm}$). We consider the quotient space $\hom$. It admits a left $\lGp$-action $$\lGp \times \hom \rightarrow \hom,\quad (g, h\Ap) \mapsto g^{-1}h\Ap,$$ and a right $\An$-action $$\hom \times \An \rightarrow \lGn\backslash \lG/\Ap,\quad (h\Ap,g)\mapsto hg\Ap.$$ The right $\An$-action is well-defined since $A$ is abelian. These actions induce infinitesimal actions $\xi^L: \lgp\rightarrow \Der\C[\hom]$, $(\phi\mapsto \phi^L)$ and $\xi^R: \abeln\rightarrow \C[\hom]$, $(\phi\mapsto \phi^R)$ respectively. In particular, $\C[\lGp/A_L+]$ becomes a differential algebra by letting $s^R$ be the differential. Set \begin{equation}\label{E} E_\al =k(e_\al|K s K^{-1})\in \C[\lGp],\quad (\al\in I \sqcup\Delta_0). \end{equation} We have $E_\al\in \C[\lGp/\Ap]$ since \begin{eqnarray*} a^R (e_\al|K s K^{-1})&=& \text{the}\ \ep\text{-linear term of}\ (e_\al|K \e^{\ep a} s (K \e^{\ep a})^{-1})\\ &=&(e_\al|K[a,s]K^{-1})=0, \end{eqnarray*} for $a\in \abelp$. Since the Poisson vertex algebra $V^k(\g_0)$ is an algebra of differential polynomials, there exists a unique homomorphism of differential algebras $$\Psi_k: V^k(\g_0)\rightarrow \C[\lGp/\Ap],\quad e_\al \mapsto E_\al.$$ \begin{theorem} \label{main2} Suppose that $(\g,f)$ satisfies (F) and $k\in\C$ is generic. Then $\Psi_k$ is an isomorphism of differential algebras and satisfies \begin{equation}\label{intertwins} \Psi_k Q_\al^W \Psi_k^{-1}=-\frac{1}{k}e_\al^L,\quad \al\in\Pi. \end{equation} \end{theorem} \begin{proof} Since $V^k(\g_0)$ and $\C[\hom]$ are polynomial rings, it suffices to show that the linear map $d_0\Psi_k: T^*_0\text{Spec}V^k(\g_0) \rightarrow T^*_{[e]}\hom$ between the cotangent spaces is an isomorphism. By the identification $\lgp^* \cong \glau_{<0}$ and $\abelp^* \cong \abeln$ induced by $\k$, we obtain $T_{[e]}^*\hom \cong (\lgp/\abelp)^* \cong \glau_{<0}/\abeln$. This isomorphism is given by $dz_\al \mapsto e_{\bar{\al}}$. Under this isomorphism, we have $d_0\Psi_k(\pd^ne_\al)=k\ad_s^{n+1} e_\al$, as proved by inductive use of the $n=0$ case: \begin{align*} d_0\Psi_k(e_\al)&=kd(e_\al|KsK^{-1})=kd(e_\al|[\sum_{|\bt|=1}z_\bt e_\bt,s])\\ &=\sum\limits_{|\bt|=1}k([s,e_\al]|e_\bt) dz_\bt=k\sum\limits_{|\bt|=1}([s,e_\al]|e_\bt) e_{\bar{\bt}}\\ &=k[s,e_\al]. \end{align*} It follows from (F2) and (F3) that $\Psi_k$ is an isomorphism. To show \eqref{intertwins}, it suffices to show $$u_\al^L E_\bt=(f|[e_\bt, e_\al]),\ [u_\al^L,s^R]=\frac{1}{k}\sum_{\begin{subarray}{c}\rho\in I\sqcup \Delta_0\\ \gam\in[\al]\end{subarray}} c_{\al,\rho}^\gam E_{\bar{\rho}} u_\gam^L,\quad \al\in \Pi_1,\ \bt\in I\sqcup \Delta_0,$$ where $u_\al=-\frac{1}{k}e_\al$ by \eqref{eq;1}, \eqref{eq;2}. For the first one, we have \begin{align*} u_\al^L E_\bt&= \text{the}\ \ep \text{-linear term of}\ k(e_\bt|\e^{-\ep e_\al/k}K s K^{-1}\e^{\ep e_\al/k})\\ &=k(e_\bt|[-\frac{1}{k}e_\al,KsK^{-1}])=([e_\bt,e_\al]|KsK^{-1})\\ &=([e_\bt,e_\al]|s)=([e_\bt,e_\al]|f). \end{align*} The second one follows from Lemma \ref{3.1} since $$[u_\al^L,s^R]=[u_\al,(K s K^{-1})_0]_+^L=\frac{-1}{k^2}\sum\limits_{\begin{subarray}{c}\rho\in I\sqcup \Delta_0\\ \gam\in[\al]\end{subarray}} c_{\al,\rho}^\gam E_{\bar{\rho}} e_\gam^L=\frac{1}{k}\sum_{\begin{subarray}{c}\rho\in I\sqcup \Delta_0\\ \gam\in[\al]\end{subarray}} c_{\al,\rho}^\gam E_{\bar{\rho}} u_\gam^L.$$ \end{proof} Note that the Lie subalgebra $\g_+=\oplus_{j>0}\g_j$ is generated by the subspace $\oplus_{\al\in\Pi}\g_\al$. Let $G_+\subset \lGp$ denote the closed subgroup whose Lie algebra is $\g_+$. \begin{cor}\label{doublecoset} The isomorphism $\Psi_k$ restricts to an isomorphism of differential algebras $\W^k(\g,f)\cong \C[\dcoset]$. \end{cor} \begin{proof} The claim follows from Theorem \ref{main2} since $$\W^k(\g,f)\cong \bigcap_{\al\in\Pi_1} \Ker\bigl(Q_\al^W:V^k(\g_0)\rightarrow V^k(\g_0)\bigr),$$ by Theorem \ref{main1} and (F1). \end{proof} \subsection{Mutually commutative derivations on $\W^k(\g,f)$} By (F3), we have a Lie algebra homomorphism $\xi^R:\abeln \rightarrow \Der(\C[\hom]),\ (a \mapsto a^R)$. \begin{proposition} \sl The action $\xi^R$ preserves $\C[\dcoset]\subset\C[\hom]$. \end{proposition} \begin{proof} By Lemma \ref{3.1}, we have, for $e_\al\in\g$, $(\al\in\Pi_1)$, and $a\in \abeln$, $$[e_\al^L,a^R]=[e_\al,(K a K^{-1})_-]_+^L=[e_\al,(K a K^{-1})_0]^L=\sum\limits_{\bt \in I\sqcup \Delta_0} F_\bt[e_\al,e_\bt]^L,$$ for some polynomials $F_\bt\in \C[\hom]$. Hence, for $G\in \C[\dcoset]$, we have $$e_\al^L(a^R G)=[e_\al^L,a^R] G=\sum\limits_{\bt \in I\sqcup\Delta_0} F_\bt[e_\al,e_\bt]^L G=0$$. \end{proof} Since $\abeln$ is abelian, its image $\xi^R(\abeln)\subset \Der\C[\dcoset]$ is also abelian. By Theorem \ref{main2}, $\xi^R(\abeln)$ is identified with a set of commutative derivations of $\W^k(\g,f)$, which we denote by $\mathcal{H}^k(\g,f)$. In the next section, we will prove that $\mathcal{H}^k(\g,f)$ is an integrable Hamiltonian hierarchy associated with the classical affine $\W$-algebra $\W^k(\g,f)$. \section{$\mathcal{H}^k(\g,f)$ as an integrable Hamiltonian hierarchy} In this section, we always assume that the condition (F) holds and $k\in\C$ is generic, and identify $V^k(\g_0)$ with $\C[\hom]$ by $\Psi_k$, (Theorem \ref{main2}). \subsection{Construction of Hamiltonians} Let $\Omega_\dr(\lGp)=\C[\lGp]\otimes \bigwedge{\lgp^*}$ denote the algebraic de Rham complex of $\lGp$. Here $\lgp^*$ is the vector space dual to the space $\lgp$ of the right invariant vector fields $\lgp^L$ on $\lGp$. We take a basis $\vp^\al$, $(\al\in\afDp)$, of $\lgp^*$ so that $\vp^\al(e_\bt)=\delta_{\al,\bt}$, $(\al,\bt\in \afDp)$, holds. Note that the complex $\Omega_\dr(\lGp)$ coincides with the Chevalley-Eilenberg complex of the Lie algebra $\lgp$ with coefficients in $(\C[\lGp],\xi^L)$. Similarly, let $\Omega_\dr(\Ap)=\C[\Ap]\otimes \bigwedge{\abelp^*}$ denote the algebraic de Rham complex of $\Ap$. Then the inclusion $\Ap \hookrightarrow \lGp$ induces the projection $\Omega_\dr(\lGp) \rightarrow \Omega_\dr(\Ap)$. It restricts to $\abelp^R$-invariant subcomplexes: \begin{equation} \label{eq;4} \pi: \C[\hom]\otimes \bigwedge{\lgp^*} \rightarrow \C \otimes \bigwedge{\abelp^*}. \end{equation} As a $\lgp$-module, $\C[\hom]$ is isomorphic to the $\lgp$-module $\text{Coind}_{\abelp}^{\lgp}\C$ coinduced from the trivial $\abelp$-module $\C$. Then $\pi$ induces an isomorphism $$H^*(\lgp; \C[\hom]) \cong H^*\bigl(\lgp; \text{Coind}_{\abelp}^{\lgp} \C\bigr) \cong H^*(\abelp;\C)$$ by Shapiro's lemma, (cf. \cite{fuks86}). The right hand side is isomorphic to $\bigwedge \abelp^*$ since $\abelp$ is abelian. The action $\xi^R$ of $\abeln$ induces $\abeln$-actions on the complexes $\Omega_\dr(\lGp)$, $\Omega_\dr(\Ap)$, which commute with $\pi$. Since $\abel$ is abelian, $\abeln$ acts on $\Omega_\dr(\Ap)$ trivially. Thus $\abeln$ acts on the cohomology $H^*(\lgp; \C[\hom])$ trivially. In particular, $s$ acts on $H^*(\lgp; \C[\hom])$ trivially. By abuse of notation, we write the above $\abeln$-actions by $\xi^R$. Consider the double complex $\mathscr{C}$ $$\begin{CD} \C @>\iota>> \Omega_\dr(\lGp) @>\pm s^R>> \Omega_\dr(\lGp) @>\ep>> \C. \end{CD}$$ Here $\iota$ is the unit morphism and $\ep$ the counit morphism. It has the following shape: $$\begin{CD} @. \C_{(0,2)} \\ @. @A\ep AA \\ 0_{(-1,1)} @>>> \C[\hom]_{(0,1)} @>d>> \C[\hom] \otimes \glau^*_{>0\ (1,1)} @>d>>\cdots\\ @. @A \pd=s^R AA @A -s^R AA \\ 0_{(-1,0)} @>>> \C[\hom]_{(0,0)} @>d>> \C[\hom] \otimes \glau_{>0\ (1,0)} @>d>>\cdots,\\ @. @A\iota AA \\ @. \C_{(0,-1)} \end{CD}$$ where the subscript $(i,j)$ denotes the bidegree of $\mathscr{C}$. Then the calculation of the cohomology $H^*(\mathscr{C})$ via spectral sequences gives the isomorphisms \begin{equation} \abelp^* \cong H^1(\mathscr{C}) \cong \Ker\Bigl(d: \frac{\C[\hom]}{\C \oplus \Im(s^R)} \rightarrow \frac{\C[\hom] \otimes \lgp^*}{\Im(s^R)}\Bigr). \end{equation} By Theorem \ref{main2}, we have an isomorphism $\C[\hom]/\C \oplus \Im(s^R)\cong \Lie(V^k(\g_0))/\C$ as vector spaces. Since $V^k(\g_0)$ is an algebra of differential polynomials, $\int: \C\rightarrow \Lie\bigl(V^k(\g_0)\bigr)$ is injective. Thus we may lift an element $\int f\in \Lie(V^k(\g_0))/\C$ to $\int \tilde{f}\in \Lie(V^k(\g_0))/\C$ which is without the constant term $\tilde f(0)=0$. Identifying $\abeln$ with $\abelp^*$ by $\k$, we obtain an isomorphism \begin{equation*} \abeln\cong \abelp^* \cong \left\{\int f\in\Lie(V^k(\g_0))| f(0)=0,\ d\int f=0\right\},\quad a\mapsto \int H(a). \end{equation*} The statement of the following proposition makes sense by Lemma \ref{2.3}. \begin{proposition} \label{4.1} \sl The image $\int H(\abeln)$ lies in $\Lie(\W^k(\g,f))$. \end{proposition} \begin{proof} We may assume that the basis $\{e_\al\}_{\al\in\afDp}$ of $\lgp$ respects the decomposition $\lgp=\Im(\ad_s)\oplus \Ker(\ad_s)$. For $a\in \abeln$, the element $\int H(a)\in \Lie(V^k(\g_0))$ satisfies, by construction, $d H(a)=s^R A$ for some element $A=\sum_{\al\in\afDp}F_\al \otimes \vp^\al \in \C[\hom]\otimes\lgp^*$. Then, \begin{align*} d H(a)&=s^R A= s^R\sum\limits_{\al\in\afDp}F_\al\otimes \vp^\al\\ &=\sum\limits_{\al\in\afDp} \pd(F_\al) \otimes \vp^\al +\sum\limits_{\al\in\afDp} F_\al \otimes s^R(\vp^\al)\\ &=\sum\limits_{\al\in\Pi_1} \left(\pd F_\al +\sum_{\begin{subarray}{c}\bt\in\Pi_1\\ \gam\in \sqcup \Delta_0\end{subarray}}F_\bt c_{\al,\gam}^{\bt}E_{\bar{\gam}} \right) \otimes \vp^\al+ \sum\limits_{\al\in\afDp\backslash\Pi_1}\widetilde{F_\al^n}\otimes \vp^\al, \end{align*} for some element $\widetilde{F_\al^n}\in\C[\hom]$. In the last equality, we have used Lemma \ref{4.2} below. By the identification $\C[\hom]\cong V^k(\g_0)$ and $\C[\hom] \vp^\al \cong \L(-\al)$, $(\al\in \Pi_1)$, we obtain $$\sum\limits_{\al\in\Pi_1}\left(\pd F_\al +\sum_{\begin{subarray}{c}\bt\in\Pi_1\\ \gam\in I\sqcup\Delta_0\end{subarray}}F_\bt c_{\al,\gam}^{\bt}E_{\bar{\gam}} \right) \otimes \vp^\al= \pd\left(\sum\limits_{\al\in\Pi_1}F_\al \otimes \vp^\al\right)=0\in \bigoplus_{\al\in \Pi_1}\Lie\bigl(\L(-\al)\bigr).$$ On the other hand, by the definition of $d$, we have $d H(a)= \sum\limits_{\al\in\afDp} e_\al^L H(a) \otimes \vp^\al$. It follows that $e_\al^L \int H(a)=0$ for all $\al\in\Pi_1$, which implies $\int H(a)\in\Lie(\W^k(\g,f))$ by Theorem \ref{main2}. \end{proof} \begin{lemma} \label{4.2} \sl The following formula holds \begin{equation}\label{eq;4.2} s^R \vp^\al=\frac{1}{k}\sum_{\begin{subarray}{c}\bt\in\afDp\\ \gam\in I\sqcup \Delta_0\end{subarray}} E_{\bar{\gam}} c_{\bt,\gam}^\al \vp^\bt+\sum_{\bt\in \afDp} c_{\bt,s}^\al \vp^\bt,\quad \al\in \afDp. \end{equation} \end{lemma} \begin{proof} By Lemma \ref{3.1}, we have \begin{eqnarray} \label{eq;5} [s^R, e_\al^L]=-[e_\al, (K s K^{-1})_-]_+^L=-\frac{1}{k}\sum_{\bt\in I\sqcup \Delta_0} E_{\bar{\bt}} [e_\al, e_\bt]^L - [e_\al, s]_+^L. \end{eqnarray} Then, from the definition of $\lgp^*$, we have $$\left(s^R \vp^\al\right) (e_\bt)=-\vp^\al(\pd e_\bt)=\frac{1}{k}\sum_{\gam\in I\sqcup \Delta_0}E_{\bar{\gam}}c_{\bt,\gam}^\al+c_{\bt,s}^\al.$$ The claim \eqref{eq;4.2} follows from it immediately. \end{proof} \subsection{Poisson vertex superalgebra structure on $\C[\hom] \otimes \bigwedge \lgp^*$} We extend the Poisson vertex algebra structure on $\C[\hom]$ given by Theorem \ref{main2}, to the whole differential superalgebra $(\C[\hom] \otimes \bigwedge \lgp^*,s^R)$. \begin{proposition} \label{4.3} \sl The differential superalgebra $(\C[\hom]\otimes\bigwedge\lgp^*,s^R)$ admits a unique Poisson vertex superalgebra structure, which satisfies \begin{enumerate} \item $\{ u_\lam v\}= [u, v]+k(u|v) \lam,\quad (u,v\in \g_0)$, \item $\{\vp^\al_\lam\vp^\bt\}=0,\quad (\vp^\al, \vp^\bt \in \glau_1^*\cup \abelp^*)$, \item $\{u_\lam \vp^\al\}=-\sum c_{u,\bt}^\al \vp^\bt,\quad (\vp^\al\in \glau_1^*\cup \abelp^*,u\in \g_0)$. \end{enumerate} It satisfies \begin{eqnarray} \label{eq;6} \{u_\lam \vp^\al\}=-\sum c_{u,\bt}^\al \vp^\bt,\quad (\vp^\al\in \glau^* ,u\in \g_0). \end{eqnarray} \end{proposition} \begin{proof} It follows from Lemma \ref{4.2} that $\C[\hom] \otimes \bigwedge \lgp^*$ is an algebra of differential polynomials in the variables given by the union of bases of $\g_0$ and $\glau_1^* \cup \abelp^*$. By Theorem \ref{1.2}, it suffices to show \eqref{skew}, \eqref{Jacobi} for these variables in order to prove that (1)-(3) define a Poisson vertex superalgebra on $\C[\hom] \otimes \bigwedge \lgp^*$. The only non-trivial ones are \eqref{Jacobi} for $\{ e_\al, e_\bt, \vp^\gam\}$. By \eqref{skew}, they reduce to the special case \begin{equation}\label{Jacobicheck} \{e_{\al\lam}\{e_{\bt\mu}\vp^\gam\}\}-\{e_{\bt\mu}\{e_{\al\lam}\vp^\gam\}\}=\{\{e_{\al\lam}e_\bt\}_{\lam+\mu}\vp^\gam\}. \end{equation} Since it coincides with the Jacobi identity $(\coad(e_\al)\coad(e_\bt)-\coad(e_\bt)\coad(e_\al))\vp^\gam=\coad([e_\al,e_\bt])\vp^\gam$ of the coadjoint $\g_0$-action on $\lgp^*$, \eqref{Jacobicheck} holds. Next, we show \eqref{eq;6}. Since $\ad_s$ is an isomorphism on $\Im(\ad_s)\subset \glau$, we have its inverse $\ad_s^{-1}$ on $\Im(\ad_s)$ and on its dual space $\Im(\ad_s)^*$. Then we have a decomposition $\glau_{>0}=\ad_s^{-1}\Im(\ad_s)_{>0}\oplus (\glau_1\cup \abelp)$. We show \eqref{eq;6} for $\vp^\al\in \Im(\ad_s)$ by induction on degree. From Lemma \ref{4.2}, we have $$\ad_s^{-1}\vp^a= \pd \vp^a -\frac{1}{k} \sum_{\al,b} c_{b,\al}^a e_{\bal} \vp^b,\quad \vp^\al\in \Im(\ad_s)_1.$$ Here the sum with respect to Greek letters (resp.\ Roman letters) is taken over $I\sqcup \Delta_0$ (resp.\ $\afDp$). We use the same rule below. Then, for $e_\al\in\g_0$ and $\vp^a\in \Im(\ad_s)_1$, we have $$\{e_{\al\lam}(\ad_s^{-1})^n\vp^a\}=-\sum\limits_{|b|=|a|} c_{\al,b}^a (\ad_s^{-1})^n\vp^b,$$ which is proved by the inductive use of the $n=1$ case: \begin{align*} \{e_{\al\lam}\ad_s^{-1}\vp^a\} &= \{e_{\al,\lam}\pd \vp^a\}-\frac{1}{k} \sum_{\bt,b} c_{b,\bt}^a \{e_{\al\lam}e_{\bbt} \vp^b\}\\ &=(\pd+\lam)\{e_{\al,\lam}\pd \vp^a\}-\frac{1}{k}\sum_{\bt, b} c_{b,\bt}^a( \{e_{\al \lam}e_{\bbt}\}\vp^b+\{e_{\al\lam}\vp^b\}e_{\bbt})\\ &=-(\pd+\lam)\sum_b c_{\al b}^a \vp^b-\frac{1}{k} \sum\limits_{\al,b} c_{b,\al}^a\left(([e_\al, e_{\bbt}]+k(e_\al|e_{\bbt})\lam) \vp^b-\sum_c c_{\al,c}^b e_{\bbt}\vp^c\right)\\ &=-\lam\left(\sum_b c_{\al,b}^a \vp^b+\sum_{\bt,b} c_{b,\bt}^a (e_\al|e_{\bbt})\vp^b\right)\\ &\qquad-\left(\sum\limits_b c_{\al,b}^a\pd \vp^b+\frac{1}{k}\sum_{\bt,b}c_{b,\bt}^a\bigl([e_\al, e_{\bbt}]\vp^b-\sum_{c}c_{\al,c}^b e_{\bbt}\vp^c\bigr)\right)\\ &=-\sum_b c_{\al,b}^a \ad_s^{-1}\vp^b. \end{align*} In the last equality, we used $(e_\al|e_{\bbt})=\delta_{\al,\bt}$ for the $\lam$-linear term and use Lemma \ref{4.2}, $c_{a,b}^c=c_{\bar{c},a}^{\bar{b}}$, and the Jacobi identity of the Lie bracket for the constant term. This implies \eqref{eq;6} for $\vp^\al\in \Im(\ad_s)^*$. Then \eqref{eq;6} follows from this and (3). \end{proof} The de Rham differential $d=\sum_{\al \in \afDp} e_\al^L \otimes \vp^\al$ on $\C[\hom]$ can be described in term of the Poisson vertex superalgebra structure in Proposition \ref{4.3}. \begin{lemma} \label{4.4} \sl The differential $d$ is determined uniquely by\\ (i) $d E_\bt=-k\sum_{\al\in\afDp}([e_\bt,s]|e_\al) \otimes \vp^\al,\quad (\bt\in I\sqcup \Delta_0)$,\\ (ii) $[d, \pd]=0$. \end{lemma} \begin{proof} (1) follows immediately from the formula $e_\bt^L E_\al=-k([e_\al,s]|e_\bt)$ (see also \eqref{E}). (2) follows from \eqref{eq;5} and Lemma \ref{4.2}. \end{proof} Let $s^*\in \lgp^*$ denote the element corresponding to $s\in\lgn$ by the identification $\kappa=\inv: \lgn\cong \lgp^*$ and $\bar{s}\in \lgp$ the element corresponding to $s^*$. \begin{proposition} \sl Under the isomorphism $\Psi_k: V^k(\g_0)\cong \C[\hom]$, $$d=-k\{s^*_\lam -\}|_{\lam=0},$$ holds on $\C[\hom]$. \end{proposition} \begin{proof} It suffices to show that $-k\{s^*_\lam -\}|_{\lam=0}$ satisfies (i), (ii) in Lemma \ref{4.4}. (i) follows from the definition of $s^*$ and Proposition \ref{4.3}, (3). (ii) follows from \eqref{sesqui}. \end{proof} The following property of the derivation $\eta_a=\eta(\int H(a))\in\Der(\C[\hom])$, $(a\in\abeln)$, will be used. \begin{lemma} \label{4.5} \sl For $a\in\abeln$, there exist some polynomials $F_\al(a)\in\C[\hom]$, ($\al\in I \sqcup\Delta_0$), which satisfies $$[u^L,\eta_a]=\sum_{\al\in I \sqcup\Delta_0} F_\al(a) [u, e_\al]^L,\quad u\in\glau_1.$$ \end{lemma} \begin{proof} By construction of $\int H(a)$, we have $d \int H(a)=\int dH(a)=0\in\Lie(\C[\hom]\otimes\bigwedge \lgp^*)$ and thus $\{dH(a)_\lam-\}|_{\lam=0}=0$. By \eqref{Jacobi}, we have \begin{align*} \{dH(a)_\lam -\}|_{\lam=0} &=\Bigl[\{s^*_\lam-\}|_{\lam=0}, \{H(a)_\mu-\}|_{\mu=0}\Bigr] =\Bigl[\sum_{\al\in\afDp} e_\al^L\otimes\vp^\al, \eta_a\Bigr]\\ &=\sum_{\al\in\afDp} \Bigl([e_\al^L, \eta_a]\otimes \vp^\al -e_\al^L\otimes\eta_a(\vp^\al)\Bigr). \end{align*} It follows from them that $$[e_\al^L, \eta_a]=\sum_{\bt\in\afDp} \langle e_\al \mid \eta_a(\vp^\bt) \rangle e_\bt^L,$$ where $\langle-\mid- \rangle: \lgp\times\lgp^*\rightarrow\C$ denotes the canonical pairing. By Proposition \ref{4.3}, we have $$\eta_a(\vp^\bt)= \{ H(a)_\lam \vp^\bt\}|_{\lam=0}=\sum\limits_{\gam\in I\sqcup \Delta_0} \{ e_{\gam\pd} \vp^\bt\}_\rightarrow \frac{\delta H(a)}{\delta e_\gam} =-\sum\limits_{\begin{subarray}{c}\gam\in I\sqcup \Delta_0\\ \rho\in \afDp\end{subarray}} c_{\gam,\rho}^\bt\vp^\rho \frac{\delta H(a)}{\delta e_\gam}.$$ Hence, for $\al \in \glau_1$ we obtain \begin{align*} [e_\al^L, \eta_a]&=\sum_{\bt\in \afDp} \langle e_\al \mid \eta_a(\vp^\bt)\rangle e_\bt^L =-\sum\limits_{\bt\in \afDp}\sum\limits_{\begin{subarray}{c}\gam\in I\sqcup \Delta_0\\ \rho\in \afDp\end{subarray}} c_{\gam,\rho}^\bt \frac{\delta H(a)}{\delta e_\gam} \langle e_\al \mid \vp^\rho\rangle e_\bt^L\\ &=\sum\limits_{\gam\in I\sqcup \Delta_0} \frac{\delta H(a)}{\delta e_\gam} [e_\al, e_\gam]^L. \end{align*} This completes the proof. \end{proof} \subsection{Integrable Hamiltonian hierarchy} Let $\Vect(\lGp)$ denote the Lie algebra of the algebraic vector fields on $\lGp$. Recall the infinitesimal $\lgp$-actions \eqref{leftaction}, \eqref{rightaction}. \begin{lemma} \label{4.6} \sl We have $\Vect(\lGp)^{\lgp^L} =\lgp^R$ and $\Vect(\lGp)^{\lgp^R} =\lgp^L$. \end{lemma} \begin{proof} Although this is well-know, we give a proof for the completeness of the paper. The pairing $$\U(\lgp) \times \C[\lGp]\rightarrow \C,\quad (X_1X_2\cdots X_n, F)\mapsto (X_1^LX_2^L\cdots X_n^LF)(e),$$ defined by is $\lgp$-invariant and nondegenerate. Here $\U(\lgp)$ denotes the universal enveloping algebra of $\lgp$. Hence we have $\C[\lGp]^{\lgp^L} \cong \U(\lgp)/\lgp\U(\lgp) \cong \C$. Since $\lgp^L$ commutes with $\lgp^R$ and $\Vect(\lGp)\cong \C[\lGp]\otimes_{\C}\lgp^R$, we obtain $$\Vect(\lGp)^{\lgp^L}\cong (\C[\lGp]\otimes_{\C} \lgp^R)^{\lgp^L} = \C[\lGp]^{\lgp^L}\otimes_{\C} \lgp^R\cong \lgp^R.$$ The latter claim is proved similarly. \end{proof} We say that an element $X\in\Vect(\lGp)$ satisfies (P) if $$[u^L,X]\in\sum_{\al\in I \sqcup\Delta_0}\C[\lGp][u, e_\al]^L,\quad u\in\glau_1.$$ Let $\liep$ denote the set of elements in $\Vect(\lGp)$ satisfying (P). It is straightforward to show that $\liep\subset \Vect(\lGp)$ form a Lie subalgebra. The Witt algebra $\witt=\C((t)) \frac{\pd}{\pd t}$ acts on $\glau$ by derivations with respect to $t$. Since the subalgebra $\witt_-=\C[t^{-1}]t\frac{\pd}{\pd t}$ preserves $\glau_{\leq0}$, it acts on $B\backslash G((t))$ infinitesimally and therefore on the open dense subset $\lGp$. The induced $\witt_-$-action on $\C[\lGp]$ is given by $$L_n \exp\left(\sum_{\al\in\afDp} z_\al e_\al\right)= \text{the}\ \ep\text{-linear term of}\ \exp\left(\sum_{\al\in\afDp} z_\al e_\al\right)e^{\ep L_n},$$ where $L_n=-t^{n+1}\frac{\pd}{\pd t}\in \witt_-$. \begin{lemma} \sl We have $\liep=\witt_-\ltimes \glau^R$. \end{lemma} \begin{proof} By Lemma \ref{3.1}, we have $\glau^R\subset \liep$. In particular, this implies that $\liep$ is a $\glau$-module. The inclusion $\witt_-\subset \liep$ is shown in the same way as in the proof of Lemma \ref{3.1}. It is obvious that these two actions give an action of their semidirect product $\witt_-\ltimes \glau^R\subset \liep$. To show its equality, consider the quotient $\glau$-module $M:=\liep/\witt_- \ltimes \glau^R$. Let us show that $M$ belongs to the category $\mathcal{O}$ of the affine Lie algebra $\afg=\gpoly \oplus \C C \oplus \C L_0$ at level 0. Let $\afg=\afn_+\oplus \afh \oplus \afn_-$ be the standard triangular decomposition of $\afg$. Indeed, the action of the Cartan subalgebra $\widetilde{\h}=\h \oplus \C C \oplus L_{0}$ on $\Vect(\lGp)$ is diagonalizable and, moreover, preserves $\liep$ and $\witt_- \ltimes \glau^R$. Hence it is diagonalizable on $M=\liep/\witt_- \ltimes \glau^R$. Let us show that the dimension $\dim \U(\afn_+) w$ is finite for any $w\in M$. Let $\tw\in\liep$ denote a lift of $w$. The decomposition $\afn_+=\g_0^+\ltimes\afn_{>0}$ where $\g_0^+=\bigoplus_{\al \in \Delta_{0+}} \g_{0\al}$ and $\afn_{>0}=\afn_+\cap \glau_{>0}$ induces a decomposition of the universal enveloping algebra $\U(\afn_+) \cong \U(\g_0^+)\otimes\U(\afn_{>0})$ as vector spaces. By definition of $\liep$, $\tw$ satisfies, for any $u\in\glau_1$, $$[u^L, w]= \sum\limits_{\al\in I\sqcup\Delta_0} F_\al(w) [e_\al, u]^L\ \text{for some element}\ F_\al(w)\in\C[\lGp].$$ Since $\afn_{>0}^L$ and $\afn_{>0}^R$ commute, for any $Z\in \U(\afn_{>0})$, $$ [u^L,Z^R(w)]=Z^R([u^L,w])=\sum\limits_{\al \in I \sqcup\Delta_0} Z^R\bigl(F_\al (w)\bigr) [e_\al, u]^L.$$ If $Z^R(F_\al(w))=0$ for all $\al\in I \sqcup\Delta_0$ and $u\in \glau_1$, then by Lemma \ref{4.6}, we have $[Z^R, w]\in\afn_{>0}^R$, which is zero in $M$. Considering the $\widetilde{\h}$-weights of $Z^R(F_\al(w))$, we conclude that $\dim\U(\afn_{>0})^R(F_\al(w))<\infty$ for each $\al$ and hence $\dim \U(\afn_{>0})^R w<\infty$. Thus it remains to show $\dim \U(\g_0^+) w<\infty$. Take $u\in\afg_1$ and $v\in \g_0^+$, we have $[u^L, v^R]=[u, v]^L$ by Lemma \ref{3.1}. Then by setting $[u^L,w]=\sum_{\al\in I\sqcup \Delta_0}G_\al(w)[u,e_\al]^L$ for some element $G_\al\in\C[\lGp]$, we have \begin{align*} [u^L, [v^R, w]]&=[[u, v]^L, w]+ [v^R, [u^L, w]]\\ &=\sum\limits_ {\al\in I\sqcup \Delta_0} G_\al(w)[[u, v], e_\al]^L +\sum\limits_{\al\in I\sqcup \Delta_0} v^R(G_\al(w)) [u, e_\al]^L. \end{align*} By induction, we obtain \begin{align*} [u^L,[v_1^R,[v_2^R,\cdots,[v_n^R,w]\cdots] =\sum_{\begin{subarray}{c}\al\in I\sqcup \Delta_0\\A\sqcup B=\{1,2,\cdots,n\}\end{subarray}} (v_A^R G_\al(w)) [[u,v_B],e_\al]^L \end{align*} for $v_1,v_2,\cdots,v_n\in \g_0^+$, where $v_A^R=v_{a_1}^Rv_{a_2}^R\cdots v_{a_{\mathrm{max}}}^R$ for $A=\{a_1,a_2,\cdots,a_{\mathrm{max}}\}$, $(a_1<a_2<\cdots <a_{\mathrm{max}})$, and $[u,v_B]=[\cdots[u,v_{b_1}],v_{b_2}],\cdots],v_{b_{\mathrm{max}}}]$ for $B=\{b_1,b_2,\cdots,$ $b_{\mathrm{max}}\}$, $(b_1<b_2<\cdots<b_{\mathrm{max}})$. Since $\ad_{\g_0^+}^N \glau_1=0$ and $((\g_0^+)^R)^N G_\al(w)=0$ for $N$ large enough, it follows that $[u^L,[v_1^R,[v_2^R,\cdots,[v_n^R,w]\cdots]=0$ for $n$ sufficiently large and hence $[v_1^R,[v_2^R,\cdots,[v_n^R,w],\cdots]\in\afn_{>0}^R$, which is zero in $M$. Therefore, we obtain $\dim\U(\g_0^+)^R w<\infty$. Thus we conclude that $M$ belongs to the category $\mathcal{O}$. Suppose $M\neq0$. Then it contains a nonzero highest weight vector $w$. Let $\tw\in\liep$ denote a lift of $w$. Then $\witt_- \ltimes \glau^R + \C \tw$ is an extension of $\afn$-module $\witt_- \ltimes \glau^R$ by a trivial 1-dimensional $\afn$-module $\C w$, which is nontrivial by construction. Thus this extension defines a nonzero element in $H^1(\afn; \witt_- \ltimes \glau^R)$. Since the action $\glau^R$ on $\C[\lGp]$ is faithful, $\glau^R\cong \glau$ as $\afn$-modules. On the other hand, the cohomology $H^n(\afn; \witt_- \ltimes \glau^R)$ is describes as $$H^i(\afn; \witt_- \ltimes \glau) \cong H^{i-1}(\afn; \C) \otimes \witt_{>0},$$ where $\witt_{>0}=\C[[t]]t^2 \frac{\pd}{\pd t}$ (\cite{feiginfrenkel95}). This isomorphism is $\widetilde{\h}$-equivariant. In particular, all the degrees of the homogeneous elements with respect to the action of $x+(d+1) L_0$ in the first cohomology $H^1(\afn; \witt_- \ltimes \glau) \cong \witt_{>0}$ are positive. (See Section 1 for the definition of $x$ and $d$.) On the other hand, the cocycle corresponding to $\C w$ is negative due to the definition of $\liep$, which is a contradiction. Therefore, we conclude $M=0$. This completes the proof \end{proof} \begin{theorem} \label{main3} Suppose that $(\g,f)$ satisfies (F) and $k\in\C$ is generic. Then, $\mathcal{H}^k(\g,f)$ is an integrable Hamiltonian hierarchy associated with $\W^k(\g,f)$. \end{theorem} \begin{proof} By abuse of notation, let $\eta_a\in \Der(V^k(\g_0))$ also denote the corresponding element in $\Vect(\hom)$ by $\Psi_k$ in Theorem \ref{main2}. Since $\lGp \cong (\hom) \times \Ap$ as affine schemes in an $\Ap^R$-equivariant way, the elements of $\Vect(\hom)$ naturally lift to elements of $\Vect(\lGp)$ which commute with $\abelp^R$. Again, let $\eta_a$ denote the lift of $\eta_a$ in $\Vect(\lGp)$. Then by Lemma \ref{4.5}, the lift $\eta_a$ lies in $\liep$ and, moreover, $[u^L,\eta_a]=\sum_{\al I\sqcup \Delta_0} F_\al(\eta_a)[u,e_\al]^L$ with $F_\al(\eta_a)\in\C[\hom]$ for $u\in\glau_1$. Since $\bar{s}\in \abelp$, we have, for $u\in\glau_1$, \begin{align*} [u^L, [\bar{s}^R, \eta_a]]&=[\bar{s}^R, [u^L, \eta_a]]=\bigl[\bar{s}^R, \sum_\al F_\al(\eta_a) [u, e_\al]^L\bigr] = \sum_\al \bar{s}^R(F_\al(\eta_a)) [u, e_\al]^L\\ &=0. \end{align*} Since $\glau_1$ generates $\lgp$, we obtain $[\bar{s}^R, \eta_a]\in \Vect(\lGp)^{\lgp^L}=\lgp^R$. The degree of $[\bar{s}^R, \eta_a]$ is non-positive and the degrees of the elements in $\lgp^R$ are positive with respect to the action of $x+(d+1) L_0$. This implies $[\bar{s}^R, \eta_a]=0$, i.e., $\eta_a\in \liep^{\bar{s}^R}$. On the other hand, we have $\liep^{\bar{s}^R}=\witt_-^{\bar{s}^R}\ltimes (\glau^{\bar{s}})^R$ since $[\bar{s}^R, \witt_-]\subset[\abel, \witt_-]^R \subset \abel^R$ and $[\bar{s}^R,\glau^R]\cap\abel^R=0$. Since $\bar{s}\in \glau$ is not an element in $\g$, we have $\witt_-^{\bar{s}^R}=0$. From the definition of $\bar{s}$ (and $s$), we obtain $\glau^{\bar{s}^R}= \abel^R$. Thus $\liep^{\bar{s}^R}=\abel^R$ holds. Therefore, $\eta_a$ lies in $\abel^R$, which means $\int H(\abeln)\subset \abel^R$. Considering the degree given by the action of $x+(d+1)L_0$, we see $\int H(\abeln)\subset \abeln^R$. By Lemma \ref{1.4}, it is easy to show that the map $\int$ on $ H(\abeln)$ is injective. Since the map $a\mapsto \int H(a)$ preserves the degree with respect to the action of $x+(d+1) L_0$, we obtain $\int H(\abeln)=\abeln^R$. \end{proof}

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\section{Introduction} \setcounter{equation}{0} \label{sec:introduction} Classical solutions of supersymmetric gauge theories and supergravities that preserve some fraction of supersymmetry play a distinguished role in these theories as they typically exhibit particular stability and non-renormalization properties, due to the rigidness of the underlying supersymmetry algebra and its representations. Moreover, in the search of new solutions to supersymmetric theories it is often technically simpler to solve the first order Killing spinor equations rather than to address the full second order field equations. The systematic study of these questions in supergravity theories goes back to the seminal work of Tod~\cite{Tod:1983pm,Tod:1995jf} (see~\cite{Gibbons:1982fy} for earlier work), and has further and systematically been developed in~\cite{Gauntlett:2002sc,Gauntlett:2002nw} and a large body of follow-up work for higher-dimensional supergravity theories. In this approach, one assumes the existence of a Killing spinor and considers all bilinear tensors that can be constructed from it. The Killing spinor equations together with standard Fierz identities translate into a set of algebraic and differential identities among these bilinear tensors which can be employed to constrain the structure of the geometry and the matter dynamics. In many cases this allows for a complete classification of the supersymmetric solutions of the theory. So far, this bilinear tensor analysis has mainly been employed and proven very useful to construct and classify supersymmetric solutions in theories with eight real supercharges, such as $N=2$ supergravity in four dimensions, see e.g.~\cite{Caldarelli:2003pb,Meessen:2006tu,Huebscher:2006mr,Cacciatori:2008ek,Klemm:2009uw,Hristov:2009uj}. In the context of half-maximal supergravities (i.e.\ sixteen real supercharges), the analysis becomes more involved due to the richer structure of the extended $R$-symmetry groups. Previous work on such theories includes \begin{itemize} \item[(i)] the pure ungauged $D=4, N=4$ supergravity~\cite{Bellorin:2005zc}, \item[(ii)] the pure ungauged $D=5, N=4$ supergravity~\cite{Liu:2006is}, \item[(iii)] the pure ungauged and the $SU(2)$ gauged $D=7, N=1$ supergravity~\cite{Cariglia:2004qi,MacConamhna:2004fb}, \end{itemize} \noindent but has throughout been restricted to theories without additional vector multiplets, leading in particular to scalar target spaces of very small dimensions ($\le 2$). In this paper, we apply the bilinear tensor techniques to the matter-coupled half-maximal $N=8$ supergravity in three dimensions~\cite{Marcus:1983hb,Nicolai:2001ac}. The three-dimensional case exhibits a number of interesting properties: in particular, the metric in three dimensions does not carry propagating degrees of freedom, and vector gauge fields can be dualized into scalar fields. As a result, the entire dynamics of these theories takes place in the scalar sector, whose target space is a coset space manifold $SO(8,n)/(SO(8)\times SO(n))$ of dimension~$8n$. The bilinear tensor analysis which we present in this paper leads to an interesting interplay of the Killing spinor equations and this target space geometry. The structures we exhibit in this paper represent a first example of the structures that will also appear in higher dimensions upon inclusion of vector multiplets with coset space geometry. The advantage of the three-dimensional setting is the fact that all matter dynamics is uniformly described in the scalar sector. Upon passing to the gauged $N=8$ supergravity, vector fields have to be introduced in three dimensions, however these do not represent additional propagating degrees of freedom, but enter with a Chern-Simons coupling and are related to the scalar fields by (a non-abelian version of) their standard first order duality equations. As such, the structure of the theory (and its Killing spinor equations) is still organized by the full group $SO(8,n)$ even though the explicit choice of a gauge group breaks this global symmetry of the theory. We will analyze the structure of the Killing spinor equations and their consequences for the matter fields in a fully $SO(8,n)$ covariant manner without specifying the three-dimensional gauge group. Of course, when further exploiting these structures to construct explicit solutions, this choice will have to be made. Some three-dimensional supersymmetric solutions of the half-maximal ungauged theory with higher-dimensional origin in heterotic string theory (compactified on a seven torus) have been constructed in \cite{Forste:1997ac,Bakas:1997jy,Bourdeau:1997gm}, see also \cite{Greene:1989ya,Sen:1994wr} for earlier work. Some supersymmetric solutions of the gauged $N=8$ theory have been identified in~\cite{Berg:2001ty,Berg:2006ng,Gava:2010rx}, among them are analytic domain wall solutions which can be interpreted as holographic RG flows in particular scenarios of AdS$_3$/CFT$_2$ dualities. In all these different contexts it would clearly be interesting and highly desirable to dispose of a more systematic approach to construct and classify the possible supersymmetric solutions. In this paper, we take a first step in this direction and apply the bilinear tensor techniques to the matter-coupled $D=3$, $N=8$ supergravity, which is reviewed in section~2. We construct in section 3 the bilinear tensors from the Killing spinor and translate the full content of the Killing spinor equations into a set of algebraic and differential identities among these tensors. As usual, these tensors provide a timelike or null Killing vector that generates a symmetry of the full solution (i.e.\ including the matter sector). In the timelike case, the $SO(8)$ $R$-symmetry breaks down to $SO(2)\times SO(6)$ and we show in section~4 that the Killing vector can be extended to a canonical tetrad in which we expand the fields and obtain the general solution for the scalar current. The $8n$ scalar fields naturally split into the $4n+4n$ eigenvectors of an antisymmetric matrix $\Omega$ related to the $SO(2)$ part of the composite connection on the coset space manifold. With the general solution for the scalar current we analyze the first order duality equation between vector and scalar fields and show that it reduces to a two-dimensional equation for the field-strength of the undetermined part of the vector fields. In particular, for the ungauged theory we obtain an explicit expression of the vector fields in terms of the scalar fields of the theory. In section 4.4, we give a general analysis to determine to which extent the equations of motion are satisfied as a consequence of the existence of a Killing spinor. In order to construct explicit solutions, the Killing spinor equations have to be amended by the integrability relations for the scalar current and we do this in section 5 for the ungauged theory, leaving the full analysis of the gauged theory for future work. In this case, the three-dimensional metric takes diagonal form with the conformal factor satisfying a Liouville-type equation. The integrability relations for the scalar current reduce to a covariant holomorphicity equation (with the composite connection of the scalar target space) for its unknown component. As an illustration, we employ a simple ansatz to derive in section 5 explicit solutions with $1\le n\le4$ (active) matter multiplets. Finally, in section~6 we analyze the case of a null Killing vector in the ungauged theory for which we derive the most general solution for metric and scalar fields which is a pp-wave. Appendix~A collects our $SO(8,n)$ conventions and some useful formulae. \section{Half-maximal supergravities in three dimensions} In this section we review the structure of half-maximal (i.e.\ $N=8$) supergravity theories in three dimensions. In three dimensions, pure (super-)gravity does not possess propagating degrees of freedom. Moreover, vector fields can be dualized into scalar fields, such that all propagating degrees of freedom may be accommodated in the scalar sector of the theory. The ungauged theory is described by an $SO(8,n)/(SO(8)\times SO(n))$ coset space sigma-model coupled to gravity~\cite{Marcus:1983hb}. The general gauged theory in which gauge fields are added with a Chern-Simons coupling has been constructed in~\cite{Nicolai:2001ac}, to which we refer for details and conventions. \subsection{Bosonic field content and Lagrangian} The Lagrangian of the general gauged $N=8$ supergravity in three dimensions is given by \cite{Nicolai:2001ac} \begin{eqnarray} {\cal L}&=& -\ft14 \,eR + {\cal L}_{\rm CS} + \ft14 e \,{\cal P}^{\mu\,Ir} {\cal P}_\mu^{\,Ir} +{\cal L}_{\rm pot} + {\cal L}_{\rm F}\;. \label{L} \end{eqnarray} Let us explain each term separately. We use signature $(+--)$, and denote the dreibein determinant by $e=\sqrt{\det\,g_{\mu\nu}}$, such that the first term is the standard Einstein-Hilbert term of three-dimensional gravity. The vector fields in three-dimensional supergravity do not represent propagating degrees of freedom, but couple in the gauged theory with a Chern-Simons kinetic term ${\cal L}_{\rm CS}$ given in (\ref{LCS}) below, such that they are related by their first order equations of motion to the scalar fields. The gauge group $G_0$ is a subgroup of $SO(8,n)$, the isometry group of the scalar sector. Its generators $\Xi_{\cal MN}$ can be represented as linear combinations of the $SO(8,n)$ generators $X^{\cal MN}=-X^{\cal NM}$ by means of the {\em embedding tensor} $\theta_{\cal MNKL}$ as \begin{eqnarray} \Xi_{\cal MN} &\equiv& \theta_{\cal MNKL}\, X^{\cal KL} \;, \label{generators} \end{eqnarray} where indices {\small ${\cal M}, {\cal N}, \dots$} label the vector representation of $SO(8,n)$. They can be raised and lowered with the $SO(8,n)$ invariant indefinite metric $\eta_{\cal MN}$. Moreover, we split these indices into $\{{\small{\cal M}}\} \rightarrow \{ I,r \}$ according to the signature of $\eta_{\cal MN}$, i.e.\ $I = 1, \dots, 8$, $r=1, \dots, n$, with $I$ labeling the vector representation ${\bf 8}_v$ of $SO(8)$. The specific form of the gauge group $G_0$, its dimension and its embedding into $SO(8,n)$ are entirely encoded in the constant tensor $\theta_{\cal MNKL}$. Supersymmetry requires $\theta_{\cal MNKL}$ to be completely antisymmetric in its four indices.\footnote{In fact, the condition which supersymmetry imposes is slightly weaker, it also allows for contributions of the type $\theta_{\cal MNKL}=\eta_{{\cal M}[{\cal K}}\xi_{{\cal L}]{\cal N}}-\eta_{{\cal N}[{\cal K}}\xi_{{\cal L}]{\cal M}}$ with a symmetric $\xi_{\cal MN}=\xi_{\cal NM}$, cf.~\cite{deWit:2003ja}, but we shall not consider these cases here.} In addition, gauge covariance requires $\theta_{\cal MNKL}$ to satisfy the bilinear relation \begin{eqnarray} 2\theta^{\cal Q}{}_{\cal KL[M}\theta_{\cal N]RSQ}&=& \theta^{\cal Q}{}_{\cal RMN}\theta_{\cal KLSQ} - \theta^{\cal Q}{}_{\cal KLR}\theta_{\cal MNSQ} \;. \label{quadcon} \end{eqnarray} A standard example of gauged ${\cal N}=8$ supergravity corresponds to the choice \begin{equation} \theta_{IJKL} ~=~ \left\{ \begin{array}{rl} \epsilon_{IJKL} & \mbox{for}\quad I,J,K,L \in \{1,2,3,4\} \\ \alpha \, \epsilon_{IJKL} & \mbox{for} \quad I,J,K,L \in \{5,6,7,8\} \\[.5ex] 0 & \mbox{otherwise} \end{array} \right. \;, \label{theta} \end{equation} which solves (\ref{quadcon}) and describes the gauging of an $SO(4)\times SO(4) \subset SO(8)$ subgroup of $SO(8,n)$, the parameter $\alpha$ corresponding to the ratio of coupling constants of the two $SO(4)$ factors. This theory describes the coupling of the spin $1/2$ multiplet in the reduction of six-dimensional supergravity on $AdS_3\times S^3$ (for $\alpha=0$), or $AdS_3\times S^3\times S^3$. More complicated gaugings have been given in \cite{Nicolai:2003ux,Hohm:2005ui} which describe the couplings of higher massive multiplets in this compactification. In the following we shall not specify the explicit gauge group, but derive the constraints on the structure of supersymmetric solutions for general choice of $\theta_{\cal MNKL}$. Explicitly, the Chern-Simons term ${\cal L}_{\rm CS}$ is given by\footnote{ Here and in the following, we denote by $\varepsilon_{\mu\nu\rho}$ the totally antisymmetric spacetime tensor, i.e.\ in particular it carries the determinant $e$ of the vielbein: $\varepsilon_{123}=e$, etc.} \begin{equation} {\cal L}_{\rm CS} ~=~ -\ft14\,e\,\varepsilon^{\mu\nu\rho}\,g\theta_{\cal KLMN}\,A_\mu{}^{\cal KL} \Big(\partial_\nu A_\rho\,{}^{\cal MN} +\ft83\,g\, \eta^{\cal MR}\,\theta_{\cal RSPQ}\, A_\nu{}^{\cal PQ} A_\rho{}^{\cal SN} \Big) \;, \label{LCS} \end{equation} where the parameter $g$ denotes the gauge coupling constant. The scalar sector in the Lagrangian \Ref{L} is described by a gauged coset space sigma-model $SO(8,n)/(SO(8)\times SO(n))$. It can be parametrized by a group-valued $SO(8,n)$ matrix ${\cal S}$ (i.e.\ a matrix satisfying ${\cal S}\eta {\cal S}^T=\eta$) which defines the left-invariant scalar currents as \begin{eqnarray} {\cal J}_\mu~\equiv~ {\cal S}^{-1}{\cal D}_\mu {\cal S} &\equiv& {\cal S}^{-1} \,\Big(\partial_\mu + g \theta_{\cal KLMN} \,A_\mu{}^{\cal KL} X^{\cal MN}\Big)\, {\cal S} ~\in~{\rm Lie}\,SO(8,n) \;, \label{current} \end{eqnarray} with the covariant derivatives carrying the gauge group generators $\Xi_{\cal KL}$ of (\ref{generators}). We split these currents according to \begin{eqnarray} {\cal Q}_\mu^{IJ} ~\equiv ~ ({\cal J}_\mu)^{IJ} \;, \quad {\cal Q}_\mu^{rs} ~\equiv ~ ({\cal J}_\mu)^{rs} \;, \quad {\cal P}_\mu^{Ir} ~\equiv ~ ({\cal J}_\mu)^{Ir} \;, \label{defPQ} \end{eqnarray} corresponding to the compact generators $X^{IJ}$, $X^{rs}$ and noncompact generators $Y^{Ir}$ of the algebra $\mathfrak{so}(8,n)\equiv{\rm Lie}\, SO(8,n)$, see appendix~\ref{app:so} for our algebra conventions. We may explicitly separate the gauge field contributions to the currents as \begin{eqnarray} {\cal Q}_\mu^{IJ}&=& Q_\mu^{IJ}+2gA_\mu{}^{\cal KL}{\cal V}^{\cal MN}{}_{IJ}\,\theta_{\cal KLMN} \;, \nonumber\\ {\cal P}_\mu^{Jr} &=& {P}_\mu^{Jr} +g A_\mu^{\cal KL} {\cal V}^{\cal MN}{}_{Jr}\,\theta_{\cal KLMN} \;, \label{PgP} \end{eqnarray} where we use the short-hand notation ${\cal V}^{\cal MN}{}_{{\cal PQ}}\equiv 2{\cal S}^{[{\cal M}}{}_{\cal P}{\cal S}^{{\cal N}]}{}_{\cal Q}$, and $Q^{IJ}_\mu \equiv {\cal Q}^{IJ}_\mu |_{g=0}$, $P^{Ir}_\mu \equiv {\cal P}^{Ir}_\mu |_{g=0}$ are defined by (\ref{current}) at $g=0$, i.e.\ represent the currents of the ungauged sigma-model, cf.~appendix~A. The scalar kinetic term in (\ref{L}) is defined in terms of the noncompact components ${\cal P}_\mu^{Jr}$ of the scalar current and thus invariant under local transformations \begin{eqnarray} {\cal S} ~\rightarrow~ {\cal S}\,H(x)\;, \label{coset} \end{eqnarray} with $H(x)\in SO(8)\times SO(n)$, taking care of the coset redundancy. The integrability equations $2\partial_{[\mu}{\cal J}_{\nu]}+[{\cal J}_\mu,{\cal J}_\nu]= {\cal S}^{-1} [{\cal D}_{\mu},{\cal D}_{\nu}] {\cal S}$ induced by the definition of (\ref{current}), translate into \begin{eqnarray} {\cal D}^{\vphantom{I}}_{[\mu} {\cal P}^{Ir}_{\nu]} &\equiv& \partial^{\vphantom{I}}_{[\mu} {\cal P}^{Ir}_{\nu]}+{\cal Q}_{[\mu}^{IJ} {\cal P}^{Jr}_{\nu]} +{\cal Q}_{[\mu}^{rs} {\cal P}^{Is}_{\nu]} ~=~ \ft12\, g\theta_{\cal KLMN}\,{\cal F}_{\mu\nu}{}^{\cal KL} {\cal V}^{\cal MN}{}_{Ir} \;, \label{intP}\\[.5ex] {\cal Q}^{IJ}_{\mu\nu} &\equiv& 2\partial_{[\mu}{\cal Q}^{IJ}_{\nu]} + 2 {\cal Q}^{IK}_{[\mu}{\cal Q}^{KJ}_{\nu]} ~=~ -2{\cal P}_{[\mu}^{Ir}{\cal P}_{\nu]}^{Jr} +g\theta_{\cal KLMN}\,{\cal F}_{\mu\nu}{}^{\cal KL} {\cal V}^{\cal MN}{}_{IJ} \;,\qquad\quad \label{intQ1}\\[.5ex] {\cal Q}^{rs}_{\mu\nu} &\equiv& 2\partial_{[\mu}{\cal Q}^{rs}_{\nu]} + 2 {\cal Q}^{rt}_{[\mu}{\cal Q}^{ts}_{\nu]} ~=~ -2{\cal P}_{[\mu}^{Ir}{\cal P}_{\nu]}^{Is} +g\theta_{\cal KLMN}\,{\cal F}_{\mu\nu}{}^{\cal KL} {\cal V}^{\cal MN}{}_{rs} \;, \label{intQ2} \end{eqnarray} with the nonabelian field strength ${\cal F}_{\mu\nu}{}^{\cal KL}$\,. Finally, the gauged theory carries a scalar potential whose form is fully determined by supersymmetry as \begin{equation} {\cal L}_{\rm pot} ~=~ -eg^2\,{V}\;,\qquad \mbox{with}\quad V~\equiv~ \ft18\,A_2^{A\dot{A} r}A_2^{A\dot{A} r}-\ft14 A_1^{AB}A_1^{AB} \;, \label{potential} \end{equation} in terms of the scalar $SO(8)$ tensors $A_1$, $A_2$, defined by contracting the embedding tensor of (\ref{generators}) with the scalar fields as \begin{eqnarray} A_{1\, AB}&=& -\ft1{48}\,\Gamma^{IJKL}_{AB}\, {\cal V}^{\cal KL}{}_{IJ}\,{\cal V}^{\cal MN}{}_{KL}\,\theta_{\cal KLMN} \;, \nonumber\\ A_{2\, A\dot{A} r}&=& -\ft1{12}\,\Gamma^{IJK}_{A\dot{A}}\, {\cal V}^{\cal KL}{}_{IJ}\,{\cal V}^{\cal MN}{}_{Kr}\,\theta_{\cal KLMN} \;. \label{A1A2} \end{eqnarray} Here $A=1, \dots, 8$, and $\dot{A}=1, \dots, 8$, label the two $SO(8)$ spinor representations ${\bf 8}_s$, ${\bf 8}_c$, and $\Gamma^I_{A\dot{A}}, \Gamma^{IJ}_{AB}$, etc.\ denote the corresponding $\Gamma$-matrices. The tensors (\ref{A1A2}) describe the Yukawa couplings in the fermionic sector that we describe explicitly in the next subsection. Up to fermionic contributions, the bosonic equations of motion derived from the Lagrangian (\ref{L}) are given by \begin{eqnarray} R_{\mu\nu}-\ft12 R g_{\mu\nu} &=& {\cal P}^{Ir}_\mu {\cal P}^{Ir}_\nu - \ft12 g_{\mu\nu}\, {\cal P}^{Ir}_\rho {\cal P}^{\rho\,Ir} + 2 g^2 V g_{\mu\nu} \;, \label{eqm:Einstein} \\[1ex] {\cal D}^\mu\,{\cal P}^{Ir}_\mu &=& \ft14\,g^2 \Gamma^{I}_{A\dot{A}} \left( A^{\dot{A}r\,\dot{B}s}_3A_2^{A\dot{B}s} -3A_1^{AB} A_2^{B\dot{A}r} \right) \;, \label{eqm:scalars} \end{eqnarray} for the metric and the scalar fields, with the scalar tensor $A^{\dot{A}r\,\dot{B}s}_3$ defined in (\ref{A3}) below. Varying the vector fields in the Lagrangian (\ref{L}) gives rise to the first order duality equation \begin{eqnarray} \theta_{\cal KLMN}\, {\cal F}_{\mu\nu}{}^{\cal MN} &=& \varepsilon_{\mu\nu\rho} \theta_{\cal KLMN} {\cal V}^{\cal MN}{}_{Ir}\,{\cal P}^{\rho\,Ir}\;, \label{duality} \end{eqnarray} which manifests the fact that the vector fields do not carry propagating degrees of freedom. Note further, that even though formally we have introduced ${\rm dim}\,SO(8,n)$ vector fields $A_\mu{}^{\cal MN}$, only their projections $\theta_{\cal KLMN}\, {A}_{\mu}{}^{\cal MN}$ appear in the equations. For example, with (\ref{theta}) only 12 vector fields appear in the action (or 6 if $\alpha=0$). \subsection{Fermions and Killing spinor equations} The fermionic couplings of the Lagrangian (\ref{L}) are given by \begin{eqnarray} e^{-1}{\cal L}_{\rm F}&=& \ft12 \varepsilon^{\mu\nu\rho} \overline{\psi}{}^A_\mu {\cal D}_\nu\psi^A_\rho -\ft12\,\mathrm{i} \overline{\chi}^{\dot{A} r}\gamma^\mu {\cal D}_\mu\chi^{\dot{A} r} - \ft12 {\cal P}_\mu^{Ir}\overline{\chi}^{\dot{A} r} \Gamma^I_{A\dot{A}}\gamma^\nu\gamma^\mu\psi^A_\nu \nonumber\\[1ex] &&{} +\ft12 g A_1^{AB} \,\overline{\psi}{}^A_\mu \gamma^{\mu\nu} \psi^B_\nu +{\rm i} g A_2^{A\dot{A} r} \,\overline{\chi}^{\dot{A} r} \gamma^{\mu} \psi^A_\mu +\ft12 gA_3^{\dot{A} r\, \dot{B} s}\, \overline{\chi}^{\dot{A} r}\chi^{\dot{B} s} \;. \label{LF} \end{eqnarray} Here $\chi^{\dot{A} r}$ and $\psi^A_\mu$ are two-component Majorana spinors, transforming in the ${\bf 8}_c$ and ${\bf 8}_s$ of $SO(8)$, respectively. Covariant derivatives ${\cal D}_\mu$ on these spinors include the spin connection, the $SO(8)$ connection ${\cal Q}_{\mu\,\dot{A}\dot{B}}\equiv\ft14\,{\cal Q}_{\mu\,IJ}\,\Gamma^{IJ}_{\dot{A}\dot{B}}$ and ${\cal Q}_{\mu\,AB}\equiv\ft14\,{\cal Q}_{\mu\,IJ}\,\Gamma^{IJ}_{AB}$, respectively, and the $SO(n)$ connection ${\cal Q}_{\mu}^{rs}$ obtained from (\ref{defPQ}). We use $\gamma^{\mu\nu\rho}=-\mathrm{i}\varepsilon^{\mu\nu\rho}$ for the three-dimensional $\gamma$-matrices. The Yukawa-type couplings are given by the scalars tensors $A_1$, $A_2$ from (\ref{A1A2}) and \begin{eqnarray} A_3^{\dot{A} r\, \dot{B} s} = \ft1{48}\,\delta^{rs}\,\Gamma^{IJKL}_{\dot{A}\dot{B}}\, {\cal V}^{\cal KL}{}_{IJ}\,{\cal V}^{\cal MN}{}_{KL}\,\theta_{\cal KLMN} +\ft1{2}\,\Gamma^{IJ}_{\dot{A}\dot{B}}\, {\cal V}^{\cal KL}{}_{IJ}\,{\cal V}^{\cal MN}{}_{rs}\,\theta_{\cal KLMN} \;, \label{A3} \end{eqnarray} which also appears in the bosonic equations of motion (\ref{eqm:scalars}). Combining their definition with (\ref{dV}) shows that these tensors are related by the differential relations \begin{eqnarray} {\cal D}_\mu A_1^{AB}&=& \ft12\left(\Gamma^I_{A\dot{A}}A_{2}^{B\dot{A}r}+ \Gamma^I_{B\dot{A}}A_{2}^{A\dot{A}r}\right)\,{\cal P}^{Ir}_\mu \;, \nonumber\\ {\cal D}_\mu A_{2}^{A\dot{A} r} &=& \ft12\Gamma^I_{B\dot{A}} A_1^{AB}\,{\cal P}_\mu^{Ir}+ \ft12\Gamma^I_{A\dot{B}} A_3^{\dot{A}r\,\dot{B}s}\,{\cal P}_\mu^{Is}+ \ft1{16} \Gamma^J_{A\dot{A}}\Gamma^{IJ}_{\dot{B}\dot{C}}A_3^{\dot{B}r\,\dot{C}s}\,{\cal P}_\mu^{Is} \;, \end{eqnarray} which play an important role in proving supersymmetry of the action (\ref{L}). The quadratic constraints (\ref{quadcon}) translate into various bilinear identities among the scalar tensors, such as the {supersymmetric Ward identity} \begin{eqnarray} A_1^{AC}A_1^{BC}-\ft12\,A_2^{A\dot{A} r}A_2^{B\dot{A} r} &=& \ft18\delta^{AB}\, \left( A_1^{CD}A_1^{CD}-\ft12\,A_2^{C\dot{A} r}A_2^{C\dot{A} r} \right) \;. \label{qc1} \end{eqnarray} The full action (\ref{L}) is $N=8$ supersymmetric. The Killing spinor equations of the theory follow as usual by imposing the vanishing of the fermionic supersymmetry variations on a given bosonic background \begin{eqnarray} \label{KS1} 0 &\equiv& \delta_\epsilon \psi_\mu^A ~=~ (\partial_\mu + \ft14 \omega_\mu{}^{ab}\gamma_{ab})\,\epsilon^A +{\cal Q}_{\mu\,AB} \,\epsilon^B + \mathrm{i} g A_1^{AB}\gamma_\mu\,\epsilon^B \;, \\[1ex] 0 &\equiv&{} \delta_\epsilon \chi^{\dot{A} r} ~=~ \ft{\mathrm{i}}{2}\Gamma^I_{A\dot{A}}\gamma^\mu {\cal P}_\mu^{Ir}\epsilon^A +g A_2^{A\dot{A} r}\epsilon^A \;, \label{KS2} \end{eqnarray} with the scalar tensors $A_{1}$, $A_{2}$ from (\ref{A1A2}) above. In the rest of the paper we will analyze the consequences of these equations for supersymmetric solutions of the theory. \section{Killing spinor bilinears} We will study in this paper the structure of supersymmetric solutions of the $N=8$ theory defined by (\ref{L}). Let us assume the existence of $p$ (commuting) Killing spinors $\epsilon^{A}_{(\alpha)}$, $\alpha=1, \dots, p$, satisfying equations (\ref{KS1}) and (\ref{KS2}). From these we can define real scalar and vector functions bilinear in the Killing spinors, as follows \begin{eqnarray} F^{\,AB}_{\alpha\beta} ~\equiv~ \bar\epsilon^{A}_{(\alpha)}\,\epsilon^{B}_{(\beta)} \;,\qquad V_{\mu}{}^{\,AB}_{\,\,\alpha\beta} ~\equiv~ \mathrm{i}\,\bar\epsilon^{A}_{(\alpha)}\,\gamma_{\mu}\,\epsilon^{B}_{(\beta)} \;. \end{eqnarray} By construction they satisfy \begin{eqnarray} F^{AB}_{\alpha\beta} ~=~- F^{BA}_{\beta\alpha} \;,\qquad V_{\mu}{}^{AB}_{\,\,\alpha\beta}~=~ V_{\mu}{}^{BA}_{\,\,\beta\alpha} \;. \end{eqnarray} In the following, we focus on the case $p=1$ and consider the real tensors built from a single commuting Killing spinor $\epsilon^{A}$ \begin{eqnarray} F^{[AB]}~=~\bar\epsilon^{A}\epsilon^{B}\;,\qquad V_{\mu}{}^{(AB)}~=~\mathrm{i}\,\bar\epsilon^{A}\gamma_{\mu}\,\epsilon^{B}\;. \label{FV} \end{eqnarray} Furthermore, we define the ${\rm SO}(8)$ invariant real combination $V_{\mu}\equiv V_{\mu}{}^{AA}$\,. In the rest of this section we will translate the full content of the Killing spinor equations (\ref{KS1}), (\ref{KS2}) into a set of algebraic and differential relations for the invariant tensors $F^{AB}$ and $V_\mu{}^{AB}$\,. \subsection{Algebraic relations} \label{subsec:algeq} {}The cubic Fierz identities for the Killing spinor $\epsilon^{A}$, together with the definitions (\ref{FV}) induce the relations \begin{eqnarray} V_{\mu}{}^{AB}\gamma^{\mu}\epsilon^{C} &=& -2\mathrm{i} F^{C(A}\epsilon^{B)} \;, \nonumber\\ V_\mu{}^{AB}\epsilon^C -\mathrm{i} \varepsilon_{\mu\nu\rho} V^{\nu\,AB}\gamma^\rho\epsilon^{B} &=& -2\mathrm{i} F^{C(A}\gamma_\mu\epsilon^{B)} \;. \end{eqnarray} Similarly, evaluating quartic Fierz identities, one obtains the identities \begin{eqnarray} V^{\mu}V_{\mu} &=& -2 V^{\mu\,AB}V_\mu{}^{AB} ~=~ 2F^{AB}F^{AB} \;, \nonumber\\ V^{\mu\,AB}V_\mu{}^{CD} &=& -2 F^{A(C}F^{D)B} \;, \nonumber\\ V_{[\mu}{}^{AB}V_{\nu]}{}^{CD} &=& \ft12\,\varepsilon_{\mu\nu\rho}\, \Big(V^{\rho\,A(C}\,F^{D)B}+V^{\rho\,B(C}\,F^{D)A}\Big) \;, \nonumber\\ V_{(\mu}{}^{AC}V_{\nu)}{}^{BC} &=& V_{(\mu}{}V_{\nu)}{}^{AB} -\ft12 g_{\mu\nu}\,V^{\rho}{}V_{\rho}{}^{AB} \;, \nonumber\\ F^{C[A}\,V_\mu{}^{B]C}&=& -\ft12\,F^{AB}\,V_\mu \;, \label{quartic1} \end{eqnarray} bilinear in the tensors $F^{AB}$, $V_\mu{}^{AB}$. From a sextic Fierz identity, one finally finds \begin{eqnarray} F^{AB}F^{BC}F^{CD} + f^2 \,F^{AD} &=& 0 \;, \label{sextic} \end{eqnarray} with $f^2=\frac12 F^{AB}F^{AB}$\,. This implies that the antisymmetric matrix $F^{AB}$ has only two nonvanishing eigenvalues $\pm \mathrm{i}f$\,. Accordingly, it will often be convenient to change to an explicit basis for the $SO(8)$ spinor indices \begin{eqnarray} A=(a,\tilde{a})\;,\qquad a=1,2\;,\quad \tilde{a}=3,\dots,8\;, \label{basis} \end{eqnarray} in which ${F}^{ab}=f\epsilon^{ab}$\,, $F^{a\tilde{a}}=0=F^{\tilde{a}\tilde{b}}$.\footnote{We use conventions $\epsilon^{12}=\epsilon_{12}=1$.} For non-vanishing $f$, this corresponds to a breaking of the $R$-symmetry according to $SO(8)\rightarrow SO(2)\times SO(6)$, under which the fundamental representations branch as \begin{eqnarray} {\bf 8}_s \rightarrow 1_{-1}+6_0+1_{+1}\;,\quad {\bf 8}_v \rightarrow 4_{-1/2}+\overline{4}_{+1/2}\;,\quad {\bf 8}_c \rightarrow \overline{4}_{-1/2}+4_{+1/2} \;, \label{branching} \end{eqnarray} with subscripts denoting $SO(2)$ charges. The second and the last equation of~\Ref{quartic1} then imply that the only nonvanishing components of $V_{\mu}{}^{AB}$ are the~$V_{\mu}{}^{ab}$. The third equation of~\Ref{quartic1} shows that the three vectors $V_\mu{}^{11}$, $V_\mu{}^{12}= V_{\mu}^{21}$, and $V_\mu{}^{22}$ satisfy the algebra \begin{eqnarray} V_{[\mu}{}^{11}V_{\nu]}{}^{22} &=& -\varepsilon_{\mu\nu\rho}\, f\,V^{\rho\,12} \;, \nonumber\\ V_{[\mu}{}^{12}V_{\nu]}{}^{11} &=& \ft12\varepsilon_{\mu\nu\rho}\, f\,V^{\rho\,11} \;, \nonumber\\ V_{[\mu}{}^{12}V_{\nu]}{}^{22} &=& -\ft12\varepsilon_{\mu\nu\rho}\, f\,V^{\rho\,22} \;. \label{alg} \end{eqnarray} According to the second equation of~\Ref{quartic1}, they are normalized as \begin{eqnarray} 0&=& V_\mu{}^{11}V^{\mu\,11} ~=~ V_\mu{}^{22}V^{\mu\,22} ~=~ V_\mu{}^{12}V^{\mu\,11} ~=~ V_\mu{}^{12}V^{\mu\,22} \;,\nonumber\\ 2f^2 &=& V_\mu{}^{11}V^{\mu\,22} ~=~ -2 V_\mu{}^{12}V^{\mu\,12} \;. \label{alg2} \end{eqnarray} For non-vanishing $f$, these vectors thus form an orthogonal basis of the three-dimensional spacetime. In the explicit basis (\ref{basis}), all algebraic relations (\ref{quartic1}) are summarized by (\ref{alg}) and (\ref{alg2}). For later use, we use the explicit basis (\ref{basis}) to define (again for non-vanishing $f$) the antisymmetric matrices \begin{eqnarray} \Omega^{IJ} \equiv \ft12 \epsilon^{ab} \Gamma^I_{a\dot{A}}\Gamma^J_{b\dot{A}} \;, \qquad \Omega_{\dot{A}\dot{B}} \equiv \ft12 \epsilon^{ab} \Gamma^I_{a\dot{A}}\Gamma^I_{b\dot{B}} \;, \label{Omegas} \end{eqnarray} satisfying $\Omega_{\dot{A}\dot{B}} \Omega_{\dot{B}\dot{C}} = -\delta_{\dot{A}\dot{C}}$\, and $\Omega^{IJ}\Omega^{JK} = -\delta^{IK}$\,. Group-theoretically, they manifest the fact that under the above branching (\ref{branching}) of the ${\bf 8}_v$ and ${\bf 8}_c$, there appears another invariant tensor in their respective tensor products. In contrast, for $f=0$ (i.e.\ $F^{AB}=0$), the first two equations of (\ref{quartic1}) state that all vectors $V_\mu{}^{AB}$ are null and mutually orthogonal, i.e.\ they are all proportional according to $V_\mu{}^{AB}=\Lambda^{AB} V_\mu$, with a symmetric matrix $\Lambda^{AB}$ of trace 1. The fourth equation of (\ref{quartic1}) then imposes \begin{eqnarray} \Lambda^2 &=& \Lambda \;, \end{eqnarray} for the matrix $\Lambda$, i.e.\ this matrix has a single non-vanishing eigenvalue. We can choose a basis in which the only non-vanishing component of $V_\mu{}^{AB}$ is \begin{eqnarray} V_\mu{}^{11} = V_\mu{}^{AA} =V_\mu \;. \end{eqnarray} Accordingly we split the index $A=(1,\tilde{a})$, corresponding to a breaking of $SO(8)$ to $SO(7)$, under which the fundamental representations branch as \begin{eqnarray} {\bf 8}_s \rightarrow 1+7\;,\quad {\bf 8}_v \rightarrow 8\;,\quad {\bf 8}_c \rightarrow 8 \;. \label{branching_null} \end{eqnarray} \subsection{Differential equations} \label{subsec:diffeq} In addition to the algebraic relations induced by Fierz identities, the Killing spinor equations impose a number of differential equations on the tensors $F^{AB}$, $V_\mu{}^{AB}$. The first of the Killing spinor equations (\ref{KS1}) implies that \begin{eqnarray} D_{\mu}\, F^{AB} &=& 2g\,A_1^{C[A}\,V_\mu{}_{\vphantom{1}}^{B]C}\;, \label{diff1}\\ D_\mu\,V_\nu{}^{AB} &=& 2g\,g_{\mu\nu}\,A_1^{C(A}\,F^{B)C} +2g\,\varepsilon_{\mu\nu\rho}\, A_1^{C(A}\,V^{\rho\,B)C} \label{diff2} \;, \end{eqnarray} where the derivative $D_\mu$ now is the full $SO(8)$ covariant derivative. From (\ref{diff1}) we thus obtain in the basis (\ref{basis}) \begin{eqnarray} \partial_{\mu}\, f &=& g\, A_1^{ca} \epsilon^{ab}\,V_\mu{}^{bc} \;, \label{df} \end{eqnarray} or equivalently, using (\ref{alg2}): $V^{\mu\, ab}\, \partial_{\mu} f = -2gf^2 \epsilon_{\vphantom{1}}^{c(a}A_1^{b)c} $\,. In particular, for vanishing $g$, i.e.\ in the ungauged theory, the function $f$ is constant. From~(\ref{diff2}) we obtain that \begin{eqnarray} D_{(\mu} V_{\nu)} &=& 0\;, \end{eqnarray} i.e.\ the vector $V_{\mu}$ is a Killing vector of the solution. Equation~(\ref{alg2}) moreover shows that this vector is either timelike (for $f\not=0$) or null (for $f=0$), in accordance with the expectations. In the following sections, we will treat the two cases separately. From (\ref{df}) together with (\ref{alg2}), we find also that $V^\mu\partial_\mu f=0$, i.e.\ the function $f$ is also constant along the Killing vector field. Contracting the dilatino equation (\ref{KS2}) with $\overline{\epsilon}^B$ and $\overline{\epsilon}^B\gamma_\nu$, respectively, gives rise to two equations for the scalar current ${\cal P}_\mu^{Ir}$: \begin{eqnarray} 0 &=&{} \ft12\Gamma^I_{B\dot{A}}V^{\mu\,AB} \,{\cal P}_\mu^{Ir} +g F^{AB}\,A_2^{B\dot{A} r} \;, \label{dilatino1}\\ 0 &=&{} - \ft12\Gamma^I_{B\dot{A}} \,F^{AB} \, {\cal P}_\mu^{Ir} + \ft12\varepsilon_{\mu\nu\rho} \Gamma^I_{B\dot{A}} V^{\rho\,AB}\,{\cal P}^{\nu\,Ir} + g A_2^{B\dot{A} r}V_\mu{}^{AB} \;. \label{dilatino2} \end{eqnarray} The first equation implies in particular (upon contracting with $\Gamma^J_{A\dot{A}}$ and using (\ref{PgP}) and the definition (\ref{A1A2})) \begin{eqnarray} V^{\mu} {P}_\mu^{Ir} &=& -g \theta_{\cal KLMN} {\cal V}^{\cal MN}{}_{Ir}\, \Big(V^{\mu}A_\mu{}^{\cal KL} + f\,{\cal V}^{\cal KL}{}_{JK}\,\Omega^{JK} \Big) \;, \label{VP} \end{eqnarray} with $\Omega^{JK}$ from (\ref{Omegas}). This shows that with the particular (Coulomb type) gauge choice \begin{eqnarray} V^{\mu}A_{\mu}{}^{\cal KL} &=& -f {\cal V}^{\cal KL}{}_{MN}\,\Omega^{MN} \;, \nonumber\\ V^{\mu}Q^{IJ}_{\mu} &=& 0 ~=~ V^{\mu}Q^{rs}_{\mu} \;, \label{VQ} \end{eqnarray} of vector and $SO(8)$ gauge freedom, all scalar fields are likewise constant along the Killing vector field \begin{eqnarray} {\cal L}_{V}\,{\cal S} ~=~ 0\;. \end{eqnarray} It requires some more work and the explicit use of the duality equation (\ref{duality}) to show that as a consequence of equations (\ref{dilatino1}), (\ref{dilatino2}) also the gauge fields are constant along the Killing vector field. We come back to this in section~\ref{subsec:vectors}. The gauge fixing (\ref{VQ}) furthermore suggests to split off the $V_\mu$ contribution in the vector fields and introduce new gauge fields $\hat{A}^{IJ}_\mu$ according to \begin{eqnarray} A_{\mu}{}^{\cal KL} &=& -\frac{1}{4f}\, {\cal V}^{\cal KL}{}_{MN}\,\Omega^{MN}\,V_\mu + \hat{A} _\mu {}^{\cal KL} \;, \label{Ahat} \end{eqnarray} for non-vanishing $f$. This split will play an important role in the following. To summarize, we have shown in this section, that the invariant tensors $F^{AB}$ and $V_\mu{}^{AB}$ satisfy the algebraic relations (\ref{alg}) and (\ref{alg2}) as a consequence of the Fierz identities of the underlying Killing spinor. Moreover, in terms of these tensors, the Killing spinor equations (\ref{KS1}), (\ref{KS2}) take the equivalent form (\ref{diff1}), (\ref{diff2}), (\ref{dilatino1}), (\ref{dilatino2}). In the following we will study these equations in more detail. \section{Timelike case: general discussion} \label{sec:general} In the previous section we have identified the Killing vector field $V_\mu$ among the tensors built from the Killing spinor. Moreover, we have translated all the constraints imposed by three-dimensional Fierz identities into the algebraic relations (\ref{alg}), (\ref{alg2}), and the full content of the Killing spinor equations into the differential equations (\ref{diff1}), (\ref{diff2}) and (\ref{dilatino1}), (\ref{dilatino2}). In the following, we will study these equations and their consequences in detail. We first discuss the case $f\not=0$ of a timelike Killing vector $V^\mu V_\mu > 0$\,. The null case is presented separately in section \ref{sec:null}. \subsection{Three-dimensional spacetime} \label{subsec:properties} Define the complex vector field \begin{eqnarray} Z_\mu &\equiv& X_\mu+\mathrm{i} Y_\mu ~\equiv~ 2\,V_\mu^{12} +\mathrm{i}( V_\mu^{11}-V_\mu^{22}) \;, \label{VZ} \end{eqnarray} with norm given by (\ref{alg2}) as \begin{eqnarray} \label{norms} Z_\mu \bar Z^\mu = -2V_\mu V^\mu = -8 f^2 \;. \end{eqnarray} Then equations (\ref{alg}) and (\ref{alg2}) translate into \begin{eqnarray} V_{[\mu}Z_{\nu]} &=& \mathrm{i} \varepsilon_{\mu\nu\rho}f\,Z^\rho \;,\qquad Z_{[\mu} \bar{Z}_{\nu]} = -2\mathrm{i}\varepsilon_{\mu\nu\rho}f\,V^\rho \;. \end{eqnarray} These equations summarize all the algebraic equations derived in section~\ref{subsec:algeq}. They encode the fact that the mutually orthogonal vector fields $V_\mu$ and $Z_\mu$ form a canonical tetrad for the three-dimensional spacetime, which we shall employ in the following. We can choose the three-dimensional vielbein as \begin{eqnarray} E_\mu{}^a &=& \frac1{2f}\, (V_\mu,X_\mu,Y_\mu) \;, \qquad E_a{}^\mu ~=~ \frac1{2f}\, (V^\mu,-X^\mu,-Y^\mu)^T \;, \label{EEfull} \end{eqnarray} consistently satisfying $E_\mu{}^a E_b{}^\mu=\delta_b^a$ and \begin{eqnarray} {\rm det}\,E_\mu{}^a ~=~\frac1{8f^3}\,e\varepsilon^{\mu\nu\rho} \, V_\mu X_\nu Y_\rho ~=~ \frac1{4f^2}\,e\, V_\mu V^\mu ~=~ e \;. \end{eqnarray} Let us now turn to the differential relations of section~\ref{subsec:diffeq}. We further denote by \begin{eqnarray} A^v\equiv A_1^{11}+A_1^{22}\;,\qquad A^z\equiv 2A_1^{12}+\mathrm{i} (A_1^{11}-A_1^{22}) \;, \label{someA} \end{eqnarray} certain components of the scalar tensor $A_1^{AB}$ in the basis (\ref{basis}). Then equation (\ref{df}) takes the form \begin{eqnarray} \partial_{\mu} f &=& \frac{\mathrm{i}g}4\, ( A^{\bar{z}} Z_\mu -A^z \bar{Z}_\mu ) \;, \label{dfbasis} \end{eqnarray} whereas (\ref{diff2}) induces \begin{eqnarray} \nabla_\mu\,V_\nu{} &=& g\,\varepsilon_{\mu\nu\rho}\, \left( A^v\,V^{\rho} +\ft12( A^z \, \bar Z^{\rho} +A^{\bar z} \, Z^{\rho}) \right) \;, \nonumber\\[1ex] \nabla_\mu Z_\nu &=& 2\mathrm{i}gf A^z\,g_{\mu\nu} +g \varepsilon_{\mu\nu\rho}\, \left( A^v\,Z^{\rho}+A^z\,V^{\rho} \right) -2\mathrm{i} \,{\cal Q}_\mu\,Z_\nu \;, \label{dVZ} \end{eqnarray} with the $SO(2)$ connection ${\cal Q}_\mu=\frac14 \Omega^{IJ} {\cal Q}^{IJ}_\mu$\,. Thereby, we have reformulated the first Killing spinor equation (\ref{KS1}). Let us note that equation (\ref{dfbasis}) follows as a consequence of (\ref{dVZ}) using that the norm of the Killing vector field is given as $V^\mu V_\mu = 4f^2$\,. Choosing the vielbein as (\ref{EEfull}), equations (\ref{dVZ}) precisely encode the spin connection $\omega_\mu{}^{ab}$ \begin{eqnarray} \omega_{\mu}{}^{01}+\mathrm{i}\, \omega_{\mu}{}^{02} &=& \frac{\mathrm{i}g}{2f}\,\left( A^z\, V_{\mu}+A^v\, Z_\mu \right) \,, \nonumber\\ \omega_{\mu}{}^{12}&=& \frac{g}{4f}\, \left(2A^v V_\mu +A^z\,Z_\mu+A^{\bar{z}}\,\bar{Z}_\mu \right) +2 {{\cal Q}}_\mu \,. \label{spinconnection} \end{eqnarray} Some more computation shows that with this explicit form of the spin connection, the Killing spinor equation (\ref{KS1}) can be explicitly integrated to \begin{eqnarray} \epsilon^A &=& \sqrt{f}\;\{\hat\epsilon,\mathrm{i}\gamma_0\, \hat\epsilon,0,0,0,0,0,0\} \;, \end{eqnarray} where $\hat\epsilon$ is a constant spinor satisfying the projection\footnote{ For the flat gamma matrices (with tangent space indices) we use the explicit representation $\gamma_0=\sigma_2, \gamma_1=-\mathrm{i}\sigma_3, \gamma_2=-\mathrm{i}\sigma_1$ in terms of Pauli matrices, so that $\gamma^{012}=\gamma_{012}=-\mathrm{i}$.} \begin{eqnarray} (\gamma_0-\gamma_2)\,\hat\epsilon &=& 0\;. \end{eqnarray} We have thus reconstructed the Killing spinor from the bilinear tensors. Note that it also satisfies $V^\mu \partial_\mu \epsilon^A =0$\,. \subsection{The scalar current} Equations (\ref{diff1}) and (\ref{diff2}) moreover determine the mixed components of the $SO(8)$ connection ${\cal Q}^{AB}_\mu$ in the basis (\ref{basis}) according to \begin{eqnarray} {\cal Q}_\mu^{a\tilde{a}} &=& -\frac{g}{f}\,A_1^{\tilde{a}c}\,V_{\mu \, bc}\,\epsilon^{ab} \;, \label{Qmixed0} \end{eqnarray} leaving unconstrained the others, which again manifests the breaking of $SO(8)$ down to $SO(2)\times SO(6)$. Explicitly, \begin{eqnarray} \hat{\cal Q}_\mu^{1\tilde{a}}+\mathrm{i} \hat{\cal Q}_\mu^{2\tilde{a}}&=& -\frac{g}{2f}\, (A_1^{1\tilde{a}}-\mathrm{i}A_1^{2\tilde{a}})\,\bar Z_\mu \;, \label{Qmixed} \end{eqnarray} where according to (\ref{Ahat}) we have defined $\hat{\cal Q}_\mu$ by splitting off its contribution in $V_\mu$\, (it is a non-trivial consistency check that this contribution as induced by (\ref{Qmixed0}) precisely coincides with the assignment of (\ref{VQ})). Accordingly, with (\ref{VQ}) the remaining components of ${\cal Q}_\mu^{IJ}$ define the $SO(2)\times SO(6)$ connection \begin{eqnarray} {\cal Q}_{\mu} &=& - \frac{g}{2f} A^v\,V_\mu + \hat{\cal Q}_{\mu} \;, \nonumber\\ {\cal Q}^{\tilde{a}\tilde{b}}_{\mu} &=& -\frac{g}{128nf} \Gamma^{IJ}_{[\tilde{a}\tilde{b}}\Gamma^{KL}_{12]} \Gamma^{IJKL}_{\dot{A}\dot{B}}\,A_3^{\dot{A}r\dot{B}r}\,V_\mu + \hat{\cal Q}^{\tilde{a}\tilde{b}}_{\mu} \;. \label{QQ} \end{eqnarray} Let us now consider the remaining part ${\cal P}_\mu^{Ir}$ of the scalar current. In the basis (\ref{VZ}) of vector fields and using (\ref{VP}) we expand ${\cal P}_\mu^{Ir}$ as \begin{eqnarray} {\cal P}_\mu^{Ir}&=& -\,\frac{g}{2f}\,\epsilon^{ab}\,\Gamma^I_{a\dot{A}}\,A_2^{b\dot{A} r}\,V_\mu ~+~\hat{\cal P}_\mu^{Ir} \nonumber\\[.5ex] &=& -\,\frac{g}{2f}\,\epsilon^{ab}\,\Gamma^I_{a\dot{A}}\,A_2^{b\dot{A} r}\,V_\mu ~+~\frac1{f}\left({\cal P}^{Ir}\,{Z}_\mu +\overline{{\cal P}^{Ir}}\,\bar{Z}_\mu \right) \;, \label{ansatzP} \end{eqnarray} with complex components ${\cal P}^{Ir}$, to be determined. Plugging (\ref{ansatzP}) into (\ref{dilatino1}) after some calculation (which makes use of the properties (\ref{alg2}) of the vector fields) leads to the compact eigenvector equation \begin{eqnarray} \left(\delta^{IJ}-\mathrm{i}\,\Omega^{IJ} \right)\left({\cal P}^{Jr} +\ft{1}{4}\mathrm{i}g\,B_+^{Jr} \right) &=& 0 \;, \label{OmegaP} \end{eqnarray} where we have defined the following combinations \begin{eqnarray} B^{Ir}_\pm&=& \mp\frac{\mathrm{i}}2(\Gamma^I_{1\dot{A}}\pm\mathrm{i} \Gamma^I_{2\dot{A}})\, (A_2^{1\dot{A}r}\pm\mathrm{i}A_2^{2\dot{A}r}) \;, \end{eqnarray} of scalar components of the tensor $A_2^{A\dot{A}r}$ in the basis (\ref{basis}). It is straightforward to verify, that these $B^{Ir}_\pm$ are eigenvectors of $\Omega^{IJ}$ from (\ref{Omegas}) according to $\Omega^{IJ} B^{Jr}_\pm = \pm\mathrm{i} B^{Ir}_\pm$\,. The general solution to (\ref{OmegaP}) is thus given by setting \begin{eqnarray} {\cal P}^{Ir} &=& {\cal P}^{Ir}_- - \frac{\mathrm{i}g}{4}\,B_+^{Ir} \;, \end{eqnarray} where ${\cal P}^{Ir}_-$ is an arbitrary eigenvector of $\Omega^{IJ}$ with eigenvalue $-\mathrm{i}$\,. The full solution (\ref{ansatzP}) then takes the form \begin{eqnarray} \hat{{\cal P}}_\mu^{Ir}&=& \frac1{f}\,({\cal P}_+^{Ir}\,\bar{Z}_\mu+{\cal P}_-^{Ir}\,{Z}_\mu) -\frac{\mathrm{i}g}{4f}\,(B_+^{Ir} Z_\mu-B_-^{Ir}\bar{Z}_\mu) \;, \label{solutionP} \end{eqnarray} where ${\cal P}_+^{Ir}\equiv\overline{{\cal P}_-^{Ir}}$ is an eigenvector of $\Omega^{IJ}$ with eigenvalue $+\mathrm{i}$. Some further calculation shows that (\ref{solutionP}) also identically solves equation (\ref{dilatino2}). The full content of the Killing spinor equations (\ref{KS1}), (\ref{KS2}) is thus contained in the form of the spin connection (\ref{spinconnection}) and the solution (\ref{ansatzP}), (\ref{solutionP}) for the scalar current. Of course the solution for the scalar current (\ref{solutionP}) is consistent only if in addition this current satisfies the integrability conditions (\ref{intP})--(\ref{intQ2}). This severely constrains the choice of the components~${\cal P}^{Ir}_\pm$. For the gauged theories $g\not=0$ these equations involve the non-abelian field strength ${\cal F}_{\mu\nu}{}^{\cal KL}$ which in turn is related to the scalar current itself by means of the duality equations (\ref{duality}). The resulting structure thus is rather intricate and will be treated in a separate publication~\cite{gauged_progress}. In this paper, we will in section~\ref{sec:ungauged} explicitly work out the integrability conditions in the ungauged case $g=0$. \subsection{Vector fields and duality} \label{subsec:vectors} The vector fields appearing in three-dimensional supergravity are not propagating but come with a Chern-Simons coupling which relates their field strength to the scalar fields by means of the duality equation (\ref{duality}) \begin{eqnarray} \theta_{\cal KLMN}\, {\cal F}_{\mu\nu}{}^{\cal MN} &=& \theta_{\cal KLMN} \,\varepsilon_{\mu\nu\rho} {\cal V}^{\cal MN}{}_{Ir}\,{\cal P}^{\rho\,Ir}\;. \label{duality0} \end{eqnarray} For the supersymmetric solutions it turned out to be natural to explicitly split off the vector field components in the direction of the Killing vector field according to (\ref{Ahat}). Accordingly, the field strength ${\cal F}_{\mu\nu}{}^{\cal KL}$ is decomposed into \begin{eqnarray} {\cal F}_{\mu\nu}{}^{\cal KL} &=& -\hat{\cal D}_{[\mu}\left(\frac1{2f}{\cal V}^{\cal KL}{}_{MN}\Omega^{MN} V_{\nu]}\right) + \hat{\cal F}_{\mu\nu}{}^{\cal KL} \;, \end{eqnarray} where $\hat{\cal D}$ refers to the covariant derivative including only the vector field $\hat{A}_\mu{}^{\cal KL}$ and its non-abelian field strength $\hat{\cal F}_{\mu\nu}{}^{\cal KL}$. The first term on the r.h.s.\ can be evaluated upon using the relations (\ref{dVZ}), (\ref{dV}) and the explicit form of (\ref{solutionP}). Without going into the details of the derivation (which also require the structure of the quadratic constraints (\ref{quadcon}) on the embedding tensor and will be discussed in a separate publication), we note that as a final result the duality equation (\ref{duality0}) takes the form (for non-vanishing $f$) \begin{eqnarray} \theta_{\cal KLMN}\, \hat{\cal F}_{\mu\nu}{}^{\cal MN} &=& \frac{g}{2f}\, \theta_{\cal KLMN} \,\varepsilon_{\mu\nu\rho} \left( {\cal V}^{\cal MN}{}_{IJ}\Omega^{IJ}A^v - {\cal V}^{\cal MN}{}_{Ir}\Omega^{IJ}B^v_{Jr} \right) V^\rho\;,\qquad\quad \label{dualityhat} \end{eqnarray} with $A^v$ from (\ref{someA}) and $B^v_{Ir}\equiv \Gamma^I_{1\dot{A}}A_2^{1\dot{A}r}+\Gamma^I_{2\dot{A}}A_2^{2\dot{A}r}$. This equation shows a few remarkable properties. Note first, that all contributions proportional to ${\cal P}_\pm^{Ir}$ have dropped out from the original equation (\ref{duality0}). As a result, the r.h.s.\ of (\ref{dualityhat}) has no contributions of order $g^0$ which has important consequences for the ungauged theory, as we will discuss below. Second, the r.h.s.\ of (\ref{dualityhat}) is entirely proportional to $\varepsilon_{\mu\nu\rho}V^\rho$. This implies in particular, that \begin{eqnarray} V^\mu \hat{\cal F}_{\mu\nu}{}^{\cal KL} &=& 0\;, \end{eqnarray} which finally shows that also the vector fields $\hat{A}_\mu{}^{\cal KL}$ are constant in the direction of the Killing vector field. While the full structure of (\ref{dualityhat}) will be analyzed elsewhere, let us discuss here its consequences for the ungauged theory. In the limit $g=0$, all vector fields consistently decouple from the Lagrangian (\ref{L}). Still the ungauged theory hosts a remnant of the duality equation which is given by the unprojected version of equation (\ref{duality0}): \begin{eqnarray} {F}_{\mu\nu}{}^{\cal KL} &=& \varepsilon_{\mu\nu\rho}\,{\cal V}^{\cal KL}{}_{Ir}\, P^{\rho\,Ir}\;, \label{duality_ab} \end{eqnarray} with abelian field strength ${F}_{\mu\nu}{}^{\cal KL}$. Even though the vector fields are no longer part of the action, they can be defined on-shell by means of this equation. In particular, the Bianchi identities for the field strength ${F}_{\mu\nu}{}^{\cal KL}$ is precisely equivalent to the scalar field equations of motion (\ref{eqm:scalars}) at $g=0$: the r.h.s.\ of (\ref{duality_ab}) is the conserved $SO(8,n)$ Noether current of the ungauged theory. This represents the standard duality between vectors and scalar fields in three dimensions. From equation (\ref{dualityhat}) we see that in this case \begin{eqnarray} \hat{F}_{\mu\nu}{}^{\cal KL} &=& 0 \;, \end{eqnarray} i.e.\ the vector fields $\hat{A}_\mu{}^{\cal KL}$ are locally flat. In other words, for supersymmetric solutions of the ungauged theory, using (\ref{Ahat}), the duality equation (\ref{duality_ab}) can be explicitly integrated to \begin{eqnarray} A_{\mu}{}^{\cal KL} &=& -\frac{1}{4f}\, {\cal V}^{\cal KL}{}_{MN}\,\Omega^{MN}\,V_\mu \;, \label{dualA} \end{eqnarray} (for non-vanishing $f$) which allows to express the dual vectors directly in terms of the scalar fields. This is a remarkable property of the supersymmetric solutions; in general, the dual vectors are nonlocal functions of the scalar fields. The relation (\ref{dualA}) is of particular importance when discussing a possible higher dimensional origin of the three-dimensional solutions. Most compactifications to three dimensions, e.g.\ the heterotic string on a seven-torus \cite{Sen:1994wr}, lead to a version of the three-dimensional theory which features propagating scalar and vector fields. It is only upon dualizing all vectors into scalars, that the $SO(8,n)$ symmetry of the theory becomes manifest and the action takes the compact form (\ref{L}). Equation (\ref{dualA}) thus gives an explicit formula for the original three-dimensional vectors which can then be lifted up to their higher-dimensional ancestors. We come back to this discussion in the conclusions. \subsection{Killing Spinor Identities} \label{sec:KSI} In this section we will apply the method of Killing spinor identities \cite{Kallosh:1993wx,Bellorin:2005hy} to determine which field equations are satisfied automatically once the Killing spinor equations are solved. As follows immediately from supersymmetry of an action $S$, if the Killing spinor equations are satisfied, the following relations hold \begin{eqnarray} \sum_{\rm b} \frac{\delta S}{\delta \phi_{\rm b}} \frac{\partial (\delta_\epsilon \phi_{\rm b})}{ \partial \phi_{\rm f}} &=& 0 \;, \label{KSI} \end{eqnarray} where $\phi_{\rm b}$ and $\phi_{\rm f}$ represent the bosonic and fermionic fields, respectively, of the theory. We denote the bosonic equations of motion of (\ref{L}) as \begin{eqnarray} {\cal E}^\mu{}_{\alpha} \equiv \frac{\delta S}{\delta e_\mu{}^\alpha}\;,\qquad {\cal E}^\mu{}_{\cal KL} \equiv \frac{\delta S}{\delta A_\mu{}^{\cal KL}} \;,\qquad {\cal E}_{Ir} \equiv \frac{\delta S}{\delta \Sigma^{Ir}} \;, \end{eqnarray} where the last derivative is taken with respect to a left invariant vector field $\Sigma^{Ir}$ along the coset manifold $SO(8,n)/(SO(8)\times SO(n))$. Equations (\ref{KSI}) thus amount to linear relations among these equations. Specifically, we find with the bosonic supersymmetry transformations given by~\cite{Marcus:1983hb,Nicolai:2001ac} \begin{equation} \begin{array}{rclrcl} {\cal S}^{-1}\delta_\epsilon {\cal S} &\!=\!& Y^{Ir}\;\overline{\epsilon}^A\,\Gamma^I_{A{\dot{A}}}\chi^{{\dot{A}} r}\;, &\quad\; \; \\[1ex] \delta_\epsilon e_\mu{}^\alpha &\!=\!& {\rm i} \overline{\epsilon}^A\,\gamma^\alpha\psi^A_\mu \;, & \; \\[1ex] \delta_\epsilon A_\mu{}^{\cal KL} &\!=\!& \multicolumn{4}{l}{-\ft12\VV{\cal KL}{IJ}\,\overline{\epsilon}^A\,\Gamma^{IJ}_{AB}\psi^B_\mu + {\rm i}\,\VV{\cal KL}{Ir}\,\overline{\epsilon}^A\,\Gamma^I_{A{\dot{A}}}\gamma_\mu\chi^{{\dot{A}} r} \;,} \end{array} \end{equation} that equations (\ref{KSI}) imply the relations \begin{eqnarray} 2 {\cal E}^\mu{}_{\alpha}\, \overline{\epsilon}^A\,\gamma^\alpha + {\rm i} {\cal E}^\mu{}_{\cal KL} \VV{\cal KL}{IJ}\,\overline{\epsilon}^B\,\Gamma^{IJ}_{BA} &=&0 \;, \nonumber\\[1ex] \label{ksi2} {\cal E}_{Ir}\;\overline{\epsilon}^A\,\Gamma^I_{A{\dot{A}}} + {\rm i} {\cal E}^\mu{}_{\cal KL} \,\VV{\cal KL}{Ir}\Gamma^I_{A{\dot{A}}}\,\overline{\epsilon}^A\,\gamma_\mu &=& 0\;, \end{eqnarray} among the bosonic equations of motion. After contracting these relations with all possible combinations of $\epsilon^B$ and $\gamma_\nu \epsilon^B$ and using the algebraic relations (\ref{alg}), (\ref{alg2}), one finds that for the timelike case (i.e.\ $f\not=0$) they imply the following relations \begin{eqnarray} {\cal E}^{\mu}{}_\alpha V^\alpha &=& f {\cal E}^\mu{}_{\cal KL} \VV{\cal KL}{IJ}\Omega^{IJ} \;,\qquad V^{[\mu}{\cal E}^{\nu]}{}_\alpha ~=~ 0\;, \nonumber\\[.5ex] 2f{\cal E}^\mu{}_{\cal KL} \VV{\cal KL}{Ir} &=& \Omega_{IJ} {\cal E}^{Jr}V^\mu \;. \label{KSItime} \end{eqnarray} In particular, we see that most of the Einstein equations ${\cal E}^{(\mu}{}_\alpha\, e^{\nu)\alpha}$ are indeed satisfied as a consequence of the Killing spinor equations, except for their component in direction $V^\mu V^\nu$ which is proportional to $V_\mu {\cal E}^\mu{}_{\cal KL} \VV{\cal KL}{IJ}\Omega^{IJ}$. Similarly, the scalar field equation is satisfied only up to a term proportional to $V_\mu {\cal E}^\mu{}_{\cal KL} \VV{\cal KL}{Ir}$. In order to ensure that a given solution of the Killing spinor equations solves all equations of motion, we thus have to impose separately the $V_\mu$ component of the duality equation (\ref{duality}). Specifically, this amounts to imposing equation (\ref{dualityhat}) that we have encountered in the previous section. For the ungauged theory, this equation is absent, i.e.\ for $f\not=0$ the full set of equations of motion is satisfied as a consequence of the Killing spinor equations. Note however, that the derivation of (\ref{KSI}) has made implicit use of the integrability relations (\ref{intP})--(\ref{intQ2}) of the scalar current, i.e.\ in all cases also these integrability relations will have to be imposed on the solution in order to ensure that it satisfies all equations of motion. \section{Timelike case: the ungauged theory} \label{sec:ungauged} From now on, we concentrate on the case of the ungauged theory (i.e.\ set $g=0$), leaving the analysis of the gauged theories for a separate publication~\cite{gauged_progress}. For $g=0$, the structure of the spin connection (\ref{spinconnection}) simplifies drastically and it has only a single non-vanishing component \begin{eqnarray} \omega_\mu{}^{12} &=& 2 Q_\mu\;. \label{spinc_ungauged} \end{eqnarray} The only non-vanishing component of its curvature thus is $R_{\mu\nu}{}^{12}= 2Q_{\mu\nu}$, such that the Ricci tensor and Ricci scalar of the three-dimensional spacetime are given by \begin{eqnarray} R_{\mu\nu} &=& -\frac1{8f^2}\,R\,Z_{(\mu} \bar{Z}_{\nu)} \;, \qquad \quad R = \frac1{f}\, \varepsilon_{\mu\nu\rho}\,Q^{\mu\nu}\,V^\rho \;. \label{RbyR} \end{eqnarray} Let us note that this equation gives rise to the interesting factorization structure \begin{eqnarray} R_{\mu\nu}+2\mathrm{i}\,Q_{\mu\nu} &=& -\frac1{8f^2}\,R\,Z_{\mu} \bar{Z}_{\nu} \;. \end{eqnarray} The general solution for the scalar current~(\ref{solutionP}) in this case takes the short form \begin{eqnarray} P_\mu^{Ir}&=& \frac{1}{f}\, \left( P_+^{Ir}\,\bar{Z}_\mu+ P_-^{Ir}\,{Z}_\mu \right) \;, \label{PinZ} \end{eqnarray} where as before the components $P_+^{Ir}=\overline{P_-^{Ir}}$ are arbitrary eigenvectors of $\Omega^{IJ}$ from (\ref{Omegas}) corresponding to eigenvalues $\pm\mathrm{i}$, respectively. \subsection{Special coordinates} Since $V_\mu$ represents a timelike Killing vector field of the solution, we can choose coordinates such that $V^\mu\partial_\mu = \partial/\partial t$ and no matter field or metric component depends on the time variable~$t$. For the spatial part of the three-dimensional spacetime, we use coordinates $x^i$, $i=1, 2$\,. Normalization then implies that $V_\mu=(4f^2,\rho_1,\rho_2)$ with functions $\rho_i$. According to (\ref{dfbasis}), $f$ is a constant and for simplification in the following we rescale $t$ such that $f=\frac12$. From (\ref{dVZ}) we find that $\partial_{[\mu}V_{\nu]}=0$, hence $\rho_i=\partial_i\rho$ for a function $\rho$ which can be absorbed by redefinition (translation) of $t$. Moreover, the remaining $2\times2$ block in the vielbein (\ref{EEfull}) can be brought into conformal gauge. Thus in these special coordinates the three-dimensional vielbein reduces to \begin{eqnarray} E_\mu{}^a ~=~ \left( \begin{array}{ccc} 1&0&0\\ 0&e^\sigma&0\\ 0&0&e^\sigma \end{array} \right) \;, \end{eqnarray} with the conformal factor $\sigma$\,. Its spin connection is given by $\omega_\mu{}^{12} = (0,-\partial_2 \sigma, \partial_1\sigma)$\,. Accordingly, we find for the Ricci tensor \begin{eqnarray} R_{11} = R_{22}= -\Box\sigma\qquad \Longrightarrow \qquad R=2e^{-2\sigma} \Box\sigma \;, \label{RRR} \end{eqnarray} with the two-dimensional flat Laplacian $\Box=\partial^i \partial_i$\,. From (\ref{spinc_ungauged}) the $SO(2)$ connection is given by $Q_\mu = \frac12 \omega_\mu{}^{12}$\,, with curvature $Q_{ij}=\frac12 \epsilon_{ij} \Box\sigma$\,. This shows that if the three-dimensional spacetime is not flat, the $SO(2)$ connection is necessarily non-vanishing. In complex coordinates $z=x^1+\mathrm{i}x^2$ and with (\ref{PinZ}), the full scalar current takes the form \begin{eqnarray} \label{QPcomplex} Q_z=-\frac{\mathrm{i}}2\,\partial_z\sigma \;,\qquad P_{z}^{Ir} ~=~ 2e^\sigma\,P_-^{Ir} \;. \end{eqnarray} Projecting for this current the integrability relation (\ref{intQ1}) onto its $SO(2)$ part gives rise to the equation \begin{eqnarray} \partial_z \partial_{\bar{z}}\,\sigma &=& -2\,e^{2\sigma} P_+^{Ir}P_-^{Ir} \;. \label{Boxsigma} \end{eqnarray} Putting this together with (\ref{RRR}) shows that $R=P^\mu P_\mu$ in precise agreement with the Einstein equations (\ref{eqm:Einstein}), as expected from the general analysis of section~\ref{sec:KSI}. In order to completely determine the solution it remains to solve the remaining part of the integrability equations (\ref{intP})--(\ref{intQ2}). Notably, the first one takes the remarkably simple form \begin{eqnarray} D_{\bar{z}} \left(e^\sigma\,P_-^{Ir}\right) &=& 0\;. \label{DPhol} \end{eqnarray} This is due to the fact that $P_+^{Ir}$ and $P_-^{Ir}$ are eigenvectors corresponding to different eigenvalues of $\Omega^{IJ}$ such that in this coordinate basis the two terms of (\ref{intP}) must vanish separately. Note however, that the derivative in (\ref{DPhol}) is covariant in that it carries the full composite connections $Q^{IJ}_\mu$ and $Q^{rs}_\mu$. In general, it is thus not sufficient to choose $e^\sigma P_-^{Ir}$ to be a holomorphic function of $z$. The remaining two integrability equations (\ref{intQ1}), (\ref{intQ2}) take the form \begin{eqnarray} Q^{IJ}_{z\bar{z}} &=& 8 e^{2\sigma} P_{+}^{r[I}P_{-}^{J]r} \;, \qquad Q^{rs}_{z\bar{z}} ~=~ 8 e^{2\sigma} P_{+}^{I[r}P_{-}^{s]I} \;. \label{intQQ0} \end{eqnarray} It is straightforward to verify that any solution to (\ref{DPhol}) also satisfies the scalar equations of motion (\ref{eqm:scalars}) in accordance with the general relations derived in section~\ref{sec:KSI}. We have thus reduced the construction of supersymmetric solutions to finding common solutions to the two-dimensional equations (\ref{Boxsigma})--(\ref{intQQ0}). In the following, we will exploit the covariantly holomorphic structure of (\ref{DPhol}) to study some explicit examples. \subsection{Explicit solutions} In this section, we will illustrate the structure of the covariantly holomorphic equation (\ref{DPhol}) by constructing in detail a few explicit solutions and showing that indeed they satisfy the full set of equations of motion. Let us denote by ${\mathbb P}{}_+^{Ii}$ the four (normalized and orthogonal) eigenvectors of $\Omega^{IJ}$ from (\ref{Omegas}) with eigenvalues $+\mathrm{i}$, such that \begin{eqnarray} ({\mathbb P}_+^\dagger {\mathbb P}_+)^{ij} &=& \delta^{ij}\;,\nonumber\\ ({\mathbb P}_+ {\mathbb P}_+^\dagger)^{IJ} &=& \ft12\, (\delta-\mathrm{i} \Omega)^{IJ} \;. \label{PPP} \end{eqnarray} Expanding $P^{Ir}_{\pm}$ in terms of these eigenvectors we write \begin{eqnarray} P^{Ir}_- &=& \mathbb{P}^{Ii}_- \, \Sigma{}^{ir} \;, \label{generalP} \end{eqnarray} with coefficients $\Sigma{}^{ir}$. We furthermore define the hermitean matrix ${H}^{rs}=(\Sigma^\dagger \Sigma\,)^{rs}$ and the antisymmetric hermitean matrices \begin{eqnarray} M^{IJ} &=& \ft12( P^{Ir}_{+} P^{Jr}_- - P^{Jr}_{+} P^{Ir}_-) ~=~ \mathrm{i}\, \Im(\mathbb{P}_+ \overline{\Sigma} \Sigma^T\,\mathbb{P}_+^\dagger)^{IJ} \;, \nonumber\\ N^{rs} &=& \ft12( P^{Ir}_{+} P^{Is}_- - P^{Is}_{+} P^{Ir}_-) ~=~ \mathrm{i}\, (\Im {H})^{rs}~=~ {H}^{[rs]} \;. \end{eqnarray} It is easy to check that $[\Omega, M]=0$\,, i.e.\ $M\in \mathfrak{so}(2)\times \mathfrak{so}(6)$\,, and $\Omega^{IJ}M^{IJ}=-\mathrm{i} {\rm Tr}\,{H}$, which implies that $\widehat{M}^{IJ}\equiv M^{IJ}+\ft{\mathrm{i}}8\Omega^{IJ} {\rm Tr}\,{H} \in \mathfrak{so}(6)$. Note that for $n=4$, the matrix $\widehat{M}^{IJ}$ vanishes. Some further calculation yields \begin{eqnarray} M^{IJ} P^{J r}_+ &=& \ft12 {H}^{rs} P_+^{Is} \;, \qquad \widehat{M}^{IJ} P^{J r}_+ ~=~ \ft12 \left({H}^{rs}-\ft14\delta^{rs}\, {\rm Tr}\,{H}\right) P_+^{Is} \;. \end{eqnarray} Integrability (\ref{intQQ0}) (projected onto its $SO(2)$ part) together with (\ref{Boxsigma}) yields the differential equation: \begin{eqnarray} \mathrm{i}\partial_z \partial_{\bar{z}} \sigma &=& \partial_z Q_{\bar{z}}- \partial_{\bar{z}} Q_z ~=~ -2\mathrm{i} e^{2\sigma}\, {\rm Tr}\,{H} \;, \label{intSO2} \end{eqnarray} which precisely coincides with the Einstein equation~(\ref{eqm:Einstein}). In the following we will study an explicit ansatz for the scalar current. We choose the coefficients in (\ref{generalP}) such that \begin{eqnarray} P^{Ir}_- &=& \mathbb{P}^{Ii}_- \, {{\cal U}}{}^{ir}\, {\zeta(z,\bar{z})} \;, \label{Pans} \end{eqnarray} with a yet unconstrained complex function $\zeta(z,\bar{z})$ and a constant matrix ${\cal U}$, satisfying $({\cal U}^\dagger {\cal U})^{rs}=\delta^{rs}$. This requires the number of matter multiplets to satisfy $n\le 4$ and implies $H^{rs}=|\zeta|^2\, \delta^{rs}$\,. For the $SO(2)\times SO(6)$ connection $Q_\mu^{IJ}$, we make the ansatz \begin{eqnarray} Q_z^{IJ} &=& -\frac{\mathrm{i}}4\,\partial_z\sigma\, \Omega^{IJ} - \overline{g(z, \bar{z})} \: \widehat{M}^{IJ} \;, \label{Qans} \end{eqnarray} in accordance with (\ref{QPcomplex}), and set $Q_\mu^{rs}=0$. Then the second equation of (\ref{intQQ0}) is trivially satisfied, whereas the first equation in addition to (\ref{intSO2}) gives the nontrivial differential equation \begin{eqnarray} \Big\{ \Re[\partial_z ( g |\zeta|^2)]-4e^{2\sigma} |\zeta|^2 \Big\}\; \widehat{M}^{IJ} &=& 0 \;. \label{int1} \end{eqnarray} Finally, it remains to impose the integrability equation (\ref{DPhol}) which yields \begin{eqnarray} \partial_{\overline{z}} {\rm ln}\, \zeta +\ft54\partial_{\overline{z}} \sigma + \ft18(n-4)\, g \, |\zeta|^2 &=& 0 \;, \label{int2} \end{eqnarray} which (for $n\not=4$) can be solved for $g$. Plugging this solution back into (\ref{int1}) and using (\ref{intSO2}) yields a differential equation for $|\zeta|^2$ that has the general solution \begin{eqnarray} |\zeta|^2 &=& e^{-2(n+1)\sigma/n} \,|\chi(z)|^2 \;, \end{eqnarray} with an arbitrary holomorphic function $\chi(z)$. Without loss of generality, we can choose to set \begin{eqnarray} \zeta &=& e^{-(n+1)\sigma/n} \,\chi(z) \;, \label{zeta} \end{eqnarray} the undetermined phase of $\zeta$ precisely corresponds to an $SO(6)$ gauge transformation. For $n=4$, equation (\ref{int2}) directly induces (\ref{zeta}). Putting everything together, we find for the scalar current \begin{eqnarray} Q_z^{IJ} = -\Big\{ \frac{\mathrm{i}}4\, \Omega^{IJ} +\frac{2}{n|\zeta|^2} \widehat{M}^{IJ} \Big\}\; \partial_z \sigma \;, \qquad P^{Ir}_z = 2\,\mathbb{P}^{Ii}_- \, {\cal U}^{ir}\, e^{-\sigma/n} \,\chi(z) \;, \end{eqnarray} while $\sigma$ should satisfy the Liouville equation \begin{eqnarray} \partial_z \partial_{\bar{z}} \sigma &=& -2n\, e^{-2\sigma/n} \,|\chi(z)|^2 \;, \label{boxsigma} \end{eqnarray} that descends from (\ref{intSO2}). Its general solution can be given in terms of an arbitrary holomorphic function $\Phi(z)$ \begin{eqnarray} e^\sigma &=& \left(\frac{\sqrt{2} |\chi|\,(1-|\Phi|^2)}{|\Phi'|}\right)^n \;. \end{eqnarray} Defining $\chi(z)=\Psi(z) \Phi'(z)$, the scalar current and the metric are thus given by \begin{eqnarray} P^{Ir}_z &=& \frac{\sqrt{2}\, e^{\mathrm{i}\, {\rm arg}\,\Psi(z)}\,\Phi'}{1-|\Phi|^2} \,\,\mathbb{P}^{Ii}_- \, {\cal U}^{ir} \;, \nonumber\\ ds^2 &=& dt^2 - 16 \Big( |\Psi|(1-|\Phi|^2) \Big)^{2n} dz \overline{dz} \;, \label{metric00} \end{eqnarray} where $1\le n\le 4$, which summarizes our supersymmetric solution of the $N=8$ theory. Its curvature is given by \begin{eqnarray} R &=& -2^{3-n} n\, |\Psi|^{-8}\, |\Phi'|^2\,\left(1-|\Phi|^2 \right)^{-2(n+1)} \;. \end{eqnarray} We can give the solution in more explicit form by explicitly integrating up the scalar current to the scalar matrix ${\cal S}$. For simplicity, we restrict to the case $n=4$ while for arbitrary $n$ this can be done in precise analogy. For $n=4$ the $SO(6)$ connection vanishes, and the full scalar current is given by \begin{eqnarray} J_z &\equiv& {\cal S}^{-1}\partial_z{\cal S} ~=~ -\frac{\mathrm{i}}8\, \Omega^{IJ}X^{IJ} \partial_z \sigma + \frac{\sqrt{2}\, e^{\mathrm{i}\, {\rm arg}\,\Psi(z)}\,\Phi'}{1-|\Phi|^2} \,\,\mathbb{P}^{Ir}_- \,Y^{Ir} \;, \label{JJ} \end{eqnarray} where we have furthermore chosen ${\cal U}^{ir}=\delta^{ir}$. The generators appearing in (\ref{JJ}) take the form of tensor products \begin{eqnarray} \ft12\Omega^{IJ}X^{IJ} = \left( \begin{array}{ccc} 0&1&0\\ -1&0&0\\ 0&0&0 \end{array} \right) \otimes I_4 \;, \quad \mathbb{P}^{Ir}_-\,Y^{Ir} = \frac{1}{\sqrt{2}} \left( \begin{array}{ccc} 0&0&1\\ 0&0& -\mathrm{i}\\ 1&-\mathrm{i}&0 \end{array} \right) \otimes I_4 \;, \end{eqnarray} such that $SO(8,4)$ splits into four copies of $SO(2,1)$. Some calculation shows that the current $J_z$ takes the matrix form \begin{eqnarray} J_z &=& {\bf J}_z + H^{-1}\partial_z H + H^{-1} {\bf J}_z H\;, \label{Jnew}\\[2ex] &&\qquad\qquad \mbox{with}\quad H={\rm exp}\Big\{ \frac{\mathrm{i}}{4}\,{\rm log}(\overline{\Psi}/\Psi) \,\Omega^{IJ} X^{IJ} \Big\} \;,\nonumber\\[1ex] &&\qquad\qquad \mbox{and}\quad {\bf J}_z = \frac{\Phi'}{1-|\Phi|^2}\; \left( \begin{array}{ccc} 0&\mathrm{i} \overline{\Phi}&-1\\ -\mathrm{i} \overline{\Phi}&0& \mathrm{i}\\ -1&\mathrm{i}&0 \end{array} \right) \otimes I_4 \;. \nonumber \end{eqnarray} In particular, this current depends on $\Psi(z)$ only via an $SO(2)$ gauge transformation. Equation (\ref{Jnew}) can then be explicitly integrated up to yield the matrix form of ${\cal S}$ \begin{eqnarray} {\cal S}\,H^{-1} &=& \frac1{1-|\Phi|^2} \left( \begin{array}{ccc} 1+\ft12(\Phi^2+\overline{\Phi}{}^2) & 2\,\Re\Phi\, \Im\Phi & -2\, \Re\Phi\\ 2\,\Re\Phi\, \Im\Phi & 1-\ft12(\Phi^2+\overline{\Phi}{}^2) &-2\, \Im\Phi \\ -2\, \Re\Phi & -2\, \Im\Phi & 1+|\Phi|^2 \end{array} \right) \otimes I_4 \;.\qquad\quad \end{eqnarray} This form of the scalar fields together with the metric (\ref{metric00}) summarizes the supersymmetric solution in terms of two unconstrained holomorphic functions $\Phi(z)$ and $\Psi(z)$. One of them could be absorbed by a conformal redefinition of the two-dimensional coordinates. Finally, for this solution we may evaluate the explicit form of the dual vector fields~(\ref{dualA}) which leads to \begin{eqnarray} A_{t}{}^{IJ} &=& - {\cal S}^{I}{}_M {\cal S}^{J}{}_N\,\Omega^{MN} ~=~ -\frac{1+|\Phi|^2}{1-|\Phi|^2} \,\Omega^{IJ} \;, \nonumber\\ A_{t}{}^{Ir} &=& - {\cal S}^{I}{}_M {\cal S}^{r}{}_N\,\Omega^{MN} ~=~ \frac{\sqrt{2}\,\mathrm{i}}{1-|\Phi|^2} \left(\,\overline{\Phi}\, \mathbb{P}^{Ir}_+ - \Phi\, \mathbb{P}^{Ir}_-\right) \;, \label{dualA0} \end{eqnarray} up to a possible flat contribution. As discussed above, when lifting this solution to a higher-dimensional theory, e.g.~the heterotic string on a seven-torus, a number of ten-dimensional fields will be triggered by these three-dimensional vectors. For example, the Kaluza-Klein vector of the ten-dimensional metric descends to the three-dimensional vector field components $A_\mu{}^{I1}$, choosing $r=1$ in the second line of (\ref{dualA0}), in accordance with the breaking of $SO(8,n)$ to $SO(7,n-1)$ manifest after reduction. The components (\ref{dualA0}) thus become part of the ten-dimensional metric. This concludes our discussion of explicit examples. With a more sophisticated ansatz for the scalar current replacing (\ref{Pans}) and (\ref{Qans}), the construction may be generalized to produce more complicated solutions of the Killing spinor equations. \section{The null case} \label{sec:null} In this last section we consider the case $f=0$, when $F^{AB}=0$ and the Killing vector $V^\mu$ is a null vector. From the differential relations (\ref{diff1}), (\ref{diff2}) we obtain in this case \begin{eqnarray} {\cal Q}_{\mu}^{1\tilde{a}}=0= A_1^{1\tilde{a}} \;, \qquad D_\mu V_\nu{} = 2g\varepsilon_{\mu\nu\rho} A_1^{11} V^\rho \;. \label{DVnull} \end{eqnarray} Choosing coordinates such that the Killing vector field is given by $V^\mu\partial_\mu = \frac{\partial}{\partial v}$, the metric can then be cast into the general form \begin{eqnarray} ds^2 &=& F du^2 - H^2 dx^2 + 2 G du\,dv \;, \label{nullmetric} \end{eqnarray} with functions $F$, $H$, and $G$ which depend on the coordinates $x$ and $u$ only. Upon computing its Christoffel symbols, the second equation of (\ref{DVnull}) translates into \begin{eqnarray} \partial_x G &=& 4gA_1^{11} HG \;, \end{eqnarray} which relates the functions $G$ and $H$. In particular, for $g=0$, we may choose coordinates such that $G=1$\,. Further fixing of coordinates then allows to also put $H=1$. In the following, we specify to the ungauged theory. The dilatino equation (\ref{dilatino2}) shows that in this case \begin{eqnarray} P^{Ir}_\mu &=& P^{Ir}\, V_\mu \;, \label{Pnull} \end{eqnarray} with yet unspecified components $P^{Ir}$. Thus in particular $P^{Ir}_{[\mu} P^{Js}_{\nu]}=0$ and the integrability equations (\ref{intQ1}), (\ref{intQ2}) show that $Q_{\mu}^{IJ}$ and $Q_\mu^{rs}$ are flat connections. The remaining integrability condition $D^{\vphantom{I}}_{[\mu}P^{Ir}_{\nu]}=\partial^{\vphantom{I}}_{[\mu}P^{Ir}_{\nu]}=0$ thus states that $P^{Ir}$ in (\ref{Pnull}) is a function of $u$ only, i.e. \begin{eqnarray} P^{Ir}_\mu &=& P^{Ir}(u) \,V_\mu \;. \label{Pu} \end{eqnarray} With (\ref{Pu}) we have solved all the algebraic and differential relations that follow from the Killing spinor equations, as well as the scalar integrability conditions. In the null case however, this is not sufficient to imply the full set of equations of motion, as may be confirmed directly from (\ref{ksi2}): the $V_\mu V_\nu$ component of the Einstein equations remains to be imposed separately. With (\ref{Pu}), the Einstein equations (\ref{eqm:Einstein}) take the form \begin{eqnarray} R_{\mu\nu} &=& P^{Ir}_{\mu} P^{Ir}_{\nu} \;, \end{eqnarray} which in particular implies that the Ricci scalar vanishes $R = P^{Ir}_{\mu} P^{\mu\,Ir}= 0$. Using the explicit metric (\ref{nullmetric}) with $G=H=1$, the Einstein equations yield \begin{eqnarray} \partial^2_x F &=& 2 P^{Ir}P^{Ir} \;, \end{eqnarray} thus $F(u,x)=x^2 P^{Ir}(u)P^{Ir}(u) + x R(u) + T(u)$, with arbitrary functions $R(u)$, $T(u)$. The latter may be absorbed by a further redefinition of coordinates. Summarizing, for $g=0$ we find as the most general supersymmetric null solution the pp-wave \begin{eqnarray} P^{Ir}_\mu &=& P^{Ir}(u)\, V_\mu \;,\nonumber\\[.5ex] ds^2 &=& \left(x^2 P^{Ir}(u)P^{Ir}(u) + x\, R(u)\right)\, du^2 - dx^2 + 2 du\,dv \;. \label{Pgnull} \end{eqnarray} The scalar current can be integrated up to a scalar matrix ${\cal S}(u)$ depending on $u$ only. In turn, any such matrix gives rise to a current of the form (\ref{Pgnull}). As in the timelike case, we may dualize some of the scalar fields back into vectors by means of (\ref{duality_ab}). In the null case and with the above metric, this equation takes the explicit form \begin{eqnarray} {F}_{xu}{}^{\cal MN} &=& 2{\cal S}^{[{\cal M}}{}_{K}(u){\cal S}^{{\cal N}]}{}_{r}(u)\,{P}^{Kr}(u)\;, \label{dual_null} \end{eqnarray} and can be integrated in $u$ to obtain the gauge field with a single non-vanishing component $A_x{}^{\cal MN}(u)$\,. Finally, we can explicitly solve the Killing spinor equations (\ref{KS1}), to reconstruct the Killing spinor \begin{eqnarray} \epsilon &=& \left( x^2 P^{Ir}(u)P^{Ir}(u) + x\, R(u)\right)^{1/4}\, \hat\epsilon \;, \end{eqnarray} where $\hat\epsilon$ is a constant spinor satisfying $\gamma^u\,\hat\epsilon=0$\,. \section{Conclusions} In this paper we have taken the first steps to constructing and classifying the supersymmetric solutions of half-maximal matter-coupled three-dimensional supergravity. We have translated the Killing spinor equations into a set of algebraic and differential relations among the bilinear tensors built from the Killing spinor. This allows to express the spacetime metric and the scalar current in terms of these tensors. For the ungauged theory, we have reduced the integrability conditions for the current to the covariant holomorphicity condition (\ref{DPhol}) and constructed particular solutions by choosing appropriate ansaetze. For the case of a null Killing vector we have given in section~6 the most general solution of the ungauged theory. An interesting aspect for the solutions found in this model is their possible higher-dimensional origin. For this it is important to recall that the dimensional reduction to three dimensions generically leads to a theory with scalar and vector fields of which the latter have to be dualized into scalars in order to bring the action into the form (\ref{L}). For example, reduction of the heterotic string on a seven-torus leads to a theory with global symmetry $SO(7,23)$ and vector fields transforming in the vector representation. Only after dualizing the 30 vector fields into scalars by means of (\ref{duality_ab}) the full symmetry $SO(8,24)$ is manifest, and the scalars parametrize the coset space $SO(8,24)/(SO(8)\times SO(24))$. In particular, before dualization only an $SO(7)$ subgroup of the full $SO(8)$ $R$-symmetry group is manifest under which the fundamental representations branch as \begin{eqnarray} {\bf 8}_s \rightarrow 8\;,\quad {\bf 8}_v \rightarrow 7+1\;,\quad {\bf 8}_c \rightarrow 8 \;. \label{branchingR} \end{eqnarray} In order to lift the above constructed solutions back to higher dimensions, first a number of scalars will have to be dualized back into the corresponding vector fields. For this, we have derived the explicit formula (\ref{dualA}) for the vector fields (or (\ref{dual_null}) in the null case). On the other hand, we have seen in the above construction, that the supersymmetric solutions of (\ref{L}) are organized by particular subgroups of the $R$-symmetry group, namely $SO(2)\times SO(6)$ and $SO(7)$ for the solutions with timelike and null Killing vector, respectively. These will be broken upon singling out particular scalars according to (\ref{branchingR}). Comparing the branchings (\ref{branching}) and (\ref{branching_null}) to (\ref{branchingR}) allows to identify the common subgroups and shows that the dualization of 30 scalars back into the original vector fields, leaves an underlying manifest $U(3)$ and $G_2$ symmetry for the solutions with timelike and null Killing vector, respectively. This group structure will have to be studied in more detail in order to systematically address the higher-dimensional origin in this context. \medskip What we have presented in this paper is the first systematic approach using the bilinear tensor analysis to the construction of supersymmetric solutions in three-dimensio\-nal supergravities. It naturally suggests a number of further research directions and generalizations. First of all, while in this paper we have restricted the explicit construction of solutions to the ungauged theory, we have shown that large parts of the structure also find their analogue in the full gauged theory. A more detailed analysis of the gauged theory, in which the non-abelian duality between vector and scalar fields plays a key role will be presented elsewhere~\cite{gauged_progress}. The three-dimensional case provides an instructive scenario for the interplay of the coset space geometry of the scalar target space with the structure of the Killing spinor equations. These structures appear in a more compact form than in higher-dimensional theories, due to the fact that in three dimensions all dynamical degrees of freedom are accommodated in the scalar sector and higher rank $p$-forms are absent. Yet, a thorough understanding of the three-dimensional case and in particular of the gauged theory will be of importance for the study of the half-maximal supergravity theories in higher dimensions coupled to $n$ vector multiplets whose scalar fields form similar coset spaces. So far, the systematic study of supersymmetric solutions in matter coupled theories has essentially been restricted to the ungauged quarter-maximal theories, where in four dimensions the scalar target spaces are described by special K\"ahler and quaternionic K\"ahler geometries, see~\cite{Meessen:2006tu,Huebscher:2006mr,Cacciatori:2008ek,Klemm:2009uw}. The gaugings of the half-maximal theories~\cite{Schon:2006kz} on the other hand are organized by symmetry groups similar to the ones studied here, such that the solutions of the Killing spinor equations will exhibit similar structures in their scalar sectors, see~$D=4$ \cite{Ellmer:dipl} for some initial discussion. An ultimate goal would be the extension of the present analysis to the maximal (gauged and ungauged) supergravities in the various dimensions, whose scalar target space geometries are given by exceptional coset space sigma models. Let us finally note that upon taking a proper flat-space limit~\cite{Bergshoeff:2008ix,Bergshoeff:2008bh}, three-dimensional $N=8$ supergravity reduces to the distinct superconformal BLG model of~\cite{Bagger:2007jr,Gustavsson:2007vu}. It would be interesting to study if techniques similar to the ones presented here can be applied to classify the BPS solutions of the BLG theory, in particular, this should relate to the structures found in~\cite{Jeon:2008bx,Jeon:2008zj}. \bigskip \bigskip \bigskip \subsection*{Acknowledgements} We would like to thank the ENS de Lyon, Bo\u{g}azi\c{c}i University and the IMBM of Istanbul for hospitality during the course of this work. A part of the calculations performed in section~\ref{sec:general} has been facilitated by use of the computer algebra system Cadabra~\cite{Peeters:2006kp,Peeters:2007wn}. The work of N.S.D. and {\"O}.S. is partially supported by the Scientific and Technological Research Council of Turkey (T{\"U}B\.{I}TAK). The work of H.S. is supported in part by the Agence Nationale de la Recherche (ANR). \bigskip \bigskip \section*{Appendix} \begin{appendix} \section{$SO(8,n)$ algebra} \label{app:so} \subsection{Commutators} The non-compact algebra $\mathfrak{so}(8,n)={\rm Lie}\,SO(8,n)$ can be described in closed form in terms of generators $X^{\cal MN}=-X^{\cal NM}$ with commutators \begin{eqnarray} [X^{\cal MN},X^{\cal KL}] &=& 2\eta^{\cal M[K}X^{\cal L]N}-2\eta^{\cal N[K}X^{L]M} \;, \end{eqnarray} with the $SO(8,n)$ invariant diagonal metric $\eta_{\cal MN} = {\rm diag}(\underbrace{1,1,\dots,1}_{8\times},\underbrace{-1,-1,\dots,-1}_{n\times})$. \noindent Upon splitting the index ${\cal M}\rightarrow (I,r)$, such that $\eta^{IJ}=\delta^{IJ}, \eta^{rs}=-\delta^{rs}$, the algebra takes the form \begin{eqnarray} {}[X^{IJ},X^{KL}] &=& 2\delta^{I[K}X^{L]J}-2\delta^{J[K}X^{L]I} \;,\qquad {}[X^{IJ},Y^{Kr}] ~=~ -2\delta^{K[I}Y^{J]r} \;,\nonumber\\ {}[X^{pq},X^{rs}] &=& 2\delta^{p[r}X^{s]q}-2\delta^{q[r}X^{s]p} \;,\qquad {}[X^{rs},Y^{Ip}] ~=~ -2\delta^{p[r}Y^{Is]} \;,\nonumber\\ {}[Y^{Ir},Y^{Js}] &=& \delta^{IJ}X^{rs}+\delta^{rs}X^{IJ} \;. \label{alg8} \end{eqnarray} where for comparison to standard conventions we have furthermore redefined the generators $X^{rs}\rightarrow - X^{rs}$. Here, $X^{IJ}$ and $X^{rs}$ denote the compact generators of $SO(8)$ and $SO(n)$ respectively, the $8n$ noncompact generators are denoted by $Y^{Ir}$. \subsection{Coset space $SO(8,n)/ \left( SO(8)\times SO(n)\right)$} The scalar fields describing the $SO(8,n)/ \left( SO(8)\times SO(n)\right)$ coset space sigma model are parametrized by a group element ${\cal S}\in SO(8,n)$ evaluated in the fundamental representation, i.e.\ by an $(8+n)\times(8+n)$ matrix satisfying \begin{eqnarray} {\cal S}\,\eta\, {\cal S}^T &=& \eta \;. \end{eqnarray} The coset structure is expressed by the invariance of the theory under local transformations (\ref{coset}). In the ungauged theory, the scalar current $J_\mu\equiv {\cal S}^{-1}\partial_\mu{\cal S}$ may be decomposed as \begin{eqnarray} J_\mu &=& \ft12Q_\mu^{IJ} X^{IJ} + \ft12Q_\mu^{rs} X^{rs} + P_\mu^{Ir} Y^{Ir} \;, \label{current0} \end{eqnarray} in terms of the generators (\ref{alg8}) and the Lagrangian of the ungauged theory is given by the $SO(8)\times SO(n)$ invariant combination $\frac14P_\mu^{Ir}P^{\mu\,Ir}$. The integrability equations $2\partial_{[\mu}J_{\nu]}+[J_\mu,J_\nu]=0$ induced by the definition (\ref{current0}) translate into \begin{eqnarray} D_{[\mu} P_{\nu]} &=& 0\;, \label{integrabilityP}\\ Q^{IJ}_{\mu\nu} &\equiv& 2\partial_{[\mu}Q^{IJ}_{\nu]} + 2 Q^{IK}_{[\mu}Q^{KJ}_{\nu]} ~=~ -2P_{[\mu}^{Ir}P_{\nu]}^{Jr} \;, \label{integrabilityPQ}\\ Q^{rs}_{\mu\nu} &\equiv& 2\partial_{[\mu}Q^{rs}_{\nu]} + 2 Q^{rt}_{[\mu}Q^{ts}_{\nu]} ~=~ -2P_{[\mu}^{Ir}P_{\nu]}^{Is} \;. \label{integrabilityQrs} \end{eqnarray} In the gauged theory, all derivatives are covariant w.r.t.\ a non-abelian gauge group according to (\ref{current}) and these relations acquire additional contributions proportional to the non-abelian field strength, as given in the main text in (\ref{intP})--(\ref{intQ2}). Let us further note that (\ref{current0}) may be rewritten as \begin{eqnarray} D_\mu\,{\cal S}^{\cal M}{}_{I}&=& {\cal S}^{\cal M}{}_{r}\,{ P}_\mu^{Ir} \;, \qquad D_\mu\,{\cal S}^{\cal M}{}_{r}~=~ {\cal S}^{\cal M}{}_{I}\,{ P}_\mu^{Ir}\;, \label{dV} \end{eqnarray} with $D_\mu$ representing the $SO(8)\times SO(n)$ covariant derivative. \subsection{$SO(8)$ $\Gamma$-matrix identities} Here, we list a number of identities for the $SO(8)$ $\Gamma$-matrices, which have proven useful in the calculations of the main text \begin{eqnarray} \Gamma^{IJ}_{AB} \Gamma^{IJ}_{CD} &=& 16 \delta_{AB}^{CD} \;, \nonumber\\[2ex] \Gamma^{IJ}_{CD} \Gamma^{IJKL}_{AB} &=& -8(\Gamma^{KL}_{C(A}\delta_{B)D}-\Gamma^{KL}_{D(A}\delta_{B)C}) +2\delta_{AB}\Gamma^{KL}_{CD} \;, \nonumber\\[2ex] \Gamma^{IJ}_{CD} \Gamma^{KL}_{EF} \Gamma^{IJKL}_{AB} &=& -128\,(\delta_{C[E}\delta_{F](A}\delta_{B)D} -\delta_{D[E}\delta_{F](A}\delta_{B)C}) +32\,\delta_{AB}\delta_{C[E}\delta_{F]D} \;, \nonumber\\[2ex] \Gamma^{IJ}_{AB} \Gamma^{KL}_{CD} \Gamma^{IJKL}_{\dot{A}\dot{B}} &=& \Gamma^{IJ}_{[AB} \Gamma^{KL}_{CD]} \Gamma^{IJKL}_{\dot{A}\dot{B}} \;, \nonumber\\[2ex] \Gamma^{IJ}_{AB}\,\Gamma^{IJKL}_{\dot{A}\dot{B}} &=& 8\, (\Gamma^{[K}_{A\dot{A}}\Gamma^{L]}_{B\dot{B}}- \Gamma^{[K}_{B\dot{A}}\Gamma^{L]}_{A\dot{B}}) -2 \delta_{\dot{A}\dot{B}}\Gamma^{KL}_{AB} \;, \nonumber\\[2ex] \Gamma^{IJ}_{AB}\Gamma^{IJK}_{C\dot{A}} &=& 2 \Gamma^{KJ}_{AB}\Gamma^{J}_{C\dot{A}}+ 16 \delta_{C[A} \Gamma^K_{B]\dot{A}} \;. \end{eqnarray} \end{appendix} \providecommand{\href}[2]{#2}\begingroup\raggedright

