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Approximate inference algorithms for two-layer Bayesian networks AndrewY. Ng Computer Science Division UC Berkeley Berkeley, CA 94720 ang@cs.berkeley.edu Michael I. Jordan Computer Science Division and Department of Statistics UC Berkeley Berkeley, CA 94720 jordan@cs.berkeley.edu Abstract We present a class of approximate inference algorithms for graphical models of the QMR-DT type. We give convergence rates for these algorithms and for the Jaakkola and Jordan (1999) algorithm, and verify these theoretical predictions empirically. We also present empirical results on the difficult QMR-DT network problem, obtaining performance of the new algorithms roughly comparable to the Jaakkola and Jordan algorithm. 1 Introduction The graphical models formalism provides an appealing framework for the design and analysis of network-based learning and inference systems. The formalism endows graphs with a joint probability distribution and interprets most queries of interest as marginal or conditional probabilities under this joint. For a fixed model one is generally interested in the conditional probability of an output given an input (for prediction), or an input conditional on the output (for diagnosis or control). During learning the focus is usually on the likelihood (a marginal probability), on the conditional probability of unobserved nodes given observed nodes (e.g., for an EM or gradient-based algorithm), or on the conditional probability of the parameters given the observed data (in a Bayesian setting). In all of these cases the key computational operation is that of marginalization. There are several methods available for computing marginal probabilities in graphical models, most of which involve some form of message-passing on the graph. Exact methods, while viable in many interesting cases (involving sparse graphs), are infeasible in the dense graphs that we consider in the current paper. A number of approximation methods have evolved to treat such cases; these include search-based methods, loopy propagation, stochastic sampling, and variational methods. Variational methods, the focus of the current paper, have been applied successfully to a number of large-scale inference problems. In particular, Jaakkola and Jordan (1999) developed a variational inference method for the QMR-DT network, a benchmark network involving over 4,000 nodes (see below). The variational method provided accurate approximation to posterior probabilities within a second of computer time. For this difficult A. Y. Ng and M. 1. Jordan 534 inference problem exact methods are entirely infeasible (see below), loopy propagation does not converge to correct posteriors (Murphy, Weiss, & Jordan, 1999), and stochastic sampling methods are slow and unreliable (Jaakkola & Jordan, 1999). A significant step forward in the understanding of variational inference was made by Kearns and Saul (1998), who used large deviation techniques to analyze the convergence rate of a simplified variational inference algorithm. Imposing conditions on the magnitude of the weights in the network, they established a 0 ( Jlog N / N) rate of convergence for the error of their algorithm, where N is the fan-in. In the current paper we utilize techniques similar to those of Kearns and Saul to derive a new set of variational inference algorithms with rates that are faster than 0 ( Jlog N / N). Our techniques also allow us to analyze the convergence rate of the Jaakkola and Jordan (1999) algorithm. We test these algorithms on an idealized problem and verify that our analysis correctly predicts their rates of convergence. We then apply these algorithms to the difficult the QMR-DT network problem. 2 Background 2.1 The QMR-DT network The QMR-DT (Quick Medical Reference, Decision-Theoretic) network is a bipartite graph with approximately 600 top-level nodes di representing diseases and approximately 4000 lower-level nodes Ii representing findings (observed symptoms). All nodes are binaryvalued. Each disease is given a prior probability P(d i = 1), obtained from archival data, and each finding is parameterized as a "noisy-OR" model: P(h = lid) = 1- e-(lio-L:jE"i (lijd j , where 7T'i is the set of parent diseases for finding h and where the parameters obtained from assessments by medical experts (see Shwe, et aI., 1991). Letting Zi = OiQ Oij are + I:jE 1l'i Oijdj , we have the following expression for the likelihood I : (1) where the sum is a sum across the approximately 2600 configurations of the diseases. Note that the second product, a product over the negative findings, factorizes across the diseases dj ; these factors can be absorbed into the priors P (dj ) and have no significant effect on the complexity of inference. It is the positive findings which couple the diseases and prevent the sum from being distributed across the product. Generic exact algorithms such as the junction tree algorithm scale exponentially in the size of the maximal clique in a moralized, triangulated graph. Jaakkola and Jordan (1999) found cliques of more than 150 nodes in QMR-DT; this rules out the junction tree algorithm. Heckerman (1989) discovered a factorization specific to QMR-DT that reduces the complexity substantially; however the resulting algorithm still scales exponentially in the number of positive findings and is only feasible for a small subset of the benchmark cases. I In this expression, the factors P( dj) are the probabilities associated with the (parent-less) disease nodes, the factors (1 - e - Zi) are the probabilities of the (child) finding nodes that are observed to be in their positive state, and the factors e -Zi are the probabilities of the negative findings. The resulting product is the joint probability P(f, d), which is marginalized to obtain the likelihood P(f). Approximate Inference Algorithms for Two-Layer Bayesian Networks 2.2 535 The Jaakkola and Jordan (JJ) algorithm Jaakkola and Jordan (1999) proposed a variational algorithm for approximate inference in the QMR-DT setting. Briefly, their approach is to make use of the following variational inequality: where Ci is a deterministic function of Ai. This inequality holds for arbitrary values of the free "variational parameter" Ai. Substituting these variational upper bounds for the probabilities of positive findings in Eq. (1), one obtains a factorizable upper bound on the likelihood. Because of the factorizability, the sum across diseases can be distributed across the joint probability, yielding a product of sums rather than a sum of products. One then minimizes the resulting expression with respect to the variational parameters to obtain the tightest possible variational bound. 2.3 The Kearns and Saul (KS) algorithm A simplified variational algorithm was proposed by Kearns and Saul (1998), whose main goal was the theoretical analysis of the rates of convergence for variational algorithms. In their approach, the local conditional probability for the finding Ii is approximated by its value at a point a small distance Ci above or below (depending on whether upper or lower bounds are desired) the mean input E[Zi]. This yields a variational algorithm in which the values Ci are the variational parameters to be optimized. Under the assumption that the weights Oij are bounded in magnitude by T / N, where T is a constant and N is the number of parent ("disease") nodes, Kearns and Saul showed that the error in likelihood for their algorithm converges at a rate of O( Vlog N / N). 3 Algorithms based on local expansions Inspired by Kearns and Saul (1998), we describe the design of approximation algorithms for QMR-DT obtained by expansions around the mean input to the finding nodes. Rather than using point approximations as in the Kearns-Saul (KS) algorithm, we make use of Taylor expansions. (See also Plefka (1982), and Barber and van de Laar (1999) for other perturbational techniques.) Consider a generalized QMR-DT architecture in which the noisy-OR model is replaced by a general function 'IjJ( z) : R -t [0, 1] having uniformly bounded derivatives, i.e., \'IjJ(i) (z) \ :::; B i . Define F(Zl, . .. , ZK) = rr~l ('IjJ(zi))fi rr~l (1 - 'IjJ(Zd)l-fi so that the likelihood can be written as (2) P(f) = E{z;}[F(Zl"" ,ZK)]. Also define /-ti = E[Zi] = Ow + 2:7=1 OijP(dj = 1). A simple mean-field-like approximation can be obtained by evaluating F at the mean values P(f) ~ F(/-t1, ... ,/-tK). We refer to this approximation as "MF(O)." (3) Expanding the function F to second order, and defining (i = Zi - /-ti, we have: K P(f) E{fi} rF(jl) + L L 11 =1 1 K Fi1 (J1)(i 1 + 21 K .L L Fid2 (J1)Eh (i2 + 11 =112=1 (4) A. Y. Ng and M I. Jordan 536 where the subscripts on F represent derivatives. Dropping the remainder term and bringing the expectation inside, we have the "MF(2)" approximation: 1 P(f) ~ F(il) + 2 K K LL Fili2(Jt)E[EilEi2] i 1 =1 i2=1 More generally, we obtain a "MF(i)" approximation by carrying out a Taylor expansion to i-th order. 3.1 Analysis In this section, we give two theorems establishing convergence rates for the MF( i) family of algorithms and for the Jaakkola and Jordan algorithm. As in Kearns and Saul (1998), our results are obtained under the assumption that the weights are of magnitude at most O(lIN) (recall that N is the number of disease nodes). For large N, this assumption of "weak interactions" implies that each Zi will be close to its mean value with high probability (by the law of large numbers), and thereby gives justification to the use of local expansions for the probabilities of the findings. Due to space constraints, the detailed proofs of the theorems given in this section are deferred to the long version of this paper, and we will instead only sketch the intuitions for the proofs here. Theorem 1 Let K (the number offindings) be fixed, and suppose IfJ ij I :::; ~ for all i, j for some fixed constantT. Then the absolute error ofthe MF(k) approximation is 0 (N(!:!1)/2) for k odd and 0 (N(k72+1) ) for k even. Proof intuition. First consider the case of odd k. Since IfJ ij I :::; ~, the quantity Ei = Zi J-Li = 2: j fJ ij (dj - E[ dj ]) is like an average of N random variables, and hence has standard deviation on the order 11m. Since MF(k) matches F up to the k-th order derivatives, we find that when we take a Taylor expansion ofMF(k)'s error, the leading non-zero term is the k + 1-st order term, which contains quantities such as 10:+1. Now because Ei has standard deviation on the order 11m, it is unsurprising that E[E:+l] is on the order 1IN(k+l)/2, which gives the error of MF(k) for odd k. For k even, the leading non-zero term in the Taylor expansion of the error is a k + 1-st order term with quantities such as 10:+1. But if we think of Ei as converging (via a central limit theorem effect) to a symmetric distribution, then since symmetric distributions have small odd central moments, E[ 10:+1] would be small. This means that for k even, we may look to the order k + 2 term for the error, which leads to MF(k) having the the same big-O error as MF(k + 1). Note this is also consistent with how MF(O) and MF(l) always give the same estimates and hence have the same absolute error. 0 A theorem may also be proved for the convergence rate of the Jaakkola and Jordan (JJ) algorithm. For simplicity, we state it here only for noisy-OR networks. 2 A closely related result also holds for sigmoid networks with suitably modified assumptions; see the full paper. Theorem 2 Let K befixed, and suppose 'Ij;(z) = 1-e- z is the noisy-ORfunction. Suppose further that 0 :::; fJ ij :::; ~ for all i, j for some fixed constant T, and that J-Li ~ J-Lmin for all i, for some fixed J-Lmin > O. Then the absolute error of the JJ approximation is 0 (~ ). 2Note in any case that 11 can be applied only when 1/J is log-concave, such as in noisy-OR networks (where incidentally all weights are non-negative). 537 Approximate Inference Algorithms for Two-Layer Bayesian Networks The condition of some Pmin lowerbounding the Pi'S ensures that the findings are not too unlikely; for it to hold, it is sufficient that there be bias ("leak") nodes in the network with weights bounded away from zero. Proof intuition. Neglecting negative findings, (which as discussed do not need to be handled variationally,) this result is proved for a "simplified" version of the JJ algorithm, that always chooses the variational parameters so that for each i, the exponential upperbound on '1/J(Zi) is tangent to '1/J at Zi = Pi. (The "normal" version of JJ can have error no worse than this simplified one.) Taking a Tay lor expansion again of the approximation's error, we find that since the upperbound has matched zeroth and first derivatives with F, the error is As discussed in the MF(k) proof outline, a second order term with quantities such as this quantity has expectation on the order 1jN, and hence JJ's error is O(ljN). 0 f.t. To summarize our results in the most useful cases, we find that MF(O) has a convergence rate of O(ljN) , both MF(2) and MF(3) have rates of O(ljN2) , and JJ has a convergence rate of O(ljN). 4 Simulation results 4.1 Artificial networks We carried out a set of simulations that were intended to verify the theoretical results presented in the previous section. We used bipartite noisy-OR networks, with full connectivity between layers and with the weights ()ij chosen uniformly in (0, 2jN). The number N of top-level ("disease") nodes ranged from 10 to 1000. Priors on the disease nodes were chosen uniformly in (0,1). The results are shown in Figure 1 for one and five positive findings (similar results where obtained for additional positive findings). 100 r--_ _ _ _ _~------___. -..._--_._---... --..... _--... -. 10?' -- - - - - - __ _ 10?' ....?. ...... .... . .. . ............... ... .. .... ... e w10' 1t th 10 100 #diseases 1000 10 100 1000 #diseases Figure 1: Absolute error in likelihood (averaged over many randomly generated networks) as a function of the number of disease nodes for various algorithms. The short-dashed lines are the KS upper and lower bounds (these curves overlap in the left panel), the long-dashed line is the 11 algorithm and the solid lines are MF(O), MF(2) and MF(3) (the latter two curves overlap in the right panel). The results are entirely consistent with the theoretical analysis, showing nearly exactly the expected slopes of -112, -1 and -2 on a loglog plot. 3 Moreover, the asymptotic results are 3The anomalous behavior of the KS lower bound in the second panel is due to the fact that the algorithm generally finds a vacuous lower bound of 0 in this case, which yields an error which is essentially constant as a function of the number of diseases. A. Y. Ng and M. 1. Jordan 538 also predictive of overall performance: the MF(2) and MF(3) algorithms perform best in all cases, MF(O) and JJ are roughly equivalent, and KS is the least accurate. 4.2 QMR-DT network We now present results for the QMR-DT network, in particular for the four benchmark CPC cases studied by Jaakkola and Jordan (1999). These cases all have fewer than 20 positive findings; thus it is possible to run the Heckerman (1989) "Quickscore" algorithm to obtain the true likelihood. Case 16 Case 32 '0' 10' ''' ,, '0? 10?:10 ,, , '0?' 10'r. '0 ,, 8'?? .c 8 ~t O?30 ,, ~to'" Qj ~ 1 0~ 10" ? '0- :---- to'" '0- ? 3 . , , 10? ?1 7 0 #Exactly treated findings 3 ? 5 #Exactly treated findings Figure 2: Results for CPC cases 16 and 32, for different numbers of exactly treated findings. The horizontal line is the true likelihood, the dashed line is J1's estimate, and the lower solid line is MF(3)'s estimate. Case 34 lO" ~-~--...___---,.---...,.--~------, Case 46 lO' ? .--~--...___---,.---...,.--~--__, ---10?"'F - - - - - - - - - - --------- ----:;;",.-""==;J lO'"F - - - - -_ _ _ _ _-=",...-=======j 1O~ ,0 ?o~-----!---~.- - 7,'--'-7'---:':lO:------!12 #Exacdy treated findings to...o~-____!_--~.- - 7 ' . - - - ! - ' - --::-----!12' #Exaclly treated findings Figure 3: Results for CPC cases 34 and 46. Same legend as above. In Jaakkola and Jordan (1999), a hybrid methodology was proposed in which only a portion of the findings were treated approximately ; exact methods were used to treat the remaining findings . Using this hybrid methodology, Figures 2 and 3 show the results of running JJ and MF(3) on these four cases. 4 4These experiments were run using a version of the 11 algorithm that optimizes the variational parameters just once without any findings treated exactly, and then uses these fixed values of the parameters thereafter. The order in which findings are chosen to be treated exactly is based on 11's estimates, as described in Jaakkola and Jordan (1999). Missing points in the graphs for cases 16 and Approximate Inference Algorithms for Two-Layer Bayesian Networks 539 The results show the MF algorithm yielding results that are comparable with the JJ algorithm. 5 Conclusions and extension to multilayer networks This paper has presented a class of approximate inference algorithms for graphical models of the QMR-DT type, supplied a theoretical analysis of convergence rates, verified the rates empirically, and presented promising empirical results for the difficult QMR-DT problem. Although the focus of this paper has been two-layer networks, the MF(k) family of algorithms can also be extended to multilayer networks. For example, consider a 3-layer network with nodes bi being parents of nodes d i being parents of nodes Ii. To approximate Pr[J] using (say) MF(2), we first write Pr[J] as an expectation of a function (F) of the Zi'S, and approximate this function via a second-order Taylor expansion. To calculate the expectation of the Taylor approximation, we need to calculate terms in the expansion such as E[d i ], E[didj ] and E[dn When di had no parents, these quantities were easily derived in terms of the disease prior probabilities. Now, they instead depend on the joint distribution of d i and d j , which we use our two-layer version of MF(k), applied to the first two (b i and di ) layers of the network, to approximate. It is important future work to carefully study the performance of this algorithm in the multilayer setting. Acknowledgments We wish to acknowledge the helpful advice of Tommi Jaakkola, Michael Kearns, Kevin Murphy, and Larry Saul. References [1] Barber, D., & van de Laar, P. (1999) Variational cumulant expansions for intractable distributions. Journal of Artificial Intelligence Research, 10,435-455. [2] Heckerman, D. (1989). A tractable inference algorithm for diagnosing multiple diseases. In Proceedings of the Fifth Conference on Uncertainty in Artificial Intelligence. [3] Jaakkola, T. S., & Jordan, M. 1. (1999). Variational probabilistic inference and the QMR-DT network. Journal of Artificial Intelligence Research, 10,291-322. [4] Jordan, M . 1., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1998). An introduction to variational methods for graphical models. In Learning in Graphical Models. Cambridge: MIT Press. [5] Kearns, M. 1., & Saul, L. K. (1998). Large deviation methods for approximate probabilistic inference, with rates of convergence. In G. F. Cooper & S. Moral (Eds.), Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence . San Mateo, CA: Morgan Kaufmann. [6] Murphy, K. P., Weiss, Y, & Jordan, M. 1. (1999). Loopy belief propagation for approximate inference: An empirical study. In Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence. [7] Plefka, T. (1982). Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model. In 1. Phys. A: Math. Gen., 15(6). [8] Shwe, M., Middleton, B., Heckerman, D., Henrion, M., Horvitz, E., Lehmann, H., & Cooper, G. (1991). Probabilistic diagnosis using a reformulation of the INTERNIST-1/QMR knowledge base 1. The probabilistic model and inference algorithms. Methods of Information in Medicine, 30, 241-255. 34 correspond to runs where our implementation of the Quickscore algorithm encountered numerical problems. 

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Correctness of belief propagation in Gaussian graphical models of arbitrary topology Yair Weiss Computer Science Division UC Berkeley, 485 Soda Hall Berkeley, CA 94720-1776 Phone: 510-642-5029 William T. Freeman Mitsubishi Electric Research Lab 201 Broadway Cambridge, MA 02139 Phone: 617-621-7527 yweiss@cs.berkeley.edu freeman @merl.com Abstract Local "belief propagation" rules of the sort proposed by Pearl [15] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation"using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes", whose decoding algorithm is equivalent to loopy belief propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of the performance of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly Gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give sufficient conditions for convergence and show that when belief propagation converges it gives the correct posterior means for all graph topologies, not just networks with a single loop. The related "max-product" belief propagation algorithm finds the maximum posterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the convergence points of the max-product algorithm in loopy networks are maxima over a particular large local neighborhood of the posterior probability. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks. Problems involving probabilistic belief propagation arise in a wide variety of applications, including error correcting codes, speech recognition and medical diagnosis. If the graph is singly connected, there exist local message-passing schemes to calculate the posterior probability of an unobserved variable given the observed variables. Pearl [15] derived such a scheme for singly connected Bayesian networks and showed that this "belief propagation" algorithm is guaranteed to converge to the correct posterior probabilities (or "beliefs"). Several groups have recently reported excellent experimental results by running algorithms 674 Y. Weiss and W T. Freeman equivalent to Pearl's algorithm on networks with loops [8, 13, 6]. Perhaps the most dramatic instance of this performance is for "Turbo code" [2] error correcting codes. These codes have been described as "the most exciting and potentially important development in coding theory in many years" [12] and have recently been shown [10, 11] to utilize an algorithm equivalent to belief propagation in a network with loops. Progress in the analysis of loopy belief propagation has been made for the case of networks with a single loop [17, 18, 4, 1] . For these networks, it can be shown that (1) unless all the compatabilities are deterministic, loopy belief propagation will converge. (2) The difference between the loopy beliefs and the true beliefs is related to the convergence rate of the messages - the faster the convergence the more exact the approximation and (3) If the hidden nodes are binary, then the loopy beliefs and the true beliefs are both maximized by the same assignments, although the confidence in that assignment is wrong for the loopy beliefs. In this paper we analyze belief propagation in graphs of arbitrary topology, for nodes describing jointly Gaussian random variables. We give an exact formula relating the correct marginal posterior probabilities with the ones calculated using loopy belief propagation. We show that if belief propagation converges, then it will give the correct posterior means for all graph topologies, not just networks with a single loop. We show that the covariance estimates will generally be incorrect but present a relationship between the error in the covariance estimates and the convergence speed. For Gaussian or non-Gaussian variables, we show that the "max-product" algorithm, which calculates the MAP estimate in singly connected networks, only converges to points that are maxima over a particular large neighborhood of the posterior probability of loopy networks. 1 Analysis To simplify the notation, we assume the graphical model has been preprocessed into an undirected graphical model with pairwise potentials. Any graphical model can be converted into this form, and running belief propagation on the pairwise graph is equivalent to running belief propagation on the original graph [18]. We assume each node X i has a local observation Yi . In each iteration of belief propagation, each node X i sends a message to each neighboring X j that is based on the messages it received from the other neighbors, its local observation Yl and the pairwise potentials Wij(Xi , Xj) and Wii(Xi, Yi) . We assume the message-passing occurs in parallel. The idea behind the analysis is to build an unwrapped tree. The unwrapped tree is the graphical model which belief propagation is solving exactly when one applies the belief propagation rules in a loopy network [9, 20, 18]. It is constructed by maintaining the same local neighborhood structure as the loopy network but nodes are replicated so there are no loops. The potentials and the observations are replicated from the loopy graph. Figure 1 (a) shows an unwrapped tree for the diamond shaped graph in (b). By construction, the belief at the root node X- I is identical to that at node Xl in the loopy graph after four iterations of belief propagation. Each node has a shaded observed node attached to it, omitted here for clarity. Because the original network represents jointly Gaussian variables, so will the unwrapped tree. Since it is a tree, belief propagation is guaranteed to give the correct answer for the unwrapped graph. We can thus use Gaussian marginalization formulae to calculate the true mean and variances in both the original and the unwrapped networks. In this way, we calculate the accuracy of belief propagation for Gaussian networks of arbitrary topology. We assume that the joint mean is zero (the means can be added-in later). The joint distri- Correctness of Belief Propagation 675 Figure 1: Left: A Markov network with mUltiple loops. Right: The unwrapped network corresponding to this structure. bution of z =(: ) is given by P(z) = ae-!zTVz, where V = (~:: ~::) . It is straightforward to construct the inverse covariance matrix V of the joint Gaussian that describes a given Gaussian graphical model [3]. Writing out the exponent of the joint and completing the square shows that the mean I-' of x, given the observations y, is given by: (1) and the covariance matrix C~IY of x given y is: C~IY = V~-;l. We will denote by C~dY the ith row of C~IY so the marginal posterior variance of Xi given the data is (72 (i) = C~i Iy (i). We will use - for unwrapped quantities. We scan the tree in breadth first order and denote by x the vector of values in the hidden nodes of the tree when so scanned. Simlarly, we denote by y the observed nodes scanned in the same order and Vn , V~y the inverse covariance matrices. Since we are scanning in breadth first order the last nodes are the leaf nodes and we denote by L the number of leaf nodes. By the nature of unwrapping, tL(1) is the mean of the belief at node Xl after t iterations of belief propagation, where t is the number of unwrappings. Similarly 0- 2 (1) = 6~1Iy(1) is the variance of the belief at node Xl after t iterations. Because the data is replicated we can write y = Oy where O(i, j) = 1 if Yi is a replica of Yj and 0 otherwise. Since the potentials W(Xi' Yi) are replicated, we can write V~yO = OV~y. Since the W(Xi, X j) are also replicated and all non-leaf Xi have the same connectivity as the corresponding Xi, we can write V~~O = OVzz + E where E is zero in all but the last L rows. When these relationships between the loopy and unwrapped inverse covariance matrices are substituted into the loopy and unwrapped versions of equation I, one obtains the following expression, true for any iteration [19]: (2) where e is a vector that is zero everywhere but the last L components (corresponding to the leaf nodes). Our choice of the node for the root of the tree is arbitrary, so this applies to all nodes of the loopy network. This formula relates, for any node of a network with loops, the means calculated at each iteration by belief propagation with the true posterior means. Similarly when the relationship between the loopy and unwrapped inverse covariance matrices is substituted into the loopy and unwrapped definitions of C~IY we can relate the Y Weiss and W T Freeman 676 0.5 0.4 ~ 0.3 ~ .~ 0.2 n; ~ 0.1 8 "t:> ? 0 -0.1 -0.2 20 0 40 60 80 100 node Figure 2: The conditional correlation between the root node and all other nodes in the unwrapped tree of Fig. 1 after eight iterations. Potentials were chosen randomly. Nodes are presented in breadth first order so the last elements are the correlations between the root node and the leaf nodes. We show that if this correlation goes to zero, belief propagation converges and the loopy means are exact. Symbols plotted with a star denote correlations with nodes that correspond to the node Xl in the loopy graph. The sum of these correlations gives the correct variance of node Xl while loopy propagation uses only the first correlation. marginalized covariances calculated by belief propagation to the true ones [19]: -2 a (1) = a 2 (1) + CZllyel - - Czt/ye2 (3) where el is a vector that is zero everywhere but the last L components while e2 is equal to 1 for all nodes in the unwrapped tree that are replicas of Xl except for Xl. All other components of e2 are zero, Figure 2 shows Cz1l Y for the diamond network in Fig. 1. We generated random potential functions and observations and calculated the conditional correlations in the unwrapped tree. Note that the conditional correlation decreases with distance in the tree - we are scanning in breadth first order so the last L components correspond to the leaf nodes. As the number of iterations of loopy propagation is increased the size of the unwrapped tree increases and the conditional correlation between the leaf nodes and the root node decreases. From equations 2-3 it is clear that if the conditional correlation between the leaf nodes and the root nodes are zero for all sufficiently large unwrappings then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. In practice the conditional correlations will not actually be equal to zero for any finite unwrapping. In [19] we give a more precise statement: if the conditional correlation of the root node and the leaf nodes decreases rapidly enough then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. We also show sufficient conditions on the potentials III (Xi, X j) for the correlation to decrease rapidly enough: the rate at which the correlation decreases is determined by the ratio of off-diagonal and diagonal components in the quadratic fonn defining the potentials [19]. How wrong will the variances be? The tenn CZllye2 in equation 3 is simply the sum of many components of Cz11y . Figure 2 shows these components. The correct variance is the sum of all the components witHe the belief propagation variance approximates this sum with the first (and dominant) tenn. Whenever there is a positive correlation between the root node and other replicas of Xl the loopy variance is strictly less than the true variance - the loopy estimate is overconfident. 677 Correctness of Belief Propagation ~07 e iDO.6 ., ";;;05 fr ~04 SOR '"~03 0.2 0.1 20 30 40 50 60 iterations (a) (b) Figure 3: (a) 25 x 25 graphical model for simulation. The unobserved nodes (unfilled) were connected to their four nearest neighbors and to an observation node (filled). (b) The error of the estimates of loopy propagation and successive over-relaxation (SOR) as a function of iteration. Note that belief propagation converges much faster than SOR. Note that when the conditional correlation decreases rapidly to zero two things happen. First, the convergence is faster (because CZdyel approaches zero faster) . Second, the approximation error of the variances is smaller (because CZ1 / y e2 is smaller). Thus we have shown, as in the single loop case, quick convergence is correlated with good approximation. 2 Simulations We ran belief propagation on the 25 x 25 2D grid of Fig. 3 a. The joint probability was: (4) where Wij = 0 if nodes Xi, Xj are not neighbors and 0.01 otherwise and Wii was randomly selected to be 0 or 1 for all i with probability of 1 set to 0.2. The observations Yi were chosen randomly. This problem corresponds to an approximation problem from sparse data where only 20% of the points are visible. We found the exact posterior by solving equation 1. We also ran belief propagation and found that when it converged, the calculated means were identical to the true means up to machine precision. Also, as predicted by the theory, the calculated variances were too small - the belief propagation estimate was overconfident. In many applications, the solution of equation 1 by matrix inversion is intractable and iterative methods are used. Figure 3 compares the error in the means as a function of iterations for loopy propagation and successive-over-relaxation (SOR), considered one of the best relaxation methods [16]. Note that after essentially five iterations loopy propagation gives the right answer while SOR requires many more. As expected by the fast convergence, the approximation error in the variances was quite small. The median error was 0.018. For comparison the true variances ranged from 0.01 to 0.94 with a mean of 0.322. Also, the nodes for which the approximation error was worse were indeed the nodes that converged slower. Y. Weiss and W T Freeman 678 3 Discussion Independently, two other groups have recently analyzed special cases of Gaussian graphical models. Frey [7] analyzed the graphical model corresponding to factor analysis and gave conditions for the existence of a stable fixed-point. Rusmevichientong and Van Roy [14] analyzed a graphical model with the topology of turbo decoding but a Gaussian joint density. For this specific graph they gave sufficient conditions for convergence and showed that the means are exact. Our main interest in the Gaussian case is to understand the performance of belief propagation in general networks with multiple loops. We are struck by the similarity of our results for Gaussians in arbitrary networks and the results for single loops of arbitrary distributions [18]. First, in single loop networks with binary nodes, loopy belief at a node and the true belief at a node are maximized by the same assignment while the confidence in that assignment is incorrect. In Gaussian networks with multiple loops, the mean at each node is correct but the confidence around that mean may be incorrect. Second, for both singleloop and Gaussian networks, fast belief propagation convergence correlates with accurate beliefs. Third, in both Gaussians and discrete valued single loop networks, the statistical dependence between root and leaf nodes governs the convergence rate and accuracy. The two models are quite different. Mean field approximations are exact for Gaussian MRFs while they work poorly in sparsely connected discrete networks with a single loop. The results for the Gaussian and single-loop cases lead us to believe that similar results may hold for a larger class of networks. Can our analysis be extended to non-Gaussian distributions? The basic idea applies to arbitrary graphs and arbitrary potentials: belief propagation is performing exact inference on a tree that has the same local neighbor structure as the loopy graph. However, the linear algebra that we used to calculate exact expressions for the error in belief propagation at any iteration holds only for Gaussian variables. We have used a similar approach to analyze the related "max-product" belief propagation algorithm on arbitrary graphs with arbitrary distributions [5] (both discrete and continuous valued nodes). We show that if the max-product algorithm converges, the max-product assignment has greater posterior probability then any assignment in a particular large region around that assignment. While this is a weaker condition than a global maximum, it is much stronger than a simple local maximum of the posterior probability. The sum-product and max-product belief propagation algorithms are fast and parallelizable. Due to the well known hardness of probabilistic inference in graphical models, belief propagation will obviously not work for arbitrary networks and distributions. Nevertheless, a growing body of empirical evidence shows its success in many networks with loops. Our results justify applying belief propagation in certain networks with mUltiple loops. This may enable fast, approximate probabilistic inference in a range of new applications. References [1] S.M. Aji, G.B. Hom, and R.J. McEliece. On the convergence of iterative decoding on graphs with a single cycle. In Proc. 1998 ISIT, 1998. [2] c. Berrou, A. Glavieux, and P. Thitimajshima. Near Shannon limit error-correcting coding and decoding: Turbo codes. In Proc. IEEE International Communications Conference '93, 1993. [3] R. Cowell. Advanced inference in Bayesian networks. In M.1. Jordan, editor, Learning in Graphical Models . MIT Press, 1998. [4] G.D. Forney, F.R. Kschischang, and B. Marcus. Iterative decoding of tail-biting trellisses. preprint presented at 1998 Information Theory Workshop in San Diego, 1998. Correctness of Belief Propagation 679 [5] W. T. Freeman and Y. Weiss. On the fixed points of the max-product algorithm. Technical Report 99-39, MERL, 201 Broadway, Cambridge, MA 02139, 1999. [6] W.T. Freeman and E.C. Pasztor. Learning to estimate scenes from images. In M.S. Kearns, S.A. SoUa, and D.A. Cohn, editors, Adv. Neural Information Processing Systems I I. MIT Press, 1999. [7] B.J. Frey. Turbo factor analysis. In Adv. Neural Information Processing Systems 12. 2000. to appear. [8) Brendan J. Frey. Bayesian Networksfor Pattern Classification, Data Compression and Channel Coding. MIT Press, 1998. [9) R.G . Gallager. Low Density Parity Check Codes. MIT Press, 1963. [10) F. R. Kschischang and B. J. Frey. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Journal on Selected Areas in Communication , 16(2):219-230, 1998. [11] R.J. McEliece, D.J .C. MackKay, and J.F. Cheng. Turbo decoding as as an instance of Pearl's 'belief propagation' algorithm. IEEE Journal on Selected Areas in Communication, 16(2): 140152,1998. [12J R.J. McEliece, E. Rodemich, and J.F. Cheng. The Turbo decision algorithm. In Proc. 33rd Allerton Conference on Communications, Control and Computing, pages 366-379, Monticello, IL, 1995. [I3J K.P. Murphy, Y. Weiss, and M.1. Jordan. Loopy belief propagation for approximate inference : an empirical study. In Proceedings of Uncertainty in AI, 1999. [14] Rusmevichientong P. and Van Roy B. An analysis of Turbo decoding with Gaussian densities. In Adv. Neural Information Processing Systems I2 . 2000. to appear. [15) Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. [16J Gilbert Strang. Introduction to Applied Mathel1Ultics. Wellesley-Cambridge, 1986. [I7J Y. Weiss. Belief propagation and revision in networks with loops. Technical Report 1616, MIT AI lab, 1997. [18J Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, to appear, 2000. [19] Y. Weiss and W. T. Freeman. Loopy propagation gives the correct posterior means for Gaussians. Technical Report UCB.CSD-99-1046, Berkeley Computer Science Dept., 1999. www.cs.berkeley.edu yweiss/. [20J N. Wiberg. Codes and decoding on general graphs. PhD thesis, Department of Electrical Engineering, U. Linkoping, Sweden, 1996. 

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Robust Learning of Chaotic Attractors Rembrandt Bakker* Chemical Reactor Engineering Delft Univ. of Technology r.bakker@stm.tudelft?nl Jaap C. Schouten Marc-Olivier Coppens Chemical Reactor Engineering Chemical Reactor Engineering Eindhoven Univ. of Technology Delft Univ. of Technology J.C.Schouten@tue.nl coppens@stm.tudelft?nl Floris Takens C. Lee Giles Cor M. van den Bleek Dept. Mathematics University of Groningen NEC Research Institute Princeton Nl Chemical Reactor Engineering Delft Univ. of Technology F. Takens@math.rug.nl giles@research.nj.nec.com vdbleek@stm.tudelft?nl Abstract A fundamental problem with the modeling of chaotic time series data is that minimizing short-term prediction errors does not guarantee a match between the reconstructed attractors of model and experiments. We introduce a modeling paradigm that simultaneously learns to short-tenn predict and to locate the outlines of the attractor by a new way of nonlinear principal component analysis. Closed-loop predictions are constrained to stay within these outlines, to prevent divergence from the attractor. Learning is exceptionally fast: parameter estimation for the 1000 sample laser data from the 1991 Santa Fe time series competition took less than a minute on a 166 MHz Pentium PC. 1 Introduction We focus on the following objective: given a set of experimental data and the assumption that it was produced by a deterministic chaotic system, find a set of model equations that will produce a time-series with identical chaotic characteristics, having the same chaotic attractor. The common approach consists oftwo steps: (1) identify a model that makes accurate shorttenn predictions; and (2) generate a long time-series with the model and compare the nonlinear-dynamic characteristics of this time-series with the original, measured time-series. Principe et al. [1] found that in many cases the model can make good short-tenn predictions but does not learn the chaotic attractor. The method would be greatly improved if we could minimize directly the difference between the reconstructed attractors of the model-generated and measured data, instead of minimizing prediction errors. However, we cannot reconstruct the attractor without first having a prediction model. Until now research has focused on how to optimize both step 1 and step 2. For example, it is important to optimize the prediction horizon of the model [2] and to reduce complexity as much as possible. This way it was possible to learn the attractor of the benchmark laser time series data from the 1991 Santa Fe *DelftChemTech, Chemical Reactor Engineering Lab, lulianalaan 136, 2628 BL, Delft, The Netherlands; http://www.cpt.stm.tudelft.nllcptlcre!researchlbakker/. 880 R. Bakker. J. C. Schouten. M.-Q. Coppens. F. Takens. C. L. Giles and C. M. v. d. Bleek time series competition. While training a neural network for this problem, we noticed [3] that the attractor of the model fluctuated from a good match to a complete mismatch from one iteration to another. We were able to circumvent this problem by selecting exactly that model that matches the attractor. However, after carrying out more simulations we found that what we neglected as an unfortunate phenomenon [3] is really a fundamental limitation of current approaches. An important development is the work of Principe et al. [4] who use Kohonen Self Organizing Maps (SOMs) to create a discrete representation of the state space of the system. This creates a partitioning of the input space that becomes an infrastructure for local (linear) model construction. This partitioning enables to verify if the model input is near the original data (i. e. , detect if the model is not extrapolating) without keeping the training data set with the model. We propose a different partitioning of the input space that can be used to (i) learn the outlines of the chaotic attractor by means of a new way of nonlinear Principal Component Analysis (PCA), and (ii) enforce the model never to predict outside these outlines. The nonlinear PCA algorithm is inspired by the work of Kambhatla and Leen [5] on local PCA: they partition the input space and perform local PCA in each region. Unfortunately, this introduces discontinuities between neighboring regions. We resolve them by introducing a hierarchical partitioning algorithm that uses fuzzy boundaries between the regions . This partitioning closely resembles the hierarchical mixtures of experts of Jordan and Jacobs [6]. In Sec. 2 we put forward the fundamental problem that arises when trying to learn a chaotic attractor by creating a short-term prediction model. In Sec. 3 we describe the proposed partitioning algorithm. In Sec. 4 it is outlined how this partitioning can be used to learn the outline of the attractor by defining a potential that measures the distance to the attractor. In Sec. 5 we show modeling results on a toy example, the logistic map, and on a more serious problem, the laser data from the 1991 Santa Fe time series competition. Section 6 concludes. 2 The attractor learning dilemma Imagine an experimental system with a chaotic attractor, and a time-series of noise-free measurements taken from this system. The data is used to fit the parameters of the model =FwC; ' ;_I"" ,Zt-m) whereF is a nonlinear function, wcontains its adjustable parameters and m is a positive constant. What happens if we fit the parameters w by nonlinear least squares regression? Will the model be stable, i.e. , will the closed-loop long term prediction converge to the same attractor as the one represented by the measurements? ;.1 Figure 1 shows the result of a test by Diks et al. [7] that compares the difference between the model and measured attractor. The figure shows that while the neural network is trained to predict chaotic data, the model quickly converges to the measured attractor 20 t-----;r-----r-.....,.1t_:_----.----.-----i-,----i-,----, (S=O), but once in a while, from one 15 .. ...... .. .... ~ ..... .. .. . -,, ..... ----. -. -- ,,... -.. .... . - - iteration to another, the match between , , , , .... ' , , , , the attractors is lost. I : ~ ;;;'10 ............ .. ~ .... . .. ? ? _ _ L __ ? -' ? ? _ ?? ? ? ? _ _ -' _ _ _ _ _ , ??? ? .- -_ .. _-- -..... .. , , , , ? ? ?? ? ?? ? . To understand what causes this . .... _5 _.. ---- -.-----: -_. - .? .,, , , instability, imagine that we try to fit the ~.L ?......... ~I Lo.. : parameters of a model = ii + B Zt 0'-? ?.. ? '. while the real system has a point 8000 attractor, Z= a, where Z is the state of training progress leg iterations] the system and a its attracting value. Figure 1: Diks test monitoring curve for a neural Clearly, measurements taken from this network model trained on data from an system contain no information to experimental chaotic pendulum [3]. ;.1 o Robust Learning of Chaotic Attractors 881 estimate both ii and B. If we fit the model parameters with non-robust linear least squares, B may be assigned any value and if its largest eigenvalue happens to be greater than zero, the model will be unstable! For the linear model this problem has been solved a long time ago with the introduction of singular value decomposition. There still is a need for a nonlinear counterpart of this technique, in particular since we have to work with very flexible models that are designed to fit a wide variety of nonlinear shapes, see for example the early work of Lapedes and Farber [8]. It is already common practice to control the complexity of nonlinear models by pruning or regularization. Unfortunately, these methods do not always solve the attractor learning problem, since there is a good chance that a nonlinear term explains a lot of variance in one part of the state space, while it causes instability of the attractor (without affecting the one-stepahead prediction accuracy) elsewhere. In Secs. 3 and 4 we will introduce a new method for nonlinear principal component analysis that will detect and prevent unstable behavior. 3. The split and fit algorithm The nonlinear regression procedure of this section will form the basis of the nonlinear principal component algorithm in Sec. 4. It consists of (i) a partitioning of the input space, (ii) a local linear model for each region, and (iii) fuzzy boundaries between regions to ensure global smoothness. The partitioning scheme is outlined in Procedure 1: Procedure 1: Partitioning the input space 1) Start with the entire set Z of input data 2) Determine the direction of largest variance of Z: perform a singular value decomposition of Z into the product ULVT and take the eigenvector (column of V) with the largest singular value (on the diagonal of EJ. 3) Split the data in two subsets (to be called: clusters) by creating a plane perpendicular to the direction of largest variance, through the center of gravity of Z. 4) Next, select the cluster with the largest sum squared error to be split next, and recursively apply 2-4 until a stopping criteria is met. Figures 2 and 3 show examples of the partitioning. The disadvantage of dividing regression problems into localized subproblems was pointed out by Jordan and Jacobs [6]: the spread of the data in each region will be much smaller than the spread of the data as a whole, and this will increase the variance of the model parameters. Since we always split perpendicular to the direction of maximum variance, this problem is minimized. The partitioning can be written as a binary tree, with each non-terminal node being a split and each terminal node a cluster. Procedure 2 creates fuzzy boundaries between the clusters. Procedure 2. Creating fuzzy boundaries 1) An input i enters the tree at the top of the partitioning tree. 2) The Euclidean distance to the splitting hyperplane is divided by the bandwidth f3 of the split, and passed through a sigmoidal function with range [0,1]. This results in i's share 0 in the subset on z's side of the splitting plane. The share in the other subset is I-a. 3) The previous step is carried out for all non-terminal nodes of the tree. 882 R. Bakker. J. C. Schouten, M.-Q. Coppens, F. Takens, C. L. Giles and C. M. v. d. Bleek z 4) The membership Pc of to subset (terminal node) c is computed by taking the product of all previously computed shares 0 along the path from the terminal node to the top of the tree. If we would make all parameters adjustable, that is (i) the orientation of the splitting hyperplanes, (ii) the bandwidths f3, and (iii) the local linear model parameters, the above model structure would be identical to the hierarchical mixtures of experts of Jordan and Jacobs [6]. However, we already fixed the hyperplanes and use Procedure 3 to compute the bandwidths: Procedure 3. Computing the Bandwidths 1) The bandwidths of the terminal nodes are taken to be a constant (we use 1.65, the 90% confidence limit of a normal distribution) times the variance of the subset before it was last split, in the direction of the eigenvector of that last split. 2) The other bandwidths do depend on the input z. They are computed by climbing upward in the tree. The bandwidth of node n is computed as a weighted sum between the fJs of its right and left child, by the implicit formula Pn=OL PL uRPR' in which uLand OR depend on Pn? Starting from initial guess Pn=PL if oL>O?5, or else Pn=PR' the formula is solved in a few iterations . This procedure is designed to create large overlap between neighboring regions and almost no overlap between non-neighboring regions. What remains to be fitted is the set of the local linear models. The j-th output of the split&fit model for a given input zp is computed: c Yj,p =L fl; {ii;zp +b/}. where iicand b contain the linear model parameters of subset c, C c=J and C is the number of clusters. We can determine the parameters of all local linear models in one global fit that is linear in the parameters. However, we prefer to locally optimize the parameters for two reasons: (i) it makes it possible to locally control the stability of the attractor and do the principal component analysis of Sec. 4; and (ii) the computing time for a linear regression problem with r regressors scales -O(~). If we would adopt global fitting, r would scale linearly with C and, while growing the model, the regression problem would quickly become intractable. We use the following iterative local fitting procedure instead. Procedure 4. Iterative Local Fitting 1) Initialize a J by N matrix of residuals R to zero, J being the number of outputs and N the number of data. 2) For cluster c, if an estimate for its linear model parameters already exists, forc each input vector p add flcJYv. l,p to the matrix of . residuals, otherwise add flpYj,p to R, Yj.P being the j-th element of the deSIred output vector for sample z p. 3) Least squares fit the linear model parameters of cluster c to predict the current residuals R, and subtract the (new) estimate, fl;Y;,p' from R. 4) Do 2-4 for each cluster and repeat the fitting several times (default: 3). From simulations we found that the above fast optimization method converges to the global minimum if it is repeated many times. Just as with neural network training, it is often better to use early stopping when the prediction error on an independent test set starts to increase. 883 Robust Learning o/Chaotic Attractors 4. Nonlinear Principal Component Analysis To learn a chaotic attractor from a single experimental time-series we use the method ofdelays: the state l consists of m delays taken from the time series. The embedding dimension m must be chosen large enough to ensure that it contains sufficient infonnation for faithful reconstruction of the chaotic attractor, see Takens [9]. Typically, this results in an mdimensional state space with all the measurents covering only a much lower dimensional, but non-linearly shaped, subspace. This creates the danger pointed out in Sec. 2: the stability of the model in directions perpendicular to this low dimensional subspace cannot be guaranteed. With the split & fit algorithm from Sec. 3 we can learn the non-linear shape of the low dimensional subspace, and, if the state of the system escapes from this subspace, we use the algorithm to redirect the state to the nearest point on the subspace. See Malthouse [10] for limitations of existing nonlinear peA approaches. To obtain the low dimensional subspace, we proceed according to Procedure 5. Procedure 5. Learning the Low-dimensional Subspace 1) Augment the output of the model with the m-dimensional statel: the model will learn to predict its own input. 2) In each cluster c, perfonn a singular value decomposition to create a set of m principal directions, sorted in order of decreasing explained variance. The result of this decomposition is also used in step 3 of Procedure 4. 3) Allow the local linear model of each cluster to use no more than mred of these principal directions. 4) Define a potential P to be the squared Euclidian distance between the state l and its prediction by the model. The potential P implicitly defines the lower dimensional subspace: if a state l is on the subspace, P will be zero. P will increase with the distance of l from the subspace. The model has learned to predict its own input with small error, meaning that it has tried to reduce P as much as possible at exactly those points in state space where the training data was sampled. In other words, P will be low if the input l is close to one of the original points in the training data set. From the split&fit algorithm we can analytically compute the gradient dPldl. Since the evaluation of the split&fit model involves a backward (computing the bandwidths) and forward pass (computing memberships), the gradient algorithm involves a forward and backward pass through the tree. The gradient is used to project states that are off the nonlinear subspace onto the subspace -2 -1 2 Figure 2. Projecting two-dimensional data on a onedimensional self-intersecting subspace. The colorscale represents the potential P, white indicates P>0.04 .. 884 R. Bakker, J C. Schouten, M.-O. Coppens, F. Takens, C. L. Giles and C. M. v. d. Bleek in one or a few Newton-Rhapson iterations. Figure 2 illustrates the algorithm for the problem of creating a one-dimensional representation of the number '8'. The training set consists of 136 clean samples, Xl and Fig. 2 shows how a set of 272 noisy inputs is projected by a 48 subset split&fit model onto the one-dimensional subspace. Note that the center of the '8' cannot be well represented by a one-dimensional space. We leave development of an algorithm that -1 o X1 automatically detects the optimum local subspace dimension for future research. Figure 3. Learning the attractor of the twoinput logistic map. The order of creation of the 5. Application Examples splits is indicated. The colorscale represents the potential P, white indicates P>O.05. First we show the nonlinear principal component analysis result for a toy example, the logistic map Zt+l =4z t (1-Zt). If we use a model Zt+l =Fw(zt) , where the prediction only depends on one previous output, there is no lower dimensional space to which the attractor is confined. However, if we allow the output to depend on more than a single delay, we create a possibility for unstable behavior. Figure 3 shows how well the split&fit algorithm learns the one-dimensional shape of the attractor after creating only five regions. The parabola is slightly deformed (seen from the white lines perpendicular to the attractor), but this may be solved by increasing the number of splits. Next we look at the laser data. The complex behavior of chaotic systems is caused by an interplay of destabilizing and stabilizing forces: the destabilizing forces make nearby points in state space diverge, while the stabilizing forces keep the state of the system bounded. This process, known as 'stretching and folding', results in the attractor of the system: the set of points that the state of the system will visit after all transients have died out. In the case of the laser data this behavior is clear cut: destabilizing forces make the signal grow exponentially until the increasing amplitude triggers a collapse that reinitiates the sequence. We have seen in neural network based models [3] and in this study that it is very hard for the models to cope with the sudden collapses. Without the nonlinear subspace correction of Sec. 4, most of the (a) train data 0.4 , - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - , (b) 1000 time Figure 4. Laser data from the Santa Fe time series competition. The 1000 sample train data set is followed by iterated prediction of the model (a). After every prediction a correction is made to keep P (see Sec. 4) small. Plot (b) shows P before this correction. Robust Learning of Chaotic Attractors 885 models we tested grow without bounds after one or more rise and collapse sequences. That is not very surprising - the training data set contains only three examples of a collapse. Figure 4 shows how this is solved with the subspace correction: every time the model is about to grow to infinity, a high potential P is detected (depicted in Fig. 3b) and the state of the system is directed to the nearest point on the subspace as learned from the nonlinear principal component analysis. After some trial and error, we selected an embedding dimension m of 12 and a reduced dimension mred of 4. The split&fit model starts with a single dataset, and was grown until 48 subsets. At that point, the error on the 1000 sample train set was still decreasing rapidly but the error on an independent 1000 sample test set increased. We compared the reconstructed attractors of the model and measurements, using 9000 samples of closed-loop generated and 9000 samples of measured data. No significant difference between the two could be detected by the Diles test [7]. 6. Conclusions We present an algorithm that robustly models chaotic attractors. It simultaneously learns (1) to make accurate short term predictions; and (2) the outlines of the attractor. In closed-loop prediction mode, the state of the system is corrected after every prediction, to stay within these outlines. The algorithm is very fast, since the main computation is to least squares fit a set of local linear models. In our implementation the largest matrix to be stored is N by C, N being the number of data and C the number of clusters. We see many applications other than attractor learning: the split&fit algorithm can be used as a fast learning alternative to neural networks and the new form of nonlinear peA will be useful for data reduction and object recognition. We envisage to apply the technique to a wide range of applications, from the control and modeling of chaos in fluid dynamics to problems in finance and biology to fluid dynamics. Acknowledgements This work is supported by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for Scientific Research (NWO). References [1] 1.e. Principe, A. Rathie, and 1.M. Kuo. "Prediction of Chaotic Time Series with Neural Networks and the Issue of Dynamic Modeling". Int. J. Bifurcation and Chaos. 2, 1992. P 989. [2] 1.M. Kuo. and 1.C. Principe. "Reconstructed Dynamics and Chaotic Signal Modeling". In Proc. IEEE Int'l Conf. Neural Networks, 5, 1994, p 3l31. [3] R Bakker, J.C. Schouten, e.L. Giles. F. Takens, e.M. van den Bleek, "Learning Chaotic Attractors by Neural Networks", submitted. [4] 1.e. Principe, L. Wang, MA Motter, "Local Dynamic Modeling with Self-Organizing Maps and Applications to Nonlinear System Identification and Control" .Proc. IEEE. 86(11). 1998. [5] N. Kambhatla, T.K. Leen. "Dimension Reduction by Local PCA", Neural Computation. 9,1997. p. 1493 [6] M.I. Jordan, RA. Jacobs. "Hierarchical Mixtures of Experts and the EM Algorithm". Neural Compution. 6. 1994. p. 181. [7] e. Diks, W.R. van Zwet. F. Takens. and 1. de Goede, "Detecting differences between delay vector distributions", PhYSical Review E. 53, 1996. p. 2169. [8] A. Lapedes. R Farber. "Nonlinear Signal Processing Using Neural Networks: Prediction and System Modelling". Los Alamos Technical Report LA-UR-87-2662. [9] F. Takens, "Detecting strange attractors in turbulence", Lecture notes in Mathematics, 898, 1981, p. 365. [10] E.C. Malthouse. "Limitations of Nonlinear PCA as performed with Generic Neural Networks. IEEE Trans. Neural Networks. 9(1). 1998. p. 165. 

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Correctness of belief propagation in Gaussian graphical models of arbitrary topology Yair Weiss Computer Science Division UC Berkeley, 485 Soda Hall Berkeley, CA 94720-1776 Phone: 510-642-5029 William T. Freeman Mitsubishi Electric Research Lab 201 Broadway Cambridge, MA 02139 Phone: 617-621-7527 yweiss@cs.berkeley.edu freeman @merl.com Abstract Local "belief propagation" rules of the sort proposed by Pearl [15] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation"using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes", whose decoding algorithm is equivalent to loopy belief propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of the performance of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly Gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give sufficient conditions for convergence and show that when belief propagation converges it gives the correct posterior means for all graph topologies, not just networks with a single loop. The related "max-product" belief propagation algorithm finds the maximum posterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the convergence points of the max-product algorithm in loopy networks are maxima over a particular large local neighborhood of the posterior probability. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks. Problems involving probabilistic belief propagation arise in a wide variety of applications, including error correcting codes, speech recognition and medical diagnosis. If the graph is singly connected, there exist local message-passing schemes to calculate the posterior probability of an unobserved variable given the observed variables. Pearl [15] derived such a scheme for singly connected Bayesian networks and showed that this "belief propagation" algorithm is guaranteed to converge to the correct posterior probabilities (or "beliefs"). Several groups have recently reported excellent experimental results by running algorithms 674 Y. Weiss and W T. Freeman equivalent to Pearl's algorithm on networks with loops [8, 13, 6]. Perhaps the most dramatic instance of this performance is for "Turbo code" [2] error correcting codes. These codes have been described as "the most exciting and potentially important development in coding theory in many years" [12] and have recently been shown [10, 11] to utilize an algorithm equivalent to belief propagation in a network with loops. Progress in the analysis of loopy belief propagation has been made for the case of networks with a single loop [17, 18, 4, 1] . For these networks, it can be shown that (1) unless all the compatabilities are deterministic, loopy belief propagation will converge. (2) The difference between the loopy beliefs and the true beliefs is related to the convergence rate of the messages - the faster the convergence the more exact the approximation and (3) If the hidden nodes are binary, then the loopy beliefs and the true beliefs are both maximized by the same assignments, although the confidence in that assignment is wrong for the loopy beliefs. In this paper we analyze belief propagation in graphs of arbitrary topology, for nodes describing jointly Gaussian random variables. We give an exact formula relating the correct marginal posterior probabilities with the ones calculated using loopy belief propagation. We show that if belief propagation converges, then it will give the correct posterior means for all graph topologies, not just networks with a single loop. We show that the covariance estimates will generally be incorrect but present a relationship between the error in the covariance estimates and the convergence speed. For Gaussian or non-Gaussian variables, we show that the "max-product" algorithm, which calculates the MAP estimate in singly connected networks, only converges to points that are maxima over a particular large neighborhood of the posterior probability of loopy networks. 1 Analysis To simplify the notation, we assume the graphical model has been preprocessed into an undirected graphical model with pairwise potentials. Any graphical model can be converted into this form, and running belief propagation on the pairwise graph is equivalent to running belief propagation on the original graph [18]. We assume each node X i has a local observation Yi . In each iteration of belief propagation, each node X i sends a message to each neighboring X j that is based on the messages it received from the other neighbors, its local observation Yl and the pairwise potentials Wij(Xi , Xj) and Wii(Xi, Yi) . We assume the message-passing occurs in parallel. The idea behind the analysis is to build an unwrapped tree. The unwrapped tree is the graphical model which belief propagation is solving exactly when one applies the belief propagation rules in a loopy network [9, 20, 18]. It is constructed by maintaining the same local neighborhood structure as the loopy network but nodes are replicated so there are no loops. The potentials and the observations are replicated from the loopy graph. Figure 1 (a) shows an unwrapped tree for the diamond shaped graph in (b). By construction, the belief at the root node X- I is identical to that at node Xl in the loopy graph after four iterations of belief propagation. Each node has a shaded observed node attached to it, omitted here for clarity. Because the original network represents jointly Gaussian variables, so will the unwrapped tree. Since it is a tree, belief propagation is guaranteed to give the correct answer for the unwrapped graph. We can thus use Gaussian marginalization formulae to calculate the true mean and variances in both the original and the unwrapped networks. In this way, we calculate the accuracy of belief propagation for Gaussian networks of arbitrary topology. We assume that the joint mean is zero (the means can be added-in later). The joint distri- Correctness of Belief Propagation 675 Figure 1: Left: A Markov network with mUltiple loops. Right: The unwrapped network corresponding to this structure. bution of z =(: ) is given by P(z) = ae-!zTVz, where V = (~:: ~::) . It is straightforward to construct the inverse covariance matrix V of the joint Gaussian that describes a given Gaussian graphical model [3]. Writing out the exponent of the joint and completing the square shows that the mean I-' of x, given the observations y, is given by: (1) and the covariance matrix C~IY of x given y is: C~IY = V~-;l. We will denote by C~dY the ith row of C~IY so the marginal posterior variance of Xi given the data is (72 (i) = C~i Iy (i). We will use - for unwrapped quantities. We scan the tree in breadth first order and denote by x the vector of values in the hidden nodes of the tree when so scanned. Simlarly, we denote by y the observed nodes scanned in the same order and Vn , V~y the inverse covariance matrices. Since we are scanning in breadth first order the last nodes are the leaf nodes and we denote by L the number of leaf nodes. By the nature of unwrapping, tL(1) is the mean of the belief at node Xl after t iterations of belief propagation, where t is the number of unwrappings. Similarly 0- 2 (1) = 6~1Iy(1) is the variance of the belief at node Xl after t iterations. Because the data is replicated we can write y = Oy where O(i, j) = 1 if Yi is a replica of Yj and 0 otherwise. Since the potentials W(Xi' Yi) are replicated, we can write V~yO = OV~y. Since the W(Xi, X j) are also replicated and all non-leaf Xi have the same connectivity as the corresponding Xi, we can write V~~O = OVzz + E where E is zero in all but the last L rows. When these relationships between the loopy and unwrapped inverse covariance matrices are substituted into the loopy and unwrapped versions of equation I, one obtains the following expression, true for any iteration [19]: (2) where e is a vector that is zero everywhere but the last L components (corresponding to the leaf nodes). Our choice of the node for the root of the tree is arbitrary, so this applies to all nodes of the loopy network. This formula relates, for any node of a network with loops, the means calculated at each iteration by belief propagation with the true posterior means. Similarly when the relationship between the loopy and unwrapped inverse covariance matrices is substituted into the loopy and unwrapped definitions of C~IY we can relate the Y Weiss and W T Freeman 676 0.5 0.4 ~ 0.3 ~ .~ 0.2 n; ~ 0.1 8 "t:> ? 0 -0.1 -0.2 20 0 40 60 80 100 node Figure 2: The conditional correlation between the root node and all other nodes in the unwrapped tree of Fig. 1 after eight iterations. Potentials were chosen randomly. Nodes are presented in breadth first order so the last elements are the correlations between the root node and the leaf nodes. We show that if this correlation goes to zero, belief propagation converges and the loopy means are exact. Symbols plotted with a star denote correlations with nodes that correspond to the node Xl in the loopy graph. The sum of these correlations gives the correct variance of node Xl while loopy propagation uses only the first correlation. marginalized covariances calculated by belief propagation to the true ones [19]: -2 a (1) = a 2 (1) + CZllyel - - Czt/ye2 (3) where el is a vector that is zero everywhere but the last L components while e2 is equal to 1 for all nodes in the unwrapped tree that are replicas of Xl except for Xl. All other components of e2 are zero, Figure 2 shows Cz1l Y for the diamond network in Fig. 1. We generated random potential functions and observations and calculated the conditional correlations in the unwrapped tree. Note that the conditional correlation decreases with distance in the tree - we are scanning in breadth first order so the last L components correspond to the leaf nodes. As the number of iterations of loopy propagation is increased the size of the unwrapped tree increases and the conditional correlation between the leaf nodes and the root node decreases. From equations 2-3 it is clear that if the conditional correlation between the leaf nodes and the root nodes are zero for all sufficiently large unwrappings then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. In practice the conditional correlations will not actually be equal to zero for any finite unwrapping. In [19] we give a more precise statement: if the conditional correlation of the root node and the leaf nodes decreases rapidly enough then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. We also show sufficient conditions on the potentials III (Xi, X j) for the correlation to decrease rapidly enough: the rate at which the correlation decreases is determined by the ratio of off-diagonal and diagonal components in the quadratic fonn defining the potentials [19]. How wrong will the variances be? The tenn CZllye2 in equation 3 is simply the sum of many components of Cz11y . Figure 2 shows these components. The correct variance is the sum of all the components witHe the belief propagation variance approximates this sum with the first (and dominant) tenn. Whenever there is a positive correlation between the root node and other replicas of Xl the loopy variance is strictly less than the true variance - the loopy estimate is overconfident. 677 Correctness of Belief Propagation ~07 e iDO.6 ., ";;;05 fr ~04 SOR '"~03 0.2 0.1 20 30 40 50 60 iterations (a) (b) Figure 3: (a) 25 x 25 graphical model for simulation. The unobserved nodes (unfilled) were connected to their four nearest neighbors and to an observation node (filled). (b) The error of the estimates of loopy propagation and successive over-relaxation (SOR) as a function of iteration. Note that belief propagation converges much faster than SOR. Note that when the conditional correlation decreases rapidly to zero two things happen. First, the convergence is faster (because CZdyel approaches zero faster) . Second, the approximation error of the variances is smaller (because CZ1 / y e2 is smaller). Thus we have shown, as in the single loop case, quick convergence is correlated with good approximation. 2 Simulations We ran belief propagation on the 25 x 25 2D grid of Fig. 3 a. The joint probability was: (4) where Wij = 0 if nodes Xi, Xj are not neighbors and 0.01 otherwise and Wii was randomly selected to be 0 or 1 for all i with probability of 1 set to 0.2. The observations Yi were chosen randomly. This problem corresponds to an approximation problem from sparse data where only 20% of the points are visible. We found the exact posterior by solving equation 1. We also ran belief propagation and found that when it converged, the calculated means were identical to the true means up to machine precision. Also, as predicted by the theory, the calculated variances were too small - the belief propagation estimate was overconfident. In many applications, the solution of equation 1 by matrix inversion is intractable and iterative methods are used. Figure 3 compares the error in the means as a function of iterations for loopy propagation and successive-over-relaxation (SOR), considered one of the best relaxation methods [16]. Note that after essentially five iterations loopy propagation gives the right answer while SOR requires many more. As expected by the fast convergence, the approximation error in the variances was quite small. The median error was 0.018. For comparison the true variances ranged from 0.01 to 0.94 with a mean of 0.322. Also, the nodes for which the approximation error was worse were indeed the nodes that converged slower. Y. Weiss and W T Freeman 678 3 Discussion Independently, two other groups have recently analyzed special cases of Gaussian graphical models. Frey [7] analyzed the graphical model corresponding to factor analysis and gave conditions for the existence of a stable fixed-point. Rusmevichientong and Van Roy [14] analyzed a graphical model with the topology of turbo decoding but a Gaussian joint density. For this specific graph they gave sufficient conditions for convergence and showed that the means are exact. Our main interest in the Gaussian case is to understand the performance of belief propagation in general networks with multiple loops. We are struck by the similarity of our results for Gaussians in arbitrary networks and the results for single loops of arbitrary distributions [18]. First, in single loop networks with binary nodes, loopy belief at a node and the true belief at a node are maximized by the same assignment while the confidence in that assignment is incorrect. In Gaussian networks with multiple loops, the mean at each node is correct but the confidence around that mean may be incorrect. Second, for both singleloop and Gaussian networks, fast belief propagation convergence correlates with accurate beliefs. Third, in both Gaussians and discrete valued single loop networks, the statistical dependence between root and leaf nodes governs the convergence rate and accuracy. The two models are quite different. Mean field approximations are exact for Gaussian MRFs while they work poorly in sparsely connected discrete networks with a single loop. The results for the Gaussian and single-loop cases lead us to believe that similar results may hold for a larger class of networks. Can our analysis be extended to non-Gaussian distributions? The basic idea applies to arbitrary graphs and arbitrary potentials: belief propagation is performing exact inference on a tree that has the same local neighbor structure as the loopy graph. However, the linear algebra that we used to calculate exact expressions for the error in belief propagation at any iteration holds only for Gaussian variables. We have used a similar approach to analyze the related "max-product" belief propagation algorithm on arbitrary graphs with arbitrary distributions [5] (both discrete and continuous valued nodes). We show that if the max-product algorithm converges, the max-product assignment has greater posterior probability then any assignment in a particular large region around that assignment. While this is a weaker condition than a global maximum, it is much stronger than a simple local maximum of the posterior probability. The sum-product and max-product belief propagation algorithms are fast and parallelizable. Due to the well known hardness of probabilistic inference in graphical models, belief propagation will obviously not work for arbitrary networks and distributions. Nevertheless, a growing body of empirical evidence shows its success in many networks with loops. Our results justify applying belief propagation in certain networks with mUltiple loops. This may enable fast, approximate probabilistic inference in a range of new applications. References [1] S.M. Aji, G.B. Hom, and R.J. McEliece. On the convergence of iterative decoding on graphs with a single cycle. In Proc. 1998 ISIT, 1998. [2] c. Berrou, A. Glavieux, and P. Thitimajshima. Near Shannon limit error-correcting coding and decoding: Turbo codes. In Proc. IEEE International Communications Conference '93, 1993. [3] R. Cowell. Advanced inference in Bayesian networks. In M.1. Jordan, editor, Learning in Graphical Models . MIT Press, 1998. [4] G.D. Forney, F.R. Kschischang, and B. Marcus. Iterative decoding of tail-biting trellisses. preprint presented at 1998 Information Theory Workshop in San Diego, 1998. Correctness of Belief Propagation 679 [5] W. T. Freeman and Y. Weiss. On the fixed points of the max-product algorithm. Technical Report 99-39, MERL, 201 Broadway, Cambridge, MA 02139, 1999. [6] W.T. Freeman and E.C. Pasztor. Learning to estimate scenes from images. In M.S. Kearns, S.A. SoUa, and D.A. Cohn, editors, Adv. Neural Information Processing Systems I I. MIT Press, 1999. [7] B.J. Frey. Turbo factor analysis. In Adv. Neural Information Processing Systems 12. 2000. to appear. [8) Brendan J. Frey. Bayesian Networksfor Pattern Classification, Data Compression and Channel Coding. MIT Press, 1998. [9) R.G . Gallager. Low Density Parity Check Codes. MIT Press, 1963. [10) F. R. Kschischang and B. J. Frey. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Journal on Selected Areas in Communication , 16(2):219-230, 1998. [11] R.J. McEliece, D.J .C. MackKay, and J.F. Cheng. Turbo decoding as as an instance of Pearl's 'belief propagation' algorithm. IEEE Journal on Selected Areas in Communication, 16(2): 140152,1998. [12J R.J. McEliece, E. Rodemich, and J.F. Cheng. The Turbo decision algorithm. In Proc. 33rd Allerton Conference on Communications, Control and Computing, pages 366-379, Monticello, IL, 1995. [I3J K.P. Murphy, Y. Weiss, and M.1. Jordan. Loopy belief propagation for approximate inference : an empirical study. In Proceedings of Uncertainty in AI, 1999. [14] Rusmevichientong P. and Van Roy B. An analysis of Turbo decoding with Gaussian densities. In Adv. Neural Information Processing Systems I2 . 2000. to appear. [15) Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. [16J Gilbert Strang. Introduction to Applied Mathel1Ultics. Wellesley-Cambridge, 1986. [I7J Y. Weiss. Belief propagation and revision in networks with loops. Technical Report 1616, MIT AI lab, 1997. [18J Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, to appear, 2000. [19] Y. Weiss and W. T. Freeman. Loopy propagation gives the correct posterior means for Gaussians. Technical Report UCB.CSD-99-1046, Berkeley Computer Science Dept., 1999. www.cs.berkeley.edu yweiss/. [20J N. Wiberg. Codes and decoding on general graphs. PhD thesis, Department of Electrical Engineering, U. Linkoping, Sweden, 1996. Gaussian Fields for Approximate Inference in Layered Sigmoid Belief Networks David Barber'" Stichting Neurale Netwerken Medical Physics and Biophysics Nijmegen University, The Netherlands barberdOaston.ac.uk Peter Sollich Department of Mathematics King's College, University of London London WC2R 2LS, U.K. peter.sollichOkcl.ac.uk Abstract Layered Sigmoid Belief Networks are directed graphical models in which the local conditional probabilities are parameterised by weighted sums of parental states. Learning and inference in such networks are generally intractable, and approximations need to be considered. Progress in learning these networks has been made by using variational procedures. We demonstrate, however, that variational procedures can be inappropriate for the equally important issue of inference - that is,? calculating marginals of the network. We introduce an alternative procedure, based on assuming that the weighted input to a node is approximately Gaussian distributed. Our approach goes beyond previous Gaussian field assumptions in that we take into account correlations between parents of nodes. This procedure is specialized for calculating marginals and is significantly faster and simpler than the variational procedure. 1 Introduction Layered Sigmoid Belief Networks [1] are directed graphical models [2] in which the local conditional probabilities are parameterised by weighted sums of parental states, see fig ( 1). This is a graphical representation of a distribution over a set of binary variables Si E {a, I}. Typically, one supposes that the states of the nodes at the bottom of the network are generated by states in previous layers. Whilst, in principle, there is no restriction on the number of nodes in any layer, typically, one considers structures similar to the "fan out" in fig(l) in which higher level layers provide an "explanation" for patterns generated in lower layers. Such graphical models are attractive since they correspond to layers of information processors, of potentially increasing complexity. Unfortunately, learning and inference in such networks is generally intractable, and approximations need to be considered. Progress in learning has been made by using variational procedures [3,4, 5]. However, another crucial aspect remains inference [2]. That is, given some evidence (or none), calculate the marginal of a variable, conditional on this evidence. This assumes that we have found a suitable network from some learning procedure, and now wish ?Present Address: NCRG, Aston University, Birmingham B4 7ET, U.K. 394 D. Barber and P. Sollich to query this network. Whilst the variational procedure is attractive for learning, since it generally provides a bound on the likelihood of the visible units, we demonstrate that it may not always be equally appropriate for the inference problem. A directed graphical model defines a distribution over a set of variables s = (S1 ... sn) that factorises into the local conditional distributions, n p(S1 . .. sn) = IIp(silll'i) (1) i=1 where lI'i denotes the parent nodes of node i . In a layered network, these are the nodes in the proceeding layer that feed into node i. In a sigmoid belief network the local probabilities are defined as P (s; = ll~;) = " ( ~ W;jSj + 0;) Figure 1: A Layered Sigmoid Belief Network =" (h;) (2) where the "field" at node i is defined as hi = 2:j WijSj + fh and er(h) = 1/(1 + e- h ). Wij is the strength of the connection between node i and its parent node j; if j is not a parent of i we set Wij O. Oi is a bias term that gives a parent-independent bias to the state of node i . = We are interested in inference - in particular, calculating marginals of the network for cases with and without evidential nodes. In section (2) we describe how to approximate the quantities p(Si 1) and discuss in section (2.1) why our method can improve on the standard variational mean field theory. Conditional marginals, such as p(Si = IISj = 1, Sk = 0) are considered in section (3). = 2 Gaussian Field Distributions Under the 0/1 coding for the variables Si, the mean of a variable, mi is given by the probability that it is in state 1. Using the fact from (2) that the local conditional distribution of node i is dependent on its parents only through its field hi, we have (3) where we use the notation ?(-)p to denote an average with respect to the distribution p. If there are many parents of node i, a reasonable assumption is that the distribution of the field hi will be Gaussian, p(hi ) ~ N (J,Li,er[). Under this Gaussian Field (GF) assumption, we need to work out the mean and variance, which are given by (4) j j err = ((Llh i )2) = L WijWikRjk (5) j,k where Rjk = (LlSjLlsk). We use the notation Ll (-) == (-) - ?(.) . The diagonal terms of the node covariance matrix are ~i = mi (1- mi)' In contrast to previous studies, we include off diagonal terms in the calculation of R [4] . From Gaussian Fields for Approximate Inference 395 (5) we only need to find correlations between parents i and j of a node. These are easy to calculate in the layered networks that we are considering, because neither i nor j is a descendant of the other: Rjj = p(Sj = 1, Sj = 1) = J (6) mjmj p(Si = Ilhj)p(Sj = Ilhj)p(hj, hj)dh - mimj = (0" (hd 0" (h j ) P (7) (8) (h J, h) - mjmj J Assuming that the joint distribution p( h j , h j ) is Gaussian, we again need its mean and covariance, given by ~ij = (D.hjD.hj) = L WjkWjl (D.skD.SI) = L kl WikWjlRkl (10) kl Under this scheme, we have a closed set of equations, (4,5,8,10) for the means mj and covariance matrix Rij which can be solved by forward propagation of the equations. That is, we start from nodes without parents, and then consider the next layer of nodes, repeating the procedure until a full sweep through the network has been completed. The one and two dimensional field averages, equations (3) and (8), are computed using Gaussian Quadrature. This results in an extremely fast procedure for approximating the marginals mi, requiring only a single sweep through the network. Our approach is related to that of [6] by the common motivating assumption that each node has a large number of parents. This is used in [6] to obtain actual bounds on quantities of interest such as joint marginals. Our approach does not give bounds. Its advantage, however, is that it allows fluctuations in the fields hi, which are effectively excluded in [6] by the assumed scaling of the weights Wij with the number of parents per node. 2.1 Relation to Variational Mean Field Theory In the variational approach, one fits a tractable approximating distribution Q to the SBN. Taking Q factorised, Q(s) = Dj m:' (1 - md l - 3 ? we have the bound In p (Sl ... sn) 2: L {-mj In mj - (1 - md In (1 - md} i The final term in (11) causes some difficulty even in the case in which Q is a factorised model. Formally, this is because this term does not have the same graphical structure as the tractable model Q. One way around around this difficulty is to employ a further bound, with associated variational parameters [7]. Another approach is to make the Gaussian assumption for the field hi as in section (2). Because Q is factorised, corresponding to a diagonal correlation matrix R, this gives [4] (12) 396 D. Barber and P Sollich where Pi = ~j Wijmj + Oi and (1[ = ~j w[jmj(l - mj). Note that this is a one dimensional integral of a smooth function. In contrast to [4] we therefore evaluate this quantity using Gaussian Quadrature. This has the advantage that no extra variational parameters need to be introduced. Technically, the assumption of a Gaussian field distribution means that (11) is no longer a bound. Nevertheless, in practice it is found that this has little effect on the quality of the resulting solution. In our implementation of the variational approach, we find the optimal parameters mi by maximising the above equation for each component mi separately, cycling through the nodes until the parameters mi do not change by more than 10- 1 This is repeated 5 times, and the solution with the highest bound score is chosen. Note that these equations cannot be solved by forward propagation alone since the final term contains contributions from all the nodes in the network. This is in contrast to the GF approach of section (2) . Finding appropriate parameters mi by the variational approach is therefore rather slower than using the GF method. ?. In arriving at the above equations, we have made two assumptions. The first is that the intractable distribution is well approximated by a factorised model. The second is that the field distribution is Gaussian. The first step is necessary in order to obtain a bound on the likelihood of the model (although this is slightly compromised by the Gaussian field assumption). In the GF approach we dispense with this assumption of an effectively factorised network (partially because if we are only interested in inference, a bound on the model likelihood is less relevant). The GF method may therefore prove useful for a broader class of networks than the variational approach . 2.2 Results for unconditional marginals We compared three procedures for estimating the conditional values p(Si = 1) for all the nodes in the network, namely the variational theory, as described in section (2.1), the diagonal Gaussian field theory, and the non-diagonal Gaussian field theory which includes correlation effects between parents. Results for small weight values Wij are shown in fig(2). In this case, all three methods perform reasonably well, although there is a significant improvement in using the GF methods over the variational procedure; parental correlations are not important (compare figs(2b) and (2c)) . In fig(3) the weights and biases are chosen such that the exact mean variables mi are roughly 0.5 with non-trivial correlation effects between parents. Note that the variational mean field theory now provides a poor solution, whereas the GF methods are relatively accurate. The effect of using the non-diagonal R terms is beneficial, although not dramatically so. 3 Calculating Conditional Marginals We consider now how to calculate conditional marginals, given some evidential nodes. (In contrast to [6], any set of nodes in the network, not just output nodes, can be considered evidential.) We write the evidence in the following manner E = {SCi = SCi' . . . Sc" = SC,.} = {ECl ... Ec,.} The quantities that we are interested in are conditional marginals which, from Bayes rule are related to the joint distribution by P (Si = liE) = = P (Si 1, E) (13) P (Si = 0, E) + P (Si = 1, E) That is, provided that we have a procedure for estimating joint marginals, we can obtain conditional marginals too. Without loss of generality, we therefore consider Gaussian Fields for Approximate Inference Em>ruoing1_ model 1ft 2Or--~ 397 Em>r using Ga_ Fiaid. Ooagonal ooyariance cowuiance O<Xll (a) Mean error = 0.0377 (b) Mean error = 0.0018 (c) Mean error 001 = 0.0017 Figure 2: Error in approximating p(Si = 1) for the network in fig(l), averaged over all the nodes in the network. In each of 100 trials, weights were drawn from a zero mean, unit variance Gaussian; biases were set to O. Note the different scale in (b) and (c). In (a) we use the variational procedure with a factorised Q, as in section (2.1). In (b) we use the Gaussian field equations, assuming a diagonal covariance matrix R. This procedure was repeated in (c) including correlations between parents. E+ = E U {Si = I}, which then contains n + 1 "evidential" variables. That is, the desired marginal variable is absorbed into the evidence set . For convenience, we then split the nodes into two sets, those containing the evidential or "clamped" nodes, C, and the remaining "free" nodes F . The joint evidence is then given by (14) 8F = I:p (ECllll'~l) ... p (En+llll'~"+l) p (sh 11I'jJ ...p (Sfm 11I'jJ 8F (15) where 11'; are the parents of node i, with any evidential parental nodes set to their values as specified in E+. In the sigmoid belief network if i is an evidential node otherwise (16) = p(Eklll'Z) is therefore determined by the distribution of the field hZ Li WkiS; +Ok . Examining (15), we see that the product over the "free" nodes defines a SBN in which the local probability distributions are given by those of the original network, but with any evidential parental nodes clamped to their evidence values . Therefore, (17) Consistent with our previous assumptions, we assume that the distribution of the fields h+ = (h~l'" h~"+l) is jointly Gaussian. We can then find the mean and covariance matrix for the distribution of h+ by repeating the calculation of section (2) in which evidential nodes have been clamped to their evidence values. Once this Gaussian has been determined, it can be used in (17) to determine p( E+). Gaussian averages of products of sigmoids are calculated by drawing 1000 samples from the Gaussian over which we wish to integrate 1 . Note that if there are evidential nodes lIn one and two dimensions (n = 0, 1), or n = 1, we use Gaussian Quadrature. D. Barber and P Sollich 398 Error uoing lado_ model 1M l00r-~--~----~~--~ eo Error uoing Ga_ian Field. Diegorel """arianee Em:>< uoing Ga .....n Field. Non Diagonal ""w"ianee EO 70,---~------~--------, 50 EO 50 40 30 30 20 20 10 10 o1 (a) Mean error = 0.4188 0.2 0.3 0.4 0.5 06 (b) Mean error = 0.0253 00 II.. 01 02 0.3 0.4 0.5 0.6 (c) Mean error = 0.0198 Figure 3: All weights are set to uniformly from 0 to 50. Biases are set to -0.5 of the summed parental weights plus a uniform random number from -2.5 to 2.5 . The root node is set to be 1 with probability 0.5. This has the effect of making all the nodes in the exact network roughly 0.5 in mean, with non-negligible correlations between parental nodes. 160 simulations were made. in different layers, we require the correlations between their fields h to evaluate (17) . Such 'inter-layer' correlations were not required in section (2) , and to be able to use the same calculational scheme we simply neglect them. (We leave a study of the effects of this assumption for future work .) The average in (17) then factors into groups, where each group contains evidential terms in a particular layer. The conditional marginal for node i is obtained from repeating the above procedure in which the desired marginal node is clamped to its opposite value, and then using these results in (13). The above procedure is repeated for each conditional marginal that we are interested in. Although this may seem computationally expensive, the marginal for each node is computed quickly, since the equations are solved by one forward propagation sweep only. Error uoing Gauosian Field, Diago".1 covarIanee (a) Mean error = 0.1534 (b) Mean error = 0.0931 Em:>< uoing Gau"ian Field. Non Diagonal """ariance (c) Mean error = 0.0865 Figure 4: Estimating the conditional marginal of the top node being in state 1, given that the four bottom nodes are in state 1. Weights were drawn from a zero mean Gaussian with variance 5, with biases set to -0.5 the summed parental weights plus a uniform random number from -2.5 to 2.5 . Results of 160 simulations. 3.1 Results for conditional marginals We used the same structure as in the previous experiments, as shown in fig(I). We are interested here in calculating the probability that the top node is in state 1, Gaussian Fields for Approximate Inference 399 given that the four bottom nodes are in state 1. Weights were chosen from a zero mean Gaussian with variance 5. Biases were set to negative half of the summed parent weights, plus a uniform random value from -2.5 to 2.5. Correlation effects in these networks are not as strong as in the experiments in section (2.2), although the improvement of the G F theory over the variational theory seen in fig (4) remains clear. The improvement from the off diagonal terms in R is minimal. 4 Conclusion Despite their appropriateness for learning, variational methods may not be equally suited to inference, making more tailored methods attractive. We have considered an approximation procedure that is based on assuming that the distribution of the weighted input to a node is approximately Gaussian. Correlation effects between parents of a node were taken into account to improve the Gaussian theory, although in our examples this gave only relatively modest improvements. The variational mean field theory performs poorly in networks with strong correlation effects between nodes. On the other hand, one may conjecture that the Gaussian Field approach will not generally perform catastrophically worse than the factorised variational mean field theory. One advantage of the variational theory is the presence of an objective function against which competing solutions can be compared. However, finding an optimum solution for the mean parameters mj from this function is numerically complex. Since the Gaussian Field theory is extremely fast to solve, an interesting compromise might be to prime the variational solution with the results from the Gaussian Field theory. Acknowledgments DB would like to thank Bert Kappen and Wim Wiegerinck for stimulating and helpful discussions. PS thanks the Royal Society for financial support. [1] R. Neal. Connectionist learning of Belief Networks. Artificial Intelligence, 56:71-113, 1992. [2] E. Castillo, J. M. Gutierrez, and A. S. Radi. Expert Systems and Probabilistic Network Models. Springer, 1997. [3] M. I. Jordan, Z. Gharamani, T. S. Jaakola, and L. K. Saul. An Introduction to Variational Methods for Graphical Models. In M. I. Jordan, editor, Learning in Graphical Models, pages 105-161. Kluwer, 1998. [4] L. Saul and M. I. Jordan. A mean field learning algorithm for unsupervised neural networks. In M. I. Jordan, editor, Learning in Graphical Models, 1998. [5] D. Barber and W Wiegerinck. Tractable variational structures for approximating graphical models. In M.S. Kearns, S.A. Solla, and D.A. Cohn, editors, Advances in Neural Information Processing Systems NIPS 11. MIT Press, 1999. [6] M. Kearns and 1. Saul. Inference in Multilayer Networks via Large Deviation Bounds. In Advances in Neural Information Processing Systems NIPS 11, 1999. [7] L. K. Saul, T. Jaakkola, and M. I. Jordan. Mean Field Theory for Sigmoid Belief Networks. Journal of Artificial Intelligence Research, 4:61-76, 1996. 

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Generalized Model Selection For Unsupervised Learning In High Dimensions Shivakumar Vaithyanathan IBM Almaden Research Center 650 Harry Road San Jose, CA 95136 Shiv@almaden.ibm.com Byron Dom IBM Almaden Research Center 650 Harry Road San Jose, CA 95136 dom@almaden.ibm.com Abstract We describe a Bayesian approach to model selection in unsupervised learning that determines both the feature set and the number of clusters. We then evaluate this scheme (based on marginal likelihood) and one based on cross-validated likelihood. For the Bayesian scheme we derive a closed-form solution of the marginal likelihood by assuming appropriate forms of the likelihood function and prior. Extensive experiments compare these approaches and all results are verified by comparison against ground truth. In these experiments the Bayesian scheme using our objective function gave better results than cross-validation. 1 Introduction Recent efforts define the model selection problem as one of estimating the number of clusters[ 10, 17]. It is easy to see, particularly in applications with large number of features, that various choices of feature subsets will reveal different structures underlying the data. It is our contention that this interplay between the feature subset and the number of clusters is essential to provide appropriate views of the data.We thus define the problem of model selection in clustering as selecting both the number of clusters and the feature subset. Towards this end we propose a unified objective function whose arguments include the both the feature space and number of clusters. We then describe two approaches to model selection using this objective function. The first approach is based on a Bayesian scheme using the marginal likelihood for model selection. The second approach is based on a scheme using cross-validated likelihood. In section 3 we apply these approaches to document clustering by making assumptions about the document generation model. Further, for the Bayesian approach we derive a closed-form solution for the marginal likelihood using this document generation model. We also describe a heuristic for initial feature selection based on the distributional clustering of terms. Section 5 describes the experiments and our approach to validate the proposed models and algorithms. Section 6 reports and discusses the results of our experiments and finally section 7 provides directions for future work. 971 Model Selection for Unsupervised Learning in High Dimensions 2 Model selection in clustering Model selection approaches in clustering have primarily concentrated on determining the number of components/clusters. These attempts include Bayesian approaches [7,10], MDL approaches [15] and cross-validation techniques [17] . As noticed in [17] however, the optimal number of clusters is dependent on the feature space in which the clustering is performed. Related work has been described in [7]. 2.1 A generalized model for clustering Let D be a data-set consisting of "patterns" {d I, .. , d v }, which we assume to be represented in some feature space T with dimension M. The particular problem we address is that of clustering D into groups such that its likelihood described by a probability model p(DTIQ), is maximized, where DT indicates the representation of D in feature space T and Q is the structure of the model, which consists of the number of clusters, the partitioning of the feature set (explained below) and the assignment of patterns to clusters. This model is a weighted sum of models {P(DTIQ, ~)I~ E [Rm} where ~ is the set of all parameters associated with Q . To define our model we begin by assuming that the feature space T consists of two sets: U - useful features and N noise features. Our feature-selection problem will thus consist of partitioning T (into U and N) for a given number of clusters. Assumption 1 The feature sets represented by independent p(DTIQ,~) U and N are conditionally = P(D N I Q, ~) P(D u I Q,~) (1) where DN indicates data represented in the noise feature space and D U indicates data represented in useful feature space. Using assumption 1 and assuming that the data is independently drawn, we can rewrite equation (1) as p(DTIQ,~) = {n p(d~ I ~N). 1=1 nn k=IJED, p(dy I ~f)} (2) where V is the number of patterns in D, p(dy I ~u) is the probability of dy given the parameter vector ~f and p(d~ I ~N) is the probability of d~ given the parameter vector ~N . Note that while the explicit dependence on Q has been removed in this notation , it is implicit in the number of clusters K and the partition of T into Nand U. 2.2 Bayesian approach to model selection The objective function, represented in equation (2) is not regularized and attempts to optimize it directly may result in the set N becoming empty - resulting in overfitting. To overcome this problem we use the marginallikelihood[2]. Assumption 2 All parameter vectors are independent. n (~) =n (~N). K n n (~f) k=1 where the n( ... ) denotes a Bayesian prior distribution. The marginal likelihood, using assumption 2, can be written as P(D T I Q)= IN [UP(d~ I ~N)]n(~N)d~N. DL [!lp(d Y I ~f)]n(~f)d~f(3) S. Vaithyanathan and B. Dom 972 where SN, SV are integral limits appropriate to the particular parameter spaces. These will be omitted to simplify the notation. 3.0 Document clustering Document clustering algorithms typically start by representing the document as a "bag-of-words" in which the features can number - 10 4 to 10 5 . Ad-hoc dimensionality reduction techniques such as stop-word removal, frequency based truncations [16] and techniques such as LSI [5] are available. Once the dimensionality has been reduced, the documents are usually clustered into an arbitrary number of clusters. 3.1 Multinomial models Several models of text generation have been studied[3]. Our choice is multinomial models using term counts as the features. This choice introduces another parameter indicating the probability of the Nand U split. This is equivalent to assuming a generation model where for each document the number of noise and useful terms are determined by a probability (}s and then the terms in a document are "drawn" with a probability ((}n or ()~ ). 3.2 Marginal likelihood / stochastic complexity To apply our Bayesian objective function we begin by substituting multinomial models into (3) and simplifying to obtain P(D I Q) = (t N;,t V ) S[((}S)tN (1- (}s)t u]n((}s)d(}S . [Ii: II ({t. tIYEU}J] S[II((}k)t,.u]n((}f)d(}f ? k==1 IEDk I,UU UEV (4) ( tf J1S[II((}n)ti .? ] n((}N) d(}N [iI j=1 {tj,nlnEN} nEN where ( (.'.\) is the multinomial coefficient, ti,u is the number of occurrences of the feature term document i u in document i, tYis the total number of all useful features (terms) in (tY =L U ti,u, t~:, and ti,n are to be interpreted similar to above but for (n noise features, = k l(~~k) ! , tNis the total number of all noise features in all patterns and tVis the total number of all useful features in all patterns. To solve (4) we still need a form for the priors {n( ... )}. The Beta family is conjugate to the Binomial family [2] and we choose the Dirichlet distribution (mUltiple Beta) as the form for both n((}f) and n((}N) and the Beta distribution for n((}s). Substituting these into equation (8) and simplifying yields P (D I Q) =[ f(Ya + Yb) f(tN + Ya)f(tV + Yb) ] ? [ f(/J) II f(/Jn + t n) ] f(Ya)f(Yb) [(tV + t N + Ya + Yb) f(/J + t N) nEN f(/Jn) [ [(0') [(0'+ v) K r(O'k + IDkl)] [K f(a) [(au + tV ] D f(lD kl) ? Df(a+ Du f(a u) tU(k) (5) Model Selection for Unsupervised Learning in High Dimensions 973 where f3, and au are the hyper-parameters of the Dirichlet prior for noise and useful features respectively, f3 = f3n , a = au, U = ukand is the "gamma" function. L L UEU neN ro L k=1 Further, Yu, Yure the hyper parameters of the Beta prior for the split probability, IDkl is the number of documents in cluster k and tU(k is computed as L tf. The results iEDk reported for our evaluation will be the negative of the log of equation (5), which (following Rissanen [14]) we refer to as Stochastic Complexity (SC). In our experiments all values of the hyper-parameters pj ,aj (Jk> Ya and Ybare set equal to 1 yielding uniform priors. 3.3 Cross-Validated likelihood To compute the cross validated likelihood using multinomial models we first substitute the multinomial functional forms, using the MLE found using the training set. This results in the following equation I QP) = ,......., N ,.....,., U VIt!\ 1 ,.....,.., K ~ [(05)t" .. (1- ( 5)1,,] IT p(evf ION) . IT IT peevy I O~i)' p(q) (6) 1=1 k=IJEDk ,..., ,..., where Os, ON and O~i) are the MLE of the appropriate parameter vectors . For our implementation of MCCV, following the suggestion in [17], we have used a 50% split of the training and test set. For the vCV criterion although a value of v = 10 was suggested therein, for computational reasons we have used a value of v = 5. P(CVT --- { 3.4 Feature subset selection algorithm for document clustering As noted in section 2.1, for a feature-set of size M there are a total of 2M partitions and for large M it would be computationally intractable to search through all possible partitions to find the optimal subset. In this section we propose a heuristic method to obtain a subset of tokens that are topical (indicative of underlying topics) and can be used as features in the bag-of-words model to cluster documents. 3.4.1 Distributional clustering for feature subset selection Identifying content-bearing and topical terms, is an active research area [9]. We are less concerned with modeling the exact distributions of individual terms as we are with simply identifying groups of terms that are topical. Distributional clustering (DC), apparently first proposed by Pereira et al [13], has been used for feature selection in supervised text classification [1] and clustering images in video sequences [9]. We hypothesize that function, content-bearing and topical terms have different distributions over the documents. DC helps reduce the size of the search space for feature selection from 2M to 2 e , where C is the number of clusters produced by the DC algorithm. Following the suggestions in [9], we compute the following histogram for each token. The first bin consists of the number of documents with zero occurrences of the token, the second bin is the number of documents consisting of a single occurrence of the token and the third bin is the number of documents that contain more two or more occurrences of the term. The histograms are clustered using reLative entropy ~(. II .) as S. Vaithyanathan and B. Dom 974 a distance measure. For two terms with probability distributions PI (.) and P2(.), this is given by [4]: ,1.(Pt(t) II P2(t)) = kt '" PI(t) log PI(t) P2(t) (7) We use a k-means-style algorithm in which the histograms are normalized to sum to one and the sum in equation (7) is taken over the three bins corresponding to counts of 0,1, and ~ 2. During the assignment-to-clusters step of k-means we compute II PCk) (where pw is the normalized histogram for term wand Pq(t) is the centroid of cluster k) and the term w is assigned to the cluster for which this is minimum [13,8]. ,1.(pw 4.0 Experimental setup Our evaluation experiments compared the clustering results against human-labeled ground truth. The corpus used was the AP Reuters Newswire articles from the TREC-6 collection. A total of 8235 documents, from the routing track, existing in 25 classes were analyzed in our experiments. To simplify matters we disregarded multiple assignments and retained each document as a member of a single class. 4.1 Mutual information as an evaluation measure of clustering We verify our models by comparing our clustering results against pre-classified text. We force all clustering algorithms to produce exactly as many clusters as there are classes in the pre-classified text and we report the mutual information[ 4] (MI) between the cluster labels and pre-classified class labels 5.0 Results and discussions After tokenizing the documents and discarding terms that appeared in less than 3 documents we were left with 32450 unique terms . We experimented with several numbers of clusters for DC but report only the best (lowest SC) for lack of space. For each of these clusters we chose the best of 20 runs corresponding to different random starting clusters. Each of these sets includes one cluster that consists of high-frequency words and upon examination were found to contain primarily function words, which we eliminated from further consideration. The remaining non-function-word clusters were used as feature sets for the clustering algorithm. Only combinations of feature sets that produced good results were used for further document clustering runs. We initialized the EM algorithm using k-means algorithm - other initialization schemes are discussed in [11]. The feature vectors used in this k-means initialization were generated using the pivoted normal weighting suggested in [16]. All parameter vectors and were estimated using Laplace's Rule of Succession[2]. Table 1 shows the best results of the SC criterion, the vCV and MCCV using the feature subsets selected by the different combinations of distributional clusters. The feature subsets are coded as FSXP where X indicates the number of clusters in the distributional clustering and P indicates the cluster number(s) used as U. For SC and MI all results reported are averages over 3 runs of the k-means+EM combination with different initialization fo k-means. For clarity, the MI numbers reported are normalized such that the theoretical maximum is 1.0. We also show comparisons against no feature selection (NF) and LSI. Of eN Model Selection for Unsupervised Learning in High Dimensions 975 For LSI, the principal 165 eigenvectors were retained and k-means clustering was performed in the reduced dimensional space. While determining the number of clusters, for computational reasons we have limited our evaluation to only the feature subset that provided us with the highest MI, i.e., FS41-3 . Feature Set FS41-3 FS52 NF LSI Useful Features 6,157 386 32,450 324501165 Table SC X 107 2.66 2.8 2.96 NA 1 Comparison vCV X 107 0.61 0.3 1.25 NA Of Results MCCV X 107 1.32 0.69 2.8 NA Ml 0.61 0.51 0.58 0.57 Figure 1 1~1.. . ? , ...... .: . " '" .. . ,.. .. .?" ....,....... Log ....IQ..... L...hood ThIwI:_P'!Ch I, 1~1.. ? . Flgur.2 ~ . " . ? ... MCCV? .... gII ..... l avl .......-ood I " 5.3 Discussion The consistency between the MI and SC (Figure 1) is striking. The monotonic trend is more apparent at higher SC indicating that bad clusterings are more easily detected by SC while as the solution improves the differences are more subtle. Note that the best value of SC and Ml coincide. Given the assumptions made in deriving equation (5), this consistency and is encouraging. The interested reader is referred to [18] for more details. Figures 2 and 3 indicate that there is certainly a reasonable consistency between the cross-validated likelihood and the MI although not as striking as the SC. Note that the MI for the feature sets picked by MCCV and vCV is significantly lower than that of the best feature-set. Figures 4,5 and 6 show the plots of SC, MCCV and vCV as the number of clusters is increased. Using SC we see that FS41-3 reveals an optimal structure around 40 clusters. As with feature selection, both MCCV and vCV obtain models of lower complexity than Sc. Both show an optimum of about 30 clusters. More experiments are required before we draw final conclusions, however, the full Bayesian approach seems a practical and useful approach for model selection in document clustering. Our choice of likelihood function and priors provide a closed-form solution that is computationally tractable and provides meaningful results. 6.0 Conclusions In this paper we tackled the problem of model structure determination in clustering. The main contribution of the paper is a Bayesian objective function that treats optimal model selection as choosing both the number of clusters and the feature subset. An important aspect of our work is a formal notion that forms a basis for doing feature selection in unsupervised learning. We then evaluated two approaches for model selection: one using this objective function and the other based on cross-validation. S. Vaithyanathan and B. Dom 976 Both approaches performed reasonably well - with the Bayesian scheme outperforming the cross-validation approaches in feature selection. More experiments using different parameter settings for the cross-validation schemes and different priors for the Bayesian scheme should result in better understanding and therefore more powerful applications of these approaches . Fig.... :: I . """. ! I te. 1: '--"_ _ _ _------' t. t WCCV ? ,..,._lIogLDiIfIood Xl01 I . ....... I la~ 1-'-_ _ __ r:: _ ----1 ..... ~.. .? .. ? .. ? - + ? ?. ' , tl. " _ _ 0.-. ? ~ ?? ",._ Flgur,' ... " : 1=' ... ~ " HI " 1 I ? . . ? ? _oliO-. . ?. .I OM ~ OM ? . ? __ "0..-. References [I] Baker, D., et aI, Distributional Clustering of Words for Text Classification, SIGIR 1998. [2] Bernardo, J. M. and Smith, A. F. M., Bayesian Theory, Wiley, 1994. [3] Church, K.W. et aI, Poisson Mixtures. Natural Language Engineering. 1(12), 1995. [4] Cover, T.M . and Thomas, J.A. Elements of Information Theory. Wiley-Interscience, 1991. [5] Deerwester,S. et aI, Indexing by Latent Semantic Analysis,JASIS, 1990. [6] Dempster, A.et aI., Maximum Likelihood from Incomplete Data Via the EM Algorithm. JRSS, 39,1977. [7] Hanson,R., et aI, Bayesian Classification with Correlation and Inheritance, IJCAI,1991 . [8] Iyengar, G., Clustering images using relative entropy for efficient retrieval, VLBV, 1998. [9] Katz, S.M . , Distribution of content words and phrases in text and language modeling, NLE, 2,1996. [10] Kontkanen, P.T. et ai, Comparing Bayesian Model Class Selection Criteria by Discrete Finite Mixtures, ISIS'96 Conference, 1996. [II] Meila, M., Heckerman, D., An Experimental Comparison of Several Clustering and Initialization Methods, MSR-TR-98-06. [12] Nigam, K et aI, Learning to Classify Text from Labeled and Unlabeled Documents, AAAI, 1998. [13] Pereira, F.C.N. et ai , Distributional clustering of English words, ACL,1993. [14] Rissanen, J., Stochastic Complexity in Statistical Inquiry. World\ Scientific, 1989. [15] Rissanen, J., Ristad E., Unsupervised classification with stochastic complexity." The US/Japan Conference on the Frontiers of Statistical Modeling,1992. [16] Singhal A. et aI, Pivoted Document Length Normalization, SIGIR, 1996. [17] Smyth, P., Clustering using Monte Carlo cross-validation, KDD, 1996. [18] Vaithyanathan, S. and Dom, B. Model Selection in Unsupervised Learning with Applications to Document Clustering. IBM Research Report RJ-I 0137 (95012) Dec. 14, 1998 . 

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On input selection with reversible jump Markov chain Monte Carlo sampling Peter Sykacek Austrian Research Institute for Artificial Intelligence (OFAI) Schottengasse 3, A-10lO Vienna, Austria peter@ai. univie. ac. at Abstract In this paper we will treat input selection for a radial basis function (RBF) like classifier within a Bayesian framework. We approximate the a-posteriori distribution over both model coefficients and input subsets by samples drawn with Gibbs updates and reversible jump moves. Using some public datasets, we compare the classification accuracy of the method with a conventional ARD scheme. These datasets are also used to infer the a-posteriori probabilities of different input subsets. 1 Introduction Methods that aim to determine relevance of inputs have always interested researchers in various communities. Classical feature subset selection techniques, as reviewed in [1], use search algorithms and evaluation criteria to determine one optimal subset. Although these approaches can improve classification accuracy, they do not explore different equally probable subsets. Automatic relevance determination (ARD) is another approach which determines relevance of inputs. ARD is due to [6] who uses Bayesian techniques, where hierarchical priors penalize irrelevant inputs. Our approach is also "Bayesian": Relevance of inputs is measured by a probability distribution over all possible feature subsets. This probability measure is determined by the Bayesian evidence of the corresponding models. The general idea was already used in [7] for variable selection in linear regression mo.dels. Though our interest is different as we select inputs for a nonlinear classification model. We want an approximation of the true distribution over all different subsets. As the number of subsets grows exponentially with the total number of inputs, we can not calculate Bayesian model evidence directly. We need a method that samples efficiently across different dimensional parameter spaces. The most general method that can do this is the reversible jump Markov chain Monte Carlo sampler (reversible jump Me) recently proposed in [4]. The approach was successfully applied by [8] to determine a probability distribution in a mixture density model with variable number of kernels and in [5] to sample from the posterior of RBF regression networks with variable number of kernels. A Markov chain that switches between different input subsets is useful for two tasks: Counting how often a particular subset was visited gives us a relevance measure of the corresponding inputs; For classification, we approximate On Input Selection with Reversible Jump MCMC 639 the integral over input sets and coefficients by summation over samples from the Markov chain. The next sections will show how to implement such a reversible jump MC and apply the proposed algorithm to classification and input evaluation using some public datasets. Though the approach could not improve the MLP-ARD scheme from [6] in terms of classification accuracy, we still think that it is interesting: We can assess the importance of different feature subsets which is different than importance of single features as estimated by ARD. 2 Methods The classifier used in this paper is a RBF like model. Inference is performed within a Bayesian framework. When conditioning on one set of inputs , the posterior over model parameters is already multimodal. Therefore we resort to Markov chain Monte Carlo (MCMC) -sampling techniques to approximate the desired posterior over both model coefficients and feature subsets. In the next subsections we will propose an appropriate architecture for the classifier and a hybrid sampler for model inference. This hybrid sampler consists of two parts: We use Gibbs updates ([2]) to sample when conditioning on a particular set of inputs and reversible jump moves that carry out dimension switching updates. 2.1 The classifier I~ order to allow input relevance determination by Bayesian model selection , the classifier needs at least one coefficient that is associated with each input: Roughly speaking, the probability of each model is proportional to the likelihood of the most probable coefficients, weighted by their posterior width divided by their prior width. The first factor always increases when using more coefficients (or input features). The second will decrease the more inputs we use and together this gives a peak for the most probable model. A classifier that satisfies these constraints is the so called classification in the sampling paradigm. We model class conditional densities and together with class priors express posterior probabilities for classes. In neural network literature this approach was first proposed in [10). We use a model that allows for overlapping class conditional densities: D p(~lk) =L d=l K WkdP(~I~) , p(~) =L PkP(~lk) (1) k=l Using Pk for the J{ class priors and p(~lk) for the class conditional densities, (1) expresses posterior probabj,Jities for classes as P(kl~) = PkP(~lk)/p(~). We choose the component densities, p(~IcI> d), to be Gaussian with restricted parametrisation: Each kernel is a multivariate normal distribution with a mean and a diagonal covariance matrix. For all Gaussian kernels together, we get 2 * D * I parameters, with I denoting the current input dimension and D denoting the number of kernels. Apart from kernel coefficients, cI>d , (1) has D coefficients per class, Wkd, indicating the prior kernel allocation probabilities and J{ class priors. Model (1) allows to treat labels of patterns as missing data and use labeled as well as unlabeled data for model inference. In this case training is carried out using the likelihood of observing inputs and targets: p(T, X18) = rrr;=lrr;::=lPkPk(~nk Ifu)rr~=lp(bnI8) , (2) where T denotes labeled and X unlabeled training data. In (2) 8 k are all coefficients the k-th class conditional density depends on. We further use 8 for all model P. Sykacek 640 coefficients together, nk as number of samples belonging to class k and m as index for unlabeled samples. To make Gibbs updates possible, we further introduce two latent allocation variables. The first one, d, indicates the kernel number each sample was generated from, the second one is the unobserved class label c, introduced for unlabeled data. Typical approaches for training models like (1), e.g. [3] and [9], use the EM algorithm, which is closely related to the Gibbs sampler introduce in the next subsection. 2.2 Fixed dimension sampling In this subsection we will formulate Gibbs updates for sampling from the posterior when conditioning on a fixed set of inputs. In order to allow sampling from the full conditional, we have to choose priors over coefficients from their conjugate family: ? Each component mean, !!!d, is given a Gaussian prior: !!!d '" Nd({di). ? The inverse variance of input i and kernel d gets a Gamma prior: u;;l '" r( a, ,Bi). ? All d variances of input i have a common hyperparameter, ,Bi, that has itself a Gamma hyperprior: ,Bi ,...., r(g, hi). ? The mixing coefficients, ~, get a Dirichlet prior: ~ '" 1J (6w , ... , 6w ). ? Class priors, P, also get a Dirichlet prior: P '" 1J(6p , ... ,6p). The quantitative settings are similar to those used in [8]: Values for a are between 1 and 2, g is usually between 0.2 and 1 and hi is typically between 1/ Rr and 10/ with Ri denoting the i'th input range. The mean gets a Gaussian prior centered at the midpoint, with diagonal inverse covariance matrix ~, with "'ii = 1/ The prior counts d w and 6p are set to 1 to give the corresponding probabilities non-informative proper Dirichlet priors. Rr, Rr. e, The Gibbs sampler uses updates from the full conditional distributions in (3). For notational convenience we use ~ for the parameters that determine class conditional densities. We use m as index over unlabeled data and Cm as latent class label. The index for all data is n, dn are the latent kernel allocations and nd the number of samples allocated by the d-th component. One distribution does not occur in the prior specification. That is Mn(l, ... ) which is a multinomial-one distribution. Finally we need some counters: ml ... mK are the counts per class and mlk .. mDk count kernel allocations of class-k-patterns. The full conditional of the d-th kernel variances and the hyper parameter ,Bi contain i as index of the input dimension. There we express each u;J separately. In the expression of the d-th kernel mean, I 641 On Input Selection with Reversible Jump MCMC illd, we use .lGt to denote the entire covariance matrix. PkP(~mlfu) Mn ( 1, { I:k PkP(~mlfu)' Mn (1, {I:,WtndP(~nl~) Wt,.dP(~nl~) }) k= l..K ,d= l..D}) p(~J ..) r (9 + p(~I???) 1) (ow + mlk, ... ,ow + mDk) p(PI???) p(illdl???) 1) (op r (a + ~d, + ~ Vnld,.=d L (~n,i -llid,i)2) Da. hi (3) + ;; ud,! ) + ml, ... , op + mK) N ((nd~l + ~)-l(ndVdl~ + ~S), (ndVd 1 + ~)-l) f3i i?,. 2.3 Moving between different input subsets The core part of this sampler are reversible jump updates, where we move between different feature subsets. The probability of a feature subset will be determined by the corresponding Bayesian model evidence and by an additional prior over number of inputs. In accordance with [7J, we use the truncated Poisson prior: p(I) = 1/ ( I jax ) c ~~ , where c is a constant and Imax the total nr. of inputs. Reversible jump updates are generalizations of conventional Metropolis-Hastings updates, where moves are bijections (x, u) H (x', u'). For a thorough treatment we refer to [4J. In order to switch subsets efficiently, we will use two different types of moves. The first consist of a step where we add one input chosen at random and a matching step that removes one randomly chosen input. A second move exchanges two inputs which allows "tunneling" through low likelihood areas. Adding an input, we have to increase the dimension of all kernel means and diagonal covariances. These coefficients are drawn from their priors. In addition the move proposes new allocation probabilities in a semi deterministic way. Assuming the ordering, Wk,d ~ Wk,d+1: op Vd ~ D/2 Beta(b a , bb + 1) = Wk,D+l-d + Wk ,dOp { W~'D+l-d w~ , d = wk ,d(1 - op) (4) The matching step proposes removing a randomly chosen input. Removing corresponding kernel coefficients is again combined with a semi deterministic proposal of new allocation probabilities, which is exactly symmetric to the proposal in (4). 642 P. Sykacek Table 1: Summary of experiments Data Ionosphere Pima Wine avg(#) 4.3 4 4.4 max(#) 9 7 8 We accept births with probability: n, min( 1, lh. rt x x (~':) RBF (%,n a ) (91.5,11) (78 .9,111 (100, 01 r MLP (%,nb) 95.5,4 79.8,8 96.8,2 p(;(;/) G, J2,; g f g(.,.~ -')"-1 exp( -Ii' exp ( - 05 ;" (I'd - <d)' ) "'~-') x dm / (I + 1) x 1 bm/(Imax - I) (~, V27i) D TID exp ( -0.5Ih(J.l~ - ed)2) x 1 (Ii;)) D TID (0"~-2)a-l exp( _(3'0"~-2) ). (5) The first line in (5) are the likelihood and prior ratio. The prior ratio results from the difference in input dimension, which affects the kernel means and the prior over number of inputs. The first term of the proposal ratio is from proposing to add or remove one input. The second term is the proposal density of the additional kernel components which cancels with the corresponding term in the prior ratio. Due to symmetry of the proposal (4) and its reverse in a death move, there is no contribution from changing allocation probabilities. Death moves are accepted with probability ad = l/ab. The second type of move is an exchange move. We select a new input and one from the model inputs and propose new mean coefficients. This gives the following acceptance probability: (*,J2;) D TID exp ( -0.5"Jb(J.ld - ed)2) min( 1, lh. ratio x (~, J2;) D ITD exp ( -0.5 Ih(J.ld - ed)2) (6) cm/ I TID N(J.ldl???) ) x---:-:--'-----..,..x cm/(Imax - I) TID N(J.l~I???) . The first line of (6) are again likelihood and prior ratio. For exchange moves, the prior ratio is just the ratio from different values in the kernel means. The first term in the proposal ratio is from proposing to exchange an input. The second term is the proposal density of new kernel mean components. The last part is from proposing new allocation probabilities. 3 Experiments Although the method can be used with labeled and unlabeled data, the following experiments were performed using only labeled data. For all experiments we set a = 2 and 9 = 0.2. The first two data sets are from the VCI repositoryl. We use 1 Available at http://www.ics.uci.edu/ mlearn/MLRepository.html. On Input Selection with Reversible Jump MCMC 643 the Ionosphere data which has 33 inputs, 175 training and 176 test samples. For this experiment we use 6 kernels and set h = 0.5. The second data is the wine recognition data which provides 13 inputs, 62 training and 63 test samples. For this data, we use 3 kernels and set h = 0.28. The third experiment is performed with the Pima data provided by B. D . Ripley2. For this one we use 3 kernels and set h = 0.16. For all experiments we draw 15000 samples from the posterior over coefficients and input subsets. We discard the first 5000 samples as burn in and use the rest for predictions. Classification accuracy, is compared with an MLP classifier using R. Neals hybrid Monte Carlo sampling with ARD priors on inputs. These experiments use 25 hidden units. Table 1 contains further details: avg( #) is the average and max(#) the maximal number of inputs used by the hybrid sampler; RBF (%, na) is the classification accuracy of the hybrid sampler and the number of errors it made that were not made by the ARD-MLP; MLP(%, nb) is the same for the ARD-MLP. We compare classifiers by testing (na, nb) against the null hypothesis that this is an observation from a Binomial Bn(na +nb , 0.5) distribution. This reveals that neither difference is significant. Although we could not improve classification accuracy on these data, this does not really matter because ARD methods usually lead to high generalization accuracy and we can compete. The real benefit from using the hybrid sampler is that we can infer probabilities telling us how much different subsets contribute to an explanation of the target variables. Figure 3 shows the occurrence probabilities of feature subsets and features. Note that table 1 has also details about how many features were used in these problems. Especially the results from Ionosphere data are interesting as on average we use only 4.3 out of 33 input features. For ionosphere and wine data the Markov chain visits about 500 different input subsets within 10000 samples. For the Pima data the number is about 60 and an order of magnitude smaller. 4 Discussion In this paper we have discussed a hybrid sampler that uses Gibbs updates and reversible jump moves to approximate the a-posteriori distribution over parameters and input subsets in nonlinear classification problems. The classification accuracy of the method could compete with R . Neals MLP-ARD implementation. However the real advantage of the method is that it provides us with a relevance measure of feature subsets. This allows to infer the optimal number of inputs and how many different explanations the data provides. Acknow ledgements I want to thank several people for having used resources they provide: I have used R.Neals hybrid Markov chain sampler for the MLP experiments; The data used for the experiments were obtained form the University at Irvine repository and from B. D. Ripley. Furthermore I want to express gratitude to the anonymous reviewers for their comments and to J.F.G . de Freitas for useful discussions during the conference. This work was done in the framework of the research project GZ 607.519/2V /B/9/98 "Verbesserung der Biosignalverarbeitung durch Beruecksichtigung von Unsicherheit und Konfidenz", funded by the Austrian federal ministry of science and transport (BMWV). 2 Available at http://www.stats.ox.ac.uk P. Sykacek 644 Probabil~les 10 of input subsets-Ionosphere Probabilities of inputs - Ionosphere 30r-----~------------~_, 9.1% :[7.14) 8 .6%: [3. 4) 7.9%: [4. 7) 4.6% : [4. 15) 5 20 10 o j, I. o j 100 J 1. 200 300 400 500 10 Probabilities of input subsets-Pima 17.3% : [2. 7j 16.1%:[2. 9.8% : [2. 5) 8.6% : [2. 5. 7) 15 10 20 30 Probabilities of inputs - Pima 20r-----~--------------TO 40r---~~--------~-----' 30 20 5 10 ?0L-~~~2~0~~~~40~~~-6~0 o~--~~--~----~----~ o Probabilities of input subsets - Wine 10r---~--~--~----~--~ 9.5?,,": [1. 13) 5.2%: [7. 13) 3,4?,,": [ 13) 3.1%: [1.12.13) 5 246 8 Probabilities of inputs - Wine 3Or-------~-------------, 20 10 0U-------~------~----~ 100 200 300 400 500 o 5 10 Figure 1: Probabilities of inputs and input subsets measuring their relevance. References [1] P. A. Devijver and J. V. Kittler. Pattern Recognition. A Statistical Approach. PrenticeHall, Englewood Cliffs, NJ, 1982. [2] S. Geman and D. Geman. Stochastic relaxation, gibbs distributions and the bayesian restoration of images. IEEE Trans. Pattn. Anal. Mach. Intel., 6:721-741, 1984. [3] Z. Ghahramani, M.1. Jordan Supervised Learning from Incomplete Data via an EM Approach In Cowan J.D., et al.(eds.), Advances in Neural Information Processing Systems 6, Morgan Kaufmann, Los Altos/Palo Alto/San Francisco, pp.120-127, 1994. [4] P. J. Green. Reversible jump markov chain monte carlo computation and bayesian model determination. Biometrika, 82:711-732, 1995. [5] C. C. Holmes and B. K. Mallick. Bayesian radial basis functions of variable dimension. Neural Computation, 10:1217-1234, 1998. [6] R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1986. [7] D. B. Phillips and A. F. M. Smith. Bayesian model comparison via jump diffusioons. In W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 215-239, London, 1996. Chapman & Hall. [8] S. Richardson and P.J. Green On Bayesian Analysis of Mixtures with an unknown number of components Journal Royal Stat. Soc. B, 59:731-792, 1997. [9] M. Stensmo, T.J . Sejnowski A Mixture Model System for Medical and Machine Diagnosis In Tesauro G., et al.(eds.), Advances in Neural Information Processing System 7, MIT Press, Cambridge/Boston/London, pp.1077-1084, 1995. [10] H. G. C. Traven A neural network approach to statistical pattern classification by "semi parametric" estimation of probability density functions IEEE Trans. Neur. Net., 2:366-377, 1991. 

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Memory Capacity of Linear vs. Nonlinear Models of Dendritic Integration Panayiota Poirazi* Biomedical Engineering Department University of Southern California Los Angeles, CA 90089 Bartlett W. Mel* Biomedical Engineering Department University of Southern California Los Angeles, CA 90089 poirazi@sc/. usc. edu mel@lnc.usc.edu Abstract Previous biophysical modeling work showed that nonlinear interactions among nearby synapses located on active dendritic trees can provide a large boost in the memory capacity of a cell (Mel, 1992a, 1992b). The aim of our present work is to quantify this boost by estimating the capacity of (1) a neuron model with passive dendritic integration where inputs are combined linearly across the entire cell followed by a single global threshold, and (2) an active dendrite model in which a threshold is applied separately to the output of each branch, and the branch subtotals are combined linearly. We focus here on the limiting case of binary-valued synaptic weights, and derive expressions which measure model capacity by estimating the number of distinct input-output functions available to both neuron types. We show that (1) the application of a fixed nonlinearity to each dendritic compartment substantially increases the model's flexibility, (2) for a neuron of realistic size, the capacity of the nonlinear cell can exceed that of the same-sized linear cell by more than an order of magnitude, and (3) the largest capacity boost occurs for cells with a relatively large number of dendritic subunits of relatively small size. We validated the analysis by empirically measuring memory capacity with randomized two-class classification problems, where a stochastic delta rule was used to train both linear and nonlinear models. We found that large capacity boosts predicted for the nonlinear dendritic model were readily achieved in practice. -http://lnc.usc.edu P. Poirazi and B. W. Mel 158 1 Introduction Both physiological evidence and connectionist theory support the notion that in the brain, memories are stored in the pattern of learned synaptic weight values. Experiments in a variety of neuronal preparations however, inQicate that the efficacy of synaptic transmission can undergo substantial fluctuations up or down, or both, during brief trains of synaptic stimuli. Large fluctuations in synaptic efficacy on short time scales seem inconsistent with the conventional connectionist assumption of stable, high-resolution synaptic weight values. Furthermore, a recent experimental study suggests that excitatory synapses in the hippocampus-a region implicated in certain forms of explicit memory-may exist in only a few long-term stable states, where the continuous grading of synaptic strength seen in standard measures of long-term potentiation (LTP) may exist only in the average over a large population of two-state synapses with randomly staggered thresholds for learning (Petersen, Malenka, Nicoli, & Hopfield, 1998). According to conventional connectionist notions, the possibility that individual synapses hold only one or two bits of long-term state information would seem to have serious implications for the storage capacity of neural tissue. Exploration of this question is one of the main themes of this paper. In a related vein, we have found in previous biophysical modeling studies that nonlinear interactions between synapses co-activated on the same branch of an active dendritic tree could provide an alternative form of long-term storage capacity. This capacity, which is largely orthogonal to that tied up in conventional synaptic weights, is contained instead in the spatial permutation of synaptic connections onto the dendritic tree-which could in principle be modified in the course of learning or development (Mel, 1992a, 1992b). In a more abstract setting, we recently showed that a large repository of model flexibility lies in the choice as to which of a large number of possible interaction terms available in high dimension is actually included in a learning machine's discriminant function, and that the excess capacity contained in this "choice flexibility" can be quantified using straightforward counting arguments (Poirazi & Mel, 1999). 2 Two Alternative Models of Dendritic Integration In this paper, we use a similar function-counting approach to address the more biologically relevant case of a neuron with mUltiple quasi-independent dendritic compartments (fig. 1). Our primary objective has been to compare the memory capacity of a cell assuming two different modes of dendritic integration. According to the linear model, the neuron's activation level aL(x) prior to thresholding is given by a weighted sum of of its inputs over the cell as a whole. According to the nonlinear model, the k synaptic inputs to each branch are first combined linearly, a static (e.g. sigmoidal) nonlinearity is applied to each of the m branch subtotals, and the resulting branch outputs are summed to produce the cell's overall activity aN{x): (1) The expressions for aL and aN were written in similar form to emphasize that the models have an identical number of synaptic weights, differing only in the presence or absence of a fixed nonlinear function g applied to the branch subtotals. Though individual synaptic weights in both models are constrained to have a value of 1, any of the d input lines may form multiple connections on the same or different 159 Memory Capacity of Linear vs. Nonlinear Models of Dendritic Integration m 3 ? .. ' , , I . Figure 1: A cell is modeled as a set of m identical branches connected to a soma, where each branch contains k synaptic contacts driven by one of d distinct input lines. branches as a means of representing graded synaptic strengths. Similarly, an input line which forms no connection has an implicit weight of O. In light of this restriction to positive (or zero) weight values, both the linear and nonlinear models are split into two opponent channels a+ and a- dedicated to positive vs. negative coefficients, respectively. This leads to a final output for each model: yL(x) = sgn [at(x) - aL(x)] YN(X) = sgn [a;t(x) - aiV(x)] (2) where the sgn operator maps the total activation level into a class label of {-I, I}. In the following, we derive expressions for the number of distinct parameter st.ates available to the linear vs. nonlinear models, a measure which we have found to be a reliable predictor of storage capacity under certain restrictions (Poirazi & Mel, 1999). Based on these expressions, we compute the capacity boost provided by the branch nonlinearity as a function of the number of branches m, synaptic sites per branch k, and input space dimensionality d. Finally, we test the predictions of the analytical model by training both linear and nonlinear models on randomized classification problems using a stochastic delta rule, and empirically measure and compare the storage capacities of the two models. 3 Results 3.1 Counting Parameter States: Linear vs. Nonlinear Model We derived expressions for BLand B N, which estimate the total number of parameter bits available to the linear vs. nonlinear models, respectively: B N = 2log2 1) (( k+d-1) k m +m - BL = 2log 2 ( S+d-1) S (3) These expressions estimate the number of non-redundant states in each neuron type, i.e., those assignments of input lines to dendritic sites which yield distinct P Poirazi and B. W Mel 160 input-output functions YL or YN? These formulae are plotted in figure 2A with d = 100, where each curve represents a cell with a fixed number of branches (indicated by m). In each case, the capacity increases steadily as the number of synapses per branch, k, is increased. The logarithmic growth in the capacity of the linear model (evident in an asymptotic analysis of the expression for B L) is shown at the bottom of the graph (circles), from which it may be seen that the boost in capacity provided by the dendritic branch nonlinearity increases steadily with the number of synaptic sites. For a cell with 100 branches containing 100 synaptic sites each, the capacity boost relative to the linear model exceeds a factor of 20. Figure 2B shows that for a given total number of synaptic sites, in this case s = m? k = 10,000, the capacity of the nonlinear cell is maximized for a specific choice of m and k. The peak of each of the three curves (computed for different values of d) occurs for a cell containing 1,250 branches with 8 synapses each. However, the capacity is only moderately sensitive to the branch count: the capacity of a cell with 100 branches of 100 synapses each, for example, lies within a factor of two of the optimal configuration. The linear cell capacities can be found at the far right edge of the plot (m = 10,000), since a nonlinear model with one synapse per branch has a number of trainable states identical to that of a linear model. 3.2 Validating the Analytical Model To test the predictions of the analytical model, we trained both linear and nonlinear cells on randomized two-class classification problems. Training samples were drawn from a 40-dimensional spherical Gaussian distribution and were randomly assigned positive or negative labels-in some runs, training patterns were evenly divided between positive and negative labels, with similar results. Each of the 40 original input dimensions was recoded using a set of 10 I-dimensional binary, nonoverlapping receptive fields with centers spaced along each dimension such that all receptive fields would be activated equally often. This manipulation mapped the original 40-dimensional learning problem into 400 dimensions, thereby increasing the discriminability of the training samples. The relative memory capacity of linear vs. nonlinear cells was then determined empirically by comparing the number of training patterns learnable at a fixed error rate of 2%. The learning rule used for both cell types was similar to the "clusteron" learning rule described in (Mel, 1992a), and involved two mechanisms known to contribute to neural development: (1) random activity-independent synapse formation, and (2) activity-dependent synapse stabilization. In each iteration, a set of 25 synapses was chosen at random, and the "worst" synapse was identified based on the correlation over the training set of (i) the input's pre-synaptic activity, (ii) the post-synaptic activity (Le. the local nonlinear branch response for the nonlinear energy model or a constant of 1 for the linear model), and (iii) a global "delta" signal with a value of a if the cell responded correctly to the input pattern, or ?l if the cell responded incorrectly. The poorest-performing synapse on the branch was then targeted for replacement with a new synapse drawn at random from the d input lines. The probability that the replacement actually occurred was given by a Boltzmann equation based on the difference in the training set error rates before and after the replacement. A "temperature" variable was gradually lowered over the course of the simulation, which was terminated when no further improvement in error rates was seen. Results of the learning runs are shown in fig. 3 where the analytical capacity (measured in bits) was scaled to the numerical capacity (measured in training patterns Memory Capacity ofLinear vs. Nonlinear Models ofDendritic Integration B Capacity of Linear VS. Nonlinear Model A Capacity of Linear vs. Nonlinear Model for Various Geometries for Different Input Space Dimensions x 10' 8.---~----~--~----~---, d = 100 ~ ? Nonlinear Model 6 14 ~ m- 000 5 >. '0!IS .t". . . . x10' Hr m=lOOO 7 161 m 12 : d= j lOO~ , '(i.) .... 10 s = 10,000 " j , Nonlinear Model , , co ....... 4 1t3 U '"'. 2 Linear ~del l~ Linear Model 2000 4000 6000 8000 10000 ~~l Syn:p:c Sires ~ o 2000 4000 6000 8000 10000 Number of Branches (m) * Figure 2: Comparison of linear vs. nonlinear model capacity as a function of branch geometry. A. Capacity in bits for linear and several nonlinear cells with different branch counts (for d = 100). For each curve indexed by branch count m, sites per branch k increases from left to right as indicated iconically beneath the x-axis. For all cells, capacity increases with an increasing number of sites, though the capacity of the linear model grows logarithmically, leading to an increasingly large capacity boost for the size-matched nonlinear cells. B. Capacity of a nonlinear model with 10,000 sites for different values of input space dimension d. Branch count m grows along the x-axis. Cells at right edge of plot contain only one synapse per branch, and thus have a number of modifiable parameters (and hence capacity) equivalent to that of the linear model. All three curves show that there exist an optimal geometry which maximizes the capacity of the nonlinear model (in this case 1,250 branches with 8 synapses each). learned at 2% error). Two key features of the theoretical curves (dashed lines) are echoed in the empirical performance curves (solid lines), including the much larger storage capacity of the nonlinear cell model, and the specific cell geometry which maximizes the capacity boost. 4 Discussion We found using both analytical and numerical methods that in the limit of lowresolution synaptic weights, application of a fixed output nonlinearity to each compartment of a dendritic tree leads to a significant boost in capacity relative to a cell whose post-synaptic integration is linear. For example, given a cell with 10,000 synaptic contacts originating from 400 distinct input lines, the analysis predicts a 23-fold increase in capacity for the nonlinear cell, while numerical simulations using a stochastic delta rule actually achieve a I5-fold boost. Given that a linear and a nonlinear model have an identical number of synaptic contacts with uniform synaptic weight values, what accounts for the capacity boost? The principal insight gained in this work is that the attachment of a fixed nonlinearity to each branch in a neuron substantially increases its underlying "model 162 P. Poirazi and B. W. Mel Figure 3: Comparison of capacity boost predicted by analysis vs. that observed empirically when linear and nonlinear models were trained using the same - _. Analytical stochastic delta rule. Dashed (Bits/14) 70 lines: analytical curves for linNumerical ear vs. nonlinear model for a cell I \ (Training Patterns) 6 ,+ \ with 10,000 sites show capacity I, \ for varying cell geometries. Solid >. 50 \. lines: empirical performance for '13 \ Nonlinear Model same two cells at 2% error cri~ 40 ____________________\ 03 terion, using a subunit nonlinearity g(x) = x lO (similar re<..) 30 '" , sults were seen using a sigmoidal ,, ,, nonlinearity, though the param, ,. 2 eters of the optimal sigmoid depended on the cell geometry). Linear Model For both analytical and numeri, 2 cal curves, peak capacity is seen oo 10 20 30 40 50 60 70 80 90 100x10 for cell with 1,000 branches (10 Number of Branches (m) synapses per branch) .. Cap~city exceeds that of same-sIzed lmear .:Jk- model by a factor of 15 at the m ~ peak, and by more than a factor of 7 for cells ranging from about 3 to 60 synapses per branch (horizontal dotted line). 1 ~, * ---I...... flexibility" , i.e. confers upon the cell a much larger choice of distinct input-output relations from which to select during learning. This may be illustrated as follows. For the linear model, branching structure is irrelevant so that YL depends only on the number of input connections formed from each of the d input lines. All spatial permutations of a set of input connections are thus interchangeable and produce identical cell responses. This massive redundancy confines the capacity of the linear model to grow only logarithmically with an increasing number of synaptic sites (fig. 1A), an unfortunate limitation for a brain in which the formation of large numbers of synaptic contacts between neurons is routine. In contrast, the model with nonlinear subunits contains many fewer redundancies: most spatial permutations of the same set of input connections lead to non-identical values of YN, since an input x swapped from branch bi to branch b2 leads to the elimination of the k - 1 interaction terms involving x on branch bi and the creation of k -1 new interaction terms on branch b2 ? Interestingly, the particular form of the branch nonlinearity has virtually no effect on the capacity of the cell as far as the counting arguments are concerned (though it can have a profound effect on the cell's "representational bias"-see below), since the principal effect of the nonlinearity in our capacity calculations is to break the symmetry among the different branches. The issue of representational bias is a critical one, however, and must be considered when attempting to predict absolute or relative performance rates for particular classifiers confronted with specific learning problems. Thus, intrinsic differences in the geometry of linear vs. nonlinear discriminant functions mean that the param- Memory Capacity ofLinear vs. Nonlinear Models ofDendritic Integration 163 eters available to the two models may be better or worse suited to solve a given learning problem, even if the two models were equated for total parameter flexibility. While such biases are not taken into account in our analysis, they could nonetheless have a substantial effect on measured error rates-and could thus throw a performance advantage to one machine or the other. One danger is that performance differences measured empirically could be misinterpreted as arising from differences in underlying model capacity, when in fact they arise from differential suitability of the two classifiers for the learning problem at hand. To avoid this difficulty, the random classification problems we used to empirically assess memory capacity were chosen to level the playing field for the linear vs. nonlinear cells, since in a previous study we found that the coefficients on linear vs. nonlinear (quadratic) terms were about equally efficient as featUres for this task. In this way, differences in measured performance on these tasks were primarily attributable to underlying capacity differences, rather than differences in representational bias. This experimental control permitted more meaningful comparisons between our analytical and empirical tests (fig. 3). The problem of representational bias crops up in a second guise, wherein the analytical expressions for capacity in eq. 1 can significantly overestimate the actual performance of the cell. This occurs when a particular ensemble of learning problems fails to utilize all of the entropy available in the cell's parameter space-for example, by requiring the cell to visit only a small subset of its parameter states relatively often. This invalidates the maximum parameter entropy assumption made in the derivation of eq. 1, so that measured performance will tend to fall below predicted values. The actual performance of either model when confronted with an ensemble of learning problems will thus be determined by (1) the number of trainable parameters available to the neuron (as measured by eq. 1), (2) the suitability of the neuron's parameters for solving the assigned learning problems, and (3) the utilization of parameters, which relates to the entropy in the joint probability of the parameter values averaged over the ensemble of learning problems. In our comparisons here of linear and nonlinear cells, we we have calculated (1), and have attempted to control for (2) and (3). In conclusion, our results build upon the results of earlier biophysical simulations, and indicate that in the limit of a large number of low-resolution synaptic weights, nonlinear dendritic processing could nonetheless have a major impact on the storage capacity of neural tissue. References Mel, B. W. (1992a). The clusteron: Toward a simple abstraction for a complex neuron. In Moody, J., Hanson, S., & Lippmann, R. (Eds.), Advances in Neural Information Processing Systems, vol. 4, pp. 35-42: Morgan Kaufmann, San Mateo, CA. Mel, B. W. (1992b). NMDA-based pattern discrimination in a modeled cortical neuron. Neural Comp., 4, 502-516. Petersen, C. C. H., Malenka, R. C., Nicoll, R. A., & Hopfield, J. J. (1998). All-ornone potentiation and CA3-CA1 synapses. Proc. Natl. Acad. Sci. USA, 95, 4732-4737. Poirazi, P., & Mel, B. W. (1999). Choice and value flexibility jointly contribute to the capacity of a subsampled quadratic classifier. Neural Comp., in press. 

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An Information-Theoretic Framework for Understanding Saccadic Eye Movements Tai Sing Lee * Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Stella X. Yu Robotics Institute Carnegie Mellon University Pittsburgh, PA 15213 tai@es.emu.edu stella@enbe.emu.edu Abstract In this paper, we propose that information maximization can provide a unified framework for understanding saccadic eye movements. In this framework, the mutual information among the cortical representations of the retinal image, the priors constructed from our long term visual experience, and a dynamic short-term internal representation constructed from recent saccades provides a map for guiding eye navigation . By directing the eyes to locations of maximum complexity in neuronal ensemble responses at each step, the automatic saccadic eye movement system greedily collects information about the external world, while modifying the neural representations in the process. This framework attempts to connect several psychological phenomena, such as pop-out and inhibition of return, to long term visual experience and short term working memory. It also provides an interesting perspective on contextual computation and formation of neural representation in the visual system. 1 Introduction When we look at a painting or a visual scene, our eyes move around rapidly and constantly to look at different parts of the scene. Are there rules and principles that govern where the eyes are going to look next at each moment? In this paper, we sketch a theoretical framework based on information maximization to reason about the organization of saccadic eye movements. ?Both authors are members of the Center for the Neural Basis of Cognition - a joint center between University of Pittsburgh and Carnegie Mellon University. Address: Rm 115, Mellon Institute, Carnegie Mellon University, Pittsburgh, PA 15213. 835 lnformation-Theoretic Framework for Understanding Saccadic Behaviors Vision is fundamentally a Bayesian inference process. Given the measurement by the retinas, the brain's memory of eye positions and its prior knowledge of the world, our brain has to make an inference about what is where in the visual scene. The retina, unlike a camera, has a peculiar design. It has a small foveal region dedicated to high-resolution analysis and a large low-resolution peripheral region for monitoring the rest of the visual field. At about 2.5 0 visual angle away from the center of the fovea, visual acuity is already reduced by a half. When we 'look' (foveate) at a certain location in the visual scene, we direct our high-resolution fovea to analyze information in that location, taking a snap shot of the scene using our retina. Figure lA-C illustrate what a retina would see at each fixation. It is immediately obvious that our retinal image is severely limited - it is clear only in the fovea and is very blurry in the surround, posing a severe constraint on the information available to our inference system. Yet, in our subjective experience, the world seems to be stable, coherent and complete in front of us. This is a paradox that have engaged philosophical and scientific debates for ages. To overcome the constraint of the retinal image, during perception, the brain actively moves the eyes around to (1) gather information to construct a mental image of the world, and (2) to make inference about the world based on this mental image. Understanding the forces that drive saccadic eye movements is important to elucidating the principles of active perception. A B C D Figure 1. A-C: retinal images in three separate fixations. D: a mental mosaic created by integrating the retinal images from these three and other three fixations. It is intuitive to think that eye movements are used to gather information. Eye movements have been suggested to provide a means for measuring the allocation of attention or the values of each kind of information in a particular context [16]. The basic assumption of our theory is that we move our eyes around to maximize our information intake from the world, for constructing the mental image and for making inference of the scene. Therefore, the system should always look for and attentively fixate at a location in the retinal image that is the most unusual or the most unexplained - and hence carries the maximum amount of information. 2 Perceptual Representation How can the brain decide which part of the retinal image is more unusual? First of all, we know the responses of VI simple cells, modeled well by the Gabor wavelet pyramid [3,7], can be used to reconstruct completely the retinal image. It is also well established that the receptive fields of these neurons developed in such a way as to provide a compact code for natural images [8,9,13,14]. The idea of compact code or sparse code, originally proposed by Barlow [2], is that early visual neurons capture the statistical correlations in natural scenes so that only a small number T. S. Lee and S. X Yu 836 of cells out of a large set will be activated to represent a particular scene at each moment. Extending this logic, we suggest that the complexity or the entropy of the neuronal ensemble response of a hypercolumn in VI is therefore closely related to the strangeness of the image features being analyzed by the machinery in that hypercolumn. A frequent event will have a more compact representation in the neuronal ensemble response. Entropy is an information measure that captures the complexity or the variability of signals. The entropy of a neuronal ensemble in a hypercolumn can therefore be used to quantify the strangeness of a particular event. A hypercolumn in the visual cortex contains roughly 200,000 neurons, dedicated to analyzing different aspects of the image in its 'visual window' . These cells are tuned to different spatial positions, orientations, spatial frequency, color disparity and other cues. There might also be a certain degree of redundancy, i.e. a number of neurons are tuned to the same feature . Thus a hypercolumn forms the fundamental computational unit for image analysis within a particular window in visual space. Each hypercolumn contains cells with receptive fields of different sizes, many significantly smaller than the aggregated 'visual window' of the hypercolumn. The entropy of a hypercolumn's ensemble response at a certain time t is the sum of entropies of all the channels, given by, H(u(R:;, t)) = - 2: 2:p(u(R:;, v, 0', B, t)) log2P(u(R:;, v, 0', B, t)) 9,<7 V where u(R:;, t) denotes the responses of all complex cell channels inside the visual window R:; of a hypercolumn at time t, computed within a 20 msec time window. u(i, 0', B, t) is the response of a VI complex cell channel of a particular scale 0' and orientation 0' at spatial location i at t. p(u(R:;, v, 0', B, t)) is the probability of cells in that channel within the visual window R:; of the hypercolumn firing v number of spikes . v can be computed as the power modulus of the corresponding simple cell channels, modeled by Gabor wavelets [see 7] . L:v p(u(R:;, v, 0', B, t)) 1. The probability p(u(R:;, v, 0', B, t)) can be computed at each moment in time because of the variations in spatial position of the receptive fields of similar cell within the hypercolumn - hence the 'same' cells in the hypercolumn are analyzing different image patches, and also because of the redundancy of cells coding similar features. = The neurons' responses in a hypercolumn are subject to contextual modulation from other hypercolumns, partly in the form of lateral inhibition from cells with similar tunings. The net observed effect is that the later part of VI neurons' response, starting at about 80 msec, exhibits differential suppression depending on the spatial extent and the nature of the surround stimulus. The more similar the surround stimulus is to the center stimuli, and the larger the spatial extent of the 'similar surround', the stronger is the suppressive effect [e.g. 6]. Simoncelli and Schwartz [15] have proposed that the steady state responses of the cells can be modeled by dividing the response of the cell (i.e. modeled by the wavelet coefficient or its power modulus) by a weighted combination ofthe responses of its spatial neighbors in order to remove the statistical dependencies between the responses of spatial neighbors. These weights are found by minimizing a predictive error between the center signal from the surround signals. In our context, this idea of predictive coding [see also 14] is captured by the concept of mutual information between the ensemble responses of the different hypercolumns as given below, I(u(R x , t); u(Ox, t - dtd) H(u(R:;, t)) - H(u(Rx , t)lu(Ox, t - dtd) 2: 2: [P(U(R:;,VR,O',B,t),u(O:;,vn,O',B,t)) <7,9 I VR,VO p(u(Rx, VR, 0', B, t), u(Ox, vn, 0', B, t)) ] og2 p(u(R:;, vR,O',B,t)),p(u(O:;,vn, O',B,t)) . Information-Theoretic Framework for Understanding Saccadic Behaviors 837 where u(Rx, t) is the ensemble response of the hypercolumn in question, and u(Ox, t) is the ensemble response of the surrounding hypercolumns. p(u(Rx , VR, (1', (), t)) is the probability that cells of a channel in the center hypercolumn assumes the response value VR and p(u(Ox, VR, (1', (), t)) the probability that cells of a similar channel in the surrounding hypercolumns assuming the response value Vn. tl is the delay by which the surround information exerts its effect on the center hypercolumn. The mutual information I can be computed from the joint probability of ensemble responses of the center and the surround. The steady state responses of the VI neurons, as a result of this contextual modulation, are said to be more correlated to perceptual pop-out than the neurons' initial responses [5,6]. The complexity of the steady state response in the early visual cortex is described by the following conditional entropy, H(u(R x , t)lu(Ox, t - dtd) = H(u(R x , t)) - I(u(R x , t); u(Ox, t - dtd). However, the computation in VI is not limited to the creation of compact representation through surround inhibition. In fact, we have suggested that VI plays an active role in scene interpretation particularly when such inference involves high resolution details [6]. Visual tasks such as the inference of contour and surface likely involve VI heavily. These computations could further modify the steady state responses of VI, and hence the control of saccadic eye movements. 3 Mental Mosaic Representation The perceptual representation provides the basic force for the brain to steer the high resolution fovea to locations of maximum uncertainty or maximum signal complexity. Foveation captures the maximum amount of available information in a location. Once a location is examined by the fovea, its information uncertainty is greatly reduced. The eyes should move on and not to return to the same spot within a certain period of time. This is called the 'inhibition of return'. How can we model this reduction of interest? We propose that the mind creates a mental mosaic of the scene in order to keep track of the information that have been gathered. By mosaic, we mean that the brain can assemble successive retinal images obtained from multiple fixations into a coherent mental picture of the scene. Figure ID provides an example of a mental mosaic created by combining information from the retinal images from 6 fixations. Whether the brain actually keeps such a mental mosaic of the scene is currently under debate. McConkie and Rayner [10] had suggested the idea of an integrative visual buffer to integrate information across multiple saccades. However, numerous experiments demonstrated we actually remember relatively little across saccades [4]. This lead to the idea that brain may not need an explicit internal representation of the world. Since the world is always out there, the brain can access whatever information it needs at the appropriate details by moving the eyes to the appropriate place at the appropriate time. The subjective feeling of a coherent and a complete world in front of us is a mere illusion [e.g. 1]. The mental mosaic represented in Figure ID might resemble McConkie and Rayner's theory superficially. But the existence of such a detailed high-resolution buffer with a large spatial support in the brain is rather biologically implausible. Rather, we think that the information corresponding to the mental mosaic is stored in an interpreted and semantic form in a mesh of Bayesian belief networks in the brain (e.g. involving PO, IT and area 46). This distributed semantic representation of T. S. Lee and S. X Yu 838 the mental mosaic, however, is capable of generating detailed (sometimes false) imagery in early visual cortex using the massive recurrent convergent feedback from the higher areas to VI. However, because of the limited support provided by VI machinery, the instantiation of mental imagery in VI has to be done sequentially one 'retinal image' frame at a time, presumably in conjunction with eye movement, even when the eyes are closed. This might explain why vivid visual dream is always accompanied by rapid eye movement in REM sleep. The mental mosaic accumulates information from the retinal images up to the last fixation and can provide prediction on what the retina will see in the current fixation. For each u(i, (T, 0) cell, there is a corresponding effective prediction signal m(i, (T, 0) fed back from the mental mosaic. This prediction signal can reduce the conditional entropy or complexity of the ensemble response in the perceptual representation by discounting the mutual information between the ensemble response to the retinal image and the mental mosaic prediction as follow, H(u(R x , t)lm(Rx, t - 6t2)) = H(u(R x , t)) - I(u(R x , t), m(Rx , t - dt2)) where 6t2 is the transmission delay from the mental mosaic back to VI. At places where the fovea has visited, the mental mosaic representation has high resolution information and m(i, (T, 0, t - 6t2) can explain u(i, (T, 0, t) fully. Hence, the mutual information is high at those hypercolumns and the conditional entropy H(u(R x , t)lm(R x , t - 6t2)) is low, with two consequences: (1) the system will not get the eyes stuck at a particular location; once the information at i is updated to the mental mosaic, the system will lose interest and move on; (2) the system will exhibit 'inhibition of return' as the information in the visited locations are fully predicted by the mental mosaic. Also, from this standpoint, the 'habituation dynamics' often observed in visual neurons when the same stimulus is presented multiple times might not be simply due to neuro-chemical fatigue, but might be understood in terms of mental mosaic being updated and then fed back to explain the perceptual representation in VI. The mental mosaic is in effect our short-term memory of the scene. It has a forgetting dynamics, and needs to be periodically updated. Otherwise, it will rapidly fade away. 4 Overall Reactive Saccadic Behaviors Now, we can combine the influence of the two predictive processes to arrive at a discounted complexity measure of the hypercolumn's ensemble response: H(u(R x , t)) -I(u(Rx , t); u(Ox, t - 6tt)) -I(u(Rx , t); m(R x , t - 6t2)) +I(u(Ox , t - 6td; m(Rx , t - 6t2)) If we can assume the long range surround priors and the mental mosaic short term memory are independent processes, we can leave out the last term, I(u(Ox, t 6td; m(Rx , t - 6t2)), of the equation. The system, after each saccade, will evaluate the new retinal scene and select the location where the perceptual representation has the maximum conditional entropy. To maximize the information gain, the system must constantly search for and make a saccade to the locations of maximum uncertainty (or complexity) computed from Information-Theoretic Framework for Understanding Saccadic Behaviors 839 the hypercolumn ensemble responses in VI at each fixation. Unless the number of saccades is severely limited, this locally greedy algorithm, coupled the inhibition of return mechanism, will likely steer the system to a relatively optimal global sampling of the world - in the sense that the average information gain per saccade is maximized, and the mental mosaic's dissonance with the world is minimized . 5 Task-dependent schema Representation However, human eye movements are not simply controlled by the generic information in a bottom-up fashion . Yarbus [16] has shown that, when staring at a face, subjects' eyes tend to go back to the same locations (eyes, mouth) over and over again. Further, he showed that when asked different questions, subjects exhibited different kinds of scan-paths when looking at the same picture. Norton and Stark [12] also showed that eye movements are not random, but often exhibit repetitive or even idiosyncratic path patterns. To capture these ideas, we propose a third representation, called task schema, to provide the necessary top-down information to bias the eye movement control. It specifies the learned or habitual scan-paths for a particular task in a particular context or assigns weights to different types of information. Given that we arenot mostly unconscious of the scan-path patterns we are making, these task-sensitive or context-sensitive habitual scan-patterns might be encoded at the levels of motor programs, and be downloaded when needed without our conscious control. These motor programs for scan-paths can be trained from reinforcement learning. For example, since the eyes and the mouths convey most of the emotional content of a facial expression, a successful interpretation of another person's emotion could provide the reward signal to reinforce the motor programs just executed or the fixations to certain facial features. These unconscious scan-path motor programs could provide the additional modulation to automatic saccadic eye movement generation. 6 Discussion In this paper, we propose that information maximization might provide a theoretical framework to understand the automatic saccadic eye movement behaviors in human. In this proposal, each hypercolumn in V 1 is considered a fundamental computational unit. The relative complexity or entropy of the neuronal ensemble response in the VI hypercolumns, discounted by the predictive effect of the surround, higher order representations and working memory, creates a force field to guide eye navigation. The framework we sketched here bridge natural scene statistics to eye movement control via the more established ideas of sparse coding and predictive coding in neural representation. Information maximization has been suggested to be a possible explanation for shaping the receptive fields in the early visual cortex according to the statistics of natural images [8,9,13,14] to create a minimum-entropy code [2,3]. As a result, a frequent event is represented efficiently with the response of a few neurons in a large set, resulting in a lower hypercolumn ensemble entropy, while unusual events provoke ensemble responses of higher complexity. We suggest that higher complexity in ensemble responses will arouse attention and draw scrutiny by the eyes, forcing the neural representation to continue adapting to the statistics of the natural scenes. The formulation here also suggests that information maximization might provide an explanation for the formation of horizontal predictive network in VI as well as higher order internal representations, consistent with the ideas of predictive coding [11, 14, 15]. Our theory hence predicts that the adaptation of the 840 T. S. Lee and S. X Yu neural representations to the statistics of natural scenes will lead to the adaptation of? saccadic eye movement behaviors. Acknowledgements The authors have been supported by a grant from the McDonnell Foundation and a NSF grant (LIS 9720350). Yu is also being supported in part by a grant to Takeo Kanade. References [1] Ballard, D. Hayhoe, M.M. Pook, P.K. & Rao, RP.N. (1997). Deictic codes for the embodiment of cognition. Behavioral and Brain Science, 20:4, December, 723-767. [2] Barlow, H.B. (1989). Unsupervised learning. Neural Computation, 1, 295-311. [3] Daugman, J.G. (1989). Entropy reduction and decorrelation in visual coding by oriented neural receptive fields. IEEE Transactions on Biomedical Engineering 36:, 107-114. [4] Irwin, D. E, 1991. Information Integration across Saccadic Eye Movements. Cognitive Psychology, 23(3):420-56. [5] Knierim, J. & Van Essen, D.C. Neural response to static texture patterns in area VI of macaque monkey. J. Neurophysiology, 67: 961-980. [6] Lee, T.S., Mumford, D., Romero R & Lamme, V.A.F. (1998). The role of primary visual cortex in higher level vision. Vision Research 38, 2429-2454. [7] Lee, T.S. (1996). Image representation using 2D Gabor wavelets. IEEE Transaction of Pattern Analysis and Machine Intelligence. 18:10, 959-971. [8] Lewicki, M. & Olshausen, B. (1998). Inferring sparse, overcomplete image codes using an efficient coding framework. In Advances in Neural Information Processing System 10, M. Jordan, M. Kearns and S. Solla (eds). MIT Press. [9] Linsker, R (1989). How to generate ordered maps by maximizing the mutual information between input and output signals. Neural Computation, 1: 402-411.0 [10] McConkie, G.W. & Rayner, K. (1976) . Identifying the span of effective stimulus in reading. Literature review and theories of reading. In H. Singer and RB. Ruddell (Eds), Theoretical models and processes of reading, 137-162. Newark, D.E.: International Reading Association. [11] Mumford, D. (1992). On the computational architecture of the neocortex II. Biological cybernetics, 66, 241-251. [12] Norton, D. and Stark, 1. (1971) Eye movements and visual perception. American, 224, 34-43. Scientific [13] Olshausen, B.A., & Field, D.J. (1996), Emergence of simple cell receptive field properties by learning a sparse code for natural images. Nature, 381: 607-609. [14] Rao R., & Ballard, D.H. (1999). Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive field effects. Nature Neuroscience, 2:1 7987. [15] Simoncelli, E.P. & Schwartz, O. (1999). Modeling surround suppression in VI neurons with a statistically-derived normalization model. In Advances in Neural Information Processing Systems 11, . M.S. Kearns, S.A. Solla, and D.A. Cohn (eds). MIT Press. [16] Yarbus, A.L. (1967). Eye movements and vision. Plenum Press. 

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Online Independent Component Analysis With Local Learning Rate Adaptation Nicol N. Schraudolph Xavier Giannakopoulos nic<Didsia.ch xavier<Didsia.ch IDSIA, Corso Elvezia 36 6900 Lugano, Switzerland http://www.idsia.ch/ Abstract Stochastic meta-descent (SMD) is a new technique for online adaptation of local learning rates in arbitrary twice-differentiable systems. Like matrix momentum it uses full second-order information while retaining O(n) computational complexity by exploiting the efficient computation of Hessian-vector products. Here we apply SMD to independent component analysis, and employ the resulting algorithm for the blind separation of time-varying mixtures. By matching individual learning rates to the rate of change in each source signal's mixture coefficients, our technique is capable of simultaneously tracking sources that move at very different, a priori unknown speeds. 1 Introduction Independent component analysis (ICA) methods are typically run in batch mode in order to keep the stochasticity of the empirical gradient low. Often this is combined with a global learning rate annealing scheme that negotiates the tradeoff between fast convergence and good asymptotic performance. For time-varying mixtures, this must be replaced by a learning rate adaptation scheme. Adaptation of a single, global learning rate, however, facilitates the tracking only of sources whose mixing coefficients change at comparable rates [1], resp. switch all at the same time [2]. In cases where some sources move much faster than others, or switch at different times, individual weights in the unmixing matrix must adapt at different rates in order to achieve good performance. We apply stochastic meta-descent (SMD), a new online adaptation method for local learning rates [3, 4], to an extended Bell-Sejnowski ICA algorithm [5] with natural gradient [6] and kurtosis estimation [7] modifications. The resulting algorithm is capable of separating and tracking a time-varying mixture of 10 sources whose unknown mixing coefficients change at different rates. N. N. Schraudo!ph and X Giannakopou!os 790 2 The SMD Algorithm Given a sequence XQ, Xl, ... of data points, we minimize the expected value of a twice-differentiable loss function fw(x) with respect to its parameters W by stochastic gradient descent: Wt+l = Wt + Pt? 8t , where It (1) and . denotes component-wise multiplication. The local learning rates P are best adapted by exponentiated gradient descent [8, 9], so that they can cover a wide dynamic range while staying strictly positive: lnpt 1 ... npt-l - a fwt (xt) I-t alnp Pt-l . exp(1-t It .Vt) , (2) where Vt and I-t is a global meta-learning rate. This approach rests on the assumption that each element of P affects f w( x) only through the corresponding element of w. With considerable variation, (2) forms the basis of most local rate adaptation methods found in the literature. In order to avoid an expensive exponentiation [10] for each weight update, we typically use the linearization etL ~ 1 + u, valid for small luI, giving (3) where we constrain the multiplier to be at least (typically) (} = 0.1 as a safeguard against unreasonably small - or negative - values. For the meta-level gradient descent to be stable, I-t must in any case be chosen such that the multiplier for P does not stray far from unity; under these conditions we find the linear approximation (3) quite sufficient. Definition of v. The gradient trace v should accurately measure the effect that a change in local learning rate has on the corresponding weight. It is tempting to consider only the immediate effect of a change in Pt on Wt+l: declaring Wt and 8t in (1) to be independent of Pt, one then quickly arrives at ... Vt+l _ aWt+l ~l ... = u npt = ... ~ (4) Pt? Ut However, this common approach [11, 12, 13, 14, 15] fails to take into account the incremental nature of gradient descent: a change in P affects not only the current update of W, but also future ones. Some authors account for this by setting v to an exponential average of past gradients [2, 11, 16]; we found empirically that the method of Almeida et al. [15] can indeed be improved by this approach [3]. While such averaging serves to reduce the stochasticity of the product ?It-l implied by (3) and (4), the average remains one of immediate, single-step effects. It By contrast, Sutton [17, 18] models the long-term effect of P on future weight updates in a linear system by carrying the relevant partials forward through time, as is done in real-time recurrent learning [19]. This results in an iterative update rule for v, which we have extended to nonlinear systems [3, 4]. We define vas an 791 Online leA with Local Rate Adaptation exponential average of the effect of all past changes in p on the current weights: ... Vt+ 1 - = ( 1- A) ~ i=O The forgetting factor 0 into (5) gives ~ d OWt+1 . npt-i L.J 1\ 0 1 ... (5) A ~ 1 is a free parameter of the algorithm. Inserting (1) (6) where H t denotes the instantaneous Hessian of fw(i!) at time t. The approximation in (6) assumes that (Vi> 0) oPt!OPt-i = 0; this signifies a certain dependence on an appropriate choice of meta-learning rate p.. Note that there is an efficient O(n) algorithm to calculate HtVt without ever having to compute or store the matrix H t itself [20]; we shall elaborate on this technique for the case of independent component analysis below. Meta-level conditioning. The gradient descent in P at the meta-level (2) may of course suffer from ill-conditioning just like the descent in W at the main level (1); the meta-descent in fact squares the condition number when v is defined as the previous gradient, or an exponential average of past gradients. Special measures to improve conditioning are thus required to make meta-descent work in non-trivial systems. Many researchers [11, 12, 13, 14] use the sign function to radically normalize the p-update. Unfortunately such a nonlinearity does not preserve the zero-mean property that characterizes stochastic gradients in equilibrium -- in particular, it will translate any skew in the equilibrium distribution into a non-zero mean change in p. This causes convergence to non-optimal step sizes, and renders such methods unsuitable for online learning. Notably, Almeida et al. [15] avoid this pitfall by using a running estimate of the gradient's stochastic variance as their meta-normalizer. In addition to modeling the long-term effect of a change in local learning rate, our iterative gradient trace serves as a highly effective conditioner for the meta-descent: the fixpoint of (6) is given by Vt = [AHt + (I-A) diag(I/Pi)]-llt (7) - a modified Newton step, which for typical values of A (i. e., close to 1) scales with the inverse of the gradient. Consequently, we can expect the product It . Vt in (2) to be a very well-conditioned quantity. Experiments with feedforward multi-layer perceptrons [3, 4] have confirmed that SMD does not require explicit meta-level normalization, and converges faster than alternative methods. 3 Application to leA We now apply the SMD technique to independent component analysis, using the Bell-Sejnowski algorithm [5] as our base method. The goal is to find an unmixing N. N. Schraudolph and X Giannakopoulos 792 matrix W t which - up to scaling and permutation - provides a good linear estimate Vt == WtXt of the independent sources St present in a given mixture signal Xt? The mixture is generated linearly according to Xt = Atst , where At is an unknown (and unobservable) full rank matrix. We include the well-known natural gradient [6] and kurtosis estimation [7] modifications to the basic algorithm, as well as a matrix Pt of local learning rates. The resulting online update for the weight matrix W t is (8) where the gradient D t is given by Dt == 8f;;~~t) = ([Vt ? tanh(Vt)] vt - 1) W t , (9) with the sign for each component of the tanh(Vt) term depending on its current kurtosis estimate. Following Pearlmutter [20], we now define the differentiation operator RVt (g(Wt? 8g(W~r+ rVt) Ir=o u == (10) which describes the effect on 9 of a perturbation of the weights in the direction of Vt. We can use RVt to efficiently calculate the Hessian-vector product (11) where "vee" is the operator that concatenates all columns of a matrix into a single column vector. Since Rv, is a linear operator, we have Rv,(Wt ) RVt (Vt) RVt (tanh(Vd) Vt, Rv, (WtXt) = VtXt, diag( tanh' (Vt?) VtXt , (12) (13) (14) and so forth (cf. [20]). Starting from (9), we apply the RVt operator to obtain Rv,[([Vt ? tanh(Vt)] ytT - 1) Wt] Ht*Vt - + RVt([ Yt ? tanh(Vt)] vt - 1) Wt + [(1 ? diag[tanh'(Vt)]) VtXt vt + [Vt ? tanh(Vd](Vtxt)T] W t ([Vt ? tanh(Vt)] ([ Vt ? tanh(Vt)] vt - 1) Vt vt - 1) Vt (15) In conjunction with the matrix versions of our learning rate update (3) (16) and gradient trace (6) (17) this constitutes our SMD-ICA algorithm. 793 Online leA with Local Rate Adaptation 4 Experiment The algorithm was tested on an artificial problem where 10 sources follow elliptic trajectories according to Xt = (Abase + Al sin(wt) + A2 cos(wt)) St (18) where Abase is a normally distributed mixing matrix, as well as Al and A 2, whose columns represent the axes of the ellipses on which the sources travel. The velocities ware normally distributed around a mean of one revolution for every 6000 data samples. All sources are supergaussian. The ICA-SMD algorithm was implemented with only online access to the data, including on-line whitening [21]. Whenever the condition number of the estimated whitening matrix exceeded a large threshold (set to 350 here), updates (16) and (17) were disabled to prevent the algorithm from diverging. Other parameters settings were It = 0.1, >. = 0.999, and p = 0.2. Results that were not separating the 10 sources without ambiguity were discarded. Figure 1 shows the performance index from [6] (the lower the better, zero being the ideal case) along with the condition number of the mixing matrix, showing that the algorithm is robust to a temporary confusion in the separation. The ordinate represents 3000 data samples, divided into mini-batches of 10 each for efficiency. Figure 2 shows the match between an actual mixing column and its estimate, in the subspace spanned by the elliptic trajectory. The singularity occurring halfway through is not damaging performance. Globally the algorithm remains stable as long as degenerate inputs are handled correctly. 5 Conclusions Once SMD-ICA has found a separating solution, we find it possible to simultaneously track ten sources that move independently at very different, a priori unknown OOr-------~------T_------~------,_------~------~ Error index cond(A)120 ---+--- 50 40 30 Figure 1: Global view of the quality of separation N. N. Schraudolph and X Giannakopoulos 794 Or---.---.,~--,----.----~---.----.----.----r---. Estimation error - -, -2 -3 -4 -5 -, -6~--~--~~--~--~----~--~----~---L----~--~ -2.5 -2 -'.5 -0.5 o 0.5 '.5 2 2.5 Figure 2: Projection of a column from the mixing matrix. Arrows link the exact point with its estimate; the trajectory proceeds from lower right to upper left. speeds. To continue tracking over extended periods it is necessary to handle momentary singularities, through online estimation of the number of sources or some other heuristic solution. SMD's adaptation of local learning rates can then facilitate continuous, online use of ICA in rapidly changing environments. Acknowledgments This work was supported by the Swiss National Science Foundation under grants number 2000-052678.97/1 and 2100-054093.98. References [1] J. Karhunen and P. Pajunen, "Blind source separation and tracking using nonlinear PCA criterion: A least-squares approach", in Proc. IEEE Int. Conf. on Neural Networks, Houston, Texas, 1997, pp. 2147- 2152. [2] N. Murata, K.-R. Milller, A. Ziehe, and S.-i. Amari, "Adaptive on-line learning in changing environments", in Advances in Neural Information Processing Systems, M. C. Mozer, M. I. Jordan, and T . Petsche, Eds. 1997, vol. 9, pp. 599- 605, The MIT Press, Cambridge, MA. [3] N. N. Schraudolph, "Local gain adaptation in stochastic gradient descent", in Proceedings of the 9th International Conference on Artificial Neural Networks, Edinburgh, Scotland, 1999, pp. 569-574, lEE, London, ftp://ftp.idsia.ch/ pub/nic/smd.ps.gz. [4] N. N. Schraudolph, "Online learning with adaptive local step sizes", in Neural Nets - WIRN Vietri-99; Proceedings of the 11th Italian Workshop on Neural Nets, M. Marinaro and R. Tagliaferri, Eds., Vietri suI Mare, Salerno, Italy, 1999, Perspectives in Neural Computing, pp. 151-156, Springer Verlag, Berlin. Online leA with Local Rate Adaptation 795 [5] A. J. Bell and T. J. Sejnowski, "An information-maximization approach to blind separation and blind deconvolution", Neural Computation, 7(6):11291159,1995. [6] S.-i. Amari, A. Cichocki, and H. H. Yang, "A new learning algorithm for blind signal separation", in Advances in Neural Information Processing Systems, D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, Eds. 1996, vol. 8, pp. 757-763, The MIT Press, Cambridge, MA. [7] M. Girolami and C. Fyfe, "Generalised independent component analysis through unsupervised learning with emergent bussgang properties", in Proc. IEEE Int. Conf. on Neural Networks, Houston, Texas, 1997, pp. 1788-179l. [8] J. Kivinen and M. K. Warmuth, "Exponentiated gradient verSus gradient descent for linear predictors", Tech. Rep. UCSC-CRL-94-16, University of California, Santa Cruz, June 1994. [9] J. Kivinen and M. K. Warmuth, "Additive versus exponentiated gradient updates for linear prediction", in Proc. 27th Annual ACM Symposium on Theory of Computing, New York, NY, May 1995, pp. 209-218, The Association for Computing Machinery. [10] N. N. Schraudolph, "A fast, compact approximation of the exponential function", Neural Computation, 11(4):853-862, 1999. [11] R. Jacobs, "Increased rates of convergence through learning rate adaptation", Neural Networks, 1:295- 307, 1988. [12] T. Tollenaere, "SuperSAB: fast adaptive back propagation with good scaling properties", Neural Networks, 3:561-573, 1990. [13] F. M. Silva and L. B. Almeida, "Speeding up back-propagation", in Advanced Neural Computers, R. Eckmiller, Ed., Amsterdam, 1990, pp. 151-158, Elsevier. [14] M. Riedmiller and H. Braun, "A direct adaptive method for faster backpropagation learning: The RPROP algorithm", in Proc. International Conference on Neural Networks, San Francisco, CA, 1993, pp. 586-591, IEEE, New York. [15] L. B. Almeida, T. Langlois, J. D. Amaral, and A. Plakhov, "Parameter adaptation in stochastic optimization", in On-Line Learning in Neural Networks, D. Saad, Ed., Publications of the Newton Institute, chapter 6. Cambridge University Press, 1999, ftp://146.193. 2 . 131/pub/lba/papers/adsteps . ps .gz. [16] M. E. Harmon and L. C. Baird III, "Multi-player residual advantage learning with general function approximation" , Tech. Rep. WL-TR-1065, Wright Laboratory, WL/ AACF, 2241 Avionics Circle, Wright-Patterson Air Force Base, OH 45433-7308, 1996, http://vvv.leemon.com/papers/sim_tech/sim_tech.ps.gz. [17] R. S. Sutton, "Adapting bias by gradient descent: an incremental version of delta-bar-delta", in Proc. 10th National Conference on Artificial Intelligence. 1992, pp. 171-176, The MIT Press, Cambridge, MA, ftp://ftp.cs.umass.edu/ pub/anv/pub/sutton/sutton-92a.ps.gz. [18] R. S. Sutton, "Gain adaptation beats least squares?", in Proc. 7th Yale Workshop on Adaptive and Learning Systems, 1992, pp. 161-166, ftp://ftp.cs. umass.edu/pub/anv/pub/sutton/sutton-92b.ps.gz. [19] R. Williams and D. Zipser, "A learning algorithm for continually running fully recurrent neural networks", Neural Computation, 1:270-280, 1989. [20] B. A. Pearlmutter, "Fast exact multiplication by the Hessian", Neural Computation, 6(1):147-160,1994. [21] J. Karhunen, E. Oja, L. Wang, R. Vigario, and J. Joutsensalo, "A class of neural networks for independent component analysis", IEEE Trans. on Neural Networks, 8(3):486-504, 1997. 

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Local probability propagation for factor analysis Brendan J. Frey Computer Science, University of Waterloo, Waterloo, Ontario, Canada Abstract Ever since Pearl's probability propagation algorithm in graphs with cycles was shown to produce excellent results for error-correcting decoding a few years ago, we have been curious about whether local probability propagation could be used successfully for machine learning. One of the simplest adaptive models is the factor analyzer, which is a two-layer network that models bottom layer sensory inputs as a linear combination of top layer factors plus independent Gaussian sensor noise. We show that local probability propagation in the factor analyzer network usually takes just a few iterations to perform accurate inference, even in networks with 320 sensors and 80 factors. We derive an expression for the algorithm's fixed point and show that this fixed point matches the exact solution in a variety of networks, even when the fixed point is unstable. We also show that this method can be used successfully to perform inference for approximate EM and we give results on an online face recognition task. 1 Factor analysis A simple way to encode input patterns is to suppose that each input can be wellapproximated by a linear combination of component vectors, where the amplitudes of the vectors are modulated to match the input. For a given training set, the most appropriate set of component vectors will depend on how we expect the modulation levels to behave and how we measure the distance between the input and its approximation. These effects can be captured by a generative probabilit~ model that specifies a distribution p(z) over modulation levels z = (Zl, ... ,ZK) and a distribution p(xlz) over sensors x = (Xl, ... ,XN)T given the modulation levels. Principal component analysis, independent component analysis and factor analysis can be viewed as maximum likelihood learning in a model of this type, where we assume that over the training set, the appropriate modulation levels are independent and the overall distortion is given by the sum of the individual sensor distortions. In factor analysis, the modulation levels are called factors and the distributions have the following form: p(Zk) = N(Zk; 0,1), p(z) = nf=lP(Zk) = N(z; 0, I), p(xnl z ) = N(xn; E~=l AnkZk, 'l/Jn), p(xlz) = n:=IP(xnlz) = N(x; Az, 'It). (1) The parameters of this model are the factor loading matrix A, with elements Ank, and the diagonal sensor noise covariance matrix 'It, with diagonal elements 'l/Jn. A belief network for the factor analyzer is shown in Fig. 1a. The likelihood is p(x) = 1 N(z; 0, I)N(x; Az, 'It)dz = N(x; 0, AA T + 'It), (2) 443 Local Probability Propagation for Factor Analysis (b) - ...., -... - 'J " ... 'r .... 1t. , E '" "' I:~ ~. '. Figure 1: (a) A belief network for factor analysis. (b) High-dimensional data (N = 560). and online factor analysis consists of adapting A and q, to increase the likelihood of the current input, such as a vector of pixels from an image in Fig. lb. Probabilistic inference - computing or estimating p{zlx) - is needed to do dimensionality reduction and to fill in the unobserved factors for online EM-type learning. In this paper, we focus on methods that infer independent factors. p(zlx) is Gaussian and it turns out that the posterior means and variances of the factors are E[zlx] = (ATq,-l A + 1)-1 AT q,-lx, diag(COV(zlx)) = diag(AT q,-l A + 1)-1). (3) Given A and q" computing these values exactly takes O(K2 N) computations, mainly because of the time needed to compute AT q,-l A. Since there are only K N connections in the network, exact inference takes at least O{K) bottom-up/top down iterations. Of course, if the same network is going to be applied more than K times for inference (e.g., for batch EM), then the matrices in (3) can be computed once and reused. However, this is not directly applicable in online learning and in biological models. One way to circumvent computing the matrices is to keep a separate recognition network, which approximates E[zlx] with Rx (Dayan et al., 1995). The optimal recognition network, R = (A Tq,-l A+I)-l ATq,-l, can be approximated by jointly estimating the generative network and the recognition network using online wakesleep learning (Hinton et al., 1995). 2 Probability propagation in the factor analyzer network Recent results on error-correcting coding show that in some cases Pearl's probability propagation algorithm, which does exact probabilistic inference in graphs that are trees, gives excellent performance even if the network contains so many cycles that its minimal cut set is exponential (Frey and MacKay, 1998; Frey, 1998; MacKay, 1999). In fact, the probability propagation algorithm for decoding lowdensity parity-check codes (MacKay, 1999) and turbocodes (Berrou and Glavieux, 1996) is widely considered to be a major breakthrough in the information theory community. When the network contains cycles, the local computations give rise to an iterative algorithm, which hopefully converges to a good answer. Little is known about the convergence properties of the algorithm. Networks containing a single cycle have been successfully analyzed by Weiss (1999) and Smyth et al. (1997), but results for networks containing many cycles are much less revealing. The probability messages produced by probability propagation in the factor analyzer network of Fig. 1a are Gaussians. Each iteration of propagation consists of passing a mean and a variance along each edge in a bottom-up pass, followed by passing a mean and a variance along each edge in a top-down pass. At any instant, the 444 B.J. Frey bottom-up means and variances can be combined to form estimates of the means and variances of the modulation levels given the input. Initially, the variance and mean sent from the kth top layer unit to the nth sensor is set to vk~ = 1 and 7]i~ = 0. The bottom-up pass begins by computing a noise level and an error signal at each sensor using the top-down variances and means from the previous iteration: s~) = 'l/Jn + 2:{:=1 A;kVk~-I) , e~) = Xn - 2: {:=1Ank7]i~-l). (4) These are used to compute bottom-up variances and means as follows: ",(i) = s(i)/A2 _ v(i-l) lI(i) = e(i)/A k + 7](i-l) (5) 'l'nk n nk kn' r'nk n n kn' The bottom-up variances and means are then combined to form the current estimates of the modulation variances and means: (i) Vk N (i) A(i) _ = 1/(1 + 2:n=1 1/?nk)' Zk - (i)"",N Vk (i)/",(i) L..Jn=lJ.tnk 'l'nk' (6) The top-down pass proceeds by computing top-down variances and means as follows: vk~ = l/(l/vii ) - l/?~l), 7]i~ = vk~(.iki) /vii ) - J.t~V?~l)? (7) Notice that the variance updates are independent of the mean updates, whereas the mean updates depend on the variance updates. 2.1 Performance of local probability propagation. We created a total of 200,000 factor analysis networks with 20 different sizes ranging from K = 5, N = 10 to K = 80, N = 320 and for each size of network we measured the inference error as a function of the number of iterations of propagation. Each of the 10,000 networks of a given size was produced by drawing the AnkS from standard normal distributions and then drawing each sensor variance 'l/Jn from an exponential distribution with mean 2:{:=1 A;k' (A similar procedure was used by Neal and Dayan (1997).) For each random network, a pattern was simulated from the network and probability propagation was applied using the simulated pattern as input. We measured the error between the estimate z(i) and the correct value E[zlx] by computing the difference between their coding costs under the exact posterior distribution and then normalizing by K to get an average number of nats per top layer unit. Fig. 2a shows the inference error on a logarithmic scale versus the number of iterations (maximum of 20) in the 20 different network sizes. In all cases, the median error is reduced below .01 nats within 6 iterations. The rate of convergence of the error improves for larger N, as indicated by a general trend for the error curves to drop when N is increased. In contrast, the rate of convergence of the error appears to worsen for larger K, as shown by a general slight trend for the error curves to rise when K is increased. For K ~ N/8, 0.1% of the networks actually diverge. To better understand the divergent cases, we studied the means and variances for all of the divergent networks. In all cases, the variances converge within a few iterations whereas the means oscillate and diverge. For K = 5, N = 10, 54 of the 10,000 networks diverged and 5 of these are shown in Fig. 2b. This observation suggests that in general the dynamics are determined by the dynamics of the mean updates. 2.2 Fixed points and a condition for global convergence. When the variance updates converge, the dynamics of probability propagation in factor analysis networks become linear. This allows us to derive the fixed point of propagation in closed form and write an eigenvalue condition for global convergence. 445 Local Probability Propagation for Factor Analysis (a) K = 5 K ~',:~ ~ 01~ = 10 K=20 K=40 K=80 g"Xl~ 10 ... II ~ 1:~ 11' 0 ~ ~ .01 ~ ',: u 2: 0, ~ ',: ~ ~ o 0' ,OO~ ~' O~ ~ 01 0 10 20 Figure 2: (a) Performance of probability propagation . Median inference error (bold curve) on a logarithmic scale as a function of the number of iterations for different sizes of network parameterized by K and N. The two curves adjacent to the bold curve show the range within which 98% of the errors lie. 99 .9% of the errors were below the fourth, topmost curve. (b) The error, bottom-up variances and top-down means as a function of the number of iterations (maximum of 20) for 5 divergent networks of size K = 5, N = 10. To analyze the system of mean updates, we define the following length K N vec. - (i) _ ( (i) (i) (i) (i ) (i))T - (i) _ ? tors 0 f means an d t he mput . TJ - 1711,1721"'" 17Kl' 1712' " . , 17KN , P, (i) (i) (i ) (i) (i) )T ( )T ( J-tll,J-t12 ' ''' ,J-tlK , J-t21'''' , J-tNK , X= Xl,Xl, .. ? ,Xl,X2, .. ? , X2 , XN, .. ? ,XN , where each Xn is repeated K times in the last vector. The network parameters are represented using K N x K N diagonal matrices, A and q,. The diagonal of A is A11, ... , AIK , A21, ... , ANK, and the diagonal of q, is '1/111, '1/121, ... , '1/INI, where 1 is the K x K identity matrix. The converged bottom-up variances are represented using a diagonal matrix ~ with diagonal ?11, ... , ?IK , ?21, .. . , ?NK. The summation operations in the propagation formulas are represented by a K N x K N matrix I: z that sums over means sent down from the top layer and a K N x K N matrix I: x that sums over means sent up from the sensory input: 1 1 ) :Ex = ' (~i 1 1 (8) 1 These are N x N matrices of K x K blocks, where 1 is the K x K block of ones and 1 is the K x K identity matrix. Using the above representations, the bottom-up pass is given by ji, (i) = A-I X _ A- I (:E z - I)Af7(i-l), (9) and the top-down pass is given by f7( i) = (I + diag(:Ex~ -1 :Ex) _ ~ -1) -1 (I: x _ I)~ -1 ji,( i ) . Substituting (10) into (9), we get the linear update for ji,(i) = A-I X _ A-I (:E z _ (10) ji,: I)A(I + diag(:Exci -l:E x) _ c) -1) -1 (:Ex _ I)ci -1 ji, (i -l). (11) B.J. Frey 446 B[]Bga~Q~g[] 1.24 1.07 1.49 1.13 1.03 1.02 1.09 1.01 1.11 1.06 Figure 3: The error (log scale) versus number of iterations (log scale. max. of 1000) in 10 of the divergent networks with K = 5. N = 10. The means were initialized to the fixed point solutions and machine round-off errors cause divergence from the fixed points. whose errors are shown by horizontal lines. The fixed point of this dynamic system, when it exists, is ji,* = ~ (A~ + (tz - I)A(I + diag(I:xc) -lt x) - ~ -1) -\t x - I)) -1 x. (12) A fixed point exists if the determinant of the expression in large braces in (12) is nonzero. We have found a simplified expression for this determinant in terms of the determinants of smaller, K x K matrices. Reinterpreting the dynamics in (11) as dynamics for Aji,(i), the stability of a fixed point is determined by the largest eigenvalue of the update matrix, (I: z - I)A (I + - - -1 - - -1 -1 - - -1 - -1 diag(Exc}) Ex)-c}) ) (Ex-I)c}) A . If the modulus ofthe largest eigenvalue is less than 1, the fixed point is stable. Since the system is linear, if a stable fixed point exists, the system will be globally convergent to this point. Of the 200,000 networks we explored, about 99.9% of the networks converged. For 10 of the divergent networks with K = 5, N = 10, we used 1000 iterations of probability propagation to compute the steady state variances. Then, we computed the modulus of the largest eigenvalue of the system and we computed the fixed point. After initializing the bottom-up means to the fixed point values, we performed 1000 iterations to see if numerical errors due to machine precision would cause divergence from the fixed point. Fig. 3 shows the error versus number of iterations (on logarithmic scales) for each network, the error of the fixed point, and the modulus of the largest eigenvalue. In some cases, the network diverges from the fixed point and reaches a dynamic equilibrium that has a lower average error than the fixed point. 3 Online factor analysis To perform maximum likelihood factor analysis in an online fashion, each parameter should be modified to slightly increase the log-probability of the current sensory input,logp(x). However, since the factors are hidden, they must be probabilistically "filled in" using inference before an incremental learning step is performed. If the estimated mean and variance of the kth factor are Zk and Vk, then it turns out (e.g., Neal and Dayan, 1997) the parameters can be updated as follows: Ank 'IjIn where 1} f- f- Ank + l}[Zk(Xn - Ef=1 AnjZj) - (l-l})'ljln + l}[(xn - VkAnk]/'ljln, Ef=1 AnjZj)2 + Ef=1 VkA~j], (13) is a learning rate. Online learning consists of performing some number of iterations of probability propagation for the current input (e.g., 4 iterations) and then modifying the parameters before processing the next input. 3.1 Results on simulated data. We produced 95 training sets of 200 cases each, with input sizes ranging from 20 sensors to 320 sensors. For each of 19 sizes of factor analyzer, we randomly selected 5 sets of parameters as described above and generated a training set. The factor analyzer sizes were K E {5, 10,20,40, 80}, Local Probability Propagation for Factor Analysis 447 Figure 4: (a) Achievable errors after the same number of epochs of learning using 4 iterations versus 1 iteration. The horizontal axis gives the log-probability error (log scale) for learning with 1 iteration and the vertical axis gives the error after the same number of epochs for learning with 4 iterations. (b) The achievable errors for learning using 4 iterations of propagation versus wake-sleep learning using 4 iterations. > K. For each factor analyzer and simulated data set, we estimated the optimal log-probability of the data using 100 iterations of EM. N E {20, 40, 80,160, 320}, N For learning, the size of the model to be trained was set equal to the size of the model that was used to generate the data. To avoid the issue of how to schedule learning rates, we searched for achievable learning curves, regardless of whether or not a simple schedule for the learning rate exists. So, for a given method and randomly initialized parameters, we performed one separate epoch of learning using each of the learning rates, 1,0.5, ... ,0.5 20 and picked the learning rate that most improved the log-probability. Each successive learning rate was determined by comparing the performance using the old learning rate and one 0.75 times smaller. We are mainly interested in comparing the achievable curves for different methods and how the differences scale with K and N. For two methods with the same K and N trained on the same data, we plot the log-probability error (optimal logprobability minus log-probability under the learned model) of one method against the log-probability error of the other method. Fig. 4a shows the achievable errors using 4 iterations versus using 1 iteration. Usually, using 4 iterations produces networks with lower errors than those learned using 1 iteration. The difference is most significant for networks with large K, where in Sec. 2.1 we found that the convergence of the inference error was slower. Fig. 4b shows the achievable errors for learning using 4 iterations of probability propagation versus wake-sleep learning using 4 iterations. Generally, probability propagation achieves much smaller errors than wake-sleep learning, although for small K wake-sleep performs better very close to the optimum log-probability. The most significant difference between the methods occurs for large K, where aside from local optima probability propagation achieves nearly optimal log-probabilities while the log-probabilities for wake-sleep learning are still close to their values at the start of learning. 4 Online face recognition Fig. 1b shows examples from a set of 30,000 20 x 28 greyscale face images of 18 different people. In contrast to other data sets used to test face recognition methods, these faces include wide variation in expression and pose. To make classification more difficult, we normalized the images for each person so that each pixel has B.J. Frey 448 the same mean and variance. We used probability propagation and a recognition network in a factor analyzer to reduce the dimensionality of the data online from 560 dimensions to 40 dimensions. For probability propagation, we rather arbitrarily chose a learning rate of 0.0001, but for wake-sleep learning we tried learning rates ranging from 0.1 down to 0.0001. A multilayer perceptron with one hidden layer of 160 tanh units and one output layer of 18 softmax units was simultaneously being trained using gradient descent to predict face identity from the mean factors. The learning rate for the multilayer perceptron was set to 0.05 and this value was used for both methods. For each image, a prediction was made before the parameters were modified. Fig. 5 shows online error curves obtained by filtering the losses. The curve for probability propagation is generally below the curves for wake-sleep learning. The figure also shows the error curves for two forms of online nearest neighbors, where only the most recent W cases are used to make a prediction. The form of nearest neighbors that performs the worst has W set so that the storage requirements are the same as for the factor analysis / multilayer perceptron method. The better form of nearest neighbors has W set so that the number of computations is the same as for the factor analysis / multilayer perceptron method. 5 Summary ~ j "' " \ ' i .. ',"--""~ \<',::,::'--, ...,... "" '. ~, '. Number of pattern presentations Figure 5: Online error curves for probability propagation (solid), wake-sleep learning (dashed), nearest neighbors (dot-dashed) and guessing (dotted). It turns out that iterative probability propagation can be fruitful when used for learning in a graphical model with cycles, even when the model is densely connected. Although we are more interested in extending this work to more complex models where exact inference takes exponential time, studying iterative probability propagation in the factor analyzer allowed us to compare our results with exact inference and allowed us to derive the fixed point of the algorithm. We are currently applying iterative propagation in multiple cause networks for vision problems. References C. Berrou and A. Glavieux 1996. Near optimum error correcting coding and decoding: Turbo-codes. IEEE TI-ans. on Communications, 44, 1261-1271. P. Dayan, G. E. Hinton, R. M. Neal and R. S. Zemel 1995. The Helmholtz machine. Neural Computation 1, 889-904. B. J. Frey and D. J. C. MacKay 1998. A revolution: Belief propagation in graphs with cycles. In M. Jordan, M. Kearns and S. Solla (eds), Advances in Neural Information Processing Systems 10, Denver, 1997. B. J. Frey 1998. Graphical Models for Machine Learning and Digital Communication. MIT Press, Cambridge MA. See http://wvv.cs.utoronto.ca/-frey . G. E. Hinton, P. Dayan, B. J. Frey and R. M. Neal 1995. The wake-sleep algorithm for unsupervised neural networks. Science 268, 1158-1161. D. J. C. MacKay 1999. Information Theory, Inference and Learning Algorithms. Book in preparation, currently available at http://wol.ra.phy.cam.ac . uk/mackay. R. M. Neal and P. Dayan 1997. Factor analysis using delta-rule wake-sleep learning. Neural Computation 9, 1781-1804. P. Smyth, R. J . McEliece, M. Xu, S. Aji and G. Horn 1997. Probability propagation in graphs with cycles. Presented at the workshop on Inference and Learning in Graphical Models, Vail, Colorado. Y. Weiss 1998. Correctness of local probability propagation in graphical models. To appear in Neural Computation. 

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537 A MASSIVELY PARALLEL SELF-TUNING CONTEXT-FREE PARSER! Eugene Santos Jr. Department of Computer Science Brown University Box 1910, Providence, RI 02912 eSj@cs.brown.edu ABSTRACT The Parsing and Learning System(PALS) is a massively parallel self-tuning context-free parser. It is capable of parsing sentences of unbounded length mainly due to its parse-tree representation scheme. The system is capable of improving its parsing performance through the presentation of training examples. INTRODUCTION Recent PDP research[Rumelhart et al .? 1986; Feldman and Ballard, 1982; Lippmann, 1987] involving natural language processtng[Fanty, 1988; Selman, 1985; Waltz and Pollack, 1985] have unrealistically restricted sentences to a fixed length. A solution to this problem was presented in the system CONPARSE[Charniak and Santos. 1987]. A parse-tree representation scheme was utilized which allowed for processing sentences of any length. Although successful as a parser. it's achitecture was strictly hand-constructed with no learning of any form. Also. standard learning schemes were not appUcable since it differed from all the popular architectures, in particular. connectionist ones. In this paper. we present the Parsing and Learning System(PALS) which attempts to integrate a learning scheme into CONPARSE. It basically allows CONPARSE to modify and improve its parsing capability. IThis research was supported in part by the Office of Naval Research under contract NOOOI4-79-C-0592, the National Science Foundation under contracts IST-8416034 and IST-8515005, and by the Defense Advanced Research Projects Agency under ARPA Order No. 4786. 538 Santos REPRESENTATION OF PARSE TREE A parse-tree Is represented by a matrix where the bottom row consists of the leaves of the tree In left-to-right order and the entries In each column above each leaf correspond to the nodes In the path from leaf to root. For example, looking at the simple parse-tree for the sentence "noun verb noun", the column entries for verb would be verb, vp. and S. (see Figure 1) (As In previous work, PALS takes part-of-speech as input, not words.) S S NP noun ,..S """'IIIl VP VP NP noun ... verb..oil Figure 1. Parse tree as represented by a collection of columns in the matrix. In addition to the columns of nontermlnals. we introduce the binder entries as a means of easily determining whether two Identical nonterminals in adjacent columns represent the same nonterminal in a parse tree (see Figure 2). , S NP noun -S VP verb / / S ~VP NP noun s ~ NP VP noun NP I verb '" noL Figure 2. Parse tree as represented by a collection of columns in the matrix plus binders. To distributively represent the matrix. each entry denotes a collection of labeled computational units. The value of the entry is taken to be the label of the unit with the largest value. A Massively Parallel Self-Tuning Context-Free Parser A nontenninal entry has units which are labeled with the nontenninals of a language plus a special label ''blank''. When the "blank" unit is largest, this indicates that the entry plays no part in representing the current parse tree. binder entry has units which are labeled from 1 to the number of rows in the matrix. A unit labeled k then denotes the binding of the nontenninal entry on its immediate left to the nontenninal entry in the kth row on its right. To indicate that no binding exists, we use a special unit label "e" called an edge. A In general, it is easiest to view an entry as a vector of real numbers where each vector component denotes some symbol. (For more infonnation see [Charntak and Santos, 1987].) In the current implementation of PALS, entry unit values range from 0 to 1. The ideal value for entry units is thus 1 for the largest entry unit and 0 for all remaining entry units. We essentially have "1" being "yes" and "0" being no. LANGUAGE RULES In order to determine the values of the computational units mentioned in the previous section, we apply a set of language rules. Each compuatatlonal unit will be detenntned by some subset of these rules. Each language rule is represented by a single node, called a rule node. A rule node takes its input from several computational units and outputs to a Single computational unit. The output of each rule node is also modified by a non-negative value called a rule-weight. This weight represents the applicability of a language rule to the language we are attempting to parse (see PARSING). In the current implementation of PALS, rule-weight values range from 0 to 1 being similar to probabilities. Basically, a rule node attempts to express some rule of grammar. As with CONPARSE, PALS breaks context-free grammars into several subrules. For example, as part of S --> NP VP, PALS would have a rule stating that an NP entry would like to have an S immediately above it in the same column. Our rule for this grammar rule will then take as input the entry's computational unit labeled NP and output to the unit labeled S in the entry immediately above(see Figure 3). 539 540 Santos Entry iJ Rule-Node Entry i-l,j Figure 3. A rule node for S above NP. a more complex example, if an entry is a NP, the NP does not continue right. I.e .. has an edge, and above is an S that continues to the right. then below the second S is a VP. As In general. to determine a unit's value, we take all the rule nodes and combine their influences. This will be much clearer when we discuss parsing in the next section. PARSING Since we are dealing with a PDP-type architecture, the size of our matrix is fixed ahead of time. However. the way we use the matrix representation scheme allows us to handle sentences of unbounded length as we shall see. The system parses a sentence by taking the first word and placing it in the lower rightmost entry; it then attempts to construct the column above the word by using its rule nodes. Mter this processing. the system shifts the first word and its column left and inserts the second word. Now both words are processed simultaneously. This shifting and processing continues until the last word is shifted through the matrix (see Figure 4). Since sentence lengths may exceed the size of the matrix. we are only processing a portion at a time. creating partial parse-trees. The complete parse-tree is the combination of these partial ones. A Massively Parallel Self-Tuning Context-Free Parser 8 NP noun notDl verb t,f8 8 8 1- 1-8 NP VP noun verb noun - ~8 / t,fVP NP VP / NP n01Dl verb notDl Figure 4. Parsing of noun verb noun. Basically, the system builds the tree in a bottom-up fashion. However, it can also build left-right, right-left, and top-down since columns may be of differing height. In general, columns on the left in the matrix will be more complete and hence possibly higher than those on the right. LEARNING The goal of PALS is to learn how to parse a given language. Given a system consisting of a matrix with a set of language rules. we learn parsing by determining how to apply each language rule. In general, when a language rule is inconsistent with the language we are learning to parse, its corresponding rule-weight drops to zero, essentially disconnecting the rule. When a language rule is consistent, its ruleweight then approaches one. In PALS, we learn how to parse a sentence by using training examples. The teacher/trainer gives to the system the complete parse tree of the sentence to be learned. Because of the restrictions imposed by our matrix, we may be unable to fully represent the complete parse tree given by the teacher. To learn how to parse the sentence, we can only utilize a portion of the complete parse tree at anyone time. Given a complete parse tree. the system simply breaks it up into manageable chunks we call snapshots. Snapshots are matrices which contain a portion of the complete parse tree. Given this sequence of snapshots, we present them to the system in a fashion Similar to parsing. The only difference is that we clamped the 541 542 Santos snapshot to the system matrix while it fires its rule nodes. From this, we can easily detenntne whether a rule node has incorrectly fired or not by seeing if it fired consistently with given snapshot. We punish and reward accordingly. As the system is trained more and more, our rule-weights contain more and more information. We would like the rule-weights of those rules used frequently during training to not change as much as those not frequently used. This serves to stabilize our knowledge. It also prevents the system from being totally corrupted when presented with an incorrect training example. As in traditional methods, we find the new rule-weights by minimizing some function which gives us our desired learning. The function which captures this learning method is Lt,j {CfJ ( Clf.,j - ~i,j )2 + [ ~,j ~i,j + ( 1 - ~,j ) ( 1 - ~iJ ) I 2 ri,j2} where i are the unit labels for some matrix entry. j are the language rules associated with units i, (l1.j are the old rule-weights. ~i,j are the new ruleweights, ci,J is the knowledge preservation coefficient which is a function of the frequency that language rule j for entry unit i has been fired during learning. ri.j is the unmodified rule output using snapshot as input, and Si,j is the measure of the correctness of language rule j for unit 1. RESULTS In the current implementation of PALS, we utilize a 7x6 matrix and an average of fifty language rules per entry unit to parse English. Obviously, our set of language rules will determine what we can and cannot learn. Currently, the system is able to learn and parse a modest subset of the English language. It can parse sentences with moderate sentence embedding and phrase attachment from the following grammar: SM S NP --> S per --> NPVP --> (det) (adj)* noun (PP)* (WHCL) (INFPI) NP PP WHCL S/NP S/NP --> INFP2 --> prep NP --> that S/NP -->VP --> NP VP/NP A Massively Parallel Self-Tuning Context-Free Parser INFPl INFI VP VP VP VP VP VP/NP INFP2 INF2 --> (NP) INFI --> to (adv) VP/NP --> (aux) tIVerb NP (PP)* --> (aux) intrverb --> (aux) copverb NP --> (aux) copverb PP --> (aux) copverb adj --> (aux) trverb (PP)* --> (NP) INF2 --> to (adv) VP We have found that sentences only require approximately two trainings. We have also found that by adding more consistent language rules, the system improved by actually generating parse trees which were more "certain" than previously generated ones. In other words, the values of the entry units in the final parse tree were much closer to the ideal. When we added inconsistent language rules, the system degraded. However, with slightly more training, the system was back to normal. It actually had to first eliminate the inconsistent rules before being able to apply the consistent ones. Finally, we attempted to train the system with incorrect training examples after being trained with correct ones. We found that even though the system degraded slightly, previous learning was not completely lost. This was basically due to the stability employed during learning. CONCLUSIONS We have presented a system capable of parsing and learning how to parse. The system parses by creating a sequence of partial parse trees and then combining them into a parse tree. It also places no limit on sentence length. Given a system consisting of a matrix and an aSSOCiated set of language rules, we attempt to learn how to parse the language described by the complete parse tree supplied by a teacher/trainer. The same set of language rules may also be able to learn a different language. Depending on the diversity of the language rules, it may also learn both simultaneously, te., parse both languages. (A simple example is two languages with distinct terminals and nontenntnals.) The system learns by being presented complete parse-trees and adding to its knowledge by modifying its rule-weights. 543 544 Santos The system requires a small number of trainings per training example. Also, incorrect training examples do not totally corrupt the system. PROBLEMS AND FUTURE RESEARCH Eventhough the PALS system has managed to integrate learning. there are still some problems. First, as in the CONPARSE system, we can only handle moderately embedded sentences. Second, the system is very positional. Something that is learned in one portion of the matrix is not generalized to other portions. There is no rule aquisition in the PALS system. Currently, all rules are assumed to be built-in to the system. PALS's ability to suppress incorrect rules would entail rule learning if the possible set of language rules was very tightly constrained so that, in effect. all rules could be tried. Some linguists have suggested quite limited schemes but if any would work with PALS is not known. REFERENCES Rumelhart, D. et ~., ParaUel Distributed Processing: Explorations in the Microstructures oj Cognition. Volume 1, The MIT Press (1986). Charniak, E. and Santos, E., "A connectionist context-free parser which is not context-free, but then it is not really connectionist either," The Ninth Annual Coriference oj the Cognitive Science Society pp. 70-77 (1987). Fanty, M., "Learning in Structured Connectionist Networks." TR 252. University of Rochester Computer Science Department (1988). Selman. Bart, "Rule-based processing in a connectionist system for natural language understanding." Technical Report CSRI-168. Computer Systems Research Institute. University of Toronto (1985). Waltz. D. and Pollack. J., "MaSsively parallel parsing: a strongly interactive model of natural language interpretation," Cognitive Science 9 pp. 51-74 (1985). Feldman. J.A. and Ballard, D.H., "Connectionist models and their properties," Cognitive Science 6 pp. 205-254 (1982). Lippmann. R, "An introduction to computing with neural nets," IEEE ASSP Magazine pp. 4-22 (April 1987). 

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A MCMC approach to Hierarchical Mixture Modelling Christopher K. I. Williams Institute for Adaptive and Neural Computation Division of Informatics, University of Edinburgh 5 Forrest Hill, Edinburgh EHI 2QL, Scotland, UK ckiw@dai.ed.ac.uk http://anc.ed.ac.uk Abstract There are many hierarchical clustering algorithms available, but these lack a firm statistical basis. Here we set up a hierarchical probabilistic mixture model, where data is generated in a hierarchical tree-structured manner. Markov chain Monte Carlo (MCMC) methods are demonstrated which can be used to sample from the posterior distribution over trees containing variable numbers of hidden units. 1 Introduction Over the past decade or two mixture models have become a popular approach to clustering or competitive learning problems. They have the advantage of having a well-defined objective function and fit in with the general trend of viewing neural network problems in a statistical framework. However, one disadvantage is that they produce a "flat" cluster structure rather than the hierarchical tree structure that is returned by some clustering algorithms such as the agglomerative single-link method (see e.g. [12]). In this paper I formulate a hierarchical mixture model, which retains the advantages of the statistical framework, but also features a tree-structured hierarchy. The basic idea is illustrated in Figure 1(a). At the root of the tree (level l) we have a single centre (marked with a x). This is the mean of a Gaussian with large variance (represented by the large circle). A random number of centres (in this case 3) are sampled from the level 1 Gaussian, to produce 3 new centres (marked with o's). The variance associated with the level 2 Gaussians is smaller. A number of level 3 units are produced and associated with the level 2 Gaussians. The centre of each level 3 unit (marked with a +) is sampled from its parent Gaussian. This hierarchical process could be continued indefinitely, but in this example we generate data from the level 3 Gaussians, as shown by the dots in Figure lea). A three-level version of this model would be a standard mixture model with a Gaussian prior on where the centres are located. In the four-level model the third level centres are clumped together around the second level means, and it is this that distinguishes the model from a flat mixture model. Another view of the generative process is given in Figure l(b), where the tree structure denotes which nodes are children of particular parents. Note also that this is a directed acyclic graph, with the arrows denoting dependence of the position of the child on that of the parent. A MCMC Approach to Hierarchical Mixture Modelling 681 In section 2 we describe the theory of probabilistic hierarchical clustering and give a discussion of related work. Experimental results are described in section 3. (a) (b) Figure 1: The basic idea of the hierarchical mixture model. (a) x denotes the root of the tree, the second level centres are denoted by o's and the third level centres by +'s. Data is generated from the third level centres by sampling random points from Gaussians whose means are the third level centres. (b) The corresponding tree structure. 2 Theory We describe in turn (i) the prior over trees, (ii) the calculation of the likelihood given a data vector, (iii) Markov chain Monte Carlo (MCMC) methods for the inference of the tree structure given data and (iv) related work. 2.1 Prior over trees We describe first the prior over the number of units in each layer, and then the prior on connections between layers. Consider a L layer hierarchical model. The root node is in levell, there are n2 nodes in level 2, and so on down to nL nodes on level L. These n's are collected together in the vector n. We use a Markovian model for P(n), so that P(n) = P(ndP(n2In1) ... P(nLlnL-1) with P(n1) = 8(n1,I). Currently these are taken to be Poisson distributions offset by 1, so that P(ni+llni) rv PO(Aini) + 1, where Ai is a parameter associated with level i. The offset is used so that there must always be at least one unit in any layer. Given n, we next consider how the tree is formed. The tree structure describes which node in the ith layer is the parent of each node in the (i + 1)th layer, for i = 1, ... , L - 1. Each unit has an indicator vector which stores the index of the parent to which it is attached. We collect all these indicator vectors together into a matrix, denoted Z(n). The probability of a node in layer (i + 1) connecting to any node in layer i is taken to be I/ni. Thus L-1 P(n, Z(n)) = P(n)P(Z(n)ln) = P(n) IT (I/ni)n + i 1? i=l We now describe the generation of a random tree given nand Z(n). For simplicity we describe the generation of points in I-d below, although everything can be extended to arbitrary dimension very easily. The mean f-Ll of the level 1 Gaussian is at the origin 1. The I It is easy to relax this assumption so that /-L 1 has a prior Gaussian distribution, or is located at some point other than the origin. 682 C. K. l. Williams level 2 means It], j = 1, ... ,n2 are generated from N (It 1 , af), where ar is the variance associated with the level ] node. Similarly, the position of each level 3 node is generated from its level 2 parent as a displacement from the position of the level 2 parent. This This process continues displacement is a Gaussian RV with zero mean and variance on down to the visible variables. In order for this model to be useful, we require that ar > > ... > i.e. that the variability introduced at successive levels declines monotonically (cj scaling of wavelet coefficients). ai. ai 2.2 aI-I' Calculation of the likelihood The data that we observe are the positions of the points in the final layer; this is denoted x. To calculate the likelihood of x under this model, we need to integrate out the locations of the means of the hidden variables in levels 2 through to L - 1. This can be done explicitly, however, we can shorten this calculation by realizing that given Z(n), the generative distribution for the observables x is Gaussian N(O, C). The covariance matrix C can be calculated as follows. Consider two leaf nodes indexed by k and i. The Gaussian RVs that generated the position of these two leaves can be denoted l _ 1 Xk - Wk + 2 Wk + ... + (L-l) Wk , XI = WI1 + WI2 + ... + WI(L-l) . To calculate the covariance between Xk and Xl, we simply calculate (XkXI). This depends crucially on how many of the w's are shared between nodes k and l (cj path analysis). For example, if Wk i- wl, i.e. the nodes lie in different branches of the tree at levell, their covariance is zero. If k = l, the variance is just the sum of the variances of each RV in the tree. In between, the covariance of Xk and XI can be determined by finding at what level in the tree their common parent occurs. Under these assumptions, the log likelihood L of x given Z (n) is 1 nL L=-"21 x T C -1 x-"2logICI-Tlog21r. (1) In fact this calculation can be speeded up by taking account of the tree structure (see e.g. [8]). Note also that the posterior means (and variances) of the hidden variables can be calculated based on the covariances between the hidden and visible nodes. Again, this calculation can be carried out more efficiently; see Pearl [11] (section 7.2) for details. 2.3 Inference for nand Z (n) Given n we have the problem of trying to infer the connectivity structure Z given the observations x. Of course what we are interested in is the posterior distribution over Z, i.e. P(Zlx, n). One approach is to use a Markov chain Monte Carlo (MCMC) method to sample from this posterior distribution. A straightforward way to do this is to use the Metropolis algorithm, where we propose changes in the structure by changing the parent of a single node at a time. Note the similarities of this algorithm to the work of Williams and Adams [14] on Dynamic Trees COTs); the main differences are Ci) that disconnections are not allowed, i.e. we maintain a single tree (rather than a forest), and (ii) that the variables in the DT image models are discrete rather than Gaussian. We also need to consider moves that change n. This can be effected with a split/merge move. In the split direction, consider a node with a parent and several children. Split this node and randomly assign the children to the two split nodes. Each of the split nodes keeps the same parent. The probability of accepting this move under the Metropolis-Hastings scheme is a . (1 = mm P(n', Z(n')lx)Q(n', Z(n'); n, Z(n))) 'P(n, Z(n)lx)Q(n, Z(n); n', Z(n')) , A MCMC Approach to Hierarchical Mixture Modelling 683 where Q(n', Z(n'); n, Z(n)) is the proposal probability of configuration (n', Z(n')) given configuration (n, Z (n)). This scheme is based on the work on MCMC model composition (MC 3 ) by Madigan and York [9], and on Green 's work on reversible jump MCMC [5]. Another move that changes n is to remove "dangling" nodes, i.e. nodes which have no children. This occurs when all the nodes in a given layer "decide" not to use one or more nodes in the layer above. An alternative to sampling from the posterior is to use approximate inference, such as mean-field methods. These are currently being investigated for DT models [1]. 2.4 Related work There are a very large number of papers on hierarchical clustering; in this work we have focussed on expressing hierarchical clustering in terms of probabilistic models. For example Ambros-Ingerson et at [2] and Mozer [10] developed models where the idea is to cluster data at a coarse level, subtract out mean and cluster the residuals (recursively). This paper can be seen as a probabilistic interpretation of this idea. The reconstruction of phylogenetic trees from biological sequence (DNA or protein) information gives rise to the problem of inferring a binary tree from the data. Durbin et al [3] (chapter 8) show how a probabilistic formulation of the problem can be developed, and the link to agglomerative hierarchical clustering algorithms as approximations to the full probabilistic method (see ?8.6 in [3]). Much of the biological sequence work uses discrete variables, which diverges somewhat from the focus of the current work. However work by Edwards (1970) [4] concerns a branching Brownian-motion process, which has some similarities to the model described above. Important differences are that Edwards' model is in continuous time, and the the variances of the particles are derived from a Wiener process (and so have variance proportional to the lifetime of the particle). This is in contrast to the decreasing sequence of variances at a given number of levels assumed in the above model. One important difference between the model discussed in this paper and the phylogenetic tree model is that points in higher levels of the phylogenetic tree are taken to be individuals at an earlier time in evolutionary history, which is not the interpretation we require here. An very different notion of hierarchy in mixture models can be found in the work on the AutoClass system [6]. They describe a model involving class hierarchy and inheritance, but their trees specify over which dimensions sharing of parameters occurs (e.g. means and covariance matrices for Gaussians). In contrast, the model in this paper creates a hierarchy over examples labelled 1, ... ,n rather than dimensions. Xu and Pearl [15] discuss the inference of a tree-structured belief network based on knowledge of the covariances of the leaf nodes. This algorithm cannot be applied directly in our case as the covariances are not known, although we note that if multiple runs from a given tree structure were available the covariances might be approximated using sample estimates. Other ideas concerning hierarchical clustering are discussed in [13] and [7]. 3 Experiments We describe two sets of experiments to explore these ideas. 3.1 Searching over Z with n fixed 100 4-level random trees were generated from the prior, using values of >'1 = 1.5, >'2 = 2, 3, and 10, (Ji 1, (J~ 0.01. These trees had between 4 and 79 leaf >'3 = (JI = = = C. K I. Williams 684 nodes, with an average of 30. For each tree n was kept the same as in the generative tree, and sampling was carried out over Z starting from a random initial configuration. A given node proposes changing its parent, and this proposal is accepted or rejected with the usual Metropolis probability. In one sweep, each node in levels 3 and 4 makes such a move. (Level 2 nodes only have one parent so there is no point in such a move there.) To obtain a representative sample of P(Z(n)ln, x), we should run the chain for as long as possible. However, we can also use the chain to find configurations with high posterior probability, and in this case running for longer only increases the chances of finding a better configuration. In our experiments the sampler was run for 100 sweeps. As P(Z(n)ln) is uniform for fixed n, the posterior is simply proportional to the likelihood term. It would also be possible to run simulated annealing with the same move set to search explicitly for the maximum a posteriori (MAP) configuration. The results are that for 76 of the 100 cases the tree with the highest posterior probability (HPP) configuration had higher posterior probability than the generative tree, for 20 cases the same tree was found and in 4 cases the HPP solution was inferior to the generative tree. The fact that in almost all cases the sampler found a configuration as good or better than the generative one in a relatively small number of sweeps is very encouraging. In Figure 2 the generative (left column) and HPP trees for fixed n (middle column) are plotted for two examples. In panel (b) note the "dangling" node in level 2, which means that the level 3 nodes to the left end up in a inferior configuration to (a). By contrast, in panel (e) the sampler has found a better (less tangled) configuration than the generative model (d). _'1150841 (a) (b) (c) (d) (e) (f) Figure 2: (a) and (d) show the generative trees for two examples. The corresponding HPP trees for fixed n are plotted in (b) and (e) and those for variable n in (c) and (f). The number in each panel is the log posterior probability of the configuration. The nodes in levels 2 and 3 are shown located at their posterior means. Apparent non-tree structures are caused by two nodes being plotted almost on top of each other. 3.2 Searching over both nand Z Given some data x we will not usually know appropriate numbers of hidden units. This motivates searching over both Z and n, which can be achieved using the split/merge moves discussed in section 2.3. In the experiments below the initial numbers of units in levels 2 and 3 (denoted n2 and A MCMC Approach to Hierarchical Mixture Modelling 685 113) were set using the simple-minded formulae 113 = rdim(x)/A31112 = r113/A21. A proper inferential calculation for 71,2 and 71,3 can be carried out, but it requires the solution of a non-linear optimization problem. Given 112 and 113, the initial connection configuration was chosen randomly. The search method used was to propose a split/merge move (with probability 0.5:0.5) in level 2, then to sample the level 2 to level 3 connections, and then to propose a split-merge move in level 3, and then update the level 3 to level 4 connections. This comprised a single sweep, and as above 100 sweeps were used. Experiments were conducted on the same trees used in section 3.1. In this case the results were that for 50 out of the 100 cases, the HPP configuration had higher posterior probability than the generative tree, for 11 cases the same tree was found and in 39 cases the HPP solution was inferior to the generative tree. Overall these results are less good than the ones in section 3.1, but it should be remembered that the search space is now much larger, and so it would be expected that one would need to search longer. Comparing the results from fixed n against those with variable n shows that in 42 out of 100 cases the variable n method gave a higher posterior probability. in 45 cases it was lower and in 13 cases the same trees were found. The rightmost column of Figure 2 shows the HPP configurations when sampling with variable n on the two examples discussed above. In panel (c) the solution found is not very dissimilar to that in panel (b), although the overall probability is lower. In Cf), the solution found uses just one level 2 centre rather than two, and obtains a higher posterior probability than the configurations in (e) and Cd). 4 Discussion The results above indicate that the proposed model behaves sensibly, and that reasonable solutions can be found with relatively short amounts of search. The method has been demonstrated on univariate data, but extending it to multivariate Gaussian data for which each dimension is independent given the tree structure is very easy as the likelihood calculation is independent on each dimension. There are many other directions is which the model can be developed. Firstly, the model as presented has uniform mixing proportions, so that children are equally likely to connect to each potential parent. This can be generalized so that there is a non-uniform vector of connection probabilities in each layer. Also, given a tree structure and independent Dirichlet priors over these probability vectors, these parameters can be integrated out analytically. Secondly, the model can be made to generate iid data by regarding the penultimate layer as mixture centres; in this case the term P(nLI71,L-l) would be ignored when computing the probability of the tree. Thirdly, it would be possible to add the variance variables to the MCMC scheme, e.g. using the Metropolis algorithm, after defining a suitable prior on the The constraint that all variances in the same level are sequence of variances equal could also be relaxed by allowing them to depend on hyperparameters set at every level. Fourthly, there may be improved MCMC schemes that can be devised. For example, in the current implementation the posterior means of the candidate units are not taken into account when proposing merge moves Ccj [5]). Fifthly, for the multivariate Gaussian version we can consider a tree-structured factor analysis model, so that higher levels in the tree need not have the same dimensionality as the data vectors. ai, ... ,ai-I' One can also consider a version where each dimension is a multinomial rather than a continuous variable. In this case one might consider a model where a multinomial parameter vector (}l in the tree is generated from its parent by (}l = 'Y(}l-l + (1- 'Y)r where 'Y E [0,1] and r is a random vector of probabilities. An alternative model could be to build a tree 686 C. K. 1. Williams structured prior on the a parameters of the Dirichlet prior for the multinomial distribution. Acknowledgments This work is partially supported through EPSRC grant GRIL 78161 Probabilistic Models for Sequences. I thank the Gatsby Computational Neuroscience Unit (UCL) for organizing the "Mixtures Day" in March 1999 and supporting my attendance, and Peter Green, Phil Dawid and Peter Dayan for helpful discussions at the meeting. I also thank Amos Storkey for helpful discussions and Magnus Rattray for (accidentally!) pointing me towards the chapters on phylogenetic trees in [3]. References [1] N. J. Adams, A. Storkey, Z. Ghahramani, and C. K. 1. Williams. MFDTs: Mean Field Dynamic Trees. Submitted to ICPR 2000,1999. [2] J. Ambros-Ingerson, R. Granger, and G. Lynch. Simulation of Paleocortex Performs Hierarchical Clustering. Science , 247:1344-1348,1990. [3] R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis. Cambridge University Press, Cambridge, UK, 1998. [4] A. w. F. Edwards. Estimation of the Branch Points of a Branching Diffusion Process. Journal of the Royal Statistical Society B, 32(2): 155-174, 1970. [5] P. J. Green. Reversible Jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4):711-732, 1995. [6] R. Hanson, J. Stutz, and P. Cheeseman. Bayesian Classification with Correlation and Inheritance. In IlCAI-91: Proceedings of the Twelfth International Joint Conference on Artificial Intelligence, 1991 . Sydney, Australia. [7] T. Hofmann and J. M. Buhmann. Hierarchical Pairwise Data Clustering by MeanField Annealing. In F. Fogelman-Soulie and P. Gallinari, editors, Proc. ICANN 95. EC2 et Cie, 1995. [8] M. R. Luettgen and A. S. Wi11sky. Likelihood Calculation for a Class of Multiscale Stochastic Models, with Application to Texture Discrimination. IEEE Trans. Image Processing, 4(2): 194-207, 1995. [9] D. Madigan and J. York. Bayesian Graphical Models for Discrete Data. International Statistical Review, 63:215-232,1995. [10] M. C. Mozer. Discovering Discrete Distributed Representations with Iterated Competitive Learning. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3. Morgan Kaufmann, 1991. [11] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, 1988. [12] B. Ripley. Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge, UK, 1996. [13] N. Vasconcelos and A. Lippmann. Learning Mixture Hierarchies. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11, pages 606-612. MIT Press, 1999. [14] C. K. 1. Williams and N. J. Adams. DTs: Dynamic Trees. In M. J. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11 . MIT Press, 1999. [15] L. Xu and J. Pearl. Structuring Causal Tree Models with Continuous Variables. In L. N. Kanal, T. S. Levitt, and J. F. Lemmer, editors, Uncertainty in Artificial Intelligence 3. Elsevier, 1989. 

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Agglomerative Information Bottleneck Noam Slonim Naftali Tishby* Institute of Computer Science and Center for Neural Computation The Hebrew University Jerusalem, 91904 Israel email: {noamm.tishby}(Qcs.huji.ac.il Abstract We introduce a novel distributional clustering algorithm that maximizes the mutual information per cluster between data and given categories. This algorithm can be considered as a bottom up hard version of the recently introduced "Information Bottleneck Method". The algorithm is compared with the top-down soft version of the information bottleneck method and a relationship between the hard and soft results is established. We demonstrate the algorithm on the 20 Newsgroups data set. For a subset of two newsgroups we achieve compression by 3 orders of magnitudes loosing only 10% of the original mutual information. 1 Introduction The problem of self-organization of the members of a set X based on the similarity of the conditional distributions of the members of another set, Y, {p(Ylx)}, was first introduced in [8] and was termed "distributional clustering" . This question was recently shown in [9] to be a special case of a much more fundamental problem: What are the features of the variable X that are relevant for the prediction of another, relevance, variable Y? This general problem was shown to have a natural in~ormation theoretic formulation: Find a compressed representation of the variable X, denoted X, such that the mutual information between X and Y, I (X; Y), is as high as possible, under a constraint on the mutual information between X and X, I (X; X). Surprisingly, this variational problem yields an exact self-consistent equations for the conditional distributions p(ylx), p(xlx), and p(x). This constrained information optimization problem was called in [9] The Information Bottleneck Method. The original approach to the solution of the resulting equations, used already in [8], was based on an analogy with the "deterministic annealing" approach to clustering (see [7]). This is a top-down hierarchical algorithm that starts from a single cluster and undergoes a cascade of cluster splits which are determined stochastically (as phase transitions) into a "soft" (fuzzy) tree of clusters. In this paper we propose an alternative approach to the information bottleneck N Slonim and N Tishby 618 problem, based on a greedy bottom-up merging. It has several advantages over the top-down method. It is fully deterministic, yielding (initially) "hard clusters", for any desired number of clusters. It gives higher mutual information per-cluster than the deterministic annealing algorithm and it can be considered as the hard (zero temperature) limit of deterministic annealing, for any prescribed number of clusters. Furthermore, using the bottleneck self-consistent equations one can "soften" the resulting hard clusters and recover the deterministic annealing solutions without the need to identify the cluster splits, which is rather tricky. The main disadvantage of this method is computational, since it starts from the limit of a cluster per each member of the set X. 1.1 The information bottleneck method The mutual information between the random variables X and Y is the symmetric functional of their joint distribution, I(X;Y) = L p(x,y) log ( ~~~'~\) L = p(x)p(ylx) log (P(Y(lx))) . PY (1) The objective of the information bottleneck method is to extract a compact representation of the variable X, denoted here by X, with minimal loss of mutual information to another, relevance, variable Y. More specifically, we want to find a (possibly stochastic) map, p(xlx), that minimizes the (lossy) coding length of X via X, I(Xi X), under a constraint on the mutual information to the relevance variable I(Xi Y). In other words, we want to find an efficient representation of the variable X, X, such that the predictions of Y from X through X will be as close as possible to the direct prediction of Y from X. xEX,yEY P PY xEX,yEY As shown in [9], by introducing a positive Lagrange multiplier 13 to enforce the mutual information constraint, the problem amounts to minimization of the Lagrangian: ?[P(xlx)] = I(Xi X) - f3I(Xi Y) , (2) with respect to p(xlx), subject to the Markov condition X -t X -t Y and normalization. This minimization yields directly the following self-consistent equations for the map p(xlx), as well as for p(Ylx) and p(x): p(xlx) = i(~~;) exp (-f3D KL [P(Ylx)lIp(Ylx)]) { p(Ylx) = 2:xp(Ylx)p(xlx)~ p(x) = 2: x p(xlx)p(x) (3) where Z(f3, x) is a normalization function. The functional DKL[Pllq] == 2: y p(y) log ~f~~ is the Kulback-Liebler divergence [3J , which emerges here from the variational principle. These equations can be solved by iterations that are proved to converge for any finite value of 13 (see [9]). The Lagrange multiplier 13 has the natural interpretation of inverse temperature, which suggests deterministic annealing [7] to explore the hierarchy of solutions in X, an approach taken already in [8J . The variational principle, Eq.(2), determines also the shape of the annealing process, since by changing 13 the mutual informations Ix == I(X; X) and Iy == I(Y; X) vary such that My = 13- 1 (4) Mx . Agglomerative Information Bottleneck 619 Thus the optimal curve, which is analogous to the rate distortion function in information theory [3], follows a strictly concave curve in the (Ix,Iy) plane, called the information plane. Deterministic annealing, at fixed number of clusters, follows such a concave curve as well, but this curve is suboptimal beyond a certain critical value of f3. Another interpretation of the bottleneck principle comes from the relation between the mutual information and Bayes classification error. This error is bounded above and below (see [6]) by an important information theoretic measure of the class conditional distributions p(XIYi), called the Jensen-Shannon divergence. This measure plays an important role in our context. The Jensen-Shannon divergence of M class distributions, Pi(X), each with a prior 1 ~ i ~ M, is defined as, [6,4]. 7ri, M M JS7r (Pl,P2, ???,PM] == H[L 7riPi(X)] - L 7ri H [Pi(X)] , i=l (5) i=l where H[P(x)] is Shannon's entropy, H[P(x)] = - Ex p(x) logp(x). The convexity of the entropy and Jensen inequality guarantees the non-negativity of the JSdivergence. 1.2 The hard clustering limit For any finite cardinality of the representation IXI == m the limit f3 -+ 00 of the Eqs.(3) induces a hard partition of X into m disjoint subsets. In this limit each member x E X belongs only to the subset x E X for which p(Ylx) has the smallest DKL[P(ylx)lIp(ylx)] and the probabilistic map p(xlx) obtains the limit values 0 and 1 only. In this paper we focus on a bottom up agglomerative algorithm for generating "good" hard partitions of X. We denote an m-partition of X, i.e. X with cardinality m, also by Zm = {Zl,Z2, ... ,Zm}, in which casep(x) =p(Zi). We say that Zm is an optimal m-partition (not necessarily unique) of X if for every other m-partition of X, Z:n, I(Zm; Y) ~ I(Z:n; Y). Starting from the trivial N-partition, with N = lXI, we seek a sequence of merges into coarser and coarser partitions that are as close as possible to optimal. It is easy to verify that in the f3 -+ 00 limit Eqs.(3) for the m-partition distributions are simplified as follows. Let x == Z = {XI,X2, ... ,xl z l} , Xi E X denote a specific component (Le. cluster) of the partition Zm, then { 1 p(zlx) = I 0 if x E Z . Vx E X oth erWlse p(ylz) = plz) El~l P(Xi, y) Vy p(z) = El~l P(Xi) E Y (6) Using these distributions one can easily evaluate the mutual information between Zm and Y, I(Zm; Y), and between Zm and X, I(Zm; X), using Eq.(l). Once any hard partition, or hard clustering, is obtained one can apply "reverse annealing" and "soften" the clusters by decreasing f3 in the self-consistent equations, Eqs.( 3). Using this procedure we in fact recover the stochastic map, p(xlx), from the hard partition without the need to identify the cluster splits. We demonstrate this reverse deterministic annealing procedure in the last section. 620 1.3 N. Slonim and N. Tishby Relation to other work A similar agglomerative procedure, without the information theoretic framework and analysis, was recently used in [1] for text categorization on the 20 newsgroup corpus. Another approach that stems from the distributional clustering algorithm was given in [5] for clustering dyadic data. An earlier application of mutual information for semantic clustering of words was used in [2]. 2 The agglomerative information bottleneck algorithm The algorithm starts with the trivial partition into N = IXI clusters or components, with each component contains exactly one element of X. At each step we merge several components of the current partition into a single new component in a way that locally minimizes the loss of mutual information leX; Y) = l(Zm; Y). Let Zm be the current m-partition of X and Zm denote the new m-partition of X after the merge of several components of Zm. Obviously, m < m. Let {Zl, Z2, ... , zd ~ Zm denote the set of components to be merged, and Zk E Zm the new component that is generated by the merge, so m = m - k + 1. To evaluate the reduction in the mutual information l(Zm; Y) due to this merge one needs the distributions that define the new m-partition, which are determined as follows. For every Z E Zm, Z f:. Zk, its probability distributions (p(z),p(ylz),p{zlx? remains equal to its distributions in Zm. For the new component, Zk E Zm, we define, p(Zk) = L~=l P(Zi) { p(yIZk) = P(~Ic) ~~=l P(Zi, y) \ly E Y . (7) (zlx) = 1f x ~i for some ~ ~ ~ k \Ix X p 0 otherw1se E {1 E 1 It is easy to verify that Zm is indeed a valid m-partition with proper probability distributions. Using the same notations, for every merge we define the additional quantities: ? The merge prior distribution: defined by ilk == (71"1,71"2, ... , 7I"k), where is the prior probability of Zi in the merged subset, i.e. 7I"i == :t::). 7I"i ? The Y -information decrease: the decrease in the mutual information l(X; Y) due to a single merge, cHy(Zl' ""Zk) == l(Zm; Y) - l(Zm; Y) ? The X-information decrease: the decrease in the mutual information l(X, X) due to a single merge, cHx (Z1, Z2, ... , Zk) == l(Zm; X) - l(Zm; X) Our algorithm is a greedy procedure, where in each step we perform "the best possible merge" , i.e. merge the components {Z1, ... , zd of the current m-partition which minimize cHy(Z1, ... , Zk). Since cHy(Zl, ... , Zk) can only increase with k (corollary 2), for a greedy procedure it is enough to check only the possible merging of pairs of components of the current m-partition. Another advantage of merging only pairs is that in this way we go through all the possible cardinalities of Z = X, from N to 1. For a given m-partition Zm = {Z1,Z2, ... ,Zm} there are m(~-1) possible pairs to merge. To find "the best possible merge" one must evaluate the reduction of information cHy(Zi, Zj) = l(Zm; Y) - l(Zm-1; Y) for every pair in Zm , which is O(m . WI) operations for every pair. However, using proposition 1 we know that cHy(Zi, Zj) = (P(Zi) + p(Zj? . JSn 2 (P(YIZi),p(Ylzj?, so the reduction in the mutual 621 Agglomerative Information Bottleneck information due to the merge of Zi and Zj can be evaluated directly (looking only at this pair) in O(IYI) operations, a reduction of a factor of m in time complexity (for every merge). Input: Empirical probability matrix p(x,y), N = Output: IX\, M = IYI Zm : m-partition of X into m clusters, for every 1 ::; m ::; N Initialization: ? Construct Z == X - For i = 1.. .N * Zi = {x;} * P(Zi) = p(Xi) * p(YIZi) = p(YIXi) for every y E Y * p(zlxj) = 1 if j = i and 0 otherwise - Z={Zl, ... ,ZN} ? for every i, j = 1.. .N, i < j, calculate di,j = (p(Zi) +p(Zj))' JSn2[P(ylzi),p(ylzj)] (every di,j points to the corresponding couple in Z) Loop: ? For t = 1... (N - 1) - Find {a,.B} = argmini ,j {di,j } (if there are several minima choose arbitrarily between them) - Merge {z"" Zj3} => z : * p(z) = p(z",) + p(Zj3) * p(ylz) = r>li) (p(z""y) +p(zj3,y)) for every y E Y * p(zlx) = 1 if x E z'" U Zj3 and 0 otherwise, for every x E X - Update Z = {Z - {z"" z,q}} U{z} (Z is now a new (N - t)-partition of X with N - t clusters) - Update di ,j costs and pointers w.r.t. z (only for couples contained z'" or Zj3). ? End For Figure 1: Pseudo-code of the algorithm. 3 Discussion The algorithm is non-parametric, it is a simple greedy procedure, that depends only on the input empirical joint distribution of X and Y. The output of the algorithm is the hierarchy of all m-partitions Zm of X for m = N, (N - 1), ... ,2,1. Moreover, unlike most other clustering heuristics, it has a built in measure of efficiency even for sub-optimal solutions, namely, the mutual information I(Zm; Y) which bounds the Bayes classification error. The quality measure of the obtained Zm partition is the fraction of the mutual information between X and Y that Zm captures. This is given by the curve II Z,;,-: vs. m = 1Zm I. We found that empirically this curve was concave. If this is always true the decrease in the mutual information at every step, given by 8(m) == I(Z7n;Y)(-:~7n-l;Y) can only increase with decreasing m. Therefore, if at some point 8(m) becomes relatively high it is an indication that we have reached a value of m with "meaningful" partition or clusters. Further N. Slonim and N. Tishby 622 merging results in substantial loss of information and thus significant reduction in the performance of the clusters as features. However, since the computational cost of the final (low m) part of the procedure is very low we can just as well complete the merging to a single cluster. 0 .8 1 >" :;:~ ? 06 - :::"0.5 >- >" ~ ~O . 4 0.2 ... NG100 -.6.+ I- o~'----;-;;;;;;------ "3000:=-- -= 121 0 .2 0 .4 0 .6 I(Z;X) / H(X) 0.8 %~--~ '0~--~2~0----~ 30~---4~ 0----~ ~ ~ IZI Figure 2: On the left figure the results of the agglomerative algorithm are shown in the "information plane", normalized I(Z ;Y) vs. normalized I(Z ; X) for the NGlOOO dataset. It is compared to the soft version of the information bottleneck via "reverse annealing" for IZI = 2,5, 10, 20, 100 (the smooth curves on the left). For IZI = 20 , 100 the annealing curve is connected to the starting point by a dotted line. In this plane the hard algorithm is clearly inferior to the soft one. On the right-hand side: I(Zm, Y) of the agglomerative algorithm is plotted vs. the cardinality of the partition m for three subsets of the newsgroup dataset . To compare the performance over the different data cardinalities we normalize I(Zm ;Y) by the value of I(Zso ;Y) , thus forcing all three curves to start (and end) at the same points. The predictive information on the newsgroup for NGlOOO and NGIOO is very similar, while for the dichotomy dataset, 2ng, a much better prediction is possible at the same IZI, as can be expected for dichotomies. The inset presents the full curve of the normalized I(Z ;Y) vs. IZI for NGIOO data for comparison. In this plane the hard partitions are superior to the soft ones. 4 Application To evaluate the ideas and the algorithm we apply it to several subsets of the 20Newsgroups dataset, collected by Ken Lang using 20, 000 articles evenly distributed among 20 UseNet discussion groups (see [1]). We replaced every digit by a single character and by another to mark non-alphanumeric characters. Following this preprocessing, the first dataset contained the 530 strings that appeared more then 1000 times in the data. This dataset is referred as NG1000 . Similarly, all the strings that appeared more then 100 times constitutes the NG100 dataset and it contains 5148 different strings. To evaluate also a dichotomy data we used a corpus consisting of only two discussion groups out of the 20Newsgroups with similar topics: alt. atheism and talk. religion. misc. Using the same pre-processing, and removing strings that occur less then 10 times, the resulting "lexicon" contained 5765 different strings. We refer to this dataset as 2ng. We plot the results of our algorithm on these three data sets in two different planes. First , the normalized information ;g~~~ vs. the size of partition of X (number of clusters) , IZI. The greedy procedure directly tries to maximize J(Z ; Y) for a given IZI, as can be seen by the strong concavity of these curves (figure 2, right). Indeed the procedure is able to maintain a high percentage of the relevant mutual information of the original data, while reducing the dimensionality of the "features" , Agglomerative Information Bottleneck 623 IZI, by several orders of magnitude. On the right hand-side of figure 2 we present a comparison between the efficiency of the procedure for the three datasets. The two-class data, consisting of 5765 different strings, is compressed by two orders of magnitude, into 50 clusters, almost without loosing any of the mutual information about the news groups (the decrease in I(Xi Y) is about 0.1%). Compression by three orders of magnitude, into 6 clusters, maintains about 90% of the original mutual information. Similar results, even though less striking, are obtained when Y contain all 20 newsgroups. The NG100 dataset was compressed from 5148 strings to 515 clusters, keeping 86% of the mutual information, and into 50 clusters keeping about 70% of the information. About the same compression efficiency was obtained for the NG1000 dataset . The relationship between the soft and hard clustering is demonstrated in the Information plane, i.e., the normalized mutual information values, :ti;; ~~ vs. Ik(~)' In this plane, the soft procedure is optimal since it is a direct maximization of I(Z; Y) while constraining I(Zi X). While the hard partition is suboptimal in this plane, as confirmed empirically, it provides an excellent starting point for reverse annealing. In figure 2 we present the results of the agglomerative procedure for NG1000 in the information plane, together with the reverse annealing for different values of IZI. As predicted by the theory, the annealing curves merge at various critical values of f3 into the globally optimal curve, which correspond to the "rate distortion function" for the information bottleneck problem. With the reverse annealing ("heating") procedure there is no need to identify the cluster splits as required in the original annealing ("cooling") procedure. As can be seen, the "phase diagram" is much better recovered by this procedure, suggesting a combination of agglomerative clustering and reverse annealing as the ultimate algorithm for this problem. References [1] L. D. Baker and A. K. McCallum. Distributional Clustering of Words for Text Classification In ACM SIGIR 98, 1998. [2] P. F . Brown, P.V. deSouza, R .L. Mercer, V.J. DellaPietra, and J.C. Lai. Class-based n-gram models of natural language. In Computational Linguistics, 18( 4}:467-479, 1992. [3] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, New York, 1991. [4] R. EI-Yaniv, S. Fine, and N. Tishby. Agnostic classification of Markovian sequences. In Advances in Neural Information Processing (NIPS'97) , 1998. [5] T. Hofmann, J . Puzicha, and M. Jordan. Learning from dyadic data. In Advances in Neural Information Processing (NIPS'98), 1999. [6] J. Lin. Divergence Measures Based on the Shannon Entropy. IEEE Transactions on Information theory, 37(1}:145- 151, 1991. [7] K. Rose. Deterministic Annealing for Clustering, Compression, Classification, Regression, and Related Optimization Problems. Proceedings of the IEEE, 86(11}:2210- 2239, 1998. [8] F.C. Pereira, N. Tishby, and L. Lee. Distributional clustering of English words. In 30th Annual Meeting of the Association for Computational Linguistics, Columbus, Ohio, pages 183-190, 1993. [9] N. Tishby, W. Bialek, and F . C. Pereira. The information bottleneck method: Extracting relevant information from concurrent data. Yet unpublished manuscript, NEC Research Institute TR, 1998. 

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Support Vector Method for Multivariate Density Estimation Vladimir N. Vapnik Royal Halloway College and AT &T Labs, 100 Schultz Dr. Red Bank, NJ 07701 vlad@research.att.com Sayan Mukherjee CBCL, MIT E25-201 Cambridge, MA 02142 sayan@ai.mit.edu Abstract A new method for multivariate density estimation is developed based on the Support Vector Method (SVM) solution of inverse ill-posed problems. The solution has the form of a mixture of densities. This method with Gaussian kernels compared favorably to both Parzen's method and the Gaussian Mixture Model method. For synthetic data we achieve more accurate estimates for densities of 2, 6, 12, and 40 dimensions. 1 Introduction The problem of multivariate density estimation is important for many applications, in particular, for speech recognition [1] [7]. When the unknown density belongs to a parametric set satisfying certain conditions one can estimate it using the maximum likelihood (ML) method. Often these conditions are too restrictive. Therefore, non-parametric methods were proposed. The most popular of these, Parzen's method [5], uses the following estimate given data Xl, ... , Xl: (1) where K'Yl(t - Xi) is a smooth function such that J K'Yl(t - xi)dt = 1. Under some conditions on "Yl and K'Yl (t - Xi), Parzen's method converges with a fast asymptotic rate. An example of such a function is a Gaussian with one free parameter "Y; (the width) The structure of the Parzen estimator is too complex: the number of terms in (1) is equal to the number of observations (which can be hundreds of thousands). V. N. Vapnik and S. Mukherjee 660 Researchers believe that for practical problems densities can be approximated by a mixture with few elements (Gaussians for Gaussian Mixture Models (GMM)). Therefore, the following parametric density model was introduced m m P(x, a,~) = 2: (}:iP(X, ai, ~i)' (}: ~ 0, 2: (}:i = 1, i=l (3) i=l where P(x, ai, ~i) are Gaussians with different vectors ai and different diagonal covariance matrices ~i; (}:i is the proportion of the i-th Gaussian in the mixture. It is known [9] that for general forms of Gaussian mixtures the ML estimate does not exist. To use the ML method two values are specified: a lower bound on diagonal elements of the covariance matrix and an upper bound on the number of mixture elements. Under these constraints one can estimate the mixture parameters using the EM algorithm. This solution, however, is based on predefined parameters. In this article we use an SVM approach to obtain an estimate in the form of a mixture of densities. The approach has no free parameters. In our experiments it performs better than the GMM method. 2 Density estimation is an ill-posed problem A density p(t) is defined as the solution of the equation i~ p(t) dt = (4) F(x), where F(x) is the probability distribution function. Estimating a density from data involves solving equation (4) on a given set of densities when the distribution function F(x) is unknown but a random i.i.d. sample Xl, ... , Xe is given. The empirical distribution function Fe(x) is a good approximation of the actual distribution, where O( u) is the step-function. In the univariate case, for sufficiently large l the distribution of the supremum error between F(x) and Ft(x) is given by the Kolmogorov-Smirnov distribution P{sup IF(x) - Fe(x)1 < c/Vi} = 1- 22:( _1)k-1 exp{ -2c 2 k 2 }. x (5) k=l Hence, the problem of density estimation can be restated as solving equation (4) but replacing the distribution function F(x) with the empirical distribution function Fe(x) which converges to the true one with the (fast) rate 0(1)' for univariate and multivariate cases. The problem of solving the linear operator equation Ap Ft(x) is ill-posed. =F with approximation In the 1960's methods were proposed for solving ill-posed problems using approximations Ft converging to F as l increases. The idea of these methods was to 661 Support Vector Method for Multivariate Density Estimation introduce a regularizing functional O(P) (a semi-continuous, positive functional for which O(p) ~ c, c > 0 is a compactum) and define the solution Pt which is a trade-off between O(p) and IIAp - Ftll. The following two methods which are asymptotically equivalent [11] were proposed by Tikhonov [8] and Phillips [6] min [IIAp - Ftll 2 p + ItO(P)] minO(p) s.t. IIAp - Fill p , It > 0, It -+ 0, < et, et > 0, et -+ O. (6) (7) For the stochastic case it can be shown for both methods that if Ft(x) converges in probability to F(x) and Ii -+ 0 then for sufficiently large f and arbitrary 1/ and J.L the following inequality holds [10] [9] [3] (8) where f > fO(1/, J.L) and PEl (p,Pi), PE2(F, Fi ) are metrics in the spaces p and F. Since Fi(X) -+ F(x) in probability with the rate O( ~), from equation (8) it follows that if Ii 3 > O( ~) the solutions of equation (4) are consistent. Choice of regularization parameters For the deterministic case the residual method [2] can be used to set the regularization parameters (,i in (6) and ei in (7)) by setting the parameter (,i or ei) such that Pi satisfies the following IIApi - Fill = IIF(x) - Fi(X) II = Ui, (9) where Ui is the known accuracy of approximation of F(x) by Fi(X). We use this idea for the stochastic case. The Kolmogorov-Smirnov distribution is used to set Ui, Ui = c/...[i, where c corresponds to an appropriate quantile. For the multivariate case one can either evaluate the appropriate quantile analytically [4] or by simulations. The density estimation problem can be solved using either regularization method (6) or (7). Using method (6) with a L2 norm in image space F and regularization functional O(p) = (Tp, Tp) where T is a convolution operator, one obtains Parzen's method [10] [9] with kernels defined by operator T 4 SVM for density estimation We apply the SVM technique to equation (7) for density estimation. We use the C norm in (7) and solve equation (4) in a set of functions belonging to a Reproducing Kernel Hilbert Space (RKHS). We use the regularization functional O(P) = IIplit = (p,p)1i. (10) A RKHS can be defined by a positive definite kernel K (x, y) and an inner product (f, g)1i in Hilbert space 1-l such that (f(x),K(x,Y))1i = f(y) "If E 1-l. (11) V. N. Vapnik and S. Mukherjee 662 Note that any positive definite function K(x,y) has an expansion 00 K(x,y) = LAi<!)i(x)<Pi(Y) (12) i=l where Ai and <Pi(X) are eigenvalues and eigenfunctions of K(x, y). Consider the set of functions 00 (13) f(x,c) = LCi<Pi(X) i=l and the inner product 00 * ** ci Ci (f( X,C *) , f( X,C **) = ""' ~~. i=l (14) t Kernel (12), inner product (14), and set (13) define a RKHS and For functions from a RKHS the functional (10) has the form 00 2 C? O(p) = ""' ~ At. , i=l (15) t where Ai is the i-th eigenvalue of the kernel K(x, y). The choice of the kernel defines smoothness properties on the solution. To use method (7) to solve for the density in equation (4) in a RKHS with a solution satisfying condition (9) we minimize O(p) = (p,p)ll subject to the constraint We look for a solution of equation (4) with the form l p(t) = L fiiKl'l (Xi, t). (16) i=l Accounting for (16) and (11) minimizing (10) is equivalent to minimizing l O(p,p) = (p,p)ll = L i,j=l fiifijKl'l(Xi,Xj) (17) Support Vector Method for Multivariate Density Estimation 663 subject to constraints (18) i f3i ~ 0, I:f3i = 1. (19) i=1 This optimization problem is closely related to the SV regression problem with an O"i-insensitive zone [9]. It can be solved using the standard SVM technique. Generally, only a few of the f3i will be nonzero, the Xi corresponding to these f3i are called support vectors. Note that kernel (2) has width parameter 'Yi. We call the value of this parameter admissible if it satisfies constraint (18) (the solution satisfies condition (9)). The admissible set 'Ymin :::; 'Yi :::; 'Ymax is not empty since for Parzen's method (which also has form (16)) such a value does exist. Among the 'Yi in this admissible set we select the one for which O(P) is smallest or the number of support vectors is minimum. Choosing other kernels (for example Laplacians) one can estimate densities using non-Gaussian mixture models which for some problems are more appropriate [1]. 5 Experiments Several trials of estimates constructed from sampling distributions were examined. Boxplots were made of the L1 (p) norm over the trials. The horizontal lines of the boxplot indicate the 5%, 25%, 50%, 75%, and 95% quantiles of the error distribution. For the SVM method we set O"i = c/V'i, where c = .36, .41, .936, and 1.75 for two, six, twelve and forty dimensions. For Parzen's method 'Yi was selected using a leave-one out procedure. The GMM method uses the EM algorithm and sets all parameters except n, the upper bound on the number of terms in the mixture [7]. Figure (1) shows plots of the SVM estimate using a Gaussian kernel and the GMM estimate when 60 points were drawn form a mixture of a Gaussian and Laplacian in two dimensions. Figure (2a) shows four boxplots of estimating a density defined by a mixture of two Laplacians in a two dimensional space using 200 observations. Each boxplot shows outcomes of 100 trials: for the SVM method, Parzen's method, and the GMM method with parameters n = 2, and n = 4. Figure (2c) shows the distribution of the number of terms for the SVM method. Figure (2b) shows boxplots of estimating a density defined by the mixture of four Gaussians in a six dimensional space using 600 observations. Each boxplot shows outcomes of 50 trials: for the SVM method, Parzen's method, and the G MM method with parameters n = 4, and n = 8. Figure (2c) shows the distribution of the number of terms for the SVM method. Figure (3a) shows boxplots of outcomes of estimating a density defined by the mixture of four Gaussians and four Laplacians in a twelve dimensional space using V. N. Vapnik and S. Mukherjee 664 400 observations. Each boxplot shows outcomes of 50 trials: for the SVM method, Parzen's method, and the GMM method with parameter n = 8. Figure (3c) shows the distribution of the number of terms for the SVM method. Figure (3b) shows boxplots of outcomes of estimating a density defined by the mixture of four Gaussians and four Laplacians in a forty dimensional space using 480 observations. Each box-plot shows outcomes of 50 trials: for the SVM method, Parzen's method, and the GMM method with parameter n = 8. Figure (3c) shows the distribution of the number of terms for the SVM method. 6 Summary A method for multivariate density estimation based on the SVM technique for solving ill-posed problems is introduced. This method has a form of a mixture of densities. The estimate in general has only a few terms. In experiments on synthetic data this method is more accurate than the GMM method. References [1] S. Basu and C.A. Micchelli. Parametric density estimation for the classification of acoustic feature vectors in speech recognition. In Nonlinear Modeling, Advanced Black-Box Techniques. Kluwer Publishers, 1998. [2] V.A. Morozov. Berlin, 1984. M ethods for solving incorrectly posed problems. Springer-Verlag, [3] S. Mukherjee and V. Vapnik. Multivariate density estimation: An svm approach. AI Memo 1653, Massachusetts Institute of Technology, 1999. [4] S. Paramasamy. On multivariate kolmogorov-smirnov distribution. Statistics ability Letters, 15:140-155, 1992. fj Prob- [5] E. Parzen. On estimation of a probability density function and mode. Ann. Math. Statis. , 33:1065-1076, 1962. [6] D.L. Phillips. A technique for the numerical solution of integral equations of the first kind. J.Assoc. Comput. Machinery, 9:84- 97, 1962. [7] D. Reynolds and R. Rose. Robust text-independent speaker identification using gaussian mixture speaker models. IEEE Trans on Speech and Audio Processing, 3(1) :1-27, 1995. [8] A. N. Tikhonov. Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl., 4:1035-1038, 1963. [9] V. N. Vapnik. Statistical learning theory. J. Wiley, 1998. [10] V.N. Vapnik and A.R. Stefanyuk. Nonparametric methods for restoring probability densities. Avtomatika i Telemekhanika, (8) :38-52, 1978. [11] V.V. Vasin. Relationship of several variational methods for the approximate solution of ill-posed problems. Math Not es, 7:161- 166, 1970. 665 Support Vector Method/or Multivariate Density Estimation , .-r ~ ,??-r? .~ (a) . (c) (b) Figure 1: (a) The true distribution (b) the GMM case with 4 mixtures (c) the Parzen case (d) the SVM case for 60 points. X 10-3 0.16 --,- 2.1 0.14 0.12 e 1.9 88 0.1 ~ ~O.08 0.06 0.04 , El9 + $ :::; I .S 0 --r ~ B , 1.5 1.4 GMM2 15z 1.6 ~ Parzen ~ 1.7 0.02 SVM ~ ~, --,-- -+SVM GMM4 (a) ~ Parzen GMM4 2-"'" GMMS 6 dirnll'l!ilQ(ls (c) (b) Figure 2: (a) Boxplots of the L1 (P) error for the mixture of two Laplacians in two dimensions for the SVM method, Parzen's method, and the GMM method with 2 and 4 Gaussians. (b) Boxplots of the L 1 (P) error for mixture of four Gaussians in six dimensions with the SVM method, Parzen's method, and the GMM method with 4 mixtures. (c) Boxplots of distribution of the number of terms for the SVM method for the two and six dimensional cases. 0.16r-~---------' 0.12 B 0.14 0.1 0.12 g0.1 ~0 .08 ~O.08 ~O.oe 25 ~ & 0.04 0.08 0.02 0.04 SVM Parzen (a) GMMS -- SVM Parzen (b) GMMS 12 Dim 40 Dim (c) Figure 3: (a) Boxplots of the Ll (p) error for the mixture of four Laplacians and four Gaussians in twelve dimensions for the SVM method , Parzen's method, and the GMM method with 8 Gaussians . (b) Boxplots of the L1 (P) error for the mixture of four Laplacians and four Gaussians in forty dimensions for the SVM method, Parzen's method, and the GMM method with 8 Gaussians. (c) Boxplots of distribution of the number of terms for the SVM method for the twelve and forty dimensional cases. 

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Semiparametric Approach to Multichannel Blind Deconvolution of Nonminimum Phase Systems L.-Q. Zhang, S. Amari and A. Cichocki Brain-style Information Systems Research Group, BSI The Institute of Physical and Chemical Research Wako shi, Saitama 351-0198, JAPAN zha@open.brain.riken.go.jp {amari,cia }@brain.riken.go.jp Abstract In this paper we discuss the semi parametric statistical model for blind deconvolution. First we introduce a Lie Group to the manifold of noncausal FIR filters. Then blind deconvolution problem is formulated in the framework of a semiparametric model, and a family of estimating functions is derived for blind deconvolution. A natural gradient learning algorithm is developed for training noncausal filters. Stability of the natural gradient algorithm is also analyzed in this framework. 1 Introduction Recently blind separation/deconvolution has been recognized as an increasing important research area due to its rapidly growing applications in various fields, such as telecommunication systems, image enhancement and biomedical signal processing. Refer to review papers [7] and [13] for details. A semi parametric statistical model treats a family of probability distributions specified by a finite-dimensional parameter of interest and an infinite-dimensional nuisance parameter [12]. Amari and Kumon [10] have proposed an approach to semiparametric statistical models in terms of estimating functions and elucidated their geometric structures and efficiencies by information geometry [1]. Blind source separation can be formulated in the framework of semi parametric statistical models. Amari and Cardoso [5] applied information geometry of estimating functions to blind source separation and derived an admissible class of estimating functions which includes efficient estimators. They showed that the manifold of mixtures is m-curvature free, so that we can design algorithms of blind separation without taking much care of misspecification of source probability functions. The theory of estimating functions has also been applied to the case of instantaneous mixtures, where independent source signals have unknown temporal correlations [3]. It is also applied to derive efficiency and superefficiency of demixing learning algorithms [4]. Most of these theories treat only blind source separation of instantaneous mixtures. It is only recently that the natural gradient approach has been proposed for multichannel blind L.-Q. Zhang, S. Amari and A. Cichocki 364 deconvolution [8], [18]. The present paper extends the geometrical theory of estimating functions to the semiparametric model of multichannel blind deconvolution. For the limited space, the detailed derivations and proofs are left to a full paper. 2 Blind Deconvolution Problem In this paper, as a convolutive mixing model, we consider a multichannel linear timeinvariant (LTI) systems, with no poles on the unit circle, of the form 00 x(k) = L Hps(k - p), (1) p=-oo where s(k) is an n-dimensional vector of source signals which are spatially mutually independent and temporarily identically independently distributed, and x(k) is an n-dimensional sensor vector at time k, k = 1,2, . . '. We denote the unknown mixing filter by H(z) = 2::-00 Hpz-p. The goal of multichannel blind deconvolution is to retrieve source signals s(k) only using sensor signals x(k), k = 1,2"", and certain knowledge of the source signal distributions and statistics. We carry out blind deconvolution by using another multichannel LTI system of the form y(k) = W(z)x(k), (2) where W(z) = 2:~= -N Wpz-P, N is the length of FIR filter W(z), y(k) [Yl (k), ... ,Yn(k)V is an n-dimensional vector of the outputs, which is used to estimate the source signals. When we apply W(z) to the sensor signal x(k), the global transfer function from s(k) to y(k) is defined by G(z) = W(z)H(z). The goal of the blind deconvolution task is to find W(z) such that G(z) = PAD(z), where P E R nxn is a permutation matrix, D(z) = diag{z-d 1 , ??? ,z- dn }, and A E R n x n is a nonsingular diagonal scaling matrix. 3 Lie Group on M (N, N) In this section, we introduce a Lie group to the manifold of noncausal FIR filters. The Lie group operations playa crucial role in the following discussion. The set of all the noncausal FIR filters W (z) of length N, having the constraint that W is nonsingular, is denoted by M(N,N) = {W(Z) I W(z) = .tN W.z - ? , det(W) # o}, (3) where W is an N x N block matrix, ... W_N+ll ... W - N+2 . . .. (4) Wo M(N, N) is a manifold of dimension n 2 (2N + 1). In general, multiplication of two filters in M(N, N) will enlarge the filter length and the result does belong to M(N, N) anymore. This makes it difficult to introduce the Riemannian structure to the manifold of noncausal FIR filters. In order to explore possible geometrical structures of M(N, N) which will lead to effective learning algorithms for W (z) , we define algebraic operations of filters in the Lie group framework. First, we introduce a novel filter decomposition of noncausal filters in M (N, N) into a product of two one-sided FIR filters [19], which is illustrated in Fig. 1. Blind Deconvolution ofNonminimum Phase Systems 365 Unknown :s(k) n: u(k) R(Z") ~ L(z) x(k) H(z) n i y(k) n i Mixing model Demixing model Figure 1: Illustration of decomposition of noncausal filters in M (N, N) Lemma 1 [19] If the matrix W is nonsingular, any noncausalfilter W(z) in M(N,N) has the decomposition W(z) = R(z)L(z-l), where R(z) = L::=o Rpz-P, L(Z-l) = L::=o LpzP are one-sided FIR filters. In the manifold M(N, N), Lie operations, multiplication as follows: For B(z), C(z) E M(N, N), B(z) * C(z) = [B(z)C(z)]N' * and inverse t, are defined Bt(z) = Lt(Z-l)Rt(z), (5) where [B(Z)]N is the truncating operator that any terms with orders higher than N in the polynomial B (z) are truncated, and the inverse of one-side FIR filters is recurrently defined by ~ = RO l , = - L::=l Rt_qRqRO l , p = 1,'" ,N. Refer to [18] for the detailed derivation. With these operations, both B(z) * C(z) and Bt (z) still remain in the manifold M (N, N). It is easy to verify that the manifold M (N, N) with the above operations forms a Lie Group. The identity element is E(z) = I. at 4 Semiparametric Approach to Blind Deconvolution We first introduce the basic theory of semiparametric models, and formulate blind deconvolution problem in the framework of the semiparametric models. 4.1 Semiparametric model Consider a general statistical model {p( Xj 6, en, where x is a random variable whose probability density function is specified by two parameters, 6 and 6 being the parameter of interest, and being the nuisance parameter. When the nuisance parameter is of infinite dimensions or of functional degrees of freedom, the statistical model is called a semiparametric model [12]. The gradient vectors of the log likelihood u(Zj 6, e) = 81ogp(z;6.e) v(z,,,'it 6 ~) -- 81ogp(z;6,e) are called the score functions of the parameter 80 ' 8( , e, e e of interest or shortly 6-score and the nuisance score or shortly -score, respectively. In the semiparametric model, it is difficult to estimate both the parameters of interest and nuisance parameters at the same time, since the nuisance parameter is of infinite degrees of freedom. The semiparametric approach suggests to use an estimating function to estimate the parameters of interest, regardless of the nuisance parameters. The estimating function is a vector function z(x, 6), independent of nuisance parameters satisfying the following conditions e e, (6) 1) Eo,dz(x,6)] = 0, 2) det(lC) i= 0, where IC = Eo,d 8z(x,6) 88 ]. (7) L.-Q. Zhang, S. Amari and A. Cichocki 366 3) Eo ,dz(x, 8)zT (x, 8)] < 00, (8) e. for all 8 and Generally speaking, it is difficult to find an estimating function. Amari and Kawanabe [9] studied the information geometry of estimating functions and provided a novel approach to find all the estimating functions. In this paper, we follow the approach to find a family of estimating functions for bind deconvolution. 4.2 Semiparametric Formulation for Blind Deconvolution Now we tum to formulate the blind deconvolution problem in the framework of semi parametric models. From the statistical point of view, the blind deconvolution problem is to estimate H(z) or H- 1 (z) from the observed data VL = {x(k), k = 1, 2", .}. The estimate includes two unknowns: One is the mixing filter H(z) which is the parameter of interest, and the other is the probability density function p(s) of sources, which is the nuisance parameter in the present case. FOIf blind deconvolution problem, we usually assume that source signals are zero-mean, E[sil' = 0, for i = 1"", n. In addition, we generally impose constraints on the recovered signals to remove the indeterminacy, (9) A typical example of the constraint is k i ( Si) = sf -1. Since the source signals are spatially mutually independent and temporally iid, the pdfr(s) can be factorized into a product form r(s) = TI~l r(si). The purpose of this paper is to find a family of estimating functions for blind deconvolution. Remarkable progress has been made recently in the theory of the semiparametric approach [9],[12]. It has been shown that the efficient score itself is an estimating function for blind separation. 5 Estimating Functions In this section, we give an explicit form of the score function matrix of interest and the nuisance tangent space, by using a local nonholonomic reparameterization. We then derive a family of estimating functions from it. 5.1 Score function matrix and its representation Since the mixing model is a matrix filter, we write an estimating function in the same matrix filter format N F(x;H(z)) = L Fp(x;H(z))z-P, (10) p= -N where F p(x; H(z)) are n x n-matrices. In orderto derive the explicit form ofthe H-score, we reparameterize the filter in a small neighborhood of H (z) by using a new variable matrix filter as H(z) * (I - X(z)), where 1 is the identity element of the manifold M(N, N). The variation X(z) represents a local coordinate system at the neighborhoodNH of H(z) on the manifold M(N, N). The variation dH(z) of H(z) is represented as dH(z) = -H(z) * dX(z). Letting W(z) = Ht(z), we have dX(z) = dW(z) * wt(z) , (11 ) which is a nonholonomic differential variable [6] since (11) is not integrable. With this representation of the parameters, we can obtain learning algorithms having the equivariant property [14] since the deviation dX(z) is independent of a specific H(z). The relative or the natural gradient of a cost function on the manifold can be automatically derived from this representation [21, [14], [18]. Blind Deconvolution ofNonminimum Phase Systems 367 {p(x;e,;)} {p(x;9,e)} Figure 2: Illustration of orthogonal decomposition of score functions The derivative of any cost function l(H(z)) with respect to a noncausal filter X(z) E:==-N Xpz-P is defined by L N 8l(H(z? _ aX(z) 8l(H(z? z-p p==-N axp (12) Now we can easily calculate the score function matrix of noncausal filter X(z), alogp(XiH(z),r) _ aX(z) where lP(y) N '"' () T(k _ ) -p L.J lP Y Y Pz , p=-N (13) = ('Pi(Yi),"', 'Pn(Yn?T, 'Pi(Yi) = dlO~;:(II/). and y = Ht(z)x. S.2 Efficient scores The efficient scores, denoted by UE(s; H(z), r), can be obtained by projecting the score function to the space orthogonal to the nuisance tangent space TJ'{z},r' which is illustrated in figure 2. In this section, we give an explicit form of the efficient scores for blind deconvolution. Lemma 2 [5} The tangent nuisance space TJ'{z),r is a linear space spanned by the nui- sance score junctions, denoted by TJ'{z),r = {E:=I CiOi(Si)} , where Ci are coefficients, and ai(si) are arbitrary junctions, satisfying the/ollowing conditions E[ai(si)2] < 00, E[sai(si)] = 0, E[k(si)ai(si?) = O. (14) We rewrite the score function (13) into the form U(s; H(z), r) = E!-N Upz-P, where Up = (cp(si(k?sj(k - P?nxn. Lemma 3 The off-diagonal elements UO,ij(S; H(z), r), i =/: j, and the delay elements Up,ij(S; H(z), r), P =/: 0, 0/ the score junctions are orthogonal to the nuisance tangent space TJ(z),r' Lemma 4 The projection 0/ UO,ii to the space orthogonal to the nuisance tangent space TJ(z),r is o/the/orm W(Si) = Co + CISi + C2k(Si), where Cj are any constants. 368 L.-Q. Zhang, S. Amari and A. Cichocki In summary we have the following theorem Theorem 1 The efficient score, UE(s; H(z), r) U: = L::=- NU: z-P, is given by <p(s)sT(k - p), for p:f. 0; (15) for off diagonal elements, for diagonal elements. U~ (16) For the instantaneous mixture case, it has been proven [9] that the semiparametric model for blind separation is information m-curvature free. This is also true in the multichannel blind deconvolution case. As a result, the efficient score function is an estimating function for blind deconvolution. Using this result, we easily derive a family of estimating functions for blind deconvolution N F(x(k); W(z)) = L c.p(y(k))y(k - pf z-P - I, (17) p=-N where y(k) = W(z)x(k), and <p is a given function vector. The estimating function is the efficient score function, when Co = Cl = 0, C2 = 1 and ki(Si) = c.pi(sdsi - 1. 6 Natural Gradient Learning and its Stability Ordinary stochastic gradient methods for parameterized systems suffer from slow convergence due to the statistical correlations of the processes signals. While quasi-Newton and related methods can be used to improve convergence, they also suffer from the mass computation and numerical instability, as well as local convergence. The natural gradient approach was developed to overcome the drawback of the ordinary gradient algorithm in the Riemannian spaces [2, 8, 15]. It has been proven that the natural gradient algorithm is an efficient algorithm in blind separation and blind deconvolution [2]. The efficient score function ( the estimating function) gives an efficient search direction for updating filter X(z) . Therefore, the updating rule for X(z) is described by Xk+l(Z) = Xk(z) -1]F(x(k), Wk(Z)), (18) where 1] is a learning rate. Since the new parameterization X(z) is defined by a nonholonomic transformation dX (z) = dW (z) * wt (z ), the deviation of W (z) is given by ~ W(z) = ~X(z) * W(z). (19) Hence, the natural gradient learning algorithm for W (z) is described as (20) Wk+l(Z) = Wk(Z) -1]F(x(k), Wk(z)) * Wk(z) , where F(x, W (z)) is an estimating function in the form (17). The stability ofthe algorithm (20) is equivalent to the one of algorithm (18). Consider the averaged version of algorithm (18) (21) ~X(z) = -1]E[F(x(k), Wk(Z))] . Analyzing the variational equation of the above equation and using the mutual independence and i.i.d. properties of source signals, we derive the stability conditions of learning algorithm (21) at vicinity of the true solution mi for i , j = 1," ' , n, where mi + 1 > 0, K.i > 0, K.iK.ja;aJ > 1, = E(c.p'(Yi (k))y;(k)], K.i = E[c.p~(Yi)], (22) a; = E[IYiI 2 ]. Therefore, we have the following theorem: Theorem 2 If the conditions (22) are satisfied, then the natural gradient learning algorithm (20) is locally stable. Blind Deconvolution of Nonminimum Phase Systems 369 References [1] S. Amari. Differential-geometrical methods in statistics, Lecture Notes in Statistics, volume 28. Springer, Berlin, 1985. [2] S. Amari. Natural gradient works efficiently in learning. Neural Computation, 10:251-276, 1998. [3 J S. Amari. ICA of temporally correlated signals - Learning algorithm. In Proceeding of 1st Inter. Workshop on Independent Component Analysis and Signal Separation, pages 37-42, Aussois, France, January, 11-15 1999. [4] S. Amari. Superefficiency in blind source separation. IEEE Trans. on Signal Processing, 47(4):936-944, April 1999. [5] S. Amari and J.-F. Cardoso. Blind source separation- semiparametric statistical approach. IEEE Trans. Signal Processing, 45:2692-2700, Nov. 1997. [6] S. Amari, T. Chen, and A. Cichocki. Nonholonomic orthogonal constraints in blind source separation. Neural Comput., to be published. [7] S. Amari and A. Cichocki. Adaptive blind signal processing- neural network approaches. Proceedings of the IEEE, 86(10):2026-2048,1998. [81 S. Amari, S. Douglas, A. Cichocki, and H. Yang. Multichannel blind deconvolution and equalization using the natural gradient. In Proc. IEEE Workshop on Signal Processing Adv. in Wireless Communications, pages 101-104, Paris, France, April 1997. [9] S. Amari and M. Kawanabe. Estimating functions in semiparametric statistical models. In I. V. Basawa, v.P. Godambe, and R.L. Taylor, editors, Estimating Functions, volume 32 of Monograph Series, pages 65-81. IMS, 1998. [10] S. Amari and M. Kumon. Estimation in the presence of infinitely many nuisance parameters in semiparametric statistical models. Ann. Statistics, 16: 1044-1068, 1988. [11] A.l. Bell and T.l. Sejnowski. An information maximization approach to blind separation and blind deconvolution. Neural Computation, 7: 1129-1159, 1995. [12] P. Bickel, C. Klaassen, Y. Ritov, and J. Wellner. Efficient and Adaptive Estimation for Semiparametric Models. The Johns Hopkins Univ. Press, Baltimore and London, 1993. [13] J.-F Cardoso. Blind signal separation: Statistical principles. Proceedings of the IEEE, 86(10):2009-2025,1998. [14] J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE Trans. Signal Processing, SP-43: 30 17-3029, Dec 1996. [15] A. Cichocki and R. Unbehauen. Robust neural networks with on-line learning for blind identification and blind separation of sources. IEEE Trans Circuits and Systems I: Fundamentals Theory and Applications, 43(11):894-906, 1996. [16] L. Tong, R.W. Liu, v.c. Soon, and Y.F. Huang. Indeterminacy and identifiability of blind identification. IEEE Trans. Circuits, Syst., 38(5):499-509, May 1991. [17] H. Yang and S. Amari. Adaptive on-line learning algorithms for blind separation: Maximum entropy and minimal mutual infonnation. Neural Comput., 9: 1457-1482, 1997. [18] L. Zhang, A. Cichocki, and S. Amari. Geometrical structures of FIR manifold and their application to multichannel blind deconvolution. In Proceeding of NNSP'99, pages 303-312, Madison, Wisconsin, August 23-25 1999. [19] L. Zhang, A. Cichocki, and S. Amari. Multichannel blind deconvolution of nonminimum phase systems using information backpropagation. In Proceedings of the Fifth International Conference on Neural Information Processing(ICONIP'99), page 210-216, Perth, Australia, Nov. 16-20 1999. 

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Learning the Similarity of Documents: An Information-Geometric Approach to Document Retrieval and Categorization Thomas Hofmann Department of Computer Science Brown University, Providence, RI hofmann@cs.brown.edu, www.cs.brown.edu/people/th Abstract The project pursued in this paper is to develop from first information-geometric principles a general method for learning the similarity between text documents. Each individual document is modeled as a memoryless information source. Based on a latent class decomposition of the term-document matrix, a lowdimensional (curved) multinomial subfamily is learned. From this model a canonical similarity function - known as the Fisher kernel - is derived. Our approach can be applied for unsupervised and supervised learning problems alike. This in particular covers interesting cases where both, labeled and unlabeled data are available. Experiments in automated indexing and text categorization verify the advantages of the proposed method. 1 Introduction The computer-based analysis and organization of large document repositories is one oftoday's great challenges in machine learning, a key problem being the quantitative assessment of document similarities. A reliable similarity measure would provide answers to questions like: How similar are two text documents and which documents match a given query best? In a time, where searching in huge on-line (hyper-)text collections like the World Wide Web becomes more and more popular, the relevance of these and related questions needs not to be further emphasized. The focus of this work is on data-driven methods that learn a similarity function based on a training corpus of text documents without requiring domain-specific knowledge. Since we do not assume that labels for text categories, document classes, or topics, etc. are given at this stage, the former is by definition an unsupervised learning problem. In fact, the general problem of learning object similarities precedes many "classical" unsupervised learning methods like data clustering that already presuppose the availability of a metric or similarity function. In this paper, we develop a framework for learning similarities between text documents from first principles. In doing so, we try to span a bridge from the foundations of statistics in information geometry [13, 1] to real-world applications in information retrieval and text learning, namely ad hoc retrieval and text categorization. Although the developed general methodology is not limited to text documents, we will for sake of concreteness restrict our attention exclusively to this domain. Learning the Similarity ofDocuments 2 915 Latent Class Decomposition Memoryless Information Sources Assume we have available a set of documents V = {d l , ..? , dN} over some fixed vocabulary of words (or terms) W = {WI, ... , WM}. In an information-theoretic perspective, each document di can be viewed as an information source, i.e. a probability distribution over word sequences. Following common practice in information retrieval, we will focus on the more restricted case where text documents are modeled on the level of single word occurrences. This means that we we adopt the bag-of- words view and treat documents as memoryless information sources. I A. Modeling assumption: Each document is a memoryless information source. This assumption implies that each document can be represented by a multinomial probability distribution P(wjldi), which denotes the (unigram) probability that a generic word occurrence in document d i will be Wj. Correspondingly, the data can be reduced to some simple sufficient statistics which are counts n(di , Wj) of how often a word Wj occurred in a document d j ? The rectangular N x M matrix with coefficients n(di , Wj) is also called the term-document matrix. Latent Class Analysis Latent class analysis is a decomposition technique for contingency tables (cf. [5, 3] and the references therein) that has been applied to language modeling [15] ("aggregate Markov model") and in information retrieval [7] ("probabilistic latent semantic analysis"). In latent class analysis, an unobserved class variable Zk E Z = {zt, ... , ZK} is associated with each observation, i.e. with each word occurrence (d i , Wj). The joint probability distribution over V x W is a mixture model that can be parameterized in two equivalent ways P(di , Wj) = K K k=l k=l 2: P(zk)P(dilzk)P(wjlzk) = P(di) 2: P( WjIZk)P(Zk Idd . (1) The latent class model (1) introduces a conditional independence assumption, namely that di and Wj are independent conditioned on the state of the associated latent variable. Since the cardinality of Zk is typically smaller than the number of documents/words in the collection, Zk acts as a bottleneck variable in predicting words conditioned on the context of a particular document. To give the reader a more intuitive understanding of the latent class decomposition, we have visualized a representative subset of 16 "factors" from a K = 64 latent class model fitted from the Reuters2I578 collection (cf. Section 4) in Figure 1. Intuitively, the learned parameters seem to be very meaningful in that they represent identifiable topics and capture the corresponding vocabulary quite well. By using the latent class decomposition to model a collection of memory less sources, we implicitly assume that the overall collection will help in estimating parameters for individual sources, an assumption which has been validated in our experiments. B. Modeling assumption: Parameters for a collection of memoryless information sources are estimated by latent class decomposition. Parameter Estimation The latent class model has an important geometrical interpretation: the parameters ?1 == P( Wj IZk) define a low-dimensional subfamily of the multinomial family, S(?) == {11" E [0; I]M : 1I"j = :Ek 1/;k?1 for some1/; E [0; I]K, :Ek 1/;k = I}, i.e. all multinomials 11" that can be obtained by convex combinations from the set of "basis" vectors {?k : 1 :::; k :::; K}. For given ?-parameters, 1 Extensions to the more general case are possible, but beyond the scope of this paper. T. Hofmann 916 government presIdent tax budget cut s pending chairman cuts vice deficit company taxes reform named billion trading director america.n exchange general futures stock options index contracts market london exchanges executive chief officer board motors chrysler gm car ford test cars motor banks debt brazil new loans dlrs bankers b .. nk payments billion tr .. de japan j .. panese ec pct january february rise rose 1986 december year fell prices oil crude united officials community energy petroleum prices bpd barrels barrel european imports exploration price states marks currency dollar german airlines bundesbank aircraft central port mark boeing west employees dollars airline dealers vs areas cts weather net area loss normal min good shr crop qtr damage revs caused profit affected note people gold steel plant mining copper unlon air workers strike ton s silver metal production ounCeS food drug study aid s prod uct bllhon dlrs year surplu s deficit foreign current trade account reserves house rea.gan president administration congress trea.tment white company environmental products approval s ecretary told volcker reagans Figure 1: 16 selected factors from a 64 factor decomposition ofthe Reuters21578 collection. The displayed terms are the 10 most probable words in the class-conditional distribution P (Wj IZk) for 16 selected states Zk after the exclusion of stop words. each 1/;i , 1/;i == P(zkldi), will define a unique multinomial distribution rri E S(?). Since S( ?) defines a submanifold on the multinomial simplex, it corresponds to a curved exponential subfamily. 2 We would like to emphasis that we propose to learn both, the parameters within the family (the 1/;'s or mixing proportions P(Zk Idi )) and the parameters that define the subfamily (the ?'s or class-conditionals P(WjIZk)). The standard procedure for maximum likelihood estimation in latent variable models is the Expectation Maximization (EM) algorithm. In the E-step one computes posterior probabilities for the latent class variables, P( Zk )P( di IZk )P( Wj IZk) 2:1 P(zI)P(dilzt)P(wjlz/) P(Zk) P( di IZk )P( Wj IZk) P(di' Wj) (2) The M-step formulae can be written compactly as P(diIZk)} N M P(WjIZk) ex n(dn, wm)P(zkldn, wm) P(Zk) n=l m=l 2: 2: { X din djm (3) 1 where 6 denotes the Kronecker delta. Related Models As demonstrated in [7], the latent class model can be viewed as a probabilistic variant of Latent Semantic Analysis [2], a dimension reduction technique based on Singular Value Decomposition. It is also closely related to the non-negative matrix decomposition discussed in [12] which uses a Poisson sampling model and has been motivated by imposing non-negativity constraints on a decomposition by PCA. The relationship of the latent class model to clustering models like distributional clustering [14] has been investigated in [8]. [6] presents yet another approach to dimension reduction for multinomials which is based on spherical models, a different type of curved exponential subfamilies than the one presented here which is affine in the mean-value parameterization. 2Notice that graphical models with latent variable are in general stratified exponential families [4], yet in our case the geometry is simpler. The geometrical view also illustrates the well-known identifiability problem in latent class analysis. The interested reader is referred to [3]. As a practical remedy, we have used a Bayesian approach with conjugate (Dirichlet) prior distributions over all multinomials which for the sake of clarity is not described in this paper since it is very technical but nevertheless rather straightforward. Learning the Similarity of Documents 3 917 Fisher Kernel and Information Geometry The Fisher Kernel We follow the work of [9] to derive kernel functions (and hence similarity functions) from generative data models. This approach yields a uniquely defined and intrinsic (i. e. coordinate invariant) kernel, called the Fisher kernel. One important implication is that yardsticks used for statistical models carryover to the selection of appropriate similarity functions. In spite of the purely unsupervised manner in which a Fisher kernel can be learned, the latter is also very useful in supervised learning, where it provides a way to take advantage of additional unlabeled data. This is important in text learning, where digital document databases and the World Wide Web offer a huge background text repository. As a starting point, we partition the data log-likelihood into contributions from the various documents. The average log-probability of a document d i , i.e. the probability of all the word occurrences in d i normalized by document length is given by, M l(dd = L F(wjld j=l K i) log L P(WjIZk)P(Zkldi), F(wj Idi ) == 2: n(d(, ~j)) n d" k=l m (4) Wm which is up to constants the negative Kullback-Leibler divergence between the empirical distribution F(wjldi ) and the model distribution represented by (1). In order to derive the Fisher kernel, we have to compute the Fisher scores u(d i ; 0), i.e. the gradient of l(dd with respect to 0, as well as the Fisher information 1(0) in some parameterization 0 [13]. The Fisher kernel at {) is then given by [9] (5) Computational Considerations For computational reasons we propose to approximate the (inverse) information matrix by the identity matrix, thereby making additional assumptions about information orthogonality. More specifically, we use a variance stabilizing parameterization for multinomials - the square-root parameterization - which yields an isometric embedding of multinomial families on the positive part of a hypersphere [11]. In this parameterization, the above approximation will be exact for the multinomial family (disregarding the normalization constraint). We conjecture that it will also provide a reasonable approximation in the case of the subfamily defined by the latent class model. c. Simplifying assumption: The Fisher information in the square-root parameterization can be approximated by the identity matrix. Interpretation of Results Instead of going through the details of the derivation which is postponed to the end of this section, it is revealing to relate the results back to our main problem of defining a similarity function between text documents. We will have a closer look at the two contributions reSUlting from different sets of parameters. The contribution which stems from (square-root transformed) parameters P(Zk) is (in a simplified version) given by L P(Zk Idi)P(Zk Id n )/ P(Zk) . (6) k J( is a weighted inner product in the low-dimensional factor representation of the documents by mixing weights P(zkldi). This part of the kernel thus computes a "topical" overlap between documents and is thereby able to capture synonyms, i.e., words with an identical or similar meaning, as well as words referring to the same T. Hofmann 918 topic. Notice, that it is not required that di and dn actually have (many) terms in common in order to get a high similarity score. The contribution due to the parameters P(WjIZk) is of a very different type. Again using the approximation of the Fisher matrix, we arrive at the inner product K:(di, do) = l( P(Wj Idi ) I'>(wj Id o) ~ P(zkldi';(2~;:; Ido , Wj) ? (7) j( has also a very appealing interpretation: It essentially computes an inner product between the empirical distributions of di and dn , a scheme that is very popular in the context of information retrieval in the vector space model. However, common words only contribute, if they are explained by the same factor(s), i.e., if the respective posterior probabilities overlap. This allows to capture words with multiple meanings, so-called polysems. For example, in the factors displayed in Figure 1 the term "president" occurs twice (as the president of a company and as the president of the US). Depending on the document the word occurs in, the posterior probability will be high for either one of the factors, but typically not for both. Hence, the same term used in different context and different meanings will generally not increase the similarity between documents, a distinction that is absent in the naive inner product which corresponds to the degenerate case of K = 1. Since the choice of K determines the coarseness of the identified "topics" and different resolution levels possibly contribute useful information, we have combined models by a simple additive combination of the derived inner products. This combination scheme has experimentally proven to be very effective and robust. D. Modeling assumption: Similarities derived from latent class decompositions at different levels of resolution are additively combined. In summary, the emergence of important language phenomena like synonymy and polysemy from information-geometric principles is very satisfying and proves in our opinion that interesting similarity functions can be rigorously derived, without specific domain knowledge and based on few, explicitly stated assumptions (A-D). Technical Derivation Define Pjk == 2v'P(wjlzk), then 8l(d j ) oP(WjIZk) = . fp(wjlzk) P(wjldi ) P(zkJdd oP(WjIZk) OPjk V P(wjld i ) P(wjlddP(Zkldi' Wj) v'P(WjIZk) Similarly we define Pk = 2v'P(Zk). Applying Bayes' rule to substitute P(zkldd in l(dd (i.e. P(zkldd = P(zk)P(di/zk)/P(di)) yields 8l(dd OPk 8l(d;) OP(Zk) = v'P(Zk) P(dilzk) ~ P(wjldd P(W ' IZk) OP(Zk) OPk P(dd ~ P(WjJdd J J P(zkJdd ~ P(wj1di)p( I ) P( zkld j ) L...J W? Zk ~ ~==== v'P(Zk) j P(wjldi) J v'P(Zk) . The last (optional) approximation step makes sense whenever P(wjld j ) ~ P(wjldi ). Notice that we have ignored the normalization constraints which would yield a (reactive) term that is constant for each multinomial. Experimentally, we have observed no deterioration in performance by making these additional simplifications. Learning the Similarity ofDocuments VSM VSM++ Medline 44.3 67.2 919 Cranfield 29.9 37.9 CACM 17.9 27.5 CISI 12.7 20.3 Table 1: Average precision results for the vector space baseline method (VSM) and the Fisher kernel approach (VSM ++) for 4 standard test collections, Medline, Cranfield, CACM, and CIS!. I earn 20x sub lOx sub 5x sub all data lOx cv SVM SVM++ kNN kNN++ SVM SVM++ kNN kNN++ SVM SVM++ kNN kNN++ SVM SVM++ kNN kNN++ 5.51 4.56 5.91 5.05 4.88 4.11 5.51 4.94 4.09 3.64 5.13 4.74 2.92 2.98 4.17 4.07 acq 7.67 5.37 9.64 7.80 5.54 4.84 9.23 7.47 4.40 4.15 8.70 6.99 3.21 3.15 6.69 5.34 money 3.25 2.08 3.24 3.11 2.38 2.08 2.64 2.42 2.10 1.78 2.27 2.22 1.20 1.21 1.78 1.73 grain 2.06 1.71 2.54 2.35 1.71 1.42 2.55 2.28 1.32 0.98 2.40 2.18 0.77 0.76 1.73 1.58 crude 2.50 1.53 2.42 1.95 1.88 1.45 2.42 1.88 1.46 1.19 2.23 1.74 0.92 0.86 1.42 1.18 average 4.20 3.05 4.75 4.05 3.27 2.78 4.47 3.79 2.67 2.35 4.14 3.57 1.81 1.79 3.16 2.78 lmprov. +27.4% - +14.7% - +15.0% +15.2% - +12 .1% +13.7% - +0.6% +12.0% Table 2: Classification errors for k-nearest neighbors (kNN) SVMs (SVM) with the naive kernel and with the Fisher kernel(++) (derived from J( = 1 and J( = 64 models) on the 5 most frequent categories of the Reuters21578 corpus (earn, acq, monex-fx, grain, and crude) at different subsampling levels. 4 Experimental Results We have applied the proposed method for ad hoc information retrieval , where the goal is to return a list of documents, ranked with respect to a given query. This obviously involves computing similarities between documents and queries. In a follow-up series of experiments to the ones reported in [7] - where kernels K(d i , dn ) = ~k P(Zk Idi)P( Zk Idn ) and JC(di ' dn ) = ~j P(Wj Idi )P(wjldn ) have been proposed in an ad hoc manner - we have been able to obtain a rigorous theoretical justification as well as some additional improvements. Average precision-recall values for four standard test collections reported in Table 1 show that substantial performance gains can be achieved with the help of a generative model (cf. [7] for details on the conducted experiments). To demonstrate the utility of our method for supervised learning problems, we have applied it to text categorization, using a standard data set in the evaluation, the Reuters21578 collections of news stories. We have tried to boost the performance of two classifiers that are known to be highly competitive for text categorization: the k- nearest neighbor method and Support Vector Machines (SVMs) with a linear kernel [10]. Since we are particularly interested in a setting, where the generative model is trained on a larger corpus of unlabeled data, we have run experiments where the classifier was only trained on a subsample (at subsampling factors 20x,10x,5x). The results are summarized in Table 2. Free parameters of the base classifiers have been optimized in extensive simulations with held-out data. The results indicate T. Hofmann 920 that substantial performance gains can be achieved over the standard k-nearest neighbor method at all subsampling levels. For SVMs the gain is huge on the subsampled data collections, but insignificant for SVMs trained on all data. This seems to indicate that the generative model does not provide any extra information, if the SVM classifier is trained on the same data. However, notice that many interesting applications in text categorization operate in the small sample limit with lots of unlabeled data. Examples include the definition of personalized news categories by just a few example, the classification and/or filtering of email, on-line topic spotting and tracking, and many more. 5 Conclusion We have presented an approach to learn the similarity of text documents from first principles. Based on a latent class model, we have been able to derive a similarity function, that is theoretically satisfying, intuitively appealing, and shows substantial performance gains in the conducted experiments. Finally, we have made a contribution to the relationship between unsupervised and supervised learning as initiated in [9] by showing that generative models can help to exploit unlabeled data for classification problems. References [1] Shun'ichi Amari. Differential-geometrical methods in statistics. Springer-Verlag, Berlin, New York, 1985. [2] S. Deerwester, S. T. Dumais, G. W. Furnas, T. K. Landauer, and R. Harshman. Indexing by latent semantic analysis. Journal of the American Society for Information Science, 41:391-407, 1990. [3] M. J. Evans, Z. Gilula, and I. Guttman. Latent class analysis of two-way contingency tables by Bayesian methods. Biometrika, 76(3):557-563, 1989. [4] D. Geiger, D. Heckerman, H. King, and C. Meek. Stratified exponential families: Graphical models and model selection. Technical Report MSR-TR-98-31, Microsoft Research, 1998. [5] Z. Gilula and S. J . Haberman. Canonical analysis of contingency tables by maximum likelihood. Journal of the American Statistical Association, 81(395):780-788, 1986. [6] A. Gous. Exponential and Spherical Subfamily Models. PhD thesis, Stanford, Statistics Department, 1998. [7] T. Hofmann. Probabilistic latent semantic indexing. In Proceedings of the 22th International Conference on Research and Development in Information Retrieval (SIGIR), pages 50-57, 1999. [8] T . Hofmann, J. Puzicha, and M. I. Jordan. Unsupervised learning from dyadic data. In Advances in Neural Information Processing Systems 11. MIT Press, 1999. [9] T. Jaakkola and D. Haussler. Exploiting generative models in discriminative classifiers. In Advances in Neural Information Processing Systems 11. MIT Press, 1999. [lO] T. Joachims. Text categorization with support vector machines: Learning with many relevant features. In International Conference on Machine Learning (ECML), 1998. [ll] R.E. Kass and P. W. Vos. Geometrical foundations of asymptotic inference. Wiley, New York, 1997. [12] D. Lee and S. Seung. Learning the parts of objects by non-negative matrix factorization. Nature, 401:788-791, 1999. [13] M. K. Murray and J. W. Rice. Differential geometry and statistics. Chapman & Hall, London, New York, 1993. [14] F.C.N. Pereira, N.Z. Tishby, and L. Lee. Distributional clustering of English words. In Proceedings of the ACL, pages 183- 190, 1993. [15] L. Saul and F . Pereira. Aggregate and mixed-order Markov models for statistical language processing. In Proceedings of the 2nd International Conference on Empirical Methods in Natural Language Processing, 1997. 

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Search for Information Bearing Components in Speech Howard Hua Yang and Hynek Hermansky Department of Electrical and Computer Engineering Oregon Graduate Institute of Science and Technology 20000 NW, Walker Rd., Beaverton, OR97006, USA {hyang,hynek}@ece.ogi.edu, FAX:503 7481406 Abstract In this paper, we use mutual information to characterize the distributions of phonetic and speaker/channel information in a timefrequency space. The mutual information (MI) between the phonetic label and one feature, and the joint mutual information (JMI) between the phonetic label and two or three features are estimated . The Miller's bias formulas for entropy and mutual information estimates are extended to include higher order terms. The MI and the JMI for speaker/channel recognition are also estimated. The results are complementary to those for phonetic classification. Our results show how the phonetic information is locally spread and how the speaker/channel information is globally spread in time and frequency. 1 Introduction Speech signals typically carry information about number of target sources such as linguistic message, speaker identity, and environment in which the speech was produced. In most realistic applications of speech technology, only one or a few information targets are important. For example, one may be interested in identifying the message in the signal regardless of the speaker or the environments in which the speech was produced, or the identification of the speaker is needed regardless of the words the targeted speaker is saying. Thus, not all components of the signal may be equally relevant for a decoding of the targeted information in the signal. The speech research community has at its disposal rather large speech databases which are mainly used for training and testing automatic speech recognition (ASR) systems. There have been relatively few efforts to date to use such databases for deriving reusable knowledge about speech and speech communication processes which could be used for improvements of ASR technology. In this paper we apply information-theoretic approaches to study a large hand-labeled data set of fluent speech to learn about the information structure of the speech signal including the distribution of speech information in frequency and in time. Based on the labeled data set, we analyze the relevancy of the features for phonetic 804 H. H. YangandH Hermansky classifications and speaker/channel variability. The features in this data set are labeled with respect to underlying phonetic classes and files from which the features come from. The phoneme labels relate to the linguistic message in the signal, and the file labels carry the information about speakers and communication channels (each file contains speech of a single speaker transmitted through one telephone channel). Thus, phoneme and file labels are two target variables for statistical inference. The phoneme labels take 19 different values corresponding to 19 broad phoneme categories in the OGI Stories database [2]. The file labels take different values representing different speakers in the OGI Stories database. The relevancy of a set of features is measured by the joint mutual information (JMI) between the features and a target variable. The phoneme target variable represents in our case the linguistic message . The file target variable represents both different speakers and different telephone channels. The joint mutual information between a target variable and the features quantifies the relevancy of the features for that target variable. Mutual information measure the statistical dependence between random variables. Morris et al (1993) used mutual information to find the critical points of information for classifying French Vowel-Plosive-Vowel utterances. Bilmes(1998) showed recently that the information appears to be spread over relatively long temporal spans. While Bilmes used mutual information between two variables on nonlabeled data to reveal the mutual dependencies between the components of the spectral energies in time and frequency, we focused on joint mutual information between the phoneme labels or file labels and one, two or three feature variables in the time-frequency plane[7, 6] and used this concept to gain insight into how information about phonemes and speaker/channel variability is distributed in the time-frequency plane. 2 Data Set and Preprocessing The data set used in this paper is 3-hour phonetically labeled telephone speech, a subset of the English portion (Stories) ofthe OGI multi-lingual database [2] containing approximately 50 seconds of extemporaneous speech from each of 210 different speakers. The speech data is labeled by a variable Y taking 19 values representing 19 most often occurring phoneme categories. The average phoneme duration is about 65 ms and the number of phoneme instances is 6542l. Acoustic features X (fk, t) for the experiments are derived from a short-time analysis of the speech signal with a 20 ms analysis window (Hamming) at the frame t advanced in 10 ms steps. The logarithmic energy at a frequency fk is computed from the squared magnitude FFT using a critical-band spaced (log-like in the frequency variable) weighting function in a manner similar to that of the computation of Perceptual Linear Prediction coefficients [3]. In particular, the 5-th, 8-th and 12th bands are centered around 0.5, 1 and 2 kHz respectively. Each feature X(fk, t) is labeled by a phoneme label YP(t) and a file label Y J (t). We use mutual information to measure the relevancy of X(/k, t - d) across all frequencies fk and in a context window - D ::; d ::; + D for the phoneme classification and the speaker/channel identification. 3 Estimation of MI and Bias Correction In this paper, we only consider the mutual information (MI) between discrete random variables. The phoneme label and the file label are discrete random variables. Search for Information Bearing Components in Speech 805 However, the feature variables are bounded continuous variables. To obtain the quantized features, we divide the maximum range of the observed features into cells of equal volume so that we can use histogram to estimate mutual information defined by " p(x, y) I(Xi Y) = L...Jp(x, y) log2 (x) ( )' x,y P PY If X and Yare jointly Gaussian, then I(Xi Y) = -~ In(1 - p2) where p is the correlation coefficient between X and Y. However, for speech data the feature variables are generally non-Gaussian and target variables are categorical type variables. Correlations involving a categorical variable are meaningless. The MI can also be written as I(XiY) = H(X) = + H(Y) - H(X, Y) H(Y) - H(YIX) = H(X) - H(XIY) (1) where H (Y IX) is a conditional entropy defined by - H(YIX) = - L:p(x) L:p(Ylx) log2P(ylx). x y The two equations in (1) mean that the MI is the uncertainty reduction about Y give X or the uncertainty reduction about X give Y. Based on the histogram, H(X) is estimated by H(X) = - L: ni log2 ni . n n ~ where ni is the number of data points in the i-th cell and n is the data size. And I(X i Y) is estimated by i(Xi Y) = H(X) + H(Y) - H(X, Y). Miller(1954)[4] has shown that H(X) is an underestimate of H(X) and i(Xi Y) is an overestimate of I (X i Y) . The biases are r - A 1 E[H(X)] - H(X) = - 2In(2)n E[i(X;Y)]-I(X;Y) 1 (2) + O( n 2 ) = (r-l)(c-l) +O(~) 2In(2)n n2 (3) where rand c are the number of cells for X and Y respectively. Interestingly, the first order terms in (2) and (3) do not depend on the probability distribution. After using these formulas to correct the'estimates, the new estimates have the same variances as the old estimates but with reduced biases. However, these formulas break down when rand n are of the same order. Extending Miller's approach, we find a high order correction for the bias. Let {pd be the probability distribution of X, then E[H(X)] - H(X) r-l 1 = - 2In(2)n + 6In(2)n2 (S( {Pi}) 1 1 --(S({pd) - 1) + 0(-) 3 4 4n n 3r + 2) (4) H. H. Yang and H. Hermansky 806 The last two terms in the bias (4) depend on the unknown probabilities {pd. In practice they are approximated by the relative frequency estimates. Similarly, we can find the bias formulas of the high order terms O(nl:2) and O(n\) for the MI estimate. When X is evenly distributed, Pi = 1/r, so S( {pd) = r2 and , r-1 E[H(X)]- H(X) = - 2ln(2)n 1 + 6ln(2)n 2 (r 2 1 2 - 3r + 2) - 4n 3 (r -1) 1 + O(n 4 ). Theoretically S( {Pi}) has no upper bound when one of the probabilities is close to zero. However, in practice it is hard to collect a sample to estimate a very small probability. For this reason, we assume that Pi is either zero or greater than 6/r where 6 > 0 is a small constant does not depend on nor r . Under this assumption S( {pd) ::; r 2/6 and the amplitUde of the last term in (4) is less than (r2 /6 - 1) . 4!3 4 MI in Speech for Phonetic Classification The three hour telephone speech in the OGI database gives us a sample size greater than 1 million, n = 1050000. To estimate the mutual information between three features and a target variable, we need to estimate the entropy H(Xl' X 2, X3, Y). Take B = 20 as the number of bins for each feature variable and C = 19 is the number of phoneme categories. Then the total number of cells is r = B3 * C. After a constant adjustment, assuming 6 = 1, the bias is O( :2) = 6ln(12)n2 (r2 - 3r + 2) = 0.005(bits). It is shown in Fig. 1(a) that X(/4,t) and X(/5,t) are most relevant features for phonetic classification. From Fig. 1(b), at 5 Bark the MI spread around the current frame is 200 ms. Given one feature Xl, the information gain due to the second feature is the difference I(Xl,X2;Y)- I(Xl;Y) = I(X 2;YIXd where I(X2; YIXd is called the information gain of X 2 given Xl. It is a conditional mutual information defined by It is shown in Fig. 1(c)-(d) that given X(/5, t) across different bands the maximum information gain is achieved by X(/g, t), and within 5 Bark band the maximum information gain is achieved by X (/5 , t - 5). The mutual informations I(X(/4' t), X(/k, t + d); Y) for k 1, ... ,15, k ? 4, and d ?1, ... ,?1O, the information gain from the second feature in the vicinity of the first one, are shown in Fig. 2. The asymmetric distribution of the MI around the neighborhood (/5, d = 0) indicates that the phonetic information is spread asymmetrically through time but localized in about 200 ms around the current frame. = = Based on our data set, we have H(Y) = 3.96 (bits). The JMI for three frequency features and three temporal features are shown in Fig. 1{e)-{f). Based on these estimates, the three frequency features give 28% reduction in uncertainty about Y while the three temporal features give 19% reduction. Search for Information Bearing Components in Speech 807 0.55 0 .5 0.5 0.45 0.4 0.4 :II .Iii :II 0 . 35 0.3 .S; :iii! :iii! 0.3 0 .2 0 . 25 0 .1 0 .2 0,'5 6 0 earl< 9 12 0 -400 15 200 )~ 0 0.9 0 . 85 O .BS 0.8 0 .8 0 . 75 0 . 75 :II .'" :iii 0.7 0 .85 07 ~~ 0 .65 0.8 0 .6 0.55 0.5 0 .55 0 3 9 8 12 0 .5 -400 15 Bark -200 0 400 (d) 1.2 1 .2 1 .1 1 .1 :II .& :iii 0.9 200 time shift In rna (C) :II .Iii :iii 400 (b) (a) 0 .9 :II . S; :iii 200 tim_ shift In rna 0.9 0 .8 0 .8 0.7 o.71=--------..,. ---------------- o . eO~-~3--~B,.------:9:----:,:'::2---:',S? ea'" (e) ~~0~0~---:2~0~0--~0-----2~0~0~-~400 tl",. shirt In rna (f) Figure 1: (a) MIs of individual features in different bands. (b) MIs of individual feature at 5 Bark with different lOms-frame shifts. (c) JMIs of two features: at 5 Bark and in other bands. (d) JMIs of two features: current frame and shifted frames, both at 5 Bark. (e) JMIs of three features: at 5 Bark, 9 Bark and in other bands. The dashed line is th~ JMI level achieved by the two features X (15, t) and X (19, t). (f) JMIs of three features: current frame, 5th frame before current frame, and other shifted frames , all at 5 Bark. The dashed line is the JMI level achieved by X (15 , t) and X (15 , t - 5) . The size of our data set is n = 1050000. Therefore, we can reliably estimate the joint H H Yang and H Hermansky 808 MI between three features and the phoneme label. However, to estimate the JMI for more than 3 features we have the problem of curse of dimensionality since for k features, r = Bk * C is exponential increasing. For example, when k 4, B = 20, and C = 19, the second order bias is O(1/n2) = 2.02 (bits) which is too high to be ignored. To extend our approach beyond the current three-feature level, we need either to enlarge our data set or to find an alternative to the histogram based MI estimation. = . .. ... 0.95 0.9 0.85 ~ 0.8 .. ...; .Ii 0.75 ~ 0.7 ~0.65 0.6 0.55 0.5 15 100 o Bark -100 frame shill In ms Figure 2: The 3-D plot of joint mutual information around X(!4 , t). An asymmetric distribution is apparent especially around 4 Bark and 5 Bark. 1 .3 .---~-~--~-~----, 1.5 .----~--~--~----, 1.4 1 .2 1.3 1 .1 1 .2 1.1 ~ 0 .9 . S; 2 0 .8 0.9 0 .7 O.B I~ '" '" , 1.,t band 15-th band ,, , \ 0 .5 3 6 9 Bark (a) 12 15 ___ _ I \ I \ 0 .6 I ;' 0 .7 I I I I I 0 .6 ~ I, ~~O~0~--~'0~0---0~--'~00~-~ 200 time shift In rna (b) Figure 3: (a) The MI between one frequency feature and the file label. (b) The JMI between two features and the file identity labels. 5 MI in Speech for Speaker/Channel Recognition The linguistic variability expressed by phoneme labels is not the only variability present in speech. We use the mutual information to evaluate relevance to other Search for Information Bearing Components in Speech 809 sources of variabilities such as speaker/channel variability. Taking the file label as a target variable , we estimated the mutual information for one and two features. It is shown in Fig. 3(a) that the most relevant features are in the very low frequency channels, which in our case of telephone speech carry only very little speech information. Fig. 3(b) shows that the second most relevant feature for speaker/channel recognition is at least 150 ms apart from the first most relevant feature. These results suggest that the information about the speaker and the communication channel is not localized in time. These results are complementary to the results for phonetic classification shown in Fig. 1 (a) and (d) . 6 CONCLUSIONS Our results have shown that the information theoretic analysis of labeled speech data is feasible and useful for obtaining reusable knowledge about speech/channel variabilities. The joint mutual information of two features for phonetic classification is asymmetric around the current frame. We also estimated the joint mutual information between the phoneme labels and three feature variables . The uncertainty about the phonetic classification is reduced by adding more features. The maximum uncertainty reductions due to three frequency features and three temporal features are 28% and 19% respectively. The mutual informations of one and two features for speaker/channel recognition are estimated. The results show that the most relevant features are in the very low frequency bands. At 1 Bark and 5 Bark, the second most relevant temporal feature for speaker/channel recognition is at least 150 ms apart from the first most relevant feature. These results suggest that the information about the speaker and the communication channel is not localized in time. These results are complementary to the results for phonetic classification for which the mutual information is generally localized with some time spread. References [1] J. A. Bilmes. Maximum mutual information based reduction strategies for crosscorrelation based joint distribution modeling . In ICASSP98 , pages 469-472, April 1998. [2] R. Cole, M. Fanty, M. Noel, and T . Lander . Telephone speech corpus development at CSLU. In ICSLP, pages 1815-1818, Yokohama, Sept. 1994. [3] H. Hermansky. Perceptual linear predictive (PLP) analysis of speech. 1. A coust. Soc. Am., 87(4):1738-1752, April 1990 . [4] G. A. Miller. Note on the bias of information estimates. In H. Quastler, editor, Information Theory and Psychology , pages 95-100. The Free Press, Illinois, 1954. [5] Andrew Morris, Jean-Luc Schwartz , and Pierre Escudier . An information theoretical investigation into the distribution of phonetic information across the auditory spectogram. Computer Speech fj Language, 7(2):121-136, April 1993. [6] H. H. Yang, S. Van Vuuren, , S. Sharma, and H. Hermansky. Relevancy of timefrequency features for phonetic classification and speaker-channel recognition. Accepted by Speech Communication , 1999 . [7] H. H . Yang , S. Van Vuuren, and H. Hermansky. Relevancy of time-frequency features for phonetic classification measured by mutual information. In ICASSP99, pages 1:225-228 , Phoenix, March 1999. PART VII VISUAL PROCESSING 

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Resonance in a Stochastic Neuron Model with Delayed Interaction Toru Ohira* Sony Computer Science Laboratory 3-14-13 Higashi-gotanda Shinagawa, Tokyo 141, Japan ohira@csl.sony.co.jp Yuzuru Sato Institute of Physics, Graduate School of Arts and Science, University of Tokyo 3-8-1 Komaba, Meguro, Tokyo 153 Japan ysato@sacral.c.u-tokyo.ac.jp Jack D. Cowan Department of Mathematics University of Chicago 5734 S. University, Chicago, IL 60637, U.S.A cowan@math.uchicago.edu Abstract We study here a simple stochastic single neuron model with delayed self-feedback capable of generating spike trains. Simulations show that its spike trains exhibit resonant behavior between "noise" and "delay". In order to gain insight into this resonance, we simplify the model and study a stochastic binary element whose transition probability depends on its state at a fixed interval in the past. With this simplified model we can analytically compute interspike interval histograms, and show how the resonance between noise and delay arises. The resonance is also observed when such elements are coupled through delayed interaction. 1 Introd uction "Noise" and "delay" are two elements which are associated with many natural and artificial systems and have been studied in diverse fields. Neural networks provide representative examples of information processing systems with noise and delay. Though much research has gone into the investigation of these two factors in the community, they have mostly been separately studied (see e.g. [1]). Neural * Affiliated also with the Laboratory for Information Synthesis, RIKEN Brain Science Institute, Wako, Saitama, Japan 315 Resonance in a Stochastic Neuron Model with Delayed Interaction models incorporating both noise and delay are more realistic [2], but their complex characteristics have yet to be explored both theoretically and numerically. The main theme of this paper is the study of a simple stochastic neural model with delayed interaction which can generate spike trains. The most striking feature of this model is that it can show a regular spike pattern with suitably "tuned" noise and delay [3]. Stochastic resonance in neural information processing has been investigated by others (see e.g. [4]). This model, however, introduces a different type of such resonance, via delay rather than through an external oscillatory signal. It can be classified with models of stochastic resonance without an external signal [5] . The novelty of this model is the use of delay as the source of its oscillatory dynamics. To gain insight into the resonance, we simplify the model and study a stochastic binary element whose transition probability depends on its state at a fixed interval in the past. With this model, we can analytically compute interspike interval histograms, and show how the resonance between noise and delay arises. We further show that the resonance also occurs when such stochastic binary elements are coupled through delayed interaction. 2 Single Delayed-feedback Stochastic Neuron Model Our model is described by the following equations: d Jl dt Vet) ?(V( t)) -Vet) + W?(V(t - r)) + eL(t) 2 -1 1 + e-1)(V(t)-9) (1) where 11 and () are constants, and V is the membrane potential of the neuron. The noise term eL has the following probability distribution. pee = u) 1 (-L 5: u 5: L) 2L o (u<-L,u>L) , (2) i.e., eL is a time uncorrelated uniformly distributed noise in the range (-L , L). It can be interpreted as a fluctuation that is much faster than the membrane relaxation time Jl. The model can be interpreted as a stochastic neuron model with delayed self-feedback of weight W, which is an extension of a model with no delay previously studied using the Fokker- Planck equation [6]. We numerically study the following discretized version: 2 Vet + 1) = 1 + e-1)(V(t-T ) - 9) - 1 + eL (3) We fix 11 and () so that this map has two basins of attractors of differing size with no delay, as shown in Figure l(A) . We have simulated the map (3) with various noise widths and delays and find regular spiking behavior as shown in Fig l(C) for tuned noise width and delay. In case the noise width is too large or too small given self-feedback delay, this rhythmic behavior does not emerge, as shown in Fig1(B) and (D). We argue that the delay changes the effective shape of the basin of attractors into an oscillatory one, just like that due to an external oscillating force which, as is well-known, leads to stochastic resonance with a tuned noise width. The analysis of the dynamics given by (1) or (3), however, is a non- trivial task, particularly with T. Ohira, Y. Sato and J. D. Cowan 316 respect to the spike trains. A previous analysis using the Fokker-Planck equation cannot capture this emergence of regular spiking behavior. This difficulty motivates us to further simplify our model, as described in the next section. (E) ,cp (A) I ... .~. (0) (b) ., ,.p (F) (8) X(t) V(t) .. , 200 ... m .oo 100 1000 t t L-- (G) (e) V(t) X(t) , , , , .. .. ,>--- I - I - L.. ~ , " I - L.. ' - - (8) (D) V(t) X(t) Figure 1: (A) The shape of the sigmoid function 4> (b) for", = 4 and 0 = 0.1. The straight line (a) is 4> = V and the crossings of the two lines indicate the stationary point of the dynamics. Also, the typical dynamics of V (t) from the map model are shown as we change noise width L. The values of L are (B) L = 0.2, (C) L = 0.4, (D) L = 0.7. The data is taken with T = 20, '" = 4.0, 0 = 0.1 and the initial condition V(t) = 0.0 for t E [-r,O]. The plots are shown between t = a to 1000. (E) Schematic view of the single binary model. Some typical dynamics from the binary model are also shown. The values of parameters are r = 10, q = 0.5, and (F) p = 0.005, (G) p = 0.05, and (H) p = 0.2. 3 Delayed Stochastic Binary N enron Model The model we now discuss is an approximation of the dynamics that retains the asymmetric stochastic transition and delay. The state X(t) of the system at time step t is either -lor 1. With the same noise the model is described as follows: X(t + 1) O[f(X(t + 1 f(n) = 2?a + b) + n(a - b?, eL, T? eLl, O[n] 1 (0 ~ n), -1 (0) n), (4) where a and b are parameters such that lal ~ L and Ibl ~ L, and r is the delay. This model is an approximate discretization of the space of map (3) into two states Resonance in a Stochastic Neuron Model with Delayed Interaction 317 with a and b controlling the bias of transition depending on the state of X r steps earlier. When a i- b, the transition between the two states is asymmetric, reflecting the two differing sized basins of attractors. We can describe this model more concisely in probability space (Figure I(E)). The formal definition is given as follows: P(I, t + 1) P(-I,t+l) p q p, X(t 1- q, X(t q, X(t 1- p, X(t 1 b 2(1 + L)' 1 a 2(1 - L)' r) = -1, - r) = 1, r) = 1, - r) = -1, (5) where P(s, t) is the probability that X(t) = s. Hence, the transition probability of the model depends on its state r steps in the past, and is a special case of a delayed random walk [7]. We randomly generate X(t) for the interval t = (-r, 0) . Simulations are performed in which parameters are varied and X(t) is recorded for up to 10 6 steps. They appear to be qualitatively similar to those generated by the map dynamics (Figure I(F),(G),(H)). ;,From the trajectory X(t), we construct a residence time histogram h( u) for the system to be in the state -1 for u consecutive steps. Some examples of the histograms are shown in Figure 2 (q = 1 - q = 0.5, r = 10). We note that with p ? 0.5, as in Figure 2(A), the model has a tendency to switch or spike to the X = 1 state after the time step interval of r. But the spike trains do not last long and result in a small peak in the histogram. For the case of Figure 2(C) where p is closer to 0.5, we observe less regular transitions and the peak height is again small. With appropriate p as in Figure 2(B), spikes tend to appear at the interval T more frequently, resulting in higher peaks in the histogram. This is what we mean by stochastic resonance (Figure 2(D)). Choosing an appropriate p is equivalent to "tuning" the noise width L, with other parameters appropriately fixed. In this sense, our model exhibits stochastic resonance. This model can be treated analytically. The first observation to make with the model is that given r, it consists of statistically independent r + 1 Markov chains. Each Markov chain has its state appearing at every r+l interval. With this property of the model, we label time step t by the two integers sand k as follows t=s(r+l)+k, (O::;s,O::;k::;r) (6) Let P?(t) == P?(s, k) be the probability for the state to be in the ?1 state at time t or (s, k). Then, it can be shown that P+(s, k) P_ (s, k) a {3 , ,s + ,s a(1 - ,S) + {3(1- ,S) p , p+q q --, p+q 1 - (p + q). P+(s = 0, k), P_(s = 0, k), (7) In the steady state, P+(s --+ oo,k) == P+ = a and P_(s --+ oo,k) == P_ = {3. The steady state residence time histogram can be obtained by computing the following T. Ohira, Y. Safo and J. D. Cowan 318 = quantity, h(u) P(+;-,Uj+), which is the probability that the system takes consecutive -1 states U times between two +1 states. With the definition of the model and statistical independence between Markov chains in the sequence, the following expression can be derived: P(+;-,Uj+) P+(P_)Up+ = (,8)U(a)2 (1 ~ U < r) (8) = p+(p-r(1- q) = (,8r(a)(1- q) (u = r) (9) = p+(p-r(q)(1- p)U-T(p) = (,8)U(p)2 (u > r) (10) With appropriate normalization, this expression reflects the shape of the histogram obtained by numerical simulations, as shown in Figure 2. Also, by differentiating equation (9) with respect to p, we derive the resonant condition for the peak to reach maximum height as (11) q=pr or, equivalently, L - a = (L + b)r. (12) In Figure 2(D), we see that maximum peak amplitude is reached by choosing parameters according to equation (11). We note that this analysis for the histogram is exact in the stationary limit, which makes this model unique among those showing stochastic resonance. (M "':L b(a) ~ Ol~ ." 'M. j ~ ~ ~ ~ l~ I~ ~ I'!. I '!.:O ? (at "...': I~ ." hCD) . . Ol~ . ~. 1 ~ , " 10 12 ~ I' 17 ~ jO ? (et ~ ..."'~ (I Ol~ b(ot .." ,', !> ~ "L I" us ? )0 UP!> 20 ? ", h(tt . "" ." .... ~. 10 10 to "'a ...." Figure 2: Residence time histogram and dynamics of X(t) as we change p. The 0.005, (B) p = 0.05, (C) p = 0.2. The solid line in the values of p are (A) p histogram is from the analytical expression given in equations (8-10). Also, in (D) we show a plot of peak height by varying p. The solid line is from equation (9). The parameters are r = 10, q = 0.5. = 4 Delay Coupled Two Neuron Case We now consider a circuit comprising two such stochastic binary neurons coupled with delayed interaction. We observe again that resonance between noise and delay Resonance in a Stochastic Neuron Model with Delayed Interaction 319 takes place. The coupled two neuron model is a simple extension of the model in the previous section. The transition probability of each neuron is dependent on the other neuron's state at a fixed interval in the past. Formally, it can be described in probability space as follows. Pl(l, t + 1) X 2(t - 72) = -1, X 2(t - 72) = 1, X 2(t - 72) = 1, ql, 1 - PI, X 2(t - 72) = -1, Xl(t - 7d = -1, P2, 1- q2, Xl(t - 7d = 1, q2, Xl(t - 71) = 1, 1- P2, Xl(t - 71) = -1 PI! 1- q!, Pl(-I,t+l) P 2(1, t + 1) P2(-I,t+l) (13) Pi(S, t) is the probability that the state of the neuron i is Xi(t) = s. We have performed simulation experiments on the model and have again found resonance between noise and delay. Though more intricate than the single neuron model, we can perform a similar theoretical analysis of the histograms and have obtained approximate results for some cases. For example, we obtain the following approximate analytical results for the peak height of the interspike histogram of Xl for the case 71 = 72 == 7. ( The peak occurs at 71 + 72 + 1.) H(Pl' P2, qI, q2) J.tl (PI, P2, qI, q2) J.t2 (PI, P2, ql, q2) J.t3 (PI, P2, ql , q2 ) J.t4 (PI! P2, qI, q2 ) II (PbP2, ql, q2) 12(PI,P2, qI, q2) S(PI,P2, ql, q2) = {J.t3(P!, P2, ql, q2 )ql + J.t4(PI, P2, ql, q2)(1 - pd Y (14) {J.tl(P!'P2, ql, q2)(qlq2PI + ql(l - q2)(1 - ql)) +J.t2(Pl,P2,ql,q2)((I- pdq2Pl + (1- Pl)(I- q2)(I- qd)} h (PI, P2, ql, q2)!2(PI, P2, ql, q2) (15) S(PI , P2 , ql , q2) II (PI ,P2, qI, q2) (16) S(PI,P2, q!, q2) !2(PI,P2, ql, q2) (17) s (PI! P2, ql, q2) 1 (18) S(PI,P2, ql, q2) PI(I - P2) + P2(1 - ql) (19) ql (1 - q2) + q2 (1 - qd P2 + PI (1 - P2 - q2) (20) q2 + ql(l - P2 -q2) h (PI, P2, ql, q2)!2(PI ,P2, ql, q2) (21) + h(PI,P2, ql, q2) + !2(PI,P2, ql, q2) + 1 These analytical results are compared with the simulation experiments, examples of which are shown in Figure 3. A detailed analysis, particularly for the case of 71 =I 72, is quite intricate and is left for the future. 5 Discussion There are two points to be noted. Firstly, although there are examples which may indicate that stochastic resonance is utilized in biological information processing, it is yet to be explored if the resonance between noise and delay has some role in T. Ohira, Y. Sato and J. D. Cowan 320 (A) (C) (8) 0 . 01 h(t) . .. h(t) ::::~ 0 . 008 ~ ..00< o.oo:z 0 . 00:3: 0 .2 pi . 0.3 0.1 0 ,' .. 0 .1 0.' 0 .5 Figure 3: A plot of peak height by varying P2. The solid line is from equation (1420). The parameters are T1 = T2 = 10, q1 = q2 = 0.5, (A)P1 = P2, (B) P1 = 0.005, (C) P1 = 0.025. neural information processing. Secondly, there are many investigations of spiking neural models and their applications (see e.g., [8]). Our model can be considered as a new mechanism for generating controlled stochastic spike trains. One can predict its application to weak signal transmission analogous to recent research using stochastic resonance with a larger number of units in series [9]. Investigations of the network model with delayed interactions are currently underway. References [1) Hertz, J. A., Krogh, A., & Palmer, R . G. (1991). Introduction to the Theory of Neural Computation. Redwood City: Addison-Wesley. [2) Foss, J., Longtin, A., Mensour, B., & Milton, J . G. (1996). Multistability and Delayed Recurrent Loops. Physical Review Letters, 76, 708-711; Pham, J., Pakdaman, K., Vibert, J.-F. (1998). Noise-induced coherent oscillations in randomly connected neural networks. Physical Review E, 58, 3610-3622; Kim, S., Park, S. H., Pyo, H.-B. (1999). Stochastic Resonance in Coupled Oscillator Systems with Time Delay. Physical Review Letters, 8!, 1620-1623; Bressloff, P. C. (1999). Synaptically Generated Wave Propagation in Excitable Neural Media. Physical Review Letters, 8!, 2979-2982. [3) Ohira, T. & Sato, Y. (1999). Resonance with Noise and Delay. Physical Review Letters, 8!, 2811-2815. [4) Gammaitoni, L., Hii.nggi, P., Jung, P., & Marchesoni, F.(1998). Stochastic Resonance. Review of Modem Physics, 70, 223-287. [5) Gang, H., Ditzinger, T., Ning, C. Z., & Haken, H.(1993) Stochastic Resonance without External Periodic Force. Physical Review Letters, 71, 807-810; Rappel, W-J. & Strogatz, S. H. (1994). Stochastic resonance in an autonomous system with a nonuniform limit cycle. Physical Review E, 50,3249-3250; Longtin, A. (1997). Autonomous stochastic resonance in bursting neurons. Physical Review E, 55, 868-876. [6) Ohira, T. & Cowan J . D. (1995). Stochastic Single Neurons, Neural Communication, 7518-528. [7) Ohira, T. & Milton, J. G. (1995) . Delayed Random Walks. Physical Review E, 5!, 3277-3280; Ohira, T. (1997). Oscillatory Correlation of Delayed Random Walks, Physical Review E, 55, RI255-1258. [8) Maas, W. (1997). Fast Sigmoidal Network via Spiking Neurons. Neural Computation, 9(2), 279-304; Maas, W. (1996). Lower Bounds for the Computational Power of Networks of Spiking Neurons. Neural Computation, 8(1), 1-40. [9) Locher, M., Cigna, D., and Hunt, E. R. (1998). Noise Sustained Propagation of a Signal in Coupled Bistable Electric Elements Physical Review Letters, 80, 5212-5215. 

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Information Capacity and Robustness of Stochastic Neuron Models Elad Schneidman Idan Segev N aftali Tishby Institute of Computer Science, Department of Neurobiology and Center for Neural Computation, Hebrew University Jerusalem 91904, Israel { elads, tishby} @cs.huji.ac.il, idan@lobster.ls.huji.ac.il Abstract The reliability and accuracy of spike trains have been shown to depend on the nature of the stimulus that the neuron encodes. Adding ion channel stochasticity to neuronal models results in a macroscopic behavior that replicates the input-dependent reliability and precision of real neurons. We calculate the amount of information that an ion channel based stochastic Hodgkin-Huxley (HH) neuron model can encode about a wide set of stimuli. We show that both the information rate and the information per spike of the stochastic model are similar to the values reported experimentally. Moreover, the amount of information that the neuron encodes is correlated with the amplitude of fluctuations in the input, and less so with the average firing rate of the neuron. We also show that for the HH ion channel density, the information capacity is robust to changes in the density of ion channels in the membrane, whereas changing the ratio between the Na+ and K+ ion channels has a considerable effect on the information that the neuron can encode. Finally, we suggest that neurons may maximize their information capacity by appropriately balancing the density of the different ion channels that underlie neuronal excitability. 1 Introduction The capacity of neurons to encode information is directly connected to the nature of spike trains as a code. Namely, whether the fine temporal structure of the spik~ train carries information or whether the fine structure of the train is mainly noise (see e.g. [1, 2]). Experimental studies show that neurons in vitro [3, 4] and in vivo [5, 6, 7], respond to fluctuating inputs with repeatable and accurate spike trains, whereas slowly varying inputs result in lower repeatability and 'jitter' in the spike timing. Hence, it seems that the nature of the code utilized by the neuron depends on the input that it encodes [3, 6]. Recently, we suggested that the biophysical origin of this behavior is the stochas- Capacity and Robustness oJStochastic Neuron Models 179 ticity of single ion channels. Replacing the average conductance dynamics in the Hodgkin-Huxley (HH) model [8], with a stochastic channel population dynamics [9, 10, 11], yields a stochastic neuron model which replicates rather well the spike trains' reliability and precision of real neurons [12]. The stochastic model also shows subthreshold oscillations, spontaneous and missing spikes, all observed experimentally. Direct measurement of membranal noise has also been replicated successfully by such stochastic models [13]. Neurons use many tens of thousands of ion channels to encode the synaptic current that reaches the soma into trains of spikes [14]. The number of ion channels that underlies the spike generation mechanism, and their types, depend on the activity of the neuron [15, 16]. It is yet unclear how such changes may affect the amount and nature of the information that neurons encode. Here we ask what is the information encoding capacity of the stochastic HH model neuron and how does this capacity depend on the densities of different of ion channel types in the membrane. We show that both the information rate and the information per spike of the stochastic HH model are similar to the values reported experimentally and that neurons encode more information about highly fluctuating inputs. The information encoding capacity is rather robust to changes in the channel densities of the HH model. Interestingly, we show that there is an optimal channel population size, around the natural channel density of the HH model. The encoding capacity is rather sensitive to changes in the distribution of channel types, suggesting that changes in the population ratios and adaptation through channel inactivation may change the information content of neurons. 2 The Stochastic HH Model The stochastic HH (SHH) model expands the classic HH model [8], by incorporating the stochastic nature of single ion channels [9, 17]. Specifically, the membrane voltage dynamics is given by the HH description, namely, dV cmTt = -gLCV - VL) - gK(V, t)(V - VK) - gNa(V, t)(V - VNa) +I (1) where V is the membrane potential, VL, VK and VNa are the reversal potentials of the leakage, potassium and sodium currents, respectively, gL, gK(V, t) and gNa(V, t) are the corresponding ion conductances, Cm is the membrane capacitance and I is the injected current. The ion channel stochasticity is introduced by replacing the equations describing the ion channel conductances with explicit voltage-dependent Markovian kinetic models for single ion channels [9, 10]. Based on the activation and inactivation variables of the deterministic HH model, each K+ channel can be in one of five different states, and the rates for transition between these states are given in the following diagram, [~ ] 4Qn ~ f3n [ ~ ] 3a n ~ 2f3n [ ~ ] 2a n ~ 3f3n [ ~ ] an ~ 4f3n [ ~ ] (2) where [nj] refers to the number of channels which are currently in the state nj. Here [n4] labels the single open state of a potassium channel, and an, i3n, are the voltage-dependent rate-functions in the HH formalism. A similar_model is used for the Na+ channel (The Na+ kinetic model has 8 states, with only one open state, see [12] for details). The potassium and sodium membrane conductances are given by, (3) gK(V, t) = ,K [114] gNa(V, t) = ,Na [mahl] where and ,Na are the conductances of an ion channel for the K+ and Na+ respectively. We take the conductance of a single channel to be 20pS [14] for both the ,K E. Schneidman. l. Segev and N. Tishby 180 K+ and Na+ channel types 1. Each of the ion channels will thus respond stochastically by closing or opening its 'gates' according to the kinetic model, fluctuating around the average expected behavior. Figure 1 demonstrates the effect of the ion A B Figure 1: Reliability of firing patterns in a model of an isopotential Hodgkin-Huxley membrane patch in response to different current inputs. (A) Injecting a slowly changing current input (low-pass Gaussian white noise with a mean TJ = 8I1A/cm 2 , and standard deviation a = 1 p,A/ cm 2 which was convolved with an 'alpha-function' with a time constant To = 3 msec, top frame), results in high 'jitter' in the timing of the spikes (raster plots of spike responses, bottom frame). (B) The same patch was again stimulated repeatedly, with a highly fluctuating stimulus (TJ = 8 p,A/cm 2 , a = 7 p,A/cm 2 and To = 3 msec, top frame) The 'jitter' in spike timing is significantly smaller in B than in A (i.e. increased reliability for the fluctuating current input). Patch area used was 200 p,m 2 , with 3,600 K+ channels and 12,000 Na+ channels. (Compare to Fig.l in see [3]). (C) Average firing rate in response to DC current input of both the HH and the stochastic HH model. (D) Coefficient of variation of the inter spike interval of the SHH model in response to DC inputs, giving values which are comparable to those observed in real neurons channel stochasticity, showing the response of a 200 J.Lm 2 SHH isopotential membrane patch (with the 'standard' SHH channel densities) to repeated presentation of suprathreshold current input. When the same slowly varying input is repeatedly presented (Fig. lA) , the spike trains are very different from each other, i.e. , spike firing time is unreliable. On the other hand, when the input is highly fluctuating (Fig. IB), the reliability of the spike timing is relatively high. The stochastic model thus replicates the input-dependent reliability and precision of spike trains observed in pyramidal cortical neurons [3] . As for cortical neurons, the Repeatability and Precision of the spike trains of the stochastic model (defined in [3]) are strongly correlated with the fluctuations in the current input and may get to sub-millisecond precision [12]. The f-I curve of the stochastic model (Fig. lC) and the coefficient of variation (CV) of the inter-spike intervals (lSI) distribution for DC inputs (Fig. ID) are both similar to the behavior of cortical neurons in vivo [18], in clear contrast to the deterministic model 2 1 The number of channels is thus the ratio between the total conductance of a single type of ion channels and the single channel conductance, and so the 'standard' SHH densities will be 60 Na+ and 18 Na+ channels per p,m 2 . 2 Although the total number of channels in the model is very large, the microscopic level ion channel noise has a macroscopic effect on the spike train reliability, since the number Capacity and Robustness ofStochastic Neuron Models 3 181 The Information Capacity of the SHH Neuron Expanding the Repeatability and Precision measures [3], we turn to quantify how much information the neuron model encodes about the stimuli it receives. We thus present the model with a set of 'representative' input current traces, and the amount of information that the respective spike trains encode is calculated. Following Mainen and Sejnowski [3], we use a set of input current traces which imitate the synaptic current that reaches the soma from the dendritic tree. We convolve a Gaussian white noise trace (with a mean current 1} and standard deviation 0') with an alpha function (with a To: = 3 msec). Six different mean current values are used (1} = 0,2,4,6,8,10 pA/cm 2 ) , and five different std values (0' = 1,3,5,7, 9pA/cm 2 ), yielding a set of 30 input current traces (each is 10 seconds long). This set of inputs is representative of the wide variety of current traces that neurons might encounter under in vivo conditions in the sense that the average firing rates for this set of inputs which range between 2 - 70 Hz (not shown). We present these input traces to the model, and calculate the amount of information that the resulting spike trains convey about each input, following [6, 19]. Each input is presented repeatedly and the resulting spike trains are discretized in D..T bins, using a sliding 'window' of size T along the discretized sequence. Each train of spikes is thus transformed into a sequence of K-letter 'words' (K = T/D..T) , consisting of O's (no spike) and l's (spike). We estimate P(W), the probability of the word W to appear in the spike trains, and then compute the entropy rate of its total word distribution, Htotal = - L P(W) log2 P(W) bits/word (4) W which measures the capacity of information that the neuron spike trains hold [20, 6, 19]. We then examine the set of words that the neuron model used at a particular time t over all the repeated presentations of the stimulus, and estimate P(Wlt), the time-dependent word probability distribution. At each time t we calculate the time-dependent entropy rate, and then take the average of these entropies Hnoise = (- LP(Wlt)lOg2 P(Wlt))t w bits/word (5) where ( .. .)t denotes the average over all times t. Hnoise is the noise entropy rate, which measures how much of the fine structure of the spike trains of the neuron is just noise. After performing the calculation for each of the inputs, using different word sizes 3, we estimate the limit of the total entropy and noise entropy rates at T --* 00, where the entropies converge to their real values (see [19] for details) . Figure 2A shows the total entropy rate of the responses to the set of stimuli, ranging from 10 to 170 bits/sec. The total entropy rate is correlated with the firing rates of the neuron (not shown). The noise entropy rate however, depends in a different way on the input parameters: Figure 2B shows the noise entropy rate of the responses to the set of stimuli, which may get up to 100 bits/sec. Specifically, for inputs with high mean current values and low fluctuation amplitude, many of the spikes are of ion channels which are open near the spike firing threshold is rather small [12). The fluctuations in this small number of open channels near firing threshold give rise to the input-dependent reliability of the spike timing. 3 t he bin size T = 2 msec has been set to be small enough to keep the fine temporal structure of the spike train within the word sizes used, yet large enough to avoid undersampling problems 182 E. Schneidman, /. Segev and N. Tishby just noise, even if the mean firing rate is high. The difference between the neuron's entropy rate (the total capacity of information of the neuron's spike train) and the noise entropy rate, is exactly the average rate of information that the neuron's spike trains encode about the input , I(stimulus , spike train) = Htotal - Hnoise [20, 6], this is shown in Figure 2C. The information rate is more sensitive to the size of A .~: 200 B ;f. ':'~ 150 150 100 100 50 50 10 0 0 10 0 a fl.LA/cm 2 ) '1 ,. 100 :~ 60 40 20 5 a lllAIcm2 ) 0 0 a 0 .~ ~ ~ i 0 a fl.LA/cm 2 ) D ~;) 2 ~ )~r . 1 0 10 3 2 .5 3 ~ 0 0 fl.LA/cm 2 ) ;;; 80 10 200 2 . 1.5 10 5 '1 [llAIcm 2 ) 0 .5 5 0 0 a lllAIcm 2 ) Figure 2: Information capacity of the SHH model. (A) The total spike train entropy rate of the SHH model as a function of 'TI, the current input mean, and a, the standard deviation (see text for details). Error bar values of this surface as well as for the other frames range between 1 - 6% (not shown). (B) Noise entropy rate as a function of the current input parameters. (C) The information rate about the stimulus in the spike trains, as a function of the input parameters, calculated by subtracting noise entropy from the total entropy (note the change in grayscale in C and D). (D) Information per spike as a function of the input parameters, which is calculated by normalizing the results shown in C by the average firing rate of the responses to each of the inputs. fluctuations in the input than to the mean value of the current trace (as expected, from the reliability and precision of spike timing observed in vitro [3] and in vivo [6] as well as in simulations [12]). The dependence of the neural code on the input parameters is better reflected when calculating the average amount of information per spike that the model gives for each of the inputs (Fig. 2D) (see for comparison the values for the Fly's HI neuron [6]). 4 The effect of Changing the Neuron Parameters on the Information Capacity Increasing the density of ion channels in the membrane compared to the 'standard' SHH densities, while keeping the ratio between the K+ and Na+ channels fixed, only diminishes the amount of information that the neuron encodes about any of the inputs in the set. However, the change is rather small: Doubling the channel density decreases the amount of information by 5 - 25% (Fig. 3A), depending on the specific input. Decreasing the channel densities of both types, results in encoding more information about certain stimuli and less about others. Figure 3B shows that having half the channel densities would result with in 10% changes in the information in both directions. Thus, the information rates conveyed by the stochastic model are robust to changes in the ion channel density. Similar robustness (not shown) has been observed for changes in the membrane area (keeping channel Capacity and Robustness ofStochastic Neuron Models 183 density fixed) and in the temperature (which effects the channel kinetics). However, A .... ., 1.2 B 1.2 jl.2 1 .1 .21.2 :5 1 1 ~ 0.9 ..s0.8 10 "1 '~o 1.s 10 5 o c 1.1 ~ 0 0 .9 10 O.S 10 5 a[~em2) '1 [pAIem2) $A O.S 5 o 0 a[~Alem2) 0.4 D >~ 3 .5 0.3 3 2.5 0.2 2 1.5 0 .1 10 0 .5 o 0 a[~Alem2) 0 Figure 3: The effect of changing the ion channel densities on the information capacity. (A) The ratio of the information rate of the SHH model with twice the density of the 'standard' SHH densities divided by the information rate of the mode with 'standard' SHH densities. (B) As in A, only for the SHH model with half the 'standard' densities. (C) The ratio of the info rate of the SHH model with twice as many Na+ channels, divided by the info rate of the standard SHH Na+ channel density, where the K+ channel density remains untouched (note the change in graycale in C and D). (D) As in C, only for the SHH model with the number of Na+ channels reduced by half. changing the density of the Na+ channels alone has a larger impact on the amount of information that the neuron conveys about the stimuli. Increasing Na+ channel density by a factor of two results in less information about most of the stimuli, and a gain in a few others (Fig. 3C). However, reducing the number of Na+ channels by half results in drastic loss of information for all of the inputs (Fig. 3D). 5 Discussion We have shown that the amount of information that the stochastic HH model encodes about its current input is highly correlated with the amplitude of fluctuations in the input and less so with the mean value of the input. The stochastic HH model, which incorporates ion channel noise, closely replicates the input-dependent reliability and precision of spike trains observed in cortical neurons. The information rates and information per spike are also similar to those of real neurons. As in other biological systems (e.g., [21]), we demonstrate robustness of macroscopic performance to changes in the cellular properties - the information coding rates of the SHH model are robust to changes in the ion channels densities as well as in the area of the excitable membrane patch and in the temperature (kinetics) of the channel dynamics. However, the information coding rates are rather sensitive to changes in the ratio between the densities of different ion channel types, suggests that the ratio between the density of the K+ channels and the Na+ channels in the 'standard' SHH model may be optimal in terms of the information capacity. This may have important implications on the nature of the neural code under adaptation and learning. We suggest that these notions of optimality and robustness may be a key biophysical principle of the operation of real neurons. Further investigations should take into account the activity-dependent nature of the channels and the 184 E. Schneidman, I. Segev and N. nshby neuron [15, 16] and the notion of local learning rules which could modify neuronal and suggest loca1learning rules as in [22]. Acknowledgements This research was supported by a grant from the Ministry of Science, Israel. References [1] Rieke F ., Warland D., de Ruyter van Steveninck R., and Bialek W. Spike: Exploring the Neural Code. MIT Press, 1997. [2] Shadlen M. and Newsome W. Noise, neural codes and cortical organization. Curro Opin. Neurobiol., 4:569-579, 1994. [3] Mainen Z. and Sejnowski T . Reliability of spike timing in neocortical neurons. Science, 268:1503-1508, 1995. [4] Nowak L., Sanches-Vives M., and McCormick D. Influence of low and high frequency inputs on spike timing in visual cortical neurons. Cerebral Cortex, 7:487-501, 1997. [5] Bair W. and Koch C. Temporal precision of spike trains in extrastriate cortex of the behaving macaque monkey. Neural Comp., 8:1185-1202, 1996. [6] de Ruyter van Steveninck R ., Lewen G., Strong S., Koberle R., and Bialek W. Reproducibility and variability in neural spike trains. Science, 275:1805-1808, 1997. [7] Reich D., Victor J., Knight B., Ozaki T., and Kaplan E. Response variability and timing precision of neuronal spike trains in vivo. J. Neurophysiol., 77:2836:2841, 1997. [8] Hodgkin A. and Huxley A. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117:500-544, 1952. [9] Fitzhugh R . A kinetic model of the conductance changes in nerve membrane. J. Cell. Comp o Physiol., 66:111-118, 1965. [10] DeFelice L. Introduction to Membrane Noise. Perseus Books, 1981. [11] Skaugen E. and Wall0e L. Firing behavior in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. Acta Physiol. Scand., 107:343-363, 1979. [12] Schneidman E ., Freedman B., and Segev I. Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Comp., 10:1679-1704, 1998. [13] White J., Klink R., Alonso A., and Kay A. Noise from voltage-gated channels may influence neuronal dynamics in the entorhinal cortex. J Neurophysiol, 80:262-9, 1998. [14] Hille B. Ionic Channels of Excitable Membrane. Sinauer Associates, 2nd ed., 1992. [15] Marder E., Abbott L., Turrigiano G., Liu Z., and Golowasch J . Memory from the dynamics of intrinsic membrane currents. Proc. Natl. Acad. Sci., 93:13481-6, 1996. [16] Toib A., Lyakhov V., and Marom S. Interaction between duration of activity and rate of recovery from slow inactivation in mammalian brain Na+ channels. J Neurosci., 18:1893-1903, 1998. [17] Strassberg A. and DeFelice L. Limits of the HH formalism: Effects of single channel kinetics on transmembrane voltage dynamics. Neural Comp., 5:843-856, 1993. [18] Softky W. and Koch C . The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci ., 13:334-350, 1993. [19] Strong S., Koberle R., de Ruyter van Steveninck R., and Bialek W. Entropy and information in neural spike trains. Phys. Rev. Lett., 80:197-200, 1998. [20] Cover T.M. and Thomas J.A. Elements of Information Theory. Wiley, 1991. [21] Barkai N. and Leibler S. Robustness in simple biochemical networks. Nature, 387:913917, 1997. [22] Stemmler M. and Koch C. How voltage-dependent conductances can adapt to maximize the information encoded by neuronal firing rate. Nat. Neurosci., 2:521-7, 1999. 

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Spike-based learning rules and stabilization of persistent neural activity Xiaohui Xie and H. Sebastian Seung Dept. of Brain & Cog. Sci., MIT, Cambridge, MA 02139 {xhxie, seung}@mit.edu Abstract We analyze the conditions under which synaptic learning rules based on action potential timing can be approximated by learning rules based on firing rates. In particular, we consider a form of plasticity in which synapses depress when a presynaptic spike is followed by a postsynaptic spike, and potentiate with the opposite temporal ordering. Such differential anti-Hebbian plasticity can be approximated under certain conditions by a learning rule that depends on the time derivative of the postsynaptic firing rate. Such a learning rule acts to stabilize persistent neural activity patterns in recurrent neural networks. 1 INTRODUCTION Recent experiments have demonstrated types of synaptic ~re 11111111111 111111111111111111 plasticity that depend on the post 11111111111 11111111? ?11. A temporal ordering of presynaptic and postsynaptic spiking. At B 0L-.i cortical [ I] and hippocampal[2] t .:oLl____ o synapses, long-term potenti1000 2000 tpost - tpre time (ms) ation is induced by repeated pairing of a presynaptic spike Figure I: (A) Pairing function for differential Heband a succeeding postsynaptic bian learning. The change in synaptic strength is plotspike, while long-term deprested versus the time difference between postsynaptic sion results when the order and presynaptic spikes. (B) Pairing function for difis reversed. The dependence ferential anti-Hebbian learning. (C) Differential antiof the change in synaptic Hebbian learning is driven by changes in firing rates. strength on the difference The synaptic learning rule of Eq. (l) is applied to two l:!..t = tpost - tpre between Poisson spike trains. The synaptic strength remains postsynaptic and presynaptic roughly constant in time, except when the postsynapspike times has been measured tic rate changes. quantitatively. This pairing function, sketched in Figure lA, has positive and negative lobes correspond to potentiation and depression. and a width of tens of milliseconds. We will refer to synaptic plasticity associated with this pairing function as differential Hebbian plasticity-Hebbian because the conditions for o~i=~=::====: _:3/_-----' ? \:;J __~ X Xie and H. S. Seung 200 potentiation are as predicted by Hebb[3], and differential because it is driven by the difference between the opposing processes of potentiation and depression. The pairing function of Figure IA is not characteristic of all synapses. For example, an opposite temporal dependence has been observed at electrosensory lobe synapses of electric fish[4]. As shown in Figure IB, these synapses depress when a presynaptic spike is followed by a postsynaptic one, and potentiate when the order is reversed. We will refer to this as differential anti-Hebbian plasticity. According to these experiments, the maximum ranges of the differential Hebbian and antiHebbian pairing functions are roughly 20 and 40 ms, respectively. This is fairly short, and seems more compatible with descriptions of neural activity based on spike timing rather than instantaneous firing rates[5, 6]. In fact, we will show that there are some conditions under which spike-based learning rules can be approximated by rate-based learning rules. Other people have also studied the relationship between spike-based and rate-based learning rules[7, 8]. The pairing functions of Figures IA and IB lead to rate-based learning rules like those traditionally used in neural networks, except that they depend on temporal derivatives of firing rates as well as firing rates themselves. We will argue that the differential antiHebbian learning rule of Figure IB could be a general mechanism for tuning the strength of positive feedback in networks that maintain a short-term memory of an analog variable in persistent neural activity. A number of recurrent network models have been proposed to explain memory-related neural activity in motor [9] and prefrontal [ 10] cortical areas, as well as the head direction system [11] and oculomotor integrator[ 12, 13, 14]. All of these models require precise tuning of synaptic strengths in order to maintain continuously variable levels of persistent activity. As a simple illustration of tuning by differential antiHebbian learning, a model of persistent activity maintained by an integrate-and-fire neuron with an excitatory autapse is studied. 2 SPIKE-BASED LEARNING RULE Pairing functions like those of Figure 1 have been measured using repeated pairing of a single presynaptic spike with a single postsynaptic spike. Quantitative measurements of synaptic changes due to more complex patterns of spiking activity have not yet been done. We will assume a simple model in which the synaptic change due to arbitrary spike trains is the sum of contributions from all possible pairings of presynaptic with postsynaptic spikes. The model is unlikely to be an exact description of real synapses, but could turn out to be approximately valid. We will write the spike train of the ith neuron as a series of Dirac delta functions, Si (t) = <5(t - Tr), where Tr is the nth spike time of the ith neuron. The synaptic weight from neuron j to i at time t is denoted by W ij (t). Then the change in synaptic weight induced by presynaptic spikes occurring in the time interval [0, Tj is modeled as Ln Wij(T + >.) - Wij(>') = [T dtj io foo dti f(ti - tj)Si(ti) Sj(tj) (1) -00 Each presynaptic spike is paired with all postsynaptic spikes produced before and after. For each pairing, the synaptic weight is changed by an amount depending on the pairing function f. The pairing function is assumed to be nonzero inside the interval [-T, Tj, and zero outside. We will refer to T as the pairing range. According to our model, each presynaptic spike results in induction of plasticity only after a latency>.. Accordingly, the arguments T + >. and >. of W ij on the left hand side of the equation are shifted relative to the limits T and 0 of the integral on the right hand side. We 201 Spike-based Learning and Stabilization ofPersistent Neural Activity will assume that the latency>. is greater than the pairing range T, so that Wi} at any time is only influenced by presynaptic and postsynaptic spikes that happened before that time, and therefore the learning rule is causal. 3 RELATION TO RATE-BASED LEARNING RULES The learning rule of Eq. (1) is driven by correlations between presynaptic and postsynaptic activities. This dependence can be made explicit by making the change of variables u = ti - t j in Eq. (I), which yields Wij(T + >.) - W ij (>.) = iTT duf(u)Cij(u) (2) where we have defined the cross-correlation Cij(u) = !aT dt Si(t + u) Sj(t) . (3) and made use of the fact that f vanishes outside the interval [-T, T]. Our immediate goal is to relate Eq. (2) to learning rules that are based on the cross-correlation between firing rates, Crre(u) = !aT dt Vi(t + u) Vj(t) (4) There are a number of ways of defining instantaneous firing rates. Sometimes they are computed by averaging over repeated presentations of a stimulus. In other situations, they are defined by temporal filtering of spike trains. The following discussion is general, and should apply to these and other definitions of firing rates. The "rate correlation" is commonly subtracted from the total correlation to obtain the "spike correlation" C:r ke = Cij - Cijate. To derive a rate-based approximation to the learning rule (2), we rewrite it as Wij(T + >.) - Wij(>') = iTT du f(u)Cijate(u) + iTT du f(u)C:r ke (u) (5) and simply neglect the second term. Shortly we will discuss the conditions under which this is a good approximation. But first we derive another form for the first term by applying the approximation Vi(t + u) ~ Vi(t) + UVi(t) to obtain j T -T duf(u)Crre(u) ~ iT dt[fiovi(t) + 131Vi(t)]VJ (t) (6) 0 where we define (7) This approximation is good when firing rates vary slowly compared to the pairing range T . The learning rule depends on the postsynaptic rate through fio Vi + 131 Vi . When the first term dominates the second, then the learning rule is the conventional one based on correlations between firing rates, and the sign of fio determines whether the rule is Hebbian or anti-Hebbian. In the remainder of the paper, we will discuss the more novel case where 130 = O. This holds for the pairing functions shown in Figures lA and IB, which have positive and negative lobes with areas that exactly cancel in the definition of 130. Then the dependence on X Xie and H. S. Seung 202 postsynaptic activity is purely on the time derivative of the firing rate. Differential Hebbian learning corresponds to /31 > 0 (Figure IA), while differential anti-Hebbian learning leads to /31 < 0 (Figure IB). To summarize the /30 = 0 case, the synaptic changes due to rate correlations are approximated by W ij ex: -ViVj (diff. anti-Hebbian) (8) for slowly varying rates. These formulas imply that a constant postsynaptic firing rate causes no net change in synaptic strength. Instead, changes in rate are required to induce synaptic plasticity. To illustrate this point, Figure lC shows the result of applying differential anti-Hebbian learning to two spike trains. The presynaptic spike train was generated by a 50 Hz Poisson process, while the postsynaptic spike train was generated by an inhomogeneous Poisson process with rate that shifted from 50 Hz to 200 Hz at 1 sec. Before and after the shift, the synaptic strength fluctuates but remains roughly constant. But the upward shift in firing rate causes a downward shift in synaptic strength, in accord with the sign of the differential anti-Hebbian rule in Eq. (8). The rate-based approximation works well for this example, because the second term of Eq. (5) is not so important. Let us return to the issue of the general conditions under which Pike (u) are this term can be neglected. With Poisson spike trains, the spike correlations zero in the limit T -7 00, but for finite T they fluctuate about zero. The integr~l over u in the second term of (5) dampens these fluctuations. The amount of dampening depends on the pairing range T, which sets the limits of integration. In Figure 1C we used a relatively long pairing range of 100 ms, which made the fluctuations small even for small T. On the other hand, if T were short, the fluctuations would be small only for large T_ Averaging over large T is relevant when the amplitUde of f is small, so that the rate of learning is slow. In this case, it takes a long time for significant synaptic changes to accumulate, so that plasticity is effectively driven by integrating over long time periods T in Eq. (l). C: In the brain, nonvanishing spike correlations are sometimes observed even in the T -7 00 limit, unlike with Poisson spike trains. These correlations are often roughly symmetric about zero, in which case they should produce little plasticity if the pairing functions are antisymmetric as in Figures lA and lB. On the other hand, if the spike correlations are asymmetric, they could lead to substantial effects[6]. 4 EFFECTS ON RECURRENT NETWORK DYNAMICS The learning rules of Eq. (8) depend on both presynaptic and postsynaptic rates, like learning rules conventionally used in neural networks. They have the special feature that they depend on time derivatives, which has computational consequences for recurrent neural networks of the form Xi + Xi = L Wiju(Xj) + bi (9) j Such classical neural network equations can be derived from more biophysically realistic models using the method of averaging[ 15] or a mean field approximation[ 16]. The firing rate of neuron j is conventionally identified with Vj = u(Xj). v; The cost function E( {Xi}; {Wij}) = ~ Li quantifies the amount of drift in firing rate at the point Xl , ... , X N in the state space of the network. If we consider Vi to be a function of Xi and W ij defined by (9), then the gradient ofthe cost function with respect to W ij is given by BE / BWij = U' (Xi)ViVj. Assuming that U is a monotonically increasing function so that u' (xd > 0, it follows that the differential Hebbian update of (8) increases the cost function, Spike-based Learning and Stabilization ofPersistent Neural Activity 203 and hence increases the magnitude of the drift velocity. In contrast, the differential antiHebbian update decreases the drift velocity. This suggests that the differential anti-Hebbian update could be useful for creating fixed points of the network dynamics (9). 5 PERSISTENT ACTIVITY IN A SPIKING AUTAPSE MODEL The preceding arguments about drift velocity were based on approximate rate-based descriptions of learning and network dynamics. It is important to implement spike-based learning in a spiking network dynamics, to check that our approximations are valid. Therefore we have numerically simulated the simple recurrent circuit of integrate-and-fire neurons shown in Figure 2. The core of the circuit is the "memory neuron," which makes an excitatory autapse onto itself. It also receives synaptic input from three input neurons: a tonic neuron, an excitatory burst neuron, and an inhibitory burst neuron. It is known that this circuit can store a shortterm memory of an analog variable in persistent activity, if the strengths of the autapse and tonic synapse are precisely INHIBITORY BURST tuned[ 17]. Here we show that this tun? ing can be accomplished by the spikebased learning rule of Eq. (1), with a dFigure 2: Circuit diagram for autapse model ifferential anti-Hebbian pairing function like that of Figure 1B. The memory neuron is described by the equations C m dr Tsyn dt dV dt = (10) (1) +r n where V is the membrane potential. When V reaches V'thres, a spike is considered to have occurred, and V is reset to Vreset. Each spike at time Tn causes a jump in the synaptic activation r of size CY.r/Tsyn, after which r decays exponentially with time constant Tsyn until the next spike. The synaptic conductances of the memory neuron are given by (12) The term W r is recurrent excitation from the autapse, where W is the strength of the autapse. The synaptic activations ro, r +, and r _ of the tonic, excitatory burst, and inhibitory burst neurons are governed by equations like (10) and (1), with a few differences. These neurons have no synaptic input; their firing patterns are instead determined by applied currents lapp,o, lapp,+ and lapp,_. The tonic neuron has a constant applied current, which makes it fire repetitively at roughly 20 Hz (Figure 3). For the excitatory and inhibitory burst neurons the applied current is normally zero, except for brief 100 ms current pulses that cause bursts of action potentials. As shown in Figure 3, if the synaptic strengths W and Wo are arbitrarily set before learning, the burst neurons cause only transient changes in the firing rate of the memory neuron. After applying the spike-based learning rule (1) to tune both W and W o, the memory X Xie and H. S. Seung 204 111111111111 I ~IIIIIIIII I IUIIIIIIII I /untuned 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 I I~ ____~I~____~______=-_____ I " tuned 1IIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIII'.tl I 1 sec 1111111111'"111111111111111111111111111 Figure 3: Untuned and tuned autapse activity. The middle three traces are the membrane potentials of the three input neurons in Figure 2 (spikes are drawn at the reset times of the integrate-and-fire neurons). Before learning, the activity of the memory neuron is not persistent, as shown in the top trace. After the spike-based learning rule (1) is applied to the synaptic weights Wand W o, then the burst inputs cause persistent changes in activity. em = 1 nF, gL = 0.025 J-lS, VL = -70 mY, VE = 0 mY, VI = -70 mY, vthres = -52 mY, Vr eset = -59 mY, a s = 1, Tsyn = 100 ms, Iapp,o = 0.5203 nA, I app ,? = 0 or 0.95 nA, Ts yn ,O = 100 ms, Tsyn,+ = Ts yn,- = 5 ms, W+ = 0.1, W_ = 0.05. neuron is able to maintain persistent activity. During the interburst intervals (from A after one burst until A before the next), we made synaptic changes using the differential antiHebbian pairing function f(t) = -Asin(7l'tjT) for spike time differences in the range [-T, T] with A = 1.5 X 10- 4 and T=A=120 ms. The resulting increase in persistence time can be seen in Figure 4A, along with the values of the synaptic weights versus time. To quantify the performance of the system at maintaining persistent activity, we determined the relationship between dv / dt and v using a long sequence of interburst intervals, where v was defined as the reciprocal of the interspike interval. If Wand Wo are fixed at optimally tuned values, there is still a residual drift, as shown in Figure 4B. But if these parameters are allowed to adapt continuously, even after good tuning has been achieved, then the residual drift is even smaller in magnitude. This is because the learning rule tweaks the synaptic weights during each interburst interval, reducing the drift for that particular firing rate. Autapse learning is driven by the autocorrelation of the spike train, rather than a crosscorrelation. The peak in the autocorrelogram at zero lag has no effect, since the pairing function is zero at the origin. Since the autocorrelation is zero for small time lags, we used a fairly large pairing range in our simulations. In a recurrent network of many neurons, a shorter pairing range would suffice, as the cross-correlation does not vanish near zero. 6 DISCUSSION We have shown that differential anti-Hebbian learning can tune a recurrent circuit to maintain persistent neural activity. This behavior can be understood by reducing the spike-based learning rule (l) to the rate-based learning rules ofEqs. (6) and (8). The rate-based approximations are good if two conditions are satisfied. First, the pairing range must be large, or the rate of learning must be slow. Second, spike synchrony must be weak, or have little effect on learning due to the shape of the pairing function. The differential anti-Hebbian pairing function results in a learning rule that uses -Vi as a negative feedback signal to reduce the amount of drift in firing rate, as illustrated by our simulations of an integrate-and-fire neuron with an excitatory autapse. More generally, the learning rule could be relevant for tuning the strength of positive feedback in networks that maintain a short-term memory of an analog variable in persistent neural activity. Spike-based Learning and Stabilization of Persistent Neural Activity A 250 B c ' 200 0.395 ~150 W WO 0.16 0385 ~ i! 0 ?100 "" 0.12 10 20 "me Is) I 6 I 4 1 5 10 15 tlme(s) 20 .~~ 2 1~ :I: ~r? 0 ~-2 ~~! 50 00 205 25 ~~~ -4f -at -8' 20 40 60 rate (Hzl '0 80 i 100 Figure 4: Tuning the autapse. (A) The persistence time of activity increases as the weights Wand Wo are tuned. Each transition is driven by pseudorandom bursts of input (B) Systematic relationship between drift dv/dt in firing rate and v, as measured from a long sequence of interburst intervals. If the weights are continuously fine-tuned ('*') the drift is less than with fixed well-tuned weights ('0'). For example, the learning rule could be used to improve the robustness of the oculomotor integrator[12, 13, 14] and head direction system[l1] to mistuning of parameters. In deriving the differential forms of the learning rules in (8), we assumed that the areas under the positive and negative lobes of the pairing function are equal, so that the integral defining 130 vanishes. In reality, this cancellation might not be exact. Then the ratio of 131 and 130 would limit the persistence time that can be achieved by the learning rule. Both the oculomotor integrator and the head direction system are also able to integrate vestibular inputs to produce changes in activity patterns. The problem of finding generalizations of the present learning rules that train networks to integrate is still open . References [1] H. Markram, J. Lubke, M. Frotscher, and B. Sakmann. Science, 275(5297):213-5, 1997. [2] G. Q. Bi and M. M. Poo. 1 Neurosci, 18(24):10464-72,1998. [3] D. O . Hebb. Organization of behavior. Wiley, New York, 1949. [4] C. C. Bell, V. Z. Han, Y. Sugawara, and K. Grant. Nature , 387(6630):278-81 , 1997. [5] w. Gerstner, R . Kempter, 1. L. van Hemmen, and H. Wagner. Nature, 383(6595):76-81, 1996. [6] L. F. Abbott and S. Song. Adv. Neural Info. Proc. Syst., 11, 1999. [7] P. D. Roberts . 1. Comput. Neurosci., 7:235-246, 1999. [8] R. Kempter, W. Gerstner, and J. L. van Hemmen. Phys. Rev. E, 59(4):4498-4514, 1999. [9] A. P. Georgopoulos, M. Taira, and A. Lukashin. Science, 260:47-52, 1993. [10] M. Camperi and X. J. Wang. 1 Comput Neurosci, 5(4):383-405, 1998. [11] K. Zhang. 1. Neurosci., 16:2112-2126, 1996. [12] S. C. Cannon, D. A. Robinson, and S. Shamma. Bio!. Cybern., 49:127-136,1983. [13] H. S. Seung. Proc. Nat!. A cad. Sci. USA, 93:13339-13344, 1996. [14] H. S. Seung, D. D. Lee, B. Y. Reis, and D. W. Tank. Neuron, 2000. [15] B. Ermentrout. Neural Comput., 6:679-695, 1994. [16] O. Shriki, D. Hansel, and H. Sompolinsky. Soc. Neurosci. Abstr., 24:143, 1998. [17] H. S. Seung, D. D. Lee, B. Y. Reis, and D. W. Tank. 1. Comput. Neurosci., 2000. III THEORY PART 

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Boosting with Multi-Way Branching in Decision Trees Yishay Mansour David McAllester AT&T Labs-Research 180 Park Ave Florham Park NJ 07932 {mansour, dmac }@research.att.com Abstract It is known that decision tree learning can be viewed as a form of boosting. However, existing boosting theorems for decision tree learning allow only binary-branching trees and the generalization to multi-branching trees is not immediate. Practical decision tree algorithms, such as CART and C4.5, implement a trade-off between the number of branches and the improvement in tree quality as measured by an index function . Here we give a boosting justification for a particular quantitative trade-off curve. Our main theorem states, in essence, that if we require an improvement proportional to the log of the number of branches then top-down greedy construction of decision trees remains an effective boosting algorithm. 1 Introduction Decision trees have been proved to be a very popular tool in experimental machine learning. Their popularity stems from two basic features - they can be constructed quickly and they seem to achieve low error rates in practice. In some cases the time required for tree growth scales linearly with the sample size. Efficient tree construction allows for very large data sets. On the other hand, although there are known theoretical handicaps of the decision tree representations, it seem that in practice they achieve accuracy which is comparable to other learning paradigms such as neural networks. While decision tree learning algorithms are popular in practice it seems hard to quantify their success ,in a theoretical model. It is fairly easy to see that even if the target function can be described using a small decision tree, tree learning algorithms may fail to find a good approximation. Kearns and, Mansour [6] used the weak learning hypothesis to show that standard tree learning algorithms perform boosting. This provides a theoretical justification for decision tree learning similar Boosting with Multi-Way Branching in Decision Trees 301 to justifications that have been given for various other boosting algorithms, such as AdaBoost [4]. Most decision tree learning algorithms use a top-down growth process. Given a current tree the algorithm selects some leaf node and extends it to an internal node by assigning to it some "branching function" and adding a leaf to each possible output value of this branching function. The set of branching functions may differ from one algorithm to another, but most algorithms used in practice try to keep the set of branching functions fairly simple. For example, in C4.5 [7], each branching function depends on a single attribute. For categorical attributes, the branching is according to the attribute's value, while for continuous attributes it performs a comparison of the attribute with some constant. Of course such top-down tree growth can over-fit the data - it is easy to construct a (large) tree whose error rate on the training data is zero. However, if the class of splitting functions has finite VC dimension then it is possible to prove that, with high confidence of the choice of the training data, for all trees T the true error rate of T is bounded by f(T) + 0 (JITI/m) where f(T) is the error rate of T on the training sample, ITI is the number of leaves of T, and m is the size of the training sample. Over-fitting can be avoided by requiring that top-down tree growth produce a small tree. In practice this is usually done by constructing a large tree and then pruning away some of its nodes. Here we take a slightly different approach. We assume a given target tree size s and consider the problem of constructing a tree T with ITI = sand f(T) as small as possible. We can avoid over-fitting by selecting a small target value for the tree size. A fundamental question in top-down tree growth is how to select the branching function when growing a given leaf. We can think of the target size as a "budget" . A four-way branch spends more of the tree size budget than does a two-way branch - a four-way branch increases the tree size by roughly the same amount as two twoway branches. A sufficiently large branch would spend the entire tree size budget in a single step. Branches that spend more of the tree size budget should be required to achieve more progress than branches spending less ofthe budget. Naively, one would expect that the improvement should be required to be roughly linear in the number of new leaves introduced - one should get a return proportional to the expense. However, a weak learning assumption and a target tree size define a nontrivial game between the learner and an adversary. The learner makes moves by selecting branching functions and the adversary makes moves by presenting options consistent with the weak learning hypothesis. We prove here that the learner achieve a better value in this game by selecting branches that get a return considerably smaller than the naive linear return. Our main theorem states, in essence, that the return need only be proportional to the log of the number of branches. 2 Preliminaries We assume a set X of instances and an unknown target function f mapping X to {O,l}. We assume a given "training set" S which is a set of pairs of the form (x, f(x)). We let 1l be a set of potential branching functions where each hE 1l is a function from X to a finite set Rh - we allow different functions in 1l to have different ranges. We require that for any h E 1l we have IRhl ~ 2. An 1l-tree is Y. Mansour and D. McAllester 302 a tree where each internal node is labeled with an branching function h E 1i and has children corresponding to the elements of the set Rh. We define ITI to be the number ofleafnodes ofT. We let L(T) be the set ofleafnodes ofT. For a given tree T, leaf node f of T and sample S we write Sl to denote the subset of the sample S reaching leaf f. For f E T we define Pl to be the fraction of the sample reaching leaf f, i.e., ISll/ISI. We define ql to be the fraction of the pairs (x, f(x? in Sl for which f(x) = 1. The training error ofT, denoted i(T), is L:lEL(T)Plmin(ql, 1- ql). 3 The Weak Learning Hypothesis and Boosting Here, as in [6], we view top-down decision tree learning as a form of Boosting [8, 3]. Boosting describes a general class of iterative algorithms based on a weak learning hypothesis. The classical weak learning hypothesis applies to classes of Boolean functions. Let 1i2 be the subset of branching functions h E 1i with IRhl = 2. For c5 > the classical c5-weak learning hypothesis for 1i2 states that for any distribution on X there exists an hE 1i2 with PrD(h(x) f f(x)) ~ 1/2-c5. Algorithms designed to exploit this particular hypothesis for classes of Boolean functions have proved to be quite useful in practice [5]. ? Kearns and Mansour show [6] that the key to using the weak learning hypothesis for decision tree learning is the use of an index function I : [0, 1] ~ [0,1] where I(q) ~ 1, I(q) ~ min(q, (1- q)) and where I(T) is defined to be L:lEL(T) PlI(ql). Note that these conditions imply that i(T) ~ I(T). For any sample W let qw be the fraction of pairs (x, f(x)) E W such that f(x) = 1. For any h E 1i let Th be the decision tree consisting of a single internal node with branching function h plus a leaf for each member of IRh I. Let Iw (Th) denote the value of I(Th) as measured with respect to the sample W. Let ~ (W, h) denote I (qW ) - Iw (Th). The quantity ~(W, h) is the reduction in the index for sample W achieved by introducing a single branch. Also note that Pt~(Sl, h) is the reduction in I(T) when the leaf f is replaced by the branch h. Kearns and Mansour [6] prove the following lemma. Lemma 3.1 (Kearns & Mansour) Assuming the c5-weak learning hypothesis for 1i2, and taking I(q) to be 2Jq(1- q), we have that for any sample W there exists an h E 1i2 such that ~(W,h) ~ ~:I(qw). This lemma motivates the following definition. Definition 1 We say that 1i2 and I satisfies the "I-weak tree-growth hypothesis if for any sample W from X there exists an hE 1i2 such that ~(W, h) ~ "II(qw). Lemma 3.1 states, in essence, that the classical weak learning hypothesis implies the weak tree growth hypothesis for the index function I(q) = 2Jq(l - q). Empirically, however, the weak tree growth hypothesis seems to hold for a variety of index functions that were already used for tree growth prior to the work of Kearns and Mansour. The Ginni index I(q) 4q(1 - q) is used in CART [1] and the entropy I(q) = -q log q - (1- q) log(l- q) is used in C4.5 [7]. It has long been empirically observed that it is possible to make steady progress in reducing I(T) for these choices of I while it is difficult to make steady progress in reducing i(T). = We now define a simple binary branching procedure. For a given training set S and target tree size s this algorithm grows a tree with ITI = s. In the algorithm Boosting with Multi-Way Branching in Decision Trees 303 odenotes the trivial tree whose root is a leaf node and Tl ,h denotes the result of replacing the leaf l with the branching function h and a new leaf for each element of Rh. T=0 WHILE (ITI l f- < s) DO argmaxl 'ftl1(til) h f- argmaxhEl?:l~(Sl' h) T f- Tl,h; END-WHILE We now define e(n) to be the quantity TI~:ll(l-;). Note that e(n) ~ TI~:/ e- 7 ~ .. - l /" e--Y Wi"'l 1 S < e--Y Inn = n--Y. = Theorem 3.2 (Kearns & Mansour) 1f1l2 and I satisfy the ,-weak tree growth hypothesis then the binary branching procedure produces a tree T with i(T) ~ I(T) ~ e(ITI) ~ ITI--Y? Proof: The proof is by induction on the number of iterations of the procedure. We have that 1(0) ~ 1 e(l) so the initial tree immediately satisfies the condition. We now assume that the condition is satisfied by T at the begining of an iteration and prove that it remains satisfied by Tl,h at the end of the iteration. Since I(T) = LlET Ih1(til) we have that the leaf l selected by the procedure is such that Pl1(til) 2: II~)? By the ,-weak tree growth assumption the function h selected by the procedure has the property that ~(Sl, h) 2: ,1(ql). We now l " This implies that have that I(T) - I(Tl,h) = Pl~(Sl' h) 2: P1I1(til) 2: I(Tl,h) ~ I(T) - rh1(T) = (1- j;)I(T) ~ (1- rh)e(ITI) = e(ITI + 1) = e(ITl,hl). = ,II?i o 4 Statement of the Main Theorem We now construct a tree-growth algorithm that selects multi-way branching functions. As with many weak learning hypotheses, the ,-weak tree-growth hypothesis can be viewed as defining a game between the learner and an adversary. Given a tree T the adversary selects a set of branching functions allowed at each leaf of the tree subject to the constraint that at each leaf l the adversary must provide a binary branching function h with ~(Sl' h) 2: ,1(til). The learner then selects a leaf land a branching function h and replaces T by Tl,h. The adversary then again selects a new set of options for each leaf subject to the ,-weak tree growth hypothesis. The proof of theorem 3.2 implies that even when the adversary can reassign all options at every move there exists a learner strategy, the binary branching procedure, guaranteed to achieves a final error rate of ITI--Y. Of course the optimal play for the adversary in this game is to only provide a single binary option at each leaf. However, in practice the "adversary" will make mistakes and provide options to the learner which can be exploited to achieve even lower error rates. Our objective now is to construct a strategy for the learner which can exploit multi-way branches provided by the adversary. We first say that a branching function h is acceptable for tree T and target size Y. Mansour and D. MeAl/ester 304 = s if either IRhl 2 or ITI < e(IRh!)s"Y/(2IRh!). We also define g(k) to be the quantity (1 - e(k?/"Y. It should be noted that g(2) = 1. It should also be noted that e( k) '" e -'Y Ink and hence for "Y In k small we have e( k) '" 1 - "Y In k and hence g(k) '" Ink. We now define the following multi-branch tree growth procedure. T=0 WHILE (ITI < s) DO l +- argm~ Ptl(qt) h +- argmaxhEll , h acceptable for T and s ~(St, h)/g(IRhl) T +- Tt,h; END-WHILE A run of the multi-branch tree growth procedure will be called "Y-boosting if at each iteration the branching function h selected has the property that ~(St, h) / g(lRh I) ~ "YI(qt). The "Y-weak tree growth hypothesis implies that ~(St,h)/g(IRhl) ~ "YI(qt)/g(2) = "YI(qt). Therefore, the "Y-weak tree growth hypothesis implies that every run of the multi-branch growth procedure is "Y-bootsing. But a run can be "Y-bootsing by exploiting mutli-way branches even when the "Y-weak tree growth hypothesis fails. The following is the main theorem of this paper. Theorem 4.1 1fT is produced by a "Y-boosting run of the multi-branch tree-growth procedure then leT) 5 ~ e(ITI) ~ ITI-'Y? Proof of Theorem 4.1 To prove the main theorem we need the concept of a visited weighted tree, or VWtree for short. A VW-tree is a tree in which each node m is assigned both a rational weight Wm E [0,1] and an integer visitation count Vm ~ 1. We now define the following VW tree growth procedure. In the procedure Tw is the tree consisting of a single root node with weight wand visitation count 1. The tree Tt.w1 .... .w/c is the result of inserting k new leaves below the leaf l where the ith new leaf has weight Wi and new leaves have visitation count 1. W +- any rational number in [0,1] T+-Tw FOR ANY NUMBER OF STEPS REPEAT THE FOLLOWING l +- argmaxt e(tI:~wl Vt +- Vt + 1 OPTIONALLY T +- Tt.Wl .. .. ,W lll WITH WI + .. . Wtll ~ e(vt)wt We first prove an analog of theorem 3.2 for the above procedure. For a VW-tree T we define ITI to be LtEL(T) Vt and we define leT) to be LtEL(T) e( Vt)Wt. Lemma 5.1 The VW procedure maintains the invariant that leT) ~ e(ITI). Proof: The proof is by induction on the number of iterations of the algorithm. The result is immediate for the initial tree since eel) 1. We now assume that leT) ~ e(IT!) at the start of an iteration and show that this remains true at the end of the iteration. = 305 Boosting with Multi- Way Branching in Decision Trees We can associate each leaf l with Vt "subleaves" each of weight e(vt)wt/Vt. We have that ITI is the total number of these subleaves and I(T) is the total weight of these subleaves. Therefore there must exist a subleaf whose weight is at least I(T)/ITI. Hence there must exist a leaf l satisfying e(vt)wt/Vt 2': I(T)/ITI. Therefore this relation must hold of the leaf l selected by the procedure. = Let T' be the tree resulting from incrementing Vt. We now have I(T) - I(T') e(vt)wt- e(vt + l)wt = e(vt)wt- (1- ;;)e(vt)wt ;;e(vt)wt 2': "/I~)' So we have I(T') ~ (1 )I(T) ~ (1 )e(ITI) = e(IT'I). ffl ffl = Finally, if the procedure grows new leaves we have that the I(T) does not increase and that ITI remains the same and hence the invariant is maintained. 0 For any internal node m in a tree T let C(m) denote the set of nodes which are children of m. A VW-tree will be called locally-well-formed if for every internal node m we have that Vm = IC(m)l, that I:nEC(m) Wn ~ e(IC(m)l)wm . A VW-tree will be called globally-safe ifmaxtEL(T) e(vt)wt/Vt ~ millmEN(T) e(vt-1)wt/(vt-1) where N(T) denotes the set of internal nodes of T. Lemma 5.2 If T is a locally well-formed and globally safe VW-tree, then T is a possible output of the VW growth procedure and therefore I(T) ~ e(ITI). Proof: Since T is locally well formed we can use T as a "template" for making nondeterministic choices in the VW growth procedure. This process is guaranteed to produce T provided that the growth procedure is never forced to visit a node corresponding to a leaf of T. But the global safety condition guarantees that any unfinished internal node of T has a weight as least as large as any leaf node of T. o We now give a way of mapping ?i-trees into VW-trees. More specifically, for any ?i-tree T we define VW(T) to be the result of assigning each node m in T the weight PmI(qm), each internal node a visitation count equal to its number of children, and each leaf node a visitation count equal to 1. We now have the following lemmas. Lemma 5.3 If T is grown by a I-boosting run of the multi-branch procedure then VW(T) is locally well-formed. Proof: Note that the children of an internal node m are derived by selecting a branching function h for the node m. Since the run is I-boosting we have ~(St, h)/g(IRhi) 2': II(qt). Therefore ~(St, h) = (I(tit) - 1St (n)) 2': I(tit)(l e(IRhl)). This implies that Ist(Th ) ~ e(IRhDI(qt). Multiplying by Pt and trans0 forming the result into weights in the tree VW(T) gives the desired result. The following lemma now suffices for theorem 4.1. Lemma 5.4 If T is grown by a I-boosting run of the multi-branch procedure then VW(T) is globally safe. Proof: First note that the following is an invariant of a I-boosting run of the multi-branch procedure. max Wt < min Wt tEL(VW(T)) - mEN(VW(T)) 306 Y. Mansour and D. MeAl/ester The proof is a simple induction on ,-boosting tree growth using the fact that the procedure always expands a leaf node of maximal weight. We must now show that for every internal node m and every leaf ? we have that Wi ~ e(k -1)w m /(k -1) where k is the number of children of m. Note that if k = 2 then this reduces to Wi ~ Wm which follows from the above invariant. So we can assume without loss of generality that k > 2. Also, since e( k) / k < e( k - 1) / (k - 1), it suffices to show that Wi ~ e(k)wm/k. Let m be an internal node with k > 2 children and let T' be the tree at the time m was selected for expansion. Let Wi be the maximum weight of a leaf in the final tree T. By the definition of the acceptability condition, in the last s/2 iterations we are performing only binary branching. Each binary expansion reduces the index by at least , times the weight of the selected node. Since the sequence of nodes selected in the multi-branch procedure has non-increasing weights, we have that in any iteration the weight of the selected node is at least Wi . Since there are at least s/2 binary expansions after the expansion of m, each of which reduces I by at least ,Wi, we have that s,wd2 ~ I(T') so Wi ~ 2I(T')/(/s). The acceptability condition can be written as 2/(/s) ~ e(k)/(kIT'1) which now yields WI ~ I(T')e(k)/(kIT'I). But we have that I(T')/IT'I ~ Wm which now yields WI ~ e(k)wm/k as desired. 0 References [1] Leo Breiman, Jerome H. Friedman, Richard A. Olshen, and Charles J. Stone. Classification and Regression Trees. Wadsworth International Group, 1984. [2] Tom Dietterich, Michael Kearns and Yishay Mansour. Applying the Weak Learning Framework to understand and improve C4.5. In Proc. of Machine Learning, 96-104, 1996. [3] Yoav Freund. Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256-285, 1995. [4] Yoav Freund and Robert E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Computational Learning Theory: Second European Conference, EuroCOLT '95, pages 23-37. SpringerVerlag, 1995. [5] Yoav Freund and Robert E. Schapire. Experiments with a new boosting algorithm. In Machine Learning: Proceedings of the Thirteenth International Conference, pages 148-156, 1996. [6] Michael Kearns and Yishay Mansour. On the boosting ability of top-down decision tree learning. In Proceedings of the Twenty-Eighth ACM Symposium on the Theory of Computing, pages 459-468,1996. [7] J. Ross Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufmann, 1993. [8] Robert E. Schapire. The strength of weak learnability. Machine Learning, 5(2):197-227, 1990. 

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712 A PROGRAMMABLE ANALOG NEURAL COMPUTER AND SIMULATOR Paul Mueller*, Jan Vander Spiegel, David Blackman*, Timothy Chiu, Thomas Clare, Joseph Dao, Christopher Donham, Tzu-pu Hsieh, Marc Loinaz *Dept.of Biochem. Biophys., Dept. of Electrical Engineering. University of Pennsylvania, Philadelphia Pa. ABSTRACT This report describes the design of a programmable general purpose analog neural computer and simulator. It is intended primarily for real-world real-time computations such as analysis of visual or acoustical patterns, robotics and the development of special purpose neural nets. The machine is scalable and composed of interconnected modules containing arrays of neurons, modifiable synapses and switches. It runs entirely in analog mode but connection architecture, synaptic gains and time constants as well as neuron parameters are set digitally. Each neuron has a limited number of inputs and can be connected to any but not all other neurons. For the determination of synaptic gains and the implementation of learning algorithms the neuron outputs are multiplexed, AID converted and stored in digital memory. Even at moderate size of 1()3 to IDS neurons computational speed is expected to exceed that of any current digital computer. OVERVIEW The machine described in this paper is intended to serve as a general purpose programmable neuron analog computer and simulator. Its architecture is loosely based on the cerebral cortex in the sense that there are separate neurons, axons and synapses and that each neuron can receive only a limited number of inputs. However, in contrast to the biological system, the connections can be modified by external control permitting exploration of different architectures in addition to adjustment of synaptic weights and neuron parameters. The general architecture of the computer is shown in Fig. 1. Themachinecontains large numbers of the following separate elements: neurons, synapses, routing switches and connection lines. Arrays of these elements are fabricated on VLSI chips which are mounted on planar chip carriers each of which forms a separate module. These modules are connected directly to neighboring modules. Neuron arrays are arranged in rows and columns and are surrounded by synaptic and axon arrays. A Programmable Analog Neural Computer and Simulator The machine runs entirely in analog mode. However, connection architectures, synaptic gains and neuron parameters such as thresholds and time constants are set by a digital computer. For determining synaptic weights in a learning mode, time segments of the outputs from neurons are multiplexed, digitized and stored in digital memory. The modular design allows expansion to any degree and at moderate to large size. to lOS neurons. operational speed would exceed that of any currently available i.e. digital computer. 103 I I I I ? I I I I ? I SWITCHES LINES D SYNAPSES NEURONS Figure 1. Layout and general architecture. The machine is composed of different modules shown here as squares. Each module contains on a VLSI chip an array of components (neurons. synapses or switches) and their control circuits. Our prototype design calls for 50 neuron modules for a total of 800 neurons each having 64 synapses. The insert shows the direction of data flow through the modules. Outputs from each neuron leave north and south and are routed through the switch modules east and west and into the synapse modules from north and south. They can also bypass the synapse modules north and south. Input to the neurons through the synapses is from east and west. Power and digital control lines run north and south. THE NEURON MODULES Each neuron chip contains 16 neurons, an analog multiplexer and control logic. (See Figs. 2 & 3.) Input-output relations of the neurons are idealized versions of a typical biological neuron. Each unit has an adjustable threshold (bias), an adjustable minimum output value at threshold and a maximum output (See Fig. 4). Output time constants are selected on the switch chips. The neuron is based on an earlier design which used discrete components (Mueller and Lazzaro, 1986). 713 714 Mueller, et al Inputs to each neuron come from synapse chips east and west (SIR, SIL), outputs (NO) go to switch chips north and south. Each neuron has a second input that sets the minimum output at threshold which is common for all neurons on the chip and selected through a separate synapse line. The threshold is set from one of the synapses connected to a fixed voltage. An analog multiplexer provides neuron output to a common line, OM, which connects to an AID converter. ~6 NJ IS CK 01\ ORI PHI2 ANALOG I\LL T I PLEXER CRO SILl SIL 2 SIR SIR I SILlS SIL I6 SIR IS SIR I6 ?I ?I NO. 2 NJ2 Figure 2. Block diagram of the neuron chip containing 16 neurons . .. . -- _ _ _ _~_ _ _ _ _ _ ~F _ _ _ _~ _ _ ? .... b-.. ~,r ~ "'1<1'" -. ,-~:""",,-- !l'r.:. 11,"" ~~ ........ ~ ~:..._ nil');~ Figure 3. Photograph of a test chip containing 5 neurons. A more recent version has only one output sign. A Programmable Analog Neural Computer and Simulator 4 o o 5 SUM OF INPUTS/VOLTS Figure 4. Transfer characteristic obtained from a neuron on the chip shown in Fig.3. Each unit has an adjustable threshold, Vt which was set here to 1.5V, a linear transfer region above threshold, an adjustable minimum output at threshold E x set to 1V and a maximum output, Emo. THE SYNAPSE MODULES Each synapse chip contains a 32 * 16 array of synapses. The synaptic gain of each synapse is set by serial input from the computer and is s tored at each synapse. Dynamic range of the synapse gains covers the range from 0 to 10 with 5 bit resolution, asixth bit determines the sign. The gains are implemented by current mirrors which scale the neuron output after it has been converted from a voltage to a current. The modifIable synapse designs reported in the literature use either analog or digital signals to set the gains (Schwartz, et. al., 1989, Raffel, et.al, 1987, Alspector and Allen, 1987). We chose the latter method because of its greater reproducibility and because direct analog setting of the gains from the neuron outputs would require a prior knowledge of and commitment to a particular learning algorithm. Layout and perfonnance of the synapse module are shown in Figs. 5-7. As seen in Fig. 7a, the synaptic transfer function is linear from 0 to 4 V. The use of current mirrors pennits arbitrary scaling of the synaptic gains (weights) with trade off between range and resolution limited to 5 bits. Our current design calls for a minimum gain of 1/32 and a maximum of 10. The lower end of the dynamic range is detennined by the number of possible inputs per neuron which when active should not drive the neuron output to its limit, whereas the high gain values are needed in situations where a single or very few synapses must be effective such as in the copying of activity from one neuron to another or for veto inhibition. The digital nature of the synaptic gain control does not allow straight forward implementation of a logarithmic gain scale. Fig. 7b. shows two possible relations between digital code and synaptic gain. In one case the total gain is the sum of 5 individual gains each controlled by one bit. This leads inevitably to jumps in the gain curve. In a second case a linear 3 bit gain is multiplied by four different constants 715 716 Mueller, et al controlled by the 4th and 5th bit. This scheme affords a better approximation to a logarithmic scale. So far we have implemented only the first scheme. Although the resolution of an individual synapse is limited to 5 bits, several synapses driven by one neuron can be combined through switching, permitting greater resolution and dynamic range. o o o NI32 DATA Figure S. Diagram of the synapse module. Each synapse gain is set by a 5 bit word stored in local memory. The memory is implemented as a quasi dynamic shift register that reads the gain data during the programming phase. Voltage to current converters transform the neuron output (N!) into a current. I Conv are current mirrors that scale the currents with 5 bit resolution. The weighted currents are summed on a common line to the neuron input (SO). Figure 6. Photograph of a synapse test chip. A Programmable Analog Neural Computer and Simulator AJOT""'T'--------.,..., B 1 iEm Weight-10 o ~u ~ ~ :J t 10 o .. o~---- o 1 Welght-1/32 ~~~~~~ 2 J Input (Volts) -J+--+-----~~~-----~-+-----+_~ o 4 8 12 18 20 24 28 J2 Code Figure 7a. Synapse transfer characteristics for three different settings. The data were obtained from the chip shown in Fig. 6. b. Digital code vs. synaptic gain, squares are current design, triangles use a two bit exponent. THE SWITCH MODULES The switch modules serve to route the signals between neurons and allow changes to the connection architecture. Each module contains a 32*32 cross point array of analog switches which are set by serial digital input. There is also a set of serial switches that can disconnect input and output lines. In addition to switches the modules contain circuits which control the time constants of the synapse transfer function (see Figs. 8 & 9). The switch perfonnance is summarized in Table 1. UIN UIN Lilli RIlIl Lilli RIll L3B R30 L31 R3t UN+RIN DIN RIN I BIT f'E.fl.Gty L I N - 0 - - RIN 2 BIT f'E.1lfJ/.Y/ LOGIC 0lIl111 0lIl1 031'1 031 Figure 8. Diagram of switching fabric. Squares and circles represent switch cells which connect the horizontal and vertical connectors or cut the conductors. The units labeled T represent adjustable time constants. 717 718 Mueller, et al TABLE 1. Switch Chip Performance Process On resistance Off resistance 3uCMOS <3 KOhm > 1 TOhm Input capacitance Array download time Memory/switch size < IpF 2us 75u x 90u ADJUSTMENT OF SYNAPTIC TIME CONSTANTS For the analysis or generation of temporal patterns as they occur in motion or speech, adjustable time constants of synaptic transfer must be available (Mueller, 1988). Low pass filtering of the input signal to the synapse with 4 bit control of the time constant over a range of 5 to 500 ms is sufficient to deal with real world data. By com bining the low passed input with a direct input of opposite sign, both originating from the same neuron, the typical "ON" and "OFF" responses which serve as measures of time after beginning and end of events and are common in biological systems can be obtained. Several designs are being considered for implementing the variable low pass filter. Since not all synapses need to have this feature, the circuit will be placed on only a limited number of lines on the switch chip. PACKAGING All chips are mounted on identical quad surface mount carriers. Input and output lines are arranged at right angles with identical leads on opposite sides. The chip carriers are mounted on boards. SOFTWARE CONTROL AND OPERATION Connections, synaptic gains and time constants are set from the central computer either manually or from libraries containing connection architectures for specific tasks. Eventually we envision developing a macro language that would generate subsystems and link them into a larger architecture. Examples are feature specific receptor fields, temporal pattern analyzers, or circuits for motion control. The connection routing is done under graphic control or through routing routines as they are used in circuit board design. A Programmable Analog Neural Computer and Simulator The primary areas of application include real-world real-time or compressed time pattern analysis, robotics, the design of dedicated neural circuits and the exploration of different learning algorithms. Input to the machine can come from sensory transducer arrays such as an electronic retina, cochlea (Mead, 1989) or tactile sensors. For other computational tasks, input is provided by the central digital computer through activation of selected neuron populations via threshold control. It might seem that the limited number of inputs per neuron restricts the computations performed by anyone neuron. However the results obtained by one neuron can be copied through a unity gain synapse to another neuron which receives the appropriate additional inputs. In performance mode the machine could exceed by orders of magnitude the computational speed of any currently available digital computer. A rough estimate of attainable speed can be made as follows: A network with 103 neurons each receiving 100 inputs with synaptic transfer time constants ranging from 1 ms to 1 s, can be described by 103 simultaneous differential equations. Assuming an average step length of 10 us and 10 iterations per step, real time numerical solutions of this system on a digital machine would require approximately 1011 FLOPS. Microsecond time constants and the computation of threshold non-linearities would require a computational speed equivalent to > 1012 FLOPS on a digital computer and this seems a reasonable estimate of the computational power of our machine. Furthermore, in contrast to digital mUltiprocessors, computational power would scale linearly with the number of neurons and connections. Acknowledgements Supported by grants from ONR (NOOOI4-89-J-1249) and NSF (EET 166685). References Alspector, J., Allen, R.B. A neuromorphic VLSI learning system. Advanced research in VLSI. Proceedings of the 1987 Stanford Conference. (1987). Mead, C. Analog VLSI and Neural Systems. Addison Wesley, Reading, Ma (1989). Mueller, P. Computation of temporal Pattern Primitives in a Neural Net for Speech Recognition. InternationalNeuralNetworkSociety. FirstAnnualMeeting,Boston Ma.? (1988). Mueller, P., Lazzaro, J. A Machine for Neural Computation of Acoustical Patterns. AlP Conference Proceedings. 151:321-326, (1986). Raffel, J.I., Mann, J.R., Berger, R., Soares, A.M., Gilbert, S., A Generic Architecture for Wafer-Scale neuromorphic Systems. IEEE First International Conference on Neural Networks. San Diego, CA. (1987). Schwartz, D., Howard, R., Hubbard, W., A Programmable Analog Neural Network Chip, J. of Solid State Circuits, (to be published). 719 

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Reconstruction of Sequential Data with Probabilistic Models and Continuity Constraints Miguel A. Carreira-Perpifian Dept. of Computer Science, University of Sheffield, UK miguel@dcs.shefac.uk Abstract We consider the problem of reconstructing a temporal discrete sequence of multidimensional real vectors when part of the data is missing, under the assumption that the sequence was generated by a continuous process. A particular case of this problem is multivariate regression, which is very difficult when the underlying mapping is one-to-many. We propose an algorithm based on a joint probability model of the variables of interest, implemented using a nonlinear latent variable model. Each point in the sequence is potentially reconstructed as any of the modes of the conditional distribution of the missing variables given the present variables (computed using an exhaustive mode search in a Gaussian mixture). Mode selection is determined by a dynamic programming search that minimises a geometric measure of the reconstructed sequence, derived from continuity constraints. We illustrate the algorithm with a toy example and apply it to a real-world inverse problem, the acoustic-toarticulatory mapping. The results show that the algorithm outperforms conditional mean imputation and multilayer perceptrons. 1 Definition of the problem Consider a mobile point following a continuous trajectory in a subset of ]RD. Imagine that it is possible to obtain a finite number of measurements of the position of the point. Suppose that these measurements are corrupted by noise and that sometimes part of, or all, the variables are missing. The problem considered here is to reconstruct the sequence from the part of it which is observed. In the particular case where the present variables and the missing ones are the same for every point, the problem is one of multivariate regression. If the pattern of missing variables is more general, the problem is one of missing data reconstruction. Consider the problem of regression. If the present variables uniquely identify the missing ones at every point of the data set, the problem can be adequately solved by a universal function approximator, such as a multilayer perceptron. In a probabilistic framework, the conditional mean of the missing variables given the present ones will minimise the average squared reconstruction error [3]. However, if the underlying mapping is one-to-many, there will be regions in the space for which the present variables do not identify uniquely the missing ones. In this case, the conditional mean mapping will fail, since it will give a compromise value-an average of the correct ones. Inverse problems, where the inverse Probabilistic Sequential Data Reconstruction 415 of a mapping is one-to-many, are of this type. They include the acoustic-to-articulatory mapping in speech [15], where different vocal tract shapes may produce the same acoustic signal, or the robot arm problem [2], where different configurations of the joint angles may place the hand in the same position. In some situations, data reconstruction is a means to some other objective, such as classification or inference. Here, we deal solely with data reconstruction of temporally continuous sequences according to the squared error. Our algorithm does not apply for data sets that either lack continuity (e.g. discrete variables) or have lost it (e.g. due to undersampling or shuffling). We follow a statistical learning approach: we attempt to reconstruct the sequence by learning the mapping from a training set drawn from the probability distribution of the data, rather than by solving a physical model of the system. Our algorithm can be described briefly as follows. First, a joint density model of the data is learned in an unsupervised way from a sample of the datal . Then, pointwise reconstruction is achieved by computing all the modes of the conditional distribution of the missing variables given the present ones at the current point. In principle, any of these modes is potentially a plausible reconstruction. When reconstructing a sequence, we repeat this mode search for every point in the sequence, and then find the combination of modes that minimises a geometric sequence measure, using dynamic programming. The sequence measure is derived from local continuity constraints, e.g. the curve length. The algorithm is detailed in ?2 to ?4. We illustrate it with a 2D toy problem in ?5 and apply it to an acoustic-to-articulatory-like problem in ?6. ?7 discusses the results and compares the approach with previous work. Our notation is as follows. We represent the observed variables in vector form as t = (tl' ... , t D) E ~D. A data set (possibly a temporal sequence) is represented as {t n } ~=l . Groups of variables are represented by sets of indices I, J E {I, ... , D}, so that if I = {I, 7, 3}, then tI = (tlt7t3). 2 Joint generative modelling using latent variables Our starting point is a joint probability model of the observed variables p( t). From it, we can compute conditional distributions of the form p( t..71 tI) and, by picking representative points, derive a (multivalued) mapping tI ~ t..7. Thus, contrarily to other approaches, e.g. [6], we adopt multiple pointwise imputation. In ?4 we show how to obtain a single reconstructed sequence of points. Although density estimation requires more parameters than mapping approximation, it has a fundamental advantage [6]: the density model represents the relation between any variables, which allows to choose any missing/present variable combination. A mapping approximator treats asymmetrically some variables as inputs (present) and the rest as outputs (missing) and can't easily deal with other relations. The existence of functional relationships (even one-to-many) between the observed variables indicates that the data must span a low-dimensional manifold in the data space. This suggests the use of latent variable models for modelling the joint density. However, it is possible to use other kinds of density models. In latent variable modelling the assumption is that the observed high-dimensional data t is generated from an underlying low-dimensional process defined by a small number L of latent variables x = (Xl, ... , xL) [1] . The latent variables are mapped by a fixed I In our examples we only use complete training data (i.e., with no missing data), but it is perfectly possible to estimate a probability model with incomplete training data by using an EM algorithm [6]. M 416 A. Carreira-Perpiful.n transformation into a D-dimensional data space and noise is added there. A particular model is specified by three parametric elements: a prior distribution in latent space p(x), a smooth mapping f from latent space to data space and a noise model in data space p(tlx). Marginalising the joint probability density function p(t, x) over the latent space gives the distribution in data space, p(t). Given an observed sample in data space {t n };;=l' a parameter estimate can be found by maximising the log-likelihood, typically using an EM algorithm. We consider the following latent variable models, both of which allow easy computation of conditional distributions of the form p( tJ ItI ): Factor analysis [1], in which the mapping is linear, the prior in latent space is unit Gaussian and the noise model is diagonal Gaussian. The density in data space is then Gaussian with a constrained covariance matrix. We use it as a baseline for comparison with more sophisticated models. The generative topographic mapping (GTM) [4] is a nonlinear latent variable model, where the mapping is a generalised linear model, the prior in latent space is discrete uniform and the noise model is isotropic Gaussian. The density in data space is then a constrained mixture of isotropic Gaussians. In latent variable models that sample the latent space prior distribution (like GTM), the mixture centroids in data space (associated to the latent space samples) are not trainable parameters. We can then improve the density model at a higher computational cost with no generalisation loss by increasing the number of mixture components. Note that the number of components required will depend exponentially on the intrinsic dimensionality of the data (ideally coincident with that of the latent space, L) and not on the observed one, D. 3 Exhaustive mode finding Given a conditional distribution p(tJltI), we consider all its modes as plausible predictions for tJ. This requires an exhaustive mode search in the space of t J . For Gaussian mixtures, we do this by using a maximisation algorithm starting from each centroid2 , such as a fixed-point iteration or gradient ascent combined with quadratic optimisation [5]. In the particular case where all variables are missing, rather than performing a mode search, we return as predictions all the component centroids. It is also possible to obtain error bars at each mode by locally approximating the density function by a normal distribution. However, if the dimensionality of tJ is high, the error bars become very wide due to the curse of the dimensionality. An advantage of multiple pointwise imputation is the easy incorporation of extra constraints on the missing variables. Such constraints might include keeping only those modes that lie in an interval dependent on the present variables [8] or discarding low-probability (spurious) modes-which speeds up the reconstruction algorithm and may make it more robust. A faster way to generate representative points of p(tJltI) is simply to draw a fixed number of samples from it-which may also give robustness to poor density models. However, in practice this resulted in a higher reconstruction error. 4 Continuity constraints and dynamic programming (D.P) search Application of the exhaustive mode search to the conditional distribution at every point of the sequence produces one or more candidate reconstructions per point. To select a 2 Actually, given a value of tz, most centroids have negligible posterior probability and can be removed from the mixture with practically no loss of accuracy. Thus, a large number of mixture components may be used without deteriorating excessively the computational efficiency. 417 Probabilistic Sequential Data Reconstrnction trajectory factor an. Average squared reconstruction error mean dpmode ..., N Mi ssing pattern h tl 0 tl or t2 -2 10% 50% 90% -4 Factor analysis 3.8902 4.3226 4.2020 1.0983 6.2914 21.4942 MLP" 0.2046 2.5126 - - GTM mean dpmode cmode 0.2044 2.4224 1.2963 0.3970 4.6530 20.7877 0.2168 0 .0522 0.1305 0.0253 0. 1176 2 .2261 0.2168 0.0522 0. 1305 0.0251 0.0771 0.0643 aThe MLP cannot be applied to varying patterns of missing data. -6 -6 -4 -2 tl Table 1: Trajectory reconstruction for a 2D problem. The table gives the average squared reconstruction error when t2 is missing (row 1), tl is missing (row 2), exactly one variable per point is missing at random (row 3) or a percentage of the values are missing at random (rows 4-6). The graph shows the reconstructed trajectory when tl is missing: factor analysis (straight, dotted line), mean (thick, dashed), dpmode (superimposed on the trajectory). single reconstructed sequence, we define a local continuity constraint: consecutive points in time should also lie nearby in data space. That is, if 8 is some suitable distance in JR.D, 8 (tn, tn+ 1) should be small. Then we define a global geometric measure ~ for a sequence {t n };;=1 as ~ ({t n };;=I) ~f ~ '2:.::118 (tn, tn+t). We take 8 as the Euclidean distance, so becomes simply the length of the sequence (considered as a polygonal line). Finding the sequence of modes with minimal ~ is efficiently achieved by dynamic programming. 5 Results with a toy problem To illustrate the algorithm, we generated a 2D data set from the curve (tl, t2) = (x, x + 3 sin(x)) for x E [-211',211'], with normal isotropic noise (standard deviation 0.2) added. Thus, the mapping tl -+ t2 is one-to-one but the inverse one, t2 -+ tl, is multivalued. One-dimensional factor analysis (6 parameters) and GTM models (21 parameters) were estimated from a 1000-point sample, as well as two 48-hidden-unit multilayer perceptrons (98 parameters), one for each mapping. For GTM we tried several strategies to select points from the conditional distribution: mean (the conditional mean), dpmode (the mode selected by dynamic programming) and cmode (the closest mode to the actual value of the missing variable). The cmode, unknown in practice, is used here to compute a lower bound on the performance of any mode-based strategy. Other strategies, such as picking the global mode, a random mode or using a local (greedy) search instead of dynamic programming, gave worse results than the dpmode. Table 1 shows the results for reconstructing a IOO-point trajectory. The nonlinear nature of the problem causes factor analysis to break down in all cases. For the one-to-one mapping case (t2 missing) all the other methods perform well and recover the original trajectory, with mean attaining the lowest error, as predicted by the theory3. For the one-to-many case (tl missing, see fig .), both the MLP and the mean are unable to track more than one branch of the mapping, but the dpmode still recovers the original mapping. For random missing 3 A combined strategy could retain the optimality of the mean in the one-to-one case and the advantage of the modes in the one-to-many case, by choosing the conditional mean (rather than the mode) when the conditional distribution is unimodal, and all the modes otherwise. M 418 Missing pattern PLP EPG 10% 50% blocks Factor analysis 0.9165 3.7177 0.2046 1.1285 0.1950 A. Carreira-Perpinan GTM mean dpmode cmode 0.6217 2.3729 0.0947 0.7540 0.1669 0.6250 2.0613 0.0903 0.6527 0.1005 0.4587 1.0538 0.0841 0.6023 0.0925 Table 2: Average squared reconstruction error for an utterance. The last row corresponds to a missing pattern of square blocks totalling 10% of the utterance. patterns4 , the dpmode is able to cope well with high amounts of missing data. The consistently low error of the cmode shows that the modes contain important information about the possible options to predict the missing values. The performance of the dpmode, close to that of the cmode even for large amounts of missing data, shows that application of the continuity constraint allows to recover that information. 6 Results with real speech data We report a preliminary experiment using acoustic and e1ectropalatographic (EPG) data5 for the utterance "Put your hat on the hatrack and your coat in the cupboard" (speaker FG) from the ACCOR database [10]. 12th-order perceptual linear prediction coefficients [7] plus the log-energy were computed at 200 Hz from its acoustic waveform. The EPG data consisted of 62-bit frames sampled at 200 Hz, which we consider as 62-dimensional vectors of real numbers. No further preprocessing of the data was carried out. Thus, the resulting sequence consisted of over 600 75-dimensional real vectors. We constructed a training set by picking, in random order, 80% of these vectors. The whole utterance was used for the reconstruction test. We trained two density models: a 9-dimensional factor analysis (825 parameters) and a two-dimensional 6 GTM (3676 parameters) with a 20 x 20 grid (resulting in a mixture of 400 isotropic Gaussians in the 75-dimensional data space). Table 2 confirms again that the linear method (factor analysis) fares worst (despite its use of a latent space of dimension L = 9). The dpmode attains almost always a lower error than the conditional mean, with up to a 40% improvement (the larger the higher the amount of missing data). When a shuffled version of the utterance (thus having lost its continuity) was reconstructed, the error of the dpmode was consistently higher than that of the mean, indicating that the application of the continuity constraint was responsible for the error decrease. 7 Discussion Using a joint probability model allows flexible construction of predictive distributions for the missing data: varying patterns of missing data and multiple pointwise imputations are possible, as opposed to standard function approximators. We have shown that the modes of the conditional distribution of the missing variables given the present ones are potentially 4Note that the nature of the missing pattern (missing at random, missing completely at random, etc. [9]) does not matter for reconstruction-although it does for estimation. 5 An EPG datum is the (binary) contact pattern between the tongue and the palate at selected locations in the latter. Note that it is an incomplete articulatory representation of speech. 6 A latent space of 2 dimensions is clearly too low for this data, but the computational complexity of GTM prevents the use of a higher one. Still, its nonlinear character compensates partly for this. Probabilistic Sequential Data Reconstruction 419 plausible reconstructions of the missing values, and that the application of local continuity constraints-when they hold-can help to recover the actually plausible ones. Previous work The key aspects of our approach are the use of a joint density model (learnt in an unsupervised way), the exhaustive mode search, the definition of a geometric trajectory measure derived from continuity constraints and its implementation by dynamic programming. Several of these ideas have been applied earlier in the literature, which we review briefly. The use of the joint density model for prediction is the basis of the statistical technique of multiple imputation [9]. Here, several versions of the complete data set are generated from the appropriate conditional distributions, analysed by standard complete-data methods and the results combined to produce inferences that incorporate missing-data uncertainty. Ghahramani and Jordan [6] also proposed the use of the joint density model to generate a single estimate of the missing variables and applied it to a classification problem. Conditional distributions have been approximated by MLPs rather than by density estimation [16], but this lacks flexibility to varying patterns of missing data and requires an extra model of the input variables distribution (unless assumed uniform). Rohwer and van der Rest [12] introduce a cost function with a description length interpretation whose minimum is approximated by the densest mode of a distribution. A neural network trained with this cost function can learn one branch of a multivariate mapping, but is unable to select other branches which may be correct at a given time. Continuity constraints implemented via dynamic programming have been used for the acoustic-to-articulatory mapping problem [15]. Reasonable results (better than using an MLP to approximate the mapping) can be obtained using a large codebook of acoustic and articulatory vectors. Rahim et al. [11] achieve similar quality with much less computational requirements using an assembly of MLPs, each one trained in a different area of the acoustic-articulatory space, to locally approximate the mapping. However, clustering the space is heuristic (with no guarantee that the mapping is one-to-one in each region) and training the assembly is difficult. It also lacks flexibility to varying missingness patterns. A number of trajectory measures have been used in the robot arm problem literature [2] and minimised by dynamic programming, such as the energy, torque, acceleration, jerk, etc. Temporal modelling It is important to remark that our approach does not attempt to model the temporal evolution of the system. The joint probability model is estimated statically. The temporal aspect of the data appears indirectly and a posteriori through the application of the continuity constraints to select a trajectory? In this respect, our approach differs from that of dynamical systems or from models based in Markovian assumptions, such as hidden Markov models or other trajectory models [13, 14]. However, the fact that the duration or speed of the trajectory plays no role in the algorithm may make it invariant to time warping (e.g. robust to fast/slow speech styles). Choice of density model The fact that the modes are a key aspect of our approach make it sensitive to the density model. With finite mixtures, spurious modes can appear as ripple superimposed on the density function in regions where the mixture components are sparsely distributed and have little interaction. Such modes can lead the DP search to a wrong trajectory. Possible solutions are to improve the density model (perhaps by increasing the number of components, see ?2, or by regularisation), to smooth the conditional distribution or to look for bumps (regions of high probability mass) instead of modes. 7However, the method may be derived by assuming a distribution over the whole sequence with a normal, Markovian dependence between adjacent frames. M. 420 A. Carreira-Perpifuin Computational cost The DP search has complexity O(N M2), where M is an average of the number of modes per sequence point and N the number of points in the sequence. In our experiments M is usually small and the DP search is fast even for long sequences. The bottleneck of the reconstruction part of the algorithm is obtaining the modes of the conditional distribution for every point in the sequence when there are many missing variables. Further work We envisage more thorough experiments using data from the Wisconsin X-ray microbeam database and comparing with recurrent MLPs or an MLP committee, which may be more suitable for multi valued mappings. Extensions of our algorithm include different geometric measures (e.g. curvature-based rather than length-based), different strategies for multiple pointwise imputation (e.g. bump searching) or multidimensional constraints (e.g. temporal and spatial). Other practical applications include audiovisual mappings for speech, hippocampal place cell reconstruction and wind vector retrieval from scatterometer data. Acknowledgments We thank Steve Renals for useful conversations and for comments about this paper. References [1] D. J. Bartholomew. Latent Variable Models and Factor Analysis. Charles Griffin & Company Ltd., London, 1987. [2] N. Bernstein. The Coordination and Regulation 0/ Movements. Pergamon, Oxford, 1967. [3] C. M. Bishop. Neural Networks/or Pattern Recognition. Oxford University Press, 1995. [4] C. M. Bishop, M. Svensen, and C. K. I. Williams. GTM: The generative topographic mapping. Neural Computation, 10(1):215-234, Jan. 1998. [5] M. A. Carreira-Perpifian. Mode-finding in Gaussian mixtures. Technical Report CS-99-03, Dept. of Computer Science, University of Sheffield, UK, Mar. 1999. Available online at http://vvv.dcs.shef.ac.uk/-miguel/papers/cs-99-03.html. [6] Z. Ghahramani and M. I. Jordan. Supervised learning from incomplete data via an EM approach. In NIPS 6, pages 120-127,1994. [7] H. Hermansky. Perceptual linear predictive (PLP) analysis of speech. 1. Acoustic Soc. Amer., 87(4):1738-1752, Apr. 1990. [8] L. Josifovski, M. Cooke, P. Green, and A. Vizinho. State based imputation of missing data for robust speech recognition and speech enhancement. In Proc. Eurospeech 99. pages 2837-2840, 1999. [9] R. 1. A. Little and D. B. Rubin. Statistical Analysis with Missing Data. John Wiley & Sons, New York, London, Sydney, 1987. [10] A. Marchal and W. J. Hardcastle. ACCOR: Instrumentation and database for the cross-language study of coarticulation. Language and Speech, 36(2, 3): 137-153, 1993. [11] M. G. Rahim, C. C. Goodyear, W. B. Kleijn, J. Schroeter, and M. M. Sondhi . On the use of neural networks in articulatory speech synthesis. 1. Acoustic Soc. Amer., 93(2): 1109-1121, Feb. 1993. [12] R. Rohwer and J. C. van der Rest. Minimum description length, regularization, and multi modal data. Neural Computation, 8(3):595-609, Apr. 1996. [13] S. Roweis. Constrained hidden Markov models. In NIPS 12 (this volume), 2000. [14] L. K. Saul and M. G . Rahim. Markov processes on curves for automatic speech recognition. In NIPS 11, pages 751-757, 1999. [15] 1. Schroeter and M. M. Sondhi. Techniques for estimating vocal-tract shapes from the speech signal. IEEE Trans. Speech and Audio Process., 2(1): 133-150, Jan. 1994. [16] V. Tresp, R. Neuneier, and S. Ahmad. Efficient methods for dealing with missing data in supervised learning. In NiPS 7, pages 689-696, 1995. 

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Inference for the Generalization Error Claude Nadeau CIRANO 2020, University, Montreal, Qc, Canada, H3A 2A5 jcnadeau@altavista.net Yoshua Bengio CIRANO and Dept. IRO Universite de Montreal Montreal, Qc, Canada, H3C 3J7 bengioy@iro.umontreal.ca Abstract In order to to compare learning algorithms, experimental results reported in the machine learning litterature often use statistical tests of significance. Unfortunately, most of these tests do not take into account the variability due to the choice of training set. We perform a theoretical investigation of the variance of the cross-validation estimate of the generalization error that takes into account the variability due to the choice of training sets. This allows us to propose two new ways to estimate this variance. We show, via simulations, that these new statistics perform well relative to the statistics considered by Dietterich (Dietterich, 1998). 1 Introduction When applying a learning algorithm (or comparing several algorithms), one is typically interested in estimating its generalization error. Its point estimation is rather trivial through cross-validation. Providing a variance estimate of that estimation, so that hypothesis testing and/or confidence intervals are possible, is more difficult, especially, as pointed out in (Hinton et aI., 1995), if one wants to take into account the variability due to the choice of the training sets (Breiman, 1996). A notable effort in that direction is Dietterich's work (Dietterich, 1998). Careful investigation of the variance to be estimated allows us to provide new variance estimates, which tum out to perform well. Let us first layout the framework in which we shall work. We assume that data are available in the form Zjl = {Z1, ... , Zn}. For example, in the case of supervised learning, Zi = (Xi,}Ii) E Z ~ RP+q, where p and q denote the dimensions of the X/s (inputs) and the }Ii 's (outputs). We also assume that the Zi'S are independent with Zi rv P(Z) . Let ?(D; Z), where D represents a subset of size nl ::; n taken from Zjl, be a function Znl X Z -t R For instance, this function could be the loss incurred by the decision that a learning algorithm trained on D makes on a new example Z. We are interested in estimating nJ.l. == E[?(Zjl; Zn+1)] where Zn+1 rv P(Z) is independent of Zjl. Subscript n stands for the size of the training set (Zjl here). The above expectation is taken over Zjl and Zn+1, meaning that we are interested in the performance of an algorithm rather than the performance of the specific decision function it yields on the data at hand. According to Dietterich's taxonomy (Dietterich, 1998), we deal with problems of type 5 through 8, (evaluating learning algorithms) rather then type 1 through 4 (evaluating decision functions). We call nJ.l. the generalization error even though it can also represent an error difference: ? Generalization error We may take ?(D; Z) = ?(D; (X, Y)) = Q(F(D)(X), Y), (1) C. Nadeau and Y. Bengio 308 where F(D) (F(D) : ]RP ~ ]Rq) is the decision function obtained when training an algorithm on D, and Q is a loss function measuring the inaccuracy of a decision. For instance, we could have Q(f), y) = I[f) 1= y], where I[ ] is the indicator function, for classification problems and Q(f), y) =11 f) - y 11 2 , where is II . II is the Euclidean norm, for "regression" problems. In that case nJ.L is what most people call the generalization error. ? Comparison of generalization errors Sometimes, we are not interested in the performance of algorithms per se, but instead in how two algorithms compare with each other. In that case we may want to consider .cCDi Z) = .c(Di (X, Y)) = Q(FA(D)CX), Y) - Q(FB(D)(X), Y), (2) where FA(D) and FB(D) are decision functions obtained when training two algorithms (A and B) on D , and Q is a loss function. In this case nJ.L would be a difference of generalization errors as outlined in the previous example. The generalization error is often estimated via some form of cross-validation. Since there are various versions of the latter, we layout the specific form we use in this paper. ? Let Sj be a random set of nl distinct integers from {I, ... , n }(nl < n). Here nl represents the size of the training set and we shall let n2 = n - nl be the size of the test set. ? Let SI, ... SJ be independent such random sets, and let Sj = {I, ... , n} \ Sj denote the complement of Sj. ? Let Z Sj = {Zi Ii E Sj} be the training set obtained by subsampling Zr according to the random index set Sj. The corresponding test set is ZSj = {Zili E Sj}. ? Let L(j, i) = .c(Zs;; Zi). According to (1), this could be the error an algorithm trained on the training set ZSj makes on example Zi. According to (2), this could be the difference of such errors for two different algorithms. ,i'k k ? Let (1,j = 2:~=1 L(j, i{) where i{, ... are randomly and independently drawn from Sj. Here we draw K examples from the test set ZS'j with replacement and compute the average error committed. The notation does not convey the fact that {1,j depends on K, nl and n2 . ? Let {1,j = limK ..... oo (1,j = ';2 2:iES~ L(j, i) denote what {1,j becomes as K increases J without bounds. Indeed, when sampling infinitely often from ZS'j' each Zi (i E Sj) is yielding the usual "average test error". The use of K is chosen with relative frequency .l.., n2 just a mathematical device to make the test examples sampled independently from Sj. Then the cross-validation estimate of the generalization error considered in this paper is ~K J I '""' ~ nl J.LJ - J L.J J.Lj. n2 _ j=1 We note that this an unbiased estimator of nlJ.L = E[.c{Zfl, Zn+r)] (not the same as nJ.L). This paper is about the estimation of the variance of ~~ {1,~. We first study theoretically this variance in section 2, leading to two new variance estimators developped in section 3. Section 4 shows part of a simulation study we performed to see how the proposed statistics behave compared to statistics already in use. 2 Analysis of Var[ ~~itr] Here we study Var[ ~~ {1,~]. This is important to understand why some inference procedures about nl J.L presently in use are inadequate, as we shall underline in section 4. This investigation also enables us to develop estimators of Var[ ~~ {1,~] in section 3. Before we proceed, we state the following useful lemma, proved in (Nadeau and Bengio, 1999). Inference for the Generalization Error 309 Lemma 1 Let U1, ... , Uk be random variables with common mean (3, common variance 6 and Cov[Ui , Uj] = "I, Vi '# j. Let1r = be the correlation between Ui and Uj (i '# j). J Let U = k- 1 2::=1 Ui and 8b = k~1 2::=1 (Ui U)2 be the sample mean and sample variance respectively. Then E[8b] = 6 - "I and Var[U] "I + (6~'Y) = 6 (11" + lk1l') . - = To study Var[ ~i j1,~] we need to define the following covariances. ? Let lio = liO(nl) = Var[L(j, i)] when i is randomly drawn from 8J. ? Let lil = lil (nl, n2) = Cov[L(j, i), L(j, i')] for i and i' randomly and independently drawn from 8j. ? Let li2 = liZ(nl, n2) = Cov[L(j, i), L(j', i')], with j independently drawn from 8j and 8jl respectively. '# j', i and i' randomly and ? Let li3 = li3(nl) = Cov[L(j, i), L(j, i')] for i, i' E 8j and i same as lil. In fact, it may be shown that ") L('), z")] -C O[L( V ) , z, lil lio + (nz nz - 1) li 3 -_ '# i'. This is not the . li3 + lio - nz li3 . (3) nz Let us look at the mean and variance of j1,j and ~i j1,~. Concerning expectations, we obviously have E[j1,j] = n1f.? and thus E[ ~ij1,~] = n1f.?. From Lemma 1, we have Var[j1,j] lil + O'?KO'I which implies Var[j1,j] = Var[ lim j1,j] = lim Var[j1,j] = lil. = K-too It can also be shown that Cov[j1,j, j1,j'] = TT var [n2 ~K] _ n1f.?J -liz+ TT [~] K-too liZ, var f.?j J j '# j', and therefore (using Lemma 1) liZ _ lil -liZ+ + 0'0-0'1 K J - liZ . (4) We shall often encounter liO, lil, liZ, li3 in the future, so some knowledge about those quantities is valuable. Here's what we can say about them. Proposition 1 For given nl and n2, we have 0 ~ liz ~ lil ~ lio and 0 ~ li3 ~ lil. Proof See (Nadeau and Bengio, 1999). A natural question about the estimator ~i j1,~ is how nl, nz, K and J affect its variance. Proposition 2 The variance of ~i j1,~ is non-increasing in J, K and nz. Proof See (Nadeau and Bengio, 1999). Clearly, increasing K leads to smaller variance because the noise introduced by sampling with replacement from the test set disappears when this is done over and over again. Also, averaging over many trainltest (increasing J) improves the estimation of nl f.?. Finally, all things equal elsewhere (nl fixed among other things), the larger the size of the test sets, the better the estimation of nl f.?. The behavior of Var[ ~i j1,~] with respect to nl is unclear, but we conjecture that in most situations it should decrease in nl. Our argument goes like this. The variability in ~i j1,~ comes from two sources: sampling decision rules (training process) and sampling testing examples. Holding n2, J and K fixed freezes the second source of variation as it solely depends on those three quantities, not nl. The problem to solve becomes: how does nl affect the first source of variation? It is not unreasonable to say that the decision function yielded by a learning algorithm is less variable when the training set is large. We conclude that the first source of variation, and thus the total variation (that is Var[ ~ij1,~]) is decreasing in nl. We advocate the use of the estimator (5) C. Nadeau and Y Bengio 310 as it is easier to compute and has smaller variance than ~~it} (J, nl, n2 held constant). Var[ n2 11 00 ] = lim Var[ nl,-J K-+oo where P -- ~ 111 n2 rl.K] nl,-J = (72 + (71 - (72 - J - (7 1 (p + -1 -Jp)- ' (6) oo r/OO] -- Corr[ll. '-j , '-j' . 3 Estimation of Var[ ~~JtJ] We are interested in estimating ~~(7J == Var[ ~~ it:f] where ~~ it:f is as defined in (5). We provide two different estimators of Var[ ~~ it:f]. The first is simple but may have a positive or negative bias for the actual variance. The second is meant to be conservative, that is, if our conjecture of the previous section is correct, its expected value exceeds the actual variance. 1st Method: Corrected Resampled t- Test. Let us recall that ~~ it:f = jj2 be the sample variance of the itj's. According to Lemma 1, E[jj 21=(71(1-p)= I-p P+ !=?(71 J ( I-P) p+~ = (p+!=?) ~ 1. + I-p Var[ ~~ iL:f]. J (J + G) jj2 is an unbiased estimator of so that (71 J'Ef=1 itj . Let Var[ ~~it:f] l+--L ' J I-p (7) The only problem is that p = p(nl,n2) = :~t~:::~~, the correlation between the itj's, is unknown and difficult to estimate. We use a naive surrogate for p as follows. Let us recall that iLj = 'EiES~, ?(ZSj; Zi). For the purpose of building our estimator, let us make the :2 approximation that ?(ZSj; Zi) depends only on Zi and nl. Then it is not hard to show (see (Nadeau and Bengio, 1999)) that the correlation between the itj's becomes nl~n2' There- Var[~~iL:fl is (J + l~~o) jj2 where Po = po(nl,n2) = nl~n2' that is (J + ~ ) jj2. This will tend to overestimate or underestimate Var[ ~~ iL:f] according to whether Po > p or Po < p. Note that this first method basically does not require fore our first estimator of any more computations than that already performed to estimate generalization error by cross-validation. 2nd Method: Conservative Z. Our second method aims at overestimating Var[ ~~ iL:f] which will lead to conservative inference, that is tests of hypothesis with actual size less than the nominal size. This is important because techniques currently in use have the opposite defect, that is they tend to be liberal (tests with actual size exceeding the nominal size), which is typically regarded as less desirable than conservative tests. Estimating ~~ (7J unbiasedly is not trivial as hinted above. However we may estimate uJ be the unbiased unbiasedly nn?1 (7J = Var[ nn?1 it:fl where n~ = L!!2 J - n2 < nl. Let n? n1 estimator, developed below, of the above variance. We argued in the previous section that Var[ ~~ it:fl ~ Var[ ~~ iL:fl. Therefore ~;uJ will tend to overestimate ~~(7J, that is E[ n2a-2] = n2(72 > n2(72 n; J n; J - nl J' Here's how we may estimate ~? (7J without bias. For simplicity, assume that n is even. 1 We have to randomly split our data into two distinct data sets, Dl and D1, of size ~ each. Let iL(1) be the statistic of interest ( ~; iL:f) computed on D 1 . This involves, among other things, drawing J train/test subsets from DI . Let iL(l) be the statistic computed on D 1? Then iL(l) and iL(l) are independent since Dl and Dl are independent data sets, so A it(I )+it(1))2 it(!)+it(I))2 I(A AC)2' b' ed . that (/-L(l) 2 + (AC J.L(I) 2 = 2" /-L(l) - /-L(l) IS an un las estImate of ~?(7J. This splitting process may be repeated M times. This yields Dm and D~, with Zr 1 Inference for the Generalization Error Dm U D~ 311 = zf, Dm n D~ = 0 for m = 1, ... , M. Each split yields a pair (it(m) , it(m?) that is such that ~(it(m) - it(m?)2 is unbiased for ~~U}. This allows us to use the following unbiased estimator of ~? U}: 1 ~2 _ n2 n~ U J - M 1 ""' (~ ~ c )2 2M L..J J-t(m) - J-t(m) . (8) m=1 Note that, according to Lemma 1, Var[ ~~oj] = t Var[(it(m) - it(m?)2] (r + IMr) with r = Corr[(it(i) - it(i?)2, (it(j) - it(j?)2] for i i- j. Simulations suggest that r is usually close to 0, so that the above variance decreases roughly like for M up to 20, say. The second method is therefore a bit more computation intensive, since requires to perform cross-validation M times, but it is expected to be conservative. k 4 Simulation study We consider five different test statistics for the hypothesis Ho : niJ-t = J-to. The first three are methods already in use in the machine learning community, the last two are the new methods we put forward. They all have the following form reject Ho if Iit ~J-to I > c. (9) Table 1 describes what they are 1. We performed a simulation study to investigate the size (probability of rejecting the null hypothesis when it is true) and the power (probability of rejecting the null hypothesis when it is false) of the five test statistics shown in Table 1. We consider the problem of estimating generalization errors in the Letter Recognition classification problem (available from www. ics. uci . edu/pub/machine-learning-databases). The learning algorithms are 1. Classification tree We used the function tree in Splus version 4.5 for Windows. The default arguments were used and no pruning was performed. The function predict with option type="class" was used to retrieve the decision function of the tree: FA (Zs)(X). Here the classification loss function LAU,i) = I[FA(Zsj)(X i ) i- Yi ] is equal to 1 whenever this algorithm misclassifies example i when the training set is Sj; otherwise it is O. 2. First nearest neighbor We apply the first nearest neighbor rule with a distorted distance metric to pun down the performance of this algorithm to the level of the classification tree (as in (Dietterich, 1998?. We have LBU, i) equal to 1 whenever this algorithm misclassifies example i when the training set is Sj; otherwise it is O. In addition to inference about the generalization errors ni J-tA and ni J-tB associated with those two algorithms, we also consider inference about niJ-tA-B = niJ-tA - niJ-tB = E[LA-B(j,i)] whereLA_B(j,i) = LAU,i) - LB(j,i). We sample, without replacement, 300 examples from the 20000 examples available in the Letter Recognition data base. Repeating this 500 times, we obtain 500 sets of data of the form {ZI,"" Z300}. Once a data set = {ZI,'" Z300} has been generated, we may zloO lWhen comparing two classifiers, (Nadeau and Bengio, 1999) show that the t-test is closely related to McNemar's test described in (Dietterich, 1998). The 5 x 2 cv procedure was developed in (Dietterich, 1998) with solely the comparison of classifiers in mind but may trivially be extended to other problems as shown in (Nadeau and Bengio, 1999). C. Nadeau and Y. Bengio 312 c II II Name n2 AOO nl/-Ll n2 AOO ~2 SV(L(I, i)) yO-:.l t n2 - 1,1-ar/2 t J - 1,1-ar/2 see (Dietterich, 1998) t S ,1-ar/2 1: conservative Z n/2/-Ll n2 AOO 2: corr. resampled t n2 AOO nl /-L J t-test (McNemar) resampled t nl/-LJ n(2 AOO Dietterich's 5 x 2 cv nl/-LJ n2 A2 n'UJ 1 Zl-ar/2 2 (!.J + ~) nl 0- tJ-l,1-ar/2 n2IT3+~ITO-IT3) ITn -IT3 I+J~ >1 >1 " ? ~~IT? < 1 n,u J l+JE l+J~ Table 1: Description of five test statistics in relation to the rejection criteria shown in (9). Zp and h,p refer to the quantile p of the N(O, 1) and Student tk distribution respectively. 0- 2 is as defined above (7) and SV (L(I, i)) is the sample variance of the L(I, i)'s involved in ~i {l{'. The ~t~~l ratio (which comes from proper application of Lemma 1, except for Dietterich's 5 x 2 cv and the Conservative Z) indicates if a test will tend to be conservative (ratio less than 1) or liberal (ratio greater than 1). perform hypothesis testing based on the statistics shown in Table 1. A difficulty arises 300 here), those methods don't aim at inference for the same however. For a given n (n generalization error. For instance, Dietterich's 5 x 2 cv test aims at n/2/-L, while the others aim at nl/-L where nl would usually be different for different methods (e.g. nl = 23n for the t test statistic, and nl = ~~ for the resampled t test statistic, for instance). In order to compare the different techniques, for a given n, we shall always aim at n/2/-L, i.e. use nl = ?-. However, for statistics involving ~ip.r with J > 1, normal usage would call for nl to be 5 or 10 times larger than n2, not nl = n2 = ?-. Therefore, for those statistics, we = p.r l~ so that ~ 5. To obtain ~~;o we simply throw out 40% also use nl = ?- and n2 of the data. For the conservative Z, we do the variance calculation as we would normally = = for instance) to obtain ~i2-n2a-J = ;~~~a-J. However, in the numerator we n/2AOO d n2 AOO n/lOAoo' d f n2 AOO I' db compute b oth n/2/-LJ an n/2/-LJ n/2 /-LJ mstea 0 n-n2/-LJ' as exp rune a ove. do (n2 = l~ Note that the rationale that led to the conservative Z statistics is maintained, that is ;~~~a-J 2A OO] . bot h TT E [n/lOA2] overestimates var [n/lOAOO] n/2 /-LJ an d TT var [n/ n/2/-LJ: 2n/s u J > _ TT var [n/lOAOO] n/2 /-LJ > [n/2 00] var n/2/-LJ . TT A Figure 1 shows the estimated power of different statistics when we are interested in /-LA and /-LA-B. We estimate powers by computing the proportion of rejections of Ho . We see that tests based on the t-test or resampled t-test are liberal, they reject the null hypothesis with probability greater than the prescribed a = 0.1, when the null hypothesis is true. The other tests appear to have sizes that are either not significantly larger the 10% or barely so. Note that Dietterich's 5 x 2cv is not very powerful (note that its curve has the lowest power on the extreme values of muo). To make a fair comparison of power between two curves, one should mentally align the size (bottom of the curve) of these two curves. Indeed, even the resampled t-test and the conservative Z that throw out 40% of the data are more powerful. That is of course due to the fact that the 5 x 2 cv method uses J = 1 instead of J = 15. This is just a glimpse of a much larger simulation study. When studying the corrected resampled t-test and the conservative Z in their natural habitat (nl = ~9 and n2 = l~)' we see that they are usually either right on the money in term of size, or slIghtly conservative. Their powers appear equivalent. The simulations were performed with J up to 25 and M up to 20. We found that taking J greater than 15 did not improve much the power of the Inference for the Generalization Error 313 Figure 1: Powers of the tests about Ho : /-LA = /-Lo (left panel) and Ho : /-LA-B = /-Lo (right panel) at level a = 0.1 for varying /-Lo. The dotted vertical lines correspond to the 95% confidence interval for the actual/-LA or /-LA-B. therefore that is where the actual size of the tests may be read. The solid horizontal line displays the nominal size of the tests. i.e. 10%. Estimated probabilities of rejection laying above the dotted horizontal line are significatively greater than 10% (at significance level 5%). Solid curves either correspond to the resampled t-test or the corrected resampled t-test. The resampled t-test is the one that has ridiculously high size. Curves with circled points are the versions of the ordinary and corrected resampled t-test and conservative Z with 40% of the data thrown away. Where it matters J = 15. M = 10 were used. statistics. Taking M = 20 instead of M = 10 does not lead to any noticeable difference in the distribution of the conservative Z. Taking M = 5 makes the statistic slightly less conservative. See (Nadeau and Bengio. 1999) for further details. 5 Conclusion This paper addresses a very important practical issue in the empirical validation of new machine learning algorithms: how to decide whether one algorithm is significantly better than another one. We argue that it is important to take into account the variability due to the choice of training set. (Dietterich. 1998) had already proposed a statistic for this purpose. We have constructed two new variance estimates of the cross-validation estimator of the generalization error. These enable one to construct tests of hypothesis and confidence intervals that are seldom liberal. Furthermore. tests based on these have powers that are unmatched by any known techniques with comparable size. One of them (corrected resampled t-test) can be computed without any additional cost to the usual K-fold crossvalidation estimates. The other one (conservative Z) requires M times more computation. where we found sufficiently good values of M to be between 5 and 10. References Breiman, L. (1996). Heuristics of instability and stabilization in model selection. Annals of Statistics, 24 (6):2350-2383. Dietterich, T. (1998). Approximate statistical tests for comparing supervised classification learning algorithms. Neural Computation, 10 (7):1895-1924. Hinton. G., Neal. R., Tibshirani, R., and DELVE team members (1995). Assessing learning procedures using DELVE. Technical report, University of Toronto, Department of Computer Science. Nadeau, C. and Bengio, Y. (1999). Inference for the generalisation error. Technical Report in preparation, CIRANO. 