BkiUbro5i7PA9LposNAO

\section{Introduction} The complex matrix simplex $\Delta_{n, d}$ is the set of the sequences $(Z^{(1)}, \cdots, Z^{(n)})$ of non negative $d\times d$ Hermitian matrices such that $\sum_{i=1}^{n} Z^{(i)} \leq \Id$, where the inequality is understood in the sense of Hermitian matrices. On the matrix simplex, there exist natural probability measures , with densities $C\det(Z^{(1)})^{a_1-1}\cdots \det(Z^{(n)})^{a_n-1}\det(\Id - \sum^{n}_{i =1}Z^{(i)})^{a_{n+1} -1}$ (see Section \ref{sec.matrix.Dirichlet.complex}). As the natural extensions of the Dirichlet measures on the simplex, they are called matrix Dirichlet measures . It turns out that on matrix simplex there exist many diffusion processes which admit matrix Dirichlet measures as reversible ones, and their generators may be diagonalized by a sequence of orthogonal polynomials whose variables are the entries of the matrices. Therefore the matrix simplex appears to be a polynomial domain as described in \cite{BOZ}, see Section \ref{sec.polynomial.models}. The purpose of this paper is to describe these diffusion processes that we call matrix Dirichlet processes, referring to their reversible measure. They appear in a natural way in many different models in probability : in the polar decomposition of Brownian matrices, in the projection of Brownian motions on $SU(N)$, and also in the projection of Wishart matrices, as we will see below. To begin with, we deal with diffusion processes on the simplex which have Dirichlet measures as reversible ones. Dirichlet measures are multivariate generalizations of beta distributions, and play an important role in statistics, such as prior distributions in Bayesian statistics, machine learning, natural language processing, etc. They also appear, together with their associated diffusions processes, in population biology, for example Wright-Fisher models. In this paper, we talk about Dirichlet processes, by which we mean that these diffusion processes on the simplex are polynomial models with Dirichlet measures as their reversible measures. We should point out that they have nothing to do with the Dirichlet processes introduced by Ferguson \cite{ferguson}, which have been widely used in statistics, or the Dirichlet processes introduced by F\"ollmer \cite{follmer}, which are related to Dirichlet form. The matrix Dirichlet measures, as an analogy of Dirichlet distributions, were first introduced by Gupta and Richards~\cite{GuptaRichards}, as special cases of matrix Liouville measures. They have been deeply studied, for example by Olkin and Rubin~\cite{olkin}, Gupta and Nagar~\cite{guptanagar}, see also Letac~\cite{Letac12, LetacMassam} and the references therein. They not only provide models for multiple random matrices, which are related with orthogonal polynomials, integration formulas, etc., but also reflect the geometry of spaces of matrices, see \cite{guptanagar, Hua}. Therefore, it is natural to consider their corresponding diffusion processes. It is worth to mention that matrix Jacobi processes, which can be considered as a one matrix case of matrix Dirichlet processes, were introduced by Doumerc in \cite{Doumerc}, and studied by Demni in \cite{demni2, demni1}. Our interest of this topic not only lies in its importances in statistics and random matrices, but also in the fact that it provides a polynomial model of multiple matrices. In a polynomial model, the diffusion operator (the generator) can be diagonalized by orthogonal polynomials, and this leads to the algebraic description of the boundary see Section~\ref{sec.polynomial.models}. Such polynomial models are quite rare : up to affine transformation, there are 3 models in $\bbR$~\cite{bakrymazet} and 11 models on compact domains in $\bbR^{2}$~\cite{BOZ}. More recently, Bakry and Bressaud~\cite{BakryB} provided new models in dimension 2 and dimension 3, by relaxing the hypothesis in \cite{BOZ} that polynomials are ranked with respect to their natural degree, and investigating the finite groups of $O(3)$ and their invariant polynomials. As we will see in this paper, the simplex and the matrix simplex are both polynomial domains, see Section~\ref{sec.polynomial.models}. Moreover, there exist many different polynomial models on these domians, which is a quite rare situation : in general, there is just one polynomial diffusion process on a given polynomial domain, up to scaling. The situation here is quite complicated, since we are dealing with a family of matrices. By applying the theory of boundary equations, as introduced in \cite{BakryZ}, we are able to describe Dirichlet processes on the simplex, and we provide two models of such matrix Dirichlet processes. Our two models are found to be realized, via various projections, through the Brownian motion on the special unitary group, the polar decomposition of complex matrices and also through the Wishart processes. This leads to some efficient ways to describe image measures in such projections. This paper is organized as follows. In Section \ref{sec.polynomial.models}, we present the basics on diffusion operators and polynomial models that we will use in this paper. In Section \ref{sec.scalar.simplex}, we introduce the various Dirichlet processes on the simplex, and describe their realizations in the special cases from spherical Laplacian or other operators on the sphere, and also from Ornstein-Uhlenbeck or Laguerre processes. In Section \ref{sec.matrix.Dirichlet.complex}, we introduce our main results, i.e. the description of two polynomial diffusion models, and their realizations through the projections from the Brownian motion on the special unitary group and Wishart processes, which are generalizations of the similar results in the scalar (i.e. the simplex) case . Finally in Appendix \ref{polar}, we provide the details of the computations on the polar decomposition of the Brownian motion on complex matrices, which are quite technical. \section{Symmetric diffusion operators and polynomial models \label{sec.polynomial.models}} We present in this section a brief introduction to symmetric diffusion processes and operators, in particular to those diffusion operators which may be diagonalized in a basis of orthogonal polynomials. Symmetric diffusion operators are described in~\cite{BGL}, which we refer the reader to for further details. Moreover, for the particular case of those diffusion associated with orthogonal polynomials, we refer to the paper~\cite{BOZ}. Although the description that we provide below is quite similar to that in ~\cite{BakryB}, we choose to present it here for completeness. Diffusion processes are Markov processes with continuous trajectories in some open set of $\bbR^n$ or on some manifold, usually given as solutions of stochastic differential equations. They are described by their infinitesimal generators, which are called diffusion operators. Diffusion operators are second order differential operators with no zero order terms. When those operators have smooth coefficients, they are given in some open subset $\Omega$ of $\bbR^d$ by their action on smooth, compactly supported function $f$ on $\Omega$, \begin{equation}\label{diffusion.gal} \LL(f) = \sum_{ij} g^{ij}(x) \partial^2_{ij} f+ \sum_i b^i(x)\partial_i f,\end{equation} where the symmetric matrix $(g^{ij})(x)$ is everywhere non negative, i.e. the operator $\LL$ is semi-elliptic. A Markov process $(\xi_t)$ is associated to such a diffusion operator through the requirement that the process $f(\xi_t)-\int_0^t \LL(f)(\xi_s) ds$ is a local martingale for any function $f$ in the domain of the operator $\LL$. In this paper we will concentrate on the elliptic case (that is when the matrix $(g^{ij})$ is everywhere non degenerate), and on the case where this operator is symmetric with respect to some probability measure $\mu$ : for any smooth functions $f,g$, compactly supported in $\Omega$, we have \begin{equation}\label{IPP} \int_\Omega f \LL(g) d\mu = \int_\Omega g \LL(f) d\mu.\end{equation} We say that $\mu$ is a reversible measure for $\LL$ when the associated stochastic process $(\xi_t)$ has a law which is invariant under time reversal, provided that the law at time $0$ of the process is $\mu$. In particular, the measure is invariant : when the associated process $(\xi_t)$ is such that the law of $\xi_0$ is $\mu$, the law of $\xi_t$ is $\mu$ for any time $t>0$. When $\mu$ has a smooth positive density $\rho$ with respect to the Lebesgue measure, the symmetry property \eqref{IPP} is equivalent to \begin{equation}\label{eq.density} b^i (x)= \sum_j \partial_j g^{ij}(x) + \sum_j g^{ij} \partial_j \log \rho,\end{equation} where $b^i(x)$ is the drift coefficient appearing in equation~\eqref{diffusion.gal}. In general, equation~\eqref{eq.density} allows to completely determine $\mu$ from the data in $\LL$, up to some normalizing constant . Now we introduce the carré du champ operator $\Gamma$. Suppose that we have some dense algebra ${\mathcal A }$ of functions in ${\mathcal L }^2(\mu)$ which is stable under the operator $\LL$ and contains the constant functions. Then for $(f,g)\in {\mathcal A }$ we define \begin{equation}\label{def.Gamma} \Gamma(f,g) = \frac{1}{2}( \LL(fg)-f\LL(g)-g\LL(f)).\end{equation} If $\LL$ is given by equation~\eqref{diffusion.gal}, and the elements of ${\mathcal A }$ are at least ${\mathcal C}^2$, we have $$\Gamma(f,g)=\sum_{ij} g^{ij} \partial_if \partial_j g,$$ so that $\Gamma$ describes in fact the second order part of $\LL$. The semi-ellipticity of $\LL$ gives rise to the fact that $\Gamma(f,f)\geq 0$, for any $f\in {\mathcal A }$. Applying formula~\eqref{IPP} with $g=1$, we obtain $\int_\Omega \LL f d\mu=0$ for any $f\in {\mathcal A }$. Then with ~\eqref{IPP} again, we see immediately that for any $(f,g)\in {\mathcal A }$ \begin{equation}\label{IPP2} \int_\Omega f\LL(g) d\mu = -\int_\Omega \Gamma(f,g) d\mu,\end{equation} so that the knowledge of $\Gamma$ and $\mu$ describes entirely the operator $\LL$. Such a triple $(\Omega, \Gamma, \mu)$ is called a Markov triple in~\cite{BGL}. By~\eqref{diffusion.gal}, we see that $\LL(x^i)= b^i$ and $\Gamma(x^i, x^j)= g^{ij}$. The operator $\Gamma$ is called the co-metric, and in our system of coordinates is described by a matrix $\Gamma= \big(\Gamma(x^i, x^j)\big)= (g^{ij})$. In our setting, we will always assume that $\Omega$ is bounded and choose ${\mathcal A }$ to be the set of polynomials. Since polynomials are not compactly supported in $\Omega$, the validity of equation~\eqref{IPP} requires extra conditions on the coefficients $(g^{ij})$ at the boundary of $\Omega$, which we will describe below. The fact that $\LL$ is a second order differential operator implies the change of variable formulas. Whenever $f=(f_1, \cdots, f_n)\in {\mathcal A }^n$ and $\Phi(f_1, \cdots, f_n)\in {\mathcal A }$, for some smooth function $\Phi : \bbR^n\mapsto \bbR$, we have \begin{equation}\label{chain.rule.L}\LL(\Phi(f))= \sum_i \partial_i \Phi(f) \LL(f_i) + \sum_{ij} \partial^2_{ij} \Phi(f) \Gamma(f_i,f_j)\end{equation} and also \begin{equation}\label{chain.rule.Gamma} \Gamma(\Phi(f),g) = \sum_i \partial_i \Phi(f) \Gamma(f_i,g).\end{equation} When ${\mathcal A }$ is the algebra of polynomials, properties~\eqref{chain.rule.L} and~\eqref{chain.rule.Gamma} are equivalent. An important feature in the examples described in this paper is the notion of image. Whenever we have a diffusion operator $\LL$ on some set $\Omega$, it may happen that we find some functions $(X_1, \cdots, X_p)$ in the algebra ${\mathcal A }$ such that $\LL(X_i)= B^i(X)$ and $\Gamma(X_i, X_j)= G^{ij}(X)$ where $X= (X_1, \cdots, X_p)$. Then we say that we have a closed system. If $(\xi_i)$ is the Markov diffusion process with generator $\LL$, $(\zeta_t)= X(\xi_t)$ is again a diffusion process, with its generator expressed in coordinates $(X_1, \cdots, X_p)$ $$\hat \LL = \sum_{ij} G^{ij}(X) \partial^2_{ij} + B^i(X)\partial_i.$$ Moreover, when $\LL$ has a reversible probability measure $\mu$, $\hat \LL$ has the image measure of $\mu$ through the map $X$ as its reversible measure . Thanks to equation~\eqref{eq.density}, this is often an efficient way to determine image measure, which will be used many times in this paper. As mentioned above, we will restrict our attention to the elliptic case. Here we expect $\LL$ to have a self adjoint extension (not unique in general), thus it has a spectral decomposition. Also we expect that the spectrum is discrete, thus that it has eigenvectors, which we will require to be polynomials in the variables $(x^i)$. Moreover, we will require that those polynomial eigenvectors to be ranked according to their degrees, i.e., if we denote by ${\mathcal H}_n$ the space of polynomials with total degree at most $n$, then for each $n$ we need that there exists an orthonormal basis of ${\mathcal H}_n$ which is made of eigenvectors for $\LL$. Equivalently, we require that $\LL$ maps ${\mathcal H}_n$ into itself. This situation is quite rare, and imposes some strong restriction on the domain $\Omega$ that we will describe below. When we have such a Markov triple $(\Omega, \Gamma, \mu)$, where $\Omega$ has a piecewise smooth (at least ${\mathcal C}^1$) boundary, $\mu$ has a smooth density with respect to the Lebesgue measure on $\Omega$. Then we call $\Omega$ a polynomial domain and $(\Omega, \Gamma, \mu)$ a polynomial model. In dimension 1 for example, up to affine transformations, there are only 3 cases of polynomial models, corresponding to the Jacobi, Laguerre and Hermite polynomials, see for example~\cite{bakrymazet}. \begin{enumerate} \item \label{Hermite.case} The Hermite case corresponds to the case where $\Omega= \bbR$, $\mu$ is the Gaussian measure $\frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx$ on $\bbR$ and $\LL$ is the Ornstein--Uhlenbeck operator \begin{equation}\label{def.OU}\LL_{OU}=\frac{d^2}{dx^2}-x\frac{d}{dx}.\end{equation} The Hermite polynomial $H_n$ of degree $n$ satisfy $\LL_{OU} P_n= -nP_n$. \item\label{Laguerre.case} The Laguerre polynomials correspond to the case where $\Omega= (0, \infty)$, the measure $\mu$ depends on a parameter $a>0$ and is $\mu_a(dx)=C_ax^{a-1}e^{-x}\,dx$ on $(0, \infty)$, and $\LL$ is the Laguerre operator \begin{equation}\label{def.Laguerre}\LL_a=x\frac{d^2}{dx^2}+ (a-x)\frac{d}{dx}.\end{equation} The Laguerre polynomial $L^{(a)}_n$ with degree $n$ satisfies $\LL_a L^{(a)}_n= -nL_n^{(a)}$. \item \label{Jacobi.case}The Jacobi polynomials correspond to the case where $\Omega= (-1,1)$, the measure $\mu$ depends on two parameters $a$ and $b$, $a,b>0$ and is is $\mu_{a,b}(dx)=C_{a,b}(1-x)^{a-1}(1+x)^{b-1}\,dx$ on $(-1,1)$, and $\LL$ is the Jacobi operator \begin{equation}\label{def.Jacobi}\LL_{a,b}=(1-x^2)\frac{d^2}{dx^2}- \big(a-b+(a+b)x\big)\frac{d}{dx}.\end{equation} The Jacobi polynomial $(J^{(a,b)}_n)_n$ with degree $n$ satisfy $$\LL_{a,b}J_n^{(a,b)}= - n(n+a+b-1)J_n^{(a,b)}.$$ \end{enumerate} For the general case, when $\Omega$ is bounded, we recall here some results in~\cite{BOZ}. \begin{proposition}\label{prop.poly.model.gal} Let $(\Omega, \Gamma, \mu)$ be a polynomial model in $\bbR^d$. Then, with $\LL$ described by equation~ \eqref{diffusion.gal}, we have \begin{enumerate} \item\label{prop1.pnt1} For $i= 1, \cdots, d$, $b^i$ is a polynomial with $\deg(b^i)\leq 1$. \item \label{prop1.pnt2} For $ i, j= 1, \cdots, d$, $g^{ij}$ is a polynomial with $\deg(g^{ij})\leq 2$. \item \label{prop1.pnt3} The boundary $\partial\Omega$ is included in the algebraic set $\{\det(g^{ij})=0\}$. \item \label{prop1.pnt4} If $\{P_1\cdots P_k=0\}$ is the reduced equation of the boundary $\partial\Omega$ (see remark~\ref{rmk.reduced.bdry.eq} below), then, for each $q= 1, \cdots k$, each $i= 1, \cdots d$, one has \begin{equation}\label{boundary.eq}\Gamma( \log P_q, x_i)= L_{i,q},\end{equation} where $L_{i,q}$ is a polynomial with $\deg(L_{i,q})\leq 1$; \item\label{prop1.pnt5} All the measures $\mu_{\alpha_1, \cdots, \alpha_k}$ with densities $C_{\alpha_1, \cdots, \alpha_k}|P_1|^{\alpha_1}\cdots |P_k|^{\alpha_k}$ on $\Omega$, where the $\alpha_i$ are such that the density is is integrable on $\Omega$, are such that $(\Omega, \Gamma, \mu_{\alpha_1, \cdots, \alpha_k})$ is a polynomial model. \item\label{prop1.pnt6} When the degree of $P_1\cdots P_k$ is equal to the degree of $\det(g^{ij})$, there are no other measures. \end{enumerate} Conversely, assume that some bounded domain $\Omega$ is such that the boundary $\partial\Omega$ is included in an algebraic surface and has reduced equation $\{P_1\cdots P_k=0\}$. Assume moreover that there exists a matrix $(g^{ij}(x))$ which is positive definite in $\Omega$ and such that each component $g^{ij}(x)$ is a polynomial with degree at most $2$. Let $\Gamma$ denote the associated carr\'e du champ operator. Assume moreover that equation~\eqref{boundary.eq} is satisfied for any $i= 1, \cdots, d$ and any $q= 1,\cdots, k$, with $L_{i,q}$ a polynomial with degree at most $1$. Let $(\alpha_1, \cdots, \alpha_k)$ be such that the $|P_1|^{\alpha_1}\cdots |P_k|^{\alpha_k}$ is integrable on $\Omega$ with respect to the Lebesgue measure, and denote $$\mu_{\alpha_1, \cdots, \alpha_k} (dx)= C_{\alpha_1, \cdots, \alpha_k}|P_1|^{\alpha_1}\cdots |P_k|^{\alpha_k}dx,$$ where $C_{\alpha_1, \cdots, \alpha_k}$ is the normalizing constant such that $\mu_{\alpha_1, \cdots, \alpha_k}$ is a probability measure. Then $(\Omega, \Gamma,\mu_{\alpha_1, \cdots, \alpha_k})$ is a polynomial model. \end{proposition} \begin{rmq}\label{rmk.reduced.bdry.eq} We say that $\{P_1\cdots P_k=0\}$ is the reduced equation of the boundary $\partial\Omega$ when \begin{enumerate} \item The polynomials $P_i$ are not proportional to each other. \item For $i= 1, \cdots k$, $P_i$ is an irreducible polynomial, both in the real and the complex field. \item For each $i= 1, \cdots, k$, there exists at least one regular point of the boundary $\partial\Omega$ such that $P_i(x)= 0$. \item For each regular point $x\in \partial\Omega$, there exist a neighborhood ${\mathcal V}$ and of $x$ and some $i$ such that $\partial\Omega\cap {\mathcal V}= \{P_i(x)=0\}\cap {\mathcal V}$. \end{enumerate} In particular, this does not mean that any point satisfying $P_i(x)=0$ for some $i$ belongs to $\partial\Omega$. \end{rmq} \begin{rmq} The determination of the polynomial domains therefore amounts to the determination of the domains $\Omega$ with an algebraic boundary, with the property that the reduced equation of $\partial\Omega$ is such that the set of equations~\eqref{boundary.eq} has a non trivial solution, for $g^{ij}$ and $L_{i,q}$. Given the reduced equation of $\partial\Omega$, equations \eqref{boundary.eq} are linear homogeneous ones in the coefficients of the polynomials $g^{ij}$ and of the polynomials $L_{i,k}$. Unfortunately, in general we need much more equations to determine the unknowns uniquely, and this requires very strong constraints on the polynomials appearing in the reduced equation of the boundary. We will see that both the simplex and the matrix simplex (in the complex and real case) are such domains where the choice of the co-metric $\Gamma$ is not unique. \end{rmq} \begin{rmq} The set of equations~\eqref{boundary.eq}, which are central in the study of polynomial models, may be reduced to less equations, when $k>1$. Indeed, if we set $P= P_1\cdots P_k$, it reduces to \begin{equation}\label{boundary.eq2}\Gamma(x_i,\log P)= L_i, \quad \deg(L_i)\leq 1.\end{equation} In fact assume that this last equation holds with some polynomial $L_i$, then on the regular part of the boundary described by $\{P_q(x)=0\}$, we have $\Gamma(x_i, P_q)= 0$ since $$\Gamma(x_i, P_q)= P_q(L_i-\sum_{l\neq q} \frac{\Gamma(x_i,P_l)}{P_l}).$$ Therefore, $\Gamma(x_i,P_q)$ vanishes on the regular part, and $P_q$ being irreducible, it divides $\Gamma(x_i, P_q)$. This leads to $\Gamma(x_i, P_q)= L_{i,q} P_q$, where $\deg(L_{i,q})\leq 1$. Thus we obtain the equation~\eqref{boundary.eq}. \end{rmq} \begin{rmq}\label{rmq.def.polynomial.domain} A bounded polynomial domain is therefore any bounded domain $\Omega$ with algebraic boundary, on which there exists a symmetric matrix $(g^{ij})$ with entries which are polynomials with degree at most 2, that is positive definite on $\Omega$ and defines on $\Omega$ an operator $\Gamma$ satisfying equation~\eqref{boundary.eq2}. \end{rmq} In~\cite{BOZ}, a complete description of all polynomial domains and models in dimension 2 is provided : 11 different cases are given up to affine transformations. This description only relies on algebraic considerations on those algebraic curves in the plane where the boundary condition~\eqref{boundary.eq} has a non trivial solution. This reflects the fundamental role played by the boundary equation. Among the 11 bounded domains provided by the classification in~\cite{BOZ}, the triangle appears to be one of the few ones (with the unit ball and a particular case of the parabolic bi-angle) where the metric is not unique. In higher dimension, the triangle may be generalized to the simplex, which appear to be also a polynomial domain. Here we extend it furthermore to the matrix simplex. \section{Dirichlet measure on the simplex\label{sec.scalar.simplex}} In this section, we recall a few facts about the simplex and provide some diffusion operators on it which shows that the simplex is a polynomial domain in the sense of Section~\ref{sec.polynomial.models}. \begin{definition} The $n$ dimensional simplex $\Delta_{n}\subset \bbR^{n}$ is the set $$ \Delta_{n} = \{0\leq x_{i} \leq 1, ~ i= 1,\cdots n, \sum^{n}_{i = 1}x_{i} \leq 1, \}. $$ Given ${\mathbf{a}} = (a_1, a_2,..., a_{n+1}) \in \mathbb{R}^{n+1}$, where $a_{i}>0$, $ i = 1,..., n+1$, the Dirichlet distribution $D_{{\mathbf{a}}}$ is the probability measure on $\Delta_n$ given by $$ \frac{1}{B_{{\mathbf{a}}}}x_{1}^{a_1 - 1}...x_{n}^{a_n - 1}(1 - x_1- ...- x_{n})^{a_{n+1} - 1}\mathbf{1}_{\Delta_{d}}(x_1, ..., x_n)dx_1...dx_{n}, $$ where $B_{{\mathbf{a}}} = \frac{\Gamma(a_1)...\Gamma(a_{n+1})}{\Gamma(a_1 + ...+a_{n +1})}$ is the normalizing constant. \end{definition} The Dirichlet measure can be considered as a $n$-dimensional generalization of the beta distribution on the real line, $$ \beta(a_1, a_2) = \frac{1}{B(a_1, a_2)}x^{a_1 -1}(1 - x)^{a_2 - 1}\mathbf{1}_{(0, 1)}(x)dx, $$ which is indeed $D_{(a_1, a_2)}$. It turns out that the simplex $\Delta_n$ is a polynomial diffusion domain in $\bbR^n$ in the sense of \cite{BOZ}, as described in Section~\ref{sec.polynomial.models}. More precisely, there exist many different polynomial models on the simplex which admit the Dirichlet measure as their reversible measure. \begin{theoreme}\label{th.Dirichlet.simplex} Let $\mathbf{A}=(A_{ij})_{i,j=1\cdots ,n+1}$ be a symmetric $(n+1)\times (n+1)$ matrix where all the coefficients $A_{ij}$ are non negative, and $A_{ii}=0$. Let $\mathbf{a}= (a_1, \cdots, a_{n+1})$ be a $(n+1)$-tuple of positive real numbers. Let $\LL_{{\mathbf{A}}, {\mathbf{a}}}$ be the symmetric diffusion operator defined on the simplex $\Delta_n$ defined by \begin{eqnarray}\label{gamma.dirichlet.simplex} \Gamma_{{\mathbf{A}}}(x_i, x_j) &=& -A_{ij}x_ix_j + \delta_{ij}\sum^{n+1}_{k=1}A_{ik}x_kx_i ,\\ \LL_{{\mathbf{A}},{\mathbf{a}}}(x_i) &=& -x_i\sum^{n+1}_{j=1} A_{ij}a_j + a_i\sum^{d+1}_{j=1}A_{ij}x_j, \end{eqnarray} where $x_{n+1} = 1 - \sum^{n}_{j=1}x_j$. Then, as soon as $A_{ij}>0$ for all $i,j= 1,\cdots, n+1$, $i\neq j$, $(\Delta_n, D_{{\mathbf{a}}}, \Gamma_{{\mathbf{A}}})$ is a polynomial model, and the operator $\LL_{{\mathbf{A}},{\mathbf{a}}}$ is elliptic on $\Delta_n$. Moreover, any polynomial diffusion model on $\Delta_n$ having $D_{{\mathbf{a}}}$ as its reversible measure has this form (however without the requirement that the coefficients $A_{ij}$ are positive). \end{theoreme} Observe that the condition $A_{ii}=0$ is irrelevant in formulas \eqref{gamma.dirichlet.simplex}, since the coefficient $A_{ii}$ vanishes in the formulas. \begin{preuve} Let us prove first that any polynomial model with the usual degree has this form. According to Proposition~\ref{prop.poly.model.gal}, to be a polynomial model, $\Gamma(x_i, x_j)$ must be a polynomial no more than degree 2 and satisfy the boundary equation \eqref{boundary.eq} : for $1 \leq i, j \leq n$, \begin{eqnarray} \label{eq1}\Gamma_{{\mathbf{A}}}(x_i, \log x_{j}) &=& L^{i,j},\\ \label{eq2}\Gamma_{{\mathbf{A}}}(x_i, \log x_{n+1}) &=& L^{i,n+1}, \end{eqnarray} where $\{L^{i,j}, 1 \leq j \leq n+1\}$ are polynomials with degree at most 1. From equation~\eqref{eq1}, we get $$ \Gamma_{{\mathbf{A}}}(x_i, x_j) = x_jx_jL^{ij}, $$ so that $\Gamma_{{\mathbf{A}}}(x_i,x_j)$ is divisible by $x_i$, and similarly by $x_j$. Therefore when $i\neq j$, there exists a constant $A_{ij}$, with $A_{ij}= A_{ij}$, such that $$ \Gamma_{{\mathbf{A}}}(x_i, x_j) = -A_{ij}x_jx_i.$$ When $i=j$, $\Gamma_{{\mathbf{A}}}(x_i,x_i)$ is divisible by $x_i$ and from equation~\eqref{eq2}, we obtain $$ \sum^{n}_{j=1}\Gamma_{{\mathbf{A}}}(x_i, x_j) = -x_{n+1}L^{i,n+1}, $$ which writes $$ \Gamma_{{\mathbf{A}}}(x_i,x_i)= x_i(\sum_{j=1, j\neq i}^{n}A_{ij}x_j) -x_{n+1}L^{i,n+1}. $$ This implies that $x_i$ divides $L^{i,n+1}$, and therefore that $L^{i,n+1}= -A_{i(n+1)} x_i$, so that $$\Gamma_{{\mathbf{A}}}(x_i,x_i)= x_i \sum_{i=1, j\neq i}^{n+1} A_{ij} x_j.$$ Conversely, it is quite immediate every operator $\Gamma_{\mathbf{A}}$ defined by equation~\eqref{eq1} satisfies the boundary condition on the simplex. The formulas for $\LL_{{\mathbf{A}}, {\mathbf{a}}}$ are just a direct consequence of the reversible measure equation~\eqref{eq.density}. The ellipticity of $\Gamma_{{\mathbf{A}}}$ when all the coefficients $A_{ij}, i\neq j$ are positive is a particular case of the real version of Theorem~\ref{matrix.dirichlet}, which will be proved in Section~\ref{sec.matrix.Dirichlet.complex}. \end{preuve} \medskip When all the coefficients $A_{ij}$ are equal to 1, and when all the parameters $a_i$ are half integers, the operator \eqref{gamma.dirichlet.simplex} is an image of the spherical Laplacian. Indeed, consider the unit sphere $\sum_{i= 1}^{N} y_i^2=1$ in $\bbR^N$, and split the set $\{1, \cdots, N\}$ in a partition of $n+1$ disjoint subsets $I_1, \cdots, I_{n+1}$ with respective size $p_1, \cdots, p_{n+1}$. For $i= 1, \cdots, n+1$, set $x_i= \sum_{j\in I_i} y_j^2$, so that $x_{n+1}= 1-\sum_{i=1}^{n} x_i$. Then it is easy to check that, for the spherical Laplace operator $\Delta_{\bbS^{N-1}}$ in $\bbR^N$ (see \cite{BGL} for details), $\Delta_{\bbS^{N-1}} (x_i)$ and $\Gamma_{\bbS^{N-1}}(x_i,x_j)$ coincide with those given in equations \eqref{gamma.dirichlet.simplex} whenever $a_i = \frac{p_i}{2}$. The next proposition generalizes this geometric interpretation for the general choice of the parameters $A_{ij}$. \begin{proposition}\label{prop.general.scalar.case} Let $I_1, \cdots, I_{n+1}$ be a partition of $\{1, \cdots, N\}$ into disjoint sets with size $p_1, \cdots, p_{n+1}$. For $i, j\in \{1, \cdots, n+1\}$, with $i\neq j$, let $\LL_{i,j} $ be the operator acting on the unit sphere $\bbS^{N-1}\subset \bbR^N$ as $$\LL_{i,j} = \sum_{p\in I_i, q\in I_j} (y_p\partial_{y_q}-y_q\partial_{y_p})^2.$$ Then, setting $x_i= \sum_{p\in I_i} y_i^2$, $\{x_1, \cdots, x_{n+1}\}$ form a closed system for any $\LL_{i,j}$, and the image of $$\frac{1}{2}\sum_{i < j} A_{ij} \LL_{i,j}$$ is the operator $\LL_{{\mathbf{A}},{\mathbf{a}}}$, where $a_i= \frac{p_i}{2}$. \end{proposition} The proof follows from a direct application of the change of variable formulas~\eqref{chain.rule.L} and~\eqref{chain.rule.Gamma} applied to the functions $\{x_p\}$. For any operator $\LL_{i,j}$, the associated carré du champ operator is $\Gamma_{i,j}(f,g)= \sum_{p\in I_i, q\in I_j} (y_p\partial_{y_q} -y_q\partial_{y_p})(f)(y_p\partial_{y_q} -y_q\partial_{y_p})(g)$. We just have to identify $\LL_{i,j} (x_p)$ and $\Gamma_{i,j}(x_p,x_q)$ through a direct and easy computation, and observe that it fits with the coefficients of $A_{ij}$ in the same expression in the definition of $\LL_{{\mathbf{A}},{\mathbf{a}}}$. It is worth to observe that the spherical Laplace operator is nothing else than $\sum_{i,j} \LL_{i,j}$. The disappearance of the operators $\LL_{i,i}$ in the general form for $\LL_{{\mathbf{A}},{\mathbf{a}}}$ comes from from the fact that the action of $\LL_{i,i}$ on any of the variables $x_p$ vanishes. To come back to the case where all the coefficients $A_{ij}$ are equal to 1, and as a consequence of the previous observation, the Dirichlet measure on the simplex is an image of the uniform measure on the sphere when the parameters are half integers, through the map that we just described $(y_1, \cdots, y_N)\mapsto (x_1, \cdots, x_{d})$ where $x_i= \sum_{j\in I_i} y_j^2$. In the same way that the uniform measure on a sphere is an image of a Gaussian measure through the map $y\in \bbR^N\mapsto \frac{y}{\|y\|}\in \bbS^{N-1}$, the Dirichlet measure may be constructed from the random variables $\sum_{j\in I_i} y_j^2$ where $y_j$ are independent standard Gaussian variables. More precisely, if $(y_1, \cdots, y_N)$ are independent real valued Gaussian variables, setting $S_i= \sum_{j\in I_i} y_j^2$ and $S= \sum_{i=1}^{d+1} S_i$, we see that $(\frac{S_1}{S},\cdots, \frac{S_d}{S} )$ follows the Dirichlet law $D_{{\mathbf{a}}}$ whenever $a_i= \frac{p_i}{2}$. Since the variables $S_i$ have $\gamma$ distribution $\frac{1}{\Gamma(\frac{p_i}{2})}x^{\frac{p_i}{2} - 1}e^{-x/2}dx$, it is not surprising that this representation of Dirichlet laws may be extended for the general case, that is when the parameters $a_i$ are no longer half integers, by replacing norms of Gaussian vectors by independent variables having $\gamma$ distribution. We quote the following proposition from~\cite{Letac12}, which gives a construction of Dirichlet random variable through gamma distributions $\gamma_{\alpha}$ on $\bbR^{+}$ given by $$\gamma_{\alpha}(dx) = \frac{1}{\Gamma(\alpha)}x^{\alpha-1}e^{-\frac{x}{\beta}}\mathbf{1}_{(0, \infty)}(x)\, dx, $$ where $\alpha> 0$ and $\Gamma(\alpha) = \int^{\infty}_0 x^{\alpha-1}e^{-x}dx$. \begin{proposition}\label{prop6} Consider independent random variables $x_1,..., x_n, x_{n+1}$ such that each $x_k$ has gamma distribution $\gamma_{\alpha_k, 2}$. Define $S = \sum^{n+1}_{k=1}x_k$. Then $S$ is independent of $\frac{1}{S}(x_1,..., x_n)$, and the distribution of $\frac{1}{S}( x_1,..., x_n)$ has the density $D_{(\alpha_1, ..., \alpha_n)}$. \end{proposition} Indeed, Proposition~\ref{prop6} is still valid at the level of the processes. Start first from an $N$-dimensional Ornstein-Uhlenbeck process, which admits the standard Gaussian measure in $\bbR^N$ as reversible measure. One should be careful here, since the spherical Brownian motion is not directly the image of an Ornstein-Uhlenbeck process through $x\mapsto \frac{x}{\|x\|}$. Indeed, writing the Ornstein-Uhlenbeck operator $\LL_{OU}= \Delta-x\cdot\nabla$ in $\bbR^N\setminus\{0\}$ in polar coordinates $(r= \|x\|, \phi= \frac{x}{\|x\|})\in (0, \infty)\times \bbS^{N-1}$, we derive $$\LL_{OU}= \partial^2_r + (\frac{N-1}{r} -r) \partial_r + \frac{1}{r^2} \Delta_{\bbS^{N-1}},$$ where $\Delta_{\bbS^{N-1}}$ is the spherical Brownian Laplace operator acting on the variable $\phi\in \bbS^{N-1}$. The structure of $\LL_{OU}$ shows that it appears as a skew product (or warped product) of a one dimensional operator with the spherical Laplace operator. In the probabilist interpretation, if $\xi_t$ is an Ornstein-Uhlenbeck process in $\bbR^N$, and if $\rho_t = \| \xi_t\|$ and $\theta_t= \frac{\xi_t}{\|\xi_t\|}$, then $\rho_t$ is a diffusion process (with generator $\partial^2_r + (\frac{N-1}{r} -r) \partial_r$), and the pair $(\rho_t, \theta_t)$ is also a diffusion process, while it is not the case for $\theta_t$ alone. We have to change the clock at which $\theta_t$ is running, namely set $\hat \theta_t= \theta_{C_t}$ where $C_t= \int_0^t \rho_s^2 ds$, such that $\hat \theta_t$ is a spherical Brownian motion which is independent of $\rho_t$. We may also consider the process $\theta_t$ alone conditioned on $\rho_t=1$. However, since the reversible law of $\hat \theta_t$ is the uniform measure on the sphere, we also see that, when the law of $\xi_0$ is the reversible law of the the process $\xi_t$ (that is the standard ${\mathcal N }(0,1)$ Gaussian law in $\bbR^N$), the law of $\theta_t$ given $\rho_t$ does not depend on $\rho_t$ for fixed time $t$, such that the law of $\theta_t$ is the uniform measure on the sphere and $\theta_t$ and $\rho_t$ are independent. Starting again with some Ornstein-Uhlenbeck process in $\xi_t= (\xi_{t}^{(1)}, \cdots, \xi_t^{(N)})$ in $\bbR^N$, and cutting the set of indices in $n+1$ parts $I_i$ as above, with $|I_i|= p_i$, we may now consider the variables $\sigma_t^{(i)}= S_i(\xi_t)$ and denote $\sigma_t= S(\xi_t)$, where $S_i(x)= \frac{1}{2}\sum_{j\in I_i} x_j^2$ and $S(x)= \sum_i S_i(x)$. Then $\sigma_t^{(i)}$ are independent Laguerre processes with generator $\LL_{a_i}(f)= 2(xf''+ (a_i-x)f')$, where $a_i= \frac{p_i}{2}$. One sees that $\sigma_t$ is again a Laguerre process with generator $\LL_a(f)= 2(xf''+ (a-x)f')$, with $a= N/2$. Then, setting $\zeta^{(i)}_t= \frac{\sigma^{(i)}_t}{\sigma_t}$ and $\zeta_t= (\zeta^{(1)}_t, \cdots, \zeta^{(n)}_t)$, $\zeta_t$ takes values in the simplex $\Delta_n$, and a simple computation shows that $(\sigma_t, \zeta_t)$ is a Markov process with generator $\LL_a + \frac{1}{S}\LL_{{\mathbf{A}},{\mathbf{a}}}$, where $\LL_{{\mathbf{A}}, {\mathbf{a}}}$ is the operator defined in Theorem \ref{th.Dirichlet.simplex}, with $A_{ij}= 2$ and $a_i= \frac{p_i}{2}$. This may be extended to the general case where the parameters $a_i$ are no longer half integers. Starting from the standard Laguerre operator, we have \begin{proposition} \label{simplex.construction1} Let $\bar a= (a_1, \cdots, a_{n+1})$ be positive integers and $\Sigma_t=(\sigma_t^{(1)}, \cdots, \sigma^{(n+1)}_t) $ be independent Laguerre processes on $\bbR_+$ with generator $\LL_{a_i}(f)= xf''+ (a_i-x)f'$. Let $\sigma_t= \sum_{i=1}^{n+1} \sigma_t^{(i)}$ and $\zeta_t= (\frac{\sigma_t^{(1)}}{\sigma_t}, \cdots , \frac{\sigma_t^{(d)}}{\sigma_t})$. Then, the pair $(\sigma_t, \zeta_t)\in \bbR_+ \times \Delta_{n}$ is a diffusion process with generator $$S\partial^2_S+ (a-S)\partial_S+ \frac{1}{S}\LL_{{\mathbf{A}},{\mathbf{a}}}$$ where $\LL_{{\mathbf{A}},{\mathbf{a}}}$ is defined in Theorem~\ref{th.Dirichlet.simplex} with $A_{ij}=1$ for $i\neq j$, ${\mathbf{a}} = \bar{a}$ and $a= \sum_{i=1}^{n+1} a_i$. \end{proposition} Here, if the coordinates that we choose for a variable in $\bbR_+ \times \Delta_n$ are $(S, x)$, where $x= (x_1, \cdots, x_n)\in \Delta_n$, then the action of the generator is $\LL_{a}$ on the variable $S$ and $\frac{1}{S}\LL_{{\mathbf{A}},{\mathbf{a}}}$ on the variable $x$. We are in the situation of a warped product, and $\sigma_t$ is a Laguerre diffusion process with generator $\LL_{a}$. Setting $C_t= \int_0^t \sigma_s ds$, $\zeta_{C_t}$ is a diffusion process on $\Delta_n$ which is independent of $\sigma_t$ and has generator $\LL_{{\mathbf{A}},{\mathbf{a}}}$. Moreover, the reversible measure of $\Sigma_t$ is the product on independent $\gamma_{a_i}$ measures on $\bbR_+$, and its image on $\bbR_+\times \Delta_n$ through the map $y= (y^{(1)}, \cdots, y^{(n+1)})\in \bbR_+^{n+1} \mapsto (S= \sum_{i=1}^{n+1} y_i, x_i= \frac{y_i}{S})$ is the product of a $\gamma_{a}$ measure on $\bbR_+$ and of a $D_{{\mathbf{a}}}$ measure on $\Delta_n$. We thus get another proof of Proposition \ref{prop6}. \begin{preuve} Following the notations of Section~\ref{sec.polynomial.models}, we consider he generator of the process $\Sigma_t$, which is $$\LL_{\bar a}=\sum_{i= 1}^{n+1} y_i\partial^2_{y_i} f+ \sum_{i= 1}^{n+1} (a_i-y_i)\partial_{y_i} f,$$ where $y= (y_1, \cdots, y_{n+1})\in \bbR_+^{(n+1)}$. Let $\Gamma_{\bar a}$ be its associated carré du champ operator. With $S= \sum_{i=1}^{n+1} y_i$ and $z_i=\frac{y_i}{S}$, $i= 1, \cdots, n$, we just have to check that \begin{enumerate} \item $\LL_{\bar a}(S)= a-S$; \item $\LL_{\bar a}(z_i)=\frac{1}{S}\big(a_i - az_i\big) $; \item $\Gamma_{\bar a}(S,S)= S$; \item $\Gamma_{\bar a}(S,z_i)=0$; \item $\Gamma_{\bar a}(z_i,z_j)=\frac{1}{S}(\delta_{ij} z_i-z_iz_j) $; \end{enumerate} This follows directly from a straightforward computation. \end{preuve} \medskip Proposition~\ref{simplex.construction1} provides a construction of the process on the simplex with generator $\LL_{{\mathbf{A}}, {\mathbf{a}}}$ for the general ${\mathbf{a}}$ but only when $A_{ij}= 1$ for $i\neq j$. To obtain a construction in the general case, we may use the same generalization that we did on the sphere with the operators $\LL_{i,j}= \sum_{k\in I_i, l\in I_j} (y_k\partial_l-y_l\partial_k)^2$, where $y= (y_i)\in \bbS^n$, i.e., consider the operator $$\LL^{{\mathbf{A}}}_{OU}= \partial^2_r + (\frac{N-1}{r} -r) \partial_r + \frac{1}{2r^2}\sum_{i < j}A_{ij}\LL_{i, j},$$ where we use $\frac{1}{2}\sum_{i < j}A_{ij}\LL_{i, j}$ instead of $\Delta_{\bbS^{N-1}}$ in the Ornstein-Uhlenbeck operator. Its reversible measure is still $e^{-\frac{1}{2}\|x\|^{2}}dx_{1}...dx_{N+1}$ on $\bbR^{N+1}$. Now let $\LL^{{\mathbf{A}}}_{OU}$ act on the variables $x^{i}= \sum_{p\in I_i } y_p^2$, and $S = \sum_{i}x_i$ $z_i=\frac{x_i}{S}$, $i= 1, \cdots, n$, then it is easy to check that the image of $\LL^{{\mathbf{A}}}_{OU}$ is $\LL_{{\mathbf{A}}, {\mathbf{a}}}$ with $a_i = \frac{p_i}{2}$. In fact this construction is essentially the same as Proposition \ref{prop.general.scalar.case}. Consider a diffusion process ${\mathbf{x}} = (x_{1}, \cdots ,x_{n+1})$ on $\bbR^{n+1}$ with a warped product symmetric diffusion generator $$ \LL = \LL_{r} + f(r)\Delta_{\bbS^{n}}, $$ where $\LL_{r}$ is a symmetric diffusion operator on $r_t = \|x_{t}\|$, $f: \bbR^{+} \rightarrow \bbR^{+}$ is a function of $r$. Define $y_{i} = x^{2}_{i}$ and $z_{i} = \frac{y_{i}}{r^{2}}$ for $1 \leq i \leq (n+1)$, then the generators of ${\mathbf{z}} = (z_{1}, \cdots ,z_{n+1})$ are indeed the image of $f(r)\Delta_{\bbS^{n}}$ through the map from ${\mathbf{x}}$ to ${\mathbf{z}}$. Following the same procedure as in Theorem \ref{th.Dirichlet.simplex}, we can always construct the Dirichlet process with a general parameter~${\mathbf{A}}$. \section{Matrix Dirichlet Processes \label{sec.matrix.Dirichlet.complex}} \subsection{The complex matrix simplex and their associated Dirichlet measures\label{subsec.matrix.Dirichlet.domain}} We now introduce the matrix simplex, the matrix Dirichlet processes and its corresponding diffusion operators. In particular, we give two families of models. The first one comes from extracted matrices from the Brownian motion on $SU(N)$, and is an analogy to the construction of the scalar Dirichlet process from spherical Brownian motion. Similarly, Proposition~\ref{simplex.construction1} may be extended to a similar construction from Wishart matrix processes, which are a natural matrix extension of Laguerre processes. Another construction arises naturally when one studies the polar decomposition of Brownian matrices, in which case the spectral projector provides a degenerate form of such matrix Dirichlet processes. Considering the real version of matrix Brownian motions, it is not hard to derive the real counterparts of our results, that we do not develop here. First we define the complex matrix generalization $\Delta_{n, d}$ of the $n$-dimensional simplex as the set of $n$-tuple of Hermitian non negative matrices $\{{\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)}\}$ such that \begin{equation}\label{matrix.simplex} \sum_{k=1}^n {\mathbf{Z}}^{(k)} \leq \Id. \end{equation} Accordingly, the complex matrix Dirichlet measure $D_{{\mathbf{a}}}$ on $\Delta_{n,d}$ is given by \begin{equation}\label{dirichlet.matrix.measure} C_{d; {\mathbf{a}}}\prod^{n}_{k=1}\det({\mathbf{Z}}^{(k)})^{a_k-1}\det(\Id - \sum^{n}_{k = 1}{\mathbf{Z}}^{(k)})^{a_{n+1}-1}\prod^{n}_{k= 1}d{\mathbf{Z}}^{(k)}, \end{equation} where ${\mathbf{a}} = (a_1, \cdots, a_{n+1})$, $\{a_{i}\}^{n+1}_{i=1}$ are all positive constants, and $d{\mathbf{Z}}^{(k)}$ is the Lebesgue measure on the entries of ${\mathbf{Z}}^{(k)}$, i.e., if ${\mathbf{Z}}^{(k)}$ has complex entries $Z^{(k)}_{pq}= X^{(k)}_{pq}+ i Y^{(k)}_{pq}$, with $Z^{(k)}_{qp}= \bar Z^{(k)}_{pq}$, $$d{\mathbf{Z}}^{(k)}= \prod_{1\leq p\leq q\leq d} dX^{(k)}_{pq}dY^{(k)}_{pq}.$$ This measure is finite exactly when the constants $a_i$ are positive, and the normalizing constant $ C_{d; {\mathbf{a}}}$ makes $D_{{\mathbf{a}}}$ a probability measure on $\Delta_{n,d}$. The normalization constant $ C_{d; {\mathbf{a}}}$ may be explicitly computed with the help of the matrix gamma function $$\Gamma_{d}(a) = \int_{A > 0} e^{-\textmd{trace}\,(A)}\det(A)^{a-1}dA = \pi^{\frac{1}{2}d(d-1)}\prod^{d}_{i=1}\Gamma(a+d -i),$$ where $\{A > 0\}$ denotes the domain of $d \times d$ positive-definite, Hermitian matrices. Then, the normalization constant may be written as $$C_{d; {\mathbf{a}}} = \frac{\prod^{n+1}_{i=1} \Gamma_{d}(a_{i})}{\Gamma_{d}(\sum^{n+1}_{i=1}a_{i})}.$$ It turns out that $\Delta_{n,d}$ is polynomial domain as described in Remark~\ref{rmq.def.polynomial.domain}, with boundary described by the equation \begin{eqnarray}\label{boundary.eq.mdp} \det({\mathbf{Z}}^{(1)}) \cdots \det({\mathbf{Z}}^{(n)}) \det(\Id- {\mathbf{Z}}^{(1)}-\cdots -{\mathbf{Z}}^{(n)})=0, \end{eqnarray} which is an algebraic equation in the coordinates $(X^{(k)}_{pq}, Y^{(k)}_{pq})$. For convenience, and as described in Section~\ref{sec.polynomial.models}, we will use complex coordinates $(Z^{(k)}_{pq}, \bar Z^{(k)}_{pq}, 1\leq k \leq n, 1\leq p <q\leq d)$ (keeping in mind that $Z^{(k)}_{pp}$ which is real), to describe the various diffusion operators acting on $\Delta_{\bbC, n,d}$. Moreover, instead of using $(Z^{(k)}_{ij}, \bar Z^{(k)}_{ij})$, $1\leq i < j\leq d$ as coordinates, in our case it is simpler to use $(Z^{(k)}_{ij}$, $i,j=1, \cdots, d)$, due to the fact that $\bar Z^{(k)}_{ij}= Z^{(k)}_{ji}$. As in the scalar case studied in Section~\ref{sec.scalar.simplex}, it turns out that there are many possible operators $\Gamma$ acting on $\Delta_{n,d}$ such that $(\Delta_{n,d}, \Gamma, D_{a_1, \cdots, a_{n+1} })$ is a polynomial model. To simplify the notations, we shall always set ${\mathbf{Z}}^{(n+1)}= \Id-\sum_{i=1}^n {\mathbf{Z}}^{(i)}$, with entries $Z^{(n+1)}_{ij}$, $i,j= 1, \cdots, d$. According to the boundary equation~\eqref{boundary.eq2}, the carr\'e du champ operator $\Gamma$ of the matrix Dirichlet process is such that each entry must be a polynomial of degree at most 2 in the variables $(Z_{ij}^{(k)}, \bar Z_{ij}^{(k)})$ and must satisfy \begin{eqnarray*}\label{beq.dirichlet} \Gamma(Z^{(p)}_{ij}, \log \det ({\mathbf{Z}}^{(k)})) = L^{p,k}, \end{eqnarray*} for every $p= 1, \cdots, n$, $k= 1, \cdots, n+1$, $i,j = 1, \cdots d$, where $L^{p,k}$ is an affine function of the entries of $\{{\mathbf{Z}}^{(1)}, ..., {\mathbf{Z}}^{(n)}\}$. However, since there are many variables in the diffusion operator $\Gamma$, we are not in the position to describe all the possible solutions for $\Gamma$ as we did in the scalar case. Therefore, we will restrict ourselves to a simpler form. Namely, we assume that for any $1 \leq p, q \leq n+1$, $$ \Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl}) = \sum_{abcdrs}(A^{p, q}_{ij, kl})^{ab, cd}_{r, s}Z^{(r)}_{ab}Z^{(s)}_{cd}, $$ with some constant coefficients $(A^{p, q}_{ij, kl})^{ab, cd}_{r, s}$. Then the tensor $({\mathbf{A}}^{p, q}_{ij, kl})$, whose entries are denoted by $\{(A^{p, q}_{ij, kl})^{ab, cd}_{r, s}\}$, should satisfy the following restrictions: \begin{itemize} \item{Since $\Gamma$ is symmetric, we have $(A^{p, q}_{ij, kl})_{ab, cd} = (A^{q, p}_{kl, ij})_{ab, cd}$, and the choice of $({\mathbf{A}}^{p, q}_{ij, kl})$ should ensure that $\Gamma$ is elliptic on the matrix simplex; } \item{The fact that the diffusions live in the matrix simplex gives rise to $$ \sum_{p=1}^{n+1} \sum_{abcd}(A^{p, q}_{ij, kl})_{ab, cd}Z^{(p)}_{ab}Z^{(q)}_{cd}= 0; $$ } \item{The boundary equation \eqref{beq.dirichlet} leads to \begin{equation}\label{eq.GammaLogDelta} \Gamma(Z^{(p)}_{ij}, \log \det {\mathbf{Z}}^{(q)}) = \sum_{abcd}(Z^{(q)})^{-1}_{lk}(A^{p, q}_{ij, kl})_{ab, cd}Z^{(p)}_{ab}Z^{(q)}_{cd} = L^{p,q}, \end{equation} for every $p= 1, \cdots, n$,$q= 1, \cdots, n+1$, where $L^{p,q}$ is an affine function in the entries of $\{{\mathbf{Z}}^{(1)}, ..., {\mathbf{Z}}^{(n)}\}$, and $(Z^{(q)})^{-1}_{ij}$ are the entries of the inverse matrix $({{\mathbf{Z}}}^{(q)})^{-1}$. This last equation comes from the diffusion property for the operator $\Gamma$ (equation~\eqref{chain.rule.Gamma}), together with the fact that, for any matrix ${\mathbf{Z}}$ with entries $Z_{ij}$, $$\partial_{Z_{ij}} \log \det({\mathbf{Z}})= Z^{-1}_{ji},$$ where $Z^{-1}_{ij}$ are the entries of the inverse matrix ${\mathbf{Z}}^{-1}$. } \end{itemize} Even under above restrictions, it is still hard to give any complete description of such tensors $\{({\mathbf{A}}^{p, q}_{ij, kl})\}$. In the following we give two models of matrix Dirichlet process, which appear quite naturally as projections from the Brownian motion on $SU(N)$, from polar decompositions of complex matrix Brownian motions, and from complex Wishart processes, as we have mentioned before. \subsection{The first polynomial diffusion model on $\Delta_{n,d}$}\label{sec.model.i} Our first model is defined in the following Theorem~\ref{matrix.dirichlet}. As we will see, it appears naturally in some projections of diffusion models on $SU(N)$ and from polar decomposition of complex matrix Brownian motions. \begin{theoreme}\label{matrix.dirichlet} Let the matrix ${\mathbf{A}} = (A_{pq}), 1 \leq p, q \leq n+1$ be a symmetric matrix. Then, consider the diffusion $\Gamma_{{\mathbf{A}}}$ operator given by \begin{equation}\label{gamma.dirichlet} \Gamma_{{\mathbf{A}}}(Z^{(p)}_{ij}, Z^{(q)}_{kl}) = \delta_{pq}\sum^{n + 1}_{s=1} A_{sp}(Z^{(s)}_{il}Z^{(p)}_{kj} +Z^{(s)}_{kj}Z^{(p)}_{il}) - A_{pq}(Z^{(q)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(q)}_{kj}), \end{equation} for $1\leq p, q \leq n$, $1\leq i,j,k,l\leq d$. Following the notations of Section~\ref{subsec.matrix.Dirichlet.domain}, in this model we have $$ ({\mathbf{A}}^{p, q}_{ij, kl})^{ab, cd}_{r, s} = A_{rs}(\delta_{pq}\delta_{pr} - \delta_{pr}\delta_{qs})(\delta_{ak}\delta_{bj}\delta_{ci}\delta_{dl} + \delta_{ai}\delta_{bl}\delta_{ck}\delta_{dj}). $$ Then $\Gamma_{\mathbf{A}}$ is elliptic if and only if the matrix $A$ has non negative entries and is irreducible. In this case, $(\Delta_{n,d}, \Gamma_{{\mathbf{A}}}, D_{{\mathbf{a}}})$ is a polynomial model. Moreover, in this case, \begin{equation}\label{l.dirichlet} \LL_{{\mathbf{A}}, {\mathbf{a}}}(Z^{(p)}_{ij})= \sum^{n + 1}_{q }2(a_{p} + d - 1)A_{pq}Z^{(q)}_{ij} - \sum^{n + 1}_{q } 2(a_{q} + d - 1)A_{pq}Z^{(p)}_{ij}. \end{equation} \end{theoreme} We recall that a matrix $A$ with non negative entries is irreducible if and only if for any $p\neq q$, there exists a path $p= p_0, p_1, \cdots, p_k= q$ such that for any $i= 0, \cdots, k-1$, $A_{p_ip_{i+1}} \neq 0$. \begin{preuve} Recall that our coordinates are the entries $Z^{(k)}_{ij}$ of the Hermitian matrices ${\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)}$ and that ${\mathbf{Z}}^{(n + 1)} = \Id - \sum^{n}_{k=1}{\mathbf{Z}}^{(k)}$. The only requirement here (apart from the ellipticity property that we willl deal with below) is that equation~\eqref{eq.GammaLogDelta} is satisfied for this particular choice of the tensor $({\mathbf{A}}^{p,q}_{ij,kl})$. Thus for $1 \leq p, q \leq n$, we have $$ \Gamma_{{\mathbf{A}}}(\log \det({\mathbf{Z}}^{(q)}), Z^{(p)}_{ij}) = \delta_{pq}\sum^{n+1}_s2A_{sp}Z^{(s)}_{ij} - 2A_{pq}Z^{(p)}_{ij} ,$$ while $$ \Gamma_{{\mathbf{A}}}(\log \det({\mathbf{Z}}^{(n+1)}), Z^{(p)}_{ij}) = -2A_{(n+1)p}Z^{(p)}_{ij}, $$ which shows that the boundary equation is satisfied for this model. We now prove that on the matrix simplex $\Delta_{n,d}$ , $\Gamma_{{\mathbf{A}}}$ given by \eqref{gamma.dirichlet} is elliptic if and only if for $p, q = 1,...., n+1$, $A_{pq} > 0$. For fixed $(p, q)$, we consider $\Gamma_{{\mathbf{A}}}(Z^{(p)}_{ij}, Z^{(q)}_{kl})$ as the $(ij, kl)$ element in a $d^{2} \times d^{2}$ matrix; and $(\Gamma_{{\mathbf{A}}}(Z^{(p)}_{ij}, Z^{(q)}_{kl}))$ is the $(p, q)$ element in a $n\times n$ matrix of $d^2\times d^2$ matrices. Notice that we may write $$ (\Gamma_{{\mathbf{A}}}) = \sum_{1\leq p<q\leq n}A_{pq}\Gamma^{(pq)} + {\mathbf{A}}_{n+1}\Gamma^{(n+1)}, $$ where \begin{enumerate} \item $(\Gamma^{pq})$ is a $nd^{2} \times nd^{2}$ block matrix with \begin{eqnarray*} (\Gamma^{(pq)})_{pq, (ij, kl)} &=& - (Z^{(q)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(q)}_{kj}),\\ (\Gamma^{(pq)})_{qp, (ij, kl)} &=& - (Z^{(q)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(q)}_{kj}),\\ (\Gamma^{(pq)})_{pp, (ij, kl)} &=& Z^{(q)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(q)}_{kj},\\ (\Gamma^{(pq)})_{qq, (ij, kl)} &=& Z^{(q)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(q)}_{kj}, \end{eqnarray*} and all other entries are $0$; \item$\Gamma^{(n+1)}$ is a diagonal block matrix with $\Gamma^{(n+1)}_{pp, (ij, kl)} = Z^{(n+1)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(n+1)}_{kj}$ and all other entries are $0$; \item ${\mathbf{A}}_{n+1}$ is a diagonal block matrix of size $nd^{2} \times nd^{2}$ satisfying $({\mathbf{A}}_{n+1})_{pp} = A_{p(n+1)}\Id_{d^{2} \times d^{2}}$. \end{enumerate} We first observe that each operator associated with $\Gamma^{(pq)}$ is non negative. To see this, we have to check that for any sequence $(\Lambda)= (\Lambda^{(1)}, \cdots, \Lambda^{(n)})$ of Hermitian matrices with entries $\lambda^{(k)}_{ij}$, $$\sum_{rs, ij, kl} \Gamma^{(pq)}_{rs, (ij, \bar{kl})}\lambda^{(r)}_{ij} \bar \lambda^{(s)}_{kl} \geq 0.$$ In fact, \begin{eqnarray*} & &\sum_{rs, ij, kl} \Gamma^{(pq)}_{rs, (ij, \bar{kl})}\lambda^{(r)}_{ij} \bar \lambda^{(s)}_{kl} \\ &=& \Lambda^{(p)}\bar{\Lambda}^{(p)}\Gamma^{(pq)}_{pp} + \Lambda^{(q)}\bar{\Lambda}^{(q)}\Gamma^{(pq)}_{qq} + \Lambda^{(p)}\bar{\Lambda}^{(q)}\Gamma^{(pq)}_{pq} + \Lambda^{(q)}\bar{\Lambda}^{(p)}\Gamma^{(pq)}_{qp}\\ &=& \Lambda^{(p)}\bar{\Lambda}^{(p)}\Gamma^{(pq)}_{pp} + \Lambda^{(q)}\bar{\Lambda}^{(q)}\Gamma^{(pq)}_{pp} - \Lambda^{(p)}\bar{\Lambda}^{(q)}\Gamma^{(pq)}_{pp} - \Lambda^{(q)}\bar{\Lambda}^{(p)}\Gamma^{(pq)}_{pp} \\ &=& (\Lambda^{(p)} - \Lambda^{(q)})\Gamma^{(pq)}_{pp}(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)})\\ &=& \sum_{ij, kl} (\lambda^{(p)} - \lambda^{(q)})_{ij}(\bar{\lambda}^{(p)} - \bar{\lambda}^{(q)})_{kl}(Z^{(q)}_{ik}Z^{(p)}_{lj} + Z^{(p)}_{ik}Z^{(q)}_{lj})\\ &=& \textmd{trace}\,({\mathbf{Z}}^{(q)}(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)}){\mathbf{Z}}^{(p)}(\Lambda^{(p)} - \Lambda^{(q)})^{t}) \\ & & \ \ \ \ + \textmd{trace}\,({\mathbf{Z}}^{(p)}(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)}){\mathbf{Z}}^{(q)}(\Lambda^{(p)} - \Lambda^{(q)})^{t}). \end{eqnarray*} For any $p$, ${\mathbf{Z}}^{(p)}$ is Hermitian and non negative-definite, so are $(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)}){\mathbf{Z}}^{(p)}(\Lambda^{(p)} - \Lambda^{(q)})^{t}$, $(\Lambda^{(p)} - \Lambda^{(q)})^{t}{\mathbf{Z}}^{(p)}(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)})$, then we have \begin{eqnarray*} \textmd{trace}\,({\mathbf{Z}}^{(q)}(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)}){\mathbf{Z}}^{(p)}(\Lambda^{(p)} - \Lambda^{(q)})^{t}) &\geq& 0, \\ \textmd{trace}\,({\mathbf{Z}}^{(p)}(\bar{\Lambda}^{(p)} - \bar{\Lambda}^{(q)}){\mathbf{Z}}^{(q)}(\Lambda^{(p)} - \Lambda^{(q)})^{t}) &\geq& 0. \end{eqnarray*} therefore $\Gamma^{(pq)}$ are all non negative-definite matrices. Similarly, we have \begin{eqnarray*} & &\sum_{p, ij, kl} A_{p(n+1)}\Gamma^{(n+1)}_{pp, (ij, \bar{kl})}\lambda^{(p)}_{ij} \bar \lambda^{(p)}_{kl} \\ &=& \sum_{p, ij, kl} A_{p(n+1)}(Z^{(n+1)}_{ik}Z^{(p)}_{lj} +Z^{(p)}_{ik}Z^{(n+1)}_{lj})\lambda^{(p)}_{ij} \bar \lambda^{(p)}_{kl} \\ &=& \sum_{p} A_{p(n+1)}(\textmd{trace}\,({\mathbf{Z}}^{(n+1)}\bar \Lambda^{(p)}{\mathbf{Z}}^{(p)}(\Lambda^{(p)})^{t}) + \textmd{trace}\,({\mathbf{Z}}^{(n+1)}(\Lambda^{(p)})^{t}{\mathbf{Z}}^{(p)}\bar \Lambda^{(p)})), \end{eqnarray*} Thus we know $\Gamma^{(n+1)} \geq 0$. We then prove the following lemma, \begin{lemme}\label{positive-definite} Let $A$, $B$ be $d \times d$ Hermitian positive definite matrices. Then for a given $d \times d$ matrix $U$, if $\textmd{trace}\,(AUBU^*)=0$, then $U=0$. \end{lemme} \begin{preuve} Suppose $A$ has the spectral decomposition $A = P^{*}DP$, where $P$ is unitary and $D = {\rm diag}\{\lambda_1, \cdots, \lambda_d\}$ with all $\lambda_{i}$ positive. Then \begin{eqnarray*} \textmd{trace}\,(AUBU^*) = \textmd{trace}\,(P^{*}DPUBU^*) = \textmd{trace}\,(DPUBU^*P^{*}). \end{eqnarray*} Notice that since $B$ is positive definite, if $PU \neq 0$, then $PUBU^*P^{*}$ is also positive definite, whose elements on the diagonal are all positive, implying $\textmd{trace}\,(DPUBU^*P^{*}) > 0$. Therefore, $\textmd{trace}\,(AUBU^*) =0$ holds only when $PU = 0$. Since $P$ is unitary, this happens only when $U = 0$. \end{preuve} \medskip Recall the boundary equation of $\Delta_{n,d}$ \eqref{boundary.eq.mdp}, then we know all ${\mathbf{Z}}^{(i)}$ are positive definite inside $\Delta_{n,d}$ . Now we claim that $\Gamma_{{\mathbf{A}}}$ is elliptic inside $\Delta_{n,d}$ if and only if ${\mathbf{A}}$ has non negative entries and is irreducible. Previous computations show that $\Gamma_{{\mathbf{A}}} \geq 0$. If there exists $\Lambda_{0}$ such that $\Lambda_{0} \Gamma_{{\mathbf{A}}} \Lambda_{0}^{*} = 0$, we have \begin{eqnarray}\label{ellipticity} & &\sum^{n}_{p < q}A_{pq}(\textmd{trace}\,({\mathbf{Z}}^{(q)}(\bar{\Lambda}_{0}^{(p)} - \bar{\Lambda}_{0}^{(q)}){\mathbf{Z}}^{(p)}(\Lambda^{(p)}_{0} - \Lambda_{0}^{(q)})^{t}) \\ \nonumber& & \ \ \ \ + \textmd{trace}\,({\mathbf{Z}}^{(p)}(\bar{\Lambda}_{0}^{(p)} - \bar{\Lambda}_{0}^{(q)}){\mathbf{Z}}^{(q)}(\Lambda_{0}^{(p)} - \Lambda^{(q)})_{0}^{t})) \\ \nonumber & & \ \ \ \ + \sum^{n}_{p}A_{p(n+1)}(\textmd{trace}\,({\mathbf{Z}}^{(n+1)}\bar \Lambda^{(p)}{\mathbf{Z}}^{(p)}(\Lambda^{(p)})^{t}) \\ \nonumber& & \ \ \ \ + \textmd{trace}\,({\mathbf{Z}}^{(n+1)}(\Lambda^{(p)})^{t}{\mathbf{Z}}^{(p)}\bar \Lambda^{(p)})) = 0. \end{eqnarray} Since all the terms in the previous sum are non-negative, they all vanish. By the fact that ${\mathbf{A}}$ is irreducible, we know that for any $1 \leq p \leq n$, there exists a path connecting $p$ and $n+1$, noted by $p, p_{1}, p_{2}, \cdots, p_{r}, (n+1)$, which leads to $A_{pp_{1}}, A_{p_{1}p_{2}}, \cdots, A_{p_{r}(n+1)} > 0$. Then by Lemma \ref{positive-definite}, we have $$ \Lambda^{(p)}_{0} = \Lambda^{(p_{1})}_{0} = \cdots = \Lambda^{(p_{r})}_{0} = 0, $$ i.e. $\Lambda_{0} = 0$, which implies that $\Gamma_{{\mathbf{A}}}$ is elliptic. Conversely, when $\Gamma_{{\mathbf{A}}}$ is elliptic, then if $\Lambda_{0} \Gamma_{{\mathbf{A}}} \Lambda_{0}^{*} = 0$, we must have $\Lambda_{0} = 0$. First we prove that ${\mathbf{A}}$ has non negative entries. For given $1 \leq p \leq (n+1)$, assume that some $A_{pq}<0$ for some $q\neq p$. Then, choose the sequence $\Lambda_{1} = (\Lambda^{(1)}_{1}, \cdots , \Lambda_{1}^{(n)})$ to be such that $\Lambda^{(p)} = (({\mathbf{Z}}^{(p)})^{-\frac{1}{2}})^{t}$ and all others are $0$, then from $\Lambda_{1}\Gamma_{{\mathbf{A}}}\Lambda_{1}^{*} > 0$, we conclude that for $({\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)}) \in \Delta_{n,d}$, \begin{eqnarray*} \sum^{n+1}_{r \neq p} 2A_{pr}\textmd{trace}\,({\mathbf{Z}}^{(r)}) > 0. \end{eqnarray*} Now, choose $Z^{(q)}= (1-\epsilon)\Id$ and $Z^{(r)}= \frac{\epsilon}{n} \Id$ for $r\neq q$, $0 < \epsilon < 1$. Then when $\epsilon < \frac{|A_{pq}|}{\sum^{n+1}_{s \neq p}|A_{ps}|}$ we obtain a contradiction. Now suppose ${\mathbf{A}}$ is not irreducible, then it has at least two strongly connected components ${\mathcal A }_{1}$, ${\mathcal A }_{2}$, such that $A_{pq} = 0$ for $p \in {\mathcal A }_{1}$, $q \in {\mathcal A }_{2}$. Suppose $(n+1) \in {\mathcal A }_{1}$, then for all $p \in {\mathcal A }_{1}$, choose $\Lambda^{(p)}_{0} = 0$, while for $q \in {\mathcal A }_{2}$, $\Lambda^{(q)}_{0} \neq 0$, such that we have $\Lambda_{0} \neq 0$ which satisfies $\Lambda_{0} \Gamma_{{\mathbf{A}}} \Lambda_{0}^{*} = 0$. Thus there is a contradiction, so we must have ${\mathbf{A}}$ to be irreducible. Finally, by a direct computation, we have \begin{eqnarray*} \LL_{{\mathbf{A}}, {\mathbf{a}}}(Z^{(p)}_{ij}) &=& \sum^{n+1}_{q=1}(a_{q} - 1)\Gamma(\log \det({\mathbf{Z}}^{(q)}), Z^{(p)}_{ij}) + \sum^{n}_{q=1}\sum_{kl}\partial_{Z^{(q)}_{kl}}\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl})\\ &=& \sum^{n+1}_{q }2(a_{p} + d - 1)A_{pq}Z^{(q)}_{ij} - \sum^{n+1}_{q } 2(a_{q} + d - 1)A_{pq}Z^{(p)}_{ij}. \end{eqnarray*} \end{preuve} \medskip In the sequel, we give two natural constructions of matrix Dirichlet process, the first one from the Brownian motion on $SU(N)$, the other from polar decomposition of Brownian complex matrices. \subsubsection{The construction from $SU(N)$}\label{construction2} In this section, we show that the matrix Dirichlet processes can be realized by the projections from Brownian motion on $SU(N)$. The construction relies on the matrix-extracting procedure, extending the construction of matrix Jacobi processes described in~\cite{Doumerc}. In the special case where all the parameters $A_{ij}$ are equal to $d$, the associated generator may be considered as an image of the Casimir operator on $SU(N)$, whenever the coefficients $a_i$ in the measure are integers. Moreover, for the general case, similar to the construction in the scalar case in Proposition~\ref{prop.general.scalar.case}, we provide a construction from more general Brownian motions on $SU(N)$, where the generator is no longer the Casimir operator on $SU(N)$. The so-called "matrix-extracting procedure" is the following: consider a matrix $u$ on $SU(N)$, then take the first $d$ lines, and split the set of all $N$ columns into $(n + 1)$ disjoint sets $I_1, ..., I_{n +1}$. For $1 \leq i \leq (n+1)$, define $d_i = |I_i|$ such that $N = d_1+\cdots + d_{n+1}$. Then we get $(n+1)$ extracted matrices $\{{\mathbf{W}}^{(i)}\}$, respectively of size $d \times d_1$, $d \times d_{2}$,..., $d \times d_{n +1}$. The matrix Jacobi process is obtained by considering the Hermitian matrix ${\mathbf{W}}^{(1)}{\mathbf{W}}^{(1)}{}^*$. Here, we extend this procedure, defining ${\mathbf{Z}}^{(i)} = {\mathbf{W}}^{(i)}({\mathbf{W}}^{(i)})^{*}$ for $i =1, ..., n+1$. Then, $\sum_{i=1}^{n+1} {\mathbf{Z}}^{(i)}= \Id$, and the process $({\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)})$ lives in the complex matrix Dirichlet simplex $\Delta_{d, n}$. The compact Lie group $SU(N)$ is semi-simple compact. There is, up to a scaling constant, a unique elliptic diffusion operator on it which commutes both with the right and the left multiplication. This operator is called the Casimir operator, see \cite{BOZ}, \cite{zribi} for more details. The Brownian motion on $SU(N)$ is the diffusion process which has the Casimir operator as its generator. It may be described by the vector fields ${\mathcal V}_{R_{ij}}$, ${\mathcal V}_{S_{ij}}$ and ${\mathcal V}_{D_{ij}}$ which are given on the entries $\{u_{ij}\}$ of ${\mathbf{u}} \in SU(N)$ matrix as \begin{eqnarray} \label{casimir.r}{\mathcal V}_{R_{ij}} &=& \sum_{k}\big(u_{kj}\partial_{u_{ki}} - u_{ki}\partial_{u_{kj}} + \bar{u}_{kj}\partial_{\bar{u}_{ki}} - \bar{u}_{ki}\partial_{\bar{u}_{kj}}\big),\\ \label{casimir.s}{\mathcal V}_{S_{ij}} &=& i\sum_{k}\big( u_{kj}\partial_{u_{ki}} + u_{ki}\partial_{u_{kj}}- \bar{u}_{kj}\partial_{\bar{u}_{ki}} - \bar{u}_{ki}\partial_{\bar{u}_{kj}} \big),\\ \label{casimir.d}{\mathcal V}_{D_{ij}} &=& i\sum_{k}\big(u_{ki}\partial_{u_{ki}} - u_{kj}\partial_{u_{kj}} - \bar{u}_{ki}\partial_{\bar{z}_{ki}} + \bar{u}_{kj}\partial_{\bar{u}_{kj}}\big). \end{eqnarray} Then for the Casimir operator $\Delta_{SU(N)}$ on $SU(N)$, we have \begin{eqnarray*}\label{casimir.opt} \Delta_{SU(N)} = \frac{1}{4N}\sum_{i < j}({\mathcal V}^{2}_{R_{ij}} + {\mathcal V}^{2}_{S_{ij}} + \frac{2}{N}{\mathcal V}^{2}_{D_{ij}}), \end{eqnarray*} and \begin{eqnarray*} \Gamma_{SU(N)}(f, g) &=& \frac{1}{4N}(\sum_{i < j}{\mathcal V}_{R_{ij}}(f){\mathcal V}_{R_{ij}}(g) + {\mathcal V}_{S_{ij}}(f){\mathcal V}_{S_{ij}}(g) + \frac{2}{N}{\mathcal V}_{D_{ij}}(f){\mathcal V}_{D_{ij}}(g)). \end{eqnarray*} From this, a simple computation provides \begin{eqnarray*} \Gamma_{SU(N)}(u_{ij}, u_{kl}) &=& -\frac{1}{2N}u_{il}u_{kj} + \frac{1}{2N^{2}}u_{ij}u_{kl},\\ \Gamma_{SU(N)}(u_{ij}, \bar{u}_{kl}) &=& \frac{1}{2N}\delta_{ik}\delta_{jl} - \frac{1}{2N^2}u_{ij}\bar{u}_{kl},\\ \Delta_{SU(N)}(u_{ij}) &=& -\frac{N^{2}-1}{2N^{2}}u_{ij}, \ \ \ \ \Delta_{SU(N)}(\bar{u}_{ij}) = -\frac{N^{2}-1}{2N^{2}}\bar{u}_{ij},\\ \end{eqnarray*} and these formulas describe entirely the Brownian motion on $SU(N)$. A simple application of the diffusion property (equations~\eqref{chain.rule.L} and \eqref{chain.rule.Gamma}) yields, for $1 \leq p, q \leq n$, for the entries $Z^{(p)}_{ij}$ of the matrices ${\mathbf{Z}}^{(p)}$, \begin{eqnarray} & &\label{gamma.ug1}\Gamma_{SU(N)}(Z^{(p)}_{ij}, Z^{(q)}_{kl}) = \frac{1}{2N}\delta_{pq}(\delta_{il}Z^{(p)}_{kj} + \delta_{kj}Z^{(p)}_{il}) - \frac{1}{2N}(Z^{(p)}_{kj}Z^{(q)}_{il} + Z^{(p)}_{il}Z^{(q)}_{kj}), \\ & &\label{l.ug2}\Delta_{SU(N)}(Z^{(p)}_{ij}) = - Z^{(p)}_{ij} + \frac{1}{N}d_{p}\delta_{ij}. \end{eqnarray} Comparing \eqref{gamma.ug1}, \eqref{l.ug2} with \eqref{gamma.dirichlet} and \eqref{l.dirichlet}, we obtain a matrix Dirichlet operator described in Theorem~\ref{matrix.dirichlet}, with $$ A_{pq}= \frac{1}{2N}, \ \ \ a_i = d_i - d + 1, $$ for any $1 \leq p, q \leq n$ and $1 \leq i \leq (n + 1)$. Therefore, the density of the reversible measure is \begin{equation} C\prod^{n}_{p = 1}\det({\mathbf{Z}}^{(p)})^{d_{p} - d}\det(\Id - \sum^{n}_{p=1}{\mathbf{Z}}^{(p)})^{d_{n+1} - d}. \end{equation} For this measure to be finite, we need $d_i > d -1$ for all $1 \leq i \leq (n+1)$, i.e., $d_i \geq d$ since these parameters are integers. It is worth to observe that this restriction is necessary for the matrices ${\mathbf{Z}}^{(i)}$ to be non degenerate. If it is not satisfied, the matrices $Z^{(i)}$ live of an algebraic manifold and their law may not have any density with respect to the Lebesgue measure. To summarize, we have \begin{proposition} The image of the Brownian motion on $SU(N)$ under the matrix-extracting procedure is a diffusion process on the complex matrix simplex $\Delta_{d,n}$ with carré du champ $\Gamma_{{\mathbf{A}}}$ and reversible measure $D_{a_1, \cdots, a_{n+1}}$ where $A_{pq}= \frac{1}{2N}, p, q= 1, \cdots, n+1, p\neq q$, $a_i = d_i -d + 1$. \end{proposition} As a corollary, we get \begin{corollaire} Whenever $d_i \geq d$, $i= 1, \cdots, n+1$, the image of the Haar measure on $SU(N)$ through the matrix-extracting procedure is the matrix Dirichlet measure $D_{d_1-d, \cdots, d_{n+1}-d}$. \end{corollaire} For the general case where the parameters $A_{ij}$ are not equal, we may follow Proposition~\ref{prop.general.scalar.case}. For $p \neq q$, we define the following $\LL^{(pq)}$ acting on the matrix simplex \\ $\{{\mathbf{Z}}^{(i)}, \sum^{n}_{i=1} {\mathbf{Z}}^{(i)} \leq \Id\}$, $$\LL^{ (pq)}= \sum_{i\in I_p, j\in I_q } {\mathcal V}_{R_{ij}}^2+ {\mathcal V}_{S_{ij}}^2+ \frac{2}{N} {\mathcal V}_{D_{ij}}^2,$$ with its corresponding carré du champ operator $\Gamma^{(p q)}$, \begin{eqnarray*} \Gamma^{(p q)}(f, g) = \sum_{i \in I_p, j \in I_q}{\mathcal V}_{R_{ij}}(f){\mathcal V}_{R_{ij}}(g) + {\mathcal V}_{S_{ij}}(f){\mathcal V}_{S_{ij}}(g) + \frac{2}{N}{\mathcal V}_{D_{ij}}(f){\mathcal V}_{D_{ij}}(g). \end{eqnarray*} \begin{lemme}\label{xeq0} For the entries $u_{ij}$ of an $SU(N)$ matrix, and denoting by $Z^{(p)}_{ij}$ the entries of the extracted matrix ${\mathbf{Z}}^{(p)}$, we have \begin{eqnarray} \label{xeq1}{\mathcal V}_{R_{ij}}Z^{(p)}_{ab} &=& {\mathbf{1}}_{i \in I_p}(u_{aj}\bar{u}_{bi} + u_{ai}\bar{u}_{bj})- {\mathbf{1}}_{j \in I_p}(u_{ai}\bar{u}_{bj} + u_{aj}\bar{u}_{bi}),\\ \label{xeq2}{\mathcal V}_{S_{ij}}Z^{(p)}_{ab} &=& \sqrt{-1}\big({\mathbf{1}}_{i \in I_p}(u_{aj}\bar{u}_{bi} - u_{ai}\bar{u}_{bj}) + {\mathbf{1}}_{j \in I_p}(u_{ai}\bar{u}_{bj} - u_{aj}\bar{u}_{bi})\big),\\ \label{xeq3}{\mathcal V}_{D_{ij}}Z^{(p)}_{ab} &=& 0, \end{eqnarray} and \begin{eqnarray} \label{xeq8}{\mathcal V}^{2}_{R_{ij}}Z^{(p)}_{ab} &=& {\mathbf{1}}_{i \in I_p}(2u_{aj}\bar{u}_{bj} - 2u_{ai}\bar{u}_{bi}) - {\mathbf{1}}_{j \in I_p}(2u_{aj}\bar{u}_{bj} - 2u_{ai}\bar{u}_{bi}),\\ \label{xeq9}{\mathcal V}^{2}_{S_{ij}}Z^{(p)}_{ab} &=& {\mathbf{1}}_{i \in I_p}(2u_{aj}\bar{u}_{bj} - 2u_{ai}\bar{u}_{bi}) - {\mathbf{1}}_{j \in I_p}(2u_{aj}\bar{u}_{bj} - 2u_{ai}\bar{u}_{bi}). \end{eqnarray} Then, \begin{eqnarray} \label{xeq4}\Gamma^{(p q)}(Z^{(p)}_{ab}, Z^{(q)}_{cd}) &=& - 2Z^{(p)}_{cb}Z^{(q)}_{ad} - 2Z^{(p)}_{ad}Z^{(q)}_{cb} , \\ \label{xeq5}\Gamma^{(p q)}(Z^{(p)}_{ab}, Z^{(p)}_{cd}) &=& 2Z^{(p)}_{cb}Z^{(q)}_{ad} + 2Z^{(p)}_{ad}Z^{(q)}_{cb}, \\ \label{xeq6}\Gamma^{(p q)}(Z^{(q)}_{ab}, Z^{(q)}_{cd}) &=& 2Z^{(p)}_{cb}Z^{(q)}_{ad} + 2Z^{(p)}_{ad}Z^{(q)}_{cb}. \end{eqnarray} For a pair $(r, s) \neq (p, q)$, we have \begin{eqnarray} \label{xeq7}\Gamma^{(pq)}(Z^{(r)}_{ab}, Z^{(s)}_{cd}) = 0. \end{eqnarray} Moreover, \begin{eqnarray} \label{xeq10}\LL^{(p q)}(Z^{(r)}_{ij}) &=& {\mathbf{1}}_{r=p}4(d_{p}Z^{(q)}_{ij} - d_{q}Z^{(p)}_{ij}) - {\mathbf{1}}_{r=q}4(d_{p}Z^{(q)}_{ij} - d_{q}Z^{(p)}_{ij}). \end{eqnarray} \end{lemme} \begin{rmq} The action of ${\mathcal V}_{D_{ij}}$ vanishes on the variables $Z^{(p)}_{ab}$ such that indeed we could replace $\LL^{(pq)}$ by $$\LL^{ (pq)}= \sum_{i\in I_p, j\in I_q } {\mathcal V}_{R_{ij}}^2+ {\mathcal V}_{S_{ij}}^2,$$ and the dimension $N$ disappears from the definition. \end{rmq} \begin{preuve} Recall \eqref{casimir.r}. Then letting ${\mathcal V}_{R_{ij}}$ act on ${\mathbf{Z}}^{p} = {\mathbf{W}}^{p}({\mathbf{W}}^{p})^{*}$ and writing $u_{ij}$ for the entries of ${\mathbf{W}}^{(p)}$, which are indeed entries of an $SU(N)$ matrix, we have \begin{eqnarray*} {\mathcal V}_{R_{ij}}Z^{(p)}_{ab} &=& \sum_{r \in I_p}{\mathcal V}_{R_{ij}}(u_{ar}\bar{u}_{br})\\ &=& {\mathbf{1}}_{i \in I_p}(u_{aj}\bar{u}_{bi} + u_{ai}\bar{u}_{bj})- {\mathbf{1}}_{j \in I_p}(u_{ai}\bar{u}_{bj} + u_{aj}\bar{u}_{bi}). \end{eqnarray*} Similarly we can prove \eqref{xeq2}, \eqref{xeq3}. By \eqref{xeq1}, \eqref{xeq2}, we obtain \begin{eqnarray*} {\mathcal V}_{R_{ij}}(u_{aj}\bar{u}_{bi} + u_{ai}\bar{u}_{bj}) &=& 2u_{aj}\bar{u}_{bj} - 2u_{ai}\bar{u}_{bi},\\ {\mathcal V}_{S_{ij}}(u_{aj}\bar{u}_{bi} - u_{ai}\bar{u}_{bj}) &=& -\sqrt{-1}(2u_{aj}\bar{u}_{bj} - 2u_{ai}\bar{u}_{bi}), \end{eqnarray*} then \eqref{xeq7}, \eqref{xeq8} follow. By \eqref{xeq1}, \eqref{xeq2} and \eqref{xeq3} we have \begin{eqnarray*} & &\Gamma^{(pq)}(Z^{(p)}_{ab}, Z^{(q)}_{cd}) \\ &=& \sum_{i \in I_p, j \in I_q} {\mathcal V}_{R_{ij}}(Z^{(p)}_{ab}){\mathcal V}_{R_{ij}}(Z^{(q)}_{cd}) + {\mathcal V}_{S_{ij}}(Z^{(p)}_{ab}){\mathcal V}_{S_{ij}}(Z^{(q)}_{cd})\\ &=& - 2Z^{(p)}_{cb}Z^{(q)}_{ad} - 2Z^{(p)}_{ad}Z^{(q)}_{cb} , \end{eqnarray*} which proves \eqref{xeq4}. In the same way, we can deduce \eqref{xeq5}, \eqref{xeq6} and \eqref{xeq7}. \eqref{xeq10} is proved by \eqref{xeq8}, \eqref{xeq9} and \begin{eqnarray*} \LL^{(p q)}(Z^{(r)}_{ij}) &=& \sum_{k \in I_p, l \in I_{q}}{\mathcal V}^{2}_{R_{kl}}Z^{(r)}_{ij} + {\mathcal V}^{2}_{S_{kl}}Z^{(r)}_{ij}\\ &=& {\mathbf{1}}_{r=p}4(d_{p}Z^{(q)}_{ij} - d_{q}Z^{(p)}_{ij}) - {\mathbf{1}}_{r=q}4(d_{p}Z^{(q)}_{ij} - d_{q}Z^{(p)}_{ij}). \end{eqnarray*} \end{preuve} \medskip Now by Lemma \ref{xeq0} we may derive the following conclusion. \begin{proposition} $(\mathbf{Z}^{(1)}, \cdots, \mathbf{Z}^{(n)})$ is a closed system for any $\LL^{(pq)}$ and the image of $$ \frac{1}{2}\sum^{n+1}_{p < q} A_{pq}\LL^{(pq)} $$ is the operator $\LL_{{\mathbf{A}}, {\mathbf{a}}}$ in Theorem \ref{matrix.dirichlet} with $a_i = d_i -d + 1$. \end{proposition} \subsubsection{The construction from polar decomposition}\label{construction1} In this section, we consider the spectral decomposition of a complex Brownian matrix ${\mathbf{m}}$. Any complex matrix ${\mathbf{m}}$ may be written as ${\mathbf{m}} = {\mathbf{V}} {\mathbf{N}}$, where ${\mathbf{V}}$ is unitary and ${\mathbf{N}}$ is non negative Hermitian, and uniquely determined as $\sqrt{{\mathbf{m}}^{*}{\mathbf{m}}}$. A complex Brownian matrix is a diffusion process whose generator may be described on the entries $m_{ij}$ of the $d \times d$ complex matrix ${\mathbf{m}}$ by $$ \Gamma(m_{ij}, m_{kl}) = 0, \ \ \Gamma(m_{ij}, \bar{m}_{kl}) = 2\delta_{ik}\delta_{jl}, \ \ \ \LL(m_{ij}) = 0. $$ This just describes the fact that the entries $m_{ij}$ are independent complex Brownian motions. The polar decomposition of a complex Brownian matrix will be discussed in Appendix~\ref{polar}. Now let ${\mathbf{H}} = {\mathbf{m}}^{*}{\mathbf{m}}$ and suppose ${\mathbf{H}}$ has the spectral decomposition ${\mathbf{H}} = {\mathbf{U}} \bf\Sigma {\mathbf{U}}^{*}$, where ${\mathbf{U}}$ is unitary and $\bf\Sigma$ is diagonal, denoted by $\bf\Sigma = {\rm diag}\{X_1, ..., X_d\}$. Since ${\mathbf{H}}$ is positive definite, we may write $\bf \Sigma = {\mathbf{D}}^{2}$, where ${\mathbf{D}} = {\rm diag} \{x_1, \cdots, x_d\}$, $x_i \geq 0$ and $X_{i} = x^{2}_{i}$ for $1 \leq i \leq d$. Write ${\mathbf{U}} = (U^{(1)}, \cdots, U^{(d)})$, where $U^{(i)}, 1\leq i \leq d$ is a $d-$column vector. Define ${\mathbf{Z}}^{(k)} = U^{(k)}(U^{(k)})^{*}$, such that for fixed $k$, ${\mathbf{Z}}^{(k)}$ is a Hermitian matrix, and also \begin{eqnarray*} \sum^{d}_{k =1}{\mathbf{Z}}^{(k)} = \Id. \end{eqnarray*} \begin{proposition} The diffusion operators of $({\mathbf{D}}, {\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(d-1)})$ are given by \begin{eqnarray*} \sum^{d}_{i=1} \big(\partial^{2}_{x_i} + (\frac{1}{x_i} + 4x_i\sum_{j \neq i}\frac{1}{x^{2}_{i} - x^{2}_{j}})\partial_{x_i} \big) + \LL_{{\mathbf{A}}}, \end{eqnarray*} where $\LL_{{\mathbf{A}}}$ is described in Theorem \ref{matrix.dirichlet} with \begin{eqnarray*} A_{pq} = 2\frac{x^{2}_{p} + x^{2}_{q}}{(x^{2}_{p} - x^{2}_{q})^2}, \ \ a_{i} = -d, \end{eqnarray*} for $1 \leq p, q, i \leq d$. Then the reversible measure is integrable only when $d=1$, which is indeed the Lebesgue measure on the simplex, corresponding to the complex scalar Dirichlet process. \end{proposition} Indeed, the matrices ${\mathbf{Z}}^{(i)}$ have rank one, and therefore their law may not have a density with respect to the Lebesgue measure. This model for matrix Dirichlet processes is in fact degenerate and lives on the boundary of the domain $\Delta_{n,d}$. \begin{preuve} The computations of Appendix~\ref{polar} provide \begin{eqnarray*} \label{ceq1}\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl}) &=& -r_{pq}(Z^{(q)}_{il}Z^{(p)}_{kj} +Z^{(p)}_{il}Z^{(q)}_{kj}) + \delta_{pq}\sum^{d}_{s = 1}r_{sp}(Z^{(s)}_{il}Z^{(p)}_{kj} +Z^{(s)}_{kj}Z^{(p)}_{il}), \\ \label{ceq2}\LL(Z^{(p)}_{ij}) &=& 2\sum^{d}_{1 \leq q \neq p}r_{pq}(Z^{(p)}_{ij} - Z^{(q)}_{ij}), \end{eqnarray*} where $r_{pq} = 2\frac{X_{p} + X_{q}}{(X_{p} - X_{q})^2}$, which the expected result. \end{preuve} \medskip In the general case, when $d>1$, we may consider the diffusion of $\{v^{(k)} = Z^{(k)}_{11}, 1 \leq k \leq d-1\}$, which is not degenerate and is indeed the complex scalar Dirichlet process. \begin{rmq} In fact, since ${\mathbf{N}} = \sqrt{{\mathbf{H}}}$, we may write ${\mathbf{N}} = {\mathbf{U}} {\mathbf{D}} {\mathbf{U}}^{*}$, where ${\mathbf{D}} = \sqrt{\bf\Sigma}$. This leads to ${\mathbf{m}} = {\mathbf{W}} {\mathbf{D}} {\mathbf{U}}^{*}$, where ${\mathbf{W}} = {\mathbf{V}} {\mathbf{U}}$. Now we may define $\{{\mathbf{Y}}^{(k)} = W^{(k)}(W^{(k)})^{*}, 1 \leq k \leq d-1 \}$, where $W^{k}$ is the $k$-th column vector in ${\mathbf{W}}$. Then following Appendix \ref{polar}, we have for the entries $Y^{(p)}_{ij}$ of ${\mathbf{Y}}^{(p)}$ \begin{eqnarray*} \Gamma(Y^{(p)}_{ij}, Y^{(q)}_{kl}) &=& -\omega_{pq}(Y^{(q)}_{il}Y^{(p)}_{kj} +Y^{(p)}_{il}Y^{(q)}_{kj}) + \delta_{pq}\sum^{d}_{s = 1}\omega_{sp}(Y^{(s)}_{il}Y^{(p)}_{kj} +Y^{(s)}_{kj}Y^{(p)}_{il}), \\ \LL(Y^{(p)}_{ij}) &=& 2\sum^{d}_{1 \leq q \neq p}\omega_{pq}(Y^{(q)}_{ij} - Y^{(p)}_{ij}), \end{eqnarray*} where $\omega_{pq} = r_{pq}$ for $p \neq q$, showing that it is exactly the same situation as $\{{\mathbf{Z}}^{(k)}, 1 \leq k \leq d-1\}$. Although the generators of ${\mathbf{W}}$ are different from those of ${\mathbf{U}}$, as we will see in Appendix A, they have no influence on the diffusion operators of ${\mathbf{Y}}^{(k)}$ and ${\mathbf{Z}}^{(k)}$, because the difference lies in $\omega_{pp}$ for $p =1, ..., d$, which make no contribution in the above formulas. It is also worth to point out that if we consider the left polar decomposition (the previous one is known as the right polar decomposition), i.e., ${\mathbf{m}} = {\mathbf{N}}' {\mathbf{V}}$, where ${\mathbf{V}}$ is the same unitary matrix and ${\mathbf{N}}' = \sqrt{{\mathbf{m}} {\mathbf{m}}^{*}}$, then ${\mathbf{W}}$ can be viewed as the unitary part in the spectral decomposition of ${\mathbf{H}}' = {\mathbf{m}} {\mathbf{m}}^{*} = {\mathbf{W}} \bf\Sigma {\mathbf{W}}^{*}$. \end{rmq} \subsection{The second polynomial diffusion model on $\Delta_{n,d}$}\label{sec.model.ii} The second model is given by the following Theorem~\ref{gamma.dirichlet2}. As in the scalar case (Section \ref{sec.scalar.simplex}), it can be naturally derived from the Ornstein-Uhlenbeck process on complex matrices. \begin{theoreme}\label{model2} Let ${\mathbf{A}}$ be a $d \times d$ positive-definite Hermitian matrix and ${\mathbf{B}}$ be a $d^{2} \times d^{2}$ positive-definite Hermitian matrix. Consider the diffusion $\Gamma_{{\mathbf{A}}, {\mathbf{B}}}$ operator given by \begin{eqnarray}\label{gamma.dirichlet2} & &\Gamma_{{\mathbf{A}}, {\mathbf{B}}}(Z^{(p)}_{ij}, Z^{(q)}_{kl}) = \delta_{pq}(A_{il}Z^{(p)}_{kj} + A_{kj}Z^{(p)}_{il}) - A_{kj}(Z^{(p)}Z^{(q)})_{il} - A_{il}(Z^{(q)}Z^{(p)})_{kj}\\ \nonumber & & \ \ \ \ + \sum_{ab}\big( B_{ia, lb}Z^{(p)}_{aj}Z^{(q)}_{kb} + B_{aj, bk}Z^{(p)}_{ia}Z^{(q)}_{bl} - B_{aj, lb}Z^{(p)}_{ia}Z^{(q)}_{kb} - B_{ia, bk}Z^{(p)}_{aj}Z^{(q)}_{bl} \big) , \end{eqnarray} for $1\leq p, q \leq n$, $1\leq i,j,k,l\leq d$. Following the notations of Section~\ref{subsec.matrix.Dirichlet.domain}, in this model we have \begin{eqnarray*} ({\mathbf{A}}^{p, q}_{ij, kl})^{ab, cd}_{r, s} &=& \delta_{pq}\delta_{pr}A_{cd}(\delta_{ak}\delta_{bj}\delta_{ci}\delta_{dl} + \delta_{ai}\delta_{bl}\delta_{ck}\delta_{dj}) - \delta_{pr}\delta_{qs}(A_{kj}\delta_{ai}\delta_{bc}\delta_{dl} + A_{il}\delta_{ck}\delta_{ad}\delta_{bj}) \\ & &+ \delta_{pr}\delta_{qs}( B_{ia, ld}\delta_{bj}\delta_{ck} + B_{bj, ck}\delta_{ai}\delta_{dl} - B_{bj, ld}\delta_{ai}\delta_{ck} - B_{ia, ck}\delta_{bj}\delta_{dl}). \end{eqnarray*} Then $(\Delta_{n,d}, \Gamma_{{\mathbf{A}}, {\mathbf{B}}}, D_{a_1, \cdots, a_{n+1} })$ is a polynomial model. Moreover, in this case, \begin{eqnarray}\label{l.dirichlet2} \LL_{{\mathbf{A}}, {\mathbf{B}}, {\mathbf{a}}}(Z^{(p)}_{ij}) &=& 2(a_p - 1 + d)A_{ij} - \sum^{n}_{q =1} (a_q - 1 + d)\big((AZ^{(p)})_{ij} + (Z^{(p)}A)_{ij}\big)\\ \nonumber & & -(a_{n+1} -1) \big((AZ^{(p)})_{ij} + (Z^{(p)}A)_{ij}\big)- 2A_{ij}\textmd{trace}\,(Z^{(p)})\\ \nonumber & &\ \ \ + \sum_{ab} \big(B_{ia, jb}Z^{(p)}_{ab} + B_{bj, ai}Z^{(p)}_{ab} - B_{ia, ba}Z^{(p)}_{bj} - B_{bj, ba}Z^{(p)}_{ia} \big), \end{eqnarray} where ${\mathbf{Z}}^{(n+1)} = \Id - \sum^{n}_{p=1}{\mathbf{Z}}^{(p)}$. \end{theoreme} \begin{preuve} First, let us show that equations~\eqref{gamma.dirichlet2} and~\eqref{l.dirichlet2} define a polynomial model. In fact, for $1 \leq p, q \leq n$, \begin{eqnarray*} \Gamma(Z^{(p)}_{ij}, \log \det {\mathbf{Z}}^{(q)}) &=& \sum_{kl}(Z^{(q)})^{-1}_{lk}\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl})\\ &=& 2A_{ij}\delta_{pq} - (AZ^{(p)})_{ij} - (Z^{(p)}A)_{ij},\\ \Gamma(Z^{(p)}_{ij}, \log \det {\mathbf{Z}}^{(n+1)}) &=& - (AZ^{(p)})_{ij} - (Z^{(p)}A)_{ij}, \end{eqnarray*} which is a polynomial model by Proposition \ref{prop.poly.model.gal}. Direct computations yield \begin{eqnarray*} \LL(Z^{(p)}_{ij}) &=& \sum^{n+1}_{q=1} (a_q - 1) \Gamma(Z^{(p)}_{ij}, \log \det Z^{(q)}) + \sum^{n}_{q} \partial_{Z^{(q)}_{kl}}\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl}) \\ &=& 2(a_p - 1 + d)A_{ij} - \sum^{n}_{q =1} (a_q - 1 + d)\big((AZ^{(p)})_{ij} + (Z^{(p)}A)_{ij}\big) \\ & &-(a_{n+1} -1) \big((AZ^{(p)})_{ij} + (Z^{(p)}A)_{ij}\big)- 2A_{ij}\textmd{trace}\,(Z^{(p)})\\ & & + \sum_{ab} \big( B_{ia, jb}Z^{(p)}_{ab} + B_{bj, ai}Z^{(p)}_{ab} - B_{ia, ba}Z^{(p)}_{bj} - B_{bj, ba}Z^{(p)}_{ia} \big). \end{eqnarray*} Now we prove that if ${\mathbf{A}}$ is a $d \times d$ Hermitian and positive-definite matrix and ${\mathbf{B}}$ is a $d^{2} \times d^{2}$ Hermitian and positive-definite matrix, then $\Gamma_{{\mathbf{A}}, {\mathbf{B}}}$ is elliptic inside the matrix simplex $\Delta_{n, d}$. In fact, consider $\Gamma_{{\mathbf{A}}, {\mathbf{B}}}$ as a $n \times n$ block matrix, and each block is of size $d^{2} \times d^{2}$, then we may write \begin{eqnarray*} \Gamma_{{\mathbf{A}}, {\mathbf{B}}} = \Gamma_{{\mathbf{A}}} + \Gamma_{{\mathbf{B}}}, \end{eqnarray*} where $\Gamma_{{\mathbf{A}}}$ is the block matrix containing ${\mathbf{A}}$ and $\Gamma_{{\mathbf{B}}}$ is the block matrix containing ${\mathbf{B}}$. Let $(\Lambda^{1}, \cdots, \Lambda^{n})$ be any sequence of $d \times d$ Hermitian matrices. Then, \begin{eqnarray*} & &\sum^{n}_{p, q = 1}\sum_{ijkl} \lambda^{p}_{ij}\bar{\lambda}^{q}_{kl}(\Gamma^{p, q}_{{\mathbf{A}}})_{ij, lk}\\ &=& \sum^{n}_{p, q =1}\sum_{ijkl} \lambda^{p}_{ij}\bar{\lambda}^{q}_{kl}(\delta_{pq}(A_{ik}Z^{(p)}_{lj} + A_{lj}Z^{(p)}_{ik}) - A_{lj}(Z^{(p)}Z^{(q)})_{ik} - A_{ik}(Z^{(q)}Z^{(p)})_{lj})\\ &=& \textmd{trace}\,\big(A (\sum^{n}_{p=1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t}) \big) \\ & & \ \ + \textmd{trace}\,\big(A (\sum^{n}_{p=1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}\bar{\Lambda}^{p} - \sum^{n}_{p, q = 1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}\bar{\Lambda}^{q}) \big) . \end{eqnarray*} Then since $A$ is positive definite, we just need to prove that \begin{eqnarray*} \sum^{n}_{p=1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t}, \ \ \sum^{n}_{p=1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}\bar{\Lambda}^{p} - \sum^{n}_{p, q = 1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}\bar{\Lambda}^{q}, \end{eqnarray*} are non negative definite. For the first one, given a vector $X = (X_1, ..., X_d)$, we have \begin{eqnarray*} & &X\big(\sum^{n}_{p=1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t}\big)X^{*}\\ &=& X\bar{\Lambda}({\mathbf{Z}}- YY^{*})\Lambda^{t}X^{*}\\ &=& X\bar{\Lambda} {\mathbf{Z}}^{\frac{1}{2}}(\Id_{nd} - ({\mathbf{Z}}^{-\frac{1}{2}}Y)({\mathbf{Z}}^{-\frac{1}{2}}Y)^{*})(X\bar{\Lambda} {\mathbf{Z}}^{\frac{1}{2}})^{*}, \end{eqnarray*} where $\Lambda$ is a vector of matrices $\Lambda = (\Lambda^{1}, ..., \Lambda^{n})$, ${\mathbf{Z}} = {\rm diag}({\mathbf{Z}}^{(1)}, ..., {\mathbf{Z}}^{(n)})$ and $Y$ is a vector of matrices such that $Y = ({\mathbf{Z}}^{(1)}, ..., {\mathbf{Z}}^{(n)})^{*}$. Then by Sylvester determinant theorem, we are able to compute the eigenvalues of $\Id - ({\mathbf{Z}}^{-\frac{1}{2}}Y)({\mathbf{Z}}^{-\frac{1}{2}}Y)^{*}$, \begin{eqnarray}\label{sylvester} & &\det\big(\lambda \Id_{nd} - (\Id_{nd} - ({\mathbf{Z}}^{-\frac{1}{2}}Y)({\mathbf{Z}}^{-\frac{1}{2}}Y)^{*})\big) \\ \nonumber&=& \det\big((\lambda - 1)\Id_{nd} + ({\mathbf{Z}}^{-\frac{1}{2}}Y)({\mathbf{Z}}^{-\frac{1}{2}}Y)^{*}\big) \\ \nonumber&=& (\lambda - 1)^{(n-1)d}\det(\lambda \Id - {\mathbf{Z}}^{(n+1)}). \end{eqnarray} Since ${\mathbf{Z}}^{(n+1)} = \Id - \sum^{n}_{p=1}{\mathbf{Z}}^{(p)}$ is also a non negative-definite Hermitian matrix, the above equation means that the eigenvalues of $\Id - ({\mathbf{Z}}^{-\frac{1}{2}}Y)({\mathbf{Z}}^{-\frac{1}{2}}Y)^{*}$ are all non negative, indicating that it is a non negative definite matrix, such that $$ X\big(\sum^{n}_{p=1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t}\big)X^{*} \geq 0, $$ thus $\sum^{n}_{p=1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t}$ is non negative definite, so is $\sum^{n}_{p=1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}\bar{\Lambda}^{p} - \sum^{n}_{p, q = 1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}\bar{\Lambda}^{q}$. Therefore from the fact that $A$ is a positive-definite Hermitian matrix, we have \begin{eqnarray*} \textmd{trace}\,\big(A (\sum^{n}_{p = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t})\big) &\geq& 0,\\ \textmd{trace}\,\big(A (\sum^{n}_{p=1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}\bar{\Lambda}^{p} - \sum^{n}_{p, q = 1}(\Lambda^{p})^{t}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}\bar{\Lambda}^{q}) \big) &\geq& 0. \end{eqnarray*} Since \begin{eqnarray*} \textmd{trace}\,\big(A (\sum^{n}_{p = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t})\big) = \textmd{trace}\, \big(A\bar{\Lambda}({\mathbf{Z}}- YY^{*})\Lambda^{t}\big), \end{eqnarray*} and from equation~\eqref{sylvester}, we know that the interior of $\Delta_{n, d}$, ${\mathbf{Z}}- YY^{*}$ is positive definite. Thus following the proof of Lemma \ref{positive-definite}, we may conclude that if $$ \textmd{trace}\,\big(A (\sum^{n}_{p = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}(\Lambda^{p})^{t} - \sum^{n}_{p, q = 1}\bar{\Lambda}^{p}{\mathbf{Z}}^{(p)}{\mathbf{Z}}^{(q)}(\Lambda^{q})^{t})\big) = 0, $$ then $\Lambda = 0$. This implies that $\Gamma_{{\mathbf{A}}}$ is elliptic inside $\Delta_{n, d}$. As for $\Gamma_{{\mathbf{B}}}$, notice that \begin{eqnarray*} & &\sum^{d}_{i,j,k,l = 1} \lambda^{p}_{ij}\bar{\lambda}^{q}_{kl}(\Gamma^{p, q}_{{\mathbf{B}}})_{ij, \bar{kl}}\\ &=& \sum^{d}_{i,j,k,l = 1} \lambda^{p}_{ij}\bar{\lambda}^{q}_{kl}( B_{ia, kb}Z^{(p)}_{aj}Z^{(q)}_{lb} + B_{aj, bl}Z^{(p)}_{ia}Z^{(q)}_{bk} - B_{aj, kb}Z^{(p)}_{ia}Z^{(q)}_{lb} - B_{ia, bl}Z^{(p)}_{aj}Z^{(q)}_{bk})\\ &=& (\Lambda^{p}\bar{{\mathbf{Z}}}^{(p)} - \bar{{\mathbf{Z}}}^{(p)}\Lambda^{p})B(\bar{{\mathbf{Z}}}^{(q)}(\Lambda^{q})^{*}- (\Lambda^{q})^{*}\bar{{\mathbf{Z}}}^{(q)}), \end{eqnarray*} Let $H^{(p)} = \Lambda^{p}\bar{{\mathbf{Z}}}^{(p)} - \bar{{\mathbf{Z}}}^{(p)}\Lambda^{p}$, since ${\mathbf{B}}$ is positive-definite, Hermitian in the sense that $B_{ij, kl} = \bar{B}_{kl, ij}$, then \begin{eqnarray*} \sum^{n}_{p, q = 1}\sum^{d}_{i,j,k,l =1} \lambda^{p}_{ij}\bar{\lambda}^{q}_{kl}(\Gamma^{p, q}_{B})_{ij, lk} &=& (\sum^{n}_{p=1}H^{(p)}) B (\sum^{n}_{q=1}(H^{(q)})^{*}) \geq 0, \end{eqnarray*} which means $\Gamma_{{\mathbf{B}}}$ is non-negative definite. Since $\Gamma_{{\mathbf{A}}, {\mathbf{B}}} = \Gamma_{{\mathbf{A}}} + \Gamma_{{\mathbf{B}}}$, we know that $\Gamma_{{\mathbf{A}}, {\mathbf{B}}}$ is elliptic inside $\Delta_{n, d}$. Then we finish the proof. \end{preuve} \medskip In what follows, we show that this model may be constructed as a projection from complex Wishart processes, which are matrix generalizations of Laguerre processes. We first recall the definition of the complex Wishart distribution, \begin{definition}\label{complex.wishart} A $d \times d$ Hermitian positive definite matrix $W$ is said to have a Wishart distribution with parameters $d$, $r \geq d$, if its distribution has a density given by \begin{equation}\label{wishart.distribution} C_{r, d} \det(W)^{r - d}e^{- \frac{1}{2}\textmd{trace}\,(W)}, \end{equation} where $C_{r, d} =(2^{rd}\pi^{\frac{1}{2}d(d-1)}\Gamma(r)\Gamma(r-1)\cdots \Gamma(r -d +1))^{-1}$ is the normalization constant. \end{definition} When $r \geq d$ is an integer, this distribution can be derived from the Gaussian distributed complex matrix. Indeed, if we consider a $d \times r$ complex matrix $X$ with its elements being independent Gaussian centered random variables, then $W = XX^{*}$ has the complex Wishart distribution with parameters $d, r$. A $d \times d$ complex Wishart process $\{W_t, t \geq 0\}$ is usually defined as a solution to the following stochastic differential equation, \begin{eqnarray*} dW_t = \sqrt{W_t}dB_t + dB^{*}_t\sqrt{W_t} + (\alpha W_t + \beta\Id_{d})dt, \ \ \ W_t = W_{0}, \end{eqnarray*} where $\{B_t, t \geq 0\}$ is a $d \times d$ complex Brownian motion, $W_0$ is a $d \times d$ Hermitian matrix. Wishart processes have been deeply studied, see \cite{Bru, demni3, donati-martin} etc. There exists more general form of Wishart processes that we will not consider here. In what follows, we extend the construction of Wishart laws from matrix Gaussian ones at the level of processes, exactly as in the scalar case where Laguerre processes may be constructed (with suitable parameters) from Ornstein-Uhlenbeck ones. We will apply the matrix extracting procedure again. The generator of an Ornstein-Uhlenbeck process on $N \times N$ complex matrices is given, on the entries $\{z_{ij}\}$ of a complex matrix ${\mathbf{z}}$, by \begin{eqnarray*} \Gamma(z_{ij}, z_{kl}) &=& 0,\\ \Gamma(z_{ij}, \bar{z}_{kl}) &=& 2\delta_{ik}\delta_{jl},\\ \LL(z_{ij}) &=& -z_{ij}. \end{eqnarray*} This describes a process on matrices where the entries are independent complex Ornstein-Uhlenbeck processes. Now we start the "matrix-extracting" procedure on ${\mathbf{z}}$, as we did before on $SU(N)$ in Section~\ref{construction2}, i.e., take the first $d$ lines and split the $N$ columns into $(n + 1)$ parts $I_{1}, \cdots, I_{n+1}$, such that $d_{i} = |I_{i}|$, for $1 \leq i \leq (n+1)$ and $N = d_1 + ... + d_{n+1}$, then we have $(n + 1)$ extracted matrices ${\mathbf{Y}}^{(1)}$, ..., ${\mathbf{Y}}^{(n+1)}$. For $1 \leq p \leq (n+1)$, define ${\mathbf{W}}^{(p)} = {\mathbf{Y}}^{(p)}({\mathbf{Y}}^{(p)})^{*}$ with entries $W^{(p)}_{ij}$. \begin{proposition} $\{{\mathbf{W}}^{(p)}, 1 \leq p \leq (n+1)\}$ form a family of independent complex Wishart processes, whose reversible measure respectively given by \eqref{wishart.distribution} with $r_{p}= d_{p}$ for $1 \leq p \leq (n+1)$. Moreover, the image of the complex Gaussian measure through the "matrix-extracting" procedure is a product of complex Wishart distributions. \end{proposition} \begin{preuve} One may check \begin{eqnarray} \label{wishart1}\Gamma(W^{(p)}_{ij}, W^{(q)}_{kl}) &=& 2\delta_{pq}(\delta_{jk}W^{(p)}_{il} + \delta_{il}W^{(p)}_{kj}),\\ \label{wishart2}\LL(W^{(p)}_{ij}) &=& 4d_{p}\delta_{ij} - 2W^{(p)}_{ij}. \end{eqnarray} Let $\rho$ be the density of the reversible measure of $\{{\mathbf{W}}^{(1)}, ..., {\mathbf{W}}^{(n+1)}\}$. Then, $$ \Gamma(\log \rho, W^{p}_{ij}) = 4(d_{p} - d)\delta_{ij} -2W^{(p)}_{ij}, $$ and we also have \begin{eqnarray*} \Gamma(\log \det(W^{(p)}), W^{(p)}_{ij}) &=& 4\delta_{ij},\\ \Gamma(\textmd{trace}\, W^{(p)}, W^{(p)}_{ij}) &=& 4W^{(p)}_{ij}, \end{eqnarray*} therefore, $$ \rho = C\prod^{n+1}_{p=1}\det(W^{(p)})^{d_p - d}e^{-\frac{1}{2}\textmd{trace}\, (\sum^{n+1}_{p=1}W^{(p)})}, $$ which shows that, under the reversible measure, we have a family of $d \times d$ independent Wishart matrices $\{{\mathbf{W}}^{(1)}, ..., {\mathbf{W}}^{(n+1)}\}$. \end{preuve} \medskip We now construct a process on the complex matrix simplex from independent Wishart processes $({\mathbf{W}}^{(1)}, \cdots {\mathbf{W}}^{(n+1)})$. As in the scalar case, we obtain a kind of warped product on the set ${\mathbf{D}} \times \Delta_{n,d}$, where ${\mathbf{D}}$ denotes the set of real diagonal matrices with positive diagonal entries. Let ${\mathbf{S}} = \sum^{n+1}_{p=1}{\mathbf{W}}^{(p)}$. Since ${\mathbf{S}}$ is a positive-definite Hermitian matrix, we may assume that it has a spectral decomposition ${\mathbf{S}} = {\mathbf{U}} {\mathbf{D}}^{2} {\mathbf{U}}^{*}$, where ${\mathbf{U}}$ is unitary and ${\mathbf{D}} = {\rm diag}\{\lambda_{1}, \cdots, \lambda_{d}\}$. Observe that ${\mathbf{U}}$ is not uniquely determined, since we may change ${\mathbf{U}}$ into ${\mathbf{U}} {\mathbf{P}}$ where ${\mathbf{P}} = {\rm diag}\{e^{i\phi_{1}}, \cdots, e^{i\phi_{d}}\}$, $0 \leq \phi_1, \cdots, \phi_d \leq 2\pi$ , and this amounts to the choice of a phase for the eigenvectors. In this paper, we choose ${\mathbf{U}}$ to be the one that has real elements on its diagonal, such that ${\mathbf{U}}$ is an analytic function of ${\mathbf{S}}$ in the Weyl chamber $\{\lambda_1< \cdots < \lambda_d\}$. This choice will be irrelevant to the construction of the process. Moreover, we introduce $({\mathbf{V}}^{(1)}, \cdots , {\mathbf{V}}^{(d)})$, whose elements are given by \begin{equation} \label{def.Vp} V^{(p)}_{ij} = U_{ip}U^{*}_{pj} \end{equation} for $1 \leq i, j, p \leq d$, and we see that the choice of phase in ${\mathbf{U}}$ has no influence on $\{{\mathbf{V}}^{(p)}\}$. Then for $1 \leq i \leq (n+1)$, write \begin{equation} \label{def.Mi.Zi} {\mathbf{M}}^{(i)} = {\mathbf{S}}^{-\frac{1}{2}}{\mathbf{W}}^{(i)}{\mathbf{S}}^{-\frac{1}{2}}, \ \ \ {\mathbf{Z}}^{(i)} = {\mathbf{U}}^{*}{\mathbf{M}}^{(i)}{\mathbf{U}}, \end{equation} for which we have the following result. \begin{theoreme}\label{th.wishart.dirichlet} Let $({\mathbf{W}}^{1}, \cdots, {\mathbf{W}}^{(n+1)})$ be $(n + 1)$ independent Wishart processes. Then with ${\mathbf{D}} = {\rm diag}\{\lambda_{1}, \cdots, \lambda_{d}\}$, $0 \leq \lambda_{1} \leq \cdots \leq \lambda_{d}$ and ${\mathbf{Z}}^{(i)}$ defined as in equation~\eqref{def.Mi.Zi}, $({\mathbf{D}}, {\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)})$ is a Markov diffusion process, where $({\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)})$ lives in the matrix simplex $\Delta_{d, n}$. The generator of the process is \begin{eqnarray}\label{opt.wishart.dirichlet} \sum^{d}_{i=1}\big(\partial^{2}_{\lambda_i} +(\frac{2(N - d)+ 1}{\lambda_i} - \lambda_i + \sum_{j \neq i}\frac{4\lambda_i}{\lambda^{2}_i - \lambda^{2}_j})\partial_{\lambda_{i}}\big) + \LL_{{\mathbf{A}}, {\mathbf{B}}, {\mathbf{a}}}, \end{eqnarray} where $\LL_{{\mathbf{A}}, {\mathbf{B}}, {\mathbf{a}}}$ is defined in Theorem~\ref{th.Dirichlet.simplex} with \begin{eqnarray*} A_{ij} &=& 2\lambda_i^{-2}\delta_{ij}, \\ B_{ij, kl} &=& \left\{ \begin{array}{ll} 2\frac{\lambda^{2}_{i} + \lambda^{2}_{j}}{(\lambda^{2}_{i} - \lambda^{2}_{j})^{2}}\delta_{ik}\delta_{jl}, & \hbox{$i \neq j$ and $k \neq l$,}\\ \frac{1}{\lambda^{2}_{i}}\delta_{ij}\delta_{ik}\delta_{jl}, & \hbox{$i = j$ or $k = l$.} \end{array} \right. \end{eqnarray*} for $1 \leq i, j, k, l \leq d$ and $a_p = d_p - d +1$ for $1 \leq p \leq (n+1)$. \end{theoreme} It is known that starting from random matrices $(W^{(1)}, \cdots, W^{(n+1)})$ distributed as independent Wishart distributions, one could get the matrix Dirichlet distribution through $M^{i} = S^{-\frac{1}{2}}W^{(i)}S^{-\frac{1}{2}}$, where $S = \sum^{n+1}_{i = 1}W^{i}$, see \cite{guptanagar}. Also from our results in the scalar case (Section \ref{sec.scalar.simplex}), it is natural to guess that $({\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ may be a matrix Dirichlet process. As we will see in the following proposition, given ${\mathbf{S}}$, $({\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ is indeed a matrix Dirichlet process; However, the operator of $({\mathbf{S}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ is much more complicated than the one of $({\mathbf{D}}, {\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)})$, since $\Gamma({\mathbf{S}}, {\mathbf{M}}^{(i)}) \neq 0$, and therefore does not have the structure of a (generalized) warped product. \begin{proposition}\label{prop.m} $({\mathbf{S}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ is a Markov diffusion process where $({\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ lives on the matrix simplex $\Delta_{n, d}$, and the generator of the process is \begin{eqnarray}\label{operator.sm} & \sum_{ijkl}2(\delta_{jk}S_{il} + \delta_{il}S_{kj})\partial_{S_{ij}}\partial_{S_{kl}} + \sum_{ij}(4N\delta_{ij} - 2S_{ij})\partial_{S_{ij}} + \LL_{{\mathbf{A}}, {\mathbf{B}}, {\mathbf{a}}}\\ \nonumber& \ \ + \sum_{ijkl, p} \sum^{d}_{a, b =1} 2\frac{\lambda_a - \lambda_b}{\lambda_a + \lambda_b}((M^{(p)}V^{(a)})_{il}V^{(b)}_{kj} - V^{(a)}_{il}(V^{(b)}M^{(p)})_{kj})\partial_{S_{ij}}\partial_{M^{(p)}_{kl}},\\ \nonumber& \end{eqnarray} where $V^{(i)}$ defined by equation~\eqref{def.Vp}, $\LL_{{\mathbf{A}}, {\mathbf{B}}, {\mathbf{a}}}$ is defined in Theorem~\ref{th.Dirichlet.simplex} with $$ A = 2S^{-1}, \ \ B_{ij, kl} = \sum^{d}_{r,s=1}\frac{4}{(\lambda_{r} + \lambda_s)^{2}}V^{(r)}_{ik}V^{(s)}_{lj}, $$ and $a_p -1 + d = d_p$. \end{proposition} Before proving Theorem \ref{th.wishart.dirichlet} and Proposition \ref{prop.m}, we first give the following lemmas regarding the action of the diffusion operators of ${\mathbf{N}} = {\mathbf{S}}^{\frac{1}{2}}$, $\{{\mathbf{M}}^{(i)}, 1 \leq i \leq n\}$ and $\{{\mathbf{Z}}^{(i)}, 1 \leq i \leq n\}$. \begin{lemme}\label{s.eigenvalues} For the generator of independent Wishart matrices \\ $({\mathbf{W}}^{(1)}, \cdots, {\mathbf{W}}^{(n+1)})$ and ${\mathbf{S}} = \sum^{n+1}_{p=1}{\mathbf{W}}^{(p)}$, we have \begin{eqnarray} \label{zeq1}\Gamma(S_{ij}, S_{kl}) &=& 2(\delta_{jk}S_{il} + \delta_{il}S_{kj}),\\ \label{zeq3}\LL(S_{ij}) &=& 4N\delta_{ij} - 2S_{ij},\\ \label{zeq2}\Gamma(S_{ij}, W^{(p)}_{kl}) &=& 2(\delta_{jk}W^{(p)}_{il} + \delta_{il}W^{(p)}_{kj}). \end{eqnarray} Moreover, suppose ${\mathbf{S}}$ has a spectral decomposition ${\mathbf{S}} = {\mathbf{U}} {\mathbf{D}}^{2} {\mathbf{U}}^{*}$, where ${\mathbf{U}}$ is unitary and ${\mathbf{D}} = {\rm diag}\{\lambda_{1}, \cdots, \lambda_{d}\}$, we have \begin{eqnarray} \label{zeq111}\Gamma(\lambda_i, \lambda_j) &=& \delta_{ij}, \\ \label{zeq222}\LL(\lambda_i) &=& \frac{2(N - d)+ 1}{\lambda_i} - \lambda_i + 4\lambda_i\sum_{j \neq i}\frac{1}{\lambda^{2}_i - \lambda^{2}_j}. \end{eqnarray} \end{lemme} \begin{preuve} Formulas \eqref{zeq1}, \eqref{zeq2} and \eqref{zeq3} are straight-forward from \eqref{wishart1} and \eqref{wishart2}. By \eqref{zeq1}, \eqref{zeq3}, we are able to compute the diffusion operators of $D = \{\lambda_{1}, ..., \lambda_{d}\}$ by the method described in Theorem~\ref{th.polar} in Appendix~\ref{polar}. Notice that the $\Gamma$ operator of ${\mathbf{S}}$ is the same as ${\mathbf{H}}$ in Theorem \ref{th.polar}, which implies that \begin{eqnarray*} \Gamma(\lambda_i, \lambda_j) = \delta_{ij}. \end{eqnarray*} Now we compute $\LL(\lambda_i)$. First, let $P(X) = \det(S - X\Id)$, then by \eqref{zeq1}, \eqref{zeq3} we have \begin{eqnarray}\label{zeq16} & &\LL(\log P(X)) \\ \nonumber&=& - 4\textmd{trace}\,((S - X\Id)^{-1})\textmd{trace}\,((S - X\Id)^{-1}S) + 4N\textmd{trace}\,((S - X\Id)^{-1})\\ \nonumber& & - 2\textmd{trace}\,((S - X\Id)^{-1}S)\\ \nonumber &=& 4(d - N)\frac{P'(X)}{P(X)} - 4\frac{X(P'(X))^{2}}{P^{2}(X)}) - 2d + 2\frac{XP'(X)}{P(X)}). \end{eqnarray} On the other hand, let $\eta_{i} = \lambda^{2}_{i}$ be the eigenvalues of ${\mathbf{S}}$, we have \begin{eqnarray*} \LL(\log P(X)) = \sum^{d}_{i=1}(-\frac{4\eta_i}{(\eta_i - X)^{2}} + \frac{\LL(\eta_{i})}{\eta_i - X}). \end{eqnarray*} Comparing it with \eqref{zeq16} leads to \begin{eqnarray*} \LL(\eta_{i}) = 8\eta_i\sum_{j \neq i}\frac{1}{\eta_i - \eta_j} + 4(N - d + 1) - 2\eta_i, \end{eqnarray*} such that \begin{eqnarray*} \LL(\lambda_i) = \frac{2(N - d)+ 1}{\lambda_i} - \lambda_i + 4\lambda_i\sum_{j \neq i}\frac{1}{\lambda^{2}_i - \lambda^{2}_j}. \end{eqnarray*} \end{preuve} \medskip \begin{lemme} Let $S$ be a positive definite, Hermitian matrix and $N$ be its positive definite square root. Suppose $S$ has the spectral decomposition $S = UD^{2}U^{*}$, where $U$ is the unitary part and $D = {\rm diag}\{\lambda_1, \cdots, \lambda_d\}$, with $\lambda_i$ all positive. With $N= UDU^*$, and $V^{(p)}_{ij} = U_{ip}U^{*}_{pj}$ for $1 \leq i, j, p \leq d$, we have for $1 \leq i, j, k, l \leq d$ \begin{eqnarray}\label{sqrt.matrix} \partial_{S_{kl}}N_{ij} = \sum^{d}_{r, s = 1}\frac{1}{\lambda_r + \lambda_s}V^{(r)}_{ik}V^{(s)}_{lj}. \end{eqnarray} \end{lemme} \begin{preuve} For fixed $k, l$, write ${\mathcal D}^{kl}_{ij} = \partial_{S_{kl}}N_{ij}$. Then from $N^{2}_{ij} = S_{ij}$, we have \begin{eqnarray*} {\mathcal D}^{kl}{\mathbf{N}} + {\mathbf{N}} {\mathcal D}^{kl} = E^{kl}, \end{eqnarray*} where $E^{kl}$ is the matrix satisfying $E^{kl}_{ab} = \delta_{ak}\delta_{bl}$. Since ${\mathbf{S}} = {\mathbf{U}} {\mathbf{D}}^{2} {\mathbf{U}}^{*}$, we may write ${\mathbf{N}} = {\mathbf{U}} {\mathbf{D}} {\mathbf{U}}^{*}$, such that \begin{eqnarray*} ({\mathbf{U}}^{*} {\mathcal D}^{kl} {\mathbf{U}}) {\mathbf{D}} + {\mathbf{D}}({\mathbf{U}}^{*} {\mathcal D}^{kl} {\mathbf{U}}) = {\mathbf{U}}^{*}E^{kl}{\mathbf{U}}. \end{eqnarray*} Therefore, \begin{eqnarray*} (U^{*}{\mathcal D}^{kl}U)_{ij} = \frac{U^{*}_{ik}U_{lj}}{\lambda_i + \lambda_j}, \end{eqnarray*} so that \begin{eqnarray*} {\mathcal D}^{kl}_{ij} &=& \sum_{r, s}\frac{1}{\lambda_r + \lambda_s}U_{ir}U^{*}_{sj}U^{*}_{rk}U_{ls}\\ &=& \sum_{r, s}\frac{1}{\lambda_r + \lambda_s}V^{(r)}_{ik}V^{(s)}_{lj}, \end{eqnarray*} which finishes the proof. \end{preuve} \medskip \begin{lemme} The following formulas hold for ${\mathbf{N}} = {\mathbf{S}}^{\frac{1}{2}}$, \begin{eqnarray} \label{zeq4}\Gamma(N_{ij}, W^{(p)}_{kl}) &=& \sum^{d}_{r, s = 1}\frac{2}{\lambda_r + \lambda_s}V^{(r)}_{il}(W^{(p)}V^{(s)})_{kj} + \sum^{d}_{r, s =1}\frac{2}{\lambda_r + \lambda_s}V^{(s)}_{kj}(V^{(r)}W^{(p)})_{il},\\ \label{zeq5}\Gamma(N_{ij}, N_{kl}) &=& \sum^{d}_{r, s = 1} 2\frac{\lambda^{2}_{r} + \lambda^{2}_{s}}{(\lambda_r + \lambda_s)^{2}}V^{(r)}_{il}V^{(s)}_{kj},\\ \label{zeq6}\LL(N_{ij}) &=& 4\sum^{d}_{r, s=1} \frac{\lambda_s}{(\lambda_r + \lambda_s)^{2}}V^{(r)}_{ij} - N_{ij} + 2(N - d)N^{-1}_{ij}. \end{eqnarray} Furthermore, \begin{eqnarray} \label{zeq7} \Gamma(N^{-1}_{ij}, N_{kl}) &=& -\sum^{d}_{r, s = 1} 2\frac{\lambda^{2}_{r} + \lambda^{2}_{s}}{\lambda_r\lambda_s(\lambda_r + \lambda_s)^{2}}V^{(r)}_{il}V^{(s)}_{kj},\\ \label{zeq8} \Gamma(N^{-1}_{ij}, W^{(p)}_{kl}) &=& - 2\sum^{d}_{r, s =1}\frac{1}{\lambda_r\lambda_s(\lambda_r + \lambda_s)}(V^{(r)}_{il}(W^{(p)}V^{(s)})_{kj} + V^{(s)}_{kj}(V^{(r)}W^{(p)})_{il}),\\ \label{zeq9} \Gamma(N^{-1}_{ij}, N^{-1}_{kl}) &=& 2\sum^{d}_{r, s =1} \frac{\lambda^{2}_{r} + \lambda^{2}_{s}}{\lambda^{2}_r\lambda^{2}_s(\lambda_r + \lambda_s)^{2}}V^{(r)}_{il}V^{(s)}_{kj},\\ \label{zeq10} \LL(N^{-1}_{ij}) &=& 4\sum^{d}_{r, s =1} \frac{1}{\lambda_s(\lambda_r + \lambda_s)^{2}}V^{(r)}_{ij} + N^{-1}_{ij} - 2(N- d)(S^{-1}N^{-1})_{ij}. \end{eqnarray} \end{lemme} \begin{preuve} By \eqref{sqrt.matrix}, we are able to compute \begin{eqnarray*} \Gamma(N_{ij}, W^{(p)}_{kl}) &=& \sum_{ab}{\mathcal D}^{ab}_{ij}\Gamma(S_{ab}, W^{(p)}_{kl})\\ &=& 2\sum^{d}_{r, s =1}\frac{1}{\lambda_r + \lambda_s}V^{(r)}_{il}(W^{(p)}V^{(s)})_{kj} + 2\sum^{d}_{r, s =1}\frac{1}{\lambda_r + \lambda_s}V^{(s)}_{kj}(V^{(r)}W^{(p)})_{il}, \end{eqnarray*} and \begin{eqnarray*} \Gamma(N_{ij}, N_{kl}) &=& \sum_{abcd} {\mathcal D}^{ab}_{ij}{\mathcal D}^{cd}_{kl}\Gamma(S_{ab}, S_{cd})\\ &=& 2\sum^{d}_{r, s=1} \frac{\lambda^{2}_{r} + \lambda^{2}_{s}}{(\lambda_r + \lambda_s)^{2}}V^{(r)}_{il}V^{(s)}_{kj},\\ \LL(N_{ij}) &=& 4\sum \frac{\lambda_s}{(\lambda_r + \lambda_s)^{2}}V^{(r)}_{ij} - N_{ij} + 2(N - d)N^{-1}_{ij}. \end{eqnarray*} Moreover, due to the fact that $$ \partial_{N_{ab}}N^{-1}_{ij} = - N^{-1}_{ia}N^{-1}_{bj}, $$ we have \begin{eqnarray*} & &\Gamma(N^{-1}_{ij}, W^{(p)}_{kl}) = \sum^{d}_{a, b=1}\partial_{N_{ab}}N^{-1}_{ij}\Gamma(N_{ab}, W^{(p)}_{kl})\\ &=& - 2\sum^{d}_{r, s =1}\frac{1}{\lambda_r\lambda_s(\lambda_r + \lambda_s)}V^{(r)}_{il}(W^{(p)}V^{(s)})_{kj} - 2\sum^{d}_{r, s=1}\frac{1}{\lambda_r\lambda_s(\lambda_r + \lambda_s)}V^{(s)}_{kj}(V^{(r)}W^{(p)})_{il}. \end{eqnarray*} Similar computations yield \begin{eqnarray*} \Gamma(N^{-1}_{ij}, N_{kl}) &=& -2\sum^{d}_{r, s=1} \frac{\lambda^{2}_{r} + \lambda^{2}_{s}}{\lambda_r\lambda_s(\lambda_r + \lambda_s)^{2}}V^{(r)}_{il}V^{(s)}_{kj},\\ \Gamma(N^{-1}_{ij}, N^{-1}_{kl}) &=& 2\sum^{d}_{r, s=1} \frac{\lambda^{2}_{r} + \lambda^{2}_{s}}{\lambda^{2}_r\lambda^{2}_s(\lambda_r + \lambda_s)^{2}}V^{(r)}_{il}V^{(s)}_{kj},\\ \LL(N^{-1}_{ij}) &=& \sum^{d}_{k, l=1}\partial_{N_{kl}}N^{-1}_{ij}\LL(N_{kl}) + \sum^{d}_{k, l, a, b=1}\Gamma(N_{kl}, N_{ab})\partial_{N_{ab}}\partial_{N_{kl}}N^{-1}_{ij}\\ &=& 4\sum^{d}_{r,s=1} \frac{1}{\lambda_s(\lambda_r + \lambda_s)^{2}}V^{(r)}_{ij} + N^{-1}_{ij} - 2(N - d)(S^{-1}N^{-1})_{ij}. \end{eqnarray*} \end{preuve} \medskip \begin{lemme}\label{lemma.m} With $V^{(i)}$ defined in equation~\eqref{def.Vp}, we have \begin{eqnarray}\label{zeq11} & &\Gamma(M^{(p)}_{ij}, M^{(q)}_{kl})\\ \nonumber&=& 2\delta_{pq}(S^{-1}_{il}M^{(p)}_{kj} + S^{-1}_{kj}M^{(p)}_{il}) - 2S^{-1}_{kj}(M^{(p)}M^{(q)})_{il} - 2S^{-1}_{il}(M^{(q)}M^{(p)})_{kj}\\ \nonumber& & - \sum^{d}_{a, b=1} \frac{4}{(\lambda_a + \lambda_b)^{2}}(V^{(b)}M^{(p)})_{kj}(V^{(a)}M^{(q)})_{il} - \sum^{d}_{a, b=1} \frac{4}{(\lambda_a + \lambda_b)^{2}}(M^{(p)}V^{(a)})_{il}(M^{(q)}V^{(b)})_{kj} \\ \nonumber& & +\sum^{d}_{a, b=1} \frac{4}{(\lambda_a + \lambda_b)^{2}}V^{(a)}_{kj}(M^{(p)}V^{(b)}M^{(q)})_{il} +\sum^{d}_{a, b =1}\frac{4}{(\lambda_a + \lambda_b)^{2}}V^{(a)}_{il}(M^{(q)}V^{(b)}M^{(p)})_{kj}, \end{eqnarray} and \begin{eqnarray}\label{zeq12} & &\LL(M^{(p)}_{ij}) \\ \nonumber &=& 4d_{p}S^{-1}_{ij} - 2(N - d)(S^{-1}M^{(p)})_{ij} - 2(N -d)(M^{(p)}S^{-1})_{ij}- 4S^{-1}_{ij}\textmd{trace}\,(M^{(p)})\\ \nonumber & &- 4\sum^{d}_{a, b=1}\frac{1}{(\lambda_a + \lambda_b)^{2}}(V^{(a)}M^{(p)})_{ij} - 4\sum^{d}_{a, b=1}\frac{1}{(\lambda_a + \lambda_b)^{2}}(M^{(p)}V^{(b)})_{ij}\\ \nonumber & & + 8\sum^{d}_{a, b=1}\frac{1}{(\lambda_a + \lambda_b)^{2}}V^{(a)}_{ij}\textmd{trace}\,(V^{(b)}M^{(p)}). \end{eqnarray} Moreover, we have \begin{eqnarray} \label{zeq20}\Gamma(M^{(p)}_{ij}, S_{kl}) &=& \sum^{d}_{a, b =1} 2\frac{\lambda_a - \lambda_b}{\lambda_a + \lambda_b}((M^{(p)}V^{(a)})_{il}V^{(b)}_{kj} - V^{(a)}_{il}(V^{(b)}M^{(p)})_{kj}). \end{eqnarray} \end{lemme} \begin{preuve} Since \begin{eqnarray*} \Gamma(M^{(p)}_{ij}, M^{(q)}_{kl}) = \Gamma((N^{-1}W^{(p)}N^{-1})_{ij}, (N^{-1}W^{(q)}N^{-1})_{kl}), \end{eqnarray*} then by \eqref{wishart1}, \eqref{zeq8} and \eqref{zeq9} we are able to prove $\eqref{zeq11}$. As for \eqref{zeq12}, direct computations yield \begin{eqnarray*} & &\LL(M^{(p)}_{ij}) = \sum^{d}_{r,s=1}\LL(N^{-1}_{ir}W^{(p)}_{rs}N^{-1}_{sj})\\ &=& \sum^{d}_{r,s=1} \big( 2\Gamma(N^{-1}_{ir}, N^{-1}_{sj})W^{p}_{rs} + 2\Gamma(N^{-1}_{ir}, W^{p}_{rs})N^{-1}_{sj} + 2N^{-1}_{ir}\Gamma(W^{p}_{rs}, N^{-1}_{sj})\\ & & + \LL(N^{-1}_{ir})W^{p}_{rs}N^{-1}_{sj} + N^{-1}_{ir}\LL(W^{p}_{rs})N^{-1}_{sj} + N^{-1}_{ir}W^{p}_{rs}\LL(N^{-1}_{sj}) \big)\\ &=& 8\sum^{d}_{a, b=1}\frac{1}{(\lambda_a + \lambda_b)^{2}}V^{(a)}_{ij}\textmd{trace}\,(V^{(b)}M^{(p)}) - 4S^{-1}_{ij}\textmd{trace}\,(M^{p})\\ & &- 4\sum^{d}_{a, b=1}\frac{1}{(\lambda_a + \lambda_b)^{2}}(V^{(a)}M^{(p)})_{ij} - 4\sum^{d}_{a, b=1}\frac{1}{(\lambda_a + \lambda_b)^{2}}(M^{(p)}V^{(b)})_{ij} \\ & & + 4d_{p}S^{-1}_{ij} - 2(N - d)(S^{-1}M^{(p)})_{ij} - 2(N-d)(M^{(p)}S^{-1})_{ij}, \end{eqnarray*} where the last equality is due to \eqref{wishart2}, \eqref{zeq8}, \eqref{zeq9} and \eqref{zeq10}. By \eqref{zeq4}, \eqref{zeq7}, we get \begin{eqnarray}\label{zeq14444} & &\Gamma(M^{(p)}_{ij}, N_{kl}) \\ \nonumber&=& \sum^{d}_{a =1}\Gamma(N_{kl}, N^{-1}_{ia})(W^{(p)}N^{-1})_{aj} + \sum^{d}_{a, b =1}N^{-1}_{ia}\Gamma(N_{kl}, W^{(p)}_{ab})N^{-1}_{bj} \\ \nonumber& & \ \ \ \ + \sum^{d}_{a =1}(N^{-1}W^{(p)})_{ia}\Gamma(N_{kl}, N^{-1}_{aj})\\ \nonumber&=& \sum^{d}_{a, b=1}2\frac{\lambda_a - \lambda_b}{(\lambda_a + \lambda_b)^{2}}(M^{(p)}V^{(a)})_{il}V^{(b)}_{kj} - 2\sum^{d}_{a, b=1}\frac{\lambda_a - \lambda_b}{(\lambda_a + \lambda_b)^{2}}V^{(a)}_{il}(V^{(b)}M^{(p)})_{kj}, \end{eqnarray} which leads to \eqref{zeq20}, showing that in fact ${\mathbf{M}}^{(p)}$ and ${\mathbf{S}}$ are not independent. \end{preuve} \medskip Now we may obtain Proposition \ref{prop.m} directly from Lemma \ref{lemma.m}. \begin{preuve} Combining \eqref{zeq1}, \eqref{zeq3} and \eqref{zeq11}, \eqref{zeq12} and \eqref{zeq20}, we derive \eqref{operator.sm} as the operator of $({\mathbf{S}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$. \end{preuve} \medskip \begin{rmq} It is natural to consider the reversible measure of \\ $({\mathbf{S}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ as soon as we have Proposition \ref{prop.m}. However, it appears quite complicated in this case, due to the fact that we need to take the unitary part ${\mathbf{U}}$ into account. More precisely, we define $\rho$ to be the density of the reversible measure of $({\mathbf{S}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ and decompose ${\mathbf{S}}$ into $({\mathbf{D}}, {\mathbf{U}})$, then we may write $\rho = \rho_{{\mathbf{D}}}\rho_{{\mathbf{U}}}D_{\bbC, {\mathbf{a}}}$, where $\rho_{{\mathbf{D}}}$ is obtained from the generators of $\{\lambda_{i}\}$, see Theorem \ref{th.wishart.dirichlet}. Again by \eqref{eq.density}, we obtain the formulas of $\rho_{{\mathbf{U}}}$, which are so complicated that we are not able to provide any explicit expression. \end{rmq} To examine the relation of ${\mathbf{N}}$ and ${\mathbf{M}}^{(p)}$ more precisely, we decompose ${\mathbf{N}}$ into ${\mathbf{D}}$ and ${\mathbf{U}}$, and explore their relations with ${\mathbf{M}}^{(p)}$ separately. The method is adapted from \cite{BakryZ}, i.e., to obtain the action of the $\Gamma$ operator of the spectrum of ${\mathbf{N}}$ and ${\mathbf{M}}^{(p)}$ by computing its action on the characteristic polynomial of ${\mathbf{N}}$ and ${\mathbf{M}}^{(p)}$. \begin{lemme}\label{lemma.wishart.eq.Gamma.L} \begin{eqnarray}\label{zeq1444} \Gamma(M^{(p)}_{ij}, \lambda_k) &=& 0, \end{eqnarray} and \begin{eqnarray} \label{zeq14}\Gamma(M^{(p)}_{ij}, U_{kl}) &=& \sum^{d}_{a=1}g_{al}(M^{(p)}U)_{il}U^{*}_{aj}U_{ka} - \sum^{d}_{a=1}g_{al}U_{il}(U^{*}M^{(p)})_{aj}U_{ka},\\ \label{zeq144}\Gamma(U^{*}_{kl}, M^{(p)}_{ij}) &=& - \sum^{d}_{a=1}g_{ka}(M^{(p)}U)_{ia}U^{*}_{al}U^{*}_{kj} + \sum^{d}_{a=1}g_{ka}U_{ia}U^{*}_{al}(U^{*}M^{(p)})_{kj}, \end{eqnarray} where \begin{eqnarray*} g_{ij} = \left\{ \begin{array}{ll} \frac{2}{(\lambda_i + \lambda_j)^{2}}, & \hbox{$i \neq j$,}\\ 0, & \hbox{$i = j$,} \end{array} \right. \end{eqnarray*} for $1 \leq i, j \leq d$. \end{lemme} \begin{preuve} Let $P_{{\mathbf{N}}}(X) = \det({\mathbf{N}} - X\Id) = \prod^{d}_{i=1}(\lambda_i - X)$ be the characteristic polynomial of ${\mathbf{N}}$. Notice that \begin{eqnarray*} & &\Gamma(\log P_{N}(X), M^{(p)}_{ij}) = \sum^{d}_{k, l=1} N^{-1}(X)_{lk}\Gamma(N_{kl}, M^{(p)}_{ij})\\ &=& \sum_{r, s}2\frac{\lambda_r - \lambda_s}{(\lambda_r + \lambda_s)^{2}}\frac{1}{\lambda_r - X}\delta_{rs}U^{*}_{sj}(M^{(p)}U)_{ir} - 2\sum_{r, s}\frac{\lambda_r - \lambda_s}{(\lambda_r + \lambda_s)^{2}}\frac{1}{\lambda_r - X}\delta_{rs}U_{ir}(U^{*}M^{(p)})_{sj} \\ &=& 0, \end{eqnarray*} from which we deduce that $$ \Gamma(M^{(p)}_{ij}, \lambda_k) = 0, $$ since $$ \Gamma(\log P_{N}(X), M^{(p)}_{ij}) = \sum^{d}_{k =1} \frac{1}{\lambda_{k} - X}\Gamma(\lambda_k, M^{(p)}_{ij}) = 0. $$ Also notice that $\Gamma(N_{kl}, M^{(p)}_{ij})$ is invariant under the unitary transformation $$({\mathbf{N}}, {\mathbf{M}}^{(p)}) \rightarrow ({\mathbf{U}}^{0}{\mathbf{N}}({\mathbf{U}}^{0})^{*}, {\mathbf{U}}^{0}{\mathbf{M}}^{(p)}({\mathbf{U}}^{0})^{*})$$ for any unitary matrix $U^{0}$, (for the details of the invariance of transformation of diffusion operators, see Appendix \ref{polar}), we may compute $\Gamma(M^{(p)}_{ij}, U_{kl})$ at $U = \Id$, then obtain it at any $U$. More precisely, by \eqref{zeq14444}, we obtain at $U = \Id$, $$ \Gamma(N_{kl}, M^{(p)}_{ij}) = 2\frac{\lambda_l - \lambda_k}{(\lambda_l + \lambda_k)^{2}}\delta_{kj}M^{(p)}_{il} - 2\frac{\lambda_l - \lambda_k}{(\lambda_l + \lambda_k)^{2}}\delta_{il}M^{(p)}_{kj}, $$ and on the other hand at $U=\Id$, $$ \Gamma(N_{kl}, M^{(p)}_{ij}) = \Gamma(U_{kl}, M^{(p)}_{ij})(\lambda_{l} - \lambda_{k}). $$ Combining the two equalities together, we have for $k \neq l$ \begin{eqnarray}\label{zeq15} \Gamma(U_{kl}, M^{(p)}_{ij})(U = \Id) = 2\frac{1}{(\lambda_l + \lambda_k)^{2}}(\delta_{kj}M^{(p)}_{il} - \delta_{il}M^{(p)}_{kj}). \end{eqnarray} Moreover, at $U = \Id$, for $1 \leq k \leq d$ $$ \Gamma(U_{kk}, \cdot) = - \Gamma(U^{*}_{kk}, \cdot). $$ Then by the fact that $U_{kk}$ is real, we have at $U = \Id$, $$ \Gamma(U_{kk}, M^{(p)}_{ij}) = 0. $$ Now write \begin{eqnarray*} g_{ij} = \left\{ \begin{array}{ll} \frac{2}{(\lambda_i + \lambda_j)^{2}}, & \hbox{$i \neq j$,}\\ 0, & \hbox{$i = j$,} \end{array} \right. \end{eqnarray*} then at any $U$, we derive \eqref{zeq14} and \eqref{zeq144} by the invariance under unitary transformations. \end{preuve} \medskip \begin{lemme}\label{lm.dirichlet2m} The diffusion operators of $\{{\mathbf{Z}}^{i}\}$ are given by \begin{eqnarray}\label{gamma.dirichlet2m} & &\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl})\\ \nonumber&=& 2\delta_{pq}(\delta_{il}\frac{1}{\lambda^{2}_{i}}Z^{(p)}_{kj} + \delta_{kj}\frac{1}{\lambda^{2}_{j}}Z^{(p)}_{il}) - 2(\delta_{kj}\frac{1}{\lambda^{2}_{j}}(Z^{(p)}Z^{(q)})_{il} + \delta_{il}\frac{1}{\lambda^{2}_{i}}(Z^{(q)}Z^{(p)})_{kj})\\ \nonumber& & + \sum^{d}_{a=1}(y_{ia}\delta_{il}Z^{(p)}_{aj}Z^{(q)}_{ka} + y_{ka}\delta_{kj}Z^{(p)}_{ia}Z^{(q)}_{al}) - y_{ik}Z^{(p)}_{kj}Z^{(q)}_{il} - y_{jl}Z^{(p)}_{il}Z^{(q)}_{kj}, \end{eqnarray} where $y_{ij} = 2\frac{\lambda^{2}_{i} + \lambda^{2}_{j}}{(\lambda^{2}_{i} - \lambda^{2}_{j})^{2}}$ when $i \neq j$ and $y_{ii} = \frac{1}{\lambda_{i}^{2}}$ for $1 \leq i \leq d$, and \begin{eqnarray}\label{l.dirichlet2m} \LL(Z^{(p)}_{ij}) &=& 4d_{p}\frac{1}{\lambda^{2}_{i}}\delta_{ij} - 2(N - d)(\frac{1}{\lambda^{2}_{i}} + \frac{1}{\lambda^{2}_{j}})Z^{(p)}_{ij} - 4\frac{1}{\lambda^{2}_{i}}\delta_{ij}\textmd{trace}\,(Z^{(p)})\\ \nonumber& & + 2\sum^{d}_{a=1}y_{ia}\delta_{ij}Z^{(p)}_{aa} - \sum^{d}_{a=1}y_{ja}Z^{(p)}_{ij} - \sum^{d}_{a =1}y_{ia}Z^{(p)}_{ij}. \end{eqnarray} Moreover, \begin{eqnarray} \label{z.diag}\Gamma(Z^{(p)}_{ij}, \lambda_k) &=& 0,\\ \label{z.unitary}\Gamma(Z^{(p)}_{ij}, U_{kl}) &=&\delta_{il}\sum^{d}_{a \neq l}d_{al}U_{ka}Z^{(p)}_{aj} - d_{jl}U_{kj}Z^{(p)}_{il}, \end{eqnarray} where $d_{ij} = 4\frac{\lambda_i\lambda_j}{(\lambda^{2}_i - \lambda^{2}_j)^{2}}$ when $i \neq j$ and $d_{ii} = 0$. \end{lemme} \begin{preuve} Notice the diffusion generator of $({\mathbf{U}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)})$ is invariant through the map $$ ({\mathbf{U}}, {\mathbf{M}}^{(1)}, \cdots, {\mathbf{M}}^{(n)}) \rightarrow ({\mathbf{U}}^{0}{\mathbf{U}}, {\mathbf{U}}^{0}{\mathbf{M}}^{(1)}({\mathbf{U}}^{0})^{*}, \cdots, {\mathbf{U}}^{0}{\mathbf{M}}^{(n)}({\mathbf{U}}^{0})^{*}). $$ Hence the diffusion generator of ${\mathbf{Z}} = ({\mathbf{Z}}^{(1)}, \cdots, {\mathbf{Z}}^{(n)})$ is also invariant. Therefore to compute $\Gamma({\mathbf{Z}}^{(p)}, {\mathbf{Z}}^{(q)})$, we may first consider the case at ${\mathbf{U}} = \Id$. By direct computations, we have \begin{eqnarray*} \Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl}) &=& \sum^{d}_{r, s =1} \big(\Gamma(M^{(p)}_{ij}, M^{(q)}_{kl}) + \Gamma(M^{(p)}_{ij}, U^{*}_{kr}U_{sl})M^{(q)}_{rs} + M^{(p)}_{rs}\Gamma(U^{*}_{ir}U_{sj}, M^{(q)}_{kl})\big)\\ & & + \sum^{d}_{u,v,r,s =1}\Gamma(U^{*}_{iu}U_{vj}, U^{*}_{kr}U_{sl})M^{(p)}_{uv}M^{(q)}_{rs} . \end{eqnarray*} The first term $\Gamma(M^{(p)}_{ij}, M^{(q)}_{kl})$ at $U = \Id$ is straightforward from \eqref{zeq11}. By \eqref{zeq15}, we get \begin{eqnarray*} & &\sum^{d}_{r,s=1}\Gamma(M^{(p)}_{ij}, U^{*}_{kr}U_{sl})M^{(q)}_{rs} \\ &=& \sum^{d}_{a=1}(- g_{ka}\delta_{kj}Z^{(p)}_{ia}Z^{(q)}_{al} - g_{al}\delta_{il}Z^{(q)}_{ka}Z^{(p)}_{aj}) + g_{jl}Z^{(p)}_{il}Z^{(q)}_{kj} + g_{ik}Z^{(p)}_{kj}Z^{(q)}_{il}, \end{eqnarray*} \begin{eqnarray*} & &\sum^{d}_{r,s=1}\Gamma(M^{(q)}_{kl}, U^{*}_{ir}U_{sj})M^{(p)}_{rs} \\ &=& \sum^{d}_{a=1}(- g_{ia}\delta_{il}Z^{(q)}_{ka}Z^{(p)}_{aj} - g_{aj}\delta_{kj}Z^{(p)}_{ia}Z^{(q)}_{al}) + g_{jl}Z^{(p)}_{il}Z^{(q)}_{kj} + g_{ik}Z^{(p)}_{kj}Z^{(q)}_{il}, \end{eqnarray*} and \begin{eqnarray*} & &\sum^{d}_{u,v,r,s =1}\Gamma(U^{*}_{iu}U_{vj}, U^{*}_{kr}U_{sl})M^{(p)}_{uv}M^{(q)}_{rs}\\ &=& \sum^{d}_{a=1}(r_{ia}\delta_{il}M^{(p)}_{aj}M^{(q)}_{ka} + r_{ka}\delta_{kj}M^{(p)}_{ia}M^{(q)}_{al}) - r_{ik}M^{(p)}_{kj}M^{(q)}_{il} - r_{jl}M^{(p)}_{il}M^{(q)}_{kj}. \end{eqnarray*} Therefore, collecting all the four terms together we obtain at $U=\Id$, \begin{eqnarray*} & &\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl})\\ &=& 2\delta_{pq}(\delta_{il}\frac{1}{\lambda^{2}_{i}}Z^{(p)}_{kj} + \delta_{kj}\frac{1}{\lambda^{2}_{j}}Z^{(p)}_{il}) - 2(\delta_{kj}\frac{1}{\lambda^{2}_{j}}(Z^{(p)}Z^{(q)})_{il} + \delta_{il}\frac{1}{\lambda^{2}_{i}}(Z^{(q)}Z^{(p)})_{kj})\\ & & - \frac{1}{\lambda_{i}^{2}}\delta_{ik}Z^{(p)}_{ij}Z^{(q)}_{il} - \frac{1}{\lambda_{j}^{2}}\delta_{jl}Z^{(p)}_{ij}Z^{(q)}_{kj} + \frac{1}{\lambda_{j}^{2}}\delta_{kj}Z^{(p)}_{ij}Z^{(q)}_{jl} + \frac{1}{\lambda_{i}^{2}}\delta_{il}Z^{(p)}_{ij}Z^{(q)}_{ki}\\ & & + \sum^{d}_{a=1}(r_{ia}\delta_{il}Z^{(p)}_{aj}Z^{(q)}_{ka} + r_{ka}\delta_{kj}Z^{(p)}_{ia}Z^{(q)}_{al}) - r_{ik}Z^{(p)}_{kj}Z^{(q)}_{il} - r_{jl}Z^{(p)}_{il}Z^{(q)}_{kj}. \end{eqnarray*} Defining \begin{eqnarray*} y_{ij} = \left\{ \begin{array}{ll} r_{ij}, & \hbox{$i \neq j$,}\\ \frac{1}{\lambda_{i}^{2}}, & \hbox{$i = j$,} \end{array} \right. \end{eqnarray*} for $1 \leq i, j \leq d$, we obtain \begin{eqnarray*} & &\Gamma(Z^{(p)}_{ij}, Z^{(q)}_{kl})\\ &=& 2\delta_{pq}(\delta_{il}\frac{1}{\lambda^{2}_{i}}Z^{(p)}_{kj} + \delta_{kj}\frac{1}{\lambda^{2}_{j}}Z^{(p)}_{il}) - 2(\delta_{kj}\frac{1}{\lambda^{2}_{j}}(Z^{(p)}Z^{(q)})_{il} + \delta_{il}\frac{1}{\lambda^{2}_{i}}(Z^{(q)}Z^{(p)})_{kj})\\ & & + \sum^{d}_{a=1}(y_{ia}\delta_{il}Z^{(p)}_{aj}Z^{(q)}_{ka} + y_{ka}\delta_{kj}Z^{(p)}_{ia}Z^{(q)}_{al}) - y_{ik}Z^{(p)}_{kj}Z^{(q)}_{il} - y_{jl}Z^{(p)}_{il}Z^{(q)}_{kj}. \end{eqnarray*} This formula is also valid at any $U$, because $Z^{(p)}$ is invariant under the map $(U, M^{p}) \rightarrow (U^{0}U, U^{0}M^{p}(U^{0})^{*})$. Thus we derive \eqref{gamma.dirichlet2m}. By \eqref{zeq15} and Theorem \ref{th.polar} (the diffusion operators of the unitary part ${\mathbf{U}}$ here are the same as in the polar decomposition case), we obtain formula~\eqref{l.dirichlet2m}. According to Theorem \ref{th.polar}, we know that $\Gamma(U, D) = 0$, together with \eqref{zeq15}, indicating that $$ \Gamma(Z^{(p)}_{ij}, \lambda_k) = 0. $$ Also by Theorem \ref{th.polar}, we have at $U=\Id$ and at any $U$, \begin{eqnarray*} \Gamma(Z^{(p)}_{ij}, U_{kl}) = 4\frac{\lambda_k\lambda_l}{(\lambda^{2}_k - \lambda^{2}_l)^{2}}(\delta_{il}Z^{(p)}_{kj} - \delta_{kj}Z^{(p)}_{il}). \end{eqnarray*} Let $d_{ij} = 4\frac{\lambda_i\lambda_j}{(\lambda^{2}_i - \lambda^{2}_j)^{2}}$ when $i \neq j$ and $d_{ii} = 0$, then at any $U$, we obtain \eqref{z.unitary}. \end{preuve} \medskip Finally we are in the position to prove Theorem \ref{th.wishart.dirichlet}. \begin{preuve} By \eqref{zeq111}, \eqref{zeq222}, we obtain the generator of ${\mathbf{D}}$. Together with \eqref{z.diag}, and comparing \eqref{gamma.dirichlet2m}, \eqref{l.dirichlet2m} with Theorem \ref{model2}, we obtain the operator \eqref{opt.wishart.dirichlet} with \begin{eqnarray*} A = 2D^{-2}, \ \ \ B_{ij, kl} = y_{ij}\delta_{ik}\delta_{jl}, \end{eqnarray*} and $a_{p} = d_{p} - d + 1$. \end{preuve} \medskip

BkiUbs-6NNjgB0Ss0sZ_

\section{todo} \section{Introduction} \label{introduction} {\it Ab initio} calculations of atomic properties can now be performed routinely, both in the framework of the MCDHF theory~\cite{GRASP2K,mcdfgme,desclaux:1975,desclaux:1993,GrantBook2007}, as well as many-body perturbation (MBPT) theory~\cite{LindgrenMorrison:86,Dzuba:CI+MBPT:2005,DzubaFlambaum:2007,% JohnsonBook:2007}. Both these methods are designed to evaluate in a systematic manner the electron-electron correlation effects, which constitute the dominant correction to all {\it ab initio} calculations based on the central-field approach. However, the complexity increases rapidly with the atomic number, and fully correlated calculations, in which all electrons are explicitly correlated, are still possible only for very light elements (see~e.g.~\cite{Bieron:Li:1996,Bieron:Be+F:1999,Yerokhin:2008} for model calculations of hyperfine constants of lithium-like systems). For heavy atoms both theories can only be applied in a limited model (one- and two-body correlation effects) or only to certain atoms (closed-shell systems or alkali-like systems). The main purpose of the present paper was to carry out an accurate calculation of hyperfine structure constants of a heavy atom within the framework of the MCDHF theory. The calculations described in the present paper are, to our knowledge, the first successful evaluation of one-, two-, and three-body electron correlation effects for a heavy, open-shell, neutral atom. The multiconfiguration model applied in the present paper is effectively equivalent to a Complete Active Space (CAS) approach, in the sense that in the calculation of the hyperfine electric quadrupole moments all non-negligible electron correlation effects were explicitly accounted for at 1~\% level of accuracy or better. The gold atom has been chosen, because the hyperfine structures~\cite{Autschbach:2002,Malkin:2004,Malkin:2006,Song:2007}, the nuclear electric quadrupole moments~\cite{Schwerdtfeger:2005,Itano:2006,Yakobi:2007,Belpassi:2007,% ThierfelderSchwerdtfegerSaue:2007}, and other properties~\cite{Pyykko:Au:2004,Pyykko:Au:2005,Pyykko:Au:2008} of gold have been the subject of much activity recently (the latest summary of nuclear quadrupole moments is given in ref.~\cite{Pyykko:08bb}). The second objective of the present paper is an evaluation of the electric quadrupole moment $Q$ of the $ ^{197} $Au isotope. \section{Theory} \label{theory} The numerical-grid wavefunctions~\cite{GRASP2K} were generated as the self-consistent solutions of the Dirac-Hartree-Fock equations~\cite{Grant:1994} in systematically increasing multiconfiguration bases (of size NCF, which is a commonly used shorthand of 'Number of Configuration Functions') of symmetry-adapted configuration state functions $ \Phi(\gamma_{k}J) $ \begin{equation} \label{ASF} \Psi(J) = \sum_{k}^{NCF} c_{k} \Phi(\gamma_{k}J), \end{equation} where $ \Psi(J) $ is an eigenfunction of even parity and of total angular momentum $J$ for each of the two states $ \Psi ( 5d^{9} 6s^{2} \,\, ^2 \! D_{3/2} ) $ and $ \Psi ( 5d^{9} 6s^{2} \,\, ^2 \! D_{5/2} ) $ of the isotope $^{197} _{\phantom{1}79}$Au. The sets $ \gamma_{k} $ describe multiconfiguration expansions, for which configuration mixing coefficients $ c_{k} $ were obtained through diagonalisation of the Dirac-Coulomb Hamiltonian \begin{equation} \label{Dirac-Coulomb-Hamiltonian} H_{DC} = \sum_{i} \left[ c {\bm{ \alpha }}_i \cdot {\bm{ p }}_i + (\beta_i -1)c^2 + V(r_i) \right] + \sum_{i>j} 1/r_{ij}. \end{equation} All calculations were done with the nucleus modelled as a sphere, where a two-parameter Fermi distribution~\cite{grasp89} was employed to approximate the radial dependence of the nuclear charge density. The nuclear magnetic dipole moment $\mu$~=~0.145746(9)~$\mu_N$ of $^{197} _{\phantom{1}79}$Au has been used in calculations of magnetic dipole hyperfine constants~\cite{Raghavan:1989,Stone:2005}. \section{Method} \label{method} The numerical wave~functions were obtained independently for the two levels of interest, $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ and $ 5d^9 6s^2 $~$ ^2 D _{5/2} $. The calculations proceeded in eight phases: \begin{enumerate} \item Spectroscopic orbitals were obtained in the Dirac-Hartree-Fock approximation. These were kept frozen in all subsequent calculations. \item Virtual orbitals were generated in an approximation (called SrD, and explained in the following subsection \ref{virtuals}), in which all single and restricted double substitutions from $3spd4spdf5spd6s$ spectroscopic orbitals to eight layers of virtual orbitals were included (see the following subsection \ref{virtuals} for definitions of spectroscopic and virtual orbital sets). \item Contributions from $1s2sp$ shells were added in the configuration-interaction calculation, i.e.,~with all orbitals frozen. Only single substitutions contributed to the expectation values. The configurations involving $1s2sp$ orbitals were carried over to the following phases. \item Unrestricted single and double substitutions (SD) were performed, in which one or two occupied orbitals from the $5spd6s$ subshells were replaced by orbitals from the virtual set '3spdf2g1h', i.e., 3 virtual orbitals of each of the 's,p,d,f' symmetries, plus 2 virtual orbitals of the 'g' symmetry, and 1 virtual orbital of the 'h' symmetry. \item Unrestricted triple substitutions (T) from $5spd6s$ valence and core orbitals to '2spdf1g' virtual set were added. \item The final series of configuration-interaction calculations were based on the multiconfiguration expansions carried over and merged from all previous phases enumerated above. \item Contributions from the Breit interaction were evaluated in the single-configuration approximation, including the full Breit operator in the self-consistent-field process. \item The values of the nuclear electric quadrupole moment $ Q( ^{197} {\rm Au} ) $ were obtained from the relation $ B(J) = 2 e Q \left< JJ | T^{(2)} | JJ \right> $, where the electronic operator $ T^{(2)} $ represents the electric field gradient at the nucleus. Expectation values of hyperfine constants $A$ and of electric field gradients were calculated~\cite{Joensson:hfs:CPC:1996} separately for both states, $ ^2 D _{3/2} $ and $ ^2 D _{5/2} $. The experimental values of the hyperfine constants $A$ and $B$ were taken from~\cite{Blachman:1967,ChildsGoodman:1966}. \end{enumerate} \subsection{Virtual orbital set} \label{virtuals} \begin{table} \caption{Calculated values of $A$ and $Q$ obtained in several approximations during the process of generation of virtual orbital set for the $D_{3/2}$ state. DHF -- uncorrelated Dirac-Hartree-Fock value; $n$ -- largest principal quantum number in the orbital set; $from$ -- spectroscopic orbitals opened for SrD substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D3-2-SrD} \begin{tabular}{lrlrdd} \colrule \multicolumn{4}{c}{experiment} & 199.8425(2) & \\ \colrule \multicolumn{1}{c}{ n } & \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{r}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule DHF & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & 1 & 218.011 & 580.807 \\ \phantom{1}7 & 5d6s & 1spdfgh & 1147 & 187.302 & 623.275 \\ \phantom{1}8 & 5spd6s & 2spdfgh & 13729 & 198.774 & 652.057 \\ % \phantom{1}9 & 4spdf...6s & 3spdfgh & 97526 & 195.492 & 547.891 \\ % 10 & 3spd...6s & 4spdfg3h & 222129 & 196.513 & 528.752 \\ 11 & 3spd...6s & 5spdfg3h & 222494 & 199.413 & 523.736 \\ 12 & 3spd...6s & 6spdfg3h & 222851 & 199.455 & 514.186 \\ % 13 & 3spd...6s & 7spdfg3h & 223212 & 200.431 & 515.489 \\ 14 & 3spd...6s & 8spdfg3h & 223573 & 199.871 & 515.495 \\ \colrule \end{tabular} \end{table} \begin{table} \caption{Calculated values of $A$ and $Q$ obtained in several approximations during the process of generation of virtual orbital set for the $D_{5/2}$ state. DHF -- uncorrelated Dirac-Hartree-Fock value; $n$ -- largest principal quantum number in the orbital set; $from$ -- spectroscopic orbitals opened for SrD substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D5-2-SrD} \begin{tabular}{lrlrdd} \colrule \multicolumn{4}{c}{experiment} & 80.236(3) & \\ \colrule \multicolumn{1}{c}{ n } & \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{r}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule \multicolumn{1}{l}{DHF} & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & 1 & 79.041 & 612.985 \\ \phantom{1}7 & 5d6s & 1spdfgh & 11984 & 69.487 & 707.216 \\ \phantom{1}8 & 5spd6s & 2spdfgh & 33291 & 72.278 & 673.387 \\ \phantom{1}9 & 4spdf...6s & 3spdfgh & 128639 & 77.761 & 558.526 \\ 10 & 3spd...6s & 4spdfg3h & 290612 & 81.020 & 532.862 \\ 11 & 3spd...6s & 5spdfg3h & 291039 & 81.045 & 534.635 \\ 12 & 3spd...6s & 6spdfg3h & 291466 & 81.248 & 520.409 \\ 13 & 3spd...6s & 7spdfg3h & 291893 & 81.214 & 520.890 \\ 14 & 3spd...6s & 8spdfg3h & 292320 & 82.136 & 520.259 \\ \colrule \end{tabular} \end{table} We generated 8 layers of virtual shells (3 layers with 'spdfgh' symmetries and 5 layers with 'spdfg' symmetries). It should be noted, that the notion of a `layer' is somewhat different when applied to occupied (also referred to as {\sl spectroscopic}) orbitals, as opposed to virtual (also referred to as {\sl correlation}) orbitals. A core `layer', i.e.,~a subset of occupied orbitals possessing the same principal quantum number (often referred to as a {\sl shell}), constitutes a set of one-electron spin-orbitals, clustered in space, and having similar one-electron energy values. On the other hand, virtual orbitals with the same principal quantum number are not necessarily spatially clustered because their one-electron energy values do not have physical meaning and may vary widely, depending on the correlation effects that a particular virtual orbital describes. Therefore a `virtual layer' usually means a subset of the virtual set, generated in one step of the procedure, as described above. Such a `layer' is often composed of orbitals with different angular symmetries. The notation used in the tables and text of the present paper reflects the above considerations, in the sense that occupied orbitals are listed by their principal and angular quantum numbers (i.e.~$5spd$ means three occupied orbitals of $s$, $p$, and $d$ symmetry with principal quantum number $n=5$), while virtual orbitals are listed by angular symmetry and quantity (i.e. '5spd' would mean fifteen virtual orbitals --- five of each of the 's', 'p', and 'd' symmetries). To avoid confusion we distinguish occupied orbitals from virtual ones in the present paper by using {\sl italics} for occupied orbitals, while virtual orbitals are enclosed in 'quotation marks'. This distinction is {\sl not} applied in the tables, since in the tables there are always headings 'from' and 'to' which clearly denote occupied and virtual orbitals, respectively. The notation should always be analysed in the proper context (see~\cite{Bieron:Au:2008} for further details). In the present calculations single and restricted double (SrD) substitutions were allowed from valence and core orbitals (starting from $5d6s$ for the first virtual layer). The restriction was applied to double substitutions in such a way that only one electron was substituted from core $3spd4spdf5spd$ shells, the other one had to be substituted from valence $6s$ shell. Each subsequent layer was generated with substitutions from deeper core shells, down to $3s$. Table~\ref{calcAQ-D3-2-SrD} shows which occupied orbitals were opened at each step, as well as composition of the virtual orbital set when subsequent layers were generated for the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ state. For instance, the line marked '10' in the first column describes the generation of the fourth virtual layer, for which the largest principal quantum number was 10; all occupied orbitals between $3s$ and $6s$ (i.e.~$3spd4spdf5spd6s$) were opened for substitutions; the virtual set was composed of 4 orbitals of symmetries 's', 'p', 'd', 'f', 'g', and 3 orbitals of 'h' symmetry. The last four layers (those with principal quantum numbers 11, 12, 13, 14) were generated with a further restriction, which allowed only single substitutions to these last layers. Table~\ref{calcAQ-D5-2-SrD} presents the analogous data obtained in the process of generation of virtual orbital layers for the $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ state. The data from both tables are also presented as crosses in figure~\ref{figure2Q2A}. \subsection{Contributions from $1s2sp$ orbitals} \label{1s2sp} After generating the virtual orbital set, all orbitals were frozen and further calculations were carried out in the configuration-interaction (CI) approach. First, the effects of $1s2sp$ orbitals were evaluated in separate CI calculations. For the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ state they are presented in the Table~\ref{calcAQ-D3-2-1s2sp}, together with the contributions of all other occupied orbitals of the gold atom. The orbitals that were open for single and restricted double substitutions to the full virtual set are listed in the first column. The contributions of individual orbitals (i.e.,~of the leftmost orbital in the first column) are listed in the fourth and sixth column and presented in graphical form in Fig.~\ref{figure-1s6s-d3}. The individual contributions of the $2p$, $2s$, and $1s$ orbitals to the total $Q$ value were of the order of 0.6~\%, 0.2~\%, and 0.02~\%, respectively. The combined contribution of $1s2sp$ shells was of the order of 0.8~\%, with respect to the total $Q$ value. The contribution to the calculated value of magnetic dipole hyperfine constant $A$ was evaluated in the same manner as for $Q$. \begin{figure} \resizebox{0.8\columnwidth}{!}{% \includegraphics{figure-1s6s-d3.eps} } \caption{ Contributions from occupied orbitals to the calculated value of $Q$ for the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ state of Au. See caption of the Table~\ref{calcAQ-D3-2-1s2sp} and Sec.~\ref{1s2sp} for further details. } \label{figure-1s6s-d3} \end{figure} \begin{table} \caption{ Contributions from occupied orbitals to the calculated values of $A$ and $Q$ for the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ state of Au; $orbitals$ = set of orbitals open for single and restricted double substitutions from all shells listed in the first column to the full virtual set; NCF = size of the multiconfiguration expansion; $\Delta$A = contribution [MHz] of the leftmost orbital from a given orbital set to the total $A$ value (i.e. the individual contribution of the $1s$ orbital is listed in the line $1s..6s$); $\Delta$Q = contribution [mb] of the leftmost orbital from a given set to the $Q$ value. } \label{calcAQ-D3-2-1s2sp} \begin{tabular}{lrdrdr} \colrule \multicolumn{1}{r}{ orbitals } & \multicolumn{1}{r}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $\Delta$A } & \multicolumn{1}{c}{ $Q$[mb] } & \multicolumn{1}{c}{ $\Delta$Q } \\ \colrule --- & 1 & 218.011 & \multicolumn{1}{c}{ --- } & 580.807 & \multicolumn{1}{c}{---} \\ 5d6s & 16457 & 189.406 & $-$28.605 & 613.418 & 32.611 \\ 5pd6s & 39808 & 153.103 & $-$36.302 & 629.418 & 16.000 \\ 5spd6s & 48129 & 190.849 & 37.746 & 625.559 & $-$3.859 \\ 4f5spd6s & 89477 & 187.661 & $-$3.188 & 623.738 & $-$1.821 \\ 4df5spd6s & 124673 & 194.614 & 6.953 & 593.938 & $-$29.800 \\ 4pdf5spd6s & 148188 & 202.721 & 8.107 & 528.442 & $-$65.496 \\ 4spdf5spd6s & 156525 & 196.476 & $-$6.245 & 529.646 & 1.204 \\ 3d4spdf5spd6s & 191721 & 199.346 & 2.870 & 525.342 & $-$4.304 \\ 3pd...6s & 215236 & 201.106 & 1.760 & 514.175 & $-$11.167 \\ 3spd...6s & 223573 & 199.872 & $-$1.234 & 515.495 & 1.320 \\ % 2p3spd...6s & 247088 & 196.564 & $-$3.308 & 518.635 & 3.140 \\ 2sp3spd...6s & 255425 & 199.576 & 3.012 & 519.539 & 0.904 \\ 1s2sp3spd...6s & 263762 & 199.554 & $-$0.022 & 519.634 & 0.095 \\ \colrule \end{tabular} \end{table} A similar procedure has been carried out for the $Q$ and $A$ values of the $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ state. The results for the $ ^2 D _{5/2} $ state are shown in table~\ref{calcAQ-D5-2-1s2sp} and in figure~\ref{figure-1s6s-d5}. The individual contributions of the $2p$, $2s$, and $1s$ orbitals to the total $Q$ value were of the order of 0.5~\%, 0.2~\%, and 0.02~\%, respectively. The combined contribution of $1s2sp$ shells was of the order of 0.7~\%, with respect to the total $Q$ value. All these contributions have been included in the $Q$ and $A$ values obtained within the SrD approximation and the configuration state functions (CSFs) involved in evaluation of these contributions were carried over to all subsequent calculations. It should be pointed out that the data in Tables~\ref{calcAQ-D3-2-1s2sp} and~\ref{calcAQ-D5-2-1s2sp} and in the Figures~\ref{figure-1s6s-d3} and~\ref{figure-1s6s-d5} were obtained with single and restricted double substitutions, i.e.,~with unrestricted double and triple substitutions excluded. Therefore the contributions of the $5psd$ and $4spdf$ shells are somewhat distorted --- if double and triple substitutions were included, the individual contributions of the $5psd$ and $4spdf$ shells would differ by a few percent. Only the $3spd$, $2sp$ and $1s$ shells are essentially insensitive to double and triple substitutions (see Sec.~\ref{sdtq} below). Therefore their contributions are approximately correct. \begin{figure} \resizebox{0.8\columnwidth}{!}{% \includegraphics{figure-1s6s-d5.eps} } \caption{ Contributions from occupied orbitals to the calculated value of $Q$ for the $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ state of Au. See caption of Table~\ref{calcAQ-D5-2-1s2sp} and Sec.~\ref{1s2sp} for further details. } \label{figure-1s6s-d5} \end{figure} \begin{table} \caption{ Contributions from occupied orbitals to the calculated values of $A$ and $Q$ for the $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ state of Au; $orbitals$ = set of orbitals open for single and restricted double substitutions from all shells listed in the first column to the full virtual set; NCF = size of the multiconfiguration expansion; $\Delta$A = contribution [MHz] of the leftmost orbital from a given orbital set to the total $A$ value (i.e. the individual contribution of the $1s$ orbital is listed in the line $1s..6s$); $\Delta$Q = contribution [mb] of the leftmost orbital from a given set to the $Q$ value. } \label{calcAQ-D5-2-1s2sp} \begin{tabular}{rrdrdr} \colrule \multicolumn{1}{r}{ orbitals } & \multicolumn{1}{r}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $\Delta$A } & \multicolumn{1}{c}{ $Q$[mb] } & \multicolumn{1}{c}{ $\Delta$Q } \\ \colrule --- & 1 & 79.041 & \multicolumn{1}{c}{ --- } & 612.985 & \multicolumn{1}{c}{---} \\ 5d6s & 21501 & 106.724 & 27.683 & 651.547 & 38.562 \\ 5pd6s & 51800 & 109.554 & 2.830 & 643.451 & $-$8.096 \\ 5spd6s & 62536 & 71.472 & $-$38.082 & 638.694 & $-$4.757 \\ 4f5spd6s & 117626 & 70.636 & $-$0.836 & 636.280 & $-$2.414 \\ 4df5spd6s & 163739 & 73.490 & 2.854 & 604.280 & $-$32.000 \\ 4pdf5spd6s & 194221 & 74.597 & 1.107 & 532.632 & $-$71.648 \\ 4spdf5spd6s & 204973 & 79.767 & 5.170 & 534.008 & 1.376 \\ 3d4spdf5spd6s & 251086 & 80.877 & 1.110 & 530.308 & $-$3.700 \\ % 3pd...6s & 281568 & 80.580 & $-$0.297 & 518.734 & $-$11.574 \\ % 3spd...6s & 292320 & 82.136 & 1.556 & 520.259 & 1.525 \\ % 2p3spd...6s & 322802 & 81.700 & $-$0.436 & 522.677 & 2.418 \\ % 2sp3spd...6s & 333554 & 78.995 & $-$2.705 & 523.757 & 1.080 \\ % 1s2sp3spd...6s & 344306 & 79.025 & 0.030 & 523.880 & 0.123 \\ \colrule \end{tabular} \end{table} \subsection{Double, triple, and quadruple substitutions} \label{sdtq} The decomposition of the electron correlation correction to the hyperfine structure into one-, two-, three-, and four-body effects can be understood from the following (simplified) analysis. The structure of the $ 5d^9 6s^2 $~$ ^2 D $ states of gold is determined to a large extent by the interaction of the valence $6s^2$ shell with a highly polarisable $5d^{9}$ shell. The direct and indirect effects of relativity bring the outer $d$ shell much closer, radially and energetically, to the valence $s$ orbital than in homologous silver and copper atoms~\cite{Pyykko:1988,Pyykko:Au:2004}. This in turn increases the polarisation of the $5d^{9}$ shell by the valence electrons. Therefore, the core-valence interaction (the leading electron correlation correction) leads to the contraction of the $6s$ orbital, which overestimates the hyperfine structure. The unrestricted double substitutions affect the hyperfine structure in two ways: directly through the configuration state functions (CSFs) themselves but also indirectly through the change of the expansion coefficients of the important configurations obtained by single substitutions. Three-particle effects in turn affect the expansion coefficients of the configurations obtained from double substitutions. In a simple picture we can describe the wave function in terms of pair-correlation functions and the three-particle effects then account for polarisation of pair-correlation functions, leading to an increase of the hyperfine structure~\cite{Joensson:hfsNa:1996}. Four-particle effects affect mostly the expansion coefficients of the configurations obtained from double substitutions. Therefore their influence on the hyperfine structure is indirect and second-order to that of the double substitutions. They are usually small and can often be neglected~\cite{Godefroid:1997}; they are discussed in Sec.~\ref{sdtq}. Tables~\ref{calcAQ-D3-2-SD} and~\ref{calcAQ-D5-2-SD} show the results of configuration-interaction calculations, where various combinations of occupied and virtual sets were tested with single and unrestricted double substitutions. The data from both tables are also presented as empty circles in Fig.~\ref{figure2Q2A}. \begin{table} \caption{Values of $Q$ and $A$ for the $D_{3/2}$ state, calculated in configuration-interaction approach, with single and unrestricted double substitutions, in several different multiconfiguration expansions; $from$ -- spectroscopic orbitals opened for substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D3-2-SD} \begin{tabular}{llrdd} \colrule \multicolumn{3}{c}{experiment} & 199.8425(2) & \\ \colrule \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule 5spd6s & 1spdfgh & 259135 & 205.426 & 521.191 \\ 4spdf5spd6s & 1spdfgh & 358019 & 205.968 & 521.503 \\ 5spd6s & 2spdf & 279559 & 210.523 & 509.839 \\ 5spd6s &2spdfg & 320545 & 211.512 & 511.461 \\ 5spd6s & 2spdfgh & 366257 & 211.480 & 512.286 \\ 5spd6s & 3spdf2g1h &459594 & 213.088 & 510.451 \\ 5spd6s & 3spdf2gh & 465794 & 213.075 & 510.402 \\ 5spd6s & 3spdfg2h & 506987 & 213.146 & 510.268 \\ 5spd6s & 4spdf2gh & 687301 & 213.200 & 510.478 \\ \colrule \end{tabular} \end{table} \begin{table} \caption{Values of $Q$ and $A$ for the $D_{5/2}$ state, calculated in configuration-interaction approach, with single and unrestricted double substitutions, in several different multiconfiguration expansions; $from$ -- spectroscopic orbitals opened for substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D5-2-SD} \begin{tabular}{rlrdd} \colrule \multicolumn{3}{c}{experiment} & 80.236(3) & \\ \colrule \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule 5spd6s & 1spdfgh & 339306 & 74.258 & 507.823 \\ 4spdf5spd6s & 1spdfgh & 467381 & 72.048 & 509.321 \\ 5spd6s & 2spdfgh & 480824 & 73.468 & 512.278 \\ 5spd6s & 3spdf2gh & 607421 & 73.494 & 512.559 \\ 5spd6s & 4spdf3gh & 898368 & 73.294 & 514.621 \\ 5spd6s & 5spdf4g3h & 1228675 & 73.212 & 514.269 \\ \colrule \end{tabular} \end{table} \begin{table} \caption{Values of $Q$ and $A$ for the $D_{3/2}$ state, calculated in configuration-interaction approach, with single and unrestricted double and triple substitutions, in several different multiconfiguration expansions; $from$ -- spectroscopic orbitals opened for substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D3-2-SDT} \begin{tabular}{rlrdd} \colrule \multicolumn{3}{c}{experiment} & 199.8425(2) & \\ \colrule \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule 5spd6s & 1spd & 265183 & 198.955 & 520.346 \\ 5spd6s & 1spdf & 386326 & 194.391 & 533.464 \\ 5spd6s & 1spdfg & 641227 & 193.744 & 536.620 \\ 5spd6s & 1spdfgh & 1012615 & 193.246 & 537.786 \\ 5spd6s & 2spd1f & 943544 & 198.752 & 522.361 \\ 5spd6s & 2spdf & 1543051 & 199.973 & 520.536 \\ 5spd6s & 3spd2f & 1200261 & 198.207 & 520.267 \\ 5spd6s & 3psdf & 1309130 & 198.254 & 520.096 \\ \colrule \end{tabular} \end{table} \begin{table} \caption{Values of $Q$ and $A$ for the $D_{5/2}$ state, calculated in configuration-interaction approach, with single and unrestricted double and triple substitutions, in several different multiconfiguration expansions; $from$ -- spectroscopic orbitals opened for substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D5-2-SDT} \begin{tabular}{rlrdd} \colrule \multicolumn{3}{c}{experiment} & 80.236~(3) & \\ \colrule \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule 5spd6s & 1spd & 341440 & 81.6955 & 514.929 \\ 5spd6s & 1spdf & 456506 & 82.1357 & 520.259 \\ 4f5spd6s & 1spdf & 1403860 & 79.9343 & 518.380 \\ 5spd6s & 1spdfg & 842883 & 80.2371 & 519.291 \\ 5spd6s & 2spdf & 1326851 & 83.0623 & 521.862 \\ \colrule \end{tabular} \end{table} \begin{table} \caption{Values of $Q$ and $A$ for the $D_{3/2}$ state, calculated in configuration-interaction approach, with single and unrestricted double, triple, and quadruple substitutions, in several different multiconfiguration expansions; $type$ -- substitution multiplicity; $from$ -- spectroscopic orbitals opened for substitutions; $to$ -- virtual orbital set; NCF -- number of configurations; see text for further details. } \label{calcAQ-D3-2-SDTQ} \begin{tabular}{lrlrdd} \colrule \multicolumn{3}{c}{experiment} & 199.8425(2) & \\ \colrule \multicolumn{1}{c}{ type } & \multicolumn{1}{c}{ from } & \multicolumn{1}{c}{ to } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule SDT & 5spd6s & 1spd & 386326 & 194.391 & 533.464 \\ SDTQ & 5spd6s & 1spd & 569497 & 194.301 & 533.653 \\% SDT & 5spd6s & 1spdf & 386326 & 194.391 & 533.464 \\ SDTQ & 5spd6s & 1spdf & 967871 & 195.376 & 531.685 \\% SDTQ & 5spd6s & 1spdf & 1089014 & 194.686 & 531.846 \\% qa3dq1f-h \colrule \end{tabular} \end{table} The second line in Tables~\ref{calcAQ-D3-2-SD} and~\ref{calcAQ-D5-2-SD} represents a calculation in which substitutions from the $4spdf$ shells were allowed to one layer of virtual orbitals. When compared with the first line, it yields the effect of $4spdf$ shells on the calculated values of $Q$ and $A$. In order to limit the size of the configuration expansions, the CSFs representing the above substitutions were not carried over to the following, higher order calculations. Instead, the corrections were included additively, as described in Sec.~\ref{corrections}. At the same time the evaluation of these corrections may be treated as a crude estimate of error arising from omitted double substitutions from occupied shells (see Sec.~\ref{error} for details). An inspection of the last column of Table~\ref{calcAQ-D3-2-SD} indicates, that three layers of virtual orbitals were necessary to reach convergence of the $Q$ and $A$ values in the {\sl single and unrestricted double substitutions} (SD) approximation for the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ state. Four layers were necessary in case of the $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ state (see Table~\ref{calcAQ-D5-2-SD}). Tables~\ref{calcAQ-D3-2-SDT} and~\ref{calcAQ-D5-2-SDT} show the results of configuration-interaction calculations, in which various combinations of occupied and virtual sets were tested with unrestricted double and triple substitutions. The data from both tables are also presented as full circles in Fig.~\ref{figure2Q2A}. Two layers of virtual orbitals were necessary to reach convergence of the $Q$ value in the {\sl single and double and triple substitutions} (SDT) approximation for both $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ and $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ states. In case of the $A$ values, convergence required three, rather than two, layers. Table~\ref{calcAQ-D3-2-SDTQ} shows the effect of quadruple substitutions. The first line represents an approximation in which single, double, and triple substitutions from $5spd6s$ orbitals to a truncated virtual layer composed of 's', 'p', and 'd' symmetries were included. The third line represents a similar approximation in which the (still truncated) virtual layer was composed of 's', 'p', 'd', and 'f' symmetries. The second, fourth, and fifth lines represent corresponding 'quadruple approximation' in which single, double, triple, and quadruple substitutions were allowed. The comparison had to be made on a reasonably small orbital set in order to be able to converge the calculation involving quadruple substitutions. The numbers of CSFs in the last two lines are different, because certain restrictions were applied in the calculation represented by the fourth line (see the comments near the end of Sec.~\ref{configuration-interaction} for details). The results presented in the table~\ref{calcAQ-D3-2-SDTQ} indicate that the correction involving quadruple substitutions is unlikely to exceed 1~\%. The CSFs representing quadruple substitutions were not carried over to the following calculations and the 'quadruple' correction was included additively, as described in Sec.~\ref{corrections}. \subsection{4-D configuration-interaction calculations} \label{configuration-interaction} A full, converged Complete Active Space (CAS) calculation for the gold atom is still unattainable due to software and hardware limitations. Based on our current calculations we estimate that the CAS approach would require configuration expansions in four 'dimensions': (1) single, double, triple, and perhaps quadruple substitutions; (2) from all core shells (or at least from $3spd4spdf5spd6s$); (3) to eight or more virtual orbital layers; (4) of $s$, $p$, $d$, $f$, $g$, $h$, and perhaps higher symmetries. One can imagine a 'space' spanned by the four 'dimensions' defined above, i.e.~'{\sl substitution} multiplicity', 'number of opened {\sl core} subshells', 'number of {\sl virtual} layers', and 'maximal {\sl symmetry} of virtual layers' dimension. In fact, this space should rather be called a 'matrix', since all four dimensions are discrete. Let's call this four-dimensional matrix a 'CAS matrix'. Each element of the matrix is represented by a multiconfiguration expansion obtained by substituting a particular number of electrons ('substitution' dimension) from specific core orbitals ('core' dimension) to a set of virtual orbitals ('virtual' dimension) of specific symmetries ('symmetry' dimension). A full CAS calculation would require several orders of magnitude larger configuration expansions than are possible even with the largest computer resources available today. \begin{table} \caption{The final configuration-interaction calculations of $Q$ and $A$ for the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ state of Au; $type$ --- description of the multiconfiguration expansions, see text for details; NCF = size of the multiconfiguration expansion. } \label{calcAQ-D3-2-F} \begin{tabular}{lrdd} \colrule \multicolumn{2}{c}{experiment} & 199.8425(2) & \\ \colrule \multicolumn{1}{c}{ type } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule SD:3hgg \!\!\! + SDT:2fd & 1182329 & 206.343 & 517.201 \\ SD:3hgf + SDT:2fd & 1144532 & 206.221 & 517.342 \\ SD:3hgf + SDT:2gd & 1711382 & 205.104 & 519.106 \\ SD:3hgf + SDT:2gf & 1847380 & 204.489 & 519.829 \\ \colrule \end{tabular} \end{table} \begin{table} \caption{The final configuration-interaction calculations of $Q$ and $A$ for the $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ state of Au; $type$ --- description of the multiconfiguration expansions, see text for details; NCF = size of the multiconfiguration expansion. } \label{calcAQ-D5-2-F} \begin{tabular}{lrdd} \colrule \multicolumn{2}{c}{experiment} & 80.236~(3) & \\ \colrule \multicolumn{1}{c}{ type } & \multicolumn{1}{c}{ NCF } & \multicolumn{1}{c}{ $A$[MHz] } & \multicolumn{1}{c}{ $Q$[mb] } \\ \colrule SD:3hgf + SDT:2fd & 1441120 & 78.2451 & 520.073 \\ SD:3hgf + SDT:2gd & 1527668 & 79.9182 & 522.066 \\ \colrule \end{tabular} \end{table} However, a computational strategy can be designed in which a considerably smaller multiconfiguration expansion yields a wave~function only marginally inferior to a full CAS wave~function, in the sense that all important electron correlation effects are included and the calculated values of $A$ and $Q$ are close to those that would result from a full, converged, CAS calculation. The strategy is based on the observation that one does not have to simultaneously push the configuration expansions to the limits of all the above mentioned 'dimensions'. Specifically, the dependence of atomic properties on the 'substitution' dimension is critical. To illustrate this approach, let's consider separately the contributions of single, double, and triple substitutions to the calculated values of $A$ and $Q$ of gold. To obtain converged results within a {\sl single substitutions} model, one has to include substitutions from all occupied shells ($1s2sp3spd4spdf5spd6s$) to eight or more virtual layers. This is illustrated in tables~\ref{calcAQ-D3-2-SrD}, \ref{calcAQ-D5-2-SrD}, \ref{calcAQ-D3-2-1s2sp}, and \ref{calcAQ-D5-2-1s2sp}, where eight virtual orbital layers were necessary to converge the series of self-consistent-field calculations. However, to obtain a converged result within a {\sl single and double substitutions} model (SD) one has to include double substitutions from $4spdf5spd6s$ occupied orbitals, not to eight, but to three or at most four virtual layers (see Tables~\ref{calcAQ-D3-2-SD} and~\ref{calcAQ-D5-2-SD}). In the {\sl single, double, and triple substitutions} model (SDT) it is enough to consider triple substitutions from $5spd6s$ occupied orbitals to two or at most three virtual layers (see Tables~\ref{calcAQ-D3-2-SDT} and~\ref{calcAQ-D5-2-SDT}). In the 'space' (or rather in the 'matrix') of the four 'dimensions' defined above, the 'core' and 'virtual' dimension sizes strongly depend on the 'substitution' dimension (in fact, all four dimensions are interdependent). Therefore, one can construct an approximation, in which all important electron correlation effects are included and the calculated values of $A$ and $Q$ are close to those that would result from a full, converged CAS calculation. In order to find a suitable approximation, we have performed a set of test calculations for several elements of the above mentioned 'matrix'. For each 'dimension', the calculations were saturated to the point where the relative change of the expectation values (i.e.,~both $A$ and $Q$) did not exceed a small fraction of a percent (usually two or three tenths of a percent). Specifically, for each 'substitution' dimension (i.e.,~for single, double, and triple substitutions) we thoroughly tested the dependence of observables on 'symmetry,' 'virtual', and 'core' spaces. When a saturated set of configuration state functions (CSFs) is obtained for a particular 'substitution' dimension, all these CSFs are carried over to the next step(s). The merged, 'final' multiconfiguration expansion represents an approximation, which is effectively equivalent to a CAS expansion, and the corresponding wavefunction is of similar quality as a CAS wavefunction, at least from the point of view of the calculated values of $A$ and $Q$. \begin{figure*}% \subfigure[ ~from electric field gradient and B($ 5d^9 6s^2 $~$ ^2 D_{3/2} $) ]{ \label{subfigureQD3} \includegraphics[width=.448\textwidth]{figureD3Q.eps}} \hspace{.3in} \subfigure[ ~from electric field gradient and B($ 5d^9 6s^2 $~$ ^2 D_{5/2} $) ]{ \label{subfigureQD5} \includegraphics[width=.44\textwidth]{figureD5Q.eps}} \\ \vspace{.3in} \subfigure[ ~A($ 5d^9 6s^2 $~$ ^2 D_{3/2} $) ]{ \label{subfigureAD3} \includegraphics[width=.448\textwidth]{figureD3A.eps}} \hspace{.3in} \subfigure[ ~A($ 5d^9 6s^2 $~$ ^2 D_{5/2} $) ]{ \label{subfigureAD5} \includegraphics[width=.44\textwidth]{figureD5A.eps}} \caption{(Color online) Nuclear quadrupole moment $Q$($^{197}\!$Au) [mb] obtained from the calculated electric field gradients, and hyperfine magnetic dipole constants $A$ [MHz] of the states $ 5d^9 6s^2 $~$ ^2 D_{3/2} $ and $ 5d^9 6s^2 $~$ ^2 D_{5/2} $, as functions of the size of the multiconfiguration expansions; line with no symbols (red online) -- 'SrD' approximation; circles (blue online) -- 'SD' approximation; triangles (black online) -- 'SDT' and final 'CAS' approximations (see text for details). Horizontal straight lines in figures (c) and (d) represent the experimental values of hyperfine constants $ A ( ^2 D_{3/2} ) $~=~199.8425(2)~MHz~\cite{Blachman:1967} and $ A ( ^2 D_{5/2} ) $~=~80.236(3)~MHz~\cite{ChildsGoodman:1966}, respectively. The small corrections described in subsection~\ref{corrections} are not included in the figures. } \label{figure2Q2A} \end{figure*} In practice there is not one single final 'CAS' expansion, but a series of such final expansions in which various sets of 'S', 'SD', and 'SDT' multiconfiguration expansions (i.e.~various sets with single, double, and triple substitutions) are merged together. Table~\ref{calcAQ-D3-2-F} shows the results obtained from a series of such final 'CAS' calculations for the $ ^2 D _{3/2} $ state, and Table~\ref{calcAQ-D5-2-F} shows the same for the $ ^2 D _{5/2} $ state. The data from both tables are also included in Fig.~\ref{figure2Q2A}. The 'CAS' expansions are composed as follows. All virtual orbitals and all CSFs generated in the SrD approximation, as described in section~\ref{virtuals}, as well as those described in section~\ref{1s2sp}, were included. The remaining CSF expansions were generated with substitutions from $5spd6s$ orbitals to virtual sets described in the first column of Tables~\ref{calcAQ-D3-2-F} and~\ref{calcAQ-D5-2-F}, where symbols before the colon represent substitution multiplicity, i.e., SD --- single and double substitutions; SDT --- single and double and triple substitutions; the symbols after the colon represent virtual orbital layers, i.e., 3hgg --- three layers (first layer with 'spdfgh' symmetries and two layers with 'spdfg' symmetries); 2fd --- two layers (first layer with 'spdf' symmetries and second layer with 'spd' symmetries); 3hgf --- three layers (first layer with 'spdfgh' symmetries, second layer with 'spdfg' symmetries; third layer with 'spdf' symmetries), etc. In the largest calculations, when single, double, and triple substitutions to two or three layers were included, we had to further limit the overall number of CSFs, due to software and hardware limitations. In those cases, the occupation number of the least important virtual orbital was restricted to single or double, thus excluding those CSFs in which this particular virtual orbital was occupied by three electrons. The difference that such a restriction brings about can always be evaluated on a smaller set of CSFs before a full calculation is performed. Therefore we always had control on the effects of the above mentioned restrictions on the calculated values of $A$ and $Q$. \section{Results} \label{results} \begin{figure} \includegraphics[width=.48\textwidth]{figureD3Q-D5Q-logarithmic.eps} \caption{(Color online) Nuclear quadrupole moment $Q$ [mb] of the $^{197}\!$Au isotope obtained from the calculated electric field gradients, as a function of the size of multiconfiguration expansions for the states $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ (triangles -- red online) and $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ (circles -- blue online), compared with other theoretical and experimental results. The small corrections described in subsection~\ref{corrections} are not included in the figure. The values which represent multiconfiguration expansions of sizes smaller than 20000 are outside the figure, except the uncorrelated Dirac-Hartree-Fock values, represented by the single triangle (red online) for $ ^2 D _{3/2} $ and the single circle (blue online) for $ ^2 D _{5/2} $. The six values with error bars are from references~\cite{Yakobi:2007,Belpassi:2007,Itano:2006,PaladeWagner:2003,% Powers:1974,ChildsGoodman:1966}; the four values without error bars, represented by pluses, are from references~\cite{ThierfelderSchwerdtfegerSaue:2007,Schwerdtfeger:2005,% Blachman:1967}; all data are arranged in reverse chronological order, with the most recent results to the left. } \label{figureD3Q-D5Q-logarithmic} \end{figure} More extensive calculations turned out to be beyond the 100 node limit for this project on the Linux cluster at the National Institute of Standards and Technology (NIST), USA. Therefore the calculations of the magnetic dipole constants $A$ did not yield converged results. As might be expected, the effects of double and triple substitutions are relatively larger for $A$ than for $Q$, therefore the calculations of the $Q$ values were essentially converged; they yield: Q($ ^2 D _{3/2} $)~=~519.829~mb, and Q($ ^2 D _{5/2} $)~=~522.066~mb, respectively. \subsection{Corrections} \label{corrections} As mentioned in section~\ref{sdtq}, the contributions arising from unrestricted double substitutions from $4spdf$ orbitals were evaluated separately and included additively in the final $Q$ values. They yield +0.312~mb and +1.498~mb for the two states $ ^2 D _{3/2} $ and $ ^2 D _{5/2} $, respectively. The effects of the quadruple substitutions were also evaluated separately, in a very limited fashion, and only for the $ ^2 D _{3/2} $ state. As explained in section~\ref{sdtq}, the correction arising from the quadruple substitutions for the $ ^2 D _{3/2} $ state lowers the $Q$ value by 1.779~mb. The dependence of the $Q$ values on double and triple substitutions indicates that the quadruple correction might be smaller for the $Q( ^2 D _{5/2})$ value than for the $Q( ^2 D _{3/2})$ value, but we were unable to evaluate the former. Therefore we assumed identical, $-$1.8~mb corrections for both states. The corrections arising from the Breit interaction were calculated at the Dirac-Hartree-Fock level with full relaxation i.e.~with a frequency-dependent Breit term \begin{eqnarray} \label{breit} B_{ij} & = & - \frac{\bm{\alpha}_{i} \cdot \bm{\alpha}_{j}} {r_{ij}} -\frac{\bm{\alpha}_{i} \cdot \bm{\alpha}_{j}} {r_{ij}} [\cos(\omega_{ij}r_{ij})-1] \nonumber \\ & & + c^2 (\bm{\alpha}_{i} \cdot \vec{\nabla}_{i}) (\bm{\alpha}_{j} \cdot \vec{\nabla}_{j}) \frac{\cos(\omega_{ij} r_{ij}/c) - 1} {\omega_{ij}^{2} r_{ij}} \end{eqnarray} included in the self-consistent-field functional, using the \emph{mcdfgme} code \cite{desclaux:1993,indelicato:1995,mcdfgme}. In the formula above, $r_{ij}=\left|\vec{r}_{i}-\vec{r}_{j}\right|$ is the inter-electronic distance, $\omega_{ij}$ is the energy of the photon exchanged between two electrons, $\bm{\alpha}_{i}$ are Dirac matrices, and $c=1/\alpha$ is the speed of light. The Breit corrections are highly state-dependent (see also~\cite{Yakobi:2007}, where the Gaunt part was evaluated) and yield 2.3~mb and 0.6~mb for the two states, $ ^2 D _{3/2} $ and $ ^2 D _{5/2} $, respectively. The quantum electrodynamics (QED) corrections to the $Q$ values are expected to be very small. We evaluated the VP (vacuum polarisation) correction with the \emph{mcdfgme} code, following Ref.~\cite{boucard:2000}, and obtained a value of the order of 0.01~\%. When all above mentioned corrections are included, the $Q$ values become: Q($ ^2 D _{3/2} $)~=~520.641~mb and Q($ ^2 D _{5/2} $)~=~522.364~mb. The average of the above two results yields Q($ ^{197}\!$Au)~=~521.5~mb. \subsection{Error estimate} \label{error} A rigorous, systematic treatment of the error bar of the calculated electric quadrupole moment $Q$ would require evaluation of the effects of: all omitted virtual orbitals, all CSFs which were not included in the configuration expansions, as well as all physical effects that were not included or were treated approximately. However, we were only able to obtain very crude estimates of certain sources of systematic errors. We believe that none exceeded 1~\%, but the calculations presented in this paper were far too extensive to permit a rigorous treatment of the error. Therefore we have to resort to a less rigorous method. One of the frequently used methods of evaluation of the accuracy of calculated electric quadrupole moments $Q$ is based on the simultaneous calculations of magnetic dipole hyperfine constants $A$, and on subsequent comparison of calculated $A$ values with their experimental counterparts. As mentioned above, the calculations of the magnetic dipole constants $A$ have not converged. However, the amplitudes of the final oscillations of the two curves representing the values of $A$ for the two states of interest are comparable to the uncertainty of $A$ arising from the accuracy of the nuclear magnetic dipole moment value $\mu$. There are currently two different $\mu$ values in the literature~\cite{Raghavan:1989,Stone:2005}, $\mu$ = 0.145746(9) and $\mu$ = 0.148158(8), which differ by about 2~\%. Taken at face value, our results seem to favor the smaller value, $\mu$~=~0.145746(9), which, as mentioned in Sec.~\ref{theory}, has been used in the present calculations. However, the overall accuracy of our calculations (in particular, the evaluation of higher-order terms) does not permit us to draw a definitive conclusion. Therefore, the difference between the two values of $\mu$ should rather be treated as a source of systematic error in the determination of $A$. Therefore, we did not push the calculations of magnetic dipole constants $A$ further beyond their current level of convergence and, consequently, the calculations of $A$ values could not be used as reliable sources of error estimate for nuclear moments. Another method to estimate the accuracy of $Q$ is to consider the differences between the final values obtained from different states. However, in the present paper we were able to converge the calculations for only two atomic levels. The difference between the results obtained for these two levels turned out to be quite small, which rendered this method useless in this particular case. Considering the computational methodology employed in this paper, it is obvious that the final value depends on the choice of the multiconfiguration expansions representing the last few points on the curves in Fig.~\ref{figureD3Q-D5Q-logarithmic}, while the accuracy of the final value is connected with convergence of these curves. Therefore, we based the estimate of the error bar on the oscillations of the tail of the two curves in Fig.~\ref{figureD3Q-D5Q-logarithmic}. The largest difference taken from the last few points on the curves representing $ ^2 D _{3/2} $ and $ ^2 D _{5/2} $ states amounts to 3~mb and 4~mb, respectively. As an additional source of uncertainty we assumed the additive corrections described in subsection~\ref{corrections}, since all of them were evaluated in a rather crude approximation. For instance, the contribution of the Breit interaction was calculated at the Dirac-Hartree-Fock level, without regard for electron correlation effects. When all above sources of uncertainly are taken into account the total error bar amounts to 5~mb, which yields our final calculated value of quadrupole moment Q($^{197}\!$Au)~=~521.5~$\pm$~{5.0}~mb. \section{Comparisons} \label{comparisons} The results of our calculation are compared with previous evaluations in Table~\ref{Qsummary} and in Fig.~\ref{figureD3Q-D5Q-logarithmic}. It is worth noting that our result is in agreement with three most recent theoretical values, obtained with three different methods, but all these recent results (including ours) are considerably smaller than other, earlier values. Yakobi~{\sl et al}~\cite{Yakobi:2007} performed calculations for the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ and $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ states of atomic gold within the four-component Dirac-Coulomb framework~\cite{Eliav:1994,Kaldor:1998}. They correlated 51 out of the 79 electrons in the large basis sets (up to 26s22p18d12f8g5h uncontracted Gaussian functions) with the relativistic Fock-space coupled cluster method, including single and double excitations (CCSD). The contribution of the Gaunt term, the main part of the Breit interaction, was also evaluated. Belpassi~{\sl et al}~\cite{Belpassi:2007} performed molecular relativistic Dirac-Coulomb-Gaunt Hartree-Fock calculations~\cite{Pernpointner:2001} for a series of molecules: AuF, XeAuF, KrAuF, ArAuF, (OC)AuF, and AuH. The electronic correlation contributions were included at CCSD(T) and CCSD-T levels. The value of the nuclear quadrupole moment $Q$ was obtained from the determinations of the electric field gradient at the gold nucleus for the above mentioned molecules, combined with experimental values of the nuclear quadrupole coupling constants. Thierfelder~{\sl et al}~\cite{ThierfelderSchwerdtfegerSaue:2007} performed four-component relativistic density-functional (DFT) calculations for diatomic compounds CuX and AuX (X = H, F, Cl, Br, and I) with and without CO attached, i.e., OC-CuX and OC-AuX (X = F, Cl, Br, and I). They employed a newly developed functional~\cite{Yanai:2004}, whose role is to correctly describe the long-range part of exchange interactions~\cite{Goll:2007}, and obtained the averaged result~Q~=~526~mb. This value is within the error bounds of our value. Our result, in turn, falls within the error bounds published by Belpassi~{\sl et al} (Q~=~510(15)~mb), as well as those by Yakobi~{\sl et al} (Q~=~521(7)~mb). The agreement with Yakobi~{\sl et al} may be somewhat accidental because particular contributions show larger differences. The two outstanding differences arise from triple substitutions and from deep core orbitals. Yakobi~{\sl et al} evaluated the effect of the triple substitutions by performing single-reference CCSD(T) calculation for the $ ^2 D _{5/2} $ level, and obtained a 0.3~\% shift. The effect of triple substitutions is indeed smaller for the $ ^2 D _{5/2} $ level, but for the $ ^2 D _{3/2} $ level our calculations indicate a shift of the order of 2~\%. However, this discrepancy may be attributed to the methodological differences in the two papers. The definition of triple substitutions in the configuration-interaction method used here differs substantially from that in the CCSD(T) approach, due to the exponential nature of the coupled-cluster operator. The coupled-cluster approximation includes a subset of the CI triple substitutions (the "unlinked" diagrams), as well as that of higher order substitutions, already at the CCSD level. The CCSD(T) yields only the "linked" part as the effect of the triple substitutions. Therefore, the contribution of the CI triple substitutions may indeed be expected to be larger than that of the CC triple substitutions. Another difference arises from contributions of deep core orbitals. The effects of $3spd$, $2sp$, and $1s$ orbitals were neglected by Yakobi~{\sl et al}, while in our calculations they were all included. Their combined effect was to lower the $Q$ value by about 2~\%. \begin{table} \caption{Comparison of the present $Q$($^{197}$Au) value (in~mb~=~10$^{-31}$~m$^2$) with other recent values and with previous (muonic) standard value. } \label{Qsummary} \begin{tabular}{llll} \colrule Reference & Source & Q($^{197} \! $Au) & \\ \colrule This work & Au atom, $^2D_{3/2}$, $^2D_{5/2}$ & \multicolumn{1}{l}{ 521.5$\pm$5.0 } \\ Yakobi et al~\cite{Yakobi:2007} & Au atom, $^2D_{3/2}$, $^2D_{5/2}$ & \multicolumn{1}{l}{ 521$\pm$7 } \\ Belpassi et al~\cite{Belpassi:2007} & AuF, LAuF molecules & \multicolumn{1}{l}{ 510$\pm$15 } \\ Thierfelder et al~\cite{ThierfelderSchwerdtfegerSaue:2007} \,\,\, & AuX, LAuX molecules \,\,\,\,\, & \multicolumn{1}{l}{ 526 } \\ Powers et al~\cite{Powers:1974} & Muonic & \multicolumn{1}{l}{ 547$\pm$16 } \\ \colrule \end{tabular} \end{table} \section*{Conclusions} \label{conclusions} The multiconfiguration Dirac-Hartree-Fock (MCDHF) model has been employed to calculate the expectation values responsible for the hyperfine splittings of the $ 5d^9 6s^2 $~$ ^2 D _{3/2} $ and $ 5d^9 6s^2 $~$ ^2 D _{5/2} $ levels of atomic gold. To our knowledge, this is the first calculation in which \mbox{one-,} two-, and three-body electron correlation effects were included and saturated for electric quadrupole hyperfine values of a heavy, open-shell, neutral atom. The correlation effects involving all 79 electrons were accounted for with a procedure that is equivalent to a full Complete Active Space calculation. All electron correlation effects were explicitly accounted for at a 1~\% level of precision or better. Calculated electric field gradients, together with experimental values of the electric quadrupole hyperfine structure constants $B$, allow us to extract a nuclear electric quadrupole moment $Q$~=~521.5(5.0)~mb of $^{197}$Au. If taken at face value, the summary in table~\ref{Qsummary} suggests that our $Q$ value, together with that of Yakobi~{\sl et al}~\cite{Yakobi:2007}, could be the new standard value. \vspace{1cm} \section*{Acknowledgements} This work was supported by the Polish Ministry of Science and Higher Education (MNiSW) in the framework of scientific grant No.~1~P03B~110~30 awarded for the years 2006-2009. PJ acknowledges support from the Swedish Research Council (Vetenskapsr{\aa}det). PP belongs to the Finnish Center of Excellence in Computational Molecular Science (CMS). The visits of JB at Helsinki were supported by The Academy of Finland. The large scale calculations were performed on the Raritan Linux cluster at the National Institute of Standards and Technology (NIST) in Gaithersburg, USA. JB would like to express his gratitude for the hospitality which was extended to him during his visits to the Chemistry Department of the University of Helsinki and the Atomic Spectroscopy Group at NIST. PJ acknowledges the support of the Helmholtz Alliance Program of the Helmholtz Association, contract HA-216 ``Extremes of Density and Temperature: Cosmic Matter in the Laboratory''. Laboratoire Kastler Brossel is ``Unit\'e Mixte de Recherche du CNRS, de l' ENS et de l'UPMC n$^{\circ}$ 8552''. We thank the (anonymous) referee for pointing our attention to the structural differences between CI and CC methods.