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Application of Blind Separation of Sources to Optical Recording of Brain Activity Holger Schoner, Martin Stetter, Ingo Schie61 Department of Computer Science Technical University of Berlin Germany {hjsch,moatl,ingos}@cs.tu-berlin.de John E. W. Mayhew University of Sheffield, UK j. e.mayhew@sheffield.ac.uk Jennifer S. Lund, Niall McLoughlin Klaus Obermayer Institute of Ophthalmology University College London, UK {j.lund,n.mcloughlin}@ucl.ac.uk Department of Computer Science, Technical University of Berlin, Germany oby@cs.tu-berlin.de Abstract In the analysis of data recorded by optical imaging from intrinsic signals (measurement of changes of light reflectance from cortical tissue) the removal of noise and artifacts such as blood vessel patterns is a serious problem. Often bandpass filtering is used, but the underlying assumption that a spatial frequency exists, which separates the mapping component from other components (especially the global signal), is questionable. Here we propose alternative ways of processing optical imaging data, using blind source separation techniques based on the spatial decorre1ation of the data. We first perform benchmarks on artificial data in order to select the way of processing, which is most robust with respect to sensor noise. We then apply it to recordings of optical imaging experiments from macaque primary visual cortex. We show that our BSS technique is able to extract ocular dominance and orientation preference maps from single condition stacks, for data, where standard post-processing procedures fail. Artifacts, especially blood vessel patterns, can often be completely removed from the maps. In summary, our method for blind source separation using extended spatial decorrelation is a superior technique for the analysis of optical recording data. 1 Introduction One approach in the attempt of comprehending how the human brain works is the analysis of neural activation patterns in the brain for different stimuli presented to a sensory system. An example is the extraction of ocular dominance or orientation preference maps from recordings of activity of neurons in the primary visual cortex of mammals. A common technique for extracting such maps is optical imaging (01) of intrinsic signals. Currently this is the imaging technique with the highest spatial resolution (~ 100 J1m) for mapping of the cortex. This method is explained e.g. in [1], for similar methods using voltage sensitive dyes see [2, 3] . 01 uses changes in light reflection to estimate spatial patterns of stimulus 950 H. Sch6ner. M Stetter. I. Schiej3l, J E. Mayhew, J Lund, N. Mcloughlin and K. Obermayer answers. The overall change recorded by a CCD or video camera is the total signal. The part of the total signal due to local neural activity is called the mapping component and it derives from changes in deoxyhemoglobin absorption and light scattering properties of the tissue. Another component of the total signal is a "global" component, which is also correlated with stimulus presentation, but has a much coarser spatial re~olution . It derives its part from changes in the blood volume with the time. Other components are blood vessel artifacts, the vasomotor signal (slow oscillations of neural activity), and ongoing activity (spontaneous, stimulus-uncorrelated activity). Problematic for the extraction of activity maps are especially blood vessel artifacts and sensor noise, such as photon shot noise. A procedure often used for extracting the activity maps from the recordings is bandpass filtering, after preprocessing by temporal , spatial , and trial averaging. Lowpass filtering is unproblematic, as the spatial resolution of the mapping signal is limited by the scattering properties of the brain tissue, hence everything above a limiting frequency must be noise. The motivation for highpass filtering, on the other hand, is questionable as there is no specific spatial frequency separating local neural activity patterns and the global signal [4]). A different approach, Blind Source Separation (BSS), models the components of the recorded image frames as independent sources, and the observations (recorded image frames) as noisy linear mixtures of the unknown sources. After performing the BSS the mapping component should ideally be concentrated in one estimated source, the global signal in another, and blood vessel artifacts, etc. in further ones. Previous work ([5]) has shown that BSS algorithms, which are based on higher order statistics ([6, 7, 8]), fail for optical imaging data, because of the high signal to noise ratio. In this work we suggest and investigate versions of the M&S algorithm [9, 10], which are robust against sensor noise, and we analyze their performance on artificial as well as real optical recording data. In section 2 we describe an improved algorithm, which we later compare to other methods in section 3. There an artificial data set is used for the analysis of noise robustness, and benchmark results are presented. Then, in section 4, it is shown that the newly developed algorithm is very well able to separate the different components of the optical imaging data, for ocular dominance as well as orientation preference data from monkey striate cortex. Finally, section 5 provides conclusions and perspectives for future work. 2 Second order blind source separation Let m be the number of mixtures and r the sample index, i.e. a vector specifying a pixel in the recorded images. The observation vectors y(r) = (Y1(r) , ... ,Ym?')f are assumed to be linear mixtures of m unknown sources s(r) = (Sl (r) , . . . ,Sm (r)) with A being the m x m mixing matrix and n describing the sensor noise: y(r) = As(r) + n (1) The goal of BSS is to obtain optimal source estimates s(r) under the assumption that the original sources are independent. In the noiseless case W = A -1 would be the optimal demixing matrix. In the noisy case, however, W also has to compensate for the added W . A . s(r) + W . n. BSS algorithms are generally only able to noise: s(r) == Wy(r) recover the original sources up to a permutation and scaling. = Extended Spatial Decorrelation (ESD) uses the second order statistics of the observations to find the source estimates. If sources are statistical independent all source crosscorrelations Ci(,~) (D.r) = (si (r)Sj(r+ D.r))r = ~ LSi (r)Sj(r+ D.r) r , i =F j (2) Application of BSS to Optical Recording of Brain Activity 951 must vanish for all shifts ~r, while the autocorrelations (i = j) of the sources remain (the variances). Note that this implies that the sources must be spatially smooth. Motivated by [to] we propose to optimize the cost function, which is the sum of the squared cross-correlations of the estimated sources over a set of shifts {~r}, E(W) = LL ((WC(~r)WT)i,jr (3) 6r i~j = L L \Si(r)Sj(r + ~r))~ , 6r i t j with respect to the demixing matrix W. The matrix Ci,j(~r) = (Yi (r)Yj(r + ~r))r denotes the mixture cross-correlations for a shift ~r. This cost function is minimized using the Polak Ribiere Conjugate Gradient technique, where the line search is substituted by a dynamic step width adaptation ([11]). To keep the demixing matrix W from converging to the zero matrix, we introduce a constraint which keeps the diagonal elements of T = W-l (in the noiseless case and for non-sphered data T is an estimate of the mixing matrix, with possible permutations) at a value of 1.0. Convergence properties are improved by sphering the data (transforming their correlation matrix for shift zero to an identity matrix) prior to decorrelating the mixtures. Note that use of multiple shifts ~r allows to use more information about the auto- and cross-correlation structure of the mixtures for the separation process. Two shifts provide just enough constraints for a unique solution ([to]). Multiple shifts, and the redundancy they introduce, additionally allow to cancel out part of the noise by approximate simultaneous diagonalization of the corresponding cross correlation matrices. In the presence of sensor noise, added after mixing, the standard sphering technique is problematic. When calculating the zero-shift cross-correlation matrix the variance of the noise contaminates the result, and sphering using a shifted cross-correlation matrix, is recommended ([12]). For spatially white sensor noise and sources with reasonable auto correlations this technique is more appropriate. In the following we denote the standard algorithm by dpaO, and the variant using noise robust sphering by dpa1. 3 Benchmarks for artificial data The artificial data set used here, whose sources are approximately uncorrelated for all shifts, is shown in the left part of figure 1. The mixtures were produced by generating a random mixing matrix (in this case with condition number 3.73), applying it to the sources, and finally adding white noise of different variances. In order to measure the performance on the artificial data set we measure a reconstruction error (RE) between the estimated and the correct sources via (see [l3]): RE(W) = od(L ?(r)sT(r)) , r od(C) 1 ~ 1 =N ~ N _ I 1 ( L J IC? ?1 maXk'I~i ,kl - ) 1 (4) The correlation between the real and the estimated sources (the argument to "od"), should be close to a permutation matrix, if the separation is successful. If the maxima of two rows are in the same column, the separation is labeled unsuccessful. Otherwise, the normalized absolute sum of non-permutation (cross-correlation) elements is computed and returned as the reconstruction error. We now compare the method based on optimization of (3) by gradient descent with the following variants of second order blind source separation: (1) standard spatial decorrelation 952 H. Schaner; M Stetter; I. Schiej3l, J. E. Mayhew, J. Lund, N. McLoughlin and K. Obermayer -~-.-- opt 0.5 g ~0.4 ? gOJ 0.5 mean . ~. <; JiO.4 cor -. '. l:l ~ hI''I, c: 0 '' ?. 80.2 " ': " H, ~02 0:: '. 10 15 20 Signal to Noise Ratio (dB) " ~ 0.1 25 00 - jacO jacl dpaO dpal % ~ , ~ :. , '!i ___ 5 , 1 b 0.1 o0 .__ . ... gO.3 -" 0:: -..., 5 -- 10 ----15 20 25 Signal to Noi se Ratio (dB) Figure 1: The set of three approximately uncorrelated source images of the artificial data set (left) . The two plots (middle, right) show the reconstruction error versus signal to noise ratio for different separation algorithms. In the right plot jac1 and dpa1 are very close together. using the optimal single shift yielding the smallest reconstruction error (opt). (2) Spatial decorrelation using the shift selected by .6.rcor = argmax{.D.r} norm (C(.6.r) - diag (C(.6.r))) norm (diag (C(.6.r)))?' (5) where "diag" sets all off-diagonal elements of its argument matrix to zero, and "norm" computes the largest singular value of its argument matrix (cor). .6.rcor is the shift for which the cross correlations are largest, i.e. whose signal to noise ratio (SNR) should be best. (3) Standard spatial decorrelation using the average reconstruction error for all successful shifts in a 61 x 61 square around the zero shift (mean). (4) A multi-shift algorithm ([12]), using several elementary rotations (Jacobi method) to build an orthogonal demixing matrix, which optimizes the cost function (3). The variants using standard sphering and noise robust sphering are denoted by dacO) and dac1). cor, opt, and mean use two shifts for their computation; but as one of those is always the zero-shift, there is only one shift to choose and they are called single-shift algorithms here. Figure 1 gives two plots which show the reconstruction error (4) versus the SNR (measured in dB) for single shift (middle) and multi-shift (right) algorithms. The error bars indicate twice the standard error of the mean (2x SEM), for 10 runs with the same mixing matrix, but newly generated noise of the given noise level. In each of these runs, the best result of three was selected for the gradient descent method. This is because, contrary to the other algorithms, the gradient descent algorithm depends on the initial choice of the demixing matrix. All multi-shift algorithms (all except opt and mean), used 8 shifts (?r, ?r), (?r, 0), and (0, ?r) for each r E {I, 3, 5, 10,20, 30}, so 48 all together. Several points are noticeable in the plots. (i) The cor algorithm is generally closer to the optimum than to the average successful shift. (ii) A comparison between the two plots shows that the multi-shift algorithms (right plot) are able to perform much better than even the optimal single-shift method. For low to medium noise levels this is even the case when using the standard sphering method combined with the gradient descent algorithm. (iii) The advantage of the noise robust sphering method, compared to the standard sphering, is obvious: the reconstruction error stays very low for all evaluated noise levels, for both the jac1 and dpa1 algoritlnns. (iv) The gradient descent technique is more robust than the Jacobi method For the standard sphering its performance is much better than that of the Jacobi method. Figure 1 shows results which were produced using a single mixing matrix. However, our simulations show that the algorithms compare qualitatively similar when using mixing ma- 953 Application of BSS to Optical Recording ofBrain Activity t = 1 sec. t = 2 sec. t = 3 sec. t = 4 sec. t = 5 sec. t = 6 sec. t = 7 sec. Figure 2: Optical imaging stacks. The top stack is a single condition stack from ocular dominance experiments, the lower one a difference stack from orientation preference experiments (images for 90? gratings subtracted from those for 0? gratings). The stimulus was present during recording images 2-7 in each row. Two large blood vessels in the top and left regions of the raw images were masked out prior to the analysis. trices with condition numbers between 2 and 10. The noise robust versions of the multishift algorithms generally yield the best separation results of all evaluated algorithms. 4 Application to optical imaging We now apply extended spatial decorrelation to the analysis of optical imaging data. The data consists of recordings from the primary visual cortex of macaque monkeys. Each trial lasted 8 seconds, which were recorded with frame rates of 15 frames per second. A visual stimulus (a drifting bar grating of varying orientation) was presented between seconds 2 and 8. Trials were separated by a recovery period of 15 seconds without stimulation. The cortex was illuminated at a wavelength of 633 nm. One pixel corresponds to about 15 J.Lm on the cortex; the image stacks used for further processing, consisting of 256 x 256 pixels, covered an area of cortex of approximately 3.7 mm 2 . Blocks of 15 consecutive frames were averaged, and averaging over 8 trials using the same visual stimulus further improved the SNR. First frame analysis (subtraction of the first, blank, frame from the others) was then applied to the resulting stack of 8 frames, followed by lowpass filtering with 14 cycles/mm. Figure 2 shows the resulting image stacks for an ocular dominance and an orientation preference experiment. One observes strong blood vessel artifacts (particularly in the top row of images), which are superimposed to the patchy mapping component that pops up over time. Figure 3 shows results obtained by the application of extended spatial decorrelation (using dpaO). Only those estimated sources containing patterns different from white noise are shown. Backprojection of the estimated sources onto the original image stack yields the amplitude time series of the estimated sources, which is very useful in selecting the mapping component: it can be present in the recordings only after the stimulus onset (starting at t = 2 sec.). The middle part shows four estimated sources for the ocular dominance single condition stack. The mapping component (first image) is separated from the global component (second image) and blood vessel artifacts (second to fourth) quite well. The time course of the mapping component is plausible as well: calculation of a plausibility index (sum of squared differences between the normalized time series and a step function, which is 0 before and 1 after the stimulus onset) gives 0.5 for the mapping component and 2.31 for the next best one. Results for the gradient descent algorithm are similar for this data set, regardless of the sphering technique used. The Jacobi method also gives similar results, but a small blood vessel artifact is remaining in the resulting map. The cor algorithm usually gives much worse separation results. In the right part of figure 3 two es- 954 H Schaner, M Stetter, I SchieJ3l, 1. E. Mayhew, 1. Lund, N McLoughlin and K. Obermayer Figure 3: Left: Summation technique for ocular dominance (aD) experiment (upper) and orientation preference (OP) experiment (lower). Middle, Right: dpaO algorithm applied to the same aD single condition (middle) and OP (right) stacks. The images show the 4 (aD) and 2 (OP) estimated components, which are visually different from white noise. In the bottom row the respective time courses of the estimated sources are given. timated sources (those different from white noise) for the orientation preference difference stack can be seen. Here the proposed algorithm (dpaO) again works very well (plausibility index is 0.56 for mapping component, compared to 3.04 for the best other component). It generally has to be applied a few times (usually around 3 times) to select the best separation result Uudging by visual quality of the separation and the time courses of the estimated sources), because of its dependence on parameter initialization; in return it yields the best results of all algorithms used, especially when compared to the traditional summation technique. The similar results when using standard and noise robust sphering, and the small differences between the gradient descent and the Jacobi algorithms indicate, that not sensor noise is the limiting factor for the quality of the extracted maps. Instead it seems that, assuming a linear mixing model, no better results can be obtained from the used image stacks. It will remain for further research to analyze, how appropriate the linear mixing model is, and whether the underlying biophysical components are sufficiently uncorrelated . In the meantime the maps obtained by the ESD algorithm are superior to those obtained using conventional techniques like summation of the image stack. 5 Conclusion The results presented in the previous sections show the advantages of the proposed algorithm: In the comparison with other spatial decorrelation algorithms the benefit in using multiple shifts compared to only two shifts is demonstrated. The robustness against sensor noise is improved, and in addition, the selection of multiple shifts is less critical than selecting a single shift, as the resulting multi-shift system of equations contains more redundancy. In comparison with the Jacobi method, which is restricted to find only orthogonal demixing matrices, the greater tolerance of demixing by a gradient descent technique concerning noise and incorrect sphering are demonstrated. The application of second order blind separation of sources to optical imaging data shows that these techniques represent an important alternative to the conventional approach, bandpass filtering followed by summation of the image stack, for extraction of neural activity maps. Vessel artifacts can be separated from the mapping component better than using classical approaches. The spatial decorrelation algorithms are very well adapted to the optical imaging task, because of their use of spatial smoothness properties of the mapping and other biophysical components. An important field for future research concerning BSS algorithms is the incorporation of prior knowledge about sources and the mixing process, e.g. that the mixing has to be causal: the mapping signal cannot occur before the stimulus is presented. Assumptions Application ofBSS to Optical Recording of Brain Activity 955 about the time course of signals could also be helpful, as well as knowledge about their spatial statistics. Smearing and scattering limit the resolution of recordings of biological components, and, depending on the wavelength of the light used for illumination, the mapping component constitutes only a certain percentage of the changes in total light reflections. Acknowledgments This work has been supported by the Wellcome Trust (050080IZJ97). References [I] T. Bonhoeffer and A. Grinvald. Optical imaging based on intrinsic signals: The methodology. In A. Toga and J. C. Maziotta, editors, Brain mapping: The methods, pages 55-97, San Diego, CA, 1996. Academic Press, Inc. [2] G. G. Blasdel and G. Salama. Voltage-sensitive dyes reveal a modular organization in monkey striate cortex. Nature, 321 :579-585, 1986. [3] G. G. Blasdel. Differential imaging of ocular dominance and orientation selectivity in monkey striate cortex. 1. Neurosci., 12:3115-3138, 1992. [4] M. Stetter, T. Otto, T. Mueller, F. Sengpiel, M . Huebener, T. Bonhoeffer, and K. Obermayer. Temporal and spatial analysis of intrinsic signals from cat visual cortex. Soc. Neurosci. Abstr., 23:455,1997. [5] I. SchieGl, M. Stetter, J. E. W. Mayhew, S. Askew, N. McLoughlin, J. B. Levitt, J. S. Lund, and K. Obermayer. Blind separation of spatial signal patterns from optical imaging records. In J .-F. Cardoso, C. Jutten, and P. Loubaton, editors, Proceedings of the lCA99 workshop, volume I, pages 179-184, 1999. [6] A. J. Bell and T. J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7: 1129-1159, 1995. [7] S. Amari . Neural learning in structured parameter spaces - natural riemannian gradient. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, volume 9, 1996. [8] A. Hyvtlrinen and E. Oja. A fast fixed point algorithm for independent component analysis. Neural Comput., 9:1483-1492,1997. [9] J. C. Platt and F. Faggin. Networks for the separation of sources that are superimposed and delayed. In 1. E. Moody, S. 1. Hanson, and R. P. Lippmann, editors, Advances in Neurallnformation Processing Systems, volume 4, pages 730--737, 1991. [10] L. Molgedey and H. G. Schuster. Separation of a mixture of independent signals using time delayed correlations. Phys. Rev. Lett., 72:3634-3637, 1994. [II] S. M. Riiger. Stable dynamic parameter adaptation. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, editors , Advances in Neural Information Processing Systems., volume 8, pages 225231 . MIT Press Cambridge, MA, 1996. [12] K.-R. Miiller, Philips P, and A. Ziehe. Jadetd: Combining higher-order statistics and temporal information for Blind Source Separation (with noise). In J.-F. Cardoso, C. Jutten, and P. Loubaton, editors, Proceedings of the 1. lCA99 Workshop, Aussois, volume I, pages 87-92, 1999. [13] B.-U. Koehler and R. Orglmeister. Independent component analysis using autoregressive models. In 1.-F. Cardoso, C. Jutten, and P. Loubaton, editors, Proceedings of the lCA99 workshop, volume I, pages 359-363, 1999. 

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Model Selection for Support Vector Machines Olivier Chapelle*,t, Vladimir Vapnik* * AT&T Research Labs, Red Bank, NJ t LIP6, Paris, France { chapelle, vlad} @research.au.com Abstract New functionals for parameter (model) selection of Support Vector Machines are introduced based on the concepts of the span of support vectors and rescaling of the feature space. It is shown that using these functionals, one can both predict the best choice of parameters of the model and the relative quality of performance for any value of parameter. 1 Introduction Support Vector Machines (SVMs) implement the following idea: they map input vectors into a high dimensional feature space, where a maximal margin hyperplane is constructed [6]. It was shown that when training data are separable, the error rate for SVMs can be characterized by (1) where R is the radius ofthe smallest sphere containing the training data and M is the margin (the distance between the hyperplane and the closest training vector in feature space). This functional estimates the VC dimension of hyperplanes separating data with a given margin M. To perform the mapping and to calculate Rand M in the SVM technique. one uses a positive definite kernel K(x, x') which specifies an inner product in feature space. An example of such a kernel is the Radial Basis Function (RBF). K(x, x') = e-llx-x'II2/20'2. This kernel has a free parameter (7 and more generally, most kernels require some parameters to be set. When treating noisy data with SVMs. another parameter. penalizing the training errors. also needs to be set. The problem of choosing the values of these parameters which minimize the expectation of test error is called the model selection problem. It was shown that the parameter of the kernel that minimizes functional (1) provides a good choice for the model: the minimum for this functional coincides with the minimum of the test error [1]. However. the shapes of these curves can be different. In this article we introduce refined functionals that not only specify the best choice of parameters (both the parameter of the kernel and the parameter penalizing training error). but also produce curves which better reflect the actual error rate. 231 Model Selection for Support Vector Machines The paper is organized as follows. Section 2 describes the basics of SVMs, section 3 introduces a new functional based on the concept of the span of support vectors, section 4 considers the idea of rescaling data in feature space and section 5 discusses experiments of model selection with these functionals. 2 Support Vector Learning We introduce some standard notation for SVMs; for a complete description, see [6]. Let (Xi, Yih <i<l be a set of training examples, Xi E jRn which belong to a class labeled by Yi E {- f, f}. The decision function given by a SVM is : (2) where the coefficients a? are obtained by maximizing the following functional: l I t W(a) = Lai - i=l 2' L aiajYiYjK(Xi,Xj) (3) i,j=l under constraints t L aiYi = 0 and 0 ~ ai ~ C i = 1, ... , f. i=l C is a constant which controls the tradeoff between the complexity of the decision function and the number of training examples misclassified. SVM are linear maximal margin classifiers in a high-dimensional feature space where the data are mapped through a non-linear function <p(x) such that <P(Xi) . <p(Xj) = K(Xi,Xj). The points Xi with ai > 0 are called support vectors. We distinguish between those with o < ai < C and those with ai = C. We call them respectively support vectors of the first and second category. 3 Prediction using the span of support vectors The results introduced in this section are based on the leave-one-out cross-validation estimate. This procedure is usually used to estimate the probability of test error of a learning algorithm. 3.1 The leave-one-out procedure The leave-one-out procedure consists of removing from the training data one element, constructing the decision rule on the basis of the remaining training data and then testing the removed element. In this fashion one tests all f elements of the training data (using f different decision rules). Let us denote the number of errors in the leave-one-out procedure by ?(Xl' Yl, .. . , Xl, Yl) . It is known [6] that the the leave-one-out procedure gives an almost unbiased estimate of the probability of test error: the expectation of test error for the machine trained on f - 1 examples is equal to the expectation of 1?(Xl' Yl, ... , Xl, Yt). We now provide an analysis of the number of errors made by the leave-one-out procedure. For this purpose, we introduce a new concept, called the span of support vectors [7]. O. Chapelle and V. N. Vapnik 232 3.2 Span of support vectors Since the results presented in this section do not depend on the feature space, we will consider without any loss of generality, linear SVMs, i.e. K (Xi, Xj) = Xi . Xj. Suppose that 0? = (a?, .. ., a~) is the solution of the optimization problem (3). For any fixed support vector xp we define the set Ap as constrained linear combinations of the support vectors of the first category (Xi)i:;t:p : .t Ai = 1, 0 t=l , t#p ~ a? + Yiypa~Ai ~ c} . (4) Note that Ai can be less than O. We also define the quantity Sp, which we call the span of the support vector xp as the minimum distance between xp and this set (see figure 1) (5) t... .. AI 2= +inf t...3 = -inf ?? ? '' - - ' ~ .. 2,, Figure 1: Three support vectors with al = a2 = a3/2. The set Al is the semi-opened dashed line. It was shown in [7] that the set Ap is not empty and that Sp = d(xp, Ap) ~ Dsv, where D sv is the diameter of the smallest sphere containing the support vectors. Intuitively, the smaller Sp = d(xp, Ap) is, the less likely the leave-one-out procedure is to make an error on the vector xp' Formally, the following theorem holds : Theorem 1 [7J If in the leave-one-out procedure a support vector xp corresponding to o < a p < C is recognized incorrectly, then the following inequality holds aO > p - 1 Sp max(D, 1/.JC)? This theorem implies that in the separable case (C = (0), the number of errors made by the leave-one-out procedure is bounded as follows: ?(Xl' Yl, .'" Xl, Yl) ~ 2: p a~ maxp SpD = maxp SpD / M2 , because 2: a~ = 1/M2 [6]. This is already an improvement compared to functional (I), since Sp ~ Dsv. But depending on the geometry of the support vectors the value of the span Sp can be much less than the diameter D sv of the support vectors and can even be equal to zero. We can go further under the assumption that the set of support vectors does not change during the leave-one-out procedure, which leads us to the following theorem: Model Selection for Support Vector Machines 233 Theorem 2 If the sets of support vectors of first and second categories remain the same during the leave-one-out procedure. then for any support vector xp. the following equality holds: yp[fO(xp) - fP(x p)] = o~S; where fO and fP are the decisionfunction (2) given by the SVM trained respectively on the whole training set and after the point xp has been removed. The proof of the theorem follows the one of Theorem 1 in [7]. The assumption that the set of support vectors does not change during the leave-one-out procedure is obviously not satisfied in most cases. Nevertheless, the proportion of points which violate this assumption is usually small compared to the number of support vectors. In this case, Theorem 2 provides a good approximation of the result of the leave-one procedure, as pointed out by the experiments (see Section 5.1, figure 2). As already noticed in [1], the larger op is, the more "important" in the decision function the support vector xp is. Thus, it is not surprising that removing a point xp causes a change in the decision function proportional to its Lagrange multiplier op . The same kind of result as Theorem 2 has also been derived in [2], where for SVMs without threshold, the following inequality has been derived: yp(f?(xp) - fP(xp)) ~ o~K(xp,xp). The span Sp takes into account the geometry of the support vectors in order to get a precise notion of how "important" is a given point. The previous theorem enables us to compute the number of errors made by the leave-oneout procedure: Corollary 1 Under the assumption of Theorem 2, the test error prediction given by the leave-one-out procedure is (6) Note that points which are not support vectors are correctly classified by the leave-one-out procedure. Therefore t/. defines the number of errors of the leave-one-out procedure on the entire training set. Under the assumption in Theorem 2, the box constraints in the definition of Ap (4) can be removed. Moreover, if we consider only hyperplanes passing through the origin, the constraint E Ai = 1 can also be removed. Therefore, under those assumptions, the computation of the span Sp is an unconstrained minimization of a quadratic form and can be done analytically. For support vectors of the first category, this leads to the closed form S~ = l/(K Mpp, where Ksv is the matrix of dot products between support vectors of the first category. A similar result has also been obtained in [3] . s In Section 5, we use the span-rule (6) for model selection in both separable and nonseparable cases. 4 Rescaling As we already mentioned, functional (1) bounds the VC dimension of a linear margin classifier. This bound is tight when the data almost "fills" the surface of the sphere enclosing the training data, but when the data lie on a flat ellipsoid, this bound is poor since the radius of the sphere takes into account only the components with the largest deviations. The idea we present here is to make a rescaling of our data in feature space such that the radius of the sphere stays constant but the margin increases, and then apply this bound to our rescaled data and hyperplane. 0. Chapelle and V. N. Vapnik 234 Let us first consider linear SVMs, i.e. without any mapping in a high dimensional space. The rescaling can be achieved by computing the covariance matrix of our data and rescaling according to its eigenvalues. Suppose our data are centered and let ('PI' ... ,'Pn) be the normalized eigenvectors of the covariance matrix of our data. We can then compute the smallest enclosing box containing our data, centered at the origin and whose edges are parallels to ('PI' ... , 'Pn)' This box is an approximation of the smallest enclosing ellipsoid. The length of the edge in the direction 'P k is J-Lk = maxi IXi . 'P k I. The rescaling consists of the following diagonal transformation: D :x --t Dx = LJ-Lk(X' 'Pk) 'Pk' k = Let us consider Xi = D-I xi and w Dw. The decision function is not changed under this transformation since w . Xi = W . xi and the data Xi fill a box of side length 1. Thus, in functional (l), we replace R2 by 1 and 1/M2 by w2 . Since we rescaled our data in a box, we actually estimated the radius of the enclosing ball using the foo-norm instead of the classical f 2 -norm. Further theoretical works needs to be done to justify this change of norm. In the non-linear case, note that even if we map our data in a high dimensional feature space, they lie in the linear subspace spanned by these data. Thus, if the number of training data f is not too large, we can work in this subspace of dimension at most f. For this purpose, one can use the tools of kernel PCA [5] : if A is the matrix of normalized eigenvectors of the Gram matrix Kij = K (Xi, Xj) and (>'d the eigenvalues, the dot product Xi . 'P k is replaced by v'XkAik and W? 'Pk becomes v'XkL:i AikYiO'i. Thus, we can still achieve the diagonal transformation A and finally functional (1) becomes L k >.~ max Ark ~ (2: A ik YiO'i)2 . i 5 Experiments To check these new methods, we performed two series of experiments. One concerns the choice of (7, the width of the RBF kernel, on a linearly separable database, the postal database. This dataset consists of 7291 handwritten digit of size 16x16 with a test set of 2007 examples. Following [4], we split the training set in 23 subsets of 317 training examples. Our task consists of separating digit 0 to 4 from 5 to 9. Error bars in figures 2a and 3 are standard deviations over the 23 trials. In another experiment, we try to choose the optimal value of C in a noisy database, the breast-cancer database! . The dataset has been split randomly 100 times into a training set containing 200 examples and a test set containing 77 examples. Section 5.1 describes experiments of model selection using the span-rule (6), both in the separable case and in the non-separable one, while Section 5.2 shows VC bounds for model selection in the separable case both with and without rescaling. 5.1 Model selection using the span-rule In this section, we use the prediction of test error derived from the span-rule (6) for model selection. Figure 2a shows the test error and the prediction given by the span for different values of the width (7 of the RBF kernel on the postal database. Figure 2b plots the same functions for different values of C on the breast-cancer database. We can see that the method predicts the correct value of the minimum. Moreover, the prediction is very accurate and the curves are almost identical. I Available from http; I Ihorn. first. gmd. del "'raetsch/da ta/breast-cancer 235 Model Selection for Support Vector Machines 40,-----~---~r=_=_-""T==es=:'t=er=ro=r= ' l l - 35 Span prediction 30 25 g20 UJ 15 i" ", 10 5 ~6 -4 -2 0 Log sigma 2 4 (a) choice of (T in the postal database 6 o 2 4 6 8 10 12 Loge (b) choice of C in the breast-cancer database Figure 2: Test error and its prediction using the span-rule (6). The computation of the span-rule (6) involves computing the span Sp (5) for every support Yp!(xp)/a~, rather vector. Note, however, that we are interested in the inequality than the exact value of the span Sp. Thus, while minimizing Sp = d(xp, Ap), if we find a point x* E Ap such that d(xp, x*)2 ::; Yp! (xp ) / a~, we can stop the minimization because this point will be correctly classified by the leave-one-out procedure. S; ::; It turned out in the experiments that the time required to compute the span was not prohibitive, since it is was about the same than the training time. There is a noteworthy extension in the application of the span concept. If we denote by one hyperparameter of the kernel and if the derivative 8K(;~'Xi) is computable, then it e is possible to compute analytically 8 ~ aiS~~y;fO(x;) , which is the derivative of an upper bound of the number of errors made by the leave-one-out procedure (see Theorem 2). This provides us a more powerful technique in model selection. Indeed, our initial approach was to choose the value of the width (T of the RBF kernel according to the minimum of the span-rule. In our case, there was only hyperparamter so it was possible to try different values of (T. But, if we have several hyperparameters, for example one (T per component, _~ (Xk- X j,)2 2<T~ , it is not possible to do an exhaustive search on all the possible values of of the hyperparameters. Nevertheless, the previous remark enables us to find their optimal value by a classical gradient descent approach. K(x, x') = e k Preliminary results seem to show that using this approach with the previously mentioned kernel improve the test error significantely. 5.2 VC dimension with rescaling In this section, we perform model selection on the postal database using functional (1) and its rescaled version. Figure 3a shows the values of the classical bound R2 / M2 for different values of (T. This bound predicts the correct value for the minimum, but does not reflect the actual test error. This is easily understandable since for large values of (T, the data in input space tend to be mapped in a very flat ellipsoid in feature space, a fact which is not taken into account [4]. Figure 3b shows that by performing a rescaling of our data, we manage to have a much tighter bound and this curve reflects the actual test error, given in figure 2a. 0. Chape/le and V. N. Vapnik 236 120 18000'---~--~--~--r=~==~=.=~ 16000 - VC Dimension with rescali 100 14000 12000 80 E E 10000 ~ 8000 ~ 60 > 6000 40 '6 4000 20 2000 ~L-==~===c~~ ....., -4 -2 ____~__~__~ 0 Log sigma 2 (a) without rescaling 4 6 ~ -4 -2 0 Log sigma 2 4 6 (b) with rescaling Figure 3: Bound on the VC dimension for different values of ~ on the postal database. The shape of the curve with rescaling is very similar to the test error on figure 2. 6 Conclusion In this paper, we introduced two new techniques of model selection for SVMs. One is based on the span, the other is based on rescaling of the data in feature space. We demonstrated that using these techniques, one can both predict optimal values for the parameters of the model and evaluate relative performances for different values of the parameters. These functionals can also lead to new learning techniques as they establish that generalization ability is not only due to margin. Acknowledgments The authors would like to thank Jason Weston and Patrick Haffner for helpfull discussions and comments. References [1] C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):121-167, 1998. [2] T. S. Jaakkola and D. Haussler. Probabilistic kernel regression models. In Proceedings of the J999 Conference on AI and Statistics, 1999. [3] M. Opper and O. Winther. Gaussian process classification and SVM: Mean field results and leave-one-out estimator. In Advances in Large Margin Classifiers. MIT Press, 1999. to appear. [4] B. SchOlkopf, J. Shawe-Taylor, A. 1. Smola, and R. C. Williamson. Kernel-dependent Support Vector error bounds. In Ninth International Conference on Artificial Neural Networks, pp. 304 309 [5] B. SchOlkopf, A. Smola, and K.-R. Muller. Kernel principal component analysis. In Artificial Neural Networks -ICANN'97, pages 583 - 588, Berlin, 1997. Springer Lecture Notes in Computer Science, Vol. 1327. [6] V. Vapnik. Statistical Learning Theory. Wiley, New York, 1998. [7] V. Vapnik and O. Chapelle. Bounds on error expectation for SVM. Neural Computation, 1999. Submitted. 

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Approximate Planning in Large POMDPs via Reusable Trajectories Michael Kearns AT&T Labs mkearns@research.att.com Yishay Mansour Tel Aviv University mansour@math.tau.ac.il AndrewY. Ng UC Berkeley ang@cs.berkeley.edu Abstract We consider the problem of reliably choosing a near-best strategy from a restricted class of strategies TI in a partially observable Markov decision process (POMDP). We assume we are given the ability to simulate the POMDP, and study what might be called the sample complexity that is, the amount of data one must generate in the POMDP in order to choose a good strategy. We prove upper bounds on the sample complexity showing that, even for infinitely large and arbitrarily complex POMDPs, the amount of data needed can be finite, and depends only linearly on the complexity of the restricted strategy class TI, and exponentially on the horizon time. This latter dependence can be eased in a variety of ways, including the application of gradient and local search algorithms. Our measure of complexity generalizes the classical supervised learning notion of VC dimension to the settings of reinforcement learning and planning. 1 Introduction Much recent attention has been focused on partially observable Markov decision processes (POMDPs) which have exponentially or even infinitely large state spaces. For such domains, a number of interesting basic issues arise. As the state space becomes large, the classical way of specifying a POMDP by tables of transition probabilities clearly becomes infeasible. To intelligently discuss the problem of planning - that is, computing a good strategy 1 in a given POMDP - compact or implicit representations of both POMDPs, and of strategies in POMDPs, must be developed. Examples include factored next-state distributions [2, 3, 7], and strategies derived from function approximation schemes [8]. The trend towards such compact representations, as well as algorithms for planning and learning using them, is reminiscent of supervised learning, where researchers have long emphasized parametric models (such as decision trees and neural networks) that can capture only limited structure, but which enjoy a number of computational and information-theoretic benefits. Motivated by these issues, we consider a setting were we are given a generative model, or lThroughout, we use the word strategy to mean any mapping from observable histories to actions, which generalizes the notion of policy in a fully observable MDP. M Kearns. Y. Mansour and A. Y. Ng 1002 simulator, for a POMDP, and wish to find a good strategy 7r from some restricted class of strategies II . A generative model is a "black box" that allows us to generate experience (trajectories) from different states of our choosing. Generative models are an abstract notion of compact POMDP representations, in the sense that the compact representations typically considered (such as factored next-state distributions) already provide efficient generative models. Here we are imagining that the strategy class II is given by some compact representation or by some natural limitation on strategies (such as bounded memory). Thus, the view we are adopting is that even though the world (POMDP) may be extremely complex, we assume that we can at least simulate or sample experience in the world (via the generative model), and we try to use this experience to choose a strategy from some "simple" class II. We study the following question: How many calls to a generative model are needed to have enough data to choose a near-best strategy in the given class? This is analogous to the question of sample complexity in supervised learning - but harder. The added difficulty lies in the reuse of data. In supervised learning, every sample (x, f(x)) provides feedback about every hypothesis function h(x) (namely, how close h(x) is to f(x)) . If h is restricted to lie in some hypothesis class 1i, this reuse permits sample complexity bounds that are far smaller than the size of 1i. For instance, only O(log(I1il)) samples are needed to choose a near-best model from a finite class 1i. If 1i is infinite, then sample sizes are obtained that depend only on some measure of the complexity of1i (such as VC dimension [9]), but which have no dependence on the complexity of the target function or the size of the input domain. In the POMDP setting, we would like analogous sample complexity bounds in terms of the "complexity" of the strategy class II - bounds that have no dependence on the size or complexity of the POMDP. But unlike the supervised learning setting, experience "reuse" is not immediate in POMDPs. To see this, consider the "straw man" algorithm that, starting with some 7r E II, uses the generative model to generate many trajectories under 7r, and thus forms a Monte Carlo estimate of V 7r (so). It is not clear that these trajectories under 7r are of much use in evaluating a different 7r' E II, since 7r and 7r' may quickly disagree on which actions to take. The naive Monte Carlo method thus gives 0(1111) bounds on the "sample complexity," rather than O(log(IIII)), for the finite case. In this paper, we shall describe the trajectory tree method of generating "reusable" trajectories, which requires generating only a (relatively) small number of trajectories - a number that is independent of the state-space size of the POMDP, depends only linearly on a general measure of the complexity of the strategy class II, and depends exponentially on the horizon time. This latter dependence can be eased via gradient algorithms such as Williams' REINFORCE [10] and Baird and Moore's more recent YAPS [1], and by local search techniques. Our measure of strategy class complexity generalizes the notion of VC dimension in supervised learning to the settings of reinforcement learning and planning, and we give bounds that recover for these settings the most powerful analogous results in supervised learning - bounds for arbitrary, infinite strategy classes that depend only on the dimension of the class rather than the size of the state space. 2 Preliminaries We begin with some standard definitions. A Markov decision process (MDP) is a tuple (S, So, A , {P (,1 s, a)}, R), where: S is a (possibly infinite) state set; So E S is a start state; A {al' . .. ,ad are actions; PC Is, a) gives the next-state distribution upon taking action a from state s; and the reward function R(s, a) gives the corresponding rewards. We assume for simplicity that rewards are deterministic, and further that they are bounded = Approximate Planning in Large POMDPs via Reusable Trajectories 1003 in absolute value by Rmax. A partially observable Markov decision process (POMDP) consists of an underlying MOP and observation distributions Q(ols) for each state s, where 0 is the random observation made at s. We have adopted the common assumption of a fixed start state,2 because once we limit the class of strategies we entertain, there may not be a single "best" strategy in the classdifferent start states may have different best strategies in II. We also assume that we are given a POMOP M in the form of a generative model for M that, when given as input any state-action pair (s, a), will output a state S' drawn according to P(?ls, a), an observation o drawn according to Q(?ls), and the reward R(s, a). This gives us the ability to sample the POMOP M in a random-access way. This definition may initially seem unreasonably generous: the generative model is giving us a fully observable simulation of a partially observable process. However, the key point is that we must still find a strategy that performs well in the partially observable setting. As a concrete example, in designing an elevator control system, we may have access to a simulator that generates random rider arrival times, and keeps track of the waiting time of each rider, the number of riders waiting at every floor at every time of day, and so on. However helpful this information might be in designing the controller, this controller must only use information about which floors currently have had their call button pushed (the observables). In any case, readers uncomfortable with the power provided by our generative models are referred to Section 5, where we briefly describe results requiring only an extremely weak form of partially observable simulation. At any time t, the agent will have seen some sequence of observations, 00,??., Ot, and will have chosen actions and received rewards for each of the t time We write its observable history as h steps prior to the current one. (( 00, ao, TO), ... , (Ot-l , at-I, Tt-l ), (Ot, _, _)). Such observable histories, also called trajectories, are the inputs to strategies. More formally, a strategy 7r is any (stochastic) mapping from observable histories to actions. (For example, this includes approaches which use the observable history to track the belief state [5].) A strategy class II is any set of strategies. We will restrict our attention to the case of discounted return,3 and we let, E [0,1) be the discount factor. We define the t::-horizon time to be HE = 10gl'(t::(1 - ,)/2Rmax ). Note that returns beyond the first HE-steps can contribute at most t::/2 to the total discounted return. Also, let Vmax = Rmax/(l - ,) bound the value function. Finally, for a POMDP M and a strategy class II, we define opt(M, II) = SUP7rEII V7r (so) to be the best expected return achievable from So using II. Our problem is thus the following: Given a generative model for a POMOP M and a strategy class II, how many calls to the generative model must we make, in order to have enough data to choose a 7r E II whose performance V7r(so) approaches opt(M, II)? Also, which calls should we make to the generative model to achieve this? 3 The Trajectory Tree Method We now describe how we can use a generative model to create "reusable" trajectories. For ease of exposition, we assume there are only two actions al and a2, but our results generalize easily to any finite number of actions. (See the full paper [6].) 2 An equivalent definition is to assume a fixed distribution D over start states, since So can be a "dummy" state whose next-state distribution under any action is D. 3The results in this paper can be extended without difficulty to the undiscounted finite-horizon setting [6]. M. Keams, Y. Mansour and A. Y. Ng 1004 A trajectory tree is a binary tree in which each node is labeled by a state and observation pair, and has a child for each of the two actions. Additionally, each link to a child is labeled by a reward, and the tree's depth will be H~, so it will have about 2H e nodes. (In Section 4, we will discuss settings where this exponential dependence on H~ can be eased.) Each trajectory tree is built as follows: The root is labeled by So and the observation there, 00 ' Its two children are then created by calling the generative model on (so, ad and (so, a2), which gives us the two next-states reached (say s~ and s~ respectively), the two observations made (say o~ and o~), and the two rewards received (r~ R(so, ad and r~ = R(so, a2). Then (s~ , aD and (s~, o~) label the root's aI-child and a2-child, and the links to these children are labeled r~ and r~. Recursively, we generate two children and rewards this way for each node down to depth H~ . = Now for any deterministic strategy tr and any trajectory tree T, tr defines a path through tr starts at the root, and inductively, if tr is at some internal node in T, then we feed to tr the observable history along the path from the root to that node, and tr selects and moves to a child of the current node. This continues until a leaf node is reached, and we define R( tr , T) to be the discounted sum of returns along the path taken. In the case that tr is stochastic, tr defines a distribution on paths in T, and R(tr , T) is the expected return according to this distribution. (We will later also describe another method for treating stochastic strategies.) Hence, given m trajectory trees T 1 , ... , T m, a natural estimate for V7r(so) is V7r(so) = ,; 2:::1 R(tr, Ti). Note that each tree can be used to evaluate any strategy, much the way a single labeled example (x , f(x)) can be used to evaluate any hypothesis h(x) in supervised learning. Thus in this sense, trajectory trees are reusable. T: Our goal now is to establish uniform convergence results that bound the error of the estimates V7r (so) as a function of the "sample size" (number of trees) m. Section 3.1 first treats the easier case of deterministic classes II; Section 3.2 extends the result to stochastic classes. 3.1 The Case of Deterministic II Let us begin by stating a result for the special case of finite classes of deterministic strategies, which will serve to demonstrate the kind of bound we seek. Theorem 3.1 Let II be any finite class of deterministic strategies for an arbitrary twoaction POMDP M. Let m trajectory trees be created using a generative modelfor M, and V7r(so) be the resulting estimates. lfm = 0 ((Vrnax /t)2(log(IIII) + log(1/8))), then with probability 1 - 8, IV7r (so) - V7r (so) I t holds simultaneously for alltr E II. :s Due to space limitations, detailed proofs of the results of this section are left to the full paper [6] , but we will try to convey the intuition behind the ideas. Observe that for any fixed deterministic tr, the estimates R( tr, Ti) that are generated by the m different trajectory trees Ti are independent. Moreover, each R(tr, T i ) is an unbiased estimate of the expected discounted H~ -step return of tr, which is in turn t/2-close to V7r(so). These observations, combined with a simple Chernoff and union bound argument, are sufficient to establish Theorem 3.1. Rather than developing this argument here, we instead move straight on to the harder case of infinite II. When addressing sample complexity in supervised learning, perhaps the most important insight is that even though a class 1i may be infinite, the number of possible behaviors of 1i on a finite set of points is often not exhaustive. More precisely, for boolean functions, we say that the set Xl, ... , Xd is shattered by 1i if every of the 2d possible labelings of Approximate Planning in Large POMDPs via Reusable Trajectories 1005 these points is realized by some h E 1i. The VC dimension of 1i is then defined as the size of the largest shattered set [9]. It is known that if the VC dimension of 1i is d, then the number <P d(m) of possible labelings induced by 1i on a set of m points is at most (em Jd)d, which is much less than 2 m for d ? m. This fact provides the key leverage exploited by the classical VC dimension results, and we will concentrate on replicating this leverage in our setting. If II is a (possibly infinite) set of deterministic strategies, then each strategy tr E II is simply a deterministic function mapping from the set of observable histories to the set {al' a2}, and is thus a boolean function on observable histories. We can therefore write VC(II) to denote the familiar VC dimension of the set of binary functions II. For example, if II is the set of all thresholded linear functions of the current vector of observations (a particular type of memoryless strategy), then VC(II) simply equals the number of parameters. We now show intuitively why a class II of bounded VC dimension d cannot induce exhaustive behavior on a set T l , ... ,Tm of trajectory trees for m ? d. Note that if trl, tr2 E II are such that their "reward labelings" (R(trl' T l ), ... ,R(trl' T m)) and (R( tr2, Tt), ... , R( tr2, T m)) differ, then R( trl, Ti) =f. R(tr2' T i ) for some 1 ::; i ::; m. But if trl and tr2 give different returns on T i , then they must choose different actions at some node in T i . In other words, every different reward labeling of the set of m trees yields a different (binary) labeling of the set of m . 2H ? observable histories in the trees. So, the number of different tree reward labelings can be at most <Pd(m? 2H<) ::; (em? 2H<Jd)d. By developing this argument carefully and applying classical uniform convergence techniques, we obtain the following theorem. (Full proof in [6].) Theorem 3.2 Let II be any class of deterministic strategies for an arbitrary two-action POMDP M, and let VC(II) denote its VC dimension. Let m trajectory trees be created using a generative model for M, and "\I7r (so) be the resulting estimates. If (1) then with probability 1 - 6, IV 7r (so) - "\I7r (so) I ::; ? holds simultaneously for alltr E II. 3.2 The Case of Stochastic II We now address the case of stochastic strategy classes. We describe an approach where we transform stochastic strategies into "equivalent" deterministic ones and operate on the deterministic versions, reducing the problem to the one handled in the previous section. The transformation is as follows: Given a class of stochastic strategies II, each with domain X (where X is the set of all observable histories), we first extend the domain to be X x [0,1]. Now for each stochastic strategy tr E II, define a corresponding deterministic transformed strategy tr' with domain X x [0,1], given by: tr'(h, r) = al if r ::; Pr[tr(h) = ad, and 7r'(h,r) = a2 otherwise (for any hEX, r E [0,1]). Let II' be the collection of these transformed deterministic strategies tr'. Since II' is just a set of deterministic boolean functions, its VC dimension is well-defined. We then define the pseudo-dimension of the original set of stochastic strategies II to be p VC(II) = VC(II').4 Having transformed the strategy class, we also need to transform the POMDP, by augmenting the state space S to be S x [0,1]. Informally, the transitions and rewards remain the same, except that after each state transition, we draw a new random variable r uniformly in [0,1], and independently of all previous events. States are now of the form (s, r), and we let r be an observed variable. Whenever in the original POMDP a stochastic strategy tr would 4This is equivalent to the conventional definition of the pseudo-dimension of IT [4], when it is viewed as a set of maps into real-valued action-probabilities. M Kearns, Y. Mansour and A. Y. Ng 1006 have been given a history h, in the transformed POMDP the corresponding deterministic transformed strategy 7r' is given (h, r), where r is the [0, l]-random variable at the current state. By the definition of 7r', it is easy to see that 7r' and 7r have exactly the same chance of choosing each action at any node (randomization over r). We are now back in the deterministic case, so Theorem 3.2 applies, with VC(II) replaced by pVC (II) = VC(II'), and we again have the desired uniform convergence result. 4 Algorithms for Approximate Planning Given a generative model for a POMDP, the preceding section's results immediately suggest a class of approximate planning algorithms: generate m trajectory trees T 1 , ... , T m, and search for a 7r E II that maximizes V7r (so) = (1/ m) L R( 7r, Ti). The following corollary to the uniform convergence results establishes the soundness of this approach. Corollary 4.1 Let II be a class of strategies in a POMDP M, and let the number m of trajectory trees be as given in Theorem 3.2. Let it = argmax 7r Err{V7r(so)} be the policy in II with the highest empirical return on the m trees. Then with probability 1 - 0, it is near-optimal within II: (2) V7T(SO) ~ opt(M, II) - 2?. If the suggested maximization is computationally infeasible, one can search for a local maximum 7r instead, and uniform convergence again assures us that V7r (so) is a trusted estimate of our true performance. Of course, even finding a local maximum can be expensive, since each trajectory tree is of size exponential in H{. However, in practice it may be possible to significantly reduce the cost of the search. Suppose we are using a class of (possibly transformed) deterministic strategies, and we perform a greedy local search over II to optimize V7r (so). Then at any time in the search, to evaluate the policy we are currently considering, we really need to look at only a single path of length Hf in each tree, corresponding to the path taken by the strategy being considered. Thus, we should build the trajectory trees lazily - that is, incrementally build each node of each tree only as it is needed to evaluate R( 7r, Ti) for the current strategy 7r. If there are parts of a tree that are reached only by poor policies, then a good search algorithm may never even build these parts of the tree. In any case, for a fixed number of trees, each step of the local search now takes time only linear in H f ? 5 There is a different approach that works directly on stochastic strategies (that is, without requiring the transformation to deterministic strategies). In this case each stochastic strategy 7r defines a distribution over all the paths in a trajectory tree, and thus calculating R( 7r, T) may in general require examining complete trees. However, we can view each trajectory tree as a small, deterministic POMDP by itself, with the children of each node in the tree being its successor nodes. So if II = {7re : E IRd} is a smoothly parameterized family of stochastic strategies, then algorithms such as William's REINFORCE [10] can be used to find an unbiased estimate of the gradient (d/ de) V7r 9 (so), which in turn can be used to e 5 See also (Ng and Jordan, in preparation) which, by assuming a much stronger model of a POMDP (a deterministic function 1 such that I(s, a, r) is distributed according to P('ls, a) when r is distributed Uniform[O,l]), gives an algorithm that enjoys uniform convergence bounds similar to those presented here, but with only a polynomial rather than exponential dependence on H,. The algorithm samples a number of vectors r(i) E [0, IjH., each of which, with I, defines an H,-step Monte Carlo evaluation trial for any policy 7r . The bound is on the number of such random vectors needed (rather than on the total number of calls to f). Approximate Planning in Large POMDPs via Reusable Trajectories 1007 perform stochastic gradient ascent to maximize V7r 8 (so). Moreover, for a fixed number of trees, these algorithms need only O(H?) time per gradient estimate; so combined with lazy tree construction, we again have a practical algorithm whose per-step complexity is only linear in the horizon time. This line of thought is further developed in the long version of the paper.6 5 The Random Trajectory Method Using a fully observable generative model of a POMDP, we have shown that the trajectory tree method gives uniformly good value estimates, with an amount of experience linear in VC(II), and exponential in H?. It turns out we can significantly weaken the generative model, yet still obtain essentially the same theoretical results. In this harder case, we assume a generative model that provides only partially observable histories generated by a truly random strategy (which takes each action with equal probability at every step, regardless of the history so far). Furthermore, these trajectories always begin at the designated start state, so there is no ability provided to "reset" the POMDP to any state other than so. (Indeed, underlying states may never be observed.) Our method for this harder case is called the Random Trajectory method. It seems to lead less readily to practical algorithms than the trajectory tree method, and its formal description and analysis, which is more difficult than for trajectory trees, are given in the long version of this paper [6]. As in Theorem 3.2, we prove that the amount of data needed is linear in VC(II), and exponential in the horizon time - that is, by averaging appropriately over the resulting ensemble of trajectories generated, this amount of data is sufficient to yield uniformly good estimates of the values for all strategies in II. References [1] L. Baird and A. W. Moore. Gradient descent for general Reinforcement Learning. In Advances in Neural Information Processing Systems 11, 1999. [2] C. Boutilier, T. Dean, and S. Hanks. Decision theoretic planning: Structural assumptions and computational leverage. Journal of Artificial Intelligence Research, 1999. [3] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In Proc. UAI, pages 33-42, 1998. [4] David Haussler. Decision theoretic generalizations of the PAC model for neural net and oter learning applications. Information and Computation, 100:78-150, 1992. [5] L. P. Kaelbling, M. L. Littman, and A. R. Cassandra. Planning and acting in partially observable stochastic domains. ArtifiCial Intelligence, 101, 1998. [6] M. Kearns, Y. Mansour, and A. Y. Ng. Approximate planning in large POMDPs via reusable trajectories. (long version), 1999. [7] D. Koller and R. Parr. Computing factored value functions for poliCies in structured MDPs. In Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999. [8] R. S. Sutton and A. G. Barto. Reinforcement Learning. MIT Press, 1998. [9] Y.N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, 1982. [10] R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256, 1992. 6In the full paper, we also show how these algorithms can be extended to find in expected O( He) time an unbiased estimate of the gradient of the true value V 8 (so) for discounted infinite horizon problems (whereas most current algorithms either only converge asymptotically to an unbiased estimate of this gradient, or need an absorbing state and "proper" strategies). 7T 

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An Environment Model for N onstationary Reinforcement Learning Samuel P. M. Choi pmchoi~cs.ust.hk Dit-Yan Yeung Nevin L. Zhang dyyeung~cs.ust.hk lzhang~cs.ust.hk Department of Computer Science, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong Abstract Reinforcement learning in nonstationary environments is generally regarded as an important and yet difficult problem. This paper partially addresses the problem by formalizing a subclass of nonstationary environments. The environment model, called hidden-mode Markov decision process (HM-MDP), assumes that environmental changes are always confined to a small number of hidden modes. A mode basically indexes a Markov decision process (MDP) and evolves with time according to a Markov chain. While HM-MDP is a special case of partially observable Markov decision processes (POMDP), modeling an HM-MDP environment via the more general POMDP model unnecessarily increases the problem complexity. A variant of the Baum-Welch algorithm is developed for model learning requiring less data and time. 1 Introduction Reinforcement Learning (RL) [7] is a learning paradigm based upon the framework of Markov decision process (MDP). Traditional RL research assumes that environment dynamics (i.e., MDP parameters) are always fixed (Le., stationary). This assumption, however, is not realistic in many real-world applications. In elevator control [3], for instance, the passenger arrival and departure rates can vary significantly over one day, and should not be modeled by a fixed MDP. Nonetheless, RL in nonstationary environments is regarded as a difficult problem. In fact, it is an impossible task if there is no regularity in the ways environment dynamics change. Hence, some degree of regularity must be assumed. Typically, nonstationary environments are presummed to change slowly enough such that online RL algorithms can be employed to keep track the changes. The online approach is memoryless in the sense that even if the environment ever revert to the previously learned dynamics, learning must still need to be started all over again. S. P. M Choi, D.-y' Yeung and N. L. Zhang 988 1.1 Our Proposed Model This paper proposes a formal model [1] for the nonstationary environments that repeats their dynamics in certain ways. Our model is inspired by the observations from the real-world nonstationary tasks with the following properties: Property 1. Environmental changes are confined to a small number of modes, which are stationary environments with distinct dynamics. The environment is in exactly one of these modes at any given time. This concept of modes seems to be applicable to many real-world tasks. In an elevator control problem, for example, the system might operate in a morning-rush-hour mode, an evening-rush-hour mode and a non-rush-hour mode. One can also imagine similar modes for other control tasks, such as traffic control and dynamic channel allocation [6]. Property 2. Unlike states, modes cannot be directly observed; the current mode can only be estimated according to the past state transitions. It is analogous to the elevator control example in that the passenger arrival rate and pattern can only be inferred through the occurrence of pick-up and drop-off requests. Property 3. Mode transitions are stochastic events and are independent of the control system's responses. In the elevator control problem, for instance, the events that change the current mode of the environment could be an emergency meeting in the administrative office, or a tea break for the staff on the 10th floor. Obviously, the elevator's response has no control over the occurrence of these events. Property 4. Mode transitions are relatively infrequent. In other words, a mode is more likely to retain for some time before switching to another one. If we consider the emergency meeting example, employees on different floors take time to arrive at the administrative office, and thus would generate a similar traffic pattern (drop-off requests on the same floor) for some period of time. Property 5. The number of states is often substantially larger than the number of modes. This is a common property for many real-world applications. In the elevator example, the state space comprises all possible combinations of elevator positions, pick-up and drop-off requests, and certainly would be huge. On the other hand, the mode space could be small. For instance, an elevator control system can simply have the three modes as described above to approximate the reality. Based on these properties, an environment model is proposed by introducing a mode variable to capture environmental changes. Each mode specifies an MDP and hence completely determines the current state transition function and reward function (property 1). A mode, however, is not directly observable (property 2), and evolves with time according to a Markov process (property 3). The model is therefore called hidden-mode model. Note that our model does not impose any constraint to satisfy properties 4 and 5. In other words, the hidden-mode model can work for environments without these two properties. Nevertheless, as will be shown later, these properties can improve learning in practice. 1.2 Related Work Our hidden-mode model is related to a non stationary model proposed by Dayan and Sejnowski [4]. Although our model is more restrictive in terms of representational power, it involves much fewer parameters and is thus easier to learn. Besides, other than the number of possible modes, we do not assume any other knowledge about 989 An Environment Model for Nonstationary Reinforcement Learning the way environment dynamics change. Dayan and Sejnowski, on the other hand, assume that one knows precisely how the environment dynamics change. The hidden-mode model can also be viewed as a special case of the hidden-state model, or partially observable Markov decision process (POMDP). As will be shown later, a hidden-mode model can always be represented by a hidden-state model through state augmentation. Nevertheless, modeling a hidden-mode environment via a hidden-state model will unnecessarily increase the problem complexity. In this paper, the conversion from the former to the latter is also briefly discussed. 1.3 Our Focus There are two approaches for RL. Model-based RL first acquires an environment model and then, from which, an optimal policy is derived. Model-free RL, on the contrary, learns an optimal policy directly through its interaction with the environment. This paper is concerned with the first part of the model-based approach, i.e., how a hidden-mode model can be learned from experience. We will address the policy learning problem in a separate paper. 2 Hidden-Mode Markov Decision Processes This section presents our hidden-mode model. Basically, a hidden-mode model is defined as a finite set of MDPs that share the same state space and action space, with possibly different transition functions and reward functions. The MDPs correspond to different modes in which a system operates. States are completely observable and their transitions are governed by an MDP. In contrast, modes are not directly observable and their transitions are controlled by a Markov chain. We refer to such a process as a hidden-mode Markov decision process (HM-MDP). An example of HM-MDP is shown in Figure l(a). Time ? Mode Action ... ... StaIC (a) A 3-mode, 4-state, I-action HM-MDP (b) The evolution of an HM-MDP. The arcs indicate dependencies between the variables Figure 1: An HM-MDP Formally, an HM-MDP is an 8-tuple (Q,S,A,X,Y,R,rr,'l'), where Q, S and A represent the sets of modes, states and actions respectively; the mode transition function X maps mode m to n with a fixed probability Xmn; the state transition function Y defines transition probability, Ym(8, a, s'), from state 8 to 8' given mode m and action a; the stochastic reward function R returns rewards with mean value rm (8, a); II and 'l1 denote the prior probabilities of the modes and of the states respectively. The evolution of modes and states over time is depicted in Figure 1 (b). S. P. M. Choi, D.-y' Yeung and N. L. Zhang 990 HM-MDP is a subclass of POMDP. In other words, the former can be reformulated as a special case of the latter. Specifically, one may take an ordered pair of any mode and observable state in the HM-MDP as a hidden state in the POMDP, and any observable state of the former as an observation of the latter. Suppose the observable states sand s' are in modes m and n respectively. These two HMMDP states together with their corresponding modes form two hidden states (m, s) and (n, s') for its POMDP counterpart. The transition probability from (m, s) to (n, s') is then simply the mode transition probability Xmn multiplied by the state transition probability Ym(s, a, s'). For an M-mode, N-state, K-action HM-MDP, the equivalent POMDP thus has N observations and M N hidden states. Since most state transition probabilities are collapsed into mode transition probabilities through parameter sharing, the number of parameters in an HM-MDP (N 2 M K + M2) is much less than that of its corresponding POMDP (M2 N 2K). 3 Learning a Hidden-Mode Model There are now two ways to learn a hidden-mode model. One may learn either an HM-MDP, or an equivalent POMDP instead. POMDP models can be learned via a variant of the Baum-Welch algorithm [2]. This POMDP Baum-Welch algorithm requires 8(M2 N 2T) time and 8(M2 N 2K) storage for learning an M-mode, Nstate, K-action HM-MDP, given T data items. A similar idea can be applied to the learning of an HM-MDP. Intuitively, one can estimate the model parameters based on the expected counts of the mode transitions, computed by a set of auxiliary variables. The major difference from the original algorithm is that consecutive state transitions, rather than the observations, are considered. Additional effort is thus needed for handling the boundary cases. This HM-MDP Baum-Welch algorithm is described in Figure 2. 4 Empirical Studies This section empirically examines the POMDP Baum-Welch 1 and HM-MDP BaumWelch algorithms. Experiments based on various randomly generated models and some real-world environments were conducted. The results are quite consistent. For illustration, a simple traffic control problem is presented. In this problem, one direction of a two-way traffic is blocked, and cars from two different directions (left and right) are forced to share the remaining road. To coordinate the traffic, two traffic lights equipped with sensors are set. The system then has two possible actions: either to signal cars from the left or cars from the right to pass. For simpliCity, we assume discrete time steps and uniform speed of the cars. The system has 8 possible states; they correspond to the combinations of whether there are cars waiting on the left and the right directions, and the stop signal position in the previous time step. There are 3 traffic modes. The first one has cars waiting on the left and the right directions with probabilities 0.3 and 0.1 respectively. In the second mode, these probabilities are reversed. For the last one, both probabilities are 0.3. In addition, the mode transition probability is 0.1. A cost of -1.0 results if lChrisman's algorithm also attempts to learn a minimal possible number of states. Our paper concerns only with learning the model parameters. 991 An Environment Model for Nonstationary Reinforcement Learning Given a collection of data and an initial model parameter vector repeat 0. 0=0 Compute forward variables (Xt. (Xl (i) = 1/;$1 (X2(i) = 1I"i 1/;$1 Yi(SI, al,S2) (Xt+l(j) = L:iEQ (Xt(i) Xii Yi(St,at,St+l) "Ii E Q "Ii E Q "Ii E Q Compute backward variables (3t . (3T(i) = 1 (3t(i) = LiEQXii Yi(St,at,St+I) (3t+I(j) (31(i) = L:iEQ 1I"j Yi(sl , al,s2) (32(j) "Ii E Q "Ii E Q "Ii E Q "I i , j E Q "Ii E Q Compute the new model parameter 0. _ .. _ L;-2 {. (i,i) Xl] - ~T . 8(a, b) = L....t=1 "'Yt (I) {01 a= b af.b 1Ti = "Yl (i) until maxi IOi - OJ I < to Figure 2: HM-MDP Baum-Welch Algorithm a car waits on either side. The experiments were run with the same initial model for data sets of various sizes. The algorithms iterated until the maximum change of the model parameters was less than a threshold of 0.0001. The experiment was repeated for 20 times with different random seeds in order to compute the median. Then the learned models were compared in their POMDP forms using the Kullback-Leibler (KL) distance [5], and the total CPU running time on a SUN Ultra I workstation was measured. Figure 3 (a) and (b) report the results. Generally speaking, both algorithms learn a more accurate environment model as the data size increases (Figure 3 (a)). This result is expected as both algorithms are statistically-based, and hence their performance relies largely on the data size. When the training data size is very small , both algorithms perform poorly. However, as the data size increases, HM-MDP Baum-Welch improves substantially faster than POMDP Baum-Welch. It is because an HM-MDP in general consists of fewer free s. P M 992 Choi. D.-y' Yeung and N. L. Zhang '0000 " /---.-.- ..-....-----.-.. ~...--.... --... ............ ...... .. -.----.".~ o ~~~~~~~~~~~~~~ !SOO 1000 1501) 2000 2&00 )000 :1500 oKIOO .&500 5000 o Wndow9tD (a) Error in transition function 10500L--'OOOJ...--'..."500-2000~~ .... ,,---:"_':::"--=_.,.,.......-:-""':':-:--,.... ...,,...,---:-!,OOO WIndowSiz. (b) Required learning time Figure 3: Empirical results on model learning parameters than its POMDP counterpart. HM-MDP Baum-Welch also runs much faster than POMDP Baum-Welch (Figure 3 (b)). It holds in general for the same reason discussed above. Note that computational time is not necessarily monotonically increasing with the data size. It is because the total computation depends not only on the data size, but also on the number of iterations executed. From our experiments, we noticed that the number of iterations tends to decrease as the data size increases. Larger models have also been tested. While HM-MDP Baum-Welch is able to learn models with several hundred states and a few modes, POMDP Baum-Welch was unable to complete the learning in a reasonable time. Additional experimental results can be found in [1]. 5 Discussions and Future Work The usefulness of a model depends on the validity of the assumptions made. We now discuss the assumptions of HM-MDP, and shed some light on its applicability to real-world nonstationary tasks. Some possible extensions are also discussed. Modeling a nonstationary environment as a number of distinct MDPs. MDP is a flexible framework that has been widely adopted in various applications. Modeling nonstationary environments by distinct MDPs is a natural extension to those tasks. Comparing to POMDP, our model is more comprehensive: each MDP naturally describes a mode of the environment. Moreover, this formulation facilitates the incorporation of prior knowledge into the model initialization step. States are directly observable while modes are not. While completely observable states are helpful to infer the current mode, it is also possible to extend the model to allow partially observable states. In this case, the extended model would be equivalent in representational power to a POMDP. This could be proved easily by showing the reformulation of the two models in both directions. An Environment Model for Nonstationary Reinforcement Learning 993 Mode changes are independent of the agent's responses. This property may not always hold for all real-world tasks. In some applications, the agent's actions might affect the state as well as the environment mode. In that case, an MDP should be used to govern the mode transition process. Mode transitions are relatively infrequent. This is a property that generally holds in many applications. Our model, however, is not limited by this condition. We have tried to apply our model-learning algorithms to problems in which this property does not hold. We find that our model still outperforms POMDP, although the required data size is typically larger for both models. Number of states is substantially larger than the number of modes. This is the key property that significantly reduces the number of parameters in HM-MDP compared to that in POMDP. In practice, introduction of a few modes is sufficient for boosting the system performance. More modes might only help little. Thus a trade-off between performance and response time must be decided. There are additional issues that need to be addressed. First, an efficient algorithm for policy learning is required. Although in principle it can be achieved indirectly via any POMDP algorithm, a more efficient algorithm based on the model-based approach is possible. We will address this issue in a separate paper. Next, the number of modes is currently assumed to be known. We are now investigating how to remove this limitation. Finally, the exploration-exploitation issue is currently ignored. In our future work, we will address this important issue and apply our model to real-world nonstationary tasks. References [1] S. P. M. Choi, D. Y. Yeung, and N. L. Zhang. Hidden-mode Markov decision processes. In IJCAI 99 Workshop on Neural, Symbolic, and Reinforcement Methods for Sequence Learnin9, 1999. [2] L. Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In AAAI-92, 1992. [3] R. H. Crites and A. G. Barto. Improving elevator performance using reinforcement learning. In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems 8, 1996. [4] P. Dayan and T. J. Sejnowski. Exploration bonuses and dual control. Machine Learning, 25(1):5- 22, Oct. 1996. [5J S. Kullback. Information Theory and Statistics. Wiley, New York, NY, USA, 1959. [6] S. Singh and D. P. Bertsekas. Reinforcement learning for dynamic channel allocation in cellular telephone systems. In Advances in Neural Information Processing Systems 9, 1997. [7] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. The MIT Press, 1998. 

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Rules and Similarity in Concept Learning Joshua B. Tenenbaum Department of Psychology Stanford University, Stanford, CA 94305 jbt@psych.stanford.edu Abstract This paper argues that two apparently distinct modes of generalizing concepts - abstracting rules and computing similarity to exemplars - should both be seen as special cases of a more general Bayesian learning framework. Bayes explains the specific workings of these two modes - which rules are abstracted, how similarity is measured - as well as why generalization should appear rule- or similarity-based in different situations. This analysis also suggests why the rules/similarity distinction, even if not computationally fundamental, may still be useful at the algorithmic level as part of a principled approximation to fully Bayesian learning. 1 Introduction In domains ranging from reasoning to language acquisition, a broad view is emerging of cognition as a hybrid of two distinct modes of computation, one based on applying abstract rules and the other based on assessing similarity to stored exemplars [7]. Much support for this view comes from the study of concepts and categorization. In generalizing concepts, people's judgments often seem to reflect both rule-based and similarity-based computations [9], and different brain systems are thought to be involved in each case [8]. Recent psychological models of classification typically incorporate some combination of rule-based and similarity-based modules [1,4]. In contrast to this currently popular modularity position, I will argue here that rules and similarity are best seen as two ends of a continuum of possible concept representations. In [11,12], I introduced a general theoretical framework to account for how people can learn concepts from just a few positive examples based on the principles of Bayesian inference. Here I explore how this framework provides a unifying explanation for these two apparently distinct modes of generalization. The Bayesian framework not only includes both rules and similarity as special cases but also addresses several questions that conventional modular accounts do not. People employ particular algorithms for selecting rules and measuring similarity. Why these algorithms as opposed to any others? People's generalizations appear to shift from similarity-like patterns to rule-like patterns in systematic ways, e.g., as the number of examples observed increases. Why these shifts? This short paper focuses on a simple learning game involving number concepts, in which both rule-like and similarity-like generalizations clearly emerge in the judgments of human subjects. Imagine that I have written some short computer programs which take as input a natural number and return as output either "yes" or "no" according to whether that number 60 J. B. Tenenbaum satisfies some simple concept. Some possible concepts might be "x is odd", "x is between 30 and 45", "x is a power of3", or"x is less than 10". For simplicity, we assume that only numbers under 100 are under consideration. The learner is shown a few randomly chosen positive examples - numbers that the program says "yes" to - and must then identify the other numbers that the program would accept. This task, admittedly artificial, nonetheless draws on people's rich knowledge of number while remaining amenable to theoretical analysis. Its structure is meant to parallel more natural tasks, such as word learning, that often require meaningful generalizations from only a few positive examples of a concept. Section 2 presents representative experimental data for this task. Section 3 describes a Bayesian model and contrasts its predictions with those of models based purely on rules or similarity. Section 4 summarizes and discusses the model's applicability to other domains. 2 The number concept game Eight subjects participated in an experimental study of number concept learning, under essentially the same instructions as those given above [11]. On each trial, subj ects were shown one or more random positive examples of a concept and asked to rate the probability that each of 30 test numbers would belong to the same concept as the examples observed. X denotes the set of examples observed on a particular trial, and n the number of examples. Trials were designed to fall into one of three classes. Figure la presents data for two representative trials of each class. Bar heights represent the average judged probabilities that particular test numbers fall under the concept given one or more positive examples X, marked by "*"s. Bars are shown only for those test numbers rated by subjects; missing bars do not denote zero probability of generalization, merely missing data. = = On class I trials, subjects saw only one example of each concept: e.g., X {16} and X {60}. To minimize bias, these trials preceded all others on which multiple examples were given. Given only one example, people gave most test numbers fairly similar probabilities of acceptance. Numbers that were intuitively more similar to the example received slightly higher ratings: e.g., for X = {16}, 8 was more acceptable than 9 or 6, and 17 more than 87; for X = {60}, 50 was more acceptable than 51, and 63 more than 43 . The remaining trials each presented four examples and occured in pseudorandom order. On class II trials, the examples were consistent with a simple mathematical rule: X = {16 , 8, 2, 64} or X {60, 80 , 10, 30}. Note that the obvious rules, "powers of two" and "multiples often", are in no way logically implied by the data. "Multiples offive" is a possibility in the second case, and "even numbers" or "all numbers under 80" are possibilities in both, not to mention other logically possible but psychologically implausible candidates, such as "all powers of two, except 32 or4". Nonetheless, subjects overwhelmingly followed an all-or-none pattern of generalization, with all test numbers rated near 0 or 1 according to whether they satisified the single intuitively "correct" rule. These preferred rules can be loosely characterized as the most specific rules (i.e., with smallest extension) that include all the examples and that also meet some criterion of psychological simplicity. = On class III trials, the examples satisified no simple mathematical rule but did have similar magnitudes: X {16, 23 , 19, 20} and X {60, 52, 57, 55} . Generalization now followed a similarity gradient along the dimension of magnitude. Probability ratings fell below 0.5 for numbers more than a characteristic distance beyond the largest or smallest observed examples - roughly the typical distance between neighboring examples ("'" 2 or 3). Logically, there is no reason why participants could not have generalized according to = = e 61 Rules and Similarity in Concept Learning various complex rules that happened to pick out the given examples, or according to very different values of~, yet all subjects displayed more or less the same similarity gradients. To summarize these data, generalization from a single example followed a weak similarity gradient based on both mathematical and magnitude properties of numbers. When several more examples were observed, generalization evolved into either an all-or-none pattern determined by the most specific simple rule, or, when no simple rule applied, a more articulated magnitude-based similarity gradient falling off with characteristic distance roughly equal to the typical separation between neighboring examples. Similar patterns were observed on several trials not shown (including one with a different value of and on two other experiments in quite different domains (described briefly in Section 4). e e) 3 The Bayesian model In [12], I introduced a Bayesian framework for concept learning in the context oflearning axis-parallel rectangles in a multidimensional feature space. Here I show that the same framework can be adapted to the more complex situation oflearning number concepts and can explain all of the phenomena of rules and similarity documented above. Formally, we observe n positive examples X = {x(1), ... , x(n)} of concept C and want to compute p(y E CIX), the probability that some new object y belongs to C given the observations X. Inductive leverage is provided by a hypothesis space 11. of possible concepts and a probabilistic model relating hypotheses h to data X. The hypothesis space. Elements ofll. correspond to subsets of the universe of objects that are psychologically plausible candidates for the extensions of concepts. Here the universe consists of numbers between 1 and 100, and the hypotheses correspond to subsets such as the even numbers, the numbers between 1 and 10, etc. The hypotheses can be thought of in terms of either rules or similarity, i.e., as potential rules to be abstracted or as features entering into a similarity computation, but Bayes does not distinguish these interpretations. Because we can capture only a fraction of the hypotheses people might bring to this task, we would like an objective way to focus on the most relevant parts of people's hypothesis space. One such method is additive clustering (ADCLUS) [6,10], which extracts a setoffeatures that best accounts for subjects' similarity judgments on a given set of objects. These features simply correspond to subsets of objects and are thus naturally identified with hypotheses for concept learning. Applications of ADCLUS to similarity judgments for the numbers 0-9 reveal two kinds of subsets [6,10]: numbers sharing a common mathematical property, such as {2, 4, 8} and {3, 6, 9}, and consecutive numbers of similar magnitude, such as {I, 2, 3, 4} and {2, 3, 4, 5, 6}. Applying ADCLUS to the full set of numbers from 1 to 100 is impractical, but we can construct an analogous hypothesis space for this domain based on the two kinds of hypotheses found in the ADCLUS solution for 0-9. One group of hypotheses captures salient mathematical properties: odd, even, square, cube, and prime numbers, multiples and powers of small numbers (~ 12), and sets of numbers ending in the same digit. A second group of hypotheses, representing the dimension of numerical magnitude, includes all intervals of consecutive numbers with endpoints between 1 and 100. Priors and likelihoods. The probabilistic model consists of a prior p( h) over 11. and a likelihood p( X Ih) for each hypothesis h E H. Rather than assigning prior probabilities to each ofthe 5083 hypotheses individually, I adopted a hierarchical approach based on the intuitive division of 11. into mathematical properties and magnitude intervals. A fraction A of the total probability was allocated to the mathematical hypotheses as a group, leaving (1 - A) for J. B. Tenenbaum 62 the magnitude hypotheses. The ,\ probability was distributed uniformly across the mathematical hypotheses. The (1 - ,\) probability was distributed across the magnitude intervals as a function of interval size according to an Erlang distribution, p( h) ex (Ihl/ li 2 )e- 1hl /0', to capture the intuition that intervals of some intermediate size are more likely than those of very large or small size. ,\ and Ii are treated as free parameters of the model. The likelihood is determined by the assumption of randomly sampled positive examples. In the simplest case, each example in X is assumed to be independently sampled from a uniform density over the concept G. For n examples we then have: p(Xlh) l/lhl n if Vj, xU) E h (1) o otherwise, where Ih I denotes the size of the subset h. For example, if h denotes the even numbers, then Ihl = 50, because there are 50 even numbers between I and 100. Equation I embodies the size principle for scoring hypotheses: smaller hypotheses assign greater likelihood than do larger hypotheses to the same data, and they assign exponentially greater likelihood as the number of consistent examples increases. The size principle plays a key role in learning concepts from only positive examples [12], and, as we will see below, in determining the appearance of rule-like or similarity-like modes of generalization. Given these priors and likelihoods, the posterior p( hlX) follows directly from Bayes' rule. Finally, we compute the probability of generalization to a new object y by averaging the predictions of all hypotheses weighted by their posterior probabilities p( h IX): p(y E GIX) =L (2) p(y E Glh)p(hIX). hE1i Equation 2 follows from the conditional independence of X and the membership of y E G, given h. To evaluate Equation 2, note that p(y E Glh) is simply 1 ify E h, and 0 otherwise. = Model results. Figure Ib shows the predictions of this Bayesian model (with'\ 1/2, Ii = 10). The model captures the main features of the data, including convergence to the most specific rule on Class II trials and to appropriately shaped similarity gradients on Class III trials. We can understand the transitions between graded, similarity-like and all-or-none, rule-like regimes of generalization as arising from the interaction of the size principle (Equation 1) with hypothesis averaging (Equation 2). Because each hypothesis h contributes to the average in Equation 2 in proportion to its posterior probability p(hIX), the degree of uncertainty in p(hIX) determines whether generalization will be sharp or graded. When p( h IX) is very spread out, many distinct hypotheses contribute significantly, resulting in a broad gradient of generalization. When p(hIX) is concentrated on a single hypothesis h*, only h* contributes significantly and generalization appears all-or-none. The degree of uncertainty in p( h IX) is in tum a consequence ofthe size principle. Given a few examples consistent with one hypothesis that is significantly smaller than the next-best competitor - such as X = {16, 8, 2, 64}, where "powers of two" is significantly smaller than "even numbers" - then the smallest hypothesis becomes exponentially more likely than any other and generalization appears to follow this most specific rule. However, given only one example (such as X {16}), or given several examples consistent with many similarly sized hypothesessuch as X = {16, 23,19, 20}, where the top candidates are all very similar intervals: "numbers between 16 and 23", "numbers between 15 and 24", etc. - the size-based likelihood favors the smaller hypotheses only slightly, p(hIX) is spread out over many overlapping hypotheses and generalization appears to follow a gradient of similarity. That the Bayesian = Rules and Similarity in Concept Learning 63 model predicts the right shape for the magnitude-based similarity gradients on Class III trials is no accident. The characteristic distance ? of the Bayesian generalization gradient varies with the uncertainty in p( h IX), which (for interval hypotheses) can be shown to covary with the intuitively relevant factor of average separation between neighboring examples. Bayes vs. rules or similarity alone. It is instructive to consider two special cases of the Bayesian model that are equivalent to conventional similarity-based and rule-based algorithms from the concept learning literature. What I call the SIM algorithm was pioneered by [5] and also described in [2,3] as a Bayesian approach to learning concepts from both positive and negative evidence. SIM replaces the size-based likelihood with a binary likelihood that measures only whether a hypothesis is consistent with the examples: p( X Ih) :::: 1 ifVj, xli) E h, and 0 otherwise. Generalization under SIM is just a count of the features shared by y and all the examples in X, independent of the frequency of those features or the number of examples seen. As Figure Ic shows, SIM successfully models generalization from a single example (Class I) but fails to capture how generalization sharpens up after multiple examples, to either the most specific rule (Class II) or a magnitude-based similarity gradient with appropriate characteristic distance ? (Class III). What I call the MIN algorithm preserves the size principle but replaces the step of hypothesis averaging with maximization: p(y E GIX) :::: 1 ify E arg maXh p(Xlh), and 0 otherwise. MIN is perhaps the oldest algorithm for concept learning [3] and, as a maximum likelihood algorithm, is asymptotically equivalent to Bayes. Its success for finite amounts of data depends on how peaked p(hIX) is (Figure Id). MIN always selects the most specific consistent rule, which is reasonable when that hypothesis is much more probable than any other (Class II), but too conservative in other cases (Classes I and III). In quantitative terms, the predictions of Bayes correlate much more highly with the observed data (R 2 :::: 0.91) than do the predictions of either SIM (R 2 :::: 0.74) or MIN (R 2 :::: 0.47). In sum, only the full Bayesian framework can explain the full range of rule-like and similarity-like generalization patterns observed on this task. 4 Discussion Experiments in two other domains provide further support for Bayes as a unifying framework for concept learning. In the context of multidimensional continuous feature spaces, similarity gradients are the default mode of generalization [5]. Bayes successfully models how the shape of those gradients depends on the distribution and number of examples; SIM and MIN do not [12]. Bayes also successfully predicts how fast these similarity gradients converge to the most specific consistent rule. Convergence is quite slow in this domain (n "" 50) because the hypothesis space consists of densely overlapping subsets - axisparallel rectangles - much like the interval hypotheses in the Class III number tasks. Another experiment engaged a word-learning task, using photographs of real objects as stimuli and a cover story oflearning a new language [11]. On each trial, subjects saw either one example of a novel word (e.g., a toy animal labeled with "Here is a blicket."), or three examples at one of three different levels of specificity: subordinate (e.g., 3 dalmatians labeled with "Here are three blickets."), basic (e.g., 3 dogs), or superordinate (e.g., 3 animals). They then were asked to pick the other instances of that concept from a set of 24 test objects, containing matches to the example(s) at all levels (e.g., other dalmatians, dogs, animals) as well as many non-matching objects. Figure 2 shows data and predictions for all three models. Similarity-like generalization given one example rapidly converged to the most specific rule after only three examples were observed, just as in the number task (Classes I and II) but in contrast to the axis-parallel rectangle task or the Class III num- , 64 J. B. Tenenbaum ber tasks, where similarity-like responding was still the norm after three or four examples. For modeling purposes, a hypothesis space was constructed from a hierarchical clustering of subjects' similarity judgments (augmented by an a priori preference for basic-level concepts) [11] . The Bayesian model successfully predicts rapid convergence from a similarity gradient to the minimal rule, because the smallest hypothesis consistent with each example set is significantly smaller than the next-best competitor (e.g., "dogs" is significantly smaller than "dogs and cats", just as with "multiples often" vs. "multiples of five"). Bayes fits the full data extremely well (R 2 = 0.98); by comparison, SIM (R 2 = 0.83) successfully accounts for only the n = 1 trials and MIN (R 2 = 0.76), the n = 3 trials. In conclusion, a Bayesian framework is able to account for both rule- and similarity-like modes of generalization, as well as the dynamics of transitions between these modes, across several quite different domains of concept learning. The key features of the Bayesian model are hypothesis averaging and the size principle. The former allows either rule-like or similarity-like behavior depending on the uncertainty in the posterior probability. The latter determines this uncertainty as a function of the number and distribution of examples and the structure ofthe learner's hypothesis space. With sparsely overlapping hypotheses - i.e., the most specific hypothesis consistent with the examples is much smaller than its nearest competitors - convergence to a single rule occurs rapidly, after just a few examples. With densely overlapping hypotheses - i.e., many consistent hypotheses of comparable size - convergence to a single rule occurs much more slowly, and a gradient of similarity is the norm after just a few examples. Importantly, the Bayesian framework does not so much obviate the distinction between rules and similarity as explain why it might be useful in understanding the brain. As Figures 1 and 2 show, special cases of Bayes corresponding to the SIM and MIN algorithms consistently account for distinct and complementary regimes of generalization. SIM, without the size principle, works best given only one example or densely overlappipg hypotheses, when Equation I does not generate large differences in likelihood. MIN, without hypothesis averaging, works best given many examples or sparsely overlapping hypotheses, when the most specific hypothesis dominates the sum over 1i in Equation 2. In light of recent brain-imaging studies dissociating rule- and exemplarbased processing [8], the Bayesian theory may best be thought of as a computational-level account of concept learning, with multiple subprocesses - perhaps subserving SIM and MIN - implemented in distinct neural circuits. I hope to explore this possibility in future work. References [1] M. Erickson & J. Kruschke (1998). Rules and exemplars in category learning. JEP: General 127, 107-140. [2] D. Haussler, M. Kearns, & R. Schapire (1994). Bounds on the sample complexity of Bayesian learning using information theory and the VC-dimension. Machine Learning 14,83-113. [3] T. Mitchell (1997). Machine Learning. McGraw-Hill. [4] R. Nosofsky & T. Palmeri (1998). A rule-plus-exception model for classifying objects in continuous-dimension spaces. Psychonomic Bull. & Rev. 5,345-369. [5] R. Shepard (1987). Towards a universal law of generalization for psychological science. Science 237, 1317-1323. [6] R. Shepard & P. Arabie (1979). Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psych. Rev. 86, 87-123. [7] S. Sloman & L. Rips (1998). Similarity and Symbols in Human Thinking. MIT Press. [8] E. Smith, A. Patalano & 1. Jonides (1998). Alternative strategies of categorization. In [6]. [9] E. Smith & S. Sloman (1994). Similarity- vs. rule-based categorization. Mem. & Cog. 22,377. [10] J. Tenenbaum (1996). Learning the structure of similarity. NIPS 8. [11] J. Tenenbaum (1999). A Bayesian Framework/or Concept Learning. Ph. D. Thesis, MIT. [12] J. Tenenbaum (1999). Bayesian modeling of human concept learning. NIPS I I. Rules and Similarity in Concept Learning 65 (a) Average generalization judgments: o.g~II~"~IIII Class I o.g [111~1I1 I. Class II Class III l o.~o ..I Jill _II I X=60 I II X=168264 I ? 1I. ! X=60 80 1030 o1~ ?? III 0.5 11 I II I IIlmll,' III I I X = 1: ? !. 1._*_111 1 II ... X=1623 1920 ? ? X=60 52 57 55 of ... 0.5 1 II I. LI~~~~ ' _ _~~~_ _~~~~ 10 20 30 40 50 60 70 80 90100 11111*11111 ? I ?? **'* I 10 20 30 40 50 60 70 80 90100 (b) Bayesian model: X= 16 o.gtllllil~'1 II ? 1 ?. ?. ?. .I X = 16 8 2 64 o1t 0.5 I IIliI a _ ~ ?1 I? I X= 1623 1920 ? o.g X=60 II L II I I ?? " ??1.1111 I I 1 o.g ..I. 1r o.g II. 10 20 30 40 50 60 70 80 90 100 X=6080103O I. I II ...L1J.. I. I .. . II ... X = 60 52 57 55 uljl. ... .. 10 20 30 40 50 60 70 80 90 100 (c) Pure similarity model (SIM): 1 t 0'8llhk~~lll 1 X= 16 I 1 II . J o.gf!II~u!wUIII . I II X=168264 o?~f"I??11 J I X=1623 1920 * ? ?J I. X=60 o.g IlL 1t I.! I ..1I1.1~IJIl !d X=60 80 10 30 o.g[ "1L 1r o.g II. 10 20 30 40 50 60 70 80 90 100 III hlllll??? 111 II X = 60 52 57 55 II I 11111.1111 I. 10 20 30 40 50 60 70 80 90 100 (d) Pure rule model (MIN): X= 16 a1f.....L * ?.... ?. 0.5 .I * 1f .I..... a.... X=168264 X=1623 1920 X = 60 1 0.5 o ??? o.g 1.1. 1 ~ o.g .1. 0.5 l!Ett* 10 20 30 40 50 60 70 80 90100 ........1..... .. I. III 1I.1..l.1~1~ro 10 30 X=60525755 .1 ...Jl. "' .. *** 10 20 30 40 50 60 70 80 90 100 Figure 1: Data and model predictions for the number concept task. (a) Average generalization judgments: (b) Bayesian model: o?~~?~~?~~?~UL (c) Pure similarity model (SIM): Training examples: I 1 I. 3 subordmate r 1 I 3 basic r 1 I. 3 superordinate r 1 0.5~.5~.5~.5 o?~IIL?~IIL?~IIL?~UL (d) Pure rule model (MIN): o?~~?~~?~~?~UL Figure 2: Data and model predictions for the word learning task. 

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Reinforcement Learning Using Approximate Belief States Andres Rodriguez * Artificial Intelligence Center SRI International 333 Ravenswood Avenue, Menlo Park, CA 94025 rodriguez@ai.sri.com Ronald Parr, Daphne Koller Computer Science Department Stanford University Stanford, CA 94305 {parr,koller}@cs.stanford.edu Abstract The problem of developing good policies for partially observable Markov decision problems (POMDPs) remains one of the most challenging areas of research in stochastic planning. One line of research in this area involves the use of reinforcement learning with belief states, probability distributions over the underlying model states. This is a promising method for small problems, but its application is limited by the intractability of computing or representing a full belief state for large problems. Recent work shows that, in many settings, we can maintain an approximate belief state, which is fairly close to the true belief state. In particular, great success has been shown with approximate belief states that marginalize out correlations between state variables. In this paper, we investigate two methods of full belief state reinforcement learning and one novel method for reinforcement learning using factored approximate belief states. We compare the performance of these algorithms on several well-known problem from the literature. Our results demonstrate the importance of approximate belief state representations for large problems. 1 Introduction The Markov Decision Processes (MDP) framework [2] is a good way of mathematically formalizing a large class of sequential decision problems involving an agent that is interacting with an environment. Generally, an MDP is defined in such a way that the agent has complete knowledge of the underlying state of the environment. While this formulation poses very challenging research problems, it is still a very optimistic modeling assumption that is rarely realized in the real world. Most of the time, an agent must face uncertainty or incompleteness in the information available to it. An extension of this formalism that generalizes MDPs to deal with this uncertainty is given by partially observable Markov Decision Processes (POMDPs) [1, 11] which are the focus of this paper. Solving a POMDP means finding an optimal behavior policy 7l'*, that maps from the agent's available knowledge of the environment, its belief state, to actions. This is usually done through a function, V, that assigns values to belief states. In the fully observable (MDP) "The work presented in this paper was done while the first author was at Stanford University. Reinforcement Learning Using Approximate Belief States 1037 case, a value function can be computed efficiently for reasonably sized domains. The situation is somewhat different for POMDPs, where finding the optimal policy is PSPACEhard in the number of underlying states [6]. To date, the best known exact algorithms to solve POMDPs are taxed by problems with a few dozen states [5]. There are several general approaches to approximating POMDP value functions using reinforcement learning methods and space does not permit a full review of them. The approach upon which we focus is the use of a belief state as a probability distribution over underlying model states. This is in contrast to methods that manipulate augmented state descriptions with finite memory [9, 12] and methods that work directly with observations [8] . The main advantage of a probability distribution is that it summarizes all of the information necessary to make optimal decisions [1]. The main disadvantages are that a model is required to compute a belief state, and that the task of representing and updating belief states in large problems is itself very difficult. In this paper, we do not address the problem of obtaining a model; our focus is on the the most effective way of using a model. Even with a known model, reinforcement learning techniques can be quite competitive with exact methods for solving POMDPs [lO]. Hence, we focus on extending the model-based reinforcement learning approach to larger problems through the use of approximate belief states. There are risks to such an approach: inaccuracies introduced by belief state approximation could give an agent a hopelessly inaccurate perception of its relationship to the environment. Recent work [4], however, presents an approximate tracking approach, and provides theoretical guarantees that the result of this process cannot stray too far from the exact belief state. In this approach, rather than maintaining an exact belief state, which is infeasible in most realistically large problems, we maintain an approximate belief state, usually from some restricted class of distributions. As the approximate belief state is updated (due to actions and observations), it is continuously projected back down into this restricted class. Specifically, we use decomposed belief states, where certain correlations between state variables are ignored. In this paper we present empirical results comparing three approaches to belief state reinforcement learning. The most direct approach is the use of a neural network with one input for each element of the full belief state. The second is the SPOVA method [lO], which uses a function approximator designed for POMDPs and the third is the use of a neural network with an approximate belief state as input. We present results for several well-known problems in the POMDP literature, demonstrating that while belief state approximation is ill-suited for some problems, it is an effective means of attacking large problems. 2 Basic Framework and Algorithms A POMDP is defined as a tuple < S, A, 0, T, R, 0 > of three sets and three functions. S is a set of states, A is a set of actions and is a set of observations. The transition function T : S x A ~ II( S) specifies how the actions affect the state of the world. It can be viewed as T( Si, a, S j) = P( S j la, sd, the probability that the agent reaches state S j if it currently is in state Si and takes action a. The reward function R : S x A ~ 1R determines the immediate reward received by the agent The observation model 0 : S x A ~ II( 0) determines what the agent perceives, depending on the environment state and the action taken. O(s, a, 0) = P( ola, s) is the probability that the agent observes 0 when it is in state s, having taken the action a. ? 1038 2.1 A. Rodriguez, R. Parr and D. Koller POMDP belief states A beliefstate, b, is defined as a probability distribution over all states S E S, where b(s), represents probability that the environment is in state s. After taking action a and observing 0, the belief state is updated using Bayes rule: 1 1 b (s ) O(S', a, 0) L.SES T(Si, a, s')b(sd = P( s I a, 0, b) = =---::::-:--:,-,:'=--~--:-:-:---:? L.sjES O(Sj, a, 0) L.siES T(Si' a, Sj)b(Si) 1 The size of an exact belief state is equal to the number of states in the model. For large problems, maintaining and manipulating an exact belief state can be problematic even if the the transition model has a compact representation [4]. For example, suppose the state space is described via a set of random variables X = {Xl, ... ,Xn }, where each Xi takes on values in some finite domain Val(Xi ), a particular S defines a value Xi E VaJ(Xi ) for each variable Xi. The full belief state representation will be exponential in n. We use the approximation method analyzed by Boyen and Koller [4], where the variables are partitioned into a set of disjoint clusters C I ... Ck and belief functions, bl ... bk are maintained over the variables in each cluster. At each time step, we compute the exact belief state, then compute the individual belief functions by marginalizing out inter-cluster correlations. For some assignment, Ci, to variables in C i , we obtain bi(Ci) = L.ygCl P(Ci' y). An approximation of the original, full belief state is then reconstructed as b( s) = n~=l bi (Ci). By representing the belief state as a product of marginal probabilities, we are projecting the belief state into a reduced space. While a full belief state representation for n state variables would be exponential in n, the size of decomposed belief state representation is exponential in the size of the largest cluster and additive in the number of clusters. For processes that mix rapidly enough, the errors introduced by approximation will stay bounded over time [4]. As discussed by Boyen and Koller [4], this type of decomposed belief state is particularly suitable for processes that can themselves be factored and represented as a dynamic Bayesian network [3]. In such cases we can avoid ever representing an exponentially sized belief state. However, the approach is fully general, and can be applied in any setting where the state is defined as an assignment of values to some set of state variables. 2.2 Value functions and policies for POMDPs If one thinks of a POMOP as an MOP defined over belief states, then the well-known fixed point equations for MOPs still hold. Specifically, V*(b) = m~x [L sES b(s)R(s, a) + 'Y L P(ola, b)V*(bl )] oED where'Y is the discount factor and b' (defined above) is the next belief state. The optimal policy is determined by the maximizing action for each belief state. In principle, we could use Q-Iearning or value iteration directly to solve POMOPs. The main difficulty lies in the fact that there are uncountably many belief states, making a tabular representation of the value function impossible. Exact methods for POMOPs use the fact that finite horizon value functions are piecewiselinear and convex [11], ensuring a finite representation. While finite, this representation can grow exponentially with the horizon, making exact approaches impractical in most settings. Function approximation is an attractive alternative to exact methods. We implement function approximation using a set of parameterized Q-functions, where Qa(b, W a ) is the reward-to-go for taking action a in belief state b. A value function is reconstructed from the Q-functions as V(b) maxa(Qa(b, W a )), and the update rule for Wa when a transition = Reinforcement Learning Using Approximate Belief States 1039 from state b to b' under action a with reward R is: 2.3 Function approximation architectures We consider two types of function approximators. The first is a two-layer feedforward neural network with sigmoidal internal units and a linear outermost layer. We used one network for each Q function. For full belief state reinforcement learning, we used networks hidden nodes. For with lSI inputs (one for each component of the belief state) and approximate belief state reinforcement learning, we used networks with one input for each assignment to the variables in each cluster. If we had two clusters, for example, each with 3 binary variables, then our Q networks would each have 2 3 + 2 3 = 16 inputs. We kept the number of hidden nodes for each network as the square root of the number of inputs. v'fSf Our second function approximator is SPOVA [10], which is a soft max function designed to exploit the piecewise-linear structure of POMDP value functions. A SPOVA Q function maintains a set of weight vectors Wal . . . W ai, and is evaluated as: In practice, a small value of k (usually 1.2) is adopted at the start of learning, making the function very smooth. This is increased during learning until SPOVA closely approximates a PWLC function of b (usually k = 8). We maintained one SPOVA Q function for each vectors to each function. This gave O(IAIISI parameters action and assigned to both SPOVA and the full belief state neural network. JiST 3 JiST) Empirical Results We present results on several problems from the POMDP literature and present an extension to a known machine repair problem that is designed to highlight the effects of approximate belief states. Our results are presented in the form of performance graphs, where the value of the current policy is obtained by taking a snapshot of the value function and measuring the discounted sum of reward obtained by the resulting policy in simulation. We use "NN" to refer to the neural network trained reinforcement learner trained with the full belief state and the term "Decomposed NN" to refer to the neural network trained with an approximate belief which is decomposed as a product of marginals. We used a simple exploration strategy, starting with a 0.1 probability of acting randomly, which decreased linearly to 0.01. Due to space limitations, we are not able to describe each model in detail. However, we used publicly available model description files from [5].1 Table 3.4 shows the running times of the different methods. These are generally much lower than what would be required to solve these problems using exact methods. 3.1 Grid Worlds We begin by considering two grid worlds, a 4 x 3 world from [10] and a 60-state world from [7]. The 4 x 3 world contains only 11 states and does not have a natural decomposition into state variables, so we compared SPOVA only with the full belief state neural network. I See hup:/Iwww.cs.brown.edu/research/ai/pomdp/index.html. Note that this file format specifies a starting distribution for each problem and our results are reported with respect to this starting distribution. A. Rodriguez, R. Parr and D. Koller 1040 S POVA NN ------Oecompo.-l NN ______/ " 01 ~ ! 05 o .0.5 / o. f? / J -. ?1 50'---"""OOOO""""""20000-'--"""".L---"-""""'"c:--:::-"--60000-'--= ' OOOO.L--",,, '"c:--90000 ,,-.--' ,OOOOO ISOOOO 1IOOOO Figure 1: a) 3 x 4 Grid World, b) 60-state maze The experimental results, which are averaged over 25 training runs and 100 simulations per policy snapshot, are presented in Figure 1a. They show that SPOVA learns faster than the neural network, but that the network does eventually catch up. The 60-state robot navigation problem [7] was amenable to a decomposed belief state approximation since its underlying state space comes from the product of 15 robot positions and 4 robot orientations. We decomposed the belief state with two clusters, one containing a position state variable and the other containing an orientation state variable. Figure 1b shows results in which SPOVA again dominates. The decomposed NN has trouble with this problem because the effects of position and orientation on the value function are not easily decoupled, i.e., the effect of orientation on value is highly state-dependent. This meant that the decomposed NN was forced to learn a much more complicated function of its inputs than the function learned by the network using the full belief state. 3.2 Aircraft Identification Aircraft identification is another problem studied in Cassandra's thesis. It includes sensing actions for identifying incoming aircraft and actions for attacking threatening aircraft. Attacks against friendly aircraft are penalized, as are failures to intercept hostile aircraft. This is a challenging problem because there is tension in deciding between the various sensors. " Better sensors tend to make the base more visible to hostile aircraft, while more stealthy sensors are less accurate. The sensors give information about both the aircraft's type and distance from the base. The state space of this problem is comprised of three main components. aircraft type - eitherthe aircraft is a friend orit is a foe; distance -how far the aircraft is currently from the base discretized into an adjustable number, d, of distinct distances; vis ibi 1 i ty - a measure of how visible the base is to the approaching aircraft, which is discretized into 5 levels. We chose d = 10, gaving this problem 104 states. The problem has a natural decomposition into state variables for aircraft type, distance and base visibility. The results for the three algorithms are shown in Figure 2(a). This is the first problem where we start to see an advantage from decomposing the belief state. For the decomposed NN, we used three separate clusters, one for each variable, which meant that the network had only 17 inputs. Not only did the simpler network learn faster, but it learned a better policy overall. We believe that this illustrates an important point: even though SPOVA and the full belief state neural network may be more expressive than the decomposed NN, the decomposed NN is able to search the space of functions it can represent much more efficiently due to the reduced number of parameters. 1041 Reinforcement Learning Using Approximate BeliefStates SPOVA, NN ----- DKompo!ll8d NN . so SPOVA NN O.:.ompo!ll8d NN ?20 ?20 O'--'-'~ OOOOO "-:--:-200000 '-'--"""""' OOO:---:-: """"" """""'SOOOOO ~-:eooooo -:":-:-:--:: 700000 """""""' 800000 ":::::-900000 """""'--' ''''' 1*IIIIonI o 10000 2IXlOO 30000 40000 50000 lteratbns 60000 70000 aoooo 90000 l00c00 Figure 2.: a) Aircraft Identification, b) Machine Maintenance 3.3 Machine Maintenance Our last problem was the machine maintenance problem from Cassandra's database. The problem assumes that there is a machine with a certain number of components. The quality of the parts produced by the machine is determined by the condition of the components. Each component can be in one of four conditions: good - the component is in good condition; fair - the component has some amount of wear, and would benefit from some maintenance; bad - the part is very worn and could use repairs; broken - the part is broken and must be replaced. The status of the components is observable only if the machine is completely disassembled. Figure 2(b) shows performance results for this problem for the 4 component version of this problem. At 256 states, it was at the maximum size for which a full belief state approach was manageable. However, the belief state for this problem decomposes naturally into clusters describing the status of each machine, creating a decomposed belief state with just four components. The graph shows the dominance of this this simple decomposition approach. We believe that this problem clearly demonstrates the advantage of belief state decomposition: The decomposed NN learns a function of 16 inputs in fraction of the time it takes for the full net or SPOVA to learn a lower-quality function of 256 inputs. 3.4 Running Times The table below shows the running times for the different problems presented above. These are generally much less than what would be required to solve these problems exactly. The full NN and SPOVA are roughly comparable, but the decomposed neural network is considerably faster. We did not exploit any problem structure in our approximate belief state computation, so the time spent computing belief states is actually larger for the decomposed NN. The savings comes from the the reduction in the number of parameters used, which reduced the number of partial derivatives computed. We expect the savings to be significantly more substantial for processes represented in a factored way [3], as the approximate belief state propagation algorithm can also take advantage of this additional structure. 4 Concluding Remarks We have a proposed a new approach to belief state reinforcement learning through the use of approximate belief states. Using well-known examples from the POMDP literature, we have compared approximate belief state reinforcement learning with two other methods A. Rodriguez, R. Parr and D. Koller 1042 Problem 3x4 Hallway Aircraft ID MachineM. SPOVA 19.1 s 32.8 min 38.3 min 2.5 h NN 13.0s 47 .1 min 49.9 min 2.6 h Decomposed NN 3.2 min 4.4 min 4.7 min Table 1: Run times (in seconds, minutes or hours) for the different algorithms that use exact belief states. Our results demonstrate that, while approximate belief states may not be ideal for tightly coupled problem features, such as the position and orientation of a robot, they are a natural and effective means of addressing some large problems. Even for the medium-sized problems we showed here, approximate belief state reinforcement learning can outperform full belief state reinforcement learning using fewer trials and much less CPU time. For many problems, exact belief state methods will simply be impractical and approximate belief states will provide a tractable alternative. Acknowledgements This work was supported by the ARO under the MURI program "Integrated Approach to Intelligent Systems," by ONR contract N66001-97-C-8554 under DARPA's HPKB program, and by the generosity of the Powell Foundation and the Sloan Foundation. References [1] K. J. Astrom. Optimal control of Markov decision processes with incomplete state estimation. l. Math. Anal. Applic., 10:174-205,1965. [2] R.E. Bellman. Dynamic Programming. Princeton University Press, 1957. [3] C. Boutilier, T. Dean, and S. Hanks . Decision theoretic planning: Structural assumptions and computational leverage. Journal of Artijiciallntelligence Research, 1999. [4] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In Proc. UAI, 1998. [5] A. Cassandra. Exact and approximate Algorithms for partially observable Markov Decision Problems. PhD thesis, Computer Science Dept., Brown Univ., 1998. [6] M. Littman. Algorithms for Sequential Decision Making. PhD thesis, Computer Science Dept., Brown Univ., 1996. [7] M. Littman, A. Cassandra, and L.P. Kaelbling. Learning policies for partially observable environments: Scaling up. In Proc. ICML, pages 362-370, 1996. [8] J. Loch and S. Singh. Using eligibility traces to find the best memory less policy in partially observable markov decision processes. In Proc. ICML. Morgan Kaufmann, 1998. [9] Andrew R. McCallum. Overcoming incomplete perception with utile distinction memory. In Proc.ICML, pages 190-196, 1993. [10] Ronald Parr and Stuart Russell. Approximating optimal policies for partially observable stochastic domains. In Proc. IlCAI, 1995. [11] R. D. Smallwood and E. J. Sondik. The optimal control of partially observable Markov processes over a finite horizon. Operations Research, 21: 1071-1088,1973. [12] M. Wiering and J. Schmidhuber. HQ-leaming: Discovering Markovian subgoals for non-Markovian reinforcement learning. Technical report, Istituo Daile Molle di Studi sull'Intelligenza Artificiale, 1996. 

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Learning sparse codes with a mixture-of-Gaussians prior Bruno A. Olshausen Department of Psychology and Center for Neuroscience, UC Davis 1544 Newton Ct. Davis, CA 95616 baolshausen@ucdavis.edu K. Jarrod Millman Center for Neuroscience, UC Davis 1544 Newton Ct. Davis, CA 95616 kjmillman@ucdavis. edu Abstract We describe a method for learning an overcomplete set of basis functions for the purpose of modeling sparse structure in images. The sparsity of the basis function coefficients is modeled with a mixture-of-Gaussians distribution. One Gaussian captures nonactive coefficients with a small-variance distribution centered at zero, while one or more other Gaussians capture active coefficients with a large-variance distribution. We show that when the prior is in such a form, there exist efficient methods for learning the basis functions as well as the parameters of the prior. The performance of the algorithm is demonstrated on a number of test cases and also on natural images. The basis functions learned on natural images are similar to those obtained with other methods, but the sparse form of the coefficient distribution is much better described. Also, since the parameters of the prior are adapted to the data, no assumption about sparse structure in the images need be made a priori, rather it is learned from the data. 1 Introduction The general problem we address here is that of learning a set of basis functions for representing natural images efficiently. Previous work using a variety of optimization schemes has established that the basis functions which best code natural images in terms of sparse, independent components resemble a Gabor-like wavelet basis in which the basis functions are spatially localized, oriented and bandpass in spatial-frequency [1, 2, 3, 4]. In order to tile the joint space of position, orientation, and spatial-frequency in a manner that yields useful image representations, it has also been advocated that the basis set be overcomplete [5], where the number of basis functions exceeds the dimensionality of the images being coded. A major challenge in learning overcomplete bases, though, comes from the fact that the posterior distribution over the coefficients must be sampled during learning. When the posterior is sharply peaked, as it is when a sparse prior is imposed, then conventional sampling methods become especially cumbersome. 842 B. A. Olshausen and K. J. Millman One approach to dealing with the problems associated with overcomplete codes and sparse priors is suggested by the form of the resulting posterior distribution over the coefficients averaged over many images. Shown below is the posterior distribution of one of the coefficients in a 4 x's overcomplete representation. The sparse prior that was imposed in learning was a Cauchy distribution and is overlaid (dashed line). It would seem that the coefficients do not fit this imposed prior very well, and instead want to occupy one of two states: an inactive state in which the coefficient is set nearly to zero, and an active state in which the coefficient takes on some significant non-zero value along a continuum. This suggests that the appropriate choice of prior is one that is capable of capturing these two discrete states. II I I ? I I . I I -2 \ 0 2 coo!IIdon1v.... Figure 1: Posterior distribution of coefficients with Cauchy prior overlaid. Our approach to modeling this form of sparse structure uses a mixture-of-Gaussians prior over the coefficients. A set of binary or ternary state variables determine whether the coefficient is in the active or inactive state, and then the coefficient distribution is Gaussian distributed with a variance and mean that depends on the state variable. An important advantage of this approach, with regard to the sampling problems mentioned above, is that the use of Gaussian distributions allows an analytical solution for integrating over the posterior distribution for a given setting of the state variables. The only sampling that needs to be done then is over the binary or ternary state variables. We show here that this problem is a tractable one. This approach differs from that taken previously by Attias [6] in that we do not use variational methods to approximate the posterior, but rather we rely on sampling to adequately characterize the posterior distribution over the coefficients. 2 Mixture-of-Gaussians model An image, I(x, y), is modeled as a linear superposition of basis functions, ?i(X, y), with coefficients ai, plus Gaussian noise II( x, y) : (1) In what follows this will be expressed in vector-matrix notation as 1= q. a + II. The prior probability distribution over the coefficients is factorial, with the distribution over each coefficient ai modeled as a mixture-of-Gaussians distribution with either two or three Gaussians (fig. 2). A set of binary or ternary state variables Si then determine which Gaussian is used to describe the coefficients. The total prior over both sets of variables, a and s, is of the form (2) 843 Learning Sparse Codes with a Mixture-of-Gaussians Prior Two Gaussians (binary state variables) Three Gaussians (ternary state variables) P(lIj) P(lIj) _,;=-1 .<;=1 ai ai Figure 2: Mixture-of-Gaussians prior. where P(Si) determines the probability of being in the active or inactive states, and P(ailsi) is a Gaussian distribution whose mean and variance is determined by the current state Si. The total image probability is then given by P(IIO) = L P(s/O) JP(I/a, O)P(als, O)da (3) s where P(Ila, 0) 1 e -~II-4>aI2 -2 (4) ZAN P(als,O) _l_e-t(a-Il(s))t Aa(s) (a-Il(s)) (5) ZAa(s) P(sIO) 1 _1 s t --e 2 ZA. A s (6) ? and the parameters 0 include AN, 4), Aa(s), f.L(s), and As . Aa(s) is a diagonal inverse covariance matrix with elements Aa(S)ii = Aa; (Si). (The notations Aa(s) and f.L(s) are used here to explicitly reflect the dependence of the means and variances of the ai on sd As is also diagonal (for now) with elements ASii = As;. The model is illustrated graphically in figure 3. Si (binary or ternary) Figure 3: Image model. B. A. Olshausen and K. J. Millman 844 3 Learning The objective function for learning the parameters of the model is the average log-likelihood: (7) ? = (log P(IIO)) Maximizing this objective will minimize the lower bound on coding length. Learning is accomplished via gradient ascent on the objective, ?. The learning rules for the parameters As, Aa (s), J-t( s) and ~ are given by: ex = {)? {)>"Si 1 2 [(Si)P(SiI 9) - (8) (si)P(sII,9)] {)? {)>"ai (u) ! [(8(Si - u))P(sII,9)_ 2 >"ai (u) (8(Si - u) (Kii(U) - 2ai(U)J-ti(U) + J-t~(u)))P(sII,9)] (9) {)? {)J-ti( u) >"ai (u) (8(Si - u) (ai(u) - J-ti(U))) (10) {)? {)~ >"N [I (a(s)) P(sII,9) - ~ (K(s)) P(SII,9)] (11) where u takes on values 0,1 (binary) or -1,0,1 (ternary) and K(s) H- 1 (s) + a(s) a(s)T. (a and H are defined in eqs. 15 and 16 in the next section.) Note that in these expressions we have dropped the outer brackets averaging over images simply to reduce clutter. Thus, for each image we must sample from the posterior P(sll, 0) in order to collect the appropriate statistics needed for learning. These statistics must be accumulated over many different images, and then the parameters are updated according to the rules above. Note that this approach differs from that of Attias [6] in that we do not attempt to sum over all states, s, or to use the variational approximation to approximate the posterior. Instead, we are effectively summing only over those states that are most probable according to the posterior. We conjecture that this scheme will work in practice because the posterior has significant probability only for a small fraction of states s, and so it can be well-characterized by a relatively small number of samples. Next we present an efficient method for Gibbs sampling from the posterior. 4 Sampling and inference In order to sample from the posterior P(sll,O), we first cast it in Boltzmann form: P(sll,O) ex e-E(s) where E(s) = -logP(s,IIO) = -logP(sIO) J P(lla,O)P(als,O)da (12) Learning Sparse Codes with a Mixture-of-Gaussians Prior 845 ~ST Ass + 10gZAa(S) + Eals(a,s) + ~IOgdetH(s) + const. (13) and (14) a = argminEals(a,s) H(s) = \7\7 aEals(a, s) (15) a = )\Nif!T if! + Aa(s) (16) Gibbs-sampling on P(sII, (}) can be performed by flipping state variables Si according to P(Si t- sa) = l+e~E(?i~?a) 1 P(Si t- sa) = (binary) (17) (ternary) (18) Where sa = Si in the binary case, and sa and sf3 are the two alternative states in the ternary case. AE(Si t- sa) denotes the change in E(s) due to changing Si to sa and is given by: (19) where ASi = sa - Si, AAai = Aai (sa) - Aai (Si), J = H- 1 , and Vi = Aai (Si) J.Li(Si). Note that all computations for considering a change of state are local and involve only terms with index i. Thus, deciding whether or not to change state can be computed quickly. However, if a change of state is accepted, then we must update J. Using the Sherman-Morrison formula, this can be kept to an O(N2) computation: J t- J - [ AAak 1 + AAak ] J k Jk (20) Jkk As long as accepted state changes are rare (which we have found to be the case for sparse distributions), then Gibbs sampling may be performed quickly and efficiently. In addition, Hand J are generally very sparse matrices, so as the system is scaled up the number of elements of a that are affected by a flip of Si will be relatively few. In order to code images under this model, a single state of the coefficients must be chosen for a given image. We use for this purpose the MAP estimator: argmaxP(aII,s, (}) (21) arg max P(sII, (}) (22) a s Maximizing the posterior distribution over s is accomplished by assigning a temperature, P(sII, (}) ex e-E(s)/T and gradually lowering it until there are no more state changes. (23) B. A. Olshausen and K. 1. Millman 846 5 5.1 Results Test cases We first trained the algorithm on a number of test cases containing known forms of both sparse and non-sparse (bi-modal) structure, using both critically sampled (complete) and 2x's overcomplete basis sets. The training sets consisted of 6x6 pixel image patches that were created by a sparse superposition of basis functions (36 or 72) with P(ISil = 1) = 0.2, Aa; (0) = 1000, and Aa; (1) = 10. The results of these test cases confirm that the algorithm is capable of correctly extracting both sparse and non-sparse structure from data, and they are not shown here for lack of space. 5.2 Natural images We trained the algorithm on 8x8 image patches extracted from pre-whitened natural images. In all cases, the basis functions were initialized to random functions (white noise) and the prior was initialized to be Gaussian (both Gaussians of roughly equal variance). Shown in figure 4a, b are the results for a set of 128 basis functions (2 x 's overcomplete) in the two-Gausian case. In the three-Gaussian case, the prior was initialized to be platykurtic (all three Gaussians of equal variance but offset at three different positions). Thus, in this case the sparse form of the prior emerged completely from the data. The resulting priors for two of the coefficients are shown in figure 4c, with the posterior distribution averaged over many images overlaid. For some of the coefficients the posterior distribution matches the mixture-of-Gaussians prior well, but for others the tails appear more Laplacian in form. Also, it appears that the extra complexity offered by having three Gaussians is not utilized: Both Gaussians move to the center position and have about the same mean. When a non-sparse, bimodal prior is imposed, the basis function solution does not become localized, oriented, and bandpass as it does with sparse priors. 5.3 Coding efficiency We evaluated the coding efficiency by quantizing the coefficients to different levels and calculating the total coefficient entropy as a function of the distortion introduced by quantization. This was done for basis sets containing 48, 64, 96, and 128 basis functions. At high SNR's the overcomplete basis sets yield better coding efficiency, despite the fact that there are more coefficients to code. However, the point at which this occurs appears to be well beyond the point where errors are no longer perceptually noticeable (around 14 dB). 6 Conclusions We have shown here that both the prior and basis functions of our image model can be adapted to natural images. Without sparseness being imposed, the model both seeks distributions that are sparse and learns the appropriate basis functions for this distribution. Our conjecture that a small number of samples allows the posterior to be sufficiently characterized appears to hold. In all cases here, averages were collected over 40 Gibbs sweeps, with 10 sweeps for initialization. The algorithm proved capable of extracting the structure in challenging datasets in high dimensional spaces. The overcomplete image codes have the lowest coding cost at high SNR levels, but at levels that appear higher than is practically useful. On the other hand, the 847 Learning Sparse Codes with a Mixture-of-Gaussians Prior a. c. : ~~::~~ . - '~ ". ~:. " 10'" . '0" -2 " " 10.... 10~ -. -1 0 I 2 -Z .\ J./ "'\. _I 0 1 2 Figure 4: An overcomplete set of 128 basis functions (a) and priors (b, vertical axis is log-probability) learned from natural images. c, Two of the priors learned from a three-Gaussian mixture using 64 basis functions, with the posterior distribution averaged over many coefficients overlaid. d, Rate distortion curve comparing the coding efficiency of different learned basis sets. sum of marginal entropies likely underestimates the true entropy of the coefficients considerably, as there are certainly statistical dependencies among the coefficients. So it may still be the case that the overcomplete bases will show a win at lower SNR's when these dependencies are included in the model (through the coupling term As). Acknowledgments This work was supported by NIH grant R29-MH057921. References [1] Olshausen BA, Field DJ (1997) "Sparse coding with an overcomplete basis set: A strategy employed by VI?" Vision Research, 37: 3311-3325. [2] Bell AJ, Sejnowski TJ (1997) "The independent components of natural images are edge filters," Vision Research, 37: 3327-3338. [3] van Hateren JH, van der Schaaff A (1997) "Independent component filters of natural images compared with simple cells in primary visual cortex," Proc. Royal Soc. Lond. B, 265: 359-366. [4] Lewicki MS , Olshausen BA (1999) "A probabilistic framework for the adaptation and comparison of image codes," JOSA A, 16(7): 1587-160l. [5] Simoncelli EP, Freeman WT, Adelson EH, Heeger DJ (1992) "Shiftable multiscale transforms," IEEE Transactions on Information Theory, 38(2): 587-607. [6] Attias H (1999) "Independent factor analysis," Neural Computation, 11: 803-852. 

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An Oscillatory Correlation Framework for Computational Auditory Scene Analysis GuyJ.Brown Department of Computer Science University of Sheffield Regent Court, 211 Portobello Street, Sheffield S 1 4DP, UK Email: g.brown@dcs.shefac.uk DeLiang L. Wang Department of Computer and Information Science and Centre for Cognitive Science The Ohio State University Columbus, OH 43210-1277, USA Email: dwang@cis.ohio-state.edu Abstract A neural model is described which uses oscillatory correlation to segregate speech from interfering sound sources. The core of the model is a two-layer neural oscillator network. A sound stream is represented by a synchronized population of oscillators, and different streams are represented by desynchronized oscillator populations. The model has been evaluated using a corpus of speech mixed with interfering sounds, and produces an improvement in signal-to-noise ratio for every mixture. 1 Introduction Speech is seldom heard in isolation: usually, it is mixed with other environmental sounds. Hence, the auditory system must parse the acoustic mixture reaching the ears in order to retrieve a description of each sound source, a process termed auditory scene analysis (ASA) [2] . Conceptually, ASA may be regarded as a two-stage process . The first stage (which we term 'segmentation') decomposes the acoustic stimulus into a collection of sensory elements. In the second stage ('grouping'), elements that are likely to have arisen from the same environmental event are combined into a perceptual structure called a stream. Streams may be further interpreted by higher-level cognitive processes. Recently, there has been a growing interest in the development of computational systems that mimic ASA [4], [1], [5]. Such computational auditory scene analysis (CASA) systems are inspired by auditory function but do not model it closely; rather, they employ symbolic search or high-level inference engines. Although the performance of these systems is encouraging, they are no match for the abilities of a human listener; also, they tend to be complex and computationally intensive. In short, CASA currently remains an unsolved problem for real-time applications such as automatic speech recognition. Given that human listeners can segregate concurrent sounds with apparent ease, computational systems that are more closely modelled on the neurobiological mechanisms of hearing may offer a performance advantage over existing CAS A systems. This observation - together with a desire to understand the neurobiological basis of ASA - has led some investigators to propose neural network models of ASA. Most recently, Brown and Wang [3] have given an account of concurrent vowel separation based on oscillatory correlation. In this framework, oscillators that represent a perceptual stream are synchronized (phase locked with zero phase lag), and are desynchronized from oscillators that represent different streams [8]. Evidence for the oscillatory correlation theory comes from neurobiological studies which report synchronised oscillations in the auditory, visual and olfactory cortices (see [10] for a review). G. J. Brown and D. L. Wang 748 In this paper, we propose a neural network model that uses oscillatory correlation as the underlying neural mechanism for ASA; streams are formed by synchronizing oscillators in a two-dimensional time-frequency network. The model is evaluated on a task that involves the separation of two time-varying sounds. It therefore extends our previous study [3], which only considered the segregation of vowel sounds with static spectra. 2 Model description The input to the model consists of a mixture of speech and an interfering sound source, sampled at a rate of 16 kHz with 16 bit resolution. This input signal is processed in four stages described below (see [10] for a detailed account). 2.1 Peripheral auditory processing Peripheral auditory frequency selectivity is modelled using a bank of 128 gammatone filters with center frequencies equally distributed on the equivalent rectangular bandwidth (ERB) scale between 80 Hz and 5 kHz [1]. Subsequently, the output of each filter is processed by a model of inner hair cell function. The output of the hair cell model is a probabilistic representation of auditory nerve firing activity. 2.2 Mid-level auditory representations Mechanisms similar to those underlying pitch perception can contribute to the perceptual separation of sounds that have different fundamental frequencies (FOs) [3]. Accordingly, the second stage of the model extracts periodicity information from the simulated auditory nerve firing patterns. This is achieved by computing a running autocorrelation of the auditory nerve activity in each channel , forming a representation known as a correlogram [1], [5]. At time step j, the autocorrelation A(iJ,'t) for channel i with time lag 't is given by: K-I I. r(i,j-k)r(i,j-k-'t)w(k) A(i, j,'t) = (1) k=O Here, r is the output of the hair cell model and w is a rectangular window of width K time steps. We use K = 320, corresponding to a window width of 20 ms. The autocorrelation lag 't is computed in L steps of the sampling period between 0 and L-1 ; we use L = 201, corresponding to a maximum delay of 12.5 ms. Equation (1) is computed for M time frames, taken at 10 ms intervals (i .e., at intervals of 160 steps of the time indexj). For periodic sounds, a characteristic 'spine' appears in the correlogram which is centered on the lag corresponding to the stimulus period (Figure 1A). This pitch-related structure can be emphasized by forming a 'pooled' correlogram s(j,'t), which exhibits a prominent peak at the delay corresponding to perceived pitch: N s(j, 't) I. A (i, j, 't) = i (2) =I It is also possible to extract harmonics and formants from the correlogram, since frequency channels that are excited by the same acoustic component share a similar pattern of periodicity. Bands of coherent periodicity can be identified by cross-correlating adjacent correlogram channels; regions of high correlation indicate a harmonic or formant [1] . The cross-correlation C(iJ) between channels i and i+ 1 at time frame j is defined as: L-I C(i,j) = IL.A(i,j, 't)A(i+l,j, 't) t=O (l~i~N-l) (3) Here, A(i, j , 't) is the autocorrelation function of (1) which has been normalized to have zero mean and unity variance. A typical cross-correlation function is shown in Figure 1A. 749 Oscillatory Correlation for CASA 2.3 Neural oscillator network: overview Segmentation and grouping take place within a two-layer oscillator network (Figure IB). The basic unit of the network is a single oscillator, which is defined as a reciprocally connected excitatory variable x and inhibitory variable y [7]. Since each layer of the network takes the form of a time-frequency grid, we index each oscillator according to its frequency channel (i) and time frame (j): Xij = 3xij-xt+2-Yij+lij+Sij+P (4a) Yij = ?(y(1 + tanh(xi/~? (4b) - Yij) Here, Ii} represents external input to the oscillator, Si} denotes the coupling from other oscillators in the network, c, 'Y and ~ are parameters, and p is the amplitude of a Gaussian noise term. If coupling and noise are ignored and Ii} is held constant, (4) defines a relaxation oscillator with two time scales. The x-nullcline, i.e. Xii' = 0, is a cubic function and the y-nullcline is a sigmoid function. If Ii" > 0, the two nul clines intersect only at a point along the middle branch of the cubic with ~ chosen small. In this case, the oscillator exhibits a stable limit cycle for small values of c, and is referred to as enabled. The limit cycle alternates between silent and active phases of near steady-state behaviour. Compared to motion within each phase, the alternation between phases takes place rapidly, and is referred to as jumping. If Ii" < 0, the two nullclines intersect at a stable fixed point. In this case, no oscillation occurs. Hence, oscillations in (4) are stimulus-dependent. 2.4 Neural oscillator network: segment layer In the first layer of the network, segments are formed - blocks of synchronised oscillators that trace the evolution of an acoustic component through time and frequency. The first layer is a two-dimensional time-frequency grid of oscillators with a global inhibitor (see Figure IB). The coupling term Sij in (4a) is defined as ~ Sij = kl E (5) Wij ,k/H(xk/-e x )- WzH(z-e z ) N(i, j) where H is the Heaviside function (i.e., H(x) = I for x ~ 0, and zero otherwise), Wij,kl is the connection weight from an oscillator (iJ) to an oscillator (k,/) and N(iJ) is the four nearest neighbors of (iJ). The threshold ex is chosen so that an oscillator has no influence on its A B 5000 'N ::z:: ';:: 2741 u <= I!.l '"i3" d::... ~I!.l 1457 U Q) <= <= ~ ..c: U i n.D i , 2.5 5.0 j , I 7.5 10.0 12.5 Autocorrelation Lag (ms) Figure I: A. Correlogram of a mixture of speech and trill telephone, taken 450 ms after the start of the stimulus. The pooled correlogram is shown in the bottom panel, and the crosscorrelation function is shown on the right. B. Structure of the two-layer oscillator network. G. J. Brown and D. L. Wang 750 neighbors unless it is in the active phase. The weight of neighboring connections along the time axis is uniformly set to 1. The connection weight between an oscillator (iJ) and its vertical neighbor (i+lJ) is set to 1 if C(iJ) exceeds a threshold Se; otherwise it is set to O. Wz is the weight of inhibition from the global inhibitor z, defined as (6) where <roo = 1 if xi} 2:: Sz for at least one oscillator (iJ), and <roo = 0 otherwise. Hence Sz is a threshold. If <roo = 1, z ~ 1. Small segments may form which do not correspond to perceptually significant acoustic components. In order to remove these noisy fragments, we introduce a lateral potential Pi} for oscillator (iJ), defined as [11]: Pij = (1 - Pij)H[ L.. H(x kl - ex) - epJ - ?Pij (7) kleNp(i,j) Here, Sp is a threshold. Nf(i J ) is called the potential neighborhood of (iJ), which is chosen to be (iJ-l) and (iJ+l). I both neighbors of (iJ) are active, Pi} approaches 1 on a fast time scale; otherwise, Pij relaxes to 0 on a slow time scale determined by c. The lateral potential plays its role by gating the input to an oscillator. More specifically, we replace (4a) with (4a') iij = 3xij-x:j +2-Yij+ lijH(pij-e) +Sij+P With Pij initialized to 1, it follows that Pij will drop below the threshold S unless the oscillator (iJ) receives excitation from its entire potential neighborhood. Given our choice of neighborhood in (5), this implies that a segment must extend for at least three consecutive time frames. Oscillators that are stimulated but cannot maintain a high potential are relegated to a discontiguous 'background' of noisy activity. An oscillator (iJ) is stimulated if its corresponding input lij > O. Oscillators are stimulated only if the energy in their corresponding correlogram channel exceeds a threshold Sa. It is evident from (1) that the energy in a correlogram channel i at time j corresponds to A(iJ,O); thus we set Ii} =0.2 if A(iJ,O) > Sa' and Iij =-5 otherwise. Figure 2A shows the segmentation of a mixture of speech and trill telephone. The network was simulated by the LEGION algorithm [8], producing 94 segments (each represented by a distinct gray level) plus the background (shown in black). For convenience we show all segments together in Figure 2A, but each actually arises during a unique time interval. B g 5000 2741 >. <.) c: <) ~ 1457 ~ !:S 1: <) 729 U 03 c: c: ~ ..c: 315 U 80 0.0 Time (seconds) 1.5 Time (seconds) Figure 2: A. Segments formed by the first layer of the network for a mixture of speech and trill telephone. B. Categorization of segments according to FO. Gray pixels represent the set P, and white pixels represent regions that do not agree with the FO. 751 Oscillatory Correlation for CASA 2.5 Neural oscillator network: grouping layer The second layer is a two-dimensional network of laterally coupled oscillators without global inhibition. Oscillators in this layer are stimulated if the corresponding oscillator in the first layer is stimulated and does not form part of the background. Initially, all oscillators have the same phase, implying that all segments from the first layer are allocated to the same stream. This initialization is consistent with psychophysical evidence suggesting that perceptual fusion is the default state of auditory organisation [2]. In the second layer, an oscillator has the same form as in (4), except that Xu is changed to: iii = 3x ij - x~ + 2 - (4a") Yij + Ii) 1 + !1H(Pij - a)] + Sij + P Here, Jl is a small positive parameter; this implies that an oscillator with a high lateral potential gets a slightly higher external input. We choose NpCiJ) and aR so that oscillators which correspond to the longest segment from the first layer are the first to jump to the active phase. The longest segment is identified by using the mechanism described in [9]. The coupling term in (4a") consists of two types of coupling: e v Sij = Sij + Sij (8) Here, S;j represents mutual excitation between oscillators within each segment. We set S~ = 4 if the active oscillators from the same segment occupy more than half of the length of the segment; otherwise S~j = 0.1 if there is at least one active oscillator from the same segment. S; denotes vertical connections between oscillators corresponding to The coupling term different frequency channels and different segments, but within the same time frame. At each time frame, an FO is estimated from the pooled correlogram (2) and this is used to classify frequency channels into two categories: a set of channels, P, that are consistent with the FO, and a set of channels that are not (Figure 2B). Given the delay 'tm at which the largest peak occurs in the pooled correlogram, for each channel i at time frame j, i E P if AU, j, 't m ) / A(i, j, 0) > ad (9) Since AUJ,O) is the energy in correlogram channel i at time j, (9) amounts to classification on the basis of an energy threshold. We use ad = 0.95. The delay 'tm can be found by using a winner-take-all network, although for simplicity we currently apply a maximum selector. A 5IMM) N N :r:'-' >. :r:'-' 2741 u >. u Q) Q) ;:::l ;:::l c: .. g' c: go 1457 .... Q) c 729 Q) U U "0 c: Q) c: ; ..c: 1457 ~ .... Q) u.. C Q) 2741 ? 315 ..c: U U 80 Time (seconds) Time (seconds) Figure 3: A. Snapshot showing the activity of the second layer shortly after the start of simulation. Active oscillators (white pixels) correspond to the speech stream. B. Another snapshot, taken shortly after A. Active oscillators correspond to the telephone stream. G. J. Brown and D. L. Wang 752 The FO classification process operates on channels, rather than segments. As a result, channels within the same segment at a particular time frame may be allocated to different FO categories. Since segments cannot be decomposed, we enforce a rule that all channels of the same frame within each segment must belong to the same FO category as that of the majority of channels. After this conformational step, vertical connections are fonned such that, at each time frame, two oscillators of different segments have mutual excitatory links if the two corresponding channels belong to the same FO category; otherwise they have mutual inhibitory links. S~ is set to -O.S if (iJ) receives an input from its inhibitory links; similarly, s~ is set to O.S if (iJ) receives an input from its vertical excitatory links. At present, our model has no mechanism for grouping segments that do not overlap in time. Accordingly, we limit operation of the second layer to the time span of the longest segment. After fonning lateral connections and trimming by the longest segment, the network is numerically solved using the singular limit method [6]. Figure 3 shows the response of the second layer to the mixture of speech and trill telephone. The figure shows two snapshots of the second layer, where a white pixel indicates an active oscillator and a black pixel indicates a silent oscillator. The network quickly forms two synchronous blocks, which desynchronize from each other. Figure 3A shows a snapshot taken when the oscillator block (stream) corresponding to the segregated speech is in the active phase; Figure 3B shows a subsequent snapshot when the oscillator block corresponding to the trill telephone is in the active phase. Hence, the activity in this layer of the network embodies the result of ASA; the components of an acoustic mixture have been separated using FO infonnation and represented by oscillatory correlation. 2.6 Resynthesis The last stage of the model is a resynthesis path. Phase-corrected output from the gammatone filterbank is divided into 20 ms sections, overlapping by 10 ms and windowed with a raised cosine. A weighting is then applied to each section, which is unity if the corresponding oscillator is in its active phase, and zero otherwise. The weighted filter outputs are summed across all channels to yield a resynthesized wavefonn. A B 70 90 'c:Q "'""' 60 '~ "'""' "0 '-' '-' .9 50 <;:; .... 0 til 0 0 .... 70 > 0 u ~ ?0 I B e;; 20 s= ft- tlO . r;; s= ?I :; 0 -10 ? F ?n,~If .i~ ::1 10 ? ? I ?? I I ..c: u 0 0 .J1l NO Nl N2 N3 N4 N5 N6 N7 N8 N9 Intrusion type 0.. CIl -,,- ",.- n I" ?' Ir r-- F 60 ? 50 f:.Il o 40 s= 0 - I, I 0 80 "0 40 r-- s= 30 100 :;-'\ H< ~ 'i itllE 30 20 Bi IO 0 ' J "f J U:, ~ ", mfi t "1 ',' il t' f l f' j: ,d NO Nl N2 N3 N4 N5 N6 N7 N8 N9 Intrusion type Figure 4: A. SNR before (black bar) and after (grey bar) separation by the model. Results are shown for voiced speech mixed with ten intrusions (NO = 1 kHz tone; Nl = random noise; N2 = noise bursts; N3 = 'cocktail party' noise; N4 = rock music; NS = siren; N6 = trill telephone; N7 = female speech; N8 = male speech; N9 = female speech). B. Percentage of speech energy recovered from each mixture after separation by the model. 

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794 NEURAL ARCHITECTURE Valentino Braitenberg Max Planck Institute Federal Republic of Germany While we are waiting for the ultimate biophysics of cell membranes and synapses to be completed, we may speculate on the shapes of neurons and on the patterns of their connections. Much of this will be significant whatever the outcome of future physiology. Take as an example the isotropy, anisotropy and periodicity of different kinds of neural networks. The very existence of these different types in different parts of the brain (or in different brains) defeats explanation in terms of embryology; the mechanisms of development are able to make one kind of network or another. The reasons for the difference must be in the functions they perform. The tasks which they solve in one case apparently refer to some space which is intrinsically isotropic, in another to a situation in which different coordinates mean different things. In the periodic case, the tasks obviously refer to some kind of modules and to their relations. The examples I have in mind are first the cerebral cortex, quite isotropic in the plane of the cortex, second the cerebellar cortex with very different sets of fibers at right angles to each other (one excitatory, as we know today, and the other inhibitory) and third some of the nerve nets behind the eye of the fly. Besides general patterns of symmetry, some simple statements of a statistical nature can be read off the histological picture. If a neuron is a device picking up excitation (and/or inhibition) on its ten to ten thousand afferent synapses and producing excitation (or inhibition) on ten to ten thousand synapses on other neurons, the density, geometrical distribution and reciprocal overlap of the clouds of afferent and of efferent synapses of individual neurons provide unquestionable constraints to neural computation. In the simple terms of the histological practitioner, this translates into the description of the shapes of dendritic and axonal trees, into counts of neurons and synapses, differential counts of synapses of the excitatory and inhibitory kind and measurements of the axonal and dendritic lengths. 

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Emergence of Topography and Complex Cell Properties from Natural Images using Extensions of ICA Aapo Hyviirinen and Patrik Hoyer Neural Networks Research Center Helsinki University of Technology P.O. Box 5400, FIN-02015 HUT, Finland aapo.hyvarinen~hut.fi, patrik.hoyer~hut.fi http://www.cis.hut.fi/projects/ica/ Abstract Independent component analysis of natural images leads to emergence of simple cell properties, Le. linear filters that resemble wavelets or Gabor functions. In this paper, we extend ICA to explain further properties of VI cells. First, we decompose natural images into independent subspaces instead of scalar components. This model leads to emergence of phase and shift invariant features, similar to those in VI complex cells. Second, we define a topography between the linear components obtained by ICA. The topographic distance between two components is defined by their higher-order correlations, so that two components are close to each other in the topography if they are strongly dependent on each other. This leads to simultaneous emergence of both topography and invariances similar to complex cell properties. 1 Introduction A fundamental approach in signal processing is to design a statistical generative model of the observed signals. Such an approach is also useful for modeling the properties of neurons in primary sensory areas. The basic models that we consider here express a static monochrome image J (x, y) as a linear superposition of some features or basis functions bi (x, y): n J(x, y) = 2: bi(x, Y)Si (1) i=l where the Si are stochastic coefficients, different for each image J(x, y). Estimation of the model in Eq. (1) consists of determining the values of Si and bi(x, y) for all i and (x, y), given a sufficient number of observations of images, or in practice, image patches J(x,y). We restrict ourselves here to the basic case where the bi(x,y) form an invertible linear system. Then we can invert Si =< Wi, J > where the Wi denote the inverse filters, and < Wi, J >= L.x,y Wi(X, y)J(x, y) denotes the dot-product. 828 A. Hyviirinen and P Hoyer The Wi (x, y) can then be identified as the receptive fields of the model simple cells, and the Si are their activities when presented with a given image patch I(x, y). In the basic case, we assume that the Si are nongaussian, and mutually independent. This type of decomposition is called independent component analysis (ICA) [3, 9, 1, 8], or sparse coding [13]. Olshausen and Field [13] showed that when this model is estimated with input data consisting of patches of natural scenes, the obtained filters Wi(X,y) have the three principal properties of simple cells in VI: they are localized, oriented, and bandpass (selective to scale/frequency). Van Hateren and van der Schaaf [15] compared quantitatively the obtained filters Wi(X, y) with those measured by single-cell recordings of the macaque cortex, and found a good match for most of the parameters. We show in this paper that simple extensions of the basic ICA model explain emergence of further properties of VI cells: topography and the invariances of complex cells. Due to space limitations, we can only give the basic ideas in this paper. More details can be found in [6, 5, 7]. First, using the method of feature subspaces [11], we model the response of a complex cell as the norm of the projection of the input vector (image patch) onto a linear subspace, which is equivalent to the classical energy models. Then we maximize the independence between the norms of such projections, or energies. Thus we obtain features that are localized in space, oriented, and bandpass, like those given by simple cells, or Gabor analysis. In contrast to simple linear filters, however, the obtained feature subspaces also show emergence of phase invariance and (limited) shift or translation invariance. Maximizing the independence, or equivalently, the sparseness of the norms of the projections to feature subspaces thus allows for the emergence of exactly those invariances that are encountered in complex cells. Second, we extend this model of independent subspaces so that we have overlapping subspaces, and every subspace corresponds to a neighborhood on a topographic grid. This is called topographic ICA, since it defines a topographic organization between components. Components that are far from each other on the grid are independent, like in ICA. In contrast, components that are near to each other are not independent: they have strong higher-order correlations. This model shows emergence of both complex cell properties and topography from image data. 2 Independent subspaces as complex cells In addition to the simple cells that can be modelled by basic ICA, another important class of cells in VI is complex cells. The two principal properties that distinguish complex cells from simple cells are phase invariance and (limited) shift invariance. The purpose of the first model in this paper is to explain the emergence of such phase and shift invariant features using a modification of the ICA model. The modification is based on combining the principle of invariant-feature subspaces [11] and the model of multidimensional independent component analysis [2]. Invariant feature subspaces. The principle of invariant-feature subspaces states that one may consider an invariant feature as a linear subspace in a feature space. The value of the invariant, higher-order feature is given by (the square of) the norm of the projection of the given data point on that subspace, which is typically spanned by lower-order features. A feature subspace, as any linear subspace, can always be represented by a set of orthogonal basis vectors, say Wi(X, y), i = 1, ... , m, where m is the dimension of the subspace. Then the value F(I) of the feature F with input vector I(x, y) is given by F(I) = L::l < Wi, I >2, where a square root 829 Emergence of VI properties using Extensions ofleA might be taken. In fact, this is equivalent to computing the distance between the input vector I (X, y) and a general linear combination of the basis vectors (filters) Wi(X, y) of the feature subspace [11]. In [11], it was shown that this principle, when combined with competitive learning techniques, can lead to emergence of invariant image features. Multidimensional independent component analysis. In multidimensional independent component analysis [2] (see also [12]), a linear generative model as in Eq. (1) is assumed. In contrast to ordinary leA, however, the components (responses) Si are not assumed to be all mutually independent. Instead, it is assumed that the Si can be divided into couples, triplets or in general m-tuples, such that the Si inside a given m-tuple may be dependent on each other, but dependencies between different m-tuples are not allowed. Every m-tuple of Si corresponds to m basis vectors bi(x, y). The m-dimensional probability densities inside the m-tuples of Si is not specified in advance in the general definition of multidimensional leA [2]. In the following, let us denote by J the number of independent feature subspaces, and by Sj,j = 1, ... , J the set of the indices of the Si belonging to the subspace of index j . Independent feature subspaces. Invariant-feature subspaces can be embedded in multidimensional independent component analysis by considering probability distributions for the m-tuples of Si that are spherically symmetric, i.e. depend only on the norm. In other words, the probability density Pj (.) of the m-tuple with index j E {1, ... , J}, can be expressed as a function of the sum of the squares of the si,i E Sj only. For simplicity, we assume further that the Pj(') are equal for all j, i.e. for all subspaces. Assume that the data consists of K observed image patches I k (x, y), k = 1, ... , K. Then the logarithm of the likelihood L of the data given the model can be expressed as K 10gL(wi(x, y), i = L.n) = L J L 10gp(L < Wi, h >2) + Klog Idet WI k=1 j=1 (2) iESj where P(LiESj sT) = pj(si,i E Sj) gives the probability density inside the j-th m-tuple of Si, and W is a matrix containing the filters Wi(X, y) as its columns. As in basic leA, prewhitening of the data allows us to consider the Wi(X, y) to be orthonormal, and this implies that log I det WI is zero [6]. Thus we see that the likelihood in Eq. (2) is a function of the norms of the projections of Ik(x,y) on the subspaces indexed by j, which are spanned by the orthonormal basis sets given by Wi(X, y), i E Sj. Since the norm of the projection of visual data on practically any subspace has a supergaussian distribution, we need to choose the probability density P in the model to be sparse [13], i.e. supergaussian [8]. For example, we could use the following probability distribution logp( L st) = -O:[L s~11/2 + {3, iESj (3) iESj which could be considered a multi-dimensional version of the exponential distribution. Now we see that the estimation of the model consists of finding subspaces such that the norms of the projections of the (whitened) data on those subspaces have maximally sparse distributions. The introduced "independent (feature) subspace analysis" is a natural generalization of ordinary leA. In fact, if the projections on the subspaces are reduced to dotproducts, i.e. projections on 1-D subs paces , the model reduces to ordinary leA A. Hyviirinen and P. Hoyer 830 (provided that, in addition, the independent components are assumed to have nonskewed distributions). It is to be expected that the norms of the projections on the subspaces represent some higher-order, invariant features. The exact nature of the invariances has not been specified in the model but will emerge from the input data, using only the prior information on their independence. When independent subspace analysis is applied to natural image data, we can identify the norms of the projections (2:iESj st)1/2 as the responses of the complex cells. If the individual filter vectors Wi(X, y) are identified with the receptive fields of simple cells, this can be interpreted as a hierarchical model where the complex cell response is computed from simple cell responses Si, in a manner similar to the classical energy models for complex cells. Experiments (see below and [6]) show that the model does lead to emergence of those invariances that are encountered in complex cells. 3 Topographic leA The independent subspace analysis model introduces a certain dependence structure for the components Si. Let us assume that the distribution in the subspace is sparse, which means that the norm of the projection is most of the time very near to zero. This is the case, for example, if the densities inside the subspaces are specified as in (3). Then the model implies that two components Si and Sj that belong to the same subspace tend to be nonzero simultaneously. In other words, and S] are positively correlated. This seems to be a preponderant structure of dependency in most natural data. For image data, this has also been noted by Simoncelli [14). s; Now we generalize the model defined by (2) so that it models this kind of dependence not only inside the m-tuples, but among all ''neighboring'' components. A neighborhood relation defines a topographic order [10). (A different generalization based on an explicit generative model is given in [5].) We define the model by the following likelihood: K n n 10gL(wi(x,y),i = 1, ... ,n) = LLG(Lh(i,j) k=I j=l < Wi,h >2) +KlogldetWI (4) i=l Here, h(i, j) is a neighborhood function, which expresses the strength of the connection between the i-th and j-th units. The neighborhood function can be defined in the same way as with the self-organizing map [10). Neighborhoods can thus be defined as one-dimensional or two-dimensional; 2-D neighborhoods can be square or hexagonal. A simple example is to define a 1-D neighborhood relation by h(i,j) = {I, 0, if Ii - ~I ~ m otherwIse. (5) The constant m defines here the width of the neighborhood. The function G has a similar role as the log-density of the independent components in classic ICA. For image data, or other data with a sparse structure, G should be chosen as in independent subspace analysis, see Eq. (3). Properties of the topographic leA model. Here, we consider for simplicity only the case of sparse data. The first basic property is that all the components Si are uncorrelated, as can be easily proven by symmetry arguments [5]. Moreover, their variances can be defined to be equal to unity, as in classic ICA. Second, components Si and S j that are near to each other, Le. such that h( i, j) is significantly non-zero, Emergence oj VI properties using Extensions ojleA 831 tend to be active (non-zero) at the same time. In other words, their energies sf and s; are positively correlated. Third, latent variables that are far from each other are practically independent. Higher-order correlation decreases as a function of distance, assuming that the neighborhood is defined in a way similar to that in (5). For details, see [5]. Let us note that our definition of topography by higher-order correlations is very different from the one used in practically all existing topographic mapping methods. Usually, the distance is defined by basic geometrical relations like Euclidean distance or correlation. Interestingly, our principle makes it possible to define a topography even among a set of orthogonal vectors whose Euclidean distances are all equal. Such orthogonal vectors are actually encountered in leA, where the basis vectors and filters can be constrained to be orthogonal in the whitened space. 4 Experiments with natural image data We applied our methods on natural image data. The data was obtained by taking 16 x 16 pixel image patches at random locations from monochrome photographs depicting wild-life scenes (animals, meadows, forests, etc.). Preprocessing consisted of removing the De component and reducing the dimension of the data to 160 by peA. For details on the experiments, see [6, 5]. Fig. 1 shows the basis vectors of the 40 feature subspaces (complex cells), when subspace dimension was chosen to be 4. It can be seen that the basis vectors associated with a single complex cell all have approximately the same orientation and frequency. Their locations are not identical, but close to each other. The phases differ considerably. Every feature subspace can thus be considered a generalization of a quadrature-phase filter pair as found in the classical energy models, enabling the cell to be selective to some given orientation and frequency, but invariant to phase and somewhat invariant to shifts. Using 4 dimensions instead of 2 greatly enhances the shift invariance of the feature subspace. In topographic leA, the neighborhood function was defined so that every neighborhood consisted of a 3 x 3 square of 9 units on a 2-D torus lattice [10]. The obtained basis vectors, are shown in Fig. 2. The basis vectors are similar to those obtained by ordinary leA of image data [13, 1]. In addition, they have a clear topographic organization. In addition, the connection to independent subspace analysis is clear from Fig. 2. Two neighboring basis vectors in Fig. 2 tend to be of the same orientation and frequency. Their locations are near to each other as well. In contrast, their phases are very different. This means that a neighborhood of such basis vectors, i.e. simple cells, is similar to an independent subspace. Thus it functions as a complex cell. This was demonstrated in detail in [5]. 5 Discussion We introduced here two extensions of leA that are especially useful for image modelling. The first model uses a subspace representation to model invariant features. It turns out that the independent subspaces of natural images are similar to complex cells. The second model is a further extension of the independent subspace model. This topographic leA model is a generative model that combines topographic mapping with leA. As in all topographic mappings, the distance in the representation space (on the topographic "grid") is related to some measure of distance between represented components. In topographic leA, the distance between represented components is defined by higher-order correlations, which gives 832 A. Hyviirinen and P Hoyer the natural distance measure in the context of leA. An approach closely related to ours is given by Kohonen's Adaptive Subspace SelfOrganizing Map [11). However, the emergence of shift invariance in [11) was conditional to restricting consecutive patches to come from nearby locations in the image, giving the input data a temporal structure like in a smoothly changing image sequence. Similar developments were given by F6ldiak [4). In contrast to these two theories, we formulated an explicit image model. This independent subspace analysis model shows that emergence of complex cell properties is possible using patches at random, independently selected locations, which proves that there is enough information in static images to explain the properties of complex cells. Moreover, by extending this subspace model to model topography, we showed that the emergence of both topography and complex cell properties can be explained by a single principle: neighboring cells should have strong higher-order correlations. References [1] A.J. Bell and T.J. Sejnowski. The 'independent components' of natural scenes are edge filters. Vision Research, 37:3327-3338, 1997. [2] J.-F. Cardoso. Multidimensional independent component analysis. In Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP'98), Seattle, WA, 1998. [3] P. Comon. Independent component analysis - a new concept? Signal Processing, 36:287-314, 1994. [4] P. Foldiak. Learning invariance from transformation sequences. Neural Computation, 3:194-200, 1991. [5] A. Hyvarinen and P. O. Hoyer. Topographic independent component analysis. 1999. Submitted, available at http://www.cis.hut.firaapo/. [6] A. Hyvarinen and P. O. Hoyer. Emergence of phase and shift invariant features by decomposition of natur:al images into independent feature subspaces. Neural Computation, 2000. (in press). [7] A. Hyvarinen, P. O. Hoyer, and M. Inki. The independence assumption: Analyzing the independence of the components by topography. In M. Girolami, editor, Advances in Independent Component Analysis. Springer-Verlag, 2000. in press. [8] A. Hyvarinen and E. Oja. A fast fixed-point algorithm for independent component analysis. Neural Computation, 9(7):1483-1492, 1997. [9] C. Jutten and J. Herault. Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture. Signal Processing, 24:1-10, 1991. [10] T. Kohonen. Self-Organizing Maps. Springer-Verlag, Berlin, Heidelberg, New York, 1995. [11] T. Kohonen. Emergence of invariant-feature detectors in the adaptive-subspace selforganizing map. Biological Cybernetics, 75:281-291, 1996. [12] J. K. Lin. Factorizing multivariate function classes. In Advances in Neural Information Processing Systems, volume 10, pages 563-569. The MIT Press, 1998. [13] B. A. Olshausen and D. J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381:607-609, 1996. [14] E. P. Simoncelli and O. Schwartz. Modeling surround suppression in VI neurons with a statistically-derived normalization model. In Advances in Neural Information Processing Systems 11, pages 153-159. MIT Press, 1999. [15] J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc. Royal Society ser. B, 265:359-366, 1998. Emergence of Vi properties using Extensions of leA - ", 833 ;II :II ? ? ? .. ? ? .. -, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '11'1 .; ? Figure 1: Independent subspaces of natural image data. The model gives Gaborlike basis vectors for image windows. Every group of four basis vectors corresponds to one independent feature subspace, or complex cell. Basis vectors in a subspace are similar in orientation, location and frequency. In contrast, their phases are very different. .- ? ? ~ I ? ? I; IiioiII iii ? ? " I . iii I , 'i ~ ." . I ? ? Figure 2: Topographic leA of natural image data. This gives Gabor-like basis vectors as well. Basis vectors that are similar in orientation, location and/or frequency are close to each other. The phases of near by basis vectors are very different, giving each neighborhood properties similar to a complex cell. 

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Training Data Selection for Optimal Generalization in Trigonometric Polynomial Networks Masashi Sugiyama*and Hidemitsu Ogawa Department of Computer Science, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo, 152-8552, Japan. sugi@cs. titeck. ac.jp Abstract In this paper, we consider the problem of active learning in trigonometric polynomial networks and give a necessary and sufficient condition of sample points to provide the optimal generalization capability. By analyzing the condition from the functional analytic point of view, we clarify the mechanism of achieving the optimal generalization capability. We also show that a set of training examples satisfying the condition does not only provide the optimal generalization but also reduces the computational complexity and memory required for the calculation of learning results. Finally, examples of sample points satisfying the condition are given and computer simulations are performed to demonstrate the effectiveness of the proposed active learning method. 1 Introduction Supervised learning is obtaining an underlying rule from training examples , and can be formulated as a function approximation problem. If sample points are actively designed, then learning can be performed more efficiently. In this paper, we discuss the problem of designing sample points, referred to as active learning, for optimal generalization. Active learning is classified into two categories depending on the optimality. One is global optimal, where a set of all training examples is optimal (e.g. Fedorov [3]) . The other is greedy optimal, where the next training example to sample is optimal in each step (e.g. MacKay [5], Cohn [2], Fukumizu [4], and Sugiyama and Ogawa [10]). In this paper, we focus on the global optimal case and give a new active learning method in trigonometric polynomial networks. The proposed method does not employ any approximations in its derivation, so that it provides exactly the optimal generalization capability. Moreover, the proposed method reduces the computational complexity and memory required for the calculation of learning results. Finally, the effectiveness of the proposed method is demonstrated through computer simulations. '' http://ogawa-www.cs.titech.ac?jprsugi. 625 Training Data Selection for Optimal Generalization 2 Formulation of supervised learning In this section, the supervised learning problem is formulated from the functional analytic point of view (see Ogawa [7]). Then, our learning criterion and model are described. 2.1 Supervised learning as an inverse problem Let us consider the problem of obtaining the optimal approximation to a target function f(x) of L variables from a set of M training examples. The training examples are made up of sample points Xm E V, where V is a subset of the Ldimensional Euclidean space R L, and corresponding sample values Ym E C: {(xm, Ym) 1Ym = f(xm) + nm}~=l' (1) where Ym is degraded by zero-mean additive noise n m . Let nand Y be Mdimensional vectors whose m-th elements are nm and Ym, respectively. Y is called a sample value vector. In this paper, the target function f(x) is assumed to belong to a reproducing kernel Hilbert space H (Aronszajn [1]). If H is unknown, then it can be estimated by model selection methods (e.g. Sugiyama and Ogawa [9]). Let K(?,?) be the reproducing kernel of H. If a function 'l/Jm(x) is defined as 'l/Jm (x) = K (x, x m ), then the value of f at a sample point Xm is expressed as f(x m ) = (I, 'l/Jm), where (-,.) stands for the inner product. For this reason, 'l/Jm is called a sampling function. Let A be an operator defined as M A= ~ (em0~, (2) m=l where em is the m-th vector of the so-called standard basis in C M and (. 0 7) stands for the Neumann-Schatten productl. A is called a sampling operator. Then, the relationship between f and Y can be expressed as Y = Af +n. (3) Let us denote a mapping from Y to a learning result fo by X: fo = Xy, (4) where X is called a learning operator. Then, the supervised learning problem is reformulated as an inverse problem of obtaining X providing the best approximation fa to f under a certain learning criterion. 2.2 Learning criterion and model As mentioned above, function approximation is performed on the basis of a learning criterion. Our purpose of learning is to minimize the generalization error of the learning result fa measured by Je = Enllfo - f11 2 , (5) where En denotes the ensemble average over noise. In this paper, we adopt projection learning as our learning criterion. Let A*, R(A*), and PR(AO) be the adjoint operator of A, the range of A*, and the orthogonal projection operator onto R(A*), respectively. Then, projection learning is defined as follows. IFor any fixed 9 in a Hilbert space HI and any fixed f in a Hilbert space H2, the Neumann-Schatten product (f ? g) is an operator from HI to H2 defined by using any hE HI as (f?g)h = (h,g)f? 626 M Sugiyama and H. Ogawa Definition 1 (Projection learning) (Ogawa !6j) An operator X is called the projection learning operator if X minimizes the functional J p [X] = En II X nll 2 under the constraint XA = Pn(A*). It is well-known that Eq.(5) can be decomposed into the bias and variance: JG = IIPn(A*)f - fl12+ En11Xn112. (6) Eq.(6) implies that the projection learning criterion reduces the bias to a certain level and minimizes the variance. Let us consider the following function space. Definition 2 (Trigonometric polynomial space) Let x = (e(I),e(2), .. ?,e(L))T. For 1 :S l :S L, let Nl be a positive integer and Vl = [-7r,7r]. Then, a function space H is called a trigonometric polynomial space of order (N1 , N 2 , ... , N L) if H is spanned by (7) (8) The dimension J.l of a trigonometric polynomial space of order (N1 , N 2 , ... , N L) is J.l = (2Nl + 1), and the reproducing kernel of this space is expressed as nf=1 L K(x, x') = II Kl(e(l), e(l)I), (9) l=1 where K, 3 (~{l) ~('}') = { , if e(l) if -=1= e(l)' , e(l) = e(l)I. (10) Active learning in trigonometric polynomial space The problem of active learning is to find a set {Xm}~=1 of sample points providing the optimal generalization capability. In this section, we give the optimal solution to the active learning problem in the trigonometric polynomial space. Let At be the Moore-Penrose generalized inverse 2 of A. Then, the following proposition holds. Proposition 1 If the noise covariance matrix Q is given as Q = a 2 I with a 2 > 0, then the projection learning operator X is expressed as X = At. Note that the sampling operator A is uniquely determined by {Xm}~=1 (see Eq.(2)). From Eq.(6), the bias of a learning result fo becomes zero for all f in H if and only if N(A) = {O}, where NO stands for the null space of an operator. For this reason, 2 An operator X is called the Moore-Penrose generalized inverse of an operator A if X satisfies AXA = A, XAX = X, (AX)'" = AX, and (XA)" = XA . 627 Training Data Selection for Optimal Generalization H Figure 1: Mechanism of noise suppression by Theorem 1. If a set {xm}~= l of sample points satisfies A* A = MI, then XAf = f, IIXntll = JMlln111, and Xn2 = o. we consider the case where a set {Xm}~=l of sample points satisfies N(A) = {o}. In this case, Eq.(6) is reduced to (11) which is equivalent to the noise variance in H. Consequently, the problem of active learning becomes the problem of finding a set {Xm }~= 1 of sample points minimizing Eq.(l1) under the constraint N(A) = {a} . First, we derive a condition for optimal generalization in terms of the sampling operator A. Theorem 1 Assume that the noise covariance matrix Q is given as Q = (721 with > o. Then, Je in Eq.{11) is minimized under the constraint N(A) = {O} if and only if (12) A*A =MI, where I denotes the identity operator on H. In this case, the minimum value of Je is (72J.L/M, where J.L is the dimension of H . (72 Eq.(12) implies that {:;k1jJm}~=l forms a pseudo orthonormal basis (Ogawa [8]) in H, which is an extension of orthonormal bases. The following lemma gives interpretation of Theorem 1. Lemma 1 When a set {Xm}~=l of sample points satisfies Eq.(12}, it holds that for all f E H, (13) XAf f IIAfl1 IIXul1 rullfll for all f E H, { *llull for u E 'R.(A), o (14) (15) for u E'R.(A).l. Eqs.(14) and (15) imply that k A becomes an isometry and VMX becomes a partial isometry with the initial space 'R.(A) , respectively. Let us decompose the noise n as n = nl + n2, where nl E 'R.(A) and n2 E 'R.(A).l. Then, the sample value vector y is rewritten as y = Af + nl + n2. It follows from Eq.(13) that the signal component Af is transformed into the original function f by X. From Eq.(15) , X suppresses the magnitude of noise nl in 'R.(A) by and completely removes the k 628 M Sugiyama and H. Ogawa -71" ? ? C .----. 211" M . -71" 71" ? C (a) Theorem 2 ? ? (b) Theorem 3 Figure 2: Two examples of sample points such that Condition (12) holds (1-? and M = 6). =3 noise n2 in R(A).l. This analysis is summarized in Fig.1. Note that Theorem 1 and its interpretation are valid for all Hilbert spaces such that K(x, x) is a constant for any x. In Theorem 1, we have given a necessary and sufficient condition to minimize Ja in terms of the sampling operator A. Now we give two examples of sample points {X m };;:[=l such that Condition (12) holds. From here on, we focus on the case when the dimension L of the input x is 1 for simplicity. However, the following results can be easily scaled to the case when L > 1. Theorem 2 Let M 2: 1-?, where 1-? is the dimension of H. Let c be an arbitrary constant such that -71" < c :::; -71" + ~. If a set {X m };;:[=l of sample points is determined as (16) then Eq.(12) holds. Theorem 3 Let M = kl-? where k is a positive integer. Let c be an arbitrary constant such that -71" :::; c :::; -71" + ~. If a set {X m };;:[=l of sample points is determined as 271" (17) Xm =c+-r, where r = m - 1 (mod 1-?), 1-? then Eq. (12) holds. Theorem 2 means that M sample points are fixed to 271"1M intervals in the domain [-71",71"] and sample values are gathered once at each point (see Fig.2 (a? . In contrast, Theorem 3 means that 1-? sample points are fixed to 271"11-? intervals in the domain and sample values are gathered k times at each point (see Fig.2 (b?. Now, we discuss calculation methods ofthe projection learning result fo(x). Let h m be the m-th column vector of the M-dimensional matrix (AA*)t. Then, for general sample points, the projection learning result fo(x) can be calculated as M (y, hm )1/Jm(x). fo(x) = L (18) m=l When we use the optimal sample points satisfying Condition (12), the following theorems hold. Theorem 4 When Eq.(12) holds, the projection learning result fo(x) can be calculated as 1 M (19) fo(x) = M LYm1/Jm(X). m=l Training Data Selection for Optimal Generalization 629 Theorem 5 When sample points are determined following Theorem 3, the projection learning result fo (x) can be calculated as 1 I-' 1 J.I. fo(x) = - LYp'I/Jp(x), p=l where YP = k k LYp+J.I.(q-l). (20) q=l In Eq.(18), the coefficient of 'l/Jm(x) is obtained by the inner product (y, h m). In contrast, it is replaced with Ym/M in Eq.(19) , which implies that the Moore-Penrose generalized inverse of AA* is not required for calculating fo(x). This property is quite useful when the number M of training examples is very large since the calculation of the Moore-Penrose generalized inverse of high dimensional matrices is sometimes unstable. In Eq.(20), the number of basis functions is reduced to I-' and the coefficient of 'l/Jp(x) is obtained by Yp/I-', where YP is the mean sample values at xp. For general sample points, the computational complexity and memory required for calculating fo(x) by Eq.(18) are both O(M2). In contrast, Theorem 4 states that if a set of sample points satisfies Eq.(12) , then both the computational complexity and memory are reduced to O(M). Hence, Theorem 1 and Theorem 4 do not only provide the optimal generalization but also reduce the computational complexity and memory. Moreover, if we determine sample points following Theorem 3 and calculate the learning result fo(x) by Theorem 5, then the computational complexity and memory are reduced to 0 (1-'). This is extremely efficient since I-' does not depend on the number M of training examples. The above results are shown in Tab.1. 4 Simulations In this section, the effectiveness of the proposed active learning method is demonstrated through computer simulations. Let H be a trigonometric polynomial space of order 100, and the noise covariance matrix Q be Q = I . Let us consider the following three sampling schemes. (A) Optimal sampling: Training examples are gathered following Theorem 3. (B) Experimental design: Eq.(2) in Cohn [2] is adopted as the active learning criterion. The value of this criterion is evaluated by 30 reference points. The sampling location is determined by multi-point-search with 3 candidates. (C) Passive learning: Training examples are given unilaterally. Fig.3 shows the relation between the number of training examples and the generalization error. The horizontal and vertical axes display the number of training examples and the generalization error Je measured by Eq.(5), respectively. The solid line shows the sampling scheme (A). The dashed and dotted lines denote the averages of 10 trials of the sampling schemes (B) and (C), respectively. When the number of training examples is 201, the generalization error of the sampling scheme (A) is 1 while the generalization errors of the sampling schemes (B) and (C) are 3.18 x 104 and 8.75 x 104 , respectively. This graph illustrates that the proposed sampling scheme gives much better generalization capability than the sampling schemes (B) and (C) especially when the number of training examples is not so large. 5 Conclusion We proposed a new active learning method in the trigonometric polynomial space. The proposed method provides exactly the optimal generalization capability and M Sugiyama and H Ogawa 630 Table 1: Computational complexity and memory required for projection learning. Calculation methods Eq.(18) Computational Complexity and Memory O(M2) 1 Or----,---:---,.-~---r---____." (jj c:: ,gtil ?M O(M) Theorem S? O(J.L) = kJ.L where J.L is the dimension of Hand k is a positive integer. I I I 6 I Cii (jj c:: CI) I .' I ..... I I 4 " Ol CI) I- , I .~ .?: Theorem 4 Optimal sampling Experimental design Passive learning ... 8 E 2 ' .. , . ..... '.;. .... ' "- ........ ........... - .... - OL-----~------~-----~-----~ 300 400 500 600 The number of training examples Figure 3: Relation between the number of training examples and the generalization error. at the same time, it reduces the computational complexity and memory required for the calculation of learning results. The mechanism of achieving the optimal generalization was clarified from the functional analytic point of view. References [1] N. Aronszajn. Theory of reproducing kernels. Transactions on American Mathematical Society, 68:337-404, 1950. [2] D. Cohn, Neural network exploration using optimal experiment design. In J. Cowan et al. (Eds.), Advances in Neural Information Processing Systems 6, pp. 679-686. Morgan-Kaufmann Publishers Inc., San Mateo, CA, 1994. [3J V. V . Fedorov. Theory of Optimal Experiments. Academic Press, New York, 1972. [4] K. Fukumizu. Active learning in multilayer perceptrons. In D. Touretzky et al. (Eds.), Advances in Neural Information Processing Systems 8, pp. 295-301. The MIT Press, Cambridge, 1996. [5] D. MacKay. Information-based objective functions for active data selection. Neural Computation, 4(4):590-604, 1992. [6] H. Ogawa, Projection filter regularization of ill-conditioned problem. In Proceedings of SPIE, 808, Inverse Problems in Optics, pp. 189-196, 1987. [7] H, Ogawa. Neural network learning, generalization and over-learning. In Proceedings of the ICIIPS'92, International Conference on Intelligent Information Processing fj System, vol. 2, pp. 1-6, Beijing, China, 1992. [8] H. Ogawa. Theory of pseudo biorthogonal bases and its application. In Research Institute for Mathematical Science, RIMS Kokyuroku, 1067, Reproducing Kernels and their Applications, pp. 24-38, 1998. [9] M. Sugiyama and H. Ogawa. Functional analytic approach to model selectionSubspace information criterion. In Proceedings of 1999 Workshop on InformationBased Induction Sciences (IBIS'99), pp. 93-98, Syuzenji, Shizuoka, Japan, 1999 (Its complete version is available at ftp://ftp.cs.titech.ac.jp/pub/TR/99/TR990009.ps.gz). [10] M. Sugiyama and H. Ogawa. Incremental active learning in consideration of bias, Technical Report of IEICE, NC99-56, pp. 15-22, 1999 (Its complete version is available at ftp://ftp.cs.titech.ac.jp/pub/TR/99/TR99-001O.ps.gz). 

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Variational Inference for Bayesian Mixtures of Factor Analysers Zoubin Ghahramani and Matthew J. Beal Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, England {zoubin,m.beal}Ggatsby.ucl.ac.uk Abstract We present an algorithm that infers the model structure of a mixture of factor analysers using an efficient and deterministic variational approximation to full Bayesian integration over model parameters. This procedure can automatically determine the optimal number of components and the local dimensionality of each component (Le. the number of factors in each factor analyser) . Alternatively it can be used to infer posterior distributions over number of components and dimensionalities. Since all parameters are integrated out the method is not prone to overfitting. Using a stochastic procedure for adding components it is possible to perform the variational optimisation incrementally and to avoid local maxima. Results show that the method works very well in practice and correctly infers the number and dimensionality of nontrivial synthetic examples. By importance sampling from the variational approximation we show how to obtain unbiased estimates of the true evidence, the exact predictive density, and the KL divergence between the variational posterior and the true posterior, not only in this model but for variational approximations in general. 1 Introduction Factor analysis (FA) is a method for modelling correlations in multidimensional data. The model assumes that each p-dimensional data vector y was generated by first linearly transforming a k < p dimensional vector of unobserved independent zero-mean unit-variance Gaussian sources, x, and then adding a p-dimensional zeromean Gaussian noise vector, n, with diagonal covariance matrix \}!: i.e. y = Ax+n. Integrating out x and n, the marginal density of y is Gaussian with zero mean and covariance AA T + \}!. The matrix A is known as the factor loading matrix. Given data with a sample covariance matrix I:, factor analysis finds the A and \}! that optimally fit I: in the maximum likelihood sense. Since k < p, a single factor analyser can be seen as a reduced parametrisation of a full-covariance Gaussian. 1 IFactor analysis and its relationship to principal components analysis (peA) and mixture models is reviewed in (10). Z. Ghahramani and M. J. Heal 450 A mixture of factor analysers (MFA) models the density for y as a weighted average of factor analyser densities s P(yjA, q,,7r) = LP(sj7r)P(yjs,AS, '11), (1) s=1 where 7r is the vector of mixing proportions, s is a discrete indicator variable, and AS is the factor loading matrix for factor analyser s which includes a mean vector for y. By exploiting the factor analysis parameterisation of covariance matrices, a mixture of factor analysers can be used to fit a mixture of Gaussians to correlated high dimensional data without requiring O(P2) parameters or undesirable compromises such as axis-aligned covariance matrices. In an MFA each Gaussian cluster has intrinsic dimensionality k (or ks if the dimensions are allowed to vary across clusters). Consequently, the mixture of factor analysers simultaneously addresses the problems of clustering and local dimensionality reduction. When '11 is a multiple of the identity the model becomes a mixture of probabilistic PCAs. Tractable maximum likelihood procedure for fitting MFA and MPCA models can be derived from the Expectation Maximisation algorithm [4, 11]. The maximum likelihood (ML) approach to MFA can easily get caught in local maxima. 2 Ueda et al. [12] provide an effective deterministic procedure for avoiding local maxima by considering splitting a factor analyser in one part of space and merging two in a another part. But splits and merges have to be considered simultaneously because the number of factor analysers has to stay the same since adding a factor analyser is always expected to increase the training likelihood. A fundamental problem with maximum likelihood approaches is that they fail to take into account model complexity (Le. the cost of coding the model parameters) . So more complex models are not penalised, which leads to overfitting and the inability to determine the best model size and structure (or distributions thereof) without resorting to costly cross-validation procedures. Bayesian approaches overcome these problems by treating the parameters 0 as unknown random variables and averaging over the ensemble of models they define: P(Y) = / dO P(YjO)P(O). (2) P(Y) is the evidence for a data set Y = {yl, .. . ,yN}. Integrating out parameters penalises models with more degrees of freedom since these models can a priori model a larger range of data sets. All information inferred from the data about the parameters is captured by the posterior distribution P(OjY) rather than the ML point estimate 0. 3 While Bayesian theory deals with the problems of overfitting and model selection/averaging, in practice it is often computationally and analytically intractable to perform the required integrals. For Gaussian mixture models Markov chain Monte Carlo (MCMC) methods have been developed to approximate these integrals by sampling [8, 7]. The main criticism of MCMC methods is that they are slow and 2 Technically, the log likelihood is not bounded above if no constraints are put on the determinant of the component covariances. So the real ML objective for MFA is to find the highest finite local maximum of the likelihood. 3We sometimes use () to refer to the parameters and sometimes to all the unknown quantities (parameters and hidden variables). Formally the only difference between the two is that the number of hidden variables grows with N, whereas the number of parameters usually does not. 451 Variational Inference for Bayesian Mixtures of Factor Analysers it is usually difficult to assess convergence. Furthermore, the posterior density over parameters is stored as a set of samples, which can be inefficient. Another approach to Bayesian integration for Gaussian mixtures [9] is the Laplace approximation which makes a local Gaussian approximation around a maximum a posteriori parameter estimate. These approximations are based on large data limits and can be poor, particularly for small data sets (for which, in principle, the advantages of Bayesian integration over ML are largest). Local Gaussian approximations are also poorly suited to bounded or positive parameters such as the mixing proportions of the mixture model. Finally, it is difficult to see how this approach can be applied to online incremental changes to model structure. In this paper we employ a third approach to Bayesian inference: variational approximation. We form a lower bound on the log evidence using Jensen's inequality: 1: == In P(Y) = In / dO P(Y, 0) ~/ dO Q(O) In P6~~~) == F, (3) which we seek to maximise. Maximising F is equivalent to minimising the KLdivergence between Q(O) and P(OIY), so a tractable Q can be used as an approximation to the intractable posterior. This approach draws its roots from one way of deriving mean field approximations in physics, and has been used recently for Bayesian inference [13, 5, 1]. The variational method has several advantages over MCMC and Laplace approximations. Unlike MCMC, convergence can be assessed easily by monitoring F. The approximate posterior is encoded efficiently in Q(O) . Unlike Laplace approximations, the form of Q can be tailored to each parameter (in fact the optimal form of Q for each parameter falls out of the optimisation), the approximation is global, and Q optimises an objective function. Variational methods are generally fast, F is guaranteed to increase monotonically and transparently incorporates model complexity. To our knowledge, no one has done a full Bayesian analysis of mixtures of factor analysers. Of course, vis-a-vis MCMC, the main disadvantage of variational approximations is that they are not guaranteed to find the exact posterior in the limit. However, with a straightforward application of sampling, it is possible to take the result of the variational optimisation and use it to sample from the exact posterior and exact predictive density. This is described in section 5. In the remainder of this paper we first describe the mixture of factor analysers in more detail (section 2). We then derive the variational approximation (section 3). We show empirically that the model can infer both the number of components and their intrinsic dimensionalities, and is not prone to overfitting (section 6). Finally, we conclude in section 7. 2 The Model Starting from (1), the evidence for the Bayesian MFA is obtained by averaging the likelihood under priors for the parameters (which have their own hyperparameters): P(Y) / d7rP(7rIa:) / dvP(vla,b) / dA P(Alv), g[.t, P(s?I1r) J dx?P(xn)p(ynlx?,sn,A', q;)]. (4) Z. Ghahramani and M. J. Beal 452 Here {a, a, b, "Ill} are hyperparameters 4 , v are precision parameters (Le. inverse variances) for the columns of A. The conditional independence relations between the variables in this model are shown graphically in the usual belief network representation in Figure 1. While arbitrary choices could be made for the priors on the first line of (4), choosing priors that are conjugate to the likelihood terms on the second line of (4) greatly simplifies inference and interpretability.5 So we choose P(7rJa) to be symmetric Dirichlet, which is conjugate to the multinomial P(sJ7r). The prior for the factor loading matrix plays a key role in this model. Each component of the mixture has a Gaussian prior P(ABJV B), where each element of the vector VB is the precision of a column of A. IT one of these precisions vi -t 00, :''!:~",~.................. 1 then the outgoing weights for factor Xl will go to zero, which allows the model to reduce the inFigure 1: Generative model for trinsic dimensionality of X if the data does not variational Bayesian mixture of factor analysers. Circles denote warrant this added dimension. This method of random variables, solid rectangles intrinsic dimensionality reduction has been used denote hyperparameters, and the by Bishop [2] for Bayesian peA, and is closely dashed rectangle shows the plate related to MacKay and Neal's method for automatic relevance determination (ARD) for inputs (i.e. repetitions) over the data. to a neural network [6]. To avoid overfitting it is important to integrate out all parameters whose cardinality scales with model complexity (Le. number of components and their dimensionalities). We therefore also integrate out the precisions using Gamma priors, P(vJa, b). 3 The Variational Approximation Applying Jensen's inequality repeatedly to the log evidence (4) we lower bound it using the following factorisation of the distribution of parameters and hidden variables: Q(A)Q(7r, v)Q(s, x). Given this factorisation several additional factorisations fallout of the conditional independencies in the model resulting in the variational objective function: F= jd-n;Q(-n;) In + t, .t, PJ7;~) + Q(s") t, j dv'Q(v') lIn P6;~~) b) + jdA'Q(A') In P6~~~') 1 [j d-n; Q(-n;) Pci~:~~) + j In + jdABQ(A B) j dx"Q(x"Js") In Q~~:~") dxnQ(xnJsn)lnp(ynJxn,sn,AB, "Ill)] The variational posteriors Q('), as given in the Appendix, are derived by performing a free-form extremisation of F w.r.t. Q. It is not difficult to show that these extrema are indeed maxima of F. The optimal posteriors Q are of the same conjugate forms as the priors. The model hyperparameters which govern the priors can be estimated in the same fashion (see the Appendix). 4We currently do not integrate out 1lJ', although this can also be done. 5Conjugate priors have the same effect as pseudo-observations. (5) Variational lriference for Bayesian Mixtures of Factor Analysers 4 453 Birth and Death When optimising F , occasionally one finds that for some s: Ln Q(sn) = O. These zero responsibility components are the result of there being insufficient support from the local data to overcome the dimensional complexity prior on the factor loading matrices. So components of the mixture die of natural causes when they are no longer needed. Removing these redundant components increases F . Component birth does not happen spontaneously, so we introduce a heuristic. Whenever F has stabilised we pick a parent-component stochastically with probability proportional to e- f3F? and attempt to split it into two; Fa is the s-specific contribution to F with the last bracketed term in (5) normalised by Ln Q(sn). This works better than both cycling through components and picking them at random as it concentrates attempted births on components that are faring poorly. The parameter distributions of the two Gaussians created from the split are initialised by partitioning the responsibilities for the data, Q(sn), along a direction sampled from the parent's distribution. This usually causes F to decrease, so by monitoring the future progress of F we can reject this attempted birth if F does not recover. Although it is perfectly possible to start the model with many components and let them die, it is computationally more efficient to start with one component and allow it to spawn more when necessary. 5 Exact Predictive Density, True Evidence, and KL By importance sampling from the variational approximation we can obtain unbiased estimates of three important quantities: the exact predictive density, the true log evidence [" and the KL divergence between the variational posterior and the true posterior. Letting 0 = {A, 7r}, we sample Oi '" Q (0). Each such sample is an instance of a mixture of factor analysers with predictive density given by (1). We weight these predictive densities by the importance weights Wi = P(Oi, Y)/Q(Oi), which are easy to evaluate. This results in a mixture of mixtures of factor analysers, and will converge to the exact predictive density, P(ylY), as long as Q(O) > 0 wherever P(OIY) > O. The true log evidence can be similarly estimated by [, = In(w), where (.) denotes averaging over the importance samples. Finally, the KL divergence is given by: KL(Q(O)IIP(OIY)) = In(w) - (In w). This procedure has three significant properties. First, the same importance weights can be used to estimate all three quantities. Second, while importance sampling can work very poorly in high dimensions for ad hoc proposal distributions, here the variational optimisation is used in a principled manner to pick Q to be a good approximation to P and therefore hopefully a good proposal distribution. Third, this procedure can be applied to any variational approximation. A detailed exposition can be found in [3]. 6 Results Experiment 1: Discovering the number of components. We tested the model on synthetic data generated from a mixture of 18 Gaussians with 50 points per cluster (Figure 2, top left). The variational algorithm has little difficulty finding the correct number of components and the birth heuristics are successful at avoiding local maxima. After finding the 18 Gaussians repeated splits are attempted and rejected. Finding a distribution over number of components using F is also simple. Experiment 2: The shrinking spiral. We used the dataset of 800 data points from a shrinking spiral from [12] as another test of how well the algorithm could 454 Z. Ghahramani and M. J. Beal Figure 2: (top) Exp 1: The frames from left to right are the data, and the 2 S.D. Gaussian ellipses after 7, 14, 16 and 22 accepted births. (bottom) Exp 2: Shrinking spiral data and 1 S.D. Gaussian ellipses after 6, 9, 12, and 17 accepted births. Note that the number of Gaussians increases from left to right. number intrinsic dlmensionalnies of points 7 per cluster 8 8 16 32 64 128 -7600 - 76OOQ 500 1000 1500 I I 1 1 1 1 1 3 4 2 2 2 2 2 2 2 2 2 2 2 4 I 6 7 7 3 4 4 3 3 3 2000 Figure 3: (left) Exp 2: :F as function of iteration for the spiral problem on a typical run. Drops in :F constitute component births. Thick lines are accepted attempts, thin lines are rejected attempts. (middle) Exp 3: Means of the factor loading matrices. These results are analogous to those given by Bishop [2] for Bayesian peA. (right) Exp 3: Table with learned number of Gaussians and dimension ali ties as training set size increases. Boxes represent model components that capture several of the clusters. escape local maxima and how robust it was to initial conditions (Figure 2, bottom). Again local maxima did not pose a problem and the algorithm always found between 12-14 Gaussians regardless of whether it was initialised with 0 or 200. These runs took about 3-4 minutes on a 500MHz Alpha EV6 processor. A plot of:F shows that most of the compute time is spent on accepted moves (Figure 3, left). Experiment 3: Discovering the local dimensionalities. We generated a synthetic data set of 300 data points in each of 6 Gaussians with intrinsic dimensionalities (7432 2 1) embedded in 10 dimensions. The variational Bayesian approach correctly inferred both the number of Gaussians and their intrinsic dimensionalities (Figure 3, middle). We varied the number of data points and found that as expected with fewer points the data could not provide evidence for as many components and intrinsic dimensions (Figure 3, right). 7 Discussion Search over model structures for MFAs is computationally intractable if each factor analyser is allowed to have different intrinsic dimensionalities . In this paper we have shown that the variational Bayesian approach can be used to efficiently infer this model structure while avoiding overfitting and other deficiencies of ML approaches. One attraction of our variational method, which can be exploited in other models, is that once a factorisation of Q is assumed all inference is automatic and exact. We can also use :F to get a distribution over structures if desired. Finally we derive 455 Variational Inference for Bayesian Mixtures ofFactor Analysers a generally applicable importance sampler that gives us unbiased estimates of the true evidence, the exact predictive density, and the KL divergence between the variational posterior and the true posterior. Encouraged by the results on synthetic data, we have applied the Bayesian mixture of factor analysers to a real-world unsupervised digit classification problem. We will report the results of these experiments in a separate article. Appendix: Optimal Q Distributions and Hyperparameters Q(xnls n ) '" N(x n ,", E S ) Q(A~) ""' N(X:, Eq ,S) In Q(sn) = [1jJ(wu s ) -1jJ(w)] + xn,s=EsxsT'lT-lyn, 1 2 In IE s I + Q(vl) ""' Q(ai,bl) (In p(ynlxn, sn, AS , 'IT)) + c X:= ['IT-l"tQ(sn)ynxn 'STEq,S] , ai=a+~, bi=b+~:t(A~12) q n=l E s - 1= (A ST 'IT -1 AS) + I, Eq ,S -~ 'IT;ql Q(1r) '" D(wu) N L Q(sn)(xnxnT) +diag(vS), n=l q=l wU s = ~ + N L Q(sn) n=l where {N, Q, D} denote Normal, Gamma and Dirichlet distributions respectively, (-) denotes expectation under the variational posterior, and 1jJ (x) is the digamma function 1jJ(x) == lnr(x). Note that the optimal distributions Q(A S) have block diagonal covariance structure; even though each AS is a p x q matrix, its covariance only has O(pq2) parameters. Differentiating:F with respect to the parameters, a and b, of the precision prior we get fixed point equations 1jJ(a) = (In v)+lnb and b = a/(v). Similarly the fixed point for the parameters of the Dirichlet prior is 1jJ(a) -1jJ(a/S) + 2: [1jJ(wu s ) -1jJ(w)]/S = o. tx References [1] H. Attias. Inferring parameters and structure of latent variable models by variational Bayes. In Proc. 15th Conf. on Uncertainty in Artificial Intelligence, 1999. [2] C.M. Bishop. Variational PCA. In Proc. Ninth Int. Conf. on Artificial Neural Networks. ICANN, 1999. [3] Z. Ghahramani, H. Attias, and M.J. Beal. Learning model structure. Technical Report GCNU-TR-1999-006, (in prep.) Gatsby Unit, Univ. College London, 1999. [4] Z. Ghahramani and G.E. Hinton. The EM algorithm for mixtures of factor analyzers. Technical Report CRG-TR-96-1 [http://~.gatsby . ucl. ac. uk/ ~zoubin/papers/tr-96-1.ps.gz], Dept. of Compo Sci. , Univ. of Toronto, 1996. [5] D.J.C. MacKay. Ensemble learning for hidden Markov models. Technical report, Cavendish Laboratory, University of Cambridge, 1997. [6] R.M. Neal. Assessing relevance determination methods using DELVE. In C.M. Bishop, editor, Neural Networks and Machine Learning, 97-129. Springer-Verlag, 1998. [7] C.E. Rasmussen. The infinite gaussian mixture model. In Adv. Neur. Inf. Pmc. Sys. 12. MIT Press, 2000. [8] S. Richardson and P.J. Green. On Bayesian analysis of mixtures with an unknown number of components. J. Roy. Stat. Soc.-Ser. B, 59(4) :731-758, 1997. [9] S.J. Roberts , D. Husmeier, 1. Rezek, and W. Penny. Bayesian approaches to Gaussian mixture modeling. IEEE PAMI, 20(11):1133- 1142, 1998. [10] S. T . Roweis and Z. Ghahramani. A unifying review of linear Gaussian models. Neural Computation, 11(2):305- 345, 1999. [11] M.E. Tipping and C.M. Bishop. Mixtures of probabilistic principal component analyzers. Neural Computation, 11(2):443- 482, 1999. [12] N. Ueda, R. Nakano, Z. Ghahramani, and G.E. Hinton. SMEM algorithm for mixture models. In Adv. Neur. Inf. Proc. Sys. 11 . MIT Press, 1999. [13] S. Waterhouse, D.J.C. Mackay, and T. Robinson. Bayesian methods for mixtures of experts. In Adv. Neur. Inf. Proc. Sys. 1. MIT Press, 1995. 

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Mixture Density Estimation Jonathan Q. Li Department of Statistics Yale University P.O. Box 208290 New Haven, CT 06520 Andrew R. Barron Department of Statistics Yale University P.O. Box 208290 New Haven, CT 06520 Qiang.Li@aya.yale. edu Andrew. Barron@yale. edu Abstract Gaussian mixtures (or so-called radial basis function networks) for density estimation provide a natural counterpart to sigmoidal neural networks for function fitting and approximation. In both cases, it is possible to give simple expressions for the iterative improvement of performance as components of the network are introduced one at a time. In particular, for mixture density estimation we show that a k-component mixture estimated by maximum likelihood (or by an iterative likelihood improvement that we introduce) achieves log-likelihood within order 1/k of the log-likelihood achievable by any convex combination. Consequences for approximation and estimation using Kullback-Leibler risk are also given. A Minimum Description Length principle selects the optimal number of components k that minimizes the risk bound. 1 Introduction In density estimation, Gaussian mixtures provide flexible-basis representations for densities that can be used to model heterogeneous data in high dimensions. We introduce an index of regularity Cf of density functions f with respect to mixtures of densities from a given family. Mixture models with k components are shown to achieve Kullback-Leibler approximation error bounded by c}/k for every k. Thus in a manner analogous to the treatment of sinusoidal and sigmoidal networks in Barron [1],[2], we find classes of density functions f such that reasonable size networks (not exponentially large as function of the input dimension) achieve suitable approximation and estimation error. Consider a parametric family G = {<pe(x) , x E X C Rd' : fJ E e c Rd} of probability density functions parameterized by fJ E e. Then consider the class C = CONV(G) of density functions for which there is a mixture representation of the form fp(x) = Ie <pe(x)P(dfJ) where <pe(x) are density functions from G and P is a probability measure on (1) e. The main theme of the paper is to give approximation and estimation bounds of arbitrary densities by finite mixture densities. We focus our attention on densities J. Q. Li and A. R. Barron 280 inside C first and give an approximation error bound by finite mixtures for arbitrary f E C. The approximation error is measured by Kullback-Leibler divergence between two densities, defined as DUllg) = J f(x) log[f(x)jg(x)]dx. (2) In density estimation, D is more natural to use than the L2 distance often seen in the function fitting literature. Indeed, D is invariant under scale transformations (and other 1-1 transformation of the variables) and it has an intrinsic connection with Maximum Likelihood, one of the most useful methods in the mixture density estimation. The following result quantifies the approximation error. THEOREM 1 Let G = {4>8(X) : 0 E 8} and C= CONV(G). Let f(x) J 4>8 (x)P(dO) E C. There exists fk' a k-component mixture of 4>8, such that (3) In the bound, we have 2 Cf and 'Y = 4[log(3Ve) = JJJ4>8 4>~(x)P(dO) (x)P(dO) dx, (4) + a], where a = sup log 81,82,X 4>81 (x) . 4> 82 (x) (5) Here, a characterizes an upper bound of the log ratio of the densities in G, when the parameters are restricted to 8 and the variable to X . Note that the rate of convergence, Ijk, is not related to the dimensions of 8 or X. The behavior of the constants, though, depends on the choices of G and the target f? For example we may take G to be the Gaussian location family, which we restrict to a set X which is a cube of side-length A. Likewise we restrict the parameters to be in the same cube. Then, dA 2 a<-2? (7 (6) The value of c} depends on the target density M components, then f. Suppose f is a finite mixture with In this case, a is linear in dimension. (7) C} ~ M, with equality if and only if those M components are disjoint. Indeed, suppose f(x) = E!l Pi 4>8; (x), then Pi 4>8; (x)j E!l Pi 4>8; (x) ~ 1and hence c} = J "",M L..-i=l CI;;4>8;.(X)) 4>8; (x) dx Ei=l Pt4>8; (x) ~ JI)I)4>8; M (x)dx = M. (8) i=l Genovese and Wasserman [3] deal with a similar setting. A Kullback-Leibler approximation bound of order IjVk for one-dimensional mixtures of Gaussians is given by them. In the more general case that f is not necessarily in C, we have a competitive optimality result. Our density approximation is nearly at least as good as any gp in C. 281 Mixture Density Estimation THEOREM 2 For every gp(x) = f ?o(x)P(d8), DUIlIk) ~ DUllgp) Here, 2 C/,P = J(Jf c2 + ~p 'Y. ?~(x)P(d8) ?o(x)P(d8))2 f(x)dx. (9) (10) In particular, we can take infimum over all gp E C, and still obtain a bound. Let DUIIC) = infgEc DUlIg). A theory of information projection shows that if there exists a sequence of fk such that DUllfk) -t DUIIC), then fk converges to a function 1*, which achieves DUIIC). Note that 1* is not necessarily an element in C. This is developed in Li[4] building on the work of Bell and Cover[5]. As a consequence of Theorem 2 we have (11) where c},* is the smallest limit of cJ,p for sequences of P achieving DUlIgp) that approaches the infimum DUIIC). We prove Theorem 1 by induction in the following section. An appealing feature of such an approach is that it provides an iterative estimation procedure which allows us to estimate one component at a time. This greedy procedure is shown to perform almost as well as the full-mixture procedures, while the computational task of estimating one component is considerably easier than estimating the full mixtures. Section 2 gives the iterative construction of a suitable approximation, while Section 3 shows how such mixtures may be estimated from data. Risk bounds are stated in Section 4. 2 An iterative construction of the approximation We provide an iterative construction of Ik's in the following fashion. Suppose during our discussion of approximation that f is given. We seek a k-component mixture fk close to f. Initialize h by choosing a single component from G to minimize DUllh) = DUII?o). Now suppose we have fk-l(X). Then let fk(X) = (1 - a)fk-l(X) + a?o(x) where a and 8 are chosen to minimize DUIIIk). More generally let Ik be any sequence of k-component mixtures, for k = 1,2, ... such that DUIIIk) ~ mina,o DUII(l - a)fk-l + a?o). We prove that such sequences Ik achieve the error bounds in Theorem 1 and Theorem 2. Those familiar with the iterative Hilbert space approximation results of Jones[6], Barron[l]' and Lee, Bartlett and Williamson[7], will see that we follow a similar strategy. The use of L2 distance measures for density approximation involves L2 norms of component densities that are exponentially large with dimension. Naive Taylor expansion of the Kullback-Leibler divergence leads to an L2 norm approximation (weighted by the reciprocal of the density) for which the difficulty remains (Zeevi & Meir[8], Li[9]). The challenge for us was to adapt iterative approximation to the use of Kullback-Leibler divergence in a manner that permits the constant a in the bound to involve the logarithm of the density ratio (rather than the ratio itself) to allow more manageable constants. J. Q. Li and A. R. Barron 282 The proof establishes the inductive relationship Dk ::; (1 - a)D k- 1 + 0. 2 B, where B is bounded and Dk = DUllfk). By choosing 0.1 = 1,0.2 thereafter ak = 2/k, it's easy to see by induction that Dk ::; 4B/k. (12) = 1/2 and tr To get (12), we establish a quadratic upper bound for -log -log ?1-0:)"',_1+0:?e). Three key analytic inequalities regarding to the logarithm will be handy for us, for r ~ -log(r) ::; -(r - 1) + [-log(ro) + ro - l](r _ 1)2 (ro - 1)2 (13) 2[ -log(r) + r - 1] ::; Iog r, (14) -log(r)+r-l<I/2 I -() (r _ 1)2 + og r (15) ro > 0, r-l and where log- (-) is the negative part of the logarithm. The proof of of inequality (13) is done by verifying that -lo(~(.:it!-1 is monotone decreasing in r. Inequalities (14) and (15) are shown by separately considering the cases that r < 1 and r > 1 (as well as the limit as r -+ 1). To get the inequalities one multiplies through by (r -1) or (r - 1)2, respectively, and then takes derivatives to obtain suitable monotonicity in r as one moves away from r = 1. Now apply the inequality (13) with r = (1-0:)"'_1 +o:?e 9 and ro = (1-0:)'k-1, 9 where 9 is an arbitrary density in C with 9 = J ?9P(d9). Note that r ~ ro in this case because o:t e ~ O. Plug in r = ro + a~ at the right side of (13) and expand the square. Then we get -log(r) < -(ro + a: _ 1) + a? I ( ) - - - og ro [-IO~~o~~fo -1][(ro - 1) + (ag?W +ro -1] 2 ?[-log(ro) +ro + a 2?2[-log(ro) + 0.2 -1] . g2 (ro - 1) 9 ro - 1 Now apply (14) and (15) respectively. We get a? 2?2 _ ? -log(r) ::; -log(ro) - - + a 2(1/2 + log (ro)) + a-Iog(ro). (16) 9 9 9 9 Note that in our application, ro is a ratio of densities in C. Thus we obtain an upper bound for log-(ro) involving a. Indeed we find that (1/2 + log-(ro)) ::; "1/4 where "I is as defined in the theorem. In the case that f is in C, we take 9 = f. Then taking the expectation with respect to f of both sides of (16), we acquire a quadratic upper bound for Dk, noting that tr. r = Also note that D k is a function of 9. The greedy algorithm chooses 9 to minimize Dk(9). Therefore Dk ::; mjnD k (9) ::; / D k (9)P(d9). (17) Plugging the upper bound (16) for Dk(9) into (17), we have Dk::; ( ([-log(ro)- a? 19 Ix 9 +a2?:("f/4)+a~log(ro)]J(x)dxP(d9). 9 9 (18) 283 Mixture Density Estimation where Thus TO = (1 - a)fk-1 (x)jg(x) and P is chosen to satisfy Ie ?>e(x)P(dO) = g(x). Dk ~ (1- a)Dk- 1 + a 2! ?>~(x)P(dO) (g(x))2 f(x)dx{rj4) + a log(l- a) - a -log(l- a). It can be shown that alog(l- a) - a -log(l - a) inductive relationship, ~ (19) O. Thus we have the desired (20) Therefore, Dk ~ 'Yc 2 f. In the case that f does not have a mixture representation of the form I ?>eP(dO), i.e. f is outside the convex hull C, we take Dk to be I f(x) log dx for any given gp(x) = I ?>e(x)P(dO). The above analysis then yields Dk = DUllfk) -DUllgp) ::; j:f:? 'Yc 2 f 3 as desired. That completes the proof of Theorems 1 and 2. A greedy estimation procedure The connection between the K-L divergence and the MLE helps to motivate the following estimation procedure for /k if we have data Xl, ... , Xn sampled from f. The iterative construction of fk can be turned into a sequential maximum likelihood estimation by changing min DUllfk) to max 2:~1 log fk (Xi) at each step. A surprising result is that the resulting estimator A has a log likelihood almost at least as high as log likelihood achieved by any density gp in C with a difference of order 1jk. We formally state it as n n 1=1 1=1 1 '~logfk(Xi) " ~ 1 '~IOg9p(Xi) " ~ ~ ~ - 2 cF P 'k (21) for all gp E C. Here Fn is the empirical distribution, for which c2Fn,P (ljn) 2:~=1 c~;,p where 2 I ?>Hx)P(dO) (22) Cx,P = (f ?>e(x)P(dO))2 . The proof of this result (21) follows as in the proof in the last section, except that now we take Dk = EFn loggp(X)j fk(X) to be the expectation with respect to Fn instead of with respect to the density f. Let's look at the computation at each step to see the benefits this new greedy procedure can bring for us. We have ik(X) = (1- a)ik-1(X) + a?>e(x) with 0 and a chosen to maximize n L log[(l - a)f~-l (Xi) + a?>e(Xi )] (23) i=l which is a simple two component mixture problem, with one of the two components, f~-l(X), fixed. To achieve the bound in (21), a can either be chosen by this iterative maximum likelihood or it can be held fixed at each step to equal ak (which as before is ak = 2jk for k > 2). Thus one may replace the MLE-computation of a kcomponent mixture by successive MLE-computations of two-component mixtures. The resulting estimate is guaranteed to have almost at least as high a likelihood as is achieved by any mixture density. J. Q. Li and A. R. Barron 284 A disadvantage of the greedy procedure is that it may take a number of steps to adequately downweight poor initial choices. Thus it is advisable at each step to retune the weights of convex combinations of previous components (and even perhaps to adjust the locations of these components) , in which case, the result from the previous iterations (with k - 1 components) provide natural initialization for the search at step k. The good news is that as long as for each k, given ik-l, the A is chosen among k component mixtures to achieve likelihood at least as large as the choice achieving maxol:~=llog[(l - ak)fk-l (Xi) + ak<Po(Xi )), that is, we require that n n L log f~(Xd ~ mt" L log[(l- ak)f~-l (Xi) + ak<Po(X i )), (24) i=l i=l then the conclusion (21) will follow. In particular, our likelihood results and risk bound results apply both to the case that A is taken to be global maximizer of the likelihood over k-component mixtures as well as to the case that ik is the result of the greedy procedure. 4 Risk bounds for the MLE and the iterative MLE The metric entropy of the family G is controlled to obtain the risk bound and to determine the precisions with which the coordinates of the parameter space are allowed to be represented. Specifically, the following Lipschitz condition is assumed: for (} E e c Rd and x E X C R d, d sup Ilog <PO (x) -log <Po' (x)1 ~ B L IOj - 0jl xEX j=l (25) where OJ is the j-th coordinate of the parameter vector. Note that such a condition is satisfied by a Gaussian family with x restricted to a cube with sidelength A and has a location parameter 0 that is also prescribed to be in the same cube. In particular, if we let the variance be 0'2, we may set B = 2AI 0'2. Now we can state the bound on the K-L risk of A. THEOREM 3 Assume the condition {25}. Also assume e to be a cube with sidelength A. Let ik(X) be either the maximizer of the likelihood over k -component mixtures or more generally any sequence of density estimates f~ satisfying {24}. We have A E(DUllfk)) - DUIIC) ~ 'Y 2 2 cf 2kd k .. + 'Y-:;;: log(nABe). (26) From the bound on risk, a best choice of k would be of order roughly Vn leading to a bound on ED(fllf~) - DUIIC) of order 1/Vn to within logarithmic factors. However the best such bound occurs with k = 'Ycf, .. VnI.j2dlog(nABe) which is not available when the value of cf, .. is unknown. More importantly, k should not be chosen merely to optimize an upper bound on risk, but rather to balance whatever approximation and estimation sources of error actually occur. Toward this end we optimize a penalized likelihood criterion related to the minimum description length principle, following Barron and Cover [10] . Let l(k) be a function of k that satisfies l:~l e-l(k) ::; 1, such as l(k) = 2Iog(k+ 1). Mixture Densiry Estimation 285 A penalized MLE (or MDL) procedure picks k by minimizing ! n t log i=l 1 h(Xi ) A + 2kd log (nABe) + 21(k)jn. (27) n Then we have E(DUllh)) - DUlle) A ~ 2 2 Cf 2kd m1nb k * + r-;-log(nABe) + 21(k)jn}. (28) A proof of these risk bounds is given in Li[4]. It builds on general results for maximum likelihood and penalized maximum likelihood procedures. Recently, Dasgupta [11] has established a randomized algorithm for estimating mixtures of Gaussians, in the case that data are drawn from a finite mixture of sufficiently separated Gaussian components with common covariance, that runs in time linear in the dimension and quadratic in the sample size. However, present forms of his algorithm require impractically large sample sizes to get reasonably accurate estimates of the density. It is not yet known how his techniques will work for more general mixtures. Here we see that iterative likelihood maximization provides a better relationship between accuracy, sample size and number of components. References [1] Barron, Andrew (1993) Universal Approximation Bounds for Superpositions of a Sigmoidal Function. IEEE Transactions on Information Theory 39, No.3: 930-945 [2] Barron, Andrew (1994) Approximation and Estimation Bounds for Artificial Neural Networks. Machine Learning 14: 115-133. [3] Genovese, Chris and Wasserman, Larry (1998) Rates of Convergence for the Gaussian Mixture Seive. Manuscript. [4] Li, Jonathan Q. (1999) Estimation of Mixture Models. Ph.D Dissertation. The Department of Statistics. Yale University. [5] Bell, Robert and Cover, Thomas (1988) Game-theoretic optimal portfolios. Management Science 34: 724-733. [6] Jones, Lee (1992) A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. Annals of Statistics 20: 608-613. [7] Lee, W.S., Bartlett, P.L. and Williamson R.C. (1996) Efficient Agnostic Learning of Neural Networks with Bounded Fan-in. IEEE Transactions on Information Theory 42, No.6: 2118-2132. [8] Zeevi, Assaf and Meir Ronny (1997) Density Estimation Through Convex Combinations of Densities: Approximation and Estimation Bounds. Neural Networks 10, No.1: 99-109. [9] Li, Jonathan Q. (1997) Iterative Estimation of Mixture Models. Ph.D. Prospectus. The Department of Statistics. Yale University. [10] Barron, Andrew and Cover, Thomas (1991) Minimum Complexity Density Estimation. IEEE Transactions on Information Theory 37: 1034-1054. [11] Dasgupta, Sanjoy (1999) Learning Mixtures of Gaussians. Pmc. IEEE Conf. on Foundations of Computer Science, 634-644. 

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Model selection in clustering by uniform convergence bounds* Joachim M. Buhmann and Marcus Held Institut flir Informatik III, RomerstraBe 164, D-53117 Bonn, Germany {jb,held}@cs.uni-bonn.de Abstract Unsupervised learning algorithms are designed to extract structure from data samples. Reliable and robust inference requires a guarantee that extracted structures are typical for the data source, Le., similar structures have to be inferred from a second sample set of the same data source. The overfitting phenomenon in maximum entropy based annealing algorithms is exemplarily studied for a class of histogram clustering models. Bernstein's inequality for large deviations is used to determine the maximally achievable approximation quality parameterized by a minimal temperature. Monte Carlo simulations support the proposed model selection criterion by finite temperature annealing. 1 Introduction Learning algorithms are designed to extract structure from data. Two classes of algorithms have been widely discussed in the literature - supervised and unsupervised learning. The distinction between the two classes depends on supervision or teacher information which is either available to the learning algorithm or missing. This paper applies statistical learning theory to the problem of unsupervised learning. In particular, error bounds as a protection against overfitting are derived for the recently developed Asymmetric Clustering Model (ACM) for co-occurrence data [6]. These theoretical results show that the continuation method "deterministic annealing" yields robustness of the learning results in the sense of statistical learning theory. The computational temperature of annealing algorithms plays the role of a control parameter which regulates the complexity of the learning machine. Let us assume that a hypothesis class 1? of loss functions h(x; a) is given. These loss functions measure the quality of structures in data. The complexity of 1? is controlled by coarsening, i.e., we define a 'Y-cover of 1?. Informally, the inference principle advocated by us performs learning by two inference steps: (i) determine the optimal approximation level l' for consistent learning (in terms of large risk deviations); (ii) given the optimal approximation level 1', average over all hypotheses in an appropriate neighborhood of the empirical minimizer. The result of the inference *This work has been supported by the German Israel Foundation for Science and Research Development (GIF) under grant #1-0403-001.06/95. Model Selection in Clustering by Uniform Convergence Bounds 217 procedure is not a single hypothesis but a set of hypotheses. This set is represented either by an average of loss functions or, alternatively, by a typical member of this set. This induction approach is named Empirical Risk Approximation (ERA) [2]. The reader should note that the learning algorithm has to return an average structure which is typical in a 'Y-cover sense but it is not supposed to return the hypothesis with minimal empirical risk as in Vapnik's "Empirical Risk Minimization" (ERM) induction principle for classification and regression [9]. The loss function with minimal empirical risk is usually a structure with maximal complexity, e.g., in clustering the ERM principle will necessarily yield a solution with the maximal number of clusters. The ERM principle, therefore, is not suitable as a model selection principle to determine the number of clusters which are stable under sample fluctuations. The ERA principle with its approximation accuracy 'Y solves this problem by controlling the effective complexity of the hypothesis class. In spirit, this approach is similar to the Gibbs-algorithm presented for example in [3]. The Gibbs-algorithm samples a random hypothesis from the version space to predict the label of the 1 + lth data point Xl+!o The version space is defined as the set of hypotheses which are consistent with the first 1 given data points. In our approach we use an alternative definition of consistency, where all hypothesis in an appropriate neighborhood of the empirical minimizer define the version space (see also [4]). Averaging over this neighborhood yields a structure with risk equivalent to the expected risk obtained by random sampling from this set of hypotheses. There exists also a tight methodological relationship to [7] and [4] where learning curves for the learning of two class classifiers are derived using techniques from statistical mechanics. 2 The Empirical Risk Approximation Principle The data samples Z = {zr E 0, 1 ~ r ~ l} which have to be analyzed by the unsupervised learning algorithm are elements of a suitable object (resp. feature) space O. The samples are distributed according to a measure J.L which is not assumed to be known for the analysis.l A mathematically precise statement of the ERA principle requires several definitions which formalize the notion of searching for structure in the data. The quality of structures extracted from the data set Z is evaluated by the empirical risk R(a; Z) := 2:~=1 h(zr; a) of a structure a given the training set Z. The function h(z; a) is known as loss function in statistics. It measures the costs for processing a generic datum z with model a. Each value a E A parameterizes an individual loss function with A denoting the set of possible parameters. The loss function which minimizes the empirical risk is denoted by &1. := arg minaEA R( a; Z). The relevant quality measure for learning is the expected risk R(a) .h(z; a) dJ.L(z). The optimal structure to be inferred from the data is a1. .argminaEA R(a). The distribution J.L is assumed to decay sufficiently fast with bounded rth moments Ell {Ih(z; a) - R(a)IT} ~ rh? r - 2 V II {h(z; an, 'Va E A (r > 2). Ell {.} and VII {.} denote expectation and variance of a random variable, respectively. T is a distribution dependent constant. ERA requires the learning algorithm to determine a set hypotheses on the basis of the finest consistently learnable cover of the hypothesis class. Given a learning accuracy 'Y a subset of parameters A-y = {al,'" ,aIA-yI-l} U {&1.} can be defined such that the hypothesis class 1i is covered by the function balls with index sets B-y(a) := {a' : Ih(z; a') - h(z; a)1 dJ.L(z) ~ 'Y}, i. e. A C UaEA-y B-y(a). The em- t In In 1 Knowledge of covering numbers is required in the following analysis which is a weaker type of information than complete knowledge of the probability measure IL (see also [5]). 218 J. M Buhmann and M Held pirical minimizer &1. has been added to the cover to simplify bounding arguments. Large deviation theory is used to determine the approximation accuracy '1 for learning a hypothesis from the hypothesis class 11.. The expected risk of the empirical minimizer exceeds the global minimum of the expected risk R(01.) by faT with a probability bounded by Bernstein's inequality [8] ~ -21 (faT - 'Y)} l(f-'Y/ aT )2) _ < P { sup IR(o) - R(o)1 aEA-y < 21A')'1 exp ( - 8 + 4r (f _ 'Y/a T) = o. (1) The complexity IA')' I of the coarsened hypothesis class has to be small enough to guarantee with high confidence small f-deviations. 2 This large deviation inequality weighs two competing effects in the learning problem, i. e. the probability of a large deviation exponentially decreases with growing sample size I, whereas a large deviation becomes increasingly likely with growing cardinality of the 'Y-cover of the hypothesis class. According to (1) the sample complexity Io (-y, f, 0) is defined by to (f - '1/ aT) 2 2 log IA')'I - 8 + 4r (f _ 'Y/aT) + log "8 = o. (2) With probability 1 - 0 the deviation of the empirical risk from the expected risk is bounded by ~ (foPta T - '1) =: 'Yapp ? Averaging over a set of functions which exceed the empirical minimizer by no more than 2'Yapp in empirical risk yields an average hypothesis corresponding to the statistically significant structure in the data, i.e., R( 01.) - R( &1.) ~ R( 01. ) + 'Yapp - (R( &1.) - 'Yapp ) ~ 2'Y app since R( 01.) ~ R( &1.) by definition. The key task in the following remains to calculate the minimal precision f( '1) as a function of the approximation '1 and to bound from above the cardinality IA')' I of the 'Y-cover for specific learning problems. 3 Asymmetric clustering model The asymmetric clustering model was developed for the analysis resp. grouping of objects characterized by co-occurrence of objects and certain feature values [6]. Application domains for this explorative data analysis approach are for example texture segmentation, statistical language modeling or document retrieval. Denote by n = X x y the product space of objects Xi EX, 1 ~ i ~ nand features Yj E y, 1 ~ j ~ j. The Xi E X are characterized by observations Z = {zr} = {(Xi(r),Yj(r)) ,T = 1, ... ,l}. The sufficient statistics of how often the object-feature pair (Xi, Yj) occurs in the data set Z is measured by the set of frequencies {'f]ij : number of observations (Xi, Yj) /total number of observations}. Derived measurements are the frequency of observi~g object Xi, i. e. 'f]i = 2:;=1 'f]ij and the frequency of observing feature Yj given object Xi, i. e. 'f]jli = 'f]ij/'f]i. The asymmetric clustering model defines a generative model of a finite mixture of component probability distributions in feature space with cluster-conditional distributions q = (qjlv) ' 1 ~ j ~ j, 1 ~ v ~ k (see [6]). We introduce indicator variables M iv E {O, 1} for the membership of object Xi in cluster v E {I, ... ,k}. 2::=1 M iv = 1 Vi : 1 ~ i ~ n enforces the uniqueness constraint for assignments. 2The maximal standard deviation (1 T := sUPaEA-y y'V {h(z; a)} defines the scale to measure deviations of the empirical risk from the expected risk (see [2]). Model Selection in Clustering by Uniform Convergence Bounds 219 Using these variables the observed data Z are distributed according to the generative model over X x y: 1 k P {xi,YjIM,q} = - ~ (3) n L--v=1 Mivqjlv' For the analysis of the unknown data source - characterized (at least approximatively) by the empirical data Z - a structure 0: = (M, q) with M E {O, I} n x k has to be inferred. The aim of an ACM analysis is to group the objects Xi as coded by the unknown indicator variables M iv and to estimate for each cluster v a prototypical feature distribution qjlv' Using the loss function h(Xi' Yj; 0:) = logn - 2:~=1 M iv logqjlv the maximization of the likelihood can be formulated as minimization of the empirical risk: R(o:; Z) = 2:~=1 2:;=11}ij h(xi, Yj; 0:), where the essential quantity to be minimized is the expected risk: R(o:) = 2:~=1 2:;=1 ptrue {Xi, Yj} h(Xi' Yj; 0:). Using the maximum entropy principle the following annealing equations are derived [6]: 2:~1 (M iv )1}ij _ ~n wi=1 (M iv ) - L--.t=1 "n A qjlv exp (Miv )1}i "n (M hv )1}j1i, wh=1 (4) [.8 2:;=1 1}jli log Q]lv ] The critical temperature: Due to the limited precision of the observed data it is natural to study histogram clustering as a learning problem with the hypothesis class 1? = {-2:vMivlogqjlv :Miv E {0,1} /\ 2:vMiv = 1/\ Qjlv E H,t, .. ? ,1}/\ 2:j qjlv = I}. The limited number of observations results in a limited precision of the frequencies 1}jli' The value Q;lv = 0 has been excluded since it causes infinite expected risk for ptrue {Yj IXi} > O. The size of the regularized hypothesis class A-y can be upper bounded by the cardinality of the complete hypothesis class divided by the minimal cardinality of a 'Y-function ball centered at a function of the 'Y-cover A-y, i. e. IA-yl ~ 11?1/!llin IB-y(&)I. oEA'T The cardinality of a function ball with radius 'Y can be approximated by adopting for x ~ 0): techniques from asymptotic analysis [1] (8 (x) = g IB-y(5)1 = L 8 L ('Y - . } M { q,lo L i ?J' ~ptrue {Yj IXi} IIOg ~~I~(i) I) (6) %Im(t) and the entropy S is given by S(q,Q,x) = 'Yx - Lv Qv (L j qjlv .!.n L--, ~ ,log ~ exp L--p -1) + (-x L-~, ptrue J {Yj IXi} IIOg _Qjlp I). (7) %Im(i) The auxiliary variables Q = {Q v } ~=1 are Lagrange parameters to enforce the normalizations 2: j qjlv = 1. Choosing %10 = qjlm(i) Vm(i) = 0:, we obtain an approximation of the integral. The reader should note that a saddlepoint approximation in J. M Buhmann and M Held 220 the usual sense is only applicable for the parameter x but will fail for the q, Q parameters since the integrand is maximal at the non-differentiability point of the absolute value function. We, therefore, expand S (q, Q,x) up to linear terms 0 (q - q) and integrate piece-wise. Using the abbreviation Kill := Lj ptrue {Yj Ixd IIog qj~:~i) I the following saddle point approximation for the integral over x is obtained: 1 I: I: , = Pij.?Kjlj.? n .t=1 j.?=1 n k ? wIth Pia ( -XKia) = Lexp (~)" j.? exp -XKij.? (8) The entropy S evaluated at q = q yields in combination with the Laplace approximation [1] an estimate for the cardinality of the ,-cover log IA')' I = n (log k - S) + -21 I:.t,p KipP ip (I: II P illKill - KiP) x2 (9) where the second term results from the second order term of the Taylor expansion around the saddle point. Inserting this complexity in equation (2) yields an equation which determines the required number of samples 10 for a fixed precision f and confidence o. This equation defines a functional relationship between the precision f and the approximation quality, for fixed sample size 10 and confidence o. Under this assumption the precision f depends on , in a non-monotone fashion, i. e. (10) using the abbreviation C = log IA')' I + log~. The minimum of the function ?(,) defines a compromise between uncertainty originating from empirical fluctuations and the loss of precision due to the approximation by a ,-cover. Differentiating with respect to , and setting the result to zero (df(T)/d, = 0) yields as upper bound for the inverse temperature: ~ 1 10 ( 10+C7"2 x <- 7" + -;:;~:;;;==~iiT (1T 2n V21 0C + 7"2C2 )-1 (11) Analogous to estimates of k-means, phase-transitions occur in ACM while lowering the temperature. The mixture model for the data at hand can be partitioned into more and more components, revealing finer and finer details of the generation process. The critical xopt defines the resolution limit below which details can not be resolved in a reliable fashion on the basis of the sample size 10 . Given the inverse temperature x the effective cardinality of the hypothesis class can be upper bounded via the solution of the fix point equation (8). On the other hand this cardinality defines with (11) and the sample size lo an upper bound on x. Iterating these two steps we finally obtain an upper bound for the critical inverse temperature given a sample size 10. Empirical Results: For the evaluation of the derived theoretical result a series of Monte-Carlo experiments on artificial data has been performed for the asymmetric clustering model. Given the number of objects n = 30, the number of groups k = 5 and the size of the histograms f = 15 the generative model for this experiments was created randomly and is summarized in fig. 1. From this generative model sample sets of arbitrary size can be generated and the true distributions ptrue {Yj IXi} can be calculated. In figure 2a,b the predicted temperatures are compared to the empirically observed critical temperatures, which have been estimated on the basis of 2000 different samples of randomly generated co-occurrence data for each 10. The expected risk (solid) Model Selection in Clustering by Uniform Convergence Bounds v 1 2 3 4 5 m(i} 221 qjlv 0.11,0.01,0.11,0.07,0.08,0.04,0.06,0,0.13,0.07, 0.08, 0.1, 0, 0.11,0.031 0.18,0.1,0.09,0.02,0.05,0.09,0.08,0.03,0.06, 0.07, 0.03, 0.02, 0.07, 0.06, 0.05} 0.17,0.05,0.05,0.06,0.06,0.05,0.03,0.11,0.09,0, 0.02,0.1,0.03,0.07, 0.11} 0.15,0.07,0.1,0.03,0.09,0.03,0.04,0.05,0.06, 0.05,0.08,0.04,0.08,0.09, 0.04} 0.09,0.09,0.07,0.1,0.07,0.06,0.06,0.11,0.07,0.07, 0.1, 0.02,0.07,0.02, O} = (5,3,2,5,2,2,5,4,2,2,2,4,1,5,3,5,3,4,1 , 2,2,3,1,1,2, 5, 5, 2, 2, 1) Figure 1: Generative ACM model for the Monte-Carlo experiments. and empirical risk (dashed) of these 2000 inferred models are averaged. Overfitting sets in when the expected risk rises as a function of the inverse temperature x. Figure 2c indicates that on average the minimal expected risk is assumed when the effective number is smaller than or equal 5, i. e. the number of clusters of the true generative model. Predicting the right computational temperature, therefore, also enables the data analyst to solve the cluster validation problem for the asymmetric clustering model. Especially for 10 = 800 the sample fluctuations do not permit the estimate of five clusters and the minimal computational temperature prevents such an inference result. On the other hand for lo = 1600 and 10 = 2000 the minimal temperature prevents the algorithm to infer too many clusters, which would be an instance of overfitting. As an interesting point one should note that for an infinite number of observations the critical inverse temperature reaches a finite positive value and not more than the five effective clusters are extracted. At this point we conclude, that for the case of histogram clustering the Empirical Risk Approximation solves for realizable rules the problem of model validation, i. e. choosing the right number of clusters. Figure 2d summarizes predictions of the critical temperature on the basis of the empirical distribution 1]ij rather than the true distribution ptrue {Xi, Yj}. The empirical distribution has been generated by a training sample set with x of eq. (11) being used as a plug-in estimator. The histogram depicts the predicted inverse temperature for 10 = 1200. The average of these plug-in estimators is equal to the predicted temperature for the true distribution. The estimates of x are biased towards too small inverse temperatures due to correlations between the parameter estimates and the stopping criterion. It is still an open question and focus of ongoing work to rigorously bound the variance of this plug- in estimator. Empirically we observe a reduction of the variance of the expected risk occurring at the predicted temperature for higher sample sizes lo . 4 Conclusions The two conditions that the empirical risk has to uniformly converge towards the expected risk and that all loss functions within an 2,&PP -range of the global empirical risk minimum have to be considered in the inference process limits the complexity of the underlying hypothesis class for a given number of samples. The maximum entropy method which has been widely employed in deterministic annealing procedures for optimization problems is substantiated by our analysis. Solutions with too many clusters clearly overfit the data and do not generalize. The condition that the hypothesis class should only be divided in function balls of size , forces us to stop the stochastic search at the lower bound of the computational temperature. Another important result of this investigation is the fact that choosing the right stopping temperature for the annealing process not only avoids overfitting but also solves the cluster validation problem in the realizable case of ACM. A possible inference of too many clusters using the empirical risk functional is suppressed. 222 J. M Buhmann and M Held 80 1-' ,/ a) ) 78 6 - - _ " ; ' ,' \ \ \ 4 / - \ \ / ! / /' 80 b) _ ... lo 0 800emP ._yo 101200lmp 78 'o'2000mp / 76 / ~/ \ '"~74 \ \ \ \ 2 \ 72 \ , ~~ ~" \ \ "- \ 0 \ 680 11 '0 , '----------------------- ,5 20 inverse temperature c) 10 -~ .~, ," 9 ,I ill 8 ~ [ i . . ..... p 26 ~ <II .;, ~ <II 30 -.--~~ ,- 35 680~---~---~ 'O---~'~5---~2~ 0 ---~25~---3O~---~35? inverse temperature .. ~ ----- --.- ..... .i 7 '0 6 ~ -- ... _----- ...... - 70 ,._-_._-- 5 4 3 2 o 5 10 15 20 inverse temperature 25 30 35 ~ Figure 2: Comparison between the theoretically derived upper bound on x and the observed critical temperatures (minimum of the expected risk vs. x curve). Depicted are the plots for 10 = 800,1200,1600,2000. Vertical lines indicate the predicted critical temperatures. The average effective number of clusters is drawn in part c. In part d the distribution of the plug- in estimates is shown for La = 1200. References [1] N. G. De Bruijn. Asymptotic Methods in Analysis. North-Holland Publishing Co., (repr. Dover), Amsterdam, 1958, (1981) . [2] J . M. Buhmann. Empirical risk approximation. Technical Report IAI-TR 98-3, Institut fur Informatik III, Universitat Bonn, 1998. [3] D. Haussler, M. Kearns, and R. Schapire. Bounds on the sample complexity of Bayesian learning using information theory and the VC dimension. Machine Learning, 14(1) :83113, 1994. [4] D. Haussler, M. Kearns , H.S. Seung, and N. Tishby. Rigorous learning curve bounds from statistical mechanics. Machine Learning, 25:195- 236, 1997. [5] D. Haussler and M. Opper. Mutual information, metric entropy and cumulative relative entropy risk. Annals of Statistics, December 1996. [6] T . Hofmann, J. Puzicha, and M.I. Jordan. Learning from dyadic data. In M. S. Kearns , S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11. MIT Press, 1999. to appear. [7] H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Physical Review A, 45(8):6056-6091 , April 1992. [8] A. W . van der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes. Springer-Verlag, New York, Berlin, Heidelberg, 1996. [9] V. N. Vapnik. Statistical Learning Theory. Wiley- Interscience, New York, 1998. 

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Image Recognition in Context: Application to Microscopic Urinalysis XuboSong* Department of Electrical and Computer Engineering Oregon Graduate Institute of Science and Technology Beaverton, OR 97006 xubosong@ece.ogi.edu Joseph Sill Department of Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 joe@busy.work.caltech.edu Yaser Abu-Mostafa Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 yase r@work.caltech.edu Harvey Kasdan International Remote Imaging Systems, Inc. Chatsworth, CA 91311 Abstract We propose a new and efficient technique for incorporating contextual information into object classification. Most of the current techniques face the problem of exponential computation cost. In this paper, we propose a new general framework that incorporates partial context at a linear cost. This technique is applied to microscopic urinalysis image recognition, resulting in a significant improvement of recognition rate over the context free approach. This gain would have been impossible using conventional context incorporation techniques. 1 BACKGROUND: RECOGNITION IN CONTEXT There are a number of pattern recognition problem domains where the classification of an object should be based on more than simply the appearance of the object itself. In remote sensing image classification, where each pixel is part of ground cover, a pixel is more likely to be a glacier if it is in a mountainous area, than if surrounded by pixels of residential areas. In text analysis, one can expect to find certain letters occurring regularly in particular arrangement with other letters(qu, ee,est, tion, etc.). The information conveyed by the accompanying entities is referred to as contextual information. Human experts apply contextual information in their decision making [2][ 6]. It makes sense to design techniques and algorithms to make computers aggregate and utilize a more complete set of information in their decision making the way human experts do. In pattern recognition systems, however, *Author for correspondence 964 X B. Song, J Sill, Y. Abu-Mostafa and H. Kasdan the primary (and often only) source of information used to identify an object is the set of measurements, or features , associated with the object itself. Augmenting this information by incorporating context into the classification process can yield significant benefits. Consider a set of N objects Ti , i = 1, ... N. With each object we associate a class label Ci that is a member of a label set n = {1 , ... , D} . Each object Ti is characterized by a set of measurements Xi E R P, which we call a feature vector. Many techniques [1][2][4J[6} incorporate context by conditioning the posterior probability of objects' identities on the joint features of all accompanying objects. i.e .? P(Cl, C2,??? , cNlxl , . . . , XN). and then maximizing it with respectto Cl, C2, . .. , CN . It can ""' (N\ given be shown thatp(cl,c2, . . . , cNlxl, . . . ,xN) ex p(cllxl) ... p(CNlxN) p(~ci 1 ?.. p CN certain reasonable assumptions. Once the context-free posterior probabilities p( Ci IXi) are known. e.g. through the use of a standard machine learning model such as a neural network, computing P(Cl, ... ,CNlxl, . . . ,XN) for all possible Cl, ... ,CN would entail (2N + 1)DN multiplications. and finding the maximum has complexity of DN. which is intractable for large Nand D. [2J Another problem with this formulation is the estimation of the high dimensional joint distribution p( Cl, ... , CN), which is ill-posed and data hungry. One way of dealing with these problems is to limit context to local regions. With this approach, only the pixels in a close neighborhood. or letters immediately adjacent are considered [4][6][7J. Such techniques may be ignoring useful information, and will not apply to situations where context doesn't have such locality, as in the case of microscopic urinalysis image recognition. Another way is to simplify the problem using specific domain knowledge [1], but this is only possible in certain domains. These difficulties motivate the efficient incorporation of partial context as a general framework, formulated in section 2. In section 3, we discuss microscopic urinalysis image recognition. and address the importance of using context for this application. Also in section 3, techniques are proposed to identify relevant context. Empirical results are shown in section 4. followed by discussions in section 5. 2 FORMULATION FOR INCORPORATION OF PARTIAL CONTEXT To avoid the exponential computational cost of using the identities of all accompanying objects directly as context, we use "partial context". denoted by A. It is called "partial" because it is derived from the class labels. as opposed to consisting of an explicit labelling of all objects. The physical definition of A depends on the problem at hand. In our application. A represents the presence or absence of certain classes. Then the posterior probability of an object Ti having class label Ci conditioned on its feature vector and the relevant context A is p(XiICi, A)P(Ci ; A) P(Xi ; A) We assume that the feature distribution of an object depends only on its own class. i.e., p(xilci, A) = p(xi lci) . This assumption is roughly true for most real world problems. Then. Image Recognition in Context: Application to Microscopic Urinalysis 965 ( .1 .)p(ciI A ) p(A)p(Xi) ( .1 . A) -- p(xilci)p(Ci; A) _ -pCtX t p(xijJ~IIA) p(Ci) P(Xi; A) pC~Xt, ()( p(cilxi) () = p(cilxi)P(Ci, A) Ci where p(Ci, A) = p~(~j~) is called the context ratio, through which context plays its role. The context-sensitive posterior probability p( Ci lXi, A) is obtained through the context-free posterior probability p(cilxi) modified by the context ratio P(Ci, A) . P The partial-context maximum likelihood decision rule chooses class label Ci for element i such that Ci = argmaxp(cilxi, A) (I) Cj A systematic approach to identify relevant context A is addressed in section 3.3. The partial-context approach treats each element in a set individually, but with additional information from the context-bearing factor A . Once p(cilxi) are known for all i = 1, ... , N, and the context A is obtained, to maximize p(cilxi, A) from D possible values that Ci can take on and for all i, the total number of multiplications is 2N, and the complexity for finding the maximum is N D. Both are linear in N. The density estimation part is also trivial since it is very easy to estimate p(cIA). 3 3.1 MICROSCOPIC URINALYSIS INTRODUCTION Urine is one of the most complex body fluid specimens: it potentially contains about 60 meaningful types of elements. Microscopic urinalysis detects the presence of elements that often provide early diagnostic information concerning dysfunction, infection, or inflammation of the kidneys and urinary tract. Thus this non-invasive technique can be of great value in clinical case management. Traditional manual microscopic analysis relies on human operators who read the samples visually and identify them, and therefore is time-consuming, labor-intensive and difficult to standardize. Automated microscopy of all specimens is more practical than manual microscopy, because it eliminates variation among different technologists. This variation becomes more pronounced when the same technologist examines increasing numbers of specimens. Also, it is less labor-intensive and thus less costly than manual microscopy. It also provides more consistent and accurate results. An automated urinalysis system workstation (The Y ellowI RI ST M, International Remote Imaging Systems, Inc.) has been introduced in numerous clinical laboratories for automated microscopy. Urine samples are processed and examined at lOOx (low power field) and 400x magnifications (high power field) with bright-field illumination. The Y ellowI RI ST M automated system collects video images of formed analytes in a stream of un centrifuged urine passing an optical assembly. Each image has one analyte in it. These images are given to a computer algorithm for automatic identification of analytes. Context is rich in urinalysis and plays a crucial role in analyte classification. Some combinations of analytes are more likely than others. For instance, the presence of bacteria indicates the presence of white blood cells, since bacteria tend to cause infection and thus trigger the production of more white blood cells. If amorphous crystals show up, they tend to show up in bunches and in all sizes. Therefore, if there are amorphous crystallook-alikes in various sizes, it is quite possible that they are amorphous crystals. Squamous epithelial cells can appear both flat or rolled up. If squamous epithelial cells in one form are detected, X B. Song, J Sill, Y. Abu-Mostafa and H. Kasdan 966 Table I: Features extracted from urine anylates images reature number reature desc:ription ( 2 4 9 10 tht: m~an or hluc distribution the mean of gn...-cn dislrihulmn 15 th paccnlile of ?ray level hislo?ram 85 th percenlile of gray level hislogmm lh~ standard devia.tion \11' gray level intensity energy of the (.aplacian lransl\)rmalion of grey level image II 12 13 14 IS 16 then it is likely that there are squamous epithelial cells in the other form. Utilizing such context is crucial for classification accuracy. The classes we are looking at are bacteria, calcium oxalate crystals, red blood cells, white blood cells, budding yeast, amorphous crystals, uric acid crystals, and artifacts. The task of automated microscopic urinalysis is, given a urine specimen that consists of up to a few hundred images of analytes, to classify each analyte into one of these classes. The automated urinalysis system we developed consists of three steps: image processing and feature extraction, learning and pattern recognition, and context incorporation. Figure 1 shows some example analyte images. Table 1 gives a list of features extracted from analyte images. 1 3.2 CONTEXT-FREE CLASSIFICATION The features are fed into a nonlinear feed-forward neural network with 16 inputs, 15 hidden units with sigmoid transfer functions, and 8 sigmoid output units. A cross-entropy error function is used in order to give the output a probability interpretation. Denote the input feature vector as x, the network outputs a D dimensional vector (D = 8 in our case) p = {p(dlx)} , d = 1, ... , D, where p(dlx) is p{dlx) = Prob( an analyte belongs to class dl feature x) The decision made at this stage is d{x) = argmax p(dlx) d 3.3 IDENTIFICATION OF RELEVANT PARTIAL CONTEXT Not all classes are relevant in terms of carrying contextual information. We propose three criteria based on which we can systematicalIy investigate the relevance of the class presence. To use these criteria, we need to know the folIowing distributions: the class prior distribution p(c) for c = 1, ... ,D; the conditional class distribution p{cIAd) for c = 1, ... ,D 1>'1 and >'2 are respectively the larger and the smaller eigenvalues of the second moment matrix of an image. 967 Image Recognition in Context: Application to Microscopic Urinalysis and d = 1, . .. ,D; and the class presence prior distribution p(Ad) for d = 1, . . . , D. Ad is a binary random variable indicating the presence of class d. Ad = 1 if class d is present, and Ad = 0 otherwise. All these distributions can be easily estimated from the database. The first criterion is the correlation coefficient between the presence of any two classes; the second one is the classical mutual information I(e; Ad) between the presence of a class Ad and the class probability pee), where I(e; Ad) is defined as I(e; Ad) = H(e) H(eIAd) where H(e) = 2:~1 p(e = i)ln(p(e = i)) is the entropy of the class priors and H(eIAd) = P(Ad = I)H(eIAd = 1)+P(Ad = O)H(eIAd = 0) is the conditional entropy of e conditioned on Ad. The third criterion is what we call the expected relative entropy D(eIIAd) between the presence ofa class Ad and the labeling probability pee) , which we define as D(eIIAd) = P(Ad = I)D(p(e)llp(eIAd = 1)) + P(Ad = O)D(p(e)llp(eIAd = 0)) where D(p(e)llp(eIAd 1)) 2:~lP(e = ilAd = l)ln(p(c;/l~t)=l)) and D(p(e)llp(eIAd = 0)) = 2:~1 p(e = ilAd = O)ln(p(C;/l~t)=O)) According to the first criterion, one type of analyte is considered relevant to another if the absolute value of their correlation coefficient is beyond a certain threshold. It shows that uric acid crystals, budding yeast and calcium oxalate crystals are not relevant to any other types even by a generous threshold of 0.10. Similarly, the bigger the mutual information between the presence of a class and the class distribution, the more relevant this class is. Ranking the analyte types in terms of I(e; Ad) in a descending manner gives rise to the following list: bacteria, amorphous crystals, red blood cells, white blood cells, uric acid crystals, budding yeast and calcium oxalate crystals. Once again, ranking the analyte types in terms of D(eIIAd) in a descending manner gives rise to the following list: bacteria, red blood cells, amorphous crystals, white blood cells, calcium oxalate crystals, budding yeast and uric acid crystals. All three criteria lead to similar conclusions regarding the relevance of class presence - bacteria, red blood cells, amorphous crystals, and white blood cells are relevant, while calcium oxalate crystals, budding yeast and uric acid crystals are not. (Baed on prior knowledge, we discard artifacts from the outset as an irrelevant class.) 3.4 ALGORITHM FOR INCORPORATING PARTIAL CONTEXT Once the M relevant classes are identified, the following algorithm is used to incorporate partial context. Step 0 Estimate the priors p(eIAd) and pee), for e E {I, 2, .. . , D} and d E {I, 2, ... , D}. Step 1 For a given Xi, compute p(edxi) for ei = 1,2, . .. , Dusing whichever base machine learning model is preferred ( in our case, a neural network). Step 2 Let the M relevant classes be R 1 , ..? , RM. According to the no-context p( ei IXi) and certain criteria for detecting the presence or absence of all the relevant classes, get A RI , ? ?? ,ARM' Step 3 Letp(ei lXi , Ao) = p(eilxi), where Ao is the null element. Incorporate context from each relevant class sequentially, i.e., for m = 1 to M, iteratively compute p(eilxi; Ao, .. . , ARm_I ' ARTn) = p(eilxi' Ao,.? . , ARTn_J p(ei IARTn)p(AR"J pee) Step 4 Recompute A RI , . . . ,ARM based on the new class labellings. Return to step 3 and repeat until algorithm converges. 2 2Hence, the algorithm has an E-M flavor, in that it goes back and forth between finding the most 968 X B. Song, J. Sill, Y. Abu-Mostafa and H Kasdan amorphous crystals artifacts calcium oxalate crystals hyaline casts Figure I: Example of some of the analyte images. Step 5 Label the objects according to the final context-contammg p(cilxi, ARI'?? ? ' ARM)' i.e., Ci = argmaxp(ciIXi, A R1 , ... , ARM) for i = 1, ... , N. Ci This algorithm is invariant with respect to the ordering of the M relevant classes in (Ai, ... , AM). The proof is omitted here. 4 RESULTS The algorithm using partia.1 context was tested on a database of 83 urine specimens, containing a total of 20,276 analyte images. Four classes are considered relevant according to the criteria described in section 3.3: bacteria, red blood cells, white blood cells and amorphous crystals. We measure two types of error: analyte-by-analyte error, and specimen diagnostic error. The average analyte-by-analyte error is reduced from 44.48% before using context to 36.66% after, resulting a relative error reduction of 17.6% (Table 2). The diagnosis for a specimen is either normal or abnormal. Tables 3 and 4 compare the diagnostic performance with and without using context, and Table 5 lists the relative changes. We can see using context significantly increases correct diagnosis for both normal and abnormal specimens, and reduces both false positives and false negatives. average element-by-element error without context 44.48 % with context 36.66 % Table 2: Comparison of using and not using contextual information for analyte-by-analyte error. probable class labels given the context and determining the context given the class labels. Image Recognition in Context: Application to Microscopic Urinalysis truly normal truly abnormal estimated normal 40.96 % 19.28 % 969 estimated abnormal 7.23 % 32.53 % Table 3: Diagnostic confusion matrix not using context truly normal truly abnormal estimated normal 42.17 % 16.87 % estimated abnormal 6.02% 34.94 % Table 4: Diagnostic confusion matrix using context truly normal truly abnormal estimated normal +2.95 % - 12.50 % estimated abnormal -16.73 % +7.41 % Table 5: Relative accuracy improvement (diagonal elements) and error reduction (off diagonal elements) in the diagnostic confusion matrix by using context. 5 CONCLUSIONS We proposed a novel framework that can incorporate context in a simple and efficient manner, avoiding exponential computation and high dimensional density estimation. The application of the partial context technique to microscopic urinalysis image recognition demonstrated the efficacy of the algorithm. This algorithm is not domain dependent, thus can be readily generalized to other pattern recognition areas. ACKNOWLEDGEMENTS The authors would like to thank Alexander Nicholson, Malik Magdon-Ismail, Amir Atiya at the Caltech Learning Systems Group for helpful discussions. References [I) Song, X.B . & SilU. & Abu-Mostafa & Harvey Kasdan, (1997) "Incorporating Contextual Information in White Blood Cell Identification", In M. Jordan, MJ. Kearns and S.A. Solla (eds.), Advances in Neural Information Processing Systems 7,1997, pp. 950-956. Cambridge, MA: MIT Press. [2] Song, Xubo (1999) "Contextual Pattern Recognition with Application to Biomedical Image Identification", Ph.D. Thesis, California Institute of Science and Technology. [3) Boehringer-Mannheim-Corporation, Urinalysis Today, Boehringer-Mannheim-Corporation, 1991. [4] Kittler, J.."Relaxation labelling", Pattern Recognition Theory and Applications, 1987, pp. 99108., Pierre A. Devijver and Josef Kittler, Editors, Springer-Verlag. [5] Kittler, J. & Illingworth, J., " Relaxation Labelling Algorithms - A Review", Image and Vision Computing, 1985, vol. 3, pp. 206-216. [6] Toussaint, G., "The Use of Context in Pattern Recognition", Pattern Recognition, 1978, vol. 10, pp. 189-204. [7] Swain, P. & Vardeman, S. & Tilton, J., "Contextual Classification of Multispectral Image Data", Pattern Recognition, 1981, Vol. 13, No.6, pp. 429-441. 

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Recurrent cortical competition: Strengthen or weaken? Peter Adorjan*, Lars Schwabe, Christian Piepenbrock* , and Klaus Obennayer Dept. of Compo Sci., FR2-I, Technical University Berlin Franklinstrasse 28/29 10587 Berlin, Germany adorjan@epigenomics.com, {schwabe, oby} @cs.tu-berlin.de, piepenbrock@epigenomics.com http://www.ni.cs.tu-berlin.de Abstract We investigate the short term .dynamics of the recurrent competition and neural activity in the primary visual cortex in terms of information processing and in the context of orientation selectivity. We propose that after stimulus onset, the strength of the recurrent excitation decreases due to fast synaptic depression. As a consequence, the network shifts from an initially highly nonlinear to a more linear operating regime. Sharp orientation tuning is established in the first highly competitive phase. In the second and less competitive phase, precise signaling of multiple orientations and long range modulation, e.g., by intra- and inter-areal connections becomes possible (surround effects). Thus the network first extracts the salient features from the stimulus, and then starts to process the details. We show that this signal processing strategy is optimal if the neurons have limited bandwidth and their objective is to transmit the maximum amount of information in any time interval beginning with the stimulus onset. 1 Introduction In the last four decades there has been a vivid and highly polarized discussion about the role of recurrent competition in the primary visual cortex (VI) (see [12] for review). The main question is whether the recurrent excitation sharpens a weakly orientation tuned feedforward input, or the feed-forward input is already sharply tuned, hence the massive recurrent circuitry has a different function. Strong cortical recurrency implements a highly nonlinear mapping of the feed-forward input, and obtains robust and sharply tuned cortical response even if only a weak or no feed-forward orientation bias is present [6, 11, 2]. However, such a competitive network in most cases fails to process mUltiple orientations within the classical receptive field and may signal spurious orientations [7]. This motivates the concept that the primary visual cortex maps an already sharply orientation tuned feed-forward input in a less competitive (more linear) fashion [9, 13]. Although these models for orientation selectivity in VI vary on a wide scale, they have one common feature: each of them assumes that the synaptic strength is constant on the short time scale on which the network operates. Given the phenomenon of fast synaptic *Current address: Epigenomics GmbH, Kastanienallee 24,0-10435 Berlin, Germany P. Adorjan, L. Schwabe, C. Piepenbrock and K. Obermayer 90 dynamics this, however, does not need to be the case. Short term synaptic dynamics, e.g., of the recurrent excitatory synapses would allow a cortical network to operate in bothcompetitive and linear-regimes. We will show below (Section 2) that such a dynamic cortical amplifier network can establish sharp contrast invariant orientation tuning from a broadly tuned feed-forward input, while it is still able to respond correctly to mUltiple orientations. We then show (Section 3) that decreasing the recurrent competition with time naturally follows from functional considerations, i.e. from the requirement that the mutual information between stimuli and representations is maximal for any time interval beginning with stimulus onset. We consider a free-viewing scenario, where the cortical layer represents a series of static images that are flashed onto the retina for a fixation period (~T = 200 - 300 ms) between saccades. We also assume that the spike count in increasing time windows after stimulus onset carries the information. The key observations are that the signal-to-noise ratio of the cortical representation increases with time (because more spikes are available) and that the optimal strength of the recurrent connections (w.r.t. information transfer) decreases with the decreasing output noise. Consequently the model predicts that the information content per spike (or the SNR for ajixed sliding time window) decreases with time for a flashed static stimulus in accordance with recent experimental studies. The neural system thus adapts to its own internal changes by modifying its coding strategy, a phenomenon which one may refer to as "dynamic coding". 2 Cortical amplifier with fast synaptic plasticity To investigate our first hypothesis, we set up a model for an orientation-hypercolumn in the primary visual cortex with similar structure and parameters as in [7]. The important novel feature of our model is that fast synaptic depression is present at the recurrent excitatory connections. Neurons in the cortical layer receive orientation-tuned feed-forward input from the LGN and they are connected via a Mexican-hat shaped recurrent kernel in orientation space. In addition, the recurrent and feed-forward excitatory synapses exhibit fast depression due to the activity dependent depletion of the synaptic transmitter [1, 14]. We compare the response of the cortical amplifier models with and without fast synaptic plasticity at the recurrent excitatory connections to single and mUltiple bars within the classical receptive field. The membrane potential V (0, t) of a cortical cell tuned to an orientation 0 decreases due to the leakage and the recurrent inhibition, and increases due to the recurrent excitation T a at V(O, t) + V(O, t) (1) where T = 15 ms is the membrane time constant and ILGN (0, t) is the input received from the LGN. The recurrent excitatory and inhibitory cortical inputs are given by r:"(O, t) (2) where ~ (Of, 0) is a 1T periodic circular difference between the preferred orientations, JCX(O, Of , t) are the excitato~ and inhibitory connection strengths (with a E {exc, inh}, J~x~x = 0.2 m V /Hz and J:::ax = 0.8m V /Hz), and f is the presynaptic firing rate. The excitatory synaptic efficacy r xc is time dependent due to the fast synaptic depression, while the efficacy of inhibitory synapses Jinh is assumed to be constant. The recurrent excitation is sharply tuned (j exc = 7.5 0 , while the inhibition has broad tuning (jinh = 90 0 ? The mapping from the membrane potential to firing rate is approximated by a linear function with a threshold at 0 (f(O) = ,6max(O, V(O)),,6 = 15Hz/mV). Gaussian-noise with variances 91 Recurrent Cortical Competition: Strengthen or Weaken? Feedforward Input ,....., 1 >E ....::l g- Static N X '-' - n Q) rJJ '-' ,--,., , = o.. --- ,, 0 rJJ Q) ~ ~90 -45 45 90 0 Orientation [deg] Depressing 15 ~90 -45 (a) j! N X 15 '-' Q) rJJ = 0 0.. rJJ Q) ~ \ 0 45 90 Orientation [deg] ~90 -45 0 45 90 Orientation [deg] (b) (c) Figure 1: The feed-forward input (a), and the response ofthe cortical amplifier model with static recurrent synaptic strength (b), and a network with fast synaptic depression (c) if the stimulus is single bar with different stimulus contrasts (40%dotted; 60%dashed; 80%solid line). The cortical response is averaged over the first 100 illS after stimulus onset. of 6 Hz and 1.6 Hz is added to the input intensities and to the output of cortical neurons. The orientation tuning curves of the feed-forward input ILGN are Gaussians (O'"LGN = 18?) resting on a strong additive orientation independent component which would correspond to a geniculo-cortical connectivity pattern with an approximate aspect ratio of 1:2. Both, the orientation dependent and independent components increase with contrast. Considering a free-viewing scenario where the environment is scanned by saccading around and fixating for short periods of 200 - 300 illS we model stationary stimuli present for 300 illS. The stimuli are one or more bars with different orientations. Feed-forward and recurrent excitatory synapses exhibit fast depression. Fast synaptic deI'ression is modeled by the dynamics of the expected synaptic transmitter or "resource" R( t) for each synapse. The amount of the available transmitter decreases proportionally to the release probability p and to the presynaptic firing rate /, and it recovers exponentially (T~~N = 120 illS, Tr~~x = 850 illS, pLGN = 0.35 and pCtx = 0.55), 1 - R(t) Tree - /(t)p(t)R(t) = - Teff R(t) (f() ()) 1 +. t, P t Tree (3) The change of the membrane potential on the postsynaptic cell at time t is proportional to the released transmitter pR(t). The excitatory connectivity strength between neurons tuned to orientations 0 and 0' is expressed as j?xe(o , 0', t) = J:::~xpR991(t). Similarly this applies to the feed-forward synapses. Fast synaptic plasticity at the feed-forward synapses has been investigated in more detail in previous studies [3, 4]. In the following, we compare the predictions of the cortical amplifier model with and without fast synaptic depression at the recurrent excitatory connections. In both cases fast synaptic depression is present at the feed-forward connections limiting the duration of the effective feed-forward input to 200 - 400 illS. Figure 1 shows the orientation tuning curves at different stimulus contrasts. The feed-forward input is noisy and broadly tuned (Fig. la). Both models exhibit contrast invariant tuning (Fig. 1b, c). If fast synaptic depression is present at the recurrent excitation, the cortical network sharpens the broadly tuned feedforward input in the initial response phase. Once sharply tuned input is established, the tuning width does not change, only the response amplitude decreases in time. The predictions of the two models differ substantially if multiple orientations are present (Fig. 2). At first, we test the cortical response to two bars separated by 60? with different intensities (Figs. 2a, b). If the recurrent synaptic weights are static and strong enough (Fig. 2a), then only one orientation is signaled. The cortical network selects the orientation 92 P Adorjim. L. Schwabe. C. Piepenbrock and K. Obermayer Feedforward Input (a) f~1 I~I~--~_~~"~,------~ ~90 -45 0 45 Activity Profile Average Cortical Response g .: S ~ 90 Orientation [deg] ,' I .b~---~#"" A I .,g 90 0 -90 90 . ~ o ?c (b) .. ,----, ... '.-'--<" ',,--,'" ;;20 1~ .. .. ???INI 'i 110 :>E c: (c) - .. ~90 ,-,,, , .... -- '. a) 0 45 Orienretionldeg) 90 ~ -90 90 0 r:S:;;1 (: Ul -45 g .~ 0 ~ ''''' 0 , " --- /', -90 ~~~_ o '" ~90- -45 . 0 '. 45 '90 -90 (d) Orientation [deg] :;a????\~~~\$~* ~'> ~ \m0 '" s m?" : 0 ~ . ~-... 150 Time [ms] 300 Figure 2: The response of the cortical amplifier model with static (a,c) and fast depressing recurrent synapses (b, d). In both models the feed-forward synapses are fast depressing. In the left column the feed-forward input is shown, that is same for both models. Two types of stimuli were applied. The first stimulus consists of a stronger (a = -30?) and a weaker bar (a +30?) (a, b); the second stimulus consists of three equal intensity bars with orientations that are separated by 60? (c, d). In the middle column the cortical response is shown averaged for different time windows ([0 .. 30] dotted; [0 .. 80] dashed; [200 .. 300] solid line). In the right column the cortical activity profile is plotted as a function of time. Gray values indicate the activity with bright denoting high activities. = with the highest amplitude in a winner-take-all fashion. In contrast, if synaptic depression is present at the recurrent excitatory synapses, both bars are signaled in parallel (at low release probability, Fig. 2b) or after each other (high release probability, data not shown). First, those cells fire which are tuned to the orientation of the bar with the stronger intensity, and a sharply tuned response emerges at a single orientation-the network operates in a winner-take-all regime. The synapses of these highly active cells then become strongly depressed and cortical competition decreases. As the network is shifted to a more linear operation regime, the second orientation is signaled too. Note that this phenomenontogether with the observed contrast invariant tuning-cannot be reproduced by simply decreasing the static synaptic weights in the cortical amplifier model. The recurrent synaptic efficacy changes inhomogeneously in the network depending on the activity. Only the synapses of the highly active cells depress strongly, and therefore a sharply tuned response can be evoked by a bar with weak intensity. Fast synaptic depression thus behaves as a local self-regulation that modulates competition with a certain delay. This delay, and therefore the delay of the rise of the response to the second bar depends on the effective time constant reff(f(t),p) = rrec/(l + pf(t)rrec) of the synaptic depression at the recurrent connections. If the depression becomes faster due to an increase in the release probability p, then the delay decreases. The delay also scales with the difference between the bar intensities. The closer to each other they are, the shorter the delay will be. In Figs. 2c, d the cortical response to three bars with equal intensities is presented. Cells tuned to the presented three orientations respond in parallel if fast synaptic depression at the recurrent excitation is present (Figs. 2d). The cortical network with strong static recurrent synapses again fails to signal faithfully its feed-forward input. Additive noise on the 93 Recurrent Cortical Competition: Strengthen or Weaken? feed-forward input introduces a slight symmetry breaking and the network with static recurrent weights responds strongly at the orientation of only one of the presented bars (Fig. 2c). In summary, our simulations revealed that a recurrent network with fast synaptic depression is capable of obtaining robust sharpening of its feed-forward input and it also responds correctly to multiple orientations. Note that other local activity dependent adaptation mechanisms, such as slow potassium current, would have similar effects as the synaptic depression on the highly orientation specific excitatory connections. An experimentally testable prediction of our model is that the response to a flashed bar with lower contrast can be delayed by masking it with a second bar with higher contrast (Fig. 2b, right). We also suggest that long range integration from outside of the classical receptive field could emerge with a similar delay. In the initial phase of the cortical response, strong local features are amplified. In the longer, second phase, recurrent competition decreases and then weak modulatory recurrent or feed-forward input has a stronger relative effect. In the following, we investigate whether this strategy is favorable from the point of view of cortical encoding. 3 Dynamic coding In the previous section we have proposed that during cortical processing a highly nonlinear phase is followed by a more linear mode if we consider a short stimulus presentation or a fixation period. The simulations demonstrated that unless the recurrent competition is modulated in time, the network fails to account for more than one feature in its input. From a strictly functional point of view the question arises, why not to use weak recurrent competition during the whole processing period. We investigate this problem in an abstract signal-encoder framework i7 = g( i) + 1] , (4) where i is the input to the "cortical network", g(i) is a nonlinear mapping and-for the sake of simplicitY-1] is additive Gaussian noise. Naturally, in a real recurrent network output noise becomes input noise because of the feedback. Here we use the simplifying assumption that only output noise is present on the transformed input signal (input noise would lead to different predictions that should be further investigated). Output noise can be interpreted as a noisy channel that projects out from, e.g., the primary visual cortex. The nonlinear transformation g(i) here is considered as a functional description of a cortical amplifier network without analyzing how actually it is "implemented". Considering orientation selectivity, the signal i can be interpreted as a vector of intensities (or contrasts) of edges with different orientations. Edges which are not present have zero intensity. The coding capacity of a realistic neural network is limited. Among several other noise sources, this limitation could arise from imprecision in spike timing and a constraint on the maximal or average firi ng rate. The input-output mapping g( i) of a cortical amplifier network is approximated with the soft-max function (5) The f3 parameter can be interpreted as the level of recurrent competition. As f3 -+ 0 the network operates in a more linear mode, while f3 -+ 00 puts it into a highly nonlinear winner-take-all mode. In all cases the average activity in the network is constrained which has been suggested to minimize metabolic costs [5]. Let us consider a factorizing input distribution, 1 p( i) = Z IIi exp (_x ---t- a) for x ~ 0, (6) P. Adorjfm, L. Schwabe, C. Piepenbrock and K. Obermayer 94 8rr===~~----~----~ --- 0.5 1.0 0--0 6 'J:;-----Q- .c!1 - - ,,-s-' 00 0.05 0.1 Noise (stdev) 0.15 Figure 3: The optimal competition parameter j3 as a function of the standard deviation of the Gaussian output noise 'f}. The optimal j3 is calculated for highly super-Gaussian, Gaussian, and sub-Gaussian stimulus densities. The sparsity parameter a is indicated in the legend. where the exponent a detennines the sparsity of the probability density function, Z is a nonnalizing constant, and ~ detennines the variance. If a = 2, the input density is the positive half of a multivariate Gaussian distribution. With a > 2 the signal distribution becomes sub-Gaussian, and with a < 2 it becomes super-Gaussian. For optimal processing in time one needs to gain the maximal infonnation about the signal for any increasing time window. Let us assume that the stimulus is static and it is presented for a limited time. As time goes ahead after stimulus onset, the time window for the encoding and the read-out mechanism increases. During a longer period more samples of the noisy network output are available, and thus the output noise level decreases with time. We suggest that the optimal competition parameter j3opt_at which the mutual infonnation between input i and output if (Eq. 4) is maximized-<iepends on the noise level. As the noise decreases with time, j3 or the recurrent cortical competition should also change during cortical processing. To demonstrate this idea, the mutual infonnation is calculated numerically for a three-dimensional state space. One might expect that at higher noise levels the highest infonnation transfer can be obtained if the typical and salient features are strongly amplified. Note that this is only true if the standard deviation of the noise scales sub-linearly with activity, which is true for an additive noise process as well as Poisson firing. As noise decreases (e.g., with increasing the time window for estimation), the level of competition should decrease distributing the available resources (e.g., spikes) among more units and letting the network respond to finer details at the input. Investigating the level of optimal competition j3 as a function of the standard deviation of the output noise (Fig. 3) this intuition is indeed justified. The optimal j3 scales with the standard deviation of the additive noise process. Comparing signal distributions with the same variance but with different sparsity exponents a, we find that the sparser the signal distribution is, the higher the optimal competition becomes, because multiple features are unlikely to be present at the same time if the input distribution is sparse. By enforcing competition, the optimal encoding strategy also generates an activity distribution where only few units fire for a presented stimulus. Since edges with different orientations fonn a sparse distributed representation of natural scenes [8], our work suggests that a strongly competitive visual cortical network could achieve a better performance on our visual environment than a simple linear network would do. We can now interpret our simulation results presented in the Section 2 from a functional point of view and give a prediction for the dynamics of the recurrent cortical competition. Noting that the output noise is decreasing with increasing time-window for encoding, the cortical competition should also decrease following a similar trajectory as presented in Fig. 3. If competition is low and static, then the cumulative mutual infonnation between input and output would converge only slowly towards the overall infonnation that is available in the stimulus. If the competition is high during the whole observation period, then after a fast rise the cumulative mutual information would saturate well below the possible Recurrent Cortical Competition: Strengthen or Weaken? 95 maximum. If the level of competition is dynamic, and it decreases from an initially highly competitive state, then the network obtains maximal information transfer in time. One may argue that the valuable information about the signals mainly depends on the interest of the observer. Considering an encoding system for one variable it has been suggested that in a highly attentive state the recurrent competition increases [10]. In the view of our results we would refine this statement by suggesting that competition increases or decreases depending on the level of visual detail the observer pays attention to. Whenever representation of small details is also required, reducing competition is the optimal strategy given enough bandwidth. In summary, using a detailed model for an orientation hypercolumn in VI we have demonstrated that sharp contrast invariant tuning and faithful representation of multiple features can be achieved by a recurrent network if the recurrent competition decreases in time after stimulus onset. The model predicts that the cortical response to weak details in the stimulus emerges with a delay if a second stronger feature is also present. The modulation from, e.g., outside of the classical receptive field also has a delayed effect on cortical activity. Our study within an abstract framework revealed that weakening the recurrent cortical competition on a fast time scale is functionally advantageous, because a maximal amount of information can be transmitted in any time window after stimulus onset. Acknowledgments Supported by the Boehringer Ingelheim Fonds (C. P.), by the German Science Foundation (DFG grant GK 120-2) and by Wellcome Trust 0500801ZJ97. References [1] L. F. Abbott, J. A. Varela, K. Sen, and S. B. Nelson. Synaptic depression and cortical gain control. Science, 275:220-224,1997. [2] P. Adoljan, J.B. Levitt, J.S. Lund, and K. Obennayer. A model for the intracortical origin of orientation preference and tuning in macaque striate cortex. Vis. Neurosci .? 16:303-318. 1999. Contrast adaptation and info[3] P. AdOljan. C. Piepenbrock. and K. Obennayer. max in visual cortical neurons. Rev. Neurosci.. 10: 181-200, 1999. ftp://ftp.cs.tuberlin.de/pub/locaVnilpapers/adp99-contrast.ps.gz. [4] O. B. Artun, H. Z. Shouval, and L. N. Cooper. The effect of dynamic synapses on spatiotemporal receptive fields in visual cortex. Proc. Natl. Acad. Sci.? 95:11999-12003, 1998. [5] R. Baddeley. An efficient code in VI? Nature. 381 :560-561,1996. [6] R. Ben-Yishai, R. Lev Bar-Or. and H. Sompolinsky. Theory of orientation tuning in visual cortex. Proc. Natl. Acad. Sci.? 92:3844-3848,1995. [7] M. Carandini and D. L. Ringach. Predictions of a recurrent model of orientation selectivity. Vision Res., 37:3061-3071.1997. [8] D. J. Field. What is the goal of sensory coding. Neural Comput.? 6:559-601, 1994. [9] D. H. Hubel and T. N. Wiesel. Receptive fields. binocular interaction and functional architecture in cat's visual cortex. 1. Physiol., 165:559-568. 1962. [10] D. K. Lee. L. Itti, C. Kock. and 1. Braun. Attention activates winner-take-all competition among visual filters. Nat. Neurosci., 2:375-381,1999. [11] D. C. Somers, S. B. Nelson. and M. Sur. An emergent model of orientation selectiVity in cat visual cortical simple cells. 1. Neurosci., 15:5448-65, 1995. [12] H. Sompolinsky and R. Shapley. New perspectives on the mechanisms for orientation selectivity. Curro Op. in Neurobiol., 7:514-522, 1997. [13] T. W. Troyer, A. E. Krukowski. N. J. Priebe, and K. D. Miller. Contrast-invariant orientation tuning in visual cortex: Feedforward tuning and correlation-based intracortical connectivity. 1. Neurosci., 18:5908-5927. 1998. [14] M. V. Tsodyks and H. Markram. The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. Proc. Natl. Acad. Sci., 94:719-723. 1997. 

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The Entropy Regularization Information Criterion Alex J. Smola Dept. of Engineering and RSISE Australian National University Canberra ACT 0200, Australia Alex.Smola@anu.edu.au John Shawe-Taylor Royal Holloway College University of London Egham, Surrey 1W20 OEX, UK john@dcs.rhbnc.ac.uk Bernhard Scholkopf Microsoft Research Limited St. George House, 1 Guildhall Street Cambridge CB2 3NH bsc@microsoft.com Robert C. Williamson Dept. of Engineering Australian National University Canberra ACT 0200, Australia Bob. Williamson @anu.edu.au Abstract Effective methods of capacity control via uniform convergence bounds for function expansions have been largely limited to Support Vector machines, where good bounds are obtainable by the entropy number approach. We extend these methods to systems with expansions in terms of arbitrary (parametrized) basis functions and a wide range of regularization methods covering the whole range of general linear additive models. This is achieved by a data dependent analysis of the eigenvalues of the corresponding design matrix. 1 INTRODUCTION Model selection criteria based on the Vapnik-Chervonenkis (VC) dimension are known to be difficult to obtain, worst case, and often not very tight. Yet they have the theoretical appeal of providing bounds, with few or no assumptions made. Recently new methods [8, 7, 6] have been developed which are able to provide a better characterization of the complexity of function classes than the VC dimension, and moreover, are easily obtainable and take advantage of the data at hand (i.e. they employ the concept of luckiness). These techniques, however, have been limited to linear functions or expansions of functions in terms of kernels as happens to be the case in Support Vector (SV) machines. In this paper we show that the previously mentioned techniques can be extended to expansions in terms of arbitrary basis functions, covering a large range of practical algorithms such as general linear models, weight decay, sparsity regularization [3], and regularization networks [4]. 343 The Entropy Regularization Information Criterion 2 SUPPORT VECTOR MACHINES Support Vector machines carry out an effective means of capacity control by minimizing a weighted sum of the training error (1) and a regularization term Q[J] = ~llwI12; i.e. they minimize the regularized risk functional Rreg[J] 1 := Remp[f] A m + AQ[f] = m ~ C(Xi, Yi, f(Xi)) + "2llwI12 . (2) t=l Here X := {Xl, ... Xm} C X denotes the training set, Y := {YI, ... Ym} C }j the corresponding labels (target values), X, }j the corresponding domains, A > a a regularization constant, C : X X }j X }j -+ JRt a cost function, and f : X -+ }j is given by f(x) := (x, w), or in the nonlinear case f(x) := (4l(x), w). (3) Here 4l : X -+ l' is a map into a feature space 1'. Finally, dot products in feature space can be written as (4l(x), 4l(X')) = k(x, x') where k is a so-called Mercer kernel. For n E N, ~n denotes the n-dimensional space of vectors x = (Xl, ... , Xn). We define spaces as follows: as vector spaces, they are identical to ~n, in addition, they are endowed with p-norms: f; We write fp = fr;:o Furthermore let Ue~ := {x: Ilxlle~ ::; fora < p < 00 forp = 00 I} be the unitf;-baU. For model selection purposes one wants to obtain bounds on the richness of the map Sx Sx : w f-t (f(xd, ... , f(xm)) = ((4l(xd, w), ... , (4l(xm), w)). (4) where w is restricted to an f2 unit ball of some radius A (this is equivalent to choosing an appropriate value of A - an increase in A decreases A and vice versa). By the "richness" of Sx specificaUy we mean the ?-covering numbers N( ?, SX (AUe;;, ), f1:J of the set Sx(AUlm). In the standard COLT notation, we mean p f: N(?, SX(AUl;;')' f:) := min { n There exists a set {Zl, ... zn} C F such that for all Z E Sx(AUem) we have min liz - zililm < ? p l::;i::;n 00 } - See [8] for further details. When carrying out model selection in this case, advanced methods [6] exploit the distribution of X mapped into feature space 1', and thus of the spectral properties of the operator Sx by analyzing the spectrum of the Gram matrix G = [gij]ij, where gij := k(Xi, Xj). All this is possible since k(Xi,Xj) can be seen as a dot product of Xi,Xj mapped into some feature space 1', i.e. k(Xi, Xj) = (4l(Xi), 4l(Xj )) . This property, whilst true for SV machines with Mercer kernels, does not hold in general case where f is expanded in terms of more or less arbitrary basis functions. 344 A. J. Smola. J. Shawe-Taylor, B. Sch61kopf and R. C. Williamson 3 THE BASIC PROBLEMS One basic problem is that when expanding 1 into n (5) i=l with Ii (x) being arbitrary functions, it is not immediately obvious how to regard 1 as a dot product in some feature space. One can show that the VC dimension of a set of n linearly independent functions is n. Hence one would intuitively try to restrict the class of admissible models by controlling the number of basis functions n in terms of which 1 can be expanded. Now consider an extreme case. In addition to the n basis functions Ii defined previously, we are given n further basis functions II, linearly independent of the previous ones, which differ from Ii only on a small domain X', i.e. Iilx\x = IIlx\xl. Since this new set of functions is linearly independent, the VC dimension of the joint set is given by 2n. On the other hand, if hardly any data occurs on the domain X', one would not notice the difference between Ii and II. In other words, the joint system of functions would behave as if we only had the initial system of n basis functions. 1 An analogous situation occurs if II = Ii + ?gi where ? is a small constant and gi was bounded, say, within [0, 1J. Again, in this case, the additional effect of the set offunctions II would be hardly noticable, but still, the joint set of functions would count as one with VC dimension 2n. This already indicates, that simply counting the number of basis functions may not be a good idea after all. .' ''~ Figure 1: From left to right: (a) initial set of functions h, ... , 15 (dots on the x-axis indicate sampling points); (b) additional set of functions IL ... , I~ which differ globally, but only by a small amount; (c) additional set offunctions IL ... , I~ which differ locally, however by a large amount; (d) spectrum of the corresponding design matrices - the bars denote the cases (a)-(c) in the corresponding order. Note that the difference is quite small. On the other hand, the spectra of the corresponding design matrices (see Figure 1) are very similar. This suggests the use of the latter for a model selection criterion. Finally we have the practical problem that capacity control, which in SV machines was carried out by minimizing the length of the "weight vector" w in feature space, cannot be done in an analogous way either. There are several ways to do this. Below we consider three that have appeared in the literature and for which there exist effective algorithms. Example 1 (Weight Decay) Define Q[IJ := ~ L:i ar . i.e. the coefficients ai of the junction expansion are constrained to an ?2 ball. In this case we can consider the following where operator S(1)? X . ?n 2 -t ?m 00' Sr): aM (f(xd, ... , I(x m )) = ((f(Xl), a), . .. , (f(Xm), a)) = Fa (6) Here I(x):= Ul(x) , .. ?In(x)), Fij := Ii(Xj), a'- (al, ... ,an ) and a E AUl'2for some A> O. 345 The Entropy Regularization Information Criterion Example 2 (Sparsity Regularization) In this case Q[J] := Li lail, i.e. the coefficients ai of the function expansion are constrained to an ?1 ball to enforce sparseness [3]. Thus sC;) :?1 -t ?~ with sC;) mapping a as in (6) except a E AUlI. This is similar to expansions encountered in boosting or in linear programming machines. Example 3 (Regularization Networks) Finally one could set Q[J] := ~a T Qa for some positive definite matrix Q. For instance, Qij could be obtainedfrom (Ph, P fj) where P is a regularization operator penalizing non-smooth functions [4J. In this case a lives inside some n-dimensional ellipsoid. By substituting a' := Q% a one can reduce this setting to the case of example 1 with a different set of basis functions (f'(x) = Q-% f(x)) and consider an evaluation operator s~) : ?2 s~): a' f-+ (f(xd, . .. , f(xm)) -t ?: given by = ((Q-% f(X1), a'), . .. , (Q-t f(xm), a')) = Q-t Fa' (7) where a' E AUl2 for some A> 0 and Fij = fi(xj) as in example 1. Example 4 (Support Vector Machines) An important special case of example 3 are Support Vector Machines where we have Qij = k(Xi,Xj) andfi(x) = k(Xi,X), henceQ = F. Hence the possible values generated by a Support Vector Machine can be written as s~): a' f-+ (f(X1), ... , f(xm)) where a' E AUl2 for some A = ((Q-% f(xd, a'), . .. , (Q-% f(xm), a')) = Ft a' (8) > o. 4 ENTROPY NUMBERS Covering numbers characterize the difficulty of learning elements of a function class. Entropy numbers of operators can be used to compute covering numbers more easily and more tightly than the traditional techniques based on VC-like dimensions such as the fat shattering dimension [1]. Knowing el (S x) = ? (see below for the definition) tells one that 10g:N(? , F,?~) ::; I, where F is the effective class of functions used by the regularised learning machines under consideration. In this section we summarize a few basic definitions and results as presented in [8] and [2]. The lth entropy number ?l (F) of a set F with a corresponding metric d is the precision up t~ whicI:! F can _be approximated by 1 elements of F; i.e. for all f E F there exists some fi E {h, ? ??, fd such that d(f, fi) ::; ?l. Hence ?1(F) is the functional inverse of the covering number of F. The entropy number of an bounded linear operator T: A -t B between normed linear spaces A and B is defined as ?1(T) := ?1(T(UA)) with the metric d being induced by II . liB. The dyadic entropy numbers el are defined by el := ?2'+1 (the latter quantity is often more convenient to deal with since it corresponds to the log of the covering number). We make use of the following three results on entropy numbers of the identity mapping from ?;1 into ?;2' diagonal operators, and products of operators. Let id;l ,P2 : ?;1 -t ?;2 ; id;1 ,P2 : x f-+ x The following result is due to Schlitt; the constants 9.94 and 1.86 were obtained in [9]. Proposition 1 (Entropy numbers for identity operators) Be mEN. Then el(id~,2) ::; 9.94 (t log (1 + T) ) 1 2 1 & el (id~,(xJ ::; 1.86 (t log (1 + T) )2 (9) A. J Smola, J Shawe-Taylor, B. SchOlkopfand R. C. Williamson 346 Proposition 2 (Carl and Stephani [2, p.11]) Let E, F, G be Banach spaces, R : F -+ G, and S: E -+ F. Then,forn, tEN, en+t-l (RS) ~ en(R)et(S), en(RS) ~ en (R)IISII and en(RS) ~ Note that the latter two inequalities follow directly from the fact that R: F -+ G by definition of the operator norm IIRII. Proposition 3 Let 0"1 ~ 0"2 ~ . .. ~ O"j ~ . .. ~ 0, 1 ~ p ~ 00 en(S)IIRII. ?l (R) (to) = IIRIlfor all and (11) for x = (Xl, X2, ... , Xj, . .. ) E f!p be the diagonal operator from f!p into itself, generated by the sequence (0" j ) j. Then for all n E N, 5 THE MAIN RESULT We can now state the main theorem which gives bounds on the entropy numbers of S~) for the first three examples of model selection described above (since Support Vector Machines are a special case of example 3 we will not deal with it separately). Proposition 4 Let! be expanded in a linear combination of basis functions as ! .L~=l adi and the coefficients a restricted to one of the convex sets as described in the examples 1 to 3. Moreover denote by Fij := !j(Xi) the design matrix on a particular sample X, and by Q the regularization matrix in the case of example 3. Then the following bound on Sx holds. 1. In the case of weight decay (ex. 1)(with h el(S~)) ~ + l2 ~ l + 1) 1.96 (llllog(1 +m/h))t 2. 1n the case of weight sparsity regularization (ex. 2) (with el(S~)) ~ 18.48 (lillog (1 + m/h)) t eI2(~)' h + l2 + l3 el 2 (~) (l3'llog (1 3. Finally, in the case of regularization networks (ex. 3) (with II el (Sr)) ~ 1.96 (lillog (1 (13) l + 2) + m/l3)) t. + l2 + m/h)) t el (~). 2 ~ ~ l (14) + 1) (15) Here ~ is a diagonal scaling operator (matrix) with (i, i) entries .j(ii and (.j(ii)i are the eigenvalues (sorted in decreasing order) of the matrix FFT in the case of examples 1 and 2, and FQ-l FT in the case of example 3. The entropy number of ~ is readily bounded in terms of (O"i)i by using (3). One can see that the first setting (weight decay) is a special case of the third one, namely when Q = 1, i.e. when Q is just the identity matrix. Proof The proofrelies on a factorization of S~) (i = 1,2,3) in the following way. First we consider the equivalent operator Sx mapping from f!~ to and perform a singular value decomposition [5] of the latter into Sx = V~W where V, W are operators of norm 1, and ~ contains the singular values of S~), i.e. the singular values of F and FQ- t f!r 347 The Entropy Regularization Information Criterion respectively. The latter, however, are identical to the square root of the eigenvalues of F FT or FQ-l FT. Consequently we can factorize S~) as in the diagram (16) Finally, in order to compute the entropy number of the overall operator one only has to use the factorization of Sx into S~) = id~oo VL:W for i E {1,3} and into S~) = id~oo VL:Wid~, 2 for example 2, and apply Proposition 2 several times. We also exploit l. ? the fact that for singular value decompositions IIVI\' IIWII s The present theorem allows us to compute the entropy numbers (and thus the complexity) of a class of functions on the current sample X. Going back to the examples of section 3, which led to large bounds on the VC dimension one can see that the new result is much less susceptible to such modifications: the addition of f{. ... f~ to h, ... f n does not change the eigenspectrum L: of the design matrix significantly (possibly only doubling the nominal value of the singular values), if the functions fi differ from fi only slightly. Consequently also the bounds will not change significantly even though the number of basis functions just doubled. Also note that the current error bounds reduce to the results of [6] in the SV case: here Q ij = Fij = k( Xi, X j) (both the design matrix F and the regularization matrix Q are determined by kernels) and therefore FQ-l F = Q. Thus the analysis of the singular values of FQ-l F leads to an analysis of the eigenvalues of the kernel matrix, which is exactly what is done when dealing with SV machines. 6 ERROR BOUNDS To use the above result we need a bound on the expected error of a hypothesis f in terms of the empirical error (training error) and the observed entropy numbers ?n(J'). We use [6, Theorem 4.1] with a small modification. Theorem 1 Let:1' be a set of linear junctions as described in the previous examples with en(Sx) as the corresponding bound on the observed entropy numbers of:1' on the dataset X. Moreover suppose thatforafixed threshold b E [?for some f E :1', sgn(f - b) correctly classifies the set X with a margin 'Y := minlSiSm If(Xi) - bl. t). Finally let U := min{ n E N with en(Sx) s 'Y /8.001} and a(U, <5) := 3.08(1 + bIn Then with confidence 1- <5 over X (drawn randomly from pm where P is some probability distribution) the expected error ofsgn(f - b) is boundedfrom above by ?(m,U,<5) =! (U(1+a(U,~)log(5t-m)log(17m)) + log (l6r)) . (17) The proof is essentially identical to that of [6, Theorem 4.1] and is omitted. [6] also shows how to compute en (S x) efficiently including an explicit formula for evaluating el (L:). 7 DISCUSSION We showed how improved bounds could be obtained on the entropy numbers of a wide class of popular statistical estimators ranging from weight decay to sparsity regularization 348 A. J Smola. J Shawe- Taylor, B. SchOllropf and R. C. Williamson (with SV machines being a special case thereof). The results are given in a way that is directly useable for practicioners without any tedious calculations of the VC dimension or similar combinatorial quantities. In particular, our method ignores (nearly) linear dependent basis functions automatically. Finally, it takes advantage of favourable distributions of data by using the observed entropy numbers as a base for stating bounds on the true entropy numbers with respect to the function class under consideration. Whilst this leads to significantly improved bounds (we achieved an improvement of approximately two orders of magnitude over previous VC-type bounds involving only the radius of the data R and the weight vector IIwll in the experiments) on the expected risk, the bounds are still not good enough to become predictive. This indicates that possibly rather than using the standard uniform convergence bounds (as used in the previous section) one might want to use other techniques such as a PAC-Bayesian treatment (as recently suggested by Herbrich and Graepel) in combination with the bounds on eigenvalues of the design matrix. Acknowledgements: This work was supported by the Australian Research Council and a grant of the Deutsche Forschungsgemeinschaft SM 62/1-1. References [1] N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler. Scale-sensitive Dimensions, Uniform Convergence, and Learnability. 1. of the ACM, 44(4):615-631,1997. [2] B. Carl and I. Stephani. Entropy, compactness, and the approximation of operators. Cambridge University Press, Cambridge, UK, 1990. [3] S. Chen, D. Donoho, and M. Saunders. Atomic decomposition by basis pursuit. Technical Report 479, Department of Statistics, Stanford University, 1995. [4] F. Girosi, M. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural Computation, 7:219-269,1995. [5] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1992. [6] B. Scholkopf, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Generalization bounds via eigenvalues of the gram matrix. Technical Report NC-TR-99-035, NeuroColt2, University of London, UK, 1999. [7] J. Shawe-Taylor and R. C. Williamson. Generalization performance of classifiers in terms of observed covering numbers. In Proc. EUROCOLT'99, 1999. [8] R. C. Williamson, A. J. Smola, and B. Scholkopf. Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators. NeuroCOLT NC-TR-98-019, Royal Holloway College, 1998. [9] R. C. Williamson, A. J. Smola, and B. SchOlkopf. A Maximum Margin Miscellany. Typescript, 1999. 

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Information Factorization in Connectionist Models of Perception Javier R. Movellan Department of Cognitive Science Institute for Neural Computation University of California San Diego James L. McClelland Center for the Neural Bases of Cognition Department of Psychology Carnegie Mellon University Abstract We examine a psychophysical law that describes the influence of stimulus and context on perception. According to this law choice probability ratios factorize into components independently controlled by stimulus and context. It has been argued that this pattern of results is incompatible with feedback models of perception. In this paper we examine this claim using neural network models defined via stochastic differential equations. We show that the law is related to a condition named channel separability and has little to do with the existence of feedback connections. In essence, channels are separable if they converge into the response units without direct lateral connections to other channels and if their sensors are not directly contaminated by external inputs to the other channels. Implications of the analysis for cognitive and computational neurosicence are discussed. 1 Introduction We examine a psychophysical law, named the Morton-Massaro law, and its implications to connectionist models of perception and neural information processing. For an example of the type of experiments covered by the Morton-Massaro law consider an experiment by Massaro and Cohen (1983) in which subjects had to identify synthetic consonant sounds presented in the context of other phonemes. There were two response alternatives, seven stimulus conditions, and four context conditions. The response alternatives were /1/ and /r/, the stimuli were synthetic sounds generated by varying the onset frequency of the third formant, followed by the vowel /i/. Each of the 7 stimuli was placed after each offour different context consonants, /v/, /s/, /p/, and /t/. Morton (1969) and Massaro independently showed that in a remarkable range of experiments of this type, the influence of stimulus and context on response probabilities can be accounted for with a factorized version of Luce's strength model (Luce, 1959) P(R = k I S = i, C = j) Tls(i, k) Tlc(j, k) 2: I TIs C1" l) TIc), C l)' . . for (l,,),k) E S x ex 'R. (1) Here S, C and R are random variables representing the stimulus, context and the subject's response, S, C and 'R are the set of stimulus, context and response al- J. R. Movellan and J. L. McClelland 46 ternatives, l1s(i, k) > 0 represents the support of stimulus i for response k, and l1c(j, k) > 0 the support of context j for response k. Assuming no strength parameter is exactly zero, (1) is equivalent to P(R P(R = k I S = i,e = j) = '(l1S(i,k)) = II S = i, e = j) l1s(i, l) (l1c(j,k)) , for all (i,j,k) E S x ex R. l1c(j, l) (2) This says that response probability ratios factorize into two components, one which is affected by the stimulus but unaffected by the context and one affected by the context but unaffected by the stimulus. 2 Diffusion Models of Perception Massaro (1989) conjectured that the Morton-Massaro law may be incompatible with feedback models of perception. This conjecture was based on the idea that in networks with feedback connections the stimulus can have an effect on the context units and the context can have an effect on the stimulus units making it impossible to factorize the influence of information sources. In this paper we analyze such a conjecture and show that, surprisiQ.gly, the Morton-Massaro law has little to do with the existence of feedback and lateral connections. We ground our analysis on continuous stochastic versions of recurrent neural networks 1. We call these models diffusion (neural) networks for they are stochastic diffusion processes defined by adding Brownian motion to the standard recurrent neural network dynamics. Diffusion networks are defined by the following stochastic differential equation dYi(t) = JLi(Y(t), X) dt + dBi(t) (J for i E {I, ... , n}, (3) where Yi(t) is a random variable representing the internal potential at time t of the unit, Y(t) = (Yl(t),??? ,Yn(t))', X represents the external input, which consists of stimulus and context, and Bi is Brownian motion, which acts as a stochastic driving term. The constant (J > 0, known as the dispersion, controls the amount of noise injected onto each unit. The function JLi, known as the drift, determines the average instantaneous change of activation and is borrowed from the standard recurrent neural network literature: this change is modulated by a matrix w of connections between units, and a matrix v that controls the influence of the external inputs onto each unit. ith JLi(Yi(t), X) 1 - = ~i(Yi(t)) (Yi(t) - Yi(t)), for all i E {I,??? , n}, (4) where 1/ ~i is a positive function, named the capacitance, controlling the speed of processing and Yi(t) = L j Zj(t) Wi,j Zj(t) + LVi,kXk, for alli E {I, .. ? ,n}, (5) k = CPi(}j(t)) = CP(O!i }j(t)) = 1/(1 + e- a ? Y;(t)). (6) Here Wi ,j, an element of the connection matrix w, is the weight from unit j to unit i, Vi,k is an element of the matrix v, cP is the logistic activation function and the O!i > 0 terms are gain parameters, that control the sharpness of the activation functions. For large values of O!i the activation function of unit i converges to a step function. The variable Zj(t) represents a short-time mean firing rate (the activation) of unit lFor an analysis grounded on discrete time networks with binary states see McClelland (1991). 47 Information Factorization j scaled in the (0,1) range. Intuition for equation (4) can be achieved by thinking of it as a the limit of a discrete time difference equation, in such case Y(t + ~t) = Yi(t) + J.'i (Yi (t), X)~t + (rli5:tNi (t), (7) where the Ni(t) are independent standard Gaussian random variables. For a fixed state at time t there are two forces controlling the change in activation: the drift, which is deterministic, and the dispersion which is stochastic. This results in a distribution of states at time t + ~t. As ~t goes to zero, the solution to the difference equation (7) converges to the diffusion process defined in (4). In this paper we focus on the behavior of diffusion networks at stochastic equilibrium, i.e., we assume the network is given enough time to approximate stochastic equilibrium before its response is sampled. 3 Channel Separability In this section we show that the Morton-Massaro is related to an architectural constraint named channel separability, which has nothing to do with the existence of feedback connections. In order to define channel separability it is useful to characterize the function of different units using the following categories: 1) Response specification units: A unit is a response specification unit, if, when the state of all the other units in the network is fixed, changing the state of this unit affects the probability distribution of overt responses. 2) Stimulus units: A unit belongs to the stimulus channel if: a) it is not a response unit, and b) when the state of the response units is fixed, the probability distribution of the activations of this unit is affected by the stimulus. 3) Context units: A unit belongs to the context channel if: a) it is not a response unit, and b) when the states of the response units are fixed, the probability distribution of the activations of this unit can be affected by the context. Given the above definitions, we say that a network has separable stimulus and context channels if the stimulus and context units are disjoint: no unit simultaneously belongs to the stimulus and context channels. In essence, channels are structurally separable if they converge into the response units without direct lateral connections to other channels and if their sensors are not directly contaminated by external inputs to the other channels (see Figure 1). In the rest of the paper we show that if a diffusion network is structurally separable the Morton-Massaro law can be approximated with arbitrary precision regardless of the existence of feedback connections. For simplicity we examine the case in which the weight matrix is symmetric. In such case, each state has an associated goodness function that greatly simplifies the analysis. In a later section we discuss how the results generalize to the non-symmetric case. Let y E IRn represent the internal potential of a diffusion network. Let Zi = cp(aiYi) for i = 1,??? , n represent the firing rates corresponding to y. Let zS, ZC and zr represent the components of z for the units in the stimulus channel, context channel and response specification module. Let x be a vector representing an input and let x S , XC be the components of x for the external stimulus and context. Let a = (a1,??? , an) be a fixed gain vector and ZO/(t) a random vector representing the firing rates at time t of a network with gain vector a. Let = limt-+oo (t), represent the firing rates at stochastic equilibrium. In Movellan (1998) it is shown that if the weights are symmetric i.e., W = w' and l/Ki(x) = dcpi(X)/dx then the equilibrium probability density of is as follows za za za PZQlx(zs,zc,zr I XS,X C ) = K ( 1 a Xs,Xc ) exp((2/0'2) Ga(zs,zr I XS,X C )) , (8) 48 J. R. Movellan and J. L. McClelland /CoDtut SdmU~ Input Figure 1: A network with separable context and stimulus processing channels. The stimulus sensor and stimulus relay units make up the stimulus channel units, and the context sensor and context channel units make up the context channel units. Note that any of the modules can be empty except the response module. where Ka(x s , xc) = / exp((2/(72) Ga(z I Xs , xc)) dz, (9) n Ga(z I x) = H(z I x) - L Sa; (Zi), (10) i=l H(z I x) Sa; (Zi) = z' w z/2 + z' V x, (11) = ai (IOg(Zi) + log(1 - Zi)) + ~i (Zi log(zi) + (1 - Zi) log(1 - Zi)) . (12) Without loss of generality hereafter we set (72 = 2. When there are no direct connections between the stimulus and context units there are no terms in the goodness function in which XS or ZS occur jointly with XC or ZC. Because of this, the goodness can be separated into three additive terms, that depend on x S , XC and a third term which depends on the response units: Ga(z\zc,zr I XS,X C) = G~(zs,zr I X + G~(zr,zc I XC) + G~(zr) S) where G~(ZS, zr I XS) = (zs),w s,szs/2 + (zS)'ws,rzr + (ZS)'vs,sx s + (zr),vr,sx s - , (13) L S(zt) , i (14) G~(ZC, zr I XS ) = (ZC)'wc,czc /2 + (zc),wc,rzr + (zc),vc,cx c + (zr),vr,cx c - L S(zf) , i (15) (16) 49 Information Factorization where ws,r is a submatrix of w connecting the stimulus and response units. Similar notation is used for the other submatrices of wand v. It follows that we can write the ratio of the jOint probability density' of two states z and z as follows: PZ.. lx(zs,zc,zr I XS,X C ) exp(G~(zS,zr I x s ) + G~(zc,zr I XC) + G~(zr? (17) pZ.. lx(zS,zC,zrlxs,x c ) - exp(G~(zS,zrlxs)+G~(zC,zrlxC)+G~(zr?' which factorizes as desired. To get probability densities for the response units, we integrate over the states of all the other units PZ;;IX(zr I XS,X C) = / / PZ.. lx(zs,zc,zr I XS,X C) dz s dz c , (18) and after rearranging terms pZ;;IX(zr I XS,X C) = Kcr(:s,x C) ( / exp( Gz(zs,zr I XS) + Gr(zr? dZ S) (19) ( / exp( G c(ZC, zr I xc? dZ C) , which also factorizes. All is left is mapping continuous states of the response units to discrete external responses. To do so we partition the space of the response specification units into discrete regions. The probability of a response becomes the integral of the probability density over the region corresponding to that response. The problem is that the integral of probability densities does not necessarily factorize even though the densities factorize at every point. Fortunately there are two important cases for which the law holds, at least as a good approximation. The first case is when the response regions are small and thus we can approximate the integral over that region by the density at a point times the volume of the region. In such a case the ratio of the integrals can be approximated by the ratio of the probability densities of those individual states. The second case applies to models, like McClelland and Rumelhart's (1981) interactive activation model, in which each response is associated with a distinct response unit. These models typically have negative connections amongst the response units so that at equilibrium one unit tends to be active while the others are inactive. In such a case a common response policy picks the response corresponding to the active unit. We now show that such a policy can approximate the Morton-Massaro law to an arbitrary level of precision as the gain parameter of the response units is increased. Let z represent the joint state of a network and let the first r components of z be the states of the response specification units. Let z(1) = (1,0,0, ... ,0)', Z(2) = (0,1,0,??? ,0)' be two r-dimensional vectors representing states of the response specification units. For i E {1,2} and ~ E (0,1) let z~) = (1 - Z(i?~ R~) = {x E IRr + (z(i?(l - : Xj ~), E ((1- ~)Z~i), ~ (20) + (1 - ~)Z~i?, for j = 1,??? , r}. (21) The sets R~) and R~) are regions of the [O,l]r space mapping into two distinct external responses. We now investigate the convergence of the probability ratio of these two responses as we let ~ 4 0, i.e., as the response regions collapse into corners of [0, l]r. I X = x) J PzrlX(u I x)du = = lim Rt . " A-+O P(Z~ E R~) I X = x) A-+O JR~) PZ;;lx(u I x)du . ~rpZ;;IX(z~) I x) . J J eG~(z~),z?,ze I z}dz dz c hm (1) = hm A-+O ~rpZ;;IX(zA I x) A-+O J J eG .. (zt. ,Z',ze I z)dz s dz c lim p(zr E cr R(2) A (2) (22) 8 (1) ? (23) J. R. Movellan and J. L. McClelland 50 Table 1: Predictions by the Morton-Massaro law (left side) versus diffusion network (square brackets) for subject 7 of Massaro and Cohen (1983) Experiment 2. Each prediction of the diffusion network is based on 100 random samples. Context Stimulus 0 1 2 3 4 5 6 0.0017 0.0126 0.1105 0.5463 0.9827 0.9999 0.9999 P S V 0.01 0.00 0.19 0.54 1.00 1.00 1.00 0.0000 0.0000 0.0008 0.0079 0.2756 0.9924 0.9924 0.00 0.00 0.00 0.00 0.30 0.99 1.00 0.0152 0.1008 0.5208 0.9133 0.9980 0.9999 0.9999 0.03 0.10 0.45 0.91 1.00 1.00 1.00 T 0.9000 0.9849 0.9984 0.9998 0.9999 1.0000 1.0000 0.91 0.97 1.00 1.00 1.00 1.00 1.00 Now note that r Go(z~), Z8, ZC I x) = H(z~), Z8, ZC I x) - L So; (Z~!i) - L So; (zt) - L SOj (zj), i=1 i j (24) and since E~=l So; (Z~!i) = E;=1 So; (Z~!i)' it follows that . P(Z~ E R~) I X = x) _ hm (1) ~-+o P(Z~ E R~ I X = x) J J eH(z~),z?,ze I x)-E;Sa;(zi)-E j Saj(zj)dz J J eH(za.. ,Z',ze I x)-E; Sai(zt)-E j Saj(Zj)dz 8 dz c 8 dzc (1) ? (25) It is easy to show that this ratio factorizes. Moreover, for all .6. 0:1 = ... = O:r = 0:, where 0: > 0 then lim P(Z~ E [.6.,1 - .6.t) 0-+00 = 0, > 0 if we let (26) since as the gain of the response units increases So; decreases very fast at the corners of (0, 1y. Thus as 0: -4 00 the random variable Z~ converges in distribution to a discrete random variable with mass at the corner of the [0, It hypercube and with factorized probability ratios as expressed on (25). Since the indexing ofthe response units is arbitrary the argument applies to all the responses. o 4 Discussion Our analysis establishes that in diffusion networks the Morton-Massaro law is not incompatible with the presence of feedback and lateral connections. Surprisingly, even though in diffusion networks with feedback connections stimulus and context units are interdependent, it is still possible to factorize the effect of stimulus and context on response probabilities. The analysis shows that the Morton-Massaro can be arbitrarily approximated as the sharpness of the response units is increased. In practice we have found very good approximations with relatively small values of the sharpness parameter (see Table 1 for an example). The analysis assumed that the weights were symmetric. Mathematical analysis of the general case with non-symmetric weights is difficult. Information Factorization 51 However useful approximations exist (Movellan & McClelland, 1995) showing that if the noise parameter (7 is relatively small or if the activation function c.p is approximately linear, symmetric weights are not needed to exhibit the Morton-Massaro law. The analysis presented here has potential applications to investigate models of perception and the functional architecture of the brain. For example the interactive activation model of word perception has a separable architecture and thus, diffusion versions of it adhere to the Morton Massaro law. The analysis also points to potential applications in computational neuroscience. It would be of interest to study whether the Morton-Massaro holds at the level of neural responses. For example, we may excite a neuron with two different sources of information and observe its short term average response to combination of stimuli. If the observed distribution of responses exhibits the Morton-Massaro law, this would be consistent with the existence of separable channels converging into that neuron. Otherwise, it would indicate that the channels from the two input areas to the response may not be structurally separable. References Luce, R. D. (1959). Individual choice behavior. New York: Wiley. Massaro, D. W. (1989). Testing between the TRACE Model and the fuzzy logical model of speech perception. Cognitive Psychology, 21, 398-42l. Massaro, D. W. (1998). Perceiving Talking Faces. Cambridge, Massachusetts: MIT Press. Massaro, D. W. & Cohen, M. M. (1983a). Phonological constraints in speech perception. Perception and Psychophysics, 94, 338-348. McClelland, J. L. (1991). Stochastic interactive activation and the effect of context on perception. Cognitive Psychology, 29, 1-44. Morton, J. (1969). The interaction of information in word recognition. Psychological Review, 76, 165-178. Movellan, J. R. (1998). A Learning Theorem for Networks at Detailed Stochastic Equilibrium. Neural Computation, 10(5), 1157-1178. Movellan, J. R. & McClelland, J. L. (1995) . Stochastic interactive processing, channel separability and optimal perceptual inference: an examination of Morton's law. Technical Report PDP.CNS.95A, Available at http://cnbc.cmu.edu, Carnegie Mellon University. 

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Modeling High-Dimensional Discrete Data with Multi-Layer Neural Networks Samy Bengio * IDIAP CP 592, rue du Simplon 4, 1920 Martigny, Switzerland bengio@idiap.ch Yoshua Bengio Dept.IRO Universite de Montreal Montreal, Qc, Canada, H3C 317 bengioy@iro.umontreal.ca Abstract The curse of dimensionality is severe when modeling high-dimensional discrete data: the number of possible combinations of the variables explodes exponentially. In this paper we propose a new architecture for modeling high-dimensional data that requires resources (parameters and computations) that grow only at most as the square of the number of variables, using a multi-layer neural network to represent the joint distribution of the variables as the product of conditional distributions. The neural network can be interpreted as a graphical model without hidden random variables, but in which the conditional distributions are tied through the hidden units. The connectivity of the neural network can be pruned by using dependency tests between the variables. Experiments on modeling the distribution of several discrete data sets show statistically significant improvements over other methods such as naive Bayes and comparable Bayesian networks, and show that significant improvements can be obtained by pruning the network. 1 Introduction The curse of dimensionality hits particularly hard on models of high-dimensional discrete data because there are many more possible combinations of the values of the variables than can possibly be observed in any data set, even the large data sets now common in datamining applications. In this paper we are dealing in particular with multivariate discrete data, where one tries to build a model of the distribution of the data. This can be used for example to detect anomalous cases in data-mining applications, or it can be used to model the class-conditional distribution of some observed variables in order to build a classifier. A simple multinomial maximum likelihood model would give zero probability to all of the combinations not encountered in the training set, i.e., it would most likely give zero probability to most out-of-sample test cases. Smoothing the model by assigning the same non-zero probability for all the unobserved cases would not be satisfactory either because it would not provide much generalization from the training set. This could be obtained by using a multivariate multinomial model whose parameters B are estimated by the maximum a-posteriori (MAP) principle, i.e., those that have the greatest probability, given the training data D, and using a diffuse prior PCB) (e.g. Dirichlet) on the parameters. A graphical model or Bayesian network [6, 5) represents the joint distribution of random variables Zl ... Zn with n P(ZI ... Zn) = II P(ZiIParentsi) i=l ?Part of this work was done while S.B. was at CIRANO, Montreal, Qc. Canada. Modeling High-Dimensional Discrete Data with Neural Networks 401 where Parentsi is the set of random variables which are called the parents of variable i in the graphical model because they directly condition Zi, and an arrow is drawn, in the graphical model, to Zi, from each of its parents. A fully connected "left-to-right" graphical model is illustrated in Figure 1 (left), which corresponds to the model n P(ZI . .. Zn) = II P(ZiIZl ... Zi-r) . (1) i= l Figure 1: Left: a fully connected "left-to-right" graphical model. Right: the architecture of a neural network that simulates a ful1y connected "left-to-right" graphical model. The observed values Zi = Zi are encoded in the corresponding input unit group. hi is a group of hidden units. gi is a group of output units, which depend on Zl ... Zi - l , representing the parameters of a distribution over Zi. These conditional probabilities P(ZiIZl . . . Zi-r) are multiplied to obtain the joint distribution. Note that this representation depends on the ordering of the variables (in that all previous variables in this order are taken as parents). We call each combination of the values of Parentsi a context. In the "exact" model (with the full table of all possible contexts) all the orders are equivalent, but if approximations are used, different predictions could be made by different models assuming different orders. In graphical models, the curse of dimensionality shows up in the representation of conditional distributions P(Zi IParentsi) where Zi has many parents. If Zj E Parentsi can take nj values, there are TI j nj different contexts which can occur in which one would like to estimate the distribution of Zi. This serious problem has been addressed in the past by two types of approaches, which are sometimes combined: 1. Not modeling all the dependencies between all the variables: this is the approach mainly taken with most graphical models or Bayes networks [6, 5] . The set of independencies can be assumed using a-priori or human expert knowledge or can be learned from data. See also [2] in which the set Parentsi is restricted to at most one element, which is chosen to maximize the correlation with Zi. 2 . Approximating the mathematicalform of the joint distribution with a form that takes only into account dependencies of lower order, or only takes into account some of the possible dependencies, e.g., with the Rademacher-Walsh expansion or multi-binomial [1,3], which is a low-order polynomial approximation of a full joint binomial distribution (and is used in the experiments reported in this paper). The approach we are putting forward in this paper is mostly of the second category, although we are using simple non-parametric statistics of the dependency between pairs of variables to further reduce the number of required parameters. In the multi-binomial model [3], the joint distribution of a set of binary variables is approximated by a polynomial. Whereas the "exact" representation of P( Zl = Z l , ... Zn = zn) as a function of Z l . . . Zn is a polynomial of degree n, it can be approximated with a lower Y. Bengio and S. Bengio 402 degree polynomial, and this approximation can be easily computed using the RademacherWalsh expansion [1] (or other similar expansions, such as the Bahadur-Lazarsfeld expansion [1]) . Therefore, instead of having 2 n parameters, the approximated model for P(Zl , . . . Zn) only requires O(nk) parameters. Typically, order k = 2 is used. The model proposed here also requires O(n 2 ) parameters, but it allows to model dependencies between tuples of variables, with more than 2 variables at a time. In previous related work by Frey [4], a fully-connected graphical model is used (see Figure 1, left) but each of the conditional distributions is represented by a logistic, which take into account only first-order dependency between the variables: 1 P(Zi = llZl ... Zi-d = ( L j<i Wj Z j )' 1 + exp -Wo In this paper, we basically extend Frey's idea to using a neural network with a hidden layer, with a particular architecture, allowing multinomial or continuous variables, and we propose to prune down the network weights . Frey has named his model a Logistic Autoregressive Bayesian Network or LARC . He argues that the prior variances on the logistic weights (which correspond to inverse weight decays) should be chosen inversely proportional to the number of conditioning variables (i.e. the number of inputs to the particular output neuron). The model was tested on a task of learning to classify digits from 8x8 binary pixel images. Models with different orderings of the variables were compared and did not yield significant differences in performance. When averaging the predictive probabilities from 10 different models obtained by considering 10 different random orderings, Frey obtained small improvements in likelihood but not in classification. The model performed better or equivalently to other models tested: CART, naive Bayes, K-nearest neighbors, and various Bayesian models with hidden variables (Helmholtz machines). These results are impressive, taking into account the simplicity of the LARC model. 2 Proposed Architecture The proposed architecture is a "neural network" implementation of a graphical model where all the variables are observed in the training set, with the hidden units playing a significant role to share parameters across different conditional distributions. Figure 1 (right) illustrates the model in the simpler case of a fully connected (Ieft-to-right) graphical model (Figure 1, left) . The neural network represents the parametrized function jo(zt, . .. , zn) = log(?O(Zl = Zl,? ? ., Zn = zn)) (2) approximating the joint distribution of the variables, with parameters 0 being the weights of the neural network. The architecture has three layers, with each layer organized in groups associated to each of the variables. The above log-probability is computed as the sum of conditional log-probabilities n jO(Zl , . .. , zn) = L 109(P(Zi = zilgi(zl, .. . , Zi-l))) i=l where gi( Zt, . .. , zi-d is the vector-valued output of the i-th group of output units, and it gives the value of the parameters of the distribution of Zi when Zl = Zl , Z2 = Z2, .. . , Zi-l = Zi-l' For example, in the ordinary discrete case, gi may be the vector of probabilities associated with each of the possible values of the multinomial random variable Zi . In this case, we have P(Zi = i'lgi) = gi ,i' In this example, a softmax output for the i-th group may be used to force these parameters to be positive and sum to 1, i.e., gi ,i' = Lil e g' i , i' Modeling High-Dimensional Discrete Data with Neural Networks 403 where g~ i' are linear combinations of the hidden units outputs, with i' ranging over the number of elements of the parameter vector associated with the distribution of Zi (for a fixed value of Zl ... Zi-l). To guarantee that the functions gi(Zl, ... , Zi-l) only depend on Zl ... Zi-l and not on any of Zi ... Zn, the connectivity struture of the hidden units must be constrained as follows: g~,i' = bi,i' + 2: j~i mj 2: Wi,i' ,j,j' hj,j' j'=1 where the b's are biases and the w's are weights of the output layer, and the hj,j' is the output of the j'-th unit (out of mj such units) in the j-th group of hidden layer nodes. It may be computed as follows: hj ,j' = tanh(cj,j' + 2: nk 2: Vj ,j' ,k ,k' Zk ,k') k<j k'=l where the c's are biases and the v's are the weights of the hidden layer, and Zk,k' is k'-th element of the vectorial input representation of the value Zk = Zk. For example, in the binary case (Zi = 0 or 1) we have used only one input node, i.e., Zi binomial -t Zi,O = Zi and in the multinomial case we use the one-hot encoding, Zi E {O, 1, ... ni - I} -t Zi ,i' = 8Zi ,i' where 8i ,i' = 1 if i = i' and 0 otherwise. The input layer has n - 1 groups because the value Zn = Zn is not used as an input. The hidden layer also has n - 1 groups corresponding to the variables j = 2 to n (since P(Z.) is represented unconditionally in the first output group, its corresponding group does not need any hidden units or inputs, but just has biases). 2.1 Discussion The number of free parameters of the model is O(n 2 H) where H = maXi mj is the maximum number of hidden units per hidden group (i.e., associated with one of the variables). This is basically quadratic in the number of variables, like the multi-binomial approximation that uses a polynomial expansion of the joint distribution. However, as H is increased, representation theorems for neural networks suggest that we should be able to approximate with arbitrary precision the true joint distribution. Of course the true limiting factor is the amount of data, and H should be tuned according to the amount of data. In our experiments we have used cross-validation to choose a value of mj = H for all the hidden groups. In this sense, this neural network representation of P(ZI ... Zn) is to the polynomial expansions (such as the multi-binomial) what ordinary multilayer neural networks for function approximation are to polynomial function approximators. It allows to capture high-order dependencies, but not all of them. It is the number of hidden units that controls "how many" such dependencies will be captured, and it is the data that "chooses" which of the actual dependencies are most useful in maximizing the likelihood. Unlike Bayesian networks with hidden random variables, learning with the proposed architecture is very simple, even when there are no conditional independencies. To optimize the parameters we have simply used gradient-based optimization methods, either using conjugate or stochastic (on-line) gradient, to maximize the total log-likelihood which is the sum of values of f (eq. 2) for the training examples. A prior on the parameters can be incorporated in the cost function and the MAP estimator can be obtained as easily, by maximizing the total log-likelihood plus the log-prior on the parameters. In our experiments we have used a "weight decay" penalty inspired by the analysis of Frey [4], with a penalty proportional to the number of weights incoming into a neuron. Y. Bengio and S. Bengio 404 However, it is not so clear how the distribution could be generally marginalized, except by summing over possibly many combinations of the values of variables to be integrated. Another related question is whether one could deal with missing values: if the total number of values that the missing variables can take is reasonably small, then one can sum over these values in order to obtain a marginal probability and maximize this probability. If some variables have more systematically missing values, they can be put at the end of the variable ordering, and in this case it is very easy to compute the marginal distribution (by taking only the product of the output probabilities up to the missing variables). Similarly, one can easily compute the predictive distribution of the last variable given the first n - 1 variables. The framework can be easily extended to hybrid models involving both continuous and discrete variables. In the case of continuous variables, one has to choose a parametric form for the distribution of the continuous variable when all its parents (i.e., the conditioning context) are fixed. For example one could use a normal, log-normal, or mixture of normals. Instead of having softmax outputs, the i-th output group would compute the parameters of this continuous distribution (e.g., mean and log-variance). Another type of extension allows to build a conditional distribution, e.g., to model P(ZI ... ZnlXl ... Xm). One just adds extra input units to represent the values of the conditioning variables Xl ... X m . Finally, an architectural extension that we have implemented is to allow direct input-tooutput connections (still following the rules of ordering which allow gi to depend only on Zl ... Zi-l). Therefore in the case where the number of hidden units is 0 (H = 0) we obtain the LARC model proposed by Frey [4]. 2.2 Choice of topology Another type of extension of this model which we have found very useful in our experiments is to allow the user to choose a topology that is not fully connected (Ieft-to-right). In our experiments we have used non-parametric tests to heuristically eliminate some of the connections in the network, but one could also use expert or prior knowledge, just as with regular graphical models, in order to cut down on the number of free parameters. In our experiments we have used for a pairwise test of statistical dependency the Kolmogorov-Smirnov statistic (which works both for continuous and discrete variables) . The statistic for variables X and Y is Jl sup IP(X :::; Xi, Y :::; Yi) - P(X :::; Xi)P(Y :::; Yi) I i where l is the number of examples and P is the empirical distribution (obtained by counting s = over the training data). We have ranked the pairs according to their value of the statistic s, and we have chosen those pairs for which the value of statistic is above a threshold value s*, which was chosen by cross-validation. When the pairs {(Zi' Zj)} are chosen to be part of the model, and assuming without loss of generality that i < j for those pairs, then the only connections that are kept in the network (in addition to those from the k-th hidden group to the k-th output group) are those from hidden group i to output group j, and from input group i to hidden group j, for every such (Zi' Zj) pair. 3 Experiments In the experiments we have compared the following models: ? Naive Bayes: the likelihood is obtained as a product of multinomials (one per variable). Each multinomial is smoothed with a Dirichlet prior. ? Multi-Binomial (using Rademacher-Walsh expansion of order 2) [3]. Since this only handles the case of binary data, it was only applied to the DNA data set. ? A simple graphical model with the same pairs of variables and variable ordering as selected for the neural network, but in which each of the conditional distribution is modeled Modeling High-Dimensional Discrete Data with Neural Networks 405 by a separate multinomial for each of the conditioning context. This works only if the number of conditioning variables is small so in the Mushroom, Audiology, and Soybean experiments we had to reduce the number of conditioning variables (following the order given by the above tests). The multinomials are also smoothed with a Dirichlet prior. ? Neural network: the architecture described above, with or without hidden units (i.e., LARC), with or without pruning. 5-fold cross-validation was used to select the number of hidden units per hidden group and the weight decay for the neural network and LARC. Cross-validation was also used to choose the amount of pruning in the neural network and LARC, and the amount of smoothing in the Dirichlet priors for the muItinomials of the naive Bayes model and the simple graphical model. 3.1 Results All four data sets were obtained on the web from the VCI Machine Learning and STATLOG databases. Most of these are meant to be for classification tasks but we have instead ignored the classification and used the data to learn a probabilistic model of all the input features. ? DNA (from STATLOG): there are 180 binary features. 2000 cases were used for training and cross-validation, and 1186 for testing. ? Mushroom (from VCI): there are 22 discrete features (taking each between 2 and 12 values). 4062 cases were used for training and cross-validation, and 4062 for testing. ? Audiology (from VCI): there are 69 discrete features (taking each between 2 and 7 values). 113 cases are used for training and 113 for testing (the original train-test partition was 200 + 26 and we concatenated and re-split the data to obtain more significant test figures). ? Soybean (from VCI): there are 35 discrete features (taking each between 2 and 8 values). 307 cases are used for training and 376 for testing. Table 1 clearly shows that the proposed model yields promising results since the pruned neural network was superior to all the other models in all 4 cases, and the pairwise differences with the other models are statistically significant in all 4 cases (except Audiology, where the difference with the network without hidden units, LARC, is not significant). 4 Conclusion In this paper we have proposed a new application of multi-layer neural networks to the modelization of high-dimensional distributions, in particular for discrete data (but the model could also be applied to continuous or mixed discrete / continuous data). Like the polynomial expansions [3] that have been previously proposed for handling such high-dimensional distributions, the model approximates the joint distribution with a reasonable (O( n 2 )) number of free parameters but unlike these it allows to capture high-order dependencies even when the number of parameters is small. The model can also be seen as an extension of the previously proposed auto-regressive logistic Bayesian network [4], using hidden units to capture some high-order dependencies. Experimental results on four data sets with many discrete variables are very encouraging. The comparisons were made with a naive Bayes model, with a multi-binomial expansion, with the LARC model and with a simple graphical model, showing that a neural network did significantly better in terms of out-of-sample log-likelihood in all cases. The approach to pruning the neural network used in the experiments, based on pairwise statistical dependency tests, is highly heuristic and better results might be obtained using approaches that take into account the higher order dependencies when selecting the conditioning variables. Methods based on pruning the fully connected network (e.g., with a "weight elimination" penalty) should also be tried. Also, we have not tried to optimize Y Bengio and S. Bengio 406 naive Bayes multi-Binomial order 2 ordinary graph. model LARC prunedLARC full-conn. neural net. pruned neural network naive Bayes multi-Binomial order 2 ordinary graph. model LARC prunedLARC full-conn. neural net. pruned neural network DNA mean (stdev) p-value 100.4 (.18) <le-9 117.8(.01) <le-9 108.1 (.06) <le-9 83.2 (.24) 7e-5 91.2(.15) <le-9 120.0 (.02) <le-9 82.9 (.21) Audiology mean (stdev) p-value 36.40 (2 .9) <le-9 16.56 (.48) 17.69 (.65) 16.69 (.41) 17 .39 (.58) 16.37 (.45) 6.8e-4 <le-9 0.20 <le-9 Mushroom mean (stdev) p-value 47 .00 (.29) <le-9 44.68 (.26) <le-9 42.51 (.16) <le-9 43.87 (.13) < le-9 33.58 (.01) <le-9 31.25 (.04) Soybean mean (stdev) p-value 34.74 (1.0) <le-9 43 .65 (.07) 16.95 (.35) 19.06 (.43) 21.65 (.43) 16.55 (.27) <le-9 5.5e-4 <le-9 <le-9 Table 1: Average out-of-sample negative log-likelihood obtained with the various models on four data sets (standard deviations of the average in parenthesis and p-value to test the null hypotheses that a model has same true generalization error as the pruned neural network). The pruned neural network was better than all the other models in in all cases, and the pair-wise difference is always statistically significant (except with respect to the pruned LARC on Audiology). the order of the variables, or combine different networks obtained with different orders, like [4] . References [1] RR Bahadur. A representation of the joint distribution of responses to n dichotomous items. In ed. H. Solomon, editor, Studies in Item Analysis and Predictdion, pages 158-168. Stanford University Press, California, 1961. [2] c.K. Chow. A recognition method using neighbor dependence. IRE Trans. Elec. Comp., EC-l1 :683-690, October 1962. [3] RO. Duda and P.E. Hart. Pattern Classification and Scene Analysis. Wiley, New York, 1973. [4] B. Frey. Graphical models for machine learning and digital communication. MIT Press, 1998. [5] Steffen L. Lauritzen. The EM algorithm for graphical association models with missing data. Computational Statistics and Data Analysis, 19:191-201,1995. [6] Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. 

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410 NEURAL CONTROL OF SENSORY ACQUISITION: THE VESTIBULO-OCULAR REFLEX. Michael G. Paulin, Mark E. Nelson and James M. Bower Division of Biology California Institute of Technology Pasadena, CA 91125 ABSTRACT We present a new hypothesis that the cerebellum plays a key role in actively controlling the acquisition of sensory infonnation by the nervous system. In this paper we explore this idea by examining the function of a simple cerebellar-related behavior, the vestibula-ocular reflex or VOR, in which eye movements are generated to minimize image slip on the retina during rapid head movements. Considering this system from the point of view of statistical estimation theory, our results suggest that the transfer function of the VOR, often regarded as a static or slowly modifiable feature of the system, should actually be continuously and rapidly changed during head movements. We further suggest that these changes are under the direct control of the cerebellar cortex and propose experiments to test this hypothesis. 1. INTRODUCTION A major thrust of research in our laboratory involves exploring the way in which the nervous system actively controls the acquisition of infonnation about the outside world. This emphasis is founded on our suspicion that the principal role of the cerebellum, through its influence on motor systems, is to monitor and optimize the quality of sensory information entering the brain. To explore this question, we have undertaken an investigation of the simplest example of a cerebellar-related motor activity that results in improved sensory inputs, the vestibulo-ocular reflex (VOR). This reflex is responsible for moving the eyes to compensate for rapid head movements to prevent retinal image slip which would otherwise significantly degrade visual acuity (Carpenter, 1977). 2. VESTIBULO-OCULAR REFLEX (VOR) The VOR relies on the vestibular apparatus of the inner ear which is an inertial sensor that detects movements of the head. Vestibular output caused by head movements give rise to compensatory eye movements through an anatomically well described neural pathway in the brain stem (for a review see Ito, 1984). Visual feedback also makes an important contribution to compensatory eye movements during slow head movements, Neural Control of Sensory Acquisition but during rapid head movements with frequency components greater than about 1Hz, the vestibular component dominates (Carpenter, 1977). A simple analysis of the image stabilization problem indicates that during head rotation in a single plane, the eyes should be made to rotate at equal velocity in the opposite direction. This implies that, in a simple feedforward control model, the VOR transfer function should have unity gain and a 1800 phase shift. This would assure stabilized retinal images of distant objects. It turns out, however, that actual measurements reveal the situation is not this simple. Furman, O'Leary and Wolfe (1982), for example, found that the monkey VOR has approximately unity gain and 1800 phase shift only in a narrow frequency band around 2Hz. At 4Hz the gain is too high by a factor of about 30% (fig. 1). 1.2 ~ -< C) 1.0 j IHf ,...., ~ 5 0 - I til ff1d 1tHf~ff 1\ lIJ (f) ~O n.. -5 0.8 2 3 4 FREQUENCY (Hz) 5 3 2 4 5 FREQUENCY (Hz) Figure 1: Bode gain and phase plots for the transfer function of the horizontal component of the VOR of the alert Rhesus monkey at high frequencies (Data from Furman et al. (1982?. Given the expectation of unity gain, one might be tempted to conclude from the monkey data that the VOR simply does not perform well at high frequencies. But 4Hz is not a very high frequency for head movements, and perhaps it is not the VOR which is performing poorly, but the simplified analysis using classical control theory. In this paper, we argue that the VOR uses a more sophisticated strategy and that the "excessive" gain in the system seen at higher frequencies actually improves VOR performance. 3. OPTIMAL ESTIMATION In order to understand the discrepancy between the predictions of simple control theory models and measured VOR dynamics, we believe it is necessary to take into account more of the real world conditions under which the VOR operates. Examples include noisy head velocity measurements, conduction delays and multiple, possibly conflicting, measurements of head velocity, acceleration, muscle contractions, etc., generated by different sensory modalities. The mathematical framework that is appropriate for analyz- 411 412 Paulin, Nelson and Bower ing problems of this kind is stochastic state-space dynamical systems theory (Davis and Vinter, 1985). This framework is an extension of classical linear dynamical systems theory that accommodates multiple inputs and outputs, nonlinearities, time-varying dynamics, noise and delays. One area of application of the state space theory has been in target tracking, where the basic principle involves using knowledge of the dynamics of a target to estimate its most probable trajectory given imprecise data. The VOR can be viewed as a target tracking system whose target is the "world", which moves in head coordinates. We have reexamined the VOR from this point of view. The Basic VOR. To begin our analysis of the VOR we have modeled the eye-head-neck system as a damped inverted pendulum with linear restoring forces (fig. 2) where the model system is driven by random (Gaussian white) torque. Within this model, we want to predict the correct compensatory "eye" movements during "head" movements to stabilize the direction in which the eye is pointing. Figure 2 shows the amplitude spectrum of head velocity for this model. In this case, the parameters of the model result in a system that has a natural resonance in the range of 1 to 2 Hz and attenuates higher frequencies. 20 ? ? ??? S I S 0.1 z , s. u ? '. ? ?? i. I 1.0 FREQUENCY Figure 2: Amplitude spectrum of model head velocity. We provide noisy measurements of "head" velocity and then ask what transfer function, or filter, will give the most accurate "eye" movement compensation? This is an estimation problem and, for Gaussian measurement error, the solution was discovered by Kalman and Bucy (1961). The optimal fIlter or estimator is often called the KalmanBucy filter. The gain and phase plots of the optimal filter for tracking movements of the inverted pendulum model are shown in figure 3. It can be seen that the gain of the optimal estimator for this system peaks near the maximum in the spectrum of "head-neck" velocity (fig. 2). This is a general feature of optimal filters. Accordingly, to accurately compensate for head movement in this system, the VOR would need to have a frequency dependent gain. Neural Control of Sensory Acquisition - - ~ 20 ~ 0 bO 0 So j 0 5 ~ ~ ~ tI) -20 ~90 0.1 1.0 .0 10.0 Figure 3: Bode gain plot (left) and phase plot (right) of an optimal estimator for tracking the inverted pendulum using noisy data. Time Varying dynamics and the VOR So far we have considered our model for VOR optimization only in the simple case of a constant head-neck velocity power spectrum. Under natural conditions, however, this spectrum would be expected to change. For example, when gait changes from walking to running, corresponding changes in the VOR transfer function would be necessary to maintain optimal performance. To explore this, we added a second inverted pendulum to our model to simulate body dynamics. We simulated changes in gait by changing the resonant frequency of the trunk. Figure 4 compares the spectra of head-neck velocity with two different trunk parameters. As in the previous example, we then computed transfer functions of the optimal filters for estimating head velocity from noisy measurements in these two cases. The gain and phase characteristics of these filters are also shown in Figure 5. These plots demonstrate that significant changes in the transfer function of the VOR would be necessary to maintain visual acuity in our model system under these different conditions. Of course, in the real situation head-neck dynamics will change rapidly and continuously with changes in gait, posture, substrate, etc. requiring rapid continuous changes in VOR dynamics rather than the simple switch implied here. -~ ~ ? 5 HEAD 20 0 .......----~~-~ ~ ~ -20 0.1 1.0 FREQUENCY 10.0 Figure 4: Head velocity spectrum during "walking" (light) and "running" (heavy). 413 414 Paulin, Nelson and Bower ~ ~ 0 ? -.8- 1----""'::3I~ 5 ~ -20 ? ? ?? , Ci. .1 D ?? 1.0 10.0 .1 1.0 10.0 Figure 5: Bode gain plots (left) and phase plots (right) for optimal estimators of head angular velocity during "walking" (light) and "running" (heavy). 4. SIGNIFICANCE TO THE REAL VOR Our results show that the optimal VOR transfer function requires a frequency dependent gain to accurately adjust to a wide range of head movements under real world conditions. Thus, the deviations from unity gain seen in actual measurements of the VOR may not represent poor, but rather optimal, performance. Our modeling similarly suggests that several other experimental results can be reinterpreted. For example, localized peaks or valleys in the VOR gain function can be induced experimentally through prolonged sinusoidal oscillations of subjects wearing magnifying or reducing lenses. However, this "frequency selectivity" is not thought to occur naturally and has been interpreted to imply the existence of frequency selective channels in the VOR control network (Lisberger, Miles and Optican, 1983). In our view there is no real distinction between this phenomenon and the "excessive" gain in normal monkey VOR; in each case the VOR optimizes its response for the particular task which it has to solve. This is testable. If we are correct, then frequency selective gain changes will occur following prolonged narrow-band rotation in the light without wearing lenses. In the classical framework there is no reason for any gain changes to occur in this situation. Another phenomenon which has been observed experimentally and that the current modeling sheds new light on is referred to as "pattern storage". After single-frequency sinusoidal oscillation on a turntable in the light for several hours, rabbits will continue to produce oscillatory eye movements when the lights are extinguished and the turntable stops. Trained rabbits also produce eye oscillations at the training frequency when oscillated in the dark at a different frequency (Collewijn, 1985). In this case the sinusoidal pattern seems to be "stored" in the nervous system. However, the effect is naturally accounted for by our optimal estimator hypothesis without relying on an explicit "pattern storage mechanism". An optimal estimator works by matching its dynamics to the dynamics of the signal generator, and in effect it tries to force an internal model to mimic the signal generator by comparing actual and expected patterns of sensory inputs. When Neural Control of Sensory Acquisition no data is available, or the data is thought to be very unreliable, an optimal estimator relies completely, or almost completely, on the model. In cases where the signal is patterned the estimator will behave as though it had memorized the pattern. Thus, if we hypothesize that the VOR is an optimal estimator we do not need an extra hypothesis to explain pattern storage. Again, our hypothesis is testable. If we are correct, then repeating the pattern storage experiments using rotational velocity waveforms obtained by driving a frequency-tuned oscillator with Gaussian white noise will produce identical dynamical effects in the VOR. There is no sinusoidal pattern in the stimulus, but we predict that the rabbits can be induced to generate sinusoidal eye movements in the dark after this training. The modeling results shown in figures 4 and 5 represent an extension of our ideas into the area of gait (or more generally "context") dependent changes in VOR which has not been considered very much in VOR research. In fact, VOR experimental paradigms, in general, are explicitly set up to produce the most stable VOR dynamics possible. Accordingly, little work has been done to quantify the short term changes in VOR dynamics that must occur in response to changes in effective head-neck dynamics. Experiments of this type would be valuable and are no more difficult technically than experiments which have already been done. For example, training an animal on a turntable which can be driven randomly with two distinct velocity power spectra, i.e. two "gaits", and providing the animal with external cues to indicate the gait would, we predict, result in an animal that could use the cues to switch its VOR dynamics. A more difficult but also more compelling demonstration would be to test VOR dynamics with impulsive head accelerations in different natural situations, using an unrestrained animal. s. SENSOR FUSION AND PREDICTION To this point, we have discussed compensatory eye movements by treating the VOR as a single input, single output system. This allowed us to concentrate on a particular aspect of VOR control: tracking a time-varying dynamical system (the head) using noisy data. In reality there are a number of other factors which make control of compensatory eye movements a somewhat more complex task than it appears to be when it is modeled using classical control theory. For example, a variety of vestibular as well as non-vestibular signals (e.g. visual, proprioceptive) relating to head movements are transmitted to the compensatory eye movement control network (Ito, 1984). This gives rise to a "sensor fusion" problem where data from different sources must be combined. The optimal solution to this problem for a multiple input - multiple output, time-varying linear, stochastic system is also given by the Kalman-Bucy filter (Davis and Vinter, 1985). Borah, Young and Curry (1988) have demonstrated that a Kalman-Bucy filter model of visualvestibular sensor fusion is able to account for visual-vestibular interactions in motion perception. Oman (1982) has also developed a Kalman-Bucy filter model of visualvestibular interactions. Their results show that the optimal estimation approach is useful 415 416 Paulin, Nelson and Bower for analyzing multivariate aspects of compensatory eye movement control, and complement our analysis of dynamical aspects. Another set of problems arises in the VOR because of small time delays in neural transmission and muscle activation. To optimize its response, the mammalian VOR needs to make up for these delays by predicting head movements about lOmsec in advance (ret). Once the dynamics of the signal generator have been identified, prediction can be performed using model-based estimation (Davis and Vinter, 1985). A neural analog of a Taylor series expansion has also been proposed as a model of prediction in the VOR (pellionisz and LUnas, 1979), but this mec.hanism is extremely sensitive to noise in the data and was abandoned as a practical technique for general signal prediction several decades ago in favor of model-based techniques (Wiener, 1948). The later approach may be more appropriate for analyzing neural mechanisms of prediction (Arbib and Amari, 1985). An elementary description of optimal estimation theory for target tracking, and its possible relation to cerebellar function, is given by Paulin (1988). 6. ROLE OF CEREBELLAR CORTEX IN VOR CONTROL To this point we have presented a novel characterization of the problem of compensatory eye movement control without considering the physical circuitry which implements the behavior. However, there are two parts to the optimal estimation problem. At each instant it is necessary to (a) filter the data using the optimal transfer function to drive the desired response and (b) determine what transfer function is optimal at that instant and adjust the filtering network accordingly. The first problem is fairly straightforward, and existing models of VOR demonstrate how a network of neurons based on known brains tern circuitry can implement a particular transfer function (Cannon and Robinson, 1985). The second problem is more difficult because requires continuous monitoring of the context in which head movements occur using a variety of sources of relevant data to tune the optimal filter for that context. We speculate that the cerebellar cortex performs this task. First, the cortex of the vestibulo-cerebellum is in a position to mflke , the required computation, since it receives detailed information from multiple sensory modalities that provide information on the state of the motor system (Ito, 1985). Second, the cerebellum projects to and appears to modulate the brain stem compensatory eye movement control network (Mackay and Murphy, 1979). We predict that the cerebellar cortex is necessary to produce rapid, context-dependent optimal state dependent changes in VOR transfer function which we have discussed. This speculation can be tested with turntable experiments similar to those described in section 4 above in the presence and absence of the cerebellar cortex. Neural Control of Sensory Acquisition 7. THE GENERAL FUNCTION OF CEREBELLAR CORTEX According to our hypothesis, the cerebellar cortex is required for making optimal compensatory eye movements during head movements. This is accomplished by continuously modifying the dynamics of the underlying control network in the brainstem, based on current sensory information. The function of the cerebellar cortex in this case can then be seen in a larger context as using primary sensory information (vestibular, visual) to coordinate the use of a motor system (the extraoccular eye muscles) to position a sensory array (the retina) to optimize the quality of sensory information available to the brain. We believe that this is the role played by the rest of the cerebellum for other sensory systems. Thus, we suspect that the hemispheres of the rat cerebellum, with their peri-oral tactile input (Bower et al., 1983), are involved in controlling the optimal use of these tactile surfaces in sensory exploration through the control of facial musculature. Similarly, the hemispheres of the primate cerebellum, which have hand and finger tactile inputs (Ito, 1984), may be involved in an analogous exploratory task in primates. These tactile sensory-motor systems are difficult to analyze, and we are currently studying a functionally analogous but more accessible model system, the electric sense of weakly electric fish (cf Rasnow et al., this volume). 8.CONCLUSION Our view of the cerebellum assigns it an important dynamic role which contrasts markedly with the more limited role it was assumed to have in the past as a learning device (Marr, 1969; Albus, 1971; Robinson, 1976). There is evidence that cerebellar cortex has some learning abilities (Ito, 1984), but it is recognized that cerebellar cortex has an important dynamic role in motor control. However, there are widely differing opinions as to the nature of that role (Ito, 1985; Miles and Lisberger, 1981; Pellionisz and Llinas, 1979). Our proposal, that the VOR is a neural analog of an optimal estimator and that the cerebellar cortex monitors context and sets reflex dynamics accordingly, should not be interpreted as a claim that the nervous system actually implements the computations which are involved in applied optimal estimation, such as the KalmanBucy filter. Understanding the neural basis of cerebellar function will require the combined power of a number of experimental, theoretical and modeling approaches (cf Wilson et al., this volume). We believe that analyses of the kind presented here have an important role in characterizing behaviors controlled by the cerebellum. Acknowledgments This work was supported by the NIH (BNS 22205), the NSF (EET-8700064), and the Joseph Drown Foundation. References Arbib M.A. and Amari S. 1985. Sensori-moto Transformations in the Brain (with a critique of the tensor theory of the cerebellum). J. Theor. BioI. 112:123-155 417 418 Paulin, Nelson and Bower Borah J., Young L.R. and Curry, R.E. 1988. Optimal Estimator Model for Human Spatial Orientation. In: Proc. N.Y. Acad. Sci. B. Cohen and V. Henn (eds.). In Press. Bower lM. and Woolston D.C. 1983. The Vertical Organization of Cerebellar Cortex. J. Nemophysiol. 49: 745-766. Carpenter R.H.S. 1977. Movements of the Eyes. Pion, London. Davis M.B.A. and Vinter R.B. 1985. Stochastic Modelling and Control. Chapman and Hall, NY. Funnan J.M., O'Leary D.P. and Wolfe lW. 1982. Dynamic Range of the Frequency Response of the Horizontal Vestibulo-Ocular Reflex of the Alert Rhesus Monkey. Acta Otolaryngol. 93: 81 Ito, M. 1984. The Cerebellum and Neural Control. Raven Press, NY. Kalman R.E. 1960. A New Approach to Linear Filtering and Prediction Problems. l Basic Eng., March 1960. Kalman R.E. and Bucy R.S. 1961. New Results in Linear Filtering and Prediction Theory. 1. Basic Eng., March 1961. Lisberger, S.G. 1988. The Nemal Basis for Learning of Simple Motor Skills. Science, 242:728735. Lisberger S.G. , Miles F.A. and Optican L.M. 1983. Frequency Selective Adaptation: Evidence for Channels in the Vestibulo-Ocular Reflex. J. Neurosci. 3:1234-1244 Mackay W.A. and Murphy J.T. 1979. Cerebellar Modulation of reflex Gain. Prog. Neurobiol. 13:361-417. Oman C.M. 1982. A heuristic mathematical Model for the Dynamics of Sensory Conflict and Motion Sickness. Acta Oto-Laryngol. S392. Paulin M.G. 1988. A Kalman Filter Model of the Cerebellum. In: Dynamic Interactions in Nemal Networks: Models and Data. M. Arbib and S. Amari (eds). Springer-Verlag, NY. pp239-261. Pellionisz A. and Llinas R. 1979. Brain Modelling by Tensor Network Theory and Computer Simulation. The Cerebellum: Distributed Processor for Predictive Coordination. Nemoscience 4:323-348. Robinson D.A. 1976. Adaptive Control of the Vestibulo-Ocular Reflex by the Cerebellum. J. Nemophys.36:954-969. Robinson D.A. 1981. The Use of Control Systems Analysis in the Neurophysiology of Eye Movements. Ann. Rev. Neurosci. 4:463-503. Wiener, N. 1948. Cybernetics: Communication and Control in the Animal and the Machine. MIT Press, Boston. 

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Broadband Direction-Of-Arrival Estimation Based On Second Order Statistics Justinian Rosca Joseph 6 Ruanaidh Alexander Jourjine Scott Rickard {rosca,oruanaidh,jourjine,rickard}@scr.siemens.com Siemens Corporate Research, Inc. 755 College Rd E Princeton, NJ 08540 Abstract N wideband sources recorded using N closely spaced receivers can feasibly be separated based only on second order statistics when using a physical model of the mixing process. In this case we show that the parameter estimation problem can be essentially reduced to considering directions of arrival and attenuations of each signal. The paper presents two demixing methods operating in the time and frequency domain and experimentally shows that it is always possible to demix signals arriving at different angles. Moreover, one can use spatial cues to solve the channel selection problem and a post-processing Wiener filter to ameliorate the artifacts caused by demixing. 1 Introduction Blind source separation (BSS) is capable of dramatic results when used to separate mixtures of independent signals. The method relies on simultaneous recordings of signals from two or more input sensors and separates the original sources purely on the basis of statistical independence between them. Unfortunately, BSS literature is primarily concerned with the idealistic instantaneous mixing model. In this paper, we formulate a low dimensional and fast solution to the problem of separating two signals from a mixture recorded using two closely spaced receivers. Using a physical model of the mixing process reduces the complexity of the model and allows one to identify and to invert the mixing process using second order statistics only. We describe the theoretical basis of the new approach, and then focus on two algorithms, which were implemented and successfully applied to extensive sets of real-world data. In essence, our separation architecture is a system of adaptive directional receivers designed using the principles ofBSS. The method bears resemblance to methods in beamforming [8] in that it works by spatial filtering. Array processing techniques [2] reduce noise by separating signal space from noise space, which necessitates more receivers than emitters. The main differences are that standard beamforming and array processing techniques [8, 2] are generally strictly concerned with processing directional narrowband signals. The difference with BSS [7, 6] is that our approach is model-based and therefore the elements of the mixing matrix are highly constrained: a feature that aids in the robust and reliable identification of the mixing process. J. Rosca, J. 776 6 Ruanaidh, A. Jourjine and S. Rickard The layout of the paper is as follows. Sections 2 and 3 describe the theoretical foundation of the separation method that was pursued. Section 4 presents algorithms that were developed and experimental results. Finally we summarize and conclude this work. 2 Theoretical foundation for the BSS solution As a first approximation to the general multi-path model, we use the delay-mixing model. In this model, only direct path signal components are considered. Signal components from one source arrive with a fractional delay between the time of arrivals at two receivers. By fractional delays, we mean that delays between receivers are not generally integer multiples of the sampling period. The delay depends on the position of the source with respect to the receiver axis and the distance between receivers. Our BSS algorithms demix by compensating for the fractional delays. This, in effect, is a form of adaptive beamforming with directional notches being placed in the direction of sources of interference [8]. A more detailed account of the analytical structure of the solutions can be found in [1]. Below we address the case of two inputs and two outputs but there is no reason why the discussion cannot be generalized to multiple inputs and multiple outputs. Assume a linear mixture of two sources, where source amplitude drops off in proportion to distance: Xi(t) = 1 - S I (t - R-I _Z ) Ril 1 + -S2(t - C Ri2 R-2 (1) _Z ) C j = 1, 2, where c is the speed of wave propagation, and Rij indicates the distance from receiver i to source j. This describes signal propagation through a uniform non-dispersive medium. In the Fourier domain, Equation 1 results in a mixing matrix A( w) given by: A(w) = [~lle-jW~ ~12e-jW~ 1 -jw~ R21e c 1 R 22 e _jw!!JJ.. 1 (2) c It is important to note that the columns can be scaled arbitrarily without affecting separation of sources because rescaling is absorbed into the sources. This implies that row scaling in the demixing matrix (the inverse of A( is arbitrary. w? Using the Cosine Rule, Rij can be expressed in terms of the distance Rj of source j to the midpoint between two receivers, the direction of arrival of source j, and the distance between receivers, d, as follows: = [HJ + (~)' + 2(-1)' R;j m r 1 Hj COS OJ (3) Expanding the right term above using the binomial expansion and preserving only zeroth and first order terms, we can express distance from the receivers to the sources as: Rij = ( Rj + 8~j) + (_l)i (~) cosOj (4) This approximation is valid within a 5% relative error when d ::; ~. With the substitution for Rij and with the redefinition of source j to include the delay due to the term within brackets in Equation 4 divided by c, Equation 1 becomes: Xi(t) = ~ ~ij .Sj (t+(-l)i?(:c).cosOj ) , i= 1,2 (5) J In the Fourier domain, equation 5 results in the simplification to the mixing matrix A( w): A(w) = [ _1_ e-jwo1 R Il ? . _1_ eJW01 R21? _1_ e-jw02 ] Rl2 . _1_ ejw02 R 22' (6) 777 Broadband DOA Estimation Based on Second Order Statistics Here phases are functions of the directions of arrival ()j (defined with respect to the midpoint between receivers), the distance between receivers d, and the speed of propagation c: Oi 2dc cos ()i ,i 1, 2. Rij are unknown, but we can again redefine sources so diagonal elements are unity: = = (7) where c), C2 are two positive real numbers. In wireless communications sources are typically distant compared to antenna distance. For distant sources and a well matched pair of receivers c) ~ C2 ~ 1. Equation 7 describes the mixing matrix for the delay model in the frequency domain, in terms of four parameters, 0) ,02, c), C2. The corresponding ideal demixing matrix W(w), for each frequency w, is given by: W(w) _) 1 = [A(w) ] = detA(w) [e jW02 (8) -c2 .ejwol The outputs, estimating the sources, are: ] _ W w [X)(W) ] _ 1 [ [ z)(w) Z2(W) () X2(W) - detA(w) _c)e- jW02 ] [ e-; WO l x)(w) ] X2(W) (9) Making the transition back to the time domain results in the following estimate of the outputs: (10) where @ is convolution, and (11) Formulae 9 and 10 form the basis for two algorithms to be described next, in the time domain and the frequency domains. The algorithms have the role of determining the four unknown parameters. Note that the filter corresponding to H (w, 0) , 02, C), C2) should be applied to the output estimates in order to map back to the original inputs. 3 Delay and attenuation compensation algorithms The estimation of the four unknown parameters 0), 02, C), C2 can be carried out based on second order criteria that impose the constraint that outputs are decorrelated ([9, 4, 6, 5]). 3.1 Time and frequency domain approaches The time domain algorithm is based on the idea of imposing the decorrelation constraint (Z) (t), Z2(t)} 0 between the estimates ofthe outputs, as a function of the delays D) and D2 and scalar coefficients c) and C2. This is equivalent to the following criterion: = (12) where F(.) measures the cross-correlations between the signals given below, representing filtered versions of the differences of fractionally delayed measurements: J Rosca. J 778 = h(t, D), D2, e), e2) 0 Z2(t) = h(t, D) , D2, e) , e2) 0 Z)(t) 6 Ruanaidh. A. Jourjine and S. Rickard + D2) - e)X2(t?) (e2X) (t + D2) - X2(i?) (X)(t F(D), D2, e), e2) (13) = (Z)(t), Z2(t)} In the frequency domain, the cross-correlation of the inputs is expressed as follows: RX(w) = A(w)Rs (w)AH(w) ( 14) The mixing matrix in the frequency domain has the form given in Equation 7. Inverting this cross correlation equation yields four equations that are written in matrix form as: ( 15) Source orthogonality implies that the off-diagonal terms in the covariance matrix must be zero: RT2(W) =0 Rf)(w) = 0 (16) For far field conditions (i.e. the distance between the receivers is much less than the distance from sources) one obtains the following equations: = The terms a e- jw1h and b = e- jwoz are functions of the time delays. Note that there is a pair of equations of this kind for each frequency. In practice, the unknowns should be estimated from data at all available frequencies to obtain a robust estimate. 3.2 Channel selection Up to this point, there was no guarantee that estimated parameters would ensure source separation in some specific order. We could not decide a priori whether estimated parameters for the first output channel correspond to the first or second source. However, the dependence of the phase delays on the angles of arrival suggests a way to break the permutation symmetry in source estimation, that is to decide precisely which estimate to present on the first channel (and henceforth on the second channel as well). The core idea is that directionality and spatial cues provide the information required to break the symmetry. The criterion we use is to sort sources in order of increasing delay. Note that the correspondence between delays and sources is unique when sources are not symmetrical with respect to the receiver axis. When sources are symmetric there is no way of distinguishing between their positions because the cosine of the angles of arrival, and hence the delay, is invariant to the sign of the angle. 4 Experimental results A robust implementation of criterion 12 averages cross-correlations over a number of windows, of given size. More precisely F is defined as follows: F( 0),02) = L Blocks I(Z) (t), Z2(t)W ( 18) 779 Broadband DOA Estimation Based on Second Order Statistics Normally q = 1 to obtain a robust estimate. Ngo and Bhadkamkar [5] suggest a similar criterion using q = 2 without making use of the determinant of the mixing matrix. After taking into account all terms from Equation 18, including the determinant of the mixing matrix A, we obtain the function to be used for parameter estimation in the frequency domain: F(01,02) = ~ ~ I I q 1 2 ? -b a Rl1 x (W) - -R22(W) b x x (w) - -bRI2(w) 1 x - abR21 (19) w { det A} + TJ a a where TJ is a (Wiener Filter-like) constant that helps prevent singularities and q is normally set to one. Computing the separated sources using only time differences leads to highpass filtered outputs. In order to implement exactly the theoretical demixing procedure presented one has to divide by the determinant of the mixing matrix. Obviously one could filter using the inverse of the determinant to obtain optimal results. This can be implemented in the form of a Wiener filter. The Wiener filter requires knowledge both ofthe signal and noise power spectral densities. This information is not available to us but a reasonable approximation is to assume that the (wideband) sources have a flat spectral density and the noise corrupting the mixtures is white. In this case, the Wiener Filter becomes: H w _ ( ( )- {detA(W)}2) { det A (w )} 2 + TJ 1 det A (w ) (20) where the parameter TJ has been empirically set to the variance of the mixture. Applying this choice of filter usually dramatically improves the quality of the separated outputs. The technique of postprocessing using the determinant of the mixing matrix is perfectly general and applies equally well to demixtures computed using matrices of FIR filters. The quality of the result depends primarily on the care with which the inverse filter is implemented. It also depends on the accuracy of the estimate for the mixing parameters. One should avoid using the Wiener filter for near-degenerate mixtures. The proof of concept for the theory outlined above was obtained using speech signals which if anything pose a greater challenge to separation algorithms because of the correlation structure of speech. Two kinds of data are considered in this paper: synthetic direct propagation delay data and synthetic mUlti-path data. Data can be characterized along two dimensions of difficulty: synthetic vs. real-world, and direct path vs. multi-path. Combinations along these dimensions represented the main type of data we used. The value of distance between receivers dictates the order of delays that can appear due to direct path propagation, which is used by the demixing algorithms. Data was generated synthetically employing fractional delays corresponding to the various positions of the sources [3]. We modeled multi-path by taking into account the decay in signal amplitude due to propagation distance as well as the absorption of waves. Only the direct path and one additional path were considered. The algorithms developed proved successful for separation of two voices from direct path mixtures, even where the sources had very similar spectral power characteristics, and for separation of one source for multi-path mixtures. Moreover, outputs were free from artifacts and were obtained with modest computational requirements. Figure 1 presents mean separation results of the first and second channels, which correspond to the first and second sources, for various synthetic data sets. Separation depends on the angles of arrival. Plots show no separation in the degenerate case of equal or closeby angles of arrival, but more than lOdB mean separation in the anechoic case and 5dB in the mUlti-path case. 780 J. Rosca, J. 50 ,. .. . Ruanaidh, A. Jourjine and S. Rickard i f .. I I \/ i" I ,. .~ .. I" ,," I~ so / 1.." . :-..\ " '" f. I t -1?0 6 ') Doma1_ AnechoicT~ Anechoic F '00 50 .... -",so .. ., =~H "-1'- ... 210 I -6. .... 50 '00 ,. ,. i I -"- If ill ... ... ... ,,., .... .... :-' .... i,. i" I. 1. I j" :\ " f? I ,. , =t~Oomal so '00 .... -"'so 210 ... " I? ' .... =t~ so '00 ... ... -"'50 Figure 1: Two sources were positioned at a relatively large distance from a pair of closely spaced receivers. The first source was always placed at zero degrees whilst the second source was moved uniformly from 30 to 330 degrees in steps of 30 degrees. The above shows mean separation and standard deviation error bars of first and second sources for six synthetic delay mixtures or synthetic mUlti-path data mixtures using the time and frequency domain algorithms. 5 Conclusions The present source separation approach is based on minimization of cross-correlations of the estimated sources, in the time or frequency domains, when using a delay model and explicitly employing dirrection of arrival. The great advantage of this approach is that it reduces source separation to a decorrelation problem, which is theoretically solved by a system of equations. Although the delay model used generates essentially anechoic time delay algorithms, the results of this work show systematic improvements even when the algorithms are applied to real multi-path data. In all cases separation improvement is robust with respect to the power ratios of sources. Acknowledgments We thank Radu Balan and Frans Coetzee for useful discussions and proofreading various versions of this document and our collaborators within Siemens for providing extensive data for testing. Broadband DOA Estimation Based on Second Order Statistics 781 References [1] A. Jourjine, S. Rickard, J. 6 Ruanaidh, and J. Rosca. Demixing of anechoic time delay mixtures using second order statistics. Technical Report SCR-99-TR-657, Siemens Corporate Research, 755 College Road East, Princeton, New Jersey, 1999. [2] Hamid Krim and Mats Viberg. Two decades of array signal processing research. IEEE Signal Processing Magazine, 13(4), 1996. [3] Tim Laakso, Vesa Valimaki, Matti Karjalainen, and Unto Laine. Splitting the unit delay. IEEE Signal Processing Magazine, pages 30-60,1996. [4] L. Molgedey and H.G. Schuster. Separation of a mixture of independent signals using time delayed correlations. Phys.Rev.Lett., 72(23):3634-3637, July 1994. [5] T. J. Ngo and N.A. Bhadkamkar. Adaptive blind separation of audio sources by a physically compact device using second order statistics. In First International Workshop on leA and BSS, pages 257-260, Aussois, France, January 1999. [6] Lucas Parra, Clay Spence, and Bert De Vries. Convolutive blind source separation based on multiple decorrelation. In NNSP98, 1988. [7] K. Torkolla. Blind separation for audio signals: Are we there yet? In First International Workshop on Independent component analysis and blind source separation, pages 239244, Aussois, France, January 1999. [8] V. Van Veen and Kevin M. Buckley. Beamforrning: A versatile approach to spatial filtering. IEEE ASSP Magazine, 5(2), 1988. [9] E. Weinstein, M. Feder, and A. Oppenheim. Multi-channel signal separation by decorrelation. IEEE Trans. on Speech and Audio Processing, 1(4):405-413, 1993. 

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Dual Estimation and the Unscented Transformation EricA. Wan ericwan@ece.ogi.edu Rudolph van der Merwe rudmerwe@ece.ogi.edu Alex T. Nelson atneison@ece.ogi.edu Oregon Graduate Institute of Science & Technology Department of Electrical and Computer Engineering 20000 N.W. Walker Rd., Beaverton, Oregon 97006 Abstract Dual estimation refers to the problem of simultaneously estimating the state of a dynamic system and the model which gives rise to the dynamics. Algorithms include expectation-maximization (EM), dual Kalman filtering, and joint Kalman methods. These methods have recently been explored in the context of nonlinear modeling, where a neural network is used as the functional form of the unknown model. Typically, an extended Kalman filter (EKF) or smoother is used for the part of the algorithm that estimates the clean state given the current estimated model. An EKF may also be used to estimate the weights of the network. This paper points out the flaws in using the EKF, and proposes an improvement based on a new approach called the unscented transformation (UT) [3]. A substantial performance gain is achieved with the same order of computational complexity as that of the standard EKF. The approach is illustrated on several dual estimation methods. 1 Introduction We consider the problem of learning both the hidden states Xk and parameters w of a discrete-time nonlinear dynamic system, F(Xk , Vk, w) H(xk, nk, w), (1) (2) where Yk is the only observed signal. The process noise Vk drives the dynamic system, and the observation noise is given by nk. Note that we are not assuming additivity of the noise sources. A number of approaches have been proposed for this problem. The dual EKF algorithm uses two separate EKFs: one for signal estimation, and one for model estimation. The states are estimated given the current weights and the weights are estimated given the current states. In the joint EKF, the state and model parameters are concatenated within a combined state vector, and a single EKF is used to estimate both quantities simultaneously. The EM algorithm uses an extended Kalman smoother for the E-step, in which forward and 667 Dual Estimation and the Unscented Transformation backward passes are made through the data to estimate the signal. The model is updated during a separate M-step. For a more thorough treatment and a theoretical basis on how these algorithms relate, see Nelson [6]. Rather than provide a comprehensive comparison between the different algorithms, the goal of this paper is to point out the assumptions and flaws in the EKF (Section 2), and offer a improvement based on the unscented transformation/filter (Section 3). The unscented filter has recently been proposed as a substitute for the EKF in nonlinear control problems (known dynamic model) [3]. This paper presents new research on the use of the UF within the dual estimation framework for both state and weight estimation. In the case of weight estimation, the UF represents a new efficient "second-order" method for training neural networks in general. 2 Flaws in the EKF Assume for now that we know the model (weight parameters) for the dynamic system in Equations 1 and 2. Given the noisy observation Yk, a recursive estimation for Xk can be expressed in the form, Xk = (optimal prediction ofxk) + Gk x [Yk - (optimal prediction ofYk)] (3) This recursion provides the optimal MMSE estimate for Xk assuming the prior estimate Xk and current observation Yk are Gaussian. We need not assume linearity of the model. The optimal terms in this recursion are given by yl: = E[H(xl:, nk)], (4) where the optimal prediction xl: is the expectation of a nonlinear function of the random variables Xk-l and Vk-l (similar interpretation for the optimal prediction of Yk). The optimal gain term is expressed as a function of posterior covariance matrices (with Yk = Yk - Yl:) ? Note these terms also require taking expectations of a nonlinear function of the prior state estimates. The Kalman filter calculates these quantities exactly in the linear case. For nonlinear models, however, the extended KF approximates these as: YI: = H(xl:,fl), (5) where predictions are approximated as simply the function of the prior mean value for estimates (no expectation taken). The covariance are determined by linearizing the dynamic equations (Xk+l ~ AXk + BVk, Yk ~ CXk + Dnk), and then determining the posterior covariance matrices analytically for the linear system. As such, the EKF can be viewed as providing "first-order" approximations to the optimal terms (in the sense that expressions are approximated using a first-order Taylor series expansion of the nonlinear terms around the mean values). While "second-order" versions of the EKF exist, their increased implementation and computational complexity tend to prohibit their use. 3 The Unscented TransformationIFilter The unscented transformation (UT) is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation [3] . Consider propagating a random variable a (dimension L) through a nonlinear function, (3 = g( a). Assume a has mean ct and covariance P Q . To calculate the statistics of {3, we form a matrix X of 2L + 1 sigma vectors Xi, where the first vector (Xo) corresponds to ct, and the rest are computed from the mean (+ )plus and (-)minus each column of the matrix square-root of P Q . These sigma E. A. Wan, R. v. d. Merwe and A. T. Nelson 668 vectors are propagated through the nonlinear function, and the mean and covariance for [3 are approximated using a weighted sample mean and covariance, /3 ~ - 1? {~g(xo) + -21 I:9(Xi)} +~ Pp" , (6) i=l L:~ {~[g(XO) - il][g(Xo) - il)T + ~ ~[g(X') - il)[g(X,) - ilf} (7) where ~ is a scaling factor. Note that this method differs substantially from general "sampling" methods (e.g., Monte-Carlo methods and particle filters [1]) which require orders of magnitude more sample points in an attempt to propagate an accurate (possibly nonGaussian) distribution of the state. The UT approximations are accurate to the third order for Gaussian inputs for all nonlinearities. For non-Gaussian inputs, approximations are accurate to at least the second-order, with the accuracy determined by the choice of ~ [3]. A simple example is shown in Figure 1 for a 2-dimensional system: the left plots shows the true mean and covariance propagation using Monte-Carlo sampling; the center plots show the performance of the UT (note only 5 sigma points are required); the right plots show the results using a linearization approach as would be done in the EKF. The superior performance of the UT is clear. Actual (sampling) mean (3 I = g(o) 1 UT Linearized (EKF) .11"=- 0 -I (3 Yi = g(X i ) 1 = g(o) P(:! = ATPaA l Figure 1: Example of the UT for mean and covariance propagation. a) actual, b) UT, c) first-order linear (EKF). The unscented filter (UF) [3] is a straightforward extension of the UT to the recursive estimation in Equation 3, where we set 0: = Xk, and denote the corresponding sigma matrix as X(klk). The UF equations are given on the next page. It is interesting to note that no explicit calculation of lacobians or Hessians are necessary to implement this algorithm. The total number of computations is only order ?2 as compared to ?3 for the EKF. I 4 Application to Dual Estimation This section shows the use of the UF within several dual estimation approaches. As an application domain for comparison, we consider modeling a noisy time-series as a nonlinear INote that a matrix square-root using the Cholesky factorization is of order L 3 /6. However, the covariance matrices are expressed recursively, and thus the square-root can be computed in only order L2 by performing a recursive update to the Cholesky factorization. 669 Dual Estimation and the Unscented Transformation UF Equations Wo = K/(L + K) WI . .. W2L = 1/2(L + fl.) , X(klk - 1) = F[X(k - 11k - 1), P!lv2 ] x"k = 2:~!o WiXi(klk - 1) P"k = 2:~!o WdXi(klk - 1) - x"k][Xi(klk - 1) - x"kf Y(klk - 1) = H[X(klk - 1), P~;] Y"k = 2:~!o WiYi(klk - 1) P XkYk = 2::!o Wi[Yi(klk 2::!o Wi[Xi(klk - Xk = x"k + PXkYkP~:h (n - P hh = 1) - Y"kJ[Yi(klk - 1) - 1) - x"k][Yi(klk - 1) - Yk"f Y"kf Y"k) P k = P"k - PX"Yk(P~:yJTP!'kYk autoregression: Xk = f(Xk-l, ... Xk-M, Yk = Xk w) + Vk + nk, Vk E {l. . . N} (8) The underlying clean signal Xk is a nonlinear function of its past M values, driven by white Gaussian process noise Vk with variance 11;. The observed data point Yk includes the additive noise nk, which is assumed to be Gaussian with variance 11;. The corresponding state-space representation for the signal Xk is given by: + B? Vk-I Yk = [1 (9) (10) 0 In this context, the dual estimation problem consists of simultaneously estimating the clean signal Xk and the model parameters w from the noisy data Yk. 4.1 Dual EKF I Dual UF One dual estimation approach is the dual extended Kalman filter developed in [8, 6]. The dual EKF requires separate state-space representation for the signal and the weights. A state-space representation for the weights is generated by considering them to be a stationary process with an identity state transition matrix, driven by process noise Uk: + Uk Wk = Wk-l Yk = f(Xk-I,Wk) +Vk +nk? (11) (12) The noisy measurement Yk has been rewritten as an observation on w. This allows the use of an EKF for weight estimation (representing a "second-order" optimization procedure) [7]. Two EKFs can now be run simultaneously for signal and weight estimation. At every time-step, the current estimate of the weights is used in the signal-filter, and the current estimate of the signal-state is used in the weight-filter. 670 E. A. Wan, R. v. d. Merwe and A. T Nelson The dual UFIEKF algorithm is formed by simply replacing the EKF for state-estimation with the UF while still using an EKF for weight-estimation. In the dual UF algorithm both state- and weight-estimation are done with the UF. Note that the state-transition is linear in the weight filter, so the nonlinearity is restricted to the measurement equation. Here, the UF gives a more exact measurement-update phase of estimation. The use of the UF for weight estimation in general is discussed in further detail in Section 5. 4.2 Joint EKF I Joint UF An alternative approach to dual estimation is provided by the joint extended Kalman filter [4,5]. In this framework the signal-state and weight vector are concatenated into a single, joint state vector: Zk = The estimation of Zk can be done recursively by writing the state-space equations for the joint state as: [xf wfV. (13) and running an EKF on the joint state-space to produce simultaneous estimates of the states Xk and w . As discussed in [6], the joint EKF provides approximate MAP estimates by maximizing the joint density of the signal and weights given the noisy data. Again, our approach in this paper is to use the UF instead of the EKF to provide more accurate estimation of the state, resulting in the joint UF algorithm. 4.3 EM - Unscented Smoothing A somewhat different iterative approach to dual estimation is given by the expectationmaximization (EM) algorithm applied to nonlinear dynamic systems [2]. In each iteration, the conditional expectation of the signal is computed, given the data and the current estimate of the model (E-step). Then the model is found that maximizes a function of this conditional mean (M-step). For linear models, the M-step can be solved in closed form . The E-step is computed with a Kalman smoother, which combines the forward-time estimated mean and covariance (x{ ,pt) of the signal given past data, with the backward-time predicted mean and covariance (xf ,pf) given the future data, producing the following smoothed statistics given all the data: (14) (15) When a MLP neural network model is used, the M-step can no longer be computed in closed-form, and a gradient-based approach is used instead. The resulting algorithm is usually referred to as generalized EM (GEM) 2. The E-step is typically approximated by an extended Kalman smoother, wherein a linearization of the model is used for backward propagation of the state estimates. We propose improving the E-step of the EM algorithm for nonlinear models by using a UP instead of an EKF to compute both the forward and backward passes in the Kalman smoother. Rather than linearize the model for the backward pass, as in [2], a neural network is trained on the backward dynamics (as well as the forward dynamics). This allows for a more exact backward estimation phase using the UF, and enables the development of an unscented smoother (US). 2 An exact M-step is possible using RBF networks [2]. Dual Estimation and the Unscented Transformation 671 4.4 Experiments We present results on two simple time-series to provide a clear illustration of the use of the UP over the EKE The first series is the Mackey-Glass chaotic series with additive WGN (SNR ~ 3dB). The second time series (also chaotic) comes from an autoregressive neural network with random weights driven by Gaussian process noise and also corrupted by additive WGN (SNR ~ 3dB). A standard 5-3-1 MLP with tanh hidden activation functions and a linear output layer was used in all the filters. The process and measurement noise variances were assumed to be known. Results on training and testing data, as well as training curves for the different dual estimation methods are shown below. The quoted numbers are normalized (clean signal variance) mean-square estimation and prediction errors. The superior performance of the UT based algorithms (especially the dual UF) are clear. Note also the more stable learning curves using the UF approaches. These improvements have been found to be consistent and statistically significant on a number of additional experiments. Mackey-Glass Algorithm Dual EKF Dual UF/EKF Dual UF Joint EKF Joint UF Train Est. Pred. 0.20 0.50 0.19 0.50 0.15 0.45 0.22 0.53 0.19 0.50 Test Est. Pred. 0.21 0.54 0.19 0.53 0.14 0.48 0.22 0.56 0.18 0.53 Chaotic AR-NN Algorithm . Dual EKF Dual UF/EKF Dual UF Joint EKF Joint UF Test Est. Pred. 0.69 0.36 0.28 0.69 0.27 0.63 0.34 0.72 0.30 0.67 Train Est. Pred. 0.32 0.62 0.26 0.58 0.23 0.55 0.29 0.58 0.25 0.55 Chaotic AR-NN Mackey-Glass .. o. 0.' 01 ~O8 Dual EKF Dual UFIEKF 0 Dual UF 0 JointEKF " ...... UF W en :::;; :::;; o. ]05 "~" . '5E035 " E go" C 03 03 0 25 02 0.1 5 0 ? 10 11 02 0 10 15 ~eration iteration 20 2S '" The final table below compares smoother performance used for the E-step in the EM algorithm. In this case, the network models are trained on the clean time-series, and then tested on the noisy data using either the standard Kalman smoother with linearized backward model (EKS 1), a Kalman smoother with a second nonlinear backward model (EKS2), and the unscented smoother (US). The forward (F), backward (B), and smoothed (S) estimation errors are reported. Again the performance benefits of the unscented approach is clear. Mackey-Glass Algorithm EKSI EKS2 US Norm. MSE B S 0.20 0.70 0.27 0.19 0.20 0.3] 0.]0 0.24 0.08 F Chaotic AR-NN Algorithm EKSI EKS2 US Norm. MSE B S 0.35 0.32 0.28 0.35 0.22 0.23 0.23 0.2] 0.16 F 5 UF Neural Network Training As part of the dual UF algorithm, we introduced the use of the UF for weight estimation. The approach can also be seen as a new method for the general problem of training neural networks (i.e., for regression or classification problems where the input x is observed and 672 E. A. Wan. R. v. d. Merwe and A. T. Nelson no state-estimation is required). The advantage of the UF over the EKF in this case is not as obvious, as the state-transition function is linear (See Equation 11). However, as pointed out earlier, the observation is nonlinear. Effectively, the EKF builds up an approximation to the expected Hessian by taking outer products of the gradient. The UF, however, may provide a more accurate estimate through direct approximation of the expectation of the Hessian. We have performed a number of preliminary experiments on standard benchmark data. The figure below shows the mean and std. oflearning curves (computed over 100 experiments with different initial weights) for the Mackay Robot Arm Mapping dataset. Note the faster convergence, lower variance, and lower final MSE performance of the UF weight training. While these results are encouraging, further study is still necessary to fully contrast differences between UF and EKF weight training. O.06n---.-----,"L"le;O;anmW ln;:;r,g1"it.;'"w lu rv:v:ec.s,..---r= I::;;U~F(;:=m.== an I r 7=>l) 0 .05 - UF(. ld) : : ~~~ ~~~n) W ~ 0 .004 c: ~ 0,03 E 0 .02 0.01 '\ ~_. -..~ __ ~ "--- __ -.. _ _ _ _ _ _ _ _ _ _ _ _ _ _ ? _ _ ...... - - .. .. 6 Conclusions The EKF has been widely accepted as a standard tool in the machine learning community. In this paper we have presented an alternative to the EKF using the unscented filter. The UF consistently achieves a better level of accuracy than the EKF at a comparable level of complexity. We demonstrated this performance gain on a number of dual estimation methods as well as standard regression modeling. Acknowledgements This work was sponsored in part by the NSF under grant IRI-9712346. References [1] J. F. G. de Freitas, M. Niranjan, A. H. Gee, and A. Doucet. Sequential Monte Carlo methods for optimisation of neural network models. Technical Report TR-328, Cambridge University Engineering Department, Cambridge, England, November 1998. [2] Z. Ghahramani and S. T. Roweis. Learning nonlinear dynamical systems using an EM algorithm. In M. J. Keams, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems II: Proceedings of the 1998 Conference. MIT Press, 1999. [3] S. J. Julier and J. K. Uhlmann. A New Extension of the Kalman Filter to Nonlinear Systems. In Proc. of AeroSense: The 11th International Symposium on Aerospace/Defence Sensing. Simulation and Controls. Orlando. Florida., 1997. [4] R. E. Kopp and R. J. Orford. Linear regression applied to system identification for adaptive control systems. AlAA 1., I :2300-06, October 1963. [5] M. B. Matthews and G. S. Moschytz. Neural-network nonlinear adaptive filtering using the extended Kalman filter algorithm. In INNC, pages 115-8, 1990. [6] A. T. Nelson. Nonlinear Estimation and Modeling of Noisy Time-Series by Dual Kalman Filtering Methods. PhD thesis, Oregon Graduate Institute, 1999. In preparation. [7] S. Singhal and L. Wu. Training multilayer perceptrons with the extended Kalman filter. In Advances in Neural Information Processing Systems 1, pages 133-140, San Mateo, CA, 1989. Morgan Kauffman. [8] E. A. Wan and A. T. Nelson. Dual Kalman filtering methods for nonlinear prediction, estimation, and smoothing. In Advances in Neural Information Processing Systems 9, 1997. 

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Independent Factor Analysis with Temporally Structured Sources Hagai Attias hagai@gatsby.ucl.ac.uk Gatsby Unit, University College London 17 Queen Square London WCIN 3AR, U.K. Abstract We present a new technique for time series analysis based on dynamic probabilistic networks. In this approach, the observed data are modeled in terms of unobserved, mutually independent factors, as in the recently introduced technique of Independent Factor Analysis (IFA). However, unlike in IFA, the factors are not Li.d.; each factor has its own temporal statistical characteristics. We derive a family of EM algorithms that learn the structure of the underlying factors and their relation to the data. These algorithms perform source separation and noise reduction in an integrated manner, and demonstrate superior performance compared to IFA. 1 Introduction The technique of independent factor analysis (IFA) introduced in [1] provides a tool for modeling L'-dim data in terms of L unobserved factors. These factors are mutually independent and combine linearly with added noise to produce the observed data. Mathematically, the model is defined by Yt = HXt + Ut, where Xt is the vector of factor activities at time t, L' x L mixing matrix, and Ut is the noise. (1) Yt is the data vector, H is the The origins of IFA lie in applied statistics on the one hand and in signal processing on the other hand. Its statistics ancestor is ordinary factor analysis (FA), which assumes Gaussian factors. In contrast, IFA allows each factor to have its own arbitrary distribution, modeled semi-parametrically by a I-dim mixture of Gaussians (MOG). The MOG parameters, as well as the mixing matrix and noise covariance matrix, are learned from the observed data by an expectation-maximization (EM) algorithm derived in [1]. The signal processing ancestor of IFA is the independent component analysis (ICA) method for blind source separation [2]-[6]. In ICA, the factors are termed sources, and the task of blind source separation is to recover them from the observed data with no knowledge of the mixing process. The sources in ICA have non-Gaussian distributions, but unlike in IFA these distributions are usually fixed by prior knowledge or have quite limited adaptability. More significant restrictions Dynamic Independent Factor Analysis 387 are that their number is set to the data dimensionality, i.e. L = L' ('square mixing'), the mixing matrix is assumed invertible, and the data are assumed noise-free (Ut = 0). In contrast, IFA allows any L, L' (including more sources than sensors, L > L'), as well as non-zero noise with unknown covariance. In addition, its use of the flexible MOG model often proves crucial for achieving successful separation [1]. Therefore, IFA generalizes and unifies FA and ICA. Once the model has been learned, it can be used for classification (fitting an IFA model for each class), completing missing data, and so on. In the context of blind separation, an optimal reconstruction of the sources Xt from data is obtained [1] using a MAP estimator. However, IFA and its ancestors suffer from the following shortcoming: They are oblivious to temporal information since they do not attempt to model the temporal statistics of the data (but see [4] for square, noise-free mixing). In other words, the model learned would not be affected by permuting the time indices of {yt}. This is unfortunate since modeling the data as a time series would facilitate filtering and forecasting, as well as more accurate classification. Moreover, for source separation applications, learning temporal statistics would provide additional information on the sources, leading to cleaner source reconstructions. To see this, one may think of the problem of blind separation of noisy data in terms of two components: source separation and noise reduction. A possible approach might be the following two-stage procedure. First, perform noise reduction using, e.g., Wiener filtering. Second, perform source separation on the cleaned data using, e.g., an ICA algorithm. Notice that this procedure directly exploits temporal (second-order) statistics of the data in the first stage to achieve stronger noise reduction. An alternative approach would be to exploit the temporal structure of the data indirectly, by using a temporal source model. In the resulting single-stage algorithm, the opemtions of source sepamtion and noise reduction are coupled. This is the approach taken in the present paper. In the following, we present a new approach to the independent factor problem based on dynamic probabilistic networks. In order to capture temporal statistical properties of the observed data, we describe each source by a hidden Markov model (HMM). The resulting dynamic model describes a multivariate time series in terms of several independent sources, each having its own temporal characteristics. Section 2 presents an EM learning algorithm for the zero-noise case, and section 3 presents an algorithm for the case of isotropic noise. The case of non-isotropic noise turns out to be computationally intractable; section 4 provides an approximate EM algorithm based on a variational approach. Notation: The multivariable Gaussian density is denoted by g(z, E) =1 27rE 1- 1 / 2 exp( -z T E- l z/2). We work with T-point time blocks denoted Xl:T = {Xt}[=I' The ith coordinate of Xt is x~. For a function f, (f(Xl:T)) denotes averaging over an ensemble of Xl:T blocks. 2 Zero Noise The MOG source model employed in IFA [1] has the advantages that (i) it is capable of approximating arbitrary densities, and (ii) it can be learned efficiently from data by EM. The Gaussians correspond to the hidden states of the sources, labeled by s. Assume that at time t, source i is in state s~ = s. Its signal x~ is then generated by sampling from a Gaussian distribution with mean JL! and variance v!. In order to capture temporal statistics of the data, we endow the sources with temporal structure by introducing a transition matrix a!,s between the states. Focusing on H. Attias 388 = 1, ... , T, the resulting probabilistic model is defined by i P( Sti=i Si St-l = S') = as's' P( Soi = S) = 7rsi , a time block t p(X~ IS~ = S) = g(x~ - J.L!,v!), P(Yl:T) =1 detG IT P(Xl:T), (2) xL where P(Xl:T) is the joint density of all sources i = 1, ... , L at all time points, and the last equation follows from Xt = GYt with G = H- 1 being the unmixing matrix. As usual in the noise-free scenario (see [2]; section 7 of [1]), we are assuming that the mixing matrix is square and invertible. The graphical model for the observed density P(Yl:T I W) defined by (2) is parametrized by W = {G ij , J.L!, v!, 7r!, a!, s}' This model describes each source as a first-order HMM; it reduces to a time-independent model if a!,s = 7r!. Whereas temporal structure can be described by other means, e.g. a moving-average [4] or autoregressive [6] model, the HMM is advantageous since it models high-order temporal statistics and facilitates EM learning. Omitting the derivation, maximization with respect to G ij results in the incremental update rule bG = ?G - 1 T ?T </>(Xt)x[G , L (3) t=l where </>(xn = Es 'Y:(s)(x~ - J.L!)/v!, and the natural gradient [3] was used; ? is an appropriately chosen learning rate. For the source parameters we obtain the update rules Et 'Yt(s)x~ Et 'Y1(s) i , as' s _ - t ~t( s' , s) i (')' u t 'Yt-l S E ~ (4) with the initial probabilities updated via 7r! = 'YA(s). We used the standard HMM notation 'Y:(s) = p(s~ = S I xLT)' ~t(s',s) = P(SLI = s',s~ = s I xLT)' These posterior densities are computed in the E-step for each source, which is given in terms of the data via x~ = E j Gijyl, using the forward-backward procedure [7]. The algorithm (3-4) may be used in several possible generalized EM schemes. An efficient one is given by the following two-phase procedure: (i) freeze the source parameters and learn the separating matrix G using (3); (ii) freeze G and learn the source parameters using (4), then go back to (i) and repeat. Notice that the rule (3) is similar to a natural gradient version of Bell and Sejnowski's leA rule [2]; in fact, the two coincide for time-independent sources where </>(Xi) = -alogp(xi)/axi. We also recognize (4) as the Baum-Welch method. Hence, in phase (i) our algorithm separates the sources using a generalized leA rule, whereas in phase (ii) it learns an HMM for each source. Remark. Often one would like to model a given L'-variable time series in terms of a smaller number L ~ L' of factors. In the framework of our noise-free model Yt = HXt, this can be achieved by applying the above algorithm to the L largest principal components of the data; notice that if the data were indeed generated by L factors, the remaining L' - L principal components would vanish. Equivalently, one may apply the algorithm to the data directly, using a non-square L x L' unmixing matrix G. Results. Figure 1 demonstrates the performance of the above method on a 4 x 4 mixture of speech signals, which were passed through a non-linear function to modify their distributions. This mixture is inseparable to leA because the source model used by the latter does not fit the actual source densities (see discussion in [1]). We also applied our dynamic network to a mixture of speech signals whose distributions Dynamic Independent Factor Analysis 389 HMM-ICA leA 3 3 0 .8 2 0 .7 0 .8 ):i"0.5 '>:! 'zs:O.4 0 0.3 -1 0 .2 -2 0.1 0 -4 -3 -3 -2 4 -2 o 2 -2 x1 0 2 x1 Figure 1: Left: Two of the four source distributions. Middle: Outputs of the EM algorithm (3-4) are nearly independent. Right: the outputs of leA (2) are correlated. were made Gaussian by an appropriate non-linear transformation. Since temporal information is crucial for separation in this case (see [4],[6]), this mixture is inseparable to leA and IFA; however, the algorithm (3-4) accomplished separation successfully. 3 Isotropic Noise We now turn to the case of non-zero noise Ut ::j:. O. We assume that the noise is white and has a zero-mean Gaussian distribution with covariance matrix A. In general, this case is computationally intractable (see section 4). The reason is that the Estep requires computing the posterior distribution P(SO:T, Xl:T I Yl:T) not only over the source states (as in the zero-noise case) but also over the source signals, and this posterior has a quite complicated structure. We now show that if we assume isotropic noise, i.e. Aij = )..6ij , as well as square invertible mixing as above, this posterior simplifies considerably, making learning and inference tractable. This is done by adapting an idea suggested in [8] to our dynamic probabilistic network. We start by pre-processing the data using a linear transformation that makes their covariance matrix unity, i.e., (YtyT) = I ('sphering'). Here (-) denotes averaging over T-point time blocks. From (1) it follows that HSHT = )..'1, where S = (XtxT) is the diagonal covariance matrix of the sources, and )..' = 1 -)... This, for a square invertible H, implies that HTH is diagonal. In fact, since the unobserved sources can be determined only to within a scaling factor, we can set the variance of each source to unity and obtain the orthogonality property HTH = )..'1. It can be shown that the source posterior now factorizes into a product over the individual sources, P(SO:T, Xl :T I Yl:T) = TIiP(sb:T, XLT I Yl:T), where P(Sb:T,xLT I Yl:T) ()( [rrg(X; -T):'aD ? v;p(s: I SLl)] vbp(sb)? (5) t=l The means and variances at time t in (5), as well as the quantities vL depend on both the data Yt and the states s~; in particular, T); = (~j Hjiyl + )..j1!)/(>..'vs +)..) and a-; = )..v!/(>..'vs + )..), using s = s1; the expression for the v; are omitted. The transition probabilities are the same as in (2). Hence, the posterior distribution (5) effectively defines a new HMM for each source, with yrdependent emission and transition probabilities. To derive the learning rule for H, we should first compute the conditional mean Xt of the source signals at time t given the data. This can be done recursively using (5) as in the forward-backward procedure. We then obtain 1 T c= T~YtXr. t=l (6) H. Attias 390 This fractional form results from imposing the orthogonality constraint HTH = >..'1 using Lagrange multipliers and can be computed via a diagonalization procedure. The source parameters are computed using a learning rule (omitted) similar to the noise-free rule (4). It is easy to derive a learning rule for the noise level ,\ as well; in fact, the ordinary FA rule would suffice. We point out that, while this algorithm has been derived for the case L = L', it is perfectly well defined (though sub-optimal: see below) for L :::; L'. Non-Isotropic Noise 4 The general case of non-isotropic noise and non-square mixing is computationally intractable. This is because the exact E-step requires summing over all possible source configurations (st, ... , SfL) at all times tl, ... , tL = 1, ... , T. The intractability problem stems from the fact that, while the sources are independent, the sources conditioned on a data vector Yl:T are correlated, resulting in a large number of hidden configurations. This problem does not arise in the noise-free case, and can be avoided in the case of isotropic noise and square mixing using the orthogonality property; in both cases, the exact posterior over the sources factorizes. The EM algorithm derived below is based on a variational approach. This approach was introduced in [9J in the context of sigmoid belief networks, but constitutes a general framework for ML learning in intractable probabilistic networks; it was used in a HMM context in [IOJ. The idea is to use an approximate but tractable posterior to place a lower bound on the likelihood, and optimize the parameters by maximizing this bound. A starting point for deriving a bound on the likelihood L is Neal and Hinton's [l1J formulation of the EM algorithm: T ~ L L Eq logp(Yt I Xt) + L Eq logp(sb:T' xi:T) - Eq logq, (7) t=l i=l where Eq denotes averaging with respect to an arbitrary posterior density over the hidden variables given the observed data, q = q(SO:T,Xl:T I Yl:T). Exact EM, as shown in [11], is obtained by maximizing the bound (7) with respect to both the posterior q (corresponding to the E-step) and the model parameters W (Mstep). However, the resulting q is the true but intractable posterior. In contrast, in variational EM we choose a q that differs from the true posterior, but facilitates a tractable E-step. L = lOgp(Yl:T) E-Step. We use q(sO:T,Xl:T I Yl :T) = parametrized as q(s~ = s I SLI = S',Yl:T) ex: q(Xt IYl :T) = TIiq(sb:T I Yl:T)TItq(Xt I Yl:T), q(sb = s I Yl :T) ex: '\!,ta!,s, Q(Xt - Pt, ~t) . ,\! ,t 7r! , (8) Thus, the variational transition probabilities in (8) are described by multiplying the original ones a!, s by the parameters '\~,t' subject to the normalization constraints. The source signals Xt at time t are jointly Gaussian with mean Pt and covariance ~t. The means, covariances and transition probabilities are all time- and datadependent, i.e., Pt = f(Yl:T, t) etc. This parametrization scheme is motivated by the form of the posterior in (5); notice that the quantities v~ ,t there become the variational parameters ~;j,,\~ t of (8). A related scheme was used in [IOJ in a different context. Since these parameters will be adapted independently of the model parameters, the non-isotropic algorithm is expected to give superior results compared to the isotropic one. pL 1]:, a-t, 391 Dynamic Independent Factor Analysis Mixing Reco nstruc tion O ~--------~------~ 5 -10 0 ~-20 ~ .L3Cfl>___~O ~ L.U -40 0 -5 0 -10 0 - 15 -5~S;----:0:------::5'-------:-::'0:------:-'? 15 -20 - 5 0 SNA (dB) 5 10 15 SNR (dB) Figure 2: Left: quality of the model parameter estimates. Right: quality of the source reconstructions. (See text). Of course, in the true posterior the Xt are correlated, both temporally among themselves and with St, and the latter do not factorize. To best approximate it, the variational parameters V = {p~, ~~j , >..! t} are optimized to maximize the bound on .c, or equivalently to minimize the KL' distance between q and the true posterior. This requirement leads to the fixed point equations Pt (HT A -lH + Bt)-l(HT A -lYt 1 [1 . --:- exp - - log V Z zZt 2 _ (pi _ t s + b t ), J-Li)2 + ~ii] s . t 2vZs ~t = (HT A-1H + Bt)-l , , (9) where Bij = Ls[rl(S)/v!]6ij , b~ = Ls ,l(s)J-L!/v!, and the factors zf ensure normalization. The HMM quantities ,f(s) are computed by the forward-backward procedure using the variational transition probabilities (8). The variational parameters are determined by solving eqs. (9) iteratively for each block Yl :T; in practice, we found that less then 20 iterations are usually required for convergence. M-Step. The update rules for W are given for the mixing parameters by 1~ T T T A = T L,)YtYt - YtPt H ), (10) t and for the source parameters by Lt ,f(s)p~ Lt ,I(s) , Lt ~f(s', s) Lt,Ll(S') , Vi s = Lt ,f(s)((p~ - J-L~)2 + ~~i) Lt ,f(s) (11) where the ~Hs' , s) are computed using the variational transition probabilities (8). Notice that the learning rules for the source parameters have the Baum-Welch form, in spite of the correlations between the conditioned sources. In our variational approach, these correlations are hidden in V, as manifested by the fact that the fixed point equations (9) couple the parameters V across time points (since ,:(s) depends on >"!,t=l:T) and sources. Source Reconstruction. From q(Xt I Yl :T) (8), we observe that the MAP source estimate is given by Xt = Pt(Yl:T), and depends on both Wand V. Results. The above algorithm is demonstrated on a source separation task in Figure 2. We used 6 speech signals, transformed by non-linearities to have arbitrary one-point densities, and mixed by a random 8 x 6 matrix Ho. Different signalto-noise (SNR) levels were used. The error in the estimated H (left, solid line) is quantified by the size ofthe non-diagonal elements of (HTH)-l HTHo relative to the H Attias 392 diagonal; the results obtained by IFA [1], which does not use temporal information, are plotted for reference (dotted line). The mean squared error of the reconstructed sources (right, solid line) and the corresponding IFA result (right, dashed line) are also shown. The estimate and reconstruction errors of this algorithm are consistently smaller than those of IFA, reflecting the advantage of exploiting the temporal structure of the data. Additional experiments with different numbers of sources and sensors gave similar results. Notice that this algorithm, unlike the previous two, allows both L ::; L' and L > L'. We also considered situations where the number of sensors was smaller than the number of sources; the separation quality was good, although, as expected, less so than in the opposite case. 5 Conclusion An important issue that has not been addressed here is model selection. When applying our algorithms to an arbitrary dataset, the number of factors and of HMM states for each factor should be determined. Whereas this could be done, in principle, using cross-validation, the required computational effort would be fairly large. However, in a recent paper [12] we develop a new framework for Bayesian model selection, as well as model averaging, in probabilistic networks. This framework, termed Variational Bayes, proposes an EM-like algorithm which approximates full posterior distributions over not only hidden variables but also parameters and model structure, as well as predictive quantities, in an analytical manner. It is currently being applied to the algorithms presented here with good preliminary results. One field in which our approach may find important applications is speech technology, where it suggests building more economical signal models based on combining independent low-dimensional HMMs, rather than fitting a single complex HMM. It may also contribute toward improving recognition performance in noisy, multispeaker, reverberant conditions which characterize real-world auditory scenes. References [1] Attias, H. (1999). Independent factor analysis. Neur. Camp. 11, 803-85l. [2] Bell, A.J. & Sejnowski, T .J. (1995). An information-maximization approach to blind separation and blind deconvolution. Neur. Camp. 7, 1129-1159. [3] Amari, S., Cichocki, A. & Yang, H.H. (1996). A new learning algorithm for blind signal separation. Adv. Neur. Info. Pmc. Sys. 8,757-763 (Ed. by Touretzky, D.S. et al). MIT Press, Cambridge, MA. [4] Pearlmutter , B.A. & Parra, L.C. (1997). Maximum likelihood blind source separation: A context-sensitive generalization of ICA. Adv. Neur. Info. Pmc. Sys. 9, 613-619 (Ed. by Mozer, M.C. et al). MIT Press, Cambridge, MA. [5] Hyviirinen, A. & Oja, E. (1997). A fast fixed-point algorithm for independent component analysis. Neur. Camp. 9, 1483-1492. [6] Attias, H. & Schreiner, C.E. (1998). Blind source separation and deconvolution: the dynamic component analysis algorithm. Neur. Camp. 10, 1373-1424. [7] Rabiner, L. & Juang, B.-H. (1993). Fundamentals of Speech Recognition. Prentice Hall, Englewood Cliffs, NJ. [8] Lee, D.D. & Sompolinsky, H. (1999) , unpublished; D.D. Lee, personal communication. [9] Saul, L.K., Jaakkola, T., and Jordan, M.L (1996). Mean field theory of sigmoid belief networks. J. Art. Int. Res. 4, 61-76. [10] Ghahramani, Z. & Jordan, M.L (1997). Factorial hidden Markov models. Mach. Learn. 29, 245-273. [11] Neal, R.M. & Hinton, G.E. (1998). A view of the EM algorithm that justifies incremental, sparse, and other variants. Learning in Graphical Models, 355-368 (Ed. by Jordan, M.L). Kluwer Academic Press. [12] Attias, H. (2000). A variational Bayesian framework for graphical models. Adv. Neur. Info. Pmc. Sys. 12 (Ed. by Leen, T. et al). MIT Press, Cambridge, MA. 

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Recognizing Evoked Potentials in a Virtual Environment * Jessica D. Bayliss and Dana H. Ballard Department of Computer Science University of Rochester Rochester, NY 14627 {bayliss,dana}@cs.rochester.edu Abstract Virtual reality (VR) provides immersive and controllable experimental environments. It expands the bounds of possible evoked potential (EP) experiments by providing complex, dynamic environments in order to study cognition without sacrificing environmental control. VR also serves as a safe dynamic testbed for brain-computer .interface (BCl) research. However, there has been some concern about detecting EP signals in a complex VR environment. This paper shows that EPs exist at red, green, and yellow stop lights in a virtual driving environment. Experimental results show the existence of the P3 EP at "go" and "stop" lights and the contingent negative variation (CNY) EP at "slow down" lights. In order to test the feasibility of on-line recognition in VR, we looked at recognizing the P3 EP at red stop tights and the absence of this signal at yellow slow down lights. Recognition results show that the P3 may successfully be used to control the brakes of a VR car at stop lights. 1 Introduction The controllability of VR makes it an excellent candidate for use in studying cognition. It expands the bounds of possible evoked potential (EP) experiments by providing complex, dynamic environments in order to study decision making in cognition without sacrificing environmental control. We have created a flexible system for real-time EEG collection and analysis from within virtual environments. The ability of our system to give quick feedback enables it to be used in brain-computer interface (BCl) research, which is aimed at helping individuals with severe motor deficits to become more independent. Recent BCl work has shown the feasibility of on-line averaging and biofeedback methods in order to choose characters or move a cursor on a computer screen with up to 95% accuracy while sitting still and concentrating on the screen [McFarland et aI., 1993; Pfurtscheller et al., 1996; Vaughn et al., 1996; Farwell and Donchin, 1988]. Our focus is to dramatically extend the BCl by allowing evoked potentials to propel the user through alternate virtual environments. For example, a *This research was supported by NIHIPHS grantl-P41-RR09283. It was also facilitated in part by a National Physical Science Consortium Fellowship and by stipend support from NASA Goddard Space Flight Center. 4 J. D. Bayliss and D. H. Ballard Figure 1: (Left) An individual demonstrates driving in the modified go cart. (Right) A typical stoplight scene in the virtual environment. user could choose a virtual living room from a menu of rooms, navigate to the living room automatically in the head-mounted display, and then choose to turn on the stereo. As shown in [Farwell and Donchin, 1988], the P3 EP may be used for a brain-computer interface that picks characters on a computer monitor. Discovered by [Chapman and Bragdon, 1964; Sutton et aI., 1965] and extensively studied (see [Polich, 1998] for a literature review), the P3 is a positive waveform occurring approximately 300-500 ms after an infrequent task-relevant stimulus. We show that requiring subjects to stop or go at virtual traffic lights elicits this EP. The contingent negative variation (CNV), an EP that happens preceding an expected stimulus, occurs at slow down lights. In order to test the feasibility of on-line recognition in the noisy VR environment, we recognized the P3 EP at red stop lights and the lack of this signal at yellow slow down lights. Results using a robust Kalman filter for off-line recognition indicate that the car may be stopped reliably with an average accuracy of 84.5% while the on-line average for car halting is 83%. 2 The Stoplight Experiments The first experiment we performed in the virtual driving environment shows that a P3 EP is obtained when subjects stop or go at a virtual light and that a CNV occurs when subjects see a slow down light. Since all subjects received the same light colors for the slow down, go, and stop conditions we then performed a second experiment with different light colors in order to disambiguate light color from the occurrence of the P3 and CNV. Previous P3 research has concentrated primarily on static environments such as the continuous performance task [Rosvold et aI., 1956]. In the visual continuous performance task (VCPT), static images are flashed on a screen and the subject is told to press a button when a rare stimulus occurs or to count the number of occurrences of a rare stimulus. This makes the stimulus both rare and task relevant in order to evoke a P3. As an example, given red and yellow stoplight pictures, a P3 should occur if the red picture is less frequent than the yellow and subjects are told to press a mouse button only during the red light. We assumed a similar response would occur in a VR driving world if certain lights were infrequent and subjects were told to stop or go at them. This differs from the VCPT in two important ways: 1. In the VCPT subjects sit passively and respond to stimuli. In the driving task, Recognizing Evoked Potentials in a Virtual Environment 5 subjects control when the stimuli appear by where they drive in the virtual world. 2. Since subjects are actively involved and fully immersed in the virtual world, they make more eye and head movements. The movement amount can be reduced by a particular experimental paradigm, but it can not be eliminated. The first difference makes the VR environment a more natural experimental environment. The second difference means that subjects create more data artifacts with extra movement. We handled these artifacts by first manipulating the experimental environment to reduce movements where important stimulus events occurred. This meant that all stoplights were placed at the end of straight stretches of road in order to avoid the artifacts caused by turning a corner. For our on-line recognition, we then used the eye movement reduction technique described in [Semlitsch et al., 1986] in order to subtract a combination of the remaining eye and head movement artifact. 2.1 Experimental Setup All subjects used a modified go cart in order to control the virtual car (see Figure 1). The virtual reality interface is rendered on a Silicon Graphics Onyx machine with 4 processors and an Infinite Reality Graphics Engine. The environment is presented to the subject through a head-mounted display (HMD). Since scalp EEG recordings are measured in microvolts, electrical signals may easily interfere during an experiment. We tested the effects of wearing a VR4 HMD containing an ISCAN eye tracker and discovered that the noise levels inside of the VR helmet were comparable to noise levels while watching a laptop screen [Bayliss and Ballard, 1998]. A trigger pulse containing information about the color of the light was sent to the EEG acquisition system whenever a light changed. While an epoch size from -100 ms to 1 sec was specified, the data was recorded continuously. Information about head position as well as gas, braking, and steering position were saved to an external file. Eight electrodes sites (FZ, CZ, CPZ, PZ, P3, P4, as well as 2 vertical EOG channels) were arranged on the heads of seven subjects with a linked mastoid reference. Electrode impedances were between 2 and 5 kohms for all subjects. Subjects ranged in age from 19 to 52 and most had no previous experiences in a virtual environment. The EEG signal was amplified using Grass amplifiers with an analog bandwidth from 0.1 to 100 Hz. Signals were then digitized at a rate of 500 Hz and stored to a computer. 2.2 Ordinary Traffic Light Color Experiment Five subjects were instructed to slow down on yellow lights, stop for red lights, and go for green lights. These are normal traffic light colors. Subjects were allowed to drive in the environment before the experiment to get used to driving in VR. In order to make slow down lights more frequent, all stoplights turned to the slow down color when subjects were further than 30 meters aways from them. When the subject drove closer than 30 meters the light then turned to either the go or stop color with equal probability. The rest of the light sequence followed normal stoplights with the stop light turning to the go light after 3 seconds and the go light not changing. We calculated the grand averages over red, green, and yellow light trials (see Figure 2a). Epochs affected by artifact were ignored in the averages in order to make sure that any existing movements were not causing a P3-like signal. Results show that a P3 EP occurs for both red and green lights. Back averaging from the green/red lights to the yellow light shows the existence of a CNV starting at approximately 2 seconds before the light changes to red or green. 6 J. D. Bayliss and D. H Ballard Stop Light ~ Slow Down Light Go Light -5 uv 1\ .~'"\ i " \ I I ! \ I \ I i: \\ .v'l,,/AI \, I I) +lOuv -8 uv .E bO ;J u ~ b) ~ t) > \"~'~ ~ ; ,: , '::1 j '-lOOms :"' ,~t r/'" " .~ '\ f I".j" "\?.....\ ".; t<iooms' '-lOOms A, f...j Vv f I lOOOms I h'-3~000ii1S;;:::::::::==::;;2;;OO~ms~1 +12 uv Figure 2: a) Grand averages for the red stop, green go, and yellow slow down lights. b) Grand averages for the yellow stop, red go, and green slow down lights. All slow down lights have been back-averaged from the occurrence of the go/stop light in order to show the existence of a CNY. 2.3 Alternative Traffic Light Colors The P3 is related to task relevance and should not be related to color, but color needed to be disambiguated as the source of the P3 in the experiment. We had two subjects slow down at green lights, stop at yellow lights, and go at red lights. In order to get used to this combination of colors, subjects were allowed to drive in the town before the experiment. The grand averages for each light color were calculated in the same manner as the averages above and are shown in Figure 2b. As expected, a P3 signal existed for the stop condition and a CNV for the slow down condition. The go condition P3 was much noisier for these two subjects, although a slight P3-like signal is still visible. 3 Single Trial Recognition Results While averages show the existence of the P3 EP at red stop lights and the absence of such at yellow slow down lights, we needed to discover if the signal was clean enough for single trial recognition as the quick feedback needed by a BCI depends on quick recognition. While there were three light conditions to recognize, there were only two distinct kinds of evoked potentials. We chose to recognize the difference between the P3 and the CNV since their averages are very different. Recognizing the difference between two kinds of EPs gives us the ability to use a BCI in any task that can be performed using a series of binary decisions. We tried three methods for classification of the P3 EP: correlation, independent component analysis (ICA), and a robust Kalman filter. Approximately, 90 slow down yellow light and 45 stop red light trials from each subject were classified. The reason we allowed a yellow light bias to enter recognition is because the yellow light currently represents an unimportant event in the environment. In a real BCI unimportant events are likely to occur more than user-directed actions, making this bias justifiable. Recognizing Evoked Potentials in a Virtual Environment 7 Table 1: Recognition Results (p Subjects S1 S2 S3 S4 S5 Correlation %Correct Red Yel Total 81 51 64 95 73 63 89 56 66 81 60 67 63 66 65 ICA Red 76 86 72 73 65 < 0.01) %Correct Yel Total 77 77 87 88 82 87 71 69 74 79 Robust Kalman Filter %Correct Total Red Yel 55 86 77 82 94 90 74 81 85 91 82 65 78 92 87 Table 2: Recognition Results for Return Subjects Subjects S4 S5 Robust K-Filter % Correct Total Red Yel 73 90 85 67 87 80 As expected, the data obtained while driving contained artifacts, but in an on-line BCI these artifacts must be reduced in order to make sure that what the recognition algorithm is recognizing is not an artifact such as eye movement. In order to reduce these artifacts, we performed the on-line linear regression technique described in [Semlitsch et aI. , 1986] in order to subtract a combination of eye and head movement artifact. In order to create a baseline from which to compare the performance of other algorithms, we calculated the correlation of all sample trials with the red and yellow light averages from each subject's maximal P3 electrode site using the following formula: correlation = (sample * aveT)/(11 sample II * II ave II) (1) where sample and ave are both 1 x 500 vectors representing the trial epochs and light averages (respectively). We used the whole trial epoch for recognition because it yielded better recognition than just the time area around the P3. If the highest correlation of a trial epoch with the red and yellow averages was greater than 0.0, then the signal was classified as that type of signal. If both averages correlated negatively with the single trial, then the trial was counted as a yellow light signal. As can be seen in Table 1, the correct signal identification of red lights was extremely high while the yellow light identification pulled the results down. This may be explained by the greater variance of the yellow light epochs. Correlations in general were poor with typical correlations around 0.25. ICA has successfully been used in order to minimize artifacts in EEG data [Jung et at. , 1997; Vigario, 1997] and has also proven useful in separating P3 component data from an averaged waveform [Makeig et aI., 1997]. The next experiment used ICA in order to try to separate the background EEG signal from the P3 signal. Independent component analysis (lCA) assumes that n EEG data channels x are a linear combination of n statistically independent signals s : x= As (2) where x and s are n x 1 vectors. We used the matlab package mentioned in [Makeig et aI. , 1997] with default learning values, which finds a matrix W by stochastic gradient descent. 8 J D. Bayliss and D. H. Ballard This matrix W performs component separation. All data was sphered in order to speed convergence time. After training the W matrix, the source channel showing the closest P3-like signal (using correlation with the average) for the red light average data was chosen as the signal with which to correlate individual epochs. The trained W matrix was also used to find the sources of the yellow light average. The red and yellow light responses were then correlated with individual epoch sources in the manner of the first experiment. The third experiment used the robust Kalman filter framework formulated by Rao [Rao, 1998]. The Kalman filter assumes a linear model similar to the one ofICA in equation 2, but assumes the EEG output x is the observable output of a generative or measurement matrix A and an internal state vector s of Gaussian sources. The output may also have an additional noise component n, a Gaussian stochastic noise process with mean zero and a covariance matrix given by ~ = E[nnTj, leading to the model expression: x = As + n. In order to find the most optimal value of s, a weighted least-squares criterion is formulated: (3) where s follows a Gaussian distribution with mean s and covariance M. Minimizing this criterion by setting ~; = 0 and using the substitution N = (AT~-lU + M-1)-1 yields the Kalman filter equation, which is basically equal to the old estimate plus the Kalman gain times the residual error. (4) In an analogous manner, the measurement matrix A may be estimated (learned) if one assumes the physical relationships encoded by the measurement matrix are relatively stable. The learning rule for the measurement matrix may be derived in a manner similar to the rule for the internal state vector. In addition, a decay term is often needed in order to avoid overfitting the data set. See [Rao, 1998] for details. In our experiments both the internal state matrix s and the measurement matrix A were learned by training them on the average red light signal and the average yellow light signal. The signal is measured from the start of the trial which is known since it is triggered by the light change. We used a Kalman gain of 0.6 and a decay of 0.3. After training, the signal estimate for each epoch is correlated with the red and yellow light signal estimates in the manner of experiment 1. We made the Kalman filter statistically robust by ignoring parts of the EEG signal that fell outside a standard deviation of 1.0 from the training signals. The overall recognition results in Table 1 suggest that both the robust Kalman filter and ICA have a statistically significant advantage over correlation (p < 0.01). The robust Kalman filter has a very small advantage over ICA (not statistically significant). In order to look at the reliability of the best algorithm and its ability to be used on-line two of the Subjects (S4 and SS) returned for another VR driving session. In these sessions the brakes of the driving simulator were controlled by the robust Kalman filter recognition algorithm for red stop and yellow slow down lights. Green lights were ignored. The results of this session using the Robust Kalman Filter trained on the first session are shown in Table 2. The recognition numbers for red and yellow lights between the two sessions were compared using correlation. Red light scores between the sessions correlated fairly highly - 0.82 for S4 and 0.69 for SS. The yellow light scores between sessions correlated poorly with both S4 and SS at approximately -0.1. This indicates that the yellow light epochs tend to correlate poorly with each other due to the lack of a large component such as the P3 to tie them together. Recognizing Evoked Potentials in a Virtual Environment 9 4 Future Work This paper showed the viability of recognizing the P3 EP in a VR environment. We plan to allow the P3 EP to propel the user through alternate virtual rooms through the use of various binary decisions. In order to improve recognition for the BCI we need to experiment with a wider and more complex variety of recognition algorithms. Our most recent work has shown a dependence between the human computer interface used in the BCI and recognition. We would like to explore this dependence in order to improve recognition as much as possible. References [Bayliss and Ballard, 1998) lD. Bayliss and D.H. Ballard, ''The Effects of Eye Tracking in a VR Helmet on EEG Recording," TR 685, University of Rochester National Resource Laboratory for the Study of Brain and Behavior, May 1998. [Chapman and Bragdon, 1964) R.M. Chapman and H.R. Bragdon, "Evoked responses to numerical and non-numerical visual stimuli while problem solving.," Nature, 203: 1155-1157, 1964. [Farwell and Donchin, 1988) L. A. Farwell and E. Donchin, "Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials," Electroenceph. Clin. Neurophysiol., pages 510-523, 1988. [Jung et al., 1997) 1'.P. Jung, C. Humphries,1'. Lee, S. Makeig, M.J. McKeown, Y. lragui, and 1'.l Sejnowski, "Extended ICA Removes Artifacts from Electroencephalographic Recordings," to Appear in Advances in Neural Information Processing Systems, 10, 1997. [Makeig et al., 1997) S. Makeig, 1'. Jung, A.J. Bell, D. Ghahremani, and 1'.J. Sejnowski, "Blind Separation of Auditory Event-related Brain Responses into Independent Components," Proc. Nat'l Acad. Sci. USA , 94:10979-10984, 1997. [McFarland et al., 1993) D.l McFarland, G.w. Neat, R.F. Read, and J.R. Wolpaw, "An EEG-based method for graded cursor control," Psychobiology, 21(1):77-81, 1993. [Pfurtscheller et al. , 1996) G. Pfurtscheller, D. Flotzinger, M. Pregenzer, J. Wolpaw, and D. McFarland, "EEG-based Brain Computer Interface (BCI)," Medical Progress through Technology, 21:111-121 , 1996. [Polich, 1998] J. Polich, "P300 Clinical Utility and Control of Variability," J. of Clinical NeurophYSiology, 15(1): 14-33, 1998. [Rao, 1998] R. P.N. Rao, "Visual Attention during Recognition," Advances in Neural Information Processing Systems, 10, 1998. [Rosvold et al., 1956] H.E. Rosvold, A.F. Mirsky, I. Sarason, E.D. Bransome Jr., and L.H. Beck, "A Continuous Performance Test of Brain Damage," 1. Consult. Psychol., 20, 1956. [SemIitschetal., 1986) H.Y. SemIitsch, P. Anderer, P Schuster, and O. Presslich, "A solution for reliable and valid reduction of ocular artifacts applied to the P300 ERP;' Psychophys., 23 :695703,1986. [Sutton et al., 1965) S. Sutton, M. Braren, J. Zublin, and E. John, "Evoked potential correlates of stimulus uncertainty," Science, 150: 1187-1188, 1965. [Vaughn et al., 1996) 1'.M. Vaughn, J.R. Wolpaw, and E. Donchin, "EEG-Based Communication: Prospects and Problems," IEEE Trans. on Rehabilitation Engineering, 4(4):425-430, 1996. [Vigario, 1997) R. Vigario, "Extraction of ocular artifacts from eeg using independent component analysis," Electroenceph. Clin. Neurophysiol., 103:395-404, 1997. 

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Bayesian Network Induction via Local Neighborhoods Dimitris Margaritis Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 D.Margaritis@cs.cmu.edu Sebastian Thrun Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 S. Thrun@cs.cmu.edu Abstract In recent years, Bayesian networks have become highly successful tool for diagnosis, analysis, and decision making in real-world domains. We present an efficient algorithm for learning Bayes networks from data. Our approach constructs Bayesian networks by first identifying each node's Markov blankets, then connecting nodes in a maximally consistent way. In contrast to the majority of work, which typically uses hill-climbing approaches that may produce dense and causally incorrect nets, our approach yields much more compact causal networks by heeding independencies in the data. Compact causal networks facilitate fast inference and are also easier to understand. We prove that under mild assumptions, our approach requires time polynomial in the size of the data and the number of nodes. A randomized variant, also presented here, yields comparable results at much higher speeds. 1 Introduction A great number of scientific fields today benefit from being able to automatically estimate the probability of certain quantities of interest that may be difficult or expensive to observe directly. For example, a doctor may be interested in estimating the probability of heart disease from indications of high blood pressure and other directly measurable quantities. A computer vision system may benefit from a probability distribution of buildings based on indicators of horizontal and vertical straight lines. Probability densities proliferate the sciences today and advances in its estimation are likely to have a wide impact on many different fields . Bayesian networks are a succinct and efficient way to represent a joint probability distribution among a set of variables. As such, they have been applied to fields such as those mentioned [Herskovits90][Agosta88]. Besides their ability for density estimation, their semantics lend them to what is sometimes loosely referred to as causal discovery, namely directional relationships among quantities involved. It has been widely accepted that the most parsimonious representation for a Bayesian net is one that closely represents the causal independence relationships that may exist. For these reasons, there has been great interest in automatically inducing the structure of Bayesian nets automatically from data, preferably also preserving the independence relationships in the process. Two research approaches have emerged. The first employs independence properties of the underlying network that produced the data in order to discover parts of its structure. This approach is mainly exemplified by the SGS and PC algorithms in [Spirtes93], as well 506 D. Margaritis and S. Thrun Figure 1: On the left, an example of a Markov blanket of variable X is shown. The members of the blanket are shown shaded. On the right, an example reconstruction of a 5 x 5 rectangular net of branching factor 3 by the algorithm presented in this paper using 20000 samples. Indicated by dotted lines are 3 directionality errors. as for restricted classes such as trees [Chow68] and poly trees [Rebane87]. The second approach is concerned more with data prediction, disregarding independencies in the data. It is typically identified with a greedy hill-climbing or best-first beam search in the space of legal structures, employing as a scoring function a form of data likelihood, sometimes penalized for network complexity. The result is a local maximum score network structure for representing the data, and is one of the more popular techniques used today. This paper presents an approach that belongs in the first category. It addresses the two main shortcomings of the prior work which, we believe, are preventing its use from becoming more widespread. These two disadvantages are: exponential execution times, and proneness to errors in dependence tests used. The former problem is addressed in this paper in two ways. One is by identifying the local neighborhood of each variable in the Bayesian net as a preprocessing step, in order to facilitate the recovery of the local structure around each variable in polynomial time under the assumption of bounded neighborhood size. The second, randomized version goes one step further, employing a user-specified number of randomized tests (constant or logarithmic) in order to ascertain the same result with high probability. The second disadvantage of this research approach, namely proneness to errors, is also addressed by the randomized version, by using multiple data sets (if available) and Bayesian accumulation of evidence. 2 The Grow-Shrink Markov Blanket Algorithm The concept of the Markov blanket of a variable or a set of variables is central to this paper. The concept itself is not new. For example, see [PearI88]. It is surprising, however, how little attention it has attracted for all its being a fundamental property of a Bayesian net. What is new in this paper is the introduction of the explicit use of this idea to effectively limit unnecessary computation, as well as a simple algorithm to compute it. The definition of a Markov blanket is as follows: denoting V as the set of variables and X HS Y as the conditional dependence of X and Y given the set S, the Markov blanket BL(X) ~ V of X E V is any set of variables such that for any Y E V - BL(X) - {X}, X ft-BL(x) Y. In other words, BL(X) completely shields variable X from any other variable in V . The notion of a minimal Markov blanket, called a Markov boundary, is also introduced in [PearI88] and its uniqueness shown under certain conditions. The Markov boundary is not unique in certain pathological situations, such as the equality of two variables. In our following discussion we will assume that the conditions necessary for its existence and uniqueness are satisfied and we will identify the Markov blanket with the Markov boundary, using the notation B (X) for the blanket of variable X from now on. It is also illuminating to mention that, in the Bayesian net framework, the Markov blanket of a node X is easily identifiable from the graph: it consists of all parents, children and parents of children of X. An example Markov blanket is shown in Fig. 1. Note that any of these nodes, say Y, is dependent with X given B (X) - {Y}. 507 Bayesian Network Induction via Local Neighborhoods 1. S t- 0. 2. While:3 Y E V - {X} such that Y HS X, do S t- S U {Y}. 3. While:3 YES such that Y ft-S-{Y} X, do S t- S - {Y}. [Growing phase] [Shrinking phase] 4. B(X) t- S. Figure 2: The basic Markov blanket algorithm. The algorithm for the recovery of the Markov blanket of X is shown in Fig. 2. The idea behind step 2 is simple: as long as the Markov blanket property of X is violated (ie. there exists a variable in V that is dependent on X), we add it to the current set S until there are no more such variables. In this process however, there may be some variables that were added to S that were really outside the blanket. Such variables would have been rendered independent from X at a later point when "intervening" nodes of the underlying Bayesian net were added to S. This observation necessitates step 3, which identifies and removes those variables. The algorithm is efficient, requiring only O( n) conditional tests, making its running time O(n IDI), where n = IVI and D is the set of examples. For a detailed derivation of this bound as well as a formal proof of correctness, see [Margaritis99]. In practice one may try to minimize the number of tests in step 3 by heuristically ordering the variables in the loop of step 2, for example by ascending mutual information or probability of dependence between X and Y (as computed using the X2 test, see section 5). 3 Grow-Shrink (GS) Algorithm for Bayesian Net Induction The recovery of the local structure around each node is greatly facilitated by the knowledge of the nodes' Markov blankets. What would normally be a daunting task of employing dependence tests conditioned on an exponential number of subsets of large sets of variables-even though most of their members may be irrelevant-can now be focused on the Markov blankets of the nodes involved, making structure discovery much faster and more reliable. We present below the plain version of the GS algorithm that utilizes blanket information for inducing the structure of a Bayesian net. At a later point of this paper, we will present a robust, randomized version that has the potential of being faster and more reliable, as well as being able to operate in an "anytime" manner. In the following N (X) represents the direct neighbors of X. [ Compute Markov Blankets ] For all X E V, compute the Markov blanket B (X) . [ Compute Graph Structure] For all X E V and Y E B(X), determine Y to be a direct neighbor of X if X and Y are dependent given S for all S ~ T, where T is the smaller of B (X) - {Y} and B(Y) - {X}. [Orient Edges] For all X E V and YEN (X), orient Y -+ X if there exists a variable Z E N (X) - N (Y) - {Y} such that Y and Z are dependent given S U {X} for all S ~ U, where U is the smaller of B (Y) - {Z} and B (Z) - {Y}. [ Remove Cycles] Do the following while there exist cycles in the graph: 1. Compute the set of edges C = {X -+ Y such that X -+ Y is part of a cycle}. 2. Remove the edge in C that is part of the greatest number of cycles, and put it in R. 508 D. Margaritis and S. Thrun [ Reverse Edges] Insert each edge from R in the graph, reversed. [ Propagate Directions] For all X E V and Y E N(X) such that neither Y ~ X nor X ~ Y, execute the following rule until it no longer applies: If there exists a directed path from X to Y, orient X ~ Y . In the algorithm description above, step 2 determines which of the members of the blanket of each node are actually direct neighbors (parents and children). Assuming, without loss of generality, that B (X) - {Y} is the smaller set, if any of the tests are successful in separating (making independent) X from Y, the algorithm determines that there is no direct connection between them. That would happen when the conditioning set S includes all parents of X and no common children of X and Y. It is interesting to note that the motivation behind selecting the smaller set to condition on stems not only from computational efficiency but from reliability as well: a conditioning set S causes the data set to be split into 21 S 1 partitions; smaller conditioning sets cause the data set to be split into larger partitions and make dependence tests more reliable. Step 3 exploits the fact that two variables that have a common descendant become dependent when conditioning on a set that includes any such descendant. Since the direct neighbors of X and Y are known from step 2, we can determine whether a direct neighbor Y is a parent of X if there exists another node Z (which, coincidentally, is also a parent) such that any attempt to separate Y and Z by conditioning on a subset of the blanket of Y that includes X, fails (assuming that B(Y) is smaller than B(Z)). If the directionality is indeed Y ~ X ~ Z, there should be no such subset since, by conditioning on X, a permanent dependency path between Y and Z is created. This would not be the case if Y were a child of X. It is straightforward to show that the algorithm requires 0 (n 2 + nb2 2 b ) conditional independence tests, where b maxx(IB(X)I). Under the assumption that b is bounded by a constant, this algorithm is O( n 2 ) in the number of conditional independence tests. It is worthwhile to note that the time to compute a conditional independence test by a pass over the data set Dis O( n IDt) and not O(2IVI). An analysis and a formal proof of correctness of the algorithm is presented in [Margaritis99]. = Discussion The main advantage of the algorithm comes through the use of Markov blankets to restrict the size of the conditioning sets. The Markov blankets may be usually wrong in the side of including too many nodes because they are represented by a disjunction of tests for all values of the conditioning set, on the same data. This emphasizes the importance of the "direct neighbors" step which removes nodes that were incorrectly added during the Markov blanket computation step by admitting variables whose dependence was shown high confidence in a large number of different tests. It is also possible that an edge direction is wrongly determined during step 3 due to nonrepresentative or noisy data. This may lead to directed cycles in the resulting graph. It is therefore necessary to remove those cycles by identifying the minimum set of edges than need to be reversed for all cycles to disappear. This problem is closely related [Margaritis99] to the Minimum Feedback Arc Set problem, which is concerned with identifying a minimum set of edges that need to be removed from a graph that possibly contains directed cycles, in order for all such cycles to disappear. Unfortunately, this problem is NP-complete in its generality [Junger85]. We introduce here a reasonable heuristic for its solution that is based on the number of cycles that an edge that is part of a cycle is involved in. Not all edge directions can be determined during the last two steps. For example, nodes with a single parent or multi-parent nodes (called colliders) whose parents are directly connected do not apply to step 3, and steps 4 and 5 are only concerned with already directed edges. Step 6 attempts to ameliorate that, through orienting edges in a way that does not introduce 509 Bayesian Network Induction via Local Neighborhoods a cycle, if the reverse direction necessarily does. It is not obvious that, for example, if the direction X -t Y produces a cycle in an otherwise acyclic graph, the opposite direction Y -t X will not also. However, this is the case. For the proof of this, see [Margaritis99]. The algorithm is similar to the SGS algorithm presented in [Spirtes93], but differs in a number of ways. Its main difference lies in the use of Markov blankets to dramatically improve performance (in many cases where the bounded blanket size assumptions hold) . Its structure is similar to SGS, and the stability (frequently referred to as robustness in the following discussion) arguments presented in [Spirtes93] apply. Increased reliability stems from the use of smaller conditioning sets, leading to greater number of examples per test. The PC algorithm, also in [Spirtes93], differs from the GS algorithm in that it involves linear probing for a separator set, which makes it unnecessarily inefficient. 4 Randomized Version of the GS Algorithm The GS algorithm, as presented above, is appropriate for situations where the maximum Markov blanket of each of a set of variables is small. While it is reasonable to assume that in many real-life problems where high-level variables are involved this may be the case, other problems such as Bayesian image retrieval in computer vision, may employ finer representations. In these cases the variables used may depend in a direct manner on many others. For example, we may choose to use variables to characterize local texture in different parts of an image. If the resolution of the mapping from textures to variables is increasingly fine, direct dependencies among those variables may be plentiful and therefore the maximum Markov blanket size may be significant. Another problem that has plagued independence-test based algorithms for Bayesian net structure induction in general is that their decisions are based on a single or a few tests ("hard" decisions), making them prone to errors due to noise in the data. This also applies to the the GS algorithm. It would therefore be advantageous to employ multiple tests before deciding on a direct neighbor or the direction of an edge. The randomized version of the GS algorithm addresses these two problems. Both of them are tackled through randomized testing and Bayesian evidence accumulation. The problem of exponential running times in the maximum blanket size of steps 2 and 3 of the plain algorithm is overcome by replacing them by a series of tests, whose number may be specified by the user, with the members of the conditioning set chosen randomly from the smallest blanket of the two variables. Each such test provides evidence for or against the direct connection between the two variables, appropriately weighted by the probability that circumstances causing that event occur or not, and due to the fact that connectedness is the conjunction of more elementary events. This version of the algorithm is not shown here in detail due to space restrictions. Its operation follows closely the one of the plain GS version. The main difference lies in the usage of Bayesian updating of the posterior probability of a direct link (or a dependence through a collider) between a pair of variables X and Y using conditional dependence tests that take into account independent evidence. The posterior probability Pi of a link between X and Y after executing i dependence tests dj, j = 1, .. . , i is Pi= Pi-ldi Pi-d(G ------------~---------- Pi-ldi + (1 - +1- dd where G == G(X, Y) = 1 - (4)ITI is a factor that takes values in the interval [0,1) and can be interpreted as the "(un)importance" of the truth of each test di , while T is the smaller of B(X) - {Y} and B(Y) - {X}. We can use this accumulated evidence to guide our decisions to the hypothesis that we feel most confident about. Besides being able to do that in a timely manner due to the user-specified number of tests, we also note how this approach also addresses the robustness problem mentioned above through the use of mUltiple weighted tests, and leaving for the end the "hard" decisions that involve a threshold (ie. comparing the posterior probability with a threshold, which in our case is ~) . D. Margaritis and S. Thrun 510 Kl-divergance verSUS number of samples 0.00015 r--~--C...-~--~;;::Pla-:-in-::G::::SCN:-_ -_--, Randomized GS8N ....- .. Hill-Clil"l"tling. score' data likelihood .. ? ,. Hill-Glirnblng, soore: BIC - .Q--- 00001 5e-05 BOOO 12000 Nurrber of sarrples 4000 __- __ l00r-----------::p~lai~nG~Sr-BN~-~ -~ PlainGSBN Randomized GSBN .... ~ ... Hill-Clirrtling, score data likelihood 125 I 100 ~ 75 .s Hill-Climbing, soore: BIC - !w ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . III- ...... . . ...... . . . . SO Randomized GSBN .... ~... Hill-Clirrbi~h~~i~~~~h~ .~_ . ,.. B - ~.~ GO 20000 Direction errors versus number of sarrplss Edge errors versus number of sarrples 1~r---_--_- 16000 1? 75 ~ 50 i 25 ~ S i5 25 o OL---~--~--~---~-~ o 4000 8000 12000 Nurrber of sarrples 18000 20000 0 4000 8000 12000 18000 20000 Number of 5a!1l)les Figure 3: Results for a 5 x 5 rectangular net with branching factor 2 (in both directions, blanket size 8) as a function of the number of samples. On the top, KL-divergence is depicted for the plain GS , randomized GS, and hill-climbing algorithms. On the bottom, the percentage of edge and direction errors are shown. Note that certain edge error rates for the hill-climbing algorithm exceed 100%. 5 Results Throughout the algorithms presented in this paper we employ standard chi-square (X 2 ) conditional dependence tests (as is done also in [Spirtes93]) in order to compare the histograms P(X) and P(X I Y). The X2 test gives us a probability of the error of assuming that the two variables are dependent when in fact they are not (type II error of a dependence test), from which we can easily derive the probability that X and Y are dependent. There is an implicit confidence threshold T involved in each dependence test, indicating how certain we wish to be about the correctness of the test without unduly rejecting dependent pairs, something that is always possible in reality due to the presence of noise. In all experiments we used T = 0.95, which corresponds to a 95% confidence test. We test the effectiveness of the algorithms through the following procedure: we generate a random rectangular net of specified dimensions and up/down branching factor. A number of examples are drawn from that net using logic sampling and they are used as input to the algorithm under test. The resulting nets can be compared with the original ones along dimensions of KL-divergence and difference in edges and edge directionality. The KLdivergence was estimated using a Monte Carlo procedure. An example reconstruction was shown in the beginning of the paper, Fig. 1. Fig. 3 shows how the KL-divergence between the original and the reconstructed net as well as edge omissions/false additions/reversals as a function of number of samples used. It demonstrates two facts. First, that typical KL-divergence for both GS and hill-climbing algorithms is low (with hill-climbing slightly lower), which shows good performance for applications where prediction is of prime concern. Second, the number of incorrect edges and the errors in the directionality of the edges present is much higher for the hill-climbing algorithm, making it unsuitable for accurate Bayesian net reconstruction. Fig. 4 shows the effects of increasing the Markov blanket through an increasing branching factor. As expected, we see a dramatic (exponential) increase in execution time of the plain Bayesian Network Induction via Local Neighborhoods 511 Edge I Direction Errors versus Branching Factor Execution Time versus Branching Factor 100r-------~~~----~~~----, Edge errors, plain GSBN ~ Edge errors, randomized GSBN ---~--Direction errors, plain GSBN Direction errors, randomized GSBN - &--- 90 80 70 ~ 60 ~ 50 ~ 40 i= 30 ___ __ ___ _ __ _...___ ___ ___ .... ___ __ _ 2?L==~==~~?----=----=-----=----~----------- ----l 10 -.,,""'-.. -----...... .. _. ? _ _" _______ .olII ___ ____ __ ? ___ O~------~--------~--------~ 2 3 4 Branching Factor 5 22000 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 Plain GSBN - Randomized GSBN ----K---- .----- O~------~--------~------~ 2 3 4 Branching Factor 5 Figure 4: Results for a 5 x 5 rectangular net from which 10000 samples were generated and used for reconstruction, versus increasing branching factor. On the left, errors are slowly increasing as expected, but comparable for the plain and randomized versions of the GS algorithm. On the right, corresponding execution times are shown. GS algorithm, though only a mild increase of the randomized version. The latter uses 200 (constant) conditional tests per decision, and its execution time increase can be attributed to the (quadratic) increase in the number of decisions. Note that the error percentages between the plain and the randomized version remain relatively close. The number of direction errors for the GS algorithm actually decreases due to the larger number of parents for each node (more "V" structures), which allows a greater number of opportunities to recover the directionality of an edge (using an increased number of tests). 6 Discussion In this paper we presented an efficient algorithm for computing the Markov blanket of a node and then used it in the two versions of the GS algorithm (plain and randomized) by exploiting the properties of the Markov blanket to facilitate fast reconstruction of the local neighborhood around each node, under assumptions of bounded neighborhood size. We also presented a randomized variant that has the advantages of faster execution speeds and added reconstruction robustness due to multiple tests and Bayesian accumulation of evidence. Simulation results demonstrate the reconstruction accuracy advantages of the algorithms presented here over hill-climbing methods. Additional results also show that the randomized version has a dramatical execution speed benefit over the plain one in cases where the assumption of bounded neighborhood does not hold, without significantly affecting the reconstruction error rate. References [Chow68] [Herskovits90] [Spirtes93] [PearI88] [Rebane87] [Verma90] [Agosta88] [Cheng97] [Margaritis99] [Jtinger85] C.K. Chow and C.N. Liu. Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, 14, 1968. E.H. Herskovits and G.F. Cooper. Kutat6: An entropy-driven system for construction of probabilistic expert systems from databases. VAI-90. P. Spirtes, C. Glymour, and R. Scheines. Causation, Prediction, and Search, Springer, 1993. 1. Pearl. Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, 1988. G. Rebane and J. Pearl. The recovery of causal poly-trees from statistical data. VAI-87. T.S. Verma, and J. Pearl. Equivalence and Synthesis of Causal Models. VAI-90. J.M. Agosta. The structure of Bayes networks for visual recognition. VAI-88. 1. Cheng, D.A. Bell, W. Liu, An algorithm for Bayesian network construction from data. AI and Statistics, 1997. D. Marg aritis , S. Thrun, Bayesian Network Induction via Local Neighborhoods. TR CMV-CS-99-134, forthcoming. M. Junger, Polyhedral combinatorics and the acyclic subdigraph problem, Heldermann, 1985. 