BkiUbs3xK03BfL1dUFi8

\section{Introduction} Mixed states, namely positive semidefinite matrices of trace one, are not easy to characterise from a mathematical point of view. The reasons are at least threefold: first, they form a convex set with infinitely many extreme points, and therefore do not admit a concise description such as a vector space, or a convex set with ``corners''. Second, positive semidefinite matrices on a tensor product space $\mathcal{M}_d\otimes \mathcal{M}_d$ are not only given by convex combinations of positive semidefinite matrices on $\mathcal{M}_d$ and $\mathcal{M}_d$. And, third, the dimension of the vector space where they live grows exponentially with the number of subsystems, that is, describing a state in $\mathcal{M}_d \otimes \cdots \otimes \mathcal{M}_d$ ($n$ times) requires roughly $ d^{2n}$ parameters. The third problem is not specific to positive semidefinite matrices, but is shared by vectors $|\psi\rangle \in \mathbb{C}^{d} \otimes \cdots \otimes \mathbb{C}^d$ (describing \emph{pure states}) and Hermitian operators (usually describing Hamiltonians) living in tensor product spaces. The latter problem has motivated the program of tensor networks, which aims at developing efficient descriptions of quantum many-body systems \cite{Or18}. One of the central ideas of this program is that locally-based descriptions of states already capture many states of physical interest. While this has worked very well for pure states, mixed states seem to be more challenging, even in one spatial dimension. Some of the reasons for that are the difficulties associated to the description of positive semidefinite matrices mentioned above. In this paper, we study local representations of positive semidefinite (psd) matrices $\rho$ with a one-dimensional spatial structure. We aim to provide a comprehensive set of results regarding one-dimensional psd matrices; to this end, we will include known results and prove new ones. Our main tool to prove new results will be a correspondence between psd matrices and nonnegative matrices that we will establish in \cref{thm:corresp}; see below for further explanations. Remark first that, physically, a one-dimensional structure means that $\rho$ describes the mixed state of a spin chain in one spatial dimension, so that, intuitively, the correlations between sites $i$ and $j$ are mediated by the sites inbetween, $i<l<j$. Mathematically, it means that $\rho$ is an element of a tensor product space with a natural order, $\mathcal{H}^{[1]} \otimes \mathcal{H}^{[2]} \otimes \mathcal{H}^{[3]}\otimes \cdots \otimes \mathcal{H}^{[n]}$, where $\mathcal{H}^{[l]}$ is the Hilbert space associated to site $l$, which is given by $\mathcal{M}_{d_l}$, the set of complex matrices of size $d_l\times d_l$. This natural order will be reflected in the fact that elements of $\mathcal{H}^{[l]}$ will share indices with $\mathcal{H}^{[l-1]}$ and $\mathcal{H}^{[l+1]}$ only. We will analyse the following decompositions of $\rho$: \begin{itemize} \item[(i)] the Matrix Product Density Operator (MPDO) form, which is the most efficient representation, \item[(ii)] the separable decomposition, which only exists for separable states, i.e.\ convex combination of positive semidefinite matrices on each tensor factor, and \item[(iii)] the local purification form, which has the advantage that the positivity is explicit in the local matrices, but which can be much more inefficient than the MPDO form \cite{De13c}, \end{itemize} as well as their translational invariant (t.i.) analogues: \begin{itemize} \item[(iv)] the t.i.\ MPDO form, \item[(v)] the t.i.\ separable form (for t.i.\ separable states) and \item[(vi)] the t.i.\ local purification. \end{itemize} For each decomposition we define a corresponding rank (see \cref{tab:decomp}), which will be given by the minimum dimension of the tensors involved in that decomposition. \begin{table}[htb]\centering \begin{tabular}{cr|l } &Decomposition of a psd matrix $\rho$& Minimal dimension\\ \hline (i) &MPDO & operator Schmidt rank ($\textrm{osr}$)\\ (ii) &separable decomposition & separable rank ($\textrm{sep-rank}$)\\ (iii) &local purification & purification rank ($\textrm{puri-rank}$)\\ (iv) &t.i.\ MPDO & t.i.\ operator Schmidt rank ($\textrm{ti-osr}$)\\ (v) &t.i.\ separable decomposition & t.i.\ separable rank ($\textrm{ti-sep-rank}$)\\ (vi) &t.i.\ local purification & t.i.\ purification rank ($\textrm{ti-puri-rank}$) \end{tabular} \caption{Decompositions for psd matrices considered in this paper, in the non-translational invariant and the translational invariant (t.i.) case, as well as their associated ranks. } \label{tab:decomp} \end{table} For bipartite states which are diagonal in the computational basis, \begin{eqnarray} \label{eq:rhoM} \rho =\sum_{i,j} m_{ij} |i,j\rangle\langle i,j|, \end{eqnarray} we establish a correspondence between decompositions (i), (ii), (iii), and a symmetric version of (iv), (v) and (vi) of \cref{tab:decomp}, and factorisations of the nonnegative matrix\footnote{That is, entrywise nonnegative. A Hermitian matrix with nonnegative eigenvalues is called positive semidefinite.} \begin{eqnarray} \label{eq:M} M = \sum_{i,j} m_{ij} |i\rangle \langle j| \end{eqnarray} presented in \cref{tab:nndecomp} (\cref{thm:corresp}). Most of these factorisations of nonnegative matrices have been defined previously, as well as their corresponding ranks (right column of \cref{tab:nndecomp}). \begin{table}[htb]\centering \begin{tabular}{cr|l } & Factorisation of a nonnegative matrix $M$ & Minimal dimension\\ \hline (i) & minimal factorisation & rank\\ (ii) &nonnegative factorisation & nonnegative rank ($\textrm{rank}_+$)\\ (iii) &positive semidefinite factorisation & psd rank ($\textrm{psd-rank}$)\\ (iv) &symmetric factorisation & symmetric rank ($\textrm{symm-rank}$)\\ (v) & completely positive factorisation & cp rank ($\textrm{cp-rank}$)\\ (vi) &completely psd transposed factorisation & cpsdt rank ($\textrm{cpsdt-rank}$) \end{tabular} \caption{Factorisations of a nonnegative matrix considered in \cref{thm:corresp}.} \label{tab:nndecomp} \end{table} In words, we see the entries of a nonnegative matrix $M$ as the diagonal elements of a psd matrix of form \eqref{eq:rhoM}, and prove a correspondence of decompositions in \cref{thm:corresp}. From our perspective, this is interesting for the study of decompositions of mixed states, as it provides a source of inspiration to generalise results (as illustrated in this paper, in \cite{De13c} and \cite{De19c}), as well as for the study of ranks of nonnegative matrices, as it provides a natural path to generalisation to the case that $\rho$ is not diagonal, and/or $\rho$ is multipartite. In this paper, we will illustrate the use of \cref{thm:corresp} for decompositions of mixed states. Specifically, in \cref{sec:bounds} and \cref{sec:tibounds}, we will analyse and characterise the decompositions of psd matrices mentioned in \cref{tab:decomp} by proving several bounds and relations among the various ranks. Many of the results will be generalisations of the corresponding results for nonnegative matrices, and some will be derived independently. We will point to more possible generalisations in the outlook (\cref{sec:concl}). The paper is meant to be accessible for researchers in quantum information and convex algebraic geometry, as we will rederive basic definitions. It is organized as follows: \begin{itemize} \item In \cref{sec:pos} we present and analyse the MPDO form, the separable decomposition, and the local purification form. \item In \cref{sec:posti} we define and analyse the t.i.\ MPDO form, the t.i.\ separable decomposition, and the t.i.\ local purification. \item In \cref{sec:matrixcones} we present factorisations of nonnegative matrices and their correspondence with decompositions of psd matrices (\cref{thm:corresp}). \item In \cref{sec:bounds} we provide bounds for the non-t.i.\ decompositions. \item In \cref{sec:tibounds} we provide bounds for the t.i.\ decompositions. \item In \cref{sec:concl} we conclude and present an outlook. \item Finally, in \cref{app:proofmain} we prove \cref{thm:corresp}. \end{itemize} \section{Decompositions of psd matrices} \label{sec:pos} In this section we present and analyse the relevant decompositions and ranks for psd matrices in one spatial dimension in the non-t.i.\ case. First we present some general notions (\cref{ssec:gen}), then the Matrix Product Density Operator (MPDO) form (\cref{ssec:mpdo}), the separable decomposition (\cref{ssec:sepdecomp}), and finally the local purification form (\cref{ssec:lp}). \subsection{General notions} \label{ssec:gen} Throughout this paper, a positive semidefinite (psd) matrix is a Hermitian matrix with nonnegative eigenvalues, and $\rho\geqslant 0$ denotes that $\rho$ is psd. Our main object of study is a psd matrix $\rho$ defined on an $n$-fold tensor product space, $$ 0 \leqslant \rho \in \mathcal{H}^{[1]} \otimes \cdots \otimes \mathcal{H}^{[n]}, $$ where $\mathcal{H}^{[l]}$ is the Hilbert space associated to site $l$, which is identified with the space of complex matrices of size $d_l\times d_l$, denoted $\mathcal{M}_{d_l}$.\footnote{Everything is finite-dimensional in our discussion.} In some cases, for simplicity, we will assume that $d_l=d$ for all $l$. The Hilbert space $\mathcal{H}^{[1]} \otimes \cdots \otimes \mathcal{H}^{[n]}$ is often called the physical space, and its dimension $d_1^2\cdots d_n^2$ is called the physical dimension. \begin{remark}[Ignoring normalisation]\label{rem:norm} In this paper we will ignore normalisation conditions on $\rho$ that are often imposed in physics, since considering $\rho/\mathrm{tr}(\rho)$ instead of $\rho$ amounts to multiplying $\rho$ by a positive number, which does not change any of the ranks that we will analyse. For this reason, in this paper we will analyse decompositions of psd matrices, rather than decompositions of states. \demo\end{remark} Let us recall some basic definitions. \begin{definition}\label{def:basic} A psd matrix $0\leqslant \rho \in \mathcal{M}_{d_1} \otimes \cdots \otimes \mathcal{M}_{d_n} $ is \begin{itemize} \item \emph{product} if it can be written as $\rho = A^{[1]} \otimes A^{[2]} \otimes \cdots \otimes A^{[n]}$ with $A^{[j]}\in \mathcal{M}_{d_j}$. \item \emph{separable} if it can be written as a sum of product psd matrices. \item \emph{entangled} if it is not separable. \item \emph{pure} if $\textrm{rank}(\rho)=1$. \end{itemize} \end{definition} \subsection{The Matrix Product Density Operator form} \label{ssec:mpdo} This subsection is devoted to the first natural way of representing a psd matrix $\rho$: the MPDO form. The results of this section are not new; they can be found, or can be easily derived from, e.g., Refs.\ \cite{Ve04d,Zw04}. \begin{definition}[MPDO]\label{def:mpdo} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. A \emph{Matrix Product Density Operator} (MPDO) form \cite{Ve04d} of $\rho$ is given by \begin{eqnarray} \rho = \sum_{\alpha_1,\ldots,\alpha_{n-1}=1}^D A^{[1]}_{\alpha_1} \otimes A^{[2]}_{\alpha_1,\alpha_2} \otimes \cdots \otimes A^{[n-1]}_{\alpha_{n-2},\alpha_{n-1}} \otimes A^{[n]}_{\alpha_{n-1}} , \label{eq:mpdo} \end{eqnarray} where $A^{[l]}_{\alpha}\in\mathcal{H}^{[l]}$ for $l=1,n$, and $A^{[l]}_{\alpha,\alpha'}\in\mathcal{H}^{[l]}$ for $1<l<n$. The minimum such $D$ is called the operator Schmidt rank of $\rho$, denoted $\textrm{osr}(\rho)$. \end{definition} Note that we are referring to it as {\it an} MPDO form instead of {\it the} MPDO form, because it is not unique (see \cref{rem:freedom}). To keep the notation simple, we will assume that each $\mathcal{H}^{[l]}$ is given by $\mathcal{M}_d$ throughout this section. \begin{remark}[Expressing the physical indices]\label{rem:phys} In \cref{def:mpdo}, for $1<l<n$, we see the tensor $A^{[l]}$ as a collection $(A_{\alpha,\beta}^{[l]}\in \mathcal{M}_d)_{\alpha,\beta=1}^D$. We can also see $A^{[l]}$ as a collection $((A^{[l]})^{i,j}\in \mathcal{M}_D)_{i,j=1}^d$, where Latin letters such as $i,j$ denote physical indices, and Greek letters such as $\alpha,\beta$ denote virtual indices. Similarly, $A^{[1]}$ and $A^{[n]}$ only have three indices, namely $i,j,\alpha$. While in \cref{def:mpdo} we see $A^{[1]}$ as a collection $\{A^{[1]}_\alpha \in \mathcal{M}_d\}_{\alpha=1}^D$, we can also see it as a set of row vectors $\{(A^{[1]})^{i,j} \in \mathbb{C}^D\}_{i,j=1}^d$. The situation is similar for $A^{[n]}$, with the only difference that $\{(A^{[n]})^{i,j}\}$ are column vectors. This allows us to write the physical indices explicitly in \eqref{eq:mpdo}, resulting in \begin{eqnarray} \rho = \sum_{i_1,\ldots,i_n,j_1,\ldots,j_n=1}^d (A^{[1]})^{i_1,j_1}(A^{[2]})^{i_2,j_2}\cdots (A^{[n]})^{i_n,j_n} \nonumber \\ |i_1,\ldots, i_n\rangle\langle j_1,\ldots,j_n| . \nonumber \end{eqnarray} \demo \end{remark} \begin{remark}[Connection to tensor rank] \label{rem:trtompdo} For $n=2$ the operator Schmidt rank is just the tensor rank, ${\rm tsr}(\rho)$, i.e.\ the minimal number of elementary tensors needed to obtain $\rho$ as their sum, since an MPDO form is just $$\rho=\sum_{\alpha=1}^D A^{[1]}_{\alpha}\otimes A^{[2]}_{\alpha}.$$ In general we have $$\textrm{osr}(\rho)\leq {\rm tsr}(\rho)\leq\textrm{osr}(\rho)^{n-1}.$$ The first inequality is obtained by starting with a decomposition $$\rho=\sum_{i=1}^{{\rm tsr}(\rho)} B_i^{[1]}\otimes B_i^{[2]}\otimes\cdots\otimes B_i^{[n]}$$ and defining $A_{\alpha}^{[l]}=B_\alpha^{[l]}$ for $l=1,n$, and $A_{\alpha,\beta}^{[l]}=\delta_{\alpha,\beta} B_{\alpha}^{[l]}$ for $l=2,\ldots, n-1.$ The second inequality is clear by counting the number of summands in (\ref{eq:mpdo}). \demo\end{remark} \begin{remark}[Computing the MPDO form] \label{rem:constructionMPDO} An MPDO form that realizes ${\rm osr}(\rho)$ can be obtained by doing successive singular value decompositions (SVD) between the linear bipartitions $[1,\ldots, l] |[l+1, \ldots, n]$ for $1\leq l<n$ of $\rho$. Namely, first we do an SVD across bipartition $[1]|[2\ldots n]$, to obtain \begin{eqnarray}\label{eqn:svd1} \rho= \sum_{\alpha_1=1}^{D_1} A^{[1]}_{\alpha_1} \otimes E^{[2\ldots n]}_{\alpha_1}, \label{eq:rho-OSR1} \end{eqnarray} where $A^{[1]}_{\alpha} \in \mathcal{H}^{[1]}$, and $E^{[2\ldots n]}_{\alpha} \in \mathcal{H}^{[2]}\otimes \cdots \otimes \mathcal{H}^{[n]}$, where we have absorbed the singular values, say, in $A^{[1]}_{\alpha_1}$. $D_1$ is thus the rank of $\rho$ across bipartition $[1]|[2,\ldots, n]$, and we have that $D_1\leq d^2$. Now we consider the tensor $E^{[2\ldots n]} = \{E^{[2\ldots n]}_{\alpha_1}\}_{\alpha_1}$. Index $\alpha_1$ is associated to the Hilbert space of the first virtual system, denoted $\mathcal{H}^{[1^{\textrm{v}}]} = \mathbb{C}^{D_1}$, and therefore $E^{[2\ldots n]}$ is an element of $\mathcal{H}^{[1^{\textrm{v}}]} \otimes \mathcal{H}^{[2]}\otimes \cdots \otimes\mathcal{H}^{[n]}$. We now do an SVD across bipartition $[1^{\textrm{v}},2]|[3,\ldots,n]$, to obtain \begin{eqnarray} \label{eq:D2} E^{[2\ldots n]} = \sum_{\alpha_2=1}^{ D_2} A^{[2]}_{\alpha_2} \otimes E^{[3\ldots n]}_{\alpha_2} , \end{eqnarray} where $A^{[2]}_{\alpha} \in \mathcal{H}^{[1^{\textrm{v}}]} \otimes \mathcal{H}^{[2]}$ and $E^{[3\ldots n]}_{\alpha} \in \mathcal{H}^{[3]}\otimes\ldots \otimes \mathcal{H}^{[n]}$. Writing down explicitly an element of $ \mathcal{H}^{[1^{\textrm{v}}]} $, labeled by index $\alpha_1$, we obtain $$ E^{[2\ldots n]}_{\alpha_1} = \sum_{\alpha_2=1}^{ D_2} A^{[2]}_{\alpha_1,\alpha_2} \otimes E^{[3\ldots n]}_{\alpha_2} . $$ Note that $D_2\leq D_1d^2$. Then we proceed similarly with the next tensor, $E^{[3\ldots n]}$, until we obtain form \eqref{eq:mpdo}. Since for subsequent decompositions we will have that $D_l\leq D_{l-1}d^2$, in general $D_l$ grows exponentially with $l$. It is quite easy to see that $D_l$ is the rank of $\rho$ across the bipartition $[1,\ldots, l] |[l+1, \ldots, n]$. This already shows that $D:=\max_l D_l$ is the minimum number such that a decomposition of the form \eqref{eq:mpdo} is possible. \demo\end{remark} \begin{remark}[Freedom in the decomposition]\label{rem:freedom} To construct the MPDO form we do not really need the SVD, but any decomposition whose intermediate dimension is the rank. Namely, given a matrix $A \in \mathbb{C}^{p\times q}$ any decomposition $A=BC$ where $B$ has $r=\textrm{rank}(A)$ columns works as well. In this paper we do not fix the freedom in the choice of $B,C$. The analogue of the MPDO form for vectors $|\psi\rangle \in \mathbb{C}^{d^n}$ gives rise to vectors in Matrix Product State form, or simply Matrix Product States \cite{Pe07,Fa92}. This freedom is well characterised in this case, where it is fixed by choosing the so-called canonical form \cite{Pe07}, or its generalisation, the irreducible form \cite{De17}. \demo \end{remark} \begin{remark}[The Hermitian MPDO form] In \cite{De19c}, a \emph{Hermitian} MPDO is introduced, which only differs from the MPDO form in the fact that the tensors $A^{[l]}_{\alpha,\beta}$ need to be Hermitian. The associated minimal number of terms is called the Hermitian operator Schmidt rank, denoted $\textrm{hosr}$. In Ref.\ \cite{De19c} it is shown that if $\rho$ is a bipartite psd matrix (i.e.\ $n=2$), then $\textrm{osr}(\rho)=\textrm{hosr}(\rho)$, but in the multipartite case, $\textrm{osr}(\rho)\leq \textrm{hosr}(\rho)\leq 2^{n-1}\textrm{osr}(\rho)$, although we do not know whether the latter inequality is tight. In this paper we will refrain from analysing the hermitian operator Schmidt rank further. We remark that one can force the local tensors to be ``nearly Hermitian" by only doubling the number of terms $D$. To see this, define a tensor $B^{[l]}\in \mathcal{M}_d ( \mathcal{M}_{2D})$ as \begin{align*} &(B^{[l]})^{i,j} = 2^{-1/n}\sum_{\alpha,\beta=1}^D \left( (A^{[l]})^{i,j}_{\alpha,\beta} |\alpha\rangle\langle \beta| + (\bar A^{[l]})^{j,i}_{\alpha,\beta} |\alpha+D\rangle\langle \beta+D| \right), \quad 1<l<n \\ & (B^{[1]})^{i,j} = 2^{-1/n}\sum_{\alpha=1}^D \left( (A^{[1]})^{i,j}_{\alpha} \langle \alpha| + (\bar A^{[1]})^{j,i}_{\alpha} \langle \alpha|\right), \end{align*} where $\bar{}$ denotes complex conjugate, and where the tensors $A^{[l]}$'s are those of \eqref{eq:mpdo}. $B^{[n]}$ is defined in the same way, and since $\rho$ is Hermitian, we have that $$ \rho = \sum_{i_1,\ldots,i_n, j_1,\ldots, j_n=1}^d\mathrm{tr}\left((B^{[1]})^{i_1,j_1} \cdots (B^{[n]})^{i_n,j_n}\right) |i_1,\ldots,i_n\rangle \langle j_1,\ldots, j_n| , $$ and $B$ is ``nearly Hermitian" with respect to the physical indices $i,j$, as we need an additional permutation of the virtual indices: $$ B^{i,j}_{\alpha,\beta } = \bar B^{j,i}_{\beta\oplus D,\alpha\oplus D} , $$ where $\oplus$ means sum modulo $2D$. \demo\end{remark} The disadvantage of the MPDO form is that the local tensors $A^{[l]}_{\alpha,\alpha'}$ are not psd. This is a challenge for the theoretical program of tensor networks, which aims at characterising the properties of $\rho$ in terms of the local tensors, such as the exponential decay of correlations \cite{Pe07,Sc10b}, symmetries \cite{Sc10b} or the existence of a continuum limit \cite{De17b}, to cite some examples. Yet, from the local tensors of the MPDO form one cannot characterise the most basic property of $\rho$, namely that it is psd. It is also problematic numerically, as a truncation of the auxiliary index (i.e.\ a replacement of $D$ by $\tilde D < D$) will generally destroy the positivity of $\rho$. Enforcing positivity in the local matrices leads to the local purification, which we discuss in \cref{ssec:lp}. \begin{remark}[The Matrix Product Operator (MPO) form] \label{rem:mpo} Consider an operator \begin{eqnarray} L\in \mathcal{M}_{d_1,d_1'}\otimes \cdots \otimes \mathcal{M}_{d_n,d_n'}, \label{eq:Lmporem} \end{eqnarray} which need not be a square matrix, and where $\mathcal{M}_{d,d'}$ denotes the space of complex matrices of size $d\times d'$. By the same construction as for the MPDO (see \cref{rem:constructionMPDO}), we can reach the so-called \emph{Matrix Product Operator} (MPO) form \begin{eqnarray} L = \sum_{\alpha_1,\ldots,\alpha_{n-1}=1}^{D} C^{[1]}_{\alpha_1}\otimes C^{[2]}_{\alpha_1,\alpha_2}\otimes\cdots \otimes C^{[n]}_{\alpha_{n-1}} , \label{eq:Lpurif} \end{eqnarray} where $C^{[1]}_{\alpha} \in \mathcal{M}_{d_1,d_1'}$, $C^{[n]}_{\alpha} \in \mathcal{M}_{d_n,d_n'}$ and $C^{[l]} \in\mathcal{M}_{d_l,d_l'}$ for all intermediate $l$'s. The minimum such $D$ is also called the operator Schmidt rank of $L$, denoted $\textrm{osr}(L)$. Indeed, the only difference between the MPO and the MPDO form is that in the latter the operator is globally psd. This highlights the fact that the construction of the MPDO form does not use that $\rho$ is psd. If $L$ is a vector (i.e.\ a column matrix), the operator Schmidt rank is called the Schmidt rank, usually. In this paper, nonetheless, we will still refer to it as the operator Schmidt rank of $L$ to avoid introducing new terminology. The same will be true for the t.i.\ operator Schmidt rank, to be introduced in \cref{def:tiMPDO}. \demo \end{remark} We conclude this section with some basic inequalities fulfilled by the operator Schmidt rank. \begin{proposition}\label{prop:osrineq} Let $\rho, \tau \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. Then \begin{itemize} \item[(i)] $\textrm{osr}(\rho+\tau)\leq \textrm{osr}(\rho)+\textrm{osr}(\tau).$ \item[(ii)] $\textrm{osr}(\rho\tau)\leq \textrm{osr}(\rho)\textrm{osr}(\tau).$ \end{itemize} \end{proposition} \emph{Proof.} If $A^{[i]}$ and $B^{[i]}$ are tensors that provide an MPO form for $\rho$ and $\tau$, respectively, then the block-diagonal sums $A^{[i]}\oplus B^{[i]}$ provide an MPO form for $\rho+\tau$, and the tensors $A^{[i]}\otimes B^{[i]}$ provide a MPO form for $\rho\tau$. \demo \subsection{The separable decomposition} \label{ssec:sepdecomp} While the MPDO form (and the later to be defined) local purification form exist for any psd matrix $\rho$, we now consider the separable decomposition, which exists only for separable psd matrices. To the best of our knowledge, the separable decomposition and the associated separable rank are introduced here, although it is a very natural definition that may have been considered before. We start by recalling the definition of separable psd matrix (\cref{def:basic}). \begin{definition}[Separable psd matrix]\label{def:sep} Let $0\leqslant \rho \in \mathcal{H}^{[1]} \otimes \cdots \otimes \mathcal{H}^{[n]} $. We say that $\rho$ is \emph{separable} if it can be written as \begin{eqnarray} \rho = \sum_{\alpha} \: A^{[1]}_\alpha \otimes A^{[2]}_\alpha \otimes \cdots \otimes A^{[n]}_\alpha, \label{eq:sepdef} \end{eqnarray} where \begin{eqnarray} A^{[1]}_\alpha \otimes A^{[2]}_\alpha \otimes \cdots \otimes A^{[n]}_\alpha \geqslant 0 \label{eq:prod} \end{eqnarray} for all $\alpha$. \end{definition} Note that separability is often defined in terms of convex combinations of psd product states. Since we are ignoring the normalisation (see \cref{rem:norm}), we can consider sums instead of convex combinations. Note also that the condition on product psd matrices [Eq.\ \eqref{eq:prod}] implies that each $A_{\alpha}^{[l]}$ (for each $\alpha,l$) is semidefinite, that is, either psd or negative semidefinite, and that an even number of them is negative semidefinite. By redefining the negative semidefinite ones as $-A_{\alpha}^{[l]} $, we can assume w.l.o.g. that each $A_{\alpha}^{[l]}$ is psd. \begin{definition}[Separable decomposition]\label{def:sepdecomp} Let $0\leqslant \rho \in \mathcal{H}^{[1]} \otimes \cdots \otimes \mathcal{H}^{[n]} $ be separable. A \emph{separable decomposition} of $\rho$ is given by \begin{eqnarray} \rho = \sum_{\alpha_1,\ldots, \alpha_{n-1}=1}^D \chi^{[1]}_{\alpha_{1}} \otimes \chi^{[2]}_{\alpha_{1},\alpha_2} \otimes \cdots \otimes \chi^{[n]}_{\alpha_{n-1}} \label{eq:sep} \end{eqnarray} where each of these matrices is psd, i.e.\ $\chi^{[1]}_{\alpha}\geqslant 0$ and $\chi^{[n]}_{\alpha}\geqslant 0$, and $\chi^{[l]}_{\alpha,\beta}\geqslant 0 $ for $1<l<n$. The minimal such $D$ is called the \emph{separable rank} of $\rho$, denoted $\textrm{sep-rank}( \rho) $. \end{definition} With the construction from \cref{rem:trtompdo} it is clear that a state is separable if and only if it admits a separable decomposition. We finish by establishing a basic inequality of the separable rank---the proof is exactly as the one of \cref{prop:osrineq}. \begin{proposition} Let $0\leqslant \rho,\rho' \in \mathcal{H}^{[1]} \otimes \cdots \otimes \mathcal{H}^{[n]} $ be separable. Then so is $\rho+\rho',$ and $$\textrm{sep-rank}(\rho+\rho')\leq\textrm{sep-rank}(\rho)+\textrm{sep-rank}(\rho').$$ \end{proposition} \subsection{The local purification form} \label{ssec:lp} In this subsection we present and analyse another natural decomposition of $\rho$, namely the local purification, whose main feature is the fact that the local tensors are psd. The local purification form was introduced in Refs.\ \cite{Ve04d,Zw04}; results presented below can thus be found or be easily derived from Refs.\ \cite{Ve04d,Zw04}. The quantum square root rank (\cref{def:qsqrtrank}) is, to the best of our knowledge, introduced here for the first time. To introduce this form, recall that the local physical space $\mathcal{H}^{[l]}$ is identified with $\mathcal{M}_d$. We denote the column space by $\mathcal{V}^{[l]}$ and the row space by $\mathcal{V}^{[l]*}$, so that $\mathcal{M}_d =\mathcal{V}^{[l]*}\otimes \mathcal{V}^{[l]}$. We also introduce an \emph{auxiliary} space associated to site $l$ as $ \mathcal{V}^{[l^\textrm{a}]} = \mathbb{C}^{r}$ with some $1\leq r$. \begin{definition}{\rm \bf Local purification form}\cite{Ve04d} \label{def:localpurif} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. A \emph{local purification form} of $\rho$ is defined as $\rho = L L^\dagger $, where $L$ is in Matrix Product Operator form (\cref{rem:mpo}), \begin{eqnarray} L = \sum_{\alpha_1,\ldots,\alpha_{n-1}=1}^{D} C^{[1]}_{\alpha_1}\otimes C^{[2]}_{\alpha_1,\alpha_2}\otimes\cdots \otimes C^{[n]}_{\alpha_{n-1}} , \label{eq:Lpurif} \end{eqnarray} where $C^{[l]}_{\alpha} \in \mathcal{V}^{[l]} \otimes \mathcal{V}^{[l^\textrm{a}]*}$ for $l=1,n$ and $C^{[l]}_{\alpha,\beta} \in \mathcal{V}^{[l]} \otimes \mathcal{V}^{[l^\textrm{a}]*}$ for $1<l<n$. The minimum such $D$ is called the \emph{purification rank}, denoted $\textrm{puri-rank}(\rho)$. Explicitly, $$ \textrm{puri-rank}(\rho) = \min\{\textrm{osr}(L)|LL^\dagger =\rho\}. $$ \end{definition} Note that if the auxiliary space has the same dimension as the physical space, then simply $C_{\alpha,\beta}^{[l]} \in \mathcal M_d$. On the other hand, if $\rho$ is a pure state and thereby has $\textrm{rank}(\rho)=1$, then $L$ is a column vector and $ \mathcal{V}^{[l^\textrm{a}]*}$ has dimension 1 for all $l^\textrm{a}$. A local purification form always exists. To see this, denote the spectral decomposition of $\rho$ by $\rho = \sum_{j=1}^r \lambda_j |\psi_j\rangle \langle\psi_j| $, where $r=\textrm{rank}(\rho)$ and define $$ L = \sum_{j=1}^r \sqrt{\lambda_j} |\psi_j\rangle \langle v_j|, $$ where $\{|v_j\rangle\}$ is some orthonormal basis. Then it is clear that $LL^\dagger =\rho$. In fact this fully characterises the set of $L$ such that $LL^\dagger=\rho$, that is, the only freedom is in the choice of the orthonormal basis $\{\langle v_j|\}$. Therefore any such $L$ can be written as $L=L_0W^\dagger$, where $L_0$ is defined with, say, the computational basis, $$ L_0 = \sum_{j=1}^r \sqrt{\lambda_j} |\psi_j\rangle \langle j|, $$ and $W$ is an isometry, $W : \mathbb{C}^{r} \to \mathbb{C}^{r'}$ with $r'\geq r$, with $W^\dagger W =I$. This allows us to rewrite the purification rank as $$ \textrm{puri-rank}(\rho) = \min_W\{\textrm{osr}(L_0W^\dagger) | W^\dagger W =I \} . $$ In words, the optimal local purification form will be given by the orthonormal basis $\{|v_j\rangle\}$ that minimises $\textrm{osr}(L)$. \begin{remark}[A purification in the physics literature] In the physics literature, a vectorised version of $L$ is called a purification. Explicitly, if $L$ is the matrix $L = \sum_{i,j}\ell_{ij}|i\rangle\langle j|$, its vectorised version is denoted $|L\rangle = \sum_{i,j} \ell_{ij}|i\rangle |j\rangle$. Denoting the second subsystem on which $|L\rangle$ is defined as $\textrm{aux}$ (the auxiliary subsystem), we find that $\rho = LL^\dagger = \mathrm{tr}_{\textrm{aux}}|L\rangle\langle L|$, where $\mathrm{tr}_{\textrm{aux}}$ is the partial trace over the auxiliary subsystem. \demo\end{remark} Note that the local purification form is not asking that $\rho$ has form \eqref{eq:mpdo} with $A^{[l]}_{\alpha,\alpha'}\geqslant 0 $ for every $\alpha,\alpha'$. The latter is precisely the separable form, which exists only if $\rho$ is separable (see \cref{ssec:sepdecomp}). Instead, the local purification form always exists, and one has to be slightly more careful to see how the local matrices are psd. Namely, in the local purification form, \begin{eqnarray} \quad\rho = \sum_{\alpha_1,\ldots,\alpha_{n-1},\beta_1,\ldots,\beta_{n-1}=1}^{D} B^{[1]}_{\alpha_1,\beta_1} \otimes B^{[2]}_{\alpha_1,\alpha_2,\beta_1,\beta_2}\otimes \cdots \otimes B^{[n]}_{\alpha_{n-1},\beta_{n-1}}, \label{eq:lp} \end{eqnarray} where the $B$'s are psd matrices with respect to the following grouping of the indices: \begin{eqnarray} \nonumber B^{[l]} &=& \sum_{i,\alpha,j,\beta} (B^{[l]})^{i,j}_{\alpha,\beta} |i,\alpha\rangle \langle j,\beta| \geqslant 0 , \quad l=1,n\\ \nonumber B^{[l]} &= & \sum_{i,\alpha,\alpha',j,\beta,\beta'} (B^{[l]})^{i,j}_{\alpha,\alpha',\beta,\beta'} |i,\alpha,\alpha'\rangle \langle j,\beta,\beta'| \geqslant 0, \quad 1<l<n. \label{eq:Bl} \end{eqnarray} Explicitly, $B^{[l]}$ is constructed as $C^{[l]} C^{[l] \dagger}$, where $C^{[l]}$ are the local matrices of $L$ (Eq.\ \eqref{eq:Lpurif}), namely \begin{align*} &B^{[l]}_{\alpha,\beta} = \sum_{i,j=1}^{d} \sum_{k=1}^{r}(C^{[l]})^{i,k}_{\alpha}(\bar C^{[l]})^{k,j}_{\beta} |i\rangle\langle j| , \quad l=1, n \\ & B^{[l]}_{\alpha,\beta,\alpha',\beta'} = \sum_{i,j=1}^{d} \sum_{k=1}^{r} (C^{[l]})^{i,k}_{\alpha,\alpha'} (\bar C^{[l]})^{k,j}_{\beta,\beta'}|i\rangle\langle j| , \quad 1<l<n . \end{align*} Thus, this form contains a local certificate of positivity, in the sense that if the tensors $C^{[l]}$ are contracted as specified above, then $\rho\geqslant 0$ by construction. The problem is that the purification rank may need to be much larger than the operator Schmidt rank, as we will see in \cref{thm:separation}. \begin{remark}{\rm \bf (Operational interpretation of the purification rank)}\cite{Ja13}\label{rem:operationalpuri} The \emph{quantum correlation complexity} of a quantum state $\rho$, $\textrm{Q}(\rho)$, is defined as the minimum size of a seed that Alice and Bob need to share in order to produce $\rho$ via local operations \cite{Ja13}. The \emph{size} of a state is defined as half the number of qubits of the state \cite{Ja13}. In our terminology, \begin{eqnarray} \textrm{Q}(\rho) = \min\{t \mid &&\exists \: 0\leqslant \tau\in \mathcal{M}_{2^t} \otimes \mathcal{M}_{2^t}\textrm{ and completely positive}\nonumber \\ && \textrm{trace preserving maps}\: \mathcal{E}_A, \mathcal{E}_B : (\mathcal{E}_A\otimes \mathcal{E}_B)(\tau) =\rho \} \nonumber \end{eqnarray} Note that in the definition of $\textrm{Q}(\rho)$, Alice and Bob are allowed to do local operations but are not allowed to communicate (even classically). Thus, $\textrm{Q}(\rho)$ is nonincreasing under local operators, but not under classical communications, and therefore is only an upper bound to the amount of entanglement. In addition, the \emph{quantum communication complexity} of a quantum state $\rho$, $\textrm{QComm}(\rho)$, is defined as the minimum number of qubits exchanged between Alice and Bob, initially sharing a product state, to produce $\rho$ at the end of the protocol. Ref.\ \cite{Ja13} shows that (without using the term purification rank) $$ \textrm{Q}(\rho) = \textrm{QComm}(\rho) = \lceil \log_2 \textrm{puri-rank}(\rho)\rceil . $$ This thus gives an operational interpretation of $\textrm{puri-rank}$. In \cref{rem:nonneg} we will comment on the operational interpretation of two related ranks, namely the nonnegative rank and the psd rank. Note also that a multipartite version of the quantum correlation and quantum communication complexity is proposed in Ref.\ \cite{Ja14}, but with a different structure than the one considered here. \demo\end{remark} A basic inequality for the purification rank is stated in the following proposition. \begin{proposition}\label{pro:submulti} For $0\leqslant \rho , \rho' \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$ we have $$\textrm{puri-rank}(\rho+\rho')\leq \textrm{puri-rank}(\rho)+ \textrm{puri-rank}(\rho').$$ \end{proposition} \emph{Proof.} Let $L$ and $L'$ be optimal local purifications of $\rho$ and $\rho'$, respectively. We can append zero columns to all local matrices in $L$ and prepend zero columns to all local matrices in $L'$ without changing $\textrm{osr}(L), \textrm{osr}(L')$ and the fact that $LL^\dagger=\rho, L'L'^{\dagger}=\rho'$. We can thus assume $LL'^\dagger=0=L'L^\dagger$. Then $L+L'$ provides a purification of $\rho+\rho'$ and we obtain \begin{align*}\textrm{puri-rank}(\rho+\rho')&\leq \textrm{osr}(L+L')\\ &\leq \textrm{osr}(L)+\textrm{osr}(L')\\ &=\textrm{puri-rank}(\rho)+\textrm{puri-rank}(\rho') \end{align*} \demo To end this section, we introduce the quantum square root rank of a psd matrix. As we will see in \cref{thm:corresp}, this is the psd analogue of the square root rank of a nonnegative matrix. \begin{definition}[The quantum square root rank]\label{def:qsqrtrank} Let $0\leqslant \rho\in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. The \emph{quantum square root rank} of $\rho$, denoted $\textrm{q-sqrt-rank}(\rho) $, is defined as $$ \textrm{q-sqrt-rank}(\rho) = \min_\tau\{\textrm{osr}(\tau) | \tau^2 = \rho, \tau\ {\rm Hermitian} \} , $$ \end{definition} Note that the minimum is over all Hermitian square roots of $\rho$. If we denote the spectral decomposition of $\rho$ by $\rho = U D U^\dagger$ with $D=\textrm{diag}(\lambda_1, \lambda_2,\ldots)$, then its square roots are of the form $$ \tau = U D^{1/2} U^\dagger, \quad D^{1/2} =\textrm{diag}(\pm\lambda_1, \pm\lambda_2,\ldots). $$ Thus, $\textrm{q-sqrt-rank}(\rho)$ is obtained by choosing the signs of the eigenvalues which minimise the operator Schmidt rank of $\tau$. It is obvious that the quantum square root rank upper bounds the purification rank, see also \cref{pro:purisqrt}. \section{Decompositions of t.i.\ psd matrices} \label{sec:posti} In this section we introduce and analyse decompositions of translationally invariant (t.i.) psd matrices in one spatial dimension. We will first present general notions of a t.i.\ state (\cref{ssec:ti}), and then present the relevant decompositions in the t.i.\ case: the t.i.\ MPDO form (\cref{ssec:tiMPDO}), the the t.i.\ separable decomposition (\cref{ssec:tisepdecomp}), and the t.i.\ local purification form (\cref{ssec:tilp}). \subsection{General notions} \label{ssec:ti} For t.i.\ psd matrices $\rho$ we will denote the local Hilbert space associated to any individual subsystem by $\mathcal{H}_{\rm l}$, which is given by $\mathcal{M}_d$, and the total Hilbert space by $\mathcal{H} = (\mathcal{H}_{\rm l})^{\otimes n}$. Sometimes we will emphasize the system size by writing $\rho_n$ instead of $\rho$. We start by defining translational invariance. \begin{definition}[Translational invariance]\label{def:ti} Let $0\leqslant \rho \in \mathcal{H} = (\mathcal{H_{\rm l}})^{\otimes n}$. We say that $\rho$ is \emph{translationally invariant} (t.i.) if $T\rho T^\dagger =\rho$, where $T$ is the translation operator, $$ T = \sum_{i_1,\ldots, i_n=1}^d |i_2,i_3,\ldots, i_1\rangle \langle i_1, i_2,\ldots, i_n|. $$ \end{definition} Note that the action of $T$ just transforms an elementary tensor $A^{[1]}\otimes\cdots\otimes A^{[n]}$ to $A^{[2]}\otimes\cdots\otimes A^{[n]}\otimes A^{[1]}.$ So $\rho$ is t.i.\ if and only if it is invariant under cyclic permutations of the indices. Except for the case $n=2$, this is a weaker condition than being invariant under arbitrary permutations $\pi$ (i.e.\ $\rho = \pi\rho\pi^\dagger$), in which case $\rho$ would be called symmetric. Note also that if $\rho$ is t.i., then $\rho$ needs to have periodic boundary conditions, whereas in \cref{sec:pos} we considered open boundary conditions. \subsection{The t.i.\ MPDO form} \label{ssec:tiMPDO} We now define the t.i.\ analogue of the MPDO form. This form was considered already in \cite{Pe07} (although it was not given the name t.i.\ MPDO form, nor its associated rank was called t.i.\ osr). We will prove some slightly more general result in \cref{pro:W}, as well as the easily derivable properties of \cref{pro:tiosr}. \begin{definition}[T.i.\ MPDO] \label{def:tiMPDO} Let $0\leqslant \rho \in (\mathcal{H}_{\rm l})^{\otimes n}$. A \emph{translationally invariant (t.i.) MPDO} form of $\rho$ is given by \begin{eqnarray} \rho = \sum_{\alpha_1,\ldots,\alpha_{n}=1}^D A_{\alpha_1,\alpha_2} \otimes A_{\alpha_2,\alpha_3} \otimes \cdots \otimes A_{\alpha_{n},\alpha_1} , \label{eq:timpdo} \end{eqnarray} where $A_{\alpha,\alpha'}\in\mathcal{H}_\mathrm{l}$. The minimal such $D$ is called the t.i.\ operator Schmidt rank of $\rho$, denoted $\textrm{ti-osr}(\rho)$. \end{definition} Note that in the case $n=2$, the t.i.\ MPDO form, as we have defined it here, is $$\rho=\sum_{\alpha,\beta=1}^D A_{\alpha,\beta}\otimes A_{\beta,\alpha}.$$ We will restrict to this kind of decomposition throughout this section, and will go back to this observation in \cref{ssec:symm}. Clearly, a t.i.\ representation such as \eqref{eq:timpdo} implies that $\rho$ is t.i. The converse is also true, but generally not at a fixed cost, i.e.\ $D$ needs to grow with $n$: \begin{remark}{\rm \bf (Imposing translational invariance)}\cite{Pe07} \label{rem:ti} If $\rho$ is t.i.\ and has a non-t.i.\ representation, say an MPDO form given by \eqref{eq:mpdo}, this can be made t.i.\ at the expense of increasing $D$ to $Dn$, in the general case. To see this, first note that we can transform \eqref{eq:mpdo} to $$\rho=\sum_{\alpha_1,\ldots, \alpha_{n}=1}^D A_{\alpha_1,\alpha_2}^{[1]}\otimes A_{\alpha_2,\alpha_3}^{[2]}\otimes\cdots\otimes A_{\alpha_{n-1},\alpha_n}^{[n]}$$ by just padding the row vector $A^{[1]}$ and the column vector $A^{[n]}$ with zeros (we express the virtual indices here). Then define $$A= n^{-1/n}\left(\begin{array}{cccc}0 & A^{[1]} & 0 & 0 \\0 & 0 & \ddots & 0 \\0 & 0 & 0 & A^{[n-1]} \\A^{[n]} & 0 & 0 & 0\end{array}\right).$$ Since $\rho$ is t.i.\ one immediately verifies that $$\rho=\sum_{\alpha_1,\ldots,\alpha_n=1}^{Dn} A_{\alpha_1,\alpha_2}\otimes \cdots\otimes A_{\alpha_{n-1},\alpha_n}.$$ \demo\end{remark} We now consider a state which is t.i.\ but in a non-trivial way. \begin{example}[The $W$ state] \label{ex:Wstate} Consider a pure state $\rho = |W\rangle\langle W|$, so that by \cref{pro:pure} it suffices to study the operator Schmidt rank of $|W\rangle$. Here $|W\rangle$ is the so-called $W$ state on $n$ sites, \begin{eqnarray} |W\rangle = n^{-1/2} \sum_{j=1}^n\sigma_x^{(j)} |0\rangle^{\otimes n} . \label{eq:W} \end{eqnarray} Here $\sigma_x^{(j)}$ denotes the operator $\sigma_{x}=|0\rangle\langle 1| + |1\rangle\langle 0|$ acting on site $j$, and $|0\rangle^{\otimes n}$ denotes the $n$-fold tensor product of $|0\rangle$. For example, for $n=3$, $$ |W\rangle = \frac{1}{\sqrt{3}}( |0,0,1\rangle+|0,1,0\rangle +|1,0,0\rangle). $$ This state has a non-t.i.\ representation with $D=2$ given by \begin{eqnarray} \label{eq:ABW} A^0 = \begin{pmatrix} 1&0\\ 0&1\end{pmatrix}, \quad A^1 = \begin{pmatrix} 0&1\\ 0&0\end{pmatrix},\quad B = \begin{pmatrix} 0&0\\ 1&0\end{pmatrix}, \end{eqnarray} namely \begin{eqnarray} |W\rangle = n^{-1/2} \sum_{i_1,\ldots, i_n=0}^1 \mathrm{tr}(B A^{i_1}A^{i_2}\cdots A^{i_n})|i_1,\ldots,i_n\rangle . \label{eq:Wnti} \end{eqnarray} We can obtain a t.i.\ representation of size $2n$ by using the idea of \cref{rem:ti}. Explicitly, for a given $n$ we define \begin{eqnarray} C^i = n^{-1/n} \begin{pmatrix} 0 & BA^{i} &0 & \ldots &0\\ 0 & 0 & A^{i} &\ldots & 0\\ \vdots & \vdots &\ddots & \ddots &\vdots\\ A^{i} & 0 & \ldots & 0& 0 \end{pmatrix} \label{eq:C} \end{eqnarray} and we have \begin{eqnarray} |W\rangle =\sum_{i_1,\ldots, i_n=0}^1 \mathrm{tr}(C^{i_1}C^{i_2}\cdots C^{i_n})|i_1,\ldots,i_n\rangle. \label{eq:Wti} \end{eqnarray} Note that the size of $C^i$ is $2n$. \demo\end{example} We now show that any t.i.\ representation of the $W$ state requires a bond dimension which grows at least as $\sqrt{n}$, where $n$ is the system size. \begin{proposition} \label{pro:W} Let $|W\rangle$ be the $W$ state on $n$ sites defined in \cref{ex:Wstate}. Then $$ \textrm{ti-osr}(|W\rangle) \geq \sqrt{n}. $$ \end{proposition} A proof that $\textrm{ti-osr}(|W\rangle) \geq \Omega(n^{1/3})$ is provided in \cite[Appendix]{Pe07} and the subsequent proof of Wielandt's Theorem \cite{Sa10} (see \cite{Mi18b} for the latest, stronger result). Here we use the results of Ref.\ \cite{De17} to prove this bound which is, to the best of our knowledge, new, and independent of the quantum Wielandt's Theorem. \emph{Proof.} First observe that $|W\rangle$ is t.i.\ in a non-trivial way, as it contains a sum of $n$ states, each of which is invariant under $T^n$. The translation operator $T$ on $|W\rangle$ thus generates a cyclic permutation of these terms. In the language of Ref.\ \cite{De17}, $|W\rangle$ is $n$-periodic. Now consider a t.i.\ representation of $|W\rangle$ such as the one of \eqref{eq:Wti}. From Ref.\ \cite{De17} it follows that the transfer matrix $E := \sum_{i=0}^1 C^i\otimes \bar C^i$ must have $n$ eigenvalues of modulus 1 of the form $$ \{e^{i2\pi r /n}\}_{r=0}^{n-1}. $$ Note that $E$ is a matrix of size $D^2$. But $n\leq D^2$ and therefore $D\geq \sqrt{n}$. \demo \begin{remark}[T.i.\ Matrix Product Operator form]\label{rem:timpo} In analogy to \cref{rem:mpo}, given a t.i.\ operator $L\in \mathcal{M}_{d}\otimes \cdots \otimes \mathcal{M}_{d}$ ($n$ times), we define its t.i.\ Matrix Product Operator form as a decomposition \begin{eqnarray} L = \sum_{\alpha_1,\ldots,\alpha_{n}=1}^{D} C_{\alpha_1,\alpha_2}\otimes C_{\alpha_1,\alpha_2}\otimes\cdots \otimes C_{\alpha_{n},\alpha_1} , \label{eq:Lpurif} \end{eqnarray} where $C_{\alpha,\beta} \in \mathcal{M}_{d}$. The minimum such $D$ is also called the t.i.\ operator Schmidt rank of $L$, $\textrm{ti-osr}(L)$. The situation is entirely parallel to the non-t.i.\ case: the only difference between the t.i.\ MPO and t.i.\ MPDO form is that in the latter the operator is globally psd. \demo\end{remark} The following result is again proven analogously to \cref{prop:osrineq}. \begin{proposition} \label{pro:tiosr} Let $\rho,\tau \in (\mathcal{H}_{\rm l})^{\otimes n}$ be t.i. Then so are $\rho+\tau, \rho\tau$ and \begin{itemize} \item[(i)] $\textrm{ti-osr}(\rho+\tau)\leq\textrm{ti-osr}(\rho)+\textrm{ti-osr}(\tau).$ \item[(ii)] $\textrm{ti-osr}(\rho\tau)\leq\textrm{ti-osr}(\rho)\textrm{ti-osr}(\tau).$ \end{itemize} \end{proposition} \subsection{The t.i.\ separable decomposition} \label{ssec:tisepdecomp} Here we introduce and characterise the t.i.\ separable decomposition, which, to the best of our knowledge, is new. Before introducing the t.i.\ separable decomposition, we start by defining and characterising t.i.\ separable psd matrices. \begin{definition}[T.i.\ separable psd matrix]\label{def:tisep} Let $0\leqslant \rho \in (\mathcal{H}_\mathrm{l})^{\otimes n}$. We say that $\rho$ is \emph{t.i.\ separable} if $\rho$ is t.i.\ (\cref{def:ti}) and $\rho$ is separable (\cref{def:sep}). \end{definition} It is easy to see that $\rho$ is t.i.\ separable if and only if it is of the form $$ \rho = \frac{1}{n} \sum_{m=1}^{n} T^m \sigma T^{\dagger m} $$ where $\sigma$ is separable. Note that if $\rho$ is a sum of t.i.\ product matrices $$ \rho = \sum_{\alpha} \sigma_\alpha \otimes \cdots \otimes \sigma_\alpha $$ where $\sigma_\alpha\geqslant 0$, then $\rho$ is t.i.\ separable, but the converse is not true, as the following example shows: \begin{eqnarray} \rho = \frac{1}{2} (|0\rangle \langle 0| \otimes |1\rangle \langle 1| + |1\rangle \langle 1|\otimes |0\rangle \langle 0|). \label{eq:mixedW2} \end{eqnarray} See \cref{ex:modifiedW} for more properties of this state. We now define the t.i.\ separable decomposition. \begin{definition}[T.i.\ separable decomposition]\label{def:tisepdecomp} Let $0\leqslant \rho \in (\mathcal{H}_\mathrm{l})^{\otimes n}$. A \emph{t.i.\ separable decomposition} of $\rho$ is a form \begin{eqnarray} \label{eq:tisepdecomp} \rho = \sum_{\alpha_1,\ldots, \alpha_n=1}^D \chi_{\alpha_{1},\alpha_2} \otimes \chi_{\alpha_{2},\alpha_3} \otimes \cdots \otimes \chi_{\alpha_{n},\alpha_1} , \end{eqnarray} where $\chi_{\alpha,\beta} \geqslant 0 $ for all $\alpha,\beta$. The minimal such $D$ is called the \emph{t.i.\ separable rank} of $\rho$, denoted $\textrm{ti-sep-rank}( \rho) $. \end{definition} This definition captures precisely the set of t.i.\ separable psd matrices: \begin{proposition} \label{lem:tiseptidecomp} Let $0\leqslant \rho \in (\mathcal{H}_\textrm{l})^{\otimes n}$. Then $\rho$ is t.i.\ separable (\cref{def:tisep}) if and only if it admits a t.i.\ separable decomposition (\cref{def:tisepdecomp}). \end{proposition} \emph{Proof.} Let $\rho$ be t.i.\ separable. If we write down a separable decomposition of $\rho$ and then apply the construction from \cref{rem:ti} we obtain the desired t.i.\ separable decomposition. The converse direction is clear. \demo The following example is a modified version of the $W$ state (\cref{ex:Wstate}). \begin{example}[Mixed state version of the $W$ state]\label{ex:modifiedW} Consider the t.i.\ separable state $$ \rho = \frac{1}{n} \sum_{i=1}^n \sigma_x^{(i)} (|0\rangle\langle 0|)^{\otimes n}\sigma_x^{(i)}. $$ Note that Eq.\ \eqref{eq:mixedW2} corresponds to this state for $n=2$, and thus this state is not a convex combination of t.i.\ product states. We want to see that \begin{eqnarray} \textrm{ti-sep-rank}(\rho)\geq \frac{\sqrt{n}}{2} \: \textrm{sep-rank}(\rho) = \sqrt{n} . \label{eq:tisepsep} \end{eqnarray} We use essentially the same argument as in \cref{pro:W}. We first provide a separable decomposition of $\rho$ with $\textrm{sep-rank}(\rho)=2$. Define the 4-tensor $\chi = \{\chi_{\alpha,\beta}^{i,j}\}$ using the definitions of \eqref{eq:ABW}: $\chi^{0,0} = A^0$, $\chi^{1,1} = A^1$, and the rest 0, as well as $B$ defined there. Then $$ (\tau^{[1]})^{i,j} = B \chi^{i,j}, \quad (\tau^{[l]})^{i,j} = \chi^{i,j}\quad 1<l\leq n $$ provide a separable decomposition of $\rho$ with $\textrm{sep-rank}(\rho)=2$. Now we want to see the first inequality of \eqref{eq:tisepsep}. So consider an optimal t.i.\ separable decomposition given by a tensor $\tau = \{\tau_{\alpha,\beta}^{i,j}\}$. By construction $\tau_{\alpha,\beta} \in \mathcal{M}_2$ is psd for all $\alpha,\beta$, and $D=\textrm{ti-sep-rank}(\rho)$. First note that since $\rho$ is diagonal in the computational basis, we can assume that $\tau_{\alpha,\beta}^{i,j} = \delta(i,j) C^i_{\alpha,\beta} $ for some $C^i_{\alpha,\beta}$. But $$ \sum_{i_1,\ldots, i_n=0}^1 \mathrm{tr}(C^{i_1}\cdots C^{i_n})|i_1,\ldots, i_n\rangle = |W\rangle $$ is the $W$ state, and \cref{pro:W} shows that the bond dimension of $C$ is $\geq \sqrt{n} $. \demo\end{example} The next result is again proven analogously to \cref{prop:osrineq}. \begin{proposition} Let $0\leqslant \rho,\rho' \in (\mathcal{H}_\mathrm{l})^{\otimes n}$ be t.i.\ separable. Then so is $\rho+\rho'$ and $$\textrm{ti-sep-rank}(\rho+\rho')\leq\textrm{ti-sep-rank}(\rho)+ \textrm{ti-sep-rank}(\rho').$$ \end{proposition} \subsection{The t.i.\ local purification form} \label{ssec:tilp} Here we characterise the t.i.\ local purification form, which had been considered, e.g. in \cite{De15} (although it was not given this name). Before introducing the t.i.\ separable decomposition, we start by defining and characterising t.i.\ separable psd matrices. In the t.i.\ MPDO form of \cref{def:tiMPDO}, the local tensors $A_{\alpha,\beta}$ need not be psd, as in the MPDO form. Enforcing positivity on the local matrices leads to the t.i.\ local purification form. To introduce it, we define $\mathcal{V}_\textrm{l} $ as the column space associated to the local, physical Hilbert space $\mathcal{H}_{\textrm{l}}$, and $\mathcal{V}^\textrm{a}_\textrm{l}$ as the column space associated to the local, auxiliary Hilbert space $\mathcal{H}^{\textrm{a}}_{\textrm{l}}$. \begin{definition}[T.i.\ local purification] \label{def:tilocalpurif} Let $0\leqslant \rho \in (\mathcal{H}_\mathrm{l})^{\otimes n}$. A \emph{t.i.\ local purification form} of $\rho$ is defined as $\rho = LL^\dagger $, where $L$ is in t.i.\ Matrix Product Operator form (\cref{rem:timpo}), \begin{eqnarray} \label{eq:Ltilp} L= \sum_{\alpha_1,\ldots,\alpha_{n}=1}^{D} C_{\alpha_1,\alpha_2}\otimes C_{\alpha_2,\alpha_3}\otimes \cdots \otimes C_{\alpha_{n},\alpha_1} , \end{eqnarray} where $C_{\alpha,\beta} \in \mathcal{V}_\textrm{l} \otimes \mathcal{V}^\textrm{a}_\textrm{l}$. The minimum such $D$ is called the t.i.\ purification rank, denoted $\textrm{ti-puri-rank}(\rho)$. Explicitly, $$ \textrm{ti-puri-rank}(\rho) = \min\{\textrm{ti-osr}(L) | LL^\dagger = \rho\}. $$ \end{definition} \begin{remark}[Existence of the t.i.\ local purification]\label{rem:exlocpuri} Note that a t.i.\ local purification exists for every t.i.\ psd matrix. The unique psd square-root of $\rho$ is a polynomial expression in $\rho$, and thus also t.i. It therefore admits a t.i.\ MPO form, as argued in \cref{rem:ti}. This provides a t.i.\ local purification of $\rho$ (which is generally not the optimal one). \demo\end{remark} To see how the local matrices of the t.i.\ local purification are psd, we proceed similarly as in \cref{ssec:lp}. Namely, $\rho$ has the form \begin{eqnarray} \qquad \rho = \sum_{\alpha_1,\ldots,\alpha_{n},\beta_1,\ldots,\beta_{n}=1}^{D} B_{\alpha_1,\alpha_2,\beta_1,\beta_2}\otimes B_{\alpha_2,\alpha_3,\beta_2,\beta_3} \otimes \cdots \otimes B_{\alpha_{n},\alpha_1,\beta_n,\beta_1} , \label{eq:lpti} \end{eqnarray} where $B$ is psd because it is constructed as $CC^\dagger$, where $C$ is given by Eq.\ \eqref{eq:Ltilp}; explicitly: $$ B = \sum_{i,j=1}^d \sum_{\alpha,\alpha',\beta,\beta'=1}^D \sum_{k=1}^{r} C^{k,i}_{\alpha,\alpha'} \bar C^{j,k}_{\beta,\beta'} |i,\alpha,\alpha'\rangle\langle j,\beta,\beta'| \geqslant 0. $$ The proof of the following inequality is similar to the one of \cref{pro:submulti}. \begin{proposition} Let $0\leqslant \rho,\rho' \in (\mathcal{H}_\mathrm{l})^{\otimes n}$ be t.i. Then so is $\rho+\rho'$ and $$\textrm{ti-puri-rank}(\rho+\rho')\leq \textrm{ti-puri-rank}(\rho)+ \textrm{ti-puri-rank}(\rho').$$ \end{proposition} \section{Correspondence with factorisations of nonnegative matrices} \label{sec:matrixcones} In this section we present a correspondence of decompositions of bipartite psd matrices which are diagonal in the computational basis with factorisations of nonnegative matrices. The results of this section are new, except when we review definitions (as in \cref{ssec:decomp}). First we will comment on the difference between t.i.\ decompositions and symmetric decompositions in the bipartite case (\cref{ssec:symm}), since in this section we will be interested in symmetric bipartite decompositions. Then we will define the factorisations of nonnegative matrices relevant for our problem (\cref{ssec:decomp}), and finally we will present the correspondence (\cref{ssec:correspondence}). \subsection{Comparison with symmetric decompositions} \label{ssec:symm} In \cref{sec:pos} we analysed the $\textrm{osr}$, the $\textrm{sep-rank}$ and the $\textrm{puri-rank}$, and in \cref{sec:posti} their t.i.\ analogues, namely the $\textrm{ti-osr}$, the $\textrm{ti-sep-rank}$ and the $\textrm{ti-puri-rank}$. Another natural decomposition is the fully symmetric version of each of the three ranks, where for example the symmetric tensor rank \cite{Co08c} is defined to be the minimal $r$ such that $$ \rho = \sum_{\alpha=1}^r A_\alpha \otimes A_\alpha \otimes \ldots \otimes A_\alpha . $$ Similarly, one could define the symmetric $\textrm{sep-rank}$ and the symmetric $\textrm{puri-rank}$. Now, although full symmetry and translational invariance coincide in the bipartite case, the symmetric and the t.i. decompositions do not coincide. Namely, the symmetric decomposition results in \begin{eqnarray} \label{eq:specialsymm} \rho=\sum_{\alpha} A_{\alpha}\otimes A_{\alpha}, \end{eqnarray} whereas the t.i.\ decomposition results in \begin{eqnarray} \label{eq:specialti} \rho=\sum_{\alpha,\beta} A_{\alpha,\beta}\otimes A_{\beta,\alpha}. \end{eqnarray} In this section we will be interested in decompositions of type \eqref{eq:specialsymm}. For this reason we now include some general existence result about symmetric decompositions. \begin{proposition}\label{prop:exspecial} Every t.i.\ $0\leqslant \rho \in \mathcal{H}_{\rm l}\otimes \mathcal{H}_{\rm l}$ admits a decomposition of the form \eqref{eq:specialsymm}, and a t.i.\ local purification of the form \eqref{eq:specialsymm}. \end{proposition} However, this is not the case for the separable decomposition, as we will see in \cref{cor:cpsdt}. \emph{Proof.} In the bipartite case, translational invariance is the same as full symmetry, so the first statement is just the well-known decomposition of symmetric matrices \cite{Co08c}. In fact, the t.i.\ operator Schmidt rank equals the operator Schmidt rank. A non-optimal decomposition can also explicitly be obtained by first choosing a decomposition on the double edge, $$ \rho=\sum_{\alpha,\beta}A_{\alpha,\beta}\otimes A_{\beta,\alpha} $$ whose existence we have seen above, and using that \begin{align*} \sum_{\alpha,\beta}(A_{\alpha,\beta}+A_{\beta,\alpha})\otimes (A_{\alpha,\beta}+A_{\beta,\alpha})+i(A_{\alpha,\beta}-A_{\beta,\alpha})\otimes i(A_{\alpha,\beta}-A_{\beta,\alpha})&=4\rho. \end{align*} The existence of the t.i.\ local purification then follows from the argument of \cref{rem:exlocpuri}. \demo \subsection{Factorisations of nonnegative matrices} \label{ssec:decomp} We now consider a rectangular nonnegative matrix $M\in \mathbb{R}_{+}^{p\times q}$, where $\mathbb{R}_{+}$ denotes the set of non-negative reals. We will consider six factorisations of $M$, and each will be associated to a roman number which we will use in \cref{thm:corresp}. For the symmetric factorisations [(iv), (v), and (vi)], $M$ will need to be square and symmetric. For every factorisation there will be a minimal dimension of the matrices involved, which defines the rank associated to that factorisation. The factorisations are the following: \begin{itemize} \item[(i)] A \emph{minimal factorisation} is an expression $M=AB$ where $A$ has $\textrm{rank}(M)$ columns. If $M$ is real, $A$ and $B$ can be chosen real without loss of generality. This factorisation can be obtained, for instance, by doing the singular value decomposition of $M=U\Sigma V$ and absorbing the singular values somewhere. \item[(ii)] The \emph{nonnegative factorisation} \cite{Ya91} is an expression $M=AB$ where $A$ and $B$ are nonnegative, i.e.\ $A\in \mathbb{R}_+^{p\times r}$ and $B\in \mathbb{R}_+^{r\times q}$. The minimal such $r$ is called the nonnegative rank, denoted $\textrm{rank}_+(M)$. \item[(iii)] The \emph{positive semidefinite (psd) factorisation} \cite{Fi11,Fa14} is an expression $M_{i,j} =\mathrm{tr}(E_iF_j^t)$, where $E_i$ and $F_j$ are psd matrices of size $r\times r$ with entries in the rationals, the real or the complex numbers. The minimal such $r$ is called the \emph{psd rank}, denoted $\textrm{psd-rank}^{\mathbb{Q}}(M)$, $\textrm{psd-rank}^{\mathbb{R}}(M)$, and $\textrm{psd-rank}^{\mathbb{C}}(M)$, respectively. \item[(iv)] A \emph{symmetric factorisation} is an expression $M= AA^t$, where $A\in \mathbb{C}^{p\times r}$. The minimal such $r$ is called the \emph{symmetric rank} of $M$, denoted $\textrm{symm-rank}(M)$. Such a decomposition can be found by diagonalisation of symmetric bilinear forms over $\mathbb{C}$ \cite{Co82}. In fact, there always exists some invertible (complex) matrix $P$ such that $PMP^t$ is diagonal, with only ones and zeros on the diagonal. This can also be understood as doing elementary row {\it and the same} column operations to $M$ over $\mathbb{C}$, to bring it to diagonal form. If the resulting diagonal matrix has $r$ ones on the diagonal, then $A:=P^{-1}\left(\begin{array}{c}I_r \\0\end{array}\right)$ provides a symmetric decomposition. From this construction it also follows that $\textrm{symm-rank}(M)=\textrm{rank}(M)$ for symmetric matrices. \item[(v)] The \emph{completely positive (cp) factorisation} \cite{Be15b} is an expression $M=AA^t$, where $A$ is a nonnegative matrix $A\in \mathbb{R}_+^{p\times r}$. The minimal such $r$ is called the \emph{cp rank} of $M$, denoted $\textrm{cp-rank}(M)$. \item[(vi)] The \emph{completely positive semidefinite transposed (cpsdt) factorisation} is an expression $M_{i,j} =\mathrm{tr}(E_iE_j^t)$, where $E_i$ are psd matrices of size $r\times r$ with entries in the rationals, the reals or the complex numbers. The minimal such $r$ is called the \emph{cpsdt rank}, denoted $\textrm{cpsdt-rank}^{\mathbb{Q}}(M)$, $ \textrm{cpsdt-rank}^{\mathbb{R}}(M)$, and $\textrm{cpsdt-rank}^{\mathbb{C}}(M)$, respectively. \end{itemize} \begin{remark} Note the transposition on the second term in the definition of a cpsdt factorisation. If the psd matrices $E_i$ have entries in $\mathbb Q$ or $\mathbb R$ it can be omitted, since such psd matrices are symmetric. So the notion of a cpsdt factorisation coincides with the notion of a \emph{completely positive semidefinite factorisation} (cpsd) \cite{La15c} in that case, which has been studied a lot recently. However, the transpose makes a difference over the complex numbers. Indeed, a cpsd factorisation does not exist for every symmetric nonnegative matrix, for example because it requires that the matrix is positive semidefinite (and even this is not sufficient in general). However, a cpsdt factorisation does always exist, as we will see in \cref{cor:cpsdt}. Also note that in the definition of the psd factorisation, one could omit the transposition at the $F_i$ without changing the notion, since $E_i$ and $F_i$ are independent anyway. \demo\end{remark} Note that the symmetric factorisation, the cp factorisation, and the cpsdt factorisation are the symmetric versions of the minimal factorisation, the nonnegative factorisation, and the psd factorisation, respectively. At the same time, the nonnegative factorisation and the cp factorisations can be obtained from the minimal factorisation and the symmetric factorisation, respectively, by imposing that the matrices are nonnegative. Moreover, the psd and the cpsdt factorisations are the non-commutative generalisations of the nonnegative factorisation and the cp factorisation, respectively. These relations are summarised in Figure \ref{fig:decomp}. \begin{figure}[htb] \[ \begin{tikzcd} \textrm{(i) minimal factorisation} \arrow{r}{\textrm{symmetric}} \arrow[swap]{d}{\textrm{nonnegative}} & \textrm{(iv) symmetric factorisation} \arrow{d}{\textrm{nonnegative}} \\ \textrm{(ii) nonnegative factorisation} \arrow{r}{\textrm{symmetric}} \arrow[swap]{d}{\textrm{non-commutative}} & \textrm{(v) cp factorisation} \arrow{d}{\textrm{non-commutative}} \\ \textrm{(iii) psd factorisation}\arrow{r}{\textrm{symmetric}}& \textrm{(vi) cpsdt factorisation} \end{tikzcd} \] \caption{The relations among the factorisations of nonnegative matrices.} \label{fig:decomp} \end{figure} Clearly the symmetric decompositions [(iv), (v), and (vi)] can only exist for symmetric matrices (i.e.\ for $M=M^t$). However, while every symmetric nonnegative matrix has a symmetric and a cpsdt factorisation, not every symmetric nonnegative matrix has a cp decomposition. One obvious reason is that if $M$ admits a cp factorisation, then $M$ needs to be psd as well---see \cite{La15c} or \cite{Fa14}. This is not the case in the symmetric factorisation, since the entries of $A$ can be complex. Finally we review the definition of the square root rank of a nonnegative matrix (see, e.g., \cite{Fa14}). We will use the \emph{Hadamard square root decomposition}, which is an expression $M = N \circ N$, where $\circ$ denotes the Hadamard product (i.e.\ entrywise multiplication, $n_{ij}^2 = m_{ij}$). Note that the square root rank is denoted $\textrm{rank}_{\sqrt{}}$ in \cite{Fa14}. \begin{definition}[The square root rank]\label{def:sqrtrank} Let $M$ be a nonnegative matrix. The \emph{square root rank} of $M$, denoted $\textrm{sqrt-rank}(M)$, is defined as $$ \textrm{sqrt-rank}(M) = \min \{\textrm{rank} (N) | N\circ N = M\}. $$ \end{definition} The minimisation is thus over all entrywise square roots of $M$, i.e.\ for each entry we can choose the positive or the negative square root. We end this section by reviewing the operational interpretation of the nonnegative and the psd rank, as given in Ref.\ \cite{Ja13}. \begin{remark}{\rm \bf (Operational interpretation of the nonnegative and the psd rank)}\label{rem:nonneg} Given a bipartite probability distribution $p=\{p(x,y)\}_{x,y}$, define its \emph{size} as half of the total number of bits. The \emph{randomized correlation complexity} $\textrm{R}(p)$ is defined as the minimum size of a joint probability distribution that Alice and Bob need to share to produce $p$ by means of only local operations \cite{Ja13}. Ref.\ \cite{Ja13} also considers a communication scenario in which Alice and Bob do not share anything from the start. The \emph{randomized communication complexity} $\textrm{RComm}(p)$ is the minimum number of bits that they need exchange to produce $p$ \cite{Ja13}. Then $$\textrm{R}(p) = \textrm{RComm}(p) = \lceil \log_2 \textrm{rank}_+ (p)\rceil .$$ Concerning the psd rank, given a bipartite quantum state $\rho$, define its \emph{size} as half of the total number of qubits. The \emph{quantum correlation complexity} $\textrm{Q}(p)$ is defined as the minimum size of a quantum state that Alice and Bob have to share to produce $p$ (Alice and Bob can then apply local operations) \cite{Ja13}. Similarly, the \emph{quantum communication complexity} $\textrm{QComm}(p)$ is the minimum number of qubits that they have to send to each other to produce $p$ \cite{Ja13}. Then $$\textrm{Q}(p) = \textrm{QComm}(p) = \lceil \log_2 \textrm{psd-rank}(p)\rceil . $$ Another operational interpretation is provided by \cite{Fi11}. Given a joint probability distribution $p$, $\log(\textrm{rank}_+ (p))$ and $\log(\textrm{psd-rank} (p))$ are used to characterise the amount of classical or, respectively, quantum communication needed to compute $p$ in expectation. A further interpretation is given in Ref.\ \cite{We08}, where the nonnegative matrix is a matrix of measurement outcomes on a quantum state, and the psd rank is used to determine the minimal dimension of the Hilbert space where the quantum state lives. \demo\end{remark} \subsection{Correspondence of decompositions} \label{ssec:correspondence} Here we present a correspondence between the factorisations of nonnegative matrices of \cref{ssec:decomp} and decompositions of bipartite psd matrices which are diagonal in the computational basis, namely \begin{eqnarray} \sigma = \sum_{i=1}^{d_1} \sum_{j=1}^{d_2} m_{ij} |i,j\rangle\langle i,j|. \label{eq:cb} \end{eqnarray} We reserve the letter $\sigma$ for such psd matrices; a general psd matrix is called $\rho$. Clearly, $\sigma$ is psd if and only if $M = (m_{ij})_{i,j} \in \mathbb{R}_+^{d_1\times d_2}$ is a nonnegative matrix. We will relate the symmetric decompositions of $\sigma$ given in \cref{ssec:symm} with the symmetric factorisations of $M$. In this case, $d_1=d_2=:d$ and we have that $\sigma$ is t.i.\ which if and only if $M$ is symmetric, i.e.\ $M=M^t$. \begin{theorem}\label{thm:corresp} Consider the psd matrix $\sigma$ of \eqref{eq:cb}, and let $$M= \sum_{i=1}^{d_1} \sum_{j=1}^{d_2} m_{ij} |i\rangle\langle j|$$ denote the nonnegative matrix containing the diagonal elements of $\sigma$. Then the following correspondence of decompositions, and consequently of ranks, holds: \begin{table}[htb]\centering \begin{tabular}{cr|l} &Decomposition of $\sigma$ & Decomposition of $M$ \\ \hline (i)&operator Schmidt decomposition & minimal factorisation \\ (ii)&separable decomposition& nonnegative factorisation \\ (iii)& local purification & complex psd factorisation \\ (iv)&t.i.\ operator Schmidt decomposition& symmetric factorisation \\ (v)&t.i.\ separable decomposition & cp factorisation\\ (vi)&t.i.\ local purification & complex cpsdt factorisation \\ \end{tabular} \label{tab:corresp} \end{table} \begin{table}[htb]\centering \begin{tabular}{crcl} &Corresponding rank of $\sigma$ && Corresponding rank of $M$ \\ \hline (i)&$\textrm{osr}( \sigma) $ &$=$& $\textrm{rank}(M)$\\ (ii)&$\textrm{sep-rank}( \sigma) $ &$=$& $\textrm{rank}_+(M)$\\ (iii)&$\textrm{puri-rank} (\sigma)$ &$=$& $ \textrm{psd-rank}^{\mathbb{C}} (M)$\\ (iv)&$\textrm{ti-osr}( \sigma) $ &$=$& $\textrm{symm-rank}(M)$\\ (v)&$ \textrm{ti-sep-rank}( \sigma)$ &$=$& $\textrm{cp-rank}(M)$\\ (vi)&$\textrm{ti-puri-rank}( \sigma)$ &$=$& $\textrm{cpsdt-rank}^{\mathbb{C}} (M)$\\ \end{tabular} \label{tab:ranks} \end{table} In addition, the following correspondence of ranks holds: \begin{table}[htb]\centering \begin{tabular}{crcl} &Corresponding rank of $\sigma$ && Corresponding rank of $M$ \\ \hline (vii) &$\textrm{q-sqrt-rank}( \sigma) $ &$=$& $\textrm{sqrt-rank}(M)$ \end{tabular} \label{tab:qsqrtrank} \end{table} \end{theorem} Note that the $\textrm{ti-osr}$, $ \textrm{ti-sep-rank}$ and $\textrm{ti-puri-rank}$ refer to decompositions of type \eqref{eq:specialsymm}, as remarked in \cref{ssec:symm}. The proof of \cref{thm:corresp} is given in \cref{app:proofmain}. We remark that correspondence (ii) was already observed in \cite{De13c} and \cite{Ja14}. \begin{corollary}\label{cor:cpsdt} (i) There are bipartite separable t.i.\ psd matrices that do not admit a symmetric separable decomposition of the type \eqref{eq:specialsymm}. (ii) There are bipartite t.i. psd matrices, diagonal in the computational basis, that do not admit a real symmetric local purification of type \eqref{eq:specialsymm}. (iii) Every symmetric nonnegative matrix admits a (complex) cpsdt-factorization. \end{corollary} \emph{Proof.} (i) It is well-known that not every symmetric nonnegative matrix has a cp factorization. For example, being psd is an additional necessary condition, but even this is not sufficient for $d\geq 5$. (ii) As the proof of \cref{thm:corresp} shows, a real symmetric local purification of type \eqref{eq:specialsymm} for $\rho$ would lead to a cpsdt factorization with real psd matrices of $M$, which is also a cpsd factorization. Such a factorization does not always exists, since being psd is an additional necessary condition (but even this is not sufficient for $d\geq 5$). (iii) follows from \cref{thm:corresp} and \cref{prop:exspecial}. \demo This situation is summarised in \cref{tab:symmti}. We refer the reader to \cite{De19d} for a more thorough study of the existence of this type of decompositions. \begin{table}[h] \begin{tabular}{l|c|c} & Of type & Does it always exist? \\ \hline \multirow{2}{*}{Operator Schmidt decomp.} & \eqref{eq:specialsymm} & Yes \\ & \eqref{eq:specialti} & Yes \\ \hline \multirow{2}{*}{Separable decomp.} & \eqref{eq:specialsymm} & No\\ & \eqref{eq:specialti} &Yes \\ \hline \multirow{2}{*}{Purification } & \eqref{eq:specialsymm} & Yes \\ & \eqref{eq:specialti} & Yes \\ \end{tabular} \caption{Decompositions of bipartite psd matrices, in the symmetric case (of type \eqref{eq:specialsymm}) and in the t.i.\ case (of type \eqref{eq:specialti}). } \label{tab:symmti} \end{table} \begin{remark}[Decompositions for completely positive maps] Any bipartite psd matrix $\rho\in \mathcal{M}_{d_1} \otimes \mathcal{M}_{d_2}$ is associated to a completely positive map $\mathcal{E}: \mathcal{M}_{d_2} \to \mathcal{M}_{d_1}$ via the the Choi--Jamio\l kowski isomorphism. For the particular case of $\sigma$ [Eq.\ \eqref{eq:cb}], the associated completely positive map is \begin{eqnarray} \mathcal{E}(X)= \sum_{i=1}^{d_1} \sum_{j=1}^{d_2}m_{ij} |i\rangle \langle j|X|j\rangle \langle i |. \end{eqnarray} In words, $\mathcal{E}$ projects a matrix $X\in \mathcal{M}_{d_2}$ to its diagonal elements, and transforms it into another diagonal matrix, thus in fact $\mathcal{E}:\mathcal{M}_{d_2} \to \mathcal{D}_{d_1}$ where $\mathcal{D}_{d}$ is the set of diagonal matrices of size $d\times d$. If a diagonal matrix is considered as a column vector, this is just matrix multiplication with $M$. That $\mathcal{E}$ is completely positive implies that, when $\mathcal{E}$ acts on diagonal matrices, it is a nonnegative map (i.e.\ it maps nonnegative diagonal matrices to nonnegative diagonal matrices). Now, the factorisations of $\sigma$ can be interpreted as the following factorisations of $\mathcal{E}$: \begin{itemize} \item[(i)] $\textrm{rank}(M)$ is the minimal $r$ such that $\mathcal{E}:\mathcal{D}_{d_2}\to \mathcal{D}_{r} \to \mathcal{D}_{d_1}$. \item[(ii)] $\textrm{rank}_+(M)$ is the minimal $r$ so that $\mathcal{E}:\mathcal{D}_{d_2}\stackrel{\mathcal{T}_2}{\to} \mathcal{D}_{r} \stackrel{\mathcal{T}_1}{\to} \mathcal{D}_{d_1}$, and each $\mathcal{T}_j$ is a nonnegative map. \item[(iii)] $ \textrm{psd-rank}^\mathbb{C} (M)$ is the minimal $r$ such that $\mathcal{E}:\mathcal{D}_{d_2}\stackrel{\mathcal{T}_2}{\to} \mathcal{M}_{r} \stackrel{\mathcal{T}_1}{\to} \mathcal{D}_{d_1}$, and each $\mathcal{T}_j$ is a completely positive map. \item[(iv)] $\textrm{symm-rank}(M)$ is the minimal $r$ so that $\mathcal{E}:\mathcal{D}_{d}\stackrel{\mathcal{T}}{\to} \mathcal{D}_{r} \stackrel{\mathcal{T}'}{\to} \mathcal{D}_{d}$, where $\mathcal{T}'$ denotes the dual of the map\footnote{The dual is defined by $\mathrm{tr}(Y\mathcal{T}(X)) = \mathrm{tr}(\mathcal{T}'(Y)X)$.}. \item[(v)] $\textrm{cp-rank}(M)$ is the minimal $r$ so that $\mathcal{E}:\mathcal{D}_{d}\stackrel{\mathcal{T}}{\to} \mathcal{D}_{r} \stackrel{\mathcal{T}'}{\to} \mathcal{D}_{d}$, and $\mathcal{T}$ is a nonnegative map. \item[(vi)] $\textrm{cpsdt-rank}^\mathbb{C} (M)$ is the minimal $r$ such that $\mathcal{E}:\mathcal{D}_{d}\stackrel{\mathcal{T}}{\to} \mathcal{M}_{r} \stackrel{\mathcal{T}'\circ\ \vartheta}{\to} \mathcal{D}_{d}$, and $\mathcal{T}$ is a completely positive map ($\vartheta$ denotes the transposition map). \end{itemize} Finally note that the $\textrm{cpsd-rank}^\mathbb{C} (M)$ is the minimal $r$ such that $$\mathcal{E}:\mathcal{D}_{d}\stackrel{\mathcal{T}}{\to} \mathcal{M}_{r} \stackrel{\mathcal{T}'}{\to} \mathcal{D}_{d}$$ and $\mathcal{T}$ is a completely positive map. \demo\end{remark} \begin{remark}{\rm \bf (Other decompositions of nonnegative matrices in quantum information)} \label{rem:otherconnections} Decompositions of nonnegative matrices appear in several contexts in quantum information. In the context of games or correlations, the nonnegative matrix is a matrix of conditional probabilities $\{p(a,b|x,y)\}_{a,b,x,y}$, and the different decompositions and ranks correspond to different strategies (e.g.\ classical versus quantum) of realising these probabilities \cite{Si17}. In that context, the normalisation conditions on the probabilities are important. In particular, in Ref.\ \cite{Si17} the authors show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of the completely positive cone, the completely positive semidefinite cone, the non-signaling cone or the nonnegative cone, respectively. \demo\end{remark} \section{Characterisation of decompositions of psd matrices} \label{sec:bounds} Here we characterise the three decompositions of psd matrices presented in \cref{sec:pos} by proving relations among their ranks, namely $\textrm{osr}$, $\textrm{sep-rank}$ and $\textrm{puri-rank}$. We will review some known results (citing the corresponding source) and also provide some new relations (without a citation). Many of the results are inspired by the correspondence of \cref{thm:corresp}: in some cases they are immediate implications of the corresponding results for nonnegative matrices, and in other cases they are generalisations thereof. A few results have been derived independently of these connections. This section is structured as follows: we characterise product psd matrices (\cref{ssec:product}), separable psd matrices (\cref{ssec:sep}), pure psd matrices (\cref{ssec:pure}), and general psd matrices (\cref{ssec:general}). Recall that they are all defined in \cref{def:basic}. We will then give further upper bounds on the purification rank (\cref{ssec:ubpuri}), and lower bounds based on the entanglement of purification (\cref{ssec:ep}). \subsection{Product psd matrices} \label{ssec:product} We start by characterising product psd matrices. The following can be seen as a generalization of \cite[Proposition 2.8]{Fa14}, by using the correspondence of \cref{thm:corresp}. \begin{proposition}[Characterisation of product psd matrices] \label{pro:product} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. The following are equivalent: \begin{itemize} \item[(i)]$\textrm{osr}(\rho) = 1 $ \item[(ii)]$\textrm{puri-rank}(\rho) = 1$ \item[(iii)]$\textrm{sep-rank}(\rho)=1$ \end{itemize} \end{proposition} \emph{Proof}. (i) $\iff$ (iii). Let the operator Schmidt decomposition be $\rho = A^{[1]}\otimes A^{[2]} \otimes \ldots \otimes A^{[n]} $, so that $\textrm{osr}(\rho)=1$. Since $\rho\geqslant 0$, each $A^{[i]}$ is either positive semidefinite or negative semidefinite, and the number of negative semidefinite matrices is even. We can redefine the negative semidefinite matrices as minus themselves without changing $\rho$, and thus obtain that each $A^{[i]}$ is psd. This form is already a separable decomposition with $\textrm{sep-rank}(\rho)=1$. The converse direction is immediate. (ii) $\iff$ (iii). Consider $\rho = A^{[1]}\otimes \cdots \otimes A^{[n]} $ with $A^{[l]}\geqslant 0$, so that $\textrm{sep-rank}(\rho)=1$. To see that $\textrm{puri-rank}(\rho) = 1$, write $A^{[l]} = C^{[l]}C^{[l]\dagger}$, and define $L = C^{[1]} \otimes \cdots \otimes C^{[n]}$. This satisfies that $\textrm{osr}(L)=1$ and $LL^\dagger = \rho$, and therefore $\textrm{puri-rank}(\rho) = 1$. The converse direction is immediate. \demo \begin{example} $\rho = \frac{I}{2} \otimes |0\rangle\langle 0| \otimes |+\rangle\langle +|$ is a product state (where $|+\rangle = (|0\rangle+|1\rangle)/\sqrt{2}$ and $I = |0\rangle\langle 0|+|1\rangle\langle 1|$). From this expression, it has $\textrm{osr}(\rho)=1$. Since every term is psd, this expression is also a separable decomposition, and thus $\textrm{sep-rank}(\rho)=1$. The matrix $L = \frac{I}{\sqrt{2}} \otimes |0\rangle\langle 0| \otimes |+\rangle\langle +| $ is such that $LL^\dagger = \rho$ and has $\textrm{osr}(L)=1$, and thus $\textrm{puri-rank}(\rho)=1$. \end{example} \subsection{Separable psd matrices} \label{ssec:sep} We now characterise separable psd matrices. \begin{proposition}{\rm \bf (Characterisation of separable psd matrices)} \label{pro:sep} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$ be separable (\cref{def:sep}). Then \begin{itemize} \item[(i)]$\textrm{osr}(\rho)\leq \textrm{sep-rank}(\rho)$. \item[(ii)]$\textrm{puri-rank}(\rho)\leq \textrm{sep-rank}(\rho)$, but $\textrm{sep-rank}(\rho)$ cannot be upper bounded by a function of $\textrm{puri-rank}(\rho)$ only. \end{itemize} \end{proposition} \emph{Proof}. (i) follows from the fact that the separable decomposition is a special case of the MPDO form. (ii) The inequality follows from the fact that the separable decomposition (\cref{def:sepdecomp}) is a special case of the local purification form [Eq.\ \eqref{eq:lp}], in which $\alpha_i=\beta_i$ for $i=1,\ldots, n-1$. We show it explicitly for the bipartite case. So consider the separable decomposition $$ \rho = \sum_{\alpha=1}^s \sigma_\alpha\otimes \tau_\alpha $$ where $s=\textrm{sep-rank}(\rho)$ and $\sigma_\alpha\geqslant 0$ and $\tau_\alpha\geqslant 0$. We write $\sigma_\alpha = A_\alpha A_\alpha^\dagger$ and $\tau_\alpha = B_\alpha B_\alpha^\dagger$, and define the matrix $$ L = \sum_{\alpha=1}^s (A_\alpha \otimes \langle \alpha|) \otimes B_\alpha $$ This verifies that $LL^\dagger = \rho$, and thus $L$ is a purification of $\rho$ with $\textrm{osr}(L)\leq s$. Thus $\textrm{puri-rank}(\rho)\leq \textrm{sep-rank}(\rho)$. The separation between $\textrm{sep-rank}(\rho)$ and $\textrm{puri-rank}(\rho)$ follows from \cref{thm:corresp}, and the fact that there is a separation between $\textrm{rank}_+$ and $\textrm{psd-rank}$ \cite{Go12}. That is, $\textrm{psd-rank}$ cannot be upper bounded by a function of $\textrm{rank}_+$ only. \demo In the special case of $n=2$, if $\textrm{osr}(\rho)=2$, then $\rho$ is separable, and we know exactly the values of the three ranks: \begin{proposition}[\cite{De19c}]\label{pro:osr2} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \mathcal{H}^{[2]}$. If $\textrm{osr}(\rho)=2$, then $\rho$ is separable with $\textrm{sep-rank}(\rho)=2$. Thus, $\textrm{puri-rank}(\rho)=2$ as well. \end{proposition} \begin{example}Consider the bipartite matrix $$ \rho = I\otimes I + \sigma_x \otimes \sigma_x. $$ This has $\textrm{osr}(\rho)=2$ and is thus separable. In this case the separable decomposition can be found by inspection, namely \begin{eqnarray} \rho = \frac{1}{2}(|+,+\rangle \langle +,+| +|-,-\rangle \langle -,-| ), \label{eq:rhosep+} \end{eqnarray} where $|\pm \rangle = (|0\rangle\pm|1\rangle)/\sqrt{2}$, which indeed has $\textrm{sep-rank}(\rho)=2$. \end{example} \subsection{Pure psd matrices} \label{ssec:pure} We now characterise pure psd matrices, that is, $\rho$ with $\textrm{rank}(\rho)=1$. In this case $\rho = LL^\dagger$ where $L$ is a column vector. \begin{proposition}[Characterisation of pure psd matrices] \label{pro:pure} $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$ be a pure psd matrix, i.e.\ $\rho = L L^\dagger$ where $L$ is a column vector. Then \begin{enumerate} \item[(i)] $\textrm{osr}(\rho) = \textrm{osr}(L)^2$. \item[(ii)]$\textrm{puri-rank}(\rho) = \textrm{osr}(L)$. \end{enumerate} \end{proposition} See \cref{rem:mpo} concerning the use of operator Schmidt rank for vectors. \emph{Proof}. We prove it for the bipartite case for ease of notation. In addition, we write $|L\rangle$ instead of $L$ in order to emphasise that $L$ is a column vector. The same will be true of other matrices appearing in this proof. (i) Consider the operator Schmidt decomposition $|L\rangle = \sum_{\alpha=1}^s |A_\alpha\rangle \otimes |B_\alpha\rangle $ where $s=\textrm{osr}(L)$, and thus $\rho = \sum_{\alpha,\beta=1}^s |A_\alpha\rangle \langle A_\beta| \otimes |B_\alpha\rangle \langle B_\beta|$. This shows that $\textrm{osr}(\rho)\leq s^2$. Since $\{|A_\alpha\rangle \}_{\alpha}$ and $\{|B_\alpha\rangle \}_{\alpha}$ are linearly independent, so are $\{|A_\alpha\rangle \langle A_{\beta}|\}_{\alpha,\beta}$ and $\{|B_\alpha\rangle \langle B_{\beta}|\}_{\alpha,\beta}$, and thus the dimension of their span is $s^2$. Thus $\textrm{osr}(\rho)=s^2$. (ii) That $\textrm{puri-rank}(\rho) \leq \textrm{osr}(L)$ is clear, since $L $ provides a purification where the auxiliary system has dimension one. Since $\{|A_\alpha\rangle \langle A_{\beta}|\}_{\alpha,\beta}$ and $\{|B_\alpha\rangle \langle B_{\beta}|\}_{\alpha,\beta}$ are linearly independent, there cannot exist another purification $L'$ with $\textrm{osr}(L')<\textrm{osr}(L)$. \demo Thus, for a pure psd matrix $\rho = LL^\dagger$ (with $L$ a column vector) we have that $$ \textrm{osr}(\rho) = \textrm{osr}(L)^2 =\textrm{puri-rank}(\rho)^2 . $$ \begin{example} Consider the pure state $\rho = |\Phi^+\rangle \langle \Phi^+|$, where $L= |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. This is a purification with $\textrm{osr}(L) =\textrm{puri-rank}(\rho) =2 $. It is immediate to see that $\textrm{osr}(\rho) = 4$. \end{example} \subsection{General psd matrices} \label{ssec:general} We now consider the general case, that is, where $\rho$ need not be product, separable or pure. First we will bound the various ranks in terms of the physical dimension of $\rho$ (\cref{pro:physdim}), and then characterise the relation between $\textrm{osr}$ and $\textrm{puri-rank} $ (\cref{pro:osrpurirank} and \cref{pro:separationosrpuri}). \begin{proposition}{\rm \bf (Bounds in terms of the physical dimension)} \label{pro:physdim} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes\mathcal{H}^{[n]}$, where each $\mathcal{H}^{[j]}$ is given by $\mathcal{M}_d$. The following relations hold: \begin{itemize} \item[(i)] $\textrm{osr}(\rho) \leq d^{2\lfloor n/2\rfloor}$. \item[(ii)] $\textrm{puri-rank}(\rho)\leq d^{2\lfloor n/2\rfloor}$. \item[(iii)] If $\rho$ is separable, then $\textrm{sep-rank}(\rho)\leq d^{2n}$. \end{itemize} \end{proposition} The bounds can be extended in a straightforward way to the case that each local Hilbert space has a different dimension, i.e.\ where $\mathcal{H}^{[j]}$ is given by $\mathcal{M}_{d_j}$. \emph{Proof}. (i) This follows from the construction of \cref{ssec:mpdo}, since there we obtained that at bipartition $[1,\ldots, l] |[l+1, \ldots, n]$, $$ D_{l}\leq \min\{d^2 D_{l-1},d^{2l},d^{2(n-l)} \}. $$ (ii) By \cref{pro:purisqrt} we have that $\textrm{puri-rank}(\rho)\leq \textrm{osr}(\sqrt{\rho})$. Since $\sqrt{\rho}\in \mathcal{M}_{d^n}$, (i) implies that $\textrm{osr}(\sqrt{\rho}) \leq d^{2\lceil n/2\rceil}$. (iii) We have $\rho\in (\textrm{Her}_d)^{\otimes n} = \textrm{Her}_{d^n}$, which is a real vector space of dimension $d^{2n}$. By Caratheodory's Theorem, $\rho$ can be written as a sum of at most $d^{2n}$ elements of product psd matrices. This is an upper bound on the ``tensor rank version" of the separable rank, which itself upper bounds the separable rank, and thus proves the claim. \demo We now turn to the relation between $\textrm{osr}(\rho)$ and $\textrm{puri-rank}(\rho)$. \begin{proposition} \label{pro:osrpurirank} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. Then $\textrm{osr}(\rho)\leq \textrm{puri-rank}(\rho)^2$, and the bound is tight for pure psd matrices. \end{proposition} \emph{Proof.} For pure psd matrices we have that $\textrm{osr}(\rho)= \textrm{puri-rank}(\rho)^2$ by \cref{pro:pure}. For a general $\rho$, consider an optimal purification $L$, i.e.\ such that $L L^\dagger =\rho $ and $\textrm{osr}(L) =\textrm{puri-rank}(\rho)$. Then \begin{equation*} \textrm{osr}(\rho) = \textrm{osr}(LL^\dagger)\leq \textrm{osr}(L)\textrm{osr}(L^\dagger) =\textrm{osr}(L)^2 =\textrm{puri-rank}(\rho)^2 \end{equation*} \demo \begin{proposition}{\rm \bf (Separation result for entangled psd matrices)}\cite{De13c,Go12} \label{pro:separationosrpuri} $\textrm{puri-rank}$ cannot be upper bounded by a function of $\textrm{osr}$ only. In particular, there is a sequence of psd matrices diagonal in the computational basis $(\rho_n \in \mathcal{M}_{2}^{\otimes (2n)} )_n$, with $\textrm{osr}(\rho_n)=3$ for all $n$, and $\textrm{puri-rank}(\rho_n)>\Omega(\log n)$. \label{thm:separation} \end{proposition} \emph{Proof}. Ref.\ \cite{Go12} shows that $\textrm{rank}_{\textrm{psd}}(M)$ cannot be upper bounded by $\textrm{rank}(M)$ only. In particular, Ref.\ \cite{Go12} shows that if $S_t$ is the slack matrix of the regular $t$-gon, then $\textrm{rank}(S_t) =3$ (for $t\geq 3$) and $\textrm{psd-rank}(S_t) > \Omega(\log t)$. Using the correspondence of \cref{thm:corresp}, this implies that there is a sequence of bipartite psd matrices diagonal in the computational basis whose purification rank cannot be upper bounded by its operator Schmidt rank. This counterexample is extended to the multipartite case in \cite{De13c}. \demo \subsection{Upper bounds on the purification rank} \label{ssec:ubpuri} Given the separation result of \cref{pro:separationosrpuri}, it is interesting to study how the purification rank can be upper bounded. We now provide several such upper bounds. We start by relating the purification rank with the quantum square root rank (\cref{def:qsqrtrank}). This result is the analogue of \cite[Theorem 2.9 (v)]{Fa14} for psd matrices. \begin{proposition} \label{pro:purisqrt} Let $0\leqslant \rho\in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$. Then $\textrm{puri-rank}(\rho)\leq \textrm{q-sqrt-rank}(\rho)$, but $\textrm{q-sqrt-rank}(\rho)$ cannot be upper bounded by a function of $\textrm{puri-rank}(\rho)$ only. \end{proposition} \emph{Proof.} Concerning the first statement, simply note that in $\textrm{q-sqrt-rank}(\rho)$ one minimises over the square roots of $\rho$, whereas in $\textrm{puri-rank}(\rho)$ (\cref{def:localpurif}) one minimizes over matrices $L$ such that $LL^\dagger =\rho$, which in particular includes the square roots of $\rho$. The second statement follows from the fact that the square root rank cannot be upper bounded by a function of the psd rank \cite{Fa14}, and from \cref{thm:corresp}. \demo The following results upper bound the purification rank by a function of the operator Schmidt rank \emph{and} the physical dimension $d^n$ \cite{De13c}. Here we prove them in the language of this paper. \begin{proposition}{\rm \bf (Bounds of $\textrm{puri-rank}$ in terms of the physical dimension and $\textrm{osr}$)}\cite{De13c} Let $0\leqslant \rho \in \mathcal{H}^{[1]}\otimes \cdots \otimes \mathcal{H}^{[n]}$ where each $\mathcal{H}^{[j]}$ is given by $\mathcal{M}_d$. The following holds: \begin{itemize} \item[(i)] \label{prop:sos} Let $m$ be the number of different eigenvalues of $\rho$ including 0, so that $m \leq \min\{\textrm{rank}(\rho)+1,d^n\}$. Then $$ \textrm{puri-rank}(\rho)\leq \textrm{q-sqrt-rank}(\rho)\leq \frac{\textrm{osr}(\rho)^m-1}{\textrm{osr}(\rho)-1}. $$ \item[(ii)] $\textrm{puri-rank}(\rho)\leq \textrm{osr}(\rho) (\textrm{rank}(\rho))^2$. \end{itemize} \end{proposition} \emph{Proof.} (i) We clearly have $$ \textrm{puri-rank}(\rho) \leq \textrm{q-sqrt-rank}(\rho) \leq \textrm{osr}(\sqrt{\rho}) $$ where $\sqrt{\rho}\geqslant 0$, and where the first inequality follows from \cref{pro:purisqrt}. There is a polynomial $p$ of degree $m-1$, such that $\sqrt{\rho} = p(\rho). $ We thus have that $$ \textrm{osr}(\sqrt{\rho}) = \textrm{osr}(p_m(\rho)) \leq \sum_{l=0}^{m-1} \textrm{osr}(\rho^l) \leq \sum_{l=0}^{m-1} \textrm{osr}(\rho)^l = \frac{\textrm{osr}(\rho)^m-1}{\textrm{osr}(\rho)-1}. $$ (ii) Consider the spectral decomposition of $\rho = \sum_{j=1}^s\lambda_j |\psi_j\rangle\langle\psi_j|$, with $s =\textrm{rank}(\rho)$. The ``standard'' choice of $L$ is $L = \sum_{j=1}^s \sqrt{\lambda_j} |\psi_j\rangle \langle j|$, as this clearly satisfies $\rho=LL^\dagger$. For this $L$ we have that $\textrm{puri-rank}(\rho)\leq \textrm{osr}(L)$, and in the following we will upper bound the latter. Consider a product state $|p_\alpha\rangle$, so that by definition $\textrm{osr}(|p_\alpha\rangle) =1$. Define $$ |\chi_\alpha\rangle :=\rho |p_\alpha\rangle = LL^\dagger |p_\alpha\rangle. $$ By construction we have that $\textrm{osr}(|\chi_\alpha\rangle)\leq \textrm{osr}(\rho)$. Now choose $s$ product states $\{|p_\alpha\rangle\}$ so that $\{L^\dagger |p_\alpha\rangle\}_{\alpha=1}^s$ are linearly independent. Thus we can express $|\psi_j\rangle = \sum_{\alpha=1}^s c_{j,\alpha} |\chi_\alpha\rangle$ for some coefficients $c_{j,\alpha}$, so that $\textrm{osr}(|\psi_j\rangle)\leq \textrm{osr}(\rho) \: s$, and finally $$ \textrm{osr}(L)\leq s \max_j [ \textrm{osr}( |\psi_j\rangle) \textrm{osr}( \langle j| ) ]\leq \textrm{osr}(\rho) s^2, $$ the desired result. \demo \subsection{Lower bounds based on the entanglement of purification} \label{ssec:ep} We now show that, in the bipartite case, we can lower bound the purification rank using the entanglement of purification \cite{Te02}. The entanglement of purification is a measure of classical and quantum correlations, which, for a bipartite state $\rho_{AB}$ is defined as \begin{eqnarray} E_p(\rho_{AB}) = \min_{\psi} \{E(|\psi\rangle_{A,A',B,B'} ))| \mathrm{tr}_{A'B'} |\psi\rangle\langle \psi| = \rho\} , \nonumber \end{eqnarray} where the entropy of entanglement is defined as $$ E(|\psi\rangle_{A,A',B,B'} ) = S_1(\rho_{AA'}), $$ where $\rho_{AA'} = \mathrm{tr}_{BB'} |\psi\rangle_{AA'BB'}\langle \psi| $ and where $S_1$ is the von Neumann entropy, $S_1(\rho) =-\mathrm{tr}(\rho\log\rho)$. \begin{proposition}{\rm \bf (Bound in terms of the entanglement purification)} Let $\rho_{AB}$ be a bipartite state. Then $E_p(\rho_{AB}) \leq \log (\textrm{puri-rank}(\rho_{AB})$). \end{proposition} \emph{Proof.} In the bipartite case the purification rank is defined as $$ \textrm{puri-rank}(\rho_{AB}) = \min_{\psi} \{ \textrm{SR}(|\psi\rangle ) | \mathrm{tr}_{A'B'} |\psi\rangle\langle \psi| = \rho\}, $$ where the $\textrm{SR}(|\psi\rangle )$ denotes the Schmidt rank of $|\psi\rangle$ across bipartition $AA'|BB'$. Noting that $\textrm{SR}(|\psi\rangle) = \textrm{rank}(\rho_{AA'})$ and recalling that the Renyi entropy with parameter $\alpha=0$ is defined as is $S_0(\rho) = \log\textrm{rank} (\rho) $ we have that $$ \log ( \textrm{puri-rank}(\rho_{AB})) = \min_{\psi} \{ S_0(\rho_{AA'}) | \mathrm{tr}_{A'B'} |\psi\rangle\langle \psi| = \rho\}. $$ Let $|\phi\rangle$ denote the state that achieves the minimisation in the previous equation. Then we have that \begin{eqnarray} \nonumber\log ( \textrm{puri-rank}(\rho_{AB})) &=& S_0(\mathrm{tr}_{BB'}(|\phi\rangle\langle \phi|))\\ \nonumber&\geq & S_1(\mathrm{tr}_{BB'}(|\phi\rangle\langle \phi|))\\ \nonumber&\geq & \min_{\psi} \{S_1(\mathrm{tr}_{BB'}(|\psi\rangle\langle \psi|) | \mathrm{tr}_{A'B'}|\psi\rangle\langle\psi|=\rho \} \\ \nonumber &=& E_p(\rho_{AB}) , \end{eqnarray} where we have used that $S_0(\rho)\geq S_1(\rho)$. \demo Thus any lower bound of the entanglement of purification also lower bounds the logarithm of the purification rank. \section{Characterisation of decompositions of t.i.\ psd matrices} \label{sec:tibounds} In this section we characterise the decompositions of t.i.\ psd matrices (\cref{def:ti}). In the bipartite case we will focus exclusively on decompositions of type \eqref{eq:specialti}. We first characterise t.i.\ product psd matrices, t.i.\ separable psd matrices and general t.i.\ psd matrices (\cref{ssec:tigeneral}), and then give relations with their non-t.i.\ counterparts (\cref{ssec:nti}). As in the previous section, we will review some results (citing the source), and mostly prove new relations (without a citation). \subsection{T.i.\ psd matrices} \label{ssec:tigeneral} First, for t.i.\ product psd matrices the analogue of \cref{pro:product} is true: \begin{proposition}[Characterisation of t.i.\ product states]\label{pro:tiproduct} Let $0\leqslant \rho \in (\mathcal{H}_{\mathrm{l}})^{\otimes n}$ be t.i. The following are equivalent: \begin{itemize} \item[(i)] $\textrm{ti-osr}(\rho) = 1$ \item[(ii)] $\textrm{ti-puri-rank}(\rho) = 1 $ \item[(iii)] $\textrm{ti-sep-rank}(\rho)=1$ \end{itemize} \end{proposition} \emph{Proof.} Analogous to the proof of \cref{pro:product}. \demo For t.i.\ separable psd matrices (\cref{def:tisep}) the analogue of \cref{pro:sep} is true: \begin{proposition}{\rm \bf (Relation between of $\textrm{ti-puri-rank}$ and $\textrm{ti-sep-rank}$)} Let $0\leqslant \rho \in (\mathcal{H}_{\mathrm{l}})^{\otimes n}$ be a t.i.\ separable psd matrix (\cref{def:tisep}). Then $\textrm{ti-puri-rank}(\rho)\leq\textrm{ti-sep-rank}(\rho)$. \end{proposition} \emph{Proof.} Analogous to the proof of \cref{pro:sep}. \demo We now characterise t.i.\ psd matrices $\rho$ which are not necessarily product or separable. Note again that any such matrix has a t.i.\ local purification. \begin{proposition} For $\rho\in (\mathcal{H}_\mathrm{l})^{\otimes n}$ and any t.i.\ $L$ with $LL^\dagger=\rho$ we have $$ \textrm{ti-puri-rank}(\rho)\leq \textrm{ti-osr}(L)\leq n\cdot \textrm{osr}(L). $$ \end{proposition} \emph{Proof.} The first inequality is trivial, the second follows from the construction of \cref{rem:ti} to $L$. \demo We now give the analogue of \cref{pro:osrpurirank}. \begin{proposition} \label{pro:tiosrpurirank} Let $0\leqslant\rho\in (\mathcal{H}_\mathrm{l})^{\otimes n}$ be t.i. Then $\textrm{ti-osr}(\rho)\leq \textrm{ti-puri-rank}(\rho)^2$. \end{proposition} \emph{Proof.} Analogous to the proof of \cref{pro:osrpurirank}. \demo The previous proposition relates $\textrm{ti-osr}$ with $\textrm{ti-puri-rank}$ for a fixed system size $n$. It is also interesting to study whether $\textrm{ti-osr}$ with $\textrm{ti-puri-rank}$ can be related in a way that is independent of $n$, i.e.\ uniform in $n$. In the following, we review a negative result in this direction \cite{De15}. In order to state our result we need a few definitions first. Given a tensor $A=\{A_{\alpha,\beta}\in \mathcal{M}_{d}\}_{\alpha,\beta=1}^D$, define $$ \rho_n(A) := \sum_{\alpha_1,\ldots, \alpha_n=1}^D A_{\alpha_1,\alpha_2} \otimes A_{\alpha_2,\alpha_3} \otimes \cdots \otimes A_{\alpha_n,\alpha_1} , $$ \begin{theorem}[\cite{De15}] \label{thm:undec} Not for every tensor $A$ such that $\rho_n(A)\geqslant 0$ for all $n\geq 1$, there is a tensor $B$ such that $$ \rho_n(A) \propto \rho_n(B)^\dagger \rho_n(B) \qquad \forall n. $$ \end{theorem} Here $\propto$ means ``proportional to'', that is, there is a positive constant $c_n$ so that $\rho_n(A)=c_n \rho_n(B)^\dagger \rho_n(B)$. The theorem says that even if there is a tensor $A$ that gives rise a family of psd matrices $\rho_n(A)$ for all $n$, there may not exist another $B$ (of any finite size) which provides a t.i.\ local purification of $\rho_n(A)$ \emph{which is valid for all system sizes}. This is true even if we allow for a different proportionality constant $c_n$ for each $n$. In other words, the theorem says that there are psd matrices that admit a t.i.\ MPDO form independent of the system size, but no t.i.\ local purification form independent of the system size. This is true even if $\rho_n(A)$ is diagonal in the computational basis, i.e.\ if $A^{i,j}_{\alpha,\alpha'}\propto \delta(i,j)$ (where $\delta(i,j)$ is the Kronecker delta) and $(d,D)\geq (7,7)$ \cite{De15}. The idea of the proof is the following. Ref.\ \cite{De15} shows that the assumption ``$\rho_n(A) \geqslant 0$ for all $n$" is in fact undecidable, i.e.\ given a tensor $A$, there is no algorithm that decides whether $\rho_n(A)\geqslant 0$ for all $n$. Now assume that there is a $B$ such that $\rho_n(A) \propto \rho_n(B)\rho_n(B)^\dagger$ for all $n$. Ref.\ \cite{De15} provides an algorithm to find this $B$. This allows to verify that $\rho_n(A)\geqslant 0$, and thus to solve an undecidable problem. Thus, this $B$ cannot exist in general. Note that \cref{thm:undec} is proven without any reference to the corresponding decompositions of nonnegative matrices (correspondences of \cref{thm:corresp}), albeit they may be related. Finally we remark that similar problems but with open boundary conditions have been shown to be undecidable \cite{Kl14}, and NP-complete for finitely many $n$ \cite{Kl14}, and similar results have been obtained for two dimensional structures of tensor networks \cite{Sc18}. \subsection{Relation to the non-t.i.\ counterparts} \label{ssec:nti} We now give relations between the t.i.\ ranks and their non-t.i.\ counterparts. We start with the relation between $\textrm{osr}$ and $\textrm{ti-osr}$. \begin{proposition}[Relation between $\textrm{osr}$ and $\textrm{ti-osr}$]\label{pro:tintiosr} Let $0\leqslant \rho \in (\mathcal{H}_\mathrm{l})^{\otimes n}$ be t.i. Then $$ \textrm{osr} (\rho ) \leq \textrm{ti-osr} (\rho )\leq n \: \textrm{osr}(\rho) . $$ \end{proposition} \emph{Proof.} The first inequality follows from the fact that the t.i.\ MPDO is a special case of the MPDO form in which the local tensors are independent of the site. The second inequality follows from the construction of \cref{rem:ti}. \demo We now turn to the relation between $\textrm{puri-rank}$ and $\textrm{ti-puri-rank}$. \begin{proposition}{\rm \bf (Relation between $\textrm{puri-rank}$ and $\textrm{ti-puri-rank}$)} \label{pro:puriranktinonti} Let $0\leqslant \rho \in (\mathcal{H}_{\mathrm{l}})^{\otimes n}$ be t.i. Then $$\textrm{puri-rank}(\rho)\leq \textrm{ti-puri-rank}(\rho).$$ \end{proposition} \emph{Proof.} Simply note that the t.i.\ local purification form is a special case of the local purification in which the local tensors are independent of the site. \demo Note that we cannot use the construction of \cref{rem:ti} to upper bound $\textrm{ti-puri-rank}$ by $\textrm{puri-rank}$, since $L$ need not be t.i., as discussed in \cref{ssec:tigeneral}. We now turn to the $\textrm{ti-sep-rank}$ and its relation to $\textrm{sep-rank}$. \begin{proposition}{\rm \bf (Relation between $\textrm{sep-rank}$ and $\textrm{ti-sep-rank}$)} \label{pro:sepranktiseprank} Let $0\leqslant \rho\in (\mathcal{H}_\mathrm{l})^{\otimes n}$ be a t.i.\ separable psd matrix (\cref{def:tisep}). Then $$ \textrm{sep-rank}(\rho)\leq \textrm{ti-sep-rank}(\rho)\leq n \: \textrm{sep-rank}(\rho) $$ and the first inequality is tight. \end{proposition} \emph{Proof.} The first inequality is obvious, since the t.i.\ separable decomposition is a special case of the separable decomposition. To see that it is tight, simply note that for a t.i.\ product state we have $\textrm{ti-sep-rank}(\rho)=1$ by \cref{pro:tiproduct} and thus also $\textrm{sep-rank}(\rho)=1$. The second inequality follows from the construction of \cref{lem:tiseptidecomp}. \demo We do not know whether the second inequality in \cref{pro:sepranktiseprank} is tight. \cref{ex:modifiedW} provides a state for which $$ \textrm{ti-sep-rank}(\rho)\geq \frac{\sqrt{n}}{2} \: \textrm{sep-rank}(\rho). $$ The previous three results (\cref{pro:tintiosr}, \cref{pro:puriranktinonti} and \cref{pro:sepranktiseprank}) imply that if we have a t.i.\ psd matrix and we enforce translational invariance in its representation, this will generally increase the cost by an amount that depends on the system size. This holds for the three t.i.\ representations considered here: the t.i.\ MPDO form, the t.i.\ local purification form (if it exists), and the t.i.\ separable form (if $\rho$ is separable). \section{Conclusions and Outlook} \label{sec:concl} In this paper we have studied several natural decompositions of positive semidefinite matrices $\rho$ with a one-dimensional structure. We have considered and characterised the MPDO form, the separable form and the local purification, as well as their corresponding ranks ($\textrm{osr}, \textrm{sep-rank},\textrm{puri-rank}$). We have also defined and characterised their translationally invariant (t.i.) analogues: the t.i.\ MPDO form, the t.i.\ separable form and the t.i.\ local purification, as well as their corresponding ranks ($\textrm{ti-osr}, \textrm{ti-sep-rank}, \textrm{ti-puri-rank}$). In the bipartite t.i.\ case, we have considered the symmetric versions of these decompositions. For bipartite states which are diagonal in the computational basis, we have presented a correspondence between these decompositions and factorisations of nonnegative matrices (\cref{thm:corresp}). We have leveraged this correspondence to derive several bounds and relations between the different ranks, presented in \cref{sec:bounds} and \cref{sec:tibounds}. Note that Ref.\ \cite{Gl19} proves some of the results of \cref{thm:corresp}, with a focus on applications on machine learning (Ref.\ \cite{Gl19} coincidentally appeared on the arxiv on the same day as this paper). We remark that the results of this paper have been greatly generalised in \cite{De19d}, where tensor decompositions not only on one-dimensional spatial structures, but on arbitrary simplicial complexes are considered. The translational invariant symmetry considered in this paper is lifted to an invariance given by an arbitrary symmetry group, and the existence and the associated rank of these invariant tensor decompositions is studied. Ref.\ \cite{De19d} also proves a much more general version of \cref{thm:corresp}, and explores some of their consequences regarding separations and generalisations. Beyond these results, some straightforward open questions are the following: \begin{itemize} \item It is known that there is a separation between $\textrm{rank}$ and $\textrm{psd-rank}$, and between $\textrm{rank}_+$ and $\textrm{psd-rank}$, which can be represented by the symbol $\ll$ (see, e.g.\ \cite{Fa14}): $$ \textrm{rank} \ll \textrm{psd-rank} \ll \textrm{rank}_+ $$ Via \cref{thm:corresp}, this immediately implies a separation between $\textrm{osr}$ and $\textrm{puri-rank}$, and between $\textrm{puri-rank}$ and $\textrm{sep-rank}$, respectively, $$ \textrm{osr} \ll \textrm{puri-rank} \ll \textrm{sep-rank} . $$ But can the separations be stronger between $\textrm{osr}$ and $\textrm{puri-rank}$, or between $\textrm{puri-rank}$ and $\textrm{sep-rank}$, than for their counterparts above? \item \cref{pro:submulti} has simply used the fact that $\textrm{psd-rank}$ is submultiplicative to conclude that $\textrm{puri-rank}$ is too. But can there be stronger differences between $\textrm{puri-rank}(\rho) \textrm{puri-rank}(\rho')$ than for the $\textrm{psd-rank}$? \end{itemize} One could also study generalisations of the upper bounds for the cp rank of Ref.\ \cite{Be03}, the lower bounds to the nonnegative and cp rank \cite{Fa16}, or lower bounds to the cpsd rank \cite{Gr17}. Another interesting direction concerns the implications of the computational complexity results of factorisations of nonnegative matrices \cite{Sh16} for decompositions of quantum states. Also, it has been recently discovered that the set of quantum correlations is not closed \cite{Sl17}, from which it follows that the set of cpsd matrices is not closed \cite{Si17}. It would be worth investigating the consequences of that for quantum states. A further perspective concerns the study of approximate versions of the various decompositions, where one could investigate whether the separations between the ranks also hold in the approximate case. Preliminary results\cite{De20} suggest that all separations between ranks disappear for approximations in the Schatten $p$-norm with $p>1$.