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Audio-Vision: Using Audio-Visual Synchrony to Locate Sounds John Hershey .. Javier Movellan jhershey~cogsci.ucsd.edu movellan~cogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Abstract Psychophysical and physiological evidence shows that sound localization of acoustic signals is strongly influenced by their synchrony with visual signals. This effect, known as ventriloquism, is at work when sound coming from the side of a TV set feels as if it were coming from the mouth of the actors. The ventriloquism effect suggests that there is important information about sound location encoded in the synchrony between the audio and video signals. In spite of this evidence, audiovisual synchrony is rarely used as a source of information in computer vision tasks. In this paper we explore the use of audio visual synchrony to locate sound sources. We developed a system that searches for regions of the visual landscape that correlate highly with the acoustic signals and tags them as likely to contain an acoustic source. We discuss our experience implementing the system, present results on a speaker localization task and discuss potential applications of the approach. Introd uction We present a method for locating sound sources by sampling regions of an image that correlate in time with the auditory signal. Our approach is inspired by psychophysical and physiological evidence suggesting that audio-visual contingencies play an important role in the localization of sound sources: sounds seem to emanate from visual stimuli that are synchronized with the sound. This effect becomes particularly noticeable when the perceived source of the sound is known to be false, as in the case of a ventriloquist's dummy, or a television screen. This phenomenon is known in the psychophysical community as the ventriloquism effect, defined as a mislocation of sounds toward their apparent visual source. The effect is robust in a wide variety of conditions, and has been found to be strongly dependent on the degree of "synchrony" between the auditory and visual signals (Driver, 1996; Bertelson, Vroomen, Wiegeraad & de Gelder, 1994). "1'0 whom correspondence should be addressed. 814 J. Hershey and J. R. Movellan The ventriloquism effect is in fact less speech-specific than first thought. For example the effect is not disrupted by an upside-down lip signal (Bertelson, Vroomen, Wiegeraad & de Gelder, 1994) and is just as strong when the lip signals are replaced by light flashes that are synchronized with amplitude peaks in the audio signal (Radeau & Bertelson, 1977). The crucial aspect here is correlation between visual and auditory intensity over time. When the light flashes are not synchronized the effect disappears. The ventriloquism effect is strong enough to produce an enduring localization bias, known as the ventriloquism aftereffect. Over time, experience with spatially offset auditory-visual stimuli causes a persistent shift in subsequent auditory localization. Exposure to audio-visual stimuli offset from each other by only 8 degrees of azimuth for 20-30 minutes is sufficient to shift auditory localization by the same amount. A corresponding shift in neural processing has been detected in macaque monkeys as early as primary auditory cortex(Recanzone, 1998). In barn owls a misalignment of visual and auditory stimuli during development causes the realignment of the auditory and visual maps in the optic tectum(Zheng & Knudsen, 1999; Stryker, 1999; Feldman & Knudsen, 1997). The strength of the psychophysical and physiological evidence suggests that audiovisual contingency may be used as an important source of information that is currently underutilized in computer vision tasks. Visual and auditory sensor systems carry information about the same events in the world, and this information must be combined correctly in order for a useful interaction of the two modalities. Audiovisual contingency can be exploited to help determine which signals in different modalities share a common origin. The benefits are two-fold: the two signals can help localize each other, and once paired can help interpret each other. To this effect we developed a system to localize speakers using input from a camera and a single microphone. The approach is based on searching for regions of the image which are "synchronized" with the acoustic signal. Measuring Synchrony The concept of audio-visual synchrony is not well formalized in the psychophysical literature, so for a working definition we interpret synchrony as the degree of mutual information between audio and spatially localized video signals. Ultimately it is a causal relationship that we are often interested in, but causes 'can only be inferred from effects such as synchrony. Let a(t) E IRn be a vector describing the acoustic signal at time t. The components of a(t) could be cepstral coefficients, pitch measurements, or the outputs of a filter bank. Let v(x, y, t) E IRm be a vector describing the visual signal at time t, pixel (x,y). The components ofv(x,y,t) could represent Gabor energy coefficients, RGB color values, etc. Consider now a set of s audio and visual vectors S = (a(tl), v(x, y, tl?l=k-s-l,. .. ,k sampled at times tk-s-l,'" ,tk and at spatial coordinates (x, y). Given this set of vectors our goal is to provide a number that describes the temporal contingency between audio and video at time tk' The approach we take is to consider each vector in S as an independent sample from a joint multivariate Gaussian process (A(tk), V(x, y, tk? and define audio-visual synchrony at time tk as the estimate of the mutual information between the audio and visual components of the process. Let A(tk) ,..., Nn(ltA(tk), ~A(tk?' and V(x,y, tk) ,..., Nm(ltv(x, y, t), ~v(x,y, tk)), where It represents means and ~ covariance matrices. Let A(tk) and V(x, y, tk) be jointly Gaussian, i.e., (A(tk), V(x, y, tk? ,..., Nn+m(ltA,V (x, y, tk), ~A,V(X, y, tk)' 815 Audio Vision: Using Audio-Vzsual Synchrony to Locate Sounds The mutual information between A(x, y, tk) and V(tk) can be shown to be as follows [(A(tk); V(x, y, tk)) = H(A(tk)) + H(V(x, y, tk)) 1 "2log(27re)nIEA(tk)1 - H(A(tk), V(x, y, tk)) 1 + "2log(27re)mIEv(x, y, tk)1 (1) 1 -"2log(27re)n+mIEA ,v(x,y, tk)1 (2) IEA(tk)IIEv(x,y,tk)1 -11og "'-----:-::::---'--'-'--;----'-----'-::-:--':"':' 2 IEA,V(X,y,tk)I' In the special case that n (3) = m = 1, then (4) where p(x, y, tk) is the Pearson correlation coefficient between A(tk) and V(x, y, tk)' For each triple (x, y, tk) we estimate the mutual information between A( tk) and V(x, y, tk) by considering each element of S as an independent sample from the random vector (A(tk), V(x, y, tk))' This amounts to computing estimates of the joint covariance matrix EA,V (x, y, tk). For example the estimate of the covariance between the ith audio component and the jLh video component would be as follows 1 SAi,v; (x, y, tk) 8-1 = s _ 1 I)ai(tk-l) - ai(tk))(Vj(X,y, tk-l) - Vj (x, y, tk)), (5) 1=0 where (6) (7) (8) These simple covariance estimates can be computed recursively in constant time with respect to the number of timepoints. The independent treatment of pixels would lend well to a parallel implementation. To measure performance, a secondary system produces a single estimate of the auditory location, for use with a database of labeled solitary audiovisual sources. Unfortunately there are many ways of producing such estimates so it becomes difficult to separate performance of the measure from the underlying system. The model used here is a centroid computation on the mutual information estimates, with some enhancements to aid tracking and reduce background noise. Implementation Issues A real time system was prototyped using a QuickCam on the Linux operating system and then ported to NT as a DirectShow filter. l'his platform provides input from real-time audio and video capture hardware as well as from static movie files. The video output could also be rendered live or compressed and saved in a movie file. The implementation was challenging in that it turns out to be rather difficult 816 J. Hershey and J. R. Movellan -2 -'.'----f:.20--"~--:':"--""=:----:':''':--~'20 F,_ (a) M is talking. ~.~~'~.~20~-~"=:---"~~~-~"-=~-~"~"? F". . . (b) J is talking. Figure 1: Normalized audio and visual intensity across sequences of frames in which a sequence of four numbers is spoken. The top trace is the contour of the acoustic energy from one of two speakers, M or J, and the bottom trace is the contour of intensity values for a single pixel, (147,100), near the mouth of J. to process precisely time-synchronized audio and video on a serial machine in real time. Multiple threads are required to read from the peripheral audio and visual devices. By the time the audio and visual streams reach the AV filter module, they are quite separate and asynchronous. The separately threaded auditory and visual packet streams must be synchronized, buffered, and finally matched and aligned by time-stamps before they can finally be processed. It is interesting that successful biologial audiovisual systems employ a parallel architecture and thus avoid this problem. Results To obtain a performance baseline we first tried the simplest possible approach: A single audio and visual feature per location: n = m = 1, v(x, y, t) E IR is the intensity of pixel (x, y) at time t, and a(t) E IR is the average acoustic energy over the interval [t - 6.t, tJ, where 6.t = 1/30 msec , the sampling period for the NTSC video signal. Figure 1 illustrates the time course of these signals for a non-synchronous and a synchronous pair of acoustic energy and pixel intensity. Notice in particular that in the synchonous pair, 1(b), where the sound and pixel values come from the same speaker, the relationship between the signals changes over time. There are regions of positive and negative covariance strung together in succession. Clearly the relationship over the entire sequence is far from linear. However over shorter time periods a linear relationship looks like a better approximation. Our window size of 16 samples (Le., s = 16 in 5 coincides approximately with this time-scale. Perhaps by averaging over many small windows we can capture on a larger scale what would be lost to the same method applied with a larger window. Of course there is a trade-off in the time-scale between sensitivity to spurious transients, and the response time of the system. We applied this mutual information measure to all the pixels in a movie, in the spirit of the perceptual maps of the brain. The result is a changing topographic map of audiovisual mutual information. Figure 2 illustrates two snapshots in which 817 Audio Vision: Using Audio-Visual Synchrony to Locate Sounds (a) Frame 206: M (at left) is talking. (b) Frame 104: J (at right) is talking. Figure 2: Estimated mutual information between pixel intensity and audio intensity (bright areas indicate greater mutual information) overlaid on stills from the video where one person is in mid-utterance. different parts of the face are synchronous (possibly with different sign) with the sound they take part in producing. It is interesting that the synchrony is shared by some parts, such as the eyes, that do not directly contribute to the sound, but contribute to the communication nonetheless. ___ To estimate the position of the speaker we computed a centroid were each point was weighted by the estimated mutual information between the correpsonding pixel and the audio signal. At each time step the mutual information was estimated using 16 past frames (Le., s = 16) In order to reduce the intrusion of spurious correlations from competing targets, once a target has been found, we employ a Gaussian influence function. (Goodall, 1983) The influence function reduces the weight given to mutual information from locations far from the current centroid when computing the next centroid. To allow for the speedy disengagement from a dwindling source of mutual information we set a threshold on the mutual information. Measurements under the threshold are treated as zero. This threshold also reduces the effects of unwanted background noise, such as camera and microphone jitter. A Sx(t) = Lx L x 8(1og(1 - f} (x , y, t)))'I/;(X, Sx(t - 1)) Y A Lx L y 8(log(1- p2(X,y,t)))'I/;(x, Sx(t -1)) (9) where Sx(t) represents the estimate of the x coordinate for the position of the speaker at time t. 8(.) is the thresholding function, and 'I/;(x, Sx(t - 1)) is the influence function , which depends upon the 'position x of the pixel being sampled and the prior estimate Sx(t-1). p2(X, y, t) is the estimate of the correlation between the intensity in pixel (x , y) and the acoustic enery, when using the 16 past video frames. -~ log(l- p2(x, y, t)) is the corresponding estimate of mutual information (the factor, -~ cancels out in the quotient after adjusting the threshold function accordingly. ) We tried the approach on a movie of two people (M and J) taking turns while saying random digits. Figure 3 shows the estimates of the actual positions of the speaker J. Hershey and J. R. Movellan 818 as a function of time. The estimates clearly provide information that could be used to localize the speaker, especially in combination with other approaches (e.g., flesh detection) . 180 180 ~ ~ 140 ~ ~ 120 i ~ 100 ~ !_ 80 40 20~----~----~----~----~----~------~--~ o 100 200 300 400 500 600 700 Frame Number Figure 3: Estimated and actual position of speaker at each frame for six hundred frames. The sources, M and J, took turns uttering a series of four digits, for three turns each. The actual positions and alternation times were measured by hand from the video recording Conclusions We have presented exploratory work on a system for localizing sound sources on a video signal by tagging regions of the image that are correlated in time with the auditory signal. The approach was motivated by the wealth of evidence in the psychophysical and physiological literature showing that sound localization is strongly influenced by synchrony with the visual signal. We presented a measure of local synchrony based on modeling the audio-visual signal as a non-stationary Gaussian process. We developed a general software tool that accepts as inputs all major video and audio file formats as well as direct input from a video camera. We tested the tool on a speaker localization task with very encouraging results. The approach could have practical applications for localizing sound sources in situations where where acoustic stereo cues are inexistent or unreliable. For example the approach could be used to help localize the actor talking in a video scene and put closed-captioned text near the audio source. The approach could also be used to guide a camera in teleconferencing applications. While the results reported here are very encouraging, more work needs to be done before practical applications are developed. For example we need to investigate more sophisticated methods for processing the audio and video signals. At this point we use average energy to represent the video and thus changes in the fundamental frequency that do not affect the average energy would not be captured by our model. Similarly local video decompositions, like spatio-temporal Gabor filtering, or approaches designed to enhance the lip regions may be helpful. The Audio Vision: Using Audio-Visual Synchrony to Locate Sounds 819 changing symmetry observed between audio and video signals might be addressed rectifying or squaring the normalized signals and derivatives. Finally, relaxing the Gaussian constraints in our measure of audio-visual contingency may help improve performance. While the work shown here is exploratory at this point, the approach is very promising: It emphasizes the idea of machine perception as a multimodal process it is backed by psychophysical evidence, and when combined with other approaches it may help improve robustness in tasks such as localization and separation of sound sources. References Bertelson, P., Vroomen, J., Wiegeraad, G., & de Gelder, B. (1994). Exploring the relation between McGurk interference and ventriloquism. In Proceedings of the 1994 International Conference on Spoken Language Processing, volume 2, pages 559-562. Driver, J. (1996). Enhancement of selective listening by illusory mislocation of speech sounds due to lip-reading. Nature, 381, 66-68. Feldman, D. E . & Knudsen, E. I. (1997). An anatomical basis for visual calibration of the auditiory space map in the barn owl's midbrain. The Journal of Neuroscience, 17(17), 6820-6837. Goodall, C. (1983). M-Estimators of Location: an outline of the theory. Wiley series in probability and mathematical statistics. Applied probability and statistics. Radeau, M. & Bertelson, P. (1977). Adaptation to auditory-visual discordance and ventriloquism in semi-realistic situations. Perception and Psychophysics, 22, 137-146. Recanzone, G. H. (1998). Rapidly induced auditory plasticity: The ventriloquism aftereffect. Proceedings of the National Academy of Sciences, USA, 95, 869- 875. Stryker, M. P. (1999) . Sensory Maps on the Move. Science, 925-926. Zheng, W. & Knudsen, E. I. (1999). Functional Selection of Adaptive Auditory Space Map by GABAA-Mediated Inhibition, 962-965. 

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A Geometric Interpretation of v-SVM Classifiers David J. Crisp Centre for Sensor Signal and Information Processing, Deptartment of Electrical Engineering, University of Adelaide, South Australia Christopher J.C. Burges Advanced Technologies, Bell Laboratories, Lucent Technologies Holmdel, New Jersey dcrisp@eleceng.adelaide.edu.au burges@lucent.com Abstract We show that the recently proposed variant of the Support Vector machine (SVM) algorithm, known as v-SVM, can be interpreted as a maximal separation between subsets of the convex hulls of the data, which we call soft convex hulls. The soft convex hulls are controlled by choice of the parameter v. If the intersection of the convex hulls is empty, the hyperplane is positioned halfway between them such that the distance between convex hulls, measured along the normal, is maximized; and if it is not, the hyperplane's normal is similarly determined by the soft convex hulls, but its position (perpendicular distance from the origin) is adjusted to minimize the error sum. The proposed geometric interpretation of v-SVM also leads to necessary and sufficient conditions for the existence of a choice of v for which the v-SVM solution is nontrivial. 1 Introduction Recently, SchOlkopf et al. [I) introduced a new class of SVM algorithms, called v-SVM, for both regression estimation and pattern recognition. The basic idea is to remove the user-chosen error penalty factor C that appears in SVM algorithms by introducing a new variable p which, in the pattern recognition case, adds another degree of freedom to the margin. For a given normal to the separating hyperplane, the size of the margin increases linearly with p. It turns out that by adding p to the primal objective function with coefficient -v, v 2: 0, the variable C can be absorbed, and the behaviour of the resulting SVM - the number of margin errors and number of support vectors - can to some extent be controlled by setting v. Moreover, the decision function produced by v-SVM can also be produced by the original SVM algorithm with a suitable choice of C. In this paper we show that v-SVM, for the pattern recognition case, has a clear geometric interpretation, which also leads to necessary and sufficient conditions for the existence of a nontrivial solution to the v-SVM problem. All our considerations apply to feature space, after the mapping of the data induced by some kernel. We adopt the usual notation: w is the normal to the separating hyperplane, the mapped 245 A Geometric Interpretation ofv-SVM Classifiers data is denoted by Xi E !RN , i = 1, ... ,1, with corresponding labels are scalars, and ~i' i = 1", ,,1 are positive scalar slack variables. 2 Yi E {?1}, b, p v-SVM Classifiers The v-SVM formulation, as given in [1], is as follows: minimize pI = 1 211w/112 - Vp' 1 + y l:~~ (1) i with respect to w', b' , p', ~i, subject to: Yi(W' . Xi + b/ ) ~ p' - ~~, ~i ~ 0, p' ~ o. (2) Here v is a user-chosen parameter between 0 and 1. The decision function (whose sign determines the label given to a test point x) is then: l' (x) = w' . x + b' . The Wolfe dual of this problem is: maximize to Ph (3) = -~ 2:ij OiOjYiYjXi . Xj subject (4) with w' given by w' = 2:i 0iYiXi . SchOlkopf et al. [1] show that v is an upper bound on the fraction of margin errors 1 , a lower bound on the fraction of support vectors, and that both of these quantities approach v asymptotically. Note that the point w' = b' = p = ~i = 0 is feasible, and that at this point, pI = O. Thus any solution of interest must have pI ::; O. Furthermore, if Vp' = 0, the optimal solution is at w' = b' = p = ~i = 02 ? Thus we can assume that v p' > 0 (and therefore v > 0) always. Given this, the constraint p' ~ 0 is in fact redundant: a negative value of p' cannot appear in a solution (to the problem with this constraint removed) since the above (feasible) solution (with p' = 0) gives a lower value for P'. Thus below we replace the constraints (2) by (5) 2.1 A Reparameterization of v-SVM We reparameterize the primal problem by dividing the objective function pI by v 2 /2, the constraints (5) by v, and by making the following substitutions: 2 w' b' p' I-' = -, w = - , b = -, p = -, ~i vl v v v ~i = -. v (6) 1 A margin error Xi is defined to be any point for which ?i > 0 (see [1]). 2In fact we can prove that, even if the optimal solution is not unique, the global solutions still all have w = 0: see Burges and Crisp, "Uniqueness of the SYM Solution" in this volume. 246 D. J. Crisp and C. J. C. Burges This gives the equivalent formulation: minimize (7) with respect to w, b, p, ~i' subject to: (8) = IT we use as decision function f(x) f'(x)/v, the formulation is exactly equivalent, although both primal and dual appear different. The dual problem is now: minimize (9) with respect to the ai, subject to: (10) with w given by w = 1 2: i aiYiXi. In the following, we will refer to the reparameterized version of v-StrM given above as J.'-SVM, although we emphasize that it describes the same problem. 3 A Geometric Interpretation of l/-SVM In the separable case, it is clear that the optimal separating hyperplane is just that hyperplane which bisects the shortest vector joining the convex hulls of the positive and negative polarity points 3 ? We now show that this geometric interpretation can be extended to the case of v-SVM for both separable and nonseparable cases. 3.1 The Separable Case We start by giving the analysis for the separable case. The convex hulls of the two classes are (11) and (12) Finding the two closest points can be written as the following optimization problem: min CIt (13) 3S ee, for example, K. Bennett, 1997, in http://www.rpi.edu/bennek/svmtalk.ps (also, to appear). 247 A Geometric Interpretation of v-SVM Classifiers subject to: L ai L = 1, ai = 1, a t' > _ 0 (14) i:y;=-l i:y;=+l Taking the decision boundary j(x) = w? x + b = 0 to be the perpendicular bisector of the line segment joining the two closest points means that at the solution, (15) and b= -w? p, where (16) Thus w lies along the line segment (and is half its size) and p is the midpoint of the line segment. By rescaling the objective function and using the class labels Yi = ?1 we can rewrite this as 4 : (17) subject to (18) The associated decision function is j( x) = w . x + b where w p = ~ L:i aiXi and b = -w.p = L:ij aiYiajXi . Xj. -t 3.2 = ~ L:i aiYiXi, The Connection with v-SVM Consider now the two sets of points defined by: H+ JJ = { '. ~ I.y;-+l aiXil .. ~ I.y.-+l ai = 1, 0 ~ ai ~ fL} (19) and (20) We have the following simple proposition: Proposition 1: H+ JJ C H+ and H-JJ C H_, and H+ JJ and H-JJ are both convex sets. Furthermore, the positions of the points H+ JJ and H-JJ with respect to the Xi do not depend on the choice of origin. Proof: Clearly, since the ai defined in H+ JJ is a subset of the ai defined in H+, H+ JJ C H+, similarly for H_. Now consider two points in H+ JJ defined by aI, a2. Then all points on the line joining these two points can be written as L:i:y;=+l ((1A)ali + Aa2i)Xi, 0 ~ A ~ 1. Since ali and a2i both satisfy 0 ~ ai ~ fL, so does (1- A)ali +Aa2i, and since also L:i:y;=+l (1- A)ali+Aa2i = 1, the set H+ JJ is convex. 4That one can rescale the objective function without changing the constraints follows from uniqueness of the solution. See also Burges and Crisp, "Uniqueness of the SVM Solution" in this volume. 248 D. J. Crisp and C. J. C. Burges The argument for H_~ is similar. Finally, suppose that every Xi is translated by Xo, i.e. Xi -+ Xi + Xo 'Vi. Then since L:i:Yi=+l ai = 1, every point in H+~ is also translated by the same amount, similarly for H-w 0 The problem of finding the optimal separating hyperplane between the convex sets H+~ and H_~ then becomes: (21) subject to (22) Since Eqs. (21) and (22) are identical to (9) and (10), we see that the v-SVM algorithm is in fact finding the optimal separating hyperplane between the convex sets H+~ and H-w We note that the convex sets H+~ and H_~ are not simply uniformly scaled versions of H + and H _. An example is shown in Figure 1. xl xl xl 1'=5112 1'=113 ...... '! 1/3 5::: xl xl 113 :"::.~ .' xl 112 116 5112 x2 xl -lo:rrrTTT17TTT17~ --t----"I---+-----. xl 112 Figure 1: The soft convex hull for the vertices of a right isosceles triangle, for various 1'. Note how the shape changes as the set grows and is constrained by the boundaries of the encapsulating convex hull. For I' < ~, the set is empty. Below, we will refer to the formulation given in this section as the soft convex hull formulation, and the sets of points defined in Eqs. (19) and (20) as soft convex hulls. 3.3 Comparing the Offsets and Margin Widths The natural value of the offset bin the soft convex hull approach, b= -w . p, arose by asking that the separating hyperplane lie halfway between the closest extremities of the two soft convex hulls. Different choices of b just amount to hyperplanes with the same normal but at different perpendicular distances from the origin. This value of b will not in general be the same as that for which the cost term in Eq. (7) is minimized. We can compare the two values as follows. The KKT conditions for the J.'-SVM formulation are (I' - ai)~i ai(Yi(w?Xi+b)-p+~i) - 0 0 Multiplying (24) by Yi, summing over i and using (23) gives (23) (24) A Geometric Interpretation ofv-SVM Classifiers 249 (25) Thus the separating hyperplane found in the J.'-SVM algorithm sits a perpendicular distance 12ifiorr l:i Yi~i I away from that found in the soft convex hull formulation. For the given w, this choice of b results in the lowest value of the cost, J.' l:i ~i. The soft convex hull approach suggests taking p = w . w, since this is the value Iii takes at the points l:Yi=+l (XiXi and l:Yi=-l (XiXi. Again, we can use the KKT conditions to compare this with p. Summing (24) over i and using (23) gives p= p+ ~ L~i. (26) i Since p = 3.4 W? w, this again shows that if p = 0 then w = ~i = 0, and, by (25), b = O. The Primal for the Soft Convex Hull Formulation By substituting (25) and (26) into the J.'-SVM primal formulation (7) and (8) we obtain the primal formulation for the soft convex hull problem: minimize (27) with respect to w, b, p, ~i, subject to: Yi (W ? Xi + b-) 2:: p_ - ~i + J.' "~ 1 + 2YiYj ~j, (28) j It is straightforward to check that the dual is exactly (9) and (10). Moreover, by summing the relevant KKT conditions, as above, we see that b = -w?p and p = w?w. Note that in this formulation the variables ~i retain their meaning according to (8). 4 Choosing v In this section we establish some results on the choices for v, using the J.'-SVM formulation. First, note that l:i (XiYi = 0 and l:i (Xi = 2 implies l:i:Yi=+l (Xi = l:i:Yi=-l (Xi = 1. Then (Xi 2:: 0 gives (Xi ~ 1, Vi. Thus choosing J.' > 1, which corresponds to choosing v < 2/1, results in the same solution of the dual (and hence the same normal w) as choosing J.' = 1. (Note that different values of J.' > 1 can still result in different values of the other primal variables, e.g. b). The equalities l:i:Yi=+l (Xi = l:i:y;=-l (Xi = 1 also show that if J.' < 2/1 then the feasible region for the dual is empty and hence the problem is insoluble. This corresponds to the requirement v < 1. However, we can improve upon this. Let 1+ (L) be the number of positive (negative) polarity points, so that 1+ + L = I. Let lmin == min{I+,L}. Then the minimal value of J.' which still results in a nonempty feasible region is J.'min = 1/lmin. This gives the condition v ~ 2Imin /l. We define a "nontrivial" solution of the problem to be any solution with w =I o. The following proposition gives conditions for the existence of nontrivial solutions. D. J. Crisp and C. J. C. Burges 250 Proposition 2: A value of v exists which will result in a nontrivial solution to the v-SVM classification problem if and only if {H+I-' : I-' = I-'min} n {H_I-' : I-' = I-'min} = 0. Proof: Suppose that {H+I-' : I-' = I-'min} n {H_I-' : I-' = I-'min} =1= 0. Then for all allowable values of I-' (and hence v), the two convex hulls will intersect, since {H+I-' : I-' = I-'min} C {H+I-' : I-' ~ I-'min} and {H_I-' : I-' = I-'min} C {H_I-' : I-' ~ I-'min}. IT the two convex hulls intersect, then the solution is trivial, since by definition there then exist feasible points z such that z = Li:Yi=+lOiXi and z = Li:Yi=_lOiXi, and hence 2w = Li 0iYiXi = Li:Yi=+lOiXi - Li:Yi=-l 0iXi = 0 (cf. (21), (22). Now suppose that {H+I-' : I-' = I-'min} n {H_I-' : I-' = I-'min} = 0. Then clearly a nontrivial solution exists, since the shortest distance between the two convex sets {H+1-' : I-' = I-'min} and {H -I-' : I-' = I-'min} is not zero, hence the corresponding w =1= o. 0 Note that when 1+ = L, the condition amounts to the requirement that the centroid of the positive examples does not coincide with that of the negative examples. Note also that this shows that, given a data set, one can find a lower bound on v, by finding the largest I-' that satisfies H_I-' n H+I-' = 0. 5 Discussion The soft convex hull interpretation suggests that an appropriate way to penalize positive polarity errors differently from negative is to replace the sum I-' Li ~i in (7) with 1-'+ Li:Yi=+l ~i + 1-'- Li:Yi=-l ~i? In fact one can go further and introduce a I-' for every train point. The I-'-SVM formulation makes this possibility explicit, which it is not in original v-SVM formulation. Note also that the fact that v-SVM leads to values of b which differ from that which would place the optimal hyperplane halfway between the soft convex hulls suggests that there may be principled methods for choosing the best b for a given problem, other than that dictated by minimizing the sum of the ~i 'so Indeed, originally, the sum of ~i 's term arose in an attempt to approximate the number of errors on the train set [21. The above reasoning in a sense separates the justification for w from that for b. For example, given w, a simple line search could be used to find that value of b which actually does minimize the number of errors on the train set. Other methods (for example, minimizing the estimated Bayes error [3]) may also prove useful. Acknowledgments C. Burges wishes to thank W. Keasler, V. Lawrence and C. Nohl of Lucent Technologies for their support. References [1] B. Scholkopf and A. Smola and R. Williamson and P. Bartlett. New support vector algorithms, neurocolt2 nc2-tr-1998-031. Technical report, GMD First and Australian National University, 1998. [2] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273-297, 1995. [3] C. J. C. Burges and B. SchOlkopf. Improving the accuracy and speed of support vector learning machines. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 375-381, Cambridge, MA, 1997. MIT Press. 

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Graded grammaticality in Prediction Fractal Machines Shan Parfitt, Peter Tiilo and Georg Dorffner Austrian Research Institute for Artificial Intelligence, Schottengasse 3, A-IOIO Vienna, Austria. { shan,petert,georg} @ai. univie. ac. at Abstract We introduce a novel method of constructing language models, which avoids some of the problems associated with recurrent neural networks. The method of creating a Prediction Fractal Machine (PFM) [1] is briefly described and some experiments are presented which demonstrate the suitability of PFMs for language modeling. PFMs distinguish reliably between minimal pairs, and their behavior is consistent with the hypothesis [4] that wellformedness is 'graded' not absolute. A discussion of their potential to offer fresh insights into language acquisition and processing follows. 1 Introduction Cognitive linguistics has seen the development in recent years of two important, related trends. Firstly, a widespread renewal of interest in the statistical, 'graded' nature of language (e.g. [2]-[4]) is showing that the traditional all-or-nothing notion of well-formedness may not present an accurate picture of how the congruity of utterances is represented internally. Secondly, the analysis of state space trajectories in artificial neural networks (ANNs) has provided new insights into the types of processes which may account for the ability of learning devices to acquire and represent language, without appealing to traditional linguistic concepts [5]-[7]. Despite the remarkable advances which have come out of connectionist research (e.g. [8]), and the now common use of recurrent networks, and Simple Recurrent Networks (SRNs) [9] especially, in the study of language (e.g. [10]), recurrent neural networks suffer from particular problems which make them imperfectly suited to language tasks. The vast majority of work in this field employs small networks and datasets (usually artificial), and although many interesting linguistic issues may be thus tackled, real progress in evaluating the potentials of state trajectories and graded 'grammaticality' to uncover the underlying processes responsible for overt linguistic phenomena must inevitably be limited whilst the experimental tasks remain so small. Nevertheless, there are certain obstacles to the scaling-up of networks trained by back-propagation (BP). Such networks tend towards ever Graded Grammaticality in Prediction Fractal Machines 53 longer training times as the sizes of the input set and of the network increase, and although Real-Time Recurrent Learning (RTRL) and Back-propagation Through Time are potentially better at modeling temporal dependencies, training times are longer still [11]. Scaling-up is also difficult due to the potential for catastrophic interference and lack of adaptivity and stability [12]-[14]. Other problems include the rapid loss of information about past events as the distance from the present increases [15] and the dependence of learned state trajectories not only on the training data, but also upon such vagaries as initial weight vectors, making their analysis difficult [16]. Other types of learning device also suffer problems. Standard Markov models require the allocation of memory for every n-gram, such that large values of n are impractical; variable-length Markov models are more memory-efficient, but become unmanageable when trained on large data sets [17]. Two important, related concerns in cognitive linguistics are thus (a) to find a method which allows language models to be scaled up, which is similar in spirit to recurrent neural networks, but which does not encounter the same problems of scale, and (b) to use such a method to evince new insights into graded grammaticality from the state trajectories which arise given genuinely large, naturally-occurring data sets. Accordingly, we present a new method of generating state trajectories which avoids most of these problems. Previously studied in a financial prediction task, the method creates a fractal map of the training data, from which state machines are built. The resulting models are known as Prediction Fractal Machines (PFMs) [18] and have some useful properties. The state trajectories in the fractal representation are fast and computationally efficient to generate, and are accurate and well-understood; it may be inferred that, even for very large vocabularies and training sets, catastrophic interference and lack of adaptivity and stability will not be a problem, given the way in which representations are built (demonstrating this is a topic for future work); training times are significantly less than for recurrent networks (in the experiments described below, the smallest models took a few minutes to build, while the largest ones took only around three hours; in comparison, all of the ANNs took longer - up to a day - to train); and there is little or no loss of information over the course of an input sequence (allowing for the finite precision of the computer). The scalability of the PFM was taken advantage of by training on a large corpus of naturally-occurring text. This enabled an assessment of what potential new insights might arise from the use of this method in truly large-scale language tasks. 2 Prediction Fractal Machines (PFMs) A brief description of the method of creating a PFM will now be given. Interested readers should consult [1], since space constraints preclude a detailed examination here. The key idea behind our predictive model is a transformation F of symbol sequences from an alphabet (here, tagset) {I, 2, ... , N} into points in a hypercube H = [0, I]D. The dimensionality D of the hypercube H should be large enough for each symbol 1, 2, ... , N to be identified with a unique vertex of H. The particular assignment of symbols to vertices is arbitrary. The transformation F has the crucial property that symbol sequences sharing the same suffix (context) are mapped close to each other. Specifically, the longer the common suffix shared by two sequences, the smaller the (Euclidean) distance between their point representations. The transformation F used in this study corresponds to an Iterative Function System [19] S. Parfitt, P TIno and G. Dorffner 54 consisting of N affine maps i : H -+ H, i = 1,2, ... , N, i(x) = ~(x + ti), tj E {a, l}D, ti =F tj for i =F j. (1) Given a sequence 5182 ... 5L of L symbols from the alphabet 1,2, .. . , N, we construct its point representation as where x* is the center {l}D of the hypercube H. (Note that as is common in the Iterative Function Systems literature, i refers either to a symbol or to a map, depending upon the context.) PFMs are constructed on point representations of subsequences appearing in the training sequence. First, we slide the window of length L > 1 over the training sequence. At each position we transform the sequence of length L appearing in the window into a point. The set of points obtained by sliding through the whole training sequence is then partitioned into several classes by k-means vector quantization (in the Euclidean space), each class represented by a particular codebook vector. The number of code book vectors required is chosen experimentally. Since quantization classes group points lying close together, sequences having point representations in the same class (potentially) share long suffixes. The quantization classes may then be treated as prediction contexts, and the corresponding predictive symbol probabilities computed by sliding the window over the training sequence again and counting, for each quantization class, how often a sequence mapped to that class was followed by a particular symbol. In test mode, upon seeing a new sequence of L symbols, the transformation F is again performed, the closest quantization center found, and the corresponding predictive probabilities used to predict the next symbol. 3 An experimental comparison of PFMs and recurrent networks The performance of the PFM was compared against that of a RTRL-trained recurrent network on a next-tag prediction task. Sixteen grammatical tags and a 'sentence start' character were used. The models were trained on a concatenated sequence (22781 tags) of the top three-quarters of each of the 14 sub-corpora of the University of Pennsylvania 'Brown' corpus 1 . The remainder was used to create test data, as follows. Because in a large training corpus of naturally-occurring data, contexts in most cases have more than one possible correct continuation, simply counting correctly predicted symbols is insufficient to assess performance, since this fails to count correct responses which are not targets. The extent to which the models distinguished between grammatical and ungrammatical utterances was therefore additionally measured by generating minimal pairs and comparing their negative log likelihoods (NLLs) per symbol with respect to the model. Likelihood is computed by sliding through the test sequence and for each window position, determining the probability of the symbol that appears immediately beyond it. As processing progresses, these probabilities are multiplied. The negative of the natural logarithm is then taken and divided by the number of symbols. Significant differences in NLLs Ihttp://www.ldc.upenn.edu/ Graded Grammaticality in Prediction Fractal Machines 55 are much harder to achieve between members of minimal pairs than between grammatical and random sequences, and are therefore a good measure of model validity. Minimal pairs generated by theoretically-motivated manipulations tend to be no longer ungrammatical given a small tagset, because the removal of grammatical sub-classes necessarily also removes a large amount of information. Manipulations were therefore performed by switching the positions of two symbols in each sentence in the test sets. Symbols switched could be any distance apart within the sentence, as long as the resulting sentence was ungrammatical under all surface instantiations. By changing as little as possible to make the sentence ungrammatical, the goal was retained that the task of distinguishing between grammatical and ungrammatical sequences be as difficult as possible. The test data then consisted of 28 paired grammatical/ungrammatical test sets (around 570 tags each), plus an ungrammatical, 'meaningless' test set containing all 17 codes listed several times over, used to measure baseline performance. Ten 1st-order randomly-initialised networks were trained for 100 epochs using RTRL. The networks consisted of 1 input and 1 output layer, each with 17 units corresponding to the 17 tags, 2 hidden layers, each with 10 units, and 1 context layer of 10 units connected to the first hidden layer. The second hidden layer was used to increase the flexibility of the maps between the hidden representations in the recurrent portion and the tag activations at the output layer. A logistic sigmoid activation function was used, the learning rate and momentum were set to 0.05, and the training sequence was presented at the rate of one tag per clock tick. The PFMs were derived by clustering the fractal representation of the training data ten times for various numbers of codebook vectors between 5 and 200. More experiments were performed using PFMs than neural networks because in the former case, experience in choosing appropriate numbers of codebook vectors was initially lacking for this type of data. The results which follow are given as averages, either over all neural networks, or else over all PFMs derived from a given number of codebook vectors. The networks correctly predicted 36.789% and 32.667% of next tags in the grammatical and ungrammatical test sets, respectively. The PFMs matched this performance at around 30 codebook vectors (37 .134% and 32.814% respectively), and exceeded it for higher numbers of vectors (39.515% and 34.388% respectively at 200 vectors). The networks generated mean NLLs per symbol of 1.966 and 2.182 for the grammatical and ungrammatical test sets, respectively (a difference of 0.216) and 4.157 for the 'meaningless' test set (the difference between NLLs for grammatical and 'meaningless' data = 2.191). The PFMs matched this difference in NLLs at 40 codebook vectors (NLL grammatical = 1.999, NLL ungrammatical = 2.217; difference = 0.218). The NLL for the 'meaningless' data at 40 codebook vectors was 6.075 (difference between NLLs for grammatical and 'meaningless' data = 4.076). The difference between NLLs for grammatical and ungrammatical, and for grammatical and 'meaningless' data sets, became even larger with increased numbers of codebook vectors. The difference in performance between grammatical and ungrammatical test sets was thus highly significant in all cases (p < .0005): all the models distinguished what was grammatical from what was not. This conclusion is supported by the fact that the mean, NLLs for the 'meaningless' test set were always noticeably higher than those for the minimal pair sets. S. Parfitt, P. Tina and G. Daiffner 56 4 Discussion The PFMs exceeded the performance of the networks for larger numbers of codebook vectors, but it is possible that networks with more hidden nodes would also do better. In terms of ease of use, however, as well as in their scaling-up potential, PFMs are certainly superior. Their other great advantage is that the representations created are dependable (see section 1), making hypothesis creation and testing not just more rapid, but also more straightforward: the speed with which PFMs may be trained made it possible to make statistically significant observations for a large number of clustering runs. In the introduction, 'graded' wellformedness was spoken of as being productive of new hypotheses about the nature of language. Our use of minimal pairs, designed to make a clear-cut distinction between grammatical and ungrammatical utterances, appears to leave this issue to one side. But in reality, our results were rather pertinent to it, as the use of the likelihood measure might indeed imply. The Brown corpus consists of subcorpora representative of 14 different discourse types, from fiction to government documents. Whereas traditional notions of grammaticality would lead us to treat all of the 'ungrammatical' sentences in the minimal pair test sets as equally ungrammatical, the NLLs in our experiments tell a different story. The grammatical versions consistently had a lower associated NLL (higher probability) than the ungrammatical versions, but the difference between these was much smaller than that between the 'meaningless' data and either the grammatical or the ungrammatical data. This supports the concept of 'graded grammaticality', and NLLs for 'meaningless' data such as ours might be seen as a sort of benchmark by which to measure all lesser degrees ofungrammaticality. (Note incidentally that the PFMs appear to associate with the 'meaningless' data a significantly higher NLL than did the networks, even though the difference between the NLLs of the grammatical and ungrammatical data was the same. This is suggestive of PFMs having greater powers of discrimination between grades of wellformedness than the recurrent networks used, but further research will be needed to ascertain the validity of this.) Moreover, the NLL varied not just between grammatical and ungrammatical test sets, but also from sentence to sentence, from word to word and from discourse style to discourse style. While it increased, often dramatically, when the manipulated portion of an ungrammatical sentence was encountered, some words in grammatical sentences exhibited a similar effect: thus, if a subsequence in a well-formed utterance occurs only rarely - or never - in a training set, it will have a high associated NLL in the same way as an ungrammatical one does. This is likely to happen even for very large corpora, since some grammatical structures are very rare. This is consistent with recent findings that, during human sentence processing, well-formedness is linked to conformity with expectation [20] as measured by CLOZE scores. Interesting also was the remarkable variation in NLL between discourse styles. Although the mean NLL across all discourse styles (test sets) is lower for the grammatical than for the ungrammatical versions, it cannot be guaranteed that the grammatical version of one test set will have a lower NLL than the ungrammatical version of another. Indeed, the grammatical and ungrammatical NLLs interleave, as may be observed in figure 1, which shows the NLLs for the three discourse styles which lie at the bottom, middle and top of the range. Even more interestingly, if the NLLs for the grammatical versions of all discourse styles are ordered according to where they lie within this range, it becomes clear that NLL is a predictor of discourse style. Styles which linguists class as 'formal', e.g. those of 57 Graded Grammaticality in Prediction Fractal Machines NU,s associaIed wi1h grammatical and ungrammalical versions of 3 discourse types 3r----------r~~------~--------~------~--~ Leamed text: grammatical - leamed text ungrammalical ....... Romantic: fiction: grammatical .?. ... Romantfc fiction: ungrammalical -+2.8 Science fiction: grammatical ? .? .? Science fiction: ungrammatical -D .? .. ....... 2.8 ae G. :::I z : 1:).. 2.4 \. _.-a ... ? ..?t ? .. ? q ?.. B:Ie...... _a?- ..... ... __ .. _ ..- - -...- __----'" .--+-'---- ._ ..----...- . ...... _....-.-.-.- -- t::::~~~::-::-:-?-=~?? ~ --"'- .......' 2.2 \'--_.. ___ --?? _.-? ? 2 ~~~~~~:::.:~~~:~~~~-::~~:?.:;:::-:::::.::::.:.:;:::.:;:::.:;;::::;;;:::;:::.::~::"::,. ~ 1.8 o 50 100 No. of codebook vectors 150 200 Figure 1: NLLs of minimal pair test sets containing different discourse styles suggest grades of wellformedness based upon prototypicality. the Learned and Government Document test sets, have the lowest NLLs, with the three Press test sets clustering just above, and the Fiction test sets, exemplifying creative language use, clustering at the high end. Similarly, that the Learned and Government test sets have the lowest NLLs conforms with the intuition that their usage lies closest to what is grammatically 'prototypical ' - even though in the training set, 6 out of the 14 test sets are fiction and thus might be expected to contribute more to the prototype. That they do not, suggests that usage varies significantly across fiction test sets. 5 Conclusion Work on the use of PFMs in language modeling is at an early stage, but as results to date show, they have a lot to offer. A much larger project is planned, which will examine further Allen and Seidenberg's hypothesis that 'graded grammaticality' (or wellformedness) applies not only to syntax, but also to other language subdomains such as semantics, an integral part of this being the use of larger corpora and tagsets, and the identification of vertices with semantic/syntactic features rather than atomic symbols. Identifying the possibilities of combining PFMs with ANNs, for example as a means of bypassing the normal method of creating state-space trajectories, is the subject of current study. Acknowledgments This work was supported by the Austrian Science Fund (FWF) within the research project "Adaptive Information Systems and Modeling in Economics and Management Science" (SFB 010). The Austrian Research Institute for Artificial Intelligence is supported by the Austrian Federal Ministry of Science and Transport. 58 S. Parfitt, P Tino and G. DorjJner References [1] P. Tino & G. Dorffner (1998). Constructing finite-context sources from fractal representations of symbolic sequences. Technical Report TR-98-18, Austrian Research Institute for AI, Vienna. [2] J. R. Taylor (1995). Linguistic categorisation: Prototypes in linguistic theory. Clarendon, Oxford. [3] J. R. Saffran, R. N. Aslin & E. L. Newport (1996) . Statistical cues in language acquisition: Word segmentation by infants. In Proc. of the Cognitive Science Society Conference, 376-380, La Jolla, CA. [4] J. Allen & M. S. Seidenberg (in press). The emergence of grammaticality in connectionist networks. In B. Macwhinney (ed.), Emergentist approaches to language: Proc . of the 28th Carnegie Symposium on cognition. Erlbaum. [5] S. Parfitt (1997). Aspects of anaphora resolution in artificial neural networks: Implications for nativism. PhD thesis, Imperial College, London. [6] D. Servan-Schreiber et al (1989). Graded state machines: The representation of temporal contingencies in Simple Recurrent Networks. In Advances in Neural Information Processing Systems, 643-652. [7] W. Tabor & M. Tanenhaus (to appear). Dynamical models of sentence processing. Cognitive Science. [8] J. L. Elman et al (1996). Rethinking innateness: A connectionist perspective on development. Bradford. [9] J . L. Elman (1990). Finding structure in time. In: Cognitive Science, 14: 179-211. [10] S. Lawrence, C. Lee Giles & S. Fong (in press). Natural language grammatical inference with recurrent neural networks. IEEE Trans. on knowledge and data engineering. [11] J. Hertz, A. Krogh & R. G. Palmer (1991) . Introduction to the theory of neural computation. Addison Wesley. [12] M. McCloskey & N. J. Cohen (1989). Catastrophic interference in connectionist networks: The sequential learning problem. In G. Bower (ed.), The psychology of learning and motivation, vol 24. Academic, NY. [13] J . K. Kruschke (1991) . ALCOVE: A connectionist model of human category learning. In R. P. Lippman et al (eds.), Advances in Neural Information Processing 9, 649-655. Kaufmann, San Mateo, CA. [14] S. Grossberg (ed.) (1988). Neural networks and natural intelligence. Bradford, MIT, Cambs, MA. [15] Y. Bengio, P. Simard & P. Frasconi (1994). Learning long-term dependencies with gradient descent is difficult. IEEE Trans. on neural networks, 5(2). [16] M. P. Casey (1996). The dynamics of discrete-time computation, with application to recurrent neural networks and finite-state machine extraction. Neural Computation, 8(6):1135-1178. [17] D. Ron, Y. Singer & N. Tishby (1996). The power of amnesia. Machine Learning, 25. [18] P. Tino, B. G. Horne, C. Lee Giles & P. C. Collingwood (1998). Finite state machines and recurrent neural networks - automata and dynamical systems approaches. In J. E. Dayhoff & O. Omidvar (eds.), Neural Networks and Pattern Recognition, 171- 220. Academic. [19] M. F. Barnsley (1988). Fractals everywhere. Academic, NY. [20] S. Coulson, J. W. King & M. Kutas (1998). Expect the unexpected: Responses to morphosyntactic violations. Language and Cognitive Processes, 13(1). 

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Some Theoretical Results Concerning the Convergence of Compositions of Regularized Linear Functions Tong Zhang Mathematical Sciences Department IBM T.1. Watson Research Center Yorktown Heights, NY 10598 tzhang@watson.ibm.com Abstract Recently, sample complexity bounds have been derived for problems involving linear functions such as neural networks and support vector machines. In this paper, we extend some theoretical results in this area by deriving dimensional independent covering number bounds for regularized linear functions under certain regularization conditions. We show that such bounds lead to a class of new methods for training linear classifiers with similar theoretical advantages of the support vector machine. Furthermore, we also present a theoretical analysis for these new methods from the asymptotic statistical point of view. This technique provides better description for large sample behaviors of these algorithms. 1 Introduction In this paper, we are interested in the generalization performance of linear classifiers obtained from certain algorithms. From computational learning theory point of view, such performance measurements, or sample complexity bounds, can be described by a quantity called covering number [11, 15, 17], which measures the size of a parametric function family. For two-class classification problem, the covering number can be bounded by a combinatorial quantity called VC-dimension [12, 17]. Following this work, researchers have found other combinatorial quantities (dimensions) useful for bounding the covering numbers. Consequently, the concept of VC-dimension has been generalized to deal with more general problems, for example in [15, 11]. Recently, Vapnik introduced the concept of support vector machine [16] which has been successful applied to many real problems. This method achieves good generalization by restricting the 2-norm of the weights of a separating hyperplane. A similar technique has been investigated by Bartlett [3], where the author studied the performance of neural networks when the I-norm of the weights is bounded. The same idea has also been applied in [13] to explain the effectiveness of the boosting algorithm. In this paper, we will extend their results and emphasize the importance of dimension independence. Specifically, we consider the following form of regularization method (with an emphasis on classification problems) which has been widely studied for regression problems both in statistics and in Convergence of Regularized Linear Functions 371 numerical mathematics: inf Ex yL(w, 2:, y) = inf Ex yl(wT 2:Y) w W I I + Ag(W), (1) where Ex ,y is the expectation over a distribution of (2:, y), and y E {-1, 1} is the binary label of data vector 2:. To apply this fonnulation for the purpose oftraining linear classifiers. we can choose I as a decreasing function, such that I (.) ~ 0, and choose 9 (w) ~ 0 as a function that penalizes large w (liIl1w~oo g( w) -4 00). A is an appropriately chosen positive parameter to balance the two tenns. The paper is organized as follows. In Section 2, we briefly review the concept of covering numbers as well as the main results related to analyzing the perfonnance of learning algorithms. In Section 3, we introduce the regularization idea. Our main goal is to construct regularization conditions so that dimension independent bounds on covering numbers can be obtained. Section 4 extends results from the previous section to nonlinear compositions of linear functions. In Section 5. we give an asymptotic fonnula for the generalization perfonnance of a learning algorithm, which will then be used to analyze an instance of SVM. Due to the space limitation, we will only present the main results and discuss their implications. The detailed derivations can be found in [18]. 2 Covering numbers We fonnulate the learning problem as to find a parameter from random observations to minimize risk: given a loss function L( a, x) and n observations Xl = {x 1, ... , x n } independently drawn from a fixed but unknown distribution D, we want to find a that minimizes the expected loss over 2: (risk): R(a) = ExL(a,x)= / L(a,x)dP(x). (2) The most natural method for solving (2) using a limited number of observations is by the empirical risk minimization (ERM) method (cf [15, 16]). We simply choose a parameter a that minimizes the observed risk: 1 n R(a,X l ) = - LL(a,xi). (3) n i=l We denote the parameter obtained in this way as a erm (Xl)' The convergence behavior of this method can be analyzed by using the VC theoretical point of view. which relies on the unifonn convergence of the empirical risk (the unifonn law of large numbers): SUPa IR(a, Xl) - R(a)l. Such a bound can be obtained from quantities that measure the size of a Glivenko-Cantelli class. For finite number of indices, the family size can be measured simply by its cardinality. For general function families, a well known quantity to measure the degree ofunifonn convergence is the covering number which can be be dated back to Kolmogrov [8, 9]. The idea is to discretize (which can depend on the data Xl) the parameter space into N values a1, . .. ,aN SO that each L(a, .) can be approximated by L( ai, .) for some i. We shall only describe a simplified version relevant for our purposes. Definition 2.1 Let B be a metric space with metric p. Given a norm p, observations Xl = [Xl, ... ,xn ]. and vectors I(a, Xl) = [/(a, Xl)"" ,/(a, x n )] E Bn parameterized by a, the covering number in p-norm, denoted as Np (I, ?, Xl)' is the minimum number of a collection o/vectors V1, ... ,Vm E B n such that Va. 3Vi: IIp(l(a,Xl),vi)lIp ::; n 1/P ?. We also denote Np(l, ?, n) = maxx~ Np(l, ?, Xl). Note that from the definition and the Jensen's inequality, we have N p ::; N q for p ::; q. We will always assume the metric on R to be IX1 - x21 if not explicitly specified otherwise. The following theorem is due to Pollard [11]: T. Zhang 372 Theorem 2.1 ([11]) \;/n, f > ? and distribution D. P(s~p IR(a, X~) - R(a)1 > ?j -nf 2 ~ 8E(Af1 (L , f/8, X~)] exp( 128M2)' where M = sUPa,:z: L(a, x) - infa,:z: L(a, x). and X~ = {Xl, . .. ,X' l } are independently drawn from D. The constants in the above theorem can be improved for certain problems; see [4. 6, 15, 16] for related results. However, they yield very similar bounds. The result most relevant for this paper is a lemma in [3] where the 1-nonn covering number is replaced by the oo-nonn covering number. The latter can be bounded by a scale-sensitive combinatorial dimension [1], which can be bounded from the I-norm covering number if this covering number does not depend on n. These results can replace Theorem 2.1 to yield better estimates under certain circumstances. Since Bartlett's lemma in [3] is only for binary loss functions, we shall give a generalization so that it is comparable to Theorem 2.1 : Theorem 2.2 Let It and 12 be two functions: R n -+ [0, 1] such that /Y1 - Y21 ~ I implies ~ h(Y2) ~ h(Y1) where h : R n -+ [0,1] is a reference separatingfunction, then It (Y1) P[s~p[E:z:It(L(a, -nf 2 x?) - Ex-;-h(L(a, x))] > f] ~ 4E[Afoo(L, I, X~)] exp( 32)' Note that in the extreme case that some choice of a achieves perfect generalization: E:z:h(L(a, x)) 0, and assume that our choices of a(X1) always satisfy the condition EXf h(L( a, x? = 0, then better bounds can be obtained by using a refined version of the Chernoffbound. = 3 Covering number bounds for linear systems In this section, we present a few new bounds on covering numbers for the following form of real valued loss functions: d L(w, x) = xT w= L XiWi ? (4) i=l As we shall see later, these bounds are relevant to the convergence properties of (1). Note that in order to apply Theorem 2.1, since Afl < Af2 , therefore it is sufficient to estimate Af2(L, ?, n) for ? > O. It is clear that Af2(L, f, ~ is not finite ifno restrictions on x and w are imposed. Therefore in the following, we will assume that each I/xil/p is bounded. and study conditions ofllw// q so that logAf(j, f, n) is independent or weakly dependent of d. Our first result generalizes a theorem of Bartlett [3]. The original results is with p = 00 and q 1, and the related technique has also appeared in [10, 13]. The proof uses a lemma that is attributed to Maurey (cf. [2, 7]). = Theorem 3.1 V/lxi/lp ~ band Ilw/lq ~ a, where lip + 1/q == 1 and 2 ~ p ~ log2 Af2(L, f, n) ~ 00, then r7a b 1Iog (2d + 1). 2 2 2 The above bound on the covering number depends logarithmically on d, which is already quite weak (as compared to linear dependency on d in the standard situation). However, the bound in Theorem 3.1 is nottightforp < 00. For example, the following theorem improves the above bound for p = 2. Our technique of proof relies on the SVD decomposition [5] for matrices, which improves a similar result in [14 J by a logarithmic factor. Convergence of Regularized Linear Functions 373 The next theorem shows that if lip + llq mdependent of dimension. = Theorem 3.3 Let L(w, x) xTw. J = lip + 1jq - 1 > 0, then > 1, then the 2-nonn covering number is also {f'llxillp :::; band Ilwllq :::; a, where 1 :::; q :::; 2 and One consequence of this theorem is a potentially refined explanation for the boosting algorithm. In [13], the boosting algorithm has been analyzed by using a technique related to results in [3] which essentially rely on Theorem 3.1 withp = 00. Unfortunately, the bound contains a logarithmic dependency on d (in the most general case) which does not seem to fully explain the fact that in many cases the perfonnance of the boosting algorithm keeps improving as d increases. However, this seemingly mysterious behavior might be better understood from Theorem 3.3 under the assumption that the data is more restricted than simply being oo-nonn bounded. For example, when the contribution of the wrong predictions is bounded by a constant (or grow very slowly as d increases), then we can regard its p-th nonn bounded for some p < 00 . In this case, Theorem 3.3 implies dimensional independent generalization. If we want to apply Theorem 2.2, then it is necessary to obtain bounds for infinity-nonn covering numbers. The following theorem gives such bounds by using a result from online learning. Theorem 3.4 lfllxillp :::; band Ilwllq :::; a, where 2 :::; p < 00 and lip + 11q = 1, then tiE> O. In the case of p = 00, an entropy condition can be used to obtain dimensional independent covering number bounds. = = Definition 3.1 Let f1. [f1.i] be a vector with positive entries such that 11f1.lll 1 (in this case, we call f1. a distribution vector). Let x = [Xi] "# 0 be a vector of the same length, then we define the weighted relative entropy of x with re5pect to f1. as: entro~(x) ~ IXil = ~ IXil ln J-Lillxlh' ? Theorem 3.5 Given a distribution vector f1., If llxi lloo :::; band Ilwlll :::; :::; c, where we assume that w has non-negative entries, then tiE> 0, entro ~ (w) log2 Noo(L, E, n) :::; a and 36b 2 ( a 2 + ac) E2 log2[2 r4ab/ E+ 21n + 1] . Theorems in this section can be combined with Theorem 4.1 to fonn more complex covering number bounds for nonlinear compositions oflinear functions. 374 4 T. Zhang Nonlinear extensions Consider the following system: + wTh(a, x)) , L([a, w], x) = I(g(a, x) (5) where x is the observation, and [a, w] is the parameter. We assume that function with bounded total variation. 1 is a nonlinear Definition 4.1 A/unction 1 : R -+ R is said to satisfy the Lipschitz condition with parameter"Y ifVx, y: I/( x) - I(y) I ~ )'Ix - yl? Definition 4.2 The total variation of a/unction 1 : R -+ R is defined as L TV(f, x) = sup :2:0<X1 L I/(xi) - I(xi-dl ? ' <Xl~X t=l We also denote TV(f, (0) as TV(f). Theorem 4.1 .if L([a, w], x) = I(g(a, x) + w T h(a, x)), where TV(f) < 00 and 1 is Lipschitz with parameter),. Assume also that w is a d-dimensional vector and Ilwllq :s; c, then VEl, E2 > 0, and n > 2(d + 1): Iog 2 Nr (L, E1 + E2, n) < (d + 1) log2[d en max(l TV(f) J, 1)] + log2 Nr([g , h], E2h, n) , - +1 where the metric o/[g, h) is defined as Ig1 - g21 2E1 + cllh1 - h211p (l/p + l/q = 1). Example 4.1 Consider classification by hyperplane: L( w, x) = J( w T x < 0) where J is the set indicator function. Let L' (w, x) = 10 (w T x) be another loss function where 1 z<0 lo(z) = { 1 - z z E [0 , 1] . o z>1 Instead of using ERM for estimating parameter that minimizes the risk of L , consider the scheme of minimize empirical risk associated with L', under the assumption that II x 112 :s; b and constraint that JJwl12 :s; a. Denote the estimated parameter by w n . It follows from the covering number bounds and Theorem 2.1 that with probability of at least 1 - 1]: n 1 / 2 ab In( nab + 2) + In 1.. _ _ _ _ _ _ _ _--'-'7 ). n If we apply a slight generalization of Theorem 2.2 and the covering number bound of Theorem 3.4, then with probability of at least 1 - T/: ExJ(w~ x ~ 0) :s; EXfJ(w~ x :s; 2)') + O( 1 a 2 b2 - ( - 2 In(abh + 2) n )' 1 + In n + In -)) T/ for all)' E (0,1]. 0 Bounds given in this paper can be applied to show that under appropriate regularization conditions and assumptions on the data, methods based on (1) lead to generalization performances of the form 0(1/ .jn), where 0 symbol (which is independent of d) is used to indicate that the hidden constant may include a polynomial dependency on Iog( n). It is also important to note that in certain cases, ,\ will not appear (or it has a small influence on the convergence) in the constant of 0, as being demonstrated by the example in the next section. 375 Convergence of Regularized Linear Functions 5 Asymptotic analysis The convergence results in the previous sections are in the form of VC style convergence in probability, which has a combinatorial flavor. However, for problems with differentiable function families involving vector parameters, it is often convenient to derive precise asymptotic results using the differential structure. Assume that the parameter a E Rm in (2) is a vector and L is a smooth function. Let a* denote the optimal parameter; "\1 ex denote the derivative with respect to a; and 'It( a, x) denote "\1 exL(a, x) . Assume that V = U= J J "\1 ex'lt(a* , x) dP(x) 'It ( a * , x) 'It ( a * , x f dP (x) . Then under certain regularity conditions, the asymptotic expected generalization error is given by 1 (6) E R(a erm ) R(a*) + 2n tr(V-1U). = More generally, for any evaluation function h( a) such that "\1 h( a*) = 0: 1 E h(a erm ) I=::j h(a*) + -tr(V- 1 "\12h? V-1U), (7) 2n where "\1 2 h is the Hessian matrix of hat a*. Note that this approach assumes that the optimal solution is unique. These results are exact asymptotically and provide better bounds than those from the standard PAC analysis. Example 5.1 We would like to study a form of the support vector machine: Consider L(a, x) = f(a T x) + ~Aa2 , z <1 z>1. Because of the discontinuity in the derivative of f , the asymptotic formula may not hold. However, if we make an assumption on the smoothness of the distribution x, then the expectation of the derivative over x can still be smooth. In this case, the smoothness of f itself is not crucial. Furthermore, in a separate report. we shall illustrate that similar small sample bounds without any assumption on the smoothness of the distribution can be obtained by using techniques related to asymptotic analysis. Consider the optimal parameter a* and letS = {x : a*Tx::; 1}. Note that Aa* = ExEsx, and U = EXES(X - ExEsx)(x - EXEsxf. Assume that 3')' > 0 S.t. P(a*T x ::; ')') = 0, then V = AI + B where B is a positive semi-definite matrix. It follows that E x2 tr(V-1U) ::; tr(U)jA ::; EXES *T Ila*I I ~::; sup Ilxll~ l la*ll~j')'. xESa X Now, consider an obtained from observations Xl risk associated with loss function L( a, x), then ExL(a emp , x) ::; inf ExL(a, x) ex = [Xl, '" ,x n ] by minimizing empirical 1 sup I lx l l~lla*ll~ + -2 ')'n asymptotically. Let A --+ 0, this scheme becomes the optimal separating hyperplane [16]. This asymptotic bound is better than typical PAC bounds with fixed A. 0 Note that although the bound obtained in the above example is very similar to the mistake bound for the perceptron online update algorithm, we may in practice obtain much better estimates from (6) by plugging in the empirical data. 376 T. Zhang References [I] N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler. Scale-sensitive dimensions, uniform convergence, and learnability. Journal of the ACM, 44(4):615-631, 1997. [2] A.R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Injormation Theory, 39(3):930-945, 1993. [3] P.L. Bartlett. The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network. IEEE Transactions on Information Theory, 44(2):525-536, 1998. [4] R.M. Dudley. A course on empirical processes, volume 1097 of Lecture Notes in Mathematics. 1984. [5] G.H. Golub and C.P. Van Loan. Matrix computations. Johns Hopkins University Press, Baltimore, MD, third edition, 1996. [6] D. Haussler. Generalizing the PAC model: sample size bounds from metric dimension-based uniform convergence results. In Proc. 30th IEEE Symposium on Foundations of Computer Science, pages 40-45, 1989. [7] Lee K. Jones. A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. Ann. Statist., 20(1) : 60~13, 1992. [8] A.N. Kolmogorov. Asymptotic characteristics of some completely bounded metric spaces. Dokl. Akad. Nauk. SSSR, 108:585-589, 1956. [9] A.N. Kolmogorov and Y.M. Tihomirov. f-entropyand f-capacity of sets in functional spaces. Amer. Math. Soc. Trans!., 17(2):277-364,1961. [10] Wee Sun Lee, P.L. Bartlett, and R.C. Williamson. Efficient agnostic learning of neural networks with bounded fan-in. IEEE Transactions on Information Theory, 42(6):2118-2132,1996. [II] D. Pollard. Convergence of stochastic processes. Springer-Verlag, New York, 1984. [12] N. Sauer. On the density of families of sets. Journal of Combinatorial Theory (Series A), 13: 145-147,1972. [13] Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the margin: a new explanation for the effectiveness of voting methods. Ann. Statist., 26(5): 1651-1686,1998. [14] 1. Shawe-Taylor, P.L. Bartlett, R.C. Williamson, and M. Anthony. Structural risk minimization over data-dependent hierarchies. IEEE Trans. In! Theory, 44(5): 19261940, 1998. [15] Y.N. Vapnik. Estimation of dependences based on empirical data. Springer-Verlag, New York, 1982. Translated from the Russian by Samuel Kotz. [16] Y.N. Vapnik. The nature of statistical learning theory. Springer-Verlag, New York, 1995. [17] Y.N. Vapnik and AJ. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and Applications, 16:264-280, 1971. [18] Tong Zhang. Analysis of regularized linear functions for classification problems. Technical Report RC-21572, IBM, 1999. PART IV ALGORITHMS AND ARCHITECTURE 

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687 AN ANALOG VLSI CHIP FOR THIN-PLATE SURFACE INTERPOLATION John G. Harris California Institute of Technology Computation and Neural Systeins Option, 216-76 Pasadena, CA 91125 ABSTRACT Reconstructing a surface from sparse sensory data is a well-known problem iIi computer vision. This paper describes an experimental analog VLSI chip for smooth surface interpolation from sparse depth data. An eight-node ID network was designed in 3J.lm CMOS and successfully tested. The network minimizes a second-order or "thinplate" energy of the surface. The circuit directly implements the coupled depth/slope model of surface reconstruction (Harris, 1987). In addition, this chip can provide Gaussian-like smoothing of images. INTRODUCTION Reconstructing a surface from sparse sensory data is a well-known problem in computer vision. Early vision modules typically supply sparse depth, orientation, and discontinuity information. The surface reconstruction module incorporates these sparse and possibly conflicting measurements of a surface into a consistent, dense depth map. The coupled depth/slope model provides a novel solution to the surface reconstruction problem (Harris, 1987). A ID version of this model has been implemented; fortunately, its extension to 2D is straightforward. Figure 1 depicts a high-level schematic of the circuit. The di voltages represent noisy and possibly sparse input data, the ZiS are the smooth output values, and the PiS are the explicitly computed slopes. The vertical data resistors (with conductance g) control the confidence in the input data. In the absence of data these resistors are open circuits. The horizontal chain of smoothness resistors of conductance ..\ forces the derivative of the data to be smooth. This model is called the coupled depth/slope model because of the coupling between the depth and slope representations provided by the subtractor elements. The subtractors explicitly calculate a slope representation of the surface. Any depth or slope node can be made into a constraint by fixing a voltage source to the proper location in the network. Intuitively, any sudden change in slope is smoothed out with the resistor mesh. 688 Harris Figure 1. The coupled depth/slope model. The tri-directional subtractor device (shown in Figure 2) is responsible for the coupling between the depth and slope representations. If nodes A and B are set with ideal voltage sources, then node C will be forced to A - B by the device. This circuit element is unusual in that all of its terminals can act as inputs or outputs. If nodes Band C are held constant with voltage sources, then the A terminal is fixed to B + C. If A and C are input, then B becomes A - C. When further constraints are added, this device dissipates a power proportional to (A - B - C)2. In the limiting case of a continuous network, the total dissipated power is (1) The three terms arise from the power dissipated in the sub tractors and in the two different types of resistors. Energy minimization techniques and standard calculus of variations have been used to formally show that the reconstructed surfaces, z, satisfy the 1D biharmonic equation between input data points (Harris, 1987). In the tw~dimensional formulation, z is a solution of (2) This interpolant, therefore, provides the same results as minimizing the energy of a thin plate, which has been commonly used in surface reconstruction algorithms on digital computers (Grimson, 1981; Terzopoulos, 1983). IMPLEMENTATION The eight-node 1D network shown in Figure 1 was designed in 3J.lm CMOS (Mead, 1988) and fabricated through MOSIS. Three important components of the model must be mapped to analog VLSI: the two different types of resistors and the subtractors. The vertical confidence resistors are built with simple transconductance An Analog VLSI Chip for Thin-Plate Surface Interpolation A B c Figure 2. Tri-directional subtract constraint device. amplifiers (transamps) connected as followers. The bias voltage of the transamp follower determines its conductance (g) and therefore signifies the certainty of the data. If there are no data for a given location, the corresponding transamp follower is turned off. The horizontal smoothness resistors are implemented with Mead's saturating resistor (Mead, 1988). Since conventional CMOS processes lack adequate resistive elements, we are forced to build resistors out of transistor elements. The bias voltage for Mead's resistor allows the effective conductance of these circuit elements to vary over many orders of magnitude. The most difficult component to implement in analog VLSI is the subtract constraint device. Its construction led to a general theory of constraint boxes which can be used to implement all sorts of constraints which are useful in early vision (Harris, 1988). The implementation of the subtract constraint device is a straightforward application of constraint box theory. Figure 3 shows a generic n terminal constraint box enforcing a constraint F on its voltage terminals. The constraints are enforced by generating a feedback current lie for each constrained voltage terminal. Suppose F can be written as One possible feedback equation which implements this constraint is given by 1.: 8F = - F8Vl: - (4) When this particular choice of feedback current is used, the constraint box minimizes the least-squares error in the constraint equation (Harris, 1989). Notice that F can be scaled by any arbitrary scaling factor. This scaling factor and the capacitance at each node determine the speed of convergence of a single constraint box. 689 690 Harris V t-------i~-.. k Figure 3. Generic n terminal constraint box. The subtract constraint box given in Figure 2 requires a constraint of A - B which leads to the following error equation: F(A,B,C) =A- B - =C, (5) C Straightforward application of constraint box theory yields 8F -F 8A B - C) -F~ = (A- B - 1B Ie =-(A - - -F :~ = (A - C) (6) B - C) where lA, I B , and Ie represent feedback currents that must be generated by the device. These current feedback equations can be implemented with two modified widerange transamps (see Figure 4). In its linear range, a single transamp produces a current proportional to the difference of its two inputs. The negative input to each transamp is indicated by an inverting circle. The transamps have been modified to produce four outputs, two positive and two negative. The negative outputs are also represented by inverting circles. Because the difference terminal C can be positive or negative, it is measured with respect to a voltage reference VREF. VREF is a global signal which defines zero slope. As seen in Figure 4, the An Analog VLSI Chip for Thin-Plate Surface Interpolation A I----V REF c B IB Ie ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _- J Figure 4. Tri-directional subtract constraint box. proper combination of positive and negative outputs from the two transamps are fed back to the voltage terminals to implement the feedback equations given in eq. (6). Analog networks which solve most regularizable early vision problems can be designed with networks consisting solely of linear resistances and batteries (Poggio and Koch, 1985). Unfortunately, many times these networks contain negative resistances that are troublesome to implement in analog VLSI. For example, the circuit shown in Figure 5 computes the same solutions as the coupled depth/slope network described in this paper. Interestingly, a 2-D implementation of this idea was implemented in the 1960s using inductors and capacitors (Volynskii and Bukhman, 1965). Proper choice of the frequency of alternating current allowed the circuit elements to act as pure positive and negative impedances. Unfortunately, negative resistances are troublesome to implement, especially in analog VLSI. One of the big advantages of using constraint boxes to implement early vision algorithms is that the resulting networks do not require negative resistances. ANALYSIS Figure 6 shows a sample output of the circuit. Data (indicated by vertical dashed lines) were supplied at nodes 2, 5, and 8. As expected, the chip finds a smooth solution (solid line) which extrapolates beyond the known data points. It is wellknown that a single resistive grid minimizes the first-order or membrane energy of a surface. Luo, Koch, and Mead (1988) have implemented a 48x48 resistive grid to perform surface interpolation. Figure 6 also shows the simulated performance of a first-order energy or membrane energy minimization. Data points are again supplied at nodes 2, 5, and 8. In contrast to the second-order chip results, the solution (dashed line) is much more jagged and does not extrapolate outside of 691 692 Harris ? g -R -R -R -R -R -R Figure 5. A negative-resistor resistor solution to the ID biharmonic equation. I .? .... .... ~ 1.7 ~ .... ~ ~ ~ ~ ~ -. ~ I I / 1.6 ~ ~ / / ~ ~ 1.5 1.4 13+------+------+------+------~----~------~----~ I 2 3 4 5 6 7 ? Figure 6. Measured data from the second-order chip (solid line) and simulated first-order result (dashed line). An Analog VLSI Chip for Thin-Plate Surface Interpolation 1.0 0.' 0.6 ,, ,, , I 0.4 I I I I / G.2 / ,, " / ..-.-----.: . ' . .. . ... o.o+r-~.~ .... .... ... ,. --..... .... 42+---~~--~----~--~~-~---~--~~--+--~--~ -5 "" -3 -2 -I 0 2 3 4 5 Figure 7. Graphical comparison of ID analytic Green's functions for first-order (dashed line), second-order (dotted line) and Gaussian (solid line). the known data points (for example, see node 1). Interestingly, psychophysics experiments support the smoother interpolant used by the second-order coupled depth/slope chip (Grimson, 1981). Unlike the second-order network, the firstorder network is not rigid enough to incorporate either orientation constraints or orientation discontinuities (Terzopoulos, 1983). Image smoothing is a special case of surface interpolation where the data are given on a dense grid. The first-order network is a poor smoothing operator. A comparison of analytic Green's function of first and second-order networks is shown in Figure 7 (the first-order shown with a dashed line and the secondorder with a solid line). Note that the analytic Green's function of the secondorder network (solid line) and that of standard Gaussian convolution (dotted line) are nearly identical. This fact was pointed out by Poggio, Voorhees, and Yuille (1986), when they suggested the use of the second-order energy to regularize the edge detection problem. Gaussian convolution has been claimed by many authors to be the "optimal" smoothing operator and is commonly used as the first stage of edge detection. Though the second-order network can be used to smooth images, Gaussian convolution cannot be used to solve the more difficult problem of interpolating from sparse data points. 693 694 Harris CONCLUSION Biharmonic surface interpolation has been successfully demonstrated in analog VLSI. To test true performance, we plan to combine a larger version of this chip with an analog stereo network. Work has already started on building the necessary circuitry for discontinuity detection during surface reconstruction. The Gaussianlike smoothing effect of this network will be further explored through building a network with photoreceptors supplying dense data input. Acknowledgements Support for this research was provided by the Office of Naval Research and the System Development Foundation. The author is a Hughes Aircraft Fellow and thanks Christof Koch and Carver Mead for their ongoing support. Additional thanks to Berthold Horn for several helpful suggestions. References Grimson, W.E.L. From Images to Surfaces, MIT Press, Cambridge, (1981). Harris, J.G. A new approach to surface reconstruction: the coupled depth/slope model, Proc. IEEE First Inti. Con! Computer Vision, pp. 277-283, London, (1987). Harris, J.G. Solving early vision problems with VLSI constraint networks, Neural Architectures for Computer Vision Workshop, AAAI-88, Minneapolis, Minnesota, Aug. 20 (1988). Harris, J .G. Designing analog constraint boxes to solve energy minimization problems in vision, submitted to INNS Neural Networks Conference, Washington D.C., June (1989) Luo, J., Koch, C., and Mead, C. An experimental subthreshold, analog CMOS tw<r dimensional surface interpolation circuit, Neural Information and Processing Systems Conference, Denver, Nov. (1988). Mead, C.A. Analog VLSI and Neural Systems, Addison-Wesley, Reading, (1989). Poggio, T. and Koch, C. Ill-posed problems in early vision: from computational theory to analogue networks, Proc. R. Soc. Lond. B 226: 303-323 (1985). Poggio, T., Voorhees, H., and Yuille, A. A regularized solution to edge detection, Arti! Intell. Lab Memo No. 833, MIT, Cambridge, (1986). Terzopoulos, D. Multilevel computational processes for visual surface reconstruction, Compo Vision Graph. Image Proc. 24: 52-96 (1983). Volynskii, B. A. and Bukhman, V. Yeo Analogues for Solution of Boundary- Value Problems, Pergamon Press, New York, (1965). 

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Bifurcation Analysis of a Silicon Neuron Girish N. Patel] , Gennady s. Cymbalyuk2,3, Ronald L. Calabrese2 , and Stephen P. DeWeerth 1 lSchool of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Ga. 30332-0250 {girish.patel, steve.deweerth} @ece.gatech.edu 2Department of Biology Emory University 1510 Clifton Road, Atlanta, GA 30322 {gcym, rcalabre}@biology.emory.edu 3Institute of Mathematical Problems in Biology RAS Pushchino, Moscow Region, Russia 142292 (on leave) Abstract We have developed a VLSI silicon neuron and a corresponding mathematical model that is a two state-variable system. We describe the circuit implementation and compare the behaviors observed in the silicon neuron and the mathematical model. We also perform bifurcation analysis of the mathematical model by varying the externally applied current and show that the behaviors exhibited by the silicon neuron under corresponding conditions are in good agreement to those predicted by the bifurcation analysis. 1 Introduction The use of hardware models to understand dynamical behaviors in biological systems is an approach that has a long and fruitful history [1 ][2]. The implementation in silicon of oscillatory neural networks that model rhythmic motor-pattern generation in animals is one recent addition to these modeling efforts [3][4]. The oscillatory patterns generated by these systems result from intrinsic membrane properties of individual neurons and their synaptic interactions within the network [5]. As the complexity of these oscillatory silicon systems increases, effective mathematical analysis becomes increasingly more important to our understanding their behavior. However, the nonlinear dynamical behaviors of the model neurons and the large-scale interconnectivity among these neurons makes it very difficult to analyze theoretically the behavior of the resulting very large-scale integrated (VLSI) systems. Thus, it is important to first identify methods for modeling the model neurons that underlie these oscillatory systems. Several simplified neuronal models have been used in the mathematical simulations of pattern generating networks [6][7][8] . In this paper, we describe the implementation of a 732 G. N Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. DeWeerth two-state-variable silicon neuron that has been used effectively to develop oscillatory networks [9][10]. We then derive a mathematical model of this implementation and analyze the neuron and the model using nonlinear dynamical techniques including bifurcation analysis [11]. Finally, we compare the experimental data derived from the silicon neuron to that obtained from the mathematical model. 2 The silicon model neuron The schematic for our silicon model neuron is shown in Figure 1. This silicon neuron is inspired by the two-state, Morris-Lecar neuron model [12][ 13]. Transistor M I ' analogous to the voltage-gated calcium channel in the Morris-Lecar model, provides an instantaneous inward current that raises the membrane potential towards V High when the membrane is depolarized. Transistor M2 ' analogous to the voltage-gated potassium channel in the Morris-Lecar model, provides a delayed outward current that lowers the membrane potential toward V Low when the membrane is depolarized. V H and V L are analogous to the half-activation voltages for the inward and outward currents, respectively. The voltages across C I and C2 are the state variables representing the membrane potential, V, and the slow "activation" variable of the outward current, W, respectively. The W -nullcline represents its steady-state activation curve. Unlike the Morris-Lecar model, our silicon neuron model does not possess a leak current. Using current conservation at node V, the net current charging C I is given by (1) where iH and iL are the output currents of a differential pair circuit, and a p and aN describe the ohmic effects of transistors M J and M2 , respectively. The net current into C 2 is given by (2) where ix is the output current of the OTA, and ~p and ~N account for ohmic effects of the pull-up and the pull-down transistors inside the OTA. V High OTA w v '------<>--- V Low Figure 1: Circuit diagram of the silicon neuron. The circuit incorporates analog building blocks including two differential pair circuits composed of a bias current, IB H , and transistors M4-M s, and a bias current, IB L , and transistors M6-M7' and a single followerintegrator circuit composed of an operational transconductance amplifier (OTA), Xl in the configuration shown and a load capacitor, C 2 . The response of the follower-integrator circuit is similar to a first-order low-pass filter. 733 Bifurcation Analysis ofa Silicon Neuron The output currents of the differential-pair and an OTA circuits, derived by using subthreshold transistor equations [2], are a Fenni function and a hyperbolic-tangent function, respectively [2]. Substituting these functions for i H , i L , and ix in (1) and (2) yields . C 1V = Iexta p + IBH e K(V-YH) / U T l+e e K(V _ YH) / U T a p - K(W - YL) / U T IBL l+e K(W _ YL) / U T aN (3) where v - V High / UT a p = I -e aN ~N = 1- e = 1- e Y Low - V I UT - W / UT (4) U T is the thennal voltage, V dd is the supply voltage, and K is a fabrication dependent parameter. The tenns a p and aN limit the range of V to within V High and V Low' and the terms ~p and ~N limit the range of W to within the supply rails (Vdd and Gnd). In order to compare the model to the experimental results, we needed to determine values for all of the model parameters. V Hi!\h' V Low' V H' V L ' and V dd were directly measured in experiments. The parameters IBH and IBL were measured by voltage-clamp experiments performed on the silicon neuron. At room temperature, U T ::::: 0.025 volts. The value of K ::::: 0.65 was estimated by measuring the slope of the steady-state activation curve of inward current. Because W was implemented as an inaccessible node, IT could only be estimated. Based on the circuit design, we can assume that the bias currents IT and IBH are of the same order of magnitude. We choose IT::::: 2.2 nA to fit the bifurcation diagram (see Figure 3). Cl and C2, which are assumed to be identical according to the physical design, are time scaling parameters in the model. We choose their values (Cl =C2 =28 pF) to fit frequency dependence on lext (see Figure 4). 3 Bifurcation analysis The silicon neuron and the mathematical model! described by (3) demonstrate various dynamical behaviors under different parametric conditions. In particular, stable oscillations and steady-state equilibria are observed for different values of the externally applied current, I ext . We focused our analysis on the influence of I ext on the neuron behavior for two reasons: (i) it provides insight about effects of synaptic currents, and (ii) it allows comparison with neurophysiological experiments in which polarizing current is used as a primary control parameter. The main results of this work are presented as the comparison between the mathematical models and the experimental data represented as bifurcation diagrams and frequency dependencies. The null clines described by (3) and for lext = 32 nA are shown in Figure 2A. In the regime that we operate the circuit, the W -null cline is an almost-linear curve and the Vnullcline is an N-shaped curve. From (3), it can be seen that when IBH + lext > IBL the nullclines cross at (V, W)::::: (V High , V High ) and the system has high voltage (about 5 volts) steady-state equilibrium. Similarly, for I ext close to zero, the system has one stable equilibrium point close to (V, W) ::::: (V Low' V Low). !The parameters used throughout the analyses of the model are V Low = 0 V , V High = 5 V, V L = V H = 2.5 V, I BH = 6.5 nA , I BL = 42 nA, IT = 2.2 nA , V dd = 5 V, V t = 0.025 mV, and K = 0.65. G. N Patel, G. S. Cymba/yule, R. L. Calabrese and S. P. De Weerth 734 A 2.85 W-nullcline / / @ 2.8 / 2.75 ~ 2.7 (5 > .....,.. ~ 2.65 V-nullcline 2.6 2.55 2.5 I 2.45~______~______~______~______~______~1 o 3 2 5 4 V (volts) 3.2 B 3 -- 2.8 In l 2.6 > 2.4 2.2 2 0 5 10 15 20 25 30 35 time (msec) Figure 2: Nullclines and trajectories in the model of the silicon neuron for lex! = 32 nA. The system exhibits a stable limit-cycle (filled circles), an unstable limit-cycle (unfilled circles), and stable equilibrium point. Unstable limit-cycle separates the basins of attraction of the stable limit-cycle and stable equilibrium point. Thus, trajectories initiated within the area bounded by the unstable limit-cycle approach the stable equilibrium point (solid line in A's inset, and "x's" in B). Trajectories initiated outside the unstable limitcycle approach the stable limit-cycle . In A, the inset shows an expansion at the intersection of the V - and W -nullclines. 735 Bifurcation Analysis ofa Silicon Neuron Experimental data A ........................... 5 .. 003 .0 x ~ ~ ? x ..?? I 1 0 ",>0< x xxxx ? ? x >t#* xxxx >2 x ? ? ? ? ? 4 2:- ~ ?? x ? ? ? ??????????????????????????? 10 0 20 ? ?? 30 40 50 40 50 lext (nAmps) Modeling data B 5 ( 0 -> 2 > .? ? ? 4 003 .- ....................... -. ?? ,J rt---------; ? .'- ....................... 1 0 0 10 20 ? ? ~ 30 Iext (nAmps) Figure 3: Bifurcation diagrams of the hardware implementation (A) and of the mathematical model (B) under variation of the externally applied current. In A, the steadystate equilibrium potential of V is denoted by "x"s. The maximum and minimum values of V during stable oscillations are denoted by the filled circles. In B, the stable and unstable equilibrium points are denoted by the solid and dashed curve, respectively, and the minimum and maximum values of the stable and unstable oscillations are denoted by the filled and unfilled circles, respectively. In B, limit-cycle oscillations appear and disappear via sub-critical Andronov-Hopf bifurcations. The bifurcation diagram (B) was computed with the LOCBIF program [14]. G. N. Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. De Weerth 736 A B Experimental data 100 ? 80 ? N ? >. 60 ? . ? ? 100~--~--~----~~ -N 80 ?? >. 60 ? ~ g . .~~.. .., ... ~ LL ? ? ??? t:T 20 -::c ? ~ 40 Modeling data c ~. ? ??? ??? ~ Q) :l t:T ... 40 Q) LL o~--~----~------~ o 0 10 20 lext (nAmps) 30 20 0 0 10 20 30 lext (nAmps) Figure 4: Frequency dependence of the silicon neuron (A) and the mathematical model (B) on the externally applied current. For moderate values of lext ([1 nA,34 nA)), the stable and unstable equilibrium points are close to (V, W) ::::: (V H' V L) (Figure 3). In experiments in which lext was varied, we observed a hard loss of the stability of the steady-state equilibrium and a transition into oscillations at lext = 7.2 nA (I ext = 27.5 nA). In the mathematical model, at the critical value of lext = 7.7 nA (lext = 27.8 nA), an unstable limit cycle appears via a subcritical Andronov-Hopf bifurcation. This unstable limit cycle merges with the stable limit cycle at the fold bifurcation at lext = 3.4 nA (lext = 32.1 nA). Similarly, in the experiments, we observed hard loss of stability of oscillations at lext = 2.0 nA (I ext = 32.8 nA). Thus, the system demonstrates hysteresis. For example. when lext = 20 nA the silicon neuron has only one stable regime, namely, stable oscillations. Then if external current is slowly increased to lext = 32.8 nA. the form of oscillations changes. At this critical value of the current, the oscillations suddenly lose stability, and only steady-state equilibrium is stable. Now, when the external current is reduced, the steady-state equilibrium is observed at the values of the current where oscillations were previously exhibited. Thus, within the ranges of externally applied currents (2.0,7.2) and (27.5,32.8), oscillations and a steady-state equilibrium are stable regimes as shown in Figure 2. 4 Discussion We have developed a two-state silicon neuron and a mathematical model that describes the behavior of this neuron. We have shown experimentally and verified mathematically that this silicon neuron has three regions of operation under the variation of its external current (one of its parameters). We also perform bifurcation analysis of the mathematical model by varying the externally applied current and show that the behaviors exhibited by the silicon neuron under corresponding conditions are in good agreement to those predicted by the bifurcation analysis. This analysis and comparison to experiment is an important step toward our understanding of a variety of oscillatory hardware networks that we and others are developing. The Bifurcation Analysis ofa Silicon Neuron 737 model facilitates an understanding of the neurons that the hardware alone does not provide. In particular for this neuron, the model allows us to determine the location of the unstable fixed points and the types of bifurcations that are exhibited. In higher-order systems, we expect that the model will provide us insight about observed behaviors and complex bifurcations in the phase space. The good matching between the model and the experimental data described in this paper gives us some confidence that future analysis efforts will prove fruitful. Acknowledgments S. DeWeerth and G. Patel are funded by NSF grant IBN-95 II 721 , G.S. Cymbalyuk is supported by Russian Foundation of Fundamental Research grant 99-04-49112, R.L. Calabrese and G.S. Cymbalyuk are supported by NIH grants NS24072 and NS34975. References [1] Van Der Pol, B (1939) Biological rhythms considered as relaxation oscillations In H. Bremmer and c.J. Bouwkamp (eds) Selected Scientific Papers, Vol 2, North Holland Pub. Co., 1960. [2] Mead, C.A. Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA, 1989. [3] Simoni, M.E, Patel, G.N., DeWeerth, S.P., & Calabrese, RL. Analog VLSI model of the leech heartbeat elemental oscillator. Sixth Annual Computational Neuroscience Meeting, 1997. in Big Sky, Montana. [4] DeWeerth, S., Patel, G., Schimmel, D., Simoni, M. and Calabrese, R (1997). In Proceedings of the Seventeenth Conference on Advanced Research in VLSI, RB. Brown and A.T. Ishii (eds), Los Alamitos, CA: IEEE Computer Society, 182-200. [5] Marder, E. & Calabrese, RL. (1996) Principles of rhythmic motor pattern generation. Physiological Reviews 76 (3): 687-717. [6] Kopell, N. & Ermentrout, B. (1988) Coupled oscillators and the design of central pattern generators. Mathematical biosciences 90: 87-109. [7] Skinner, EK., Turrigiano, G.G., & Marder, E. (1993) Frequency and burst duration in oscillating neurons and two-cell networks. Biological Cybernetics 69: 375-383. [8] Skinner, EK., Gramoll, S., Calabrese, R.L., Kopell, N. & Marder, E. (1994) Frequency control in biological half-center oscillators. In EH. Eeckman (ed.), Computation in neurons and neural systems, pp. 223-228, Boston: Kluwer Academic Publishers. [9] Patel, G. Holleman, J., DeWeerth, S. Analog VLSI model of intersegmental coordination with nearest-neighbor coupling. In , 1997. [10] Patel, G. A neuromorphic architecture for modelling intersegmental coordination. Ph.D. dissertation, Georgia Institute of Technology, 1999. [11] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Applied Mathematical Sciences, 42. Springer-Verlag, New York, New York, Heidelberg, Berlin, 1983. [12] Morris, C. and Lecar, H. (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. J, 35: 193-213. [13] Rinzel, J. & Ermentrout, G.B. (1989) Analysis of Neural Excitability and Oscillations. In C. Koch and I. Segev (eds) Methods in Neuronal Modeling from Synapses to Networks. MIT press, Cambridge, MA. [14] Khibnik, A. I. , Kuznetsov, Yu.A., Levitin, v.v., Nikolaev, E.V. (1993) Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. Physica D 62 (1-4): 360-367. 

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Greedy importance sampling Dale Schuurmans Department of Computer Science University of Waterloo dale@cs.uwaterloo.ca Abstract I present a simple variation of importance sampling that explicitly searches for important regions in the target distribution. I prove that the technique yields unbiased estimates, and show empirically it can reduce the variance of standard Monte Carlo estimators. This is achieved by concentrating samples in more significant regions of the sample space. 1 Introduction It is well known that general inference and learning with graphical models is computationally hard [1] and it is therefore necessary to consider restricted architectures [13], or approximate algorithms to perform these tasks [3, 7]. Among the most convenient and successful techniques are stochastic methods which are guaranteed to converge to a correct solution in the limit oflarge samples [10, 11, 12, 15]. These methods can be easily applied to complex inference problems that overwhelm deterministic approaches. The family of stochastic inference methods can be grouped into the independent Monte Carlo methods (importance sampling and rejection sampling [4, 10, 14]) and the dependent Markov Chain Monte Carlo (MCMC) methods (Gibbs sampling, Metropolis sampling, and "hybrid" Monte Carlo) [5, 10, 11, 15]. The goal of all these methods is to simulate drawing a random sample from a target distribution P (x) (generally defined by a Bayesian network or graphical model) that is difficult to sample from directly. This paper investigates a simple modification of importance sampling that demonstrates some advantages over independent and dependent-Markov-chain methods. The idea is to explicitly search for important regions in a target distribution P when sampling from a simpler proposal distribution Q. Some MCMC methods, such as Metropolis and "hybrid" Monte Carlo, attempt to do something like this by biasing a local random search towards higher probability regions, while preserving the asymptotic "fair sampling" properties of the exploration [11, 12]. Here I investigate a simple direct approach where one draws points from a proposal distribution Q but then explicitly searches in P to find points from significant regions. The main challenge is to maintain correctness (i.e., unbiased ness) of the resulting procedure, which we achieve by independently sampling search subsequences and then weighting the sample points so that their expected weight under the proposal distribution Q matches their true probability under the target P. 597 Greedy Importance Sampling Importance sampling ? Draw Xl , ... , X n independently from Q. ? Weight each point Xi by W(Xi) = ~I::l. ? For a random variable, f, estimate E p (.,) f(x) I ",n by f = n L.Ji=I f(Xi)W(Xi). "Indirect" importance sampling ? Draw Xl, ... ,xn independently from Q. .Weighteachpointxibyu(xi) = (3~~i/. ? For a random variable, f, estimate Ep(.,)f(x) A by A f = ",n L.Ji=I f(Xi)U(Xi)/ ",n L.Ji=I U(Xi). Figure 1: Regular and "indirect" importance sampling procedures 2 Generalized importance sampling Many inference problems in graphical models can be cast as determining the expected value of a random variable of interest, f, given observations drawn according to a target distribution P. That is, we are interested in computing the expectation Ep(x) f(x). Usually the random variable f is simple, like the indicator of some event, but the distribution P is generally not in a form that we can sample from efficiently. Importance sampling is a useful technique for estimating Ep(x) f (x) in these cases. The idea is to draw independent points xl, .. " Xn from a simpler "proposal" distribution Q, but then weight these points by w(x) P(x)/Q(x) to obtain a "fair" representation of P. Assuming that we can efficiently evaluate P(x) at each point, the weighted sample can be used to estimate desired expectations (Figure 1). The correctness (i.e., unbiasedness) of this procedure is easy to establish, since the expected weighted value of f under Q is just Eq(x)f(x)w(x) = = EXEX [f(x)w(x)] Q(x) = Ex EX [f(x)~t:n Q(x) = EXEX f(x)P(x) = Ep(x)f(x), This technique can be implemented using "indirect" weights u( x) = f3P (x) / Q ( x) and an alternative estimator (Figure 1) that only requires us to compute a fixed multiple of P (x). This preserves asymptotic correctness because ~ E7=1 f(xdu(xd and ~ E?=l U(Xi) converge to f3Ep(x)f(x) and f3 respectively, which yields j -t Ep(x)f(x) (generally [4]). It will always be possible to apply this extended approach below, but we drop it for now. Importance sampling is an effective estimation technique when Q approximates P over most of the domain, but it fails when Q misses high probability regions of P and systematically yields samples with small weights. In this case, the reSUlting estimator will have high variance because the sample will almost always contain unrepresentative points but is sometimes dominated by a few high weight points. To overcome this problem it is critical to obtain data points from the important regions of P. Our goal is to avoid generating systematically under-weight samples by explicitly searching for significant regions in the target distribution P. To do this, and maintain the unbiased ness of the resulting procedure, we develop a series of extensions to importance sampling that are each provably correct. The first extension is to consider sampling blocks of points instead of just individual points. Let B be a partition of X into finite blocks B, where UBE8 B = X, B n B' = 0, and each B is finite. (Note that B can be infinite.) The "block" sampling procedure (Figure 2) draws independent blocks of points to construct the final sample, but then weights points by their target probability P(x) divided by the total block probability Q(B (x)), For discrete spaces it is easy to verify that this procedure yields unbiased estimates, since Eq(x) [EXjEB(X) f(xj)w(Xj)] EBE8 EXiEB EBE8 [EXjEB [EXjEB = EXEX f(xj)w(Xj)] Q(Xi) = f(xj) ~?~n Q(B) [L:XjEB(X) f(xj)w(Xj)] Q(x) EBE8 [EXjEB = f(xj)w(Xj)] Q(B) = = EBE8 [EXjEB f(xj )P(Xj)] = L:xEx f(x)P(x). 598 D. Schuurmans "Block" importance sampling ? Draw Xl , ... , Xn independently from Q. ? For Xi, recover block Bi = {Xi,l, ... ,Xi,bJ. ? Create a large sample out of the blocks Xl ,1, ... , Xl ,bl , X2 ,1, ... , X2,b2' ??? , Xn,l, ... , Xn ,b" . ? Weighteachx I' ,}' byw(x.') = ",oiP(zi,j) I,} Q(z,,)' L.Jj=l ' ,J ? For a random variable, f, estimate Ep(z) f(x) by j = ~ 2:~=1 2:~~1 f(Xi ,j)W(Xi,j) . "Sliding window" importance sampling ? Draw Xl, ... , xn independently from Q. ? For Xi , recover block Bi , and let Xi ,l = Xi : - Get Xi,l 'S successors Xi,l, Xi,2, ... , Xi ,m by climbing up m - 1 steps from Xi ,l . - Get predecessorsxi,_m+l" ... ,Xi,-l , Xi,O by climbing down m - 1 steps from Xi,l . - Weight W(Xi ,j)= P(Xi ,i)/2:!=i_m+lQ (x; ,k) ? Create final sample from successor points XI , I, ... , Xl , m, X2,1 , ??. , X2 ,m, ?.. , Xn ,l, ... , Xn ,m. ? For a random variable, f, estimate Ep(z) f(x) 1 ",n ",m by f = n 6i=1 6j=1 f(Xi,j)W(Xi,j) . A Figure 2: "Block" and "sliding window" importance sampling procedures Crucially, this argument does not depend on how the partition of X is chosen. In fact, we could fix any partition, even one that depended on the target distribution P, and still obtain an unbiased procedure (so long as the partition remains fixed) . Intuitively, this works because blocks are drawn independently from Q and the weighting scheme still produces a "fair" representation of P . (Note that the results presented in this paper can all be extended to continuous spaces under mild technical restrictions. However, for the purposes of clarity we will restrict the technical presentation in this paper to the discrete case.) The second extension is to allow countably infinite blocks that each have a discrete total order . . . < Xi -1 < Xi < Xi +1 < .. . defined on their elements. This order could reflect the relative probability of Xi and X j under P, but for now we just consider it to be an arbitrary discrete order. To cope with blocks of unbounded length, we employ a "sliding window" sampling procedure that selects a contiguous sub-block of size m from within a larger selected block (Figure 2). This procedure builds each independent subsample by choosing a random point Xl from the proposal distribution Q, determining its containing block B(xt), and then climbing up m - 1 steps to obtain the successors Xl, X2, ??. , X m , and climbing down m - 1 steps to obtain the predecessors X- m +1 , ... , X-I, Xo . The successor points (including Xl) appear in the final sample, but the predecessors are only used to determine the weights of the sample points. Weights are determined by the target probability P (x) divided by the probability that the point X appears in a random reconstruction under Q. This too yields an unbiased estimator [2:~lf(xj)w(Xj)] = 2: XtEX [2:~;~-1 f(xj)2:~=j~~:: Q(Xk)] Q(Xl ) = '" 2: 2: l + m - 1 f(xj)P(Xj)Q(xd - ' " 2: 2: j f(xj)P(Xj)Q(xt} 6BEB x t EB j=l "'J. Q(Xk) - 6BEB Xj EB l=j-m+1 "'J . Q(Xk) 6k=J-m+l 6k=J-m+l 2:BEB2: x j EBf(xj )P(Xj )Ei::=::: ~;::~ = 2:BEB2:xjEBf(xj )P(Xj)= 2: xEx f(x)P(x). sinceEQ(x) (The middle line breaks the sum into disjoint blocks and then reorders the sum so that instead of first choosing the start point Xl and then XL'S successors Xl, .. ? , Xl+m-l. we first choose the successor point Xj and then the start points Xj-m+1 , ... , Xj that could have led to Xj). Note that this derivation does not depend on the particular block partition nor on the particular discrete orderings, so long as they remain fixed. This means that, again, we can use partitions and orderings that explicitly depend on P and still obtain a correct procedure. 599 Greedy Importance Sampling "Greedy" importance sampling (I-D) e Draw Xl , ... , Xn independently from Q. eForeachxi , letxi, l =Xi : - Compute successors Xi ,l, Xi ,2, ... ,Xi,m by taking m - 1 size ? steps in the direction of increase. - Compute predecessors Xi,-m+l, ... ,Xi ,-l ,Xi ,Oby taking m -1 size ? steps in the direction of decrease. - If an improper ascent or descent occurs, truncate paths as shown on the upper right. - Weightw(xi,j) = P(Xi ,j)/L:~=j_m+l Q(Xi ,k) . e Create the final sample from successor points collision If(x)P(x)1 k x? x" ~ if 6 "\, XI ,l, ??? , XI ,m , X'2 , I , ??? , X2 ,m , ??? ,Xn ,l, ??? , ::tn ,m- e For a random variable, f, estimate Ep(z) f(x) A by f ",m = -;;1 ",n L..ti=l L.Jj=l f(Xi ,j)W(Xi ,j) . merge Figure 3: "Greedy" importance sampling procedure; "colliding" and "merging" paths. 3 Greedy importance sampling: I-dimensional case Finally, we apply the sliding window procedure to conduct an explicit search for important regions in X. It is well known that the optimal proposal distribution for importance sampling isjust Q* (x) = If(x )P(x)11 EXEX If(x )P(x) 1 (which minimizes variance [2]). Here we apply the sliding window procedure using an order structure that is determined by the objective If(x )P(x )1 . The hope is to obtain reduced variance by sampling independent blocks of points where each block (by virtue of being constructed via an explicit search) is likely to contain at least one or two high weight points. That is, by capturing a moderate size sample of independent high weight points we intuitively expect to outperform standard methods that are unlikely to observe such points by chance. Our experiments below verify this intuition (Figure 4). The main technical issue is maintaining unbiasedness, which is easy to establish in the 1dimensional case. In the simple I-d setting, the "greedy" importance sampling procedure (Figure 3) first draws an initial point Xl from Q and then follows the direction of increasing If(x)P(x)l, taking fixed size ? steps, until either m - 1 steps have been taken or we encounter a critical point. A single "block" in our final sample is comprised of a complete sequence captured in one ascending search. To weight the sample points we account for all possible ways each point could appear in a subsample, which, as before, entails climbing down m-l steps in the descent direction (to calculate the denominators). The unbiasedness of the procedure then follows directly from the previous section, since greedy importance sampling is equivalent to sliding window importance sampling in this setting. The only nontrivial issue is to maintain disjoint search paths. Note that a search path must terminate whenever it steps from a point x? to a point x** with lower value; this indicates that a collision has occurred because some other path must reach x? from the "other side" of the critical point (Figure 3). At a collision, the largest ascent point x? must be allocated to a single path. A reasonable policy is to allocate x? to the path that has the lowest weight penultimate point (but the only critical issue is ensuring that it gets assigned to a single block). By ensuring that the critical point is included in only one of the two distinct search paths, a practical estimator can be obtained that exhibits no bias (Figure 4). To test the effectiveness of the greedy approach I conducted several I-dimensional experiments which varied the relationship between P, Q and the random variable f (Figure 4). In 600 D. Schuurmans these experiments greedy importance sampling strongly outperformed standard methods, including regular importance sampling and directly sampling from the target distribution P (rejection sampling and Metropolis sampling were not competitive). The results not only verify the unbiasedness of the greedy procedure, but also show that it obtains significantly smaller variances across a wide range of conditions. Note that the greedy procedure actually uses m out of 2m - 1 points sampled for each block and therefore effectively uses a double sample. However, Figure 4 shows that the greedy approach often obtains variance reductions that are far greater than 2 (which corresponds to a standard deviation reduction of V2). 4 Multi-dimensional case Of course, this technique is worthwhile only if it can be applied to multi-dimensional problems. In principle, it is straightforward to apply the greedy procedure of Section 3 to multi-dimensional sample spaces. The only new issue is that discrete search paths can now possibly "merge" as well as "collide"; see Figure 3. (Recall that paths could not merge in the previous case.) Therefore, instead of decomposing the domain into a collection of disjoint search paths, the objective If(x)P(x)1 now decomposes the domain into a forest of disjoint search trees. However, the same principle could be used to devise an unbiased estimator in this case: one could assign a weight to a sample point x that is just its target probability P (x) divided by the total Q-probability of the subtree of points that lead to x in fewer than m steps. This weighting scheme can be shown to yield an unbiased estimator as before. However, the resulting procedure is impractical because in an N-dimensional sample space a search tree will typically have a branching factor of n(N); yielding exponentially large trees. Avoiding the need to exhaustively examine such trees is the critical issue in applying the greedy approach to multi-dimensional spaces. The simplest conceivable strategy is just to ignore merge events. Surprisingly, this turns out to work reasonably well in many circumstances. Note that merges will be a measure zero event in many continuous domains. In such cases one could hope to ignore merges and trust that the probability of "double counting" such points would remain near zero. I conducted simple experiments with a version of greedy importance sampling procedure that ignored merges. This procedure searched in the gradient ascent direction of the objective If{x)p{x)1 and heuristically inverted search steps by climbing in the gradient descent direction. Figures 5 and 6 show that, despite the heuristic nature of this procedure, it nevertheless demonstrates credible performance on simple tasks. The first experiment is a simple demonstration from [12, 10] where the task is to sample from a bivariate Gaussian distribution P of two highly correlated random variables using a "weak" proposal distribution Q that is standard normal (depicted by the elliptical and circular one standard deviation contours in Figure 5 respectively). Greedy importance sampling once again performs very well (Figure 5); achieving unbiased estimates with lower variance than standard Monte Carlo estimators, including common MCMC methods. To conduct a more significant study, I applied the heuristic greedy method to an inference problem in graphical models: recovering the hidden state sequence from a dynamic probabilistic model, given a sequence of observations. Here I considered a simple Kalman filter model which had one state variable and one observation variable per time-step, and used theconditionaldistributionsXtlXt _ l "-' N(Xt_l,O';), ZtlXt "" N(xt,O'~) and initial distribution Xl "-' N(O,O';) . The problem was to infer the value of the final state variable Xt given the observations Zl, Z2, "', Zt. Figure 6 again demonstrates that the greedy approach Greedy Importance Sampling 601 has a strong advantage over standard importance sampling. (In fact, the greedy approach can be applied to "condensation" [6, 8] to obtain further improvements on this task, but space bounds preclude a detailed discussion.) Overall, these preliminary results show that despite the heuristic choices made in this section, the greedy strategy still performs well relative to common Monte Carlo estimators, both in terms of bias and variance (at least on some low and moderate dimension problems). However, the heuristic nature of this procedure makes it extremely unsatisfying. In fact, merge points can easily make up a significant fraction of finite domains. It turns out that a rigorously unbiased and feasible procedure can be obtained as follows. First, take greedy fixed size steps in axis parallel directions (which ensures the steps can be inverted). Then, rather than exhaustively explore an entire predecessor tree to calculate the weights of a sample point, use the well known technique of Knuth [9] to sample a single path from the root and obtain an unbiased estimate of the total Q-probability of the tree. This procedure allows one to formulate an asymptotically unbiased estimator that is nevertheless feasible to implement. It remains important future work to investigate this approach and compare it to other Monte Carlo estimation methods on large dimensional problems-in particular hybrid Monte Carlo [11, 12]. The current results already suggest that the method could have benefits. References [1] P. Dagum and M. Luby. Approximating probabilistic inference in Bayesian belief networks is NP-hard. Artif Intell, 60: 141-153, 1993. [2] M. Evans. Chaining via annealing. Ann Statist, 19:382-393, 1991. [3] B. Frey. Graphical Models for Machine Learning and Digital Communication. MIT Press, Cambridge, MA, 1998. [4] J. Geweke. Baysian inference in econometric models using Monte Carlo integration. Econometrica, 57:1317-1339, 1989. [5] W. Gilks, S. Richardson, and D. Spiegelhalter. Markov chain Monte Carlo in practice. Chapman and Hall, 1996. [6] M. Isard and A. Blake. Coutour tracking by stochastic propagation of conditional density. In ECCV, 1996. [7] M. Jordan, Z. Ghahramani, T. Jaakkola, and L. Saul. An introduction to variational methods for graphical models. In Learning in Graphical Models. Kluwer, 1998. [8] K. Kanazawa, D. Koller, and S. Russell. Stochastic simulation algorithms for dynamic probabilistic networks. In UAl, 1995. [9] D. Knuth. Estimating the efficiency of backtracking algorithms. Math. Comput., 29(129): 121136,1975. [10] D. MacKay. Intro to Monte Carlo methods. In Learning in Graphical Models. Kluwer, 1998. [11] R. Neal. Probabilistic inference using Markov chain Monte Carlo methods. 1993. [12] R. Neal. Bayesian Learning for Neural Networks . Springer, New York, 1996. [13] J. Pearl. Probabilistic Reasoning in Intelligence Systems. Morgan Kaufmann, 1988. [14] R. Shacter and M. Peot. Simulation approaches to general probabilistic inference in belief networks. In Uncertainty in Artificial Intelligence 5. Elsevier, 1990. [15] M. Tanner. Tools for statistical inference: Methods for exploration of posterior distributions and likelihoodfunctions. Springer, New York, 1993. 602 D. Schuurmans . f' ._.11 ? Direc 0.779 0.001 0.071 mean bias stdev Greed Imprt 0.777 0.003 0.065 0.781 0.001 0.Q38 Direc Greed 1m rt Direc Greed 1.038 0.002 0.088 1.044 0.003 0.049 1.032 0.008 0.475 0.258 0.049 0.838 0.208 0.000 0.010 Imrt 0.209 0.001 0.095 . .. {/... Direc 6.024 0.001 0.069 Greed Imprt 6.028 0.004 0.037 6.033 0.009 0.094 Figure 4: I-dimensional experiments: 1000 repetitions on estimation samples of size 100. Problems with varying relationships between P, Q, I and IIPI. / / .. :~:. .. .. .... .. ., 'I:'::'~" ,1 :.~ mean bias stdev Direct Greedy Importance Rejection 0.1884 0.0022 0.07 0.1937 0.0075 0.1374 0.1810 0.0052 0.1762 0.1506 0.0356 0.2868 Gibbs 0.3609 0.1747 0.5464 Metropolis 8.3609 8.1747 22.1212 Figure 5: 2-dimensional experiments: 500 repetitions on estimation samples of size 200. Pictures depict: direct, greedy importance, regular importance, and Gibbs sampling, showing 1 standard deviation countours (dots are sample points, vertical lines are weights). mean bias stdev Importance Greedy 5.2269 2.7731 1.2107 6.9236 1.0764 0.1079 Figure 6: A 6-dimensional experiment: 500 repetitions on estimation samples of size 200. Estimating the value of Xt given the observations Zl, "" Zt. Pictures depict paths sampled by regular versus greedy importance sampling. 

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Lower Bounds on the Complexity of Approximating Continuous Functions by Sigmoidal Neural Networks Michael Schmitt Lehrstuhl Mathematik und Informatik FakuWit ftir Mathematik Ruhr-Universitat Bochum D-44780 Bochum, Germany mschmitt@lmi.ruhr-uni-bochum.de Abstract We calculate lower bounds on the size of sigmoidal neural networks that approximate continuous functions. In particular, we show that for the approximation of polynomials the network size has to grow as O((logk)1/4) where k is the degree of the polynomials. This bound is valid for any input dimension, i.e. independently of the number of variables. The result is obtained by introducing a new method employing upper bounds on the Vapnik-Chervonenkis dimension for proving lower bounds on the size of networks that approximate continuous functions. 1 Introduction Sigmoidal neural networks are known to be universal approximators. This is one of the theoretical results most frequently cited to justify the use of sigmoidal neural networks in applications. By this statement one refers to the fact that sigmoidal neural networks have been shown to be able to approximate any continuous function arbitrarily well. Numerous results in the literature have established variants of this universal approximation property by considering distinct function classes to be approximated by network architectures using different types of neural activation functions with respect to various approximation criteria, see for instance [1, 2, 3, 5, 6, 11, 12, 14, 15]. (See in particular Scarselli and Tsoi [15] for a recent survey and further references.) All these results and many others not referenced here, some of them being constructive, some being merely existence proofs, provide upper bounds for the network size asserting that good approximation is possible if there are sufficiently many network nodes available. This, however, is only a partial answer to the question that mainly arises in practical applications: "Given some function, how many network nodes are needed to approximate it?" Not much attention has been focused on establishing lower bounds on the network size and, in particular, for the approximation of functions over the reals. As far as the computation of binary-valued Complexity ofApproximating Continuous Functions by Neural Networks 329 functions by sigmoidal networks is concerned (where the output value of a network is thresholded to yield 0 or 1) there are a few results in this direction. For a specific Boolean function Koiran [9] showed that networks using the standard sigmoid u(y) = 1/(1 + e- Y ) as activation function must have size O(nl/4) where n is the number of inputs. (When measuring network size we do not count the input nodes here and in what follows.) Maass [13] established a larger lower bound by constructing a binary-valued function over IRn and showing that standard sigmoidal networks require O(n) many network nodes for computing this function. The first work on the complexity of sigmoidal networks for approximating continuous functions is due to DasGupta and Schnitger [4]. They showed that the standard sigmoid in network nodes can be replaced by other types of activation functions without increasing the size of the network by more than a polynomial. This yields indirect lower bounds for the size of sigmoidal networks in terms of other network types. DasGupta and Schnitger [4] also claimed the size bound AO(I/d) for sigmoidal networks with d layers approximating the function sin(Ax). In this paper we consider the problem of using the standard sigmoid u(y) = 1/(1 + e- Y ) in neural networks for the approximation of polynomials. We show that at least O?logk)1/4) network nodes are required to approximate polynomials of degree k with small error in the loo norm. This bound is valid for arbitrary input dimension, i.e., it does not depend on the number of variables. (Lower bounds can also be obtained from the results on binary-valued functions mentioned above by interpolating the corresponding functions by polynomials. This, however, requires growing input dimension and does not yield a lower bound in terms of the degree.) Further, the bound established here holds for networks of any number of layers. As far as we know this is the first lower bound result for the approximation of polynomials. From the computational point of view this is a very simple class of functions; they can be computed using the basic operations addition and multiplication only. Polynomials also play an important role in approximation theory since they are dense in the class of continuous functions and some approximation results for neural networks rely on the approximability of polynomials by sigmoidal networks (see, e.g., [2, 15]). We obtain the result by introducing a new method that employs upper bounds on the Vapnik-Chervonenkis dimension of neural networks to establish lower bounds on the network size. The first use of the Vapnik-Chervonenkis dimension to obtain a lower bound is due to Koiran [9] who calculated the above-mentioned bound on the size of sigmoidal networks for a Boolean function. Koiran's method was further developed and extended by Maass [13] using a similar argument but another combinatorial dimension. Both papers derived lower bounds for the computation of binary-valued functions (Koiran [9] for inputs from {O, 1}n, Maass [13] for inputs from IRn). Here, we present a new technique to show that and how lower bounds can be obtained for networks that approximate continuous functions. It rests on two fundamental results about the Vapnik-Chervonenkis dimension of neural networks. On the one hand, we use constructions provided by Koiran and Sontag [10] to build networks that have large Vapnik-Chervonenkis dimension and consist of gates that compute certain arithmetic functions. On the other hand, we follow the lines of reasoning of Karpinski and Macintyre [7] to derive an upper bound for the VapnikChervonenkis dimension of these networks from the estimates of Khovanskil [8] and a result due to Warren [16]. In the following section we give the definitions of sigmoidal networks and the VapnikChervonenkis dimension. Then we present the lower bound result for function approximation. Finally, we conclude with some discussion and open questions. 330 2 M Schmitt Sigmoidal Neural Networks and VC Dimension We briefly recall the definitions of a sigmoidal neural network and the VapnikChervonenkis dimension (see, e.g., [7, 10]). We consider /eed/orward neural networks which have a certain number of input nodes and one output node. The nodes which are not input nodes are called computation nodes and associated with each of them is a real number t, the threshold. Further, each edge is labelled with a real number W called weight. Computation in the network takes place as follows: The input values are assigned to the input nodes. Each computation node applies the standard sigmoid u(y) = 1/(1 + e- V ) to the sum W1Xl + ... + WrXr - t where Xl, .?. ,X r are the values computed by the node's predecessors, WI, ??? ,W r are the weights of the corresponding edges, and t is the threshold. The output value of the network is defined to be the value computed by the output node. As it is common for approximation results by means of neural networks, we assume that the output node is a linear gate, i.e., it just outputs the sum WIXI + ... + WrXr - t. (Clearly, for computing functions on finite sets with output range [0, 1] the output node may apply the standard sigmoid as well.) Since u is the only sigmoidal function that we consider here we will refer to such networks as sigmoidal neural networks. (Sigmoidal functions in general need to satisfy much weaker assumptions than u does.) The definition naturally generalizes to networks employing other types of gates that we will make use of (e.g. linear, multiplication, and division gates). The Vapnik-Chervonenkis dimension is a combinatorial dimension of a function class and is defined as follows: A dichotomy of a set S ~ IRn is a partition of S into two disjoint subsets (So, Sl) such that So U SI = S. Given a set F offunctions mapping IRn to {O, I} and a dichotomy (So, Sd of S, we say that F induces the dichotomy (So, Sd on S if there is some f E F such that /(So) ~ {O} and f(Sd ~ {I}. We say further that F shatters S if F induces all dichotomies on S. The VapnikChervonenkis (VC) dimension of F, denoted VCdim(F), is defined as the largest number m such that there is a set of m elements that is shattered by F. We refer to the VC dimension of a neural network, which is given in terms of a "feedforward architecture", i.e. a directed acyclic graph, as the VC dimension of the class of functions obtained by assigning real numbers to all its programmable parameters, which are in general the weights and thresholds of the network or a subset thereof. Further, we assume that the output value of the network is thresholded at 1/2 to obtain binary values. 3 Lower Bounds on Network Size Before we present the lower bound on the size of sigmoidal networks required for the approximation of polynomials we first give a brief outline of the proof idea. We will define a sequence of univariate polynomials (Pn)n>l by means of which we show how to construct neural architectures N n consistmg of various types of gates such as linear, multiplication, and division gates, and, in particular, gates that compute some of the polynomials. Further, this architecture has a single weight as programmable parameter (all other weights and thresholds are fixed). We then demonstrate that, assuming the gates computing the polynomials can be approximated by sigmoidal neural networks sufficiently well, the architecture Nn can shatter a certain set by assigning suitable values to its programmable weight. The final step is to reason along the lines of Karpinski and Macintyre [7] to obtain via Khovanskil's estimates [8] and Warren's result [16] an upper bound on the VC dimension of N n in terms of the number of its computation nodes. (Note that we cannot directly apply Theorem 7 of [7] since it does not deal with division gates.) Comparing this bound with the cardinality of the shattered set we will then be able 331 Complexity ofApproximating Continuous Functions by Neural Networks (3) W 1 n P3 (1) (2) W1 W1 (3) Wi (3) W1 n P2 (2) Wj (1) Wk (1) (2) Wn Wn n P1 Wn j --------------------------------~ k--------------------------------------------------~ Figure 1: The network N n with values k, j, i, 1 assigned to the input nodes Xl, X2, X3, X4 respectively. The weight W is the only programmable parameter of the network. to conclude with a lower bound on the number of computation nodes in N n and thus in the networks that approximate the polynomials. Let the sequence (Pn)n2: l of polynomials over IR be inductively defined by Pn(X) = { 4x(1 - x) P(Pn-dx)) n = 1, n 2:: 2 . Clearly, this uniquely defines Pn for every n 2:: 1 and it can readily be seen that Pn has degree 2n. The main lower bound result is made precise in the following statement. Theorem 1 Sigmoidal neural networks that approximate the polynomials (Pn)n >l on the interval [0,1] with error at most O(2- n ) in the 100 norm must have at least n(nl/4) computation nodes. Proof. For each n a neural architecture N n can be constructed as follows: The network has four input nodes Xl, X2, X3, X4. Figure 1 shows the network with input values assigned to the input nodes in the order X4 = 1, X3 = i, X2 = j, Xl = k. There is one weight which we consider as the (only) programmable parameter of N n . It is associated with the edge outgoing from input node X4 and is denoted by w. The computation nodes are partitioned into six levels as indicated by the boxes in Figure 1. Each level is itself a network. Let us first assume, for the sake of simplicity, that all computations over real numbers are exact. There are three levels labeled with II, having n + 1 input nodes and one output node each, that compute so-called projections 7r : IRnH -+ IR where 7r(YI,"" Yn, a) = Ya for a E {I, ... , n}. The levels labeled P3 , P2 , PI have one input node and n output nodes each. Level P3 receives the constant 1 as input and thus the value W which is the parameter of the network. We define the output values of level P A for>. = 3,2, 1 by (A) wb = Pbon"'-l ( v) , b= 1, ... ,n where v denotes the input value to level P A. This value is equal to w for>. = 3 and (A+l) , .?. , Wn()..+l) ,XA+l ) oth erWlse. . OUT (A) can b id vve observe t h at wb+l e calcu ate f rom 7r (WI 332 M Schmitt w~A) as Pn>'_l(W~A?). Therefore, the computations of level P A can be implemented using n gates each of them computing the function Pn>.-l. We show now that Nn can shatter a set of cardinality n 3 ? Let S = {I, ... ,n p. It has been shown in Lemma 2 of [10] that for each (/31 , ... , /3r) E {O, 1Y there exists some W E [0,1] such that for q = 1, ... ,T pq(w) E [0,1/2) if /3q = 0, and pq(w) E (1/2,1] if /3q = 1. This implies that, for each dichotomy (So, Sd of S there is some that for every (i, j, k) E S Pk (pj.n (Pi.n 2(w))) Pk(Pj.n(Pi.n2(w))) < 1/2 > 1/2 if if W E [0,1] such (i, j, k) E So , (i,j,k)ES1' Note that Pk(Pj.n(Pi.n2 (w))) is the value computed by N n given input values k, j, i, 1. Therefore, choosing a suitable value for w, which is the parameter of Nn , the network can induce any dichotomy on S. In other words, S is shattered by Nn . An such that for each E > weights can be chosen for An such that the function in,? computed by this network satisfies lim?~o in,?(Yl, ... ,Yn, a) = Ya. Moreover, this architecture consists of O(n) computation nodes, which are linear, multiplication, and division gates. (Note that the size of An does not depend on E.) Therefore, choosing E sufficiently small, we can implement the projections 1r in N n by networks of O(n) computation nodes such that the resulting network N~ still shatters S. Now in N~ we have O(n) computation nodes for implementing the three levels labeled II and we have in each level P A a number of O(n) computation nodes for computing Pn>.-l, respectively. Assume now that the computation nodes for Pn>.-l can be replaced by sigmoidal networks such that on inputs from S and with the parameter values defined above the resulting network N:: computes the same functions as N~. (Note that the computation nodes for Pn>.-l have no programmable parameters.) ? It has been shown in Lemma 1 of [10] that there is an architecture N::. We estimate the size of According to Theorem 7 of Karpinski and Macintyre [7] a sigmoidal neural network with I programmable parameters and m computation nodes has VC dimension O((ml)2). We have to generalize this result slightly before being able to apply it. It can readily be seen from the proof of Theorem 7 in [7] that the result also holds if the network additionally contains linear and multiplication gates. For division gates we can derive the same bound taking into account that for a gate computing division, say x/y, we can introduce a defining equality x = z . Y where z is a new variable. (See [7] for how to proceed.) Thus, we have that a network with I programmable parameters and m computation nodes, which are linear, multiplication, division, and sigmoidal gates, has VC dimension O((ml)2). In particular, if m is the number of computation nodes of N::, the VC dimension can shatter a set is O(m 2 ). On the other hand, as we have shown above, of cardinality n 3 ? Since there are O(n) sigmoidal networks in computing the functions Pn>.-l, and since the number of linear, multiplication, and division gates is bounded by O(n), for some value of A a single network computing Pn>.-l must have size at least O(fo). This yields a lower bound of O(nl/4) for the size of a sigmoidal network computing Pn. N:: N:: Thus far, we have assumed that the polynomials Pn are computed exactly. Since polynomials are continuous functions and since we require them to be calculated only on a finite set of input values (those resulting from S and from the parameter values chosen for w to shatter S) an approximation of these polynomials is sufficient. A straightforward analysis, based on the fact that the output value of the network has a "tolerance" close to 1/2, shows that if Pn is approximated with error O(2- n ) Complexity ofApproximating Continuous Functions by Neural Networks 333 in the loo norm, the resulting network still shatters the set S. This completes the proof of the theorem. D The statement of the previous theorem is restricted to the approximation of polynomials on the input domain [0,1]. However, the result immediately generalizes to any arbitrary interval in llt Moreover, it remains valid for multivariate polynomials of arbitrary input dimension. Corollary 2 The approximation of polynomials of degree k by sigmoidal neural networks with approximation error O(ljk) in the 100 norm requires networks of size O((log k)1/4). This holds for polynomials over any number of variables. 4 Conclusions and Open Questions We have established lower bounds on the size of sigmoidal networks for the approximation of continuous functions. In particular, for a concrete class of polynomials we have calculated a lower bound in terms of the degree of the polynomials. The main result already holds for the approximation of univariate polynomials. Intuitively, approximation of multivariate polynomials seems to become harder when the dimension increases. Therefore, it would be interesting to have lower bounds both in terms of the degree and the input dimension. Further, in our result the approximation error and the degree are coupled. Naturally, one would expect that the number of nodes has to grow for each fixed function when the error decreases. At present we do not know of any such lower bound. We have not aimed at calculating the constants in the bounds. For practical applications such values are indispensable. Refining our method and using tighter results it should be straightforward to obtain such numbers. Further, we expect that better lower bounds can be obtained by considering networks of restricted depth. To establish the result we have introduced a new method for deriving lower bounds on network sizes. One of the main arguments is to use the functions to be approximated to construct networks with large VC dimension. The method seems suitable to obtain bounds also for the approximation of other types of functions as long as they are computationally powerful enough. Moreover, the method could be adapted to obtain lower bounds also for networks using other activation functions (e.g. more general sigmoidal functions, ridge functions, radial basis functions). This may lead to new separation results for the approximation capabilities of different types of neural networks. In order for this to be accomplished, however, an essential requirement is that small upper bounds can be calculated for the VC dimension of such networks. Acknowledgments I thank Hans U. Simon for helpful discussions. This work was supported in part by the ESPRIT Working Group in Neural and Computational Learning II, NeuroCOLT2, No. 27150. References [1] A. Barron. Universal approximation bounds for superposition of a sigmoidal function. IEEE Transactions on Information Theory, 39:930--945, 1993. 334 M Schmitt [2J C. K. Chui and X. Li. Approximation by ridge functions and neural networks with one hidden layer. Journal of Approximation Theory, 70:131-141,1992. [3J G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2:303-314, 1989. [4J B. DasGupta and G. Schnitger. The power of approximating: A comparison of activation functions. In C. L. Giles, S. J. Hanson, and J. D. Cowan, editors, Advances in Neural Information Processing Systems 5, pages 615-622, Morgan Kaufmann, San Mateo, CA, 1993. [5] K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4:251-257, 1991. [6] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2:359-366, 1989. [7] M. Karpinski and A. Macintyre. Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks. Journal of Computer and System Sciences, 54:169-176, 1997. [8] A. G. Khovanskil. Fewnomials, volume 88 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1991. [9] P. Koiran. VC dimension in circuit complexity. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity CCC'96, pages 81-85, IEEE Computer Society Press, Los Alamitos, CA, 1996. [10] P. Koiran and E. D. Sontag. Neural networks with quadratic VC dimension. Journal of Computer and System Sciences, 54:190-198, 1997. [11] V. Y. Kreinovich. Arbitrary nonlinearity is sufficient to represent all functions by neural networks: A theorem. Neural Networks, 4:381-383, 1991. [12] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6:861-867, 1993. [13] W. Maass. Noisy spiking neurons with temporal coding have more computational power than sigmoidal neurons. In M. Mozer, M. 1. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 211-217. MIT Press, Cambridge, MA, 1997. [14] H. Mhaskar. Neural networks for optimal approximation of smooth and analytic functions. Neural Computation, 8:164-177, 1996. [15J F. Scarselli and A. C. Tsoi. Universal approximation using feedforward neural networks: A survey of some existing methods and some new results. Neural Networks, 11:15-37, 1998. [16] H. E. Warren. Lower bounds for approximation by nonlinear manifolds. Transactions of the American Mathematical Society, 133:167-178, 1968. 

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Dynamics of Supervised Learning with Restricted Training Sets and Noisy Teachers A.C.C. Coolen Dept of Mathematics King's College London The Strand, London WC2R 2LS, UK tcoolen@mth.kc1.ac.uk C.W.H.Mace Dept of Mathematics King's College London The Strand, London WC2R 2LS, UK cmace@mth.kc1.ac.uk Abstract We generalize a recent formalism to describe the dynamics of supervised learning in layered neural networks, in the regime where data recycling is inevitable, to the case of noisy teachers. Our theory generates reliable predictions for the evolution in time of training- and generalization errors, and extends the class of mathematically solvable learning processes in large neural networks to those situations where overfitting can occur. 1 Introduction Tools from statistical mechanics have been used successfully over the last decade to study the dynamics of learning in layered neural networks (for reviews see e.g. [1] or [2]). The simplest theories result upon assuming the data set to be much larger than the number of weight updates made, which rules out recycling and ensures that any distribution of relevance will be Gaussian. Unfortunately, both in terms of applications and in terms of mathematical interest, this regime is not the most relevant one. Most complications and peculiarities in the dynamics of learning arise precisely due to data recycling, which creates for the system the possibility to improve performance by memorizing answers rather than by learning an underlying rule. The dynamics of learning with restricted training sets was first studied analytically in [3] (linear learning rules) and [4] (systems with binary weights). The latter studies were ahead of their time, and did not get the attention they deserved just because at that stage even the simpler learning dynamics without data recycling had not yet been studied. More recently attention has moved back to the dynamics of learning in the recycling regime. Some studies aimed at developing a general theory [5, 6, 7], some at finding exact solutions for special cases [8]. All general theories published so far have in common that they as yet considered realizable scenario's: the rule to be learned was implementable by the student, and overfitting could not yet occur. The next hurdle is that where restricted training sets are combined with unrealizable rules. Again some have turned to non-typical but solvable cases, involving Hebbian rules and noisy [9] or 'reverse wedge' teachers [10]. More recently the cavity method has been used to build a general theory [11] (as yet for batch learning only). In this paper we generalize the general theory launched in [6,5,7], which applies to arbitrary learning rules, to the case of noisy teachers. We will mirror closely the presentation in [6] (dealing with the simpler case of noise-free teachers), and we refer to [5, 7] for background reading on the ideas behind the formalism. A. C. C. Coolen and C. W. H. Mace 238 2 Definitions As in [6, 5] we restrict ourselves for simplicity to perceptrons. A student perceptron operates a linear separation, parametrised by a weight vector J E iRN : S:{-I,I}N -t{-I,I} S(e) = sgn[J?e] It aims to emulate a teacher o~erating a similar rule, which, however, is characterized by a variable weight vector BE iR ,drawn at random from a distribution P(B) such as P(B) = >'6[B+B*] output noise: + (1->')6[B-B*] (1) P(B) = [~~/NrN e- tN (B-B')2/E2 (2) The parameters>. and ~ control the amount of teacher noise, with the noise-free teacher B = B* recovered in the limits>. -t 0 and ~ -t O. The student modifies J iteratively, using examples of input vectors which are drawn at random from a fixed (randomly composed) E {-I, I}N with a> 0, and the corresponding training set containing p = aN vectors values of the teacher outputs. We choose the teacher noise to be consistent, i.e. the answer will remain the same when that particular question given by the teacher to a question re-appears during the learning process. Thus T(e?) = sgn[BJL . e], with p teacher weight vectors BJL, drawn randomly and independently from P(B), and we generalize the training l , B l ), . .. , (e, BP)}. Consistency of teacher noise is natural set accordingly to jj = in terms of applications, and a prerequisite for overfitting phenomena. Averages over the training set will be denoted as ( ... ) b; averages over all possible input vectors E {-I, I}N as ( ... )e. We analyze two classes of learning rules, of the form J (? + 1) = J (?) + f).J (?): Gaussian weight noise: e e e He e = 11 {e(?) 9 [J(?)?e(?), B(?)?e(?)] - ,J(?) } f).J(?) = 11 {(e 9 [J(?)?e, B?eDl> - ,J(m) } on-line: f).J(?) batch : (3) In on-line learning one draws at each step ? a question/answer pair (e (?), B (?)) at random from the training set. In batch learning one iterates a deterministic map which is an average over all data in the training set. Our performance measures are the training- and generalization errors, defined as follows (with the step function O[x > 0] = 1, O[x < 0] = 0): Et(J) = (O[-(J ?e)(B ?em b Eg(J) = (O[-(J ?e)(B* ?e)])e (4) We introduce macroscopic observables, taylored to the present problem, generalizing [5, 6]: Q[J]=J 2, R[J]=J?B*, P[x,y,z;J]=(6[x-J?e]6[y-B*?e]6[z-B?eDl> (5) As in [5, 6] we eliminate technical subtleties by assuming the number of arguments (x, y, z) for which P[x, y, z; J] is evaluated to go to infinity after the limit N -t 00 has been taken. 3 Derivation of Macroscopic Laws Upon generalizing the calculations in [6, 5], one finds for on-line learning: ! ! Q = 2'f} !dXdydZ P[x, y, z] xg[x, z] - 2'f},Q + 'f}2!dXdYdZ P[x, y, z] g2[x, z] (6) R = 'f} !dXdydZ P[x, y, z] y9[x, z]- 'f},R (7) :t P[x, y, z] = ~ ! dx' P[x', y, z] {6[x-x' -'f}G[x', z]] -6[x-x']} -'f}! / dx'dy'dz' / dx'dy'dz'9[x', z]A[x, y, z; x',y', z'] 1 +'i'f}2 ! + 'f}, :x EP2P[x, y, z] dx'dy'dz' P[x', y', z']92[x', z'] 8x {xP[x , y, z]} (8) Supervised Learning with Restricted Training Sets 239 The complexity of the problem is concentrated in a Green's function: A[x, y, Zj x', y', z'] = lim N-+oo (( ([1-6ee , ]6[x-J?e]6[y-B*?e]6[z-B?e] (e?e')6[x' -J?e']6[y' - B*?e']6[y' - B?e'])i?i> )QW;t J It involves a conditional average of the form (K[J])QW;t = dJ Pt(JIQ,R,P)K[J], with Pt(J) 6[Q-Q[J]]6[R- R[J]] nXYZ 6[P[x, y, z] -P[x, y, Zj J]] Pt(JIQ,R,P) JdJ Pt(J) 6[Q - Q[J]]6[R- R[J]] nXYZ 6[P[x, y, z] - P[x, y, z; J]] = in which Pt (J) is the weight probability density at time t. The solution of (6,7,8) can be used to generate the N -+ 00 performance measures (4) at any time: Et =/ dxdydz P[x, y, z]O[-xz] Eg = 11"-1 arccos[RIVQ] (9) Expansion of these equations in powers of"" and retaining only the terms linear in "" gives the corresponding equations describing batch learning. So far this analysis is exact. 4 Closure of Macroscopic Laws As in [6, 5] we close our macroscopic laws (6,7,8) by making the two key assumptions underlying dynamical replica theory: (i) For N -+ 00 our macroscopic observables obey closed dynamic equations. (ii) These equations are self-averaging with respect to the specific realization of D. (i) implies that probability variations within {Q, R, P} subshells are either absent or irrelevant to the macroscopic laws. We may thus make the simplest choice for Pt (J IQ, R, P): Pt(JIQ,R,P) -+ 6[Q-Q[J]] 6[R-R[J]] II 6[P[x,y,z]-P[x,y,ZjJ]] (10) xyz The procedure (10) leads to exact laws if our observables {Q, R, P} indeed obey closed equations for N -+ 00. It is a maximum entropy approximation if not. (ii) allows us to average the macroscopic laws over all training sets; it is observed in simulations, and proven using the formalism of [4]. Our assumptions (10) result in the closure of (6,7,8), since now the Green's function can be written in terms of {Q, R, Pl. The final ingredient of dynamical replica theory is doing the average of fractions with the replica identity / JdJ W[JID]GIJID]) \ JdJ W[JID] = lim sets /dJ I ??? dJn (G[J 1 ID] n-+O IT W[JO<ID])sets a=1 Our problem has been reduced to calculating (non-trivial) integrals and averages. One finds that P[x, y, z] P[x, zly]P[y] with Ply] (211")-!exp[-!y 21With the short-hands Dy = P[y]dy and (f(x, y, z)) = Dydxdz P[x, zly]f(x, y, z) we can write the resulting macroscopic laws, for the case of output noise (1), in the following compact way: = d dt Q = 2",(V - ,Q) [) [)tP[x,zly] = = J + rJ2 Z d dtR = ",(W - ,R) (11) 1 [)x[)22P[x,zIY] a1/dx'P[x',zly] {6[x-x'-",G[x',z]]-6[x-x'] }+2",2Z -",:x {P[x,zly] [U(x-RY)+Wy-,x+[V-RW-(Q-R2)U]~[x,y,z])} (12) with U = (~[x, y, z]9[x, z]), v = (x9[x, z]), W = (y9[x, z]), Z = (9 2[x, z]) The solution of (12) is at any time of the following form: P[x,zly] = (1-,x)6[y-z]P+[xly] + ,x6[y+z]P-[xly] (13) A. C. C. Coolen and C. W. H. Mace 240 Finding the function <I> [x, y, z] (in replica symmetric ansatz) requires solving a saddle-point problem for a scalar observable q and two functions M?[xly]. Upon introducing B = . . :. V. .,. .q.,-Q___R,-2 Q(I-q) (with Jdx M?[xly] Jdx M?[xly]eBxs J[x, y] Jdx M?[xly]eBxs (f[x, y])? = * = 1 for all y) the saddle-point equations acquire the fonn p?[Xly] = for all X, y : ((x-Ry)2) + (qQ-R 2)[I-!:.] a ! Ds (O[X -xl); 2 !DYDS S[(I-A)(X); + A(X);] = qQ+Q-2R ..jqQ_R2 (14) (15) The equations (14) which detennine M?[xly] have the same structure as the corresponding (single) equation in [5, 6], so the proofs in [5, 6] again apply, and the solutions M?[xly], given a q in the physical range q E [R2/Q, 1], are unique. The function <I> [x, y, z] is then given by <I> [X, y, z] =! Ds s {(I-A)O[Z-y](o[X -x)); + AO[Z+Y](o[X -xl);} ..jqQ_R2 P[X, zly] (16) Working out predictions from these equations is generally CPU-intensive, mainly due to the functional saddle-point equation (14) to be solved at each time step. However, as in [7] one can construct useful approximations of the theory, with increasing complexity: (i) Large a approximation (giving the simplest theory, without saddle-point equations) (ii) Conditionally Gaussian approximation for M[xly] (with y-dependent moments) (iii) Annealed approximation of the functional saddle-point equation 5 Benchmark Tests: The Limits a --+ 00 and ,\ --+ 0 We first show that in the limit a --+ 00 our theory reduces to the simple (Q, R) formalism of infinite training sets, as worked out for noisy teachers in [12]. Upon making the ansatz p?[xly] = P[xly] = [27r(Q-R 2)]-t e- t [x- Rv]2/(Q-R 2) (17) one finds <I>[x,y,Z] = (x-Ry)/(Q-R 2) M?[xly] = P[xly], Insertion of our ansatz into (12), followed by rearranging of terms and usage of the above expression for <I> [x, y, z], shows that (12) is satisfied. The remaining equations (11) involve only averages over the Gaussian distribution (17), and indeed reduce to those of [12]: ~! Q = (I-A) { 2(x9[x, y)) 1 d --d R 1} t + 1}{92[x, y)) } + A {2(x9[x,-y)) + 1}(92[x,-y)) } - 2,Q = (I-A)(y9[x,y)) + A(y9[x,-yl) -,R Next we turn to the limit A --+ 0 (restricted training sets & noise-free teachers) and show that here our theory reproduces the fonnalism of [6,5]. Now we make the following ansatz: P+[xly] = P[xly], P[x, zly] = o[z-y]P[xIY] (18) Insertion shows that for A = 0 solutions of this fonn indeed solve our equations, giving <p[x, y, z]--+ <I> [x, y] and M+[xly] M[xly), and leaving us exactly with the fonnalism of [6, 5] describing the case of noise-free teachers and restricted training sets (apart from some new tenns due to the presence of weight decay, which was absent in [6, 5]). = 241 Supervised Learning with Restricted Training Sets 0. , r------~--__, 0..4 ~-------_____I 0..4 11>=0.' 0..3 a=4 0. , 0..0. -- , 0. 0.2 _ __ ___ _____ _ a= 1 0;=1 ------- ---- -- --- - 0. 0;=2 =-= - 0;=2 - - ----- - a=4 a=4 = =-= --=-=--=-=--=-=-=-- -=-=-_oed a=4 , 0;=2 ':::::========:::j 0..3 -- - ---- 0;=1 :::---- - -----1 0;=2 0..2 11>=0.' ~-------~ 0;=1 0., 11>=0, " , no. I 0. , 0. " Figure 1: On-line Hebbian learning: conditionally Gaussian approximation versus exact solution in [9] (.,., = 1, ,X = 0.2). Left: "I = 0.1, right: "I = 0.5. Solid lines: approximated theory, dashed lines: exact result. Upper curves: Eg as functions of time (here the two theories agree), lower curves: E t as functions of time. 6 Benchmark Tests: Hebbian Learning The special case of Hebbian learning, i.e. Q[x, z] = sgn(z), can be solved exactly at any time, for arbitrary {a, ,x, "I} [9], providing yet another excellent benchmark for our theory. For batch execution of Hebbian learning the macroscopic laws are obtained upon expanding (11,12) and retaining only those terms which are linear in.,.,. All integrations can now be done and all equations solved explicitly, resulting in U =0, Z = 1, W = (I-2,X)J2/7r, and Q = Qo e-2rryt + 2Ro(I-2'x) e-17"Yt[I_e-rrrt] "I f{ + [~(I-2,X)2+.!.] V:; 7r a [I-e- 17 "Y tF "12 R = Ro e- 17"Y t +(I-2'x)J2/7r[I-e- 17"Y t ]/"I q = [aR2+(I_e- 17"Yt)2 i'l]/aQ p?[xIY] = [27r(Q-R2)] -t e-tlz-RH sgn(y)[1-e-"..,t]/a"Y]2/(Q-R2) (19) From these results, in tum, follow the performance measures Eg = 7r- 1 arccos[ R/ JQ) and E = ! - !(1-,X)!D 2 t 2 erf[IYIR+[I-e- 77"Y t ]/a"l] + !,X!D erf[IYIR-[I-e- 17"Y t ]/a"l] Y J2(Q-R2) 2 y J2(Q-R2) Comparison with the exact solution, calculated along the lines of [9] or, equivalently, obtained upon putting t ? in [9], shows that the above expressions are all exact. .,.,-2 For on-line execution we cannot (yet) solve the functional saddle-point equation in general. However, some analytical predictions can still be extracted from (11,12,13): Q = Qo e-217"Yt + 2Ro(I-2,X) e-77"Yt[I_e-17"Yt] "I R = Ro e- 17"Y t + (I-2,X)J2/7r[I-e- 17"Y t ]/"I J f{ + [~(I-2,X)2+.!.] V:; 7r a [I_e- 17"Y t ]2 "12 + !L[I_e- 217"Y t ] 2"1 dx xP?[xIY] = Ry ? sgn(y)[I-e- 17"Y t ]/a"l with U =0, W = (I-2,X)J2/7r, V = W R+[I-e- 17"Y t ]/a"l, and Z = 1. Comparison with the results in [9] shows that the above expressions, and thus also that of E g , are all fully exact, at any time. Observables involving P[x, y, z] (including the training error) are not as easily solved from our equations. Instead we used the conditionally Gaussian approximation (found to be adequate for the noiseless Hebbian case [5, 6, 7]). The result is shown in figure 1. The agreement is reasonable, but significantly less than that in [6]; apparently teacher noise adds to the deformation of the field distribution away from a Gaussian shape. 242 A. C. C. Coolen and C. W H. Mac ~ 0.6 000000 0.4 0.4 E ~ 0.2 I i 0.0 0 4 2 6 10 0.0 -3 -2 -I 0 X 0.6 f 0.4 0.4 [ E 0.2 0.2 0.0 L-o!i6iIII."""""',-"--~_~~_ _--' -3 -2 -I 0 2 3 X ,= Figure 2: Large a approximation versus numerical simulations (with N = 10,000), for 0 and A = 0.2. Top row: Perceptron rule, with.,., = ~. Bottom row: Adatron rule, with.,., = ~. Left: training errors E t and generalisation errors Eg as functions of time, for aE {~, 1, 2}. Lines: approximated theory, markers: simulations (circles: E t , squares: Eg) . Right: joint distributions for student field and teacher noise p?[x] = dy P[x, y, z = ?y] (upper: P+[x], lower: P-[x]). Histograms: simulations, lines: approximated theory. J 7 Non-Linear Learning Rules: Theory versus Simulations In the case of non-linear learning rules no exact solution is known against which to test our formalism, leaving numerical simulations as the yardstick. We have evaluated numerically the large a approximation of our theory for Perceptron learning, 9[x, z] = sgn(z)O[-xz], and for Adatron learning, 9[x, z] = sgn(z)lzIO[-xz]. This approximation leads to the following fully explicit equation for the field distributions: 1/ d -p?[xly] = dt a . With U= ' +1 dx' p?[x'ly]{o[x-x'-.,.,.1'[x', ?y]] -o[x-x]} _ ~ {P[ I ] [W _ .,., 8 x y y J X ~ p?[xly] _.,.,2 Z!:I 2 2 uX ,X + U[X?(y)-RY]+(V-RW)[X-X?(y)]]} Q _ R2 Dydx {(I-A)P+[xly][x-P(y)]9[x,Y]+AP-[xly][x-x-(y)]9[x,-y]) V = W= Z= ! 1 1 Dydx x {(I-A)P+[xly]9[x, Y]+AP-[xly]9[x,-y]) Dydx y {(1-A)P+[xly]9[x, Y]+AP-[xly]9[x,-y]) Dydx {(I-A)P+[xly]92[x, Y]+AP-[xly]9 2[x,-yJ) Supervised Learning with Restricted Training Sets 243 J and with the short-hands X?(y) = dx xP?[xly). The result of our comparison is shown in figure 2. Note: E t increases monotonically with a, and Eg decreases monotonically with a, at any t. As in the noise-free formalism [7], the large a approximation appears to capture the dominant terms both for a -7 00 and for a -7 O. The predicting power of our theory is mainly limited by numerical constraints. For instance, the Adatron learning rule generates singularities at x = 0 in the distributions P?[xly) (especially for small "I) which, although predicted by our theory, are almost impossible to capture in numerical solutions. 8 Discussion We have shown how a recent theory to describe the dynamics of supervised learning with restricted training sets (designed to apply in the data recycling regime, and for arbitrary online and batch learning rules) [5, 6, 7] in large layered neural networks can be generalized successfully in order to deal also with noisy teachers. In our generalized approach the joint distribution P[x, y, z) for the fields of student, 'clean' teacher, and noisy teacher is taken to be a dynamical order parameter, in addition to the conventional observables Q and R. From the order parameter set {Q, R, P} we derive the generalization error Eg and the training error E t . Following the prescriptions of dynamical replica theory one finds a diffusion equation for P[x, y, z], which we have evaluated by making the replica-symmetric ansatz. We have carried out several orthogonal benchmark tests of our theory: (i) for a -7 00 (no data recycling) our theory is exact, (ii) for A -7 0 (no teacher noise) our theory reduces to that of [5, 6, 7], and (iii) for batch Hebbian learning our theory is exact. For on-line Hebbian learning our theory is exact with regard to the predictions for Q, R, Eg and the y-dependent conditional averages Jdx xP?[xly), at any time, and a crude approximation of our equations already gives reasonable agreement with the exact results [9] for E t . For non-linear learning rules (Perceptron and Adatron) we have compared numerical solution of a simple large a aproximation of our equations to numerical simulations, and found satisfactory agreement. This paper is a preliminary presentation of results obtained in the second stage of a research programme aimed at extending our theoretical tools in the arena of learning dynamics, building on [5, 6, 7]. Ongoing work is aimed at systematic application of our theory and its approximations to various types of non-linear learning rules, and at generalization of the theory to multi-layer networks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Mace C.W.H. and Coolen AC.C (1998), Statistics and Computing 8, 55 Saad D. (ed.) (1998), On-Line Learning in Neural Networks (Cambridge: CUP) Hertz J.A., Krogh A and Thorgersson G.I. (1989), J. Phys. A 22, 2133 HomerH. (1992a), Z. Phys. B 86, 291 and Homer H. (1992b), Z. Phys. B 87,371 Coolen A.C.C. and Saad D. (1998), in On-Line Learning in Neural Networks, Saad D. (ed.), (Cambridge: CUP) Coolen AC.C. and Saad D. (1999), in Advances in Neural Information Processing Systems 11, Kearns D., Solla S.A., Cohn D.A (eds.), (MIT press) Coolen A.C.C. and Saad D. (1999), preprints KCL-MTH-99-32 & KCL-MTH-99-33 Rae H.C., Sollich P. and Coolen AC.C. (1999), in Advances in Neural Information Processing Systems 11, Kearns D., Solla S.A., Cohn D.A. (eds.), (MIT press) Rae H.C., Sollich P. and Coolen AC.C. (1999),J. Phys. A 32, 3321 Inoue J.I. (1999) private communication Wong K.YM., Li S. and Tong YW. (1999),preprint cond-mat19909004 Biehl M., Riegler P. and Stechert M. (1995), Phys. Rev. E 52, 4624 

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Efficient Approaches to Gaussian Process Classification Lehel Csato, Ernest Fokoue, Manfred Opper, Bernhard Schottky Neural Computing Research Group School of Engineering and Applied Sciences Aston University Birmingham B4 7ET, UK. {opperm,csatol}~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, Solvegatan 14 A, S-223 62 Lund, Sweden winther~thep.lu.se Abstract We present three simple approximations for the calculation of the posterior mean in Gaussian Process classification. The first two methods are related to mean field ideas known in Statistical Physics. The third approach is based on Bayesian online approach which was motivated by recent results in the Statistical Mechanics of Neural Networks. We present simulation results showing: 1. that the mean field Bayesian evidence may be used for hyperparameter tuning and 2. that the online approach may achieve a low training error fast. 1 Introduction Gaussian processes provide promising non-parametric Bayesian approaches to regression and classification [2, 1]. In these statistical models, it is assumed that the likelihood of an output or target variable y for a given input x E RN can be written as P(Yla(x)) where a : RN --+ R are functions which have a Gaussian prior distribution, i.e. a is (a priori) assumed to be a Gaussian random field. This means that any finite set of field variables a(xi), i = 1, ... ,l are jointly Gaussian distributed with a given covariance E[a(xi)a(xj)] = K(Xi' Xj) (we will also assume a zero mean throughout the paper). Predictions on a(x) for novel inputs x, when a set D of m training examples (Xi, Yi) i = 1, . . . , m , is given, can be computed from the posterior distribution of the m + 1 variables a(x) and a(xd, ... ,a(xm). A major technical problem of the Gaussian process models is the difficulty of computing posterior averages as high dimensional integrals, when the likelihood is not Gaussian. This happens for example in classification problems. So far, a variety of approximation techniques have been discussed: Monte Carlo sampling [2], the MAP approach [4], bounds on the likelihood [3] and a TAP mean field approach [5]. In this paper, we will introduce three different novel methods for approximating the posterior mean of the random field a(x), which we think are simple enough to be used in practical applications. Two of the techniques L. Csato, E. Fokoue, M Opper, B. Schottky and 0. Winther 252 are based on mean field ideas from Statistical Mechanics, which in contrast to the previously developed TAP approach are easier to implement. They also yield simple approximations to the total likelihood of the data (the evidence) which can be used to tune the hyperparameters in the covariance kernel K (The Bayesian evidence (or MLII) framework aims at maximizing the likelihood of the data). We specialize to the case of a binary classification problem, where for simplicity, the class label Y = ? 1 is assumed to be noise free and the likelihood is chosen as P(Yla) = 8(ya) , (1) where 8(x) is the unit step function, which equals 1 for x > 0 and zero else. We are interested in computing efficient approximations to the posterior mean (a(x)), which we will use for a prediction of the labels via Y = sign(a(x)), where (.. .) denotes the posterior expectation. If the posterior distribution of a(x) is symmetric around its mean, this will give the Bayes optimal prediction. Before starting, let us add two comments on the likelihood (1). First, the MAP approach (i.e. predicting with the fields a that maximize the posterior) would not be applicable, because it gives the trivial result a(x) = O. Second, noise can be easily introduced within a probit model [2], all subsequent calculations will only be slightly altered. Moreover, the Gaussian average involved in the definition of the probit likelihood can always be shifted from the likelihood into the Gaussian process prior, by a redefinition of the fields a (which does not change the prediction), leaving us with the simple likelihood (1) and a modified process covariance [5] . 2 Exact Results At first glance, it may seem that in order to calculate (a(x)) we have to deal with the joint posterior of the fields ai = a(xi) ' i = 1, ... , m together with the field at the test point a(x) . This would imply that for any test point, a different new m + 1 dimensional average has to be performed. Actually, we will show that this is not the case. As above let E denote the expectation over the Gaussian prior. The posterior expectation at any point, say x (a(x)) E [a (x) TI7=l P(Yj laj)] E [TI7=l P(Yj laj)] = --=-~---~--"- (2) can by integration by parts-for any likelihood-be written as (a(x )) ~ K(x, Xj )ajYj = L...J and j )) ",.=y . (CHnp(Yj1a .... J J aa . j (3) J showing that aj is not dependent on the test point x. It is therefore not necessary to compute a m + 1 dimensional average for every prediction. We have chosen the specific definition (3) in order to stress the similarity to predictions with Support Vector Machines (for the likelihood (1), the aj will come out nonnegative) . In the next sections we will develop three approaches for an approximate computation of the aj. 3 Mean Field Method I: Ensemble Learning Our first goal is to approximate the true posterior distribution 1 1 1 T -1 m p( alD ) - e-"2 a K a P(Y ?la?) m - Z J(27r)m detK j=l J J II (4) 253 Efficient Approaches to Gaussian Process Classification of a == (al,"" am) by a simpler, tractable distribution q. Here, K denotes the covariance matrix with elements Kij = K (Xi, Xj). In the variational mean field approach-known as ensemble learning in the Neural Computation Community,the relative entropy distance K L(q,p) = da q(a) In :~:~ is minimized in the family of product distributions q(a) = TI,7=l qj(aj). This is in contrast to [3], where a variational bound on the likelihood is computed. We get J KL(q,p) = ! qi(ai) daiqi(ai) In P(Yilai) L [K-1Lj (ai)O(aj)o ~ , ~ i,j,i#j + +~L [K-1Li (a~)o i where (.. ')0 denotes expectation w.r.t. q. By setting the functional derivative of KL(q,p) with respect to qi(a) equal to zero, we find that the best product distribution is a Gaussian prior times the original Likelihood: 1 (o-"'i)2 qi(a) ex: P(Yila) ~e- hi (5) where mi = -Ai :Ej,#i(K-l)ij(aj)o and Ai = [K-l]:l. Using this specific form for the approximated posterior q(a), replacing the average over the true posterior in (3) by the approximation (5), we get (using the likelihood (I)) a set of m nonlinear equations in the unknowns aj: where D(z) = e- z2 / 2 /..f2i and <I>(z) = J~oo dt D(t). As a useful byproduct of the variational approximation, an upper bound on the Bayesian evidence P(D) = J da 7r(a)P{Dla) can be derived. (71' denotes the Gaussian process prior and P{Dla) = TI,7=l P(Yjlaj)). The bound can be written in terms of the mean field 'free energy' as -lnP(D) < Eqlnq(a) -Eqln[7r(a)P(Dla)] - '''In<l> ~ , (Y', JXi m j ) + ~2 "'y ?a ?(K?'3? -8" A?)y3?a?3 ~ " 'J' (7) ~ which can be used as a yardstick for selecting appropriate hyperparameters in the covariance kernel. The ensemble learning approach has the little drawback, that it requires inversion of the covariance matrix K and, for the free energy (7) one must compute a determinant. A second, simpler approximation avoids these computations. 4 Mean Field Theory II: A 'Naive' Approach The second mean field theory aims at working directly with the variables aj. As a starting point, we consider the partition function (evidence), Z = P(D) = ! dze-tzTKz IT P(Yjlzj) , ;=1 (8) 254 L. Csato. E. Fokoue. M. Opper. B. Schottky and O. Winther which follows from (4) by a standard Gaussian integration, introducing the Fourier transform of the Likelihood .P(Ylz) = I g~ eiaz P(Yla) with i being the imaginary unit. It is tempting to view (8) as a normalizing partition function for a Gaussian process Zi having covariance matrix K-l and likelihood P. Unfortunately, P is not a real number and precludes a proper probabilistic interpretation. N evertheless, dealing formally with the complex measure defined by (8), integration by parts shows that one has YjCl!j = -i(zj)., where the brackets (... ). denote a average over the complex measure. This suggests a simple approximation for calculating the Cl!j. One may think of trying a saddle-point (or steepest descent) approximation to (8) and replace (Zj). by the value of Zj (in the complex Z plane) which makes the integrand stationary thereby neglecting the fluctuations of the Zj. Hence, this approximation would treat expectations of products as (ZiZj). as (Zi).(Zj)*, which may be reasonable for i i= j, but definitely not for the self-correlation i = j. According to the general formalism of mean field theories (outlined e.g. in [6]), one can improve on that idea, by treating the 'self-interactions' separately. This can be done by replacing all Zi (except in the form zi) by a new variable J.Li by inserting a Dirac 8 function representation 8(z - J.L) = J ~r;:e-im(z-J1.) into (8) and integrate over the Z and a variables exactly (the integral factorizes), and finally perform a saddle-point integration over the m and J.L variables. The details of this calculation will be given elsewhere. Within the saddle-point approximation, we get the system of nonlinear equations z; i,i=l=j i,i=l=j which is of the same form as (6) with Aj replaced by the simpler K jj . These equations have also been derived by us in [5] using a Callen identity, but our present derivation allows also for an approximation to the evidence. By plugging the saddlepoint values back into the partition function, we get -lnP(D) ~ - ~ln~ (Y'~) ~Y'Cl!'(K"tJ ~ VKii + ~2 ~ t t t t 8?tJ?K 11.. )y?Cl!? J J ~ which is also simpler to compute than (7) but does not give a bound on the true evidence. 5 A sequential Approach Both previous algorithms do not give an explicit expression for the posterior mean, but require the solution of a set of nonlinear equations. These must be obtained by an iterative procedure. We now present a different approach for an approximate computation of the posterior mean, which is based on a single sequential sweep through the whole dataset giving an explicit update of the posterior. The algorithm is based on a recently proposed Bayesian approach to online learning (see [8] and the articles of Opper and Winther& Solla in [9]). Its basic idea applied to the Gaussian process scenario, is as follows: Suppose, that qt is a Gaussian approximation to the posterior after having seen t examples. This means that we approximate the posterior process by a Gaussian process with mean (a(x))t and covariance Kt(x, y), starting with (a(x))o = 0 and Ko(x, y) = K(x, y). After a new data point Yt+l is observed, the posterior is updated according to Bayes rule. The new non-Gaussian posterior qt is projected back into the family of Gaussians by choosing the closest Gaussian qt+l minimizing the relative entropy K L(qt, qt+d Efficient Approaches to Gaussian Process Classification 255 in order to keep the loss of information small. This projection is equivalent to a matching of the first two moments of lit and qt+1 ' E.g., for the first moment we get (a(x))t+l = (a(x) P(Yt+1la(Xt+d))t (P( I ( ))) Yt+l a Xt+l t = (a(x))t + Kl (t)Kt(x, Xt+l) where the second line follows again from an integration by parts and Kt{t) 1!:!?!. and ~2(t) -- K t (x t+l, x t+l ) . This recursion and u ~ tJ>(zt} with z t -- Yt?l (a(Xt?l)t u(t) the corresponding one for K t can be solved by the ansatz t (a(x))t L K(x, xj)Yjaj(t) (10) j=l L K(x, Xi)Cij (t)K(x, Xj) + K(x, y) (11) i,j where the vector a(t) = (al,"" at, 0, 0, ... ) and the matrix C(t) (which has also only txt nonzero elements) are updated as a(t + 1) a(t) + Kl(t) (C(t)kt+l + et+l) C(t + 1) C(t) where K2 (t) = ~ { ~~~:?) @y + K2(t) (C(t)kt+l + et+1) (C(t)kt+1 + et+1f - (~gt\) r}, (12) k t is the vector with elements K tj , j = 1 ... , t and @ denotes the element-wise product between vectors. The sequen- tial algorithm defined by (10)-(12) has the advantage of not requiring any matrix inversions. There is also no need to solve a numerical optimization problem at each time as in the approach of [11] where a different update of a Gaussian posterior approximation was proposed. Since we do not require a linearization of the likelihood, the method is not equivalent to the extended Kalman Filter approach. Since it is possible to compute the evidence of the new datapoint P(Yt+1) = (P(Yt+1lat+d)t based on the old posterior, we can compute a further approximation to the log evidence for m data via In P(Dm) = 2:~lln(P(Yt+llat+1)k 6 Simulations We present two sets of simulations for the mean field approaches. In the first, we test the Bayesian evidence framework for tuning the hyperparameters of the covariance function (kernel). In the second, we test the ability of the sequential approach to achieve low training error and a stable test error for fixed hyperparameters. For the evidence framework, we give simulation results for both mean field free energies (7) and (10) on a single data set, 'Pima Indian Diabetes (with 200/332 training/test-examples and input dimensionality d = 7) [7]. The results should therefore not be taken as a conclusive evidence for the merits of these approaches, but simply as an indication that they may give reasonable results. We use the radial basis function covariance function K(x,x') = exp (-~ 2:~WI(XI - XD2) . A diagonal term v is added to the covariance matrix corresponding to a Gaussian noise added to the fields with variance v [5]. The free energy, -lnP(D) is minimized by gradient descent with respect to v and the lengthscale parameters WI, ? .. , Wd and the mean field equations for aj are solved by iteration before each update of the hyperparameters (further details will be given elsewhere). Figure 1 shows the evolution of the naive mean free energy and the test error starting from uniform L. Csat6, E. Fokoue, M. Opper, B. &hottky and 0. Winther 256 ws. It typically requires of the order of 10 iteration steps of the a;-equations between each hyperparameter update. We also used hybrid approaches, where the free energy was minimized by one mean field algorithm and the hyperparameters used in the other. As it may be seen from table 1, the naive mean field theory can overestimate the free energy (since the ensemble free energy is an upper bound to the free energy). The overestimation is not nearly as severe at the minimum of the naive mean field free energy. Another interesting observation is that as long as the same hyperparameters are used the actual performance (as measured by the test error) is not very sensitive to the algorithm used. This also seems to be the case for the TAP mean field approach and Support Vector Machines [5]. 74..-----~----~-------,-----__, 115..-----~----~-------,-----__, 0114 \\ Ii:" ' 70 g .E 1. 113 >. ~ ,, W68 ~ Q) c: ~112 1-66 \ , ~ u... 64 '. "'----- 111 o 20 ~ Iterations 60 62 ~ 0~----2~0----~~~--~~~--~~ Iterations Figure 1: Hyperparameter optimization for the Pima Indians data set using the naive mean field free energy. Left figure: The free energy as a function of the number of hyperparameter updates. Right figure: The test error count (out of 332) as a function of the number of hyperparameter updates. Table 1: Pima Indians dataset. Hyperparameters found by free energy minimization. Left column gives the free energy -lnP(D) used in hyperparameter optimization. Test error counts in range 63- 75 have previously been reported [5] Free Energy minimization Ensemble Mean Field, eq. (7) Naive Mean Field, eq. (10) Ensemble MF Error -lnP(D) 72 62 100.6 107.0 Naive MF Error -lnP(D) 70 62 183.2 110.9 For the sequential algorithm, we have studied the sonar [10] and crab [7] datasets. Since we have not computed an approximation to the evidence so far, a simple fixed polynomial kernel was used. Although a probabilistic justification of the algorithm is only valid, when a single sweep through the data is used (the independence of the data is assumed), it is tempting to reuse the same data and iterate the procedure as a heuristic. The two plots show that in this way, only a small improvement is obtained, and it seems that the method is rather efficient in extracting the information from the data in a single presentation. For the sonar dataset, a single sweep is enough to achieve zero training error. Acknowledgements: BS would like to thank the Leverhulme Trust for their support (F /250/K). The work was also supported by EPSRC Grant GR/L52093. 257 Efficient Approaches to Gaussian Process Classification so - Training Error - - Test Error 45 40 35 35 30 ... 30 25 g25 I 20 1,- w 20 ~ 15 15 10 10 \ -" 00 20 40 60 80 1'\ .." \ 140 ___ 160 ,- - .. - - - - - .. 5 ?0L----5~0---F=::::lIQ.-A.1SO'::-----::-2~OO Heration. Figure 2: Training and test errors during learning for the sonar (left) and crab dataset (right). The vertical dash-dotted line marks the end of the training set and the starting point of reusing of it. The kernel function used is K(x, x') (1 + x . x' jm)k with order k = 2 (m is the dimension of inputs) . References [1] Williams C.K.I. and Rasmussen C.E., Gaussian Processes for Regression, in Neural Information Processing Systems 8, Touretzky D.s, Mozer M.C. and Hasselmo M.E. (eds.), 514-520, MIT Press (1996) . [2] Neal R .M, Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification, Technical Report 9702, Department of Statistics, University of Toronto (1997). [3] Gibbs M.N. and Mackay D.J.C., Variational Gaussian Process Classifiers, Preprint Cambridge University (1997). [4] Williams C.K.I. and Barber D, Bayesian Classification with Gaussian Processes, IEEE Trans Pattern Analysis and Machine Intelligence, 20 1342-1351 (1998). [5] Opper M. and Winther O. Gaussian Processes for Classification: Mean Field Algorithms, Submitted to Neural Computation, http://www.thep.lu.se /tf2/staff/winther/ (1999). [6] Zinn-Justin J, Quantum Field Theory and Critical Phenomena, Clarendon Press Oxford (1990) . [7] Ripley B.D, Pattern Recognition and Neural Networks, Cambridge University Press (1996). [8] Opper M., Online versus Offline Learning from Random Examples: General Results, Phys. Rev. Lett. 77, 4671 (1996). [9] Online Learning in Neural Networks, Cambridge University Press, D. Saad (ed.) (1998). [10] Gorman R.P and Sejnowski T .J, Analysis of hidden units in a layered network trained to classify sonar targets, Neural Networks 1, (1988). [11] Jaakkola T . and Haussler D. Probabilistic kernel regression, In Online Proceedings of 7-th Int. Workshop on AI and Statistics (1999) , http://uncertainty99.microsoft.com/proceedings.htm. 

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A generative model for attractor dynamics Richard S. Zemel Department of Psychology University of Arizona Tucson, AZ 85721 Michael C. Mozer Department of Computer Science University of Colorado Boulder, CO 80309-0430 zemel@u.arizona.edu mozer@colorado.edu Abstract Attractor networks, which map an input space to a discrete output space, are useful for pattern completion. However, designing a net to have a given set of attractors is notoriously tricky; training procedures are CPU intensive and often produce spurious afuactors and ill-conditioned attractor basins. These difficulties occur because each connection in the network participates in the encoding of multiple attractors. We describe an alternative formulation of attractor networks in which the encoding of knowledge is local, not distributed. Although localist attractor networks have similar dynamics to their distributed counterparts, they are much easier to work with and interpret. We propose a statistical formulation of localist attract or net dynamics, which yields a convergence proof and a mathematical interpretation of model parameters. Attractor networks map an input space, usually continuous, to a sparse output space composed of a discrete set of alternatives. Attractor networks have a long history in neural network research. Attractor networks are often used for pattern completion, which involves filling in missing, noisy, or incorrect features in an input pattern. The initial state of the attractor net is typically determined by the input pattern. Over time, the state is drawn to one of a predefined set of states-the attractors. Attractor net dynamics can be described by a state trajectory (Figure 1a). An attractor net is generally implemented by a set of visible units whose activity represents the instantaneous state, and optionally, a set of hidden units that assist in the computation. Attractor dynamics arise from interactions among the units. In most formulations of afuactor nets,2,3 the dynamics can be characterized by gradient descent in an energy landscape, allowing one to partition the output space into attractor basins. Instead of homogeneous attractor basins, it is often desirable to sculpt basins that depend on the recent history of the network and the arrangement of attractors in the space. In psychological models of human cognition, for example, priming is fundamental: after the model visits an attractor, it should be faster to fall into the same attractor in the near future, i.e., the attractor basin should be broadened. 1 ,6 Another property of attractor nets is key to explaining behavioral data in psychological and neurobiological models: the gang effect, in which the strength of an attractor is influenced by other attractors in its neighborhood. Figure 1b illustrates the gang effect: the proximity of the two rightmost afuactors creates a deeper attractor basin, so that if the input starts at the origin it will get pulled to the right. 81 A Generative Model for Attractor Dynamics k, " , ,/ " "--- ---- .. - Figure 1: (a) A two-dimensional space can be carved into three regions (dashed lines) by an attractor net. The dynamics of the net cause an input pattern (the X) to be mapped to one of the attractors (the O's). The solid line shows the temporal trajectory of the network state. (b) the actual energy landscape for a localist attractor net as a function of y, when the input is fixed at the origin and there are ((-1,0), (1,0), (1, -A)), with a uniform prior. The shapes of three attractors, W attractor basins are influenced by the proximity of attractors to one another (the gang effect). The origin of the space (depicted by a point) is equidistant from the attractor on the left and the attractor on the upper right, yet the origiri clearly lies in the basin of the right attractors. = 1bis effect is an emergent property of the distribution of attractors, and is the basis for interesting dynamics; it produces the mutually reinforcing or inhibitory influence of similar items in domains such as semantics,9 memory,lO,12 and olfaction.4 Training an attract or net is notoriously tricky. Training procedures are CPU intensive and often produce spurious attractors and ill-conditioned attractor basins. 5,11 Indeed, we are aware of no existing procedure that can robustly translate an arbitrary specification of an attractor landscape into a set of weights. These difficulties are due to the fact that each connection participates in the specification of multiple attractors; thus, knowledge in the net is distributed over connections. We describe an alternative attractor network model in which knowledge is localized, hence the name localist attractor network. The model has many virtues, including: a trivial procedure for wiring up the architecture given an attractor landscape; eliminating spurious attractors; achieving gang effects; providing a clear mathematical interpretation of the model parameters, which clarifies how the parameters control the qualitative behavior of the model (e.g., the magnitude of gang effects); and proofs of convergence and stability. A Iocalist attractor net consists of a set of n state units and m attractor units. Parameters associated with an attractor unit i encode the location of the attractor, denoted Wi, and its "pull" or strength, denoted 7ri, which influence the shape of the attractor basin. Its activity at time t, qi(t), reflects the normalized distance from the attractor center to the current state, y(t), weighted by the attractor strength: g(y, w, 0") 7rjg(y(t) , Wi, O"(t)) L:j 7rjg(y(t) , Wj, O"(t)) (1) exp( -\y - w\2/20"2) (2) Thus, the attractors form a layer of normalized radial-basis-function units. The input to the net, &, serves as the initial value of the state, and thereafter the state is pulled toward attractors in proportion to their activity. A straightforward 82 R. S. Zemel and M. C. Mozer expression of this behavior is: (3) where a(l) = Ion the first update and a(t) = 0 fort> 1. More generally, however, one might want to gradually reduce a over time, allowing for a persistent effect of the external input on the asymptotic state. The variables O"(t) and a(t) are not free parameters of the model, but can be derived from the formalism we present below. The localist attractor net is motivated by a generative model of the input based on the attractor distribution, and the network dynamics corresponds to a search for a maximum likelihood interpretation of the observation. In the following section, we derive this result, and then present simulation studies of the architecture. 1 A MAXIMUM LIKELIHOOD FORMULATION The starting point for the statistical formulation of a localist attractor network is a mixture of Gaussians model. A standard mixture of Gaussians consists of m Gaussian density functions in n dimensions. Each Gaussian is parameterized by a mean, a covariance matrix, and a mixture coefficient. The mixture model is generative, i.e., it is considered to have produced a set of observations. Each observation is generated by selecting a Gaussian based on the mixture coefficients and then stochastically selecting a point from the corresponding density function. The model parameters are adjusted to maximize the likelihood of a set of observations. The Expectation-Maximization (EM) algorithm provides an efficient procedure for estimating the parameters.The Expectation step calculates the posterior probability qi of each Gaussian for each observation, and the Maximization step calculates the new parameters based on the previous values and the set of qi . The mixture of Gaussians model can provide an interpretation for a localist attractor network, in an unorthodox way. Each Gaussian corresponds to an attractor, and an observation corresponds to the state. Now, however, instead of fixing the observation and adjusting the Gaussians, we fix the Gaussians and adjust the observation. If there is a single observation, and a = 0 and all Gaussians have uniform spread 0", then Equation 1 corresponds to the Expectation step, and Equation 3 to the Maximization step in this unusual mixture model. Unfortunately, this simple characterization of the localist attractor network does not produce the desired behavior. Many situations produce partial solutions, in which the observation does not end up at an attractor. For example, if two unidimensional Gaussians overlap significantly, the most likely value for the observation is midway between them rather than at the mean of either Gaussian. We therefore extend this mixture-of-Gaussians formulation to better characterize the localist attractor network. As in the simple model, each of the m attractors is a Gaussian generator, the mean of which is a location in the n-dimensional state space. The input to the net, e, is considered to have been generated by a stochastic selection of one of the attractors, followed by the addition of zero-mean Gaussian noise with variance specified by the attractor. Given a particular observation e, the an attractor's posterior probability is the normalized Gaussian probability of e, weighted by its mixing proportion. This posterior distribution for the attractors corresponds to a distribution in state space that is a weighted sum of Gaussians. We then consider the attractor network as encoding this distribution over states implied by the attractor posterior probabilities. At anyone time, however, the attractor network can only represent a single position in state space, rather than 83 A Generative Model for Attractor Dynamics the entire distribution over states. This restriction is appropriate when the state is an n-dimensional point represented by the pattern of activity over n state units. To accommodate this restriction, we change the standard mixture of Gaussians generative model by interjecting an intermediate level between the attractors and the observation. The first generative level consists of the discrete attractors, the second is the state space, and the third is the observation. Each observation is considered to have been generated by moving down this hierarchy: 1. select an attractor x = i from the set of attractors 2. select a state (i.e., a pattern of activity across the state units) based on the preferred location of that attractor: y = Wi + Ny 3. select an observation z = yG + N z The observation z produced by a particular state y depends on the generative weight matrix G. In the networks we consider here, the observation and state spaces are identical, so G is the identity matrix, but the formulation allows for z to lie in some other space. Ny and N z describe the zero-mean, spherical Gaussian noise introduced at the two levels, with deviations (1 y and (1z, respectively. In comparison with the 2-level Gaussian mixture model described above, this 3level model is more complicated but more standard: the observation & is preserved as stable data, and rather than the model manipulating the data here it can be viewed as iteratively manipulating an internal representation that fits the observation and attractor structure. The attractor dynamics correspond to an iterative search through state space to find the most likely single state that: (a) was generated by the mixture of Gaussian attractors, and (b) in tum generated the observation. Under this model, one could fit an observation & by finding the posterior distribution over the hidden states (X and Y) given the observation: (X i Y = ylZ = &) = p(&ly, i)p(y, i) p(&IY) 1riP(yli) (4) p, p(&) Jy p(&IY)L:i1riP(Yli)dy = = where the conditional distributions are Gaussian: p(Y = ylX = i) = 9(ylwi, (1y) and p(&IY = y) = 9(&ly, (1z). Evaluating the distribution in Equation 4 is tractable, because the partition function is a sum of a set of Gaussian integrals. Due to the restriction that the network cannot represent the entire distribution, we do not directly evaluate this distribution but instead adopt a mean-field approach, in which we approximate the posterior by another distribution Q(X, YI&). Based on this approximation, the network dynamics can be seen as minimizing an objective function that describes an upper bound on the negative log probability of the observation given the model and mean-field parameters. In this approach, one can choose any form of Q to estimate the posterior distribution, but a better estimate allows the network to approach a maximum likelihood solution. 13 We select a simple posterior: Q(X, Y) = qid(Y = y), where qi = Q(X = i) is the responsibility assigned to attractor i, and y is the estimate of the state that accounts for the observation. The delta function over Y is motivated by the restriction that the explanation of an input consists of a single state. Given this posterior distribution, the objective for the network is to minimize the free energy F, described here for a particular input example &: F(q,yl&) = J (. , ~ ~ Q(X = i, Y = y) QX=t,Y=y)lnp(&,X=i,Y=y)dy L,. qi In 1r,qi. -lnp(&IY) - L,. qi lnp(yli) R. S. Zemel and M C. Mozer 84 where 7r; is the prior probability (mixture coefficient) associated with attractor i. These priors are parameters of the generative model, as are (fy, u z , and w. F(q, yl&) = 2: qdn 7r,qi. + 2 \ 1& - Yl2 + 2 \ Uz i (f Y 2: qi!y - wd 2 + n In((fy(fz) (5) i Given an observation, a good set of mean-field parameters can be determined by alternating between updating the generative parameters and the mean-field parameters. The update procedure is guaranteed to converge to a minimum of F, as long as the updates are done asynchronously and each update minimizes F with respect to a parameter. s The update equations for the mean-field parameters are: y = qi - 2&+ (f z2E i q,.w,. (f2 + (f2 y z (f y 7riP(Y Ii) E j 7rjp(yli) (6) (7) In our simulations, we hold most of the parameters of the generative model constant, such as the priors 7r, the weights w, and the generative noise in the observation, (f z. The only aspect that changes is the generative noise in the state, (f Y' which is a single parameter shared by all attractors: 2= d 1~ L..Jqi I'Y -Wi 12 (fy (8) i The updates of Equations 6-8 can be in any order. We typically initialize the state y to & at time 0, and then cyclically update the qi, (fy, then y. This generative model avoids the problem of spurious attractors described above for the standard Gaussian mixture model. Intuition into how the model avoids spurious attractors can be gained by inspecting the update equations. These equations effectively tie together two processes: moving y closer to some Wi than the others, and increasing the corresponding responsibility qi. As these two processes evolve together, they act to descrease the noise (fy, which accentuates the pull of the attractor. Thus stable points that do not correspond to the attractors are rare. 2 SIMULATION STUDIES To create an attractor net, we specify the parameters (7ri' w;) associated with the attractors based on the desired structure of the energy landscape (e.g., Figure Ib). The only remaining free parameter, (f z, plays an important role in determining how responsive the system is to the external input. We have conducted several simulation studies to explore properties of localist attractor networks. Systematic investigations with a 200-dimensional state space and 200 attractors, randomly placed at corners of the 200-D hypercube, have demonstrated that spurious responses are exceedingly rare unless more than 85% of an input's features are distorted (Figure 2), and that manipulating parameters such as noise and prior probabilities has the predicted effects. We have also conducted studies of localist attractor networks in the domain of visual images of faces. These simulations have shown that gang effects arise when there is structure among the attractors. For example, when the attractor set consists of a single view of several different faces, and multiple views of one face, then an input that is a morphed face-a linear combination of one of the single-view faces and one view of the gang face-will end up in the gang attractor even when the initial weighting assigned to the gang face was less than 40%. A Generative Modelfor Attractor Dynamics 85 100 ~~~==~~~~--------~~loo % Missing features Figure 2: The input must be severely corrupted before the net makes spurious (final state not at an attractor) or adulterous (final state at a neighbor of the generating attractor) responses. (a) The percentage of spurious responses increases as (Tz is increased. (b) The percentage of adulterous responses increases as (Tz is decreased. To test the architecture on a larger, structured problem, we modeled the domain of three-letter English words. The idea is to use the attractor network as a content addressable memory which might, for example, be queried to retrieve a word with P in the third position and any letter but A in the second position, a word such as HIP. The attractors consist of the 423 three-letter English words, from ACE to ZOO. The state space of the attractor network has one dimension for each of the 26 letters of the English alphabet in each of the 3 positions, for a total of 78 dimensions. We can refer to a given dimension by the letter and position it encodes, e.g., P3 denotes the dimension corresponding to the letter P in the third position of the word. The attractors are at the comers of a [-1, +1]18 hypercube. The attractor for a word such as HIP is located at the state having value -Ion all dimensions except for HI, h, and P3 which have value +1. The external input specifies a state that constrains the solution. For example, one might specify lip in the third position" by setting the external input to +1 on dimension P3 and to -Ion dimensions ll'3, for all letters ll' other than P. One might specify the absence of a constraint in a particular letter position, p, by setting the external input to a on dimensions ll'p, for all letters ll'. The network's task is to settle on a state corresponding to one of the words, given soft constraints on the letters. The interactive-activation model of word perception7 performs a similar computation, and our implementation exhibits the key qualitative properties of their model. If the external input specifies a word, of course the attractor net will select that word. Interesting queries are those in which the external input underconstrains or overconstrains the solution. We illustrate with one example of the network's behavior, in which the external input specifies D 1 , E2 , and G3 ? Because DEG is a nonword, no attractor exists for that state. The closest attractors share two letters with DEG, e.g., PEG, BEG, DEN, and DOG. Figure 3 shows the effect of gangs on the selection of a response, BEG. 3 CONCLUSION Localist attractor networks offer an attractive alternative to standard attractor networks, in that their dynamics are easy to control and adapt. We described a statistical formulation of a type of localist attractor, and showed that it provides a Lyapunov function for the system as well as a mathematical interpretation for the network parameters. The dynamics of this system are derived not from intuitive arguments but from this formal mathematical model. Simulation studies show that the architecture achieves gang effects, and spurious attractors are rare. This approach is inefficient if the attractors have compositional structure, but for many applications of pattern recognition or associative memory, the number of items R. S. Zemel and M C. Mozer 86 Iteration I ~?;s~., ~?; ~: f.? DUG In ~ ~ . . ' j:. . h' f.j.~?.;:: .~:)~.: BEG ;';;,"',' ~.:;?:( ;: _""- '.~ ,'<};~ '';'Uoli :.':(; l;,;)f f:.:: ~+t: ~. <::~'i . ;;,.'.. ,:~;.:,,: >.~~~-t".- r::';"' ~ '.~r.': ,' :: "; ;.U'> .?..?? :; " 3: LEG '~~', u:':,: ; '~: .F? DEN DEW F,.; ~,': :? ' :. :.:':. . ~:. :.: :.< . ,:;'.:: ?'>:'?J 1:/-:: 1:S~ ':?:: DIG ,,':i_ 'a, ~ .... ~ , ~ ~~::, DOG ': L ?<~ ; ,~.: , ;.:~: ! ~ '<f" ' /:.\~~ ~.!.;, ~:, ~ '.~':";:, ;. ':.' ~y'.)' :;',~ :_'>":'.? ':.';.:; ~;.;:: :. ;.. ( ~ .. '.':~ ",j' ":r.:r :.~.!. PEa -'::'.? ,.y Iteration 2 LEG PEG Iteration 3 lEG lEG PEG Iteration 4 lEG Iteration 5 lEG Figure 3: Simulation of the 3-letter word attractor network, queried with DEC. Each frame shows the relative activity of attractor units at various points in processing. Activity in each frame is normalized such that the most active unit is printed in black ink; the lighter the ink color, the less active the unit. Only attractor units sharing at least one letter with DEC are shown. The selection, BEG, is a product of a gang effect. The gangs in this example are formed by words sharing two letters. The most common word beginnings are PE- (7 instances) and DI- (6); the most common word endings are -AC (10) and -ET (10); the most common first-last pairings are B-G (5) and D-G (3). One of these gangs supports B1, two support E2 , and three support G3 , hence BEG is selected. being stored is small. The approach is especially useful in cases where attractor locations are known, and the key focus of the network is the mutual influenre of the attractors, as in many cognitive modelling studies. References [1) Becker, s., Moscovitch, M, Belumann,M, & Joordens, S. (1997). Long-term semantic priming: A computational account and empirical evidence. TOIlTtUIl of Experimental Psychology: Learning, Memory, & Cognition, 23(5), 10591082. [2) Golden, R (1988). Probabilistic characterization ofneura1modelcomputations. In D. Z. Anderson (Ed.), Neural Infrmnation Processing Systems (pp. 310-316). American Institute of PhysiCS. [3) Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the NationlU Aaulemy of Sciences, 79,2554-2558. [4) Kay, LM., Lancaster. L.R, & Freeman W.J. (1996). Reafference and attractors in the olfactory system during odor recognition. Int JNeural Systems, 7(4),489-95. [5) Mathis, D. (1997). A computational theory of consciousness in cognition. Unpublished Doctoral Dissertation. Boulder. CO: Department of Computer Science, University of Colorado. [6) Mathis, D., & Mozer, M. C. (1996). Conscious and unconscious perception: A computational theory. In G. Cottrell (Ed.), Proceedings of the Eighteenth Annual Conference of the Cognitive Science Society (pp. 324-328). Erlbaum. [7) McC1e11and, J. L. & Rumelhart, D. E. (1981). An interactive activation model of context effects in letter perception: Part L An account of basic findings. Psychological Reuiew, 88,375-407. [8) Neal. R M & Hinton, G. E. (1998). A view of the EM algorithm that justifies incremental, sparse, and other variants. In M I. Jordan (Ed.), Learning in Graphical Models. Kluwer Academic Press. [9) McRae, K., de Sa, V. R, & SeidenbeIg, M S. (1997) On the nature and scope of featural representations of word meaning. TournaI of Experimental Psychology; General, 126(2),99-130. [10) Redish, A. D. & Touretzky, D. S. (1998). The role of the hippocampus in solving the Morris water maze. Neural Computation, 10(1), 73-111. [11) Rodrigues, N. c., & Fontanari, J. F. (1997). Multiva1ley structure of attractor neural networks. Journal of Physics A (Mathematical and General), 30, 7945-7951 . [12) Samsonovich, A. & McNaughton, B. L. (1997) Path integration and cognitive mapping in a continuous attractor neural network model TournaI ofNeumscience, 17(15),5900-5920. [13) Saul, L.K., Jaakkola, T., & Jordan, ML (1996). Mean field theory for sigmoid belief networks. Tounud of AI Resetm:h, 4, 61-76. PART II NEUROSCIENCE 

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Robust Recognition of Noisy and Superimposed Patterns via Selective Attention Soo-Young Lee Brain Science Research Center Korea Advanced Institute of Science & Technology Yusong-gu, Taejon 305-701 Korea Michael C. Mozer Department of Computer Science University of Colorado at Boulder Boulder, CO 80309 USA sylee@ee.kaist.ac.kr mozer@cs.colorado.edu Abstract In many classification tasks, recognition accuracy is low because input patterns are corrupted by noise or are spatially or temporally overlapping. We propose an approach to overcoming these limitations based on a model of human selective attention. The model, an early selection filter guided by top-down attentional control, entertains each candidate output class in sequence and adjusts attentional gain coefficients in order to produce a strong response for that class. The chosen class is then the one that obtains the strongest response with the least modulation of attention. We present simulation results on classification of corrupted and superimposed handwritten digit patterns, showing a significant improvement in recognition rates. The algorithm has also been applied in the domain of speech recognition, with comparable results. 1 Introduction In many classification tasks, recognition accuracy is low because input patterns are corrupted by noise or are spatially or temporally overlapping. Approaches have been proposed to make classifiers more robust to such perturbations, e.g., by requiring classifiers to have low input-to-output mapping sensitivity [1]. We propose an approach that is based on human selective attention. People use selective attention to focus on critical features of a stimulus and to suppress irrelevant features. It seems natural to incorporate a selective-attention mechanism into pattern recognition systems for noisy real world applications. Psychologists have for many years studied the mechanisms of selective attention (e.g., [2]-[4]). However, controversy still exists among competing theories, and only a few models are sufficiently well defmed to apply to engineering pattern recognition problems. Fukushima [5] has incorporated selective attention and attention-switching algorithms into his Neocognitron model, and has demonstrated good recognition performance on superimposed digits. However, the Neocognitron model has many unknown parameters which must be determined heuristically, and its performance is sensitive to the parameter values. Also, its computational requirements are prohibitively expensive for many realtime applications. Rao [6] has also recently introduced a selective attention model based 32 s. -Y. Lee and M C. Mozer on Kalman filters and demonstrated classifications of superimposed patterns. However, his model is based on linear systems, and a nonlinear extension is not straightforward. There being no definitive approach to incorporating selective attention into pattern recognition, we propose a novel approach and show it can improve recognition accuracy. 2 Psychological Views of Selective Attention The modern study of selective attention began with Broadbent [7]. Broadbent presented two auditory channels to subjects, one to each ear, and asked subjects to shadow one channel. He observed that although subjects could not recall most of what took place in the unshadowed channel, they could often recall the last few seconds of input on that channel. Therefore, he suggested that the brain briefly stores incoming stimuli but the stimulus information fades and is neither admitted to the conscious mind nor is encoded in a way that would permit later recollection, unless attention is directed toward it. This view is known as an early filtering or early selection model. Treisman [8] proposed a modification to this view in which the filter merely attenuates the input rather than absolutely preventing further analysis. Although late-selection and hybrid views of attention have been proposed, it is clear that early selection plays a significant role in human information processing [3]. The question about where attention acts in the stream of processing is independent of another important issue: what factors drive attention to select one ear or one location instead of another. Attention may be directed based on low-level stimulus features, such as the amplitude of a sound or the color of a visual stimulus. This type of attentional control is often called bottom up. Attention may also be directed based on expectations and object knowledge, e.g., to a location where critical task-relevant information is expected. This type of attentional control is often called top down. 3 A Multilayer Perceptron Architecture for Selective Attention We borrow the notion of an early selection filter with top-down control and integrate it into a multilayer perceptron (MLP) classifier, as depicted in Figure 1. The dotted box is a standard MLP classifier, and an attention layer with one-to-one connectivity is added in front of the input layer. Although we have depicted an MLP with a single hidden layer, our approach is applicable to general MLP architectures. The kth element of the input vector, denoted xk, is gated to the kth input of the MLP by an attention gain or filtering coefficient ak. Previously, the first author has shown a benefit of treating the ak's like ordinary adaptive parameters during training [9]-[12]. In the present work, we fix the attention gains at 1 during training, causing the architecture to behave as an ordinary MLP. However, we allow the gains to be adjusted during classification of test patterns. Our basic conjecture is that recognition accuracy may be improved if attention can suppress noise along irrelevant dimensions and enhance a weak signal along relevant dimensions. "Relevant" and "irrelevant" are determined by topdown control of attention. Essentially, we use knowledge in the trained MLP to determine which input dimensions are critical for classifying a test pattern. To be concrete, consider an MLP trained to classify handwritten digits. When a test pattern is presented, we can adjust the attentional gains via gradient descent so as to make the input as good an example of the class "0" as possible. We do this for each of the different output classes, "0" through "9", and choose the class for which the strongest response is obtained with the smallest 33 Robust Pattern Recognition via Selective Attention attentional modulation (the exact quantitative rule is presented below). The conjecture is that if the net can achieve a strong response for a class by making a small attentional modulation, that class is more likely to be correct than whichever class would have been selected without applying selective attention . .........................................................................................................................................................._................,: ! ~ ~ Xl~----~~~~~--~~~~~~~~~~~~~yl 1 1 ~ ~--~-+~~~~~~~~~~~~~~ ~ ~ li ~------~~>E~~~~~~~~~~~~~~~~~. Y3 ! ! ~ ? ? x ~ i: a ; 'l: h w(l) vJ2) /M 1 i ................................................ __ ..........................................................................................................._... _ .....! Figure 1: MLP architecture for selective attention The process of adjusting the attentional gains to achieve a strong response from a particular class-<:all it the attention class-proceeds as follows. First, a target output vector t S= [1'1 1'2? ??t'M f is defmed. For bipolar binary output representations, t,S = 1 is for the attention class and -1 for the others. Second, the attention gain ak's are set to I. Third, the attention gain ak's are adapted to minimize error E S == ~ L (t,S - y,)2 with the given , input x = [XI X2??? XN f and pre-trained and frozen synaptic weights W. The update rule is based on a gradient-descent algorithm with error back-propagation. At the (n+ J)'th iterative epoch, the attention gain a k is updated as (Ia) (lb) where E denotes the attention output error, ojl) thej'th attribute of the back-propagated error at the first hidden-layer, and WJ~) the synaptic weight between the input x k and the j'th neuron at the first hidden layer. Finally, " is a step size. The attention gains are thresholded to lie in [0, 1]. The application of selective attention to a test example is summarized as follows: Step 1: Apply a test input pattern to the trained MLP and compute output values. Step 2: For each of the classes with top m activation values, (I) Initialize all attention gain ak' s to 1 and set the target vector tS. (2) Apply the test pattern and attention gains to network and compute output. (3) Apply the selective attention algorithm in Eqs.( 1) to adapt the attention gains. (4) Repeat steps (2) and (3) until the attention process converges. (5) Compute an attention measure M on the asymptotic network state. s.-Y. Lee and M. C. Mozer 34 Step 3: Select the class with a minimum attention measure M as the recognized class. The attention measure is defined as (2a) Dj =:f(Xk -xk )2/2N = Eo 2 :fxk(l-ak) =4[/i 2 (2b) /2N - y;(i)]2/2M , (2c) where D/ is the square of Euclidean distance between two input patterns before and after the application of selective attention and Eo is the output error after the application of selective attention. Here, D/ and Eo are normalized with the number of input pixels and number of output classes, respectively. The superscript s for attention classes is omitted for simplicity. To make the measure M a dimensionless quantity, one may normalize the D/ and Eo with the input energy (~kX~ ) and the training output error, respectively. However, it does not affect the selection process in Step 3. One can think of the attended input i as the minimal deformation of the test input needed to trigger the attended class, and therefore the Euclidean distance between x and i is a good measure for the classification confidence. In fact, D/ is basically the same quantity minimized by Rao [6]. However, the MLP classifier in our model is capable of nonlinear mapping between the input and output patterns. A nearest-neighbor classifier, with the training data as examples, could also be used to find the minimum-distance class. Our model with the MLP classifier computes a similar function without the large memory and computational requirements. The proposed selective attention algorithm was tested on recognition of noisy numeral patterns. The numeral database consists of samples of the handwritten digits (0 through 9) collected from 48 people, for a total of 480 samples. Each digit is encoded as a 16x16 binary pixel array. Roughly 16% of the pixels are black and coded as 1; white pixels are coded as O. Four experiments were conducted with different training sets of 280 training patterns each. A one hidden-layer MLP was trained by back propagation. The numbers of input, hidden, and output neurons were 256, 30, and to, respectively. Three noisy test patterns were generated from each training pattern by randomly flipping each pixel value with a probability Pt, and the 840 test patterns were presented to the network for classification. In Figure 2, the false recognition rate is plotted as a function of the number of candidates considered for the attentional manipulation, m. (Note that the run time of the algorithm is proportional to m, but that increasing m does not imply a more lax classification criterion, or additional external knowledge playing into the classification.) Results are shown for three different pixel inversion probabilities, Pt =0.05, 0.1, and 0.15. Considering the average 16% of black pixels in the data, the noisy input patterns with Pt= 0.15 correspond to a SNR of approximately 0 dB. For each condition in the figure, the false recognition rates for the four different training sets are marked with an '0', and the means are connected by the solid curve. 35 Robust Pattern Recognition via Selective Attention A standard MLP classifier corresponds to m = I (i.e., only the most active output of the MLP is considered as a candidate response). The false recognition rate is clearly lower when the attentional manipulation is used to select a response from the MLP (m > I). It appears that performance does not improve further by considering more than the top three candidates. 4 Attention Switching Superimposed Patterns 2 I/) Q) a; a: c:: 1.5 o -..::; ?c Ol o o Q) a: 3l ~ 0.5 o for Suppose that we superimpose the binary input patterns for two different handwritten digits using the logical OR operator (the pixels corresponding to the black ink have logical value 1). Can we use attention to recognize the two patterns in sequence? This is an extreme case of a situation that is common in visual pattern recognition-where two patterns are spatially overlapping. We explore the following algorithm. First, one pattern is recognized with the selective attention process used in Section 3. Second, attention is switched from the recognized pattern to the remaining pixels in the image. Switching is accomplished by removing attention from the pixels of the recognized pattern: the attentional gain of an input is clamped to 0 following switching if and only if its value after the first-stage selective attention process was 1 (Le., that input was attended during the recognition of the first pattern); all other gains are set to 1. Third, the recognition process with selective attention is performed again to recognize the second pattern. The proposed selective attention and attention switching algorithm was tested for recognition of 2 superimposed numeral data. Again, four experiments were conducted with l\ r---.. 2 3 4 5 Number of Candidates (a) Pr=O.05, 20r-~----~----~----~----~-. I/) Q) Cii a: c:: 15 ,g ?c g> o 10 Q) a: 3l ~ 5 OL-~----~----~----~----~~ 2 3 4 5 Number of Candidates (b) Pr=O.IO, 40r-~----~----~----~----~-. I/) Q) a; a: c:: 30 o :t: c:: Ol 8 20 Q) a: 3l ~ 10 OL-~----~----~----~----~~ 2 3 4 5 Number of Candidates (c) Pr=O.15, Figure 2: False recognition rates for noisy patterns as a function of the number of top candidates. Each binary pixel of training patterns is randomly inverted with a probability Pr. s.-Y. 36 Lee and M C. Mozer different training sets. For each experiment, 40 patterns were selected from 280 training patterns, and 720 test patterns were generated by superimposing pairs of patterns from different output classes. The test patterns were still binary. .-. ~ I??? ~ ~~ ... ~ ~ [kJ ~ [9] [9] Et][3] ~[2J0[J ~~g]m Figure 3: Examples of Selective Attention and Attention Switching Figure 3 shows six examples of the selective attention and attention switching algorithm in action, each consisting of four panels in a horizontal sequence. The six examples were fonned by superimposing instances of the following digit pairs: (6,3), (9,0), (6,4), (9,3), (2,6), and (5,2). The fIrst panel for each example shows the superimposed pattern. The second panel shows the attended input i for the fIrst round classifIcation; because this input has continuous values, we have thresholded the values at 0.5 to facilitate viewing in the fIgure. The third panel shows the masking pattern for attention switching, generated by thresholding the input pattern at 1.0. The fourth panel sho~s the residual input pattern for the second round classifIcation. The attended input x has analog values, but thresholded by 0.5 to be shown in the second rectangles. Figure 3 shows that attention switching is done effectively, and the remaining input patterns to the second classifIer are quite visible. We compared perfonnance for three different methods. First, we simply selected the two MLP outputs with highest activity; this method utilizes neither selective attention. Second, we perfonned attention switching but did not apply selective attention (i.e., m=I). Third, we perfonned both attention switching and selective attention (with m=3). Table I summarizes the recognition rates for the fITst and the second patterns read out of the MLP for the three methods. As hypothesized, attention switching increases the recognition rate for the second pattern, and selective attention increases the recognition rate for both the fITSt and the second pattern. Table I: Recognition Rates (%) of Two Superimposed Numeral Patterns No selective attention or switching Switching only Switching & selective attention First Pattern Second Pattern 91. 3 91. 3 95.9 62.7 75.4 77.4 Robust Pattern Recognition via Selective Attention 37 5 Conclusion In this paper, we demonstrated a selective-attention algorithm for noisy and superimposed patterns that obtains improved recognition rates. We also proposed a simple attention switching algorithm that utilizes the selective-attention framework to further improve performance on superimposed patterns. The algorithms are simple and easily implemented in feedforward MLPs. Although our experiments are preliminary, they suggest that attention-based algorithms will be useful for extracting and recognizing multiple patterns in a complex background. We have conducted further simulation studies supporting this conjecture in the domain of speech recognition, which we will integrate into this presentation if it is accepted at NIPS. Acknowledgements S.Y. Lee acknowledges supports from the Korean Ministry of Science and Technology. We thank Dr. Y. Le Cun for providing the handwritten digit database. References [1] Jeong D.G., and Lee, S.Y. (1996). Merging backpropagation and Hebbian learning rules for robust classification, Neural Networks, 9:1213-1222. [2] Cowan, N. (1997). Attention and Memory: An Integrated Framework, Oxford Univ. Press. [3] Pashler, H.E. (1998). The Psychology ofAttention, MIT Press. [4] Parasuraman, R. (ed.) (1998). The Attentive Brain, MIT Press. [5] Fukushima, K. (1987). Neural network model for selective attention in visual pattern recognition and associative recall, Applied Optics, 26:4985-4992. [6] Rao, R.P.N. (1998). Correlates of attention in a model of dynamic visual recognition. In Neural Information Processing Systems 10, MIT Press. [7] Broadbent, D.E. (1958). Perception and Communication. Pergamon Press. [8] Treisman, A. (1960). Contextual cues in selective listening, Quarterly Journal of Experimental Psychology, 12:242-248. [9] Lee, H.J., Lee, S.Y. Lee, Shin, S.Y., and Koh, B.Y. (1991). TAG: A neural network model for large-scale optical implementation, Neural Computation, 3:135-143. [1O]Lee, S.Y., Jang, J.S., Shin, S.Y., & Shim, C.S. (1988). Optical Implementation of Associative Memory with Controlled Bit Significance, Applied Optics, 27:19211923. [11 ] Kruschke, J.K. (1992). ALCOVE: An Examplar-Based Connectionist Model of Category Learning, Psychological Review, 99:22-44. [12]Lee, S.Y., Kim, D.S., Abn, K.H., Jeong, J.H., Kim, H., Park, S.Y., Kim, L.Y., Lee, J.S., & Lee, H.Y. (1997). Voice Command II: a DSP implementation of robust speech recognition in real-world noisy environments, International Conference on Neural Information Processing, pp. 1051-1054, Dunedin, New Zealand. 

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Statistical Dynamics of Batch Learning s. Li and K. Y. Michael Wong Department of Physics, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong {phlisong, phkywong}@ust.hk Abstract An important issue in neural computing concerns the description of learning dynamics with macroscopic dynamical variables. Recent progress on on-line learning only addresses the often unrealistic case of an infinite training set. We introduce a new framework to model batch learning of restricted sets of examples, widely applicable to any learning cost function, and fully taking into account the temporal correlations introduced by the recycling of the examples. For illustration we analyze the effects of weight decay and early stopping during the learning of teacher-generated examples. 1 Introduction The dynamics of learning in neural computing is a complex multi-variate process. The interest on the macroscopic level is thus to describe the process with macroscopic dynamical variables. Recently, much progress has been made on modeling the dynamics of on-line learning, in which an independent example is generated for each learning step [1, 2]. Since statistical correlations among the examples can be ignored, the dynamics can be simply described by instantaneous dynamical variables. However, most studies on on-line learning focus on the ideal case in which the network has access to an almost infinite training set, whereas in many applications, the collection of training examples may be costly. A restricted set of examples introduces extra temporal correlations during learning, and the dynamics is much more complicated. Early studies briefly considered the dynamics of Adaline learning [3, 4, 5], and has recently been extended to linear perceptrons learning nonlinear rules [6, 7}. Recent attempts, using the dynamical replica theory, have been made to study the learning of restricted sets of examples, but so far exact results are published for simple learning rules such as Hebbian learning, beyond which appropriate approximations are needed [8]. In this paper, we introduce a new framework to model batch learning of restricted sets of examples, widely applicable to any learning rule which minimizes an arbitrary cost function by gradient descent. It fully takes into account the temporal correlations during learning, and is therefore exact for large networks. 287 Statistical Dynamics ofBatch Learning 2 Formulation Consider the single layer perceptron with N ? 1 input nodes {~j} connecting to a single output node by the weights {Jj }. For convenience we assume that the inputs ~j are Gaussian variables with mean 0 and variance 1, and the output state 5 is a function f(x) of the activation x at the output node, i.e. 5=f(x); x=J?{. (1) = The network is assigned to "learn" p aN examples which map inputs {{j} to the outputs {5~} (p = 1, ... ,p). 5~ are the outputs generated by a teacher percept ron {Bj }, namely (2) Batch learning by gradient descent is achieved by adjusting the weights {Jj } iteratively so that a certain cost function in terms of the student and teacher activations {x~} and {y~} is minimized. Hence we consider a general cost function (3) The precise functional form of g(x, y) depends on the adopted learning algorithm. For the case of binary outputs, f(x) = sgnx. Early studies on the learning dynamics considered Adaline learning [3, 4, 5], where g(x, y) = -(5 - x)2/2 with 5 = sgny. For recent studies on Hebbian learning [8], g(x,y) = x5. To ensure that the perceptron is regularized after learning, it is customary to introduce a weight decay term. Furthermore, to avoid the system being trapped in local minima, noise is often added in the dynamics. Hence the gradient descent dynamics is given by dJj (t) -_ N1 " ", ( ) ) I' ) ( ) (4) -;u~g (x~ t ,yl' ~j -).Jj(t +"1j t, ~ where, here and below, g' (x, y) and gil (x, y) respectively represent the first and second partial derivatives of g(x, y) with respect to x. ). is the weight decay strength, and "1j(t) is the noise term at temperature T with (5) 3 The Cavity Method Our theory is the dynamical version of the cavity method [9, 10, 11]. It uses a self-consistency argument to consider what happens when a new example is added to a training set. The central quantity in this method is the cavity activation, which is the activation of a new example for a perceptron trained without that example. Since the original network has no information about the new example, the cavity activation is stochastic. Specifically, denoting the new example by the label 0, its cavity activation at time t is ho(t) = J(t) . f1. (6) For large N and independently generated examples, ho(t) is a Gaussian variable. Its covariance is given by the correlation function G(t, s) of the weights at times t and s, that is, (7) (ho(t)ho(s?) = J(t). J(s) == G(t,s), 288 S. Li and K. Y. M. Wong where ~J and ~2 are assumed to be independent for j i- k. The distribution is further specified by the teacher-student correlation R(t), given by (ho(t)yo) = j(t) . jj = R(t). (8) Now suppose the perceptron incorporates the new example at the batch-mode learning step at time s. Then the activation of this new example at a subsequent time t > s will no longer be a random variable. Furthermore, the activations of the original p examples at time t will also be adjusted from {xJl(t)} to {x~(t)} because of the newcomer, which will in turn affect the evolution of the activation of example 0, giving rise to the so-called Onsager reaction effects. This makes the dynamics complex, but fortunately for large p '" N, we can assume that the adjustment from xJl(t) to x2(t) is small, and perturbative analysis can be applied. Suppose the weights of the original and new perceptron at time t are {Jj (t)} and {JJ(t)} respectively. Then a perturbation of (4) yields (! + ,x) (.tj(t) - Jj(t? ~g'(xo(t),yo)~J = + ~ 2: ~fgll(XJl(t), YJl)~r(J'2(t) - Jk(t?. (9) Jlk The first term on the right hand side describes the primary effects of adding example the training set, and is the driving term for the difference between the two perceptrons. The second term describes the secondary effects due to the changes to the original examples caused by the added example, and is referred to as the On sager reaction term. One should note the difference between the cavity and generic activations of the added example. The former is denoted by ho(t) and corresponds to the activation in the perceptron {Jj (t) }, whereas the latter, denoted by Xo (t) and corresponding to the activation in the percept ron {.tj (t)}, is the one used in calculating the gradient in the driving term of (9). Since their notations are sufficiently distinct, we have omitted the superscript 0 in xo(t), which appears in the background examples x~(t). o to The equation can be solved by the Green's function technique, yielding .tj(t) - Jj(t) = 2:! dsGjk(t, s) (~g~(s)~2) , (10) k where g~(s) = g'(xo(s),yo) and Gjk(t, s) is the weight Green's function satisfying Gjk(t,S) = G(O)(t - S)6jk + ~ ~! dt'G(O)(t - t')~fg~(t')~rGik(t' - s), (11) Jl\ = G(O)(t - s) e(t - s) exp( -,x(t - s? is the bare Green's function, and e is the step function. The weight Green's function describes how the effects of example 0 propagates from weight Jk at learning time s to weight Jj at a subsequent time t, including both primary and secondary effects. Hence all the temporal correlations have been taken into account. For large N, the equation can be solved by a diagrammatic approach similar to [5]. The weight Green's function is self-averaging over the distribution of examples and is diagonal, i.e. limN-+ooGjk(t,s) = G(t,s)6jk , where G(t, s) = G(O)(t - s) +a ! ! dt 1 dt2G(O)(t - td(g~(tdDJl(tl' t2))G(t2' s). (12) 289 Statistical Dynamics ofBatch Learning D ~ (t, s) is the example Green's function given by D~(t,s) = c5(t - s) + Jdt'G(t,t')g~(t')D~(t',s). (13) This allows us to express the generic activations of the examples in terms of their cavity counterparts. Multiplying both sides of (10) and summing over j, we get xo(t) - ho(t) = J dsG(t, s)g~(s). (14) This equation is interpreted as follows. At time t, the generic activation xo{t) deviates from its cavity counterpart because its gradient term g&(s) was present in the batch learning step at previous times s. This gradient term propagates its influence from time s to t via the Green's function G(t, s). Statistically, this equation enables us to express the activation distribution in terms of the cavity activation distribution, thereby getting a macroscopic description of the dynamics. To solve for the Green's functions and the activation distributions, we further need the fluctuation-response relation derived by linear response theory, C(t, s) =a J dt'G(O) (t - t')(g~(t')x~(s? + 2T J dt'G(O)(t - t')G(s, t'). (15) Finally, the teacher-student correlation is given by R(t) = a 4 J dt'G(O)(t - t')(g~(t')y~}. (16) A Solvable Case The cavity method can be applied to the dynamics of learning with an arbitrary cost function. When it is applied to the Hebb rule, it yields results identical to the exact results in [8]. Here we present the results for the Adaline rule to illustrate features of learning dynamics derivable from the study. This is a common learning rule and bears resemblance with the more common back-propagation rule. Theoretically, its dynamics is particularly convenient for analysis since g" (x) = -1, rendering the weight Green's function time translation invariant, Le. G(t, s) = G(t - s). In this case, the dynamics can be solved by Laplace transform. To monitor the progress of learning, we are interested in three performance measures: (a) Training error ft, which is the probability of error for the training examples. It is given by ft = (9 (-xsgnY?:q, , where the average is taken over the joint distribution p(x, y) of the training set. (b) Test error ftest, which is the probof the training examples are corrupted by an ability of error when the inputs additive Gaussian noise of variance ~ 2 . This is a relevant performance measure when the percept ron is applied to process data which are the corrupted versions of the training data. It is given by ftest = (H(xsgny/~JC(t,t?):r;y. When ~2 = 0, the test error reduces to the training error. (c) Generalization error fg, which is the probability of error for an arbitrary input ~j when the teacher and student outputs are compared. It is given by fg = arccos[R(t)/ JC(t, t?)/7r. e; Figure l(a) shows the evolution of the generalization error at T = O. When the weight decay strength varies, the steady-state generalization error is minimized at the optimum 7r (17) Aopt = '2 - 1, S. Li and K. Y. M Wong 290 which is independent of Q. It is interesting to note that in the cases of the linear percept ron , the optimal weight decay strength is also independent of Q and only determined by the output noise and unlearn ability of the examples [5, 7]. Similarly, here the student is only provided the coarse-grained version of the teacher's activation in the form of binary bits. For A < Aopt, the generalization error is a non-monotonic function in learning time. Hence the dynamics is plagued by overtraining, and it is desirable to introduce early stopping to improve the perceptron performance. Similar behavior is observed in linear perceptrons [5, 6, 7]. To verify the theoretical predictions, simulations were done with N = 500 and using 50 samples for averaging. As shown in Fig. l(a), the agreement is excellent. Figure 1 (b) compares the generalization errors at the steady-state and the early stopping point. It shows that early stopping improves the performance for A < Aopt, which becomes near-optimal when compared with the best result at A = Aopt. Hence early stopping can speed up the learning process without significant sacrifice in the generalization ability. However, it cannot outperform the optimal result at steadystate. This agrees with a recent empirical observation that a careful control of the weight decay may be better than early stopping in optimizing generalization [12]. 0.40 0.38 ,0.3s l 0.36 w~ 1..=10 e~ 0.32 L~:::::::::=:::;::;;;;;;;mEE:=::: 1..=0.1 1..=1.. .... a=O.5 c .Q ? ~ a=O.5 t ~ 0.32 oc " ~ 0.30 iii N ~ I ~~0.34~ 0.28 i .t:! l.-..a~=.....I~.2.....-.............................~.....-.................._ 1..=10 ~ 0.28 I..=Ql ~ 0.26 Q) c 1..=\", 0.24 0.24 00 a=1.2 [--r-hh-e; 0.20 o -----c... 2 - '--'--- ~---"--'--~-'--- 4 6 timet 8 10 12 0.22 0.0 ---'- - _.- 0.5 1.0 1.5 weight decay ).. Figure 1: (a) The evolution of the generalization error at T = 0 for Q = 0.5,1.2 and different weight decay strengths A. Theory: solid line, simulation: symbols. (b) Comparing the generalization error at the steady state (00) and at the early stopping point (t es ) for Q = 0.5,1.2 and T = O. In the search for optimal learning algorithms, an important consideration is the environment in which the performance is tested. Besides the generalization performance, there are applications in which the test examples have inputs correlated with the training examples. Hence we are interested in the evolution of the test error for a given additive Gaussian noise 6, in the inputs. Figure 2(a) shows, again, that there is an optimal weight decay parameter Aopt which minimizes the test error. Furthermore, when the weight decay is weak, early stopping is desirable. Figure 2(b) shows the value of the optimal weight decay as a function of the input noise variance 6, 2 . To the lowest order approximation, Aopt ex: 6, 2 for sufficiently large 6, 2 . The dependence of Aopt on input noise is rather general since it also holds in the case of random examples [13]. In the limit of small 6,2, Aopt vanishes as 6,2 for Q < 1, whereas Aopt approaches a nonzero constant for Q > 1. Hence for 291 Statistical Dynamics of Batch Learning a < 1, weight decay is not necessary when the training error is optimized, but when the percept ron is applied to process increasingly noisy data, weight decay becomes more and more important in performance enhancement. Figure 2(b) also shows the phase line Aot(~2) below which overtraining occurs. Again, to the lowest order approximation, Aot ex ~2 for sufficiently large ~2 . However, unlike the case of generalization error, the line for the onset of overtraining does not coincide exactly with the line of optimal weight decay. In particular, for an intermediate range of input noise, the optimal line lies in the region of overtraining, so that the optimal performance can only be attained by tuning both the weight decay strength and learning time. However, at least in the present case, computational results show that the improvement is marginal. 0.30 , 20 I 028 i ~A=Ol c . l a=1.2 ... 0.26 Q) 0.24 g ~ , :--s r A=lO A=3.6 0 I t t 15 .-< 10 l o.oo ~ 0.22 ~ 5 -0.05 ' - - - - - - - - o 0.18 0---'--~2~~~ 3----"4~--'5----'6 Time Figure 2: (a) The evolution of the test error for ~ 2 = 3, T = 0 and different weight decay strengths A (Aopt ::::: 1.5,3.6 for a = 0.5, 1.2 respectively). (b) The lines of the optimal weight decay and the onset of overtraining for a = 5. Inset: The same data with Aot - Aopt (magnified) versus ~2. 5 Conclusion Based on the cavity method, we have introduced a new framework for modeling the dynamics of learning, which is applicable to any learning cost function, making it a versatile theory. It takes into full account the temporal correlations generated by the use of a restricted set of examples, which is more realistic in many situations than theories of on-line learning with an infinite training set. While the Adaline rule is solvable by the cavity method, it is still a relatively simple model approachable by more direct methods. Hence the justification of the method as a general framework for learning dynamics hinges on its applicability to less trivial cases. In general, g~(t') in (13) is not a constant and DJl(t, s) has to be expanded as a series. The dynamical equations can then be considered as the starting point of a perturbation theory, and results in various limits can be derived, e.g. the limits of small a, large a, large A, or the asymptotic limit. Another area for the useful application of the cavity method is the case of batch learning with very large learning steps. Since it has been shown recently that such learning converges in a few steps [6], the dynamical equations remain simple enough for a meaningful study. Preliminary results along this direction are promising and will be reported elsewhere. S. Li and K. Y. M Wong 292 An alternative general theory for learning dynamics, the dynamical replica theory, has recently been developed [8]. It yields exact results for Hebbian learning, and approximate results for more non-trivial cases. Based on certain self-averaging assumptions, the theory is able to approximate the dynamics by the evolution of single-time functions , at the expense of having to solve a set of saddle point equations in the replica formalism at every learning instant. On the other hand, our theory retains the functions G(t,s) and C(t, s) with double arguments, but develops naturally from the stochastic nature of the cavity activations. Contrary to a suggestion [14], the cavity method can also be applied to the on-line learning with restricted sets of examples. It is hoped that by adhering to an exact formalism, the cavity method can provide more fundamental insights when the studies are extended to more sophisticated multilayer networks of practical importance. The method enables us to study the effects of weight decay and early stopping. It shows that the optimal strength of weight decay is determined by the imprecision in the examples, or the level of input noise in anticipated applications. For weaker weight decay, the generalization performance can be made near-optimal by early stopping. Furthermore, depending on the performance measure, optimality may only be attained by a combination of weight decay and early stopping. Though the performance improvement is marginal in the present case, the question remains open in the more general context. We consider the present work as the beginning of an in-depth study of learning dynamics. Many interesting and challenging issues remain to be explored. Acknowledgments We thank A. C. C. Coolen and D. Saad for fruitful discussions during NIPS. This work was supported by the grant HKUST6130/97P from the Research Grant Council of Hong Kong. References [1] D. Saad and S. Solla, Phys. Rev. Lett. 74,4337 (1995) . [2] [3] [4] [5] [6] [7] [8] D. Saad and M. Rattray, Phys. Rev. Lett. 79, 2578 (1997). J. Hertz, A. Krogh and G. I. Thorbergssen, J. Phys. A 22, 2133 (1989) . M. Opper, Europhys. Lett. 8, 389 (1989) . A. Krogh and J. A. Hertz, J. Phys. A 25, 1135 (1992) . S. Bos and M. Opper, J. Phys. A 31, 4835 (1998). S. Bos, Phys. Rev. E 58, 833 (1998) . A. C. C. Coolen and D. Saad, in On-line Learning in Neural Networks, ed. D. Saad (Cambridge University Press, Cambridge, 1998). [9] M. Mezard, G. Parisi and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore) (1987). [10] K. Y. M. Wong, Europhys. Lett. 30, 245 (1995). [11] K. Y. M. Wong, Advances in Neural Information Processing Systems 9, 302 (1997) . [12] L. K. Hansen, J. Larsen and T. Fog, IEEE Int. Conf. on Acoustics, Speech, and Signal Processing 4,3205 (1997). [13] Y. W. Tong, K. Y. M. Wong and S. Li, to appear in Proc. of IJCNN'99 (1999) . [14] A. C. C. Cool en and D. Saad, Preprint KCL-MTH-99-33 (1999). 

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Bayesian Reconstruction of 3D Human Motion from Single-Camera Video Nicholas R. Howe Department of Computer Science Cornell University Ithaca, NY 14850 nihowe@cs.comell.edu Michael E. Leventon Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 leventon@ai.mit.edu William T. Freeman MERL - a Mitsubishi Electric Research Lab 201 Broadway Cambridge, MA 02139 freeman@merL.com Abstract The three-dimensional motion of humans is underdetermined when the observation is limited to a single camera, due to the inherent 3D ambiguity of 2D video. We present a system that reconstructs the 3D motion of human subjects from single-camera video, relying on prior knowledge about human motion, learned from training data, to resolve those ambiguities. After initialization in 2D, the tracking and 3D reconstruction is automatic; we show results for several video sequences. The results show the power of treating 3D body tracking as an inference problem. 1 Introduction We seek to capture the 3D motions of humans from video sequences. The potential applications are broad, including industrial computer graphics, virtual reality, and improved human-computer interaction. Recent research attention has focused on unencumbered tracking techniques that don't require attaching markers to the subject's body [4, 5], see [12] for a survey. Typically, these methods require simultaneous views from multiple cameras. Motion capture from a single camera is important for several reasons. First, though underdetermined, it is a problem people can solve easily, as anyone viewing a dancer in a movie can confirm. Single camera shots are the most convenient to obtain, and, of course, apply to the world's film and video archives. It is an appealing computer vision problem that emphasizes inference as much as measurement. This problem has received less attention than motion capture from multiple cameras. Goncalves et.al. rely on perspective effects to track only a single arm, and thus need not deal with complicated models, shadows, or self-occlusion [7]. Bregler & Malik develop a body tracking system that may apply to a single camera, but performance in that domain is Bayesian Reconstruction of 3D Human Motion from Single-Camera Video 821 not clear; most of the examples use multiple cameras [4]. Wachter & Nagel use an iterated extended Kalman filter, although their body model is limited in degrees of freedom [12l Brand [3] uses an learning-based approach, although with representational expressiveness restricted by the number of HMM states. An earlier version of the work reported here [10] required manual intervention for the 2D tracking. This paper presents our system for single-camera motion capture, a learning-based approach, relying on prior information learned from a labeled training set. The system tracks joints and body parts as they move in the 2D video, then combines the tracking information with the prior model of human motion to form a best estimate of the body's motion in 3D. Our reconstruction method can work with incomplete information, because the prior model allows spurious and distracting information to be discarded . The 3D estimate provides feedback to influence the 2D tracking process to favor more likely poses. The 2D tracking and 3D reconstruction modules are discussed in Sections 3 and 4, respectively. Section 4 describes the system operation and presents performance results. Finally, Section 5 concludes with possible improvements. 2 2D Tracking The 2D tracker processes a video stream to determine the motion of body parts in the image plane over time. The tracking algorithm used is based on one presented by Ju et. al. [9], and performs a task similar to one described by Morris & Rehg [11]. Fourteen body parts are modeled as planar patches, whose positions are controlled by 34 parameters. Tracking consists of optimizing the parameter values in each frame so as to minimize the mismatch between the image data and a projection of the body part maps. The 2D parameter values for the first frame must be initialized by hand, by overlaying a model onto the 2D image of the first frame. We extend Ju et. al.'s tracking algorithm in several ways. We track the entire body, and build a model of each body part that is a weighted average of several preceding frames, not just the most recent one. This helps eliminate tracking errors due to momentary glitches that last for a frame or two. We account for self-occlusions through the use of support maps [4, 1]. It is essential to address this problem, as limbs and other body parts will often partly or wholly obscure one another. For the single-camera case, there are no alternate views to be relied upon when a body part cannot be seen. The 2D tracker returns the coordinates of each limb in each successive frame. These in tum yield the positions of joints and other control points needed to perform 3D reconstruction. 3 3D Reconstruction 3D reconstruction from 2D tracking data is underdetermined. At each frame, the algorithm receives the positions in two dimensions of 20 tracked body points, and must to infer the correct depth of each point. We rely on a training set of 3D human motions to determine which reconstructions are plausible. Most candidate projections are unnatural motions, if not anatomically impossible, and can be eliminated on this basis. We adopt a Bayesian framework, and use the training data to compute prior probabilities of different 3D motions. We model plausible motions as a mixture of Gaussian probabilities in a high-dimensional space. Motion capture data gathered in a professional studio provide the training data: frame-by-frame 3D coordinates for 20 tracked body points at 20-30 frames per second. We want to model the probabilities of human motions of some short duration, long enough be N. R. Howe, M. E. Leventon and W T. Freeman 822 informative, but short enough to characterize probabilistically from our training data. We assembled the data into short motion elements we caJled snippets of 11 successive frames, about a third of a second. We represent each snippet from the training data as a large column vector of the 3D positions of each tracked body point in each frame of the snippet. We then use those data to build a mixture-of-Gaussians probability density model [2]. For computational efficiency, we used a clustering approach to approximate the fitting of an EM algorithm. We use k-means clustering to divide the snippets into m groups, each of which will be modeled by a Gaussian probability cloud. For each cluster, the matrix M j is formed, where the columns of M j are the nj individual motion snippets after subtracting the mean J.l j. The singular value decomposition (SVD) gives M j = U j Sj where Sj contains the singular values along the diagonal, and Uj contains the basis vectors. (We truncate the SVD to include only the 50 largest singular values.) The cluster can be modeled by a multidimensional Gaussian with covariance Aj = UjSJUJ. The prior probability of a snippet x over all the models is a sum of the Gaussian probabilities weighted by the probability of each model. VI, ;j m P(x) = Lk7fje- !(x-llj)TA- 1 (X-llj) (1) j=1 Here k is a normalization constant, and 7fj is the a priori probability of model j, computed as the fraction of snippets in the knowledge base that were originally placed in cluster j . Given this approximately derived mixture-of-factors model [6], we can compute the prior probability of any snippet. To estimate the data term (likelihood) in Bayes' law, we assume that the 2D observations include some Gaussian noise with variance (T. Combined with the prior, the expression for the probability of a given snippet x given an observation ybecomes p(x,e,s,vly) = k' (e-IIY-R6 ,?.v(XlII 2/(2tr2)) (f k7fj e-!(X-llj)T A _l(X-llj)) (2) J=l In this equation, Rn ,s,ii(X) is a rendering function which maps a 3D snippet x into the image coordinate system, performing scaling s, rotation about the vertical axis and image-plane translation v. We use the EM algorithm to find the probabilities of each Gaussian in the mixture and the corresponding snippet x that maximizes the probability given the observations [6]. This allows the conversion of eleven frames of 2D tracking measurements into the most probable corresponding 3D snippet. In cases where the 2D tracking is poor, the reconstruction may be improved by matching only the more reliable points in the likelihood term of Equation 2. This adds a second noise process to explain the outlier data points in the likelihood term. e, To perform the full 3D reconstruction, the system first divides the 2D tracking data into snippets, which provides the y values of Eq. 2, then finds the best (MAP) 3D snippet for each of the 2D observations. The 3D snippets are stitched together, using a weighted interpolation for frames where two snippets overlap. The result is a Bayesian estimate of the subject's motion in three dimensions. 4 Performance The system as a whole will track and successfully 3D reconstruct simple, short video clips with no human intervention, apart from 2D pose initialization. It is not currently reliable enough to track difficult footage for significant lengths of time. However, analysis of short clips demonstrates that the system can successfully reconstruct 3D motion from ambiguous Bayesian Reconstruction of 3D Human Motion from Single-Camera Video 823 2D video. We evaluate the two stages of the algorithm independently at first, and then consider their operation as a system. 4.1 Performance of the 3D reconstruction The 3D reconstruction stage is the heart of the system. To our knowledge, no similar 2D to 3D reconstruction technique relying on prior infonnation has been published. ([3], developed simultaneously, also uses an inference-based approach). Our tests show that the module can restore deleted depth infonnation that looks realistic and is close to the ground truth, at least when the knowledge base contains some examples of similar motions. This makes the 3D reconstruction stage itself an important result, which can easily be applied in conjunction with other tracking technologies. To test the reconstruction with known ground truth, we held back some of the training data for testing. We artificially provided perfect 2D marker position data, yin Eq. 2, and tested the 3D reconstruction stage in isolation. After removing depth information from the test sequence, the sequence is reconstructed as if it had come from the 2D tracker. Sequences produced in this manner look very much like the original. They show some rigid motion error along the line of sight. An analysis of the uncertainty in the posterior probability predicts high uncertainty for the body motion mode of rigid motion parallel to the orthographic projection [10]. This slipping can be corrected by enforcing ground-contact constraints. Figure 1 shows a reconstructed running sequence corrected for rigid motion error and superimposed on the original. The missing depth information is reconstructed well, although it sometimes lags or anticipates the true motion slightly. Quantitatively, this error is a relatively small effect. After subtracting rigid motion error, the mean residual 3D errors in limb position are the same order of magnitude as the small frame-to frame changes in those positions. - ' ~ . -, " _. ~ .~ ..? Figure 1: Original and reconstructed running sequences superimposed (frames 1, 7, 14, and 21). 4.2 Performance of the 2D tracker The 2D tracker performs well under constant illumination, providing quite accurate results from frame to frame. The main problem it faces is the slow accumulation of error. On longer sequences, the errors can build up to the point where the module is no longer tracking the body parts it was intended to track. The problem is worsened by low contrast, occlusion and lighting changes. More careful body modeling [5], lighting models, and modeling of the background may address these issues. The sequences we used for testing were several seconds long and had fairly good contrast. Although adequate to demonstrate the operation of our system, the 2D tracker contains the most open research issues. 4.3 Overall system performance Three example reconstructions are given, showing a range of different tracking situations. The first is a reconstruction of a stationary figure waving one arm, with most of the motion 824 N. R. Howe. M E. Leventon and W. T. Freeman in the image plane. The second shows a figure bringing both arms together towards the camera, resulting in a significant amount of foreshortening. The third is a reconstruction of a figure walking sideways, and includes significant self-occlusion Figure 2: First clip and its reconstruction (frames 1, 2S, SO, and 7S). The first video is the easiest to track because there is little or no occlusion and change in lighting. The reconstruction is good, capturing the stance and motion of the arm. There is some rigid motion error, which is corrected through ground friction constraints. The knees are slightly bent; this may be because the subject in the video has different body proportions than those represented in the training database. Figure 3: Second clip and its reconstruction (frames 1, 2S, SO, and 7S). The second video shows a figure bringing its arms together towards the camera. The only indication of this is in the foreshortening of the limbs, yet the 3D reconstruction correctly captures this in the right arm. (Lighting changes and contrast problems cause the 2D tracker to lose the left arm partway through, confusing the reconstruction of that limb, but the right arm is tracked accurately throughout.) The third video shows a figure walking to the right in the image plane. This clip is the hardest for the 2D tracker, due to repeated and prolonged occlusion of some body parts. The tracker loses the left arm after IS frames due to severe occlusion, yet the remaining tracking information is still sufficient to perform an adequate reconstruction. At about frame 4S, the left leg has crossed behind the right several times and is lost, at which point the reconstruction quality begins to degrade. The key to a more reliable reconstruction on this sequence is better tracking. Bayesian Reconstruction of 3D Human Motion from Single-Camera Video .. .. ,t 825 :'."...., ~ 1. Figure 4: Third clip and its reconstruction (frames 6, 16, 26, and 36). 5 Conclusion We have demonstrated a system that tracks human figures in short video sequences and reconstructs their motion in three dimensions. The tracking is unassisted, although 2D pose initialization is required. The system uses prior information learned from training data to resolve the inherent ambiguity in going from two to three dimensions, an essential step when working with a single-camera video source. To achieve this end, the system relies on prior knowledge, extracted from examples of human motion. Such a learning-based approach could be combined with more sophisticated measurement-based approaches to the tracking problem [12, 8, 4]. References [1] J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani. Hierarchical model-based motion estimation. In European Conference on Computer Vision, pages 237-252, 1992. [2] C. M. Bishop. Neural networks for pattern recognition. Oxford, 1995. [3] M. Brand. Shadow puppetry. In Proc. 7th IntI. Con! on Computer Vision, pages 1237-1244. IEEE, 1999. [4] c. Bregler and 1. Malik. Tracking people with twists and exponential maps. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Santa Barbera, 1998. [5] D . M. Gavrila and L. S. Davis. 3d model-based tracking of humans in action: A multi-view approach. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Francisco, 1996. [6] Z. Ghahramani and G. E. Hinton. The EM algorithm for mixtures offactor analyzers. Technical report, Department of Computer Science, University of Toronto, May 21 1996. (revised Feb. 27, 1997). [7] L. Goncalves, E. Di Bernardo, E. Ursella, and P. Perona. Monocular tracking of the human arm in 3D. In Proceedings of the Third International Conference on Computer Vision, 1995. [8] M. Isard and A. Blake. Condensation - conditional density propagation for visual tracking. International Journal of Computer Vision, 29( 1):5-28, 1998. [9] S. X. Ju, M. J. Black, and Y. Yacoob. Cardboard people: A parameterized model of articulated image motion. In 2nd International Conference on Automatic Face and Gesture Recognition, 1996. 826 N. R. Howe, M. E. Leventon and W T. Freeman [10] M. E. Leventon and W. T. Freeman. Bayesian estimation of 3-d human motion from an image sequence. Technical Report TR98-06, Mitsubishi Electric Research Lab, 1998. [11] D. D. Morris and 1. Rehg. Singularity analysis for articulated object tracking. In IEEE Computer Societ), Conference on Computer Yz'sion and Pattern Recognition, Santa Barbera, 1998. [12] S. Wachter and H.-H. Nagel. Tracking of persons in monocular image sequences. In Nonrigid and ArticuLated Motion Workshop, 1997. 

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Transductive Inference for Estimating Values of Functions Olivier Chapelle*, Vladimir Vapnik*,t, Jason Westontt.t,* * AT&T Research Laboratories, Red Bank, USA. t Royal Holloway, University of London, Egham, Surrey, UK. tt Barnhill BioInformatics.com, Savannah, Georgia, USA. { chapelle, vlad, weston} @research.att.com Abstract We introduce an algorithm for estimating the values of a function at a set of test points Xe+!, ... , xl+m given a set of training points (XI,YI), ... ,(xe,Ye) without estimating (as an intermediate step) the regression function . We demonstrate that this direct (transducti ve) way for estimating values of the regression (or classification in pattern recognition) can be more accurate than the traditionalone based on two steps, first estimating the function and then calculating the values of this function at the points of interest. 1 Introduction Following [6] we consider a general scheme of transductive inference. Suppose there exists a function y* = fo(x) from which we observe the measurements corrupted with noise ((Xl, YI)," . (xe, Ye)), Yi = Y; + ~i' (1) Find an algorithm A that using both the given set of training data (1) and the given set of test data (2) (Xl+!,' .. , XHm) selects from a set of functions {x t--+ f (x)} a function Y = f(x) = fA(xlxl,YI, ... ,Xl,Yl,XHI"",XHm) and minimizes at the points of interest the functional R(A) = E (~ (y; - fA(Xilxl,Yl, ... ,Xl,Ye,Xl+l, . .. ,Xl+m))2) (3) (4) i=l+l where expectation is taken over X and~. For the training data we are given the vector X and the value Y, for the test data we are only given x. Usually, the problem of estimating values of a function at points of interest is sol ved in two steps: first in a given set of functions f (x, a), a E A one estimates the regression, i.e the function which minimizes the functional R(a) = J ((y - f(x, a))2dF(x, Y), (5) 422 0. Chapelle, V. N. Vapnik and J. Weston (the inductive step) and then using the estimated function Y = f(x,al) we calculate the values at points of interest yi = (6) f(x;, ae) (the deductive step). Note, however, that the estimation of a function is equivalent to estimating its values in the continuum points of the domain of the function. Therefore, by solving the regression problem using a restricted amount of information, we are looking for a more general solution than is required. In [6] it is shown that using a direct estimation method one can obtain better bounds than through the two step procedure. In this article we develop the idea introduced in [5] for estimating the values of a function only at the given points. The material is organized as follows. In Section 1 we consider the classical (inductive) Ridge Regression procedure, and the leave-one--out technique which is used to measure the quality of its solutions. Section 2 introduces the transductive method of inference for estimation of the values of a function based on this leave-one- out technique. In Section 3 experiments which demonstrate the improvement given by transductive inference compared to inductive inference (in both regression and pattern recognition) are presented. Finally, Section 4 summarizes the results. 2 Ridge Regression and the Leave-One-Out procedure In order to describe our transductive method, let us first discuss the classical twostep (inductive plus deductive) procedure of Ridge Regression. Consider the set of functions linear in their parameters n f(x, a) = L aicPi(x). (7) i=1 To minimize the expected loss (5), where F(x, y) is unknown, we minimize the following empirical functional (the so-called Ridge Regression functional [1]) l Remp(a) = 1~ eL)Yi - f(Xi, a)) 2 + 1'110.11 2 (8) i=1 where l' is a fixed positive constant, called the regularization parameter. The minimum is given by the vector of coefficients ae = a(xl, Yl, ... , Xl, Yl) = (KT K + 1'1)-1 KTy (9) where y = (Y1, ... ,Ylf, and K is a matrix with elements: Kij=cPj(Xi), i=I, ... ,?, j=I, ... ,n. (10) (11) The problem is to choose the value l' which provides small expected loss for training on a sample Sl = {(Xl,Yl), .. . ,(Xl,Yl)}. For this purpose, we would like to choose l' such that f"f minimizing (8) also minimizes R= J (Y* - f"f(x* I Sl))2dF(x*, y*)dF(Se). (12) 423 Transductive Inference for Estimating Values of Functions Since F(x, y) is unknown one cannot estimate this minimum directly. To solve this problem we instead use the leave-one-out procedure, which is an almost unbiased estimator of (12). The leave-one-out error of an algorithm on the training sample Sf. is (13) The leave-one-out procedure consists of removing from the training data one element (say (Xi, Yi)), constructing the regression function only on the basis of the remaining training data and then testing the removed element. In this fashion one tests all f elements of the training data using f different decision rules. The minimum over, of (13) we consider as the minimum over, of (12) since the expectation of (13) coincides with (12) [2]. For Ridge Regression, one can derive a dosed form expression for the leave- one- out error. Denoting (14) the error incurred by the leave-one-out procedure is [6] -_1 T. loo(r) - f f. L (Y'_k TA- 1KTy)2 ~ ~=1 ~ 'Y (15) 1 _ kT A- 1k. ~ ~ 'Y where (16) k t = (i>I(xd??? ,l/>n(Xt)f? Let, = ,0 be the minimum of (15). Then the vector yO = K*(KT K +,0 I)-I KTy (17) where I/>(XHI) K*- ( . (18) 1/>1 (XHm) is the Ridge Regression estimate of the unknown values (Ye+l' ... ,Ye+m)' 3 Leave-One-Out Error for Transductive Inference In transductive inference, our goal is to find an algorithm A which minimizes the functional (4) using both the training data (1) and the test data (2). We suggest the following method: predict (Ye+l' ... 'Ye+m) by finding those values which minimize the leave-one-out error of Ridge Regression training on the joint set (Xl, yd,?? . ,(Xl, Yl), (Xl+l, ye+l),?? ., (XHm, Ye+m)' (19) This is achieved in the following way. Suppose we treat the unknown values (Ye+l" .. ,Ye+m) as variables and for some fixed value of these variables we minimize the following empirical functional Remp(aly;, .. ?, y~) = f: m ( ~(Yi f. ~=l f(xi,a))2 Hm + .L (y; - f(xi, a))2 ) +,llaI1 2 . ~=l+1 (20) This functional differs only in the second term from the functional (8) and corresponds to performing Ridge Regression with the extra pairs (21) 424 O. Chapel/e, V. N. Vapnik and J. Weston Suppose that vector Y" = (Yi, ... , y:n) is taken from some set Y" E Y such that the pairs (21) can be considered as a sample drawn from the same distribution as the pairs (Xl, yi), ... , (Xl, yi)? In this case the leave-one-out error of minimizing (20) over the set (19) approximates the functional (4). We can measure this leaveone-out error using the same technique as in Ridge Regression. Using the closed form (15) one obtains (Y:'~ _kTt~TA-I~-1~kTY) 2 L +m 1 - k A-y k 1 7loo(rly~, .. ?,y~) = -f-- l+m i=l where we denote x = (Xl, ... , Xl+ m), Y = (YI, ... , Yl, Yi+1" (22) i i .. , Yi+m)T, and (23) Kij=<pj(Xi), i=I, ... ,i+m, j=I, ... ,n. (24) kt = (<PI(Xt} ... ,<Pn(xt)f? (25) Now let us rewrite the expression (22) in an equivalent form to separate the terms with Y from the terms with x. Introducing (26) and the matrix M with elements l+m M tJ.. -_ "~ k=l CikC kj Cu 2 (27) we obtain the equivalent expression of (22) (28) In order for the Y" which minimize the leave-one-out procedure to be valid it is required that the pairs (21) are drawn from the same distribution as the pairs (Xl, yi), ... , (Xl, yi)? To satisfy this constraint we choose vectors Y" from the set Y = {Y" : IIY" - y011 -s: R} (29) where the vector yO is the solution obtained from classical Ridge Regression. To minimize (28) under constraint (29) we use the functional (30) where 'Y" is a constant depending on R. Now, to find the values at the given points of interest (2) all that remains is to find the minimum of (30) in Y". Note that the matrix M is obtained using only the vectors X. Therefore, to find the minimum of this functional we rewrite Equation (30) as where (32) Transductive Inference for Estimating Values of Functions 425 and Mo is a ex e matrix, Ml is a e x m matrix and M2 is a m x m matrix. Taking the derivative of (31) in y* we obtain the condition for the solution 2M1 Y + 2M2 Y* - 2),*Y o + 2),*Y* = 0 (33) which gives the predictions Y* = ()'* 1+ M 2)-1 (-MIY + ),*yO) . (34) In this algorithm (which we will call Transductive Regression) we have two parameters to control: )' and )'*. The choice of )' can be found using the leave-one-out estimator (15) for Ridge Regression. This leaves )'* as the only free parameter. 4 Experiments To compare our one- step transductive approach with the classical two- step approach we performed a series of experiments on regression problems. We also describe experiments applying our technique to the problem of pattern recognition . 4.1 Regression We conducted computer simulations for the regression problem using two datasets from the DELVE repository: boston and kin-32th. The boston dataset is a well- known problem where one is required to estimate house prices according to various statistics based on 13 locational, economic and structural features from data collected by the U.S Census Service in the Boston Massachusetts area. The kin-32th dataset is a realistic simulation of the forward dynamics of an 8 link all-revolute robot arm. The task is to predict the distance of the end-effector from a target, given 32 inputs which contain information on the joint positions, twist angles and so forth. Both problems are nonlinear and contain noisy data. Our objective is to compare our transductive inference method directly with the inductive method of Ridge Regression. To do this we chose the set of basis functions ?i(X) = exp (-llx - xiI 12/2(2), i = 1, ... , e, and found the values of )' and a for Ridge Regression which minimized the leave-one-out bound (15). We then used the same values of these parameters in our transductive approach , and using the basis functions ?i(X) = exp (-llx - XiW /2(72) , i = 1, . . . , e+ m, we then chose a fixed value of)'* . For the boston dataset we followed the same experimental setup as in [4], that is, we partitioned the training set of 506 observations randomly 100 times into a training set of 481 observations and a testing set of 25 observations. We chose the values of)' and a by taking the minimum average leave- one-out error over five more random splits of the data stepping over the parameter space. The minimum was found at )' = 0.005 and log a = 0.7. For our transductive method, we also chose the parameter),* = 10. In Figure la we plot mean squared error (MSE) on the test set averaged over the 100 runs against log a for Ridge Regression and Transductive Regression. Transductive Regression outperforms Ridge Regression, especially at the minimum. To observe the influence of the number of test points m on the generalization ability of our transductive method, we ran further experiments, setting )'* = e/2m for 0. Chapel/e, V. N. Vapnik and J. Weston 426 1- 9.2 ,, 9 Tranaductive Regression - _ . Ridge Regress.,n 1 8r---~----~----~---------, 7.95 " 8.8 7.9 8.6 g7.85 UJ 8.4 W ~ 7.8 8.2 7.75 8 7.7 7.8 7.6 0.4 0.6 0.8 7.65;L--~5:--=~10===-15==:"2=0-----:!25 1.2 Log sigma Test Set Size (a) (b) 015 0.14 ,, 1- ,, 1:' ~0. 12 Q) O.13,r---~--=--- ,, ~ 0.1 0.12 t: ~0. 115 ;l ,, ,, ---- -------- ------ ----- -- ---- 0.125 ,, ,, , I- 0.11 0.135 Transductive Regression - - - RIdge RegreSSion / 0.13 g , ,, ,, , I- " 0.11 0.105 0.1 , - --- , 0.095 0.09 1 1.5 2 2.5 Log sigma 3 1 0~ 0----1~50,------.,.200 ~---:2=-=-' 50 0.09''-------:'':50-----.,. Test Set Size (c) (d) Figure 1: A comparison of Transductive Regression to Ridge Regression on the boston dataset: (a) error rates for varying (J', (b) varying the test set size, m, and on the kin-32fh dataset: (c) error rates for varying (J', (d) varying the test set size. different values of m . In Figure 1b we plot m against MSE on the testing set, at log (J' = 0.7. The results indicate that increasing the test set size gives improved performance in Transductive Regression. For Ridge Regression, of course, the size of the testing set has no influence on the generalization ability. We then performed similar experiments on the kin-32fh dataset. This time, as we were interested in large testing sets giving improved performance for Transductive Regression we chose 100 splits where we took a subset of only 64 observations for training and 256 for testing. Again the leave-one-out estimator was used to find the values, = 0.1 and log (J' = 2 for Ridge Regression, and for Transductive Regression we also chose the parameter = 0.1. We plotted MSE on the testing set against = 50/m) log (J' (Figure 1c) and the size of the test set m for log (J' = 2.75 (also, (Figure 1d) for the two algorithms. For large test set sizes our method outperforms Ridge Regression. ,* 4.2 ,* Pattern Recognition This technique can also be applied for pattern recognition problems by solving them based on minimizing functional (8) with y = ?1. Such a technique is known as a Linear Discriminant (LD) technique. Transductive Inference for Estimating Values of Functions Postal Banana Diabetes Titanic Breast Cancer Heart Thyroid AB ABR - - 12.3 26.5 22.6 30.4 20.3 4.4 10.9 23.8 22.6 26.5 16.6 4.6 427 SVM 5.5 11.5 23.5 22.4 26.0 16.0 4.8 TLD 4.7 1l.4 23.3 22.4 25.7 15.7 4.0 Table 1: Comparison of percentage test error of AdaBoost (A B) , Regularized AdaBoost (ABR), Support Vector Machines (SVM) and Tmnsductive Linear Discrimination (TLD) on seven datasets. Table 1 describes results of experiments on classification in the following problems: 2 class digit recognition (0 - 4 versus 5 - 9) splitting the training set into 23 runs of 317 observations and considering a testing set of 2000 observations, and six problems from the UCI database. We followed the same experimental setup as in [3]: the performance of a classifier is measured by its average error over one hundred partitions of the datasets into training and testing sets. Free parameter(s) are chosen via validation on the first five training datasets. The performance of the transductive LD technique was compared to Support Vector Machines, AdaBoost and Regularized AdaBoost [3]. It is interesting to note that in spite of the fact that LD technique is one of the simplest pattern recognition techniques, transductive inference based upon this method performs well compared to state of the art methods of pattern recognition . 5 Summary In this article we performed transductive inference in the problem of estimating values of functions at the points of interest. We demonstrate that estimating the unknown values via a one- step (transductive) procedure can be more accurate than the traditional two-step (inductive plus deductive) one. References [1] A. Hoerl and R. W. Kennard . Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1):55-67, 1970. [2] A. Luntz and V. Brailovsky. On the estimation of characters obtained in statistical procedure of recognition,. Technicheskaya Kibernetica, 1969. [In Russian]. [3] G. Witsch, T. Onoda, and K.-R. Muller. Soft margins for adaboost. Technical report, Royal Holloway, University of London, 1998. TR-98-2l. [4] C. Saunders, A. Gammermann, and V. Vovk. Ridge regression learning algorithm in dual variables. In Proccedings of the 15th International Conference on Machine Learning, pages 515-52l. Morgan Kaufmann, 1998. [5] V. Vapnik. Estimating of values of regression at the point of interest. In Method of Pattern Recognition. Sovetskoe Radio, 1977. [In Russian]. [6] V. Vapnik. Estimation of Dependences Based on Empirical Data. SpringerVerlag, New York, 1982. 

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622 LEARNING A COLOR ALGORITHM FROM EXAMPLES Anya C. Hurlbert and Tomaso A. Poggio Artificial Intelligence Laboratory and Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA ABSTRACT A lightness algorithm that separates surface reflectance from illumination in a Mondrian world is synthesized automatically from a set of examples, pairs of input (image irradiance) and desired output (surface reflectance). The algorithm, which resembles a new lightness algorithm recently proposed by Land, is approximately equivalent to filtering the image through a center-surround receptive field in individual chromatic channels. The synthesizing technique, optimal linear estimation, requires only one assumption, that the operator that transforms input into output is linear. This assumption is true for a certain class of early vision algorithms that may therefore be synthesized in a similar way from examples. Other methods of synthesizing algorithms from examples, or "learning", such as backpropagation, do not yield a significantly different or better lightness algorithm in the Mondrian world. The linear estimation and backpropagation techniques both produce simultaneous brightness contrast effects. The problems that a visual system must solve in decoding two-dimensional images into three-dimensional scenes (inverse optics problems) are difficult: the information supplied by an image is not sufficient by itself to specify a unique scene. To reduce the number of possible interpretations of images, visual systems, whether artificial or biological, must make use of natural constraints, assumptions about the physical properties of surfaces and lights. Computational vision scientists have derived effective solutions for some inverse optics problems (such as computing depth from binocular disparity) by determining the appropriate natural constraints and embedding them in algorithms. How might a visual system discover and exploit natural constraints on its own? We address a simpler question: Given only a set of examples of input images and desired output solutions, can a visual system synthesize. or "learn", the algorithm that converts input to output? We find that an algorithm for computing color in a restricted world can be constructed from examples using standard techniques of optimal linear estimation. The computation of color is a prime example of the difficult problems of inverse optics. We do not merely discriminate betwN'n different wavelengths of light; we assign @ American Institute of Physics 1988 623 roughly constant colors to objects even though the light signals they send to our eyes change as the illumination varies across space and chromatic spectrum. The computational goal underlying color constancy seems to be to extract the invariant surface spectral reflectance properties from the image irradiance, in which reflectance and iI-" lumination are mixed 1 ? Lightness algorithms 2-8, pioneered by Land, assume that the color of an object can be specified by its lightness, or relative surface reflectance, in each of three independent chromatic channels, and that lightness is computed in the same way in each channel. Computing color is thereby reduced to extracting surface reflectance from the image irradiance in a single chromatic channel. The image irra.diance, s', is proportional to the product of the illumination intensity e' and the surface reflectance r' in that channel: s' (x, y) = r' (x, y )e' (x, y). (1 ) This form of the image intensity equation is true for a Lambertian reflectance model, in which the irradiance s' has no specular components, and for appropriately chosen color channels 9. Taking the logarithm of both sides converts it to a sum: s(x, y) = rex, y) + e(x,y), (2) where s = loges'), r = log(r') and e = log(e'). Given s(x,y) alone, the problem of solving Eq. 2 for r(x,y) is underconstrained. Lightness algorithms constrain the problem by restricting their domain to a world of Mondrians, two-dimensional surfaces covered with patches of random colors 2 and by exploiting two constraints in that world: (i) r'(x,y) is unifonn within patches but has sharp discontinuities at edges between patches and (ii) e' (x, y) varies smoothly across the Mondrian. Under these constraints, lightness algorithms can recover a good approximation to r( x, y) and so can recover lightness triplets that label roughly constant colors 10. We ask whether it is possible to synthesize from examples an algorithm that ex? tracts reflectance from image irradiance. and whether the synthesized algorithm will resemble existing lightness algorithms derived from an explicit analysis of the constraints. We make one assumption, that the operator that transforms irradiance into reflectance is linear. Under that assumption, motivated by considerations discussed later, we use optimal linear estimation techniques to synthesize an operator from examples. The examples are pairs of images: an input image of a Mondrian under illumination that varies smoothly across space and its desired output image that displays the reflectance of the Mondrian without the illumination. The technique finds the linear estimator that best maps input into desired output. in the least squares sense. For computational convenience we use one-dimensional "training vectors" that represent vertical scan lines across the ~londrian images (Fig. 1). We generate many 624 1S0t? -~ ? ~ 100 , '. a , ~~~--------------~~--~--~~ SO 100 110 100 llO 100 lilt., Input d.t. i;[:2:=:0 a SO 101 ISO ZOO UI :~kfhJfEirQ b 0 )01 II 100 ISO 100 110 p/Jte' f~l o 50 100 UO ZOO ZSO 100 I"'" c )00 .1 p,xe' 100 ISO 100 110 lot p'." OlltPllt lIlll.l . .ll.a Fig. 1. (a) The input data, a one-dimensional vector 320 pixels long. Its random Mondrian reflectance pattern is superimposed on a linear illumination gradient with a random slope and offset. (b) shows the corresponding output solution, on the left the illumination and on the right reBectance. We used 1500 such pairs of inputoutput examples (each different from the others) to train the operator shown in Fig. 2. (c) shows the result obtained by the estimated operator when it acts on the input data (a), not part of the training set. On the left is the illumination and on the right the reflectance, to be compared with (b). This result is fairly typical: in some cases the prediction is even better, in others it is worse. different input vectors s by adding together different random T and e vectors, according to Eq. 2. Each vector r represents a pattern of step changes across space, corresponding to one column of a reHectance image. The step changes occur at random pixels and are of random amplitude between set minimum and maximum values. Each vector t represents a smooth gradient across space with a random offset and slope, correspondin~ to one column of an illumination image. We th~n arrange the training vectors sand r as the columns of two matrices Sand R, resp~ti?.. ely. Our goal is then to compute the optimal solution L of LS =R where L is a linear operator represented as a matrix. 625 It is well known that the solution of this equation that is optimal in the least squares sense is ( 4) where S+ is the Moore-Penrose pseudoinverse 11. We compute the pseudoinverse by overconstraining the problem - using many more training vectors than there are number of pixels in each vector - and using the straightforward formula that applies in the overconstrained case 12: S+ ST(SST)-l. The operator L computed in this way recovers a good approximation to the correct output vector r when given a new s, not part of the training set, as input (Fig. Ic). A second operator, estimated in the same way, recovers the illumination e. Acting on a random two-dimensional Mondrian L also yields a satisfactory approximation to the correct output image. Our estimation scheme successfully synthesizes an algorithm that performs the lightness computation in a Mondrian world. What is the algorithm and what is its relationship to other lightness algorithms? To answer these questions we examine the structure of the matrix L. We assume that, although the operator is not a convolution operator, it should approximate one far from the boundaries of the image. That is, in its central part, the operator should be space-invariant, performing the same action on each point in the image. Each row in the central part of L should therefore be the same as the row above but displaced by one element to the right. Inspection of the matrix confirmes this expectation. To find the form of L in its center, we thus average the rows there, first shifting them appropriately. The result, shown in Fig. 2, is a space-invariant filter with a narrow positive peak and a broad, shallow, negative surround. Interestingly, the filter our scheme synthesizes is very similar to Land's most recent retinex operator 5, which divides the image irradiance at each pixel by a weighted average of the irradiance at all pixels in a large surround and takes the logarithm of that result to yield lightness 13. The lightness triplets computed by the retinex operator agree well with human perception in a Mondrian world. The retinex operator and our matrix L both differ from Land's earlier retinex algorithms, which require a non-linear thresholding step to eliminate smooth gradients of illumination. The shape of the filter in Fig. 2, particularly of its large surround, is also suggestive of the "nonclassical" receptive fields that have been found in V4, a cortical area implicated in mechanisms underlying color constancy 14-17. The form of the space-invariant filter is similar to that derived in our earlier formal analysis of the lightness problem 8. It is qualitatively the same as that which results from the direct application of regularization methods exploiting the spatial constraints on reflectance and illumination described above 9.18.19. The Fourier transform of the filter of Fig. 2 is approximately a bandpass filter that cuts out low frequencies due = 626 ~ 0 ( .) ~ C' C - -80 s::. 0 Pi xe Is .2' ~ a -80 ----------- o +80 Pixels Fig. 2. The space-invariant part of the estimated operator, obtained by shifting and averaging the rows of a 160-pixel-wide central square of the matrix L, trained on a set of 1500 examples with linear illumination gradients (see Fig. 1). When logarithmic illumination gradients are used , a qualitatively similar receptive field is obtained. In a separate experiment we use a training set of one-dimensional Mondrians with either linear illumination gradients or slowly varying sinusoidal illumination components with random wavelength, phase and amplitude. T he resulting filter is shown in the inset. The surrounds of both filters extend beyond the range we can estimate reliably, the range we show here. to slow gradients of illumination and preserves intennediate frequencies due to step changes in reflectance. In contrast, the operator that recovers the illumination, e. takes the form of a low-pass filter. \Ve stress that the entire operator L is not a space-invariant filter. In this context, it is clear that the shape of the estimated operator should vary with the type of illumination gradient in the training set. We synthesize a second operator using a new set of examples that contain equal numbers of vectors with random, sinusoidally varying illumination components and VE"(tors with random, linear illumination gradients. Whereas the first operator, synthE.>Sized from examples with strictly linear illumination gradients, has a broad negative surround that remains virtually constant throughout its extent, the new operator's surround (Fig . 2, inset) has a smaller ext(,111 627 and decays smoothly towards zero from its peak negative value in its center. We also apply the operator in Fig. 2 to new input vectors in which the density and amplitude of the step changes of reflectance differ greatly from those on which the operator is trained. The operator performs well, for example, on an input vector representing one column of an image of a small patch of one reflectance against a uniform background of a different reflectance, the entire image under a linear illumination gradient. This result is consistent with psychophysical experiments that show that color constancy of a patch holds when its Mondrian background is replaced by an equivalent grey background 20. The operator also produces simultaneous brightness contrast, as expected from the shape and sign of its surround. The output reflectance it computes for a patch of fixed input reflectance decreases linearly with increasing average irradiance of the input test vector in which the patch appears. Similarly, to us, a dark patch appears darker when against a light background than against a dark one. This result takes one step towards explaining such illusions as the Koffka Ring 21. A uniform gray annulus against a bipartite background (Fig. 3a) appears to split into two halves of different lightnesses when the midline between the light and dark halves of the background is drawn across the annulus (Fig. 3b). The estimated operator acting on the Koffka Ring of Fig. 3b reproduces our perception by assigning a lower output reflectance to the left half of the annulus (which appears darker to us) than to the right half 22. Yet the operator gives this brightness contrast effect whether or not the midline is drawn across the annulus (Fig. 3c). Becau~e the opf'rator can perform only a linear transformation between the input and output images, it is not surprising that the addition of the midline in the input evokes so little change in the output. These results demonstrate that the linear operator alone cannot compute lightness in all worlds and suggest that an additional operator might be necessary to mark and guide it within bounded regions. Our estimation procedure is motivated by our previous observation 9.23,18 that standard regularization algorithms 19 in early vision define linear mappings between input and output and therefore can be estimated associatively under certain condi? tions. The technique of optimal linear estimation that we use is closely related to optimal Bayesian estimation 9. If we were to assume from the start that the optimal linear operator is space-invariant, we could considerably simplify (and streamline) the computation by using standard correlation te<:hniques 9.24. How does our estimation technique compare with other methods of "learning" a lightness algorithm? We can compute the r~ularized pseudoinverse using gradient descent on a "neural" network 25 with linf'ar units. Since the pseudoinverse is lhf" unique best linear approximation in the L1 norm. a gradient descent method that 628 minimizes the square error between the actual output and desired output of a fully connected linear network is guaranteed to converge, albeit slowly. Thus gradient descent in weight space converges to the same result as our first technique, the global minimum. b a c .n sa 0 .It .sa _ ut _ input data pixel ...~~ ~:== :iu II ... ,, e. sa . ~ - ~ . _ . ut _ output reflectance - with edge ~'~ ...:i'I~~ I' _ . .' - .. =.. e. " ~. . _ I ~ ~ ( ut , _ output reflectance - without edge Fig. 3. (a) Koffka Ring. (b) Koftka Ring with midline drawn across annulus. (c) Horizontal scan lines across Koffka Ring. Top: Scan line starting at arrow in (b). Middle: Scan line at corresponding location in the output of linear operator acting on (b). Bottom: Scan line at same location in the output of operator acting on (a). 629 We also compare the linear estimation technique with a "backpropagation" network: gradient descent on a 2-layer network with sigmoid units 25 (32 inputs, 32 "hidden units", and 32 linear outputs), using training vectors 32 pixels long. The network requires an order of magnitude more time to converge to a stable configuration than does the linear estimator for the same set of 32-pixel examples. The network's performance is slightly, yet consistently, better, measured as the root-mean-square error in output, averaged over sets of at least 2000 new input vectors. Interestingly, the backpropagation network and the linear estimator err in the same way on the same input vectors. It is possible that the backpropagation network may show considerable inprovement over the linear estimator in a world more complex than the Mondrian one. We are presently examining its performance on images with real-world features such as shading, shadows, and highlights26. We do not think that our results mean that color constancy may be learned during a critical period by biological organisms. It seems more reasonable to consider them simply as a demonstration on a toy world that in the course of evolution a visual system may recover and exploit natural constraints hidden in the physics of the world. The significance of our results lies in the facts that a simple statistical technique may be used to synthesize a lightness algorithm from examples; that the technique does as well as other techniques such as backpropagation; and that a similar technique may be used for other problems in early vision. Furthermore, the synthesized operator resembles both Land's psychophysically-tested retinex operator and a neuronal nonclassical receptive field. The operator's properties suggest that simultaneous color (or brightness) contrast might be the result of the visual system's attempt to discount illumination gradients 27 REFERENCES AND NOTES 1. Since we do not have perfect color constancy, our visual system must not extract reflectance exactly. The limits on color constancy might reveal limits on the underlying computation. 2. 3. 4. and S. 5. 6. E.H. Land, Am. Sci. 52,247 (1964). E.H. Land and J.J. McCann, J. Opt. Soc. Am. 61, 1 {1971}. E.H. Land, in Central and Peripheral Mechanisms of Colour Vision, T. Ottoson Zeki, Eds., (Macmillan, New York, 1985), pp. 5-17. E.H. Land, Proc. Nat. Acad. Sci. USA 83, 3078 (1986). B.K.P. Hom, Computer Graphics and Image Processing 3, 277 (1974). 630 7. A. Blake, in Central and Peripheral Mechanisms of Colour Vision, T. Ottoson and S. Zeki, Eds., (Macmillan, New York, 1985), pp. 45-59. 8. A. Hurlbert, J. Opt. Soc. Am. A 3,1684 (1986). 9. A. Hurlbert and T. Poggio, ArtificiaLIntelligence Laboratory Memo 909, (M.LT., Cambridge, MA, 1987). 10. r'{x,y) can be recovered at best only to within a constant, since Eq. 1 is invariant under the transformation of r' int.o ar' and e' into a-ie', where a is a constant. 11. A. Albert, Regression and the Moore-Penrose Pseudoinllerse, (Academic Press, New York, 1972). 12. The pseudoinverse, and therefore L, may also be computed by recursive techniques that improve its form as more data become available l l . 13. Our synthesized filter is not exactly identical with Land's: the filter of Fig. 2 subtracts from the value at each point the average value of the logarithm of irradiance at all pixels, rather than the logarithm of the average values. The estimated operator is therefore linear in the logarithms, whereas Land's is not. The numerical difference between the outputs of the two filters is small in most cases (Land, personal communication), and both agree well with psychophysical results. 14. R. Desimone, S.J. Schein, J. Moran and L.G. Ungerleider, Vision Res. 25, 441 (1985). 15. H.M. Wild, S.R. Butler, D. Carden and J.J. Kulikowski, Nature (London) 313, 133 (1985). 16. S.M. Zeki, Neuroscience 9, 741 (1983). 17. S.M. Zeki, Neuroscience 9, 767 (1983). 18. T. Poggio, et. al, in Proceedings Image Understanding Workshop, L. Baumann, Ed., (Science Applications International Corporation, McLean, VA, 1985), pp.' 25-39. 19. T. Poggio, V. Torre and C. Koch, Nature (London) 317,314 (1985). 20. A. Valberg and B. Lange-Malecki, Investigative Ophthalmology and Visual Science Supplement 28, 92 (1987). 21. K. Koffka, Principles of Gestalt Psychology, (Harcourt, Brace and Co., New York, 1935). 22. Note that the operator achieves this effect by subtracting a non-existent illumination gradient from the input signal. 23. T. Poggio and A. Hurlbert, Artificial Intelligence Laboratory Working Paper 264, (M.LT., Cambridge, MA, 1984). 24. Estimation of the operator on two-dimensional examples is possible, but computationally very expensive if done in the same way. The present computer simulations require several hours when run on standard serial computers. The two-dimensional case 631 will need much more time (our one-dimensional estimation scheme runs orders of magnitude faster on a CM-1 Connection Machine System with 16K-processors). 25. D. E. Rumelhart, G.E. Hinton and R.J. Williams, Nature (London) 323, 533 (1986 ). 26. A. Hurlbert, The Computation of Color, Ph.D. Thesis, M.l. T., Cambridge, MA, in preparation. 2i. We are grateful to E. Land, E. Hildreth, .J. Little, F. Wilczek and D. Hillis for reading the draft and for useful discussions. A. Rottenberg developed the routines for matrix operations that we used on the Connection Machine. T. Breuel wrote the backpropagation simulator. 

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795 SONG LEARNING IN BIRDS M. Konishi Division of Biology California Institute of Technology Birds sing to communicate. Male birds use song to advertise their territories and attract females. Each bird species has a unique song or set of songs. Song conveys both species and individual identity. In most species, young birds learn some features of adult song. Song develops gradually from amorphous to fixed patterns of vocalization as if crystals form out of liquid. Learning of a song proceeds in two steps; birds commit the song to memory in the first stage and then they vocally reproduce it in the second stage. The two stages overlap each other in some species, while they are separated by several months in other species. The ability of a bird to commit a song to memory is restricted to a period known as the sensitive phase. Vocal reproduction of the memorized song requires auditory feedback. Birds deafened before the second stage cannot reproduce the memorized song. Birds change vocal output until it matches with the memorized song, which thus serves as a template. Birds use a built-in template when a tutor model is not available. Exposure to a tutor model modifies this innate template. A series of brain nuclei controls song production and patterning. Recording multiand single neurons from this nuclei in the singing bird is possible. The learned temporal pattern of song is recognizable in the neural discharge of these nuclei. The need for auditory feedback for song learning suggests the presence of links between the auditory and vocal control systems. One such link is found in the HVc, one of the forebrain song nuclei. This nucleus contains neurons sensitive to sound in addition to those which control song production. In the white-crowned sparrow, the HVc contains neurons selective for the bird's own individual song. The stimulus selectivity of these neurons are thus shaped by the bird's hearing of its own voice during song development. [1] Konishi, M. (1985) Birdson: from behavior to neuron. Ann. Rev. Neurosci. 8:125-170. [2] Konishi, M. (1985) The role of auditory feedback in the control of vocalization in the white-crowned sparrow. Z. Tierpsychol. 22:770-783. [3] McCasland, J. S. (1987) Neuronal control of bird song production. J. Neurosci., 723-739. [4] Margoliash, D. (1983) Acoustic parameters underlying the responses of songspecific neurons in the white-crowned sparrow. J. Neurosci. 3:10389-1057. [5] Nottebohm, F. T., Stokes, M., & Leonard, C. M. (1976) Central control of song in the canary Serinus canarius. J. Compo Neurol. 165:457-486. 

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Regular and Irregular Gallager-type Error-Correcting Codes Y. Kabashirna and T. Murayarna Dept. of Compt. IntI. & Syst. Sci. Tokyo Institute of Technology Yokohama 2268502, Japan D. Saad and R. Vicente Neural Computing Research Group Aston University Birmingham B4 7ET, UK Abstract The performance of regular and irregular Gallager-type errorcorrecting code is investigated via methods of statistical physics. The transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of non-zero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity may be saturated in equilibrium for many of the regular codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding. We show that irregular codes may saturate Shannon's capacity but with improved dynamical properties. 1 Introduction The ever increasing information transmission in the modern world is based on reliably communicating messages through noisy transmission channels; these can be telephone lines, deep space, magnetic storing media etc. Error-correcting codes play a significant role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission and decoding the corrupted received code-word for retrieving the original message. In his ground breaking papers, Shannon[l] analyzed the capacity of communication channels, setting an upper bound to the achievable noise-correction capability of codes, given their code (or symbol) rate, constituted by the ratio between the number of bits in the original message and the transmitted code-word. Shannon's bound is non-constructive and does not provide a recipe for devising optimal codes. The quest for more efficient codes, in the hope of saturating the bound set by Shannon, has been going on ever since, providing many useful but sub-optimal codes. One family of codes, presented originally by Gallager[2]' attracted significant interest recently as it has been shown to outperform most currently used techniques[3]. Gallager-type codes are characterized by several parameters, the choice of which defines a particular member of this family of codes. Current theoretical results[3] 273 Regular and Irregular Gallager-type Error-Correcting Codes offer only bounds on the error probability of various architectures, proving the existence of very good codes under some restrictions; decoding issues are examined via numerical simulations. In this paper we analyze the typical performance of Gallager-type codes for several parameter choices via methods of statistical mechanics. We then validate the analytical solution by comparing the results to those obtained by the TAP approach and via numerical methods. 2 The general framework In a general scenario, a message represented by an N dimensional Boolean vector JO which is transmitted through a noisy channel with some flipping probability p per bit (other noise types may also be studied). The received message J is then decoded to retrieve the original message. eis encoded to the M dimensional vector In this paper we analyze a slightly different version of Gallager-type codes termed the MN code[3] that is based on choosing two randomly-selected sparse matrices A and B of dimensionality M x N and M x M respectively; these are characterized by K and L non-zero unit elements per row and C and L per column respectively. The finite numbers K, C and L define a particular code; both matrices are known to both sender and receiver. Encoding is carried out by constructing the modulo 2 inverse of B and the matrix B- 1 A (mod 2); the vector JO = B- 1 A (mod 2, Boolean vector) constitutes the codeword. Decoding is carried out by taking the product of the matrix B and the received message J = JO +( (mod 2), corrupted by the Boolean noise vector (, resulting in Ae + B (. The equation e Ae + B( = AS + B'T (mod 2) e (1) is solved via the iterative methods of Belief Propagation (BP)[3] to obtain the most probable Boolean vectors Sand 'T; BP methods in the context of error-correcting codes have recently been shown to be identical to a TAP[4] based solution of a similar physical system[5]. The similarity between error-correcting codes of this type and Ising spin systems was first pointed out by Sourlas[6], who formulated the mapping of a simpler code, somewhat similar to the one presented here, onto an Ising spin system Hamiltonian. We recently extended the work of Sourlas, that focused on extensively connected systems, to the finite connectivity case[5] as well as to the case of MN codes [7]. To facilitate the current investigation we first map the problem to that of an Ising model with finite connectivity. We employ the binary representation (?1) of the dynamical variables Sand 'T and of the vectors J and JO rather than the Boolean (0,1) one; the vector JO is generated by taking products of the relevant binary = TIiE/.' ~i' where the indices J.L = (h, ... iK) correspond to the message bits non-zero elements of B-1 A, producing a binary version of JO. As we use statistical mechanics techniques, we consider the message and codeword dimensionality (N and M respectively) to be infinite, keeping the ratio between them R = N 1M, which constitutes the code rate, finite. Using the thermodynamic limit is quite natural as Gallager-type codes are usually used for transmitting long (10 4 - 105 ) messages, where finite size corrections are likely to be negligible. To explore the system's capabilities we examine the Hamiltonian J2 Y. Kabashima, T. Murayama, D. Saad and R. Vicente 274 The tensor product DlJ.uJ,.J.{Tl where JlJ.u = TIiEIJ. ~i TI jEu (j and u = (jl,'" iL), is the binary equivalent of Ae + B(, treating both signal (8 and index i) and noise (7" and index j) simultaneously. Elements of the sparse connectivity tensor D IJ.U take the value 1 if the corresponding indices of both signal and noise are chosen (Le., if all corresponding indices of the matrices A and Bare 1) and 0 otherwise; it has C unit elements per i-index and L per j-index representing the system's degree of connectivity. The f> function provides 1 if the selected sites' product TIiEIJ. Si TIjEu Tj is in disagreement with the corresponding element JIJ.U, recording an error, and 0 otherwise. Notice that this term is not frustrated, as there are M +N degrees of freedom and only M constraints from Eq.(l), and can therefore vanish at sufficiently low temperatures. The last two terms on the right represent our prior knowledge in the case of sparse or biased messages Fs and of the noise level Fr and require assigning certain values to these additive fields. The choice of f3 -+ 00 imposes the restriction of Eq.(l), limiting the solutions to those for which the first term of Eq.(2) vanishes, while the last two terms, scaled with f3, survive. Note that the noise dynamical variables 7" are irrelevant to measuring the retrieval success m = Jr (~~1 ~i sign (Si)!3 ) ~ . The latter monitors the normalized mean overlap between the Bayes-optimal retrieved message, shown to correspond to the alignment of (Si)!3 to the nearest binary value[6], and the original message; the subscript f3 denotes thermal averaging. Since the first part of Eq.(2) is invariant under the map Si -+ Si~i, Tj -+ Tj(j and JIJ.U -+ JIJ.U TIiEIJ. ~i TI jEu (j = 1, it is useful to decouple the correlation between the vectors 8, 7" and Rewriting Eq.(2) one obtains a similar expression apart from the last terms on the right which become Fs / f3 L:k Sk ~k and Fr / f3 ~k Tk (k. e,(. The random selection of elements in D introduces disorder to the system which is treated via methods of statistical physics. More specifically, we calculate the partition function Z(D , J) = Tr{8 ,7"} exp[-f31i] averaged over the disorder and the statistical properties of the message and noise, using the replica method[5, 8, 9]. Taking f3 -+ 00 gives rise to a set of order parameters (~ tZi Sf Sf ,.. ,S7) q"",(3 ?..? "Y = .=1 T"".(3, ..,"Y = (~ ty; rj rf, .. ,r?) .=1 (3400 (3400 (2) where a, f3, .. represent replica indices, and the variables Zi and 1j come from enforcing the restriction of C and L connections per index respectively[5]: f> ( "D . . (. L .) . <t,t2 ,?? ,JL> '2 ,?? ,'tK - c) i = 0 21T dZ ZL: h .... i K f<i.i 2 ? .. ?h 2 >-(C+l) 7r ' (3) and similarly for the restriction on the j indices. To proceed with the calculation one has to make an assumption about the order parameters symmetry. The assumption made here, and validated later on, is that of replica symmetry in the following representation of the order parameters and the related conjugate variables Qa ,!3 ..-y aq / dx 7r(X) xl , Qa,!3 .. -y = aq- / dx 1?(x) Xl r a,!3 .. -y ar / dy p(y) yl , r a,!3 .. -y = a; / dy p(Y) yl , (4) where l is the number of replica indices, a. are normalization coefficients, and 7r(x) , 1?(x) , p(y) and p(Y) represent probability distributions. Unspecified integrals Regular and Irregular Gallager-type Error-Correcting Codes 275 are over the range [-1, + 1]. One then obtains an expression for the free energy per spin expressed in terms of these probability distributions liN (In Z)~,(,'D The free energy can then be calculated via the saddle point method. Solving the equations obtained by varying the free energy w.r.t the probability distributions 1T(X), 1?(x), p(y) and p(y), is difficult as they generally comprise both delta peaks and regular[9] solutions for the ferromagnetic and paramagnetic phases (there is no spin-glass solution here as the system is not frustrated). The solutions obtained in the case of unbiased messages (the most interesting case as most messages are compressed prior to transmission) are for the ferromagnetic phase: 1T(X) = 8(x - 1) , 1?(x) = 8(x - 1) , p(y) = 8(y - 1) , p(Y) = 8(Y - 1), (5) and for the paramagnetic phase: = 8(x) = 8(Y) 1T(X) 8(x) , 1?(x) p(y) 1 + tanh Fr r( _ h F ) 1 - tanh Fr r( hF ) 2 u y tan r + 2 u Y + tan r? , p(Y) (6) These solutions obey the saddle point equations. However, it is unclear whether the contribution of other delta peaks or of an additional continuous solution will be significant and whether the solutions (5) and (6) are stable or not. In addition, it is also necessary to validate the replica symmetric ansatz itself. To address these questions we obtained solutions to the system described by the Hamiltonian (2) via TAP methods of finitely connected systems[5]; we solved the saddle point equations derived from the free energy numerically, representing all probability distributions by up to 104 bin models and by carrying out the integrations via Monte-Carlo methods; finally, to show the consistency between theory and practice we carried out large scale simulations for several cases, which will be presented elsewhere. 3 Structure of the solutions The various methods indicate that the solutions may be divided to two different categories: K =L =2 and either K ~ 3 or L ~ 3. We therefore treat them separately. For unbiased messages and either K ~ 3 or L ~ 3 we obtain the solutions (5) and (6) both by applying the TAP approach and by solving the saddle point equations numerically. The former was carried out at the value of Fr which corresponds to the true noise and input bias levels (for unbiased messages Fa = 0) and thus to Nishimori's condition[lO], where no replica symmetry breaking effects are expected. This is equivalent to having the correct prior within the Bayesian framework[6] and enables one to obtain analytic expressions for some observables as long as some gauge requirements are obeyed [10] . Numerical solutions show the emergence of stable dominant delta peaks, consistent with those of (5) and (6). The question of longitudinal mode stability (corresponding to the replica symmetric solution) was addressed by setting initial conditions for the numerical solutions close to the solutions (5) and (6), showing that they converge back to these solutions which are therefore stable. The most interesting quantity to examine is the maximal code rate, for a given corruption process, for which messages can be perfectly retrieved. This is defined in the case of K,L~3 by the value of R=KIC=NjM for which the free energy of the ferromagnetic solution becomes smaller than that of the paramagnetic solution, constituting a first order phase transition. A schematic description of the solutions obtained is shown in the inset of Fig.1a. The paramagnetic solution (m = 0) has a lower free energy than the ferromagnetic one (low Ihigh free energies are denoted 276 Y. Kabashima, T. Murayama, D. Saad and R. Vicente by the thick and thin lines respectively, there are no axis lines at m = 0,1) for noise levels P > Pc and vice versa for P ~ Pc; both solutions are stable. The critical code rate is derived by equating the ferromagnetic and paramagnetic free energies to obtain Rc = 1-H2(p) = 1+(plog2P+(1- p)log2(1- p)) . This coincides with Shannon's capacity. To validate these results we obtained TAP solutions for the unbiased message case (K = L = 3, C = 6) as shown in Fig.1a (as +) in comparison to Shannon's capacity (solid line). Analytical solutions for the saddle point equations cannot be obtained for biased patterns and we therefore resort to numerical methods ana the TAP approach. The maximal information rate (Le., code-rate xH2 (fs = (1 + tanh Fs)/2) - the source redundancy) obtained by the TAP method (0) and numerical solutions of the saddle point equations (0), for each noise level, are shown in Fig.1a. Numerical results have been obtained using 103 _10 4 bin models for each probability distribution and had been run for 105 steps per noise level point. The various results are highly consistent and practically saturate Shannon's bound for the same noise level. The MN code for K , L ~ 3 seems to offer optimal performance. However, the main drawback is rooted in the co-existence of the stable m = 1 and m = 0 solutions, shown in Fig.1a (inset), which implies that from some initial conditions the system will converge to the undesired paramagnetic solution. Moreover, studying the ferromagnetic solution numerically shows a highly limited basin of attraction, which becomes smaller as K and L increase, while the paramagnetic solution at m = 0 always enjoys a wide basin of attraction. Computer simulations (see also [3]) show that as initial conditions for the decoding process are typically of close-to-zero magnetization (almost no prior information about the original message is assumed) it is likely that the decoding process will converge to the paramagnetic solution. While all codes with K, L ~ 3 saturate Shannon's bound in their equilibrium properties and are characterized by a first order, paramagnetic to ferromagnetic, phase transition, codes with K = L = 2 show lower performance and different physical characteristics. The analytical solutions (5) and (6) are unstable at some flip rate levels and one resorts to solving the saddle point equations numerically and to TAP based solutions. The picture that emerges is sketched in the inset of Fig.1b: The paramagnetic solution dominates the high flip rate regime up to the point PI (denoted as 1 in the inset) in which a stable, ferromagnetic solution, of higher free energy, appears (thin lines at m = ?1). At a lower flip rate value P2 the paramagnetic solution becomes unstable (dashed line) and is replaced by two stable sub-optimal ferromagnetic (broken symmetry) solutions which appear as a couple of peaks in the various probability distributions; typically, these have a lower free energy than the ferromagnetic solution until P3, after which the ferromagnetic solution becomes dominant. Still, only once the sub-optimal ferromagnetic solutions disappear, at the spinodal point Ps, a unique ferromagnetic solution emerges as a single delta peak in the numerical results (plus a mirror solution). The point in which the sub-optimal ferromagnetic solutions disappear constitutes the maximal practical flip rate for the current code-rate and was defined numerically (0) and via TAP solutions (+) as shown in Fig.1b. Notice that initial conditions for TAP and the numerical solutions were chosen almost randomly, with a slight bias of 0(10- 12 ), in the initial magnetization. The TAP dynamical equations are identical to those used for practical BP decoding[5], and therefore provide equivalent results to computer simulations with the same parameterization, supporting the analytical results. The excellent convergence results obtained point out the existence of a unique pair of global solutions to which the system converges (below Ps) from practically all initial conditions. This observation and the practical implications of using K = L = 2 code have not been obtained by Regular and Irregular Gallager-type Error-Correcting Codes 277 information theory methods (e.g.[3]}j these prove the existence of very good codes for C = L ~ 3, and examine decoding properties only via numerical simulations. 4 Irregular Constructions Irregular codes with non-uniform number of non-zero elements per column and uniform number of elements per row were recently introduced [11, 12] and were found to outperform regular codes. It is relatively straightforward to adapt our methods to study these particular constructions. The restriction of the number of connections per index can be replaced by a set of N restrictions of the form (1), enforcing Cj non-zero elements in the j-th column of the matrix A, and other M restrictions enforcing Ll non-zero elements in the l-th column of the matrix B. By construction these restrictions must obey the relations E.7=l C j = MK and E~l Ll = ML. One can assume that a particular set of restrictions is generated independently by the probability distributions P(C) and P(L). With that we can compute average properties of irregularly constructed codes generated by arbitrary distributions. Proceeding along the same lines to those of the regular case one can find a very similar expression for the free energy which can be interpreted as a mixture of regular codes with column weights sampled with probabilities P(C) and P(L). As long as we choose probability distributions which vanish for C, L = 0 (avoiding trivial non-invertible matrices) and C, L = 1 (avoiding single checked bits), the solutions to the saddle point equations are the same as those obtained in the regular case (Eqs.5, 6) leading to exactly the same predictions for the maximum performance. The differences between regular and irregular codes show up in their dynamical behavior. In the irregular case with K > 2 and for biased messages the basin of attraction is larger for higher noise levels [13]. 5 Conclusion In this paper we examined the typical performance of Gallager-type codes. We discovered that for a certain choice of parameters, either K ~ 3 or L ~ 3, one potentially obtains optimal performance, saturating Shannon's bound. This comes at the expense of a decreasing basin of attraction making the decoding process increasingly impractical. Another code, K = L = 2, shows "close to optimal performance with a very large basin of attraction, making it highly attractive for practical purposes. The decoding performance of both code types was examined by employing the TAP approach, an iterative method identical to the commonly used BP. Both numerical and TAP solutions agree with the theoretical results. The equilibrium properties of regular and irregular constructions is shown to be the same. The improved performance of irregular codes reported in the literature can be explained as consequence of dynamical properties. This study examines the typical performance of these increasingly important error-correcting codes, from which optimal parameter choices can be derived, complementing the bounds and empirical results provided in the information theory literature. Important aspects that are yet to be investigated include other noise types, finite size effects and the decoding dynamics itself. Acknowledgement Support by the JSPS RFTF program (YK), The Royal Society and EPSRC grant GR/N00562 (DS) is acknowledged. Y. Kabashima. T. Murayama. D. Saad and R. Vicente 278 1 1 Ferro 0.8 0.8 ~ 0.6 0.6 a: I a: 0.4 0.4 p 0.2 0.2 0 0.1 0.2 0.3 P 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 P Figure 1: Critical code rate as a function of the flip rate p, obtained from numerical solutions and the TAP approach (N = 104 ), and averaged over 10 different initial conditions with error bars much smaller than the symbols size. (a) Numerical solutions for K = L = 3, C = 6 and varying input bias fs (0) and TAP solutions for both unbiased (+) and biased (0) messages; initial conditions were chosen close to the analytical ones. The critical rate is multiplied by the source information content to obtain the maximal information transmission rate, which clearly does not go beyond R = 3/6 in the case of biased messages; for unbiased patterns H 2 (fs) = 1. Inset: The ferromagnetic and paramagnetic solutions as functions of p; thick and thin lines denote stable solutions of lower and higher free energies respectively. (b) For the unbiased case of K = L = 2; initial conditions for the TAP (+) and the numerical solutions (0) are of almost zero magnetization. Inset: The ferromagnetic (optimal/sub-optimal) and paramagnetic solutions as functions of p; thick and thin lines are as in (a), dashed lines correspond to unstable solutions. References [1] C.E. Shannon, Bell Sys. Tech.J., 27, 379 (1948); 27, 623 (1948). [2] R.G. Gallager, IRE Trans.Info . Theory, IT-8, 21 (1962). [3] D.J.C. MacKay, IEEE Trans.IT, 45, 399 (1999) . [4] D. Thouless, P.W. Anderson and R.G. Palmer, Phil. Mag., 35 , 593 (1977). [5] Y. Kabashima and D. Saad, Europhys.Lett., 44 668 (1998) and 45 97 (1999). [6] N. Sourlas, Nature, 339, 693 (1989) and Euro.Phys.Lett., 25 , 159 (1994). [7] Y. Kabashima, T. Murayama and D. Saad, Phys.Rev.Lett., (1999) in press. [8] K.Y.M. Wong and D. Sherrington, J.Phys.A, 20, L793 (1987). [9] C. De Dominicis and P.Mottishaw, J.Phys.A, 20, L1267 (1987). [10] H. Nishimori, Prog. Theo.Phys., 66, 1169 (1981). [11] M. Luby et. ai, IEEE proceedings of ISIT98 (1998) and Analysis of Low Density Codes and Improved Designs Using Irregular Graphs, preprint. [12] D.J.C. MacKay et. al, IEEE Trans.Comm., 47, 1449 (1999). [13] R. Vicente et. ai, http://xxx.lanl.gov/abs/cond-mat/9908358 (1999). 

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Differentiating Functions of the Jacobian with Respect to the Weights Gary William Flake NEC Research Institute 4 Independence Way Princeton, NJ 08540 jiake@research.nj.nec.com Barak A. Pearlmutter Dept of Computer Science, FEC 313 University of New Mexico Albuquerque, NM 87131 bap@cs.unm.edu Abstract For many problems, the correct behavior of a model depends not only on its input-output mapping but also on properties of its Jacobian matrix, the matrix of partial derivatives of the model's outputs with respect to its inputs. We introduce the J-prop algorithm, an efficient general method for computing the exact partial derivatives of a variety of simple functions of the Jacobian of a model with respect to its free parameters. The algorithm applies to any parametrized feedforward model, including nonlinear regression, multilayer perceptrons, and radial basis function networks. 1 Introduction Let f (x, w) be an n input, m output, twice differentiable feedforward model parameterized by an input vector, x, and a weight vector w. Its Jacobian matrix is defined as J= [ ~ ()xl : aim a~" ~l aim = df(x, w) . dx ax" The algorithm we introduce can be used to optimize functions of the form aXI (1) or Ev(w) 2 1 = 211Jv - bll (2) where u, v, a, and b are user-defined constants. Our algorithm, which we call J-prop, can be used to calculate the exact value of both Eu / or Ev / in 0 (1) times the time required to calculate the normal gradient. Thus, I-prop is suitable for training models to have specific first derivatives, or for implementing several other well-known algorithms such as Double Backpropagation [1] and Tangent Prop [2]. a aw a aw Clearly, being able to optimize Equations 1 and 2 is useful; however, we suspect that the formalism which we use to derive our algorithm is actually more interesting because it allows us to modify J-prop to easily be applicable to a wide-variety of model types and G. W. Flake and B. A. Pear/mutter 436 objective functions. As such, we spend a fair portion of this paper describing the mathematical framework from which we later build J-prop. This paper is divided into four more sections. Section 2 contains background information and motivation for why optimizing the properties of the Jacobian is an important problem. Section 3 introduces our formalism and contains the derivation of the J-prop algorithm. Section 4 contains a brief numerical example of J-prop. And, finally, Section 5 describes further work and gives our conclusions. 2 Background and motivation Previous work concerning the modeling of an unknown function and its derivatives can be divided into works that are descriptive or prescriptive. Perhaps the best known descriptive result is due to White et al. [3,4], who show that given noise-free data, a multilayer perceptron (MLP) can approximate the higher derivatives of an unknown function in the limit as the number of training points goes to infinity. The difficulty with applying this result is the strong requirements on the amount and integrity of the training data; requirements which are rarely met in practice. This problem was specifically demonstrated by Principe, Rathie and Kuo [5] and Deco and Schiirmann [6], who showed that using noisy training data from chaotic systems can lead to models that are accurate in the input-output sense, but inaccurate in their estimates of quantities related to the Jacobian of the unknown system, such as the largest Lyapunov exponent and the correlation dimension. MLPs are particularly problematic because large weights can lead to saturation at a particular sigmoidal neuron which, in tum, results in extremely large first derivatives at the neuron when evaluated near the center of the sigmoid transition. Several methods to combat this type of over-fitting have been proposed. One of the earliest methods, weight decay [7], uses a penalty term on the magnitude of the weights. Weight decay is arguably optimal for models in which the output is linear in the weights because minimizing the magnitude of the weights is equivalent to minimizing the magnitude of the model's first derivatives. However, in the nonlinear case, weight decay can have suboptimal performance [1] because large (or small) weights do not always correspond to having large (or small) first derivatives. The Double Backpropagation algorithm [1] adds an additional penalty term to the error function equal to II E / 112. Training on this function results in a form of regularization that is in many ways an elegant combination of weight decay and training with noise: it is strictly analytic (unlike training with noise) but it explicitly penalizes large first derivatives ofthe model (unlike weight decay). Double Backpropagation can be seen as a special case of J-prop, the algorithm derived in this paper. a ax As to the general problem of coercing the first derivatives of a model to specific values, Simard, et at., [2] introduced the Tangent Prop algorithm, which was used to train MLPs for optical character recognition to be insensitive to small affine transformations in the character space. Tangent Prop can also be considered a special case of J-prop. 3 Derivation We now define a formalism under which J-prop can be easily derived. The method is very similar to a technique introduced by Pearlmutter [8] for calculating the product of the Hessian of an MLP and an arbitrary vector. However, where Pearlmutter used differential operators applied to a model's weight space, we use differential operators defined with respect to a model's input space. Our entire derivation is presented in five steps. First, we will define an auxiliary error Differentiating Functions of the Jacobian 437 function that has a few useful mathematical properties that simplify the derivation. Next, we will define a special differential operator that can be applied to both the auxiliary error function, and its gradient with respect to the weights. We will then see that the result of applying the differential operator to the gradient of the auxiliary error function is equivalent to analytically calculating the derivatives required to optimize Equations 1 and 2. We then show an example of the technique applied to an MLP. Finally, in the last step, the complete algorithm is presented. To avoid confusion, when referring to generic data-driven models, the model will always be expressed as a vector function y = f (x, w), where x refers to the model input and w refers to a vector of all of the tunable parameters of the model. In this way, we can talk about models while ignoring the mechanics of how the models work internally. Complementary to the generic vector notation, the notation for an MLP uses only scalar symbols; however, these symbols must refer to internal variable of the model (e.g., neuron thresholds, net inputs, weights, etc.), which can lead to some ambiguity. To be clear, when using vector notation, the input and output of an MLP will always be denoted by x and y, respectively, and the collection of all of the weights (including biases) map to the vector w. However, when using scalar arithmetic, the scalar notation for MLPs will apply. 3.1 Auxiliary error function Our auxiliary error function, E, is defined as E(x, w) = u T f(x, w). (3) Note that we never actually optimize with respect E; we define it only because it has the property that aE/ax = u T J, which will be useful to the derivation shortly. Note that a E / ax appears in the Taylor expansion of E about a point in input space: -T E(x + Ax, w) = E(x, w) + ~! Ax + 0 (1IAXI12) . (4) Thus, while holding the weights, w, fixed and letting Ax be a perturbation of the input, x, Equation 4 characterizes how small changes in the input of the model change the value of the auxiliary error function. Be setting Ax = rv, with v being an arbitrary vector and r being a small value, we can rearrange Equation 4 into the form: ~ [E(x+rv,w) -E(x,w)] +O(r) lim -1 [E(x r~O r a- + rv, w) aE(x + rv,w) r I -] - E(x, w) . (5) r=O This final expression will allow us to define the differential operator in the next subsection. 3.2 Differential operator Let h(x, w) be an arbitrary twice differentiable function. We define the differentiable operator (6) G. W Flake and B. A. Pearlmutter 438 which has the property that Rv{E(x, w)} = u T Jv. Being a differential operator, Rv{-} obeys all of the standard rules for differentiation: Rv{c} Rv{ c? h(x, w)} Rv{h(x,w) + g(x,w)} Rv{h(x, w) . g(x, w)} Rv{h(g(x, w), w)} o = Rv{!h(X,W) } c? Rv{h(x, w)} Rv{ h(x, w)} + Rv{g(x, w)} Rv{h(x,w)}? g(x,w) + h(x,w)? Rv{g(x,w)} h'(g(x,w))? Rv{g(x,w)} d dt Rv{h(x, w)} The operator also yields the identity Rv{ x} = v. 3.3 Equivalence We will now see that the result of calculating Rv{ a E / aw} can be used to calculate both aEu/aw and aEv/aw. Note that Equations 3-5 all assume that both u and v are independent of x and w. To calculate aEu/aw and aEv/aw, we will actually set u or v to a value that depends on both x and w; however, the derivation still works because our choices are explicitly made in such a way that the chain rule of differentiation is not supposed to be applied to these terms. Hence, the correct analytical solution is obtained despite the dependence. To optimize with respect to Equation 1, we use: II a 1 J T u - a 112 = (au T J) T (J T u - a) aw'2 ~ = Rv { aE } ' aw (7) with v = (JT U - a). To optimize with respect to Equation 2, we use: ~! IIJv aw2 bll 2 = (Jv _ b)T (aJv) aw = Rv{ aE} , aw (8) with u = (Jv - b). 3.4 Method applied to MLPs We are now ready to see how this technique can be applied to a specific type of model. Consider an MLP with L + 1 layers of nodes defined by the equations: y~ = g(x~) N/ Xl t L 1-1 I Yj Wij - (9) rii' (10) j In these equations, superscripts denote the layer number (starting at 0), subscripts index over terms in a particular layer, and NI is the number of input nodes in layer l . Thus, y~ is the output of neuron i at node layer l, and xi is the net input coming into the same neuron. Moreover, yf is an output of the entire MLP while y? is an input going into the MLP. The feedback equations calculated with respect to E are: aE ayf (11) 439 Differentiating Functions of the Jacobian 8E (12) 8y~ 8E (13) 8x't 8E (14) 8w!j 8E 8E 8()! 8x lt J (15) ' where the Ui term is a component in the vector u from Equation 1. Applying the R v {?} operator to the feedforward equations yields: (16) Rv{Y?} Rv{yD g'(x~)Rv{ xD (for 1 > 0) (17) N/ L Rv{y~-l } W~j' Rv{ x~} (18) j where the Vi term is a component in the vector v from Equation 2. As the final step, we apply the R v {?} operator to the feedback equations, which yields: Rv{:~ } (19) Rv{ :~} (20) Rv{ :~} (21) :!,} (22) Rv{ ::;} (23) Rv{ 3.5 o Complete algorithm Implementing this algorithm is nearly as simple as implementing normal gradient descent. For each type of variable that is used in an MLP (net input, neuron output, weights, thresholds, partial derivatives, etc.), we require that an extra variable be allocated to hold the result of applying the R v {?} operator to the original variable. With this change in place, the complete algorithm to compute 8Eu /8w is as follows : ? Set u and a to the user specified vectors from Equation 1. ? Set the MLP inputs to the value of x that J is to be evaluated at. ? Perform a normal feedforward pass using Equations 9 and 10. ? Set 8E/8yf to Ui. G. W. Flake and B. A. Pearlmutter 440 (a) (b) Figure 1: Learning only the derivative: showing (a) poor approximation of the function with (b) excellent approximation of the derivative. ? Perform the feedback pass with Equations 11-15. Note that values in the aEjay? terms are now equal to JT U. ? Set v to (JT u - a) ? Perform a Rv{ .} forward pass with Equations 16-18. ? Set the Rv{ 8Ej8yf} terms to O. ? Perform a Rv{?} backward pass with Equations 19-23. After the last step, the values in the Rv{ 8E j 8w!j} and Rv{ 8 E j aeD terms contain the required result. It is important to note that the time complexity of the "J?forward" and "J. backward" calculations are nearly identical to the typical output and gradient evaluations (i.e., the "forward" and "backward" passes) of the models used. A similar technique can be used for calculating 8Evj8w. The main difference is that the Rv{ .} forward pass is performed between the normal forward and backward passes because u can only be determined after the Rv{ f (z, w)} has been calculated. 4 Experimental results To demonstrate the effectiveness and generality of the J-prop algorithm, we have implemented it on top of an existing neural network library [9] in such a way that the algorithm can be used on a large number of architectures, including MLPs, radial basis function net? works, and higher order networks. We trained an MLP with ten hidden tanh nodes on 100 points with conjugate gradient. The training exemplars consisted of inputs in [-1, 1] and a target derivative from 3 cos( 3x) + 5cos(lOx). Our unknown function (which the MLP never sees data from) is sin(3x) + sin(lOx). The model quickly converges to a solution in approximately 100 iterations. l Figure 1 shows the performance of the MLP. Having never seen data from the unknown function, the MLP yields a poor approximation of the function, but a very accurate approximation of the function's derivative. We could have trained on both outputs and derivatives, but our goal was to illustrate that J?prop can target derivatives alone. Differentiating Functions of the Jacobian 5 441 Conclusions We have introduced a general method for calculating the weight gradient of functions of the Jacobian matrix of feedforward nonlinear systems. The method can be easily applied to most nonlinear models in common use today. The resulting algorithm, J-prop, can be easily modified to minimize functionals from several application domains [10]. Some possible uses include: targeting known first derivatives, implementing Tangent Prop and Double Backpropagation, enforcing identical VO sensitivities in auto-encoders, deflating the largest eigenvalue and minimizing all eigenvalue bounds, optimizing the determinant for blind source separation, and building nonlinear controllers. While some special cases of the J-prop algorithm have already been studied, a great deal is unknown about how optimization of the Jacobian changes the overall optimization problem. Some anecdotal evidence seems to imply that optimization of the Jacobian can lead to better generalization and faster training. It remains to be seen if J-prop used on a nonlinear extension of linear methods will lead to superior solutions. Acknowledgements We thank Frans Coetzee, Yannis Kevrekidis, Joe O'Ruanaidh, Lucas Parra, Scott Rickard, Justinian Rosca, and Patrice Simard for helpful discussions. GWF would also like to thank Eric Baum and the NEC Research Institute for funding the time to write up these results. References [1] H. Drucker and Y. Le Cun. Improving generalization performance using double backpropagation. IEEE Transactions on Neural Networks, 3(6), November 1992. [2] P. Simard, B. Victorri, Y. Le Cun, and J. Denker. Tangent prop-A formalism for specifying selected invariances in an adaptive network. In John E. Moody, Steve J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems, volume 4, pages 895-903. Morgan Kaufmann Publishers, Inc., 1992. [3] H. White and A. R. Gallant. On learning the derivatives of an unknown mapping with multilayer feedforward networks. In Halbert White, editor, Artificial Neural Networks, chapter 12, pages 206-223. Blackwell, Cambridge, Mass., 1992. [4] H. White, K. Hornik, and M. Stinchcombe. Universal approximation of an unknown mapping and its derivative. In Halbert White, editor, Artificial Neural Networks, chapter 6, pages 55-77. Blackwell, Cambridge, Mass., 1992. [5] J. Principe, A. Rathie, and J. Kuo. Prediction of chaotic time series with neural networks and the issues of dynamic modeling. Bifurcations and Chaos, 2(4), 1992. [6] G. Deco and B. Schiirmann. Dynamic modeling of chaotic time series. In Russell Greiner, Thomas Petsche, and Stephen Jose Hanson, editors, Computational Learning Theory and Natural Learning Systems, volume IV of Making Learning Systems Practical, chapter 9, pages 137-153. The MIT Press, Cambridge, Mass., 1997. [7] G. E. Hinton. Learning distributed representations of concepts. In Proc. Eigth Annual Con! Cognitive Science Society, pages 1-12, Hillsdale, NJ, 1986. Erlbaum. [8] Barak A. Pearlmutter. Fast exact multiplication by the Hessian. Neural Computation, 6(1):147-160,1994. [9] G. W. Flake. Industrial strength modeling tools. Submitted to NIPS 99, 1999. [10] G. W. Flake and B. A. Pearl mutter. Optimizing properties of the Jacobian of nonlinear feedforward systems. In preperation, 1999. 

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Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly David Horn Nir Levy School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel horn~neuron.tau.ac.il nirlevy~post.tau.ac.il Isaac Meilijson Eytan Ruppin School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel isaco~math.tau.ac.il ruppin~math.tau.ac.il Abstract We investigate the behavior of a Hebbian cell assembly of spiking neurons formed via a temporal synaptic learning curve. This learning function is based on recent experimental findings . It includes potentiation for short time delays between pre- and post-synaptic neuronal spiking, and depression for spiking events occuring in the reverse order. The coupling between the dynamics of the synaptic learning and of the neuronal activation leads to interesting results. We find that the cell assembly can fire asynchronously, but may also function in complete synchrony, or in distributed synchrony. The latter implies spontaneous division of the Hebbian cell assembly into groups of cells that fire in a cyclic manner. We invetigate the behavior of distributed synchrony both by simulations and by analytic calculations of the resulting synaptic distributions. 1 Introduction The Hebbian paradigm that serves as the basis for models of associative memory is often conceived as the statement that a group of excitatory neurons (the Hebbian cell assembly) that are coupled synaptically to one another fire together when a subset of the group is being excited by an external input. Yet the details of the temporal spiking patterns of neurons in such an assembly are still ill understood. Theoretically it seems quite obvious that there are two general types of behavior: synchronous neuronal firing, and asynchrony where no temporal order exists in the assembly and the different neurons fire randomly but with the same overall rate. Further subclassifications were recently suggested by [BruneI, 1999]. Experimentally this question is far from being settled because evidence for the associative 130 D. Hom, N. Levy, 1. Meilijson and E. Ruppin memory paradigm is quite scarce. On one hand, one possible realization of associative memories in the brain was demonstrated by [Miyashita, 1988] in the inferotemporal cortex. This area was recently reinvestigated by [Yakovlev et al., 1998] who compared their experimental results with a model of asynchronized spiking neurons. On the other hand there exists experimental evidence [Abeles, 1982] for temporal activity patterns in the frontal cortex that Abeles called synfire-chains. Could they correspond to an alternative type of synchronous realization of a memory attractor? To answer these questions and study the possible realizations of attractors in cortical-like networks we investigate the temporal structure of an attractor assuming the existence of a synaptic learning curve that is continuously applied to the memory system. This learning curve is motivated by the experimental observations of [Markram et al., 1997, Zhang et al., 1998] that synaptic potentiation or depression occurs within a critical time window in which both pre- and post-synaptic neurons have to fire. If the pre-synaptic neuron fires first within 30ms or so, potentiation will take place. Depression is the rule for the reverse order. The regulatory effects of such a synaptic learning curve on the synapses of a single neuron that is subjected to external inputs were investigated by [Abbott and Song, 1999] and by [Kempter et al., 1999]. We investigate here the effect of such a rule within an assembly of neurons that are all excited by the same external input throughout a training period, and are allowed to influence one another through their resulting sustained activity. 2 The Model We study a network composed of N E excitatory and NJ inhibitory integrate-and-fire neurons. Each neuron in the network is described by its subthreshold membrane potential Vi{t) obeying . Vi{t) 1 =- - Vi{t) + R1i(t) (1) Tn where Tn is the neuronal integration time constant. A spike is generated when Vi{t) reaches the threshold Vrest + fJ, upon which a refractory period of TRP is set on and the membrane potential is reset to Vreset where Vrest < Vreset < Vrest + fJ. Ii{t) is the sum of recurrent and external synaptic current inputs. The net synaptic input charging the membrane of excitatory neuron i at time t is R1i(t) = L J~E{t) L 0 (t - t~ - I j Td) - L j Ji~J L 0 (t - tj - Td) +r xt (2) m summing over the different synapses of j = 1, ... , NE excitatory neurons and of j = 1, ... ,NJ inhibitory neurons, with postsynaptic efficacies J~E{t) and Ji~J respectively. The sum over 1 (m) represents a sum on different spikes arriving at synapse j, at times t t; + Td (t tj + Td), where t~ (tj) is the emission time of the l-th (m-th) spike from the excitatory (inhibitory) neuron j and Td is the synap- = = tic delay. Iext, the external current, is assumed to be random and independent at each neuron and each time step, drawn from a Poisson distribution with mean Aext. Analogously, the synaptic input to the inhibitory neuron i at time tis j j m We assume full connectivity among the excitatory neurons, but only partial connectivity between all other three types of possible connnections, with connection Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly 131 probabilities denoted by eEl, e l E and C l I. In the following we will report simulation results in which the synaptic delays Td were assigned to each synapse, or pair of neurons, randomly, chosen from some finite set of values. Our analytic calculation will be done for one fixed value of this delay parameter. The synaptic efficacies between excitatory neurons are assumed to be potentiated or depressed according to the firing patterns of the pre- and post-synaptic neurons. In addition we allow for a uniform synaptic decay. Thus each excitatory synapse obeys (4) where the synaptic decay constant Ts is assumed to be very large compared to the membrane time constant Tn. J/JE(t) are constrained to vary in the range [0, Jma:~ ]. The change in synaptic efficacy is defined by Fij (t), as Fij(t) = L [6(t - t:)Kp(t; - t:) + 6(t - t;)KD(t; - t:)] (5) k ,l where Kp and KD are the potentiation and depression branches of the kernel function K(6) = -cO exp [- (a6 + b)2] (6) plotted in Figure 1. Following [Zhang et al., 1998] we distinguish between the situation where the postsynaptic spike, at t~, appears after or before the presynaptic spike, at t~, using the asymmetric kernel that captures the essence of their experimental observations. .. -o !.'--~-~~--' o-~-~~-----' k /I =t'_t , I Figure 1: The kernel function whose left part, Kp, leads to potentiation of the synapse, and whose right branch, KD, causes synaptic depression. 3 Distributed Synchrony of a Hebbian Assembly We have run our system with synaptic delays chosen randomly to be either 1, 2, or 3ms, and temporal parameters Tn chosen as 40ms for excitatory neurons and 20ms for inhibitory ones. Turning external input currents off after a while we obtained sustained firing activities in the range of 100-150 Hz. We have found, in addition to synchronous and asynchronous realizations of this attractor, a mode of distributed synchrony. A characteristic example of a long cycle is shown in Figure 2: The 100 excitatory neurons split into groups such that each group fires at the same frequency and at a fixed phase difference from any other group. The J/JE synaptic efficacies D. Horn, N Levy, 1. Meilijson and E. Ruppin 132 ':~ :tI ':f ': f j j r I : I: : I : r : I : r r :I :1 I :I I :I : I ] :1 I :1 : 1 1 :1 : I :1 , I : " : I : "I 1 : 1: I ": " r n ~ JO Figure 2: Distributed synchronized firing mode. The firing patterns of six cell assemblies of excitatory neurons are displayed vs time (in ms). These six groups of neurons formed in a self-organized manner for a kernel function with equal potentiation and depression. The delays were chosen randomly from three values, 1 2 or 3ms, and the system is monitored every 0.5ms . are initiated as small random values. The learning process leads to the self-organized synaptic matrix displayed in Figure 3(a). The block form of this matrix represents the ordered couplings that are responsible for the fact that each coherent group of neurons feeds the activity of groups that follow it. The self-organized groups form spontaneously. When the synapses are affected by some external noise, as can come about from Hebbian learning in which these neurons are being coupled with other pools of neurons, the groups will change and regroup, as seen in Figure 3(b) and 3(c). (a) (b) (c) Figure 3: A synaptic matrix for n = 6 distributed synchrony. The synaptic matrix between the 100 excitatory neurons of our system is displayed in a grey-level code with black meaning zero efficacy and white standing for the synaptic upper-bound. (a) The matrix that exists during the distributed synchronous mode of Figure 2. Its basis is ordered such that neurons that fire together are grouped together. (b) Using the same basis as in (a) a new synaptic matrix is shown, one that is formed after stopping the sustained activity of Figure 2, introducing noise in the synaptic matrix, and reinstituting the original memory training. (c) The same matrix as (b) is shown in a new basis that exhibits connections that lead to a new and different realization of distributed synchrony. A stable distributed synchrony cycle can be simply understood for the case of a single synaptic delay setting the basic step, or phase difference , of the cycle. When several delay parameters exist, a situation that probably more accurately represents the a-function character of synaptic transmission in cortical networks, distributed Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly 133 synchrony may still be obtained, as is evident from Figure 2. After some time the cycle may destabilize and regrouping may occur by itself, without external interference. The likelihood of this scenario is increased because different synaptic connections that have different delays can interfere with one another. Nonetheless, over time scales of the type shown in Figure 2, grouping is stable. 4 Analysis of a Cycle In this section we analyze the dynamics of the network when it is in a stable state of distributed synchrony. We assume that n groups of neurons are formed and calculate the stationary distribution of JffE(t) . In this state the firing pattern of every two neurons in the network can be characterized by their frequency l/(t) and by their relative phase 8. We assume that 8 is a random normal variable with mean J.Lo and standard deviation 0'0 . Thus, Eq. 4 can be rewritten as the following stochastic differential equation dJi~E(t) = [J.LFij(t) - :s J!jE(t)] dt+O'Fij(t)dW(t) (7) where Fij (t) (Eq. 5) is represented here by a drift term J.LFij (t) and a diffusion term O'Fij (t) which are its mean and standard deviation. W(t) describes a Wiener process. Note that both J.LFij (t) and O'Fij (t) are calculated for a specific distribution of 8 and are functions of J.Lo and 0'0. The stochastic process that satisfies Eq. 7 will satisfy the Fokker-Plank equation for the probability distribution f of JIfE, 2 8 [(J.LF.] (t) _ _1) JPlE f(JPlE t) ] 0'2Fij (t) 8 f(JPlE + 8JEE2 8f(JPlE t) = ___ .J' 8t 8JPlE 1J .. T'J tJ' 2 'J' t) (8) ij S with reflecting boundary conditions imposed by the synaptic bounds, 0 and Jmax . Since we are interested in the stable state of the process we solve the stationary equation. The resulting density function is [1 (2J.LFij J EE 1J EE2) 1 EE N f(Jij ,J.Lt5, 0'15) = O'}ij (t) exp O'}ij (t) ij - Ts ij (9) where (10) Eq. 9 enables us to calculate the stationary distribution of the synaptic efficacies between the presynaptic neuron i and the post-synaptic neuron j given their frequency l/ and the parameters J.Lo and 0'15. An example of a solution for a 3-cycle is shown in Figure 4. In this case all neurons fire with frequency l/ = (3Td)-1 and J.Lt5 takes one of the values -Td, 0, Td. Simulation results of a 3-cycle in a network of excitatory and inhibitory integrateand-fire neurons described in Section 2 are given in Figure 5. As can be seen the results obtained from the analysis match those observed in the simulation. 5 Discussion The interesting experimental observations of synaptic learning curves [Markram et al., 1997, Zhang et al., 1998] have led us to study their implications for the firing patterns of a Hebbian cell assembly. We find that, in addition D. Horn, N. Levy, 1. Meilijson and E. Ruppin 134 (a) (b) 70 60 so 40 30 20 ) ,0 01 0 .2 0.3 0 04 0 0.' 0.3 0.2 0.4 0. 5 J EE " Figure 4: Results of the analysis for n = 3, a6 = 2ms and Td = 2.5ms. (a) The synaptic matrix. Each of the nine blocks symbolizes a group of connections between neurons that have a common phase-lag J..l6 . The mean of Ji~E was calculated for each cell by Eq. 9 and its value is given by the gray scale tone. (b) The distribution of synaptic values between all excitatory neurons. (a) (b) 5o0 0 , - - - - - - - - - - - - - , 4500 4000 3500 3000 2500 2000 1500 500 0. ' 0 .2 0.3 0.4 0. 5 o 0" - - 0 ..'..- - '... 0.2:--~0.3::----::"" 0.4c--'"-:" 0. 5 , JEE Figure 5: Simulation results for a network of N E = 100 and NJ = 50 integrateand-fire neurons, when the network is in a stable n = 3 state. Tn = 10ms for both excitatory and inhibitory neurons. The average frequency of the neurons is 130 Hz. (a) The excitatory synaptic matrix. (b) Histogram of the synaptic efficacies. to the expected synchronous and asynchronous modes, an interesting behavior of distributed synchrony can emerge. This is the phenomenon that we have investigated both by simulations and by analytic evaluation. Distributed synchrony is a mode in which the Hebbian cell assembly breaks into an n-cycle. This cycle is formed by instantaneous symmetry breaking, hence specific classification of neurons into one of the n groups depends on initial conditions , noise, etc. Thus the different groups of a single cycle do not have a semantic invariant meaning of their own. It seems perhaps premature to try and identify these cycles with synfire chains [Abeles, 1982] that show recurrence of firing patterns of groups of neurons with periods of hundreds of ms. Note however , that if we make such an identification , it is a different explanation from the model of [Herrmann et al., 1995J , which realizes the synfire chain by combining sets of preexisting patterns into a cycle. The simulations in Figures 2 and 3 were carried out with a learning curve that possessed equal potentiation and depression branches , i.e. was completely antisymmetric in its argument. In that case no synaptic decay was allowed . Figure 5, on the other hand, had stronger potentiation than depression, and a finite synaptic Distributed Synchrony ofSpiking Neurons in a Hebbian Cell Assembly 135 decay time was assumed. Other conditions in these nets were different too, yet both had a window of parameters where distributed synchrony showed up . Using the analytic approach of section 4 we can derive the probability distribution of synaptic values once a definite cyclic pattern of distributed synchrony is formed. An analytic solution of the combined dynamics of both the synapses and the spiking neurons is still an open challenge. Hence we have to rely on the simulations to prove that distributed synchrony is a natural spatiotemporal behavior that follows from combined neuronal dynamics and synaptic learning as outlined in section 2. To the extent that both types of dynamics reflect correctly the dynamics of cortical neural networks, we may expect distributed synchrony to be a mode in which neuronal attractors are being realized. The mode of distrbuted synchrony is of special significance to the field of neural computation since it forms a bridge between the feedback and feed-forward paradigms. Note that whereas the attractor that is formed by the Hebbian cell assembly is of global feedback nature, i.e. one may regard all neurons of the assembly as being connected to other neurons within the same assembly, the emerging structure of distributed synchrony shows that it breaks down into groups. These groups are connected to one another in a self-organized feed-forward manner, thus forming the cyclic behavior we have observed. References [Abbott and Song, 1999] L. F. Abbott and S. Song. Temporally asymmetric hebbian learning, spike timing and neuronal response variability. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11: Proceedings of the 1998 Conference, pages 69 - 75. MIT Press, 1999. [Abeles, 1982] M. Abeles. Local Cortical Circuits. Springer, Berlin, 1982. [BruneI, 1999] N. BruneI. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 1999. [Herrmann et al., 1995] M. Herrmann, J . Hertz, and A. Prugel-Bennet. Analysis of synfire chains. Network: Compo in Neural Systems, 6:403 - 414, 1995. [Kempter et al. , 1999] R. Kempter, W. Gerstner, and J . Leo van Hemmen. Spikebased compared to rate-based hebbian learning. In M. S. Kearns , S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11: Proceedings of the 1998 Conference, pages 125 - 131. MIT Press, 1999. [Markram et al., 1997] H. Markram, J. Lubke, M. Frotscher, and B. Sakmann. Regulation of synaptic efficacy by coincidence of postsynaptic aps and epsps. Science, 275(5297):213 - 215 , 1997. [Miyashita, 1988] Y. Miyashita. Neuronal correlate of visual associative long-term memory in the primate temporal cortex. Nature, 335:817 - 820, 1988. [Yakovlev et al., 1998] V. Yakovlev, S. Fusi, E . Berman, and E . Zohary. Inter-trial neuronal activity in inferior temporal cortex: a putative vehicle to generate longterm visual associations. Nature Neurosc ., 1(4) :310 - 317, 1998. [Zhang et al., 1998] L. I. Zhang, H. W. Tao, C. E. Holt, W . A. Harris, and M. Poo. A critical window for cooperation and competition among developing retinotectal synapses. Nature, 395:37 - 44, 1998. 

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Constructing Heterogeneous Committees Using Input Feature Grouping: Application to Economic Forecasting Yuansong Liao and John Moody Department of Computer Science, Oregon Graduate Institute, P.O.Box 91000, Portland, OR 97291-1000 Abstract The committee approach has been proposed for reducing model uncertainty and improving generalization performance. The advantage of committees depends on (1) the performance of individual members and (2) the correlational structure of errors between members. This paper presents an input grouping technique for designing a heterogeneous committee. With this technique, all input variables are first grouped based on their mutual information. Statistically similar variables are assigned to the same group. Each member's input set is then formed by input variables extracted from different groups. Our designed committees have less error correlation between its members, since each member observes different input variable combinations. The individual member's feature sets contain less redundant information, because highly correlated variables will not be combined together. The member feature sets contain almost complete information, since each set contains a feature from each information group. An empirical study for a noisy and nonstationary economic forecasting problem shows that committees constructed by our proposed technique outperform committees formed using several existing techniques. 1 Introduction The committee approach has been widely used to reduce model uncertainty and improve generalization performance. Developing methods for generating candidate committee members is a very important direction of committee research. Good candidate members of a committee should have (1) good (not necessarily excellent) individual performance and (2) small residual error correlations with other members. Many techniques have been proposed to reduce residual correlations between members. These include resampling the training and validation data [3], adding randomness to data [7], and decorrelation training [8]. These approaches are only effective for certain models and problems. Genetic algorithms have also been used to generate good and diverse members [6]. Input feature selection is one of the most important stages of the model learning process. It has a crucial impact both on the learning complexity and the general- Y. Liao and J. Moody 922 ization performance. It is essential that a feature vector gives sufficient information for estimation. However, too many redundant input features not only burden the whole learning process, but also degrade the achievable generalization performance. Input feature selection for individual estimators has received a lot of attention because of its importance. However, there has not been much research on feature selection for estimators in the context of committees. Previous research found that giving committee members different input features is very useful for improving committee performance [4], but is difficult to implement [9]. The feature selection problem for committee members is conceptually different than for single estimators. When using committees for estimation, as we stated previously, committee members not only need to have reasonable performance themselves, but should also make decisions independently. When all committee members are trained to model the same underlying function, it is difficult for committee members to optimize both criteria at the same time. In order to generate members that provide a good balance between the two criteria, we propose a feature selection approach, called input feature grouping, for committee members. The idea is to give each member estimator of a committee a rich but distinct feature sets, in the hope that each member will generalize independently with reduced error correlations. The proposed method first groups input features using a hierarchical clustering algorithm based on their mutual information, such that features in different groups are less related to each other and features within a group are statistically similar to each other. Then the feature set for each committee member is formed by selecting a feature from each group. Our empirical results demonstrate that forming a heterogeneous committee using input feature grouping is a promising approach. 2 Committee Performance Analysis There are many ways to construct a committee. In this paper, we are mainly interested in heterogeneous committees whose members have different input feature sets. Committee members are given different subsets of the available feature set. They are trained independently, and the committee output is either a weighted or unweighted combination of individual members' outputs. In the following, we analyze the relationship between committee errors and average member errors from the regression point of view and discuss how the residual correlations between members affect the committee error. We define the training data V = {(X.B, y.B);;:3 = 1,2, . . . N} and the test data T = {(XI', YI'); JL = 1,2, ... oo}, f. '" where both are assumed to be generated by the model: Y = t(X) + f. , N(o, (72) . The data V and T are independent, and inputs are drawn from an unknown distribution. Assume that a committee has K members. Denote the available input features as X = [Xl, X2, . .. ,X m ], the feature sets for the ith and jth members as Xi = [Xiu Xi2' . .. , xm;] and Xj = [Xjl) Xj2' . .. ,x mj ] respectively, where Xi EX, Xj E X and Xi =I X j , and the mapping function of the ith and lh member models trained on data from V as fi(Xd and fi(Xj ). Define the model error ef = tl' - h(Xn , for all JL = 1,2,3, ... ,00 and i = 1,2, ... , K. 923 Constructing Heterogeneous Committees for Economic Forecasting The MSE of a committee is K = K ~2 L ?11 [(en2] + ~2 L ?11 [efej] i=l , (1) i#j and the average MSE made by the committee members acting individually is K Eave = ~ L ?11 [(en 2 1 , (2) i=l where ?[.] denotes the expectation over all test data T. Using Jensen's inequality, we get Ec ~ Eave, which indicates that the performance of a committee is always equal to or better than the average performance of its members. We define the average model error correlation as C = K(i -1) l:~j ?11 [efejl , and then have 1 K-1 1 K-1 (3) Ec = KEave + ~C = (K + ~q)Eave , where q = Be . We consider the following four cases of q: ave ? Case 1: - K~l ~ q < O. In this case, the model errors between members are anti-correlated, which might be achieved through decorrelation training. ? Case 2: q = O. In this case, the model errors between members are uncorrelated, and we have: Ec = Eave. That is to say, a committee can do much better than the average performance of its members. k <q< 1. If Eave is bounded above, when the committee size we have Ec = qEave . This gives the asymptotic limit of a committee's performance. As the size of a committee goes to infinity, the committee error is equal to the average model error correlation C. The difference between Ec and Eave is determined by the ratio q. ? Case 3: 0 K -t 00, ? Case 4: q = 1. In this case, Ec is equal to Eave. This happens only when ei = ej, for all i,j = 1, ... ,K. It is obvious that there is no advantage to combining a set of models that act identically. It is clear from the analyses above that a committee shows its advantage when the ratio q is less than one. The smaller the ratio q is, the better the committee performs compared to the average performance of its members. For the committee to achieve substantial improvement over a single model, committee members not only should have small errors individually, but also should have small residual correlations between each other. 3 Input Feature Grouping One way to construct a feature subset for a committee member is by randomly picking a certain number of features from the original feature set. The advantage of this method is that it is simple. However, we have no control on each member's performance or on the residual correlation between members by randomly selecting subsets. Y. Liao and J. Moody 924 Instead of randomly picking a subset of features for a member , we propose an input feature grouping method for forming committee member feature sets. The input grouping method first groups features based on a relevance measure in a way such that features between different groups are less related to one another and features within a group are more related to one another. After grouping, there are two ways to form member feature sets. One method is to construct the feature set for each member by selecting a feature from ? each group. Forming a member's feature set in this way, each member will have enough information to make decision , and its feature set has less redundancy. This is the method we use in this paper. Another way is to use each group as the feature set for a committee member. In this method each member will only have partial information. This is likely to hurt individual member's performance. However, because the input features for different members are less dependent, these members tend to make decisions more independently. There is always a trade-off between increasing members ' independence and hurting individual members' performance. If there is no redundancy among input feature representations, removing several features may hurt individual members' performance badly, and the overall committee performance will be hurt even though members make decisions independently. This method is currently under investigation. The mutual information [(Xi; X j) between two input variables Xi and X j is used as the relevance measure to group inputs. The mutual information [(Xi; Xj) , which is defined in equation 4, measures the dependence between the two random variables. (4) If features Xi and X j are highly dependent, [(Xi; X j) will be large. Because the mutual information measures arbitrary dependencies between random variables , it has been effectively used for feature selections in complex prediction tasks [1], where methods bases on linear relations like the correlation are likely to make mistakes. The fact that the mutual information is independent of the coordinates chosen permits a robust estimation. 4 Empirical Studies We apply the input grouping method to predict the one-month rate of change of the Index of Industrial Production (IP), one of the key measures of economic activity. It is computed and published monthly. Figure 4 plots monthly IP data from 1967 to 1993. Nine macroeconomic time series , whose names are given in Table 1, are used for forecasting IP. Macroeconomic forecasting is a difficult task because data are usually limited, and these series are intrinsically very noise and nonstationary. These series are preprocessed before they are applied to the forecasting models. The representation used for input series is the first difference on one month time scales of the logged series. For example, the notation IP.L.Dl represents IP.L.Dl == In(IP(t)) -In(IP(t-l)). The target series is IP.L.FDl , which is defined as IP.L.FDI == In(IP(t+l)) - In(IP(t)). The data set has been one of our benchmarks for various studies [5, 10]. Constructing Heterogeneous Committees for Economic Forecasting 925 Index of Industrial Production: 1967 ? 1993 Figure 1: U.S. Index of Industrial Production (IP) for the period 1967 to 1993. Shaded regions denote official recessions, while unshaded regions denote official expansions. The boundaries for recessions and expansions are determined by the National Bureau of Economic Research based on several macroeconomic series. As is evident for IP, business cycles are irregular in magnitude, duration, and structure, making prediction of IP an interesting challenge. Series IP SP DL M2 CP CB HS TB3 Tr Description Index of Industrial Production Standard & Poor's 500 Index of Leading Indicators Money Supply Consumer Price Index Moody's Aaa Bond Yield HOUSing Starts 3-month Treasury Bill Yield Yield Curve Slope: (10-Year Bond Composite)-(3-Month Treasury Bill) Table 1: Input data series. Data are taken from the Citibase database. During the grouping procedure, measures of mutual information between all pairs of input variables are computed first. A simple histogram method is used to calculate these estimates. Then a hierarchical clustering algorithm [2] is applied to these values to group inputs. Hierarchical clustering proceeds by a series of successive fusions of the nine input variables into groups. At any particular stage, the process fuses variables or groups of variables which are closest, base on their mutual information estimates. The distance between two groups is defined as the average of the distances between all pairs of individuals in the two groups. The result is presented by a tree which illustrates the fusions made at each successive level (see Figure 2). From the clustering tree, it is clear that we can break the input variables into four groups: (IP.L.Dl, DL.L.Dl) measure recent economic changes, (SP.L .Dl) reflects recent stock market momentum, (CB.D1, TB3 .D1, Tr.D1) give interest rate information, and (M2.L.D1, CP.L.D1, HS.L.D1) provide inflation information. The grouping algorithm meaningfully clusters the nine input series. Y. Liao and J. Moody 926 ~ :; ~ ::l ::I ~ ? ~ ~ :;; ~ S ~ es ~ Figure 2: Variable grouping based on mutual information. Y label is the distance. Eighteen differept subsets of features can be generated from the four groups by selecting a feature from each group. Each subset is given to a committee member. For example, the subsets (IP.L.Dl, SP.L.Dl, CB.Dl, M2.L.Dl) and (DL.L.Dl, SP.L.Dl, TB3.Dl, M2.L.Dl) are used as feature sets for different committee members. A committee has totally eighteen members. Data from Jan. 1950 to Dec. 1979 is used for training and validation, and from Jan. 1980 to Dec. 1989 is used for testing. Each member is a linear model that is trained using neural net techniques. We compare the input grouping method with three other committee member generating methods: baseline, random selection, and bootstrapping. The baseline method is to train a committee member using all the input variables. Members are only different in their initial weights. The bootstrapping method also trains a member using all the input features, but each member has different bootstrap replicates of the original training data as its training and validation sets. The random selection method constructs a feature set for a member by randomly picking a subset from the available features. For comparison with the grouping method, each committee generated by these three methods has 18 members. Twenty runs are performed for each of the four methods in order to get reliable performance measures. Figure 3 shows the boxplots of normalized MSE for the four methods. The grouping method gives the best result, and the performance improvement is significant compared to other methods. The grouping method outperforms the random selection method by meaningfully grouping of input features. It is interesting to note that the heterogeneous committee methods, grouping and random selection, perform better than homogeneous methods for this data set. One of the reasons for this is that giving different members different input sets increases their model independence. Another reason could be that the problem becomes easier to model because of smaller feature sets. 5 Conclusions The performance of a committee depends on both the performance of individual members and the correlational structure of errors between members. An empirical study for a noisy and nonstationary economic forecasting problem has demonstrated that committees constructed by input variable grouping outperform committees formed by randomly selecting member input variables. They also outperform committees without any input variable manipulation. 927 Constructing Heterogeneous Committees Jor Economic Forecasting Commltt.. Performanc. Comp.rteon (20 rurw) 0.84 8; I I 0.82 UJ ~ 0.8 ) J 0.78 ! 9 0.75 I -L 1 0.74 8 I -I1 1 : Gr~lng. 2 3 2:Random ??lectlon. 3.BaM"n..... Bootatrtpplng Figure 3: Comparison between four different committee member generating methods. The proposed grouping method gives the best result, and the performance improvement is significant compared to the other three methods. References [1] R . Battiti. Using mutual information for selecting features in supervised neural net learning. IEEE TI-ans. on Neural Networks, 5(4), July 1994. [2] B.Everitt. Cluster Analysis. Heinemann Educational Books, 1974. [3] L. Breiman. Bagging predictors. Machine Learning, 24(2):123-40, 1996. [4] K.J. Cherkauer. Human expert-level performance on a scientific image analysis task by a system using combined artificai neural networks. In P. Chan, editor, Working Notes of the AAAI Workshop on Integrating Multiple Learned Models, pages 15-2l. 1996. [5] J. Moody, U. Levin, and S. Rehfuss. Predicting the U.S. index of industrial production. In proceedings of the 1993 Parallel Applications in Statistics and Economics Conference, Zeist, The Netherlands . Special issue of Neural Network World, 3(6):791794, 1993. [6] D. Opitz and J. Shavlik. Generating accurate and diverse members of a neuralnetwork ensemble. In D . Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems 8. MIT Press, Cambridge, MA, 1996. [7] Y. Raviv and N. Intrator. Bootstrapping with noise: An effective regularization technique. Connection Science, 8(3-4):355-72, 1996. [8] B. E. Rosen. Ensemble learning using decorrelated neural networks. Science, 8(3-4):373-83, 1996. Connection [9] K. Tumer and J. Ghosh. Error correlation and error reduction in ensemble classifiers. Connection Science, 8(3-4):385-404, December 1996. [10] L. Wu and J . Moody. A smoothing regularizer for feedforward and recurrent neural networks. Neural Computation, 8.3:463- 491, 1996. 

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A recurrent model of the interaction between Prefrontal and Inferotemporal cortex in delay tasks ALFONSO RENART, NESTOR PARGA Departamento de F{sica Te6rica Universidad Aut6noma de Madrid Canto Blanco, 28049 Madrid, Spain http://www.ft.uam.es/neurociencialGRUPO/grup0.1!nglish.html and EDMUND T. ROLLS Oxford University Department of Experimental Psychology South Parks Road, Oxford OX] 3UD, England Abstract A very simple model of two reciprocally connected attractor neural networks is studied analytically in situations similar to those encountered in delay match-to-sample tasks with intervening stimuli and in tasks of memory guided attention. The model qualitatively reproduces many of the experimental data on these types of tasks and provides a framework for the understanding of the experimental observations in the context of the attractor neural network scenario. 1 Introduction Working memory is usually defined as the capability to actively hold information in memory for short periods of time. In primates, visual working memory is usually studied in experiments in which, after the presentation of a given visual stimulus, the monkey has to withhold its response during a certain delay period in which no specific visual stimulus is shown. After the delay, another stimulus is presented and the monkey has to make a response which depends on the interaction between the two stimuli. In order to bridge the temporal gap between the stimuli, the first one has to be held in memory during the delay. Electrophysiological recordings in primates during the performance of this type of tasks has revealed that some populations of neurons in different brain areas such as prefrontal (PF), inferotemporal (IT) or posterior parietal (PP) cortex, maintain approximately constant firing rates during the delay periods (for a review see [1]) and this delay activity states have been postulated as the internal representations of the stimuli provoking them [2]. Although up to now most of the modeling effort regarding the operation of networks able to support stable delay activity states has been put in the study of un i-modular (homogeneous) networks, there is evidence that in order for the monkey to solve the tasks satisfactorily, the interaction of several different neural structures is needed. A number of studies of delay match-to-sample tasks with intervening stimuli in primates performed by Desimone and 172 A. Renart, N Parga and E. TRolls colleagues has revealed that although IT cortex supports delay activity states and shows memory related effects (differential responses to the same, fixed stimulus depending on its status on the trial, e.g. whether it matches or not the sample), it cannot, by itself, provide the information necessary to solve the task, as the delay activity states elicited by each of the stimuli in a sequence are disrupted by the input information associated with each new stimulus presented [3, 4, 5]. Another structure is therefore needed to store the information for the whole duration of the trial. PF cortex is a candidate, since it shows selective delay activity maintained through entire trials even with intervening stimuli [6]. A series of parallel experiments by the same group on memory guided attention [7, 8] have also shown differential firing of IT neurons in response to the same visual stimulus shown after a delay (an array of figures), depending on previous information shown before the delay (one of the figures in the array working as a target stimulus). This evidence suggests a distributed memory system as the proper scenario to study working memory tasks as those described above. Taking into account that both IT and PF cortex are known to be able to support delay activity states, and that they are bi-directionally connected, in this paper we propose a simple model consisting of two reciprocally connected attractor neural networks to be identified with IT and PF cortex. Despite its simplicity, the model is able to qualitatively reproduce the behavior of IT and PF cortex during delay match-to-sample tasks with intervening stimuli, the behavior of IT cells during memory guided attention tasks, and to provide an unified picture of these experimental data in the context of associative memory and attractor neural networks. 2 Model and dynamics The model network consists of a large number of (excitatory) neurons arranged in two modules. Following [9, 10], each neuron is assumed to be a dynamical element which transforms an incoming afferent current into an output spike rate according to a given transduction function. A given afferent current Iai to neuron i (i = 1, ... ,N) in module a (a IT, PF) decays with a characteristic time constant T but increases proportionally to the spike rates Vbj of the rest of the neurons in the network (both from inside and outside its module) connected to it, the contribution of each presynaptic neuron, e.g. neuron j from module b, being proportional to the synaptic efficacy between the two. This can be expressed through the following equation = Jt/ d1ai(t) = _ Iai(t) dt T + '" J~~,b) ~ ~J . + h(~xt) VbJ bj a~ (1) . An external current h~~xt) from outside the network, representing the stimuli, can also be imposed on every neuron. Selective stimuli are modeled as proportional to the stored patterns, i.e. h~~ezt) = haTJ~i' where ha is the intensity of the external current to module a. The transduction function of the neurons transforming currents into rates has been chosen as a threshold hyperbolic tangent of gain G and threshold O. The synaptic efficacies between the neurons of each module and between the neurons in different modules are respectively [11, 12] p o "'(TJai I-' - I) (I-' I) 1(1 _J J)Nt ~ TJaj - J(a,a) ij - i.../- J' r a = IT,PF (2) p j (a,b) - ij - "'( I-' I) (I-' I) 1(1 _9 J)Nt ~ TJai TJbj - \.J" v ~,J a .../r b . (3) 173 Recurrent Model of IT-PF Interactions in Delay Tasks The intra-modular connections express the learning of P binary patterns {17~i = 0,1, f.L = 1, ... , P} by each module, each of them signaling which neurons are active in each of the sustained activity configurations. Each variable Tl~i is supposed to take the values 1 and 0 with probabilities f and (1 - f) respectively, independently across neurons and across patterns. The inter-modular connections reflect the temporal associations between the sustained activity states of each module. In this way, every stored pattern f.L in the IT module has an associated pattern in the PF module which is labelled by the same index. The normalization constant Nt = N(Jo + g) has been chosen so that the sum of the magnitudes of the inter- and the intra-modular connections remains constant and equal to 1 while their relative values are varied. When this constraint is imposed the strength of the connections can be expressed in terms of a single independent parameter 9 measuring the relative intensity of the inter- vs. the intra-modular connections (Jo can be set equal to 1 everywhere). We will limit our study to the case where the number of stored patterns per module P does not increase proportionally to the size of the modules N since a large number of stored patterns does not seem necessary to describe the phenomenology of the delay match-to-sample experiments. Since the number of neurons in a typical network one may be interested in is very large, e.g. '" 10 5 - 106 , the analytical treatment of the set of coupled differential equations (1) becomes intractable. On the other hand, when the number of neurons is large, a reliable description of the asymptotic solutions of these equations can be found using the techniques of statistical mechanics [13, 9]. In this framework, instead of characterizing the states of the system by the state of every neuron, this characterization is performed in terms of macroscopic quantities called order parameters which measure and quantify some global properties of the network as a whole. The relevant order parameters appearing in the description of our system are the overlaps of the state of each module with each of the stored patterns m~, defined as: 1 m~ = N? 2)17~i - f)Vai X i where the symbol ? ?1/ , (4) ... ?1/ stands for an average over the stored patterns. Using the free energy per neuron of the system at zero temperature :F (which we do not write explicitly to reduce the technicalities to a minimum) we have modeled the experiments by giving the order parameters the following dynamics: (5) This dynamics ensures that the stationary solutions, corresponding to the values of the order parameters at the attractors, correspond also to minima of the free energy, and that, as the system evolves, the free energy is always minimized through its gradient. The time constant of the macroscopic dynamics is a free parameter which has been chosen equal to the time constant of the individual neurons, reflecting the assumption that neurons operate in parallel. Its value has been set to T = 10 ms. Equations (5) have been solved by a simple discretizing procedure (first order Runge-Kutta method). Since not all neurons in the network receive the same inputs, not all of them behave in the same way, i.e. have the same firing rates. In fact, the neurons in each of the module can be split into different sub-populations according to their state of activity in each of the stored patterns. The mean firing rate of the neurons in each SUb-population depends on the particular state realized by the network (characterized by the values of the order parameters). Associated to each pattern there are two larger sub-populations, to be denoted as foreground (all active neurons) and background (all inactive neurons) of that pattern. 174 A. Renart, N. Parga and E. T. Rolls The overlap with a given pattern can be expressed as the difference between the mean firing rate of the neurons in its foreground and its background. The average is performed over all other sub-populations to which each neuron in the foreground (background) may belong to, where the probability of a given sub-population is equal to the fraction of neurons in the module belonging to it (determined by the probability distribution of the stored patterns as given above). This partition of the neurons into sub-populations is appealing since, in experiments, cells are usually classified in terms of their response properties to a set of fixed stimuli, i.e. whether each stimulus is effective or ineffective in driving their response. The modeling of the different experiments proceeded according to the macroscopic dynamics (5), where each stimulus was implemented as an extra current for a desired period of time. 3 Sequence with intervening stimuli In order to study delay match-to-sample tasks with intervening stimuli [5, 6], the module to be identified with IT was sequentially stimulated with external currents proportional to some of the stored patterns with a delay between them. To take into account the large fraction of PF neurons with non-selective responses to the visual stimuli (which may be involved in other aspects of the task different from the identification of the stimuli), and since the neurons in our modules are, by definition, stimulus selective (although they are probably connected to the non-selective neurons) a constant, non-selective current of the same intensity as the selective input to the IT module was applied (during the same time) equally to all sub-populations of the PF module. The external current to the IT module was stimulus selective because the fraction of IT neurons with non-selective responses to the visual stimuli is very small [6]. The results can be seen in Figure 1 where the sequence ABA with A as the sample stimulus and B as a non-matching stimulus has been studied. The values of the model parameters are listed in the caption. In Figure 1a, the mean firing rates of the foreground populations of patterns AIT and BIT of the IT module have been plotted as a function of time. The main result is that, as observed in the experiments, the delay activity in the IT module is determined by the last stimulus presented. The delay activity provoked by a given stimulus is disrupted by the next, unless it corresponds to the same stimulus, in which case the effect of the stimulus is to increase the firing rate of the neurons in its foreground. We have checked that no noticeable effects occur if more nonmatching stimuli are presented (they are all equivalent with respect to the sample) or if a non-match stimulus is repeated. If the coupling g between the modules is weak enough [12] the behavior in the PF module is different. This can be seen in Figure 1b, where the time evolution of the mean firing rates of the foreground of the two associated patterns ApF and BpF stored in the PF module are shown. In agreement with the findings of Desimone and colleagues, the neurons in the PF module remain correlated with the sample for the whole trial, despite the non-selective signal received by all PF neurons (not only those in the foreground of the sample) and the fact that the selective current from the IT module tends to activate the pattern associated with the current stimulus. Desimone and colleagues [5, 6] report that the response of some neurons (not necessarily those with sample selective delay activity or with stimulus selective responses) in both IT and PF cortex to some stimuli, is larger if those stimuli are matches in their trials than if the same stimuli are non-matches. This has been denoted as match enhancement. In the present scenario the explanation is straightforward: when a stimulus is a non-match, IT and PF are in different states and therefore send inconsistent signals to each other. The firing rate of the neurons of each module is maintained in that case solely by the contribution to the total current coming from the recurrent collaterals. On the other hand, when the stimulus is the match, both modules find themselves in states associated in the synapses 175 Recurrent Model of IT-PF Interactions in Delay Tasks (a) (b) 0.8 0.7 Ir- 0.5 C> 0.4 a: .'"Ii:c: i 0.7 !i '\ 0.6 ., a; r- ' ''I ! 0.6 ! , i I 0.3 \ 0.2 0.1 0.5 C> 0.4 a: I \ "'---._.._-- ., a; c: ?c Ii: \. 0.3 0.2 r.~ .... . . _ .._ .-J~_ _ _--,?L.__ O'---'--~~-~~~-"""""'-'---' o 2 3 4 56789 Time (s) . \ fL... i\..... ?? "l. . ____J \. ___.___._.J l.__._.___. 0.1 ...........-,---, OL--'--~~-~~~- o 234 5 6 7 8 9 Time (s) Figure 1: (a) Mean rates in the foreground of patterns AIT (solid line) and BIT (dashed line) in the IT module as a function of time. (b) Same but for patterns A pF and BpF of the PF module. Model parameters are G = 1.3, () = 10- 3 , f = 0.2, 9 = 10- 2 , h = 0.13. Stimuli are presented during 500 ms at seconds 0, 3, and 6 following the sequence ABA. between the neurons connecting them, PF because it has remained that way the whole trial, and IT because it is driven by the current stimulus. When this happens, the contribution to the total current from the recurrent collaterals and from the long range afferents add up consistently, and the firing rate increases. In order for this explanation to hold there should be a correlation between the top-down input from PF and the sensory bottom-up signal to IT. Indeed, experimental evidence for such a correlation has very recently been found [14]. This is an important experimental finding which supports our theory. Looking at Figure 1, one sees that the effect is not evident in the model during the time of stimulus presentation, which is the period where it has been reported. The effect is, in fact, present, although its magnitude is too small to be noticeable in the figure. We would argue, however, that this quantitative difference is an artifact of the model. This is because the enhancement effect is very noticeable on the delay periods, where essentially the same neurons are active as during the stimulus presentations (i.e., where the same correlations between the top-down and bottom-up signals exist) but with lower firing rates. During stimulus presentations the firing rates are closer to the saturation regime, and therefore the dynamical response range of the neurons is largely reduced. 4 Memory guided attention To test the differential response of cells as a function of the contents of memory, we have followed [7, 8] and studied a sub-population of IT cells which are simultaneously in the foreground of one of the patterns (AIT) and in the background of another (BIT) in the same conditions as the previous section (same model parameters). In Figure 2a the response of this sub-population as a function of time has been plotted in two different situations. In the first one, the effective stimulus AIT was shown first (throughout this section non selective stimulation of PF proceeded as in the last section) and after a delay, a stimulus array equal to the sum of AIT and BIT was presented. The second situation is exactly equal, except for the fact that the cue stimulus shown first was the ineffective stimulus BIT. The response of the same sub-population to the same stimulus array is totally different and determined by the cue stimulus: If the sub-population is in the background of the cue, its response is null during the trial except for the initial period of the presentation of the array. In accordance with the experimental observations [7, 8], its response grows initially (as one would expect, since during the array presentation time, stimulation is symmetric with A. Renart, N. Parga and E. T. Rolls 176 respect of A and B) but is later suppressed by the top-down signal being sent by the PF module. This suppression provides a clear example of a situation in which the contents of memory (in the form of an active PF activity state) are explicitly gating the access of sensory information to IT, implementing a non-spatial attentional mechanism. (a) 0.8 (b) .---~~~-~-~-~---, r 0.7 r- 0.6 0.5 Q) iii cr: 0.4 cr: 0.4 c ."'" ?c u: *'" c u: 0.2 r\ 0.1 _.. _--_..-----_...._....! \ .._----_ .._"- 2 3 Time (s) 4 5 I ! ? I i i l.-\ 0.5 0.3 0 r-?1 0.7 0.6 l_. _ _. _ ._._. 0.3 0.2 0.1 r--\ 0 -0.1 6 0 2 3 4 5 6 Time (s) Figure 2: (a) Mean rates as a function of time in IT neurons which are both in the foreground AIT and in the background of BIT when the cue stimulus is AIT (solid line) or BIT (dashed line). (b) Mean rates of the same neurons when CIT is the cue stimulus and the array is AIT alone (long dashed line), BIT alone (short dashed line) or the sum of AIT and BIT (solid line). Cue present until 500 ms. Array present from 3000 ms to 3500 ms. Model parameters as in Figure 1 In the model, the PF module remains in a state correlated with the cue during the whole trial (to our knowledge there are no measurements of PF activity during memory guided attention tasks) and therefore provides a persistent signal 'in the direction' ofthe cue which biases the competition between AIT and BIT established at the onset of the array. This is how the gating mechanism is implemented. The competitive interactions between the stimuli in the array are studied in Figure 2b, which is an emulation of the target-absent trials of [8]. In this figure, the same sub-population is studied under situations in which the cue stimulus is not present in the array (another one of the stored patterns, i.e. CIT) The three curves correspond to different arrays: The effective stimulus alone, the ineffective stimulus alone, and a sum of the two as in the previous experiment. In all three, the PF module remains in a sustained activity state correlated with CIT the whole trial and therefore, since the patterns are independent, the signal it sends to IT is symmetric with respect of A and B. Thus, the response of the sub-population during the array is in this case unbiased, and the effect of the competitive interactions can be isolated. The result is that, as observed experimentally, the response to the complex array is intermediate between the one to the effective stimulus alone and the one to the ineffective stimulus alone. The nature of the competition in an attractor network like the one under study here is based on the fact that complex stimulus combinations are not stored in the recurrent collaterals of each module. These connections tend to stabilize the individual patterns which, being independent, tend to cancel each other when presented together. After the array is presented, the state of the IT module, which is correlated with CIT in the initial delay, becomes correlated with AIT or BIT if they are presented alone. When the array contains both of them in a symmetric fashion, since the sum of the patterns is not a stored pattern itself, the IT module remains correlated with pattern CIT due to the signal from the PF module. 177 Recurrent Model of IT-PF Interactions in Delay Tasks 5 Discussion We have proposed a toy model consisting of two reciprocally connected attractor modules which reproduces nicely experimental observations regarding intra-trial data in delay match-to-sample and memory guided attention experiments in which the interaction between IT and PF cortex is relevant. Several important issues are taken into account in the model: a complex interaction between the PF and IT modules resultant from the association of frequent patterns of activity in both modules, delay activity states in each module which exert mutually modulatory influences on each other, and a common substrate (we emphasize that the results on Sections 3 and 4 where obtained with exactly the same model parameters, just by changing the type of task) for the explanation of apparently diverse phenomena. Perception is clearly an active process which results from the complex interactions between past experience and incoming sensory information. The main goal of this model was to show that a very simple associational (Hebbian) pattern of connectivity between a perceptual module and a 'working memory' module can provide the basic ingredients needed to explain coherently different experimentally found neural mechanisms related to this process. The model has clear limitations in terms of 'biological realism' which will have to be improved in order to use it to make quantitative predictions and comparisons, and does not provide a complete an exhaustive account of the very complex and diverse phenomena in which temporo-frontal interactions are relevant (there is, for example, the issue of how to reset PF activity in between trials [15]). However, it is precisely the simplicity of the mechanism it provides and the fact that it captures the essential features of the experiments, despite being so simple, what makes it likely that it will remain relevant after being refined. Acknowledgements This work was funded by a Spanish grant PB96-0047. We acknowledge the Max Planck Institute for Physics of Complex Systems in Dresden, Germany, for the hospitality received by A.R. and N.P. during the meeting held there from March 1 to 26, 1999. References [1] J. M. Fuster. Memory in the cerebral cortex. Cambridge, MA: MIT Press (1995) [2] D. J. Amit. Belulvioral and Brain Sciences 18, 617-657 (1995) [3] G. C. Baylis & E. T. Rolls. Exp. Brain Res. 65,614-622 (1987) [4] E. K. Miller, L. Li & R. Desimone. 1. Neurosci. 13, 1460-1478 (1993) [5] E. K. Miller & R. Desimone. Science 263,520-522 (1994) [6] E. K. Miller, C. A. Erickson & R. Desimone. 1. Neurosci. 16,5154-5167 (1996) [7] L. Chelazzi, E. K. Miller, J. Duncan & R. Desimone. Nature 363,345-347 (1993) [8] L. Chelazzi, J. Duncan, E. K. Miller & R. Desimone. 1. Neurophysiol. 80,2918-2940 (1998) [9] R. Kuhn. In Statistical Meclulnics of Neural Networks. (ed. L. Garrido), 19-32. Berlin: Springer-Verlag (1990) [10] D. J. Amit & M. V. Tsodyks. Network 2,259-273 (1991) [11] A. Renart, N. Parga & E. T. Rolls. Neural Computation 11, 1349-1388 (1999). [12] A. Renart, N. Parga & E. T. Rolls. Network 10,237-255 (1999). [13] M. Mezard, G. Parisi & M. Virasoro. Spin glass theory and beyond. Singapore: World Scientific (1987) [14] H. Tomita, M Ohbayashi, K. Nakahara, I. Hasegawa & Y. Miyashita. Nature 401, 699-703 (1999) [15] D. Durstewitz, M. Kelc & o. Giintiirkiin. 1. Neurosci. 19,2807-2822 (1999) 

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From Coexpression to Coregulation: An Approach to Inferring Transcriptional Regulation among Gene Classes from Large-Scale Expression Data Eric Mjolsness Jet Propulsion Laboratory California Institute of Technology Pasadena CA 91109-8099 mjolsness@jpl.nasa.gov Tobias Mann Jet Propulsion Laboratory California Institute of Technology Pasadena CA 91109-8099 mann@aigjpl.nasa.gov Rebecca Castaiio Jet Propulsion Laboratory California Institute of Technology Pasadena CA 91109-8099 becky@aigjpl.nasa.gov Barbara Wold Division of Biology California Institute of Technology Pasadena CA 91125 woldb@its.caltech.edu Abstract We provide preliminary evidence that eXlstmg algorithms for inferring small-scale gene regulation networks from gene expression data can be adapted to large-scale gene expression data coming from hybridization microarrays. The essential steps are (1) clustering many genes by their expression time-course data into a minimal set of clusters of co-expressed genes, (2) theoretically modeling the various conditions under which the time-courses are measured using a continious-time analog recurrent neural network for the cluster mean time-courses, (3) fitting such a regulatory model to the cluster mean time courses by simulated annealing with weight decay, and (4) analysing several such fits for commonalities in the circuit parameter sets including the connection matrices. This procedure can be used to assess the adequacy of existing and future gene expression time-course data sets for determ ining transcriptional regulatory relationships such as coregulation . 1 Introduction In a cell, genes can be turned "on" or "off' to varying degrees by the protein products of other genes. When a gene is "on" it is transcribed to produce messenger RNA (mRNA) which can subsequently be translated into protein molecules. Some of these proteins are transcription factors which bind to DNA at specific sites and thereby affect which genes are transcribed and how often. This trancriptional Inferring Transcriptional Regulation among Gene Classes 929 regulation feedback circuitry provides a fundamental mechanism for information processing in the cell. It governs differentiation into diverse cell types and many other basic biological processes. Recently, several new technologies have been developed for measuring the "expression" of genes as mRNA or protein product. Improvements in conventional f1uorescently labeled antibodies against proteins have been coupled with confocal microscopy and image processing to partially automate the simultaneous measurement of small numbers of proteins in large numbers of individual nuclei in the fruit fly Drosophila melanogaster [1]. In a complementary way, the mRNA levels of thousands of genes, each averaged over many cells, have been measured by hybridization arrays for various species including the budding yeast Saccharomyces cerevisiae [2] . The high-spatial-resolution protein antibody data has been quantitatively modeled by "gene regulation network" circuit models [3] which use continuous-time, analog, recurrent neural networks (Hopfield networks without an objective function) to model transcriptional regulation [4][5] . This approach requires some machine learning technique to infer the circuit parameters from the data, and a particular variant of simulated annealing has proven effective [6][7]. Methods in current biological use for analysing mRNA hybridization data do not infer regulatory relationships, but rather simply cluster together genes with similar patterns of expression across time and experimental conditions [8][9] . In this paper, we explore the extension of the gene circuit method to the mRNA hybridization data which has much lower spatial resolution but can currently assay a thousand times more genes than immunofluorescent image analysis. The essential problem with using the gene circuit method, as employed for immunoflourescence data, on hybridization data is that the number of connection strength parameters grows between linearly and quadratically in the number of genes (depending on sparsity assumptions) . This requires more data on each gene, and even if that much data is available, simulated annealing for circuit inference does not seem to scale well with the number of unknown parameters. Some form of dimensionality reduction is called for. Fortunately dimensionality reduction is available in the present practice of clustering the large-scale time course expression data by genes, into gene clusters. In this way one can derive a small number of cluster-mean time courses for "aggregated genes", and then fit a gene regulation circuit to these cluster mean time courses. We will discuss details of how this analysis can be performed and then interpreted. A similar approach using somewhat different algorithms for clustering and circuit inference has been taken by Hertz [10]. In the following , we will first summarize the data models and algorithms used, and then report on preliminary experiments in applying those algorithms to gene expression data for 2467 yeast genes [9][11]. Finally we will discuss prospects for and limitations of the approach. 2 Data Models and Algorithms The data model is as follows . We imagine that there is a small, hidden regulatory network of " aggregate genes" which regulate one another by the analog neural network dynamics [3] T . dv; I dt = g(~ T.v + h) ~ 1/ J / I Xv . I I E. Mjolsness, T Mann, R. Castano and B. Wold 930 In which Vi is the continuous-valued state variable for gene product i, ~j is the matrix of positive, zero, or negative connections by which one transcription factor can enhance or repress another, and gO is a nonlinear monotonic sigmoidal activation function. When a particular matrix entry ~j is nonzero, there is a regulatory "connection" from gene product} to gene i. The regulation is enhancing if T is positive and repressing if it is negative. If ~j is zero there is no connection . This network is run forwards from some initial condition and time-sampled to generate a wild-type time course for the aggregate genes. In addition, various other time courses can be generated under alternative experimental conditions by manipulating the parameters. For example an entire aggregate gene (corresponding to a cluster of real genes) could be removed from the circuit or otherwise modified to represent mutants. External input conditions could be modeled as modifications to h . Thus we get one or several time courses (trajectories) for the aggregate genes. From such aggregate time courses, actual gene data is generated by addition of Gaussian-distributed noise to the logarithms of the concentration variables. Each time point in each cluster has its own scalar standard deviation parameter (and a mean arising from the circuit dynamics). Optionally, each gene's expression data may also be multiplied by a time-independent proportionality constant. Regulatory aggregate genes (large circles) and cluster member genes (small circles). T Given this data generation model and suitable gene expression data, the problem is to infer gene cluster memberships and the circuit parameters for the aggregate genes' regulatory relationships. Then, we would like to use the inferred cluster memberships and regulatory circuitry to make testable biological predictions. This data model departs from biological reality in many ways that could prove to be important, both for inference and for prediction. Except for the Gaussian noise model, each gene in a cluster is models as fully coregulated with every other one they are influenced in the same ways by the same regulatory connection strengths. Also, the nonlinear circuit model must not only reflect transcriptional regulation, but all other regulatory circuitry affecting measured gene expression such as kinasephosphatase networks. Under this data model, one could formulate a joint Bayesian inference problem for the clustering and circuit inference aspects of fitting the data. But given the highly provisional nature of the model, we simply apply in sequence an existing mixtureof-Gaussians clustering algorithm to preprocess the data and reduce its dimensionality , and then an existing gene circuit inference algorithm . Presumably a joint optimization algorithm could be obtained by iterating these steps. 2.1 Clustering A widely used clustering algorithm for mixure model estimation is ExpectationMaximization (EM)[12]. We use EM with a diagonal covariance in the Gaussian, so that for each feature vector component a (a combination of experimental condition Inferring Transcriptional Regulation among Gene Classes 931 and time point in a time course) and cluster a there is a standard deviation parameter G aa . In preprocessing, each concentration data point is divided by its value at time zero and then a logarithm taken. The log ratios are clustered using EM. Optionally, each gene's entire feature vector may be normalized to unit length and the cluster centers likewise normalized during the iterative EM algorithm. In order to choose the number of clusters, k, we use the cross-validation algorithm described by Smyth [13]. This involves computing the likelihood of each optimized fit on a test set and averaging over runs and over divisions of the data into training and test sets. Then, we can examine the likelihood as a function of k in order to choose k. Normally one would pick k so as to maximize cross-validated likelihood. However, in the present application we also want to reward small values of k which lead to smalIer circuits for the circuit inference phase of the algorithm. The choice of k will be discussed in the next section. 2.2 Circuit Inference We use the Lam-Delosme variant of simulated annealing (SA) to derive connection strengths T, time constants t, and decay rates f..., as in previous work using this gene circuit method [4][5]. We set h to zero. The score function which SA optimizes is S(T,r,A) = AI(v;(t;T,r,A)-vi (t?)2 + WI7;j2 h ij +exp[B(I7;i2 + I A7 + Ir;)]-l ij The first term represents the fit to data Vi. The second term is a standard weight decay term. The third term forces solutions to stay within a bounded region in weight space. We vary the weight decay coefficient W in order to encourage relatively sparse connection matrix solutions. 3 3.1 Results Data We used the Saccharomyces cerevisiae data set of [9]. It includes three longer time courses representing different ways to synchronize the normal cell cycle [II], and five shorter time courses representing altered conditions. We used all eight time courses for clustering, but just 8 time points of one of the longer time courses (alpha factor synchronized cell cycle) for the circuit inference. It is likely that multiple long time courses under altered conditions will be required before strong biological predictions can be made from inferred regulatory circuit models. 3.2 Clustering We found that the most likely number of classes as determined by cross validation was about 27, but that there is a broad plateau of high-likelihood cluster numbers from 15 to 35 (Figure I). This is similar to our results with another gene expression data set for the nematode worm Caenorhabditis e/egans supplied by Stuart Kim; these more extensive clustering experiments are summarized in Figure 2. Clustering experiments with synthetic data is used to understand these results. These experiments show that the cross-validated log likelihood curve can indicate the number of clusters present in the data, justifying the use of the curve for that E. Mjolsness, T. Mann, R. Castano and B. Wold 932 purpose. In more detail, synthetic data generated from 14 20-dimensional spherical Gaussian clusters were clustered using the EMlCV algorithm. The likelihoods showed a sharp peak at k=14 unlike Figures 1 or 2. In another experiment, 14 20dimensional spherical Gaussian superclusters were used to generate second-level clusters (3 subclusters per supercluster), which in turn generated synthetic data points. This two-level hierarchical model was then clustered with the EMlCV method. The likelihood curves (not shown) were quite similar to Figures 1 and 2, with a higher-likelihood plateau from roughly 14 to 40. x 10" ~, :- ~ :1; ,i ./ I l ~'.~ ; --~--~,~ ~ ---7.1~--~~~ ' ---=a--~~7---~~--~~ ero.V.....d~ Figure 1. Cross-validated log-likelihood scores, displayed and averaged over 5 runs, for EM clustering of S. cerevisiae gene expression data [9]. Horizontal axis: k, the "requested" or maximal number of cluster centers in the fit. Some cluster centers go unmatched to data. Vertical axis: log likelihood score for the fit, scatterplotted and' averaged. Likelihoods have not been integrated over any range of parameters for hypothesis testing. k ranges from 2 to 40 in increments of 1. Solid line shows average likelihood value for each k. ~, + _u + 7---!" ~10o-----:':-'.-----::20o----,. c!=----~",-----;.. Number of Clusters ++ + +++ ~~---~~--~,~~--~,~~--~~ ~--~* Number of Clusters Figure 2. Cross-validated log-likelihood scores, averaged over 13 runs, for EM clustering of C. elegans gene expression data from S. Kim's lab. Horizontal axis: k, the "requested" or maximal number of cluster centers in the fit. Some cluster centers go unmatched to data. Vertical axis: log likelihood score for the fit, as an average over 13 runs plus or minus one standard deviation. (Left) Fine-scale plot, k =2 to 60 in increments of 2. (Right) Coarsescale plot, k=2 to 202 in increments of 10. Both plots show an extended plateau of relatively likely fits between roughly k =14 and k =40. From Figures 1 and 2 and the synthetic data experiments mentioned above, we can guess at appropriate values for k which take into account both the measured likelihood of clustering and the requirements for few parameters in circuit-fitting. For example choosing k=15 clusters would put us at the beginning of the plateau, losing very little cluster likelihood in return for reducing the aggregate genes circuit size from 27 to 15 players. The interpretation would be that there are about 15 superclusters in hierarchically clustered data, to which we will fit a 15-player Inferring Transcriptional Regulation among Gene Classes 933 regulatory circuit. Much more aggressive would be to pick k=7 or 8 clusters, for a relatively significant drop in log-likelihood in return for a further substantial decrease in circuit size. An acceptable range of cluster numbers (and circuit sizes) would seem to be k=8 to 15. 3.3 Gene Circuit Inference It proved possible to fit the k= 15 time course using weight decay W=1 but without using hidden units. W=O and W=3 gave less satisfactory results. Four of the 15 clusters are shown in Figure 3 for one good run (W= 1). Scores for our first few (unselected) runs at the current parameter settings are shown in Table 1. Each run took between 24 and 48 hours on one processor of an Sun UItrasparc 60 computer. Even with weight decay, it is possible that successful fits are really overfits with this particular data since there are about twice as many parameters as data points. Weight Decay W <Score> 0 3 <Simulated Annealing Moves>/1 0/\6 Number of runs 1.536 +/- 0.134 2.803 +/- 0.437 3 0.787 +/- 0.394 2.782 +/- 0.200 10 1.438 +/- 0.037 2.880 +/- 0.090 4 Table 1. Score function parameters were A= 1.0. B=O.O 1. Annealing runs statistics are reported when the temperature dropped below 0.0001. All the best scores and visually acceptable fits occurred in W=I runs. The average values of the data fit, weight decay, and penalty terms in the score function for W=1 were {0.378, 0.332, 0.0667} after slightly more annealing. There were a few significant similarities between the connection matrices computed in the two lowest-scoring runs. The most salient feature in the lowest-scoring network was a set of direct feedback loops among its strongest connections: cluster 8 both excited and was inhibited by cluster 10, and cluster 10 excited and was inhibited by cluster 15. This feature was preserved in the second-best run. A systematic search for "concensus circuitry" shows convergence towards a unique connection matrix for the 8-point time series data used here, but more complete 16time-point data gives mUltiple "clusters" of connection matrices. From parametercounting one might expect that making robust and unique regulatory predictions will require the use of more trajectory data taken under substantially different conditions. Such data is expected to be forthcoming. 4 Discussion We have illustrated a procedure for deriving regulatory models from large-scale gene expression data. As the data becomes more comprehensive in the number and nature of conditions under which comparable time courses are measured, this procedure can be used to determine when biological hypotheses about gene regulation can be robustly derived from the data. Acknowledgments This work was supported in part by the Whittier Foundation, the Office of Naval Research under contract NOOO 14-97-1-0422, and the NASA Advanced Concepts Program. Stuart Kim (Stanford University) provided the C. elegans gene expression array data. The GRN simulation and inference code is due in part to Charles Garrett and George Marnellos. The EM clustering code is due in part to Roberto Manduchi. 934 E. Mjolsness, T. Mann, R. Castano and B. Wold 't ~ : : : : ~i ::v:: ===:=;; J J ; : ; ; ; :~ 0.5,'-- ---'2-- , --'-3- -- .' - ----'-5- - -6'-------'-7 - - - ' 8 - 2 3 4 5 6 7 8 :~ , 2 3 4 5 6 7 8 ongIIlIII doll JIIII,1<od,.11h an ? . gm ftl JIIIIrkeel WIth an 0 Figure 3. Four clusters (numbers 9-12) of a 15-c1uster mixture of Gaussians model of 2467 genes each assayed over an eight-point time course; cluster means (shown as x) are fit to a gene regulation network model (shown as 0). References [I] D. Kosman, J. Reinitz, and D. H. Sharp, "Automated Assay of Gene Expression at Cellular Resolution " Pacific Symposium on Biocomputing '98. Eds. R. Altman, A. K. Dunker, L. Hunter, and T. E. Klein" World Scientific 1998. [2] J. L. DeRisi, V. R. IyeT, and P. O. Brown, "Exploring the Metabolic and Genetic Control of Gene Expreession on a Genomic Scale". Science 278, 680-686. [3] A Connectionist Model of Development, E. Mjolsness, D. H. Sharp, and J. Reinitz, Journal of Theoretical Biology 152:429-453, 1991. [4] J. Reinitz, E. Mjolsness, and D. H. Sharp, "Model for Cooperative Control of Positional Information in Drosophila by Bicoid and Maternal Hunchback". 1. Experimental Zoology 271:47-56, 1995. Los Alamos National Laboratory Technical Report LAUR-92-2942 1992. [5] 1. Reinitz and D. H. Sharp, "Mechanism of eve Stripe Formation". Mechanisms of Development 49: 133-158, 1995. [6] [7] J. Lam and 1. M. Delosme. "An Efficient Simulated Annealing Schedule: Derivation" and" ... Implementation and Evaluation". Technical Reports 8816 and 8817, Yale University Electrical Engineering Department, New Haven CT 1988. [8] x. Wen, S. Fuhrman, G. S. Michaels, D. B. Carr, S. Smith, J. L. Barker, and R. Somogyi, "Large-Scale Temporal Gene Expression Mapping of Central Nervous System Development", Proc. Natl. Acal. Sci. USA 95:334-339, January 1998. [9] M. B. Eisen, P. T. Spellman, P. O. Brown, and D. Botstein, "Cluster Analysis and Display of Genome-Wide Expression Patterns", Proc. Natl. Acad. Scie. USA 95:1486314868, December 1998. [10] 1. Hertz, lecture at Krogerrup Denmark computational biology summer school, July 1998. [11] Spellman PT, Sherlock G, Zhang MQ, et aI., "Comprehensive identification of cell cycle-regulated genes of the yeast Saccharomyces cerevisiae by microarray hybridization", Mol. Bio. Cell. 9: (12) 3273-3297 Dec 1998. [12] Dempster, A. P., Laird, N. M. and Rubin, D. B. "Maximum likelihood from incomplete data via the EM algorithm," 1. Royal Statistical Society, Series B, 39:1-38,1977. [13] P. Smyth, "Clustering using Monte Carlo Cross-Validation", Proceedings of the 2 nd International Conference on Knowledge Discovery and Data Mining, AAAI Press, 1996. 

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An MEG Study of Response Latency and Variability in the Human Visual System During a Visual-Motor Integration Task Akaysha C. Tang Dept. of Psychology University of New Mexico Albuquerque, NM 87131 akaysha@unm.edu Barak A. Pearlmutter Dept. of Computer Science University of New Mexico Albuquerque, NM 87131 bap@cs. unm. edu Tim A. Hely Santa Fe Institute 1399 Hyde Park Road Santa Fe, NM 87501 Michael Zibulevsky Dept. of Computer Science University of New Mexico Albuquerque, NM 87131 Michael P. Weisend VA Medical Center 1501 San Pedro SE Albuquerque, NM 87108 timhely@santafe. edu michael@cs.unm.edu mweisend@unm.edu Abstract Human reaction times during sensory-motor tasks vary considerably. To begin to understand how this variability arises, we examined neuronal populational response time variability at early versus late visual processing stages. The conventional view is that precise temporal information is gradually lost as information is passed through a layered network of mean-rate "units." We tested in humans whether neuronal populations at different processing stages behave like mean-rate "units". A blind source separation algorithm was applied to MEG signals from sensory-motor integration tasks. Response time latency and variability for multiple visual sources were estimated by detecting single-trial stimulus-locked events for each source. In two subjects tested on four visual reaction time tasks, we reliably identified sources belonging to early and late visual processing stages. The standard deviation of response latency was smaller for early rather than late processing stages. This supports the hypothesis that human populational response time variability increases from early to late visual processing stages. 1 Introduction In many situations, precise timing of a motor output is essential for successful task completion. Somehow the reliability in the output timing is related to the reliability of the underlying neural systems associated with different stages of processing. Recent literature from animal studies suggests that individual neurons from different brain regions and different species can be surprising reliable [1, 2, 5, 7-9, 14, 17, 18], A. C. Tang, B. A. Pearlmutter. T. A. Hely, M. Zibulevsky and M P. Weisend 186 on the order of a few milliseconds. Due to the low spatial resolution of electroencephalography (EEG) and the requirement of signal averaging due to noisiness of magnetoencephalography (MEG), in vivo measurement of human populational response time variability from different processing stages has not been available. In four visual reaction time (RT) tasks, we estimated neuronal response time variability at different visual processing stages using MEG. One major obstacle that has prevented the analysis of response timing variability using MEG before is the relative weakness of the brain's magnetic signals (lOOf!') compared to noise in a shielded environment (magnetized lung contaminants: 106 f!'j abdominal currents lO5f!'j cardiogram and oculogram: 104 f!'j epileptic and spontaneous activity: lO3f!') and in the sensors (10fT) [13]. Consequently, neuronal responses evoked during cognitive tasks often require signal averaging across many trials, making analysis of singletrial response times unfeasible. Recently, Bell-Sejnowski Infomax [1995] and Fast ICA [10] algorithms have been used successfully to isolate and remove major artifacts from EEG and MEG data [11, 15, 20] . These methods greatly increase the effective signal-to-noise ratio and make single-trial analysis of EEG data feasible [12]. Here, we applied a SecondOrder Blind Identification algorithm (SOBI) [4] (another blind source separation, or BSS, algorithm) to MEG data to find out whether populational response variability changes from early to late visual processing stages. 2 2.1 Methods Experimental Design Two volunteer normal subjects (females, right handed) with normal or correctedto-normal visual acuity and binocular vision participated in four different visual RT tasks. Subjects gave informed consent prior to the experimental procedure. During each task we recorded continuous MEG signals at a 300Hz sampling rate with a band-pass filter of I-100Hz using a 122 channel Neuromag-122. In all four tasks, the subject was presented with a pair of abstract color patterns, one in the left and the other in the right visual field. One of the two patterns was a target pattern. The subject pressed either a left or right mouse button to indicate on which side the target pattern was presented. When a correct response was given, a low or high frequency tone was presented binaurally following respectively a correct or wrong response. The definition of the target pattern varied in the four tasks and was used to control task difficulty which ranged from easy (task 1) to more difficult (task 4) with increasing RTs. (The specific differences among the four tasks are not important for the analysis which follows and are not discussed further.) In this study we focus on the one element that all tasks have in common, Le. activation of multiple visual areas along the visual pathways. Our goal is to identify visual neuronal sources activated in all four visual RT tasks and to measure and compare response time variability between neuronal sources associated with early and later visual processing stages. Specifically, we test the hypothesis that populational neuronal response times increase from early to later visual processing stages. 2.2 Source Separation Using SOBI In MEG, magnetic activity from different neuronal populations is observed by many senSOrs arranged around the subject's head. Each sensor responds to a mixture of the signals emitted by multiple sources. We used the Second-Order Blind Identi- MEG Study ofResponse Latency and Variability 187 fication algorithm (SOBI) [4] (a BSS algorithm) to simultaneously separate neuromagnetic responses from different neuronal populations associated with different stages of visual processing. Responses from different neuronal populations will be referred to as source responses and the neuronal populations that give rise to these responses will be referred to as neuronal sources or simply sources. These neuronal sources often, but not always, consist of a spatially contiguous population of neurons. BSS separates the measured sensor signals into maximally independent components, each having its own spatial map. Previously we have shown that some of these BSS separated components correspond to noise sources, and many others correspond to neuronal sources [19]. To establish the identity of the components, we analyzed both temporal and spatial properties of the BSS separated components. Their temporal properties are displayed using MEG images, similar to the ERP images described by [12] but without smoothing across trials. These MEG images show stimulus or response locked responses across many trials in a map, from which response latencies across all displayed trials can be observed with a glance. The spatial properties of the separated components are displayed using a field map that shows the sensor projection of a given component. The intensity at each point on the field map indicates how strongly this component influences the sensor at this location. The correspondence between the separated components and neuronal populational responses at different visual processing stages were established by considering both spatial and temporal properties of the separated components [19]. For example, a component was identified as an early visual neuronal source if and only if (1) the field pattern, or the sensor projection, of the separated component showed a focal response over the occipital lobe, and (2) the ERP image showed visual stimulus locked responses with latencies shorter than all other visual components and falling within the range of early visual responses reported in studies using other methods. Only those components consistent both spatially and temporally with known neurophysiology and neuroanatomy were identified as neuronal sources. 2.3 Single Event Detection and Response Latency Estimation For all established visual components we calculated the single-trial response latency as follows. First, a detection window was defined using the stimulus-triggered average (STA). The beginning of the detection window was defined by the time at which the STA first exceeded the range of baseline fluctuation. Baseline fluctuation was estimated from the time of stimulus onset for approximately 50ms (the visual response occurred no earlier than 60ms after stimulus onset.) The detection window ended when the STA first returned to the same level as when the detection window began. The detection threshold was determined using a control window with the same width as the detection window, but immediately preceding the detection window. The threshold was adjusted until no more than five false detections occurred within the control window for each ninety trials. We estimated RTs using the leading edge of the response, rather than the time of the peak as this is more robust against noise. 3 Results In both subjects across all four visual RT tasks, SOBI generated components that corresponded to neuronal populational responses associated with early and late stages of visual processing. In both subjects, we identified a single component with a sensor projection at the occipital lobe whose latency was the shortest among all 188 1 2 3 4 1 2 3 4 A. C. Tang, B. A. Pearlmutter, T. A. Hely, M Zibulevsky and M P. Weisend e ??? e ??? e ??? e ??? e ??? e ??? e ??? e ??? late source ? ??? e ??? -- . a? ? ??? e ??? e ??? e ??? e ??? Figure 1: MEG images and field maps for an early and a late source from each task, for subject 1 (top) and subject 2 (bottom). MEG image pixels are brightnesscoded source strength. Each row of a bitmap is one trial, running 1170ms from left to right. Vertical bars mark stimulus onset, and 333ms of pre-stimulus activity is shown. Each panel contains 90 trials. Field map brightness indicates the strength with which a source activates each of the 61 sensor pairs. visual stimulus locked components within task and subject (Fig. 1 left). We identified multiple components that had sensor projections either at occipital-parietal, occipital-temporal, or temporal lobes, and whose response latencies are longer than early-stage components within task and subject (Fig. 1 right). Fig. 2a shows examples of detected single-trial responses for one early and one late visual component (left: early; right: late) from one task. To minimize false positives, the detection threshold was set high (allowing 5 false detections out of 90 trials) at the expense of a low detection rate (15%- 67%.) When Gaussian filters were applied to the raw separated data, the detection rates were increased to 22- 91% (similar results hold but not shown) . Fig. 2b shows such detected response time histograms superimposed on the stimulus triggered average using raw separated data. One early (top row) and two late visual components (middle and bottom rows) are plotted for each of the four experiments in subject one. The histogram width is smallest for early visual components (short mean response latency) and larger for late visual components (longer latency.) We computed the standard deviation of component response times as a measure of response variability. Fig. 2c shows the response variability as a function of mean response latency for subject one. Early components (solid boxes, shorter mean latency) have smaller variability (height of the boxes) while late components (dashed boxes, longer mean latency) have larger variability (height of the boxes) . Multiple 189 MEG Study of Response Latency and Variability :bd:Qlld~~ -'~[d llJ l~J l~J -- -- -- -l~J ~~ I!] ' : ~ -- -- -. -. a 100 200 300 40Q 0 100 200 300 40Q a 100 ~- 200 300 400 0 100 ~~ 200 300 400 -~ o 100 200 300 40CI C 100 200 300 40CI 0 100 200 300 400 0 100 200 300 .wo o 100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 40 \ i : : ! : .-----------: ,------, i 1 1. ____ 1 100 150 200 latency (ms) Figure 2: (a, left) Response onset was estimated for each trial via threshold crossing within a window of eligibility. (b, top right) The stimulus-locked averages for a number of sources overlaid on histograms of response onset times. (c, bottom right) Scatter plot of visual components from all experiments on subject 1 showing the standard deviation of the latency (y axis) versus the mean latency (x axis), with the error bars in each direction indicating one standard error in the respective measurement. Lines connect sources from each task. visual components from each task are connected by a line. Four tasks were shown here. There is a general trend of increasing standard deviation of response times as a function of early-to-late processing stages (increasing mean latency from left to right). For the early visual components the standard deviation ranges from 6.6?0.63ms to 13.4?1.23ms, and for the late visual components , from 9.9?0.86ms to 38.8?3.73ms (t = 3.565, p = 0.005.) 4 Discussion By applying SOBl to MEG data from four visual RT tasks, we separated components corresponding to neuronal populational responses associated with early and 190 A. C. Tang, B. A. Pearlmutter; T. A. Hely, M Zibulevsky and M P. Weisend later stage visual processing in both subjects across all tasks. We performed singletrial RT detection on these early- and late-stage components and estimated both the mean and stdev of their response latency. We found that variability of the populational response latency increased from early to late processing stages. These results contrast with single neuron recordings obtained previously. In early and late visual processing stages, the rise time of mean firing rate in single units remained constant, suggesting an invariance in RT variability [16]. Characterizing the precise relationship between single neuron and populational response reliability is difficult without careful simulations or simultaneous single unit and MEG recording. However, some major differences exist between the two types of studies. While MEG is more likely to sample a larger neuronal population, single unit studies are more likely to be selective to those neurons that are already highly reliable in their responses to stimulus presentation. It is possible that the most reliable neurons at both the early and late processing stages are equally reliable while large differences exist between the early and late stages for the low reliability neurons. Previously, ICA algorithms have been used successfully to separate out various noise and neuronal sources in MEG data [19, 20J. Here we show that SOBI can also be used to separate different neuronal sources, particularly those associated with different processing stages. The SOBI algorithm assumes that the components are independent across multiple time scales and attempts to minimize the temporal correlation at these time scales. Although neuronal sources at different stages of processing are not completely independent as assumed in SOBl's derivation, BSS algorithms of this sort are quite robust even when the underlying assumptions are not fully met [6J, i.e. the goodness of the separation is not significantly affected. The ultimate reality check should come from satisfying physiological and anatomical constraints derived from prior knowledge of the neural system under study. This was carried out for our analysis. Firstly, the average response latencies of the separated components fell within the range of latencies reported in MEG studies using conventional source modeling methods. Secondly, the spatial patterns of sensor responses to these separated components are consistent with the known functional anatomy of the visual system. We have attempted to rule out many confounding factors. Our observed results cannot be accounted for by a higher signal to noise ratio in the early visual responses. The increase in measured onset response time variability from early to late visual processing stages was actually accompanied by an slightly lower signalto-noise ratio among the early components. The number of events detected for the later components were also slightly greater than the earlier components. The higher signal-to-noise ratio at later components should reduce noise-induced variability in the later components, which would bias against the hypothesis that later visual responses have greater response time variability. We also found that response duration and detection window size cannot account for the observed differential variabilities. Later visual responses also had gentler onset slopes (as measured by the stimulustriggered average). Sensor noise unavoidably introduces noise into the response onset detection process. We cannot rule out the possibility that the interaction of the noise with the response onset profiles might give rise to the observed differential variabilities. Similarly, we cannot rule out the possibility that even greater control of the experimental situation, such as better fixation and more effective head restraints, would differentially reduce the observed variabilities. In general, all measured variabilities can only be upper bounds, subject to downward revision as improved instrumentation and experiments become available. It is with this caution in mind that we conclude that response time variability of neuronal populations increases from early to late processing stages in the human visual system. MEG Study of Response Latency and Variability 191 Acknowledgments This research was supported by NSF CAREER award 97-02-311, and by the National Foundation for Functional Brain Imaging. References [1] M. Abeles, H. Bergman, E. Margalit, and E Vaadia. Spatiotemporal firing patterns in the frontal cortex of behaving monkeys. J. Neurophys., 70:1629-1638, 1993. [2] W. Bair and C. Koch. Temporal precision of spike trains in extrastriate cortex of the behaving macaque monkey. Neural Computation, 8(6):1184-1202, 1996. [3] A. J. Bell and T. J . Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6):1129-1159, 1995. [4] A. Belouchrani, K. A. Meraim, J.-F. Cardoso, and E. Moulines. Second-order blind separation of correlated sources. In Proc. Int. ConJ. on Digital Sig. Proc., pages 346-351, Cyprus, 1993. [5) M. J. Berry, W. K. Warland, and M. Meister. The structure and precision of retinal spike trains. Proc. Natl. Acad. Sci. USA, 94:5411-5416, 1997. [6] J .-F. Cardoso. Blind signal separation: statistical principles. Proceedings of the IEEE, 9(10):2009-2025, October 1998. [7] R. R . de Ruyter van Steveninck, G. D. Lewen, S. P. Strong, R. Koberle, and W. Bialek. Reproducibility and variability in neural spike trains. Science, 275:1805-1808, 1997. [8] R. C. deCharms and M. M. Merzenich. Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381:610-3, 1996. [9] M. Gur, A. Beylin, and D. M. Snodderly. Response variability of neurons in primary visual cortex (VI) of alert monkeys. J. Neurosci., 17(8):2914-2920, 1997. [10] A. Hyvarinen and E. Oja. A fast fixed-point algorithm for independent component analysis. Neural Computation, 9(7), October 1997. [11] T.-P. Jung, C. Humphries, T.-W. Lee, M. J. McKeown, V. Iragui, S. Makeig, and T. J. Sejnowski. Removing electroencephalographic artifacts by blind source separation. Psychophysiology, 1999. In Press. [12] T.-P. Jung, S. Makeig, M. Westerfield, J. Townsend, E. Courchesne, and T. J. Sejnowski. Analyzing and visualizing single-trial event-related potentials. In Advances in Neural Infonnation Processing Systems 11, pages 118-124. MIT Press, 1999. [13] J. D. Lewine and W. W. Orrison, II. Magnetoencephalography and magnetic source imaging. In Functional Brain Imaging, pages 369-417. Mosby, St. Louis, 1995. [14] Z. F. Mainen and T. J . Sejnowski. Reliability of spike timing in neocortical neurons. Science, 268:1503-1506, 1995. [15] S. Makeig, T.-P. Jung, A. J. Bell, D. Ghahremani, and T . J . Sejnowski. Blind separation of auditory event-related brain responses into independent components. Proc. Nat. A cad. Sci., 94:10979-84, 1997. [16] P. Marsalek, C. Koch, and J. Maunsell. On the relationship between synaptic input and spike output jitter in individual neurons. Proc. Natl. Acad. Sci., 94:735-40, 1997. [17] D. S. Reich, J. D. Victor, B. W. Knight, and T . Ozaki. Response variability and timing precision of neuronal spike trains in vivo. J. Neurophys., 77:2836-2841, 1997. [18] A. C. Tang, A. M. Bartels, and T. J . Sejnowksi. Effects of cholinergic modulation on responses of neocortical neurons to fluctuating inputs. Cereb. Cortex, 7:502-9, 1997. [19] A. C. Tang, B. A. Pearlmutter, M. Zibulevsky, and R. Loring. Response time variability in the human sensory and motor systems. In Computational Neuroscience, 1999. To appear as a special issue of Neurocomputing. [20] R. Vigario, V. Jousmaki, M. Hamruainen, R. Hari, and E. Oja. Independent component analysis for identification of artifacts in magnetoencephalographic recordings. In Advances in Neural Infonnation Processing Systems 10. MIT Press, 1998. 

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The Parallel Problems Server: an Interactive Tool for Large Scale Machine Learning Charles Lee Isbell, Jr. Parry Husbands isbell @research.att.com AT&T Labs 180 Park Avenue Room A255 Florham Park, NJ 07932-0971 PIRHusbands@lbl.gov Lawrence Berkeley National LaboratorylNERSC 1 Cyclotron Road, MS 50F Berkeley, CA 94720 Abstract Imagine that you wish to classify data consisting of tens of thousands of examples residing in a twenty thousand dimensional space. How can one apply standard machine learning algorithms? We describe the Parallel Problems Server (PPServer) and MATLAB*P. In tandem they allow users of networked computers to work transparently on large data sets from within Matlab. This work is motivated by the desire to bring the many benefits of scientific computing algorithms and computational power to machine learning researchers. We demonstrate the usefulness of the system on a number of tasks. For example, we perform independent components analysis on very large text corpora consisting of tens of thousands of documents, making minimal changes to the original Bell and Sejnowski Matlab source (Bell and Sejnowski, 1995). Applying ML techniques to data previously beyond their reach leads to interesting analyses of both data and algorithms. 1 Introduction Real-world data sets are extremely large by the standards of the machine learning community. In text retrieval, for example, we often wish to process collections consisting of tens or hundreds of thousands of documents and easily as many different words. Naturally, we would like to apply machine learning techniques to this problem; however, the sheer size of the data makes this difficult. This paper describes the Parallel Problems Server (PPServer) and MATLAB *P. The PPServer is a "linear algebra server" that executes distributed memory algorithms on large data sets. Together with MATLAB*P, users can manipulate large data sets within Matlab transparently. This system brings the efficiency and power of highly-optimized parallel computation to researchers using networked machines but maintain the many benefits of interactive environments. We demonstrate the usefulness of the PPServer on a number of tasks. For example, we perform independent components analysis on very large text corpora consisting of tens of thousands of documents with minimal changes to the original Bell and Sejnowski Matlab source (Bell and Sejnowski, 1995). Applying ML techniques to datasets previously beyond C. L. Isbell, Jr. and P. Husbands 704 Libraries Computational & Interface Routines Machine) Machin~, MatlabS Local Variables ml . . . [ ??? .. ? ??????l m, ... [ ........ ..... m, ... [ ...... .. m, ... [............. Figure 1: Use of the PPServer by Matlab is almost completely transparent. PPServer variables are tied to the PPServer itself while Matlab maintains handles to the data. Using Matlab's object system, functions using PPServer variables invoke PPServer commands implicitly. their reach, we discover interesting analyses of both data and algorithms. 2 The Parallel Problems Server The Parallel Problems Server (PPServer) is the foundation of this work. The PPServer is a realization of a novel client-server model for computation on very large matrices. It is compatible with any Unix-like platform supporting the Message Passing Interface (MPI) library (Gropp, Lusk and Skjellum, 1994). MPI is the standard for multi-processor communication and is the most portable way for writing parallel code. The PPServer implements functions for creating and removing distributed matrices, loading and storing them from/to disk using a portable format, and performing elementary matrix operations. Matrices are two-dimensional single or double precision arrays created on the PPServer itself (functions are provided for transferring matrix sections to and from a client) . The PPServer supports both dense and sparse matrices. The PPServer communicates with clients using a simple request-response protocol. A client requests an action by issuing a command with the appropriate arguments, the server executes that command, and then notifies the client that the command is complete. The PPServer is directly extensible via compiled libraries called packages. The PPServer implements a robust protocol for communicating with packages. Clients (and other packages) can load and remove packages on-the-fty, as well as execute commands within packages. Package programmers have direct access to information about the PPServer and its matrices. Each package represents its own namespace, defining a set of visible function names. This supports data encapsulation and allows users to hide a subset of functions in one package by loading another that defines the same function names. Finally, packages support common parallel idioms (eg applying a function to every element of a matrix), making it easier to add common functionality. All but a few PPServer commands are implemented in packages, including basic matrix operations. Many highly-optimized public libraries have been realized as packages using appropriate wrapper functions. These packages include ScaLAPACK (Blackford et a1., 1997), S3L (Sun's optimized version of ScaLAPACK), PARPACK (Maschhoff and Sorensen, 1996), and PETSc (PETSc, ). Large Scale Machine Learning Using The Parallel Problems Server 1 function H=hilb(n) l:nj 2 J 3 J = J (ones (n, 1) , : ) I J' j 4 5 E = ones (n, n) j 6 H = E. / (I+J-1) j 705 j Figure 2: Matlab code for producing Hilbert matrices. When n is influenced by P, each of the constructors creates a PPServer object instead of a Matlab object. 3 MATLAB*P By directly using the PPServer's client communication interface, it is possible for other applications to use the PPServer's functionality. We have implemented a client interface for Matlab, called MA1LAB*P. MATLAB*P is a collection of Matlab 5 objects, Matlab m-files (Matlab's scripting language) and Matlab MEX programs (Matlab's extemallanguage API) that allows for the transparent integration of Matlab as a front end for the Parallel Problems Server. The choice of Matlab was influenced by several factors. It is the de facto standard for scientific computing, enjoying wide use in industry and academia. In the machine learning community, for example, algorithms are often written as Matlab scripts and made freely available. In the scientific computing community, algorithms are often first prototyped in Matlab before being optimized for languages such as Fortran. We endeavor to make interaction with the PPServer as transparent as possible for the user. In principle, a typical Matlab user should never have to make explicit calls to the PPServer. Further, current Matlab programs should not have to be rewritten to take advantage of the PPServer. Space does not permit a complete discussion of MA1LAB*P (we refer the reader to (Husbands and Isbell, 1999)); however, we will briefly discuss how to use prewritten Matlab scripts without modification. This is accomplished through the simple but innovative P notation. We use Matlab 5's object oriented features to create PPServer objects automatically. P is a special object we introduce in Matlab that acts just like the integer 1. A user typing a=ones (1000*P, 1000) or b=rand( 1000, 1000*P) obtains two 1000-by-lOOO matrices distributed in parallel. The reader can guess the use of P here: it indicates that a is distributed by rows and b by columns. To a user, a and b are matrices, but within Matlab, they are handles to special distributed types that exist on the PPServer. Any further references to these variables (e.g. via such commands as eig, svd, inv, *, +, -) are recognized as a call to the PPServer rather than as a traditional Matlab command. Figure 2 shows the code for Matlab's built in function hilb. The call hilb (n) produces the n x n Hilbert matrix (H ij = i+}-l)' When n is influenced by P, a parallel array results: ? J=l: n in line 2 creates the PPServer vector 1,2, ? . . , n and places a handle to it in J. Note that this behavior does not interfere with the semantics of for loops (for i=l: n) as Matlab assigns to i the value of each column of 1: n: the numbers 1,2, ... , n. ? ones (n, 1) in line 3 produces a PPServer matrix. ? Emulation of Matlab's indexing functions results in the correct execution of line 3. C. L. Isbell, Jr. and P. Husbands 706 ? Overloading of ' (the transpose operator) executes line 4 on the PPServer. ? In line 5, E is generated on the PPServer because ofthe overloading of ones. ? Overloading elementary matrix operations makes H a PPServer matrix (line 6). The Parallel Problems Server and MA1LAB *p have been tested extensively on a variety of platforms. They currently run on Cray supercomputers! , clusters of symmetric multiprocessors from Sun Microsystems and DEC as well as on clusters of networked Intel PCs. The PPServer has also been tested with other clients, including Common LISP. Although computational performance varies depending upon the platform, it is clear that the system provides distinct computational advantages. Communication overhead (in our experiments, roughly two milliseconds per PPServer command) is negligible compared to the computational and space advantage afforded by transparent access to highly-optimized linear algebra algorithms. 4 Applications in Text Retrieval In this section we demonstrate the efficacy of the PPServer on real-world machine learning problems. In particular we explore the use of the PPServer and MA1LAB*P in the text retrieval domain. The task in text retrieval is to find the subset of a collection of documents relevant to a user's information request. Standard approaches are based on the Vector Space Model (VSM). A document is a vector where each dimension is a count of the occurrence of a different word. A collection of documents is a matrix, D, where each column is a document vector di. The similarity between tw~ documents is their inner product, dj. Queries are just like documents, so the relevance of documents to a query, q, is DT q. Jf Typical small collections contain a thousand vectors in a ten thousand dimensional space, while large collections may contain 500,000 vectors residing in hundreds of thousands of dimensions. Clearly, well-understood standard machine learning techniques may exhibit unpredictable behavior under such circumstances, or simply may not scale at all. Classically, ML-like approaches try to construct a set of linear operators which extract the underlying "topic" structure of documents. Documents and queries are projected into that new (usually smaller) space before being compared using the inner product. The large matrix support in MA1LAB*P enables us to use matrix decomposition techniques for extracting linear operators easily. We have explored several different algorithms(Isbell and Viola, 1998). Below, we discuss two standard algorithms to demonstrate how the PPServer allows us to perform interesting analysis on large datasets. 4.1 Latent Semantic Indexing Latent Semantic Indexing (LSI) (Deerwester et al., 1990) constructs -a smaller document matrix by using the Singular Value Decomposition (SVD): D = U SV T . U contains the eigenvectors of the co-occurrence matrix while the diagonal elements of S (referred to as singular values) contain the square roots of their corresponding eigenValUes. The eigenvectors with the largest eigenvalues capture the axes of largest variation in the data. LSI projects documents onto the k-dimensional subspace spanned by the first k columns of U (denoted Uk) so thatthe documents are now: V[ = S;;lUk. Queries are similarly projected. Thus, the document-query scores for LSI can be obtained with simple Matlab code: 1 Although there is no Matlab for the Cray, we are still able to use it to "execute" Matlab code -in parallel. 707 Large Scale Machine Learning Using The Parallel Problems Server '" .. . IG . ,, ------------ - ---- - - Figure 3: The first 200 singular values of a collection of about 500,000 documents and 200,000 terms, and singular values for half of that collection. Computation for on the full collection took only 62 minutes using 32 processors on a Cray TIE. D=dsparse('term-doc'); %D SPARSE reads a sparse matrix Q=dsparse('queries'); [U,S,VJ=svds(D,k); % compute the k-SVD of D sc=getlsiscores(U,S,V,Q); % computes v*(l/s)*u'*q The scores that are returned can then be combined with relevance judgements to obtain precision/recall curves that are displayed in Matlab: r=dsparse('judgements'); [pr,reJ=precisionrecall(sc,r); plot (re ( , @' ) , pr ( , @' ) ) ; In addition to evaluating the performance of various techniques, we can also explore characteristics of the data itself. For example, many implementations of LSI on large collections use only a subset of the documents for computational reasons. This leads one to question how the SVD is affected. Figure 3 shows the first singular values for one large collection as well as for a random half of that collection. It shows that the shape of the curves are remarkably similar (as they are for the other half). This suggests that we can derive a projection matrix from just half of the collection. An evaluation of this technique can easily be performed using our system. Prernlinary experiments show nearly identical retrieval performance. 4.2 What are the Independent Components of Documents? Independent components analysis (ICA)(Bell and Sejnowski, 1995) also recovers linear projections from data. Unlike LSI, which finds principal components, ICA finds axes that are statistically independent. ICA's biggest success is probably its application to the blind source separation or cocktail party problem. In this problem, one observes the output of a number of microphones. Each microphone is assumed to be recording a linear mixture of a number of unknown sources. The task is to recover the original sources. There is a natural embedding of text retrieval within this framework. The words that are observed are like microphone signals, and underlying ''topics'' are the source signals that give rise to them. Figure 4 shows a typical distribution of words projected along axes found by ICA. 2 Most words have a value close to zero. The histogram shows only the words large positive or 2These results are from a collection containing transcripts of White House press releases from 1993. There are 1585 documents and 18,675 distinct words. C. L. Isbell, Jr. and P. Husbands 708 africa apartheid I -1 anc transition mandela continent elite ethiopia saharan -0.75 0.5 0.75 Figure 4: Distribution of words with large magnitude an ICA axis from White House text. negative values. One group of words is made up of highly-related terms; namely, "africa," "apartheid," and "mandela." The other group of words are not directly related, but each cooccurs with different individual words in the first group. For example, "saharan" and "africa" occur together many times, but not in the context of apartheid and South Africa; rather, in documents concerning US policy toward Africa in general. As it so happens, "saharan" acts as a discriminating word for these SUbtopics. As observed in (Isbell and Viola, 1998), it appears that ICA is finding a set of words, S, that selects for related documents, H, along with another set of words, T, whose elements do not select for H, but co-occur with elements of S. Intuitively, S selects for documents in a general subject area, and T removes a specific subset of those documents, leaving a small set of highly related documents. This suggests a straightforward algorithm to achieve the same goal directly. This local clustering approach is similar to an unsupervised version of Rocchio with Query Zoning (Singhal, 1997). Further analysis of ICA on similar collections reveals other interesting behavior on large datasets. For example, it is known that ICA will attempt to find an unmixing matrix that is full rank. This is in conflict with the notion that these collections actually reside in a much smaller subspace. We have found in our experiments with ICA that some axes are highly kurtotic while others produce gaussian-like distributions. We conjecture that any axis that results in a gaussian-like distribution will be split arbitrarily among all "empty" axes. For all intents and purposes, these axes are uninformative. This provides an automatic noisereduction technique for ICA when applied to large datasets. For the purposes of comparison, Figure 5 illustrates the performance of several algorithms (including ICA and various clustering techniques) on articles from the Wall Street Journal.3 5 Discussion We have shown that MATLAB *p enables portable, high-performance interactive supercomputing using the Parallel Problems Server, a powerful mechanism for writing and accessing optimized algorithms. Further, the client communication protocol makes it possible to implement transparent integration with sufficiently powerful clients, such as Matlab 5. With such a tool, researchers can now use Matlab as something more than just a way for prototyping algorithms and working on small problems. MATLAB*P makes it possible to interactively operate on and visualize large data sets. We have demonstrated this last claim by using the PPServer system to apply ML techniques to large datasets, allowing for analyses of both data and algorithms. MATLAB*P has also been used to implement versions of Diverse Density(Maron, 1998), MIMIC(DeBonet, Isbell and Viola, 1996), and gradient descent. 3The WSJ collection contains 42,652 documents and 89,757 words Large Scale Machine Learning Using The Parallel Problems Server - - - _ . - - T.... " " - 709 .71~~~-~;:=======il - 0.0 - --- l.SI ~_a.... ....... DocuMnIIt_ a....... teA ", Recall Figure 5: A comparison of different algorithms on the Wall Street Journal References Bell, A. and Sejnowski, T. (1995). An infonnation-maximizaton approach to blind source separation and blind deconvolution. Neural Computation, 7:1129-1159. Blackfor<~, L. S., Choi, J., Cleary, A, D' Azevedo, E., Demmel, J., Dhilon, I., Dongarra, 1., Hammarling, S. , Henry, G., Petitet, A, Stanley, K, Walker, D., and Whaley, R. (1997). ScaLAPACK Users' Guide. http://www.netlib.orglscalapacklsluglscalapack..slug.htrnl. DeBonet, J., Isbell, C., and Viola, P. (1996). Mimic: Finding optima by estimating probability densities. In Advances in Neural Information Processing Systems. Deerwester, S., Dumais, S. T., Landauer, T. K, Furnas, G. w., and Harshman, R. A (1990). Indexing by latent semantic analysis. Journal of the Society for Information Science, 41(6):391-407. Frakes, W. B. and Baeza-Yates, R., editors (1992). Information Retrieval: Data Structures and Algorithms. Prentice-Hall. Gropp, W., Lusk, E., and Skjellum, A (1994). Using MPI: Portable Parallel Programming with the Message-Passing Interface. The MIT Press. Husbands, P. and Isbell, C. (1999). MITMatlab: A tool for interactive supercomputing. In Proceedings of the Ninth SIAM Conference on Parallel Processingfor Scientific Computing. Isbell, C. and Viola, P. (1998). Restructuring sparse high dimensional data for effective retrieval. In Advances in Neural Information Processing Systems. Kwok, K L. (1996). A new method of weighting query tenns for ad-hoc retrieval. In Proceedings of the 19th ACMlSIGIR Conference,pages 187-195. Maron, O. (1998). A framework for multiple-instance learning. In Advances in Neural Information Processing Systems . Maschhoff, K J. and Sorensen, D. C. (1996). A Portable Implementation of ARPACK for Distributed Memory Parallel Computers. In Preliminary Proceedings of the Copper Mountain Conference on Iterative Methods. O'Brien, G. W. (1994). Infonnation management tools for updating an svd-encoded indexing scheme. Technical Report UT-CS-94-259, University of Tennessee. PETSc. The Portable, Extensible http://www.mcs.anl.gov/home/group/petsc.htrnl. Toolkit for Scientific Computation. PPServer. The Parallel Problems Server Web Page. http://www.ai.mit.edulprojects/ppserver. Sahami, M ., Hearst, M., and Saund, E. (1996). Applying the multiple cause mixture model to text categorization. In Proceedings of the 13th International Machine Learning Conference. Singhal, A (1997). Learning routing queries in a query zone. In Proceedings of the 20th International Conference on Research and Development in Information Retrieval. 

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Bayesian averaging is well-temperated Lars Kai Hansen Department of Mathematical Modelling Technical University of Denmark B321 DK-2800 Lyngby, Denmark lkhansen@imm .dtu.dk Abstract Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. While a simple variational argument shows that Bayes averaging is generalization optimal given that the prior matches the teacher parameter distribution the situation is less clear if the teacher distribution is unknown. I define a class of averaging procedures, the temperated likelihoods, including both Bayes averaging with a uniform prior and maximum likelihood estimation as special cases. I show that Bayes is generalization optimal in this family for any teacher distribution for two learning problems that are analytically tractable: learning the mean of a Gaussian and asymptotics of smooth learners. 1 Introduction Learning is the stochastic process of generalizing from a random finite sample of data. Often a learning problem has natural quantitative measure of generalization. If a loss function is defined the natural measure is the generalization error, i.e., the expected loss on a random sample independent of the training set. Generalizability is a key topic of learning theory and much progress has been reported. Analytic results for a broad class of machines can be found in the litterature [8, 12, 9, 10] describing the asymptotic generalization ability of supervised algorithms that are continuously parameterized. Asymptotic bounds on generalization for general machines have been advocated by Vapnik [11]. Generalization results valid for finite training sets can only be obtained for specific learning machines, see e.g. [5]. A very rich framework for analysis of generalization for Bayesian averaging and other schemes is defined in [6]. A veraging has become popular as a tool for improving generalizability of learning machines . In the context of (time series) forecasting averaging has been investigated intensely for decades [3]. Neural network ensembles were shown to improve generalization by simple voting in [4] and later work has generalized these results to other types of averaging. Boosting, Bagging, Stacking, and Arcing are recent examples of averaging procedures based on data resampling that have shown useful see [2] for a recent review with references. However, Bayesian averaging in particular is attaining a kind of cult status. Bayesian averaging is indeed provably optimal in a L. K. Hansen 266 number various ways (admissibility, the likelihood principle etc) [1]. While it follows by construction that Bayes is generalization optimal if given the correct prior information, i.e., the teacher parameter distribution, the situation is less clear if the teacher distribution is unknown. Hence, the pragmatic Bayesians downplay the role of the prior. Instead the averaging aspect is emphasized and "vague" priors are invoked. It is important to note that whatever prior is used Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. In this contribution I analyse two scenarios where averaging can improve generalizability and I show that the vague Bayes average is in fact optimal among the averaging schemes investigated. Averaging is shown to reduce variance at the cost of introducing bias, and Bayes happens to implement the optimal bias-variance trade-off. 2 Bayes and generalization Consider a model that is smoothly parametrized and whose predictions can be described in terms of a density function 1 . Predictions in the model are based on a given training set: a finite sample D = {Xa}~=l of the stochastic vector x whose density - the teacher - is denoted p(xIOo). In other words the true density is assumed to be defined by a fixed, but unknown, teacher parameter vector 00 . The model, denoted H, involves the parameter vector and the predictive density is given by p(xID, H) = ! ? p(xIO, H)p(OID, H)dO (1) p(OID, H) is the parameter distribution produced in training process. In a maximum likelihood scenario this distribution is a delta function centered on the most likely parameters under the model for the given data set. In ensemble averaging approaches, like boosting bagging or stacking, the distribution is obtained by training on resampled traning sets. In a Bayesian scenario, the parameter distribution is the posterior distribution, p(DIO, H)p(OIH) p(OID, H) = f p(DIO', H)p(O'IH)dO' (2) where p(OIH) is the prior distribution (probability density of parameters if D is empty). In the sequel we will only consider one model hence we suppress the model conditioning label H. The generalization error is the average negative log density (also known as simply the "log loss" - in some applied statistics works known as the "deviance") r(DIOo) = ! -logp(xID)p(xIOo)dx, (3) The expected value of the generalization error for training sets produced by the given teacher is given by f(Oo) = !! -logp(xID)p(xIOo)dxp(DIOo)dD. (4) lThis does not limit us to conventional density estimation; pattern recognition and many functional approximations problems can be formulated as density estimation problems as well. 267 Bayesian Averaging is Well-Temperated Playing the game of "guessing a probability distribution" [6] we not only face a random training set, we also face a teacher drawn from the teacher distribution p( Bo) . The teacher averaged generalization must then be defined as r= J f(Bo)p(Bo)dBo . (5) This is the typical generalization error for a random training set from the randomly chosen teacher - produced by the model H. The generalization error is minimized by Bayes averaging if the teacher distribution is used as prior. To see this, form the Lagrangian functional ?[q(xID)] = JJJ -logq(xID)p(xIBo)dxp(DIBo)dDp(Bo)dBo+A J q(xID)dx (6) defined on positive functions q(xID). The second term is used to ensure that q(xID) is a normalized density in x . Now compute the variational derivative to obtain 6? 1 6q(xID) = - q(xID) J p(xIBo)p(DIBo)p(Bo)dBo + A. (7) Equating this derivative to zero we recover the predictive distribution of Bayesian averagmg, p(DIB)p(B) (8) q(xID) = p(xIB) Jp(DIB')p(B')dB' dB, J =J where we used that A p(DIB)p(B)dB is the appropriate normalization constant. It is easily verified that this is indeed the global minimum of the averaged generalization error. We also note that if the Bayes average is performed with another prior than the teacher distribution p( Bo), we can expect a higher generalization error . The important question from a Bayesian point of view is then: Are there cases where averaging with generic priors (e.g. vague or uniform priors) can be shown to be optimal? 3 Temperated likelihoods To come closer to a quantative statement about when and why vague Bayes is the better procedure we will analyse two problems for which some analytical progress is possible. We will consider a one-parameter family of learning procedures including both a Bayes and the maximum likelihood procedure, v(DIB) p(BI!3,D,H) = Jpf3(DIB')dB" (9) where !3 is a positive parameter (plying the role of an inverse temperature). The family of procedures are all averaging procedures, and !3 controls the width of the average. Vague Bayes (here used synonymously with Bayes with a uniform prior) is recoved for !3 = 1, while the maximum posterior procedure is obtained by cooling to zero width !3 --+ 00 . In this context the generalization design question can be frased as follows : is there an optimal temperature in the family of the temperated likelihoods? 3.1 Example: ID normal variates Let the teacher distribution be given by p(xIBo) = ~exp (-~(X 211"<72 2<7 - Bo)2) (10) L.K. Hansen 268 The model density is of the same form with (J unknown and u 2 assumed to be known. For N examples the posterior (with a uniform prior) is, p(OID) = J 2:U2 exp (- ::2 (11) (x - (J)2) , = with x 1/ N Eo: Xo:. The temperated likelihood is obtained by raising to the ,8'th power and normalizing, p((JID,,8) = Vf7iN ~ exp (,8N - 2u 2 (x - (J)2 ) . (12) The predictive distribution is found by integrating w.r.t. (J, p(xID,,8) = !P(ZIB)P(BID,~)dB; ~exp (--212 (x- X)2) 21!'u$ uf3 , (13) with u~ = u 2(1+1/,8N). We note that this distribution is wider for all the averaging procedures than it is for maximum likelihood (,8 -T (0), i.e., less variant. For very small ,8 the predictive distribution is almost independent of the data set, hence highly biased. It is straightforward to compute the generalization error of the predictive distribution for general,8. First we compute the generalization error for the specific training set D, r(D,,8, (Jo) = ! -logp(xID, ,8)p(xl(Jo)dx = log )21!'u$ + 2~~ ((x - (JO)2 + ( 2) , (14) The average generalization error is then found by averaging w.r.t the sampling distribution using x"" N((Jo, u 2/N)., r(,8) = ! r(D, ,8)dDp(DI(Jo) = log )21!'u$ + 2:$ (~ + 1) , (15) We first note that the generalization error is independent of the teacher (Jo parameter, this happened because (J is a "location" parameter. The ,8-dependency of the averaged generalization error is depicted in Figure 1. Solving 8r(,8) /8,8 = 0 we find that the optimal ,8 solves u$=U2(,8~+I)=U2(~+I) :=} ,8=1 (16) Note that this result holds for any N and is independent of the teacher parameter. The Bayes averaging at unit temperature is optimal for any given value of (Jo, hence, for any teacher distribution. We may say that the vague Bayes scheme is robust to the teacher distribution in this case. Clearly this is a much stronger optimality than the more general result proven above. 3.2 Bias-variance tradeoff It is interesting to decompose the generalization error in Eq. 15 in bias and variance components. We follow Heskes [7] and define the bias error as the generalization error of the geometric average distribution, B(,8) ! = -logp(x)p(xl(Jo)dx, (17) 269 Bayesian Averaging is Wel/-Temperated 0.7 0.& GENERALIZATION 0.5 A 1? 04 ?, v 03 02 0.1 0 0 0.5 15 2 25 3 35 45 TEMPERATURE Figure 1: Bias-variance trade-off as function of the width of the temperated likelihood ensemble (temperature = 1/(3) for N = 1. The bias is computed as the generalization error of the predictive distribution obtained from the geometric average distribution w.r.t. training set fluctuations as proposed by Heskes. The predictive distribution produced by Bayesian averaging corresponds to unit temperature (vertical line) and it achieves the minimal generalization error. Maximum-likelihood estimation for reference is recovered as the zero width/temperature limit. with p(x) = Z-l exp ( / 10g(P(X 1D)]P(D I 80 )dD) . (18) Inserting from Eq. (13), we find p(z) = ~exp (-~(X 27r0'~ 0'f3 - 0)2) . 8 (19) Integrating over the teacher distribution we find, 1 B(f3) = -2 log 27r0'~ 0'2 + -2 20'f3 (20) The variance error is given by V(f3) = r(f3) - B(f3) , 0'2 V (f3) = 2N O'~ (21) We can now quantify the statements above. By averaging a bias is introduced -the predictive distribution becomes wider- which decrease the variance contribution initially so that the generalization error being the sum of the two decreases. At still higher temperatures the bias becomes too strong and the generalization error start to increase. The Bayes average at unit temperature is the optimal trade-off within the given family of procedures. 270 3.3 L. K. Hansen Asymptotics for smoothly parameterized models We now go on to show that a similar result also holds for general learning problems in limit of large data sets. We consider a system parameterized by a finite dimensional parameter vector O. For a given large training set and for a smooth likelihood function, the temperated likelihood is approximately Gaussian centered at the maximum posterior parameters[13]' hence the normalized temperated posterior reads P(OI(3D,H) = I(3NA(~OML) lexp (_(3; 60'A(D,OML)60) (22) where 60 = O-OML, with OML = OML(D) denoting the maximum likelihood solution for the given training sample. The second derivative or Hessian matrix is given by 1 N A(D,O) N LA(xa,O) (23) a=l {)2 A(x,O) = ()O{)O' - log p( x 10 ) (24) p(xIO)p(OI(3, D)dO (25) The predictive distribution is given by p(xl(3, D) = ! we write p(xIO) = exp(-f(xIO)) and expand f(xIO) around OML to second order, we find p(xIO) ~ p(XIOML) exp (-a(xIOML)'60 - ~60' A(xIOML)60) . (26) We are then in position to perform the integration over the posterior to find the normalized predictive distribution, p(xl(3, D) = p(XIOML) I(3N A(D)I 1 , I(3NA(D) + A(x)1 exp ( 2'a(xIOML) A(xIOML)a(xIOML)). (27) Proceeding as above, we compute the generalization error f((3, ( 0 ) = !! -logp(xl(3, D)p(xIOo)dxp(DIOo)dD (28) For sufficiently smooth likelihoods, fluctuations in the maximum likelihood parameters will be asymptotic normal, see e.g. [8], and furthermore fluctuations in A(D) can be neglected, this means that we can approximate, A(x) + A(D) ~ (~ + l)Ao, Ao = ! A(xIOo)p(xIOo)dx (29) where Ao is the averaged Fisher information matrix. With these approximations (valid as N --+ (0) the generalization error can be found, 1) d 1+ ~ d ( f((3, ( 0 ) ~ f(oo) + 2 log 1 + (3N - 21 + (3N' (30) with d = dim(O) denoting the dimension of the parameter vector. Like in the ID example (Eq. (15)) we find the generalization error is asymptotically independent of the teacher parameters. It is minimized for (3 1 and we conclude that Bayes is well-temperated in the asymptotics and that this holds for any teacher distribution. In the Bayes literature this is refered to as the prior is overwhelmed by data [1]. Decomposing the errors in bias and variance contributions we find similar results as for in ID example, Bayes introduces the optimal bias by averaging at unit temperature. = Bayesian Averaging is Well-Temperated 4 271 Discussion We have seen two examples of Bayes averaging being optimal, in particular improving on maximum likelihood estimation. We found that averaging introduces a bias and reduces variance so that the generalization error (being the sum of bias and variance) initially decrease. Bayesian averaging at unit temperature is the optimal width of the averaging distribution. For larger temperatures (widths) the bias is too strong and the generalization error increases. Both examples were special in the sense that they lead to generalization errors that are independent of the random teacher parameter. This is not generic, of course, rather the generic case is that a mis-specified prior can lead to arbitrary large learning catastrophes. Acknowledgments I thank the organizers of the 1999 Max Planck Institute Workshop on Statistical Physics of Neural Networks Michael Biehl, Wolfgang Kinzel and Ido Kanter, where this work was initiated. I thank Carl Edward Rasmussen, Jan Larsen, and Manfred Opper for stimulating discussions on_Bayesian averaging. This work was funded by the Danish Research Councils through the Computational Neural Network Center CONNECT and the THOR Center for Neuroinformatics. References [1] C.P. Robert: The Bayesian Choice - A Decision- Theoretic Motivation. Springer Texts in Statistics, Springer Verlag, New York (1994). A. Ohagan: Bayesian Inference. Kendall's Advanced Theory of Statistics. Vol 2B. The University Press, Cambridge (1994). [2] L. Breiman: Using adaptive bagging to debias regressions. Technical Report 547, Statistics Dept. U.C. Berkeley, (1999) . [3] R.T. Clemen Combining forecast: A review and annotated bibliography. Journal of Forecasting 5, 559 (1989). [4] L.K. Hansen and P. Salamon: Neural Network Ensembles. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 993-1001 (1990). [5] L.K . Hansen: Stochastic Linear Learning: Exact Test and Training Error Averages. Neural Networks 6, 393-396, (1993) [6] D. Haussler and M. Opper: Mutual Information, Metric Entropy, and Cumulative Relative Entropy Risk Annals of Statistics 25 2451-2492 (1997) [7] T . Heskes: Bias/Variance Decomposition for Likelihood-Based Estimators. Neural Computation 10, pp 1425-1433, (1998) . [8] L. Ljung: System Identification: Theory for the User. Englewood Cliffs, New Jersey: Prentice-Hall, (1987). [9] J . Moody: "Note on Generalization, Regularization, and Architecture Selection in Nonlinear Learning Systems," in B.H. Juang, S.Y. Kung & C.A. Kamm (eds.) Proceedings of the first IEEE Workshop on Neural Networks for Signal Processing, Piscataway, New Jersey: IEEE, 1-10, (1991). [10] N . Murata, S. Yoshizawa & S. Amari: Network Information Criterion - Determining the Number of Hidden Units for an Artificial Neural Network Model. IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 865-872 , 1994. [11] V. Vapnik: Estimation of Dependences Based on Empirical Data. Springer-Verlag New York (1982). [12] H . White, "Consequences and Detection of Misspecified Nonlinear Regression Models," Journal of the American Statistical Association, 76(374), 419-433, (1981). [13] D .J .C MacKay: Bayesian Interpolation, Neural Computation 4, 415-447, (1992) . 