BkiUbwU5qsNCPSsgaVWK

\section{BCS-BEC Crossover Theory and the Physical Effects of Temperature} \label{sec:I} The study of ultracold trapped fermionic gases is a rapidly exploding subject \cite{Jin3,Grimm,Jin4,Ketterle3,KetterleV, Thomas2,Grimm3,ThermoScience,Salomon3,Hulet4} which is defining new directions in condensed matter and atomic physics. It has also captured the attention of physicists who study color superconducting quark matter as well as nuclear matter \cite{Wilczek,LW03,FGLW05}. Indeed, it is hard, in recent times, to find a subfield of physics which appeals this broadly to the research community. As we come to understand the gases, experimentalists will move to address fermionic atoms in optical lattices; this will provide important insight to condensed matter physicists as analogue systems for ``solving" intractable many body problems. What makes these gases (and lattices) so important is their remarkable tunability and controllability. Using a Feshbach resonance, one can tune the attractive two-body interaction from weak to strong, and thereby make a smooth crossover from a BCS superfluid to a Bose-Einstein condensation (BEC) \cite{Leggett,Eagles}. Not only does this allow high transition temperatures $T_c$ (relative to the Fermi energy $E_F$) but it may also provide insights into the high temperature cuprate superconductivity \cite{Chen2,ourreview,ReviewJLTP,LeggettNature}. Furthermore, one can tune the population of the two spin states, essentially at will, in this way, allowing exploration \cite{ZSSK06,PLKLH06,ZSSK206} of exotic polarized phases such as the FFLO \cite{FFLO,LOFF_Review,Combescot} superfluids, of interest to condensed matter, nuclear and particle physicists. One will be able to tune the lattice parameters such as bandwidth, on-site repulsion, even random disorder, etc and thereby study the famous, and as yet, unsolved Hubbard model Hamiltonian. This paper will concentrate on those issues relating to the effects which are of current interest to experimentalists. In particular, we will study BCS-BEC crossover in atomic Fermi gases, looking at a wide range of different experiments. Our group has been principally interested in the effects of finite temperature \cite{Chen1,ourreview,ReviewJLTP} and in this review, we will discuss how temperature $T$ enters into the standard crossover theory, and how temperature can be measured, and how temperature changes the character of the gas from a superfluid at low $T$ to an unusual (but strongly interacting) normal fluid at moderate $T$, and to ultimately a Fermi gas at high $T$. Temperature, of course, plays an important role in these Fermi gas experiments. As we shall see in this Review, the earliest evidence for superfluidity is generally based \cite{Jin3,Grimm,Jin4,Ketterle3,KetterleV,Thomas2,Grimm3,ThermoScience} on a comparison of the behavior of the gas at two different temperatures, presumably, one above and one below $T_c$. Experimentally, one is never strictly at $T=0$, and as a result there are thermal excitations of the gas which need to be characterized both theoretically and experimentally. Before we begin with the ultracold gases, it is useful to discuss to what extent the BCS-BEC crossover scenario relates to the high temperature copper oxide superconductors \cite{ourreview,ReviewJLTP}. While there is much controversy in the field of high $T_c$ superconductivity, there is a school of thought \cite{ourreview,PS05,TDLee1,Deutscher,Uemura,Randeriareview,LeggettNature} which argues that these systems are somewhere intermediate between BCS and BEC. These arguments are based on the observation that $T_c$ is anomalously high, and therefore the attractive interaction is presumably stronger than in conventional superconductors. These are quasi-two-dimensional (2D) materials which means that they have a tendency to exhibit ``pre-formed" pairs, that is, pairing in advance of Bose condensation. This is often referred to as fluctuation effects, but pre-formed pairs are naturally associated with a more BEC-like scenario. Importantly these pairs can be used to explain the widespread and anomalous ``pseudogap" effects \cite{Chen2,JS2} which are the focus \cite{LeeReview} of most of the current attention in the field. The existence of ``pre-formed" pairs means that a characteristic (pseudogap) energy must be supplied in the normal state to break the pairs and create fermionic excitations. Hence we say that there is a gap or, more precisely, a pseudogap in the fermionic excitation spectrum. The case that the crossover scenario is relevant to the cuprates was made quite eloquently by A.~J. Leggett in a recent status report \cite{LeggettNature} on high $T_c$ superconductivity. In this article he summarized the eight salient ``facts" about the cuprates. It is worth quoting one here, which relates to their anomalously short coherence length. \textit{ ``The size of the pairs is somewhere in the range 10-30\,\AA\ -- from measurements of the upper critical field, Fermi velocity and $T_c$. This means that the pair size is only moderately greater than the inter-conduction electron in-plane spacing, putting us in the intermediate regime of the so-called Bose-Einstein condensate to BCS superconductor (BEC-BCS) crossover, and leading us to expect very large effects of fluctuations (they are indeed found)."} The field of BCS-BEC crossover is built around early observations by Eagles \cite{Eagles} and Leggett \cite{Leggett} that the BCS ground state, proposed by Bardeen, Cooper, and Schrieffer in 1957, is much more general than was originally thought. If one increases the strength of the attraction and self-consistently solves for the fermionic chemical potential $\mu$, this wave function will correspond to a more BEC-like form of superfluidity. Knowing the ground state what is the nature of the superfluidity at finite $T$? That is the central question we will address in this Review. Even without a detailed theoretical framework we can make three important observations. \begin{itemize} \item As we go from BCS to BEC, pairs will form above $T_c$ without condensation. In the normal state, it pays in general to take advantage of the attractive pairing interaction. Only in the extreme BCS limit do pairs form exactly at $T_c$. \item The fundamental statistical entities in these superfluids are fermions. We can think of pairs of fermions as a form of ``boson", although the statistics are not precisely the same. We measure these ``bosonic" or pair-degrees of freedom indirectly via the fermionic gap parameter $\Delta(T)$. In the fermionic regime this parameter is the minimum energy which must be supplied to create fermionic excitations. It tells us about bosons indirectly through the binding together of two fermions. \item In general there will be two types of excitations in these BCS-BEC crossover systems. Only in strict BCS theory are the excitations of solely fermionic character, and only in the strict BEC limit will the excitations be solely of bosonic character. More generally in the intermediate case (often called the ``unitary" regime) the excitations consist of a mix of both fermions and bosons. \end{itemize} These observations are illustrated by Figs.~\ref{fig:Delta_Deltasc} and \ref{fig:3}. In Fig.~\ref{fig:Delta_Deltasc} we schematically plot the gap parameter $\Delta(T)$ as a function of $T$, along with the superfluid order parameter $\Delta_{sc}(T)$. The former, which represents the ``bosonic" degrees of freedom, shows that pairs continuously form once temperature is less than a crossover temperature $T^*$, while the order parameter turns on as in a second order phase transition at $T_c$. The height of the shaded region reflects the number of noncondensed pairs. This number increases monotonically with decreasing $T$, until $T_c$ is reached. A $T$ further decreases below $T_c$ the number of noncondensed pairs begins to decrease monotonically due to the condensation of zero momentum pairs. In Fig.~\ref{fig:3} we present a schematic plot of the excitation type, which shows that between BCS and BEC (i.e., in the unitary regime) there will be a mix of fermions and bosons. These bosons and fermions are not separate fluids, but rather they are strongly inter-connected. Indeed, the gap in the fermionic spectrum (related to $\Delta$) is a measure of the number of bosons in the system. \begin{figure} \centerline{\includegraphics[width=3.5in,clip]{Fig1.eps}} \caption{Contrasting behavior of the excitation gap $\Delta(T)$ and superfluid order parameter $\Delta_{sc}(T)$ versus temperature. The height of the shaded region roughly reflects the density of noncondensed pairs at each temperature.} \label{fig:Delta_Deltasc} \end{figure} \begin{figure} \centerline{\includegraphics{Fig2.eps}} \caption{Schematic illustration of excitations in the BCS, unitary and BEC regimes. The black discs represent fermionic excitations. Pair excitations become progressively dominant as the system evolves from the BCS to BEC regime.} \label{fig:3} \end{figure} \begin{figure} \centerline{\includegraphics[]{Fig3.eps}} \caption{Typical behavior of the chemical potential $\mu$ at $T=0$ in the three regimes, as a function of the interaction strength $U/U_c$, or, equivalently, $1/k_Fa$. As $U/U_c$ increases from 0, the chemical potential $\mu$ starts to decrease and then becomes negative. The character of the system changes from fermionic ($\mu>0$) to bosonic ($\mu < 0$). The pseudogap (PG) or unitary regime corresponds to non-Fermi liquid based superconductivity, and $U_c (<0)$ corresponds to critical coupling for forming a two fermion bound state in vacuum \cite{ourreview}, i.e., the unitary point where the two-body $s$-wave scattering length $a$ diverges.} \label{fig:2a} \end{figure} \section{Theory Outline} In this paper we will presume the ground state is given by the standard BCS-like wavefunction, \begin{equation} \Psi_0=\Pi_{\bf k}(u_{\mathbf{k}}+v_{\mathbf{k}} c_{\bf k,\uparrow}^{\dagger} c_{\bf -k,\downarrow}^{\dagger})|0\rangle \,, \label{eq:1a} \end{equation} where $ c^\dag_{\bf k,\sigma}$ and $ c^{}_{\bf k,\sigma}$ are the creation and annihilation operators for fermions of momentum ${\bf k}$ and spin $\sigma=\uparrow,\downarrow$. The variational parameters $v_{\mathbf{k}}$ and $u_{\bf k}$ are usually represented by the two more directly accessible parameters $\Delta_{sc}(T=0)$ and $\mu$, which characterize the fermionic system. Here $\Delta_{sc}(T=0)$ is the zero temperature superconducting order parameter. These fermionic parameters are uniquely determined in terms of the attractive interaction $U$ and the fermionic density $n$. The variationally determined self consistency conditions are given by two BCS-like equations which we refer to as the ``gap" and ``number" equations, respectively. At $T=0$, the effects of BEC-BCS crossover are most directly reflected in the behavior of the fermionic chemical potential $\mu$, which is associated with the number equation. We plot the behavior of $\mu$ in Fig.~\ref{fig:2a}, which indicates the BCS and BEC regimes. In the weak coupling regime $\mu = E_F$ and ordinary BCS theory results. With increasing $|U|$, $\mu$ begins to decrease, eventually crossing zero and then ultimately becoming negative in the BEC regime. We generally view $\mu = 0 $ as a crossing point. For positive $\mu$ the system has a remnant of a Fermi surface, and we say that it is ``fermionic". For negative $\mu$, the Fermi surface is gone and the material is ``bosonic". One can debate whether other ground states ought to be considered. Indeed the work of the Camerino group \cite{Strinati2,Strinati5} is based on a finite $T$ approach first introduced by Nozieres and Schmitt-Rink (NSR) \cite{NSR,Randeriareview}. This leads to a different, and not as readily characterized ground state. We list some arguments in support of the ground state in Eq.~(\ref{eq:1a}). (i) This is the basis for the widely studied Bogoliubov-de Gennes approach, which can be applied \cite{our_vortex,Machida,Ho_vortex,Kinnunen} to the BCS-BEC crossover problem at $T=0$. (ii) At arbitrarily strong coupling (and $T=0$), this ground state can be shown to coincide with a Gross-Pitaevskii (GP) description \cite{PSP03} of the boson system. (iii) This ground state is the basis for the rapidly proliferating theoretical literature \cite{Kinnunen,SR06,SM06,HS06, Tsinghuagroup, PWY05,PS05a} on spin polarized Fermi superfluids. In addition to our own work there have been some additional studies which include the effects of temperature \cite{Machida2,YD05,LatestStoof}, albeit at a lower order mean field theory than considered here. \begin{table} \caption{BCS theory by way of BEC analogy. Here we compare condensation in composite and point bosons; $\mu_B$ is the bosonic chemical potential, $N_0$ and $N_T$ are the number of condensed and noncondensed bosons, respectively. We define $\mu_{pair}$ as the chemical potential for the noncondensed pairs. Here $\Delta(T)$ is the total fermionic gap which contains contributions from the noncondensed ($\Delta_{pg}^2$) and condensed terms ($\Delta_{sc}^2$). In the strict BCS limit $\Delta_{pg} =0$, so that the order parameter and gap are identical. } \begin{tabular}{|p{1.35in}|p{2.05in}|p{1.3in}|} \hline & \parbox[c][5mm][c]{2.in}{\centering{Composite bosons}} & \parbox[c][5mm][c]{1.3in}{\centering{Point bosons}} \\ \hline \parbox[c][5mm][c]{1.3in}{\centering{Pair chemical potential}} & {{\parbox[c][8mm][c]{2.in}{\centering{$\mu_{pair}=0$, $T\leq T_c$} \\ Leads to BCS gap equation for $\Delta(T)$}}} & \parbox[c][5mm][c]{1.3in}{\centering{$\mu_{B}=0$, $T\leq T_c$}} \\ \hline \parbox[c][5mm][c]{1.3in}{\centering{Total ``number'' of pairs}} & \parbox[c][5mm][c]{2.in} {\centering{$\Delta^{2}(T)=\Delta_{sc}^{2}(T)+\Delta_{pg}^{2}(T)$}} & \parbox[c][5mm][c]{1.3in}{\centering{$N=N_{0}+N_{T}$}} \\ \hline \parbox[c][5mm][c]{1.3in}{\centering{Noncondensed pairs}} & \parbox[c][5mm][c]{2.in}{\centering{$Z\Delta_{pg}^{2}=\sum_{\mathbf{q}\neq 0}b(\Omega_{q})$}} & \parbox[c][5mm][c]{1.3in}{\centering{$N_{T}=\sum_{\mathbf{q}\neq 0}b(\Omega_{q})$}} \\ \hline \end{tabular} \end{table} We begin at the more physical level by stressing the analogy between condensation in this composite boson or fermionic superfluid and condensation in a gas of ideal point bosons. Our microscopic theory treats self-consistently two-particle and one-particle Green's functions on an equal footing. Because the physics is so simple and clear, we can fairly readily anticipate the form of the central equations of this BCS-BEC generalization of BCS theory. It is important to stress, however, that these equations can be derived more rigorously from a truncated series of equations of motion for the appropriate Green's functions \cite{Chen2,ChenThesis}. There are three principal equations which govern Bose condensation: the vanishing of the bosonic chemical potential at all $ T \leq T_c$ is the first. We will refer to this condition as the ``BEC condition". It is related to the usual Thouless criterion for superconductivity, but the latter is generally associated only with the temperature $T_c$. The second equation is the boson number equation. All ``bosons" must be accounted for as either condensed or noncondensed. The third equation is the number of noncondensed ``bosons", which are created by thermal excitations. This is determined simply by inserting the known excitation spectrum of the excited pairs or bosons, into the Bose distribution function $b(x)$. With this equation, and the second equation, one can then deduce the number of condensed bosons. These three central equations for bosons are indicated in Table I, on the far right, for true point bosons, and in the second column for the composite bosons which appear in fermionic superfluids. For these composite bosons the quantity which provides a measure of the ``number" of bosons ($N$) is given by $\Delta^2(T)$ (up to a constant coefficient, $Z$). How does one quantitatively establish the appropriate ``boson number" for the fermionic case? This is determined via the self consistent gap equation for $\Delta(T)$, which, in turn, is determined using the first condition: that the pair chemical potential $\mu_{pair}$ is zero at and below $T_c$. How does one compute the number of excited pairs? Once the gap equation is interpreted in terms of the appropriate noncondensed pair propagator (see below), then one knows the related excitation spectrum $\Omega_q$ of this propagator. The quantity $Z$ which appears in the last equation of the Table (for the composite bosons) gives the relation between the gap associated with noncondensed pairs ($\Delta_{pg}^2$) and the number of all pairs [$\sum b(\Omega_q)$]. It can be readily calculated in this theory; once one has the noncondensed pair propagator, $Z$ appears as the inverse residue. (Deep in the BEC regime \cite{JS3}, $Z$ is relatively simple to compute, for here the boson number density approaches the asymptote $n/2$). More precisely, the total number of bosons in the present case has to be determined self-consistently through the gap equation for $\Delta(T)$. It also involves the fermion number equation through the related fermionic chemical potential. To be consistent with the ground state variational conditions, the vanishing of the pair chemical potential is associated with a particular form for the pair propagator involving dressed Green's functions. These, in turn, determine the fermionic chemical potential through the fermion number equation. \subsection{Microscopic T-matrix Scheme} Next, we implement this picture microscopically via a $T$-matrix approximation. We include spin indices throughout so that it will be clear how to apply this scheme to spin polarized superfluids. This means that we consider the coupled equations between the particles (with propagator $G$) and the pairs [with propagator $t(P)$] and drop all higher order terms. This theory does not include direct ``boson-boson" interactions, although the pairs do interact indirectly via the fermions, in an averaged or mean field sense. Here, for all $T\leq T_{c}$, the BEC condition is interpreted as requiring that the pair chemical potential $\mu_{pair}$ associated with the noncondensed pairs, vanish. Within a $T$-matrix scheme, the pair propagator is given by \begin{equation} t_{pg}^{-1}(P)=U^{-1}+\chi(P)\,, \label{eq:1} \end{equation} where $\chi$ is the pair susceptibility. The function $\chi(P)$ is, in many ways, the most fundamental quantity we introduce. We will show that one obtains consistent answers between $T$-matrix based approaches and the BCS-Leggett ground state equations, provided the components of the pair susceptibility, defined by \begin{equation} \chi(P)=\frac{1}{2}\big[\chi_{\uparrow\downarrow}(P)+ \chi_{\downarrow\uparrow}(P)\big] \label{eq:2} \end{equation} are given by the product of one dressed and one bare Green's function \begin{equation} \label{eq:3} \hspace*{1cm}\chi_{\uparrow\downarrow}(P)=\sum_{K}G_{0\uparrow}(P-K)G_{\downarrow}(K), \qquad \chi_{\downarrow\uparrow}(P)=\sum_{K}G_{0\downarrow}(P-K)G_{\uparrow}(K)\,, \end{equation} where $P=(i\Omega_l,\mathbf{p})$, and $G$ and $G_0$ are the full and bare Green's functions respectively. Here $G_{0,\sigma}^{-1} (K) = i \omega_{n} - \xi_{\mathbf{k},\sigma}$, $\xi_{\mathbf{k},\sigma} = \epsilon_{\mathbf{k}}-\mu_\sigma$, $\epsilon_{\mathbf{k}}=\hbar^2k^2/2m$ is the kinetic energy of fermions, and $\mu_\sigma$ is the fermionic chemical potential for spin $\sigma=\uparrow,\downarrow$. Throughout this paper, we take $\hbar=1$, $k_B=1$, and use the four-vector notation $K\equiv (i\omega_n, \mathbf{k})$, $P\equiv (i\Omega_l, \mathbf{q})$, $\sum_K \equiv T\sum_n \sum_{\bf k}$, etc, where $\omega_n = (2n+1)\pi T$ and $\Omega_l = 2l\pi T$ are the standard odd and even Matsubara frequencies \cite{Fetter} (where $n$ and $l$ are integers). We now evaluate the BEC condition \begin{equation} t_{pg}^{-1} (0) = 0 = U^{-1}+\chi(0)\,. \label{eq:5} \end{equation} The one-particle Green's function for fermions with spin $\sigma$ is \begin{eqnarray} G^{-1}_{\sigma}(K)&=&G^{-1}_{0\sigma}(K)-\Sigma_{\sigma}(K) = i\omega_{n} -\xi_{k\sigma}-\Sigma_{\sigma}(K) \,, \label{eq:13} \end{eqnarray} where $\bar{\sigma}\equiv -\sigma$, and the self-energy $\Sigma_{\sigma}$ is of the BCS-like form \begin{equation} \Sigma_{\sigma}(K)=-\Delta^{2}G_{0\bar{\sigma}}(-K) = \frac{\Delta^{2}}{i\omega+\xi_{k\bar{\sigma}}} \,. \label{eq:14} \end{equation} It should be noted that we use a contact potential so that the symmetry factor $\varphi_{\mathbf{k}}$ associated with the pairing interaction is trivially $\varphi_{\mathbf{k}} = 1$. For a nontrivial $\varphi_{\mathbf{k}}$, one only needs to replace $\Delta$ with $\Delta\varphi_{\mathbf{k}}$ in Eq.~(\ref{eq:14}). We will see below how this form for the self energy very naturally arises (below $T_c$) in a $T$-matrix approach. Thus \begin{equation} G^{-1}_{\sigma}(K)= i\omega-\xi_{k\sigma}-\frac{\Delta^{2}} {i\omega+\xi_{k\bar{\sigma}}}\,. \label{eq:15} \end{equation} Now we are in position to calculate the pair susceptibility at $P=0$ \begin{eqnarray} \chi(0) &=& \chi_{\uparrow\downarrow}(0) = \chi_{\downarrow\uparrow}(0) =-\sum_{K}\frac{1}{(i\omega_{n}-E_{k\downarrow})(i\omega_{n}+E_{k\uparrow})} \,. \label{eq:17} \end{eqnarray} Substituting this expression into our BEC condition Eq.~(\ref{eq:5}), we obtain the gap equation \begin{eqnarray} 0 &=&\frac{1}{U}+\sum_{\mathbf{k}} \left[\frac{1-f(E_{k\downarrow})-f(E_{k\uparrow})}{2E_{k}} \right] = \frac{1}{U}+\sum_{\mathbf{k}}\frac{1-2\bar{f}(E_{k})}{2E_{k}} \,. \label{eq:10} \end{eqnarray} after carrying out the Matsubara summation. Here $\mu=(\mu_{\uparrow}+\mu_{\downarrow})/2$ and $h=(\mu_{\uparrow} -\mu_{\downarrow})/2$, $E_{\mathbf{k}} = \sqrt{\xi_{\mathbf{k}}^2 +\Delta^2}$, $E_{k\uparrow}=-h+E_{k}$ and $E_{k\downarrow}=h+E_{k}$, where $\xi_{k}=\epsilon_{k}-\mu$. In addition, we define the average $ \bar{f}(x) \equiv [f(x+h)+f(x-h)]/2, $ where $f(x)$ is the Fermi distribution function. The coupling constant $U$ can be replaced in favor of the dimensionless parameter, $1/k_Fa$, via the relationship $m/(4\pi a) = 1/U + \sum_{\mathbf{k}}(2\epsilon_{k})^{-1}$, where $a$ is the two-body $s$-wave scattering length, and $k_F$ is the noninteracting Fermi wave vector for the same total number density in the absence of population imbalance. Therefore the gap equation can be rewritten as \begin{equation} -\frac{m}{4\pi a}=\sum_{\mathbf{k}}\left[\frac{1-2\bar{f}(E_{k})} {2E_{k}}-\frac{1}{2\epsilon_{k}} \right] \,. \label{eq:11} \end{equation} Here the ``unitary scattering" limit corresponds to resonant scattering where $ a \rightarrow \infty$. This scattering length is tunable by magnetic field application and we say that we are on the BCS or BEC side of resonance, depending on whether the fields are higher or lower than the resonant field, or alternatively whether $a$ is negative or positive, respectively. Finally, in terms of Green's functions, we readily arrive at the number equations: $n_{\sigma}=\sum_{K}G_{\sigma}(K)$, which are consistent with their ground state counterparts \begin{eqnarray} n_{\sigma}&=&\sum_{\mathbf{k}}[f(E_{k\sigma})u_{\mathbf{k}}^{2}+ f(E_{k\bar{\sigma}}) v_{\mathbf{k}}^{2}] \,, \label{eq:12} \end{eqnarray} where the coherence factors $u_\mathbf{k}^2, v_\mathbf{k}^2 = (1\pm \xi_\mathbf{k}/E_{\mathbf{k}})/2$. Next we use this $T$-matrix scheme to derive Eq.~(\ref{eq:14}) and separate the contribution from condensed and noncondensed pairs. The diagrammatic representation of our $T$-matrix scheme is shown in Fig.~\ref{fig:T-matrix}. The first line indicates $t_{pg}$, and the second the total self energy. One can see throughout the combination of one dressed and one bare Green's function, as represented by the thick and thin lines. The self energy consists of two contributions from the noncondensed pairs or pseudogap ($pg$) and from the condensate ($sc$). There are, analogously, two contributions in the full $T$-matrix \begin{eqnarray} t &=& t_{pg} + t_{sc} \,, \label{t-matrix}\\ t_{pg}(P)&=& \frac{U}{1+U \chi(P)}, \qquad P \neq 0 \,, \label{t-matrix_pg}\\ t_{sc}(P)&=& -\frac{\Delta_{sc}^2}{T} \delta(P) \,, \label{t-matrix_sc} \label{eq:43} \end{eqnarray} where we write $\Delta_ {sc}=-U \sum _{\bf k}\langle c_{-{\bf k}\downarrow}c_{{\bf k}\uparrow}\rangle$. Similarly, we have for the fermion self energy \begin{equation} \Sigma_\sigma (K) = \Sigma_\sigma ^{sc}(K) + \Sigma_\sigma ^ {pg} (K) = \sum_P t(P) G_{0,\bar{\sigma}} (P-K) \,. \label{eq:sigma2} \end{equation} We can see at once that \begin{equation} \Sigma_\sigma ^{sc}(K) = \sum_P t_{sc}(P) G_{0,\bar{\sigma}}(P-K) = -G_{0,\bar{\sigma}} (-K) \Delta_{sc}^2 \,. \label{eq:70} \end{equation} The vanishing of the pair chemical potential implies that \begin{equation} t_{pg}^{-1} (0) = U^{-1} + \chi(0) = 0, \qquad T \leq T_c \;. \label{eq:3e} \end{equation} Moreover, a vanishing chemical potential means that $t_{pg}(P)$ diverges $P=0$. Thus, we may approximate \cite{Maly1} Eq.~(\ref{eq:sigma2}) to yield \begin{equation} \Sigma_{\sigma} (K)\approx -G_{0,\bar{\sigma}} (-K) \Delta^2 \,, \label{eq:sigma3} \end{equation} where \begin{equation} \Delta^2 (T) \equiv \Delta_{sc}^2(T) + \Delta_{pg}^2(T) \,, \label{eq:sum} \end{equation} Importantly, we are led to identify the quantity $\Delta_{pg}$ \begin{equation} \Delta_{pg}^2 \equiv -\sum_{P\neq 0} t_{pg}(P). \label{eq:delta_pg} \end{equation} Note that in the normal state (where $\mu_{pair}$ is non-zero) Eq. (\ref{eq:sigma3}) is no longer a good approximation. We now have a closed set of equations for addressing the ordered phase. This approach can be readily generalized \cite{LOFFlong} to treat more exotic polarized phases such as the LOFF state \cite{FFLO,Combescot,LOFF_Review}. We can similarly extend this approach to temperatures somewhat above $T_c$, by self consistently including a non-zero pair chemical potential. This is a necessary step in addressing a trap as well \cite{ChienRapid}. Additionally, the propagator for noncondensed pairs can now be quantified, using the self consistently determined pair susceptibility. At small four-vector $P$, we may expand the inverse of $t_{pg}$, after analytical continuation ($i\Omega_l \rightarrow \Omega+i0^+$), to obtain \begin{equation} t_{pg}^{-1} \approx a_1\Omega^2 + Z\left(\Omega - \frac{p^2}{2 M^*} + \mu _{pair} +i \Gamma^{}_P\right), \label{Omega_q:exp} \end{equation} where the imaginary part $\Gamma^{}_P \rightarrow 0$ rapidly as $p\rightarrow 0$ below $T_c$. Because we are interested in the moderate and strong coupling cases, we drop the $a_1 \Omega^2$ term in Eq. (\ref{Omega_q:exp}), and hence \begin{equation} t_{pg}(P) = \frac { Z^{-1}}{\Omega - \Omega^{}_p +\mu_{pair} + i \Gamma^{}_P}, \label{eq:expandt} \end{equation} where we associate \begin{equation} \Omega_{\mathbf{p}} \approx \frac{p^2} {2 M^*} \,. \label{eq:53} \end{equation} This establishes a quadratic pair dispersion and defines the effective pair mass, $M^*$. They can be calculated via a small $p$ expansion of $\chi(P)$, \begin{equation} Z=\left.\frac{\partial \chi}{\partial\Omega}\right|_{\Omega=0,\mathbf{p}=0}, \qquad \frac{1}{2M^*} =-\left.\frac{1}{6Z}\frac{\partial^{2} \chi}{\partial p^{2}}\right|_{\Omega=0,\mathbf{p}=0} \,. \end{equation} Finally, one can rewrite Eq. (\ref{eq:delta_pg}) as \begin{equation} \Delta_{pg}^2 (T) = Z^{-1}\sum_{\mathbf{p}} b(\Omega_p) \,. \label{eq:81} \end{equation} \begin{figure} \centerline{\includegraphics[]{Fig4.eps}} \caption{$T$-matrix and self-energy diagrams for the present $T$-matrix scheme. The self-energy comes from contributions of both condensed ($\Sigma_{sc}$) and noncondensed ($\Sigma_{pg}$) pairs. Note that there is one dressed and full Green's function in the $T$-matrix. Here $t_{pg}$ represents the propagator for the noncondensed pairs.} \label{fig:T-matrix} \end{figure} We now return to the strong analogies between our $T$-matrix approach and Bose condensation of point bosons, as summarized in Table I. We have three central equations. 1. The pair chemical potential must vanish at and below $T_c$, \begin{equation} \mu_{pair} = 0, \quad (T \leq T_c). \label{eq:57} \end{equation} Importantly this condition leads to the mean field gap equation derived in Eq.~(\ref{eq:10}). 2. There must be a conservation of the total number of (composite) ``bosons" in the system. For this condition, our central equation is Eq.~(\ref{eq:sum}). Here it is understood that the number of "bosons" is effectively represented by the parameter $\Delta^2 (T)$. Unlike the point boson case, the ``total boson number" is temperature dependent and has to be self-consistently determined. 3. The number of noncondensed pairs is readily computed in terms of the pair dispersion, just as in conventional BEC. For this condition our central equation is Eq.~(\ref{eq:81}). Then, just as in conventional BEC, the number of condensed bosons (proportional to $\Delta_{sc}^2$) is determined by the difference between $\Delta^2(T)$ and $\Delta_{pg}^2 (T)$. This, in turn, determines $T_c$ as the lowest temperature(s) in the normal state at which noncondensed pairs exhaust the total weight of $\Delta^2$ so that $\Delta_{pg}^2 = \Delta^2$. Solving for the ``transition temperature" in the absence of pseudogap effects \cite{Machida2,YD05,LatestStoof} leads to the quantity $T_c^{MF}$. More precisely, $T_c^{MF}$ should be thought of as the temperature at which the excitation gap $\Delta(T)$ vanishes. This provides a reasonable estimate, for the pairing onset temperature $T^*$, (when a stable superfluid phase exists). This is distinguished from the transition temperature. We note that $T^*$ represents a smooth crossover rather than an abrupt phase transition. It should be stressed that the dispersion relation for the noncondensed pairs is quadratic. While \textit{one will always find a linear dispersion in the collective mode spectrum \cite{Kosztin2}}, within the present class of BCS-BEC crossover theories, the restriction to a $T$-matrix scheme means that there is no feedback from the collective modes onto the pair excitation spectrum. In effect, the $T$-matrix approximation does not incorporate pair-pair interactions at a level needed to arrive at this expected linear dispersion in the pair excitation spectrum. Nevertheless, this level of approximation is consistent with the underlying ground state wavefunction. \section{Behavior of $T_c$ and Trap Effects} Before turning to experiment, it is important to discuss the behavior of $T_c$ which is plotted as a function of scattering length in the left panel of Fig.~\ref{fig:Tc} for the homogeneous case, presuming $s$-wave pairing. Starting from the BCS regime this figure shows that $T_c$ initially increases as the interaction strength increases. However, this increase competes with the opening of a pseudogap or excitation gap $\Delta(T_c)$. Technically, the pairs become effectively heavier before they form true bound states. Eventually $T_c$ reaches a maximum (very near unitarity) and then decreases slightly until field strengths corresponding to the point where $\mu$ becomes zero. At this field value (essentially where $T_c$ is minimum), the system becomes a ``bosonic" superfluid, and beyond this point $T_c$ increases slightly to reach the asymptote $0.218E_F$ corresponding to an ideal Bose gas. \begin{figure} \centerline{\includegraphics[]{Fig5.eps}} \caption{Typical behavior of $T_c$ as a function of $1/k_Fa$ in a homogeneous system (left panel) and in a trapped Fermi gas (right panel). $T_c$ follows the BCS predictions and approaches the BEC asymptote $0.218T_F$ and $0.518T_F$ in the homogeneous and trapped cases, respectively. In contrast to the homogeneous case, the BEC asymptote in a trap is much higher due to a compressed profile for trapped bosons. In the homogeneous case, $T_c$ reaches a maximum around $1/k_Fa=0$ and a minimum around where $\mu=0$. In the trapped case, this maximum/minimum behavior is washed out largely by the shrinking cloud size as $1/k_Fa$ increases.} \label{fig:Tc} \end{figure} Trap effects change these results only quantitatively as seen in the right panel of Fig.~\ref{fig:Tc}. Here the maximum in $T_c$ may no longer be visible. The calculated value of $T_c$ ($\sim 0.3 T_F$) at unitarity is in good agreement with experiment \cite{ThermoScience,Thomasnew} and other theoretical estimates \cite{Strinati4}. To treat these trap effects one introduces the local density approximation (LDA) in which $T_c$ is computed under the presumption that the chemical potential $\mu \rightarrow \mu - V(r)$ . Here we consider a spherical trap with $V(r)=\frac{1}{2}m\omega^2 r^2$. The Fermi energy $E_F$ is determined by the total atom number $N$ via $E_F \equiv k_BT_F = \hbar\omega (3N)^{1/3} \equiv \hbar^2k_F^2/2m$, where $k_F$ is the Fermi wavevector at the center of the trap. It can be seen that the homogeneous curve is effectively multiplied by an ``envelope" curve when a trap is present. This envelope, with a higher BEC asymptote, reflects the fact that the particle density at the center of the trap is higher in the bosonic, relative to the fermionic case. In this way $T_c$ is relatively higher in the BEC regime, as compared to its counterpart in the homogeneous case. Figure \ref{fig:denstrap} is a central one, for it prepares us for understanding various experiments. It presents a plot of the position dependent excitation gap $\Delta(r)$ and particle density $n(r)$ profile over the extent of the trap. An important point needs to be made: because the gap is largest at the center of the trap, bosonic excitations will be dominant there. At the edge of the trap, by contrast, where fermions are only weakly bound (since $\Delta(r)$ is small), the excitations will be primarily fermionic. We will see the implications of these observations as we examine thermodynamic \cite{ChenThermo} and radio frequency (RF) spectra data \cite{heyan} in the ultracold gases. \begin{figure} \centerline{\includegraphics[width=2.8in,clip]{Fig6.eps}} \caption{Typical spatial profile for density $n(r)$ and fermionic excitation gap $\Delta(r)$ of a Fermi gas in a trap. The curves are computed at $T=T_F/4$ and at unitarity, where $1/k_Fa=0$. Here $R_{TF}$ is the Thomas-Fermi radius, which gives the cloud size at $T=0$ in the noninteracting limit.} \label{fig:denstrap} \end{figure} One should appreciate that temperature is not straightforward to measure in these cold gases; it is most readily measured at the Fermi gas endpoint \cite{Jin4} or in the deep BEC regime \cite{Grimm4}. At unitarity, the physical temperature can be extracted using phenomenological fits to the particle density profiles based on the universality hypothesis \cite{JasonHo,Thomasnew,ThomasVirial} with proper re-calibration \cite{ThermoScience,JS5} below $T_c$. For a more general magnetic field one has to resort to adiabatic sweep thermometry \cite{ChenThermo,Williams}. Here, the magnetic field of interest is accessed via a slow, adiabatic, or isentropic, magnetic field sweep starting from either the BCS or BEC endpoints, where the temperature (and the entropy \cite{Carr}) are known. A finite temperature theory of BCS-BEC crossover is required to calculate the entropy \cite{ChenThermo} at general magnetic fields. In this way, the physical temperature can be associated with the measured endpoint temperature. Indeed, we will see below that the temperature which appears in the measured superfluid phase diagram \cite{Jin4} or in the RF pairing gap experiments \cite{Grimm4} is given in terms of the endpoint temperature. \section{Experimental Evidence for a Pseudogap in Cold Gases} Our finite temperature generalization of the BCS-like ground state has introduced the concept of a ``pseudogap". This pseudogap in the fermionic spectrum should be viewed as synonymous with the concept of noncondensed pairs, or with pairs which have a finite center of mass momentum. They are important both above and below $T_c$. In this section we want to explore the evidence for these noncondensed pairs using three different experiments: density profiles, normal state thermodynamics and RF pairing gap spectroscopy. In this section we will consider the case of an unpolarized gas. \begin{figure} \centerline{\includegraphics[width=4.5in,clip]{Fig7.eps}} \caption{Decomposition of density profiles at various temperatures at unitarity. Here green (light gray) refers to the condensate, red (dark gray) to the noncondensed pairs and blue (black) to the excited fermionic states. $T_c = 0.27T_F$, and $R_{TF}$ is the Thomas-Fermi radius. The presence of noncondensed pairs is essential \cite{JS5} for explaining why there are no sharp features in these profiles, associated with the interface of the normal and superfluid regions. Here $n_s$, $n_p$, and $n_{QP}$ denote density of superfluid, incoherent pairs, and fermioinic quasiparticle, respectively.} \label{fig:23new} \end{figure} In Fig. \ref{fig:23new} we plot a decomposition of the particle density profiles \cite{JS5} for various temperatures above and below $T_c$. The various color codes (or gray scales) indicate the condensate along with the noncondensed pairs and the fermions. This decomposition is based on the superfluid density so that all atoms participate in the condensation at $T=0$. An important observation should be noted. The noncondensed pairs are responsible for smoothing out what otherwise would be a discontinuity between the fermionic and condensate contributions. This leads to a featureless profile, in agreement with experiment \cite{Thomas,Grimm2}. Indeed, these experimental observations presented a challenge for previous theories \cite{Chiofalo,JasonHo} which ignored noncondensed pairs, and therefore predicted an effectively bimodal profile with a kink at the edge of the superfluid core. One can see from the figure that even at $T_c$, the system is different from a Fermi gas. That is, noncondensed pairs are present in the central region of the trap when the condensate is gone. Even at $T/T_c = 1.5$ there is a considerable fraction of noncondensed pairs. It is not until around $T^* = 2T_c$ for this unitary case, that noncondensed pairs have finally disappeared. \begin{figure} \centerline{\includegraphics[width=3.5in,clip]{Fig8.eps}} \caption{Pseudogap effects as apparent from thermodynamics. From Ref. \cite{ThermoScience}. The fact that the experimental data (symbols) (and the two theoretical curves) for noninteracting and unitary Fermi gases do not merge until higher $T^* > T_c$ is consistent with the presence of a normal state pseudogap.} \label{fig:232new} \end{figure} We next turn to a detailed comparison of theory and experiment for thermodynamics. Figure \ref{fig:232new} presents a plot of energy $E$ as a function of $T$ comparing the unitary and non-interacting regimes. The solid curves are theoretical while the data points are measured in $^6$Li \cite{ThermoScience}. There has been a recalibration of the experimental temperature scale in order to plot theory and experiment in the same figure. The latter was determined via Thomas-Fermi fits to the density profiles. To arrive at the calibration, we applied the same fits to the theoretically produced density profiles, examples of which appear in Fig.~\ref{fig:23new}. Good agreement between theory and experiment is apparent in Fig.~\ref{fig:232new}. In the figure, the temperature dependence of $E$ reflects primarily fermionic excitations at the edge of the trap, although there is a small bosonic contribution as well. Importantly one can see the effect of a pseudogap in the unitary case. The temperature $T^*$ is visible from the plots as that at which the non-interacting and unitary curves merge. This corresponds roughly to $T^* \approx 2 T_c$. Measurements \cite{Grimm4} of the excitation gap $\Delta$ can be made more directly, and, in this way one can further probe the existence of a pseudogap. This pairing gap spectroscopy is based on using a third atomic level, called $|3 \rangle$, which does not participate in the superfluid pairing. Under application of RF fields, one component of the Cooper pairs, called $|2 \rangle$, is excited to state $|3\rangle$. If there is no gap $\Delta$ then the energy it takes to excite $|2 \rangle$ to $|3 \rangle$ is the atomic level splitting $\omega_{23}$. In the presence of pairing (either above or below $T_c$) an extra energy $\Delta$ must be input to excite the state $|2 \rangle$, as a result of the breaking of the pairs. Figure \ref{fig:24Cheng} shows a plot of the spectra for $^6$Li near unitarity for four different temperatures, which we discuss in more detail below. In general for this case, as well as for the BCS and BEC limits, there are two peak structures which appear in the data and in the theory \cite{Torma2,heyan}: the sharp peak at $\omega_{23} \equiv 0$ which is associated with ``free" fermions at the trap edge and the broader peak which reflects the presence of paired atoms; more precisely, this broad peak derives from the distribution of $\Delta$ in the trap. At high $T$ (compared to $\Delta$), only the sharp feature is present, whereas at low $T$ only the broad feature remains. The sharpness of the free atom peak can be understood as coming from a large phase space contribution associated with the $2 \rightarrow 3$ excitations \cite{heyan}. These data alone do not directly indicate the presence of superfluidity, but rather they provide strong evidence for pairing. It is interesting to return to discuss the temperatures in the various panels. What is measured experimentally are temperatures $T'$ which correspond to the temperature at the start of an adiabatic sweep from the BEC limit to unitarity. Here fits to the BEC-like profiles are used to deduce $T'$ from the shape of the Gaussian tails in the trap. Based on knowledge \cite{ChenThermo} about thermodynamics (energy $E$ in the previous figure or, equivalently, entropy $S$), and given $T'$, one can then compute the final temperature in the unitary regime, assuming $S$ is constant. Indeed, this adiabaticity has been confirmed experimentally in related work \cite{Grimm2}. We find that the four temperatures are as indicated in the figures. Importantly, one can conclude that the first two cases correspond to a normal state, albeit not far above $T_c$. In this way, these figures suggest that a pseudogap is present as reflected by the broad shoulder above the narrow free atom peak. \begin{figure} \centerline{\includegraphics[width=5.in,clip]{Fig9.eps}} \caption{Experimental RF Spectra for $^6$Li at unitarity at 834\,G. The temperatures labeled in the figure were computed theoretically at unitarity based on adiabatic sweeps from BEC. The two top curves, thus, correspond to the normal phase, thereby, indicating pseudogap effects. Here $E_F = 2.5 \mu $K, or 52\,kHz. From Ref.~\cite{Grimm4}.} \label{fig:24Cheng} \end{figure} \section{Establishing Superfluidity in Cold Fermi Gases} From the time of the earliest discoveries \cite{Kinast2,Jin3,Grimm,Jin4,Ketterle3,KetterleV,Thomas2,Grimm3,ThermoScience} there was a drive to establish the existence of superfluidity which is more difficult on the BCS than on the BEC side \cite{Grimm,Jin3} of the resonance. There has been a sequence of experiments which have effectively made this case, beginning first with fast sweep experiments \cite{Jin4,Ketterle3}, then thermodynamical measurements \cite{ThermoScience}, and finally detection of quantized vortices \cite{KetterleV}. We discuss the first two methodologies here in the context of our theoretical framework. We limit our discussion first to the case of unpolarized superfluids. \begin{figure}[tb] \centerline{\includegraphics[]{Fig10.eps}} \caption{(Color) Earliest evidence for superfluidity: phase diagram of $^{40}$K as a function of $(T/T_F)^0$ and $1/k_F^0a$. This compares both experiment (left panel) and theoretical computations (right panel). In the left panel, a contour plot of the measured condensate fraction $N_0/N$ as a function of $1/k_F^0a$ and effective temperature $(T/T_F)^0$ is compared with theoretically calculated contour lines at $N_s/N=0$ (at the superfluid transition, black curve) and 0.01 (red curve). The experimental data have an overall systematic uncertainty of approximately 0.1 in $1/k_F^0a$. The experimental contour at $N_0/N=0.01$ and the theoretical line for $N_s/N=0.01$ are in good agreement. The dashed line represents the naive BCS result $T_c/T_F^0 \approx 0.615 e^{\pi/2k_F^0a}$. The right panel represents a more complete theoretically computed equilibrium phase diagram, with contour lines for $N_s/N$. Here all temperatures are measured in the Fermi gas regime. From Ref.~\cite{Jin_us}.} \label{fig:Phase} \end{figure} The left panel in Fig.~\ref{fig:Phase} is a plot of the first phase diagram representing the condensate fraction \textit{vs} $1/k_F^0a$, as obtained in Ref.~\cite{Jin4} for $^{40}$K. Subsequently, similar studies \cite{Ketterle3} were undertaken for $^6$Li. The figure is based on starting the system off in the free Fermi gas regime where it can be associated with an initial known temperature $(T/T_F)^0$, and then adiabatically sweeping closer to unitarity. [Here $T_F=T_F^0$, $k_F=k_F^0$ and $(T/T_F)^0$ are all measured in the noninteracting Fermi gas limit]. Once the near-unitary gas (of fixed entropy) is obtained, a fast sweep is made to the BEC regime, where the condensate fraction can be read off from a bi-modal profile. The presumption here, for which there is considerable experimental support \cite{Jin4,Ketterle3}, is that, even if the condensate fraction is not conserved upon a fast sweep to BEC, the presence or absence of a condensate will be preserved. The time frame for the sweep will not allow a condensate to form in the BEC if there were none present near unitarity, nor will it allow a condensate to disappear if it was present initially. The lines drawn on top of the experimental contour plots are the calculated \cite{Jin_us} condensate fraction contours as a function of the adiabatic sweep-projected temperatures $(T/T_F)^0$ for a 0\% and 1\% condensate fraction. These essentially correspond to the normal-superfluid phase boundary which is expected to be rather well measured in these sweep experiments. The figure to the right presents a full plot of the theoretical condensation fraction, importantly, measured with respect to the adiabatic sweep-projected temperatures $(T/T_F)^0$. This, then, is the theoretical phase diagram. For the 1\% case, the overall trends yield good agreement between theory and experiment, except for the small ``overshoot" (of unknown origin) which appears in the BEC side of the data. \begin{figure} \centerline{\includegraphics[width=2.8in,clip]{Fig11.eps}} \caption{Evidence for a superfluid phase via plots of the energy $E$ vs physical temperature $T$. The upper curve (long dashed line) and data points correspond to the BCS or essentially free Fermi gas case, and the lower curve and data correspond to unitarity. The latter provide indications for a phase transition via a slope change. The thin dashed line represents a power law fit to the unitary data below the transition. The inset shows how temperature must be recalibrated below $T_c$. Here $\beta$ is a parameter which characterizes the interaction energy at unitarity for $T=0$, and $\tilde{T}$ is a empirical temperature scale. From Ref.~\cite{ThermoScience}.} \label{fig:231new} \end{figure} The second generation experiments which helped to establish superfluidity were based on thermodynamical measurements. In Fig.~\ref{fig:231new} we show data from Ref.~\cite{ThermoScience}. What is plotted is the measured energy as a function of temperature on a log-log scale. This temperature represents a theoretical recalibration of an effective profile-measured temperature $\tilde{T}$. The key feature here is that the data (indicated by the points) show an abrupt change at a temperature one can call $T_c$. This abrupt change occurs for the unitary scattering case. No such feature is seen for the noninteracting Fermi gas, also plotted in the figure. Furthermore, this abrupt change in $E$ vs $T$ is seen \textit{even in the raw data points} (not shown here), without appealing to a theoretical re-calibration. All of this is very suggestive of a specific heat discontinuity, which is to be associated with a phase change -- presumably to a superfluid phase. The inset in the figure shows how the effective temperature $\tilde{T}$ which is obtained from a Thomas-Fermi fit to the calculated density profile compares with the physical temperature $T$. The inset shows that a recalibration is necessary only below $T_c$, to account for the presence of a condensate. We stress that the observation of a phase change made by the Duke group \cite{ThermoScience} is not dependent on this recalibration. Hence these experiments provide good evidence for a transition between a normal and superfluid phase. The last generation experiment to make the case for superfluidity was the observation of quantized vortices by the MIT group \cite{KetterleV}. These very convincing experiments will be discussed elsewhere in this lecture series. \section{Fermi Gases with Imbalanced Spin Population} The latest excitement in the field of trapped fermions pertains to gases with deliberately imbalanced spin populations \cite{ZSSK06,PLKLH06,ZSSK206}. In large part this is motivated by interest from theorists in other disciplines such as dense QCD and (isospin asymmetric) nuclear matter \cite{Wilczek,LW03,FGLW05,Sedrakian}. From the condensed matter viewpoint there has been an underlying interest in exotic and intriguingly elusive phases associated with Zeeman effects in superconductors, such as that proposed by Larkin and Ovchinnikov and by Fulde and Ferrell (LOFF). In the LOFF state \cite{FFLO} the condensate has a net momentum of a fixed $\mathbf{q}$ or of $+\mathbf{q}$ and $-\mathbf{q}$. Even more elaborate crystalline lattices of various $\mathbf{q}_i$ have also been contemplated \cite{LOFF_Review}. An additional, and very important motivation for these studies is associated with the recent claims \cite{ZSSK206} that when there is a population imbalance, the density profiles will indicate whether or not superfluidity is present and they, moreover, provide an internal mechanism for thermometry. Thus, because these experiments are claimed to identify $T_c$ itself, and because they pertain to thermometry, it should be clear that a theoretical analysis of these experiments requires an understanding of the effects of $T \neq 0$. We begin our discussion by summarizing some key experimental observations \cite{ZSSK06,PLKLH06,ZSSK206}. In a trapped cloud there appears to be a superfluid core, which, at the lowest temperatures, is unpolarized. Outside of this core there is a normal region, in which both spin components are present and this carries a significant fraction of the polarization. Beyond this ``mixed, normal region" there is a free Fermi gas composed of the majority spin species, which carries additional polarization. At suffiently low temperatures, there appears to be a form of phase separation in which the superfluid and normal phases are associated with zero and finite polarization, respectively. We turn now to a theoretical understanding \cite{ChienRapid,Chien_prl} of these experiments at $T \neq 0$ first for the case of a homogeneous gas, and then later in the trapped configuration. There has been some work along these same lines elsewhere in the literature \cite{YD05}, although without incorporating noncondensed pair effects. We focus on the Sarma or breached pair phase \cite{Sarma63}. The figures we present do not include the more exotic LOFF or phase separated states. The formal structure for addressing the former \cite{LOFFlong} is very similar to that of the Sarma state; we will defer a brief discussion of these until the end. To help with the clarity of the presentation, we state our major conclusions for the homogeneous and trapped configurations in the unitary regime, at the outset. \begin{itemize} \item In the \textit{homogeneous} case temperature serves to stabilize the polarized superfluid phase. Such a phase is unstable in the ground state. \item In a \textit{trap} at low $T$, the superfluid core remains unpolarized. Except at very low $T$, spin polarization will be mostly carried by fermions coexisting with strongly interacting, thermally excited noncondensed pairs which appear outside the core. Thus, pseudogap effects, which have been emphasized throughout, are very important here. \item With increasing $T$, polarization tends to continuously penetrate into the superfluid core of the trap, until at, or even below $T_c$, the polarization is uniformly distributed throughout the cloud. \item This superfluid Sarma or breached pair phase is limited in the amount of polarization it can accommodate, especially near unitarity. This applies to both traps and homogeneous sytems. \end{itemize} \begin{figure} \centerline{\includegraphics[clip,width=3.8in]{Fig12.eps}} \caption{Typical behavior of $T_c$ as a function of $1/k_Fa$ for spin polarizations $p=0.05$, 0.1, 0.16, and 0.3. The $T_c$ curve splits into two disconnected curves around $p=0.14$. This figure should be compared with left panel of Fig.~\ref{fig:Tc} for the unpolarized case. From Ref.~\cite{Chien_prl}.} \label{fig:TcImb} \end{figure} Figure \ref{fig:TcImb} presents a plot of $T_c$ for a homogeneous system as a function of $1/k_Fa$ for various polarizations. This figure should be compared with the left panel of Fig.~\ref{fig:Tc}. With this comparison, one sees at once that there can be no superfluidity in the deep BCS regime, once the polarization is different from zero. And when superfluidity first appears on the BCS side of resonance it is associated with two $T_c's$ at given $1/k_Fa$. The larger the polarization $p$, the harder it is for a homogeneous system to support superfluidity, except in the BEC regime. To understand the meaning of these two $T_c$'s we plot the superfluid density $n_s(T)$ as a function of $T$ in Fig.~\ref{fig:ns}, for several different values of $1/k_Fa$. If one focuses on the unitary case, for definiteness, one can see that $n_s$ vanishes at two different temperatures. The lower $T_c$ corresponds to the onset of superfluidity. At temperatures below this, the (breached pair or Sarma) state is unstable. Similarly, at the upper $T_c$, superfluidity disappears in the usual way; it is destroyed by thermal excitations. For positive values of $1/k_Fa$, the figure shows that the $n_s(T)$ curves stop abruptly. This is indicated by the dotted segments of the curves which represent thermodynamically unstable solutions. \begin{figure} \centerline{\includegraphics[clip,width=3.in]{Fig13.eps}} \caption{ Normalized superfluid density $n_s/n$ as a function of $T/T_F$ at $p=0.1$ for various $1/k_Fa$ from BCS to BEC. The dotted (segments of the) curves represent unstable solutions. The fact that at resonance and on the BCS-side there are 2 $T_c's$ is consistent with the previous Figure. From Ref.~\cite{Chien_prl}.} \label{fig:ns} \end{figure} The fact that $T_c$ may be double-valued could have been anticipated in a simpler set of calculations performed at the strict mean field level, and discussed elsewhere in the literature \cite{Sedrakian}. Here one solves Eqs.(\ref{eq:11}) and (\ref{eq:12}) only, without imposing (\ref{eq:sum}) and (\ref{eq:delta_pg}). In Fig.~\ref{fig:TcMF}, we present a plot of $T_c^{MF}$ as a function of $1/k_Fa$ for a range of $p$. This quantity can be viewed as the pairing onset temperature $T^*$, when there is a stable superfluid phase. In the inset of Fig.~\ref{fig:TcMF} we plot $\Delta(T)$ at different $1/k_{F}a$ for $p=0.3$. For $p<0.9$ and sufficiently low $T_c^{MF}$, we find that there are two $T_c^{MF}$ values. This structure implies that $\Delta$ is nonmonotonic \cite{Sedrakian} in $T$, as indicated by the bottom curve in the inset of Fig.~\ref{fig:TcMF}. The two zeroes of $\Delta$ represent the two values of $T_c^{MF}$. In contrast to the more conventional behavior (shown in the top curve for stronger pairing interaction), $\Delta$ \textit{increases} initially with $T$ at low temperature when $1/k_{F}a$ is sufficiently small. \begin{figure} \centerline{\includegraphics[clip,width=3.in]{Fig14.eps}} \caption{Mean-field behavior of $T_c^{MF}$ as a function of $1/k_Fa$ for different spin polarizations $p$. Shown in the inset is the pairing gap $\Delta(T)$ at different $1/k_Fa$ for $p=0.3$ which can vanish at two distinct temperatures. Here $E_F\equiv k_B T_F \equiv\hbar^2 k_F^2/2m$ is the noninteracting Fermi energy in the absence of polarization. The quantity $T_c^{MF}$ represents the pairing onset temperature $T^*$.} \label{fig:TcMF} \end{figure} We summarize by noting that these results indicate that \textit{temperature enables pairing associated with the breached pair or Sarma state}. This was also inferred in Refs.~\cite{YD05} and \cite{Sedrakian}. In general superfluids, one would argue that these two effects compete. We may view, then, this unusual polarized phase as an ``intermediate temperature superfluid". We now turn to the behavior of these superfluids in a trap. The same calculations are applied to the Sarma or breached pair state using the LDA to incorporate trap effects. Figure \ref{fig:15} shows the resulting behavior at unitarity for polarization of 15\% and for various temperatures from below to just above $T_c$. The upper panels plot the order parameter $\Delta_{sc}$ and the (total) gap parameter $\Delta$. Superposed on these plots is the polarization $\delta n$. The lower panels present the density profiles for each spin state. Several important features can be gleaned from the upper panels. At the lowest temperatures the bulk of the polarization is in a region where $\Delta_{sc} =0$, but $\Delta \neq0$; thus polarization is excluded at low $T$ from the superfluid core. Moreover, it can be seen that an excitation gap $\Delta$ is present throughout most of the cloud. Whenever $\Delta \neq 0$ one can infer that both spin states are present. For non-zero $\Delta$, the particle profiles are necessarily different from those of a non-interacting gas. The bulk of the polarization appears in the ``normal, mixed region", and within this portion of the trap there are strong interactions between the two spin states. Only at the very edge of the trap is there an exclusively majority component (and here $\Delta =0$). This region is occupied by a non-interacting Fermi gas and can thereby be used to set the temperature scale for the trapped cloud. \begin{figure} \centerline{\includegraphics[width=4.9in,clip]{Fig15.eps}} \caption{Spatial profiles of the Gap ($\Delta$), order parameter ($\Delta_{sc}$), and the density of the up ($n_\uparrow$) and down ($n_\downarrow$) spin components and their difference ($\delta n$) for a unitary gas in a trap at $T/T_F=0.1$, 0.14, 0.26, and 0.28, from left to right. Except at very low $T$, most of the polarization is carried by fermions co-existing with noncondensed pairs, \textit{i.e.}, in the pseudogap regime where $\Delta_{sc} = 0$, but $\Delta \neq 0$. Here $T_c/T_F\approx 0.27$.} \label{fig:15} \end{figure} We end this Section with some comparisons with experiment. We address qualitative effects in Fig.~\ref{fig:8} and semi-quantitative effects in Fig.~\ref{fig:1}. The former is for the unitary case and the latter is in the BEC regime. The left panels in Fig.~\ref{fig:8} show data from the Rice group Ref.~\cite{Rice_new}. The upper figure plots the density profiles for each spin state and the lower panel, their difference. The unpolarized core is evident, as is the sharp edge beyond which polarization abruptly appears. This behavior has been interpreted \cite{PLKLH06,Rice_new} as suggesting phase separation. Because the aspect ratios for the trap are not maintained in the profiles, it has been argued \cite{PLKLH06,Rice_new} that the LDA scheme may not be appropriate for addressing experiments on these highly anisotropic traps. In the experimental data, non-LDA effects lead to a narrower distribution for $n_\downarrow$ along the axial direction while it is broadened in the radial direction. Despite this caveat we plot our counterpart theoretical results at $T=0.06T_F$ (right panels) for qualitative comparison. This plot is designed primarily to introduce theoretical observations (which can be superposed, in effect, on the experimental plot) concerning where the superfluid phase resides in the trap. This knowledge cannot be directly gleaned from these particular experiments. These low $T$ theoretical results are rather striking, for they make it clear that when the order parameter is present the polarization is excluded. Rather, the polarization appears in the Fermi gas region outside the condensate core. At higher $T$ and lower $p$ (e.g, $T=0.1T_F$, $p=0.15$), the polarization is carried largely within the ``pseudogap" regime, where there are strong pairing correlations, but no long range order. \begin{figure} \centerline{\includegraphics[width=4.4in,clip]{Fig16.eps}} \caption{Comparison of theoretically computed density distributions of the up and down spin components (upper row) and their difference (lower row) at unitarity with the Rice group data. The central issue here is not the quantitative comparison, but rather that (as seen in the theoretical curves) polarization is confined to outside the superfluid core. In the Rice data, $n_\downarrow$ has a narrower distribution due to non-LDA effects. Data from Ref.~\cite{Rice_new}.} \label{fig:8} \end{figure} We turn finally to a comparison of data from the MIT group \cite{ZSSK206}, and in the BEC regime. The experimental data (for polarizations near 0.6) are plotted on the two right panels which show the particle density profiles for each spin state and their difference $\delta n$. The upper panels correspond to temperatures which are believed to be in the normal state, whereas, the lower panels are for the superfluid phase. The lightly dotted curves in the experiment represent extrapolations of the Thomas-Fermi fits to the curves at large radii. In the superfluid phase, the data show that the polarization at the core center is considerably smaller than it is for the normal state. One apparent difference between above and below $T_c$ experiments, is that at low $T$ (i) the minority component has contracted into the center of the trap. (ii) Another signature is a kink in the majority profile. Yet another signature is that (iii) there is a clear bi-modality in the minority component. This behavior can be compared with the theoretical results plotted in the left two panels above and below $T_c$ for a polarization $ p=0.6$. The kink in the majority component at low $T$ can be clearly attributed to the edge of the condensate as denoted by the vertical arrow (where $\Delta_{sc}=0$). The bi-modality in the minority component is amplified in the lower inset to make it more evident. These theoretical plots, at a qualitative level, exhibit the three central features of the data noted above. \begin{figure} \centerline{\includegraphics[]{Fig17.eps}} \caption{Comparison of theoretically computed density distributions of the up and down spin components and their difference $\delta n$ at different temperatures in the BEC regime with MIT data from Ref.~\cite{ZSSK206}, for $p=0.6$. Upper/lower panels are above/below $T_c$. The arrow in the bottom theoretical curve for $\delta n$ indicates the condensate edge which is reflected in features in both $n_{\uparrow}$ and $n_{\downarrow}$, much as in the data.} \label{fig:1} \end{figure} \section{Conclusions} Throughout this Review we have stressed that temperature is important in understanding and characterizing the ultracold Fermi gases. Experimentally, one is never strictly at $T=0$; as a result there are thermal excitations of the gas which need to be characterized both theoretically and experimentally. At a formal level, we have discussed how temperature enters into the standard BCS-BEC crossover theory by leading to a distinction between the superconducting order parameter $\Delta_{sc}$ and the gap parameter $\Delta$. This difference reflects the existence of pre-formed pairs above $T_c$ and noncondensed pairs below. At a more physical level we have shown how temperature changes the character of the gas from a superfluid at low $T $ to an unusual (but strongly interacting) normal fluid at moderate $T $, and to ultimately a free Fermi gas at high $T$. This strongly interacting normal fluid is most interesting, for unlike the Landau Fermi liquid (or gas) which exists just above $T_c$ in a strictly BCS superfluid, here there is a normal state excitation gap or pseudogap. By looking at three different experiments, we have provided evidence for this normal state excitation gap (in RF pairing gap spectroscopy \cite{Grimm4} and in thermodynamics \cite{ThermoScience} and for the below-$T_c$ counterpart (noncondensed pairs), via the shape of the particle density profiles \cite{Thomas,Grimm2}. In this Review we have discussed how temperature can be measured through adiabatic sweep thermometry \cite{Jin4} and through Thomas-Fermi fits to density profiles \cite{Thomas,ThermoScience}. We have shown how temperature played an important role in establishing the first generation and earliest evidence for superfluidity -- based on fast sweep experiments which yield \cite{Jin4} a phase diagram in the temperature vs $1/k_Fa$ plane. It also played an important role in the second generation indications for superfluidity associated with thermodynamical measurements \cite{ThermoScience}. Finally, we have addressed a new and exciting class of experiments \cite{ZSSK06,PLKLH06,ZSSK206} involving spin population imbalance and shown that here too temperature plays a critical role. From an experimental point of view, the most exciting features of these experiments are that they show how the density profiles can be used to establish the transition temperature $T_c$. Moreover, through the wings of the profiles, they provide a theory-independent mechanism for thermometry. From our theoretical perspective, which focuses on temperature \cite{Chien_prl,ChienRapid}, what is also exciting is that (i) the stability and character of these polarized superfluids is very sensitive to temperature and (ii) even in the normal state one sees strong interactions between the two spin components, which we associate with the finite $T$ pseudogap effects we have been discussing in this Review. As stated in Ref.~\cite{ZSSK206}, ``Already at high temperatures, above the phase transition, the larger [majority] cloud's profile is strongly deformed in the presence of the smaller [minority] cloud, a direct signature of interaction." Our discussion, thus far, has not addressed phase separation \cite{SR06} or the exotic LOFF states \cite{FFLO,Combescot} associated with polarized superfluids. The latter, which correspond to a condensate with a finite momentum $\mathbf{q}$, may well be ground states of these spin polarized gases for some range of scattering lengths $a$, since the polarized BCS like phase (with $\mathbf{q}=0$) is only stable at intermediate temperatures. These LOFF phases are discussed elsewhere in this volume. There are arguments in the literature that in some more general form they play an important role near unitarity \cite{Machida2,Kinnunen} in trapped gases. At a theoretical level, LOFF states may be included \cite{LOFFlong} following the formalism we have outlined in this Review. Underlying the interest in this general field of ultracold gases is the possibility that they may shed light on the high temperature cuprate superconductors either in the context \cite{ourreview,LeggettNature} of the BCS-BEC crossover scenario or in the context of optical lattices and Hubbard model simulations. For the former, one can return to the question of where they would fit on a phase diagram plot (such as that presented in Fig.~\ref{fig:Tc} which addresses $T_c$ vs $1/k_Fa$). When the calculations are properly redone for $d$-wave pairing on nearly two dimensional lattices, the values and shape of the $T_c$ curve are in quite good agreement with experiment \cite{Chen2,ourreview,ReviewJLTP}. Moreover, $T_c$ vanishes well before the BEC limit is reached \cite{Chen1}; one can, then, deduce that in this scenario the cuprates are close to the unitary regime, just as are the ultracold gases. Whether or not this crossover picture turns out to be correct for high $T_c$, another very important rationale for its study is the possibility of generalizing what is undoubtedly the most successful theory in condensed matter physics: BCS theory. We now understand from the ultracold gases that the nature-made superconductors to which the original theory has been applied, are only a very special case of a much more general class of superfluids. And it is most fitting on this fiftieth anniversary of BCS that we pay homage to this most remarkable of theories by recognizing its even greater generality. \acknowledgments We wish to warmly thank all our past collaborators who have contributed to the work and figures presented here: Jelena Stajic, Yan He and Chih-Chun Chien, as well as John Thomas, Joe Kinast and Andrey Turlapov, as well as Murray Holland, Marilu Chiofalo and Josh Milstein, as well as Debbie Jin, Cindy Regal and Markus Greiner. This work was supported by NSF PHY-0555325 and NSF-MRSEC Grant No.~DMR-0213745.

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\section{Introduction} We consider the conductivity problem in the presence of an inclusion with eccentric core-shell geometry which consists of a core and a surrounding shell, where the core and shell have circular boundaries that are nearly touching. Specifically, we let $B_i$ and $B_e$ be two disks such that $B_e$ contains $B_i$, as drawn in Figure \ref{fig:geometry}, where the core $B_i$ and shell $B_e\setminus B_i$ have the constant conductivities $k_i$ and $k_e$ satisfying $0<k_i,k_e\neq 1<\infty$. The aim of the paper is to derive an asymptotic formula for the solution to the conductivity problem \begin{equation} \label{Eqn} \left\{ \begin{array}{ll} \displaystyle \nabla \cdot \sigma \nabla u = 0 \qquad& \textrm{in } \mathbb{R}^2, \\ \displaystyle u(\bold{x}) - H(\bold{x}) = O(|\bold{x}|^{-1}) \qquad& \textrm{as } |\bold{x}| \to \infty, \end{array} \right. \end{equation} as the distance between $\partial B_i$ and $\partial B_e$ degenerates to zero. Here, $\sigma$ denotes the conductivity profile $\sigma = k_i \chi(B_i) + k_e \chi(B_e \setminus B_i) + \chi(\mathbb{R}^2 \setminus B_e)$, $H$ is an entire harmonic function, and $\chi(D)$ represents the characteristic function of a set $D$. The electrostatic interaction problem has been studied for a long time, including \cite{Max}. For two separated spheres, the series expression by separation of variables in bispherical coordinates and the method of image charges were developed \cite{Dav,Jef,Smy}. The electric field may blow up as the distance tends toward zero \cite{keller}, and it causes difficulty in the numerical computation. There have been many attempts to overcome this difficulty and to compute the asymptotic behavior of the electric field \cite{Bat,keller,McP2,McP,Pol,Pol2}. For two dimensions, this problem can be also considered in relation to the stress concentration in composite materials. In composite structures, which consist of a matrix and inclusions, it is common for inclusions to be closely located. Having extreme material parameters is a necessary condition for inclusions to induce the electric field or the stress tensor blow-up. Li and Vogelius showed that the electric field is bounded if the inclusions' conductivities are finite and strictly positive \cite{LV}, and Li and Nirenberg extended this result to elliptic systems \cite{LN}. In recent years, the gradient blow-up feature of the electric field in the small area between the inclusions' boundaries has attracted much attention due to its application to various imaging modalities, for example \cite{lassiter2008close,Rom}. In particular, the electric field concentration in the presence of two nearly touching separated inclusions has been intensively studied. We denote by $\epsilon$ the distance between the inclusions. The generic blow-up rate for conductors is $\epsilon^{-1/2}$ in two dimensions \cite{AKL,bab,BC,BT,keller,Y,Y2}, and $|\epsilon\ln\epsilon|^{-1}$ in three dimensions \cite{BLY,BLY2,LY}. In two dimensions, Ammari et al. deduced an optimal bound for two disks with arbitrary constant conductivities \cite{AKLLL,AKL}, and Yun obtained the blow-up rate for perfect conductors of general shapes \cite{Y,Y2}. The asymptotic behavior of the gradient blow-up solution was studied for perfect conductors of circular shapes by Kang et al. \cite{KLY}, and of {arbitrary shapes by Ammari et al. \cite{ACKLY} and Kang et al. \cite{KLeeY}. For disks with arbitrary constant conductivities, Lim and Yu obtained an asymptotic formula for the electric field \cite{LYu2D}. The blow-up feature for perfect conductors of a bow-tie structure was investigated by Kang and Yun in \cite{kang2017optimal}. % In three dimensions, the blow-up rate for perfect conductors of general shapes was obtained by Bao et al. \cite{BLY,BLY2}. For perfect conductors of spherical shapes, Ammari et al. showed that the potential difference in the conductors is necessary for the gradient blow-up \cite{ammari2007estimates}, an optimal bound for the electric field concentration was obtained by Lim and Yun \cite{LY}, and the asymptotic feature of the blow-up solution was investigated by Kang et al. \cite{KLY2} and by Lim and Yu \cite{LYu3D}. Lekner calculated the field enhancement rate by using the bispherical coordinates \cite{lekner11}. Recently, Ammari and Yu developed a numerical method to compute the blow-up solution \cite{AmmariYu}. % The blow-up rate for insulating inclusions is the same as perfectly conducting ones in two dimensions, but is different in three dimensions. Yun showed that the rate in three dimensions is $1/\epsilon^{\frac{2-\sqrt{2}}{2}}$ \cite{Y3}. For the elasticity problem, Bao et al. obtained an upper bound in two dimensions \cite{bao2015gradient}, and this result was generalized to higher dimensions \cite{bao2017gradient}. Recently, the gradient blow-up term of the solution in two dimensions was verified for {convex hard inclusions} by Kang and Yu \cite{kang2017quantitative} and for circular holes by Lim and Yu \cite{lim2017stress}. An inclusion with core-shell geometry can also induce the gradient blow-up for the solution to the conductivity problem as the distance between the boundaries of the core and shell degenerates to zero. In two dimensions, Ammari et al. obtained the blow-up rate $\epsilon^{-1/2}$ for core-shell geometry with circular boundaries \cite{AKLLL}. Recently, Li et al. obtained an optimal bound for inclusions of general shapes in two dimensions and higher \cite{li2017optimal}, and Bao et al. extended the result to the linear elasticity problem \cite{bao2017optimal}. It is worth mentioning that a plasmonic structure of core-shell geometry induces the shielding effect on a region away from the structure \cite{YL17}. Our goal in this paper is to characterize the electric field concentration for core-shell geometry with circular boundaries. We first show that only the linear term in $H(\mathbf{x})$ contributes to the gradient blow-up. We then show that the gradient blow-up happens only in the shell area $B_e\setminus B_i$. We finally express $u$ by the single and double layer potentials with image line charges as follows: for $\mathbf{x}\in B_e\setminus B_i$, \begin{align*} & u(\mathbf{x})-H(\mathbf{x})\\ &= \frac{C_x r_*^2}{2} \left[ \int_{((-\infty,0), \bold{p_1}]} \ln|\mathbf{x}-\mathbf{s}|\varphi_1(\mathbf{s}) \,d\sigma(\mathbf{s}) -\int_{[\bold{p}_2,\bold{c_i}]} \ln|\mathbf{x}-\mathbf{s}| \varphi_2(\mathbf{s})\, d\sigma(\mathbf{s})\right] \\ & + \frac{C_y r_*^2}{2} \left[ \int_{((-\infty,0), \bold{p_1}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2\rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_1(\mathbf{s})\, d\sigma(\mathbf{s}) -\int_{[\bold{p}_2,\bold{c_i}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2 \rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_2(\mathbf{s})\, d\sigma(\mathbf{s})\right] + r(\mathbf{x}), \end{align*} where $C_x,\, C_y,\, r_*^2$ are some constants depending on the background potential function $H$ and the geometry of the inclusion, and $|\nabla r|$ is uniformly bounded independently of $\epsilon,\, \mathbf{x}$. The points $\mathbf{p}_1,\mathbf{p}_2$ and $\mathbf{c}_i$ are defined according to $B_i$ and $B_e$; see Theorem \ref{asymptotic} for further details. In the derivation, we use the bipolar coordinate system due to its convenience to solve the conductivity problem in the presence of two circular interfaces. We also deduce an integral expression with image line charges for the problem relating to two separated disks. The paper is organized as follows. In section \ref{sec:bipolar}, we explain the core-shell geometry and the associated bipolar coordinate system. We decompose the solution $u$ into a singular term and a regular term in section \ref{sec:series_bipolar} and derive an asymptotic formula for the solution in terms of bipolar coordinates in section \ref{sec:bipolarapprox}. Section \ref{sec:cartesian} is about the image line charge formula, and we conclude with some discussion. \section{The bipolar coordinate system} \label{sec:bipolar} \subsection{Geometry of the core-shell structure} We let $B_i$ and $B_e$ be the disks centered at $\mathbf{c}_i$ and $\mathbf{c}_e$ with the radii $r_i$ and $r_e$, respectively. As stated before, we assume that $B_e$ contains $B_i$ and that the distance between the boundary circles, say $\epsilon=\mbox{dist}(\partial B_i, \partial B_e)$, is small. After appropriate shifting and rotation, the disk centers are $\mathbf{c}_i=(c_i,0)$ and $\mathbf{c}_e=(c_e,0)$ with \begin{equation} \label{disks:center} \displaystyle c_i = \frac{r_e ^2 - r_i ^2 - (r_e - r_i - \epsilon)^2}{2(r_e-r_i-\epsilon)}, \quad \displaystyle c_e = c_i +r_e- r_i - \epsilon . \end{equation} Note that $r_e>r_i>0$ and $c_e>c_i>0$ as shown in Figure \ref{fig:geometry}. The closest points on $\partial B_i$ and $\partial B_e$ between $\partial B_i$ and $\partial B_e$ are, respectively, $\mathbf{x}_i=(x_i,0)$ and $\mathbf{x}_e=(x_e,0)$, with \begin{align*} x_i =c_i-r_i=\frac{2\epsilon r_e-\epsilon^2}{2(r_e-r_i-\epsilon)},\quad x_e=c_e-r_e=\frac{2\epsilon r_i+\epsilon^2}{2(r_e-r_i-\epsilon)}. \end{align*} Note that $x_i,x_e=O(\epsilon)$. % \begin{figure}[!h] \begin{center} \includegraphics[width=6.5cm]{bilayer}\hskip 1cm \includegraphics[width=6.5cm]{bipolar2} \caption{An inclusion with eccentric core-shell geometry (left) and the coordinate level curves of associated bipolar coordinates (right).}\label{fig:geometry} \end{center} \end{figure} For a disk $D$ centered at $\bold{c}$ with radius $r$, the reflection of $\mathbf{x}=(x,y)\in\mathbb{R}^2$ across $\partial D$ is $$ R_{\partial D} (\bold{x}) = \frac{r^2 (\bold{x}-\bold{c})}{|\bold{x}-\bold{c}|^2}+\bold{c}. $$ We also define $R_{\partial D}(z)\in\mathbb{C}$ for $z=x+\rm{i}y$ such that $R_{\partial D}(\mathbf{x})=(\Re\{R_{\partial D} (z)\},\Im\{R_{\partial D} (z)\})$. Here the symbols $\Re\{w\}$ and $\Im\{w\}$ indicate the real and imaginary parts of a complex number $w$, respectively. We then define the reflection of a function $f$ across $\partial D$ as $$R_{\partial D}[f] (\bold{x}) = f\left( R_{\partial D}(\bold{x})\right)\qquad\mbox{for }\mathbf{x}\in\mathbb{R}^2,$$ and the combined reflections as $$R_{\partial D_1}R_{\partial D_2}\cdots R_{\partial D_n}[f](\mathbf{x})=f\left((R_{\partial D_n}\circ R_{\partial D_{n-1}}\cdots \circ R_{\partial D_1})(\mathbf{x})\right).$$ Let $R_i$ and $R_e$ denote the reflections across the circles $\partial B_i$ and $\partial B_e$, respectively. Then, one can easily show that the combined reflections $R_i \circ R_e$ and $R_e \circ R_i$ have the same fixed points $\mathbf{p}_1 = (-\alpha,0)$ and $\mathbf{p}_2=(\alpha,0)$ with \begin{equation} \label{defalp} \alpha = \frac{\sqrt{\epsilon(2r_i+\epsilon)(2r_e-\epsilon)(2r_e-2r_i-\epsilon)}}{2(r_e-r_i-\epsilon)}. \end{equation} One can also easily show that \begin{equation} \alpha= r_* \sqrt{ \epsilon} + O(\epsilon \sqrt{\epsilon}),\quad \mbox{where } r_* = \sqrt{\frac{2r_i r_e}{r_e-r_i}}.\label{def:rstar} \end{equation} The boundaries of $B_i$ and $B_e$ are then the coordinate level curves $\{\xi= \xi_i\}$ and $\{\xi = \xi_e\}$, respectively, in the bipolar coordinate system $(\xi,\theta)$ that will be defined below, with \begin{equation}\label{xi_ie} \xi_i = \ln |c_i + \alpha| - \ln (r_i),\quad \xi_e = \ln |c_e + \alpha| - \ln (r_e). \end{equation} % For later use we set \begin{equation}\label{def:xik} \begin{cases} \displaystyle\xi_{i,k}=2k(\xi_i-\xi_e)+\xi_i,\\ \displaystyle\xi_{e,k} = 2k(\xi_i-\xi_e)+\xi_e \end{cases} \end{equation} for each $k\in\mathbb{N}$. One can show $$\xi_i>\xi_e>0,$$ so that $\xi_{i,k}\geq\xi_i$ and $\xi_{e,k+1}\geq \xi_e$. \subsection{The bipolar coordinate system associated with the eccentric core-shell structure} We set the bipolar coordinate system $(\xi, \theta) \in \mathbb{R} \times (-\pi, \pi]$ with respect to the two poles located at $\mathbf{p}_1=(-\alpha,0)$ and $\mathbf{p}_2=(\alpha,0)$: for any point $\mathbf{x}=(x,y)$ in the Cartesian coordinate system, the coordinates $(\xi,\theta)$ are defined such that \begin{equation}\label{def:bipolar}e^{\xi+\rm{i}\theta}=\frac{\alpha+z}{\alpha-z}\quad\mbox{with } z=x+\rm{i}y.\end{equation} In other words \begin{align}\label{bipolar:z} z&=\alpha \frac{e^{\xi+{\rm i}\theta}-1}{e^{\xi+{\rm i}\theta}+1}= \alpha \frac{\sinh \xi}{\cosh \xi + \cos \theta}+{\rm i} \alpha \frac{\sin \theta}{\cosh \xi + \cos \theta}. \end{align} We write either $\mathbf{x}=\mathbf{x}(\xi,\theta)$ or $z=z(\xi,\theta)$ to specify bipolar coordinates if necessary. For a function $u=u(\mathbf{x})$, we may write $u(\xi,\theta)$ to indicate $u(\mathbf{x}(\xi,\theta))$ for notational simplicity. Figure \ref{fig:geometry} illustrates the coordinate level curves of the bipolar coordinate system. Note that the two level curves $\{\theta=\frac{\pi}{2}\}$ and $\{\theta=-\frac{\pi}{2}\}$ form one circle that is centered at the origin with radius $\alpha$. Therefore, we have that \begin{equation}\label{thetalevel} \begin{cases} \displaystyle |z(\xi,\theta)|\leq \alpha\qquad\mbox{if }|\theta|\leq \frac{\pi}{2},\\[2mm] \displaystyle |z(\xi,\theta)|>\alpha\qquad\mbox{if }|\theta|>\frac{\pi}{2}. \end{cases} \end{equation} One can easily see that the two scale factors coincide, {\it i.e.}, $$\left|\pd{\mathbf{x}}{\xi}\right|=\left|\pd{\mathbf{x}}{\theta}\right|=\frac{1}{h(\xi,\theta)},$$ where $$h(\xi, \theta) = \frac{\cosh \xi + \cos \theta }{\alpha}.$$ In particular, the components of $\partial \mathbf{x}/\partial \xi$ \begin{equation}\label{eqn:xdxi} \frac{\partial x}{\partial \xi} = \alpha \frac{1+\cos \theta \cosh\xi}{(\cosh\xi + \cos \theta)^2},\quad\frac{\partial y}{\partial \xi} = -\alpha \frac{\sin \theta \sinh\xi}{(\cosh\xi + \cos \theta)^2} \end{equation} satisfy that $\big| \frac{\partial x}{\partial \xi} \big|, \big| \frac{\partial y}{\partial \xi} \big|\leq \frac{1}{h(\xi,\theta)}.$ The gradient for a function $u(x,y)$ can be written as \begin{equation}\label{eqn:nabla}\nabla u=h(\xi,\theta)\pd{u}{\xi}\hat{\mathbf{e}}_\xi+h(\xi,\theta)\pd{u}{\theta}\hat{\mathbf{e}}_\theta, \quad\mbox{with }\hat{\mathbf{e}}_\xi=\frac{\partial \mathbf{x}/\partial\xi}{|\partial \mathbf{x}/\partial\xi|},\ \hat{\mathbf{e}}_\theta=\frac{\partial \mathbf{x}/\partial\theta}{|\partial \mathbf{x}/\partial\theta|}. \end{equation} For any $\xi_0\neq 0$, the level curve $\{\xi=\xi_0\}$ is the circle centered at $\mathbf{c}=(c,0)$ with radius $r$ such that \begin{equation}\label{bipolar:circle}c = \alpha \frac{e^{2\xi_0}+1}{e^{2\xi_0}-1},\quad r=\frac{\alpha }{\sinh \xi_0}.\end{equation} From \eqnref{bipolar:z}, the center $\mathbf{c}$ has the bipolar coordinates $(2\xi_0,\pi)$. On this circle, the normal and tangential derivatives of a function $u$ satisfy that \begin{align}\label{nortan} \frac{\partial u}{\partial \nu} \Big|_{\xi = \xi_0} & = -\mbox{sgn} (\xi_0) h(\xi_0,\theta) \frac{\partial u}{\partial \xi} \Big|_{\xi = \xi_0},\\\notag \frac{\partial u}{\partial T} \Big|_{\xi = \xi_0} &= -\mbox{sgn} (\xi_0) h(\xi_0,\theta) \frac{\partial u}{\partial \theta} \Big|_{\xi = \xi_0}, \end{align} where $\nu$ is the outward unit normal vector and $T$ is the rotation of $\nu$ by $\frac{\pi}{2}$-radian angle. We may simply write $\frac{\partial u}{\partial\nu}(\xi_0,\theta)$ for $\frac{\partial u}{\partial\nu}|_{\xi=\xi_0}(\xi_0,\theta)$. The reflection across the circle $\{\xi=\xi_0\}$ is actually a reflection with respect to $\xi$-variable, as follows. \begin{lemma}\label{rbip} For a disk $D$ whose boundary is the coordinate level curve $\{\xi=\xi_0\}$ with some $\xi_0\neq 0$, we have $$R_{\partial D}(\mathbf{x}(\xi, \theta)) = \mathbf{x}(2\xi_0 - \xi,\, \theta),\qquad\mbox{for all }(\xi,\theta)\neq(2\xi_0,\pi).$$\end{lemma} % \begin{proof} Assume that $D$ has the center $\mathbf{c}=(c,0)$ and the radius $r$. From \eqnref{bipolar:circle}, we deduce that $\mathbf{c}=\mathbf{x}(2\xi_0,\pi)$ and $$(c+\alpha)(c-\alpha)=\frac{\alpha^2}{\sinh^2\xi_0}=r^2,\quad e^{2\xi_0}=\frac{c+\alpha}{c-\alpha}.$$ For $z=x+\rm{i}y\neq c+\rm{i}0$, we have \begin{align*} \frac{\alpha + R_{\partial D}(z)}{\alpha - R_{\partial D}(z)}& = \frac{\alpha + \frac{r^2(z-c)}{|z-c|^2} + c}{\alpha - \frac{r^2(z-c)}{|z-c|^2} - c } = - \frac{c+\alpha}{c-\alpha}\, \frac{\frac{(z-c)(c-\alpha)}{|z-c|^2} +1}{\frac{(z-c)(c+\alpha)}{|z-c|^2} +1} \\ & = -\frac{c+\alpha}{c-\alpha}\, \frac{(c-\alpha) + (\bar{z} -c)}{(c+\alpha) + (\bar{z}-c)} = \frac{c+\alpha}{c-\alpha}\, \frac{\alpha-\bar{z}}{\alpha+\bar{z}} =e^{2\xi_0 - \xi +{\rm i}\theta}. \end{align*} We prove the lemma by use of \eqnref{def:bipolar}. \qed \end{proof} \section{Decomposition of the solution into singular and regular terms}\label{sec:series_bipolar} The gradient blow-up of the solution to \eqnref{Eqn} may or may not blow up depending on the applied field $H$. In this section, we show that only the linear term of $H$ contributes to the blow-up feature. This property naturally leads to decomposition of the solution into a singular term (that blows up as $\epsilon$ tends to zero) and a regular term (that remains bounded independently of $\epsilon$); see Theorem \ref{main} for further details. \subsection{Layer potential formulation}\label{sec:layer} Let $D$ be a bounded domain in $\mathbb{R}^2$ with a Lipschitz boundary $\partial D$. We define the single layer potential $\mathcal{S}_{\partial D}$ and the Neumann-Poincar\'{e} operator $\mathcal{K}_{\partial D}^*$ as \begin{align*} \displaystyle\mathcal{S}_{\partial D }[\varphi] (\mathbf{x}) &= \frac{1}{2\pi} \int_{\partial D} \ln |\mathbf{x}-\mathbf{y}| \varphi (\mathbf{y})\, d\sigma (\mathbf{y}), \qquad \mathbf{x} \in \mathbb{R}^2, \\ \displaystyle\mathcal{K}_{\partial D } ^* [\varphi](\mathbf{x})&= \frac{1}{2\pi}\, p.v. \int_{\partial D} \frac{\langle \mathbf{x}-\mathbf{y}, \nu_\mathbf{x} \rangle}{|\mathbf{x}-\mathbf{y}|^2} \varphi(\mathbf{y})\, d\sigma (\mathbf{y}), \qquad \mathbf{x}\in \partial D, \end{align*} where $\varphi \in L^2 (\partial D)$. Here {\it p.v.} means the Cauchy principal value. They satisfy the relation $$\frac{\partial}{\partial \nu} \mathcal{S}_{\partial D} [\varphi]\Big|^\pm = \left( \pm \frac{1}{2} + \mathcal{K}_{\partial D} ^* \right) \varphi\quad \mbox{on }\partial D,$$ where $\nu$ is the outward normal vector on $\partial D$. For a $\mathcal{C}^{1,\alpha}$-domain $D$, the operator $\mathcal{K}^*_{\partial D}$ is compact on $L^2(\partial D)$ and the eigenvalues of $\mathcal{K}^*_{\partial D}$ on $L^2(\partial D)$ lie in $(-1/2, 1/2]$; see \cite{kellog}. The operator $\lambda I-\mathcal{K}^*_{\partial D}$ is invertible on $L^2_0(\partial\Omega)$ for $|\lambda|\geq 1/2$ \cite{book}. For more properties of the Neumann-Poincar\'{e}, see \cite{book2,ammari2007polarization}. The solution $u$ to the conductivity equation \eqnref{Eqn} satisfies the transmission conditions \begin{align*} \pd{u}{\nu}\Big|^+=k_e \pd{u}{\nu}\Big|^-\qquad&\mbox{on }\partial B_e,\\ k_e\pd{u}{\nu}\Big|^+=k_i \pd{u}{\nu}\Big|^-\qquad&\mbox{on }\partial B_i, \end{align*} and it can be expressed as \begin{equation}\label{eqn:BIexpression} u(\bold{x}) = H(\bold{x}) + \mathcal{S}_{\partial B_i} [\varphi_i](\mathbf{x}) + \mathcal{S}_{\partial B_e} [\varphi_e](\mathbf{x}), \end{equation} where $(\varphi_i,\varphi_e)$ are mean-zero functions satisfying \begin{equation} \label{eqn:integraleqn0} \left(\begin{array} {cc} \displaystyle\lambda_i I-\mathcal{K}_{\partial B_i}^* & \displaystyle -\frac{\partial \mathcal{S}_{\partial B_e}}{\partial \nu_i} \\[2mm] \displaystyle -\frac{\partial \mathcal{S}_{\partial B_i}}{\partial \nu_e} & \lambda_e I-\mathcal{K}^*_{\partial B_e}\end{array} \right) \left( \begin{array}{c} \varphi_i \\ \varphi_e \end{array} \right)= \left(\begin{array} {c} \displaystyle \frac{\partial H}{\partial \nu} \Bigr|_{\partial B_i} \\[3mm] \displaystyle\frac{\partial H}{\partial \nu} \Bigr|_{\partial B_e} \end{array} \right) \end{equation} with \begin{equation}\label{lambda_ie} \lambda_i = \frac{k_i + k_e}{2(k_i - k_e)},\quad\displaystyle \lambda_e = \frac{k_e+1}{2(k_e-1)}. \end{equation} Here, $\nu_i$ and $\nu_e$ denote the outward unit normal vectors on $\partial B_i$ and $\partial B_e$, respectively. Note that $|\lambda_i|,|\lambda_e|>\frac{1}{2}$ due to the positive assumption on the conductivities $k_i,k_e$. The problem \eqnref{eqn:BIexpression} is solvable for $|\lambda_i|,|\lambda_e|>\frac{1}{2}$ as shown in \cite{ammari2013spectral}. Since $B_i$ and $B_e$ are disks, $\mathcal{K}^*_{\partial B_i}$ and $\mathcal{K}^*_{\partial B_e}$ are zero operators on $L^2_0(\partial B_i)$ and $L^2_0(\partial B_e)$, respectively. So, the system of boundary integral equations \eqnref{eqn:integraleqn0} becomes \begin{equation} \label{eqn:intdisks} \left(\begin{array} {cc} \displaystyle\lambda_i I & \displaystyle -\frac{\partial \mathcal{S}_{\partial B_e}}{\partial \nu_i} \\[2mm] \displaystyle - \frac{\partial \mathcal{S}_{\partial B_i}}{\partial \nu_e} & \lambda_e I\end{array} \right) \left( \begin{array}{c} \varphi_i \\ \varphi_e \end{array} \right)= \left(\begin{array} {c} \displaystyle ~\frac{\partial H}{\partial \nu} \Bigr|_{\partial B_i} \\[3mm] \displaystyle \frac{\partial H}{\partial \nu} \Bigr|_{\partial B_e} \end{array} \right). \end{equation} We can relate the tangential component and the normal component of the electric field on $\partial B_i$ and $\partial B_e$. The following lemma will be used later. One can prove the lemma by slightly modifying the proof of \cite[Theorem 2]{AKL}. \begin{lemma}\label{lemma:tangential}Let $u$ be the solution to \eqnref{Eqn} with an entire harmonic function $H$. Let $\widetilde{H}$ be the harmonic conjugate of $H$ and $\tilde{\sigma}= \frac{1}{k_i} \chi(B_i) + \frac{1}{k_e} \chi(B_e \setminus B_i) + \chi(\mathbb{R}^2 \setminus B_e)$. Then, the solution $v$ to the problem \begin{equation*} \left\{ \begin{array}{ll} \displaystyle \nabla \cdot \tilde{\sigma} \nabla v = 0 \qquad& \textrm{in } \mathbb{R}^2, \\ \displaystyle v(\bold{x}) - \widetilde{H}(\bold{x}) =O(|\bold{x}|^{-1})\qquad & \textrm{as } |\bold{x}| \to \infty \end{array} \right. \end{equation*} satisfies that $$ \frac{\partial u}{\partial T} = - \frac{\partial v}{\partial \nu} \Big|^{+} \textrm { on } \partial B_e, ~~~ \frac{\partial u}{\partial T} = -\frac{1}{ k_e} \frac{\partial v}{\partial \nu}\Big|^{+} \textrm { on } \partial B_i. $$ \end{lemma} \subsection{Series expansions for the density functions in bipolar coordinates}\label{sec:densities} The single layer potential associated with a disk can be written in terms of the reflection with respect to the disk's boundary. % \begin{lemma}[\cite{AKL}]\label{SLDisk}Let $D$ be a disk centered at $\mathbf{c}$ and $\nu$ be the outward unit normal vector on $\partial D$. If $v$ is harmonic in $D$ and continuous on $\overline{D}$, then we have $$\mathcal{S}_{\partial D} \left[\frac{\partial v}{\partial \nu} \Big|_{\partial D}^- \right] (\bold{x}) = \begin{cases} \displaystyle -\frac{1}{2} v(\bold{x}) + \frac{v(\mathbf{c})}{2} \quad& \mbox{for }\bold{x} \in D,\\[3mm] \displaystyle -\frac{1}{2} R_{\partial D} [v ](\bold{x}) + \frac{v(\mathbf{c})}{2} \quad&\mbox{for } \bold{x} \in \mathbb{R}^2 \setminus \overline{D}. \end{cases} $$ If $v$ is harmonic in $\mathbb{R}^2 \setminus \overline{D}$, continuous on $\mathbb{R}^2 \setminus D$ and $v(\bold{x}) \to 0$ as $|\bold{x}| \to \infty$, then we have $$\mathcal{S}_{\partial D} \left[\frac{\partial v}{\partial \nu} \Big|_{\partial D}^+ \right] (\bold{x}) = \left\{ \begin{array}{ll} \displaystyle\frac{1}{2} R_{\partial D}[ v ] (\bold{x})\quad&\mbox{for } \bold{x} \in D,\\[3mm] \displaystyle\frac{1}{2} v(\bold{x}) \quad& \mbox{for }\bold{x} \in \mathbb{R}^2 \setminus \overline{D}. \end{array} \right. $$ \end{lemma} % % We now express the density functions $\varphi_i$ and $\varphi_e$ in \eqnref{eqn:BIexpression} in series form by use of Lemma \ref{rbip}. \begin{lemma}\label{lemma:density} The density functions in \eqnref{eqn:BIexpression} can be expressed as \begin{equation} \displaystyle \label{expan:phii} \begin{cases} \displaystyle\lambda_i\varphi_i(\theta) = \left(1-\frac{1}{2\lambda_e}\right) \sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}\frac{\partial H}{\partial\nu}(\xi_{i,m},\theta),\\[3mm] \displaystyle \lambda_e\varphi_e(\theta) = \frac{\partial H}{\partial \nu}(\xi_e,\theta) +\frac{1}{2\lambda_i} \left(1-\frac{1}{2\lambda_e}\right) \sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{h(\xi_e,\theta)}{h(\xi_{e,m+1},\theta)}\frac{\partial H}{\partial \nu} (\xi_{e,m+1},\theta). \end{cases} \end{equation} \end{lemma} % \begin{proof} First, we formally expand the solution $(\varphi_i,\varphi_e)$ to \eqnref{eqn:intdisks} as follows (uniform and absolute convergence will be proved later): \begin{equation} \label{phii:reflection} \begin{cases} \displaystyle\lambda_i\varphi_i = \left(1-\frac{1}{2\lambda_e}\right)\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{\partial}{\partial \nu_i}\big( (R_e R_i)^m H \big)\Big|_{\partial B_i}, \\[3mm] \displaystyle\lambda_e \varphi_e =\frac{\partial H}{\partial \nu_e} \Big|_{\partial B_e} - \frac{1}{2\lambda_i} \left(1-\frac{1}{2\lambda_e}\right)\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{\partial}{\partial \nu_e} \big((R_i R_e)^m R_i H \big)\Big|_{\partial B_e}. \end{cases} \end{equation} Using Lemma \ref{SLDisk}, we compute \begin{align*} \displaystyle\frac{\partial \mathcal{S}_{\partial B_i} [\varphi_i]}{\partial \nu_e} & = \frac{1}{\lambda_i} \left(1-\frac{1}{2\lambda_e}\right)\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{\partial}{\partial \nu_e} \mathcal{S}_{\partial B_i}\left[ \frac{\partial}{\partial \nu_i} \big((R_e R_i)^m H \big)\Big|_{\partial B_i}\right] \\ \displaystyle&= -\frac{1}{2 \lambda_i} \left(1-\frac{1}{2\lambda_e}\right)\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{\partial}{\partial \nu_e} \big(R_i(R_e R_i)^m H\big) \Big|_{\partial B_i}\\ \displaystyle&= \lambda_e \varphi_e - \frac{\partial H}{\partial \nu_e}\Big|_{\partial B_e}, \end{align*} and \begin{align*} \displaystyle\frac{\partial \mathcal{S}_{\partial B_e} [\varphi_e]}{\partial \nu_i} & = \frac{1}{\lambda_e} \frac{\partial}{\partial \nu_i} \mathcal{S}_{\partial B_e} \left[\frac{\partial H}{\partial \nu_e} \Big|_{\partial B_e} \right] \\& \quad- \frac{1}{2\lambda_i \lambda_e} \left(1-\frac{1}{2\lambda_e}\right)\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{\partial}{\partial \nu_i} \mathcal{S}_{\partial B_e} \left[ \frac{\partial}{\partial \nu_e}\big((R_i R_e)^m R_i H\big) \Big|_{\partial B_e}\right] \\ \displaystyle&= -\frac{1}{2\lambda_e} \frac{\partial H}{\partial \nu_i} \Big|_{\partial B_i} - \frac{1}{4\lambda_i \lambda_e} \left(1-\frac{1}{2\lambda_e}\right)\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{\partial}{\partial \nu_i}\big((R_i R_e)^{m+1} H\big) \Big|_{\partial B_i}\\ \displaystyle&= \lambda_i \varphi_i - \frac{\partial H}{\partial \nu_i}\Big|_{\partial B_i}. \end{align*} Therefore the series solutions in \eqnref{phii:reflection} satisfy \eqnref{eqn:intdisks}. For the entire harmonic function $H$, we have from Lemma \ref{rbip} that \begin{align*} \displaystyle R_e R_i H(\xi,\theta) &= H(R_i \circ R_e(\xi,\theta)) = H(2(\xi_i - \xi_e)+\xi,\theta), \\ \displaystyle R_i R_e R_i H(\xi,\theta) &= H(R_i \circ R_e\circ R_i(\xi,\theta)) = H(2(\xi_i - \xi_e)+2\xi_i-\xi,\theta). \end{align*} Repeating the reflections, it follows that \begin{align*} \displaystyle (R_e R_i)^m H(\xi,\theta)&= H(2m(\xi_i-\xi_e)+\xi,\theta),\\ \displaystyle (R_i R_e)^m R_i H(\xi,\theta) &= H(2m(\xi_i-\xi_e)+2\xi_i-\xi,\theta). \end{align*} Recall that $\xi_{i,m}$ and $\xi_{e,m+1}$ are given as \eqnref{def:xik}. By use of \eqnref{nortan} we deduce that \begin{align*} \frac{\partial}{\partial \nu_i}\big( (R_e R_i)^m H \big)\Big|_{\partial B_i}& = -h(\xi_i,\theta)\frac{\partial}{\partial \xi}\left(H\left(2m(\xi_i-\xi_e)+\xi,\theta\right)\right)\Big|_{\xi=\xi_i} \\ &= \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}\frac{\partial H}{\partial\nu} (\xi_{i,m},\theta),\\[1mm] \frac{\partial}{\partial \nu_e}\big((R_i R_e)^m R_i H \big)\Big|_{\partial B_e} &= -h(\xi_e,\theta)\frac{\partial}{\partial \xi}\left(H\left(2m(\xi_i-\xi_e)+2\xi_i-\xi,\theta\right)\right)\Big|_{\xi=\xi_e} \\ &= -\frac{h(\xi_e,\theta)}{h(\xi_{e,m+1},\theta)}\frac{\partial H}{\partial \nu} (\xi_{e,m+1},\theta). \end{align*} Therefore, the equation \eqnref{phii:reflection} becomes \eqnref{expan:phii}. Now, we show uniform and absolute convergence for the series expansions. Recall that $\xi_{i,m}\geq\xi_i$ and $\xi_{e,m+1}\geq \xi_e$. Since $\mathbf{x}(\xi_{i,m},\theta),\mathbf{x}(\xi_{e,m+1},\theta)\in B_e$ for all $m$ and $$ \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}\leq 1,\quad \frac{h(\xi_e,\theta)}{h(\xi_{e,m+1},\theta)}\leq1,$$ we have \begin{align*} \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)} \left|\frac{\partial H}{\partial \nu}(\xi_{i,m},\theta)\right| &\le \Vert\nabla H(\bold{x})\Vert_{L^\infty(B_e)},\\[1mm] \frac{h(\xi_e,\theta)}{h(\xi_{e,m+1},\theta)} \left|\frac{\partial H}{\partial \nu}(\xi_{e,m+1},\theta)\right|& \le \Vert\nabla H(\bold{x})\Vert_{L^\infty(B_e)}. \end{align*} Note that $$\left| \frac{1}{4\lambda_i \lambda_e } \right| < 1$$ due to $|\lambda_i|,|\lambda_e|>\frac{1}{2}$. Therefore, each term in \eqnref{phii:reflection} (in other words, each term in \eqnref{expan:phii}) is bounded by $\Vert \nabla H(\bold{x})\Vert_{L^\infty(B_e)}\left| \frac{1}{4\lambda_i \lambda_e } \right|^m$. This implies convergence and so the proof is completed. \end{proof} \begin{cor}\label{boundedness1} If $k_i$ and $k_e$ are bounded but not small, then $|\nabla(u-H)|$ is uniformly bounded in $\mathbb{R}^2$ independently of $\epsilon$. \end{cor} \begin{proof} From Lemma \ref{lemma:density} and the analysis in its proof, we have $$\|\varphi_i\|_{L^\infty(\partial B_i)}, \|\varphi_e\|_{L^\infty(\partial B_e)} \leq 6\frac{\|\nabla H\|_{L^\infty(B_e)}}{1-\frac{1}{|4\lambda_i\lambda_e|}}.$$ The jump formula for the single layer potential and \eqnref{eqn:intdisks} implies \begin{equation}\label{jump} \left\Vert \frac{\partial u}{\partial \nu} \Big|^{\pm} \right\Vert_{L^\infty (\partial B_i)} = \left|\lambda_i \pm \frac{1}{2} \right| \Vert \varphi_i \Vert_{L^\infty (\partial B_i)}. \end{equation} Therefore we have uniform boundedness for normal derivatives of $u$ on $\partial B_i$ and $\partial B_e$. We can also prove uniform boundedness for tangential derivatives by using a harmonic conjugate of $u$, in view of Lemma \ref{lemma:tangential}. Since $(u-H)(\mathbf{x})$ satisfies the decay condition as $|\mathbf{x}|\rightarrow\infty$, $|\nabla(u-H)|$ is uniformly bounded in $\mathbb{R}^2$ independently of $\epsilon$ by the maximum principle. \end{proof} \subsection{Blow-up contribution term in $H$ and decomposition of the solution}\label{subsec:decomposition} We prove the following upper bounds of two series sums related with the scale factor function $h$, which is essential for showing that only the linear term of $H$ contributes to the gradient blow-up. \begin{lemma} \label{coshsum} Let $N_\epsilon = \frac{r_*}{\sqrt{\epsilon}}$, then we have \begin{equation*} \begin{cases} \displaystyle\sum_{m=0}^{N_\epsilon} \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)} \le \frac{C}{|z(\xi_i,\theta)|} \qquad\mbox{for all }|\theta|>\frac{\pi}{2},\\[4mm] \displaystyle\sum_{m=0}^{N_\epsilon} \frac{h(\xi_e,\theta)}{h(\xi_{e,m+1},\theta)} \le \frac{C}{|z(\xi_e,\theta)|} \qquad\mbox{for all }|\theta|>\frac{\pi}{2}, \end{cases} \end{equation*} where $C$ is a constant independent of $\epsilon,\,\theta$. \end{lemma} \begin{proof} We prove the first inequality only. The second one can be shown in the same way. Note that \begin{align} \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}&=\frac{\cosh \xi_i + \cos \theta}{\cosh \xi_{i,m} + \cos \theta} = \frac{(\cosh \xi_i + \cos \theta)}{(\cosh \xi_i - \cos \theta)}\frac{(\cosh \xi_{i,m} - \cos \theta)}{(\cosh \xi_{i,m} + \cos \theta)}\frac{(\cosh \xi_i - \cos \theta)}{(\cosh \xi_{i,m} - \cos \theta)} \notag\\\label{eqn:ratio} & \le \frac{(\cosh \xi_i + \cos \theta)}{\alpha^2 (\cosh \xi_i - \cos \theta)}\frac{\alpha^2 (\cosh \xi_{i,m} - \cos \theta)}{(\cosh \xi_{i,m} + \cos \theta)} = \left|\frac{z(\xi_{i,m},\theta)}{z(\xi_i,\theta)}\right|^2 . \end{align} We use \eqnref{def:bipolar}, to estimate the last term in the above equation, and obtain $$\frac{\alpha+z(\xi_{i,m},\theta)}{\alpha-z(\xi_{i,m},\theta)} =e^{\xi_{i,m}+\rm{i}\theta}=e^{2m(\xi_i-\xi_e)}e^{\xi_i+\rm{i}\theta}=\gamma^m\frac{\alpha+z(\xi_i,\theta) }{\alpha-z(\xi_i,\theta) }$$ with $\gamma =e^{2(\xi_i - \xi_e)}$. This implies \begin{align*} z(\xi_{i,m},\theta) &=\alpha\frac{z(\xi_i,\theta)(\gamma^m+1)+\alpha(\gamma^m-1)}{z(\xi_i,\theta)(\gamma^m-1)+\alpha(\gamma^m+1)}= \frac{\displaystyle 1+\frac{\alpha}{z(\xi_i,\theta)}\frac{\gamma^m - 1}{\gamma^m + 1}}{\displaystyle \frac{1}{\alpha}\frac{\gamma^m -1}{\gamma^m +1} + \frac{1}{z(\xi_i,\theta)}}. \end{align*} From \eqnref{def:rstar} and \eqnref{bipolar:circle}, we deduce $\xi_i = \frac{r_*}{r_i}\sqrt{\epsilon}+O(\epsilon\sqrt{\epsilon})$ and $\xi_e = \frac{r_*}{r_e}\sqrt{\epsilon}+O(\epsilon\sqrt{\epsilon})$ so that \begin{equation}\label{xidiff} \xi_i-\xi_e=\frac{2}{r_*}\sqrt{\epsilon}+O(\epsilon\sqrt{\epsilon}) \end{equation} and $$1+\frac{3}{r_*} \sqrt{\epsilon} < \gamma = e^{2(\xi_i - \xi_e)}< 1+\frac{5}{r_*} \sqrt{\epsilon}.$$ Hence, by using \eqnref{thetalevel} as well, we have for $1 \le m < \frac{r_*}{\sqrt{\epsilon}}$ and $|\theta|>\frac{\pi}{2}$ that $$1+\frac{2m}{r_*}\sqrt{\epsilon}<\gamma^m < \left( 1+\frac{5}{r_*}\sqrt{\epsilon}\right)^{ \frac{r_*}{\sqrt{\epsilon}}}< e^5$$ and $$|z(\xi_{i,m},\theta)| \leq\frac{1+\frac{\alpha}{|z(\xi_i,\theta)|}\frac{ e^5-1}{2}}{\left|\frac{2m\sqrt{\epsilon}/r_*}{\alpha(e^5+1)} + \frac{1}{z(\xi_i,\theta)} \right|} \leq \frac{C_1}{C_2 m+ \frac{1}{|z(\xi_i,\theta)|}}$$ with some positive constants $C_1$ and $C_2$ independent of $\epsilon$, $\theta$ and $m$. Hence, we have \begin{equation}\label{ineq:zxi} \left|\frac{z(\xi_{i,m},\theta)}{z(\xi_i,\theta)}\right|\leq \frac{C}{ m|z(\xi_i,\theta)| + 1}\qquad \mbox{for }1 \le m < \frac{r_*}{\sqrt{\epsilon}},\ |\theta|>\frac{\pi}{2}. \end{equation} Note that for any $t>0$, the following holds: \begin{align*} \sum_{m=0}^\infty \frac{1}{ (m t + 1)^2} &\le 1+\int_0^\infty\frac{1}{(xt+1)^2}\,dx=1+\frac{1}{t}. \end{align*} This completes the proof because of \eqnref{eqn:ratio}, \eqnref{ineq:zxi} and the boundedness of $|z(\xi_i,\theta)|$. \end{proof} \smallskip Now, we show that $|\nabla (u-H)|$ is bounded regardless of $\epsilon$ if $\nabla H(\mathbf{x})=O(|\mathbf{x}|)$ near the origin. % \begin{lemma}\label{lemma:boundedness} If $\nabla H(\mathbf{x})=O(|\mathbf{x}|)$ in a bounded region containing $B_e$, then we have $$\lVert \nabla (u-H)\rVert_{L^\infty (\mathbb{R}^2)} \le C$$ for some constant $C$ independent of $k_i,\, k_e$ and $\epsilon$. \end{lemma} % \begin{proof} Let $\nabla H$ denote the gradient with respect to the Cartesian coordinates, and $\xi_{i,m},\xi_{e,m+1}$ are defined as \eqnref{def:xik}. We first show that \begin{equation}\label{IandII}\sum_{m=0} ^{\infty} \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)} \big| \nabla H(\xi_{i,m},\theta)\big|\leq C. \end{equation} Here and after, $C$ denotes a constant independent of $\epsilon$, $\theta$ and $m$. For notational simplicity we decompose the series in \eqnref{IandII} as $$\sum_{m=0} ^{\infty} \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)} \big| \nabla H(\xi_{i,m},\theta)\big| = \sum_{m \le \frac{r_*}{\sqrt{\epsilon}}} + \sum_{m >\frac{r_*}{\sqrt{\epsilon}}}:= ~ I + II$$ and separately prove the uniform boundedness. For $|\theta|\leq\frac{\pi}{2}$, we have from \eqnref{thetalevel} and the assumption $\nabla H(\mathbf{x})=O(|\mathbf{x}|)$ in $B_e$ that $$I\leq C\frac{r_*}{\sqrt{\epsilon}}\big|z(\xi_{i,m},\theta)\big|\leq C\frac{r_*}{\sqrt{\epsilon}}\alpha=O(1).$$ For $|\theta|>\frac{\pi}{2}$, we have from Lemma \ref{coshsum} and \eqnref{ineq:zxi} $$I\leq \frac{C}{|z(\xi_i,\theta)|}\max\left\{\big|z(\xi_{i,m},\theta)\big|:m \le \frac{r_*}{\sqrt{\epsilon}}\right\}=O(1).$$ We now estimate $II$. For $m>\frac{r_*}{\sqrt{\epsilon}}$, we have $\xi_{i,m}>1$ due to \eqnref{xidiff} so that $$\big|z(\xi_{i,m},\theta)\big| < C\alpha$$ and $$\frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}=\frac{\cosh\xi_i+\cos\theta}{\cosh \xi_{i,m}+\cos\theta} \leq \frac{\cosh\xi_i}{\frac{1}{2}\cosh\xi_{i,m}}\leq 4 e^{\xi_i-\xi_{i,m}}=4e^{-2m(\xi_i-\xi_e)}.$$ Therefore, it follows that $$ I \leq \sum_{m>\frac{r_*}{\sqrt{\epsilon}}}4e^{-2m(\xi_i-\xi_e)}C\big|z(\xi_{i,m},\theta)\big|\leq \frac{C\alpha}{\xi_i-\xi_e}= O(1). $$ In the same way, one can easily show that \begin{equation}\label{IandII2} \sum_{m=0} ^{\infty} \frac{h(\xi_i,\theta)}{h(\xi_{e,m+1},\theta)} \left| \nabla H(\xi_{e,m+1},\theta)\right|\leq C. \end{equation} From \eqnref{IandII}, \eqnref{IandII2} and Lemma \ref{lemma:density} we obtain \begin{align*} |\varphi_i| < \frac{C}{\lambda_i} \left( 1-\frac{1}{2\lambda_e} \right), \quad |\varphi_e| < \frac{C}{\lambda_e} \end{align*} and, thus, \begin{align*} \left\Vert \frac{\partial u}{\partial \nu} \bigg|_{\pm} \right\Vert_{L^\infty (\partial B_i)} &= \left|\lambda_i \pm \frac{1}{2} \right| \Vert \varphi_i \Vert_{L^\infty (\partial B_i)} \le C. \end{align*} The same result holds on $\partial B_e$. We can also prove uniform boundedness for tangential derivatives by using a harmonic conjugate for the harmonic function; see Lemma \ref{lemma:tangential}. Following the same argument in the proof of cor \ref{boundedness1}, we have uniform boundedness for $|\nabla(u-H)|$ in $\mathbb{R}^2$ independently of $\epsilon$. \end{proof} % % As a direct consequence of Lemma \ref{lemma:boundedness}, we can decompose the solution into a singular term and a regular term as follows. \begin{theorem}\label{main} Let $H$ be an arbitrary entire harmonic function. Set $H_1(\mathbf{x}) = \nabla H(\mathbf{0})\cdot \mathbf{x}$ and $H_2(\mathbf{x})=H(\mathbf{x})-H_1(\mathbf{x})$. Then, the solution $u$ to the transmission problem \eqnref{Eqn} has the decomposition \begin{equation} u(\mathbf{x})= u_1(\mathbf{x})+ u_2(\mathbf{x}),\quad\mathbf{x}\in\mathbb{R}^2, \end{equation} where $u_1$ and $u_2$ are the solutions to \eqnref{Eqn} with $H_1$ and $H_2$ in the place of $H$, respectively, and $\Vert \nabla (u_2 -H_2)(\mathbf{x})\Vert_{L^\infty(\mathbb{R}^2)} < C$ for some constant $C$ independent of $\epsilon$. \end{theorem} \begin{cor} If $\lambda_i\lambda_e>\frac{1}{4}$ (equivalently, either $k_i>k_e>1$ or $k_i<k_e<1$), then for any entire function $H$ we have $$\lVert \nabla (u-H)\rVert_{L^\infty (\mathbb{R}^2)} \le C$$ with some positive constant $C$ independent of $\epsilon$. \end{cor} \begin{proof} Thanks to Theorem \ref{main}, we may consider only the linear function for $H$, say $H(x,y)=ax+by$. Then, from \eqnref{nortan} we have \begin{align*} &\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}\frac{\partial H}{\partial\nu} (\xi_{i,m},\theta) \\ =&-\sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m h(\xi_i,\theta)\left(a\pd{x}{\xi}+b\pd{y}{\xi}\right)\Big|_{\xi=\xi_{i,m}}. \end{align*} Note that from \eqnref{eqn:xdxi}, $\pd{x}{\xi}$, $\pd{y}{\xi}$ and their derivatives in $\xi$ have at most a finite number of changes in sign when $\theta$ is fixed. Moreover, $$h(\xi_i,\theta)\Big|\pd{x}{\xi}(\xi_{i,m},\theta)\Big|, \, h(\xi_i,\theta)\Big|\pd{y}{\xi}(\xi_{i,m},\theta)\Big|\leq \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}\leq 1.$$ Therefore, from the convergence property of alternating series we derive that \begin{equation}\notag \left| \sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{h(\xi_i,\theta)}{h(\xi_{i,m},\theta)}\frac{\partial H}{\partial\nu} (\xi_{i,m},\theta) \right| \le C \left| \nabla H(\bold{0}) \right| \end{equation} and, in the same way, \begin{equation}\notag \left| \sum_{m=0} ^{\infty} \left(-\frac{1}{4\lambda_i \lambda_e } \right)^m \frac{h(\xi_e,\theta)}{h(\xi_{e,m+1},\theta)}\frac{\partial H}{\partial\nu} (\xi_{e,m+1},\theta) \right| \le C \left| \nabla H(\bold{0}) \right|. \end{equation} Hence, $\varphi_i, \varphi_e$ are bounded independent of $\epsilon$. Following the same argument in cor \ref{boundedness1} we have uniform boundedness for $|\nabla(u-H)|$. \end{proof} For the case of separated disks, a boundedness result comparable to Lemma \ref{lemma:boundedness} was derived in \cite{AKLLZ}, based on the series expression in the Cartesian coordinates that is similar to \eqnref{phii:reflection}. We emphasize that in this paper, we provide a much simpler proof thanks to Lemma \ref{coshsum}. \section{Asymptotics of the singular term in bipolar coordinates}\label{sec:bipolarapprox} From the analysis in the previous section, the gradient blow-up does not occur either when $\nabla H(\mathbf{x}) = O(|\mathbf{x}|)$ or when $\lambda_i \lambda_e>\frac{1}{4}$ (equivalently, either $k_i>k_e>1$ or $k_i<k_e<1$). In this section, we first derive an asymptotic expansion for the solution to \eqnref{Eqn} assuming that $H$ is a linear function and $\lambda_i \lambda_e<-\frac{1}{4}$ (equivalently, either $k_i>k_e,\, 1>k_e$ or $k_i<k_e,\,1<k_e$). We then characterize the gradient blow-up. We follow the derivation in \cite{LYu2D}, where the asymptotic behavior for the electric field was derived in the presence of two separated circular conductors. The gradient blow-up feature is essentially related to the following quantities: \begin{equation} \label{lambdabeta} \tau_i=\frac{k_i -k_e}{k_i+k_e},\quad \tau_e = \frac{1-k_e}{1+k_e},\quad \tau = \tau_i\tau_e,\quad \beta = \frac{r_*(-\ln \tau)}{4\sqrt{\epsilon}}, \end{equation} and \begin{equation}\label{eqn:CxCy} C_x = \left( \nabla H(\bold{0}) \cdot \mathbf{e}_1 \right),\quad C_y = \left( \nabla H(\bold{0}) \cdot \mathbf{e}_2\right), \end{equation} where $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$. Note that $\tau_i = \frac{1}{2\lambda_i},\ \tau_e=-\frac{1}{2\lambda_e}$ and $\beta>0$. \subsection{Series solution by the separation of variables} In bipolar coordinates, harmonic functions admit the general solution expansion $$f(\xi, \theta) = a_0 + b_0 \xi + c_0 \theta + \sum_{n=1} ^{\infty} \left[(a_n e^{n\xi} + b_n e^{-n\xi} )\cos n\theta + (c_n e^{n\xi} + d_n e^{-n\xi}) \sin n\theta\right],$$ where $a_n, b_n, c_n$ and $d_n$ are some constants. The linear functions $x$ and $y$ can be expanded as \begin{align*} x &= \mbox{sgn}(\xi) \alpha \left[ 1+ 2 \sum_{n=1} ^{\infty} (-1)^n e^{-n|\xi|}\cos n\theta \right], \\ y &= -2 \alpha \sum_{n=1} ^{\infty} (-1)^n e^{-n |\xi|} \sin n \theta. \end{align*} % \begin{lemma}\label{lemma:transmissionseries} We define \begin{equation*} U(\bold{x}) := C+ \begin{cases} \displaystyle \sum_{n=1} ^{\infty} \left(A_n e^{n(\xi - \rm{i}\theta -2\xi_i )} + B_n e^{n(\xi - \rm{i}\theta-2\xi_e )} \right)\quad& \mbox{for }\xi<\xi_e,\\[1mm] \displaystyle \sum_{n=1} ^{\infty} \left(A_n e^{n(\xi - \rm{i}\theta -2\xi_i )} + B_n e^{n(-\xi - \rm{i}\theta)} \right) \quad& \mbox{for }\xi_e<\xi<\xi_i,\\[1mm] \displaystyle \sum_{n=1} ^{\infty} \left(A_n e^{n(- \xi - \rm{i}\theta)} + B_n e^{n(-\xi - \rm{i}\theta)}\right) \quad& \mbox{for }\xi>\xi_i \end{cases} \end{equation*} with \begin{align*} A_n& =\frac{2\alpha (-1)^{n}}{\tau^{-1} e^{2n(\xi_i - \xi_e)}-1}\left( - 1 -\tau_e^{-1} \right) e^{2n(\xi_i-\xi_e)}, \\ B_n& = \frac{2\alpha (-1)^{n}}{\tau^{-1} e^{2n(\xi_i - \xi_e)}-1}\left(1 + \tau_i^{-1}{ e^{2n(\xi_i - \xi_e)}}\right), \end{align*} and $ C=-\sum_{n=1} ^\infty A_n e^{n(-2\xi_i-{\rm i}\pi)} + B_n e^{-{\rm i}n\pi}$. Then, the functions $x+\Re\{U(\bold{x})\}$ and $y+\Im\{U(\bold{x})\}$ are the solutions to \eqnref{Eqn} for $H(\bold{x})=x$ and $H(\bold{x})=y$, respectively. \end{lemma} % \begin{proof} One can easily show that $x+\Re\{U(\bold{x})\}$ and $y+\Im\{U(\bold{x})\}$ satisfy the transmission condition in the equation \eqnref{Eqn} and demonstrate uniform convergence. We can prove the decay property in the same way as in Lemma 3.1 of \cite{LYu2D}. \end{proof} \begin{definition} The Lerch transcendent function is defined as \begin{equation}\label{eqn:Lerch} L(z;\beta)= -\int_0 ^\infty \frac{ze^{-(\beta+1)t}}{1+z e^{-t}} dt \qquad \mbox{for } z\in \mathbb{C}, |z| <1. \end{equation} % We also define $$ P(z;\beta)= (-z) \frac{\partial}{\partial z} L(z;\beta) = \int_0 ^\infty \frac{ze^{-(\beta+1)t}}{(1+z e^{-t})^2} dt.$$ \end{definition} Let us derive some properties on $P$ for later use. By the integration by parts it follows that for all $s>0$, \begin{align} P(e^{-s-\rm{i}\theta};\beta) &=\int_0^\infty e^{-\beta t}\frac{e^{-s-t-\rm{i}\theta}}{(1+e^{-s-t-\rm{i}\theta})^2} \,dt\notag\\\label{P:integral} &=\int_0^\infty e^{-\beta t}\frac{1+\cosh (s+t) \cos \theta - \rm{i} \sinh (s+t) \sin \theta}{2(\cosh (s+t) + \cos \theta)^2}\,dt. \end{align} One can easily show that \begin{align} \label{Pproperty1} \left|P(e^{-s+\rm{i}\theta }) \right|& \le \int_0^\infty e^{-\beta t} \frac{1}{2(\cosh (s+t) + \cos \theta)} \,dt\leq \frac{1}{2\beta(\cosh s + \cos \theta)}. \end{align} For any $s_2>s_1>0,\, \beta>0$ and $|\theta|\leq\pi$, the following holds: \begin{equation}\label{Pproperty2} \left|P(e^{-s_2+\rm{i}\theta};\beta)-P(e^{-s_1+\rm{i}\theta};\beta)\right|\leq \frac{s_2-s_2}{2(\cosh s_1+\cos\theta)}. \end{equation} The following integral approximation for an infinite series sum is crucial to obtaining the gradient blow-up term of $u$. \begin{lemma}[\cite{LYu2D}]\label{lemma:LYu2D:series} Let $0<a<a_0$ and $0<\tau<1$. For arbitrary $\theta \in (-\pi, \pi]$, we have $$\left| a_0 \sum_{m=1} ^{\infty}\left( \tau^{m-1} \frac{e^{-ma_0 + a + \rm{i}\theta}}{(1+e^{-ma_0 + a + \rm{i}\theta})^2}\right) - P\bigr(e^{-(a_0 - a)+\rm{i}\theta};\, \frac{-\ln{\tau}}{a_0}\big) \right| \le \frac{4a_0}{\cosh(a_0-a) + \cos \theta}.$$ \end{lemma} \subsection{Integral expression of the singular term in bipolar coordinates} We verify the asymptotic behavior of $u$ in bipolar coordinates as follows. We will later express the gradient blow-up term of $u$ in terms of the Cartesian coordinates in section \ref{sec:cartesian}. % \begin{theorem} \label{blowupterm} Let $u$ be the solution to \eqnref{Eqn}. Then we have the following: \begin{itemize} \item[\rm(a)] The solution $u$ admits that \begin{equation}\label{u:singular} u(\mathbf{x}) = H(\mathbf{x}) + C_x r_*^2\Re\big\{q\big(\mathbf{x};\beta,\tau_i, \tau_e\big)\big\} + C_y r_*^2\Im\big\{q\big(\mathbf{x};\beta,\tau_i, \tau_e\big)\big\} + r(\mathbf{x}),\quad \mathbf{x}\in\mathbb{R}^2, \end{equation} where {\it the singular function} $q$ is \begin{align*} \label{eqn:asymptotic} &q(\bold{x};\beta,\tau_i, \tau_e) \\ &:= \frac{1}{2} \begin{cases} \displaystyle-\big(\tau+ \tau_i\big)L(e^{\xi-\rm{i}\theta-2\xi_i} ;\beta)+ \big(\tau+\tau_e\big)L(e^{\xi-\rm{i}\theta-2\xi_e};\beta) \quad& \textrm{in } \mathbb{R}^2 \setminus B_e, \\[.5mm] \displaystyle- \big(\tau+ \tau_i\big)L(e^{\xi-\rm{i}\theta-2\xi_i} ;\beta)+ \big(\tau+\tau_e\big)L(e^{-\xi-\rm{i}\theta};\beta) \quad& \textrm{in } B_e \setminus B_i, \\[.5mm] \displaystyle- \big(\tau+ \tau_i\big)L(e^{-\xi-\rm{i}\theta} ;\beta) + \big(\tau+\tau_e\big)L(e^{-\xi-\rm{i}\theta};\beta) \quad& \textrm{in } B_i, \end{cases} \end{align*} and $\Vert \nabla r(\mathbf{x}) \Vert_{L^\infty(\mathbb{R}^2)} < C$ for some constant $C$ independent of $\epsilon$.\\ \item[\rm(b)] For any $k_i,k_e$, we have that $\|\nabla (u-H)\|_{L^\infty(\mathbb{R}^2\setminus\overline{(B_e\setminus B_i)})}\leq C$ for some constant $C$ independent of $\epsilon$. The gradient blow-up occurs only when $\tau_i,\tau_e$ are similar to $1$ (equivalently, $0<k_e\ll 1,k_i$). For such a case, we have that in $B_e\setminus B_i$, \begin{align*} &\nabla (u-H)(\bold{x})\\ &= \displaystyle \frac{C_x r_*(\tau_i+\tau_e+2\tau)}{2 \sqrt{\epsilon}}\big(\cosh \xi + \cos\theta\big) \Re\left\{P(e^{-\xi - \rm{i}\theta};\beta) \right\}\hat{\bold{e}}_\xi \\ \displaystyle \quad &+\frac{C_y r_*( \tau_i+\tau_e+2\tau )}{2\sqrt{\epsilon}}\big(\cosh \xi + \cos\theta\big) \Im\left\{ P(e^{-\xi - \rm{i}\theta};\beta) \right\}\hat{\bold{e}}_\xi + O(1), \end{align*} where $O(1)$ is uniformly bounded independently of $\epsilon$ and $\mathbf{x}$. \end{itemize} \end{theorem} % \begin{proof} In view of Theorem \ref{main}, we may consider only the linear function for $H$. In other words, $H(\mathbf{x})=(\nabla H(\bold{0}) \cdot (1,0)) x +(\nabla H(\bold{0}) \cdot (0,1)) y$. Moreover, we can assume $\tau=\tau_i\tau_e>0$ since the blow-up happens only in such a case. One can derive a series expansion for $\nabla U(\mathbf{x})$ by differentiating the formula in Lemma \ref{lemma:transmissionseries}. % From the definition of $P$, we have \begin{align*} \frac{\partial q}{\partial \xi}& = \frac{1}{2} \begin{cases} (\tau+ \tau_i )P(e^{\xi-\rm{i}\theta-2\xi_i} ;\beta) - (\tau+\tau_e)P(e^{\xi-\rm{i}\theta-2\xi_e};\beta)\quad & \textrm{in } \mathbb{R}^2 \setminus \overline{B_e}, \\[.5mm] (\tau+ \tau_i)P(e^{\xi-\rm{i}\theta-2\xi_i} ;\beta) + (\tau+\tau_e)P(e^{-\xi-\rm{i}\theta};\beta) \quad& \textrm{in } B_e \setminus\overline{ B_i}, \\[.5mm] -(\tau+ \tau_i)P(e^{-\xi-\rm{i}\theta} ;\beta) + (\tau+\tau_e)P(e^{-\xi-\rm{i}\theta};\beta) \quad& \textrm{in } B_i, \end{cases}\\[2mm] \frac{\partial q}{\partial \theta}& = \frac{\rm{i}}{2} \begin{cases} -(\tau+ \tau_i)P(e^{\xi-\rm{i}\theta-2\xi_i} ;\beta) + (\tau+\tau_e)P(e^{\xi-\rm{i}\theta-2\xi_e};\beta)\quad & \textrm{in } \mathbb{R}^2 \setminus \overline{B_e}, \\[.5mm] -(\tau+ \tau_i)P(e^{\xi-\rm{i}\theta-2\xi_i} ;\beta) + (\tau+\tau_e)P(e^{-\xi-\rm{i}\theta};\beta)\quad & \textrm{in } B_e \setminus \overline{B_i}, \\[.5mm] -(\tau+ \tau_i)P(e^{-\xi-\rm{i}\theta} ;\beta) + (\tau+\tau_e)P(e^{-\xi-\rm{i}\theta};\beta)\quad & \textrm{in } B_i. \end{cases} \end{align*} Thanks to Lemma \ref{lemma:LYu2D:series}, one can show that \begin{equation}\label{estimate:nablaU} \nabla U(\mathbf{x}) = r_* ^2 \nabla q(\mathbf{x};\beta,\tau_i, \tau_e) + O(1), \quad \mathbf{x} \in \mathbb{R}^2, \end{equation} in a similar way as in the proof of \cite[Lemma 5.1]{LYu2D}. Since the functions $x+\Re\{U(\bold{x})\}$ and $y+\Im\{U(\bold{x})\}$ are, respectively, the solutions to \eqnref{Eqn} with $H(\mathbf{x})=x$ and $H(\mathbf{x})=y$, we complete the proof of (a). Now we prove (b). From \eqnref{estimate:nablaU} it follows that \begin{align*} \nabla (u-H) (\bold{x})= & C_x r_*^2 (\Re\{\nabla q(\bold{x};\beta,\tau_i , \tau_e) \} \cdot \hat{\bold {e}}_\xi)\hat{\bold {e}}_\xi + C_x r_*^2 (\Re\{\nabla q(\bold{x};\beta,\tau_i , \tau_e) \} \cdot \hat{\bold {e}}_\theta)\hat{\bold {e}}_\theta \\ + &C_y r_*^2 (\Im\{\nabla q(\bold{x};\beta,\tau_i ,\tau_e) \} \cdot \hat{\bold {e}}_\xi)\hat{\bold {e}}_\xi + C_y r_*^2 (\Im\{\nabla q(\bold{x};\beta, \tau_i ,\tau_e) \} \cdot \hat{\bold {e}}_\theta)\hat{\bold {e}}_\theta+O(1). \end{align*} Using \eqnref{eqn:nabla} and \eqnref{Pproperty2}, we can show in a similar way as in \cite[section 4.1]{LYu2D} that \begin{align}\label{eqn:nablaq_core_shell} \nabla q \cdot \hat{\bold {e}}_\xi & = {O}(1) + \frac{1}{2}h(\xi,\theta) \begin{cases} (\tau_i-\tau_e)P(e^{-\xi_e-\rm{i}\theta};\beta) & \textrm{in } \mathbb{R}^2 \setminus \overline{B_e}, \\[.5mm] (\tau_i+\tau_e+2\tau)P(e^{-\xi-\rm{i}\theta};\beta) & \textrm{in } B_e \setminus \overline{B_i}, \\[.5mm] (\tau_e-\tau_i)P(e^{-\xi-\rm{i}\theta};\beta) & \textrm{in } B_i, \end{cases}\\ \notag \nabla q \cdot \hat{\bold {e}}_\theta & = O(1) + \frac{\rm{i}}{2} h(\xi,\theta)(\tau_e-\tau_i) P(e^{-\xi-\rm{i}\theta};\beta) \hspace{1.8cm} \textrm{in } \mathbb{R}^2. \end{align} By use of \eqnref{Pproperty1}, we obtain that $$\left|h(\xi,\theta) P(e^{-\xi-\rm{i}\theta};\beta)\right| \le \frac{1}{\alpha\beta}\le \frac{C}{|\ln \tau|}.$$ Note that we have \begin{align*} |\tau_i|-|\tau_e | & \le \big| |\tau_i| - 1 \big|+ \big|1-|\tau_e|\big| \le -\ln |\tau_i|- \ln|\tau_e| \le |\ln \tau| \end{align*} owing to $\tau> 0$ and $\left| \tau_i \right|, \left| \tau_e \right| < 1$. For $\tau_i, \tau_e<0$, it follows that $$ \left| \nabla q \right| \le C \quad \textrm{in } \mathbb{R}^2,$$ so that the blow-up does not happen. For $\tau_i,\tau_e>0$, we have that \begin{align*} \left| \nabla q \cdot \hat{\bold {e}}_\xi\right|&\le C \textrm{ in }(\mathbb{R}^2 \setminus \overline{B_e})\cup B_i, \\ \left| \nabla q \cdot \hat{\bold {e}}_\theta\right| &\le C \textrm{ in } \mathbb{R}^2. \end{align*} From \eqnref{eqn:nablaq_core_shell} and the definition of $h$, $\alpha$, we can derive the formula in (b) and complete the proof. \end{proof} Figure \ref{fig:result} shows the graphs of the normal and tangential components of $\nabla u$ on $\partial B_e$. As stated in Theorem \ref{blowupterm}(b), only the normal component blows up as $\epsilon$ tends to zero. \begin{figure}[!h] % \centering \begin{subfigure}{.32\textwidth} \centering \adjincludegraphics[height=4cm, width=5cm,trim={2.5cm 2.3cm 2cm 1.5cm}, clip]{color_grad_x_colorbar} \caption{$\epsilon=0.1$, $H(\mathbf{x})=x$} \end{subfigure} % \begin{subfigure}{.32\textwidth} \centering \adjincludegraphics[height=4cm, width=5cm,trim={1.0cm 0.5cm 0.7cm 0.6cm}, clip]{B2_x_normal} \caption{$\nabla (u-H) \cdot \hat{\bold {e}}_\xi$, $\epsilon=0.001$} \end{subfigure} % \begin{subfigure}{.32\textwidth} \centering \adjincludegraphics[height=4cm, width=5cm,trim={1.0cm 0.5cm 0.7cm 0.6cm}, clip]{B2_x_tan} \caption{$\nabla (u-H) \cdot \hat{\bold {e}}_\theta$, $\epsilon=0.001$} \end{subfigure} % % \begin{subfigure}{.32\textwidth} \centering \adjincludegraphics[height=4cm, width=5cm,trim={2.5cm 2.3cm 2cm 1.5cm}, clip]{color_grad_y_colorbar} \caption{$\epsilon=0.1$, $H(\mathbf{x})=y$} \end{subfigure} % \begin{subfigure}{.32\textwidth} \centering \adjincludegraphics[height=4cm, width=5cm,trim={1.0cm 0.5cm 0.7cm 0.6cm}, clip]{B2_y_normal} \caption{$\nabla (u-H) \cdot \hat{\bold {e}}_\xi$, $\epsilon=0.001$} \end{subfigure} \begin{subfigure}{.32\textwidth} \centering \adjincludegraphics[height=4cm, width=5cm,trim={1.0cm 0.5cm 0.7cm 0.6cm}, clip]{B2_y_tan} \caption{$\nabla (u-H) \cdot \hat{\bold {e}}_\theta$, $\epsilon=0.001$} \end{subfigure} \caption{Electric field for the core-shell structure with $r_i=1$, $r_e=2$, $k_i=1$, and $k_e=0.01$, where $\epsilon=0.1$ or $0.001$. The figures are contour plots of $|\nabla (u-H)|$ (the first column) and the graphs of $\nabla (u-H) \cdot \hat{\bold {e}}_\xi$ and $\nabla (u-H) \cdot \hat{\bold {e}}_\theta$ on $\partial B_e$ (the second and third column, solid curve). Dashed curves are computed by using the singular term in Theorem \ref{blowupterm}(a). The background potential is $H(\mathbf{x})=x$ (the first row) or $H(\mathbf{x})=y$ (the second row). }\label{fig:result} \end{figure} \subsection{Optimal bounds} Assume that $k_i= 1$ and $k_e \ll 1$. Then, by using \eqnref{P:integral}, one can easily deduce the upper and lower bounds for $\big| (\cosh\xi_i + \cos\theta) P(e^{-\xi_i - \rm{i}\theta};\beta) \big|$. Specifically, we have that \begin{align} \bigg|\big(\cosh \xi_i + \cos\theta\big) \Re\left\{P(e^{-\xi_i - \rm{i}\theta};\beta) \right\} \bigg| &\geq \frac{C_1}{\beta+1}\quad \mbox{ for }\theta=0,\\ \bigg|\big(\cosh \xi_i + \cos\theta\big) \Im\left\{P(e^{-\xi_i - \rm{i}\theta};\beta) \right\} \bigg| &\geq \frac{C_1}{(\beta+1)(\beta+3)}\quad \mbox{for }\theta=\frac{\pi}{2} \end{align} with some positive constant $C_1$ independent of $\epsilon,\beta$. Indeed, for $\theta = \frac{\pi}{2}$ we have \begin{align*} \bigg|\Im\left\{P(e^{-\xi_i - \frac{\pi}{2}{\rm i}};\beta) \right\}\bigg| &= \int_0 ^\infty e^{-\beta t} \frac{\sinh (\xi_i+t)}{2\cosh^2(\xi_i+t)} dt = e^{\beta \xi_i}\int_{\xi_i} ^\infty e^{-\beta t} \frac{e^t-e^{-t}}{(e^t + e^{-t})^2} dt\\ &\ge e^{\beta \xi_i}\int_{\xi_i} ^\infty e^{-\beta t} \frac{e^t-e^{-t}}{(2e^{t})^2} dt\\ &\ge \frac{1}{4} \left(\frac{e^{-\xi_i}}{\beta + 1} - \frac{e^{-3\xi_i}}{\beta + 3} \right)\\ &\ge \frac{C_1}{(\beta+1)(\beta+3)}. \end{align*} Also, we can show that \begin{align} \Big| (\cosh\xi_i + \cos\theta) P(e^{-\xi_i - \rm{i}\theta};\beta) \Big| \le \frac{C_2}{\beta + 1} \end{align} for some positive constant $C_2$ independent of $\epsilon,\beta,\theta$. Note that we have $$\frac{1}{\sqrt \epsilon (\beta + 1)} \approx \frac{1}{r_* k_e + \sqrt \epsilon} $$ owing to $$\beta = \frac{r_*}{4\sqrt \epsilon}\big(1-\tau + O((1-\tau)^2)\big) \approx \frac{r_*}{\sqrt \epsilon} k_e.$$ In total, assuming $k_i= 1$ and $k_e \ll 1$, the following holds with some positive constant $C_1,C_2$ independent of $k_e,k_i,\epsilon$: \begin{itemize} \item[$\bullet$] If $C_x \ne 0$ and $C_y =0$, then we have $$ C_1 \frac{C_x}{k_e + \frac{\sqrt \epsilon}{r_*}} \le \Vert \nabla (u-H) \Vert_{L^\infty(B_e \setminus B_i)} \le C_2 \frac{C_x}{k_e + \frac{\sqrt \epsilon}{r_*}}. $$ \item[$\bullet$] If $C_x = 0$, $C_y \ne 0$ and $\frac{r_* k_e}{\sqrt \epsilon} < C$ for some constant $C$ independent of $\epsilon$ ({\it i.e.}, $\beta$ is bounded), then we have $$ C_1 \frac{C_y}{k_e + \frac{\sqrt \epsilon}{r_*}} \le \Vert \nabla (u-H) \Vert_{L^\infty(B_e \setminus B_i)} \le C_2 \frac{C_y}{k_e + \frac{\sqrt \epsilon}{r_*}}. $$ \end{itemize} \section{Image line charge formula}\label{sec:cartesian} We obtained an asymptotic formula for the electric field in bipolar coordinates in the previous section, where the singular term is a linear combination of the Lerch transcendent functions. In this section, we rewrite the formula in terms of the Cartesian coordinates. It turns out that the singular term is actually the single and double layer potentials with line charges on the $x$-axis. Also, we derive an asymptotic formula for the case of two separated disks. As well as the core-shell geometry, the singular term for two separated disks can be expressed as the single and double layer potentials with line charges on the $x$-axis. \subsection{Line charge formula of the Lerch transcendent function}\label{sec:lerch} We first express each component of the singular function $q$, that is, a linear combination of the Lerch transcendent function, in the Cartesian coordinates. Before stating the formulas, let us define some notations. Here and after, we denote by $[ \mathbf{x}_1, \mathbf{x}_2 ]$ the line segment connecting the two points $\mathbf{x}_1, \mathbf{x}_2$. We also write $f(s,0)=f(s)$ for a function $f$ defined on $\mathbb{R}$. % \begin{lemma} \label{Lerch:xy} Assume $\xi_0\geq0$. We denote by $\bold{c_0}=(c_0,0)$ and $r_0$ the center and radius of the level curve $\{\xi=\xi_0\}$ for $\xi_0\neq0$. We denote by $\bold{c_0}=(+\infty,0)$ and $r_0=\infty$ for $\xi_0=0$. We set the charge distribution functions as \begin{align*} \displaystyle\varphi_+(s)&= 2\alpha \beta e^{2\beta\xi_0}\frac{(s-\alpha)^{\beta-1}}{(s+\alpha)^{\beta+1}} \displaystyle,\quad \psi_+(s)=e^{2\beta\xi_0} \left(\frac{s-\alpha}{s+\alpha}\right)^\beta \quad \mbox { for } \alpha<s<c_0,\\ \displaystyle\varphi_-(s)& = 2\alpha \beta e^{2\beta\xi_0} \frac{(s+\alpha)^{\beta-1}}{(s-\alpha)^{\beta+1}}, \displaystyle\quad\psi_-(s) = -e^{2\beta\xi_0}\left(\frac{s+\alpha}{s-\alpha}\right)^\beta \quad \mbox { for } -c_0<s<-\alpha. \end{align*} Then, the following holds: \begin{itemize} \item[\rm(a)] For $\xi<2\xi_0$, we have $$L(e^{\xi +\rm{i}\theta - 2\xi_0};\beta)= \int_{[\bold{p}_2,\,\bold{c_0}]} \ln|\mathbf{x}-\mathbf{s}| \varphi_+(\mathbf{s})\,d\sigma(\mathbf{s}) + r_+(\mathbf{x}) -\rm{i}\int_{[\bold{p}_2,\,\bold{c_0}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \,\mathbf{e}_2 \rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_+(\mathbf{s})\,d\sigma(\mathbf{s}), $$ where the remainder term satisfies that $|\nabla r_+(\mathbf{x}) |\leq\frac{1}{r_0}$ for all $\mathbf{x}$. % \item[\rm(b)] For $\xi>-2\xi_0$, we have $$L(e^{-\xi -\rm{i}\theta - 2\xi_0};\beta)=\int_{[-\bold{c_0}, \,\bold{p_1}]} \ln|\mathbf{x}-\mathbf{s}| \varphi_-(\mathbf{s})\,d\sigma(\mathbf{s}) + r_-(\mathbf{x})-\rm{i}\int_{[-\bold{c_0},\, \bold{p_1}]} \frac{ \langle \mathbf{x}-\mathbf{s},\, \mathbf{e}_2\rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_-(\mathbf{s})\,d\sigma(\mathbf{s}),$$ where the remainder term satisfies that $|\nabla r_-(\mathbf{x}) |\leq\frac{1}{r_0}$ for all $\mathbf{x}$. \end{itemize} \end{lemma} % \begin{proof} By integration by parts we have % \begin{align*} L(z;\beta)& = -\int_0 ^\infty \frac{ze^{-t}e^{-\beta t} }{1+z e^{-t}}\,dt \\ & = -\ln (1+z) + \beta \int_0 ^{\infty} \ln(1+ze^{-t}) e^{-\beta t}\, dt \\ &= \beta \int_0 ^{\infty} \ln \left(\frac{1+ze^{-t}}{1+z}\right) e^{-\beta t} \,dt. \end{align*} Here, we used the fact that $\beta>0$ to evaluate the integrand at $t=\infty$. % Owing to $\overline{L(z;\beta)} = L(\overline{z};\beta)$, it follows that % \begin{align*} \Re \left\{L(z;\beta)\right\} &= \frac{1}{2} \left(L(z;\beta)+L(\overline{z};\beta) \right) = \beta \int_0 ^{\infty} \ln \left|\frac{1+ze^{-t}}{1+z}\right| e^{-\beta t} \,dt. \end{align*} The imaginary part of $L(z;\beta)$ becomes \begin{align}\notag \displaystyle\Im \big\{L(z;\beta)\big\} &= \frac{1}{2\rm{i}} \left(L(z;\beta)-L(\overline{z};\beta) \right)\notag \\\notag &= -\frac{1}{2\rm{i}} \int_0 ^\infty \frac{ze^{-(\beta+1)t}}{1+z e^{-t}} - \frac{\overline{z} e^{-(\beta+1)t}}{1+\overline{z} e^{-t}} \,dt\\ \displaystyle&= -\frac{1}{2\rm{i}} \int_0 ^\infty \frac{(z-\overline{z}) e^{-(\beta+1)t}}{|1+z e^{-t}|^2}\,dt = - \int_0 ^\infty \frac{\Im\{z\} e^{-(\beta+1)t}}{|1+z e^{-t}|^2}\,dt.\label{linecharge:imaginary} \end{align} % To prove (a), we now assume $\xi<2\xi_0$ and set $z=e^{\xi +\rm{i}\theta - 2\xi_0}$. We first consider $\Re \left\{L(z;\beta)\right\}$. Note that $|z| \le 1$. We compute \begin{align} 1+ze^{-t} & = 1+e^{\xi+\rm{i}\theta} e^{-(2\xi_0 +t)}\notag \\ &= (1-e^{-(2\xi_0+t)})(1+e^{\xi+\rm{i}\theta})\left( \frac{1}{1+e^{\xi+\rm{i}\theta}} + \frac{e^{-(2\xi_0+t)}}{1-e^{-(2\xi_0+t)}} \right) \notag \\ & = (1-e^{-(2\xi_0+t)})\left(\frac{2\alpha}{\alpha-z(\xi,\theta)}\right) \left(\frac{\alpha - z(\xi,\theta)}{2\alpha} + \frac{e^{-(2\xi_0+t)}}{1-e^{-(2\xi_0+t)}} \right) \notag \\ \label{coordchange} & = \frac{1-e^{-(2\xi_0+t)}}{\alpha - z(\xi,\theta)}\left(-z(\xi,\theta) + \alpha \frac{1+e^{-(2\xi_0+t)}}{1-e^{-(2\xi_0+t)}} \right). \end{align} For ease of estimation, we set $$ s(t)=\alpha \frac{1+e^{-(2\xi_0+t)}}{1-e^{-(2\xi_0+t)}}.$$ In other words, $$e^{-t} = e^{2\xi_0}\frac{s(t)-\alpha}{s(t)+\alpha}. $$ From \eqnref{coordchange} it follows that $$1+ze^{-t}=\frac{1-e^{-(2\xi_0+t)}}{\alpha - z(\xi,\theta)}(-z(\xi,\theta)+s(t)).$$ It is immediately clear that $s(t)$ is the center of circle $\{\xi=\xi_0+\frac{t}{2}\}$ owing to \eqnref{bipolar:circle}. In particular, $s(0)=c_0$. Hence we have for $\xi_0 > 0$ \begin{align*} &\Re \left\{L(e^{\xi +\rm{i}\theta - 2\xi_0};\beta)\right\} \\ &=\beta \int_0 ^{\infty} \ln \left|\frac{1+ze^{-t}}{1+ze^{-0}}\right| e^{-\beta t}\, dt\\ &= \beta \int_0 ^{\infty} \ln \left| \frac{(1-e^{-(2\xi_0+t)})\left(z(\xi,\theta) - s(t) \right)}{\left(1-e^{-2\xi_0} \right) \left( z(\xi,\theta)-s(0)\right)}\right| e^{-\beta t}\,dt\\ &=\beta \int_0 ^{\infty} \ln\big|z(\xi,\theta)-s(t)\big| e^{-\beta t} \,dt -\beta \int_0 ^{\infty} \ln\big|z(\xi,\theta)-s(0)\big|e^{-\beta t} \,dt + L(-e^{-2\xi_0};\beta). \end{align*} Due to the fact that $$ ds =\frac{-2\alpha e^{-(2\xi_0+t)}}{\big(1-e^{-(2\xi_0+t)}\big)^2}dt = -\frac{1}{2\alpha}(s+\alpha)(s-\alpha)dt, $$ we have \begin{align*} \Re \left\{L(e^{\xi +\rm{i}\theta - 2\xi_0};\beta)\right\} &= \beta \int_0 ^{\infty} \ln\big|z(\xi,\theta)-s(t)\big| e^{-\beta t}\, dt - \ln|z(\xi,\theta)-c_0|+L(-e^{-2\xi_0};\beta)\\ &= 2\alpha \beta e^{2\beta \xi_0} \int_{\alpha} ^{c_0} \ln\big|z(\xi,\theta)-s\big| \frac{(s-\alpha)^{\beta-1}}{(s+\alpha)^{\beta+1}} \,ds + r_+(z(\xi,\theta)) \end{align*} with $$r_+(z(\xi,\theta)) = - \ln\big|z(\xi,\theta)-c_0\big|+L(-e^{-2\xi_0};\beta).$$ One can easily see that $$\big|\nabla r(z(\xi,\theta))\big|\leq\frac{1}{r_0}\quad\mbox{for all }\xi<2\xi_0.$$ Similar estimates hold for $\xi_0 = 0$. We now consider $\Im \left\{L(z;\beta)\right\}$. Note that \begin{align*} \Im \big\{e^{\xi+\rm{i}\theta}\big\} &= \Im \left\{ \frac{\alpha+z(\xi,\theta)}{\alpha - z(\xi,\theta)}\right\}\\ & = \Im \left\{ \frac{(\alpha+z(\xi,\theta))(\alpha-\overline{z(\xi,\theta)})}{\big|\alpha-z(\xi,\theta)\big|^2}\right\}\\ &=\frac{2\alpha}{\big|\alpha-z(\xi,\theta)\big|^2} \Im \big\{z(\xi,\theta)\big\}. \end{align*} Using \eqnref{linecharge:imaginary} and \eqnref{coordchange}, we have \begin{align*} \Im \left\{L(e^{\xi +\rm{i}\theta - 2\xi_0};\beta)\right\} &=- \int_0 ^\infty \frac{\Im\{e^{\xi +\rm{i}\theta - 2\xi_0}\} e^{-(\beta+1)t}}{\big|1+e^{\xi +\rm{i}\theta - 2\xi_0-t}\big|^2}\,dt\\[2mm] &= \int_0 ^\infty \frac{\Im\big\{z(\xi,\theta)\big\}e^{-\beta t} }{\big|z(\xi,\theta) - s(t) \big|^2}\frac{-2\alpha e^{-(2\xi_0+t)}}{\big(1-e^{-(2\xi_0+t)}\big)^2} \,dt\\[2mm] &= -e^{2\beta \xi_0} \int_{\alpha} ^{c_0} \frac{\Im \big\{z(\xi,\theta)\big\}}{\big|z(\xi,\theta)-s\big|^2}\left(\frac{s-\alpha}{s+\alpha}\right)^\beta\, ds. \end{align*} Note that $\Im \big\{ z(\xi,\theta) \big\} = \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2\rangle$. Hence we complete the proof of (a). We can prove (b) in a similar way, where the remainder term is $r_-(\mathbf{x}) = - \ln\big|z(\xi,\theta)+c_0\big|+L(-e^{-2\xi_0};\beta)$. \end{proof} \subsection{Asymptotic formula for the core-shell geometry} We now express the singular part of $u$ in terms of the single and double layer potential with image line charges on the $x$-axis. \begin{theorem} \label{asymptotic} Let $u$ be the solution to \eqnref{Eqn} corresponding to an entire harmonic function $H$. For any $0<k_i,k_e\neq 1<\infty$, we have that $\|\nabla (u-H)\|_{L^\infty(\mathbb{R}^2\setminus\overline{(B_e\setminus B_i)})}\leq C$ for some constant $C$ independent of $\epsilon$. The gradient blow-up for $u$ may occur only when $\tau_i,\tau_e$ are similar to $1$ (that is, equivalently, $0<k_e\ll 1,k_i$). For $k_i, k_e$ such that $0<k_e<1$ and $k_e<k_i$, we have that in $B_e\setminus B_i$, \begin{align*} u(\mathbf{x})&= H(\mathbf{x})+ \frac{C_x r_*^2}{2} \left[ \int_{((-\infty,0),\, \bold{p_1}]} \ln|\mathbf{x}-\mathbf{s}|\varphi_1(\mathbf{s}) \,d\sigma(\mathbf{s}) -\int_{[\bold{p}_2,\,\bold{c_i}]} \ln|\mathbf{x}-\mathbf{s}| \varphi_2(\mathbf{s})\, d\sigma(\mathbf{s}) \right] \\ & + \frac{C_yr_*^2}{2}\left[ \int_{((-\infty,0), \,\bold{p_1}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2\rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_1(\mathbf{s})\, d\sigma(\mathbf{s}) -\int_{[\bold{p}_2,\,\bold{c_i}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2 \rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_2(\mathbf{s})\, d\sigma(\mathbf{s})\right] + r(\mathbf{x}), \end{align*} where the charge distributions are \begin{align*} \varphi_1(s) &= \left(\tau +\tau_e \right)2\alpha\beta \, \frac{(s+\alpha)^{\beta-1}}{(s-\alpha)^{\beta+1}}\,\chi_{(-\infty,-\alpha]}(s),\\ \psi_1(s) &= \left(\tau+\tau_e\right) \left(\frac{s+\alpha}{s-\alpha}\right)^\beta\chi_{(-\infty,-\alpha]}(s),\\ \varphi_2(s) &= \left(\tau + \tau_i\right)2\alpha \beta e^{2\beta \xi_i} \, \frac{(s-\alpha)^{\beta-1}}{(s+\alpha)^{\beta+1}}\,\chi_{[\alpha,\,c_i]}(s),\\ \psi_2(s) &= \left(\tau+\tau_i\right) e^{2\beta \xi_i}\left(\frac{s-\alpha}{s+\alpha}\right)^\beta\chi_{[\alpha,\,c_i]}(s), \end{align*} and $|\nabla r(\mathbf{x})|$ is uniformly bounded independently of $\epsilon$ and $\mathbf{x}$. \end{theorem} \begin{proof} Applying Lemma \ref{Lerch:xy} to Theorem \ref{blowupterm} (a) we prove the theorem. \end{proof} \begin{figure}[!] \centering \includegraphics[width=7cm,height=4.2cm,trim={2cm 3cm 2cm 2.5cm}, clip]{coreshell_density_sup} \caption{Bold intervals indicate the support of line charges for the core-shell structure.} \label{fig:support:core_shell} \vskip .2cm % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=6cm, height=4.5cm, trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_x_beta_g1} \hskip 1cm \adjincludegraphics[width=6cm, height=4.5cm, trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_y_beta_g1} \caption{ $k_i = 1,\, k_e=0.03,\, \epsilon = 0.001$ ($\beta = 1.8979, \tau = 0.8869$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=6cm,height=4.5cm,trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_x_beta_equal1} \hskip 1cm \adjincludegraphics[width=6cm,height=4.5cm,trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_y_beta_equal1} \caption{$k_i = 1,\, k_e= 0.0158,\, \epsilon = 0.001$ ($\beta=1, \tau=0.9387$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=6cm,height=4.5cm,trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_x_beta_l1} \hskip 1cm \adjincludegraphics[width=6cm,height=4.5cm,trim={.8cm .3cm .5cm .4cm}, clip]{coreshell_y_beta_l1} \caption{$k_i = 1,\, k_e=0.01,\, \epsilon = 0.001$ ($\beta = 0.6325, \tau = 0.9608$)} \end{subfigure} % \caption{Image line charges for the core-shell structure with $r_i=1,\, r_e=2,\, k_i=1,\, \epsilon=0.001$ and various $k_e$-values. The left column shows $\frac{1}{2}(\varphi_1-\varphi_2)$, and the right column shows $\frac{1}{2}(\psi_1-\psi_2)$ in Theorem \ref{asymptotic}. As $k_e$ decreases, $\beta$ approaches to $0$. The density $\frac{1}{2}(\varphi_1-\varphi_2)$ is bounded if $\beta\leq 1$, but diverges to $\infty$ near $p_1$ and $p_2$ if $\beta<1$.}\label{fig:density:core_shell} \end{figure} \begin{figure}[!] \begin{center} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=6cm, height=4.5cm, trim={.8cm .3cm .5cm .4cm}, clip]{coreshell_eps_x_beta_l1} \hskip 1cm \adjincludegraphics[width=6cm, height=4.5cm,trim={.8cm .3cm .5cm .4cm}, clip]{coreshell_eps_y_beta_l1} \caption{$k_i = 1,\, k_e = 0.03,\, \epsilon=0.01$ ($\beta = 0.6002, \tau = 0.8869$)} \end{subfigure} % % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=6cm,height=4.5cm,trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_eps_x_beta_equal1} \hskip 1cm \adjincludegraphics[width=6cm,height=4.5cm,trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_eps_y_beta_equal1} \caption{$k_i = 1,\, k_e = 0.03,\,\epsilon=0.0036$ ($\beta=1, \tau = 0.8869$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=6cm,height=4.5cm, trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_eps_x_beta_g1} \hskip 1cm \adjincludegraphics[width=6cm,height=4.5cm, trim={.8cm .5cm .5cm .4cm}, clip]{coreshell_eps_y_beta_g1} \caption{$k_i = 1,\, k_e = 0.03,\,\epsilon=0.001$ ($\beta = 1.8979, \tau = 0.8869$)} \end{subfigure} \caption{Image line charges for the core-shell structure with $r_i=1,\, r_e=2,\, k_i=1,\,k_e=0.03$ and various $\epsilon$-values. The left column shows $\frac{1}{2}(\varphi_1-\varphi_2)$, and the right column shows $\frac{1}{2}(\psi_1-\psi_2)$ in Theorem \ref{asymptotic}. The support of line charges for the core-shell structure is drawn in Figure \ref{fig:support:core_shell}. As $\epsilon$ decreases, $\beta$ increases so that $\frac{1}{2}(\varphi_1-\varphi_2)$ becomes bounded.}\label{fig:density:core_shell_2} \end{center} \end{figure} Figure \ref{fig:support:core_shell} shows the support of the density functions in Theorem \ref{asymptotic}. One can easily see that $\frac{\partial \psi_j (s)}{\partial s} = \varphi_j (s)$ for $j=1,2$ and from \eqnref{bipolar:circle}, \begin{gather*} \int \varphi_1(s)\,ds=\tau+\tau_e,\quad \int\varphi_2(s)\,ds=\tau+\tau_i. \end{gather*} The density functions show a different feature depending on $\beta$. The functions $\varphi_1$ and $\varphi_2$ are bounded if $\beta\leq 1$, but they diverge to $\pm\infty$ near $p_1$ or $p_2$ if $\beta<1$. See Figure \ref{fig:density:core_shell} and Figure \ref{fig:density:core_shell_2}. \begin{remark} % When $\beta=0$ (equivalently $\tau_i=\tau_e=1$), which is the limiting case of $|\ln \tau|\ll \sqrt{\epsilon}$, we have that \begin{align*} L(e^{\xi+\rm{i}\theta - 2\xi_0}; 0) &= \ln|\mathbf{x}-\mathbf{p}_2| + \widetilde{r}_+ (\mathbf{x}) - {\rm i}\int_{[\bold{p}_2,\,\bold{c_0}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2 \rangle}{|\mathbf{x}-\mathbf{s}|^2}\, d\mathbf{s},\quad \xi<2\xi_0,\\ L(e^{-\xi-\rm{i}\theta - 2\xi_0}; 0) &= \ln|\mathbf{x}-\mathbf{p}_1| + \widetilde{r}_- (\mathbf{x}) + {\rm i}\int_{[-\bold{c_0}, \,\bold{p_1}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \mathbf{e}_2\rangle}{|\mathbf{x}-\mathbf{s}|^2}\, d\mathbf{s}, \quad \xi>-2\xi_0, \end{align*} where $|\nabla \widetilde{r}_+(\mathbf{x})|$ and $|\nabla \widetilde{r}_-(\mathbf{x})|$ are bounded independently of $\epsilon,\mathbf{x}$. Hence, the singular term of $u$ in \eqnref{u:singular} satisfies \begin{align} &C_x r_*^2\Re\big\{q\big(\mathbf{x};0,1, 1\big)\big\} + C_y r_*^2\Im\big\{q\big(\mathbf{x};0,1, 1\big)\big\}\notag \\[1mm]\label{beta_zero} =\ & C_x r_*^2 \ln\left|\frac{\mathbf{x}-\mathbf{p}_1}{\mathbf{x}-\mathbf{p}_2}\right| + C_y r_*^2 \int_{[\bold{p}_2,\,\bold{c_0}] \cup [-\bold{c_0}, \,\bold{p_1}]} \frac{ \langle \mathbf{x}-\mathbf{s}, \, \mathbf{e}_2 \rangle}{|\mathbf{x}-\mathbf{s}|^2} \,d\mathbf{s} +\widetilde{r}(\mathbf{x}), \quad \mathbf{x}\in B_e \setminus B_i, \end{align} where $|\nabla \widetilde{r}(\mathbf{x})|$ are bounded independently of $\epsilon,\mathbf{x}$. \smallskip Recall that we denote by $\mathbf{x}_i=(x_i,0)$ and $\mathbf{x}_e=(x_e,0)$ the closest points on $\partial B_i$ and $\partial B_e$ between $\partial B_i$ and $\partial B_e$, respectively. The gradient of the function in \eqnref{beta_zero} blows up at $\mathbf{x}_e$ as $\epsilon$ tends zero, since the distance between $\mathbf{p}_1$ and $\mathbf{p}_2$ is $O(\sqrt{\epsilon})$. However, the solution to the following problem does not blow up: \begin{equation} \label{Eqn_insul} \left\{ \begin{array}{ll} \displaystyle \Delta u_0 = 0 \qquad& \textrm{in } \mathbb{R}^2\setminus \overline{B_e}, \\ \displaystyle \pd{u_0}{\nu}=0\quad&\mbox{on }\partial B_e,\\ \displaystyle u_0(\bold{x}) - H(\bold{x}) = O(|\bold{x}|^{-1}) \qquad& \textrm{as } |\bold{x}| \to \infty. \end{array} \right. \end{equation} These facts imply that $\nabla u(\mathbf{x}_e)$ for the solution $u$ to \eqnref{Eqn} does not converge to $\nabla u_0(\mathbf{x}_e)$ when $k_e\rightarrow\infty$ and $\epsilon\rightarrow 0$, satisfying that $k_i$ is fixed and $|\ln \tau|\ll \sqrt{\epsilon}$. See \cite[section 5.3]{LYu2D} for a non-convergence result for $\nabla u$ in the case of separated disks. % \end{remark} \subsection{Asymptotic formula for the case of two separated disks} We now consider the gradient blow-up feature for the solution $u$ to \eqnref{Eqn} in the presence of two separated disks, say $B_1$ and $B_2$. In other words, the conductivity profile is given by \begin{equation}\label{sigma:separated} \sigma = k_1 \chi(B_1) + k_2 \chi(B_2) + \chi \big(\mathbb{R}^2 \setminus (B_1 \cup B_2)\big). \end{equation} As before we assume that $0<k_i,k_e\neq 1<\infty$. {For two separated disks with arbitrary conductivities, the asymptotic formula for the potential function in bipolar coordinates was derived in \cite{LYu2D}. In this section, we express the singular term by the single and double layer potential with line charges.} We set $B_j$ to be the disk centered at $(c_j,0)$ with radius $r_j$, $j=1,2$. We denote by $\epsilon$ the distance between $B_1$ and $B_2$; see Figure \ref{fig:support:separated} for the geometry of the two separated disks. We set $R_j$ to be the reflection with respect to $\partial B_j$, $j=1,2$. We can assume that, by appropriate shifting and rotation, the fixed points of the combined reflection $R_1 \circ R_2$ are $\mathbf{p}_1=(-\alpha,0),~ \mathbf{p}_2 = (\alpha,0)$ with $$ \alpha = \frac{\sqrt{\epsilon(2r_1+\epsilon)(2r_2+\epsilon)(2r_1+2r_2+\epsilon)}}{2(r_1+r_2+\epsilon)}. $$ As for the core-shell geometry, we denote by $(\xi,\theta)$ the bipolar coordinates defined with respect to the poles $\mathbf{p}_1$ and $\mathbf{p}_2$. We now set $$r_* = \sqrt{\frac{2r_1r_2}{r_1+r_2}}$$ and $$\tau_1=\frac{k_1-1}{k_1+1}, \quad \tau_2=\frac{k_2-1}{k_2+1}, \quad \tau=\tau_1\tau_2, \quad \beta = \frac{r_*(-\ln \tau)}{4\sqrt{\epsilon}}.$$ \begin{lemma}[\cite{LYu2D}] Let $u$ be the solution to \eqnref{Eqn} with $\sigma$ given by \eqnref{sigma:separated}. Then $u$ satisfies that \begin{equation}\label{u:singular_disjoint} u(\mathbf{x}) = H(\mathbf{x}) + C_x r_*^2\Re\big\{\widetilde{q}\big(\mathbf{x};\beta,\tau_i, \tau_e\big)\big\} + C_y r_*^2\Im\big\{\widetilde{q}\big(\mathbf{x};\beta,-\tau_i, -\tau_e\big)\big\} + r(\mathbf{x}),\quad \mathbf{x}\in\mathbb{R}^2, \end{equation} where \begin{align*} &\widetilde{q}(\mathbf{x};\beta,\tau_1,\tau_2) \\ &:= \frac{1}{2} % \begin{cases} \displaystyle(\tau + \tau_1) L(e^{-(\xi+\rm{i}\theta)-2\xi_1};\beta) - (\tau + \tau_2) L(e^{(\xi+\rm{i}\theta)-2\xi_2};\beta) & \mbox{in } \mathbb{R}^2 \setminus \overline{(B_1 \cup B_2)}, \\[.5mm] \displaystyle(\tau + \tau_1) L(e^{\xi-\rm{i}\theta};\beta) - (\tau + \tau_2) L(e^{(\xi+\rm{i}\theta)-2\xi_2};\beta) & \mbox{in } B_1 ,\\[.5mm] \displaystyle(\tau + \tau_1) L(e^{-(\xi+\rm{i}\theta)-2\xi_1};\beta) - (\tau + \tau_2) L(e^{-\xi+\rm{i}\theta};\beta) & \mbox{in } B_2, \end{cases} \end{align*} and $\Vert \nabla r(\mathbf{x}) \Vert_{L^\infty(\mathbb{R}^2)} < C$ for some constant $C$ independent of $\epsilon$. \end{lemma} Using the line charge formula of the Lerch transcendent function in section \ref{sec:lerch}, one can easily show the following. % \begin{theorem}\label{thm:imagecharge:separate} Let $u$ be the solution to \eqnref{Eqn} with $\sigma$ as given by \eqnref{sigma:separated}. The gradient blow-up for $u$ may occur only when $\tau$ is similar to $1$ (equivalently, $k_1,k_2\gg 1$ or $0<k_1,k_1\ll 1$). Moreover, $u$ satisfies the following integral formulas in $\mathbb{R}^2 \setminus (B_1 \cup B_2)$: \begin{itemize} \item[\rm(a)] For $1<k_1, k_2 < \infty$, we have \begin{align*} u(\mathbf{x}) = H(\mathbf{x}) + \frac{C_x r_*^2}{2} \left[ \int_{[\mathbf{c}_1, \mathbf{p}_1]} \ln|\mathbf{x}-\mathbf{s}| \varphi_1(\mathbf{s})\, d\sigma(\mathbf{s}) -\int_{[\bold{p}_2,\mathbf{c}_2]} \ln|\mathbf{x}-\mathbf{s}| \varphi_2(\mathbf{s})\, d\sigma(\mathbf{s}) \right] + r(\mathbf{x}), \end{align*} where the charge distributions are \begin{align*} \varphi_1 (s) & = (\tau + \tau_1) 2\alpha \beta e^{2\beta\xi_1} \, \frac{(s+\alpha)^{\beta-1}}{(s-\alpha)^{\beta+1}}\,\chi_{[c_1,-\alpha]}(s),\\ \varphi_2 (s) & = (\tau + \tau_2) 2\alpha \beta e^{2\beta\xi_2} \, \frac{(s-\alpha)^{\beta-1}}{(s+\alpha)^{\beta+1}}\,\chi_{[\alpha,\,c_2]}(s), \end{align*} and $\Vert \nabla r(\mathbf{x}) \Vert_{L^\infty(\mathbb{R}^2)} < C$ for some constant $C$ independent of $\epsilon$. \item[\rm(b)] For $0<k_1, k_2 < 1$, we have \begin{align*} u(\mathbf{x}) = H(\mathbf{x}) + \frac{C_y r_*^2}{2} \left[ \int_{[\mathbf{c}_1, \mathbf{p}_1]} \frac{ \langle \mathbf{x}-\mathbf{s}, \hat{\bold{e}}_y \rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_1 (\mathbf{s})\, d\sigma(\mathbf{s}) + \int_{[\bold{p}_2,\mathbf{c}_2]} \frac{ \langle \mathbf{x}-\mathbf{s}, \hat{\bold{e}}_y \rangle}{|\mathbf{x}-\mathbf{s}|^2} \psi_2 (\mathbf{s})\, d\sigma(\mathbf{s}) \right]+ r(\mathbf{x}) , \end{align*} where the charge distributions are \begin{align*} \psi_1 (s) & = (\tau + \tau_1) e^{2\beta\xi_1} \left(\frac{s+\alpha}{s-\alpha}\right)^\beta \chi_{[c_1,-\alpha]}(s),\\ \psi_2 (s) & = (\tau + \tau_2) e^{2\beta\xi_2} \left(\frac{s-\alpha}{s+\alpha}\right)^\beta \chi_{[\alpha,\,c_2]}(s), \end{align*} and $\Vert \nabla r(\mathbf{x}) \Vert_{L^\infty(\mathbb{R}^2)} < C$ for some constant $C$ independent of $\epsilon$. \end{itemize} \end{theorem} One can derive a similar line charge formula for $u$ in $B_1\cup B_2$ by use of the line charge formula of the Lerch transcendent function in section \ref{sec:lerch}. \begin{figure}[!] \centering \includegraphics[width=7cm,height=4cm,trim={2cm 3cm 2cm 2.5cm}, clip]{disjointdisk} \caption{Geometry of two separated disks. Bold intervals indicate the support of line charges.}\label{fig:support:separated} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_x_beta_g1} \hskip .5cm \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_y_beta_g1} \caption{$k_1 = 10,\, k_2 = 15$ (left) and $k_1=1/10,\, k_2 = 1/15$ (right). $ \epsilon = 0.005$ ($\beta = 1.5471, \tau = 0.7159$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_x_beta_equal1} \hskip .5cm \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_y_beta_equal1} \caption{$k_1= 15.4471,\, k_2 = 23.1707$ (left) and $k_1=1/ 15.4471,\, k_2 =1/ 23.1707$ (right). $\epsilon = 0.005$ ($\beta=1, \tau = 0.8057$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_x_beta_l1} \hskip .5cm \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_y_beta_l1} \caption{$k_1=20,\, k_2=30$ (left) and $k_1=1/20,\, k_2=1/30$ (right). $\epsilon = 0.005$ ($\beta = 0.7720, \tau = 0.8464 $)} \end{subfigure} % \caption{Image line charges for the two separated disks with $r_1=1.5,\, r_2 =2,\, \epsilon=0.005$ and various $k_1,k_2$-values. The left column shows $\frac{1}{2}(\varphi_1-\varphi_2)$, and the right column shows $\frac{1}{2}(\psi_1-\psi_2)$ in Theorem \ref{thm:imagecharge:separate}. }\label{fig:separated} \end{figure} \begin{figure}[!] \begin{center} \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_eps_x_beta_l1} \hskip .5cm \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_eps_y_beta_l1} \caption{$\epsilon=0.01$ ($\beta = 0.5459, \tau = 0.8464$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_eps_x_beta_equal1} \hskip .5cm \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_eps_y_beta_equal1} \caption{$\epsilon= 0.0030$ ($\beta=1, \tau = 0.8464$)} \end{subfigure} % \begin{subfigure}{1\textwidth} \centering \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_eps_x_beta_g1} \hskip .5cm \adjincludegraphics[width=5cm, height=3.75cm, trim={.8cm .5cm .5cm .4cm}, clip]{disjointdisk_eps_y_beta_g1} \caption{$ \epsilon=0.001$ ($\beta = 1.7263, \tau = 0.8464$)} \end{subfigure} % \caption{Image line charges for the two separated disks with $r_1=1.5,\, r_2 =2$ and various $\epsilon$-values, where $k_1 = 20,\, k_2 = 30$ (left column) and $k_1 = 1/20,\, k_2 = 1/30$ (right column). The left column shows $\frac{1}{2}(\varphi_1-\varphi_2)$, and the right column shows $\frac{1}{2}(\psi_1-\psi_2)$ in Theorem \ref{thm:imagecharge:separate}. As $\epsilon$ decreases, $\beta$ increases so that $\frac{1}{2}(\varphi_1-\varphi_2)$ becomes bounded.} \end{center} \end{figure} \section{Conclusion} We have analyzed the gradient blow-up of the solution to the conductivity problem in the presence of an inclusion with core-shell geometry. We showed that the gradient blow-up happens only when the conductivities of the core and shell satisfy the conditions $k_e<1$ and $k_e<k_i$. We then derived an asymptotic formula for the potential function in terms of the bipolar coordinates. We finally expressed the asymptotic formula in terms of the single and double layer potentials with line charges. It will be of interest to generalize the result to core-shell geometry with arbitrary shapes, or to the elasticity problem.