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366 NEURONAL MAPS FOR SENSORY-MOTOR CONTROL IN THE BARN OWL C.D. Spence, J.C. Pearson, JJ. Gelfand, and R.M. Peterson David Sarnoff Research Center Subsidiary of SRI International CN5300 Princeton, New Jersey 08543-5300 W.E. Sullivan Department of Biology Princeton University Princeton, New Jersey 08544 ABSTRACT The bam owl has fused visual/auditory/motor representations of space in its midbrain which are used to orient the head so that visual or auditory stimuli are centered in the visual field of view. We present models and computer simulations of these structures which address various problems, inclu<lln~ the construction of a map of space from auditory sensory information, and the problem of driving the motor system from these maps. We compare the results with biological data. INTRODUCTION Many neural network models have little resemblance to real neural structures, partly because the brain's higher functions, which they attempt to imitate, are not yet experimentally accessible. Nevertheless, some neural-net researchers are finding that the accessible structures are interesting, and that their functions are potentially useful. Our group is modeling a part of the barn owl's nervous system which orients the head to visual and auditory stimuli. The bam owl's brain stem and midbrain contains a system that locates visual and auditory stimuli in space. The system constructs an auditory map of spatial direction from the non-spatial information in the output of the two cochlea. This map is in the external nucleus of the inferior colliculus, or ICx [Knudsen and Konishi, 1978]. The ICx, along with the visual system, projects to the optic tectum, producing a fused visual and auditory map of space [Knudsen and Knudsen, 1983]. The map in the tectum is the source of target position used by the motor system for orienting the head. In the last futeen years, biologists have determined all of the structures in the system which produces the auditory map of space in the ICx. This system provides sev- Neuronal Maps for Sensory-Motor Coritrol in the Barn Owl eral examples of neuronal maps, regions of tissue in which the response properties of neurons vary continuously with position in the map. (For reviews, see Knudsen, 1984; Knudsen, du Lac, and Esterly, 1987; and Konishi, 1986.) Unfortunately, the motor system and the projections from the tectum are not well known, but experimental study of them has recently begun [Masino and Knudsen, 1988]. We should eventually be able to model a complete system, from sensory input to motor output In this paper we present several models of different parts of the head orientation system. Fig. 1 is an overview of the structures we'll discuss. In the second section of this paper we discuss models for the construction of the auditory space map ~ the lex. In the third section we discuss how the optic tectum might drive the motor system. CONSTRUCTION OF AN AUDITORY MAP OF SPACE The bam owl uses two binaural cues to locate objects in space: azimuth is derived from inter-aural time or phase delay (not onset time difference), while elevation is derived from inter-aural intensity difference (due to vertical asymmetries in sensitivAcoustic Core of the ICc ~ Lateral Shell of the ICc ~ External Nucleus of the IC (ICx) Optic Tectum ~ Retina Figure 1. Overview of the neuronal system for target localization in the barn owl (head orients towards potential targets for closer scrutiny). The illustration focuses on the functional representations of the neuronal computation, and does not show all of the relevant connections. The grids represent the centrally synthesized neuronal maps and the patterns within them indicate possible patterns of neuronal activation in response to acoustic stimuli. 367 368 Spence, Pearson, Gelfand, Peterson and Sullivan ity). Corresponding to these two cues are two separate processing streams which produce maps of the binamal cues, which are shown in Figs. 2-5. The information on these maps must be merged in order to construct the map of space in the ICx. lTD ---. f lex :~:~: .~ ~ tiS ~-----~ ~ t _ _ _.-...IID Azimuth Figure 2. Standard Model for the construction of an auditory map of space from maps of the binaural cues. Shading represents activity level. lID is Inter-aural Intensity Difference, lTD is Inter-aural Time Delay. A simple model for combining the two maps is shown in Fig. 2. It has not been described explicitly in the literature, but it has been hinted at [Knudsen, et ai, 1987]. For this reason we have called it the standard model. Here all of the neurons representing a given time delay or azimuth in the lTD vs. frequency map project to all of the neurons representing that azimuth in the space map. Thus a stimulus with a certain lTD would excite a strip of cells representing the associated azimuth and all elevations. Similarly, all of the neurons representing a given intensity difference or elevation in the lID vs. frequency map project to all of the neurons representing that elevation in the space map. (TIle map of lID vs. frequency is constructed in the nucleus ventralis lemnisci lateralis pars posterior, or VLVp. VLVp neurons are said to be sensitive to intensity difference, that is they fire if the intensity difference is great enough. Neurons in the VLVp are spatially organized by their intensity difference threshold [Manley. et ai, 1988]. Thus, intensity difference has a bar-chart-like representation, and our model needs some mechanism to pick out the ends of the bars.) Only the neurons at the intersection of the two strips will fire if lateral inhibition allows only those neurons receiving the most stimulation to fue. In the third section we will present a model for connections of inhibitory inter-neurons which can be applied to this model. Part of the motivation for the standard model is the problem with phase ghosts. Phase ghosts occur when the barn owl's nervous system incorrectly associates the wave fronts arriving at the two ears at high frequency. In this case, neurons in the map of lTD vs. frequency become active at locations representing a time delay which Neuronal Maps for Sensory-Motor Control in the Barn Owl differs from the true time delay by an integer multiple of the period of the sound. Because the period varies inversely with the frequency, these phase ghosts will have apparent time delays that vary with frequency. Thus, for stimuli that are not pure tones, if the bam owl can compare the activities in the map at different frequencies, it can eliminate the ghosts. The standard model does this (Fig. 2). In the lTD vs. frequency map there are more neurons fIring at the position of the true lTD than at the ghost positions, so space map neurons representing the true position will receive the most stimulation. Only those neurons representing the true position will fIre because of the lateral inhibition. There is another kind of ghost which we call the multiple-source ghost (Fig. 3). If two sounds occur simultaneously, then space map neurons representing the time delay of one source and the intensity difference of the other will receive a large amount of stimulation. Lateral inhibition may suppress these ghosts, but if so, the owl should only be able to locate one source at a time. In addition, the true sources might be suppressed. The bam owl may actually suffer from this problem, although it seems unlikely if the owl has to function in a noisy environment The relevant behavioral experiments have not yet been done. Experimental evidence does not support the standard model. The ICx receives most of its input from the lateral shell of the central nucleus of the inferior colliculus lTD --.. ?t . 4~ ::. - / / ., ~Cx 1- "' ~" i/" /: ~ " ',/ I t liD Azimuth Figure 3. Multiple-source ghosts in the standard model for the construction of the auditory space map. For clarity, only two pure tone stimuli are represented, and their frequencies and locations are such that the "phase ghost" problem is not a factor. The black squares represent regions of cells that are above threshold. The circled regions are those that are fIring in response to the lTD of one stimulus and the lID of another. These regions correspond to phantom targets. 369 370 Spence, Pearson, Gelfand, Peterson and Sullivan (lateral shell of the ICc) [Knudsen, 1983]. Neurons in the lateral shell are tuned to frequency and time delay, and these parameters are mapped [Wagner. Takahashi. and Konishi, 1987]. However, they are also affected by intensity difference [Knudsen and Konishi. 1978, I. Fujita, private communication]. Thus the lateral shell does not fit the picture of the input to the ICx in the standard model. rather it is some intermediate stage in the processing. We have a model, called the lateral shell model. which does not suffer from multiple-source ghosts (Fig. 4). In this model, the lateral shell of the ICc is a tbreedimensional map of frequency vs. intensity difference vs. time delay. A neuron in the map of time delay vs. frequency in the ICc core projects to all of the neurons in lTD lID lex Azimuth Figure 4. Lateral shell model for the construction of the auditory map of space in the ICx. f: frequency. lTD: inter-aural time delay. lID: inter-aural intensity difference. Neuronal Maps for Sensory-Motor Control in the Barn Owl the three-dimensional map which represent the same time delay and frequency. As in the standard model, a strip of neurons is stimulated, but now the frequency tuning is preserved. The map of intensity difference vs. frequency in the nucleus ventralis lemnisci lateralis pars posterior (VLVp) [Manley, et al, 1988] projects to the threedimensional map in a similar fashion. Lateral inhibition picks out the regions of intersection of the strips. Neurons in the space map in the ICx receive input from the strip of neurons in the three-dimensional map which represent the appropriate time delay and intensity difference, or equivalently azimuth and elevation. Phase ghosts will be present in the three-dimensional map, but in the ICx lateral inhibition will suppress them. Multiple-source ghosts are eliminated in the lateral shell model because the sources are essentially tagged by their frequency spectra. If two sources with no common frequencies are present, there are no neurons which represent the time delay of one source, the intensity difference of another, and a frequency common to both. In the more likely case in which some frequencies are common to both sources, there will be fewer neurons fIring at the ghost positions than at the real positions, so again lateral inhibition in the ICx can suppress fIring at the ghost positions, exactly as it suppresses the phase ghosts. The fact that intensity and time delay information is combined before frequency tuning is lost in the lex suggests that the owl handles multiple-source ghosts by frequency tagging. A three dimensional map is not essential, but it is conceptually simple. Before ending this section, we should mention that others have independently thought of this model. In particular, M. Konishi and co-workers have looked for a spatial organization or mapping of intensity response properties in the lateml shell, but they have not found it They also have said that they can't yet rule it out [M. Konishi, I. Fujita, private communication]. DRMNG THE MOTOR SYSTEM FROM MAPS As mentioned before, all of the parts of the auditory system in the bmin stem and midbrain are known, up to the optic tectum. The optic tectum has a map of spatial direction which is common to both the visual and auditory systems. In addition, it drives the motor system, so if the tectum is stimulated at a point, the bam owl's head will move to face a certain direction. The new orientation is mapped in register with the auditory/visual map of spatial direction, e.g., stimulating a location which represents a stimulus eight degrees to the right of the current orientation of the head will cause the head to turn eight degrees to the right Little is known about the projections of the tectum, although work has started [Masino and Knudsen, 1988]. There was one earlier experiment. Two electrodes were placed in one of the tecta at positions representing sensory stimuli eight degrees and sixty degrees toward the side of the head opposite to this tectum. When either position was stimulated by itself, the alert owl moved its head as expected, by eight or 371 372 Spence, Pearson, Gelfand, Peterson and Sullivan sixty degrees. When both were stimulated together. the head moved about forty degrees [du Lac and Knudsen. 1987]. The averaging of activity in the tectum is easy to explain. In some motor models it should be produced naturally by the activation of an agonist-antagonist muscle pair (see. for example. Grossberg and Kuperstein. 1986). In the presence of two stimuli. the tension in each muscle is the sum of the tensions it would have for either stimulus alone. so the equilibrium position should be about the average position. We have a different model. which produces a map of the nection strengths from tectal cells to an averaging map with the difference in represented direction. A quadratic is very broad. however. so lateral inhibition is required fairly narrow. ? 1-D tectum - . 25 average position. The concell decrease quadratically distribution of stimulation to make the active region ? . . .... .... '" . ".;- . -.rI''....' .... I.' . o 0 .... . ' ' .-.' ..' . . . 0 . -.- -. . 00 , .......... ,.. -.. .. . .. . . o o oo. ? _ 0 ? . ? '. 0 .. . . o 0 . '. ......'" . '.. . . .. .- . .. _". I...... '. _ ? ??? ??? o~----------------~~~------------------------~ Position Figure 5. Averaging map simulation. The upper thin rectangle is a one-dimensional version of the space map in the optic tectum. The two marks represent the position of stimulating electrodes that are simultaneously active. The lower rectangle is the averaging map. with position represented horizontally and time increasing vertically. The squares represent cell ftrings. Note that the activity quickly becomes centered at the average position of the two active positions in the tectum. We have simulated a one-dimensional version of this model. with 128 cells in the tectum. and in the averaging map. 128 excitatory and 128 inhibitory cells. The excitatory cells and the inhibitory cells both receive the same quadratically weighted input from the tectum. Each inhibitory cell inhibits all of the other cells in the averaging map. both excitatory and inhibitory. except for those in a small local neighborhood. (The weights are actually proportional to one minus a gaussian with a maximum of one.) An excitatory cell receives exactly the same input as the inhibitory Neuronal Maps for Sensory-Motor Control in the Barn Owl cell at the same location, so their voltages are the same. Because of this we only show the excitatory cells in Fig. 5. This figure shows cell position horizontally and time increasing vertically. The black squares are plotted at the time a given cell frres. We are interested in whether an architecture like this is biologically plausible. For this reason we have tried to be fairly realistic in our model. The cells obey a membrane equation: dv - = -gl (v - VI) - ge (v - v ) - g. (v - v.) C dt el I in which C is the capacitance, the g's are conductances, I refers to leakage quantities. e refers to excitatory quantities, and i refers to inhibitory quantities. The output of a cell is not a real number, but spikes that occur randomly at an average rate that is a monotonic function of the voltage. We used the usual sigmoidal function in this simulation, although the membrane equation automatically limits the voltage and hence the firing rate. A cell that spikes on a given time step affects other cells by affecting the appropriate conductance. To get the effect of a post synaptic potential, the conductances obey the equation of a damped harmonic oscillator: d 2g _y dg -2 -- (If-ro g - dt 2 When a spike from an excitatory cell arrives, we increment the time derivative of g by some amount If the oscillator is overdamped or critically damped, the conductance goes up for a time and then decreases, approaching zero exponentially. We are not suggesting that a damped harmonic oscillator exists in the membranes of neurons, but it efficiently models the dynamics of synaptic transmission. The equations for the conductances also have the nice property that the effects of multiple spikes at different times add. With values for the cell parameters that agree well with experimental data, it takes about twenty milliseconds for the simulated map to settle into a fairly steady state, which is a reasonable time for the function of this map. Also, there was no need to frne tune the parameters; within a fairly wide range the effect of changing them is to change the width of the region of activity. We tried another architecture for the inhibitory intemeurons, in which they received their input from the excitatory neurons and did not inhibit other inhibitory neurons. The voltages in this architecture oscillated for a very long time, without picking out a maximum. The architecture we are now using is apparently superior. Since it is quick to pick out a maximum of a broad distribution of stimulation, it should work very well in other models requiring lateral inhibition. such as the lateral shell model discussed earlier. 373 374 Spence, Pearson, Gelfand, Peterson and Sullivan CONCLUSION We have presented models for two parts of the barn owl's visual/auditory localization and head orientation system. These models make experimentally testable predictions. and suggest architectures for artificial systems. One model constructs a map of stimulus position from maps of inter-aural intensity and timing differences. This model solves potential problems with ghosts. i.e.. the representation of false sources in the presence of certain kinds of real sources. Another model computes the average value of a quantity represented on a neural map when the activity on the map has a complex distribution. This model explains recent physiological experiments. A simulation with fairly realistic model neurons has shown that a biological structure could perform this function in this way. A common feature of these models is the use of neuronal maps. We have only mentioned a few of the maps in the barn owl. and they are extremely common in other nervous systems. We think this architecture shows great promise for applications in artificial processing systems. Acknowledgments This work was supported by internal funds of the David Sarnoff Research Center. References Grossberg. S. and M. Kuperstein (1986) Neural dynamics of adaptive motor control. North-Holland. Knudsen. E.I. (1983) J. Compo Neurology. 218:174-186. Knudsen. E.I. (1984) in Dynamical Aspects of Neocortical Function. G.M. Edelman, W.E. Gall, and W.M. Cowan, editors, Wiley, New York. Knudsen, E.I. andP.F. Knudsen (1983) J. Compo Neurology, 218:187-196. Knudsen, E.I. and M. Konishi (1978) J. Neurophys .? 41:870-884. Knudsen. E.!.. S. du Lac. and S. Esterly (1987) Ann. Rev. Neurosci., 10:41-56. Konishi. M. (1986) Trends in Neuroscience. April. du Lac. S. and E.!. Knudsen (1987) Soc. for Neurosci. Abstr., 112.10. Manley. G.A.? A.C. Koeppl. and M. Konishi (1988) J. Neurosci. 8:2665-2676 Masino. T .? and E.I. Knudsen (1988) Soc. for Neurosci. Abstr., 496.16. Wagner. H.? T. Takahashi. and M. Konishi (1987) J. Neurosci. 10:3105-3116 

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Learning to Parse Images Geoffrey E. Hinton and Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London London, United Kingdom WC1N 3AR {hinton,zoubin}@gatsby.ucl.ac.uk Vee Whye Tah Department of Computer Science University of Toronto Toronto, Ontario, Canada M5S 3G4 ywteh@cs.utoronto.ca Abstract We describe a class of probabilistic models that we call credibility networks. Using parse trees as internal representations of images, credibility networks are able to perform segmentation and recognition simultaneously, removing the need for ad hoc segmentation heuristics. Promising results in the problem of segmenting handwritten digits were obtained. 1 Introd uction The task of recognition has been the main focus of attention of statistical pattern recognition for the past 40 years. The paradigm problem is to classify an object from a vector of features extracted from the image. With the advent of backpropagation [1], the choice of features and the choice of weights to put on these features became part of a single, overall optimization and impressive performance was obtained for restricted but important tasks such as handwritten character identification [2]. A significant weakness of many current recognition systems is their reliance on a separate preprocessing stage that segments one object out of a scene and approximately normalizes it. Systems in which segmentation precedes recognition suffer from the fact that the segmenter does not know the shape of the object it is segmenting so it cannot use shape information to help it. Also, by segmenting an image, we remove the object to be recognized from the context in which it arises. Although this helps in removing the clutter present in the rest of the image, it might also reduce the ability to recognize an object correctly because the context in which an object arises gives a great deal of information about the nature of the object. Finally, each object can be described in terms of its parts, which can also be viewed as objects in their own right. This raises the question of how fine-grained the segmentations should be. In the words of David Marr: "Is a nose an object? Is a head one? ... What about a man on a horseback?" [3]. G. E. Hinton, Z. Ghahramani and Y. W. Teh 464 The successes of structural linguistics inspired an alternative approach to pattern recognition in which the paradigm problem was to parse an image using a hierarchical grammar of scenes and objects. Within linguistics, the structural approach was seen as an advance over earlier statistical approaches and for many years linguists eschewed probabilities, even though it had been known since the 1970's that a version of the EM algorithm could be used to fit stochastic context free grammars. Structural pattern recognition inherited the linguists aversion to probabilities and as a result it never worked very well for real data. With the advent of graphical models it has become clear that structure and probabilities can coexist. Moreover, the "explaining away" phenomenon that is central to inference in directed acyclic graphical models is exactly what is needed for performing inferences about possible segmentations of an image. In this paper we describe an image interpretation system which combines segmentation and recognition into the same inference process. The central idea is the use of parse trees of images. Graphical models called credibility networks which describe the joint distribution over the latent variables and over the possible parse trees are used. In section 2 we describe some current statistical models of image interpretation. In section 3 we develop credibility networks and in section 4 we derive useful learning and inference rules for binary credibility networks. In section 5 we demonstrate that binary credibility networks are useful in solving the problem of classifying and segmenting binary handwritten digits. Finally in section 6 we end with a discussion and directions for future research. 2 Related work Neal [4] introduced generative models composed of multiple layers of stochastic logistic units connected in a directed acyclic graph. In general, as each unit has multiple parents, it is intractable to compute the posterior distribution over hidden variables when certain variables are observed. However, Neal showed that Gibbs sampling can be used effectively for inference [4]. Efficient methods of approximating the posterior distribution were introduced later [5, 6, 7] and these approaches were shown to yield good density models for binary images of handwritten digits [8] . The problem with these models which make them inappropriate for modeling images is that they fail to respect the 'single-parent' constraint: in the correct interpretation of an image of opaque objects each object-part belongs to at most one object - images need parse trees, not parse DAGs. Multiscale models [9] are interesting generative models for images that use a fixed tree structure. Nodes high up in the tree control large blocks of the image while bottom level leaves correspond to individual pixels. Because a tree structure is used, it is easy to compute the exact posterior distribution over the latent (non-terminal) nodes given an image. As a result, the approach has worked much better than Markov random fields which generally involve an intractable partition function . A disadvantage is that there are serious block boundary artifacts, though overlapping trees can be used to smooth the transition from one block to another [10]. A more serious disadvantage is that the tree cannot possibly correspond to a parse tree because it is the same for every image. Zemel, Mozer and Hinton [11] proposed a neural network model in which the activities of neurons are used to represent the instantiation parameters of objects or their parts, Le. the viewpoint-dependent coordinate transformation between an object's intrinsic coordinate system and the image coordinate system. The weights on connections are then used to represent the viewpoint-invariant relationship between the instantiation parameters of a whole, rigid object and the instantiation parame- Learning to Parse Images 465 ters of its parts. This model captures viewpoint invariance nicely and corresponds to the way viewpoint effects are handled in computer graphics, but there was no good inference procedure for hierarchical models and no systematic way of sharing modules that recognize parts of objects among multiple competing object models. Simard et al [12] noted that small changes in object instantiation parameters result in approximately linear changes in (real-valued) pixel intensities. These can be captured successfully by linear models. To model larger changes, many locally linear models can be pieced together. Hinton, Dayan and Revow [13] proposed a mixture of factor analyzers for this. Tipping and Bishop have recently shown how to make this approach much more computationally efficient [14]. To make the approach really efficient, however, it is necessary to have multiple levels of factor analyzers and to allow an analyzer at one level to be shared by several competing analyzers at the next level up. Deciding which subset of the analyzers at one level should be controlled by one analyzer at the level above is equivalent to image segmentation or the construction of part of a parse tree and the literature on linear models contains no proposals on how to achieve this. 3 A new approach to image interpretation We developed a class of graphical models called credibility networks in which the possible interpretations of an image are parse trees, with nodes representing objectparts and containing latent variables. Given a DAG, the possible parse trees of an image are constrained to be individual or collections of trees where each unit satisfies the single-parent constraint, with the leaves being the pixels of an image. Credibility networks describe a joint distribution over the latent variables and possible tree structures. The EM algorithm [15] can be used to fit credibility networks to data. Let i E I be a node in the graph. There are three random variables associated with i. The first is a multinomial variate ,xi = {.xij hEpa(i) which describes the parent of i from among the potential parents pa(i) : ,x . . _ 1J - {I if parent of i is j, 0 if parent of i is not j. (1) The second is a binary variate Si which determines whether the object i l is present (Si = 1) or not (Si = 0). The third is the latent variables Xi that describe the pose and deformation of the object. Let A = {,xi : i E I}, S = {Si : i E I} and X = {Xi: i E I}. Each connection j -+ i has three parameters also. The first, Cij is an unnormalized prior probability that j is i's parent given that object j is present. The actual prior probability is 7rij = C" S ' 1J J (2) 2:kEPa(i) CikSk We assume there is always a unit 1 E pa(i) such that SI = 1. This acts as a default parent when no other potential parent is present and makes sure the denominator in (2) is never O. The second parameter, Pij, is the conditional probability that object i is present given that j is i's parent (,xij = 1). The third parameter tij characterizes the distribution of Xi given ,xij = 1 and Xj' Let 0 = {Cij,Pij, tij : i E I,j E pa(i)}. Using Bayes' rule the joint distribution over A, S and X given 0 is peA, S, XIO) = peA, SIO)p(XIA, S, 0). Note that A and S together define a parse tree for the image. Given the parse tree the distribution over latent variables p(XIA, S, 0) can be ITechnically this should be the object represented by node i. G. E. Hinton, Z. Ghahramani and Y. W. Teh 466 efficiently inferred from the image. The actual form of p(XIA, S, 0) is unimportant. The joint distribution over A and S is = II P(A,SIO) II (3) (1I'ijP:j(1- Pij)I-8 i )Ai j iEi jEpa(i) 4 Binary credibility networks The simulation results in section 5 are based on a simplified version of credibility networks in which the latent variables X are ignored. Notice that we can sum out A from the joint distribution (3), so that P(SIO) = II L 1I'ijP:j (1 - (4) Pij)I-8 i iEi jEpa(i) Using Bayes' rule and dividing (3) by (4), we have II II ( P(AIS, 0) = iEi jEpa(i) CijSjP:j (1: Pij)I-8 i I:kEPa(i) cik s kPik(l - Pik) 1-8') (5) Aij ? )1-8 .. Let rij = CijP:j (1 - Pij We can view rij as the unnormalized posterior probability that j is i's parent given that object j is present. The actual posterior is the fraction in (5) : (6) Given some observations 0 c S, let 11 = S \ 0 be the hidden variables. We approximate the posterior distribution for 11 using a factored distribution Q(ll) = II O':i (1 - O'i)1-8 i (7) iEi EQ[-log P(SIO) + log Q(S)] is L(EQ[IOg L cijsj-Iog L The variational free energy, F(Q,O) = F(Q,O) = Ci jSjP:j(1-Pi j )1-8i]) iEi L jEpa(i) (O'i log O'i + (1 - + jEpa(i) O'i) log (1 - (8) O'i)) iEi The negative of the free energy -F is a lower bound on the log likelihood of generating the observations O. The variational EM algorithm improves this bound by iteratively improving -F with respect to Q (E-step) and to 0 (M-step). Let ch(i) be the possible children of i. The inference rules can be derived from (8) : EQ [log L O'i CijSjPij -log jEpa(i) = sigmoid + L EQ [log L lEch( i) L cijsj(l - Pij)] jEpa(i) jEpa(l) rljSj -log L J:::: (9) CljSj jEpa(I)' Let D be the training set and Qd be the mean field approximation to the posterior distribution over 11 given the training data (observation) dE D. Then the learning Learning to Parse Images 467 Figure 1: Sample images from the test set. The classes of the two digits in each image in a row are given to the left. rules are (10) (11) For an efficient implementation of credibility networks using mean field approximations, we still need to evaluate terms of the form E[logx] and ~[l/x] where x is a weighted sum of binary random variates. In our implementation we used the simplest approximations: E[logx] ~ 10gE[x] and E[l/x] ~ l/E[x]. Although biased the implementation works well enough in general. 5 Segmenting handwritten digits Hinton and Revow [16] used a mixture of factor analyzers model to segment and estimate the pose of digit strings. When the digits do not overlap, the model was able to identify the digits present and segment the image easily. The hard cases are those in which two or more digits overlap significantly. To assess the ability of credibility networks at segmenting handwritten digits, we used superpositions of digits at exactly the same location. This problem is much harder than segmenting digit strings in which digits partially overlap. The data used is a set of 4400 images of single digits from the classes 2, 3, 4 and 5 derived from the CEDAR CDROM 1 database [17]. Each image has size 16x16. The size of the credibility network is 256-64-4. The 64 middle layer units are meant to encode low level features, while each of the 4 top level units are meant to encode a digit class. We used 700 images of single digits from each class ? to train the network. So it was not trained to segment images. During training we clamped at 1 the activation of the top layer unit corresponding to the class of the digit in the current image while fixing the rest at O. After training, the network was first tested on the 1600 images of single digits not in the training set. The predicted class of each image was taken to be the G. E. Hinton, Z. Ghahramani and Y. W. Teh 468 Figure 2: Segmentations of pairs of digits. (To make comparisons easier we show the overlapping image in both columns of a)-I).) class corresponding to the top layer unit with the highest activation. The error rate was 5.5%. We then showed the network 120 images of two overlapping digits from distinct classes. There were 20 images per combination of two classes. Some examples are given in Figure 1. The predicted classes of the two digits are chosen to be the corresponding classes of the 2 top layer units with the highest activations. A human subject (namely the third author) was tested on the same test set. The network achieved an error rate of 21.7% while the author erred on 19.2% of the images. We can in fact produce a segmentation of each image into an image for each class present. Recall that given the values of S the posterior probability of unit j being pixel i's parent is Wij. Then the posterior probability of pixel i belonging to digit class k is L: j EQ[WijWjk). This gives a simple way to segment the image. Figure 2 shows a number of segmentations. Note that for each pixel, the sum of the probabilities of the pixel belonging to each digit class is 1. To make the picture clearer, a white pixel means a probability of :::; .1 of belonging to a class, while black means ~ .6 probability, and the intensity of a gray pixel describes the size of the probability if it is between .1 and .6. Figures 2a) to 2f) shows successful segmentations, while Figure 2g) to 21) shows unsuccessful segmentations. 6 Discussion Using parse trees as the internal representations of images, credibility networks avoid the usual problems associated with a bottom-up approach to image interpretation. Segmentation can be carried out in a statistically sound manner, removing the need for hand crafted ad hoc segmentation heuristics. The granularity problem for segmentation is also resolved since credibility networks use parse trees as internal representations of images. The parse trees describe the segmentations of the image at every level of granularity, from individual pixels to the whole image. We plan to develop and implement credibility networks in which each latent variable Xi is a multivariate Gaussian, so that a node can represent the position, orientation and scale of a 2 or 3D object, and the conditional probability models on the links can represent the relationship between a moderately deformable object and its parts. Learning to Parse Images 469 Acknowledgments We thank Chris Williams, Stuart Russell and Phil Dawid for helpful discussions and NSERC and ITRC for funding. References [1] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing: Explorations in The Microstructure of Cognition. Volume 1 : Foundations. The MIT Press, 1986. [2] Y. Le Cun, B. Boser, J. S. Denker, S. SoH a, R. E. Howard, and L. D. Jackel. Back-propagation applied to handwritten zip code recognition. Neural Computation, 1(4):541-551, 1989. [3] D. Marr. Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. W. H. Freeman and company, San Francisco, 1980. [4] R. M. Neal. Connectionist learning of belief networks. Artificial Intelligence, 56:71113, 1992. [5] P. Dayan, G. E. Hinton, R. M. Neal, and R. S. Zemel. Helmholtz machines. Neural Computation, 7:1022-1037, 1995. [6] G. E. Hinton, P. Dayan, B. J. Frey, and R. M. Neal. The wake-sleep algorithm for self-organizing neural networks. Science, 268:1158-1161, 1995. [7] L. K. Saul and M. I. Jordan. Attractor dynamics in feedforward neural networks. Submitted for publication. [8] B. J. Frey, G. E. Hinton, and P. Dayan. Does the wake-sleep algorithm produce good density estimators? In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 8. The MIT Press, 1995. [9] M. R. Luettgen and A. S. Willsky. Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination. IEEE Transactions on Image Processing, 4(2):194-207, 1995. [10] W. W. Irving, P. W. Fieguth, and A. S. Willsky. An overlapping tree approach to multiscale stochastic modeling and estimation. IEEE Transactions on Image Processing, 1995. [11] R. S. Zemel, M. C. Mozer, and G. E. Hinton. TRAFFIC: Recognizing objects using hierarchical reference frame transformations. In Advances in Neural Information Processing Systems, volume 2. Morgan Kaufmann Publishers, San Mateo CA, 1990. [12] P. Simard, Y. Le Cun, and J. Denker. Efficient pattern recognition using a new transformation distance. In S. Hanson, J. Cowan, and L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufmann Publishers, San Mateo CA, 1992. [13] G. E. Hinton, P. Dayan, and M. Revow. Modeling the manifolds of images of handwritten digits. IEEE Transactions on Neural Networks, 8:65-74, 1997. [14] M. E. Tipping and C. M. Bishop. Mixtures of probabilistic principal component analysis. Technical Report NCRG/97/003, Aston University, Department of Computer Science and Applied Mathematics, 1997. [15] A.P. Dempster, N.M. Laird, and D.B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B, 39:1-38, 1977. [16] G. E. Hinton and M. Revow. Using mixtures of factor analyzers for segmentation and pose estimation, 1997. [17] J. J. Hull. A database for handwritten text recognition research. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(5):550-554, 1994. 

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Probabilistic methods for Support Vector Machines Peter Sollich Department of Mathematics, King's College London Strand, London WC2R 2LS, U.K. Email: peter.sollich@kcl.ac.uk Abstract I describe a framework for interpreting Support Vector Machines (SVMs) as maximum a posteriori (MAP) solutions to inference problems with Gaussian Process priors. This can provide intuitive guidelines for choosing a 'good' SVM kernel. It can also assign (by evidence maximization) optimal values to parameters such as the noise level C which cannot be determined unambiguously from properties of the MAP solution alone (such as cross-validation error) . I illustrate this using a simple approximate expression for the SVM evidence. Once C has been determined, error bars on SVM predictions can also be obtained. 1 Support Vector Machines: A probabilistic framework Support Vector Machines (SVMs) have recently been the subject of intense research activity within the neural networks community; for tutorial introductions and overviews of recent developments see [1, 2, 3]. One of the open questions that remains is how to set the 'tunable' parameters of an SVM algorithm: While methods for choosing the width of the kernel function and the noise parameter C (which controls how closely the training data are fitted) have been proposed [4, 5] (see also, very recently, [6]), the effect of the overall shape of the kernel function remains imperfectly understood [1]. Error bars (class probabilities) for SVM predictions important for safety-critical applications, for example - are also difficult to obtain. In this paper I suggest that a probabilistic interpretation of SVMs could be used to tackle these problems. It shows that the SVM kernel defines a prior over functions on the input space, avoiding the need to think in terms of high-dimensional feature spaces. It also allows one to define quantities such as the evidence (likelihood) for a set of hyperparameters (C, kernel amplitude Ko etc). I give a simple approximation to the evidence which can then be maximized to set such hyperparameters. The evidence is sensitive to the values of C and Ko individually, in contrast to properties (such as cross-validation error) of the deterministic solution, which only depends on the product CKo. It can thfrefore be used to assign an unambiguous value to C, from which error bars can be derived. P. Sollich 350 I focus on two-class classification problems. Suppose we are given a set D of n training examples (Xi, Yi) with binary outputs Yi = ?1 corresponding to the two classes. The basic SVM idea is to map the inputs X onto vectors c/>(x) in some high-dimensional feature space; ideally, in this feature space, the problem should be linearly separable. Suppose first that this is true. Among all decision hyperplanes w?c/>(x) + b = 0 which separate the training examples (Le. which obey Yi(W'c/>(Xi) + b) > 0 for all Xi E D x , Dx being the set of training inputs), the SVM solution is chosen as the one with the largest margin, Le. the largest minimal distance from any of the training examples. Equivalently, one specifies the margin to be one and minimizes the squared length of the weight vector IIwI1 2 [1], subject to the constraint that Yi(W'c/>(Xi) + b) 2:: 1 for all i. If the problem is not linearly separable, 'slack variables' ~i 2:: 0 are introduced which measure how much the margin constraints are violated; one writes Yi(W'c/>(Xi) + b) 2:: 1 - ~i' To control the amount of slack allowed, a penalty term C Ei ~i is then added to the objective function ~ IIwI1 2, with a penalty coefficient C. Training examples with Yi(w ?c/>(xd + b) 2:: 1 (and hence ~i = 0) incur no penalty; all others contribute C[l - Yi(W ' c/>(Xi) + b)] each. This gives the SVM optimization problem: Find wand b to minimize ~llwl12 + C Ei l(Yi[W ' c/>(Xi) + b]) (1) where l(z) is the (shifted) 'hinge loss', l(z) = (1- z)8(1- z). To interpret SVMs probabilistically, one can regard (1) as defining a (negative) log-posterior probability for the parameters wand b of the SVM, given a training set D. The first term gives the prior Q(w,b) "" exp(-~llwW - ~b2B-2). This is a Gaussian prior on W; the components of W are uncorrelated with each other and have unit variance. I have chosen a Gaussian prior on b with variance B2; the flat prior implied by (1) can be recovered! by letting B -+ 00. Because only the 'latent variable' values O(x) = w?c/>(x) + b - rather than wand b individually - appear in the second, data dependent term of (1), it makes sense to express the prior directly as a distribution over these. The O(x) have a joint Gaussian distribution because the components ofw do, with covariances given by (O(x)O(x')) = ((c/>(x) ?w) (w?c/>(x'))) + B2 = c/>(x)?c/>(x') + B2. The SVM prior is therefore simply a Gaussian process (GP) over the functions 0, with covariance function K(x,x') = c/>(x) ?c/>(x') + B2 (and zero mean). This correspondence between SVMs and GPs has been noted by a number of authors, e.g. [6, 7, 8, 9, 10J . The second term in (1) becomes a (negative) log-likelihood if we define the probability of obtaining output Y for a given X (and 0) as Q(y =?llx, 0) = ~(C) exp[-Cl(yO(x))] (2) We set ~(C) = 1/[1 + exp(-2C)] to ensure that the probabilities for Y ?1 never add up to a value larger than one. The likelihood for the complete data set is then Q(DIO) = It Q(Yilxi, O)Q(Xi), with some input distribution Q(x) which remains essentially arbitrary at this point. However, this likelihood function is not normalized, because lI(O(x)) = Q(llx, 0) + Q( -llx, 0) = ~(C){ exp[ -Cl(O(x))] + exp[-Cl( -O(x))]} < 1 lIn the probabilistic setting, it actually makes more sense to keep B finite (and small); for B -+ 00, only training sets with all Yi equal have nonzero probability. 351 Probabilistic Methods for Support Vector Machines except when IO(x)1 = 1. To remedy this, I write the actual probability model as P(D,9) = Q(DI9)Q(9)/N(D) . (3) Its posterior probability P(9ID) '" Q(DI9)Q(9) is independent Qfthe normalization factor N(D); by construction, the MAP value of 9 is therefore the SVM solution. The simplest choice of N(D) which normalizes P(D, 9) is D-independent: N = Nn = Jd9Q(9)Nn(9), N(9) = JdxQ(x)lI(O(x)). (4) Conceptually, this corresponds to the following procedure of sampling from P(D, 9): First, sample 9 from the GP prior Q(9) . Then, for each data point, sample x from Q(x). Assign outputs Y = ?1 with probability Q(ylx,9), respectively; with the remaining probability l-lI(O(x)) (the 'don't know' class probability in [11]), restart the whole process by sampling a new 9. Because lI(O(x)) is smallest 2 inside the 'gap' IO(x)1 < 1, functions 9 with many values in this gap are less likely to 'survive' until a dataset of the required size n is built up. This is reflected in an n-dependent factor in the (effective) prior, which follows from (3,4) as P(9) '" Q(9)Nn(9). Correspondingly, in the likelihood P(ylx,9) = Q(ylx, 9)/1I(O(x)), P(xI9) '" Q(x) lI(O(x)) (5) (which now is normalized over y = ?1), the input density is influenced by the function 9 itself; it is reduced in the 'uncertainty gaps' IO(x)1 < 1. To summarize, eqs. (2-5) define a probabilistic data generation model whose MAP solution 9* = argmax P(9ID) for a given data set D is identical to a standard SVM. The effective prior P(9) is a GP prior modified by a data set size-dependent factor; the likelihood (5) defines not just a conditional output distribution, but also an input distribution (relative to some arbitrary Q(x)). All relevant properties of the feature space are encoded in the underlying GP prior Q(9), with covariance matrix equal to the kernel K(x, Xl). The log-posterior of the model In P(9ID) = -t J dx dxl O(X)K-l(X, Xl) O(XI) - C 'Ei l(YiO(xi)) + const (6) is just a transformation of (1) from wand b to 9. By differentiating w.r.t. the O(x) for non-training inputs, one sees that its maximum is of the standard form O*(x) = Ei (}:iYiK(X, Xi); for YiO*(Xi) > 1, < 1, and = lone has (}:i = 0, (}:i = C and (}:i E [0, C] respectively. I will call the training inputs Xi in the last group marginal; they form a subset of all support vectors (the Xi with (}:i > 0). The sparseness of the SVM solution (often the number of support vectors is ? n) comes from the fact that the hinge loss l(z) is constant for z > 1. This contrasts with other uses of GP models for classification (see e.g. [12]), where instead of the likelihood (2) a sigmoidal (often logistic) 'transfer function' with nonzerO gradient everywhere is used. Moreover, in the noise free limit, the sigmoidal transfer function becomes a step function, and the MAP values 9* will tend to the trivial solution O*(x) = O. This illuminates from an alternative point of view why the margin (the 'shift' in the hinge loss) is important for SVMs. Within the probabilistic framework, the main effect of the kernel in SVM classification is to change the properties of the underlying GP prior Q(9) in P(9) '" 2This is true for C > In 2. For smaller C, v( O( x? is actually higher in the gap, and the model makes less intuitive sense. P. Sollich 352 (e) (h) Figure 1: Samples from SVM priors; the input space is the unit square [0,1]2. 3d plots are samples 8(x) from the underlying Gaussian process prior Q(8). 2d greyscale plots represent the output distributions obtained when 8(x) is used in the likelihood model (5) with C = 2; the greyscale indicates the probability of y = 1 (black: 0, white: 1). (a,b) Exponential (Ornstein-Uhlenbeck) kernel/covariance function Koexp(-Ix - x/l/l), giving rough 8(x) and decision boundaries. Length scale l = 0.1, Ko = 10. (c) Same with Ko = 1, i.e. with a reduced amplitude of O(x); note how, in a sample from the prior corresponding to this new kernel, the grey 'uncertainty gaps' (given roughly by 18(x)1 < 1) between regions of definite outputs (black/white) have widened. (d,e) As first row, but with squared exponential (RBF) kernel Ko exp[-(x - X I )2/(2l 2)], yielding smooth 8(x) and decision boundaries. (f) Changing l to 0.05 (while holding Ko fixed at 10) and taking a new sample shows how this parameter sets the typical length scale for decision regions. (g,h) Polynomial kernel (1 + x?xl)P, with p = 5; (i) p = 10. The absence of a clear length scale and the widely differing magnitudes of 8(x) in the bottom left (x = [0,0]) and top right (x = [1,1]) corners of the square make this kernel less plausible from a probabilistic point of view. 353 Probabilistic Methods for Support Vector Machines Q(O)Nn(o). Fig. 1 illustrates this with samples from Q(O) for three different types of kernels. The effect of the kernel on smoothness of decision boundaries, and typical sizes of decision regions and 'uncertainty gaps' between them, can clearly be seen. When prior knowledge about these properties of the target is available, the probabilistic framework can therefore provide intuition for a suitable choice of kernel. Note that the samples in Fig. 1 are from Q(O), rather than from the effective prior P(O). One finds, however, that the n-dependent factor Nn(o) does not change the properties of the prior qualitatively3. 2 Evidence and error bars Beyond providing intuition about SVM kernels, the probabilistic framework discussed above also makes it possible to apply Bayesian methods to SVMs. For example, one can define the evidence, i.e. the likelihood of the data D, given the model as specified by the hyperparameters C and (some parameters defining) K(x, x'). It follows from (3) as P(D) = Q(D)/Nn, Q(D) = JdO Q(DIO)Q(O). (7) The factor Q(D) is the 'naive' evidence derived from the unnormalized likelihood model; the correction factor Nn ensures that P(D) is normalized over all data sets. This is crucial in order to guarantee that optimization of the (log) evidence gives optimal hyperparameter values at least on average (M Opper, private communication). Clearly, P(D) will in general depend on C and K(x,x') separately. The actual SVM solution, on the other hand, i.e. the MAP values 0*, can be seen from (6) to depend on the product C K (x, x') only. Properties of the deterministically trained SVM alone (such as test or cross-validation error) cannot therefore be used to determine C and the resulting class probabilities (5) unambiguously. I now outline how a simple approximation to the naive evidence can be derived. Q(D) is given by an integral over all B(x), with the log integrand being (6) up to an additive constant. After integrating out the Gaussian distributed B( x) with x ? Dx , an intractable integral over the B(Xi) remains. However, progress can be made by expanding the log integrand around its maximum B*(Xi)' For all non-marginal training inputs this is equivalent to Laplace's approximation: the first terms in the expansion are quadratic in the deviations from the maximum and give simple Gaussian integrals. For the remaining B(Xi), the leading terms in the log integrand vary linearly near the maximum. Couplings between these B(Xi) only appear at the next (quadratic) order; discarding these terms as subleading, the integral factorizes over the B(xd and can be evaluated. The end result of this calculation is: InQ(D) ~ -! LiYi<liB*(Xi) - CLil(YiB*(xd) - nln(l + e- 2C ) - ! Indet(LmKm) (8) The first three terms represent the maximum of the log integrand, In Q(DIO*); the last one comes from the integration over the fluctuations of the B(x). Note that it only contains information about the marginal training inputs: Km is the corresponding submatrix of K(x, x'), and Lm is a diagonal matrix with entries 3Quantitative changes arise because function values with IO(x)1 < 1 are 'discouraged' for large nj this tends to increase the size of the decision regions and narrow the uncertainty gaps. I have verified this by comparing samples from Q(O) and P(O). P. Sollich 354 0 -0.1 -0.2 -0.3 -0.4 -0.5 9(x) 2 1 o 1 2 0.8 P(y=llx) I 0.6 I 0.4 I \ 0.2 -1 -2 0.2 0.4 x 0.6 0.8 1 o I I I I I \. o 0.2 C J 0.4 x 0.6 4 3 I I I I , \. }; 0.8 1 Figure 2: Toy example of evidence maximization. Left: Target 'latent' function 8(x) (solid line). A SVM with RBF kernel K(x, Xl) = Ko exp[-(x - X I )2 /(2[2)], [ = 0.05, CKo = 2.5 was trained (dashed line) on n = 50 training examples (circles). Keeping CKo constant, the evidence P(D) (top right) was then evaluated as a function of C using (7,8). Note how the normalization factor Nn shifts the maximum of P(D) towards larger values of C than in the naive evidence Q(D). Bottom right: Class probability P(y = 11x) for the target (solid), and prediction at the evidence maximum C ~ 1.8 (dashed) . The target was generated from (3) with C=2 . 27r[ai(C -ai)/C]2. Given the sparseness ofthe SVM solution, these matrices should be reasonably small, making their determinants amenable to numerical computation or estimation [12]. Eq. (8) diverges when ai -+ a or -+ C for one of the marginal training inputs; the approximation of retaining only linear terms in the log integrand then breaks down. I therefore adopt the simple heuristic of replacing det(LmKm) by det(1 + LmKm), which prevents these spurious singularities (I is the identity matrix) . This choice also keeps the evidence continuous when training inputs move in or out of the set of marginal inputs as hyperparameters are varied. Fig. 2 shows a simple application of the evidence estimate (8) . For a given data set, the evidence P(D) was evaluated4 as a function of C. The kernel amplitude Ko was varied simultaneously such that C Ko and hence the SVM solution itself remained unchanged. Because the data set was generated artificially from the probability model (3), the 'true' value of C = 2 was known; in spite of the rather crude approximation for Q(D), the maximum of the full evidence P(D) identifies C ~ 1.8 quite close to the truth . The approximate class probability prediction P(y = 11x, D) for this value of C is also plotted in Fig. 2; it overestimates the noise in the target somewhat. Note that P(ylx, D) was obtained simply by inserting the MAP values 8*(x) into (5). In a proper Bayesian treatment, an average over the posterior distribution P(OID) should of course be taken; I leave this for future work. 4The normalization factor Nn was estimated, for the assumed uniform input density Q(x) of the example, by sampling from the GP prior Q(9) . If Q(x) is unknown, the empirical training input distribution can be used as a proxy, and one samples instead from a multivariate Gaussian for the 9(xd with covariance matrix K(Xi , Xj). This gave very similar values of In Nn in the example, even when only a subset of 30 training inputs was used. Probabilistic Methods for Support Vector Machines 355 In summary, I have described a probabilistic framework for SVM classification. It gives an intuitive understanding of the effect of the kernel, which determines a Gaussian process prior. More importantly, it also allows a properly normalized evidence to be defined; from this, optimal values of hyperparameters such as the noise parameter C, and corresponding error bars, can be derived. Future work will have to include more comprehensive experimental tests of the simple Laplacetype estimate of the (naive) evidence Q(D) that I have given, and comparison with other approaches. These include variational methods; very recent experiments with a Gaussian approximation for the posterior P(9ID), for example, seem promising [6]. Further improvement should be possible by dropping the restriction to a 'factor-analysed' covariance form [6]. (One easily shows that the optimal Gaussian covariance matrix is (D + K- 1 )-1, parameterized only by a diagonal matrix D.) It will also be interesting to compare the Laplace and Gaussian variational results for the evidence with those from the 'cavity field' approach of [10]. Acknowledgements It is a pleasure to thank Tommi Jaakkola, Manfred Opper, Matthias Seeger, Chris Williams and Ole Winther for interesting comments and discussions, and the Royal Society for financial support through a Dorothy Hodgkin Research Fellowship. References [1] C J C Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2:121-167, 1998. [2] A J Smola and B Scholkopf. A tutorial on support vector regression. 1998. Neuro COLT Technical Report TR-1998-030; available from http://svm.first.gmd.de/. [3] B Scholkopf, C Burges, and A J Smola. Advances in Kernel Methods: Su.pport Vector Machines . MIT Press, Cambridge, MA , 1998. [4) B Scholkopf, P Bartlett, A Smola, and R Williamson. Shrinking the tube: a new support vector regression algorithm. In NIPS 11. [5] N Cristianini, C Campbell, and J Shawe-Taylor. Dynamically adapting kernels in support vector machines. In NIPS 11. [6] M Seeger. Bayesian model selection for Support Vector machines, Gaussian processes and other kernel classifiers. Submitted to NIPS 12. [7] G Wahba. Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV. Technical Report 984, University of Wisconsin, 1997. [8] T S Jaakkola and D Haussler. Probabilistic kernel regression models. In Proceedings of The 7th International Workshop on Artificial Intelligence and Statistics. To appear. [9] A J Smola, B Scholkopf, and K R Muller. The connection between regularization operators and support vector kernels. Neu.ral Networks, 11:637-649, 1998. [10] M Opper and 0 Winther. Gaussian process classification and SVM: Mean field results and leave-one-out estimator. In Advances in Large Margin Classifiers. MIT Press. To appear. [11] P Sollich. Probabilistic interpretation and Bayesian methods for Support Vector Machines. Submitted to ICANN 99. [12] C K I Williams. Prediction with Gaussian processes: From linear regression to linear prediction and beyond. In M I Jordan, editor, Learning and Inference in Graphical Models, pages 599-621. Kluwer Academic, 1998. 

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Bayesian Transduction Thore Graepel, Ralf Herbrich and Klaus Obermayer Department of Computer Science Technical University of Berlin Franklinstr. 28/29, 10587 Berlin, Germany {graepeI2, raith, oby} @cs.tu-berlin.de Abstract Transduction is an inference principle that takes a training sample and aims at estimating the values of a function at given points contained in the so-called working sample as opposed to the whole of input space for induction. Transduction provides a confidence measure on single predictions rather than classifiers - a feature particularly important for risk-sensitive applications. The possibly infinite number of functions is reduced to a finite number of equivalence classes on the working sample. A rigorous Bayesian analysis reveals that for standard classification loss we cannot benefit from considering more than one test point at a time. The probability of the label of a given test point is determined as the posterior measure of the corresponding subset of hypothesis space. We consider the PAC setting of binary classification by linear discriminant functions (perceptrons) in kernel space such that the probability of labels is determined by the volume ratio in version space. We suggest to sample this region by an ergodic billiard. Experimental results on real world data indicate that Bayesian Transduction compares favourably to the well-known Support Vector Machine, in particular if the posterior probability of labellings is used as a confidence measure to exclude test points of low confidence. 1 Introduction According to Vapnik [9], when solving a given problem one should avoid solving a more general problem as an intermediate step. The reasoning behind this principle is that in order to solve the more general task resources may be wasted or compromises may have to be made which would not have been necessary for the solution of the problem at hand. A direct application of this common-sense principle reduces the more general problem of inferring a functional dependency on the whole of input space to the problem of estimating the values of a function at given points (working sample), a paradigm referred to as transductive inference. More formally, given a probability measure PXY on the space of data X x y = X x {-I, +1}, a training sample S = {(Xl, yd, ... ,(Xl, Yi)} is generated i.i.d. according to PXY. Additional m data points W = {Xl+I, ... ,Xi+ m } are drawn: the working sample. The goal is to label the objects of the working sample W using a fixed set 1{ of functions 457 Bayesian Transduction f : X Jo--ot {-I, +I} so as to minimise a predefined loss. In . contrast, inductive inference, aims at choosing a single function It E 1i best suited to capture the dependency expressed by the unknown P XY . Obviously, if we have a transductive algorithm A (W, S, 1i) that assigns to each working sample W a set of labels given the training sample S and the set 1i offunctions, we can define a function fs : X Jo--ot {-I, +1} by fs (x) A ({x} ,S, 1i) as a result ofthe transduction algorithm. There are two crucial differences to induction, however: i) A ({ x} , S, 1i) is not restricted to select a single decision function f E 1i for each x, ii) a transduction algorithm can give performance guarantees on particular labellings instead of functions. In practical applications this difference may be of great importance. = After all, in risk sensitive applications (medical diagnosis, financial and critical control applications) it often matters to know how confident we are about a given prediction. In this case a general confidence measure of the classifier w.r. t. the whole input distribution would not provide the desired warranty at all. Note that for linear classifiers some guarantee can be obtained by the margin [7] which in Section 4 we will demonstrate to be too coarse a confidence measure. The idea of transduction was put forward in [8], where also first algorithmic ideas can be found . Later [1] suggested an algorithm for transduction based on linear programming and [3] highlighted the need for confidence measures in transduction. The paper is structured as follows: A Bayesian approach to transduction is formulated in Section 2. In Section 3 the function class of kernel perceptrons is introduced to which the Bayesian transduction scheme is applied . For the estimation of volumes in parameter space we present a kernel billiard as an efficient sampling technique. Finally, we demonstrate experimentally in Section 4 how the confidence measure for labellings helps Bayesian Transduction to achieve low generalisation error at a low rejection rate of test points and thus to outperform Support Vector Machines (SVMs). 2 Bayesian Transductive Classification Suppose we are given a training sample S = {(Xl, YI) , . .. , (Xl, Yl )} drawn i.i.d. from PXY and a working sample W = {XHI,' " , XHm} drawn i.i.d. from P x . Given a prior PH over the set 1i of functions and a likelihood P (Xy)lIH=f we obtain a posterior probability PHI(Xy)l=s ~f PHIS by Bayes' rule. This posterior measure induces a probability measure on labellings b E {-I, +l}m of the working sample byl (1) For the sake of simplicity let us assume a PAC style setting, i.e. there exists a in the space 1i such that PYlx=x (y) = 6 (y (x)). In this case one function can define the so-called version-space as the set of functions that is consistent with the training sample r r (2) outside which the posterior PHIS vanishes. Then Pymls,w (b) represents the prior measure of functions consistent with the training sample S and the labelling b on the working sample W normalised by the prior measure of functions consistent with S alone. The measure PH can be used to incorporate prior knowledge into 1 Note that the number of different labellings b implement able by 1l is bounded above by the value of the growth function IIu (JWI) [8, p . 321]. T. Graepe/, R. Herbrich and K. Obennayer 458 the inference process. If no such knowledge is available, considerations of symmetry may lead to "uninformative" priors. Given the measure PYFnIS,W over labellings, in order to arrive at a risk minimal decision w.r.t. the labelling we need to define a loss function I : ym X ym I---t IR+ between labellings and minimise its expectation, R (b, S, W) = EYFnIS,W [I (b, ym)] = I (b, b/) PYFnIS,W (b /) , (3) 2: {b'} 2m where the summation runs over all the sample. Let us consider two scenarios: possible labellings b ' of the working 1. A 0-1-loss on the exact labelling b, i.e. for two labellings band b ' m Ie (b, b/) = 1- II 6 (b i - bD ?} i=l Re (b, S, W) =1- PYFnIS,W (b) . (4) = In this case choosing the labelling be argminb Re (b, S, W) of the highest joint probability Pymls,w (b) minimises the risk. This non-labelwise loss is appropriate if the goal is to exactly identify a combination of labels, e.g. the combination of handwritten digits defining a postal zip code. Note that classical SVM transduction (see, e.g. [8, 1]) by maximising the margin on the combined training and working sample approximates this strategy and hence does not minimise the standard classification risk on single instances as intended. 2. A 0-1-10ss on the single labels bi, i.e. for two labellings band b ' 1 m 1$ (b, b/) = (1- 6 (b i - bD) , (5) 2: m i=l R$ (b, S, W) ! f 2: (1- 6 (bi - b~)) Pymls,w (b /) i=l {b'} 1 m - 2: (1- PHIs ({f: f(Xl+i) = bd)) . m i=l Due to the independent treatment of the loss at working sample points the risk R$ (b, S, W) is minimised by the labelling of highest marginal probability of the labels, i.e. bi = argmaXyEY PHIs ({f: f(Xl+i) = y}). Thus in the case of the labelwise loss (5) a working sample of m > 1 point does not offer any advantages over larger working samples w.r. t. the Bayes-optimal decision. Since this corresponds to the standard classification setting, we will restrict ourselves to working samples of size m = 1, i.e. to one working point Xl+1. 3 Bayesian Transduction by Volume 3.1 The Kernel Perceptron We consider transductive inference for the class of kernel perceptrons. The decision functions are given by f (x) = sign ?w, q, (xl) >') = sign (t a;k (x;, X)) l w = 2: (?itP (xd E :F , i=l 459 Bayesian Transduction Figure 1: Schematic view of data space (left) and parameter space ( right) for a classification toy example. Using the duality given by (w , 4> (x)):F = 0 data points on the left correspond to hyperplanes on the right, while hyperplanes on the left can be thought of as points on the right . where the mapping 4> : X t--+ :F maps from input space X to a feature space :F completely determined by the inner product function (kernel) k : X x X t--+ IR (see [9 , 10]) . Given a training sample S = {(Xi , Yi)}~=l we can define the version space - the set of all perceptrons compatible with the training data - as in (2) having the additional constraint Ilwll:F = 1 ensuring uniqueness. In order to obtain a prediction on the label b1 of the working point Xl+l we note that Xl+l may bisects the volume V of version space into two sub-volumes V+ and V-, where the perceptrons in V+ would classify Xl+l as b1 = +1 and those in V- as b1 = -l. The ratio p+ = V+ IV is the probability of the labelling b1 = +1 given a uniform prior PH over wand the class of kernel perceptrons, accordingly for b1 = -1 (see Figure 1) . Already Vapnik in [8, p. 323] noticed that it is troublesome to estimate sub- volumes of version space. As the solution to this problem we suggest to use a billiard algorithm . 3.2 Kernel Billiard for Volume Estimation The method of playing billiard in version space was first introduced by Rujan [6] for the purpose of estimating its centre of mass and consequently refined and extended to kernel spaces by [4]. For Bayesian Transduction the idea is to bounce the billiard ball in version space and to record how much time it spends in each of the sub-volumes of interest. Under the assumption of ergodicity [2] w.r .t. the uniform measure in the limit the accumulated flight times for each sub-volume are proportional to the sub-volume itself. Since the trajectory is located in :F each position wand direction v of the ball can be expressed as linear combinations of the 4> (xd , i.e. l l W= L Q:i4> (Xi) i=l v = L ,Bi4> (Xi) i=l l (w, v):F =L Q:i,Bjk (Xi, Xj) i,j=l where 0:, {3 are real vectors with f components and fully determine the state of the billiard. The algorithm for the determination of the label b1 of Xl+l proceeds as follows : 1. Initialise the starting position Wo in V (S) using any kernel perceptron algorithm that achieves zero training error (e .g . SVM [9]) . Set V+ = V- = O. T. Graepel, R. Herbrich and K. Obennayer 460 2. Find the closest boundary of V (S) starting from current w into direction v, where the flight times Tj for all points including Xl+1 are determined using (w,tP(Xj?:r (v,tP(Xj)):r . The smallest positive flight time Tc = minj :T;>o Tj in kernel space corresponds to the closest data point boundary tP (xc) on the hypersphere. Note, that if Tc -7 00 we randomly generate a direction v pointing towards version space, i.e. y (v, tP (x)):r > 0 assuming the last bounce was at tP (x). 3. Calculate the ball's new position w' according to W , = + TcV Ilw + Tcvll:r w . (1 - /2) Calculate the distance tf = Ilw - w'llsphere = arccos Ilw - w'lI;' on the hypersphere and add it to the volume estimate VY corresponding to the current label y = sign (w + w', tP (Xl+d):r)? If the test point tP (xl+d was hit, i.e. c = l + 1, keep the old direction vector v. Otherwise update to the reflection direction v', v' = v - 2 (v, tP (xc) ):r tP (xc) . Go back to step 2 unless the stopping criterion (8) is met. Note that in practice one trajectory can be calculated in advance and can be used for all test points. The estimators of the probability of the labellings are then given by p+ = V+ /(V+ + V-) and p = V- /(v+ + V-). Thus, the algorithm outputs b1 with confidence Ctrans according to ~ Ctrans def argmaXyEY iY' def (2 . max (pi" ,p) - 1) E [0, 1] . , (6) (7) Note that the Bayes Point Machine (BPM) [4] aims at an optimal approximation of the transductive classification (6) by a single function f E 1{ and that the well known SVM can be viewed as an approximation of the BPM by the centre of the largest ball in version space. Thus, treating the real valued output If(xl+1) I ~f G;nd of SVM classifiers as a confidence measure can be considered an approximation of (7). The consequences will be demonstrated experimentally in the following section. Disregarding the issue of mixing time [2] and the dependence of trajectories we assume for the stopping criterion that the fraction pt of time tt spent in volume V+ on trajectory i of length (tt + f;) is a random variable having expectation p+ . Hoeffding's inequality [5] bounds the probability of deviation from the expectation p+ by more than f, P (!; t ! - p+ <: ,) ~ exp (-2n,2) ~ ~. p (8) Thus if we want the deviation f from the true label probability to be less than f < 0.05 with probability at least 1 - T} = 0.99 we need approximately n R:j 1000 bounces. The computational effort of the above algorithm for a working set of size m is of order 0 (nl (m + l)). 461 Bayesian Transduction 1= 100--1 2 o . . o ~~----~----~--~----~--~ 0.00 0.05 0.10 0 lei 020 rejeclion rate (a) o~ 000 _ _~_ _~_ _~~_ _~_ _~~ 0.05 0 .10 0 lei 020 0.25 030 rejection rate (b) Figure 2: Generalisation error vs. rejection rate for Bayesian Transduction and SVMs for the thyroid data set (0' = 3) (a) and the heart data set (0' = 10). The error bars in both directions indicate one standard deviation of the estimated means. The upper curve depicts the result for the SVM algorithm; the lower curve is the result obtained by Bayesian Transduction. 4 Experimental Results We focused on the confidence Ctrans Bayesian Transduction provides together with the prediction b1 of the label. If the confidence Ctrans reflects reliability of a label estimate at a given test point then rejecting those test points whose predictions carry low confidence should lead to a reduction in generalisation error on the remaining test points . In the experiments we varied a rejection threshold () between [0, 1] thus obtaining for each () a rejeection rate together with an estimate of the generalisation error at non-rejected points. Both these curves were linked by their common ()-axis resulting in a generalisation error versus rejection rate plot. We used the UCI 2 data sets thyroid and heart because they are medical applications for which the confidence of single predictions is particularly important. Also a high rejection rate due to too conservative a confidence measure may incur considerable costs. We trained a Support Vector Machine using RBF kernels k (x, x') = exp ( -llx - x'1l2 /20'2) with 0' chosen such as to insure the existence of a version space. We used 100 different training samples obtained by random 60%:40% splits of the whole data set. The margin Clnd of each test point was calculated as a confidence measure of SVM classifications. For comparison we determined the labels b1 and resulting confidences Ctrans using the Bayesian Transduction algorithm (see Section 3) with the same value of the kernel parameter. Since the rejection for the Bayesian Transduction was in both cases higher than for SVMs at the same level () we determined ()max which achieves the same rejection rate for the SVM confidence measures as Bayesian Transduction achieves at () = 1 (thyroid: ()max = 2.15, heart: ()max 1.54). The results for the two data sets are depicted in Figure 2. = In the thyroid example Figure 2 (a) one can see that Ctrans is indeed an appropriate indicator of confidence: at a rejection rate of approximately 20% the generalisation error approaches zero at minimal variance. For any desired generalisation error Bayesian Transduction needs to reject significantly less examples of the test set as compared to SVM classifiers, e.g. 4% less at 2.3% generalisation error. The results of the heart data set show even more pronounced characteristics w.r.t. to the rejection 2UCI University of California at Irvine: Machine Learning Repository T. Graepe/, R. Herbrich and K. Obermayer 462 rate. Note that those confidence measures considered cannot capture the effects of noise in the data which leads to a generalisation error of 16.4% even at maximal rejection () = 1 corresponding to the Bayes error under the given function class. 5 Conclusions and FUture Work In this paper we a presented a Bayesian analysis of transduction. The required volume estimates for kernel perceptrons in version space are performed by an ergodic billiard in kernel space. Most importantly, transduction not only determines the label of a given point but also returns a confidence measure of the classification in the form of the probability of the label under the model. Using this confidence measure to reject test examples then lead to improved generalisation error over SVMs. The billiard algorithm can be extended to the case of non-zero training error by allowing the ball to penetrate walls, a property that is captured by adding a constant>. to the diagonal of the kernel matrix [4] . Further research will aim at the discovery of PAC-Bayesian bounds on the generalisation error of transduction. Acknowledgements We are greatly indebted to U. Kockelkorn for many interesting suggestions and discussions . This project was partially funded by Technical University of Berlin via FIP 13/41. References [1] K. Bennett. Advances in Kernel Methods - Support Vector Learning, chapter 19, Combining Support Vector and Mathematical Programming Methods for Classification, pages 307-326. MIT Press, 1998. [2] I. Cornfeld, S. Fomin, and Y. Sinai. Ergodic Theory. Springer Verlag, 1982. [3] A. Gammerman, V. Vovk, and V. Vapnik. Learning by transduction. In Proceedings of Uncertainty in AI, pages 148-155, Madison, Wisconsin, 1998. [4] R. Herbrich, T. Graepel, and C. Campbell. Bayesian learning in reproducing kernel Hilbert spaces. Technical report, Technical University Berlin, 1999. TR 99-1l. [5] W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13-30, 1963. [6] P. Rujan. Playing billiard in version space. Neural Computation, 9:99-122, 1997. [7] J. Shawe-Taylor. Confidence estimates of classification accuracy on new examples. Technical report, Royal Holloway, University of London, 1996. NC2-TR-1996-054. [8] V. Vapnik. Estimation of Dependences Based on Empirical Data. Springer, 1982. [9] V. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995. [10] G. Wahba. Spline Models for Observational Data. Society for Industrial and Applied Mathematics, Philadelphia, 1990. 

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Policy Gradient Methods for Reinforcement Learning with Function Approximation Richard S. Sutton, David McAllester, Satinder Singh, Yishay Mansour AT&T Labs - Research, 180 Park Avenue, Florham Park, NJ 07932 Abstract Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and determining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly represented by its own function approximator, independent of the value function, and is updated according to the gradient of expected reward with respect to the policy parameters. Williams's REINFORCE method and actor-critic methods are examples of this approach. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. Using this result, we prove for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy. Large applications of reinforcement learning (RL) require the use of generalizing function approximators such neural networks, decision-trees, or instance-based methods. The dominant approach for the last decade has been the value-function approach, in which all function approximation effort goes into estimating a value function, with the action-selection policy represented implicitly as the "greedy" policy with respect to the estimated values (e.g., as the policy that selects in each state the action with highest estimated value). The value-function approach has worked well in many applications, but has several limitations. First, it is oriented toward finding deterministic policies, whereas the optimal policy is often stochastic, selecting different actions with specific probabilities (e.g., see Singh, Jaakkola, and Jordan, 1994). Second, an arbitrarily small change in the estimated value of an action can cause it to be, or not be, selected. Such discontinuous changes have been identified as a key obstacle to establishing convergence assurances for algorithms following the value-function approach (Bertsekas and Tsitsiklis, 1996). For example, Q-Iearning, Sarsa, and dynamic programming methods have all been shown unable to converge to any policy for simple MDPs and simple function approximators (Gordon, 1995, 1996; Baird, 1995; Tsitsiklis and van Roy, 1996; Bertsekas and Tsitsiklis, 1996). This can occur even if the best approximation is found at each step before changing the policy, and whether the notion of "best" is in the mean-squared-error sense or the slightly different senses of residual-gradient, temporal-difference, and dynamic-programming methods. In this paper we explore an alternative approach to function approximation in RL. 1058 R. S. Sutton, D. McAl/ester. S. Singh and Y. Mansour Rather than approximating a value function and using that to compute a deterministic policy, we approximate a stochastic policy directly using an independent function approximator with its own parameters. For example, the policy might be represented by a neural network whose input is a representation of the state, whose output is action selection probabilities, and whose weights are the policy parameters. Let 0 denote the vector of policy parameters and p the performance of the corresponding policy (e.g., the average reward per step). Then, in the policy gradient approach, the policy parameters are updated approximately proportional to the gradient: ap ~O~CtaO' (1) where Ct is a positive-definite step size. If the above can be achieved, then 0 can usually be assured to converge to a locally optimal policy in the performance measure p. Unlike the value-function approach, here small changes in 0 can cause only small changes in the policy and in the state-visitation distribution. In this paper we prove that an unbiased estimate of the gradient (1) can be obtained from experience using an approximate value function satisfying certain properties. Williams's (1988, 1992) REINFORCE algorithm also finds an unbiased estimate of the gradient, but without the assistance of a learned value function. REINFORCE learns much more slowly than RL methods using value functions and has received relatively little attention. Learning a value function and using it to reduce the variance of the gradient estimate appears to be ess~ntial for rapid learning. Jaakkola, Singh and Jordan (1995) proved a result very similar to ours for the special case of function approximation corresponding to tabular POMDPs. Our result strengthens theirs and generalizes it to arbitrary differentiable function approximators. Konda and Tsitsiklis (in prep.) independently developed a very simialr result to ours. See also Baxter and Bartlett (in prep.) and Marbach and Tsitsiklis (1998). Our result also suggests a way of proving the convergence of a wide variety of algorithms based on "actor-critic" or policy-iteration architectures (e.g., Barto, Sutton, and Anderson, 1983; Sutton, 1984; Kimura and Kobayashi, 1998). In this paper we take the first step in this direction by proving for the first time that a version of policy iteration with general differentiable function approximation is convergent to a locally optimal policy. Baird and Moore (1999) obtained a weaker but superficially similar result for their VAPS family of methods. Like policy-gradient methods, VAPS includes separately parameterized policy and value functions updated by gradient methods. However, VAPS methods do not climb the gradient of performance (expected long-term reward), but of a measure combining performance and valuefunction accuracy. As a result, VAPS does not converge to a locally optimal policy, except in the case that no weight is put upon value-function accuracy, in which case VAPS degenerates to REINFORCE. Similarly, Gordon's (1995) fitted value iteration is also convergent and value-based, but does not find a locally optimal policy. 1 Policy Gradient Theorem We consider the standard reinforcement learning framework (see, e.g., Sutton and Barto, 1998), in which a learning agent interacts with a Markov decision process (MDP). The state, action, and reward at each time t E {O, 1, 2, . . .} are denoted St E S, at E A, and rt E R respectively. The environment's dynamics are characterized by state transition probabilities, P:SI = Pr {St+ 1 = Sf I St = s, at = a}, and expected rewards 'R~ = E {rt+l 1st = s, at = a}, 'r/s, Sf E S, a E A. The agent's decision making procedure at each time is characterized by a policy, 1l'(s, a, 0) = Pr {at = alst = s, O}, 'r/s E S,a E A, where 0 E ~, for l ? lSI, is a parameter vector. We assume that 1l' is diffentiable with respect to its parameter, i.e., that a1f~~a) exists. We also usually write just 1l'(s, a) for 1l'(s, a, 0). Policy Gradient Methods for RL with Function Approximation 1059 With function approximation, two ways of formulating the agent's objective are useful. One is the average reward formulation, in which policies are ranked according to their long-term expected reward per step, p(rr): p(1I") = n-+oon lim .!.E{rl +r2 + ... +rn 11I"} = '" ?ff(s) "'1I"(s,a)'R.:, ~ ~ II Q where cP (s) = limt-+oo Pr {St = slso, 11"} is the stationary distribution of states under 11", which we assume exists and is independent of So for all policies. In the average reward formulation, the value of a state-action pair given a policy is defined as 00 Q1r(s,a) p(1I") I So = LE {rt - = s,ao = a,1I"}, Vs E S,a E A. t=l The second formulation we cover is that in which there is a designated start state So, and we care only about the long-term reward obtained from it. We will give our results only once, but they will apply to this formulation as well under the definitions p(1I") = E{t. "(t-lrt I 8 0 ,1I"} and Q1r(s,a) = E{t. "(k-lrt+k 1St = s,at = a, 11" }. where,,( E [0,1] is a discount rate ("( = 1 is allowed only in episodic tasks). In this formulation, we define d1r (8) as a discounted weighting of states encountered starting at So and then following 11": cP(s) = E:o"(tpr{st = slso,1I"}. Our first result concerns the gradient of the performance metric with respect to the policy parameter: Theorem 1 (Policy Gradient). For any MDP, in either the average-reward or start-state formulations, ap = "'.ftr( )'" a1l"(s,a)Q1r( ao ~u II s ~ ao ) s, a . (2) Q Proof: See the appendix. This way of expressing the gradient was first rtiscussed for the average-reward formulation by Marbach and Tsitsiklis (1998), based on a related expression in terms of the state-value function due to Jaakkola, Singh, and Jordan (1995) and Coo and Chen (1997). We extend their results to the start-state formulation and provide simpler and more direct proofs. Williams's (1988, 1992) theory of REINFORCE algorithms can also be viewed as implying (2). In any event, the key aspect of both expressions for the gradient is that their are no terms of the form adiJII): the effect of policy changes on the distribution of states does not appear. This is convenient for approximating the gradient by sampling. For example, if 8 was sampled from the distribution obtained by following 11", then Ea a1r~~,a) Q1r (s, a) would be an unbiased estimate of ~. Of course, Q1r(s, a) is also not normally known and must be estimated. One approach is to use the actual returns, R t = E~l rt+k - p(1I") (or Rt = E~l "(k-lrt+k in the start-state formulation) as an approximation for each Q1r (St, at). This leads to Williams's episodic REINFORCE algorithm, t::..Ot oc a1r~~,at2 Rt 7r (1 ) (the ~a St,at 7r\St,Ut) corrects for the oversampling of actions preferred by 11"), which is known to follow ~ in expected value (Williams, 1988, 1992). 2 Policy Gradient with Approximation Now consider the case in which Q1r is approximated by a learned function approximator. If the approximation is sufficiently good, we might hope to use it in place of Q1r R. S. Sutton, D. MeAl/ester, S. Singh and Y. Mansour 1060 in (2) and still point roughly in the direction of the gradient. For example, Jaakkola, Singh, and Jordan (1995) proved that for the special case of function approximation arising in a tabular POMDP one could assure positive inner product with the gradient, which is sufficient to ensure improvement for moving in that direction. Here we extend their result to general function approximation and prove equality with the gradient. Let fw : S x A - ~ be our approximation to Q7f, with parameter w. It is natural to learn f w by following 1r and updating w by a rule such as AWt oc I,u [Q7f (St, at) fw(st,at)]2 oc [Q7f(st,at) - fw(st,at)]alw~~,ad, where Q7f(st,at) is some unbiased estimator of Q7f(st, at), perhaps Rt. When such a process has converged to a local optimum, then LcF(s):E 1r(s,a)[Q7f (s,a) - fw(s,a)] 8f~~,a) = o. (3) a /I Theorem 2 (Policy Gradient with Function Approximation). If fw satisfies (3) and is compatible with the policy parameterization in the sense that l then 8fw(s, a) 81r(s, a) 1 = 8w 80 1r(s, a) , (4) 8p ~ ~ 81r(s, a) ao = ~cF(s) ~ ao fw(s,a). (5) a II Proof: Combining (3) and (4) gives Ld7f (s) L II 87r1~a) [Q 7f (s,a) - fw(s,a)] = 0 (6) a which tells us that the error in fw(s, a) is orthogonal to the gradient of the policy parameterization. Because the expression above is zero, we can subtract it from the policy gradient theorem (2) to yield ap ao = L cF(s) II L a1r1~ a) Q7f(s, a) - :E cF(s) :E a1r1~ a) [Q 7f (s, a) 11 ~ ~ ~cF(s)~ /I = 3 a1r(s,a) ao [Q7f(s,a)-Q7f(s,a)+fw(s,a)] a ~ ~ ~ cF(s) ~ II fw(s, a)] a II a1r(s,a) ao fw(s, a). Q.E.D. a Application to Deriving Algorithms and Advantages Given a policy parameterization, Theorem 2 can be used to derive an appropriate form for the value-function parameterization. For example, consider a policy that is a Gibbs distribution in a linear combination of features: 'is E S,s E A, ITsitsiklis (personal communication) points out that /w being linear in the features given on the righthand side may be the only way to satisfy this condition. 1061 Policy Gradient Methods for RL with Function Approximation where each <Psa is an i-dimensional feature vector characterizing state-action pair s, a. Meeting the compatibility condition (4) requires that ofw(s,a) _ o1r(s,a) 1 _ 00 ( ) OW 7rS,a A 'l'sa _ L (S, b)A'l'sb, b 1r so that the natural parameterization of fw is fw(s,a) ~w T ["',. - ~"(S,b)""bl ? In other words, fw must be linear in the same features as the policy, except normalized to be mean zero for each state. Other algorithms can easily be derived for a variety of nonlinear policy parameterizations, such as multi-layer backpropagation networks. The careful reader will have noticed that the form given above for f w requires that it have zero mean for each state: l:a 1r(s, a)fw(s, a) = 0, Vs E S . In this sense it is better to think of f w as an approximation of the advantage function, A7r(s,a) = Q7r(s,a) - V7r(s) (much as in Baird, 1993), rather than of Q7r . Our convergence requirement (3) is really that fw get the relative value of the actions correct in each state, not the absolute value, nor the variation from state to state. Our results can be viewed as a justification for the special status of advantages as the target for value function approximation in RL. In fact, our (2), (3), and (5), can all be generalized to include an arbitrary function of state added to the value function or its approximation. For example, (5) can be generalized to ~ = l:s d7r (s) l:a 87r~~,a) [fw(s, a) + v(s)] ,where v : S ---+ R is an arbitrary function. (This follows immediately because l:a 87r~~a) = 0, Vs E S.) The choice of v does not affect any of our theorems, but can substantially affect the variance of the gradient estimators. The issues here are entirely analogous to those in the use of reinforcement baselines in earlier work (e.g., Williams, 1992; Dayan, 1991; Sutton, 1984). In practice, v should presumably be set to the best available approximation of V7r. Our results establish that that approximation process can proceed without affecting the expected evolution of fw and 1r . 4 Convergence of Policy Iteration with Function Approximation Given Theorem 2, we can prove for the first time that a form of policy iteration with function approximation is convergent to a locally optimal policy. Theorem 3 (Policy Iteration with Function Approximation). Let 1r and fw be any differentiable function approximators for the policy and value function respectively that satisfy the compatibility condition (4) and for which 18;~~9;) I < B < 00. Let {Ok}~o be any step-size sequence such that limk-+oo Ok = 0 and l:k Ok = 00. Then, for any MDP with bounded rewards, the maxe,s,a ,i,j sequence {p(1rk)}r=o, defined by any 00, 1rk = 1r(.,., Ok), and w such that '"' ~crk(S) '"' ~ 1rk(s,a) [Q7r k (s,a) - fw(s,a) ]ofw(s,a) ow Wk s Ok+l = '"' = ? a '"' 01rk(S, 00 a) fWk(s,a), Ok+Ok~crk(S)~ s a converges such that limk-+oo 8P~;k) = o. Proof: Our Theorem 2 assures that the Ok update is in the direction of the gradient. 8 2 7r(s a) ....?..?....The bounds on 89;89j and on the MDP's rewards together assure us that 89i89j R. S. Sutton, D. MeAl/ester. S. Singh and Y. Mansour 1062 is also bounded. These, together with the step-size requirements, are the necessary conditions to apply Proposition 3.5 from page 96 of Bertsekas and Tsitsiklis (1996), which assures convergence to a local optimum. Q.E.D. Acknowledgements The authors wish to thank Martha Steenstrup and Doina Precup for comments, and Michael Kearns for insights into the notion of optimal policy under function approximation. References Baird, L. C. (1993) . Advantage Updating. Wright Lab. Technical Report WL-TR-93-1l46. Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. Proc. of the Twelfth Int. Co,:4. on Machine Learning, pp. 30-37. Morgan Kaufmann. Baird, L. C., Moore, A. W . (1999) . Gradient descent for general reinforcement learning. NIPS 11. MIT Press. Barto, A. G., Sutton, R. S., Anderson, C. W. (1983). Neuronlike elements that can solve difficult learning control problems. IEEE 1rans. on Systems, Man, and Cybernetics 19:835. Baxter, J ., Bartlett, P. (in prep.) Direct gradient-based reinforcement learning: I. Gradient estimation algorithms. Bertsekas, D. P., Tsitsiklis, J. N. (1996). Neuro-Dynamic Programming. Athena Scientific. Cao, X.-R., Chen, H.-F. (1997) . Perturbation realization, potentials, and sensitivity analysis of Markov Processes, IEEE 1hlns. on Automatic Control 42{1O):1382-1393. Dayan, P. (1991). Reinforcement comparison. In D. S. Touretzky, J. L. Elman, T. J. Sejnowski, and G. E. Hinton (eds.), Connectionist Models: Proceedings of the 1990 Summer School, pp. 45-51. Morgan Kaufmann. Gordon, G. J. (1995). Stable function approximation in dynamic programming. Proceedings of the Twelfth Int. Conf. on Machine Learning, pp. 261-268. Morgan Kaufmann. Gordon, G. J. (1996). Chattering in SARSA(A). CMU Learning Lab Technical Report. Jaakkola, T., Singh, S. P., Jordan, M. I. (1995) Reinforcement learning algorithms for partially observable Markov decision problems, NIPS 7, pp. 345-352. Morgan Kaufman. Kimura, H., Kobayashi, S. (1998). An analysis of actor/critic algorithms using eligibility traces: Reinforcement learning with imperfect value functions. Proc. ICML-98, pp. 278-286. Konda, V. R., Tsitsiklis, J. N. (in prep.) Actor-critic algorithms. Marbach, P., Tsitsiklis, J. N. (1998) Simulation-based optimization of Markov reward processes, technical report LIDS-P-2411, Massachusetts Institute of Technology. Singh, S. P., Jaakkola, T., Jordan, M. I. (1994) . Learning without state-estimation in partially observable Markovian decision problems. Proc. ICML-94, pp. 284-292. Sutton, R. S. (1984). Temporal Credit Assignment in Reinforcement Learning. Ph.D. thesis, University of Massachusetts, Amherst. Sutton, R. S., Barto, A. G. (1998) . Reinforcement Learning: An Introduction. MIT Press. Tsitsiklis, J. N. Van Roy, B. (1996) . Feature-based methods for large scale dynamic programming. Machine Learning 22:59-94. Williams, R. J . (1988) . Toward a theory of reinforcement-learning connectionist systems. Technical Report NU-CCS-88-3, Northeastern University, College of Computer Science. Williams, R. J. (1992) . Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning 8:229-256. Appendix: Proof of Theorem 1 We prove the theorem first for the average-reward formulation and then for the startstate formulation. 8V1I'(s) def 8fJ a = ~ [87r(S, 8 Q11'( s,a) ] ~ 80 a) Q11'() s,a +7r(s,a)80 a 1063 Policy Gradient Methods for RL with Function Approximation Therefore, = '""" ap ae L- [a1r(S,a)Q1T( ) ( ),"""pa aV1T(S')]_ aV1T(s) ao s, a + 1r s, a L- ss' ao ae a ~ Summing both sides over the stationary distribution d1T , = '""" L- d 1T ( s, a) Q1T (s, a) + '""" a aV1T (s') s) '""" L- a1r(ae L- U..nr ( s) '""" L- 7r ( s, a) '""" L- Pss' ae a s s _ a s' L~(s)av;O(s), s but since ~ is stationary, _ Ld1T (s) av;o(s) s a1r~~a) Q1T(s,a). :: = Ld1T (s) L Q.E.D. a s For the start-state formulation: aV 1T ( s) def a '""" 1T ae = ae L- 1r(s, a)Q (s, a) 'risE S a [a1r~~ a) Q1T(S, a) + 1r(s, a) :e Q1T (s, a)] = L a ~ ~ [inr~~ a) Q'(s,a) +--(s,a) :0 ['R~ + ~ ~P:., V'(S')]] ~ ~ [inr~~ a) Q'(s,a) +--(s,a) ~ ~P:.,! V'(S')] = '"""~ LL- 'Y It Pr ( s x -+ (7) x, k, 1r ),"""a1r(x,a)Q1T() L- ae x, a , It=o a after several steps of unrolling (7), where Pr(s -+ x, k, 1r) is the probability of going from state s to state x in k steps under policy 1r. It is then immediate that ap ae I } a {~t-l a 71' = aoE 'Y rt So,1r = ae v (so) ti = '"""~ L- L- 'Y It Pr ( So s = k=O s, k, 1r) ,"",,87r(s,a)Q7I'( L- ae s, a) a L ~(s) L s -+ a a1r~~ a) Q7I'(s, a). Q.E.D. 

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U nmixing Hyperspectral Data Lucas Parra, Clay Spence, Paul Sajda Sarnoff Corporation, CN-5300, Princeton, NJ 08543, USA {lparra, cspence,psajda} @sarnoff.com Andreas Ziehe, Klaus-Robert Miiller GMD FIRST.lDA, Kekulestr. 7, 12489 Berlin, Germany {ziehe,klaus}@first.gmd.de Abstract In hyperspectral imagery one pixel typically consists of a mixture of the reflectance spectra of several materials, where the mixture coefficients correspond to the abundances of the constituting materials. We assume linear combinations of reflectance spectra with some additive normal sensor noise and derive a probabilistic MAP framework for analyzing hyperspectral data. As the material reflectance characteristics are not know a priori, we face the problem of unsupervised linear unmixing. The incorporation of different prior information (e.g. positivity and normalization of the abundances) naturally leads to a family of interesting algorithms, for example in the noise-free case yielding an algorithm that can be understood as constrained independent component analysis (ICA). Simulations underline the usefulness of our theory. 1 Introduction Current hyperspectral remote sensing technology can form images of ground surface reflectance at a few hundred wavelengths simultaneously, with wavelengths ranging from 0.4 to 2.5 J.Lm and spatial resolutions of 10-30 m. The applications of this technology include environmental monitoring and mineral exploration and mining. The benefit of hyperspectral imagery is that many different objects and terrain types can be characterized by their spectral signature. The first step in most hyperspectral image analysis systems is to perform a spectral unmixing to determine the original spectral signals of some set of prime materials. The basic difficulty is that for a given image pixel the spectral reflectance patterns of the surface materials is in general not known a priori. However there are general physical and statistical priors which can be exploited to potentially improve spectral unmixing. In this paper we address the problem of unmixing hyperspectral imagery through incorporation of physical and statistical priors within an unsupervised Bayesian framework. We begin by first presenting the linear superposition model for the reflectances measured. We then discuss the advantages of unsupervised over supervised systems. Unmixing Hyperspectral Data 943 We derive a general maximum a posteriori (MAP) framework to find the material spectra and infer the abundances. Interestingly, depending on how the priors are incorporated, the zero noise case yields (i) a simplex approach or (ii) a constrained leA algorithm. Assuming non-zero noise our MAP estimate utilizes a constrained least squares algorithm. The two latter approaches are new algorithms whereas the simplex algorithm has been previously suggested for the analysis of hyperspectral data. Linear Modeling To a first approximation the intensities X (Xi>.) measured in each spectral band A = 1, ... , L for a given pixel i = 1, ... , N are linear combinations of the reflectance characteristics S (8 m >.) of the materials m = 1, ... , M present in that area. Possible errors of this approximation and sensor noise are taken into account by adding a noise term N (ni>'). In matrix form this can be summarized as X = AS + N, subject to: AIM = lL, A ~ 0, (1) where matrix A (aim) represents the abundance of material m in the area corresponding to pixel i, with positivity and normalization constraints. Note that ground inclination or a changing viewing angle may cause an overall scale factor for all bands that varies with the pixels. This can be incorporated in the model by simply replacing the constraint AIM = lL with AIM ~ lL which does does not affect the discussion in the remainder of the paper. This is clearly a simplified model of the physical phenomena. For example, with spatially fine grained mixtures, called intimate mixtures, multiple reflectance may causes departures from this first order model. Additionally there are a number of inherent spatial variations in real data, such as inhomogeneous vapor and dust particles in the atmosphere, that will cause a departure from the linear model in equation (1). Nevertheless, in practical applications a linear model has produced reasonable results for areal mixtures. Supervised vs. Unsupervised techniques Supervised spectral un mixing relies on the prior knowledge about the reflectance patterns S of candidate surface materials, sometimes called endmembers, or expert knowledge and a series of semiautomatic steps to find the constituting materials in a particular scene. Once the user identifies a pixel i containing a single material, i.e. aim = 1 for a given m and i, the corresponding spectral characteristics of that material can be taken directly from the observations, i.e., 8 m >. = Xi>. [4]. Given knowledge about the endmembers one can simply find the abundances by solving a constrained least squares problem. The problem with such supervised techniques is that finding the correct S may require substantial user interaction and the result may be error prone, as a pixel that actually contains a mixture can be misinterpreted as a pure endmember. Another approach obtains endmembers directly from a database. This is also problematic because the actual surface material on the ground may not match the database entries, due to atmospheric absorption or other noise sources. Finding close matches is an ambiguous process as some endmembers have very similar reflectance characteristics and may match several entries in the database. Unsupervised unmixing, in contrast, tries to identify the endmembers and mixtures directly from the observed data X without any user interaction. There are a variety of such approaches. In one approach a simplex is fit to the data distribution [7, 6, 2]. The resulting vertex points of the simplex represent the desired endmembers, but this technique is very sensitive to noise as a few boundary points can potentially change the location of the simplex vertex points considerably. Another approach by Szu [9] tries to find abundances that have the highest entropy subject to constraints that the amount of materials is as evenly distributed as possible - an assumption L. Parra, C. D. Spence, P Sajda, A. Ziehe and K.-R. Muller 944 which is clearly not valid in many actual surface material distributions. A relatively new approach considers modeling the statistical information across wavelength as statistically independent AR processes [1]. This leads directly to the contextual linear leA algorithm [5]. However, the approach in [1] does not take into account constraints on the abundances, noise, or prior information. Most importantly, the method [1] can only integrate information from a small number of pixels at a time (same as the number of endmembers). Typically however we will have only a few endmembers but many thousand pixels. 2 2.1 The Maximum A Posterior Framework A probabilistic model of unsupervised spectral unmixing Our model has observations or data X and hidden variables A, S, and N that are explained by the noisy linear model (1). We estimate the values of the hidden variables by using MAP (A SIX) p , = p(XIA, S)p(A, S) = Pn(XIA, S)Pa(A)ps(S) p(X) p(X) (2) with Pa(A), Ps(S), Pn(N) as the a priori assumptions of the distributions. With MAP we estimate the most probable values for given priors after observing the data, A MAP , SMAP = argmaxp(A, SIX) (3) A,S Note that for maximization the constant factor p(X) can be ignored. Our first assumption, which is indicated in equation (2) is that the abundances are independent of the reflectance spectra as their origins are completely unrelated: (AO) A and S are independent. The MAP algorithm is entirely defined by the choices of priors that are guided by the problem of hyperspectral unmixing: (AI) A represent probabilities for each pixel i. (A2) S are independent for different material m. (A3) N are normal i.i.d. for all i, A. In summary, our MAP framework includes the assumptions AO-A3. 2.2 Including Priors Priors on the abundances be represented as, Positivity and normalization of the abundances can (4) eo where 60 represent the Kronecker delta function and the step function. With this choice a point not satisfying the constraint will have zero a posteriori probability. This prior introduces no particular bias of the solutions other then abundance constraints. It does however assume the abundances of different pixels to be independent. Prior on spectra Usually we find systematic trends in the spectra that cause significant correlation. However such an overall trend can be subtracted and/or filtered from the data leaving only independent signals that encode the variation from that overall trend. For example one can capture the conditional dependency structure with a linear auto-regressive (AR) model and analyze the resulting "innovations" or prediction errors [3]. In our model we assume that the spectra represent independent instances of an AR process having a white innovation process em.>. distributed according to Pe(e). With a Toeplitz matrix T of the AR coefficients we 945 Unmixing Hyperspectral Data can write, em = Sm T. The AR coefficients can be found in a preprocessing step on the observations X. If S now represents the innovation process itself, our prior can be represented as, M Pe (S) <X Pe(ST) = L L II II Pe( L sm>.d>.>.,) , (5) m=1 >.=1 >.'=1 Additionally Pe (e) is parameterized by a mean and scale parameter and potentially parameters determining the higher moments of the distributions. For brevity we ignore the details of the parameterization in this paper. Prior on the noise As outlined in the introduction there are a number of problems that can cause the linear model X = AS to be inaccurate (e.g. multiple reflections, inhomogeneous atmospheric absorption, and detector noise.) As it is hard to treat all these phenomena explicitly, we suggest to pool them into one noise variable that we assume for simplicity to be normal distributed with a wavelength dependent noise variance a>., L p(XIA, S) = Pn(N) = N(X - AS,~) = II N(x>. - As>., a>.l) , (6) >.=1 where N (', .) represents a zero mean Gaussian distribution, and 1 the identity matrix indicating the independent noise at each pixel. 2.3 MAP Solution for Zero Noise Case Let us consider the noise-free case. Although this simplification may be inaccurate it will allow us to greatly reduce the number of free hidden variables - from N M + M L to M2 . In the noise-free case the variables A, S are then deterministically dependent on each other through a N L-dimensional 8-distribution, Pn(XIAS) = 8(X - AS). We can remove one of these variables from our discussion by integrating (2). It is instructive to first consider removing A p(SIX) <X I dA 8(X - AS)Pa(A)ps(S) = IS- 1IPa(XS- 1 )Ps(S). (7) We omit tedious details and assume L = M and invertible S so that we can perform the variable substitution that introduces the Jacobian determinant IS-II . Let us consider the influence of the different terms. The Jacobian determinant measures the volume spanned by the endmembers S. Maximizing its inverse will therefore try to shrink the simplex spanned by S. The term Pa(XS- 1 ) should guarantee that all data points map into the inside of the simplex, since the term should contribute zero or low probability for points that violate the constraint. Note that these two terms, in principle, define the same objective as the simplex envelope fitting algorithms previously mentioned [2]. In the present work we are more interested in the algorithm that results from removing S and finding the MAP estimate of A. We obtain (d. Eq.(7)) p(AIX) oc I dS 8(X - AS)Pa(A)ps(S) = IA -llps(A- 1 X)Pa(A). (8) For now we assumed N = M. 1 If Ps (S) factors over m , i.e. endmembers are independent, maximizing the first two terms represents the leA algorithm. However, lIn practice more frequently we have N > M. In that case the observations X can be mapped into a M dimensional subspace using the singular value decomposition (SVD) , X = UDV T , The discussion applies then to the reduced observations X = u1x with U M being the first M columns of U . L. Parra. C. D. Spence. P Sajda. A. Ziehe and K.-R. Muller 946 the prior on A will restrict the solutions to satisfy the abundance constraints and bias the result depending on the detailed choice of Pa(A), so we are led to constrained ICA. In summary, depending on which variable we integrate out we obtain two methods for solving the spectral unmixing problem: the known technique of simplex fitting and a new constrained ICA algorithm. 2.4 MAP Solution for the Noisy Case Combining the choices for the priors made in section 2.2 (Eqs.(4), (5) and (6)) with (2) and (3) we obtain AMAP, SMAP = "''i~ax ft {g N(x", - a,s" a,) ll. P,(t. 'm,d",) } , (9) subject to AIM = lL, A 2: O. The logarithm of the cost function in (9) is denoted by L = L(A, S). Its gradient with respect to the hidden variables is = _AT nm diag(O')-l - fs(sm) 88L Sm (10) where N = X - AS, nm are the M column vectors of N, fs(s) = - olnc;(s). In (10) fs is applied to each element of Sm. The optimization with respect to A for given S can be implemented as a standard weighted least squares (L8) problem with a linear constraint and positivity bounds. Since the constraints apply for every pixel independently one can solve N separate constrained LS problems of M unknowns each. We alternate between gradient steps for S and explicit solutions for A until convergence. Any additional parameters of Pe(e) such as scale and mean may be obtained in a maximum likelihood (ML) sense by maximizing L. Note that the nonlinear optimization is not subject to constraints; the constraints apply only in the quadratic optimization. 3 Experiments 3.1 Zero Noise Case: Artificial Mixtures In our first experiment we use mineral data from the United States Geological Survey (USGS)2 to build artificial mixtures for evaluating our unsupervised unmixing framework. Three target endmembers where chosen (Almandine WS479 , Montmorillonite+Illi CM42 and Dickite NMNH106242). A spectral scene of 100 samples was constructed by creating a random mixture of the three minerals. Of the 100 samples, there were no pure samples (Le. no mineral had more than a 80% abundance in any sample). Figure 1A is the spectra of the endmembers recovered by the constrained ICA technique of section 2.3, where the constraints were implemented with penalty terms added to the conventional maximum likelihood ICA algorithm. These are nearly identical to the spectra of the true endmembers, shown in figure 1B, which were used for mixing. Interesting to note is the scatter-plot of the 100 samples across two bands. The open circles are the absorption values at these two bands for endmembers found by the MAP technique. Given that each mixed sample consists of no more than 80% of any endmember, the endmember points on the scatter-plot are quite distant from the cluster. A simplex fitting technique would have significant difficulty recovering the endmembers from this clustering. 2see http://speclab.cr .usgs.gov /spectral.lib.456.descript/ decript04.html 947 Unmixing Hyperspectral Data found endmembers observed X and found S target endmembers o g 0.8 ~ ~0.6 ., ~ ~ 0.4 o O~------' 50 100 150 wavelength O~------' 200 A 50 100 150 wavelength B 200 0.2'---~------' 0.4 0.6 0.8 wavelength=30 C Figure 1: Results for noise-free artificial mixture. A recovered endmembers using MAP technique. B "true" target endmembers. C scatter plot of samples across 2 bands showing the absorption of the three endmembers computed by MAP (open circles). 3.2 Noisy Case: Real Mixtures To validate the noise model MAP framework of section 2.4 we conducted an experiment using ground truthed USGS data representing real mixtures. We selected lOxl0 blocks of pixels from three different regions 3 in the AVIRIS data of the Cuprite, Nevada mining district. We separate these 300 mixed spectra assuming two endmembers and an AR detrending with 5 AR coefficients and the MAP techniques of section 2.4. Overall brightness was accounted for as explain in the linear modeling of section 1. The endmembers are shown in figure 2A and B in comparison to laboratory spectra from the USGS spectral library for these minerals [8J . Figure 2C shows the corresponding abundances, which match the ground truth; region (III) mainly consists of Muscovite while regions (1)+(I1) contain (areal) mixtures of Kaolinite and Muscovite. 4 Discussion Hyperspectral unmixing is a challenging practical problem for unsupervised learning. Our probabilistic approach leads to several interesting algorithms: (1) simplex fitting, (2) constrained ICA and (3) constrained least squares that can efficiently use multi-channel information. An important element of our approach is the explicit use of prior information. Our simulation examples show that we can recover the endmembers, even in the presence of noise and model uncertainty. The approach described in this paper does not yet exploit local correlations between neighboring pixels that are well known to exist. Future work will therefore exploit not only spectral but also spatial prior information for detecting objects and materials. Acknowledgments We would like to thank Gregg Swayze at the USGS for assistance in obtaining the data. 3The regions were from the image plate2.cuprite95.alpha.2um.image.wlocals.gif in ftp:/ /speclab.cr.usgs.gov /pub/cuprite/gregg.thesis.images/, at the coordinates (265,710) and (275,697), which contained Kaolinite and Muscovite 2, and (143,661), which only contained Muscovite 2. L. Parra, C. D, Spence, P Sajda, A. Ziehe and K-R. Muller 948 Muscovite Kaolinite 0.8 0.7 0 .65 0.6 0.6 0.55 0.4 0.5 'c .?.? ", "'0 .. ' ., 0.45 0.3 0.4,--~--:-:-:-"-~----:--:--~ 160 190 200 waveleng1h A 210 220 180 190 200 wavelength B 210 220 C Figure 2: A Spectra of computed endmember (solid line) vs Muscovite sample spectra from the USGS data base library. Note we show only part of the spectrum since the discriminating features are located only between band 172 and 220. B Computed endmember (solid line) vs Kaolinite sample spectra from the USGS data base library. C Abundances for Kaolinite and Muscovite for three regions (lighter pixels represent higher abundance). Region 1 and region 2 have similar abundances for Kaolinite and Muscovite, while region 3 contains more Muscovite. References [1] J. Bayliss, J. A. Gualtieri, and R. Cromp. Analyzing hyperspectral data with independent component analysis. In J. M. Selander, editor, Proc. SPIE Applied Image and Pattern Recognition Workshop, volume 9, P.O. Box 10, Bellingham WA 98227-0010, 1997. SPIE. [2] J.W. Boardman and F.A. Kruse. Automated spectral analysis: a geologic example using AVIRIS data, north Grapevine Mountains, Nevada. In Tenth Thematic Conference on Geologic Remote Sensing, pages 407-418, Ann arbor, MI, 1994. Environmental Research Institute of Michigan. [3] S. Haykin. Adaptive Filter Theory. Prentice Hall, 1991. [4] F. Maselli, , M. Pieri, and C. Conese. Automatic identification of end-members for the spectral decomposition of remotely sensed scenes. Remote Sensing for Geography, Geology, Land Planning, and Cultural Heritage (SPIE) , 2960:104109,1996. [5] B. Pearlmutter and L. Parra. Maximum likelihood blind source separation: A context-sensitive generalization ofICA. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 613-619, Cambridge MA, 1997. MIT Press. [6] J.J. Settle. Linear mixing and the estimation of ground cover proportions. International Journal of Remote Sensing, 14:1159-1177,1993. [7] M.O. Smith, J .B. Adams, and A.R. Gillespie. Reference endmembers for spectral mixture analysis. In Fifth Australian remote sensing conference, volume 1, pages 331-340, 1990. [8] U.S. Geological Survey. USGS digital spectral library. Open File Report 93-592, 1993. [9] H. Szu and C. Hsu. Landsat spectral demixing a la superresolution of blind matrix inversion by constraint MaxEnt neural nets. In Wavelet Applications IV, volume 3078, pages 147-160. SPIE, 1997. 

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U nmixing Hyperspectral Data Lucas Parra, Clay Spence, Paul Sajda Sarnoff Corporation, CN-5300, Princeton, NJ 08543, USA {lparra, cspence,psajda} @sarnoff.com Andreas Ziehe, Klaus-Robert Miiller GMD FIRST.lDA, Kekulestr. 7, 12489 Berlin, Germany {ziehe,klaus}@first.gmd.de Abstract In hyperspectral imagery one pixel typically consists of a mixture of the reflectance spectra of several materials, where the mixture coefficients correspond to the abundances of the constituting materials. We assume linear combinations of reflectance spectra with some additive normal sensor noise and derive a probabilistic MAP framework for analyzing hyperspectral data. As the material reflectance characteristics are not know a priori, we face the problem of unsupervised linear unmixing. The incorporation of different prior information (e.g. positivity and normalization of the abundances) naturally leads to a family of interesting algorithms, for example in the noise-free case yielding an algorithm that can be understood as constrained independent component analysis (ICA). Simulations underline the usefulness of our theory. 1 Introduction Current hyperspectral remote sensing technology can form images of ground surface reflectance at a few hundred wavelengths simultaneously, with wavelengths ranging from 0.4 to 2.5 J.Lm and spatial resolutions of 10-30 m. The applications of this technology include environmental monitoring and mineral exploration and mining. The benefit of hyperspectral imagery is that many different objects and terrain types can be characterized by their spectral signature. The first step in most hyperspectral image analysis systems is to perform a spectral unmixing to determine the original spectral signals of some set of prime materials. The basic difficulty is that for a given image pixel the spectral reflectance patterns of the surface materials is in general not known a priori. However there are general physical and statistical priors which can be exploited to potentially improve spectral unmixing. In this paper we address the problem of unmixing hyperspectral imagery through incorporation of physical and statistical priors within an unsupervised Bayesian framework. We begin by first presenting the linear superposition model for the reflectances measured. We then discuss the advantages of unsupervised over supervised systems. Unmixing Hyperspectral Data 943 We derive a general maximum a posteriori (MAP) framework to find the material spectra and infer the abundances. Interestingly, depending on how the priors are incorporated, the zero noise case yields (i) a simplex approach or (ii) a constrained leA algorithm. Assuming non-zero noise our MAP estimate utilizes a constrained least squares algorithm. The two latter approaches are new algorithms whereas the simplex algorithm has been previously suggested for the analysis of hyperspectral data. Linear Modeling To a first approximation the intensities X (Xi>.) measured in each spectral band A = 1, ... , L for a given pixel i = 1, ... , N are linear combinations of the reflectance characteristics S (8 m >.) of the materials m = 1, ... , M present in that area. Possible errors of this approximation and sensor noise are taken into account by adding a noise term N (ni>'). In matrix form this can be summarized as X = AS + N, subject to: AIM = lL, A ~ 0, (1) where matrix A (aim) represents the abundance of material m in the area corresponding to pixel i, with positivity and normalization constraints. Note that ground inclination or a changing viewing angle may cause an overall scale factor for all bands that varies with the pixels. This can be incorporated in the model by simply replacing the constraint AIM = lL with AIM ~ lL which does does not affect the discussion in the remainder of the paper. This is clearly a simplified model of the physical phenomena. For example, with spatially fine grained mixtures, called intimate mixtures, multiple reflectance may causes departures from this first order model. Additionally there are a number of inherent spatial variations in real data, such as inhomogeneous vapor and dust particles in the atmosphere, that will cause a departure from the linear model in equation (1). Nevertheless, in practical applications a linear model has produced reasonable results for areal mixtures. Supervised vs. Unsupervised techniques Supervised spectral un mixing relies on the prior knowledge about the reflectance patterns S of candidate surface materials, sometimes called endmembers, or expert knowledge and a series of semiautomatic steps to find the constituting materials in a particular scene. Once the user identifies a pixel i containing a single material, i.e. aim = 1 for a given m and i, the corresponding spectral characteristics of that material can be taken directly from the observations, i.e., 8 m >. = Xi>. [4]. Given knowledge about the endmembers one can simply find the abundances by solving a constrained least squares problem. The problem with such supervised techniques is that finding the correct S may require substantial user interaction and the result may be error prone, as a pixel that actually contains a mixture can be misinterpreted as a pure endmember. Another approach obtains endmembers directly from a database. This is also problematic because the actual surface material on the ground may not match the database entries, due to atmospheric absorption or other noise sources. Finding close matches is an ambiguous process as some endmembers have very similar reflectance characteristics and may match several entries in the database. Unsupervised unmixing, in contrast, tries to identify the endmembers and mixtures directly from the observed data X without any user interaction. There are a variety of such approaches. In one approach a simplex is fit to the data distribution [7, 6, 2]. The resulting vertex points of the simplex represent the desired endmembers, but this technique is very sensitive to noise as a few boundary points can potentially change the location of the simplex vertex points considerably. Another approach by Szu [9] tries to find abundances that have the highest entropy subject to constraints that the amount of materials is as evenly distributed as possible - an assumption L. Parra, C. D. Spence, P Sajda, A. Ziehe and K.-R. Muller 944 which is clearly not valid in many actual surface material distributions. A relatively new approach considers modeling the statistical information across wavelength as statistically independent AR processes [1]. This leads directly to the contextual linear leA algorithm [5]. However, the approach in [1] does not take into account constraints on the abundances, noise, or prior information. Most importantly, the method [1] can only integrate information from a small number of pixels at a time (same as the number of endmembers). Typically however we will have only a few endmembers but many thousand pixels. 2 2.1 The Maximum A Posterior Framework A probabilistic model of unsupervised spectral unmixing Our model has observations or data X and hidden variables A, S, and N that are explained by the noisy linear model (1). We estimate the values of the hidden variables by using MAP (A SIX) p , = p(XIA, S)p(A, S) = Pn(XIA, S)Pa(A)ps(S) p(X) p(X) (2) with Pa(A), Ps(S), Pn(N) as the a priori assumptions of the distributions. With MAP we estimate the most probable values for given priors after observing the data, A MAP , SMAP = argmaxp(A, SIX) (3) A,S Note that for maximization the constant factor p(X) can be ignored. Our first assumption, which is indicated in equation (2) is that the abundances are independent of the reflectance spectra as their origins are completely unrelated: (AO) A and S are independent. The MAP algorithm is entirely defined by the choices of priors that are guided by the problem of hyperspectral unmixing: (AI) A represent probabilities for each pixel i. (A2) S are independent for different material m. (A3) N are normal i.i.d. for all i, A. In summary, our MAP framework includes the assumptions AO-A3. 2.2 Including Priors Priors on the abundances be represented as, Positivity and normalization of the abundances can (4) eo where 60 represent the Kronecker delta function and the step function. With this choice a point not satisfying the constraint will have zero a posteriori probability. This prior introduces no particular bias of the solutions other then abundance constraints. It does however assume the abundances of different pixels to be independent. Prior on spectra Usually we find systematic trends in the spectra that cause significant correlation. However such an overall trend can be subtracted and/or filtered from the data leaving only independent signals that encode the variation from that overall trend. For example one can capture the conditional dependency structure with a linear auto-regressive (AR) model and analyze the resulting "innovations" or prediction errors [3]. In our model we assume that the spectra represent independent instances of an AR process having a white innovation process em.>. distributed according to Pe(e). With a Toeplitz matrix T of the AR coefficients we 945 Unmixing Hyperspectral Data can write, em = Sm T. The AR coefficients can be found in a preprocessing step on the observations X. If S now represents the innovation process itself, our prior can be represented as, M Pe (S) <X Pe(ST) = L L II II Pe( L sm>.d>.>.,) , (5) m=1 >.=1 >.'=1 Additionally Pe (e) is parameterized by a mean and scale parameter and potentially parameters determining the higher moments of the distributions. For brevity we ignore the details of the parameterization in this paper. Prior on the noise As outlined in the introduction there are a number of problems that can cause the linear model X = AS to be inaccurate (e.g. multiple reflections, inhomogeneous atmospheric absorption, and detector noise.) As it is hard to treat all these phenomena explicitly, we suggest to pool them into one noise variable that we assume for simplicity to be normal distributed with a wavelength dependent noise variance a>., L p(XIA, S) = Pn(N) = N(X - AS,~) = II N(x>. - As>., a>.l) , (6) >.=1 where N (', .) represents a zero mean Gaussian distribution, and 1 the identity matrix indicating the independent noise at each pixel. 2.3 MAP Solution for Zero Noise Case Let us consider the noise-free case. Although this simplification may be inaccurate it will allow us to greatly reduce the number of free hidden variables - from N M + M L to M2 . In the noise-free case the variables A, S are then deterministically dependent on each other through a N L-dimensional 8-distribution, Pn(XIAS) = 8(X - AS). We can remove one of these variables from our discussion by integrating (2). It is instructive to first consider removing A p(SIX) <X I dA 8(X - AS)Pa(A)ps(S) = IS- 1IPa(XS- 1 )Ps(S). (7) We omit tedious details and assume L = M and invertible S so that we can perform the variable substitution that introduces the Jacobian determinant IS-II . Let us consider the influence of the different terms. The Jacobian determinant measures the volume spanned by the endmembers S. Maximizing its inverse will therefore try to shrink the simplex spanned by S. The term Pa(XS- 1 ) should guarantee that all data points map into the inside of the simplex, since the term should contribute zero or low probability for points that violate the constraint. Note that these two terms, in principle, define the same objective as the simplex envelope fitting algorithms previously mentioned [2]. In the present work we are more interested in the algorithm that results from removing S and finding the MAP estimate of A. We obtain (d. Eq.(7)) p(AIX) oc I dS 8(X - AS)Pa(A)ps(S) = IA -llps(A- 1 X)Pa(A). (8) For now we assumed N = M. 1 If Ps (S) factors over m , i.e. endmembers are independent, maximizing the first two terms represents the leA algorithm. However, lIn practice more frequently we have N > M. In that case the observations X can be mapped into a M dimensional subspace using the singular value decomposition (SVD) , X = UDV T , The discussion applies then to the reduced observations X = u1x with U M being the first M columns of U . L. Parra. C. D. Spence. P Sajda. A. Ziehe and K.-R. Muller 946 the prior on A will restrict the solutions to satisfy the abundance constraints and bias the result depending on the detailed choice of Pa(A), so we are led to constrained ICA. In summary, depending on which variable we integrate out we obtain two methods for solving the spectral unmixing problem: the known technique of simplex fitting and a new constrained ICA algorithm. 2.4 MAP Solution for the Noisy Case Combining the choices for the priors made in section 2.2 (Eqs.(4), (5) and (6)) with (2) and (3) we obtain AMAP, SMAP = "''i~ax ft {g N(x", - a,s" a,) ll. P,(t. 'm,d",) } , (9) subject to AIM = lL, A 2: O. The logarithm of the cost function in (9) is denoted by L = L(A, S). Its gradient with respect to the hidden variables is = _AT nm diag(O')-l - fs(sm) 88L Sm (10) where N = X - AS, nm are the M column vectors of N, fs(s) = - olnc;(s). In (10) fs is applied to each element of Sm. The optimization with respect to A for given S can be implemented as a standard weighted least squares (L8) problem with a linear constraint and positivity bounds. Since the constraints apply for every pixel independently one can solve N separate constrained LS problems of M unknowns each. We alternate between gradient steps for S and explicit solutions for A until convergence. Any additional parameters of Pe(e) such as scale and mean may be obtained in a maximum likelihood (ML) sense by maximizing L. Note that the nonlinear optimization is not subject to constraints; the constraints apply only in the quadratic optimization. 3 Experiments 3.1 Zero Noise Case: Artificial Mixtures In our first experiment we use mineral data from the United States Geological Survey (USGS)2 to build artificial mixtures for evaluating our unsupervised unmixing framework. Three target endmembers where chosen (Almandine WS479 , Montmorillonite+Illi CM42 and Dickite NMNH106242). A spectral scene of 100 samples was constructed by creating a random mixture of the three minerals. Of the 100 samples, there were no pure samples (Le. no mineral had more than a 80% abundance in any sample). Figure 1A is the spectra of the endmembers recovered by the constrained ICA technique of section 2.3, where the constraints were implemented with penalty terms added to the conventional maximum likelihood ICA algorithm. These are nearly identical to the spectra of the true endmembers, shown in figure 1B, which were used for mixing. Interesting to note is the scatter-plot of the 100 samples across two bands. The open circles are the absorption values at these two bands for endmembers found by the MAP technique. Given that each mixed sample consists of no more than 80% of any endmember, the endmember points on the scatter-plot are quite distant from the cluster. A simplex fitting technique would have significant difficulty recovering the endmembers from this clustering. 2see http://speclab.cr .usgs.gov /spectral.lib.456.descript/ decript04.html 947 Unmixing Hyperspectral Data found endmembers observed X and found S target endmembers o g 0.8 ~ ~0.6 ., ~ ~ 0.4 o O~------' 50 100 150 wavelength O~------' 200 A 50 100 150 wavelength B 200 0.2'---~------' 0.4 0.6 0.8 wavelength=30 C Figure 1: Results for noise-free artificial mixture. A recovered endmembers using MAP technique. B "true" target endmembers. C scatter plot of samples across 2 bands showing the absorption of the three endmembers computed by MAP (open circles). 3.2 Noisy Case: Real Mixtures To validate the noise model MAP framework of section 2.4 we conducted an experiment using ground truthed USGS data representing real mixtures. We selected lOxl0 blocks of pixels from three different regions 3 in the AVIRIS data of the Cuprite, Nevada mining district. We separate these 300 mixed spectra assuming two endmembers and an AR detrending with 5 AR coefficients and the MAP techniques of section 2.4. Overall brightness was accounted for as explain in the linear modeling of section 1. The endmembers are shown in figure 2A and B in comparison to laboratory spectra from the USGS spectral library for these minerals [8J . Figure 2C shows the corresponding abundances, which match the ground truth; region (III) mainly consists of Muscovite while regions (1)+(I1) contain (areal) mixtures of Kaolinite and Muscovite. 4 Discussion Hyperspectral unmixing is a challenging practical problem for unsupervised learning. Our probabilistic approach leads to several interesting algorithms: (1) simplex fitting, (2) constrained ICA and (3) constrained least squares that can efficiently use multi-channel information. An important element of our approach is the explicit use of prior information. Our simulation examples show that we can recover the endmembers, even in the presence of noise and model uncertainty. The approach described in this paper does not yet exploit local correlations between neighboring pixels that are well known to exist. Future work will therefore exploit not only spectral but also spatial prior information for detecting objects and materials. Acknowledgments We would like to thank Gregg Swayze at the USGS for assistance in obtaining the data. 3The regions were from the image plate2.cuprite95.alpha.2um.image.wlocals.gif in ftp:/ /speclab.cr.usgs.gov /pub/cuprite/gregg.thesis.images/, at the coordinates (265,710) and (275,697), which contained Kaolinite and Muscovite 2, and (143,661), which only contained Muscovite 2. L. Parra, C. D, Spence, P Sajda, A. Ziehe and K-R. Muller 948 Muscovite Kaolinite 0.8 0.7 0 .65 0.6 0.6 0.55 0.4 0.5 'c .?.? ", "'0 .. ' ., 0.45 0.3 0.4,--~--:-:-:-"-~----:--:--~ 160 190 200 waveleng1h A 210 220 180 190 200 wavelength B 210 220 C Figure 2: A Spectra of computed endmember (solid line) vs Muscovite sample spectra from the USGS data base library. Note we show only part of the spectrum since the discriminating features are located only between band 172 and 220. B Computed endmember (solid line) vs Kaolinite sample spectra from the USGS data base library. C Abundances for Kaolinite and Muscovite for three regions (lighter pixels represent higher abundance). Region 1 and region 2 have similar abundances for Kaolinite and Muscovite, while region 3 contains more Muscovite. References [1] J. Bayliss, J. A. Gualtieri, and R. Cromp. Analyzing hyperspectral data with independent component analysis. In J. M. Selander, editor, Proc. SPIE Applied Image and Pattern Recognition Workshop, volume 9, P.O. Box 10, Bellingham WA 98227-0010, 1997. SPIE. [2] J.W. Boardman and F.A. Kruse. Automated spectral analysis: a geologic example using AVIRIS data, north Grapevine Mountains, Nevada. In Tenth Thematic Conference on Geologic Remote Sensing, pages 407-418, Ann arbor, MI, 1994. Environmental Research Institute of Michigan. [3] S. Haykin. Adaptive Filter Theory. Prentice Hall, 1991. [4] F. Maselli, , M. Pieri, and C. Conese. Automatic identification of end-members for the spectral decomposition of remotely sensed scenes. Remote Sensing for Geography, Geology, Land Planning, and Cultural Heritage (SPIE) , 2960:104109,1996. [5] B. Pearlmutter and L. Parra. Maximum likelihood blind source separation: A context-sensitive generalization ofICA. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 613-619, Cambridge MA, 1997. MIT Press. [6] J.J. Settle. Linear mixing and the estimation of ground cover proportions. International Journal of Remote Sensing, 14:1159-1177,1993. [7] M.O. Smith, J .B. Adams, and A.R. Gillespie. Reference endmembers for spectral mixture analysis. In Fifth Australian remote sensing conference, volume 1, pages 331-340, 1990. [8] U.S. Geological Survey. USGS digital spectral library. Open File Report 93-592, 1993. [9] H. Szu and C. Hsu. Landsat spectral demixing a la superresolution of blind matrix inversion by constraint MaxEnt neural nets. In Wavelet Applications IV, volume 3078, pages 147-160. SPIE, 1997. Invariant Feature Extraction and Classification in Kernel Spaces Sebastian Mika l , Gunnar Ratsch l , Jason Weston 2 , Bernhard Sch8lkopf3, Alex Smola4 , and Klaus-Robert Muller l 1 GMD FIRST, Kekulestr. 7,12489 Berlin, Germany 2 Barnhill BioInformatics, 6709 Waters Av., Savannah, GR 31406, USA 3 Microsoft Research Ltd., 1 Guildhall Street, Cambridge CB2 3NH, UK 4 Australian National University, Canberra, 0200 ACT, Australia {mika, raetsch, klaus }@first.gmd.de, jasonw@dcs.rhbnc.ac.uk bsc@microsoft.com, Alex.Smola.anu.edu.au Abstract We incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination. Employing a unified framework in terms of a nonlinear variant of the Rayleigh coefficient, we propose non-linear generalizations of Fisher's discriminant and oriented PCA using Support Vector kernel functions . Extensive simulations show the utility of our approach. 1 Introduction It is common practice to preprocess data by extracting linear or nonlinear features. The most well-known feature extraction technique is principal component analysis PCA (e.g. [3]). It aims to find an orthonormal, ordered basis such that the i-th direction describes as much variance as possible while maintaining orthogonality to all other directions. However, since PCA is a linear technique, it is too limited to capture interesting nonlinear structure in a data set and nonlinear generalizations have been proposed, among them Kernel PCA [14], which computes the principal components of the data set mapped nonlinearly into some high dimensional feature space F. Often one has prior information, for instance, we might know that the sample is corrupted by noise or that there are invariances under which a classification should not change. For feature extraction, the concepts of known noise or transformation invariance are to a certain degree equivalent, i.e. they can both be interpreted as causing a change in the feature which ought to be minimized. Clearly, invariance alone is not a sufficient condition for a good feature, as we could simply take the constant function. What one would like to obtain is a feature which is as invariant as possible while still covering as much of the information necessary for describing the particular data. Considering only one (linear) feature vector wand restricting to first and second order statistics of the data one arrives at a maximization of the so called Rayleigh coefficient (1) 527 Invariant Feature Extraction and Classification in Kernel Spaces where w is the feature vector and Sf, SN are matrices describing the desired and undesired properties of the feature , respectively (e.g. information and noise). If S/ is the data covariance and SN the noise covariance, we obtain oriented PCA [3J . If we leave the field of data description to perform supervised classification, it is common to choose S / as the separability of class centers (between class variance) and SN to be the within class variance. In that case , we recover the well known Fisher Discriminant [7J. The ratio in (1) is maximized when we cover much of the information coded by S/ while avoiding the one coded by SN . The problem is known to be solved, in analogy to PCA , by a generalized symmetric eigenproblem S/w = >"SNW [3], where>.. E ~ is the corresponding (biggest) eigenvalue. In this paper we generalize this setting to a nonlinear one. In analogy to [8, 14J we first map the data via some nonlinear mapping <l> to some high-dimensional feature space F and then optimize (1) in F . To avoid working with the mapped data explicitly (which might be impossible if F is infinite dimensional) we introduce support vector kernel functions [11], the well-known kernel trick. These kernel functions k(x , y) compute a dot product in some feature space F , i.e. k(x , y) = (<l>(x)? <l>(y)) . Formulating the algorithms in Fusing <l> only in dot products , we can replace any occurrence of a dot product by the kernel function k. Possible choices for k which have proven useful e.g. in Support Vector Machines [2] or Kernel PCA [14J are Gaussian RBF, k(x , y) = exp( -llx - yI12/ c), or polynomial kernels , k(x , y) = (x? y)d , for some positive constants c E ~ and dEN, respectively. The remainder of this paper is organized as follows: The next section shows how to formulate the optimization problem induced by (1) in feature space. Section 3 considers various ways to find Fisher's Discriminant in F; we conclude with extensive experiments in section 4 and a discussion of our findings. 2 Kernelizing the Rayleigh Coefficient To optimize (1) in some kernel feature space F we need to find a formulation which uses only dot products of <l>-images. As numerator and denominator are both scalars this can be done independently. Furthermore, the matrices S/ and SN are basically covariances and thus the sum over outer products of <l>-images. Therefore, and due to the linear nature of (1) every solution W E F can be written as an expansion in terms of mapped training datal, i.e. l W= L Cti<l>(Xi). (2) i =l To define some common choices in F let X = {Xl , .. . ,xe} be our training sample and, where appropriate, Xl U X2 = X , Xl n X2 = 0, two subclasses (with IXi I = ?i). We get the full covariance of X by C= 1 f L (<l>(x) - m)(<l>(x) - m)T with m = f1 L <l>(x) , ~EX (3) ~EX I SB and Sw are operators on a (finite-dimensional) subspace spanned by the CP(Xi ) (in a possibly infinite space). Let w = VI + V2, where VI E Span(CP(Xi) : i = 1, .. . , f) and V2 1. Span(CP(xi) : i = 1, ... , f) . Then for S Sw or S SB (which are both symmetric) = (w , Sw) ((VI ((VI = + V2) , S(VI + V2)) + V2)S, VI) (VI , SVI) As VI lies in the span of the cp(Xi) and S only operates on this subspace there exist an expansion of w which maximizes J(w) . S. Mika. G. Riitsch. J. Weston. B. Scholkopj, A. J. Smola and K.-R. Muller 528 which could be used as Sf in oriented Kernel PCA. For SN we could use an estimate of the noise covariance, analogous to the definition of C but over mapped patterns sampled from the assumed noise distribution. The standard formulation of the Fisher discriminant in F, yielding the Kernel Fisher Discriminant (KFD) [8] is given by Sw = L L (cJ>(x) - mi)(cJ>(x) - mdT and SB = (m2 - mt}(m2 - ml)T, i=I,2 xEX; the within-class scatter Sw (as SN), and the between class scatter SB ( as Sf). Here mi is the sample mean for patterns from class i. To incorporate a known invariance e.g. in oriented Kernel PCA, one could use the tangent covariance matrix [12], T 1 = ft 2 L (cJ>(x) - cJ>(?tx))(cJ>(x) - cJ>(?tx))T for some small t> O. (4) :IlEX Here ?t is a local I-parameter transformation. T is a finite difference approximation t of the covariance of the tangent of ?t at point cJ>(x) (details e.g. in [12]). Using Sf = C and SN = T in oriented Kernel PCA, we impose invariance under the local transformation ?t. Crucially, this matrix is not only constructed from the training patterns X. Therefore, the argument used to find the expansion (2) is slightly incorrect. Neverthless, we can assume that (2) is a reasonable approximation for describing the variance induced by T. Multiplying either of these matrices from the left and right with the expansion (2), we can find a formulation which uses only dot products. For the sake of brevity, we only give the explicit formulation of (1) in F for KFD (cf. [8] for details) . Defining (I-'i)j = L:IlEXi k(xj,x) we can write (1) for KFD as t J(a) = (aTI-') 2 aTNa aTMa aTNa' (5) where N = KKT - Li=1,2fil-'iI-'T, I-' = 1-'2 - 1-'1 ' M = I-'I-'T, and Kij = k(xi,xj). The results for other choices of Sf and SN in F as for the cases of oriented kernel PCA or transformation invariance can be obtained along the same lines. Note that we still have to maximize a Rayleigh coefficient. However, now it is a quotient in terms of expansion coefficients a, and not in terms of w E F which is a potentially infinite-dimensional space. Furthermore, it is well known that the solution for this special eigenproblem is in the direction of N- 1(1-'2 - 1-'1) [7), which can be solved using e.g. a Cholesky factorization of N. The projection of a new pattern x onto w in F can then be computed by l (w? cJ>(x)) = LQik(xi'x). (6) i=1 3 Algorithms Estimating a covariance matrix with rank up to f from f samples is ill-posed. Furthermore, by performing an explicit centering in F each covariance matrix loses one more dimension, i.e. it has only rank f - 1 (even worse, for KFD the matrix N has rank f - 2). Thus the ratio in (1) is not well defined anymore, as the denominator might become zero. In the following we will propose several ways to deal with this problem in KFD. Furthermore we will tackle the question how to solve the optimization problem of KFD more efficiently. So far, we have an eigenproblem of size .e x .e. If .e becomes large this is numerically demanding. Reformulations of the original problem allow to overcome some of these limitations. Finally, we describe the connection between KFD and RBF networks. Invariant Feature Extraction and Classification in Kernel Spaces 3.1 529 Regularization and Solution on a Subspace As noted before, the matrix N has only rank ? - 2. Besides numerical problems which can cause the matrix N to be not even positive, we could think of imposing some regularization to control capacity in F. To this end, we simply add a mUltiple of the identity matrix to N, Le. replace N by NJ1. where NJ1. := N + /-LI. (7) This can be viewed in different ways: (i) for /-L > 0 it makes the problem feasible and numerically more stable as NJ1. becomes positive; (ii) it can be seen as decreasing the bias in sample based estimation of eigenvalues (cf. [6)); (iii) it imposes a regularization on 11011 2, favoring solutions with small expansion coefficients. furthermore, one could use other regularization type additives to N, e.g. penalizing IIwl1 2 in analogy to SVM (by adding the kernel matrix Kij = k(xi' Xj)). To optimize (5) we need to solve an ? x ? eigenproblem, which might be intractable for large ?. As the solutions are not sparse one can not directly use efficient algorithms like chunking for Support Vector Machines (cf. [13]). To this end, we might restrict the solution to lie in a subspace, Le. instead of expanding w by (2) we write (8) i=l with m < l. The patterns Zi could either be a subset of the training patterns X or e.g. be estimated by some clustering algorithm. The derivation of (5) does not change, only K is now m x ? and we end up with m x m matrices N and M. Another advantage is, that it increases the rank of N (relative to its size) although there still might be some need for regularization. 3.2 Quadratic optimization and Sparsification Even if N has full rank, maximizing (5) is underdetermined: if 0 is optimal, then so is any multiple thereof. Since 0T M 0 = (0T J..L)2, M has rank one. Thus we can seek for a vector 0, such that oTNo is minimal for fixed OTJ..L (e.g. to 1). The solution is unique and we can find the optimal 0 by solving the quadratic optimization problem: (9) Although the quadratic optimization problem is not easier to solve than the eigenproblem, it has an appealing interpretation. The constraint 0 T J..L = 1 ensures, that the average class distance, projected onto the direction of discrimination, is constant, while the intra class variance is minimized, i.e. we maximize the average margin. Contrarily, the SVM approach [2] optimizes for a large minimal margin. Considering (9) we are able to overcome another shortcoming of KFD . The solutions 0 are not sparse and thus evaluating (6) is expensive. To solve this we can add an h-regularizer >'110111 to the objective function, where>. is a regularization parameter allowing us to adjust the degree of sparseness. 3.3 Connection to RBF Networks Interestingly, there exists a close connection between RBF networks (e.g. [9, 1)) and KFD. If we add no regularization and expand in all training patterns, we find that an optimal 0 is given by 0 = K- 1 y, where K is the symmetric, positive matrix of all kernel elements k(xi' Xj) and y the ?1 label vector2. A RBF-network with the 2To see this, note that N can be written as N = KDK where D = I -YIyT -Y2Y; has rank e- 2, while Yi is the vector of l/Vli's for patterns from class i and zero otherwise. 530 S. Mika, G. Ratsch, J. Weston, B. SchOlkopf, A. J. Smola and K.-R. Muller RBF AB ABR SVM KFD Banana 10.8?O.06 12.3?O.07 10.9?0.04 1l.5?O.07 10.8?O.05 B.Cancer 27.6?0.47 30.4?0.47 26.5?0.45 26.o?O.4725.8?0.46 Diabetes 24.3?O.19 26.5?O.23 23.8?O.18 23.5?0.17 23.2?O.16 24.7?O.24 27.5?O.25 24.3?O.21 23.6?O.21 23.1?0.22 German Heart 17.6?O.33 20.3?O.34 16.5?O.35 16.0?O.33 16.1?0.34 Image 3.3?O.06 2.1?O.01 2.1?O.06 3.o?O.06 4.8?O.06 1.7?O.02 1.9?O.03 1.6?0.01 1.7?O.01 1.5?O.01 Ringnorm 34.4?O.20 35.7?O.18 34.2?O.22 32.4?O.18 33.2?0.11 F.Sonar Splice 10.o?O.10 10.1?O.05 9 .5?O.01 10.9?O.07 10.5?O.06 Thyroid 4.5?O.21 4.4?O.22 4.6?O.22 4.8?O.22 4.2?O.21 23.3?O.13 22.6?O.12 22.6?O.12 22.4?O.10 23.2?O.20 Titanic Twonorm 2.9?O.03 3.0?O.03 2.1?O.02 3.0?O.02 2.6?O.02 Waveform 10.7?O.1l 10.8?O.06 9.8?O.08 9. 9? o. 04 9.9?O.04 Table 1: Comparison between KFD, single RBF classifier, AdaBoost (AB), regul. Ada(ABR) Boost and SVMs (see text) . Best result in bold face, second best in italics. same kernel at each sample and fixed kernel width gives the same solution, if the mean squared error between labels and output is minimized. Also for the case of restricted expansions (8) there exists a connection to RBF networks with a smaller number of centers (cf. [4]) . 4 Experiments Kernel Fisher Discriminant Figure 1 shows an illustrative comparison of the features found by KFD, and Kernel PCA. The KFD feature discriminates the two classes, the first Kernel PCA feature picks up the important nonlinear structure. To evaluate the performance of the KFD on real data sets we performed an extensive comparison to other state-of-the-art classifiers, whose details are reported in [8j.3 We compared the Kernel Fisher Discriminant and Support Vector Machines, both with Gaussian kernel, to AdaBoost [5], and regularized AdaBoost [10] (cf. table 1). For KFD we used the regularized within-class scatter (7) and computed projections onto the optimal direction w E :F by means of (6). To use w for classification we have to estimate a threshold. This can be done by e.g. trying all thresholds between two outputs on the training set and selecting the median of those with the smallest empirical error, or (as we did here) by computing the threshold which maximizes the margin on the outputs in analogy to a Support Vector Machine, where we deal with errors on the trainig set by using the SVM soft margin approach. A disadvantage of this is, however , that we have to control the regularization constant for the slack variables. The results in table 1 show the average test error and the standard If K has full rank, the null space of D , which is spanned by Yl and Y2' is the null space of N . For a = K- 1Y we get aTN a = 0 and aTJ.? =I O. As we are free to fix the constraint aT J.? to any positive constant (not just 1), a is also feasible. 3The breast cancer domain was obtained from the University Medical Center, Inst. of Oncology, Ljubljana, Yugoslavia. Thanks to M. Zwitter and M. Soklic for the data. All data sets used in the experiments can be obtained via http://www.first.gmd.de/-raetsch/ . Figure 1: Comparison of feature found by KFD (left) and first Kernel PCA feature (right). Depicted are two classes (information only used by KFD) as dots and crosses and levels of same feature value. Both with polynomial kernel of degree two, KFD with the regularized within class scatter (7) (/1 = 10- 3 ) . Invariant Feature Extraction and Classification in Kernel Spaces 531 deviation of the averages' estimation, over 100 runs with different realizations of the datasets. To estimate the necessary parameters, we ran 5-fold cross validation on the first five realizations of the training sets and took the model parameters to be the median over the five estimates (see [10] for details of the experimental setup). Using prior knowledge. A toy example (figure 2) shows a comparison of Kernel PCA and oriented Kernel PCA, which used S[ as the full covariance (3) and as noise matrix SN the tangent covariance (4) of (i) rotated patterns and (ii) along the x-axis translated patterns. The toy example shows how imposing the desired invariance yields meaningful invariant features. In another experiment we incorporated prior knowledge in KFD. We used the USPS database of handwritten digits, which consists of 7291 training and 2007 test patterns, ~ach 2?6 .dimensional gray scale ima~es of the digits 0 ... 9: We use? the regulanzed withm-class scatter (7) (p, = 10- ) as SN and added to It an multiple A of the tangent covariance (4), i.e. SN = NJj + AT. As invariance transformations we have chosen horizontal and vertical translation, rotation, and thickening (cf. [12]), where we simply averaged the matrices corresponding to each transformation. The feature was extracted by using the restricted expansion (8), where the patterns Zi were the first 3000 training samples. As kernel we have chosen a Gaussian of width 0.3?256, which is optimal for SVMs [12]. For each class we trained one KFD which classified this class against the rest and computed the 10-class error by the winnertakes-all scheme. The threshold was estimated by minimizing the empirical risk on the normalized outputs of KFD. Without invariances, i.e. A = 0, we achieved a test error of 3.7%, slightly better than a plain SVM with the same kernel (4.2%) [12]. For A = 10- 3 , using the tangent covariance matrix led to a very slight improvement to 3.6%. That the result was not significantly better than the corresponding one for KFD (3.7%) can be attributed to the fact that we used the same expansion coefficients in both cases. The tangent covariance matrix, however, lives in a slightly different subspace. And indeed, a subsequent experiment where we used vectors which were obtained by clustering a larger dataset, including virtual examples generated by the appropriate invariance transformation, led to 3.1 %, comparable to an SVM using prior knowledge (e.g. [12]; best SVM result 2.9% with local kernel and virtual support vectors). 5 Conclusion In the task of learning from data it is equivalent to have prior knowledge about e.g. invariances or about specific sources of noise. In the case of feature extraction, we seek features which are sufficiently (noise-) invariant while still describing interesting structure. Oriented PCA and, closely related, Fisher's Discriminant, use particularly simple features, since they only consider first and second order statistics for maximizing the Rayleigh coefficient (1). Since linear methods can be too restricted in many real-world applications, we used Support Vector Kernel functions to obtain nonlinear versions of these algorithms, namely oriented Kernel PCA and Kernel Fisher Discriminant analysis. Our experiments show that the Kernel Fisher Discriminant is competitive or in Figure 2: Comparison of first features found by Kernel PCA and oriented Kernel PCA (see text); from left to right: KPCA, OKPCA with rotation and translation invariance; all with Gaussian kernel. 532 S. Mika, G. Riitsch, J. Weston, B. SchOlkopf, A. J. Smola and K.-R. Muller some cases even superior to the other state-of-the-art algorithms tested. Interestingly, both SVM and KFD construct a hyperplane in :F which is in some sense optimal. In many cases, the one given by the solution w of KFD is superior to the one of SVMs. Encouraged by the preliminary results for digit recognition, we believe that the reported results can be improved, by incorporating different invariances and using e.g. local kernels [12]. Future research will focus on further improvements on the algorithmic complexity of our new algorithms, which is so far larger than the one of the SVM algorithm, and on the connection between KFD and Support Vector Machines (cf. [16, 15]) . Acknowledgments This work was partially supported by grants of the DFG (JA 379/5-2,7-1,9-1) and the EC STORM project number 25387 and carried out while BS and AS were with GMD First. References [1) C.M. Bishop. Neural Networks for Pattern Recognition. Oxford Univ. Press, 1995. [2] B. Boser, 1. Guyon, and V.N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proc. COLT, pages 144- 152. ACM Press, 1992. [3) K.I . Diamantaras and S.Y. Kung. Principal Component Neural Networks. Wiley, New York,1996. [4] B.Q. Fang and A.P. Dawid. Comparison of full bayes and bayes-least squares criteria for normal discrimination. Chinese Journal of Applied Probability and Statistics, 12:401- 410, 1996. [5] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In EuroCOLT 94. LNCS, 1994. [6] J.H. Friedman. Regularized discriminant analysis. Journal of the American Statistical Association, 84(405):165- 175, 1989. [7] K Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press, San Diego, 2nd edition, 1990. [8) S. Mika, G. Ratsch, J. Weston, B. Scholkopf, and K-R. Muller. Fisher discriminant analysis with kernels. In Y.-H. Hu, J . Larsen, E. Wilson, and S. Douglas, editors, Neural Networks for Signal Processing IX, pages 41-48. IEEE, 1999. [9] J. Moody and C. Darken. Fast learning in networks of locally-tuned processing units. Neural Computation, 1(2):281-294, 1989. [10] G. Ratsch, T. Onoda, and K-R. Muller. Soft margins for adaboost. Technical Report NC-TR-1998-021, Royal Holloway College, University of London, UK, 1998. [11] S. Saitoh. Theory of Reproducing Kernels and its Applications. Longman Scientific & Technical, Harlow, England, 1988. [12] B. Scholkopf. Support vector learning. Oldenbourg Verlag, 1997. [13) B. Scholkopf, C.J.C. Burges, and A.J. Smola, editors. Advances in Kernel Methods Support Vector Learning. MIT Press, 1999. [14] B. Scholkopf, A.J. Smola, and K-R. Muller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299-1319, 1998. [15] A. Shashua. On the relationship between the support vector machine for classification and sparsified fisher's linear discriminant. Neural Processing Letters, 9(2):129- 139, April 1999. [16) S. Tong and D. Koller. Bayes optimal hyperplanes --+ maximal margin hyperplanes. Submitted to IJCA1'99 WorkshOp on Support Vector Machines (robotics. stanford. edurkoller/), 1999. 

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Bayesian Map Learning in Dynamic Environments Kevin P. Murphy Computer Science Division University of California Berkeley, CA 94720-1776 murphyk@cs.berkeley.edu Abstract We consider the problem of learning a grid-based map using a robot with noisy sensors and actuators. We compare two approaches: online EM, where the map is treated as a fixed parameter, and Bayesian inference, where the map is a (matrix-valued) random variable. We show that even on a very simple example, online EM can get stuck in local minima, which causes the robot to get "lost" and the resulting map to be useless. By contrast, the Bayesian approach, by maintaining multiple hypotheses, is much more robust. We then introduce a method for approximating the Bayesian solution, called Rao-Blackwellised particle filtering. We show that this approximation, when coupled with an active learning strategy, is fast but accurate. 1 Introduction The problem of getting mobile robots to autonomously learn maps of their environment has been widely studied (see e.g., [9] for a collection of recent papers). The basic difficulty is that the robot must know exactly where it is (a problem called localization), so that it can update the right part of the map. However, to know where it is, the robot must already have a map: relying on dead-reckoning alone (Le., integrating the motor commands) is unreliable because of noise in the actuators (slippage and drift). One obvious solution is to use EM, where we alternate between estimating the location given the map (the E step), and estimating the map given the location (the M step). Indeed, this approach has been successfully used by several groups [8, 11, 12]. However, in all of these works, the trajectory of the robot was specified by hand, and the map was learned off-line. For fully autonomous operation, and to cope with dynamic environments, the map must be learned online. We consider two approaches to online learning: online EM, and Bayesian inference, K. P Murphy 1016 a c Figure 1: (a) The POMDP represented as a graphical model. L t is the location, Mt(i) is the label of the i'th grid cell, At is the action, and Zt is the observation. Dotted circles denote variables that EM treats as parameters. (b) A one-dimensional grid with binary labels (white = 0, black = 1). (c) A two-dimensional grid, with four labels (closed doors, open doors, walls, and free space). where we treat the map as a random variable. In Section 3, we show that the Bayesian approach can lead to much better results than online EM; unfortunately, it is computationally intractable, so in Section 4, we discuss an approximation based on Rao-BIackwellised particle filtering. 2 The model We now precisely define the model that we will use in this paper; it is similar to, but much simpler than, the occupancy grid model in [12]. The map is defined to be a grid, where each cell has a label which represents what the robot would see at that point. More formally, the map at time t is a vector of discrete random variables, M t (i) E {I, ... , No}, where 1 ::; i ::; N L. Of course, the map is not observed directly, and nor is the robot's location, L t E {I, ... , NL}. What is observed is Zt E {l, ... ,No}, the label of the cell at the robot's current location, and At E {I, ... ,NA}, the action chosen by the robot just before time t. The conditional independence assumptions we are making are illustrated in Figure l(a). We start by considering the very simple one-dimensional grid shown in Figure l(b), where there are just two actions, move right (-+) and move left (f-), and just two labels, off (0) and on (1). This is sufficiently small that we can perform exact Bayesian inference. Later, we will generalize to two dimensions. The prior for the location is a delta function with all its mass on the first (left-most) cell, independent of AI. The transition model for the location is as follows. Pa P r ( L t = J?1 Lt-I =~,. A t =-+ ) = { 11 - Pa o if j = i + 1, j < N if j = i, j < N if j = i = N otherwise where Pa is the probability of a successful action, i.e., 1 - Pa is the probability that the robot's wheels slip. There is an analogous equation for the case when At =f-. Note that it is not possible to pass through the "rightmost" cell; the robot can use this information to help localize itself. The prior for the map is a product of the priors for each cell, which are uniform. (We could model correlation between neighboring cells using a Markov Random Field, although this is computationally expensive.) The transition model for the map is a product of the transition models for each cell, which are defined as follows: Bayesian Map Learning in Dynamic Environments 1017 the probability that a 0 becomes a 1 or vice versa is Pc (probability of change), and hence the probability that the cell label remains the same is 1 - Pc. Finally, the observation model is Pr(Zt = klMt = (mI , ... , mNL), L t = i) = { Po 1- Po if mi = k otherwise where Po is the probability of a succesful observation, Le. , 1 - Po is the probability of a classification error. Another way of writing this, that will be useful later, is to introduce the dummy deterministic variable, which has the following distribution: Pr(Z: = klMt = (mI, ... ,mNL) , L t = i) = 8(k,mi) , where 8(a, b) = 1 if a = b and is 0 otherwise. Thus acts just like a multiplexer, selecting out a component of M t as determined by the "gate" Lt. The output of the multiplexer is then passed through a noisy channel, which flips bits with probability 1 - Po, to produce Zt. Z:, Z: 3 Bayesian learning compared to EM For simplicity, we assume that the parameters Po, Pa and Pc, are all known. (In this section, we use Po = 0.9, Pa = 0.8 and Pc = 0, so the world is somewhat "slippery", but static in appearance.) The state estimation problem is to compute the belief state Pr(L t , MtIYl:t), where Yt = (Zt, At) is the evidence at time t; this is equivalent to performing online inference in the graphical model shown in Figure 1(a). Unfortunately, even though we have assumed that the components of M t are a priori independent, they become correlated by virtue of sharing a common child, Zt. That is, since the true location of the robot is unknown, all of the cells are possible causes of the observation, and they "compete" to "explain" the data. Hence all of the hidden variables become coupled, and the belief state has size O(NL2NL). If the world is static (Le. , Pc = 0) , we can treat M as a fixed , but unknown, parameter; this can then be combined with the noisy sensor model to define an HMM with the following observation matrix: B(i , k) ~ Pr(Zt = kiLt = i; M) = L Pr(Zt = klZ: = j)8(M(i),j) j We can then learn B using EM, as in [8, 11, 12]. (We assume for now that the HMM transition matrix is independent of the map, and encodes the known topology of the grid, Le., the robot can move to any neighboring cell, no matter what its label is. We will lift this restriction in the 2D example. We can formulate an online version of EM as follows. We use fixed-lag s"moothing with a sliding window of length W, and compute the expected sufficient statistics (ESS) for the observation matrix within this window as follows: Ot(i, k) = 2:~=t-W : Z," =k LT1t(i) , where LTlt(i) = Pr(LT = iIYl:t)? We can compute L using the forwards-backwards algorithm, using Lt-W-Ilt-I as the prior. (The initial condition is L = 11", where 11" is the (known) prior for Lo.) Thus the cost per time step is O(2W Nl). In the M step, we normalize each row of Ot + d x Ot-l, where 0 < d < 1 is a decay constant, to get the new estimate of B . We need to downweight the previous ESS since they were computed using out-of-date parameters; in addition, exponential forgetting allows us to handle dynamic environments. [1] discuss some variations on this algorithm. 1018 K. P. Murphy _al_. .,. . . ., '. . __'. . ......,.'I!'. a ~Ir! ' . lei b c d Figure 2: (a) The full joint posterior on P(Mt !Yl:t). 0 and 255, on the axis into the page, represent the maps where every cell is off and every cell is on, respectively; the mode at t = 16 is for map 171, which corresponds to the correct pattern 01010101. (b-d) Estimated map. Light cells are more likely to contains Os, so the correct pattern should have light bars in the odd rows. (b) The marginals of the exact joint. (c) Online EM. (d) Omine EM. As the window length increases, past locations are allowed to look at more and more future data, and hence their estimates become more accurate; however, the space and time requirements increase. Nevertheless, there are occasions when even the maximum window size (i.e., looking all the way back to 'T = 0) will perform poorly, because of the greedy hill-climbing nature of EM. For a simple example of this, consider the environment shown in Figure 1 (b). Suppose the robot starts in cell 1, keeps going right until it comes to the end of the "corridor", and then heads back "home". Suppose further that there is a single slippage error at t = 4, so the actual path and observation sequence of the robot is as follows: t Lt Zt At 1 1 0 2 2 1 3 4 3 o --7 --7 4 1 5 6 7 456 101 8 7 0 --7 --7 --7 --7 --7 9 10 11 876 101 +- +- +- 12 13 14 15 16 54321 01010 +- +- +- +- +- To study the effect of this sequence, we computed Pr(Mt , L t !Yl:t) by applying the junction tree algorithm to the graphical model in Figure l(a). We then marginalized out L t to compute the posterior P(Mt ): see Figure 2(a). At t = 1, there are 27 modes, corresponding to all possible bit patterns on the unobserved cells. At each time step, the robot thinks it is moving one step to the right. Hence at t = 8, the robot thinks it is in cell 8, and observes O. When it tries to move rightf it knows it will remain in cell 8 (since the robot knows where the boundaries are). Hence at t = 9, it is almost 70% confident that it is in cell 8. At t = 9, it observes a 1, which contradicts its previous observation of O. There are two possible explanations: this is a sensor error, or there was a motor error. Which of these is more likely depends on the relative values of the sensor noise, Po, and the system noise, Pa. In our experiments, we found that the motor error hypothesis is much more likely; hence the mode of the posterior jumps from the wrong map (in which M(5) = 1) to the right map (in which M(5) = 0). Furthermore, as the robot returns to "familiar territory", it is able to better localize itself (see Figure 3(a)), and continues to learn the map even for far-away cells, because they are all correlated (in Figure 2(b), the entry for cell 8 becomes sharper even as the robot returns to cell 1) We now compare the Bayesian solution with EM. Online EM with no smoothing was not able to learn the correct map. Adding smoothing with the maximum window size of Wt = t did not improve matters: it is still unable to escape the local Bayesian Map Learning in Dynamic Environments a b 1019 I c Figure 3: Estimated location. Light cells are more likely to contain the robot. (a) Optimal Bayes solution which marginalizes out the map. (b) Dead-reckoning solution which ignores the map. Notice how "blurry" it is. (c) Online EM solution using fixed-lag smoothing with a maximal window length. minimum in which M(5) = 1, as shown in Figure 2(c). (We tried various values of the decay rate d, from 0.1 to 0.9, and found that it made little difference.) With the wrong map, the robot "gets lost" on the return journey: see Figure 3(c). Offline EM, on the other hand, does very well, as shown in Figure 2(d); although the initial estimate oflocation (see Figure 3(b)) is rather diffuse, as it updates the map it can use the benefit of hindsight to figure out where it must have been. 4 Rao-Blackwellised particle filtering Although the Bayesian solution exhibits some desirable properties, its running time is exponential in the size of the environment. In this section, we discuss a sequential Monte Carlo algorithm called particle filtering (also known as sm filtering, the bootstrap filter, the condensation algorithm, survival of the fittest, etc; see [10, 4] for recent reviews). Particle filtering (PF) has already been successfully applied to the problem of (global) robot localization [5]. However, in that case, the state space was only of dimension 3: the unknowns were the position of the robot, (x, y) E lR?, and its orientation, () E [0,211"]. In our case, the state space is discrete and of dimension 0(1 + NL), since we need to keep track of the map as well as the robot's location (we ignore orientation in this paper). Particle filtering can be very inefficient in high-dimensional spaces. The key observation which makes it tractable in this context is that, if Ll:t were known, then the posterior on M t would be factored; hence M t can be marginalized out analytically, and we only need to sample Lt. This idea is known in the statistics literature as RaoBlackwellisation [10, 41. In more detail, we will approximate the posterior at time t using a set of weighted particles, where each particle specifies a trajectory L 1 :t , and the corresponding conditionally factored representation of P(Mt ) = TIi P(Mt(i)); we will denote the j'th particle at time t as bF). Note that we do not need to actually store the complete trajectories Ll:t: we only need the most recent value of L. The approach we take is essentially the same as the one used in the conditional linear Gaussian models of [4, 3], except we replace the Kalman filter update with one which exploits the conditionally factored representation of P(Mt ). In particular, the algorithm is as follows: For each particle j = 1, ... , N s , we do the following: 1. Sample L~~l from a proposal distribution, which we discuss below. 2. Update each component of the map separately using L~~l and Zt+1 Pr(Mt~lIL~~l = i,bP),Zt+l) oc Pr(zt+1IMt~l(i)) rrPr(Mi~l(k)IMF)(k)) k K. P. Murphy 1020 IIK _ a b _ ... '~I ~ I. I I I d c Figure 4: (a-b) Results using 50 particles. (c-d) Results using BK. . 3. Update the weIghts: (j) Wt+l (j) (j) = u t + 1 w t ,where (j) Ut+l . IS defined below. We then res ample Ns particles from the normalised weights, using Liu's residual resampling algorithm [10], and set = 1/Ns for all j. We consider two proposal distributions. The first is a simple one which just uses the transition model to predict the new location: Pr(Lt+1lb~j), at+1) . In this case, the incremental weight is U~~l <X P(zt+1IL~~l,b~j)). The optimal proposal distribution (the one which minimizes the variance of the importance weights) takes the most recent evidence into account, and can be shown to have the form Pr(Lt+1lb~j), at+l, Zt+l) with incremental weight Ut+1 <X P(Zt+1lb~j)) . Computing this requires marginalizing out Mt+l and Lt+l' which can be done in O(NL) time (details omitted). WWl In Figure 4, we show the results of applying the above algorithm to the same problem as in Section 3; it can be seen that it approximates the exact solution- very closely, using only 50 particles. The results shown are for a particular random number seed; other seeds produce qualitatively very similar results, indicating that 50 particles is in fact sufficient in this case. Obviously, as we increase the number of particles, the error and variance decrease, but the running time increases (linearly). The question of how many particles to use is a difficult one: it depends both on the noise parameters and the structure of the environment (if every cell has a unique label, localization is easy). Since we are sampling trajectories, the number of hypotheses, and hence the number of particles needed, grows exponentially with time. In the above example, the robot was able to localize itself quite accurately when it reached the end of the corridor, where most hypotheses "died off". In general, the number of particles will depend on the length of the longest cycle in the environment, so we will need to use active learning to ensure tractability. In the dynamic two-dimensional grid world of Figure l(c), we chose actions so as to maximize expected discounted reward (using policy iteration), where the reward for visiting cell i is where H(?) is the normalized entropy. Hence, if the robot is "lost", so H(L t ) ~ 1, the robot will try to visit a cell which it is certain about (see [6] for a better approach); otherwise, it will try to explore uncertain cells. After learning the map, the robot spends its time visiting each of the doors, to keep its knowledge of their state (open or closed) up-to-date. We now briefly consider some alternative approximate inference algorithms. Examining the graphical structure of our model (see Figure l(a)) , we see that it is identical 1021 Bayesian Map Learning in Dy namic Environments to a Factorial HMM [7] (ignoring the inputs). Unfortunately, we cannot use their variational approximation, because they assume a conditional Gaussian observation model, whereas ours is almost deterministic. Another popular approximate inference algorithm for dynamic Bayes nets (DBNs) is the "BK algorithm" [2, 1]. This entails projecting the joint posterior at time t onto a product-of-marginals representation P(Lt, Mt (1) , . . . , Mt(NdIYl:t) = P(Lt IYl :t) II P(Mt(i)IYl :t) i and using this as a factored prior for Bayesian updating at time t + 1. Given a factored prior, we can compute a factored posterior in O(NL) time by conditioning on each L t +1, and then averaging. We found that the BK method does very poorly on this problem (see Figure 4), because it ignores correlation between the cells. Of course, it is possible to use pairwise or higher order marginals for tightly coupled sets of variables. Unfortunately, the running time is exponential in the size of the largest marginal , and in our case, all the Mt(i) variables are coupled. Acknowledgments I would like to thank Nando de Freitas for helping me get particle filtering to work, Sebastian Thrun for an interesting discussion at the conference, and Stuart Russell for encouraging me to compare to EM . This work was supported by grant number ONR N00014-97-1-0941. References [1) X. Boyen and D. Koller. Approximate learning of dynamic models. In NIPS, 1998 . [2) X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In UAI, 1998. [3) R. Chen and S. Liu. Mixture Kalman filters . Submitted, 1999. [4) A. Doucet, S. Godsill, and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 1999. [5) D. Fox, W. Burgard, F. Dellaert, and S. Thrun. Monte carlo localization: Efficient position estimation for mobile robots. In AAAI, 1999. [6) D. Fox, W . Burgard, and S. Thrun. Active Markov localization for mobile robots. Robotics and Autonomous Systems, 1998. [7] Z. Ghahramani and M. Jordan. Factorial Hidden Markov Models. Machine Learning, 29:245- 273, 1997. [8) S. Koenig and R. Simmons. Unsupervised learning of probabilistic models for robot navigation. In ICRA, 1996. [9] D. Kortenkamp , R. Bonasso, and R. Murphy, editors. Artificial Intelligence and Mobile Robots: case studies of successful robot systems. MIT Press, 1998. [10] J . Liu and R. Chen. Sequential monte carlo methods for dynamic systems. JASA , 93:1032-1044, 1998. [11) H. Shatkay and L. P. Kaelbling. Learning topological maps with weak local odometric information. In IlCAI, 1997. [12) S. Thrun, W . Burgard, and D. Fox. A probabilistic approach to concurrent mapping and localization for mobile robots. Machine Learning, 31:29- 53, 1998. 

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An Improved Decomposition Algorithm for Regression Support Vector Machines Pavel Laskov Department of Computer and Information Sciences University of Delaware Newark, DE 19718 laskov@asel. udel. edu Abstract A new decomposition algorithm for training regression Support Vector Machines (SVM) is presented. The algorithm builds on the basic principles of decomposition proposed by Osuna et. al ., and addresses the issue of optimal working set selection. The new criteria for testing optimality of a working set are derived. Based on these criteria, the principle of "maximal inconsistency" is proposed to form (approximately) optimal working sets. Experimental results show superior performance of the new algorithm in comparison with traditional training of regression SVM without decomposition. Similar results have been previously reported on decomposition algorithms for pattern recognition SVM. The new algorithm is also applicable to advanced SVM formulations based on regression, such as density estimation and integral equation SVM. 1 Introd uction The increasing interest in applications of Support Vector Machines (SVM) to largescale problems ushers in new requirements for computational complexity of their training algorithms. Requests have been recently made for algorithms capable of handling problems containing 105 - 106 examples [1]. Training an SVM constitutes a quadratic programming problem, and a typical SVM package uses an off-the-shelf optimization software to obtain a solution to it. The number of variables in the optimization problem is equal to the number of training data points (for the pattern recognition SVM) or twice that number (for the regression SVM). The speed of general-purpose optimization methods is insufficient for problems containing more than a few thousand examples. This has motivated a quest for special-purpose training algorithms to take advantage of the particular structure of SVM training problems. The main avenue of research in SVM training algorithms is decomposition. The key idea of decomposition, due to Osuna et. al. [2], is to freeze all but a small number of optimization variables, and to solve a sequence of small fixed-size problems. The set of variables whose values are optimized at a current iteration is called the working set. Complexity of re-optimizing the working set is assumed to be constant-time. An Improved Decomposition Algorithm for Regression Support Vector Machines 485 In order for a decomposition algorithm to be successful) the working set must be selected in a smart way. The fastest known decomposition algorithm is due to Joachims [3]. It is based on Zoutendijk)s method of feasible directions proposed in the optimization community in the early 1960)s. However Joachims) algorithm is limited to pattern recognition SVM because it makes use of labels being ?l. The current article presents a similar algorithm for the regression SVM. The new algorithm utilizes a slightly different background from optimization theory. The Karush-Kuhn-Tucker Theorem is used to derive conditions for determining whether or not a given working set is optimal. These conditions become the algorithm)s termination criteria) as an alternative to Osuna)s criteria (also used by Joachims without modification) which used conditions for individual points. The advantage of the new conditions is that knowledge of the hyperplane)s constant factor b) which in some cases is difficult to compute) is not required. Further investigation of the new termination conditions allows to form the strategy for selecting an optimal working set. The new algorithm is applicable to the pattern recognition SVM) and is provably equivalent to Joachims) algorithm. One can also interpret the new algorithm in the sense of the method of feasible directions. Experimental results presented in the last section demonstrate superior performance of the new method in comparison with traditional training of regression SVM. 2 General Principles of Regression SVM Decomposition The original decomposition algorithm proposed for the pattern recognition SVM in [2] has been extended to the regression SVM in [4]. For the sake of completeness I will repeat the main steps of this extension with the aim of providing terse and streamlined notation to lay the ground for working set selection. Given the training data of size I) training of the regression SVM amounts to solving the following quadratic programming problem in 21 variables: Maximize subject to: W(a) 1y-T0: - -0: TD 0: 2 eTa 0 a-Ct < 0 a > 0 (1) where K -K The basic idea of decomposition is to split the variable vector a into the working set aB of fixed size q and the non-working set aN containing the rest ofthe variables. The corresponding parts of vectors e and y will also bear subscripts Nand B . The matrix D is partitioned into D BB ) DBN = D~B and D NN . A further requirement is that) for the i-th element of the training data) both 0i and 0; are either included in or omitted from the working set.l The values of the variables in the non-working set are frozen for the iteration) and optimization is only performed with respect to the variables in the working set. Optimization of the working set is also a quadratic program. This can be seen by re-arranging the terms of the objective function and the equality constraint in IThis rule facilitates formulation of sub-problems to be solved at each iteration. P. Laskov 486 (1) and dropping the terms independent of o.B from the objective. The resulting quadratic program (sub-problem) is formulated as follows: -T Maximize subject to: -T - (YB - QNDNB)QB - - a T TCBQB +cNQN o.B - 1 -T 'iQBDBBQB (2) < 0 > 0 Cl The basic decomposition algorithm chooses the first working set at random, and proceeds iteratively by selecting sub-optimal working sets and re-optimizing them, by solving quadratic program (2), until all subsets of size q are optimal. The precise formulation of termination conditions will be developed in the following section. 3 Optimality of a Working Set In order to maintain strict improvement of the objective function, the working set must be sub-optimal before re-optimization. The classical Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for optimality of a quadratic program. I will use these conditions applied to the standard form of a quadratic program, as described in [5], p. 36. The standard form of a quadratic program requires that all constraints are of equality type except for non-negativity constraints. To cast the refression SVM quadratic program (1) into the standard form, the slack variables s = (81 , '" ,82l) corresponding to the box constraints, and the following matrices are introduced: I 0]I ' o E= [f], z= m' (3) where 1 is a vector of length I , C is a vector of length 21. The zero element in vector z reflects the fact that a slack variable for the equality constraint must be zero. In the matrix notation all constraints of problem (1) can be compactly expressed as: ETz f z > 0 (4) In this notation the Karush-Kuhn-Tucker Theorem can be stated as follows: Theorem 1 (Karush-Kuhn-Tucker Theorem) The primal vector z solves the quadratic problem (1) if and only if it satisfies (4) and there exists a dual vector u T = (ITT w T ) = (ITT (I' yT? such that: IT = DO. + Ew Y ~ 0 uT z =a Y> 0 (5) (6) (7) It follows from the Karush-Kuhn-Tucker Theorem that if for all u satisfying conditions (6) - (7) the system of inequalities (5) is inconsistent then the solution of problem (1) is not optimal. Since the objective function of sub-problem (2) was obtained by merely re-arranging terms in the objective function of the initial problem (1), the same conditions guarantee that the sub-problem (2) is not optimal. Thus, the main strategy for identifying sub-optimal working sets will be to enforce inconsistency of the system (5) while satisfying conditions (6) - (7) . 487 An Improved Decomposition Algorithm for Regression Support Vector Machines Let us further analyze inequalities in (5). Each inequality has one of the following forms: 7T'i -rPi + E + Vi + J.L > 0 rPi + E - v; - J.L > 0 (8) (9) where I rPi = Yi - 2:)aj - a;)Kij j=l Consider the values ai can possible take: 1. ai = O. In this case Si = C, and, by complementarity condition (7), Vi Then inequality (8) becomes: 7T'i = -rPi + E + J.L ~ 0 ~ J.L ~ = C. By complementarity condition (7), 7T'i becomes: 2. ai -rPi + E + J.L + Vi ~ = 0 rPi - = O. J.L::; = O. E Then inequality (8) rPi - E < ai < C. By complementarity condition (7), Vi = 0, 7T'i = O. Then inequality (8) becomes: 3. 0 - rPi + E + J.L Similar reasoning for 0 ~ J.L = rPi - E a; and inequality (9) yields the following results: 1. a; = O. Then 2. a; = C. 3. 0 < = Then a; < C . Then As one can see, the only free variable in system (5) is J.L. Each inequality restricts J.L to a certain interval on a real line. Such intervals will be denoted as J.L-sets in the rest of the exposition. Any subset of inequalities in (5) is inconsistent if the intersection of the corresponding J.L-sets is empty. This provides a lucid rule for determining optimality of any working set: it is sub-optimal if the intersection of J.L-sets of all its points is empty. A sub-optimal working set will also be denoted as "inconsistent". The following summarizes the rules for calculation of J.L-sets, taking into account that for regression SVM aia; = 0: [rPi - E, rPi + E], if ai = 0, a; = 0 [rPi - E, rPi - E], if 0 < ai < C, a; = 0 (-00, rPi - E], if ai = C, a; = 0 [rPi + E, rPi + E], if ai = 0, 0 < a; < C [rPi + E, +(0), (10) 488 4 P. Laskov Maximal Inconsistency Algorithm While inconsistency of the working set at each iteration guarantees convergence of decomposition, the rate of convergence is quite slow if arbitrary inconsistent working sets are chosen. A natural heuristic is to select "maximally inconsistent" working sets, in a hope that such choice would provide the greatest improvement of the objective function. The notion of "maximal inconsistency" is easy to define: let it be the gap between the smallest right boundary and the largest left boundary of p-sets of elements in the training set: G=L-R L = O<i<l max pL R = O<i<l min pr 1 where p!, pi are the left and the right boundaries respectively (possibly minus or plus infinity) of the p-set Mi. It is convenient to require that the largest possible inconsistency gap be maintained between all pairs of points comprising the working set. The obvious implementation of such strategy is to select q/2 elements with the largest values of pi and q/2 elements with the smallest values of pr. The maximal inconsistency strategy is summarized in Algorithm 1. Algorithm 1 Maximal inconsistency SVM decomposition algorithm. Let S be the list of all samples. while (L > R) ? compute Mi according to the rules (10) for all elements in S ? select q/2 elements with the largest values of pi ("left pass") ? select q /2 elements with the smallest values of p r ("right pass") ? re-optimize the working set Although the motivation provided for the maximal inconsistency algorithm is purely heuristic, the algorithm can be rigorously derived, in a similar fashion as Joachims' algorithm, from Zoutendijk's feasible direction problem. Details of such derivation cannot be presented here due to space constraints. Because of this relationship I will further refer to both algorithms as "feasible direction" algorithms. 5 Experimental Results Experimental evaluation of the new algorithm was performed on the modThe original data set is available under ified KDD Cup 1998 data set. http:j/www.ics.uci.edu/"-'kdd/databases/kddcup98/kddcup98.html. The following modifications were made to obtain a pure regression problem: ? All 75 character fields were eliminated. ? Numeric fields CONTROLN, ODATEDW, TCODE and DOB were elimitated. The remaining 400 features and the labels were scaled between 0 and 1. Initial subsets of the training database of different sizes were selected for evaluation of the scaling properties of the new algorithm. The training times of the algorithms, with and without decomposition, the numbers of support vectors, including bounded support vectors, and the experimental scaling factors, are displayed in Table 1. An Improved Decomposition Algorithm for Regression Support Vector Machines Table 1: Training time (sec) and number Examples no dcmp 500 39 226 1000 2000 1490 3000 5744 27052 5000 scaling factor: 2.84 SV-scaling factor: 3.06 489 of SVs for the KDD Cup problem dcmp total SV BSV 10 274 0 41 518 3 158 970 5 1429 397 7 1252 2349 15 2.08 2.24 Table 2: Training time (sec) and number of SVs for the KDD Cup problem, reduced feature space. Examples no dcmp dcmp total SV BSV 500 170 18 56 30 346 1000 62 44 374 2000 1768 198 144 510 4789 729 366 3000 222 22115 1139 5000 863 354 scaling factor: 2.55 1.72 SV-scaling factor: 3.55 2.35 The experimental scaling factors are obtained by fitting lines to log-log plots of the running times against sample sizes, in the number of examples and the number of unbounded support vectors respectively. Experiments were run on SGI Octane with 195MHz clock and 256M RAM. RBF kernel with, = 10, C = 1, termination accuracy 0.001, working set size of 20, and cache size of 5000 samples were used. A similar experiment was performed on a reduced feature set consisting of the first 50 features selected from the full-size data set. This experiment illustrates the behavior of the algorithms when the large number of support vectors are bounded. The results are presented in Table 2. 6 Discussion It comes at no surprise that the decomposition algorithm outperforms the conventional training algorithm by an order of magnitude. Similar results have been well established for pattern recognition SVM. Remarkable is the co-incidence of scaling factors of the maximal inconsistency algorithm and Joachims' algorithm: his scaling factors range from 1.7 to 2.1 [3]. I believe however, that a more important performance measure is SV -scaling factor, and the results above suggest that this factor is consistent even for problems with significantly different compositions of support vectors. Further experiments should investigate properties of this measure. Finally, I would like to mention other methods proposed in order to speed-up training of SVM, although no experimental results have been reported for these methods with regard to training of the regression SVM. Chunking [6], p. 366, iterates through 490 P. Laskov the training data accumulating support vectors and adding a "chunk" of new data until no more changes to a solution occur. The main problem with this method is that when the percentage of support vectors is high it essentially solves the problem of almost the same size more than once. Sequential Minimal Optimization (SMO), proposed by Platt [7] and easily extendable to the regression SVM [1], employs an idea similar to decomposition but always uses the working set of size 2. For such a working set, a solution can be calculated "by hand" without numerical optimization. A number of heuristics is applied in order to choose a good working set. It is difficult to draw a comparison between the working set selection mechanisms of SMO and the feasible direction algorithms but experimental results of Joachims [3] suggest that SMO is slower. Another advantage of feasible direction algorithms is that the size of the working set is not limited to 2, as in SMO. Practical experience shows that the optimal size of the working set is between 10 and 100. Lastly, traditional optimization methods, such as Newton's or conjugate gradient methods, can be modified to yield the complexity of 0(s3), where s is the number of detected support vectors [8]. This can be a considerable improvement over the methods that have complexity of 0(13), where 1 is the total number of training samples. The real challenge lies in attaining sub-0(s3) complexity. While the experimental results suggest that feasible direction algorithms might attain such complexity, their complexity is not fully understood from the theoretical point of view. More specifically, the convergence rate, and its dependence on the number of support vectors, needs to be analyzed. This will be the main direction of the future research in feasible direction SVM training algorithms. References [1] Smola, A., Sch61kopf, B. (1998) A Tutorial on Support Vector Regression. NeuroCOLT2 Technical Report NC2- TR-1998-030. [2] Osuna, E., Freund, R., Girosi, F. (1997) An Improved Training Algorithm for Support Vector Machines. Proceedings of IEEE NNSP'97. Amelia Island FL. [3] Joachims, T. (1998) Making Large-Scale SVM Learning Practical. Advances in Kernel Methods - Support Vector Learning. B. Sch61kopf, C. Burges, A. Smola, (eds.) MIT-Press. [4] Osuna, E. (1998) Support Vector Machines: Training and Applications. Ph. D. Dissertation. Operations Research Center, MIT. [5] Boot, J. (1964) Quadratic Programming. Algorithms - Anomalies - Applications. North Holland Publishing Company, Amsterdam. [6] Vapnik, V. (1982) Estimation of Dependencies Based on Empirical Data. Springer-Verlag. [7] Platt, J. (1998) Fast Training of Support Vector Machines Using Sequential Minimal Optimization. Advances in Kernel Methods - Support Vector Learning. B. Sch5lkopf, C. Burges, A. Smola, (eds.) MIT-Press. [8] Kaufman, L. (1998) Solving the Quadratic Programming Problem Arising in Support Vector Classification. Advances in Kernel Methods - Support Vector Learning. B. Sch5lkopf, C. Burges, A. Smola, (eds.) MIT-Press. 

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A Multi-class Linear Learning Algorithm Related to Winnow Chris Mesterhann* Rutgers Computer Science Department 110 Frelinghuysen Road Piscataway, NJ 08854 mesterha@paul.rutgers.edu Abstract In this paper, we present Committee, a new multi-class learning algorithm related to the Winnow family of algorithms. Committee is an algorithm for combining the predictions of a set of sub-experts in the online mistake-bounded model oflearning. A sub-expert is a special type of attribute that predicts with a distribution over a finite number of classes. Committee learns a linear function of sub-experts and uses this function to make class predictions. We provide bounds for Committee that show it performs well when the target can be represented by a few relevant sub-experts. We also show how Committee can be used to solve more traditional problems composed of attributes. This leads to a natural extension that learns on multi-class problems that contain both traditional attributes and sub-experts. 1 Introduction In this paper, we present a new multi-class learning algorithm called Committee. Committee learns a k class target function by combining information from a large set of sub-experts. A sub-expert is a special type of attribute that predicts with a distribution over the target classes. The target space of functions are linear-max functions. We define these as functions that take a linear combination of sub-expert predictions and return the class with maximum value. It may be useful to think of the sub-experts as individual classifying functions that are attempting to predict the target function. Even though the individual sub-experts may not be perfect, Committee attempts to learn a linear-max function that represents the target function. In truth, this picture is not quite accurate. The reason we call them sub-experts and not experts is because even though a individual sub-expert might be poor at prediction, it may be useful when used in a linear-max function. For example, some sub-experts might be used to add constant weights to the linear-max function. The algorithm is analyzed for the on-line mistake-bounded model oflearning [Lit89]. This is a useful model for a type of incremental learning where an algorithm can use feedback about its current hypothesis to improve its performance. In this model, the algorithm goes through a series of learning trials. A trial is composed of three steps. First, the algorithm ?Part of this work was supported by NEe Research Institute, Princeton, NJ. C. Mesterharm 520 receives an instance, in this case, the predictions of the sub-experts. Second, the algorithm predicts a label for the instance; this is the global prediction of Committee. And last, the algorithm receives the true label of the instance ; Committee uses this information to update its estimate of the target. The goal of the algorithm is to minimize the total number of prediction mistakes the algorithm makes while learning the target. The analysis and performance of Committee is similar to another learning algorithm, Winnow [Lit89] . Winnow is an algorithm for learning a linear-threshold function that maps attributes in [0, 1] to a binary target. It is an algorithm that is effective when the concept can be represented with a few relevant attributes, irrespective of the behavior of the other attributes. Committee is similar but deals with learning a target that contains only a few relevant sub-experts. While learning with sub-experts is interesting in it's own right, it turns out the distinction between the two tasks is not significant. We will show in section 5 how to transform attributes from [0, 1] into sub-experts. Using particular transformations, Committee is identical to the Winnow algorithms, Balanced and WMA [Lit89]. Furthermore, we can generalize these transformations to handle attribute problems with multi-class targets. These transformations naturally lead to a hybrid algorithm that allows a combination of sub-experts and attributes for multi-class learning problems. This opens up a range of new practical problems that did not easily fit into the previous framework of [0, 1J attributes and binary classification. 2 Previous work Many people have successfully tried the Winnow algorithms on real-world tasks. In the course of their work, they have made modifications to the algorithms to fit certain aspects of their problem. These modifications include multi-class extensions. For example, [DKR97] use Winnow algorithms on text classification problems. This multiclass problem has a special form ; a document can belong to more than one class. Because of this property, it makes sense to learn a different binary classifier for each class. The linear functions are allowed, even desired, to overlap. However, this paper is concerned with cases where this is not possible. For example, in [GR96] the correct spelling of a word must be selected from a set of many possibilities. In this setting, it is more desirable to have the algorithm select a single word. The work in [GR96] presents many interesting ideas and modifications of the Winnow algorithms. At a minimum, these modification are useful for improving the performance of Winnow on those particular problems. Part of that work also extends the Winnow algorithm to general multi-class problems. While the results are favorable, the contribution ofthis paper is to give a different algorithm that has a stronger theoretical foundation for customizing a particular multi-class problem. Blum also works with multi-class Winnow algorithms on the calendar scheduling problem of [MCF+94] . In [Blu95], a modified Winnow is given with theoretical arguments for good performance on certain types of multi-class disjunctions. In this paper, these results are extended, with the new algorithm Committee, to cover a wider range of multi-class linear functions. Other related theoretical work on multi-class problems includes the regression algorithm EG ?. In [KW97], Kivinen and Warmuth introduce EG ?, an algorithm related to Winnow but used on regression problems. In general, while regression is a useful framework for many multi-class problems, it is not straightforward how to extend regression to the concepts learned by Committee. A particular problem is the inability of current regression techniques to handle 0-1 loss. A Multi-class Linear Learning Algorithm Related to Winnow 521 3 Algorithm This section of the paper describes the details of Committee. Near the end of the section, we will give a formal statement of the algorithm. 3.1 Prediction scheme Assume there are n sub-experts. Each sub-expert has a positive weight that is used to vote for k different classes; let Wi be the weight of sub-expert i. A sub-expert can vote for several classes by spreading its weight with a prediction distribution. For example, if k = 3, a sub-expert may give 3/5 of its weight to class 1, 1/5 of its weight to class 2, and 1/5 of its weight to class 3. Let Xi represent this prediction distribution, where x~ is the fraction of the weight sub-expert i gives to class j . The vote for class j is L~=I WiX~. Committee predicts the class that has the highest vote. (On ties, the algorithm picks one of the classes involved in the tie.) We call the function computed by this prediction scheme a linear-max function, since it is the maximum class value taken from a linear combination of the SUb-expert predictions. 3.2 Target function The goal of Committee is to mInimIZe the number of mistakes by quickly learning sub-expert weights that correctly classify the target function. Assume there exists fL, a vector of nonnegative weights that correctly classifies the target. Notice that fL can be multiplied by any constant without changing the target. To remove this confusion, we will normalize the weights to sum to 1, i.e., L~=Il-1i = 1. Let ((j) be the target's vote for class j. n ((j) = L l-1iX~ t=I Part of the difficulty of the learning problem is hidden in the target weights. Intuitively, a target function will be more difficult to learn ifthere is a small difference between the (votes of the correct and incorrect classes. We measure this difficulty by looking at the minimum difference, over all trials, of the vote of the correct label and the vote of the other labels. Assume for trial t that Pt is the correct label. 8= min tETnals (min(((pt)-((j))) rlpt Because these are the weights of the target, and the target always makes the correct prediction, 8 > o. One problem with the above assumptions is that they do not allow noise (cases where 8 <::; 0). However, there are variations of the analysis that allow for limited amounts of noise [Lit89, Lit91]. Also experimental work [Lit95, LM] shows the family of Winnow algorithms to be much more robust to noise than the theory would predict. Based on the similarity of the algorithm and analysis, and some preliminary experiments, Committee should be able to tolerate some noise. 3.3 Updates Committee only updates on mistakes using multiplicative updates. The algorithm starts by initializing all weights to 1 In. During the trials, let P be the correct label and .x be the predicted label of Committee. When .x =1= P the weight of each sub-expert i is multiplied by aX; -x;. This corresponds to increasing the weights of the sub-experts who predicted the C. Mesterharm 522 correct label instead of the label Committee predicted. The value of 0' is initialized at the start of the algorithm. The optimal value of 0' for the bounds depends on 6. Often 6 is not known in advance, but experiments on Winnow algorithms suggest that these algorithms are more flexible, often performing well with a wider range of 0' values [LM). Last, the weights are renormalize to sum to 1. While this is not strictly necessary, normalizing has several advantages including reducing the likelyhood of underflow/overflow errors. 3.4 Committee code Initialization 'Vi E {l , . .. , n} W i:= lin. Set 0' > 1. Trials Instance sub-experts (Xl , . .. , xn) . Prediction >. is the first class c such that for all other classes J, "n c L...- i =1 W i X i > - "n j L...-i=1 W i X t ? Update Let p be the correct label. If mistake (>' fori:=l ton Wi := p O'X i >. -x, Wi . Normalize weights, L:~= l 3.5 # p) Wt =1 Mistake bound We do not have the space to give the proof for the mistake bound of Committee, but the technique is similar to the proof of the Winnow algorithm, Balanced, given in [Lit89). For the complete proof, the reader can refer to [Mes99). Theorem 1 Committee makes at most 2ln (n) 162mistakes when the target conditions in section 3.2 are satisfied and 0' is set to (1 - 6) - 1/ 2. Surprisingly, this bound does not refer to the number of classes. The effects of larger values of k show up indirectly in the 6 value. While it is not obvious, this bound shows that Committee performs well when the target can be represented by a small fraction of the sub-experts. Call the sub-experts in the target the relevant sub-experts. Since 6 is a function of the target, 6 only depends on the relevant sub-experts. On the other hand, the remaining sub-experts have a small effect on the bound since they are only represented in the In( n) factor. This means that the mistake bound of Committee is fairly stable even when adding a large number of additional sub-experts. In truth, this doesn ' t mean that the algorithm will have a good bound when there are few relevant sub-experts. In some cases, a small number of sub-experts can give an arbitrarily small 6 value. (This is a general problem with all the Winnow algorithms.) What it does mean is that, given any problem, increasing the number of irrelevant sub-experts will only have a logarithmic effect on the mistake bound. 4 Attributes to sub-experts Often there are no obvious sub-experts to use in solving a learning problem. Many times the only information available is a set of attributes. For attributes in [0,1]' we will show how to use Committee to learn a natural kind of k class target function , a linear machine. To learn this target, we will transform each attribute into k separate sub-experts. We will use some of the same notion as Committee to help understand the transformation. 523 A Multi-class Linear Learning Algorithm Related to Winnow 4.1 Attribute target (linear machine) A linear machine [DH73] is a prediction function that divides the feature space into disjoint convex regions where each class corresponds to one region. The predictions are made by a comparing the value of k different linear functions where each function corresponds to a class. More formally, assume there are m - 1 attributes and k classes. Let Zi E [0,1] be attribute i. Assume the target function is represented using k linear functions of the attributes. Let ((j) = L::II-L{ Zi be the linear function for class j where I-Li is the weight of attribute i in class j. Notice that we have added one extra attribute. This attribute is set to 1 and is needed for the constant portion of the linear functions. The target function labels an instance with the class of the largest ( function. (Ties are not defined.) Therefore, ((j) is similar to the voting function for class j used in Committee. 4.2 Transforming the target One difficulty with these linear functions is that they may have negative weights. Since Committee only allows targets with nonnegative weights, we need transform to an equivalent problem that has nonnegative weights. This is not difficult. Since we are only concerned with the relative difference between the ( functions, we are allowed to add any function to the (functions as long as we add it to all (functions. This gives us a simple procedure to remove negative weights. For example, if ((1) = 3Z1 - 2Z2 + 1z3 -4, we can add 2Z2 +4 to every ( function to remove the negative weights from ((1). It is straightforward to extend this and remove all negative weights. We also need to normalize the weights. Again, since only the relative difference between the ( functions matter, we can divide all the ( functions by any constant. We normalize the weights to sum to 1, i.e., L:~=1L:~11-L{ = 1. At this point, without loss of generality, assume that the original ( functions are nonnegative and normalized. The last step is to identify a 8 value. We use the same definition of 8 as Committee substituting the corresponding ( functions of the linear machine. Assume for trial t that Pt is the correct label. 8= 4.3 min tETrwls (min( ((Pt) - ((j))) ji-P, Transforming the attributes The transformation works as follows: convert attribute Zi into k sub-experts. Each sub-expert will always vote for one of the k classes with value Zi. The target weight for each of these sub-experts is the corresponding target weight of the attribute, label pair in the ( functions . Do this for every attribute. Notice that we are not using distributions for the sub-expert predictions. A sub-expert's prediction can be converted to a distribution by adding a constant amount to each class prediction. For example, a sub-expert that predicts Zl = .7, Z2 = 0, Z3 = can be changed to Zl = .8, Z2 = .1, Z3 = .1 by adding .1 to each class. This conversion does not affect the predicting or updating of Committee. ? 524 C. Mesterharm Theorem 2 Committee makes at most 2In(mk)/8 2 mistakes on a linear machine, as defined in this section, when 0 is set to (1 - 8)-1/2. Proof: The above target transformation creates mk normalized target sub-experts that vote with the same ( functions as the linear machine. Therefore, this set of sub-experts has the same 8 value. Plugging these values into the bound for Committee gives the result. This transformation provides a simple procedure for solving linear machine problems. While the details of the transformation may look cumbersome, the actual implementation of the algorithm is relatively simple. There is no need to explicitly keep track of the sub-experts. Instead, the algorithm can use a linear machine type representation. Each class keeps a vector of weights, one weight for each attribute. During an update, only the correct class weights and the predicted class weights are changed. The correct class weights are multiplied by O Zi; the predicted class weights are multiplied by o -z' . The above procedure is very similar to the Balanced algorithm from [Lit89] , in fact, for k = 2, it is identical. A similar transformation duplicates the behavior of the linear-threshold learning version ofWMA as given in [Lit89]. While this transformation shows some advantages for k = 2, more research is needed to determine the proper way to generalize to the multi-class case. For both of these transformations, the bounds given in this paper are equivalent (except for a superficial adjustment in the 8 notation of WMA) to the original bounds given in [Lit89] . 4.4 Combining attributes and sub-experts These transformations suggest the proper way to do a hybrid algorithm that combines sub-experts and attributes: use the transformations to create new sub-experts from the attributes and combine them with the original sub-experts when running Committee. It may even be desirable to break original sub-experts into attributes and use both in the algorithm because some sub-experts may perform better on certain classes. For example, if it is felt that a sub-expert is particularly good at class 1, we can perform the following transformation. Now, instead of using one weight for the whole sub-expert, Committee can also learn based on the sub-expert's performance for the first class. Even if a good target is representable only with the original sub-experts, these additional sub-experts will not have a large effect because of the logarithmic bound. In the same vein, it may be useful to add constant attributes to a set of sub-experts. These add only k extra SUb-experts, but allow the algorithm to represent a larger set of target functions . 5 Conclusion In this paper, we have introduced Committee, a multi-class learning algorithm. We feel that this algori thm will be important in practice, extending the range of problems that can be handled by the Winnow family of algorithms. With a solid theoretical foundation, researchers can customize Winnow algorithms to handle various multi-class problems. A Multi-class Linear Learning Algorithm Related to Winnow 525 Part of this customization includes feature transformations. We show how Committee can handle general linear machine problems by transforming attributes into sub-experts. This suggests a way to do a hybrid learning algorithm that allows a combination of sub-experts and attributes. This same techniques can also be used to add to the representational power on a standard sub-expert problem. In the future, we plan to empirically test Committee and the feature transformations on real world problems. Part of this testing will include modifying the algorithm to use extra information, that is related to the proof technique [Mes99), in an attempt to lower the number of mistakes. We speculate that adjusting the multiplier to increase the change in progress per trial will be useful for certain types of multi-class problems. Acknowledgments We thank Nick Littlestone for stimulating this work by suggesting techniques for converting the Balanced algorithm to multi-class targets. Also we thank Haym Hirsh, Nick Littlestone and Warren Smith for providing valuable comments and corrections. References [Blu95] Avrim Blum. Empirical support for winnow and weighted-majority algorithms: results on a calendar scheduling domain. In ML-95, pages 64-72, 1995. [DH73) R. O. Duda and P. Hart. Pattern Classification and Scene Analysis. Wiley, New York,1973 . [DKR97] I. Dagan, Y. Karov, and D. Roth. Mistake-driven learning in text categorization. In EMNLP-97, pages 55-63,1997. [GR96) A. R. Golding and D. Roth. Applying winnow to context-sensitive spelling correction . In ML-96, 1996. [KW97) Jyrki Kivinen and Manfred K. Warmuth. Additive versus exponentiated gradient updates for linear prediction. Information and Computation, 132(1): 1-64, 1997. [Lit89] Nick Littlestone. Mistake bounds and linear-threshold learning algorithms. PhD thesis, University of California, Santa Cruz, 1989. Technical Report UCSC-CRL-89-11. [Lit91) Nick Littlestone. Redundant noisy attributes, attribute errors, and linearthreshold learning using winnow. In COLT-91 , pages 147-156,1991. [Lit95] Nick Littlestone. Comparing several linear-threshold learning algorithms on tasks involving superfluous attributes. In ML-95, pages 353-361 , 1995. [LM) Nick Littlestone and Chris Mesterharm. A simulation study of winnow and related algorithms . Work in progress. [MCF+94) T. Mitchell, R. Caruana, D. Freitag, 1. McDermott, and D. Zabowski. Experience with a personal learning assistant. CACM, 37(7):81-91, 1994. [Mes99) Chris Mesterharm. A multi-class linear learning algorithm related to winnow with proof. Technical report, Rutgers University, 1999. 

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The Relevance Vector Machine Michael E. Tipping Microsoft Research St George House, 1 Guildhall Street Cambridge CB2 3NH, U.K. mtipping~microsoft.com Abstract The support vector machine (SVM) is a state-of-the-art technique for regression and classification, combining excellent generalisation properties with a sparse kernel representation. However, it does suffer from a number of disadvantages, notably the absence of probabilistic outputs, the requirement to estimate a trade-off parameter and the need to utilise 'Mercer' kernel functions. In this paper we introduce the Relevance Vector Machine (RVM), a Bayesian treatment of a generalised linear model of identical functional form to the SVM. The RVM suffers from none of the above disadvantages, and examples demonstrate that for comparable generalisation performance, the RVM requires dramatically fewer kernel functions. 1 Introd uction In supervised learning we are given a set of examples of input vectors {Xn}~=l along with corresponding targets {tn}~=l' the latter of which might be real values (in regression) or class labels (classification). From this 'training' set we wish to learn a model of the dependency of the targets on the inputs with the objective of making accurate predictions of t for previously unseen values of x. In real-world data, the presence of noise (in regression) and class overlap (in classification) implies that the principal modelling challenge is to avoid 'over-fitting' of the training set. A very successful approach to supervised learning is the support vector machine (SVM) [8]. It makes predictions based on a function of the form N 2:: (1) wnK(x, x n ) + Wo, n=l where {w n } are the model 'weights' and K(?,?) is a kernel function. The key feature of the SVM is that, in the classification case, its target function attempts to minimise the number of errors made on the training set while simultaneously maximising the 'margin' between the two classes (in the feature space implicitly defined by the kernel). This is an effective 'prior' for avoiding over-fitting, which leads to good generalisation, and which furthermore results in a sparse model dependent only on a subset of kernel functions: those associated with training examples Xn that lie either on the margin or on the 'wrong' side of it. State-of-the-art results have been reported on many tasks where SVMs have been applied. y(x) = 653 The Relevance Vector Machine However, the support vector methodology does exhibit significant disadvantages: ? Predictions are not probabilistic. In regression the SVM outputs a point estimate, and in classification, a 'hard' binary decision. Ideally, we desire to estimate the conditional distribution p(tlx) in order to capture uncertainty in our prediction. In regression this may take the form of 'error-bars', but it is particularly crucial in classification where posterior probabilities of class membership are necessary to adapt to varying class priors and asymmetric misclassification costs. ? Although relatively sparse, SVMs make liberal use of kernel functions, the requisite number of which grows steeply with the size of the training set. ? It is necessary to estimate the error/margin trade-off parameter 'e' (and in regression, the insensitivity parameter I f' too). This generally entails a cross-validation procedure, which is wasteful both of data and computation. ? The kernel function K(?,?) must satisfy Mercer's condition. In this paper, we introduce the 'relevance vector machine' (RVM), a probabilistic sparse kernel model identical in functional form to the SVM. Here we adopt a Bayesian approach to learning, where we introduce a prior over the weights governed by a set of hyperparameters, one associated with each weight, whose most probable values are iteratively estimated from the data. Sparsity is achieved because in practice we find that the posterior distributions of many of the weights are sharply peaked around zero. Furthermore, unlike the support vector classifier, the nonzero weights in the RVM are not associated with examples close to the decision boundary, but rather appear to represent 'prototypical' examples of classes. We term these examples 'relevance' vectors, in deference to the principle of automatic relevance determination (ARD) which motivates the presented approach [4, 6J. The most compelling feature of the RVM is that, while capable of generalisation performance comparable to an equivalent SVM, it typically utilises dramatically fewer kernel functions. Furthermore, the RVM suffers from none of the other limitations of the SVM outlined above. In the next section, we introduce the Bayesian model, initially for regression, and define the procedure for obtaining hyperparameter values, and thus weights. In Section 3, we give brief examples of application of the RVM in the regression case, before developing the theory for the classification case in Section 4. Examples of RVM classification are then given in Section 5, concluding with a discussion. 2 Relevance Vector Regression Given a dataset of input-target pairs {xn, tn}~=l' we follow the standard formulation and assume p(tlx) is Gaussian N(tIY(x), a 2 ). The mean ofthis distribution for a given x is modelled by y(x) as defined in (1) for the SVM. The likelihood of the dataset can then be written as - ~w)1I2 } , N x (N + 1) p(tlw, a 2 ) = (27ra 2 )-N/2 exp { - 2: 2 lit (2) where t = (tl ... tN), W = (wo .. . WN) and ~ is the 'design' matrix with ~nm = K(x n , Xm - l) and ~nl = 1. Maximum-likelihood estimation of wand a 2 from (2) will generally lead to severe overfitting, so we encode a preference for smoother functions by defining an ARD Gaussian prior [4, 6J over the weights: N p(wla) = II N(wiIO,ai i=O 1 ), (3) ME. Tipping 654 with 0 a vector of N + 1 hyperparameters. This introduction of an individual hyperparameter for every weight is the key feature of the model, and is ultimately responsible for its sparsity properties. The posterior over the weights is then obtained from Bayes' rule: p(wlt, 0,0'2) = (21r)-(N+l)/21:E1- 1 / 2 exp { -~(w - J.lY:E-1(w - JL)}, (4) with :E = (q,TBq, JL = + A)-I, (5) :Eq, TBt, (6) where we have defined A = diag(ao,al, ... ,aN) and B = 0'-2IN. Note that 0'2 is also treated as a hyperparameter, which may be estimated from the data. By integrating out the weights, we obtain the marginal likelihood, or evidence [2], for the hyperparameters: p(tIO,0'2) = (21r)-N/2IB- 1 + q,A -1q, TI- 1 / 2 exp { -~e(B-l + q,A -lq,T)-lt} . (7) For ideal Bayesian inference, we should define hyperpriors over 0 and 0'2, and integrate out the hyperparameters too. However, such marginalisation cannot be performed in closed-form here, so we adopt a pragmatic procedure, based on that of MacKay [2], and optimise the marginal likelihood (7) with respect to 0 and 0'2, which is essentially the type II maximum likelihood method [1] . This is equivalent to finding the maximum of p(o, 0'2It), assuming a uniform (and thus improper) hyperprior. We then make predictions, based on (4), using these maximising values. 2.1 Optimising the hyperparameters Values of 0 and 0'2 which maximise (7) cannot be obtained in closed form, and we consider two alternative formulae for iterative re-estimation of o . First, by considering the weights as 'hidden' variables, an EM approach gives: new 1 1 (8) ai = -( 2) 2' Wi p(wlt,Q,u2) Eii + J-Li Second, direct differentiation of (7) and rearranging gives: 'Yi ' ainew = 2 (9) J-Li where we have defined the quantities 'Yi = 1 - aiEii, which can be interpreted as a measure of how 'well-determined' each parameter Wi is by the data [2]. Generally, this latter update was observed to exhibit faster convergence. For the noise variance, both methods lead to the same re-estimate: (10) In practice, during re-estimation, we find that many of the ai approach infinity, and from (4), p(wilt,0,0'2) becomes infinitely peaked at zero - implying that the corresponding kernel functions can be 'pruned'. While space here precludes a detailed explanation, this occurs because there is an 'Occam' penalty to be paid for smaller values of ai, due to their appearance in the determinant in the marginal likelihood (7). For some ai, a lesser penalty can be paid by explaining the data with increased noise 0'2, in which case those ai -+ 00. The Relevance Vector Machine 3 3.1 655 Examples of Relevance Vector Regression Synthetic example: the 'sine' function The function sinc(x) = Ixl- 1 sin Ixl is commonly used to illustrate support vector regression [8], where in place of the classification margin, the f.-insensitive region is introduced, a 'tube' of ?f. around the function within which errors are not penalised. In this case, the support vectors lie on the edge of, or outside, this region. For example, using linear spline kernels and with f. = 0.01, the approximation ofsinc(x) based on 100 uniformly-spaced noise-free samples in [-10, 10J utilises 39 support vectors [8]. By comparison, we approximate the same function with a relevance vector model utilising the same kernel. In this case the noise variance is fixed at 0.012 and 0 alone re-estimated. The approximating function is plotted in Figure 1 (left), and requires only 9 relevance vectors. The largest error is 0.0087, compared to 0.01 in the SV case. Figure 1 (right) illustrates the case where Gaussian noise of standard deviation 0.2 is added to the targets. The approximation uses 6 relevance vectors, and the noise is automatically estimated, using (10), as (7 = 0.189. 1.2 0.8 0.6 . 0.4 '. .- . 0.2 , 0 -0.2 -0.4 5 10 -10 -5 10 Figure 1: Relevance vector approximation to sinc(x): noise-free data (left), and with added Gaussian noise of (]" = 0.2 (right) . The estimated functions are drawn as solid lines with relevance vectors shown circled, and in the added-noise case (right) the true function is shown dashed. 3.2 Some benchmarks The table below illustrates regression performance on some popular benchmark datasets - Friedman's three synthetic functions (results averaged over 100 randomly generated training sets of size 240 with a lOOO-example test set) and the 'Boston housing' dataset (averaged over 100 randomised 481/25 train/test splits). The prediction error obtained and the number of kernel functions required for both support vector regression (SVR) and relevance vector regression (RVR) are given. Dataset Friedman #1 Friedman #2 Friedman #3 Boston Housing _ errors_ SVR RVR _ kernels _ SVR RVR 2.92 4140 0.0202 8.04 116.6 110.3 106.5 142.8 2.80 3505 0.0164 7.46 59.4 6.9 11.5 39.0 656 4 M E. TIpping Relevance Vector Classification We now extend the relevance vector approach to the case of classification - Le. where it is desired to predict the posterior probability of class membership given the input x. We generalise the linear model by applying the logistic sigmoid function a(y) = 1/(1 + e- Y ) to y(x) and writing the likelihood as N P(tlw) = II a{y(xn)}tn [1 - a{Y(Xn)}]l-tn . (11) n==l However, we cannot integrate out the weights to obtain the marginal likelihood analytically, and so utilise an iterative procedure based on that of MacKay [3]: 1. For the current, fixed, values of a we find the most probable weights WMP (the location of the posterior mode). This is equivalent to a standard optimisation of a regularised logistic model, and we use the efficient iterativelyreweighted least-squares algorithm [5] to find the maximum. 2. We compute the Hessian at WMP: \7\7logp(t, wla)1 WMP = _(<)TB<) + A), (12) where Bnn = a{y(x n )} [1 - a{y(x n )}], and this is negated and inverted to give the covariance I: for a Gaussian approximation to the posterior over weights, and from that the hyperparameters a are updated using (9). Note that there is no 'noise' variance a 2 here. This procedure is repeated until some suitable convergence criteria are satisfied. Note that in the Bayesian treatment of multilayer neural networks, the Gaussian approximation is considered a weakness of the method if the posterior mode is unrepresentative of the overall probability mass. However, for the RVM, we note that p(t, wla) is log-concave (i.e. the Hessian is negative-definite everywhere), which gives us considerably more confidence in the Gaussian approximation. 5 5.1 Examples of RVM Classification Synthetic example: Gaussian mixture data We first utilise artificially generated data in two dimensions in order to illustrate graphically the selection of relevance vectors. Class 1 (denoted by 'x ') was sampled from a single Gaussian, and overlaps to a small degree class 2 ('.'), sampled from a mixture of two Gaussians. A relevance vector classifier was compared to its support vector counterpart, using the same Gaussian kernel. A value of C for the SVM was selected using 5-fold crossvalidation on the training set. The results for a typical dataset of 200 examples are given in Figure 2. The test errors for the RVM (9.32%) and SVM (9.48%) are comparable, but the remarkable feature of contrast is the complexity of the classifiers. The support vector machine utilises 44 kernel functions compared to just 3 for the relevance vector method. It is also notable that the relevance vectors are some distance from the decision boundary (in x-space). Given further analysis, this observation can be seen to be consistent with the hyperparameter update equations. A more qualitative explanation is that the output of a basis function lying on or near the decision boundary is a poor indicator of class membership, and such basis functions are naturally 'penalised' under the Bayesian framework. 657 The Relevance Vector Machine SVM: error=9.48% vectors=44 , \ \ @. ~ " , x x \ X x x X x x x ~x ., .... ? . ? II ? 'C' ? "I. x ?\ ? x x ~ ?? _~ x x~ .. x.":X~x x ~ x :xx xx'f. RVM: error=9.32% vectors=3 ? ? xxx x x x ,, ,, ? I I , \ Figure 2: Results of training functionally identical SVM (left) and RVM (right) classifiers on a typical synthetic dataset. The decision boundary is shown dashed, and relevance/support vectors are shown circled to emphasise the dramatic reduction in complexity of the RVM model. 5.2 Real examples In the table below we give error and complexity results for the 'Pima Indian diabetes' and the 'U.S.P.S. handwritten digit' datasets. The former task has been recently used to illustrate Bayesian classification with the related Gaussian Process (GP) technique [9], and we utilised those authors' split of the data into 200 training and 332 test examples and quote their result for the GP case. The latter dataset is a popular support vector benchmark, comprising 7291 training examples along with a 2007-example test set, and the SVM result is quoted from [7]. Dataset Pima Indians U.S.P.S. ___ errors ___ SVM GP RVM 67 68 65 4.4% 5.1% _ _ kernels _ _ SVM GP RVM 109 2540 200 4 316 In terms of prediction accuracy, the RVM is marginally superior on the Pima set, but outperformed by the SVM on the digit data. However, consistent with other examples in this paper, the RVM classifiers utilise many fewer kernel functions. Most strikingly, the RVM achieves state-of-the-art performance on the diabetes dataset with only 4 kernels. It should be noted that reduced set methods exist for subsequently pruning support vector models to reduce the required number of kernels at the expense of some increase in error (e.g. see [7] for some example results on the U.S.P.S. data). 6 Discussion Examples in this paper have effectively demonstrated that the relevance vector machine can attain a comparable (and for regression, apparently superior) level of generalisation accuracy as the well-established support vector approach, while at the same time utilising dramatically fewer kernel functions - implying a considerable 658 ME. Tipping saving in memory and computation in a practical implementation. Importantly, we also benefit from the absence of any additional nuisance parameters to set, apart from the need to choose the type of kernel and any associated parameters. In fact, for the case of kernel parameters, we have obtained improved (both in terms of accuracy and sparsity) results for all the benchmarks given in Section 3.2 when optimising the marginal likelihood with respect to multiple input scale parameters in Gaussian kernels (q. v. [9]). Furthermore, we may also exploit the Bayesian formalism to guide the choice of kernel itself [2], and it should be noted that the presented methodology is applicable to arbitrary basis functions, so we are not limited, for example, to the use of 'Mercer' kernels as in the SVM. A further advantage of the RVM classifier is its standard formulation as a probabilistic generalised linear model. This implies that it can be extended to the multiple-class case in a straightforward and principled manner, without the need to train and heuristically combine multiple dichotomous classifiers as is standard practice for the SVM. Furthermore, the estimation of posterior probabilities of class membership is a major benefit, as these convey a principled measure of uncertainty of prediction, and are essential if we wish to allow adaptation for varying class priors, along with incorporation of asymmetric misclassification costs. However, it must be noted that the principal disadvantage of relevance vector methods is in the complexity of the training phase, as it is necessary to repeatedly compute and invert the Hessian matrix, requiring O(N2) storage and O(N3) computation. For large datasets, this makes training considerably slower than for the SVM. Currently, memory constraints limit us to training on no more than 5,000 examples, but we have developed approximation methods for handling larger datasets which were employed on the U.S.P.S. handwritten digit database'. We note that while the case for Bayesian methods is generally strongest when data is scarce, the sparseness of the resulting classifier induced by the Bayesian framework presented here is a compelling motivation to apply relevance vector techniques to larger datasets. Acknowledgements The author wishes to thank Chris Bishop, John Platt and Bernhard Sch5lkopf for helpful discussions, and JP again for his Sequential Minimal Optimisation code. References [1) J. O. Berger. Statistical decision theory and Bayesian analysis. Springer, New York, second edition , 1985. [2) D. J. C. Mackay. Bayesian interpolation. Neural Computation, 4(3):415-447, 1992. [3) D. J . C. Mackay. The evidence framework applied to classification networks. Neural Computation, 4(5):720-736, 1992. (4) D . J . C. Mackay. Bayesian non-linear modelling for the prediction competition. In ASHRAE Transactions, vol. 100, pages 1053- 1062. ASHRAE, Atlanta, Georgia, 1994. (5) 1. T . Nabney. Efficient training of RBF networks for classification. In Proceedings of ICANN99, pages 210-215, London, 1999. lEE. [6) R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1996. (7) B. Sch6lkopf, S. Mika, C. J. C. Burges, P. Knirsch, K.-R. Miiller, G . Ratsch , and A. J. Smola. Input space versus feature space in kernel-based methods. IEEE Transactions on Neural Networks, 10(5) :1000- 1017, 1999. [8) V. N. Vapnik. Statistical Learning Theory. Wiley, New York, 1998. [9) C. K. 1. Williams and D. Barber. Bayesian classification with Gaussian processes. IEEE Trans. Pattern Analysis and Machine Intelligence, 20(12) :1342-1351, 1998. 

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678 ALOW-POWER CMOS CIRCUIT WHICH EMULATES TEMWORALELECTIDCALPROPERTIES OF NEURONS Jack L. Meador and Clint S. Cole Electrical and Computer Engineering Dept. Washington State University Pullman WA. 99164-2752 ABSTRACf This paper describes a CMOS artificial neuron. The circuit is directly derived from the voltage-gated channel model of neural membrane, has low power dissipation, and small layout geometry. The principal motivations behind this work include a desire for high performance, more accurate neuron emulation, and the need for higher density in practical neural network implementations. INTRODUCTION Popular neuron models are based upon some statistical measure of known natural behavior. Whether that measure is expressed in terms of average firing rate or a firing probability, the instantaneous neuron activation is only represented in an abstract sense. Artificial electronic neurons derived from these models represent this excitation level as a binary code or a continuous voltage at the output of a summing amplifier. While such models have been shown to perform well for many applications, and form an integral part of much current work, they only partially emulate the manner in which natural neural networks operate. They ignore, for example, differences in relative arrival times of neighboring action potentials -- an important characteristic known to exist in natural auditory and visual networks {Sejnowski, 1986}. They are also less adaptable to fme-grained, neuron-centered learning, like the post-tetanic facilitation observed in natural neurons. We are investigating the implementation and application of neuron circuits which better approximate natural neuron function. BACKGROUND The major temporal artifacts associated with natural neuron function include the spacio-temporal integration of synaptic activity, the generation of an action potential (AP), and the post-AP hyperpolarization (refractory) period (Figure 1). Integration, manifested as a gradual membrane depolarization, occurs when the neuron accumulates sodium ions which migrate through pores in its cellular membrane. The rate of ion migration is related to the level of presynaptic AP bombardment, and is also known to be a non-linear function of transmembrane potential. Efferent AP generation occurs when the voltage-sensitive membrane of the axosomal hillock reaches some threshold potential whereupon a rapid increase in sodium permeability leads to A Low-Power CMOS Circuit Which Emulates Neurons complete depolarization. Immediately thereafter, sodium pores "close" simultaneously with increased potassium permeability, thereby repolarizing the membrane toward its resting potential. The high potassium permeability during AP generation leads to the transient post-AP hyperpolarization state known as the refractory period. v Actl vat Ion Threshold ( 1) ( 3) Figure 1. Temporal artifacts associated with neuron function. (1) gradual depolarization, (2) AP generation, (3) refractory period. Several analytic and electronic neural models have been proposed which embody these characteristics at varying levels of detail. These neuromimes have been used to good advantage in studying neuron behavior. However, with the advent of artificial neural networks (ANN) for computing, emphasis has switched from modeling neurons for physiologic studies to developing practical neural network implementations. As the desire for high performance ANNs grows, models amenable to hardware implementation become more attractive. The general idea behind electronic neuromimes is not new. Beginning in 1937 with work by Harmon {Harmon, 1937},lIectronic circuits have been used to model and study neuronal behavior. In the late 196(Ys, Lewis {Lewis, 1968} developed a circuit which simulated the Hodgkin-Huxley model for a single neuron, followed by MacQregor's circuit {MacGregor, 1973} in the early 1970's which modelled a group of 50 neurons. With the availability of VLSI in the 1980's, electronic neural implementations have largely moved to the realm of integrated circuits. Two different strategies have been documented: analog implementations employing operational amplifiers {Graf, et at, 1987,1988; Sivilotti, et at, 1986; Raffel, 1988; Schwartz, et al, 1988}; and digital implementations such as systolic arrays {Kung, 1988}. More recently, impulse neural implementations are receiving increased attention. like other models, these neuromimes generate outputs based on some non-linear function of the weighted net inputs. However, interneuron communication is realized through impulse streams rather than continuous voltages or binary numbers {Murray, 1988; N. El-Leithy, 1987}. Impulse networks communicate neuron activation as variable pulse repetition rates. The impulse neuron circuits which shall be discussed offer both small geometry and low power dissipation as well as a closer approximation to natural neuron function. 679 680 Meador and Cole - A CMOS IMPULSE NEURON An impulse neuron circuit developed for use in CMOS neural networks is shown in Figure 2. In this circuit, membrane ion current is modeled by charge flowing to and from Ca. Potassium and sodium influx is represented by current flow from Vdd to the capacitor, and ion efflux by flow from the capacitor to ground. The Field EffectTransistors (FETs) connected between Vdd, Vsr , and the capacitor emulate voltageand chemically-gated ion channels found in natural neural membrane. In the Figure, PET 1 corresponds to the post-synaptic chemicaIly-gated ion channels associated with one synapse. PETs 2, 3, and 4 emulate the voltage-gated channels distributed throughout a neuron membrane. The following equations summarize circuit operation: Ca dVa/dt=/31E (Vr,Va)+/3:uF(Va)-/34G (Va) (1) E(Vr,Va) = (Vr-Va-V",)(Vdd-Va)-(Vdd-Va)2 /2 (2) F(Y. ) ={(Vdd-Vtp) (Va-Vdd)-(Va-Vdd)2 /2 if g(t) ~O (3) 0 a otherwlSe G(V)={(Vdd-V",)Va-Va2/2 if h(t~)=O 0 a otherwlSe g(t) =h (t)(l-h (t-C)) (4) (5) 0 if Va(t) > Vth; 1 h()Vtl<Va(t)<Vth and h (Va(t-e))=O t - 1 if Va(t)<Vtl ; Vtl<Va(t)<Vth and h (~(t-e))=l (6) Vdd {52 Exci tator y Synapse 10 {53 {51 + Axon Vs + Va Ca 1 /5. Figure 2. A CMOS impulse neuron with one excitatory synapse-PET. A Low-Power CMOS Circuit Which Emulates Neurons Equation (1) expresses how changes in Va (which emulates instantaneous neuron excitation) depend upon the sum of three current components controlled by these PETs. E, F, and G in equations (2) through (4) express PET drain-source currents as functions terminal voltages. Equations (3) and (5) rely upon the assumption that PET 2 and PET 3 are implemented as a single dual-gate device where the transconductance f3n=fJ2f33/f.P2+/33). Non-saturated PET operation is assumed for these equations even though the PETs will momentarily pass through saturation at the onset of conduction in the actual circuit. The Schmitt trigger circuit establishes a nonlinear positive feedback path responsible for action potential initiation. The upper threshold of the trigger (VIII) emulates the natural neuron activation threshold while the lower threshold (Va) emulates the maximum hyperpolarization voltage. Equation (6) expresses the hysterisis present in the Schmitt trigger transfer characteristic. When Vs reaches the upper Schmitt threshold, PET 3 turns on, creating a current path from Vtid to Cs , and emulating the upswing of a natural action potential spike. A moment later, PET 2 turns off, starting the action potential downswing. Simultaneously, PET 4 turns on, begining the absolute refractory period where Cs is discharged toward the maximum hyperpolarization potential. When that potential is reached, the Schmitt trigger turns off PET 4 and the impulse firing cycle is complete. The capacitor terminal voltage Va emulates all gross temporal artifacts associated with membrane potential, including spacio-temporal integration, the action potential spike, and a refractory period. The instantaneous net excitation to the neuron is represented by the total current flowing into the summing node on the floating plate of the capacitor. Charge packets are transferred from Vtid to the capacitor by the excitatory synapse PET. Excitatory packet magnitude is dependent upon the transconductance PI. Inhibitory synapses (not shown) operate similarly, but instead reduce capacitor voltage by drawing charge to Va. A buffered action potential signal useful for driving many synapse PETs is available at the axon output. The membrane potential components (E,F,and G) of the circuit equations describe nonlinear relationships between post-synaptic excitation (E), membrane potential (F and G), and membrane ion currents. The functional forms of these components are equivalent to those found between terminal voltages and currents in non-saturated PETs. It is notable that natural voltage-gated channels do not necessarily follow the same current-voltage relationship of a PET. Even though more accurate models and emulations of natural membrane conductance exist, it seems unlikely at this time that they would help further improve neural network implementation. There is little doubt that more complex circuitry would be required to better approximate the true nonlinear relationship found in the biochemistry of natural neural membrane. That need conflicts directly with the goal of high-density integration. IMPULSE NEURAL NE1WORKS ~ Organizing a collection of neuron circuits into a useful network confIguration requires some weight specification method. Weight values can be either directly specified by the designer or learned by the network. A method particularly suited for use with the fIXed PET-synapses of the foregoing circuit is to fust learn weights using an "off-line" 681 682 Meador and Cole simulation, then translate the numerical results to physical FET transconductances. To do this, the activation function of an impulse neuron is derived and used in a modified back-propagation learning procedure. IMPULSE NEURON ACfIVATION FUNCTION Learning algorithms typically require some expression of the neuron activation function. Neuron activation can be expressed as a numerical value, a binary pattern, or a circuit voltage. In an impulse neuron, activation is expressed in terms of firing rate. The more frequently an impulse neuron circuit fues, the greater its activation. Impulse neuron activation is a nonlinear function of the excitation imparted through its synapse connections. An analytical expression of this nonlinear function can be derived using a rectangular approximation of neuron impulse waveforms. It is fust necessary to defme a unit-impulse as one impulse conducted by a synapse FET having some pre-determined reference transconductance (f3~). In Figure 3, To represents an invariant activation impulse width which is assumed to be identical for all neurons. T 1 represents the variable time period required for the neuron to accumulate the equivalent of K unit-impulses input excitation prior to firing. It can be assumed that net input comes from a single excitatory synapse with no other excitation. It shall also be assumed that impulses arrive at a constant rate, so (7) T 1 =K /W;jR; where R/ is the firing rate of the source neuron and connecting neuron; and neuron j. The firing rate of the receiving neuron will be Rj this becomes: W;j is the weight of the synapse =1/(To + T 1). Substituting for T1 (8) F"tgure 3 compares this function with the logistic activation function. The impulse activation function approaches zero at the rate of 1/K when R; approaches zero. The function also approaches an asymptote of Rj =l/T0 as R; increases without bound. Any non-synaptic source which causes current flow from Vtit to Co will shift the curve to the left, and reflect a spontaneous firing rate at zero input excitation. A similar current source to VoU will shift the function to the right, reflecting a positive firingonset threshold. Circuit-level simulations show a clear correspondence to these analytical results. This functional form is also evident in activation curves experimentally observed with natural neurons {Guyton, 1986}. Various natural neurons are known to exhibit both spontaneous firing and fuing-onset thresholds as well. The impulse activation function constant, K, is determined by several factors including fJ~, Co, and To. Assuming that To? T h no leakage current exists, and that a FET conducting in its non-saturated region can be approximated by a resistor, the following expression for K is obtained: (9) A Low-Power CMOS Circuit Which Emulates Neurons where Rchon =l/pteJ<Vdd -V".), Ca is the summing capacitance, Va Vth are the low and high threshold voltages of the Schmitt trigger, and V". is the gate threshold voltage for an excitatory PET-synapse. A more accurate K value can be obtained by using the non-saturated PET current equation and solving a nonlinear differential equation. 1.0 rj~ ~ tTo~ 1, 1 ri j rectangUlar Impulse Train J 0 .5 Logistic Activation Impulse Activation o ~--~----------------------------------~---------------------------------------------ri o Figure 3. Rectangular impulse train approximation for impulse activation function derivation. Unlike the logistic function which asymptotically approaches zero, impulse activation is equal to zero over a range of net excitation. BACK?PROPAGATION IN IMPULSE NE1WORKS A back-propagation algorithm has been used to learn connection weights for impulse neural networks. At this time, weight values are non-adaptive (they are fixed at circuit fabrication) because they are implemented as invariant PET transconductances. Adaptive synapses compatible with impulse neuron circuits are in the early stages of development, but are not available at this time. Much can be learned about these networks using non-adaptive prototypes, however. As a result, weight learning is performed offline as part of the network design process. The back-propagation procedure used to learn weights for impulse networks differs from the generalized delta rule {Rumelhart, 1986} in two ways. The fust difference is the use of the impulse activation function instead of the logistic function. Any activation nonlinearity is a viable candidate for use with the generalized delta rule as long as it is differentiable. This is where difficulties mount with the impulse activation function. First of all, it is not differentiable at zero. What seems to be more important, however, is that its first derivative equals zero over a range of 683 684 Meador and Cole inputs. Examination of the generalized delta rule (which performs gradient-descent) reveals that when the fust derivative of neuron activation becomes zero, connections associated with that-neuron will cease to adapt. Once this happens, the procedure will most probably never arrive at a problem solution. To work around this problem, a second deviation from the generalized delta rule was implemented. This involves a departure from using the true first derivative when the impulse activation becomes zero. A small constant can be used to guarantee that learning continues even though the associated neuron activation is zero: Act =l/(To +K /Net) A ,_{(l/(To+K/Net)' if Net >0 ct - e otherwise (10) (11) The use of these equations yields a back-propagation algorithm for impulse networks which does not perform true gradient descent, yet which so far has been observed to learn solutions to logic problems such as XOR and the 4-2-4 encoder. Investigation of other offline learning algorithms for impulse networks continues. Currently, this algorithm fulfills the immediate need for an offline procedure which can be used in the design of multi-layer impulse neural networks. IMPLEMENTATION Two requirements for high density integration are low power dissipation and small circuit geometry. CMOS impulse neurons use switching circuits having no continuous power dissipation. A conventional op-amp circuit must draw constant current to achieve linear bias. An op-amp also requires larger circuit geometries for gain accuracy over typical fabrication process variations. Such is not the case for nonlinear switching circuits. As a result, these neurons and others like them are expected to help improve analog neural network integration density. An impulse neuron circuit has been designed which eliminates FETs 2 and 3 of Figure 2 in exchange for reduced layout area. In this circuit, Va no longer exhibits an activation potential spike. This spike seems irrelevant given the buffered impulse available at the axon output. The modified neuron circuit occupies 200 X 25 lambda chip area. A fIXed PET-synapse occupies a 16 by 18 lambda rectangle. With these dimensions a full-interconnect layout containing 40 neurons and 1600 fIXed connections will fit on a MOSIS 2-micron tiny chip. XOR and 4-2-4 networks of these circuits are being developed for 2-micron CMOS. CONCLUSION The motivation of this work is to improve neural network implementation technology by designing CMOS circuits derived from the temporal characteristics of natural neurons. The results obtained thus far include: Two CMOS circuits which closely correspond to the voltage-gat ed-channel model of natural neural membrane. A Low-Power CMOS Circuit Which Emulates Neurons Simulations which show that these impulse neurons emulate gross artifacts of natural neuron function. Initial work on a back-propagation algorithm which learns logic solutions using the impulse neuron activation function. The development of prototype impulse network I.Cs. Future goals involve extending this investigation to plastic synapse and neuron circuits, alternate algorithms for both offline and online learning, and practical implementations. Rererences H. P. Graf W. Hubbard L. D. Jackel P. G. N. deVegvar. A CMOS associative Memory Chip. IEEE ICNN Con. Proc., pp. 461-468, (1987). H. P. Graf L. D. Jackel W. E. Hubbard. VLSI Implementation of a Neural Network Model. IEEE Computer, pp. 41-49, (1988). A.C. Guyton. Chapt. 10. Organization of the Nervous System: Basic Functions of Synapses. Textbook o[ Physiology, p.l36. (1986) N. EI-Liethy, R.W. Newcomb, M. Zaghlou. A Basic MOS Neural-Type Junction A Perspective on Neural-Type Microsystems. IEEE ICNN Con. Proc., pp. 469-477, (1987). E. R. Lewis. Using Electronic Circuits to Model Simple Neuroelectric Interactions. Proc. IEEE 56, pp. 931-949, (1968). R. J. MacGregor R. M. Oliver. A General-Purpose Electronic Model for Arbitrary Configurations of Neurons. 1. Theor. Bioi. 38, pp. 527-538 (1973). S. Y. Kung. and J. N. Hwang. Parallel Achitectures for Artificial Neural Nets. IEEE ICNN Con. Proc., pp. 11-165 to 11-172, (1988). J. Mann R. Lippman B. Berger J. Raffel. A Self-Organizing Neural Net Chip. IEEE Cust.Integr. Ckts. Conf" pp. 10.3.1-10.3.5 (1988). A. F. Murray A. V. W. Smith. Asynchronous VLSI Neural Networks Using PulseStream Arithmetic. IEEE lnl. of Sol. St. Phys. 23, pp. 688-697, (1988). J. I. Raffel. Electronic Implementation of Neuromorphic Systems. IEEE Cust. Integr. Ckts. Conf" pp. 10.1.1-10.1.7, (1988). D. Rumelhart, G.E. Hinton, and RJ. Williams. Learning Internal Representations by Error Propagation. Parallel Distributed Processing, Vol 1: Foundations, pp. 318-364, (1986). O. H. Schmitt. Mechanical Solution of the Equations of Nerve Impulse Propagation. Am. 1. Physiol. 119, pp. 399-400, (1937). D. B. Schwartz R. E. Howard. A Programmable Analog Neural Network Chip. IEEE Cust. Integr. Ckts. Conf., pp. 10.2.1-1.2.4, (1988). 685 686 Meador and Cole T J. Sejnowski. Open Questions About Computation in Cerebral Cortex. Parallel Distributed Processing Vol. 2:Psychological and Biological Models, pp. 378-385, (1986). M. A. Sivilotti M. R. Emerling C. A. Mead. VISI Architectures for Implementation of Neural Networks. Am. Ins. of Phys., 408-413, (1986). 