BkiUby_xK0-nUKDhpf0w

\section{Phrase localization} \label{supp_phrase_loc} Our work makes advances on phrase detection. The problem stands in contrast to localization-only tasks where detection is not required (\ie ground truth phrases are given at test time). As prior work \cite{Plummer_2020} demonstrated, it is too easy to overfit to the localization-only task. In other words, localization performance and detection performance are not causally related, so localization performance is of very limited use for evaluating our detection approach. However, to be complete, we report localization performance of our model without PPA in Table~\ref{loc_no_ppa} and with PPA in Table~\ref{loc_ppa}. We note that our methods' localization numbers are on par with previous work, with the only exception of when PPA is applied on Flickr30K. We would argue that this is likely due, in part, to the fact that the PPA augmentations are noisy, as \cite{Plummer_2020} themselves acknowledged. For example, using PPA augmentation on Flickr30K, any region labeled as "swimmers head cap" would also be labeled as a positive for "roof" or "jacket," although these are clearly incorrect associations. Since our work more accurately distinguishes between phrases, we would more likely label the correct image region for such spurious associations. \section{Concept generation ablations} \label{supp_concept_generate_abl} In our work, we make use of visually coherent groups (concepts) to provide negative samples that cover a broad spectrum of the training distribution and minimize false negatives. We argue that minimizing visual coherence noise in our concepts is key to minimizing false negatives. As noted in Section 3.1 in our paper, DBSCAN + ViCo result in concepts with the least amount of visual coherence noise. Thus, we expect this combination to have the best performance. To verify this claim, we provide an ablation study that document the impact of varying the embedding/clustering algorithm combination on the CR component of our PFP model. The component is responsible for using the concepts as negative samples. For each combination, we select the clustering algorithm hyper parameters that produce the least visual coherence noise. As we can see in Table~\ref{cr_ab}, DBSCAN+ViCo perform the best on both datasets. While other combinations have equal performance on Flickr30k entities, their performance align in RefCOCO+ dataset with our visual coherence noise metric. More concretely, K-means overall perform worse than DBSCAN and ViCO improves performance on either of the clustering algorithms when compared to GLoVE. \begin{table*}[t] \setlength{\tabcolsep}{1pt} \centering \begin{tabular}{|l|ccc|c|ccc|c|} \hline & \multicolumn{4}{|c|}{Flickr30K Entities} & \multicolumn{4}{|c|}{RefCOCO+} \\ \hline \#Train Samples &zero shot &few shot & common &mean&zero shot &few shot & common &mean\\ \hline Baseline [SimNet w/CCA~\cite{Plummer_2020}] & 9.7 & 11.2 & 17.3 & 12.7 & 6.0 & 10.2 & 20.1 & 12.1 \\ K-means + GLoVE & 10.0 & 11.8 & 18.5 & 13.4 & 6.0 & 9.9 & 20.4 & 12.1 \\ K-means + ViCo & 10.1 & 11.9 & 18.3 & 13.4 & 6.0 & 9.8 & 20.7 & 12.2 \\ DBSCAN + GLoVE & 9.9 & 11.9 & 18.4 & 13.4 & 6.2 & 10.4 & 20.7 & 12.4 \\ DBSCAN + ViCo & \textbf{10.4} & \textbf{11.8}& \textbf{18.3} & \textbf{13.5} & \textbf{6.1} & \textbf{10.4} & \textbf{21.0} & \textbf{12.6} \\ \hline \end{tabular} \vspace{-2mm} \caption{mAP Split by frequency of training instances. The table contains ablations of our model that compares the performance of its CR component using different concept generation methods (\ie clustering algorithm/embedding combinations) to state of the art (SimNet w/CCA). See Section~\ref{supp_concept_generate_abl} for discussion.} \label{cr_ab} \end{table*} \section{Evaluation dataset selection} \label{supp_datasets} The sparsity of phrase detection datasets annotations poses significant challenges on evaluation. Real world datasets annotations simply can not cover all the possible positive cases. For example, while a region might be annotated with only the phrase \textit{blue shirt}, it can also be correctly labeled with \textit{clothing}. Thus, if a model assigns a region the phrase \textit{clothing}, the evaluation process will incorrectly classify the case as negative (\ie false negative). Prior work \cite{Plummer_2020} attempted to mitigate this problem by introducing Positive Phrase Augmentations (PPA) where structures like WordNet \cite{10.1145/219717.219748} are used to derive additional positive samples for a given annotation. However, this problem is not limited to issues with synonyms. Phrases might have different structures but can convey the same meaning (\eg frisbee that is round vs a round frisbee), thus resulting in the same false negatives problem in evaluation. While the authors of datasets like Flickr30K \cite{7410660} limited the structure of their annotations such that this problem would not arise, this is not the case for datasets like Visual Genome \cite{krishnavisualgenome} or Referit \cite{KazemzadehOrdonezMattenBergEMNLP14}. Moreover, both datasets were used in prior work to evaluate phrase detection algorithms \cite{Plummer_2020}, but given the aforementioned problem with false negatives, their validity for phrase detection evaluation is not clear. To quantitatively document this issue, we sampled 30 random phrases from each dataset and considered the top 5 most similar phrases using the visual based language representation ViCo \cite{gupta2019vico}. For each of these top 5 phrases, we manually counted the number of false negatives. We report the average results in Table~\ref{false_neg}. As we can observe, both Referit and Visual Genome suffer from significantly higher false negative rates when compared to Flickr30k Entities. Thus, they are not viable evaluation datasets for phrase detection and so we choose not to use them in our paper. We instead use Flickr30k Entities as well as RefCOCO+ \cite{10.1007/978-3-319-46475-6_5}. RefCOCO+ was collected using the same underlying game as Referit but the authors improved the data collection standards. For example, they disallowed players from using location words in their referring expressions by adding "taboo" words to the ReferIt Game which earlier would result in many false negatives. Thus, as the authors note, the resulting expressions were more appearance focused and more concise. This is further evident in the lower false negative rate in Table~\ref{false_neg} when compared to that of Referit. Even though Flickr30k and RefCOCO+ are significantly less noisy in terms of false negatives than Referit and Visual Genome, they still exhibit a significant amount of noise. Thus, future work can benefit from developing better evaluation procedures and datasets to improve the the evaluation accuracy of phrase detection models. \section{Additional qualitative results} \label{supp_qualt} \begin{table}[t!] \setlength{\tabcolsep}{2pt} \centering \begin{tabular}{|l|c|} \hline & \multicolumn{1}{|c|}{False Negative Rate} \\ \hline Referit \cite{KazemzadehOrdonezMattenBergEMNLP14} & 74\%\\ Visual Genome \cite{krishnavisualgenome} & 72\%\\ Flickr30k Entities \cite{7410660} & 31\%\\ RefCOCO+ \cite{10.1007/978-3-319-46475-6_5}& 40\%\\ \hline \end{tabular} \caption{\textbf{Dataset annotations' false negative rate.} From each of the following datasets: Flickr30k, Visual Genome, RefCOCO+ and Referit, we sample 30 random phrases and manually calculate the average percentage of false negatives in each phrase top 5 most similar phrases. See Section~\ref{supp_datasets} for more details and discussion.} \label{false_neg} \end{table} \begin{figure*}[h] \begin{subfigure}{\textwidth} \centering \includegraphics[width=0.99\linewidth]{images/qualt_full_2.png} \end{subfigure} \caption[]% {\small Comparison between our model vs prior work model on positive (ground truth) and randomly sampled negative phrases scores. Our model is significantly better able to separate positive vs negative phrases scores than prior work. Refer to section~\ref{supp_qualt} for discussion.} \label{fig:qualt_supp_1} \end{figure*} \begin{figure*}[h] \begin{subfigure}{\textwidth} \centering \includegraphics[width=0.99\linewidth]{images/qualt_full_3.png} \end{subfigure} \caption[]% {\small Comparison between our model vs prior work model on positive (ground truth) and randomly sampled negative phrases scores. Our model is significantly better able to separate positive vs negative phrases scores than prior work. Refer to section~\ref{supp_qualt} for discussion.} \label{fig:qualt_supp_2} \end{figure*} In Figures~\ref{fig:qualt_supp_1} and~\ref{fig:qualt_supp_2}, we provide additional qualitative results. As we discuss in the paper, the figures demonstrate how our model is better able at expanding the gap between the positive phrases (\ie groundtruth) and the negative phrases scores. For example, consider Figure~\ref{fig:qualt_supp_2} (a) middle image, prior work model assigns 0.995 score for the phrase \textit{edge mountain} while our model in Figure~\ref{fig:qualt_supp_2} (b) middle image assigns 0.39 score while maintaining high confidence in the ground truth phrases. This improvement is a direct result of using our concepts in both the CR and EC modules. The concepts expose the model to a wider scope of the language distribution while minimizing false negatives, and hence increase the model confidence against negative phrases. We also note that this increased discriminative power translate to tighter and better fit boxes. For example, consider the box around the waterfall in Figure~\ref{fig:qualt_supp_2} right image (a) [prior work] compared to (b) [our work]. Our work model boxes fit the target phrase \textit{beautiful waterfall} more tightly. This improvement can be due to the model's increase confidence in the phrase with respect to other negative phrases. \section{Introduction} \begin{figure}[t] \centering \includegraphics[width=\linewidth]{images/coverage.png} \caption{\textbf{Comparing training strategies}. (a) prior work used hard-negative mining to make their model more discriminative (\eg,~\cite{hinami-satoh-2018-discriminative}). However, their negatives often contain false negatives due to annotation sparsity. (b) training with our PFP-Net involves creating a set of visually coherent concepts, and then training a phrase detector to discriminate between them. As a result, our minibatches better represent the entirety of the dataset, and careful construction of our concepts helps greatly reduce the false negatives seen when using hard-negative mining.} \label{fig:prior_our_work} \end{figure} \begin{figure*}[h!] \centering \includegraphics[width=0.95\linewidth]{images/model.png} \caption{\textbf{PFP-Net overview}. Our model follows a Faster R-CNN framework~\cite{ren2016faster}. Given an image, we encode it with a ResNet-101~\cite{he2015deep} base encoder and generate a set of candidate boxes using a region proposal network. In addition to learning the similarity between regions and phrases (Section \ref{sec:phrase-region}), our approach introduces a way of automatically identify a set of coarse visually-related concepts (Section \ref{sec:mining})), and train our model to discriminate between these concepts and joint phrase-image representations as well as regularize our language embeddings using an entity-concept loss (see Section \ref{sec:ec_cr}). We also introduce a fine-grained reasoning module (Section~\ref{sec:fgm}) that forces the PFP-Net to discriminate between mutually-exclusive set of fine-grained tokens (see Figure~\ref{fig:model_inference} for an illustration for how FGM is used at test time). Note that our paper's contributions are labeled with dotted boxes. } \label{fig:model} \end{figure*} In phrase detection the goal is to identify regions from a database of images that are relevant to a phrase. This task is significantly more challenging than the localization-only task addressed in prior work, \eg, \cite{10.1145/3078971.3078976, 8237357, fukui-etal-2016-multimodal, 7780862,7780378, kazemzadeh-etal-2014-referitgame, 10.1007/978-3-030-01258-8_16, plummer2017phrase, 7410660, wang2018learning, 10.1007/978-3-319-46484-8_42, yeh2018interpretable, Wang_2019_ICCV}, that only has to localize a phrase within a ground truth image (\ie, ground truth image-phrase pairs are used at test time). Since phrase detectors can distinguish between images that are relevant to a phrase, they are more useful than localization methods in downstream tasks. For example, a phrase detector could tell you if a boat is \emph{a yacht} or \emph{a canoe} and indicate its location, but a phrase localization model would only point to the most likely regions the two phrases may appear in. However, distinguishing between semantically similar phrases that refer to different entities is challenging~\cite{Plummer_2020}. Methods for related phrase grounding tasks have proposed hard-negative mining methods to make their models more discriminative~\cite{hinami-satoh-2018-discriminative}, but these methods suffer from a high rate of false negatives which harms detection performance~\cite{Plummer_2020}. This is because these methods, in part, rely on existing structured datasets like WordNet~\cite{10.1145/219717.219748} to filter out false negatives, but many semantically similar phrases are not accounted for as illustrated in Figure~\ref{fig:prior_our_work}(a). To address these issues, we introduce Phrase-concepts to Fine-grained Phrase-tokens Net (PFP-Net). To avoid the false negatives from hard-negative mining methods used in prior work (\eg~\cite{hinami-satoh-2018-discriminative}) we automatically construct coarse concepts: visually coherent groups of entities (\eg animals vs automobiles). The model then is trained to discriminate between a phrase and its unrelated concepts as shown in Figure~\ref{fig:prior_our_work}(b). This presents a significant challenge, since we need to create concept groups that are fine-grained enough to provide informative negatives to improve the discriminative power of our model, while being coarse enough to minimize false negatives. We find that simply using K-Means clustering over a language embedding like BERT~\cite{devlin2019bert} or GLoVE~\cite{pennington-etal-2014-glove} results in noisy concept groups. This is likely because these language embeddings were trained to embed words used in similar contexts near each other, but not to distinguish between visually different entities. For example, they would embed \emph{school} and \emph{teacher} nearby since they are related, even though they are not visually similar which is important for phrase detection. In addition, K-means forces all entities to belong to a cluster, even the outliers, which we found often results in incoherent groups. To avoid these issues, we use visually discriminative language features to represent words. Then we use a density-based clustering to create a set of visually coherent concepts, which we assign to phrases based on their semantic similarity. Using our concepts provide two advantages over prior work. First, by creating coarse concepts we minimize false negatives. Second, comparing a phrase to all of the concepts ensures each image region sees a balanced distribution of easy and hard negative samples, which can improve performance over focusing solely on hard-negatives~\cite{wu2018sampling}. However, while knowing when two semantically similar words are mutually exclusive may be challenging to automate in general, we find that we can mine sets of words that typically refer to difficult fine-grained differences (\eg \emph{a red shirt} vs.\ \emph{a blue shirt}). We take advantage of these sets of mutually exclusive words adding a fine-grained module to our model. The module differentiates between fine grained words and then augment the main model score with its prediction. An overview of PFP-Net is provided in Figure~\ref{fig:model}. Our contributions can be summarized as: \begin{itemize} \setlength\itemsep{0em} \item A novel model, PFP-Net, that achieves a 1-1.5 point gain over SOTA on phrase detection over by mining visually coherent concepts and then learning to discriminate between them. \item A fine grained reasoning module that boosts performance by 1-4 points over affected phrases by learning critical visual cues to differentiate between otherwise visually similar instances. \item A novel method for automatically mining visually coherent groups that improves the distribution of our minibatches to better represent the training data. \end{itemize} \section{Related work} Most prior work in phrase grounding has focused on the localization-only tasks, where you are provided a ground truth image-phrase pair and have to identify the relevant image region (\eg,~\cite{Bajaj_2019_ICCV,hu2015natural,8237357,kazemzadeh-etal-2014-referitgame,plummer2017phrase,10.1007/978-3-030-01258-8_16,yang2019fast}). However, Plummer~\etal~\cite{Plummer_2020} demonstrated that these methods tend to overfit to the localization task, \ie, they improve performance on localization, but reduce detection performance. This is partly because distinguishing between similar phrases in the localization-only task is unnecessary since most images only contain a reference to a single object of the same type~\cite{Plummer_2020}. Thus, a localization model that is given the phrase \emph{a young teenager} as input could look for the object category ``person'' to identify the right object most of the time, whereas a detector also has to determine if it exists at all. Some work falls between detection and prior work in localization, where a ground truth image-phrase pair is not provided~\cite{hinami-satoh-2018-discriminative,zhang2017discriminative}, but they severely limit the number of negative phrases at test time. Thus, methods from these tasks also often do not generalize to phrase detection~\cite{Plummer_2020}. Also relevant to our work is Gupta~\etal\cite{gupta2020contrastive}, who introduce a sampling method for weakly-supervised phrase grounding. They use the language model BERT \cite{devlin2019bert} to sample negatives that preserve the phrase context (\eg a walking [Blank], where Blank can not be a car). This limits its application since it requires phrase grounding datasets to be annotated with this context. However, this approach also relies on noisy external language databases like WordNet~\cite{10.1145/219717.219748} to identify false negatives. In contrast, our approach creates coherent concepts that boost performance without relying on contextual annotations. \section{Visually coherent concept mining} \label{sec:mining} Vision-language datasets are only annotated with a few examples of the nearly infinite valid annotations. Several methods have been proposed to increase the number of annotations (\eg~\cite{gupta2020contrastive,hinami-satoh-2018-discriminative,Plummer_2020,VendrovICLR2016order}), but prior work in identifying good negative examples leads to significant noise that hurts performance~\cite{Plummer_2020}. As discussed in the Introduction, this is in part because the structured language datasets used by prior work also contain incomplete annotations, resulting in many false positives. Instead, we automatically identify a set of concepts, sets of entities like clothing or buildings, and then use the concepts that a phrase is not related to as negatives during training (discussed in Section~\ref{sec:model}). We define our concepts as $C = \{c_j = \{e_{k}\}_{k=0}^{k=L_j}\}_{j=0}^{j=M}$ where $e$ represents an entity (a noun referring to object) in the dataset vocabulary, $M$ is the total number of concepts, and $L_j$ is the total number of entities in concept $c_j$. Note also that $c_1 \cup c_2 .. \cup c_M = \emptyset$. We describe how we obtain these concepts in Section~\ref{sec:concepts}. Then, we describe how we assign each phrase to its set of negative concepts in Section~\ref{sec:assignment}. \begin{figure}[t] \centering \includegraphics[width=0.95\linewidth]{images/kmeansvsdbscan.png} \caption{\textbf{Qualitative comparison of visual concept clustering methods using ViCO~\cite{gupta2019vico} embeddings.} As seen in \textbf{(a)} using K-means creates many noisy concept entries (denoted in red) since it requires all words to be cluster members. In contrast, in \textbf{(b)} we show concept groups created by the density-based clustering approach DBSCAN, which contains far less noise since it will only group words that are tightly coupled. See Figure~\ref{fig:visual_noise} for a quantitative comparison of concept group quality.} \label{fig:concepts} \end{figure} \subsection{Obtaining concept groups} \label{sec:concepts} We start by extracting the entities (\eg, t-shirt, dog, etc) from our phrases. A straightforward approach for grouping semantically similar phrases would be to perform K-means clustering over some pretrained language model like GLoVE \cite{pennington-etal-2014-glove} or BERT\cite{devlin2019bert}. However, text-only representations are not trained to capture visual-semantic relationships (\eg, they typically consider a \emph{teacher} and \emph{school} to be very similar, even though they are visually different). Thus, prior work has demonstrated that training word embeddings so that they capture visual-semantic relationships is key to good performance on many tasks~\cite{burns2019iccv}. While several transformer-based vision-language embeddings have been introduced (\eg,~\cite{chen2020uniter,lu2019vilbert,su2019vlbert}), they are trained in a weakly-supervised setting where related, but semantically different entities which are important for our method are difficult to capture. Therefore, we selected ViCo \cite{gupta2019vico} as it was trained to discriminate between visually different entities. Simply selecting good language features is insufficient to get good concepts, however. Figure~\ref{fig:concepts}(a) shows that using K-means clustering over ViCo features produces noisy concepts. This is because K-means does not impose a constraint over each cluster density. \Ie, outliers in the embedding space are still clustered with other concepts even though there is little evidence they belong to the same concept. Instead, we leverage DBSCAN, \cite{9356727} whose clusters formed from entities that fall within a density threshold $\epsilon$. Thus, only tightly knit clusters are produced and noisy entities are discarded, creating more visually coherent concepts as shown in Figure~\ref{fig:concepts}(b). \smallskip \noindent\textbf{Concept group quality.} We evaluated different methods of creating concept groups via manual inspection. For each set of concepts, we counted the number of incoherent concepts (where 50\% of the concept entities or more are not visually coherent). Figure~\ref{fig:visual_noise} compares different methods of clustering, language embeddings, and hyperparameters affecting the number of clusters ($\epsilon$ for DBSCAN or $K$ for K-means). Our results show that DBSCAN over ViCo embeddings creates the most visually-coherent concepts. \subsection{Assigning phrases to concepts} \label{sec:assignment} After creating our concepts in Section~\ref{sec:concepts}, we now need to assign each phrase to its set of relevant concepts. A simple assignment process is to pair a phrase with a concept with which it shares an entity. We call this simple entity-phrase matching. This approach, however, disregards phrase-concept pairs that don't share entities but are nevertheless viable pairs. Thus, we make these assignments using semantic similarity as detailed below. First, given a phrase $p$, and phrase entities $p_e$, we consider the phrase concepts $p_{c1}$ which are the result of a simple entity-phrase matching: $p_{c1} = \{c : c \cap p_{e} \neq \emptyset, c \in C \}$. We also consider concepts $p_{c2}$ which are the result of a similarity based assignment process. This method is designed for when a phrase is still relevant to a concept but they do not explicitly share any entities (\eg \emph{hoody} with concept: \emph{shirt and sweatshirt})). More concretely, for each entity $e \in p_{e}$, we look up the entity ViCo representation. We will denote this representation as $\widetilde{e}$. We also compute a concept $c$ representation by averaging its entities ViCo feature vectors. We will denote this as $\widetilde{c}$. Now assume $\textnormal{sim}(\widetilde{e}, \widetilde{c}) = \frac{\widetilde{e} \textnormal{ }\widetilde{c}}{||\widetilde{e}|| \textnormal{ } ||\widetilde{c}||}$, we compute an "association" score between the entity $e$ and a concept $c$ as follows: \begin{equation} \textnormal{Assoc}(e, c) =\small \frac{\textnormal{exp}(\textnormal{sim}(\widetilde{e}, \widetilde{c})/\tau)}{\sum_{j=0}^{M} \textnormal{exp}(\textnormal{sim}(\widetilde{e}, \widetilde{c})/\tau)} \label{eq:assoc} \end{equation} \begin{figure}[t!] \includegraphics[width=\linewidth]{images/noise.drawio.png} \caption{\textbf{Quantitative comparison of concept creation methods}. For each algorithm/embedding combination, we produce several sets of clusters with varying size (\ie total number of resulting clusters). For each set, we calculate the visual coherence noise (see Section ~\ref{sec:concepts} for noise definition). We observe that DBSCAN exhibits consistently less noise than KMeans. See Section~\ref{sec:concepts} for further discussion.} \label{fig:visual_noise} \end{figure} From here, $p_{c2} = \{ c : \textnormal{Assoc}(e, c) > \gamma, e \in p_{e}, c \in C\}$. This way, the phrase is assigned to concepts with which it shares the most statistical ViCo based similarity. The final set of phrase concepts is therefore, $p_{c} = p_{c1} \cup p_{c2}$. Thus, a phrase that is near multiple concepts will be assigned to all of them, and so we will only contrast phrases against groups that we are confident are negatives. \section{Learning From General Phrase-concepts to Fine-grained Phrase-tokens} \label{sec:model} In phrase detection the task is to determine what images regions $r$ can be described by phrase $p$. Prior work addressing phrase localization only selects image regions from the ground truth image, but in phrase detection regions from all images in the dataset are considered. To address this problem, we introduce a neural network that learns from Phrase-concepts to Fine-grained Phrase-tokens (PFP-net). For a fair comparison to prior work~\cite{hinami-satoh-2018-discriminative,Plummer_2020}, PFP-net adopts a Faster R-CNN architecture~\cite{ren2016faster} to obtain a set of image regions, which we describe briefly in Section~\ref{sec:phrase-region}. Then, we use the concept mining procedure described in Section~\ref{sec:mining} in two ways. First, in Section~\ref{sec:ec_cr} we use the mined entity based concepts to identify the general object type of a phrase which provides a broad coverage informative negatives. Second, we obtain the initial set of concepts described in Section~\ref{sec:concepts} over adjectives, where the concepts typically denote mutually-exclusive fine-grained differences in phrases (\eg, \emph{a red truck} vs.\ \emph{a green truck}). Thus, in Section~\ref{sec:fgm} we use these adjective-concepts in our model to differentiate between mutually exclusive phrases. \subsection{Encoding phrases and image regions} \label{sec:phrase-region} Following~\cite{hinami-satoh-2018-discriminative,Plummer_2020}, we adopt a Faster R-CNN architecture \cite{ren2016faster} using a ResNet-101~\cite{he2015deep} encoder as a backbone in order to encode regions and mean-pooled HGLMM Fisher vectors \cite{hglmm} to represent our phrases. However, we note that the contributions of our approach, namely the concept-conditioned training procedure and fine-grained module , described in the subsequent sections, are adaptable to any detection architecture. We obtain a final region and phrase representation using a pair of 2-layer multi-layer perceptrons (MLP), one that takes the HGLMM features as input and the other that takes the ROI-pooled features from the Faster R-CNN encoder. After obtaining a good feature representation for the image regions $r$ and phrase (or entity/concept) features $p$ from their respective MLPs, we compute the similarity between them via: \begin{equation} PR_{Sim}(r, p) = \text{MLP}_{3\text{Layer}}(r \odot p), \label{eq:sim} \end{equation} \noindent where $\odot$ denotes an elementwise product. Note that this function is represented as the ``Joint MLP'' in Figure \ref{fig:model}. Following Plummer~\etal~\cite{Plummer_2020}, partway through training we re-initialize the 2-layer region and text MLPs via canonical correlation analysis (CCA)~\cite{Hotelling1992}, which Plummer~\etal found greatly improved phrase detection performance. We also apply an L2 regularization ($L_{reg}$) with the initial CCA weights when fine-tuning these MLPs to avoid catastrophic forgetting. Our model is trained using a \textbf{phrase-region similarity loss (PR)}. Formally, let $K$ be the number of image region-query pairs in a batch, $s^{p}$ be a region-phrase pair score, $l^p$ be its $-1/1$ label indicating whether it is a negative/positive pair, $a$ be the conditional embedding weights before the softmax, then the base loss is: \begin{equation} \label{eqn:base} L_{base} = \sum_{i=1}^{K} \log(1 + \exp(-l^{p}_i s^{p}_i)) + \lambda_{reg} L_{reg} \end{equation} \noindent where $\lambda_{reg}$ is a scalar parameter. \subsection{Expanding batch coverage using concepts} \label{sec:ec_cr} One of the primary contributions is using the generated concepts described in Section~\ref{sec:mining} to provide a set of informative negatives that represent our entire training dataset. We use these concepts in two ways: \textbf{(1)} as additional samples for the output of the Eq~\ref{eq:sim} computed between the regions and concepts (referred to as ``Concept-region Loss in Figure~\ref{fig:model}), and \textbf{(2)} to regularize the MLP textual representation (referred to as ``Entity-Concept Loss in Figure~\ref{fig:model}). Note that our concepts are encoded using text features. See our detailed explanations below. \smallskip \noindent\textbf{Concept-region Loss (CR)} Given a phrase $p$, and its set of related concepts $p_c$, we use the unrelated concepts $\overline{p_c} = \{c : c \not\in p_c, c \in C \}$ as negative samples. We then pair those concepts with the region $r$ associated with phrase $p$ to obtain negative concept-region pairs $\overline{r_c}$. We then simply concatenate the concept-region pairs to the phrase-region pairs and then feed their scores through the logistic loss described in Eq~\ref{eqn:base}. Let $s^{c}$ be region-negative concept score, $L$ be the total number of negative concept-region pair, then the loss: \begin{equation} \label{eqn:cr} L_{CR} = \sum_{i=1}^{L} \log(1 + \exp(s^{c}_i)) \end{equation} \noindent\textbf{Entity-concept Loss (EC).} In addition to ensuring that each image region is embedded near its related concepts, we also encourage phrases to embed near its associated concepts and far from others. For each phrase in our batch, we extract their entities (nouns). We randomly sample one entity for each phrase. We then compute an entity-concept score $s^{e}$ by taking the dot product between the $\ell_2$ normalized entity features and the concept representations followed by a Sigmoid activation function. Now assuming $l^e$ is the entity-concept -1/1 label indicating whether it is a negative/positive pair, and $S$ is the total number of entity-concept samples, then the entity regularization loss can be computed as: \begin{equation} \label{eq:ec} L_{EC} = \sum_{i=1}^{S} \log(1 + \exp(-s^{e}_i l^{e}_i)) \end{equation} \subsection{Fine-Grained reasoning Module (FGM)} \label{sec:fgm} After obtaining our initial phrase-region similarity scores using Eq~\ref{eq:sim}, we modify it by taking into account fine-grained differences between phrases. In particular, we have the model predict the scores of sets of mutually exclusive fine-grained tokens, and then use them to augment the main model region-phrase scores. More formally, given a set of tokens sets $F$ where each $f \in F$ is itself a set of fine grained tokens, our novel fine grain module learns to discriminate between each element of $f$. We obtain $f$ using the same bottom up approach used to mine for concepts over adjectives. However, in this case, we train the model to discriminate between each concept tokens rather than the concepts themselves, since adjectives typically group together words that denote fine-grained differences between phrases, as mentioned earlier. While other parts-of-speech could also be used (\eg, verbs also denote a mutually-exclusive state of an object), we found that our datasets did not contain enough samples for training a FGM module over other concept types. The FGM module encodes image regions using a set of convolutional layers and then performs multi-label classification on each set of fine grained tokens $f$. Now, assume $t$ is a fine grained token such that $t \in f$, $R$ be the number of region-fine grained token pairs, $s^t$ be the region-token score, and $l^t$ be its 0/1 label indicating whether it is a positive/negative region-token pair, then: \begin{equation} \label{eq:fgm} L_{FGM} = \sum_{i}^{R} l^{t}_{i} \log{s^{t}_{i}} + (1 - l^{t}_{i})\log(1-s^{t}_{i})). \end{equation} \noindent With this, the final loss for PFP-net is: \begin{equation} \label{eq:final_loss} L_{final} = L_{base} + \lambda_{CR} L_{CR} + \lambda_{EC} L_{EC} + L_{FGM} \end{equation} \subsection{Model inference} At test time the FGM module's scores are augmented with the base model's phrase-region scores. Given a phrase $p$, the phrase tokens $p_t$ and $t\in p_t$, a phrase-region pair score $s^p$, and token-region score $s^{t}$, then the final score $s^f$: \begin{equation} s^{f} = (1-\lambda_{f})s^{p} + \lambda_{f}s^{t}, \label{eq:final_score} \end{equation} \noindent where $\lambda_{f}$ is a scalar parameter that applies for every $t \in f$. See Figure~\ref{fig:model_inference} for an illustration. \begin{figure}[t!] \centering \includegraphics[width=\linewidth]{images/inference.png} \caption{\textbf{PFP-Net inference overview} During inference, the model takes in three inputs: the image, the phrase, and any applicable fine grained token within the phrase. The model then produces two scores: a phrase-region score and a fine grained token-region score. Finally, the weighted sum of the scores is then computed following Eq~\ref{eq:final_score} to produce the final score $s^f$.} \label{fig:model_inference} \end{figure} \section{Experiments} \label{experiments} \noindent\textbf{Datasets:} We evaluate our PFP-net on the two phrase grounding datasets which are the best suited for benchmarking phrase detection. First, we use Flickr30K Entities \cite{7410660} that consists of 276K bounding boxes in 32K images for the noun phrases associated with each image's descriptive captions (5 per image) from the Flickr30K dataset \cite{young-etal-2014-image}. We use the official splits \cite{7410660} that consist of 30K/1K/1K train/test/validation images. Second, we evaluate on RefCOCO+ \cite{10.1007/978-3-319-46475-6_5}, which consists of 19,992 images from the COCO dataset \cite{10.1007/978-3-319-10602-1_48} that have been labeled with 141,564 region descriptions. We use the official split \cite{10.1007/978-3-319-46475-6_5}, which splits the train/val and testing sets 16K/1.5K/1.5K. Both datasets are licensed under creative commons. \smallskip \noindent\textbf{Metrics:} We follow the evaluation protocols of Plummer~\etal~\cite{Plummer_2020}. For every image we obtain the most likely region and confidence score for every phrase in our test split. For any ground truth phrases in an image, we consider a phrase successfully localized if the predicted bounding box has at least 0.5 intersection-over-union with its associated ground truth bounding box. Then, we compute average precision (AP) for each phrase and then split them into zero-shot, few-shot, and common sets, based on if they didn't occur in our training split, if they had between 1-100 occurrences, or if they occurred more than 100 times, respectively. We then report an overall mAP for each set of phrases, as well as the average of them for an overall performance score. This procedure ensures that the zero-shot and few-shot phrases are not over represented compared to the common phrases, since the zero- and few-shot sets have more unique phrases, but represent a smaller portion of overall instances. \smallskip \noindent\textbf{Implementation details.} We train PFP-net with an ADAM \cite{kingma2017adam} optimizer using the hyperparameter settings of Plummer~\etal~\cite{Plummer_2020} for a fair comparison except where stated. We set all hyperparameters introduced by our work via grid search. Specifically, when creating our concept groups using DBSCAN we set $\epsilon$ (the threshold to determine clustering tolerance), and the $\gamma, \tau$ in Eq ~\ref{eq:assoc}. The used values were $\gamma = 0.2, \tau = 0.01$ for both datasets. $\epsilon = 0.43$ for Flickr30K and $\epsilon = 0.53$ for RefCOCO+. For entity regularization, we set $\lambda_{ER} = 0.1$ and $\lambda_{CR} = 1$ for both Flickr30K Entities and RefCOCO+. We trained our model using a single NVIDIA RTX 8000 GPU using an internal cluster. \smallskip \begin{figure*}[t!] \centering \includegraphics[width=0.99\linewidth]{images/qualt_full.png} \caption[]% {\small Comparison between our model vs prior work model on positive (ground truth) and randomly sampled negative phrases scores. Our model is significantly better able to separate positive vs negative phrases scores than prior work. This result is key in enhancing model disriminative ability which in turn translates to improvement on the mAP scores as documented in Table ~\ref{tb:main_results}. See Section~\ref{sec:model_analysis} for additional discussion.} \label{fig:qualt} \end{figure*} \begin{figure}[h!] \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_groups_flickr.png} \caption{Flickr30K Entities} \end{subfigure} \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_groups_refcoco.png} \caption{RefCOCO+} \end{subfigure} \caption{Effect of the concept set size $|C|$ has on detection performance. See Section~\ref{sec:model_analysis} for discussion.} \label{fig:hyper_groups} \end{figure} \begin{figure}[h!] \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_gamma_flickr.png} \caption{Flickr30K Entities} \end{subfigure} \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_gamma_refcoco.png} \caption{RefCOCO+} \end{subfigure} \caption{Effect of changing $\gamma$ in Eq~\ref{eq:assoc} has on detection performance. See Section~\ref{sec:model_analysis} for discussion.} \label{fig:hyper_gamma} \end{figure} \begin{figure}[h!] \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_lambda_flickr.png} \caption{Flickr30K Entities} \end{subfigure} \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_lambda_refcoco.png} \caption{RefCOCO+} \end{subfigure} \caption{Effect of changing $\lambda_{EC}$ in Eq~\ref{eq:final_loss} has on detection performance. See Section~\ref{sec:model_analysis} for discussion.} \label{fig:hyper_lambda_ec} \end{figure} \subsection{Results} \label{sec:results_discussion} Table \ref{tb:main_results} compares ablations of PFP-Net with the current state-of-the-art on phrase detection. Comparing the last line of Table \ref{tb:main_results}(b) to the results from prior work in Table \ref{tb:main_results}(a) we get 1-1.5 point gain in mAP over the state of the art. Much of this gain came from common phrases, where we achieved a 2 point gain over prior work. In Table \ref{tb:main_results}(b) we also note the contribution of each component of our model, with the biggest gain coming from using our concept-region loss (CR). Note that in Table \ref{tb:main_results}(a) the hard-negative mining approach of Hinami and Satoh~\cite{hinami-satoh-2018-discriminative} achieved a much more modest 0.2 point gain. Note that the results in our tables are the results of 5 different runs to ensure our reported gains are not due to model variance. We note that CR produces consistent improvements on both datasets across all phrase types (zero shot, few shot, common). EC results in small, but largely consistent improvements on both datasets. These results indicate that our negative samples are effective in improving our model discriminative ability. In addition to the gains from using our concepts, we note that Table \ref{tb:main_results}(b) reports that the FGM module from Section~\ref{sec:fgm} further boosts performance by a small but consistent gain for both datasets. Moreover, in Table \ref{fgm_results}, we report performance on only the phrases affected by our FGM module, which reports a significant 3.8 point improvement on Flickr30K Entities, while also obtaining a 1.5 point improvement on RefCOCO+. Finally, Table~\ref{tb:main_results_ppa} reports the performance of PFP-Net using positive phrase augmentation (PPA)~\cite{Plummer_2020}, which reduces annotation sparsity by pairing ground truth phrases with plausible positive phrases using WordNet~\cite{10.1145/219717.219748}. We note that PPA does not change the relative gains of the detection methods, but obtains higher absolute performance. \subsection{PFP-Net Analysis} \label{sec:model_analysis} To further understand the behaviour of our model, we examine several qualitative images in Figure ~\ref{fig:qualt}. Compared to prior work, our model maintains its confidence in positive ground truth phrases and correctly localize them while also having significantly lower scores for negative phrases. For example, in Figure~\ref{fig:qualt}(b) the negative phrase "sofa" in the left image has a significantly lower score than prior work in Figure ~\ref{fig:qualt}(a). This is evidence that simply optimizing for phrase localization only does not translate to discriminative detection power. As shown in the center image of Figure \ref{fig:qualt} prior work fails to differentiate between white and black, resulting in an incorrect localization of the "white dog". Our PFP-Net, on the other hand, with the help of our FGM module, correctly localizes the phrase. We further investigate our model behaviour via sensitively analysis of its hyperparameters. First, we examine the effect of changing the number of used concepts in Figure~\ref{fig:hyper_groups}. For each dataset, we progressively increase DBSCAN's density threshold until we can not generate more concepts. We observe that overall more concepts improve performance on both datasets. This is probably because more concepts split the language distribution into finer portions, thus improving the accuracy of our phrase-concept assignment in Section~\ref{sec:assignment} which translate to better performance overall. Second, we examine the effect of changing Eq \ref{eq:assoc} $\gamma$ parameter in Figure~\ref{fig:hyper_gamma} that controls whether a given concept is assigned to a phrase (\ie they are related). We vary the parameter between 0 (a phrase is assigned to every given concept) and 1 (a phrase is only assigned to concepts that share tokens with). We note that performance is at best for values of $\gamma$ between (0.2, 0.6). Performance drop for $\gamma < 0.2$ as we assign wrong concepts to phrases. Performance also drops when $\gamma > 0.8$ which indicates the importance of our similarity based matching component. In other words, simply assigning phrases to concepts that only share tokens with is not sufficient. This is because, as noted in Section~\ref{sec:assignment}, there are many concepts that are still relevant to certain phrases but also do not share tokens with. Third, and finally, we study the effect of changing $\lambda_{EC}$ used in Eq \ref{eq:final_loss} in Figure~\ref{fig:hyper_lambda_ec} that controls the contribution of our Concept-Entity loss. Overall, we note that performance for both datasets is best at 0.1. We note that both the $\gamma$ from Eq \ref{eq:assoc} and $\lambda_{EC}$ from Eq \ref{eq:final_loss} achieved the highest performance at the same parameter value, demonstrating that these hyperparameter val generalize across datasets, whereas PFP-Net tends to favor larger number of concepts. \section{Conclusion} In this work, we introduced a new phrase detection mode (PFP-NET) that significantly improves performance by 1-1.5 points on two phrase detection data-sets. The model does so by incorporating visually coherent clusters (concepts) to sample negative concept-region as well concept-entity pairs that effectively improve the model discriminative abilities when compared to prior work. Our model further improves performance by incorporating a novel fine grained module that learns to discriminate between adjective fine grained tokens. Note that although our experiments used the Faster R-CNN framework to fairly compare to prior work, the contributions made by our paper (concept-based sampling and fine grained module) are modular and so can be adapted to any underlying detection framework. \section{Limitations and societal impact} \label{broader-impacts} Phrase detection methods can help us understand the content of images. This is useful for downstream tasks like image captioning and visual question answering. This is useful for some tasks like answering questions about images posed by people who have visual impairments. However, these benefits also come with some risks, such as enabling some applications in tasks like surveillance by allowing a user to quickly locate specific entities in a database of images. Although our approach provides significant performance improvements, absolute performance on this task is still poor. 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The camera-ready copy should not contain a ruler. (\LaTeX\ users may use options of cvpr.sty to switch between different versions.) Reviewers: note that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly. Just use fractional references (\eg, this line is $087.5$), although in most cases one would expect that the approximate location will be adequate. \subsection{Paper ID} Make sure that the Paper ID from the submission system is visible in the version submitted for review (replacing the ``*****'' you see in this document). If you are using the \LaTeX\ template, \textbf{make sure to update paper ID in the appropriate place in the tex file}. \subsection{Mathematics} Please number all of your sections and displayed equations as in these examples: \begin{equation} E = m\cdot c^2 \label{eq:important} \end{equation} and \begin{equation} v = a\cdot t. \label{eq:also-important} \end{equation} It is important for readers to be able to refer to any particular equation. Just because you did not refer to it in the text does not mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin's description of how to write mathematics: \url{http://www.pamitc.org/documents/mermin.pdf}. \subsection{Blind review} Many authors misunderstand the concept of anonymizing for blind review. Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available. Blind review means that you do not use the words ``my'' or ``our'' when citing previous work. That is all. (But see below for tech reports.) Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith; it says that you are building on her work. If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work. An example of a bad paper just asking to be rejected: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Removed for blind review \end{quote} An example of an acceptable paper: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of the paper of Smith \etal [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213. \end{quote} If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work. In such cases, include the anonymized parallel submission~\cite{Authors14} as supplemental material and cite it as \begin{quote} [1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as supplemental material {\tt fg324.pdf}. \end{quote} Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report. For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a tech report for further details. Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''. Then submit the tech report as supplemental material. Again, you may not assume the reviewers will read this material. Sometimes your paper is about a problem which you tested using a tool that is widely known to be restricted to a single institution. For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal. You can handle this paper like any other. Do not write ``We show how to improve our previous work [Anonymous, 1968]. This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''. That would be silly, and would immediately identify the authors. Instead write the following: \begin{quotation} \noindent We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems [Zeus et al. 1968] did not handle case B properly. Ours handles it by including a foo term in the bar integral. ... The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know. It displayed the following behaviours, which show how well we solved cases A and B: ... \end{quotation} As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. \medskip \noindent FAQ\medskip\\ {\bf Q:} Are acknowledgements OK?\\ {\bf A:} No. Leave them for the final copy.\medskip\\ {\bf Q:} How do I cite my results reported in open challenges? {\bf A:} To conform with the double-blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\medskip\\ \begin{figure}[t] \centering \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}} \caption{Example of caption. It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.} \label{fig:onecol} \end{figure} \subsection{Miscellaneous} \noindent Compare the following:\\ \begin{tabular}{ll} \verb'$conf_a$' & $conf_a$ \\ \verb'$\mathit{conf}_a$' & $\mathit{conf}_a$ \end{tabular}\\ See The \TeX book, p165. The space after \eg, meaning ``for example'', should not be a sentence-ending space. So \eg is correct, {\em e.g.} is not. The provided \verb'\eg' macro takes care of this. When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word). If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal. However, use it only when there are three or more authors. Thus, the following is correct: ``Frobnication has been trendy lately. It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.'' This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors. \begin{figure*} \centering \begin{subfigure}{0.68\linewidth} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \caption{An example of a subfigure.} \label{fig:short-a} \end{subfigure} \hfill \begin{subfigure}{0.28\linewidth} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \caption{Another example of a subfigure.} \label{fig:short-b} \end{subfigure} \caption{Example of a short caption, which should be centered.} \label{fig:short} \end{figure*} \section{Formatting your paper} \label{sec:formatting} All text must be in a two-column format. The total allowable size of the text area is $6\frac78$ inches (17.46 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be $1\frac{1}{8}$ inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately $1\frac{5}{8}$ inches (4.13 cm) from the bottom edge of the page. \subsection{Margins and page numbering} All printed material, including text, illustrations, and charts, must be kept within a print area $6\frac{7}{8}$ inches (17.46 cm) wide by $8\frac{7}{8}$ inches (22.54 cm) high. Page numbers should be in the footer, centered and $\frac{3}{4}$ inches from the bottom of the page. The review version should have page numbers, yet the final version submitted as camera ready should not show any page numbers. The \LaTeX\ template takes care of this when used properly. \subsection{Type style and fonts} Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access. MAIN TITLE. Center the title $1\frac{3}{8}$ inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title. AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines. The ABSTRACT and MAIN TEXT are to be in a two-column format. MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx.~$\frac{1}{6}$ inch or 0.422 cm). Make sure your text is fully justified---that is, flush left and flush right. Please do not place any additional blank lines between paragraphs. Figure and table captions should be 9-point Roman type as in \cref{fig:onecol,fig:short}. Short captions should be centred. \noindent Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings. FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after. SECOND-ORDER HEADINGS. (For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after. If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line. \subsection{Footnotes} Please use footnotes\footnote{This is what a footnote looks like. It often distracts the reader from the main flow of the argument.} sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. \subsection{Cross-references} For the benefit of author(s) and readers, please use the {\small\begin{verbatim} \cref{...} \end{verbatim}} command for cross-referencing to figures, tables, equations, or sections. This will automatically insert the appropriate label alongside the cross-reference as in this example: \begin{quotation} To see how our method outperforms previous work, please see \cref{fig:onecol} and \cref{tab:example}. It is also possible to refer to multiple targets as once, \eg~to \cref{fig:onecol,fig:short-a}. You may also return to \cref{sec:formatting} or look at \cref{eq:also-important}. \end{quotation} If you do not wish to abbreviate the label, for example at the beginning of the sentence, you can use the {\small\begin{verbatim} \Cref{...} \end{verbatim}} command. Here is an example: \begin{quotation} \Cref{fig:onecol} is also quite important. \end{quotation} \subsection{References} List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include page numbers and the name(s) of editors of referenced books. When you cite multiple papers at once, please make sure that you cite them in numerical order like this \cite{Alpher02,Alpher03,Alpher05,Authors14b,Authors14}. If you use the template as advised, this will be taken care of automatically. \begin{table} \centering \begin{tabular}{@{}lc@{}} \toprule Method & Frobnability \\ \midrule Theirs & Frumpy \\ Yours & Frobbly \\ Ours & Makes one's heart Frob\\ \bottomrule \end{tabular} \caption{Results. Ours is better.} \label{tab:example} \end{table} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. In \LaTeX, avoid using the \texttt{center} environment for this purpose, as this adds potentially unwanted whitespace. Instead use {\small\begin{verbatim} \centering \end{verbatim}} at the beginning of your figure. Please ensure that any point you wish to make is resolvable in a printed copy of the paper. Resize fonts in figures to match the font in the body text, and choose line widths that render effectively in print. Readers (and reviewers), even of an electronic copy, may choose to print your paper in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.pdf} \end{verbatim} } \subsection{Color} Please refer to the author guidelines on the CVPR\ CVPR 2021\ web page for a discussion of the use of color in your document. If you use color in your plots, please keep in mind that a significant subset of reviewers and readers may have a color vision deficiency; red-green blindness is the most frequent kind. Hence avoid relying only on color as the discriminative feature in plots (such as red \vs green lines), but add a second discriminative feature to ease disambiguation. \section{Final copy} You must include your signed IEEE copyright release form when you submit your finished paper. We MUST have this form before your paper can be published in the proceedings. Please direct any questions to the production editor in charge of these proceedings at the IEEE Computer Society Press: \url{https://www.computer.org/about/contact}. {\small \bibliographystyle{ieee_fullname} \section{Introduction} \begin{figure}[t] \centering \includegraphics[width=\linewidth]{images/coverage.png} \caption{\textbf{Comparing training strategies}. (a) prior work used hard-negative mining to make their model more discriminative (\eg,~\cite{hinami-satoh-2018-discriminative}). However, their negatives often contain false negatives due to annotation sparsity. (b) training with our PFP-Net involves creating a set of visually coherent concepts, and then training a phrase detector to discriminate between them. As a result, our minibatches better represent the entirety of the dataset, and careful construction of our concepts helps greatly reduce the false negatives seen when using hard-negative mining.} \label{fig:prior_our_work} \end{figure} \begin{figure*}[h!] \centering \includegraphics[width=0.95\linewidth]{images/model.png} \caption{\textbf{PFP-Net overview}. Our model follows a Faster R-CNN framework~\cite{ren2016faster}. Given an image, we encode it with a ResNet-101~\cite{he2015deep} base encoder and generate a set of candidate boxes using a region proposal network. In addition to learning the similarity between regions and phrases (Section \ref{sec:phrase-region}), our approach introduces a way of automatically identify a set of coarse visually-related concepts (Section \ref{sec:mining})), and train our model to discriminate between these concepts and joint phrase-image representations as well as regularize our language embeddings using an entity-concept loss (see Section \ref{sec:ec_cr}). We also introduce a fine-grained reasoning module (Section~\ref{sec:fgm}) that forces the PFP-Net to discriminate between mutually-exclusive set of fine-grained tokens (see Figure~\ref{fig:model_inference} for an illustration for how FGM is used at test time). Note that our paper's contributions are labeled with dotted boxes. } \label{fig:model} \end{figure*} In phrase detection the goal is to identify regions from a database of images that are relevant to a phrase. This task is significantly more challenging than the localization-only task addressed in prior work, \eg, \cite{10.1145/3078971.3078976, 8237357, fukui-etal-2016-multimodal, 7780862,7780378, kazemzadeh-etal-2014-referitgame, 10.1007/978-3-030-01258-8_16, plummer2017phrase, 7410660, wang2018learning, 10.1007/978-3-319-46484-8_42, yeh2018interpretable, Wang_2019_ICCV}, that only has to localize a phrase within a ground truth image (\ie, ground truth image-phrase pairs are used at test time). Since phrase detectors can distinguish between images that are relevant to a phrase, they are more useful than localization methods in downstream tasks. For example, a phrase detector could tell you if a boat is \emph{a yacht} or \emph{a canoe} and indicate its location, but a phrase localization model would only point to the most likely regions the two phrases may appear in. However, distinguishing between semantically similar phrases that refer to different entities is challenging~\cite{Plummer_2020}. Methods for related phrase grounding tasks have proposed hard-negative mining methods to make their models more discriminative~\cite{hinami-satoh-2018-discriminative}, but these methods suffer from a high rate of false negatives which harms detection performance~\cite{Plummer_2020}. This is because these methods, in part, rely on existing structured datasets like WordNet~\cite{10.1145/219717.219748} to filter out false negatives, but many semantically similar phrases are not accounted for as illustrated in Figure~\ref{fig:prior_our_work}(a). To address these issues, we introduce Phrase-concepts to Fine-grained Phrase-tokens Net (PFP-Net). To avoid the false negatives from hard-negative mining methods used in prior work (\eg~\cite{hinami-satoh-2018-discriminative}) we automatically construct coarse concepts: visually coherent groups of entities (\eg animals vs automobiles). The model then is trained to discriminate between a phrase and its unrelated concepts as shown in Figure~\ref{fig:prior_our_work}(b). This presents a significant challenge, since we need to create concept groups that are fine-grained enough to provide informative negatives to improve the discriminative power of our model, while being coarse enough to minimize false negatives. We find that simply using K-Means clustering over a language embedding like BERT~\cite{devlin2019bert} or GLoVE~\cite{pennington-etal-2014-glove} results in noisy concept groups. This is likely because these language embeddings were trained to embed words used in similar contexts near each other, but not to distinguish between visually different entities. For example, they would embed \emph{school} and \emph{teacher} nearby since they are related, even though they are not visually similar which is important for phrase detection. In addition, K-means forces all entities to belong to a cluster, even the outliers, which we found often results in incoherent groups. To avoid these issues, we use visually discriminative language features to represent words. Then we use a density-based clustering to create a set of visually coherent concepts, which we assign to phrases based on their semantic similarity. Using our concepts provide two advantages over prior work. First, by creating coarse concepts we minimize false negatives. Second, comparing a phrase to all of the concepts ensures each image region sees a balanced distribution of easy and hard negative samples, which can improve performance over focusing solely on hard-negatives~\cite{wu2018sampling}. However, while knowing when two semantically similar words are mutually exclusive may be challenging to automate in general, we find that we can mine sets of words that typically refer to difficult fine-grained differences (\eg \emph{a red shirt} vs.\ \emph{a blue shirt}). We take advantage of these sets of mutually exclusive words adding a fine-grained module to our model. The module differentiates between fine grained words and then augment the main model score with its prediction. An overview of PFP-Net is provided in Figure~\ref{fig:model}. Our contributions can be summarized as: \begin{itemize} \setlength\itemsep{0em} \item A novel model, PFP-Net, that achieves a 1-1.5 point gain over SOTA on phrase detection over by mining visually coherent concepts and then learning to discriminate between them. \item A fine grained reasoning module that boosts performance by 1-4 points over affected phrases by learning critical visual cues to differentiate between otherwise visually similar instances. \item A novel method for automatically mining visually coherent groups that improves the distribution of our minibatches to better represent the training data. \end{itemize} \section{Visually coherent concept mining} \label{sec:mining} Vision-language datasets are only annotated with a few examples of the nearly infinite valid annotations. Several methods have been proposed to increase the number of annotations (\eg~\cite{gupta2020contrastive,hinami-satoh-2018-discriminative,Plummer_2020,VendrovICLR2016order}), but prior work in identifying good negative examples leads to significant noise that hurts performance~\cite{Plummer_2020}. As discussed in the Introduction, this is in part because the structured language datasets used by prior work also contain incomplete annotations, resulting in many false positives. Instead, we automatically identify a set of concepts, sets of entities like clothing or buildings, and then use the concepts that a phrase is not related to as negatives during training (discussed in Section~\ref{sec:model}). We define our concepts as $C = \{c_j = \{e_{k}\}_{k=0}^{k=L_j}\}_{j=0}^{j=M}$ where $e$ represents an entity (a noun referring to object) in the dataset vocabulary, $M$ is the total number of concepts, and $L_j$ is the total number of entities in concept $c_j$. Note also that $c_1 \cup c_2 .. \cup c_M = \emptyset$. We describe how we obtain these concepts in Section~\ref{sec:concepts}. Then, we describe how we assign each phrase to its set of negative concepts in Section~\ref{sec:assignment}. \begin{figure}[t] \centering \includegraphics[width=0.95\linewidth]{images/kmeansvsdbscan.png} \caption{\textbf{Qualitative comparison of visual concept clustering methods using ViCO~\cite{gupta2019vico} embeddings.} As seen in \textbf{(a)} using K-means creates many noisy concept entries (denoted in red) since it requires all words to be cluster members. In contrast, in \textbf{(b)} we show concept groups created by the density-based clustering approach DBSCAN, which contains far less noise since it will only group words that are tightly coupled. See Figure~\ref{fig:visual_noise} for a quantitative comparison of concept group quality.} \label{fig:concepts} \end{figure} \subsection{Obtaining concept groups} \label{sec:concepts} We start by extracting the entities (\eg, t-shirt, dog, etc) from our phrases. A straightforward approach for grouping semantically similar phrases would be to perform K-means clustering over some pretrained language model like GLoVE \cite{pennington-etal-2014-glove} or BERT\cite{devlin2019bert}. However, text-only representations are not trained to capture visual-semantic relationships (\eg, they typically consider a \emph{teacher} and \emph{school} to be very similar, even though they are visually different). Thus, prior work has demonstrated that training word embeddings so that they capture visual-semantic relationships is key to good performance on many tasks~\cite{burns2019iccv}. While several transformer-based vision-language embeddings have been introduced (\eg,~\cite{chen2020uniter,lu2019vilbert,su2019vlbert}), they are trained in a weakly-supervised setting where related, but semantically different entities which are important for our method are difficult to capture. Therefore, we selected ViCo \cite{gupta2019vico} as it was trained to discriminate between visually different entities. Simply selecting good language features is insufficient to get good concepts, however. Figure~\ref{fig:concepts}(a) shows that using K-means clustering over ViCo features produces noisy concepts. This is because K-means does not impose a constraint over each cluster density. \Ie, outliers in the embedding space are still clustered with other concepts even though there is little evidence they belong to the same concept. Instead, we leverage DBSCAN, \cite{9356727} whose clusters formed from entities that fall within a density threshold $\epsilon$. Thus, only tightly knit clusters are produced and noisy entities are discarded, creating more visually coherent concepts as shown in Figure~\ref{fig:concepts}(b). \smallskip \noindent\textbf{Concept group quality.} We evaluated different methods of creating concept groups via manual inspection. For each set of concepts, we counted the number of incoherent concepts (where 50\% of the concept entities or more are not visually coherent). Figure~\ref{fig:visual_noise} compares different methods of clustering, language embeddings, and hyperparameters affecting the number of clusters ($\epsilon$ for DBSCAN or $K$ for K-means). Our results show that DBSCAN over ViCo embeddings creates the most visually-coherent concepts. \subsection{Assigning phrases to concepts} \label{sec:assignment} After creating our concepts in Section~\ref{sec:concepts}, we now need to assign each phrase to its set of relevant concepts. A simple assignment process is to pair a phrase with a concept with which it shares an entity. We call this simple entity-phrase matching. This approach, however, disregards phrase-concept pairs that don't share entities but are nevertheless viable pairs. Thus, we make these assignments using semantic similarity as detailed below. First, given a phrase $p$, and phrase entities $p_e$, we consider the phrase concepts $p_{c1}$ which are the result of a simple entity-phrase matching: $p_{c1} = \{c : c \cap p_{e} \neq \emptyset, c \in C \}$. We also consider concepts $p_{c2}$ which are the result of a similarity based assignment process. This method is designed for when a phrase is still relevant to a concept but they do not explicitly share any entities (\eg \emph{hoody} with concept: \emph{shirt and sweatshirt})). More concretely, for each entity $e \in p_{e}$, we look up the entity ViCo representation. We will denote this representation as $\widetilde{e}$. We also compute a concept $c$ representation by averaging its entities ViCo feature vectors. We will denote this as $\widetilde{c}$. Now assume $\textnormal{sim}(\widetilde{e}, \widetilde{c}) = \frac{\widetilde{e} \textnormal{ }\widetilde{c}}{||\widetilde{e}|| \textnormal{ } ||\widetilde{c}||}$, we compute an "association" score between the entity $e$ and a concept $c$ as follows: \begin{equation} \textnormal{Assoc}(e, c) =\small \frac{\textnormal{exp}(\textnormal{sim}(\widetilde{e}, \widetilde{c})/\tau)}{\sum_{j=0}^{M} \textnormal{exp}(\textnormal{sim}(\widetilde{e}, \widetilde{c})/\tau)} \label{eq:assoc} \end{equation} \begin{figure}[t!] \includegraphics[width=\linewidth]{images/noise.drawio.png} \caption{\textbf{Quantitative comparison of concept creation methods}. For each algorithm/embedding combination, we produce several sets of clusters with varying size (\ie total number of resulting clusters). For each set, we calculate the visual coherence noise (see Section ~\ref{sec:concepts} for noise definition). We observe that DBSCAN exhibits consistently less noise than KMeans. See Section~\ref{sec:concepts} for further discussion.} \label{fig:visual_noise} \end{figure} From here, $p_{c2} = \{ c : \textnormal{Assoc}(e, c) > \gamma, e \in p_{e}, c \in C\}$. This way, the phrase is assigned to concepts with which it shares the most statistical ViCo based similarity. The final set of phrase concepts is therefore, $p_{c} = p_{c1} \cup p_{c2}$. Thus, a phrase that is near multiple concepts will be assigned to all of them, and so we will only contrast phrases against groups that we are confident are negatives. \section{Learning From General Phrase-concepts to Fine-grained Phrase-tokens} \label{sec:model} In phrase detection the task is to determine what images regions $r$ can be described by phrase $p$. Prior work addressing phrase localization only selects image regions from the ground truth image, but in phrase detection regions from all images in the dataset are considered. To address this problem, we introduce a neural network that learns from Phrase-concepts to Fine-grained Phrase-tokens (PFP-net). For a fair comparison to prior work~\cite{hinami-satoh-2018-discriminative,Plummer_2020}, PFP-net adopts a Faster R-CNN architecture~\cite{ren2016faster} to obtain a set of image regions, which we describe briefly in Section~\ref{sec:phrase-region}. Then, we use the concept mining procedure described in Section~\ref{sec:mining} in two ways. First, in Section~\ref{sec:ec_cr} we use the mined entity based concepts to identify the general object type of a phrase which provides a broad coverage informative negatives. Second, we obtain the initial set of concepts described in Section~\ref{sec:concepts} over adjectives, where the concepts typically denote mutually-exclusive fine-grained differences in phrases (\eg, \emph{a red truck} vs.\ \emph{a green truck}). Thus, in Section~\ref{sec:fgm} we use these adjective-concepts in our model to differentiate between mutually exclusive phrases. \subsection{Encoding phrases and image regions} \label{sec:phrase-region} Following~\cite{hinami-satoh-2018-discriminative,Plummer_2020}, we adopt a Faster R-CNN architecture \cite{ren2016faster} using a ResNet-101~\cite{he2015deep} encoder as a backbone in order to encode regions and mean-pooled HGLMM Fisher vectors \cite{hglmm} to represent our phrases. However, we note that the contributions of our approach, namely the concept-conditioned training procedure and fine-grained module , described in the subsequent sections, are adaptable to any detection architecture. We obtain a final region and phrase representation using a pair of 2-layer multi-layer perceptrons (MLP), one that takes the HGLMM features as input and the other that takes the ROI-pooled features from the Faster R-CNN encoder. After obtaining a good feature representation for the image regions $r$ and phrase (or entity/concept) features $p$ from their respective MLPs, we compute the similarity between them via: \begin{equation} PR_{Sim}(r, p) = \text{MLP}_{3\text{Layer}}(r \odot p), \label{eq:sim} \end{equation} \noindent where $\odot$ denotes an elementwise product. Note that this function is represented as the ``Joint MLP'' in Figure \ref{fig:model}. Following Plummer~\etal~\cite{Plummer_2020}, partway through training we re-initialize the 2-layer region and text MLPs via canonical correlation analysis (CCA)~\cite{Hotelling1992}, which Plummer~\etal found greatly improved phrase detection performance. We also apply an L2 regularization ($L_{reg}$) with the initial CCA weights when fine-tuning these MLPs to avoid catastrophic forgetting. Our model is trained using a \textbf{phrase-region similarity loss (PR)}. Formally, let $K$ be the number of image region-query pairs in a batch, $s^{p}$ be a region-phrase pair score, $l^p$ be its $-1/1$ label indicating whether it is a negative/positive pair, $a$ be the conditional embedding weights before the softmax, then the base loss is: \begin{equation} \label{eqn:base} L_{base} = \sum_{i=1}^{K} \log(1 + \exp(-l^{p}_i s^{p}_i)) + \lambda_{reg} L_{reg} \end{equation} \noindent where $\lambda_{reg}$ is a scalar parameter. \subsection{Expanding batch coverage using concepts} \label{sec:ec_cr} One of the primary contributions is using the generated concepts described in Section~\ref{sec:mining} to provide a set of informative negatives that represent our entire training dataset. We use these concepts in two ways: \textbf{(1)} as additional samples for the output of the Eq~\ref{eq:sim} computed between the regions and concepts (referred to as ``Concept-region Loss in Figure~\ref{fig:model}), and \textbf{(2)} to regularize the MLP textual representation (referred to as ``Entity-Concept Loss in Figure~\ref{fig:model}). Note that our concepts are encoded using text features. See our detailed explanations below. \smallskip \noindent\textbf{Concept-region Loss (CR)} Given a phrase $p$, and its set of related concepts $p_c$, we use the unrelated concepts $\overline{p_c} = \{c : c \not\in p_c, c \in C \}$ as negative samples. We then pair those concepts with the region $r$ associated with phrase $p$ to obtain negative concept-region pairs $\overline{r_c}$. We then simply concatenate the concept-region pairs to the phrase-region pairs and then feed their scores through the logistic loss described in Eq~\ref{eqn:base}. Let $s^{c}$ be region-negative concept score, $L$ be the total number of negative concept-region pair, then the loss: \begin{equation} \label{eqn:cr} L_{CR} = \sum_{i=1}^{L} \log(1 + \exp(s^{c}_i)) \end{equation} \noindent\textbf{Entity-concept Loss (EC).} In addition to ensuring that each image region is embedded near its related concepts, we also encourage phrases to embed near its associated concepts and far from others. For each phrase in our batch, we extract their entities (nouns). We randomly sample one entity for each phrase. We then compute an entity-concept score $s^{e}$ by taking the dot product between the $\ell_2$ normalized entity features and the concept representations followed by a Sigmoid activation function. Now assuming $l^e$ is the entity-concept -1/1 label indicating whether it is a negative/positive pair, and $S$ is the total number of entity-concept samples, then the entity regularization loss can be computed as: \begin{equation} \label{eq:ec} L_{EC} = \sum_{i=1}^{S} \log(1 + \exp(-s^{e}_i l^{e}_i)) \end{equation} \subsection{Fine-Grained reasoning Module (FGM)} \label{sec:fgm} After obtaining our initial phrase-region similarity scores using Eq~\ref{eq:sim}, we modify it by taking into account fine-grained differences between phrases. In particular, we have the model predict the scores of sets of mutually exclusive fine-grained tokens, and then use them to augment the main model region-phrase scores. More formally, given a set of tokens sets $F$ where each $f \in F$ is itself a set of fine grained tokens, our novel fine grain module learns to discriminate between each element of $f$. We obtain $f$ using the same bottom up approach used to mine for concepts over adjectives. However, in this case, we train the model to discriminate between each concept tokens rather than the concepts themselves, since adjectives typically group together words that denote fine-grained differences between phrases, as mentioned earlier. While other parts-of-speech could also be used (\eg, verbs also denote a mutually-exclusive state of an object), we found that our datasets did not contain enough samples for training a FGM module over other concept types. The FGM module encodes image regions using a set of convolutional layers and then performs multi-label classification on each set of fine grained tokens $f$. Now, assume $t$ is a fine grained token such that $t \in f$, $R$ be the number of region-fine grained token pairs, $s^t$ be the region-token score, and $l^t$ be its 0/1 label indicating whether it is a positive/negative region-token pair, then: \begin{equation} \label{eq:fgm} L_{FGM} = \sum_{i}^{R} l^{t}_{i} \log{s^{t}_{i}} + (1 - l^{t}_{i})\log(1-s^{t}_{i})). \end{equation} \noindent With this, the final loss for PFP-net is: \begin{equation} \label{eq:final_loss} L_{final} = L_{base} + \lambda_{CR} L_{CR} + \lambda_{EC} L_{EC} + L_{FGM} \end{equation} \subsection{Model inference} At test time the FGM module's scores are augmented with the base model's phrase-region scores. Given a phrase $p$, the phrase tokens $p_t$ and $t\in p_t$, a phrase-region pair score $s^p$, and token-region score $s^{t}$, then the final score $s^f$: \begin{equation} s^{f} = (1-\lambda_{f})s^{p} + \lambda_{f}s^{t}, \label{eq:final_score} \end{equation} \noindent where $\lambda_{f}$ is a scalar parameter that applies for every $t \in f$. See Figure~\ref{fig:model_inference} for an illustration. \begin{figure}[t!] \centering \includegraphics[width=\linewidth]{images/inference.png} \caption{\textbf{PFP-Net inference overview} During inference, the model takes in three inputs: the image, the phrase, and any applicable fine grained token within the phrase. The model then produces two scores: a phrase-region score and a fine grained token-region score. Finally, the weighted sum of the scores is then computed following Eq~\ref{eq:final_score} to produce the final score $s^f$.} \label{fig:model_inference} \end{figure} \section{Experiments} \label{experiments} \noindent\textbf{Datasets:} We evaluate our PFP-net on the two phrase grounding datasets which are the best suited for benchmarking phrase detection. First, we use Flickr30K Entities \cite{7410660} that consists of 276K bounding boxes in 32K images for the noun phrases associated with each image's descriptive captions (5 per image) from the Flickr30K dataset \cite{young-etal-2014-image}. We use the official splits \cite{7410660} that consist of 30K/1K/1K train/test/validation images. Second, we evaluate on RefCOCO+ \cite{10.1007/978-3-319-46475-6_5}, which consists of 19,992 images from the COCO dataset \cite{10.1007/978-3-319-10602-1_48} that have been labeled with 141,564 region descriptions. We use the official split \cite{10.1007/978-3-319-46475-6_5}, which splits the train/val and testing sets 16K/1.5K/1.5K. Both datasets are licensed under creative commons. \smallskip \noindent\textbf{Metrics:} We follow the evaluation protocols of Plummer~\etal~\cite{Plummer_2020}. For every image we obtain the most likely region and confidence score for every phrase in our test split. For any ground truth phrases in an image, we consider a phrase successfully localized if the predicted bounding box has at least 0.5 intersection-over-union with its associated ground truth bounding box. Then, we compute average precision (AP) for each phrase and then split them into zero-shot, few-shot, and common sets, based on if they didn't occur in our training split, if they had between 1-100 occurrences, or if they occurred more than 100 times, respectively. We then report an overall mAP for each set of phrases, as well as the average of them for an overall performance score. This procedure ensures that the zero-shot and few-shot phrases are not over represented compared to the common phrases, since the zero- and few-shot sets have more unique phrases, but represent a smaller portion of overall instances. \smallskip \noindent\textbf{Implementation details.} We train PFP-net with an ADAM \cite{kingma2017adam} optimizer using the hyperparameter settings of Plummer~\etal~\cite{Plummer_2020} for a fair comparison except where stated. We set all hyperparameters introduced by our work via grid search. Specifically, when creating our concept groups using DBSCAN we set $\epsilon$ (the threshold to determine clustering tolerance), and the $\gamma, \tau$ in Eq ~\ref{eq:assoc}. The used values were $\gamma = 0.2, \tau = 0.01$ for both datasets. $\epsilon = 0.43$ for Flickr30K and $\epsilon = 0.53$ for RefCOCO+. For entity regularization, we set $\lambda_{ER} = 0.1$ and $\lambda_{CR} = 1$ for both Flickr30K Entities and RefCOCO+. We trained our model using a single NVIDIA RTX 8000 GPU using an internal cluster. \smallskip \begin{figure*}[t!] \centering \includegraphics[width=0.99\linewidth]{images/qualt_full.png} \caption[]% {\small Comparison between our model vs prior work model on positive (ground truth) and randomly sampled negative phrases scores. Our model is significantly better able to separate positive vs negative phrases scores than prior work. This result is key in enhancing model disriminative ability which in turn translates to improvement on the mAP scores as documented in Table ~\ref{tb:main_results}. See Section~\ref{sec:model_analysis} for additional discussion.} \label{fig:qualt} \end{figure*} \begin{figure}[h!] \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_groups_flickr.png} \caption{Flickr30K Entities} \end{subfigure} \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_groups_refcoco.png} \caption{RefCOCO+} \end{subfigure} \caption{Effect of the concept set size $|C|$ has on detection performance. See Section~\ref{sec:model_analysis} for discussion.} \label{fig:hyper_groups} \end{figure} \begin{figure}[h!] \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_gamma_flickr.png} \caption{Flickr30K Entities} \end{subfigure} \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_gamma_refcoco.png} \caption{RefCOCO+} \end{subfigure} \caption{Effect of changing $\gamma$ in Eq~\ref{eq:assoc} has on detection performance. See Section~\ref{sec:model_analysis} for discussion.} \label{fig:hyper_gamma} \end{figure} \begin{figure}[h!] \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_lambda_flickr.png} \caption{Flickr30K Entities} \end{subfigure} \begin{subfigure}{0.49\columnwidth} \centering \includegraphics[width=\textwidth]{images/hyper_lambda_refcoco.png} \caption{RefCOCO+} \end{subfigure} \caption{Effect of changing $\lambda_{EC}$ in Eq~\ref{eq:final_loss} has on detection performance. See Section~\ref{sec:model_analysis} for discussion.} \label{fig:hyper_lambda_ec} \end{figure} \subsection{Results} \label{sec:results_discussion} Table \ref{tb:main_results} compares ablations of PFP-Net with the current state-of-the-art on phrase detection. Comparing the last line of Table \ref{tb:main_results}(b) to the results from prior work in Table \ref{tb:main_results}(a) we get 1-1.5 point gain in mAP over the state of the art. Much of this gain came from common phrases, where we achieved a 2 point gain over prior work. In Table \ref{tb:main_results}(b) we also note the contribution of each component of our model, with the biggest gain coming from using our concept-region loss (CR). Note that in Table \ref{tb:main_results}(a) the hard-negative mining approach of Hinami and Satoh~\cite{hinami-satoh-2018-discriminative} achieved a much more modest 0.2 point gain. Note that the results in our tables are the results of 5 different runs to ensure our reported gains are not due to model variance. We note that CR produces consistent improvements on both datasets across all phrase types (zero shot, few shot, common). EC results in small, but largely consistent improvements on both datasets. These results indicate that our negative samples are effective in improving our model discriminative ability. In addition to the gains from using our concepts, we note that Table \ref{tb:main_results}(b) reports that the FGM module from Section~\ref{sec:fgm} further boosts performance by a small but consistent gain for both datasets. Moreover, in Table \ref{fgm_results}, we report performance on only the phrases affected by our FGM module, which reports a significant 3.8 point improvement on Flickr30K Entities, while also obtaining a 1.5 point improvement on RefCOCO+. Finally, Table~\ref{tb:main_results_ppa} reports the performance of PFP-Net using positive phrase augmentation (PPA)~\cite{Plummer_2020}, which reduces annotation sparsity by pairing ground truth phrases with plausible positive phrases using WordNet~\cite{10.1145/219717.219748}. We note that PPA does not change the relative gains of the detection methods, but obtains higher absolute performance. \subsection{PFP-Net Analysis} \label{sec:model_analysis} To further understand the behaviour of our model, we examine several qualitative images in Figure ~\ref{fig:qualt}. Compared to prior work, our model maintains its confidence in positive ground truth phrases and correctly localize them while also having significantly lower scores for negative phrases. For example, in Figure~\ref{fig:qualt}(b) the negative phrase "sofa" in the left image has a significantly lower score than prior work in Figure ~\ref{fig:qualt}(a). This is evidence that simply optimizing for phrase localization only does not translate to discriminative detection power. As shown in the center image of Figure \ref{fig:qualt} prior work fails to differentiate between white and black, resulting in an incorrect localization of the "white dog". Our PFP-Net, on the other hand, with the help of our FGM module, correctly localizes the phrase. We further investigate our model behaviour via sensitively analysis of its hyperparameters. First, we examine the effect of changing the number of used concepts in Figure~\ref{fig:hyper_groups}. For each dataset, we progressively increase DBSCAN's density threshold until we can not generate more concepts. We observe that overall more concepts improve performance on both datasets. This is probably because more concepts split the language distribution into finer portions, thus improving the accuracy of our phrase-concept assignment in Section~\ref{sec:assignment} which translate to better performance overall. Second, we examine the effect of changing Eq \ref{eq:assoc} $\gamma$ parameter in Figure~\ref{fig:hyper_gamma} that controls whether a given concept is assigned to a phrase (\ie they are related). We vary the parameter between 0 (a phrase is assigned to every given concept) and 1 (a phrase is only assigned to concepts that share tokens with). We note that performance is at best for values of $\gamma$ between (0.2, 0.6). Performance drop for $\gamma < 0.2$ as we assign wrong concepts to phrases. Performance also drops when $\gamma > 0.8$ which indicates the importance of our similarity based matching component. In other words, simply assigning phrases to concepts that only share tokens with is not sufficient. This is because, as noted in Section~\ref{sec:assignment}, there are many concepts that are still relevant to certain phrases but also do not share tokens with. Third, and finally, we study the effect of changing $\lambda_{EC}$ used in Eq \ref{eq:final_loss} in Figure~\ref{fig:hyper_lambda_ec} that controls the contribution of our Concept-Entity loss. Overall, we note that performance for both datasets is best at 0.1. We note that both the $\gamma$ from Eq \ref{eq:assoc} and $\lambda_{EC}$ from Eq \ref{eq:final_loss} achieved the highest performance at the same parameter value, demonstrating that these hyperparameter val generalize across datasets, whereas PFP-Net tends to favor larger number of concepts. \section{Conclusion} In this work, we introduced a new phrase detection mode (PFP-NET) that significantly improves performance by 1-1.5 points on two phrase detection data-sets. The model does so by incorporating visually coherent clusters (concepts) to sample negative concept-region as well concept-entity pairs that effectively improve the model discriminative abilities when compared to prior work. Our model further improves performance by incorporating a novel fine grained module that learns to discriminate between adjective fine grained tokens. Note that although our experiments used the Faster R-CNN framework to fairly compare to prior work, the contributions made by our paper (concept-based sampling and fine grained module) are modular and so can be adapted to any underlying detection framework. \section{Related work} Most prior work in phrase grounding has focused on the localization-only tasks, where you are provided a ground truth image-phrase pair and have to identify the relevant image region (\eg,~\cite{Bajaj_2019_ICCV,hu2015natural,8237357,kazemzadeh-etal-2014-referitgame,plummer2017phrase,10.1007/978-3-030-01258-8_16,yang2019fast}). However, Plummer~\etal~\cite{Plummer_2020} demonstrated that these methods tend to overfit to the localization task, \ie, they improve performance on localization, but reduce detection performance. This is partly because distinguishing between similar phrases in the localization-only task is unnecessary since most images only contain a reference to a single object of the same type~\cite{Plummer_2020}. Thus, a localization model that is given the phrase \emph{a young teenager} as input could look for the object category ``person'' to identify the right object most of the time, whereas a detector also has to determine if it exists at all. Some work falls between detection and prior work in localization, where a ground truth image-phrase pair is not provided~\cite{hinami-satoh-2018-discriminative,zhang2017discriminative}, but they severely limit the number of negative phrases at test time. Thus, methods from these tasks also often do not generalize to phrase detection~\cite{Plummer_2020}. Also relevant to our work is Gupta~\etal\cite{gupta2020contrastive}, who introduce a sampling method for weakly-supervised phrase grounding. They use the language model BERT \cite{devlin2019bert} to sample negatives that preserve the phrase context (\eg a walking [Blank], where Blank can not be a car). This limits its application since it requires phrase grounding datasets to be annotated with this context. However, this approach also relies on noisy external language databases like WordNet~\cite{10.1145/219717.219748} to identify false negatives. In contrast, our approach creates coherent concepts that boost performance without relying on contextual annotations. \section{Introduction} Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. 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This system is new because it handles the following cases: A, B. Previous systems [Zeus et al. 1968] didn't handle case B properly. Ours handles it by including a foo term in the bar integral. ... The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know. It displayed the following behaviours which show how well we solved cases A and B: ... \end{quotation} As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. FAQ: Are acknowledgements OK? No. Leave them for the final copy. \begin{figure}[t] \begin{center} \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}} \end{center} \caption{Example of caption. 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This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors. If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal. For this citation style, keep multiple citations in numerical (not chronological) order, so prefer \cite{Alpher03,Alpher02,Authors14} to \cite{Alpher02,Alpher03,Authors14}. \begin{figure*} \begin{center} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \end{center} \caption{Example of a short caption, which should be centered.} \label{fig:short} \end{figure*} \section{Formatting your paper} All text must be in a two-column format. The total allowable width of the text area is $6\frac78$ inches (17.5 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1.0 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1.0 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page. \subsection{Margins and page numbering} All printed material, including text, illustrations, and charts, must be kept within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm) high. Page numbers should be in footer with page numbers, centered and .75 inches from the bottom of the page and make it start at the correct page number rather than the 4321 in the example. To do this fine the line (around line 23) \begin{verbatim} \setcounter{page}{4321} \end{verbatim} where the number 4321 is your assigned starting page. Make sure the first page is numbered by commenting out the first page being empty on line 46 \begin{verbatim} \end{verbatim} \subsection{Type-style and fonts} Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access. MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title. AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines. The ABSTRACT and MAIN TEXT are to be in a two-column format. MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422 cm). Make sure your text is fully justified---that is, flush left and flush right. Please do not place any additional blank lines between paragraphs. Figure and table captions should be 9-point Roman type as in Figures~\ref{fig:onecol} and~\ref{fig:short}. Short captions should be centred. \noindent Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings. FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after. SECOND-ORDER HEADINGS. (For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after. If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line. \subsection{Footnotes} Please use footnotes\footnote {This is what a footnote looks like. It often distracts the reader from the main flow of the argument.} sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. \subsection{References} List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include the name(s) of editors of referenced books. \begin{table} \begin{center} \begin{tabular}{|l|c|} \hline Method & Frobnability \\ \hline\hline Theirs & Frumpy \\ Yours & Frobbly \\ Ours & Makes one's heart Frob\\ \hline \end{tabular} \end{center} \caption{Results. Ours is better.} \end{table} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. Please ensure that any point you wish to make is resolvable in a printed copy of the paper. Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print. Many readers (and reviewers), even of an electronic copy, will choose to print your paper in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage[dvips]{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.eps} \end{verbatim} } \subsection{Color} Please refer to the author guidelines on the CVPR 2018 web page for a discussion of the use of color in your document. \section{Final copy} You must include your signed IEEE copyright release form when you submit your finished paper. We MUST have this form before your paper can be published in the proceedings. {\small \bibliographystyle{ieee} \section{Introduction} Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version. \subsection{Language} All manuscripts must be in English. \subsection{Dual submission} Please refer to the author guidelines on the CVPR 2021~web page for a discussion of the policy on dual submissions. \subsection{Paper length} Papers, excluding the references section, must be no longer than eight pages in length. The references section will not be included in the page count, and there is no limit on the length of the references section. For example, a paper of eight pages with two pages of references would have a total length of 10 pages. {\bf There will be no extra page charges for CVPR 2021.} Overlength papers will simply not be reviewed. This includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this \LaTeX\ guide already sets figure captions and references in a smaller font. The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts. The reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven. \subsection{The ruler} The \LaTeX\ style defines a printed ruler which should be present in the version submitted for review. The ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution. If you are preparing a document using a non-\LaTeX\ document preparation system, please arrange for an equivalent ruler to appear on the final output pages. The presence or absence of the ruler should not change the appearance of any other content on the page. The camera ready copy should not contain a ruler. (\LaTeX\ users may use options of cvpr.cls to switch between different versions.) Reviewers: note that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly. Just use fractional references (e.g.\ this line is $095.5$), although in most cases one would expect that the approximate location will be adequate. \subsection{Mathematics} Please number all of your sections and displayed equations. It is important for readers to be able to refer to any particular equation. Just because you didn't refer to it in the text doesn't mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin's description of how to write mathematics: \url{http://www.pamitc.org/documents/mermin.pdf}. \subsection{Blind review} Many authors misunderstand the concept of anonymizing for blind review. Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available. Blind review means that you do not use the words ``my'' or ``our'' when citing previous work. That is all. (But see below for techreports.) Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith; it says that you are building on her work. If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work. An example of a bad paper just asking to be rejected: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Removed for blind review \end{quote} An example of an acceptable paper: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of the paper of Smith \etal [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213. \end{quote} If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work. In such cases, include the anonymized parallel submission~\cite{Authors14} as additional material and cite it as \begin{quote} [1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as additional material {\tt fg324.pdf}. \end{quote} Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report. For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a techreport for further details. Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''. Then submit the techreport as additional material. Again, you may not assume the reviewers will read this material. Sometimes your paper is about a problem which you tested using a tool which is widely known to be restricted to a single institution. For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal. You can handle this paper like any other. Don't write ``We show how to improve our previous work [Anonymous, 1968]. This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''. That would be silly, and would immediately identify the authors. Instead write the following: \begin{quotation} \noindent We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems [Zeus et al. 1968] didn't handle case B properly. Ours handles it by including a foo term in the bar integral. ... The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know. It displayed the following behaviours which show how well we solved cases A and B: ... \end{quotation} As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. \medskip \noindent FAQ\medskip\\ {\bf Q:} Are acknowledgements OK?\\ {\bf A:} No. Leave them for the final copy.\medskip\\ {\bf Q:} How do I cite my results reported in open challenges? {\bf A:} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\medskip\\ \begin{figure}[t] \begin{center} \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}} \end{center} \caption{Example of caption. It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.} \label{fig:long} \label{fig:onecol} \end{figure} \subsection{Miscellaneous} \noindent Compare the following:\\ \begin{tabular}{ll} \verb'$conf_a$' & $conf_a$ \\ \verb'$\mathit{conf}_a$' & $\mathit{conf}_a$ \end{tabular}\\ See The \TeX book, p165. The space after \eg, meaning ``for example'', should not be a sentence-ending space. So \eg is correct, {\em e.g.} is not. The provided \verb'\eg' macro takes care of this. When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word.) However, use it only when there are three or more authors. Thus, the following is correct: `` Frobnication has been trendy lately. It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.'' This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors. If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal. For this citation style, keep multiple citations in numerical (not chronological) order, so prefer \cite{Alpher03,Alpher02,Authors14} to \cite{Alpher02,Alpher03,Authors14}. \begin{figure*} \begin{center} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \end{center} \caption{Example of a short caption, which should be centered.} \label{fig:short} \end{figure*} \section{Formatting your paper} All text must be in a two-column format. The total allowable width of the text area is $6\frac78$ inches (17.5 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1.0 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1.0 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page. \subsection{Margins and page numbering} All printed material, including text, illustrations, and charts, must be kept within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm) high. Page numbers should be in footer with page numbers, centered and .75 inches from the bottom of the page and make it start at the correct page number rather than the 4321 in the example. To do this fine the line (around line 20) \begin{verbatim} \setcounter{page}{4321} \end{verbatim} where the number 4321 is your assigned starting page. \subsection{Type-style and fonts} Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access. MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title. AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines. The ABSTRACT and MAIN TEXT are to be in a two-column format. MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422 cm). Make sure your text is fully justified---that is, flush left and flush right. Please do not place any additional blank lines between paragraphs. Figure and table captions should be 9-point Roman type as in Figures~\ref{fig:onecol} and~\ref{fig:short}. Short captions should be centred. \noindent Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings. FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after. SECOND-ORDER HEADINGS. (For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after. If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line. \subsection{Footnotes} Please use footnotes\footnote {This is what a footnote looks like. It often distracts the reader from the main flow of the argument.} sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. \subsection{References} List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include the name(s) of editors of referenced books. \begin{table} \begin{center} \begin{tabular}{|l|c|} \hline Method & Frobnability \\ \hline\hline Theirs & Frumpy \\ Yours & Frobbly \\ Ours & Makes one's heart Frob\\ \hline \end{tabular} \end{center} \caption{Results. Ours is better.} \end{table} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. Please ensure that any point you wish to make is resolvable in a printed copy of the paper. Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print. Many readers (and reviewers), even of an electronic copy, will choose to print your paper in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage[dvips]{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.eps} \end{verbatim} } \subsection{Color} Please refer to the author guidelines on the CVPR 2021~web page for a discussion of the use of color in your document. \section{Final copy} You must include your signed IEEE copyright release form when you submit your finished paper. We MUST have this form before your paper can be published in the proceedings. Please direct any questions to the production editor in charge of these proceedings at the IEEE Computer Society Press: \url{https://www.computer.org/about/contact}. {\small \bibliographystyle{ieee_fullname} \section{Introduction} After receiving paper reviews, authors may optionally submit a rebuttal to address the reviewers' comments, which will be limited to a {\bf one page} PDF file. Please follow the steps and style guidelines outlined below for submitting your author response. Note that the author rebuttal is optional and, following similar guidelines to previous CVPR conferences, it is meant to provide you with an opportunity to rebut factual errors or to supply additional information requested by the reviewers. It is NOT intended to add new contributions (theorems, algorithms, experiments) that were not included in the original submission. You may optionally add a figure, graph or proof to your rebuttal to better illustrate your answer to the reviewers' comments. Per a passed 2018 PAMI-TC motion, reviewers should not request additional experiments for the rebuttal, or penalize authors for lack of additional experiments. This includes any experiments that involve running code, e.g., to create tables or figures with new results. \textbf{Authors should not include new experimental results in the rebuttal}, and reviewers should discount any such results when making their final recommendation. Authors may include figures with illustrations or comparison tables of results reported in the submission/supplemental material or in other papers. The rebuttal must adhere to the same blind-submission as the original submission and must comply with this rebuttal-formatted template. \subsection{Response length} Author responses must be no longer than 1 page in length including any references and figures. Overlength responses will simply not be reviewed. This includes responses where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this \LaTeX\ guide already sets figure captions and references in a smaller font. \section{Formatting your Response} {\bf Make sure to update the paper title and paper ID in the appropriate place in the tex file.} All text must be in a two-column format. The total allowable width of the text area is $6\frac78$ inches (17.5 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The top margin should begin 1.0 inch (2.54 cm) from the top edge of the page. The bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page. Please number all of your sections and any displayed equations. It is important for readers to be able to refer to any particular equation. Wherever Times is specified, Times Roman may also be used. Main text should be in 10-point Times, single-spaced. Section headings should be in 10 or 12 point Times. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422 cm). Figure and table captions should be 9-point Roman type as in Figure~\ref{fig:onecol}. List and number all bibliographical references in 9-point Times, single-spaced, at the end of your response. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include the name(s) of editors of referenced books. \begin{figure}[t] \begin{center} \fbox{\rule{0pt}{1in} \rule{0.9\linewidth}{0pt}} \end{center} \caption{Example of caption. It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.} \label{fig:long} \label{fig:onecol} \end{figure} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. Please ensure that any point you wish to make is resolvable in a printed copy of the response. Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print. Many readers (and reviewers), even of an electronic copy, will choose to print your response in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage[dvips]{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.eps} \end{verbatim} } {\small \bibliographystyle{ieee_fullname} \section{Introduction} \label{sec:intro} Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version. \subsection{Language} All manuscripts must be in English. \subsection{Dual submission} Please refer to the author guidelines on the CVPR\ CVPR 2021\ web page for a discussion of the policy on dual submissions. \subsection{Paper length} Papers, excluding the references section, must be no longer than eight pages in length. The references section will not be included in the page count, and there is no limit on the length of the references section. For example, a paper of eight pages with two pages of references would have a total length of 10 pages. {\bf There will be no extra page charges for CVPR\ CVPR 2021.} Overlength papers will simply not be reviewed. This includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this \LaTeX\ guide already sets figure captions and references in a smaller font. The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts. The reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven. \subsection{The ruler} The \LaTeX\ style defines a printed ruler which should be present in the version submitted for review. The ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution. If you are preparing a document using a non-\LaTeX\ document preparation system, please arrange for an equivalent ruler to appear on the final output pages. The presence or absence of the ruler should not change the appearance of any other content on the page. The camera-ready copy should not contain a ruler. (\LaTeX\ users may use options of cvpr.sty to switch between different versions.) Reviewers: note that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly. Just use fractional references (\eg, this line is $087.5$), although in most cases one would expect that the approximate location will be adequate. \subsection{Paper ID} Make sure that the Paper ID from the submission system is visible in the version submitted for review (replacing the ``*****'' you see in this document). If you are using the \LaTeX\ template, \textbf{make sure to update paper ID in the appropriate place in the tex file}. \subsection{Mathematics} Please number all of your sections and displayed equations as in these examples: \begin{equation} E = m\cdot c^2 \label{eq:important} \end{equation} and \begin{equation} v = a\cdot t. \label{eq:also-important} \end{equation} It is important for readers to be able to refer to any particular equation. Just because you did not refer to it in the text does not mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin's description of how to write mathematics: \url{http://www.pamitc.org/documents/mermin.pdf}. \subsection{Blind review} Many authors misunderstand the concept of anonymizing for blind review. Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available. Blind review means that you do not use the words ``my'' or ``our'' when citing previous work. That is all. (But see below for tech reports.) Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith; it says that you are building on her work. If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work. An example of a bad paper just asking to be rejected: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Removed for blind review \end{quote} An example of an acceptable paper: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of the paper of Smith \etal [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213. \end{quote} If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work. In such cases, include the anonymized parallel submission~\cite{Authors14} as supplemental material and cite it as \begin{quote} [1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as supplemental material {\tt fg324.pdf}. \end{quote} Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report. For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a tech report for further details. Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''. Then submit the tech report as supplemental material. Again, you may not assume the reviewers will read this material. Sometimes your paper is about a problem which you tested using a tool that is widely known to be restricted to a single institution. For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal. You can handle this paper like any other. Do not write ``We show how to improve our previous work [Anonymous, 1968]. This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''. That would be silly, and would immediately identify the authors. Instead write the following: \begin{quotation} \noindent We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems [Zeus et al. 1968] did not handle case B properly. Ours handles it by including a foo term in the bar integral. ... The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know. It displayed the following behaviours, which show how well we solved cases A and B: ... \end{quotation} As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. \medskip \noindent FAQ\medskip\\ {\bf Q:} Are acknowledgements OK?\\ {\bf A:} No. Leave them for the final copy.\medskip\\ {\bf Q:} How do I cite my results reported in open challenges? {\bf A:} To conform with the double-blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\medskip\\ \begin{figure}[t] \centering \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}} \caption{Example of caption. It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.} \label{fig:onecol} \end{figure} \subsection{Miscellaneous} \noindent Compare the following:\\ \begin{tabular}{ll} \verb'$conf_a$' & $conf_a$ \\ \verb'$\mathit{conf}_a$' & $\mathit{conf}_a$ \end{tabular}\\ See The \TeX book, p165. The space after \eg, meaning ``for example'', should not be a sentence-ending space. So \eg is correct, {\em e.g.} is not. The provided \verb'\eg' macro takes care of this. When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word). If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal. However, use it only when there are three or more authors. Thus, the following is correct: ``Frobnication has been trendy lately. It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.'' This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors. \begin{figure*} \centering \begin{subfigure}{0.68\linewidth} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \caption{An example of a subfigure.} \label{fig:short-a} \end{subfigure} \hfill \begin{subfigure}{0.28\linewidth} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \caption{Another example of a subfigure.} \label{fig:short-b} \end{subfigure} \caption{Example of a short caption, which should be centered.} \label{fig:short} \end{figure*} \section{Formatting your paper} \label{sec:formatting} All text must be in a two-column format. The total allowable size of the text area is $6\frac78$ inches (17.46 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be $1\frac{1}{8}$ inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately $1\frac{5}{8}$ inches (4.13 cm) from the bottom edge of the page. \subsection{Margins and page numbering} All printed material, including text, illustrations, and charts, must be kept within a print area $6\frac{7}{8}$ inches (17.46 cm) wide by $8\frac{7}{8}$ inches (22.54 cm) high. Page numbers should be in the footer, centered and $\frac{3}{4}$ inches from the bottom of the page. The review version should have page numbers, yet the final version submitted as camera ready should not show any page numbers. The \LaTeX\ template takes care of this when used properly. \subsection{Type style and fonts} Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access. MAIN TITLE. Center the title $1\frac{3}{8}$ inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title. AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines. The ABSTRACT and MAIN TEXT are to be in a two-column format. MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx.~$\frac{1}{6}$ inch or 0.422 cm). Make sure your text is fully justified---that is, flush left and flush right. Please do not place any additional blank lines between paragraphs. Figure and table captions should be 9-point Roman type as in \cref{fig:onecol,fig:short}. Short captions should be centred. \noindent Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings. FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after. SECOND-ORDER HEADINGS. (For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after. If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line. \subsection{Footnotes} Please use footnotes\footnote{This is what a footnote looks like. It often distracts the reader from the main flow of the argument.} sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. \subsection{Cross-references} For the benefit of author(s) and readers, please use the {\small\begin{verbatim} \cref{...} \end{verbatim}} command for cross-referencing to figures, tables, equations, or sections. This will automatically insert the appropriate label alongside the cross-reference as in this example: \begin{quotation} To see how our method outperforms previous work, please see \cref{fig:onecol} and \cref{tab:example}. It is also possible to refer to multiple targets as once, \eg~to \cref{fig:onecol,fig:short-a}. You may also return to \cref{sec:formatting} or look at \cref{eq:also-important}. \end{quotation} If you do not wish to abbreviate the label, for example at the beginning of the sentence, you can use the {\small\begin{verbatim} \Cref{...} \end{verbatim}} command. Here is an example: \begin{quotation} \Cref{fig:onecol} is also quite important. \end{quotation} \subsection{References} List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include page numbers and the name(s) of editors of referenced books. When you cite multiple papers at once, please make sure that you cite them in numerical order like this \cite{Alpher02,Alpher03,Alpher05,Authors14b,Authors14}. If you use the template as advised, this will be taken care of automatically. \begin{table} \centering \begin{tabular}{@{}lc@{}} \toprule Method & Frobnability \\ \midrule Theirs & Frumpy \\ Yours & Frobbly \\ Ours & Makes one's heart Frob\\ \bottomrule \end{tabular} \caption{Results. Ours is better.} \label{tab:example} \end{table} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. In \LaTeX, avoid using the \texttt{center} environment for this purpose, as this adds potentially unwanted whitespace. Instead use {\small\begin{verbatim} \centering \end{verbatim}} at the beginning of your figure. Please ensure that any point you wish to make is resolvable in a printed copy of the paper. Resize fonts in figures to match the font in the body text, and choose line widths that render effectively in print. Readers (and reviewers), even of an electronic copy, may choose to print your paper in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.pdf} \end{verbatim} } \subsection{Color} Please refer to the author guidelines on the CVPR\ CVPR 2021\ web page for a discussion of the use of color in your document. If you use color in your plots, please keep in mind that a significant subset of reviewers and readers may have a color vision deficiency; red-green blindness is the most frequent kind. Hence avoid relying only on color as the discriminative feature in plots (such as red \vs green lines), but add a second discriminative feature to ease disambiguation. \section{Final copy} You must include your signed IEEE copyright release form when you submit your finished paper. We MUST have this form before your paper can be published in the proceedings. Please direct any questions to the production editor in charge of these proceedings at the IEEE Computer Society Press: \url{https://www.computer.org/about/contact}. {\small \bibliographystyle{ieee_fullname} \section{Limitations and societal impact} \label{broader-impacts} Phrase detection methods can help us understand the content of images. This is useful for downstream tasks like image captioning and visual question answering. This is useful for some tasks like answering questions about images posed by people who have visual impairments. However, these benefits also come with some risks, such as enabling some applications in tasks like surveillance by allowing a user to quickly locate specific entities in a database of images. Although our approach provides significant performance improvements, absolute performance on this task is still poor. While this suggests there are ample opportunity for researchers to improve performance on this task, it also indicates that the results of these systems should not be trusted blindly. We also note that Flickr30K Entities and COCO may contain personally identifiable information, but they remain standard benchmarks making them an important comparison. \section{Phrase localization} \label{supp_phrase_loc} Our work makes advances on phrase detection. The problem stands in contrast to localization-only tasks where detection is not required (\ie ground truth phrases are given at test time). As prior work \cite{Plummer_2020} demonstrated, it is too easy to overfit to the localization-only task. In other words, localization performance and detection performance are not causally related, so localization performance is of very limited use for evaluating our detection approach. However, to be complete, we report localization performance of our model without PPA in Table~\ref{loc_no_ppa} and with PPA in Table~\ref{loc_ppa}. We note that our methods' localization numbers are on par with previous work, with the only exception of when PPA is applied on Flickr30K. We would argue that this is likely due, in part, to the fact that the PPA augmentations are noisy, as \cite{Plummer_2020} themselves acknowledged. For example, using PPA augmentation on Flickr30K, any region labeled as "swimmers head cap" would also be labeled as a positive for "roof" or "jacket," although these are clearly incorrect associations. Since our work more accurately distinguishes between phrases, we would more likely label the correct image region for such spurious associations. \section{Concept generation ablations} \label{supp_concept_generate_abl} In our work, we make use of visually coherent groups (concepts) to provide negative samples that cover a broad spectrum of the training distribution and minimize false negatives. We argue that minimizing visual coherence noise in our concepts is key to minimizing false negatives. As noted in Section 3.1 in our paper, DBSCAN + ViCo result in concepts with the least amount of visual coherence noise. Thus, we expect this combination to have the best performance. To verify this claim, we provide an ablation study that document the impact of varying the embedding/clustering algorithm combination on the CR component of our PFP model. The component is responsible for using the concepts as negative samples. For each combination, we select the clustering algorithm hyper parameters that produce the least visual coherence noise. As we can see in Table~\ref{cr_ab}, DBSCAN+ViCo perform the best on both datasets. While other combinations have equal performance on Flickr30k entities, their performance align in RefCOCO+ dataset with our visual coherence noise metric. More concretely, K-means overall perform worse than DBSCAN and ViCO improves performance on either of the clustering algorithms when compared to GLoVE. \begin{table*}[t] \setlength{\tabcolsep}{1pt} \centering \begin{tabular}{|l|ccc|c|ccc|c|} \hline & \multicolumn{4}{|c|}{Flickr30K Entities} & \multicolumn{4}{|c|}{RefCOCO+} \\ \hline \#Train Samples &zero shot &few shot & common &mean&zero shot &few shot & common &mean\\ \hline Baseline [SimNet w/CCA~\cite{Plummer_2020}] & 9.7 & 11.2 & 17.3 & 12.7 & 6.0 & 10.2 & 20.1 & 12.1 \\ K-means + GLoVE & 10.0 & 11.8 & 18.5 & 13.4 & 6.0 & 9.9 & 20.4 & 12.1 \\ K-means + ViCo & 10.1 & 11.9 & 18.3 & 13.4 & 6.0 & 9.8 & 20.7 & 12.2 \\ DBSCAN + GLoVE & 9.9 & 11.9 & 18.4 & 13.4 & 6.2 & 10.4 & 20.7 & 12.4 \\ DBSCAN + ViCo & \textbf{10.4} & \textbf{11.8}& \textbf{18.3} & \textbf{13.5} & \textbf{6.1} & \textbf{10.4} & \textbf{21.0} & \textbf{12.6} \\ \hline \end{tabular} \vspace{-2mm} \caption{mAP Split by frequency of training instances. The table contains ablations of our model that compares the performance of its CR component using different concept generation methods (\ie clustering algorithm/embedding combinations) to state of the art (SimNet w/CCA). See Section~\ref{supp_concept_generate_abl} for discussion.} \label{cr_ab} \end{table*} \section{Evaluation dataset selection} \label{supp_datasets} The sparsity of phrase detection datasets annotations poses significant challenges on evaluation. Real world datasets annotations simply can not cover all the possible positive cases. For example, while a region might be annotated with only the phrase \textit{blue shirt}, it can also be correctly labeled with \textit{clothing}. Thus, if a model assigns a region the phrase \textit{clothing}, the evaluation process will incorrectly classify the case as negative (\ie false negative). Prior work \cite{Plummer_2020} attempted to mitigate this problem by introducing Positive Phrase Augmentations (PPA) where structures like WordNet \cite{10.1145/219717.219748} are used to derive additional positive samples for a given annotation. However, this problem is not limited to issues with synonyms. Phrases might have different structures but can convey the same meaning (\eg frisbee that is round vs a round frisbee), thus resulting in the same false negatives problem in evaluation. While the authors of datasets like Flickr30K \cite{7410660} limited the structure of their annotations such that this problem would not arise, this is not the case for datasets like Visual Genome \cite{krishnavisualgenome} or Referit \cite{KazemzadehOrdonezMattenBergEMNLP14}. Moreover, both datasets were used in prior work to evaluate phrase detection algorithms \cite{Plummer_2020}, but given the aforementioned problem with false negatives, their validity for phrase detection evaluation is not clear. To quantitatively document this issue, we sampled 30 random phrases from each dataset and considered the top 5 most similar phrases using the visual based language representation ViCo \cite{gupta2019vico}. For each of these top 5 phrases, we manually counted the number of false negatives. We report the average results in Table~\ref{false_neg}. As we can observe, both Referit and Visual Genome suffer from significantly higher false negative rates when compared to Flickr30k Entities. Thus, they are not viable evaluation datasets for phrase detection and so we choose not to use them in our paper. We instead use Flickr30k Entities as well as RefCOCO+ \cite{10.1007/978-3-319-46475-6_5}. RefCOCO+ was collected using the same underlying game as Referit but the authors improved the data collection standards. For example, they disallowed players from using location words in their referring expressions by adding "taboo" words to the ReferIt Game which earlier would result in many false negatives. Thus, as the authors note, the resulting expressions were more appearance focused and more concise. This is further evident in the lower false negative rate in Table~\ref{false_neg} when compared to that of Referit. Even though Flickr30k and RefCOCO+ are significantly less noisy in terms of false negatives than Referit and Visual Genome, they still exhibit a significant amount of noise. Thus, future work can benefit from developing better evaluation procedures and datasets to improve the the evaluation accuracy of phrase detection models. \section{Additional qualitative results} \label{supp_qualt} \begin{table}[t!] \setlength{\tabcolsep}{2pt} \centering \begin{tabular}{|l|c|} \hline & \multicolumn{1}{|c|}{False Negative Rate} \\ \hline Referit \cite{KazemzadehOrdonezMattenBergEMNLP14} & 74\%\\ Visual Genome \cite{krishnavisualgenome} & 72\%\\ Flickr30k Entities \cite{7410660} & 31\%\\ RefCOCO+ \cite{10.1007/978-3-319-46475-6_5}& 40\%\\ \hline \end{tabular} \caption{\textbf{Dataset annotations' false negative rate.} From each of the following datasets: Flickr30k, Visual Genome, RefCOCO+ and Referit, we sample 30 random phrases and manually calculate the average percentage of false negatives in each phrase top 5 most similar phrases. See Section~\ref{supp_datasets} for more details and discussion.} \label{false_neg} \end{table} \begin{figure*}[h] \begin{subfigure}{\textwidth} \centering \includegraphics[width=0.99\linewidth]{images/qualt_full_2.png} \end{subfigure} \caption[]% {\small Comparison between our model vs prior work model on positive (ground truth) and randomly sampled negative phrases scores. Our model is significantly better able to separate positive vs negative phrases scores than prior work. Refer to section~\ref{supp_qualt} for discussion.} \label{fig:qualt_supp_1} \end{figure*} \begin{figure*}[h] \begin{subfigure}{\textwidth} \centering \includegraphics[width=0.99\linewidth]{images/qualt_full_3.png} \end{subfigure} \caption[]% {\small Comparison between our model vs prior work model on positive (ground truth) and randomly sampled negative phrases scores. Our model is significantly better able to separate positive vs negative phrases scores than prior work. Refer to section~\ref{supp_qualt} for discussion.} \label{fig:qualt_supp_2} \end{figure*} In Figures~\ref{fig:qualt_supp_1} and~\ref{fig:qualt_supp_2}, we provide additional qualitative results. As we discuss in the paper, the figures demonstrate how our model is better able at expanding the gap between the positive phrases (\ie groundtruth) and the negative phrases scores. For example, consider Figure~\ref{fig:qualt_supp_2} (a) middle image, prior work model assigns 0.995 score for the phrase \textit{edge mountain} while our model in Figure~\ref{fig:qualt_supp_2} (b) middle image assigns 0.39 score while maintaining high confidence in the ground truth phrases. This improvement is a direct result of using our concepts in both the CR and EC modules. The concepts expose the model to a wider scope of the language distribution while minimizing false negatives, and hence increase the model confidence against negative phrases. We also note that this increased discriminative power translate to tighter and better fit boxes. For example, consider the box around the waterfall in Figure~\ref{fig:qualt_supp_2} right image (a) [prior work] compared to (b) [our work]. Our work model boxes fit the target phrase \textit{beautiful waterfall} more tightly. This improvement can be due to the model's increase confidence in the phrase with respect to other negative phrases. \section{Introduction} Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version. \subsection{Language} All manuscripts must be in English. \subsection{Dual submission} Please refer to the author guidelines on the CVPR 2021~web page for a discussion of the policy on dual submissions. \subsection{Paper length} Papers, excluding the references section, must be no longer than eight pages in length. The references section will not be included in the page count, and there is no limit on the length of the references section. For example, a paper of eight pages with two pages of references would have a total length of 10 pages. {\bf There will be no extra page charges for CVPR 2021.} Overlength papers will simply not be reviewed. This includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this \LaTeX\ guide already sets figure captions and references in a smaller font. The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts. The reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven. \subsection{The ruler} The \LaTeX\ style defines a printed ruler which should be present in the version submitted for review. The ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution. If you are preparing a document using a non-\LaTeX\ document preparation system, please arrange for an equivalent ruler to appear on the final output pages. The presence or absence of the ruler should not change the appearance of any other content on the page. The camera ready copy should not contain a ruler. (\LaTeX\ users may use options of cvpr.cls to switch between different versions.) Reviewers: note that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly. Just use fractional references (e.g.\ this line is $095.5$), although in most cases one would expect that the approximate location will be adequate. \subsection{Mathematics} Please number all of your sections and displayed equations. It is important for readers to be able to refer to any particular equation. Just because you didn't refer to it in the text doesn't mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin's description of how to write mathematics: \url{http://www.pamitc.org/documents/mermin.pdf}. \subsection{Blind review} Many authors misunderstand the concept of anonymizing for blind review. Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available. Blind review means that you do not use the words ``my'' or ``our'' when citing previous work. That is all. (But see below for techreports.) Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith; it says that you are building on her work. If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work. An example of a bad paper just asking to be rejected: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Removed for blind review \end{quote} An example of an acceptable paper: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of the paper of Smith \etal [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213. \end{quote} If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work. In such cases, include the anonymized parallel submission~\cite{Authors14} as additional material and cite it as \begin{quote} [1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as additional material {\tt fg324.pdf}. \end{quote} Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report. For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a techreport for further details. Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''. Then submit the techreport as additional material. Again, you may not assume the reviewers will read this material. Sometimes your paper is about a problem which you tested using a tool which is widely known to be restricted to a single institution. For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal. You can handle this paper like any other. Don't write ``We show how to improve our previous work [Anonymous, 1968]. This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''. That would be silly, and would immediately identify the authors. Instead write the following: \begin{quotation} \noindent We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems [Zeus et al. 1968] didn't handle case B properly. Ours handles it by including a foo term in the bar integral. ... The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know. It displayed the following behaviours which show how well we solved cases A and B: ... \end{quotation} As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. \medskip \noindent FAQ\medskip\\ {\bf Q:} Are acknowledgements OK?\\ {\bf A:} No. Leave them for the final copy.\medskip\\ {\bf Q:} How do I cite my results reported in open challenges? {\bf A:} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\medskip\\ \begin{figure}[t] \begin{center} \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}} \end{center} \caption{Example of caption. It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.} \label{fig:long} \label{fig:onecol} \end{figure} \subsection{Miscellaneous} \noindent Compare the following:\\ \begin{tabular}{ll} \verb'$conf_a$' & $conf_a$ \\ \verb'$\mathit{conf}_a$' & $\mathit{conf}_a$ \end{tabular}\\ See The \TeX book, p165. The space after \eg, meaning ``for example'', should not be a sentence-ending space. So \eg is correct, {\em e.g.} is not. The provided \verb'\eg' macro takes care of this. When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word.) However, use it only when there are three or more authors. Thus, the following is correct: `` Frobnication has been trendy lately. It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.'' This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors. If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal. For this citation style, keep multiple citations in numerical (not chronological) order, so prefer \cite{Alpher03,Alpher02,Authors14} to \cite{Alpher02,Alpher03,Authors14}. \begin{figure*} \begin{center} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \end{center} \caption{Example of a short caption, which should be centered.} \label{fig:short} \end{figure*} \section{Formatting your paper} All text must be in a two-column format. The total allowable width of the text area is $6\frac78$ inches (17.5 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1.0 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1.0 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page. \subsection{Margins and page numbering} All printed material, including text, illustrations, and charts, must be kept within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm) high. Page numbers should be in footer with page numbers, centered and .75 inches from the bottom of the page and make it start at the correct page number rather than the 4321 in the example. To do this fine the line (around line 20) \begin{verbatim} \setcounter{page}{4321} \end{verbatim} where the number 4321 is your assigned starting page. \subsection{Type-style and fonts} Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access. MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title. AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines. The ABSTRACT and MAIN TEXT are to be in a two-column format. MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422 cm). Make sure your text is fully justified---that is, flush left and flush right. Please do not place any additional blank lines between paragraphs. Figure and table captions should be 9-point Roman type as in Figures~\ref{fig:onecol} and~\ref{fig:short}. Short captions should be centred. \noindent Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings. FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after. SECOND-ORDER HEADINGS. (For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after. If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line. \subsection{Footnotes} Please use footnotes\footnote {This is what a footnote looks like. It often distracts the reader from the main flow of the argument.} sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. \subsection{References} List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include the name(s) of editors of referenced books. \begin{table} \begin{center} \begin{tabular}{|l|c|} \hline Method & Frobnability \\ \hline\hline Theirs & Frumpy \\ Yours & Frobbly \\ Ours & Makes one's heart Frob\\ \hline \end{tabular} \end{center} \caption{Results. Ours is better.} \end{table} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. Please ensure that any point you wish to make is resolvable in a printed copy of the paper. Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print. Many readers (and reviewers), even of an electronic copy, will choose to print your paper in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage[dvips]{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.eps} \end{verbatim} } \subsection{Color} Please refer to the author guidelines on the CVPR 2021~web page for a discussion of the use of color in your document. \section{Final copy} You must include your signed IEEE copyright release form when you submit your finished paper. We MUST have this form before your paper can be published in the proceedings. Please direct any questions to the production editor in charge of these proceedings at the IEEE Computer Society Press: \url{https://www.computer.org/about/contact}. {\small \bibliographystyle{ieee_fullname} \section{Introduction} Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press. This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version. \subsection{Language} All manuscripts must be in English. \subsection{Dual submission} Please refer to the author guidelines on the CVPR 2018 web page for a discussion of the policy on dual submissions. \subsection{Paper length} Papers, excluding the references section, must be no longer than eight pages in length. The references section will not be included in the page count, and there is no limit on the length of the references section. For example, a paper of eight pages with two pages of references would have a total length of 10 pages. {\bf There will be no extra page charges for CVPR 2018.} Overlength papers will simply not be reviewed. This includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this \LaTeX\ guide already sets figure captions and references in a smaller font. The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts. The reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven. \subsection{The ruler} The \LaTeX\ style defines a printed ruler which should be present in the version submitted for review. The ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution. If you are preparing a document using a non-\LaTeX\ document preparation system, please arrange for an equivalent ruler to appear on the final output pages. The presence or absence of the ruler should not change the appearance of any other content on the page. The camera ready copy should not contain a ruler. (\LaTeX\ users may uncomment the \verb'\cvprfinalcopy' command in the document preamble.) Reviewers: note that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly. Just use fractional references (e.g.\ this line is $095.5$), although in most cases one would expect that the approximate location will be adequate. \subsection{Mathematics} Please number all of your sections and displayed equations. It is important for readers to be able to refer to any particular equation. Just because you didn't refer to it in the text doesn't mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin's description of how to write mathematics: \url{http://www.pamitc.org/documents/mermin.pdf}. \subsection{Blind review} Many authors misunderstand the concept of anonymizing for blind review. Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available. Blind review means that you do not use the words ``my'' or ``our'' when citing previous work. That is all. (But see below for techreports.) Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith; it says that you are building on her work. If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work. An example of a bad paper just asking to be rejected: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Removed for blind review \end{quote} An example of an acceptable paper: \begin{quote} \begin{center} An analysis of the frobnicatable foo filter. \end{center} In this paper we present a performance analysis of the paper of Smith \etal [1], and show it to be inferior to all previously known methods. Why the previous paper was accepted without this analysis is beyond me. [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213. \end{quote} If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work. In such cases, include the anonymized parallel submission~\cite{Authors14} as additional material and cite it as \begin{quote} [1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as additional material {\tt fg324.pdf}. \end{quote} Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report. For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a techreport for further details. Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''. Then submit the techreport as additional material. Again, you may not assume the reviewers will read this material. Sometimes your paper is about a problem which you tested using a tool which is widely known to be restricted to a single institution. For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your solution. The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal. You can handle this paper like any other. Don't write ``We show how to improve our previous work [Anonymous, 1968]. This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''. That would be silly, and would immediately identify the authors. Instead write the following: \begin{quotation} \noindent We describe a system for zero-g frobnication. This system is new because it handles the following cases: A, B. Previous systems [Zeus et al. 1968] didn't handle case B properly. Ours handles it by including a foo term in the bar integral. ... The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know. It displayed the following behaviours which show how well we solved cases A and B: ... \end{quotation} As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors. A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess. He or she would have to be sure that no other authors could have been contracted to solve problem B. FAQ: Are acknowledgements OK? No. Leave them for the final copy. \begin{figure}[t] \begin{center} \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}} \end{center} \caption{Example of caption. It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.} \label{fig:long} \label{fig:onecol} \end{figure} \subsection{Miscellaneous} \noindent Compare the following:\\ \begin{tabular}{ll} \verb'$conf_a$' & $conf_a$ \\ \verb'$\mathit{conf}_a$' & $\mathit{conf}_a$ \end{tabular}\\ See The \TeX book, p165. The space after \eg, meaning ``for example'', should not be a sentence-ending space. So \eg is correct, {\em e.g.} is not. The provided \verb'\eg' macro takes care of this. When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word.) However, use it only when there are three or more authors. Thus, the following is correct: `` Frobnication has been trendy lately. It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.'' This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors. If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal. For this citation style, keep multiple citations in numerical (not chronological) order, so prefer \cite{Alpher03,Alpher02,Authors14} to \cite{Alpher02,Alpher03,Authors14}. \begin{figure*} \begin{center} \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}} \end{center} \caption{Example of a short caption, which should be centered.} \label{fig:short} \end{figure*} \section{Formatting your paper} All text must be in a two-column format. The total allowable width of the text area is $6\frac78$ inches (17.5 cm) wide by $8\frac78$ inches (22.54 cm) high. Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first page) should begin 1.0 inch (2.54 cm) from the top edge of the page. The second and following pages should begin 1.0 inch (2.54 cm) from the top edge. On all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page. \subsection{Margins and page numbering} All printed material, including text, illustrations, and charts, must be kept within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm) high. Page numbers should be in footer with page numbers, centered and .75 inches from the bottom of the page and make it start at the correct page number rather than the 4321 in the example. To do this fine the line (around line 23) \begin{verbatim} \setcounter{page}{4321} \end{verbatim} where the number 4321 is your assigned starting page. Make sure the first page is numbered by commenting out the first page being empty on line 46 \begin{verbatim} \end{verbatim} \subsection{Type-style and fonts} Wherever Times is specified, Times Roman may also be used. If neither is available on your word processor, please use the font closest in appearance to Times to which you have access. MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of the first page. The title should be in Times 14-point, boldface type. Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word). Leave two blank lines after the title. AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and printed in Times 12-point, non-boldface type. This information is to be followed by two blank lines. The ABSTRACT and MAIN TEXT are to be in a two-column format. MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-spacing. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422 cm). Make sure your text is fully justified---that is, flush left and flush right. Please do not place any additional blank lines between paragraphs. Figure and table captions should be 9-point Roman type as in Figures~\ref{fig:onecol} and~\ref{fig:short}. Short captions should be centred. \noindent Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize only the first word of section titles and first-, second-, and third-order headings. FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after. SECOND-ORDER HEADINGS. (For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after. If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line. \subsection{Footnotes} Please use footnotes\footnote {This is what a footnote looks like. It often distracts the reader from the main flow of the argument.} sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. \subsection{References} List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example~\cite{Authors14}. Where appropriate, include the name(s) of editors of referenced books. \begin{table} \begin{center} \begin{tabular}{|l|c|} \hline Method & Frobnability \\ \hline\hline Theirs & Frumpy \\ Yours & Frobbly \\ Ours & Makes one's heart Frob\\ \hline \end{tabular} \end{center} \caption{Results. Ours is better.} \end{table} \subsection{Illustrations, graphs, and photographs} All graphics should be centered. Please ensure that any point you wish to make is resolvable in a printed copy of the paper. Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print. Many readers (and reviewers), even of an electronic copy, will choose to print your paper in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic. When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below {\small\begin{verbatim} \usepackage[dvips]{graphicx} ... \includegraphics[width=0.8\linewidth] {myfile.eps} \end{verbatim} } \subsection{Color} Please refer to the author guidelines on the CVPR 2018 web page for a discussion of the use of color in your document. \section{Final copy} You must include your signed IEEE copyright release form when you submit your finished paper. We MUST have this form before your paper can be published in the proceedings. {\small \bibliographystyle{ieee}

BkiUc-E5qsBC5C3p7Qn-

\section{Introduction} Several methods exist for solving supervised learning problems of regression and classification~\citep{HTF09,Bis06}. The main goal is to estimate a model of the data generation process to \emph{predict} at best the target value corresponding to a combination of features not seen before. However, not all methods are suitable to \emph{optimize} on top of the estimated model, i.e., to solve a mathematical programming problem that contains the estimated model as part of the constraints and/or the objective function. For example, to find the best combination of features providing a desired target, possibly under constraints on the features one can choose. In this case, the model is used as a surrogate of the underlying (and unknown) features-to-target mapping to formulate the decision problem. Applications range from derivative-free black-box optimization~\citep{Kus64,Jon01,BCD10,Bem20,BP21}, to engineering design~\citep{QHSGVT05}, and control engineering, in particular model predictive control~\citep{CB99,MRD18,BBM17}, where actuation commands are decided in real-time by a numerical optimization algorithm based on a dynamical model of the controlled process that is learned from data~\citep{Lju99,SL19}, see for instance the approach proposed recently in~\citep{MB21}. When optimizing over a learned model is a goal, a clear tradeoff exists between the accuracy of the model on test data and the complexity of the model, which ultimately determines the complexity of the mathematical programming problem resulting from using the model. On one extreme, we have linear regression models, which are very simple to represent as linear relations among optimization variables but have limited expressiveness. On the other extreme, random forests and other ensemble methods, k-nearest neighbors, kernel support vector machines, and other methods, can capture the underlying model very accurately but are difficult to encode in an optimization problem. Neural networks and Gaussian processes can be a good compromise between the compactness of the model and the representation of the feature-to-target relation, but are nonlinear models leading to nonconvex optimization problems that are possibly difficult to solve to global optimality. In this paper, we advocate the use of \emph{piecewise linear} (PWL) models as a good tradeoff between their simplicity, due to the linearity of the model on polyhedral regions of the feature-vector space, and expressiveness, due to the good approximation properties of piecewise linear functions~\citep{Bre93,LU92,CD88,JDD00,BOPS11}. We refer to such models with the more appropriate, although less common, term \emph{piecewise affine} (PWA), to highlight the presence of an intercept in each submodel. PWA models can be easily encoded into optimization problems by using mixed-integer linear inequalities~\citep{BM99}, and hence optimize over them to reach a global minimum by using mixed-integer programming~\citep{Lod10}, for which excellent public domain and commercial packages exist. Many classical machine learning methods have an underlying PWA structure: ridge classification, logistic (and more generally softmax) regression, hinging hyperplanes~\citep{Bre93}, and neural networks with ReLU activation functions, they all require evaluating the maximum of linear functions to predict target values; the predictor associated with a decision tree is a piecewise constant (PWC) function over a partition of the feature-vector space in boxes; $k$-nearest neighbor classifiers can be also expressed as PWC functions over polyhedral partitions (the comparison of squared Euclidean norms $\|x-x_i\|_2^2\leq \|x-x_j\|_2^2$ used to determine the nearest neighbors of $x$ is equivalent to the linear relation $2(x_j-x_i)'x\leq \|x_j\|_2^2-\|x_i\|_2^2$), although the number of polyhedra largely grows with the number of training samples. Different piecewise affine regression methods have been proposed in the system identification literature for getting switching linear dynamical models from data~\citep{FMLM03a,RBL04a,BGPV05,NTK05,HLCXV15}. See also the survey paper~\citep{PKFV07} and the recursive PWA regression algorithms proposed in~\citep{BBD11,BPB16a}. Most of such methods identify a prescribed number of linear models and associate one of them to each training datapoint, therefore determining a clustering of the data. As a last step, a multicategory discrimination problem is solved to determine a function that piecewise-linearly separates the clusters~\citep{BM94}. For instance, the approach of~\cite{NTK05} consists of first clustering the feature+target vectors by using a Gaussian mixture model, then use support vector classification to separate the feature-vector space. In~\citep{FMLM03a}, the authors propose instead to cluster the vectors whose entries are the coefficients of local linear models, one model per datapoint, then piecewise-linearly separate the clusters. In~\citep{BPB16a}, $K$ recursive least-squares problems for regression are run in parallel to cluster data in on-line fashion, based on both quality of fit obtained by each linear model and proximity to the current centroids of the clusters, and finally the obtained clusters are separated by a PWL function. \subsection{Contribution} This paper proposes a general supervised learning method for regression and/or classification of multiple targets that results in a PWA predictor over a single PWA partition of the feature space in $K$ polyhedral cells. In each polyhedron, the predictor is either affine (for numeric targets) or given by the max of affine functions, i.e., convex piecewise affine (for categorical targets). Our goal is to obtain an overall predictor that admits a simple encoding with binary and real variables, to be able to solve optimization problems involving the prediction function via mixed-integer linear or quadratic programming. The number $K$ of linear predictors is therefore limited by the tolerated complexity of the resulting mixed-integer encoding of the PWA predictor. Rather than first clustering the training data and fitting $K$ different linear predictors, and then finding a PWL separation function to get the PWA partition, we simultaneously cluster, PWL-separate, and fit by solving a block-coordinate descent problem, similarly to the K-means algorithm~\citep{Llo57}, where we alternate between fitting models/separating clusters and reassigning training data to clusters. We call the algorithm PARC (Piecewise Affine Regression and Classification) and show that it converges in a finite number of iterations by showing that the sum of the loss functions associated with regression, classification, piecewise linear separation errors decreases at each iteration. PWL separation is obtained by solving softmax regression problems or, as a simpler alternative, by taking the Voronoi partition induced by the cluster centroids. We test the PARC algorithm on different synthetic and real-world datasets. After showing that PARC can reconstruct an underlying PWA function from its samples, we investigate the effect of $K$ in reconstructing a nonlinear function, also showing how to optimize with respect to the feature vector so that the corresponding target is as close as possible to a given reference value. Then we test PARC on many real-world datasets proposed for regression and classification, comparing its performance to alternative regression and classification techniques that admit a mixed-integer encoding of the predictor of similar complexity, such as simple neural networks based on ReLU activation functions and small decision trees. A Python implementation of the PARC algorithm is available at \url{http://cse.lab.imtlucca.it/~bemporad/parc}. \subsection{Outline} After formulating the multivariate PWL regression and classification problem in Section~\ref{sec:statement}, we describe the proposed PARC algorithm and prove its convergence properties in Section~\ref{sec:PARC}. In Section~\ref{sec:predictor} we define the PWA prediction function for regression and classification, showing how to encode it using mixed-integer linear inequalities using big-M techniques. Section~\ref{sec:examples} presents numerical tests on synthetic and real-world datasets. Some conclusions are finally drawn in Section~\ref{sec:conclusions}. \subsection{Notation and definitions} Given a finite set $\CC$, $\card \CC$ denotes its number of elements (cardinality). Given a vector $a\in\rr^n$, $\|a\|_2$ is the Euclidean norm of $a$, $[a]_i$ denotes the $i$th component of $a$. Given two vectors $a,b\in\rr^n$, we denote by $[a=b]$ the binary quantity that is $1$ if $a=b$ or $0$ otherwise. Given a matrix $A\in\rr^{m\times n}$, $\|A\|_F$ denotes the Frobenius norm of $A$. Given a polyhedron $P\subseteq\rr^n$, $\mathring{P}$ denotes its interior. Given a finite set $S$ of real numbers $\{s_1,\ldots,s_K\}$ we denote by \begin{equation} \arg\min_{s\in S}=\min_{h}\{h\in\{1,\ldots,K\}: s_h\leq s_j,\ \forall j\in\{1,\ldots,K\}\} \label{eq:argmin} \end{equation} Taking the smallest index $h$ in~\eqref{eq:argmin} breaks ties in case of multiple minimizers. The $\arg\max$ function of a set $S$ is defined similarly by replacing $ s_h\leq s_j$ with $s_h\geq s_j$ in~\eqref{eq:argmin}. \begin{definition} A collection $\PP$ of sets $\{P_1,\ldots,P_K\}$ is said a \emph{polyhedral partition} of $\rr^n$ if $P_i$ is a polyhedron, $P_i\subseteq\rr^n$, $\forall i=1,\ldots,K$, $\cup_{i=1}^K P_i=\rr^n$, and $\mathring{P}_i\cap\mathring{P}_j=\emptyset$, $\forall i,j=1,\ldots,K$, $i\neq j$. \end{definition} \begin{definition} \label{def:PWA} A function $j:\rr^n\rightarrow\{1,\ldots,K\}$ is said \emph{integer piecewise constant} (IPWC)~\citep{CB17} if there exist a polyhedral partition $\PP=\{P_1,\ldots,P_K\}$ of $\rr^n$ such that \begin{equation} j(x) = \arg\min_{h} \{h\in\{1,\ldots,K\}:\ x\in P_h\} \label{eq:ipwc-fun} \end{equation} for all $x\in\rr^n$. \end{definition} The ``$\arg\min$'' in~\eqref{eq:ipwc-fun} prevents possible multiple definitions of $j(x)$ on overlapping boundaries $P_i\cap P_j\neq \emptyset$. \begin{definition} \label{def:PWA2} A function $f:\rr^n\to\rr^m$ is said \emph{piecewise affine} (PWA) if there exists an IPWC function $j:\rr^n\to\{1,\ldots,K\}$ defined over a polyhedral partition $\PP$ and $K$ pairs $(a^i,b^i)$, $a^i\in\rr^{m\times n}$, $b^i\in\rr^m$, such that \begin{equation} f(x) = a^{j(x)}x+b^{j(x)} \label{eq:pwa-fun} \end{equation} for all $x\in\rr^n$. It is said \emph{piecewise constant} if $a^i=0$, $\forall i\in\{1,\ldots,K\}$. \end{definition} \begin{definition} A \emph{piecewise linear (PWL) separation} function $\Phi:\rr^n\to\rr$~\citep{BM94} is defined by \begin{subequations} \beqar \Phi(x)&=&\omega^{j(x)}x+\gamma^{j(x)}\label{eq:PWL-sep-a}\\ j(x)&=&\displaystyle{\min\left\{\arg\max_{j=1,\ldots,K}\{\omega^jx+\gamma^j\}\right\}} \label{eq:PWL-sep-b} \eeqar \label{eq:PWL-sep \end{subequations} where $\omega^j\in\rr^n$, $\gamma^j\in\rr$, $\forall j=1,\ldots,K$. \end{definition} A PWL separation function is convex~\citep{Sch87} and PWA over the polyhedral partition $\PP=\{P_1,\ldots,P_K\}$ where \begin{equation} P_j=\{x\in\rr^n:\ (\omega^h-\omega^j)x\leq \gamma^j-\gamma^h,\ \forall h=1,\ldots,K,\ h\neq j\},\ j=1,\ldots,K \label{eq:PWL-partition} \end{equation} \section{Problem statement} \label{sec:statement} We have a training dataset $(x_k,y_k)$, $k=1,\ldots,N$, where $x_k$ contains $n_c$ numerical and $n_d$ categorical features, each one of the latter containing $n_i$ possible values $\{v^i_1,\ldots,v^i_{n_i}\}$, $i=1,\ldots,n_d$, and $y_k$ contains $m_c$ numerical targets and $m_d$ categorical targets, each one containing $m_i$ possible values $\{w^i_1,\ldots,w^i_{m_i}\}$, $i=1,\ldots,m_d$. We assume that categorical features have been one-hot encoded into $n_i-1$ binary values, so that $x_k\in\XX$, $\XX=\rr^{n_c}\times\{0,1\}^{s_x}$, $s_x=\sum_{i=1}^{n_d}(n_i-1)$. By letting $n=n_c+s_x$ we have $x_k\in\rr^n$. Moreover, let $y_k=\smallmat{y_{ck}\\y_{dk}}$, $y_{ck}\in\rr^{m_c}$, $[y_{dk}]_i\in\{w^i_1,\ldots,w^i_{m_i}\}$, $\forall i=1,\ldots,m_d$, and define $\YY=\rr^{m_c}\times \{w^1_1,\ldots,w^1_{m_1}\}\times\ldots\times\{w^{m_d}_1,\ldots,w^{m_d}_{m_{m_d}}\}$, so that we have $y_k\in\YY$. Several approaches exist to solve regression problems to predict the numerical components $y_c$ and classification problems for the categorical target vector $y_d$. In this paper, we are interested in generalizing linear predictors for regression and classification to \emph{piecewise linear} predictors $\hat y:\rr^n\to\YY$ over a single \emph{polyhedral partition} $\PP=\{P_1,\ldots,P_{K}\}$ of $\rr^n$. More precisely, we want to solve the posed multivariate regression and classification problem by finding the following predictors \begin{subequations} \beqar [\hat y_c(x)]_i&=&a^{j(x)}_ix+b^{j(x)}_i,\ i=1,\ldots,m_c\label{eq:PWA-regression}\\\null [\hat y_d(x)]_i&=&w^i_{h},\ h=\arg\max_{t\in I(i)}\{a^{j(x)}_tx+b^{j(x)}_t\},\ i=1,\ldots,m_d \label{eq:PWA-classifier \eeqa \label{eq:PWA-predictors \end{subequations} where $j(x)$ is defined as in~\eqref{eq:ipwc-fun} and the coefficient/intercept values $a^j\in\rr^n$, $b^j\in\rr$ define a PWA function $f:\rr^n\to\rr^m$ as in~\eqref{eq:pwa-fun}, in which $m=m_c+\sum_{i=1}^{m_d}m_i$. In~\eqref{eq:PWA-predictors}, $I(i)$ denotes the set of indices corresponding to the $i$th categorical target $[y_d]_i$, $I(i)=\{t(i)+1,\ldots,t(i)+m_i\}$, $t(i)=m_c+\sum_{h=1}^{i-1}m_h$. Note that subtracting the same quantity $\bar ax+\bar b$ from all the affine terms in~\eqref{eq:PWA-classifier} does not change the maximizer, for any arbitrary $\bar a\in\rr^n$, $\bar b\in\rr$. To well-pose $\hat y_d$, according to~\eqref{eq:argmin} we also assume that the smallest index is taken in case ties occur when taking the maximum in~\eqref{eq:PWA-classifier}. We emphasize that all the components of $\hat y(x)$ in~\eqref{eq:PWA-predictors} share the \emph{same polyhedral partition} $\PP$. A motivation for this requirement is to be able to efficiently solve optimization problems involving the resulting predictor $\hat y$ using mixed-integer programming, as we will detail in Section~\ref{sec:MIP}. Clearly, if this is not a requirement, by treating each target independently the problem can be decomposed in $m_c$ PWA regression problems and $m_d$ PWA classification problems. Our goal is to jointly separate the training dataset in $K$ clusters $\CC_1,\ldots,\CC_K$, $\CC_1=\{x_k: k\in J_j\}$, where $\cup_{i=1}^KJ_i=\{1,\ldots,N\}$, $J_i\cap J_j=\emptyset$, $\forall i,j\in\{1,\ldots,N\}$, $i\neq j$, and to find optimal coefficients/intercepts $a^j$, $b^j$ for~\eqref{eq:PWA-predictors}. In particular, if the clusters were given, for each numerical target $[y_c]_i$, $i=1,\ldots,m_c$, we solve the ridge regression problem \begin{equation} \min_{a^j_i,b^j_i} \alpha_j(\|a^j_i\|_2^2+(b^j_i)^2)+\sum_{k\in J_j} (y_{ki}-a^j_ix_k-b^j_i)^2 \label{eq:ridge} \end{equation} with respect to the vector $a^j_i\in\rr^n$ of coefficients and intercept $b^j_i\in\rr$, where $\alpha_j=\frac{\card{J_j}}{N}\alpha$ and $\alpha>0$ is an $\ell_2$-regularization parameter. For each binary target $[y_d]_i$, $i=1,\ldots,m_d$, we solve the regularized softmax regression problem, a.k.a. Multinomial Logistic Regression (MLR) problems~\citep{Cox66,Thi69}, \begin{eqnarray} \min_{\scriptsize\ba{c}\{a^j_h,b^j_h\}\\h\in I(i)\ea}&& \sum_{h\in I(i)}\alpha_j(\|a^j_h\|_2^2+(b^j_h)^2)-\sum_{h=1}^{m_i} \sum_{\scriptsize \ba{c} k\in J_j:\\\null [y_{dk}]_i=w^i_h\ea} \log\frac{e^{a^j_{h+t(i)} x_k+b^j_{h+t(i)}}} {\sum_{t\in I(i)} e^{a^j_tx_k+b^j_t}} \label{eq:softmax-regression} \end{eqnarray} Note that, by setting $\alpha>0$, both~\eqref{eq:ridge} and~\eqref{eq:softmax-regression} are strictly convex problems, and therefore their optimizers are unique. It is well known that in the case of binary targets $[y_d]_i\in\{0,1\}$, problem~\eqref{eq:softmax-regression} is equivalent to the regularized logistic regression problem \begin{equation} \min_{a^j_h,b^j_h} \alpha_j(\|a^j_h\|_2^2+(b^j_h)^2)+\sum_{k\in J_j} \log\left(1+e^{(1-2[y_{dk}]_i)(a^j_ix_k+b^j_i)}\right) \label{eq:logistic} \end{equation} where $h=t(i)+1$. Similarly, for preparing the background for what will follow in the next sections, we can rewrite~\eqref{eq:softmax-regression} as \begin{eqnarray} \min_{\scriptsize\ba{c}\{a^j_h,b^j_h\}\\h\in I(i)\ea}&&\hspace{-.2cm} \sum_{h\in I(i)}\alpha_j(\|a^j_h\|_2^2+(b^j_h)^2)+\sum_{h=1}^{m_i} \sum_{\scriptsize \ba{c} k\in J_j:\\\null [y_{dk}]_i=w^i_h\ea} \log\left(\sum_{t\in I(i)} e^{a^j_tx_k+b^j_t}\right) \nonumber\\ &&\hspace*{-1.5cm}-a^j_{h+t(i)}x_k-b^j_{h+t(i)}=\hspace*{-.2cm}\min_{\scriptsize\ba{c}\{a^j_h,b^j_h\}\\h\in I(i)\ea}\hspace{-.2cm} \sum_{h\in I(i)}\alpha_j(\|a^j_h\|_2^2+(b^j_h)^2)+\sum_{k\in J_j} \log\left(\sum_{t\in I(i)} e^{a^j_tx_k+b^j_t}\right)\nonumber\\ &&\hspace*{-1.5cm}-\sum_{h=1}^{m_i}[[y_{dk}]_i=w^i_h](a^j_{h+t(i)}x_k+b^j_{h+t(i)}) \label{eq:softmax-regression-2} \end{eqnarray} \subsection{Piecewise linear separation} \label{sec:PWL-separation} Clustering the feature vectors $\{x_k\}$ in $\CC_1,\ldots,\CC_K$ should be based on two goals. On the one hand, we wish to have all the data values $(x_k,y_k)$ that can be best predicted by $(a^j,b^j)$ in the same cluster $\CC_j$. On the other hand, we would like the clusters $\CC_1,\ldots,\CC_K$ to be piecewise linearly separable, i.e., that there exist a PWL separation function $\Phi:\rr^n\to\rr$ as in~\eqref{eq:PWL-sep} such that $\CC_i\subseteq P_i$. The above goals are usually conflicting (unless $y_k$ is a piecewise linear function of $x_k$), and we will have to trade them off. Several approaches exist to find a PWL separation function $\Phi$ of given clusters $\CC_1,\ldots,\CC_K$, usually attempting at minimizing the number of misclassified feature vectors $x_k$ (i.e., $x_k\in\CC_i$ and $x_k\not\in P_i$) in case the clusters are not piecewise-linearly separable. Linear programming was proposed in~\citep{BM94} to solve the following problem \[ \min_{\omega,\gamma} \sum_{j=1}^K\sum_{\scriptsize\ba{c}h=1\\h\neq j\ea}^K \sum_{k:\ x_k\in\CC_j}^N \frac{1}{\card \CC_j}\max\{(\omega^h-\omega^j)x_k+\gamma^j-\gamma^j+1,0\} \] Other approaches based on the piecewise smooth optimization algorithm of~\citep{BBP15} and averaged stochastic gradient descent \citep{Bot12} were described in \citep{BPB16a}. In this paper, we use instead $\ell_2$-regularized softmax regression \begin{subequations} \begin{equation} \ba{rl} \min_{\omega,\gamma} & \beta(\|\omega\|_F^2+\|\gamma\|_2^2)+\displaystyle{\sum_{j=1}^K\sum_{k:\ x_k\in\CC_j} -\log\frac{e^{\omega^jx_k+\gamma^j}}{\sum_{i=1}^K e^{\omega^ix_k+\gamma^i}}} \ea \label{eq:softmax-cost} \end{equation} with $\beta\geq 0$, whose solution $\omega,\gamma$ provides the PWL separation function~\eqref{eq:PWL-sep} as \begin{equation} j(x)=\arg\max_{j=1,\ldots,K} \frac{e^{\omega^jx+\gamma^j}}{\sum_{i=1}^K e^{\omega^ix+\gamma^i}}= \arg\max_{j=1,\ldots,L} \omega^jx+\gamma^j \label{eq:PWL-softmax} \end{equation \label{eq:softmax \end{subequations} and hence a polyhedral partition $\PP$ of the feature vector space as in~\eqref{eq:PWL-partition}. Note that, as observed earlier, there are infinitely many PWL functions $\Phi(x)$ as in~\eqref{eq:PWL-sep-a} providing the same piecewise-constant function $j(x)$. Hence, as it is customary, one can set one pair $(\omega^i,\gamma^i)=(0,0)$, for instance $\omega^K=0$, $\gamma^K=0$ (this is equivalent to dividing both the numerator and denominator in the first maximization in~\eqref{eq:PWL-softmax} by $e^{\omega^Kx+\gamma^K}$), and solve the reduced problem \begin{equation} \ba{rl} \min_{\{\omega^j,\gamma^j\}_{j=1}^{K-1}} &\beta(\|\omega\|_F^2+\|\gamma\|_2^2)+ \displaystyle{\sum_{j=1}^K\sum_{k:\ x_k\in\CC_j} -\log\frac{e^{\omega^jx_k+\gamma^j}}{1+\sum_{i=1}^{K-1} e^{\omega^ix_k+\gamma^i}}} \ea \label{eq:softmax-2} \end{equation} An alternative approach to softmax regression is to obtain $\PP$ from the Voronoi diagram of the centroids \begin{equation} \bar x_j= \arg\min_{x}\sum_{k\in J_j}\|x_k-x\|_2^2 =\frac{1}{\card{\CC_j}}\sum_{k\in J_j}x_k \label{eq:centroids} \end{equation} of the clusters, inducing the PWL separation function as in~\eqref{eq:PWL-sep} with \begin{subequations} \begin{eqnarray} j(x)&=&\arg\min_{j=1,\ldots,K}\|x-\bar x_j\|_2^2= \arg\max_{j=1,\ldots,K}\omega^j x+\gamma^j\\ \omega^j&=&\bar x_j',\ \gamma^j=-\frac{1}{2}\|\bar x_j\|_2^2 \end{eqnarray} \label{eq:Voronoi-partition \end{subequations} Note that the Voronoi partitioning~\eqref{eq:Voronoi-partition} has $Kn$ degrees of freedom (the centroids $\bar x_j$), while softmax regression~\eqref{eq:PWL-softmax} has $Kn+(K-n-1)$ degrees of freedom. \section{Algorithm} \label{sec:PARC} In the previous section, we have seen how to get the coefficients $a^j,b^j$ by ridge~\eqref{eq:ridge} or softmax~\eqref{eq:softmax-regression} regression when the clusters $\CC$ are given, and how to get a PWL partition of $\CC$. The question remains on how to determine the clusters $\CC_1,\ldots,\CC_K$. Let us assume that the coefficients $a^j,b^j$ have been fixed. Following~\eqref{eq:ridge} and~\eqref{eq:softmax-regression-2} we could assign each training vector $x_k$ to the corresponding cluster $\CC_{j}$ such that the following weighted sum of losses \begin{eqnarray} V^y(a^j,b^j,x_k,y_k)&=&\sum_{i=1}^{m_c}\mu_{ci}(y_{ki}-a^j_ix_k-b^j_i)^2\\ &&\hspace{-3cm}+\sum_{i=1}^{m_d}\mu_{di}\log\left(\sum_{t\in I(i)} e^{a^j_tx_k+b^j_t}\right) -\sum_{h=1}^{m_i}[[y_{dk}]_i=w^i_h](a^j_{h+t(i)}x_k+b^j_{h+t(i)}) \label{eq:cost_k-y} \end{eqnarray} is minimized, where $\mu_c\in\rr^{m_c}$, $\mu_d\in\rr^{m_d}$ are vectors of relative weights on fit losses. Besides the average quality of prediction~\eqref{eq:cost_k-y}, we also want to consider the location of the feature vectors $x_k$ to promote PWL separability of the resulting clusters using the two approaches (softmax regression and Voronoi diagrams) proposed in Section~\ref{sec:PWL-separation}. Softmax regression induces the criterion \begin{subequations} \begin{equation} V^x_s(\omega^j,\gamma^j,x_k)=- \log\frac{e^{\omega^jx_k+\gamma^j}}{1+\sum_{i=1}^{K-1} e^{\omega^ix_k+\gamma^i}}= \log\left(1+\sum_{i=1}^{K-1} e^{\omega^ix_k+\gamma^i}\right)-\omega^jx_k-\gamma^j \label{eq:cost_k-x-2} \end{equation} for $j=1,\ldots,K$, where $\omega^K=0$, $\gamma^K=0$. Note that the last logarithmic term in~\eqref{eq:cost_k-x-2} does not depend on $j$, so that it might be neglected in case $V^x_s$ gets minimized with respect to $j$. Alternatively, because of~\eqref{eq:Voronoi-partition}, Voronoi diagrams suggest penalizing the distance between $x_k$ and the centroid $\bar x_j$ of the cluster \begin{equation} V^x_v(\bar x_j,x_k)=\|x_k-\bar x_j\|_2^2 \label{eq:cost_k-x-1} \end{equation} Criteria~\eqref{eq:cost_k-x-2} and~\eqref{eq:cost_k-x-1} can be combined as follows: \begin{equation} V^x(\omega^j,\gamma^j,x_k)= \left\{\ba{ll} V^x_s(\omega^j,\gamma^j,x_k) & \mbox{if PWL partitioning~\eqref{eq:PWL-partition} is used}\\ V^x_v((\omega^j)',x_k)+0\cdot\gamma_j & \mbox{if Voronoi partitions~\eqref{eq:Voronoi-partition} are used} \ea\right. \end{equation} \label{eq:cost_k \end{subequations} Then, each training vector $x_k$ is assigned to the cluster $\CC_{j_k}$ such that \begin{equation} j_k=\arg\min_{j=1,\ldots,K} V^y(a^j,b^j,x_k,y_k)+\sigma V^x(\omega^j,\gamma^j,x_k) \label{eq:assignment} \end{equation} where $\sigma\geq 0$ is a relative weight that allows trading off between target fitting and PWL separability of the clusters. Note that, according to the definition in~\eqref{eq:argmin}, in the case of multiple minima the optimizer $j_k$ in~\eqref{eq:assignment} is always selected as the smallest index among optimal indices. The idea described in this paper is to alternate between fitting linear predictors as in~\eqref{eq:ridge}--\eqref{eq:softmax-regression} and reassigning vectors to clusters as in~\eqref{eq:assignment}, as described in Algorithm~\ref{algo:PARC} that we call PARC (Piecewise Affine Regression and Classification). The following theorem proves that indeed PARC is an algorithm, as it terminates in a finite number of steps to a local minimum of the problem of finding the $K$ best linear predictors. \begin{theorem} \label{th:convergence} Algorithm~\ref{algo:PARC} converges in a finite number of steps to a local minimum of the following mixed-integer optimization problem \begin{subequations} \beqar \min_{a,b,\omega,\gamma,z} && V(a,b,\omega,\gamma,z)\nonumber\\ \st&& \displaystyle{\sum_{j=1}^Kz_{kj}=1},\ \forall k=1,\ldots,N \label{eq:PARC-optim}\\ \hspace*{-1cm}V(a,b,\omega,\gamma,z)&=&\sigma\beta(\|\omega\|_F^2+\|\gamma\|_2^2)+ \sum_{j=1}^K\sum_{k=1}^Nz_{kj}\left( \frac{\alpha}{N}(\|a^j\|_F^2+\|b^j\|_2^2)+\right.\nonumber\\ && \left. V^y(a^j,b^j,x_k,y_k)+\sigma V^x(\omega^j,\gamma^j,x_k)\right) \label{eq:PARC-problem-cost} \eeqar \label{eq:PARC-problem \end{subequations} where $a^j\in\rr^{m\times n}$, $b^j\in\rr^{m}$, $\omega^j\in\rr^{K\times n}$,$\gamma^j\in\rr^{K}$, $\forall j=1,\ldots,K$, $z\in\{0,1\}^{N\times K}$, and with either $\omega^K=0$, $\gamma^K=0$, and $\beta\geq 0$ if PWL partioning~\eqref{eq:PWL-partition} is used, or $\gamma^j=-\frac{1}{2}\|\omega^j\|_2^2$, $\forall j=1,\ldots,K$, and $\beta=0$ if Voronoi partions~\eqref{eq:Voronoi-partition} are used. \end{theorem} \proof We prove the theorem by showing that Algorithm~\ref{algo:PARC} is a block-coordinate descent algorithm for problem~\eqref{eq:PARC-problem}, alternating between the minimization with respect to $(a,b,\omega,\gamma)$ and with respect to $z$. The proof follows arguments similar to those used to prove convergence of unsupervised learning approaches like K-means. The binary variables $z_{kj}$ are hidden variables such that $z_{kj}=1$ if and only if the target vector $y_k$ is predicted by $j(x_k)=j$ as in~\eqref{eq:PWA-predictors}. The initial clustering $\CC_1,\ldots,\CC_K$ of $\{x_k\}$ determines the initialization of the latent variables, i.e., $z_{kj}=1$ if and only if $x_k\in\CC_j$, or equivalently $k\in J_j$. Let us consider $z$ fixed. Since \begin{eqnarray*} \sum_{j=1}^K\sum_{k=1}^Nz_{kj}\left(\frac{\alpha}{N}(\|a^j\|_F^2+\|b^j\|_2^2)\right)& =&\sum_{j=1}^K\frac{\card{J_j}}{N}\alpha(\|a^j\|_F^2+\|b^j\|_2^2)\\ &=&\sum_{j=1}^K\sum_{i=1}^{m_c+m_d}\alpha_j(\|a^j_i\|_2^2+(b^j_i)^2) \end{eqnarray*} problem~\eqref{eq:PARC-problem} becomes separable into ($i$) $Km_c$ independent optimization problems of the form~\eqref{eq:ridge}, ($ii$) $Km_d$ softmax regression problems as in~\eqref{eq:softmax-regression}, and ($iii$) either a softmax regression problem as in~\eqref{eq:softmax-cost} or $K$ optimization problems as in~\eqref{eq:centroids}. Let $a^j$, $b^j$, $\omega^j$, $\gamma^j$ be the solution to such problems and consider now them fixed. In this case, problem~\eqref{eq:PARC-problem} becomes \begin{equation} \ba{rl} \min_{z\in\{0,1\}^{N\times K}} & \displaystyle{\sum_{k=1}^N\sum_{j=1}^Kz_{kj}\left(V^y(a^j,b^j,x_k,y_k)+\sigma V^x(a^j,b^j,\omega^j,\gamma^j,x_k)\right)}\\ \st& \displaystyle{\sum_{j=1}^Kz_{kj}=1},\ \forall k=1,\ldots,N \ea \label{eq:all-assignments} \end{equation} which is separable with respect to $k$ into $N$ independent binary optimization problems. The solution of~\eqref{eq:all-assignments} is given by computing $j_k$ as in~\eqref{eq:assignment} and by setting $z_{j_k}=1$ and $z_j=0$ for all $j=1,\ldots,K$, $j\neq j_k$. Having shown that PARC is a coordinate-descent algorithm, the cost $V(a,b,\omega$, $\gamma$, $z)$ in~\eqref{eq:PARC-problem} is monotonically non-increasing at each iteration of Algorithm~\ref{algo:PARC}. Moreover, since all the terms in the function are nonnegative, the sequence of optimal cost values is lower-bounded by zero, so it converges asymptotically. Moreover, as the number of possible combinations $\{z_{kj}\}$ are finite, Algorithm~\ref{algo:PARC} always terminates after a finite number of steps, since we have assumed that the smallest index $j_k$ is always taken in~\eqref{eq:assignment} in case of multiple optimizers. The latter implies that no chattering between different combinations $z_{kj}$ having the same cost $V$ is possible. \qed \begin{algorithm}[t] \caption{PARC (Piecewise Affine Regression and Classification)} \label{algo:PARC} ~~\textbf{Input}: Training dataset $(x_k,y_k)$, $k=1,\ldots,N$; number $K$ of desired linear predictors; $\ell_2$-regularization parameters $\alpha>0$, $\beta\geq 0$; fitting/separation tradeoff parameter $\sigma\geq 0$; output weight vector $\mu\in\rr^m$, $\mu\geq 0$; initial clustering $\CC_1,\ldots,\CC_K$ of $\{x_k\}$. \vspace*{.1cm}\hrule\vspace*{.1cm} \begin{enumerate}[label*=\arabic*., ref=\theenumi{}] \item $i\leftarrow 1$; \item \textbf{Repeat} \begin{enumerate}[label=\theenumi{}.\arabic*., ref=\theenumi{}.\arabic*] \item \textbf{For all} $j=1,\ldots,K$ \textbf{do} \label{algo:PARC-for} \begin{enumerate}[label=\theenumii{}.\arabic*., ref=\theenumii{}.\arabic*] \item Solve the ridge regression problem~\eqref{eq:ridge}, $\forall i=1,\ldots, m_c$; \label{algo:PARC-ridge} \item Solve the softmax regression problem~\eqref{eq:softmax-regression}, $\forall i=m_c+1,\ldots, m$; \label{algo:PARC-classification} \end{enumerate} \item PWL separation: either compute the cluster centroids $\omega^j=\bar x_j'$~\eqref{eq:centroids} and set $\gamma_j=0$, $j=1,\ldots,K$ (Voronoi partitioning), or $\omega^j,\gamma^j$ as in~\eqref{eq:softmax-cost} (general PWL separation); \label{algo:PARC-separation} \item \textbf{For all} $k=1,\ldots,N$ \textbf{do} \label{algo:PARC-assign} \begin{enumerate}[label=\theenumii{}.\arabic*., ref=\theenumii{}.\arabic*] \item Evaluate $j_k$ as in~\eqref{eq:assignment}; \item Reassign $x_k$ to cluster $\CC_{j_k}$; \label{algo:PARC-assignment} \end{enumerate} \end{enumerate} \item \textbf{Until} convergence; \label{algo:PARC-until} \item \textbf{End}. \end{enumerate} \vspace*{.1cm}\hrule\vspace*{.1cm} ~~\textbf{Output}: Final number $K_f$ of clusters; coefficients $a_j$ and intercepts $b_j$ of linear functions, and $\omega^j,\gamma^j$ of PWL separation function, $j=1,\ldots,K_f$, final clusters $\CC_1,\ldots,\CC_{K_f}$. \end{algorithm} Theorem~\ref{th:convergence} proved that PARC converges in a finite number of steps. Hence, a termination criterion for Step~\ref{algo:PARC-until} of Algorithm~\ref{algo:PARC} is that $z$ does not change from the previous iteration. An additional termination criterion is to introduce a tolerance $\epsilon>0$ and stop when the optimal cost $V(a,b,\omega,\gamma,z)$ has not decreased more than $\epsilon$ with respect to the previous iteration. In this case, as the reassignment in Step~\ref{algo:PARC-assignment} may have changed the $z$ matrix, Steps~\ref{algo:PARC-ridge}--\ref{algo:PARC-classification} must be executed before stopping, in order to update the coefficients/intercepts $(a,b)$ accordingly. Note that PARC is only guaranteed to converge to a local minimum; whether this is also a global one depends on the provided initial clustering $\CC_1,\ldots,\CC_K$, i.e., on the initial guess on $z$. In this paper, we initialize $z$ by running the K-means++ algorithm~\citep{AV07} on the set of feature vectors $x_1,\ldots,x_N$. For solving single-target regression problems, an alternative approach to get the initial clustering could be to associate to each datapoint $x_k$ the coefficients $c_k$ of the linear hyperplane fitting the $K_n$ nearest neighbors of $x_k$ (cf.~\cite{FMLM03a}), for example, by setting $K_n=2(n+1)$, and then run K-means on the set $c_1,\ldots,c_n$ to get an assignment $\delta_k$. This latter approach, however, can be sensitive to noise on measured targets and is not used in the numerical experiments reported in Section~\ref{sec:examples}. As in the K-means algorithm, some clusters may become empty during the iterations, i.e., some indices $j$ are such that $z_{kj}=0$ for all $k=1,\ldots,N$. In this case, Step~\ref{algo:PARC-for} of Algorithm~\ref{algo:PARC} only loops on the indices $j$ for which $z_{kj}=1$ for some $k$. Note that the values of $a^j$, $b^j$, $\omega^j$, and $\gamma^j$, where $j$ is the index of an empty cluster, do not affect the value of the overall function $V$ as their contribution is multiplied by 0 for all $k=1,\ldots,N$. Note also that some categories may disappear from the subset of samples in the cluster in the case of multi-category targets. In this case, still~\eqref{eq:softmax-regression} provides a solution for the coefficients $a_j^h,b^j_h$ corresponding to missing categories $h$, so that $V^y$ in~\eqref{eq:cost_k-y} remains well posed. After the algorithm stops, clusters $\CC_j$ containing less than $c_{\rm min}$ elements can be also eliminated, interpreting the corresponding samples as outliers (alternatively, their elements could be reassigned to the remaining clusters). We mention that after the PARC algorithm terminates, for each numeric target $[y_c]_i$ and cluster $\CC_j$ one can further fine-tune the corresponding coefficients/intercepts $a_i^j$, $b_i^j$ by choosing the $\ell_2$-regularization parameter $\alpha^j$ in each region via leave-one-out cross-validation on the subset of datapoints contained in the cluster. In case some features or targets have very different ranges, the numeric components in $x_k$, $y_k$ should be scaled. Note that purely solving $m_c$ ridge and $m_d$ softmax regression on the entire dataset corresponds to the special case of running PARC with $K=1$. Note also that, when $\sigma\rightarrow +\infty$, PARC will determine a PWL separation of the feature vectors, then solve $m_c$ ridge and $m_d$ softmax regression on each cluster. In this case, if the initial clustering $\CC$ is determined by K-means, PARC stops after one iteration. We remark that evaluating~\eqref{eq:cost_k-y} and \eqref{eq:cost_k-x-2} (as well as solving softmax regression problems) requires computing the logarithm of the sum of exponential, see, e.g., the recent paper \citep{BHH19} for numerically accurate implementations. When the PWL separation~\eqref{eq:softmax-cost} is used, or in case of classification problems, most of the computation effort spent by PARC is due to solving softmax regression problems. In our implementation, we have used the general L-BFGS-B algorithm~\citep{BLBZ95}, with warm-start equal to the value obtained from the previous PARC iteration for the same set of optimization variables. Other efficient methods for solving MLR problems have been proposed in the literature, such as iteratively reweighted least squares (IRLS), that is a Newton-Raphson method~\citep{Lea90}, stochastic average gradient (SAG) descent~\citep{SLB17}, the alternating direction method of multipliers (ADMM)~\citep{BPCPE11}, and methods based on majorization-minimization (MM) methods~\citep{KCFH90,FSS15,JB20}. We remark that PARC converges even if the softmax regression problem~\eqref{eq:softmax-cost} is not solved to optimality. Indeed, the proof of Theorem~\ref{th:convergence} still holds as long as the optimal cost in~\eqref{eq:softmax-cost} decreases with respect to the last computed value of $\omega,\gamma$. This suggests that during intermediate PARC iterations, in case general PWL separation is used, to save computations one can avoid using tight optimization tolerances in Step~\ref{algo:PARC-separation}. Clearly, loosening the solution of problem~\eqref{eq:softmax-cost} can impact the total number of PARC iterations; hence, there is a tradeoff to take into account. We finally remark that Steps~\ref{algo:PARC-for} and~\ref{algo:PARC-assign} can be parallelized for speeding computations up. \section{Predictor} \label{sec:predictor} After determining the coefficients $a^j$, $b^j$ by running PARC, we can define the prediction functions $\hat y_c$, $\hat y_d$, and hence the overall predictor $\hat y$ as in~\eqref{eq:PWA-predictors}. This clearly requires defining $j(x)$, i.e., a function that associates to any vector $x\in\rr^n$ the corresponding predictor out of the $K$ available. Note that the obtained clusters $\CC_j$ may not be piecewise-linearly separable. In principle any classification method on the dataset $\{x_k,\delta_k\}$, where $\delta_k=j$ if and only if $x_k\in\CC_j$, can be used to define $j(x)$. For example, nearest neighbors ($j(x)=\arg\min_{k=1,\ldots,N}\|x-x_k\|_2^2$), decision trees, na\"ive Bayes, or one-to-all neural or support vector classifiers to mention a few. In this paper, we are interested in defining $j(x)$ using a polyhedral partition $\PP=\{P_1,\ldots,P_K\}$ as stated in Section~\ref{sec:statement}, that is to select $j(x)$ such that it is IPWC as defined in~\eqref{eq:ipwc-fun}. Therefore, the natural choice is to use the values of $(\omega^j,\gamma^j)$ returned by PARC to define a PWL separation function by setting $j(x)$ as in~\eqref{eq:PWL-softmax}, which defines $P_j$ as in~\eqref{eq:PWL-partition}, or, if Voronoi partitioning is used in PARC, set $j(x)=\arg\min_{j=1,\ldots,K_f}\|x-\bar x_j\|_2^2$, which leads to polyhedral cells $P_j$ as in~\eqref{eq:Voronoi-partition}. As the clusters $\CC_1,\ldots,\CC_{K_f}$ may not be piecewise-linearly separable, after defining the partition $\PP=\{P_1,\ldots,P_{K_f}\}$, one can cluster the datapoints again by redefining $\CC_j=\{x_k:\ x_k\in P_j,\ k=1,\ldots,N\}$ and then execute one last time Steps~\ref{algo:PARC-ridge}--\ref{algo:PARC-classification} of the PARC algorithm to get the final coefficients $a,b$ defining the predictors $\hat y_c$, $\hat y_d$. Note that these may not be continuous functions of the feature vector $x$. Finally, we remark that the number of floating point operations (flops) required to evaluate the predictor $\hat y(x)$ at a given $x$ is roughly $K$ times that of a linear predictor, as it involves $K$ scalar products $[\omega^j\ \gamma^j]\smallmat{x\\ 1}$ as in~\eqref{eq:PWL-softmax} or~\eqref{eq:Voronoi-partition} ($2K(n_x+1)$ flops), taking their maximum, and then evaluate a linear predictor (another $2(n_x+1)$ flops per target in case of regression~\eqref{eq:PWA-regression} and $2m_i(n_x+1)$ flops and a maximum for multi-category targets~\eqref{eq:PWA-classifier}). \subsection{Mixed-integer encoding} \label{sec:MIP} To optimize over the estimated model $\hat y$ we need to suitably encode its numeric components $\hat y_c$ and categorical components $\hat y_d$ by introducing binary variables. First, let us introduce a binary vector $\delta\in\{0,1\}^K$ to encode the PWL partition induced by~\eqref{eq:PWL-sep} \begin{subequations} \begin{eqnarray} \hspace*{-.5cm}(\omega^i-\omega^j)x&\leq& \gamma^j-\gamma^i+M_{ji}(1-\delta_j),\ \forall i=1,\ldots,K,\ i\neq j,\ \forall j=1,\ldots,K \label{eq:mip-pwa-bigM}\\ \sum_{j=1}^K\delta_j&=&1\label{eq:mip-pwa-xor} \end{eqnarray} \label{eq:mip-partition \end{subequations} where $\omega^j,\gamma^j$ are the coefficients optimized by the PARC algorithm when PWL separation~\eqref{eq:PWL-sep} is used (with $\omega^K=0$, $\gamma^K=0$), or $\omega^j=\bar x_j'$ and $\gamma=-\|\bar x_j\|_2^2$ if Voronoi partitioning~\eqref{eq:Voronoi-partition} is used instead. The constraint~\eqref{eq:mip-pwa-bigM} is the ``big-M'' reformulation of the logical constraint $[\delta_{j}=1]\rightarrow[x\in P_i]$, that, together with the exclusive-or (SOS-1) constraint~\eqref{eq:mip-pwa-xor} models the constraint $[\delta_{j}=1]\leftrightarrow[x\in P_i]$. The values of $M_{ji}$ are upper-bounds that need to satisfy \begin{equation} M_{ji}\geq \max_{x\in\BB} (\omega^i-\omega^j)x-\gamma^j+\gamma^i,\ \forall i,j=1,\ldots,K,\ i\neq j \label{eq:Mij} \end{equation} where $\BB\subset\rr^n$ is a compact subset of features of interest. For example, given the dataset $\{x_k\}_{k=1}^N$ of features, we can set $\BB$ as a box containing all the sample feature vectors so that the values $M_{ij}$ in~\eqref{eq:Mij} can be easily computed by solving $K(K-1)$ linear programs. A simpler way to estimate the values $M_{ji}$ is given by the following lemma \citep[Lemma 1]{LK00}: \begin{lemma} \label{lemma:bigM} Let $\BB=\{x\in\rr^n:\ x_{\rm min}\leq x\leq x_{\rm max}$ and $v\in\rr^n$. Let $v^+=\max\{v,0\}$, $v^-=\max\{v,0\}$. Then \begin{equation} \sum_{i=1}^n v_i^+x_{\rm min,i}-v_i^-x_{\rm max,i}\leq v'x\leq \sum_{i=1}^n v_i^+x_{\rm max,i}-v_i^-x_{\rm min,i} \label{eq:big-M-lemma} \end{equation} \end{lemma} \proof Since $x_{\rm min,i}\leq x_i\leq x_{\rm max,i}$ and $v=v^+-v^-$, we get \[ v'x=\sum_{i=1}^nv_ix_i=\sum_{i=1}^n (v_i^+-v_i^-)x_i\leq \sum_{i=1}^n v_i^+x_{\rm max,i}-v_i^-x_{\rm min,i} \] and similarly $v'x\geq \sum_{i=1}^n v_i^+x_{\rm min,i}-v_i^-x_{\rm max,i}$. \qed By applying Lemma~\ref{lemma:bigM} for $v=\omega^i-\omega^j$,~\eqref{eq:Mij} is satisfied by setting \begin{equation} M_{ji}=\gamma^i-\gamma^j+\sum_{h=1}^n \max\{\omega^i_h-\omega^j_h,0\} x_{\rm max,h}-\max\{\omega^j_h-\omega^i_h,0\}x_{\rm min,h} \label{eq:big-Mij-lemma} \end{equation} for all $i,j=1,\ldots,K,\ i\neq j$. Having encoded the PWL partition, the $i$th predictor is given by \begin{subequations} \begin{equation} [\hat y_c(x)]_i=\sum_{j=1}^K p_{ij} \label{eq:ychat-1} \end{equation} where $p_{ij}\in\rr$ are optimization variables representing the product $p_{ji}=\delta_j(a^j_ix+b_i^j)$. This is modeled by the following mixed-integer linear inequalities \begin{equation} \ba{rcl} p_{ji}&\leq&a^j_ix+b_i^j-M^{c-}_{ji}(1-\delta_j)\\ p_{ji}&\geq&a^j_ix+b_i^j-M^{c+}_{ji}(1-\delta_j)\\ p_{ji}&\leq&M^{c+}_{ji}\delta_j\\ p_{ji}&\geq&M^{c-}_{ji}\delta_j \ea \label{eq:ychat-2} \end{equation} The coefficients $M^{c-}_{ji}$, $M^{c+}_{ji}$ need to satisfy $M^{c-}_{ji}\leq \min_{x\in\BB}a^j_ix+b_i^j\leq \max_{x\in\BB}a^j_ix+b_i^j\leq M^{c+}_{ji}$ and can be obtained by linear programming or, more simply, by applying Lemma~\ref{lemma:bigM}. Regarding the $m_d$ classifiers $\hat y_{di}$, to model the ``$\arg\max$'' in~\eqref{eq:PWA-classifier} we further introduce $s_y$ binary variables $\nu_{ih}\in\{0,1\}$, $h=1,\ldots,m_i$, $i=1,\ldots,m_d$, satisfying the following big-M constraints \begin{eqnarray} &&\hspace*{-1.5cm}(a^j_h-a^j_t)x\geq b^j_t-b^j_h-M^d_{ht}(2-\nu_{ih}-\delta_j),\ \forall h,t\in I(i),\ h\neq t,\ \forall j=1,\ldots,K\label{eq:bigM-max}\\ &&\hspace*{-1.5cm}\sum_{h=1}^{m_i}\nu_{ih}=1,\ \forall i=1,\ldots,m_d \label{eq:ydhat} \end{eqnarray} where the coefficients $M^d_{ht}$ must satisfy $\displaystyle{ M^{d}_{ht}\geq \max_{j=1,\ldots,K}\{\max_{x\in\BB}(a^j_t-a^j_h)x+b^j_t-b^j_h\}} $. Note that the constraints in~\eqref{eq:bigM-max} become redundant when $\delta_j=0$ or $\nu_{ih}=0$ and lead to $a^j_hx+b^j_h\geq a^j_t+b^j_t$ for all $t\in I(i)$, $t\neq h$, when $\nu_{ih}=1$ and $\delta_j=1$, which is the binary equivalent of $[\hat y_d(x)]_i=w^i_h$ for $x\in P_j$. Then, the $i$th classifier is given by \begin{equation} [\hat y_d(x)]_i=\sum_{h=1}^{m_i}w^i_{h}\nu_{ih} \label{eq:ydhat-3} \end{equation} \label{eq:yhat-bigM \end{subequations} In conclusion,~\eqref{eq:mip-partition} and~\eqref{eq:yhat-bigM} provide a mixed-integer linear reformulation of the predictors $\hat y_c,\hat y_d$ as in~\eqref{eq:PWA-predictors} returned by the PARC algorithm. This enables solving optimization problems involving the estimated model, possibly under linear and logical constraints on features and targets. For example, given a target vector $y_{\rm ref}$, the problem of finding the feature vector $x^*$ such that $\hat y_c(x^*)\approx y_{\rm ref}$ can be solved by minimizing $\|\hat y_c(x)-y_{\rm ref}\|_\infty$ as in the following mixed-integer linear program (MILP) \begin{eqnarray} \min_{x,p,\delta,\epsilon}&&\epsilon\nonumber\\ \st && \epsilon\geq \pm \left(\sum_{j=1}^K p_{ij}-y_{{\rm ref},i}\right) \label{eq:MILP-tracking}\\ &&\mbox{Constraints~\eqref{eq:mip-partition},~\eqref{eq:ychat-1},~\eqref{eq:ychat-2}} \nonumber \end{eqnarray} The benefit of the MILP formulation~\eqref{eq:MILP-tracking} is that it can be solved to global optimality by very efficient solvers. Note that if a more refined nonlinear predictor $\hat y_{NL}$ is available, for example, a feedforward neural network trained on the same dataset, the solution $x^*$ can be used to warm-start a nonlinear programming solver based on $\hat y_{NL}$, which would give better chances to find a global minimizer. \section{Examples} \label{sec:examples} We test the PARC algorithm on different examples. First, we consider synthetic data generated from sampling a piecewise affine function and see whether PARC can recover the function. Second, we consider synthetic data from a toy example in which a nonlinear function generates the data, so to test the effect of the main hyper-parameters of PARC, namely $K$ and $\sigma$, also optimizing over the model using mixed-integer linear programming. In Section~\ref{sec:real-datasets} we will instead test PARC on several regression and classification examples on real datasets from the PMLB repository~\citep{Olson2017PMLB}. All the results have been obtained in Python 3.8.3 on an Intel Core i9-10885H CPU @2.40GHz machine. The \textsf{scikit-learn} package~\citep{scikit-learn} is used to solve ridge and softmax regression problems, the latter using L-BFGS to solve the nonlinear programming problem~\eqref{eq:softmax-regression}. \subsection{Synthetic datasets} \label{sec:toy-datasets} \subsubsection{Piecewise affine function} We first test whether PARC can reconstruct targets generated from the following randomly-generated PWA function \begin{eqnarray} f(x)&=&\max\left\{ \smallmat{0.8031\\0.0219\\-0.3227}'\smallmat{x_1\\x_2\\1} , \smallmat{0.2458\\-0.5823\\-0.1997}'\smallmat{x_1\\x_2\\1} , \smallmat{0.0942\\-0.5617\\-0.1622}'\smallmat{x_1\\x_2\\1} , \smallmat{0.9462\\-0.7299\\-0.7141}'\smallmat{x_1\\x_2\\1} , \right.\nonumber\\&&\left. \smallmat{-0.4799\\0.1084\\-0.1210}'\smallmat{x_1\\x_2\\1} , \smallmat{0.5770\\0.1574\\-0.1788}'\smallmat{x_1\\x_2\\1} \right\} \label{eq:PWA-example-fun} \end{eqnarray} We generate a dataset of 1000 random samples uniformly distributed in the box $[-1,1]\times[-1,1]$, plotted in Figures~\ref{fig:PWA_function_3} and~\ref{fig:PWA_function_1}, from which we extract $N=800$ training samples and leave the remaining $N=200$ samples for testing. Figure~\ref{fig:PWA_function_2} shows the partition generated by the PWL function~\eqref{eq:PWA-example-fun} as in~\eqref{eq:PWL-partition}. We run PARC with $K=6$, $\sigma=0$, PWL partitioning~\eqref{eq:softmax} and $\beta=10^{-3}$, stopping tolerance $\epsilon=10^{-4}$ on $V(a,b,\omega,\gamma,z)$, which converges in $2.2$~s after 8 iterations. The sequence of function values $V$ is reported in Figure~\ref{fig:PWA_function_4}. The final polyhedral partition obtained by PARC is shown in Figure~\ref{fig:PWA_function_2}. In this ideal case, PARC can recover the underlying function generating the data quite well. \begin{figure}[th!] \centerline{ \begin{tabular}[t]{c} \subfigure[True PWA function~\eqref{eq:PWA-example-fun} and dataset]{\parbox{.45\hsize}{\includegraphics[width=\hsize]{PWA_function_3}\\~} \label{fig:PWA_function_3}} \hspace{.1\hsize} \subfigure[Cost function during PARC iterations] {\parbox{.45\hsize}{\includegraphics[width=\hsize]{PWA_function_4}\\~} \label{fig:PWA_function_4}}\\ \subfigure[True PWA partition induced by~\eqref{eq:PWA-example-fun}]{\parbox{.45\hsize}{\includegraphics[width=\hsize]{PWA_function_1}\\~} \label{fig:PWA_function_1}} \hspace{.1\hsize} \subfigure[PWA partition generated by PARC] {\parbox{.45\hsize}{\includegraphics[width=\hsize]{PWA_function_2}\\~} \label{fig:PWA_function_2}}\\ \end{tabular} } \label{fig:PWA_example} \caption{PARC algorithm for regression on training data generated by the PWA function~\eqref{eq:PWA-example-fun}.} \end{figure} \subsubsection{Nonlinear function} We solve another simple regression example on a dataset of $N=1000$ randomly-generated samples of the nonlinear function \begin{equation} y(x_1,x_2) = \sin\left(4x_1-5\left(x_2 - \frac{1}{2}\right)^2\right) + 2x_2 \label{eq:NL-example-function} \end{equation} Again we use 80\% of the samples as training data and the remaining 20\% for testing. The function and the training dataset are shown in Figures~\ref{fig:NL_function},~\ref{fig:NL_function_3d}. We run PARC with $\sigma=1$, $\epsilon=10^{-4}$, PWL partitioning~\eqref{eq:softmax} with $\beta=10^{-3}$, and different values of $K$. The level sets and training data are reported in Figure~\ref{fig:NL_example-1}. The resulting piecewise linear regression functions are shown in Figure~\ref{fig:NL_example-2}, which also shows the solution obtained by solving the MILP~\eqref{eq:MILP-tracking} for $y_{\rm ref}=3$. The results obtained by running PARC for different values of $K$, $\sigma$ and the two alternative separation criteria (Voronoi partitioning and softmax regression with $\beta=10^{-3}$) are reported in Table~\ref{tab:NL_R2train} (R$^2$-score on training data), Table~\ref{tab:NL_R2test} (R$^2$-score on test data), Table~\ref{tab:NL_CPUtrain} (CPU time [s] to execute PARC), Table~\ref{tab:NL_Iters} (number of PARC iterations). The best results are usually obtained for $\sigma=1$ using softmax regression (S) for PWL partitioning as in~\eqref{eq:softmax-cost}. The CPU time spent to solve the MILP~\eqref{eq:MILP-tracking} using the CBC solver\footnote{\url{https://github.com/coin-or/Cbc}} through the Python MIP package~\footnote{\url{https://github.com/coin-or/python-mip}} for $K$ = 3, 5, 8, 12, and 30 is, respectively, 8, 29, 85, 251, and 1420~ms. Note that the case $K=1$ corresponds to ridge regression on the entire dataset, while $\sigma=10000$ approximates the case $\sigma\rightarrow +\infty$, corresponding to pure PWL separation + ridge regression on each cluster. \begin{figure}[t] \centerline{ \begin{tabular}[t]{c} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_function}\\~} \label{fig:NL_function}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_3_sigma_1.0000_Softmax_1}\\~} \label{fig:NL_function_3}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_5_sigma_1.0000_Softmax_1}\\~} \label{fig:NL_function_5}} \\ \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_8_sigma_1.0000_Softmax_1}\\~} \label{fig:NL_function_8}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_12_sigma_1.0000_Softmax_1}\\~} \label{fig:NL_function_12}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_30_sigma_1.0000_Softmax_1}\\~} \label{fig:NL_function_30}} \end{tabular} } \caption{Training data and results of PARC for regression: nonlinear function~\eqref{eq:NL-example-function}.} \label{fig:NL_example-1} \end{figure} \begin{figure}[th!] \centerline{ \begin{tabular}[t]{c} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_function_3D}\\~} \label{fig:NL_function_3d}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_3_sigma_1.0000_Softmax_3}\\~} \label{fig:NL_function_3_3d}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_5_sigma_1.0000_Softmax_3}\\~} \label{fig:NL_function_5_3d}} \\ \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_8_sigma_1.0000_Softmax_3}\\~} \label{fig:NL_function_8_3d}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_12_sigma_1.0000_Softmax_3}\\~} \label{fig:NL_function_12_3d}} \hspace{.05\hsize} \subfigure[ {\parbox{.3\hsize}{\includegraphics[width=\hsize]{NL_K_30_sigma_1.0000_Softmax_3}\\~} \label{fig:NL_function_30_3d}} \end{tabular} } \caption{Training data and results of PARC for regression: nonlinear function~\eqref{eq:NL-example-function}. The result of the MILP optimization~\eqref{eq:MILP-tracking} is represented by the red dot.} \label{fig:NL_example-2} \end{figure} \begin{table}[t]\begin{center} \scalebox{.8}[0.8]{ \begin{tabular}{r|r|r|r|r|r|r} $\sigma$ & $K=1$~~~~ & $K=3$~~~~ & $K=5$~~~~ & $K=8$~~~~ & $K=12$~~~~ & $K=30$~~~~ \\[.5em]\hline (S) \hfill 0 & 0.565 (1.4\%)& 0.899 (2.0\%)& 0.979 (0.2\%)& 0.991 (0.2\%)& 0.995 (0.2\%)& 0.998 (0.1\%) \\ (V) \hfill 0 & 0.565 (1.4\%)& 0.886 (3.1\%)& 0.974 (0.3\%)& 0.986 (0.2\%)& 0.993 (0.2\%)& 0.998 (0.0\%) \\ (S) \hfill 0.01 & 0.565 (1.4\%)& 0.899 (2.2\%)& 0.979 (0.2\%)& 0.991 (0.2\%)& 0.995 (0.1\%)& 0.999 (0.1\%) \\ (V) \hfill 0.01 & 0.565 (1.4\%)& 0.887 (3.1\%)& 0.973 (0.3\%)& 0.986 (0.2\%)& 0.993 (0.1\%)& 0.998 (0.0\%) \\ (S) \hfill 1 & 0.565 (1.4\%)& 0.895 (2.3\%)& 0.982 (0.2\%)& 0.994 (0.2\%)& 0.998 (0.0\%)& 0.999 (0.0\%) \\ (V) \hfill 1 & 0.565 (1.4\%)& 0.881 (3.0\%)& 0.974 (0.3\%)& 0.986 (0.2\%)& 0.994 (0.1\%)& 0.999 (0.0\%) \\ (S) \hfill 100 & 0.565 (1.4\%)& 0.908 (0.9\%)& 0.977 (0.5\%)& 0.986 (0.2\%)& 0.994 (0.1\%)& 0.999 (0.0\%) \\ (V) \hfill 100 & 0.565 (1.4\%)& 0.887 (3.6\%)& 0.972 (0.4\%)& 0.989 (0.3\%)& 0.995 (0.0\%)& 0.999 (0.0\%) \\ (S) \hfill 10000 & 0.565 (1.4\%)& 0.834 (2.1\%)& 0.969 (0.3\%)& 0.985 (0.2\%)& 0.994 (0.1\%)& 0.999 (0.0\%) \\ (V) \hfill 10000 & 0.565 (1.4\%)& 0.865 (3.6\%)& 0.971 (0.3\%)& 0.985 (0.2\%)& 0.994 (0.1\%)& 0.999 (0.0\%) \\ \hline \end{tabular}} \caption{PARC regression on targets from nonlinear function~\eqref{eq:NL-example-function}: R$^2$ score on training data, mean (std). PWL separation: (S) = softmax regression, (V) for Voronoi partitioning.} \label{tab:NL_R2train} \end{center} \end{table} \begin{table}[t]\begin{center} \scalebox{.8}[0.8]{ \begin{tabular}{r|r|r|r|r|r|r} $\sigma$ & $K=1$~~~~ & $K=3$~~~~ & $K=5$~~~~ & $K=8$~~~~ & $K=12$~~~~ & $K=30$~~~~ \\[.5em]\hline (S) \hfill 0 & 0.548 (6.5\%)& 0.889 (2.6\%)& 0.976 (0.5\%)& 0.989 (0.3\%)& 0.994 (0.2\%)& 0.997 (0.1\%) \\ (V) \hfill 0 & 0.548 (6.5\%)& 0.872 (3.6\%)& 0.970 (0.7\%)& 0.985 (0.4\%)& 0.993 (0.2\%)& 0.998 (0.1\%) \\ (S) \hfill 0.01 & 0.548 (6.5\%)& 0.894 (2.5\%)& 0.976 (0.5\%)& 0.989 (0.4\%)& 0.994 (0.1\%)& 0.998 (0.1\%) \\ (V) \hfill 0.01 & 0.548 (6.5\%)& 0.877 (3.3\%)& 0.969 (0.6\%)& 0.985 (0.3\%)& 0.992 (0.3\%)& 0.997 (0.1\%) \\ (S) \hfill 1 & 0.548 (6.5\%)& 0.883 (2.8\%)& 0.981 (0.3\%)& 0.993 (0.2\%)& 0.997 (0.1\%)& 0.999 (0.0\%) \\ (V) \hfill 1 & 0.548 (6.5\%)& 0.868 (3.5\%)& 0.970 (0.7\%)& 0.985 (0.3\%)& 0.993 (0.2\%)& 0.998 (0.1\%) \\ (S) \hfill 100 & 0.548 (6.5\%)& 0.898 (1.6\%)& 0.970 (1.0\%)& 0.982 (0.2\%)& 0.992 (0.2\%)& 0.998 (0.0\%) \\ (V) \hfill 100 & 0.548 (6.5\%)& 0.874 (4.1\%)& 0.967 (0.8\%)& 0.987 (0.4\%)& 0.993 (0.2\%)& 0.998 (0.1\%) \\ (S) \hfill 10000 & 0.548 (6.5\%)& 0.816 (3.2\%)& 0.963 (0.8\%)& 0.980 (0.3\%)& 0.992 (0.1\%)& 0.998 (0.0\%) \\ (V) \hfill 10000 & 0.548 (6.5\%)& 0.846 (4.4\%)& 0.965 (0.8\%)& 0.982 (0.3\%)& 0.993 (0.2\%)& 0.998 (0.0\%) \\ \hline \end{tabular}} \caption{PARC regression on targets from nonlinear function~\eqref{eq:NL-example-function}: R$^2$ score on test data, mean (std). PWL separation: (S) = softmax regression, (V) for Voronoi partitioning.} \label{tab:NL_R2test} \end{center} \end{table} \begin{table}[t]\begin{center} \scalebox{.8}[0.8]{ \begin{tabular}{r|r|r|r|r|r|r} $\sigma$ & $K=1$~~~~ & $K=3$~~~~ & $K=5$~~~~ & $K=8$~~~~ & $K=12$~~~~ & $K=30$~~~~ \\[.5em]\hline (S) \hfill 0 & 1.0 (0.0\%)& 18.9 (44.6\%)& 13.1 (27.3\%)& 16.9 (28.4\%)& 18.9 (22.8\%)& 13.0 (17.9\%) \\ (V) \hfill 0 & 1.0 (0.0\%)& 20.2 (39.7\%)& 12.8 (26.3\%)& 17.1 (30.4\%)& 20.1 (27.4\%)& 13.9 (20.5\%) \\ (S) \hfill 0.01 & 1.0 (0.0\%)& 17.7 (42.4\%)& 13.3 (37.0\%)& 17.4 (28.3\%)& 20.5 (39.6\%)& 12.3 (19.6\%) \\ (V) \hfill 0.01 & 1.0 (0.0\%)& 17.8 (43.0\%)& 13.8 (32.7\%)& 14.3 (26.3\%)& 19.7 (31.5\%)& 14.5 (33.8\%) \\ (S) \hfill 1 & 1.0 (0.0\%)& 19.2 (46.0\%)& 11.2 (27.2\%)& 15.5 (27.3\%)& 14.2 (17.8\%)& 7.9 (14.9\%) \\ (V) \hfill 1 & 1.0 (0.0\%)& 19.4 (41.4\%)& 13.0 (39.1\%)& 15.1 (23.1\%)& 18.9 (36.4\%)& 12.5 (33.2\%) \\ (S) \hfill 100 & 1.0 (0.0\%)& 19.4 (24.1\%)& 8.2 (36.3\%)& 5.8 (32.5\%)& 4.0 (37.8\%)& 5.2 (24.0\%) \\ (V) \hfill 100 & 1.0 (0.0\%)& 17.4 (49.1\%)& 11.4 (41.7\%)& 17.2 (42.6\%)& 12.8 (23.5\%)& 8.9 (22.7\%) \\ (S) \hfill 10000 & 1.0 (0.0\%)& 3.0 (31.2\%)& 3.1 (26.8\%)& 3.4 (30.0\%)& 4.6 (27.0\%)& 5.5 (29.9\%) \\ (V) \hfill 10000 & 1.0 (0.0\%)& 11.7 (53.7\%)& 5.9 (45.8\%)& 4.5 (29.7\%)& 4.2 (37.1\%)& 3.5 (64.5\%) \\ \hline \end{tabular}} \caption{PARC regression on targets from nonlinear function~\eqref{eq:NL-example-function}: number of PARC iterations, mean (std). PWL separation: (S) = softmax regression, (V) for Voronoi partitioning.} \label{tab:NL_Iters} \end{center} \end{table} \begin{table}[t]\begin{center} \scalebox{.8}[0.8]{ \begin{tabular}{r|r|r|r|r|r|r} $\sigma$ & $K=1$~~~~ & $K=3$~~~~ & $K=5$~~~~ & $K=8$~~~~ & $K=12$~~~~ & $K=30$~~~~ \\[.5em]\hline (S) \hfill 0 & 0.12 (9.8\%)& 1.33 (43.6\%)& 1.46 (24.6\%)& 2.90 (28.0\%)& 5.18 (21.1\%)& 7.14 (17.3\%) \\ (V) \hfill 0 & 0.04 (11.4\%)& 0.78 (36.1\%)& 0.77 (24.9\%)& 1.58 (30.7\%)& 2.64 (26.3\%)& 4.44 (19.5\%) \\ (S) \hfill 0.01 & 0.12 (7.7\%)& 1.25 (40.3\%)& 1.48 (36.4\%)& 2.95 (30.1\%)& 5.47 (40.9\%)& 7.22 (18.7\%) \\ (V) \hfill 0.01 & 0.04 (13.2\%)& 0.65 (41.2\%)& 0.82 (31.4\%)& 1.31 (25.1\%)& 2.58 (30.8\%)& 4.61 (32.8\%) \\ (S) \hfill 1 & 0.12 (10.4\%)& 1.36 (43.8\%)& 1.42 (24.7\%)& 3.36 (26.0\%)& 4.14 (15.9\%)& 4.26 (14.9\%) \\ (V) \hfill 1 & 0.04 (13.5\%)& 0.71 (38.6\%)& 0.78 (35.7\%)& 1.36 (22.7\%)& 2.48 (36.0\%)& 3.98 (31.1\%) \\ (S) \hfill 100 & 0.12 (8.9\%)& 1.45 (24.2\%)& 1.11 (34.5\%)& 1.10 (30.4\%)& 1.04 (35.6\%)& 2.68 (22.1\%) \\ (V) \hfill 100 & 0.04 (10.6\%)& 0.64 (44.9\%)& 0.69 (38.0\%)& 1.56 (43.5\%)& 1.70 (21.9\%)& 2.90 (21.0\%) \\ (S) \hfill 10000 & 0.12 (9.2\%)& 0.24 (27.1\%)& 0.41 (22.6\%)& 0.66 (26.9\%)& 1.14 (24.2\%)& 2.80 (27.2\%) \\ (V) \hfill 10000 & 0.04 (11.5\%)& 0.45 (49.7\%)& 0.38 (39.7\%)& 0.45 (24.8\%)& 0.62 (31.4\%)& 1.21 (61.8\%) \\ \hline \end{tabular}} \caption{PARC regression on targets from nonlinear function~\eqref{eq:NL-example-function}: training time [s], mean (std). PWL separation: (S) = softmax regression, (V) for Voronoi partitioning.} \label{tab:NL_CPUtrain} \end{center} \end{table} \subsection{Real-world datasets} \label{sec:real-datasets} We test the PARC algorithm on real-world datasets for regression and classification from the PMLB repository~\citep{Olson2017PMLB}. The features containing four or less distinct values are treated as categorical and one-hot encoded, all the remaining features as numerical. In all tests, $N_{\rm tot}$ denotes the total number of samples in the dataset, whose 80\% is used for training and the rest 20\% for testing. PARC is run with $\sigma=1$, softmax regression for PWL partitioning~\eqref{eq:softmax-cost}, $\epsilon=10^{-4}$, $\alpha=0.1$, $\beta=10^{-3}$. The minimum size of a cluster not to be discarded is 1\% of the number $N$ of training samples. Prediction quality is measured in terms of $R^2$ score (in case of regression problems), or accuracy score $a$ (for classification), respectively defined as \[ R^2=1-\frac{\sum_{k=1}^N(y_k-\hat y(x_k))^2}{\sum_{k=1}^N(y_k-\frac{1}{N}\sum_{k=1}^Ny_k)^2},\quad a=\frac{1}{N}\sum_{k=1}^N [\hat y(x_k)=y_k] \] The neural networks and decision trees used for comparison are trained using scikit-learn~\citep{scikit-learn} functions. The stochastic optimizer Adam~\citep{KB15} is used for training the coefficient and bias terms of the neural network. \subsubsection{Regression problems} We extracted all the datasets from the PMLB repository with numeric targets containing between $N_{\rm tot}=500$ and $5000$ samples and between $n_x=2$ and $20$ features (before one-hot encoding categorical features). Five-fold cross-validation is run on the training dataset for all values of $K$ between $2$ and $15$ to determine the value $K^*$ that is optimal in terms of average $R^2$ score. For comparison, we run PARC with fixed values of $K$ and compare against other methods providing piecewise linear partitions, particularly a neural network with ReLU activation function with a single layer of $K^*$ neurons and a decision tree with ten non-leaf nodes. Note that the neural network requires $K^*$ binary variables to encode the ReLU activation functions in a MIP, the same number as the PWL regressor determined by PARC, as described in Section~\ref{sec:MIP}. In contrast, the decision tree requires ten binary variables. The $R^2$ scores obtained on the datasets are shown in Table~\ref{tab:dataset-regression-training} (training data) and in Table~\ref{tab:dataset-regression-test} (test data). The CPU time spent on solving the training problems is reported in Table~\ref{tab:dataset-CPUTIME-regression}. The results show that PARC often provides better fit on training data, especially for large values of $K$. On test data, PARC and neural networks with $K^*$ ReLU neurons provide the best results. Some poor results of PARC on test data for large values of $K$ are usually associated with overfitting the training dataset, see for example \textsf{522\_pm10}, \textsf{547\_no2}, \textsf{627\_fri\_C1\_1000\_5}. \subsubsection{Classification problems} We extracted all datasets from the PMLB repository with categorical targets with at most $m_1=10$ classes, containing between $N_{\rm tot}=1000$ and 5000 samples, and between $n_x=2$ and 20 features (before one-hot encoding categorical features). We compare PARC with $K=2$, $3$, $5$ to softmax regression (corresponding to setting $K=1$ in PARC), a neural network (NN) with ReLU activation function and a single layer of $K=5$ neurons, and a decision tree (DT) with $5$ non-leaf nodes. Encoding the PARC classifier as an MIP requires $K+m_1$ binary variables as described in Section~\ref{sec:MIP}, the NN requires $5+m_1$ binary variables for MIP encoding, the DT requires $5+m_1$ binary variables ($m_1$ variables are required to encode the $\arg\max$ selecting the class with highest score). In this test campaign, computing $K^*$ by cross-validation has not shown to bring significant benefits and is not reported. The accuracy scores obtained on the datasets are shown in Table~\ref{tab:dataset-classification-training} (training data) and in Table~\ref{tab:dataset-classification-test} (test data). The CPU time spent on solving the training problems is reported in Table~\ref{tab:dataset-CPUTIME-classification}. On training data, PARC with $K=5$ provides the best accuracy in about 75\% of the datasets, with neural networks based on $5$ ReLU neurons better performing in the remaining cases. On test data, most methods perform similarly, with neural networks providing slightly superior accuracy. \section{Conclusions} \label{sec:conclusions} The regression and classification algorithm proposed in this paper generalizes linear regression and classification approaches, in particular, ridge regression and softmax regression, to a piecewise linear form. Such a form is amenable for mixed-integer encoding, particularly beneficial when the obtained predictor becomes part of an optimization model. Results on synthetic and real-world datasets show that the accuracy of the method is comparable to that of alternative approaches that admit a piecewise linear form of similar complexity. A possible drawback of PARC is its computation time, mainly due to solving a sequence of softmax regression problems. This makes PARC applicable to datasets whose size, in terms of number of samples and features, is such that standard softmax regression is a feasible approach. Other regression and classification methods, such as deep neural networks, more complex decision trees, and even random forests may achieve better scores on test data and reduced training time. However, they would return predictors that are more complicated to optimize over the predictor than the proposed piecewise linear models. The proposed algorithm can be extended in several ways. For example, $\ell_1$-penalties can be introduced in~\eqref{eq:cost_k-y} to promote sparsity of $a,b$. The proof of Theorem~\ref{th:convergence} can be easily extended to cover such a modification. Moreover, basis functions $\phi_i(x)$ can be used instead of $x$ directly, such as canonical piecewise linear functions~\citep{LU92,CD88,JDD00} to maintain the PWL nature of the predictor, with possibly different basis functions chosen for partitioning the feature space and for fitting targets. The proposed algorithm is also extendable to other regression, classification, and separation methods than linear ones, as long as we can associate a suitable cost function $V^y$/$V^x$. As an example, neural networks with ReLU activation functions might be used instead of ridge regression for extended flexibility, for which we can define $V_y(a^j,b^j,x_k,y_k)$ as the loss computed on the training data of cluster \#$j$. Ongoing research is devoted to alternative methods to obtain the initial assignment of datapoints to clusters, as this is a crucial step that affects the quality of the minimum PARC converges to, and to applying the proposed method to data-driven model predictive control of hybrid dynamical systems. \include{datasets_regression_training} \include{datasets_regression_test} \include{datasets_regression_CPUTIME} \include{datasets_classification_training} \include{datasets_classification_test} \include{datasets_classification_CPUTIME}