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A Neuromorphic VLSI System for Modeling the Neural Control of Axial Locomotion Girish N. Patel girish@ece.gatech.edu Edgar A. Brown ebrown@ece.gatech.edu Stephen P. DeWeerth steved@ece.gatech.edu School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Ga. 30332-0250 Abstract We have developed and tested an analog/digital VLSI system that models the coordination of biological segmental oscillators underlying axial locomotion in animals such as leeches and lampreys. In its current form the system consists of a chain of twelve pattern generating circuits that are capable of arbitrary contralateral inhibitory synaptic coupling. Each pattern generating circuit is implemented with two independent silicon Morris-Lecar neurons with a total of 32 programmable (floating-gate based) inhibitory synapses, and an asynchronous address-event interconnection element that provides synaptic connectivity and implements axonal delay. We describe and analyze the data from a set of experiments exploring the system behavior in terms of synaptic coupling. 1 Introduction In recent years, neuroscientists and modelers have made great strides towards illuminating structure and computational properties in biological motor systems. For example, much progress has been made toward understanding the neural networks that elicit rhythmic motor behaviors, including leech heartbeat, crustacean stomatogastric mill and lamprey swimming (a good review on these is in [1] and [2]). It is thought that these same mechanisms form the basis for more complex motor behaviors. The neural substrate for these control mechanisms are called central pattern generators (CPG). In the case of locomotion these circuits are distributed along the body (in the spinal cord of vertebrates or in the ganglia of invertebrates) and are richly interactive with sensory input and descending connections from the brain, giving rise to a highly distributed system as shown in Figure 1. In cases in which axial locomotion is involved, such as leech and lamprey swimming, synaptic interconnection patterns among autonomous segmental oscillators along the animal's axis produce coordinated motor patterns. These intersegmental coordination architectures have been well studied through both physiological experimentation and mathematical modeling. In addition, undulatory gaits in snakes have also been studied from a robotics perspective [3]. However, a thorough understanding of the computational principles in these systems is still lacking. A Neuromorphic System for Modeling Axial Locomotion 725 Figure 1: Neuroanatomy of segmented animals. In order to better understand the computational paradigms that mediate intersegmental coordination and the resulting neural control of axial locomotion (and other motor patterns), we are using neuromorphic very large-scale integrated (VLSI) circuits to develop models of these biological systems. The goals in our research are (i) to study how the properties of individual neurons in a network affect the overall system behavior; (ii) to facilitate the validation of the principles underlying intersegmental coordination; and (iii) to develop a real-time, low power, motion control system. We want to exploit these principles and architectures both to improve our understanding of the biology and to design artificial systems that perform autonomously in various environments. Parameter Input Embedded Controller Event Output Address-Event Communication Network " / 12 segments Figure 2: Block-level diagram of the implemented system. The intersegmental communications network facilitates communication among the intrasegmental units with pipelined stages. In this paper, we present a VLSI model of intersegmental coordination as shown in Figure 2. Each segment in our system is implemented with a custom Ie containing a CPG consisting of two silicon model neurons, each one with 16 inhibitory synapses whose values are stored on chip and are continuously variable; an asynchronous address event communications IC that implements the queuing and delaying of events providing synaptic connectivity and thus simulating axonal properties; and a microcontroller (with internal AID converter and timer) that facilitates the modification of individual parameters through a serial bus. The entire system consists of twelve such segments linked to a computer on which a graphical user interface (GUI) is implemented. By using the GUI, we are able to control all of the synaptic connections in the system and to measure the result- G. N Patel, E. A. Brown and S. P. DeWeerth 726 ing neural outputs. We present the system model, and we investigate the role of synaptic coupling in the establishment of phase lags along this chain of neural oscillators. 2 Pattern generating circuits The smallest neural system capable of generating the basic alternating activity that characterizes the swimming ePGs is the half-center oscillator, essentially two bursting neurons with reciprocally inhibitory connections [1] as shown in Figure 3a. In biological systems, the associated neurons have both slow and fast time constants to facilitate the fast spiking (action potentials) and the slower bursting oscillations that control the elicited movements as shown in Figure 3b. To simplify the parameter space of our system, we use reduced two-state silicon neurons [4]. The output of each silicon neuron is an oscillation that represents the envelope of the bursting activity (i.e. the spiking activity and corresponding fast time constants are eliminated) as shown in Figure 3c. Each neuron also has 16 analog synapses that receive off-chip input. The synaptic parameters are stored in an array of floating-gate transistors [5] that provide nonvolatile analog memory. CPG B 1 JW~WID~,J,U~l.v 2 111 11111 1111111 11111 11111111111 11 111 111111 1111 1111 III 1 C 2 A Figure 3: Half-center oscillator and the generation of events in spiking and nonspiking silicon neurons. Events are generated by detecting rapid rises in the membrane potential of spiking neurons or by detecting rapid rises and falls in nonspiking neurons. 3 Intersegmental communication Our segmented system consists of an array of ePG circuits interconnected via an communication network that implements an asynchronous, address-event protocol [6][7]. Each ePG is connected to one node of this address-event intersegmental communication system as illustrated in Figure 2. This application-specific architecture uses a pipelined broadcast scheme that is based upon its biological counterpart. The principal advantage of using this custom scheme is that requisite addresses and delays are generated implicitly based upon the system architecture. In particular the system implements distance-dependent delays and relative addressing. The delays, which are thought to be integral to the network computation, replicate the axonal delays that result as action potentials propagate down an animal's body [2]. The relative addressing greatly simplifies the implementation of synaptic spread [8], the hypothesized translational invariance in the intersegmental connectivity in biological axial locomotion systems. Thus , we can set the synaptic parameters identically at every segment, greatly reducing system complexity. In this architecture (which is described in more depth in [4]), each event is passed from segment to neighboring segment bidirectionally down the length of the one-dimensional 727 A Neuromorphic System for Modeling Axial Locomotion communications network. By delaying each event at every segment, the pipeline architecture facilitates the creation of distance-dependent delays. The other primary advantage of this architecture is that it can easily generate a relative addressing scheme. Figure 4 illustrates the event-passing architecture with respect to the relative addressing and distancedependent delays. Each event, generated at a particular node (the center node, in this example), is transmitted bidirectionally down the length of the network. It is delayed by time /). T at each segment, not including the initiating segment. t=to+2/).T t=to+/).T t=to+/).T t=t o+2/).T = = V '8"'O"'8~8~c;( A -2 A = -1 A= 0 A 1 Figure 4: Relative addressing and distance-dependent delays. The events are generated by the neurons in each segment. Because these are not spiking neurons, we could not use the typical scheme of generating one event per action potential. Instead, we generate one event at the beginning and end of each burst (as illustrated in Figure 3) and designate the individual events as rising or falling . In each segment the events are stored in a queue (Figure 5), which implements delay based upon uniform conduction velocities. As an event arrives at each new segment, it is time stamped, its relative address is incremented (or decremented), and then it is stored in the queue for the /). T interval. As the event exits the queue, its data is decoded by the intrasegmental units, and synaptic inputs are applied to the appropriate intrasegmental neurons. events from rostral segment (closer to head) ..... .. 9: ev ents from events from intrasegmental unit -.. QUEUE (event storage and processing) l... 9: ... .. ... to ro stral and cau dal segments and intrasegmental unit caudal segment (closer to tail) Figure 5: Block-level diagram of a communications node illustrating how events enter and exit each stage of the pipeline. 4 Experiments and Discussion We have implemented the complete system shown in Figure 2, and have performed a number of experiments on the system. In Figure 6, we show the behaviors the system exhibits when it is configured with asymmetrical nearest-neighbor connections. The system displays traveling waves whose directions depend on the direction of the dominant coupling. Note that the intersegmental phase lags vary for different swim frequencies. One important set of experiments focussed on the role of long-distance connections on the system behaviors. In these experiments, we configured the system with strong descending (towards the tail) connections such that robust rearward traveling waves (forward swimming) are observed. The long-distance connections are weak enough to avoid any bifurcations in behavior (different type of behavior). Thus , the traveling wave solution resulting from the nearest-neighbor connections persists as we progressively add long-distance connections. In Figure 7 we show the dependency of the swim frequency and the total phase lag (summation of the normalized intersegmental phase lags, where 1 == 360 0 ) on the extent of the connections. The results show a clear difference in behav- G. N. Patel, E. A. Brown and S. P. DeWeerth 728 stronger descending coupling stronger ascending coupling . . . . . . . . . . .. , c: . ..... 1O&-" 8 1 6 4 0.5 o -0.2 -0.1 2 o 0.1 0.2 time (sec) ~ (l) CI) 1 o.~ ~t=~~~~~ -0.2 -0.1 o 0.1 0.2 time (sec) B 1 0.5 o time (sec) 0.05 o .L.fI==::1.L:;:::1l:::=:,:..:-.J o 0.05 -0.05 time (sec) Figure 6: Traveling waves in the system with asymmetrical, nearest-neighbor connections. Plots are cross-correlations between rising edge events generated by a neuron in segment six and events generated by homolog neurons in each segment. Stronger ascending connections (A & B) produce forward traveling waves (backward swimming) and stronger descending connections (C & D) produce rearward traveling waves (forward swimming) . An externally applied current (lext) controls the swim frequency. At small values of lext (6.7 nA) the periods of the swim cycles are approximately 0.180 ms and 0.150 ms for A & C, respectively; for large values of lext (32.8 nA), the periods of the swim cycles are approximately 36 ms and 33 ms for B & D, respectively. iors between the lowest tonic drive (lext = 21.9 nA) and the two higher tonic drives. (By tonic drive, we mean a constant dc current is applied to all neurons.) In the former, the sensitivity of long-distance connections on frequency and intersegmental phase lags is considerably greater than in the latter. The demarcation in behavior may be attributed to different behaviors at different tonic drives. For lower tonic drive, the long-distance connections tend to synchronize the system (decrease the intersegmental phase lags). At the higher tonic drives, long-distance connections do not affect the system considerably. For lext = 32.8 nA, connections that span up to four segments aid in producing uniformity in the intersegmental phase lags. Although this does not hold for lext = 48.1 nA, long-distance connections playa more significant role in preserving the total phase difference. At lext = 32.8 nA and lext = 48.1 nA, the system with short-distance connections produces a total phase difference of 1.19 and 1.33, respectively. In contrast, for lext = 32.8 nA and lext = 48.1 nA, the system with long-distance connections that span up to seven segments produces a total phase difference of 1.20 and 1.25, respectively. In the above experiments, we have demonstrated that, in a specific parameter regime, weak long-distance connections can affect the intersegmental phase lags. However, these weight profiles should not be construed as a possible explanation on what the weight profiles in a biological system might be. The parameter regime in which we observed this behavior is small; at moderate strengths of coupling, the traveling wave solutions disappear and move towards synchronous behavior. Recent experiments done on spinalized lampreys reveal that long-distance connections are moderately strong [10]. Thus, our CUTrent model is unable to replicate this aspect of intersegmental coordination. There are several explanations that may account for this discrepancy. 729 A Neuromorphic System for Modeling Axial Locomotion 40 r---~-~-~----' B 1.5 .---~-~-~--, A 30 m i)' ~ c .c ~ 20 c. c- -.... 1 ~ O> :? 0.5 10 o~--~----~--~--~ o 2 4 extent 6 8 o'----~----~--~----' o 2 4 6 8 extent Figure 7: Effects of weak long-distance connections on swimming frequency (A), on the total phase difference (summation of the normalized intersegmental phase lags) (B), and on the standard deviation of the intersegmental phase lags (C). 5 < = denote Iext = 48.1 nA, 32.8 nA, and 21 nA, respectively. In the segmental CPG network of the animal, there are many classes of neurons that send projections to many other classes of neurons. The phase a connection imposes is determined by which neuron class connects with which other neuron class. In our system, the segmental CPG network has only a single class of neurons upon which the long-distance connections can impose their phase. Depending on where in parameter space we operate our system, the long-distance connections have too little or too great an effect on the behavior of the system. At high tonic drives, the sensitivity of the weak long-distance connections on the intersegmental phase lags is small, whereas for small tonic drives, the long-distance connections have a great effect on the intersegmental phase lags. It has been shown that if the waveform of the oscillators is sinusoidal (i.e., the time scales of the two state variables are not too different), traveling wave solutions exist and have a large basin of attraction [11]. However, as the disparity between the two time scales is made larger (i.e., the neurons are stiff and the waveform of the oscillations appears square-wave like), the system will move towards synchrony. In our implementation, to facilitate accurate communication of events, we bias the neurons with relatively large differences in the time scales. Thus, this restriction reduces the parameter regime in which we can observe stable traveling waves. Another factor that determines the range of parameters in which stable traveling waves are observed is the slope of our synaptic coupling function. When the slope of the coupling function is steep, the total synaptic current over a cycle can increase significantly, causing weak connections to appear strong. This has an overall effect of synchronizing the network [11]. For coupling functions whose slopes are shallow, the total synaptic current over a cycle is reduced; therefore, the connections appear weak and larger intersegmental phase lags are possible. Thus, the sharp synaptic coupling function in our implementation, which is necessary for communication, is another factor that diminishes the parameter regime in which we can observe stable traveling waves. The above factors limit the parameter range in which we observe traveling waves. However, all of these issues can be addressed by improving our CPG network. The first issue can be addressed by increasing the number of neuron classes or adding more segments. The second and third issues can be addressed by adding spiking neurons in our CPG network so that the form of the oscillations can be coded in the spike train and the synaptic coupling functions can be implemented on the receiving side of the CPG chip. The fourth G. N. Patel, E. A. Brown and S. P. DeWeerth 730 issue can be addressed by designing self-adapting neurons that tune their internal parameters so that their waveforms and intrinsic frequencies are matched. Although weak coupling may not be biologically plausible, producing traveling waves based on phase oscillators would be an interesting research direction. 5 Conclusions and Future Work In this paper, we described a functional, neuromorphic VLSI system that implements an array of neural oscillators interconnected by an address-event communication network. This system represents our most ambitious neuromorphic VLSI effort to date, combining 24 custom ICs, a special-purpose asynchronous communication architecture designed analogously to its biological counterpart, large-scale synaptic interconnectivity with parameters stored using floating-gate devices, and a computer interface for setting the parameters and for measuring the neural activity. The working system represents the culmination of a four-year effort, and now provides a testbed for exploring a variety of biological hypotheses and theoretical predictions. Our future directions in the development of this system are threefold. First, we will continue to explore, in depth, the operation of the present system, comparing it to theoretical predictions and biological hypotheses. Second, we are implementing a segmented mechanical system that will provide a moving output and will facilitate the implementation of sensory feedback. Third, we are developing new CPG model centered around sensory feedback and motor learning. The modular design of the system, which puts all of the neural and synaptic specificity on the CPG IC, allows us to design a completely new CPG and to replace it in the system without changing the communication architecture. References [1] E. Marder & R.L. Calabrese. Principles of rhythmic motor pattern generation. Physiological Reviews 76 (3): 687-717,1996. [2] A.H. Cohen, G.B. Ermentrout, T Kiemel, N. Kopell, K.A. Sigvardt, & TL. Williams. Modeling of intersegmental coordination in the lamprey central pattern generator for locomotion. TINS 15:434-438, 1992. [3] S. Hirose. Biologically Inspired Robots: Snake-like Locomotors and Manipulators. Oxford University Press, 1993. [4] S. DeWeerth, G. Patel, D. Schimmel, M. Simoni, & R.L. Calabrese. A VLSI Architecture for Modeling Intersegmental Coordination. In Proceedings of the Seventeenth Conference on Advanced Research in VLSI, R.B. Brown and A.T Ishii (eds), Los Alamitos, CA: IEEE Computer Society, 182-200, 1997. [5] P. Hasler, B.A. Minch, and C. Diorio. Adaptive circuits using pFet floating-gate devices. In Scott Wills and Stephen DeWeerth editors, 20th Conference of Advanced Research in VLSI, pages 215-230, Los Alamitos, California, CA: IEEE Computer Society, 1999. [6] M.A. Mahowald. VLSI Analogs of Neuronal Visual Processing: A Synthesis of Form and Function. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1992. [7] K.A. Boahen. Communicating Neuronal Ensembles between Neuromorphic Chips. Analog Integrated Circuits and Signal Processing, 1997. [8] T Wiliams. Phase Coupling and Synaptic Spread in Chains of Coupled Neuronal Oscillators. Science, vol. 258, 1992. [9] G. Patel. A Neuromorphic Architecture for Modeling Intersegmental Coordination. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1999. [10] A. H. Cohen. Personal communication. [11] D. Somers & N. Kopell. Waves and synchrony in networks of oscillators of relaxation and nonrelaxation type. Phyica D, 89:169-183,1995. [12] N. Kopell & G.B. Ermentrout. Coupled oscillators and the design of central pattern generators. Mathematical Biosciences, 90:87-109, 1988. 

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v-Arc: Ensemble Learning in the Presence of Outliers t G. Ratsch t , B. Scholkopf1, A. Smola", K.-R. Miillert, T. Onodatt , and S. Mikat t GMD FIRST, Rudower Chaussee 5,12489 Berlin, Germany Microsoft Research, 1 Guildhall Street, Cambridge CB2 3NH, UK * Dep. of Engineering, ANU, Canberra ACT 0200, Australia tt CRIEPI, 2-11-1, Iwado Kita, Komae-shi, Tokyo, Japan {raetsch, klaus, mika}~first.gmd.de,bsc~microsoft.com, Alex.Smola~anu.edu.au,onoda~criepi.denken.or.jp Abstract AdaBoost and other ensemble methods have successfully been applied to a number of classification tasks, seemingly defying problems of overfitting. AdaBoost performs gradient descent in an error function with respect to the margin, asymptotically concentrating on the patterns which are hardest to learn. For very noisy problems, however, this can be disadvantageous. Indeed, theoretical analysis has shown that the margin distribution, as opposed to just the minimal margin, plays a crucial role in understanding this phenomenon. Loosely speaking, some outliers should be tolerated if this has the benefit of substantially increasing the margin on the remaining points. We propose a new boosting algorithm which allows for the possibility of a pre-specified fraction of points to lie in the margin area Or even on the wrong side of the decision boundary. 1 Introduction Boosting and related Ensemble learning methods have been recently used with great success in applications such as Optical Character Recognition (e.g. [8, 16]). The idea of a large minimum margin [17] explains the good generalization performance of AdaBoost in the low noise regime. However, AdaBoost performs worse on noisy tasks [10, 11], such as the iris and the breast cancer benchmark data sets [1]. On the latter tasks, a large margin on all training points cannot be achieved without adverse effects on the generalization error. This experimental observation was supported by the study of [13] where the generalization error of ensemble methods was bounded by the sum of the fraction of training points which have a margin smaller than some value p, say, plus a complexity term depending on the base hypotheses and p. While this bound can only capture part of what is going on in practice, it nevertheless already conveys the message that in some cases it pays to allow for some points which have a small margin, or are misclassified, if this leads to a larger overall margin on the remaining points. To cope with this problem, it was mandatory to construct regularized variants of AdaBoost, which traded off the number of margin errors and the size of the margin 562 G. Riitsch, B. Sch6lkopf, A. J. Smola, K.-R. Muller, T. Onoda and S. Mika [9, 11]. This goal, however, had so far been achieved in a heuristic way by introducing regularization parameters which have no immediate interpretation and which cannot be adjusted easily. The present paper addresses this problem in two ways. Primarily, it makes an algorithmic contribution to the problem of constructing regularized boosting algorithms. However, compared to the previous efforts, it parameterizes the above trade-off in a much more intuitive way: its only free parameter directly determines the fraction of margin errors. This, in turn, is also appealing from a theoretical point of view, since it involves a parameter which controls a quantity that plays a crucial role in the generalization error bounds (cf. also [9, 13]). Furthermore, it allows the user to roughly specify this parameter once a reasonable estimate of the expected error (possibly from other studies) can be obtained, thus reducing the training time. 2 Boosting and the Linear Programming Solution Before deriving a new algorithm, we briefly discuss the properties of the solution generated by standard AdaBoost and, closely related, Arc-GV (2], and show the relation to a linear programming (LP) solution over the class of base hypotheses G. Let {gt(x) : t = 1, ... ,T} be a sequence of hypotheses and a = [al ... aT] their weights satisfying at ~ O. The hypotheses gt are elements of a hypotheses class G = {g: x 14 [-1, In, which is defined by a base learning algorithm. The ensemble generates the label which is the weighted majority of the votes by sign(f(x)) In order to express that of notation we define where f(x) = ~ lI:ill gt(x). (1) f and therefore also the margin p depend on a and for ease p(z, a) := yf(x) where z := (x, y) and f is defined as in (1). (2) Likewise we use the normalized margin: p(a):= min P(Zi, a) , l~t~m (3) Ensemble learning methods have to find both, the hypotheses gt E G used for the combination and their weights a. In the following we will consider only AdaBoost algorithms (including Arcing). For more details see e.g. (4, 2]. The main idea of AdaBoost is to introduce weights Wt(Zi) on the training patterns. They are used to control the importance of each single pattern in learning a new hypothesis (Le. while repeatedly running the base algorithm). Training patterns that are difficult to learn (which are misclassified repeatedly) become more important. The minimization objective of AdaBoost can be expressed in terms of margins as m (4) i=1 In every iteration, AdaBoost tries to minimize this error by a stepwise maximization of the margin. It is widely believed that AdaBoost tries to maximize the smallest margin on the training set [2, 5, 3, 13, 11]. Strictly speaking, however, a general proof is missing. It would imply that AdaBoost asymptotically approximates (up to scaling) the solution of the following linear programming problem over the complete hypothesis set G (cf. [7], assuming a finite number of basis hypotheses): maximize subject to p p( Zi, a) ~ p for all 1 < i < m at, P ~ 0 for all 1 ~ t ~ IGI lIalil = 1 (5) 563 v-Arc: Ensemble Learning in the Presence o/Outliers Since such a linear program cannot be solved exactly for a infinite hypothesis set in general, it is interesting to analyze approximation algorithms for this kind of problems. Breiman [2] proposed a modification of AdaBoost - Arc-GV - making it possible to show the asymptotic convergence of p(a t ) to the global solution pIP: Theorem 1 (Breiman [2]). Choose at in each iteration as at := argmin Lexp [-llatlll (p(Zi' at) - p(a t - I ))], aE[O,I] (6) i and assume that the base learner always finds the hypothesis 9 E G which minimizes the weighted training error with respect to the weights. Then lim p( at) = pIp. t-HX> Note that the algorithm above can be derived from the modified error function 9 g v(a t ):= Lexp [-llatlll (p(Zi' at) - p(a t - I ))]. (7) The question one might ask now is whether to use AdaBoost or rather Arc-GV in practice. Does Arc-GV converge fast enough to benefit from its asymptotic properties? In [12] we conducted experiments to investigate this question. We empirically found that (a) AdaBoost has problems finding the optimal combination if pIp < 0, (b) Arc-GV's convergence does not depend on pIp, and (c) for pIp> 0, AdaBoost usually converges to the maximum margin solution slightly faster than Arc-GV. Observation (a) becomes clear from (4): 9(a) will not converge to and lIal11 can be bounded by some value. Thus the asymptotic case cannot be reached, whereas for Arc-GV the optimum is always found. Moreover, the number of iterations necessary to converge to a good solution seems to be reasonable, but for a near optimal solution the number of iterations is rather high. This implies that for real world hypothesis sets, the number of iterations needed to find an almost optimal solution can become prohibitive, but we conjecture that in practice a reasonably good approximation to the optimum is provided by both AdaBoost and Arc-GV. 3 v- Algorithms For the LP-AdaBoost [7] approach it has been shown for noisy problems that the generalization performance is usually not as good as the one of AdaBoost [7, 2, 11]. From Theorem 5 in [13] (cf. Theorem 3 on page 5) this fact becomes clear, as the minimum of the right hand side of inequality (cf. (13)) need not necessarily be achieved with a maximum margin. We now propose an algorithm to directly control the number of margin errors and therefore also the contribution of both terms in the inequality separately (cf. Theorem 3). We first consider a small hypothesis class and end up with a linear program - v-LP-AdaBoost. In subsection 3.2 we then combine this algorithm with the ideas from section 2 and get a new algorithm v-Arc - which approximates the v-LP solution. ? 3.1 v-LP-AdaBoost Let us consider the case where we are given a (finite) set G = {g: x I-t [-1, 1n ofT hypotheses. To find the coefficients a for the combined hypothesis f(x) we extend the LP-AdaBoost algorithm [7, 11] by incorporating the parameter v [15] and solve the following linear optimization problem: maximize pE::'I ~i subject to P(Zi' a) ::::: p - ~i for all 1 ~ i ~ m (8) ~i' at, P ::::: for all 1 ~ t ~ T and 1 ~ i ~ m lIalh = 1 v!n ? G. Riitsch, B. SchOlkopf, A. J. Smola, K.-R. Muller, T. Onoda and S. Mika 564 This algorithm does not force all margins to be beyond zero and we get a soft margin classification (cf. SVMs) with a regularization constant The following proposition shows that v has an immediate interpretation: Proposition 2 (Ratsch et al. [12]). Suppose we run the algorithm given in (8) on some data with the resulting optimal P > o. Then v!n. 1. v upper bounds the fraction of margin errors. 2. 1 - v upper bounds the fraction of patterns with margin larger than p. Since the slack variables ~i only enter the cost function linearly, their absolute size is not important. Loosely speaking, this is due to the fact that for the optimum of the primal objective function, only derivatives wrt. the primal variables matter, and the derivative of a linear function is constant. In the case of SVMs [14], where the hypotheses can be thought of as vectors in some feature space, this statement can be translated into a precise rule for distorting training patterns without changing the solution: we can move them locally orthogonal to a separating hyperplane. This yields a desirable robustness property. Note that the algorithm essentially depends on the number of outliers, not on the size of the error [15]. 3.2 The v-Arc Algorithm Suppose we have a very large (but finite) base hypothesis class G. Then it is difficult to solve (8) as (5) directly. To this end, we propose a new algorithm - v-Arc - that approximates the solution of (8). The optimal p for fixed margins P(Zi' a) in (8) can be written as a?+) . (9) P(Zi' a?+ and subtracting pv(a) := argmax (p - _1 f)p - p(Zi' pE[O,I] vm i=1 where (~)+ := max(~, 0). Setting ~i := (pv(a) I:~l ~i from the resulting inequality on both sides yields (for all 1 ~ i ~ m) v!n P(Zi' a) 1 + ~i - - vm m L~i i=1 ~ 1 pv(a) - vm m L~i . i=1 (10) Two more substitutions are needed to transform the problem into one which can be solved by the AdaBoost algorithm. In particular we have to get rid of the slack variables ~i again by absorbing them into quantities similar to P(Zi' a) and p(a). This works as follows: on the right hand side of (10) we have the objective function (cf. (8? and on the left hand side a term that depends nonlinearly on a. Defining _ pv(a) 1 := pv(a) - - m L ~i vm. _ and Pv(Zi' a) := P(Zi' a) + ~i - 1 - m '"' vm~ ,=1 ~i, (11) i=l which we substitute for p(a) and p(z,a) in (5), respectively, we obtain a new optimization problem. Note that ,ov (a) and ,ov (Zi' a) play the role of a corrected or virtual margin. We obtain a nonlinear min-max problem maximize subject to ,o( Zi, a) ~ ,o(a) ,o(a) at lIallt for all 1 ~ i > 0 for all 1 ~ t ~1 ~ ~ m T ' (12) which Arc-GV can solve approximately (cf. section 2). Hence, by replacing the margin p(Z, a) by ,o(z,a) in equation (4) and the other formulas for Arc-GV (cf. [2]), 565 v-Arc: Ensemble Learning in the Presence o/Outliers we obtain a new algorithm which we refer to as v-Arc. We can now state interesting properties for v-Arc by using Theorem 5 of [13] that bounds the generalization error R(f) for ensemble methods. In our case Rp(f) ~ v by construction (i.e. the number of patterns with a margin smaller than p, cf. Proposition 2), thus we get the following simple reformulation of this bound: Theorem 3. Let p(x, y) be a distribution over X x [-1,1]' and let X be a sample of m examples chosen iid according to p. Suppose the base-hypothesis space G has VC dimension h, and let [) > 0. Then with probability at least 1 - [) over the random choice of the training set X, Y, every function f generated by v-Arc, where v E (0,1) and pv > 0, satisfies the following bound: R(f) ~ v + ~ l h) (hIO g2 (m 2 Pv m + Iog (!)) . ~ u (13) So, for minimizing the right hand side we can tradeoff between the first and the second term by controlling an easily interpretable regularization parameter v. 4 Experiments We show a set of toy experiments to illustrate the general behavior of v-Arc. As base hypothesis class G we use the RBF networks of [11], and as data a two-class problem generated from several 2D Gauss blobs (cf. Banana shape dataset from http://www.first.gmd.derdata/banana .html.). We obtain the following results: ? v-Arc leads to approximately vm patterns that are effectively used in the training of the base learner: Figure 1 (left) shows the fraction of patterns that have high average weights during the learning process (i.e. Ei=l Wt(Zi) > 112m). We find that the number of the latter increases (almost) linearly with v. This follows from (11) as the (soft) margin of patterns with p(z, a) < Pv is set to pv and the weight of those patterns will be the same. ? The (estimated) test error, averaged over 10 training sets, exhibits a rather flat minimum in v (Figure 1 (lower)). This indicates that just as for vSVMs, where corresponding results have been obtained, v is a well-behaved parameter in the sense that a slight misadjustment it is not harmful. ? v-Arc leads to the fraction v of margin errors (cf. dashed line in Figure 1) exactly as predicted in Proposition 2. ? Finally, a good value of v can already be inferred from prior knowledge of the expected error. Setting it to a value similar to the latter provides a good starting point for further optimization (cf. Theorem 3). Note that for v = 1, we recover the Bagging algorithm (if we used bootstrap samples), as the weights of all patterns will be the same (Wt(Zi) = 11m for all i = 1, . . . ,m) and also the hypothesis weights will be constant (at'" liT for all t = 1, .. . ,T) . Finally, we present a small comparison on ten benchmark data sets obtained from the VCI [1] benchmark repository (cf. http://ida.first.gmd.de/-raetsch/data/benchmarks.html). We analyze the performance of single RBF networks, AdaBoost, v-Arc and RBF-SVMs. For AdaBoost and v-Arc we use RBF networks [11] as base hypothesis. The model parameters of RBF (number of centers etc.), v-Arc (v) and SVMs (0', C) are optimized using 5-fold cross-validation. More details on the experimental setup can 566 0.8 ~ "- 0.16 number of important panems ~ ~ G. Riitsch, B. SchO/kopf, A. J. Smola, K.-R. Muller, T. Onoda and S. Mika 0. 15 Arc-GV ~ 0 .6 w 0. '4 Bagging '0 c ~ 0 .4 training error ~ 0.12 0.11 o o. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 1: Toy experim~nt (0' = 0): the left graph shows the average Yfraction of important patterns, the avo fraction of margin errors and the avo training error for different values of the regularization constant v for v-Arc. The right graph shows the corresponding generalization error. In both cases, the parameter v allows us to reduce the test errors to values much lower than for the hard margin algorithm (for v = 0 we recover ArcGV / AdaBoost, and for v = 1 we get Bagging.) be found in [11]. Fig. 1 shows the generalization error estimates (after averaging over 100 realizations of the data sets) and the confidence interval. The results of the best classifier and the classifiers that are not significantly worse are set in bold face . To test the significance, we used at-test (p = 80%). On eight out of the ten data sets, v-Arc performs significantly better than AdaBoost. This clearly shows the superior performance of v-Arc for noisy data sets and supports this soft margin approach for AdaBoost. Furthermore, we find comparable performances for v-Arc and SVMs. In three cases the SVM performs better and in two cases v-Arc performs best. Summarizing, AdaBoost is useful for low noise cases, where the classes are separable. v-Arc extends the applicability of boosting to problems that are difficult to separate and should be applied if the data are noisy. 5 Conclusion We analyzed the AdaBoost algorithm and found that Arc-GV and AdaBoost are efficient for approximating the solution of non-linear min-max problems over huge hypothesis classes. We re-parameterized the LP Reg-AdaBoost algorithm (cf. [7, 11]) and introduced a new regularization constant v that controls the fraction of patterns inside the margin area. The new parameter is highly intuitive and has to be optimized only on a fixed interval [0,1] . Using the fact that Arc-GV can approximately solve min-max problems, we found a formulation of Arc-G V - v-Arc - that implements the v-idea for Boosting by defining an appropriate soft margin. The present paper extends previous work on regularizing boosting (DOOM [9], AdaBoostReg [11]) and shows the utility and flexibility of the soft margin approach for AdaBoost. Banana B.Cancer Diabetes German Heart Ringnorm F .Sonar Thyroid Titanic Waveform RBF 10.8 ? 0.06 27.6 ? 0.47 24.3 ? 0.19 24.7 ? 0.24 17.6 ? 0.33 1.7 ? 0.02 34.4 ? 0.20 4.5 ? 0.21 23.3 ? 0.13 10.7 ? 0.11 AB 12.3 ? 0.07 30.4 ? 0.47 26.5 ? 0.23 27.5 ? 0.25 20.3 ? 0.34 1.9 ? 0.03 35.7 ? 0.18 4.4 ? 0.22 22.6 ? 0.12 10.8 ? 0.06 v-Arc 10.6 ? 0.05 25.8 ? 0.46 23.7 ? 0.20 24.4 ? 0.22 16.5 ? 0.36 1.7 ? 0.02 34.4 ? 0.19 4.4 ? 0.22 23.0 ? 0.14 10.0 ? 0.07 SVM 11.5 ? 0.07 26.0 ? 0.47 23.5 ? 0.17 23.6 ? 0.21 16.0 ? 0.33 1.7 ? 0.01 32.4 ? 0.18 4.8 ? 0.22 22.4 ? 0.10 9.9 ? 0.04 Table 1: Generalization error estimates and confidence intervals. The best classifiers for a particular data set are marked in bold face (see text). v-Arc: Ensemble Learning in the Presence of Outliers 567 We found empirically that the generalization performance in v-Arc depends only slightly on the choice of the regularization constant. This makes model selection (e.g. via cross-validation) easier and faster. Future work will study the detailed regularization properties of the regularized versions of AdaBoost, in particular in comparison to v-LP Support Vector Machines . Acknowledgments: Partial funding from DFG grant (Ja 379/52) is gratefully acknowledged. This work was done while AS and BS were at GMD FIRST. References [1] C. Blake, E. Keogh, and C. J. Merz. UCI repository of machine learning databases, 1998. http://www.ics.uci.edu/ ",mlearn/MLRepository.html. [2] L. Breiman. Prediction games and arcing algorithms. Technical Report 504, Statistics Department, University of California, December 1997. [3] M. Frean and T. Downs. A simple cost function for boosting. Technical report, Dept. of Computer Science and Electrical Eng., University of Queensland, 1998. [4] Y. Freund and R. E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Computational Learning Theory: Eurocolt '95, pages 23-37. Springer-Verlag, 1995. [5] Y. Freund and R. E . Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. J. of Compo fj Syst. Sc. , 55(1):119- 139, 1997. [6] J . Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. Technical report, Stanford University, 1998. [7) A. Grove and D. Schuurmans. Boosting in the limit: Maximizing the margin of learned ensembles. In Proc. of the 15th Nat. Conf. on AI, pages 692- 699 , 1998. [8] Y. LeCun, L. D . Jackel, L. Bottou, C. Cortes, J . S. Denker, H. Drucker, I. Guyon, U. A. Muller, E. Sackinger, P. Simard, and V. Vapnik. Learning algorithms for classification: A comparison on handwritten digit recognition. Neural Networks, pages 261-276, 1995. [9) L. Mason, P. L. Bartlett, and J. Baxter. Improved generalization through explicit optimization of margins. Machine Learning, 1999. to appear. (10) J. R. Quinlan. Boosting first-order learning (invited lecture). Lecture Notes in Computer Science, 1160:143, 1996. (11) G. Ratsch , T. Onoda, and K.-R. Muller. Soft margins for AdaBoost. Technical Report NC-TR-1998-021, Department of Computer Science, Royal Holloway, University of London, Egham, UK , 1998. To appear in Machine Learning. (12) G. Ratsch, B. Schokopf, A. Smola, S. Mika, T. Onoda, and K.-R. Muller. Robust ensemble learning. In A.J . Smola, P.L. Bartlett, B. Scholkopf, and D. Schuurmans, editors, Advances in LMC, pages 207-219. MIT Press, Cambridge, MA , 1999. [13] R. Schapire, Y. Freund, P. L. Bartlett, and W . Sun Lee. Boosting the margin: A new explanation for the effectiveness of voting methods. Annals of Statistics, 1998. (Earlier appeared in: D. H. Fisher, Jr. (ed.), Proc. ICML97, M. Kaufmann). [14] B. Scholkopf, C. J. C. Burges, and A. J. Smola. Advances in Kernel Methods Support Vector Learning. MIT Press, Cambridge, MA, 1999. (15) B. Scholkopf, A. Smola, R. C. Williamson, and P. L. Bartlett. New support vector algorithms. Neural Computation, 12:1083 - 1121, 2000. (16) H. Schwenk and Y. Bengio. Training methods for adaptive boosting of neural networks. In Michael I. Jordan, Michael J. Kearns, and Sara A. Solla, editors, Advances in Neural Inf. Processing Systems, volume 10. The MIT Press, 1998. [17) V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995. 

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Bayesian model selection for Support Vector machines, Gaussian processes and other kernel classifiers Matthias Seeger Institute for Adaptive and Neural Computation University of Edinburgh 5 Forrest Hill, Edinburgh EHI 2QL seeger@dai.ed.ac.uk Abstract We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes. The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility to use sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models. 1 Introduction Bayesian techniques have been widely and successfully used in the neural networks and statistics community and are appealing because of their conceptual simplicity, generality and consistency with which they solve learning problems. In this paper we present a new method for applying the Bayesian methodology to Support Vector machines. We will briefly review Gaussian Process and Support Vector classification in this section and clarify their relationship by pointing out the common roots. Although we focus on classification here, it is straightforward to apply the methods to regression problems as well. In section 2 we introduce our algorithm and show relations to existing methods. Finally, we present experimental results in section 3 and close with a discussion in section 4. Let X be a measure space (e.g. X = ~d) and D = (X,t) = {(Xl,tt), ... , (Xn,t n )), Xi E X, ti E {-l,+l} a noisy LLd. sample from a latent function y : X -+ lR, where P(tly) denotes the noise distribution. Given further points X. we wish to predict t. so as to minimize the error probability P(tlx., D), or (more difficult) to estimate this probability. Generative Bayesian methods attack this problem by placing a stochastic process prior P(y(?)) over the space of latent functions and 604 M Seeger then compute posterior and predictive distributions P(yID), P(y.lx., D) as P(yID) = P(Dly)P(y) P(D) , P(y.ID,x.) = f (1) P(y.ly)P(yID) dy where y = (y(Xi?i, y. = y(x.), the likelihood P(Dly) = TIi P(tiIYi) and P(D) is a normalization constant. P(tlx., D) can then be obtained by averaging P(tly.) over P(y.lx., D). Gaussian process (GP) or spline smoothing models use a Gaussian process prior on y(.) which can be seen as function of X into a set of random variables such that for each finite XI C X the corresponding variables are jointly Gaussian (see [15] for an introduction). A GP is determined by a mean function 1 x 1-4 E[y(x)] and a positive definite covariance kernel K(x,x). Gaussian process classification (GPC) amounts to specifying available prior knowledge by choosing a class of kernels K(x, xIO), 0 E e, where 0 is a vector of hyperparameters, and a hyperprior P(O). Usually, these choices are guided by simple attributes of y(.) such as smoothness, trends, differentiability, but more general approaches to kernel design have also been considered [5]. For 2-class classification the most common noise (1 + exp( _U?-1 the distribution is the binomial one where P(tly) (j(ty), (j(u) logistic function, and y is the logit 10g(P( +llx)/ P( -llx? of the target distribution. For this noise model the integral in (1) is not analytically tractable, but a range of approximative techniques based on Laplace approximations [16], Markov chain Monte Carlo [7], variational methods [2] or mean field algorithms [8] are known. = = We follow [16]. The Laplace approach to GPC is to approximate the posterior P(yID,O) by the Gaussian distribution N(y, 1i- 1) where y = argmaxP(yID, 0) is the posterior mode and 1i = \7~\7y(-logP(YID,O?, evaluated at y. Then it is easy to show that the predictive distribution is Gaussian with mean k(x.),K-l y and variance k. - k(x.)'K- 1k(x.) where K is the covariance matrix (K(Xi,Xj?ij, k(?) = (K(Xi, '?i, k. = K(x., x.) and the prime denotes transposition. The final discriminant is therefore a linear combination of the K (Xi, .). The discriminative approach to the prediction problem is to choose a loss function get, y), being an approximation to the misclassification loss2 I{tY:5o} and then to search for a discriminant y(.) which minimizes E [get, y(x.?] for the points x. of interest (see [14]). Support Vector classification (SVC) uses the c-insensitive loss (SVC loss) get, y) = [1 - ty]+, [u]+ = uI{u~o} which is an upper bound on the misclassification loss, and a reproducing kernel Hilbert space (RKHS) with kernel K(x,xIO) as hypothesis space for y(.). Indeed, Support Vector models and the Laplace method for Gaussian processes are special cases of spline smoothing models in RKHS where the aim is to minimize the functional n ~9(ti'Yi) + AllyOIl~ (2) i=l where II . 11K denotes the norm of the RKHS. It can be shown that the minimizer of (2) can be written as k(?), K-1y where y maximizes n - ~9(ti'Yi) - Ay'K-ly. (3) i=l All these facts can be found in [13]. Now (3) is, up to terms not depending on y, the log posterior in the above GP framework if we choose g(t,y) = -logP(tly) and lW.l.O.g. we only consider GPs with mean function 0 in what follows. denotes the indicator function of the set A c lR.. 2IA 605 Bayesian Model Selection for Support Vector Machines absorb A into O. For the SVC loss, (3) can be transformed into a dual problem via y = Ka, where a is a vector of dual variables, which can be efficiently solved using quadratic programming techniques. [12] is an excellent reference. Note that the SVC loss cannot be written as the negative log of a noise distribution, so we cannot reduce SVC to a special case of a Gaussian process classification model. Although a generative model for SVC is given in [11], it is easier and less problematic to regard SVC as efficient approximation to a proper Gaussian process model. Various such models have been proposed (see [8],[4]). In this work, we simply normalize the SVC loss pointwise, i.e. use a Gaussian process model with the normalized BVe loss g(t, y) = [1 - ty]+ + log Z(y), Z(y) = exp( -[1 - y]+) + exp( -[1 + y]+). Note that g(t, y) is a close approximation of the (unnormalized) SVC loss. The reader might miss the SVM bias parameter which we dropped here for clarity, but it is straightforward to apply this semiparametric extension to GP models to0 3 . 2 A variational method for kernel classification The real Bayesian way to deal with the hyperparameters 0 is to average P(y.lx., D, 0) over the posterior P( OlD) in order to obtain the predictive distribution P(y.lx., D). This can be approximated by Markov chain Monte Carlo methods [7], [16] or simply by P(y.lx.,D,9), 9 = argmaxP(OID). The latter approach, called maximum a-posteriori (MAP), can be justified in the limit of large n and often works well in practice. The basic challenge of MAP is to calculate the evidence P(DI9) = ! P(D,yI9)dy = ! exp (- t.9(ti,Yi?) N(yIO,K(9))dy. (4) Our plan is to attack (4) by a variational approach. Let P be a density from a model class r chosen to approximate the posterior P(yID, 0). Then: -logP(DIO) = -JP(Y)lOg (P(D'YIO)~(Y)) P(y ID, O)P(y) - !- = F(P, 0) - dy (5) P (ID, y) P(y) log ( P(y 0)) dy where we call F(P, 0) = Ep[-log P(D, yIO)] +Ep[logP(y)] the variational free energy. The second term in (5) is the well-known Kullback-Leibler divergence between P and the posterior which is nonnegative and equals zero iff P(y) = P(yID,O) almost everywhere with respect to the distribution P. Thus, F is an upper bound on - log P (D I0), and changing (P, 0) to decrease F enlarges the evidence or decreases the divergence between the posterior and its approximation, both being favourable. This idea has been introduced in [3] as ensemble learning4 and has been successfully applied to MLPs [1]. The latter work also introduced the model class r we use here, namely the class of Gaussians with mean IL and factor-analyzed covariance ~ = V + L,~1 Cjcj, V diagonal with positive elements 5 . Hinton and 3This is the "random effects model with improper prior" of [13], p.19, and works by placing a flat improper prior on the bias parameter. 4We average different discriminants (given by y) over the ensemble P. 5 Although there is no danger of overfitting, the use of full covariances would render the optimization more difficult, time and memory consuming. M Seeger 606 van Camp [3] used diagonal covariances which would be M = 0 in our setting. By choosing a small M, we are able to track the most important correlations between the components in the posterior using O( M n) parameters to represent P. Having agreed on r, the criterion F and its gradients with respect to (J and the parameters of P can easily and efficiently be computed except for the generic term (6) a sum of one-dimensional Gaussian expectations which are, depending on the actual g, either analytically tractable or can be approximated using a quadrature algorithm. For example, the expectation for the normalized SVC loss can be decomposed into expectations over the (unnormalized) SVC loss and over log Z(y) (see end of section 1). While the former can be computed analytically, the latter expectation can be handled by replacing log Z (y) by a piecewise defined tight bound such that the integral can be solved analytically. For the GPC loss (6) cannot be solved analytically and was in our experiments approximated by Gaussian quadrature. We can optimize F using a nested loop algorithm as follows. In the inner loop we run an optimizer to minimize F w.r.t. P for fixed (J. We used a conjugate gradients optimizer since the number of parameters of P is rather large. The outer loop is an optimizer minimizing F w.r.t. (J, and we chose a Quasi-Newton method here since the dimension of e is usually rather small and gradients w.r.t. (J are costly to evaluate. We can use the resulting minimizer (P,O) of F in two different ways. The most natural is to discard P, plug 0 into the original architecture and predict using the mode of P(y ID, 0) as an approximation to the true posterior mode, benefitting from a kernel now adapted to the given data. This is particularly interesting for Support Vector machines due to the sparseness of the final kernel expansion (typically only a small fraction of the components in the weight vector K-1iJ is non-zero, the corresponding datapoints are termed Support Vectors) which allows very efficient predictions for a large number of test points. However, we can also retain P and use it as a Gaussian approximation of the posterior P(yID, 0). Doing so, we can use the variance of the approximative predictive distribution P(y.lx., D) to derive error bars for our predictions, although the interpretation of these figures is somewhat complicated in the case of kernel discriminants like SVM whose loss function does not correspond to a noise distribution. 2.1 Relations to other methods Let us have a look at alternative ways to maximize (4). If the loss get, y) is twice differentiable everywhere, progress can be made by replacing g by its second order Taylor expansion around the mode of the integrand. This is known as Laplace approximation and is used in [16] to maximize (4) approximately. However, this technique cannot be used for nondifferentiable losses of the c-insensitive type 6 ? Nevertheless, for the SVC loss the evidence (4) can be approximated in a Laplacelike fashion [11], and it will be interesting to compare the results of this work with ours. This approximation can be evaluated very efficiently, but is not continuous 7 6The nondifferentiabilities cannot be ignored since with probability one a nonzero number of the ih sit exactly at these margin locations. 7 Although continuity can be accomplished by a further modification, see [11]. 607 Bayesian Model Selection for Support Vector Machines w.r .t. (J and difficult to optimize if the dimension of e is not small. Opper and Winther [8] use mean field ideas to derive an approximate leave-one-out test error estimator which can be quickly evaluated, but suffers from the typical noisiness of cross-validation scores. Kwok [6] applies the evidence framework to Support Vector machines, but the technique seems to be restricted to kernels with a finite eigenfunction expansion (see [13] for details) . It is interesting to compare our variational method to the Laplace method of [16] and the variational technique of [2]. Let g(t, y) be differentiable and suppose that for given (J we restrict ourselves to approximate (6) by replacing g(ti' Yi) by the expansion 2g g A)(Yi - JLi )2 , g(ti' JLi) + 8 By (ti, JLi) (Yi - JLi ) + 2.18 8y2 (ti, Yi (7) where fj is the posterior mean . This will change the criterion F to Fapproz, say. Then it is easy to show that the Gaussian approximation to the posterior employed by the Laplace method, namely N(fj, (K- 1 + W)-l), W = diag(u(Yi)(1-u(Yi?) , minimizes Fapproz w.r.t. P if full covariances ~ are used, and plugging this minimizer into Fapproz we end up with the evidence approximation which is maximized by the Laplace method. The latter is not a variational technique since the approximation (7) to the loss function is not an upper bound, and works only for differentiable loss functions . If we upper bound the loss function g(t,y) by a quadratic polynomial and add the variational parameters of this bound to the parameters of P, our method becomes broadly similar to the lower bound algorithm of [2]. Indeed, since for fixed variational parameters of the polynomials we can easily solve for the mean and covariance of P, the former parameters are the only essential ones. However, the quadratic upper bound is poor for functions like the SVC loss , and in these cases our bound is expected to be tighter. 3 Experiments We tested our variational algorithm on a number of datasets from the UCI machine learning repository and the DELVE archive of the University of Toront08 : Leptograpsus crabs, Pima Indian diabetes, Wisconsin Breast Cancer, Ringnorm, Twonorm and Waveform (class 1 against 2). Descriptions may be found on the web. In each case we normalized the whole set to zero mean , unit variance in all input columns, picked a training set at random and used the rest for testing. We chose (for X = JRd) the well-known squared-exponential kernel (see [15]): K(x,xI9) = C (exp ( - 2~ t Wi (Xi - Xi)') + v), 9 = ?Wi):'C,V)'. (8) All parameters are constrained to be positive, so we chose the representation ()i = v'f. We did not use a prior on (J (see comment at end of this section). For comparison we trained a Gaussian Process classifier with the Laplace method (also without hyperprior) and a Support Vector machine using lO-fold cross-validation to select the free parameters. In the latter case we constrained the scale parameters Wi to be equal (it is infeasible to adapt d + 2 hyperparameters to the data using crossvalidation) and dropped the v parameter while allowing for a bias parameter. As mentioned above, within the variational method we can use the posterior mode fj 8See http://vvv . cs. utoronto. cal ...... del ve and http://vvv.ics.uci.edu/ ......mlearn/MLRepository.html . M Seeger 608 Name crabs pima wdbc twonorm ringnorm waveform train size 80 200 300 300 400 800 test size 120 332 269 7100 7000 2504 Var.GP IL 4 3 66 66 11 11 233 224 119 124 206 204 I y GP Lapl. 4 68 8 297 184 221 Var. SVM y IL 4 4 64 66 10 10 260 223 129 126 211 206 SVM 10-CV 4 67 9 163 160 197 Lin. discr. 3 67 19 207 1763 220 Table 1: Number of test errors for various methods. as well as the mean IL of P for prediction, and we tested both methods. Error bars were not computed. The baseline method was a linear discriminant trained to minimize the squared error. Table 1 shows the test errors the different methods attained. These results show that the new algorithm performs equally well as the other methods we considered. They have of course to be regarded in combination with how much effort was necessary to produce them. It took us almost a whole day and a lot of user interactions to do the cross-validation model selection. The rule-of-thumb that a lot of Support Vectors at the upper bound indicate too large a parameter C in (8) failed for at least two of these sets, so we had to start with very coarse grids and sweep through several stages of refinement. An effect known as automatic relevance determination (ARD) (see [7]) can be nicely observed on some of the datasets, by monitoring the length scale parameters Wi in (8). Indeed, our variational SVC algorithm almost completely ignored (by driving their length scales to very small values) 3 of the 5 dimensions in "crabs", 2 of 7 in "pima" and 3 of 21 in "waveform". On "wdbc", it detected dimension 24 as particularly important with regard to separation, all this in harmony with the GP Laplace method. Thus, a sensible parameterized kernel family together with a method of the Bayesian kind allows us to gain additional important information from a dataset which might be used to improve the experimental design. Results of experiments with the methods tested above and hyperpriors as well as a more detailed analysis of the experiments can be found in [9]. 4 Discussion We have shown how to perform model selection for Support Vector machines using approximative Bayesian variational techniques. Our method is applicable to a wide range of loss functions and is able to adapt a large number of hyperparameters to given data. This allows for the use of sophisticated kernels and Bayesian techniques like automatic relevance determination (see [7]) which is not possible using other common model selection criteria like cross-validation. Since our method is fully automatic, it is easy for non-experts to use 9 , and as the evidence is computed on the training set, no training data has to be sacrificed for validation. We refer to [9] where the topics of this paper are investigated in much greater detail. A pressing issue is the unfortunate scaling of the method with the training set 9 As an aside, this opens the possibility of comparing SVMs against other fullyautomatic methods within the DELVE project (see section 3) . Bayesian Model Selection for Support Vector Machines 609 size n which is currently O(n 3)1O. We are currently explori~g the applicability of the powerful approximations of [10] which might bring us very much closer to the desired O(n2) scaling (see also [2]). Another interesting issue would be to connect our method with the work of [5] who use generative models to derive kernels in situations where the "standard kernels" are not applicable or not reasonable. Acknowledgments We thank Chris Williams, Amos Storkey, Peter Sollich and Carl Rasmussen for helpful and inspiring discussions. This work was partially funded by a scholarship of the Dr. Erich Muller foundation. We are grateful to the Division of Informatics for supporting our visit in Edinburgh, and to Chris Williams for making it possible. References [1] David Barber and Christopher Bishop. Ensemble learning for multi-layer networks. In Advances in NIPS, number 10, pages 395-401. MIT Press, 1997. [2] Mark N. Gibbs. Bayesian Gaussian Processes for Regression and Classification. PhD thesis, University of Cambridge, 1997. [3] Geoffrey E. Hinton and D. Van Camp. Keeping neural networks simple by minimizing the description length of the weights. In Proceedings of the 6th annual conference on computational learning theory, pages 5- 13, 1993. [4] Tommi Jaakkola, Marina Meila, and Tony Jebara. Maximum entropy discrimination. In Advances in NIPS, number 13. MIT Press, 1999. [5] Tommi S. Jaakkola and David Haussler. Exploiting generative models in discriminative classifiers. In Advances in NIPS, number 11, 1998. [6] James Tin-Tau Kwok. Integrating the evidence framework and the Support Vector machine. Submitted to ESANN 99, 1999. [7] Radford M. Neal. Monte Carlo implementation of Gaussian process models for Bayesian classification and regression. Technical Report 9702, Department of Statistics, University of Toronto, January 1997. [8] Manfred Opper and Ole Winther. GP classification and SVM: Mean field results and leave-one-out estimator. In Advances in Large Margin Classifiers. MIT Press, 1999. [9] Matthias Seeger. Bayesian methods for Support Vector machines and Gaussian processes. Master's thesis, University of Karlsruhe, Germany, 1999. Available at http://vvw.dai.ed.ac.uk/-seeger. [10] John Skilling. Maximum entropy and Bayesian methods. Cambridge University Press, 1988. [11] Peter Sollich. Probabilistic methods for Support Vector machines. In Advances in NIPS, number 13. MIT Press, 1999. [12] Vladimir N. Vapnik. Statistical Learning Theory. Wiley, 1998. [13] Grace Wahba. Spline Models for Observational Data. CBMS-NSF Regional Conference Series. SIAM, 1990. [14] Grace Wahba. Support Vector machines, reproducing kernel Hilbert spaces and the randomized GACV. Technical Report 984, University of Wisconsin, 1997. [15] Christopher K. 1. Williams. Prediction with Gaussian processes: From linear regression to linear prediction and beyond. In M. 1. Jordan, editor, Learning in Graphical Models. Kluwer, 1997. [16] Christopher K.I. Williams and David Barber. Bayesian classification with Gaussian processes. IEEE 7rans. PAMI, 20(12):1342-1351, 1998. laThe running time is essentially the same as that of the Laplace method, thus being comparable to the fastest known Bayesian GP algorithm. 

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Support Vector Method for Novelty Detection Bernhard Scholkopf*, Robert Williamson?, Alex Smola?, John Shawe-Taylor t , John Platt* ? * Microsoft Research Ltd., 1 Guildhall Street, Cambridge, UK Department of Engineering, Australian National University, Canberra 0200 t Royal Holloway, University of London, Egham, UK * Microsoft, 1 Microsoft Way, Redmond, WA, USA bsc/jplatt@microsoft.com, Bob.WilliamsoniAlex.Smola@anu.edu.au, john@dcs.rhbnc.ac.uk Abstract Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified l/ between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. We provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled data. 1 INTRODUCTION During recent years, a new set of kernel techniques for supervised learning has been developed [8]. Specifically, support vector (SV) algorithms for pattern recognition, regression estimation and solution of inverse problems have received considerable attention. There have been a few attempts to transfer the idea of using kernels to compute inner products in feature spaces to the domain of unsupervised learning. The problems in that domain are, however, less precisely specified. Generally, they can be characterized as estimating junctions of the data which tell you something interesting about the underlying distributions. For instance, kernel PCA can be characterized as computing functions which on the training data produce unit variance outputs while having minimum norm in feature space [4] . Another kernel-based unsupervised learning technique, regularized principal manifolds [6], computes functions which give a mapping onto a lower-dimensional manifold minimizing a regularized quantization error. Clustering algorithms are further examples of unsupervised learning techniques which can be kernelized [4] . An extreme point of view is that unsupervised learning is about estimating densities. Clearly, knowledge of the density of P would then allow us to solve whatever problem can be solved on the basis of the data. The present work addresses an easier problem: it Support Vector Method for Novelty Detection 583 proposes an algorithm which computes a binary function which is supposed to capture regions in input space where the probability density lives (its support), i.e. a function such that most of the data will live in the region where the function is nonzero [5]. In doing so, it is in line with Vapnik's principle never to solve a problem which is more general than the one we actually need to solve. Moreover, it is applicable also in cases where the density of the data's distribution is not even well-defined, e.g. if there are singular components. Part of the motivation for the present work was the paper [1]. It turns out that there is a considerable amount of prior work in the statistical literature; for a discussion, cf. the full version of the present paper [3]. 2 ALGORITHMS We first introduce terminology and notation conventions. We consider training data Xl, ... , Xl E X, where fEN is the number of observations, and X is some set. For simplicity, we think of it as a compact subset of liN. Let ~ be a feature map X -t F, i.e. a map into a dot product space F such that the dot product in the image of ~ can be computed by evaluating some simple kernel [8] k(x, y) = (~(x) . ~(y)), (1) such as the Gaussian kernel (2) Indices i and j are understood to range over 1, ... ,f (in compact notation: 't, J E [fD. Bold face greek letters denote f-dimensional vectors whose components are labelled using normal face typeset. In the remainder of this section, we shall develop an algorithm which returns a function f that takes the value + 1 in a "small" region capturing most of the data points, and -1 elsewhere. Our strategy is to map the data into the feature space corresponding to the kernel, and to separate them from the origin with maximum margin. For a new point X, the value f(x) is determined by evaluating which side of the hyperplane it falls on, in feature space. Via the freedom to utilize different types of kernel functions, this simple geometric picture corresponds to a variety of nonlinear estimators in input space. To separate the data set from the origin, we solve the following quadratic program: ~llwl12 min wEF,eEiRt,PEiR subject to (w? ~(Xi)) + ;l L i ei 2:: P - ei, (3) P ei 2:: o. (4) Here, 1/ E (0, 1) is a parameter whose meaning will become clear later. Since nonzero slack variables ei are penalized in the objective function, we can expect that if wand p solve this problem, then the decision function f(x) = sgn((w . ~(x)) - p) will be positive for most examples Xi contained in the training set, while the SV type regularization term Ilwll will still be small. The actual trade-off between these two goals is controlled by 1/. Deriving the dual problem, and using (1), the solution can be shown to have an SV expansion f(x) = 'gn ( ~ a;k(x;, x) - p) (5) (patterns Xi with nonzero ll!i are called SVs), where the coefficients are found as the solution of the dual problem: main ~ L ll!ill!jk(Xi, Xj) ij subject to 0 ~ ll!i ~ :f' L ll!i = 1. (6) B. ScMlkop/, R. C. Williamson, A. J Smola, J Shawe-Taylor and J C. Platt 584 This problem can be solved with standard QP routines. It does, however, possess features that sets it apart from generic QPs, most notably the simplicity of the constraints. This can be exploited by applying a variant of SMO developed for this purpose [3]. The offset p can be recovered by exploiting that for any ll:i which is not at the upper or lower bound, the corresponding pattern Xi satisfies p = (w . <.P(Xi)) L:j ll:jk(Xj, Xi) . = Note that if v approaches 0, the upper boundaries on the Lagrange multipliers tend to infinity, i.e. the second inequality constraint in (6) becomes void. The problem then resembles the corresponding hard margin algorithm, since the penalization of errors becomes infinite, as can be seen from the primal objective function (3). It can be shown that if the data set is separable from the origin, then this algorithm will find the unique supporting hyperplane with the properties that it separates all data from the origin, and its distance to the origin is maximal among all such hyperplanes [3]. If, on the other hand, v approaches I, then the constraints alone only allow one solution, that where all ll:i are at the upper bound 1/ (v?). In this case, for kernels with integral I, such as normalized versions of (2), the decision function corresponds to a thresholded Parzen windows estimator. To conclude this section, we note that one can also use balls to describe the data in feature space, close in spirit to the algorithms of [2], with hard boundaries, and [7], with "soft margins." For certain classes of kernels, such as Gaussian RBF ones, the corresponding algorithm can be shown to be equivalent to the above one [3]. 3 THEORY In this section, we show that the parameter v characterizes the fractions of SVs and outliers (Proposition 1). Following that, we state a robustness result for the soft margin (Proposition 2) and error bounds (Theorem 5). Further results and proofs are reported in the full version of the present paper [3]. We will use italic letters to denote the feature space images of the corresponding patterns in input space, i.e. X i := <.P(Xi). Proposition 1 Assume the solution of (4) satisfies p =1= 0. The following statements hold: (i) v is an upper bound on the fraction of outliers. (ii) v is a lower bound on the fraction of SVs. (iii) Suppose the data were generated independently from a distribution P(x) which does not contain discrete components. Suppose, moreover, that the kernel is analytic and nonconstant. With probability 1, asymptotically, v equals both the fraction of SVs and the fraction of outliers. The proof is based on the constraints of the dual problem, using the fact that outliers must have Lagrange multipliers at the upper bound. Proposition 2 Local movements of outliers parallel to w do not change the hyperplane. We now move on to the subject of generalization. Our goal is to bound the probability that a novel point drawn from the same underlying distribution lies outside of the estimated region by a certain margin. We start by introducing a common tool for measuring the capacity of a class :r of functions that map X to lit Definition 3 Let (X, d) be a pseudo-metric space, I let A be a subset of X and f. > 0. A set B ~ X is an f.-cover for A if, for every a E A, there exists b E B such that d( a , b) ::::; f.. The f.-covering number of A, Nd(f., A), is the minimal cardinality of an f.-cover for A (if there is no such finite cover then it is defined to be 00). I i.e. with a distance function that differs from a metric in that it is only semidefinite 585 Support Vector Method for Novelty Detection The idea is that B should be finite but approximate all of A with respect to the pseudometric d. We will use the loo distance over a finite sample X = (Xl, .. ? , Xl) for the pseudometric in the space of functions, dx(f, g) = m~E[lllf(xd - g(xi)l. Let N(E,~, f) = SUPXEXI Ndx (E, ~). Below, logarithms are to base 2. Theorem 4 Consider any distribution P on X and any 0 E lR. Suppose Xl, ?.. , Xl are generated U.d. from P. Then with probability 1 - 6 over such an f-sample, if we find f E ~ such that f(Xi) ~ 0 for all i E [f), +, P{x : f(x) < 0 -,} :s; t(k + log 2(l), where k = rlog:Nb,~, 2f)1- We now consider the possibility that for a small number of points f(Xi) fails to exceed 0+,. This corresponds to having a non-zero slack variable ~i in the algorithm, where we take 0 + , = p / II w II and use the class of linear functions in feature space in the application of the theorem. There are well-known bounds for the log covering numbers of this class. Let f be a real valued function on a space X. Fix 0 E lR. For X E X, define d(x,J, ,) = max{O,O +, - f(x)}. Similarly for a training sequence X, we define 'D(X, f, ,) = L:xEX d(x, f, ,). Theorem 5 Fix 0 E lR. Consider a fixed but unknown probability distribution P on the input space X and a class of real valued functions ~ with range [a, b). Then with probability 1 - 6 over randomly drawn training sequences X of size f, for all, > 0 and any f E ~, P {x: f(x) < 0 - , and X ~ X} :s; t(k + log ~l ), where k = rlogN('V/2 ~ , U) I , + 64(b-a)'D(X,J,'y) -y2 log ( 8'D(X,J,-y) ell ) log (32l(b-a)2)1. -y2 The theorem bounds the probability of a new point falling in the region for which f(x) has value less than 0 - " this being the complement of the estimate for the support of the distribution. The choice of, gives a trade-off between the size of the region over which the bound holds (increasing, increases the size of the region) and the size of the probability with which it holds (increasing, decreases the size of the log covering numbers). The result shows that we can bound the probability of points falling outside the region of estimated support by a quantity involving the ratio of the log covering numbers (which can be bounded by the fat shattering dimension at scale proportional to ,) and the number of training examples, plus a factor involving the I-norm of the slack variables. It is stronger than related results given by [I], since their bound involves the square root of the ratio of the Pollard dimension (the fat shattering dimension when, tends to 0) and the number of training examples. The output of the algorithm described in Sec. 2 is a function f(x) = 2:::i aik(xi' x) which is greater than or equal to p - ~i on example Xi. Though non-linear in the input space, this function is in fact linear in the feature space defined by the kernel k . At the same time the 2-norm of the weight vector is given by B = J aT K a, and so we can apply the theorem with the function class ~ being those linear functions in the feature space with 2-norm bounded by B . If we assume that 0 is fixed, then, = p - 0, hence the support of the distribution is the set {x : f (x) ~ 0 - , = 20 - p}, and the bound gives the probability of a randomly generated point falling outside this set, in terms of the log covering numbers of the function class ~ and the sum of the slack variables ~i. Since the log covering numbers 586 B. SchOlkopj R. C. Williamson, A. 1. Smola, 1. Shawe-Taylor and 1. C. Platt 2 f) this gives a bound in terms at scale, /2 of the class ~ can be bounded by O( B:!F-Iog "Y of the 2-norm of the weight vector. Ideally, one would like to allow () to be chosen after the value of p has been determined, perhaps as a fixed fraction of that value. This could be obtained by another level of structural risk minimisation over the possible values of p or at least a mesh of some possible values. This result is beyond the scope of the current preliminary paper, but the form of the result would be similar to Theorem 5, with larger constants and log factors. Whilst it is premature to give specific theoretical recommendations for practical use yet, one thing is clear from the above bound. To generalize to novel data, the decision function to be used should employ a threshold TJ ? p, where TJ < 1 (this corresponds to a nonzero I)' 4 EXPERIMENTS We apply the method to artificial and real-world data. Figure 1 displays 2-D toy examples, and shows how the parameter settings influence the solution. Next, we describe an experiment on the USPS dataset of handwritten digits. The database contains 9298 digit images of size 16 x 16 = 256; the last 2007 constitute the test set. We trained the algorithm, using a Gaussian kernel (2) of width c = 0.5 . 256 (a common value for SVM classifiers on that data set, cf. [2]), on the test set and used it to identify outliers - it is folklore in the community that the USPS test set contains a number of patterns which are hard or impossible to classify, due to segmentation errors or mislabelling. In the experiment, we augmented the input patterns by ten extra dimensions corresponding to the class labels of the digits. The rationale for this is that if we disregarded the labels, there would be no hope to identify mislabelled patterns as outliers. Fig. 2 shows the 20 worst outliers for the USPS test set. Note that the algorithm indeed extracts patterns which are very hard to assign to their respective classes. In the experiment, which took 36 seconds on a Pentium II running at 450 MHz, we used a 11 value of 5%. Figure I: First two pictures: A single-class SVM applied to two toy problems; 11 = C = 0.5, domain: [-1, 1F. Note how in both cases, at least a fraction of 11 of all examples is in the estimated region (cf. table). The large value of 11 causes the additional data points in the upper left comer to have almost no influence on the decision function. For smaller values of 11, such as 0.1 (third picture), the points cannot be ignored anymore. Alternatively, one can force the algorithm to take these 'outliers' into account by changing the kernel width (2): in the fourth picture, using c = 0.1,11 = 0.5, the data is effectively analyzed on a different length scale which leads the algorithm to consider the outliers as meaningful points. Support Vector Method/or Novelty Detection 587 ~"1r~ftC ~"'J.. ~()nl~) 9 -507 1 -4580 -377 1 -282 7 -2162 -2003 -1869 -179 5 - -153 3 -143 6 -1286 - 3 0 -1177 -93 5 -78 0 -58 7 -52 6 -48 3 Figure 2: Outliers identified by the proposed algorithm, ranked by the negative output of the SVM (the argument of the sgn in the decision function). The outputs (for convenience in units of 10- 5 ) are written underneath each image in italics, the (alleged) class labels are given in bold face. Note that most of the examples are "difficult" in that they are either atypical or even mislabelled. 5 DISCUSSION One could view the present work as an attempt to provide an algorithm which is in line with Vapnik's principle never to solve a problem which is more general than the one that one is actually interested in. E.g., in situations where one is only interested in detecting novelty, it is not always necessary to estimate a full density model of the data. Indeed, density estimation is more difficult than what we are doing, in several respects. Mathematically speaking, a density will only exist if the underlying probability measure possesses an absolutely continuous distribution function. The general problem of estimating the measure for a large class of sets, say the sets measureable in Borel's sense, is not solvable (for a discussion, see e.g. [8]). Therefore we need to restrict ourselves to making a statement about the measure of some sets. Given a small class of sets, the simplest estimator accomplishing this task is the empirical measure, which simply looks at how many training points fall into the region of interest. Our algorithm does the opposite. It starts with the number of training points that are supposed to fall into the region, and then estimates a region with the desired property. Often, there will be many such regions - the solution becomes unique only by applying a regularizer, which in our case enforces that the region be small in a feature space associated to the kernel. This, of course, implies, that the measure of smallness in this sense depends on the kernel used, in a way that is no different to any other method that regularizes in a feature space. A similar problem, however, appears in density estimation already when done in input space. Let p denote a density on X. If we perform a (nonlinear) coordinate transformation in the input domain X, then the density values will change; loosely speaking, what remains constant is p(x) . dx, while dx is transformed, too. When directly estimating the probability measure of regions, we are not faced with this problem, as the regions automatically change accordingly. An attractive property of the measure of smallness that we chose to use is that it can also be placed in the context of regularization theory, leading to an interpretation of the solution as maximally smooth in a sense which depends on the specific kernel used [3]. The main inspiration for our approach stems from the earliest work of Vapnik and collaborators. They proposed an algorithm for characterizing a set of unlabelled data points by separating it from the origin using a hyperplane [9]. However, they quickly moved on to two-class classification problems, both in terms of algorithms and in the theoretical development of statistical learning theory which originated in those days. From an algorithmic point of view, we can identify two shortcomings of the original approach which may have caused research in this direction to stop for more than three decades. Firstly, the original 588 B. Scholkopf, R. C. Williamson, A. J Smola, J Shawe-Taylor and J C. Platt algorithm in was limited to linear decision rules in input space, secondly, there was no way of dealing with outliers. In conjunction, these restrictions are indeed severe - a generic dataset need not be separable from the origin by a hyperplane in input space. The two modifications that we have incorporated dispose of these shortcomings. Firstly, the kernel trick allows for a much larger class of functions by nonlinearly mapping into a high-dimensional feature space, and thereby increases the chances of separability from the origin. In particular, using a Gaussian kernel (2), such a separation exists for any data set Xl, ... , Xl: to see this, note that k(Xi, Xj) > 0 for all i, j, thus all dot products are positive, implying that all mapped patterns lie inside the same orthant. Moreover, since k(Xi, Xi) = 1 for all i, they have unit length. Hence they are separable from the origin. The second modification allows for the possibility of outliers. We have incorporated this 'softness' of the decision rule using the v-trick and thus obtained a direct handle on the fraction of outliers. We believe that our approach, proposing a concrete algorithm with well-behaved computational complexity (convex quadratic programming) for a problem that so far has mainly been studied from a theoretical point of view has abundant practical applications. To turn the algorithm into an easy-to-use black-box method for practicioners, questions like the selection of kernel parameters (such as the width of a Gaussian kernel) have to be tackled. It is our expectation that the theoretical results which we have briefly outlined in this paper will provide a foundation for this formidable task. Acknowledgement. Part of this work was supported by the ARC and the DFG (# Ja 37919-1), and done while BS was at the Australian National University and GMD FIRST. AS is supported by a grant of the Deutsche Forschungsgemeinschaft (Sm 62/1-1). Thanks to S. Ben-David, C. Bishop, C. Schnorr, and M. Tipping for helpful discussions. References [1] S. Ben-David and M. Lindenbaum. Learning distributions by their density levels: A paradigm for learning without a teacher. Journal of Computer and System Sciences, 55:171-182,1997. [2] B. SchOlkopf, C. Burges, and V. Vapnik. Extracting support data for a given task. In U. M. Fayyad and R. Uthurusamy, editors, Proceedings, First International Conference on Knowledge Discovery & Data Mining. AAAI Press, Menlo Park, CA, 1995. [3] B. SchOlkopf, J. Platt, J. Shawe-Taylor, AJ. Smola, and R.c. Williamson. Estimating the support of a high-dimensional distribution. TR MSR 99 - 87, Microsoft Research, Redmond, WA, 1999. [4] B. Scholkopf, A. Smola, and K.-R. Muller. Kernel principal component analysis. In B. SchOlkopf, C. Burges, and A. Smola, editors, Advances in Kernel Methods - Support Vector Learning. MIT Press, Cambridge, MA, 1999. 327 - 352. [5] B. SchOlkopf, R. Williamson, A. Smola, and J. Shawe-Taylor. Single-class support vector machines. In J. Buhmann, W. Maass, H. Ritter, and N. Tishby, editors, Unsupervised Learning, Dagstuhl-Seminar-Report 235, pages 19 - 20, 1999. [6] A. Smola, R. C. Williamson, S. Mika, and B. Scholkopf. Regularized principal manifolds. In Computational Learning Theory: 4th European Conference, volume 1572 of Lecture Notes in Artificial Intelligence, pages 214 - 229. Springer, 1999. [7] D.MJ. Tax and R.P.W. Duin. Data domain description by support vectors. In M. Verleysen, editor, Proceedings ESANN, pages 251 - 256, Brussels, 1999. D Facto. [8] V. Vapnik. Statistical Learning Theory. Wiley, New York, 1998. [9] V. Vapnik and A. Lerner. Pattern recognition using generalized portraits. Avtomatika i Telemekhanika, 24:774 -780, 1963. 

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Spiking Boltzmann Machines Geoffrey E. Hinton Gatsby Computational Neuroscience Unit University College London London WCIN 3AR, UK hinton@gatsby. ucl. ac. uk Andrew D. Brown Department of Computer Science University of Toronto Toronto, Canada andy@cs.utoronto.ca Abstract We first show how to represent sharp posterior probability distributions using real valued coefficients on broadly-tuned basis functions. Then we show how the precise times of spikes can be used to convey the real-valued coefficients on the basis functions quickly and accurately. Finally we describe a simple simulation in which spiking neurons learn to model an image sequence by fitting a dynamic generative model. 1 Population codes and energy landscapes A perceived object is represented in the brain by the activities of many neurons, but there is no general consensus on how the activities of individual neurons combine to represent the multiple properties of an object. We start by focussing on the case of a single object that has multiple instantiation parameters such as position, velocity, size and orientation. We assume that each neuron has an ideal stimulus in the space of instantiation parameters and that its activation rate or probability of activation falls off monotonically in all directions as the actual stimulus departs from this ideal. The semantic problem is to define exactly what instantiation parameters are being represented when the activities of many such neurons are specified. Hinton, Rumelhart and McClelland (1986) consider binary neurons with receptive fields that are convex in instantiation space. They assume that when an object is present it activates all of the neurons in whose receptive fields its instantiation parameters lie. Consequently, if it is known that only one object is present, the parameter values of the object must lie within the feasible region formed by the intersection of the receptive fields of the active neurons. This will be called a conjunctive distributed representation. Assuming that each receptive field occupies only a small fraction of the whole space, an interesting property of this type of "coarse coding" is that the bigger the receptive fields, the more accurate the representation. However, large receptive fields lead to a loss of resolution when several objects are present simultaneously. When the sensory input is noisy, it is impossible to infer the exact parameters of objects so it makes sense for a perceptual system to represent the probability distribution across parameters rather than just a single best estimate or a feasible region. The full probability distribution is essential for correctly combining infor- Spiking Boltzmann Machines E(x) P(X) 123 Figure 1: a) Energy landscape over a onedimensional space. Each neuron adds a dimple (dotted line) to the energy landscape (solid line). b) The corresponding probability density. Where dimples overlap the corresponding probability density becomes sharper. Since the dimples decay to zero, the location of a sharp probability peak is not affected by distant dimples and multimodal distributions can be represented. mation from different times or different Sources. One obvious way to represent this distribution (Anderson and van Essen, 1994) is to allow each neuron to represent a fairly compact probability distribution over the space of instantiation parameters and to treat the activity levels of neurons as (unnormalized) mixing proportions. The semantics of this disjunctive distributed representation is precise, but the percepts it allows are not because it is impossible to represent distributions that are sharper than the individual receptive fields and, in high-dimensional spaces, the individual fields must be broad in order to cover the space. Disjunctive representations are used in Kohonen's self-organizing map which is why it is restricted to very low dimensional latent spaces. The disjunctive model can be viewed as an attempt to approximate arbitrary smooth probability distributions by adding together probability distributions contributed by each active neuron. Coarse coding suggests a multiplicative approach in which the addition is done in the domain of energies (negative log probabilities). Each active neuron contributes an energy landscape over the whole space of instantiation parameters. The activity level of the neuron multiplies its energy landscape and the landscapes for all neurons in the population are added (Figure 1). If, for example, each neuron has a full covariance Gaussian tuning function, its energy landscape is a parabolic bowl whose curvature matrix is the inverse of the covariance matrix. The activity level of the neuron scales the inverse covariance matrix. If there are k instantiation parameters then only k + k(k + 1)/2 real numbers are required to span the space of means and inverse covariance matrices. So the real-valued activities of O(k2) neurons are sufficient to represent arbitrary full covariance Gaussian distributions over the space of instantiation parameters. Treating neural activities as multiplicative coefficients on additive contributions to energy landscapes has a number of advantages. Unlike disjunctive codes, vague distributions are represented by low activities so significant biochemical energy is only required when distributions are quite sharp. A central operation in Bayesian inference is to combine a prior term with a likelihood term or to combine two conditionally independent likelihood terms. This is trivially achieved by adding two energy landscapes l . lWe thank Zoubin Ghahramani for pointing out that another important operation, convolving a probability distribution with Gaussian noise, is a difficult non-linear operation on the energy landscape. 124 2 G. E. Hinton and A. D. Brown Representing the coefficients on the basis functions To perform perception at video rates, the probability distributions over instantiation parameters need to be represented at about 30 frames per second. This seems difficult using relatively slow spiking neurons because it requires the real-valued multiplicative coefficients on the basis functions to be communicated accurately and quickly using all-or-none spikes. The trick is to realise that when a spike arrives at another neuron it produces a postsynaptic potential that is a smooth function of time. So from the perspective of the postsynaptic neuron, the spike has been convolved with a smooth temporal function. By adding a number of these smooth functions together, with appropriate temporal offsets, it is possible to represent any smoothly varying sequence of coefficient values on a basis function, and this makes it possible to represent the temporal evolution of probability distributions as shown in Figure 2. The ability to vary the location of a spike in the single dimension of time thus allows real-valued control of the representation of probability distributions over multiple spatial dimensions. b) a) neuron 2 time .~ >" I ~ 'iii OOiS Q) II: , , Encoded Value Time Figure 2: a)Two spiking neurons centered at 0 and 1 can represent the time-varying mean and standard deviation on a single spatial dimension. The spikes are first convolved with a temporal kernel and the resulting activity values are treated as exponents on Gaussian distributions centered at 0 and 1. The ratio of the activity values determines the mean and the sum of the activity values determines the inverse variance. b) The same method can be used for two (or more) spatial dimensions. Time flows from top to bottom. Each spike makes a contribution to the energy landscape that resembles an hourglass (thin lines). The waist of the hourglass corresponds to the time at which the spike has its strongest effect on some post-synaptic population. By moving the hourglasses in time, it is possible to get whatever temporal cross-sections are desired (thick lines) provided the temporal sampling rate is comparable to the time course of the effect of a spike. Our proposed use of spike timing to convey real values quickly and accurately does not require precise coincidence detection, sub-threshold oscillations, modifiable time delays, or any of the other paraphernalia that has been invoked to explain how the brain could make effective use of the single, real-valued degree of freedom in the timing of a spike (Hopfield, 1995). The coding scheme we have proposed would be far more convincing if we could show how it was learned and could demonstrate that it was effective in a simulation. There are two ways to design a learning algorithm for such spiking neurons. We could work in the relatively low-dimensional space of the instantiation parameters and design the learning to produce the right representations and interactions between representations in this space. Or we could treat this space as an implicit emergent property of the network and design the learning algorithm to optimize Spiking Boltzmann Machines 125 some objective function in the much higher-dimensional space of neural activities in the hope that this will create representations that can be understood using the implicit space of instantiation parameters. We chose the latter approach. 3 A learning algorithm for restricted Boltzmann machines Hinton (1999) describes a learning algorithm for probabilistic generative models that are composed of a number of experts. Each expert specifies a probability distribution over the visible variables and the experts are combined by multiplying these distributions together and renormalizing. (1) where d is a data vector in a discrete space, Om is all the parameters of individual model m, Pm(d\Om) is the probability of d under model m, and i is an index over all possible vectors in the data space. The coding scheme we have described is just a product of experts in which each spike is an expert. We first summarize the Product of Experts learning rule for a restricted Boltzmann machine (RBM) which consists of a layer of stochastic binary visible units connected to a layer of stochastic binary hidden units with no intralayer connections. We then extend RBM's to deal with temporal data. In an RBM, each hidden unit is an expert. When it is off it specifies a uniform distribution over the states of the visible units . When it is on, its weight to each visible unit specifies the log odds that the visible unit is on. Multiplying together the distributions specified by different hidden units is achieved by adding the log odds. Inference in an RBM is much easier than in a causal belief net because there is no explaining away. The hidden states, S j, are conditionally independent given the visible states, Si, and the distribution of Sj is given by the standard logistic function a: p(Sj = 1) = a(L:i WijSi). Conversely, the hidden states of an RBM are marginally dependent so it is easy for an RBM to learn population codes in which units may be highly correlated. It is hard to do this in causal belief nets with one hidden layer because the generative model of a causal belief net assumes marginal independence. An RBM can be trained by following the gradient of the log likelihood of the data: (2) where < SiSj >0 is the expected value of SiSj when data is clamped on the visible units and the hidden states are sampled from their conditional distribution given the data, and < SiSj >00 is the expected value of SiSj after prolonged Gibbs sampling that alternates between sampling from the conditional distribution of the hidden states given the visible states and vice versa. This learning rule not work well because the sampling noise in the estimate of < SiSj >00 swamps the gradient. It is far more effective to maximize the difference between the log likelihood of the data and the log likelihood of the one-step reconstructions of the data that are produced by first picking binary hidden states from their conditional distribution given the data and then picking binary visible states from their conditional distribution given the hidden states. The gradient of the log G. E. Hinton and A. D. Brown 126 likelihood of the one-step reconstructions is complicated because changing a weight changes the probability distribution of the reconstructions: + (3) where Ql is the distribution of the one-step reconstructions of the training data and Qoo is the equilibrium distribution (i.e. the stationary distribution of prolonged Gibbs sampling). Fortunately, the cumbersome third term is sufficiently small that ignoring it does not prevent the vector of weight changes from having a positive cosine with the true gradient of the difference of the log likelhoods so the following very simple learning rule works much better than Eq. 2. (4) 4 Restricted Boltzmann machines through time Using a restricted Boltzmann machine we can represent time by spatializing it, i.e. taking each visible unit, i, and hidden unit, j, and replicating them through time with the constraint that the weight WijT between replica t of i and replica t + T of j does not depend on t. To implement the desired temporal smoothing, we also force the weights to be a smooth function of T that has the shape of the temporal kernel, shown in Figure 3. The only remaining degree of freedom in the weights between replicas of i and replicas of j is the scale of the temporal kernel and it is this scale that is learned. The replicas of the visible and hidden units still form a bipartite graph and the probability distribution over the hidden replicas can be inferred exactly without considering data that lies further into the future than the width of the temporal kernel. One problem with the restricted Boltzmann machine when we spatialize time is that hidden units at one time step have no memory of their states at previous time steps; they only see the data. If we were to add undirected connections between hidden units at different time steps, then the architecture would return to a fully connected Boltzmann machine in which the hidden units are no longer conditionally independent given the data. A useful trick borrowed from Elman nets is to allow the hidden units to see their previous states, but to treat these observations like data that cannot be modified by future hidden states. Thus, the hidden states may still be inferred independently without resorting to Gibbs sampling. The connections between hidden layer weights also follow the time course of the temporal kernel. These connections act as a predictive prior over the hidden units. It is important to note that these forward connections are not required for the network to model a sequence, but only for the purposes of extrapolating into the future. Figure 3: The form of the temporal kernel. 127 Spiking Boltzmann Machines Now the probability that Sj(t) = 1 given the states of the visible units is, P(Sj(t) = 1) = u (~W,jh,(t) + ~ W,;h,(t)) . where hi(t) is the convolution of the history of visible unit i with the temporal kernel, 00 T=O and hk(t), the convolution of the hidden unit history, is computed similarly. 2 Learning the weights follows immediately from this formula for doing inference. In the positive phase the visible units are clamped at each time step and the posterior of the hidden units conditioned on the data is computed (we assume zero boundary conditions for time before t = 0). Then in the negative phase we sample from the posterior of the hidden units, and compute the distribution over the visible units at each time step given these hidden unit states. In each phase the correlations between the hidden and visible units are computed and the learning rule is, 00 AWij = 00 L L r(7) ((Sj(t)Si(t - 7))0 - (Sj(t)Si(t - 7))1) . t=O T=O 5 Results We trained this network on a sequence of 8x8 synthetic images of a Gaussian blob moving in a circular path. In the following diagrams we display the time sequence of images as a matrix. Each row of the matrix represents a single image with its pixels stretched out into a vector in scanline order, and each column is the time course of a single pixel. The intensity f the pixel is represented by the area of the white patch. We used 20 hidden units. Figure 5a shows a segment (200 time steps) of the time series which was used in training. In this sequence the period of the blob is 80 time steps. Figure 5b shows how the trained model reconstructs the data after we sample from the hidden layer units. Once we have trained the model it is possible to do forecasting by clamping visible layer units for a segment of a sequence and then doing iterative Gibbs sampling to generate future points in the sequence. Figure 5c shows that given 50 time steps from the series, the model can predict reasonably far into the future, before the pattern dies out. One problem with these simulations is that we are treating the real valued intensities in the images as probabilities. While this works for the blob images, where the values can be viewed as the probabilities of pixels in a binary image being on, this is not true for more natural images. 6 Discussion In our initial simulations we used a causal sigmoid belief network (SBN) rather than a restricted Boltzmann machine. Inference in an SBN is much more difficult than in an RBM. It requires Gibbs sampling or severe approximations, and even if a temporal kernel is used to ensure that a replica of a hidden unit at one time 2Computing the conditional probability distribution over the visible units given the hidden states is done in a similar fashion, with the caveat that the weights in each direction must be symmetric. Thus, the convolution is done using the reverse kernel. G. E. Hinton and A. D. Brown 128 a) c) b) Figure 4: a) The original data, b) reconstruction of the data, and c) prediction of the data given 50 time steps of the sequence. The black line indicates where the prediction begins. has no connections to replicas of visible units at very different times, the posterior distribution of the hidden units still depends on data far in the future. The Gibbs sampling made our SBN simulations very slow and the sampling noise made the learning far less effective than in the RBM. Although the RBM simulations seem closer to biological plausibility, they too suffer from a major problem. To apply the learning procedure it is necessary to reconstruct the data from the hidden states and we do not know how to do this without interfering with the incoming datastream. In our simulations we simply ignored this problem by allowing a visible unit to have both an observed value and a reconstructed value at the same time. Acknowledgements We thank Zoubin Ghahramani, Peter Dayan, Rich Zemel, Terry Sejnowski and Radford Neal for helpful discussions. This research was funded by grants from the Gatsby Foundation and NSERC. References Anderson, C.H. & van Essen, D.C (1994). Neurobiological computational systems. In J.M Zureda, R.J. Marks, & C.J. Robinson (Eds.), Computational Intelligence Imitating Life 213-222. New York: IEEE Press. Hinton, G. E. (1999) Products of Experts. ICANN 99: Ninth international conference on Artificial Neural Networks, Edinburgh, 1-6. Hinton, G. E., McClelland, J. L., & Rumelhart, D. E. (1986) Distributed representations. In Rumelhart, D. E. and McClelland, J. L., editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, MIT Press, Cambridge, MA. Hopfield, J. (1995). Pattern recognition computation using action potential timing for stimulus representation. Nature, 376, 33-36. 

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A Winner-Take-All Circuit with Controllable Soft Max Property Shih-Chii Lin Institute for Neuroinformatics, ETHjUNIZ Winterthurstrasse 190, CH-8057 Zurich Switzerland shih@ini.phys.ethz.ch Abstract I describe a silicon network consisting of a group of excitatory neurons and a global inhibitory neuron. The output of the inhibitory neuron is normalized with respect to the input strengths. This output models the normalization property of the wide-field directionselective cells in the fly visual system. This normalizing property is also useful in any system where we wish the output signal to code only the strength of the inputs, and not be dependent on the number of inputs. The circuitry in each neuron is equivalent to that in Lazzaro's winner-take-all (WTA) circuit with one additional transistor and a voltage reference. Just as in Lazzaro's circuit, the outputs of the excitatory neurons code the neuron with the largest input. The difference here is that multiple winners can be chosen. By varying the voltage reference of the neuron, the network can transition between a soft-max behavior and a hard WTA behavior. I show results from a fabricated chip of 20 neurons in a 1.2J.Lm CMOS technology. 1 Introduction Lazzaro and colleagues (Lazzaro, 1988) were the first to implement a hardware model of a winner-take-all (WTA) network. This network consists of N excitatory cells that are inhibited by a global signal. Improvements of this network with addition of positive feedback and lateral connections have been described (Morris, 1998; Indiveri, 1998). The dynamics and stability properties of networks of coupled excitatory and inhibitory neurons have been analyzed by many (Amari, 1982; Grossberg, 1988). Grossberg described conditions under which these networks will exhibit WTA behavior. Lazzaro's network computes a single winner as reflected by the outputs of the excitatory cells. Several winners can be chosen by using more localized inhibition. In this work, I describe two variants of a similar architecture where the outputs of the excitatory neurons code the relative input strengths as in a soft-max computation. The relative values of the outputs depend on the number of inputs, their relative strengths and two parameter settings in the network. The global inhibitory 8.-c. Liu 718 Figure 1: Network model of recurrent inhibitory network. signal can also be used as an output. This output saturates with increasing number of active inputs, and the saturation level depends on the input strengths and parameter settings. This normalization property is similar to the normalization behavior of the wide-field direction-selective cells in the fly visual system. These cells code the temporal frequency of the visual inputs and are largely independent of the stimulation size. The circuitry in each neuron in the silicon network is equivalent to that in Lazzaro et. al.'s hard WTA network with an additional transistor and a voltage reference. By varying the voltage reference, the network can transition between a soft-max computation and a hard WTA computation. In the two variants, the outputs of the excitatory neurons either code the strength of the inputs or are normalized with respect to a constant bias current. Results from a fabricated network of 20 neurons in a 1.2J.Lm AMI CMOS show the different regimes of operation. 2 Network with Global Inhibition The generic architecture of a recurrent network with excitatory neurons and a single inhibitory neuron is shown in Figure 1. The excitatory neurons receive an external input, and they synapse onto a global inhibitory neuron. The inhibitory neuron, in turn, inhibits the excitatory neurons. The dynamics of the network is described as follows: dYi dt N = -Yi + ei - ~ g(~ WjYj) (1) j=l where Wj is the weight of the synapse between the jth excitatory neuron and the inhibitory neuron, and Yj is the state of the jth neuron. Under steady-state conditions, Yi = ei - YT, where YT = g(L:~l WjYj)? Assume a linear relationship between YT and Yj, and letting Wj = W, _ N ~ _ YT - W ~ Yj - "N W L..Jj=l ej 1 + wN j=l As N increases, YT = L:;l ej ? If all inputs have the same level, e, then YT = e. 719 A Winner-Take-All Circuit with Controllable Soft Max Property Figure 2: First variant of the architecture. Here we show the circuit for two excitatory neurons and the global inhibition neuron, M 4 ? The circuit in each excitatory neuron consists of an input current source, h, and transistors, M1 to M 3 . The inhibitory transistor is a fixed current source, lb . The inputs to the inhibitory transistor, 101 and I~2 are normalized with respect to lb. 3 First Variant of Network with Fixed Current Source In Sections 3 and 4, I describe two variants of the architecture shown in Figure 1. The two variants differ in the way that the inhibition signal is generated. The first network in Figure 2 shows the circuitry for two excitatory neurons and the inhibition neuron. Each excitatory neuron is a linear threshold unit and consists of an input current, h, and transistors, Ml, M 2 , and M 3 . The state of the neuron is represented by the current, I r1 . The diode-connected transistor, M 2 , introduces a rectifying nonlinearity into the system since Ir1 cannot be negative. The inhibition current, Ir, is sunk by M 1 , and is determined by the gate voltage, VT. The inhibition neuron consists of a current source, Ib, and VT is determined by the corresponding current, Ir1 and the corresponding transistor, M3 in each neuron. Notice that IT cannot be greater than the largest input to the network and the inputs to this network can only be excitatory. The input currents into the transistor, M 4 , are defined as 101 and 102 and are normalized with respect to the current source, h. In the hard WTA condition, the output current of the winning neuron is equal to the bias current, h. This network exhibits either a soft-maximum behavior or a hard WTA behavior depending on the value of an external bias, Va. The inhibition current, IT, is derived as: (2) where N is the number of "active" excitatory neurons (that is, neurons whose Ii > IT), Ii is the same input current to each neuron, and Ia = Ioe",vQ/uT. In deriving the above equation, we assumed that K, = 1. The inhibition current, IT, is Iri x Ial h? a linear combination of the states of the neurons because Ir = 2:f Figure 3(a) shows the response of the common-node voltage, VT, as a function of the number of inputs for different input values measured from a fabricated silicon network of 20 neurons. The input current to each neuron is provided by a pFET transistor that is driven by the gate voltage, Yin. All input currents are equal in this figure. The saturation behavior of the network as a function of the number s.-c. 720 Liu 0.8,---~-~-~-~-~----, Vin=3.9V -' . 0.7 0.6 Vin=4.3V .II'~''''--.'' 5 ......... ?? ' .. - - - ... ... ..... .. 10 15 20 Number of inputs (a) 25 30 5 10 15 20 25 30 Number of inputs (b) Figure 3: (a) Common-node voltage, VT, as a function of the number of input stimuli. Va = O.8V. (b) Common-node voltage, VT, as a function of the number of inputs with an input voltage of 4.3V and Vb = O.7V. The curves correspond to different values of Va . of inputs can be seen in the different traces and the saturation level increases as Vin decreases. As seen in Equation 2, the point at which the response saturates is dependent on the ratio, h / I a. In Figure 3(b), I show how the curve saturates at different points for different values of Va and a fixed hand Vin. In Figure 4, I set all inputs to zero except for two inputs, Vin1 and Vin2 that are set to the same value. I measured 101 and 101 as a function of Va as shown in Figure 4(a). The four curves correspond to four values of Vin. Initially both currents 101 and 102 are equal as is expected in the soft-max condition. As Va increases, the network starts exhibiting a WTA behavior. One of the output currents finally goes to zero above a critical value of Va. This critical value increases for higher input currents because of transistor backgate effects. In Figure 4(b), I show how the output currents respond as a function of the differential voltage between the two inputs as shown in Figure 4. Here, I fixed one input at 4.3V and swept the second input differentially around it. The different curves correspond to different values of Va. For a low value of Va, the linear differential input range is about lOOmV. This linear range decreases as Va is increased (corresponding to the WTA condition). 4 Second Variant with Diode-Connected Inhibition Transistor In the second variant shown in Figure 5, the current source, M4 is replaced by a diode-connected transistor and the output currents, 10i' follow the magnitude of the input currents. The inhibition current, Ir, can be expressed as follows: (3) where la is defined in Section 3. We sum Equation 3 over all neurons and assuming equal inputs, we get Ir = J'LJri x la. This equation shows that the feedback signal has a square root dependence on the neuron states. As we will see, this causes the feedback signal to saturate quickly with the number of inputs. 721 A Winner-Take-All Circuit with Controllable Soft Max Property 6 r-... 5 2.5 Va=O ~ ~4 ~ , 2 -: 1.5 ".\'0" 5~'~\ /~. " I Va=O 6V Va=O.7V Al\',/' ..s 2 < -0.2 l~ ' ~ ,. <\ / ) .. \ \> .,.~J \ \ '" , 0.5 Va=O.4V ,i -0.1 ,,~ 0 0.1 Vio2-Viol (V) 0.2 0.3 (b) (a) Figure 4: (a) Output currents, 101 and 1 02 , as a function of Va: for a subthreshold bias current and Yin = 4.0V to 4.3V. (b) Outputs, 101 and 1 02 , as a function of the differential input voltage, ~ Vin, with Yinl = 4.3V. Figure 5: Second variant of network. The schematic shows two excitatory neurons with diode-connected inhibition transistor. Substituting lri = Ii - IT in Equation 3, we solve for Jr, N IT = -1a: N + (Ia: N )2 + 41a: L Ii (4) From measurements from a fabricated circuit with 20 neurons, I show the dependence of VT (the natural logarithm of Jr) on the number of inputs in Figure 6(a). The output saturates quickly with the number of inputs and the level of saturation increases with increased input strengths. All the inputs have the same value. The network can also act as a WTA by changing Va:. Again, all inputs are set to zero except for two inputs whose gate voltages are both set at 4.2V. As shown in Figure 6(b), the output currents, 101 and 1 02 , are initially equal, and as Va: increases above 0.6V, the output currents split apart and eventually, 102 = OA. The final value of 101 depends on the maximum input current. This data shows that the network acts as a WTA circuit when Va: > 0.73V. If I set Vin2 = 4.25V instead, the output currents split at a lower value of Va:. s.-c. 722 Liu 0.45.--~-~-~-~-~------, 5 Vinl=4.2V. Vin2=4.25V 4 $ ~3 2 0.20'---:-2-~4-~6:--8=---1-:':0:----:'12 Number of inputs (a) 0.6 0.7 0.8 0.9 Va (V) (b) Figure 6: (a) Common-node voltage, VT, as a function of the number of inputs for input voltages, 3.9V, 4.06V, and 4.3V for Va = O.4V. (b) Outputs, 101 and 1 02 , as a function of Va for Vinl = 4.2V, Vin2 = 4.25V for the 2 curves with asterisks and for Vinl = Vin2 = 4.2V for the 2 curves with circles. 5 Inhibition The WTA property arises in both variants of this network if the gain parameter, Va, is increased so that the diode-connected transistor, M 2 , can be ignored. Both variants then reduce to Lazzaro's network. In the first variant, the feedback current (Ir) is a linear combination ofthe neuron states. However, when the gain parameter is increased so that M2 can be ignored, the feedback current is now a nonlinear combination of the input states so the WTA behavior is exhibited by these reduced networks. Under hard WTA conditions, if Ir is initially smaller than all the input currents, the capacitances C at the nodes Vr1 and Vr2 are charged up by the difference between the individual input current and IT, i.e., d~t = liCIT. Since the inhibition current is a linear combination of Iri and Iri is exponential in Vri , we can see that IT is a sum of the exponentials of the input currents, h Hence the feedback current is nonlinear in the input currents. Another way of viewing this condition in electronic terms is that in the soft WTA condition, the output node of each neuron is a softimpedance node, or a low-gain node. In the hard WTA case, the output node is now a high-impedance node or a high-gain node. Any input differences are immediately amplified in the circuit. 6 Discussion Hahnloser (Hahnloser, 1998) recently implemented a silicon network of linear threshold excitatory neurons that are coupled to a global inhibitory neuron. The inhibitory signal is a linear combination of the output states of the excitatory neurons. This network does not exhibit WTA behavior unless the excitatory neurons include a self-excitatory term. The inhibition current in his network is also generated via a diode-connected transistor. The circuitry in two variants described here is more compact than the circuitry in his network. Recurrent networks with the architecture described in this paper have been proposed by Reichardt and colleagues (Reichardt, 1983) in modelling the aggregation property A Winner-Take-All Circuit with Controllable Soft Max Property 723 of the wide-field direction-selective cells in flies. The synaptic inputs are inhibited by a wide-field cell that pools all the synaptic inputs. Similar networks have also been used to model cortical processing, for example, orientation selectivity (Douglas, 1995). The network implemented here can model the aggregation property of the directionselective cells in the fly. By varying a voltage reference, the network implements either a soft-max computation or a hard WTA computation. This circuitry will be useful in hardware models of cortical processing or motion processing in invertebrates. Acknowledgments I thank Rodney Douglas for supporting this work, and the MOSIS foundation for fabricating this circuit. I also thank Tobias Delbriick for proofreading this document. This work was supported in part by the Swiss National Foundation Research SPP grant and the U.S. Office of Naval Research. References Amari, S., and Arbib, M. A., "Competition and cooperation in neural networks," New York, Springer-Verlag, 1982. Grossberg, W., "Nonlinear neural networks: Principles, mechanisms, and architectures," Neural Networks, 1, 17-61, 1988. Hanhloser, R., "About the piecewise analysis of networks of linear threshold neurons," Neural Networks, 11,691- 697, 1988. Hahnloser, R., "Computation in recurrent networks of linear threshold neurons: Theory, simulation and hardware implementation," Ph.D. Thesis, Swiss Federal Institute of Technology, 1998. Lazzaro, J., Ryckebusch, S. Mahowald, M.A., and Mead. C., "Winner-take-all networks of O(n) complexity," In Tourestzky, D. (ed), Advances in Neural Information Processing Systems 1, San Mateo, CA: Morgan Kaufman Publishers, pp. 703-711, 1988. Morris, T .G., Horiuchi, T. and Deweerth, S.P., "Object-based selection within an analog VLSI visual attention system," IEEE Trans. on Circuits and Systems II, 45:12, 1564-1572, 1998. Indiveri, G., "Winner-take-all networks with lateral excitation," Neuromorphic Systems Engineering, Editor, Lande, TS., 367-380, Kluwer Academic, Norwell, MA, 1998. Reichardt, W., Poggio, T., and Hausen, K., "Figure-ground discrimination by relative movement in the visual system of the fly," BioI. Cybern., 46, 1-30, 1983. Douglas, RJ., Koch, C., Mahowald, M., Martin, KAC., and Suarez, HH., "Recurrent excitation in neocortical circuits," Science, 269:5226,981-985, 1995. 

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A Variational Bayesian Framework for Graphical Models Hagai Attias hagai@gatsby.ucl.ac.uk Gatsby Unit, University College London 17 Queen Square London WC1N 3AR, U.K. Abstract This paper presents a novel practical framework for Bayesian model averaging and model selection in probabilistic graphical models. Our approach approximates full posterior distributions over model parameters and structures, as well as latent variables, in an analytical manner. These posteriors fall out of a free-form optimization procedure, which naturally incorporates conjugate priors. Unlike in large sample approximations, the posteriors are generally nonGaussian and no Hessian needs to be computed. Predictive quantities are obtained analytically. The resulting algorithm generalizes the standard Expectation Maximization algorithm, and its convergence is guaranteed. We demonstrate that this approach can be applied to a large class of models in several domains, including mixture models and source separation. 1 Introduction A standard method to learn a graphical model 1 from data is maximum likelihood (ML). Given a training dataset, ML estimates a single optimal value for the model parameters within a fixed graph structure. However, ML is well known for its tendency to overfit the data. Overfitting becomes more severe for complex models involving high-dimensional real-world data such as images, speech, and text. Another problem is that ML prefers complex models, since they have more parameters and fit the data better. Hence, ML cannot optimize model structure. The Bayesian framework provides, in principle, a solution to these problems. Rather than focusing on a single model, a Bayesian considers a whole (finite or infinite) class of models. For each model, its posterior probability given the dataset is computed. Predictions for test data are made by averaging the predictions of all the individual models, weighted by their posteriors. Thus, the Bayesian framework avoids overfitting by integrating out the parameters. In addition, complex models are automatically penalized by being assigned a lower posterior probability, therefore optimal structures can be identified. Unfortunately, computations in the Bayesian framework are intractable even for lWe use the term 'model' to refer collectively to parameters and structure. H. Attias 210 very simple cases (e.g. factor analysis; see [2]). Most existing approximation methods fall into two classes [3]: Markov chain Monte Carlo methods and large sample methods (e.g., Laplace approximation). MCMC methods attempt to achieve exact results but typically require vast computational resources, and become impractical for complex models in high data dimensions. Large sample methods are tractable, but typically make a drastic approximation by modeling the 'posteriors over all parameters as Normal, even for parameters that are not positive definite (e.g., covariance matrices). In addition, they require the computation ofthe Hessian, which may become quite intensive. In this paper I present Variational Bayes (VB), a practical framework for Bayesian computations in graphical models. VB draws together variational ideas from intractable latent variables models [8] and from Bayesian inference [4,5,9], which, in turn, draw on the work of [6]. This framework facilitates analytical calculations of posterior distributions over the hidden variables, parameters and structures. The posteriors fall out of a free-form optimization procedure which naturally incorporates conjugate priors, and emerge in standard forms, only one of which is Normal. They are computed via an iterative algorithm that is closely related to Expectation Maximization (EM) and whose convergence is guaranteed. No Hessian needs to be computed. In addition, averaging over models to compute predictive quantities can be performed analytically. Model selection is done using the posterior over structure; in particular, the BIC/MDL criteria emerge as a limiting case. 2 General Framework We restrict our attention in this paper to directed acyclic graphs (DAGs, a.k.a. Bayesian networks). Let Y = {y., ... ,YN} denote the visible (data) nodes, where n = 1, ... , N runs over the data instances, and let X = {Xl, ... , XN} denote the hidden nodes. Let e denote the parameters, which are simply additional hidden nodes with their own distributions. A model with a fixed structure m is fully defined by the joint distribution p(Y, X, elm). In a DAG, this joint factorizes over the nodes, i.e. p(Y,X I e,m) = TIiP(Ui I pai,Oi,m), where Ui E YUX, pai is the set of parents of Ui, and Oi E e parametrize the edges directed toward Ui. In addition, we usually assume independent instances, p(Y, X Ie, m) = TIn p(y n, Xn Ie, m). We shall also consider a set of structures m E M, where m controls the number of hidden nodes and the functional forms of the dependencies p( Ui I pai , 0i, m), including the range of values assumed by each node (e.g., the number of components in a mixture model). Associated with the set of structures is a structure prior p(m). Marginal likelihood and posterior over parameters. For a fixed structure m, we are interested in two quantities. The first is the parameter posterior distribution p(e I Y,m). The second is the marginal likelihood p(Y I m), also known as the evidence assigned to structure m by the data. In the following, the reference to m is usually omitted but is always implied. Both quantities are obtained from the joint p(Y, X, elm). For models with no hidden nodes the required computations can often be performed analytically. However, in the presence of hidden nodes, these quantities become computationally intractable. We shall approximate them using a variational approach as follows. Consider the joint posterior p(X, elY) over hidden nodes and parameters. Since it is intractable, consider a variational posterior q(X, elY), which is restricted to the factorized form q(X, elY) = q(X I Y)q(e I Y) , (1) wher"e given the data, the parameters and hidden nodes are independent. This A Variational Baysian Frameworkfor Graphical Models 211 restriction is the key: It makes q approximate but tractable. Notice that we do not require complete factorization, as the parameters and hidden nodes may still be correlated amongst themselves. We compute q by optimizing a cost function Fm[q] defined by Fm[q] = ! dE> q(X)q(E? log ~~i~(:j ~ logp(Y I m) , (2) where the inequality holds for an arbitrary q and follows from Jensen's inequality (see [6]); it becomes an equality when q is the true posterior. Note that q is always understood to include conditioning on Y as in (1). Since Fm is bounded from above by the marginal likelihood, we can obtain the optimal posteriors by maximizing it w.r.t. q. This can be shown to be equivalent to minimizing the KL distance between q and the true posterior. Thus, optimizing Fm produces the best approximation to the true posterior within the space of distributions satisfying (1), as well as the tightest lower bound on the true marginal likelihood. Penalizing complex models. To see that the VB objective function Fm penalizes complexity, it is useful to rewrite it as Fm = (log p(Y, X I E? q(X) )X,9 - KL[q(E? II p(E?] , (3) where the average in the first term on the r.h.s. is taken w.r.t. q(X, E?. The first term corresponds to the (averaged) likelihood. The second term is the KL distance between the prior and posterior over the parameters. As the number of parameters increases, the KL distance follows and consequently reduces Fm. This penalized likelihood interpretation becomes transparent in the large sample limit N -7 00, where the parameter posterior is sharply peaked about the most probable value E> = E>o. It can then be shown that the KL penalty reduces to (I E>o 1/2) log N, which is linear in the number of parameters I E>o I of structure m. Fm then corresponds precisely the Bayesian information criterion (BIC) and the minimum description length criterion (MDL) (see [3]). Thus, these popular model selection criteria follow as a limiting case of the VB framework. Free-form optimization and an EM-like algorithm. Rather than assuming a specific parametric form for the posteriors, we let them fall out of free-form optimization of the VB objective function. This results in an iterative algorithm directly analogous to ordinary EM. In the E-step, we compute the posterior over the hidden nodes by solving 8Fm/8q(X) = 0 to get q(X) ex e(log p(Y,XI9?e , (4) where the average is taken w.r.t. q(E?. In the M-step, rather than the 'optimal' parameters, we compute the posterior distribution over the parameters by solving 8Fm/8q(E? = 0 to get q(E? ex e(IOgp(y,X I9?x p (E? , (5) where the average is taken w.r.t. q(X). This is where the concept of conjugate priors becomes useful. Denoting the exponential term on the r.h.s. of (5) by f(E?, we choose the prior p(E? from a family of distributions such that q(E? ex f(E?p(E? belongs to that same family. p(E? is then said to be conjugate to f(E?. This procedure allows us to select a prior from a fairly large family of distributions (which includes non-informative ones as limiting cases) H. Attias 212 and thus not compromise generality, while facilitating mathematical simplicity and elegance. In particular, learning in the VB framework simply amounts to updating the hyperparameters, i.e., transforming the prior parameters to the posterior parameters. We point out that, while the use of conjugate priors is widespread in statistics, so far they could only be applied to models where all nodes were visible. Structure posterior. To compute q(m) we exploit Jensen's inequality once again to define a more general objective function, .1'[q] l:mEM q(m) [.1'm + logp(m)jq(m)] ~ 10gp(Y), where now q = q(X I m, Y)q(8 I m, Y)q( m I Y) . After computing .1'm for each m EM, the structure posterior is obtained by free-form optimization of .1': q(m) ex e:Frnp(m) . (6) Hence, prior assumptions about the likelihood of different structures, encoded by the prior p(m), affect the selection of optimal model structures performed according to q( m), as they should. Predictive quantities. The ultimate goal of Bayesian inference is to estimate predictive quantities, such as a density or regression function. Generally, these quantities are computed by averaging over all models, weighting each model by its posterior. In the VB framework, exact model averaging is approximated by replacing the true posterior p(8 I Y) by the variational q(8 I Y). In density estimation, for example, the density assigned to a new data point Y is given by p(y I Y) = J d8 p(y I 8) q(8 I Y) . In some situations (e.g. source separation), an estimate of hidden node values x from new data y may be required. The relevant quantity here is the conditional p(x I y, Y), from which the most likely value of hidden nodes is extracted. VB approximates it by p(x I y, Y) ex J d8 p(y, x I 8) q(8 I Y). 3 Variational Bayes Mixture Models Mixture models have been investigated and analyzed extensively over many years. However, the well known problems of regularizing against likelihood divergences and of determining the required number of mixture components are still open. Whereas in theory the Bayesian approach provides a solution, no satisfactory practical algorithm has emerged from the application of involved sampling techniques (e.g., [7]) and approximation methods [3] to this problem. We now present the solution provided by VB. We consider models of the form m P(Yn I 8,m) = LP(Yn I Sn = s,8) p(sn = s I 8), (7) s=1 where Yn denotes the nth observed data vector, and Sn denotes the hidden component that generated it. The components are labeled by s = 1, ... , m, with the structure parameter m denoting the number of components. Whereas our approach can be applied to arbitrary models, for simplicity we consider here Normal component distributions, P(Yn I Sn = s, 8) = N(JJ. s' r 8), where I-Ls is the mean and r 8 the precision (inverse covariance) matrix. The mixing proportions are P(Sn = S I 8) = 'Tr s' In hindsight, we use conjugate priors on the parameters 8 = {'Trs, I-Ls' rs}. The mixing proportions are jointly Dirichlet, p( {'Trs}) = V(..~O), the means (conditioned on the preCisions) are Normal, p(l-Ls Irs) = N(pO, f3 0 r s), and the precisions are Wishart, p(r s) = W(v O, ~O). We find that the parameter posterior for a fixed m A Variational Baysian Framework/or Graphical Models 213 factorizes into q(8) = q({1I"s})flsq(J.?s,rs). The posteriors are obtained by the following iterative algorithm, termed VB-MOG. E-step. Compute the responsibilities for instance n using (4): = S I Yn) ex ffs r!/2 e-(Yn-P,)Tr,(Yn-P,)/2 e- d / 2f3? , (8) here X = S and q(S) = TIn q(Sn). This expression resembles the re- ,: == q(sn noting that sponsibilities in ordinary MLj the differences stem from integrating out the parameters. The special quantities in (8) are logffs == (log1l"s) = 1/1()..s) -1/1CLJs' )..s,), d logrs == (log I rs I) = Li=l1/1(lIs + 1 - i)/2) - log 1 ~s 1 +dlog2, and i\ == (r s) = IIs~;l, where 1/1(x) = dlog r(x)/dx is the digamma function, and the averages (-} are taken w.r.t. q(8). The other parameters are described below. M-step. Compute the parameter posterior in two stages. First, compute the quantities 1I"s 1 N " n , = N 'L..J's J.?s = 1 N " n Yn , N 'L..J's ~ Lis N = " n enS ' N1 'L..J's (9) n=l where C~ = (Yn - f.ts)(Yn - f.ts)T and fls = N7r s. This stage is identical to the M-step in ordinary EM where it produces the new parameters. In VB, however, the quantities in (9) only help characterize the new parameter posteriors. These posteriors are functionally identical to the priors but have different parameter values. The mixing proportions are jointly Dirichlet, q( {11" s}) = D( {)..s}), the means are Normal, q(J..t s Irs) = N(ps' /3srs), and the precisions are Wishart, p(rs) = W(lI s , ~s). The posterior parameters are updated in the second stage, using the simple rules )..s lis fls +)..0, = - Ns + 0 II, s n=1 /3s = fls + /30 , (10) aT a ~s = NsEs + N s/3 (J.?s - P )(I's - P ) /(Ns + f3 ) + ~ . Ps = (flsf.ts + /3opO)/(Ns +~) , s n=l - - - 0 - 0- ? The final values of the posterior parameters form the output of the VB-MOG. We remark that (a) Whereas no specific assumptions have been made about them, the parameter posteriors emerge in suitable, non-trivial (and generally non-Normal) functional forms. (b) The computational overhead of the VB-MOG compared to EM is minimal. (c) The covariance of the parameter posterior is O(l/N), and VBMOG reduces to EM (regularized by the priors) as N ~ 00. (d) VB-MOG has no divergence problems. (e) Stability is guaranteed by the existence of an objective function. (f) Finally, the approximate marginal likelihood F m , required to optimize the number of components via (6), can also be obtained in closed form (omitted). Predictive Density. Using our posteriors, we can integrate out the parameters and show that the density assigned by the model to a new data vector Y is a mixture of Student-t distributions, m (11) s=1 where component S has Ws = lis + 1 - d d.o.f., mean Ps' covariance As = ?/3s + 1)//3sws)~s, and proportion 7rs = )..s/ Ls' )..s" (11) reduces to a MOG as N ~ 00. Nonlinear Regression. We may divide each data vector into input and output parts, Y = (yi,y o ), and use the model to estimate the regression function yO = f(yi) and error spheres. These may be extracted from the conditional p(yO I yt, Y) = L:n=l Ws tw~ (yO I p~, A~), which also turns out to be a mixture of Student-t distributions, with means p~ being linear, and covariances A~ and mixing proportions Ws nonlinear, in yi, and given in terms of the posterior parameters. H Attias 214 Buffalo post offIce digits Misclasslflcation rate histogram 1 , - - - - -- -- - - - - -- , 0.8 0.6 0 .4 0 .2 0 0 ( ,, ,, -, - _1 - 0 .05 ~ , ,, ,, 0 .1 Figure 1: VB-MOG applied to handwritten digit recognition. VB-MOG was applied to the Boston housing dataset (UCI machine learning repository), where 13 inputs are used to predict the single output, a house's price. 100 random divisions of the N = 506 dataset into 481 training and 25 test points were used, resulting in an average MSE of 11.9. Whereas ours is not a discriminative method, it was nevertheless competitive with Breiman's (1994) bagging technique using regression trees (MSE=11.7). For comparison, EM achieved MSE=14.6. Classification. Here, a separate parameter posterior is computed for each class c from a training dataset yc. Test data vector y is then classified according to the conditional p(c I y, {yC}), which has a form identical to (11) (with c-dependent parameters) multiplied by the relative size of yc. VB-MOG was applied to the Buffalo post office dataset, which contains 1100 examples for each digit 0 - 9. Each digit is a gray-level 8 x 8 pixel array (see examples in Fig. 1 (left)). We used 10 random 500-digit batches for training, and a separate batch of 200 for testing. An average misclassification rate of .018 was obtained using m = 30 components; EM achieved .025. The misclassification histograms (VB=solid, EM=dashed) are shown in Fig. 1 (right). 4 VB and Intractable Models: a Blind Separation Example The discussion so far assumed that a free-form optimization of the VB objective function is feasible. Unfortunately, for many interesting models, in particular models where ordinary ML is intractable, this is not the case. For such models, we modify the VB procedure as follows: (a) Specify a parametric functional form for the posterior over the hidden nodes q(X) , and optimize w.r.t. its parameters, in the spirit of [8J. (b) Let the parameter posterior q(8) fall out of free-form optimization, as before. We illustrate this approach in the context of the blind source separation (BSS) problem (see, e.g., [1]). This problem is described by Yn = HX n + Un , where Xn is an unobserved m-dim source vector at instance n, H is an unknown mixing matrix, and the noise Un is Normally distributed with an unknown precision >'1. The task is to construct a source estimate xn from the observed d-dim data y. The sources are independent and non-Normally distributed. Here we assume the high-kurtosis distribution p(xi) ex: cosh-\xf /2) , which is appropriate for modeling speech sources. One important but heretofore unresolved problem in BSS is determining the number m of sources from data. Another is to avoid overfitting the mixing matrix. Both problems, typical to ML algorithms, can be remedied using VB. It is the non-Normal nature ofthe sources that renders the source posterior p(X I Y) intractable even before a Bayesian treatment. We use a Normal variational posterior q(X) = TIn N(x n lPn' r n) with instance-dependent mean and precision. The mixing matrix posterior q(H) then emerges as Normal. For simplicity, >. is optimized rather than integrated out. The reSUlting VB-BSS algorithm runs as follows: A Variational Baysian Framework for Graphical Models log PrIm) o source reconstruction errOr o ' -1000 -5 -2000 -10 -3000 -4000 215 -15 2 4 6 8 m 10 12 -roO~--~5~--~1-0--~ 15 SNR(dB) Figure 2: Application of VB to blind source separation algorithm (see text). E-step. Optimize the variational mean Pn by iterating to convergence, for each n, the fixed-point equation XfI:T(Yn - HPn) - tanhpn /2 = C- I Pn , where C is the source covariance conditioned on the data. The variational precision matrix turns out to be n-independent: r n = A. T AA. + 1/2 + C- I . M-step. Update the mean and precision of the posterior q(H) (rules omitted). This algorithm was applied to ll-dim data generated by linearly mixing 5 lOOmseclong speech and music signals obtained from commercial CDs. Gaussian noise were added at different SNR levels. A uniform structure prior p( m) = 1/ K for m ~ K was used. The resulting posterior over the number of sources (Fig. 2 (left)) is peaked at the correct value m = 5. The sources were then reconstructed from test data via p(x I y, Y). The log reconstruction error is plotted vs. SNR in Fig. 2 (right, solid). The ML error (which includes no model averaging) is also shown (dashed) and is larger, reflecting overfitting. 5 Conclusion The VB framework is applicable to a large class of graphical models. In fact, it may be integrated with the junction tree algorithm to produce general inference engines with minimal overhead compared to ML ones. Dirichlet, Normal and Wishart posteriors are not special to models treated here but emerge as a general feature. Current research efforts include applications to multinomial models and to learning the structure of complex dynamic probabilistic networks. Acknowledgements I thank Matt Beal, Peter Dayan, David Mackay, Carl Rasmussen, and especially Zoubin Ghahramani, for important discussions. References [1) Attias, H. (1999). Independent Factor Analysis. Neural Computation 11, 803-85l. [2) Bishop, C.M. (1999). Variational Principal Component Analysis. Proc. 9th ICANN. [3) Chickering, D.M. & Heckerman, D. (1997) . Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables. Machine Learning 29, 181-212. [4) Hinton, G.E. & Van Camp, D. (1993). Keeping neural networks simple by minimizing the description length of the weights. Proc. 6th COLT, 5-13. [5) Jaakkola, T. & Jordan, M.L (1997). Bayesian logistic regression: A variational approach. Statistics and Artificial Intelligence 6 (Smyth, P. & Madigan, D., Eds). [6) Neal, R.M. & Hinton, G.E. (1998). A view of the EM algorithm that justifies incremental, sparse, and other variants. Learning in Graphical Models, 355-368 (Jordan, M.L, Ed). Kluwer Academic Press, Norwell, MA. [7) Richardson, S. & Green, P.J. (1997). On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society B, 59, 731-792. [8) Saul, L.K., Jaakkola, T., & Jordan, M.I. (1996). Mean field theory of sigmoid belief networks. Journal of Artificial Intelligence Research 4, 61-76. [9) Waterhouse, S., Mackay, D., & Robinson, T. (1996). Bayesian methods for mixture of experts. NIPS-8 (Touretzky, D.S. et aI, Eds). MIT Press. 

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The Relaxed Online Maximum Margin Algorithm Yi Li and Philip M. Long Department of Computer Science National University of Singapore Singapore 119260, Republic of Singapore {liyi,p/ong}@comp.nus.edu.sg Abstract We describe a new incremental algorithm for training linear threshold functions: the Relaxed Online Maximum Margin Algorithm, or ROMMA. ROMMA can be viewed as an approximation to the algorithm that repeatedly chooses the hyperplane that classifies previously seen examples correctly with the maximum margin. It is known that such a maximum-margin hypothesis can be computed by minimizing the length of the weight vector subject to a number of linear constraints. ROMMA works by maintaining a relatively simple relaxation of these constraints that can be efficiently updated. We prove a mistake bound for ROMMA that is the same as that proved for the perceptron algorithm. Our analysis implies that the more computationally intensive maximum-margin algorithm also satisfies this mistake bound; this is the first worst-case performance guarantee for this algorithm. We describe some experiments using ROMMA and a variant that updates its hypothesis more aggressively as batch algorithms to recognize handwritten digits. The computational complexity and simplicity of these algorithms is similar to that of perceptron algorithm , but their generalization is much better. We describe a sense in which the performance of ROMMA converges to that of SVM in the limit if bias isn't considered. 1 Introduction The perceptron algorithm [10, 11] is well-known for its simplicity and effectiveness in the case of linearly separable data. Vapnik's support vector machines (SVM) [13] use quadratic programming to find the weight vector that classifies all the training data correctly and maximizes the margin, i.e. the minimal distance between the separating hyperplane and the instances. This algorithm is slower than the perceptron algorithm, but generalizes better. On the other hand, as an incremental algorithm, the perceptron algorithm is better suited for online learning, where the algorithm repeatedly must classify patterns one at a time, then finds out the correct classification, and then updates its hypothesis before making the next prediction. In this paper, we design and analyze a new simple online algorithm called ROMMA (the Relaxed Online Maximum Margin Algorithm) for classification using a linear threshold 499 The Relaxed Online Maximum Margin Algorithm function. ROMMA has similar time complexity to the perceptron algorithm, but its generalization performance in our experiments is much better on average. Moreover, ROMMA can be applied with kernel functions. We conducted experiments similar to those performed by Cortes and Vapnik [2] and Freund and Schapire [3] on the problem of handwritten digit recognition. We tested the standard perceptron algorithm, the voted perceptron algorithm (for details, see [3]) and our new algorithm, using the polynomial kernel function with d = 4 (the choice that was best in [3]). We found that our new algorithm performed better than the standard perceptron algorithm, had slightly better performance than the voted perceptron. For some other research with aims similar to ours, we refer the reader to [9,4,5,6]. The paper is organized as foIlows. In Section 2, we describe ROMMA in enough detail to determine its predictions, and prove a mistake bound for it. In Section 3, we describe ROMMA in more detail. In Section 4, we compare the experimental results of ROMMA and an aggressive variant of ROMMA with the perceptron and the voted perceptron algorithms. 2 2.1 A mistake-bound analysis The online algorithms For concreteness, our analysis will concern the case in which instances (also called patterns) and weight vectors are in R n . Fix n EN. In the standard online learning model [7], learning proceeds in trials. In the tth trial, the algorithm is first presented with an instance it ERn . Next, the algorithm outputs a prediction Yt of the classification of it. Finally, the algorithm finds out the correct classification Yt E {-1 , 1}. If Yt =I=- Yt, then we say that the algorithm makes a mistake. It is worth emphasizing that in this model, when making its prediction for the tth trial, the algorithm only has access to instance-classification pairs for previous trials. All of the online algorithms that we will consider work by maintaining a weight vector WI which is updated between trials, and predicting Yt sign( Wt . it), where sign( z) is 1 if z is positive, -1 if z is negative, and 0 otherwise.! = The perceptron algorithm. The perceptron algorithm, due to Rosenblatt [10, 11], starts off with Wi = O. When its prediction differs from the label Yt, it updates its weight vector by Wt+i = Wt + Ytit. If the prediction is correct then the weight vector is not changed. The next three algorithms that we will consider assume that all of the data seen by the online algorithm is collectively linearly separable, i.e. that there is a weight vector u such that for all each trial t, Yt = sign( u . xd. When kernel functions are used, this is often the case in practice. The ideal online maximum margin algorithm. On each trial t, this algorithm chooses a weight vector Wt for which for all previous trials s ::; t, sign( Wt . is) = Ys, and which maximizes the minimum distance of any is to the separating hyperplane. It is known [1, 14] that this can be implemented by choosing Wt to minimize Ilwdl subject to the constraints that Ys (Wt . xs ) ;::: 1 for all s ::; t. These constraints define a convex polyhedron in weight space which we will refer to as Pt. The relaxed online maximum margin algorithm. This is our new algorithm. The first difference is that trials in which mistakes are not made are ignored. The second difference 'The prediction of 0, which ensures a mistake, is to make the proofs simpler. The usual mistake bound proof for the perceptron algorithm goes through with this change. Y. Li and P. M Long 500 is in how the algorithm responds to mistakes. The relaxed algorithm starts off like the ideal algorithm. Before the second trial, it sets W2 to be the shortest weight vector such that Yl (W2 . i l ) 2:: 1. If there is a mistake on the second trial, it chooses W3 as would the ideal algorithm, to be the smallest element of (1) However, if the third trial is a mistake, then it behaves differently. Instead of choosing W4 to be the smallest element of {w: yI(w? i l ) 2:: I} n {w: Y2(W. i 2 ) 2:: I} n {w: Y3(W? i3) 2:: I} , it lets W4 be the smallest element of {w: W3 . W 2:: JJw3112} n {w: Y3(W. i3) 2:: I}. This can be thought of as, before the third trial, replacing the polyhedron defined by (1) with the halfspace {w : W3 ? W 2:: JJW3JJ2} (see Figure 1). Note that this halfspace contains the polyhedron of (1); in fact, it contains any convex set whose smallest element is W3. Thus, it can be thought of as the least restrictive convex constraint for which the smallest satisfying weight vector is W3. Let us call this halfspace H 3 . The algorithm continues in this manner. If the tth trial is a mistake, then Wt+l is chosen to be the smallest element of H t n {w : Yt(w? it) 2:: I}, and Ht+l is set to be {w : Wt+l . W 2:: IIwt+lJJ2}. If the tth trial is not a Wt and Ht+l H t . We will mistake, then Wt+l call H t the old constraint, and {w : Yt (w . it) 2: I} the new constraint. Figure 1: In ROMMA, a convex polyhedron in weight space is replaced with the halfspace with the same smallest element. Note that after each mistake, this algorithm needs only to solve a quadratic programming problem with two linear constraints. In fact, there is a simple closed-form expression for Wt+l as a function of Wt, it and Yt that enables it to be computed incrementally using time similar to that of the perceptron algorithm. This is described in Section 3. = = The relaxed online maximum margin algorithm with aggressive updating. The algorithm is the same as the previous algorithm, except that an update is made after any trial in which yt{Wt . it} < 1, not just after mistakes. 2.2 Upper bound on the number of mistakes made Now we prove a bound on the number of mistakes made by ROMMA. As in previous mistake bound proofs (e.g. [8]), we will show that mistakes result in an increase in a "measure of progress", and then appeal to a bound on the total possible progress. Our proof will use the squared length of Wt as its measure of progress. First we will need the following lemmas. Lemma 1 On any run of ROMMA on linearly separable data, if trial t was a mistake, then the new constraint is binding at the new weight vector; i.e. Yt (Wt+l . it) = 1. Proof: For the purpose of contradiction suppose the new constraint is not binding at the new weight vector Wt+l. Since Wt fails to satisfy this constraint, the line connecting Wt+l and Wt intersects with the border hyperplane of the new constraint, and we denote the aWt + (l-a)Wt+l, 0 < a < 1. intersecting point q. Then Wq can be represented as Wq w = 501 The Relaxed Online Maximum Margin Algorithm Since the square of Euclidean length II . 1\2 is a convex function, the following holds: IIwql\2 ~ allwtll 2 + (1 - a) IIwt+d2 Since Wt is the unique smallest member of H t and Wt+1 which implies i= Wt, we have IIwtl12 < IIwt+11l2, (2) Since Wt and Wt+1 are both in H t , Wq is too, and hence (2) contradicts the definition of 0 Wt+1? Lemma 2 On any run of ROMMA on linearly separable data, if trial t was a mistake, and not the first one, then the old constraint is binding at the new weight vector, i.e. Wt + 1 . Wt = IIwtV Proof: Let At be the plane of weight vectors that make the new constraint tight, i.e. At = {tV : Yt(w? xd = I}. By Lemma 1, Wt+1 E At . Let at = Ytxtlllxtll2 be the element of At that is perpendicular to it. Then each wE At satisfies IIwII 2 lIatll2 + IIw - at 1 2, and therefore the length of a vector W in At is minimized when W = at and is monotone in the distance from W to at. Thus, if the old constraint is not binding, then Wt+1 = at. since otherwise the solution could be improved by moving Wt+1 a little bit toward at. But the old constraint requires that (Wt . Wt+d 2: IIwtll2, and if Wt+1 = at Ytxtlllxtll2, this means that Wt . (YtxtlllxtIl2) 2: Ilwtll2. Rearranging, we get Yt(Wt . xd 2: IIxtll211wtlI2 > 0 (IIXtll > 0 follows from the fact that the data is linearly separable, and IIwt!\ > 0 follows from the fact that there was at least one previous mistake). But since trial t was a mistake, Yt (Wt . Xt) ~ 0, a contradiction. 0 = = Now we're ready to prove the mistake bound. Theorem 3 Choose mEN, and a sequence (Xl, Yd,???, (xm , Ym) of patternclassijicationpairsinRn x {-1,+1}. LetR = maxtl\xtli. Ifthereisaweightvector ii such that Yt (ii . Xt) 2: 1 for all 1 ~ t ~ m, then the number of mistakes made by ROMMA on (Xl, yd, .. . , (xm, Ym) is at most R211ii1l 2. Proof: First, we claim that for all t, ii E H t . This is easily seen since ii satisfies all the constraints that are ever imposed on a weight vector, and therefore all relaxations of such constraints. Since Wt is the smallest element of Ht. we have IIwtll ~ lliill. = = We have W2 Ylxdllid 2, and therefore IIw211 1/lIx1\\ 2:: 1/ R which implies IIw2112 2: 1/ R2. We claim that if any trial t > 1 is a mistake, then IIWt+1112 2: IIwtlI 2 + 1/ R2. This will imply by induction that after M mistakes, the squared length of the algorithm's weight vector is at least M / R2, which, since all of the algorithm's weight vectors are no longer than ii, will complete the proof. Choose an index t > 1 of a trial in which a mistake is made. Let At = {tV : Yt (w . it) = I} and B t = {w : (w . Wt) = IIwtIl2}. By Lemmas 1 and 2, Wt+1 EAt n B t . B The distance from Wt to At (call?it pe) satisfies IYt(xt . we) -11 Pt = I xtll 2: Figure 2: At, B t , and Pt 1 lIitll 2: 1 R' (3) since the fact that there was a mistake in trial t implies Yt(Xt . Wt) ~ O. Also, since Wt+1 E At. (4) 502 Y. Li and P. M. Long Because Wt is the normal vector of B t and Wt+1 E B t , we have + I Wt+1 - Wt1l 2. IIWt+d2 - IIWt 112 = IIWt+! IIWt+1112= IIWtll2 Thus, applying (3) and (4), we have which, as discussed above, completes the proof. - welI2 2: p; 2: 1/ R2, 0 Using the fact, easily proved using induction, that for all t, Pt ~ H t , we can easily prove the following, which complements analyses of the maximum margin algorithm using independence assumptions [1, 14, 12]. Details are omitted due to space constraints. Theorem 4 Choose mEN, and a sequence (x\, yd,"', (im , Ym) of patternclassification pairs in R n x {-I , +1}. Let R = maXt lIitli. If there is a weight vector ii such that Yt (ii . it) 2: 1 for all 1 ::; t ::; m, then the number of mistakes made by the ideal online maximum margin algorithm on (Xl, yd, .. " (xm, Ym) is at most R211ii1l 2. In the proof of Theorem 3, if an update is made and Yt (Wt . id < 1 - 0 instead of Yt (Wt . it) ::; 0, then the progress made can be seen to be at least 02/ R2. This can be applied to prove the following. Theorem 5 Choose 0 > 0, mEN, and a sequence (Xl, Y1) , ... , (X m, Ym) of patternclassification pairs in R n x {-I, + I}. Let R = maXt lIiell. If there is a weight vector ii such that Yt (ii . Xt) 2: 1 for aliI::; t ::; m, then if (i1' yI), ... , (im, Ym) are presented on line the number of trials in which aggressive ROMMA has Yt (Wt . it) < 1 - 0 is at most R2I1iiIl 2/0 2. Theorem 5 implies that, in a sense, repeatedly cycling through a dataset using aggressive ROMMA will eventually converge to SVM; note however that bias is not considered. 3 An efficient implementation When the prediction of ROMMA differs from the expected label, the algorithm chooses Wt+! to minimize IIWt+!1I subject to AWt+! = b, where A = (~f) and b = ( 11~:"2 ) . Simple calculation shows that Wt+! AT (AAT)-lb ("wtIl2(Yt - (Wt ? it)) ) ~ ( IIxtII211Wtll2 - Yt(Wt . it)) ~ lIitll211Wtll2 - (Wt . ie)2 Wt + IIxtll211Wtll2 _ (Wt . XtP Xt? If on trials t in which a mistake is made, Ct (5) = Since the computations required by ROMMA involve inner products together with a few operations on scalars, we can apply the kernel method to our algorithm, efficiently solving the original problem in a very high dimensional space. Computationally, we only need to modify the algorithm by replacing each inner product computation (ii . Xj) with a kernel function computation IC (ii, Xj). To make a prediction for the tth trial, the algorithm must compute the inner product between Xt and prediction vector Wt. In order to apply the kernel function, as in [1, 3], we store each prediction vector Wt in an implicit manner, as the weighted sum of examples on which 503 The Relaxed Online Maximum Margin Algorithm mistakes occur during the training. In particular. each Wt is represented as Wt = n t-l (t_ l ) (t-l) n IT WI + L djxj Cj J=1 J=1 C n=J+l (Wt+l The above formula may seem daunting; however, making use of the recurrence ?x) = Ct (Wt . x) + dt (Xt . x). it is obvious that the complexity of our new algorithm is similar to that of perceptron algorithm. This was born out by our experiments. The implementation for aggressive ROMMA is similar to the above. 4 Experiments We did some experiments using the ROMMA and aggressive ROMMA as batch algorithms on the MNIST OCR database. 2 We obtained a batch algorithm from our online algorithm in. the usual way, making a number of passes over the dataset and using the final weight vector to classify the test data. Every example in this database has two parts, the first is a 28 x 28 matrix which represents the image of the corresponding digit. Each entry in the matrix takes value from {O, . . . , 255}. The second part is a label taking a value from {O,? .. , g} . The dataset consists of 60, 000 training examples and 10,000 test examples. We adopt the following polynomial kernel: K(Xi, Xj) = (1 + (Xi? Xj))d. This corresponds to using an expanded collection of features including all products of at most d components of the original feature vector (see [14]). Let us refer to the mapping from the original feature vector to the expanded feature vector as <1>. Note that one component of <I> (x) is always 1, and therefore the component of the weight vector corresponding to that component can be viewed as a bias. In our experiments, we set WI = <1>(6') rather than (5 to speed up the learning of the coefficient corresponding to the bias. We chose d = 4 since in experiments on the same problem conducted in [3 , 2], the best results occur with this value. To cope with multiclass data, we trained ROMMA and aggressive ROMMA once for each of the 10 labels. Classification of an unknown pattern is done according to the maximum output of these ten classifiers. As every entry in the image matrix takes value from {O , ? .. , 255}, the order of magnitude of K(x , x) is at least 10 26 , which might cause round-off error in the computation of Ci and di . We scale the data by dividing each entry with 1100 when training with ROMMA. Table 1: Experimental results on MNIST data T=l Err MisNo percep voted-percep ROMMA agg-ROMMA agg-ROMMACNC) 2.84 2.26 2.48 2.14 2.05 7970 7970 7963 6077 5909 T=2 Err MisNo 2.27 1.88 1.96 1.82 1.76 10539 10539 9995 7391 6979 T=3 Err MisNo 1.99 1.76 1.79 1.71 1.67 11945 11945 10971 7901 7339 T =4 Err MisNo 1.85 1.69 1.77 1.67 1.63 12800 12800 11547 8139 7484 Since the performance of online learning is affected by the order of sample sequence, all the results shown in Table 1 average over 10 random permutations. The columns marked 2National Institute for Standards and Technology, special database http://www.research.att.com/... yanniocr for information on obtaining this dataset. 3. See 504 Y Li and P M. Long "MisNo" in Table 1 show the total number of mistakes made during the training for the 10 labels. Although online learning would involve only one epoch, we present results for a batch setting until four epochs (T in Table 1 represents the number of epochs). To deal with data which are linearly inseparable in the feature space, and also to improve generalization, Friess et al [4] suggested the use of quadratic penalty in the cost function, which can be implemented using a slightly different kernel function [4, 5]: iC(Xk ' Xj) = K(Xk, Xj) + c5kj ).., where c5kj is the Kronecker delta function. The last row in Table 1 is the result of aggressive ROMMA using this method to control noise ().. = 30 for 10 classifiers). We conducted three groups of experiments, one for the perceptron algorithm (denoted "percep"), the second for the voted perceptron (denoted "voted-percep") whose description is in [3], the third for ROMMA, aggressive ROMMA (denoted "agg-ROMMA"), and aggressive ROMMA with noise control (denoted "agg-ROMMA(NC)"). Data in the third group are scaled. All three groups set 'lih = <1>(0). The results in Table 1 demonstrate that ROMMA has better performance than the standard perceptron, aggressive ROMMA has slightly better performance than the voted perceptron. Aggressive ROMMA with noise control should not be compared with perceptrons without noise control. Its presentation is used to show what performance our new online algorithm could achieve (of course it's not the best, since all 10 classifiers use the same )"). A remarkable phenomenon is that our new algorithm behaves very well at the first two epochs. References [1] B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. Proceedings of the 1992 Workshop on Computational Learning Theory, pages 144-152, 1992. [2] C. Cortes and V. Vapnik. Support-vector networks. Machine Learning, 20(3):273-297,1995. [3] y. Freund and R. E. Schapire. Large margin classification using the perceptron algorithm. Proceedings of the 1998 Conference on Computational Learning Theory, 1998. [4] T. T. Friess, N. Cristianini, and C. Campbell. The kernel adatron algorithm: a fast and simple learning procedure for support vector machines. In Proc. 15th Int. Con! on Machine Learning. Morgan Kaufman Publishers, 1998. [5] S. S. Keerthi, S. K. Shevade, C. Bhattacharyya, and K. R. K. Murthy. A fast iterative nearest point algorithm for support vector machine c1assiifer design. Technical report, Indian Institute of Science, 99. TR-ISL-99-03 . [6] Adam Kowalczyk. Maximal margin perceptron. In Smola, Bartlett, Scholkopf, and Schuurmans, editors, Advances in Large Margin Classifiers, 1999. MIT-Press. [7] N. Littlestone. Learning quickly when irrelevant attributes abound: a new linear-threshold algorithm. Machine Learning, 2:285-318, 1988. [8] N. Littlestone. Mistake Bounds and Logarithmic Linear-threshold Learning Algorithms. PhD thesis, UC Santa Cruz, 1989. [9] John C. Platt. Fast training of support vector machines using sequential minimal optimization. In B. Scholkopf, C. Burges, A. Smola, editors, Advances in Kernel Methods: Support Vector Machines , 1998. MIT Press. [10] F. Rosenblatt. The perceptron: A probabilistic model for information storage and organization in the brain. Psychological Review, 65:386-407, 1958. [11] F. Rosenblatt. Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms . Spartan Books, Washington, D. C., 1962. [12] J. Shawe-Taylor, P. Bartlett, R. Williamson, and M. Anthony. A framework for structural risk minimization. In Proc. of the 1996 Conference on Computational Learning Theory, 1996. [13] V. N. Vapnik. Estimation of Dependencies based on Empirical Data. Springer Verlag, 1982. [14] V. N. Vapnik. The Nature of Statistical Learning Theory . Springer, 1995. 

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Wiring optimization in the brain Dmitri B. Chklovskii Sloan Center for Theoretical Neurobiology The Salk Institute La Jolla, CA 92037 mitya@salk.edu Charles F. Stevens Howard Hughes Medical Institute and Molecular Neurobiology Lab The Salk Institute La Jolla, CA 92037 stevens@salk.edu Abstract The complexity of cortical circuits may be characterized by the number of synapses per neuron. We study the dependence of complexity on the fraction of the cortical volume that is made up of "wire" (that is, ofaxons and dendrites), and find that complexity is maximized when wire takes up about 60% of the cortical volume. This prediction is in good agreement with experimental observations. A consequence of our arguments is that any rearrangement of neurons that takes more wire would sacrifice computational power. Wiring a brain presents formidable problems because of the extremely large number of connections: a microliter of cortex contains approximately 105 neurons, 109 synapses, and 4 km ofaxons, with 60% of the cortical volume being taken up with "wire", half of this by axons and the other half by dendrites. [ 1] Each cortical neighborhood must have exactly the right balance of components; if too many cell bodies were present in a particular mm cube, for example, insufficient space would remain for the axons, dendrites and synapses. Here we ask "What fraction of the cortical volume should be wires (axons + dendrites)?" We argue that physiological properties ofaxons and dendrites dictate an optimal wire fraction of 0.6, just what is actually observed. To calculate the optimal wire fraction, we start with a real cortical region containing a fixed number of neurons, a mm cube, for example, and imagine perturbing it by adding or subtracting synapses and the axons and dendrites needed to support them. The rules for perturbing the cortical cube require that the existing circuit connections and function remain intact (except for what may have been removed in the perturbation), that no holes are created, and that all added (or subtracted) synapses are typical of those present; as wire volume is added, the volume of the cube of course increases. The ratio of the number of synapses per neuron in the perturbed cortex to that in the real cortex is denoted by 8, a parameter we call the relative complexity. We require that the volume of non-wire components (cell bodies, blood vessels, glia, etc) is unchanged by our perturbation and use 4> to denote the volume fraction of the perturbed cortical region that is made up of wires (axons + dendrites; 4> can vary between zero and one), with the fraction for the real brain being 4>0. The relation between relative complexity 8 and wire volume fraction 4> is given by the equation (derived in Methods) 1 (1-4?2/34> -,\5 1 - 4>0 4>0? 8-- (I) D. B. Chklovskii and C. F Stevens 104 2 -- :\ = .9 ( I) ?1 )( ..!! Q.. :\=1 1 E o u O~----r---~-----.----~--~ 0.0 0.2 0.4 0.6 0.8 1.0 Wire fraction (+) Figure I: Relative complexity (0) as a function of volume wire fraction (e/?. The graphs are calculated from equation (1) for three values of the parameter A as indicated; this parameter determines the average length of wire associated with a synapse (relative to this length for the real cortex, for which (A = 1). Note that as the average length of wire per synapse increases, the maximum possible complexity decreases. For the following discussion assume that A = 1; we return to the meaning of this parameter later. To derive this equation two assumptions are made. First, we suppose that each added synapse requires extra wire equal to the average wire length and volume per synapse in the unperturbed cortex. Second, because adding wire for new synapses increases the brain volume and therefore increases the distance axons and dendrites must travel to maintain the connections they make in the real cortex, all of the dendrite and unmyelinated axon diameters are increased in proportion to the square of their length changes in order to maintain the intersynaptic conduction times[2] and dendrite cable lengths[3] as they are in the actual cortex. If the unmyelinated axon diameters were not increased as the axons become longer, for example, the time for a nerve impulse to propagate from one synapse to the next would be increased and we would violate our rule that the existing circuit and its function be unchanged. We note that the vast majority of cortical axons are unmyelinated.[l] The plot of as a function of e/> is parabolic-like (see Figure 1) with a maximum value at e/> = 0.6, a point at which dO/de/> = O. This same maximum value is found for any possible value of e/>o, the real cortical wire fraction. o Why does complexity reach a maximum value at a particular wire fraction? When wire and synapses are added, a series of consequences can lead to a runaway situation we call the wiring catastrophe. If we start with a wire fraction less than 0.6, adding wire increases the cortical volume, increased volume makes longer paths for axons to reach their targets which requires larger diameter wires (to keep conduction delays or cable attenuation constant from one point to another), the larger wire diameters increase cortex volume which means wires must be longer, etc. While the wire fraction e/> is less than 0.6, increasing complexity is accompanied by finite increases in e/>. At e/> = 0.6 the rate at which wire fraction increases with complexity becomes infinite de/>/dO --t 00); we have reached the wiring catastrophe. At this point, adding wire becomes impossible without decreasing complexity or making other changes - like decreasing axon diameters - that alter cortical function. The physical cause of the catastrophe is a slow growth of conduction velocity and dendritic cable length with diameter combined with the requirement that the conduction times between synapses (and dendrite cable lengths) be unchanged in the perturbed cortex. We assumed above that each synapse requires a certain amount of wire, but what if we could 105 Wiring Optimization in the Brain add new synapses using the wire already present? We do not know what factors determine the wire volume needed to support a synapse, but if the average amount of wire per synapse could be less (or more) than that in the actual cortex, the maximum wire fraction would still be 0.6. Each curve in Figure 1 corresponds to a different assumed average wire length required for a synapse (determined by A), and the maximum always occurs at 0.6 independent of A. In the following we consider only situations in which A is fixed. For a given A, what complexity should we expect for the actual cortex? Three arguments favor the maximum possible complexity. The greatest complexity gives the largest number of synapses per neuron and this permits more bits of information to be represented per neuron. Also, more synapses per neuron decreases the relative effect caused by the loss or malfunction of a single synapse. Finally, errors in the local wire fraction would minimally affect the local complexity because d(} / dqJ = 0 at if> = 0.6. Thus one can understand why the actual cortex has the wire fraction we identify as optimal. [ 1] This conclusion that the wire fraction is a maximum in the real cortex has an interesting consequence: components of an actual cortical circuit cannot be rearranged in a way that needs more wire without eliminating synapses or reducing wire diameters. For example, if intermixing the cell bodies of left and right eye cells in primate primary visual cortex (rather than separating them in ocular dominance columns) increased the average length of the wire[4] the existing circuit could not be maintained just by a finite increase in volume. This happens because a greater wire length demanded by the rearrangement of the same circuit would require longer wire per synapse, that is, an increased A. As can be seen from Figure I, brains with A > 1 can never achieve the complexity reached at the maximum of the A = 1 curve that corresponds to the actual cortex. Our observations support the notion that brains are arranged to minimize wire length. This idea, dating back to Cajal[5], has recently been used to explain why retinotopic maps exist[6],[7], why cortical regions are separated, why ocular dominance columns are present in primary visual cortex[4],[8],[9] and why the cortical areas and flat worm ganglia are placed as they are. [ 10-13] We anticipate that maximal complexity/minimal wire length arguments will find further application in relating functional and anatomical properties of brain. Methods The volume of the cube of cortex we perturb is V, the volume of the non-wire portion is W (assumed to be constant), the fraction of V consisting of wires is if>, the total number of synapses is N, the average length of axonal wire associated with each synapse is s, and the average axonal wire volume per unit length is h; the corresponding values for dendrites are indicated by primes (s' and h'). The unperturbed value for each variable has a 0 subscript; thus the volume of the cortical cube before it is perturbed is Vo = Wo + No(soho + s~h~). (2) We now define a "virtual" perturbation that we use to explore the extent to which the actual cortical region contains an optimal fraction of wire. If we increase the number of synapses by a factor () and the length of wire associated with each synapse by a factor A, then the perturbed cortical cube's volume becomes Vo = Wo + A(} (Nosoho ~ + Nos~h~ ~~) (V/VO)1/3 . (3) This equation allows for the possibility that the average wire diameter has been perturbed and increases the length of all wire segments by the "mean field" quantity (V/VO)1/3 to take account of the expansion of the cube by the added wire; we require our perturbation disperses the added wire as uniformly as possible throughout the cortical cube. D. B. Chklovskii and C. F. Stevens 106 To simplify this relation we must eliminate h/ ho and h' / h&; we consider these terms in tum. When we perturb the brain we require that the average conduction time (s/u, where u is the conduction velocity) from one synapse to the next be unchanged so that s/u so/uo, or = ~ uo = !.... = '\so(V/VO)1/3 = '\(V/VO)1/3. So (4) So Because axon diameter is proportional to the square of conduction velocity u and the axon volume per unit length h is proportional to diameter squared, h is proportional to u 4 and the ratio h/ ho can be written as (5) For dendrites, we require that their length from one synapse to the next in units of the cable length constant be unchanged by the perturbation. The dendritic length constant is proportional to the square root of the dendritic diameter d, so 8/.fd = 80/ ViIO or ~ = (;~)2 = (,\(V/VO)1/3f =,\2(V/Vo)2/3. (6) Because dendritic volumes per unit length (h and h') vary as the square of the diameters, we have that ~~ = (~) 2 =,\4 (V/VO)4/3. (7) The equation (2) can thus be rewritten as V = Wo + No(soh o + s~h~)B,\5 (V/Vo)5/3 . (8) Divide this equation by Vo, define v = VIVo, and recognize that Wo/Vo that 4>0 = No(soho + s&h&)/Vo ; the result is = (1 - 4>0) and (9) Because the non-wire volume is required not to change with the perturbation, we know that 4?; substitute this in equation (9) and rearrange to give Wo = (1 - 4>o)Vo = (1 - 4?V which means that v = (1 - 4>0)/(1 - B = ~ ( 1,\5 4> ) 2/3 .!t 1 - 4>0 4>0 . (1) the equation used in the main text. We have assumed that conduction velocity and the dendritic cable length constant vary exactly with the square root of diameter[2],[ 14] but if the actual power were to deviate slightly from 112 the wire fraction that gives the maximum complexity would also differ slightly from 0.6. Acknowledgments This work was supported by the Howard Hughes Medical Institute and a grant from NIH to C.F.S. D.C. was supported by a Sloan Fellowship in Theoretical Neurobiology. Wiring Optimization in the Brain 107 References [1] Braitenberg. V. & Schuz. A. Cortex: Statistics and Geometry of Neuronal Connectivity (Springer. 1998). [2] Rushton. W.A.H. A Theory of the Effects of Fibre Size in Medullated Nerve. J. Physiol. 115. 101-122 (1951). [3] Bekkers. J.M. & Stevens. C.F. Two different ways evolution makes neurons larger. Prog Brain Res 83. 37-45 (1990). [4] Mitchison. G. Neuronal branching patterns and the economy of cortical wiring. Proc R Soc Lond B Bioi Sci 245.151-158 (1991). [5] Cajal. S.R.Y. Histology of the Nervous System 1-805 (Oxford University Press, 1995). [6] Cowey. A. Cortical maps and visual perception: the Grindley Memorial Lecture. Q J Exp Phychol 31. 1-17 (1979). [7] Allman J.M. & Kaas J.H. The organization of the second visual area (V II) in the owl monkey: a second order transformation of the visual hemifield. Brain Res 76: 247-65 (1974). [8] Durbin. R. & Mitchison. G. A dimension reduction framework for understanding cortical maps. Nature 343. 644-647 (1990). [9] Mitchison. G. Axonal trees and cortical architecture. Trends Neurosci 15. 122-126 (1992). [10] Young. M.P. Objective analysis of the topological organization of the primate cortical visual system. Nature 358. 152-154 (1992). [11] Cherniak. C. Loca1 optimization of neuron arbors. Bioi Cybern 66. 503-510 (1992). [12] Cherniak. C. Component placement optimization in the brain. J Neurosci 14.2418-2427 (1994). [13J Cherniak. C. Neural component placement. Trends Neurosci 18.522-527 (1995). [14J Rall. W. in Handbook of Physiology, The Nervous Systems, Cellular Biology of Neurons (ed. Brookhart. J.M.M .? V.B.) 39-97 (Am. Physiol. Soc .? Bethesda. MD. 1977). 

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Topographic Transformation as a Discrete Latent Variable Nebojsa Jojic Beckman Institute University of Illinois at Urbana www.ifp.uiuc.edu/",jojic Brendan J. Frey Computer Science University of Waterloo www.cs.uwaterloo.ca/ ... frey Abstract Invariance to topographic transformations such as translation and shearing in an image has been successfully incorporated into feedforward mechanisms, e.g., "convolutional neural networks", "tangent propagation". We describe a way to add transformation invariance to a generative density model by approximating the nonlinear transformation manifold by a discrete set of transformations. An EM algorithm for the original model can be extended to the new model by computing expectations over the set of transformations. We show how to add a discrete transformation variable to Gaussian mixture modeling, factor analysis and mixtures of factor analysis. We give results on filtering microscopy images, face and facial pose clustering, and handwritten digit modeling and recognition. 1 Introduction Imagine what happens to the point in the N-dimensional space corresponding to an N-pixel image of an object, while the object is deformed by shearing. A very small amount of shearing will move the point only slightly, so deforming the object by shearing will trace a continuous curve in the space of pixel intensities. As illustrated in Fig. la, extensive levels of shearing will produce a highly nonlinear curve (consider shearing a thin vertical line ), although the curve can be approximated by a straight line locally. Linear approximations of the transformation manifold have been used to significantly improve the performance of feedforward discriminative classifiers such as nearest neighbors (Simard et al., 1993) and multilayer perceptrons (Simard et al., 1992). Linear generative models (factor analysis, mixtures of factor analysis) have also been modified using linear approximations of the transformation manifold to build in some degree of transformation invariance (Hinton et al., 1997). In general, the linear approximation is accurate for transformations that couple neighboring pixels, but is inaccurate for transformations that couple nonneighboring pixels. In some applications (e.g., handwritten digit recognition), the input can be blurred so that the linear approximation becomes more robust. For significant levels of transformation, the nonlinear manifold can be better modeled using a discrete approximation. For example, the curve in Fig. 1a can be N. Jojic and B. J. Frey 478 (b) (c) (e) (d) p(z) ~ Figure 1: (a) An N-pixel greyscale image is represented by a point (unfilled disc) in an Ndimensional space. When the object being imaged is deformed by shearing. the point moves along a continuous curve. Locally. the curve is linear. but high levels of shearing produce a highly nonlinear curve. which we approximate by discrete points (filled discs) indexed bye. (b) A graphical model showing how a discrete transformation variable can be added to a density model p(z) for a latent image z to model the observed image x . The Gaussian pdf p(xle, z) captures the eth transformation plus a small amount of pixel noise. (We use a box to represent variables that have Gaussian conditional pdfs.) We have explored (c) transformed mixtures of Gaussians. where c is a discrete cluster index; (d) transformed component analysis (TeA). where y is a vector of Gaussian factors. some of which may model locally linear transformation perturbations; and (e) mixtures of transformed component analyzers. or transformed mixtures of factor analyzers. e represented by a set of points (filled discs). In this approach, a discrete set of possible transformations is specified beforehand and parameters are learned so that the model is invariant to the set of transformations. This approach has been used to design "convolutional neural networks" that are invariant to translation (Le Cun et al., 1998) and to develop a general purpose learning algorithm for generative topographic maps (Bishop et al., 1998) . We describe how invariance to a discrete set of known transformations (like translation) can be built into a generative density model and we show how an EM algorithm for the original density model can be extended to the new model by computing expectations over the set of transformations. We give results for 5 different types of experiment involving translation and shearing. 2 Transformation as a Discrete Latent Variable We represent transformation f by a sparse transformation generating matrix G e that operates on a vector of pixel intensities. For example, integer-pixel translations of an image can be represented by permutation matrices. Although other types of transformation matrix may not be accurately represented by permutation matrices, many useful types of transformation can be represented by sparse transformation matrices. For example, rotation and blurring can be represented by matrices that have a small number of nonzero elements per row (e.g., at most 6 for rotations). The observed image x is linked to the nontransformed latent image z and the transformation index f E {I, ... , L} as follows: p(xlf, z) = N(x; Gez, w), (1) where W is a diagonal matrix of pixel noise variances. Since the probability of a transformation may depend on the latent image, the joint distribution over the latent image z, the transformation index f and the observed image x is p(x, f, z) = N(x; Gez, w)P(flz)p(z). (2) The corresponding graphical model is shown in Fig. lb. For example, to model noisy transformed images of just one shape, we choose p(z) to be a Gaussian distribution. Topographic Transformation as a Discrete Latent Variable 479 2.1 Transformed mixtures of Gaussians (TMG). Fig. lc shows the graphical model for a TMG, where different clusters may havp. different transformation probabilities. Cluster c has mixing proportion 7rc , mean /-t c and diagonal covariance matrix ~ c. The joint distribution is (3) where the probability of transformation f for cluster c is Plc. Marginalizing over the latent image gives the cluster/transformation conditional likelihood, (4) which can be used to compute p(x) and the cluster/transformation responsibility P(f, clx). This likelihood looks like the likelihood for a mixture of factor analyzers (Ghahramani and Hinton, 1997). However, whereas the likelihood computation for N latent pixels takes order N 3 time in a mixture of factor analyzers, it takes linear time, order N, in a TMG, because Gl~cG'I + W is sparse. 2.2 Transformed component analysis (TCA). Fig. Id shows the graphical model for TCA (or "transformed factor analysis"). The latent image is modeled using linearly combined Gaussian factors, y. The joint distribution is p(x, f, z, y) = N(x; Glz, w)N(z; /-t + Ay, ~ )N(y; 0, I)Pl, (5) where /-t is the mean of the latent image, A is a matrix of latent image components (the factor loading matrix) and ~ is a diagonal noise covariance matrix for the latent image. Marginalizing over the factors and the latent image gives the transformation conditional likelihood, p(xlf) = N(x; Gl/-t, Gl(AA T + ~)G'I + w), (6) which can be used to compute p(x) and the transformation responsibility p(flx). + ~)G'I is not sparse, so computing this likelihood exactly takes N 3 time. However, the likelihood can be computed in linear time if we assume IGl(AA T + f))G'I + wi ~ IGl(AAT + ~)G'II, which corresponds to assuming that the observed noise is smaller than the variation due to the latent image, or that the observed noise is accounted for by the latent noise model, ~. In our experiments, this approximation did not lead to degenerate behavior and produced useful models. Gl(AA T By setting columns of A equal to the derivatives of /-t with respect to continuous transformation parameters, a TCA can accommodate both a local linear approximation and a discrete approximation to the transformation manifold. 2.3 Mixtures of transformed component analyzers (MTCA). A combination of a TMG and a TCA can be used to jointly model clusters, linear components and transformations. Alternatively, a mixture of Gaussians that is invariant to a discrete set of transformations and locally linear transformations can be obtained by combining a TMG with a TCA whose components are all set equal to transformation derivatives. The joint distribution for the combined model in Fig. Ie is p(x, f, z, c, y) = N(x; GlZ, w)N(z; /-t c + AcY, ~c)N(y; 0, I)Plc7rc. The cluster/transformation likelihood is p(xlf,c) = N(X;Gl/-tc,Gl(AcA; ~c)G'I + w), which can be approximated in linear time as for TCA. (7) + 480 3 N. Jojic and B. J. Frey Mixed Transformed Component Analysis (MTCA) We present an EM algorithm for MTCA; EM algorithms for TMG or TCA emerge by setting the number of factors to 0 or setting the number of clusters to 1. Let 0 represent a parameter in the generative model. For Li.d. data, the derivative of the log-likelihood of a training set Xl, ... ,XT with respect to 0 can be written 810gp(XI80, ... ,XT) IJ)I ] = ~ E [8 80 logp(xt, c, (., Z, Y Xt , T '"' (8) t=l where the expectation is taken over p(c, f, z, ylxt). The EM algorithm iteratively solves for a new set of parameters using the old parameters to compute the expectations. This procedure consistently increases the likelihood of the training data. By setting (8) to 0 and solving for the new parameter values, we obtain update equations based on the expectations given in the Appendix. Notation: (-) = .!. Ei=l (.) is a sufficient statistic computed by averaging over the training set; diag(A) gives a vector containing the diagonal elements of matrix A; diag(a) gives a diagonal matrix whose diagonal contains the elements of vector a; and a 0 h gives the element-wise product of vectors a and h. Denoting the updated parameters by "-", we have ire = (P(cJXt)), he = (P(flxt, c)), (9) _ (P(clxt)E[z - AeyIXt,c]) J.?e = (P(cIXt)) , (10) diag( (P( cIXt)E[(z - J.?e - AeY) 0 (z - J.?e - Aey)IXt, cD) ~e = (P(cIXt)) , (11) ~ = diag( (E[(Xt -GiZ)O(Xt - Giz)IXtD), (12) - Ae = (P(cJxdE[(z - J.?e)yTlxtl)(P(cIXt)E[yyTlxtD-I. (13) To reduce the number of parameters, we will sometimes assume Pic does not depend on c or even that Pic is held constant at a uniform distribution. 4 Experiments 4.1 Filtering Images from a Scanning Electron Microscope (SEM). SEM images (e.g., Fig. 2a) can have a very low signal to noise ratio due to a high variance in electron emission rate and modulation of this variance by the imaged material (Golem and Cohen, 1998). To reduce noise, multiple images are usually averaged and the pixel variances can be used to estimate certainty in rendered structures. Fig. 2b shows the estimated means and variances of the pixels from 230 140 x 56 SEM images like the ones in Fig. 2a. In fact, averaging images does not take into account spatial uncertainties and filtering in the imaging process introduced by the electron detectors and the high-speed electrical circuits. We trained a single-cluster TMG with 5 horizontal shifts and 5 vertical shifts on the 230 SEM images using 30 iterations of EM. To keep the number of parameters almost equal to the number of parameters estimated using simple averaging, the transformation probabilities were not learned and the pixel variances in the observed image were set equal after each M step. So, TMG had 1 more parameter. Fig. 2c shows the mean and variance learned by the TMG. Compared to simple averaging, the TMG finds sharper, more detailed structure. The variances are significantly lower, indicating that the TMG produces a more confident estimate of the image. 481 Topographic Transformation as a Discrete Latent Variable (a) (b) (e) Figure 2: (a) 140 x 56 pixel SEM images. (b) The mean and variance of the image pixels. (c) The mean and variance found by a TMG reveal more structure and less uncertainty. (a) -.- f I .~ ,. !. I\:~ .. --- f= , ... VI -t., -, --i..--.-~.. -.. .... , ~ -~_1~_. "'t~ - . ~ r f :s.- .,f~ ,.. -' tr I. .. ~ ~ <,.\ Sf' ~ ~'-,fll" .."iIIio . . !~V '. ~ .. ~;'''; ..,,,,-L ~- r . .. . -.~ ~ ~ . ,.?:" ~I ,.". , .:p. ~ f" ...-" . . . ,-,;I iI' '.j (d) t l~ ijIt . , . .l ...... . :or -. ,. ,,~: y: .", . .'-?, FO ,.,io ~ ~ ! .. .~ .<t --.,j .jIo ;.'~. ....# ... r; ~. ..,. .,. ..... ~ .~. ...... O ' J .. ...,.. ... ........ ~ . .', , ,:.~ ? (e) (b) (f) (e) (g) Figure 3: (a) Frontal face images of two people. (b) Cluster means learned by a TMG and (c) a mixture of Gaussians. (d) Images of one person with different poses. (e) Cluster means learned by a TMG. (f) Less detailed cluster means learned by a mixture of Gaussians. (g) Mean and first 4 principal components of the data. which mostly model lighting and translation. 4.2 Clustering Faces and Poses. Fig. 3a shows examples from a training set of 400 jerky images of two people walking across a cluttered background. We trained a TMG with 4 clusters, 11 horizontal shifts and 11 vertical shifts using 15 iterations of EM after initializing the weights to small, random values. The loop-rich MATLAB script executed in 40 minutes on a 500MHz Pentium processor. Fig. 3b shows the cluster means, which include two sharp representations of each person's face, with the background clutter suppressed. Fig. 3c shows the much blurrier means for a mixture of Gaussians trained using 15 iterations of EM. Fig. 3d shows examples from a training set of 400 jerky images of one person with different poses. We trained a TMG with 5 clusters, 11 horizontal shifts and 11 vertical shifts using 40 iterations of EM. Fig. 3e shows the cluster means, which capture 4 poses and mostly suppress the background clutter. The mean for cluster 4 includes part of the background, but this cluster also has a low mixing proportion of 0.1. A traditional mixture of Gaussians trained using 40 iterations of EM finds blurrier means, as shown in Fig. 3f. The first 4 principal components mostly try to account for lighting and translation, as shown in Fig. 3g. 482 N Jojic and B. J. Frey (d) (e) (f) Figure 4: Modeling handwritten digits. (a) Means and components and (b) the sheared + translated means (dimmed transformations have low probability) for each of 10 TCA models trained on 200 examples of each digit. (c) Means and components of 10 FA models trained on the same data. (d) Digits generated from the 10 TCA models and (e) the 10 FA models. (f) The means for a mixture of 10 Gaussians, a mixture of 10 factor analyzers and a 10-ciuster TMG trained on all 2000 digits. In each case, the best of 10 experiments was selected. 4.3 Modeling Handwritten Digits. We performed both supervised and unsupervised learning experiments on 8 x 8 greyscale versions of 2000 digits from the CEDAR CDROM (Hull, 1994). Although the preprocessed images fit snugly in the 8 x 8 window, there is wide variation in "writing angle" (e.g., the vertical stroke of the 7 is at different angles). So, we produced a set of 29 shearing+translation transformations (see the top row of Fig. 4b) to use in transformed density models. In our supervised learning experiments, we trained one 10-component TCA on each class of digit using 30 iterations of EM. Fig. 4a shows the mean and 10 components for each of the 10 models. The lower 10 rows of images in Fig. 4b show the sheared and translated means. In cases where the transformation probability is below 1%, the image is dimmed. We also trained one lO-component factor analyzer on each class of digit using 30 iterations of EM. The means and components are shown in Fig. 4c. The means found by TCA are sharper and whereas the components found by factor analysis often account for writing angle (e.g., see the components for 7) the components found by TCA tend to account for line thickness and arc size. Fig. 4d and e show digits that were randomly generated from the TCAs and the factor analyzers. Since different components in the factor analyzers account for different stroke angles, the simulated digits often have an extra stroke, whereas digits simulated from the TCAs contain fewer spurious strokes. To test recognition performance, we trained 10-component factor analyzers and TCAs on 200 examples of each digit using 50 iterations of EM. Each set of models used Bayes rule to classify 1000 test patterns and while factor analysis gave an error rate of 3.2%, TCA gave an error rate of only 2.7%. In our unsupervised learning experiments, we fit 10-cluster mixture models to the entire set of 2000 digits to see which models could identify all 10 digits. We tried a mixture of 10 Gaussians, a mixture of 10 factor analyzers and a lO-cluster TMG. In each case, 10 models were trained using 100 iterations of EM and the model with 483 Topographic Transformation as a Discrete Latent Variable the highest likelihood was selected and is shown in Fig. 4f. Compared to the TMG, the first two methods found blurred and repeated classes. After identifying each cluster with its most prevalent class of digit, we found that the first two methods had error rates of 53% and 49%, but the TMG had a much lower error rate of 26%. 5 Summary In many learning applications, we know beforehand that the data includes transformations of an easily specified nature (e.g., shearing of digit images). If a generative density model is learned from the data, the model must extract a model of both the transformations and the more interesting and potentially useful structure. We described a way to add transformation invariance to a generative density model by approximating the transformation manifold with a discrete set of points. This releases the generative model from needing to model the transformations. 5 different types of experiment show that the method is effective and quite efficient. Although the time needed by this method scales exponentially with the dimensionality of the transformation manifold, we believe that it will be useful in many practical applications and that it illustrates what is possible with a generative model that incorporates a latent transformation variable. We are exploring the performance of a faster variational learning method and extending the model to time series. Acknowledgements. We used CITO, NSERC, NSF and Beckman Foundation grants. References C. M. Bishop, M. Svensen and C. K. I. Williams 1998. GTM: The generative topographic mapping. Neural Computation 10:1, 215- 235 . G. E. Hinton, P. Dayan and M. Revow 1997. Modeling the manifolds of images of handwritten digits. IEEE 1rans. on Neural Networks 8, 65- 74. Z. Ghahramani and G. E. Hinton 1997. The EM algorithm for mixtures of factor analyzers. University of Toronto Technical Report CRG-TR-96-1. Available at www.gatsby.ucl.ac.uk/ ... zoubin. R. Golem and I. Cohen 1998. Scanning electron microscope image enhancement. School of Computer and Electrical Engineering project report, Ben-Gurion University. J. J . Hull 1994. A database for handwritten text recognition research. IEEE 1rans. on Pattern Analysis and Machine Intelligence 16:5, 550-554. Y. Le Cun, L. Bottou, Y. Bengio and P. Haffner 1998. Gradient-based learning applied to document recognition. Proceedings of the IEEE 86:11, November, 2278-2324. P. Y. Simard, B. Victorri, Y. Le Cun and J. Denker 1992. Tangent Prop - A formalism for specifying selected invariances in an adaptive network. In Advances in Neural Information Processing Systems 4, Morgan Kaufmann, San Mateo, CA. P. Y. Simard, Y. Le Cun and J. Denker 1993. Efficient pattern recognition using a new transformation distance. In S. J. Hanson, J . D. Cowan and C. L. Giles, Advances in Neural Information Processing Systems 5, Morgan Kaufmann, San Mateo, CA. Appendix: The Sufficient Statistics Found in the E-Step The sufficient statistics for the M-Step are computed in the E-Step using sparse linear algebra during a single pass through the training set. Before making this pass, the following matrices are computed: Ot,c = COV(zlx,y,l,c) = (~;l +G~-?-lGt)-\ (3t,c = COV(ylx,l,c) = (I+A~~;JAc? A~~;lOt.c~;lAc)-l. For each case in the training set, P(c,llxt) is first computed for each combination of c, l, before computing E[ylxt,l, c] = {3l,cA~ ~;;-l [Ot ,cGi-?-lXt - (I-Ot,c~;;-l )I-'c]' E[zlxt, l, c] = I-'c +Ot,cG~-?-l (Xt - Gtl-'c) +Ot~-l Ac{3t,cA~ ~;;-lOt , cG~-?-J (Xt - Gtl-'c), E[(2J-I.'c}:(2J-I.'c) IXt ,l, c] = (E[zlxt ,l,cH.&c}c(E[zlxt ,l,cH.&c)+diag(Ot,c)+diag(Ol , c~;l Ac{3t,cA~ ~;lOt,c), E[(z-l-'c)y'lxt , l, c] = (E[zlxt,l, c]-l-'c)E[Ylxt,l, c]' + Ot,c~;l A c{3t ,c' The expectations needed in (10)-(13) are then computed from P(clxt}E[z - AcylXt, c] = Et P(c, lIXt)(E[zIXt, l, c] - AcE[Ylxt,l, cD, P(clxt}E[(z-l-'cAcy)o(z-l-'c-AcY) IXt, c] = Et P(c, llxt} {E[(z-l-'c)o(z-I-'C>lxt,l, c] +diag(Acl3t,cA~) - 2diag(AcE[(zl-'C>y'lxt, l, +(AcE[Ylxt, e, c])o(AcE[ylxt, e, cD}, E[(Xt-Gtz)o(Xt-Gtz)lxtl Ec,t P(c,lIXt) {(XtGtE[zlxt, l, c]) 0 (Xt - GtE[zIXt, l, c]) + diag(GtOt,cGi) + diag(GtOt,c~;l Ac{3t,cA~~;lOt,cG~)}, P(clxt}E[(z-l-')y'lxt, c] = El P(c,llxt}E[(z-l-')y'lxt, l, c], P(clxt)E[yy'lxt, c] = Et P(c,llxt}{3t,c + Et P(c, llxt)E[ylxt.e, c]E[ylxt.l, c]'. en = 

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356 USING BACKPROPAGATION WITH TEMPORAL WINDOWS TO LEARN THE DYNAMICS OF THE CMU DIRECT-DRIVE ARM II K. Y. Goldberg and B. A. Pearlmutter School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 ABSTRACT Computing the inverse dynamics of a robot ann is an active area of research in the control literature. We hope to learn the inverse dynamics by training a neural network on the measured response of a physical ann. The input to the network is a temporal window of measured positions; output is a vector of torques. We train the network on data measured from the first two joints of the CMU Direct-Drive Arm II as it moves through a randomly-generated sample of "pick-and-place" trajectories. We then test generalization with a new trajectory and compare its output with the torque measured at the physical arm. The network is shown to generalize with a root mean square error/standard deviation (RMSS) of 0.10. We interpreted the weights of the network in tenns of the velocity and acceleration filters used in conventional control theory. INTRODUCTION Dynamics is the study of forces. The dynamic response of a robot arm relates joint torques to the position, velocity, and acceleration of its links. In order to control an ann at high spee<L it is important to model this interaction. In practice however, the dynamic response is extremely difficult to predict. A dynamic controller for a robot ann is shown in Figure 1. Feedforward torques for a desired trajectory are computed off-line using a model of arm dynamics and applied to the joints at every cycle in an effort to linearize the resulting system. An independent feedback loop at each joint is used to correct remaining errors and compensate for external disturbances. See ([3]) for an introduction to dynamiC control of robot arms. Conventional control theory has difficulty addressing physical effects such as friction [I], backlash [2], torque non-linearity (especially dead zone and saturation) [2], highfrequency dynamics [2], sampling effects [7], and sensor noise [7]. We propose to use a three-layer backpropagation network [4] with sigmoid thresholds to fill the box marked "inverse arm" in Figure 1. We will treat the arm as an unknown non-linear transfer function to be represented by weights of the network. 

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Perceptual Organization Based on Temporal Dynamics Xiuwen Liu and DeLiang L. Wang Department of Computer and Information Science Center for Cognitive Science The Ohio State University, Columbus, OR 43210-1277 Email: {liux, dwang}@cis.ohio-state.edu Abstract A figure-ground segregation network is proposed based on a novel boundary pair representation. Nodes in the network are boundary segments obtained through local grouping. Each node is excitatorily coupled with the neighboring nodes that belong to the same region, and inhibitorily coupled with the corresponding paired node. Gestalt grouping rules are incorporated by modulating connections. The status of a node represents its probability being figural and is updated according to a differential equation. The system solves the figure-ground segregation problem through temporal evolution. Different perceptual phenomena, such as modal and amodal completion, virtual contours, grouping and shape decomposition are then explained through local diffusion. The system eliminates combinatorial optimization and accounts for many psychophysical results with a fixed set of parameters. 1 Introduction Perceptual organization refers to the ability of grouping similar features in sensory data. This, at a minimum, includes the operations of grouping and figure-ground segregation, which refers to the process of determining relative depths of adjacent regions in input data and thus proper occlusion hierarchy. Perceptual organization has been studied extensively and many of the existing approaches [5] [4] [8] [10] [3] start from detecting discontinuities, i.e. edges in the input; one or several configurations are then selected according to certain criteria, for example, non-accidental ness [5] . Those approaches have several disadvantages for perceptual organization. Edges should be localized between regions and an additional ambiguity, the ownership of a boundary segment, is introduced, which is equivalent to figure-ground segregation [7]. Due to that, regional attributions cannot be associated with boundary segments. Furthermore, because each boundary segment can belong to different regions, the potential search space is combinatorial. To overcome some of the problems, we propose a laterally-coupled network based on a boundary-pair representation to resolve figure-ground segregation. An occluding boundary is represented by a pair of boundaries of the two associated regions, and 39 Perceptual Organization Based on Temporal Dynamics (a) Figure 1: On- and off-center cell responses. (a) On- and off-center cells. (b) Input image. (c) On-center cell responses. (d) Off-center cell responses (e) Binarized on- and off-center cell responses, where white regions represent on-center response regions and black off-center regions. initiates a competition between the regions. Each node in the network represents a boundary segment. Regions compete to be figural through boundary-pair competition and figure-ground segregation is resolved through temporal evolution. Gestalt grouping rules are incorporated by modulating coupling strengths between different nodes within a region , which influences the temporal dynamics and determines the percept of the system. Shape decomposition and grouping are then implemented through local diffusion using the results from figur e-ground segregation. 2 Figure-Ground Segregation Network The central problem in perceptual organization is to determine relative depths among regions. As figure reversal occurs in certain circumstances, figure-ground segregation cannot be resolved only based on local attributes. 2.1 The Network Architecture The boundary-pair representation is motivated by on- and off-center cells, shown in Fig. l(a). Fig. l(b) shows an input image and Fig. l(c) and (d) show the on- and off-center responses. Without zero-crossing, we naturally obtain double responses for each occluding boundary, as shown in Fig. l(e). In our boundary-pair representation, each boundary is uniquely associated with a region. In this paper, we obtain closed region boundaries from segmentation and form boundary segments using corners and junctions, which are detected through local corner and junction detectors. A node i in the figure-ground segregation network represents a boundary segment, and Pi represents its probability being figural, which is set to 0.5 initially. Each node is laterally coupled with neighboring nodes on the closed boundary. The connection weight from node i to j, Wij, is 1 and can be modified by T-junctions and local shape information. Each occluding boundary is represented by a pair of boundary segments of the involved regions . For example, in Fig. 2(a), nodes 1 and 5 form a boundary pair , where node 1 belongs to the white region and node 5 belongs to the black region. Node i updates its status by: T dP dt i = ILL "~ Wki(Pk - Pi) kEN(i) + ILJ(l - P i) "~ H(Qli) lEJ (i) + ILB(l B i ) (1) - Pi) exp( - K B Here N(i) is the set of neighboring nodes of i, and ILL , ILJ , and ILB are parameters to determine the influences from lateral connections , junctions , and bias. J(i) is 40 2 X Liu and D. L. Wang ----?--<@r--- 8 6 ---oo{[D--@---- 4 I 3 (a) (b) Figure 2: (a) The figure-ground segregation network for Fig. l(b). Nodes 1, 2, 3 and 4 belong to the white region; nodes 5, 6, 7, and 8 belong to the black region; and nodes 9 and 10, and nodes 11 and 12 belong to the left and right gray regions respectively. Solid lines represent excitatory coupling while dashed lines represent inhibitory connections. (b) Result after surface completion. Left and right gray regions are grouped together. the set of junctions that are associated with i and Q/i is the junction strength of node i of junction l. H(x) is given by H(x) = tanh(j3(x - OJ )), where j3 controls the steepness and OJ is a threshold. In (1), the first term on the right reflects the lateral influences. When nodes are strongly coupled, they are more likely to be in the same status, either figure or background. The second term incorporates junction information. In other words, at a T-junction, segments that vary more smoothly are more likely to be figural. The third term is a bias, where Bi is the bias introduced to simulate human perception. The competition between paired nodes i and j is through normalization based on the assumption that only one of the paired nodes should be figural at a given time: p(Hl) = pt/(P~ + pt) and p(tH) = P~/(pt + P~) t t t J J J t J' 2.2 Incorporation of Gestalt Rules To generate behavior that is consistent with human perception, we incorporate grouping cues and some Gestalt grouping principles. As the network provides a generic model, additional grouping rules can also be incorporated. T-junctions T-junctions provide important cues for determining relative depths [7] [10]. In Williams and Hanson's model [10], T-junctions are imposed as topological constraints. Given aT-junction l, the initial strength for node i that is associated with lis: Q exp( -Ci(i,C(i?/ KT) Ii = 1/2 LkENJ(I) exp( -Ci(k ,c(k?)/ K T ) , where K T is a parameter, N J (l) is a set of all the nodes associated with junction l, c( i) is the other node in N J (l) that belongs to the same region as node i, and Ci(ij) is the angle between segments i and j. Non-accidentalness Non-accidentalness tries to capture the intrinsic relationships among segments [5]. In our system, an additional connection is introduced to node i if it is aligned well with a node j from the same region and j rf. N(i) initially. The connection weight Wij is a function of distance and angle between the involved ending points. This can be viewed as virtual junctions, resulting in virtual contours and conversion of a corner into a T-junction if involved nodes become figural. This corresponds to an organization criterion proposed by Geiger et al [3}. 41 Perceptual Organization Based on Temporal Dynamics Time Time Time Figure 3: Temporal behavior of each node in the network shown in Fig. 2(a). Each plot shows the status of the corresponding node with respect to time. The dashed line is 0.5. Shape information Shape information plays a central role in Gestalt principles and is incorporated through enhancing lateral connections. In this paper, we consider local symmetry. Let j and k be two neighboring nodes of i: Wij = 1 + C exp( -Iaij - akil/ KaJ * exp( -(Lj / Lk + Lk/ Lj - 2)/ K L )), where C, KQ:, and KL are parameters and L j is the length of segment j. Essentially the lateral connections are strengthened when two neighboring segments of i are symmetric. Preferences Human perceptual systems often prefer some organizations over others. Here we incorporated a well-known figure-ground segregation principle, called closeness. In other words, the system prefers filled regions over holes. In current implementation, we set Bi == 1.0 if node i is part of a hole and otherwise Bi == o. 2.3 Temporal Properties of the Network After we construct the figure-ground segregation network, each node is updated according to (1). Fig. 3 shows the temporal behavior of the network shown in Fig. 2(a). The system approaches to a stable solution. For figure-ground segregation, we can binarize the status of each node using threshold 0.5. Thus the system generates the desired percept in a few iterations. The black region occludes other regions while gray regions occlude the white region. For example, P5 is close to 1 and thus segment 5 is figural, and PI is close to 0 and thus segment 1 is in the background. 2.4 Surface Completion After figure-ground segregation is resolved, surface completion and shape decomposition are implemented through diffusion [3]. Each boundary segment is associated with regional attributes such as the average intensity value because its ownership is known. Boundary segments are then grouped into diffusion groups based on similarities of their regional attributes and if they are occluded by common regions. In Fig. 1(b), three diffusion groups are formed, namely, the black region, two gray regions, and the white region. Segments in one diffusion group are diffused simultaneously. For a figural segment, a buffer with a given radius is generated. Within the buffer, the values are fixed to 1 for pixels belonging to the region and 0 otherwise. Now the problem becomes a well-defined mathematical problem. We need to solve 42 X Liu and D. L. Wang (c) Figure 4: Images with virtual contours. In each column, the top shows the input image and the bottom the surface completion result, where completed surfaces are shown according to their relative depths and the bottom one is the projection of all the completed surfaces. (a) Alternate pacman. (b) Reverse-contrast pacman. (c) Kanizsa triangle. (d) Woven square. (e) Double pacman. the heat equation with given boundary conditions. Currently, the heat equation is solved through local diffusion. The results from diffusion are then binarized using threshold 0.5. Fig. 2(b) shows the results for Fig. l(b) after surface completion. Here the two gray regions are grouped together through surface completion because occluded boundaries allow diffusion. The white region becomes the background, which is the entire image. 3 Experimental Results Given an image, the system automatically constructs the network and establishes the connections based on the rules discussed in Section 2.2. For all the experiments shown here, a fixed set of parameters is used. 3.1 Modal and Amodal Completion We first demonstrate that the system can simulate virtual contours and modal completion. Fig. 4 shows the input images and surface completion results. The system correctly solves figure-ground segregation problem and generates the most probable percept. Fig. 4 (a) and (b) show two variations of pacman images [9] [4]. Even though the edges have opposite contrast, the virtual rectangle is vivid. Through boundary-pair representation, our system can handle both cases using the same network. Fig. 4(c) shows a typical virtual image [6] and the system correctly simulates the percept. In Fig. 4( d) [6], the rectangular-like frame is tilted, making the order between the frame and virtual square not well-defined. Our system handles that in the temporal domain. At any given time, the system outputs one of the Perceptual Organization Based on Temporal Dynamics 43 (a) ~'---I (b) (c ) (d) (e) (f) Figure 5: Surface completion results. (a) and (b) Bregman figures [1]. (c) and (d) Surface completion results for (a) and (b). (e) and (f) An image of some groceries and surface completion result. completed surfaces. Due to this, the system can also handle the case in Fig. 4(e) [2], where the percept is bistable, as the order between the two virtual squares is not well defined. Fig. 5(a) and (b) show the well-known Bregman figures [1]. In Fig. 5(a), there is no perceptual grouping and parts of B's remain fragmented. However, when occlusion is introduced as in Fig. 5(b), perceptual grouping is evident and fragments of B's are grouped together. Our results, shown in Fig. 5 (c) and (d), are consistent with the percepts. Fig. 5(e) shows an image of groceries, which is used extensively in [8]. Even though the T-junction at the bottom is locally confusing, our system gives the most plausible result through lateral influences of the other two strong T-junctions. Without search and parameter tuning, our system gives the optimal solution shown in Fig. 5(f). 3.2 Comparison with Existing Approaches As mentioned earlier, at the minimum, figure-groud segregation and grouping need to be addresssed for perceptual organization. Edge-based approaches [4] [10] attempt to solve both problems simultaneously by prefering some configurations over combinatorially many ones according to certain creteria. There are several difficulties common to those approaches. First it cannot account for different human percepts of cases where edge elements are similar. Fig. 5 (a) and (b) are wellknown examples in this regard. Another example is that the edge-only version of Fig. 4( c) does not give rise to a vivid virtual contour as in Fig. 4( c) [6]. To reduce the potential search space, often contrast signs of edges are used as additional contraints [10J. However, both Fig. 4 (a) and (b) give rise to virtual contours despite the opposite edge contrast signs. Essentially based on Fig. 4(b), Grossberg and Mingolla [4] claimed that illusory contours can join edges with different directions of contrast , which does not hold in general. As demonstrated through experiments, our approach does offer a common principle underlying these examples. Our approach shares some similarities with the one by Geiger et al [3]. In both approaches , perceptual organization is solved in two steps. In [3], figure-ground segregation is encoded implicitly in hypotheses which are defined at junction points. Because potential hypotheses are combinatorial, only a few manually chosen ones are tested in their experiments, which is not sufficient for a general computational 44 X Liu and D. L. Wang model. In our approach, by resolving figure-ground segregation, there is no need to define hypotheses explicitly. In both methods, grouping is implemented through diffusion. In [3], "heat" sources for diffusion are given manually for each hypothesis whereas our approach generates "heat" sources automatically using the figure-ground segregation results. Finally, in our approach, local ambiguities can be resolved through lateral connections using temporal dynamics, resulting in robust behavior. To obtain good results for Fig. 5(e), Nitzberg et al [8] need to tune parameters and increase their search space substantially due to the misleading T-junction at the bottom of Fig. 5(e). 4 Conclusion In this paper we have proposed a network for perceptual organization using temporal dynamics. The pair-wise boundary representation resolves the ownership ambiguity inherent in an edge-based representation and is equivalent to a surface representation through diffusion, providing a unified edge- and surface-based representation. Through temporal dynamics, our model allows for interactions among different modules and top-down influences can be incorporated. Acknowledgments Authors would like to thank S. C. Zhu and M. Wu for their valuable discussions. This research is partially supported by an NSF grant (IRI-9423312) and an ONR Young Investigator Award (N00014-96-1-0676) to DLW. References [1] A. S. Bregman, "Asking the 'What for' question in auditory perception," In Perceptual Organization, M. Kubovy and J R. Pomerantz, eds., Lawrence Erlbaum Associates, Publishers, Hillsdale, New Jersey, pp. 99-118, 1981. [2] M. Fahle and G. Palm, "Perceptual rivalry between illusory and real contours," Biological Cybernetics, vol. 66, pp. 1-8, 1991. [3] D. Geiger, H. Pao, and N . Rubin, "Salient and multiple illusory surfaces," In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 118-124, 1998. [4] S. Grossberg and E . Mingolla, "Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations," Perception & Psychophysics, vol. 38, pp. 141-170, 1985. [5] D. G. Lowe, Perceptual Organization and Visual Recognition, Kluwer Academic Publishers, Boston, 1985. [6] G. Kanizsa, Organization in Vision, Praeger, New York, 1979. [7] K. Nakayama, Z. J . He, and S. Shimojo, "Visual surface representation: a critical link between lower-level and higher-level vision," In Visual Cognition, S. M. Kosslyn and D. N. Osherson, eds. , The MIT Press, Cambridge, Massachusetts, vol. 2, pp. 1-70, 1995. [8] M. Nitzberg, D. Mumford, and T. Shiota, Filtering, Segmentation and Depth, Springer-Verlag, New York, 1993. [9] R. Shapley and J. Gordon, "The existence of interpolated illusory contours depends on contrast and spatial separation," In The Perception of Illusory Contours, S. Petry and G. E. Meyer, eds., Springer-Verlag, New York, pp. 109-115, 1987. [10] L. R. Williams and A. R. Hanson, "Perceptual Completion of Occluded Surfaces," Computer Vision and Image Understanding, vol. 64, pp. 1-20, 1996. 

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Spectral Cues in Human Sound Localization Craig T. Jin Department of Physiology and Department of Electrical Engineering, Univ. of Sydney, NSW 2006, Australia Anna Corderoy Department of Physiology Univ. of Sydney, NSW 2006, Australia Simon Carlile Department of Physiology and Institute of Biomedical Research Univ. of Sydney, NSW 2006, Australia Andre van Schaik Department of Electrical Engineering, Univ. of Sydney, NSW 2006, Australia Abstract The differential contribution of the monaural and interaural spectral cues to human sound localization was examined using a combined psychophysical and analytical approach. The cues to a sound's location were correlated on an individual basis with the human localization responses to a variety of spectrally manipulated sounds. The spectral cues derive from the acoustical filtering of an individual's auditory periphery which is characterized by the measured head-related transfer functions (HRTFs). Auditory localization performance was determined in virtual auditory space (VAS). Psychoacoustical experiments were conducted in which the amplitude spectra of the sound stimulus was varied independentlyat each ear while preserving the normal timing cues, an impossibility in the free-field environment. Virtual auditory noise stimuli were generated over earphones for a specified target direction such that there was a "false" flat spectrum at the left eardrum. Using the subject's HRTFs, the sound spectrum at the right eardrum was then adjusted so that either the true right monaural spectral cue or the true interaural spectral cue was preserved. All subjects showed systematic mislocalizations in both the true right and true interaural spectral conditions which was absent in their control localization performance. The analysis of the different cues along with the subjects' localization responses suggests there are significant differences in the use of the monaural and interaural spectral cues and that the auditory system's reliance on the spectral cues varies with the sound condition. 1 Introduction Humans are remarkably accurate in their ability to localize transient, broadband noise, an ability with obvious evolutionary advantages. The study of human auditory localization has a considerable and rich history (recent review [I]) which demonstrates that there are three general classes of acoustical cues involved in the localization process: (1) interaural time differences, ITDs; (2) interaurallevel differences, ILDs; and (3) the spectral cues resulting Spectral Cues in Human Sound Localization 769 from the auditory periphery. It is generally accepted that for humans, the lTD and ILD cues only specify the location of the sound source to within a "cone of confusion" [I], i.e., a locus of points approximating the surface of a cone symmetric with respect to the interaural axis. It remains, therefore, for the localization system to extract a more precise sound source location from the spectral cues. The utilization of the outer ear spectral cues during sound localization has been analyzed both as a statistical estimation problem, (e.g., [2]) and as optimization problem, often using neural networks, (e.g., [3]). Such computational models show that sufficient localization information is provided by the spectral cues to resolve the cone of confusion ambiguity which corroborates the psychoacoustical evidence. Furthermore, it is commonly argued that the interaural spectral cue, because of its natural robustness to level and spectral variations, has advantages over the monaural spectral cues alone. Despite these observations, there is still considerable contention as to the relative role or contribution of the monaural versus the interaural spectral cues. In this study, each subject's spectral cues were characterized by measuring their head related transfer functions (HRTFs) for 393 evenly distributed positions in space. Measurements were carried out in an anechoic chamber and were made for both ears simultaneously using a "blocked ear" technique [I]. Sounds filtered with the HRTFs and played over earphones, which bypass the acoustical filtering of the outer ear, result in the illusion of free-field sounds which is known as virtual auditory space (VAS). The HRTFs were used to generate virtual sound sources in which the spectral cues were manipulated systematically. The recorded HRTFs along with the Glasberg and Moore cochlear model [4] were also used to generate neural excitation patterns (frequency representations of the sound stimulus within the auditory nerve) which were used to estimate the different cues available to the subject during the localization process. Using this analysis, the interaural spectral cue was characterized and the different localization cues have been correlated with each subjects' VAS localization responses. 2 VAS Sound Localization The sound localization performance of four normal hearing subjects was examined in VAS using broadband white noise (300 - 14 000 Hz). The stimuli were filtered under three differing spectral conditions. (1) control: stimuli were filtered with spectrally correct left and right ear HRTFs for a given target location, (2) veridical interaural: stimuli at the left ear were made spectrally flat with an appropriate dB sound level for the given target location, while the stimuli at the right ear were spectrally shaped to preserve the correct interaural spectrum, (3) veridical right monaural: stimuli at the left ear were spectrally flat as in the second condition, while the stimuli at the right ear were filtered with the correct HRTF for the given target location, resulting in an inappropriate interaural spectral difference. For each condition, a minimum-phase filter spectral approximation was made and the interaural time difference was modeled as an all-pass delay [5]. Sounds were presented at approximately 70 dB SPL and with duration 150 ms (with 10 ms raised-cosine onset and offset ramps). Each subject performed five trials at each of 76 test positions for each stimulus condition. Detailed sound localization methods can be found in [1]. A short summary is presented below. 2.1 Sound Localization Task The human localization experiments were carried out in a darkened anechoic chamber. Virtual auditory sound stimuli were presented using earphones (ER-2, Etymotic Research, with a flat frequency response, within 3 dB, between 200-16 000 Hz). The perceived location of the virtual sound source was indicated by the subject pointing hislher nose in 770 C. T. lin, A. Corderoy, S. Carlile and A. v. Schaik the direction of the perceived source. The subject's head orientation and position were monitored using an electromagnetic sensor system (Polhemus, Inc.). 2.2 Human Sound Localization Performance The sound localization performance of two subjects in the three different stimulus conditions are shown in Figure 1. The pooled data across 76 locations and five trials is presented for both the left (L) and right (R) hemispheres of space from the viewpoint of an outside observer. The target location is shown by a cross and the centroid of the subjects responses for each location is shown by a black dot with the standard deviation indicated by an ellipse. Front-back confusions are plotted, although, they were removed for calculating the standard deviations. The subjects localized the control broadband sounds accurately (Figure 1a). In contrast, the subjects demonstrated systematic mislocalizations for both the veridical interaural and veridical monaural spectral conditions (Figures I b,c). There is clear pulling of the localization responses to particular regions of space with evident intersubject variations. (8) Subject 1: Broadband Control Subject 2: Broadband Control Ellipse: Standard Deviation L:i;;:~ses ~JC~~ ~~i.iif. e'??.R ,':.,;':::::::.><" (b) Subject 1: Veridicallnteraural Spectrum L R Subject 2: Veridical Interaural Spectrum L ,.c._:.. . . ~...:. . .." ~I~;> (c) Le Subject 1: Veridical Right Monaural Spectrum '... ~l~. :.:. - ' ., " _ ? ....-.L . .;.' ... .,. ~~.j;~;" ":':"$.. '~-, . ' ?' /...., R " ~ .~ ? . .?. ~...... .t ' ~ i ~: ? ......~?.;;.:J;...;,1"" ',. .-. ';'-"'" , ' ;::? Subject 2: Veridical Right Monaural Spectrum L~~R V~ Figure 1: Localization performance for two subjects in the three sound conditions: (a) control broadband; (b) veridical interaural; (c) veridical monaural. See text for details. 3 Extraction of Acoustical Cues With accurate measurements of each individual's outer ear filtering, the different acoustical cues can be compared with human localization performance on an individual basis. In order to extract the different acoustical cues in a biologically plausible manner, a model of peripheral auditory processing was used. A virtual source sound stimulus was prepared as described in Secion 2 for a particular target location. The stimulus was then filtered using a cochlear model based on the work of Glasberg and Moore [4]. This cochlear model consisted of a set of modified rounded-exponential auditory filters. The width and shape of the auditory filters change as a function of frequency (and sound level) in a manner Spectral Cues in Human Sound Localization 771 consistent with the known physiological and psychophysical data. These filters were logarithmically spaced on the frequency axis with a total of 200 filters between 300 Hz and 14 kHz. The cochlea's compressive non-linearity was modelled mathematically using a logarithmic function. Thus the logarithm of the output energy ofa given filter indicated the amount of neural activity in that particular cochlear channel. The relative activity across the different cochlear channels was representative of the neural excitation pattern (EP) along the auditory nerve and it is from this excitation pattern that the different spectral cues were estimated. For a given location, the left and right EPs themselves represent the monaural spectral cues. The difference in the total energy (calculated as the area under the curve) between the left and right EPs was taken as a measure of the interaural level difference and the interaural spectral shape cue was calculated as the difference between the left and right EPs. The fourth cue, interaural time difference, is a measure of the time lag between the signal in one ear as compared to the other and depends principally upon the geometrical relationship between the sound source and the human subject. This time delay was calculated using the acoustical impulse response for both ears as measured during the HRTF recordings. 4 Correlation of Cues and Location For each stimulus condition and location, the acoustical cues were calculated as described above for all 393 HRTF locations. Locations at which a given cue correlates well with the stimulus cue for a particular target location were taken as analytical predictions of the subject's response locations according to that cue. As the spectral content of the signal is varied, the cue(s) available may strongly match the cue(s) normally arising from locations other than the target location. Therefore the aim of this analysis is to establish for which locations and stimulus conditions a given response most correlated with a particular cue. The following analyses (using a Matlab toolbox developed by the authors) hinge upon the calculation of "cue correlation values". To a large extent, these calculations follow the examples described by [6] and are briefly described here. For each stimulus condition and target location, the subject performed five localizations trials. For each of the subject's five response locations, each possible cue was estimated (Section 3) assuming a flat-spectrum broadband Gaussian white noise as the stimulus. A mathematical quantity was then calculated which would give a measure of the similarity of the response location cues with the corresponding stimulus cues. The method of calculation depended on the cue and several alternative methods were tried. Generally, for a given cue, these different methods demonstrated the same basic pattern and the term "cue correlation value" has been given to the mathematical quantity that was used to measure cue similarity. The methods are as follows. For the ITO cue, the negative of the absolute value of the difference between possible response location ITDs and the stimulus ITO was used as the ITO cue correlation values (the more positive a value, the higher its correlation). The ILD cue correlation value was calculated in a similar fashion. The cue correlation values for the left and right monaural spectral cues (in this case, the shape of the neural excitation pattern) was calculated by taking the difference between the stimulus EP and the possible response location EPs and then summing across frequency the variation of this difference about its mean value. For the interaural spectral cue, the vector difference between the left and right EPs was calculated for both the stimulus and the possible response locations. The dot product between the stimulus and the possible response location vectors gave the ISO cue correlation values. The cue correlation values were normalized in order to facilitate meaningful comparisons across the different acoustical cues. Following Middlebrooks [6], a "z-score normalized" cue value, for each response location corresponding to a given target location, was obtained by subtracting the mean correlation value (across all possible locations) and dividing by the 772 C. T. Jin, A. Corderoy, S. Carlile and A. v. Schaik standard deviation. For these new cue values, termed the cue z-score values, a score of 1.0 or greater indicates p'oocl correlation. 5 Relationship between the ISD and the Cone-of-Confusion The distribution of a given cue's z-score values around the sphere of space surrounding the subject reveals the spatial directions for that cue that correlate best with the given stimulus and target location being examined. An examination of the interaural spectral cue indicated that, unlike the other cues, the range of its cue z-score variation was relatively restricted on the ipsilateral hemisphere of space relative to the sound stimulus (values on the ipsilateral side were approximately 1.0, those on the contralateral side, -1.0). This was the first indication of the more moderate variation of the ISO cue across space as compared with the monaural spectral cues. Closer examination of the ISO cue revealed more detailed variational properties. In order to facilitate meaningful comparisons with the other cues, the ISO cue z-score values were adjusted such that all negative values (i.e., those values at locations generally contralateral to the stimulus) were set to 0.0 and the cue z-score values recalculated. The spatial distribution of the rescaled ISO cue z-score values, as compared with the cue z-score values for the other cues, is shown in Figure 2. The cone of confusion described by the ITO and ILO is clearly evident (Fig. 2a,b) and it can be seen that the ISO cue is closely aligned with these cues (Fig. 2c). Furthermore, the ISO cue demonstrates significant asymmetry along the front-back dimensions. These novel observations demonstrate that while previous work [3, 2] indicates that the ISO cue provides sufficient information to determine a sound's 10eation exactly along the cone of confusion, the variation of the cue z-score values along the cone is substantially less than that for the monaural spectral cues (Fig. 2d), suggest ;:-:3 perhaps that this acts to make the monaural spectral cue a more salient cue. (8) Interaural Time Difference (c) 40 40 1.5 o o c: .40 ,g Interaural Spectrum -40 0 90 180 0 90 180 ~ (b) Interaurai Level Difference (d) Ipsilateral Monaural Spectrum iIi 40 40 o o -40 o -40 90 180 0 90 180 1.1 Azimuth Figure 2: Spatial plot of the cue z-score values for a single target location (46 0 azimuth, 20 0 elevation) and broadband sound condition. Gray-scale color values indicate the cue's correlation in different spatial directions with the stimulus cue at the target location. (Zscore values for the ISO cue have been rescaled, see text.) 6 Analysis of Subjects Responses using Cue Z-score Values A given cue's z-score values for the subject's responses across all 76 test locations and five trials were averaged. The mean and standard deviation are presented in a bar graph (Fig. 3). The subjects' response locations correlate highly with the ITO and ILD cue and Spectral Cues in Human Sound Localization 773 the standard deviation of the correlation was low (Fig. 3a,b). In other words, subjects' responses stayed on the cone of confusion of the target location. A similar analysis of the more restricted, rescaled version of the interaural spectral cue shows that despite the spectral manipulations and systematic mislocalizations, subject's were responding to locations which were highly correlated with the interaural spectral cue (Fig. 3c). The bar graphs for the monaural spectral cues ipsilateral and contralateral to the target location show the average correlation of the subjects' responses with these cues varied considerably with the stimulus condition (Fig. 3d-g) and to a lesser extent across subjects. 2 rn (ij 2 ~ 1 If 0 > 8 N Q) :::l U (e) Right Contralateral Spectrum 2 Control Veridical VeridiCal Broadband Interaural Right Mona....1 1 Q) :::l (d) Left Contralateral Spectrum 2 ILD =~= o Control VeridiCal VeridiCal Broadband Interaural Right Monaural 2 (f) Left Ipsilateral Spectrum o Control Veridical Veridical Broadband InterauraJ Right Monaural (g) Right Ipsilateral Spectrum 2 ISD o o Control Veridical Veridical Broadband Interaural Right Monaural Control VeridiCal VeridiCal Broadband Interaurai Right Monaural o Control V9!idical VeridiCal Broadband Interaural Right Monaural Control Veridical Veridical Broadband Interaural Right Monaural Figure 3: Correlation of the four subjects' (indicated by different gray bars) localization responses with the different acoustical cues for each stimulus condition. The bar heights indicates the mean cue z-score value, while the error bars indicate standard deviation. 7 Spatial Plots of Correlation Regions As the localization responses tended to lie along the cone of confusion, the relative importance of the spectral cues along the cone of confusion was examined. The correlation values for the spectral cues associated with the subjects' responses were recalculated as a z-score value using only the distribution of values restricted to the cone of confusion. This demonstrates whether the spectral cues associated with the subjects' response locations were better correlated with the stimulus cues, than for any random location on the cone of confusion. Spatial plots of the recalculated response cue z-score values for the spectral cues of one subject (similar trends across subjects), obtained for each stimulus location and across the three different sound conditions, is shown in Figure 4. Spatial regions of both high and low correlation are evident that vary with the stimulus spectrum. The z-score values for the ISD cue shows greater bilateral correlation across space in the veridical interaural condition (Fig. 4d) than for the veridical monaural condition (Fig. 4g), while the right monaural spectral cue demonstrates higher correlation in the right hemisphere of space for the veridical monaural condition (Fig. 4i) as opposed to the veridical interaural condition (Fig. 4t). This result (although not surprising) demonstrates that the auditory system is extracting cues to source location in a manner dependent on the input sound spectrum and in a manner consistent with the spectral infonnation available in the sound spectrum. Figures 4e,h clearly demonstrate that the flat sound spectrum in the left ear was strongly correlated with and influenced the subject's localization judgements for specific regions of space. 774 C. T. lin, A. Corderoy, S. Carlile and A. v. Schaik Broadband Veridical Interaural Veridical Monaural (a) (d) (9) ? a 180 Ol- (e) 40 40 ~ a a iIi -40 ?180 -40 180 ?180 a ". (1) 40 . '~'t;' ,1 ?180 ~ a E -=:2 Ol- -IalCl. a U) 180 4 180 c: -co 0u 1 >- Ol 0 ~ Ol 8 c: I/) 0 , u 0.5 N 0 40 Ol::J Ol () .~ a g (i) f ., ::J (h) c: .Q .~ .5 ?40 ?180 15 o 180 Azimuth Figure 4: Spatial plot of the spectral cue z-score values for one subject's localization responses across the three different sound conditions. 8 Conclusions The correlation of human sound localization responses with the available acoustical cues across three spectrally.. different sound conditions has provided insights into the human auditory system and its integration of cues to produce a coherent percept of spatial location. These data suggest an interrelationship between the interaural spectral cue and the cone of confusion. The ISO cue is front-back asymmetrical along the cone and its cue correlation values vary more moderately as a function of space than those of the monaural spectral cues. These data shed light on the relative role and importance of the interaural and monaural spectral cues. Acknowledgments This research was supported by the ARC, NHMRC, and Dora Lush Scholarship to CJ. References [1] S. Carlile, Virtual auditory space: Generation and applications. New York: Chapman and Hall, 1996. [2] R. O. Duda, "Elevation dependence of the interaural transfer function," in Binaural and spatial hearing in real and virtual environments (R. H. Gilkey and T. R. Anderson, eds.), ch. 3, pp. 4975, Mahwah, New Jersey: Lawrence Erlbaum Associates, 1997. [3] J. A. Janko, T. R. Anderson, and R. H. Gilkey, "Using neural networks to evaluate the viability of monaural and interaural cues for sound localization," in Binaural and Spatial Hearing in real and virtual environments (R. H. Gilkey and T. R. Anderson, eds.), ch. 26, pp. 557-570, Mahwah, New Jersey: Lawrence Erlbaum Associates, 1997. [4] B. Glasberg and B. Moore, "Derivation of auditory filter shapes from notched-noise data," Hearing Research, vol. 47, no. 1-2, pp. 103-138, 1990. [5] F. Wightman and D. Kistler, "The dominant role oflow-frequency interaural time differences in sound localization," 1. Acoust. Soc. Am., vol. 91, no. 3, pp. 1648-1661, 1992. [6] J. Middlebrooks, "Narrow-band sound localization related to external ear acoustics," 1. Acoust. Soc. Am., vol. 92, no. 5, pp. 2607-2624, 1992. 

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