BkiUc1jxK6Ot9TMSqdlF

\section{Presentation of the continuous model} The Richards equation \cite{Richards31} is one of the most well-known simplified models for water filtration in unsaturated soils. While it has been extensively studied in the case of a homogeneous domain, the heterogeneous case seems to have received less attention in the literature, at least from the numerical perspective. The purpose of this paper is to investigate a class of discretization scheme for a special instance of heterogeneous domains, namely, those with piecewise-uniform physical properties. Before stating our objectives in a precise manner, a few prerequisites must be introduced regarding the model in \S\ref{sse:richards}--\S\ref{ssec:cont.stability} and the scheme in \S\ref{ssec:mesh}--\S\ref{ssec:scheme}. The goal of the paper is fully described in \S\ref{sse:goals}, in relation with other works. Practical aspects related to the numerical resolution are detailed in \S\ref{sec:numerics} and results on illustrative test cases are shown in \S\ref{sec:results}. A summary of our main results is provided in \S\ref{ssec:main}, together with the outline of the paper. \subsection{Richards' equation in heterogeneous porous media}\label{sse:richards} Let $\Omega \subset \mathbb{R}^d$, where $d\in \{2,3\}$, be a connected open polyhedral domain with Lipschitz boundary $\partial\Omega$. A porous medium defined over the region $\Omega$ is characterized by \begin{itemize}[noitemsep] \item[--] the porosity $\phi : \Omega \rightarrow (0,1]$; \item[--] the permeability $\lambda : \Omega \rightarrow \mathbb{R}_+^*$; \item[--] the mobility function $\eta : [0,1]\times \Omega \rightarrow \mathbb{R}_+$; \item[--] the capillary pressure law $\mathcal{S} : \mathbb{R} \times \Omega \rightarrow [0,1]$. \end{itemize} The conditions to be satisfied by $\phi$, $\lambda$, $\eta$ and $\mathcal{S}$ will be elaborated on later. In a homogeneous medium, these physical properties are uniform over $\Omega$, i.e., \[ \phi(x) = \phi_0, \qquad \lambda(x) = \lambda_0, \qquad \eta(s,x)= \eta_0(s), \qquad \mathcal{S}(p,x) = \mathcal{S}_0(p) \] for all $x\in \Omega$. In a heterogeneous medium, the dependence of $\phi$, $\lambda$, $\eta$ and $\mathcal{S}$ on $x$ must naturally be taken into account. The quantity $s$, called saturation, measures the relative volumic presence of water in the medium. The quantity $p$ is the pressure. Let $T>0$ be a finite time horizon. We designate by $Q_T = (0,T) \times \Omega$ the space-time domain of interest. Our task is to find the saturation field $s : Q_T \rightarrow [0,1]$ and the pressure field $p : Q_T \rightarrow \mathbb{R}$ so as to satisfy \begin{itemize}[noitemsep] \item the interior equations \begin{subequations}\label{eq:bigsyst} \begin{alignat}{2} \phi(x) \, \partial_t s + {\text{div}}\, F & = 0 & \qquad & \text{in}\; Q_{T}, \label{eq:massbal}\\ F + \lambda(x) \,\eta(s,x) \, \nabla (p - \varrho g\cdot x) & = 0 & \qquad & \text{in}\; Q_{T},\label{eq:darcymuskat}\\ s - \mathcal{S}(p,x) & = 0 & \qquad & \text{in}\; Q_{T} ; \label{eq:spcapillar} \end{alignat} \item the boundary conditions \begin{alignat}{2} F\cdot n(x) & = 0 &\qquad & \text{on} \; (0,T) \times \Gamma^{\rm N}, \label{eq:cont.Neumann}\\ p(t,x) & = p^{\rm D}(x) & \qquad & \text{on}\; (0,T)\times \Gamma^{\rm D}; \label{eq:cont.Dirichlet} \end{alignat} \item the initial data \begin{alignat}{2} s(0,x) & = s^0(x) & \qquad & \text{in} \; \Omega. \label{eq:cont.init} \end{alignat} \end{subequations} \end{itemize} The partial differential equation \eqref{eq:massbal} expresses the water volume balance. The flux $F$ involved in this balance is given by the Darcy-Muskat law \eqref{eq:darcymuskat}, in which $g$ is the gravity vector and $\varrho$ is the known constant density of water, assumed to be incompressible. It is convenient to introduce \begin{equation} \psi = - \varrho g\cdot x, \qquad \vartheta = p+ \psi, \end{equation} referred to respectively as gravity potential and hydraulic head. In this way, the Darcy-Muskat law \eqref{eq:darcymuskat} can be rewritten as \[ F + \lambda(x) \,\eta(s,x) \, \nabla (p + \psi) = F + \lambda(x) \,\eta(s,x) \, \nabla \vartheta = 0 . \] Equation \eqref{eq:spcapillar} connecting the saturation $s$ and the pressure $p$ is the capillary pressure relation. The boundary $\partial\Omega$ is split into two non-overlapping parts, viz., \begin{equation} \partial\Omega = \Gamma^{\rm N} \cup \Gamma^{\rm D}, \qquad \Gamma^{\rm N} \cap \Gamma^{\rm D} = \emptyset, \end{equation} where $\Gamma^{\rm N}$ is open and $\Gamma^{\rm D}$ is closed, the latter having a positive $(d-1)$-dimensional Hausdorff measure $\nu^{d-1}(\Gamma^{\rm D})>0$. The no-flux Neumann condition \eqref{eq:cont.Neumann} is prescribed on $(0,T)\times\Gamma^{\rm N}$, where $n(x)$ is the outward normal unit vector at $x\in \Gamma^{\rm N}$. The Dirichlet condition \eqref{eq:cont.Dirichlet} with a known Lipschitz function $p^{\rm D} \in W^{1,\infty}(\Omega)$ is imposed on $(0,T)\times\Gamma^{\rm D}$. Note that, in our theoretical development, the function $p^{\rm D}$ is assumed to be defined over the whole domain $\Omega$, which is stronger than a data $p^{\rm D}\in L^{\infty}(\Gamma^{\rm D})$ given only on the boundary. The assumption that $p^{\rm D}$ does not depend on time can be removed by following the lines of~\cite{CNV}, but we prefer here not to deal with time-dependent boundary data in order to keep the presentation as simple as possible. Finally, the initial data $s^0 \in L^{\infty}(\Omega ; [0,1])$ in \eqref{eq:cont.init} is also a given data. In this work, we restrict ourselves to a specific type of heterogeneous media, defined as follows. We assume that the domain $\Omega$ can be partitioned into several connected polyhedral subdomains $\Omega_i$, $1 \leq i \leq I$. Technically, this means that if $\Gamma_{i,j}$ denotes the interface between $\Omega_i$ and $\Omega_j$ (which can be empty for some particular choices of $\{i,j\}$), then \begin{equation}\label{eq:geom} \Omega_i \cap \Omega_j = \emptyset, \quad \ov \Omega_i \cap \ov \Omega_j = \Gamma_{i,j}, \; \text{if}\; i \neq j, \quad \Omega = \Big(\bigcup_{1\leq i\leq I} \Omega_i\Big) \cup \Gamma, \end{equation} with $\Gamma = \bigcup_{i\neq j} \Gamma_{i,j}$. Each of these subdomains corresponds to a distinctive rocktype. Inside each $\Omega_i$, the physical properties are homogeneous. In other words, \[ \phi(x) = \phi_i, \qquad \lambda(x) = \lambda_i, \qquad \eta(s,x)= \eta_i(s), \qquad \mathcal{S}(p,x) = \mathcal{S}_i(p) \] for all $x\in \Omega_i$. Therefore, system \eqref{eq:bigsyst} is associated with \begin{subequations}\label{eq:subdomdata} \begin{alignat}{2} \phi(x) & = \sum_{1\leq i\leq I} \phi_i \, \mathbf{1}_{\Omega_i}(x), & \qquad \eta(s,x) & = \sum_{1\leq i\leq I} \eta_i(s) \, \mathbf{1}_{\Omega_i}(x), \label{eq:phieta}\\ \lambda(x) & = \sum_{1\leq i\leq I} \lambda_i \, \mathbf{1}_{\Omega_i}(x), & \qquad \mathcal{S}(p,x) & = \sum_{1\leq i\leq I} \mathcal{S}_i(p) \, \mathbf{1}_{\Omega_i}(x), \label{eq:lambdaSs} \end{alignat} \end{subequations} where $\mathbf{1}_{\Omega_i}$ stands for the characteristic function of $\Omega_i$. For all $i\in\{1,\ldots, I\}$, we assume that $\phi_i \in (0,1]$ and $\lambda_i >0$. Furthermore, we require that \begin{subequations}\label{eq:EtaSs} \begin{equation}\label{eq:Eta.1} \eta_i \text{ is increasing on } [0,1], \qquad \eta_i(0)= 0, \qquad \eta_i(1) = \frac{1}{\mu}, \end{equation} where $\mu>0$ is the (known) viscosity of water. In addition to the assumption that $\mathcal{S}(\cdot,x)$, defined in \eqref{eq:lambdaSs}, is absolutley continuous and nondecreasing, the functions $\mathcal{S}_i$ are also subject to some generic requirements commonly verified the models available in the literature: for each $i\in\{1,\ldots , I\}$, there exists $\ov p_i \leq 0$ such that \begin{equation}\label{eq:Ss.1} \mathcal{S}_i \text{ is increasing on } (-\infty, \ov p_i], \qquad \lim_{p\to-\infty} \mathcal{S}_i(p)= 0, \qquad \mathcal{S}_i \equiv 1 \text{ on } [\ov p_i, +\infty). \end{equation} \end{subequations} This allows us to define an inverse $\mathcal{S}_i^{-1}: (0,1] \to (-\infty, \ov p_i]$ such that $\mathcal{S}_i \circ \mathcal{S}_i^{-1}(s) = s$ for all $s \in (0,1]$. We further assume that for all $i \in \{1, \dots, I\}$ the function $\mathcal{S}_i$ is bounded in $L^1(\mathbb{R}_-)$, or equivalently, that $\mathcal{S}_i^{-1} \in L^1(0,1)$. It thus makes sense to consider the capillary energy density functions $\text{\EUR{}}_i: \mathbb{R} \times \Omega_i \to \mathbb{R}_+$ defined by \begin{equation}\label{eq:defenergy} \text{\EUR{}}_i(s,x) = \int_{\mathcal{S}_i(p^{\rm D}(x))}^{s} \phi_i (\mathcal{S}_i^{-1}(\varsigma)- p^{\rm D}(x)) \, {\rm d} \varsigma. \end{equation} For all $x \in \Omega_i$, the function $\text{\EUR{}}_i(\cdot, x)$ is nonnegative, convex since $\mathcal{S}_i^{-1}$ is monotone, and bounded on $[0,1]$ as a consequence of the integrability of $\mathcal{S}_i$. For technical reasons that will appear clearly later on, we further assume that \begin{equation}\label{eq:hyp.xi} \sqrt{\eta_i \circ \mathcal{S}_i} \in L^1(\mathbb{R}_-), \qquad \forall i \in \{1,\dots, I\}. \end{equation} Let $Q_{i,T} = (0,T)\times \Omega_i$ be the space-time subdomains for $1\leq i\leq I$. The interior equations \eqref{eq:massbal}--\eqref{eq:spcapillar} then boil down to \begin{subequations}\label{eq:locsyst} \begin{alignat}{2} \phi_i \, \partial_t s + {\text{div}}\, F & = 0 & \qquad & \text{in}\; Q_{i,T}, \label{eq:cont.mass}\\ F + \lambda_i \,\eta_i \, \nabla (p + \psi) & = 0 & \qquad & \text{in}\; Q_{i,T},\label{eq:cont.flux}\\ s - \mathcal{S}_i(p) & = 0 & \qquad & \text{in}\; Q_{i,T} . \label{eq:Ss} \end{alignat} \end{subequations} At the interface $\Gamma_{i,j}$ between $\Omega_i$ and $\Omega_j$, $i\neq j$, any solution of \eqref{eq:massbal}--\eqref{eq:spcapillar} satisfies the matching conditions \begin{subequations}\label{eq:matchcond} \begin{alignat}{2} F_i\cdot n_i + F_j\cdot n_j &= 0 & \qquad & \text{on}\; (0,T)\times \Gamma_{i,j}, \label{eq:cont.flux.interf}\\ p_i - p_j & = 0 & \qquad & \text{on}\; (0,T)\times \Gamma_{i,j} . \label{eq:cont.p.interf} \end{alignat} \end{subequations} In the continuity of the normal fluxes \eqref{eq:cont.flux.interf}, which is enforced by the conservation of water volume, $n_i$ denotes the outward normal to $\partial\Omega_i$ and $F_i\cdot n_i$ stands for the trace of the normal component of $F_{|_{Q_{i,T}}}$ on $(0,T)\times\partial\Omega_i$. In the continuity of pressure \eqref{eq:cont.p.interf}, which also results from \eqref{eq:massbal}--\eqref{eq:spcapillar}, $p_i$ denotes the trace on $(0,T)\times \partial\Omega_i$ of the pressure $p_{|_{Q_{i,T}}}$ in the $i$-th domain. \subsection{Stability features and notion of weak solutions}\label{ssec:cont.stability} We wish to give a proper sense to the notion of weak solution for problem \eqref{eq:bigsyst}. To achieve this purpose, we need a few mathematical transformations the definition of which crucially relies on a fundamental energy estimate at the continuous level. The calculations below are aimed at highlighting this energy estimate and will be carried out in a formal way, in constrast to those in the fully discrete setting. Multiplying~\eqref{eq:cont.mass} by $p-p^{\rm D}$, invoking \eqref{eq:defenergy}, integrating over $\Omega_i$ and summing over $i$, we end up with \begin{equation}\label{eq:NRG.0} \frac{{\rm d}}{{\rm d} t} \, \sum_{i=1}^I\int_{\Omega_i} \text{\EUR{}}_i(s,x) \, {\rm d} x + \sum_{i=1}^I \int_{\Omega_i}{\text{div}}\, F (p- p^{\rm D} ) \, {\rm d} x = 0. \end{equation} We now integrate by parts the second term. Thanks to the matching conditions \eqref{eq:matchcond} and the regularity of $p^{\rm D}$, we obtain \[ \mathtt{A} := \sum_{i=1}^I \int_{\Omega_i}{\text{div}}\, F (p-p^{\rm D}) \, {\rm d} x = - \sum_{i=1}^I \int_{\Omega_i} F \cdot \nabla ( p - p^{\rm D} ) \, {\rm d} x. \] It follows from the flux value \eqref{eq:cont.flux} that \begin{align*} \mathtt{A} & = \sum_{i=1}^I \int_{\Omega_i} \lambda_i \eta_i(s) \nabla ( p +\psi ) \cdot \nabla ( p - p^{\rm D} ) \, {\rm d} x \\ & \, = \sum_{i=1}^I \int_{\Omega_i} \lambda_i \eta_i(s) |\nabla p |^2 \, {\rm d} x - \sum_{i=1}^I \int_{\Omega_i} \lambda_i \eta_i(s) \nabla \psi \cdot \nabla p^{\rm D} \, {\rm d} x \\ & \,+ \; \sum_{i=1}^I \int_{\Omega_i} \lambda_i \eta_i(s) \nabla p \cdot \nabla (\psi - p^{\rm D} ) \, {\rm d} x. \end{align*} Young's inequality, combined with the boundedness of $\nabla p^{\rm D}$, $\nabla \psi$, $\lambda$ and $\eta$, yields \[ \mathtt{A} \, \geq \, \frac12 \sum_{i=1}^I \int_{\Omega_i} \lambda_i \eta_i(s) |\nabla p |^2 \, {\rm d} x - C \] for some $C\geq 0$ depending only on $\lambda$, $\eta$, $\psi$, $\mu$, $\Omega$ and $p^{\rm D}$. Let us define the energy $\mathfrak{E}: [0,T] \to \mathbb{R}_+$ by \[ \mathfrak{E}(t) = \sum_{i=1}^I\int_{\Omega_i} \text{\EUR{}}_i(s(t,x),x) \, {\rm d} x, \qquad 0 \leq t \leq T . \] Integrating~\eqref{eq:NRG.0} w.r.t. time results in \begin{equation}\label{eq:NRG.1} \mathfrak{E}(T) + \frac12 \sum_{i=1}^I \int\!\!\!\int_{Q_{i,T}} \lambda_i \eta_i(s) |\nabla p |^2 \, {\rm d} x \, {\rm d} t \leq \mathfrak{E}(0) + CT. \end{equation} Estimate~\eqref{eq:NRG.1} is the core of our analysis. However, it is difficult to use in its present form since $\eta_i(s) = \eta_i(\mathcal{S}_i(p))$ vanishes as $p$ tends to $-\infty$, so that the control of $\nabla p$ degenerates. To circumvent this difficulty, we resort to the nonlinear functions (customarily referred to as the Kirchhoff transforms) $\Theta_i:\mathbb{R} \to \mathbb{R}$, $\Phi_i: \mathbb{R} \to \mathbb{R}$, and $\Upsilon: \mathbb{R} \times \Omega \to \mathbb{R}$ respectively defined by \begin{subequations} \begin{alignat}{2} \Theta_i(p) & = \int_{0}^p \sqrt{\lambda_i \eta_i \circ\mathcal{S}_i(\pi)} \, {\rm d} \pi, & \qquad p & \in \mathbb{R}, \label{eq:xi.def} \\ \Phi_i(p) & = \int_{0}^p {\lambda_i \eta_i \circ\mathcal{S}_i(\pi)} \, {\rm d} \pi, & \qquad p & \in \mathbb{R},\\ \Upsilon(p) & = \int_0^p \min_{1\leq i \leq I} \sqrt{\lambda_i \eta_i \circ \mathcal{S}_i(\pi)} \, {\rm d} \pi, & \qquad p & \in \mathbb{R}, \label{eq:Upsilon} \end{alignat} \end{subequations} the notion of $\Upsilon$ being due to \cite{EEM06}. Bearing in mind that $\mathfrak{E}(T) \geq 0$, estimate~\eqref{eq:NRG.1} implies that \begin{equation}\label{eq:NRG.2} \sum_{i=1}^I \int\!\!\!\int_{Q_{i,T}} |\nabla \Theta_i(p)|^2 \, {\rm d} x \, {\rm d} t \leq 2 (\mathfrak{E}(0) + CT) < +\infty. \end{equation} As $\Phi_i \circ \Theta_i^{-1}$ is Lipschitz continuous, this also gives rise to a $L^2(Q_{i,T})$-estimate on $\nabla \Phi_i(p)$. The functions $\sum_i \Theta_i(p)\mathbf{1}_{\Omega_i}$ and $\sum_i \Phi_i(p)\mathbf{1}_{\Omega_i}$ are in general discontinuous across the interfaces $\Gamma_{i,j}$, unlike $\Upsilon(p)$. Since the functions $\Upsilon\circ \Theta_i^{-1}$ are Lipschitz continuous, we can readily infer from \eqref{eq:NRG.2} that \begin{equation}\label{eq:NRG.Upsilon} \int\!\!\!\int_{Q_T} |\nabla \Upsilon(p)|^2 \, {\rm d} x \leq C \end{equation} for some $C$ depending on $T$, $\Omega$, $\lVert\nabla p^{\rm D}\rVert_\infty$, the $\lVert\mathcal{S}_i\rVert_{L^1(\mathbb{R}_-)}$'s and \[ \ov{\lambda} = \lVert\lambda\rVert_{L^{\infty}(\Omega)} = \max_{1\leq i\leq I}\lambda_i, \qquad \ov{\eta} = \lVert\eta\rVert_{L^{\infty}(\Omega)} = \max_{1\leq i\leq I} \lVert\eta_i\rVert_{L^{\infty}(\Omega)} = \frac{1}{\mu}, \] the last equality being due to \eqref{eq:Eta.1}. Moreover, $\Upsilon(p) - \Upsilon(p^{\rm D})$ vanishes on $(0,T)\times \Gamma^{\rm D}$. Poincar\'e's inequality provides a $L^2(Q_T)$-estimate on $\Upsilon(p)$ since $\Gamma^{\rm D}$ has positive measure and since $\Upsilon(p^{\rm D})$ is bounded in $\Omega$. In view of assumption~\eqref{eq:hyp.xi}, the functions $\Theta_i$ and $\Upsilon$ are bounded on $\mathbb{R}_-$. Besides, for $p\geq 0$, $\eta_i\circ \mathcal{S}_i(p) = 1/\mu$, so that $ \Theta_i(p) = p \sqrt{\lambda_i/\mu} $ and $\Upsilon(p) = \min_{1\leq i \leq I} p \sqrt{\lambda_i/\mu}$. It finally comes that \begin{equation}\label{eq:xi_Upsilon} \Theta_i(p) \leq C (1+\Upsilon(p)), \qquad \forall p \in \mathbb{R}, \; 1 \leq i \leq I, \end{equation} from which we infer a $L^2(Q_{i,T})$-estimate on $\Theta_i(p)$. Putting \[ V = \big\{ u \in H^1(\Omega)\; | \; u_{|_{\Gamma^{\rm D}}} = 0\big\}, \] the above estimates suggest the following notion of weak solution for our problem. \begin{deff}\label{def:weak} A measurable function $p:Q_T \to \mathbb{R}$ is said to be a weak solution to the problem~\eqref{eq:cont.mass}--\eqref{eq:Ss} if \begin{subequations} \begin{align} \Theta_i(p) & \in L^2((0,T);H^1(\Omega_i)), \qquad \text{for } \; 1 \leq i \leq I,\\ \qquad \Upsilon(p) - \Upsilon(p^{\rm D}) & \in L^2((0,T);V) \end{align} and if for all $\varphi \in C^\infty_c([0,T)\times (\Omega\cup \Gamma^{\rm N}))$, there holds \begin{equation}\label{eq:weak.mass} \int\!\!\!\int_{Q_{T}} \phi \, \mathcal{S}(p,x) \partial_t \varphi \, {\rm d} x \, {\rm d} t + \int_{\Omega} \phi \, s^0 \varphi(\cdot, 0) \, {\rm d} x + \int\!\!\!\int_{Q_{T}} F \cdot \nabla \varphi \, {\rm d} x \, {\rm d} t = 0, \end{equation} with \begin{equation}\label{eq:weak.flux} F = - \nabla \Phi_i(p) + \lambda_i \eta_i(\mathcal{S}_i(p)) \, \varrho g \qquad \text{in }\; Q_{i,T}, \; 1 \leq i \leq I. \end{equation} \end{subequations} \end{deff} The expression~\eqref{eq:weak.flux} is a reformulation of the original one~\eqref{eq:cont.flux} in a quasilinear form which is suitable for analysis, even though the physical meaning of the Kirchhoff transform $\Phi_i(p)$ is unclear. While the formulation~\eqref{eq:weak.mass} should be thought of as a weak form of~\eqref{eq:cont.mass}, \eqref{eq:cont.flux.interf}, \eqref{eq:cont.init}, and \eqref{eq:cont.Neumann}, the condition $\Upsilon(p) - \Upsilon(p^{\rm D}) \in L^2((0,T);V)$ contains~\eqref{eq:cont.p.interf} and~\eqref{eq:cont.Dirichlet}. \subsection{Goal and positioning of the paper}\label{sse:goals} We are now in a position to clearly state the two objectives of this paper. The first objective is to put forward a rigorous proof that, for problem \eqref{eq:bigsyst} with heterogeneous data \eqref{eq:subdomdata}, cell-centered finite-volume schemes with upstream mobility such as described in \S\ref{ssec:scheme}, do converge towards a weak solution (in the sense of Definition \ref{def:weak}) as the discretization parameters tend to $0$. Such mathematically assessed convergence results are often dedicated to homogeneous cases: see for instance \cite{AWZ96, EGH99,RPK04} for schemes involving the Kirchhoff transforms for Richards' equation, \cite{Ahmed_M2AN} for a upstream mobility CVFE approximation of Richards' equation in anisotropic domains, \cite{CJ86, CE97, CE01} for schemes for two-phase flows involving the Kirchhoff transform, and \cite{EHM03,GRC_HAL} for upstream mobility schemes for two-phase porous media flows. For flows in highly heterogeneous porous media, rigorous mathematical results have been obtained for schemes involving the introduction of additional interface unknowns and Kirchhoff's transforms (see for instance~\cite{EEM06, NoDEA, FVbarriere, BCH13}), or under the non-physical assumption that the mobilities are strictly positive~\cite{EGHGH01, EGHM14_zamm}. It was established very recently in~\cite{BDMQ_HAL} that cell-centered finite-volumes with (hybrid) upwinding also converge for two-phase flows in heterogeneous domains, but with a specific treatment of the interfaces located at the heterogeneities. Here, the novelty lies in the fact that we do not consider any specific treatment of the interface in the design of the scheme. The second objective is of more practical nature. Even though our analysis still holds without any specific treatment of the interface, it is well-known that cell-centered upstream mobility finite-volumes can be inaccurate in the presence of heterogeneities. This observation motivated several contributions (see for instance \cite{EEN98, EEM06, HF08, EGHM14_zamm}) where skeletal (i.e., edge or vertex) unknowns where introduced in order to enforce the continuity of the pressures at the interfaces $\Gamma_{i,j}$. By means of extensive numerical simulations in \S\ref{sec:results}, we will show that without local refinement of the grid at the interface, the method still converges, but with a degraded order. Our ultimate motivation is to propose an approach which consists in adding very thin cells on both sides of the interface before using the cell centered scheme under study. Then the scheme appears to behave better, with first-order accuracy. Moreover, one can still make use of the parametrized cut-Newton method proposed in \cite{BCET_FVCA9} to compute the solution to the nonlinear system corresponding to the scheme. This method appears to be very efficient, while it avoids the possibly difficult construction of compatible parametrizations at the interfaces as in~\cite{BKJMP17, BMQ20, BDMQ_HAL}. \section{Finite-volume discretization} The scheme we consider in this paper is based on two-point flux approximation (TPFA) finite-volumes. Hence, it is subject to some restrictions on the mesh~\cite{Tipi, GK_Voronoi}. We first review the requirements on the mesh in \S\ref{ssec:mesh}. Next, we construct the upstream mobility finite-volume scheme for Richards' equation in \S\ref{ssec:scheme}. The main mathematical results of the paper, which are the well-posedness of the nonlinear system corresponding to the scheme and the convergence of the scheme, are then summarized in \S\ref{ssec:main}. \subsection{Admissible discretization of $Q_T$}\label{ssec:mesh} Let us start by discretizing w.r.t. space. \begin{deff}\label{def:meshAdmissible} An \emph{admissible mesh of $\Omega$} is a triplet $(\mathscr{T}, \mathscr{E}, {(x_K)}_{K\in\mathscr{T}})$ such that the following conditions are fulfilled: \begin{enumerate}[noitemsep] \item[(i)] Each control volume (or cell) $K\in\mathscr{T}$ is non-empty, open, polyhedral and convex, with positive $d$-dimensional Lebesgue measure $m_K > 0$. We assume that \[ K \cap L = \emptyset \quad \text{if}\; K, L \in \mathscr{T} \; \text{with}\; K \neq L, \qquad \text{while}\quad \bigcup_{K\in\mathscr{T}}\ov K = \ov \Omega. \] Moreover, we assume that the mesh is adapted to the heterogeneities of $\Omega$, in the sense that for all $K \in \mathscr{T}$, there exists $i\in \{1,\dots, I\}$ such that $K \subset \Omega_i$. \item[(ii)] Each face $\sigma \in \mathscr{E}$ is closed and is contained in a hyperplane of $\mathbb{R}^d$, with positive $(d-1)$-dimensional Hausdorff measure $\nu^{d-1}(\sigma)= m_\sigma >0$. We assume that $\nu^{d-1}(\sigma \cap \sigma') = 0$ for $\sigma, \sigma' \in \mathscr{E}$ unless $\sigma' = \sigma$. For all $K \in \mathscr{T}$, we assume that there exists a subset $\mathscr{E}_K$ of $\mathscr{E}$ such that $\partial K = \bigcup_{\sigma \in \mathscr{E}_K} \sigma$. Moreover, we suppose that $\bigcup_{K\in\mathscr{T}} \mathscr{E}_K = \mathscr{E}$. Given two distinct control volumes $K,L\in\mathscr{T}$, the intersection $\ov K \cap \ov L$ either reduces to a single face $\sigma \in \mathscr{E}$ denoted by $K|L$, or its $(d-1)$-dimensional Hausdorff measure is $0$. \item[(iii)] The cell-centers $(x_K)_{K\in\mathscr{T}}$ are pairwise distinct with $x_K \in K$, and are such that, if $K, L \in \mathscr{T}$ share a face $K|L$, then the vector $x_L-x_K$ is orthogonal to $K|L$. \item[(iv)] For the boundary faces $\sigma \subset \partial\Omega$, we assume that either $\sigma \subset \Gamma^{\rm D}$ or $\sigma \subset \ov \Gamma{}^{\rm N}$. For $\sigma \subset \partial\Omega$ with $\sigma \in \mathscr{E}_K$ for some $K\in \mathscr{T}$, we assume additionally that there exists $x_\sigma \in \sigma$ such that $x_\sigma - x_K$ is orthogonal to $\sigma$. \end{enumerate} \end{deff} In our problem, the standard Definition \ref{def:meshAdmissible} must be supplemented by a compatibility property between the mesh and the subdomains. By ``compatbility'' we mean that each cell must lie entirely inside a single subregion. Put another way, \begin{equation}\label{eq:indexK} \forall K\in\mathscr{T}, \quad \exists ! \; i(K)\in \{ 1, \ldots, I\} \;\; | \;\; K\subset \Omega_{i(K)} . \end{equation} This has two consequences. The first one is that, if we define \begin{equation} \mathscr{T}_i = \{K\in \mathscr{T}\; | \; K \subset \Omega_i\}, \qquad 1 \leq i \leq I, \end{equation} then $\mathscr{T} = \bigcup_{i=1}^I \mathscr{T}_i$. The second one is that the subdomain interfaces $\Gamma_{i,j}$ for $i\neq j$ coincide necessarily with some edges $\sigma \in \mathscr{E}$. To express this more accurately, let $\mathscr{E}_\Gamma = \{\sigma \in \mathscr{E}\; | \; \sigma \subset \Gamma\}$ be the set of the interface edges, $\mathscr{E}_{\rm ext}^{\rm D} = \{\sigma \in \mathscr{E}\; | \; \sigma \subset \Gamma^{\rm D}\}$ be the set of Dirichlet boundary edges, and $\mathscr{E}_{\rm ext}^{\rm N} = \{\sigma \in \mathscr{E}\; | \; \sigma \subset \ov \Gamma{}^{\rm N}\}$ be the set of Neumann boundary edges. Then, $\Gamma = \bigcup_{\sigma \in \mathscr{E}_\Gamma} \sigma$, while $\Gamma^{\rm D}= \bigcup_{\sigma \in \mathscr{E}_{\rm ext}^{\rm D}} \sigma$ and $\ov \Gamma{}^{\rm N}= \bigcup_{\sigma \in \mathscr{E}_{\rm ext}^{\rm N}} \sigma$. For later use, it is also convenient to introduce the subset $\mathscr{E}_i \subset\mathscr{E}$ consisting of those edges that correspond to cells in $\mathscr{T}_i$ only, i.e., \begin{subequations} \begin{equation}\label{eq:EdgesOfI} \mathscr{E}_i = \bigg(\bigcup_{K\in\mathscr{T}_i} \mathscr{E}_K\bigg)\setminus \mathscr{E}_\Gamma, \qquad 1 \leq i \leq I, \end{equation} and the subset $\mathscr{E}_{\rm int}$ of the internal edges, i.e., \begin{equation} \mathscr{E}_{\rm int} = \mathscr{E} \setminus(\mathscr{E}_{\rm ext}^{\rm D} \cup \mathscr{E}_{\rm ext}^{\rm N}) = \bigcup_{K,L\in \mathscr{T}} \{ \sigma = K|L\}. \end{equation} \end{subequations} Note that $\mathscr{E}_\Gamma \subset \mathscr{E}_{\rm int}$. To each edge $\sigma\in\mathscr{E}$, we associate a distance $d_\sigma$ by setting \begin{equation}\label{eq:distance} d_\sigma = \begin{cases} \, |x_K - x_L| & \text{if}\; \sigma = K|L \in \mathscr{E}_{\rm int}, \\ \, |x_K - x_\sigma| & \text{if}\; \sigma \in \mathscr{E}_K \cap (\mathscr{E}_{\rm ext}^{\rm D} \cup \mathscr{E}_{\rm ext}^{\rm N}). \end{cases} \end{equation} We also define $d_{K\sigma} = {\rm dist}(x_K,\sigma)$ for all $K \in \mathscr{T}$ and $\sigma \in \mathscr{E}_K$. The transmissivity of the edge $\sigma\in \mathscr{E}$ is defined by \begin{equation}\label{eq:transmissive} a_\sigma = \frac{m_\sigma}{d_\sigma}. \end{equation} Throughout the paper, many discrete quantities ${\boldsymbol u}$ will be defined either in cells $K\in\mathscr{T}$ or on Dirichlet boundary edges $\sigma \in \mathscr{E}_{\rm ext}^{\rm D}$, i.e. ${\boldsymbol u} = ( \left(u_K\right)_{K\in\mathscr{T}}, \left(u_\sigma\right)_{\sigma \in \mathscr{E}_{\rm ext}^{\rm D}} ) \in \mathbb{X}^{\mathscr{T}\cup\mathscr{E}_{\rm ext}^{\rm D}}$, where $\mathbb{X}$ can be either $\mathbb{R}^\ell$, $\ell \geq 1$, or a space of functions. Then for all $K\in\mathscr{T}$ and $\sigma \in \mathscr{E}_K$, we define the mirror value $u_{K\sigma}$ by \begin{equation}\label{eq:mirror} u_{K\sigma} = \begin{cases} \, u_L & \text{if}\; \sigma = K|L \in \mathscr{E}_{\rm int}, \\ \, u_K &\text{if}\; \sigma \in \mathscr{E}_K \cap \mathscr{E}_{\rm ext}^{\rm N}, \\ \, u_\sigma &\text{if}\; \sigma \in \mathscr{E}_K \cap \mathscr{E}_{\rm ext}^{\rm D}. \end{cases} \end{equation} The diamond cell $\Delta_\sigma$ corresponding to the edge $\sigma$ is defined as the convex hull of $\{x_K, x_{K\sigma},\sigma\}$ for $K$ such that $\sigma \in \mathscr{E}_K$, while the half-diamond cell $\Delta_{K\sigma}$ is defined as the convex hull of $\{x_K,\sigma\}$. Denoting by $m_{\Delta_\sigma}$ the Lebesgue measure of $\Delta_\sigma$, the elementary geometrical relation $m_{\Delta_\sigma} = d\, m_\sigma d_\sigma$ where $d$ stands for the dimension will be used many times in what follows. Another notational shorthand is worth introducing now, since it will come in handy in the sequel. Let \begin{subequations}\label{eq:fsubdom} \begin{equation} f(\cdot,x) = \sum_{1\leq i\leq I} f_i(\cdot) \mathbf{1}_{\Omega_i}(x) \end{equation} be a scalar quantity or a function whose dependence of $x\in\Omega$ is of the type \eqref{eq:subdomdata}. Then, for $K\in\mathscr{T}$, we slightly abuse the notations in writing \begin{equation}\label{eq:finK} f_K(\cdot) := f(\cdot,x_K) = f_{i(K)}(\cdot), \end{equation} \end{subequations} where the index $i(K)$ is defined in \eqref{eq:indexK}. The last equality in the above equation holds by virtue of the compatibility property. For example, we will have not only $\phi_K= \phi(x_K)$, $\lambda_K= \lambda(x_K)$, $\eta_K(s)= \eta(s,x_K)$, $\mathcal{S}_K(p) = \mathcal{S}(p,x_K)$ but also $\text{\EUR{}}_K(s)=\text{\EUR{}}(s,x_K)$. Likewise, we shall be writing $f_{K\sigma}(\cdot) = f(\cdot,x_{K\sigma})$ for the mirror cell without any ambiguity: if $\sigma\in\mathscr{E}_{\rm int}\cup \mathscr{E}_{\rm ext}^{\rm N}$, then $x_{K\sigma}$ is a cell-center; if $\sigma\in\mathscr{E}_{\rm ext}^{\rm D}$, then $x_{K\sigma}$ lies on the boundary but does not belong to an interface between subdomains. The size $h_\mathscr{T}$ and the regularity $\zeta_\mathscr{T}$ of the mesh are respectively defined by \begin{equation}\label{eq:diamreg} h_\mathscr{T} = \max_{K\in\mathscr{T}} \, {\rm diam}(K), \qquad \zeta_\mathscr{T} = \min_{K\in\mathscr{T}} \; {\bigg( \frac1{\operatorname{Card}\, \mathscr{E}_K}}\, \min_{\sigma \in \mathscr{E}_K} \frac{d_{K\sigma}}{{\rm diam}(K)} {\bigg)}. \end{equation} The time discretization is given by $\left(t^n\right)_{0\leq 1 \leq N}$ with $0 = t^0 < t^1 < \dots < t^N = T$. We denote by ${\Delta t^n} = t^n - t^{n-1}$ for all $n \in \{1,\dots, N\}$ and by ${\boldsymbol{{\Delta t}}} = \left({\Delta t^n}\right)_{1\leq n \leq N}$. \subsection{Upstream mobility TPFA Finite Volume scheme}\label{ssec:scheme} Given a discrete saturation profile $(s_K^{n-1})_{K\in\mathscr{T}} \in [0,1]^\mathscr{T}$ at time $t^{n-1}$, $n\in \{1,\dots, N\}$, we seek for a discrete pressure profile $(p_K^{n})_{K\in\mathscr{T}} \in \mathbb{R}^\mathscr{T}$ at time $t^n$ solution to the following nonlinear system of equations. Taking advantage of the notational shorthand \eqref{eq:finK}, we define \begin{equation}\label{eq:scheme.capi} s_K^n = \mathcal{S}_K(p_K^n), \qquad K\in\mathscr{T}, \; n\geq 1. \end{equation} The volume balance~\eqref{eq:cont.mass} is then discretized into \begin{equation}\label{eq:scheme.mass} m_K \phi_K \frac{s_K^n - s_K^{n-1}}{{\Delta t}^n} + \sum_{\sigma \in \mathscr{E}_K} m_\sigma F_{K\sigma}^n = 0, \qquad K \in \mathscr{T}, \; n \geq 1, \end{equation} using the approximation \begin{subequations} \begin{equation}\label{eq:scheme.flux} F_{K\sigma}^n = \frac{1}{d_\sigma}\lambda_\sigma \eta_\sigma^n (\vartheta_K^n - \vartheta_{K\sigma}^n ), \quad \sigma \in \mathscr{E}_K, \; K \in \mathscr{T}, \; n\geq 1, \end{equation} for the flux \eqref{eq:darcymuskat}, with \begin{equation} \vartheta_K^n = p_K^n + \psi_K, \qquad \vartheta_{K\sigma}^n = p_{K\sigma}^n + \psi_{K\sigma} , \end{equation} where the mirror values $p_{K\sigma}^n$ and $\psi_{K\sigma}$ are given by \eqref{eq:mirror}. In the numerical flux \eqref{eq:scheme.flux}, the edge permeabilities $(\lambda_\sigma)_{\sigma \in \mathscr{E}}$ are set to \[ \lambda_\sigma = \begin{cases} \, \displaystyle\frac{\lambda_K \lambda_L d_{\sigma}}{\lambda_K d_{L,\sigma}+ \lambda_L d_{K,\sigma}} & \text{if}\; \sigma = K|L \in \mathscr{E}_{\rm int}, \\ \, \lambda_K & \text{if}\; \sigma \in \mathscr{E}_{K} \cap \mathscr{E}_{\rm ext}, \end{cases} \] while the edge mobilities are upwinded according to \begin{equation}\label{eq:scheme.upwind} \eta_\sigma^n = \begin{cases} \, \eta_K(s_K^n) & \text{if}\; \vartheta_K^n > \vartheta_{K\sigma}^n , \\ \frac{1}{2}(\eta_K(s_K^n)+\eta_{K\sigma}(s_{K\sigma}^n)) & \text{if}\; \vartheta_K^n = \vartheta_{K\sigma}^n ,\\ \, \eta_{K\sigma}(s_{K\sigma}^n) & \text{if}\; \vartheta_K^n < \vartheta_{K\sigma}^n. \end{cases} \end{equation} \end{subequations} In practice, the definition of $\eta_\sigma^n$ when $\vartheta_K^n = \vartheta_{K\sigma}^n$ has no influence on the scheme. We choose here to give a symmetric definition that does not depend on the orientation of the edge $\sigma$ in order to avoid ambiguities. The boundary condition $p^{\rm D}$ is discretized into \begin{equation}\label{eq:scheme.pD} \begin{cases} \, p_K^{\rm D} = \frac{1}{m_K} \int_K p^{\rm D}(x) \, {\rm d} x & \text{for}\; K \in \mathscr{T}, \\ \, p_\sigma^{\rm D} = \;\, \frac{1}{m_\sigma} \int_\sigma p^{\rm D}(x) \, {\rm d} \nu^{d-1}(x)&\text{for}\; \sigma \in \mathscr{E}_{\rm ext}^{\rm D}, \end{cases} \end{equation} whereas the initial condition is discretized into \begin{equation}\label{eq:scheme.init} s_K^0 = \frac1{m_K} \int_K s^0(x) \, {\rm d} x, \qquad \text{for}\; K \in \mathscr{T}. \end{equation} The Dirichlet boundary condition is encoded in the fluxes~\eqref{eq:scheme.flux} by setting \begin{equation}\label{eq:p_Dir} p_\sigma^n = p^{\rm D}_\sigma, \qquad \forall \sigma \in \mathscr{E}_{\rm ext}^{\rm D}, \; n \geq 1. \end{equation} Bearing in mind the definition~\eqref{eq:mirror} of the mirror values for $\sigma \in \mathscr{E}_{\rm ext}^{\rm N}$, the no-flux boundary condition across $\sigma\in\mathscr{E}_{\rm ext}^{\rm N}$ is automatically encoded, i.e., $F_{K\sigma}^n = 0$ for all $\sigma \in \mathscr{E}_{K}\cap \mathscr{E}_{\rm ext}^{\rm N}$, $K\in\mathscr{T}$ and $n\geq 1$. In what follows, we denote by ${\boldsymbol p}^n = \left(p_K^n\right)_{K\in\mathscr{T}}$ for $1 \leq n \leq N$, and by ${\boldsymbol{s}}^n = \left(s_K^n\right)_{K\in\mathscr{T}}$ for $0 \leq n \leq N$. Besides, we set ${\boldsymbol p}^{\rm D} = ((p_K^{\rm D})_{K\in\mathscr{T}}, (p_\sigma^{\rm D})_{\sigma\in\mathscr{E}^{\rm D}} )$. \subsection{Main results and organization of the paper}\label{ssec:main} The theoretical part of this paper includes two main results. The first one, which emerges from the analysis at fixed grid, states that the schemes admits a unique solution $({\boldsymbol p}^n)_{1 \leq n \leq N}$. \begin{thm}\label{thm:1} For all $n\in \{1, \dots, N\}$, there exists a unique solution ${\boldsymbol p}^n$ to the scheme~\eqref{eq:scheme.capi}--\eqref{eq:scheme.upwind}. \end{thm} With Theorem~\ref{thm:1} at hand, we define the approximate pressure $p_{\mathscr{T}, {\Delta t}}$ by \begin{subequations} \begin{equation}\label{eq:approx_sol} p_{\mathscr{T},{\boldsymbol{{\Delta t}}}}(t,x) = p_K^n \qquad \text{for}\; (t,x) \in (t^{n-1}, t^n] \times K. \end{equation} We also define the approximate saturation as \begin{equation}\label{eq:approx_sat} s_{\mathscr{T}, {\boldsymbol{{\Delta t}}}} = \mathcal{S}(p_{\mathscr{T}, {\boldsymbol{{\Delta t}}}}, x) . \end{equation} \end{subequations} The second main result guarantees the convergence towards a weak solution of the sequence of approximate solutions as the mesh size and the time steps tend to $0$. Let $(\mathscr{T}_m, \mathscr{E}_m, \left(x_K\right)_{K\in\mathscr{T}_m})_{m\geq 1}$ be a sequence of admissible discretizations of the domain $\Omega$ in the sense of Definition~\ref{def:meshAdmissible} such that \begin{equation}\label{eq:mesh.m} h_{\mathscr{T}_m} \underset{m\to\infty}\longrightarrow 0, \qquad \sup_{m\geq 1} \; \zeta_{\mathscr{T}_m} =: \zeta < +\infty, \end{equation} where the size $h_{\mathscr{T}_m}$ and the regularity $\zeta_{\mathscr{T}_m}$ are defined in \eqref{eq:diamreg}. Let $({\boldsymbol{{\Delta t}}}_m)_{m\geq 1}$ be time discretizations of $(0,T)$ such that \begin{equation}\label{eq:dt.m} \lim_{m\to \infty} \; \max_{1 \leq n \leq N_m} {\Delta t}^n_{m} = 0. \end{equation} \begin{thm}\label{thm:2} There exists a weak solution $p: Q_T \to \mathbb{R}$ in the sense of Definition~\ref{def:weak} such that, up to a subsequence, \begin{subequations} \begin{alignat}{2} s_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m} & \underset{m\to\infty} \longrightarrow \mathcal{S}(p,x) & \qquad & \text{a.e. in}\; Q_T, \\ \Upsilon(p_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m}) & \underset{m\to\infty} \longrightarrow \Upsilon(p) & \qquad & \text{weakly in}\; L^2(Q_T). \end{alignat} \end{subequations} \end{thm} The rest of this paper is outlined as follows. Section \S\ref{sec:fixed} is devoted to the numerical analysis at fixed grid. This encompasses the existence and uniqueness result stated in Theorem~\ref{thm:1} as well as a priori estimates that will help proving Theorem~\ref{thm:2}. The convergence of the scheme, which is taken up in \S\ref{sec:convergence}, relies on compactness arguments, which require a priori estimates that are uniform w.r.t. the grid. These estimates are mainly adaptations to the discrete setting of their continuous counterparts that arised in the stability analysis sketched out in \S\ref{ssec:cont.stability}. These estimates are shown in \S\ref{ssec:compact} to provide some compactness on the sequence of approximate solutions. In \S\ref{ssec:identify}, we show that these compactness properties together with the a priori estimates are sufficient to identify any limit of an approximate solution as a weak solution to the problem. In \S\ref{sec:numerics}, we provide some details about the practical numerical resolution by laying emphasis on the switch of variable for selecting the primary unknown and on the mesh refinement at an interface in order to better enforce pressure continuity. Finally, in \S\ref{sec:results}, numerical experiments on two configurations (drying and filling cases) for two capillary pressure models (Brooks-Corey and van Genuchten-Mualem) testify to the relevance of the local refinement strategy as a simple technique to preserve accuracy. \begin{rmk} Theorem~\ref{thm:2} only states the convergence of the scheme up to a subsequence. In the case where the weak solution is unique, then the whole sequence of approximate solutions would converge towards this solution. As far as we know, uniqueness of the weak solutions to Richards' equation is in general an open problem for heterogeneous media where $x \mapsto \mathcal{S}(p,x)$ is discontinuous. Uniqueness results are however available in the one-dimensional setting for a slightly more restrictive notion of solutions, cf. \cite{FVbarriere}, or under additional assumptions on the nonlinearities $\eta_i, \mathcal{S}_i$, cf. \cite{NoDEA}. \end{rmk} \section{Analysis at fixed grid}\label{sec:fixed} \subsection{Some uniform a priori estimates}\label{ssec:apriori} In this section, our aim is to derive a priori estimates on the solutions to the scheme~\eqref{eq:scheme.capi}--\eqref{eq:scheme.init}. These estimates will be at the core of the existence proof of a solution to the scheme. They will also play a key role in proving the convergence of the scheme. The main estimate on which our analysis relies is a discrete counterpart of~\eqref{eq:NRG.1}. We recall that $a_\sigma$ is the transmissivity introduced in \eqref{eq:transmissive}. \begin{prop}\label{prop:dissip} There exist two constants $\ctel{c:main}$, $\ctel{c:dissip.2}$ depending only on $\lambda$, $\mu$, $p^{\rm D}$, $\psi$, $\zeta$, $\Omega$, $T$, $\phi$, and $\lVert\mathcal{S}_i\rVert_{L^1(\mathbb{R}_-)}$ such that \begin{subequations}\label{eq:dissconts} \begin{align} \sum_{n=1}^N {\Delta t^n} \sum_{\sigma \in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (p_K^n - p_{K\sigma}^n)^2 & \leq \cter{c:main}, \label{eq:dissip}\\ \sum_{n=1}^N {\Delta t^n} \sum_{\sigma \in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (\vartheta_K^n - \vartheta_{K\sigma}^n )^2 & \leq \cter{c:dissip.2}. \label{eq:Flux.L2} \end{align} \end{subequations} \end{prop} In \eqref{eq:dissconts}, the relationship between $\sigma$ and $K$ is to be understood as follows. For an inner edge $\sigma \in \mathscr{E}_{\rm int}$, although it can be written as $\sigma = K|L$ or $L|K$, only one of these contributes to the sum. For a boundary edge $\sigma \in \mathscr{E}_{\rm ext}$, there is only one cell $K$ such that $\sigma\in\mathscr{E}_K$, so there is no ambiguity in the sum. \begin{proof} Multiplying \eqref{eq:scheme.mass} by $\Delta t^n (p_K^n -p_K^{\rm D})$, summing over $K \in \mathscr{T}$ and $n\in\{1,\dots, N\}$, and carrying out discrete integration by parts yield \begin{equation}\label{eq:dissip.A+B} \mathtt{A} + \mathtt{B} =0, \end{equation} where we have set \begin{subequations} \begin{align} \mathtt{A} & = \sum_{n=1}^N \sum_{K\in \mathscr{T}} m_K \phi_K (s_K^n- s_K^{n-1})(p_K^n -p_K^{\rm D}), \label{eq:uni.estim.A}\\ \mathtt{B} & = \sum_{n=1}^N \Delta t^n \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_{\sigma}^n (\vartheta_K^n - \vartheta_{K\sigma}^n ) (p_K^n-p_K^{\rm D} -p_{K\sigma}^n + p_{K\sigma}^{\rm D}). \label{eq:uni.estim.B} \end{align} \end{subequations} The discrete energy density function $\text{\EUR{}}_K: [0,1] \to \mathbb{R}_+$, defined by means of the notation \eqref{eq:fsubdom} from the functions $f_i=\text{\EUR{}}_i$ introduced in \eqref{eq:defenergy}, is convex by construction. Consequently, \[ \text{\EUR{}}_K(s_K^{n-1}) - \text{\EUR{}}_K(s_K^n) \geq \text{\EUR{}}'_K(s_K^n) (s_K^{n-1} - s_K^n) = \phi_K (p_K^n -p_K^{\rm D}) (s_K^{n-1} - s_K^n) . \] Therefore, the quantity $\mathtt{A}$ of \eqref{eq:uni.estim.A} can be bounded below by \begin{multline} \qquad\qquad \mathtt{A} \, \geq \, \sum_{n=1}^N \sum_{K\in \mathscr{T}} m_K (\text{\EUR{}}_K(s_K^n)- \text{\EUR{}}_K(s_K^{n-1})) \\ = \sum_{K\in \mathscr{T}} m_K (\text{\EUR{}}_K(s_K^N)- \text{\EUR{}}_K(s_K^{0})) \, \geq \, - C_{\mathtt{A}}, \qquad\qquad \label{eq:dissp.A} \end{multline} the last inequality being a consequence of the boundedness of $\text{\EUR{}}_K$ on $[0,1]$. Writing $\vartheta = p + \psi$ and expanding each summand of \eqref{eq:uni.estim.B}, we can split $\mathtt{B}$ into $$\mathtt{B}= \mathtt{B}_1+ \mathtt{B}_2+ \mathtt{B}_3,$$ with \begin{align*} \mathtt{B}_1 & = \phantom{-} \sum_{n=1}^N \Delta t^n \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (p_K^n - p_{K\sigma}^n)^2,\\ \mathtt{B}_2 & = \phantom{-} \sum_{n=1}^N \Delta t^n \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (p_K^n - p_{K\sigma}^n)(\psi_K - \psi_{K\sigma}- p_K^{\rm D} + p_{K\sigma}^{\rm D} ),\\ \mathtt{B}_3 & = -\sum_{n=1}^N \Delta t^n \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (\psi_K - \psi_{K\sigma}) (p_K^{\rm D} - p_{K\sigma}^{\rm D}). \end{align*} It follows from \cite[Lemma 9.4]{eymardEtAl2000finiteVolumeMethod} and from the boundedness of $\eta$ that there exists a constant $C$ depending only on $\lambda$, $\mu$, $\zeta_{\mathscr{T}}$ and $\Omega$ such that \begin{subequations} \begin{align} \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (p_K^{\rm D} - p_{K\sigma}^{\rm D})^2 & \leq C \, \lVert\nabla p^{\rm D}\rVert^2_{L^2(\Omega)^d}, \\ \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (\psi_K - \psi_{K\sigma})^2 & \leq C \, \lVert\nabla \psi\rVert^2_{L^\infty(\Omega)^d} . \end{align} \end{subequations} Thanks to these estimates and to the Cauchy-Schwarz inequality, we have \[ \mathtt{B}_3\geq -C T \,\lVert\nabla p^{\rm D}\rVert_{L^2(\Omega)^d} \, \lVert\nabla \psi\rVert_{L^\infty(\Omega)^d} . \] On the other hand, Young's inequality provides $$\mathtt{B}_2\geq -\frac{1}{2} \mathtt{B}_1 - C T \, \big( \lVert\nabla p^{\rm D}\rVert^2_{L^2(\Omega)^d} + \lVert\nabla \psi\rVert^2_{L^\infty(\Omega)^d} \big).$$ Hence, \begin{equation}\label{eq:dissip.B} \mathtt{B} \, \geq \, \frac{1}{2}\sum_{n=1}^N \Delta t^n \sum_{\sigma\in \mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (p_K^n - p_{K\sigma}^n)^2 - C_{\mathtt{B}}, \end{equation} by setting $C_{\mathtt{B}} = C T \,(\lVert\nabla p^{\rm D}\rVert^2_{L^2(\Omega)^d} + \lVert\nabla \psi\rVert^2_{L^\infty(\Omega)^d} + \lVert\nabla p^{\rm D}\rVert_{L^2(\Omega)^d} \, \lVert\nabla \psi\rVert_{L^\infty(\Omega)^d} )$. Inserting \eqref{eq:dissp.A} and \eqref{eq:dissip.B} into \eqref{eq:dissip.A+B}, we recover \eqref{eq:dissip} with $\cter{c:main} = 2(C_{\mathtt{A}} + C_{\mathtt{B}})$. From \eqref{eq:dissip}, we can deduce \eqref{eq:Flux.L2} by elementary manipulations. \end{proof} So far, we have not used the upwind choice~\eqref{eq:scheme.upwind} for the mobilities $\eta_\sigma^n$. This will be done in the next lemma, where we derive a more useful variant of estimate \eqref{eq:dissip}, in which $\eta_\sigma^n$ is replaced by $\ov{\eta}_\sigma^n$ defined below. In a homogeneous medium, $\ov{\eta}_\sigma^n \geq \eta_\sigma^n$ so that the new estimate \eqref{eq:dissip.max} seems to be stronger than \eqref{eq:dissip}. We begin by introducing the functions $\widecheck{\eta}_\sigma: \mathbb{R} \to (0, 1/\mu]$ defined for $\sigma \in \mathscr{E}$ by \begin{subequations} \begin{equation} \widecheck{\eta}_\sigma(p) = \min\big\{\eta_K \circ \mathcal{S}_K(p), \eta_{K\sigma} \circ \mathcal{S}_{K\sigma}(p) \big\}, \qquad \forall p \in \mathbb{R}. \end{equation} By virtue of assumptions \eqref{eq:EtaSs}, each argument of the minimum function is nondecreasing and positive function of $p\in\mathbb{R}$. As a result, $\widecheck{\eta}_\sigma$ is also a nondecreasing and positive function of $p\in\mathbb{R}$. Note that $\widecheck{\eta}_\sigma = \eta_i \circ \mathcal{S}_i$ for all $\sigma \in \mathscr{E}_i$, while for interface edges $\sigma \subset \Gamma_{i,j}$, the mere inequality $\widecheck{\eta}_\sigma \leq \eta_i \circ \mathcal{S}_i$ holds. Next, we consider the intervals \begin{equation} \mathfrak{J}_{\sigma}^n = [ p_K^n \bot p_{K\sigma}^n , \, p_K^n \top p_{K\sigma}^n], \qquad \text{for } \, \sigma \in \mathscr{E}_K, \; K \in \mathscr{T}, \; 1 \leq n \leq N, \end{equation} with the notations $a\bot b = \min(a,b)$ and $a\top b = \max(a,b)$. At last, we set \begin{equation}\label{eq:ov_eta_sig} \ov{\eta}_\sigma^n = \max_{p \in \mathfrak{J}_\sigma^n} \widecheck{\eta}_\sigma(p), \qquad \text{for } \, \sigma \in \mathscr{E}, \; 1 \leq n \leq N. \end{equation} \end{subequations} \begin{lem}\label{lem:dissip.max} There exists a constant $\ctel{c:dissip.max}$ depending on the same data as $\cter{c:main}$ such that \begin{equation}\label{eq:dissip.max} \sum_{n=1}^N {\Delta t}^n \sum_{\sigma \in\mathscr{E}} a_\sigma \ov\eta_\sigma^n \left(p_K^n - p_{K\sigma}^n\right)^2 \leq \cter{c:dissip.max} . \end{equation} \end{lem} \begin{proof} We partition the set $\mathscr{E}$ of edges into three subsets, namely, \[ \mathscr{E}^{n}_{+} = \big\{\sigma \, |\; \vartheta_K^n > \vartheta_{K\sigma}^n \big\},\quad \mathscr{E}^{n}_{-} = \big\{\sigma \, |\; \vartheta_K^n < \vartheta_{K\sigma}^n \big\},\quad \mathscr{E}^{n}_{0} = \big\{\sigma \, |\; \vartheta_K^n = \vartheta_{K\sigma}^n \big\}. \] Invoking $\widecheck{\eta}_\sigma = \min(\eta_K \circ \mathcal{S}_K, \,\eta_{K\sigma}\circ \mathcal{S}_{K\sigma})$, we can minorize the left-hand side of~\eqref{eq:dissip} to obtain \begin{multline*} \sum_{n = 1}^N {\Delta t}^n \Big[ \sum_{\sigma \in \mathscr{E}^{n}_{+}} a_\sigma \lambda_\sigma \widecheck{\eta}_\sigma(p_K^n) (p_K^n - p_{K\sigma}^n)^2 + \sum_{\sigma \in \mathscr{E}^{n}_{-}} a_\sigma \lambda_\sigma \widecheck{\eta}_\sigma(p_{K\sigma}^n) (p_K^n - p_{K\sigma}^n)^2 \\ + \sum_{\sigma \in \mathscr{E}^{n}_{0}} a_\sigma \lambda_\sigma \textstyle\frac{1}{2} (\widecheck{\eta}_\sigma(p_K^n) + \widecheck{\eta}_\sigma(p_{K\sigma}^n)) (p_K^n - p_{K\sigma}^n)^2 \Big] \leq \cter{c:main}. \end{multline*} Starting from this inequality and using the boundedness of $\eta_i$ and $\psi$, we can readily show that there exists a constant $C$ depending on the same data as $\cter{c:main}$ such that \begin{multline*} \mathtt{D}_1 := \sum_{n = 1}^N {\Delta t}^n \Big[ \sum_{\sigma \in \mathscr{E}^{n}_{+}} a_\sigma \lambda_\sigma \widecheck{\eta}_\sigma(p_K^n) (p_K^n - p_{K\sigma}^n) (\vartheta_K^n - \vartheta_{K\sigma}^n) \\ + \sum_{\sigma \in \mathscr{E}^{n}_{-}} a_\sigma \lambda_\sigma \widecheck{\eta}_\sigma(p_{K\sigma}^n) (p_K^n - p_{K\sigma}^n) (\vartheta_K^n - \vartheta_{K\sigma}^n )\Big] \leq C, \end{multline*} in which the sum over $\mathscr{E}_0^n$ was omitted because all of its summands vanish. Simlarly to what was pointed out in equation 2.9 in~\cite{Ahmed_M2AN}, we notice that since $\eta_\sigma$ is nondecreasing w.r.t. $p$, it is straightforward to check that the definition \begin{equation} \label{eq:laurel} \widecheck{\eta}_\sigma^n := \begin{cases} \; \widecheck{\eta}_\sigma(p_K^n) & \text{ if } \vartheta^n_K > \vartheta^n_{K\sigma},\\ \; \textstyle\frac{1}{2} (\widecheck{\eta}_\sigma(p_K^n) + \widecheck{\eta}_\sigma(p_{K\sigma}^n)) & \text{ if } \vartheta^n_K = \vartheta^n_{K\sigma},\\ \; \widecheck{\eta}_\sigma(p_{K\sigma}^n) & \text{ if } \vartheta^n_K < \vartheta^n_{K\sigma} \end{cases} \end{equation} exactly amounts to \begin{equation} \label{eq:hardy} \widecheck{\eta}_\sigma^n = \begin{cases} \; \max_{p\in \mathfrak{J}_\sigma^n}\widecheck{\eta}_\sigma(p) & \text{ if } (p^n_K - p^n_{K\sigma})(\vartheta^n_K - \vartheta^n_{K\sigma}) > 0,\\ \; \textstyle\frac{1}{2} (\widecheck{\eta}_\sigma(p_K^n) + \widecheck{\eta}_\sigma(p_{K\sigma}^n)) & \text{ if } (p^n_K - p^n_{K\sigma})(\vartheta^n_K - \vartheta^n_{K\sigma}) = 0,\\ \; \min_{p\in \mathfrak{J}_\sigma^n}\widecheck{\eta}_\sigma(p) & \text{ if } (p^n_K - p^n_{K\sigma})(\vartheta^n_K - \vartheta^n_{K\sigma}) < 0. \end{cases} \end{equation} Taking advantage of this equivalence, we can transform $\mathtt{D}_1$ into \begin{multline} \mathtt{D}_1 = \sum_{n = 1}^N {\Delta t}^n \Big[ \sum_{\sigma \in \mathscr{E}^{n}_{>}} a_\sigma \lambda_\sigma \max_{\mathfrak{J}_\sigma^n}\widecheck{\eta}_\sigma (p_K^n - p_{K\sigma}^n) (\vartheta_K^n - \vartheta_{K\sigma}^n) \\ + \sum_{\sigma \in \mathscr{E}^{n}_{<}} a_\sigma \lambda_\sigma \min_{\mathfrak{J}_\sigma^n}\widecheck{\eta}_\sigma (p_K^n - p_{K\sigma}^n) (\vartheta_K^n - \vartheta_{K\sigma}^n ) \Big] \leq C , \end{multline} where $\mathscr{E}^n_{>} = \{ \sigma \,|\, (p^n_K - p^n_{K\sigma})(\vartheta^n_K - \vartheta^n_{K\sigma}) > 0 \}$ and $\mathscr{E}^n_{<} = \{ \sigma \,|\, (p^n_K - p^n_{K\sigma})(\vartheta^n_K - \vartheta^n_{K\sigma}) < 0 \}$. The second sum over $\mathscr{E}^n_{<}$ contains only negative summands and can be further minorized if $\min_{\mathfrak{J}^n_\sigma}\widecheck{\eta}_\sigma$ is replaced by $\max_{\mathfrak{J}^n_\sigma}\widecheck{\eta}_\sigma$. In other words, \[ \mathtt{D}_2 := \sum_{n = 1}^N {\Delta t}^n \sum_{\sigma\in\mathscr{E}} a_\sigma \lambda_\sigma \ov \eta_\sigma^n (p_K^n - p_{K\sigma}^n) (\vartheta_K^n - \vartheta_{K\sigma}^n ) \leq \mathtt{D}_1 \leq C. \] Writing $\vartheta = p+\psi$, expanding each summand of $\mathtt{D}_2$ and applying Young's inequality, we end up with \[ \frac{1}{2}\sum_{n = 1}^N {\Delta t}^n \sum_{\sigma\in\mathscr{E}} a_\sigma \lambda_\sigma \ov \eta_\sigma^n \, [ (p_K^n - p_{K\sigma}^n)^2 - (\psi_K^n - \psi_{K\sigma}^n)^2 ] \leq \mathtt{D}_2 \leq C . \] Estimate \eqref{eq:dissip.max} finally follows from the boundedness of $\eta$, $1/\lambda$ and $\psi$. \end{proof} The above lemma has several important consequences for the analysis. Let us start with discrete counterparts to estimations~\eqref{eq:NRG.2} and \eqref{eq:NRG.Upsilon}. \begin{cor}\label{cor:xi_Upsilon} Let $\cter{c:dissip.max}$ be the constant in Lemma~\ref{lem:dissip.max}. Then, \begin{subequations}\label{eq:dissip.xiUpsilon} \begin{align} \sum_{n=1}^N {\Delta t^n} \sum_{i=1}^I \sum_{\sigma \in \mathscr{E}_i} a_\sigma (\Theta_i(p_K^n) - \Theta_i(p_{K\sigma}^n))^2 & \leq \cter{c:dissip.max}, \label{eq:dissip.xi}\\ \sum_{n=1}^N {\Delta t^n} \sum_{\sigma \in \mathscr{E}} a_\sigma (\Upsilon(p_K^n) - \Upsilon(p_{K\sigma}^n))^2 & \leq \cter{c:dissip.max}. \label{eq:dissip.Upsilon} \end{align} \end{subequations} Moreover, there exists two constants $\ctel{c:L2.Upsilon}$, $\ctel{c:L2.xi}$ depending on the same data as $\cter{c:main}$ and additionnally on $\lVert\sqrt{\eta_i \circ \mathcal{S}_i}\rVert_{L^1(\mathbb{R}_-)}$, $1\leq i\leq I$, such that \begin{subequations}\label{eq:L2.Upsilonxi} \begin{align} \sum_{n=1}^N {\Delta t}^n \sum_{K\in\mathscr{T}} m_K |\Upsilon(p_K^n)|^2 \leq \cter{c:L2.Upsilon},\label{eq:L2.Upsilon} \\ \sum_{n=1}^N {\Delta t}^n \sum_{i=1}^I \sum_{K\in\mathscr{T}_i} m_K |\Theta_i(p_K^n)|^2 \leq \cter{c:L2.xi}.\label{eq:L2.xi} \end{align} \end{subequations} \end{cor} \begin{proof} Consider those edges $\sigma \in \mathscr{E}_i$ ---defined in \eqref{eq:EdgesOfI}--- corresponding to some fixed $i\in\{1,\ldots,I\}$, for which $\widecheck{\eta}_\sigma = \eta_i \circ \mathcal{S}_i = |\Theta'_i|^2$ and $\ov{\eta}^n_\sigma = \max_{\mathfrak{J}^n_\sigma} |\Theta'_i|^2$ due to \eqref{eq:xi.def}. By summing the elementary inequality \[ (\Theta_i(p_K^n) - \Theta_i(p_{K\sigma}^n))^2 \leq \ov{\eta}^n_{\sigma} \; (p_K^n - p_{K\sigma}^n)^2, \] over $\sigma\in\mathscr{E}_i$, $i\in\{1,\ldots,I\}$ and $n\in\{1,\ldots,N\}$ using appropriate weights, we get \[ \sum_{n=1}^N {\Delta t^n} \sum_{i=1}^I \sum_{\sigma \in \mathscr{E}_i} a_\sigma (\Theta_i(p_K^n) - \Theta_i(p_{K\sigma}^n))^2 \leq \sum_{n=1}^N {\Delta t^n} \sum_{i=1}^I \sum_{\sigma \in \mathscr{E}_i} a_\sigma \ov{\eta}^n_{\sigma} \; (p_K^n - p_{K\sigma}^n)^2 , \] whose right-hand side is obviously less than $\cter{c:dissip.max}$, thanks to~\eqref{eq:dissip.max}. This proves \eqref{eq:dissip.xi}. Similarly, the respective definitions of $\ov{\eta}_\sigma^n$ and $\Upsilon$ have been tailored so that $\max_{\mathfrak{J}^n_\sigma} |\Upsilon'|^2 \leq \ov{\eta}^n_\sigma$ for all $\sigma\in\mathscr{E}$. As a consequence, \[ (\Upsilon(p_K^n) - \Upsilon(p_{K\sigma}^n))^2 \leq \ov{\eta}_\sigma^n (p_K^n - p_{K\sigma}^n )^2. \] Summing these inequalities over $\sigma\in\mathscr{E}$ and $n\in \{1,\ldots,N\}$ with appropriate weights and invoking \eqref{eq:dissip.max}, we prove~\eqref{eq:dissip.Upsilon}. The argument for \eqref{eq:L2.Upsilon} is subtler. Starting from the basic inequality \begin{multline*} \qquad (\Upsilon(p_K^n) - \Upsilon(p_K^{\rm D})- \Upsilon(p_{K\sigma}^n) + \Upsilon(p_{K\sigma}^{\rm D}))^2\\ \leq 2 (\Upsilon(p_K^n) - \Upsilon(p_{K\sigma}^n))^2 + 2 (\Upsilon(p_K^{\rm D}) - \Upsilon(p_{K\sigma}^{\rm D}))^2 , \qquad \end{multline*} we apply the discrete Poincar\'e inequality of~\cite[Lemma 9.1]{eymardEtAl2000finiteVolumeMethod} ---which is legitimate since $\Gamma^{\rm D}$ has positive measure--- followed by \cite[Lemma 9.4]{eymardEtAl2000finiteVolumeMethod} to obtain \[ \sum_{n=1}^N {\Delta t}^n \sum_{K\in\mathscr{T}} m_K ( \Upsilon(p_K^n) - \Upsilon(p_K^{\rm D}) )^2 \\ \leq 2 C_{{\rm P},\mathscr{T}} \big( \cter{c:dissip.max} + C_\zeta T \lVert \Upsilon'\rVert_\infty \lVert \nabla p^{\rm D} \rVert^2 \big) , \] where $C_{{\rm P},\mathscr{T}}$ denotes the discrete Poincar\'e constant, and $C_\zeta$ is the quantity appearing in \cite[Lemma 9.4]{eymardEtAl2000finiteVolumeMethod} and only depends on $\zeta_{\mathscr{T}}$. $\cter{c:L2.Upsilon} = 4 C_{{\rm P},\mathscr{T}} \big( \cter{c:dissip.max} + C_\zeta T \lVert \Upsilon'\rVert_\infty \lVert \nabla p^{\rm D} \rVert^2 \big) + 2 m_\Omega T \lVert\Upsilon(p^{\rm D})\rVert_\infty^2.$ The last estimate \eqref{eq:L2.xi} results from the comparison~\eqref{eq:xi_Upsilon} of the nonlinearities $\Theta_i$ and $\Upsilon$. \end{proof} The purpose of the next lemma is to work out a weak estimate on the discrete counterpart of $\partial_t s$, which will lead to compactness properties in \S\ref{ssec:compact}. For $\varphi \in C^\infty_c( Q_T)$, let \[ \varphi_K^n = \frac1{m_K} \int_K \varphi(t^n,x) \, {\rm d} x, \qquad \forall K \in \mathscr{T}, \; 1 \leq n \leq N. \] \begin{lem}\label{lem:boundS} There exists a constant $\ctel{c:s2}$ depending on the same data as $\cter{c:main}$ such that \begin{equation}\label{boundS} \sum_{n=1}^N \sum_{K\in\mathscr{T}} m_K \phi_K (s_K^n -s_K^{n-1}) \varphi_K^n \leq \cter{c:s2} \lVert \nabla \varphi \rVert_{L^\infty(Q_T)^d}, \quad \forall\varphi\in C^\infty_c( Q_T). \end{equation} \end{lem} \begin{proof} Multiplying \eqref{eq:scheme.mass} by ${\Delta t^n} ~\varphi_K^n$, summing over $K\in\mathscr{T}$ and $n\in\{1, \cdots, N\}$ and carrying out discrete integration by parts, we end up with \[ \mathtt{A}:= \! \sum_{n=1}^N \! \sum_{K\in\mathscr{T}} m_K \phi_K (s_K^n -s_K^{n-1})\varphi_K^n = - \! \sum_{n=1}^N {\Delta t^n} \! \sum_{\sigma\in\mathscr{E}} a_\sigma \lambda_\sigma \eta_\sigma^n (\vartheta_K^n - \vartheta_{K\sigma}^n)(\varphi_K^n-\varphi_{K\sigma}^n). \] Applying the Cauchy-Schwarz inequality and using~\eqref{eq:Flux.L2}, we get \begin{equation} \mathtt{A}^2 \leq \cter{c:dissip.2} \frac{\max_i \lambda_i}\mu \sum_n {\Delta t^n} \sum_{\sigma\in\mathscr{E}} a_\sigma (\varphi_K^n-\varphi_{K\sigma}^n)^2. \end{equation} The conclusion \eqref{boundS} is then reached by means of the property (see \cite[Section 4.4]{ACM17}) \[ \sum_{n=1}^N {\Delta t^n} \sum_{\sigma\in\mathscr{E}} a_\sigma (\varphi_K^n-\varphi_{K\sigma}^n)^2 \leq C \lVert \nabla \varphi \rVert_{L^\infty(Q_T)^d}^2 \] for some $C$ depending only on $\Omega$, $T$ and the mesh regularity $\zeta_{\mathscr{T}}$. \end{proof} \subsection{Existence of a solution to the scheme} The statements of the previous section are all uniform w.r.t. the mesh and are meant to help us passing to the limit in the next section. In contrast, the next lemma provides a bound on the pressure that depends on the mesh size and on the time-step. This property is needed in the process of ensuring the existence of a solution to the numerical scheme. \begin{lem}\label{lem:Boundedness} There exist two constants $\ctel{c:pb}$, $\ctel{c:pb2}$ depending on $\mathscr{T}$, ${\Delta t}^n$ as well as on the data of the continuous model $\lambda$, $\mu$, $p^{\rm D}$, $\psi$, $\zeta$, $\Omega$, $T$, $\phi$, $\lVert \mathcal{S}_i \rVert_{L^1(\mathbb{R}_-)}$ and $\lVert \sqrt{\eta_i \circ \mathcal{S}_i} \rVert_{L^1(\mathbb{R}_-)}$, $1\leq i\leq I$, such that \begin{equation}\label{eq:Boundedness} -\cter{c:pb} \leq p_K^n \leq \cter{c:pb2}, \quad \forall K \in \mathscr{T}, \; n \in \{1, \dots, N\}. \end{equation} \end{lem} \begin{proof} From~\eqref{eq:L2.Upsilon} and from $\Upsilon(p) = p \sqrt{\min_i{\lambda_i}/\mu}$ for $p\geq 0$, we deduce that \[ p_K^n \leq \sqrt{\frac{\mu \cter{c:L2.Upsilon}}{{\Delta t}^n m_K \min_i \lambda_i}}, \qquad \forall K\in\mathscr{T}, \; 1 \leq n \leq N . \] Hence, the upper-bound $\cter{c:pb2}$ is found by maximizing the right-hand side over $K \in \mathscr{T}$ and $n \in \{1,\dots, N\}$. To show that $p_K^n$ is bounded from below, we employ a strategy that was developed in~\cite{cancesGuichard2016EntropyScheme} and extended to the case of Richards' equation in~\cite[Lemma 3.10]{Ahmed_M2AN}. From~\eqref{eq:scheme.pD}, \eqref{eq:p_Dir} and the boundedness of $p^{\rm D}$, it is easy to see that \[ p_\sigma^n \geq \inf_{x \in \partial\Omega} p^{\rm D}(x), \quad \forall \sigma \in \mathscr{E}_{\rm ext}^{\rm D}. \] Estimate~\eqref{eq:dissip.max} then shows that for all $K\in\mathscr{T}$ such that $\mathscr{E}_K \cap \mathscr{E}_{\rm ext}^{\rm D} \neq \emptyset$, we have \[ p_K^n \geq p_\sigma^n - \sqrt{\frac{\cter{c:dissip.max}}{{\Delta t}^n a_\sigma \widecheck{\eta}_\sigma (p_\sigma^n)}}=: \pi_K^n, \quad \forall \sigma \in \mathscr{E}_K \cap \mathscr{E}_{\rm ext}^{\rm D}. \] The quantity $\pi_K^n$ is well-defined, since $\widecheck{\eta}_\sigma(p^n_\sigma) > 0$ for $p^n_\sigma > -\infty$, and does not depend on time, as $p^{\rm D}$ does not either. Furthermore, if $p_K^n$ is bounded from below by some $\pi_K$, then the pressure in all its neighboring cells $L\in\mathscr{T}$ such that $\sigma = K|L \in \mathscr{E}_K$ is bounded from below by \[ p_L^n \geq \pi_K^n - \sqrt{\frac{\cter{c:dissip.max}}{{\Delta t}^n a_\sigma \widecheck{\eta}_\sigma (\pi_K^n)}} =: \pi_L^n. \] Again, $\pi_L^n$ is well-defined owing to $\widecheck{\eta}_\sigma(\pi_K^n) > 0$. Since the mesh is finite and since the domain is connected, only a finite number of edge-crossings is required to create a path from a Dirichlet boundary edge $\sigma \in \mathscr{E}_{\rm ext}^{\rm D}$ to any prescribed cell $K \in \mathscr{T}$. Hence, the lower bound $\cter{c:pb}$ is found by minimizing $\pi^n_K$ over $K\in\mathscr{T}$ and $n\in\{1, \ldots,N\}$. \end{proof} Lemma~\ref{lem:Boundedness} is a crucial step in the proof of the existence of a solution ${\boldsymbol p}^n = (p_K^n)_{K\in\mathscr{T}}$ to the scheme~\eqref{eq:scheme.capi}--\eqref{eq:p_Dir}. \begin{prop}\label{prop:existence} Given ${\boldsymbol{s}}^{n-1} = (s_K^{n-1})_{K\in\mathscr{T}} \in [0,1]^\mathscr{T}$, there exists a solution ${\boldsymbol p}^n \in \mathbb{R}^\mathscr{T}$ to the scheme~\eqref{eq:scheme.capi}--\eqref{eq:p_Dir}. \end{prop} The proof relies on a standard topological degree argument and is omitted here. However, we make the homotopy explicit for readers' convenience. Let $\gamma \in [0,1]$ be the homotopy parameter. We define the nondecreasing functions $\eta_i^{(\gamma)}:[0,1] \to \mathbb{R}_+$ by setting $\eta_i^{(\gamma)}(s) = (1-\gamma)/\mu + \gamma \eta_i(s)$ for $s \in [0,1]$, and we seek a solution ${\boldsymbol p}^{(\gamma)} = {(p_K^{(\gamma)})}_{K\in\mathscr{T}}$ to the problem \begin{subequations}\label{eq:homotopy} \begin{equation} \gamma m_K \phi_K \frac{\mathcal{S}_K(p_K^{(\gamma)}) - s_K^{n-1}}{{\Delta t}_n} + \sum_{\sigma \in \mathscr{E}_K} m_\sigma F_{K\sigma}^{(\gamma)} = 0, \qquad K \in \mathscr{T}, \; \gamma \in [0,1], \end{equation} where the fluxes $F_{K\sigma}^{(\gamma)}$ are defined by \begin{equation} F_{K\sigma}^{(\gamma)}= \frac{1}{d_\sigma}\lambda_\sigma \eta_\sigma^{(\gamma)} \big( \vartheta_K^{(\gamma)} - \vartheta_{K\sigma}^{(\gamma)} \big), \qquad \sigma \in \mathscr{E}_K, \; K \in \mathscr{T}, \;\gamma \in [0,1] \end{equation} with $\vartheta^{(\gamma)} = p^{(\gamma)} + \psi$ and using the upwind mobilities \begin{equation} \eta_\sigma^{(\gamma)} = \begin{cases} \, \eta_K^{(\gamma)}(\mathcal{S}_K(p_K^{(\gamma)})) & \text{if }\; \vartheta_K^{(\gamma)} > \vartheta_{K\sigma}^{(\gamma)} , \\ \, \frac{1}{2}(\eta_K^{(\gamma)}(\mathcal{S}_K(p_K^{(\gamma)})) + \eta_{K\sigma}^{(\gamma)}(\mathcal{S}_{K\sigma}(p_K^{(\gamma)}))) & \text{if }\; \vartheta_K^{(\gamma)} = \vartheta_{K\sigma}^{(\gamma)} ,\\ \, \eta_{K\sigma}^{(\gamma)}(\mathcal{S}_{K\sigma}(p_K^{(\gamma)})) & \text{if }\; \vartheta_K^{(\gamma)} < \vartheta_{K\sigma}^{(\gamma)} . \end{cases} \end{equation} \end{subequations} At the Dirichlet boundary edges, we still set $p_\sigma^{(\gamma)} = p_\sigma^{\rm D}$. For $\gamma = 0$, the system is linear and invertible, while for $\gamma = 1$, system \eqref{eq:homotopy} coincides with the original system \eqref{eq:scheme.capi}--\eqref{eq:p_Dir}. A priori estimates on ${\boldsymbol p}^{(\gamma)}$ that are uniform w.r.t. $\gamma \in [0,1]$ (but not uniform w.r.t. $\mathscr{T}$ nor ${\Delta t}^n$) can be derived on the basis of what was exposed previously, so that one can unfold Leray-Schauder's machinery~\cite{LS34, Dei85} to prove the existence of (at least) one solution to the scheme. \subsection{Uniqueness of the discrete solution}\label{ssec:uniqueness} To complete the proof of Theorem~\ref{thm:1}, it remains to show that the solution to the scheme is unique. This is the purpose of the following proposition. \begin{prop}\label{prop:uniqueness} Given ${\boldsymbol{s}}^{n-1} = (s_K^{n-1})_{K\in\mathscr{T}} \in [0,1]^\mathscr{T}$, the solution ${\boldsymbol p}^n \in \mathbb{R}^\mathscr{T}$ to the scheme~\eqref{eq:scheme.capi}--\eqref{eq:p_Dir} is unique. \end{prop} \begin{proof} The proof heavily rests upon the monotonicity properties inherited from the upwind choice \eqref{eq:scheme.upwind} for the mobilities. Indeed, due to the upwind choice of the mobility, the flux $F_{K\sigma}^n$ is a function of $p_K^n$ and $p_{K\sigma}^n$ that is nondecreasing w.r.t. $p_K^n$ and nonincreasing w.r.t. $p_{K\sigma}^n$. Moreover, by virtue of the monotonicity of $\mathcal{S}_K$, the discrete volume balance~\eqref{eq:scheme.mass} can be cast under the abstract form \begin{equation}\label{eq:Hh} \Hh_K^n(p_K^n, (p_{K\sigma}^n)_{\sigma \in \mathscr{E}_K}) = 0, \qquad \forall K\in \mathscr{T}, \end{equation} where $\Hh_K^n$ is nondecreasing w.r.t its first argument $p_K^n$ and nonincreasing w.r.t each of the remaining variables $(p_{K\sigma}^n)_{\sigma \in \mathscr{E}_K}$. Let $\wt{{\boldsymbol p}}^n = \left(\wt p_K^n\right)_{K\in\mathscr{T}}$ be another solution to the system \eqref{eq:scheme.capi}--\eqref{eq:p_Dir}, i.e., \begin{equation}\label{eq:Hh.wt} \Hh_K^n (\wt p_K^n, (\wt p_{K\sigma}^n)_{\sigma \in \mathscr{E}_K} ) = 0, \qquad \forall K\in \mathscr{T} . \end{equation} The nonincreasing behavior of $\Hh_K^n$ w.r.t. all its variables except the first one implies that \[ \Hh_K^n (p_K^n, (p_{K\sigma}^n \top \wt{p}_{K\sigma}^n)_{\sigma \in \mathscr{E}_K} ) \leq 0, \qquad \Hh_K^n (\wt{p}_K^n, (p_{K\sigma}^n \top \wt{p}_{K\sigma}^n)_{\sigma \in \mathscr{E}_K} ) \leq 0, \] for all $K\in \mathscr{T}$, where $a\top b = \max(a,b)$. Since $p_K^n\top \wt p_K^n$ is either equal to $p_K^n$ or to $\wt p_K^n$, we infer from the above inequalities that \begin{equation}\label{eq:Hh.top} \Hh_K^n (p_K^n\top \wt p_K^n, (p_{K\sigma}^n \top \wt p_{K\sigma}^n)_{\sigma \in \mathscr{E}_K} ) \leq 0, \qquad \forall K \in \mathscr{T}. \end{equation} By a similar argument, we can show that \begin{equation}\label{eq:Hh.bot} \Hh_K^n (p_K^n\bot \wt p_K^n, (p_{K\sigma}^n \bot \wt p_{K\sigma}^n)_{\sigma \in \mathscr{E}_K}) \geq 0, \qquad \forall K \in \mathscr{T}, \end{equation} where $a\bot b = \min(a, b)$. Subtracting~\eqref{eq:Hh.bot} from \eqref{eq:Hh.top} and summing over $K\in\mathscr{T}$, we find \begin{equation}\label{eq:uniqueness.1} \sum_{K\in\mathscr{T}} m_K \phi_K \frac{|s_K^n - \wt s_K^n|}{\Delta t^n} + \sum_{\sigma \in \mathscr{E}_{\rm ext}^{\rm D}} a_\sigma \lambda_\sigma \mathtt{R}_\sigma^n \leq 0, \end{equation} where $s_K^n = \mathcal{S}_K(p_K^n)$, $\wt{s}_K^n = \mathcal{S}_K(\wt{p}_K^n)$ and \begin{align} \mathtt{R}_\sigma^n & = \eta_K(s_K^n \top \wt{s}_K^n) (\vartheta_K^n \top \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ - \eta_K(s_\sigma^n)(\vartheta_\sigma^n - \vartheta_K^n\top \wt \vartheta_K^n )^+ \nonumber\\ & - \eta_K(s_K^n \bot \wt{s}_K^n) (\vartheta_K^n \bot \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ + \eta_K(s_\sigma^n) (\vartheta_\sigma^n - \vartheta_K^n\bot \wt \vartheta_K^n)^+ , \label{eq:residue1} \end{align} with $s_\sigma^n = \mathcal{S}_K(p^n_\sigma)$. The top line of \eqref{eq:residue1} expresses the upwinded flux of \eqref{eq:Hh.top}, while the bottom line of \eqref{eq:residue1} is the opposite of the upwinded flux of \eqref{eq:Hh.bot}. Note that, since $p^n_\sigma = p^{\rm D}_\sigma$ is prescribed at $\sigma\in\mathscr{E}^{\rm D}_{\rm ext}$, we have $\vartheta^n_\sigma = \vartheta^n_\sigma \top \wt{\vartheta}^n_\sigma = \vartheta^n_\sigma \bot \wt{\vartheta}^n_\sigma$. Upon inspection of the rearrangement \begin{align} \mathtt{R}_\sigma^n & = [\eta_K(s_K^n \top \wt{s}_K^n) - \eta_K(s_K^n \bot \wt{s}_K^n)] (\vartheta_K^n \top \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ \nonumber\\ & + \eta_K(s_K^n \bot \wt{s}_K^n) [ (\vartheta_K^n \top \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ - (\vartheta_K^n \bot \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ ] \nonumber\\ & +\eta_K(s_\sigma^n) \big[ (\vartheta_\sigma^n - \vartheta_K^n\bot \wt{\vartheta}_K^n)^+ - (\vartheta_\sigma^n - \vartheta_K^n\top \wt{\vartheta}_K^n )^+\big] , \label{eq:residue} \end{align} it is trivial that $\mathtt{R}_\sigma^n \geq 0$. As a consequence, \eqref{eq:uniqueness.1} implies that $\mathtt{R}_\sigma^n = 0$ for all $\sigma \in \mathscr{E}_{\rm ext}^{\rm D}$ and that $s_K^n = \wt{s}_K^n$ for all $K\in\mathscr{T}$. At this stage, however, we cannot yet claim that $p_K^n = \wt{p}_K^n$, as the function $\mathcal{S}_K$ is not invertible. Taking into account $s_K^n = \wt{s}_K^n$, the residue \eqref{eq:residue} becomes \begin{align} \mathtt{R}_\sigma^n & = \eta_K(s_K^n) [ (\vartheta_K^n \top \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ - (\vartheta_K^n \bot \wt{\vartheta}_K^n - \vartheta_\sigma^n )^+ ] \nonumber\\ & +\eta_K(s_\sigma^n) \big[ (\vartheta_\sigma^n - \vartheta_K^n\bot \wt{\vartheta}_K^n)^+ - (\vartheta_\sigma^n - \vartheta_K^n\top \wt{\vartheta}_K^n )^+\big] , \end{align} which can be lower-bounded by \begin{equation}\label{eq:residuelower} \mathtt{R}_\sigma^n \geq \min(\eta_K(s_K^n),\, \eta_K(s_\sigma^n)) |\vartheta_K^n - \wt \vartheta_K^n| \end{equation} thanks to the algebraic identities $a^+ - (-a)^+=a$ and $a\top b - a\bot b = |a-b|$. In view of the lower-bound on the discrete pressures of Lemma~\ref{lem:Boundedness}, we deduce from \eqref{eq:Ss.1} that $s_K^n>0$ and $\wt s_K^n >0$. The increasing behavior of $\eta_K$ implies, in turn, that $\eta_K(s_K^n)>0$ and $\eta_K(\wt s_K^n)>0$. Therefore, the conjunction of $\mathtt{R}_\sigma^n=0$ and \eqref{eq:residuelower} yields $\vartheta_K^n = \wt{\vartheta}_K^n$ and hence $p_K^n = \wt{p}_K^n$ for all cells $K$ having a Dirichlet boundary edge, i.e., $\mathscr{E}_K \cap \mathscr{E}_{\rm ext}^{\rm D} \neq \emptyset$. It remains to check that $p_K^n = \wt{p}_K^n$, or equivalently $\vartheta_K^n = \wt{\vartheta}_K^n$ for those cells $K\in\mathscr{T}$ that are far away from the Dirichlet part of the boundary. Subtracting~\eqref{eq:Hh.wt} from~\eqref{eq:Hh} and recalling that $s_K^n = \wt{s}_K^n$, we arrive at \begin{align} \sum_{\sigma \in \mathscr{E}_{K}} a_\sigma\lambda_\sigma \Big\{ \eta_K(s_K^n) & \big[ \left( \vartheta_K^n - \vartheta_{K\sigma}^n \right)^+ - ( \wt \vartheta_K^n - \wt \vartheta_{K\sigma}^n )^+ \big] \nonumber\\ + \eta_{K\sigma}(s_{K\sigma}^n) & \big[ ( \wt \vartheta_{K\sigma}^n - \wt \vartheta_K^n )^+ - ( \vartheta_{K\sigma}^n - \vartheta_K^n )^+\big] \Big\} = 0. \label{eq:HJ} \end{align} Consider a cell $K\in\mathscr{T}$ where $\vartheta_K^n - \wt \vartheta_K^n$ achieves its maximal value, i.e., \begin{equation}\label{eq:HJ.1} \vartheta_K^n - \wt \vartheta_K^n \geq \vartheta_L^n - \wt \vartheta_L^n, \qquad \forall L\in\mathscr{T}. \end{equation} This entails that \[ \vartheta_K^n - \vartheta_{K\sigma}^n \geq \wt \vartheta_K^n - \wt \vartheta_{K\sigma}^n, \qquad \forall \sigma \in \mathscr{E}_K, \] so that the two brackets in the right-hand side of \eqref{eq:HJ} are nonnegative. In fact, they both vanish by the positivity of $\eta_K(s_K^n)$ and $\eta_{K\sigma}(s_{K\sigma}^n)$. As a result, $\vartheta_K^n - \vartheta_{K\sigma}^n = \wt{\vartheta}_K^n - \wt{\vartheta}_{K\sigma}^n$ for all $\sigma \in \mathscr{E}_K$. This implies that $\vartheta_K^n - \wt \vartheta_K^n = \vartheta_L^n - \wt \vartheta_L^n$ for all the cells $L\in\mathscr{T}$ sharing an edge $\sigma=K|L$ with $K$, and thus that the cell $L$ also achieves the maximality condition~\eqref{eq:HJ.1}. The process can then be repeated over and over again. Since $\Omega$ is connected, we deduce that $\vartheta_K^n - \wt \vartheta_K^n$ is constant over $K\in\mathscr{T}$. The constant is finally equal to zero since $\vartheta_K^n = \wt \vartheta_K^n$ on the cells having a Dirichlet edge. \end{proof} \section{Convergence analysis}\label{sec:convergence} Once existence and uniqueness of the discrete solution have been settled, the next question to be addressed is the convergence of the discrete solution towards a weak solution of the continuous problem, as the mesh-size and the time-step are progressively refined. In accordance with the general philosophy expounded in~\cite{eymardEtAl2000finiteVolumeMethod}, the proof is built on compactness arguments. We start by highlighting compactness properties in \S\ref{ssec:compact}, before identifying the limit values as weak solutions in \S\ref{ssec:identify}. \subsection{Compactness properties}\label{ssec:compact} Let us define $G_{\mathscr{E}_m,{\boldsymbol{{\Delta t}}}_m}: Q_T \to \mathbb{R}^d$ and $J_{\mathscr{E}_m,{\boldsymbol{{\Delta t}}}_m} : Q_T \to \mathbb{R}^d$ by \begin{equation} G_{\mathscr{E}_m,{\boldsymbol{{\Delta t}}}_m}(t,x) = \begin{cases} \; d \displaystyle\frac{\Theta_i(p_{K\sigma}^n) - \Theta_i(p_K^n)}{d_\sigma} n_{K\sigma} , & \text{if }\; (t,x) \in (t_m^{n-1},t_m^n] \times \Delta_{\sigma} , \; \\ \; 0 & \text{otherwise}, \end{cases} \end{equation} for $\sigma \in \mathscr{E}_{i,m}$, $1 \leq n \leq N_m$ and, respectively, \begin{equation} J_{\mathscr{E}_m,{\boldsymbol{{\Delta t}}}_m}(t,x) = d \displaystyle\frac{\Upsilon(p_{K\sigma}^n) - \Upsilon(p_K^n)}{d_\sigma} n_{K\sigma}, \quad \text{if }\; (t,x) \in (t_m^{n-1},t_m^n] \times \Delta_{\sigma}, \end{equation} for $\sigma \in \mathscr{E}_{m}$, $1 \leq n \leq N_m$. We remind that $s_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m} = \mathcal{S}(p_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m},x)$ is the sequence of approximate saturation fields computed from that of approximate pressure fields $p_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m}$ by \eqref{eq:approx_sat}. \begin{prop}\label{prop:compact} There exists a measurable function $p:Q_T \to \mathbb{R}$ such that $\Upsilon(p)-\Upsilon(p^{\rm D}) \in L^2((0,T);V)$ and $\Theta_i(p) \in L^2((0,T);H^1(\Omega_i))$, $1\leq i \leq I$, such that, up to a subsequence, \begin{subequations}\label{eq:compact} \begin{alignat}{2} s_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m} & \underset{m\to+\infty} \longrightarrow \mathcal{S}(p,x) & \qquad & \text{a.e. in }\; Q_T, \label{eq:compact.s}\\ G_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} & \underset{m\to+\infty} \longrightarrow \nabla \Theta_i(p) & \qquad & \text{weakly in }\; L^2(Q_{i,T})^d, \label{eq:compact.xi}\\ J_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} & \underset{m\to+\infty} \longrightarrow \nabla \Upsilon(p) & \qquad & \text{weakly in }\; L^2(Q_{T})^d . \label{eq:compact.Upsilon} \end{alignat} \end{subequations} \end{prop} \begin{proof} We know from Corollary~\ref{cor:xi_Upsilon} that $\Theta_i(p_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m})$ and $\Upsilon(p_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m})$ are bounded w.r.t. $m$ in $L^2(Q_{i,T})$ and $L^2(Q_T)$ respectively, while $G_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ and $J_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ are respectively bounded in $L^2(Q_{i,T})^d$ and $L^2(Q_T)^d$. In particular, there exist $\wh \Theta_i \in L^2(Q_{i,T})$, $\wh \Upsilon \in L^2(Q_T)$, $J \in L^2(Q_{i,T})^d$, and $J \in L^2(Q_{T})^d$ such that \begin{subequations}\label{eq:conv.L2.weak} \begin{alignat}{4} \Theta_i(p_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m}) & \underset{m\to+\infty} \longrightarrow \wh \Theta_i & \qquad & \text{weakly in}\; L^2(Q_{i,T}), \\ \Upsilon(p_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m}) & \underset{m\to+\infty} \longrightarrow \wh \Upsilon & \qquad & \text{weakly in}\; L^2(Q_{T}), \\ G_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} & \underset{m\to+\infty} \longrightarrow G & \qquad & \text{weakly in }\; L^2(Q_{i,T})^d, \\ J_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} & \underset{m\to+\infty} \longrightarrow J & \qquad & \text{weakly in }\; L^2(Q_{T})^d. \end{alignat} \end{subequations} Establishing that $\wh \Theta_i \in L^2((0,T); H^1(\Omega_i))$ and $\wh \Upsilon\in L^2((0,T);H^1(\Omega))$ with $G = \nabla \wh \Theta_i$ and $J = \nabla \wh \Upsilon$ is now classical, see for instance \cite[Lemma 2]{EG03} or \cite[Lemma 4.4]{CHLP03}. The key points of this proof are the identification $\wh \Theta_i = \Theta_i(p)$ and $\wh \Upsilon=\Upsilon(p)$ for some measurable $p$, as well as the proofs of the almost everywhere convergence property~\eqref{eq:compact.s}. The identification of the limit and the almost everywhere convergence can be handled simultaneously by using twice~\cite[Theorem 3.9]{ACM17}, once for $\Theta_i(p)$ and once for $\Upsilon(p)$. More precisely, Lemma~\ref{lem:boundS} provides a control on the time variations of the approximate saturation $s_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m}$, whereas Corollary~\ref{cor:xi_Upsilon} provides some compactness w.r.t. space on $\Theta_i(p_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m})$ and $\Upsilon(p_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m})$. Using further that $s_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m} = \mathcal{S}_i \circ \Theta_i^{-1}\left(\Theta_i(p_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m})\right)$ with $\mathcal{S}_i \circ \Theta_i^{-1}$ nondecreasing and continuous, then one infers from \cite[Theorem 3.9]{ACM17} that \[ s_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m} \underset{m\to+\infty} \longrightarrow \mathcal{S}_i\circ \Theta_i^{-1}(\wh \Theta_i) \quad \text{a.e. in}\; Q_{i,T}. \] Let $p = \Theta_i^{-1}(\wh \Theta_i)$. Then, \eqref{eq:compact.s} and \eqref{eq:compact.xi} hold. Proving~\eqref{eq:compact.s} and \eqref{eq:compact.Upsilon} is similar, and the properties \eqref{eq:compact} can be assumed to hold for the same function $p$ up to the extraction of yet another subsequence. Finally, by applying the arguments developed in \cite[\S4.2]{BCH13}, we show that $\Upsilon(p)$ and $\Upsilon(p^{\rm D})$ share the same trace on $(0,T) \times \Gamma^{\rm D}$, hence $\Upsilon(p) - \Upsilon(p^{\rm D}) \in L^2((0,T);V)$. \end{proof} Let us now define \begin{equation} \eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}(t,x) = \eta_\sigma^n \quad \text{if }\; (t,x) \in (t_m^{n-1}, t_m^n]\times \Delta_\sigma \end{equation} for $\sigma \in \mathscr{E}_m$, $1 \leq n \leq N_m$. \begin{lem}\label{lem:eta_E} Up to a subsequence, the function $p$ whose existence is guaranteed by Proposition~\ref{prop:compact} satisfies \begin{equation} \eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} \underset{m\to\infty}\longrightarrow \eta(\mathcal{S}(p,x)) \qquad \text{in }\; L^q(Q_T), \; 1 \leq q < +\infty. \end{equation} \end{lem} \begin{proof} Because of~\eqref{eq:compact.s}, $\eta_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m} = \eta(s_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m},x)$ converges almost everywhere to $\eta(\mathcal{S}(p,x),x)$. Since $\eta$ is bounded, Lebesgue's dominated convergence theorem ensures that the convergence holds in $L^q(Q_T)$ for all $q\in [1,+\infty)$. The reconstruction $\eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ of the mobility is also uniformly bounded, so we have just to show that $\lVert \eta_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m} -\eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} \rVert_{L^1(Q_T)} \rightarrow 0$ as $m \rightarrow +\infty$. Letting $\Delta_{K\sigma}=K \cap \Delta_\sigma$ denote the half-diamond cell, we have \begin{multline*} \lVert \eta_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m} -\eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} \rVert_{L^1(Q_T)} \leq \; \sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{K\in\mathscr{T}_m} \sum_{\sigma\in\mathscr{E}_K} m_{\Delta_{K\sigma}} |\eta_K(s_K^n) - \eta_\sigma^n|\\ \leq \;\sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{\sigma\in\mathscr{E}_m} m_{\Delta_{\sigma}} |\eta_K(s_K^n) - \eta_{K\sigma}(s_{K\sigma}^n)| \, \leq \;\sum_{i=1}^I\mathtt{R}_{i,m} + \mathtt{R}_{\Gamma,m}, \end{multline*} where \begin{align*} \mathtt{R}_{i,m} = \;& \sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{\sigma \in \mathscr{E}_{i,m}} m_{\Delta_{\sigma}} |\eta_K(s_K^n) - \eta_{K\sigma}(s_{K\sigma}^n)|, \\ \mathtt{R}_{\Gamma,m} = \;& \sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{\sigma \in \mathscr{E}_{\Gamma,m}} m_{\Delta_{\sigma}} |\eta_K(s_K^n) - \eta_{K\sigma}(s_{K\sigma}^n)|. \end{align*} Let us define \[ r_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}(t,x)= |\eta_K^n -\eta_{K\sigma}^n|= r_\sigma^n \quad \text{if}\; (t,x) \in (t_m^{n-1}, t_m^n]\times \Delta_{\sigma}, \] then $r_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ is uniformly bounded by $\lVert \eta \rVert_\infty = 1/\mu$. Therefore, \[ \mathtt{R}_{\Gamma,m} \leq \frac T\mu \sum_{\sigma \in \mathscr{E}_{\Gamma,m}} m_{\Delta_{\sigma}} \leq \frac {2T \,\nu^{d-1}(\Gamma)}{\mu d} \,h_{\mathscr{T}_m} \] where $h_{\mathscr{T}_m}$ is the size of $\mathscr{T}_m$ as defined in \eqref{eq:diamreg}. Besides, for $i \in \{1,\dots, I\}$, $\eta_i \circ \mathcal{S}_i \circ \Theta_i^{-1}$ is continuous, monotone and bounded, hence uniformly continuous. This provides the existence of a modulus of continuity $\varpi_i:\mathbb{R}_+ \to \mathbb{R}_+$ with $\varpi_i(0) = 0$ such that \begin{equation}\label{eq:rnsig} r_\sigma^n := |\eta \circ \mathcal{S} \circ \Theta_i^{-1}(\Theta_K^n) - \eta \circ \mathcal{S} \circ \Theta_i^{-1}(\Theta_{K\sigma}^n)| \leq \varpi_i(|\Theta_{K}^n - \Theta_{K\sigma}^n|) \end{equation} for $\sigma \in \mathscr{E}_{i,m}$. Therefore, if the function \begin{subequations} \begin{equation}\label{eq:qeimdtm} q_{\mathscr{E}_{i,m}, {\boldsymbol{{\Delta t}}}_m}(t,x) = \begin{cases} \, |\Theta_i(p_K^n) - \Theta_i(p_{K\sigma}^n)| & \text{if}\; (t,x) \in (t_m^{n-1},t_m^n] \times \Delta_\sigma, \\ \, 0 & \text{otherwise}, \end{cases} \end{equation} for $\sigma \in \mathscr{E}_{i,m}$, $1 \, \leq \, n \leq N_m$, \end{subequations} could be proven to converge to $0$ almost everywhere in $Q_{i,T}$, then it would also be the case for $r_{\mathscr{E}_m,{\boldsymbol{{\Delta t}}}_m}$ and $\mathtt{R}_{i,m}$ as $m\rightarrow+\infty$, thanks to Lebesgue's dominated convergence theorem. Now, it follows from \eqref{eq:dissip.xi} and from the elementary geometric relation \[ m_{\Delta_\sigma} = \frac{a_\sigma}{d} d_\sigma^2 \, \leq \, 4 \frac{a_\sigma}{d} h_{\mathscr{T}_m}^2 , \] that \[ \lVert q_{\mathscr{E}_{i,m}, {\boldsymbol{{\Delta t}}}_m} \rVert^2_{L^2(Q_{i,T})} = \sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{\sigma \in \mathscr{E}_{i,m}} m_{\Delta_\sigma} |\Theta_i(p_K^n) - \Theta_i(p_{K\sigma}^n)|^2 \leq \frac{4 \cter{c:dissip.max}}d h_{\mathscr{T}_m}^2. \] Therefore, $q_{\mathscr{E}_{i,m}, {\boldsymbol{{\Delta t}}}_m}\rightarrow 0$ in $L^2(Q_{i,T})$, thus also almost everywhere up to extraction of a subsequence. This provides the desired result. \end{proof} \subsection{Identification of the limit}\label{ssec:identify} So far, we have exhibited some ``limit'' value $p$ for the approximate solution $p_{\mathscr{T}_m, {\boldsymbol{{\Delta t}}}_m}$ in Proposition~\ref{prop:compact}. Next, we show that the scheme is consistent with the continuous problem by showing that any limit value is a weak solution. \begin{prop}\label{prop:identify} The function $p$ whose existence is guaranteed by Proposition~\ref{prop:compact} is a weak solution of the problem~\eqref{eq:cont.mass}--\eqref{eq:Ss} in the sense of Definition~\ref{def:weak}. \end{prop} \begin{proof} Let $\varphi \in C_{c}^{\infty}(\{\Omega\cup \Gamma^{\rm N}\} \times [0, T))$ and denote by $\varphi_{K}^{n}=\varphi(t^n_m,x_K)$, for all $K \in \mathscr{T}_m$ and all $n\in\{0,\dots,N_m\}$. We multiply \eqref{eq:scheme.mass} by ${\Delta t}^n_m \varphi_K^{n-1}$ and sum over $n\in\{1,\dots,N_m\}$ and $K\in\mathscr{T}_m$ to obtain \begin{equation}\label{eq:A+B_m} \mathtt{A}_m + \mathtt{B}_m=0, \qquad m \geq 1, \end{equation} where we have set \begin{subequations} \begin{align} \mathtt{A}_m =& \sum_{n=1}^{N_m} \sum_{K\in\mathscr{T}_m} m_K \phi_K (s_K^n - s_K^{n-1}) \varphi_K^{n-1}, \label{eq:ident.Am}\\ \mathtt{B}_m = & \sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{\sigma \in \mathscr{E}_m} a_\sigma \lambda_\sigma \eta_\sigma^n (\vartheta_K^n - \vartheta_{K\sigma}^n ) (\varphi_K^{n-1} - \varphi_{K\sigma}^{n-1} ). \label{eq:ident.Bm} \end{align} \end{subequations} The quantity $\mathtt{A}_m$ in \eqref{eq:ident.Am} can be rewritten as \begin{align*} \mathtt{A}_m&=-\sum_{n=1}^{N_m} {\Delta t}^n_m \sum_{K\in\mathscr{T}_m} m_K \phi_K s_K^n \frac{\varphi_K^n - \varphi_K^{n-1}}{{\Delta t}_m^n}- \sum_{K\in\mathscr{T}_m} m_K \phi_K s_K^0 \varphi_K^0 \\ &=- \int\!\!\!\int_{Q_T}\phi\, s_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m} \delta \varphi_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m} \, {\rm d} x \, {\rm d} t-\int_\Omega \phi\, s_{\mathscr{T}_m}^0 \varphi_{\mathscr{T}_m}^0 \, {\rm d} x \end{align*} where \begin{alignat*}{2} \delta \varphi_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_m}(t,x) & = \frac{\varphi_K^n - \varphi_K^{n-1}}{{\Delta t}_m^n},&\qquad\text{if }\; & (t,x) \in (t_m^{n-1},t_m^{n}) \times K,\\ \varphi_{\mathscr{T}_m}^0 &= \varphi(0,x_K) & \qquad \text{if } \; & x \in K. \end{alignat*} Thanks to the regularity of $\varphi$, the function $\delta \varphi_{\mathscr{T}_m,{\Delta t}_m}$ converges uniformly to $\partial_t \varphi$ on $\Omega \times [0,T]$. Moreover, by virtue of \eqref{eq:compact.s} and the boundedness of $s_{\mathscr{T}_m,{\boldsymbol{{\Delta t}}}_n}$ we can state that \[ \int\!\!\!\int_{Q_T} \phi \, s_{\mathscr{T}_m,{\Delta t}_m} \delta \varphi_{\mathscr{T}_m,{\Delta t}_m} \, {\rm d} x \, {\rm d} t \underset{m\to+\infty}\longrightarrow \int\!\!\!\int_{Q_T} \phi \,\mathcal{S}(p,x) \partial_t \varphi \, {\rm d} x \, {\rm d} t, \] and, in view of the definition~\eqref{eq:scheme.init} of $s^0_{\mathscr{T}_m}$ and of the uniform convergence of $\varphi_{\mathscr{T}_m}^0$ towards $\varphi(0,\cdot)$, \[\int_\Omega \phi_{\mathscr{T}_m} s_{\mathscr{T}_m}^0 \varphi^0_{\mathscr{T}_m} \, {\rm d} x \underset{m\to+\infty}\longrightarrow \int_\Omega \phi \, s^0 \varphi(0,\cdot) \, {\rm d} x.\] From the above, we draw that \begin{equation}\label{eq:A_m} \lim_{m\to+\infty} \mathtt{A}_m = -\int\!\!\!\int_{Q_T} \phi \, \mathcal{S}(p,x) \partial_t \varphi \, {\rm d} x \, {\rm d} t - \int_\Omega \phi \, s^0 \varphi(0,\cdot) \, {\rm d} x. \end{equation} Let us now turn our attention to the quantity $\mathtt{B}_m$ of \eqref{eq:ident.Bm}, which can be split into $\mathtt{B}_m= \mathtt{B}_m^1 + \mathtt{B}_m^2$ using \begin{align*} \mathtt{B}_m^1=& \sum_{n=1}^{N_m} {\Delta t}_m^n \sum_{\sigma \in \mathscr{E}_m} a_\sigma \lambda_\sigma \eta_\sigma^n (p_K^n-p_{K\sigma}^n)(\varphi_K^{n-1}-\varphi_{K\sigma}^{n-1}), \\ \mathtt{B}_m^2=&\sum_{n=1}^{N_m} {\Delta t}_m^n \sum_{\sigma \in \mathscr{E}_m} a_\sigma \lambda_\sigma \eta_\sigma^n (\psi_K - \psi_{K\sigma})(\varphi_K^{n-1}-\varphi_{K\sigma}^{n-1}). \end{align*} Consider first the convective term $\mathtt{B}_m^2$. It follows from the definition of the discrete gravitational potential \[ \psi_K = -\varrho g\cdot x_K, \quad \psi_\sigma = -\varrho g\cdot x_\sigma, \qquad K\in\mathscr{T}_m, \; \sigma \in \mathscr{E}_{{\rm ext},m}^{\rm D} \] and from the orthogonality of the mesh that \[ \psi_K - \psi_{K\sigma} = d_\sigma \varrho g\cdot n_{K\sigma}, \qquad \forall \sigma \in \mathscr{E}_K \setminus \mathscr{E}_{\rm ext}^{\rm N}, \; K \in \mathscr{T}_m. \] Therefore, $\mathtt{B}^2_m$ can be transformed into \begin{align} \mathtt{B}_m^2 = & \sum_{n=1}^{N_m} {\Delta t}_m^n \sum_{\sigma \in \mathscr{E}_m} m_{\Delta_\sigma} \lambda_\sigma \eta_\sigma^n d \frac{\varphi_K^{n-1} - \varphi_{K\sigma}^{n-1}}{d_\sigma} n_{K\sigma} \cdot \varrho g \nonumber\\ =& - \int\!\!\!\int_{Q_T}\lambda_{\mathscr{E}_m}\eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} H_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} \cdot \varrho g\, \, {\rm d} x \, {\rm d} t, \label{eq:Bm2.1} \end{align} where \begin{alignat*}{2} \lambda_{\mathscr{E}_m}(x) &= \lambda_\sigma & \qquad \text{if }\; & x \in \Delta_\sigma, \; \sigma \in \mathscr{E}_m,\\ H_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}(t,x) &= (d/d_\sigma) (\varphi_{K\sigma}^{n-1} - \varphi_{K}^{n-1}) n_{K\sigma} & \qquad \text{if } \; & (t,x) \in [t^{n-1}_m, t^n_m) \times \Delta_\sigma. \end{alignat*} After~\cite[Lemma 4.4]{CHLP03}, $H_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ converges weakly in $L^2(Q_T)^d$ towards $\nabla \varphi$, while $\lambda_{\mathscr{E}_m}$ and $\eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ converge strongly in $L^4(\Omega)$ and $L^4(Q_T)$ towards $\lambda$ and $\eta(\mathcal{S}(p,x))$ respectively (cf. Lemma~\ref{lem:eta_E}). Thus, we can pass to the limit in~\eqref{eq:Bm2.1} and \begin{equation}\label{eq:Bm2.2} \lim_{m\to+\infty} \mathtt{B}_m^2 = - \int\!\!\!\int_{Q_T}\lambda \eta(\mathcal{S}(p,x)) \varrho g \cdot \nabla \varphi\, {\rm d} x \, {\rm d} t. \end{equation} The capillary diffusion term $\mathtt{B}_m^1$ appears to be the most difficult one to deal with. Taking inspiration from~\cite{cancesGuichard2016EntropyScheme}, we introduce the auxiliary quantity \begin{align*} \wt{\mathtt{B}}{}_m^1 &= \sum_{i=1}^I \wt \mathtt{B}_{i,m}^1 \\ & = \sum_{i=1}^I \sum_{n=1}^{N_m} {\Delta t}_m^n \!\! \sum_{\sigma \in \mathscr{E}_{i,m}} \!\! a_\sigma \sqrt{\lambda_i \eta_\sigma^n} (\Theta_i(p_K^n)-\Theta_i(p_{K\sigma}^n))(\varphi_K^{n-1}-\varphi_{K\sigma}^{n-1}). \end{align*} Analogously to~\cite{DE_FVCA8}, {we can define a piecewise-constant vector field $\ov{H}_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ such that \[ \ov{H}_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}(t,x) \cdot n_{K\sigma} = \varphi_{K\sigma}^{n-1} - \varphi_K^{n-1}, \qquad \text{if } \; (t,x) \in [t^{n-1}_m, t^n_m) \times \Delta_\sigma, \; \sigma \in \mathscr{E}_m, \] and such} that $\ov{H}_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}$ converges uniformly towards $\nabla \varphi$ on $\ov Q_T$. Under these circumstances, $\wt{\mathtt{B}}_{i,m}^1$ reads \[ \wt \mathtt{B}_{i,m}^1 =\int_0^T \!\! \int_{\Omega_{i,m}} \sqrt{\lambda_i \eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}} \, G_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}\cdot \ov H_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} \, {\rm d} x \, {\rm d} t \] where $\Omega_{i,m} = \bigcup_{\sigma \in \mathscr{E}_{i,m}} \Delta_\sigma \subset \Omega_i$. The strong convergence of $\sqrt{\eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}}$ in $L^2(Q_{i,T})$ towards $\sqrt{\eta_i(\mathcal{S}_i(p))}$ directly follows from the boundedness of $\eta_i$ combined with~\eqref{eq:compact.s}. Combining this with~\eqref{eq:compact.xi} results in \begin{equation}\label{eq:wtBim1} \wt{\mathtt{B}}_{i,m}^1 \underset{m\to + \infty} \longrightarrow \int\!\!\!\int_{Q_{i,T}} \!\!\!\!\!\! \sqrt{\lambda_i \eta_i(\mathcal{S}_i(p))} \nabla \Theta_i(p) \cdot \nabla \varphi \, {\rm d} x \, {\rm d} t = \int\!\!\!\int_{Q_{i,T}} \!\!\!\!\!\! \nabla \Phi_i(p) \cdot \nabla \varphi \, {\rm d} x \,{\rm d} t. \end{equation} Therefore, to finish the proof of Proposition~\ref{prop:identify}, it only remains to check that $B^1_m$ and $\wt B^1_m$ share the same limit. To this end, we observe that, by the triangle inequality, we have \begin{equation}\label{eq:diff.B} | \mathtt{B}_m^1 - \wt \mathtt{B}_{m}^1 |\leq \mathtt{R}_{\Gamma,m} + \sum_{i = 1}^I \mathtt{R}_{i,m}, \end{equation} where \begin{align*} \mathtt{R}_{\Gamma,m} & = \sum_{n=1}^{N_m}{\Delta t}_m^n \!\!\! \sum_{\sigma \in \mathscr{E}_{\Gamma,m}} \!\!\! a_\sigma \lambda_\sigma \eta_\sigma^n |p_K^n-p_{K\sigma}^n| |\varphi_K^{n-1}-\varphi_{K\sigma}^{n-1}|, \\ \mathtt{R}_{i,m} & = \sum_{n=1}^{N_m} {\Delta t}_m^n \!\!\! \sum_{\sigma \in \mathscr{E}_{i,m}} \!\!\! a_\sigma \sqrt{\lambda_i \eta_\sigma^n} | \Theta_i(p_K^n) - \Theta_i(p_{K\sigma}^n) - \sqrt{\lambda_i \eta_\sigma^n}(p_K^n - p_{K\sigma}^n) | |\varphi_K^{n-1}-\varphi_{K\sigma}^{n-1}|. \end{align*} Applying the Cauchy-Schwarz inequality and using Proposition~\ref{prop:dissip}, we find \[ |\mathtt{R}_{\Gamma,m}|^2 \leq \cter{c:main} \sum_{n=1}^{N_m} {\Delta t}_m^n \!\!\sum_{\sigma \in \mathscr{E}_{\Gamma, m}} \!\! a_\sigma \lambda_\sigma \eta_\sigma^n |\varphi_{K}^{n-1} - \varphi_{K\sigma}^{n-1}|^2 \leq 2 \cter{c:main} T \lVert \nabla \varphi \rVert_\infty^2 \frac{\max_i \lambda_i}{\mu}\nu^{d-1}(\Gamma)h_{\mathscr{T}_m}, \] so $\mathtt{R}_{\Gamma,m}\rightarrow 0$ as $m\rightarrow +\infty$. Besides, we also apply the Cauchy-Schwarz inequality to $\mathtt{R}_{i,m}$ in order to obtain \begin{align*} |\mathtt{R}_{i,m}|^2 & \leq \cter{c:main} \sum_{n=1}^{N_m} {\Delta t}_m^n \!\! \sum_{\sigma \in \mathscr{E}_{i, m}} \!\! a_\sigma \, \lambda_i \big| \sqrt{\eta_\sigma^n} - \sqrt{\wt \eta_\sigma^n} \, \big|^2 \, \big|\varphi_K^{n-1} - \varphi_{K\sigma}^{n-1}\big|^2 \\ & \leq d \lambda_i \cter{c:main} \lVert \nabla\varphi \rVert_\infty^2 \sum_{n=1}^{N_m} {\Delta t}_m^n \!\! \sum_{\sigma \in \mathscr{E}_{i, m}} \!\! m_{\Delta_\sigma} \left|{\eta_\sigma^n} -{\wt \eta_\sigma^n}\right| , \end{align*} where we have set \[ \wt \eta_\sigma^n = \begin{cases} \, \eta_i(s_K^n) & \text{if}\; p_K^n = p_{K\sigma}^n,\\ \, \displaystyle\bigg[\frac{\Theta_i(p_{K}^n) - \Theta_i(p_{K\sigma}^n)}{\sqrt{\lambda_i} (p_K^n - p_{K\sigma}^n)}\bigg]^2 & \text{otherwise}. \end{cases} \] Define \[ \wt \eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m}(t,x) = \begin{cases} \, \wt \eta_\sigma^n & \text{if}\; (t,x) \in (t_m^{n-1}, t_m^n] \times \Delta_\sigma, \; \sigma \in \bigcup_{i=1}^I \mathscr{E}_{i,m}, \\ \, 0 & \text{otherwise}. \end{cases} \] Reproducing the proof of Lemma~\ref{lem:eta_E}, we can show that \[ \wt \eta_{\mathscr{E}_m, {\boldsymbol{{\Delta t}}}_m} \underset{m\to\infty}\longrightarrow \eta(\mathcal{S}(p,x)) \qquad \text{in }\; L^q(Q_T), \; 1 \leq q < +\infty. \] Therefore, $\mathtt{R}_{i,m}\rightarrow 0$ as $m\rightarrow +\infty$. Putting things together in~\eqref{eq:diff.B}, we conclude that $\mathtt{B}_m^1$ and $\wt \mathtt{B}_{m}^1$ share the same limit, which completes the proof of Proposition~\ref{prop:identify}. \end{proof} \section{Practical aspects of numerical resolution} \label{sec:numerics} We provide some details on the resolution strategy for the discrete problem \eqref{eq:scheme.capi}--\eqref{eq:scheme.upwind}. It is based on a parametrization technique to automatically choose the most convenient variable during the Newton iterations (\S\ref{ssec:variableSwitch}) and on the addition of cells on the interfaces between different rock types (\S\ref{ssec:BCET}) to improve the pressure continuity. \subsection{Switch of variable and parametrization technique} \label{ssec:variableSwitch} A natural choice to solve the nonlinear system \eqref{eq:scheme.capi}--\eqref{eq:scheme.upwind} is to select the pressure $(p_K)_{K\in\mathscr{T}}$ as primary unknown and to solve it via an iterative method such as Newton's one. Nevertheless, the pressure variable is known to be an inefficient choice for $s \ll 1$ because of the degeneracy of Richards' equation. For dry soils, this strategy is outperformed by schemes in which saturation is the primary variable. On the other hand, the knowledge of the saturation is not sufficient to describe the pressure curve in saturated regions where the pressure-saturation relation cannot be inverted. This motivated the design of schemes involving a switch of variable \cite{DP99}--\cite{FWP95}. In this work, we adopt the technique proposed by Brenner and Cancès \cite{BC17}, in which a third generic variable $\tau$ is introduced to become the primary unknown of the system. Then the idea is to choose a parametrization of the graph $\{p, \mathcal{S}(p)\}$, i.e., to construct two functions $\mathfrak{s}: I \rightarrow [s_{\mathrm{rw}}, 1-s_{\mathrm{rn}}]$ and $\mathfrak{p}: I \rightarrow \mathbb{R}$ such that $\mathfrak{s}(\tau)=\mathcal{S}(\mathfrak{p}(\tau))$ and $\mathfrak{s}'(\tau)+\mathfrak{p}'(\tau)>0$ for all $\tau \in I \subset \mathbb{R}$. Such a parametrization is not unique, for instance one can take $I=\mathbb{R}$, $\mathfrak{p}=Id$ which amounts to solving the system always in pressure, but this is not recommended as explained before. Here, we set $I=(s_{\mathrm{rw}}, + \infty)$ and \begin{align*} \mathfrak{s}(\tau) = \begin{cases} \tau & \text{if}\; \tau \leq s_{\rm s}, \\ \mathcal{S}\left(p_{\rm s} + \displaystyle\frac{\tau-s_{\rm s}}{\mathcal{S}'(p_{\rm s}^-)}\right) & \text{if}\; \tau \geq s_{\rm s}, \\ \end{cases} \qquad \mathfrak{p}(\tau) = \begin{cases} \mathcal{S}^{-1}(\tau) & \text{if}\; \tau \leq s_{\rm s}, \\ p_{\rm s} + \displaystyle\frac{\tau-s_{\rm s}}{\mathcal{S}'(p_{\rm s}^-)}& \text{if}\; \tau \geq s_{\rm s}, \end{cases} \end{align*} where $\mathcal{S}'(p_{\rm s}^-)$ denotes the limit as $p$ tends to $p_{\rm s}=\mathcal{S}(s_{\rm s})$ from below of $\mathcal{S}'(p)$. Since the switch point $s_{\rm s}$ is taken as the inflexion point of $\mathcal{S}$, both $\mathfrak{s}$ and $\mathfrak{p}$ are $C^1$ and concave, and even $C^2$ if $\mathcal{S}$ is given by the Van Genuchten model. Moreover, for all $p\in \mathbb{R}$, there exists a unique $\tau \in (s_{\rm rw},+\infty)$ such that $(p,\mathcal{S}(p)) = (\mathfrak{p}(\tau), \mathfrak{s}(\tau))$. The resulting system $\mathcal F_n(\boldsymbol{ \tau}^n) = \mathbf{0}$ made up of $N_\mathscr{T} = {\rm Card}(\mathscr{T})$ nonlinear equations admits a unique solution $\boldsymbol{\tau}^n$, since it is fully equivalent to \eqref{eq:scheme.capi}--\eqref{eq:scheme.upwind}. More details about the practical resolution of this nonlinear system via the Newton method can be found in \cite{BCET_FVCA9}. \subsection{Pressure continuity at rock type interfaces} \label{ssec:BCET} Physically, the pressure should remain continuous on both sides of an interface between two different rock types. But this continuity is here not imposed at the discrete level. The two-point flux approximation based on the cell unknowns is strongly dependent on the mesh resolution and can induce a large error close to the rock type interface. We here propose a very simple method to improve this continuity condition in pressure. It consists in adding two thin cells of resolution $\delta$ around the rock-type interface with $\delta \ll \Delta x$ as shown in Figure \ref{fig:vt2}. \begin{figure}[htb] \centering \includegraphics[width=6cm]{img/vt2.pdf} \caption{Mesh refinement on both sides of an interface face for a 2D case. } \label{fig:vt2} \end{figure} The idea is here to add two unknowns in the neighborhood of the interface to have a more precise approximation of the pressure gradient on each side of the faces where changes of rock types occur. In this way, we avoid the introduction of face unknowns in our solver which remains unchanged. Alternatives that strictly impose the pressure continuity will be studied and compared with this approach in a forthcoming work. \section{Numerical results}\label{sec:results} In this section, we present the results obtained for different test cases. For all these cases, we consider a two-dimensional layered domain $\Omega=[0\mathrm{m}, 5\mathrm{m}]\times [-3 \mathrm{m}, 0\mathrm{m}]$ made up of two rock types denoted by RT0 and RT1 respectively, RT0 being less permeable than RT1. Using these two lithologies, the domain $\Omega$ is partitioned into three connected subdomains: $\Omega_1=[1\mathrm{m}, 4\mathrm{m}] \times [-1\mathrm{m}, 0\mathrm{m}]$, $\Omega_2= [0 \mathrm{m},5 \mathrm{m}] \times [-3 \mathrm{m}, -2\mathrm{m}]$ and $\Omega_3= \Omega ~\setminus~ (\Omega_1 \cup \Omega_2)$, {as depicted in Figure} \ref{fig:testDomain}. \begin{figure}[htb] \centering \includegraphics[width=6.4cm]{./img/testDomainbase.jpg} \caption{Simulation domain $\Omega=[0\mathrm{m},5\mathrm{m}]\times[-3\mathrm{m},0\mathrm{m}]$.} \label{fig:testDomain} \end{figure} The Brooks-Corey \cite{BROOKS} and van Genuchten-Mualem \cite{VANGENUCHTEN} petro-physical models are used to model the flow characteristics of both rock types. In these models, the water saturation and the capillary pressure are linked pointwise by the relation $s=\mathcal{S}(p)$ where $\mathcal{S}: \mathbb{R} \rightarrow [0,1]$ is nondecreasing and satisfies $\mathcal{S}(p)=1 - s_{\mathrm{rn}}$ if $p \geq p_b$ and $\mathcal{S}(p)\rightarrow s_{\mathrm{rw}}$ as $p \rightarrow -\infty$, $s_{\mathrm{rw}}$ being the residual wetting saturation, $s_{\mathrm{rn}}$ the residual non-wetting saturation and $p_b$ the entry pressure. More precisely, we have, \begin{itemize} \item for the Brooks-Corey model, \begin{align*} &s= \mathcal S(p)=\begin{cases} s_\mathrm{rw} + (1-s_\mathrm{rn}-s_\mathrm{rw})\left( \frac{p}{p_b} \right)^{- n} & \text{if}~p \leq p_b,\\ 1-s_\mathrm{rn} & \text{if}~p > p_b, \end{cases} \nonumber\\ &\nonumber\\ &k_r(s)=s_{\mathrm{eff}}^{3+ \frac{2}{n}}, \qquad\qquad s_{\mathrm{eff}}= \frac{s-s_\mathrm{rw}}{1-s_\mathrm{rn}-s_\mathrm{rw}};\nonumber \end{align*} \item for the van Genuchten-Mualem model, \begin{align*} &s= \mathcal S(p)=\begin{cases} s_\mathrm{rw} + (1-s_\mathrm{rn}-s_\mathrm{rw}) \left[1 + \left|\frac{\alpha}{\rho g} p \right|^n\right ]^{-m} & \text{if}~p \leq 0,\\ 1-s_\mathrm{rn} & \text{if}~p > 0, \end{cases} \nonumber \\ &\nonumber\\ &k_r(s)=s_{\mathrm{eff}}^{\frac{1}{2}} \{1 - [1- s_{\mathrm{eff}}^{\frac{1}{m}}]^m \}^2 , \qquad s_{\mathrm{eff}}= \frac{s-s_\mathrm{rw}}{1-s_\mathrm{rn}-s_\mathrm{rw}},\qquad\qquad m= 1 -\frac{1}{n};\nonumber \end{align*} \end{itemize} where $\eta(\cdot)= k_r(\cdot)/\mu$, $\mu= 10^{-3}~\textrm{Pa}\cdot\textrm{s}$ being water viscosity, is the relative permeability. The parameters used for both rock types are given in Table \ref{table:SoilsBC} for the Brooks-Corey model and in Table \ref{table:SoilsVG} for the van Genuchten-Mualem model. {With these choices of parameters, water is more likely to be in RT1 than in RT0, in the sense tha, at a given pressure, the water saturation is higher in RT1 than in RT0, as it can be seen on the plots of the capillary-pressure functions depicted in Figures \ref{img:lawsBC}--\ref{img:lawsVG} for these two petro-physical models. Figures \ref{img:lawsBC}--\ref{img:lawsVG} also show the relative permeability functions. Note the non-Lipschitz character of the relative permeability in the van Genuchten-Mualem framework.} {For the numerical tests, in order to avoid infinite values for the derivative of $k_r(s)$ when $s\rightarrow 1-s_\mathrm{rn}$, we approximate it for $s\in[s_{lim},1-s_\mathrm{rn}]$ using a second degree polynomial $\widetilde{k_r}(s)$. Such a polynomial satisfies the following constraints: $k_r(s_{lim})=\widetilde{k_r}(s_{lim})$ and $\widetilde{k_r}(1-s_\mathrm{rn})=1$. The value $s_{lim}$ {corresponds to $s_{\text{eff}}=0.998$.} } \begin{table}[htp!] \centering \begin{tabular}{p{0.8cm}p{1.2cm}p{1cm}p{2.2cm}p{1cm}p{1cm}p{1cm}} \hline & $1-s_{rn}$ & $s_{rw}$ & $p_b [\textrm{Pa}]$ & $n$ & $\lambda [\mathrm{m}^2]$ & $\phi$ \\ \hline RT0 & $1.0$ & $0.1$ & $-1.4708 \cdot 10^3$ & $3.0$ & $10^{-11}$ & $0.35$ \\ RT1 &$1.0$ &$0.2$& $-3.4301 \cdot 10^3$ & $1.5$ & $10^{-13}$ & $0.35$\\ \hline\\ \end{tabular} \caption{Parameters used for the Brooks-Corey model} \label{table:SoilsBC} \end{table} \begin{figure}[htb] \centering \includegraphics[height=4cm]{img/pcBC.jpg} \includegraphics[height=4cm]{img/krBC.jpg} \caption{Capillary pressure and relative permeability curves for the Brooks-Corey model} \label{img:lawsBC} \end{figure} \begin{table}[htp!] \centering \begin{tabular}{p{1.8cm}p{1.2cm}p{1cm}p{1cm}p{2.2cm}p{1.1cm}p{1cm}p{1cm}} \hline & $1-s_{\rm rn}$ & $s_{\rm rw}$ & $n$ & $\lambda~[\mathrm{m}^2]$ & $\alpha~[\mathrm{m}^{-1}]$ & $\phi$ \\ \hline RT0 (Sand) & $1.0$ & $0.0782$ & $2.239$ & $6.3812 \cdot 10^{-12}$ & $2.8 $ & $0.3658$\\ RT1 (Clay) & $1.0$ & $ 0.2262 $ & $1.3954$ & $ 1.5461 \cdot 10^{-13}$ & $1.04$ & $0.4686$\\ \hline\\ \end{tabular} \caption{Parameters used for the van Genuchten-Mualem model} \label{table:SoilsVG} \end{table} \begin{figure}[htp] \centering \includegraphics[height=4cm]{img/pcVG.jpg} \includegraphics[height=4cm]{img/krVG.jpg} \caption{Capillary pressure and relative permeability curves for the van Genuchten-Mualem model} \label{img:lawsVG} \end{figure} \subsection{Configurations {of the test cases}} For both petro-physical models, we consider two configurations further referred as filling and drainage cases, which are described in the following. \subsubsection{Filling case} {The filling test case has already been considered in~\cite{KHW92,FWP95, McBride06,CZ10}}. Starting from an initially dry domain $\Omega$, whose layers' composition is reported in Figure \ref{fig:fillingTest}, water flows from a part of the top boundary during one day. A no-flow boundary condition is applied elsewhere. More precisely, the initial capillary pressure is set to $-47.088 \cdot 10^5\textrm{Pa}$ and the water flux rate to $0.5 \textrm{m/day}$ through $\Gamma_N=\{(x,y) \,| \, x \in [1\mathrm{m}, 4\mathrm{m}], y=0\mathrm{m}\}$. For this simulation an homogeneous time-step $\Delta t= 1000\mathrm{s}$ is prescribed for the test using the Brooks-Corey model and $\Delta t= 500\mathrm{s}$ for the one using the van Genuchten-Mualem model. \begin{figure}[htb] \centering \includegraphics[height=5cm]{./img/testDomainFill.jpg} \caption{Boundary condition for the filling case} \label{fig:fillingTest} \end{figure} {The test case follows the following dynamics. Water starts invading the void porous space in $\Omega_1$. When it reaches the interface with $\Omega_3$, capillarity involves a suction force on water from $\Omega_1$ to $\Omega_3$. Since clay (RT1) has low permeability, water encounters difficulties to progress within $\Omega_3$. This yields a front moving downward in $\Omega_1$ which is stiffer for the Brooks-Corey model than for the van Genuchten-Mualem one. In both cases, the simulation is stopped before water reaches the bottom part corresponding to $\Omega_2$. In Figure \ref{img:solutionFillBC} we can observe the evolution of the saturation profile during the simulation performed on a $50\times30$ cells mesh with Brooks and Corey model, whereas the evolution corresponding to van Genuchten-Mualem nonlinearities is depicted in Figure~\ref{img:solutionFillVG}.} \newline \begin{minipage}{0.05\textwidth} \includegraphics[height=6cm]{./img/solutionFillBC/leg.jpg} \end{minipage} \begin{minipage}{0.9\textwidth} \begin{center} \includegraphics[width=4.4cm]{./img/solutionFillBC/fillBC0.jpg} \includegraphics[width=4.4cm]{./img/solutionFillBC/fillBC20.jpg} \includegraphics[width=4.4cm]{./img/solutionFillBC/fillBC40.jpg} \includegraphics[width=4.4cm]{./img/solutionFillBC/fillBC60.jpg} \includegraphics[width=4.4cm]{./img/solutionFillBC/fillBC87.jpg} \captionof{figure}{Evolution of the saturation profile for $t\in\{0s,20\cdot 10^3s, 40\cdot 10^3s, 60\cdot10^3s, 86\cdot10^3s\}$ for filling case, using Brooks and Corey model, Method A and the $50\times30$ cells mesh.} \label{img:solutionFillBC} \end{center} \end{minipage} \begin{minipage}{0.05\textwidth} \includegraphics[height=6cm]{./img/solutionFillVG/leg.jpg} \end{minipage} \begin{minipage}{0.9\textwidth} \begin{center} \includegraphics[width=4.4cm]{./img/solutionFillVG/fillVG0.jpg} \includegraphics[width=4.4cm]{./img/solutionFillVG/fillVG40.jpg} \includegraphics[width=4.4cm]{./img/solutionFillVG/fillVG80.jpg} \includegraphics[width=4.4cm]{./img/solutionFillVG/fillVG120.jpg} \includegraphics[width=4.4cm]{./img/solutionFillVG/fillVG172.jpg} \captionof{figure}{Evolution of the saturation profile for $t\in\{0s,20\cdot 10^3s, 40\cdot 10^3s, 60\cdot10^3s, 86\cdot10^3s\}$ for filling case using Van Genuchten model, Method A and the $50\times30$ cells mesh.} \label{img:solutionFillVG} \end{center} \end{minipage} \subsubsection{{Drainage case}} {This test case is designed as a two-dimensional extension of a one-dimensional test case proposed by \cite{MD99} and addressed in \cite{McBride06,CZ10}.} We simulate a vertical drainage starting from initially and boundary saturated conditions during $105 \cdot 10^4~\mathrm{s}$. At the initial time, the pressure varies with depth with $p^0(z)=-\rho g z$. A Dirichlet boundary condition $p_D=0~\textrm{Pa}$ is imposed on the bottom of the domain, more precisely on $\Gamma_D=\{(x,y) \,| \, x\in[0\mathrm{m},5\mathrm{m}], y=-3\mathrm{m}\}$. The layers' composition of $\Omega$ is reported in Figure \ref{fig:drainageTest}. For this simulation an homogeneous time-step $\Delta t= 2000\mathrm{s}$ is used for the test with the Brooks-Corey model and $\Delta t= 800\mathrm{s}$ for the one with the van Genuchten-Mualem model. \begin{figure}[htp] \centering \includegraphics[height=5cm]{./img/testDomainDry.jpg} \caption{Boundary condition for the drainage test} \label{fig:drainageTest} \end{figure} {At the top interface between $\Omega_1$ and $\Omega_3$, capillarity acts in opposition to gravity and to the evolution of the system into a dryer configuration. The interface between $\Omega_2$ and $\Omega_3$ acts in the reverse way: suction accelerates the gravity driven drainage of RT0.} In Figure \ref{img:solutionDryBC} we can observe the evolution of the saturation profile during the simulation performed on a $50\times30$ cells mesh with Brooks and Corey model, whereas the evolution corresponding to van Genuchten-Mualem nonlinearities is depicted in Figure~\ref{img:solutionDryVG}. \newline \begin{minipage}{0.025\textwidth} \vspace{-1.8cm} \includegraphics[height=5cm]{./img/solutionDryBC/leg.jpg} \end{minipage} \begin{minipage}{0.95\textwidth} \begin{center} \includegraphics[width=4.4cm]{./img/solutionDryBC/solDryBC0.jpg} \includegraphics[width=4.4cm]{./img/solutionDryBC/solDryBC131.jpg} \includegraphics[width=4.4cm]{./img/solutionDryBC/solDryBC262.jpg} \includegraphics[width=4.4cm]{./img/solutionDryBC/solDryBC393.jpg} \includegraphics[width=4.4cm]{./img/solutionDryBC/solDryBCFinal.jpg} \captionof{figure}{Evolution of the saturation profile for $t\in\{0s,26.2\cdot 10^4s, 52.4\cdot 10^4s, 78.6\cdot10^4s, 105\cdot10^4s\}$ for drying case, using Brooks and Corey model, Method A and the $50\times30$ cells mesh.} \label{img:solutionDryBC} \end{center} \end{minipage} \vspace{1cm} \begin{minipage}{0.05\textwidth} \vspace{-1.8cm} \includegraphics[height=5cm]{./img/solutionDryVG/leg.jpg} \end{minipage} \begin{minipage}{0.95\textwidth} \begin{center} \vspace{-1cm} \includegraphics[width=4.4cm]{./img/solutionDryVG/solDryVG0.jpg} \includegraphics[width=4.4cm]{./img/solutionDryVG/solDryVG327.jpg} \includegraphics[width=4.4cm]{./img/solutionDryVG/solDryVG655.jpg} \includegraphics[width=4.4cm]{./img/solutionDryVG/solDryVG982.jpg} \includegraphics[width=4.4cm]{./img/solutionDryVG/solDryVGFinalTime.jpg} \captionof{figure}{Evolution of the saturation profile for $t\in\{0s,26.16\cdot 10^4s, 52.4\cdot 10^4s, 78.56\cdot10^4s, 105\cdot10^4s\}$ for drying case using Van Genuchten model, Method A and the $50\times30$ cells mesh.} \label{img:solutionDryVG} \end{center} \end{minipage} \subsection{Comparisons {of the numerical treatments of the interfaces}}\label{ssec:num.results} For each petro-physical model and configuration, a numerical convergence analysis is carried out for the schemes with (method A) or without (method B) thin cells, whose thickness is fixed to $\delta=10^{-6}\mathrm{m}$, at rock type interfaces. Five structured meshes with the following resolutions are considered for this analysis: $50 \times 30 $, $100 \times 60$, $200 \times 120$, $400 \times 240$, $800 \times 480$. The evolution of the error is measured using the $L^2([0,T], \Omega)$-norm of the relative difference between the saturations obtained on a given mesh and a reference solution obtained with Method A and the mesh $800\times 480$. The number of Newton iterations obtained with both methods is also compared. \subsubsection{Brooks-Corey model: drainage case} For the drainage case with the Brooks-Corey model, the convergence error is given in Figure \ref{img:convBCdry}. First we notice that, for all meshes, the error is smaller with method A than with method B and that we have a linear rate of convergence with the first one whereas this rate is smaller with the latter one. The total, average and maximal number of Newton iterations are also given in Table \ref{table:newtonBCdry}. Method A appears to be slightly more expensive. \begin{figure}[htb] \centering \begin{tikzpicture} \begin{axis}[grid=major, width=0.7\textwidth, height=0.5\textwidth, xlabel={Mesh}, ylabel={L2 relative error}, ymode=log, xmode=log, xticklabels={50x30, 100x60, 200x120, 400x240}, xtick={1500,6000,24000, 96000}, cycle list name=MyCyclelist1, legend style={at={(0,0)},anchor=south west},nodes={scale=0.6, transform shape}] \addplot table[x=mesh, y=L2]{errorBCdry.txt}; \addlegendentry{Method B }; \addplot table[x=mesh, y=L2]{errorBCdryV2.txt}; \addlegendentry{Method A }; \addplot[red,dotted,domain=15e+2:96e+3,line width= 1.2pt] {0.460214823/x^0.34553081}; \addlegendentry{order $0.7$}; \addplot[blue,dotted,domain=15e+2:96e+3,line width= 1.2pt] {0.92503662/x^0.553083133}; \addlegendentry{order $1.11$}; \end{axis} \end{tikzpicture} \caption{$L^2(Q_T)$ relative error in saturation for the drainage case using Brooks and Corey model.}\label{img:convBCdry} \end{figure} \begin{table}[htp!] \centering \begin{tabular}{p{1.6cm}p{1.4cm}p{1.4cm}p{1.2cm}} \hline & $\sharp$ total & $\sharp$ avg & $\sharp$ max \\ \hline Method A & $2038$ & $3$ & $29$ \\ Method B & $1927$ & $3$ & $29$ \\ \hline\\ \end{tabular} \caption{Newton's iterations for the mesh $200 \times 120$ for the drainage case using Brooks and Corey model.} \label{table:newtonBCdry} \end{table} Let us now evaluate the saturation absolute error between results obtained with Method A and Method B. In Figure \ref{img:errDryBC} we plot the absolute-error distribution over the domain at three different times: when the cells line in $\Omega_1$ above the interface between $\Omega_1$ and $\Omega_3$ starts drying, when the cells line in $\Omega_2$ below the interface between $\Omega_3$ and $\Omega_2$ starts drying and at final time. \newline \begin{minipage}{0.03\textwidth} \vspace{-1.2cm} \includegraphics[height=3cm]{./img/satAbsErrorFillBC/leg.jpg} \end{minipage} \begin{minipage}{0.9\textwidth} \begin{center} \includegraphics[width=4.4cm]{./img/satAbsErrorDryBC/dryBC115.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorDryBC/dryBC266.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorDryBC/dryBCFinal.jpg} \captionof{figure}{Saturation absolute error between Method A and Method B for the drainage case with Brooks and Corey model at $t\in\{23\cdot 10^4s, 53.2\cdot 10^4s, 105\cdot 10^4s\}$.} \label{img:errDryBC} \end{center} \end{minipage} \subsubsection{Brooks-Corey model: filling case} For the filling case with the Brooks-Corey model, the convergence error is given in Figure \ref{img:convBCfill}. As for the previous case, Method A enables to recover a linear convergence rate. Except for the first two meshes where the error obtained with Method A is slightly larger, for all other meshes, this error is smaller than the one obtained with method B. The total, average and maximal number of Newton iterations are given in Table \ref{table:newtonBCfill}. The algorithm behaves here in the same way as before. \begin{figure}[htp] \centering \begin{tikzpicture} \begin{axis}[grid=major, width=0.7\textwidth, height=0.5\textwidth, xlabel={Mesh}, ylabel={L2 relative error}, ymode=log, xmode=log, xticklabels={50x30, 100x60, 200x120, 400x240}, xtick={1500,6000,24000, 96000}, cycle list name=MyCyclelist1, legend style={at={(0,0)},anchor=south west},nodes={scale=0.6, transform shape}] \addplot table[x=mesh, y=L2]{errorBCfill.txt}; \addlegendentry{Method B }; \addplot table[x=mesh, y=L2]{errorBCfillv2.txt}; \addlegendentry{Method A }; \addplot[red,dotted,domain=15e+2:96e+3,line width= 1.2pt] {0.29025352/x^0.150333304}; \addlegendentry{order $0.3$}; \addplot[blue,dotted,domain=15e+2:96e+3,line width= 1.2pt] {3.0039635/x^0.399990833}; \addlegendentry{order $0.8$}; \end{axis} \end{tikzpicture} \caption{$L^2(Q_T)$ relative error in saturation for the filling case using Brooks and Corey model.}\label{img:convBCfill} \end{figure} \begin{table}[htp!] \centering \begin{tabular}{p{1.6cm}p{1.4cm}p{1.4cm}p{1.2cm}} \hline & $\sharp$ total & $\sharp$ avg & $\sharp$ max \\ \hline Method A & $788$ & $9$ & $32$ \\ Method B & $659$ & $7$ & $31$ \\ \hline\\ \end{tabular} \caption{Newton's iterations for the mesh $200 \times 120$ for the filling case using Brooks and Corey model.} \label{table:newtonBCfill} \end{table} Let us now evaluate the saturation absolute error between results obtained with Method A and Method B. In Figure \ref{img:errFillBC} we plot the absolute-error distribution over the domain at three different times: when water crosses the {interface} between $\Omega_1$ and $\Omega_3$, when cells around this interface are almost saturated and at final time. \newline \begin{minipage}{0.03\textwidth} \vspace{-1.2cm} \includegraphics[height = 3.5cm]{./img/satAbsErrorFillBC/leg.jpg} \end{minipage} \begin{minipage}{0.9\textwidth} \begin{center} \includegraphics[width=4.4cm]{./img/satAbsErrorFillBC/dt20.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorFillBC/dt30.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorFillBC/finalTime.jpg} \captionof{figure}{Saturation absolute error between Method A and Method B for the filling case with Brooks and Corey model at $t\in\{20\cdot 10^3s, 30\cdot 10^3s, 86\cdot 10^3s\}$.} \label{img:errFillBC} \end{center} \end{minipage} \subsubsection{Van Genuchten-Mualem model: filling case} For the filling case with the Van Genuchten model, the convergence error is given in Figure \ref{img:convVGfill}. Both methods exhibit a linear rate of convergence. On the other hand, the error is slightly larger with method A than with method B. The total, average and maximal number of Newton iterations are given in Table \ref{table:newtonVGfill}. \begin{figure}[htp] \centering \begin{tikzpicture} \begin{axis}[grid=major, width=0.7\textwidth, height=0.5\textwidth, xlabel={Mesh}, ylabel={L2 relative error}, ymode=log, xmode=log, xticklabels={50x30, 100x60, 200x120, 400x240}, xtick={1500,6000,24000, 96000}, cycle list name=MyCyclelist1, legend style={at={(0,0)},anchor=south west},nodes={scale=0.6, transform shape}] \addplot table[x=mesh, y=L2]{errorVGfill.txt}; \addlegendentry{Method B }; \addplot table[x=mesh, y=L2]{errorVGfillv2.txt}; \addlegendentry{Method A }; \addplot[blue,dotted,domain=15e+2:96e+3,line width= 1.2pt] {2.3431041/(x)^0.388801424}; \addlegendentry{order $0.78$}; \addplot[red,dotted,domain=15e+2:96e+3,line width= 1.2pt] {1.9330889/(x)^0.383939001}; \addlegendentry{order $0.77$}; \end{axis} \end{tikzpicture} \caption{$L^2(Q_T)$ relative error in saturation for the filling case using the Van Genuchten model.}\label{img:convVGfill} \end{figure} \begin{table}[htp!] \centering \begin{tabular}{p{1.6cm}p{1.4cm}p{1.4cm}p{1.2cm}} \hline & $\sharp$ total & $\sharp$ avg & $\sharp$ max \\ \hline Method A & $959$ & $5$ & $15$ \\ Method B & $782$ & $4$ & $15$ \\ \hline\\ \end{tabular} \caption{Newton's iterations for the mesh $200 \times 120$ for the filling case using the Van Genuchten model.} \label{table:newtonVGfill} \end{table} {Figure~\ref{img:errFillVG} shows the localization of the difference for the numerical solutions provided by the two methods A and B. Unsurprisingly, the difference is located in the neighborhood of the interfaces. Moreover, as suggested by Figures~\ref{img:convBCfill} and~\ref{img:convVGfill}, the influence of the introduction of additional interface unknowns (method A) has a lower impact for van Genuchten-Mualem nonlinearities than for Brook-Corey nonlinearities.} \begin{minipage}{0.03\textwidth} \vspace{-1.2cm} \includegraphics[height=3cm]{./img/satAbsErrorFillVG/leg.jpg} \end{minipage} \begin{minipage}{0.9\textwidth} \begin{center} \includegraphics[width=4.4cm]{./img/satAbsErrorFillVG/dt55.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorFillVG/dt90.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorFillVG/finalTime.jpg} \captionof{figure}{Saturation absolute error between Method A and Method B for the filling case with the Van Genuchten model at $t\in\{27.5\cdot 10^3s, 45\cdot 10^3s, 86\cdot 10^3s\}$.} \label{img:errFillVG} \end{center} \end{minipage} \subsubsection{Van Genuchten-Mualem model: drainage case} For the drainage case with the Van Genuchten model, the convergence error is given in Figure \ref{img:convVGdry}. Both methods exhibit a linear rate of convergence. Moreover, the error is slightly larger with method B than with method A. The total, average and maximal number of Newton iterations are given in Table \ref{table:newtonVGdry}. \begin{figure}[htp] \centering \begin{tikzpicture} \begin{axis}[grid=major, width=0.7\textwidth, height=0.5\textwidth, xlabel={Mesh}, ylabel={L2 relative error}, ymode=log, xmode=log, xticklabels={50x30, 100x60, 200x120, 400x240}, xtick={1500,6000,24000, 96000}, cycle list name=MyCyclelist1, legend style={at={(0,0)},anchor=south west},nodes={scale=0.6, transform shape}] \addplot table[x=mesh, y=L2]{errorVGdry.txt}; \addlegendentry{Method B }; \addplot table[x=mesh, y=L2]{errorVGdryv2.txt}; \addlegendentry{Method A }; \addplot[blue,dotted,domain=15e+2:96e+3,line width= 1.2pt] {0.74683451/(x)^0.637271218}; \addlegendentry{order $1.27$}; \addplot[red,dotted,domain=15e+2:96e+3,line width= 1.2pt] {0.72299086/(x)^0.528197755}; \addlegendentry{order $1.05$}; \end{axis} \end{tikzpicture} \caption{$L^2(Q_T)$ relative error in saturation for the drainage case using the Van Genuchten model.}\label{img:convVGdry} \end{figure} \begin{table}[htp!] \centering \begin{tabular}{p{1.6cm}p{1.4cm}p{1.4cm}p{1.2cm}} \hline & $\sharp$ total & $\sharp$ avg & $\sharp$ max \\ \hline Method A & $3523$ & $2$ & $20$ \\ Method B & $2845$ & $2$ & $29$ \\ \hline\\ \end{tabular} \caption{Newton's iterations for the mesh $200 \times 120$ for the filling case using the Van Genuchten model.} \label{table:newtonVGdry} \end{table} Let us now evaluate the saturation absolute error between results obtained with Method A and Method B. In Figure \ref{img:errDryVG} we plot the absolute-error distribution over the domain at three different times: when the cells line in $\Omega_1$ above the interface between $\Omega_1$ and $\Omega_3$ starts drying, when the cells line in $\Omega_2$ below the interface between $\Omega_3$ and $\Omega_2$ starts drying and at final time. \newline \begin{figure} \centering \includegraphics[width=4.4cm]{./img/satAbsErrorDryVG/dryVG14.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorDryVG/dryVG132.jpg} \includegraphics[width=4.4cm]{./img/satAbsErrorDryVG/dryVGFinal.jpg} \captionof{figure}{Saturation absolute error between Method A and Method B for the drainage case with the van Genuchten-Mualem model at $t\in\{1.12\cdot 10^4s, 10.56\cdot 10^4s, 105\cdot 10^4s\}$.} \label{img:errDryVG} \end{figure} \subsubsection{{Influence of the parameter $\delta$}} Let us now analyze how the thickness of the thin cells employed in Method A affects the accuracy of the solution obtained with this method. We consider the filling and drainage cases along with the Brooks and Corey model and evaluate the relative {$L^2(Q_T)$} error between the solution obtained on the $200\times120$ cells mesh using $\delta\in\{10^{-2}m,10^{-4}m,10^{-6}m\}$ with respect to the reference solution obtained on the $800\times480$ cells mesh using $\delta_{ref}=10^{-6}m$. {As shown in Figure \ref{img:convVirtualCellBC}, the value of $\delta$ does not have a significant influence on the overall error as soon as $\delta$ is small enough. We also observe a moderate influence on the robustness of the non-linear solver for the values considered here.} \begin{figure}[htp] \centering \begin{tikzpicture} \begin{axis}[grid=major, width=0.7\textwidth, height=0.3\textwidth, xlabel={Thickness of thin cells, Method A}, ylabel={L2 relative error}, ymode=log, xmode=log, xticklabels={1e-6, 1e-4, 1e-2}, xtick={1e-6, 1e-4, 1e-2}, cycle list name=MyCyclelist1, legend style={at={(1,0)},anchor=south east},nodes={scale=0.6, transform shape}] \addplot table[x=delta, y=L2]{compErrorBCfill.txt}; \addlegendentry{Filling case}; \addplot table[x=delta, y=L2]{compErrBCdry.txt}; \addlegendentry{drainage case}; \end{axis} \end{tikzpicture} \caption{$L^2(Q_T)$ relative error in saturation as a function of the thickness $\delta$ of the thin cells with Method A using the $200\times120$ cells mesh.}\label{img:convVirtualCellBC} \end{figure} \section{Conclusions and perspectives} {This article aimed at proving that standard upstream mobility finite volume schemes for variable saturated porous media flows still converge in highly heterogeneous contexts without any specific treatment of the rock type discontinuities. The scheme is indeed shown to satisfy some energy stability which provides enough a priori estimates to carry out its numerical analysis. First, the existence of a unique solution to the nonlinear system stemming from the scheme is established thanks to a topological degree argument and from the monotonicity of the scheme. Besides, a rigorous mathematical convergence proof is conducted, based on compactness arguments. No error estimate can then be deduced from our analysis.} {Because of the choice of a backward Euler in time discretization and from the upwind choice of the mobilities, a first order in time and space accuracy is expected in the case of homogeneous computational domains. We show in numerical experiments that without any particular treatment of the interfaces at rock discontinuities, this first order accuracy can be lost, especially in the case of Brooks-Corey nonlinearities. This motivates the introduction of a specific treatment of the interfaces. The approach we propose here is based on the introduction of additional unknowns located in fictitious small additional cells on both sides of each interface. Even though the rigorous convergence proof of this approach is not provided here in the multidimensional setting -- such a proof can for instance be done by writing the scheme with the specific treatment of the interface (method A) as a perturbation of the scheme without any particular treatment of the interface (method B)--, the numerical experiments show that it allows to recover the first order accuracy without having major impacts on the implementation and on the behavior of the numerical solver.} For future researches, we suggest to test the so-called method A on a two-phase flow test {and to compare it to the approaches presented in~\cite{BDMQ_HAL}}. Moreover, in a forthcoming work, we propose two other methods to really impose the pressure continuity condition at interfaces. A comparison between all methods will be shown. {\small